Higher-order plate theory based finite element analysis of the frequency-temperature relations of quartz crystal resonators

The frequency-temperature characteristics of quartz crystal resonators, particularly the frequency stability in a specific temperature range in which the vibration modes could be strongly coupled, has been an important requirement in most applications. The analytical work on the frequency-temperature relations has been done over last decades in many aspects, ranging from fundamental theory of the thermal effect to the simplified plate equations of a few strongly coupled vibration modes. However, it has been clearly observed that due to the complication of resonator structures, such as the presence of mounting structure and asymmetric electrodes, simple and analytical solutions will not be able to consider all the factors which will have inevitable and noticeable effects on the resonators. In this paper, we incorporate the frequency-temperature theory for crystal plates based on incremental thermal field theory by Lee and Yong (1986) into our finite element analysis program, which can analyze the free vibrations of crystal plates with higher-order plate theory. The effect of the electrodes on the frequency-temperature relations is also studied in detail. The computational results are pared with experimental ones from real products. Satisfactory agreement demonstrates the precise prediction of the frequency-temperature behavior and practical applications of the current finite element analysis in product modeling and development.