 # Comparison of Time-Periodic Steady-State Nonlinear EM Field Analysis Method

The harmonic balance technique was first introduced to analyze low-frequency EM field problems in the late 1980s . Harmonic balance techniques were combined with FEM to accurately solve the problems arising from time-periodic steady-state nonlinear magnetic fields. The harmonic balance FEM (HB-FEM) uses a linear combination of sinusoids to build the solution, and represents waveforms using the sinusoid; coefficients are 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 and mild nonlinear behavior). The method can be used for weak and strong nonlinear time-periodic EM fields, including DC-biased transformer problems.

The harmonic balance FEM consists of approximating the time-periodic solution (magnetic potentials, currents, voltages, etc.) by a truncated Fourier series. Besides the frequency components of the excitation (e.g. applied voltages), the solution contains harmonics due to nonlinearity (magnetic saturation and nonlinear lumped electrical components) and due to movement (e.g. rotation). The HBFEM leads to a single, very large system of algebraic equations. Depending on the problem at hand, it may be much more efficient than the time domain approach (time stepping). Indeed, the latter inevitably requires stepping through the transient phenomenon before reaching the quasisteady-state.

The global HBFEM system of algebraic equations is derived in an original way. The Galerkin approach is applied to both the space and time discretization. The time harmonic basis functions are used for approximating the periodic time variation, as well as for weighing the time domain equations in the fundamental period. Magnetic saturation and nonlinear electrical circuit coupling are thus easily accounted for, by means of the Newton-Raphson method. Rotation in 2D FEM models of rotating machines, using the moving band technique, can also be considered. The HBFEM has been validated by applying it to several test cases (transformer feeding a rectifier bridge, various synchronous and asynchronous machines, etc.) [9-11]. The harmonic waveforms of the magnetic field, currents and voltages, and so on, are shown to converge well, compared with those obtained with time stepping, as the spectrum of the HBFEM analysis is extended.

The major differences between HBFEM and the traditional time-domain and transient analysis based FEM are shown in Table 3.1.

Table 3.1 Comparison of time-periodic steady-state nonlinear EM field analysis method

 Frequency domain method Step-by-step method Time-periodic method Harmonic balance method Computation domain Single frequency domain Time domain Time domain Multiple frequency domain For nonlinear and harmonic problems Yes, for weak nonlinear fields, but not for harmonic problems Yes, for weak nonlinear fields and harmonic problems Yes, for weak nonlinear fields and harmonic problems Yes, for weak and strongly nonlinear fields and harmonic problems Widely separated harmonics Cannot compute Difficult to compute Difficult to compute Easy to compute (e.g. PWM) Computation time depending on: Number of degrees of freedom Time step and number of degrees of freedom Time step and number of degrees of freedom Harmonic number and number of degrees of freedom Computation accuracy Large error at high frequency harmonics Truncation error Truncation error Number of harmonics considered Calculation of harmonic components Impossible Calculate from computation results Calculate from computation results Calculate several harmonics simultaneously (the result itself) Postprocessing of harmonics Impossible Indirectly Indirectly Directly