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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Precision Measurement of Microwave Comprehensive resource covering the foundations and analysis of precision noise measurements with a detailed treatment of their uncertainties Precision Measurement of Microwave Thermal Noise presents the basics of precise measurements of thermal noise at microwave frequencies and guides readers through how to evaluate the uncertainties in such measurement. The focus is on measurement methods used at the U.S. National Institute of Standards and Technology (NIST), but the general principles and methods are useful in a wide range of applications. Readers will learn how to perform accurate microwave noise measurements using the respected author's expertise of calculations to aid understanding of the challenges and solutions. The text covers the background required for the analysis of the measurements and the standards employed to calibrate radiofrequency and microwave radiometers. It also covers measurements of noise temperature (power) and the noise characteristics of amplifiers and transistors. In addition to the usual room-temperature two-port devices, cryogenic devices and multiport amplifiers are also discussed. Finally, the connection of these lab-based measurements to remote-sensing measurement (especially from space) is considered, and possible contributions of the lab-based measurements to remote-sensing applications are discussed. Specific topics and concepts covered in the text include: * Noise-temperature standards, covering ambient standards, hot (oven) standards, cryogenic standards, and other standards and noise sources * Amplifier noise, covering definition of noise parameters, measurement of noise parameters, uncertainty analysis for noise-parameter measurements, and simulations and strategies * On-wafer noise measurements, covering on-wafer microwave formalism, noise temperature, on-wafer noise-parameter measurements, and uncertainties * Multiport amplifiers, covering formalism and noise matrix, definition of noise figure for multiports, and degradation of signal-to-noise ratio Containing some introductory material, Precision Measurement of Microwave Thermal Noise is an invaluable resource on the subject for advanced students and all professionals working in (or entering) the field of microwave noise measurements, be it in a standards lab, a commercial lab, or academic research.
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Table of Contents
Cover
Title Page
Copyright
Preface
References
1 Background
1.1 Nyquist's Theorem and Noise Temperature
1.2 Microwave Networks
References
Note
2 Noise‐Temperature Standards
2.1 Introduction
2.2 Ambient Standards
2.3 Hot (Oven) Standards
2.4 Cryogenic Standards
2.5 Other Standards and Noise Sources
References
3 Noise‐Temperature Measurement
3.1 Background
3.2 Total‐Power Radiometer
3.3 Total‐Power Radiometer Uncertainties
3.4 Other Radiometer Designs
3.5 Measurements through Adapters
3.6 Traceability and Inter‐laboratory Comparisons
References
4 Amplifier Noise
4.1 Noise Figure, Effective Input Noise Temperature
4.2 Noise‐Temperature Definition Revisited
4.3 Noise Figure Measurement, Simple Case
4.4 Definition of Noise Parameters
4.5 Measurement of Noise Parameters
4.6 Uncertainty Analysis for Noise‐Parameter Measurements
4.7 Simulations and Strategies
References
5 On‐Wafer Noise Measurements
5.1 Introduction
5.2 On‐Wafer Microwave Formalism
5.3 Noise‐Temperature Measurements
5.4 On‐Wafer Noise‐Parameter Measurements
5.5 Uncertainties
References
6 Noise‐Parameter Checks and Verification
6.1 Measurement of Passive or Previously Measured Devices
6.2 Physical Bounds and Model Predictions
6.3 Tandem or Hybrid Measurements
References
7 Cryogenic Amplifiers
7.1 Background
7.2 Measurement of the Matched Noise Figure
7.3 Noise‐Parameter Measurement
References
8 Multiport Amplifiers
8.1 Introduction
8.2 Formalism and Noise Matrix
8.3 Definition of Noise Figure for Multiports
8.4 Degradation of Signal‐to‐Noise Ratio
8.5 Three‐Port Example – Differential Amplifier with Reflectionless Terminations
8.6 Four‐Port Example with Reflectionless Terminations
References
9 Remote Sensing Connection
9.1 Introduction
9.2 Theory for Standard Radiometer
9.3 Standard‐Radiometer Measurements
9.4 Standard‐Target Design
9.5 Target Reflectivity Effects
References
Index
End User License Agreement
List of Tables
Chapter 6
Table 6.1 Summary of properties of different check and verification methods....
List of Illustrations
Chapter 1
Figure 1.1 Available power spectral density as a function of frequency; (a) ...
Figure 1.2 Waves on a lossless transmission line.
Figure 1.3 A linear one‐port.
Figure 1.4 A linear two‐port.
Figure 1.5 Power delivered to a load.
Figure 1.6 A two‐port between source
G
and load
L
.
Figure 1.7 Configuration for computing
T
2
, knowing
T
1
.
Chapter 2
Figure 2.1 Depiction of elements of a primary noise standard.
Figure 2.2 Sketch of the design of a coaxial cryogenic standard..
Figure 2.3 Load, transmission line, and output used in coaxial cryogenic sta...
Figure 2.4 Design of a waveguide cryogenic standard [7].
Figure 2.5 a Silicon carbide wedges used in cryogenic waveguide standard; b ...
Chapter 3
Figure 3.1 Overview of a total‐power radiometer.
Figure 3.2 Reference planes for an isolated total‐power radiometer.
Figure 3.3 Block diagram of a total‐power radiometer.
Figure 3.4 Photo of the NIST coaxial radiometer NFRad.
Figure 3.5 Photo of the disassembled switch head of the NIST coaxial radiome...
Figure 3.6 Details of receiver circuit.
Figure 3.7 Extrapolation involved when measuring hot noise source with cryog...
Figure 3.8 Standard fractional uncertainty as a function of DUT noise temper...
Figure 3.9 Results of noise‐temperature measurements across a range of frequ...
Figure 3.10 Essential components of a Dicke or switching radiometer for labo...
Figure 3.11 Noise‐temperature measurement through an adapter.
Figure 3.12 Results from a comparison of noise‐temperature measurements amon...
Chapter 4
Figure 4.1 Input and output signal and noise of an amplifier.
Figure 4.2 Measurement of noise factor or
T
e
.
Figure 4.3 Equivalent circuit for an amplifier with noise.
Figure 4.4 Measurement setup for measurement of noise parameters.
Figure 4.5 Source and amplifier in the wave representation.
Figure 4.6 Configuration for measurement of reverse noise.
Figure 4.7 Generic measurement setup for amplifier noise parameters.
Figure 4.8 (a) Discrete terminations for input states. (b) Use of a tuner to...
Figure 4.9 (a) General form for the output measurement system. (b) RF receiv...
Figure 4.10 General shape of
T
e
(
Γ
S
)
, from Eq. (4.33).
Figure 4.11 (a) Configuration for measuring the noise parameters of the rece...
Figure 4.12 Measurement results and uncertainties for wave‐representation no...
Figure 4.13 Measurement results and uncertainties for IEEE noise parameters ...
Chapter 5
Figure 5.1 Configuration for on‐wafer noise‐temperature measurement.
Figure 5.2 (a) Standard uncertainty in on‐wafer noise‐temperature measuremen...
Figure 5.3 Definition of reference planes for a transistor.
Figure 5.4 General setup for measuring noise parameters of an on‐wafer trans...
Figure 5.5 On‐wafer reference planes: (a) from above and (b) from the side....
Figure 5.6 Outline of radiometer‐based system for on‐wafer noise‐parameter m...
Figure 5.7 Wave‐representation results for noise parameters of a particular ...
Figure 5.8 Reflection coefficients of input terminations at (a) 2 GHz, (b) 7...
Figure 5.9 Comparison of noise and VNA determinations of
G0 = |S21|2
...
Figure 5.10 (a) System calibration. (b) On‐wafer calibration configuration. ...
Figure 5.11 On‐wafer reference planes.
Figure 5.12 (a) Comparison of measurement results at reference plane D for
F
Figure 5.13 (a) Transistor with parasitic resistances extracted. (b) Theveni...
Figure 5.14 Simple model for a noisy intrinsic FET.
Chapter 6
Figure 6.1 Application of the
T
rev
test, demonstrating problems at low frequ...
Figure 6.2 The three measurement configurations for the tandem test: (a) amp...
Figure 6.3 (a) Application of the tandem test with an isolator to
ReX
12
of a...
Figure 6.4 Application of the tandem test with a mismatched transmission lin...
Chapter 7
Figure 7.1 (a) Two options for treatment of
T
vac
. (b) Vacuum contribution as...
Figure 7.2 Reference planes for noise‐figure measurement on a cryogenic ampl...
Figure 7.3 (a) Noise‐figure measurement for cryogenic amplifier in the cold‐...
Figure 7.4 Auxiliary measurements used to determine
α10, α32, α32, ΔT1′
...
Figure 7.5 (a) Measured gain of a particular cryogenic amplifier. (b) Measur...
Chapter 8
Figure 8.1 Illustration of notation.
Chapter 9
Figure 9.1 Standard‐radiometer configuration.
Figure 9.2 Antenna viewing calibration target.
p
e
represents the effective i...
Guide
Cover Page
Table of Contents
Title Page
Copyright
Dedication
Preface
Begin Reading
Index
Wiley End User License Agreement
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Precision Measurement of Microwave Thermal Noise
James Randa
Spectrum Technology and Research DivisionNational Institute of Standards and TechnologyBoulder, USA
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Published simultaneously in Canada.
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For Susan
Preface
Precision noise measurements are among the most difficult measurements in the microwave realm. The difficulty stems from the miniscule powers that must be measured, and it is compounded by the necessity of accurately measuring multiple reflection coefficients and scattering parameters in order to make the requisite corrections to the power measurements. Fortunately, in many circumstances, simplifying assumptions is justified and imprecise noise measurements suffice. Even in such cases, however, it is necessary to understand and evaluate the full, unsimplified case (at least approximately) in order to determine whether the simplifying approximations are justified.
This book attempts to present the basics of precise measurements of thermal noise at microwave frequencies. The focus is on measurement methods used at the U.S. National Institute of Standards and Technology (NIST), but it is hoped that the general principles and methods will be useful in a wide range of applications. For the most part, the material will comprise methods and theoretical underpinnings, rather than details of instrumentation. In part, this reflects the author's own expertise, but it also permits an emphasis on the general procedures and overall measurement framework. If we consider a qualitative “comprehensibility spectrum” (SI units not yet defined), the target spectrum for this book would be roughly represented by this sketch.
The presentation will be light on first principles and details of instrumentation, instead emphasizing calculations and measurement methods.
Just what constitutes a “precision measurement” varies with the field. Time and frequency measurements are made to 18 significant figures, whereas in other fields uncertainties may be measured in decibels. We will use the criterion that the measurements and methods of interest are those with the lowest uncertainties, such as those made at National Measurement Institutes (NMIs). For thermal noise measurements, this is typically on the order of tenths of a percent. The emphasis will be on accuracy, with only secondary consideration (if that) given to the time required for the measurement. Since “accurate” measurements are defined to be those with the smallest uncertainties, considerable attention will be paid to uncertainty analysis of the various measurement methods. Included in that discussion will be some mention of traceability, which underpins the accuracy of the underlying standards. In some cases, the highest accuracy may be attained by commercially available instrumentation and methods. In such cases, we will try to present the principles underlying the methods rather than the design and use of the instrumentation. The material covered is not groundbreaking or new; however, we hope to present a solid basis for this somewhat arcane but nevertheless important field.
Microwave radiometry originated in remote sensing of celestial bodies, and many or most major advances occurred in that field. This book, however, is concerned with laboratory noise measurements. We do not cover remote‐sensing applications, but we do discuss the relevance of laboratory noise measurements to them.
Much of this book is based on previously published work by the author, referenced in the relevant sections, and from various lectures at Automatic Radio Frequency Techniques Group (ARFTG) short courses and at other laboratories. The material of Chapters 4–6 is based on two earlier review articles [1, 2].
No one works or learns in a vacuum, and I owe a debt of gratitude to numerous colleagues at NIST. These include (but are not limited to) Dave Wait and Bill Daywitt, from whom I learned much about the foundations of the field; Dazhen Gu and Dave Walker, with whom I subsequently learned much new material and who provided fabrication and measurement expertise; and two exceptional technicians, Jack Rice and the late Rob Billinger, who provided fabrication support and precise, reliable measurement results on which any theoretical developments depended. The remote‐sensing work benefitted greatly from the efforts of Amanda Cox as a post‐doc and Derek Houtz as a student. To them and the many other contributors to the NIST Noise Project as well as collaborators elsewhere, thank you.
I am grateful to the editors at Wiley, first for choosing to publish this work and then for their efforts and assistance in overcoming the various hurdles that stand between a submitted manuscript and a real, material book.
On a personal note, I appreciate greatly the companionship and support of family and friends throughout my life. In particular, I am grateful to my parents, John and Catherine (Kay) Randa, for providing life lessons and a firm, stable basis from which to launch myself into the wide world. And, of course, I am grateful to my wife Susan for her continuing support, tolerance, and love through the years.
References
1
J. Randa, “Amplifier and transistor noise‐parameter measurements,” in
Wiley Encyclopedia of Electrical and Electronics Engineering
, Wiley; ed. J. Webster (2014). doi:
10.1002/047134608X.W8219
.
2
J. Randa, “Numerical modeling and uncertainty analysis of transistor noise‐parameter measurements,
The International Journal of Numerical Modelling
, Wiley Online Library (wileyonlinelibrary.com) (2014). doi:
10.1002/jnm.2039
.
1Background
1.1 Nyquist's Theorem and Noise Temperature
1.1.1 Nyquist's Theorem
Conduction electrons in a physical resistor at nonzero temperature are in continual thermal motion, and at any instant this motion induces a voltage v across the terminals of the resistor. Because the motion is random, the induced voltage averages to zero, 〈v(t)〉 = 0, but the average squared voltage is nonzero, 〈v2(t)〉 ≠ 0, and therefore electrical power can be extracted from the resistor. This phenomenon was first measured by Johnson [1] and was explained by Nyquist [2]. The entire field of thermal noise measurement rests on Nyquist's theorem,1 which relates the mean square voltage across a resistor due to thermal motion of its electrons to the physical temperature of the resistor,
where v(f) is the voltage in the differential frequency interval df centered at frequency f, R(f) is the real part of the impedance at f, h is Planck's constant, and kB is Boltzmann's constant.
For microwave frequencies, it is convenient to cast Eq. (1.1) in the form of an equation for the power available from the resistor,
where Pavail(f) is the available power in the interval Δf centered at f, and pavail(f) is the spectral available power density. In general, we will use upper case P to refer to power and lower‐case p to refer to spectral power density. The brackets in Eqs. (1.2) indicate an ensemble or time average (assumed to be the same).
Nyquist's original derivation relied on an analysis of propagation modes in a lossless transmission line. He first derived the classical result (those were the early days of quantum mechanics), which assumed that the total energy per degree of freedom was equal to kBT. That led to
or
He then noted that if instead of kBT the total energy per degree of freedom was hf/(ehf/kT−1) then the result was Eq. (1.1), and thus Eqs. (1.2). We have taken the liberty of using the approximately equal sign in Eqs. (1.3) in order to indicate that the true result is given by Eqs. (1.1) and (1.2).
Nyquist's treatment and result are reminiscent of the problem of black‐body radiation from a heated object, and the thermal noise in an electrical circuit is in fact the one‐dimensional version of black‐body radiation. As in the black‐body radiation case, the classical result, given by Eqs. (1.3), is plagued by the “ultraviolet catastrophe,” the fact that the total power available is infinite when one integrates over all frequencies. The quantum factors provide the necessary damping at high frequency keeping the total energy available finite. A modern, full‐quantum (i.e. second‐quantized) treatment of Nyquist's theorem can be found in [3]. An interesting property that emerges in the full quantum treatment is that an auxiliary field (the noise field) is actually required for a linear two‐port in order for the quantum commutation relations to be consistent (unless the S matrix is the identity matrix or the temperature is zero).
1.1.2 Limits and Numbers
It is instructive to consider the general behavior of the function in Eqs. (1.2). Figure 1.1a,b plots the available power spectral density as a function of frequency on a logarithmic and a linear scale, respectively, for different values of the physical temperature. There is a broad plateau that extends up to high frequency, where the spectral power density drops off precipitously. The low‐frequency behavior is given by expanding Eqs. (1.2) for small f,
which is a constant, independent of frequency, depending only on the physical temperature. Furthermore, that constant is very small; even for a temperature of 10 000 K, the power density in a 1 MHz bandwidth is only 0.138 pW. The high‐frequency behavior of the available power density is dominated by the exponential in the denominator, which drives the power rapidly to zero once hf/(kBT) becomes sizable. The “knee” in the graphs, where the behavior transitions from the low‐frequency constant to the high‐frequency damping, occurs at around f(GHz) ≈ 20× T(K). This transition occurs when quantum effects become important, which is governed by the value of h/kB = 0.04799 K/GHz. Thus departures from the simple constant behavior of the available power become important for very high frequency and/or very low temperature. For example, at 290 K it is a 1% effect at 116 GHz; at 100 K it is a 1% effect at 40 GHz and a 0.1% effect at 4 GHz; at 30 K and 40 GHz it is a 6.4% effect (about 0.26 dB).
Figure 1.1 Available power spectral density as a function of frequency; (a) logarithmic scale, (b) linear scale.
1.1.3 Definition of Noise Temperature
Equations (1.2) relate the available noise power from a passive device to its physical temperature. But microwave circuitry involves more than passive components. It is therefore convenient to define a “noise temperature” for active devices. Many variations have been suggested (and used), but there are two principal ways to do this [4]. The first is to use Eq. (1.2b) and define the noise temperature to be the physical temperature of a passive device that would result in the observed available power density. We will refer to this definition as the “equivalent‐physical‐temperature” definition,
The average in Eq. (1.5a) is taken over the frequency interval Δf, centered at f. This definition is popular in the remote‐sensing community, where the received power is used to measure the physical temperature of the object under observation. This definition has the appealing property that for a passive object or device, the noise temperature is simply the physical temperature. Inverting Eq. (1.5a) to obtain the equations for Tnoise yields a rather complicated expression, a point to which we shall return when considering amplifier noise measurements in Chapter 4.
The other common choice [5], which we adopt here, is to define the noise temperature as the available power spectral density divided by the Boltzmann constant times the frequency interval, which we will call the “Power Definition,”
With this definition, the noise temperature is just a surrogate for the noise power spectral density, which makes this the natural choice when dealing with microwave circuits. With the power definition, the noise temperature of a passive device or object is only approximately equal to the physical temperature,
The approximation of Eq. (1.6) is known as the Rayleigh–Jeans approximation.
Due to the approximate equality in Eq. (1.6), there is usually little difference “in every‐day life” between the power definition and the equivalent‐physical‐temperature definition. However, in precision measurements it is not uncommon to encounter a combination of high frequency, low temperature, and high precision that requires a specific choice of definition. Since this book deals with microwave precision noise measurements, we adopt the power definition, Eq. (1.5b), for the noise temperature. The considerations of Section 1.1.2 above explain when the high‐frequency, low‐temperature corrections become important, and thus when the distinction between different noise‐temperature definition starts to matter.
1.1.4 Excess Noise Ratio and T0
In dealing with noise temperatures and powers of greatly differing magnitudes, it is sometimes useful to define a decibel quantity for noise temperature. The quantity that is commonly used is the Excess Noise Ratio (ENR), defined by
where the reference temperature T0 is taken to be T0 = 290 K. Just to be clear, we treat T0 as a noise‐temperature constant; it is not a noise temperature that corresponds to a physical temperature by virtue of Eq. (1.6). It is just a number.
Some variants of the ENR definition can be found either in the literature or in casual use. Some practitioners use the power delivered to a matched (i.e. reflectionless) load in place of the available power implicit in the Tnoise in Eq. (1.7). This has the effect of introducing a mismatch factor (see Section 1.2.3) into the first term in the numerator of Eq. (1.7). There is also the question [6] of whether the reference temperature should be 290 K, or should it be the noise temperature of a passive load at a physical temperature of 290 K. We take it to be T0 = 290 K, just for simplicity. We are venturing into the hair‐splitting realm here, but in precision measurements sometimes hairs must be split.
1.2 Microwave Networks
1.2.1 Notation
Noise measurements are just a type of power measurements, and in precision noise measurements it is imperative to carefully account for all sources and flows of power. Therefore, before delving into noise measurements, it is necessary to review some basic microwave network theory and to establish the conventionsand notation that will be used in this work. Many books cover microwave circuit theory in some detail, including [7–12]. Here we just summarize the results that will be used. Our approach is similar to that of [13], with appropriate generalizations to include active devices. Unless specifically stated, we assume that all reference planes are in lossless transmission lines which support only a single propagation mode at the frequencies considered. We will typically use a and b to refer to the amplitudes of the traveling waves propagating in the two directions on a transmission line, with the normalization such that the spectral power density of the wave is given by the magnitude of its amplitude squared, |a|2 or |b|2. The net power spectral density delivered to the right in Figure 1.2 is thus given by
Figure 1.2 Waves on a lossless transmission line.
A linear one‐port, as shown in Figure 1.3, is then described by
where ΓG is the reflection coefficient of the one‐port G, and cG is the wave emanating from G due to its intrinsic noise. A linear two‐port, as depicted in Figure 1.4, is described by
where c1 and c2 are the waves due to the intrinsic noise sources in the two‐port.
Figure 1.3 A linear one‐port.
Figure 1.4 A linear two‐port.
1.2.2 Noise Correlation Matrix and Bosma's Theorem
In terms of the wave amplitudes introduced above, the noise correlation matrix of a two‐port device is defined as
This describes the noise properties of the device in a circuit, where there will in general be incident noise waves a. If we are interested in the intrinsic properties of the device itself, we want to know the noise correlation matrix for the case in which there are no incident waves. That is given by the intrinsic noise correlation matrix,
In this book, we will be interested in properties of devices themselves, rather than in their use in circuits, and so we will deal primarily with the intrinsic noise matrix of Eq. (1.12). The exception will occur in Chapter 8, where we analyze multiport amplifiers.
An important result for the intrinsic noise matrix of a passive device is Bosma's theorem [12]. This states that for a passive device at noise temperature Ta the intrinsic noise‐correlation matrix is related to the scattering matrix S by the equation
where I is the identity matrix. This means that the noise properties of a passive device are entirely determined by its S matrix (and its temperature). Although it is used mostly for two‐ports, Eq. (1.13) applies for any number of ports. For a passive two port, the elements of the intrinsic noise‐correlation matrix are given explicitly by
1.2.3 Power Ratios
The power delivered to a load L, Figure 1.5, is given by |a1|2 − |b1|2. If we use b1 = ΓLa1, we can write
Figure 1.5 Power delivered to a load.
For the available power from a source G, we refer to Eq. (1.9) and Figure 1.5 and write the expression for the power delivered from the source to the load L,
