Two Dimensional Theoretical Analysis Of Partially Electroded Circular Quartz Crystal Resonator

Quartz crystal resonator(QCR)is a kind of acoustic component which is produced based on the piezoelectric effect of the quartz crystal and it is widely used in communication and sensor.As the electric communication devices towards miniaturization,high-frequency and low-power consumption,better performance of quartz crystal resonator is requested and it is essential for us to study the vibration of the QCR including resonance frequency and vibration modes.The present paper studies the thickness-shear vibration of the circular AT-cut quartz resonator whose electrodes on both top and bottom surfaces are ring or circular.Numerical calculation and FEM(finite element method)simulation are adopted in this thesis to discuss the TSh(thickness-shear)modes of AT-cut QCRs.In numerical calculation the 2-D scalar differential equations derived by Tiersten are separated into Mathieu and modified Mathieu equations using coordinate transform and variable separation.Theoretical displacement solutions constituting Mathieu and modified Mathieu functions are substituted into the boundary conditions by using the collocation method and the resonance frequencies and vibration modes are achieved in the end.COMSOL is also used in the thesis to simulate the TSh vibration of quartz resonator based on the 2D scalar differential equations.The resonance frequencies and corresponding modes are obtained.The shapes of vibration modes obtained from two methods shows that most of the thickness-shear vibration is under electrodes and this phenomenon is known as energy trap.By comparing the results of theoretical calculation with that of FEM simulation it is obvious that the results are almost same which has validated the correctness of both methods.In addition the influence on resonance frequencies and vibration modes of the parameters of the electrodes as well as mass ratio is discussed in the paper.