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.

Figure 1

Values of the feedback phase shift $\Delta$ giving zero phase noise $(P=0)$ for constant feedback and $\eta =0$. Solid curve, $g_s=4/\pi$; dashed curve, $g_s=9/\pi$; dotted curve, $g_s=300/\pi$. These values correspond to $g_s\simeq g_c·(0.88,2,66)$, respectively.

Figure 2

Phase noise elimination in an oscillator using an unsaturated amplifier. The white area in the figures is the parameter range for which there are zero noise points $(P=0)$ for $x_s=3$ and $r=0.5$ (corresponding to $g_s=9/\pi$ in the large-gain limit). (a) $\eta =0$; (b) $\eta =3$.