# HBFEM Used in Nonlinear EM Field Problems and Power Systems

Nonlinear phenomena in EM fields are caused by nonlinear materials used in electric machines. The nonlinear materials are normally field strength-dependent, which can cause harmonics. Therefore, when the time-periodic quasi-static EM field is applied to the nonlinear material, the electromagnetic properties of the material will be functions of the EM field, which is also time-dependent. On the other hand, harmonics can also be generated by power electronic devices and drives, which are largely used in power systems, renewable energy systems and microgrids. These power electronic devices and drives are used for power rectification, power conversion (e.g. DC/DC converter) and inversion (e.g. DC/AC inverter). In fact, HBFEM can be used to effectively solve these harmonic problems in nonlinear EM fields and power systems.

Since the harmonic balance FEM technique was introduced to analyze low-frequency electromagnetic (EM) field problems in the late 1980s [11, 12], various harmonic problems in nonlinear EM fields and power systems have been investigated and solved by using HBFEM [18-24]. Harmonic balance techniques were combined with the finite element method (FEM) to accurately solve the problems arising from time-periodic, steady-state nonlinear magnetic fields. The method can be used for weak and strong nonlinear time-periodic EM fields, as well as harmonic problems in renewable energy systems and microgrids with distributed energy resources.

The harmonic balance FEM (HBFEM) method uses a linear combination of sinusoids to build the solution, and represents waveforms using the sinusoid - coefficients combined with the finite element method. It can directly solve the steady-state response of the EM field in the multi-frequency domain. Thus, the method is often considerably more efficient and accurate in capturing coupled nonlinear effects than the traditional FEM time-domain approach when the field exhibits widely separated harmonics in the frequency spectrum domain (e.g. pulse width modulation (PWM)-based power electronic devices and drives). The HBFEM consists of approximating the time-periodic solution (magnetic potentials, currents, voltages, etc.) with a truncated Fourier series. Besides the frequency components of the excitation (e.g. applied voltages or current), the solution contains harmonics due to nonlinearity (magnetic saturation and nonlinear lumped electrical components), movement (e.g. rotation in electric machines), and power electronic devices and drives.

In order to solve time-periodic nonlinear magnetic field problems, a novel numerical computation method called HBFEM was developed, which is the combination of FEM and the harmonic balance method. The principle of a new approach of HBFEM is to drive a basic formulation of the harmonic balance finite element method (HBFEM). For simplicity of fundamental formulation, a time-periodic nonlinear magnetic field is assumed as two-dimensional in the (x, y) plane, and is quasi-stationary. Therefore, the vector potential A = (0, 0, A) satisfies in the region of interest surrounded with some boundary conditions. To calculate such a quasi-static magnetic field, the following equation (1-2) can be used:

where *v* and *a* are magnetic reluctivity and conductivity.

Based on the harmonics balance theory, the governing equations of the quasi-static field containing harmonics can also be solved by using a FEM-based numerical approach. Assuming Уф = 0 in the two-dimensional case, and using Galerkin’s method to discretize, the governing equation for two dimensional problems can be written in an integral form that is given as:

where *N(x, y)* is the interpolating function.

When the applied voltage waveform is a sinusoidal signal, the current can be considered as a non-sinusoidal waveform. The waveform may be distorted due to nonlinear load or power electronic devices. Therefore, the current excitation source will include harmonic components, and the resultant magnetic field will contain all harmonic components. The vector potential *A* and current density *J* are approximated as a summation of all harmonic solutions. According to the harmonic balance method, all variables (i.e. vector potentials, flux densities and applied current) are approximated as a summation of all harmonic solutions. Therefore, the time-periodic solution (harmonic problem) can be found when an alternating magnetizing current is applied. In fact, HBFEM has been successfully used to solve various nonlinear magnetic field problems, and the computation results have been verified by experimental results listed below (detailed results will be discussed in Chapter 3 and 4).