Table of Contents

Cover

Preface

List of Symbols

Introduction

1 Background Noise in Electronics

1.1. Introduction
1.2. Spontaneous fluctuations in electronic components
1.3. Noise factor
1.4. Noise in two-ports
1.5. Characterization of noise in a two-port
1.6. Conclusion

2 Friis Formula

2.1. Introduction
2.2. Calculation method
2.3. Calculation of the admittance parameters of the Q1, Q2 association in cascade
2.4. Contribution of noise generators e_1, i₁, e₂ and i₂ to l₁ and I₄
2.5. e_Tot and i_Tot identification
2.6. Calculation of F₁₂(YS)
2.7. Friis Formula
2.8. Conclusion

3 Adapted Attenuator and Noise Factor

3.1. Introduction
3.2. Calculation of Y and S parameters
3.3. General representation of noise in two-ports
3.4. Equivalent noise generators at the input of the adapted attenuator
3.5. Noise factor on 50 Ω of the adapted attenuator
3.6. Using Bosma’s theorem
3.7. Consequences on the structure of a receiver
3.8. Equivalent noise resistance and noise conductance at the input
3.9. Conclusion

4 Noise Factor Measurement on 50 Ω

4.1. Introduction
4.2. Noise factor measurement by Y factor
4.3. Second stage correction
4.4. Measurement procedure and calculation of the noise factor of the DUT
4.5. Available power gain and insertion power gain of the DUT
4.6. Sample results
4.7. Conclusion

5 Characterization in Noise

5.1. Introduction
5.2. The tuner
5.3. Characterization in noise of the noise measurement chain
5.4. Characterization in S parameters of the device under test
5.5. Noise characterization of the device under test
5.6. Validation of a noise characterization bench
5.7. Conclusion

6 Exercises and Answers

6.1. Exercises
6.2. Solutions

Conclusion: Electronic Noise and its Measurement

Appendix 1: Admittance Parameters of a Two-Port

A1.1. Introduction
A1.2. Definition of admittance parameters
A1.3. Interest of admittance parameters
A1.4. Equivalent diagram of a two-port known by its admittance matrix
A1.5. Flow graph of a two-port known by its admittance matrix
A1.6. Generator + two-port + charge
A1.7. Conclusion on the admittance matrix

Appendix 2: S Parameters of a Two-Port

A2.1. Introduction
A2.2. Definition of S parameters
A2.3. Passage equations of power waves and current / voltage
A2.4. Physical meaning of S parameters
A2.5. Flow graph of a two-port known by its S matrix
A2.6. Generator + two-port + load
A2.7. Conclusion on the S matrix

Appendix 3: Flow Graph and Mason’s Rule

A3.1. Introduction
A3.2. Statement of Mason’s Rule
A3.3. Application of Mason’s rule to a two-port
A3.4. Conclusion and limits of Mason’s rule

Appendix 4: Noise Power Waves

A4.1. Concept of noise power waves
A4.2. Expression of the noise factor
A4.3. Passage relations of b_n1, b_n2 with e,i
A4.4. Noise power waves correlation matrix
A4.5. Inverse relations
A4.6. Conclusion

Bibliography

Index

End User License Agreement

List of Tables

Chapter 1

Table 1.1. BER as a function of NF for a QAM4 with =16 dB

Chapter 4

Table 4.1. Measurement results on a 6 dB attenuator

Chapter 5

Table 5.1. 2.5 dB attenuator, LRM calibration, measured values
Table 5.2. 2.5 dB attenuator, LRM calibration, calculated values
Table 5.3. 2.5 dB attenuator at 3 GHz, LRM calibration, measured values
Table 5.4. STORM cable, LRM calibration, measured values
Table 5.5. STORM cable, LRM calibration, calculated values

Chapter 6

Table 6.1. Noise power measurements at 1 GHz
Table 6.2. Specifications of the ZHL-1010 + Amplifier (Mini-Circuits 2019)
Table 6.3. Figure of minimum noise according to the connection resistance
Table 6.4. Gain and Noise Figure of the amplifier Mini-Circuits ZHL-1010 + (manu...

Appendix 4

Table A4.1. Positive root(s) of a second degree equation

List of Illustrations

Chapter 1

Figure 1.1. Temporal representation of thermal noise
Figure 1.2. Base access resistance of a bipolar transistor
Figure 1.3. Modeling a resistance for thermal noise
Figure 1.4. Calculation of the available thermal noise power
Figure 1.5. Correlation between thermal noises
Figure 1.6. Spectral power density: exce ss noise and thermal noise
Figure 1.7. Signal processing and signal-to-noise ratio
Figure 1.8. Synoptic of the receiver of a digital wireless transmission
Figure 1.9. Noise representation by two noise voltage generators
Figure 1.10. Noise representation by two noise current generators
Figure 1.11. Noise representation by a noise current generator and a noise volta...
Figure 1.12. Calculation of the noise factor of a two-port
Figure 1.13. Noise characterization bench of a two-port. For a color version of ...
Figure 1.14. ESIGELEC noise characterization bench. For a color version of this ...

Chapter 2

Figure 2.1. Two-port at 0K with voltage sources and equivalent noise current
Figure 2.2. Two-ports Q1 and Q2 in cascade at 0K with voltage sources and equiva...
Figure 2.3. Equivalent diagram of Q1 and Q2 in cascade: contribution from genera...
Figure 2.4. Equivalent diagram of Q1 and Q2 in cascade: contribution of the gene...
Figure 2.5. Equivalent diagram of Q1 and Q2 in cascade: contribution of the gene...
Figure 2.6. Equivalent diagram of Q1 and Q2 in cascade: contribution of the gene...
Figure 2.7. Flow graph for the connection between a source E,YS and a charge YL
Figure 2.8. Flow graph of the connection between a source E, YS, a two-port Q kn...

Chapter 3

Figure 3.1. Resistive Π attenuator
Figure 3.2. Two-port at 0K with voltage sources and equivalent noise current
Figure 3.3. Resistive attenuator in π with noise current sources for each conduc...
Figure 3.4. Noise and power waves
Figure 3.5. Flow graph for ΓS = ΓL = 0

Chapter 4

Figure 4.1. Synopsis of the noise measurement bench
Figure 4.2. Calibration of the noise measurement chain
Figure 4.3. Measurement of noise of the DUT + measurement chain
Figure 4.4. Flow graph during calibration
Figure 4.5. Flow graph of the measurement chain during measurement
Figure 4.6. Processing of measurement results by Excel. For a color version of t...

Chapter 5

Figure 5.1. Noise characterization bench of a two-port. For a color version of t...
Figure 5.2. Diagram of a tuner. For a color version of this figure, see www.iste...
Figure 5.3. Flow graph of the noise diode + tuner assembly
Figure 5.4. Simplified flow graph of the noise diode + tuner assembly (ΓD = 0)
Figure 5.5. Tuner calibration. For a color version of this figure, see www.iste....
Figure 5.6. Characterization in noise of the noise measurement chain. For a colo...
Figure 5.7. An SMA 2.5 dB attenuator m / f. For a color version of this figure, ...

Chapter 6

Figure 6.1. Modeling of the receiver
Figure 6.2. Flow graph of DUT and receiver. For a color version of this figure, ...
Figure 6.3. Representation of a noisy MOSFET. For a color version of this figure...
Figure 6.4. Magnitude of the optimum source reflection coefficient for noise as ...
Figure 6.5. 50 Ω Noise Figure as a function of the number of MOSFETs
Figure 6.6. Minimum 50 Ω Noise Figure
Figure 6.7. Evolution of the minimum noise figure as a function of the normalize...
Figure 6.8. Examples of topologies of MOSFETs with multiple gate fingers
Figure 6.9. Extraction results in Excel. For a color version of this figure, see...
Figure 6.10. Two-port known by its admittance matrix loaded at the input by a so...
Figure 6.11. Evolution of as a function of frequency for different gate lengths...
Figure 6.12. Arg (ΓSopt) as a function of the frequency (Asgaram et al. 2004)
Figure 6.13. Arg (ΓSopt) as a function of the frequency (Gao 2004)

Appendix 1

Figure A1.1. Reppresentation of a linear two-port
Figure A1.2. Equivalent diagram of a two-port of known admittance matrix [Y]
Figure A1.3. Flow graph of a two-port of known admittance matrix [Y]
Figure A1.4. Flow graph of generator, two-port and charge association

Appendix 2

Figure A2.1. Representation of the power waves of a linear two-port
Figure A2.2. Generator connection
Figure A2.3. Flowgraph of a reflection coefficient load ΓL
Figure A2.4. Flowgraph of a generator E, Γs
Figure A2.5. Flow graph of a two-port known by its S matrix
Figure A2.6. Flow graph of generator, two-port and load association

Appendix 3

Figure A3.1. Flow graph of a loaded two-port on a port. For a color version of t...
Figure A3.2. Flow graph of matrix two-port Q admittance [Y] fed by source E, Zs,...
Figure A3.3. Flow graph of a two-port Q inserted between a source and a load. Fo...
Figure A3.4. Flow graph of the connection between a source and a load. For a col...

Appendix 4

Figure A4.1. Flow graph of a two-port by noise power waves
Figure A4.2. Two-port inserted between a source Γs and a load ΓL
Figure A4.3. Two-port inserted between a source ΓS = 0 and a load ΓL = 0
Figure A4.4. Equivalent noise diagram of matrix two-port admittance [Y] loaded a...
Figure A4.5. Simplified two-port noise equivalent diagram for the calculation of...
Figure A4.6. Simplified two-port noise equivalent diagram for the calculation of...

First published 2020 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of François Fouquet to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2019953869

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-78630-532-9

Preface

As a professor/researcher at ESIGELEC since 1982, I have been confronted with “noise in radiofrequency electronics” as part of my teaching activities, as also during my research work.

For my teaching activities, it was in the second half of the 1980s, that I created course materials, tutorials and practical work on this subject for students of ESIGELEC pursuing courses in the field of radiofrequencies and microwaves and for technicians and engineers working for companies in these fields.

As far as my research activities are concerned, my first “in-depth” contact with “electronic noise” dates back to the time of my DEA diploma and then my doctorate, when I had to quantify “by hand” the impact of the use of “active load polarization” on the noise factor of a “common gate MESFET”.

In both cases, I was confronted with the same problem: unsuitable bibliographic sources. The reason being either:

– we read course documents, where only the basics were presented and often briefly, sometimes omitting notation details, making the information difficult to use and where long demonstrations were omitted too because they can be a bother but could allow an understanding of what was really going on; or
– we read articles in scientific journals, and there the prerequisites were high level and the long demonstrations, vital for the correct use of the information, had to be reconstituted by the reader from the assumptions provided and the result given. It could take days, weeks or even months, depending on the reader’s level.

It is to fill a part of the gap that exists between these two worlds that I decided to write this document to summarize what I have essentially learned about noise during the last 40 years.

On this occasion, I would particularly like to thank Mr. J.L. Gautier and Mr. D. Pasquet who initiated me to study this subject during my DEA diploma studies and my doctorate, while they were professors at ENSEA.

I would also like to thank Mr. M. Rivette and Mr. J.B. Dioux, alumni of ESIGELEC, who inspired me to do this work while I was correcting their report on the noise figure of adapted attenuator.

François FOUQUET

November 2019

List of Symbols

*	Conjugate operator on a complex, c = a + jb,c = a — jb*
a_i	Incident power wave on port i of a two-port, equal to a_t = with and
b_i	Reflected power wave on port i of a two-port, equal to b_i = with and
b_ni	Noise power wave outputted from port i of a two-port
B_Cor	Imaginary part of the correlation admittance Y_Cor, between e and i of normalized value b_Cor
B_Sopt	Imaginary part of the optimal admittance for the Y_Sopt noise, of normalized value b_Sopt
e	Equivalent noise voltage source at the input of a two-port, see Figure 1.11
ENR	Excess noise ratio of a noise source at two temperatures T_c(cold temperature) and T_H (hot temperature), equal to ENR = and expressed in dB: ENR_dB = 10 · log₁₀(ENR)
F	Noise factor of a two-port, information on the degradation of the signal / noise ratio between the input and the output of the two-port, depends on the load presented at the input of the two-port, F >1
F_eMin	The minimum excess noise factor of a two-port, equal to F_{eMin =}F_Min– 1
F_Min	The minimum noise factor of a two-port, obtained for the optimal load for noise presented at the input of the two-port
Γ_sopt	Optimum reflection coefficient for the noise to present at the input of the two-port to obtain F = F_Min, equal to Γ_Sopt =
G_Avai(Ys)	Gain in Available power gain of a two-port for a source admittance Y_s, equal to the value of the transducer power gain G_T(Y_S, Y_L) for Y_L = Y_OUT*
G_n	Equivalent noise conductance at the input of a two-port, of normalized value g_n = G_n· R₀
G_sopt	Real part of the optimal admittance for the Y_soptnoise, of normalized value g_sopt
G_T(Y_S, Y_l)	Transducic power gain of a two-port for a source admittance Y_sand a load admittance Y_L, equal to the ratio of the power collected in Y_L and the power available at the source
G₀	Normalization admittance equal to 1/R₀or 20 mS
h	Planck’s constant 6.23x10^-34J s
i	Equivalent noise current generator at the input of a two-port, see Figure 1.11
Im(c)	“Imaginary part” operator on the complex c
i_n	Equivalent noise current generator at the input of a two-port not correlated with e
i₁	Noise current generator equivalent to the input of a two-port, see Figure 1.10
i₂	Noise current generator equivalent to the input of a two-port, see Figure 1.10
j	Square root of - 1 which allows to describe a complex in the form c = Re(c) + j · Im(c)
k	Boltzmann’s constant 1.38×10^-23 J K^-1
N_C	Noise power measured at the output of a two-port when the noise source is at cold temperature T_c
N_H	Noise power measured at the output of a two-port when the noise source is at hot temperature T_H
NF	Noise Figure of a two-port, equal to 10 · log₁₀(F), NF > 0dB
μ_n	Electron mobility in cm² s^-1 V^-1
P	Power dissipated in a load Y, equal to Re(V · I) with I = Y -V*
P_sAvai	Available power of a source E,Y_S = G_s + jB_s, equal to
Re(c)	“Real part” operator on the complex c
R_N	Equivalent noise resistance at the input of a two-port, = 4kT · df · R_N , of normalized value
R₀	Normalization resistance equal to 50 Ω
S_ij	S parameters of a two-port of ports i,j
T	Temperature in K
T_e	Equivalent noise temperature of a two-port equal to T_e = T₀ · (F — 1), expressed in K
T_eMin	Equivalent minimum noise temperature of a two-port equal to T_eMin = T₀ · (F_Min — 1) , expressed in K
T₀	Reference temperature for the noise factor equal to 290 K
Y	“Y” factor, used by the measurement of the factor, equal to
Y_Cor	Correlation admittance between e and i, defined by i = Y_Cor ·
Y_ij	Admittance parameters of a two-port of ports i, j
Y_sopt	Optimum Admittance for the noise to present at the input of the two-port to obtain F = F_Min, of normalized value y_Sopt