Effects of air resistance on AT-cut quartz thickness-shear resonators

We study theoretically the effects of air resistance on an AT-cut quartz plate thickness-shear mode resonator. Mindlin´s two-dimensional equations for coupled thickness-shear and flexural motions of piezoelectric plates are employed for the crystal resonator. The equations of a Newtonian fluid and the equations of linear acoustics are used for the shear and compressive waves in the air surrounding the resonator, respectively. Solutions for free and electrically forced vibrations are obtained. The impedance of the resonator is calculated. The effects of air resistance are examined. It is found that air viscosity causes a relative frequency shift of the order of ppm. When the material quality factor of quartz Q = 10⁵, the air viscosity and compressibility both have significant effects on resonator impedance. For resonators with larger aspect ratios the effects of air resistance are weaker, and the effect of air compressibility is weaker than air viscosity.