About

This Java Applet simulates a MEMS switch using non-linear dynamic models adapted from "Numerical Simulations and Novel Constant-Charge Biasing Method for Capacitive RF MEMS Switch" by JB Lee and Charles L GoldSmith and "Nonlinear Electro-Mechanical Modeling of MEMS Switches" by Jeremy Muldavin and Gabrial Rebeiz.

**1.0 Summary**

**2.0 Variables and Constants**

** 2.1 User Inputs**

g, Height of the bridge over the electrode (µm)

w, Width of the bridge (µm)

W, Length of the electrode (µm)

E, Young's Modulus (GPa)

t, Thickness of the bridge (µm)

l, Length of bridge (µm)

v, Poisson Ratio (unitless)

V, Applied Voltage (V)

m, Mass of the membrane (kg)

** 2.2 Constants**

µ_{air} = 1.8E10-5 kg/m^{3}

ε_{0} = 8.85 E-12 F/m

** 2.3 Calculated Values**

k, Spring contant (N/kg)

k = 32Ew(t^{3})/(l^{3}) * (27/49)

F, Force (N)

F = ε_{0}*w*W*(V^{2}) / 2(g^{2})

Vp, Pull-down voltage

Vp = Sqrt( 8k*(g^{3}) / 27ε_{0}*W*w )

b, Damping coefficient

b = Sqrt(2)*µ_{air}*l*(w / g_{0}-z)^{3}

**3.) Switch motion equation**

m(d^{2}z/dt^{2}) + b(dz/dt) + kz = F

Where z(t) is the displacement of the bridge over the electrode.

Force (F) is in Newtons, and time (t) is in seconds

**4.) Solving the differential equation**

The differential equation is solved as follows:

** 4.1 Finding the roots**

Initial conditions: z(0)=0, dz(0)/dt=0, since the bridge is initially not in motion

Divide each side by m: (d^{2}z/dt)+(b/m)(dz/dt)+(k/m)z = F/m

Find the roots using a quadratic -(b/m) ± sqrt( (b/m)^{2} - 4(k/m) ) / 2

** 4.2 Homogenous and Particular Solutions**

Homogenous Equation: z_{h}(t) = C1 e^{a1*t} + C2 e^{a1*t}

Particular Solution: z_{p}(t) = A, where A is a constant

Thus, z(t) = C1 e^{a1*t} + C2 e^{a1*t} + A

Take derivatives to produce the following three equations:

1.) z(0) = C1+C2+A=0

2.) z'(0) = a1*C1 + a2*C2 = 0

3.) z"(0) = a1*a1*C1 + a2*a2*C2 = F/m (because z"(0)+(b/m)z'(0)+(k/m)z(0)=F/m)

** 4.3 Equations for Constants**

After some algebra:

C2 = (F/m) / (a2*a2-a1*a2)

C1 = (-a2/a1)*C2

A = -C1 - C2

** 4.4 Final Solution**

z(t) = ((-a2/a1)*((F/m) / (a2*a2-a1*a2)))*e^{a1*t} + ((F/m) / (a2*a2-a1*a2))*e^{a1*t} -((-a2/a1)*(F/m) / (a2*a2-a1*a2)) - ((F/m) / (a2*a2-a1*a2))

**© The University of Texas at Dallas, 2005 Written by Katie Roberts-Hoffman and Justin Marcus**