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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: IEEE Press Series on Digital & Mobile Communication
- Sprache: Englisch
An in-depth look at the theory and applications of frequency stability An understanding of the acquisition of stable frequency is essential for anyone who needs to solve noise problems in wireless communications. This book offers a thorough introduction to the principles and applications of frequency stability, arming practicing engineers with the tools they need to minimize noise in systems and devices that affect everyday communications for millions of people. With an emphasis on both practical and scientific points of view, Frequency Stability: Introduction and Applications examines frequency and time fluctuations in resonators, as well as the stability of both standard and practical microwave oscillators. It explains noise properties of building circuit blocks, introducing time domain properties and how they relate to noise spectral densities. Including a special chapter devoted to the design and properties of phase locked loops--a crucial topic for frequency synthesizers--the book also: * Examines in detail L/F noise, showing how power losses in the propagation material extend over a long period of time * Covers sapphire, optoelectronics, MW, and ring oscillators with the discussion of noise in delay-line oscillators with lasers * Offers an extended treatment of phase noise in semiconductors and amplifiers based on Van der Ziel investigations * Emphasizes the modified Allan variance in the time domain, including exact computations * Outlines the relationship between resonator frequency and output phase noises via the feedback theory Featuring numerous tables with actual data, Frequency Stability: Introduction and Applications is an invaluable guide for engineers wishing to rein in acoustic and electromagnetic interference in modern communications.
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Seitenzahl: 285
Veröffentlichungsjahr: 2012
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Contents
Cover
Half Title page
Title page
Copyright page
Preface
Symbols
Chapter 1: Noise and Frequency Stability
1.1 White Noise
1.2 Colored Noises
1.3 Small And Band Limited Perturbations of Sinusoidal Signals
1.4 Statistical Approach
1.5 Power Spectra of Stochastic Processes
References
Appendix
Chapter 2: Noise in Resonators and Oscillators
2.1 Noise Generated in Resonators
2.2 Phase Noise of Resonators: Experimental Results
2.3 Noise in Oscillators
2.4 Leeson Model
References
Chapter 3: Noise Properties of Practical Oscillators
3.1 Precision Oscillators
3.2 Practical Oscillators
3.3 Practical RC Oscillators
References
Chapter 4: Noise of Building Elements
4.1 Resistors
4.2 Inductances
4.3 Capacitance
4.4 Semiconductors
4.5 Amplifiers
4.6 Mixers
4.7 Frequency Dividers
4.8 Frequency Multipliers
References
Chapter 5: Time Domain Measurements
5.1 Basic Properties of Sample Variances
5.2 Transfer Functions of Several Time Domain Frequency Stability Measures
5.3 Time Jitter
References
Chapter 6: Phase-Locked Loops
6.1 PLL Basics
6.2 PLL Design
6.3 Stability of the PLL
6.4 Tracking
6.5 Working Ranges of PLL
6.6 Digital PLL
6.7 PLL Phase Noise
6.8 PLL Time Jitter
6.9 Spurious Signals
6.10 Synchronized Oscillators
References
Index
Frequency Stability
IEEE PRESS SERIES ON DIGITAL AND MOBILE COMMUNICATION
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Forthcoming TitlesFundamentals of Convolutional Coding, Second Edition
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Technical ReviewerProfessor Yuriy S. Shmaliy, Guanajuato University
Copyright © 2012 by the Institute of Electrical and Electronics Engineers, Inc.
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Library of Congress Cataloging-in-Publication Data:
Kroupa, Venceslav F., 1923–Frequency stability : introduction and applications / Venceslav F. Kroupa.p. cm.Includes bibliographical references.ISBN 978-1-118-15912-5 (hardback)1. Oscillators, Electric–Design and construction. 2. Frequency stability. I. Title.TK7872.O7K76 2012621.381’323—dc232011048135
To my wife Magda,for her encouragementin starting and finishing this work
Preface
Quid est ergo tempus?Si nemo ex me quaerat, scio;si quaerenti explicare velim,nescio.
What then is time?If no one asks me, I know,if I want to explain it to someonewho asks, I do not know.
S. Augustine Confessions, Book XI
The flow of time is smooth and imperceptible—fluctuations originate in the measurement systems: Earth rotation is defined as a day–night cycle; Earth motion around the Sun is defined in years; astronomical observations of the motion in the universe last for centuries or for millennia, and so on. All these measurement systems introduce unpredictable imperfections and even errors designated as noise. The difficulty is also present in modern systems based on the atomic time definition.
The time fluctuations did not cause problems with ideas or their use until the twentieth century, with the introduction of modern technologies and the advent of the importance of the rapid delivery of messages, goods, even of people.
In today’s methods of communications, our delivery channels are generally based on electromagnetic media that provide a sort of the common property that cannot be expanded. Only the channels perfected by technology and by reducing mutual interference or by extension of frequency ranges can be used (see Fig. P1). In this context, the time and frequency stability is of prime importance, and study of noise problems still proceeds. There are many papers, books, and even libraries about this topic of frequency and time stability.1 Why a new one?
Fig. P1. The state of the art of frequency synthesis from the lowest frequency ranges to the optical frequencies; frequency division and multiplication.2
The intent of this book is twofold: to refresh students’ memory of the field and provide additional information for engineers and practitioners in neighboring fields without recourse to complicated mathematics.
The first noise studies on frequency stability were based on earlier results of the probability of events and were connected with LC oscillators. However, their short-term time stability was soon insufficient even for the simplest radio traffic. Introduction of very stable crystal oscillators provided better stability (they even proved irregularities in the Earth’s rotation), but their application on both transmitting and receiving positions, often with the assistance of frequency synthesizers, was soon insufficient for the increasing number of needed dependent communication channels.
To alleviate the situation, there were investigations of the frequency stability of local generators from both theoretical and practical points of approach on one hand, and the progress of technology toward the microwaves and miniaturization on the other. This situation is depicted in Fig. P1.
These problems are connected with the time and frequency stability, the extent of the used frequencies into microwave ranges, and the technology of application of miniaturization of integrated circuits on a large scale. In accordance with the intent of this book to refresh the memory of students in the field and provide additional information for engineers in neighboring fields, we divided the subject matter into six chapters.
Chapter 1 introduces the basic concepts of noise. It begins with the term power spectral density (PSD), that is, with the magnitude of fluctuations (phase, frequency, power, etc.) in the 1-Hz frequency span in the Fourier frequency ranges. The most important noises are:
In the second part of Chapter 1, we investigate fluctuations from the probability point of view. The simplest is the rectangular distribution of events. A larger number of rectangular distributions results in the central limit or Gaussian distribution. Another approach provides the binomial distribution near to the Gaussian distribution or into the Poisson distribution. We mention the stochastic processes, the stationary processes invariant with respect to the time shift, and fractional integration, resulting in the possibility of explaining the flicker frequency phenomena and the random walk processes.
Chapter 2 investigates the noise generated in resonators, particularly, in quartz crystal resonators, with the assumption that 1/f noise is generated by material (dielectric) losses. It concludes that the product of the quality factor Q times the resonant frequency fo is a constant. Its validity was verified experimentally in the laboratory and by referring to the published noise data of crystal resonators in the entire frequency range from 5 MHz to nearly 1 GHz. Further, we discuss oscillating conditions and conclude, with the assistance of a sampling model, that the small phase error (e.g., noise) is compensated for by the integrated frequency shift of the resonant frequency. Vice versa, this experience is applied to the Leeson model.
Chapter 3 is dedicated to noise properties of very stable oscillators, quartz crystals, and new sapphire resonators (cryocooled), which are often used as a secondary frequency standard. We also include discussion of noise properties in oscillators stabilized by large-Q dielectric or optoelectronic resonators. Finally, the noise of integrated microwave oscillators in ranges designed for both LC- and RC-ring resonators is addressed. The advantage of the latter is simplicity and the problems are the large noise and power consumption.
Chapter 4 is dedicated to noises generated in individual elements and circuit blocks: resistors, inductances, capacitors, semiconductors, and amplifiers. A detailed discussion is dedicated to the different types of mixers, diode rings, and CMOS balanced and double-balanced systems. The spurious modulation signals with two-tone performance are investigated (third-order intercept points, IIP3), together with the expected noise performance.
Discussion of dividers is extended to the synchronized systems in the gigahertz ranges, to their noise, and to the regenerative division systems that provide the lowest additional noise.
Chapter 5 is devoted to the time measurements performed via the Allan variance. It discusses the reliability of the measurement and the connection of the slope of the measured characteristic with the type of investigated noise. Special attention is dedicated to the so-called modified Allan variance. The second part of this chapter deals with time jitter evaluations. Further, the probability of the time error, the bit error ratio (BER), is investigated. Eye diagrams and histograms are briefly discussed. Finally, we pay attention to the time jitter evaluation from the frequency and time domain measurements.
The last chapter, Chapter 6, deals in some detail with phase-locked loop (PLL) problems. It starts with a short introduction, proceeds to design, and stresses the importance of the design factors, such as natural frequency and damping. Further, we deal with the order and the type of PLLs, and their transients and working ranges. We discuss the properties of digital loops and tristate-phase detectors in greater detail, and investigate the noises generated in individual parts of the PLL. Finally, we investigate the synchronized oscillators, on a PLL basis, for frequency division and multiplication applications.
1 IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology—IEEE Std 1139/1988. IEEE Standard Definitions of Physical Quantities for Fundamental Frequency and Time Metrology—Random Instabilities IEEE Std 1139/1999. IEEE Std 1139/2008.
2 From V.F. Kroupa Phase Lock Loops and Frequency Synthesis, Wiley, 2003. Reproduced with permission.
Symbols
CHAPTER 1
Noise and Frequency Stability
Frequency stability or instability is a very important parameter in both modern terrestrial and space communications, in high-performance computers, in GPS (global positional system), and many other digital systems. In this connection, even very small frequency or phase changes of steering frequency generators (exciting oscillators, clock generators, frequency synthesizers, amplifiers, etc.) are of fundamental importance. Since all physical processes are subject to some sort of uncertainties due to fluctuations of individual internal or external parameters, generally designated as noise, the investigation of the overall noise properties is of the highest importance for the analysis of frequency stability.
In practice, we encounter three fundamental types of noises that differ by the power in the time or frequency unit S(f) (generally in the 1 Hz bandwidth), the latter being called the power spectral density (PSD—see Fig. 1.1).
Fig. 1.1 Fundamental types of noises [1.1, 1.2].
(Copyright © IEEE. Reprinted with permission.)
There are three major types of noises:
The last two noise processes with their integrals [generating PSD S(f) proportional to ~f3, ~f4, etc.] are often called colored noises.
1.1 WHITE NOISE
Typical representation of white noise consists of black body radiation or the thermal noise of resistors, or shot noise, in electronic devices.
1.1.1 Thermal Noise
In 1928, Johnson [1.3] and Nyquist [1.4] published a theory explaining the existence of thermal noise in conductors. It is caused by short current pulses generated by collisions of a large number of electrons. The result is such that a noiseless conductor is connected in series with a generator with a root mean square (rms) noise voltage, en (see Fig. 1.2)
Fig. 1.2 Thermal noises of conductors [1.2].
(Copyright © IEEE. Reprinted with permission.)
1.1
where k is the Boltzmann constant, T the absolute temperature (see Table 1.1), R is the resistance of the conductor (Ω), and Δf is the frequency bandwidth (in Hz) used for the appreciation of the noise action.
Table 1.1 Several physical constants
After dividing (1.1) by the frequency bandwidth Δf, we arrive at the PSD in 1-Hz bandwidth, that is,
1.2
1.3
1.4
1.1.2 Shot Noise
In all cases where the output current is composed of random arrivals of a large number of particles, we again witness fluctuations of the white noise type [1.5].
By considering an idealized transition in Fig. 1.3, where electrons flow randomly from A to B and holes flow from B to A, in a negligible transit time, each particle arrival is connected with transport of a current pulse. Consequently, in a time unit τ (S) the number of n charges generates the current
Fig. 1.3 Circuit model for the shot noise [1.2].
(Copyright © IEEE. Reprinted with permission.)
1.6
1.7
where <n> is the mean value of the number of carriers in the time unit. In such cases, the variance is equal to
1.8
By reverting to (1.6), we get for the mean current
1.9
and for its variance value,
1.10
To arrive at the PSD, we use a bit heuristic approach with the assistance of the autocorrelation [cf. (1.94)]
1.11
EXAMPLE 1.2
Find the PSD Si,n(f) of the shot noise for the transistor current
For PSD of other currents, see Fig. 1.4.
Fig. 1.4 Noise current through a semiconductor junction [1.2].
(Copyright © IEEE. Reprinted with permission.)
1.2 COLORED NOISES
Until now we have discussed PSD of noises generated by slow time-independent fluctuations, that is, with constant PSD over a large Fourier frequency range. However, with oscillators and other frequency generators, we encounter phase fluctuations with frequency-dependent PSDs proportional to 1/f, 1/f2, or even to 1/f3, 1/f4, at very low Fourier frequencies that are often called colored noises.
In the mid-1920s, Johnson [1.3] found that at very low frequencies the shot noise in vacuum tubes did not follow white noise at low frequencies and he introduced for the additive noise the name flicker noise. This name is still used. Subsequent observations proved the 1/f law for a much larger set of physical phenomena on one hand and its validity at very low frequencies on the other hand. Some years later, Bernamont [1.6] suggested a law for its PSD:
1.12
where the power of α was in the vicinity of one. In electronic devices, the higher order noises are often generated by integration in the corresponding Fourier transform division by s (cf. Table 1.2). The only exception presents 1/f noise fluctuations encountered both in crystal resonators and oscillators, and in many other physical systems [1.7] (dispersions of cars on highways [1.8], frequency change around 50 or 60 Hz in power line systems [ 1.9], or even flooding in the Nile river valley [1.1]; the latter reference is based on the time dependence of generating fluctuations). Note that all are based on the time.
Table 1.2 Fourier and Laplace transform pairs for important time functions, encountered in practice
The problem of colored noises was investigated by many authors in the past from different points of approach and often with different results; particularly, with the ever-present 1/f noise. For example, Keshner [1.10] investigated noises with different slopes, 1/fα, and arrived at a number of variables needed for generation of the desired colored noise (cf. Fig. 1.5). His finding for 1/f noise is one degree of freedom per decade.
1.2.1 Mathematical Models of 1/fα Processes
Characterization of the frequency stability of all types of generators, inclusive of phase-locked loops (PLLs), is important for applications; in the first place for their designers, and vice versa for users. In the mid-1960s, theoretical principles of the phase noise theory in frequency generators were established [1.11] and later a number of practical papers were published (e.g. [1.12, 1.13]). Here, we will briefly recall the corresponding theory.
Solution of the noise problems is performed with the assistance of statistics by investigating correlations and by the transformation of the time domain processes into the (complex) frequency domain via the Laplace transform (cf. Appendix at the end of this chapter).
1.13
In instances where the lower bound of the above integral is −∞ the Laplace transform changes into the Fourier transform, the corresponding pairs for important time functions encountered in practice are summarized in Table 1.2.
Reverting to the investigated time function, f(t) in (1.13), the process can be represented as a power series:
1.14
By retaining only the first two terms, we arrive at the exponential approximation that represents a large set of actual situations of the time domain fluctuations,
1.15
with the respective Fourier transform (cf. Table 1.2),
1.16
After multiplication with the complex conjugate of F(s), we arrive at the so-called Lorenzian PSD (cf. Section 1.5.1, Brownian Motion):
1.17
1.2.2 1/f Noise (Flicker Noise)
To generate the PSD of the 1/f slope, so often observed in practice, we encounter a large number of approaches. McWorther [1.14] suggested the mathematical model (for the flicker noise generated in semiconductors) as a multistep process composed of single events (cf. Fig. 1.6a):
Fig. 1.6 (a) Flicker phase noise generated by a set of several 1/f2 noises. Their summation (the solid line) presents the ideal slope 1/f [1.14]. (b) The simulated slope 1/f, with five 1/f2 noise characteristics providing the background set.
1.18
By assuming that in the time domain we have a set of events of the type in equation (1.15), the PSD will retain the shape as in equation (1.17) as long as the time constants, a, do not change appreciably from one. The final amplitude of the PSD is then still a2o at low Fourier frequencies. To arrive at the flicker noise behavior, we start with inspection of the PSD, S(f), in (1.17) and find that in the neighborhood of the corner frequency, 2πf ≈ 1/τ, its slope is approximately proportional to 1/f. Evidently, by proportionally increasing the time constant and decreasing the amplitude in the corresponding series,
1.19
The summation reveals a slope of 1/f (see the example in Fig. 1.6b), where we have chosen τi/τi+1 ≈ 10 and arrived at a nearly perfect slope of 1/f. This finding is in a good agreement with a discussion by Keshner [ 1.10]. However, note a rather forceful, not random, condition on the amplitudes and time constants in the set of the Lorentzian noise characteristics (1.19) needed for the generation of the flicker noise system. The difficulty is that this is true for voltage or current fluctuations (e.g., [1.15, 1.16]), however, in instances of other physical quantities (transfer of power, flow of cars on a highway, etc.) the rms of (1.19) must be used. Effectively, we face a fractional integration discussed in connection with the 1/f fluctuations by Halford [1.17] or suggested by Radeka [1.18].
In another approach, let us again consider a flow (e.g., of power) with losses during defined time periods (cf. Fig. 1.7). In that case, we introduce a sampling process with the dissipated energy, Pdiss, during one sampling period To. In the next period, we encounter nearly the same energy losses, and so on. Generalization reveals a sampling process whose noise model in the z-transform is (cf. [1.19, 1.20])
Fig. 1.7 Generation of the phase noise in the sampled form reduced to a set of pulses.
1.20
where Po is the energy of the flux. To get the corresponding Fourier transform, we have to replace z−1 with e−sTo and multiply by the transfer function H(s)
1.21
with the result
1.22
