GSO002 Problem 7

1. Electrostatic Force and Force Equilibrium

Consider the state where voltage $V$ is applied and the plate separation is $x$ ( $0 < x < d$ ) in equilibrium.

Since the initial equilibrium (at zero voltage) already balances gravity and the spring force, we can treat the system effectively as a spring with natural length $d$ .

The capacitance at separation $x$ :

C = \frac{\varepsilon_0 S}{x}

The electrostatic attractive force (derived from virtual work principle at constant voltage):

F_e = \frac{1}{2}\frac{\varepsilon_0 S V^2}{x^2}

The spring restoring force acts upward with magnitude $k(d - x)$ . The force balance equation is:

k(d - x) = \frac{1}{2}\frac{\varepsilon_0 S V^2}{x^2}

2. Limiting Condition for Equilibrium to Exist

Solving for $V^2$ :

V^2 = \frac{2k}{\varepsilon_0 S} x^2(d - x)

Equilibrium is lost when the required $V^2$ exceeds the maximum of $f(x) = x^2(d - x)$ .

Differentiating:

f'(x) = 2x(d - x) - x^2 = x(2d - 3x)

For $0 < x < d$ , $f'(x) = 0$ gives $x = \frac{2}{3}d$ .

f\left(\frac{2}{3}d\right) = \frac{4}{9}d^2 \cdot \frac{1}{3}d = \frac{4}{27}d^3

Thus, the square of the pull-in voltage:

V_0^2 = \frac{2k}{\varepsilon_0 S} \cdot \frac{4}{27}d^3 = \frac{8 k d^3}{27 \varepsilon_0 S}

3. Numerical Calculation

Using $C_0 = \frac{\varepsilon_0 S}{d}$ , so $\varepsilon_0 S = C_0 d$ :

V_0^2 = \frac{8 k d^2}{27 C_0}

Substituting:

V_0^2 = \frac{8 \times 1.35 \times (6.4 \times 10^{-6})^2}{27 \times 2.4 \times 10^{-12}} = \frac{10.8 \times 40.96 \times 10^{-12}}{64.8 \times 10^{-12}}

V_0^2 = \frac{40.96}{6} = \frac{4096}{600}

Dividing numerator and denominator by $\gcd(4096, 600) = 8$ :

$4096 \div 8 = 512$
$600 \div 8 = 75$

Since $512 = 2^9$ and $75 = 3 \times 5^2$ are coprime, the irreducible fraction is $\frac{512}{75}$ .

$A = 512$ , $B = 75$ :

A + B = 512 + 75 = 587

Final Answer: 587

GSO002 Problem 7

Pull-In Effect Analysis in a MEMS Actuator

Problem Statement

Constraints

Input Format

Solution

1. Electrostatic Force and Force Equilibrium

2. Limiting Condition for Equilibrium to Exist

3. Numerical Calculation