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.

Figure 1

The amplifier profile for $k=2$ and $s=3$.

Figure 2

Phase space of the complex amplitude $A$. (a) In a rotating frame, the noise-free orbit is a circle of fixed points. (b) Closeup view showing local coordinate axes $(\alpha ,\varphi )$ and the vector $V_{☆}$ along which perturbations relax back to the original fixed point. (c) The hatched region represents the two-dimensional (typically anisotropic) noise distribution. (d) In the strict saturation limit, the noise cloud collapses to one dimension. (e) Parameter tuning can align the noise with $V_{☆}$, thus eliminating phase diffusion. (f) More generally, the noise cloud stays two dimensional, and parameter tuning results in a partial reduction of phase diffusion.

Figure 3

(a) Response of an oscillator driven by a nonsaturated amplifier for $s=3$ and $k=2$. (b) The phase noise sensitivity coefficients $P_R^2$ (thick blue) and $P_I^2$ (thin green) for the same amplifier gain values. Note the cancellation of amplifier phase noise at small amplitudes for $G=1.01$.

Figure 4

Oscillation characteristics for constant feedback sweeps for $k=2$, $a_s\simeq 1.73$, and two feedback drive levels $d=0.1$ (dashed) and $d=2$ (solid): (a) Oscillation amplitude vs oscillation frequency; (b) oscillation frequency vs phase shift; (c) scaled phase noise sensitivity coefficient ${\left(P_{I}d\right)}^2$ vs phase shift.

Figure 5

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].