Oscillation modes in symmetrical wireless-locked systems
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Time synchronization of multiple elements of a wireless network can be achieved through the wireless coupling of their oscillator circuits. Most previous works on wireless locking of oscillators analyze the system in an idealized manner, representing the oscillator elements with phase models and describing the propagation effects with constant scalar coefficients and time delays. Here, a realistic analysis of the wireless system is presented, which relies on the extraction of the oscillator models from harmonic-balance (HB) simulations and takes into account the antenna gains and propagation effects. The most usual network configurations, corresponding to ring, fully connected, and star topologies, are investigated in detail. In symmetric conditions, the oscillation modes are detected through an eigenvalue/eigenvector calculation of an equivalent coupling matrix. For each particular mode, the system is analyzed in the following manners: by means of an analytical formulation, able to provide all the coexistent solutions, and through a circuit-level HB simulation of an equivalent system with a reduced number of oscillator elements. The stability properties are determined by means of a perturbation system of general application to any coupled structure. A specific formulation is also derived to predict the impact of discrepancies between the oscillator elements. All the results have been validated with independent circuit-level simulations and measurements.