# Movable comb fingers constitute differential capacitance C1 (C2) with left (right) fixed fingers. Calculate the static differential capacitance of the device C1=C2=C0=?

Figure 1. Structure diagram of poly-Si surface-micromachined comb accelerometer

(1). Briefly explain the working principle of this accelerometer device. (2). Calculate the total sensitive mass Ms of the accelerometer, which consists of the movable mass, plus the mass of 40 movable fingers.

X

Y

Acceleration a

anchor

connector

folded beam

left fixed fingers

right fixed fingers

movable fingers

Lb

Wb

Wm Lm mass

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(3). Use equations to find the spring constant kb1 of one section of a folded-beam. (4). Calculate the spring constant of each folded beam. (5). Derive the total spring constant ktot of the whole device. (6). Derive the displacement sensitivity of the whole device. That is, the lateral displacement Sdx in response to unit gravity acceleration input (1g) in X direction. (7). Movable comb fingers constitute differential capacitance C1 (C2) with left (right) fixed fingers. Calculate the static differential capacitance of the device C1=C2=C0=? (8). Find out the resonant frequency of the device for the working mode (vibration along X direction). 2.(15’) A bulk-micromachined Si MEMS piston micromirror is shown in Figure 2. The micromirror is supported by two beams connected to anchors. Aluminum bottom electrode was pre-deposited on substrate right underneath the mirror. Both the Al bottom electrode and the mirror have the same size and they are perfectly aligned to each other. A DC driving voltage Vd is applied between the mirror and Al electrode to activate the mirror to move perpendicular to substrate. It can be used to modulate the phase of incident light. Given the width and length of the square micromirror as: W=L=120µm, initial static capacitance gap (when Vd=0) d0=6µm, total spring constant of both beams: Ktot=0.16N/m, In order to achieve micromirror displacement as Δx=1.5µm, what is the required driving voltage Vd=? What is the pull- down threshold voltage Vth=? What is the maximum controllable displacement of the mirror without pull-down effect?

Figure 2. An electrostatic actuated MEMS piston micromirror

3. (20’) A surface-micromachined poly-silicon comb resonator device is shown in Figure 3. Electrostatic push-pull driving is used to activate the device. Driving voltages V1 and V2 are applied to the left and right fixed comb fingers separately. Assume N is the number of left (right) movable driving fingers, t is device thickness, ε is dielectric constant of air (ε=8.85×10-12F/m), d is capacitance gap, V1 and V2 are driving voltages. If N=40, t=2µm, d=2µm, and we apply the driving voltages as: )cos(12221 tVVV  , )cos(12222 tVVV  1). Calculate the electrostatic forces F1 and F2 experienced by the central mass toward left and right. 2). What is the total driving force Ftot=F1-F2? Is Ftot a constant force or periodic force? Will the central mass vibrate? 3). Will the movable mass move along vertical (Y) direction? Why? 4). Assume for each segment of beam, beam width Wb=4μm, beam length Lb=200μm, beam thickness tb=2μm. Young’s modulus of poly-Si is E=170GPa. What is the spring constant kb1 of each segment of beam? What is the spring constant kbf of one folded-beam? What is the spring constant ktot of the whole resonator? 6). What is the maximum displacement of the movable fingers under above push-pull driving? 7). Assume the total mass of the movable part is Ms=10μg, what is the resonant frequency f0 of the comb resonator?