ON THE TRACKING OF RESONANCE AND ANTIRESONANCE OF A PIEZOELECTRIC RESONATOR

A quantitative analysis on the tracking of zero-phase resonance frequency f(r), and antiresonance frequency, f(a), which has been reported previously by the author, is presented. Integral and integrodifferential equations for the system angular frequency omega (t) are derived, and numerical calculations are carried out for a model in which the voltage-controlled oscillator is of first order and the phase angle of a piezoelectric resonator is represented by a single square-well function of omega. In order to confirm the plausibility of the model, the resulting omega(t)- and VCO input-waveforms and phase-plane trajectories are compared with the corresponding results for the exact phase angle function. Lock-in conditions and lock-acquisition time are expressed in terms of dimensionless quantities involving the difference between initial frequency and either of f(r) or f(a), the difference f(a) - f(r), the loop gain, the VCO delay time constant, and the loop filter time constant. The results are consistent with our experimental data for a sample PZT renonator.