**Evaluating the Spectrum of Unlocked Injection Frequency Dividers in Pulling Mode**

### **Antonio Buonomo \* and Alessandro Lo Schiavo**

Seconda Università di Napoli, Via Roma 29, Aversa (CE) 81031, Italy; E-Mail: alessandro.loschiavo@unina2.it

**\*** Author to whom correspondence should be addressed; E-Mail: antonio.buonomo@unina2.it; Tel.: +39-081-5010222; Fax: +39-081-5037042.

*Received: 12 August 2013; in revised form: 17 September 2013 / Accepted: 18 September 2013/ Published: 25 September 2013* 

**Abstract:** We study the phenomenon of periodic pulling which occurs in certain integrated microcircuits of relevant interest in applications, namely the injection-locked frequency dividers (ILFDs). They are modelled as second-order driven oscillators working in the subharmonic (secondary) resonance regime, *i.e.*, when the selfoscillating frequency is close (resonant) to an integer submultiple *n* of the driving frequency. Under the assumption of weak injection, we find the spectrum of the system's oscillatory response in the unlocked mode through closed-form expressions, showing that such spectrum is double-sided and asymmetric, unlike the single-sided spectrum of systems with primary resonance (*n*  1). An analytical expression for the amplitude modulation of the oscillatory response is also presented. Numerical results are presented to support theoretical relations derived.

**Keywords:** Injection pulling; analog frequency dividers; injection-locked frequency dividers (ILFDs); nonlinear oscillators; synchronization; averaging method

### **1. Introduction**

It is known that periodic pulling (or frequency pulling) is a general phenomenon that happens in any system involving the injection locking of self-sustained oscillations when the frequency of the periodic forcing is just outside the locking region (Arnold's tongue) [1–3]. The occurrence of the periodic pulling is easily recognized by the characteristic aspect of the pulled oscillations, usually called beats, which exhibit a simultaneous modulation of amplitude and frequency with a pulse-like envelope of the amplitude. A theoretical investigation of the oscillatory response in the pulling mode of driven oscillators is given in a number of papers (see [1–12] and references therein) starting from the pioneering investigation of Rjasin [6], who first performed a harmonic analysis of beats to establish the spectral composition. Later on, an approximate, but physically insightful, treatment of the pulling was given in a celebrated paper of Adler [7], who obtained an analytical expression for the phase difference between the forcing and the system response neglecting the amplitude modulation. Based on that approximation, valid for the regime of so-called weak injections, the spectrum of beats was derived analytically many years later by Armand [9], by using an appealing method as simple as effective.

The features of the spectrum of beats have therefore already been known for a long time and can be summarized as follows: unlike the single-line spectrum in a locked mode, or the two-lines spectrum in a quasi-periodic mode far from the locking region, in a pulling mode the spectrum has a single sideband and is spread over many frequencies, starting from the free-running frequency, in the opposite side to that of the injected frequency. This result [9] has been reported in literature to explain experimental observations of pulling in microwave solid-state oscillators [10], in a unijunction transistor based oscillator [11], and in many papers dealing with the study of plasma instabilities and with periodically driven oscillating plasma systems (see [11,12], and references therein). These systems are well modeled by the van der Pol equation and exhibit a variety of dynamical phenomena observed in forced oscillators of van der Pol type [13–15]. In particular, mode locking and periodic pulling, bifurcations between quasi-periodic and frequency entrained states have been observed, as well as period-doubling bifurcations as a route to deterministic chaos [12], for which the study of chaotic dynamics and the derivation of lower bounds on their topological entropy is yet an attractive problem [13–17].

The pulling is observable in many electronic systems containing on-chip differential LC oscillators, and its occurrence is generally undesirable and harmful [18,19]. It is produced as a consequence of the unavoidable coupling of parts of the circuit, through the supply and the common substrate, or through parasitic paths [18,19]. It can therefore happen that an oscillator is subject to the action of an undesired periodic signal and, depending on its frequency, can operate in a locked-mode or in a pulling mode. Attempts to analytically calculate the simultaneous amplitude and frequency modulation in the pulling modes were recently made in [20,21] in the more simple case that the driving frequency is close to the self-oscillating frequency (primary resonance). The pulling phenomenon in injection-locked frequency dividers (ILFDs) is even more worrying, and its onset is to be avoided for a proper circuit operation as a divider. This imposes from one hand a reliable prediction of the locking range [22,23] and, on the other hand, a thorough understanding of the spectral properties of the oscillatory response during the pulling to avoid its effects. However, as far as is known to the authors, the pulling phenomenon in the frequency dividers, which operate in subharmonic resonance regime (secondary resonance), has never been investigated and some facets of the phenomenon yet are not known.

The present paper is devoted to the study of the pulling in subharmonic resonant systems, which is not only of theoretical but also of practical interest. By widening the analysis method in [9], we derive an analytical procedure for finding the spectral components of the unlocked oscillation in the pulling mode of injection-locked frequency dividers. The procedure is simple and straightforward, and allows us to calculate such components in the form of series taking into

account both the amplitude and frequency modulation of the unlocked oscillation. We show that the power spectrum of the unlocked signal in the pulling mode is double-sided, and asymmetric, with respect to the natural frequency of the free-running oscillator, in contrast to the single-sided spectrum of systems with primary resonance [20,21]. Numerical results are presented to support theoretical relations derived.
