(a) A DC voltage Vis applied to the microelectromechanical systems (MEMS)misaligned double-plate electrodes shown in Figure Q 1(a). The corresponding electrostatic force in Fd,direction is -epsilon0epsilonrWLV2)/ (2d2)(epsilon0 and epsilonr denote permittivity in vacuum and relative permittivity respectively. W, L, d are width, length, and gap between the two plates respectively). A linear scaling for the voltage is used, i.e. V x l',where I is the linear scale of the electrode. What happens to Fd if all dimensions of the two plates are reduced to 50% of their initial values? (b) For the three differential equations below describing MEMS dynamic systems, write their transfer functions based on the Laplace Transform.Work out fundamental frequency for each case (neglect the damping effect). 3 \ddot{x}(t)+11 \dot{x}(t)+300 x=3 f(t) a \dot{x}(t)+25 a x(t)=c f(t) ; \quad \text { a and } c \text { are non-zero constants } \frac{1}{3} \ddot{x}(t)+27 x(t)=5 f(t) (c) There is a parallel plate MEMS actuator shown in Figure Q 3(c). The initial gap go is 12 µm, and the area of the plate A is 200 µm x 100 µm. (Note:free space permittivity is 8.85 x 10-12 F/m) (1) Derive the formula for the electrostatic force F; (2) Assume a 5 volts potential is applied to the actuator. Calculate F when the top plate is in its initial position. (3) Derive the equation for Vvs. g and draw a graph showing V vs. g (g varies from 0 to 12 um with 2 µm step size; given spring constant kis 30 N/m). Give a brief explanation about the pull-in voltage. (d) Describe the working principle of a strain sensor that is made of a piezoresistive material and a piezoelectric material respectively.

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