Asymptotic modelling of a skin effect in magnetic conductors

TL;DR

This paper develops an asymptotic expansion for the electric field in magnetic conductors with high permeability, providing insights into skin depth behavior and deriving higher-order impedance boundary conditions with error estimates.

Contribution

It introduces a detailed asymptotic expansion for Maxwell solutions in high-permeability materials and derives improved impedance boundary conditions with quantifiable errors.

Findings

Asymptotic expansion of electric field in powers of 1/√μ_r

Explicit first terms of the expansion and decay profiles

Derivation of third-order impedance boundary conditions with error bounds

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.