Thermal transient model of a crystal resonator employing thickness-shear vibrations

This paper presents an improved model of thermally induced frequency transients in vacuum-enclosed thickness-shear mode quartz crystal resonators. The response times to temperature changes for different parts of the resonator and resulting thermal dynamic coefficients are examined and are related to Ballato's coefficient through a function defined by the resonator design, dependent on thermal response times only. A method is worked out for response time calculations for the different contributions to the static and dynamic temperature behavior of general and anharmonic modes. The model has been used to examine thermally induced frequency transients of the AT-cut resonator h(513) anharmonic mode excited by the modulational method within an ovenized Colpitts oscillator. A good agreement is shown between the predicted curves and experimental data over a variety of temperature ranges.