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.     

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 5 GHz quadrature relaxation oscillator with mixing for improved testability or compact front‐end implementation

We demonstrate by measurements on a test circuit that a 5 GHz relaxation oscillator with accurate quadrature outputs and low phase‐noise can be obtained, and that these favorable properties can be preserved while the mixing function is performed by this oscillator. This is useful either to measure the quadrature error at a low frequency, or to implement a low‐intermediate frequency (IF) or zero‐IF (homodyne) radio frequency front‐end. Copyright © 2008 John Wiley & Sons, Ltd.     

基于JACOBI-DAVI DSON方法的小干扰稳定性分析中关键特征值计算

提出了一种大规模电力系统小干扰稳定性分析的有效方法。应用Jacobi-Davidson方法求取系统状态矩阵的关键特征子集。该方法在搜索子空间中挑选出想要的特征值和特征向量的近似值，然后用与当前近似特征向量正交的子空间上的修正方程的解扩展搜索子空间，从而得到想要的特征值和特征向量的更好近似。算法中使用了电力系统线性化模型中的增广状态矩阵进行相应面向稀疏的计算，可准确求解修正方程，以保证算法具有渐进二次收敛速度。将提出的方法在46机系统上进行了试验，结果表明该方法灵活，稳定性好，能有效地求出系统的关键特征子集。     

Numerical harmonic modeling of long coupled transmission lines using matrix series theory and recursive approach

To improve the efficiency of the harmonic analysis, this paper presents a direct phase‐domain harmonic model for long coupled transmission lines, which can take the parameter frequency dependence, the configuration asymmetry, and the untransposition of the line into consideration. The steps of the proposed modeling method are as follows. First, in the direct phase domain and on the basis of the travelling wave equation, the coupled transmission line model was derived in a nodal admittance matrix form. Then the matrix series theory was adopted to realize the numerical calculation of the nodal admittance matrix without relying on the eigenvalue and eigenvector calculations of the propagation matrix. Finally, a recursive approach was proposed to handle the nodal equation of the network containing long transmission lines. Both the accuracy and the efficiency of the proposed approach were verified by comparing with the exact method in which it is necessary to calculate the eigenvalues and eigenvectors of the propagation matrix. The computation time and memory consumption analysis indicates that the proposed approach excels the exact method in saving computation time and memory. It should be emphasized that, by utilizing the recursive approach, the proposed approach is always numerically stable and can also be applied to DC transmission lines. Copyright © 2012 John Wiley & Sons, Ltd.     

A series‐coupled LC quadrature oscillator using phase/amplitude calibration

In this paper, an analytic approach for the estimation of the phase and amplitude error in series coupled LC quadrature oscillator (SC‐QO) is proposed. The analysis results show that imbalances in source voltage of coupling transistor because of mismatches between LC tanks are the main source of the phase and amplitude error in this oscillator. For compensation of the phase and amplitude error, a phase and amplitude‐tunable series coupled quadrature oscillator is designed in this paper. A phase shift generation circuit, designed using an added coupling transistor, can control the coupling transistor source voltage. The phase and amplitude error can simply be controlled and removed by tuning the phase shifter, while this correction does not have undesirable impact on phase noise. In fact, the proposed SC‐QO generates a phase shift in the output current, which reduces the resonator phase shift (RPS) and improves phase noise. The phase and amplitude tunable SC‐QO is able to correct the phase error up to ±12°, while amplitude imbalances are reduced as well. To evaluate the proposed analysis, a 4.5‐GHz CMOS SC‐QO is simulated using the practical 0.18‐μm TSMC CMOS technology with a current consumption of 2 mA at 1.8‐V supply voltage. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis and design of LC oscillators associated with frequency multipliers: a phase noise perspective

When a local oscillator signal generation system is based on an LC oscillator and a frequency multiplier, the question of determining the optimal multiplication factor is a key issue. In this paper, the problem is addressed in order to minimize the 1/f ² phase noise within a tuning range constraint. The analysis, with a practical graphical representation, reveals the oscillator phase noise dependence on the oscillating frequency in the transition from two different regimes, named the inductor‐limited quality factor and the capacitor‐limited quality factor. The results obtained enable the evaluation of the phase noise performance of systems based on a sub‐harmonic and super‐harmonic oscillators and how they compare with an oscillator in the fundamental mode. Crucial questions like the phase noise improvement that these systems can achieve are analytically answered. A design methodology is thus proposed and verified through measurements on a frequency source at 31 GHz, composed by a sub‐harmonic voltage‐controlled oscillator followed by an injection‐locked frequency tripler, dedicated to backhauling applications, designed on a BiCMOS process technology. The tuning range is 10%, and the phase noise at a 1‐MHz offset is −112 dBc/Hz. Copyright © 2017 John Wiley & Sons, Ltd.     

Exact and explicit expressions for the small‐signal dynamics and frequency response of switching converters

In this paper, exact and explicit expressions for the dynamics and small‐signal responses of piecewise linear switching converters are derived. The results are very useful for the exact simulation and analysis of converter circuits. These expressions can also be used to develop simplified (approximate) models of converters for practical design purposes. An example is given to show that the well‐known state‐space averaging model is in fact the first‐order approximation of our exact model. Copyright © 2004 John Wiley & Sons, Ltd.     

Extension of the variational equation to analog/digital circuits: numerical and experimental validation

Conventional shooting methods cannot be used to determine the steady‐state solution of circuits whose model is characterised by a vector field exhibiting zero order discontinuities. In the analysis of circuits working on a stable limit cycle, this limitation prevents the use of methods that exploit Floquet theory to compute the variational model and thus the properties of the fundamental matrix and, for example, phase noise in oscillators. In this paper, we use an improved shooting method that solves this drawback by resorting to saltation matrices and show how this method makes possible the correct computation of the first left eigenfunction (known as ppv ) of the fundamental matrix. ppv s are a key aspect in determining phase noise. The results obtained through numerical simulations are compared with measurements on a relaxation oscillator serving as a simple but significant comparison vehicle. 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.     

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.     

Carrier-envelope offset phase-locking with attosecond timing jitter

Inside a femtosecond laser oscillator, no coupling mechanism between the propagation speeds of the carrier and the pulse envelope exists. Therefore, the relative delay between carrier and envelope of a femtosecond oscillator will exhibit irregular fluctuations unless this jitter is actively suppressed. Both intensity and beam pointing fluctuations in the laser can introduce carrier-envelope phase changes. Based on our analysis, we are capable of reducing or avoiding certain mechanisms by proper design of the laser cavity. We use such an optimized cavity to stabilize the carrier envelope-phase to an external reference oscillator with a long-term residual jitter corresponding to only 10 attoseconds in a (100 kHz-0.01 Hz) bandwidth. This is the smallest long-term timing jitter of a femtosecond laser oscillator demonstrated to date. However, it is important to note that this stabilization was obtained with an f-to-2f heterodyne technique using additional external spectral broadening in a microstructure fiber which introduces additional carrier-envelope phase noise. We present a direct heterodyne measurement of this additional carrier-envelope phase noise due to the continuum generation process.     

Mixed‐mode analysis of the sensitivity of a radiofrequency oscillator disturbed by parasitic signals

The first step of this work is to study the susceptibility of a radiofrequency oscillator to deterministic disturbance sources. A Colpitts oscillator, working around a 4 GHz frequency, contains a heterojunction bipolar transistor with a silicon–germanium base as an active device. A mixed‐mode analysis is involved, applying a microscopic drift diffusion model to the device, whereas the rest of the circuit used is governed by Kirchhoff's laws. We assume that this tool is very relevant to grasp the influence of intrinsic or extrinsic noisy sources of the oscillator. Our first simulation raw results motivate us to discuss, and perhaps extend, via some analytical models, the so‐called impulse sensitivity function model. In this paper, we try to develop quantitative predictions about the phase noise of such oscillators, and to give some new tracks on this field. Copyright © 2008 John Wiley & Sons, Ltd.     

Phase noise characterization of oscillators through Ito calculus

Recent phase noise analysis techniques of oscillators mainly rely on solving a stochastic differential equation governing the phase noise process. This equation has been solved in the literature using a number of mathematical tools from probability theory like deriving the Fokker–Planck equation governing the phase noise probability density function. Here, a completely different approach for solving this equation in presence of white noise sources is introduced that is based on the Ito calculus for stochastic differential equations. Time‐domain analytical expressions for the correlation of the noisy variables of the oscillator are derived that in asymptotically large times give the steady‐state stochastic correlations as well as the power spectral densities of the variables. The validity of the new approach is verified by comparing its results against extensive Monte‐Carlo simulations. This approach is applied to an oscillator with a dielectric resonator at 4.127 GHz, and a very good agreement between its results with those of the Monte‐Carlo simulations and the previous approaches is observed. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Coupled transmission lines revisited

The author presents a very satisfying and disarmingly simple way of introducing students to the parallel line coupler, which derives its S matrix directly from a knowledge of its eigenvectors and eigenvalues. For a pair of symmetrical coupled lines, the eigenvectors can be found by inspection, and the eigenvalues determined easily by means of flow graphs. The elements of the S matrix are then known, being linear combinations of the eigenvalues     

Voltage stability evaluation using modal analysis

The authors discuss the voltage stability analysis of large power systems by using a modal analysis technique. The method computes, using a steady-state system model, a specified number of the smallest eigenvalues and the associated eigenvectors of a reduced Jacobian matrix. The eigenvalues, each of which is associated with a mode of voltage/reactive power variation, provide a relative measure of proximity to voltage instability. The eigenvectors are used to describe the mode shape and to provide information about the network elements and generators which participate in each mode. A simultaneous iteration method, which is well suited to applications involving large power systems, is used for selective calculation of appropriate eigenvalues. Results obtained using a 3700 bus test system are presented illustrating the applicability of the approach     

THE SVD OF SYSTEM MATRICES AND MODAL PROPERTIES IN A TWO-AREA SYSTEM

ABSTRACT

In this paper we present results of a comparison of participation factors with a new method that uses the singular value decomposition ( SVD) of complex matrices which are functions of the state matrix A, the input matrix B, the output matrix C and the eigenvalues as measures of modal controllability and observability. The agreement between the two approaches is remarkable. We also give some results about the conditioning of single eigenvalues and eigenvectors and apply them to the same system mentioned above. We find that in some region in the parameter space the eigenvalues corresponding to the area modes approach each other coalescing into a double eigenvalue. At this point and in a region surrounding it in the parameter space the eigenvalues and the eigenvectors become very ill-conditioned rendering the information obtained from the eigenvectors meaningless.     

快速求解多机系统机电振荡模式的新算法

提出了一种快速求解多机系统机电振荡模式的新方法,既能避免维数灾问题又能防止出现丢根现象。首先,通过建立发电机的经典模型,计算出多机系统的各振荡模式(特征根虚部)和振荡模态(特征向量);再用等值阻尼系统表示发电机的励磁控制作用对机电振荡模式的影响,推导出只用机组惯性常数M和阻尼系数D及相应的特征向量表示的机电模式的实部估算公式,由此公式估算出相应的实部,从而得到各振荡模式的特征根的初值;根据Heffron-Phillips全系统的状态矩阵和已经估算出的特征根,用反幂法快速求出系统状态矩阵的根,即为此多机系统的全部机电模式特征根。最后,用ENGLAND10机系统算例证明了该方法的有效性。