Cycle slipping in nonlinear circuits under periodic nonlinearities and time delays

Phase-locked loops (PLL), Costas loops and other synchronizing circuits are featured by the presence of a nonlinear phase detector, described by a periodic nonlinearity. In general, nonlinearities can cause complex behavior of the system such as multi-stability and chaos. Even if the phase locking is guaranteed for any initial conditions, the transient behavior of the circuit can still be unsatisfactory due to the cycle slipping. Growth of the phase error caused by cycle slipping is undesirable, leading e.g. to demodulation and decoding errors. This makes the problem of estimating the phase error oscillations and number of slipped cycles in nonlinear PLL-based circuits extremely important for modern telecommunications. Most mathematical results in this direction, available in the literature, focus on the phase jitter and cycle slipping under random noise and examine the relations between the probabilistic characteristics of the noise and of the phase error, e.g. the expected number of slipped cycles. At the same time, cycle slipping occurs also in deterministic systems with periodic nonlinearities, depending on the initial conditions, properties of the linear part and the periodic nonlinearity and other factors such as delays in the loop. In the present paper we give analytic estimates for the number of slipped cycles in PLL-based systems, governed by integro-differential equations, allowing to capture effects of high-order dynamics, discrete and distributed delays. We also consider the effects of singular small-parameter perturbations on the cycle slipping behavior

Variable domain transformation for linear PAC analysis of mixed-signal systems

This paper describes a method to perform linear AC analysis on mixed-signal systems which appear strongly nonlinear in the voltage domain but are linear in other variable domains. Common circuits like phase/delay-locked loops and duty-cycle correctors fall into this category, since they are designed to be linear with respect to phases, delays, and duty-cycles of the input and output clocks, respectively. The method uses variable domain translators to change the variables to which the AC perturbation is applied and from which the AC response is measured. By utilizing the efficient periodic AC (PAC) analysis available in commercial RF simulators, the circuit’s linear transfer function in the desired variable domain can be characterized without relying on extensive transient simulations. Furthermore, the variable domain translators enable the circuits to be macromodeled as weakly-nonlinear systems in the chosen domain and then converted to voltage-domain models, instead of being modeled as strongly-nonlinear systems directly

Constructive Estimates of the Pull-In Range for Synchronization Circuit Described by Integro-Differential Equations

The pull-in range, known also as the acquisition or capture range, is an important characteristics of synchronization circuits such as e.g. phase-, frequency- and delay-locked loops (PLL/FLL/DLL). For PLLs, the pull-in range characterizes the maximal frequency detuning under which the system provides phase locking (mathematically, every solution of the system converges to one of the equilibria). The presence of periodic nonlinearities (characteristics of phase detectors) and infinite sequences of equilibria makes rigorous analysis of PLLs very difficult in spite of their seeming simplicity. The models of PLLs can be featured by multi-stability, hidden attractors and even chaotic trajectories. For this reason, the pull-in range is typically estimated numerically by e.g. using harmonic balance or Galerkin approximations. Analytic results presented in the literature are not numerous and primarily deal with ordinary differential equations. In this paper, we propose an analytic method for pull-in range estimation, applicable to synchronization systems with infinite-dimensional linear part, in particular, for PLLs with delays. The results are illustrated by analysis of a PLL described by second-order delay equations

Self-oscillation

Physicists are very familiar with forced and parametric resonance, but usually not with self-oscillation, a property of certain dynamical systems that gives rise to a great variety of vibrations, both useful and destructive. In a self-oscillator, the driving force is controlled by the oscillation itself so that it acts in phase with the velocity, causing a negative damping that feeds energy into the vibration: no external rate needs to be adjusted to the resonant frequency. The famous collapse of the Tacoma Narrows bridge in 1940, often attributed by introductory physics texts to forced resonance, was actually a self-oscillation, as was the swaying of the London Millennium Footbridge in 2000. Clocks are self-oscillators, as are bowed and wind musical instruments. The heart is a "relaxation oscillator," i.e., a non-sinusoidal self-oscillator whose period is determined by sudden, nonlinear switching at thresholds. We review the general criterion that determines whether a linear system can self-oscillate. We then describe the limiting cycles of the simplest nonlinear self-oscillators, as well as the ability of two or more coupled self-oscillators to become spontaneously synchronized ("entrained"). We characterize the operation of motors as self-oscillation and prove a theorem about their limit efficiency, of which Carnot's theorem for heat engines appears as a special case. We briefly discuss how self-oscillation applies to servomechanisms, Cepheid variable stars, lasers, and the macroeconomic business cycle, among other applications. Our emphasis throughout is on the energetics of self-oscillation, often neglected by the literature on nonlinear dynamical systems.Comment: 68 pages, 33 figures. v4: Typos fixed and other minor adjustments. To appear in Physics Report

The sunflower equation: novel stability criteria

In this paper, we consider a delayed counterpart of the mathematical pendulum model that is termed sunflower equation and originally was proposed to describe a helical motion (circumnutation) of the apex of the sunflower plant. The “culprits” of this motion are, on one hand, the gravity and, on the other hand, the hormonal processes within the plant, namely, the lateral transport of the growth hormone auxin. The first mathematical analysis of the sunflower equation was conducted in the seminal work by Somolinos (1978) who gave, in particular, a sufficient condition for the solutions’ boundedness and for the existence of a periodic orbit. Although more than 40 years have passed since the publication of the work by Somolinos, the sunflower equation is still far from being thoroughly studied. It is known that a periodic solution may exist only for a sufficiently large delay, whereas for small delays the equation exhibits the same qualitative behavior as a conventional pendulum, and every solution converges to one of the equilibria. However, necessary and sufficient conditions for the stability of the sunflower equation (ensuring the convergence of all solutions) are still elusive. In this paper, we derive a novel condition for its stability, which is based on absolute stability theory of integro-differential pendulum-like systems developed in our previous work. As will be discussed, our estimate for the maximal delay, under which the stability can be guaranteed, improves the existing estimates and appears to be very tight for some values of the parameters

Nonisochronous Oscillations in Piezoelectric Nanomechanical Resonators

Nanoelectromechanical systems (NEMS) have proven an excellent test bed for exploring nonlinear dynamics due to short decay times, weak nonlinearities, and large quality factors. In contrast to previous research in nonlinear dynamics involving driven or phase fixed NEMS, where time is referenced by an external source, we describe phenomena classified by phase free phenomena. Here we describe NEMS embedded into feedback oscillators with weak nonlinearities. We make measurements of this mechanical nonlinearity by developing a transduction scheme, the piezoelectric/piezoresistive (PZE/PZR) transduction, which emphasizes the detector dynamic range over absolute sensitivity. Using these measurements, projections on quantum nondemolition schemes involving the mechanical nonlinearity as a detector are made. These measurements also are important for understanding the detection limits of NEMS sensor technology, which uses a mechanical resonator as a frequency reference in a phase locked loop (PLL). This work identifies ways to reduce noise within ‘nonlinear’ feedback oscillators, and these results have implications for sensing systems using nonlinear mechanical resonators embedded in PLLs. Since the mechanical nonlinearity of PZE/PZR resonators can be accurately calibrated, we make predictions for the behavior of these dynamical systems based on the given mechanical and electrical parameters. We show, theoretically, that local isochronicity above critical nonlinear amplitudes can create special operating points in feedback oscillators at which parametric fluctuations may cause less phase noise in the oscillator than in feedback oscillators driven below critical amplitudes. For these predictions, we present data that show quantitative agreement for the amplitude and frequency, and qualitative agreement for the phase noise. Finally, we show synchronization, assisted by nonisochronicity, between two feedback NEMS oscillators. We develop a general theoretical framework for two saturated feedback oscillators which use resonators with nonlinear stiffness. In the limit of small coupling, we show that the system obeys the Adler equation with analytical predictions for the oscillators’ individual amplitudes and net frequency difference. We develop an experiment in which the three important parameters of the system (detuning, nonisochronicity, and coupling) can be tuned, and show data that agrees with the predictions for a large range of coupling. We include data on phase slipping between two oscillators in which the aperiodic frequency difference is clearly observed. Finally, we present data on phase noise in synchronized oscillators.</p