The harmonic balance method truncates the Fourier series in a finite number of terms. In this paper we show that the truncated residues may be important to determine the stability of the approximated solution and that the truncated residues in the stability analysis can fully be considered without increasing the number of equations in the original solution. Therefore, the high order superharmonic and subharmonic responses and the cascade of bifurcations to irregular attractor can be accurately approximated by just the first few terms of the Fourier series so that analytical prediction is possible. A harmonically driven oscillator with quadratic nonlinearity is taken as examples. The explicitly analytical solutions are obtained for the steady state solutions and for the high order superharmonic approximation. The stabilities of the solutions are determined by the Floquet theory. It is shown that the predicted stability of the solution can be qualitatively different with and without the consideration of the feed forward residues. The second-, fourth- and eighth-order subharmonic analytical bifurcation solutions are calculated to obtain the cascades of bifurcations to irregular attractor. The improved analytical harmonic approximations are compared with other results and with numerical solutions. It is proved that a two superharmonic expansion with appropriate subharmonic is sufficient for determining the characteristics of the solutions of a harmonically driven oscillator with quadratic nonlinearity.
Optimal Design Methodology for High-Order Continuous-Time Wideband Delta-Sigma Converters
