  Resonant sensors determine a sensed parameter by measuring the resonance
frequency of a resonator. For fast continuous sensing, it is desirable to
operate resonant sensors in a closed-loop configuration, where a feedback loop
ensures that the resonator is always actuated near its resonance frequency, so
that the precision is maximized even in the presence of drifts or fluctuations
of the resonance frequency. However, in a closed-loop configuration, the
precision is not only determined by the resonator itself, but also by the
feedback loop, even if the feedback circuit is noiseless. Therefore, to
characterize the intrinsic precision of resonant sensors, the open-loop
configuration is often employed. To link these measurements to the actual
closed-loop performance of the resonator, it is desirable to have a relation
that determines the closed-loop precision of the resonator from open-loop
characterisation data. In this work, we present a methodology to estimate the
closed-loop resonant sensor precision by relying only on an open-loop
characterization of the resonator. The procedure is beneficial for fast
performance estimation and benchmarking of resonant sensors, because it does
not require actual closed-loop sensor operation, thus being independent on the
particular implementation of the feedback loop. We validate the methodology
experimentally by determining the closed-loop precision of a mechanical
resonator from an open-loop measurement and comparing this to an actual
closed-loop measurement.
