An H-ψ formulation for the three-dimensional eddy current problem in laminated structures

It is a very challenging problem for the direct simulation of the three-dimensional eddy currents in grain-oriented (GO) silicon steel laminations since the coating film is only several microns thick over each lamination and the magnetic permeability is nonlinear and anisotropic. In addition, the system of GO silicon steel laminations has multiple scales and the ratio of the largest scale to the smallest scale can be up to 106. In this paper, we study an H-ψ formulation for the nonlinear eddy current problem in laminated conductors. By omitting the insulating films between neighboring laminations, we propose an approximate but effective H-ψ formulation for the nonlinear eddy current problem, which reduces the scale ratio by 2–3 orders of magnitude. The well-posedness of the original problem and the approximate problem are established by examining their weak formulations. The convergence is proved for the solution of the approximate problem to the solution of the original problem as the thickness of coating films approaches zero.