An Analytic Theory of Multi-Stream Electron Beams in Traveling Wave Tubes

499 Pages·2020·39.308 MB·English

by  Alexander Figotin

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A catalogue record for this book is available from the British Library.
AN ANALYTIC THEORY OF MULTI-STREAM ELECTRON BEAMS IN TRAVELING
WAVE TUBES
Copyright © 2021 by World Scientific Publishing Co. Pte. Ltd.
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ISBN 978-981-120-919-2 (hardcover)
ISBN 978-981-120-920-8 (ebook for institutions)
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For Tamar and Siranush

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Contents
Dedication v
Preface xv
List of Symbols and Acronyms xvii
I Review of the Theory and Its Key Elements 1
1 Introduction 3
2 Summary of the TWT-system Features and Effects 7
3 e-Beam and Multi-transmission Line Parameters 11
3.1 e-Beam Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 The MTL Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Recovering the MTL Matrices from Its Significant Parameters . . . . . 17
4 System Lagrangian, Field equations, Characteristic
Equation and Eigenmodes 19
4.1 Matrix Polynomial Eigenvalue Problem. . . . . . . . . . . . . . . . . . 21
4.2 Characteristic Functions and Equations . . . . . . . . . . . . . . . . . 23
4.3 Characteristic Equations for Uncoupled e-Beam . . . . . . . . . . . . . 27
4.4 Eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.5 Characteristic and Admissible Velocities . . . . . . . . . . . . . . . . . 31
4.6 Values of the TWT Parameters Used for Plots. . . . . . . . . . . . . . 34
4.7 Infinite-Frequency Limit Approximation . . . . . . . . . . . . . . . . . 36
5 Wave-particle Interactions and Origins of Instability 39
5.1 Origins of Instability and Amplification . . . . . . . . . . . . . . . . . 40
5.1.1 Slow Wave and Its Negative Energy . . . . . . . . . . . . . . . 41
vii
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viii AnAnalytic Theory of Multi-Stream Electron Beams inTraveling Wave Tubes
5.2 Our Theory on Wave-ParticleInteractions and Origins
of Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.3 Our Views on Non-Fluid Aspects of Plasma . . . . . . . . . . . . . . . 44
6 Energy Transfer and Stream Velocities 47
6.1 Energy Transfer from the e-Beam to the MTL and Power
Gain Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.2 Energy Balance and Stream Velocities . . . . . . . . . . . . . . . . . . 49
7 Instability Concepts and Their Graphical Representation 53
7.1 Dispersion-Instability Graph . . . . . . . . . . . . . . . . . . . . . . . . 55
7.2 Instability Concepts and Quantities . . . . . . . . . . . . . . . . . . . . 56
7.3 Space Charge, Debunching Effects . . . . . . . . . . . . . . . . . . . . 58
8 Instability Branches of the Characteristic Function 61
9 Circular Approximations to Sets of Admissible Phase Velocities 65
9.1 Type I Circular Approximations. . . . . . . . . . . . . . . . . . . . . . 65
9.2 Type II, III and IV circular approximations . . . . . . . . . . . . . . . 67
10 All-frequency Modal Branches 71
10.1 afm-Branches Involving Instability . . . . . . . . . . . . . . . . . . . . 73
10.1.1 afm-branches with negative wavenumbers . . . . . . . . . . . . 73
10.1.2 afm-branches with positive wavenumbers . . . . . . . . . . . . 74
11 Instability Phases via the Dispersion-Instability Graph 77
11.1 Instability Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
11.2 Instability Phase Transitions. . . . . . . . . . . . . . . . . . . . . . . . 78
12 Instability Structure of All-Frequency Modal Branches 81
12.1 Instability Structure for Negative Wavenumbers . . . . . . . . . . . . . 82
12.2 Instability Structure for Positive Wavenumbers . . . . . . . . . . . . . 83
12.3 Instability for Large Values of TWT-System Parameter . . . . . . . . 85
12.3.1 Negative wavenumbers . . . . . . . . . . . . . . . . . . . . . . 85
12.3.2 Positive wavenumbers . . . . . . . . . . . . . . . . . . . . . . . 85
13 Instability Nodes and the Degeneracy of the Characteristic
Function 87
13.1 Onset of and Transition to Instability at its Nodes . . . . . . . . . . . 89
13.2 Instability for Small Values of TWT System Parameter. . . . . . . . . 91
13.2.1 Positive phase velocities . . . . . . . . . . . . . . . . . . . . . . 92
13.2.2 Negative phase velocities . . . . . . . . . . . . . . . . . . . . . 92
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Contents ix
14 Instability at Critical States - The Third-Order Degeneracy 95
14.1 Third-Order Degeneracy of the Dispersion Relations
at Critical States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
14.2 Dispersion-instability at critical states . . . . . . . . . . . . . . . . . . 100
15 Almost-Linear Unstable Modal Branches 107
16 Wave-packet Propagation and Amplification 111
16.1 Wave-PacketRepresentation in Dimensionless Variables . . . . . . . . 113
16.2 Wave-PacketRepresentation in Frequency Domain . . . . . . . . . . . 115
16.3 Features Special to an Exponentially Growing Wave-Packet . . . . . . 118
17 Single-Stream e-Beam and MTL 121
18 Multi-Stream e-Beam Coupled to a Single TL 127
19 MTL as an Approximation and TWT Observables 133
19.1 Recovering the MTL Coefficients through Nodal Velocities . . . . . . . 134
19.2 Recovering the MTL Coefficients through Modal Frequencies
and Velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
20 Instability - Possibilities and Limitations 139
20.1 Effect of TWT Features on the Instability . . . . . . . . . . . . . . . . 141
20.2 Multi-Stream e-Beam TWT-System for small and large values
of its parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

II Traveling Wave Tube Components 147
21 Multi-transmission Line 149
22 Multi-Stream e-Beam 153
22.1 Justification of the One-Dimensional Model . . . . . . . . . . . . . . . 155
22.2 Potential Form of the One-Dimensional Linear Model. . . . . . . . . . 156
22.3 Multi-stream e-Beam as a Dielectric Medium . . . . . . . . . . . . . . 158
22.4 Electron Beam Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 160
22.5 e-Beam Eigenmodes, Dispersion Relations and Instability . . . . . . . 163
22.6 Bounds on e-Beam Characteristic Velocities . . . . . . . . . . . . . . . 165
22.7 e-Beam Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . 167
22.8 Energy Balance for Eigenmodes . . . . . . . . . . . . . . . . . . . . . . 168
22.9 e-Beam Energy Balance when the Electric Potential is an
Independent Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170