The calculation of electrical circuit parameters of quartz crystal resonators with the consideration of equivalent viscous dissipation

We perform a theoretical analysis on electrically forced, coupled thickness-shear, flexural and face-shear vibrations of a rectangular, AT-cut quartz resonator which is electroded in its central part only. Mindlin's first-order theory with the consideration of equivalent viscous dissipation for piezoelectric plates is used, simplified by the assumption of straight-crested waves depending on one spatial variable only. Electrical circuit parameters such as resistance and capacitance of the resonator under a time-harmonic driving voltage, which are important properties in resonator design, are calculated and examined. It is found that the numerical results are consistent with earlier measurements. The results are useful to the understanding and design optimization of crystal resonators.