Capacitance of a Three Phase Line with Equilateral Spacing

Capacitance of a Three Phase Line with Equilateral Spacing

Consider the three conductors a, b and c of 3 phase overhead transmission line having the charges q_a, q_b and q_c respectively as shown in Fig. 1.29.1. Let the conductors be separated from each other by a distance of d from each other and placed on the vertices of equilateral triangle.

The radius of each conductor is say r. The voltage V_ab of the three phase line due to only charges on conductors a and b is given by,

Voltage V_ab due to only charge qc is zero as uniform charge distribution over the surface of the conductor is equivalent to a concentrated charge at the centre of conductor.

Considering all the three charges in writing the voltage equation we have,

The voltages are sinusoidal and expressed as the phasors. In absence of other charges in the vicinity the sum of the charges is zero i.e.

The Fig. 1.29.2 shows phasor diagram of balanced voltages of three phase line.

It can be seen that capacitance to neutral for single phase and equilaterally spaced three phase lines is same.

The current associated with capacitance of a transmission line is termed as charging current. In case of single phase circuits, the charging current is the product of line to line voltage and line to line susceptance.

I_C = j ω C_ab V_ab

In case of three phase circuits, the charging current is found by product of voltage to neutral and capacitive susceptance to neutral. The charging current obtained is for one phase. The current in any phase is given by,

Ic = j ω C_nV_an

The charging current is not same everywhere as the rms voltage along line varies. For obtaining the charging current the value of voltage used is that for which the line is designed which may not be actual voltage at either generating station or a load.

Example 1.29.1 A 3 phase, 50 Hz, 110 kV overhead line conductors are placed in a horizontal plane as shown in the Fig. 1.29.3. The conductor diameter is 1.5 cm. If the line length is 100 km. Calculate i) Capacitance/phase ii) Charging current/phase assuming complete transposition of the line. Assume line transposition.

Solution : Equivalent equilateral spacing is given by,

Example 1.29.2 Calculate the capacitance of a 100 km long 3-phase, 50 Hz overhead transmission line consisting of 3 conductors each of diameter 2 cm and spaced 2.5 m at the comers of an equilateral triangle.

Solution :

Example 1.29.3 A 3-phase, 50 Hz, 132 kV overhead line has conductors placed in a horizontal plane 4 m apart. Conductor diameter is 2 cm. If the line length is 100 km, calculate the charging current per phase assuming complete transposition.

AU : Dec.-08, May-18, Marks 8

Solution :

Review Questions

1. Derive the expression for the capacitance to neutral of a three phase line with equilateral spacing.

May-06, 07, Dec.-07, Marks 8

2. A three-phase, 50 Hz, 132 kV overhead line has conductor placed in horizontal plane 4 m apart. Conductor diameter is 2 cm. If the line length is 100 km. Calculate :

i) Capacitance of each conductor to neutral ii) Charging current per phase.

Assuming complete transposition.     

[Ans.: 0.8940 x10“6 F, 21.40 A]

3. A 200 km, 3 phase transmission line has its conductors placed at comers of equilateral triangle of 2.5 m side. The radius of each conductor is 1 cm calculate

i) Line to neutral capacitance of line

ii) Charging current per phase for voltage of 66 kV, 50 Hz.    

[Ans.: 2.02 µF, 24.2 A]

4. Three conductors A, B and C of a 3 phase line are arranged in a horizontal plane with D_AB = 2 m and D_BC = 2.5 m. Find line to neutral capacitance per km if diameter of each conductor is 1.24 cm. The conductors are transposed at regular intervals

[Ans.: 0.0091 µF/km]