Faltas Trifásicas Assimétricas - Método da Matriz $\small Z_{\rm{Barra}}$ - Exemplo 4.2.2

Considere uma falta na Barra 3 do circuito da figura acima. Sabendo que as componentes de sequência da matriz $\small Z_{\rm{Barra}}$ são as dadas acima, calcular as correntes de falta na Barra 3 ($\small {I}_{\rm{3}}^{\rm{a}}$, $\small {I}_{\rm{3}}^{\rm{b}}$ e $\small {I}_{\rm{3}}^{\rm{c}}$), assim como as tensões de falta nas três barras ($\small {V}_{\rm{1}}^{\rm{abc}}$, $\small {V}_{\rm{2}}^{\rm{abc}}$ e $\small {V}_{\rm{3}}^{\rm{abc}}$), 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

$\small \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}} -j0,9174\\ -j0,9174\\ -j0,9174 \end{array}} \right] $ pu

As componentes de sequência das tensões durante a falta são:

$\scriptsize {V}_{\rm{1}}^{\rm{012}} = \left[ {\begin{array}{*{20}{c}} - {Z}_{\rm{13}}^{\rm{0}} {I}_{\rm{3}}^{\rm{0}} \\ {V}_{\rm{1}}^{\rm{1}}(0) - {Z}_{\rm{13}}^{\rm{1}} {I}_{\rm{3}}^{\rm{1}} \\ - {Z}_{\rm{13}}^{\rm{2}} {I}_{\rm{3}}^{\rm{2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} -j0,140(-j0,9174)\\ 1,0 - j0,130(-j0,9174)\\ -j0,130(-j0,9174) \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} -0,1284\\ 0,8807\\ -0,1193 \end{array}} \right] $ pu

$\scriptsize {V}_{\rm{2}}^{\rm{012}} = \left[ {\begin{array}{*{20}{c}} - {Z}_{\rm{23}}^{\rm{0}} {I}_{\rm{3}}^{\rm{0}} \\ {V}_{\rm{2}}^{\rm{1}}(0) - {Z}_{\rm{23}}^{\rm{1}} {I}_{\rm{3}}^{\rm{1}} \\ - {Z}_{\rm{23}}^{\rm{2}} {I}_{\rm{3}}^{\rm{2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} -j0,065(-j0,9174)\\ 1,0 - j0,120(-j0,9174)\\ -j0,120(-j0,9174) \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} -0,0596\\ 0,8899\\ -0,1101 \end{array}} \right] $ pu

$\scriptsize {V}_{\rm{3}}^{\rm{012}} = \left[ {\begin{array}{*{20}{c}} - {Z}_{\rm{33}}^{\rm{0}} {I}_{\rm{3}}^{\rm{0}} \\ {V}_{\rm{3}}^{\rm{1}}(0) - {Z}_{\rm{33}}^{\rm{1}} {I}_{\rm{3}}^{\rm{1}} \\ - {Z}_{\rm{33}}^{\rm{2}} {I}_{\rm{3}}^{\rm{2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} -j0,350(-j0,9174)\\ 1,0 - j0,220(-j0,9174)\\ -j0,220(-j0,9174) \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} -0,3211\\ 0,7982\\ -0,2018 \end{array}} \right] $ pu

Logo as tensões durante a falta são:

$\small {V}_{\rm{1}}^{\rm{abc}} = \left[ {\begin{array}{*{20}{c}} {1}&{1}&{1}\\ {1}&{a^2}&{a}\\ {1}&{a}&{a^2} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} -0,1284\\ 0,8807\\ -0,1193 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0,633\angle{0^\circ}\\ 1,0046\angle{-120,45^\circ}\\ 1,0046\angle{120,45^\circ} \end{array}} \right] $ pu

$\small {V}_{\rm{2}}^{\rm{abc}} = \left[ {\begin{array}{*{20}{c}} {1}&{1}&{1}\\ {1}&{a^2}&{a}\\ {1}&{a}&{a^2} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} -0,0596\\ 0,8899\\ -0,1101 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0,7207\angle{0^\circ}\\ 0,9757\angle{-117,43^\circ}\\ 0,9757\angle{117,43^\circ} \end{array}} \right] $ pu

$\small {V}_{\rm{3}}^{\rm{abc}} = \left[ {\begin{array}{*{20}{c}} {1}&{1}&{1}\\ {1}&{a^2}&{a}\\ {1}&{a}&{a^2} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} -0,3211\\ 0,7982\\ -0,2018 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0,2752\angle{0^\circ}\\ 1,0647\angle{-125,56^\circ}\\ 1,0647\angle{125,56^\circ} \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

$\small \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\\ -j1,8519\\ j1,8519 \end{array}} \right] $ pu

As componentes de sequência das tensões durante a falta são:

$\scriptsize {V}_{\rm{1}}^{\rm{012}} = \left[ {\begin{array}{*{20}{c}} 0 \\ {V}_{\rm{1}}^{\rm{1}}(0) - {Z}_{\rm{13}}^{\rm{1}} {I}_{\rm{3}}^{\rm{1}} \\ - {Z}_{\rm{13}}^{\rm{2}} {I}_{\rm{3}}^{\rm{2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0\\ 1,0 - j0,130(-j1,8519)\\ -j0,130(j1,8519) \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0\\ 0,7593\\ 0,2407 \end{array}} \right] $ pu

$\scriptsize {V}_{\rm{2}}^{\rm{012}} = \left[ {\begin{array}{*{20}{c}} 0 \\ {V}_{\rm{2}}^{\rm{1}}(0) - {Z}_{\rm{23}}^{\rm{1}} {I}_{\rm{3}}^{\rm{1}} \\ - {Z}_{\rm{23}}^{\rm{2}} {I}_{\rm{3}}^{\rm{2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0\\ 1,0 - j0,120(-j1,8519)\\ -j0,120(j1,8519) \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0\\ 0,7778\\ 0,2222 \end{array}} \right] $ pu

$\scriptsize {V}_{\rm{3}}^{\rm{012}} = \left[ {\begin{array}{*{20}{c}} 0 \\ {V}_{\rm{3}}^{\rm{1}}(0) - {Z}_{\rm{33}}^{\rm{1}} {I}_{\rm{3}}^{\rm{1}} \\ - {Z}_{\rm{33}}^{\rm{2}} {I}_{\rm{3}}^{\rm{2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0\\ 1,0 - j0,220(-j1,8519)\\ -j0,220(j1,8519) \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0\\ 0,5926\\ 0,4074 \end{array}} \right] $ pu

Logo as tensões durante a falta são:

$\small {V}_{\rm{1}}^{\rm{abc}} = \left[ {\begin{array}{*{20}{c}} {1}&{1}&{1}\\ {1}&{a^2}&{a}\\ {1}&{a}&{a^2} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} 0\\ 0,7593\\ 0,2407 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1,0\angle{0^\circ}\\ 0,672\angle{-138,07^\circ}\\ 0,672\angle{138,07^\circ} \end{array}} \right] $ pu

$\small {V}_{\rm{2}}^{\rm{abc}} = \left[ {\begin{array}{*{20}{c}} {1}&{1}&{1}\\ {1}&{a^2}&{a}\\ {1}&{a}&{a^2} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} 0\\ 0,7778\\ 0,2222 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1,0\angle{0^\circ}\\ 0,6939\angle{-136,10^\circ}\\ 0,6939\angle{136,10^\circ} \end{array}} \right] $ pu

$\small {V}_{\rm{3}}^{\rm{abc}} = \left[ {\begin{array}{*{20}{c}} {1}&{1}&{1}\\ {1}&{a^2}&{a}\\ {1}&{a}&{a^2} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} 0\\ 0,5923\\ 0,4074 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1,0\angle{0^\circ}\\ 0,5251\angle{-162,21^\circ}\\ 0,5251\angle{162,21^\circ} \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:

$\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\\ 4,058 \angle{165,93^\circ}\\ 4,058 \angle{14,07^\circ} \end{array}} \right] $ pu

$\small \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}} 0,6579\\ -j2,6017\\ j1,9438 \end{array}} \right] $ pu

As componentes de sequência das tensões durante a falta são:

$\scriptsize {V}_{\rm{1}}^{\rm{012}} = \left[ {\begin{array}{*{20}{c}} - {Z}_{\rm{13}}^{\rm{0}} {I}_{\rm{3}}^{\rm{0}} \\ {V}_{\rm{1}}^{\rm{1}}(0) - {Z}_{\rm{13}}^{\rm{1}} {I}_{\rm{3}}^{\rm{1}} \\ - {Z}_{\rm{13}}^{\rm{2}} {I}_{\rm{3}}^{\rm{2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} - j0,140(-j0,6579)\\ 1,0 - j0,130(-j2,6017)\\ -j0,130(j1,9438) \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0,0921\\ 0,6618\\ 0,2527 \end{array}} \right] $ pu

$\scriptsize {V}_{\rm{2}}^{\rm{012}} = \left[ {\begin{array}{*{20}{c}} - {Z}_{\rm{23}}^{\rm{0}} {I}_{\rm{3}}^{\rm{0}} \\ {V}_{\rm{2}}^{\rm{1}}(0) - {Z}_{\rm{23}}^{\rm{1}} {I}_{\rm{3}}^{\rm{1}} \\ - {Z}_{\rm{23}}^{\rm{2}} {I}_{\rm{3}}^{\rm{2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} - j0,065(-j0,6579)\\ 1,0 - j0,120(-j2,6017)\\ -j0,120(j1,9438) \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0,0428\\ 0,6878\\ 0,2333 \end{array}} \right] $ pu

$\scriptsize {V}_{\rm{3}}^{\rm{012}} = \left[ {\begin{array}{*{20}{c}} - {Z}_{\rm{33}}^{\rm{0}} {I}_{\rm{3}}^{\rm{0}} \\ {V}_{\rm{3}}^{\rm{1}}(0) - {Z}_{\rm{33}}^{\rm{1}} {I}_{\rm{3}}^{\rm{1}} \\ - {Z}_{\rm{33}}^{\rm{2}} {I}_{\rm{3}}^{\rm{2}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} - j0,350(-j0,6579)\\ 1,0 - j0,220(-j2,6017)\\ -j0,220(j1,9438) \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0,2303\\ 0,4276\\ 0,4276 \end{array}} \right] $ pu

Logo as tensões durante a falta são:

$\small {V}_{\rm{1}}^{\rm{abc}} = \left[ {\begin{array}{*{20}{c}} {1}&{1}&{1}\\ {1}&{a^2}&{a}\\ {1}&{a}&{a^2} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} 0,0921\\ 0,6618\\ 0,2527 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1,0066\angle{0^\circ}\\ 0,5088\angle{-135,89^\circ}\\ 0,5088\angle{135,89^\circ} \end{array}} \right] $ pu

$\small {V}_{\rm{2}}^{\rm{abc}} = \left[ {\begin{array}{*{20}{c}} {1}&{1}&{1}\\ {1}&{a^2}&{a}\\ {1}&{a}&{a^2} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} 0,0428\\ 0,6878\\ 0,2333 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0,9638\angle{0^\circ}\\ 0,5740\angle{-136,70^\circ}\\ 0,5740\angle{136,70^\circ} \end{array}} \right] $ pu

$\small {V}_{\rm{3}}^{\rm{abc}} = \left[ {\begin{array}{*{20}{c}} {1}&{1}&{1}\\ {1}&{a^2}&{a}\\ {1}&{a}&{a^2} \end{array}} \right] \left[ {\begin{array}{*{20}{c}} 0,2303\\ 0,4276\\ 0,4276 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1,0855\angle{0^\circ}\\ 0,1974\angle{-180^\circ}\\ 0,1974\angle{180^\circ} \end{array}} \right] $ pu