Analytical–Numerical Solution for the Skin and Proximity Effects in Two Parallel Round Conductors

Abstract

This paper describes an analytical-numerical method for the skin and proximity effects in a system of two parallel conductors of circular cross section—a system very frequently encountered in various applications. The magnetic field generated by the current applied on each conductor is expressed by means of vector magnetic potential and expanded into Fourier series. Using the Laplace and Helmholtz equations, as well as the classical boundary conditions, the current density induced due to the proximity and skin effect is determined in each conductor. The resulting current density is expressed as a series of successive reactions. The results obtained are compared with those obtained via finite elements. Although the paper is theoretical, the considered problem has a practical significance, because transmission lines with round conductors are universally used. Besides, the results can be used to estimate errors when only the first reaction is taken into account, which gives relatively simple formulas.