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.
We numerically analyze the influence of viscosity loss in quartz on the unwanted coupling between fundamental thicknessshear (TS) and spurious modes. Classical mode-matching is used to solve the two-dimensional coupled vibrations in an ATcut quartz plate with an inverted-mesa shape. Using this technique, we calculate the loci of admittance near the main TS response as a function of the loss factor. The temperature behavior of frequency and conductance is also examined for the main TS resonance. The results reveal that, for two strongly coupled vibrations of TS and spurious modes, two resonances on the real frequency-axis are combined into one with increasing loss, although there are still two resonances in the complex frequency plane. We also find that the conductance-temperature behavior of the main TS response is more sensitive to mode coupling than the frequency-temperature behavior, and it therefore works well as a detector of mode coupling.
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