4F-6 Frequency-Temperature Behavior of Quartz Resonators Affected by Transient and Steady State Temperature Changes

Abstract

A new method which accurately predicts the frequency-temperature (f-T) behavior of quartz resonators affected by transient and steady state temperature changes is presented. Most practical resonators are subjected to thermal stresses. Conventional finite element analytical tools such as ANSYS cannot provide a sufficiently accurate model for the f-T curves of quartz resonators under thermal stresses. In our previous paper [Yong, Y-K, et al., 2005], we had proposed a method which employed a superposition of two f-T curves: one due to a stress-free, homogeneous thermal strain field, and the other due to the nonhomogeneous thermal stresses. The assumptions underlying the two f-T curves were not satisfactorily consistent. This paper presents a consistent and superior method for the problem: the constitutive equations for the Lagrangean, incremental displacements (small vibrational displacements) incorporate the temperature derivatives of the material constants. The incremental equations of small vibrations superposed on initial thermal stresses and strains are solved, and there was no need for the superposition of two f-T curves. Numerical results are compared with experimental results for a 50 MHz AT-cut quartz resonator mounted on a glass package. Good comparison between the experimental results and numerical results from our new method is shown. The difference between the thermal expansion coefficients of glass and quartz give rise to the thermal stresses that have an adverse effect on the f-T curves of AT-cut resonators. Different optimal crystal cut angles of quartz, and resonator geometry were found to achieve stable frequency-temperature behavior of the resonator in a glass package