Eect of anisotropy and piezoelectricity on the force-frequency coecient of AT-cut quartz crystals

Abstract

Abstract. A clear understanding of force-frequency e ect 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 e ect. 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 quanti ed the e ect of anisotropy and piezoelectricity of ATcut quartz on the force-frequency coe cient. To this end, besides using the mathematical models, a nite element code, based on linear and nonlinear Lagrangian formulations, was developed. The FEM showed more reliable results than di erent analytical approaches. We found that the force-frequency coe cient is more sensitive to the anisotropy than to the piezoelectricity of quartz. By considering the anisotropy, the standard error of the force-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 .

Experimental and numerical investigation of force-frequency effect in crystal resonators

“…The final state of the resonator is then defined as the superposition of the initial and incremental response of the resonator. Also, the incremental model uses an eigen-frequency solver, to obtain the resonant frequencies [3]. The simulation process is laid out in Fig.…”

“…In our previous paper [3], we showed that considering the anisotropic properties of quartz is vital for accurate modelling of force-frequency effect. However, the piezoelectricity may be neglected.…”

“…To obtain high frequencies, usually the thickness shear vibration mode is used. If the resonator is subjected to external perturbations like thermal and mechanical loads, due to some non-linear effects like force-frequency effect, its thickness shear mode resonance frequency is changed, and the stability of the output frequency is reduced [3]. The change in the resonance frequency of the crystal due to application of a pair of diametrically opposite forces is named force-frequency effect [4].…”