Capacitance of Overhead Transmission Lines

In this article we will discuss about the capacitance of overhead transmission lines.

We know that any two conductors separated by an insulating medium constitute a condenser or a capacitor. In case of an overhead line two conductors form the two plates of a capacitor and the air between the conductors behaves as the dielectric medium. Thus an overhead line can be assumed to have capacitance between conductors throughout the length of the line. The capacitance is uniformly distributed over the total length of the line and may be regarded as a uniform series of condensers connected between the conductors, as illustrated in Fig. 4.25.

When an alternating pd is applied across a transmission line, as shown in Fig. 4.26, it draws a leading current, even when supplying no load. This leading current is in quadrature with the applied voltage and is termed as the charging current.

It must be noted that the charging current is due to the capacitive effect between the conductors of the line and is not in any way dependent on the load. The strength of the charging current depends upon the voltage of transmission, the capacitance of the line and the frequency of ac supply and is given by the expression,

Charging current, I_C = 2fCV … (4.34)

where f is the frequency of supply, C is capacitance of the line and V is the voltage.

If the capacitance of an overhead line is high, the line draws more charging current, which compensates or cancels the lagging component (reactive component) of load current (under normal load conditions the load is inductive one). Hence the resultant current flowing in the line is reduced.

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The reduction in resultant current flowing in the line results in:

(I) Reduction of line losses and so increase of transmission efficiency,

(ii) Reduction in voltage drop or improvement of voltage regulation.

The other advantages of a transmission line having high capacitance are increased load capacity and improved power factor.

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Electric Field and Potential Difference:

An electric charge causes an electric field around it which theoretically extends up to infinity. If any charge is introduced in this electric field, it will be attracted or repelled according to the nature of the charge and when it will be moved, the work will be done against or by the force acting on the charge due to electric field. Hence potential in an electric field is exactly the same as potential in the gravitational field.

In general electric potential or potential at any point in an electric field is defined as the work done in moving a unit positive charge from infinity to that point.

The concept of electric potential is very important for the determination of capacitance as the latter is defined as the charge per unit potential.

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Now we will discuss in detail the electric potential owing to various important conductor arrangements:

1. Potential at a Charged Single Conductor:

Consider a long straight cylindrical conductor A of radius r metres and having a charge of q coulombs per metre of its length.

The electric field intensity at a distance x from the centre of conductor,

Taking air as medium i.e., e_r = 1

The potential difference between conductor A and infinity distant neutral plane (a plane where E and therefore, potential is zero) will be equal to work done in bringing a unit positive charge against E from infinity to conductor surface and is given as,

2. Potential at a Charged Conductor in a Group of Charged Conductors:

Consider a group of long straight conductors A, B, C, D, E … N having charges of q₁, q₂, q₃, q₄, q₅ … q_n coulombs per metre length respectively (Fig. 4.28).

Potential of conductor A due to its own charge q₁

Potential of conductor A due to charge q₂

since the field due to the charge q₂ extends from infinity up to a distance d₁ from conductor A.

Similarly potential of conductor A, due to charge q₃ of conductor C placed at a distance of d₂ metres from conductor A

Potential of conductor A due to charge q_n of conductor N placed at a distance of d_{n – 1}metres from conductor A

So overall potential difference between conductor A and infinite distant neutral plane,

Capacitance of a Single Phase Overhead Line:

Consider a single phase overhead line with two parallel conductors, each of radius r metres placed at a distance of d metres in air. It is assumed that the distance d’ between the conductors is large in comparison to the radii of the conductors. Therefore the density of charge on either conductor will be practically unaffected by the charge on the other conductor and will, therefore, be uniform throughout the length.

A uniformly distributed charge on a conductor acts as though it is concentrated on the conductor axis. Therefore, for the purpose of our present analysis it is assumed that the charge + q coulombs on conductor A and – q coulombs on conductor B are concentrated at the centres of the two conductors which are separated from each other by d metres.

PD between conductor A and neutral ‘infinite’ plane,

Similarly p d between conductor B and neutral ‘infinite’ plane,

PD between conductor A and B,

Capacitance of the line,

The above Eq. (4.37) is for capacitance between two conductors. The capacitance for each conductor, C_n (or phase to neutral) will be double of this value.

Example:

Find out capacitance of a single phase line 30 km long consisting of two parallel wires each 15 mm diameter and 1.5 m apart.

Solution:

Radius of each conductor, r = 15/2 = 7.5 mm

Spacing between conductors, d = 1.5 m = 1,500 mm

Capacitance of 3-Phase Overhead Lines:

Unsymmetrically Spaced Line:

For an untransposed unsymmetrical 3-phase line the capacitances between conductors to neutral of the three conductors are different. Suppose that the line is, as shown in Fig. 4.9, and that voltages V_A, V_B, V_C are applied to the conductors with the result that the charges per metre length are q₁, q₂, and q₃ respectively.

Potential of conductor A, (w.r.t to neutral infinite plane),

Subtracting Eq. (4.44) from Eq. (4.43) we have,

Capacitance of conductor A to neutral,

Similarly capacitance of conductor B to neutral,

Unsymmetrical Line with Transposed Conductors:

Assuming that the charge per unit length is same in every part of the transposed cycle [Figs. 4.30 (1), (2) and (3)], average value of voltage of conductor A, will be,

Voltage of conductor A in positions (1), (2), and (3) we have,

So average value of voltage of conductor A,

Substituting q₂ + q₃ = – q₁ in above equation we have,

Capacitance of conductor A to neutral,

Similarly, expressions for C_BN and C_CN can be obtained and we have,

Equilaterally Spaced Line:

For the equilateral spacing d₁ = d₂ = d₃ = d (say),

Capacitance of Double Circuit Three Phase Overhead Lines:

Normally used conductor configurations are of hexagonal spacing (Fig. 4.17) and flat vertical spacing (Fig. 4.18). It has been found that modified GMD method holds good for determination of capacitance of transposed double circuit 3-phase overhead lines with equilateral spacing (conductors at the vertices of a regular hexagon) and with flat vertical spacing. It is reasonable to assume that the modified GMD method can be used for determination of capacitance of a line with any configuration intermediate between these two configurations.

In the case of calculations of inductance, determination of self GMD (or GMR) of conductor is necessary because of internal flux linkages of the conductor. But in case of calculations of capacitance, since all charges reside on the surface of the conductor, actual radius of the conductor is used.

Symmetrically Spaced Line:

Consider a 3-phase double circuit connected in parallel—conductors A, B and C forming one circuit and conductors A’, B’ and C’ forming another circuit (conductors symmetrically spaced).

Let the charge over conductors A, B and C is q₁, q₂ and q₃ coulombs per metre length. Then charge over conductors A’, B’ and C’ will obviously be q₁, q₂ and q₃ coulombs per metre length and q₁ + q₂ + q₃ = 0.

Potential of conductor A w.r.t neutral infinite plane (Fig. 4.17),

Capacitance of conductor A to neutral,

Similarly, expressions for C_BN and C_CN can be obtained and we have,

This is because the conductors of different phases are symmetrically placed.

Equation (4.52) gives the capacitance of conductor A alone, whereas there are two conductors per phase A and A’. Therefore the capacitance of the system per phase will be twice of the capacitance of one conductor to neutral, i.e.,

Flat Vertically Spaced Line:

Consider conductors arranged, as shown in Figs. 4.18 (1), (2) and (3), corresponding to different positions in the transposition positions.

Potential of conductor A w.r.t ‘infinite’ neutral plane [Fig. 4.18 (1)],

Similarly, expressions for capacitances C_BN and C_CN can be obtained which are same as above.

Capacitance per phase will be double of C_AN i.e.

Capacitance of Bundled Conductor Line:

A bundled conductor line is shown in Fig. 4.34. The conductors of any one bundle are in parallel and it is assumed that the charge per bundle divides equally among the conductors of the bundle as D_ab >> s. Also D_ab – s ≃ D_ab + s ≃ D_ab for the same reason. The results obtained with these assumptions are fairly accurate for usual spacings. Thus if charge on phase A is q_A, then a and a’ have a charge of q_A/2 each. Similarly the charge is equally divided for phases B and C.