Exact thickness-shear resonance frequency of electroded piezoelectric crystal plates

Abstract

The determination of the precise thickness-shear frequency of electroded crystal plates has practical importance in quartz crystal resonator design and fabrication, especially when the high fundamental thickness-shear frequency has reduced the crystal plate thickness to such a degree that proper consideration of the effect of electrodes is very important. The electrodes effect as mass loading in the estimation of the resonance frequency has to be modified to consider the stiffness of electrodes, as the relative strength is increasingly noticeable. By following a known procedure in the determination of the thickness-shear frequency of an infinite AT-cut crystal plate, frequency equations of crystal plate without and with piezoelectric effect are obtained in terms of elastic constants and the electrode material density. After solving these equations for the usual design parameters of crystal resonators, the design process can be optimized to pinpoint the precise configuration to avoid time-consuming trial and reduction steps. Since these equations and solutions are presented for widely used materials and parameters, they can be easily integrated into the existing crystal resonator design and manufacturing processes.