Impedance boundary condition for vector potentials on thin layers and its application to integral equations

Thin layers of magnetic substance are often used in magnetic shieldings. Since the scale of the spatial change in electromagnetic fields in the direction of the thickness of such a thin layer is considerably different from that in the transverse directions, the numerical treatment of the interior electromagnetic fields is formidable. In this paper, it is shown that the impedance boundary conditions on the surfaces of the thin magnetic layer, which have been written in former papers by the scalar potentials, can be expressed in terms of the vector potentials. Moreover, a new numerical method for analysis of eddy currents on thin magnetic layers is introduced, in which the quasi-static magnetic field in an air region ambient the thin layer is analyzed by solving the boundary integral equations under the impedance boundary conditions without any numerical treatment of the interior field. This formulation has no difficulties even when the skin depth is very short compared with the thickness of the layer. It is shown that the numerical results obtained by the present method in two-dimensional and axisymmetric systems agree well with the results analytically obtained or computed by a conventional numerical method