On the tracking of resonance and antiresonance of a piezoelectric resonator

A quantitative analysis of the tracking of the zero-phase resonance frequency, f/sub r/, and antiresonance frequency, f/sub a/, of a piezoelectric resonator reported previously by the author (1990) 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 (VCO) is of first order and the phase angle of the 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/sub r/ or f/sub a/, the difference f/sub a/-f/sub r/, the loop gain, the VCO delay time constant, and the loop filter time constant. The results are consistent with the experimental data for a sample PZT resonator.<>