Homework 5: 

Individual Assignments:

Electrostatic Actuator Modeling

Consider an electrostatic sensor/actuator, as we have discussed in class.  The actuator is a cantilevered beam, suspended above the substrate by a gap, g.

The effective stiffness, k,  of a cantilever can be shown to be

        k = 2 E W H³ / (3 L³)

where E is the modulus, W the width, H the beam thickness and L the beam length. The effective mass, m,  of the cantilever can be shown to be

        m  = 0.5 r HWL

where  r is the density of the beam

As a starting point, consider a "baseline" design in which H = 2, W = 50, L  = 500 and g = 2  (all units in microns).

Develop BOTH the full non-linear SIMULINK  model, and the lumped-element linearized system for this device and address the following questions:

STATIC ANALYSIS:

1. Plot the deflection versus voltage of this beam. (as an actuator). 
2. What is the pull in voltage of this device?
3. Plot the voltage versus deflection for this beam (as a sensor).  Do this for two different bias voltages  - one close to zero, and one close to the pull-in voltage.

DYNAMIC ANALYSIS

1. Plot the displacement frequency response (amplitude and phase) for this device, forced sinusoidally over a wide range of frequencies.  Do this for two different conditions - zero bias voltage and a bias voltage close to the pull in voltage.  Compare the nonlinear result to the lumped parameter version - what is the source of any discrepancies that you observe?
2. Plot the voltage frequency response (amplitude and phase) for this device.  As before, do this for two different bias voltages - one close to zero, and one closer to the pull-in voltage. Compare the nonlinear result to the lumped parameter version - what is the source of any discrepancies that you observe?

(You will need to assume a value of the damping - choose a value that seems reasonable, not to large, but large enough to avoid crazy oscillations).

For ALL the cases examined above, discuss (using quantitative, numerical  examples obtained from your numerical and analytical models) how you can improve the performance of the sensor or actuator, by changing the process parameters (length, width, thickness, gap).  "Improve" can mean different things, but usually means a higher force or displacement for a given voltage in the case of an actuator, and a higher voltage for a given displacement