## Transmission line Parameters

### Introduction

Transmission line has four parameters-

1. Resistance- Resistance causes power loss in terms of heating.
2. Inductance– The power transfer capability is mainly limited by series inductance.
3. Capacitance
1. Capacitance is developed between conductor and earth.
2. Capacitance of a transmission line causes charging current.
3. Capacitance is significant in medium and long transmission line.
4. Conductance- When the AC voltage is applied across the conductor, some current flow in the dielectric medium (because of dielectric imperfections). Such current is called leakage current. Shunt conductance is due to leakage current. Leakage current depends on the atmospheric condition and pollution like moisture and surface deposits. Shunt conductance is distributed uniformly along the whole length of the line. The leakage current is very very small as compared to main current therefore, it is usually neglected.

Note: Transmission line is example of distributed circuit but for analysis purpose we will consider transmission line parameters as lumped elements.

### Transmission line Resistance

• DC resistance is given by –
$R_{dc} = \rho\dfrac{l}{a}$
•  For stranded conductors the DC resistance is greater than the above equation due to spiralling of strands.
• Distribution of current in DC is uniform.
• Distribution of current in AC is non-uniform.
• Some modification in DC resistance formula-

${ R }_{ dc }=\rho \dfrac { l }{ a } =\rho \dfrac { { l }^{ 2 } }{ al } =\rho \dfrac { { l }^{ 2 } }{ v }$

${ R }_{ dc }=\rho \dfrac { l }{ a } =\rho \dfrac { la }{ aa } =\rho \dfrac { v }{ { a }^{ 2 } }$
• Resistance can be calculated by power loss formula –
$R=\dfrac { { P }_{ l } }{ { I }^{ 2 } }$
• Factors affecting transmission line resistance-
• Resistivity of a material

• Length

• Area and volume of a conductor

• Frequency (skin and proximity effect)

• Temperature (Resistance of conductor increases with increase in temperature)

$R_{final}=R_{initial}(1 + \alpha_{initial}\Delta t)$

## Skin Effect and Proximity Effect.

### What is Skin Effect in transmission line?

• Cause- magnetic field set up by AC current.

• Flux linkage at the centre of conductor is more and it decreases as we move towards the surface of conductor.

• Hence inner strands have more inductance than the outer strands and that’s why more current flows through the outer strands.

• At radio frequency (3 kilo hertz to 300 kilo Hertz) skin effect is very significant.

• Transmission line resistance increases due to skin effect and skin effect increases with increase in conductor cross section and frequency.

• In case of a 50 hertz frequency increase in resistance due to skin effect is 3 to 5%.

• As the current due to skin effect tends to concentrate near the surface of the conductor the internal flux linkages are reduced. Consequently, the skin effect decreases the effective internal reactance.
• You can read more on Wikipedia

### What is Proximity Effect in transmission line?

• Current distribution is also disturbed by the presence of other conductor in its vicinity known as proximity effect.
• The alternating current flowing through the conductor produces an alternating magnetic field which induces Eddy current in adjacent conductors. This Eddy current causes a redistribution of the main current.
• Transmission line resistance increases due to proximity effect.
• Proximity effect is considerable in underground cables and negligible in overhead transmission line.

#### Case 1: Both conductors are carrying current in the same direction.

• Consider a conductor A and B as shown in figure both conductors are carrying the current in the same direction. (say going away from you)
• According to right hand rule, conductor A will produce a magnetic field in a clockwise direction and the same magnetic field will link to conductor B.
• According to Faraday’s law of electromagnetic induction, Eddy current will be induced in conductor B, in planes perpendicular to a magnetic field.
• The magnetic field produced by the Eddy current opposes the main magnetic field. (Lenz’s Law)
• Eddy currents flow in long loops along the conductor, in the same direction as the main alternating current on the side of the conductor facing away from the other conductor, and in opposite direction on the side of the conductor facing the other conductor therefore current density will be maximum in the region AO and BO.
• A similar explanation is applicable for conductor B, conductor B will produce a varying magnetic field and this field will link to conductor A, which will redistribute the current density in conductor A.

#### Case 2: Conductors are carrying current in the opposite direction.

• Consider current in conductor A is going away from you and current of conductor B is coming towards you in this case current density will be more on the side of the conductor facing the other conductor.

Let us briefly discuss, what is happening?

• According to the right hand rule, the magnetic field of conductor A will be clockwise and this magnetic field will induce Eddy current in conductor B as discussed in case 1.
• In this case, the only difference is that the direction of the main current of conductor B. This main current is coming towards you and therefore on the left-hand side of conductor B direction of the main current and direction of the loop current are the same. Therefore current density will be more on the side of the conductor facing the other conductor.

### Factors affecting Skin effect and Proximity effect.

• Conductor size
• Frequency
• Resistivity
• Diameter and shape of conductor
• Relative permeability of material.

Note: Effective resistance of transmission line conductor having diameter d is proportional to $\frac{d^2f\mu_r}{\rho}$

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