The phase frame matrix can be used to accurately determine the voltage drops on the feeder line
segments once the currents flowing have been determined. Since no approximations (transposition, for
example) have been made regarding the spacing between conductors, the effect of the mutual coupling
between phases is accurately taken into account. The application of Carson’s equations and the phase
frame matrix leads to the most accurate model of a line segment. Figure 21.3 shows the equivalent circuit
of a line segment.
The voltage equation in matrix form for the line segment is given by the following equation:
Vag
Vbg
Vcg2
43
5
n¼Vag
Vbg
Vcg2
43
5
mþZaa Zab Zac
Zba Zbb Zbc
Zca Zcb Zcc2
43
5Ia
Ib
Ic2
43(^5) (21:14)
whereZij¼zijlength
The phase impedance matrix is defined in Eq. (21.15). The phase impedance matrix for single-phase
and V-phase lines will have a row and column of zeros for each missing phase
½¼Zabc
Zaa Zab Zac
Zba Zbb Zbc
Zca Zcb Zcc
2
4
3
(^5) (21:15)
Equation (21.14) can be written in condensed form as
½VLGabcn¼½VLGabcmþ½Zabc½Iabc (21:16)
This condensed notation will be used throughout the document.
21.1.1.5 Sequence Impedances
Many times the analysis of a feeder will use the positive and zero sequence impedances for the line
segments. There are basically two methods for obtaining these impedances. The first method incorpor-
ates the application of Carson’s equations and the Kron reduction to obtain the phase frame impedance
matrix. The 33 ‘‘sequence impedance matrix’’ can be obtained by
½¼z 012 ½As^1 ½zabc½As V=mile (21:17)
where
½¼As
11 1
1 a^2 s as
1 as a^2 s
2
4
3
(^5) (21:18)
as¼ 1 : 0 ff 120 a^2 s¼ 1 : 0 ff 240
Node n Node m
Vagn
Vbgn
Vcgn Vcgm
Vbgm
Zab Zca Vagm
Zcc Zbc
Zbb
Ia Zaa
Ib
Ic
−−−−−−
FIGURE 21.3 Three-phase line segment.