石英振盪器在現今電子產品中，扮演著舉足輕重的地位，主要用於頻率控制，其功能為穩定訊號的頻率，應用範圍由一般的石英鐘錶，衍生至通訊、資訊、工業、軍事等領域，石英振盪器頻率的精準及穩定相當重要。石英振盪器的品質好壞取決於振盪器的穩定度，石英晶片的尺寸變化、整體結構以及環境溫度都是影響振盪品質的重要參考因素。為了求解石英自然頻率，本文中使用有限元素軟體給予適當的邊界條件藉由頻域分析計算。
本論文將完成下列事項：(1) 建立石英振盪器結構之有限元素模型；(2)以MATLAB撰寫黃金比例法及擬牛頓法腳本，並命令COMSOL執行計算之程式，以尋找石英之自然頻率；(3)以模擬所得之電抗-頻率曲線，估算石英等效電路中之電容、電感與電阻數值；以及(4)探討各參數對石英振盪器自然頻率的影響。

Quartz oscillators, mainly used for frequency control, play an important role in electronic products. The scope of applications of Quartz oscillators is from quartz clocks, communications, industry, military and other fields. The quality of the quartz oscillator depends on the frequency stability which can be affected by the temperature, the size of the quartz crystal, and the size of the electrodes. For solving the natural frequency of quartz, finite element analysis software were used to calculate the natural frequency of the quartz oscillator.
This thesis completes the following items.
(1) Establish the finite element model of the quartz oscillator structure.
(2) Use MATLAB to write a script based on the Golden Section Method and Quasi - Newton Method for controlling the finite element software to calculate the admittance of the quartz system. The frequency corresponding to maximum admittance is the natural frequency.
(3) Estimate the capacitance, inductance and resistance of the quartz equivalent circuit from the reactance-frequency curve obtained by simulation.
(4) Discuss the influence of varying system parameters on the natural frequency of a quartz oscillator.

[1] Mindlin, R. D., "Thickness-Shear and Flexural Vibrations of Crystal Plates," Journal of Applied Physics, Vol. 22, pp.316-323, 1951.
[2] Mindlin, R. D. and Deresiewicz, H., "Thickness-Shear and Flexural Vibrations of Rectangular Crystal Plates," Journal of Applied Physics 26, p.1435-1442, 1955.
[3] Wang, J., Yu ,J., Yong, Y. K., Imai, T., "A New Theory for Electroded Piezoelectric Plates and its Finite Element Application for the Forced Vibrations of Quartz Crystal Resonators," International Journal of Solids and Structures, 2000.
[4] 李繼亮, 靳澤中, 王驥, 石英晶體諧振器結構高頻振動的有限元分析, Symposium on Piezoelectricity, Acoustic Waves and Device Applications (SPAWDA), 浙江寧波, 2011.
[5] Yong, Y. K., Mihir, S. P., Srivastava, S., Tanaka, M., and Imai, T., "The Impact of Finite Element Analysis on the Design of Quartz Resonators," IEEE International Frequency Control Symposium and Exposition, 2006.
[6] 黃自貴, 陳政宇, 小型化石英振盪器之設計分析, 行政院國家科學委員會專題研究計畫成果報告, 2010.
[7] F.N. Goodall, and C.A. Wallace, "The Analysis of Unwanted-Mode Vibration Patterns in AT-cut Quartz Oscillator Crystals, Revealed By X-Ray Diffraction Topography: 11. A Partial Theoretical Description of the Unwanted Mode," Journal of Applied Physics, Vol. 8, 1975.
[8] Morio Onoe, Takeru Mutoh, and Takashi Yamaguchi, "Suppression of Unwanted B-mode of SC-cut Quartz Resonator," Proceedings of Symposium on Ultrasonic Electronics, Vol. 34, pp. 175-176, 20-22 November, 2013.
[9] Chen, Y. Y., Wu, R. X., Wang, L. H., Jing, H. M., Du, J. K., Hu, Y. T., Li, G. Q., and Wang, J., "An Analysis of Nonlinear Thickness-Shear Vibrations of Quartz Crystal Plates Using the Two-Dimensional Finite Element Method," mechanics of advanced materials and structures, 2017.
[10] Wang, J., Wu, R. X., Yong, Y. K., Du, J., and Huang, D., "An Analysis of Vibrations of Quartz Crystal Plates with Nonlinear Mindlin Plate Equations," IEEE International Frequency Control Symposium Joint with the 22nd European Frequency and Time forum, 2009.
[11] Bechmann, R., "Higher-Order Temperature Coefficients of the Elastic Stiffnesses and Compliances of Alpha-Quartz," Proceedings of the IRE, 1962.
[12] Bechmann, R., "Elastic and Piezoelectric Constants of Alphaquartz", Phys. Rev., vol. 110, pp. 1060-1061, 1958.
[13] Lee , P. C. Y. and Yong, Y. K., "Frequency-Temperature Behavior of Thickness Vibrations of Doubly Rotated Quartz Plates Affected by Plate Dimensions and Orientations," Journal of Applied Physics 60, 2327, 1986.
[14] Bechmann, R., "Frequency-Temperature-Angle Characteristics of AT- and BT-type quartz Oscillators in An Extended Temperature Range," Proceedings of the IRE, vol. 48, P. 1494, 1960.
[15] 張力生, 石英諧振器, 中國計量科學研究院.
[16] COMSOL Software Corporation, "Thickness Shear Mode Quartz Oscillator," 2020.
[17] COMSOL Software Corporation, Live Link for MATLAB User's Guide, 2012.
[18] COMSOL Software Corporation, Live Link for MATLAB簡介, 2009-2018.
[19] S. RAO, Engineering Optimization Theory and Pratice, Four Edition, Wiley, USA, 2013.
[20] S. RAO, Mechanical Vibration, Fifth Edition, Pearson, UK, 2011.
[21] "Quartz," 2020. [Online]. Available:
https://en.wikipedia.org/wiki/Quartz.
[22] "Sauerbrey equation," 2020. [Online]. Available:
https://en.wikipedia.org/wiki/Sauerbrey_equation.
[23] Mihir S. Patel, Nonlinear Behavior in Quartz Resonators and its Stability, Ph.D. dissertation, The State University of New Jersey, 2008.
[24] "TXC技術辭典," 2020. [Online]. Available:
http://www.txccorp.com/index_tw.php?action=c_technology_1&cid=1.
[25] 吳朗, 電子陶瓷-壓電, 全新科技出版社, 台北市, 1994.
[26] "漫談壓電效應與超聲波", 2017. [Online]. Available:
http://funphysicsstudio.blogspot.com/2017/08/blog-post_28.html.
[27] "壓電效應", 2020. [Online]. Available:
https://zh.wikipedia.org/wiki/%E5%A3%93%E9%9B%BB%E6%95%88%E6%87%89.
[28] 張仁豪, 表面聲波影響石墨烯的電傳導研究, 國立中興大學物理學研究所碩士論文, 2015.
[29] "COMSOL博客-壓電材料標準簡介", 2014. [Online]. Available:
https://cn.comsol.com/blogs/piezoelectric-materials-understanding-standards/.
[30] IEEE Standard on Piezoelectricity, ANSI/IEEE Standard 176-1978, 1978.
[31] "Standards on Piezoelectric Crystals", Proceedings of the I. R. E., vol. 37, no. 12, pp. 1378-1395, 1949.
[32] "ST微控制器振盪器電路設計指南", 2009. [Online]. Available:
https://www.st.com/content/st_com/zh.html.
[33] "Electrical impedance", 2020. [Online]. Available:
https://en.wikipedia.org/wiki/Electrical_impedance?fbclid=IwAR3H50vxxa_RC-fX4Nvoou0lmbJ8lza4q6tLT9vOM0cM85uHF3nEOJZRtCU.
[34] 黃以玫, 高等振動學講義, 中央大學噪音與控制實驗室, 桃園市, 2020.
[35] "常用高分子材料介電常數表", 2019. [Online]. Available:
https://kknews.cc/zh-tw/tech/6q6o2ll.html.