Flux Linkages of One Conductor in a Group

Consider a composite conductor which is made up of two or more strands which are in parallel. For simplicity let us assume that all the strands are identical and share the current equally. The sum of the currents in all the conductors is zero. Such a group of conductors is shown in the Fig. 1.

Fig. 1

     
The conductors 1, 2, 3, ... n carry the currents , , .... n carry the currents I₁, I₂, I₃... I_n. Let the distances of the conductors from a point P be D_1p, D_2p , D_3p ... D_nprespectively. Let ψ_1p1 be flux linkages of conductor due to its own current I₁ due to internal and external flux. The flux beyond point P is excluded.

       Now ψ_1p2 is the flux linkages of conductor 1 due to current is equal to flux produced by I₂ between the point P and conductor 1. Again flux beyond P is neglected

       Similarly the flux linkages ψ_1p with conductor 1 due to all the conductors in the group but the flux beyond point P is neglected.

       By expanding the logarithmic terms and rearranging the terms

       The sum of all currents is zero.

       Substituting this value in above equation

       If point P is at infinite distance so that ln (D_1p/D_np) ln (D_2p/D_np) etc will approach to zero (since ln 1 = 0) then we have,

       All the flux linkages of conductor 1 are included in the above derivation. The above expression is valid only only if sum of the currents is zero.