3. A planar electrostatic actuation system will drive a levitated mass m as shown in Fig. 3. Derive the expression for the instantaneous acceleration of the mass. Assume only the capacitance between the mass and one pad will change at any time in the direction of motion, and the gap thickness between the levitated mass and the xed electrodes does not change. 7"? "W Position-n0 sw'm' Positional\" v (b) Fig. 3. Schematic of a planar electrostatic actuation system. (a) Top View. (b) Side View. Solution: The total capacitance seen by the voltage source (assume C1 is constant with respect to x) C1(Z)C2(X,Z) 511:2 = C0 + (71(2) +C2(x,z) C , =C (x2) 0+ mi 3 1100:11/2 3C(x'z)=lr 5L 2 a: 2 z(x+L)2 So the instantaneous acceleration of the mass is A(x) _ F(x) _ 1 31/213 _ m _ 2 mz(x+l',)2 A(0) = = 1 51/215 m 2mz