The inductance of a circuit is defined as the flux linkage per unit current. Therefore, in order to find the inductance of a circuit, the determination of flux linkage is of primary importance. We shall discuss two important cases of flux linkages.
1.Flux linkage due to a Single Current Carrying Conductor:
Consider a long straight cylindrical conductor of radirus r metres and carrying a current I amperes (r.m.s.) as shown in Fig. 9.4 (i). This current will set up magnetic field. The magnetic lines of force will exist inside the conductor as well as outside the conductor. Both these fluxes will contribute to the inductance of the conductor.
Flux linkages due to internal flux: Refer to Fig. 9.4 (ii) where the X-section of the conductor is shown magnified for clarity. The magnetic field intensity at a point x metres from the centre is given by;
Assuming a uniform current density,
If μ ( = μ0μr ) is the permeability of the conductor, then flux density at the considered point is given by;
Now, flux dΦ through a cylindrical shell of radial thickness dx and axial length 1 m is given by;
This flux links with current
Therefore, flux linkages per metre length of the
Total flux linkages from centre upto the conductor surface is
Flux linkages due to external flux: Now let us calculate the flux linkages of conductor due to external flux. The external flux extends from the surface of the conductor to infinity. Referring to Fig. 9.5, the field intensity at a distance x metres (from centre) outside the conductor is given by ;
Now, flux dΦ though a cylindrical shell of thickness dx and axial length 1 metre is
The flux dΦ links all the current in the conductor once and only once.
Total flux linkages of the conductor from surface to infinity,
Overall flux linkages,
2.Flux linkage in Parallel Current Carrying Conductors:
We shall now determine the flux linkages in a group of parallel current carrying conductors. Fig. 9.6 shows, the conductors A,B,C etc. carrying currents IA,IB,IC etc. Let us consider the flux linkages With one conductor, say conductor A. There will be flux linkages with conductor A due to its own current as discussed previously. Also there Will be flux linkages with this conductor due to the mutual inductance effects of IB,IC,ID etc. We shall now determine the total flux linkages with conductor A.
Flux linkages with conductor A due to its own current
Flux linkages with conductor A due to current IB
Flux linkages with conductor A due to current IC
Total flux linkages with conductor A
Similarly, flux linkage with other conductors can be determined. The above relation provides the basis for evaluating inductance of any circuit.