Effect of Earth on capacitances of Transmission Line

Effect of Earth on capacitances of Transmission Line

The effect of earth can be neglected while calculating the capacitances and conductors are assumed to be situated in free space. But actually the conductors run parallel to the ground. The earth can be assumed to behave like an infinite, perfectly conducting plane. Due to this, electric field of the line is changed. To take into account the effect of earth, the method of images as suggested by Lord Kelvin can be used to calculate the capacitance.

If two conductors carry equal and opposite charges, then there exists a zero potential plane half way between the conductors. Using this fact, it can be assumed that the field between a single overhead charged conductor and the perfectly conducting plane, the earth is identical with the field which would be produced by replacing the conducting plane by the hypothetical conductor similar to original conductor.

The capacitance of transmission line is affected by the presence of earth. Because of earth, electric field of a line is reduced. If we assume that the earth is a perfect conductor in the form of a horizontal plane of infinite extent, we realize that the electric field of charged conductors above the earth is not the same as it would be if the equipotential surface of earth were not present.

The method of images is used while considering this type of problems. For this consider a single phase line having 2 conductors as shown in the Fig. 1.28.1.

It uses the fact that a zero potential plane exists at midway between the two long parallel and oppositely charged conductors (say +q and -q resp.). If a conducting sheet of infinite dimension is placed at this zero potential plane, the electric field between the conductors is not affected. Also if the conductor with charge -q is now removed, then the electric field above the conducting sheet remains undisturbed while the electric field below it vanishes.

A fictitious conductor is placed below each conductor of the same size and shape as the overhead conductor lying directly below the original conductor at a distance equal to twice the distance of the conductor above the plane of ground. If the earth is removed and a charge equal and opposite to that an overhead conductor is assumed on the fictitious conductor, the plane midway between conductor and its image is an equipotential surface and occupies the same position as the equipotential surface of earth. This fictitious conductor is called image conductor having the charge opposite to that of overhead conductor.

Comparing above equation with expression for capacitance of single phase line without considering the effect of earth, we can see that earth tries to increase the capacitance of line by small amount But the effect is negligible if the conductors are high above ground compared to distances between them.

The above is the special case when the conductors a and b are at the same height h from the ground.

Let us consider the general case of a two wire single phase line having conductors a and b. The spacing between the conductors is D. a' and b' are the images of conductors a and b respectively. Let the charges on conductors a and b be +q and -q respectively

Example 1.28.1 Find the capacitance of a single phase line between conductors and between phase and neutral of 10 km long consisting of 2 parallel wires each 2 cm in diameter and 4 m apart.

Determine the capacitance of the same line taking into account effect of ground between conductors and between phase and neutral. The height of conductors above ground is 6 m. 

Solution :

Neglecting the presence of ground.

Review Questions

1. What is the method of images ? How it can be used to take into account the presence of ground in calculating the capacitance of a single phase line.

2. A single-phase line is having 2 single stranded conductor and radius 0.328 cm. The conductors are spaced 4 m apart and 8 m above the ground. Calculate the capacitance to neutral per km :

1) Considering earth's effect 2) Neglecting earth's effect.

[Ans.: i) C_an = 7.8620 nF/km ii) C_an = 7.8285 nF/km]

3. A single phase line is having two single standard conductors and radius of each 0.328 cm. The conductors are separated 3 m, apart and 7.5 m above ground. Calculate :

i) Capacitance to neutral/km without considering earth effect.

ii) Capacitance to neutral/km with considering earth effect.

[Ans.: i) Can = 8.1588 nF/km, ii) Can = 8.1823 nF/km]

4. Derive an expression for capacitance of single phase transmission system and discuss the effect of earth on capacitance with suitable equation.