Finite Element Method for Nonlinear Eddy Current Problems in Power Transformers

Abstract

An efficient finite element method to take account of the nonlinearity of the magnetic materials when analyzing three dimensional eddy current problems is presented in this paper. The problem is formulated in terms of vector and scalar potentials approximated by edge and node based finite element basis functions. The application of Galerkin techniques leads to a large, nonlinear system of ordinary differential equations in the time domain. The excitations are assumed to be time-periodic and the steady state periodic solution is of interest only. This is represented in the frequency domain as a Fourier series for each finite element degree of freedom and a finite number of harmonics is to be determined, i.e. a harmonic balance method is applied. Due to the nonlinearity, all harmonics are coupled to each other, so the size of the equation system is the number of harmonics times the number of degrees of freedom. The harmonics would be decoupled if the problem were linear, therefore, a special nonlinear iteration technique, the fixed-point method is used to linearize the equations by selecting a timeindependent permeability distribution, the so called fixed-point permeability in each nonlinear iteration step. This leads to uncoupled harmonics within these steps resulting in two advantages. One is that each harmonic is obtained by solving a system of algebraic equations with only as many unknowns as there are finite element degrees of freedom. A second benefit is that these systems are independent of each other and can be solved in parallel. The appropriate selection of the fixed point permeability accelerates the convergence of the nonlinear iteration. The method is applied to the analysis of a large power transformer. The solution of the electromagnetic field allows the computation of various losses like eddy current losses in the massive conducting parts (tank, clamping plates, tie bars, etc.) as well as the specific losses in the laminated parts (core, tank shielding, etc.). The effect of the presence of higher harmonics on these losses is investigated.