The emergence and analysis of Kuramoto-Sakaguchi-like models as an effective description for the dynamics of coupled Wien-bridge oscillators

TL;DR

This paper derives a Kuramoto-Sakaguchi model from Wien-bridge oscillator circuits, providing a theoretical framework for understanding synchronization phenomena in such coupled oscillators, supported by numerical and experimental validation.

Contribution

It presents a novel derivation of the Kuramoto-Sakaguchi model directly from circuit equations of Wien-bridge oscillators, linking circuit dynamics to phase synchronization models.

Findings

Derived phase-amplitude equations match numerical simulations

Experimental results confirm theoretical predictions

Identified conditions for stable synchronization

Abstract

We derive the Kuramoto-Sakaguchi model from the basic circuit equations governing two coupled Wien-bridge oscillators. A Wien-bridge oscillator is a particular realization of a tunable autonomous oscillator that makes use of frequency filtering (via a RC band-pass filter) and positive feedback (via an Op-Amp). In the last few years, such oscillators have started to be utilized in synchronization studies. We first show that the Wien-bridge circuit equations can be cast in the form of a coupled pair of Duffing - Van der Pol equations. Subsequently, by applying the method of multiple time scales, we derive the differential equations that govern the slow evolution of the oscillator phases and amplitudes. These equations are directly reminiscent of the Kuramoto-Sakaguchi type models for the study of synchronization. We analyze the resulting system in terms of existence and stability of various coupled oscillator solutions and explain on that basis how their synchronization emerges. The phase-amplitude equations are also compared numerically to the original circuit equations, and good agreement is found. Finally, we report on experimental measurements on two coupled Wien-bridge oscillators and relate the results back to the theoretical predictions.