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A Molokovsky Sergey I
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Springer Series in Advanced Microelectronics 61

Andrey D. Grigoriev
Vyacheslav A. Ivanov
Sergey I. Molokovsky

Microwave
Electronics
Edited by Professor A. D. Grigoriev
Springer Series in Advanced Microelectronics

Volume 61

Series editors
Kukjin Chun, Seoul, Korea, Republic of (South Korea)
Kiyoo Itoh, Tokyo, Japan
Thomas H. Lee, Stanford, CA, USA
Rino Micheloni, Vimercate (MB), Italy
Takayasu Sakurai, Tokyo, Japan
Willy M. C. Sansen, Leuven, Belgium
Doris Schmitt-Landsiedel, München, Germany
The Springer Series in Advanced Microelectronics provides systematic information
on all the topics relevant for the design, processing, and manufacturing of
microelectronic devices. The books, each prepared by leading researchers or
engineers in their fields, cover the basic and advanced aspects of topics such as
wafer processing, materials, device design, device technologies, circuit design,
VLSI implementation, and subsystem technology. The series forms a bridge
between physics and engineering and the volumes will appeal to practicing
engineers as well as research scientists.

More information about this series at http://www.springer.com/series/4076

Andrey D. Grigoriev Vyacheslav A. Ivanov
•

Sergey I. Molokovsky

Microwave Electronics
Edited by Professor A. D. Grigoriev

123
Andrey D. Grigoriev Sergey I. Molokovsky (deceased)
Saint-Petersburg Electrotechnical Saint-Petersburg Electrotechnical
University “LETI” University “LETI”
Saint-Petersburg Saint-Petersburg
Russia Russia

Vyacheslav A. Ivanov
Saint-Petersburg Electrotechnical
University “LETI”
Saint-Petersburg
Russia

ISSN 1437-0387 ISSN 2197-6643 (electronic)

Springer Series in Advanced Microelectronics
ISBN 978-3-319-68890-9 ISBN 978-3-319-68891-6 (eBook)
https://doi.org/10.1007/978-3-319-68891-6
Library of Congress Control Number: 2017957671

Copyright for the English version is with the authors.

© Springer International Publishing AG 2018
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are exempt from
the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained herein or
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper

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The registered company is Springer International Publishing AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Nature is simple in its laws,
but immeasurably rich and diverse in their
applications!
G. W. Leibniz
Preface

Microwave electronics is currently a large and rapidly developing branch of science

and technology, which has a huge impact on the country’s economy and defense
capability. Most telecommunication systems, radio astronomy, accelerative tech-
niques, thermonuclear machines, technology and medicine, and practically all types
of weapons use microwave electronics technology to some extent. These
achievements are mostly due to the use of new interaction mechanisms of charged
particle fluxes with electromagnetic fields, use of new materials, and new manu-
facturing technologies of microwave electronics devices and systems.
The physical laws and phenomena used in microwave electronic devices, and the
operating principles of these devices, along with their design, characteristics and
parameters form the subject of the “Microwave Electronics” discipline. These laws,
phenomena and devices are studied more deeply in the masters programs of the
corresponding field.
Unfortunately, the textbooks and manuals on this discipline existing at the time
of preparation of this manuscript were either published a long time ago and are out
of date, or do not cover all the discipline’s content. In particular, there are no
modern manuals considering the processes taking place in a vacuum and in
semiconductor microwave devices from the unified position.
The proposed textbook describes in reasonable detail the main interaction
mechanisms of the microwave electromagnetic field with charged particles in a
vacuum and in a solid. Much attention is paid to the peculiarities of these mech-
anisms using vacuum and solid-state microwave electronic devices. This theoretical
part is supplemented by a description of the design of the devices, their parameters
and characteristics.
The authors hope that the proposed textbook will assist students studying
microwave electronics and related areas.
The book will also be useful to postgraduates and specialists engaged in the
development and application of microwave electronics products.
The textbook is based on the lecture course taught by the authors to bachelors
and masters in St. Petersburg State Electrotechnical University “LETI”.

vii
viii Preface

The authors express also their gratitude to V. B. Yancevich for valuable advices
and assistance in preparing the book for publication. The authors are also sincerely
grateful to V. B. Yankevich, the Head of the LETI Radio-technical Electronics
Department, for valuable advice and invaluable assistance in preparing the manu-
script for publication.
The Introduction, Chapters 1, 3, 5... 8, Chapter 10 and Appendix C was written by
A. D. Grigoriev, Chapter 12... 14 and Appendix A was written by V. A. Ivanov,
Chapters 2, 4 and 11 was written together by A. D. Grigoriev and V. A. Ivanov,
Chapter 9 and Appenix B was written together by A. D. Grigoriev and S. I.
Molokiovsky.

Saint-Petersburg, Russia Vyacheslav A. Ivanov

Contents

Part I Microwave Electronics Physical Foundations

1 Main Stages of Microwave Electronics Development . . . . . . . . . . . 3
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Microwave Vacuum Electronics . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Semiconductor Microwave Electronics . . . . . . . . . . . . . . . . . . . 7
1.4 Comparative Characteristics of Vacuum and Semiconductor
Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 8
1.5 Prospects for the Development of Microwave Electronics . . ... 9
2 Interaction of Charged Particles with an Alternating
Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Radiation of Individual and Collective Charged Particles . . . . . 11
2.2 Macroscopic Equations of Microwave Electronics . . . . . . . . . . . 16
2.3 Motion Equations of Charged Particles . . . . . . . . . . . . . . . . . . 18
2.3.1 Motion of a Single Particle in Vacuum . . . . . . . . . . . 18
2.3.2 The Particles Ensemble Motion in Vacuum . . . . . . . . 20
2.3.3 The Particles Ensemble Motion in Solid . . . . . . . . . . 22
2.4 Material Parameters and Relaxation Processes . . . . . . . . . . . . . 24
2.5 Noises in Microwave Devices . . . . . . . . . . . . . . . . . . . . . . . . . 32
Advancement Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Oscillations and Waves in Charged Particle Beams . . . . . . . . . . . . 43
3.1 Space Charge Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 Space Charge Waves in Electron Beams . . . . . . . . . . . . . . . . . 45
3.3 Charge Carrier Waves in Semiconductors . . . . . . . . . . . . . . . . . 49
Advancement Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

ix
x Contents

4 Interaction of Charged Particle Fluxes with a High-Frequency

Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 53
4.1 Interaction Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 53
4.2 Interaction with Quasi-Static Field, the Induced Current.
The Shokley-Ramo Theorem . . . . . . . . . . . . . . . . . . . . . . . ... 57
4.3 Current in the Flat Interelectrode Gap
and Its External Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 59
4.4 Electric Gap Field Effect on the Motion
of Charged Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 63
4.5 Energy Exchange Between Electrons and the Gap Field . . . ... 66
4.6 Interaction of Charged Particles with a Travelling Wave
Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 70
Advancement Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 71
5 A Microwave Device as a Circuit Element . . . . . . . . . . . . . . . . . . . 73
5.1 Microwave Devices Requirements . . . . . . . . . . . . . . . . . . . . . . 73
5.2 Classification of Microwave Devices . . . . . . . . . . . . . . . . . . . . 74
5.3 The Basic Functional Components of Electron Devices . . . . . . . 76
5.4 Parameters and Characteristics of Microwave Devices . . . . . . . . 78
5.4.1 Device Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.4.2 Characteristics of Microwave Devices . . . . . . . . . . . . 79
Advancement Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Part II Microwave Vacuum Electron Devices

6 Devices with Quasi-static Control . . . . . . . . . . . . . . . . .......... 87
6.1 General Characteristics and Parameters of Devices
with Quasi-static Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 The Monotron and Diode Admittance . . . . . . . . . . . . . . . . . . . 90
6.3 Operating Modes of Electron Tubes . . . . . . . . . . . . . . . . . . . . . 92
6.4 Amplifier Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.5 The Influence of Cathode Contact Inductance . . . . . . . . . . . . . . 96
6.6 The Influence of Space Charge and Displacement
Current in the Cathode-Grid Space . . . . . . . . . . . . . . . . . . . . . 98
6.7 Motion of Electrons in the Grid-Anode Space . . . . . . . . . . . . . 100
6.8 Modern Medium and High Power Tetrodes . . . . . . . . . . . . . . . 101
6.9 Microwave Vacuum Microelectronics Devices . . . . . . . . . . . . . 104
Advancement Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7 O-Type Microwave Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.1 General Characteristics of O-Type Devices . . . . . . . . . . . . . . . . 109
7.2 Klystrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.2.1 The Structure and Operating Principle
of the Double-Cavity Transit-Time Klystron . . . . . . . 110
7.2.2 Velocity Modulation in the Interaction Gap . . . . . . . . 111
7.2.3 The Kinematic Theory of Bunching . . . . . . . . . . . . . . 113
7.2.4 Effect of Longitudinal Electron Repulsion . . . . . . . . . 118
Contents xi

7.2.5 The Extraction of Energy from the Bunched Electron

Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7.2.6 Multi-Cavity Klystrons . . . . . . . . . . . . . . . . . . . . . . . 125
7.2.7 Extended Interaction Klystrons . . . . . . . . . . . . . . . . . 134
7.2.8 Multi-Beam and Multi-Barrel Klystrons . . . . . . . . . . . 136
7.2.9 Sheet Beam Klystrons . . . . . . . . . . . . . . . . . . . . . . . . 140
7.2.10 Structure, Parameters and Characteristics of Modern
Klystrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.2.11 Other Types of Klystrons . . . . . . . . . . . . . . . . . . . . . 147
7.3 Travelling Wave Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.3.1 Operating Principle of Travelling Wave Tubes . . . . . . 154
7.3.2 The Linear Theory of O-Type TWTs . . . . . . . . . . . . . 157
7.3.3 Elements of the Nonlinear Theory of TWTs . . . . . . . . 171
7.3.4 Methods of Increasing TWT Efficiency . . . . . . . . . . . 177
7.3.5 TWT Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.3.6 Parameters and Application Regions of TWTOs . . . . . 182
7.4 Backward-Wave Oscilators . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.4.1 Operating Principle of Backward-Wave Tubes . . . . . . 184
7.4.2 Linear Theory of BWOs . . . . . . . . . . . . . . . . . . . . . . 186
7.4.3 Electronic Tuning of BWOs . . . . . . . . . . . . . . . . . . . 189
7.4.4 Electronic Efficiency of BWOs . . . . . . . . . . . . . . . . . 190
7.4.5 Resonance BWOs . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
7.4.6 Design and Parameters of BWOs . . . . . . . . . . . . . . . 191
7.5 O-Type Hybrid Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.5.1 Hybridization Advantages . . . . . . . . . . . . . . . . . . . . . 193
7.5.2 The TWYSTRON . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.5.3 The Klystrode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
7.5.4 The Orotron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Advancement Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
8 M-Type Microwave Electron Devices . . . . . . . . . . . . . . . . . . . . . . . 203
8.1 General Characteristics of M-type Devices . . . . . . . . . . . . . . . . 203
8.2 Interaction of Electrons with the High-Frequency Field
in M-type Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
8.2.1 Motion of Electrons in Constant Crossed Fields . . . . . 204
8.2.2 Interaction of Electrons with the Slow Wave . . . . . . . 208
8.2.3 Linear Interaction Theory in M-type Devices . . . . . . . 210
8.3 M-type Devices with an Open Electron Beam . . . . . . . . . . . . . 217
8.3.1 The Traveling-Wave Tube of M-type . . . . . . . . . . . . . 217
8.3.2 The M-type Backward-Wave Oscillator . . . . . . . . . . . 221
8.4 M-type Devices with a Re-entrant Beam . . . . . . . . . . . . . . . . . 223
8.4.1 The Multi-cavity Magnetron . . . . . . . . . . . . . . . . . . . 223
8.4.2 Other Types of Magnetron . . . . . . . . . . . . . . . . . . . . 242
8.4.3 The Platinotron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Advancement Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
xii Contents

9 Gyro-resonant Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

9.1 The Operating Principle of Gyro-resonant Devices . . . . . . . . . . 255
9.2 Electron Beam Interaction with the High-Frequency
Electrical Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
9.2.1 Cyclotron Resonance . . . . . . . . . . . . . . . . . . . . . . . . 256
9.2.2 Azimuthal Bunching . . . . . . . . . . . . . . . . . . . . . . . . . 258
9.2.3 Equations of Electron Motion . . . . . . . . . . . . . . . . . . 261
9.2.4 Abridged Motion Equations . . . . . . . . . . . . . . . . . . . 263
9.2.5 Field and Electrons Interaction on Cyclotron
Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
9.3 The Gyrotron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
9.3.1 The Design and Operating Principle
of the Gyrotron . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
9.3.2 Electronic Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 266
9.3.3 Total Efficiency and Output Power . . . . . . . . . . . . . . 269
9.3.4 Gyrotron Starting Current . . . . . . . . . . . . . . . . . . . . . 271
9.3.5 Influence of the Spread of Electron Velocities
on Gyrotron Operation . . . . . . . . . . . . . . . . . . . . . . . 271
9.3.6 Large-Orbit Gyrotrons . . . . . . . . . . . . . . . . . . . . . . . 272
9.3.7 Parameters and Applications of Gyrotrons . . . . . . . . . 273
9.4 Gyroklystrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
9.4.1 Gyroklystron Design . . . . . . . . . . . . . . . . . . . . . . . . . 274
9.4.2 Azimuthal Bunching in Gyroklystrons . . . . . . . . . . . . 276
9.4.3 Parameters and Applications of Gyroklystrons . . . . . . 278
9.5 The Gyro-TWT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.5.1 Gyro-TWT Design . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.5.2 Features of Beam and Field Interaction . . . . . . . . . . . 280
9.6 The Gyro-BWO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
Advancement Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
10 Relativistic Microwave Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
10.1 General Characteristics of Relativistic Microwave Devices . . . . 285
10.2 Classical Relativistic Devices . . . . . . . . . . . . . . . . . . . . . . . . . . 286
10.2.1 Relativistic Klystrons . . . . . . . . . . . . . . . . . . . . . . . . 286
10.2.2 Relativistic TWTs and BWOs . . . . . . . . . . . . . . . . . . 288
10.2.3 Relativistic Magnetrons . . . . . . . . . . . . . . . . . . . . . . . 290
10.3 Free-Electron Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
10.3.1 Working Principle of Free-Electron Lasers . . . . . . . . . 292
10.3.2 The Ubitron—The Predecessor of the FEL . . . . . . . . . 293
10.3.3 The FEL—Relativistic Ubitron-Self-Oscillator . . . . . . 297
10.3.4 Analysis of Radiation Processes in the FEL . . . . . . . . 299
10.3.5 FEL-Scattertron . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
10.3.6 High-Current FEL . . . . . . . . . . . . . . . . . . . . . . . . . . 301
10.3.7 X-Ray Free-Electron Laser . . . . . . . . . . . . . . . . . . . . 302
Contents xiii

10.4 Vircators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

10.4.1 Virtual Cathode Effect . . . . . . . . . . . . . . . . . . . . . . . 307
10.4.2 Types and Parameters of Vircators . . . . . . . . . . . . . . 308
10.4.3 Low-Voltage Vircators . . . . . . . . . . . . . . . . . . . . . . . 311
10.5 Gyrocons and Magnicons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Advancement Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

Part III Semiconductor Microwave Devices

11 Key Functional Elements of Semiconductor Microwave Devices . . . 319
11.1 Elements of the Electronic Band Structure . . . . . . . . . . . . . . . . 319
11.2 Semiconductor Materials for Microwave Electronics . . . . . . . . . 323
11.2.1 Common Semiconductor Materials . . . . . . . . . . . . . . 323
11.2.2 Graphene as a Semiconductor for the Microwave
Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
11.3 Functional Elements of Microwave Semiconductor Devices
(MSD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
11.3.1 Features of the MSD Functional Scheme . . . . . . . . . . 328
11.3.2 Uniformly Doped Semiconductors . . . . . . . . . . . . . . . 329
11.3.3 Metal-Semiconductor Contact Properties . . . . . . . . . . 330
11.3.4 Properties of the p-n Junction . . . . . . . . . . . . . . . . . . 335
11.3.5 Ohmic Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
11.4 Classification of Microwave Semiconductor Devices . . . . . . . . . 343
Advancement Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
12 Diodes with Positive Dynamic Resistance . . . . . . . . . . . . . . . . . . . . 345
12.1 Detector Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
12.1.1 Designation and Design of Detector Diodes . . . . . . . . 345
12.1.2 Static and Dynamic Characteristics . . . . . . . . . . . . . . 349
12.1.3 Dynamic Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 350
12.1.4 Circuit Application . . . . . . . . . . . . . . . . . . . . . . . . . . 355
12.2 Mixer Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
12.2.1 Functional Designation and Usage Principle
of the Mixer Diode . . . . . . . . . . . . . . . . . . . . . . . . . . 356
12.2.2 Mixer Diode Schemes . . . . . . . . . . . . . . . . . . . . . . . . 359
12.3 p-i-n Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
12.3.1 Structure, Principle of Operation and Equivalent
Circuit of the p-i-n Diode . . . . . . . . . . . . . . . . . . . . . 361
12.3.2 Peculiarities of the Use of p-i-n Diodes
in Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
12.4 Varactor Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
12.4.1 Structure, Equivalent Circuit and Applications
of Varactor Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . 368
12.4.2 Varactor Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 369
12.4.3 Heterostructure Barrier Varactor (HBV diode) . . . . . . 372
xiv Contents

12.4.4 Applications of Varactor Diodes . . . . . . . . . . . . . . . . 373

12.4.5 Manley-Rowe Relations . . . . . . . . . . . . . . . . . . . . . . 376
12.4.6 Parametric Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . 378
Advancement Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
13 Diodes with Negative Dynamic Resistance . . . . . . . . . . . . . . . . . . . 385
13.1 General Characteristics of Diodes with Negative Dynamic
Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
13.2 Analysis of Semiconductor Sample Dynamic Resistance . . . . . . 387
13.3 Ways to Obtain an Alternating Convection Current
in a Diode Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
13.4 IMPATT Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
13.4.1 Structure and Operation Principle of the IMPATT
Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
13.4.2 Analysis of the Processes in the Avalanche
Zone. Equivalent Resistance . . . . . . . . . . . . . . . . . . . 401
13.4.3 Small-Signal Impedance of the IMPATT Diode . . . . . 406
13.4.4 Nonlinear Operating Mode of the IMPATT
Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
13.4.5 IMPATT Diodes Operating in Trapped Plasma
Transit Mode (TRAPATT) . . . . . . . . . . . . . . . . . . . . 412
13.4.6 IMPATT Diode Structure and Design . . . . . . . . . . . . 415
13.4.7 Structure and Parameters of IMPATT Diode
Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
13.5 Injection-and-Transit-Time Diodes . . . . . . . . . . . . . . . . . . . . . . 420
13.6 Transferred Electron Devices . . . . . . . . . . . . . . . . . . . . . . . . . . 421
13.6.1 The Gunn Effect. The Running High-Field
Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
13.6.2 Distribution of Static Field in the Gunn Diode . . . . . . 426
13.7 Tunnel Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428
13.7.1 Structure and Operating Principle . . . . . . . . . . . . . . . 428
13.7.2 Equivalent Circuit. Features of Use in the
Microwave Band . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
13.7.3 Resonance Tunnel Diode (RTD) . . . . . . . . . . . . . . . . 432
Advancement Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
14 Microwave Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
14.1 Field Effect Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
14.1.1 Structure of the Schottky Field Effect Transistor . . . . . 437
14.1.2 Static Characteristics of Schottky Field Effect
Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440
14.1.3 Small-Signal Parameters and Equivalent MESFET
Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443
14.1.4 Modelling of Field Effect Transistors . . . . . . . . . . . . . 450
14.1.5 Peculiarities of Mathematical Modeling of Field
Effect Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
Contents xv

14.1.6
Quasi-Two-Dimensional Temperature Model
of MESFET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
14.1.7 Noise Characteristics of Field Effect Transistors . . . . . 463
14.1.8 Noise Parameters of the Transistor as a Function
of the Working Regime . . . . . . . . . . . . . . . . . . . . . . 466
14.1.9 High Electron Mobility Field Effect Transistor . . . . . . 467
14.1.10 Developmental Prospects of Microwave Field Effect
Transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
14.2 Microwave Bipolar Transistors . . . . . . . . . . . . . . . . . . . . . . . . 473
14.2.1 Structure and Operating Principle . . . . . . . . . . . . . . . 473
14.2.2 Equivalent Circuits and HF Parameters of BT . . . . . . 475
14.2.3 Heterojunction Bipolar Transistors . . . . . . . . . . . . . . . 478
14.3 Microwave Transistor Specifics . . . . . . . . . . . . . . . . . . . . . . . . 480
14.3.1 Physical and Technological Limitations of Creating
Microwave Transistors . . . . . . . . . . . . . . . . . . . . . . . 480
14.3.2 Transistor “Family Tree” . . . . . . . . . . . . . . . . . . . . . . 481
14.3.3 Comparison of Transistor Speeds . . . . . . . . . . . . . . . 484
14.3.4 New Type of Transistors: Graphene FET . . . . . . . . . . 485
14.4 Using Transistors in Hybrid and Monolithic IC in the
Microwave Band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
Advancement Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
Appendix A: Time and Space Intervals Defining the Behavior
of Charged Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
Appendix B: Electron-Optical Systems of Microwave Devices . . . . . . . . . 499
Appendix C: Electrodynamic Systems of Microwave Electron
Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
Bibliography List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
Notations

Scalar values are denoted by Latin letters, in italic, and Greek letters typed in a
direct font: a, v, A, u, w, etc. Vectors and tensors are denoted by Latin and Greek
letters in bold direct type: A; B; W. When necessary, cases the notation of vectors,
tensors, and matrices are enclosed in direct brackets: jAj; jBj or are over lined: e; l
.
Complex quantities, when necessary, are marked with a dot above the symbol:
_ A.
_ A;
q; _ Constants are typed in a direct font: e, i. The scalar product is denoted by a
point: A
B, and the vector product by a sidelong cross: A  B. To denote the
differential operations on vectors, the Hamiltonian operator r is used.

a Acceleration, m/s2
A Vector potential, V
s/m
B Magnetic induction, V
s/m2
B Susceptance, S
c = 2.9979  108 Speed of light in a vacuum, m/s
C Capacitance, F
D Electric displacement vector, А
s/m2
D Diffusion coefficient, m2/s
e = 2.71828 Base of the natural logarithms
e = 1.602  10−19 Electron charge absolute value, C
E Electric field strength, V/m
f Frequency, Hz
F Force, N
G Conductance, S
G Amplification factor
h = 6.626  10−34 Planck constant, J
s
H Magnetic field intensity, А/m
j Imaginary unit
i, I Current, А
j, J Current density, А/m2
k, k Wave number, wave vector, 1/m

xvii
xviii Notations

k = 1.38  10−23 Boltzmann constant, J/K

L Inductance, H
M Mutual inductance, H
M Beam coupling coefficient
n Concentration, m−3
nm Medium refractive index
n p, n g Deceleration of wave phase and group velocity
P Power, W
Pw Power density (energy-flux density), W/m2
p Momentum, kg
m/s
Q Q-factor
q Electric charge, C
R Resistance, Ohm
Rc Interaction impedance, Ohm
Re Resonator equivalent resistance, Ohm
T Temperature, К
U Voltage (electric potential), V
v, v Velocity, m/s
W Energy, J
w Energy density, J/m3
X Reactance, Ohm
Y Admittance, S
Yg Transmission line wave admittance, S
Z Complex impedance, X
Z0 Resonator wave impedance, X
Zg Transmission line wave impedance, X
ɑ Attenuation constant, m−1
b Phase constant, m−1
c = b − iɑ Propagation constant, m−1
d Field penetration depth, m
er Relative dielectric constant
e0 = 107/(4pc2) Dielectric constant, А
s/(V
m)
η Efficiency
η0 = 120p Intrinsic impedance of free space, Ohm
h Transit angle, rad
k Electromagnetic wavelength in a vacuum, m
kc Cutoff wavelength in transmission line, m
kg Wavelength in transmission line, m
l Charge carrier mobility, m2/(V
s)
lr Relative permeability
l0 = 4p  10–7 Magnetic constant, V
s/(А
m)
P Poynting vector, W/m2
q Volume electric charge density, C/m3
q Characteristic impedance, Ohm
r Conductivity, S/m
Notations xix

s Relaxation constant, s
u Phase, rad
U Scalar potential, V
w Probability function
W Magnetic flux, Wb
x Angular frequency, rad/s
Introduction

The subject of microwave electronics is the study of the physical processes

occurring in electronic devices intended for generating, amplifying and converting
microwave band electromagnetic oscillations, as well as the development of
methods for the design and engineering of these devices and recommendations for
their application.
In accordance with the recommendations of the International Electrotechnical
Commission the microwave range is the part of the electromagnetic oscillations
spectrum from 3  108 to 3  1011 Hz (300 MHz to 300 GHz), that corresponds to
wavelengths in a vacuum from 1 to 1 mm. The microwave range is divided into
several bands:
ultrahigh frequencies (UHF or decimeter waves)—the frequency band from 300
MHz to 3 GHz (wavelengths are 1 m to 10 cm);
superhigh frequencies (SHF or centimeter waves)—the frequency band from 3 to
30 GHz (wavelengths are 10–1 cm);
extremely high frequencies (EHF or millimeter waves)—the frequency band
from 30 to 300 GHz (wavelengths are 1 cm to 1 mm).
Often the microwave range includes adjacent frequency bands. For the low
frequencies it is the very high frequencies (VHF) band or meter waves (MW)—the
frequency range is 30–300 MHz (wavelengths are 10–1 m), and for the high
frequency side—the hyperhigh frequency band (HHF) or decimillimeter waves
(DMMW)—the frequency range is 300 MHz to 3 THz (wavelengths are 1 mm to
100 lm). This range is often called the terahertz range, and waves in this range are
called T-rays. Currently researchers are paying special attention to this range.
As can be seen, microwaves occupy a large portion of the spectrum with an
extreme frequency ratio of 1:1000 (or 1:100,000, if we add the adjacent bands).
They are located on the frequency scale between radio and optical waves.
Historically, people began to use the optical range foremost when for the first time
an ancient man lit a fire in a cave to warm and light it. At the end of the nineteenth
century thanks to the works of H. Hertz, A. S. Popov, G. Marconi and many other
scientists, the radio range’s turn came.

xxi
xxii Introduction

With the development of radio engineering, the advantages of using higher

frequencies became increasingly evident. Therefore, in the mid-1930s the mastering
of the microwave range began. Both researchers and equipment designers met with
great difficulties, since the methods of generating, amplifying, detecting, and
canalization of the electromagnetic radiation developed for neighboring ranges
proved to be unsuitable for microwaves. The use of optical methods was prob-
lematic because microwave quantum energy is small compared to the energy
of thermal motion. The transfer of methods from the radio range was hindered by
the long electron-transit time in the active area of devices compared to the period of
oscillations, as well as large parasitic capacitances and inductances of the design
elements of devices. As a result, it was necessary to develop new mechanisms of
charged particles interaction with the electromagnetic field, as well as new designs
of devices and wave guiding structures. The first practically realized mechanisms
and constructions of microwave devices appeared at the end of the 1930s.
Radar creation and improvement prior to the Second World War and then during
the war gave a powerful boost to the development of microwave technology and
electronics. After the end of the war, the development of microwave electronics
continued at a heightened pace; there appeared new areas of its application—radio
astronomy, radio spectroscopy, charged particle accelerators, thermonuclear fusion
reactors, medical apparatus, and microwave heating facilities. By the end of the last
century, the microwave systems of telecommunications including cellular and
satellite communications and global positioning systems began to develop at a rapid
pace. The use of microwaves in biology and medicine, in chemistry, in the food
manufacturing industry, in new materials production technology, in logistics and in
other fields of science and industry is expanding.
The formation and development of microwave electronics became possible
thanks to the works of numerous scientists and engineering teams from many
countries.
Today, microwave electronics is a synthetic field of knowledge that unites
fundamental sciences (electrodynamics, plasma and solid state physics, mathe-
matical modeling), engineering, increasingly complex production technologies, and
modern measuring and testing equipment.
Successful work in this area requires deep theoretical knowledge and practical
skills.
In this textbook, the authors attempted to consider the processes of electro-
magnetic field interaction with fluxes of charged particles in a vacuum, in plasma,
and in a solid, from a unified position. This approach allowed the identification of
common features and differences in the designs, characteristics and parameters of
various microwave devices.
The textbook contains three parts. In the first part the mechanisms of individual
and collective electron emission, wave and oscillating processes in electron fluxes
are considered, and the basic concepts of microwave electronics are introduced.
The second part is devoted to vacuum microwave devices. Electron optical and
electrodynamics systems of these devices are considered, and their operation
Introduction xxiii

principles, designs, characteristics and parameters are expounded on. Along with
“ordinary” devices, relativistic microwave electronic devices are also considered.
The third part describes the principle of operation, design and parameters of
semiconductor microwave devices. Much attention is paid to the devices with new
wide-band materials, namely silicon carbide and gallium nitride.
In conclusion, the problems facing the developers of microwave devices and
identified and discussed, along with their possible solutions.
 Part I
Microwave Electronics Physical
Foundations
Chapter 1
Main Stages of Microwave Electronics
Development

1.1 Background

The era of radio began with proof of the existence of electromagnetic waves by H.
Hertz, professor of Karlsruhe Polytechnic University, in 1888. He created the first
primitive spark generator and receiver of these waves. The wavelength of the
radiation, which he investigated, was about 3 m (frequency of 100 MHz). In 1890
E. Branly invented a more sophisticated device for receiving electromagnetic
waves, it was a tube filled with metal filings. Under the influence of electromagnetic
radiation, the resistance of the tube sharply decreased due to micro-breakdowns of
oxide films covering the surface of the filings. Essentially, this was the first solid-
state electronic device. In 1894 O. Lodge improved this device and gave it the name
“coherer”. With its help Lodge showed wireless transmission and reception of
Morse code signals, namely the transmission of signals by radio over a distance of
about 40 m. It happened on August 14, 1894 at the Royal Institute in London.
However, Lodge did not patent his device. Later A.S. Popov and G. Marconi kick
started the wide use of radio communication.
In these first experiments, radio waves of comparatively short length were used
—lying in the meter wave band. However, the need to increase radio communi-
cation coverage, voice and music transmission necessitated the invention of new
types of generators—arc oscillators and machine generators, operating in contin-
uous wave mode with a wavelength of several kilometers.
Subsequently, the coherers in receivers were replaced by “crystal detectors ”—
semiconductor devices with a Schottky barrier, invented by the German professor
K. Brown in 1898 (although at that time there were no such concepts as “semi-
conductor” and “Schottky barrier”).
In 1906, the American scientist Lee de Forest invented a three-electrode tube
audion (triode) capable of amplifying radio signals. At that moment, vacuum
electronics began. In the 1920s, amplifiers and generators on vacuum devices

© Springer International Publishing AG 2018 3

A. D. Grigoriev et al., Microwave Electronics, Springer Series in Advanced
Microelectronics 61, https://doi.org/10.1007/978-3-319-68891-6_1
4 1 Main Stages of Microwave Electronics Development

reached the power that allowed their use in radio stations transmitters. However,
radio transmitters still used long and ultra-long waves.
In 1921, an employee of the Nizhny Novgorod Radio Laboratory O. Losev
discovered that the contact of zincite (ZnO) with a steel wire has negative dynamic
resistance. Essentially, it was the first tunnel diode . Using this device, Losev
assembled a receiver with high sensitivity, which he called a crystadyne. However,
his research did not go any further.
In the late 1920s to early 1930s, radio amateurs found out that using short waves
(wavelength of 10–50 m) it was possible to establish communication over long
distances using low-power transmitters. At that point, there began a race to master
ever higher frequencies, which continues to the present day (Fig. 1.1). During the
last 100 years, the top operating frequency of communication systems has increased
by almost a million times! This race received a boost in the late 30 s, when the task
was to detect fast-flying aircraft, and the development of radar began.
As early as 1895, A.S. Popov noticed the possibility of detecting objects (ships)
with the help of radio waves when he noticed a weakening of the signal transmitted
by one ship to another as a third ship passed between them. However, no attempts
were made to implement this observation. “The use of radio waves to detect remote
metal objects” was demonstrated by C. Huelsmeyer in 1904 (finding a ship in dense
fog), but the distance to the ship was not determined. The first pulse radar (RAdio
Detection and Ranging) was demonstrated in the United States by Robert Page in
1934.
A similar system was developed by Rudolf Kuhnhold in Germany in 1935 and at
the same time by Robert Watt in the UK. In 1943, Paige significantly increased the
accuracy and interference immunity of radars by proposing a monopulse system
which is still in use.

Fig. 1.1 Change in top operating frequency of communication systems

1.1 Background 5

In the USSR, the first radio detector of aircraft “Rapid” was created in the
Leningrad Radiophysical Institute in 1934, under the direction of A.I.
Merzheevsky. It operated at a wavelength of 5 m and had a transmitter power of
about 200 W. The structure of “Rapid” included one transmitting and three
receiving antennas. It detected aircraft at a distance of up to 5 km. The first pulse
radio locator was created by Y.B. Kobzarev at the Leningrad Physicotechnical
Institute in 1938. It allowed detection of aircraft at distances of up to 50 km and
simultaneous determination of the distance to the target.
The English were the first to begin the widespread use of radars for defense
against air attacks. The invention of the magnetron allowed them to install radar on
fighter planes. “The Battle for England” in 1940–1941, when the United Kingdom
fought against Germany alone, was to a large extent won thanks to radar. Later,
during the Second World War, radar stations (RS) were widely used by all bel-
ligerents on land, at sea and in the air. In the course of the war the USSR received
445 detecting and gun-laying RS through lend-lease.
After the war, RS development continued at a rapid pace. New fields of
microwaves application appeared: radio spectroscopy, radio astronomy, household
and industrial heating plants, plasma heating and diagnostics in thermonuclear
fusion machines, terrestrial and space communications with high speed information
transition, hidden object detection systems, biology and medicine, and many others.
In the development of the first RS designers faced the problem of lack of
sufficiently powerful and high-frequency power supply units and low-noise
amplifiers. It quickly became clear that the electron tubes at that time could not
work effectively at ultrahigh frequencies (more than 300 MHz). It was necessary to
develop devices that were free from limitations of interelectrode capacitance and the
transit time of electrons in the interaction distance.

1.2 Microwave Vacuum Electronics

The R. and S. Varian brothers’ invention of the drift klystron in Stanford University
in 1937 was the first leap in this direction. The device had output power up to 10 W
at the frequency of 1 GHz. A substantial contribution to the development of these
devices was made by Hansen, who designed the first types of rhumbatron (cavity
resonator). After the war the klystrons were improved, and today they provide power
of up to several tens of MW in the frequency range of 1–30 GHz.
The first example of a magnetron with a slit anode was created by A. Hall in
1920. He also proposed the term “magnetron”. However, this and subsequent
examples of magnetrons with a slit anode were not suitable for practical use.
A multi-cavity resonant magnetron was invented by Hollmann in Berlin in
1935, but the Germans did not appreciate the value of this device. In 1939, in the
USSR, Alekseev and Mulyarov developed the design of a multi-cavity pumped
magnetron capable of generating power of up to 300 W at the frequency of 3 GHz.
In 1940 J. Randall and H. Boot at the University of Birmingham created a compact
6 1 Main Stages of Microwave Electronics Development

multi-cavity magnetron, which provided power 100 times greater than any micro-
wave radiation source known at that time. In September 1940, Churchill agreed to
Tizard’s proposal about the transfer of a magnetron sample to the US in exchange
for financial and industrial assistance. During Tizard’s mission, the magnetron
sample with an output power of 6 kW at the frequency of 3 GHz was transferred to
the government of the USA, where the mass production of these devices was
developed. The disadvantage of the first multi-cavity magnetrons was instability
and frequency “hopping” but in 1941 Randall and Boot solved this problem by
introducing straps between the resonators. Other types of sources (klystrons)
available in the USA and Germany at that time had the power of not more than
10 W in this range.
Magnetrons became the basis of Second World War radar stations. However, in
the radar receivers crystal detectors were still used since the limiting frequency of
the vacuum diodes did not exceed 400 MHz.
The rapid development of vacuum microwave electronics in the 1940s and
1950s was characterized both by the devices appearance and by rapid parameters
improvement of the already known ones. This period can be called the golden age
of vacuum microwave electronics.
The O-type traveling-wave tube—a broad-band low-noise amplifier, which the
radar designers lacked, was invented in 1942 in the laboratory of the British
Admiralty by R. Kompfner. The theory of this device was developed by Pierce in
1950–1952 in the Bell laboratory (the USA). It allowed improvement of the device
parameters significantly.
The O-type backward-wave tube is an electrically tunable low-power gener-
ator operating in the centimeter and millimeter wavelength ranges. It was invented
by M.F. Stelmach (USSR) in 1948. A similar device was demonstrated by
Kompfner in 1951. To date, this is a device capable of generating the record high
frequences up to 1 THz.
In the 1950s, the relatives of the magnetron were invented, called M-type
electronic devices, which also used crossed fields. In 1949, D. Wilbur and P. Peters
(USA) developed a mitron, in 1950 Warnecke in France created the M-type TWT,
and then in 1952 Epstein in France invented the M-type BWT, and Brown (the
USA) in the same year created a platinotron. These devices significantly expanded
the capabilities of radar designers.
Electron cyclotron resonance masers were proposed in 1959 independently by
Gaponov-Grekhov in the USSR, and Schneider and Pantell in Australia. The first
working gyrotrons were created in 1965 at the Institute of Applied Physics of the
USSR Academy of Sciences. At present gyrotrons have output power up to
several MW in pulsed operation mode and hundreds of kW in continuous operation
mode in the millimeter wavelength range with an efficiency of up to 30–40%.
Gyroklistrons and gyro-TWTs have also been developed. In most gyroresonance
devices, superconducting magnets are used that sharply increase their weight and
dimensions.
In the 1960s, a new direction arose—relativistic microwave electronics based on
the use of electrons moving with relativistic velocities. However, the first devices of
1.2 Microwave Vacuum Electronics 7

this type had too low efficiency and bad signal quality which made their practical
use impossible.
The situation changed in the 1980s after the creation of an ubitron, a free
electron laser (FEL), and relativistic TWT-BWO. These devices (together with
gyrotrons) allowed the filling of the so-called “terahertz dip” by generating power
from a few mW to tens of W at frequencies of 0.1–1 THz.
Unfortunately, FEL and relativistic TWT-BWT are not essentialy devices but
rather facilities that together with the electron accelerator occupy a large production
building. They cannot be used on moving platforms. Therefore the task of creating
powerful sources of the coherent radiation in the range of 0.3–3 GHz is still
relevant.

1.3 Semiconductor Microwave Electronics

In 1947, W. Shockley, D. Bardeen and W. Brattain in the Bell laboratory created

the first bipolar transistor. From this moment the rapid development of semi-
conductor electronics began. However, the Brattain transistors had a limiting
amplification frequency not exceeding a few MHz.
The requirements for the microminiaturization of radioelectronic aids and the
increase in their operating frequency resulted in the development of a number of
semiconductor devices and circuits based on them.
Initially these were passive diode elements, in particular, detector and mixer
diodes on the Schottky barrier produced using a new technology, as well as varactors
and p-i-n diodes. On this basis, circuits were created that converted the signal’s
spectrum: frequency rectifiers, mixers and multipliers, as well as switches, limiters
and phase shifters. During this period of time, the creation of active elements for the
amplification and generation of microwave oscillations on transistor basis was
impossible because of technological difficulties in the fabrication of structures with
micron-sized active areas. The way of creating active (amplifying and generating)
devices based on diode structures having negative dynamic resistance in the
microwave range appeared to be simpler. One of the first devices designed for these
purposes was the tunnel diode created by L. Esaki in Japan in 1957.
In 1958, W.T. Read showed that a diode operating under avalanche breakdown
conditions can have negative dynamic resistance and, consequently, can be used to
generate or amplify electromagnetic oscillations. In 1959 A.S. Tager, A.I. Melnikov
and others experimentally discovered the effect of microwave radiation generation at
the avalanche breakdown of a semiconductor diode. This was the invention of an
impact avalanche and transit diode (IMPATT—IMPact Avalanche and Transit Time).
The IMPATT generators allowed hundreds of times more power than tunnel diodes.
In 1963, J. Gunn discovered the effect of negative dynamic resistance appearance in
GaAs samples, and the creation of devices based on diode structures with negative
differential mobility (Gunn diodes) began. Another name for such devices is devices
with intervalley electron transfer, which reflects the physical principle of their
8 1 Main Stages of Microwave Electronics Development

operation. The promising prospects of a simple diode using arsenide-gallium structures

led to an unprecedented worldwide increase in the number of studies and developments
in this field. However, the practice of using these devices revealed instability of
operation, the need to withstand rigid tolerances for manufacturing technology, and the
complexity of circuit solutions. All this led to a decrease in interest in Gunn diodes
against the background of progress in the creation of microwave transistors.
Nevertheless, Gunn diodes are still used today to generate microwave oscillations,
especially in the millimeter wavelength range. Prospects of advancement into the
terahertz range are opened for Gunn diodes with the use of new materials such as GaN.
In the early 1970s, due to the progress of semiconductor manufacturing technology
it became possible to create the first transistors in the lower part of the microwave
range. However, the exponential growth in the production of these devices and
integrated circuits based on them became possible with the use of a new, more
expensive material, gallium arsenide, instead of the widely used cheap silicon.
In the late 1970s, the metal semiconductor field-effect transistors (MESFET)
began to be commercially manufactured. The history of the birth and application of
MESFET is an example of a discovery that was much ahead of its time. Invented in
1930, it experienced a rebirth in the 1970s. It was the appearance of MESFET that
made it possible to create amplifiers and generators in the microwave range. Further
progress in the creation of not only discrete devices but also monolithic integrated
circuits was based on the use of AlGaAs heterostructures. In the 2000s, successes in
the technology of submicron heterostructures created on wide-gap semiconductors
allowed the creation of devices operating in the upper part of the microwave range
of 100–200 GHz. Such materials include silicon carbide SiC, gallium nitride GaN,
and diamond C. It is important to note that the use of new materials and reduction
of the active area dimensions to sizes of 0.05–0.1 lm led to the possibility of
obtaining the vacuum (ballistic) carrier transfer similar to the transfer process in
vacuum devices. The appearance of field-effect transistors with high electron
mobility (HEMT) in the 2000s allowed the creation of devices operating in the
upper part of the microwave range of 100–200 GHz, and the use of new wide-band
materials (GaN, SiC) increased the output power of semiconductor amplifiers and
generators in the microwave range by an order of magnitude. Currently GaN
HEMT, capable of putting out power up to 100 W in the frequency range up to
10 GHz (Cree Company, USA) are being developed. These transistors can be used
to design amplifiers with an output power of up to several kW.

1.4 Comparative Characteristics of Vacuum

and Semiconductor Devices

At present both vacuum and semiconductor devices are widely used in microwave
devices and systems. The optimal choice of a particular device is determined, first
of all, by the required power and frequency. Figure 1.2 shows the attained power
levels of different microwave device classes versus operating frequency. The
1.4 Comparative Characteristics of Vacuum and Semiconductor Devices 9

Fig. 1.2 Attained power level of different microwave device classes

notations on the figure are: GaN HEMT—GaN field-effect transistors with high
electron mobility, Si BT—silicon bipolar transistors, PHEMT—pseudomorphic
field-effect transistors with high electron mobility, GD—Gunn diode, GaAs
MESFET—Schottky barrier gallium-arsenide field-effect transistors, and TWT—
the traveling-wave tube. Apparently, vacuum devices (klystrons, TWT, gyrotrons)
are significantly (by several orders) ahead of semiconductor devices, both in
maximum output power and maximum operating frequency.
The main application areas of vacuum devices are radar station transmitters,
transmitters of high-speed communication lines, power systems for charged particle
accelerators, plasma heating in thermonuclear reactors, and microwave technology
facilities. Progress in the development of microwave semiconductor devices is
associated, first of all, with the rapid development of radar stations with Active
Electronically Scanned Arrays (AESA) and mobile communication systems.
Currently, powerful GaN-based transistors are beginning to replace vacuum
devices in the output cascades of AESA radar transmitters since such transmitters
do not require much power—it is obtained by adding the powers of hundreds and
even thousands of individual radiators in the radio beam.

1.5 Prospects for the Development of Microwave

Electronics

At present, the development of microwave electronics is proceeding in several direc-

tions. The struggle for mastering ever higher frequencies—the terahertz range—con-
tinues. The task here is to develop small-sized powerful sources of coherent radiation,
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upon the scene in their absence. The minute she saw Billy she made
a rush at him, flung her arms around him, and kissed him heartily
again and again.

"Oh, the dear little fellow!" she cried, hugging him and half crying. "To
think of all he's gone through—the poor, motherless lamb!"

"Elizabeth," said Mrs. Brown sternly, "show more sense! If you go on

like that you'll upset him! This is my daughter, Mrs. Dingle, Billy."

"Aunt Elizabeth to you, my dear!" said Mrs. Dingle, kissing the little
boy once more before she released him.

Billy looked at her with glowing eyes. He liked her, he had no doubt
about that. She had a fresh, rosy face, and eyes as deeply blue as
her little daughter's; but what won his heart so quickly was her
expression—it was so motherly and kind.

"Well, tea's ready!" exclaimed Mrs. Brown, rather impatiently. "If you
won't stay, Elizabeth—"

"I'd best go at once," interrupted Mrs. Dingle. "All right, mother! Oh,
you've put May on her coat and hat! Ready, my birdie?"

May nodded. She kissed her grandmother and "dear grandfer" as

she called William Brown, then came to Billy, put her arms around
his neck and kissed him, too.

"The poor little boy's lost his mother, mummy," she said, as her
mother took her by the hand to lead her away.

"I know, dear, I know!" Mrs. Dingle answered. "Come!"

She hurried the child out of the kitchen, and shut the door quickly.

Mrs. Brown was already seated at the head of the table. She
motioned Billy to a chair on her left, whilst her husband took one on
her right. William Brown said grace very reverently, and the meal
began.
After tea Mrs. Brown took Billy upstairs with her, and unpacked his
box. She showed him where he was to keep his belongings, and told
him she would be seriously displeased if he was not tidy. Then, as
he was very tired, she advised him to go to bed, and left him,
returning later to take away his candle. He was just going to get into
bed.

"Good-night, Billy," she said. "I'll call you in the morning."

"Thank you," Billy answered. "Good-night!"

As soon as she was gone he crept into bed. A sense of utter

loneliness had taken possession of him, and, putting the bedclothes
over his head, he gave way to a fit of weeping.

"Oh, mother, mother, mother!" he sobbed, "it's dreadful to think I

shall never see you again."

Then suddenly he remembered how Tom Turpin had reminded him

that he would be with his mother through all Eternity, and the load of
desolation and grief was lifted from his heart.

"'The gift of God is eternal life through Jesus Christ our Lord,'" he
whispered to himself, and was comforted.

CHAPTER III.
BILLY HAS A FRIGHT.
BILLY'S grandfather was a prosperous market-gardener now-a-days,
but before his second marriage he had been only a farm labourer.
He had married the widow of the former tenant of Rowley Cottage,
and together they had worked hard to save money, and were now in
a comfortable position. Billy's father had not got on with his
stepmother, so he had never gone home after he had settled in
London and married.

Rowley Cottage, which was really a fair-sized house, was situated

far down the side of a steep hill, with a hill equally steep facing it.
Before the house sloped a flower garden, at the end of which was a
shallow, rippling stream, spanned by a wooden footbridge, and
beyond the stream was a large vegetable and fruit garden.
Surrounding the house and gardens were apple orchards.

Billy's first morning in his new home was a dull one. It rained hard,
so he had to stay indoors. After breakfast his grandfather, clad in
oilskins, went out, and did not return till dinner-time. He then said
that there was a prospect of the weather clearing.

"If it does I'll show you about a bit," he said to Billy. "We might get as
far as the post office—Elizabeth will give us some tea. Won't you
come with us, Maria?" he asked his wife.

"No, thank you," answered Mrs. Brown. "I've work to do at home if

you haven't. Besides, I've no liking for traipsing about in the mud."

About three o'clock the rain began to cease, and a little later the sun
shone out. Billy and his grandfather left the house by the front door.
They stood for a minute under the porch, whilst William Brown
pointed out a house—the only human habitation in sight—almost on
the summit of the opposite hill.

"That's Mount Farm," he said, "farmer Turpin's place. You can see
Exeter from there. I used to work for farmer Turpin's father when I
was a lad. Ah, the wind's rising! We shall have no more rain for a bit!
Come along, Billy!"
He led the way to a little green gate in the garden hedge, by which
they passed into an orchard. There was a footpath through the
orchard to steep ploughed fields beyond, and a footpath through the
fields to a gateway which led into the high road.

Billy was panting when at length the high road was reached, so that
his grandfather had to wait for him to regain his breath.

"Oh, look at my boots!" exclaimed the little boy, as they moved on

again; "they're all over mud!"

William Brown laughed.

"You'll get accustomed to mud," he said; "but you must have thicker
boots. I must take you to Exeter one day and get you fitted out
properly for bad weather."

"Oh, thank you!" Billy answered, gratefully. "Shall I have leather

leggings like yours, Grandfather?" he asked.

"We'll see!" was the smiling response.

A ten minutes' walk brought them to the village—a few thatched

cottages dotted around the church and churchyard. The railway-
station, Billy learnt, was half a mile distant in the valley, and the
vicarage was midway between the church and the railway-station.

"That's the post office," said William Brown, pointing at a semi-

detached cottage with several bottles of sweets and some groceries
in the window. "And there's Elizabeth!" he added, as a stout figure, in
a dark stuff gown nearly covered by a big white apron, appeared in
the doorway.

Mrs. Dingle nodded to her stepfather, and kissed Billy, telling them
she had been on the look-out for them ever since dinner.

"And here's Uncle John!" she cried, pulling Billy inside the door and
presenting him to a little dark man wearing spectacles, who came
from behind the shop counter and peered at him in a near-sighted
way.

"Very glad to make your acquaintance, my boy!" declared John

Dingle, shaking Billy's hand heartily. "Yes," he said, "I see he's like
his father, Elizabeth; but he looks very pale—"

"He's been through enough to make him pale!" broke in his wife.
"Come into the parlour, Billy, and talk to me whilst I get tea."

Leaving his grandfather with the postmaster, Billy followed Mrs.

Dingle into a tiny parlour behind the shop. It was divided from the
shop by a door, the top half of which was of glass with a lace curtain
across it. Mrs. Dingle put the kettle on the fire and laid the table for
tea. The children were at school, she said, but would be home very
shortly, and she did hope he and her boy, Harold, would be friends.
Very soon Billy felt quite at ease with her, and began telling her
about himself and how sadly he missed his mother. She shed tears
when he spoke of his mother, whilst an expression of deep regret
settled on her rosy face.

"I wish I'd known her!" she sighed. "Often I used to think I'd write to
her, but I never did—not being much of a hand with my pen. And
now it's too late! Hark! The children are out of school!"

Billy listened. He heard a babel of children's voices mingled with

merry laughter in the road outside the shop. A few minutes later the
door between the shop and parlour opened softly, and little May
came in. The instant she caught sight of her mother's visitor her look
became eager.

"Have you found her?" she cried, her blue eyes fixed anxiously on
Billy's face.

"Found who?" Billy inquired, not understanding.

"She's thinking of your poor mother," Mrs. Dingle explained hastily;

"she doesn't realise she's dead." Then, addressing her little
daughter, she asked: "Where's Harold?"

"In the road, mummy," was the reply.

"Run and fetch him, there's a dear!"

After the child had gone Mrs. Dingle said—

"You mustn't mind if she questions you about your mother. May is
backward for her age—there are many things she can't understand,
though she's sharp enough in some ways. She learns hardly
anything at school. She can't read, or write, or do sums. The
mistress doesn't bother her to learn, for she knows she can't. Still, it's
good for her to be with other children. By-and-by, perhaps, but God
only knows—"

She broke off abruptly, May having returned, followed by her brother.

Harold was very like his mother in appearance, being a stout, rosy-
cheeked boy. His blue eyes had a merry twinkle in them, and he
looked full of fun.

Tea now being quite ready the two men were called from the shop,
the lace curtain was pulled back from the glass-top door, and, grace
having been said, the meal began.

"Now, make yourself at home, my boy," the postmaster said to Billy,

"and let me tell you once for all that you'll always find a welcome
here."

"Thank you, Mr. Dingle!" Billy replied, his eyes alight with gratitude.

"Uncle John, please!" corrected Mr. Dingle.

Billy smiled, and flushed with pleasure.

"Thank you, Uncle John!" he said, adding: "Oh, I wish mother knew
how kind you all are to me!"
Twice during tea customers came to the shop, and the postmaster
had to go to serve them. On the second occasion Billy thought he
recognised the customer's voice, and glanced quickly at his
grandfather.

"Yes!" nodded William Brown, "it's Master Tom! Why, here he

comes!"

A smiling face peeped around the half-open glass-top door, whilst its
owner said—

"What a jolly tea-party! Mrs. Dingle, won't you please give me a cup
of tea?"

Mrs. Dingle was answering that she would be delighted, when there
was the sound of a loud report at no great distance, and Billy sprang
to his feet with a terrified shriek.

"Oh! Oh!" he gasped, horror-stricken; "it's a bomb!—it's a bomb!"

"No, no, no!" Tom Turpin assured him, "nothing of the kind! It's
blasting—that is, blowing up rock with dynamite—at the stone
quarry. Don't be frightened! Really, there's nothing to be alarmed at.
You won't hear the noise, this afternoon, again."

Billy sank into his chair. He was white to the lips, and shaking. The
elders of the party looked at him with sympathy and much concern.
May's eyes expressed only wonderment, but Harold's sparkled with
amusement and scorn.

"The stone quarry's only a couple of miles away, so you'll get

accustomed to the sound of blasting," Tom Turpin continued, "and
you'll not be scared another time, for you'll know what the sound
means."

"Yes—oh, yes!" murmured Billy. He was ashamed of the terror he

had felt and exhibited. Everyone would consider him such a coward.
His lips quivered, whilst tears rose to his eyes.
"Did you think the Germans were coming?" asked Harold, with a
wide grin.

"I thought a German airship might be dropping bombs," admitted

Billy. "I—I'm ashamed of myself."

"You've no call to be that!" Mrs. Dingle told him. "It's no wonder

you're nervy, I'm sure. There, you're all right now, aren't you? Have
another cup of tea, won't you? It'll do you good."

Billy shook his head. It was with difficulty he kept from crying. He sat
in miserable silence whilst Tom Turpin talked with the others and
took his tea, and, when the young soldier left, his voice was
unsteady as he said "good-bye" to him. He was sure Tom must
despise him for having shown such fear.

It was dark long before Billy and his grandfather started for home. A
walk in complete darkness was a novel experience for the little boy,
but he was not timid, because his grandfather was with him. He said
so, adding, as the hand which held his tightened its clasp—

"I know you'll look after me, Grandfather!"

"Aye," William Brown assented, "to the best of my power. And there's
One above, Billy, Who'll look after us both. You'll soon learn to find
your way about in the darkness, and won't mind it—why, even little
May doesn't."

"Doesn't she?" cried Billy in surprise. "How brave of her!"

"You know it says in one of the psalms, 'The Lord my God shall
make my darkness to be light,'" his grandfather said thoughtfully;
"and I think that, though there's a sort of cloud over May's mind,
behind the cloud there's God's own light. The soul that has that light
knows no fear."
 CHAPTER IV.
SUNDAY.

BILLY'S first Sunday in Devonshire was a beautiful day, with

sunshine and a soft westerly breeze. The little boy accompanied
"Grandfer," as he had decided to call his grandfather in imitation of
the Dingle children, to church in the morning, and, after the service,
lingered with him in the churchyard to speak to the Dingles, all of
whom had been to church, too. Then Tom Turpin, his mother on one
side of him, his father on the other, came out of church, and stopped
and spoke, afterwards introducing Billy to his parents.

"I hope to call at Rowley Cottage to-morrow," the young soldier told
William Brown; "I want to go around your garden and see everything.
Father tells me you're doing your 'bit' to help win the war."

On their way home Billy asked his grandfather what Tom Turpin had
meant by this remark. William Brown explained that food was likely
to be very short on account of the German submarines, which were
torpedoing so many food ships, and that he was doing his "bit" to
help win the war by cultivating every inch of his garden, and growing
as many vegetables as he could.

"The worst of it is I can get so little help," he said; "there isn't a fit
man left in the village for me to employ. That means that I shall have
to work doubly hard during the coming winter and spring."

"Don't you think I could help you, Grandfer?" Billy inquired eagerly.
"You?" William Brown looked at his grandson with a slightly amused
smile. "Well, I don't know about that," he said doubtfully. "Harold
helps his father in his allotment garden, but he's very strong for his
age, whilst you're such a delicate little chap—"

"Oh, Grandfer," Billy burst in, "I do believe I'm stronger than I look!
Oh, let me help you! Let me try, at any rate! I want so much to do
something to help win the war!"

"Well, we'll see what you're fit to do," was the cautious response.

With that Billy had to be satisfied for the time. They were descending
the hill to Rowley Cottage by way of the pathfields now, and a few
minutes later found them in the orchard, where Jenny was browsing
contentedly. She allowed Billy to put his arm around her neck and
caress her. His grandfather looked on, rather anxiously at first, then
with great satisfaction.

"She's taken to you very well, Billy," he said. "You'll be able to do

anything with her, you'll find."

"Shall I?" cried Billy, delighted. "Do you think she'd let me ride her,
Grandfer?"

"I shouldn't wonder! You shall try one of these days, perhaps!"

They entered the house by the back door. Mrs. Brown was in the
kitchen, dishing dinner. She was very hot, and looked exceedingly ill-
tempered.

"Well, Maria, my dear!" her husband said cheerfully.

"Oh, it's well for you, I daresay," she retorted, "you who've had an
easy morning; but what about me who's been cooking all the time
you've been at church? There, take your seats! Dinner's ready!"

"I often wish you'd manage to do your cooking on a Saturday and

have a cold dinner on Sunday like Elizabeth," William Brown
remarked; "then you'd be able to go to church—we'd such a nice
service this morning, and—"

"Oh, no doubt Elizabeth's a better manager than her mother!"

interrupted his wife sarcastically. "I've always cooked on Sundays,
and I always shall."

It was a very good dinner, but Billy did not enjoy it, for Mrs. Brown,
who carved, gave him a thick slice of fat mutton which he could not
eat. Noting this, his grandfather remarked that he was not getting on,
and he admitted that he did not like fat meat.

"Can't you give him a cut of lean, Maria?" William Brown suggested.

"No, I can't—not without disfiguring the joint, and I'm certainly not
going to do that," Mrs. Brown answered. "Billy must learn not to be
so particular. If we can eat fat meat he can."

Her husband looked troubled, but said no more. As soon as the meal
was over he rose and went out, while Mrs. Brown began to put
together the dinner things with a clatter of plates and dishes. Billy
watched her in silence for a minute, then asked timidly: "Can I help
you, Granny?"

"Help me? You?" cried Mrs. Brown, raising her eyebrows in a

contemptuous fashion. "What can you do to help me, I should like to
know?"

"I could wash up," Billy answered, flushing, "or I could wipe the
things as you wash them—I always did that for mother. If you'll say
what you'd like me to do—"

"I'd like you to keep out of my way and leave me to do my work as I

please!" Mrs. Brown interrupted. "Stay, though, you can give these
scraps to the fowls."

The little boy took the plate of scraps she offered him and went out
into the yard. When he returned with the plate empty Mrs. Brown had
cleared the table and was washing up.

"So your mother used to make you useful?" she remarked

inquiringly.

"Yes, Granny," he answered, "and I liked helping her. She used to be

so tired sometimes—she worked very hard, you know."

"Humph! She'd have been wiser if she'd gone into a situation when
your father died instead of starting a business of her own."

Billy was silent. His mother—she had been a milliner's apprentice

before her marriage—had opened a little business of her own when
his father, who had been employed in a warehouse, had died. She
had earned enough to support her child and herself, but there had
been nothing over.

"Mother didn't want to be parted from me," the little boy said, in a
faltering voice; "and now—and now—oh, I can't bear it! Oh, what
shall I do?"

He flung himself on the settle by the fire, covered his face with his
hands, and wept.

"Don't go on like that, child," Mrs. Brown said hastily; "perhaps we'd
better not talk of your mother any more. Come, stop crying, like a
sensible boy! Why, here's May! You don't want to upset her, do you?"

Billy sat up, struggling to regain composure. He was wiping his eyes
with his pocket-handkerchief when May, entering by the back door,
appeared upon the scene. She ran to her grandmother and kissed
her, then, turning to Billy, was struck with dismay at his woe-begone
look.

"Billy's been crying," she said, in an awed tone. "Why, Billy, why?"
she asked, stealing softly to his side. Then, as the little boy's only
answer was a suppressed sob, she cried, "I know! You haven't found
your mother yet!"
"Oh, May, you don't understand!!" Billy exclaimed, with a wail of grief
in his voice. "Mother's dead!"

"Dead?" May echoed, a faintly troubled look disturbing the usual

sweet serenity of her face. "But I thought Granny said she was lost?"

"That's often said of folks who are dead," explained Mrs. Brown.

"But it isn't true, Granny," May said gravely. "If people are good and
love Jesus they go to Jesus for always when they die, don't they?"

"Oh, yes," agreed Billy. "I know my mother's safe with Jesus, May."

"Then," said May, triumphantly, the faintly troubled expression

passing from her face, "she can't be lost!"

At that minute Harold came in, looking flushed and heated. Mrs.
Brown immediately accused him of having been teasing Jenny. He
did not admit it, only laughed, and hastened to tell her that he and
May had come to take Billy to church with them.

"Yes, he shall go," Mrs. Brown decided. "Hurry and wash your hands
and brush your hair, Billy."

The Vicar of Ashleigh always held a children's service in the church

on Sunday afternoons. This afternoon the service had commenced
before the Dingle children and Billy got there. They slipped
noiselessly into a back seat and joined in the hymn which was being
sung. After the hymn the Vicar—an old man with a kind, gentle face
—gave an address, and then moved about the church, questioning
the children. More than once Billy saw his eyes fixed on him with
sympathy and interest.

"I like the Vicar very much," he said to Harold in the churchyard
afterwards.

"So does May," Harold replied; "she thinks there's no one like Mr.
Singleton. Can you find your way home by yourself, Billy?"
"Oh, yes," assented Billy, "of course I can."

"That's all right, then," smiled Harold, adding: "you'll meet nothing
you need be afraid of, and hear nothing—being Sunday there's no
blasting going on at the stone quarry to-day."

CHAPTER V.
BILLY'S PRESENT.

NEXT morning Billy came downstairs looking heavy-eyed and poorly.

He had had bad dreams, he said, when his grandfather asked him if
he had not slept well; but he did not say that in them he had lived
again through the night of the air raid and the grievous time which
had followed, so that the hours of darkness had been a horror to
him.

"You'd better spend the morning out-of-doors," remarked Mrs.

Brown. "It couldn't be finer weather—a good thing, too, as it's
washing-day. I hope Mrs. Varcoe will come early, then we shall get
the clothes dried during the day."

Mrs. Varcoe was a woman from the village, Billy learnt, who came to
Rowley Cottage every Monday morning to do the washing. He met
her in the yard, after breakfast, where he was waiting for his
grandfather, who was getting his wheel-barrow and gardening tools
from an out-house, and she paused to look at him. She was a tall,
muscular, red-headed woman, with a big freckled face and small
greenish eyes.
"Good morning!" he said politely, thinking that she was certainly the
ugliest woman he had ever seen.

"Good morning," she answered gruffly, turning towards the house.

"Mrs. Varcoe is very ugly, Grandfer," Billy remarked, as, his

grandfather having joined him, they went around the house towards
the vegetable garden.

"Aye," William Brown agreed, "but she's a good sort—a widow who's
brought up a family of boys and made men of 'em!—men of the right
kind, I mean. Four are serving their country—two in the Navy, one in
Mesopotamia, and one in France. There was another, but he was
killed in action at the beginning of the war. The eldest he was. His
death must have been a big blow to his mother; but I've never heard
her mention it except once."

"What did she say?" Billy asked, much interested.

"She said, 'It's a grief, but there's no bitterness with it. My boy died
fighting for the right, and I shan't be ashamed of him when I meet
him before God.' It was a brave speech, wasn't it?"

"Yes," agreed Billy, "I think it was."

"Now the first thing I mean to do this morning is to make a bonfire,"

William Brown said, as they entered the vegetable garden. "You can
help me collect all the dead leaves and rubbish lying about. We'll
make the bonfire in this corner where there's nothing growing at
present."

So Billy set to work with his grandfather. It took them more than an
hour to make the bonfire—a huge one. The little boy was allowed to
light it, and gave a shout of pleasure as the flames leaped up
followed by a volume of smoke.

"Oh, this is splendid!" he cried, "splendid!" A tinge of colour had

come into his pale cheeks, and his eyes were sparkling.
"It's burning very well," his grandfather said, smiling at his
excitement, "and the smoke's blowing right away from the house—
fortunately. I'd forgotten till this moment about the washing—it's
always hung out in the orchard at the right side of the house. If the
wind had not been blowing the smoke away from that direction the
clean clothes might have had smuts on them by this time, and I don't
know what Granny would have said—not more than I should have
deserved, though, of course. Ah, here comes Master Tom!"

Billy looked at Tom Turpin rather shyly as he greeted him. He wished

he had not shown himself such a coward before this young soldier,
who, he imagined, did not know what fear meant. He was very quiet
as he followed him and his grandfather about the garden, but he
listened with the greatest attention to all that was said. William
Brown showed where he intended sowing his various crops in the
spring, and the bit of orchard he meant to take into the garden.

"I don't know how I'm going to do all I want to," he remarked, "but I
shall just plod on bit by bit from day to day and do my best."

"That's what we're doing across in France and Flanders," Tom

replied gravely.

"I want to help Grandfer," Billy said eagerly. "I do wish I was bigger
and stronger. I tried just now to use Grandfer's spade, but I couldn't
—I couldn't drive it more than an inch or two into the ground." He
sighed, looking at his thin arms ruefully.

"I've some light garden tools at home my father gave me when I was
a boy no bigger than you, and you shall have them," Tom told him.
"I'd like to know they were being used. I'll give them to you, Billy, if
you'll accept them."

"Oh, Mr. Turpin!" cried the little boy. He could say no more for a
minute, so overcome was he with surprise and gratitude; then he
added earnestly: "Oh, thank you—thank you!"
"It's too kind of you, Master Tom, really, but if you'll lend the tools to
him—" William Brown was beginning, when he was interrupted.

"No, no!" Tom Turpin said decidedly, "I wish him to have them for his
own—I'm sure he'll make good use of them."

"Oh, yes, yes!" cried Billy, his face aglow with delight and
excitement.

Tom Turpin had stopped to see William Brown's garden on his way
to the village. When he left, Billy went with him through the pathfields
to the gate leading into the high road. There they were to part.

"I don't suppose I shall see you again this time I'm home," the young
man said, as he looked back at Rowley Cottage, then let his eyes
wander to his home on the opposite hill, "so this will be 'good-bye,'
Billy. I'll send the garden tools this evening by one of our men who
lives in the village."

"Oh, thank you!" cried Billy. Then, suddenly, his face, which had
been bright, clouded. "Mr. Turpin," he said, "you weren't ever afraid
of anything, were you?"

"What a question!" smiled the young man. "Why, yes, indeed," he

answered, becoming serious as he saw this was a serious matter to
his little companion. "The first night I spent in the trenches, for
instance, I was afraid," he admitted. "Oh, God knows I was awfully
afraid!"

Billy gazed at the soldier with amazement. "I should never have
thought it!" he declared; "I wouldn't have believed it if anyone but
yourself had told me! But you didn't show you were afraid?"

"I don't think I did."

"You didn't scream as I did when I heard the blasting?"

"No. I asked God to strengthen me and take my fear away. I prayed,
'Be not Thou far from me, O Lord,' and by-and-by I began to feel His
presence, and then wasn't afraid any more."

Billy drew a deep breath. "I couldn't help being afraid when I heard
the blasting," he said in an ashamed tone.

"No, nor could I help being afraid that first night in the trenches. But I
found help in my weakness, and that same help is for you if you ask
it. Now I must really be off. Good-bye!"

The young soldier vaulted over the gate, greatly to Billy's admiration,
waved his hand, and disappeared from view.

Billy hurried back through the path fields, intending to return to his
grandfather immediately; but in the orchard, hanging out clean
clothes, was Mrs. Brown, and the thought struck him that he would
tell her about the gardening tools.

"Oh, Granny," he began, running up to her, "I've had a present—at

least I'm to have it by-and-by. A set of gardening tools!"

"Oh!" exclaimed Mrs. Brown. "And who's going to give you that?" she
asked sharply. "Not your grandfather, I hope?"

"No, Mr. Turpin—Mr. Tom Turpin," Billy replied. "It's a set he had
when he was a boy. Now I shall be able to help grandfather, shan't
I?"

Mrs. Brown looked at Billy without answering, and smiled. There was
something so contemptuous in her smile that the little boy turned
from her with reddening cheeks. Of course she thought he was too
small and weak to do gardening, he told himself.

Tom Turpin sent the tools in the evening, as he had promised. Mrs.
Brown barely gave them a glance, but her husband pronounced
them to be "first-rate" and just the right weight for his grandson's use.
"I may start using them to-morrow, mayn't I, Grandfer?" asked Billy.

"Yes, if all's well," William Brown answered, smiling; "that means if

you sleep well, and come down looking better to-morrow morning
than you did to-day."

That night Billy had no bad dreams to disturb him. He added the
young soldier's prayer—"Be not Thou far from me, O Lord—" to his
usual evening prayers, and fell asleep very quickly. He did not awake
till morning—the morning of another beautiful day.

CHAPTER VI.
GARDENING.

"THINK you've been at it long enough, Billy; you'd better rest a bit."

Billy was having his first lesson in gardening. His grandfather had
shown him the proper way to use his spade, and for the last half
hour he had been labouring on a patch of ground which had to be
dug up and prepared for spring tillage. Now, as his grandfather
spoke, he ceased work and stood leaning on his spade, viewing the
freshly turned soil with great satisfaction.

"It's very warm," he remarked, "but it's grand weather, isn't it,
Grandfer?"

"That it is!" agreed William Brown. "We often get fine weather like
this hereabouts in November; it gives one an opportunity of
preparing for the winter. Golden days I call these, and one must
make the most of them, for there are days coming when there'll be
no working on the land. The leaves are hanging late on the trees this
year, but the first night's sharp frost will bring them down in a hurry—
they're ripe to fall. Why, who's this I see?"

As if he did not recognise the little figure that had entered the garden
and was hastening towards them with light, tripping steps.

"It's May," said Billy. "Do you think she has come all the way from the
village by herself?"

"Yes," nodded his grandfather. "I thought she might be here to-day,
for I knew the fine weather would make her restless and long to be
out-of-doors. When she's like that she doesn't want to go to school,
and the teacher agrees it's better not to send her. Well, May, my pet!
Come and look at Billy's beautiful tools. Show them to her, Billy."

Billy was very proud and pleased to do so. May examined each tool
separately with the greatest interest.

"Are they your very, very, own, Billy?" she inquired.

"Yes," he answered, "my very, very own. And I can use them quite
easily—they're so light. Mr. Tom Turpin gave them to me. Wasn't it
kind of him? I turned up that ground—look!"

"I think you've done enough for this morning," remarked William
Brown. "You'd better clean off your spade, and put your tools away."

Billy obeyed. His arms and shoulders were aching, but he had no
intention of admitting that. Accompanied by May he left the garden,
and put his tools in the out-house where he had been told to keep
them. He intended returning at once to his grandfather, but May took
him by the hand and led him into the orchard, saying that she
wanted to speak to Jenny and he must come with her. When Jenny
saw the children she began to bray and walk towards them.