The Analysis Of Vibrations Of Quartz Crystal Resonators With The Consideration Of Thermal Effect And Optimal Cuts

Quartz crystals are anisotropic crystal materials which physical properties change as the angle of rotations of the quartz crystal change.For quartz crystal resonators,temperature is the most important factor affecting the frequency stability.We hope that the frequency change of the resonator will be as small as possible within the operating temperature range,making the resonator highly stable in frequency.In order to study the frequency temperature characteristics of the different cuts of quartz crystal resonators and the optimal operating ambient temperature,the incremental thermal field theory is used in this thesis to analyze the vibrations of quartz crystal resonators so that we could use these results to find out the optimal cuts of quartz resonators with wider temperature range.The object of this research is a rectangular quartz crystal resonator of arbitrary double rotations.First,the thesis briefly introduces the thermal properties of quartz crystals and the incremental thermal field equations.Then,based on the three-dimensional equations of incremental thermal field,the precise procedure of frequency calculation and the frequencytemperature relationship of the infinite doubly rotated quartz crystal plate without considering the piezoelectric effect were deduced.Using the Mindlin first-order plate theory under the incremental temperature field,the dispersion and spectral equations of arbitrary doubly rotated quartz crystal resonators were also deduced.Because of the quartz resonator works at thicknessshear mode,we can obtain the frequency-temperature curves with selected the appropriate length-thickness ratio.The resonators with excellent frequency-temperature characteristics that the curve generally is a cubic curve,so that the resonator will be stable over a wide temperature range.By solving the equations with the first and second derivatives setting to zero from the frequency-temperature relationship of the quartz crystal plate,the frequency-temperature relationship represent cubic function which greatly expand the operating temperature range of the resonator.At the same time,for each specific cut,there is a unique zero temperature coefficient point,that is,the inflection point of the function,for different cuts,we can accurately know the appropriate operating temperature range to allow the resonator to work in extreme environments.This study provides a theoretical basis for the development of a highly stable quartz crystal resonator at a specified temperature.