A Study of Injection Locking in Dual-Band CMOS Frequency Dividers

We present a study of dual-band injection locking frequency dividers (ILFDs), based on a nonlinear analysis. We develop a quasi-normal model of these dividers suitable for applying the method of averaging, which allowed us to derive in a simple and expressive manner the first-approximation equations for the amplitudes and phases of the locked modes, both in transient and in steady state. The phase equations have the same form of the Adler's equation, and represent the generalization of that well-known equation to higher-order frequency dividers. These equations allowed us to derive the locking ranges in a simple explicit form, useful for design purposes. The theoretical results are validated by Spice simulations, and by measurements on a circuit prototype.