[23]
Parameters Dynamic pull-in voltage b1=20μm b1=4μm The presented model.
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[25]
15 Numeric method [26].
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[25]
626 Experimental Data[26].
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[4]
Impact of the bottom electrode position on pull-in parameters Based on above analysis, it can be concluded that common factors that affect the stability of the travel range is geometry parameters of cantilever beam, such as g/b, t/g, which are relevant to the thickness(Z) direction and usually constrained by MEMS processing technology, but the bottom electrode position is only influenced by the size of the parameter in XY plane, so it can be freely adjusted in the layout design stage, which will not have any impact on the fabrication process. (a) (b) (the scatterplot indicates the results calculated by present model, the surface plot indicates the fitting function described below) (c) Figure 6 Impact of the bottom electrode position on pull-in position The bottom electrode position is defined by two variables, the end positionβand the length of electrode w, both of which are normalized for the convenience of analysis. Impact of these two parameters on the pull-in position can be expressed as 3D diagram shown in Figure 6 (a)(b). It can be seen that bottom electrode position has a quite large control range of pull-in position. When the end position of lower electrode β is less than 0. 7, the normalized pull-in displacement is greater than 1 in Figure 6(a), that is, no pull-in phenomenon occurs in all the travel range no matter what the electrode width w is. This has been demonstrated more clearly in Figure 6(c). When the width of bottom electrode w is more than 0. 5 in Figure 6(b), almost no change can be observed in the pull-in voltage. So the distribution of bottom electrode near the fixed end can increase the pull-in stable region, and oppositely, the bottom electrode near the tip will help reduce the voltage. In actual design usually both pull-in voltage and displacement need to comply with certain requirement, then it is essential to perform trial calculation so that β and w can be set for overall consideration. For convenience of that, use MATLAB Curve Fitting Toolbox [24]. The MathWorks, Inc. 2012 Curve Fitting ToolboxTM, User's Guide. ] to obtain the fitting function of pull-in point. The following function form is selected as a fitting model of pull-in displacement and voltage, (23) Table 4 The fitting results of the function of pull-in point Function name Function Coefficients Imitative Effect b1 b2 m1 m2 R2 RMSE Pull-in displacement -1. 475.
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01877 Pull-in voltage -0. 1606.
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1716 Fitting results are shown with desirable imitative effect in Table 4. With these fitting functions, trial calculation can be performed easily and quickly so that pull-in parameters can be satisfied through the distribution of bottom electrode.
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[5]
Conclusion In order to study the impact of electrode position on stable pull-in range, we use the Rayleigh-Ritz energy method to establish a dynamic drop-in model of the cantilever, and take into account the effect of fringing capacitance and variable cross-section. For narrow and thick cantilever which has a small ratio of b/g and large ratio of t/g, the pull-in displacement increases and the pull-in voltage decreases because of the fringing capacitance effect. Modified trial function is derived in the case of partially distributed uniform load. Subsection integral is used to solve the pull-in point of variable cross-sectional cantilever, which expands the application scope of the energy method. By comparison with the published numerical model, the present model is more concise, and the deviation from experimental data is admissible. Due to the processing limitations in the thickness direction of MEMS structures, it is more convenient to control the pull-in stability through the location of the bottom electrode. This paper deduces the fitting function of pull-in parameters, which can offer quantitive references for the design of MEMS devices. It should be pointed out that trial function is vital to the model validity. Because of geometric nonlinear problem and defects in the assumptions of Euler Bernoulli beam, the approximate polynomial this paper uses as trial function could introduce some slight error. In addition, the present model is based on a single eigenmode of model, so it cannot be used to solve the structural frequency response of higher modes. References.
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