有限元素分析於石英振盪器之聲子晶體應用 與非線性效應

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姓名

奴卡旺(Nugraha Merdekawan) 查詢紙本館藏

畢業系所

機械工程學系

論文名稱

有限元素分析於石英振盪器之聲子晶體應用與非線性效應
(Finite Element Analysis in Phononic Crystal Application and Nonlinear Effects of Quartz Oscillator)

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至系統瀏覽論文 (2028-5-15以後開放)

摘要(中)

石英振盪器是5G通信電子元器件的核心元件。由於5G應用必須具備高速率和低時延，因此對頻率規範的要求比以往更加嚴格。本論文之研究方向集中在以有限元素方法進行石英振盪器的數值模擬分析。論文中首先探討聲子晶體的應用，目標為減少振盪器的能量損失，獲得高Q值；當諧振器的尺寸變小時，高驅動功率和溫差會引起非線性效應，例如使振盪頻率偏移，從而導致頻率不穩定。故本論文亦分析減少能量逸散的方法，包含副波接近主模態的影響、切面角度的影響、以及聲子晶體之應用。論文第二部分析非線性特性，包含激勵功率依賴性 (DLD) 效應以及石英諧振器的溫度-頻率效應。
本研究中首先進行坐標系的轉換，推導由於石英的大變形和非線性特性引起的應力和應變的公式，以及建立非線性運動方程，再其後進行石英振盪器之模型建立，最後用有限元軟體 COMSOL執行計算以獲得模擬結果。本研究的的數值結果與先前研究的實驗數據一致性頗高，顯示其正確性。此技術可以提供更方便的模擬程序和更準確的數值數據。

摘要(英)

The quartz oscillator, also called the quartz resonator, is one of core elements of 5G communication electronic components. Because 5G applications require high speed and low latency, the accuracy of frequency component specifications is more stringent than before. This research thus concentrated on the numerical analysis, using finite element software COMSOL, of the quartz oscillator, especially on application of the phononic crystal structure and nonlinear effects. While the size of the resonator becomes smaller, the resonant region will be closer to the mounting glue; and this results in higher dissipating energy. Energy loss can be prevented by choice of dimensions to discard unwanted modes, beveled profile, and phononic crystal application. The high drive power and the temperature difference can induce nonlinear effects such as shifting the oscillation frequency and causing frequency instability. This study also worked on simulation of nonlinear effects of the quartz resonator, including drive level dependency (DLD) effect and the temperature-frequency effect. Before analysis of nonlinearity using COMSOL, the following work were done: transformation of coordinates, derivation of weak form governing equations, and acquisition of nonlinear coefficient values of quartz material.
The numerical results of the present models were verified using experimental and simulation data from previous studies. Compared to numerical data provided by previous researches, our numerical results accord better with the experimental data. This simulation technique can provide more accurate numerical data and more convenient calculation procedures.

關鍵字(中)

★ 石英振盪器；聲子晶體；激勵功率依賴性(DLD)；溫度-頻率效應；有限元法； COMSOL 多物理場
★ 石英振盪器
★ 聲子晶體
★ 激勵功率依賴性(DLD)
★ 溫度-頻率效應
★ 有限元法
★ COMSOL 多物理場

關鍵字(英)

★ quartz oscillator
★ phononic crystal
★ drive level dependency (DLD)
★ temperature-frequency effect
★ finite element method
★ COMSOL Multiphysics
★ quartz oscillator
★ phononic crystal
★ drive level dependency (DLD)
★ temperature-frequency effect
★ finite element method
★ COMSOL Multiphysics

論文目次

摘要 .................................................................................................................................. i
Abstract ......................................................................................................................... ii
Table of Contents .......................................................................................................... iii
List of Figures ................................................................................................................ v
List of Tables .............................................................................................................. viii
Chapter 1. Introduction ................................................................................................ 1
1.1 Introduction to the quartz oscillator .......................................................................... 1
1.2 Quartz crystallography and coordinate system ........................................................ 2
1.3 Quartz cutting angle and coordinate transformation ................................................. 4
1.4 Modes of vibration .................................................................................................... 6
1.5 Nonlinear characteristic of quartz ............................................................................. 7
1.6 Phononic crystal and dispersive medium .................................................................. 8
1.7 Literature review ..................................................................................................... 10
Chapter 2. Theoretical Formulations ......................................................................... 13
2.1 Piezoelectric effect ................................................................................................. 13
2.2 Large deformation theory ....................................................................................... 14
2.2.1 Deformation and strain analysis .................................................................. 14
2.2.2 Stress analysis ............................................................................................... 15
2.3 Finite element method and weak formulation ......................................................... 17
2.3.1 Finite element method .................................................................................... 17
2.3.2 Weak formulation using Galerkin’s method ................................................... 17
2.4 Quartz oscillator governing equations ..................................................................... 18
2.4.1 Linear governing equations ............................................................................. 18
2.4.2 Nonlinear governing equations ........................................................................ 20
2.5 Quartz oscillator equivalent circuit ......................................................................... 24
2.5.1 The composition of equivalent circuits ........................................................... 24
2.5.2 Series and parallel resonant frequencies .......................................................... 25
iv
Chapter 3. COMSOL Multiphysics Settings .............................................................. 27
3.1 Finite element model of quartz oscillator ............................................................... 27
3.1.1 Geometries ....................................................................................................... 27
3.1.2 Materials .......................................................................................................... 28
3.1.3 Physics ............................................................................................................. 30
3.1.4 Boundary conditions and the domain .............................................................. 31
3.1.5 Meshes ............................................................................................................. 32
3.1.6 Studies ............................................................................................................. 34
3.2 Phononic crystal structure ....................................................................................... 34
3.2.1 Lattice structure ............................................................................................... 34
3.2.2 Floquet periodicity ........................................................................................... 35
3.2.3 Dispersion diagram .......................................................................................... 35
3.3 Nonlinear analysis of drive level dependency effect .............................................. 37
3.4 Nonlinear analysis of temperature-frequency effect ............................................... 47
Chapter 4. Results and Discussions ............................................................................. 52
4.1 Influence of changing design parameters of the quartz oscillator .......................... 52
4.2 Energy trapping method utilizing bevel and phononic crystal ................................ 58
4.2.1 Effects of beveling on quality factor and displacements ratio......................... 58
4.2.2 Introduction of phononic crystal ..................................................................... 60
4.2.2.1 Effects of lattice length (a) and radius ratio (r/a) on the band gap ....... 60
4.2.2.2 Effects of number of holes (n), x-positions (x), arrangements on quality factor and displacements ratio .......................................................................... 62
4.3 Nonlinear analysis of drive level dependency (DLD) effect ................................... 67
4.4 Nonlinear analysis of temperature-frequency effect ............................................... 69
Chapter 5. Conclusions and Future Works .............................................................. 72
5.1 Conclusions ............................................................................................................ 72
5.2 Future works ........................................................................................................... 73
References...................................................................................................................... 74
Appendix A – Material Property of Y-Cut Quartz ................................................... 79

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指導教授

黃以玫(Yi-Mei Huang)

審核日期

2023-5-11

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