A new method for the determination of the locking range of oscillators

A time-domain method for the determination of the injection-locking range of oscillators is presented. The method involves three time dimensions: the first and the second are warped time scales used for the free-running frequency and the external excitation, respectively and the third is to account for slow transients to reach a steady-state regime. The locking range is determined by tuning the frequency of the external excitation until the oscillator locks. The locking condition is determined by analyzing the Jacobian matrix of the system. The method is advantageous in that the computational effort is independent of the presence of widely separated time constants in the oscillator. Numerical results for a Van Der Pol oscillator are presented

Periodically Disturbed Oscillators

By controlling the timing of events and enabling the transmission of data over long distances, oscillators can be considered to generate the "heartbeat" of modern electronic systems. Their utility, however, is boosted significantly by their peculiar ability to synchronize to external signals that are themselves periodic in time. Although this fascinating phenomenon has been studied by scientists since the 1600s, models for describing this behavior have seen a disconnect between the rigorous, methodical approaches taken by mathematicians and the design-oriented, physically-based analyses carried out by engineers. While the analytical power of the former is often concealed by an inundation of abstract mathematical machinery, the accuracy and generality of the latter are constrained by the empirical nature of the ensuing derivations. We hope to bridge that gap here. In this thesis, a general theory of electrical oscillators under the influence of a periodic injection is developed from first principles. Our approach leads to a fundamental yet intuitive understanding of the process by which oscillators lock to a periodic injection, as well as what happens when synchronization fails and the oscillator is instead injection pulled. By considering the autonomous and periodically time-varying nature that underlies all oscillators, we build a time-synchronous model that is valid for oscillators of any topology and periodic disturbances of any shape. A single first-order differential equation is shown to be capable of making accurate, quantitative predictions about a wide array of properties of periodically disturbed oscillators: the range of injection frequencies for which synchronization occurs, the phase difference between the injection and the oscillator under lock, stable vs. unstable modes of locking, the pull-in process toward lock, the dynamics of injection pulling, as well as phase noise in both free-running and injection-locked oscillators. The framework also naturally accommodates superharmonic injection-locked frequency division, subharmonic injection-locked frequency multiplication, and the general case of an arbitrary rational relationship between the injection and oscillation frequencies. A number of novel insights for improving the performance of systems that utilize injection locking are also elucidated. In particular, we explore how both the injection waveform and the oscillator's design can be modified to optimize the lock range. The resultant design techniques are employed in the implementation of a dual-moduli prescaler for frequency synthesis applications which features low power consumption, a wide operating range, and a small chip area. For the commonly used inductor-capacitor (LC) oscillator, we make a simple modification to our framework that takes the oscillation amplitude into account, greatly enhancing the model's accuracy for large injections. The augmented theory uniquely captures the asymmetry of the lock range as well as the distinct characteristics exhibited by different types of LC oscillators. Existing injection locking and pulling theories in the available literature are subsumed as special cases of our model. It is important to note that even though the veracity of our theoretical predictions degrades as the size of the injection grows due to our framework's linearization with respect to the disturbance, our model's validity across a broad range of practical injection strengths are borne out by simulations and measurements on a diverse collection of integrated LC, ring, and relaxation oscillators. Lastly, we also present a phasor-based analysis of LC and ring oscillators which yields a novel perspective into how the injection current interacts with the oscillator's core nonlinearity to facilitate injection locking.</p

Phase Noise Reduction in an Oscillator Using Harmonic Mixing Technique

This thesis presents a new technique of injection of multi-harmonics into an oscillator, study the behavior and response of the oscillator. First general locking equations are derived using the feedback model of an oscillator. Second, using the Laplace domain modeled phase noise analysis giving insight into the phase noise model of an oscillator under multiple harmonics are analytically studied and experimentally validated with good agreement. From the locking process and phase noise analysis we can observe when compared to single tone injection there is a substantial improvement of phase noise under multi-harmonic injection, enhancement in the locking range. We present Injection Locked Frequency Divider (ILFD) circuit under multi-harmonic injection (2nd and 3rd harmonics injected) with simulation, theoretical validation. The proposed circuit is designed using the UMC library by using Op-Amp based feedback current reference circuit. The amplitude dependency of the injection signals on the phase noise is observed as increase in the injection levels decreases the phase noise

Mathematical Modeling of Electronic Systems: From Oscillators to Multipliers

The ubiquity of electronics in modern technology is undeniable. Although it is not feasible to design or analyze circuits in an exhaustively detailed fashion, it is still imperative that circuit design engineers understand the pertinent physical tradeoffs and are able to think at the appropriate level of mathematical abstraction. This thesis presents several mathematical modeling techniques of common electronic systems. First, we derive, ab initio, a general analytical model for the behavior of electrical oscillators under injection without making any assumptions about the type of oscillator or the size or shape of the injection. This model provides novel insights into the phenomena of injection locking and pulling while subsuming existing theories found in the literature. Next, we focus on the familiar scenario of an inductor-capacitor (LC) oscillator locked to a sinusoidal signal. An exact analysis of this circuit is carried out for an arbitrary injection strength and frequency, a task which has not been executed to fruition in the existing literature. This analysis intuitively illuminates the fundamental physics underlying the synchronization of electrical harmonic oscillators, and it generalizes the notion of the lock range for such oscillators into separate necessary and sufficient conditions. We then turn to the classical estimate of the bandwidth of a linear time-invariant (LTI) system via the sum of its zero-value time constants (ZVTs), and we show that this sum can actually be used to tightly bound the bandwidth—both from above and from below—in addition to simply estimating it. Finally, we look at a natural generalization of the Gilbert cell topology: an analog multiplier for an arbitrary number of inputs; we then analyze its large- and small-signal characteristics as well as its frequency response. Throughout, we will demonstrate how infusing physical intuition with mathematical rigor whilst seeking a balance between detailed analysis and abstract modularity results in models that are conceptually insightful, sufficiently accurate, and computationally feasible.</p

Injection locked ring oscillator design for application in Direct Time of Flight LIDAR

Diplomová práce přibližuje systémy LIDAR přímo měřící čas průletu a časově digitální převodníky určené k použití v těchto systémech. Představuje problematiku distribuce hodinových signálů napříč soubory časově digitálních převodníků v LIDAR systémech a věnuje se jednomu z nových řešení této problematiky, které je založené na injekcí zavěšených oscilátorech. Technika injekčního zavěšení oscilátorů je důkladně matematicky popsána. V programu Matlab byl vytvořen simulační model injekcí zavěšeného kruhového oscilátoru, který potvrzuje správnost uvedených analytických predikcí. Ve výrobní technologii ONK65 byl navržen injekcí zavěšený kruhový oscilátor stabilizovaný pomocí smyčky závěsu zpoždění, určený pro implementaci časově digitálního převodníku pro systém LIDAR. Navržený injekcí zavěšený kruhový oscilátor byl verifikován počítačovými simulacemi zohledňujícími vliv procesních, napěťových i teplotních variací. Oscilátor poskytuje specifikované časové rozlišení 50 pikosekund a dosahuje dvakrát nižší hodnoty fázového neklidu než ekvivalentní volnoběžný oscilátor v dané technologii.The diploma thesis provides an introduction to Direct Time of Flight LIDAR systems and Time to Digital Converters used in these systems. It discusses the problem of clock distribution in LIDAR Time to Digital Converter arrays, and examines one of the possible solutions to this problem based on injection locked oscillators. The injection locking phenomenon is thoroughly mathematically described and a Matlab model of an injection locked ring oscillator is presented, confirming the analytic predictions. In ONK65 processing technology, an injection locked ring oscillator biased by a delay locked loop meant specifically for application in Time to Digital Converters for LIDAR systems has been designed. The designed oscillator has been verified by computer simulations taking process, voltage and temperature variations into account and offers specified time resolution of 50 picosecond as well as two times less clock jitter than an equivalent free-running oscillator in the given processing technology.

Dynamic synchronization of sympathetic oscillators

Synchronous activity of single postganglionic sympathetic neurones (PGNs) underlies rhythmical or semi-rhythmical burst discharges recorded from peripheral sympathetic nerves. It is still controversial whether this rhythmicity is generated by an autonomous sympathetic oscillator. Previous studies have demonstrated that activity of single PGNs innervating the caudal ventral artery (CVA) of the rat's tail has a dominant rhythm (T-rhythm). The frequency of T- rhythm is different from the cardiac frequency and can be different from those of ventilatory and respiratory rhythms, suggesting that T-rhythm is generated by an oscillator independent of periodic drives originating from the arterial baroreceptors, the ventilation afferents and the respiratory network. Using the rat's tail circulation as a model, the purpose of the present study is: 1) to determine whether activity from different single PGNs is generated by multiple oscillators. 2) to establish whether synchronization of single PGNs is an obligatory feature and if not, how it is regulated. 3) to determine whether periodically driven single PGN oscillators exhibit dynamics as predicted by the theory of nonlinear coupled oscillators. 4) to explain the discharge behaviour of whole nerve activity based on the findings at single PGN level. The experiments were conducted in anaesthetized Sprague-Dawley rats. Population PGN activity was recorded from the ventral collector nerve (VCN) of the tail. Single PGN activity was recorded focally from the surface of the CVA. The interaction between two single PGNs was studied by recording two units simultaneously. The discharge behaviours of PGNs in response to a periodic input were studied using the central respiratory drive (CRD) and lung-inflation cycle (LlC)-related activity as the driving forces. The findings from the present study suggest that: 1) Activity of CVA PGNs is generated by multiple oscillators independent of CRD, LIC-related activity and cardiac activity. 2) The multiple PGN oscillators are capable of dynamic synchronization. 3) When subjected to frequency changes of LICs, single PGNs exhibit dynamics, such as 1:1 entrainment, relative coordination, high order rational frequency-lock, asynchrony, characterising nonlinear coupled oscillators. 4) Population PGN activity should be considered as output activity from a pool of dynamically interactive multiple oscillators rather than that from a single oscillator

LOW-POWER FREQUENCY SYNTHESIS BASED ON INJECTION LOCKING

Ph.DDOCTOR OF PHILOSOPH

Computationally Efficient Innovative Techniques for the Design-Oriented Simulation of Free-Running and Driven Microwave Oscillators

Analysis techniques for injection-locked oscillators/amplifiers (ILO) can be broadly divided into two classes. To the first class belong methods with a strong and rigorous theoretical basis, that can be applied to rather general circuits/systems but which are very cumbersome and/or time-consuming to apply. To the second class belong methods which are very simple and fast to apply, but either lack of validity/accuracy or are applicable only to very simple or particular cases. In this thesis, a novel method is proposed which aims at combining the rigorousness and broad applicability characterizing the first class of analysis techniques above cited with the simplicity and computational efficiency of the second class. The method relies in the combination of perturbation-refined techniques with a fundamental frequency system approach in the dynamical complex envelope domain. This permits to derive an approximate, but first-order exact, differential model of the phase-locked system useable for the steady-state, transient and stability analysis of ILOs belonging to the rather broad (and rigorously identified) class of nonlinear oscillators considered. The hybrid (analytical-numerical) nature of the formulation developed is suited for coping with all ILO design steps, from initial dimensioning (exploiting, e.g., the simplified semi-analytical expressions stemming from a low-level injection operation assumption) to accurate prediction (and fine-tuning, if required) of critical performances under high-injection signal operation. The proposed application examples, covering realistically modeled low- and high-order ILOs of both reflection and transmission type, illustrate the importance of having at one's disposal a simulation/design tool fully accounting for the deviation observed, appreciable for instance in the locking bandwidth of high-frequency circuits with respect to the simplified treatments usually applied, for a quick arrangement, in ILO design optimization procedures.Analysis techniques for injection-locked oscillators/amplifiers (ILO) can be broadly divided into two classes. To the first class belong methods with a strong and rigorous theoretical basis, that can be applied to rather general circuits/systems but which are very cumbersome and/or time-consuming to apply. To the second class belong methods which are very simple and fast to apply, but either lack of validity/accuracy or are applicable only to very simple or particular cases. In this thesis, a novel method is proposed which aims at combining the rigorousness and broad applicability characterizing the first class of analysis techniques above cited with the simplicity and computational efficiency of the second class. The method relies in the combination of perturbation-refined techniques with a fundamental frequency system approach in the dynamical complex envelope domain. This permits to derive an approximate, but first-order exact, differential model of the phase-locked system useable for the steady-state, transient and stability analysis of ILOs belonging to the rather broad (and rigorously identified) class of nonlinear oscillators considered. The hybrid (analytical-numerical) nature of the formulation developed is suited for coping with all ILO design steps, from initial dimensioning (exploiting, e.g., the simplified semi-analytical expressions stemming from a low-level injection operation assumption) to accurate prediction (and fine-tuning, if required) of critical performances under high-injection signal operation. The proposed application examples, covering realistically modeled low- and high-order ILOs of both reflection and transmission type, illustrate the importance of having at one's disposal a simulation/design tool fully accounting for the deviation observed, appreciable for instance in the locking bandwidth of high-frequency circuits with respect to the simplified treatments usually applied, for a quick arrangement, in ILO design optimization procedures