Resonance-tracking capacitive sensing beyond the linear and adiabatic limits

Capacitive sensing has been extensively utilized in high-precision measurements, including gravitational-wave detection. Compared to conventional open-loop schemes, the closed-loop resonance-tracking method significantly enhances the measurement dynamic range. Previous analyses focused primarily on the linear and adiabatic regimes. Here, we extend the theoretical framework into two unexplored regimes: (i) the linear yet nonadiabatic regime, and (ii) the nonlinear but adiabatic regime. These extensions are crucial for applications requiring both high sensitivity to minute displacements and the capability to track large-scale, slow variations. Beyond theoretical modeling, we implement a proof-of-principle experiment utilizing a hybrid analog-digital circuit, validating the predictions of our extended theoretical model. The developed modeling and design principles are generalizable to other resonance-enhanced measurement systems, potentially broadening their applicability in precision metrology.