Inductance

Inductance

An alternating flux is produced by the alternating current when flowing through a conductor. This flux links with the conductor. The conductor possesses inductance due to these flux linkage. The flux linkages per ampere is called the inductance.

Thus inductance is given by,

L = ψ / I

where          ψ = Flux linkage in weber-turns and I = Current in amperes.

The inductance of a transmission line is also a distributed parameter over the length of line. For convenience in analysis it is taken as lumped as shown in the Fig. 1.7.2.

The fundamental equation used to define inductance is given by,

e = d τ / dt .... (1.9.1)

where e is induced voltage in volts and r is number of flux linkage of the circuit in weber-tums. The number of weber turns is the product of each weber of flux and the number of turns of the circuit linked. Each line of flux is multiplied by the number of turns it links and these products are added to determine total flux linkages.

If constant permeability is assumed for the medium in which the magnetic field is set up then we have,

Nτ ∝ i   i.e.  Nτ = Li

The constant of proportionality is called inductance

τ = Li / N

Substituting this value in the fundamental equation we have,

e = d . dt [Li / N] = L/N di/dt

For N number of truns, e = L (di/dt)  .... (1.9.2)

If permeability is not constant then above equation may also be used but then the inductance is not constant.

Solving equations (1.9.1) and (1.9.2) we get, 

L = (dτ / di) H

With the flux linkage varying linearly with current then the magnetic circuit has constant permeability.

L = τ/i  H i.e τ = Li Wb-turns

In this equation, i is the instantaneous value of current. So - represents instantaneous flux linkages. For sinusoidal alternating current, flux linkages are also sinusoidal. Hence we have,

ψ = LI

The voltage drop due to flux linkage is,

V = j ω LI volts  i.e. V = j ω ψ volts

The mutual inductance between the two circuits is defined as the flux linkages of one circuit due to current in the second circuit per ampere of current in the second circuit.

If current I₂ produces ψ₁₂ linkages with circuit 1, the mutual inductance is,

M₁₂ = ψ₁₂ / I₂

The voltage drop in circuit 1 is given by,

V₁ = j ω M₁₂ I₂ = j ω ψ₁₂ volts

Mutual inductance is important in considering the influence of power lines on telephone lines and the coupling between parallel power lines.

Review Question

1. Write a note on inductance of a transmission line.