Electric Power Generation, Transmission, and Distribution

The GMDs as defined above are used inEqs. (21.9)and (21.10)to determine the various self and mutual
impedances of the line resulting in

^zzii¼riþ 0 : 0953 þj 0 :12134 ln

1

GMRi



þ 7 : 93402



(21:26)

^zznn¼rnþ 0 : 0953 þj 0 :12134 ln

1

GMRn



þ 7 : 93402



(21:27)

^zzij¼ 0 : 0953 þj 0 :12134 ln

1

Dij



þ 7 : 93402



(21:28)

^zzin¼ 0 : 0953 þj 0 :12134 ln

1

Din



þ 7 : 93402



(21:29)

Equations (21.26) through (21.29) will define a matrix of order ncondncond, where ncond is
the number of conductors (phases plus neutrals) in the line segment. Application of the Kron reduction
[Eq. (21.13)] and the sequence impedance transformation [Eq. (21.23)] lead to the following expres-
sions for the zero, positive, and negative sequence impedances:

z 00 ¼^zziiþ 2 ^zzij 3

^zzi^2 n

^zznn



V=mile (21:30)

z 11 ¼z 22 ¼^zzii^zzij

z 11 ¼z 22 ¼riþj 0 : 12134 ln

Dij

GMRi



V=mile (21:31)

Equation (21.31) is recognized as the standard equation for the calculation of the line impedances when
a balanced three-phase system and transposition are assumed.

Example 21.1

The spacings for an overhead three-phase distribu-
tion line are constructed as shown in Fig. 21.4. The
phase conductors are 336,400 26=7 ACSR (Linnet)
and the neutral conductor is 4= 06 =1 ACSR.

a. Determine the phase impedance matrix.

b. Determine the positive and zero sequence

impedances.

Solution

From the table of standard conductor data, it is
found that

336,400 26=7 ACSR: GMR¼ 0 :0244 ft

Resistance¼ 0 : 306 V=mile

4 = 06 =1 ACSR: GMR¼ 0 :00814 ft

Resistance¼ 0 : 5920 V=mile

a bc

n

4.5

3.0

4.0

2.5

FIGURE 21.4 Three-phase distribution line spacings.