Time domain surface impedance concept for low frequency electromagnetic problems—Part II: Application to transient skin and proximity effect problems in cylindrical conductors

Abstract

A time domain boundary element formulation employing the surface impedance boundary conditions (SIBCs) is developed for the 3-dimensional transient eddy current problem of cylindrical conductors. SIBCs of different orders of approximation are implemented using the perturbation technique in the small parameter proportional to the ratio of the skin depth and characteristic size of the conductor cross-section. The formulation consists of a set of time domain surface integral equations that have identical left-hand sides and can be solved using the same program procedure. The number of equations is determined by the order of approximation of the SIBC, namely: solutions in the perfect electrical conductor (PEC) limit (lowest order) and in the so-called Rytov approximation (highest order) are given by one and four equations, respectively. It is demonstrated that each equation admits separation of variables into space and time components, a property that significantly reduces computational costs compared with traditional time domain formulations that require the integral equations to be solved at each time step. For the purpose of validation, a test problem is solved by the proposed formulation and by the ‘original’ BEM based on the time-dependent fundamental solution. Conditions of applicability are discussed and the effect of such factors as the shape of the incident current pulse and proximity effect are considered.