3-D FEM Eigenvalue analysis of relative impedance and energy trapping of resonant modes in AT-cut resonators

There is currently a need to miniaturize AT-cut resonators. However, the size reduction sometimes sacrifices the resonator's Q-values and crystal impedances. Hence, there is a requirement to develop new resonator designs by changing the cut angles or electrode configurations. For these purposes, the 3-D finite element method is employed as a promising design/analysis and prototyping tool for new quartz resonators. There are generally two types of analyses: Eigenvalue analysis and forced vibration analysis. The eigenvalue analysis of resonant modes in AT-cut resonators has been shown to be accurate in predicting the resonant frequencies as a function of the resonator and electrode geometry. The forced vibration (steady state) analysis is used to calculate the motional impedance and capacitance in the resonator. In this paper we propose using the Eigenvalue analysis for comparing the impedance and energy trapping of the fundamental thickness shear mode of an AT-cut resonator as a function of the resonator and electrode geometries. By comparing and calibrating the numerical results with experimental data, the eigenvalue analysis can efficiently estimate Q-values, crystal impedances, strength of activity dips and frequency- temperature stabilities. The eigenvalue analysis could further generate information useful for choosing resonator and electrode geometry that have higher dimensional tolerance in a fabrication process. The proposed method could be employed to develop a new resonator with a different cut angle and with different electrode configurations. The forced vibration analysis is more cumbersome for these types of analyses.