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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. In the literature, high order surface impedance boundary conditions, which take into account the curvature of the conductor surface, are derived and expressed 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 behaviour 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

The two-dimensional problem of the magnetic field distribution inside and outside an imperfectly conducting, 90-degree edge is solved using the perturbation technique. The well-known solution of the problem for a perfectly conducting edge is used as an initial approximation. The surface impedance near the edge is represented in the form of asymptotic expansions in the small parameter proportional to the ratio of the skin depth and characteristic size of the conductor surface. An analytical solution for the electric field in the conductor near the edge is obtained. The solution behavior is investigated and the approximate boundary conditions relating the tangential components of the electric and magnetic fields on the conductor surface near the edge are proposed in a form suitable for numerical implementation.

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Abstract-The inverse eddy-current problem of fast transients is solved by a new boundary element formulation employing time domain surface impedance boundary conditions. The integral equation is transformed to invariant form and is solved only once for a given geometry of the problem. Numerical results are in good agreement with measured data.Index Terms-Boundary element calculations, eddy currents, magnetic sensors, time dependent magnetic fields.