Análise de Sistemas de Energia Elétrica
Departamento de Engenharia Elétrica
Unidade 4
Faltas Trifásicas Assimétricas: Método Básico
Faltas Trifásicas Assimétricas - Curto-circuito entre Linha e Terra
$\scriptsize \left. {\begin{array}{*{20}c} { {I}_{\rm{a}}}, \ \ {{{I}_{\rm{b}} }}, \ \ {{I}_{\rm{c}} } \\ { {V}_{\rm{a}}}, \ \ {{{V}_{\rm{b}} }}, \ \ {{V}_{\rm{c}} } \\ \end{array}} \right\rbrace {\large ?} $
$\scriptsize {\large \downarrow} $
$\scriptsize \left. {\begin{array}{*{20}c} { {I}_{\rm{0}}}, \ \ {{{I}_{\rm{1}} }}, \ \ {{I}_{\rm{2}} } \\ { {V}_{\rm{0}}}, \ \ {{{V}_{\rm{1}} }}, \ \ {{V}_{\rm{2}} } \\ \end{array}} \right\rbrace {\large ?} $
$\scriptsize {{I}_{\rm{a}} = {I}_{\rm{0}} + {I}_{\rm{1}} +{I}_{\rm{2}} } \\ \scriptsize {I}_{\rm{b}} = {I}_{\rm{0}} + a^2 {I}_{\rm{1}} + a {I}_{\rm{2}} \\ \scriptsize {I}_{\rm{c}} = {I}_{\rm{0}} + a {I}_{\rm{1}} + a^2 {I}_{\rm{2}} $
$\scriptsize {{V}_{\rm{a}} = {V}_{\rm{0}} + {V}_{\rm{1}} +{V}_{\rm{2}} } \\ \scriptsize {V}_{\rm{b}} = {V}_{\rm{0}} + a^2 {V}_{\rm{1}} + a {V}_{\rm{2}} \\ \scriptsize {V}_{\rm{c}} = {V}_{\rm{0}} + a {V}_{\rm{1}} + a^2 {V}_{\rm{2}} $
Faltas Trifásicas Assimétricas - Curto-circuito entre Linha e Terra
$\scriptsize {V}_{\rm{a}} = {Z}_{F} {I}_{\rm{a}} $
$\scriptsize { {I}_{\rm{b}} = {I}_{\rm{c}} = 0 } $
$\scriptsize {I}_{\rm{0,1,2}} = {\rm{A~}}^{-1} {I}_{\rm{a,b,c}} $
$\tiny \left[ {\begin{array}{*{20}{c}} {{I}_0}\\ {{I}_1}\\ {{I}_2} \end{array}} \right] = \frac{1}{3}\left[ {\begin{array}{*{20}{c}} {1}&{1}&{1}\\ {1}&{a}&{a^2}\\ {1}&{a^2}&{a} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{I}_{\rm{a}}}\\ {0}\\ {0} \end{array}} \right] $
$\tiny \boxed{ {{I}_0} = {{I}_1} = {{I}_2} = \frac{1}{3} {{I}_a} } $
Faltas Trifásicas Assimétricas - Curto-circuito entre Linha e Terra
$\scriptsize {V}_{\rm{a}} = {V}_{\rm{0}} + {V}_{\rm{1}} + {V}_{\rm{2}} $
$\scriptsize \boxed{ {V}_{\rm{0}} = - {Z}_{\rm{0}} {I}_{\rm{0}} }$ $\scriptsize \boxed{ {V}_{\rm{1}} = {E}_{\rm{a}} - {Z}_{\rm{1}} {I}_{\rm{1}} }$ $\scriptsize \boxed{ {V}_{\rm{2}} = - {Z}_{\rm{2}} {I}_{\rm{2}} }$
$\scriptsize {V}_{\rm{a}} = {E}_{\rm{a}} - ( {Z}_{\rm{0}} + {Z}_{\rm{1}} + {Z}_{\rm{2}} ) {I}_{\rm{0}} $
$\scriptsize {V}_{\rm{a}} = {Z}_{F} {I}_{\rm{a}} $
$\scriptsize \boxed{ {{I}_0} = {{I}_1} = {{I}_2} = \frac{1}{3} {{I}_a} } $
$\scriptsize {V}_{\rm{a}} = 3{Z}_{F} {I}_{\rm{0}} $
$\scriptsize 3{Z}_{F} {I}_{\rm{0}} = {E}_{\rm{a}} - ( {Z}_{\rm{0}} + {Z}_{\rm{1}} + {Z}_{\rm{2}} ) {I}_{\rm{0}} $
$\tiny \boxed{ {I}_{\rm{0}} = \frac{{E}_{\rm{a}}}{{Z}_{\rm{0}} + {Z}_{\rm{1}} + {Z}_{\rm{2}} + 3{Z}_{F}} }$
Faltas Trifásicas Assimétricas - Curto-circuito entre Linha e Terra
$\scriptsize \boxed{ {{I}_0} = {{I}_1} = {{I}_2} = \frac{1}{3} {{I}_a} } $
$\scriptsize \boxed{ {I}_{\rm{0}} = \frac{{E}_{\rm{a}}}{{Z}_{\rm{0}} + {Z}_{\rm{1}} + {Z}_{\rm{2}} + 3{Z}_{F}} }$
Faltas Trifásicas Assimétricas - Curto-circuito entre Duas Linhas
$\scriptsize \left. {\begin{array}{*{20}c} { {I}_{\rm{a}}}, \ \ {{{I}_{\rm{b}} }}, \ \ {{I}_{\rm{c}} } \\ { {V}_{\rm{a}}}, \ \ {{{V}_{\rm{b}} }}, \ \ {{V}_{\rm{c}} } \\ \end{array}} \right\rbrace {\large ?} $
$\scriptsize {\large \downarrow} $
$\scriptsize \left. {\begin{array}{*{20}c} { {I}_{\rm{0}}}, \ \ {{{I}_{\rm{1}} }}, \ \ {{I}_{\rm{2}} } \\ { {V}_{\rm{0}}}, \ \ {{{V}_{\rm{1}} }}, \ \ {{V}_{\rm{2}} } \\ \end{array}} \right\rbrace {\large ?} $
$\scriptsize {{I}_{\rm{a}} = {I}_{\rm{0}} + {I}_{\rm{1}} +{I}_{\rm{2}} } \\ \scriptsize {I}_{\rm{b}} = {I}_{\rm{0}} + a^2 {I}_{\rm{1}} + a {I}_{\rm{2}} \\ \scriptsize {I}_{\rm{c}} = {I}_{\rm{0}} + a {I}_{\rm{1}} + a^2 {I}_{\rm{2}} $
$\scriptsize {{V}_{\rm{a}} = {V}_{\rm{0}} + {V}_{\rm{1}} +{V}_{\rm{2}} } \\ \scriptsize {V}_{\rm{b}} = {V}_{\rm{0}} + a^2 {V}_{\rm{1}} + a {V}_{\rm{2}} \\ \scriptsize {V}_{\rm{c}} = {V}_{\rm{0}} + a {V}_{\rm{1}} + a^2 {V}_{\rm{2}} $
Faltas Trifásicas Assimétricas - Curto-circuito entre Duas Linhas
$\scriptsize {V}_{\rm{b}} - {V}_{\rm{c}} = {Z}_{\rm{F}} {I}_{\rm{b}} $
$\scriptsize { {I}_{\rm{b}} + {I}_{\rm{c}} = 0 } $
$\scriptsize { {I}_{\rm{a}} = 0 } $
$\tiny \left[ {\begin{array}{*{20}{c}} {{I}_0}\\ {{I}_1}\\ {{I}_2} \end{array}} \right] = \frac{1}{3}\left[ {\begin{array}{*{20}{c}} {1}&{1}&{1}\\ {1}&{a}&{a^2}\\ {1}&{a^2}&{a} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {0}\\ {{I}_{\rm{b}}}\\ {-{I}_{\rm{b}}} \end{array}} \right] $
$\scriptsize { {I}_{\rm{0}} = 0 } $
$\scriptsize {I}_{\rm{1}} = (1/3) (a - a^2) {I}_{\rm{b}} $
$\scriptsize {I}_{\rm{2}} = (1/3) (a^2 - a) {I}_{\rm{b}} $
$\scriptsize \boxed{ {I}_{\rm{1}} = - {I}_{\rm{2}} } $
Faltas Trifásicas Assimétricas - Curto-circuito entre Duas Linhas
$\tiny \left[ {\begin{array}{*{20}{c}} {{V}_a}\\ {{V}_b}\\ {{V}_c} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {1}&{1}&{1}\\ {1}&{a^2}&{a}\\ {1}&{a}&{a^2} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{V}_0}\\ {{V}_1}\\ {{V}_2} \end{array}} \right] $
$\scriptsize {V}_{\rm{b}} - {V}_{\rm{c}} = {Z}_{\rm{F}} {I}_{\rm{b}} \\ \scriptsize \ \ \ \ \ \ \ \ \ \ \ \ \ = (a^2 - a)( {V}_{\rm{1}} - {V}_{\rm{2}} ) $
$\scriptsize \boxed{ {V}_{\rm{1}} = {E}_{\rm{a}} - {Z}_{\rm{1}} {I}_{\rm{1}} } $
$\scriptsize \boxed{ {V}_{\rm{2}} = - {Z}_{\rm{2}} {I}_{\rm{2}} } $
$\scriptsize \boxed{ {I}_{\rm{2}} = - {I}_{\rm{1}} } $
$\scriptsize ( a^2 - a ) [ {E}_{\rm{a}} - ( {Z}_{\rm{1}} + {Z}_{\rm{2}} ) {I}_{\rm{1}} ] = {Z}_{\rm{F}} {I}_{\rm{b}} $
$\scriptsize \boxed{ {I}_{\rm{1}} = (1/3) (a - a^2) {I}_{\rm{b}}}; \ \tiny (a - a^2)(a^2 - a) = 3 $
$\tiny \boxed{ {I}_{\rm{1}} = \frac{E_{\rm{a}}}{{Z}_{\rm{1}} + {Z}_{\rm{2}} + {Z}_{\rm{F}} } } $
Faltas Trifásicas Assimétricas - Curto-circuito entre Duas Linhas
$\scriptsize \boxed{ {I}_{\rm{1}} = - {I}_{\rm{2}} } $
$\scriptsize \boxed{ {I}_{\rm{1}} = \frac{E_{\rm{a}}}{{Z}_{\rm{1}} + {Z}_{\rm{2}} + {Z}_{\rm{F}} } } $
Faltas Trifásicas Assimétricas - Curto-circuito entre Duas Linhas e Terra
$\scriptsize \left. {\begin{array}{*{20}c} { {I}_{\rm{a}}}, \ \ {{{I}_{\rm{b}} }}, \ \ {{I}_{\rm{c}} } \\ { {V}_{\rm{a}}}, \ \ {{{V}_{\rm{b}} }}, \ \ {{V}_{\rm{c}} } \\ \end{array}} \right\rbrace {\large ?} $
$\scriptsize {\large \downarrow} $
$\scriptsize \left. {\begin{array}{*{20}c} { {I}_{\rm{0}}}, \ \ {{{I}_{\rm{1}} }}, \ \ {{I}_{\rm{2}} } \\ { {V}_{\rm{0}}}, \ \ {{{V}_{\rm{1}} }}, \ \ {{V}_{\rm{2}} } \\ \end{array}} \right\rbrace {\large ?} $
$\scriptsize {{I}_{\rm{a}} = {I}_{\rm{0}} + {I}_{\rm{1}} +{I}_{\rm{2}} } \\ \scriptsize {I}_{\rm{b}} = {I}_{\rm{0}} + a^2 {I}_{\rm{1}} + a {I}_{\rm{2}} \\ \scriptsize {I}_{\rm{c}} = {I}_{\rm{0}} + a {I}_{\rm{1}} + a^2 {I}_{\rm{2}} $
$\scriptsize {{V}_{\rm{a}} = {V}_{\rm{0}} + {V}_{\rm{1}} +{V}_{\rm{2}} } \\ \scriptsize {V}_{\rm{b}} = {V}_{\rm{0}} + a^2 {V}_{\rm{1}} + a {V}_{\rm{2}} \\ \scriptsize {V}_{\rm{c}} = {V}_{\rm{0}} + a {V}_{\rm{1}} + a^2 {V}_{\rm{2}} $
Faltas Trifásicas Assimétricas - Curto-circuito entre Duas Linhas e Terra
$\scriptsize {V}_{\rm{b}} = {V}_{\rm{c}} = {Z}_{\rm{F}} ( {I}_{\rm{b}} + {I}_{\rm{c}} ) $
$\scriptsize {I}_{\rm{b}} = {I}_{\rm{0}} + a^2 {I}_{\rm{1}} + a {I}_{\rm{2}}$ $\scriptsize {I}_{\rm{c}} = {I}_{\rm{0}} + a {I}_{\rm{1}} + a^2 {I}_{\rm{2}} $
$\scriptsize {V}_{\rm{b}} = 3 {Z}_{\rm{F}} {I}_{\rm{0}} $
$\scriptsize {V}_{\rm{b}} = {V}_{\rm{0}} + a^2 {V}_{\rm{1}} + a {V}_{\rm{2}}$ $\scriptsize {V}_{\rm{c}} = {V}_{\rm{0}} + a {V}_{\rm{1}} + a^2 {V}_{\rm{2}} $ $\scriptsize {V}_{\rm{1}} = {V}_{\rm{2}} $ $\scriptsize 3 {Z}_{\rm{F}} {I}_{\rm{0}} = {V}_{\rm{0}} + ( a^2 + a ) {V}_{\rm{1}} $ $\scriptsize 3 {Z}_{\rm{F}} {I}_{\rm{0}} = {V}_{\rm{0}} - {V}_{\rm{1}} $
$\scriptsize \boxed{ {V}_{\rm{0}} = - {Z}_{\rm{0}} {I}_{\rm{0}} } $
$\scriptsize \boxed{ {V}_{\rm{1}} = {E}_{\rm{a}} - {Z}_{\rm{1}} {I}_{\rm{1}} } $
$\tiny \boxed{ {I}_{\rm{0}} = - \frac{{E}_{\rm{a}} - {Z}_{\rm{1}}{I}_{\rm{1}}}{ {Z}_{\rm{0}} + 3{Z}_{\rm{F}} } } $
Faltas Trifásicas Assimétricas - Curto-circuito entre Duas Linhas e Terra
$\scriptsize \boxed{ {V}_{\rm{1}} = {V}_{\rm{2}} } $
$\scriptsize \boxed{ {V}_{\rm{1}} = {E}_{\rm{a}} - {Z}_{\rm{1}} {I}_{\rm{1}} } $
$\scriptsize \boxed{ {V}_{\rm{2}} = - {Z}_{\rm{2}} {I}_{\rm{2}} } $
$\scriptsize { {I}_{\rm{2}} = - \frac{{E}_{\rm{a}} - {Z}_{\rm{1}}{I}_{\rm{1}}}{ {Z}_{\rm{2}} } } $
$\scriptsize \boxed{ {I}_{\rm{0}} = - \frac{{E}_{\rm{a}} - {Z}_{\rm{1}}{I}_{\rm{1}}}{ {Z}_{\rm{0}} + 3{Z}_{\rm{F}} } } $
$\scriptsize {I}_{\rm{a}} = {I}_{\rm{0}} +{I}_{\rm{1}} + {I}_{\rm{2}} = 0$
$\tiny \boxed{ {I}_{\rm{1}} = \frac{{E}_{\rm{a}}}{ {Z}_{\rm{1}} + \frac{ {Z}_{\rm{2}} ({Z}_{\rm{0}} + 3 {Z}_{\rm{F}} ) }{ {Z}_{\rm{2}} + {Z}_{\rm{0}} + 3{Z}_{\rm{F}} } } } $
Faltas Trifásicas Assimétricas - Curto-circuito entre Duas Linhas e Terra
$\scriptsize \boxed{ {I}_{\rm{0}} = - \frac{{E}_{\rm{a}} - {Z}_{\rm{1}}{I}_{\rm{1}}}{ {Z}_{\rm{0}} + 3{Z}_{\rm{F}} } } $
$\scriptsize \boxed{ {I}_{\rm{1}} = \frac{{E}_{\rm{a}}}{ {Z}_{\rm{1}} + \frac{ {Z}_{\rm{2}} ({Z}_{\rm{0}} + 3 {Z}_{\rm{F}} ) }{ {Z}_{\rm{2}} + {Z}_{\rm{0}} + 3{Z}_{\rm{F}} } } } $
Faltas Trifásicas Assimétricas - Método Básico - Exemplo 4.5.1
Considere uma falta na Barra 3 do circuito da figura acima. Sabendo que as impedâncias de sequência zero, positiva e negativa ($\small {Z}_{\rm{3}}^{\rm{0}}$, $\small {Z}_{\rm{3}}^{\rm{1}}$ e $\small {Z}_{\rm{3}}^{\rm{2}}$), vistas desde a Barra 3, são iguais a $\small j$0,35, $\small j$0,22 e $\small j$0,22, respectivamente, calcular as correntes de falta na própria Barra 3 ($\small {I}_{\rm{3}}^{\rm{a}}$, $\small {I}_{\rm{3}}^{\rm{b}}$ e $\small {I}_{\rm{3}}^{\rm{c}}$) para os três tipos de faltas assimétricas.
Falta Linha-Terra:
As compoentes de sequência da corrente de falta são:
$\small {I}_{\rm{3}}^{\rm{0}} = {I}_{\rm{3}}^{\rm{1}} = {I}_{\rm{3}}^{\rm{2}} = \frac{{{V}_{\rm{3}}^{\rm{a}}(0) }}{{Z}_{\rm{33}}^{\rm{1}} + {Z}_{\rm{33}}^{\rm{2}} + {Z}_{\rm{33}}^{\rm{0}} + 3{Z}_{F}}$
$\small {I}_{\rm{3}}^{\rm{0}} = {I}_{\rm{3}}^{\rm{1}} = {I}_{\rm{3}}^{\rm{2}} = \frac{{1,0 }}{j0,22 + j0,22 + j0,35 + 3(j0,1)}$
$\small {I}_{\rm{3}}^{\rm{0}} = {I}_{\rm{3}}^{\rm{1}} = {I}_{\rm{3}}^{\rm{2}} = -j0,9174$ pu
Logo as correntes de falta são:
$\small \left[ {\begin{array}{*{20}{c}} {I}_{\rm{3}}^{\rm{a}}\\ {I}_{\rm{3}}^{\rm{b}}\\ {I}_{\rm{3}}^{\rm{c}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {1}&{1}&{1}\\ {1}&{a^2}&{a}\\ {1}&{a}&{a^2} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {I}_{\rm{3}}^{\rm{0}}\\ {I}_{\rm{3}}^{\rm{1}}\\ {I}_{\rm{3}}^{\rm{2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} -j2,7523\\ 0\\ 0 \end{array}} \right] $ pu
Falta Linha-Linha:
As compoentes de sequência da corrente de falta são:
$\small {I}_{\rm{3}}^{\rm{0}} = 0 $
$\small {I}_{\rm{3}}^{\rm{1}} = - {I}_{\rm{3}}^{\rm{2}} = \frac{{{V}_{\rm{3}}^{\rm{a}}(0) }}{{Z}_{\rm{33}}^{\rm{1}} + {Z}_{\rm{33}}^{\rm{2}} + {Z}_{F}}$
$\small {I}_{\rm{3}}^{\rm{1}} = - {I}_{\rm{3}}^{\rm{2}} = \frac{{1,0 }}{j0,22 + j0,22 + j0,1}$
$\small {I}_{\rm{3}}^{\rm{1}} = - {I}_{\rm{3}}^{\rm{2}} = -j1,8519$ pu
Logo as correntes de falta são:
$\small \left[ {\begin{array}{*{20}{c}} {I}_{\rm{3}}^{\rm{a}}\\ {I}_{\rm{3}}^{\rm{b}}\\ {I}_{\rm{3}}^{\rm{c}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {1}&{1}&{1}\\ {1}&{a^2}&{a}\\ {1}&{a}&{a^2} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {I}_{\rm{3}}^{\rm{0}}\\ {I}_{\rm{3}}^{\rm{1}}\\ {I}_{\rm{3}}^{\rm{2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0\\ -3,2075\\ 3,2075 \end{array}} \right] $ pu
Falta Linha-Linha-Terra:
As compoentes de sequência da corrente de falta são:
$\small {I}_{\rm{3}}^{\rm{1}} = \frac{{V}_{\rm{3}}^{\rm{a}}(0)}{ {Z}_{\rm{33}}^{\rm{1}} + \frac{ {Z}_{\rm{33}}^{\rm{2}} \left( { {Z}_{\rm{33}}^{\rm{0}} + 3 {Z}_{\rm{F}} } \right) }{ {Z}_{\rm{33}}^{\rm{2}} + {Z}_{\rm{33}}^{\rm{0}} + 3{Z}_{\rm{F}} } } = \frac{1,0}{ j0,22 + \frac{ j0,22 \left( { j0,35 + 3 (j0,1) } \right) }{ j0,22 + j0,35 + 3(j0,1) } } = -j2,6017 $ pu
$\small {I}_{\rm{3}}^{\rm{2}} = - \frac{{V}_{\rm{3}}^{\rm{a}}(0) - {Z}_{\rm{33}}^{\rm{1}}{I}_{\rm{3}}^{\rm{1}}}{ {Z}_{\rm{33}}^{\rm{2}}} = - \frac{1,0 - (j0,22)(-j2,6017)}{ j0,22 } = j1,9438 $ pu
$\small {I}_{\rm{3}}^{\rm{0}} = - \frac{{V}_{\rm{3}}^{\rm{a}}(0) - {Z}_{\rm{33}}^{\rm{1}}{I}_{\rm{3}}^{\rm{1}}}{ {Z}_{\rm{33}}^{\rm{0}} + 3{Z}_{\rm{F}} } = - \frac{1,0 - (j0,22)(-j2,6017)}{ j0,35 + 3(j0,1) } = j0,6579 $ pu
Logo as correntes de falta são:
$\scriptsize \left[ {\begin{array}{*{20}{c}} {I}_{\rm{3}}^{\rm{a}}\\ {I}_{\rm{3}}^{\rm{b}}\\ {I}_{\rm{3}}^{\rm{c}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {1}&{1}&{1}\\ {1}&{a^2}&{a}\\ {1}&{a}&{a^2} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {I}_{\rm{3}}^{\rm{0}}\\ {I}_{\rm{3}}^{\rm{1}}\\ {I}_{\rm{3}}^{\rm{2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0\\ 4,058 \angle{165,93^\circ}\\ 4,058 \angle{14,07^\circ} \end{array}} \right] $ pu
Análise de Sistemas de Energia Elétrica
Departamento de Engenharia Elétrica