To achieve this goal basic functions must be selected

To achieve this goal, basic functions must be selected first. It is known that some basic functions have been used in ω′(x, y) po1 to capture the deformed nonlinear behavior of the diaphragms [12–15]. R1(x), R2(x) and R3(x) are some of them that have been employed in previous researches in square po1 cases
Investigations have been conducted to determine whether these basic functions are still available for rectangular membranes. Two membranes with different geometry dimensions are employed to represent each case (n<4 and ). Figs. 3 and 4 show comparisons of curve shapes between the basic functions and the FEA results when n<4 and , respectively. Both figures reveal that the existing basic functions, R1 in particular, are still able to describe the general shape of the deformed membrane along y-axis. It can be observed from Fig. 3(a) that they are also available for x-axis when n<4. However, in the case of , as shown in Fig. 4(a), all the three function fail to capture the deformed curve along x-axis. For further investigation, method of least squares is employed to estimate the errors between the basic functions and the FEA results. The values of S1, S2 and S3, which are the summed squares of residuals between the FEA results and functions R1, R2 and R3, and calculated by 21 data points, respectively, are shown in Table 3. Apparently, the same conclusion can be drawn from Table 3 as that from the curve comparisons. Therefore, a new basic function should be found for x-axis when , while function R1 could serve as the basic function for other cases. By comparing the deformed membrane curve with an exponential function in Fig. 5, it can be observed that the curve of the new function is close to that of the FEA results. Besides, the summed square of residuals between the new function and the FEA results is calculated as 0.36 along x-axis, much less than previous estimated values. Therefore, the exponential functionis proposed as the new basic function to capture the deformed shape of the membrane along x-axis when . After selecting the basic functions, the next step is establishing deflection shape function. Although function (6) is quite accurate when , the parameters and c4 have to be tuned every time the geometry dimension of the diaphragm changes. So the following analytical model is proposed to adjust to the changing geometry dimensionswhere , , and are functions of n and h, which have a form as function (12). are corresponding coefficients. 30 and 70 sets of data are used to determine the coefficients of and , respectively. For each set of data and are firstly calculated as constants using curve fitting tool in MATLAB. Subsequently, the constants are employed to determine the coefficients by surface fitting tool in MATLAB. The corresponding coefficients are given in Table 4.
When , basic function is modified by combining with function and adding two terms to improve the accuracy of predicting the deformed curve shape along -axis, representing aswhere are functions of and , which also have a form as function (12). And the overall deflection shape function when is proposed as follows:40 sets of data are used to determine functions , and the corresponding coefficients are listed in Table 4.
Eqs. (11) and (14) can be combined by using a piecewise function to derive an analytical deflection shape function for rectangular membranes whenever aswhere the piecewise function is defined as , and is a positive constant infinitely approaching 4 but less than 4.
Subsequently, the analytical deflection model for rigidly clamped rectangular membranes can be achieved by multiplying and as

Deflection model validation
The validity of analytical deflection model (16) can be verified in two aspects: similarity of deformed curves and error in capacitance values between model calculated results and FEA results. The gap thickness is given as 2μm in all the tests. As a result, the maximum deflections of membranes are near 0.7μm in the non-collapse mode. As small and large deflection are relative concept, deflections near 0.7μm are considered as large deflections in this chapter, while deflections less than 20% of membrane thicknesses are treated as small deflections. Similarly, membrane thicknesses near 0.6μm are seen as thin membranes and those near 1.5μm are considered as thick ones. Different cases will be investigated in the following.