Inductance of Three-Phase Lines with Asymmetrical Spacing

It is rather difficult to maintain symmetrical spacing as shown in Fig. 1.6 while constructing a transmission line. With asymmetrical spacing between the phases, the voltage drop due to line inductance will be unbalanced even when the line currents are balanced. Consider the three-phase asymmetrically spaced line shown in Fig. 1.7 in which the radius of each conductor is assumed to be r . The distances between the phases are denoted by D_ab, D_bcand D_ca. We then get the following flux linkages for the three phases

	(1.31)
	(1.32)
	(1.33)

Fig. 1.7 Three-phase asymmetrically spaced line.

Let us define the following operator

(1.34)

Note that for the above operator the following relations hold

(1.35)

Let as assume that the current are balanced. We can then write

Substituting the above two expressions in (1.31) to (1.33) we get the inductance of the three phases as

	(1.36)
	(1.37)
	(1.38)

It can be seen that the inductances contain imaginary terms. The imaginary terms will vanish only when D_ab= D_bc= D_ca. In that case the inductance will be same as given by (1.30).