Phase drift on networks of coupled crystal oscillators for precision timing

Precise time dissemination and synchronization have been some of the most important technological tasks for several centuries. Since the early 1800s, it was realized that precise time-keeping devices having the same stable frequency and precisely synchronized can have important applications in navigation. In modern times, satellite-based global positioning and navigation systems such as the GPS use the same principle. However, even the most sophisticated satellite navigation equipment cannot operate in every environment. In response to this need, we present a computational and analytical study of a network-based model of a high-precision, inexpensive, coupled crystal oscillator system and timing (CCOST) device. A bifurcation analysis (carried out by the authors in a related publication) [Buono et al., SIAM J. Appl. Dyn. Syst. 17, 1310 (2018)] of the network dynamics shows a wide variety of collective patterns, mainly various forms of discrete rotating waves and synchronization patterns. Results from computer simulations seem to indicate that, among all patterns, the standard traveling wave pattern in which consecutive crystals oscillate out of phase by $2\pi /N$, where $N$ is the network size, leads to phase drift error that decreases as $1/N$ as opposed to $1/\sqrt{N}$ for an uncoupled ensemble. The results should provide guidelines for future experiments, design, and fabrication tasks.

Figure 1

Two-mode crystal oscillator circuit. A second set of spurious RLC components $(R_2,L_2,C_2)$ are introduced by parasitic elements [9, 10].

Figure 2

CCOST concept with unidirectionally coupled crystal oscillators.

Figure 3

This figure displays 22 MHz rotating wave solution, $R{W}_1^1$, of the unidirectional CCOST model, with three nodes and coupling strength $\lambda =0.99$. (Top) The current, $X_i$, is plotted over a period of $1.0\times {10}^{-7}$ s. (Bottom) Noise function is displayed over the same duration.

Figure 4

The uncoupled-averaged $(\lambda =0)$ log scaled phase error as a function of array size. This figure displays the average taken over 50 samples for each $N$. A least squares regression line shows that the scaling exponent is $-0.5$, corresponding to a phase error reduction of $1/\sqrt{N}$.

Figure 5

The log scaled phase errors for the synchronized, the $T/N$ rotating wave, and the $T/2$ rotating wave solutions. $\lambda =-0.99$ for the synchronized solution and $\lambda =0.99$ for the rotating waves. The average of 50 samples taken for each $N$ is shown. A least squares regression is fitted to the logged data to illustrate the scaling exponent, $m$, of each pattern.

Figure 6

Phase error scaling for the $R{W}_1$ pattern as a function of $\lambda$ such that $\lambda \in (0,1)$, with dashed lines indicating $1/\sqrt{N}$ and $1/N$ scalings.

Figure 7

Phase error scaling for a larger number of oscillators in an unidirectionally coupled array. Coupling strength is held fixed at $\lambda =0.95$. The observed variations from the $1/N$ scaling are due to transient behavior in the system.

Figure 8

Discrete rotating wave predicted by symmetry-breaking Hopf bifurcations in a network of five crystal oscillators coupled unidirectionally.

Figure 9

Log scale phase error for the synchronized and $R{W}_2$ solutions in a bidirectionally coupled ring. The average of 50 simulations is plotted for each value of $N$. The synchronized solution performs slightly worse than the uncoupled scaling, while the $R{W}_2$ pattern is fitted to a reduction exponent of $m-0.47611$.

Figure 10

(Top) Experimental realization of a network of coupled crystal oscillators implemented via PIC boards. Each PIC board includes the oscillator circuitry and a port for coupling to create a network. RLC components guarantee that one mode oscillates at the desired frequency of 32 KHz while the other (parasitic) mode is unstable. The external small box contains the potentiometer to control the gain of the operational amplifiers and, in this way, manipulate the desired coupling strength across the network. (Bottom-left) Experimental measurements for $N=2$ and $N=3$ reveal, as expected, a traveling wave pattern among the oscillations. (Bottom-right) When the oscillators are uncoupled the pattern disappears.

Figure 11

Experimental phase error measured by averaging the phase error of a control group, i.e., an uncoupled ensemble of experimental crystal oscillators. The data points are plotted as well as least squares regression. The experimental phase error shows a reduction exponent of $m=-0.41254$.

Figure 12

Experimental phase error for a unidirectional CCOST device with the $R{W}_1$ pattern selected on a log 10 scale. The scaling exponent from the experimental data is $m=-0.8947$.