326 research outputs found

    Optimal Subharmonic Entrainment

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    For many natural and engineered systems, a central function or design goal is the synchronization of one or more rhythmic or oscillating processes to an external forcing signal, which may be periodic on a different time-scale from the actuated process. Such subharmonic synchrony, which is dynamically established when N control cycles occur for every M cycles of a forced oscillator, is referred to as N:M entrainment. In many applications, entrainment must be established in an optimal manner, for example by minimizing control energy or the transient time to phase locking. We present a theory for deriving inputs that establish subharmonic N:M entrainment of general nonlinear oscillators, or of collections of rhythmic dynamical units, while optimizing such objectives. Ordinary differential equation models of oscillating systems are reduced to phase variable representations, each of which consists of a natural frequency and phase response curve. Formal averaging and the calculus of variations are then applied to such reduced models in order to derive optimal subharmonic entrainment waveforms. The optimal entrainment of a canonical model for a spiking neuron is used to illustrate this approach, which is readily extended to arbitrary oscillating systems

    LOW-POWER FREQUENCY SYNTHESIS BASED ON INJECTION LOCKING

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    Ph.DDOCTOR OF PHILOSOPH

    Oscillator stabilization through feedback with slow wave structures

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    This article presents a new formulation to predict the steady-state, stability, and phase-noise properties of oscillator circuits, including either a self-injection network or a two-port feedback network for phase-noise reduction. The additional network contains a slow wave structure that stabilizes the oscillation signal. Its long delay inherently gives rise to multivalued solutions in some parameter intervals, which should be avoided for a reliable operation. Under a two-port feedback network, the circuit is formulated extracting two outer-tier admittance functions, which depend on the node-voltage amplitudes, phase shift between the two nodes, and excitation frequency. Then, the effect of the slow wave structure is predicted through an analytical formulation of the augmented oscillator, which depends on the numerical oscillator model and the structure admittance matrix. The solution curves are obtained in a straightforward manner by tracing a zero-error contour in the plane defined by the analysis parameter and the oscillation frequency. The impact of the slow-wave structure on the oscillator stability and noise properties is analyzed through a perturbation method, applied to the augmented oscillator. The phase-noise dependence on the group delay is investigated calculating the modulation of the oscillation carrier. The various analysis and design methods have been applied to an oscillator at 2.73 GHz, which has been manufactured and measured, obtaining phase-noise reductions of 13 dB, under a one-port load network, and 18 dB, under a feedback network.This work was supported by the Spanish Ministry of Economy ans Competitiveness through the European Regional Development Fund(ERDf)/ Fondo Europeo de Desarrollo Regional (FEDER) and under Project TEC2017-88242-C3-(1/2)-R

    Colored noise in oscillators. Phase-amplitude analysis and a method to avoid the Ito-Stratonovich dilemma

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    We investigate the effect of time-correlated noise on the phase fluctuations of nonlinear oscillators. The analysis is based on a methodology that transforms a system subject to colored noise, modeled as an Ornstein-Uhlenbeck process, into an equivalent system subject to white Gaussian noise. A description in terms of phase and amplitude deviation is given for the transformed system. Using stochastic averaging technique, the equations are reduced to a phase model that can be analyzed to characterize phase noise. We find that phase noise is a drift-diffusion process, with a noise-induced frequency shift related to the variance and to the correlation time of colored noise. The proposed approach improves the accuracy of previous phase reduced models

    Analysis and simulation methods for free-running, injection-locked and super-regenerative oscillators

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    RESUMEN: En los últimos años, muchos esfuerzos han sido dedicados al desarrollo de técnicas complementarias para el análisis de circuitos autónomos de microondas. Estas técnicas están pensadas para su uso en combinación con balance armónico, ampliamente usado para el análisis a frecuencias de microondas. De hecho, balance armónico sufre de restricciones cuando se utiliza para el análisis de circuitos autónomos, en su mayoría debidos a su falta de sensibilidad a las propiedades de estabilidad de la solución que se genera o se extingue mediante bifurcaciones. En esta tesis doctoral se presentan nuevos métodos de simulación y análisis para la caracterización y modelado de osciladores libres, sincronizados y superregenerativos. Todos los resultados obtenidos mediante los nuevos métodos de simulación y análisis han sido comparados satisfactoriamente con otras técnicas de simulación y con medidas.ABSTRACT: In the last years, numerous efforts have been devoted to the development of complementary analysis tools for autonomous microwave circuits. They are intended to be applied in combination with the harmonic-balance (HB) method, widely used at microwave frequencies. In fact, HB suffers from a number of shortcomings when dealing with autonomous circuits, mostly due the fact that it is insensitive to the stability properties of the solution, generated and extinguished through bifurcation phenomena. Here, new simulation and analysis methodologies for the characterization and modeling of free-running, injection-locked and super-regenerative oscillators have been proposed to overcome these problems when using commercial software. Results from the different new analysis methodologies have been successfully compared with independent simulations and with measurements

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

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    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

    Periodically Disturbed Oscillators

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    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
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