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Sufficient conditions are derived for global asymptotic synchronization in asystem of identical nonlinear electrical circuits coupled through lineartime-invariant (LTI) electrical networks. In particular, the conditions wederive apply to settings where: i) the nonlinear circuits are composed of aparallel combination of passive LTI circuit elements and a nonlinearvoltage-dependent current source with finite gain; and ii) a collection ofthese circuits are coupled through either uniform or homogeneous LTI electricalnetworks. Uniform electrical networks have identical per-unit-lengthimpedances. Homogeneous electrical networks are characterized by having thesame effective impedance between any two terminals with the others opencircuited. Synchronization in these networks is guaranteed by ensuring thestability of an equivalent coordinate-transformed differential system thatemphasizes signal differences. The applicability of the synchronizationconditions to this broad class of networks follows from leveraging recentresults on structural and spectral properties of Kron reduction---amodel-reduction procedure that isolates the interactions of the nonlinearcircuits in the network. The validity of the analytical results is demonstratedwith simulations in networks of coupled Chua's circuits.

Synchronization of Nonlinear Circuits in Dynamic Electrical Networks with General Topologies