Electrical Solitons: Theory, Design, and Applications

252 Pages·2010·45.159 MB·English

by  David S. Ricketts, Donhee Ham

Electrical
Solitons
Theory, Design, and Applications
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Electrical
Solitons
Theory, Design, and Applications
David S. Ricketts
Donhee Ham

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Library of Congress Cataloging‑in‑Publication Data
Ricketts, David Shawn
Electrical solitons : theory, design, and applications / David S. Ricketts, Donhee Ham.
p. cm. -- (Devices, circuits, and systems)
Includes bibliographical references and index.
ISBN 978-1-4398-2980-6 (hardback)
1. Solitons. I. Ham, Donhee. II. Title.
QC174.26.W28R53 2010
530.12’4--dc22 2010042045
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Contents
List of Figures ix
List of Tables xiii
Preface xv
I Electrical Solitons: Theory 1
1 Introduction 3
1.1 The “Solitons” . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 A Brief Overview and History of the Soliton . . . . . . 7
2 The KdV Soliton 13
2.1 The Solitary Wave Solution . . . . . . . . . . . . . . . . 13
2.2 The Periodic Soliton: The Cnoidal Wave Solution . . . 15
2.2.1 Asymptotics of the Cnoidal Wave. . . . . . . . . 19
2.2.2 The Cnoidal Wave vs. the Periodic Sech2 Wave . 20
2.3 Transient Dynamics of the KdV . . . . . . . . . . . . . 20
2.3.1 Solitary Wave (cid:54)= Soliton . . . . . . . . . . . . . . 20
2.3.2 Soliton Collisions . . . . . . . . . . . . . . . . . . 22
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.A EllipticIntegrals,FunctionsandTheirLinktoDifferential
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.A.1 Arclength of a Circle . . . . . . . . . . . . . . . . 28
2.A.2 Arclength of an Ellipse. . . . . . . . . . . . . . . 30
2.A.3 Elliptic Functions. . . . . . . . . . . . . . . . . . 31

2.A.4 Example PDEs . . . . . . . . . . . . . . . . . . . 32
3 The Heart of the Soliton: Inverse Scattering 35
3.1 Inverse Scattering Method . . . . . . . . . . . . . . . . 36
3.2 A Math Problem . . . . . . . . . . . . . . . . . . . . . . 36
3.3 KdV Solution via the Inverse Scattering Method . . . . 38
3.4 Solution of the KdV Initial Value Problem . . . . . . . 42
3.4.1 The Eigenvalue Problem Using the Schr¨odinger
Operator . . . . . . . . . . . . . . . . . . . . . . 42
3.5 Asymptotic Solution to the Inverse Scattering Method . 47
v
vi
3.5.1 Reflectionless Potentials . . . . . . . . . . . . . . 48
3.5.2 Non-Reflectionless Potentials . . . . . . . . . . . 52
3.6 Soliton Definition . . . . . . . . . . . . . . . . . . . . . 55
3.7 Transient Solutions of the KdV . . . . . . . . . . . . . . 56
3.7.1 Hirota’s Direct Method . . . . . . . . . . . . . . 56
3.7.2 Transient Solution Summary . . . . . . . . . . . 58
3.8 The Three Faces of the KdV Soliton . . . . . . . . . . . 58
4 Conservative and Dissipative Soliton Systems 61
4.1 Conservation Laws . . . . . . . . . . . . . . . . . . . . . 61
4.2 The Lossy KdV . . . . . . . . . . . . . . . . . . . . . . 63
II Electrical Solitons: Design 69
5 Electrical Nonlinear Transmission Line and Electrical
Solitons 71
5.1 The Nonlinear Transmission Line . . . . . . . . . . . . 71
5.2 Toda Lattice . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3 NLTL Lattice . . . . . . . . . . . . . . . . . . . . . . . 74
5.4 KdV Approximation of the NLTL . . . . . . . . . . . . 77
5.5 The Lossy NLTL . . . . . . . . . . . . . . . . . . . . . . 80
6 The Electrical Soliton in the Lab 85
Michael W. Chen and En Shi
6.1 Toda Lattice, NLTL Lattice and KdV Solitons . . . . . 86
6.2 Scaling and Transformations: Lab → NLTL → KdV . . 87
6.3 NLTL Characterization . . . . . . . . . . . . . . . . . . 93
6.4 Inverse Scattering on the NLTL . . . . . . . . . . . . . 96
6.5 Soliton Damping on the NLTL . . . . . . . . . . . . . . 97
6.6 Numerical Accuracy . . . . . . . . . . . . . . . . . . . . 100
III Electrical Solitons: Application 103
7 NLTL as a Two-Port System 105

Xiaofeng Li and Michael W. Chen
7.1 Pulse Compression and Tapered NLTL . . . . . . . . . 105
7.2 Shockwave NLTL . . . . . . . . . . . . . . . . . . . . . 110
7.3 Harmonic Generation . . . . . . . . . . . . . . . . . . . 117
7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8 The Soliton Oscillator 119
8.1 Basic Topology . . . . . . . . . . . . . . . . . . . . . . . 119
8.2 Instability Mechanisms . . . . . . . . . . . . . . . . . . 121
8.2.1 Case I — Voltage Limiting Amplifier . . . . . . . 121
vii
8.2.2 Case II — Linear Amplifier . . . . . . . . . . . . 122
8.3 Identification of Three Instability Mechanisms . . . . . 122
8.4 NLTL Soliton Oscillator — Working Model . . . . . . . 125
8.4.1 Operating Principles . . . . . . . . . . . . . . . . 125
8.4.1.1 Bias adjustment . . . . . . . . . . . . . 126
8.4.1.2 Amplifier operation . . . . . . . . . . . 126
8.4.2 Stability Mechanisms — Solution . . . . . . . . . 127
8.4.2.1 Distortion reduction . . . . . . . . . . . 127
8.4.2.2 Perturbation rejection . . . . . . . . . . 127
8.4.2.3 Single mode selection . . . . . . . . . . 129
8.5 System Design and Amplifier Dynamics . . . . . . . . . 129
8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 140
9 The Circular Soliton Oscillator 141
9.1 CMOS, Low MHz Prototype . . . . . . . . . . . . . . . 141
9.1.1 Oscillator Implementation . . . . . . . . . . . . . 142
9.1.1.1 Amplifier design . . . . . . . . . . . . . 143
9.1.1.2 NLTL design . . . . . . . . . . . . . . . 144
9.1.1.3 Termination . . . . . . . . . . . . . . . 144
9.1.2 Experimental Verification . . . . . . . . . . . . . 146
9.1.2.1 Adaptive bias control . . . . . . . . . . 146
9.1.2.2 Startup soliton dynamics . . . . . . . . 147
9.1.2.3 Perturbation rejection . . . . . . . . . . 147
9.1.2.4 Steady-state soliton oscillation . . . . . 151
9.1.2.5 Soliton propagation in steady-state . . 151
9.2 Bipolar, Microwave Prototype . . . . . . . . . . . . . . 154
9.2.1 Oscillator Implementation . . . . . . . . . . . . . 157
9.2.2 Experimental Results . . . . . . . . . . . . . . . 162
9.3 CMOS, Chip-scale, GHz Prototype . . . . . . . . . . . 165
9.3.1 Oscillator Implementation . . . . . . . . . . . . . 165
9.3.2 Test and Measurement . . . . . . . . . . . . . . . 167
9.3.3 Experimental Results . . . . . . . . . . . . . . . 167
9.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 170

10 The Reflection Soliton Oscillator 173
O. Ozgur Yildirim
10.1 Operating Principle . . . . . . . . . . . . . . . . . . . . 175
10.1.1 Reflection at the Amplifier End . . . . . . . . . . 176
10.1.2 Reflection at the Open End . . . . . . . . . . . . 179
10.2 Amplifier Design . . . . . . . . . . . . . . . . . . . . . . 181
10.2.1 Need for an Adaptive Bias Scheme . . . . . . . . 181
10.2.2 Reflection Amplifier with an Adaptive Bias . . . 181
10.2.3 Improved R-C Network . . . . . . . . . . . . . . 186
viii
10.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . 186
10.4 Comparison with Haus’s Oscillator . . . . . . . . . . . . 193
10.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 194
11 ChaoticSolitonOscillatorandChaoticCommunications 197
O. Ozgur Yildirim, Nan Sun, and Xiaofeng Li
11.1 Chaos and Chaotic Communications . . . . . . . . . . . 198
11.2 Chaotic Soliton Oscillator . . . . . . . . . . . . . . . . . 200
11.3 Simulation of the Chaotic Soliton Oscillator . . . . . . 202
11.4 Simulation of Chaotic Binary Communication . . . . . 203
11.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 208
12 Phase Noise of Soliton Oscillators 211
Xiaofeng Li
12.1 Phase Noise Fundamentals . . . . . . . . . . . . . . . . 212
12.2 Phase Noise Due to Direct Phase Perturbation . . . . . 215
12.2.1 Distributed Noise Sources . . . . . . . . . . . . . 215
12.2.2 Lumped Noise Sources . . . . . . . . . . . . . . . 223
12.3 Amplitude-to-Phase Noise Conversion . . . . . . . . . . 225
12.3.1 Distributed Noise Sources . . . . . . . . . . . . . 227
12.3.2 Lumped Noise Sources . . . . . . . . . . . . . . . 228
12.3.3 Indirect vs. Direct Phase Perturbations . . . . . 229
12.4 Experimental Verification . . . . . . . . . . . . . . . . . 229
12.4.1 Oscillator Prototypes . . . . . . . . . . . . . . . 229
12.4.2 Phase Noise Measurement . . . . . . . . . . . . . 230
12.4.3 Intensity of Noise Sources . . . . . . . . . . . . . 232
12.4.4 Measurement–Theory Comparison . . . . . . . . 235
12.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 238
Bibliography 239
Index 247
List of Figures
1.1 Time evolution of solitons during collision. . . . . . . . 4
1.2 Solitons in high-speed electronics. . . . . . . . . . . . . 6
1.3 The classical solitons. . . . . . . . . . . . . . . . . . . . 8
2.1 The cnoidal wave. . . . . . . . . . . . . . . . . . . . . . 18
2.2 The cnoidal wave versus periodic sech2 wave.. . . . . . 21
2.3 Soliton collision: Lax case I. . . . . . . . . . . . . . . . 23
2.4 Soliton collision: Lax case II. . . . . . . . . . . . . . . . 24
2.5 Soliton collision: Lax case III. . . . . . . . . . . . . . . 25
2.6 Soliton amplitude and phase modulation during colli-
sion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
A.1 Trigonometric and elliptic functions. . . . . . . . . . . 29
3.1 Inverse scattering method. . . . . . . . . . . . . . . . . 40
3.2 Inverse scattering method using delta function potential. 43
3.3 Soliton collision: reflectionless potential. . . . . . . . . 50
3.4 Time evolution of solitons, eigenvalues and eigenfunc-
tions for reflectionless potentials. . . . . . . . . . . . . 51
3.5 Soliton collision: Non-reflectionless potential.. . . . . . 53
3.6 Breakup of a square pulse into multiple solitons. . . . . 54
4.1 KdV soliton damping. . . . . . . . . . . . . . . . . . . 67
5.1 Nonlinear transmission line (NLTL). . . . . . . . . . . 72
5.2 Coefficients of the NLTL lattice and the KdV approxi-

mation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Loss mechanisms in the NLTL. . . . . . . . . . . . . . 81
6.1 NLTL: Toda lattice, NLTL lattice and KdV approxima-
tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Laboratory coordinates. . . . . . . . . . . . . . . . . . 89
6.3 Transformation from laboratory coordinates to KdV co-
ordinates. . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.4 Capacitance versus bias voltage. . . . . . . . . . . . . . 94
6.5 Inductor equivalent series resistance versus frequency. . 95
ix
x
6.6 NLTL dispersion measurement. . . . . . . . . . . . . . 95
6.7 Experimental inverse scattering: Fixed A. . . . . . . . 98
6.8 Experimental inverse scattering: Fixed W. . . . . . . . 99
6.9 Soliton damping on the NLTL. . . . . . . . . . . . . . . 101
6.10 Simulation inaccuracies. . . . . . . . . . . . . . . . . . 102
7.1 Monolithic NLTL. . . . . . . . . . . . . . . . . . . . . . 106
7.2 Dispersion due to structural periodicity. . . . . . . . . 107
7.3 Pulse compression using tapered NLTLs. . . . . . . . . 109
7.4 Loss mechanisms in NLTL. . . . . . . . . . . . . . . . . 110
7.5 Shockwave formation. . . . . . . . . . . . . . . . . . . . 111
7.6 Shockwave of Burgers equation. . . . . . . . . . . . . . 113
7.7 Effects of dispersion on shockwave. . . . . . . . . . . . 115
7.8 Shockwave formation on the lossy NLTL. . . . . . . . . 116
7.9 Harmonic generation with discrete device. . . . . . . . 117
7.10 Second harmonic generation with NLTL. . . . . . . . . 118
8.1 Ring NLTL and multi-mode solutions. . . . . . . . . . 120
8.2 Soliton oscillator topology. . . . . . . . . . . . . . . . . 121
8.3 Soliton oscillator topology: Saturating amplifier. . . . . 123
8.4 Soliton oscillator topology: Linear amplifier. . . . . . . 124
8.5 Stabilizing amplifier. . . . . . . . . . . . . . . . . . . . 126
8.6 Soliton oscillator: dc output during start-up. . . . . . . 127
8.7 Soliton oscillator amplifier: Operating principles. . . . 128
8.8 Soliton oscillator amplifier: Mode-dependent gain. . . . 130
8.9 Q-switched instability. . . . . . . . . . . . . . . . . . . 131
8.10 Simulation of the adaptive bias transient response. . . 134
8.11 Saturable absorber instability. . . . . . . . . . . . . . . 135
8.12 Simulation of the adaptive bias transient response and
saturable absorber. . . . . . . . . . . . . . . . . . . . . 138
8.13 Behavioral simulation of the Q-switched instability. . . 139
9.1 Low MHz prototype schematic. . . . . . . . . . . . . . 142
9.2 NLTL design. . . . . . . . . . . . . . . . . . . . . . . . 145
9.3 NLTL termination: tapered-loss NLTL. . . . . . . . . . 145
9.4 Low MHz soliton oscillator prototype. . . . . . . . . . . 146
9.5 Experimental results: Amplifier bias adjustment. . . . 148
9.6 Experimental results: Startup transient. . . . . . . . . 149
9.7 Experimental results: Perturbation rejection. . . . . . . 150
9.8 Experimentalresults: Steady-stateoscillationofmode1
and mode 2. . . . . . . . . . . . . . . . . . . . . . . . . 152
9.9 Experimental results: Robustness to disturbance. . . . 153
9.10 Steady-state soliton propagation. . . . . . . . . . . . . 154