微機電系統發展至今已過數十年。相對於傳統加工製造，使用近似半導體製程所製作的微機電(MEMS)元件能有更高的精度、更小的體積，並具有能大量製造進而降低生產成本的優勢。然而，在MEMS製程上容易在元件內部造成殘餘應力殘留，且成品與設計間也可能存有些微的尺寸誤差，而這些都將會對元件操作時之精度產生影響。本篇文章中將以電容式微機電加速度感測器為例，希望藉由一些簡單的量測，對受製程不確定性影響MEMS元件之等效機械特性進行定量分析。
在MEMS元件中，最常被使用之材料為單晶矽，由於其晶體結構使楊氏模數(Young's Modulus)具有方向性。在對結構進行力學分析時經常會簡化問題，以等向性(Isotropic)材料參數取代非等向性(Anisotropy)材料參數進行分析。本文中將使用商用有限元素軟體ANSYS®，在微機電三軸加速度感測器的三維模型中，分別給予等向性與非等向性兩種不同的材料參數，觀察其對自然頻率造成的影響，進而確認使用等向性取代非等向性材料參數是否為合理近似。
在文中後半，本文提出了利用量測自然頻率、吸合電壓、電容值與特定條件下量測獲得的電壓Vb，計算電容式微機電加速度計質量塊質量與電極間間距的關係式。在MEMS元件的製程中，元件的實際幾何尺寸與設計值間可能因蝕刻誤差的存在而有些許差異。如利用本文中提出之質量量測公式所得到的實際質量，取代設計上之理論值進行彈簧剛度係數的運算，則可對封裝等製程中殘餘應力造成的影響作更加準確的評斷。另一方面，電極間間距的量測則可預估電容值可能產生的變化，並能確認蝕刻結果與設計值間的差異，進而推估加速度計在平面上的幾何外型。上述理論由ANSYS®有限元素模型獲得驗證，所建立的模型為單軸微機電加速度感測器，模擬過程中利用模態分析求得其共振頻率，並加入電壓負載以機電耦合模擬求得吸合電壓。

The development of MEMS has passed several decades. Compared with the traditional one, the manufacturing of MEMS is similar to the semiconductor processing that make the device become smaller and more accurate. However, there are some uncertainty issues in the manufacturing process, which will influence the accuracy and reliability of the device. This article will take MEMS capacitive type acceleration sensor as an example to discuss the effect of these errors. As expected the influence can be quantified by some simple electrical measurement.
Since silicon forms a crystal structure, the Young's modulus is different in each direction. In general, it will simplify the problem by substituting anisotropic material parameters to the isotropic one during the mechanical analysis. This article will make use of commercial finite element software ANSYS® to calculate the natural frequencies of the devices which uses these two kinds of material properties respectively. In this analysis, three-dimensional finite element models of tri-axial MEMS acceleration sensors are built to verify that under what condition replacing the anisotropic material parameters by the isotropic one is a reasonable approximation.
In the latter half of this paper, formulas for estimating the effective mass and the distance between the electrodes by measuring natural frequency, pull-in voltage, capacitance values and the voltage under certain conditions is developed. In the manufacturing process of MEMS devices, there may be some difference between manufactured devices and design due to the process uncertainty. By using the measured mass that is obtained from the proposed formula instead of the ideal design value, the calculation of spring stiffness will be more accurate so that the influences of residual stress due to the packing process can have a well assessment. In addition, the difference between the etching result and the design value can be confirmed by measuring distance between the electrodes, so that the geometric shape of the manufactured accelerometer can also be estimated. All the theory is verified by an ANSYS® model of single-axis MEMS acceleration sensor. During the simulation, modal analysis is used to determine the resonant frequency; and by adding the simulation of electromechanical coupling, the pull-in voltage can be found. Finally, the calculation result is compared with the theoretical values to prove the feasibility of these formulas.

誌謝 I
摘要 II
Abstract IV
目錄 VI
表目錄 IX
圖目錄 X
第一章 緒論 1
1.1 微機電加速度計簡介 1
1.2 研究動機 3
1.3 文獻回顧 4
1.4 研究目標 13
第二章 基礎理論 14
2.1 電容式加速度計原理 14
2.1.1 梳狀電容式加速度計 14
2.1.2 扭轉電容式加速度計 17
2.2 晶體方向與材料特性之關係 18
2.3 摺疊樑剛度係數與等效質量分析 19
2.3.1 摺疊樑剛度係數計算 20
2.3.2 摺疊樑等效質量計算 23
2.4 靜電力與吸合電壓 25
2.5 有限元素法理論 29
2.5.1 線彈性有限元素法理論[27] 30
2.6 模態分析理論 33
第三章 等向、非等向性材料參數選擇對數值模擬之影響 35
3.1 三軸電容式加速度計之數值模擬模型建構 35
3.1.1 三軸電容式加速度計之幾何概述 35
3.1.2 等向性與非等向性材料參數設定 37
3.1.3 有限元素模型與邊界條件設定 38
3.2 模態分析之結果 39
3.3 模擬結果之討論分析 41
第四章 微機電加速度計機械特性之量測分析 45
4.1 微機電加速度計機械特性之量測分析 45
4.2 機械特性量測理論之有限元素模型驗證 48
4.2.1 單軸電容式加速度計有限元素模型之建立 48
4.2.2 單軸電容式加速度計之模擬結果 50
4.3 不同蝕刻誤差下質量與電極間間距計算結果 54
第五章 結論與未來工作 56
參考資料 59

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