ABSTRACT
The initial transient response of oscillators with high
quality factor Q such as quartz crystal oscillators is orders of
magnitudes larger than the period of oscillation. Therefore numerical
solution by standard techniques of the underlying system
of ordinary differential algebraic equations (DAEs) resulting from
Kirchhoff’s current and voltage laws is run time inefficient. In
this paper numerical techniques for the calculation of the initial
transient response and steady state solution are investigated. The
efficiency results from reformulating the underlying system of
ordinary DAEs by a suitable system of partial DAEs, known as
multirate PDE, and from suitable finite difference time domain
(FDTD) methods with small numerical dissipation of energy.
Unlike Harmonic Balance the waveforms are free of spurious
oscillations, caused by the non-compactness of the trigonometric
polynomials.
Keywords: Oscillator simulation, Quartz crystal oscillators,
Initial transient response and steady state, Multirate PDE method,
Trigonometric BDF methods, Optimal estimation of instantaneous
frequency, Hilbert transformation.
