High-Q capacitive-gap transduced micromechanical resonators constructed via MEMS technology have recently taken center-stage among potential next generation timing and frequency reference devices that might satisfy present and future applications. Notably, oscillators referenced to very high Q capacitive-gap transduced MEMS resonators have already made inroads into the low-end timing market, and research devices have been reported to satisfy GSM phase noise requirements while only consuming less than 80 µW of power. Meanwhile, such devices have also posted some impressively low acceleration sensitivities, with measured sensitivity vectors less than 0.5 ppb/g. Interestingly, theory predicts that the acceleration sensitivity of these devices should be even better than this, if not for frequency instability due to electrical stiffness. Indeed, electrical stiffness is predicted to set lower limits on not only short-term stability, but long-term as well, especially when one considers frequency variations due to charging or temperature-induced geometric shifts. Pursuant to circumventing electrical stiffness-based instability, this work introduces a more circuit design-friendly equivalent circuit model that uses negative capacitance to capture the influence of electrical stiffness on device and circuit behavior. This new circuit model reveals that capacitive-gap transduced micromechanical resonators can offer better stability against electrical-stiffness-based frequency instability when used in large mechanically-coupled arrays. Measurements confirm that a 215-MHz 50-resonator disk array achieves a 3.5× enhancement in frequency stability against dc-bias voltage variation over a stand-alone single disk counterpart. The new equivalent circuit predicts the measurement data and its trends quite well, creating good confidence for using this circuit to guide new oscillator and filter designs that, depending on the application, can enhance or suppress electrical stiffness.





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