以有限元素法分析石英振盪器的共振特性與激勵功率依賴性性質

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姓名

黃醴萬(Li-Wan Huang) 查詢紙本館藏

畢業系所

機械工程學系

論文名稱

以有限元素法分析石英振盪器的共振特性與激勵功率依賴性性質
(Determining Resonance Characteristics and Drive Level Dependency of Quartz Oscillators via the Finite Element Method)

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至系統瀏覽論文 (2026-8-31以後開放)

摘要(中)

石英振盪器懬泛出現於各種電子產品，其主要功能為提供設備穩定的時脈訊號使設備可以進行量測、通訊等功能，是不可或缺的電子元件。而隨著5G時代的來臨，愈來愈多的通訊需要達到高速度、低時延等需求，也因此對於石英振盪器的工作條件也更加嚴苛，此時石英振盪器的振動型態也會逐漸變得複雜，因此成為製造商設計商目前所面臨的問題。本研究由理論出發，以有限元素軟體COMSOL針對以厚度剪切模態作為主模態的AT Cut石英振盪器進行模擬與分析。本研究包含石英振盪器的設計參數分析，找出各個設計參數對於模態的影響；此外，本研究也在電極上加入鑽孔，藉此分析鑽孔的各個設計參數對於系統響應的影響；最後，藉由加入石英的各個非線性特性進入模型中進而分析石英的激勵功率依賴性，以分析石英在高功率驅動時對於系統響應的影響，此外也藉由修改材料常數為溫度函數進行溫度效應分析。數值結果顯示，除了石英與電極的厚度對於系統的頻率與Q值有最高的敏感度外，電極的長度與寬度對於頻率也有不可忽略的影響。鑽孔的深度達到石英時對於主模態的Q值會有顯著的影響，而當鑽孔的位置愈遠離中心時其對於系統的影響也會愈小。而非線性的數值結果顯示當電壓大於一定值時，系統在同一頻率下會出現多解。溫度效應分析結果與文獻的數值有差距。

摘要(英)

Quartz oscillators are commonly found in various electronic devices. Their primary function is to provide the clock signal for the devices performing measurement, communication, and other functions. With the advent of the 5G era, the requirements of high speed and low latency rely on quartz oscillators with higher frequency. To find better designs of quartz oscillators, this research used the FEM software COMSOL for simulating the dynamic characteristics and performing parametric studies of mainly the dimensions of the oscillator. Our findings also include the influence of unwanted modes and the effect of drilling holes on the electrode. Nevertheless, nonlinear finite element models have been developed to analyze the Drive Level Dependency (DLD) of quartz oscillators. Finally, analysis of temperature effect is performed by modified the material constants to a function of temperature. The numerical results shows that not only the changes of the thickness of quartz blank and electrode strongly impact either frequency or Q factor, the length and width of electrode also have a significant influence on the frequency of the oscillator. Next, the diameter of the hole affect frequency drift positively. The depth of hole affect Q-factor significantly when penetrating the quartz. Moreover, the result of DLD analysis shows that multiple resolutions is found on certain frequency when applied voltage is too high. Finally, the result of the analysis of temperature effect shows that, comparing to reference, there is a nonneglected difference on value is observed.

關鍵字(中)

★ 石英振盪器
★ 有限元素法
★ 激勵功率依賴性

關鍵字(英)

★ Quartz crystal
★ Finite element method
★ Drive level dependency

論文目次

摘要 i
Abstract ii
誌謝 iii
目錄 iv
圖目錄 viii
表目錄 xiv
第一章. 緒論 1
1.1 研究背景與目的 1
1.2 文獻回顧 2
1.3 論文架構 4
第二章. 石英振盪器簡介 5
2.1 石英振盪器的運作原理 5
2.2 石英的晶體學與座標系統 7
2.3 石英的切割角與材料座標轉換 9
2.4 石英的非線性特性 13
2.4.1 溫度效應 13
2.4.2 激勵功率依賴性 15
第三章. 基本理論 17
3.1 壓電效應 17
3.1.1 壓電效應簡介 17
3.1.2 機電耦合方程式 18
3.2 大變形理論 22
3.2.1 變形分析 22
3.2.2 應力的描述 24
3.3 石英振盪器的統御方程式 27
3.3.1 線性統御方程式 28
3.3.2 非線性統御方程式 29
3.3.3 溫度效應之統御方程式 31
3.4 有限元素法與弱型式 39
3.4.1 弱型式與蓋勒肯法 39
3.4.2 石英振盪器線性統御方程式的弱型式 41
3.5 石英振盪器的等效電路分析 43
3.5.1 等效電路的組成 43
3.5.2 串聯諧振頻率與並聯諧振頻率 44
3.6 擬牛頓法 47
3.7 黃金比例法 49
第四章. 研究方法 51
4.1 石英振盪器的有限元素模型 51
4.1.1 材料設定 54
4.1.2 邊界條件設定 56
4.1.3 網格結果收斂性 56
4.2 設計參數對頻率的影響 58
4.3 鑽孔對頻率的分析 59
4.3.1 鑽孔尺寸對響應的影響 60
4.3.2 鑽孔位置對響應的影響 61
4.4 以數值方法求取系統參數 62
4.4.1 以複數特徵值尋找自然頻率與Q值 64
4.4.2 以數值方法求取等效電路參數 65
4.5 激勵功率依賴性之非線性分析 66
4.6 溫度效應分析 73
4.6.1 修改石英的材料常數為溫度函數 73
4.6.2 更改電極的材料 76
第五章. 數值結果與討論 78
5.1 線性分析結果 78
5.2 設計參數對頻率的影響 81
5.3 副波模態的影響 86
5.4 鑽孔對頻率的影響 88
5.4.1 鑽孔孔徑的影響 88
5.4.2 鑽孔深度的影響 90
5.4.3 鑽孔位置的影響 93
5.5 以數值方法求取等效電路的參數 96
5.6 激勵功率依賴性之非線性分析 97
5.7 溫度效應分析 106
5.7.1 修改石英的材料常數為溫度函式 106
5.7.2 加入熱膨脹係數 108
5.7.1 更改電極材料係數 108
5.7.2 修改石英的密度為溫度函數 109
第六章. 結論與未來展望 112
6.1 結論 112
6.2 未來展望 114
參考文獻 115
附錄A Y-Cut石英的材料係數列表 121
附錄B 石英的3階彈性常數張量中之各項係數計算公式 131
附錄C 以COMSOL求取等效電路元件數值 133

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指導教授

黃以玫(Yi-Mei Huang)

審核日期

2021-8-16

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