考慮靜電力驅動之周圍固定微型圓板從事動態及靜態崩潰電壓預測與分析

Abstract

本研究主要在探討、預測微機電感測器、致動器或微麥克風元件中圓形薄板的崩潰電壓與崩潰位置，藉由考慮非線性的連續方程式可以求得較之前相關研究更為準確的崩潰電壓與崩潰位置，並且可得到相關靜態、動態崩潰電壓之封閉式方程式。崩潰現象的發生是在於發生在兩平行板之間的靜電力大於薄板本身的彈性回復力，因此兩力量之臨界平衡點即為崩潰位置。本研究考慮的感測器與致動器為利用兩平行圓板之間靜電力作用，一薄板振動以反映外在壓力另一背板則相對固定靜止，並建立起包含薄板彈性、殘留應力以及靜電力的動態偏微分統馭方程式，且假設圓板的邊界條件是固定於四周邊界。應用Galerkin method 在偏微分方程式使之成為離散之系統，並利用低階mode與Taylor五階展開近似靜電力去解之，進而可獲得靜態崩潰電壓的位置和封閉式方程式。另外，我們也根據所建立的非線性統馭方程式來探討微型圓板的動態特性，進而再利用能量法以及有限階數近似法去作系統的動態崩潰電壓的預測和分析。在比較系統的靜態和動態崩潰電壓的情況下，我們可獲得動態崩潰電壓大約是靜態崩潰電壓的92%，但如果額外再考慮動態中微型圓板間的空氣擠壓效應(Squeezed-film effect)後，這百分比92%會再稍為增加一點點。

This study is devoted to predict and analyze the precise pull-in voltage/position of the circular micro-plates of the MEMS sensor, actuator or microphone. It can be more precise to predict the pull-in phenomenon by analyzing the nonlinear continuous model of the device. Therefore, the closed form function of the static and dynamic pull-in voltages can be obtained. The pull-in is a phenomenon that the electrostatic force induced by the applied voltage across two plates of the device exceeds the elastic, restoring force exerted by the deformed plate, leading to a contact between the two plates. To offer a precise prediction on the pull-in, a dynamic model in the form of the partial different equation (P.D.E) is established based on the equilibrium among plate flexibility, residual stress and distributed electrostatic forces. Galerkin method is employed used to decompose the established P.D.E into discrete modal equations. Considering lower-order modes and fifth-order Taylor series expanded on the electrostatic force, the static pull-in position and voltage in closed forms are obtained; furthermore, the dynamic pull-in voltage are also found based on the analysis on the P.D.E of the dynamic modal and then using the energy method and finite-order approximation method. The ratio of the dynamic pull-in voltage to the static pull-in voltage is about 92%. If considering the squeezed-film effect of the air gap to the dynamic pull-in voltage, the ratio present 92% will slightly increased.