Simulation of the steady state of oscillators in the time domain

The numerical calculation of the limit cycle of oscillators with resonators exhibiting a high quality factor such as quartz crystals is a difficult task in the time domain. Time domain integration formulas introduce numerical damping which leads asympotically to erroneous limit cycles or spurious oscillations. The numerical problems for solving the underlying differential algebraic equations are studied in detail. Based on these results a class of novel integration formulas is derived and the results are compared with the well-known Harmonic Balance (HB) technique. The discretized system is sparser than that of the HB method, therefore easier to solve and slower run time.