Mathematical modeling and computations for electromagnetic problems

Abstract

Three-dimensional eddy current problems are widely used in the iron loss computations and the magnetic flux simulation of large power transformers. GO silicon steel laminations in large transformers have multiple scales and the ratio of the largest scale to the smallest scale can be up to 10 6 . Direct solution of threedimensional nonlinear Maxwells equations is very challenging and unrealistic for large electromagnetic devices. Based on the magnetic vector potential and the magnetic field respectively, we propose two macro-scale models for the quasi-static Maxwells equations. We establish the wellposedness of the homogenized model, and prove that the micro-scale solutions converge to the solutions of the macro-scale models weakly in H ( curl ; Ω) and strongly in L 2 (Ω) as the thickness of lamination tends to zero. We also propose a new eddy current model for the nonlinear Maxwells equations with laminated conductors. The new model omits coating films and thus reduces the scale ratio by 3 orders of magnitude. We establish the well-posedness of the new model and prove the convergence of the solution of the original problem to the solution of the new model as the thickness of coating films tends to zero. Both the macro-scale model and the new model are validated by finite element computations of an engineering benchmark problem—TEAM Workshop Problem 21 c -M1.