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研究生： 陳旻谷
Chen, Min-Ku
論文名稱： Matlab於一維至三維有限元素法（FEM）開發與壓電薄膜聲波共振器（FBAR）模擬之研究
Development of 1D-to-3D Finite Element Method (FEM) and Simulation of Film Bulk Acoustic Resonators (FBAR) Using Matlab
指導教授： 李炳鈞
Li, Bing-Jing
學位類別： 碩士
Master
系所名稱： 電機資訊學院 - 電機工程學系
Department of Electrical Engineering
論文出版年： 2025
畢業學年度： 113
語文別： 中文
論文頁數： 165
中文關鍵詞： 壓電薄膜聲波共振器 、有限元素法 、程式實現 、多維數值模擬
外文關鍵詞： FBAR, FEM, Programmatic Implementation, Multi-dimensional Numerical Simulation
相關次數： 點閱：62  下載：3 
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    • 壓電薄膜聲波共振器（FBAR）廣泛應用於射頻通訊、感測器與高頻電子元件領域，其特性深受幾何結構、材料性質及邊界條件影響。為有效地進行元件設計與分析，數值模擬已成為重要工具。本論文旨在利用有限元素法（Finite Element Method, FEM），對不同維度（1D、2D、3D）的壓電薄膜聲波共振器進建立了一套相對完整且系統化的數值模擬架構，探討不同維度模型間的差異與特性，並進一步探討邊界條件改變與雜散模態之生成與影響。
    在理論架構方面，本研究採用線性壓電理論及彈性力學的控制方程式，透過Galerkin加權殘值法推導有限元素之矩陣形式與邊界條件處理。進而撰寫FEM程式，分別建立1D、2D、3D模型，並實現數值計算與特性分析。針對各維度模型，本研究明確說明了幾何網格生成方式、有限元素矩陣之組裝細節與求解流程，以及特性分析與後處理之計算方法。
    在結果分析方面，本研究首先比較了不同維度模型的諧振特性與模態分布，發現隨維度提高，模型可描述的現象更完整，但計算成本亦顯著增加。特別是在高維度模擬中，雜散模態會更加明顯且複雜。進一步研究探討電極邊界條件對共振特性的影響，證實有限元素法在模擬電極側面邊界時有所限制。此外，本研究亦詳細驗證了不同維度模擬結果與理論值及文獻數據間之準確性，確認所提出之數值模型具有良好的可靠性。
    最後，本研究之貢獻在於提供一套全面且清晰的壓電薄膜聲波共振器有限元素數值分析與程式實現架構，並系統性比較1D、2D、3D模型差異。未來可朝更精細的3D模型建構、材料參數之優化，以及進一步考慮實際元件中可能的熱與損耗效應進行研究，期望更貼近實際應用需求，提升FBAR設計與分析的準確性與實用性。

    Film Bulk Acoustic Resonators (FBARs) are piezoelectric MEMS devices widely used in RF filters and sensors, whose resonant behavior is governed by device geometry, material properties, and electrode boundary conditions. This thesis employs linear piezoelectric theory and the finite element method (FEM) to simulate FBARs in one, two, and three dimensions using MATLAB, systematically comparing multi-dimensional models. The FEM formulation is derived via Galerkin weighted-residual integration; custom code generates 1D–3D meshes, assembles stiffness matrices, and solves for resonant modes and impedance. Results show that higher-dimensional models capture richer mode structures but require much greater computational cost. In 2D and 3D simulations, spurious lateral modes become clearly visible. Electrode-side boundary conditions significantly affect these modes, highlighting the limitations of simple FEM boundary treatments. All simulation results agree with theory and literature data, confirming the reliability of the numerical models. Overall, this work establishes a comprehensive FEM framework for FBAR analysis and provides guidance for future higher-fidelity modeling.

    摘要I 誌謝XII 目錄XIII 圖目錄XVIII 第一章 緒論 22 1-1 研究背景與動機 22 Pre-processing 23 Main body 24 Post-processing 24 1-2 研究方法與目標 27 第二章 基本理論 30 2-1 FBAR基本理論 30 2-1-1 架構 30 固態堆疊型 31 背向蝕刻 31 氣隙型 32 2-1-2 理論分析架構 32 聲學理論 32 壓電耦合 37 控制微分方程 46 2-1-3 諧振原理 47 阻抗 48 諧振與反諧振 50 2-1-4 FBAR特性表徵 53 2-2 FEM數值分析方法 54 2-2-1 FEM之數學架構 54 2-2-2 Galerkin加權殘值法 56 2-2-3 插值函數與單元離散化 57 2-2-4 單元剛度矩陣推導 60 2-2-5 全域系統組裝 61 2-2-6 多維度 FEM 模型比較 62 2-2-7 FEM在FBAR上的應用 63 2-3 Matlab 在 FEM 上的應用 65 第三章 FEM運作模式之實現與程式碼 67 3-1 3D FEM程式架構 67 3-1-1 執行環境Mingw64設定 67 3-1-2 3D FEM 主程式 70 前處理：環境設定與幾何網格生成 71 有限元素矩陣組裝與求解 77 後處理：特性計算與場分佈繪製 80 3-1-3 3D FEM各函式實作效用 82 前處理函式集合 82 FEMProcess函式集合 91 後處理函式集合 106 3-2 2D FEM程式架構 115 3-2-1 MESH的更改 115 3-2-2 形狀函數導數的不同 118 3-2-3 有限元素矩陣組裝原理的差異 119 3-2-4 後處理計算差異 122 3-3 1D FEM程式架構 127 3-3-1 MESH的更改 127 3-3-2 形狀函數導數的不同 127 3-3-3 有限元素矩陣組裝原理的差異 128 3-3-4 後處理計算差異 130 第四章 程式碼驗證與討論 135 4-1 2D FEM 模擬真空層有無與邊界條件探討 135 4-2 3D、2D、1D計算的連續與參考性 137 4-2-1 3D結果與文獻比對 137 4-2-2 3D、2D、1D結果比對與參考 138 4-3 2D、3DFEM對雜散模態的計算 141 4-3-1 2DFEM結果與討論 141 4-3-2 3DFEM結果與討論 143 4-4 針對上電極邊界條件改變的探討 147 4-4-1 2DFEM結果與討論 147 4-4-2 3DFEM結果與討論 149 4-4-3 2D側面電極條件置於真空的探討 153 第五章 結論與未來展望 156 5-1 結論 156 5-2 未來方向 158 參考文獻 160

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