Schemes for Tracking Resonance Frequency for Micro- and Nanomechanical Resonators

Nanomechanical resonators can serve as high-performance detectors and have the potential to be widely used in industry for a variety of applications. Most nanomechanical-sensing applications rely on detecting changes of resonance frequency. In commonly used frequency-tracking schemes, the resonator is driven at or close to its resonance frequency. Closed-loop systems can continually check whether the resonator is at resonance and adjust the frequency of the driving signal accordingly. In this work, we study three resonance-frequency-tracking schemes, a feedback-free (FF) scheme, a self-sustaining oscillator (SSO) scheme, and a phase-locked loop oscillator (PLLO) scheme. We improve and extend the theoretical models for the FF and the SSO tracking schemes and test the models experimentally with a nanoelectromechanical system (NEMS) resonator. We employ an SSO architecture with a pulsed positive-feedback topology and compare it to the commonly used PLLO and FF schemes. We show that all tracking schemes are theoretically equivalent and that they all are subject to the same speed versus accuracy trade-off characteristics. In order to verify the theoretical models, we present experimental steady-state measurements for all of the tracking schemes. The frequency stability is characterized by computing the Allan deviation. We obtain almost perfect correspondence between the theoretical models and the experimental measurements. These results show that the choice of the tracking scheme is dictated by cost, robustness, and usability in practice as opposed to fundamental theoretical differences in performance.