EE245 Homework 7

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1. You have a comb drive designed to move in direction x.  Assume that it has N_f comb fingers on the moving structure, and that there are N_f+1 fingers fixed on the substrate, so every moving finger has a fixed finger on either side, and the gap-closing forces in the z direction, perpendicular to the x direction, all balance.  Assume a film thickness t, an initial gap g₀, an overlap x₀, and an applied voltage V.  Calculate and plot the electrostatic force in the z direction as a function of z displacement for N_f=50, t=20um, g₀=2um, x₀=20um, and V=15 V, for z= -1.9 to 1.9 um. Put a URL for the plot here (or combine with the next question) 
2. Calculate the electrostatic spring constant K_e in the z direction at z=0 for these dimensions.  Is this a positive spring constant (unstable) or a negative spring constant (stable)?  Plot the total force in the z direction as a function of z displacement from -1.5 to 1.5 um assuming that you have a mechanical spring in the z direction with a spring constant with magnitude twice K_e, and again with a mechanical spring constant with magnitude one half of K_e.  Identify the equilibria of each system, and their stability.  Put the results on the web and post the URL here:  
3.  You have designed an electrostatic test structure with the shape shown in figure 1.  Assume that the dimensions W and L can be accurately reproduced on the wafer, but that the dimensions a and g₀ can not (why can we make this assumption?  is it true?):  Assume that due to processing issues, all of the beams in your layout are reproduced on the wafer with an under or over etch of daand the gaps are reproduced with an under or over etch of dg. You have tested the pull-in voltage of your structures, and the results are shown in the table below (and available here so that you can "load" them in matlab) .  Your job is to use the data to estimate da, dg, and the Young's modulus of the material that you are working with.  Put graphs of V_PI² vs a³ and g³ and your estimates for da, dg, and E here:  
   Figure 1  A simple gap-closing electrostatic actuator

a	g0	L	W	VPI
2	2	400	400	4.16
3	2	400	400	8.71
4	2	400	400	13.3
5	2	400	400	19
2	3	400	400	7.22
2	4	400	400	10.2
2	5	400	400	13.8
2	2	300	400	6.42
2	2	200	400	12.2
2	2	100	400	33.5
2	2	400	300	5
2	2	400	200	6.13
2	2	400	100	8.48

4. Extra for masochists: for the problem above, calculate the torsional stiffness of the support springs to rotation about the center of the actuator, and calculate the electrostatic torsional spring constant for the gap.  Will some designs become rotationally unstable before linear pull-in?  Answer here, if you dare: 
5. Do the cadence homework.
 

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