The inductance coefficients of a part of a circuit, and their applications

Abstract

Problems in connection with cables and their sheaths can in some cases be usefully discussed by means of certain coefficients which are called in this paper partial inductance coefficients, so as to distinguish them from the ordinary self and mutual-inductance coefficients, the formulæ for which apply to complete circuits onlyu. For example, formulæ are sometimes given for the “mutual” inductance between two conductors per unit length. The nomenclature assumes that the inductive effect of the first on the second is the same as that of the second on the first. This, although true for complete circuits, is not necessarily true for parts of circuits. In this paper a list of the formulae for the partial inductance coefficients is given for two hollow cylindrical conductors, both when they are external to one another and when one surrounds the other. These formulae verify Maxwell's and Rayleigh's formulæ for the complete coefficients.The formulæ for a concentric main show at once that the action of the inductive force of the inner current on the outer produces a condenser effect on the outer current.The problem of a triple-concentric main is discussed and it is shown how the theory explains the action of the inductive forces in tending to concentrate the current on the outer and inner surfaces of the inner and outer conductor of an ordinary concentric main. Instead of calling this the “skin effect” or the “proximity effect” or the “eddy-current effect” the author calls it the “Heaviside effect”.Although sheath losses can in certain cases be found by means of partial inductance coefficients, this method fails when the sheath surrounds two or more cores carrying single-phase or polyphase currents. In this case the algebraical sum of the currents flowing through a cross-section of the sheath at any instant may be zero. This shows that some of the currents through the cross-section flow in one direction and some in the other. The current density varies both in magnitude and direction as we go round the cross section. A formula for the sheath losses in a 2-core cable is given.