Description and Operation:
The clamped-clamped beam resonator (Figure 3.32) consists of a doubly supported cantilever beam disposed over a bottom electrode. The beam has length Lr , width Wr , and thickness h, and is made up of a material with Young?s modulus E and mass density r.
The bottom electrode has width We and is separated from the beam by a gap d. In operation, a dc voltage VP and an input ac voltage vi, applied across the capacitor, C = e0WrWe /d0, defined by the area of overlap between the beam and the bottom electrode, induce an electrostatic force fc on the beam, given by  that causes it to vibrate vertically, exhibiting displacement xc. In this expression, the derivative dC/dx represents the change in electrode-to-resonator capacitance per unit displacement of the resonator and is given by where d 0 is the unbiased beam-to-electrode gap. The displacement of the beam in response to vi induces a capacitive current given by and is largest when the excitation frequency is closed to the mechanical resonance frequency of the beam, which is given by where k is a scaling factor that models the effects of surface topography (e.g., the anchor step-up and its corresponding finite elasticity).
In general, the resonance frequency of the clamped-clamped beam resonator is made to deviate from (3.18) as a result of the magnitude of the polarization voltage VP, which induces a dynamic  stiffness ke that subtracts from the unbiased stiffness km. This dependence is taken into account phenomenologically by expressing the resonance frequency by [1, 46]┬á where g models the effect of the electrical spring stiffness. It shows the measured resonance frequency dependence on VP.