Superposition Theorem 

    Superposition Theorem state that in any linear bilateral network having more than one source, the response in any one of the element is equal to algebraic sum of the response caused by individual source while rest of the sources are replaced by their internal resistances.

1.We consider one independent source at a time while all other independent sources are turned off. This implies that we replace voltage source by short circuit and current source by open circuit. This way we obtain a simpler and more manageable circuit.

2.Turn off all independent sources except one source. Find the output (voltage or current) due to that active source.

3.Repeat  the above step for each of the other independent sources.

4.Find the total contribution by adding algebraically all the contributions due to the independent sources.

Consider the following linear circuit with two sources: one current source and one voltage source. The two sources are the inputs to the function. For this problem we happen to want to find two outputs, currents $i_1$i, start subscript, 1, end subscript and $i_2$i, start subscript, 2, end subscript.

Let’s analyze this circuit using superposition.

First, we suppress the current source and analyze the circuit with just the voltage source acting alone. To suppress the current source, we replace it with an open circuit.

With just the voltage source, the two output currents are:

$i_{1V} = 0 \qquad i_{2V} = \dfrac{\text{Vs}}{\text R2}$i, start subscript, 1, V, end subscript, equals, 0, i, start subscript, 2, V, end subscript, equals, start fraction, start text, V, s, end text, divided by, start text, R, end text, 2, end fraction

Where $i_{1V}$i, start subscript, 1, V, end subscript and $i_{2V}$i, start subscript, 2, V, end subscript are the currents in $\text R1$start text, R, end text, 1 and $\text R2$start text, R, end text, 2 caused by the voltage source.

Next, we restore the current source and suppress the voltage source, to calculate the contribution of the current source acting alone.

With just the current source, the two output currents are:

$i_{1I} = \text{Is} \qquad i_{2I} = 0$

Where $i_{1I}$i, start subscript, 1, I, end subscript and $i_{2I}$i, start subscript, 2, I, end subscript are the currents in $\text R1$start text, R, end text, 1 and $\text R2$start text, R, end text, 2 caused by the current source. 

[zero current in R2?]

We complete the analysis by adding the contributions from each source:

$i_1 = i_{1V} + i_{1I} = 0 + \text{Is}= \text{Is}$i, start subscript, 1, end subscript, equals, i, start subscript, 1, V, end subscript, plus, i, start subscript, 1, I, end subscript, equals, 0, plus, start text, I, s, end text, equals, start text, I, s, end text

$i_2 = i_{2V} + i_{2I} = \dfrac{\text{Vs}}{\text R2} + 0 = \dfrac{\text{Vs}}{\text R2}$

Transposition of conductors in Power Transmission Lines

 

Transposition of conductors in Power Transmission Lines

Parameters of Transmission Line:

A transmission line has four parameters, namely resistance, inductance, capacitance and conductance. The resistance ‘R’ of a line is because of conductor resistance, series inductance ‘L’ is due to the magnetic field surrounding the conductors, shunt capacitance ‘C’ is due to the electric field between conductors, and shunt conductance, ‘G’ is because of the leakage current between phases and ground.

What is Transposition of Conductors?

The interchange of conductor positions of a transmission line at regular intervals along the route is known as Transposition of Conductors.

Why transposition is needed?

In the power transmission line when the line conductors are asymmetrically spaced i.e. not equally spaced, the inductance of each phase is different causing voltage drops of different magnitudes in the three phases even if the system is operating under balanced condition (load currents are balanced in the three phases). Also the magnetic field external to the conductors is not zero thereby inducing voltages in adjacent communication lines and causing what is known as “telecommunication interference”. This can be overcome by the interchange of conductor positions at regular intervals along the route and this practice is known as “transposition of conductors”.

How transposition is done?

In a transposed transmission line each of the three conductors occupies all the three positions relative to other conductors (position 1, position 2, and position 3) for one-third of the total length of the transmission line. Transposition also balances out the line capacitance so that electro-statically induced voltages are also balanced. Figure shows the transposition of conductors over a complete cycle.

A complete cycle of transposition of line conductors.

Complications of Conductor Transposition:

Frequent transposition usually leads to complication of support structures (as can be seen by the picture below), increase the cost because of increased number of insulator strings and total weight of supports. 

Transposition on 400 kV, double circuit transmission line, near Bhopal, M.P. 

Disadvantage of Transposition

Frequently changing the position of conductors weakens the supportive structure which increases the cost of the system.