Estimation of Energy Loss and Electrical Resistance of Piezoelectric Resonator Structures by an Energy Sink Method

Abstract

The quartz resonator is the most common piezoelectric resonating structure designed to provide a stable frequency source for electronic devices and applications. An important figure of merit for the quartz resonator structure is the quality factor Q which measures the “sharpness” of its frequency response curve and its frequency stability. This quality factor Q is inversely proportional to the energy loss per cycle of oscillation. The Q is hence also inversely proportional to the resonator resistance or impedance. Currently there are no analytical tools for estimating the Q without an apriori assumption of the resonator damping or impedance. In order to get good numerical agreement with the measured data, all the current finite element software requires an assumption of either the resonator resistance or resonator Q value. We propose and present a new analytical tool for estimating the quartz resonator Q and other figures of merit by an energy sink method. Results for the thickness shear mode, AT-cut quartz resonators are presented. Experimentally measured material constants of mechanical dissipation and conductivity of quartz was included in our models. Our energy sink method is more realistic than assuming the crystal impedance or Q. One of the most important factors affecting the Q in the design of a new quartz resonator structure of a given frequency is the loss of energy from the electrode area to the base via the mounting supports. The acoustical characteristics of the plate resonator are changed when the plate is mounted onto a base. This is analogous to a soil-structure interaction problem with a semi-infinite boundary. The substrate base affects the frequency spectra of the plate resonator. A resonator with a high Q may not have a similarly high Q when mounted on a substrate base; hence the base is an energy sink. We present the frequency spectra of the quartz resonator with and without the substrate base. A scaled boundary finite element method is employed to model a semi-infinite base. Since a semi-infinite base will absorb all acoustical energies radiated from the resonator, a forced vibration analysis of such a model will provide the lower bound Q values. The model with a semi infinite base could be used for evaluating resonator and mountings designs. The effect of the mountings, plate and electrode geometries on the resonator Q and other figures of merit are presented. Comparisons of the experimental measurements of the admittance and Q with the simulated results using the semi-infinite energy sink method were performed. The measured admittance compared very well with the simulated results. The measured Q showed the same trend as the simulated Q.