Two-Dimensional Analysis of Coupled Vibrations of UHF AT-Cut Quartz Plates with Electrodes of Plano-Mesa Shape

Abstract

We numerically analyze the influence of both plate and electrode edges on unwanted coupling between fundamental thickness-shear (TS) and spurious modes. Classical mode-matching is utilized to solve the two-dimensional coupled vibrations of an AT-cut quartz plate with strip electrodes of plano-mesa shape. Using this technique, we calculate the minimum frequency differences between TS and flexural (F) or extensional (E) resonances as a function of electrode thickness. The results reveal that, with increasing electrode thickness, the origin of unwanted coupling between TS and F modes shifts from the plate edges to the electrode edges, while TS- and E-mode coupling is caused only by the electrode edges and it therefore closely correlates to the inverse of quality factor Q derived from modeling an infinite plate. We also find that there is an index of electrode thickness that suppresses both F and E components coupled with the TS mode.

Influence of Viscosity Loss on Coupled Vibrations of Ultrahigh Frequency AT-Cut Quartz Plates

“…7) (2) An analysis assuming lossless UHF plates has found strong coupling between the TS and F or E modes. 8) These items suggested that the finite quality (Q) factor of AT-cut plates could obscure the coupling phenomena in UHF resonators. Our objective is to verify this qualitative interpretation by analyzing the coupled vibrations of lossy plates.…”

Identification of Vibration Modes of Quartz Crystal Plates with Proportion of Strain and Kinetic Energies

“…The material parameters of quartz crystal and its AT-and SC-cut can be found in Refs. 3,5,8,11,14 The matrix M and N in Eq. (6) and Eq.…”

“…2 The normalized dispersion relations and frequency spectra, as effective representations of the vibration analysis results, were widely used for the optimal selection of plate parameters. Of course, other researchers also proposed similar or novel two-dimensional theory in resonator vibration analysis, like Lee, et al 10 and Sekimoto, et al 11 In recent years, applications of Mindlin plate theory have been further improved. Based on the high-order Mindlin plate theory, Yong and Zhang 12 and Wang, et al 13 developed the finite element method in analyzing high-frequency vibrations of quartz crystal resonators.…”

Three-Dimensional Analysis of Coupled Vibrations of Rectangular AT-Cut Quartz Plates with Tab Electrodes

“…The method used for 3D analysis conforms to conventional mode-matching. 2,3,[6][7][8] The structure of a plate is separated into three regions Àa x 1 Àe 1 , jx 1 j e 1 , and e 1 x 1 a. By applying a two-dimensional expansion of Legendre polynomials to the crosssection, (jx 2 j b and jx 3 j c), 9,10) eigenwaves guided along the x 1 -axis are found to construct a free solution for each region.…”