Mutual inductance between partially wound tori and a bifilar conductor

The solution of the three dimensional magnetostatic problem in cylindrical and toroidal coordinates is used to develop analytic expressions for the mutual inductance between bifilar current carrying conductors and partially wound tori of rectangular and circular cross sections. The double series containing associated Legendre functions obtained for the mutual inductance in the case of toroid with circular cross section was evaluated for certain dimensions of the toroid, after introducing a gemetrical factor dependent only on the dimensions of the toroid. To solve Laplace equations using only cylindrical coordinates, the rectangular toroid was viewed as one element of an infinite number of tori located symmetrically to the axis rotation and placed parallel to the flat surfaces of the original toroid. The mutual inductance in the case of the toroid with the rectangular cross-section was calculated using matrix formulation of the solution to the potential problem and the double series obtained for the mutual inductance was evaluated for the rectangular toroid dimensioned in such a way that both types of tori have the same average diameter and cross sectional area. To check the results from the two boundary value problems involving circular and rectangular tori, various special configurations of the rectangular toroid in the field of the bifilar lines were studied. The first configuration was that of an infinitely long toroid, on the surface of which the magnetic field was calculated and compared with the values obtained from the general solution. For two tori with almost equal cross sections the induced voltages in the windings were measured and calculated, whereby the results are found totally well within the experimental errors. For rough estimation of the induced voltage, some approximate expressions were given and numerically and experimentally verified.