Wave Propagation Over Parallel Tubular Conductors: The Alternating Current Resistance

Abstract

On the basis of Maxwell's laws and the conditions of continuity of electric and magnetic forces at the surfaces of the conductor, the fundamental equations are established for the axial electric force and the tangential magnetic force in a non-magnetic tubular conductor with parallel return. The alternating current resistance per unit length is then derived as the mean dissipation per unit length divided by the mean square current. The general formula is expressed as the product of the alternating current resistance of the conductor with concentric return and a factor, termed the “proximity effect correction factor,” which formulates the effect of the proximity of the parallel return conductor. The auxiliary functions which appear in the general formula are each given by the product of the corresponding function for the case of a solid wire and a factor involving the variable inner boundary of the conductor. In general, the resistance may be calculated from this formula, using tables of Bessel functions. The most important practical cases, however, usually involve only the limiting forms of the Bessel functions. Special formulae of this kind are given for the case of relatively large conductors, with high impressed frequencies, and for thin tubes. A set of curves illustrates the application of the formulae.