Nonlinear dynamics of divide‐by‐two injection‐locked frequency dividers in locked operation mode

Abstract

SUMMARY A method for analyzing the nonlinear dynamics of the injection‐locked frequency dividers in synchronized operation mode is presented, including the stability analysis of locked states. We use a specific divide‐by‐two circuit, namely a differential LC CMOS divider with a complementary topology, as a guideline for presentation, showing that the sizing of the devices significantly affects the synchronization mechanism of the divider, which exhibits a very rich dynamical behavior. We provide closed‐form expressions to determine the amplitude and the phase in the locked state, as well as the locking range, leading to accurate results, which are validated by numerical simulations. The presented analysis of the frequency divider dynamics enables us to establish that stable locked oscillations occur on the whole locking range predicted by the well‐known Adler's equation and that these are possible also beyond that range. Copyright © 2013 John Wiley & Sons, Ltd.

Investigation on Locking and Pulling Modes in Analog Frequency Dividers

“…It is interesting to observe that, even if the system (27) can be practically handled, only if the assumptions in (29) are made, the results are sufficiently accurate, as shown in Figure 4, where analytical results are compared with numerical results obtained by SPICE simulation of the circuit in Figure 1. From the aforementioned analysis, we conclude that the approach in [12] allows us not only to obtain the same results as in [10,11] for the LR, but it allows us also to obtain closed form solutions for the amplitude and phase as well as for the stability analysis [13] and the transient behaviour using firstorder dynamical equations.…”

“…This is the case of injection-locked Colpitts oscillators [10], of ILFDs with feedthrough, and of CMOS differential ILFDs with direct injection [12,13]. The model used in [10] for a divide-by-2 Colpitts ILFD is shown in Figure 1 with a nonlinearity of cubic type; that is,…”

“…This phenomenon has recently been considered in a number of papers aimed at obtaining useful design guidelines for analog integrated dividers based on the injection locking of LC differential oscillators in MOS technology [3][4][5][6][7][8][9][10][11][12][13]. An ILFD operates properly as a frequency divider only if the basic LC oscillator tracks the input signal, and this occurs over a limited range of frequency, called locking range (LR); its prediction is the main issue of investigation.…”

An LC CMOS injection-locked frequency divider for divide-by-two and divide-by-three operation

“…ILFD circuits, particularly divide-by-2 prescalers, have been studied extensively in the literature [14]- [24]. LC CMOS oscillator circuits with multiple inductors was previously presented by Jang et al [25] and Lee et al [26].…”

Modeling, Analysis, and Experimental Validation of Frequency Dividers with Direct Injection

“…Among these, mention should be made of papers [9,10] where a comprehensive methodology is provided for the analysis of most commonly used topologies of ILFDs, namely, with injection through a tail device and with direct injection.…”