Chapter 1: Modelling Power System Components

Inductance of Three-Phase Lines with Symmetrical Spacing

Consider the three-phase line shown in Fig. 1.6. Each of the conductors has a radius of r and their centers form an equilateral triangle with a distance D between them. Assuming that the currents are balanced, we have

 (1.23)

Consider a point P external to the conductors. The distance of the point from the phases a, b and c are denoted by Dpa, Dpband Dpcrespectively.

Fig. 1.6 Three-phase symmetrically spaced conductors and an external point P.

Let us assume that the flux linked by the conductor of phase-a due to a current Ia includes the internal flux linkages but excludes the flux linkages beyond the point P . Then from (1.18) we get

 (1.24)

The flux linkage with the conductor of phase-a due to the current Ib , excluding all flux beyond the point P , is given by (1.17) as

 (1.25)

Similarly the flux due to the current Ic is

 (1.26)

Therefore the total flux in the phase-a conductor is

The above expression can be expanded as

 (1.27)

From (1.22) we get

Substituting the above expression in (1.27) we get

 (1.28)

Now if we move the point P far away, then we can approximate Dpa » Dpb » Dpc.. Therefore their logarithmic ratios will vanish and we can write (1.28) as

 (1.29)

Hence the inductance of phase-a is given as

 (1.30)

Note that due to symmetry, the inductances of phases b and c will be the same as that of phase-a given above, i.e., Lb= Lc = L a .