Homework #4, Due Oct. 14, 2010

Note!!! Do not wait until near the deadline to start this homework. First, it will require time and possibly learning how to use some software, so you may not finish if you do not start early. Second, since the exam is the following Monday, solutions will be posted right away - you will not be able to turn it in late.

1. 7.1, Senturia. This is the spring-mass-damper that we have done before. Senturia writes out the equations. The point of problems 1-3 is to learn how to solve this problem three different ways. Choose values of the parameters that you think would be typical of a MEMS device.

You may use software other than MATLAB to solve the system of differential equations.

Note that if you are using MATLAB, and you want to pass parameters to the ode solver (such as ode45) , you need to define an extra parameter (flag, here) in the m-file. For example, the m-file would look like:

function dx = msd(t,x,flag,m,k,b)

and you would solve the system of state equations defined in the function using something like:

[t,y]=ode45('msd',[0 30],[0 0],[],m,k,b);

Where msd is the name of your function, the first matrix defines the range of times to be solved for, the second matrix contains the initial conditions for the two state variables, the options are default (empty matrix), and m,k, and b are the parameters passed to the function. Information under the help index in MATLAB for "differential equation solvers" is useful.

2. Repeat using Simulink..

Look at page 25 of the text for some hints on how to do this, and see the example below. Remember that everything will need to be scaled as the mass and frequency change by orders of magnitude in moving to the MEMS world. I found that for some conditions, I had to increase the simulation accuracy (under simulation-> parameters in the menu). Note that you will also have to adjust the parameters of the low-pass filter on the force input. Experiment to see why it is present.

3. Use PSpice or any other version of spice (see download information on the course web site) to analyze a system with the same spring constant, mass and damping constant as in problem 2. Use an effort source (force/voltage) to drive the circuit. Perform a transient analysis to see (hopefully) output waveforms similar to those you saw in problem 2. Also use an ac analysis covering the frequency range of the resonance to confirm that the resonant frequency is as predicted by theory.

When drawing the circuit, you must add a ground connection before Spice will analyze the circuit, as we know from Homework 3. You will probably want to use the pulse source.

You need to set up the analysis. Under analysis, pick transient, and pick your time range. Also under analysis/options, I found that I needed to decrease the RELTOL parameter by a factor of 10 to get a good solution. Finally, you will need to edit the source properties. Double click on the source. The relevant parameters are V1, V2, TR, TF, PW, PER, which are the two voltage states of the excitation the risetime of the pulse, the fall time of the pulse, the pulse width, and the period of the pulse.

After the circuit is analyzed currents and voltages are available. If you want to get the position, note that the position is related to the voltage on the capacitor (both are integrals of the current/velocity). You need to make certain that you have the capacitor voltage. Depending on where your ground node is (the zero volt reference) you will have to subtract two node voltages (spice will do this for you).

4. 6.2, Senturia.

5. a. Implement the simulink model shown in Fig. 7.8, Senturia. For the values given below, examine the time to contact (contact occurs at a gap of 50% of go). Yours does not have to look exactly like the one in the text, it just has to work correctly.
    b. Implement the simulink model shown in Fig. 7.9, Senturia. Note that some of the switches and the reset on the integrator may be connected differently than shown in Fig. 7.9, depending on how your elements are defined. In particular your switches may be completely different, or you may be able to completely eliminate them because we stop at half the gap with the contact. Here the idea is to also observe the opening behavior when the actuation voltage is removed.
    c. Put values into the models corresponding to a parallel-plate electrostatic actuator for a MEMS device (perhaps a microrelay). Actuator area=70x30microns, g ₀ =0.6 microns, volume of mass 50x70x6 microns, material Au, spring constant=300 N/m, pull-in stopped at g ₀ =0.3 microns by contact. Introduce just enough damping so that the device is somewhat underdamped.

What is the time to contact for voltages 1%, 10%, and 100% greater than the pull-in voltage? Note: The resistor in series with the voltage source must be fairly large for the simulation to converge. Changing the choice of solver under simulation -> parameters can help the simulator converge for smaller resistances (for example ODE23s).

Observe the opening behavior with the model from part b. You should see damped resonant behavior at the expected frequency is working correctly.