On the Method of Harmonic Balance for Lumped-Element Transformer Models

Incorporating the magnetic core material in a lumped-element transformer model generally results in a nonlinear network with hysteresis. Such a network can be solved subsequently in the time domain using time-stepping methods. The transient response reveals the inrush, whose decay time is generally enormous compared to the time constant of the excitation period (e.g., 20 ms for a voltage source with a frequency of 50 Hz). To directly obtain the steady-state solution of the lumped-element transformer model, the harmonic balance method is superior to such an analysis, because it is a frequency domain approach inherently yielding the steady-state solution. The great advantage of harmonic balance is that it can handle nonlinearities by evaluating them in the time domain and using their spectral components in the frequency domain. This paper theoretically derives the harmonic balance algorithm and further solves a lumped-element single-phase transformer model based on a mutual and leakage flux approach utilizing Hopkinson's law. The nonlinear network element is represented by an energy-based dry friction-like hysteresis model depicting the core material of the transformer. The hysteresis parameters are fitted to no-load current measurements, and the no-load case, the short-circuit case, and a chosen load case are simulated and compared to measurements. The algorithm's efficiency in terms of iterations and computational time is demonstrated by simulating with several excitation frequencies and for several load cases and comparing it to a transient analysis revealing the transformer's inrush.