Stochastic Analysis of Cycle Slips in Injection-Locked Oscillators and Analog Frequency Dividers

A detailed investigation of cycle slips in injection-locked oscillators (ILOs) and analog frequency dividers is presented. This nonlinear phenomenon gives rise to a temporal desynchronization between the injected oscillator and the input source due to noise perturbations. It involves very different time scales so even envelope-transient-based Monte Carlo analyses may suffer from high computational cost. The analysis method is based on an initial extraction of a reduced-order nonlinear model of the injected oscillator based on harmonic-balance simulations. This model has been improved with a more accurate description of oscillation dependence on the input source either at the fundamental frequency or, in the case of a frequency divider, at a given harmonic frequency. The reduced-order model enables an efficient stochastic analysis of the system based on the use of the associated Fokker-Planck equation in the phase probability density function. Several methods for the solution of the associated Fokker-Planck equation are compared with one of them being applicable under a wider range of system specifications. The analysis enables the prediction of the parameter-space regions that are best protected against cycle slips. The technique has been applied to two microwave ILOs and has been validated through commercial software envelope simulations in situations where the computational cost of the envelope simulations was acceptable, and through measurements. The measurement procedure of the cycle slipping phenomenon has been significantly improved with respect to previous work.