Locking

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—I: REGULAR PAPERS, VOL. 58, NO. 4, APRIL 2011

765

Observations Concerning the Locking Range in a Complementary Differential Injection-Locked Frequency Divider—Part II: Design Methodology

Saeid Daneshgar, Student Member, IEEE, Oscar De Feo, and Michael Peter Kennedy, Fellow, IEEE

Abstract—An intuitive approach to analyze the behavior of an Injection-Locked Frequency Divider was presented in Part I of this work; that paper provided insight into the locking behavior in the valid design area of the circuit. In this paper, we present a rigorous design methodology which provides a closed form equation showing where the locking range is wider. Theoretical predictions of the locked regions are verified by simulations of the circuit in Spectre RF using 0.35- m CMOS technology models. Index Terms—Analog frequency dividers, bifurcation theory, injection locked frequency dividers, locking range prediction, nonlinear analysis, nonlinear oscillators.

I. INTRODUCTION N PART I [1], we introduced a nonlinear analysis method which characterizes the locking behavior of an injection-locked frequency divider (ILFD). The method quantifies the qualitative widths and shapes of the locked regions as well as their center frequencies. It provides insight into where the locking range of the circuit is wider rather than calculating the width of the locked region exactly. ILFD architectures with different locking A variety of ranges has been proposed in the literature. However, few papers report both analytical predictions and experimental validation [2]–[11]. Although these papers provide methodologies for analyzing the circuit and predicting the approximate width of the locked region, none of them predicts the precise location of the center frequency of the locked region. In [2] and [3], locking results are presented for direct injection tank in conventional or complementary differenacross the tial oscillators, such as that shown in Fig....