Classical Noise. V. Noise in Self-Sustained Oscillators

Abstract

A spectrally pure self-sustained oscillator, described by a positive- and negative-impedance series circuit $Z={Z}_{n}+{Z}_{p}$ such that $Z{e}^{i\ensuremath{\omega}t}={e}^{i\ensuremath{\omega}t}Z(\ensuremath{\omega}, D)$, yields a single frequency output depending nonlinearly on a parameter $D$ related to an instantaneous or recent-time-averaged power. The oscillator operates at a point $D={D}_{0}$ at which gains and losses cancel, ${R}_{p}+{R}_{n}\ensuremath{\equiv}R({\ensuremath{\omega}}_{0}, {D}_{0})=0$, and at a frequency ${\ensuremath{\omega}}_{0}$ determined by ${X}_{p}+{X}_{n}\ensuremath{\equiv}X({\ensuremath{\omega}}_{0}, {D}_{0})=0$. Since $R=0$, the oscillator linewidth vanishes in the absence of noise. We endow the resistances ${R}_{p}$ and ${R}_{n}$ with Langevin noise sources. Amplitude fluctuations produce a broad additive background. The oscillator is unstable against phase fluctuations, which broaden the signal into finite width. A quasilinear treatment, well above threshold, demonstrates that the phase executes a Brownian motion. If $\frac{\ensuremath{\partial}R}{\ensuremath{\partial}\ensuremath{\omega}}\ensuremath{\ne}0$ or $\frac{\ensuremath{\partial}X}{\ensuremath{\partial}D}\ensuremath{\ne}0$ at the operating point, phase and amplitude fluctuations are coupled. Nevertheless, we succeed in calculating the linewidth and proving that it is independent of the rate at which power (or $D$) relaxes. A comparison is made with the "linear" treatment of oscillators as amplified noise. A reduced random process is set up, valid for time intervals obeying ${\ensuremath{\omega}}_{0}\ensuremath{\Delta}t\ensuremath{\gg}1$. Although the phase of the oscillator involves a nonlinear, nonstationary action on the Gaussian input noise, it is shown that well above threshold, for the reduced process, the phase is again properly described as a Gaussian variable subject to the expected Brownian-motion diffusion. For all well-designed oscillators, even near threshold, we establish that the reduced random process is that of a rotating-wave van der Pol oscillator. A comparison is made between quasilinear solutions of the rotating-wave van der Pol oscillator and exact solutions of the Fokker-Planck equation computed in the next paper, VI, in this series. For intensity fluctuations it is demonstrated that quasilinear methods are quantitatively valid away from the threshold region and qualitatively valid near threshold, provided that the quasilinear approximation is made in the correct variable.