Locking and Pulling in Injection-Locked LC-CMOS Dividers

In this paper, we present a comprehensive analysis of LC analog frequency dividers obtained by using an analytical approach based on the well-known asymptotic method of slowly-varying amplitude and phase. The methodology of analysis allows us not only to estimate the locking range as a function of the circuit parameters, but also the amplitude and the phase of the output voltage in steady-state and in transient, as well as to study the stability properties of the locked oscillations. Further, we investigate the transition from the locking mode to the periodic pulling mode, which happens at boundaries of locking region, showing that there is not a borderline between the two operation modes, but the two modes of operation can coexist. We highlight that there exists a border region where the occurrence of one mode or the other depends only on the initial conditions, as validated by numerical simulations.