25 research outputs found
Frequency Precision of Oscillators Based on High-Q Resonators
We present a method for analyzing the phase noise of oscillators based on
feedback driven high quality factor resonators. Our approach is to derive the
phase drift of the oscillator by projecting the stochastic oscillator dynamics
onto a slow time scale corresponding physically to the long relaxation time of
the resonator. We derive general expressions for the phase drift generated by
noise sources in the electronic feedback loop of the oscillator. These are
mixed with the signal through the nonlinear amplifier, which makes them
{cyclostationary}. We also consider noise sources acting directly on the
resonator. The expressions allow us to investigate reducing the oscillator
phase noise thereby improving the frequency precision using resonator
nonlinearity by tuning to special operating points. We illustrate the approach
giving explicit results for a phenomenological amplifier model. We also propose
a scheme for measuring the slow feedback noise generated by the feedback
components in an open-loop driven configuration in experiment or using circuit
simulators, which enables the calculation of the closed-loop oscillator phase
noise in practical systems
Eliminating 1/f noise in oscillators
We study 1/f and narrow-bandwidth noise in precision oscillators based on high-quality factor resonators and feedback. The dynamics of such an oscillator are well described by two variables, an amplitude and a phase. In this description we show that low-frequency feedback noise is represented by a single noise vector in phase space. The implication of this is that 1/f and narrow-bandwidth noise can be eliminated by tuning controllable parameters, such as the feedback phase. We present parameter values for which the noise is eliminated and provide specific examples of noise sources for further illustration
Intrinsic localized modes in parametrically driven arrays of nonlinear resonators
We study intrinsic localized modes (ILMs), or solitons, in arrays of parametrically driven nonlinear resonators with application to microelectromechanical and nanoelectromechanical systems (MEMS and NEMS). The analysis is performed using an amplitude equation in the form of a nonlinear Schrƶdinger equation with a term corresponding to nonlinear damping (also known as a forced complex Ginzburg-Landau equation), which is derived directly from the underlying equations of motion of the coupled resonators, using the method of multiple scales. We investigate the creation, stability, and interaction of ILMs, show that they can form bound states, and that under certain conditions one ILM can split into two. Our findings are confirmed by simulations of the underlying equations of motion of the resonators, suggesting possible experimental tests of the theory
Pattern selection in parametrically-driven arrays of nonlinear resonators
We study the problem of pattern selection in an array of
parametrically-driven nonlinear resonators with application to
microelectromechanical and nanoelectromechanical systems (MEMS & NEMS), using
an amplitude equation recently derived by Bromberg, Cross, and Lifshitz [PRE
73, 016214 (2006)]. We describe the transitions between standing-wave patterns
of different wave numbers as the drive amplitude is varied either
quasistatically, abruptly, or as a linear ramp in time. We find novel
hysteretic effects, which are confirmed by numerical integration of the
original equations of motion of the interacting nonlinear resonators
Phase noise of oscillators with unsaturated amplifiers
We study the role of amplifier saturation in eliminating feedback noise in self-sustained oscillators. We extend previous works that use a saturated amplifier to quench fluctuations in the feedback magnitude, while simultaneously tuning the oscillator to an operating point at which the resonator nonlinearity cancels fluctuations in the feedback phase. We consider a generalized model which features an amplitude-dependent amplifier gain function. This allows us to determine the total oscillator phase noise in realistic configurations due to noise in both quadratures of the feedback, and to show that it is not necessary to drive the resonator to large oscillation amplitudes in order to eliminate noise in the phase of the feedback
Building better oscillators using nonlinear dynamics and pattern formation
Frequency and time references play an essential role in modern technology and in living systems. The precision of self-sustained oscillations is limited by the effects of noise, which becomes evermore important as the sizes of the devices become smaller. In this paper, we review our recent theoretical results on using nonlinear dynamics and pattern formation to reduce the effects of noise and improve the frequency precision of oscillators, with particular reference to ongoing experiments on oscillators based on nanomechanical resonators. We discuss using resonator nonlinearity, novel oscillator architectures and the synchronization of arrays of oscillators, to improve the frequency precision
A Nanoscale Parametric Feedback Oscillator
We describe and demonstrate a new oscillator topology, the parametric feedback oscillator (PFO). The PFO paradigm is applicable to a wide variety of nanoscale devices and opens the possibility of new classes of oscillators employing innovative frequency-determining elements, such as nanoelectromechanical systems (NEMS), facilitating integration with circuitry and system-size reduction. We show that the PFO topology can also improve nanoscale oscillator performance by circumventing detrimental effects that are otherwise imposed by the strong device nonlinearity in this size regime
Homoclinic orbits and chaos in a pair of parametrically-driven coupled nonlinear resonators
We study the dynamics of a pair of parametrically-driven coupled nonlinear
mechanical resonators of the kind that is typically encountered in applications
involving microelectromechanical and nanoelectromechanical systems (MEMS &
NEMS). We take advantage of the weak damping that characterizes these systems
to perform a multiple-scales analysis and obtain amplitude equations,
describing the slow dynamics of the system. This picture allows us to expose
the existence of homoclinic orbits in the dynamics of the integrable part of
the slow equations of motion. Using a version of the high-dimensional Melnikov
approach, developed by Kovacic and Wiggins [Physica D, 57, 185 (1992)], we are
able to obtain explicit parameter values for which these orbits persist in the
full system, consisting of both Hamiltonian and non-Hamiltonian perturbations,
to form so-called Shilnikov orbits, indicating a loss of integrability and the
existence of chaos. Our analytical calculations of Shilnikov orbits are
confirmed numerically