Home Computer Science Harmonic Balance Finite Element Method: Applications in Nonlinear Electromagnetics and Power Systems

# HBFEM for Nonlinear Magnetic Field Problems

## HBFEM for a Nonlinear Magnetic Field with Current-Driven Source

Numerical modeling techniques are becoming more popular for electromagnetic field analysis of high-frequency, high-density switching power supplies. Other numerical techniques for the solution of structural stress and thermal analysis are also finding favor with the design engineer. This paper discusses mainly the numerical modeling and analysis of electromagnetic fields in switching power supplies. The advantages of such power supplies are high efficiency, small size and lower weight [1-3].

These advantages do not come free of charge, because electromagnetic interference (EMI), leakage inductance, skin and proximity effects, winding self-capacitance and inter-winding capacitance can present serious problems [4]. Furthermore, the non-linear nature and hysteresis of the core material can cause waveform distortion. These waveform distortions cause harmonics, which will increase power losses in both the winding and magnetic core, and causes a loss of efficiency, as well as the possibility of parasitic resonance in the system, unless properly designed. Here, we use harmonic balance analysis, combined with the finite element method, to solve problems arising from non-linear, harmonic, eddy-current and power loss problems of transformers used in switching power supplies [5].

HBFEM differs from traditional finite element time-domain methods, transient analysis and other time harmonic methods [6]. The harmonic balance uses a linear

Harmonic Balance Finite Element Method: Applications in Nonlinear Electromagnetics and Power Systems, First Edition. Junwei Lu, Xiaojun Zhao and Sotoshi Yamada.

© 2016 John Wiley & Sons Singapore Pte. Ltd. Published 2016 by John Wiley & Sons Singapore Pte. Ltd. Companion website: www.wiley.com/go/lu/HBFEM combination of sinusoids to generate a solution and represents waveforms using the coefficients of the sinusoids. It is combined with the finite element method to solve time-periodic, steady-state, nonlinear electromagnetic field problems. The HBFEM directly solves the steady-state response of the electromagnetic field in the frequency domain, and so is often considerably more efficient than traditional time-domain methods when fields exhibit widely separated time constants and mildly nonlinear behavior [7-8]. The electromagnetic field with harmonics satisfies Maxwell’s equations. The magnetic core of a converter which has nonlinear characteristics and hysteresis, and is excited by a current source of current density Js is considered. The detailed derivation has been introduced in Chapter 3.

In switching mode power supplies, some DC-DC converter topologies, such as zero voltage switching resonant converters and push-pull current source converters, can be considered as current-source to magnetic field systems, as shown in Figure 4.1. The HBFEM numerical modeling and computation results can provide more detailed information of harmonic distribution in magnetic core and eddy current distribution in windings which, compared with experimental results, are discussed in this section.

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