雙端固定壓電石英音叉樑振盪器之自然頻率分析

Abstract

本文主要研究對象為(ZYw)+2°雙端固定音叉式石英振盪器，分析壓電效應對於共振頻率的影響，以及外加電極排列與石英壓電樑運動型態之關係。本文針對異相振盪行為進行分析，中央音叉樑視為尤拉樑，兩端質量塊則依據音叉樑異相振盪對質量塊所造成的力矩來建立翹曲形變的模型；再利用組成律與模擬結果對極化電位進行三角函數的假設，並以自由振盪方式求得共振頻率。耦合部分，假設其結構為彈性體，分別討論各自結構自由振動行為，再利用漢米頓定理推導出運動統御方程式及邊界條件，用以耦合機械與電性質之連續關係；並討論振動模態振形與極化電場分佈形態，以此作為激發共振器特定模態電極設計之依據。 為瞭解製程誤差對於共振頻率影響程度，分別以石英理想截面與濕蝕刻實驗截面進行討論，並分析尺寸對於共振頻率之敏感度，以作為製程誤差分析及光罩設計之參考，於頻率準確度與製程難度中取得平衡，以達到良率提升之目的。

The thesis mainly studied about double-ended tuning fork quartz resonator, the impact of the piezoelectric effect on a natural frequency, and the movement pattern of the relationship between an electrode arrangement and a quartz piezoelectric beam. Focusing on the analysis of the anti-phase mode, the study regards a central tuning fork as the Euler beam, and two ends of the proof masses built the warping model according to the moment caused by the anti-phase mode from the tuning fork to the proof masses; then using the constitutive law and the simulation result to make trigonometric assumption to the polarized electric potential, and obtaining the piezoelectric effect by the way of the free vibration. Assuming the coupling structure as an elastic body, and discussing the free vibration to each structure; using the Hamilton's principle to derive the governing equation and the boundary condition for coupling the continuous relationship between the mechanics and the electrical properties. The mode shape of the vibration and the distribution patterns of the polarized electric field are used to be the design basis of exciting the specific electrode of the resonator. For understanding the impact of the process error to the resonant frequency, the study discusses the ideal cross-section of the quartz and the wet etching cross-section respectively; furthermore, the study has analyzed the impact of the dimension to the sensitivity of the resonant frequency for being the references of the process error analysis and the design of the mask; in order to improve the yield, it must get the balance between the frequency accuracy and the difficulty during the process.