On the Difference between Two Widely Publicized Methods for Analyzing Oscillator Phase Behavior Piet Vanassche, Georges Gielen and Willy Sansen Katholieke Universiteit Leuven - ESAT/MICAS Kasteelpark Arenberg 10, B-3001 Leuven, Belgium piet.vanassche@esat.kuleuven.ac.be ABSTRACT This paper describes the similarities and differences between two widely publicized methods for analyzing oscillator phase behav- ior. The methods were presented in [3] and [6]. It is pointed out that both methods are almost alike. While the one in [3] can be shown to be, mathematically, more exact, the approximate method in [6] is somewhat simpler, facilitating its use for purposes of anal- ysis and design. In this paper, we show that, for stationary input noise sources, both methods produce equal results for the oscilla- tor’s phase noise behavior. However, when considering injection locking, it is shown that both methods yield different results, with the approximation in [6] being unable to predict the locking behav- ior. In general, when the input signal causing the oscillator phase perturbations is non-stationary, the exact model produces the cor- rect results while results obtained using approximate model break down. 1. INTRODUCTION Oscillators are key building blocks in almost all of today’s com- munication systems. Their behavior, however, is often hard to an- alyze, since their functioning inherently relies upon nonlinear be- havior. One of the most important characteristics of an oscillator is the way its phase responds to external signals. These external sig- nals could be both unwanted, e.g. noise sources causing the phase noise, or wanted, e.g. sine waves injected for locking purposes. In recent years, much research has been devoted to the analysis of oscillator phase behavior. Circuit simulation [10] offers the most simple solution. This approach, however, is time-consuming, espe- cially for the Monte Carlo methods needed to deal with noisy in- puts, and the results are not straightforward to interpret, obscuring analysis. More compact and insightful methods have been devel- oped in both [3, 4, 8] and [6, 9]. Both, quite popular, approaches model the oscillator phase behavior using a 1-dimensional differ- ential or integral equation, which is much easier to solve than the full set of circuit equations. Using the original notation, [3] models the phase noise behavior as d θ dt (t ) = ǫv(t + θ(t )) n(t ) (1) while [6] starts from θ(t ) = ǫ  t 0 Ŵ(τ)n(τ)d τ (2) In both equations, n(t ) is an external source, noise or otherwise, θ(t ) is the oscillator phase and v(t ),Ŵ(t ) are functions depending upon the oscillator’s topology. They are respectively called the per- turbation projection vector (PPV) and the impulse sensitivity func- tion (ISF). The oscillator’s output is then determined by V osc (t ) = V s (t + θ(t )) (3) where V osc (t ) represents the actual oscillator output signal while V s (t ) is a T -periodic solution of the input-free (noiseless) oscilla- tor. Furthermore, ǫ ≪ 1 is a perturbation variable used to indicate the fact that θ(t ) varies slowly as compared to the oscillator period T . Observing both equations (1) and (2), it is seen that, essentially, they differ only slightly from each other. The question hence rises whether one can expect any significant differences in results when comparing the phase behaviors they predict. In this paper, we show that, for some classes of applications, the models (1) and (2) predict similar results, while for other classes, results are widely different. More precisely, it is shown that for n(t ) a stationary (noise) source, equations (1) and (2) will, up to 0-th order in ǫ , predict the same output phase noise. On the other hand, when n(t ) is no longer stationary, results diverge. A notewor- thy example is given by an oscillator’s injection locking behavior [1, 7]. Here, the input source n(t ) = N cos (2π ft ) is a single sine wave with f near the oscillator’s free-running frequency f 0 . A har- monic oscillator, for example, will lock both its frequency and its phase to that of n(t ). It will be shown that the model (1) is capable of predicting this behavior, while (2) is not. Related to injection locking is the behavior of the phase differences θ within sets of coupled oscillators. Since the coupling effect can be considered as a mutual injection phenomenon, (1) yields correct results while (2) breaks down. The main tool used for obtaining the results mentioned above is the averaging transformation as introduced in [2, 5]. In this paper, we extend this transformation to its most general setting, allowing us to deal with both deterministic signals, white noise and colored noise. Using the averaging transformation, it becomes possible to separate the slow-varying components of the oscillator’s phase be- havior from the fast-varying ones. These slow-varying components typically contain those characteristics of the oscillator’s behavior which are of greatest interest, like phase noise (wander) and lock- ing. The remainder of this paper is organized as follows. In section 2, we briefly discuss both the origin and properties of both models (1) and (2) for describing oscillator phase behavior. Section 3 in- troduces averaging and its use for analyzing an oscillator’s phase behavior. In section 4, we apply this principle to analyze oscilla- tor phase noise, showing that, under certain conditions, the results obtained from both model equations (1) and (2) are equivalent. Sec- tion 5 discusses the injection locking phenomenon and shows how both models predict widely different results. Finally, in section 6, we demonstrate these differences with some circuit-level simula- tions. Conclusions are presented in section 7. 0-7803-7607-2/02/$17.00 ©2002 IEEE