Investigation on Locking and Pulling Modes in Analog Frequency Dividers

We compare the main analytical results available to estimate the locking range, which is the key figure-of-merit of LC frequency dividers based on the injection locking phenomenon. Starting from the classical result by Adler concerning injection-locked oscillators, we elucidate the merits and the shortcomings of the different approaches to study injection-locked frequency dividers, with particular emphasis on divider-by-2. In particular, we show the potential of a perturbation approach which enables a more complete analysis of frequency dividers, making it possible to calculate not only the amplitude and the phase of the locked oscillation, but also the region where it exists and is stable, which defines the locking region. Finally, we analyze the dynamical behaviour of the dividers in the vicinity of the boundary of the locking region, showing that there exists a border region where the occurrence of the locking or the pulling operation mode is possible, depending on the initial conditions of the system. 1. Introduction The phenomenon of injection locking [1], or frequency entrainment, of an oscillator through an external signal underlies the operation of injection-locked frequency dividers (ILFDs), which are nowadays realized on-chip, in a number of ways suited for RF integrated circuits. The self-oscillation that characterizes the operation of ILFDs is responsible for the low power consumption, which makes them a valuable alternative to digital frequency dividers, and to Miller’s type dividers, in high-frequency low-power applications. The first, and more known, formulations of the injection locking phenomenon date back to Van der Pol and to a bright intuition of Adler, who obtained results very useful for applications by using a pragmatic approach in the study of a particular vacuum tube oscillator [2]. 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 [3–13]. 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. Predicting the LR can be accurately performed through numerical simulations [14], but analytical methods based on approximate models are able to provide a better insight into the synchronization mechanism which is useful for their design. As a lot of formulas for predicting the locking range as