Synchronization of nonlinear circuits in dynamic electrical networks

Synchronization of coupled oscillators is a pervasive theme of multi-disciplinary research. Focused on circuit-theoretic applications, in this paper, we derive sufficient conditions for global asymptotic synchronization in a system of identical nonlinear circuits coupled through linear time-invariant (LTI) electrical networks. The nonlinear circuits are composed of a parallel combination of passive LTI circuit elements and a nonlinear voltage-dependent current source with finite gain. The terminals of the nonlinear circuits are coupled through LTI networks characterized by either identical per-unit-length impedances or identical effective impedances between any two terminals. This setup is motivated by a recently proposed control strategy for inverters in microgrids. We analyze synchronization by means of an input-output analysis of a coordinate-transformed system that emphasizes signal differences. To apply the synchronization analysis to a broad class of networks, we leverage recent results on Kron reduction - a circuit reduction and transformation procedure that reveals the interactions of the nonlinear circuits. We illustrate our results with simulations in networks of coupled Chua's circuits.