Generalized Theory of Electro-Mechanical Resonators with Fractal Electrodes

In this paper, we study the operation of electrostatically actuated electromechanical resonators having fractal electrodes. Using the spring-mass model, we establish the scaling relationships for pull-in instability , voltage dependent resonance frequency 1 , and pull-in voltage , where is the fractal dimension of the electrode, is the air-gap, and are the resonance frequency at zero and applied bias respectively, and is the steady state position of the moving electrode. These scaling relationships reduce to the well known pull-in instability, and resonance frequency of resonators based on planar 2 and cylindrical 1 electrodes. The generality of the scaling relationships are confirmed by using electrodes based on array of cylinders, that allow us to continuously scan all fractal dimension in between planar and cylindrical electrodes (1 2) simply by tuning the pitch of the array.