Finite element solution of nonlinear eddy current problems with periodic excitation and its industrial applications

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 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.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 time-independent permeability distribution, the so-called fixed-point permeability in each nonlinear iteration step. This leads to uncoupled harmonics within these steps.As industrial applications, analyses of large power transformers are presented. The first example is the computation of the electromagnetic field of a single-phase transformer in the time domain with the results compared to those obtained by traditional time-stepping techniques. In the second application, an advanced model of the same transformer is analyzed in the frequency domain by the harmonic balance method with the effect of the presence of higher harmonics on the losses investigated. Finally a third example tackles the case of direct current (DC) bias in the coils of a single-phase transformer.

Comparison of 2 methods for the finite element steady‐state analysis of nonlinear 3D periodic eddy‐current problems using the A,V− formulation

“…In case of the TPFP method, second-order elements are implemented. Using the TPFP technique, 6 executing a discrete Fourier transformation to (6), the equation system becomes as follows:…”

“…In case of the parallel TPFEM formulation, the backward Euler method is adopted for the time integration scheme applied to (6). For the benefit of working with a highly sparse system matrix, first-order elements are implemented in this technique.…”

“…This will lead to a discrete harmonic balance method to achieve an efficient parallelization of the so called time periodic fixed-point (TPFP) technique. [6][7][8][9]…”

Finite element simulation of eddy current problems using magnetic scalar potentials

“…Since our aim is mainly focused on emphasizing the topological aspects of the problem, we deal indeed with a simplified linear isotropic problem, a situation that does not describe real devices (for numerical simulations in a more general situation, see, e.g., Preis et al [19], Bíró et al [20]). We consider two different geometrical models: a classical five-leg transformer and the Hexaformer transformer (see Tidblad Lundmark [16]).…”

Design and Damping Analysis of a New Eddy Current Damper for Aerospace Applications

“…Laborenz et al (2011) offer special software and elaborate its algorithm for transient finite element (FE) analysis of coupled electromagnetic-thermal problems in a squirrel cage induction motor. Bíró et al (2014) provide an efficient finite element method to take account of the nonlinearity of the magnetic materials when analyzing three-dimensional eddy current problems. It is shown that using FEM together with the continuous/discrete harmonic balance method is capable of providing the steady-state solution to large, complex real-world eddy current problems when nonlinearity of ferromagnetic media is taken into account.…”