# Phase-Locked Loop based Resonant Sensors: A Rigorous Theory and General Analysis Framework for Deciphering Fundamental Sensitivity Limitations due to Noise

## TL;DR

This paper develops a rigorous, first-principles theory and analysis framework for PLL-based nanomechanical resonant sensors, enabling precise understanding of fundamental noise limitations and guiding performance improvements.

## Contribution

It introduces a comprehensive, mathematically rigorous analysis framework for PLL-based sensors, accounting for various noise sources and sensor configurations, validated by stochastic simulations.

## Key findings

- The framework accurately predicts sensitivity limits due to thermomechanical noise.
- Simulation results agree with theoretical predictions.
- The approach can be extended to other noise sources and sensor setups.

## Abstract

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.