Effect of anisotropy and piezoelectricity on the force-frequency coefficient of AT-cut quartz crystals

A clear understanding of force-frequency effect is essential for enhancing the accuracy of quartz crystal resonators and sensors. Several analytical models have been developed to predict the frequency change of piezoelectric crystals like quartz due to forcefrequency effect. According to these models, frequency change is due to initial mechanical stresses caused by the application of diametrical forces. Usually, the anisotropy and piezoelectricity of quartz are neglected in determination of stress bias and in force-frequency equation. In this research, we quantified the effect of anisotropy and piezoelectricity of ATcut quartz on the force-frequency coefficient. To this end, besides using the mathematical models, a finite element code, based on linear and nonlinear Lagrangian formulations, was developed. The FEM showed more reliable results than different analytical approaches. We found that the force-frequency coefficient is more sensitive to the anisotropy than to the piezoelectricity of quartz. By considering the anisotropy, the standard error of theforce-frequency model decreased from 12.4 E-15 (ms/N) to 5.75 E-15 (ms/N). On the other hand, anisotropy can modify the shape of force-frequency curve at force azimuth angles close to 0°.