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 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.     

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.     

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.     

Discontinuity correction in piecewise‐linear models of oscillators for phase noise characterization

Decomposition of noise perturbation along Floquet eigenvectors has been extensively used in order to achieve a complete analysis of phase noise in oscillator. Piecewise‐linear approximation of nonlinear devices is usually adopted in numerical calculation based on multi‐step integration method for the determination of unperturbed oscillator solution. In this case, exact determination of the monodromy matrix can be hampered by the presence of discontinuities between models introduced by the approximation. In this paper we demonstrate that, without the proper corrections, relevant errors occur in the determination of eigenvalues and eigenvectors, if adjacent linear models presents discontinuities. We obtain this result by the analysis of a simple 2‐D oscillator with piecewise‐linear parameter. We also demonstrate that a correct calculation can be achieved introducing properly calculated state vector boundary conditions by the use of interface matrices. This correction takes into account the effects of discontinuities between the linear models, leading to exact calculation of eigenvalues and eigenvectors, and, consequently, of the phase noise spectrum. Copyright © 2006 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.     

Experimental comparison of phase‐noise in cross‐coupled RC‐ and LC‐oscillators

Relaxation RC‐oscillators are notorious for their poor phase‐noise performance. However, there are reasons to expect a phase‐noise reduction in quadrature oscillators obtained by cross‐coupling two relaxation oscillators. We present measurements on 5 GHz oscillators, which show that in RC‐oscillators the coupling reduces both the phase‐noise and quadrature error, whereas in LC‐oscillators the coupling reduces the quadrature error, but increases the phase‐noise. A comparison using standard figures of merit indicates that quadrature RC‐oscillators may be a viable alternative to LC‐oscillators when area and cost are to be minimized. Copyright © 2009 John Wiley & Sons, Ltd.     

Low phase noise L-band oscillators based on novel general Chebyshev bandpass filters

Oscillators act as signal sources for wireless communication systems, radars, wireless charging, and other industrial applications, whose phase noises directly determine system performance. In this paper, two low-phase noise microwave oscillators based on novel microstrip second-order band-pass filters are presented. These filters are characterized by good frequency selectivity owing to two transmission zeros near to the passbands. As a result, relatively high group delay can be achieved at center frequencies, which is vital to the phase noise performance of the oscillators to be designed. Simultaneously, the bandwidths can be made narrow enough so that actual oscillating frequency is very close to the designed one. In addition, the introduction of stepped impedance makes these filters capable of harmonic suppression, which efficiently suppresses the harmonics in the output such as the second harmonic. These filters are compact and also of easy design. Finally, two microwave oscillators at 2.0 GHz are designed on these microstrip filters. As the measured results show, their output frequencies are 2.002 GHz. The phase noises are −127.21 and −127.03 dBc/Hz@100 kHz, respectively. The second harmonic suppressions are as great as 30.45 and 39.21 dBc, respectively.     

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.     

Evaluation and comparison of GHz‐range LC oscillators using time‐varying root‐locus

This paper presents a comprehensive comparison between complementary metal‐oxide‐semiconductor (CMOS) LC‐oscillator topologies often used in GHz‐range transceivers. The comparison utilizes the time‐varying root‐locus (TVRL) method to add new insights into the operation of different oscillators. The paper focuses on the treatment of the TVRL trajectories obtained for different oscillators and establishes links between the trajectories and physical phenomena in oscillators. The evaluation of the root trajectories shows the advantages of the TVRL method for comparing oscillator topologies, which is also extended towards the analysis of voltage‐controlled oscillators. The necessary circuit simplifications required in closed‐form root‐locus analysis are avoided by the TVRL, which allows precise oscillator comparison and reveals details on the topology specifics. The derived conclusions have been verified by the Cadence Spectre‐RF simulator on 130‐nm CMOS process. Copyright © 2011 John Wiley & Sons, Ltd.     

Sinusoidal shaping of the ISF in LC oscillators

A new method to decrease the phase noise of the sinusoidal oscillators is proposed. The proposed method is based on using a dynamic transistor biasing in a typical oscillator topology. This method uses the oscillator impulse sensitivity function (ISF) shaping to reduce the sensitivity of the oscillator to the transistor noise and as a result reducing the oscillator phase noise. A 1.8 GHz, 1.8 V designed oscillator based on the proposed method shows a phase noise of ?130.3dBc/Hz at 1 MHz offset frequency, thereby showing about 6 dB phase noise decreasing in comparison with the typical constant bias topology. This result is obtained from the simulation based on 0.18u CMOS technology and on‐chip spiral inductor with a quality factor equal to 8. Copyright © 2007 John Wiley & Sons, Ltd.     

无位置传感器无刷直流电动机的锁相稳速系统

介绍了使用通用集成芯片实现的无位置传感器无刷直流电动机的稳速控制系统,此系统已投入实际应用,稳速精度小于千分之一。     

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.     

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.     

A study of superharmonic injection locking in multiband frequency dividers

A superharmonic voltage‐controlled injection‐locked frequency divider, implemented using a modified Colpitts oscillator operating at 2.5, 5 and 10 GHz and a cross‐coupled LC oscillator operating at 1.25, 2.5 and 5 GHz, is demonstrated. The proposed triple‐band operation is achieved by employing a novel technique that uses pin‐diodes and negative power supply. The discrete dividers, built with low noise hetero‐junction FETs and high‐frequency SiGe BJTs, are described theoretically while their functionality is proven experimentally. Additionally, a short phase noise analysis, which is missing in the literature, is given. Phase noise, frequency range of operation, and locking range measurement results are presented. Finally, post‐layout simulation results of a 5 GHz fully differential injection‐locked frequency divider, implemented in a 0.25µm SiGe process are provided. Copyright © 2010 John Wiley & Sons, Ltd.     

低相位噪声参考信号发生器的设计

参考信号发生器是频率合成系统的重要组成部分,是实现低噪声频率合成的基础。本文介绍了产生低相位噪声参考信号常用的技术方法,重点讨论了如何基于锁相环频率合成法实现极低相位噪声参考发生器的设计。最后基于本文讨论的技术方法,设计出了一种具有极低相位噪声的参考信号发生器,并给出了实验结果。该参考信号发生器应用于某高纯微波信号源中,取得了很好的效果。     

Divide‐by‐2 LC injection‐locked frequency divider with wide locking range at low and high injection powers

A wide locking range nMOS divide‐by‐2 RLC injection‐locked frequency divider (ILFD) was designed and implemented in the TSMC 0.18‐µm BiCMOS process. The ILFD is based on a cross‐coupled oscillator with one direct injection MOSFET and a RLC resonator. The RLC resonator is used to extend the locking range so that dual‐band locking ranges can be merged in one locking range at both low and high injection powers. At the drain‐source bias of 0.9 V for switching transistors, and at the incident power of 0 dBm the locking range of the divide‐by‐2 ILFD is 7.24 GHz, from the incident frequency 2.65 to 9.89 GHz, the locking range percentage is 115.47%. The power consumption of ILFD core is 8.685 mW. The die area is 0.726 × 0.930 mm². Copyright © 2016 John Wiley & Sons, Ltd.     

Performance characterization of a non‐linear system as both an adaptive notch filter and a phase‐locked loop

The behaviour of a non‐linear dynamical system is described. The system may be characterized as an adaptive notch filter, or alternatively, as a phase‐locked loop. Either way, the system has the inherent capability of directly providing estimates of the parameters of the extracted sinusoidal component of its input signal, namely its amplitude, phase and frequency. The structure and mathematical properties of the system are presented for two cases of fixed‐frequency and varying‐frequency operation. The effects of parameter setting of the system on its performance are studied in detail using computer simulations. Transient and steady‐state behaviour of the system are studied in the presence of noise. Simplicity of structure, high noise immunity and robustness and the capability of direct estimation of amplitude, phase and frequency are the salient features of the system when envisaged as an adaptive notch filter or a phase‐locked loop. Copyright © 2004 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.     

基于锁相环和混沌振子的微弱信号检测

为精确检测纳伏级微弱正弦信号的频率和幅值,构建了基于锁相环和Duffing振子的微弱信号混合检测系统。首先通过锁定输入锁相环的待测信号,完成信号频率的检测;然后利用锁相环输出的已知频率信号作为混沌系统的内置策动力信号,将输入到混沌系统的待测信号用Duffing振子进行幅值检测。仿真结果表明,混合系统可同时完成纳伏级微弱正弦信号的频率和幅值的检测,检测信号的最低信噪比为-22.23dB,且操作简单,工作量小,易于实现。