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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 [46] 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 [49] stiffness ke that subtracts from the unbiased stiffness km. This dependence is taken into account phenomenologically by expressing the resonance frequency by [1, 46]  where g models the effect of the electrical spring stiffness. It shows the measured resonance frequency dependence on VP.