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. 

Kirchhoffs Circuit Law

 

Kirchhoffs First Law – The Current Law, (KCL)

Kirchhoffs Current Law or KCL, states that the “tthe algebraic sum of flowing currents through a point (or junction) in a network is Zero (0) or in any electrical network, the algebraic sum of the currents meeting at a point (or junction) is Zero (0)“. I(exiting) + I(entering) = 0. This idea by Kirchhoff is commonly known as the Conservation of Charge.

Kirchhoffs Current Law

 

Here, the three currents entering the node, I1, I2, I3 are all positive in value and the two currents leaving the node, I4 and I5 are negative in value. Then this means we can also rewrite the equation as;

I1 + I2 + I3 – I4 – I5 = 0

The term Node in an electrical circuit generally refers to a connection or junction of two or more current carrying paths or elements such as cables and components. Also for current to flow either in or out of a node a closed circuit path must exist. We can use Kirchhoff’s current law when analysing parallel circuits.

Kirchhoffs Second Law – The Voltage Law, (KVL)

Kirchhoffs Voltage Law or KVL, states that “in any closed loop network, the total voltage rise in the loop is equal to the sum of all the voltage drops within the same loop”. In other words the algebraic sum of all voltages within the loop must be equal to zero. This idea by Kirchhoff is known as the Conservation of Energy.

Kirchhoffs Voltage Law

 

Starting at any point in the loop continue in the same direction noting the direction of all the voltage drops, either positive or negative, and returning back to the same starting point. It is important to maintain the same direction either clockwise or anti-clockwise or the final voltage sum will not be equal to zero. We can use Kirchhoff’s voltage law when analysing series circuits.

Determination of signs while solving problems-

Voltage Source: A positive sign is assigned to the

 EMF when going from the negative terminal of the 

voltage source to the positive terminal since there is 

an increase in potential. On the other hand, when moving

 from the positive terminal to the negative terminal,

 potential drops and the EMF is marked as negative.

2.  Resistance: The potential drop is negative when passing through

 the resistance in the same direction as the current flow because

 the potential decreases. However, if the resistance is traversed

 in the opposite direction to the current flow, the potential increases,

resulting in a positive voltage drop.

Corona

 

What is corona?

There are two types of transmission lines, overhead lines, and underground lines. We all know that; the overhead lines operate at high voltage to reduce the transmission loss. therefore, a very high amount of electric field produced between the conductors.

In overhead transmission lines, the conductors place in free air. Hence, the air acts as the dielectric medium between the conductors.

The dielectric strength of air at normal temperature and pressure is 30 kV/cm.

If the voltage between conductors is high, it causes the potential gradient between the conductors to exceed this value. In this condition, the breakdown of air will take place. And the air is ionized and current will flows through it.

The potential gradient value between conductors reaches 30 kV/cm, the air in the vicinity of the conductor becomes conducting and a hissing sound heard and some vibration produced in the conductor.

Around the conductor, a dark violet glow occurs with a hissing noise. During this process, ozone gas produced.

This entire process is known as the corona.

Corona loss in transmission line

Corona appears in the transmission line when the surface voltage gradient at the line conductor reaches the breakdown stress. Due to corona, heat and bluish light produce. There is a loss of power and energy dissipation. This loss is known as the corona loss.

The efficiency of the transmission line decreased due to corona loss.

There is also a minor effect on the voltage regulation of the transmission line. But this is always negligible.

In normal and fair atmospheric conditions, the corona loss can vary between few kW/km lengths of the conductor in Extra high voltage transmission line.

Particularly in a thunderstorm, the corona loss is very high. It can be hundreds of kW/km in bad weather conditions.

Factors affecting corona loss

1) Atmospheric condition

The corona loss occurs due to the ionization of air between the conductors. In stormy weather conditions, the rainy condition number of ions in the air between the conductors will be more. Hence, chances of corona occurrence will increase.

2) The physical condition of the conductor

• Line voltage: If the line voltage is more, the electric field between the conductor is more. Hence the chance of corona occurrence is more. In a low voltage transmission line, there is less chance of corona.
• Ratio D/r: ‘D’ is the distance between the conductors and ‘r’ is the radius of the conductor. If the distance between the conductor is more compared to the radius, the possibility of the corona is reduced. The strength of the electrostatic field reduced because of the large spacing between the conductors.
• Nature of conductor surface: The rough and irregular surface of the conductor gives rise to more corona. Stranded conductors give rise to more corona compare to the circular conductor.
• The roughness of conductor: Surface causes field distortion and high voltage gradient developed in the local area of the conductor. Thus, the chance of the occurrence of the corona is more.

3) Effect of the frequency

Corona is directly proportional to the supply frequency.

4) Effect of density of air

Corona is intentionally proportional to the density of air. Density is lower in the hilly area. Hence, in this area more chance of the corona.

5) Effect of air conductivity

Higher conductivity leads to higher corona.

Methods to reduce the corona effect

• The corona effect can be reduced by increasing the radius of the conductor or by increasing the distance between the conductors. The spacing between the conductor cannot increase beyond a certain level. Therefore, increase the radius of the conductor to reduce the corona effect.
• The open parts like a clamp, support, etc. are designed with a smooth surface.
• Operating the line at lower voltage reduces the corona loss.

Skin Effect

 Skin Effect

The non-uniform distribution of electric current over the surface or skin of the conductor carrying a.c is called the skin effect. In other words, the concentration of charge is more near the surface as compared to the core of the conductor. The ohmic resistance of the conductor is increased due to the concentration of current on the surface of the conductor.

Skin effect increases with the increase in frequency. At low frequency, such as 50Hz, there is a small increase in the current density near the surface of the conductor; but, at high frequencies, such as radio frequency, practically the whole of the currents flows on the surface of the conductor. If d.c current (frequency=0) is passed in a conductor, the current is uniformly distributed over the cross-section of the conductors.

Why skin effect occurs?

Let us consider the conductor is made up of a number of concentric cylinders. When a.c is passed in a conductor, the magnetic flux induces in it. The magnetic flux linking a cylindrical element near the center is greater than that linking another cylindrical element near the surface of the conductor. This is due to the fact that the center cylindrical element is surrounded by both the internal as well as the external flux, while the external cylindrical element is surrounded by the external flux only.

The self-inductance in the inner cylindrical element is more and, therefore, will offer a greater inductive reactance than the outer cylindrical element. This difference in the inductive reactance gives a tendency to the current to concentrate towards the surface or skin of the conductor.

The current density is maximum at the surface of the conductor and minimum at the center of the conductor. The effect is equivalent to a reduction of the cross-section area of the conductor and, therefore the effective resistance of the conductor is increased.

Factors affecting skin effect

1. Frequency – Skin effect increases with the increase in frequency.
2. Diameter – It increases with the increase in diameter of the conductor.
3. The shape of the conductor – Skin effect is more in the solid conductor and less in the stranded conductor because the surface area of the solid conductor is more.
4. Type of material – Skin effect increase with the increase in the permeability of the material (Permeability is the ability of material to support the formation of the magnetic field).

Points-to-remember

1. The Skin effect is negligible if the frequency is less than the 50Hz and the diameter of the conductor is less than the 1cm.
2. In the stranded conductors like ACSR (Aluminium Conductor Steel Reinforced) the current flows mostly in the outer layer made of aluminum, while the steel near the center carries no current and gives high tensile strength to the conductor. The concentration of current near the surface enabled the use of ACSR conductor.