The Complex World of Oscillator Noise: Modern Approaches to Oscillator (Phase and Amplitude) Noise Analysis

The study of fluctuations in oscillators has been a classical research topic in mathematics, physics, and engineering since the first half of the 20th century [1]-[4]. Besides the intellectual fascination for mathematically difficult problems, the importance of the topic is deeply rooted in practical applications, mainly in the fields of RF and microwave electronics and also telecommunications. In fact, defining a precise frequency reference is fundamental for many applications, both electrical (e.g., transmitters and receivers) and optical (e.g., lasers). The broadening of the generated spectral line is mainly due to the phase-noise component of oscillator fluctuations, which consequently is the most commonly studied feature of oscillator noise (see [5] for a recent and exhaustive review). In a dual perspective, the definition of a precise time reference is also extremely important for digital applications, thus implying the necessity to keep under control the time jitter in clocked and in sampled systems. From a theoretical standpoint, phase noise and time jitter are simply two sides of the same coin, a manifestation of the oscillator's noisy behavior. As the microwave engineer is more often interested in the phase-noise characterization, we discuss only the latter. The time-jitter estimation is discussed, for instance, in [6]

Generalized nonlinear timing/phase macromodeling: Theory, numerical methods and applications

Abstract—We extend the concept of timing/phase macromodels, pre-viously established rigorously only for oscillators, to apply to gen-eral systems, both non-oscillatory and oscillatory. We do so by first establishing a solid foundation for the timing/phase response of any nonlinear dynamical system, then deriving a timing/phase macromodel via nonlinear perturbation analysis. The macromodel that emerges is a scalar, nonlinear time-varying equation that accurately characterizes the system’s phase/timing responses. We establish strong links of this technique with projection frameworks for model order reduction. We then present numerical methods to compute the phase model. The computation involves a full Floquet decomposition – we discuss numerical issues that arise if direct computation of the monodromy matrix is used for Floquet analysis, and propose an alternative method that are numerically superior. The new method has elegant connections to the Jacobian matrix in harmonic balance method (readily available in most RF simulators). We validate the technique on several highly nonlinear systems, in-cluding an inverter chain and a firing neuron. We demonstrate that the new scalar nonlinear phase model captures phase responses under various types of input perturbations, achieving accuracies consider-ably superior to those of reduced models obtained using LTI/LPTV MOR methods. Thus, we establish a powerful new way to extract timing models of combinatorial/sequential systems and memory (e.g., SRAMs/DRAMs), synchronization systems based on oscillator enslaving (e.g., PLLs, injection-locked oscillators, CDR systems, neural processing, energy grids), signal-processing blocks (e.g., ADCs/DACs, FIR/IIR filters), etc.. I

A modified multiphase oscillator with improved phase noise performance

This paper investigates the factors that influence the phase noise performance of an oscillator and proposes a modified structure for improved phase noise performance. A single and multiphase oscillator analysis using the harmonic balance method is presented. The modified structure increases the oscillation amplitude without increasing the bias current and leads to improved phase noise performance as well as decreased power consumption. The modification is analyzed and the figure of merit of the oscillator shows a significant improvement of 21 dB. Numerical and analytical solutions are presented to predict the oscillation frequency and phase noise. The analytical solution is used to approximate the first harmonic and can be combined with numerical simulations to extrapolate phase noise performance.The measurements relating to this work were enabled through the support of SAAB Electronic Defence Systems (EDS). Funding was also received from the National Research Foundation (NRF), Department of Science and Technology, South Africa. NRF funding was for measurement equipment – a millimeter-wave vector network analyzer (under grant ID: 72321) and wafer-prober (under grant ID: 78580). NRF funding (under grant ID: 72321) also allowed collaboration with Prof Luca Larcher, Università degli studi di Modena e Reggio Emilia, Italy.http://www.elsevier.com/locate/mejo2018-04-30am2017Electrical, Electronic and Computer Engineerin

Event-Driven Simulation Methodology for Analog/Mixed-Signal Systems

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2015. 8. 김재하.Recent system-on-chip's (SoCs) are composed of tightly coupled analog and digital components. The resulting mixed-signal systems call for efficient system-level behavioral simulators for fast and systematic verifications. As the system-level verifications rely heavily on digital verification tools, it is desirable to build the mixed-signal simulator based on a digital simulator. However, the existing solutions in digital simulators suffer from a trade-off between simulation speed and accuracy. This work breaks down the trade-off and realizes a fast and accurate analog/mixed-signal behavior simulation in a digital simulator SystemVerilog. The main difference of the proposed methodology from existing ones is its way of representing continuous-time signals. Specifically, a clock signal expresses accurate timing information by carrying an additional real-value time offset, and an analog signal represents its continuous-time waveform in a functional form by employing a set of coefficients. With these signal representations, the proposed method accurately simulates mixed-signal behaviors independently of a simulator's time-step and achieves a purely event-driven simulation without involving any numerical iteration. The speed and accuracy of the proposed methodology are examined for various types of analog/mixed-signal systems. First, timing-sensitive circuits (a phase-locked loops and a clock and data recovery loop) and linear analog circuits (a channel and linear equalizers) are simulated in a high-speed I/O interface example. Second, a switched-linear-behavior simulation is demonstrated through switching power supplies, such as a boost converter and a switched-capacitor converter. Additionally, the proposed method is applied to weakly nonlinear behaviors modeled with a Volterra series for an RF power amplifier and a high-speed I/O linear equalizer. Furthermore, the nonlinear behavior simulation is extended to three different types of injection-locked oscillators exhibiting time-varying nonlinear behaviors. The experimental results show that the proposed simulation methodology achieved tens to hundreds of speed-ups while maintaining the same accuracy as commercial analog simulators.ABSTRACT I CONTENTS III LIST OF FIGURES V LIST OF TABLES XII CHAPTER 1 INTRODUCTION 1 1.1 BACKGROUND 1 1.2 MAIN CONTRIBUTION 6 1.3 THESIS ORGANIZATION 8 CHAPTER 2 EVENT-DRIVEN SIMULATION OF ANALOG/MIXED-SIGNAL BEHAVIORS 9 2.1 PROPOSED CLOCK AND ANALOG SIGNAL REPRESENTATIONS 10 2.2 SIGNAL TYPE DEFINITIONS IN SYSTEMVERILOG 14 2.3 EVENT-DRIVEN SIMULATION METHODOLOGY 16 CHAPTER 3 HIGH-SPEED I/O INTERFACE SIMULATION 21 3.1 CHARGE-PUMP PHASE-LOCKED LOOP 23 3.2 BANGBANG CLOCK AND DATA RECOVERY 37 3.3 CHANNEL AND EQUALIZERS 45 3.4 HIGH-SPEED I/O SYSTEM SIMULATION 52 CHAPTER 4 SWITCHING POWER SUPPLY SIMULATION 55 4.1 BOOST CONVERTER 57 4.2 TIME-INTERLEAVED SWITCHED-CAPACITOR CONVERTER 66 CHAPTER 5 VOLTERRA SERIES MODEL SIMULATION 72 5.1 VOLTERRA SERIES MODEL 74 5.2 CLASS-A POWER AMPLIFIER 79 5.3 CONTINUOUS-TIME EQUALIZER 84 CHAPTER 6 INJECTION-LOCKED OSCILLATOR SIMULATION 89 6.1 PPV-BASED ILO MODEL 91 6.2 LC OSCILLATOR 99 6.3 RING OSCILLATOR 104 6.4 BURST-MODE CLOCK AND DATA RECOVERY 109 CONCLUSION 116 BIBLIOGRAPHY 118 초 록 126Docto

Stochastic analysis of cycle slips in injection-locked oscillators and analog frequency dividers

A detailed investigation of cycle slips in injection-locked oscillators (ILOs) and analog frequency dividers is presented. This nonlinear phenomenon gives rise to a temporal desynchronization between the injected oscillator and the input source due to noise perturbations. It involves very different time scales so even envelope-transient-based Monte Carlo analyses may suffer from high computational cost. The analysis method is based on an initial extraction of a reduced-order nonlinear model of the injected oscillator based on harmonic-balance simulations. This model has been improved with a more accurate description of oscillation dependence on the input source either at the fundamental frequency or, in the case of a frequency divider, at a given harmonic frequency. The reduced-order model enables an efficient stochastic analysis of the system based on the use of the associated Fokker-Planck equation in the phase probability density function. Several methods for the solution of the associated Fokker-Planck equation are compared with one of them being applicable under a wider range of system specifications. The analysis enables the prediction of the parameter-space regions that are best protected against cycle slips. The technique has been applied to two microwave ILOs and has been validated through commercial software envelope simulations in situations where the computational cost of the envelope simulations was acceptable, and through measurements. The measurement procedure of the cycle slipping phenomenon has been significantly improved with respect to previous work.This work was supported by the Spanish Ministry of Economy and Competitiveness under Contract TEC2011-29264-C03-01

Frequency Precision of Oscillators Based on High-Q Resonators

We present a method for analyzing the phase noise of oscillators based on feedback driven high quality factor resonators. Our approach is to derive the phase drift of the oscillator by projecting the stochastic oscillator dynamics onto a slow time scale corresponding physically to the long relaxation time of the resonator. We derive general expressions for the phase drift generated by noise sources in the electronic feedback loop of the oscillator. These are mixed with the signal through the nonlinear amplifier, which makes them {cyclostationary}. We also consider noise sources acting directly on the resonator. The expressions allow us to investigate reducing the oscillator phase noise thereby improving the frequency precision using resonator nonlinearity by tuning to special operating points. We illustrate the approach giving explicit results for a phenomenological amplifier model. We also propose a scheme for measuring the slow feedback noise generated by the feedback components in an open-loop driven configuration in experiment or using circuit simulators, which enables the calculation of the closed-loop oscillator phase noise in practical systems

SCEE 2008 book of abstracts : the 7th International Conference on Scientific Computing in Electrical Engineering (SCEE 2008), September 28 – October 3, 2008, Helsinki University of Technology, Espoo, Finland

This report contains abstracts of presentations given at the SCEE 2008 conference.reviewe

Optimal Control and Synchronization of Dynamic Ensemble Systems

Ensemble control involves the manipulation of an uncountably infinite collection of structurally identical or similar dynamical systems, which are indexed by a parameter set, by applying a common control without using feedback. This subject is motivated by compelling problems in quantum control, sensorless robotic manipulation, and neural engineering, which involve ensembles of linear, bilinear, or nonlinear oscillating systems, for which analytical control laws are infeasible or absent. The focus of this dissertation is on novel analytical paradigms and constructive control design methods for practical ensemble control problems. The first result is a computational method %based on the singular value decomposition (SVD) for the synthesis of minimum-norm ensemble controls for time-varying linear systems. This method is extended to iterative techniques to accommodate bounds on the control amplitude, and to synthesize ensemble controls for bilinear systems. Example ensemble systems include harmonic oscillators, quantum transport, and quantum spin transfers on the Bloch system. To move towards the control of complex ensembles of nonlinear oscillators, which occur in neuroscience, circadian biology, electrochemistry, and many other fields, ideas from synchronization engineering are incorporated. The focus is placed on the phenomenon of entrainment, which refers to the dynamic synchronization of an oscillating system to a periodic input. Phase coordinate transformation, formal averaging, and the calculus of variations are used to derive minimum energy and minimum mean time controls that entrain ensembles of non-interacting oscillators to a harmonic or subharmonic target frequency. In addition, a novel technique for taking advantage of nonlinearity and heterogeneity to establish desired dynamical structures in collections of inhomogeneous rhythmic systems is derived