Exact analysis of spontaneous phase synchronization of two identical coupled electrical linear LC oscillators

This paper provides contributions to the study of the phase synchronization transient of two identical linear LC oscillators, coupled through a resistor. The main contribution is obtaining an exact equation that represents, as a function of time, the evolution of the phase angle between the two oscillators, from the initial condition until the synchronization. It is demonstrated that the system has two equilibrium points, one unstable when the phase angle is equal to π, and the other stable when it is equal to zero and to which the phase converges for any initial value other than π. It is also demonstrated that the dynamics of the two coupled oscillators can be described by the Kuramoto model. An analysis of the energy dissipated during synchronization is presented and the concept of synchronization energy is introduced. The results of the theoretical analysis are verified with the use of computer simulation.