矩形AT切石英谐振器寄生振动模式的有限元分析

随着实际应用中的谐振器向小型化与高频发展，AT切石英晶体谐振器中寄生振动模式增多，寄生振动模式与主模式厚度切为模式之间的耦合问题越来越突出，因此有必要与晶片X和Z′尺寸均有关的寄生模式进行理论分析，以Mindlin的二维晶片方程为基础，运用有元法求解AT切石英晶体谐振器的振动模式，得到了晶片频率随A长度变化的频谱图，其结果与Koga的实验结果比较，符合得比较好，与通常用解析解的方法分析的结果相比较，有限元法的优越性在于可得到更多的非直行波模式（非晶片长、宽方向），还可以模拟出各模式的主要位移分量的波形图，对抑制寄生模的大小有减少寄生模与主模的耦合提供了更详细的分析资料。

Analysis of Spurious Vibration of Rectangular AT-cut Quartz Plates Using FEM

Rectangular AT-cut resonators have been recently developed and widely used in frequency control because of their small sizes and excellent frequency-temperature stability.More unwanted spurious modes will be excited in rectangular quartz plates as the size of resonator substrate becomes smaller and smaller.It causes the serious coupling between the fundamental mode and the spurious modes, which may result in degradation of resonator performance, such as jump of frequencies and decreasing of the value of Q.Lots of authors addressed on the spurious modes along the X or Z' crystal axis only, called straight-crested vibration modes.However from famous Koga's experimental frequency spectrum, we can see that except straight-crested modes, more resonant modes are in existence which have phase reversals along both the X and Z' axis and are electrically weak.However, even a weak mode can affect the fundamental thickness-shear mode strongly when the frequencies of the spurious modes approach the fundamental mode.Because of this, it is necessary to further study spurious vibrations in detail.However, it is intractable to find a general solution to these vibrations using the analytical method because of the complication of the plate equations and boundary conditions.The merit of FEM(Finite Element Method), which can convert a problem of complicated boundary conditions into one of numerical computation, makes it a naturally powerful tool for analysis of crystal resonator.In this paper we derived the finite element formulation of the vibrations of piezoelectric quartz resonators based on Mindlin's first-order plate equations,and the convergence study was given.The resonant frequencies of different vibration modes were obtained as a function of X-length and the results were in good agreement with the well-known Koga's experimental results.In order to identify the various resonance modes,the mode shapes of predominant displacement components were calculated.Though many of the mode shapes were very complicated due to the strong coupling of all displacements,some could be distinguished for the relatively simple shapes.The theoretical mode shapes also showed that most of spurious resonances in the frequency spectrum were composed of the vibrations which depended on both the X and Z' axis,and only very few spurious modes were associated with the straight-crested waves.