Electric Power Generation, Transmission, and Distribution

With nine different inductances in a simple three-phase system the analysis could be a little
more complicated. However, a single inductance per phase can be obtained if the three conductors
are arranged with the same separation among them (symmetrical arrangement), where
D¼DAB¼DBC¼DCA. For a balanced three-phase system (IAþIBþIC¼0, orIA¼IBIC), the flux
linkage of each conductor, per unit length, will be the same. From Eq. (13.50) we have

lA¼

m 0

2 p

ðÞIBIC ln

1

GMRA



þIBln

1

D



þICln

1

D



lA¼

m 0

2 p

IBln

D

GMRA



ICln

D

GMRA



lA¼

m 0

2 p

IAln

D

GMRA



ðÞWb=m

(13:54)

If GMR value is the same for all conductors (either single or bundle GMR), the total flux linkage
expression is the same for all phases. Therefore, the equivalent inductance per phase is

Lphase¼

m 0

2 p

ln

D

GMRphase



ðÞH=m (13:55)

13.4.6 Inductance of Transposed Three-Phase Transmission Lines

In actual transmission lines, the phase conductors cannot maintain symmetrical arrangement along the
whole length because of construction considerations, even when bundle conductor spacers are used.
With asymmetrical spacing, the inductance will be different for each phase, with a corresponding
unbalanced voltage drop on each conductor. Therefore, the single-phase equivalent circuit to represent
the power system cannot be used.
However, it is possible to assume symmetrical arrangement in the transmission line by transposing the
phase conductors. In a transposed system, each phase conductor occupies the location of the other two
phases for one-third of the total line length as shown in Fig. 13.13. In this case, the average distance
geometrical mean distance (GMD) substitutes distanceD, and the calculation of phase inductance
derived for symmetrical arrangement is still valid.
The inductance per phase per unit length in a transmission line becomes

Lphase¼

m 0

2 p

ln

GMD

GMRphase



ðÞH=m (13:56)

Once the inductance per phase is obtained, the inductive reactance per unit length is

XLphase¼ 2 pfLphase¼m 0 fln

GMD

GMRphase



ðÞV=m (13:57)

A

B

C

C

B

A

A

C

B

l/ 3 l/ 3 l/ 3

FIGURE 13.13 Arrangement of conductors in a transposed line.