Asymptotic Description of the Magnetic Potential near a Corner Singularity in the Bidimensional Eddy-Current Problem

Accurate knowledge of eddy currents is of great interest for the design of many electromagnetic devices such as used in electrothermics. Taking advantage of the fast decrease of the electromagnetic field inside the conductor, impedance conditions are usually considered to reduce the computational domain. However, whereas impedance conditions can be derived up to the required precision for any conductor with a smooth surface, these conditions are not valid near corners. Few authors have proposed heuristic impedance modifications close to the corners [1, 2] but these modifications are neither satisfactory nor proved. In particular in [2], the modified impedance seems to blow up near the corner, which does not seem valid for non-magnetic materials with finite conductivity as presented in [3]. In the present work, we do not consider the derivation of the impedance condition, which is an asymptotic expansion in high frequency (or high conductivity). We are rather interested in the description of the magnetic potential in the vicinity of the corner, considering the conductivity, as well as the frequency as given parameters. We emphasize that these considerations will be helpful to understanding the behavior of the impedance condition near corner singularities that will be considered in a forthcoming paper. [1] E.M. Deeley. Surface impedance near edges and corners in three-dimensional media. IEEE Trans. on Mag., 26(2):712-714, 1990. [2] S. Yuferev, L. Proekt, and N. Ida. Surface impedance boundary conditions near corner and edges: Rigorous consideration. IEEE Trans. on Mag., 37(5):3465-3468, 2001. [3] F. Buret, M. Dauge, P. Dular, L. Krähenbühl, V. Péron, R. Perrussel, C. Poignard, and D. Voyer. Eddy currents and corner singularities. IEEE Transactions on Magnetics 48, 2 (2012) 679-68