# Figure (a) below shows the construction of a tunable capacitor that is actuated by using electrostatic force between the high-aspect ratio, inter digitated fingers. All the fingers connected to port-1 are free to move in the direction shown in Figure (a), and all the fingers that are connected to port-2 are stationery. Port-2 fingers are suspended bya set of springs, and an overall spring constant of k is defined for lumped modeling. Figure (b) shows the details ofa stationery-movable finger arrangement with the design geometries. Note that the overall tunable capacitor is the arca modulated capacitor (CI) and there is an overall parasitic capacitor at the cnd of the finger modeled with Cp.Number of finger pairs is N, the device is operating in air, heavily damped and all the fringing capacitances can be ignored. Find symbolic expressions in (a), (b), (c) as a function of geometries and k.

(a) Find expressions for Cp and C1. Assuming stationery values for these caps as CPo and C10, find the tendency of these caps with changes in displacement in r-direction. (b) Assume a voltage drive between ports I and 2. What is the maximum stable displacement xmax of the device? (c) Find the maximum value of the tuning capacitance for stable actuation. (d) Use N = 300, one-side length of one of the springs' length (i.e. from the anchor to the truss) L = 500 um,spring width w = 1 µm, d= 2 µm, h = 50 um, wf = 4 µm, xf = 30 um, re = 10 µm. Calculate the values of Cpo and C10, the maximum stable displacement and the maximum value of the tuning capacitance Cimaz-