Fully nonlinear oscillator noise analysis: an oscillator with no asymptotic phase

Oscillators exist in many systems. Detailed and correct characterization and comprehension of noise in autonomous systems such as oscillators is of utmost importance. Previous approaches to oscillator noise analysis are based on some kind of perturbation analysis, some linear and some nonlinear. However, the derivations of the equations for perturbation analysis are all based on information that is produced by a linearization of the oscillator equations around the periodic steady‐state solution, where it is assumed that the oscillator is orbitally stable and it has the so‐called asymptotic phase property. In this paper, we first discuss these notions from a qualitative perspective, and demonstrate that the asymptotic phase property is crucial in validating all of the previous approaches. We then present the case of a simple oscillator that is orbitally stable but without asymptotic phase, for which previous approaches fail. We then present a fully nonlinear noise analysis of this oscillator. We derive and compute nonlinear, non‐stationary and non‐Gaussian stochastic characterizations for both amplitude and phase noise. We arrive at results that are distinctly different when compared with the ones obtained previously for oscillators with asymptotic phase. We compare and verify our analytical results against extensive Monte Carlo simulations. Copyright © 2006 John Wiley & Sons, Ltd.     

A CMOS DCO design using delay programmable differential latches and a novel digital control scheme

A novel 16-bit CMOS digitally controlled oscillator (DCO) is described. This CMOS DCO design is based on a delay programmable differential latch and a novel digital control scheme which yields improved phase noise characteristics. Simulations of a 4-stage CMOS DCO using the 0.5 μm Agilent CMOS process parameters achieved a controllable frequency range of 750 MHz–1.6 GHz with a monotone tuning range of around 1 GHz. Monte Carlo simulations indicate that the time-period jitter due to random supply voltage fluctuations is under 250 ps for worst-case considerations. Also, phase noise was found to be in the range of −175 dBc at a frequency of 600 KHz from the carrier at 1.5 GHz (for digital control word of 1512 H) after numerous iterations of Monte Carlo simulations. FFT analysis indicate a total harmonic distortion (THD) of around − 57 dB for the DCO output signal. This CMOS design would thus provide considerable performance enhancement in digital PLL applications.     

Phase noise analysis in CMOS differential Armstrong oscillator topology

This paper reports a phase noise analysis in a differential Armstrong oscillator circuit topology in CMOS technology. The analytical expressions of phase noise due to flicker and thermal noise sources are derived and validated by the results obtained through SpectreRF simulations for oscillation frequencies of 1, 10, and 100 GHz. The analysis captures well the phase noise of the oscillator topology and shows the impact of flicker noise contribution as the major effect leading to phase noise degradation in nano‐scale CMOS LC oscillators. Copyright © 2016 John Wiley & Sons, Ltd.     

Noise covariance estimation for Kalman filter tuning using Bayesian approach and Monte Carlo

Linear time‐invariant systems play significant role in the control field. A number of methods have been published for identification of the deterministic part of a process. However, identification of the stochastic part has had much less attention. This paper deals with estimation of covariance matrices of the noise entering a linear system. The process and measurement noise covariance matrices are tuning parameters of the Kalman filter, and they affect the quality of the state estimation. The noise covariance matrices are generally not known, and their estimation from the measured data is a challenging task. This paper introduces a method for estimation of the noise covariance matrices using Bayesian approach along with Monte Carlo numerical methods. Performance of the approach is tested on various systems and noise properties. The second part of the paper compares Monte Carlo approach with the recently published methods. The speed of convergence is compared with the Cramér–Rao bounds. Copyright © 2012 John Wiley & Sons, Ltd.     

Identification of the inertia matrix of a rotating body based on errors‐in‐variables models

This paper proposes a procedure for identifying the inertia matrix of a rotating body. The procedure based on Euler's equation governing rotational motion assumes errors‐in‐variables models in which all measurements, torque as well as angular velocities, are corrupted by noises. In order for consistent estimation, we introduce an extended linear regression model by augmenting the regressors with constants and the parameters with noise‐contributed terms. A transformation, based on low‐pass filtering, of the extended model cancels out angular acceleration terms in the regressors. Applying the method of least correlation to the model identifies the elements of the inertia matrix. Analysis shows that the estimates converge to the true parameters as the number of samples increases to infinity. Monte Carlo simulations demonstrate the performance of the algorithm and support the analytical consistency. Copyright © 2009 John Wiley & Sons, Ltd.     

Thermal and flicker phase noise analysis in rotary traveling-wave oscillator

This paper analyzes the thermally induced phase noise and the up-conversion of flicker noise into phase noise of rotary traveling-wave oscillator (RTWO). Based on the analyses, this paper extracts the closed-form formulas for the thermal and flicker phase noise of the RTWO. This paper compares the theoretical results with appropriate simulations to evaluate the accuracy of the derived closed-form formulas. Comparisons confirm the accuracy of the extracted phase noise formulas. By using the presented straightforward approach along with accurate phase noise formulas, the designers can understand the RTWO ' s design tradeoffs. Also, they can design the RTWO for a specific phase noise without needing lengthy simulations.     

基于随机微分方程的振荡器相位噪声研究

提出直接从理想振荡器的输出形式出发,得到了理想振荡器所满足的自治微分方程,并通过引入随机项来描述振荡器的内部白噪声,由此建立了用以描述含白噪声振荡器系统的非线性随机微分方程模型。采用随机平均与确定性平均的方法研究了该随机微分方程,得到了描述振荡器系统相位噪声的降阶随机微分方程,从而获得了白噪声情形下,振荡器相位噪声的分析方法。并以VanderPol振荡器为例,用该方法得出了相位噪声曲线,与振荡器系统数值求解所得相位噪声的数值结果符合较好,表明该方法是有效的,为振荡器相位噪声的研究提供了一种新思路。     

Analyses and techniques for phase noise reduction in CMOS Colpitts oscillator topology

This paper reports the analyses of three techniques for phase noise reduction in the complementary metal‐oxide semiconductor (CMOS) Colpitts oscillator circuit topology. Namely, the three techniques are inductive degeneration, noise filter, and optimum current density. The design of the circuit topology is carried out in 28‐nm bulk CMOS technology. The analytical expression of the oscillation frequency is derived and validated through circuit simulations. Moreover, the theoretical analyses of the three techniques are carried out and verified by means of circuit simulations within a commercial design environment. The results obtained for the inductive degeneration and noise filter show the existence of an optimum inductance for minimum phase noise. The results obtained for the optimum bias current density technique applied to a Colpitts oscillator circuit topology incorporating either inductive degeneration or noise filter show the existence of an optimum bias current density for minimum phase noise. Overall, the analyses show that the adoption of these techniques may lead to a potential phase noise reduction up to 19 dB at a 1‐MHz frequency offset for an oscillation frequency of 10 GHz. © 2015 The Authors International Journal of Circuit Theory and Applications Published by John Wiley & Sons Ltd.     

Time‐domain analysis of phase‐noise and jitter in oscillators due to white and colored noise sources

This paper presents an original time‐domain analysis of the phase‐diffusion process, which occurs in oscillators due to the presence of white and colored noise sources. It is shown that the method supplies realistic quantitative predictions of phase‐noise and jitter and provides useful design‐oriented closed‐form expressions of such phenomena. Analytical expressions and numerical simulations are verified through measurements performed on a relaxation oscillator whose behavior is perturbed by externally controlled noise sources. Copyright © 2011 John Wiley & Sons, Ltd.     

Analysis of flicker noise conversion to phase noise in CMOS differential LC oscillators

An analysis of the flicker noise conversion to close‐in phase noise in complementary metal‐oxide semiconductor (CMOS) differential inductance‐capacitance (LC)‐voltage controlled oscillator is presented. The contribution of different mechanisms responsible for flicker noise to phase noise conversion is investigated from a theoretical point of view. Impulse sensitivity function theory is exploited to quantify flicker noise to phase noise conversion process from both tail and core transistors. The impact of different parasitic capacitances inside the active core on flicker noise to phase noise conversion is investigated. Also, it is shown how different flicker noise models for core metal‐oxide semiconductor (MOS) transistors may result in different close‐in phase noise behaviors. Based on the developed analysis, design guidelines for reducing the close‐in phase noise are introduced. Copyright © 2015 John Wiley & Sons, Ltd.     

A linear response Monte Carlo algorithm for inversion layers and magnetotransport

Certain simulation problems require the solution of the Boltzmann equation for small driving forces. Due to the bad signal to noise ratio of the standard Monte Carlo algorithm in the linear regime these simulations are extremely CPU intensive. A linear response Monte Carlo algorithm is proposed which is orders of magnitude faster than the standard approach. It is used to extract transport parameters from magnetotransport measurements and to match semiempirical inversion layer models to experimental data.     

Enhancements to the Cumulant Method for Probabilistic Optimal Power Flow Studies

This paper introduces two extenstions to the cumulant method (CM) for probabilistic optimal power flow (P-OPF) studies; the first is an enhancement to provide improved handling of limits within the P-OPF problems and the second is a way to include correlated variables. The first enhancement is termed the Limit corrected cumulant method (LCCM) which specifically addresses errors in the existing CM when limits, away from the mean solution, are encountered while solving a P-OPF problem. The LCCM approach relies on the CM to produce multiple probability density functions (PDFs) and then combines these PDFs together to generate final PDFs. The second enhancement for incorporating correlated variables into P-OPF problems is based on the composition of correlated variables from several independent ones. The proposed approaches are verified against Monte Carlo simulations (MCS) consisting of 10 000 samples and demonstrate significant improvements when compared with traditional CM results.     

Residential Load Management Under Stochastic Weather Condition in Developing Countries

Abstract—This article focuses on offline residential load management in a developing country. This load management is based on scheduling linear load models under the stochastic weather conditions. The weather condition is modeled using the probability theory and Monte Carlo simulation. The seasonality effect, the type of day, and the stochastic hourly variation of weather conditions are considered as factors governing load management.     

Shot noise in single open ion channels: A computational approach based on atomistic simulations

This paper presents a computational analysis of the noise associated with ion current in single open ion channels. The study is performed by means of a coupled Molecular Dynamics/Monte Carlo approach able to simulate the conduction process on the basis of all microscopic information today available from protein structural data and atomistic simulations. The case of potassium ions permeating the KcsA channel is considered in the numerical calculations. Results show a noise spectrum different from what is theoretically predicted for uncorrelated ion-exit events (Poisson noise), confirming the existence of correlation in ion motion within the channel, already evinced by atomistic structural analyses.     

Transformer‐coupled π‐network differential CMOS oscillator circuit topology

This paper reports a novel oscillator circuit topology based on a transformer‐coupled π‐network. As a case study, the proposed oscillator topology has been designed and studied for 60 GHz applications in the frame of the emerging fifth generation wireless communications. The analytical expression of the oscillation frequency is derived and validated through circuit simulations. The root‐locus analysis shows that oscillations occur only at that resonant frequency of the LC tank. Moreover, a closed‐form expression for the quality factor (Q) of the LC tank is derived which shows the enhancement of the equivalent quality factor of the LC tank due to the transformer‐coupling. Last, a phase noise analysis is reported and the analytical expressions of phase noise due to flicker and thermal noise sources are derived and validated by the results obtained through SpectreRF simulations in the Cadence design environment with a 28 nm CMOS process design kit commercially available. Copyright © 2016 John Wiley & Sons, Ltd.     

Survey of integrated‐circuit‐oscillator phase‐noise analysis

This tutorial distills the salient phase‐noise analysis concepts and key equations developed over the last 75 years relevant to integrated circuit oscillators. Oscillator phase and amplitude fluctuations have been studied since at least 1938 when Berstein solved the Fokker–Planck equations for the phase/amplitude distributions of a resonant oscillator. The principal contribution of this work is the organized, unified presentation of eclectic phase‐noise analysis techniques, facilitating their application to integrated circuit oscillator design. Furthermore, we demonstrate that all these methods boil down to obtaining three things: (1) noise modulation function; (2) noise transfer function; and (3) current‐controlled oscillator gain. For each method, this paper provides a short background explanation of the technique, a step‐by‐step procedure of how to apply the method to hand calculation/computer simulation, and a worked example to demonstrate how to analyze a practical oscillator circuit with that method. This survey article chiefly deals with phase‐noise analysis methods, so to restrict its scope, we limit our discussion to the following: (1) analyzing integrated circuit metal–oxide–semiconductor/bipolar junction transistor‐based LC, delay, and ring oscillator topologies; (2) considering a few oscillator harmonics in our analysis; (3) analyzing thermal/flicker intrinsic device‐noise sources rather than environmental/parametric noise/wander; (4) providing mainly qualitative amplitude‐noise discussions; and (5) omitting measurement methods/phase‐noise reduction techniques. Copyright © 2013 John Wiley & Sons, Ltd.     

Towards minimum achievable phase noise of relaxation oscillators

A relaxation oscillator design is described, which has a phase noise rivaling ring oscillators, while also featuring linear frequency tuning. We show that the comparator in a relaxation‐oscillator loop can be prevented from contributing to 1/f² colored phase noise and degrading control linearity. The resulting oscillator is implemented in a power efficient way with a switched‐capacitor circuit. The design results from a thorough analysis of the fundamental phase noise contributions. Simple expressions modeling the theoretical phase noise performance limit are presented, as well as a design strategy to approach this limit. To verify theoretical predictions, a relaxation oscillator is implemented in a baseline 65 nm CMOS process, occupying 200 µm × 150 µm. Its frequency tuning range is 1–12 MHz, and its phase noise is L(100kHz) = ?109dBc/Hz at f_osc = 12MHz, while consuming 90 μW. A figure of merit of ?161dBc/Hz is achieved, which is only 4 dB from the theoretical limit. Copyright © 2012 John Wiley & Sons, Ltd.     

A study of amplitude‐to‐phase noise conversion in planar oscillators

This paper explores the many interesting implications for oscillator design, with optimized phase‐noise performance, deriving from a newly proposed model based on the concept of oscillator conjugacy. For the case of 2‐D (planar) oscillators, the model prominently predicts that only circuits producing a perfectly symmetric steady‐state can have zero amplitude‐to‐phase (AM‐PM) noise conversion, a so‐called zero‐state. Simulations on standard industry oscillator circuits verify all model predictions and, however, also show that these circuit classes cannot attain zero‐states except in special limit‐cases which are not practically relevant. Guided by the newly acquired design rules, we describe the synthesis of a novel 2‐D reduced‐order LC oscillator circuit which achieves several zero‐states while operating at realistic output power levels. The potential future application of this developed theoretical framework for implementation of numerical algorithms aimed at optimizing oscillator phase‐noise performance is briefly discussed.     

A new methodology for probabilistic short-circuit evaluation with applications in power quality analysis

The main aim of this paper is to propose a probabilistic short-circuit approach to generate the probability distributions of the System Average RMS Variation Index (SARFI) index. The proposed methodology is based on the combination of the admittance summation method in phase coordinates with the Monte Carlo method. This new approach has been tested in a feeder belonging to an electricity distribution company in the Northeast of Brazil to generate the probability distributions of the SARFI index. The tests results demonstrated that the proposed model is a powerful tool to stochastic prediction of voltage sags.     

Phase noise analysis of source injection coupled quadrature oscillator

This paper analyzes the thermally induced phase noise and the up-conversion of flicker noise into phase noise of source injection coupled quadrature oscillator (SIC-QOSC), for the first time. Furthermore, this paper provides a complete analysis for the injection current of the SIC-QOSC and extracts the closed-form expressions for it for the first time, too. These expressions lead to obtaining the harmonics of the injection current as well as the oscillation amplitude, which is necessary for the phase noise analysis. To evaluate the extracted equations, this paper compares the calculated results with appropriate simulations. Comparisons confirm the accuracy of the proposed injection current expressions and the phase noise formulas. Using the closed-form equations of phase noise, designers can understand the SIC-QOSC's design tradeoffs and design the oscillator for given phase noise.