The Effects of Nonlinearity on Parametric Amplifiers

Mechanical and electromechanical parametric amplifiers have garnered significant interest, as of late, due to the increased need for low-noise signal amplification in resonant micro/nanosystems. While these devices, which are traditionally designed to operate in a linear range, potentially represent an elegant, on-chip amplification solution, it is not readily apparent that this technical approach will suffice in all micro/nanoresonator implementations, due to the scale-dependent nature of a mechanical or electromechanical amplifier’s dynamic range. The present work investigates whether the aforementioned linear dynamic range constraint is truly a practical limitation, by considering the behavior of a representative degenerate parametric amplifier driven within a nonlinear frequency response regime. The work adopts a comparatively simple lumped-mass model for analysis and proceeds with the characterization of pertinent performance metrics, including gain/pump and gain/phase behaviors. Ultimately, the work concludes that parametric amplification can be realized in a nonlinear context, but such implementations generally lead to inferior amplifier performance. INTRODUCTION Recently, mechanical and electromechanical parametric amplifiers have garnered significant interest, due to the increased need for low-noise signal amplification in resonant micro/nanotransducers. While these devices utilize the same fun∗Address all correspondence to this author. damental mode of operation as their classical, purely-electrical counterparts, mechanical and electromechanical parametric amplifiers typically feature a scale-dependent dynamic range. Because of this, many micro/nanotransducers feature, at best, a narrow window of forcing amplitude in which a parametricallyamplified, linear (Lorentzian) frequency response can exist. Accordingly, if robust, low-noise signal amplification is to be realized in practical application, it may need to be done in a nonlinear context. The present work investigates the feasibility of implementing parametric amplification in a nonlinear, micro/nanoscale, mechanical or electromechanical resonator by characterizing the effects of a cubic nonlinearity on a classical degenerate parametric amplifier. The work adopts a comparatively-simple, lumpedmass model for analysis and utilizes standard perturbation methods to characterize and evaluate pertinent performance metrics, including gain/pump and gain/phase behaviors. The paper ultimately concludes with a brief overview and a discussion of ongoing research, including planned experimental investigations at both the microand nanoscales. 1 SYSTEM MODELING AND ANALYSIS Though micro/nanomechanical parametric amplifiers vary greatly in form, most are typically modeled as simple linear resonators driven by combined, parametric and direct, excitations (see, for example, Refs. [1–4]). While such a model suffices for resonators with an appreciable dynamic range, in practice 1 Copyright c © 2008 by ASME Information in Engineering Conference IDETC/CIE 2008 August 3-6, 2008, Brooklyn, New York, USA Proceedings of the ASME 2008 International Design Engineering Technical Conferences & Computers and