Shunt Parameters Of Transmission Lines

Introduction

Capacitance in a transmission line results due to the potential difference between the conductors. Capacitance between two parallel conductors depends on the size and the spacing between the conductors.

Shunt parameters of Transmission Lines

In this section we shall derive the line capacitance of different line configuration.

Usually the capacitance is neglected for the transmission lines that are less than 50 miles (80 km) long. T

The capacitance becomes significant for longer lines with higher voltage.

Capacitance in a transmission line results due to the potential difference between the conductors. The conductors get charged in the same way as the parallel plates of a capacitor.

Capacitance between two parallel conductors depends on the size and the spacing between the conductors.

Shunt parameter of transmission line is divided into four parts.

o Capacitance of a Straight Conductor.

o Capacitance of a Single-Phase Line.

o Capacitance of a Three-Phase Transposed Line.

o Effect of Earth on the Calculation of Capacitance.

Title: < Shunt parameters of Transmission Lines>

Description: < Capacitance in a transmission line results due to the potential difference between the conductors, Capacitance between two parallel conductors depends on the size and the spacing between the conductors >

Question:-

1. Write a short note on Shunt of Transmission Lines?2. Explain the parameters of shunt transmission line?

Capacitance of a Straight Conductor

Introduction

Straight conductors have some capacitance, which is known as capacitance of a straight conductor.

Capacitance of a Straight Conductor

Consider the round conductor shown in Fig. the conductor has a radius of r and carries a charge of q coulombs. The capacitance C is the ratio of charge q of the conductor to the impressed voltage, i.e.

The charge on the conductor gives rise to an electric field with radial flux lines where the total electric flux is equal to the charge on the conductor.

By Gauss's law, the electric flux density at a cylinder of radius x when the conductor has a length of 1 m is

The electric filed intensity is defined as the ratio of electric flux density to the permittivity of the medium, Therefore

Figure:- cylindrical conductor with radial flux lines

Now consider the long straight conductor of Figure shown below that is carrying a positive charge q C/m. Let two points P1 and P2 be located at distances D1 and D2 respectively from the center of the conductor. The conductor is an equipotential surface in which we can assume that the uniformly distributed charge is concentrated

at the center of the conductor. The potential difference V12 between the points P1 and P2 is the work done in moving a unit of charge from P2 to P1 . Therefore the voltage drop between the two points can be computed by integrating the field intensity over a radial path

between the equipotential surfaces, i.e.,

Figure:-Path of integration between two points external to a round straight conductor.

Eq. (1.57)

Eq. (1.58)

Eq. (1.59)

C/m2

V/m

Eq. (1.60)

Title: < Capacitance of a Straight Conductor >

Description: < Straight conductors have some capacitance, which is known as capacitance of a straight conductor >

Question:-

1. Explain Capacitance of a Straight Conductor?2. Describe the expression of capacitance of straight conductor?

Capacitance of a 1- transmission line

Introduction

It is an electric device which consists of two conductor separated by dielectric medium and it is used for storing electrostatic

energy is known as capacitor.

The capacitance of capacitor is the ration of the charge on one of its conductors to the potential difference between the

conductors.

Capacitance of a 1- transmission line:-

The separation between the conductors is D . Let us assume that conductor 1 carries a charge of q1 C/m while conductor 2

carries a charge q2 C/m. The presence of the second conductor and the ground will disturb field of the first conductor.

However we assume that the distance of separation between the conductors is much larger compared to the radius of the

conductor and the height of the conductor is much larger than D for the ground to disturb the flux. Therefore the distortion is

small and the charge is uniformly distributed on the surface of the conductor.

Assuming that the conductor 1 alone has the charge q1 , the voltage between the conductors is

Similarly if the conductor 2 alone has the charge q2 , the voltage between the conductors is

The above equation implies that

From the principle of superposition we can write

For a single-phase line let us assume that q1 (= -q2 ) is equal to q . We

therefore have

Assuming r1 = r2= r3, we can rewrite as

Therefore the capacitance between the conductors is given by

The above equation gives the capacitance between two conductors. For the purpose of transmission line modeling, the

capacitance is defined between the conductor and neutral. Therefore the value of the capacitance is given.

Title: < Capacitance of a 1- transmission line >

Description : < It is an electric device which consists of two conductor separated by dielectric medium and it is used for storing

electrostatic energy is known as capacitor, The capacitance of capacitor is the ration of the charge on one of its conductors to

the potential difference between the conductors>

Question:-

1. What do you know about capacitor Explain?

2. Give the explanation about Capacitance?

3. Derive the Expression for the Capacitance of a 1- transmission line?

Capacitance of a Three Phase line with equilateral spacing

IntroductionThere are three capacitance is given and having equal space between them now we have to calculate the capacitance of these three phase line of three identical conductors.

Capacitance of a Three Phase line with equilateral spacing:-

As show in figure given below a three phase line composed of three identical conductor of radius r placed in equilateral configuration.

As we know that Er

We can write from the above equation for Vab and Vac as

Sum of the charge on the three conductors is zero. Thus qb + qc = -qa

Substitute the value in 3.10

From the phasor diagram of given below figure that

Substituting for (Vab + Vac) from Eq. (3.12) in Eq. (3.11) we get

The capacitance of the line to natural immediately follows as

For air medium (Kr = 1).

The line charge current of phase a is Ia (line charging) = jCnVan

Phasor diagram of balanced three phase voltage

Title :< Capacitance of a Three Phase line with equilateral spacing >

Description : < There are three capacitance is given and having equal space between them now we have to calculate the capacitance of these three phase line of three identical conductors>

Question

1. Explain the Capacitance of a Three Phase line with equilateral spacing?2. Derive the expression for Capacitance of a Three Phase line with equilateral spacing?

Capacitance of three phase unsymmetrical space transmission line-1

Introduction

The calculation of capacitance in case of conductors in three phase system which are not equally spaced is difficult. If the line is untransposed the capacitances of each phase to neutral is not same.

Capacitance of three phase unsymmetrical space transmission line:-

The calculation of capacitance in case of conductors in three phase system which are not equally spaced is difficult. If the line

is untransposed the capacitances of each phase to neutral is not same.

In case of transposed line the average capacitance of each line to neutral over a complete transposition cycle is same as the

average capacitance to neutral of any other phase. Each conductor occupies the same position of every other conductor after

equal distance. The effect of unsymmetry between the lines is small and calculations are carried out by considering

transposition of lines.

The Fig. 1 shows three phase line with unsymmetrical spacing. The radius of each conductor is r.

When phase 'a' is in position 1, 'b' in position 2 and 'c' is in position 3.

When phase 'a' is in position 2, 'b' in position 3 and 'c' is in position 1.

When phase 'a' is in position 3, 'b' in position 1 and 'c' is in position 2.

Average voltage between conductors 'a' and 'b' is given by

Capacitance of three phase unsymmetrical space transmission line-2

Similarly average voltage drop between a and c is given by,

We have, Vab + Vac = 3 Van

For balanced circuit (qa + qb + qc ) = 0

The line charging current for a three phase line in phasor line in phasor for is Ia (line charging) = jCnVan A/Km.

Title : < Capacitance of three phase unsymmetrical space transmission line >

Description : < The calculation of capacitance in case of conductors in three phase system which are not equally spaced is difficult. If the line is untransposed the capacitances of each phase to neutral is not same>

Question

1. Give the explanation in brief of Capacitance of three phase unsymmetrical space transmission line?2. Derive the Expression of Capacitance of three phase unsymmetrical space transmission line?

Capacitance of a double circuit line, part-2

Equation (3.32) represent the capacitance of conductor a alone, however there are two conductors per phase a and a . Therefore the capacitance of the system per phase will be twice the capacitance of an conductor to neutral, i.e.

The expression for capacitance for other phases will also be the same

The potential of conductor a in position 2

The potential of conductor a in position 3

The average potential of conductor a in three positions

Capacitance of a double circuit line, part-3

The capacitance C per phase of the system

Flat Vertical Spacing:-

As shown in figure, The conductors of different phases are not asymmetrically placed, therefore the derivation of capacitance expression will require the transposition of conductors as shown below

Double circuit flat vertical spacing , Transposed line

Now, we require to find out the average voltage of conductor a in the three different positions due to the charge on conductor a and the conductors b ,c , a ,b and c. suppose a point is so far from the conductors then Da = Db = Dc = Da = Db = Dc .The potential of point p is approximately zero. The potential of conductor a in position 1

Title : Description:

Question1. Explain hexagonal spacing and flat vertical spacing?2. Describe the Capacitance of a double circuit line in detail?3. Derive the Expression of Capacitance of a double circuit line?

Capacitance of a double circuit line, part 1

Introduction

There are two conductors concepts are used (1) hexagonal Spacing (2) Flat Vertical Spacing.

Hexagonal Spacing

When the same phase conductors are connected in parallel the charge per unit length is same as shown in figure For the symmetrical arrangement phase are balanced and the conductors of each phase are also balanced if the effect of ground is neglected. Therefore the transposition of the conductor is not required.

Double circuit line-Hexagonal Spacing

Let us assume that the p is very far from the system of conductor and at point P potential is almost Zero, the potential of conductor a with respect to point P due to the charge on the conductor itself and the charges on conductors b, c, a, b is given by

Effect of Earth on Capacitance of Transmission Line, Part-2

a) Effect of earth on capacitance of three phase transmission line

Figure is given below shows that the asymmetrically spaced of capacitance.Suppose that a point P so far from the system of potential of conductor a in position 1.

Three phase transmission line and its diagram

Potential of a in position 2

Similarly potential of a in position 3

The average voltage of conductor a,

From the above equation, the effect of earth is to increase the capacitance, but the height of the conductor is larger as Compared to the separation between the conductors so practically effect of earth on the capacitance can be neglected.

Title : < Effect of Earth on Capacitance of Transmission Line >

Description : < 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 effect of earth is observed on both single phase and three phase line.>

Question

1. Explain Effect of Earth on Capacitance of Transmission Line?2. Derive the expression for Effect of Earth on Capacitance of single phase Transmission Line?3. Derive the expression for Effect of Earth on Capacitance of three phase Transmission Line?

Effect of Earth on Capacitance of Transmission Line, Part-1Introduction

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 effect of earth is observed on both single phase and three phase line.

Effect of Earth on Capacitance of Transmission Line :-

a) Effect of earth on capacitance of single phase transmission line

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.

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.

The voltage Vab is given as,

But qb = - qa

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.

1. introduction to Shunt Parameters Of Transmission Lines2. Capacitance of a Straight Conductor3. Capacitance of a 1- F transmission line4. Capacitance of a Three Phase line with equilateral spacing5. Capacitance of three phase unsymmetrical space transmission line6. Capacitance of a double circuit line7. Effect of Earth on Capacitance of Transmission Line

Shunt Parameters Of Transmission Lines

Introduction

Capacitance in a transmission line results due to the potential difference between the conductors. Capacitance between two parallel conductors depends on the size and the spacing between the conductors.

Shunt parameters of Transmission Lines

In this section we shall derive the line capacitance of different line configuration.

Usually the capacitance is neglected for the transmission lines that are less than 50 miles (80 km) long. T

The capacitance becomes significant for longer lines with higher voltage.

Capacitance in a transmission line results due to the potential difference between the conductors. The conductors get charged in the same way as the parallel plates of a capacitor.

Capacitance between two parallel conductors depends on the size and the spacing between the conductors.

Shunt parameter of transmission line is divided into four parts.

Capacitance of a Straight Conductor.

Capacitance of a Single-Phase Line.

Capacitance of a Three-Phase Transposed Line.

Effect of Earth on the Calculation of Capacitance.

Title: < Shunt parameters of Transmission Lines>

Description: < Capacitance in a transmission line results due to the potential difference between the conductors, Capacitance between two parallel conductors depends on the size and the spacing between the conductors >

Question:-

1. Write a short note on Shunt of Transmission Lines?

2. Explain the parameters of shunt transmission line?