Comparison of TimePeriodic SteadyState Nonlinear EM Field Analysis Method
The harmonic balance technique was first introduced to analyze lowfrequency EM field problems in the late 1980s [8]. Harmonic balance techniques were combined with FEM to accurately solve the problems arising from timeperiodic steadystate nonlinear magnetic fields. The harmonic balance FEM (HBFEM) 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 steadystate response of the EM field in the multifrequency domain. Thus, the method is often considerably more efficient and accurate in capturing coupled nonlinear effects than the traditional FEM timedomain 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 timeperiodic EM fields, including DCbiased transformer problems.
The harmonic balance FEM consists of approximating the timeperiodic 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 quasisteadystate.
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 NewtonRaphson 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.) [911]. 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 timedomain and transient analysis based FEM are shown in Table 3.1.
Table 3.1 Comparison of timeperiodic steadystate nonlinear EM field analysis method
Frequency domain method 
Stepbystep method 
Timeperiodic 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 