The effect of electrode on the thickness-shear resonance frequency of piezoelectric crystal plates and resonator design

The determination of the precise thickness-shear frequency of electroded crystal plates have practical importance in quartz crystal resonator design and fabrication, especially when the high fundamental thickness-shear frequency has reduced the thickness of crystal plate to such a degree that the proper consideration of the effect of electrodes has been 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 standard 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 electrode elastic constants and density. Solving these equations for the usual design parameters of crystal resonators, the design process can be optimized to pin point to the precise configuration to avoid time-consuming trial 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.