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Electro-Static Membrane Model in CAD

M. Zubert, M. Napieralska, A. Napieralski
Department of Microelectronic and Computer Science, PL

silicon membrane, diaphragm, micromachine, model design

(Please see the PDF) The model simplification is one of the most important problems in its design process. Every so often this approach tends to the physical phenomena violation, exceeding assumed domains or published models (and commercial software) application without taking into consideration there fundamental assumptions. Unfortunately, there are a lot of publications confirming this problem. The paper will present the often appearing mistakes introduced in the modeling of micromachined pressure sensor based on silicon membrane (diaphragm) and taking into account electrostatic phenomena. Both mechanical and electrostatic phenomena can be wrongly described but electrical one is the worst. 1) Mechanical domain The mechanical model of this device can be described using small and large deflection theory of the clamped thin plate under uniform pressure and with (or without) additional build-in stress [1][2]. Unfortunately, published analytical solutions of large deflection include several minors’ mistakes. In most of cases it can be visible in the comparison of FEA and proposed model. The simplified model of the membrane for small and large deflection has been presented in the paper [3]. Another solution of this problem can be obtained using function approximation. 2) Electrical domain Additional problem occurs in the electro-static phenomena model. The most of authors divide the membrane plate for several parts and describe it as the parallel connected capacitances, assuming charge density invariability on the membrane surface (for each and ever elementary capacitor) - assumption A1. This problem can be simply presented for small deflected clamped circular membrane. In the particularly case the membrane deflection can be described by the following equation [1] (see Figure 1): The charge density on the membrane surface is proportional to its curvature (for simplification we assume ), therefore where c- constant. Obtained equation allows to charge density estimation (, see Figure 2), it can be simply proof that curve has similar shape for various pressures. From the other case the charge density on the second capacitance sheet is typically described by the equation, hence assumption A1 is wrong and the some published results are erroneous. As we see the mistakes can be occurred in both of domain and can lead to error commutation and model miss functional. Taking into account the proper shape of charge density and membrane deflection can lead to obtain the fine accordance the simulation with measured results.

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