微元件之邊緣電容研究

Abstract

這份研究目標在於發展一能夠準確計算邊緣電容的方程式解。現今電容的感測以及靜電力驅動的元件已成熟地應用在微機電領域。因此，電容的設計以及預測準確度足以影響一元件的特性，所以可以得知邊緣電容的計算的挑戰性及重要性。目前許多研究已應用有限元素分析，取代準確度不足的分析模型，而這些有限元素法的應用早已植入在商用的套裝軟體當中，如ANSYS，COMSOL，CovnetorWare，和Intellisuite。雖然有限元素模型已成主流，但其尚有需要改進的缺失，如計算時間的需求長、計算機硬體的需求高等。更重要地，有限元素分析的解答並不能帶來任何物理上的意義，換句話說，這個電容值不能做其他物理上的運算，例如微分、積分等步驟，這也就是發展分析解的最主要目的及優勢。在研究中，先以簡單的平行板電容作為研究的對象，利用保角映射轉換以及一些方程式的近似法，推導一可以表達完整平行板電容的分析解。這個結果與數值模擬(Ansys)結果比較，只有不到百分之一的誤差，而與實驗值相比，也是非常接近。這個高精確度的分析解，相較於其他的方式更為便利，可以快速的計算出正確的結果。

This research aims at developing a general solution for calculating the capacitance of fringe field precisely. Capacitive sensing and electrostatic actuating devices are commonly used on microdevices. Therefore, determination of the capacitance is critical in the design to determine the performances of such devices. Accurate determination of capacitance is very challenging in virtue of the fringing field. The finite element method (FEM) is often used for calculating the capacitance as lacking precision analytical model. FEM has been implemented in various commercial MEMS simulation software such as ANSYS, COMSOL, CovnetorWare, and Intellisuite. However, FEM has the disadvantages of un-explicit physical meaning and requiring massive numerical calculations, and therefore is not easy to carry out the parametric study of capacitive devices. This research develops an analytical solution for calculating the three-dimensional fringe field capacitance of paralleled-plate-type capacitors, which is the commonest structure. The analytical solution is derived from the conformal mapping method and simplified by some approximate approaching techniques. The present analytical solution shows very high accuracy within one-percentage error comparing with the experimental results and the numerical simulation by ANSYS. By the present high precision analytical solution, one can easily evaluate the capacitance in few seconds by manual work.