Asymptotic modelling of a skin effect in magnetic conductors

Abstract

We consider the time-harmonic Maxwell equations set on a domain made of two subdomains $\Omega_{-}$ and $\Omega_{+}$, such that $\Omega_{-}$ represents a magnetic conductor and $\Omega_{+}$ represents a non-magnetic material, and the relative magnetic permeability $\mu_{r}$ between the two materials is very high. Assuming smoothness for the interface between the subdomains and regularity of the data, the electric field solution of the Maxwell equations possesses an asymptotic expansion in powers of the parameter $\eps=1/ \sqrt{\mu_{r}}$ with profile terms rapidly decaying inside the magnetic conductor. We make explicit the first terms of this expansion. As an application of the asymptotic expansion we obtain the asymptotic behavior of a skin depth function that allows to measure the boundary layer phenomenon at large relative permeability. As another application of this expansion we give elements of proof for the derivation of impedance boundary conditions (IBCs) up to the third order of approximation with respect to the parameter $\varepsilon$, and we prove error estimates for the IBCs.