Motional capacitance of layered piezoelectric thickness-mode resonators

The Butterworth-Van Dyke equivalent circuit for description of the electrical behavior of piezoelectric bulk resonators is considered. The motional capacitance, C/sub 1/, in the circuit characterizes the strength of piezoelectric excitability of a vibration mode. For layered one-dimensional (1-D) structures this parameter can be calculated from the admittance given by the transfer matrix description of H. Nowotny and E. Benes (1987). Introducing the equivalent area of a vibration mode, the calculation is generalized for the three-dimensional (3-D) case of thickness-mode vibration amplitudes varying only slowly in the lateral directions. Detailed formulae are given for the case of singly rotated quartz crystals or ultrasonic transducers with additional layers on one or two sides. Good agreement of the calculated C/sub 1/ with experimental data is shown for mass-loaded planoconvex AT-cut quartz crystals.<>