Asymptotic Models for Ferromagnetic Materials in Eddy Current Problem

The eddy current problem has crucial applications in the industrial field such as inductive heating or braking of heavy vehicles. In this context, we intend to analyze the eddy current problem for the electromagnetic field in magnetically soft materials. These materials are used for the cores of transformers in order to reduce the energy losses resulting from the alternating currents. The problem under consideration lies in two highly heterogeneous materials with a common interface: we are interested to study the electromagnetic field in a soft magnetic conducting body with a high constant permeability mu = mu−, embedded in a magnetic dielectric medium with a constant permeability mu=mu+, in a three-dimensional setting. Therefore, we propose to investigate the behavior of the electromagnetic field when the magnetic permeability is “dominant” that is the ferromagnetic case. However, in the ferromagnetic materials, we may have a small skin depth and the amplitude of the magnetic field decays rapidly with depth inside the magnetic conducting domain. As a result, the numerical simulation of such a problem seems challenging although prohibitive. In order to overcome this difficulty, we introduce for instance a small parameter epsilon, then we develop an asymptotic approach based on a multiscale expansion. This asymptotic method was also applied previously for highly conductive nonmagnetic materials in eddy current problem. Therefore, we derive asymptotic models that reflect the skin effect and can be solved numerically by using a coarse mesh. The aim of our present work is twofold. First, we derive an asymptotic expansion of the electromagnetic field as the parameter epsilon tends to zero when the surface of the conducting body is smooth. Then, as an application, we measure the magnetic skin effect by presenting the asymptotic behavior of the characteristic depth. Finally, we compare the latter result with the previous work corresponding to the skin effect in a bidimensional eddy current problem, when the contrast of permeability is predominant