Experimental study of synchronization of coupled electrical self-oscillators and comparison to the Sakaguchi-Kuramoto model

We explore the collective phase dynamics of Wien-bridge oscillators coupled resistively. We carefully analyze the behavior of two coupled oscillators, obtaining a transformation from voltage to effective phase. From the phase dynamics we show that the coupling can be quantitatively described by Sakaguchi's modification to the Kuramoto model. We also examine an ensemble of oscillators whose frequencies are taken from a flat distribution within a fixed frequency interval. We characterize in detail the synchronized cluster, its initial formation, as well as its effect on unsynchronized oscillators, all as a function of a global coupling strength.

Figure 1

Circuit diagram of two coupled Wien-bridge oscillators. The intrinsic frequency is determined by the values of $r,R$, and $C$. The resistor $R_i$ receives the voltage from the shared voltage divider, whereas the resistor $R_o$ transmits to the voltage divider. For our experiments we have $R_1=27\mathrm{k}\mathrm{\Omega },R_2=3\mathrm{k}\mathrm{\Omega },C=0.1\mu \mathrm{F},R=4.7\mathrm{k}\mathrm{\Omega },r$ represents a $5\mathrm{k}\mathrm{\Omega }$ potentiometer, $R_i=62\text{k}\mathrm{\Omega }$, and $R_o=1.1\text{k}\mathrm{\Omega }$.

Figure 2

A schematic of the $N$ Wien-bridge oscillators. We incorporate two sets of switches, one to prevent particular oscillators from “talking,” and the other to prevent them from “listening” to the others. The oscillators operate at frequencies between 200 and $400\text{Hz}$, and their voltages are sampled every $0.1\text{ms}$.

Figure 3

(a) Average oscillator frequencies as we vary one of the oscillators' natural frequencies, indicating the range of frequency mismatch over which two oscillators will still synchronize. These are based on the average period of oscillation over the full 500-ms time series. The dashed black and dotted red lines indicate the self-coupled frequencies of oscillator 1 and 2, respectively; notice that we vary the frequency of the first, but not the second. (b–d) The dynamic frequencies of the two oscillators when coupled, corresponding to points enumerated in (a). The coupling is initially set to zero and then abruptly turned on at $t=0.05\text{s}$. For (b), the two oscillators quickly come into stable synchronization, but for (c) and (d) this is not the case, as dephasing occurs in bursts. These bursts occur more frequently with larger frequency mismatch.

Figure 4

(a) A comparison between the theoretical prediction of the Sakaguchi-Kuramoto model, Eqs. (2) and (3), for the two-oscillator synchronization-frequency (solid line) and the experimental data points shown previously (black squares and red circles, explained in Fig. 3). Good quantitative agreement is found across the whole set of experiments. (b) Just above the highest phase-locked frequency, rapid phase-slippage events at regular time intervals were observed in the experiment: black squares and red circles with connecting lines represent the measured dynamic frequencies of the faster and slower oscillator, respectively. The phenomenon is faithfully reproduced in simulations of the Sakaguchi-Kuramoto model, for which the black solid line and red dashed line represent the faster and slower oscillator, respectively. (In the graph, the trace obtained by numerical simulation is shifted lower for visual clarity.)

Figure 5

(a) Plot of long-time average dynamic frequencies vs. self-coupled frequencies for three coupling strengths. The strength decreases from A to C, as indicated by $y$ value of the same letter in the next figure. (b) The $y$ axis represents the coupling strength between oscillators as given by the value of $R_B$. The $x$ axis indicates the number of talkers. As we move from left the right, incrementally more low-frequency oscillators are silenced (by opening switch $S_2$). The color indicates the size of the largest cluster, ranging from 0 (red, no sync) to 19 (blue, full sync). The line and dots indicates the theoretically predicted threshold for full synchronization. (c) Plot similar to (a) for only five talkers. As with (a), the strength is indicated by the $y$ value of the same letter in figure (b).

Figure 6

Capturing the transients on route to synchronization. Switches, $S_1$ and $S_2$, are closed for all oscillators. (a) Full coupling strength $\left(R_B=4\mathrm{k}\mathrm{\Omega }\right)$ and (b) reduced coupling strength $\left(R_B=3.5\mathrm{k}\mathrm{\Omega }\right)$. The dashed red line marks the approximate end of the transient and the full formation of the synchronized cluster.

Figure 7

Transients for the case where the switches S2 are open for all but the top six oscillators. (a) Full coupling strength $\left(R_B=4\mathrm{k}\mathrm{\Omega }\right)$ and (b) reduced coupling strength $\left(R_B=1.5\mathrm{k}\mathrm{\Omega }\right)$. Notice the oscillator with the highest natural frequency (depicted in the dotted red line) experiences periodic bursts in frequency, corresponding to times when its phase detaches from that of the cluster. A few lower-frequency oscillators form a desynchronized random background.