Inductance

An alternative flux is produced by the alternating current when flowing through a conductor. This flux links with the conductor. The conductor possesses inductance due to 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

                             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.

       The fundamental equation used to define inductance is given by,

                         e = dτ/dt                                                             ...................... (1)

       where e is induced voltage in volts and is number of flux linkage of the circuit in weber-turns. The number of weber turns is the product of each weber of flux and the number of 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    α  i

.^..             Nτ =Li

       The constant of proportionality is called inductance

.^..             τ = Li/N

       Substituting this value in the fundamental equation we have,

       For N number of turns,

                e = L di/dt                                                                                    .............. (2)

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

       Solving equations (1) and (2) we get,

         L = dτ/dt H

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

         L = τ/i H

.^..       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
            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 per ampere of current in the second circuit.

       If current produces ψ₁₂  flux linkages with circuit 1, The mutual inductance is
            M₁₂ = ψ₁₂/I₂  
       The voltage drop in circuit 1 is given by
            V₂ = 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.