Effect of Earth on Capacitance of Single Circuit Three Phase Line with Transposition

Effect of Earth on Capacitance of Single Circuit Three Phase Line with Transposition

Consider a three phase single circuit line with general spacing. The method of images can be applied for determining the capacitance of this line to take into consideration the effect of earth. Let us assume that the line is fully transposed. The conductors a, b and c carry the charges qa, qb and qc respectively and occupy the positions 1, 2 and 3 in the first portion of the transposition cycle while they occupy positions 2, 3 and 1 in the second portion of the transposition cycle and so on. Let a', b' and c' be images of conductors a, b, c respectively with charges - q_a, - q_b and - q_c. The transposition cycle is as shown in the Fig. 1.31.1.

The equations for the three sections of the transposition cycle can be written for the voltage drop Vab with three charged conductors and their images.

When phase 'a' is in position 1, 'b' in position 2 and 'c' in position 3.

When phase 'a' is in position 2, 'b' in position 3 and 'c' in position 1

The special case of above is the three conductors which are equilaterally spaced at a distance of d from each other as shown in the Fig. 1.31.2.

Here, D₁₂ = D₂₃ = D₃₁ = d

Substituting this in the expression for C_an obtained above, the capacitance to neutral of three phase equilaterally spaced conductors taking into account the effect of earth can be computed. 

Example 1.31.1 Find the capacitance per km per phase to neutral of a three phase line arranged as shown in the Fig. 1.31.3. The outside diameter of ACSR conductor is 2.50 cm. The line is transposed. Take the effect of ground into consideration.

Solution : Here the method of images is to be applied for the calculation of capacitance. Consider the three conductors A, B, C carrying changes q_B and q_C respectively. The line is transposed. This is shown in the Fig. 1.31.3 (a).

The plane of the earth is as shown in the Fig. 1.31.3 (a) below the ground there are image conductors A',B',C' carrying the charges q_A ,q_B and q_C, respectively.

Writing the voltage drop V_AB for a complete transposition cycle. Similar equation can be written for voltage drop

V_AC 311(1 then usinS V_AB + V_AC = 3 V_AN the capacitance to neutral value can be obtained.

With conductor A in position 1, B in position 2, C in position 3 we have, 

Similar equations for V_AB are written for the other parts of the transposition cycle.

Simplifying the equations as explained earlier we get,

Example 1.31.2 Find the capacitance per km per phase of a 3 phase line arranged in a horizontal plane spaced 8 meters apart. The height of all conductors above the earth is 13 m. The diameter of each conductor is 2.6 cm. The line is completely transposed. Take the effect of ground into account.

Solution : Here the method of images is to be applied for the calculation of capacitance. Consider the three conductors A, B, C carrying changes q_A, q_B and q_C respectively. The line is transposed. This is shown in Fig. 1.31.4 (a).

The plane of the earth is as shown in Fig. 1.31.4 (a). Below the ground there are image conductors A', B', C' carrying the charges q_A', q_B' and q_C' respectively.

Writing the voltage drop V_AB for a complete transposition cycle similar equation can be written for voltage drop V_AC then using V_AB + V_AC = 3 V_AN the capacitance to neutral value can be obtained.

With conductor A in position 1, B in position 2, C in position 3 we have,

Similar equations for V_AB are written for the other parts of the transposition cycle. Simplifying the equations as explained earlier we get,

Review Question

1. Derive the expression for the capacitance per phase of a 3-phase overhead transmission line with unsymmetrical spacing of conductors taking into account the effect of earth. Assume complete transposition. Comment on the effect of earth on the capacitance 0/ the transmission line.