Finite Element Approach to Frequency-Temperature Characteristic Prediction of Rotated Quartz Crystal Plate Resonators

Abstract

The finite element technique is utilized for calculating the natural frequencies and frequency-temperature coefficients of thin anisotropic quartz crystal plates. The computer programs developed can accommodate a plate cut in any direction in the crystallographic coordinate. Examples of X-cut rectangular plates and DT-cut plano- convex plates are shown and numerical results are compared with mea- sured ones as well as with theoretical predictions if available. UARTZ crystal resonators present well-known problems but are still increasingly used as time-keeping devices not only for telephone and telecommunication systems but also for watches and microprocessors. A finite element method (FEM) has been applied to the calculation of the nat- ural frequencies and their associated modes in thin piezoelec- tric plates in plane motion together with their admittance characteristic at the electrical terminals (l). The temperature characteristic is another important factor of the resonator, particularly when it is utilized as a frequency control device. The prediction of the temperature dependence of the natural frequency and Q factor of a composite circular transducer (Langevin-type) has been demonstrated (2), in which the ad- hesive layers are assumed to be responsible for the temperature characteristic so that only their Young's modulus and loss factor are considered to be temperature-dependent and no geometrical expansion is taken into account. This paper is concerned with the prediction of the natural frequencies of thin quartz crystal plates and their temperature characteristics. The finite element technique is employed to calculate them, taking into effect the geometrical change of the plates due to thermal expansion and the temperature dependence of the elastic stiffness tensor. The plates are as- sumed to be thm and of plane-stress motion. No electro- mechanical coupling effect is included as it is small in quartz crystal. The computer program was developed to calculate the natural frequencies of various modes and their frequency- temperature coefficients (up to the third order) of a thin anisotropic plate cut in any direction of its crystallographic