Abstract-Phase noise is a topic of theoretical and practical interest in electronic circuits, as well as in other fields, such as optics. Although progress has been made in understanding the phenomenon, there still remain significant gaps, both in its fundamental theory and in numerical techniques for its characterization. In this paper, we develop a solid foundation for phase noise that is valid for any oscillator, regardless of operating mechanism. We establish novel results about the dynamics of stable nonlinear oscillators in the presence of perturbations, both deterministic and random. We obtain an exact nonlinear equation for phase error, which we solve without approximations for random perturbations. This leads us to a precise characterization of timing jitter and spectral dispersion, for computing which we develop efficient numerical methods. We demonstrate our techniques on a variety of practical electrical oscillators and obtain good matches with measurements, even at frequencies close to the carrier, where previous techniques break down. Our methods are more than three orders of magnitude faster than the brute-force Monte Carlo approach, which is the only previously available technique that can predict phase noise correctly.
Nanomechanical resonators are used as high performance sensors of physical stimuli such as force and mass changes. Any such physical stimulus produces a shift in the resonance frequency of the nanomechanical structure, which can be measured accurately by using a feedback system that locks the frequency of a signal generator to the resonance. Closed-loop frequency tracking is the most prevalent technique in the fields of nanomechanical sensors and non-contact atomic force microscopy. Ultimate performance of sensors is limited by various nonideal effects such as temperature variations, radiation, electromagnetic interference, and noise arising from inherent physical mechanisms. Here, we consider the noise performance of nanomechanical resonant sensors, which has so far eluded explanation with conflicting results reported in the literature. We present a precise theory for these ubiquitous sensors based on nanomechanical resonators under feedback in order to decipher the fundamental sensitivity limitations due to thermomechanical noise. The results we obtain, when the performance is limited by the thermomechanical noise of the resonator, are in complete agreement with the ones from stochastic simulations. Our findings shed light on recent results in the literature and resolve a critical problem regarding the frequency noise of nanomechanical sensors under feedback. Our results have applications in nanomechanics, atomic force microscopy, microwave and suspended microchannel resonators.
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