General Formulation for the Analysis of Injection-Locked Coupled-Oscillator Systems

The use of an external injection-locking source prevents undesired frequency shifts in coupled-oscillator systems. Here an explicit formulation for the analysis of these systems is presented, based on a numerical model of the oscillator elements in free-running regime, obtained with harmonic balance. It demonstrates the opposed effects of oscillator coupling and injection locking in a realistic manner and brings to light the influence of the oscillator characteristics and coupling-network parameters. The explicit expressions provide the minimum injection-locking power required for a full variation of the inter-stage phase shift and enable the locking bandwidth to be fixed or maximized through a suitable choice of the coupling networks. Two different cases are considered: a basic system of three oscillator elements and a general system with any number of elements. In the latter case, the influence of the particular element chosen for the signal injection on the synchronization bandwidth is studied. The stability analysis is based on a perturbation system, which, unlike previous works, relies on a direct analytical calculation of the steady-state solutions, instead of a numerical one. This enables the full coverage of all possible solutions and hence an in-depth understanding of the influence of the coupling network and injection source on the stable phase-shift ranges. The phase noise analysis, applied to the same explicit formulation, demonstrates the injection locking effects and enables the identification of different regions as the offset frequency increases.