靜電致動微型樑之崩潰電壓預測與分枝現象分析

Abstract

本研究主要探討微機電(MEMS)感測器或致動器元件中微型樑之動態與靜態崩潰電壓，藉由考慮非線性的連續方程式進一步求得崩潰電壓的封閉式(Closed form)。本研究考慮的感測器與致動器為利用兩平行板之間的靜電力驅動，一微型樑變形以反應外力而另一背板電極則固定靜止。所謂靜態崩潰是指當樑上由外加電壓產生的靜電力超過其本身的彈性恢復力時，造成樑與電極之間接觸的現象；而動態崩潰則是更進一步考慮微型樑的慣性力與空氣擠壓阻尼效應。為了達成預測崩潰電壓的目的，本研究首先建立包含微型樑彈性、殘留應力、空氣擠壓阻尼效應以及考慮邊緣電場效應之靜電力的動態偏微分統馭方程式。並假設變形的微型樑固定於兩端，Galerkin method 則應用於偏微分方程式使之成為離散之系統，使用所建立的連續模型即可對動態和靜態崩潰電壓做分析。此外，本研究同時利用分枝現象與相位圖的概念來探討崩潰電壓。由本研究所提出的數學模型分析討論系統參數與崩潰電壓之間的關係，並可得知動態崩潰電壓約為靜態崩潰電壓的91% ~ 92%。為了確認系統之數學模型正確，本研究最後將使用微機電模擬軟體IntelliSuite與已知實驗數據和數學模型模擬結果互相驗證比較，確認數學模型有效。

This study is devoted to provide precise predictions of the static and dynamic pull-in voltage of a general clamped-clamped micro-beam based on a continuous model. The pull-in is a phenomenon which occurs when the electrostatic force on the micro-beam exceeds the elastic restoring exerted by beam deformation, leading to a contact between the actuated beam and the bottom electrode. To derive pull-in voltage, a dynamic model in partial differential equations is established based on the equilibrium among beam flexibility, inertia, residual stress, squeeze film, distributed electrostatic forces and its electrical field fringing effects. The method of Galerkin decomposition is next employed to convert the established system partial differential equations into reduced discrete modal equations. Considering lower-order modes and approximating the deflection by a different order series, the bifurcation based on phase portraits are conducted to derive static and dynamic pull-in voltages. It is found that the pull-in phenomenon follows well a known generalized homoclinic bifurcation, and the dynamic pull-in voltage is around 91 to 92 % of the static counterpart. However, the derived dynamic pull-in voltage is found dependent on the varied beam parameters, different from a fixed predicted value derived in past works, where only lumped models are assumed. Furthermore, accurate closed-form predictions are provided for the cases of non-narrow beams. The predictions are finally validated by finite element analysis and existing experimental data.