Electric Power Generation, Transmission, and Distribution

SolveEq. (21.71)for the load voltages:

½ŠVLGabcm¼½ŠVLGabcn 1 : 5 ½ŠZabc½ŠIabcn¼

6761 : 10 ff 2 : 32

6877 : 7 ff 122 : 43

6836 : 33 ff 117 : 21

2

6

4

3

7

5

The voltage unbalance at the load using the NEMA definition is

Vunbalance¼

max (Vdeviation)

Vavg

100 ¼ 0 :937%

The point of Example 21.7 is to demonstrate that even though the system is perfectly balanced at the
substation, the unequal mutual coupling between the phases results in a significant voltage unbalance at
the load; significant because NEMA requires that induction motors be derated when the voltage
unbalance is 1% or greater.

21.1.3.2 Approximate Line Segment Model

Many times the only data available for a line segment will be the positive and zero sequence impedances.
An approximate three-phase line segment model can be developed by applying the ‘‘reverse impedance
transformation’’ from symmetrical component theory.
Using the known positive and zero sequence impedances, the ‘‘sequence impedance matrix’’ is given by

Zseq



¼

Z 0 00

0 Zþ 0

00 Zþ

2

4

3

(^5) (21:74)
The reverse impedance transformation results in the following ‘‘approximate phase impedance matrix:’’
Zapprox

¼½ŠAs Zseq

½ŠAs^1 ¼
1
3
ðÞ 2 ZþZ 0 ðÞZ 0 Zþ ðÞZ 0 Zþ
ðÞZ 0 Zþ ðÞ 2 ZþZ 0 ðÞZ 0 Zþ
ðÞZ 0 Zþ ðÞZ 0 Zþ ðÞ 2 ZþZ 0
2
4
3
(^5) (21:75)
Notice that the approximate phase impedance matrix is characterized by the three diagonal terms being
equal and all mutual terms being equal. This is the same result that is achieved if the line is assumed to
be transposed. Substituting the approximate phase impedance matrix into Eq. (21.71) results in
Van
Vbn
Vcn
2
4
3
5
n
¼
Van
Vbn
Vcn
2
4
3
5
m
þ
1
3
ðÞ 2 ZþZ 0 ðÞZ 0 Zþ ðÞZ 0 Zþ
ðÞZ 0 Zþ ðÞ 2 ZþZ 0 ðÞZ 0 Zþ
ðÞZ 0 Zþ ðÞZ 0 Zþ ðÞ 2 ZþZ 0
2
4
3
5
Ia
Ib
Ic
2
4
3
5
n
(21:76)
Equation (21.76) can be expanded and an equivalent circuit for the approximate line segment model can
be developed. This approximate model is shown inFig. 21.12.
The errors made by using this approximate line segment model are demonstrated in the following
example.
Example 21.8
For the line of Example 21.7, the positive and zero sequence impedances were determined to be
Zþ¼ 0 : 3061 þj 0 : 6270 V=mile
Z 0 ¼ 0 : 7735 þj 1 : 9373 V=mile