Electric Power Generation, Transmission, and Distribution

The resulting sequence impedance matrix is of the form:

½Š¼z 012

z 00 z 01 z 02

z 10 z 11 z 12

z 20 z 21 z 22

2

4

3

(^5) V=mile (21:19)
wherez 00 ¼the zero sequence impedance
z 11 ¼the positive sequence impedance
z 22 ¼the negative sequence impedance
In the idealized state, the off-diagonal terms of Eq. (21.19) would be zero. When the off-diagonal terms
of the phase impedance matrix are all equal, the off-diagonal terms of the sequence impedance matrix
will be zero. For high-voltage transmission lines, this will generally be the case because these lines are
transposed, which causes the mutual coupling between phases (off-diagonal terms) to be equal.
Distribution lines are rarely if ever transposed. This causes unequal mutual coupling between phases,
which causes the off-diagonal terms of the phase impedance matrix to be unequal. For the nontran-
sposed line, the diagonal terms of the phase impedance matrix will also be unequal. In most cases, the
off-diagonal terms of the sequence impedance matrix are very small compared to the diagonal terms and
errors made by ignoring the off-diagonal terms are small.
Sometimes the phase impedance matrix is modified such that the three diagonal terms are equal and
all of the off-diagonal terms are equal. The usual procedure is to set the three diagonal terms of the phase
impedance matrix equal to the average of the diagonal terms of Eq. (21.15) and the off-diagonal terms
equal to the average of the off-diagonal terms of Eq. (21.15). When this is done, the self and mutual
impedances are defined as
zs¼
1
3
ðÞzaaþzbbþzcc (21:20)
zm¼
1
3
ðÞzabþzbcþzca (21:21)
The phase impedance matrix is now defined as
½Š¼zabc
zs zm zm
zm zs zm
zm zm zs
2
4
3
(^5) (21:22)
When Eq. (21.17) is used with this phase impedance matrix, the resulting sequence matrix is diagonal
(off-diagonal terms are zero). The sequence impedances can be determined directly as
z 00 ¼zsþ 2 zm
z 11 ¼z 22 ¼zszm
(21:23)
A second method that is commonly used to determine the sequence impedances directly is to employ
the concept of geometric mean distances (GMDs). The GMD between phases is defined as
Dij¼GMDij¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(^3) DabDbcDca
p
(21:24)
The GMD between phases and neutral is defined as
Din¼GMDin¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(^3) DanDbnDcn
p
(21:25)