Electric Power Generation, Transmission, and Distribution

Substitute intoEq. (21.54):

Yts¼j

77 : 586

ln

Rb

Ra

¼j^77 :^586

ln

0 : 0452

0 : 0153

¼j 71 : 8169 mS=mile

21.1.3 Line Segment Models

21.1.3.1 Exact Line Segment Model

The exact model of a three-phase line segment is shown in Fig. 21.11. For the line segment in Fig. 21.11,
the equations relating the input (noden) voltages and currents to the output (nodem) voltages and
currents are

½ŠVLGabcn¼½Ša½ŠVLGabcmþ½Šb½ŠIabcm (21:55)

½ŠIabcn¼½Šc½ŠVLGabcmþ½Šd½ŠIabcm (21:56)

where

½Š¼a ½ŠU

1

2

½ŠZabc½ŠYabc (21:57)

½Š¼b ½ŠZabc (21:58)

½Šc¼½ŠYabc

1

4

½ŠZabc½ŠYabc^2 (21:59)

½Š¼d ½ŠU

1

2

½ŠZabc½ŠYabc (21:60)

In Eqs. (21.57) through (21.60), the impedance matrix [Zabc] and the admittance matrix [Yabc] are
defined earlier in this document.
Sometimes it is necessary to determine the voltages at nodemas a function of the voltages at noden
and the output currents at nodem. The necessary equation is

½ŠVLGabcm¼½ŠA½ŠVLGabcn½ŠB½ŠIabcm (21:61)

where

½Š¼A ½ŠþU

1

2

½ŠZabc½ŠYabc

 1

(21:62)

Node NNNNNN n

Vagn

Vcgn

Zaa

Zbb

Zcc

Zca

Zbc

[Yabc]

[ICabc]n [ICabc]m

1

2

Zab

Icn

Ibn

Ian

Icm

Ibm

Iam

Vbgn

Node m

Vagm

Vcgm

Vbgm

1

2 [Yabc]

+

+

+

+

+

+

FIGURE 21.11 Three-phase line segment model.