Time domain surface impedance concept for low frequency electromagnetic problems-Part I: Derivation of high order surface impedance boundary conditions in the time domain

The surface impedance boundary conditions (SIBCs) for the tangential component of the electric field and normal component of the magnetic field on the smooth curved surface of a homogeneous non-magnetic conductor are derived in time- and frequency-domain. Scale factors for the basic variables are introduced in such a way that, a small parameter, equal to the ratio of the penetration depth and the body´s characteristic size, appears in the dimensionless Maxwell´s equations for the conducting region. The perturbation method is then used to represent the SIBCs in the form of power series in this small parameter and the first four terms of the expansions are derived. The zero-order, first-order, second-order and third-order terms of the expansions are the solution of the problem in the perfect electrical conductor limit, the Leontovich approximation, the Mitzner approximation and in the high order approximation (referred to as Rytov´s approximation), respectively. Therefore, the accuracy of the proposed conditions exceeds the accuracy of the SIBC for planar surfaces (Leontovich´s approximation) that are usually used in the time-domain analysis, by two orders of magnitude. In Part II of this paper, the formulation of the SIBCs developed here in conjunction with a boundary element method is demonstrated and applied to the problem of transient skin and proximity effect problems in cylindrical conductors.