A Modified Maxwell Model for the Aging of AT-Cut Quartz Crystal Resonators

Quartz crystal resonators have been widely used in electronic communication products due to their advantages of frequency stability and strong anti-interference ability. However, in the long-term usage, the frequency of the quartz crystal resonator shifts with time, which is known as the aging phenomenon. In the cases when high stability of frequency is required, the frequency shift has a large impact on the applications. Therefore, it is important to establish a feasible model to predict the aging of quartz crystal resonators. In this paper, the potential of the viscoelastic theory in the description of the aging of quartz crystal resonator is discussed. Since the thickness vibration theory suggested the relation between the frequency of quartz crystal resonator and the elastic constants, the frequency of quartz crystal resonator is time-dependent if the elastic constant is replaced by an effective elastic constant, which is time-dependent. The definition of the effective elastic constants is proposed via stress relaxation of the Maxwell mode in viscoelasticity. Furthermore, the time-temperature superposition principle is adopted to predict the aging of quartz crystal resonator at different temperatures. The elastic parameters of the Maxwell mode are the elastic constants of quartz, and the viscous parameters of the Maxwell mode are determined by fitting the proposed theory with the experiment data. Similarly, the movement factors are determined. The results proves that the frequency aging of quartz resonators can be well described by the newly proposed theory.