Surface Impedance Boundary Conditions in Terms of Various Formalisms

When a surface impedance boundary condition is implemented in a numerical formulation, it must be represented in terms of the state variables of that formulation. The usual practice is to derive high order surface impedance boundary conditions, which take into account the curvature of the conductor surface, and express them as relations between tangential components of the electric and magnetic fields or between tangential and normal components of the magnetic field. In the present paper, general expressions of high order surface impedance boundary conditions are obtained by solving the diffusion equation using perturbation techniques, under the hypothesis that the skin depth is much smaller than the characteristic size of the conductor's surface. The obtained formulas are applicable to various electromagnetic vector quantities and describe the behavior of the fields not only on the surface of the conductor but also in the whole skin layer. A simple numerical example is used to validate the derived expressions.