  Nanomechanical resonators are used in building ultra-sensitive mass and force
sensors. In a widely used resonator based sensing paradigm, each modal
resonance frequency is tracked with a phase-locked loop (PLL) based system.
There is great interest in deciphering the fundamental sensitivity limitations
due to inherent noise and fluctuations in PLL based resonant sensors to improve
their performance. In this paper, we present a precise, first-principles based
theory for the analysis of PLL based resonator tracking systems. Based on this
theory, we develop a general, rigorously-derived noise analysis framework for
PLL based sensors. We apply this framework to a setting where the sensor
performance is mainly limited by the thermomechanical noise of the
nanomechanical resonator. The results that are deduced through our analysis
framework are in complete agreement with the ones we obtain from extensive,
carefully run stochastic simulations of a PLL based sensor system. We compare
the conclusions we derive with the recent results in the literature. Our theory
and analysis framework can be used in assessing PLL based sensor performance
with other sources of noise, e.g., from the electronic components, actuation
and sensing mechanisms, and due to the signal generator, as well as for a
variety of PLL based sensor configurations such as multi-mode and nonlinear
sensing.
