Shift in the Location of Resonant Frequencies Caused by Large Electromechanical Coupling in Thickness-Mode Resonators

The fundamental and overtone resonant frequencies of a thickness-mode, piezoelectric resonator depart from an integral-multiple relationship when the electromechanical coupling of the resonator material becomes large. These frequencies can be calculated from the equation tanX = X/k2, where X is a normalized frequency and k is an electromechanical coupling factor. The ratios of measured resonant frequencies of both thickness-dilatational and thickness-shear mode resonators made from barium titanate, PZT-5, and PZT-7 ceramics were found to show good agreement with the theory. An important practical application of this anharmonicity is in the determination of the coupling factor from the ratios of the measured fundamental and overtone resonant frequencies. For this purpose, roots of the frequency equation are tabulated for values of mode number ranging from 1 to 4 and of coupling coefficient ranging from 0 to 0.99 in intervals of 0.01. These tables can be used in the determination of three different coupling coefficients: kt, k15, and k33. These coupling factors along with the printed tables can be used to determine elastic wave velocities from measured resonant frequencies. An experimental procedure using an admittance bridge is discussed in some detail. An application of the techniques developed in this paper to the study of depoling effects due to thermal and mechanical disturbances of a ceramic material is also discussed.