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11.7 APPENDIX VIII. DERIVATION OF PHASE FAULT DETECTION USING THE NEGATIVE SEQUENCE IMPEDANCE PLANE Relating the symmetrical component current to the phase currents. 0 2 2 1 0 2 1 2 0 2 1 I I a I a Ic I I a I a Ib I I I Ia = = = Equation A11.8.1 Assume I 0 is zero for phase-to-phase faults. Compute the phase-to-phase currents as the difference between any two of the three phase currents in Equation A11.8.1 through Equation A11.8.4. ( ( ( ( 2 2 1 2 1 2 0 2 1 2 0 2 1 I 30 3 2 Iab I 30 3 60 I 60 1 30 3 Iab I a 1 I a 1 Iab , I I a I a I I I Ib Ia Iab ° - = ° - ° ° - ° = - - = - = - = Equation A11.8.2 2 2 2 1 2 I a 3 j 2 I 330 3 2 Iab faults AB for 60 1 I I : where = ° = ° = ( ( ( ( 2 2 1 2 0 2 2 1 0 2 1 2 I a a I a a Ibc I I a I a I I a I a Ic Ib Ibc - - = - = - = Equation A11.8.3 ( ( 2 2 1 2 2 1 I 3 j 2 I I a a Ibc faults BC for I I : where = - - = - = ( 3 j a a : where 2 = - ( ( ( ( 2 2 1 2 2 1 0 2 1 0 2 2 1 I 30 3 2 Ica I 210 3 60 I 60 1 150 3 Ica I 1 a I 1 a Ica I I I I I a I a Ia Ic Ica ° - = ° ° - ° ° = - - = - = - = Equation A11.8.4 2 2 1 2 I a 3 j 2 I 210 3 2 Ica faults CA for 60 1 I I : where = ° = ° - = Substitute the following identities into Equation A11.8.2 through Equation A11.8.4.

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  • 11.7 APPENDIX VIII. DERIVATION OF PHASE FAULT DETECTION USING THE NEGATIVE SEQUENCE IMPEDANCE PLANE

    Relating the symmetrical component current to the phase currents.

    022

    1

    0212

    021

    IIaIaIc

    IIaIaIb

    IIIIa

    ++=

    ++=

    ++=

    Equation A11.8.1

    Assume I0 is zero for phase-to-phase faults. Compute the phase-to-phase currents as the difference between any two of the three phase currents in Equation A11.8.1 through Equation A11.8.4.

    ( ) ( )( ) ( )

    2

    21

    212

    0212

    021

    I3032Iab

    I30360I601303Iab

    Ia1Ia1Iab

    ,IIaIaIIIIbIaIab

    -=

    -+-=

    -+-=

    ++-++=-=

    Equation A11.8.2

    22

    2

    12

    Ia3j2I33032Iab

    faultsABfor601II

    :where

    ==

    =

    ( ) ( )( ) ( ) 2212

    022

    10212

    IaaIaaIbc

    IIaIaIIaIaIcIbIbc

    -+-=

    ++-++=-= Equation A11.8.3

    ( ) ( ) 221221

    I3j2IIaaIbc

    faultsBCforII

    :where

    =--=

    -=

    ( ) 3jaa:where

    2 =-

    ( ) ( )( ) ( )

    2

    21

    22

    1

    021022

    1

    I3032Ica

    I210360I6011503Ica

    I1aI1aIca

    IIIIIaIaIaIcIca

    -=

    +-=

    -+-=

    ++-++=-=

    Equation A11.8.4

    22

    12

    Ia3j2I21032Ica

    faultsCAfor601II

    :where

    ==

    -=

    Substitute the following identities into Equation A11.8.2 through Equation A11.8.4.


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