Optimal operating points of oscillators using nonlinear resonators

Abstract

We demonstrate an analytical method for calculating the phase sensitivity of a class of oscillators whose phase does not affect the time evolution of the other dynamic variables. We show that such oscillators possess the possibility for complete phase noise elimination. We apply the method to a feedback oscillator which employs a high Q weakly nonlinear resonator and provide explicit parameter values for which the feedback phase noise is completely eliminated and others for which there is no amplitude-phase noise conversion. We then establish an operational mode of the oscillator which optimizes its performance by diminishing the feedback noise in both quadratures, thermal noise, and quality factor fluctuations. We also study the spectrum of the oscillator and provide specific results for the case of 1/f noise sources.

Phase noise of oscillators with unsaturated amplifiers

“…This condition has solutions for providing s > (4/3) 5/4 [6]. Note that this condition is identical to the condition for having bistability in the open-loop system driven at the saturation value s [9].…”

“…The total sensitivity to both quadratures of amplifier noise, as given by the expression (36), for s = 3 and k = 2. As the gain level grows, the phase noise approaches the saturated amplifier behavior having two zero phase noise points [6].…”

“…This connection is counterintuitive since, quite generally, one associates bifurcation points with enhanced noise sensitivity. More recently, Kenig et al [5,6] showed that this phenomenon is not restricted to the specific Duffing-like system studied by Greywall et al, by reformulating the dynamical problem in a more general setting.…”

Building better oscillators using nonlinear dynamics and pattern formation

“…The amplitude equation (10) falls into a class of systems where the noise projection method is vastly simplified [1]. These are the ones for which the dynamical variables…”

“…Our focus is on oscillators built from nanomechanical devices, but the ideas apply generally. This paper is a summary of work published in a number of papers [1][2][3][4][5][6]; more details can be found in those publications. There is also experimental work supporting some of the results [7,8].…”