Inductance of Three Phase Lines with Equilateral and Symmetrical Spacing

Inductance of Three Phase Lines with Equilateral and Symmetrical Spacing

Consider a three phase line consisting of three conductors a, b and c as shown in the Fig. 1.17.1. These three conductors are equally spaced at the comers of an equilateral triangle having radius r.

The flux linkages of conductor a are given by,

If the currents are assumed to be balanced then

Because of symmetry, conductors b and c will have same inductance as that of conductor a. Each phase consists of only one conductor. So the above equation gives inductance per phase of the three phase lines. For stranded conductor we write Ds whereas for single conductor it is replaced by r'.

Example 1.17.1 A three phase transmission line 100 km long has its conductors of 1.63 cm diameter spaced at the corners of an equilateral triangle of 100 cm side. The arrangement is as shown in the Fig. 1.17.2. Find the inductance per phase of the system.

Solution : D = Distance between the conductors = 3 m

Example 1.17.2 A three phase line with equilateral spacing of 3 m is to be rebuilt with horizontal spacing (D₁₃ = 2 D₁₂ = 2 D₃₂). The conductors are to be fully transposed. Determine the spacing between adjacent conductors such that the new line has the same inductance as the original line. 

Solution : Consider the three phase line with equilateral spacing. Let us find the inductance of this line.

D = Distance between the conductors = 3 m

Let r be the radius of each of the conductors.

The inductance of the line per phase per km is given by,

This line is to be rebuilt with horizontal spacing such that

The radius of the conductor remains same as r

The inductance per phase per km with this arrangement is given by,

Equating equations (I) and (II) as the inductance in both the arrangements remains same.

Review Questions

1. Derive an expression for the inductance of a three phase overhead transmission line when conductors are symmetrically spaced.

2. Find the inductance per km of a three phase transmission line using 1.24 cm diameter conductors when these are placed at the corners of an equilateral triangle of each side 2 m.

[Ans.: 1.2052 × 10^-3 H/km]

3. A three phase transmission line 100 km long has its conductors of 0.6 cm diameter spaced at the corners of an equilateral triangle of 100 cm side. The arrangement is as shown in the Fig. 1.17.4. Find the inductance per phase of the system.  

[Ans.: 0.1211 H]