Steady state solution of periodically excited nonlinear eddy current problems

In many cases the steady state periodic solution of eddy current problems with periodic excitation is needed only. If the problem is linear, it is straightforward to obtain this solution either in the frequency domain applying harmonic balance techniques or in the time domain using discrete Fourier transforms. For nonlinear problems, this cannot be done, since the harmonics are all coupled. Using a fixed-point method to solve the nonlinear equations, however, enables using harmonic balance or discrete Fourier transforms within each nonlinear iteration. The method is described in the presentation and examples of its application are given. Introduction The most straightforward method of solving nonlinear electromagnetic field problems in the time domain by the method of finite elements (FEM) is using time-stepping techniques. This requires the solution of a large nonlinear equation system at each time step and is, therefore, very time consuming, especially if a three-dimensional problem is being treated. If the excitations are non-periodic or if, in case of periodic excitations, the transient solution is required, one cannot avoid time-stepping. In many cases however, the excitations of the problem are periodic, and it is only the steady-state periodic solution which is needed. Then, it is wasteful to step through several periods to achieve this by the “brute force” method [1] of time stepping. A successful method to avoid stepping through several periods in such a case is the time-periodic finite element method introduced in [2]. To accelerate the originally slow convergence of the method a singular-decomposition technique has been introduced in [3] and it has even been parallelized in [4]. A time domain technique using the fixed-point method to decouple the time steps has been introduced in [5] and applied to two-dimensional eddy current problems described by a single component vector potential. The optimal choice of the fixed point permeability for such problems has been presented in [6] both in the time domain and using harmonic balance principles. The method has been applied to three-dimensional problems in terms of a magnetic vector potential and an electric scalar potential (A,V-A formulation) in [7] and, employing a current vector potential and a magnetic scalar potential $(T$, Φ-Φ formulation), in [8] and [9]. In contrast to the time-periodic finite element method, the periodicity condition is directly present in the formulation instead of being satisfied iteratively. The aim of this work is to present a review of the fixed-point based method and to show its application to industrial problems. A detailed version has been published in [10]. Summary of the method 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 either in the frequency domain as a finite Fourier series or in the time domain as a set of discrete time values within one period for each finite element degree of freedom. The former approach is the (continuous) harmonic balance method and, in the latter one, discrete Fourier transformation will be shown to lead to a discrete harmonic balance method. Due to the nonlinearity, all harmonics, both continuous and discrete, are coupled to each other, so the size of the equation system is the number of harmonics times the number of degrees of freedom. In the time domain approach, the number of discrete harmonics is equal to the number of the time values within one period. The harmonics would be decoupled if the problem were linear, therefore, a special nonlinear iteration technique, the fixed-point method is used to linearise the equations by selecting a time-in-dependent 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 applications presented concern the simulation of the steady state of large power transformers with time-harmonic excitations. In addition to taking account of the nonsinusoidal time variation of the electromagnetic field due to saturation, direct current bias in the magnetizing currents can also be allowed for.