Fast identification of the pull-in voltage of a nano/micro-electromechanical system

Abstract

The pull-in voltage is crucial in designing an optimal nano/micro-electromechanical system (N/MEMS). It is vital to have a simple formulation to calculate the pull-in voltage with relatively high accuracy. Two simple and effective methods are suggested for this purpose; one is an ancient Chinese algorithm and the other is an extension of He’s frequency formulation.

Introduction

The pull-in instability^1–6 is an inherent property of a nano/micro-electromechanical system (N/MEMS) when the applied voltage reaches a threshold (see Figure 1), and it plays a significant role in electrostatically actuated sensors for their effective and reliable operation. The MEMS system opens a broad road for microfluidics, ⁷ energy harvester, ⁸ drug delivery device, ⁹ timing and frequency control, ¹⁰ and portable devices. ¹¹ The pull-in voltage can be easily obtained for the linear case; however, it is highly intricate and challenging for the nonlinear case.

Figure 1.

The MEMS system.

Since the problem of static pull-in (as a consequence of a saddle-node bifurcation)^12–16 and dynamic pull-in (as a consequence of a homoclinic bifurcation)^17–20 in the MEMS structure is a highly decisive factor, fast estimation of the pull-in voltage is much needed in practical applications.

We consider the following dynamical pull-in problem ²¹

$$x^{″}+x-\frac{\lambda }{1-x}=0,x^{\prime }\left(0\right)=0,x\left(0\right)=0,\lambda >0$$

(1)

where x is the dimensionless distance as shown in Figure 1 and $\lambda$ is a voltage-related parameter.

The pull-in behavior occurs when $\lambda >\lambda \text{*}$ , where $\lambda \text{*}=0.203632188$ ; it is the solution of the following transcendental equation ²¹

$$\frac{{\left(1+\sqrt{1-4\lambda }\right)}^2}{4}+2\lambda \ln \left|\left(1-\frac{1+\sqrt{1-4\lambda }}{2}\right)\right|=0$$

(2)

This implicit function cannot clearly see the effect of the system’s parameters on the pull-in voltage. An analytical closed-form solution is much needed for MEMS applications. ¹⁶ There are many experimental, numerical, and analytical methods for this purpose; for example, the finite-difference method, ²² the finite element method, ²³ the modal expansion method, ²⁴ the generalized differential quadrature method, ²⁵ the Ritz method,^26,27 and the variational iteration method.^28,29

To solve $\lambda$ simply and effectively, we suggest two methods, one is the ancient Chinese method^30,31 and the other is He’s frequency formulation.^32,33

Ancient Chinese Algorithm

We begin with the well-known Newton iteration method. If we choose the initial guess as ${\lambda }_0=0.25$ , the method becomes invalid entirely. The convergence depends strongly upon the initial guess; a good one always leads to an ideal result, while an inappropriate one might result in a divergent outcome.

Consider a simple example

$$f\left(x\right)=\sin x=0$$

(3)

We want to find a solution between 0 and $2\pi$ , so we begin with $x_0=3\text{.}14/2$ and $3\times 3\text{.}15/2$ , respectively; the Newton iteration processes are shown in Tables 1 and 2. Both cases cannot lead to the needed result.

Table 1.

The Newton iteration process.

Iteration no	$x_n$	$x_{n+1}$
0	$3.14/2$	−1254.1956
1	−1254.1956	−1253.3531
2	−1253.3531	−1253.4964
3	−1253.4964	−1253.4955
4	−1253.4955	−1253.4955

Table 2.

The Newton iteration process.

Iteration no	$x_n$	$x_{n+1}$
0	$3\times 3\text{.}15\text{/}2$	84.016526
1	84.016526	85.059595
2	85.059595	84.818486
3	84.818486	84.823002
4	84.823002	84.823002

To solve the transcendental equation given in equation (2), we introduce the ancient Chinese method.^30,31 Consider an algebraic equation in the form

$$f\left(x\right)=0$$

(4)

The ancient Chinese algorithm begins with two trials, ³⁰ $x_1$ and $x_2$ , which lead to two residuals $f\left(x_1\right)$ and $f\left(x_2\right)$ . The approximate solution of equation (4) is ³⁰

$$x=\frac{x_{1}f\left(x_2\right)-x_{2}f\left(x_1\right)}{f\left(x_2\right)-f\left(x_1\right)}$$

(5)

The iteration process of the ancient Chinese algorithm is illustrated in Figure 2. For the above example, the two residuals are

$f\left(x_1=3\text{.}14/2\right)=\text{0}\text{.}999999683$ and $f\left(x_2=3\times 3\text{.}15/2\right)=-\text{0}\text{.}9999205$ ;

according to equation (5), we estimate that $x=3\text{.}14756246$ . This is an approximate solution with a relative error of 0.19%.

Figure 2.

The iteration process of the ancient Chinese algorithm.

To solve equation (2), we introduce a residual function

$$R\left(\lambda \right)=\frac{{\left(1+\sqrt{1-4\lambda }\right)}^2}{4}+2\lambda \ln \left|\left(1-\frac{1+\sqrt{1-4\lambda }}{2}\right)\right|$$

(6)

We choose two trials ${\lambda }_1=0.25$ and ${\lambda }_2=0.17$ , the residuals are $R_1=-\text{0}\text{.}096574$ and $R_2=\text{0}\text{.}0936174$ , respectively, so the pull-in voltage can be calculated as

$${\lambda }^{\ast }=\frac{R_2{\lambda }_1-R_1{\lambda }_2}{R_2-R_1}=\frac{0.0936174\times 0.25+0.096574\times 0.17}{0.0936174+0.096574}=0.209378$$

(7)

This has a relative error of 2.82%. The accuracy can be improved using the other two trials ${\lambda }_1=0.25$ and ${\lambda }_2=0.209378$ ; and the two residuals are $R_1=-0.096574$ and $R_2=-\text{0}\text{.}014168$ , respectively, using the ancient Chinese algorithm, we have ${\lambda }^{\ast }=0.202394$ with a relative error of 0.60%. Alternatively, we use the two trials ${\lambda }_1=0.209378$ and ${\lambda }_2=0.17$ , we have two residuals $R_1=-\text{0}\text{.}014168$ and $R_2=0.093617$ , and the pull-in voltage is ${\lambda }^{\ast }=0.204202$ with a relative error of 0.27%. A modification of the ancient Chinese algorithm was given in Ref. 30, and it is called Chun-Hui He algorithm in the literature. ³¹

He's frequency formulation

He’s frequency formulation^32,33 is to calculate the frequency of a nonlinear oscillator in the form

$$x^{″}+q\left(x\right)=0$$

(8)

where q(x) is the nonlinear function of x. The formulation is^32,33

$${\omega }^2=\frac{q\left(x_{\text{0}}\right)}{x_{\text{0}}}$$

(9)

where $\omega$ is the frequency and $x_{\text{0}}$ is a location point; generally, we recommend $x_{\text{0}}$ =0.8 A for fast estimation, where A is the amplitude. Equation (9) and its modifications were widely used in the literature to find a periodic solution of a nonlinear oscillator.^34,35

For equation (1), we have

$${\omega }^2=1-\frac{\lambda }{x_{\text{0}}\left(1-x_{\text{0}}\right)}$$

(10)

The pull-in threshold is to change the periodic solution ( ${\omega }^2>0$ ) to the non-periodic one ( ${\omega }^2<0$ ), so the pull-in voltage can be determined from the condition

$${\omega }^2=0$$

(11)

That is

$$\lambda \text{*}=x_{\text{0}}(1-x_{\text{0}})$$

(12)

We have a maximum $\lambda {\text{*}}_{\max }$ = 1/4 when $x_{\text{0}}=1/2$ , this leads to a relative error of 22.78%, too high to be used in practical applications. We choose multiple points to calculate $\lambda \text{*}$ ; then, an average is used. We choose $x_{\text{0}}=1/2$ , 1/3, 1/4, and 1/5, and we can determine $\lambda \text{*}$ =1/4, 2/9,3/16, and 4/25, respectively. Its average value is

$$\lambda \text{*}=0.2049$$

(13)

Now, the relative error reduces to 0.62%.

Discussion and Conclusion

Newton’s iteration method is sensitive to the initial guess. A good guess always leads to a good result, while a not-good guess results in a non-convergent solution, or the convergent solution is a wrong one. To overcome the shortcoming of the Newton’s iteration method, we suggest two simple but effective methods in this paper to determine the pull-in voltage. A simple method with relatively high accuracy is welcome in many practical applications.