Optimal operating points of oscillators using nonlinear resonators

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.

Figure 1

The amplitude vs frequency curve of the oscillator described by Eq. (10). The low- and high-amplitude red dots are ${\Delta }_{1,2}$, respectively, for which $d{\Omega }_0/d\Delta =d{\Omega }_0/da=0$. The star is ${\Delta }_{a\mid \varphi }$ for which $\partial {\Omega }_0/\partial a=0$. The bottom curve is for $s=3$, and the top one is for $s=10$, indicating that ${\Delta }_1$ and ${\Delta }_{a\mid \varphi }$ approach each other as saturation level increases.

Figure 2

Special phase-shift values as a function of the drive amplitude. ${\Delta }_{1,2}$, which eliminate the feedback phase noise, bifurcate at the critical Duffing drive amplitude $s={(4/3)}^{(5/4)}$. On increasing saturation level, ${\Delta }_2$ approaches $\pi /2$, and the other three special phase-shift values approach $\pi$.

Figure 3

Phase sensitivity as a function of the phase shift. (a) Different contributions ${\left({\mathbf{v}}_{\perp }^T{\mathbf{v}}_{noise}\right)}^2$ to the phase noise for $s=3$. We see that the feedback phase noise curve is zero at ${\Delta }_{1,2}$ and that those of the real component of thermal noise and quality factor noise are zero at ${\Delta }_{a\mid \varphi }$, which is slightly to the right of ${\Delta }_1$. (b) The total phase sensitivity, which is a sum of all the curves in (a), for different values of the drive level. We see that as saturation level grows, the optimal operating point approaches ${\Delta }_1$, denoted with a red asterisk, and the noise level at this point decreases.

Figure 4

The spectrum of an oscillator subjected to $1/f$ noise. The exact solution is $\mathcal{F}\left[e^{-V\left(\tau \right)/2}\right]$ with $V\left(\tau \right)$ given by Eq. (31). For very weak noise, small offset frequencies are described by the Lorentzian (26); however, for stronger noise, this frequency regime is approximated by the Gaussian (33) as shown in (a) and (b), respectively. Both curves approach the expression (25) at large frequencies. Other parameters are ${\omega }_c=0.1$, $P^2=1$, and ${\omega }_m=10$.