Bechmann’s Number for Harmonic Overtones of Thickness-Shear Vibrations of Rotated Y-Cut Quartz Crystal Plates

Han Zhang,Ji Wang,Yumei Chen

doi:10.3390/coatings10070667

Abstract

A procedure based on approximate solutions of three-dimensional equations of wave propagation is utilized for calculating Bechmann’s number for the harmonic overtones of thickness-shear modes in the rotated Y-cut quartz crystal plates. Bechmann’s number is used for the optimization and improvement of electrodes to yield superior performance in the design of quartz crystal resonators. Originally, Bechmann’s number is found through practical experiences, and analytical results were provided afterward to enable optimal design of novel resonator structures. The outcomes in this study are from a simplified theoretical prediction and they are consistent with known empirical results, making it is possible to design optimal quartz crystal resonators for cases without adequate experimental data for a higher frequency and smaller size.

Highlights

Pure thickness-shear waves without couplings to other modes in a plate have displacement in parallel to the middle plane and surfaces of the plate and perpendicular to the direction of wave propagation
The simple reason for even x is because the modes will include the lowest anharmonic series, this is currently the only interest for resonator optimization; odd in y is because vibration modes which are even in y does not contribute to the electric current through the plate if the piezoelectricity is included from the device functioning point of view
By approximating the thickness-shear vibrations of quartz crystal plates with and without coatings of electrodes by a simple displacement function, analytical solutions of vibration frequency and wavelengths are obtained from the ideal structure for the specific mode

Summary

Introduction

Pure thickness-shear waves without couplings to other modes in a plate have displacement in parallel to the middle plane and surfaces of the plate and perpendicular to the direction of wave propagation. There is a noticeable frequency gap between ωm and ωm , resulting in real and imaginary wavenumbers in different regions, which make waves propagate in the plated portion but not in the unplated portion This is the unique functioning principle of quartz crystal resonators in preserving the energy input through electrodes. The thickness‐shear waves for the optimal design of quartz crystal resonators [1,2] At such a frequency, the thickness-shear will propagate under the electrode but not in the unplated regions, satisfying the functioning waves will propagate under the electrode but not in the unplated regions, satisfying the functioning condition of a resonator.

Vibration Equations of Quartz Crystal Plates and Approximation

Plates with Electrode Coatings

Bechamann’s Number of a Resonator

Conclusions