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We derive the Kuramoto-Sakaguchi model from the basic circuit equationsgoverning two coupled Wien-bridge oscillators. A Wien-bridge oscillator is aparticular realization of a tunable autonomous oscillator that makes use offrequency filtering (via a RC band-pass filter) and positive feedback (via anOp-Amp). In the last few years, such oscillators have started to be utilized insynchronization studies. We first show that the Wien-bridge circuit equationscan 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 thedifferential equations that govern the slow evolution of the oscillator phasesand amplitudes. These equations are directly reminiscent of theKuramoto-Sakaguchi type models for the study of synchronization. We analyze theresulting system in terms of existence and stability of various coupledoscillator solutions and explain on that basis how their synchronizationemerges. The phase-amplitude equations are also compared numerically to theoriginal circuit equations, and good agreement is found. Finally, we report onexperimental measurements on two coupled Wien-bridge oscillators and relate theresults back to the theoretical predictions.

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