Homework #4
Assigned: Feb. 4th, 2009
Due: Feb. 10th, 2009
Points: 30

• Problem 1: 10 points
  Consider a parallel-plate capacitor of area A in which one plate is fixed and the other is restrained by a spring of stiffness k. Take the air gap between them as g₀.
  Add an electrical capacitor (a) in series, and (b) in parallel with the parallel-plate capacitor. Let this electrical capacitance be equal to alpha*(epsilon₀*A)/g₀. Find alpha for each case, if possible, such that the stable region of the travel of the moving plate is g₀ before pull-in occurs for increasing values of the applied voltage, V.
• Problem 2: 20 points
  Take the above parallel-plate capacitor. Now, consider the mass to be m and damping to be b. Asume that there is an insulation layer of thickness t_d of relative permittivity of 5.
  (a) Write a program (preferably in Matlab) to compute the pull-in and pull-up voltages and times.
  (b) Apply large DC and sufficiently small AC voltages and observe the behavior. By introducing a small perturbation to the plate after it reaches the steady state and show that the frequency of its natural oscillations due to the perturbation approaches zero as the DC voltage approaches the pull-in voltage. For extra credit, plot this frequency as a function of the DC voltage.