Nonlinear behavior in quartz resonators and its stability

Piezoelectric materials form an integral part of frequency control devices. With the advancement in technology smaller frequency control devices [MEMS: Micro-Electrical Mechanical Systems] are manufactured. However, with the decrease in the size of these devices, the nonlinearities of the piezoelectric materials become predominant and it results in frequency instability. Hence, even though smaller resonators are manufactured they are found lacking in frequency stability. Thus, the need for smaller ultra-stable high frequency resonators and oscillators requires an extensive study of the nonlinear behavior of piezoelectric material. Quartz is mainly used as the piezoelectric material for manufacturing ultra-stable high frequency devices. Thus, an attempt has been made to study the nonlinear behavior of quartz with respect to its different cut angles. Three dimensional finite element models are developed to calculate the effect of nonlinearities on the thickness shear mode resonant frequency.;The intrinsic nonlinearities affecting the quartz resonators at high frequencies (>10 MHz) are drive level dependency (DLD), acceleration sensitivity effect and the frequency-temperature behavior. The effect of all these nonlinearities on the stability of quartz resonator was studied in terms of Quality (Q) factor values. Frequency spectrum charts with dissipation and having no prior assumption of loss factor values were developed for the calculation of Q-factor values. The energy sink method was developed for evaluating the lower bound Q-factor value when the quartz resonator was mounted on substrate. The quartz resonators operating at 10 MHz and 40 MHz were used to validate the energy sink method with the measurement data. The comparisons showed a very good agreement between the calculated and the measured Q-factor values. The method was used to calculate the Q-factor values for doubly rotated cut angles of quartz.;The iterative algorithm developed for studying the drive level dependency effect in quartz resonators was rigorously fortified by comparing it with the measured data for 40 MHz AT-Cut, SC-cut and BT-Cut quartz resonators. The doubly rotated quartz cut with an angle of &phis; = 8· &phis; = 34.93· and &phis; = 12· &phis; = 34.93· were predicted to have the lowest DLD sensitivity. Superposed equation of motion was derived for the acceleration effect in quartz resonators. 10 MHz circular AT-cut and SC-cut quartz resonators modeled for the acceleration sensitivity effect showed a good comparison with the measurement data. The method showed that applying a DC-bias of 0.092 ppm/volt had counteracted the acceleration sensitivity of SC-cut quartz resonators. The frequency-temperature (f-T) behavior of quartz resonators was classified into static f-T behavior and dynamic f-T behavior. The finite element models for the behaviors showed excellent agreement with the measured data for 50 MHz AT-cut quartz resonator when mounted on a glass substrate. An optimum cut angle of 38· was found to have a stable f-T behavior when enclosed in a glass package. Based on this study, designs of ultra-stable thin film quartz resonators operating in the frequency range of 3.4 GHz and solidly mounted resonators operating in the frequency range of 1.9 GHz, for different configurations are mentioned.