Analytical Approach to the Study of Injection-Locked Frequency Dividers

An analytical method for the nonlinear analysis of injection-locked frequency dividers (ILFDs) is presented based on the method of the slowly-varying amplitude and phase. We introduce a general model of ILFDs and derive the associated averaging equations describing their first-order dynamical behavior. These equations are solved in closed form for two typical injection techniques of ILFDs, namely the injection via tail device and the direct injection. The oscillation amplitude and phase in the locked states and during the transient, as well as the locking range are obtained in explicit form. Numerical simulations validate the presented formulas, which provide useful design insights.