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Reduction of harmonic balance equations through Galerkin's method

Harmonic balance suffers of the main drawback of requiring prohibitive numerical resources when the simulated circuit leads to waveforms with very sharp variations, since a large number of harmonics has to be used. The main reason is due to the fact that the numerical effort increases as a polynomial function of the number of harmonics. This drawback can be mitigated if peculiar solvers that need to neither store nor LU-factorise the Jacobian matrix related to the harmonic balance problem are used (e.g., GMRES). Another approach is to adopt oversampling, i.e. still “well” evaluating the characteristics of the highly nonlinear devices while using a limited number of harmonics. This aspect is investigated in this paper and, by showing how the harmonic balance is a peculiar version of the more general Galerkin's method, a new version of the latter is introduced that potentially lowers the problem dimensionality. Furthermore it allows us to rigorously introduce and describe an oversampling technique that aids convergence by pushing aliasing effects to the high frequency portion of the spectrum. Its effectiveness is shown through a test circuit and numerical simulations.