3D Modeling, Analysis, and Design of a Traveling-Wave Tube

3D MODELING, ANALYSIS, AND DESIGN OF A TRAVELING-WAVE TUBE

USING A MODIFIED RING-BAR STRUCTURE WITH RECTANGULAR

TRANSMISSION LINES GEOMETRY

SADIQ ALI ALHUWAIDI

B.S., University of Colorado, Boulder, 2011

M.S., University of Colorado, Colorado Springs, 2014

A dissertation submitted to the Graduate Faculty of the

University of Colorado Colorado Springs

in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

Department of Electrical and Computer Engineering

2017

SADIQ ALI ALHUWAIDI

This dissertation for the Doctor of Philosophy degree by

Sadiq Ali Alhuwaidi

has been approved for the

Department of Electrical and Computer Engineering

Heather Song, Chair

T.S. Kalkur

Charlie Wang

John Lindsey

Zbigniew Celinski

Date 12/05/2017

Alhuwaidi, Sadiq Ali (Ph.D. Engineering - Electrical Engineering)

3D Modeling, Analysis, and Design of a Traveling-Wave Tube Using a Modified Ring-

Bar Structure with Rectangular Transmission Lines Geometry

Dissertation directed by Associate Professor Heather Song.

ABSTRACT

A novel slow-wave structure of the traveling-wave tube consisting of rings and rectangular coupled transmission lines is modeled, analyzed, and designed in the frequency range of 1.89-2.72 GHz. The dispersion and interaction impedance characteristics are investigated using High Frequency Structure Simulator, HFSS, and a power run is carried out using Finite-Difference Time-Domain (FDTD) code, VSim. The performance of the design providing a better output power, gain, bandwidth, and efficiency is compared to the conventional and existing designs by implementing cold- and hot-test simulations. In addition, an electron gun and periodic permanent magnet, PPM, is designed using EGUN code and ANSYS Maxwell, respectively. The electron beam has a beam voltage of 262 kV, beam current of 12 A, cathode emission density of 5.968 , and minimum radius of A 2.0 mm. The required gun parameters and magnetic field levels,c including the geometrical quantities, are calculated to produce the appropriate electron flow and achieve adequate beam stability. Iterations and analysis of those quantities are provided to properly understand the procedure of the design.

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DEDICATION

This dissertation is dedicated to the memory of my grandmother, Zainab, who always prayed for me, to my beloved parents, Ali and Balkess, without whom none of this work would be possible, to my wife, Maryam, and son, Jafar, for supporting me in all my endeavors, to my sister and brothers for standing by me, and to the memory of my uncle,

Naeem.

ACKNOWLEDGMENTS

I owe thanks to many people for helping me prepare this work. Unfortunately, limited space dictates that only a few of them can receive a formal acknowledgment. But this is not taken as a disparagement of those whose contributions remain anonymous. My gratitude is immeasurable.

My foremost appreciation goes to my academic advisor Dr. Heather Song for her fundamental role in my doctoral work. I am deeply indebted to her for the non-stop accompaniment of my progress during the research and providing all conditions to keep my work running. I would like to thank Dr. T.S. Kalkur for his excellent guidance throughout my degree, and particularly the courses taken with him. I would like to express my gratitude to Dr. John Lindsey for the substantial influence that his courses have had on my knowledge. In addition, I gratefully acknowledge my Ph.D. committee members, Dr.

Charlie Wang and Dr. Zbigniew Celinski, for their time and valuable suggestions of the dissertation. I am grateful to Tech-X Corporation for giving me VSim software to pursue my research towards my doctoral degree. Finally, this work would not be accomplished without my parents, brothers, sister, and wife, who cheered me up, supported me academically and emotionally through the rough road to finish this dissertation, and stood by me.

TABLE OF CONTENTS

CHAPTER

I. INTRODUCTION...... 1

1.1 Early Milestones of Traveling-Wave Tube...... 1

1.2 Classical Types of Electronics ...... 4

1.2.1 Solid State Devices ...... 4

1.2.2 Vacuum Devices ...... 5

1.3 Domain of Vacuum Tubes ...... 10

1.4 Literature Work ...... 11

1.5 Novelty of Proposed Work ...... 14

1.6 Overview of Dissertation ...... 16

II. BACKGROUND AND THEORY ...... 17

2.1 Basic Operation of Traveling-Wave Tube ...... 17

2.2 Electron Dynamics ...... 23

2.2.1 Electric Field ...... 23

2.2.2 Magnetic Field ...... 29

2.3 Source of Electrons ...... 30

2.3.1 Cathode ...... 31

2.3.2 Thermionic Emission ...... 32

2.3.3 Schottky Effect...... 39

2.3.4 Space Charge Limitation...... 42

2.3.5 Life Expectancy ...... 45

2.4 Electron Gun and Focusing Structure ...... 46

2.4.1 Electron Guns...... 47

2.4.2 Focusing Structure ...... 60

2.4.2.1 Uniform-Field Focusing...... 61

2.4.2.2 Periodic Permanent Magnet (PPM) Focusing...... 70

2.5 Traveling Wave Interaction ...... 79

2.5.1 Electronic, Circuit, and Determinantal Equations ...... 79

2.5.1.1 Electronic Equation ...... 80

2.5.1.2 Circuit Equation ...... 82

2.5.1.3 Determinantal Equation ...... 85

2.5.2 Synchronous Condition ...... 86

2.5.3 Nonsynchronous Condition ...... 90

2.6 TWT Slow-Wave Circuits ...... 91

2.6.1 Wave Velocities ...... 92

2.6.2 Dispersion ...... 94

2.6.2.1 Coaxial Transmission Line ...... 94

2.6.2.2 Rectangular Waveguide ...... 96

2.6.3 Bandwidth ...... 102

2.6.4 Power ...... 109

2.6.4.1 Backward Wave Oscillations and Suppression to Peak Power ...... 109

2.6.4.2 Typical Support Techniques to Average Power ...... 113

2.6.5 Attenuators and Severs ...... 118

2.6.6 Ring-Bar and Ring-Loop TWT ...... 120

2.7 Collector ...... 123

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2.8 Transmission Line Fundamentals ...... 128

III. ELECTRON GUN AND FOCUSING STRUCTURE DESIGNS ...... 133

3.1 Overview ...... 133

3.2 Design Specifications...... 135

3.2.1 First Electron Gun Design ...... 135

3.2.2 Electron Gun Design of the Proposed Novel Slow-Wave Structure ...... 135

3.3 Calculations...... 135

3.3.1 Electron Gun Parameters ...... 136

3.3.1.1 First Electron Gun Design ...... 138

3.3.1.2 Electron Gun Design of the Proposed Novel Slow-Wave Structure .. 139

3.3.2 Periodic Permanent Magnet Parameters ...... 140

3.3.2.1 Electron Gun Design of the Proposed Novel Slow-Wave Structure .. 141

3.3.3 Iterations ...... 142

3.3.3.1 First Electron Gun Design ...... 143

3.3.3.2 Electron Gun Design of the Proposed Novel Slow-Wave Structure .. 144

3.3.4 Parameter Analysis ...... 146

3.4 Electron Gun Simulations and Designs ...... 161

3.4.1 First Electron Gun Design ...... 162

3.4.2 Electron Gun Design of the Proposed Novel Slow-Wave Structure ...... 165

3.5 Periodic Permanent Magnet Simulations and Designs ...... 166

3.5.1 Magnet of Electron Gun Design of the Proposed Novel Slow-Wave Structure ...... 167

3.6 Electron Gun Design of the Proposed Novel Slow-Wave Structure with Magnet ...... 172

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3.7 Discussion ...... 173

3.7.1 First Electron Gun Design ...... 173

3.7.2 Electron Gun Design of the Proposed Novel Slow-Wave Structure with Magnet ...... 174

IV. A NOVEL SLOW-WAVE CIRCUIT STRUCTURE WITH COLD-TEST SIMULATIONS ...... 176

4.1 Mutual Inductance and Capacitance ...... 176

4.1.1 Mutual Inductance ...... 177

4.1.2 Mutual Capacitance ...... 179

4.2 Coupled Wave Equations ...... 181

4.3 Coupled Line Analysis ...... 185

4.4 High Power Slow-Wave Circuit Structure ...... 185

4.4.1 Early Stage of ANSYS High Frequency Structure Simulator (HFSS) ...... 186

4.4.2 Final Design of ANSYS High Frequency Structure Simulator (HFSS) .... 215

V. A NOVEL SLOW-WAVE CIRCUIT STRUCTURE WITH HOT-TEST SIMULATIONS ...... 224

5.1 Finite-Difference Time-Domain (FDTD) Code, VSim ...... 224

5.2 Comparison between the Novel Slow-Wave Circuit Structure, Ring-Bar Structure, Half-Ring Helical Structure, Ring-Loop Structure, Curved Ring-Bar Structure, and Wave-Ring Helical Structure ...... 241

5.3 Future Work ...... 243

REFERENCES ...... 246

APPENDICES ...... 255

• APPENDIX A ...... 255

A.1 First Electron Gun Design with current density of 2 A/cm2 ...... 255

• APPENDIX B ...... 257

B.1 Electron Gun of the Proposed Novel Slow-Wave Structure ...... 257

• APPENDIX C ...... 259

C.1 Electron Gun Plots and Analysis ...... 259

• APPENDIX D ...... 279

D.1 First Electron Gun with current density of 2 A/cm2 ...... 279

• APPENDIX E ...... 289

E.1 Current Density of 5.968 A/cm2 with Magnet for the Proposed Slow-Wave Structure Design...... 289

• APPENDIX F...... 313

F.1 Parameters of Novel Slow-Wave Structure to Perform Hot Test Simulations Using VSim ...... 313

LIST OF TABLES

TABLE

1.1: Comparison between the existing designs of the traveling wave tube including ring- bar structure, half-ring helical structure, ring loop structure, curved ring-bar structure, and wave-ring helical structure...... 13

2.1: Work functions at room temperature and their melting temperature for various metals [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 37

2.2: Characteristics of control electrodes [80]...... 59

3.1: Specifications of the first electron gun design derived from [2] with a beam voltage of 10 kV, beam current of 1 A, minimum beam radius of 1 mm, and cathode emission density of 2 A/cm2...... 135

3.2: Specifications of electron gun design of the proposed novel slow-wave structure of the TWT with a beam voltage of 262 kV, beam current of 12 A, minimum beam radius of 2 mm, and cathode emission density of 5.968 A/cm2...... 135

3.3: Calculated electron gun parameters for the first design with a beam voltage of 10 kV, beam current of 1 A, minimum beam radius of 1 mm, and cathode emission density of 2 A/cm2...... 138

3.4: Calculated electron gun parameters of the proposed novel slow-wave structure of the TWT with a beam voltage of 262 kV, beam current of 12 A, minimum beam radius of 2 mm, and cathode emission density of 5.968 A/cm2...... 139

3.5: Calculated magnet stack parameters used in the electron gun design of the proposed novel slow-wave structure of the traveling wave tube...... 142

3.6: Initial iteration for the first electron gun design...... 143

3.7: Final iteration for the first electron gun design...... 144

3.8: Initial iteration for the electron gun design of the proposed novel slow-wave structure of the traveling wave tube...... 145

3.9: Final iteration for the electron gun design of the proposed novel slow-wave structure of the traveling wave tube...... 145

3.10: Materials used for each geometry in the periodic permanent magnet...... 168

3.11: PPM design parameters dimensions...... 169

3.12: Maximum field levels in iron and air along one cell of the magnet stack...... 170

3.13: Maximum field levels in iron and air along the periodic permanent magnet stack...... 171

3.14: Results of the electron gun trajectory using EGUN code for the first electron gun design with a current density of 2 A/cm2...... 173

3.15: Results of the electron gun trajectory using EGUN code for the proposed novel slow-wave structure of the traveling wave tube with a beam voltage of 262 kV, beam current of 12 A, and cathode emission density of 5.968 A/cm2 with the magnet...... 174

4.1: Dimensions of the geometrical structure of the novel slow-wave circuit structure of the TWT at the early stage...... 189

4.2: Calculated parameters of the chose design of the novel structure of the TWT whose dimensions are L = 16.0, W = 10.5, and p = 22.0 [in mm]...... 218

5.1: Specifications of the helix slow-wave circuit structure of the compact lightweight traveling wave tube [72]...... 227

5.2: Electron gun parameters of the compact lightweight traveling wave tube [72]...... 228

5.3: Specifications and calculations of the periodic permanent magnet of the compact lightweight traveling wave tube [72]...... 228

5.4: Simulated recorded output power, input power, and gain of the compact lightweight traveling wave tube in the frequency range of 2.0-4.0 GHz...... 229

5.5: Design specifications of the design of novel slow-wave structure of the TWT with L = 16.0, W = 10.5, p = 22.0 [in mm]...... 231

5.6: Simulated recorded output power, input power, and gain of the novel slow-wave structure of the TWT using VSim with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 in the frequency range of 1.85-2.80 GHz...... 238

5.7: Output power and input power of the novel slow-wave structure of the TWT using VSim with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 at 2.40 GHz...... 240

5.8: Comparison between the designed novel slow-wave circuit structure of the traveling wave tube with L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm and existing designs including ring-bar structure, half-ring helical structure, ring loop, curved ring-bar structure, and wave-ring helical structure...... 242

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LIST OF FIGURES

FIGURE

1.1: System implementation with electronics throughout the microwave frequency range and beyond...... 4

1.2: Categories of vacuum tubes throughout the microwave frequency range and beyond...... 6

1.3: Basic configuration of a klystron [1]...... 6

1.4: Basic configuration of a traveling wave tube [1]...... 7

1.5: Basic configuration of a magnetron [1]...... 8

1.6: Basic configuration of a crossed-field amplifier [1]...... 9

1.7: Basic configuration of a gyrotron oscillator [1]...... 9

1.8: Average power and frequency range of vacuum and solid-state devices throughout the microwave frequency range and beyond [1]...... 10

1.9: Ring-bar structure [30]...... 11

1.10: Half-ring helical structure [32]...... 12

1.11: Ring-loop structure [33]...... 12

1.12: Ring-loop and curved ring-bar structures [33]...... 12

1.13: Wave-ring helical structure [34]...... 12

2.2: Patterns of electric field and RF charge for a single-wire transmission line above an existing ground plane [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 18

2.3: Patterns of electric field and RF charge for a helix [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 19

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2.4: When the beam enters the circuit, energy is bunched and extracted from the beam due to the existing axial field [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 20

2.5: When the interaction between the electron beam and circuit occurs, energy is bunched and extracted from the beam due to the existing axial field [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 20

2.6: Basic coupled cavity TWT...... 21

2.7: Basic coupled cavity circuit [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 22

2.8: Vector diagram of circuit voltage [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 23

2.9: Electron gun of TWT [1]...... 25

2.10: Deflection of electron by a magnetic field [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 29

2.11: Energy level diagram for electrons near the surface of a metal between a cathode and vacuum [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 32

2.12: Two electrons with sufficient energies to be emitted, but moving in different directions [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 33

2.13: Fermi-Dirac distribution function for T = 0 and 1273 K...... 34

2.14: Electric field pattern established by an electron and its image [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 40

2.15: Energy-band diagram between a metal and surface and a vacuum [44]...... 41

2.16: Potential distribution with and without electrons from cathode to anode in a parallel-plane diode [Reproduced by permission from Author A. S. Gilmour, Jr.,

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2.17: Potential near the cathode surface [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 43

2.18: Current-voltage relationship with one microperveance...... 45

2.19: Electron gun design components with identified three regions [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 47

2.20: Parallel electron flow achieved by focusing the electrodes [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 48

2.21: Electron trajectories divergence with (solid lines) and without (dashed lines) electrons [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 48

2.22: Parallel flow beam due to the focused electrode at cathode potential [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 49

2.23:A spherical diode, where inner and outer diameters represent the cathode and anode, respectively [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 50

2.24: Conical diode with half angle [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 51

2.25: Low perveance increases the distortion near the anode aperture [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 52

2.26: A higher perveance increases the size of the anode and decreases the distance between the cathode and anode resulting in some distortion near the cathode [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 52

2.27: A modified focused electrode to improve the electron gun design by reducing the distortion of equipotential profiles and improving the electron focusing and cathode

2.28: Quantities used in the analysis of effect of anode aperture to calculate the gun parameters [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 53

2.29: Electron beam shape in region 3 [1]...... 55

2.30: Cathode-to-anode voltage way to control the beam current in the electron gun [1]...... 56

2.31: Modulating anode way to control the beam current in the electron gun [1]...... 57

2.32: Focusing electrode way to control the beam current in the electron gun [1]...... 57

2.33: Grid way to control the beam current in the electron gun [1]...... 58

2.34: Gun parameters to determine grid-cathode spacing [1]...... 58

2.35: The effects of space charge and focusing forces on the electron beam [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 60

2.36: The use of a solenoid to generate a magnetic field and focus the beam in linear beam tubes [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 61

2.37: Configuration of magnetic flux lines as the electron beam enters the solenoid [1]. 61

2.38: Electron trajectory in the axial field [1]...... 62

2.39: Brillouin flow condition [1]...... 63

2.40: Resulted beam dynamics when the electron beam enters the magnetic field [1]. ... 63

2.41: Magnetic field configuration for Brillouin flow at the entrance going to the focusing structure [1]...... 67

2.42: Obtained electron beam if the used magnetic flux density is less than Brillouin flux density [1]...... 68

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2.43: Beam shape as the magnetic flux density is varied compared to the Brillouin flux density [1]...... 69

2.44: Beam shape as the magnetic flux density is varied compared to the Brillouin flux density when db/dz is larger than zero [1]...... 69

2.45: A system of periodic permanent magnet with a periodic focusing [1]...... 70

2.46: Difference between a beam ripple and scalloping [1]...... 71

2.47: Beam envelop curves for three cases of the magnetic field with different values of α and β [89]...... 73

2.48: Focusing conditions as α and β are varied [1]...... 73

2.49: Unstable conditions for the normalized beam radius equation based on α values [1]...... 74

2.50: A series of convergent lenses demonstrating the PPM [1]...... 74

2.51: Focusing conditions in terms of optical rays for different focal lengths [1]...... 75

2.52: Intensification factor versus radius compression ratio of the PPM field...... 78

2.53: Normalized focusing factor versus radius compression ratio of the PPM field...... 79

2.54: Transmission model for the RF circuit of the TWT...... 83

2.55: Transmission line model for the RF circuit of the TWT to determine current at point A...... 83

2.56: Transmission line model for the RF circuit of the TWT to determine the voltages around loop ABCD...... 84

2.57: Power gain as a function of CN for the synchronous condition in a traveling wave tube...... 90

2.58: Difference between group and phase velocity [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 92

2.59: Opposite directions of the group and phase velocities [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 93

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2.60: Illustration of dispersion characteristics between the phase velocity and frequency [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 94

2.61: Electric and magnetic fields' lines of a coaxial transmission line in the fundamental TEM mode [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 95

2.62: Brillouin diagram for a coaxial transmission line in the TEM mode [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 95

2.θ3: Two plane waves at angles ±α in the z-direction [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 96

2.64: Group and phase velocities inside a waveguide [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 97

2.65: Wave configurations inside the waveguide for frequencies f1 > f2 > f3 [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 98

2.66: Quantities used to derive the dispersion characteristics of a waveguide [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 98

2.67: Brillouin diagram for a rectangular waveguide [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 99

2.68: Changes in the propagation constant when the angular frequency is varied [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 100

2.69: Group velocity from the Brillouin diagram for different wave configurations [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 101

2.70: Electric field distributions in the dominant mode in the rectangular waveguide [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling

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2.71: Brillouin diagram in the rectangular waveguide for propagating waves in either direction [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 102

2.72: Saturated output power versus frequency for a helix TWT [1]...... 103

2.73: Effect of harmonic injection on the saturated output power for a helix [1]...... 103

2.74: Helix being cut at points x and is being straightened [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 104

2.75: Ideal Brillouin diagram without dispersion for a helix [1]...... 104

2.76: Electric field pattern with two different frequencies [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 105

2.77: Magnetic flux cancellation between the helix turns [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 106

2.78: Brillouin diagram for a helix with a 10° pitch angle [1]...... 106

2.79: Common techniques used to control the dispersion [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 107

2.80: Normalized phase velocity and Pierce’s velocity parameter as a function of frequency for the suggested techniques to control dispersion [98]...... 108

2.81: Small signal gain as a function of frequency for the suggested techniques to control dispersion [98]...... 108

2.82: Backward wave oscillations on a helix for two turns [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 110

2.83: Suppressing BWO with the use of resonant loss to produce attenuation [99]...... 111

2.84: Saturated output power of a 10 kW helix TWT with a resonant loss at 8 GHz [100]...... 111

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2.85: Technique of pitch change to suppress backward wave oscillations [99]...... 112

2.86: Peak output power versus midband frequency with BWO suppression techniques and without them [100]...... 113

2.87: A typical use of support rods with a helix...... 113

2.88: Interaction impedance of a helix with and without the use of support rods [1]. ... 114

2.89: Thermal conductivities of some dielectric and metal materials [1]...... 114

2.90: Temperature drop between helix and support rods and between support rods and barrel [99]...... 115

2.91: Thermal interface conductivities versus contact pressure for some dielectrics interfaced with a helix made of tungsten [1]...... 115

2.92: Pressure or hot insertion technique [1]...... 116

2.93: Comparison between rod support and block support structures [102]...... 117

2.94: Comparison of the helix temperature with respect to the input power between the triangulation, pressure or hot insertion, and brazing techniques [103]...... 118

2.95: Quantities used in the analysis of oscillations [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 119

2.96: Lossy filum attenuator used with a helix [1]...... 119

2.97: Use of two severs with a helix to suppress the backward wave and obtain a better efficiency than the attenuator [1]...... 120

2.98: Ring bar and contrawound helix circuits [1]...... 120

2.99: Backward wave interactions for a single and bifilar helix [1]...... 121

2.100: Brillouin diagram for the ring bar structure [1]...... 122

2.101: Normalized phase velocity for the ring bar structure in the Ka-band frequency range with 18 kV beam voltage [1]...... 122

2.102: Power and bandwidth of ring bar structure in the X-band frequency range [1]. . 123

2.103: Power flow in a linear beam flow [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 124

2.104: Collector for a linear beam tube [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 126

2.105: Depressed collector circuit to recover the beam power [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 126

2.106: Power supply configuration for a multistage depressed collector [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 127

3.1: Quantities used in the analysis of effect of anode aperture to calculate the gun parameters [[Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]...... 136

3.2: Disc radius of cathode versus cathode emission density for a beam current of 12 A...... 140

3.3: Sectional view of magnet stack consisting of two magnets and iron pole pieces. .. 141

3.4: A diagram describing the procedure to iterate the angle values until achieving the appropriate electron gun parameters’ calculations...... 143

3.5: Disc radius of cathode versus cathode emission density relationship from equation (3.1) with a beam current of 50 mA...... 146

3.6: Disc radius of cathode versus cathode emission density relationship from equation (3.1) with beam currents of 50 mA in red and 1 A in blue...... 147

3.7: Disc radius of cathode versus beam current relationship from equation (3.1) with cathode emission densities of 2 A/cm2 in red, 10 A/cm2 in blue, 50 A/cm2 in green, and 100 A/cm2 in cyan...... 147

3.8: Theta versus alpha from equation (2.101) with a beam voltage of 18.2 kV and current of 50 mA...... 148

3.9: Theta versus alpha from equation (2.101) with a beam voltage of 18.2 kV and current of 50 mA in red, and beam voltage of 10 kV and current of 1 A in blue...... 148

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3.10: Perveance versus alpha from equations (2.101), (2.84), and (3.2) with different theta values of 30 degrees in red, 20 degrees in blue, 10 degrees in green and 5 degrees in cyan with beam voltage of 18.2 kV and current of 50 mA...... 149

3.11: Beam voltage versus alpha from equation (2.101) with different theta values of 30 degrees in red, 20 degrees in blue, 10 degrees in green and 5 degrees in cyan with a beam current of 50 mA...... 149

3.12: Beam current versus alpha from equation (2.101) with different theta values of 30 degrees in red, 20 degrees in blue, 10 degrees in green and 5 degrees in cyan with a beam voltage of 18.2 kV...... 150

3.13: Gamma versus alpha constants from equations (3.3-3.4)...... 150

3.14: Gamma versus its derivative constants from equation (3.6)...... 151

3.15: Slope of trajectory for Region 2 versus alpha from equation (3.5) with different values of theta and gamma derivative...... 151

3.16: Slope of trajectory for Region 2 versus Ra from equation (3.5) with different values of bo, alpha, and gamma derivative...... 152

3.17: Slope of trajectory for Region 2 versus Ra from equation (3.5) with different values of correction factor, bo, alpha, and gamma derivative...... 152

3.18: bo versus disc radius of cathode from equation (3.7) with different values of gamma...... 153

3.19: bo versus gamma from equation (3.7) with different values of disc radius of cathode...... 153

3.20: Slope of trajectory for Region 3 versus minimum beam diameter from equation (3.7) with different values of bo...... 154

3.21: Slope of trajectory for Region 3 versus bo from equation (3.7) with a beam voltage of 18.2 kV, beam current of 50 mA, and minimum beam diameter of 0.0375 mm...... 154

3.22: Slope of trajectory for Region 3 versus perveance from equation (3.7) with different values of bo...... 155

3.23: Spherical radius versus disc radius of cathode from equation (3.9) with different values of theta...... 155

3.24: Spherical radius versus theta from equation (3.9) with different values of disc radius of cathode...... 156

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3.25: Ra versus spherical radius from equation (3.10) with different values of gamma. 156

3.26: Ra versus gamma from equation (3.10) with different values of spherical radius. 157

3.27: ra versus bo from equation (3.11)...... 157

3.28: za versus ra from equation (3.12) with different values of spherical radius and Ra.158

3.29: za versus Ra from equation (3.12) with different values of spherical radius and ra...... 158

3.30: za versus spherical radius from equation (3.12) with different values of Ra and ra...... 159

3.31: zm versus minimum beam diameter from equation (3.13) with different values of za and bo...... 159

3.32: zm versus perveance from equations (3.7) and (3.13) with different values of za and bo...... 160

3.33: zm versus bo from equation (3.13) with different values of za...... 160

3.34: zm versus za from equation (3.13) with different values of bo...... 161

3.35: Diagram representing the overall method used in the gun codes [123]...... 161

3.36: Electron gun trajectory of the first design for the first electron gun with a beam voltage of 10 kV, beam current of 1 A, and cathode emission density of 2 A/cm2...... 163

3.37: A zoomed in plot of the electron gun trajectory of the first design for the first electron gun with a beam voltage of 10 kV, beam current of 1 A, and cathode emission density of 2 A/cm2...... 163

3.38: Electron gun trajectory of the second design for the first electron gun with a beam voltage of 10 kV, beam current of 1 A, and cathode emission density of 2 A/cm2...... 164

3.39: A zoomed in plot of the electron gun trajectory of the second design for the first electron gun with a beam voltage of 10 kV, beam current of 1 A, and cathode emission density of 2 A/cm2...... 164

3.40: Electron gun trajectory of the third design for the first electron gun with a beam voltage of 10 kV, beam current of 1 A, and cathode emission density of 2 A/cm2...... 165

3.41: A zoomed in plot of the electron gun trajectory of the third design for the first electron gun with a beam voltage of 10 kV, beam current of 1 A, and cathode emission density of 2 A/cm2...... 165

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3.42: Electron gun trajectory for the proposed novel slow-wave structure of the traveling wave tube with a beam voltage of 262 kV, beam current of 12 A, and cathode emission density of 5.968 A/cm2...... 166

3.43: Uniform and permanent periodic magnets with respect to the magnetic field entrance in the placement of the beam waist [95]...... 167

3.44: A single period periodic permanent magnet focusing structure [69]...... 167

3.45: One cell magnet structure consisting of a magnet block, pole pieces, and hubs using ANSYS Maxwell...... 168

3.46: Parameters of the one cell periodic permanent magnet using ANYSYS Maxwell...... 169

3.47: Magnetic field profile along one cell of the magnet stack using ANSYS Maxwell...... 169

3.48: Magnetic field profile along one cell of periodic permanent magnet using ANSYS Maxwell...... 170

3.49: Periodic permanent magnet with an array of magnet blocks...... 170

3.50: Magnetic field profile along the periodic permanent magnet stack using ANSYS Maxwell...... 171

3.51: Magnetic field profile along the array of periodic permanent magnet using ANSYS Maxwell...... 171

3.52: Final magnetic field profile along the array of periodic permanent using ANSYS Maxwell...... 172

3.53: Electron gun trajectory and magnetic field plot for the proposed novel slow-wave structure of the traveling wave tube a beam voltage of 262 kV, beam current of 12 A, and cathode emission density of 5.968 A/cm2...... 173

4.1: A simple coupled inductor circuit...... 177

4.2: A circuit with three coupled inductors...... 178

4.3: A simple coupled capacitor circuit...... 179

4.4: A circuit with three coupled capacitors...... 181

4.5: A lossless transmission line...... 182

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4.6: Two lossless transmission lines...... 183

4.7: Side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT...... 187

4.8: Perspective view of one-cell of the modeled slow-wave circuit structure of the TWT surrounded by a circular waveguide...... 187

4.9: Dimensions of the modeled one-cell slow-wave circuit structure of the TWT...... 188

4.10: Other dimensions of the modeled one-cell slow-wave circuit structure of the TWT surrounded by a circular waveguide...... 188

4.11: Available solution types from HFSS menu...... 190

4.12: Master boundary condition...... 191

4.13: Slave boundary condition...... 191

4.14: Assigning the phase delay in the slave boundary condition...... 192

4.15: Transparent view of the novel slow-wave circuit structure of the TWT design with applied master/slave boundaries...... 192

4.16: Eigenmode solution setup...... 193

4.17: Setup sweep analysis...... 193

4.18: Dispersion diagram of the novel slow-wave circuit structure of the TWT for the early stage designs with the x-axes being in degrees and circular waveguide radius of 54.61 mm...... 194

4.19: Dispersion diagram of the novel slow-wave circuit structure of the TWT for the early stage designs with the x-axes being in radians and circular waveguide radius of 54.61 mm...... 194

4.20: Dispersion diagram of the novel slow-wave circuit structure of the TWT for the early stage designs with the x-axes being in degrees and circular waveguide radius of 127.0 mm...... 196

4.21: Dispersion diagram of the novel slow-wave circuit structure of the TWT for the early stage designs with the x-axes being in radians and circular waveguide radius of 127.0 mm...... 196

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4.22: Propagation constant versus frequency of the novel slow-wave circuit structure of the TWT for the early stage designs with a circular waveguide radius of 127.0 mm. ... 197

4.23: Normalized phase velocity versus frequency of the novel slow-wave circuit structure of the TWT for the early stage designs with a circular waveguide radius of 127.0 mm...... 198

4.24: Normalized phase velocity and interaction impedance versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0, W = 13.0, p = 22.0 [in mm]...... 200

4.25: Normalized phase velocity and interaction impedance versus frequency of the novel slow-wave circuit structure of the TWT for L = 14.0, W = 13.0, p = 20.0 [in mm]...... 201

4.26: Normalized phase velocity and interaction impedance versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0, W = 15.0, p = 22.0 [in mm]...... 202

4.27: Normalized phase velocity and interaction impedance versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0, W = 20.0, p = 22.0 [in mm]...... 203

4.28: Normalized phase velocity and interaction impedance versus frequency of the novel slow-wave circuit structure of the TWT for L = 15.0, W = 13.0, p = 21.0 [in mm]...... 204

4.29: Side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT with L = 15.0, W = 0.0, and p = 21.0 [in mm]...... 205

4.30: Perspective view of one-cell of the modeled slow-wave circuit structure of the TWT surrounded by a circular waveguide with L = 15.0, W = 0.0, and p = 21.0 [in mm]. .... 205

4.31: Dispersion diagram of the slow-wave circuit structure of the TWT with L = 15.0, W = 0.0, and p = 21.0 [in mm] with the x-axes being in degrees and circular waveguide radius of 127.0 mm...... 206

4.32: Dispersion diagram of the slow-wave circuit structure of the TWT with L = 15.0, W = 0.0, and p = 21.0 [in mm] with the x-axes being in radians and circular waveguide radius of 127.0 mm...... 206

4.33: Propagation constant versus frequency of the slow-wave circuit structure of the TWT with L = 15.0, W = 0.0, p = 21.0, and circular waveguide radius of 127.0 [in mm]...... 207

4.34: Normalized phase velocity versus frequency of the slow-wave circuit structure of the TWT with L = 15.0, W = 0.0, p = 21.0, and circular waveguide radius of 127.0 [in mm]...... 207

4.35: Interaction impedance versus frequency of the slow-wave circuit structure of the TWT with L = 15.0, W = 0.0, and p = 21.0 [in mm]...... 208

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4.36: Comparison between the total area of the slow-wave structure when the width of the transmission lines is not zero at one time and zero at another time...... 209

4.37: Two parallel transmission lines...... 210

4.38: Side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT for L = 16.0, W = 10.5, and p = 22.0 [in mm]...... 216

4.39: Side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT for L = 16.0, W = 10.5, p = 22.0, and circular waveguide radius of 127.0 [in mm]...... 216

4.40: Side and Perspective views of one-cell of the modeled slow-wave circuit structure of the TWT for L = 16.0, W = 10.5, p = 22.0 [in mm] with another pair of shifted transmission line by 90°...... 218

4.41: Side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT surrounded by a circular waveguide for L = 16.0, W = 10.5, p = 22.0 [in mm] with another pair of shifted transmission line by 90°...... 219

4.42: Side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT surrounded by a circular waveguide for L = 16.0 mm, W = 10.5 mm, p = 22.0 mm with one and two pairs of transmission lines...... 219

4.43: Dispersion diagram of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm of both designs and beam line with the x-axes being in degrees and circular waveguide radius of 127.0 mm...... 220

4.44: Dispersion diagram of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm of both designs and beam line with the x-axes being in radians and circular waveguide radius of 127.0 mm...... 220

4.45: Propagation constant versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm of both designs with a circular waveguide radius of 127.0 mm...... 221

4.46: Normalized phase velocity versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm of both designs...... 221

4.47: Interaction impedance versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm of both designs...... 222

4.48: Gain parameter versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm of both designs...... 223

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5.1: Parameters of one-cell of the periodic permanent magnet of the compact lightweight traveling wave tube [72]...... 228

5.2: Simulated output power and gain of the compact lightweight traveling wave tube in the frequency range of 2.0-4.0 GHz...... 230

η.3: Authors’ work of the simulated output power and gain of the compact lightweight traveling wave tube in the frequency range of 2.0-4.0 GHz [72]...... 230

5.4: Exporting a geometry from HFSS...... 232

5.5: Perspective and side views of the geometry of the novel slow-wave structure of the TWT inside HFSS with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 ...... 232

5.6: Side and perspective views of the imported geometry of the novel slow-wave structure of the TWT inside HFSS with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 without the circular waveguide...... 233

5.7: Geometry of the novel slow-wave structure of the TWT inside VSim with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 without the tube...... 234

5.8: Geometry of the novel slow-wave structure of the TWT inside VSim with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 with the tube...... 234

5.9: Menu inside the 'Setup' window...... 235

5.10: Menu inside the 'Run' window to run the simulations...... 236

η.11: Menu inside the 'Visualize' window to view the results from ‘History’...... 237

5.12: Menu inside the 'Analyze' window to apply the low pass filter...... 237

5.13: Menu inside the 'Visualize’ window to view the results after applying the low pass filter from ‘1-D Fields’...... 238

5.14: Simulated saturated output power and gain of the novel slow-wave structure of the TWT with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 in the frequency range of 1.85-2.80 GHz...... 239

5.15: Output power versus input power of novel slow-wave structure of TWT using VSim with L = 16.0, W = 10.5, p = 22.0 [in mm], and N = 20 at 2.40 GHz...... 240

5.16: Output power versus number of periods of novel slow-wave structure of TWT using VSim with L = 16.0, W = 10.5, p = 22.0 [in mm], and N = 20 at 2.40 GHz...... 241

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5.17: Novel slow-wave structure of the TWT with unidentical transmission lines...... 244

5.18: Novel slow-wave structure of the TWT with unidentical periods resulted due to the difference in lengths...... 245

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CHAPTER I

INTRODUCTION

The traveling-wave tube (TWT), categorized as one of two major microwave devices besides klystron, is considered to be an O-type or linear beam tubes. It is capable of generating power ranging from watts to megawatts based on the radio-frequency (RF) circuits and can be used from frequencies below 1 GHz to over 100 GHz. The helix RF circuit is recognized to be used for wideband applications, but with limited power. The coupled cavity circuit is common for high power applications, but with limited bandwidth.

TWTs have been of interest in a variety of applications reaching over 50% of the sales volume among all microwave tubes. Various laboratories, industries, and research organizations are conducting research and development in TWTs in communications, satellites, radar systems, and electronic countermeasures systems as a final or high power amplifier transmitting RF pulse or a driver for other amplifiers. This chapter covers a firm grasp of the early history over the 20th century, classical types of vacuum tubes, and domain of vacuum tubes. The remainder of the chapter is devoted to an overview of the dissertation.

1.1 Early Milestones of Traveling-Wave Tube

The first developed TWT was invented by an Austrian born engineer named R.

Kompfner in early 1943 [1-7]. He worked at a government based radar British laboratory and summarized the operation of the first TWT as:

“When the radio frequency power emerging from the helix with the beam switched on was compared with the radio frequency power without the beam, it was found that, at a beam voltage of 2440 volts, there was an increase of 49%, while at a beam voltage of 2200 volts, there was a decrease of 40%.”

In late 1942, Kompfner stated that “the basic growing wave principle of the magnetron could be used for amplification of RF signals” [1]. Accordingly, his plan was developing an amplifier considering the sensitivity and noise factor. Such design was compared with the best available crystal-mixer receivers at that time. The first TWT was built and tested with an electron beam current and voltage of 110 μA and 1.83 kV, respectively. The resulted amplified power was 6 at a frequency of 3.3 GHz with a noise factor of 14 dB.

The design was improved later to reach an amplification of 14 in addition to reduce the noise factor by 3 dB to reach 11 dB.

However, in his patent, A. Haeff [8, 9], a Russian electrical engineer, earlier introduced the electron beam and RF circuit interaction in October, 1933. He indicated that a hollow electron beam deflected as a nearby RF signal propagated on a helical structure.

Haeff also stated that the velocity of the electron beam was equal to the velocity of the wave on the RF circuit. Such condition results in an existing amplification in the TWT.

However, his recognition lacked to interpret such amplification of the RF wave as it traveled.

In 1935, K. Posthumus [10], a Dutch electrical engineer, pointed out the conversion of the electron energy into an amplification of the RF wave by designing a cavity-type magnetron oscillator. He described such amplification to be caused as a result of the interaction between the tangential component of the RF wave as it traveled at a velocity equal to the velocity of electrons.

N. Lindenblad [11], working at Radio Corporation of America, obtained some amplification over a 30 MHz band at a frequency of 390 MHz in May, 1940, by applying a signal to the beam once and to the helix in other experiments. He indicated that the

interaction between the electron beam and RF wave on a helix produced a signal amplification on the helix. Lindenblad modified Haeff’s inductive output tube by replacing the cavity resonator with a helix and extending the vacuum envelope of Haeff’s tube. He also introduced the pitch helix and recognized its value such that synchronism is maintained and the velocity of the wave on the helix was equal to the velocity of the electron beam being inside the envelope. The amplification was then resulted as the velocity was reduced. In addition, Lindenblad introduced the use of a helical waveguide acting as a slow-wave circuit.

It was not until June 27 and 28, 1946, when the helix traveling wave tube was first announced in public. J. Pierce and L. Field, working at Bell Telephone Laboratories, participated at the Fourth Institute of Radio Engineer’s Electron Tube Conference at Yale.

Besides, the British wartime described the work on the helix wave traveling tubes at the same conference. Pierce and Field indicated the unique features of the helix traveling wave tubes [12]. In order to support and fix the helix structure, longitudinal insulating rods were used and positioned accurately. Furthermore, a uniform magnetic field, produced by a system of solenoids, was used to focus the electron beam. Moreover, layers of colloidal graphite on the rods were inserted as a technique to suppress backward traveling waves and oscillations for providing the appropriate loss. Besides, the gain was sacrificed with reduction at a minimum level by increasing the conductivity of the coating at the midpoint of helix. That conductivity delivered a dissipation of the unwanted reflected energy [13-

14].

Through the 20th century, the development and exposition of theories and operation of the TWT had become a research territory, especially between 1946 and 1950, leading to

have a unified coordination of the traveling wave tube. Some of those noteworthy contributions are in [15-16].

1.2 Classical Types of Electronics

The electronics-based sources have gained interest throughout the microwave frequencies and beyond due to the compactness in size and affordability in systems with integrated devices and circuits. Figure 1.1 shows the approaches to implement systems and devices with electronics throughout the microwave frequency range and beyond.

Figure 1.1: System implementation with electronics throughout the microwave frequency range and beyond. In devices, there are two main groups of the electronics-based source: solid-state and vacuum.

1.2.1 Solid State Devices

The solid-state devices are active or passive depending on the device implemented.

Examples of active devices are transistors. Recently, two different paths categorize the modern semiconductor active devices: Si technologies and III/V compound technologies.

Some examples of the Si technologies devices are SiGe Heterojunction Bipolar Transistor

(HBT) and Si Metal–Oxide–Semiconductor Field Effect Transistor (MOSFET). Some examples of III/V technologies include Heterojunction Bipolar Transistor (HBT) and High

Electron Mobility Transistor (HEMT). On the other hand, examples of passive devices are 4

diodes. Compared to the active devices, the passive devices work at higher frequencies and are used for generating and detecting signals. However, they are limited in applications.

Examples of passive devices used for generating signals include resonant tunneling diodes

(RTDs), IMPAct ionization Transit Time (IMPATT), and Gunn diodes. Examples of passive devices used for detecting signals include Schottky Barrier Diodes (SBDs), superconductor-insulator-superconductor (SIS) tunnel junction mixer, and hot electron bolometer (HEB) [17].

1.2.2 Vacuum Devices

Instead of using transistors or diodes, the vacuum electron devices include the use of vacuum tubes within which the electron beam travels. The kinetic and flow of electrons are controlled in the tube. The vacuum devices are classified based on the configurations of the tube as either fast-wave or slow-wave. Examples of fast-wave devices include gyrotrons and free electron lasers (FEL). In contrast, the electrons travel slower than the speed of light, c, in the slow-wave devices to synchronize with the wave velocity. Examples of slow-wave devices include klystrons, magnetrons, traveling wave tubes (TWTs), and backward oscillators (BWOs) [17].

Other resources classify the vacuum tube types differently based on the electric and magnetic fields produced by the electrons [1]. Figure 1.2 shows the categories of vacuum tubes in the microwave frequency range and beyond.

Figure 1.2: Categories of vacuum tubes throughout the microwave frequency range and beyond. As shown in Figure 1.2, the vacuum tubes are divided into three categories: linear-beam, crossed-field, and fast-wave tubes. The operating principles of all tubes are the same. They involve an electron beam passing through the tube and a circuit with an electromagnetic field. Amplifications or oscillations are produced when the electron beam and circuit interact with each other. Examples of the linear-beam tubes are klystrons and traveling wave tubes. Figure 1.3 illustrates the basic configuration of the klystron.

Figure 1.3: Basic configuration of a klystron [1]. As shown in Figure 1.3, the electron beam is formed in the electron gun and linearly travels to the collector passing through the RF circuit. Resonant cavities form the RF circuit without an electromagnetic coupling between them. The RF input accelerates and decelerates the electrons existing in the beam. An RF current in the beam is resulted, which is proportional to the distance the beam travels. Such current is coupled to the intermediate cavities inducing a signal and producing a field. The coupling is then followed to the output cavities producing the RF output power. The electrons are bunched as fast electrons catch

up with the slow electrons. The klystron can achieve an output power level of tens of megawatts or more and gain of 60 dB or more. However, its bandwidth is limited between a few percent and 10%.

If a broadband device is desired, the traveling wave tube replaces the klystron.

Figure 1.4 illustrates the basic configuration of the traveling wave tube.

Figure 1.4: Basic configuration of a traveling wave tube [1]. As shown in Figure 1.4, the RF circuit in the traveling wave tube is continuous. Behaving like a transmission line, the signal moves along the circuit continuously, but at a targeted velocity near to the velocity of the electron beam passing through it. The bunches of electrons are formed when the electric and magnetic fields decelerate and accelerate the electrons. An RF current in the circuit is resulted when the electron bunches pass by the circuit. Such current causes the amplitude of the RF field to become larger, which in turn, increases the intensity of electron bunching in the beam. As far as the velocity of the electron beam continues to be the same as the velocity of the signal, the bunching continues to grow and becomes more intense. The TWT can achieve an output power level of tens of watts for broadband devices and hundreds of kilowatts to megawatts for narrowband devices. The gain can reach up to 50 dB or more. Its bandwidth is between 20% and over

2 octaves.

The second category of vacuum tubes is crossed-field. Basically, the cathode in the crossed-field tubes is cylindrical and is in the center. The electron beam travels outward 7

toward the RF circuit. The magnetic field is perpendicular to the electric field, which results in a circular electron path moving around the cathode. The RF circuit acts as an anode. The electrons are bunched into spoke-like configurations whenever there is an RF field.

Examples of the cross-field tubes are magnetrons and cross-field amplifiers. Figure 1.5 illustrates the basic configuration of the magnetron.

Figure 1.5: Basic configuration of a magnetron [1]. The magnetron is an oscillator. As shown in Figure 1.5, resonant cavities form the RF circuit with an electromagnetic coupling between them. The cavity structure resonates only at a single frequency. The RF electric field in adjacent cavities is 180° out of phase. The

RF magnetic field magnetic is coupled to adjacent cavities. The oscillation is reinforced when a current in the cavity is induced as the electron spoke arrives at each gap. This occurs when “the electron spoke circles about the cathode in synchronism with the rotating field pattern on the anode” [1]. The magnetron can achieve an output power level of multimegawatt range. It can reach an efficiency as high as 88%.

The other example of the cross-field tubes is the crossed-field amplifier. It operates the same way the traveling wave tube does. However, instead of the formed electron bunches in the TWT, the electron spokes are formed. Figure 1.6 illustrates the basic configuration of the crossed-field amplifier.

Figure 1.6: Basic configuration of a crossed-field amplifier [1]. The electron spokes circle around the cathode. As the wave travels from the input and output, it grows due to the electric field from the circuit enhancing the bunches in the spoke resulting in an induced current in the circuit. Such current enhances the electric field. The crossed-field amplifier can achieve an output power level of tens of megawatts, but the gain is less than 20 dB.

The third category of vacuum tubes is fast-wave devices. The interaction between the wave and electron beam in the fast wave devices is different from the other two categories. In the linear beam and cross-field devices, the operating frequency is determined by the circuit whose dimensions are determined by the frequency. Thus, the generated power is inversely proportional to frequency. In the fast-wave devices, the operating frequency is determined by the magnetic field and cyclotron frequency. The circuit dimensions are independent of frequency. Thus, the generated power is proportional to frequency. Examples of the fast wave tubes are gyro-monotrons and gyro-amplifiers.

Figure 1.7 illustrates the basic configuration of the gyrotron oscillator.

Figure 1.7: Basic configuration of a gyrotron oscillator [1]. 9

As shown in Figure 1.7, the electron beam is hollow and electrons are spiral in shape. The velocity of the electrons is 1.5 to 2 times larger than the axial velocities. The energy of electrons plays a role in amplifying the electric field.

1.3 Domain of Vacuum Tubes

A significant factor to consider in many applications is the power level. Throughout the microwave frequency range and beyond, the vacuum tubes prevail the high power high frequency applications while the solid-state devices are used at low power and frequencies.

Figure 1.8 compares the average power and frequency range of vacuum and solid-state devices throughout the microwave frequency range and beyond.

Figure 1.8: Average power and frequency range of vacuum and solid-state devices throughout the microwave frequency range and beyond [1]. Other factors are taken into consideration in applications to compare between the vacuum tubes and solid-state devices such as efficiency, temperature, reliability, and bandwidth [18]. The vacuum devices, with the appropriate collector technique, is more efficient than the solid-state devices. Some tubes can exceed an efficiency of 70%. In addition, the operating temperature of the vacuum devices is higher than the solid-state devices. Further, most of the satellite applications use the TWT as the amplifier because the vacuum devices are more reliable than the solid-state devices. Finally, the conventional 10

helix slow-wave structure can achieve a bandwidth of over 2 octaves for the TWT, resulting in a preferred choice when a large bandwidth is desired.

1.4 Literature Work

Tremendous efforts have been performed earlier to model, design, and fabricate slow-wave structures of TWT. Some of which are known to be ring-bar structures or modified versions of ring-bar structures [19-29]. Such studies have been analyzed by different resources in a variety of aspects. In general, the ring-bar structure provides a high operating power level compared to the existing other structures such as the helix and suppresses the backward wave oscillations. However, its bandwidth capability is limited to

10-20% [1-2]. Figure 1.9 shows a conventional ring-bar structure.

Figure 1.9: Ring-bar structure [30]. As shown in Figure 1.9, one-cell consists of two rings connected once by a bar. The structure has a period, p, and thickness of the ring. It produces a high interaction impedance and efficiency and requires a large beam radius for high voltages and currents. Later, a modified ring-bar structures have been implemented such as half-ring helical structure, ring-loop structure [31], curved ring-bar structure, and wave-ring helical structure. Figures

1.10-1.13 show some of the modified ring bar structures.

Figure 1.10: Half-ring helical structure [32].

Figure 1.11: Ring-loop structure [33].

Figure 1.12: Ring-loop and curved ring-bar structures [33].

Figure 1.13: Wave-ring helical structure [34].

As shown in Figure 1.10, one-cell of the half-ring helical structure consists of two-half loops separated by a distance d. It exhibits the same dispersion characteristics as the conventional helix, but obtains a higher gain. That is, the maximum saturated power achieved for this structure in [32] is 1 kW and a higher gain than the conventional helix designs by 10 dB. Thus, this structure can be used for low power applications. As shown in Figure 1.11, one-cell of the ring-loop structure consists of two rings connected by an elliptic bar. Its normalized phase velocity is below 0.25c, which indicates the use of this structure for low power TWTs. For the curved ring-bar structure in Figure 1.12, one cell consists of two rings and two curved transmission lines classifying it as a modified ring- loop structure. It produces a high normalized phase velocity and moderate interaction impedance, which indicate the use of such structure for high power TWTs. Such structure produces the highest reported output power of 1.02 MW in the S-band frequency range and provides a bandwidth of 33%. For the wave-ring helical structure, the output and gain of the structure are increased compared to the standard helix by increasing the path motion of the wave and without changing the length and radius of helix. Such structure can be used for low power TWTs. Table 1.1 states the comparison between the existing designs of the traveling wave tube including ring-bar structure, half-ring helical structure, ring loop structure, curved ring-bar structure, and wave-ring helical structure.

Table 1.1: Comparison between the existing designs of the traveling wave tube including ring-bar structure, half-ring helical structure, ring loop structure, curved ring-bar structure, and wave-ring helical structure.

Half-Ring Ring-Bar Ring-Loop Curved Ring- Wave-Ring Parameters Helical Structure Structure Bar Structure Helical Structure Structure Vary (e.g. X- Vary (e.g. 32- Frequency[GHz] Band, Q- 2.5-3.25 1.8-2.4 2.0-4.0 38) Band) Number of Vary Helix Vary Unknown 26 Elements, N Structure Area 814x57x57 Vary Vary 740x62.5x62.5 140.5x18.0x16.0 [mm3] (p=8.0 mm)

457x37x37 (p=4.0 mm) 1.0 k (p=8.0 Peak Output mm), Low (e.g. Vary 1.02 M 39.8 Power [W] 220 1300) (p=4.0 mm) 28.0 (p=8.0 mm), Gain [dB] Vary Vary (e.g. 45) 29.0 28.0 46.0 (p = 4.0 mm) 25.00 (p=8.0 mm), Bandwidth [%] 10-20 - 33.0 - 23.43 (p=4.0 mm) 38.7 (p=8.0 mm), Efficiency [%] - Vary (e.g. 6.1) 25.0 26.5 37.0 (p = 4.0 mm) 814.0 (p=8.0 Circuit Length mm), - - 740.0 140.5 (Size) [mm] 457.0 (p = 4.0 mm) Magnetic Field - - - Yes Yes Focusing Loss Pattern - No - No - Rods Yes - - No Yes Bar One Straight One Straight One Elliptic Pair Elliptic One Straight 0.30-0.45c (7.0-12.0 Vary (e.g. GHz), Phase Velocity 0.27-0.32c 0.15-0.22c for 0.70-0.78 0.11-0.12c 0.31-0.33c 2.0-3.0 GHz (38.0-44.0 GHz) 13-35 (7.0- Interaction 12.0 GHz), Low (e.g. 20- 30-80 43-65 50-130 Impedance [Ω] 21-25 (38.0- 25) 44.0 GHz) Vary (e.g. Software Used CST - CST CST CST)

Table 1.1 will be revisited and restated in Chapter 5 when the novel slow-wave structure design is implemented.

1.5 Novelty of Proposed Work

The main objective of this research is to design a novel slow-wave structure of a

TWT, considered as a modified ring-bar structure design, and investigate its performance.

The approach is achieved by modeling the geometry, studying the characteristics based on carrying out cold-test simulations using ANSYS HFSS [35-36] and hot-test simulations

using VSim code [37]. One-cell of the novel slow-wave structure of the traveling wave tube consists of two rings connected by two pairs of transmission lines. For the cold-test simulations, the dispersion behavior, normalized phase velocity, and interaction impedance of the modeled design are investigated. Such study is described in Chapter 4. For the hot- test simulations, VSim code is used to compare the output power results to the conventional structures based on the ease of manufacturing, bandwidth, gain, and efficiency. Such study is described in Chapter 5.

Neither of the conducted studies, mentioned in Section 1.4, nor ring-bar or modified ring-bar structures have been reported with the use of VSim code. Such high performance code, described in details in Chapter 5, computationally runs intensive electromagnetic, electrostatic, magnetostatic, and plasma simulations of complex shapes by using 3D conformal Finite-Difference Time-Domain (FDTD) particle-in-cell (PIC) simulations as implemented in 3D PIC code. It uses multiprocessor parallelization allowing to obtain high level simulations. Besides, the physical behavior of the TWT is investigated through the visualization and postprocessing software. The mode spectrum and mode profile data are accurately obtained with the mode analysis tool. Also, the user can take advantage of other features such as the time-history postprocessing to examine the fast Fourier Transform and the instantaneous amplitude and frequency calculations.

Before proceeding to the novel work, an electron gun and periodic permanent magnet designs are implemented using EGUN code and ANSYS Maxwell, respectively.

Such task is described in Chapter 3. For this design, the specifications and constraints require creative POLYGON boundary inputs and electrode contours. At one stage, an electron gun is designed with a beam voltage of 262 kV, beam current of 12 A, and

minimum radius of 2 mm. Such electron gun fits the proposed novel slow-wave circuit structure of the TWT.

1.6 Overview of Dissertation

The dissertation is organized to develop a logical sequence and appropriate working knowledge of the conducted research. Background and theory of the TWTs are introduced in Chapter 2. This includes the discussions of the required components to form the electrons into a beam including, but not limited to the electron gun and focusing structures. Further, the scope in the same chapter moves to the analysis of interaction between the electron beam and RF signal. Perhaps the most primary focus of the study is investigating the performance of the TWTs through the dispersion curve, normalized phase velocity, interaction impedance, saturated output power, and gain. Other sections cover the collector and transmission line fundamentals. The remainder of this chapter is devoted to the novelty of the proposed work and motivations. In Chapter 3, high voltage, low-perveance electron guns for TWT device are designed in addition to their periodic permanent magnet focusing structure. Further, the electron gun trajectories for plenty of designs are simulated. Chapter

4 proposes a high power novel slow-wave structure design with promising characteristics and results based on cold-test simulations. The normalized phase velocity and interaction impedance of the structure are investigated. Hot-test simulations of the high power novel slow-wave structure design are carried out in Chapter 5. The output power, gain, efficiency and bandwidth is the structure are obtained. In addition, the behavior of the slow-wave circuit structure is examined in a variety of trends by obtaining the output power versus the number of periods and transfer characteristics.

CHAPTER II

BACKGROUND AND THEORY

The basis discussions and their underlying principles are significant to understand the operation of traveling wave tubes. This chapter reviews some of the fundamental topics in TWT with a general knowledge of vital theories beginning with the basic operation of

TWT. After that, Poisson’s equation and Gauss’s law are employed to discuss the electron dynamics, evaluate the static fields produced by electrons, and analyze the influence of electron motion in such fields. Next, the components which form the electrons into a beam are discussed including the source of electrons, followed by studying the interaction between the electron beam and RF signal. Such investigation opens the door to examine the dispersion characteristics and performance of TWT in addition to the conventional

TWT circuits. In order to fulfill and cover a working knowledge of TWT, the collector component, at a higher level, is discussed. Next, transmission line fundamentals are described.

2.1 Basic Operation of Traveling-Wave Tube

There are two different basic technical approaches of TWTs: the helix TWT and coupled-cavity TWT. Both types have the same operating principles and they both include an electron gun, electron beam, and collector. The differences between both types mainly occur in the RF circuits. Figure 2.1 illustrates the main elements of the basic helix TWT.

Figure 2.1: Basic helix TWT [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. As shown in Figure 2.1, the basic helix TWT consists of an electron gun, RF input, electron beam, attenuator, magnetic focusing field, helix slow-wave circuit, RF output, and collector. It is capable of producing tens to hundreds of watts classifying it as a low power device. However, over two octaves is possible to obtain classifying the helix TWT as a broadband device.

In order to study the behavior of the helix TWT, Figure 2.2 shows the electric field and RF charge patterns for a single-wire transmission line.

Figure 2.2: Patterns of electric field and RF charge for a single-wire transmission line above an existing ground plane [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. The transmission line is nondispersive, which means that the velocity is independent of frequency, propagating at the speed of light. In Figure 2.2, the electric field force on the

electron beam is significant, whereas the magnetic field force is neglected. Based on the orientation of the transmission line, the charges and electric field patterns move opposite to the RF generator at a constant amplitude.

When the single-wire transmission line is formed into a helical path, the applied RF signal travels at a velocity near that of light along the helical conductor reduced by the pitch of the helix. Figure 2.3 shows the electric field and RF charge patterns for a helix.

Figure 2.3: Patterns of electric field and RF charge for a helix [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. The polarity of the signal changes every two complete turns corresponding to every half- wavelength, . Similar to the single-wire transmission line, the electric field lines for the helix extend from the positive charge regions to the negative charge regions. The electric field components accelerate and decelerate the injected electrons along the axis of the helix.

Unlike the single-wire transmission line, there is an available electric field inside the helix.

In general, the sinusoidal field patterns are formed in the axial direction. When the velocity of the electric field is equal to the velocity of the electron beam, bunching electrons are resulted from the force the electrons experience as the beam travels through the helical path. Such bunching electrons in the beam produce the fields causing the electrons to move from regions (1) to regions (2). Consequently, the induced waveform becomes larger

compared to the initial waveform as the interaction continues to occur between the electron beam and RF wave. Further, the currents flowing to the left and right side of regions (1) produce positive and negative voltages, respectively. The phase of the induced voltage waveform to the left of the initial voltage waveform is shifted by 90°. Figure 2.4 illustrates the bunched and extracted energy in regions (1) due to the axial field when the electron beam enters the circuit.

Figure 2.4: When the beam enters the circuit, energy is bunched and extracted from the beam due to the existing axial field [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. Figure 2.5 illustrates the bunched and extracted energy due to the axial field when the interaction between the electron beam and circuit occurs.

Figure 2.5: When the interaction between the electron beam and circuit occurs, energy is bunched and extracted from the beam due to the existing axial field [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

As shown in Figures 2.4 and 2.5, the decelerating fields move toward regions (1) where the bunching electrons are. The amplification is obtained from the extracted energy of the decelerating fields moving to the circuit. Energy is extracted from the circuit field by the electrons in the accelerating fields. As saturation is met, where the bunches of electrons continue to increase and “fall back in phase,” the applied signal must be removed from the circuit. The saturation point occurs when the supplied energy is equal to the extracted energy. In such case, the wave stops amplifying.

The other basic technical approach of the TWTs is the coupled-cavity. Figure 2.6 illustrates the main elements of the basic coupled cavity TWT.

Figure 2.6: Basic coupled cavity TWT. As shown in Figure 2.6, the basic coupled cavity TWT consists of an electron gun, RF input, electron beam, sever/attenuator, periodic permanent magnet focusing, coupled cavity circuit, RF output, and multistage collector. It is capable of producing watts to megawatts classifying it as a high-power device. However, the common bandwidth to obtain is 10-20% classifying the coupled cavity TWT as a narrowband device. For the slow-wave structure, klystronlike cavities are used where the electromagnetically coupled cavities are used except at the sever regions. The coupling techniques are formed differently. However, it is significant to choose the appropriate cavity technique and cavity

dimensions to achieve the desired characteristics including the power level and electron beam velocity. The amplification can be obtained when the electron beam passes through the structure and the phase velocity is equal to the velocity of the beam. Most of the coupled cavity structures are metallic resulting in a high-power level and low thermal resistances.

It is possible to treat the coupled cavity circuit as a folded waveguide. Such circuit shares the most common properties with the folded waveguide including the dispersion characteristics. Curnow [38-39] and Gittins [40] explained the equivalent circuits of coupled cavity structures.

In order to study the behavior of the coupled cavity TWT, Figure 2.7 shows the basic concept of the coupled cavity circuit.

Figure 2.7: Basic coupled cavity circuit [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. As shown in Figure 2.7, a signal is supplied to the first cavity by the input circuit. There are two ways of coupling the structure: impedance matching section from a waveguide or magnetic field loop. An alternating current component is produced as the first voltage across the first gap, V1, modulates the electron beam velocity entering the circuit. The signal is divided into two “equal wavelets” [2] as it is induced by the beam into the second cavity. The wavelets travel in opposite directions. Such process is repeated at all voltage

gaps resulting in an increase in the ac current and velocity modulation enhancement. The voltage at a certain voltage gap is equal to the “combination of the voltage coupled from the previous gap and the sum of all the components induced by the ac beam current. The ac beam current at any point is the sum of all the currents produced by previous gaps” [2].

Such combinations can be represented as a vector diagram with the voltage vectors rotating clockwise. Figure 2.8 shows the vectors representing the circuit voltage.

Figure 2.8: Vector diagram of circuit voltage [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

2.2 Electron Dynamics

The electron beam in a TWT produces two static fields: electric and magnetic fields. Both fields, discussed next, influence the electron motion.

2.2.1 Electric Field

To estimate the electric field produced by a charge distribution, the solutions of

Poisson’s equation and Gauss’s law are recognized. If the charge is not present, Poisson’s equation becomes Laplace’s equation defined as

(Laplace) (2.1) ∇ = (Poisson) (2.2) = −

, where V is the potential, is the charge density, and o is the permittivity of vacuum being

. The Laplacian operator, , can be written in rectangular, cylindrical, − and. spherical× coordinates, �/m respectively, as

(rectangular) (2.3) ∇ = + + (cylindrical) (2.4) ∇ = + + + (spherical) (2.5) ∇ = + i sin + i � Gauss’s law can be written in integral or differential forms as written below [2],

(2.6) ∮ ∙ = (2.7) ∙ = , where Q is the total positive charge. Gauss’s law relates the electric field to the charge distribution. Its integral form states that the electric field over a closed surface area is equal to the positive charge divided by the permittivity of vacuum contained within that area.

The differential form can be used in terms of the wave equation.

Consider applying Gauss’s law to the electron beam in Region 3 of the electron gun

(discussed in Section 2.4), shown in Figure 2.9. Gauss’s law becomes

(2.8) ∫ ∙ = − , where the term is the volume and Er is the radial electric field component being equal to

(2.9) = −

Figure 2.9: Electron gun of TWT [1]. The minus sign in (2.9) shows that the electric field is in the -r direction indicating that the field is pointed inward toward the center of the beam. If the electric field is integrated inside the beam, the potential, V, is resulted as

(2.10) = The current, which is the charge per unit time, is

(2.11) = = , where b is the radius of the outer edge of the beam and ve is the electron beam velocity.

Thus, the charge density becomes

(2.12) = The current density, J, can be written as

(2.13)

From (2.11) and (2.9), the electric field can = be written as

(2.14) = − Also, the beam velocity is related to the beam voltage, Vb, by

(2.15)

= √

, where is the electron charge-to-mass ratio. The potential at the center of the beam is obtained by combining (2.10), (2.12), and (2.15) to become

(2.16) = .× √ To examine the electron motion in a static electric field, assume that an electron is moving in the +y direction and the electric field is in the -y direction. Then, the force on the electron can be written as

(2.17) If the electric field in the -y direction is constant, = =the velocit =y − of the electron moving in the

+y direction is

(2.18) = − , where vo is the initial velocity. If the initial electron velocity is zero, (2.18) becomes

(2.19) The position of the electron is = −

(2.20) = + − , where y0 is the initial position of the electron. If the initial position of the electron is zero and is at rest, (2.20) becomes

(2.21) The kinetic energy of the electron is = −

(2.22) The electrons are accelerated through= a potential of= −

(2.23)

Substituting (2.23) in (2.22), the kinetic energy = − becomes

(2.24) Equation (2.24) is only valid when the velocities = are low compared to the velocity of light, c. This is due to the assumption made that the mass is constant. However, the mass of a particle changes with the velocity according to the theory of relativity, which also defines the energy by

(2.25) When applying a force, F, over a distance, = dy , the electron mass increases and energy expands as

(2.26) Thus, Newton’s second law is = =

(2.27) At high velocities, (2.27) becomes =

(2.28) Combining (2.26) and (2.28) yields = = +

(2.29) or = + = + =

(2.30) − Integrating (2.30) gives = − = − −

(2.31) , where C is a constant and is estimatedln = −to beln − +

(2.32) = ln + ln , where mo is the mass at rest. Plugging (2.32) into (2.31) yields 27

(2.33) ln = ln [ − ] or

(2.34) = [−] The electron velocity and voltage relationship can be found from (2.26). If the electric field supplies the accelerating force, the energy obtained is

(2.35) , where dV is the voltage increment when= the= electron =was accelerated. By integration,

(2.36) Substituting (2.34) in (2.36) yields = −

(2.37) = [− − ] Rearranging (2.37),

(2.38) � = [ − + ] This equation can be written as

(2.39) � = [ − +� ] , where Vn is defined as being equal to 5.11 V. Substituting Vn in (2.34) yields × (2.40) = +

2.2.2 Magnetic Field

To estimate the magnetic field produced by the electron beam in the TWT,

Ampere’s law is recognized as

(2.41)

⃗⃗ The integral form of Ampere’s law states∮ that∙ the= magnetic field around a closed surface area is equal to the current within that area. Such field is small enough that it can be neglected. Referring to Figure 2.9, the magnetic field is

(2.42) , where b is the radius of the outer edge of =the beam. The magnetic flux density is estimated by

(2.43) = = , where o is the permeability of vacuum being . − To examine the electron motion in a static ×magnetic �/m field, assume that an electron is moving at an angle to the direction of the flux density, B, at a velocity ve. Figure 2.10 shows the deflection of electron by the magnetic field.

Figure 2.10: Deflection of electron by a magnetic field [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. The inward force on the electron is

(2.44)

The speed of the electron is constant since the= force is perpendicular to the electron trajectory. Thus, the force is constant resulting in a circular path motion to the electron.

Such circular path creates an outward centrifugal force on the electron being

(2.45) , where r is the radius of the electron path. Setting= (2.44) equal to (2.45),

(2.46) Solving for the radius of the electron path, =

(2.47) The radius of the electron path and voltage = relationship can be written as

(2.48) √ − = = .× The time it takes to complete one rotation is

(2.49) = = The frequency of such rotation becomes

(2.50) = = The angular frequency is called the cyclotron frequency, c, given by

(2.51) = =

2.3 Source of Electrons

All tube devices use the cathode as the source of electrons for the electron beam with a current density ranging between milliamperes and tens of amperes per square

centimeter of cathode area. To emit electrons from the cathode, there are two common mechanisms: thermionic emission and secondary emission. The thermionic emission technique is used in the linear beams including the TWT. The secondary emission technique is used in the cross-field beams including cross-field amplifiers and magnetron.

2.3.1 Cathode

An ideal cathode provides the main characteristics it is responsible for and delivers the expected functions properly. One of those tasks is being able to emit electrons freely without considering the influence of heat and electron leakage occurring in the device.

Another task is the ability of the cathode to emit electrons abundantly by providing an infinite current density. The third ideal case is the ability of the cathode to live infinitely besides persisting to emit electrons forever. The fourth characteristics is emitting electrons uniformly at a negligible velocity. Such features in real cathodes never exist. The working environment of the cathode operates at a high temperature. It is heated to a temperature of

1000 °C to emit electrons adequately. Such high temperature prevents the current density to be unlimited and causes some elements of the cathode to evaporate. Consequently, the cathode life reaches its depletion ending up the life of the cathode. In addition, the electron emission in real cathodes fluctuates and velocities vary resulting in noise occurring in the output signal from the noise currents in the electron beam. Also, the velocity variations of electron emission are an obstacle to confine the electrons in a well-defined beam.

There have been tremendous efforts to improve the cathode capabilities over the years [41]. Such attempts made a progress to understand the physics and chemistry of emitter surfaces including, but not limited to providing the appropriate background materials.

2.3.2 Thermionic Emission

The temperature and surface of the cathode are two constituents which strongly effect the emission of electrons. To have the electrons escape from the surface of the cathode, sufficient energy is obtained by heating up the surface of the cathode. The temperature is proportional to the number of electrons with sufficient energy to escape.

Figure 2.11 shows the energy level diagram for the electrons between the cathode and vacuum.

Figure 2.11: Energy level diagram for electrons near the surface of a metal between a cathode and vacuum [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. As shown in Figure 2.11, the electron energy levels in the cathode are represented as parabolic curves being adjacent to atoms. The energy of electrons is at Fermi level in the cathode at temperature of 0 K, denoted by Eo, representing the top of the conduction band. eϕ is the work function, which is the energy difference between the top of the conduction band in the cathode area and the vacuum level adjacent to the cathode. As temperature increases, the electron energy becomes greater than Eo. Such energy causes the electron emission to occur when the energy becomes greater than or equal to Eo + e . However, the electrons move in random directions within the cathode. The only emitted ϕelectrons are

the ones moving towards the surface. Figure 2.12 shows the electron trajectories with sufficient energies to be emitted, but move in different directions.

Figure 2.12: Two electrons with sufficient energies to be emitted, but moving in different directions [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. As shown in Figure 2.12, the electrons are represented in terms of the momentum, P, with a critical momentum, Pxc, directed toward the surface in the x-direction. If the momentum in the x-direction, Px, is greater than or equal to than the critical momentum, the electron is emitted. If the momentum in the x-direction, Px, is less than the critical momentum, the electron is reflected into the cathode. The critical momentum in terms of the work function is defined as

(2.52) The emission current density is found as = = + �

(2.53)

, where ne is the number of electrons per unit = volume.= To compute the density of electrons, the density of states is

(2.54) ℎ

, where dPx, dPy, and dPz represent the increment of momentum in the indicated directions, and h is Planck’s constant being equal to . The probability of the states − being occupied is represented by the Fermi-Dirac.× distribution function as

(2.55) −/ = + , where k is Boltzmann’s constant being equal to , and T is the absolute − temperature in Kelvin. Figure 2.13 shows the Fermi-Dirac.× distribution function for T = 0 and 1273 K.

Figure 2.13: Fermi-Dirac distribution function for T = 0 and 1273 K. The Fermi level is the energy at which the probability of occupation is 0.5. The exponential term in (2.55) becomes larger than 20 if the energy is 3kT above the Fermi energy and it becomes smaller than 0.05 if the energy is 3kT below the Fermi energy. Since the energy needed to emit electrons is greater than or equal to Fermi energy level plus the work function, the energy is in the order of 1 eV or greater leading to have

(2.56) − Thus, ≫

(2.57) − − The energy-momentum relationship is =

(2.58) Plugging (2.58) in (2.57) yields = + +

(2.59) ++− − The number of electrons with momentum = in the increment dPx, dPy, and dPz is

(2.60) ++− − The increment of the =current ℎ density is = ℎ

(2.61) Therefore, the total emission current density = is

(2.62) ++− ∞ ∞ ∞ − = ℎ =−∞ =−∞ = Two integrals in (2.62) are∫ solved ∫using the∫ definite integral given by

(2.63) ∞ − −∞ The third integral is evaluated by∫ defining a variable,= u, as

(2.64) − Rearranging terms, =

(2.65)

, where du is the derivative of (2.64). Next, the = integral � can be evaluated as

(2.66) � ∞ � − − ∫− � = � Thus, solving for (2.62) by applying (2.63) and (2.66), the current density becomes

(2.67) � � � − − − = ℎ � � = ℎ � = � , where Ao is a constant being equal to . Equation (2.67) is known as the Richardson-Dushman equation for the thermionic.× emission [42-45].

Referring to (2.67), the current emission density varies with temperature. However, such variation is negligible. For example, when 1% change is applied to the operating temperature of 1000 °C, the exponential term changes by 70%, but T2 changes by 2%.

Compared to theory, the experiments differ due to two reasons influencing (2.67). The first reason is that (2.67) assumes that the work function is independent of temperature. In experiments, the work function varies linearly with temperature. This can be stated as

(2.68)

, where R and are the Richardson work� = function � + �and temperature coefficient of work functionϕ change, respectively. Equation (2.68) leads to rewrite (2.67) as

(2.69) � − − The term in parentheses indicates that the = variation of� the work function with temperature corrects the value of the universal constant, Ao. The second reason is that (2.67) assumes that the work function in independent of the emitting surface. In experiments, the work function varies based on the crystal surface. That is, if the emitter area has two different work functions, e 1 and e 2, with the second work function being larger than the first one, the exponential termϕ in (2.69)ϕ for the first work function dominates compared to the second one and the measured constant, Ao, becomes only one half of the theoretical value.

The Richardson-Dushman equation suggests that high current emission values are obtained when the work function is low and temperature is high. However, pure metals

have does not satisfy such conditions. That is, the work function is proportional to the melting temperature. Table 2.1 states the work functions at room temperature and their melting temperature for various materials.

Table 2.1: Work functions at room temperature and their melting temperature for various metals [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

Metal Work Function [eV] Melting Temperature [°C] Barium 2.7 725 Calcium 2.9 839 Carbon 5.0 ~3550 Cesium 2.1 28 Hafnium 3.9 2227 Iridium 5.2 2410 Molybdenum 4.5 2620 Nickel 5.2 1455 Osmium 5.4 3045 Platinum 5.3 1773 Rhenium 5.1 3180 Scandium 3.5 1539 Sodium 2.7 97 Strontium 2.6 769 Tantalum 4.2 2996 Thorium 3.4 1750 Tungsten 4.6 3410 Zirconium 4.1 1852

As stated in Table 2.1, most of the materials have low work function, but low melting temperature. Barium has a low work function of 2.7 eV and a moderate melting temperature of 725 °C resulting in a preferred choice as the material to use in thermionic cathodes. One advantage is that it can interact with other elements such as Tungsten,

Nickel, Alloy, Type B, Scandate, and Oxide to produce a work function lower than 2.7 eV

[2].

The first thermionic cathodes were made from Tungsten and used in radio tubes.

The pure tungsten has a high work function of 4.6 eV and a melting temperature of 3410

°C. For other applications, tungsten was mixed with thorium dioxide to reduce the work function. Earlier at that time, cathodes were heated directly by passing the current through the filament using batteries as the power supply with a dc filament voltage. As time went on, the filaments were heated with ac current.

Over the 20th century, a variety of cathodes made of different metals were attempted and used including, but not limited to thoriated tungsten (ThW), oxide-coated, reservoir,

MK, controlled porosity dispenser (CPD), impregnated dispenser cathode, S-type, B-type,

M-type, mixed metal matrix (MMM), controlled doping (CD), and scandate [46-52].

To heat up the cathode, a pure tungsten and a small amount of rhenium are the materials used. The cathode is electrically connected to the heater from one side. Such way prevents the voltage difference to be large enough to cause an electrical breakdown. The heater is operated with either ac or dc supplies. Using ac supply generates a magnetic flux from the heater current, which extends to the cathode surface and causes the electron trajectories to be disturbed. Using dc supply requires additional circuitry and needs to consider the polarity to avoid the transfer of the emission current backwards from the cathode to the heater. The positive side of the heater is connected to the cathode [53-55].

An insulating material of aluminum oxide with a thin thickness of 0.002-0.003 coats the heaters. It is applied by either spraying or an electrolytic process, known as cataphoretic coating. For B-type dispenser cathodes, the insulating material is embedded at the back of the cathode to enhance the heat transfer between the heater and cathode. In addition, it can be potted as a powder to support the heater structure, though it takes more time for the heater to warm up. Such warm-up penalty time is not favorable for some missiles and space defense applications, which suggest the use of fast warm-up heater-

cathode assemblies. The fast warm-up heaters use the low thermal mass components, such as the bombarder-type assembly in [57]. Such assembly achieved the warm-up within one second and was made of thoriated tungsten. For large diameter cathodes, heater wafers are used and are fabricated by anisotropic pyrolytic graphite deposited onto anisotropic boron nitride [58]. To minimize radiation and conduction losses, heat shields and supported members are used which conserve heater power in dispenser cathodes.

To test the heaters and verify the proper temperature of the cathode, the heater current and voltage are examined. When the heater current is excessive, the heater assembly is at a lower temperature than it should be.

2.3.3 Schottky Effect

The presence of the electric field or voltage is not considered in the Richardson-

Dushman equation, discussed previously. In experiments, the applied electric field to the surface of a cathode increases the emission. This can be understood with the method of images, also known as Schottky effect, or image-force lowering. Figure 2.14 illustrates the electric field lines established by an electron and its image.

Figure 2.14: Electric field pattern established by an electron and its image [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. As shown in Figure 2.14, the image of the electron has a positive charge. The attractive force between the two charges is called the image force and is given by

(2.70) = This equation is also the force between the electron and surface of the metal. The potential energy at position x from the metal surface when the electron moves from infinity is

(2.71) = ∞ = ∞ − When an electric field is applied between∫ the cathode∫ and anode, the potential energy of an electron decreases to zero at the cathode, but the kinetic energy increases. Figure 2.15 demonstrates the energy-band diagram between a metal surface and a vacuum.

Figure 2.15: Energy-band diagram between a metal and surface and a vacuum [44].

As shown in Figure 2.15, the increase in the kinetic energy becomes -eEax at position x.

The total potential energy of the electron is the sum of the potential energies and is given by

(2.72) = − − The maximum potential energy occurs at xm by the condition dPE/dx = 0 and is given by

(2.73) = √ The image force lowering, , is found to be

ϕ (2.74) ∆� = √ = Next, the increase in the current can be estimated. The Richardson-Dushman equation can be modified as

(2.75) − (�− ) −�−� √ � � Equation (2.75) can be rewritten = � as = �

(2.76) √ � � =

, where Jo is the Richardson-Dushman current density or zero field current density [2, 44,

58].

2.3.4 Space Charge Limitation

Without electrons, the potential increases linearly from the cathode to the anode.

Once electrons exist, the potential is reduced by the effect of the negative electron charge.

Figure 2.16 demonstrates the effect of electrons on the potential from the cathode to the anode in a parallel-plane diode.

Figure 2.16: Potential distribution with and without electrons from cathode to anode in a parallel-plane diode [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. When the temperature is increased, the rate of electron emission is increased. This persists to occur until the electrons near the surface of the cathode become large enough to push the potential below zero. When the potential becomes negative, the electric field forces the electrons back to the cathode to bring the potential back to be positive. Figure 2.17 illustrates the potential near the cathode surface.

Figure 2.17: Potential near the cathode surface [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. As shown in Figure 2.17, electrons are emitted due to the applied electric field for (a). For

(b), equilibrium occurs and the electric field at the cathode is zero causing the potential adjacent to the cathode surface to be zero. When the potential is positive, more electrons are emitted. When the potential is negative, less electrons are emitted to increase the potential. For (c), the potential becomes negative, but the electrons are forced back by the electric field to increase the potential. The emission is space charge limited “when the electric field at the cathode surface is forced to zero by the electron cloud near the cathode surface.” This causes the emission to be independent of cathode temperature and surface condition. Thus, it “eliminates the necessity for extreme uniformity of temperature across the surface of a cathode” and “eliminates the necessity for precise control of the voltage and current to the heater in the cathode” [2].

To relate between the current and voltage in a parallel-plane, space charge limited diode, apply Poisson’s law in (2.2). For a parallel plane diode extending to infinity in the transverse direction,

(2.77) = Plugging (2.13) and (2.15) in (2.77) yields

(2.78) = = √ Multiplying the left and right sides in (2.78) by 2 dV/dx yields

(2.79) − = √ Integrating (2.79),

(2.80) = √ + , where C1 is constant. The voltage is zero at x = 0. Also, without an initial electron velocity, dV/dx = 0, which yields C1 = 0. Therefore,

(2.81) = Integrating (2.81),

(2.82) = + , where C2 is constant. The voltage is zero at x = 0, which yields C2 = 0. Therefore,

(2.83) − , which is known as Child-Langmuir = law. It describes = . the× “flow of electrons in a parallel- plane, space charge limited diode” [2]. Equation (2.83) is modified for a diode with a cathode-to-anode spacing x = d to become

(2.84) , where P is the perveance of the diode, defined = as

(2.85) − = = .×

, where A is the area of the cathode. The current, I, is the emission current density multiplied by the area of the cathode [2, 59-60]. Figure 2.18 shows the current-voltage with a perveance of , representing the slope. − ×

Figure 2.18: Current-voltage relationship with one microperveance. 2.3.5 Life Expectancy

In general, long life is a critical factor in most tube devices’ applications to minimize redundancy and replacement costs and ensure reliability. The life expectancy of the tube devices relies on the cathode whose expected lifetime ranges from 10,000 to over

100,000 hours [2]. Other recent resources state that the power TWT amplifiers can last for

15-20 years corresponding to 131400-175200 hours making it a preferred choice in the satellite applications [61]. Still, the cathode design is one of the primary reasons for most occurring failures. There are three cathode degradation mechanisms causing such failures

[50]. The first mechanism is that the perveance gradually decreases with time. The fully space charge limited region (FSCL) decreases with time due to the barium depletion in the tungsten pellet causing the work function to increase [41, 62-65]. The second mechanism is that the work function distribution changes with respect to time only and neglects the

changes in cathode temperature. Such changes occur in coated cathodes due to the change in the base metal work function, which relies on the thickness, composition, and diffusion rate. The third mechanism is that the work function unpredictably changes with respect to time and changes in cathode temperature. Such changes occur due to the inadequate supply of barium, external poisoning, or change in the “sticking coefficient of the barium to the cathode surface” [2].

The end of the cathode life can be predicted in a variety of methods. An example is when the current density reaches the operating temperature and emission current drops by about 10%. Another way of predicting the cathode life is by the depletion model from the following equation [62-65]:

(2.86) = � + � , where JOP is the observed current density, JTL is the observe current density under temperature limited, JFSCL is the observed current density under full space charge limited.

Equation (2.86) was modified by Vaughan in [66] to

(2.87) = � + � , where n is adjusted to satisfy fit to data. According to [66], n is appropriate to be between

6 and 10.

2.4 Electron Gun and Focusing Structure

All linear beam tubes including the TWT have the same operating principles.

However, they differ in the operating voltage and current, and size. This section discusses the electron gun and magnetic focusing of a beam.

2.4.1 Electron Guns

The electron gun forms the electrons from the cathode to create the electron beam, which interacts with a microwave circuit. There are three main components in the electron gun: the cathode, anode, and focusing electrode [67-73]. Figure 2.19 shows those components existing in three regions.

Figure 2.19: Electron gun design components with identified three regions [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. The electrons flow toward the center curvature of the cathode beginning from Region 1, where the spherical cathode disc and focus electrode exist. The electrode produces spherical equipotential surfaces representing the electrical potential and are perpendicular to the electric field. The electrons pass through the equipotential surfaces in Region 2 into the anode aperture which contains a hole. Due to the space charge forces, the electrons escape from the cathode-anode regions and drift away to Region 3.

There are two common problems associated with designing the electron gun: beam divergence and cathode capability to handle the current density. The beam diverges due to the electrostatic repulsion forces between electrodes. It can be avoided by focusing the

electrodes for parallel flow. Figure 2.20 illustrates the parallel electron flow achieved by focusing the electrodes.

Figure 2.20: Parallel electron flow achieved by focusing the electrodes [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. Without electrons, the equipotential profiles are normally parallel and spaced equal between each other. With electrons, these equipotential profiles become deflected to the right. The electrons move perpendicular to the equipotential profiles causing the electron trajectories to diverge. Figure 2.21 demonstrates the equipotential profiles with and without electrons.

Figure 2.21: Electron trajectories divergence with (solid lines) and without (dashed lines) electrons [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

By focusing the electrodes at the cathode potential, the equipotential profiles become straightened. Figure 2.22 shows equipotential profiles of the electron beam when the electrodes are focused.

Figure 2.22: Parallel flow beam due to the focused electrode at cathode potential [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. To configure the electrodes for parallel electron flow, mathematical solutions are solved inside and outside the electron beam. Inside the electron beam, Poisson’s equation is the same as (2.78). Outside the electron beam, Laplace’s equation applies as in (2.1). Applying the boundary conditions at y = 0,

(2.88) Previously, the Child-Langmuir equation= = , was = defined for parallel plane, space charge limited in (2.83), which can be rearranged as

(2.89) − Plugging (2.89) into (2.88) yields =to solve.× inside the beam at y = 0 as

= (2.90) − = = .× =

To solve this outside the beam at y > 0, it is assumed that the solution of the complex function can be written as

(2.91) , where only the real part is, considered. = Changing + =(2.91) to polar + coordinates by letting

(2.92) Plugging (2.92) into (2.91) yields + = ||

(2.93) When V = 0, the angle ( becomes , = || cos

(2.94) , which is referred to as Pierce Angle. The = focusing = electrodes.° at such angle are referred to as Pierce electrodes providing a parallel electron flow.

When the current density is higher than the cathode emission density, the cathode is not capable to last within an acceptable life expectancy. A cathode with a large area, compressing the flow of electrons, can solve such problem. To analyze this, consider a spherical diode, shown in Figure 2.23, whose inner and outer diameters are the cathode and anode, respectively [74].

Figure 2.23:A spherical diode, where inner and outer diameters represent the cathode and anode, respectively [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

In spherical coordinates, Poisson’s equation is

(2.95) ρ ∇ V = = ε The charge density is

(2.96) = = √ = √ , where Is is the total current of the spherical diode. Plugging (2.96) into (2.95) yields

(2.97) = √ To form the electron gun, consider the conical portion with half angle as shown in Figure

2.24.

Figure 2.24: Conical diode with half angle [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. To solve for the total current of the conical portion, let

(2.98) = ln [ ] = ln [−] (2.99) Solving− = for I +s yields . + . + . + . + .

(2.100) − − − The current of the conical = shape of√ the spherical= diode.× becomes

(2.101) −c − −c = = .51× −

The equipotential profiles are distorted close to the anode aperture. Figure 2.25 shows the equipotential profiles near the anode aperture for a low perveance [75].

Figure 2.25: Low perveance increases the distortion near the anode aperture [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. At low perveance, distortion exists near the anode aperture as shown in 2.25. As the perveance is increased, the size of the anode increases and the separation between the cathode and anode decreases causing some distortion near the cathode as shown in Figure

2.26.

Figure 2.26: A higher perveance increases the size of the anode and decreases the distance between the cathode and anode resulting in some distortion near the cathode [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

Thus, a modified focused electrode is suggested in [75] which reduces the distortion of equipotential profiles and improves the electron focusing and cathode emission uniformity as shown in Figure 2.27.

Figure 2.27: A modified focused electrode to improve the electron gun design by reducing the distortion of equipotential profiles and improving the electron focusing and cathode emission uniformity [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. The electric field near the cathode edge is reduced due to the modified electrode as shown in Figure 2.27 causing the perveance to decrease.

To analyze the effect of the anode aperture, consider the quantities in Figure 2.28.

Figure 2.28: Quantities used in the analysis of effect of anode aperture to calculate the gun parameters [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. Equation (2.101) can be rewritten as

(2.102) − = [.× −c ] − Finding dV/dR,

(2.103) − − = [.× −c ] Since

(2.104) − − − = = − Substituting (2.101)

(2.105) − − − = − − = − At the anode, V = Va, R = Ra, and dV/dR = -E1. Thus,

(2.106) − = − The focal length, F, becomes

(2.107) − = [ − − ] From (2.99),

(2.108) − = From (2.98),

(2.109) = = Equation (2.107) becomes

(2.110) = [ − − ] At the edge of the beam, the slope of trajectory is

(2.111) tan � = = [ − − ] From (2.99), the derivative is

(2.112) According to =[76], + (2.111) . + can . be written + . as + . + .

(2.113) tan � = = [ − − ] , where is the correction factor, which “depends on the size of the anode aperture relative to the spacing between the cathode and anode” [2]. A variety of resources use different values of , commonly 1.1 and 1.25 [68, 77-79].

In region 3, the minimum beam radius, bm, and axial position of the minimum radius, zm, exist. Figure 2.29 illustrates the electron beam shape in region 3.

Figure 2.29: Electron beam shape in region 3 [1]. The axial electric field is almost zero when the electron beam leaves the anode aperture.

The only force existing on the electrons is the space charge. From Figure 2.29,

(2.114) ta � − − − = = = = Solving for the minimum beam radius,

(2.115) ta � − , where . The axial= position of the minimum radius is

= ×√ 55

= + [. [ − ] + . [ − ] + . [ − ] − . [ (2.−116) ] ] There are four ways of controlling the current in the electron gun [2, 80]. The first method is the cathode-to-anode voltage, known as “brute force” way. Figure 2.30 shows the cathode-to-anode voltage way to control the beam current in the electron gun.

Figure 2.30: Cathode-to-anode voltage way to control the beam current in the electron gun [1]. The cathode-to-anode voltage way is the simplest way to turn the beam current on and off.

The cathode pulsing is turned negative to turn on the beam. The cathode pulsing is grounded potentially to turn off the beam. The disadvantage of this technique is that all beam power is switched based on the cathode pulsing causing the power modulator to be heavy, large, and inefficient. The second method of controlling the beam current is the modulating anode. It is achieved by placing a second anode between the cathode and final anode. Figure 2.31 shows the modulating anode technique to control the beam current in the electron gun.

Figure 2.31: Modulating anode way to control the beam current in the electron gun [1]. This technique is common for intermediate voltage levels without distorting the beam focusing. The intercepted current by the modulating anode is small resulting in a small power. The third method of controlling the beam current is the focusing electrode. This technique is common for only low perveance values. To reduce the beam current to zero, a negative voltage is applied to the focusing electrode. Another focusing electrode can be placed at the center of the cathode to reduce the beam current. Figure 2.32 shows the focusing electrode technique to control the beam current in the electron gun.

Figure 2.32: Focusing electrode way to control the beam current in the electron gun [1]. The beam is turned on by setting the voltage equal to the cathode potential. Setting the voltage below the cathode potential turn off the beam. The fourth method of controlling the beam current is the grid. Most applications in TWT prefer the grid technique due to its small modulator size. It is achieved by placing one or more grids near the cathode surface.

Figure 2.33 shows the grid technique to control the beam current in the electron gun. 57

Figure 2.33: Grid way to control the beam current in the electron gun [1]. The grid technique degrades the beam focusing, and affects both the beam dynamics and electron trajectories. There are different grid placement methods [81]. As its name suggests, the single grid element is the simplest structure. However, it requires to use a shadow grid, invented by Drees [82], between the cathode surface and grid control to avoid heating the grid since the intercepted cathode current can exceed 15%. The grid cathode region is approximated as a parallel-plane diode, whose area is

(2.117) −c = , where the variables Rc and are the same in Figure 2.28. To find the grid voltage as a function of the grid position, let’s define d as the electrode spacing, and dgc as the grid- cathode spacing. Figure 2.34 shows the gun parameters to determine the grid-cathode spacing.

Figure 2.34: Gun parameters to determine grid-cathode spacing [1].

The current flowing in the electron gun, I, must be equal the cathode current, Ic. That is, 58

(2.118)

The Child-Langmuir equation in (2.83) defines = the current from the cathode to grid.

Plugging (2.83), (2.101) and (2.117) into (2.118) yields

(2.119) − − .× � .× −c −c − = � or

(2.120) − � � = , where Va is the anode-to-cathode voltage, Vg is the grid-to-cathode voltage, and - a is the same constant in (2.99) evaluated at the anode. The grid voltage must not be large to cause an electrical breakdown. The grid wire spacing and area of grid wires control its cutoff value. Those dimensions can be estimated using a planar triode based on the screening fraction, which is the ratio of the grid area to the cathode area [83]. It is based on the cutoff amplification factor for a planar triode, μc, which is the “ratio of anode voltage to the cutoff grid voltage” [2].

Table 2.2 summarizes the beam control electrode characteristics with some types, which control the beam.

Table 2.2: Characteristics of control electrodes [80].

Grid Focusing at Low Type c Capacitance Interception Voltage Modulating Anode 1-3 - 50 pF 0 Good Control Focus 2-10 2-10 100 pF 0 Poor Electrode Intercepting Grid 50 100 50 pF 15% Fair Shadow Grid 30 300 50 pF 0.1% Fair

The ratio of the beam to electrode voltage is the total amplification factor, μ, and the ratio of the beam voltage to the negative electrode voltage is the cutoff amplification factor, μc.

These factors are higher in the grids’ types. The capacitance is used to drive the control electrode structure. Its value is the same for all types except the control focus electrode.

The grid interception is higher in the interception grid as expected. Overall, the modulating anode has a good focusing compared to the other types.

Molybdenum is used to fabricate the grids. It has a high work function, which decreases as the evaporation of the barium from the cathode occurs causing a grid emission.

That is why hafnium and zirconium are used to prevent the grid emission by coating the molybdenum [84]. To serve the purpose better by extending the life of zirconium film, a combination of zirconium, tungsten, and molybdenum is used [85]. Other grid structures can be developed with two control grids and a shadow grid, known as dual-mode operation, operating at two different power levels [86]. Each grid shares the responsibility of controlling the emission.

2.4.2 Focusing Structure

The electron beam in the electron gun diverges due to the space charge forces. It can be prevented with equal and opposite focusing forces, which are provided by a magnetic field aligned with the axis of the electron beam. Figure 2.35 illustrates the effects of space charge and focusing forces on the electron beam.

Figure 2.35: The effects of space charge and focusing forces on the electron beam [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. To focus the beam by generating the magnetic field, a solenoid is used. Figure 2.36 shows how the solenoid is used to focus the beam in linear beam tubes including the TWT.

Figure 2.36: The use of a solenoid to generate a magnetic field and focus the beam in linear beam tubes [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. 2.4.2.1 Uniform-Field Focusing

When the electron beam enters the solenoid and crosses magnetic flux lines, the magnetic focusing force is initiated. Figure 2.37 illustrates the configuration of magnetic flux lines as the electron beam enters the solenoid.

Figure 2.37: Configuration of magnetic flux lines as the electron beam enters the solenoid [1]. When the electron is above the axis, the magnetic force is out of the paper. When the electron is below the axis, the magnetic force is into the paper. The electron beam rotates clockwise as it enters with the flux lines. Reversing the direction of the flux lines results in a counterclockwise direction of the electron beam. The electron beam “crosses radially directed flux lines upon entering the field” [2]. The magnetic field force is produced as the

rotated beam interacts with the axial components of the flux. Figure 2.38 shows the electron trajectory in the axial field.

Figure 2.38: Electron trajectory in the axial field [1]. The equal and opposite constraining forces compresses the beam. When the charge density is uniform and there is no magnetic flux through the cathode, “the magnetic flux level that produces a magnetic force that exactly balances the space charge and centrifugal force is called Brillouin flux level,” denoted by BB [2]. As the beam rotates, the electron motion is constant in diameter and is called Brillouin flow. Once the electron beam reaches the collector, the beam exits the magnetic field and the space charge force influences the beam causing it to expand again.

When the magnetic forces balance the centrifugal and space charge forces, the beam has a radius, a, for Brillouin flow called an equilibrium radius as

(2.121) = √ , where B is the axis flux density and I is the beam current. Figure 2.39 shows the Brillouin flow condition.

Figure 2.39: Brillouin flow condition [1]. From (2.121), the equilibrium radius is inversely proportional to the flux density and magnetic force. The current and charge density are directly proportional to the equilibrium radius and space charge force.

In experiment, it is a challenging task to achieve the perfect balance to satisfy

Brillouin flow. Figure 2.40 illustrates the resulted beam dynamics when the beam enters the magnetic field.

Figure 2.40: Resulted beam dynamics when the electron beam enters the magnetic field [1]. As shown in Figure 2.40, the magnetic focusing force is smaller than the space charge force at plane z1. Therefore, the electron beam expands. At plane z2, the electrons move outward and the magnetic focusing forces and space charge force are equal. Thus, “the electrons

overshoot the equilibrium radius” [2]. At plane z3, the magnetic focusing force is larger than the space charge and centrifugal forces causing the beam to become smaller and the radial electron motion to stop. At plane z4, the electrons move inward and “the electrons overshoot the equilibrium radius” [2]. At plane z5, the beam radius is the same as plane z1.

That is, the magnetic focusing force is smaller than the space charge and centrifugal forces causing the radial motion of electrons to stop. Since the process repeats, an oscillation occurs and the variation of the periodicity is called scalloping, which can be eliminated by adjusting the magnetic field level.

The beam equation for the uniform-field focusing can be determined. The radial force equation is

(2.122) , where Busch’s theorem is used to define − as = −( + ) (2.123) = − , where rc is the position at the origin for the electrons at the cathode whose flux density is

Bc. It is assumed that electron trajectories do not cross, which means that the electron flow in the beam is laminar. This results in a constant since rc/r is constant. The radial position is proportional to the space charge, centrifugal, and magnetic forces. This also justifies that the radial position is proportional to radial acceleration and motions of the electrons. From the relationship found using Gauss’s law in (2.14), the electric field at the beam boundary where r = b is

(2.124) = − Combining Gauss’s law, Busch theorem, and force equation together, the motion of the electrons on the outer edge of the beam is referred to as the beam equation defined as

(2.125) + [ − ] − = , where L being the Larmor frequency. The cyclotron frequency is two times the

Larmor frequency.= B/ From (2.121), the beam equation can be rewritten as

(2.126) � = − [ [ − ] − ] The net radial force on the electron is zero if the right-hand side in (2.126) is zero. This means that the magnetic force, which plays a role in compressing the beam, is equal and opposite to the space charge and centrifugal forces, which play a role in expanding the beam. The left-hand side in (2.126) is proportional to the radial force on the electrons since it is proportional to the radial acceleration. When the radial forces are zero, the desired beam shape is obtained since the beam would be in equilibrium. Such equilibrium case does not allow the beam to expand and forms a cylindrical beam with a constant beam diameter. If the forces are zero, the right-hand side of (2.126) becomes

(2.127) [ − ] − = , where be is the equilibrium radius. To simplify (2.127), the expression of the flux density can be found in terms of Bc and Brillouin flux density, BB. For Brillouin flow at equilibrium where the radial acceleration is zero,

(2.128) , where is the same as , but for= Brillouin − flow. + For confined flow or immersed flow focusing where scalloping of the beam is avoided by using a larger magnetic field value than the Brillouin value, equation (2.128) becomes

(2.129) = − +

The circuit of the magnetic field for the confined flow is designed in a way to allow the magnetic flux lines to pass through the cathode. In general, the confined flow has a better beam control than the Brillouin flow. However, the magnetic field in the confined flow focusing needs to be larger and heavier compared to the Brillouin focusing. Combining

(2.128 - 2.129) and Busch theorem yields

(2.130) = + , where Ac and Ae are the area of the cathode and cross-sectional area of the equilibrium beam, respectively. The ratio in (2.130) is called the area compression ratio and it can reach a value of 25 or higher. Such high compression ratio corresponds to have a flux density of a few tens of gauss at the cathode surface being equivalent to a Brillouin flux density of

500 to 1000 gauss or more. This can be expressed as

(2.131)

, where m is the confinement factor. From = (2.131), the equilibrium beam radius can be written as

(2.132) The actual beam radius is related to the equilibrium= beam radius as

(2.133)

, where . The beam equation becomes = +

<< (2.134) , where + =

(2.135) The general solution of (2.134) is = √ −

(2.136)

Plugging (2.136) into (2.133) and = substituting sin + t = z/v cose yields

(2.137)

= + sin + cos Equation (2.137) can be rewritten as

(2.138) = [ + sin + �] , where examining the beam at z = 0 into the magnetic field yields to determine both C and

. Equation (2.138) states that for a frequency, s, and equilibrium radius, be, the beam

ϕradius scallops. The wavelength of such scallop is

(2.139) − .× = = [−] Figure 2.41 shows the magnetic field configuration for Brillouin flow at the entrance going to the focusing structure.

Figure 2.41: Magnetic field configuration for Brillouin flow at the entrance going to the focusing structure [1]. The confinement factor is 1 when the Brillouin flow is used. This means that the cathode region does not have a magnetic flux. The beam radius is

(2.140) � From (2.140), the beam radius is = constant [ + and sin is √equal to +a at � z = 0. Also, the slope of the beam radius with respect to the z axis, db/dz, is zero resulting in a C value of zero. The 67

Brillouin flux density is “the magnetic flux density that produces a beam of constant radius when no magnetic flux links (passes through) the cathode, usually denoted by BB” [1]. This quantity results in the Brillouin flow and beam. The Brillouin flux density can be written as

(2.141) − = .× A disadvantage of the Brillouin flow is the sensitivity to the magnetic field quantity used, misalignments, and RF drive. For example, the electron beam oscillates about the equilibrium diameter if the used magnetic flux density is less than the Brillouin flux density as shown in Figure 2.42.

Figure 2.42: Obtained electron beam if the used magnetic flux density is less than Brillouin flux density [1]. For Brillouin flow, the wavelength of the scallop is

(2.142) − Figure 2.43 illustrates the conditions of= the . electron× beam as the flux density is varied compared to the Brillouin flux density.

Figure 2.43: Beam shape as the magnetic flux density is varied compared to the Brillouin flux density [1].

If the slope of the electron beam radius with respect to the z axis, db/dz, is larger than zero, the electron beam shape varies with different magnetic flux densities as shown in

Figure 2.44.

Figure 2.44: Beam shape as the magnetic flux density is varied compared to the Brillouin flux density when db/dz is larger than zero [1].

For confined flow, the flow focusing can be understood by using Busch’s theorem.

Considering the rate of the rotation of the beam,

(2.143) = − − , where is the rate rotation of the beam, and is the rate of rotation of the Brillouin beam. Comparing the beam dynamics between the confined focusing and Brillouin

focusing, the scallop amplitude on the confined flow to the scallop of amplitude on the

Brillouin flow is

(2.144) = − , where C is the resulting scallop amplitude and CB is the scallop amplitude on the Brillouin beam. Comparing the effect radius of the beam when the current changes due to use of a grid, the change in the beam radius for the confined and Brillouin flows can be expressed as

(2.145) = − , where and B are the changes of beam radii for confined and Brillouin flows, respectively [87].

2.4.2.2 Periodic Permanent Magnet (PPM) Focusing

A periodic permanent magnet can replace the use of a solenoid to focus the electron beam. It reduces the size and weight of a TWT by one to two orders of magnitude. Figure

2.45 shows a system of a periodic permanent magnet.

Figure 2.45: A system of periodic permanent magnet with a periodic focusing [1].

A radial force, which compresses the beam, is produced when the axial field interacts with the rotational motion. Once the beam exits the magnet section, the space charge force influences the beam causing it to expand again. Then, the beam enters the magnet section again in the opposite direction from the previous one causing the beam to rotate in the opposite direction. Such rotation oscillates back and forth producing periods, beam expansion and beam ripple. It is worth noting that this ripple is different from the scalloping discussed previously. Figure 2.46 illustrates the difference between a beam ripple and beam scalloping.

Figure 2.46: Difference between a beam ripple and scalloping [1]. It is assumed that the magnetic field is a function of the axial position, z, and

(2.146) = cos , where Bp is the peak field, and L is the magnet period represented in Figure 2.45. Such relationship indicates that the magnetic field varies sinusoidally with distance. The beam equation can be written as

(2.147) + cos − = If the magnitudes of forces at the maximum and minimum are set equal, the relationship between the peak field and Brillouin field is found as

(2.148)

, where (2.148) suggests that the root= √ mean square= √ value of the field is the same as Brillouin field. In Chapter 3, the BB term in (2.148) is changed to another term. To examine the stability condition of the PPM, let

(2.149) � = (2.150) as indicated in [88-89]. Equation (2.1η0) = is the angular frequency “at which the field appears to the moving electrons to alternate” [1]. Then, the beam equation can be rewritten as

(2.151) � + cos � − = , where LP is the peak value of Larmor frequency and p is the average plasma frequency.

Let be the magnetic field coefficient and be the space charge coefficient defined as

(2.152) � = (2.153) The normalized beam radius is obtained =by letting as

σ = b/a (2.154) � Figure 2.47 shows the beam envelop + curves + cos for � three � − cases = of the magnetic field with different values of and [88-89].

Figure 2.47: Beam envelop curves for three cases of the magnetic field with different values of α and β [89]. As shown in Figure 2.47, cases (a) and (c) illustrate scalloping and ripples. They need to be adjusted since the focusing is not appropriate. Case (b) achieves the optimum focusing by setting

(2.155) or =

(2.156)

Figure 2.48 shows the focusing conditions =as and are varied.

Figure 2.48: Focusing conditions as α and β are varied [1]. As shown in Figure 2.48, the larger is, the more ripple is obtained. The beam diverges when becomes larger than 0.66. In addition, becomes unstable when

(2.157)

= . 73

Figure 2.49 illustrates the unstable conditions for the normalized beam radius equation in

(2.154) based on values.

Figure 2.49: Unstable conditions for the normalized beam radius equation based on α values [1]. Based on those coefficients, the unstable conditions are shifted to the frequency variables.

That is, the normalized beam radius equation in (2.154) is unstable when

(2.158) and = .

(2.159)

In general, the PPM can be thought = .of as a lens system. Figure 2.50 shows the illustration of PPM by a series of convergent lenses.

Figure 2.50: A series of convergent lenses demonstrating the PPM [1].

The beam diverges when the lenses are far apart from each other or the field is too strong.

Figure 2.51 shows the focusing conditions in terms of optical rays for different focal lengths.

Figure 2.51: Focusing conditions in terms of optical rays for different focal lengths [1].

The strength of the lens is proportional to LP2 or B2. The focal length is inversely proportional to LP2. As shown in Figure 2.51, If the focal length is one half of the lens separation, the system is stable and the ray is focused. If the focal length is less than one fourth of the lens separation, the system is unstable and the ray diverges. In experiments, it is found that

(2.160) − < .× Another relationship is called the beam stiffness factor, defined as

(2.161) This relationship differs from resources to> another . based on the right-hand side, but recent resources consider 0.87 as the appropriate value to use [1]. The stopband voltage is the adjusted beam voltage such that “beam transmission stops” [1]. It is expressed as

(2.162) = .

In (2.162), the period L is in cm, Brms in kGauss, and resulted Vs is in kV. Confined flow focusing with PPM fields can also be achieved [90-92]. It shares common advantages as the confined flow with uniform fields such as improving the control of beams from gridded guns. Another advantage that the confined flow with PPM fields has, but the confined flow with uniform fields does not have is that “the rotational energy of the beam increases as the beam becomes more confined” [1]. Thus, the efficiency and energy of the beam decrease limiting the confined flow PPM focusing to low-perveance designs [93]. It does not allow the beam to expand when RF is operated. This beam expansion is inversely proportional to the magnetic field and slightly changes when the beam stiffness factor is increased. As the stiffness factor is increased, the ripple decreases, which is defined as

(2.163) −�

% = +� × True [94-96] investigated the PPM focused beams in details. He stated that the beam enters the magnetic focusing field with normalized transverse velocities of the electrons of

(2.164) ̅ � = Those transverse velocities must be controlled within the structure. True described them as

+ an additional force that can be controlled by finding the effective microperveance, Pμ , for the beam as

(2.165) + , where P is the normal beam microperveance = + and P is the scattering microperveance

s resulting from the transverse velocity. True also suggested that the Brillouin field is the required field level to focus the beam and it can be related to the perveance as

(2.166) + + � = � , where B+ is the increasing focusing field. Such ratio is known as the magnetic field intensification ratio. The normalized transverse velocity in (2.164) is a combination of thermal and nonthermal parts as

(2.167) , where nT and T are the nonthermal �and = thermal � + parts, � respectively. The nonthermal part is resultedσ fromσ grid scattering and aberrations and is found using computer simulations.

The thermal part is resulted from the thermal velocity and is defined as

(2.168) � = 9 , where rc is the disc radius of the cathode, r95 is the beam radius with 95% of the beam current. However, this part is neglected for gridded guns since it is too small compared to the nonthermal part. Thus, equation (2.167) for gridded guns can be approximated as

(2.169)

, which True found it to be approximately� ≈ �

(2.170)

, where is the half angle of the electron� = gun. . This means that the normalized transverse velocity is proportional to the half angle of the electron gun. The scattering microperveance in (2.165) is related to the normalized transverse velocity by

(2.171) Plugging (2.171) into (2.165) yields = . �

(2.172) + = + . �

The term inside the square root of (2.166) can be multiplied by 1.65 to reduce the ripple of the beam. This value represents F in the following equation and was specified based on experimental observations as

(2.173) + + � = � Thus, the magnetic field intensification ratio is related to the normal beam microperveance and normalized transverse velocity by

(2.174) + . � = [ ( + � ) ] Also, another relationship can be derived between the normal beam microperveance, half angle of the electron gun, and disc radius of the cathode as

(2.175) 9 = . Plugging (2.175) into (2.174) yields

(2.176) + = [ . + . 9 ] , where B+/BB is called the intensification factor. Figure 2.52 shows the relationship between the intensification factor and radius compression ratio.

Figure 2.52: Intensification factor versus radius compression ratio of the PPM field.

The normalized focusing factor can be expressed as

+ (2.177) 9 = , where a is the tunnel radius. The ratio r95/a represents the tunnel fraction filled out by the beam. Figure 2.53 shows the relationship between the normalized focusing factor and radius compression ratio.

Figure 2.53: Normalized focusing factor versus radius compression ratio of the PPM field.

2.5 Traveling Wave Interaction

The small signal theory, given by Pierce [15], describes the traveling wave interaction. This section discusses the Pierce analysis by examining synchronous and nonsynchronous conditions. This includes determining the propagation constants from the combination of electronic and circuit equations without the space charge effect to obtain the determinantal equation, which is significant in investigating the interacting of wave with the electron beam to produce gain.

2.5.1 Electronic, Circuit, and Determinantal Equations

2.5.1.1 Electronic Equation

Maxwell’s equations describe the electric and magnetic fields with the existing electron charge and current as

(2.178) ∙ = − (2.179) � × = − = − (2.180) ×� = −� + = −� + (2.181)

Also, ∙ =

(2.182) From equations (2.178-2.182),.×�= the continuity −.�+ equation is ∙defined = as

(2.183) From (2.179-2.180), another equation can ∙ � be = −derived as

(2.184) � The vector identity in (2.184)×× can = be − simplified× = as � −

(2.185) Using the vector identity in (2.185)×× and =(2.178), . equation − (2.184) becomes

(2.186) − = − � − , known as the wave equation. Such equation analyzes the electron beams with space charge effects. The parameters in (2.186) are ac quantities in the beam.

Two small signal assumptions can be considered. First, the quantities vary sinusoidally with time as e-jωt. In this assumption, the wave equation can be written as

(2.187) + = −� − , known as the force equation. The constant k is the propagation constant of a wave traveling at c, which is the speed of light. Both constants, k and c, can be expressed as

(2.188) = (2.189) = The quantities in (2.187) exist only in the z direction. Thus,

(2.190) + = − − The second assumption is the quantities vary sinusoidally with distance as e-jβz, where is the propagation constant whose value is based on the circuit and beam parameters. The wave equation becomes

(2.191) − = − Equation (2.13), which defines the current density, can be expanded as ac and dc quantities in the beam as

(2.192)

Two quantities in (2.192) = are neglected+ = because+ they are too+ small: dc ve(dc) and ac ve(ac). So,

(2.193)

From the continuity = equation +in (2.183), = + =

(2.194)

Thus, − = −

(2.195) = 81

Plugging (2.195) in (2.193) yields

(2.196) = − � The force equation eliminates the velocity as

(2.197) As a result, = + = − = −

(2.198) = − The current density becomes

(2.199) = − (−)[− � ] = (− ) (2.200) = (−) , known as the electronic equation, which is the relationship between the electric field and beam current density. The plasma frequency, p, in (2.200) can be expressed as

(2.201) = √ The electronic equation can be rewritten as

(2.202) = − , where e e(dc) representing the propagation constant of the electron beam. The current densities = /� were replaced by their corresponding currents and other parameters were replaced by their equivalent expressions.

2.5.1.2 Circuit Equation

The slow-wave circuit of the TWT can be thought of as a transmission line consisting of distributed inductor and capacitor per unit length. Figure 2.54 shows the transmission line model of the slow-wave circuit.

Figure 2.54: Transmission model for the RF circuit of the TWT.

The current in the circuit is I = iz and the incremental length is whose current is

z. To determine the current at point A, consider Figure 2.55 shownz below: � =

Figure 2.55: Transmission line model for the RF circuit of the TWT to determine current at point A. Following the convention,

(2.203) � , where ∆ + = + ∆ +

(2.204) = − = −∆ Plugging (2.204) into (2.203) yields

(2.205) � ∆ = ∆ −83 ∆

As , equation (2.205) becomes

z → (2.206) � It is assumed that this current equation = +varies as ej(ωt-βz). Thus, the current equation can be written as

(2.207)

To determine the voltages around− loop = ABCD, − + consider Figure 2.56 shown below:

Figure 2.56: Transmission line model for the RF circuit of the TWT to determine the voltages around loop ABCD. The voltages can be summed as

(2.208) Equation (2.208) can be simplified = to − ∆ + + ∆

(2.209) It is assumed that this voltage equation varies= as ej(ωt-βz). Thus, the voltage equation can be written as

(2.210)

The current through the inductor, IL, can− be eliminated = in (2.210) and (2.207) to have

(2.211) = − When the electrons exist, the electric field is

(2.212) = − = = − 84

Without electrons, the phase velocity and interaction impedance can be expressed as

(2.213) = √ (2.214) = √ Also, the propagation constant can be written as

(2.215) or = =

(2.216)

, where c represents the propagation constant= of the circuit. The electric field in (2.212) can be expressed as

(2.217) = − , known as the circuit equation, which relates the electric field, Ez, to the current, iz.

2.5.1.3 Determinantal Equation

To obtain the propagation constants, the electronic equation (2.202) and circuit equation (2.217) are combined as

(2.218) [− ][ − ] = , known as the determinantal equation. The determinantal equation can also be written as

(2.219) [− ][ − ] + = , where C is a constant called the gain parameter and is found as

(2.220) � = [ ]

The gain parameter, C3, is one quarter of the circuit impedance, K, being in the order of tens of ohms to the beam impedance, V/I., being in the order of 1000-10000 of ohms. Such ratio results in a gain parameter, C, in the order of 0.01-0.1.

The determinantal equation (2.219) provides four roots resulting in four solutions to the propagation constants, , along the circuit and electron beam. Those solutions require four boundary conditions. Two of these boundary conditions indicate the two voltages at the two ends of the circuit. The other two boundaries indicate the beam current and RF velocity. To solve for the propagation constants, two assumptions are made. The first assumption is that the electron velocity is equal to the velocity of wave without electrons represented as a synchronous condition. The second assumption is that the electron velocity is different from the velocity of wave without electrons represented as a nonsynchronous condition.

2.5.2 Synchronous Condition

When the velocity of the electron is equal to the velocity of the wave, the propagation constants are written as

(2.221)

It is assumed that differs from e and =c by , which is a small quantity as

ξ (2.222)

Plugging (2.222) into (2.219) yields = + � = + �

(2.223) + �+ � [ � +� ] + = or

(2.224) +�+� [ � �+� ] + =

Since ξ is a small quantity compared to either or e, the terms e and 2 are neglected.

Thus, equation (2.224) becomes ξ ξ

(2.225) or � = −

(2.226) There are three roots in the term (-1)1/3�. =Such − terms can be expanded as

(2.227) − , where n = 0, 1,− and 2.= The[ solutions] =of costhe three − roots +are sin −

(2.228) √ √ Their corresponding propagation = + constants , = are − , = −

(2.229) √ = + = + + (2.230) √ = + = + − (2.231)

These three propagation constants= + are forward = waves.− The wave in the first root grows due to the positive imaginary part. Its eC/2 term indicates that the speed of the wave is less than the speed of the electron beam. The wave in the second root decays due to the negative imaginary part. Its eC/2 term indicates that the speed of the wave is less than the speed of the electron beam. The wave in the third root neither grows nor decays. Its - eC term indicates that the speed of the wave is greater than the speed of the electron beam. The fourth propagation constant is a backward wave given by

(2.232) = − + 87

The wave in the fourth root neither grows nor decays. Its - e term indicates that the speed of the wave is slightly greater than the speed of the electron beam.

Since it was assumed that the waves vary as ej(ωt-βz), the resulted propagation constants can be written as

(2.233) √ − [−+ ] = (2.234) √ − [−+ ] − = (2.235) − [−−] = (2.236) − [−−+ ] To estimate how much the wave has= grown, first root, R1, is considered whose propagation constant is 1. It can be estimated with respect to the wavelengths as

(2.237) The power gain for the lossless = case = is = =

(2.238) √ √ lo� ( ) = lo� ( ) = . [] This power gain in (2.238) does not consider the amplitude of the three forward waves. To do so, the RF electric field, current, and velocity are added for each forward wave at the beginning as

(2.239)

= + + (2.240)

= + + (2.241)

From (2.202), the current for each =launched + forward+ wave is

(2.242) = − = 88

(2.243) = − = (2.244) = − = Plugging (2.242-2.244) into (2.240) yields

(2.245) = + + = [ + + ] From (2.187), the velocity for each launched forward wave is

(2.246) = − − = − = −− = − (2.247) = − − = − = −− = − (2.248) = − − = − = −− = − Plugging (2.246-2.248) into (2.241) yields

(2.249) = + + = − [ + + ] Since the current and velocity at the input of the traveling wave tube is zero. Equations

(2.245) and (2.249) are set equal to zero. Taking E1, E2, and E3 terms from (2.239), (2.245), and (2.249),

(2.250)

= + + (2.251) + + = (2.252) + + = Combining (2.250-2.252) and solving for E1,

(2.253) � = −− Plugging the roots’ values in (2.2η3) yields

(2.254) = = = 89

Thus, the power gain is found to be

(2.255) √ = lo� ( ) = −. + . [] Figure 2.57 shows the power gain as a function of CN for the synchronous condition in the traveling wave tube.

Figure 2.57: Power gain as a function of CN for the synchronous condition in a traveling wave tube. The electric field with respect to the distance becomes

√ √ − − − − = [ + + ]

√ √ − − − − = [ + + ] (2.256) −− − √ 2.5.3 Nonsynchronous= Condition[ + ℎ ]

When the velocity of the electron is not equal to the velocity of the wave, numerical solutions are used to obtain the propagation constants. The electron velocity can be expressed as

(2.257)

= + 90

, where b is Pierce’s velocity parameter. The propagation constant is found to be

(2.258)

There are three conditions considered. When= b < + 0, the speed of the wave is greater than the speed of the electron beam. When b > 0, the speed of the wave is less than the speed of the electron beam. When b = 0, the speed of the wave is equal to the speed of the electron beam and the synchronous condition is met.

The propagation constants for the nonsynchronous condition take the form of real and imaginary parts. This is accomplished by using (2.239-2.241) and (2.258). The root of the solution has the form of

(2.259)

The complex propagation constant is found = to− be

(2.260)\

Since it was assumed that the waves vary = as ej(ωt −-βz), the + resulted propagation constants can be written as

(2.261) − [−−] When x is positive in eβeCxz, the wave = grows. Otherwise, the wave decays when x is negative.

2.6 TWT Slow-Wave Circuits

When the velocity of the electrons in the beam is equal or close to the velocity of the RF circuit, amplification occurs in the TWT and gain is obtained. The gain decreases as the velocities move away from each other due the frequency variation. Such study can

be investigated when the circuit is dispersive, which indicates the variation of velocity as the frequency changes.

2.6.1 Wave Velocities

Two different velocities are discussed: group and phase velocities. The difference between the velocities is illustrated in Figure 2.58.

Figure 2.58: Difference between group and phase velocity [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

The phase velocity, vp, is the velocity moving in a vector parallel to the beach. Any point on the wave indicates the phase velocity whether the person is walking along the beach and trying to keep up with the wave crest or surfboarding at a speed of the phase velocity and trying to ride the crest of the wave in parallel to the beach. To estimate the phase velocities in Figure 2.58, note the difference in the angle between the two figures indicating that the angle in the left figure is bigger than the right figure. This means that the wave crest in the left figure curves along the beach slower than one in the right figure. Thus, the magnitude of the phase velocity in the left figure is smaller than the right figure since it depends on the direction.

The group velocity, vg, is the velocity moving in any direction with respect to the phase velocity. It represents the “component of the wave velocity in the direction of the

phase velocity” [2]. For example, if the energy in the wave moves from point A to point B in the left figure of Figure 2.58, the distance is represented as in the parallel direction of the beam from point A to point C. Likewise, the direction is the critical factor in estimating the magnitude of the group velocity. Thus, the group velocity in the left figure of Figure

2.58 is smaller than the group velocity in the right figure.

The directions of the group and phase velocities can be opposite as shown in Figure

2.59.

Figure 2.59: Opposite directions of the group and phase velocities [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. It is assumed that the person walking along the beach is able to see the wave crest only when passing the seawall openings. The magnitude of the velocities in Figure 2.59 is the same as the magnitude of velocities in the left figure of Figure 2.58 if the angle is the same.

However, the directions are reversed. If the person walks from 1 to 2 and the wave crest moves from 3 to 2 at the same time, the phase velocity is to the right and group velocity is to the left. When the group and phase velocities have the same direction, the wave interaction is continuous. Otherwise, the wave interaction is periodic and is called a backward wave interaction.

2.6.2 Dispersion

When the velocity of the circuit varies with frequency, dispersion occurs. There are two ways of investigating the dispersion characteristics. The first one is a straightforward relationship between the phase velocity, vp, and frequency, f. Figure 2.60 illustrates the dispersion characteristics between the phase velocity and frequency.

Figure 2.60: Illustration of dispersion characteristics between the phase velocity and frequency [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. The second way of representing the dispersion characteristics is with Brillouin diagram or

ω-β diagram, which represents the relationship between the angular frequency, ω, and the propagation constant, β. Two different simple circuits in the microwave components are characterized: coaxial transmission line and rectangular waveguide.

2.6.2.1 Coaxial Transmission Line

The coaxial transmission line in the TEM mode is not dispersive. The phase velocity does not vary with frequency. Both E and H lines are perpendicular to the axis of transmission line. Figure 2.θ0 shows the electric and magnetic fields’ lines of a coaxial transmission line in the fundamental TEM mode.

Figure 2.61: Electric and magnetic fields' lines of a coaxial transmission line in the fundamental TEM mode [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. The group and phase velocities are equal to each other and are equal to the speed of light, c, for all frequencies. The propagation constant is

(2.262) Figure 2.62 shows the Brillouin diagram = for = the coaxial transmission line in the TEM mode.

Figure 2.62: Brillouin diagram for a coaxial transmission line in the TEM mode [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

When the slope of the Brillouin diagram is constant, being c in Figure 2.62, and is a straight line, it becomes equivalent to the representation of the phase velocity being constant and the circuit is not dispersive in the straightforward plot.

2.6.2.2 Rectangular Waveguide

The rectangular waveguide is dispersive. That is, the phase velocity varies with frequency. This can be analyzed when two plane waves with equal amplitudes propagate at angles ±α and travel in the z-direction in free space as shown in Figure 2.63.

Figure 2.63: Two plane waves at angles ±α in the z-direction [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. In Figure 2.62, the solid and dashed lines represent the maximum, being the position of the wave crests, and minimum E-field, being the position of the wave troughs, values, respectively. The E-fields’ lines are parallel to the y-z plane and the H-fields’ lines are parallel to the x-z plane. When the E-fields intersect, the E-field value becomes zero at all times the wave propagates at the associated angles and there is only the H-field. Conductors with charges and currents are inserted in the y-z and x-z planes to form a waveguide, which contains the two plane waves. The waves bounce back and forth from one side to the other.

Inside the waveguide, only one wave exists and it reflects back and forth.

To determine the group and phase velocities inside the waveguide, consider Figure

2.64.

Figure 2.64: Group and phase velocities inside a waveguide [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

The wave travels at the speed of light at an angle α in the direction of the axis of the waveguide. The group velocity in Figure 2.64 is smaller than c and is found as

(2.263)

The phase velocity in Figure 2.64 is found = as cos

(2.264) Referring to (2.264), the phase velocity = canc be higher than the speed of light if the denominator term is less than zero, which is not possible. The circuits of the TWT are designed to behave as slow-wave circuits being smaller than c, vp < c.

As frequency changes, the wave configuration inside the waveguide differs resulting in a change in the dispersion characteristics. Figure 2.65 shows the wave configurations inside the waveguide for frequencies f1 > f2 > f3.

Figure 2.65: Wave configurations inside the waveguide for frequencies f1 > f2 > f3 [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. The frequency is inversely proportional to the wavelength. As shown in Figure 2.65, as the frequency increases, the wavelength and angle, , decrease. When = 90°, the wave propagation stops and the cutoff condition occurs. In addition, the phase velocity is inversely proportional to the frequency. Thus, the phase velocity is higher in f3 than f1 and reaches ∞ at the cutoff.

To derive the dispersion characteristics for the waveguide, Figure 2.66 shows the quantities used to derive the relationship between and .

Figure 2.66: Quantities used to derive the dispersion characteristics of a waveguide [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

Based on Figure 2.66, the cutoff frequency, c, is

(2.265) or = + cos

(2.266) = + cos or

(2.267) cos = , but

(2.268) = = and

(2.269) = = Thus,

(2.270) So, the relationship between and = becomes +

(2.271) Figure 2.67 shows the relationship = between√ + and for a rectangular waveguide.

Figure 2.67: Brillouin diagram for a rectangular waveguide [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

As shown in Figure 2.67, the slope represents the phase velocity for all frequencies due to the fact that = /vp. To determine the group velocity, Figure 2.68 shows the changes in the propagation constant occurring when the angular frequency is varied.

Figure 2.68: Changes in the propagation constant when the angular frequency is varied [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. From Figure 2.66,

(2.272) / / cos = = / = / = When the angular frequency, , is reduced by , angle, , increases by , propagation constant, , decreases by , and k decreases by as

k (2.273) ∆ ∆ ≈ cos Equation (2.273) occurs for small and . From Figure 2.67,

(2.274)

Thus, = cos

(2.275) ∆ = ∆ Figure 2.69 shows the group velocity from the Brillouin diagram for different wave configurations including the cutoff condition.

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Figure 2.69: Group velocity from the Brillouin diagram for different wave configurations [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

The dominant mode of the rectangular waveguide is TE10. Figure 2.70 shows the electric field distributions of the dominant mode in the rectangular waveguide.

Figure 2.70: Electric field distributions in the dominant mode in the rectangular waveguide [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

The guided wavelength in Figure 2.70 can be expressed as

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(2.276) , which is longer than the wavelength of= the free space. In the waveguide, the Brillouin diagram is symmetrical, which indicates propagating waves in either direction. Figure 2.71 shows the Brillouin diagram in the rectangular waveguide in either direction of propagating waves.

Figure 2.71: Brillouin diagram in the rectangular waveguide for propagating waves in either direction [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

2.6.3 Bandwidth

One of the most important properties of the slow-wave structures is the bandwidth.

The conventional helix has been the dominant choice of the circuit when a wide bandwidth is desired. The helix TWT can provide a bandwidth of over two octaves. Figure 2.72 shows the saturated output power versus frequency for a helix TWT.

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Figure 2.72: Saturated output power versus frequency for a helix TWT [1]. However, high power harmonics are generated at the low frequencies due to the wide bandwidth and can be fundamental. The harmonics can be reduced by harmonic injection.

Figure 2.72 shows the effect of the harmonic injection on the saturated output power for a helix.

Figure 2.73: Effect of harmonic injection on the saturated output power for a helix [1].

Another harmonic signal is injected with appropriate amplitude and 180˚ out of phase to increase the fundamental power and decrease the saturated output power.

The helix TWT can be thought of as a single wire transmission line as indicated earlier in Figure 2.54 whose dispersion is zero. Figure 2.74 shows the helix being cut at points x and are being straightened.

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Figure 2.74: Helix being cut at points x and is being straightened [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. When the wave propagates along the equivalent wire in Figure 2.74, the phase velocity is found to be

(2.277)

The angle, , is the helix pitch angle and = can sinbe expressed as

(2.278) , where a and p are the radius and pitchcot of the = helix, respectively. The Brillouin diagram becomes ideal without dispersion and the phase velocity becomes a straight line. Figure

2.75 shows the Brillouin diagram without dispersion for a helix.

Figure 2.75: Ideal Brillouin diagram without dispersion for a helix [1].

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The electron velocity, ve, is slightly away from the phase velocity and is represented by the dashed line. Such ideal case produces unlimited bandwidth. However, there are factors playing a role in limiting the bandwidth experimentally. The first factor is that the gain decreases since gain is proportional to the circuit length, but frequency is inversely proportional to the circuit length. The electric field pattern shrinks when the frequency increases resulting in a decrease in the gain. Figure 2.76 shows the electric field pattern with two different frequencies.

Figure 2.76: Electric field pattern with two different frequencies [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. The second factor is the dispersion. The helix itself and the technique used to support the structure provide the dispersion. The last factor is the backward wave oscillations (BWO) occurring at = in Figure 2.75. As a result, some techniques are used to control the dispersion and� suppress the BWO to provide the desired bandwidth.

To obtain dispersion, the phase velocity changes as the frequency is varied. This is accomplished in some of the slow-wave circuits such as the helix, but some do not have this feature such as the coaxial transmission line in the TEM mode. The first primary reason behind dispersion is the generated coupling of the electric and magnetic field from one turn

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to the other in the helix. The magnetic flux between the turns cancels each other because the current flows in the same direction. Figure 2.77 shows the magnetic flux cancellation between the helix turns.

Figure 2.77: Magnetic flux cancellation between the helix turns [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. As the wavelength increases, the flux per unit area decreases. Also, the frequency and inductance decrease and the phase velocity increases. Figure 2.78 shows the Brillouin diagram for a helix with a 10° pitch angle.

Figure 2.78: Brillouin diagram for a helix with a 10° pitch angle [1]. Three dielectric rods were used in Figure 2.78 to support the structure. They were spaced equally and made of an insulating material. The phase velocity is inversely proportional to

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the square root of the dielectric constant of the material resulting in a decrease in the velocity.

The second reason behind dispersion is the concentration of both electric and magnetic fields between the helix turns at high frequencies. This was represented in Figure

2.76. The electric field extends farther for f2 compared to f1 since f2 < f1. A portion of these fields is intercepted by the metal barrel, especially at low frequencies. The closer the metal barrel is to the helix, the more portion of the fields is intercepted. However, bringing the barrel closer to the helix reduces the interaction impedance due to the increase in the inductive coupling and capacitive loading, which decreases the gain and efficiency.

There are suggested techniques for the dispersion control by a conducting shell.

Figure 2.79 shows the common techniques used to control the dispersion [97].

Figure 2.79: Common techniques used to control the dispersion [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. The unloaded technique has only the outer conducting shell and the three dielectric rods supporting the helix. The other three techniques have metallic anisotropic loading elements.

The normalized phase velocity, Pierce’s velocity parameter, and small signal gain were investigated with the suggested techniques in [98]. Figures 2.80-2.81 show the

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normalized phase velocity, velocity parameter, and small signal gain using the suggested techniques.

Figure 2.80: Normalized phase velocity and Pierce’s velocity parameter as a function of frequency for the suggested techniques to control dispersion [98].

Figure 2.81: Small signal gain as a function of frequency for the suggested techniques to control dispersion [98]. As shown in Figures 2.80-2.81, when the frequency increases, the normalized phase velocity decreases and Pierce’s velocity parameter increases for the unloaded technique.

Its maximum small signal gain occurs between 6 and 7 GHz indicating four helix turns when corresponding the frequency to the wavelength. The gain decreases before and after the range of 6-7 GHz because the length of the helix decreases before 6 GHz and the

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electric field shrinks after 7 GHz. For the vanes technique, the normalized phase velocity is nearly constant, and Pierce’s velocity parameter increases as the frequency increases. Its maximum small signal gain occurs in the same region the unloaded technique has, but higher by 20 dB. The gain decreases before and after the range of 6-7 GHz for the same reasons the unloaded waveguide technique does. For the “T” shaped technique, the normalized phase velocity and Pierce’s velocity parameter increase as the frequency increases. One of their small signal bandwidth exceeds 2 octaves.

2.6.4 Power

The amount of power the slow-wave structure is capable of must be investigated to provide reasonable justifications to the use of such circuit. The peak power is limited by the backward wave oscillations and the frequency of operation is a significant factor in estimating the average power of the slow-wave structure.

2.6.4.1 Backward Wave Oscillations and Suppression to Peak Power

The backward wave oscillations limit the bandwidth and output peak power capability of helix. They occur when the electron transits between two turns 180˚ in angle.

Such transition produces two turns per unit length. Figure 2.82 shows the concept of backward wave oscillations on a helix for two turns.

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Figure 2.82: Backward wave oscillations on a helix for two turns [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. As shown in the top of Figure 2.82, the electron represented by the force F is accelerated and moving to the right and a backward wave is moving to the left. At 180˚ later, the backward wave moves to the left 180˚ and there is not any force existing (F = 0) as shown in the middle of Figure 2.82. The electric field nulls meet the electrons in the same side.

At another 180˚ later, the electron is accelerated since the wave moves 180˚ to the left as shown in the bottom of Figure 2.82. The same process happens, but with deceleration instead of acceleration and BWO occur.

The oscillation frequency becomes tunable with the beam voltage. The mode of the oscillations depends on some factors such as the beam current, and distance between beam and helix [98]. Also, “Beam current is limited to avoid oscillations. Current is also limited by the need to avoid an excessive current density and excessive space charge forces that limit the effectiveness of the bunching process and the efficiency of operation” [1].

To suppress the BWO, a resonant loss is used to produce attenuation into the helix.

It is placed between the helix turns on the support rods. This technique was developed by

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Varian Associates. The length of the line is at the BWO frequency. Figure 2.83 shows the technique of suppressing BWO using a resonant loss to produce attenuation.

Figure 2.83: Suppressing BWO with the use of resonant loss to produce attenuation [99]. Figure 2.84 shows the saturated output power of a 10 kW helix TWT with a resonant loss existing at 8 GHz.

Figure 2.84: Saturated output power of a 10 kW helix TWT with a resonant loss at 8 GHz [100]. As shown in Figure 2.84, the sharp drop in the power is due to the resonant loss providing an attenuation. However, the use of the resonant loss might produce a harmonic signal, which gets amplified by the traveling wave tube at the frequency of attenuation. Thus, this harmonic can destroy the resonant loss structure.

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Another way of suppressing the BWO is to vary the pitch size between the helix turns resulting in a change in the phase velocity. This can take the form of a step change.

Figure 2.85 shows the pitch change technique to suppress the BWO.

Figure 2.85: Technique of pitch change to suppress backward wave oscillations [99]. In the right side of the helix, the pitch size is small resulting in an energy transferring from the beam into the backward wave. Thus, oscillations occur. As the backward wave moves to the left, it faces a helix turn with a larger pitch than the previous one. Thus, the wave would travel at a different phase velocity resulting in an energy transferring from the backward wave into the beam. Thus, attenuation occurs to the backward wave. If the structure is not a helix, the period size replaces the pitch. Other ways to suppress the BWO is to change the diameter of the helix or the number of support rods. Similarly, these suppression techniques change the phase velocity. Figure 2.86 shows the resulted peak power versus frequency with BWO suppression techniques and without them.

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Figure 2.86: Peak output power versus midband frequency with BWO suppression techniques and without them [100]. 2.6.4.2 Typical Support Techniques to Average Power

Some slow-wave structures must be supported by metallic materials to operate at high average power levels. Such support affects the RF characteristics of the circuit. When the cylinder surrounding the helix is a dielectric, it provides cooling to the system besides the ability to support the structure well. However, the interaction impedance becomes very low due to the low dielectric constant of the ceramic materials. Thus, minimizing the materials of the ceramics can support the structure and provides a reasonable interaction impedance. Figure 2.87 shows a typical use of the support rods with a helix.

Figure 2.87: A typical use of support rods with a helix.

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Figure 2.88 shows the interaction impedance of a helix with and without the use of support rods.

Figure 2.88: Interaction impedance of a helix with and without the use of support rods [1]. There are two common dielectric materials used to support the helix whose thermal conductivities are high: beryllium oxide (BeO) and anisotropic boron nitride (APBN).

Figure 2.89 shows the thermal conductivities of these two materials and other materials.

Figure 2.89: Thermal conductivities of some dielectric and metal materials [1]. The temperature drop in the interfaces between the helix and support rods and between the support rods and barrel produces a problem in the thermal resistance. The temperature

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reduction varies based on the applied pressure of the interfacing. Figure 2.90 shows the temperature drop in the interfaces between the helix and support rods and between the support rods and barrel.

Figure 2.90: Temperature drop between helix and support rods and between support rods and barrel [99]. Figure 2.91 shows the thermal interface conductivities versus contact pressure for some dielectrics interfaced with a helix made of tungsten.

Figure 2.91: Thermal interface conductivities versus contact pressure for some dielectrics interfaced with a helix made of tungsten [1]. To minimize the temperature drops, there are four different techniques used to assemble the support rods for TWT. The first technique is triangulation: For this technique, three

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equally spaced support rods are used and “the metal barrel that is used to surround the helix is slightly distorted into a triangular shape” [1]. The barrel is a thin tube allowing the pole pieces to slide over. The thermal resistance is minimized by maximizing the resulting forces at the interfaces between the helix and support rods and between the support rods and barrel until reaching a point where the forces are high enough, but do not distort the helix; a reason to use a helix made of strong materials like tungsten or molybdenum.

The second technique is pressure or hot insertion. Figure 2.92 shows the pressure or hot insertion technique.

Figure 2.92: Pressure or hot insertion technique [1].

There are a variety of ways to achieve the pressure or hot insertion technique. The barrel can be heated and expanded to allow the helix and support rods to be inserted. Another way is fitting the helix and support rods with a machine and inserting them inside the barrel under high pressure. The tungsten and molybdenum can be used in the pressure or hot insertion technique. The third technique is wire wrapping. Four support rods are used in this technique and they are wrapped with wire tensely. Those rods hold the helix tightly from every side [101]. The fourth technique is brazing. Although this technique is

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challenging to implement, it provides the optimum reduction of the thermal interface resistance among the four techniques. “The support rods must be brazed to every turn of the helix. There may be several tens of contact points of one support rod with a helix and several support rods (usually three)” [1]. To protect the helix, each contact point of the supporting rods is brazed. Besides tungsten and molybdenum, the copper can be used in the brazing technique.

The use of the block-support structure is also suggested to increase the interaction impedance, reduce the dielectric loading, reduce the effect of dielectric on the phase velocity, and eliminate the thermal expansion [102]. Those blocks are made of ceramics.

Figure 2.93 shows a comparison between rod support and block support structures.

Figure 2.93: Comparison between rod support and block support structures [102]. Figure 2.94 shows a comparison of the helix temperature with respect to the input power between the triangulation, pressure or hot insertion, and brazing techniques.

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Figure 2.94: Comparison of the helix temperature with respect to the input power between the triangulation, pressure or hot insertion, and brazing techniques [103]. A dc current was passed by the helix for each technique to obtain the results in Figure 2.94.

The temperature rise for the brazing technique is the lowest as expected.

2.6.5 Attenuators and Severs

One of the existing four waves travel backwards from the collector to the electron gun. The oscillations occur when

(2.279)

, where G is the gain of the tube in − dB, − L is− the loss> of the circuit in dB, o is the reflection coefficient at the output in dB, and i is the reflection coefficient at the input in dB. Figure

2.95 shows the quantities used in the analysis of oscillations.

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Figure 2.95: Quantities used in the analysis of oscillations [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. The attenuators and severs are used commonly in the TWTs. Figure 2.96 shows a lossy film attenuator used with a helix.

Figure 2.96: Lossy filum attenuator used with a helix [1]. As shown in Figure 2.96, the thickness of the film was gradually reduced to zero at the ends to match the impedance for a wideband frequency range. All waves whether forward or backwards are attenuated by this lossy film resulting in a reduction in the gain. To avoid this reduction for the forward wave, the length of the circuit can be extended. However, if the lossy film extends along the circuit where impedance is matched, the velocity spreads out faster allowing the efficiency to decrease. To avoid this, severs are used to suppress the backward wave and obtain a desired efficiency. Figure 2.97 shows the use of two severs with a helix to obtain a better efficiency than the use of an attenuator.

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Figure 2.97: Use of two severs with a helix to suppress the backward wave and obtain a better efficiency than the attenuator [1]. The forward wave is lost when it reaches the sever. However, the electron beam carries the signal across the region with its associated current and velocity. The sever region needs to be short to minimize the degradation of efficiency. In addition, the gain is determined in

Figure 2.97 based on G1, G2, Li, Lo, i, s, and o. For maximum efficiency, G1 should be low, but G2 should be high.

2.6.6 Ring-Bar and Ring-Loop TWT

When high power TWTs are desired, the ring bar, described by [104], is a suggested design for the slow-wave structure. It consists of rings connected by bars or loops and took the form of contrawound helix. Figure 2.98 shows the ring bar and the contrawound helix circuits.

Figure 2.98: Ring bar and contrawound helix circuits [1].

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Such designs are suggested to suppress the backward wave oscillations. They operate at high voltage, high current, and large beam specifications. However, the bandwidth is limited. Figure 2.99 helps understanding how these circuits work with a bifilar helix.

Figure 2.99: Backward wave interactions for a single and bifilar helix [1]. With the bifilar helix, the electron represented by the force F is accelerated and moving to the right and a backward wave is moving to the left. At 180˚ later, the electron is decelerated as shown in the middle right of Figure 2.99. At another 180˚ later, the electron is accelerated as shown in the bottom right of Figure 2.99. This means that acceleration and deceleration cancel each other in one complete cycle. As a result, the ring bar and countrahelix designs can operate with high voltage, high current, and large beam, and without suppressing the backward wave. Figure 2.100 shows the Brillouin diagram for the ring bar structure [105].

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Figure 2.100: Brillouin diagram for the ring bar structure [1]. The phase velocity varies with frequency for the ring bar structure more than the helix does. In other words, the ring bar structure is more dispersive than the helix. The intersection between the beam velocity and forward wave indicates the bandwidth of the circuit being in the range of 10-30%. It is important to have a high voltage such that the backward wave is not excited at fπ.

The length and width of the bar connecting the rings of the ring bar structure controls the phase velocity [106]. Figure 2.101 shows the normalized phase velocity of the ring bar structure in the Ka-band frequency range with 18 kV beam voltage [105].

Figure 2.101: Normalized phase velocity for the ring bar structure in the Ka-band frequency range with 18 kV beam voltage [1].

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Figure 2.102 shows the power and bandwidth of ring bar structure in the X-band frequency range.

Figure 2.102: Power and bandwidth of ring bar structure in the X-band frequency range [1]. In general, the ring bar and ring loop circuits are optimum when high power designs are desired. Raytheon made high power structures, QKW1617 and QKW1818, obtaining an average power of 160 kW and 10 kW in [107].

2.7 Collector

At a specified frequency, the dc input power converts to RF output power. Such conversion is referred to as the overall efficiency playing a role in choosing the microwave tubes for most applications. In addition, the electronic efficiency, e, is one part of the overall efficiency. It is the conversion of the electron beam power to RF power. Also, the circuit efficiency, cir, is the “efficiency of the RF circuit in delivering the generated RF power at the desired frequency to the output connector of the tube” [2].

To study the overall efficiency with the collector, consider the power flow in a linear beam shown in Figure 2.52.

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Figure 2.103: Power flow in a linear beam flow [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. As shown in Figure 2.52, there are four different powers coming in. These are for the electron beam (Po), heater (Ph), solenoid (Psol), and RF input power. However, the RF input power is not shown and neglected because it is commonly at least 30 dB less than the RF output power (Pout). In addition, there are three different powers lost. Such losses occur due to two reasons. The first reason is that the electrons are being intercepted by the existing circuit and RF losses such as the interception loss (Pint) and circuit loss (Pcir). The second reason is that the generated RF power (PRF) is not at the desired frequency such as the harmonics and intermodulation products. The last portion where power exists occurs after the RF power leaves the tube. One power is collected by the collector, which is the spent beam (Psp). Still, another power is converted to heat, known as the heat in collector

(Pheat), once the electrons strike the collector. This power can be recovered by the collector

(Prec) when the collector is designed in a way that enables the electrons to slow down before reaching the collector surface [108]. The total input power of the tube, Pin, is

(2.280)

ℎ The total power dealing with generating = + and focusing+ − the beam, Pot, is

(2.281)

Plugging (2.181) into (2.180) yields = + ℎ +

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(2.282)

The ratio of the total RF output power, Pout=, to the− total input power is the overall efficiency of the tube, ov, defined as

(2.283) = � Plugging (2.282) into (2.283) leads to

(2.284) = − The recovered power by the collector is defined as

(2.285)

, where coll is the efficiency of the collector = and the spent beam power is

(2.286)

Thus, equation (2.285) becomes = − − − −

) (2.287)

The electronic efficiency defined= earlier − can be− expressed − as−

(2.288) � + This efficiency is commonly from a few =percent to over 50%. Also, the circuit efficiency can be expressed as

(2.289) = +� = This efficiency is commonly from 75 to 90%. Combining (2.289), (2.287), and (2.284), the overall efficiency becomes

(2.290) � �+ = − − −

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When the electron beam reaches the collector, the beam expands again due to the space charge effect since the magnetic focusing field is removed. Figure 2.53 shows the collector for the linear beam tube.

Figure 2.104: Collector for a linear beam tube [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. Heat is generated at the collector surface when the electrons strike the collector at a high velocity. This occurs when the body of the tube has the same potential as the collector.

Reducing the potential on the collector to be smaller than the potential of the tube causes the velocity to decrease. This concept is known to be as depressing the collector. Thus, the generated heat at the collector surface is reduced and the recovered power is increased.

Therefore, the overall efficiency increases. Figure 2.54 illustrates the depressed collector circuit for a linear beam tube.

Figure 2.105: Depressed collector circuit to recover the beam power [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

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As shown in Figure 2.54, the power supply configuration for the depressed collector operation has two different power supplies. The collector supply differs from the cathode supply. They have different voltages when the amount of the depressed collector voltage,

Vcoll, is not zero. When the voltage of the collector supply is reduced, the electrons slow down before they are collected. Nevertheless, there is a tradeoff for depressing the collector further and improving the efficiency. When the electrons become slower, they get reflected by the collector. Such traveling back induces noise in the RF circuit and results in a feedback signal path. Also, the body current increases and causes damage to the RF structure. Electric fields are inserted in the collector to prevent the reflected electrons.

Those fields cause the electrons to stop once they reach the collector surface. They are shaped properly to prevent the acceleration of the secondary electrons.

Recently, multistage depressed collectors (MDCs) have been used. They are highly efficient and “use several electrodes at different potentials to selectively collect electrons at low energy levels” [2]. The electrons are collected on an electrode at the appropriate voltage that recovers most of its energy by sorting electrons into various energy classes.

Figure 2.55 shows the power supply configuration for a multistage depressed collector.

Figure 2.106: Power supply configuration for a multistage depressed collector [Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.].

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Choosing the optimum number of stages is based on trading off between the increase in the overall efficiency and complexity of the tube structure and its power supply. Most applications use three or four to be the optimum number of stages. The NASA Lewis

Research Center in Ohio has carried out plenty of research and development for the collectors in a variety of applications including airborne electronic countermeasures systems [109].

A disadvantage of the depressed collectors is the production of secondary electron emission, which causes noise, signal distortion, and heat. In addition, the secondary electron emission decreases the collector efficiency due to the current flow between the collector electrodes and heating of electrodes. There has been plenty of studies about such effect and how to reduce it with the use of some techniques including texturing materials for collector surfaces by bombarding the surface with argon ions or using sputtered molybdenum as a textured-inducing masking film [110-112].

2.8 Transmission Line Fundamentals

Transmission lines are structures with different shapes and sizes guiding the electromagnetic waves from one end to another. It consists of an “insulating layer of dielectric material sandwiched between two layers of metal” [113]. The printed circuit boards and multichip modules have manufactured transmission lines. An example of a transmission line is a coaxial cable and the most common material used is copper.

When applying a voltage, the electric field is resulted. The magnetic field can be obtained using Faraday’s and Ampère’s laws knowing the electric and magnetic fields are orthogonal. These laws are written as

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(Faraday’s Law) (2.291) ⃗ ∇ × ⃗ = − (Ampère’s Law) (2.292) ⃗⃗ ⃗⃗⃗ Taking the curl of (2.291) yields∇ × to = +

(2.293) ⃗ ∇ × (∇ × ⃗ ) = −∇ × Plugging (2.292) into (2.293) by knowing that ,

⃗ = ⃗⃗ (2.294) ⃗ (∇ × ⃗⃗ ) ⃗⃗ ∇ × (∇ × ⃗ ) = −∇ × = − = − + Knowing that yields to rewrite (2.294) as

⃗⃗ = ⃗ (2.295) ⃗ ⃗ Assuming that the wave propagates∇ × (∇ × freely ) = in− the region indicating that Gauss’s law becomes , equation (2.295) becomes

∇ . ⃗ = (2.296) ⃗ ⃗ , where and . Equation∇ (2.296)− is= known as the wave equation for the electric field = freely. Similarly, = the wave equation for the magnetic field is

(2.297) ⃗⃗ ⃗⃗ The electric and magnetic fields in (2.296-2.297)∇ − have= components in four dimensions: x, y, z, and time. For simplicity, the four dimensions are reduced to

(2.298)

In (2.298), the electric field varies⃗ , only , , with = z . Thus,, the curl of E becomes

(2.299) + + = + + =

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A change in the electric field produces a magnetic field and a change in the magnetic field produces an electric field. Such changes play a role in the propagation of the electromagnetic wave. Grouping the nonzero components in (2.299) together yields to

(2.300) = − (2.301) Waves traveling in this way are indicated to be= −in the transverse electromagnetic mode

(TEM).

The applied voltage to generate the electric field between a signal conductor and transmission line can be calculated using the line integral as

(2.302) , where a and b are the points on the conductor = − ∫ ⃗ and. transmission line, respectively. This voltage can be represented as the work done per unit charge written as

(2.303) [ ] = − = − ∫ ⃗ . The electric field between the conducting→ material and dielectric is found by calculating the boundary conditions using the integral form of Gauss’s law as

(2.304)

, where dV is the volume. To study∮ ⃗ the. properties= ∫ of= the electric and magnetic fields of the transmission lines, telegrapher’s equations are used. These equations are obtained by calculating the equivalent circuit parameters of the transmission line. The partial derivative of (2.302) is equal to the partial derivative of the left-hand part of (2.300), which can be written as

(2.305) ∫ ⃗ . , → = 130

In addition, the inductance is related to the right-hand side of (2.300). The self-inductance is expressed as

(2.306) = , where 1 is the magnetic flux written as

(2.307)

Thus, the circuit parameters are obtained = using∫ ⃗ .the right-hand side of (2.300) as

(2.308) ∫ ⃗ . , where L is the inductance per unit length → of the transmission= = line. The overall circuit parameters of (2.300) is

(2.309) , , , known as one of the telegrapher’s equations= −of a lossless transmission line. The same manner is applied for (2.301) to find the second telegrapher’s equation. From (2.304), the left-hand side of (2.301) is expressed in terms of the charge as

(2.310) ⃗ The voltage in (2.302) is calculated∮ .as = ∫ → =

(2.311) ⃗ , where d is the distance between = the∫ conductor . = and =transmission line and A is the area of the conductor. Substituting Q = CV in (2.310-2.311) yields

(2.312) ⃗ , where C is the capacitance = per =unit length.= ∫ Thus, . the left-hand= =side of (2.301) in terms of the capacitance becomes

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(2.313) Since the current I = dQ/dt = C dv/dt, the overall→ circuit parameters of (2.301) is

(2.314) , , , known as the other telegrapher’s equation= of− a lossless transmission line. Equations

(2.309) and (2.314) “describe the electrical characteristics of a transmission line” [113].

The properties of the transmission line are previously derived in (2.213-2.215).

Such characteristics are revisited in Chapter 4 with an edited perspective.

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CHAPTER III

ELECTRON GUN AND FOCUSING STRUCTURE DESIGNS

This chapter investigates an approach to the design of a high voltage, low- perveance electron gun and its magnet system for traveling-wave tube (TWT) device. The electron gun is driven by a 262 kV, 12 A electron beam whose minimum should be 2.0 mm. At earlier stage, different emission densities are studied ending up with involving the periodic permanent magnet (PPM) in the electron gun design whose cathode emission density is 5.968 . The electron trajectory calculation and Poisson’s equation are iterated A c to find a solution. The electron gun and magnet parameters’ calculations, plots’ analysis, electron gun design using EGUN code, and magnet stack using ANSYS Maxwell are presented.

3.1 Overview

TWT is a high-power vacuum device used in a variety of applications including radar systems, broadcasting, and satellite communications. It consists of an electron gun,

RF input/output circuits, slow-wave circuit, attenuator, collector, and magnet system. The basic components of the electron gun are cathode, anode, and focusing electrode. The slow- wave structure extracts the energy from the electrons. When applying a high voltage in the cathode, electrons with a high dc energy will be emitted. Such electrons transfer their energy to the RF circuits resulting in an amplified signal.

To design the electron gun, EGUN code, written by William. B. Hermannsfeldt

[114-116] can be used to simulate the “dynamics of charged particle beams in electron guns” [117]. It enables the user to solve a 2D Poisson’s equation with boundary conditions.

The finite solver considers the magnetic field and space charge of the beam to produce an electron flow by giving the gun geometry and applied voltage. The electric and magnetic fields influence the design to calculate the electron trajectories. With a given beam voltage

(Vb) and current (I), cathode emission density (J), and minimum beam radium (bm), the gun geometry was established by running the program electrostatically after calculating the desired parameters including, but not limited to the cathode disc and spherical radii, rc and

Rc, respectively. Some of these parameters may be varied until achieving the design specifications [118].

A periodic permanent magnet structure was also designed using ANSYS Maxwell

[119-121] to focus the electron beam of the electron gun as it travels through the axis of the slow-wave structure. ANSYS Maxwell uses the finite element analysis, FEA, to solve electric and magnetostatic problems by applying Maxwell’s equations over a finite region of space with given boundaries and appropriate materials. The PPM stack was established after calculating Brillouin field level, focusing structure inner diameter, peak field, and beam tunnel length and was assembled as a magnet, pole pieces, and hubs [122].

A variety of electron gun simulations were designed using EGUN code with different specifications. The targeted design was then created with the magnet system. The organization of this chapter is as follows: Section 3.2 provides the design specifications.

Calculations of the electron gun and magnet system parameters are performed in Section

3.3 involving some iteration and plots’ analysis. Then, Section 3.4, 3.η, and 3.θ describe the simulations and magnet system details for different designs. A discussion in Section

3.7 summarizes the overall results for each design.

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3.2 Design Specifications

Two different designs are mainly considered at the early stage. Tables 3.1 and 3.2 summarize the specifications of the two designs.

3.2.1 First Electron Gun Design

The first design was derived from [2]. It was considered to get familiar with calculating the electron gun parameters and learn simulating with EGUN code. Table 3.1 summarizes the design specifications of first electron gun.

Table 3.1: Specifications of the first electron gun design derived from [2] with a beam voltage of 10 kV, beam current of 1 A, minimum beam radius of 1 mm, and cathode emission density of 2 A/cm2.

Beam Voltage 10 kV Beam Current 1 A Minimum Beam Radius, bm 1 mm Cathode Emission Density, J 2 A The periodic permanent magnet was not involved in the first cdesign.

3.2.2 Electron Gun Design of the Proposed Novel Slow-Wave Structure

In the second design, the electron gun of the proposed novel slow-wave structure was designed whose specifications are summarized in Table 3.2.

Table 3.2: Specifications of electron gun design of the proposed novel slow-wave structure of the TWT with a beam voltage of 262 kV, beam current of 12 A, minimum beam radius of 2 mm, and cathode emission density of 5.968 A/cm2.

Beam Voltage 262 kV Beam Current 12 A Minimum Beam Radius, bm 2 mm Cathode Emission Density, J 5.968 A c 3.3 Calculations

Calculations were managed for all suggested designs based on the given design specifications and basic methods and principles of Pierce electron gun design described by

Vaughan [68]. The physical dimensions of the electron gun including the boundaries were

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estimated for each design to properly simulate and obtain the electron flow [1-2, 68-71].

This includes electron gun parameters, PPM parameters, iterations, and plots’ analysis.

3.3.1 Electron Gun Parameters

The electron gun design starts by defining the desired parameters needed. The quantities calculated in the analysis of effect of anode aperture is shown in Figure 3.1.

Figure 3.1: Quantities used in the analysis of effect of anode aperture to calculate the gun parameters [[Reproduced by permission from Author A. S. Gilmour, Jr., Principles of Traveling Wave Tubes, Norwood, MA: Artech House, Inc., 1994. © 1994 by Artech House, Inc.]. The electrode configuration for parallel electron flow beam is solved inside and outside using Laplace’s and Poisson’s equations in (2.1) and (2.2), respectively. To focus the electrode with respect to the beam edge, the Pierce angle is suggested located near the edge of the cathode as in (2.94).

For a given beam voltage (Vb), beam current (I), minimum beam radius (bm), and cathode emission density (J), the disc radius of the cathode, rc, can be calculated as

(3.1) = √ = √ Next, the angle, , is assumed initially to calculate the slope of trajectory, tan ( ). This angle is adjusted later such that the two trajectories become nearly the same. By doingϕ so,

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the electron beam will leave the lens region to the drift region with an obtained proper beam radius. By assuming an initial angle value, the constant can be found as

(3.2) − .× −c − = √ , where the constant is related to the other constant as

(3.3) Then,− can= be + determined . + . by organizing + . (3.3) as + . + .

(3.4) After that, the corresponding = − − slope . of− the trajectory+ . −is estimated − . for− Region 2, , at the edge of the beam as �

, where , and (3.5) tan � = = [ − − ] � = . (3.6) = + . + . + . + . + . Also, the corresponding slope of the trajectory for Region 3, , is estimated at the edge of the beam as �

, where and (3.7) − tan � = √ln = = = √ Afterward, the value of is revised such that becomes close to . This can be represented as � �

(3.8) a � = √a � Iterations, discussed in Section 3.3.3, are performed until determining the final angle value,

(final). Finally, the remaining gun parameters of the design are calculated as

(3.9) i = (3.10) − = 137

(3.11)

= . (3.12) = − √ − = + [. [ − ] + . [ − ] + . [ − ] − . [ (3.− ]13) , where zm is the axial position of the beam minimum. Matlab scripts in the Appendices were created to calculate the electron gun parameters stated in (3.1-3.13) for each design.

3.3.1.1 First Electron Gun Design

A Matlab script in Appendix A was created to calculate the electron gun parameters for the first design whose specifications are stated in Table 3.1. By running the script, the parameters in (3.1-3.13) were computed. Table 3.3 summarizes the obtained results of the first electron gun design.

Table 3.3: Calculated electron gun parameters for the first design with a beam voltage of 10 kV, beam current of 1 A, minimum beam radius of 1 mm, and cathode emission density of 2 A/cm2.

Parameter Cathode Emission Density, Disc Radius of Cathode, 0.4 [ ] Initial Theta, 30 � = Initial [] 1.4019 Initial � [°] 1.0036 − Initial 1.8998 �� Correction Factor, 1.25 � Initial 0.2177 Initial � 0.1462 Initial �� 0.1073 � New Theta, [ ] 21.0594 Final Theta,�� 22.58 Final � [°] 1.0604 Final � [°] 0.8152 − Final 1.6752 �� Final 0.1312 � Final 0.1462 Final �� 0.1312 � [] �� 138

1.039 0.4598 � [] 0.2119 � [] 0.6309 [] 1.6982 [] 3.3.1.2 Electron Gun [Design] of the Proposed Novel Slow-Wave Structure

Another Matlab script was created to calculate the electron gun parameters for the electron gun design of the proposed novel slow-wave structure of the TWT whose specifications are stated in Table 3.2. This script is included in Appendix B. By running the script, the parameters in (3.1-3.13) were computed. Table 3.4 summarizes the obtained electron gun parameters of the proposed novel slow-wave structure of the TWT.

Table 3.4: Calculated electron gun parameters of the proposed novel slow-wave structure of the TWT with a beam voltage of 262 kV, beam current of 12 A, minimum beam radius of 2 mm, and cathode emission density of 5.968 A/cm2.

Parameter Cathode Emission Density, Disc Radius of Cathode, 0.8 [ ] Initial Theta, 10 � = . Initial [] 1.5782 Initial � [°] 1.0919 − Initial 2.0166 �� Correction Factor, 1.25 � Initial 0.0812 Initial � 0.2685 Initial �� 0.0282 � New Theta, [ ] 5.8977 Final Theta,�� 6.675 Final � [°] 1.0542 Final � [°] 0.8115 − Final 1.6711 �� Final 0.0395 � Final 0.3554 Final �� 0.0395 � [ ] 6.8826 �� 3.0573 � [] 0.4265 � [] 3.8552 [] [] 139

11.0490 The cathode emission density was chosen such that it provides the appropriate compression [] flow and used magnetic field focusing. This factor is important in determining the operating life, which depends on the type of emitter used such as oxide coated and operating conditions such as pulsed or CW. Figure 5.2 Shows the disc radius of cathode versus cathode emission density for a beam current of 12 A.

Figure 3.2: Disc radius of cathode versus cathode emission density for a beam current of 12 A. 3.3.2 Periodic Permanent Magnet Parameters

The Brillouin field level, BB, is the required magnetic field to focus the electron beam. This is achieved by creating a periodic permanent magnet stack structure. Figure 3.3 illustrates a sectional view of magnet stack consisting of two magnets and iron pole pieces.

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Figure 3.3: Sectional view of magnet stack consisting of two magnets and iron pole pieces. The design parameters of the PPM stack are calculated for the targeted design whose current density is 5.968 . A c 3.3.2.1 Electron Gun Design of the Proposed Novel Slow-Wave Structure

To achieve adequate beam stability, the beam stiffness factor, , is chosen such that

(3.14) , where and L are the plasma wavelength > . and period, respectively. The plasma

wavelength is calculated as

(3.15) . . = (�) , where P is the normal beam microperveance calculated to be 0.08948 . Next, the Brillouin field level, , is estimated to be

(3.16) . �. √ Due to thermal velocity and grid scattering, = the practical level of the magnetic field is higher than the Brillouin level. Thus, the increased focusing field level is chosen to be the necessary field level to focus the electron, which can be expressed as

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, where (3.17) . � + = [ ( + � ) ] F = 1.65 and (3.18) . � = From (3.17), the maximum B-field in air, Bp, becomes

(3.19) + Also, the magnetic field intensification ratio= is√

(3.20) + Table 3.5 summarizes the calculated results of the magnet stack used in the electron gun design of the proposed novel slow-wave structure of the traveling wave tube.

Table 3.5: Calculated magnet stack parameters used in the electron gun design of the proposed novel slow-wave structure of the traveling wave tube.

Parameter Value Plasma Wavelength, 24.14 cm Period, L 3.09 cm Plasma Wavelength / Period, 7.812 Brillouin Field, 0.0634 T Increased Focusing Field, 0.0815 T Maximum B-field in Air, + 0.1153 T 1.285 + 3.3.3 Iterations

The basic principle of the iteration used in calculating the electron gun parameters is to revise the value of until achieving = . It can be done by obtaining a new angle value based on the old one as expressed � in �(3.8). The following diagram in

Figure 3.4 describes the procedure.

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Figure 3.4: A diagram describing the procedure to iterate the angle values until achieving the appropriate electron gun parameters’ calculations. This iteration method was done for each design to come up with the appropriate electron gun parameters.

3.3.3.1 First Electron Gun Design

For this design, two initial angles were chosen. The second initial value led to obtain the final within three attempts. Tables 3.6 and 3.7 summarize the iteration procedure for the first electron gun design.

Table 3.6: Initial iteration for the first electron gun design.

Cathode Emission Density, Step Parameter

Initial 0.1462 Initial �0.2177 = � Initial [] 0.1073 New Theta,�� 21.0594 (1) Initial Theta, Final Based�� on New 0.1129 Theta � [°] Final�� Based on 0.1842 � = ° New Theta Final � [ Based] on New 0.1360 Theta ��Initial � 0.1899

� [] 143

Initial 0.1001 Initial 0.1393 New Theta,�� 23.5961 Final Based�� on New (2) Initial Theta, 0.1434 Theta � [°]

Final�� Based on 0.1718 � = ° New Theta Final � [ Based] on New 0.1280 Theta ��

Table 3.7: Final iteration for the first electron gun design.

Cathode Emission Step Parameter Density, Final 0.1434 � = Final Theta, Final 0.1718 �� Initial Theta, � = Final � 0.1280 . ° [] Initial � = Final ° 0.1278 �� Initial � = Final Theta, Final . 0.1780 �� Initial �� = � = Final . � 0.1321 . ° [] New Theta,�� = Final . 0.1312 �� = Final Theta, Final . ° 0.1766 �� = Final � 0.1312 . ° []

3.3.3.2 Electron Gun Design of the Proposed Novel�� Slow-Wave � Structure

For this design, two initial angles were chosen. The second initial value led to obtain the final within three attempts. Tables 3.8 and 3.9 summarize the iteration procedure for the electron gun design of the proposed novel slow-wave structure of the traveling wave tube.

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Table 3.8: Initial iteration for the electron gun design of the proposed novel slow-wave structure of the traveling wave tube.

Cathode Emission Density, Step Parameter

� = Initial 0.2685 Initial .0.0812 � Initial [] 0.0282 New Theta,�� 5.8977 (1) Initial Theta, Final �� Based � on 0.0296 New Theta� [°] Final �� Based on 0.3829 � = ° New Theta Final � [] Based on 0.0419 New Theta Initial�� 0.3791 Initial 0.0309 � Initial [] 0.0416 New Theta,�� 6.9599 (2) Initial Theta, Final �� Based � on 0.0431 New Theta� [°] Final �� Based on 0.3461 � = ° New Theta Final � [] Based on 0.0385 New Theta �� Table 3.9: Final iteration for the electron gun design of the proposed novel slow-wave structure of the traveling wave tube.

Cathode Emission Step Parameter Density, Final 0.0431 � = . Initial Theta, Final Theta, Final 0.3461 �� Initial � = � = Final ° � 0.0385 . ° [] Initial � = Final . 0.0383 �� Initial �� = Final Theta, Final . 0.3584 �� New Theta,�� = � = Final . � 0.0398 . ° [] � = Final . ° 0.0395 ��

�� 145

Final Theta, Final 0.3554

� = Final � 0.0395 . ° []

�� 3.3.4 Parameter Analysis

There are various means of studying the electron gun design. A further study can be implemented to improve understanding the calculations and analyzing the equations. A

Matlab script in Appendix C was created to plot 30 different curves based on (3.1-3.13).

Figures 3.5-3.7 show the relationship between the disc radius of cathode, cathode emission density, and current derived from (3.1).

Figure 3.5: Disc radius of cathode versus cathode emission density relationship from equation (3.1) with a beam current of 50 mA.

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Figure 3.6: Disc radius of cathode versus cathode emission density relationship from equation (3.1) with beam currents of 50 mA in red and 1 A in blue.

Figure 3.7: Disc radius of cathode versus beam current relationship from equation (3.1) with cathode emission densities of 2 A/cm2 in red, 10 A/cm2 in blue, 50 A/cm2 in green, and 100 A/cm2 in cyan. Figures 3.8-3.12 show the relationship between theta, alpha, beam voltage, beam current, and perveance derived from (2.101), (2.84), and (3.2).

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Figure 3.8: Theta versus alpha from equation (2.101) with a beam voltage of 18.2 kV and current of 50 mA.

Figure 3.9: Theta versus alpha from equation (2.101) with a beam voltage of 18.2 kV and current of 50 mA in red, and beam voltage of 10 kV and current of 1 A in blue.

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Figure 3.10: Perveance versus alpha from equations (2.101), (2.84), and (3.2) with different theta values of 30 degrees in red, 20 degrees in blue, 10 degrees in green and 5 degrees in cyan with beam voltage of 18.2 kV and current of 50 mA.

Figure 3.11: Beam voltage versus alpha from equation (2.101) with different theta values of 30 degrees in red, 20 degrees in blue, 10 degrees in green and 5 degrees in cyan with a beam current of 50 mA.

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Figure 3.12: Beam current versus alpha from equation (2.101) with different theta values of 30 degrees in red, 20 degrees in blue, 10 degrees in green and 5 degrees in cyan with a beam voltage of 18.2 kV. Figure 3.13 shows the relationship between gamma and alpha constants derived from (3.3-

3.4).

Figure 3.13: Gamma versus alpha constants from equations (3.3-3.4).

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Figure 3.14 shows the relationship between gamma and its derivative constants derived from (3.6).

Figure 3.14: Gamma versus its derivative constants from equation (3.6). Figures 3.15-3.17 show the relationship between the slope of trajectory for Region 2, alpha,

Ra, and correction factor derived from (3.5).

Figure 3.15: Slope of trajectory for Region 2 versus alpha from equation (3.5) with different values of theta and gamma derivative.

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Figure 3.16: Slope of trajectory for Region 2 versus Ra from equation (3.5) with different values of bo, alpha, and gamma derivative.

Figure 3.17: Slope of trajectory for Region 2 versus Ra from equation (3.5) with different values of correction factor, bo, alpha, and gamma derivative. 152

Figures 3.18-3.22 show the relationship between the slope of trajectory for Region 3, bo, disc radius of cathode, gamma, minimum beam diameter, and perveance derived from

(3.7).

Figure 3.18: bo versus disc radius of cathode from equation (3.7) with different values of gamma.

Figure 3.19: bo versus gamma from equation (3.7) with different values of disc radius of cathode.

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Figure 3.20: Slope of trajectory for Region 3 versus minimum beam diameter from equation (3.7) with different values of bo.

Figure 3.21: Slope of trajectory for Region 3 versus bo from equation (3.7) with a beam voltage of 18.2 kV, beam current of 50 mA, and minimum beam diameter of 0.0375 mm.

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Figure 3.22: Slope of trajectory for Region 3 versus perveance from equation (3.7) with different values of bo. Figures 3.23-3.24 show the relationship between spherical and disc radius of cathode radii, and theta derived from (3.9).

Figure 3.23: Spherical radius versus disc radius of cathode from equation (3.9) with different values of theta.

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Figure 3.24: Spherical radius versus theta from equation (3.9) with different values of disc radius of cathode.

Figures 3.25-3.26 show the relationship between spherical radius, Ra, and gamma derived from (3.10).

Figure 3.25: Ra versus spherical radius from equation (3.10) with different values of gamma.

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Figure 3.26: Ra versus gamma from equation (3.10) with different values of spherical radius.

Figure 3.27 shows the relationship between ra and bo derived from (3.11).

Figure 3.27: ra versus bo from equation (3.11).

Figures 3.28-3.30 show the relationship between za, ra, Ra, and spherical radius derived from (3.12).

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Figure 3.28: za versus ra from equation (3.12) with different values of spherical radius and Ra.

Figure 3.29: za versus Ra from equation (3.12) with different values of spherical radius and ra.

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Figure 3.30: za versus spherical radius from equation (3.12) with different values of Ra and ra.

Figures 3.31-3.34 show the relationship between zm, minimum beam diameter, perveance, bo, and za derived from equations (3.7) and (3.13).

Figure 3.31: zm versus minimum beam diameter from equation (3.13) with different values of za and bo.

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Figure 3.32: zm versus perveance from equations (3.7) and (3.13) with different values of za and bo.

Figure 3.33: zm versus bo from equation (3.13) with different values of za.

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Figure 3.34: zm versus za from equation (3.13) with different values of bo.

3.4 Electron Gun Simulations and Designs

To solve Poisson’s and Laplace’s equations by matrix methods, trace the trajectories, and deposit the space charge, computer codes have been written to cover all essential aspects in the electron gun design. Regardless, the basic principles to apply the gun codes are the same in all codes which follow the flow chart illustrated in Figure 3.35.

Figure 3.35: Diagram representing the overall method used in the gun codes [123].

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By entering the input data, the meshes are set by the code. Laplace’s equation is initially solved with no space charge existing initially.

The designs have been implemented using EGUN code, which inverts the matrix of the potential array. The program POLYGON [124] sets up the boundaries, provides the valid input files of EGN2W, converts the polygonal input to the appropriate mesh oriented input, and reads the input parameter lists in NAMELIST format. The simulation tool is the

EGN2W. Four input file parts exist in the file. INPUT 1 includes the definition of the basic parameters. INPUT 3 has the axial values for the magnetic field. Column 2 includes the input of boundary points existing in the POLYGON syntax, which is described by three numbers: an integer number representing the potential, and two floating point numbers representing the values of r and z coordinates. Adding 1000 to the potential number indicates inserting a circle to the POLYGON. The run time variables are defined in

INPUT5. Neumann boundary points are inserted automatically by POLYOGN. By running the POLYGON file, an input file is generated for EGN2W resulting in obtaining a plot of trajectories and equipotential lines [51].

3.4.1 First Electron Gun Design

Three different designs have been implemented in the EGUN code for the first electron gun design whose specifications are stated in Table 3.1. The general shape of the boundaries look the same. Only minor differences exist. Appendix D includes the boundary and input files resulted from the three designs. Figure 3.36 shows the resulted electron flow of the first design for the first electron gun with a beam voltage of 10 kV, beam current of

1 A, and cathode emission density of 2 . A c

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Figure 3.36: Electron gun trajectory of the first design for the first electron gun with a beam voltage of 10 kV, beam current of 1 A, and cathode emission density of 2 A/cm2. A zoomed in area to further study the plot of Figure 3.36 is shown in Figure 3.37.

Figure 3.37: A zoomed in plot of the electron gun trajectory of the first design for the first electron gun with a beam voltage of 10 kV, beam current of 1 A, and cathode emission density of 2 A/cm2. Figure 3.38 shows the resulted electron flow of the second design for the first electron gun with a beam voltage of 10 kV, beam current of 1 A, and cathode emission density of 2 . A c

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Figure 3.38: Electron gun trajectory of the second design for the first electron gun with a beam voltage of 10 kV, beam current of 1 A, and cathode emission density of 2 A/cm2. A zoomed in area to further study the plot of Figure 3.38 is shown in Figure 3.39.

Figure 3.39: A zoomed in plot of the electron gun trajectory of the second design for the first electron gun with a beam voltage of 10 kV, beam current of 1 A, and cathode emission density of 2 A/cm2. Figure 3.40 shows the resulted electron flow of the third design for the first electron gun with a beam voltage of 10 kV, beam current of 1 A, and cathode emission density of 2 . A c 164

Figure 3.40: Electron gun trajectory of the third design for the first electron gun with a beam voltage of 10 kV, beam current of 1 A, and cathode emission density of 2 A/cm2. A zoomed in area to further study the plot of Figure 3.40 is shown in Figure 3.41.

Figure 3.41: A zoomed in plot of the electron gun trajectory of the third design for the first electron gun with a beam voltage of 10 kV, beam current of 1 A, and cathode emission density of 2 A/cm2. 3.4.2 Electron Gun Design of the Proposed Novel Slow-Wave Structure

One design has been implemented in the EGUN code for the electron gun design of the proposed novel slow-wave structure whose specifications are stated in Table 3.2.

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Appendix E includes the boundary and input file resulted from the design. Figure 3.42 shows the resulted electron flow of the electron gun design for proposed the novel slow- wave structure of the traveling wave tube with a beam voltage of 262 kV, beam current of

12 A, and cathode emission density of 5.968 . A c

Figure 3.42: Electron gun trajectory for the proposed novel slow-wave structure of the traveling wave tube with a beam voltage of 262 kV, beam current of 12 A, and cathode emission density of 5.968 A/cm2.

3.5 Periodic Permanent Magnet Simulations and Designs

A popular way of focusing an electron beam through the drift tube of a beam tube is to use PPM. It does not consume any power and easier and lighter than a uniform field permanent magnet. It uses the increased focusing field value as the strength of magnetic field focusing at the beam envelope [1-2, 69]. The difference between the permanent and uniform magnets’ systems is shown in Figure 3.43.

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Figure 3.43: Uniform and permanent periodic magnets with respect to the magnetic field entrance in the placement of the beam waist [95]. This difference indicates the beam waist placement with respect to the magnetic field entrance. Figure 3.44 shows s single period periodic permanent magnet focusing structure.

Figure 3.44: A single period periodic permanent magnet focusing structure [69]. The PPM structure was involved in only the electron gun of the proposed slow-wave structure with a current density of 5.968 , which is the targeted design. ANSYS A Maxwell was the software tool used to perform c such design.

3.5.1 Magnet of Electron Gun Design of the Proposed Novel Slow-Wave Structure

Figure 3.45 shows the cross-sectional view of the designed one cell magnet structure consisting of a magnet block, pole pieces, and hubs.

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Figure 3.45: One cell magnet structure consisting of a magnet block, pole pieces, and hubs using ANSYS Maxwell. The magnet block is made of Samarium Cobalt, and both pole pieces and hubs are made of iron as stated in Table 3.10.

Table 3.10: Materials used for each geometry in the periodic permanent magnet.

Geometry Material Magnet Samarium Cobalt (SmCo24) Pole Piece Iron Hub Iron

Some parameters were calculated to determine their dimensions, but the others were estimated by trial and error to meet the calculated field levels. Figure 3.46 shows the dimensions considered to design the magnet.

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Figure 3.46: Parameters of the one cell periodic permanent magnet using ANYSYS Maxwell. These dimensions of the one cell periodic permanent magnet in Figure 3.46 are stated in

Table 3.11.

Table 3.11: PPM design parameters dimensions.

Geometry Dimension Magnet Thickness, Mt 0.1215 cm Magnet Length, Ml 0.289 cm Pole Piece Thickness, Pt 0.033 cm Pole Piece Length, Pl 0.346 cm Hub Thickness, Ht 0.02 cm Hub Length, Hl 0.03225 cm

Figure 3.47 illustrates the magnetic field profile along one cell of the magnet stack.

Figure 3.47: Magnetic field profile along one cell of the magnet stack using ANSYS Maxwell. 169

Table 3.12 states the results of the maximum field levels in the iron and air along one cell of the magnet stack.

Table 3.12: Maximum field levels in iron and air along one cell of the magnet stack.

Maximum B-field in Iron 1.7632 T Maximum B-field in Air 1.05 T

The magnetic field profile along one cell of periodic permanent magnet is shown in Figure

3.48.

Figure 3.48: Magnetic field profile along one cell of periodic permanent magnet using ANSYS Maxwell. The one cell magnet was duplicated five more times to have a total of six magnets in the design as a stack, demonstrated in Figure 3.49.

Figure 3.49: Periodic permanent magnet with an array of magnet blocks.

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Figure 3.50 illustrates the magnetic field profile along the periodic permanent magnet stack.

Figure 3.50: Magnetic field profile along the periodic permanent magnet stack using ANSYS Maxwell. Table 3.13 states the results of the maximum field levels in the iron and air along the periodic permanent magnet stack.

Table 3.13: Maximum field levels in iron and air along the periodic permanent magnet stack.

Maximum B-field in Iron 3.5903 T Maximum B-field in Air 1.035 T

The magnetic field profile along the array of periodic permanent magnet is shown in Figure

3.51.

Figure 3.51: Magnetic field profile along the array of periodic permanent magnet using ANSYS Maxwell.

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The dimensions of the periodic permanent magnet have been modified to confine the beam properly. Figure 3.52 shows the final magnetic profile along the array of periodic permanent magnet using ANSYS Maxwell.

Figure 3.52: Final magnetic field profile along the array of periodic permanent using ANSYS Maxwell.

3.6 Electron Gun Design of the Proposed Novel Slow-Wave Structure with Magnet

The data of Figure 3.52 represents the magnetic field profile in Gauss versus the axial distance with respect to 0.4 mm. Those were extracted to be included in the electron gun design for the electron gun design of the proposed novel slow-wave structure with a current density of 5.968 . Particularly, the field levels were converted into Gauss and A are inserted inside the input c file under &INPUT3. Those field levels are located on the axis starting at z = -6 to z = ZLIM + 6 and represent BZA, which is an output of a code that the user supplies. Appendix E includes the boundary and input file of the electron gun design of the proposed novel slow-wave structure with current density of 5.968 including the A magnet inputs. Figure 3.53 shows the resulted electron flow and magnetic c field plot of

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proposed novel slow-wave structure of the traveling wave tube with a beam voltage of 262 kV, beam current of 12 A, and cathode emission density of 5.968 . A c

Figure 3.53: Electron gun trajectory and magnetic field plot for the proposed novel slow-wave structure of the traveling wave tube a beam voltage of 262 kV, beam current of 12 A, and cathode emission density of 5.968 A/cm2. The previous figure can be compared with Figure 3.42 where no magnet was used.

3.7 Discussion

3.7.1 First Electron Gun Design

Table 3.14 summarizes the results of the electron gun trajectory using EGUN code for the first electron gun design with a current density of 2 . These results are extracted A from Figures 3.36-3.41. c

Table 3.14: Results of the electron gun trajectory using EGUN code for the first electron gun design with a current density of 2 A/cm2.

EGUN Code Design Specifications / Parameter Design Design Design Calculations (1) (2) (3) Minimum Beam 1.00 1.16 1.00 1.22 Radius, bm [mm]

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Axial Position of Beam 17.08 20.00 20.00 17.80 Minimum, zm [mm] Current, I [A] 1.00 0.944 0.905 1.00 Micro-perveance, P 1.00 0.944 0.905 1.00 − [× �] As stated in the Table 3.14, the three designs, particularly Design (2), meet the minimum beam radius of 1 mm. Although it is not important to consider the axial position of beam minimum, the calculated value is very close to the resulted value with a difference of 0.72 mm in Design (3). Also, the three designs, particularly Design (3), pretty much meet the beam current of 1 A and perveance of specifications. − .× 3.7.2 Electron Gun Design of the Proposed Novel Slow-Wave Structure with Magnet

Table 3.15 summarizes the results of the electron gun trajectory using EGUN code for the proposed novel slow-wave structure of the traveling wave tube with a beam voltage of 262 kV, beam current of 12 A, and cathode emission density of 5.968 with the A magnet. c

Table 3.15: Results of the electron gun trajectory using EGUN code for the proposed novel slow-wave structure of the traveling wave tube with a beam voltage of 262 kV, beam current of 12 A, and cathode emission density of 5.968 A/cm2 with the magnet.

Design Parameter Electron Gun Design Specifications/Calculations Minimum Beam Radius, 2.0 2.0 bm [mm] Maximum B Field [T] 0.1153 0.9763 Current, I [A] 12.0 12.0 Micro-perveance, 0.08948 0.08948 − [× ] As stated in the Table� 3.15, the design meets the exact minimum beam radius of 2.0 mm.

The calculated maximum B field is 0.1153 T, while the design has a maximum B field of

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0.9763 T with a difference of 0.861 T. This is due to the resulted simulations of the periodic permanent magnet in ANSYS Maxwell. However, the provided magnetic focusing field confines the beam properly and focuses the beam. Also, the design meets the beam current of 12 A and perveance of . − .× To summarize, an electron gun and periodic permanent magnets TWT was described and designed using EGUN code and ANSYS Maxwell. The specifications of the electron gun design included a beam voltage of 262 kV, beam current of 12 A, current density of 5.968 , and 2.0 mm minimum beam radius. Such study went successfully by A meeting the requirements. c This design can be used for the proposed novel slow-wave structure of the design. The mathematical details were presented to calculate the electron gun parameters and determine the field levels needed to focus the electron flow. Iterations were involved in such calculations until reaching the correct values. The calculations were checked by creating Matlab scripts to generate curves and relate the parameters to each other. Then, the EGUN code was utilized to plot the electron trajectory of each design by defining the calculated parameters and coding the boundaries to generate input and output files with POLYGON.

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CHAPTER IV

A NOVEL SLOW-WAVE CIRCUIT STRUCTURE WITH COLD-TEST SIMULATIONS

Although the coupling mechanisms of the transmission lines cause an unintended and undesired crosstalk in some systems, the same coupled models can impact the system positively, especially in traveling wave tubes. Such advantage is obtained when guidelines of the coupled transmission lines are studied, investigated, and controlled. This chapter describes a 3D modeling, analysis, and design of a high power slow-wave circuit structure using ANSYS high frequency structure simulator (HFSS) [35-36], whose Eigen solver provides the dispersion characteristics and interaction impedance. The dimensions of the structure are varied to enhance the performance of the TWT. Cold-test simulations are carried out to analyze the operating principle of the design besides comparing the characteristics of the design to the conventional ring-bar structures. Such comparison is covered after performing the hot-test simulations in Chapter 5.

4.1 Mutual Inductance and Capacitance

When an electromagnetic wave propagates between two neighboring transmission lines, the electric and magnetic field may interact with each other and induce a coupling energy between the two transmission lines. Such energy is called crosstalk, which is caused by the mutual inductance and mutual capacitance. When crosstalk occurs in printed circuit boards, packages, and connectors, the performance of the system is impacted in two ways.

First is the change in propagation characteristics, which correspond to the propagation velocity and characteristic impedance due to the signal integrity and timing factors. Second is the coupled noise on the transmission lines which adversely affect the signal integrity.

4.1.1 Mutual Inductance

The mutual inductance is caused by the coupled energy between the transmission lines via the magnetic field. It induces a current from one transmission line to the other, which results in a coupled voltage on the other line. This coupled voltage is expressed as

(4.1) ∆ = , where L is the coupled voltage, Lm is the mutual inductance, and i is the transient current. Figure� 4.1 shows a simple coupled inductor circuit.

Figure 4.1: A simple coupled inductor circuit.

Using Faraday’s law, the voltages v1 and v2 are written as

(4.2) = + (4.3) = + , where L is the self-inductance, and Lm is the mutual inductance between both lines. In matrix form, equations (4.2-4.3) can be written as

(4.4) ⁄ [ ] = [ ][ ] When the currents i1 and i2 are equal and in the same direction,⁄ equations (4.2-4.3) become

(4.5) = = +

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Equation (4.5) is referred to a system being in the even mode. It indicates an increasing inductance by Lm. When the currents i1 and i2 are equal, but opposite in direction, equations

(4.2-4.3) become

(4.6) Equation (4.6) is referred to a system = − being= in −the odd mode. It indicates a decreasing inductance by Lm. Such relationships yield

(4.7)

For n inductive lines, the matrix form in (4.4) becomes> >

⁄ (4.8) [ ] = [ ⋱ ][ ] , where Lii and Lij are the self and mutual inductances, ⁄ respectively. The mutual inductances, Lij and Lji, are equal since the inductance matrix is symmetric or a direction independent. Figure 4.2 shows an example of a circuit with three coupled inductors.

Figure 4.2: A circuit with three coupled inductors. For the three coupled inductors in Figure 4.2, equation (4.8) becomes

(4.9) ⁄ [ ] = [ ][⁄] ⁄

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4.1.2 Mutual Capacitance

The mutual capacitance is caused by the coupled energy between the transmission lines via the electric field. When the electric field on the transmission line intersects with the other transmission line, a current is induced on the other line. This coupled current through the mutual capacitance is expressed as

(4.10) ∆ = , where C is the coupled current, Cm is the mutual capacitance, and v is the voltage signal.

Figure 4.3i shows a simple coupled capacitor circuit.

Figure 4.3: A simple coupled capacitor circuit.

The currents i1 and i2 are written as

(4.11) = + − = + − (4.12) = + − = + − , where C is the self-capacitance, and Cm is the mutual capacitance between both lines. In matrix form, equations (4.11-4.12) can be written as

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(4.13) + − ⁄ [ ] = [ ][ ] When the voltages v1 and v2 are equal,− equations + (4.10-4.11)⁄ become

(4.14) Equation (4.14) is referred to a system being= in= the even mode. It indicates a decreasing capacitance by Cm. When the voltages v1 and v2 are equal, but opposite, equations (4.11-

4.12) become

(4.15) Equation (4.15) is referred to a system= − being= in + the odd mode. It indicates an increasing capacitance by Cm. Such relationships yield

(4.16)

, where Ctotal = C + Cm. For n capacitive > lines, >the matrix form in (4.13) becomes

(4.17) ⁄ − [ ] = [ ⋱ ][ ] − , where Lii and Lij are the total and mutual capacitances, ⁄ respectively. The capacitance matrix is negative everywhere except at the total capacitances, Cii. The mutual capacitances, Cij and Cji, are equal since the capacitance matrix is symmetric or a direction independent. Figure 4.4 shows an example of a circuit with three coupled capacitors.

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Figure 4.4: A circuit with three coupled capacitors. For the three coupled inductors in Figure 4.4, equation (4.17) becomes

(4.18) − − ⁄ [] = [− −][⁄] The available two-dimensional − (2D)− full-wave solver ⁄ computes the inductance and capacitance matrices by using Laplace’s equation. In addition, the available three- dimensional (3D) full-wave solver computes the inductance and capacitance matrices by using Maxwell’s equations.

4.2 Coupled Wave Equations

To derive the wave equations for a lossless transmission line, Kirchhoff’s voltage law (KVL) and Kirchhoff’s current law (KCL) are applied. Figure 4.5 shows a differential lossless transmission line.

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Figure 4.5: A lossless transmission line. The voltage drop across the inductor is

(4.19)

The change in the current is − = − +

(4.20)

As stated in (4.19-4.20), the left-hand− sides, which= include + either− the inductor or capacitor, are frequency dependent based on the angular frequency. Dividing (4.19-4.20) by dz yields to

(4.21) = − (4.22) Differentiating (4.21-4.22) with respect to= z −gives

(4.23) = − (4.24) Plugging (4.21) into (4.24) and plugging (4.22)= − into (4.23) yields

(4.25) + = (4.26) , known as the voltage and current wave equations+ for = a uniform lossless transmission line.

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The same equations are used for n-coupled and symmetrical transmission lines.

Figure 4.6 shows differential two-lossless coupled and symmetrical transmission lines.

Figure 4.6: Two lossless transmission lines. To derive the wave equations for the two-lossless transmission line, KVL and KCL are also applied. The voltage drops across the inductors in the first and second lines, respectively, are

(4.27)

− − = − + (4.28)

The changes in the− currents in the− first and second = lines, respectively,− + are

(4.29)

− + + = − + (4.30)

Equations (4.27-−4.30) are + known as telegrapher’s − equations = for coupled − and + symmetrical transmission lines. They can be written in matrices. The voltage drops and changes in current in matrix form are

(4.31) � = −�� (4.32) , where � = −�

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(4.33) − + = [ ] − + (4.34) � = [ ] (4.35) , � = [ ] , (4.36) − + = [ ] − + (4.37) + − = [ ] − + (4.38) � = [ ] Differentiating (4.31-4.32) with respect to z gives

(4.39) � � = − (4.40) � � Substituting (4.32) into (4.39) and plugging = (4.31) − into (4.40) yield to

(4.41) � = �� (4.42) � In addition, telegrapher’s equations for coupled= transmission�� lines can be written in a variety of forms. In Maxwellian form, the four equations are

(4.43) − = + (4.44) − = + (4.45) − = + (4.46) − = + 184

, where L11 = L22 = L, L12 = L21 = Lm, C11 = C22 = C + Cm, and C12 = C21 = -Cm for two symmetrical lines. In physical form, the four equations are

(4.47) − = + (4.48) − = + (4.49) − = + − (4.50) − = − + + 4.3 Coupled Line Analysis

When coupling exists between the transmission lines, the characteristics of the system change. That is, the velocity and impedance are affected depending on the propagation mode. For two-transmission lines, the impedance and velocity are calculated as

(4.51) +

= √ (4.52) − = √+ (4.53)

, = √+ (4.54)

, = √−+ 4.4 High Power Slow-Wave Circuit Structure

The slow-wave circuit, described in Sections 2.5-2.6, can be analyzed in several distinct parts. To achieve a high power slow-wave circuit of the traveling wave tube design, three factors are considered. First, the current flowing in the structure needs to be strong

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enough. Second, the diameter of the electron beam should be large. Third, the velocity of the beam needs to be high since the kinetic energy is related to the electron velocity as

(4.55) , where m is the mass and v is the velocity. = At the same time, synchronism must be maintained between the beam velocity and RF velocity.

4.4.1 Early Stage of ANSYS High Frequency Structure Simulator (HFSS)

A novel high power slow-wave circuit structure of the traveling wave tube was modeled and designed using ANSYS HFSS [35-36] in the frequency range of 1.89-2.78

GHz. The dispersion and interaction impedance characteristics of the design were obtained.

One-cell of the design consists of two rings and two/four coupled transmission lines categorizing the structure as a modified ring bar design. The structure is surrounded by an outer conducting circular waveguide. The two rings are derived from the existing ring-bar structures known to generate a high output power, but limited bandwidth. The two/four coupled transmission lines connect the rings and produce mutual lumped elements between the two lines. Figure 4.7 shows the side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT.

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Figure 4.7: Side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT. Figure 4.8 shows the perspective view of one-cell of the modeled slow-wave circuit structure of the TWT surrounded by a circular waveguide.

Figure 4.8: Perspective view of one-cell of the modeled slow-wave circuit structure of the TWT surrounded by a circular waveguide. As shown in Figures 4.7-4.8, there are eight different geometrical dimensions in the design.

Those dimensions are illustrated and labeled in Figures 4.9-4.10.

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Figure 4.9: Dimensions of the modeled one-cell slow-wave circuit structure of the TWT.

Figure 4.10: Other dimensions of the modeled one-cell slow-wave circuit structure of the TWT surrounded by a circular waveguide.

As shown in Figures 4.9-4.10, the radius and thickness of the ring are denoted “a” and “δ”, respectively. The transmission lines are modeled in an intentional rectangular shape whose length, width, and thickness, are denoted “L”, “W”, and “t”, respectively. The period of the one-cell design is denoted “p” and “rcw” represents the radius of the circular waveguide.

The diameter plus the thickness of the ring is denoted “dr”. The dispersion characteristics, output power, gain, bandwidth, and efficiency are affected by such parameters in a variety of ways. They change the values of the capacitance and inductance per unit lengths of the

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transmission lines, which can be calculated using static field analysis from Gauss’s divergence theorem and Ampere’s circuit law. Specifically, the length and width including their ratio of the transmission lines control the phase velocity of the structure if the period of the one-cell design does not change. Also, the inductance per unit length is increased due to the rectangular shape of the transmission lines represented by “L” and “W”, which results in an increase in the impedance.

The existing rings in the structure strengthen the coupling between the transmission lines. Mutual inductance and capacitance are produced in the structure. These two mutual elements reduce the velocity of the wave based on equation (4.52). Knowing the geometry, the mutual elements can be calculated and analyzed. However, the rings suppress the mutual capacitance in the design. Thus, it is neglected in the structure and Lm is the parameter, which controls the propagation constant, phase velocity, and interaction impedance.

A variety of designs were created for the novel structure by modifying the dimensions of the geometry and examining the dispersion results. At all times, “a”, “dr”,

“d”, “t”, and “δ” were fixed. The length, L, width, W, of the transmission lines, and the period were varied. Copper was the assigned material for all elements in the structure.

Table 4.1 states the early stage attempts and specifications of the novel slow-wave circuit structure of the TWT design.

Table 4.1: Dimensions of the geometrical structure of the novel slow-wave circuit structure of the TWT at the early stage.

Design# a dr rcw δ t L W p [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm] 1 4.5 11.0 54.61 2.0 2.0 16.0 13.0 22.0 2 4.5 11.0 54.61 2.0 2.0 14.0 13.0 20.0 3 4.5 11.0 54.61 2.0 2.0 16.0 4.0 22.0 4 4.5 11.0 54.61 2.0 2.0 16.0 8.0 22.0 5 4.5 11.0 54.61 2.0 2.0 16.0 15.0 22.0

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6 4.5 11.0 54.61 2.0 2.0 24.0 13.0 30.0 7 4.5 11.0 54.61 2.0 2.0 16.0 20.0 22.0 8 4.5 11.0 54.61 2.0 2.0 15.0 13.0 21.0

The dispersion diagram was obtained from ANSYS HFSS. It is generated by applying periodic boundary conditions (PBCs) appropriately. The phase velocity and interaction impedance were found using (4.2-4.3), respectively.

The Eigenmode solver type was chosen from HFSS of the menu item as

HFSS>Solution type. Figure 4.11 shown the available solution types from HFSS menu item.

Figure 4.11: Available solution types from HFSS menu. The periodic boundary conditions were performed through the master and slave boundaries in ANSYS HFSS by selecting the negative y-z face and assigning it as a master boundary condition. The U vector was defined for this face. Next, the positive y-z axis was selected and assigned as a slave boundary condition. The V vector was defined for this face and the input phase delay was set to a variable ‘px’. The input initial value was set to 90°. Figures

4.12-4.14 show the performed steps to set up the master/slave boundaries.

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Figure 4.12: Master boundary condition.

Figure 4.13: Slave boundary condition.

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Figure 4.14: Assigning the phase delay in the slave boundary condition. Figure 4.15 shows the transparent view of the novel slow-wave circuit structure of the

TWT design using ANSYS HFSS with applied master/slave boundaries.

Figure 4.15: Transparent view of the novel slow-wave circuit structure of the TWT design with applied master/slave boundaries. To perform the analysis and optometrics setup, the solution type was added and the values were entered as shown in Figure 4.16.

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Figure 4.16: Eigenmode solution setup. An angle parametric sweep with defined steps was defined from HFSS>Optometrics

Analysis>Add Parametric. A linear step sweep was added for the variable ‘px’ from 0 to

180° with 10 steps increment. Figure 4.17 shows the setup sweep analysis for the variable

‘px’.

Figure 4.17: Setup sweep analysis.

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Next, a report was created with the given Eigenmode parameters and rectangular plot. The variable ‘px’ was chosen from choose sweep Design and Project. As a result, the dispersion curve was obtained and exported. Figures 4.18-4.19 show the resulted dispersion diagram of the novel slow-wave circuit structure of the TWT for the early stage designs with the x- axes being in degrees and radians, respectively, and circular waveguide radius of 54.61 mm.

Figure 4.18: Dispersion diagram of the novel slow-wave circuit structure of the TWT for the early stage designs with the x-axes being in degrees and circular waveguide radius of 54.61 mm.

Figure 4.19: Dispersion diagram of the novel slow-wave circuit structure of the TWT for the early stage designs with the x-axes being in radians and circular waveguide radius of 54.61 mm.

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As shown in Figures 4.18-4.19, the dispersion diagram spreads out until approximately

2.78 GHz for L = 16.0 mm, W = 13.0 mm, p = 22.0 mm, L = 14.0 mm, W = 13.0 mm, p =

20.0 mm, and L = 15.0, W = 13.0, p = 21.0 mm , 2.56 GHz for L = 16.0 mm, W = 15.0 mm, p = 22.0 mm, 2.64 GHz for L = 24.0 mm, W = 13.0 mm, p = 30.0 mm, and 2.18 GHz for L = 1θ.0 mm, W = 20.0 mm, p = 22.0 mm in between 0 and 180 degrees or 0 and π radians. It spreads further for L = 16.0 mm, W = 13.0 mm, p = 22.0 mm until 3.39 GHz, and even more for L = 16.0 mm, W = 4.0 mm, p = 22.0 mm until 4.06 GHz. This can be controlled out by changing the ratio of the length and width of the transmission lines and radius of the circular waveguide. For example, one can reduce the width to 8.0 mm and keep L = 16.0 mm and p = 22.0 mm, and the dispersion curve would extend up to 3.39

GHz. Also, if the width is 4.0 mm, L = 16.0 mm, and p = 22.0 mm, then the dispersion curve would extend up to 4.06 GHz. When the radius of the tube is increased to 127.0 mm instead of 54.61 mm, the cutoff frequency becomes smaller. Thus, the starting frequency point of the dispersion curve becomes lower by passing frequencies lower than 1.0 GHz.

This is expected since the cutoff frequency of a circular electromagnetic waveguide is inversely proportional to the radius of the waveguide. Figure 4.20-4.21 show the dispersion diagram of the novel slow-wave circuit structure of the TWT for the early stage designs with the x-axes being in degrees and radians, respectively, and circular waveguide radius of 127.0 mm.

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Figure 4.20: Dispersion diagram of the novel slow-wave circuit structure of the TWT for the early stage designs with the x-axes being in degrees and circular waveguide radius of 127.0 mm.

Figure 4.21: Dispersion diagram of the novel slow-wave circuit structure of the TWT for the early stage designs with the x-axes being in radians and circular waveguide radius of 127.0 mm. Figure 4.22 shows the propagation constant versus frequency of the novel slow-wave circuit structure of the TWT for the early stage designs with a circular waveguide radius of

127.0 mm.

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Figure 4.22: Propagation constant versus frequency of the novel slow-wave circuit structure of the TWT for the early stage designs with a circular waveguide radius of 127.0 mm. As shown in Figure 4.22, the propagation constant is between approximately 7 and above

100 rad/m within the tested frequency range for each design. The propagation constant depends on the period of the structure or pitch of the helix. This propagation constant can be obtained theoretically. The determinantal equation in (2.219) provides four solutions to the propagation constant, β, along the circuit and electron beam. In (2.219), the determinantal equation was varied as ej(ωt-βz). For dielectric loaded waveguides, the propagation constant can be written as

(4.56) = ±√ − . where , , and is the nth root of the Bessel function Jm(r) of the order m. It = represents = the propagation constant along the x-axis of the (m, n) mode. The equivalent circuit parameters of the term is . By incorporating the equivalent circuit parameters of the structure including the mutual inductance, equation (4.56) is equivalent to

(4.57) = ±√ + −

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, which represents the forward and backward waves of the propagation constants. The dispersion relation for the structure becomes

(4.58) Solving for β in (4.η8), − − =

(4.59)

= √ + (4.60)

= −√ + (4.61)

= √ − (4.62)

From (4.57-4.62), it is indicated = that − the√ mutual − capacitance is neglected because the existing rings suppress this mutual term. Figure 4.23 shows the normalized phase velocity versus frequency of the novel slow-wave circuit structure of the TWT for the early stage designs with a circular waveguide radius of 127.0 mm.

Figure 4.23: Normalized phase velocity versus frequency of the novel slow-wave circuit structure of the TWT for the early stage designs with a circular waveguide radius of 127.0 mm. As shown in Figure 4.23, the normalized phase velocity is between below 0.5 and 1 within the tested frequency for each design, which indicates a difference of more than 0.5. Such difference indicates that this structure, unlike the helix, is more dispersive than many

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existing slow-wave circuits. This normalized phase velocity can be obtained theoretically.

From (4.56-4.60), the phase velocities are

(4.63) ′ ′ ′ = = √( +) < (4.64) ′ ′ ′ = = − √( +) < (4.65) ′ ′ ′ = = √( −) > (4.66) ′ ′ ′ = = − √( −) > Equations (4.61-4.62) are solved since their resulted phase velocities are less than c, which indicate the formation of slow-wave structure. Equations (4.63-4.64) indicate resulted phase velocities which are larger than c. However, the intended design is to slow down the phase velocity to have the electrons move at a speed equal to that phase velocity.

To obtain the interaction impedance, the axial electric field along the axis of the structure and total power flow down the structure were calculated using the field calculator in ANSYS HFSS. Each design was analyzed separately since the interaction impedance differs for each design and cannot be combined in a single plot. Some designs are not appropriate in the impedance transition. Figure 4.24 shows the normalized phase velocity versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W

= 13.0 mm, p = 22.0 mm.

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Figure 4.24: Normalized phase velocity and interaction impedance versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0, W = 13.0, p = 22.0 [in mm]. As shown in Figure 4.24, the normalized phase velocity becomes 0.89 at 1.70 GHz. It drops to 0.80 at about 2.38 GHz, which is a difference of 0.09 in velocity within 0.68 GHz in frequency. The chosen interaction impedance range between 30 and 80 ohms was reasonable in terms of providing a proper transition from the input to the output. It is expected to have impedance matches within such range. The interaction impedance varies between 34 and 80 ohms between 1.73 and 2.40 GHz, which indicates a difference of 46 ohm in impedance within 0.67 GHz in frequency. Figure 4.25 shows the normalized phase velocity versus frequency of the novel slow-wave circuit structure of the TWT for L = 14.0 mm, W = 13.0 mm, p = 20.0 mm.

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Figure 4.25: Normalized phase velocity and interaction impedance versus frequency of the novel slow-wave circuit structure of the TWT for L = 14.0, W = 13.0, p = 20.0 [in mm]. As shown in Figure 4.25, the normalized phase velocity is 0.75 at 2.20 GHz. It drops to

0.53 at about 2.70 GHz, which is a difference of 0.22 in velocity and 0.50 GHz in frequency. The chosen interaction impedance range between 30 and 80 ohms was reasonable in terms of providing a proper transition from the input to the output. It is expected to have impedance matches within such range. The resulted interaction impedance varies between 30 and 80 ohms between approximately 2.26 and 2.67 GHz, which indicates a difference of 50 ohm in impedance within 0.41 GHz in frequency. Figure

4.26 shows the normalized phase velocity versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 15.0 mm, p = 22.0 mm.

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Figure 4.26: Normalized phase velocity and interaction impedance versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0, W = 15.0, p = 22.0 [in mm]. As shown in Figure 4.26, the normalized phase velocity is 0.67 at 2.4 GHz. It drops to 0.4 at about 2.59 GHz, which is a difference of 0.27 in velocity and 0.15 GHz in frequency.

The chosen interaction impedance range between 30 and 80 ohms was reasonable in terms of providing a proper transition from the input to the output. It is expected to have impedance matches within such range. The resulted interaction impedance varies between

30 and 80 ohms between 2.41 and 2.57 GHz, which indicates a difference of 50 ohm in impedance within 0.16 GHz in frequency. Figure 4.27 shows the normalized phase velocity versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W

= 20.0 mm, p = 22.0 mm.

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Figure 4.27: Normalized phase velocity and interaction impedance versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0, W = 20.0, p = 22.0 [in mm]. As shown in Figure 4.27, the normalized phase velocity is approximately 0.85 at 1.60 GHz.

It drops to 0.54 at about 2.10 GHz, which is a difference of 0.31 in velocity and 0.5 GHz in frequency. The chosen interaction impedance range between 30 and 80 ohms was reasonable in terms of providing a proper transition from the input to the output. It is expected to have impedance matches within such range. The resulted interaction impedance varies between 30 and 80 ohms between 1.68 and 2.06 GHz, which indicates a difference of 50 ohm in impedance within 0.38 GHz in frequency. Figure 4.28 shows the normalized phase velocity versus frequency of the novel slow-wave circuit structure of the

TWT for L = 15.0 mm, W = 13.0 mm, p = 21.0 mm.

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Figure 4.28: Normalized phase velocity and interaction impedance versus frequency of the novel slow-wave circuit structure of the TWT for L = 15.0, W = 13.0, p = 21.0 [in mm]. As shown in Figure 4.28, the normalized phase velocity is around 0.83 at 1.90 GHz. It drops to 0.66 at about 2.60 GHz, which is a difference of 0.17 in velocity and 0.7 GHz in frequency. The chosen interaction impedance range between 30 and 80 ohms was reasonable in terms of providing a proper transition from the input to the output. It is expected to have impedance matches within such range. The resulted interaction impedance varies between 30 and 80 ohms between approximately 1.93 and 2.58 GHz, which indicates a difference of 50 ohm in impedance within 0.65 GHz in frequency.

If the width of the transmission line is set to 0, the length to width ratio, which controls the phase velocity, does not exist. Thus, the transmission line is straight and acting as a straight ring bar. In such case, the period needs to be minimized to obtain a valid impedance transition. Otherwise, the design is not appropriate to consider as shown next.

Figure 4.29 shows the side and perspective views of one-cell of the slow-wave circuit structure of the TWT with L = 15.0 mm, W = 0.0, and p = 21.0 mm.

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Figure 4.29: Side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT with L = 15.0, W = 0.0, and p = 21.0 [in mm]. Figure 4.30 shows the perspective view of one-cell of the modeled slow-wave circuit structure of the TWT surrounded by a circular waveguide with L = 15.0 mm, W = 0.0, and p = 21.0 mm.

Figure 4.30: Perspective view of one-cell of the modeled slow-wave circuit structure of the TWT surrounded by a circular waveguide with L = 15.0, W = 0.0, and p = 21.0 [in mm]. Figures 4.31-4.32 show the resulted dispersion diagram of the slow-wave circuit structure of the TWT with L = 15.0 mm, W = 0.0, and p = 21.0 mm with the x-axes being in degrees and radians, respectively, and circular waveguide radius of 127.0 mm.

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Figure 4.31: Dispersion diagram of the slow-wave circuit structure of the TWT with L = 15.0, W = 0.0, and p = 21.0 [in mm] with the x-axes being in degrees and circular waveguide radius of 127.0 mm.

Figure 4.32: Dispersion diagram of the slow-wave circuit structure of the TWT with L = 15.0, W = 0.0, and p = 21.0 [in mm] with the x-axes being in radians and circular waveguide radius of 127.0 mm. As shown in Figures 4.31-4.32, the dispersion diagram spreads out until approximately

4.θ2 GHz between 0 and 180 degrees or 0 and π radians. Figure 4.33 shows the propagation constant versus frequency of the slow-wave circuit structure of the TWT with L = 15.0 mm, W = 0, p = 21.0 mm, and circular waveguide radius of 127.0 mm.

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Figure 4.33: Propagation constant versus frequency of the slow-wave circuit structure of the TWT with L = 15.0, W = 0.0, p = 21.0, and circular waveguide radius of 127.0 [in mm]. As shown in Figure 4.33, the propagation constant is between approximately 8 and 140 rad/m within the whole dispersion behavior. Figure 4.34 shows the normalized phase velocity versus frequency of the slow-wave circuit structure of the TWT with L = 15.0 mm,

W = 0, p = 21.0 mm, and circular waveguide radius of 127.0 mm.

Figure 4.34: Normalized phase velocity versus frequency of the slow-wave circuit structure of the TWT with L = 15.0, W = 0.0, p = 21.0, and circular waveguide radius of 127.0 [in mm]. As shown in Figure 4.34, the normalized phase velocity is between 0.7 and 1, which indicates a difference of 0.30 within the frequency of 1.30-4.62 GHz. Such difference indicates that this structure is more dispersive than the other slow-wave structures.

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To obtain the interaction impedance, the axial electric field along the axis of the structure and total power flow down the structure were calculated using the field calculator in ANSYS HFSS. Figure 4.35 shows the interaction impedance versus frequency of the slow-wave circuit structure of the TWT with L = 15.0 mm, W = 0, and p = 21.0 mm.

Figure 4.35: Interaction impedance versus frequency of the slow-wave circuit structure of the TWT with L = 15.0, W = 0.0, and p = 21.0 [in mm]. As shown in Figure 4.35, the interaction impedance is in the order of thousands and hundreds of ohms within the frequency of interest. Such impedance range is not valid in the design since the transition would not be appropriate to provide a reasonable gain for the design. Thus, the dimensions of the structure need to be modified to obtain a proper transition.

The current of the transmission line is inversely proportional to the current path length. That is, the current path in the transmission line for the straight ring bars is smaller than the current path in the transmission line for the rectangular bars.

The inductance per unit length, capacitance per unit length, and mutual inductance per unit length are calculated using field static analysis. The inductance is the ratio of the magnetic flux of the circuit to the current as

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(4.67) � , where is the magnetic flux defined as =

ϕ (4.68)

As stated in (4.66), the magnetic flux �is =proportional∫ ⃗ . to the area. When the width of the rectangular transmission lines is not zero, as in Figure 4.7, the flux linkage area is higher than the area when the width of the transmission lines is zero. Figure 4.36 shows a comparison between the total area of the slow-wave structure when the width of the transmission lines is not zero at one time and zero at another time.

Figure 4.36: Comparison between the total area of the slow-wave structure when the width of the transmission lines is not zero at one time and zero at another time. As shown in Figure 4.36, the area of the slow-wave structure when the width of the transmission line is not zero is

(4.69)

The area of the slow-wave structure when = the + width + of the transmission line is zero is

(4.70)

= +

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The 2LW term is added because of incorporating the width being not zero for the transmission lines at the top and bottom sections. The line to line inductance per unit length is obtained by considering Figure 4.37.

Figure 4.37: Two parallel transmission lines. As shown in Figure 4.37, the two parallel transmission lines are in the x-z plane. The two currents are through the +z and -z directions. The magnetic fields at P due to the first and second transmission lines, respectively, are

(4.71) = �̂ = ̂ (4.72) = ̂ − For non-magnetic dielectric, r = 1.0. Note that the currents in the transmission lines are in opposite directions. Summing (4.69-4.70) yields to

(4.73) + = ̂ + ̂ − = ̂ − The flux between two parallel wires over the surface of length l is found by integrating

(4.73) as

+ − − � = ∬ . = ∫ ∫ (̂ ) . ̂ = ( ln )| = = 210 − − =

(4.74) − − − The inductance� = of lnlength l becomes − ln − = × ln = ln

(4.75) � − The inductance per unit length is = = ln

(4.76) ′ − The approximation in (4.76) states = that = the ln spacing ≈between ln the two transmission lines is large compared to the radius of the transmission line.

Similarly, the line to line capacitance per unit length is obtained by considering

Figure 4.37. As shown in Figure 4.37, the two parallel transmission lines are in the x-z plane. The electric field between two parallel transmission lines for a cylindrical geometry is

(4.77) The electric fields at P due to the first and = second transmission lines, respectively, are

(4.78) = ̂ = ̂ (4.79) = ̂ − For vacuum, r = 1.0. Summing (4.78-4.79) yields to

(4.80) + = ̂ + ̂ − = ̂ − The potential difference between two parallel wires over the surface of length l is found by integrating (4.80) as

− − ∆ = ∫ . = ∫ ( + ) −

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(4.81) − − − The capacitance∆ = of lengthln l becomes − ln − = × ln = ln

(4.82) − = ∆ = The capacitance per unit length is

(4.83) ′ − = = ≈ The approximation in (4.83) states that the spacing between the two transmission lines is large compared to the radius of the transmission line.

For the straight ring bars whose dimensions are L = 15.0 mm, W = 0.0 mm, p =

21.0 mm, the spacing between the two transmission lines is

(4.84) The radius of the transmission = + +line is = . + + =

(4.85) Thus, the inductance and capacitance = =per unit length= . are

− (4.86) × ′ − −. = ln = ln . = .

− (4.87) .× ′ − . −. = = . = . The total inductance and capacitance are

− (4.88) − × . −. = ln = ln . = .

− (4.89) .× . . −. = = . = .

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For another design, considered to be the second, whose dimensions are L = 15.0 mm, W = 13.0 mm, p = 21.0 mm, the spacing between the two transmission lines is

(4.90) The radius =of the transmission + + +=.++ line is + = .

(4.91) Thus, the inductance and capacitance = =per unit length= . are

− (4.92) × ′ − . −. = ln = ln . = .

− (4.93) .× ′ − . −. = = . = . , where the L and C are used to distinguish between the total inductance and capacitance ’ ’ and their corresponding values per unit lengths. The total inductance and capacitance are

− (4.94) − × . −. = ln = ln . = .

− (4.95) .× . . −. = = . = . The inductance per unit length for the second design is 1.48 μH/m and the inductance per unit length for the straight ring bar design is 1.0θ μH/m, which indicates a higher inductance per unit length for the second design by 0.42 μH/m. The capacitance per unit length for the second design is 7.54 pF/m and the capacitance per unit length for the straight ring bar design is 10.54 pF/m, which indicates a higher capacitance per unit length for the straight ring bar design by 3.0 pF/m. The total inductance for the second design is 0.0221

μH and the total inductance for the straight ring bar design is 0.01η8 μH, which indicates a higher total inductance for the second design by 0.00θ3 μH. The total capacitance for the

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second design is 0.113 pF and the total capacitance for the straight ring bar design is 0.158 pF, which indicates a higher total capacitance for the straight ring bar design by 0.045 pF.

The mutual inductance between the two transmission lines of lengths l1 and l2, radius a, and separation d is obtained using

(4.96) − √ + − √ + + √ − + − ln + √ + = − ln − + √ + + − + ln − + √ + + , where the derivation for this equation is beyond the scope of this paper. When the currents travel in the same direction, the mutual inductance is positive. Else, the mutual inductance is negative. When the lengths l1 and l2 are the same, (4.94) can be expressed as

(4.97) = ln + √ + − √ + + For the straight ring bars whose dimensions are L = 15.0 mm, W = 0.0 mm, p =

21.0 mm, the mutual inductance is

− × . . . = ln . + √ + . − √ + . + (4.98) . . = . The mutual inductance per unit length is

(4.99) ′ . For the second design whose = dimensions = . are= L .= 15.0 mm, W = 13.0 mm, p = 21.0 mm, the mutual inductance is

− × . . . = ln . + √ + . − √ + . + (4.100) . . = . 214

The mutual inductance per unit length is

(4.101) ′ . . The mutual inductance per= unit length= for the= . second design is 0.θ1 μH/m and the mutual inductance per unit length for the straight ring bar design is 0.7θ μH/m, which indicates a smaller mutual inductance per unit length for the second design by 0.1η μH/m.

Next, the expected phase velocity can be calculated using (4.61). For the straight ring bars whose dimensions are L = 15.0 mm, W = 0.0 mm, p = 21.0 mm, the calculated phase velocity is

(4.102) ′ ′ ′ � � = = √( +) = √. +. . = .× = . For the second design whose dimensions are L = 15.0 mm, W = 13.0 mm, p = 21.0 mm, the calculated phase velocity is

(4.103) ′ ′ ′ � � = = √( +) = √. +. . = .× = . The calculated phase velocity for the second design is 0.84c and calculated phase velocity for the straight ring bar design is 0.76c, which indicates a higher calculated phase velocity for the second design by 0.08c.

4.4.2 Final Design of ANSYS High Frequency Structure Simulator (HFSS)

After plenty of attempts to come up with the optimum design to be analyzed further, the design whose dimensions are L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm was selected to be the chosen one. Figure 4.38 shows the side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm.

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Figure 4.38: Side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT for L = 16.0, W = 10.5, and p = 22.0 [in mm]. Figure 4.39 shows the side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT surrounded by a circular waveguide whose radius is 127.0 mm for L = 16.0 mm, W = 10.5 mm, p = 22.0 mm.

Figure 4.39: Side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT for L = 16.0, W = 10.5, p = 22.0, and circular waveguide radius of 127.0 [in mm].

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The spacing between the two transmission lines is

(4.104) The radius = of the +transmission + +=.++ line is + . = .

(4.105) Thus, the inductance and capacitance = = per unit length= . are

− (4.106) × ′ − . −. = ln = ln . = .

− (4.107) .× ′ − . −. = = . = . The total inductance and capacitance are

− (4.108) − × . −. = ln = ln . = .

− (4.109) .× . . −. = = . = . For the chosen design whose dimensions are L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, the mutual inductance is

− × . . . = ln . + √ + . − √ + . + (4.110) . . = . The mutual inductance per unit length is

(4.111) ′ . The calculated phase velocity is = = . = .

(4.112) ′ ′ ′ � � = = √( +) = √. +. . = .× = .

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Table 4.2 states the calculated parameters of the chosen design of the novel slow-wave structure of the TWT.

Table 4.2: Calculated parameters of the chose design of the novel structure of the TWT whose dimensions are L = 16.0, W = 10.5, and p = 22.0 [in mm]. Parameter Value 1.42 ′ 0.0228 [ ] [] 7.82 ′ 0.125 [ ] [] 0.63 ′ 10.08 [ ] 0.83c [] Next, another pair of transmission line, shifted by 90°, was inserted in the structure to investigate the dispersion characteristics with such change. Figures 4.40 shows the side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm with another pair of shifted transmission line by 90°.

Figure 4.40: Side and Perspective views of one-cell of the modeled slow-wave circuit structure of the TWT for L = 16.0, W = 10.5, p = 22.0 [in mm] with another pair of shifted transmission line by 90°.

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Figure 4.41 shows the side and perspective views of the modeled one-cell structure of the slow-wave circuit structure of the TWT surrounded by a circular waveguide whose dimension is 127.0 mm for L = 16.0 mm, W = 10.5 mm, p = 22.0 mm with another pair of shifted transmission line by 90°.

Figure 4.41: Side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT surrounded by a circular waveguide for L = 16.0, W = 10.5, p = 22.0 [in mm] with another pair of shifted transmission line by 90°. Figure 4.42 shows the side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT surrounded by a circular waveguide for L = 16.0 mm, W =

10.5 mm, p = 22.0 mm with one and two pairs of transmission lines.

Figure 4.42: Side and perspective views of one-cell of the modeled slow-wave circuit structure of the TWT surrounded by a circular waveguide for L = 16.0 mm, W = 10.5 mm, p = 22.0 mm with one and two pairs of transmission lines.

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Figure 4.43-4.44 show the resulted dispersion diagram of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm of both designs with the x-axes being in degrees and radians, respectively, and circular waveguide radius of 127.0 mm.

Figure 4.43: Dispersion diagram of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm of both designs and beam line with the x-axes being in degrees and circular waveguide radius of 127.0 mm.

Figure 4.44: Dispersion diagram of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm of both designs and beam line with the x-axes being in radians and circular waveguide radius of 127.0 mm. As shown in Figures 4.43-4.44, the dispersion behavior for both designs intersect with the beam line, which indicates the operating region of the designs. When the two designs deviate from each other at around 80° or 1.4 radians, the two pairs of transmission lines

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became closer to the beam line than the 1 pair of transmission lines. This indicates a possibility of achieving a better bandwidth of the two pairs than the one pair of transmission lines. Figure 4.45 shows the propagation constant versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm of both designs with a circular waveguide radius of 127.0 mm.

Figure 4.45: Propagation constant versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm of both designs with a circular waveguide radius of 127.0 mm. As shown in Figure 4.45, the propagation constant is between approximately 21 and above

140 rad/m within the frequency range of 1.0-3.21 GHz. Figure 4.46 shows the normalized phase velocity versus frequency of the novel slow-wave circuit structure of the TWT for L

= 16.0 mm, W = 10.5 mm, p = 22.0 mm of both designs.

Figure 4.46: Normalized phase velocity versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm of both designs.

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As shown in Figure 4.46, the normalized phase velocity is between 0.86 and 0.73 for the one pair of transmission lines and between 0.86 and 0.78 for the two pairs of transmission lines in the frequency range of 1.8-2.8 GHz. Since the kinetic energy is proportional to velocity, the design which has two pairs of transmission lines is expected to continue generating a higher power as frequency increases. Figure 4.47 shows the interaction impedance versus frequency of the novel slow-wave circuit structure of the TWT for L =

16.0 mm, W = 10.5 mm, p = 22.0 mm of both designs.

Figure 4.47: Interaction impedance versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm of both designs. As shown in Figure 4.47, the interaction impedance range between 30 and 80 ohms was reasonable in terms of providing a proper transition from the input to the output. It is expected to have impedance matches within such range. The resulted interaction impedance varies between 30 and 80 ohms for the two pair transmission lines design between approximately 1.85 and 2.78 GHz, which indicates a difference of 50 ohm in impedance within 0.93 GHz in frequency. The resulted interaction impedance varies between 30 and 80 ohms for the one pair transmission lines design between approximately

1.69 and 2.58 GHz, which indicates a difference of 50 ohm in impedance within 0.89 GHz

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in frequency. Figure 4.48 shows the coupling parameter, C, versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, p = 22.0 mm of both designs.

Figure 4.48: Gain parameter versus frequency of the novel slow-wave circuit structure of the TWT for L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm of both designs. As shown in Figure 4.48, the coupling parameter varies between approximately 0.062 and

0.1 in the frequency of interest for one pair of transmission lines and between 0.068 and

0.1 in the frequency of interest for two pairs of transmission lines. This curve was determined using equation (2.220). The used beam voltage and current are 262 kV and 12

A, respectively.

Such resulted features, presented earlier, for the chosen design justify a better expectation to achieve a better bandwidth and probably gain than the rest of the designs.

This can be verified after performing the hot-test simulations using VSim covered in

Chapter 5.

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CHAPTER V

A NOVEL SLOW-WAVE CIRCUIT STRUCTURE WITH HOT-TEST SIMULATIONS

After optimizing the slow-wave circuit structure of the traveling wave tube and performing cold-test simulations, the hot-test simulations can be carried out to investigate the performance of the structure. This chapter describes a 3D modeling, analysis, and design of a high power slow-wave circuit structure using conformal finite-difference time- domain (FDTD) code, VSim [37]. The novel design, consisting of two pairs of coupled transmission lines, is imported into VSim. Sequential power runs are executed to analyze the operating principle of the design. The output power, gain, bandwidth, and efficiency are obtained besides investigating the transfer characteristics of the TWT and output power versus number of periods. The simulated results are compared with the existing ring-bar structures.

5.1 Finite-Difference Time-Domain (FDTD) Code, VSim

To characterize the performance of the novel slow-wave structure, VSim, which is

“a high performance electromagnetic code using the FDTD numerical method,” is used. It

“solves for particle motion and plasma physical kinetic equations,” and enables the user to simulate a variety of designs and “problems on regular, structured, orthogonal meshes with embedded boundaries for complex geometries,” using Finite-Difference Time-Domain

(FDTD) methods. Maxwell equations are used to provide electromagnetic, electrostatics magnetostatics, plasma runs in different ways and shapes. Depending on the application areas, there are a variety of VSim packages such as Basic Simulations, Electromagnetics

EM), Microwave Devices (MD), Plasma Discharges (PD), Semiconductor Devices (DS)

and Plasma Acceleration (PA). In addition, Tech-X corporation provides other scientific software such as USim and PSim simulations and their packages for different applications and high-performance modeling tools like GPUlib. Such software can generate plenty of functions and simulations including, but not limited to “cut-cells, electron emission, dispersion control, sputtering, collisions, field ionization, and quasiparticle propagation”

[125].

If the user is willing to run a basic electromagnetic, particle, trajectory, and plasma physics, VSim Basic Simulations can be used. In addition, the user can take the advantage of VSim electromagnetics to obtain multiple outputs such as S-parameters and antenna gain, efficiency, beam width, axial ratio, side-level lobes, and directivity when willing to perform electromagnetic simulations for a variety of materials including their properties and devices such as antennas, radars, radar cross-section, specific absorption rate (SAR), waveguides, near-field and far-field radiation patterns, Q-factors in resonators and cavities, and optical devices. Also, when the user is willing to model and run microwave devices including magnetrons, gyrotrons, traveling wave-tubes and klystrons with electron beams using different types of electron emissions like Child-Langmuir, Fowler-Nordheim,

Richardson-Dushman, and Furman-Pivi, VSim Microwave Devices can be used. It also allows the user to design electron guns, collectors, striplines, and couplers and generate different outputs including, but not limited to power runs, gain, multipacting, shunt impedance, and S-matrix coefficients. Furthermore, VSim Plasma Discharges can be used to study the “effects of elastic, excitation, and ionization collisions between electrons, ions, and neutral particles using Particle-In-Cell Monte Carlo (PIC-MCC) and PIC-DSMC methods and also including charge exchange, electron recombination, sputtering, and

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secondary emission” when the user is willing to model plasma discharges and perform kinetic simulations of pressure gas discharges. Moreover, if the user is willing to perform beam and laser plasma acceleration simulations such as field-ionization induced injection,

VSim Plasma Acceleration is used for colliding laser pulses or Lorentz boosted frame

[125].

VSim consists of two major components: VSim computational engine, known as

Vorpal, and VSimComposer, which is the graphical user interface (GUI). The VSim computational engine “runs both as a serial (vorpalser) and parallel (vorpal) application for multi-processor/multi-core systems that support MPI” [12η]. To edit input files, run simulations, and visualize results, VSimComposer is used. The simulations can be carried out in two different ways: text-based or visual-based. The text based way relies on using

Python to process the input files based on math functions, macros, and variable substitutions. The VSimComposer creates input files with .pre extension. The visual-based way uses a visual representation via VisIt, embedded inside VSimComposer, of the simulations by building in graphical components or functions, or importing geometries with a valid extension. It invokes the VSim engine with defined number of steps, dumps, and restart file. To communicate between processes executed in parallel, MPI, Message

Passing Interface, programming is utilized.

Since VSim is developed to be used for high-performance and challenging problems, many simulations require to use supercomputers to handle the 3D calculations and computational methods, which might require at least 12GB RAM or more to run.

Besides, the use of complex designs and many components take a large memory of the disk space, especially the visualization-based way in VSim.

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VSim allows the user to import geometries in the visual setup with Step Files (.stp,

.step, .p12), STereoLithography Files (.stl), Visualization Toolkit Files (.vtk), and Polygon

File Format (.ply). However, the only file extension that Vorpal engine handles is .stl. The geometries inside the visual setup with defined materials are converted to .stl files. After that, the Vorpal engine imports and reads these files. Materials are imported with .vmat extension.

To get started with VSim, a compact and lightweight traveling wave tube was reproduced in the frequency range of 2.0-4.0 GHz using VSim [72]. It uses a conventional helix slow-wave circuit structure and the same specifications in [72] in the S-band frequency range. A power run of the design was obtained. The helix circuit parameters are the dimensions of the helix, circuit length, support rods, and loss pattern with attenuations.

Table 5.1 provides the specifications of the helix slow-wave circuit structure of the compact lightweight traveling wave tube.

Table 5.1: Specifications of the helix slow-wave circuit structure of the compact lightweight traveling wave tube [72].

Parameters Duplicated Design Helix Inner Diameter [mm] 3.30 Helix Outer Diameter [mm] 4.04 Tape Width [mm] 0.θη Tube Shell Inner Diameter [mm] η.71 Tube Shell Outer Diameter [mm] θ.22 Rod Shape Circular Rod Dielectric Constant BeO = θ.4 Rod Diameter [mm] 0.84 Loss Pattern [mm] η0.8 Circuit Length [mm] 152.4

The traveling wave tube was driven by a 6-kV beam voltage, 0.9 A beam current of the electron gun whose beam radius is 1.1155 mm. Table 5.2 states the electron gun parameters of the compact lightweight traveling wave tube.

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Table 5.2: Electron gun parameters of the compact lightweight traveling wave tube [72].

Parameters Duplicated Design Frequency [GHz] 2.0-4.0 Beam Voltage [kV] 6 Beam Current [A] 0.9 Beam Radius [mm] 1.11ηη

In addition, the electron beam was focused by a periodic permanent magnet whose parameters are shown in Figure 5.1.

Figure 5.1: Parameters of one-cell of the periodic permanent magnet of the compact lightweight traveling wave tube [72]. Table 5.3 states the specifications of the periodic permanent magnet of the compact lightweight traveling wave tube.

Table 5.3: Specifications and calculations of the periodic permanent magnet of the compact lightweight traveling wave tube [72].

Parameters Duplicated Design Magnet Thickness (Mt) [mm] 3.30 Magnet Inner Diameter (Mi) [mm] 7.η4 Magnet Outer Diameter (Mo) [mm] 13.97 Magnet Material Samarium Cobalt Pole Piece Thickness (Pt) [mm] 0.97 Pole Piece Inner Diameter (Pi) [mm] θ.2η Pole Piece Outer Diameter (Po) [mm] 13.97 Pole Piece Material Iron Hub Material Iron Hub Thickness (Ht) [mm] 0.θ0 Hub Outer Diameter (Ho) [mm] 7.η2 Peak Field (Bpeak) [Tesla] 0.1η3 Brillouin Field (BB) [Tesla] 0.078 Brms/BB 1.384 Plasma wavelength (p)/Magnet Period (L) 3.4θ Period (L) [mm] 8.η3

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Plasma Wavelength (p) [mm] 29.η4 Maximum B-Field in Iron [Tesla] 1.η4θ

The power readings were recorded for each frequency. Table 5.4 states the simulated output power, input power, and gain of the compact lightweight traveling wave tube in the frequency range of 2.0-4.0 GHz.

Table 5.4: Simulated recorded output power, input power, and gain of the compact lightweight traveling wave tube in the frequency range of 2.0-4.0 GHz.

Input Frequency Output Output Input Gain Power [GHz] Power [W] Power [dBm] Power [W] [dB] [dBm] 2.00 621.53 57.93 7.83 38.94 18.99 2.10 551.67 57.42 5.16 37.12 20.29 2.20 653.62 58.15 9.23 39.65 18.50 2.30 948.88 59.77 11.85 40.74 19.04 2.40 769.38 58.86 10.08 40.03 18.83 2.50 741.18 58.70 7.61 38.81 19.89 2.60 569.53 57.56 7.21 38.58 18.98 2.70 619.60 57.92 10.44 40.19 17.74 2.80 765.51 58.84 13.56 41.32 17.52 2.90 840.10 59.24 9.62 39.83 19.41 3.00 1001.62 60.01 7.34 38.66 21.35 3.10 387.85 55.89 17.44 42.42 13.47 3.20 782.75 58.94 12.92 41.11 17.82 3.30 550.37 57.41 16.30 42.12 15.29 3.40 500.82 57.00 8.70 39.40 17.60 3.50 503.29 57.02 6.83 38.34 18.67 3.60 525.45 57.21 17.81 42.51 14.70 3.70 642.84 58.08 12.94 41.12 16.96 3.80 842.47 59.26 10.34 40.14 19.11 3.90 649.13 58.12 17.03 42.31 15.81 4.00 857.80 59.33 16.01 42.04 17.29

Figure 5.2 shows the simulated output power and gain of the compact lightweight traveling wave tube in the frequency range of 2.0-4.0 GHz.

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Figure 5.2: Simulated output power and gain of the compact lightweight traveling wave tube in the frequency range of 2.0-4.0 GHz. As shown in Figure 5.2, the highest output power was 1001.01 W at 3.0 GHz, which corresponds to 60.01 dBm and the highest gain was 21.35 dB at 3.0 GHz. Clearly, the bandwidth is limited due to the obtained reading at 3.1 GHz and the efficiency is

(5.113) . = � ×% = . ×% = .% The authors’ saturated output power and saturated gain in [72] differ from the obtained results as shown in Figure 5.3.

Figure 5.3: Authors’ work of the simulated output power and gain of the compact lightweight traveling wave tube in the frequency range of 2.0-4.0 GHz [72].

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As shown in Figure 5.3, the highest power was 1000 W, which corresponds to 60.0 dBm and the highest gain was 25 dB. Their resulted bandwidth is wider than Figure 5.2. Such difference in Figures 5.2-5.3 is resulted due to the missing information needed to reproduce the same plot, which could not be found in [72]. Thus, some assumptions are made.

The design of the novel-slow structure of the TWT was implemented in VSim Basic

Simulations with L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm. It was carried out with N

= 20, which corresponds to the number of elements being used. Table 5.5 states the main specifications of the first design of novel slow-wave structure of the TWT with L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm.

Table 5.5: Design specifications of the design of novel slow-wave structure of the TWT with L = 16.0, W = 10.5, p = 22.0 [in mm].

Frequency [GHz] 1.89-2.72 Number of Elements, N 20 Beam Voltage [kV] 262 Beam Current [A] 12 Beam Radius [mm] 2 Magnetic Field Focusing [Tesla] 1.4 Tube Length [mm] 446

As stated in Table 5.5, the frequency of interest was between 1.89 and 2.72 GHz. Such frequency was considered because the dispersion diagram curve was varied between such range. The phase velocity showed a slow wave at such frequencies, where the interaction impedance was between 30 and 80 ohms. The beam voltage was 262 kV, beam current was

12 A, and beam radius was 2 mm. The magnetic field focusing was 1.4 Tesla and the length of the tube was 446 mm.

The geometrical design was created in High Frequency Structure Simulator, HFSS, and was imported to VSim as a .step file. This was done in HFSS by selecting Modeler in

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the menu bar and choosing Export as Modeler>Export. Figure 5.4 shows the appropriate procedure to export a geometry from HFSS.

Figure 5.4: Exporting a geometry from HFSS. Figure 5.5 shows the side and perspective views of the geometry of the novel slow-wave structure of the TWT inside HFSS with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N

= 20.

Figure 5.5: Perspective and side views of the geometry of the novel slow-wave structure of the TWT inside HFSS with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20

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The circular waveguide was excluded before importing the geometry into VSim. Such tube was inserted using VSim to allow proper properties and manipulations to the tube. Figure

5.6 shows the side and perspective views of the imported geometry of the novel slow-wave structure of the TWT inside HFSS with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N

= 20 without the circular waveguide.

Figure 5.6: Side and perspective views of the imported geometry of the novel slow-wave structure of the TWT inside HFSS with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 without the circular waveguide. Figure 5.7 shows the final geometry of the novel slow-wave structure of the TWT inside

VSim with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 without the tube.

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Figure 5.7: Geometry of the novel slow-wave structure of the TWT inside VSim with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 without the tube. Figure 5.8 shows the final geometry of the novel slow-wave structure of the TWT inside

VSim with L = 16.0 mm, W = 10.2 mm, p = 22.0 mm, and N = 20 with the tube.

Figure 5.8: Geometry of the novel slow-wave structure of the TWT inside VSim with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 with the tube.

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The copper was assigned as the used material for each imported geometry. After that, constant, parameters, functions, space time functions, grids, dynamics, and histories were assigned in the ‘Setup’ window for visual set-up simulations. Appendix F includes the constants, parameters, functions, space time functions, grids, dynamics, and histories used inside VSim for the novel slow-wave structure of the TWT with L = 16.0 mm, W =

10.η mm, p = 22.0 mm, and N = 20. Figure η.9 shows the menu inside ‘Setup’ window.

Figure 5.9: Menu inside the 'Setup' window.

Next, the ‘Run’ window was used after finishing the setup work inside the ‘Setup’ window. The power run was performed for the novel slow-wave structure of the TWT with

N = 20. A single run for one frequency was used with the calculated time steps [in seconds], being edited later to provide an adequate run, number of times steps, dump periodicity

(time steps). Figure η.10 shown the menu inside ‘Run’ window.

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Figure 5.10: Menu inside the 'Run' window to run the simulations.

When using the personal laptop, whose processor is Intel(R) Core™ iη-4210U CPU @

1.70 GHz 2.40 GHz with 8.00 GB RAM, the number of cores used was 6 inside the MPI tab of the run window. A single power run with the personal laptop took about 16 days to finish. After obtaining a better PC from the school, whose processor is Intel(R) Core™ iη-

4210U CPU @ 1.70 GHz 2.40 GHz with 8.00 GB RAM, the number of cores used was 12 inside the MPI tab of the run window. A single power run with the given PC from school took about 8 days to finish. For N = 20. A single power run file took 184 GB of the disk space. This yields to a total of 2392 GB for 13 points of the power run. After the simulation was finished for each frequency, the results were viewed in the ‘History’ from ‘Data View’ inside the ‘Visualize’ window. The proper name of the history was chosen to view the desired result, which shows power in Watt versus time. Also, the fast Fourier transform can be done by checking the ‘FFT’ box. Figure η.11 shows the ‘History’ from ‘Data View’ inside ‘Visualize’ window.

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Figure 5.11: Menu inside the 'Visualize' window to view the results from ‘History’. A low pass filter was added to the script and performed to the results in the

‘Analyze’ window by checking 'performLowPassFilter.py' from “Set Allowed Analyzers” which is the last installation analyzer before the magenta beta analyzers. The ‘frequency’ and ‘historyName’ need to be inputted properly to apply the low pass filter by selecting

‘Run’. Figure η.12 shows the 'performLowPassFilter.py' from “Set Allowed Analyzers” inside ‘Analyze’ window.

Figure 5.12: Menu inside the 'Analyze' window to apply the low pass filter.

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After that, the results would be filtered out and can be visualized by selecting

‘Reload Data’ inside ‘Visualize’ window and choosing ‘1-D fields’ from ‘Data View’.

Figure 5.13 shows the ‘1-D fields’ from ‘Data View’ inside ‘Visualize’ window after applying the low pass filter to the result.

Figure 5.13: Menu inside the 'Visualize’ window to view the results after applying the low pass filter from ‘1-D Fields’. The power readings were recorded for each frequency. Table 5.6 states the simulated saturated output power, input power, and gain of the novel slow-wave structure of the TWT using VSim with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 in the frequency range of 1.85-2.80 GHz.

Table 5.6: Simulated recorded output power, input power, and gain of the novel slow-wave structure of the TWT using VSim with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 in the frequency range of 1.85-2.80 GHz.

Output Input Frequency Output Power Input Gain Power Power [GHz] [W] Power [W] [dB] [dBm] [dBm] 1.85 852400.72 89.31 3087.90 64.90 24.41 1.89 948182.61 89.77 2249.86 63.52 26.25 1.90 882066.40 89.46 1993.77 63.00 26.46 2.00 965726.22 89.85 2539.94 64.05 25.80 2.10 960426.53 89.82 2594.80 64.14 25.68 2.20 984029.19 89.93 2546.15 64.06 25.87 2.30 924688.56 89.66 1717.91 62.35 27.31 2.40 1065500.90 90.28 1634.24 62.13 28.14 2.50 944641.13 89.75 1794.55 62.54 27.21

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2.60 1007359.71 90.03 1993.77 63.00 27.04 2.70 1036647.14 90.16 2114.52 63.25 26.90 2.72 1002635.92 90.01 2533.07 64.04 25.97 2.80 745850.63 88.73 2892.60 64.61 24.11

Figure 5.14 shows the simulated saturated output power and gain of the novel slow-wave structure of the TWT using VSim with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N

= 20 in the frequency range of 1.85-2.80 GHz.

Figure 5.14: Simulated saturated output power and gain of the novel slow-wave structure of the TWT with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 in the frequency range of 1.85-2.80 GHz. As shown in Figure 5.14, the output power and gain are saturated in the wide band frequency range of 1.89-2.72 GHz. The highest output power was 1.07 MW at 2.40 GHz, which corresponds to 90.29 dBm and the highest gain was 28.14 dB at 2.40 GHz. The bandwidth was estimated to be 34.58% as

(4.114) [.−.] The efficiency was calculated = to .be ×% = .%

(4.115) . = � ×% = ×% = .%

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Furthermore, the transfer characteristics of the traveling wave tube was obtained by changing the input power and obtaining the small signal gain region. This was done after calibrating the input voltage. Table 5.7 states the output power and input power of the novel slow-wave structure of the TWT using VSim with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 at 2.40 GHz.

Table 5.7: Output power and input power of the novel slow-wave structure of the TWT using VSim with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 at 2.40 GHz.

Output Power Output Power Input Power Input Power [W] [dBm] [W] [dBm] 3027 64.81 1 30.00 37984 75.80 10 40.00 293383 84.67 100 50.00 937885 89.72 1000 60.00 1065500 90.28 1634 62.13

Figure 5.15 shows the simulated transfer characteristics of the novel slow-wave structure of the TWT using VSim with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 at

2.40 GHz where the highest gain occurs.

Figure 5.15: Output power versus input power of novel slow-wave structure of TWT using VSim with L = 16.0, W = 10.5, p = 22.0 [in mm], and N = 20 at 2.40 GHz.

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As shown in Figure 5.15, the linear gain was around 35 dB before reaching the saturation region where the gain becomes 28.1 dB. Figure 5.16 shows the output power versus the number of periods of the novel slow-wave structure of TWT using VSim with L = 16.0 mm, W = 10.5 mm, p = 22.0 mm, and N = 20 at 2.40 GHz where the highest gain occurs.

Figure 5.16: Output power versus number of periods of novel slow-wave structure of TWT using VSim with L = 16.0, W = 10.5, p = 22.0 [in mm], and N = 20 at 2.40 GHz. As shown in Figure 5.16, the output power is proportional to the number of periods. When

N = 33, the output power is around 91 dBm indicating, corresponding to 1.31 MW, and indicating an efficiency of 41.7% although there is a tradeoff between the circuit length and output power.

5.2 Comparison between the Novel Slow-Wave Circuit Structure, Ring-Bar Structure, Half-Ring Helical Structure, Ring-Loop Structure, Curved Ring-Bar Structure, and Wave-Ring Helical Structure

According to the obtained results, the novel slow-wave circuit structure of the traveling wave tube is expected to be a promising design towards the development of the vacuum tubes. Table 5.8 states the comparison between the designed novel slow-wave circuit structure of the traveling wave tube with L = 16.0 mm, W = 10.5 mm, and p = 22.0

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mm and existing designs including ring-bar structure, half-ring helical structure, ring loop, curved ring-bar structure, and wave-ring helical structure.

Table 5.8: Comparison between the designed novel slow-wave circuit structure of the traveling wave tube with L = 16.0 mm, W = 10.5 mm, and p = 22.0 mm and existing designs including ring-bar structure, half-ring helical structure, ring loop, curved ring-bar structure, and wave-ring helical structure.

Half-Ring Wave-Ring Novel Slow- Ring-Bar Ring-Loop Curved Ring- Parameters Helical Helical Wave Circuit Structure Structure Bar Structure Structure Structure Vary (e.g. Vary (e.g. Frequency[GHz] 1.89-2.72 X-Band, Q- 2.5-3.25 1.8-2.4 2.0-4.0 32-38) Band) Number of 20 Vary Helix Vary Unknown 26 Elements, N 814x57x57 Structure Area (p=8.0 mm) 446x54.61x54.61 Vary Vary 740x62.5x62.5 140.5x18.0x16.0 [mm3] 457x37x37 (p=4.0 mm) 1.0 k (p=8.0 Peak Output mm), Low (e.g. 1.07 M Vary 1.02 M 39.8 Power [W] 220 1300) (p=4.0 mm) 28.0 (p=8.0 mm), Vary (e.g. Gain [dB] 28.1 Vary 29.0 28.0 46.0 (p = 4.0 45) mm) 25.00 (p=8.0 mm), Bandwidth [%] 34.6 10-20 - 33.0 - 23.43 (p=4.0 mm) 38.7 (p=8.0 mm), Vary (e.g. Efficiency [%] 34.0 - 25.0 26.5 37.0 (p = 4.0 6.1) mm) 814.0 (p=8.0 Circuit Length mm), 446.0 - - 740.0 140.5 (Size) [mm] 457.0 (p = 4.0 mm) Magnetic Field Yes - - - Yes Yes Focusing Loss Pattern No - No - No - Rods No Yes - - No Yes One Bar 2 Pairs Straight One Straight One Elliptic Pair Elliptic One Straight Straight 0.30-0.45c (7.0-12.0 Vary (e.g. GHz), 0.15-0.22c Phase Velocity 0.78-0.86c 0.27-0.32c 0.70-0.78 0.11-0.12c 0.31-0.33c for 2.0-3.0 (38.0-44.0 GHz GHz) 13-35 (7.0- 12.0 GHz), Interaction Low (e.g. 30-78 21-25 30-80 43-65 50-130 20-25) Impedance [Ω] (38.0-44.0 GHz) Vary (e.g. Software Used VSim CST - CST CST CST)

From Table 5.8, the obtained results of the novel slow-wave structure are better than the previous reported designs. It is important to consider that some of these designs do not

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have the same specifications compared to the novel one. However, the comparison is taken as an overall discussion about the existing ones. It can be concluded that the novel slow- wave structure of the TWT, like some of the other ring-bar structures, has the combination of properties of the ring generating a high output power and gain, and helix, providing a wide bandwidth. Among the designs in the table, it provides the highest reported power, bandwidth, and efficiency although its circuit length is shorter than some other designs.

With N = 33, the circuit length of the novel structure becomes 740 mm and it generates an even higher power. The cost of the design is cheap as copper was assigned as the material of the slow-wave circuit structure. The novel structure of the TWT needs neither a loss pattern to suppress backward wave oscillations since such structure like bifilar helix cancels the BWO, as discussed in Chapter 2, nor support rods techniques. Instead, two pairs of transmission lines are utilized. Also, its normalized phase velocity is higher than the other designs allowing to have more energy. Last, the resulted interaction impedance for the novel structure is moderate ranging between 30 and 78 ohms. Such characteristics of the novel slow-wave structure of the traveling wave tube favor its use in a variety of applications such as small antennas, small TWTs, couplers, RF signal dividers, and low and high BWOs.

5.3 Future Work

The research work has indicated the significance of the dispersion characteristics towards the slow-wave circuit structure of traveling wave tube. In particular, the lumped elements in the designed novel structure, inductor and capacitor, and mutual elements leading to slow wave formation change the dispersion behavior. Such elements can be

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varied between the two pairs of transmission lines by modeling a geometry which has unidentical transmission lines. That is, their dimensions including the length to width ratio are different from each other. This results in unequal forward and backward waves which have different propagation constants and phase velocities. In addition, it can enhance the performance of the overall structure to improve the gain and increase bandwidth, output power, and efficiency. Figure 5.17 shows the novel slow-wave structure of the TWT with unidentical transmission lines.

Figure 5.17: Novel slow-wave structure of the TWT with unidentical transmission lines. Furthermore, although the ring-bar structure is known to suppress the backward wave oscillations (BWOs), there can still be a resulted small percentage that can reduce the gain. To optimize the design, two mechanisms can be used in the structure to suppress the

BWO. First, a step change in the period of the structure can be used between the elements to suppress the backward wave. This can be done at by reducing the period size at the output area, which corresponds to the right-hand side of the structure and enlarging the period size at the input area, which corresponds to the left-hand side of the structure. Such change results in a change in the phase velocity of the structure. Figure 5.18 shows the novel slow-wave structure of the TWT with unidentical periods.

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Figure 5.18: Novel slow-wave structure of the TWT with unidentical periods resulted due to the difference in lengths. When the wave travels through the period with a small size, energy is transferred from the beam into the backward wave and oscillations occur. As the backward wave faces the period with a larger size, it travels faster to a velocity of the space charge wave. With this phenomenon, the backward wave gets attenuated. Second, the frequency-sensitive attenuation can be used in the structure to suppress the backward wave oscillations by providing the appropriate loss. The attenuation is placed between the elements in the structure. The length of the resonant loss is half wavelength of the BWO frequency. This loss pattern was covered in Section 2.6.5.

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254

APPENDICES

• APPENDIX A

A.1 First Electron Gun Design with current density of 2 A/cm2

Inputs

V = 10000; % Voltage [Volts] I = 1; % Current [Amperes) P = 1E-6; % Perveance bm = 1E-1; % Beam radius [cm] J = 2; % Cathode emission density [A/cm^2] A = 174*sqrt(P);

Electron Gun Parameters rc = sqrt(I/(pi*J)); % Disc radius of cathode [cm]

%Step 1 Theta_Initial = 30; % Initial assumption of angle [Degrees] Alpha = sqrt((14.67*1E-6*(1-cosd(Theta_Initial))*V^1.5)/I);

%Step 2 Gamma = Alpha-(0.275*Alpha^2)+(0.06*Alpha^3)-(0.006*Alpha^4);

%Step 3 Derivative_Gamma = 1+(0.6*Gamma)+(0.225*Gamma^2)+(0.0573*Gamma^3)+(0.0108*Gamma^4)+(0.0021*Gamma^5 ); Big_Gamma = 1.25; % Correction factor tangent_phi2 = sind(Theta_Initial)*(1- ((Big_Gamma/(3*Alpha))*(Derivative_Gamma)));

%Step 4 bo = exp(-Gamma)*rc; % [cm] tangent_phi3 = A*sqrt(log(bo/bm));

%Step 5 Theta_New = Theta_Initial*sqrt(tangent_phi3/tangent_phi2); % [Degrees]

%Step 6 Theta_Final = 22.58; % [Degrees] % Theta_Final = Theta_New; % [Degrees]

Alpha_Final = sqrt((14.67*1E-6*(1-cosd(Theta_Final))*V^1.5)/I); Gamma_Final = Alpha_Final-(0.275*Alpha_Final^2)+(0.06*Alpha_Final^3)- (0.006*Alpha_Final^4); Derivative_Gamma_Final = 1+(0.6*Gamma_Final)+(0.225*Gamma_Final^2)+(0.0573*Gamma_Final^3)+(0.0108*Gamma_ Final^4)+(0.0021*Gamma_Final^5); tangent_phi2_Final = sind(Theta_Final)*(1- ((Big_Gamma/(3*Alpha_Final))*(Derivative_Gamma_Final))); bo_Final = exp(-Gamma_Final)*rc; % [cm] tangent_phi3_Final = A*sqrt(log(bo_Final/bm));

%Step 7 Rc =rc/sind(Theta_Final); % [cm] Ra = Rc*exp(-Gamma_Final); % [cm] ra = 1.2*bo_Final; % [cm] za = Rc-sqrt(Ra^2-ra^2); % [cm] zm = za+((bm/A)*((1.914*sqrt((bo_Final/bm)-1))+(0.230*((bo_Final/bm)- 1))+(0.0107*((bo_Final/bm)-1)^2)-(0.000291*((bo_Final/bm)-1))^3)); % Axial position of the beam minimum [cm]

Published with MATLAB® R2013b

256

• APPENDIX B

B.1 Electron Gun of the Proposed Novel Slow-Wave Structure

Inputs

V = 262000; % Voltage [Volts] I = 12; % Current [Amperes) P = I/V^1.5; % Perveance bm = 2E-1; % Beam radius [cm] J = 5.968; % Cathode emission density [A/cm^2] A = 174*sqrt(P);

Electron Gun Parameters rc = sqrt(I/(pi*J)); % Disc radius of cathode [cm]

%Step 1 Theta_Initial = 10; % Initial assumption of angle [Degrees] Alpha = sqrt((14.67*1E-6*(1-cosd(Theta_Initial))*V^1.5)/I);

%Step 2 Gamma = Alpha-(0.275*Alpha^2)+(0.06*Alpha^3)-(0.006*Alpha^4);

%Step 4 bo = exp(-Gamma)*rc; % [cm] tangent_phi3 = A*sqrt(log(bo/bm));

%Step 5 Theta_New = Theta_Initial*sqrt(tangent_phi3/tangent_phi2); % [Degrees]

%Step 6 Theta_Final = 6.675; % [Degrees] % Theta_Final = Theta_New; % [Degrees]

Alpha_Final = sqrt((14.67*1E-6*(1-cosd(Theta_Final))*V^1.5)/I); Gamma_Final = Alpha_Final-(0.275*Alpha_Final^2)+(0.06*Alpha_Final^3)- (0.006*Alpha_Final^4);

257

Derivative_Gamma_Final = 1+(0.6*Gamma_Final)+(0.225*Gamma_Final^2)+(0.0573*Gamma_Final^3)+(0.0108*Gamma_ Final^4)+(0.0021*Gamma_Final^5); tangent_phi2_Final = sind(Theta_Final)*(1- ((Big_Gamma/(3*Alpha_Final))*(Derivative_Gamma_Final))); bo_Final = exp(-Gamma_Final)*rc; % [cm] tangent_phi3_Final = A*sqrt(log(bo_Final/bm));

Published with MATLAB® R2013b

258

• APPENDIX C

C.1 Electron Gun Plots and Analysis

Inputs

V = 18.2E3; % Voltage [Volts] I = 0.050; % Current [Amperes) P = I/V^1.5; % Perveance bm = 0.0375E-1; % Beam radius [cm] J = 2; % Cathode emission density [A/cm^2] A = 174*sqrt(P);

Plot of the disc radius of the cathode versus the cathode emission density [rc vs. J]

J1 = 1:0.001:200; %Range of cathode emission density I2 = 1; rc = sqrt(I./(pi*J1)); % Disc radius of cathode [cm] rc1 = rc; rc2 = sqrt(I2./(pi*J1)); figure(1) plot(J1,rc, 'r'); legend('I = 0.050 Amp'); xlabel('J [A/cm^{2}]'); ylabel('r_c [cm]'); grid on; title 'Disc Radius of Cathode Vs. Cathode Emission Density';

figure(2) plot(J1,rc1, 'r',J1,rc2,'b'); legend('I = 0.050 Amp','I = 1 Amp'); xlabel('J [A/cm^{2}]'); ylabel('r_c [cm]'); grid on; title 'Disc Radius of Cathode Vs. Cathode Emission Density';

Plot of the disc radius of the cathode versus the Beam Current [rc vs. I]

I3 = 0.001:0.00001:1; %Range of Current J2 = 2; J10 = 10; J50 = 50;

259

J100 = 100; rc3 = sqrt(I3./(pi*J2)); % Disc radius of cathode [cm] rc4 = sqrt(I3./(pi*J10)); % Disc radius of cathode [cm] rc5 = sqrt(I3./(pi*J50)); % Disc radius of cathode [cm] rc6 = sqrt(I3./(pi*J100)); % Disc radius of cathode [cm] figure(3) plot(I3,rc3, 'r',I3,rc4, 'b',I3,rc5, 'g',I3,rc6, 'c'); legend('J = 2 A/cm^{2}','J = 10 A/cm^{2}','J = 50 A/cm^{2}','J = 100 A/cm^{2}','Location','northwest'); xlabel('I [A]'); ylabel('r_c [cm]'); grid on; title 'Disc Radius of Cathode Vs. Beam Current';

Plot of alpha constant Vs. theta

Theta_Final = 0:0.001:10; % Range of Perveance

P2 =1E-6; % Perveance for 10000 V and 1 Amp

Alpha1 = sqrt((14.67*1E-6*(1-cosd(Theta_Final)))./P);

figure(4) plot(Alpha1,Theta_Final, 'r'); legend('V = 18.2 kV, I = 0.050 Amp','Location','northwest'); xlabel('(-\alpha)'); ylabel('\theta [^{o}]'); grid on; title '\theta Vs. \alpha';

Theta_Final1 = 0:0.001:30; % Range of Perveance

Alpha2 = sqrt((14.67*1E-6*(1-cosd(Theta_Final1)))./P); Alpha3 = sqrt((14.67*1E-6*(1-cosd(Theta_Final1)))./P2); figure(5) plot(Alpha2,Theta_Final1, 'r', Alpha3,Theta_Final1, 'b');

260

legend('V = 18.2 kV, I = 0.050 Amp','V = 10.0 kV, I = 1 Amp','Location','northwest'); xlabel('(-\alpha)'); ylabel('\theta [^{o}]'); grid on; title '\theta Vs. \alpha';

Plot of alpha constant Vs. voltage

V1 = 0:1000:20000; % Range of Voltage Theta_Final1 = 30; Theta_Final2 = 20; Theta_Final3 = 10; Theta_Final4 = 5;

Alpha4 = sqrt((14.67*1E-6*(1-cosd(Theta_Final1)))*V1.^1.5./I); Alpha5 = sqrt((14.67*1E-6*(1-cosd(Theta_Final2)))*V1.^1.5./I); Alpha6 = sqrt((14.67*1E-6*(1-cosd(Theta_Final3)))*V1.^1.5./I); Alpha7 = sqrt((14.67*1E-6*(1-cosd(Theta_Final4)))*V1.^1.5./I);

figure(6) plot(Alpha4,V1, 'r',Alpha5,V1, 'b',Alpha6,V1, 'g',Alpha7,V1, 'c'); legend('\theta = 30^{o}','\theta = 20^{o}','\theta = 10^{o}','\theta = 5^{o}'); xlabel('(-\alpha)'); ylabel('Voltage [V]'); grid on; title 'Voltage Vs. \alpha';

Plot of alpha constant Vs. perveance

P2 = 1E-9:0.000000001:100E-9; % Range of Perveance Theta_Final1 = 30; Theta_Final2 = 20; Theta_Final3 = 10; Theta_Final4 = 5;

Alpha8 = sqrt((14.67*1E-6*(1-cosd(Theta_Final1)))./P2); Alpha9 = sqrt((14.67*1E-6*(1-cosd(Theta_Final2)))./P2); Alpha10 = sqrt((14.67*1E-6*(1-cosd(Theta_Final3)))./P2); Alpha11 = sqrt((14.67*1E-6*(1-cosd(Theta_Final4)))./P2);

261

figure(7) plot(Alpha8,P2, 'r',Alpha9,P2, 'b',Alpha10,P2, 'g',Alpha11,P2, 'c'); legend('\theta = 30^{o}','\theta = 20^{o}','\theta = 10^{o}','\theta = 5^{o}'); xlabel('(-\alpha)'); ylabel('P'); xlim ([0 10]) grid on; title 'Perveance Vs. \alpha';

Plot of alpha constant Vs. current

I1 = 1E-3:0.0001:100E-3; % Range of Current Theta_Final1 = 30; Theta_Final2 = 20; Theta_Final3 = 10; Theta_Final4 = 5;

Alpha12 = sqrt((14.67*1E-6*(1-cosd(Theta_Final1)))*V.^1.5./I1); Alpha13 = sqrt((14.67*1E-6*(1-cosd(Theta_Final2)))*V.^1.5./I1); Alpha14 = sqrt((14.67*1E-6*(1-cosd(Theta_Final3)))*V.^1.5./I1); Alpha15 = sqrt((14.67*1E-6*(1-cosd(Theta_Final4)))*V.^1.5./I1);

figure(8) plot(Alpha12,I1, 'r',Alpha13,I1, 'b',Alpha14,I1, 'g',Alpha15,I1, 'c'); legend('\theta = 30^{o}','\theta = 20^{o}','\theta = 10^{o}','\theta = 5^{o}'); xlabel('(-\alpha)'); ylabel('I [A]'); xlim ([0 10]) grid on; title 'Current Vs. \alpha';

Plot of alpha constant Vs. gamma

Alpha16 = 0:0.01:7; % Range of Alpha

Gamma1 = Alpha16-(0.275*Alpha16.^2)+(0.06*Alpha16.^3)-(0.006*Alpha16.^4);

figure(9) plot(Alpha16,Gamma1, 'r'); xlabel('(-\alpha)'); ylabel('\gamma'); grid on;

262

title '\gamma Vs. \alpha';

Plot of bo Vs. gamma

Gamma2= 0.0:0.001:3; % Range of Gamma rc3 = 0.0892; rc4 = 0.0399; rc5 = 0.0178; rc6 = 0.0126;

bo1 = exp(-Gamma2).*rc3; bo2 = exp(-Gamma2).*rc4; bo3 = exp(-Gamma2).*rc5; bo4 = exp(-Gamma2).*rc6;

figure(10) plot(Gamma2,bo1, 'r',Gamma2,bo2, 'b',Gamma2,bo3, 'g',Gamma2,bo4, 'c'); legend('r_c = 0.0892 cm','r_c = 0.0399 cm','r_c = 0.0178 cm','r_c = 0.0126 cm'); xlabel('\gamma'); ylabel('b_o [cm]'); grid on; title 'b_o Vs. \gamma';

Plot of bo Vs. rc rc7= 0.0:0.0001:0.1; % Range of disc radius of cathode Gamma3 = 1.0563; Gamma4 = 0.9656; Gamma5 = 0.8445; Gamma6 = 0.7738;

bo5 = exp(-Gamma3).*rc7; bo6 = exp(-Gamma4).*rc7; bo7 = exp(-Gamma5).*rc7; bo8 = exp(-Gamma6).*rc7;

figure(11) plot(rc7,bo5, 'r',rc7,bo6, 'b',rc7,bo7, 'g',rc7,bo8, 'c'); legend('\gamma = 1.0563','\gamma = 0.9656','\gamma = 0.8445','\gamma = 0.7738', 'Location','northwest');

263

xlabel('r_c [cm]'); ylabel('b_o [cm]'); grid on; title 'b_o Vs. Disc Radius of Cathode';

Plot of Rc Vs. rc rc8= 0.0:0.0001:0.1; % Range of disc radius of cathode Theta_Final5 = 4.5473; Theta_Final6 = 4.0140; Theta_Final7 = 3.3533; Theta_Final8 = 2.9936;

Rc1 =rc8./sind(Theta_Final5); % [cm] Rc2 =rc8./sind(Theta_Final6); % [cm] Rc3 =rc8./sind(Theta_Final7); % [cm] Rc4 =rc8./sind(Theta_Final8); % [cm]

figure(12) plot(rc8,Rc1, 'r',rc8,Rc2, 'b',rc8,Rc3, 'g',rc8,Rc4, 'c'); legend('\theta = 4.5473^{o}','\theta = 4.0140^{o}','\theta = 3.3533^{o}','\theta = 2.9936^{o}', 'Location','northwest'); xlabel('r_c [cm]'); ylabel('R_c [cm]'); grid on; title 'R_c Vs. Disc Radius of Cathode';

Plot of Rc Vs. theta_final

Theta_Final9= 0.0:0.001:10; % Range of Final Angle rc9 = 0.0892; rc10 = 0.0399; rc11 = 0.0178; rc12 = 0.0126;

Rc5 =rc9./sind(Theta_Final9); % [cm] Rc6 =rc10./sind(Theta_Final9); % [cm] Rc7 =rc11./sind(Theta_Final9); % [cm] Rc8 =rc12./sind(Theta_Final9); % [cm]

figure(13)

264

plot(Theta_Final9,Rc5, 'r',Theta_Final9,Rc6, 'b',Theta_Final9,Rc7, 'g',Theta_Final9,Rc8, 'c'); legend('r_c = 0.0892 cm','r_c = 0.0399 cm','r_c = 0.0178 cm','r_c = 0.0126 cm'); xlabel('\theta [^{o}]'); ylabel('R_c [cm]'); ylim ([0 3]) grid on; title 'R_c Vs. Angle \theta';

Plot of Ra Vs. gamma

Gamma7= 0.0:0.001:3; % Range of Gamma Rc9 = 1.1252; Rc10 = 0.5699; Rc11 = 0.3050; Rc12 = 0.2416;

Ra1 = exp(-Gamma7).*Rc9; Ra2 = exp(-Gamma7).*Rc10; Ra3 = exp(-Gamma7).*Rc11; Ra4 = exp(-Gamma7).*Rc12;

figure(14) plot(Gamma7,Ra1, 'r',Gamma7,Ra2, 'b',Gamma7,Ra3, 'g',Gamma7,Ra4, 'c'); legend('R_c = 1.1252 cm','R_c = 0.5699 cm','R_c = 0.3050 cm','R_c = 0.2416 cm'); xlabel('\gamma'); ylabel('R_a [cm]'); grid on; title 'R_a Vs. \gamma';

Plot of Ra Vs. Rc

Rc13= 0.0:0.0001:2.0; % Range of Rc Gamma8 = 1.0563; Gamma9 = 0.9656; Gamma10 = 0.8445; Gamma11 = 0.7738;

Ra5 = exp(-Gamma8).*Rc13; Ra6 = exp(-Gamma9).*Rc13;

265

Ra7 = exp(-Gamma10).*Rc13; Ra8 = exp(-Gamma11).*Rc13;

figure(15) plot(Rc13,Ra5, 'r',Rc13,Ra6, 'b',Rc13,Ra7, 'g',Rc13,Ra8, 'c'); legend('\gamma = 1.0563','\gamma = 0.9656','\gamma = 0.8445','\gamma = 0.7738', 'Location','northwest'); xlabel('R_c [cm]'); ylabel('R_a [cm]'); grid on; title 'R_a Vs. R_c';

Plot of ra Vs. bo bo9= 0.0:0.0001:0.15; % Range of disc radius of cathode ra1 = 1.2*bo9; % [cm]

figure(16) plot(bo9,ra1, 'r'); xlabel('b_o [cm]'); ylabel('r_a [cm]'); grid on; title 'r_a Vs. b_o';

Plot of za Vs. ra ra2= 0.0:0.0001:0.05; % Range of ra

Rc13 = 1.1252; Rc14 = 0.5699; Rc15 = 0.3050; Rc16 = 0.2416;

Ra9 = 0.3913; Ra10 = 0.2170; Ra11 = 0.1311; Ra12 = 0.1114; za1 = Rc13-sqrt(Ra9.^2-ra2.^2); % [cm] za2 = Rc14-sqrt(Ra10.^2-ra2.^2); % [cm]

266

za3 = Rc15-sqrt(Ra11.^2-ra2.^2); % [cm] za4 = Rc16-sqrt(Ra12.^2-ra2.^2); % [cm]

figure(17) plot(ra2,za1, 'r',ra2,za2, 'b',ra2,za3, 'g',ra2,za4, 'c'); legend('R_c = 1.1252 cm and R_a = 0.3913 cm ','R_c = 0.5699 cm and R_a = 0.2170 cm','R_c = 0.3050 cm and R_a = 0.1311 cm','R_c = 0.2416 cm and R_a = 0.1114 cm', 'Location','northoutside'); xlabel('r_a [cm]'); ylabel('z_a [cm]'); ylim ([0 1]) grid on; title 'z_a Vs. r_a';

Plot of za Vs. Ra

Ra13= 0.0:0.0001:0.5; % Range of Ra

Rc13 = 1.1252; Rc14 = 0.5699; Rc15 = 0.3050; Rc16 = 0.2416;

ra3 = 0.0372; ra4 = 0.0182; ra5 = 0.0092; ra6 = 0.0070; za5 = Rc13-sqrt(Ra13.^2-ra3.^2); % [cm] za6 = Rc14-sqrt(Ra13.^2-ra4.^2); % [cm] za7 = Rc15-sqrt(Ra13.^2-ra5.^2); % [cm] za8 = Rc16-sqrt(Ra13.^2-ra6.^2); % [cm]

figure(18) plot(Ra13,za5, 'r',Ra13,za6, 'b',Ra13,za7, 'g',Ra13,za8, 'c'); legend('R_c = 1.1252 cm and r_a = 0.0372 cm ','R_c = 0.5699 cm and r_a = 0.0182 cm','R_c = 0.3050 cm and r_a = 0.0092 cm','R_c = 0.2416 cm and r_a = 0.0070 cm', 'Location','northoutside'); xlabel('R_a [cm]');

267

ylabel('z_a [cm]'); ylim ([0 1.5]) grid on; title 'z_a Vs. R_a';

Plot of za Vs. Rc

Rc17= 0.0:0.001:1.5; % Range of Rc

Ra9 = 0.3913; Ra10 = 0.2170; Ra11 = 0.1311; Ra12 = 0.1114; ra3 = 0.0372; ra4 = 0.0182; ra5 = 0.0092; ra6 = 0.0070; za9 = Rc17-sqrt(Ra9.^2-ra3.^2); % [cm] za10 = Rc17-sqrt(Ra10.^2-ra4.^2); % [cm] za11 = Rc17-sqrt(Ra11.^2-ra5.^2); % [cm] za12 = Rc17-sqrt(Ra12.^2-ra6.^2); % [cm]

figure(19) plot(Rc17,za9, 'r',Rc17,za10, 'b',Rc17,za11, 'g',Rc17,za12, 'c'); legend('R_a = 0.3913 cm and r_a = 0.0372 cm ','R_a = 0.2170 cm and r_a = 0.0182 cm','R_a = 0.1311 cm and r_a = 0.0092 cm','R_a = 0.1114 cm and r_a = 0.0070 cm', 'Location','northoutside'); xlabel('R_c [cm]'); ylabel('z_a [cm]'); grid on; title 'z_a Vs. R_c';

Plot of zm Vs. bo bo10= 0.005:0.0001:0.05; % Range of bo za13 = 0.7357; za14 = 0.3537; za15 = 0.1742; za16 = 0.1304;

268

zm1 = za13+((bm/A)*((1.914*sqrt((bo10/bm)-1))+(0.230*((bo10/bm)- 1))+(0.0107*((bo10/bm)-1).^2)-(0.000291*((bo10/bm)-1)).^3)); % Axial position of the beam minimum [cm] zm2 = za14+((bm/A)*((1.914*sqrt((bo10/bm)-1))+(0.230*((bo10/bm)- 1))+(0.0107*((bo10/bm)-1).^2)-(0.000291*((bo10/bm)-1)).^3)); % Axial position of the beam minimum [cm] zm3 = za15+((bm/A)*((1.914*sqrt((bo10/bm)-1))+(0.230*((bo10/bm)- 1))+(0.0107*((bo10/bm)-1).^2)-(0.000291*((bo10/bm)-1)).^3)); % Axial position of the beam minimum [cm] zm4 = za16+((bm/A)*((1.914*sqrt((bo10/bm)-1))+(0.230*((bo10/bm)- 1))+(0.0107*((bo10/bm)-1).^2)-(0.000291*((bo10/bm)-1)).^3)); % Axial position of the beam minimum [cm]

figure(20) plot(bo10,zm1, 'r',bo10,zm2, 'b',bo10,zm3, 'g',bo10,zm4, 'c'); legend('z_a = 0.7357 cm','z_a = 0.3537 cm','z_a = 0.1742 cm','z_a = 0.1304 cm', 'Location','northwest'); xlabel('b_o [cm]'); ylabel('z_m [cm]'); xlim ([0.005 0.05]) grid on; title 'Axial Position of Beam Minimum Vs. b_o';

Plot of zm Vs. bm bm1= 0.001:0.0001:0.01; % Range of bm za13 = 0.7357; za14 = 0.3537; za15 = 0.1742; za16 = 0.1304; bo11 = 0.0310; bo12 = 0.0152; bo13 = 0.0077; bo14 = 0.0058;

269

zm5 = za13+((bm1./A).*((1.914.*sqrt((bo11./bm1)-1))+(0.230.*((bo11./bm1)- 1))+(0.0107.*((bo11./bm1)-1).^2)-(0.000291.*((bo11./bm1)-1)).^3)); % Axial position of the beam minimum [cm] zm6 = za14+((bm1./A).*((1.914.*sqrt((bo12./bm1)-1))+(0.230.*((bo12./bm1)- 1))+(0.0107.*((bo12./bm1)-1).^2)-(0.000291.*((bo12./bm1)-1)).^3)); % Axial position of the beam minimum [cm] zm7 = za15+((bm1./A).*((1.914.*sqrt((bo13./bm1)-1))+(0.230.*((bo13./bm1)- 1))+(0.0107.*((bo13./bm1)-1).^2)-(0.000291.*((bo13./bm1)-1)).^3)); % Axial position of the beam minimum [cm] zm8 = za16+((bm1./A).*((1.914.*sqrt((bo14./bm1)-1))+(0.230.*((bo14./bm1)- 1))+(0.0107.*((bo14./bm1)-1).^2)-(0.000291.*((bo14./bm1)-1)).^3)); % Axial position of the beam minimum [cm]

figure(21) plot(bm1,zm5, 'r',bm1,zm6, 'b',bm1,zm7, 'g',bm1,zm8, 'c'); legend('z_a = 0.7357 cm and b_o = 0.0310 cm ','z_a = 0.3537 cm and b_o = 0.0152 cm','z_a = 0.1742 cm and b_o = 0.0077 cm','z_a = 0.1304 cm and b_o = 0.0058 cm', 'Location','northoutside'); xlabel('b_m [cm]'); ylabel('z_m [cm]'); grid on; title 'Axial Position of Beam Minimum Vs. Minimum Beam Radius';

Plot of zm Vs. za za17= 0.00:0.001:1; % Range of za

bo11 = 0.0310; bo12 = 0.0152; bo13 = 0.0077; bo14 = 0.0058;

zm9 = za17+((bm./A).*((1.914.*sqrt((bo11./bm)-1))+(0.230.*((bo11./bm)- 1))+(0.0107.*((bo11./bm)-1).^2)-(0.000291.*((bo11./bm)-1)).^3)); % Axial position of the beam minimum [cm] zm10 = za17+((bm./A).*((1.914.*sqrt((bo12./bm)-1))+(0.230.*((bo12./bm)- 1))+(0.0107.*((bo12./bm)-1).^2)-(0.000291.*((bo12./bm)-1)).^3)); % Axial position of the beam minimum [cm]

270

zm11 = za17+((bm./A).*((1.914.*sqrt((bo13./bm)-1))+(0.230.*((bo13./bm)- 1))+(0.0107.*((bo13./bm)-1).^2)-(0.000291.*((bo13./bm)-1)).^3)); % Axial position of the beam minimum [cm] zm12 = za17+((bm./A).*((1.914.*sqrt((bo14./bm)-1))+(0.230.*((bo14./bm)- 1))+(0.0107.*((bo14./bm)-1).^2)-(0.000291.*((bo14./bm)-1)).^3)); % Axial position of the beam minimum [cm]

figure(22) plot(za17,zm9, 'r',za17,zm10, 'b',za17,zm11, 'g',za17,zm12, 'c'); legend('b_o = 0.0310 cm ','b_o = 0.0152 cm','b_o = 0.0077 cm','b_o = 0.0058 cm','Location','northwest'); xlabel('z_a [cm]'); ylabel('z_m [cm]'); grid on; title 'Axial Position of Beam Minimum Vs. z_a';

Plot of zm Vs. perveance

P3= 1E-9:0.00000000001:50E-9; % Range of P3

za13 = 0.7357; za14 = 0.3537; za15 = 0.1742; za16 = 0.1304; bo11 = 0.0310; bo12 = 0.0152; bo13 = 0.0077; bo14 = 0.0058;

zm13 = za13+((bm./(174.*sqrt(P3))).*((1.914.*sqrt((bo11./bm)- 1))+(0.230.*((bo11./bm)-1))+(0.0107.*((bo11./bm)-1).^2)-(0.000291.*((bo11./bm)- 1)).^3)); % Axial position of the beam minimum [cm] zm14 = za14+((bm./(174.*sqrt(P3))).*((1.914.*sqrt((bo12./bm)- 1))+(0.230.*((bo12./bm)-1))+(0.0107.*((bo12./bm)-1).^2)-(0.000291.*((bo12./bm)- 1)).^3)); % Axial position of the beam minimum [cm] zm15 = za15+((bm./(174.*sqrt(P3))).*((1.914.*sqrt((bo13./bm)- 1))+(0.230.*((bo13./bm)-1))+(0.0107.*((bo13./bm)-1).^2)-(0.000291.*((bo13./bm)- 1)).^3)); % Axial position of the beam minimum [cm]

271

zm16 = za16+((bm./(174.*sqrt(P3))).*((1.914.*sqrt((bo14./bm)- 1))+(0.230.*((bo14./bm)-1))+(0.0107.*((bo14./bm)-1).^2)-(0.000291.*((bo14./bm)- 1)).^3)); % Axial position of the beam minimum [cm]

figure(23) plot(P3,zm13, 'r',P3,zm14, 'b',P3,zm15, 'g',P3,zm16, 'c'); legend('z_a = 0.7357 cm and b_o = 0.0310 cm ','z_a = 0.3537 cm and b_o = 0.0152 cm','z_a = 0.1742 cm and b_o = 0.0077 cm','z_a = 0.1304 cm and b_o = 0.0058 cm', 'Location','northoutside'); xlabel('P'); ylabel('z_m [cm]'); ylim ([0 2]) xlim ([1E-9 50E-9]) grid on; title 'Axial Position of Beam Minimum Vs. Perveance';

Plot of tangentphi3 Vs. bo bo15= 0.005:0.0001:0.05; % Range of bo

tangent_phi3_1 = A*sqrt(log(bo15./bm));

figure(24) plot(bo15,tangent_phi3_1, 'r'); legend('Beam Voltage = 18.2 kV, Beam Current = 0.050 A, b_m = 0.00375 cm', 'Location','northoutside'); xlabel('b_o [cm]'); ylabel('tan{\phi_3}'); grid on; title 'tan{\phi_3} Vs. b_o ';

Plot of tangentphi3 Vs. bm bm2= 0.001:0.0001:0.01; % Range of bm bo16 = 0.0310; bo17 = 0.0152; bo18 = 0.0077; bo19 = 0.0058;

272

tangent_phi3_2 = A*sqrt(log(bo16./bm2)); tangent_phi3_3 = A*sqrt(log(bo17./bm2)); tangent_phi3_4 = A*sqrt(log(bo18./bm2)); tangent_phi3_5 = A*sqrt(log(bo19./bm2));

figure(25) plot(bm2,tangent_phi3_2, 'r',bm2,tangent_phi3_3, 'b',bm2,tangent_phi3_4, 'g',bm2,tangent_phi3_5, 'c' ); legend('b_o = 0.0310 cm','b_o = 0.0152 cm','b_o = 0.0077 cm','b_o = 0.0058 cm', 'Location','northoutside'); xlabel('b_m [cm]'); ylabel('tan{\phi_3}'); grid on; title 'tan{\phi_3} Vs. Minimum Beam Radius ';

Plot of tangentphi3 Vs. perveance

P3= 0:0.00000000001:50E-9; % Range of P3

bo16 = 0.0310; bo17 = 0.0152; bo18 = 0.0077; bo19 = 0.0058; tangent_phi3_6 = (174.*sqrt(P3))*sqrt(log(bo16./bm)); tangent_phi3_7 = (174.*sqrt(P3))*sqrt(log(bo17./bm)); tangent_phi3_8 = (174.*sqrt(P3))*sqrt(log(bo18./bm)); tangent_phi3_9 = (174.*sqrt(P3))*sqrt(log(bo19./bm));

figure(26) plot(P3,tangent_phi3_6, 'r',P3,tangent_phi3_7, 'b',P3,tangent_phi3_8, 'g',P3,tangent_phi3_9, 'c' ); legend('b_o = 0.0310 cm','b_o = 0.0152 cm','b_o = 0.0077 cm','b_o = 0.0058 cm', 'Location','northoutside'); xlabel('P'); ylabel('tan{\phi_3}'); grid on; title 'tan{\phi_3} Vs. Perveance ';

273

Plot of derivative_gamma Vs. gamma

Gamma12= 0.0:0.001:3; % Range of Gamma

Derivative_Gamma1 = 1+(0.6*Gamma12)+(0.225*Gamma12.^2)+(0.0573*Gamma12.^3)+(0.0108*Gamma12.^4)+(0.0 021*Gamma12.^5);

figure(27) plot(Gamma12,Derivative_Gamma1, 'r'); xlabel('\gamma'); ylabel('df(\gamma)/d\gamma'); grid on; title 'df(\gamma)/d\gamma Vs. \gamma ';

Plot of tangentphi2 Vs. Ra

Ra14= 0.0:0.0001:0.5; % Range of Ra

Big_Gamma = 1.25; % Correction factor bo11 = 0.0310; bo12 = 0.0152; bo13 = 0.0077; bo14 = 0.0058;

Derivative_Gamma_2 = 1.9686; Derivative_Gamma_3 = 1.8519; Derivative_Gamma_4 = 1.7080; Derivative_Gamma_5 = 1.6300;

Alpha17 = 1.5059; Alpha18 = 1.3293; Alpha19 = 1.1106; Alpha20 = 0.9915; tangent_phi2_1 = (bo11./Ra14)*(1- ((Big_Gamma/(3*Alpha17))*(Derivative_Gamma_2))); tangent_phi2_2 = (bo12./Ra14)*(1- ((Big_Gamma/(3*Alpha18))*(Derivative_Gamma_3)));

274

tangent_phi2_3 = (bo13./Ra14)*(1- ((Big_Gamma/(3*Alpha19))*(Derivative_Gamma_4))); tangent_phi2_4 = (bo14./Ra14)*(1- ((Big_Gamma/(3*Alpha20))*(Derivative_Gamma_5)));

% tangent_phi2_1 = sind(Theta_Initial)*(1- ((Big_Gamma/(3*Alpha))*(Derivative_Gamma)));

figure(28) plot(Ra14,tangent_phi2_1, 'r',Ra14,tangent_phi2_2, 'b',Ra14,tangent_phi2_3, 'g',Ra14,tangent_phi2_4, 'c' ); legend('b_o = 0.0310 cm, (-\alpha) = 1.5059 and df(\gamma)/(d\gamma) = 1.9689','b_o = 0.0152 cm, (-\alpha) = 1.3293, and df(\gamma)/(d\gamma) = 1.8519','b_o = 0.0077 cm, (-\alpha) = 1.1106, and df(\gamma)/(d\gamma) = 1.7080','b_o = 0.0058 cm, (-\alpha) = 0.9915, and df(\gamma)/(d\gamma) = 1.6300', 'Location','northoutside'); xlabel('R_a [cm]'); ylabel('tan{\phi_2} (b_o/F)'); ylim ([0 0.05]) grid on; title 'Slope of Trajectory Vs. R_a ';

Plot of tangentphi2 Vs. alpha

Alpha21 = 0:0.001:1.6; % Range of Alpha

Big_Gamma = 1.25; % Correction factor

Theta_Final5 = 4.5473; Theta_Final6 = 4.0140; Theta_Final7 = 3.3533; Theta_Final8 = 2.9936;

Derivative_Gamma_2 = 1.9686; Derivative_Gamma_3 = 1.8519; Derivative_Gamma_4 = 1.7080; Derivative_Gamma_5 = 1.6300;

tangent_phi2_5 = sind(Theta_Final5)*(1- ((Big_Gamma./(3*Alpha21))*(Derivative_Gamma_2)));

275

tangent_phi2_6 = sind(Theta_Final6)*(1- ((Big_Gamma./(3*Alpha21))*(Derivative_Gamma_3))); tangent_phi2_7 = sind(Theta_Final7)*(1- ((Big_Gamma./(3*Alpha21))*(Derivative_Gamma_4))); tangent_phi2_8 = sind(Theta_Final8)*(1- ((Big_Gamma./(3*Alpha21))*(Derivative_Gamma_5)));

figure(29) plot(Alpha21,tangent_phi2_5, 'r',Alpha21,tangent_phi2_6, 'b',Alpha21,tangent_phi2_7, 'g',Alpha21,tangent_phi2_8, 'c' ); legend('\theta = 4.5473^{o} and df(\gamma)/(d\gamma) = 1.9689','\theta = 4.0140^{o}, and df(\gamma)/(d\gamma) = 1.8519','\theta = 3.3533^{o}, and df(\gamma)/(d\gamma) = 1.7080','\theta = 2.9936^{o}, and df(\gamma)/(d\gamma) = 1.6300', 'Location','northoutside'); xlabel('(-\alpha)'); ylabel('tan{\phi_2} (b_o/F)'); ylim ([0 0.05]) grid on; title 'Slope of Trajectory Vs. (-\alpha) ';

Plot of tangentphi2 Vs. Ra for different big_gammas

Ra15= 0.0:0.0001:0.5; % Range of Ra

Big_Gamma1 = 1.25; % Correction factor Big_Gamma2 = 1.0; % Correction factor Big_Gamma3 = 1.1; % Correction factor bo11 = 0.0310; bo12 = 0.0152; bo13 = 0.0077; bo14 = 0.0058;

Derivative_Gamma_2 = 1.9686; Derivative_Gamma_3 = 1.8519; Derivative_Gamma_4 = 1.7080; Derivative_Gamma_5 = 1.6300;

Alpha17 = 1.5059; Alpha18 = 1.3293; Alpha19 = 1.1106; Alpha20 = 0.9915;

276

tangent_phi2_9 = (bo11./Ra15)*(1- ((Big_Gamma1/(3*Alpha17))*(Derivative_Gamma_2))); tangent_phi2_10 = (bo12./Ra15)*(1- ((Big_Gamma1/(3*Alpha18))*(Derivative_Gamma_3))); tangent_phi2_11 = (bo13./Ra15)*(1- ((Big_Gamma1/(3*Alpha19))*(Derivative_Gamma_4))); tangent_phi2_12 = (bo14./Ra15)*(1- ((Big_Gamma1/(3*Alpha20))*(Derivative_Gamma_5))); tangent_phi2_13 = (bo11./Ra15)*(1- ((Big_Gamma2/(3*Alpha17))*(Derivative_Gamma_2))); tangent_phi2_14 = (bo12./Ra15)*(1- ((Big_Gamma2/(3*Alpha18))*(Derivative_Gamma_3))); tangent_phi2_15 = (bo13./Ra15)*(1- ((Big_Gamma2/(3*Alpha19))*(Derivative_Gamma_4))); tangent_phi2_16 = (bo14./Ra15)*(1- ((Big_Gamma2/(3*Alpha20))*(Derivative_Gamma_5))); tangent_phi2_17 = (bo11./Ra15)*(1- ((Big_Gamma3/(3*Alpha17))*(Derivative_Gamma_2))); tangent_phi2_18 = (bo12./Ra15)*(1- ((Big_Gamma3/(3*Alpha18))*(Derivative_Gamma_3))); tangent_phi2_19 = (bo13./Ra15)*(1- ((Big_Gamma3/(3*Alpha19))*(Derivative_Gamma_4))); tangent_phi2_20 = (bo14./Ra15)*(1- ((Big_Gamma3/(3*Alpha20))*(Derivative_Gamma_5)));

% tangent_phi2_1 = sind(Theta_Initial)*(1- ((Big_Gamma/(3*Alpha))*(Derivative_Gamma)));

figure(30) plot(Ra14,tangent_phi2_9, 'r',Ra14,tangent_phi2_10, 'b',Ra14,tangent_phi2_11, 'g',Ra14,tangent_phi2_12, 'c',Ra14,tangent_phi2_13, 'r--',Ra14,tangent_phi2_14, 'b--',Ra14,tangent_phi2_15, 'g--',Ra14,tangent_phi2_16, 'c-- ',Ra14,tangent_phi2_17, 'r:',Ra14,tangent_phi2_18, 'b:',Ra14,tangent_phi2_19, 'g:',Ra14,tangent_phi2_20, 'c:' ); legend('\Gamma = 1.25, b_o = 0.0310 cm, (-\alpha) = 1.5059 and df(\gamma)/(d\gamma) = 1.9689','\Gamma = 1.25,b_o = 0.0152 cm, (-\alpha) = 1.3293, and df(\gamma)/(d\gamma) = 1.8519','\Gamma = 1.25,b_o = 0.0077 cm, (- \alpha) = 1.1106, and df(\gamma)/(d\gamma) = 1.7080','\Gamma = 1.25,b_o = 0.0058 cm, (-\alpha) = 0.9915, and df(\gamma)/(d\gamma) = 1.6300','\Gamma = 1.00, b_o = 0.0310 cm, (-\alpha) = 1.5059 and df(\gamma)/(d\gamma) = 1.9689','\Gamma = 1.00,b_o = 0.0152 cm, (-\alpha) = 1.3293, and

277

df(\gamma)/(d\gamma) = 1.8519','\Gamma = 1.00,b_o = 0.0077 cm, (-\alpha) = 1.1106, and df(\gamma)/(d\gamma) = 1.7080','\Gamma = 1.00,b_o = 0.0058 cm, (- \alpha) = 0.9915, and df(\gamma)/(d\gamma) = 1.6300','\Gamma = 1.10, b_o = 0.0310 cm, (-\alpha) = 1.5059 and df(\gamma)/(d\gamma) = 1.9689','\Gamma = 1.10,b_o = 0.0152 cm, (-\alpha) = 1.3293, and df(\gamma)/(d\gamma) = 1.8519','\Gamma = 1.10,b_o = 0.0077 cm, (-\alpha) = 1.1106, and df(\gamma)/(d\gamma) = 1.7080','\Gamma = 1.10,b_o = 0.0058 cm, (-\alpha) = 0.9915, and df(\gamma)/(d\gamma) = 1.6300','Location','northoutside'); xlabel('R_a [cm]'); ylabel('tan{\phi_2} (b_o/F)'); ylim ([0 0.05]) grid on; title 'Slope of Trajectory Vs. R_a for Different Correcto Factors';

Published with MATLAB® R2013b

278

• APPENDIX D

D.1 First Electron Gun with current density of 2 A/cm2

- Design 1: • Boundary File

EGUN 10 kV, 1 A SAA 11-15-15 &INPUT1 RLIM=100, ZLIM=200, POTN=5, POT=0.0,10000.0,0.0, 0.0,0.0, &END 1 0 0 1 20.0 4.0 4 20.0 4.0 4 28.45 7.5 4 35.0 35.0 4 35.0 57.5 1004 37.0 57.5 2 4 37.7 59.5 2 37.7 82.5 2 15.0 82.5 2 15.0 31.5 2 11.65 31.5 2 9.0 51.95 2 9.0 200.0 0 0 200.0 1 0 0

• Input File EGUN 10 kV, 1 A SAA 11-15-15 &INPUT1 RLIM =100, ZLIM = 200, NBND = 1100, POTN = 4, POT( 2) = 1.0000E+04, MAGSEG = 0, IAX = 0, AQUAD = 0.0, CSYS = 1, MI = 3, SX = 7.50, SY = 1.41, SCALE =' ', PASS = 2, XR =0.9950, ERROR =0.1000E-02, TYME= 300.0, LSTPOT = 3, LSTMAG = 1, LSTBND = 1, INTPA = 0, POIS = 0, &END 1 0 1 0.0000000 -0.9990000 1 1 1 2.0000000 -0.9899561 1 2 1 2.0000000 -0.9609181 1 3 1 2.0000000 -0.9126363 1 4 1 2.0000000 -0.8450589 1 5 1 2.0000000 -0.7581131 1 6 1 2.0000000 -0.6517058

279

1 7 1 2.0000000 -0.5257229 1 8 1 2.0000000 -0.3800282 1 9 1 2.0000000 -0.2144640 1 10 1 0.1423941 -0.0284857 1 11 2 2.0000000 -0.8219936 1 12 2 2.0000000 -0.5949731 1 13 2 2.0000000 -0.3471650 1 14 2 0.2769613 -0.0782831 1 15 3 2.0000000 -0.7880137 1 16 3 2.0000000 -0.4756808 1 17 3 0.4013538 -0.1406045 1 18 4 2.0000000 -0.7829304 1 19 4 0.9990000 -0.4022095 4 20 5 2.0000000 -0.9990000 4 21 5 2.0000000 -0.5840955 4 22 5 0.4096279 -0.1697340 4 23 6 2.0000000 -0.7553730 4 24 6 0.8229809 -0.3410115 4 25 7 2.0000000 -0.9266500 4 26 7 2.0000000 -0.5122886 4 27 7 0.2363319 -0.0979271 4 28 8 0.5646572 -0.6835656 4 28 9 0.8027840 2.0000000 4 29 10 0.0409107 -0.1717987 4 29 11 0.2790356 2.0000000 4 29 12 0.5171623 2.0000000 4 29 13 0.7552891 2.0000000 4 29 14 0.9990000 2.0000000 4 30 15 0.2315407 -0.9723473 4 30 16 0.4696674 2.0000000 4 30 17 0.7077942 2.0000000 4 30 18 0.9459209 2.0000000 4 31 19 0.1840458 -0.7728958 4 31 20 0.4221725 2.0000000 4 31 21 0.6602993 2.0000000 4 31 22 0.8984261 2.0000000 4 32 23 0.1365509 -0.5734444 4 32 24 0.3746796 2.0000000 4 32 25 0.6128044 2.0000000 4 32 26 0.8509293 2.0000000 4 33 27 0.0890579 -0.3739910 4 33 28 0.3271828 2.0000000 4 33 29 0.5653114 2.0000000 4 33 30 0.8034363 2.0000000 4 34 31 0.0415611 -0.1745396 4 34 32 0.2796898 2.0000000

280

4 34 33 0.5178146 2.0000000 4 34 34 0.7559433 2.0000000 4 34 35 0.9990000 2.0000000 4 34 56 0.9990000 2.0000000 4 34 57 0.9990000 2.0000000 4 35 58 0.0644150 -0.4434128 4 35 59 0.6756935 2.0000000 4 36 60 2.0000000 -0.7670364 4 37 60 0.0000000 -0.4920502 0 37 61 0.0000000 2.0000000 0 37 80 0.0000000 2.0000000 0 37 81 0.0000000 2.0000000 2 37 82 0.0000000 0.5107422 2 36 82 2.0000000 0.5107422 2 17 82 2.0000000 0.5107422 2 16 82 2.0000000 0.5107422 2 15 82 -0.0010000 0.5107422 2 15 81 -0.0010000 2.0000000 2 15 33 -0.0010000 2.0000000 2 15 32 -0.0010000 2.0000000 2 14 31 2.0000000 0.5041103 2 13 31 2.0000000 0.5041103 2 12 31 2.0000000 0.5041103 2 11 32 0.5834789 2.0000000 2 11 33 0.4539261 2.0000000 2 11 34 0.3243723 2.0000000 2 11 35 0.1948195 2.0000000 2 11 36 0.0652657 0.5037766 2 10 37 0.9357128 2.0000000 2 10 38 0.8061600 2.0000000 2 10 39 0.6766062 2.0000000 2 10 40 0.5470533 2.0000000 2 10 41 0.4174995 2.0000000 2 10 42 0.2879467 2.0000000 2 10 43 0.1583939 2.0000000 2 10 44 0.0288401 0.2226143 2 9 45 0.8992872 2.0000000 2 9 46 0.7697334 2.0000000 2 9 47 0.6401806 2.0000000 2 9 48 0.5106268 2.0000000 2 9 49 0.3810740 2.0000000 2 9 50 0.2515211 2.0000000 2 9 51 0.1219673 0.9414482 2 8 52 0.9990000 2.0000000 2 8 198 0.9990000 2.0000000 2 8 199 0.9990000 2.0000000

281

2 8 200 0.9990000 0.0000000 0 7 200 2.0000000 0.0000000 0 2 200 2.0000000 0.0000000 0 1 200 2.0000000 0.0000000 0 0 200 0.0000000 0.0000000 0 0 199 0.0000000 2.0000000 0 0 4 0.0000000 2.0000000 0 0 3 0.0000000 2.0000000 0 0 2 0.0000000 2.0000000 888 &INPUT5 START ='SPHERE', RMAX = 20.000, RAD = 0.5200E+02, MAXRAY = - 20, LSTRH = 1,UNIT =0.2E-03 &END

- Design 2: • Boundary File EGUN 10 kV, 1 A SAA 11-15-15 &INPUT1 RLIM=900, ZLIM=900, POTN=5, POT=0.0,10000.0,0.0, 0.0,0.0, &END 1 0 0 1 20.0 4.0 4 20.0 4.0 4 28.45 7.5 4 33.5 35.0 4 33.5 57.5 1004 35.5 57.5 2 4 36.2 59.5 2 36.2 82.5 2 15.0 82.5 2 15.0 31.5 2 11.65 31.5 2 9.0 51.95 2 9.0 900.0 0 0 900.0 1 0 0

• Input File EGUN 10 kV, 1 A SAA 11-15-15 &INPUT1 RLIM =100, ZLIM = 900, NBND = 3200, POTN = 4, POT( 2) = 1.0000E+04, MAGSEG = 0, IAX = 0, AQUAD = 0.0, CSYS = 1, MI = 3, SX = 7.50, SY = 0.30, SCALE =' ', PASS = 2, XR =0.9950, ERROR =0.1000E-02, TYME= 300.0, LSTPOT = 3, LSTMAG = 1, LSTBND = 1, INTPA = 0, POIS = 0, &END

282

1 0 1 0.0000000 -0.9990000 1 1 1 2.0000000 -0.9899561 1 2 1 2.0000000 -0.9609181 1 3 1 2.0000000 -0.9126363 1 4 1 2.0000000 -0.8450589 1 5 1 2.0000000 -0.7581131 1 6 1 2.0000000 -0.6517058 1 7 1 2.0000000 -0.5257229 1 8 1 2.0000000 -0.3800282 1 9 1 2.0000000 -0.2144640 1 10 1 0.1423941 -0.0284857 1 11 2 2.0000000 -0.8219936 1 12 2 2.0000000 -0.5949731 1 13 2 2.0000000 -0.3471650 1 14 2 0.2769613 -0.0782831 1 15 3 2.0000000 -0.7880137 1 16 3 2.0000000 -0.4756808 1 17 3 0.4013538 -0.1406045 1 18 4 2.0000000 -0.7829304 1 19 4 0.9990000 -0.4022095 4 20 5 2.0000000 -0.9990000 4 21 5 2.0000000 -0.5840955 4 22 5 0.4096279 -0.1697340 4 23 6 2.0000000 -0.7553730 4 24 6 0.8229809 -0.3410115 4 25 7 2.0000000 -0.9266500 4 26 7 2.0000000 -0.5122886 4 27 7 0.2363319 -0.0979271 4 28 8 0.5374565 -0.6835656 4 28 9 0.7210484 2.0000000 4 28 10 0.9046421 2.0000000 4 29 11 0.0882359 -0.4806061 4 29 12 0.2718296 2.0000000 4 29 13 0.4554234 2.0000000 4 29 14 0.6390171 2.0000000 4 29 15 0.8226109 2.0000000 4 30 16 0.0062046 -0.0337915 4 30 17 0.1897984 2.0000000 4 30 18 0.3733902 2.0000000 4 30 19 0.5569839 2.0000000 4 30 20 0.7405777 2.0000000 4 30 21 0.9241714 2.0000000 4 31 22 0.1077652 -0.5869770 4 31 23 0.2913589 2.0000000 4 31 24 0.4749527 2.0000000 4 31 25 0.6585464 2.0000000

283

4 31 26 0.8421402 2.0000000 4 32 27 0.0257339 -0.1401615 4 32 28 0.2093277 2.0000000 4 32 29 0.3929214 2.0000000 4 32 30 0.5765152 2.0000000 4 32 31 0.7601089 2.0000000 4 32 32 0.9436989 2.0000000 4 33 33 0.1272926 -0.6933479 4 33 34 0.3108864 2.0000000 4 33 35 0.4944801 2.0000000 4 33 36 0.4953194 2.0000000 4 33 56 0.4953194 2.0000000 4 33 57 0.4953194 2.0000000 4 33 58 0.5645790 2.0000000 4 34 59 0.1758575 -0.1767426 4 35 60 2.0000000 -0.5623360 4 36 60 0.0000000 -0.4920502 0 36 61 0.0000000 2.0000000 0 36 80 0.0000000 2.0000000 0 36 81 0.0000000 2.0000000 2 36 82 0.0000000 0.5107422 2 35 82 2.0000000 0.5107422 2 17 82 2.0000000 0.5107422 2 16 82 2.0000000 0.5107422 2 15 82 -0.0010000 0.5107422 2 15 81 -0.0010000 2.0000000 2 15 33 -0.0010000 2.0000000 2 15 32 -0.0010000 2.0000000 2 14 31 2.0000000 0.5041103 2 13 31 2.0000000 0.5041103 2 12 31 2.0000000 0.5041103 2 11 32 0.5834789 2.0000000 2 11 33 0.4539261 2.0000000 2 11 34 0.3243723 2.0000000 2 11 35 0.1948195 2.0000000 2 11 36 0.0652657 0.5037766 2 10 37 0.9357128 2.0000000 2 10 38 0.8061600 2.0000000 2 10 39 0.6766062 2.0000000 2 10 40 0.5470533 2.0000000 2 10 41 0.4174995 2.0000000 2 10 42 0.2879467 2.0000000 2 10 43 0.1583939 2.0000000 2 10 44 0.0288401 0.2226143 2 9 45 0.8992872 2.0000000 2 9 46 0.7697334 2.0000000

284

2 9 47 0.6401806 2.0000000 2 9 48 0.5106268 2.0000000 2 9 49 0.3810740 2.0000000 2 9 50 0.2515211 2.0000000 2 9 51 0.1219673 0.9414482 2 8 52 0.9990000 2.0000000 2 8 898 0.9990000 2.0000000 2 8 899 0.9990000 2.0000000 2 8 900 0.9990000 0.0000000 0 7 900 2.0000000 0.0000000 0 2 900 2.0000000 0.0000000 0 1 900 2.0000000 0.0000000 0 0 900 0.0000000 0.0000000 0 0 899 0.0000000 2.0000000 0 0 4 0.0000000 2.0000000 0 0 3 0.0000000 2.0000000 0 0 2 0.0000000 2.0000000 888 &INPUT5 START ='SPHERE', RMAX = 20.000, RAD = 0.5200E+02, MAXRAY = - 20, LSTRH = 1,UNIT =0.2E-03 &END

- Design 3: • Boundary File EGUN 10 kV, 1 A SAA 11-15-15 &INPUT1 RLIM=100, ZLIM=900, POTN=5, POT=0.0,10000.0,0.0, 0.0,0.0, &END 1 0 0 1 20.0 4.0 4 20.0 4.0 4 28.45 7.5 4 30.5 35.0 4 30.5 57.5 1004 32.5 57.5 2 4 33.2 59.5 2 33.2 82.5 2 18.0 82.5 2 18.0 31.5 2 11.65 31.5 2 9.0 51.95 2 9.0 900.0 0 0 900.0 1 0 0

• Input File 285

EGUN 10 kV, 1 A SAA 11-15-15 &INPUT1 RLIM =100, ZLIM = 900, NBND = 3200, POTN = 4, POT( 2) = 1.0000E+04, MAGSEG = 0, IAX = 0, AQUAD = 0.0, CSYS = 1, MI = 3, SX = 7.50, SY = 0.28, SCALE =' ', PASS = 2, XR =0.9950, ERROR =0.1000E-02, TYME= 300.0, LSTPOT = 3, LSTMAG = 1, LSTBND = 1, INTPA = 0, POIS = 0, &END 1 0 1 0.0000000 -0.9990000 1 1 1 2.0000000 -0.9899561 1 2 1 2.0000000 -0.9609181 1 3 1 2.0000000 -0.9126363 1 4 1 2.0000000 -0.8450589 1 5 1 2.0000000 -0.7581131 1 6 1 2.0000000 -0.6517058 1 7 1 2.0000000 -0.5257229 1 8 1 2.0000000 -0.3800282 1 9 1 2.0000000 -0.2144640 1 10 1 0.1423941 -0.0284857 1 11 2 2.0000000 -0.8219936 1 12 2 2.0000000 -0.5949731 1 13 2 2.0000000 -0.3471650 1 14 2 0.2769613 -0.0782831 1 15 3 2.0000000 -0.7880137 1 16 3 2.0000000 -0.4756808 1 17 3 0.4013538 -0.1406045 1 18 4 2.0000000 -0.7829304 1 19 4 0.9990000 -0.4022095 4 20 5 2.0000000 -0.9990000 4 21 5 2.0000000 -0.5840955 4 22 5 0.4096279 -0.1697340 4 23 6 2.0000000 -0.7553730 4 24 6 0.8229809 -0.3410115 4 25 7 2.0000000 -0.9266500 4 26 7 2.0000000 -0.5122886 4 27 7 0.2363319 -0.0979271 4 28 8 0.4830532 -0.6835656 4 28 9 0.5575809 2.0000000 4 28 10 0.6321087 2.0000000 4 28 11 0.7066364 2.0000000 4 28 12 0.7811642 2.0000000 4 28 13 0.8556919 2.0000000 4 28 14 0.9302216 2.0000000 4 29 15 0.0047493 -0.0637159 4 29 16 0.0792770 2.0000000 4 29 17 0.1538048 2.0000000 4 29 18 0.2283325 2.0000000

286

4 29 19 0.3028603 2.0000000 4 29 20 0.3773880 2.0000000 4 29 21 0.4519176 2.0000000 4 29 22 0.5264454 2.0000000 4 29 23 0.6009731 2.0000000 4 29 24 0.6755009 2.0000000 4 29 25 0.7500286 2.0000000 4 29 26 0.8245564 2.0000000 4 29 27 0.8990841 2.0000000 4 29 28 0.9736137 2.0000000 4 30 29 0.0481415 -0.6459446 4 30 30 0.1226692 2.0000000 4 30 31 0.1971970 2.0000000 4 30 32 0.2717247 2.0000000 4 30 33 0.3462524 2.0000000 4 30 34 0.4207802 2.0000000 4 30 35 0.4953098 2.0000000 4 30 36 0.4956493 2.0000000 4 30 45 0.4956493 2.0000000 4 30 46 0.4956493 2.0000000 4 30 47 0.4956474 2.0000000 4 30 56 0.4956474 2.0000000 4 30 57 0.4956474 2.0000000 4 30 58 0.5649090 2.0000000 4 31 59 0.1761894 -0.1771431 4 32 60 2.0000000 -0.5624390 4 33 60 0.0000000 -0.4920502 0 33 61 0.0000000 2.0000000 0 33 80 0.0000000 2.0000000 0 33 81 0.0000000 2.0000000 2 33 82 0.0000000 0.5107422 2 32 82 2.0000000 0.5107422 2 20 82 2.0000000 0.5107422 2 19 82 2.0000000 0.5107422 2 18 82 -0.0010000 0.5107422 2 18 81 -0.0010000 2.0000000 2 18 33 -0.0010000 2.0000000 2 18 32 -0.0010000 2.0000000 2 17 31 2.0000000 0.5041103 2 13 31 2.0000000 0.5041103 2 12 31 2.0000000 0.5041103 2 11 32 0.5834789 2.0000000 2 11 33 0.4539261 2.0000000 2 11 34 0.3243723 2.0000000 2 11 35 0.1948195 2.0000000 2 11 36 0.0652657 0.5037766

287

2 10 37 0.9357128 2.0000000 2 10 38 0.8061600 2.0000000 2 10 39 0.6766062 2.0000000 2 10 40 0.5470533 2.0000000 2 10 41 0.4174995 2.0000000 2 10 42 0.2879467 2.0000000 2 10 43 0.1583939 2.0000000 2 10 44 0.0288401 0.2226143 2 9 45 0.8992872 2.0000000 2 9 46 0.7697334 2.0000000 2 9 47 0.6401806 2.0000000 2 9 48 0.5106268 2.0000000 2 9 49 0.3810740 2.0000000 2 9 50 0.2515211 2.0000000 2 9 51 0.1219673 0.9414482 2 8 52 0.9990000 2.0000000 2 8 898 0.9990000 2.0000000 2 8 899 0.9990000 2.0000000 2 8 900 0.9990000 0.0000000 0 7 900 2.0000000 0.0000000 0 2 900 2.0000000 0.0000000 0 1 900 2.0000000 0.0000000 0 0 900 0.0000000 0.0000000 0 0 899 0.0000000 2.0000000 0 0 4 0.0000000 2.0000000 0 0 3 0.0000000 2.0000000 0 0 2 0.0000000 2.0000000 888 &INPUT5 START ='SPHERE', RMAX = 20.000, RAD = 0.5200E+02, MAXRAY = - 20, LSTRH = 1,UNIT =0.20E-03 &END

288

• APPENDIX E

E.1 Current Density of 5.968 A/cm2 with Magnet for the Proposed Slow-Wave Structure Design

- Design: • Boundary File EGUN 262 kV, 12 A SAA 11-28-17 &INPUT1 RLIM=500, ZLIM=791, POTN=5, POT=0.0,262000,0.0, 0.0,0.0, &END 1 0 0 1 20.0 1.17 4 20.0 1.17 4 36.45 7.98 4 52.0 13.00 4 52.0 39.67 1004 54.0 39.67 2 4 54.7 41.67 2 54.7 279.67 2 17.06 279.67 2 17.06 94.38 2 12.66 94.38 2 12.0 172.06 2 12.0 791.0 0 0 791.0 1 0 0

• Input File EGUN 262 kV, 12 A SAA 11-28-17 &INPUT1 RLIM =500, ZLIM = 791, NBND = 4473, POTN = 4, POT( 2) = 2.6200E+05, MAGSEG = -1, IAX = 0, AQUAD = 0.0, CSYS = 1, MI = 3, SX = 7.50, SY = 0.52, SCALE =' ', PASS = 2, XR =0.9950, ERROR =0.1000E-02, TYME= 300.0, LSTPOT = 3, LSTMAG = 1, LSTBND = 1, INTPA = 0, POIS = 0, &END

&INPUT3

BZA = 28.42235824 , 28.51550106 , 28.60872917 , 28.70204261 , 289

28.79544139 , 28.88892555 , 28.98249511 , 29.07615009 , 29.16989054 , 29.26371647 , 29.35762792 , 29.45162493 , 29.54570753 , 29.63987576 , 29.73412965 , 29.82846924 , 29.92289458 , 30.0174057 , 30.11200265 , 30.20668548 , 30.30145422 , 30.39630893 , 30.49287571 , 30.59429612 , 30.69620576 , 30.79860428 , 30.90149134 , 31.00486658 , 31.10872968 , 31.21308029 , 31.31791809 , 31.42324275 , 31.52905395 , 31.63535137 , 31.7421347 , 31.84940363 , 31.95715785 , 32.06539707 , 32.17412099 , 32.28332931 , 32.39302174 , 32.50319801 , 32.61385784 , 32.72500094 , 32.83662705 , 32.9487359 , 33.06132723 ,

290

33.17440078 , 33.28795629 , 33.40199352 , 33.51651222 , 33.63151215 , 33.74699306 , 33.86295473 , 33.98822042 , 34.11618566 , 34.24465285 , 34.37362207 , 34.50309341 , 34.63306696 , 34.76354281 , 34.89452107 , 35.02600184 , 35.15798522 , 35.29047133 , 35.42346027 , 35.55695217 , 35.69094715 , 35.82544534 , 35.96044685 , 36.09595183 , 36.23196042 , 36.36847274 , 36.50548895 , 36.64300919 , 36.78103361 , 36.91956238 , 37.05859564 , 37.19813356 , 37.33817631 , 37.47872405 , 37.61977696 , 37.76133522 , 37.90339901 , 38.04596851 , 38.18904391 , 38.33262541 , 38.47671319 , 38.62130746 , 38.76640843 ,

291

38.91201629 , 39.05813127 , 39.20475357 , 39.35188341 , 39.49952101 , 39.64766661 , 39.79632043 , 39.9454827 , 40.09515366 , 40.24533354 , 40.39602261 , 40.54722109 , 40.69892925 , 40.85114734 , 41.00387562 , 41.15711435 , 41.3108638 , 41.46512424 , 41.61989594 , 41.77517918 , 41.93097425 , 42.08728142 , 42.24410098 , 42.40143324 , 42.54925907 , 42.69251153 , 42.83618433 , 42.98027678 , 43.12478821 , 43.26971799 , 43.41506548 , 43.5608301 , 43.70701127 , 43.85360843 , 44.00062106 , 44.14804864 , 44.29589071 , 44.44414679 , 44.59281645 , 44.74189927 , 44.89139486 , 45.04130285 , 45.1916229 ,

292

45.34235468 , 45.49349788 , 45.64505225 , 45.7970175 , 45.94939343 , 46.10217981 , 46.25537645 , 46.40898321 , 46.56299992 , 46.71742649 , 46.8722628 , 47.02750879 , 47.18316442 , 47.33922964 , 47.49570446 , 47.6525889 , 47.809883 , 47.96758683 , 48.12570046 , 48.28422402 , 48.44716489 , 48.61520975 , 48.78355576 , 48.95220346 , 49.1337693 , 49.31606024 , 49.49642833 , 49.66821002 , 49.84020246 , 50.01240534 , 50.18481836 , 50.35744123 , 50.53027369 , 50.70331548 , 50.87656638 , 51.05002616 , 51.22369462 , 51.39757159 , 51.57165689 , 51.74595037 , 51.92045189 , 52.09516135 , 52.27007864 ,

293

52.44520368 , 52.6205364 , 52.79607674 , 52.97182468 , 53.1477802 , 53.3239433 , 53.50031398 , 53.6768923 , 53.85367829 , 54.03067202 , 54.20787357 , 54.38528305 , 54.56290057 , 54.74072626 , 54.91876027 , 55.09700276 , 55.27545393 , 55.45411397 , 55.63298309 , 55.81206153 , 55.99134954 , 56.18473752 , 56.40679308 , 56.63006542 , 56.85454676 , 57.08023403 , 57.30712891 , 57.53523783 , 57.76457196 , 57.9951472 , 58.22698419 , 58.46010834 , 58.69454977 , 58.93034336 , 59.16752871 , 59.40615019 , 59.6462569 , 59.88790267 , 60.13114608 , 60.37605046 , 60.62268387 , 60.87111912 , 61.12143376 ,

294

61.37371006 , 61.62803507 , 61.88450055 , 62.14320302 , 62.40424373 , 62.66772868 , 62.9337686 , 63.20247898 , 63.40005442 , 63.5692112 , 63.73621897 , 63.90109515 , 64.06385789 , 64.22452607 , 64.3831193 , 64.53965791 , 64.69416299 , 64.84665632 , 64.99716045 , 65.14569863 , 66.31330322 , 68.09438521 , 70.00119528 , 72.03584976 , 74.20059655 , 76.49781514 , 78.93001656 , 81.49984343 , 84.21006991 , 87.06360176 , 90.06347629 , 93.21286238 , 95.16138756 , 96.66088384 , 98.28145075 , 100.0254716 , 101.8955217 , 103.8943679 , 106.0249693 , 108.2904765 , 110.6942321 , 118.0565666 , 128.2968064 ,

295

140.9050566 , 156.1545712 , 174.2806425 , 195.3152288 , 214.6016459 , 233.9791675 , 253.4176937 , 272.8830413 , 297.7252604 , 333.1870979 , 372.2197308 , 433.5067909 , 525.3978182 , 611.6824528 , 691.6125996 , 764.7491352 , 846.8037952 , 952.9537131 , 1073.778685 , 1203.865328 , 1345.090134 , 1500.101435 , 1668.884266 , 1851.461739 , 2066.545326 , 2316.45922 , 2566.768169 , 2791.994069 , 3019.584336 , 3249.48171 , 3481.584254 , 3741.391602 , 4004.440946 , 4264.240945 , 4520.739375 , 4776.187223 , 5034.603245 , 5291.363503 , 5557.731307 , 5792.686178 , 6031.158544 , 6272.767925 , 6517.528818 ,

296

6765.372028 , 7016.144666 , 7269.610152 , 7414.881628 , 7553.511081 , 7688.550597 , 7819.977173 , 7947.771313 , 8071.917027 , 8192.40183 , 8309.216741 , 8422.356287 , 8531.818499 , 8637.604914 , 8739.720575 , 8838.17403 , 8951.665174 , 9068.361597 , 9145.915888 , 9197.244414 , 9227.235982 , 9260.240952 , 9307.949386 , 9353.872498 , 9398.010535 , 9440.363683 , 9480.932076 , 9519.715793 , 9556.714857 , 9591.929235 , 9625.358841 , 9657.003531 , 9686.86311 , 9714.937322 , 9741.225862 , 9746.908316 , 9749.511478 , 9752.200836 , 9754.976349 , 9757.837972 , 9760.785651 , 9762.524043 , 9758.450092 ,

297

9753.962558 , 9745.132072 , 9732.149514 , 9719.481888 , 9707.129746 , 9695.093749 , 9683.37467 , 9671.973389 , 9642.987785 , 9607.606351 , 9569.891885 , 9529.84351 , 9487.460409 , 9442.741824 , 9395.687056 , 9346.295466 , 9294.566474 , 9240.499558 , 9184.094257 , 9125.350168 , 9064.266948 , 9004.253278 , 8945.164 , 8858.190461 , 8739.328959 , 8609.691008 , 8464.381034 , 8305.871648 , 8143.379476 , 7976.895297 , 7806.415169 , 7631.940437 , 7453.47773 , 7271.03896 , 7084.641325 , 6894.307305 , 6700.064665 , 6894.307305 , 7084.641325 , 7271.03896 , 7453.47773 , 7631.940437 , 7806.415169 ,

298

7976.895297 , 8143.379476 , 8305.871648 , 8464.381034 , 8609.691008 , 8739.328959 , 8858.190461 , 8945.164 , 9004.253278 , 9064.266948 , 9125.350168 , 9184.094257 , 9240.499558 , 9294.566474 , 9346.295466 , 9395.687056 , 9442.741824 , 9487.460409 , 9529.84351 , 9569.891885 , 9607.606351 , 9642.987785 , 9671.973389 , 9683.37467 , 9695.093749 , 9707.129746 , 9719.481888 , 9732.149514 , 9745.132072 , 9753.962558 , 9758.450092 , 9762.524043 , 9758.450092 , 9753.962558 , 9745.132072 , 9732.149514 , 9719.481888 , 9707.129746 , 9695.093749 , 9683.37467 , 9671.973389 , 9642.987785 , 9607.606351 ,

299

9569.891885 , 9529.84351 , 9487.460409 , 9442.741824 , 9395.687056 , 9346.295466 , 9294.566474 , 9240.499558 , 9184.094257 , 9125.350168 , 9064.266948 , 9004.253278 , 8945.164 , 8858.190461 , 8739.328959 , 8609.691008 , 8464.381034 , 8305.871648 , 8143.379476 , 7976.895297 , 7806.415169 , 7631.940437 , 7453.47773 , 7271.03896 , 7084.641325 , 6894.307305 , 6700.064665 , 6894.307305 , 7084.641325 , 7271.03896 , 7453.47773 , 7631.940437 , 7806.415169 , 7976.895297 , 8143.379476 , 8305.871648 , 8464.381034 , 8609.691008 , 8739.328959 , 8858.190461 , 8945.164 , 9004.253278 , 9064.266948 ,

300

9125.350168 , 9184.094257 , 9240.499558 , 9294.566474 , 9346.295466 , 9395.687056 , 9442.741824 , 9487.460409 , 9529.84351 , 9569.891885 , 9607.606351 , 9642.987785 , 9671.973389 , 9683.37467 , 9695.093749 , 9707.129746 , 9719.481888 , 9732.149514 , 9745.132072 , 9753.962558 , 9758.450092 , 9762.524043 , 9758.450092 , 9753.962558 , 9745.132072 , 9732.149514 , 9719.481888 , 9707.129746 , 9695.093749 , 9683.37467 , 9671.973389 , 9642.987785 , 9607.606351 , 9569.891885 , 9529.84351 , 9487.460409 , 9442.741824 , 9395.687056 , 9346.295466 , 9294.566474 , 9240.499558 , 9184.094257 , 9125.350168 ,

301

9064.266948 , 9004.253278 , 8945.164 , 8858.190461 , 8739.328959 , 8609.691008 , 8464.381034 , 8305.871648 , 8143.379476 , 7976.895297 , 7806.415169 , 7631.940437 , 7453.47773 , 7271.03896 , 7084.641325 , 6894.307305 , 6700.064665 , 6894.307305 , 7084.641325 , 7271.03896 , 7453.47773 , 7631.940437 , 7806.415169 , 7976.895297 , 8143.379476 , 8305.871648 , 8464.381034 , 8609.691008 , 8739.328959 , 8858.190461 , 8945.164 , 9004.253278 , 9064.266948 , 9125.350168 , 9184.094257 , 9240.499558 , 9294.566474 , 9346.295466 , 9395.687056 , 9442.741824 , 9487.460409 , 9529.84351 , 9569.891885 ,

302

9607.606351 , 9642.987785 , 9671.973389 , 9683.37467 , 9695.093749 , 9707.129746 , 9719.481888 , 9732.149514 , 9745.132072 , 9753.962558 , 9758.450092 , 9762.524043 , 9758.450092 , 9753.962558 , 9745.132072 , 9732.149514 , 9719.481888 , 9707.129746 , 9695.093749 , 9683.37467 , 9671.973389 , 9642.987785 , 9607.606351 , 9569.891885 , 9529.84351 , 9487.460409 , 9442.741824 , 9395.687056 , 9346.295466 , 9294.566474 , 9240.499558 , 9184.094257 , 9125.350168 , 9064.266948 , 9004.253278 , 8945.164 , 8858.190461 , 8739.328959 , 8609.691008 , 8464.381034 , 8305.871648 , 8143.379476 , 7976.895297 ,

303

7806.415169 , 7631.940437 , 7453.47773 , 7271.03896 , 7084.641325 , 6894.307305 , 6700.064665 , 6894.307305 , 7084.641325 , 7271.03896 , 7453.47773 , 7631.940437 , 7806.415169 , 7976.895297 , 8143.379476 , 8305.871648 , 8464.381034 , 8609.691008 , 8739.328959 , 8858.190461 , 8945.164 , 9004.253278 , 9064.266948 , 9125.350168 , 9184.094257 , 9240.499558 , 9294.566474 , 9346.295466 , 9395.687056 , 9442.741824 , 9487.460409 , 9529.84351 , 9569.891885 , 9607.606351 , 9642.987785 , 9671.973389 , 9683.37467 , 9695.093749 , 9707.129746 , 9719.481888 , 9732.149514 , 9745.132072 , 9753.962558 ,

304

9758.450092 , 9762.524043 , 9758.450092 , 9753.962558 , 9745.132072 , 9732.149514 , 9719.481888 , 9707.129746 , 9695.093749 , 9683.37467 , 9671.973389 , 9642.987785 , 9607.606351 , 9569.891885 , 9529.84351 , 9487.460409 , 9442.741824 , 9395.687056 , 9346.295466 , 9294.566474 , 9240.499558 , 9184.094257 , 9125.350168 , 9064.266948 , 9004.253278 , 8945.164 , 8858.190461 , 8739.328959 , 8609.691008 , 8464.381034 , 8305.871648 , 8143.379476 , 7976.895297 , 7806.415169 , 7631.940437 , 7453.47773 , 7271.03896 , 7084.641325 , 6894.307305 , 6700.064665 , 6700.064665 , 6894.307305 , 7084.641325 ,

305

7271.03896 , 7453.47773 , 7631.940437 , 7806.415169 , 7976.895297 , 8143.379476 , 8305.871648 , 8464.381034 , 8609.691008 , 8739.328959 , 8858.190461 , 8945.164 , 9004.253278 , 9064.266948 , 9125.350168 , 9184.094257 , 9240.499558 , 9294.566474 , 9346.295466 , 9395.687056 , 9442.741824 , 9487.460409 , 9529.84351 , 9569.891885 , 9607.606351 , 9642.987785 , 9671.973389 , 9683.37467 , 9695.093749 , 9707.129746 , 9719.481888 , 9732.149514 , 9745.132072 , 9753.962558 , 9758.450092 , 9762.524043 , 9758.450092 , 9753.962558 , 9745.132072 , 9732.149514 , 9719.481888 , 9707.129746 , 9695.093749 ,

306

9683.37467 , 9671.973389 , 9642.987785 , 9607.606351 , 9569.891885 , 9529.84351 , 9487.460409 , 9442.741824 , 9395.687056 , 9346.295466 , 9294.566474 , 9240.499558 , 9184.094257 , 9125.350168 , 9064.266948 , 9004.253278 , 8945.164 , 8858.190461 , 8739.328959 , 8609.691008 , 8464.381034 , 8305.871648 , 8143.379476 , 7976.895297 , 7806.415169 , 7631.940437 , 7453.47773 , 7271.03896 , 7084.641325 , 6894.307305 , 6700.064665 , 6501.946455 , 6299.991008 , 6094.241941 , 5790.924042 , 5457.761088 , 5139.905549 , 4838.241442 , 4553.240298 , 4284.961165 , 4033.035001 , 3760.339109 , 3359.285387 ,

307

3054.429192 , 2819.722251 , 2599.244934 , 2386.855949 , 2182.479791 , 1986.067614 , 1797.597227 , 1617.073093 , 1444.526333 , 1280.014722 , 1123.622692 , 975.4613293 , 835.668378 , 785.3148674 , 736.9756563 , 689.9185111 , 643.853926 , 598.4937537 , 553.5512053 , 508.7408504 , 463.7786168 , 418.3817909 , 372.6106338 , 329.6696783 , 289.4397286 ,

&END

1 0 1 0.0000000 -0.9990000 1 1 1 2.0000000 -0.9990000 1 2 1 2.0000000 -0.9880325 1 3 1 2.0000000 -0.9734231 1 4 1 2.0000000 -0.9529910 1 5 1 2.0000000 -0.9267343 1 6 1 2.0000000 -0.8946502 1 7 1 2.0000000 -0.8567358 1 8 1 2.0000000 -0.8129873 1 9 1 2.0000000 -0.7634006 1 10 1 2.0000000 -0.7079707 1 11 1 2.0000000 -0.6466076 1 12 1 2.0000000 -0.5793414 1 13 1 2.0000000 -0.5062131

308

1 14 1 2.0000000 -0.4272154 1 15 1 2.0000000 -0.3423406 1 16 1 2.0000000 -0.2515803 1 17 1 2.0000000 -0.1549256 1 18 1 0.4895020 -0.0523670 1 19 2 2.0000000 -0.9438946 4 20 2 2.0000000 -0.8288306 4 21 2 2.0000000 -0.4147686 4 22 2 0.0017071 -0.0010000 4 23 3 2.0000000 -0.5866444 4 24 3 0.4168034 -0.1725824 4 25 4 2.0000000 -0.7585201 4 26 4 0.8318996 -0.3444581 4 27 5 2.0000000 -0.9303961 4 28 5 2.0000000 -0.5163341 4 29 5 0.2469959 -0.1022720 4 30 6 2.0000000 -0.6882100 4 31 6 0.6620941 -0.2741475 4 32 7 2.0000000 -0.8600855 4 33 7 2.0000000 -0.4460235 4 34 7 0.0771904 -0.0319614 4 35 8 2.0000000 -0.6178994 4 36 8 0.5050621 -0.2038374 4 37 9 2.0000000 -0.8401604 4 38 9 2.0000000 -0.5172119 4 39 9 0.6015282 -0.1942635 4 40 10 2.0000000 -0.8713150 4 41 10 2.0000000 -0.5483656 4 42 10 0.6979980 -0.2254171 4 43 11 2.0000000 -0.9024687 4 44 11 2.0000000 -0.5795202 4 45 11 0.7944679 -0.2565718 4 46 12 2.0000000 -0.9336233 4 47 12 2.0000000 -0.6106749 4 48 12 0.8909340 -0.2877264 4 49 13 2.0000000 -0.9647779 4 50 13 2.0000000 -0.6418295 4 51 13 0.9874039 -0.3188801 4 51 14 0.9990000 2.0000000 4 51 38 0.9990000 2.0000000 4 51 39 0.9990000 2.0000000 4 52 40 0.0254631 -0.2569809 4 52 41 0.5054245 2.0000000

309

4 53 42 2.0000000 -0.5983658 4 54 42 0.0000000 -0.3243713 0 54 43 0.0000000 2.0000000 0 54 277 0.0000000 2.0000000 0 54 278 0.0000000 2.0000000 2 54 279 0.0000000 0.7064209 2 53 279 2.0000000 0.7064209 2 20 279 2.0000000 0.7064209 2 19 279 2.0000000 0.7064209 2 18 279 -0.9408627 0.7064209 2 18 278 -0.9408627 2.0000000 2 18 96 -0.9408627 2.0000000 2 18 95 -0.9408627 2.0000000 2 17 94 2.0000000 0.3922882 2 14 94 2.0000000 0.3922882 2 13 94 2.0000000 0.3922882 2 12 95 0.6524601 2.0000000 2 12 96 0.6439657 2.0000000 2 12 97 0.6354713 2.0000000 2 12 98 0.6269770 2.0000000 2 12 99 0.6184826 2.0000000 2 12 100 0.6099882 2.0000000 2 12 101 0.6014938 2.0000000 2 12 102 0.5929995 2.0000000 2 12 103 0.5845051 2.0000000 2 12 104 0.5760107 2.0000000 2 12 105 0.5675163 2.0000000 2 12 106 0.5590219 2.0000000 2 12 107 0.5505276 2.0000000 2 12 108 0.5420332 2.0000000 2 12 109 0.5335388 2.0000000 2 12 110 0.5250444 2.0000000 2 12 111 0.5165501 2.0000000 2 12 112 0.5080557 2.0000000 2 12 113 0.4995623 2.0000000 2 12 114 0.4910679 2.0000000 2 12 115 0.4825735 2.0000000 2 12 116 0.4740791 2.0000000 2 12 117 0.4655848 2.0000000 2 12 118 0.4570904 2.0000000 2 12 119 0.4485960 2.0000000 2 12 120 0.4401016 2.0000000 2 12 121 0.4316072 2.0000000

310

2 12 122 0.4231129 2.0000000 2 12 123 0.4146185 2.0000000 2 12 124 0.4061241 2.0000000 2 12 125 0.3976297 2.0000000 2 12 126 0.3891354 2.0000000 2 12 127 0.3806410 2.0000000 2 12 128 0.3721466 2.0000000 2 12 129 0.3636522 2.0000000 2 12 130 0.3551579 2.0000000 2 12 131 0.3466635 2.0000000 2 12 132 0.3381691 2.0000000 2 12 133 0.3296747 2.0000000 2 12 134 0.3211803 2.0000000 2 12 135 0.3126860 2.0000000 2 12 136 0.3041916 2.0000000 2 12 137 0.2956972 2.0000000 2 12 138 0.2872028 2.0000000 2 12 139 0.2787085 2.0000000 2 12 140 0.2702141 2.0000000 2 12 141 0.2617197 2.0000000 2 12 142 0.2532253 2.0000000 2 12 143 0.2447309 2.0000000 2 12 144 0.2362366 2.0000000 2 12 145 0.2277422 2.0000000 2 12 146 0.2192478 2.0000000 2 12 147 0.2107534 2.0000000 2 12 148 0.2022591 2.0000000 2 12 149 0.1937647 2.0000000 2 12 150 0.1852703 2.0000000 2 12 151 0.1767759 2.0000000 2 12 152 0.1682816 2.0000000 2 12 153 0.1597872 2.0000000 2 12 154 0.1512928 2.0000000 2 12 155 0.1427984 2.0000000 2 12 156 0.1343040 2.0000000 2 12 157 0.1258097 2.0000000 2 12 158 0.1173153 2.0000000 2 12 159 0.1088209 2.0000000 2 12 160 0.1003275 2.0000000 2 12 161 0.0918331 2.0000000 2 12 162 0.0833387 2.0000000 2 12 163 0.0748444 2.0000000 2 12 164 0.0663500 2.0000000

311

2 12 165 0.0578556 2.0000000 2 12 166 0.0493612 2.0000000 2 12 167 0.0408669 2.0000000 2 12 168 0.0323725 2.0000000 2 12 169 0.0238781 2.0000000 2 12 170 0.0153837 2.0000000 2 12 171 0.0068893 0.8110199 2 11 172 0.9990000 2.0000000 2 11 789 0.9990000 2.0000000 2 11 790 0.9990000 2.0000000 2 11 791 0.9990000 0.0000000 0 10 791 2.0000000 0.0000000 0 2 791 2.0000000 0.0000000 0 1 791 2.0000000 0.0000000 0 0 791 0.0000000 0.0000000 0 0 790 0.0000000 2.0000000 0 0 4 0.0000000 2.0000000 0 0 3 0.0000000 2.0000000 0 0 2 0.0000000 2.0000000 888 &INPUT5 START ='SPHERE', RMAX = 20.000, RAD = 0.1715E+03, MAXRAY = 100, LSTRH = 1,UNIT =0.4E-03 &END

312

• APPENDIX F

F.1 Parameters of Novel Slow-Wave Structure to Perform Hot Test Simulations Using VSim

Description Parameter Value/Expression for N = 20 PI 3.1415926535898 PIO2 1.5707963267949 TWOPI 6.2831853071796 LIGHTSPEED 299792458 MU0 1.2566370614359e-6 ELEMCHARGE 1.602176487e-19 ELECMASS 9.10938215e-31 PROTMASS 1.672621637e-27 MUONMASS 1.8835313e-28 KB 1.3806504e-23 EPSILON0 8.854187e-12 C2 8.98755179e+16 ELECCHARGE -1.602176487e-19 ELECMASSEV 510998.90984764 BGNX_RING -0.001 RMID_RING 0.026761 RWIRE_RING 0.001 PERIOD 0.022 Constants NPERIOD 20 BGNX_TUBE -0.0102 RADIUS_TUBE 0.05461 SPACING_COLLECT 0.0075 OR RWIRE_COAX 0.002 RADIUS_COAX 0.0045 ENDY_COAX 0.06911 FREQ 1.87-2.78 RISETIME-CYCLES 3.5 BEAM_VOLTAGE 262 XCELLS 1000 YCELLS 62 ZCELLS 50 CFLNUM 0.8 NPARTICLES_PER_T 1 IME_STEP BSTATIC 1.4 Parameters LENGTH_RING NPERIOD*PERIOD

313

BGNX_RING+LENGTH_RING+SPACING_ ENDX_TUBE COLLECTOR+0.0102 DZ_EST 2.*RADIUS_TUBE/ZCELLS THICKNESS_WALL 2.5*DZ_EST ENDX_TUBE_OUTE ENDX_TUBE+THICKNESS_WALL R BGNX_TUBE_OUTE BGNX_TUBE-THICKNESS_WALL R RADIUS_TUBE_OUT RADIUS_TUBE+THICKNESS_WALL ER LENGTH_TUBE_OU ENDX_TUBE_OUTER- TER BGNX_TUBE_OUTER BGNX_TUBE_INNER BGNX_TUBE RADIUS_TUBE_ ENDX_TUBE INNER LENGTH_TUBE_ ENDX_TUBE_INNER- INNER BGNX_TUBE_INNER _BGNY_COAX RMID_RING BOXRADIUS_COAX 1.414*RADIUS_COAX LENGTH_COAX ENDY_COAX-BGNY_COAX X_COAX_IN BGNX_RING X_COAX_OUT X_COAX_IN+LENGTH_RING _BGNY_COAX_INN BGNY_COAX-THICKNESS_WALL ER LENGTH_COAX_IN LENGTH_COAX+3*THICKNESS_WALL NER RADIUS_COAX_OU BOXRADIUS_COAX+THICKNESS_WALL TER BGNX_COAX_IN_O X_COAX_IN - RADIUS_COAX_OUTER UTER BGNX_COAX_OUT_ X_COAX_OUT - RADIUS_COAX_OUTER OUTER BGNZ_COAX_OUTE -RADIUS_COAX_OUTER R BGNZ_COAX_INNE -RADIUS_COAX R ENDX_COAX_IN_O X_COAX_IN + RADIUS_COAX_OUTER UTER ENDX_COAX_OUT_ X_COAX_OUT + RADIUS_COAX_OUTER OUTER THICKNESS_PAINT 0.99*THICKNESS_WALL X_EMIT_PLANE BGNX_TUBE ENDZ RADIUS_TUBE_OUTER*1.0001 ENDY ENDY_COAX-2*DZ_EST BGNZ -ENDZ

314

BGNY BGNZ (ENDX_TUBE_OUTER- DX BGNX_TUBE_OUTER)/XCELLS DY (ENDY-BGNY)/YCELLS DZ (ENDZ-BGNZ)/ZCELLS DL (ENDZ-BGNZ)/ZCELLS DMFRAC 0.25 DTCFL DMFRAC*DL/LIGHTSPEED DT CFLNUM*DTCFL BEAM_RADIUS 0.002 BEAM_CURRENT 12 BEAM_AREA PI*BEAM_RADIUS^2 JDENS BEAM_CURRENT/BEAM_AREA BEAM_FLUX BEAM_CURRENT/ELEMCHARGE FLUXDENS BEAM_FLUX/BEAM_AREA sqrt(2.0*BEAM_VOLTAGE/511.0e3)*LIGH VEMIT TSPEED NOMINAL_DENSITY JDENS/(VEMIT*1.602176487e-19) NSTEPS_ACROSS_C DX/(VEMIT*DT) ELL NOMINAL_PARTICL NPARTICLES_PER_CELL_PER_TIME_ST ES_PER_CELL EP*NSTEPS_ACROSS_CELL XA_HISTORY BGNX_RING+0.2*LENGTH_RING XB_HISTORY BGNX_RING+0.3*LENGTH_RING XC_HISTORY BGNX_RING+0.4*LENGTH_RING XD_HISTORY BGNX_RING+0.75*LENGTH_RING XBGN_HISTORY BGNX_RING+0.003 XEND_HISTORY BGNX_RING+LENGTH_RING-0.001 surface meshing tolerance = DMFRAC cfl number = CFLNUM time step = DT number of steps = 100000 steps between dumps = 2000 dimensionality = 3 grid spacing = uniform reuse geometry files on restart = true Basic Settings length limit = meter precision = double use GPU (if found)= false verbosity = information coordinate system = cartesian field solver = electromagnetic cerenkov filter = none particles = include particles estimated max electron density = 1.e18

315

estimated min electron temperature (eV)= 1.0 moving window = no moving window periodic directions = no periodicity H(x-(X_EMIT_PLANE- SpaceTimeF maskJ DX))*H(BEAM_RADIUS**2-y**2-z**2)- unctions 0.5 Heat capacity = 100000, resistance = 0, PEC Materials thermal conductivity = 0 absorbium Heat capacity = 0, thermal conductivity = 0 inputCoaxSolid outputCoaxSolid driftTubeSolid inputCoaxVoid outputCoaxVoid driftTubeVoid CSG emitterDisk inAndOutCoaxVoids tubeVoid inAndOutCoaxSolids tubeSolid tube Ring1 Ring2 Ring3 Ring4 Ring5 Ring6 Geometries Ring7 Ring8 Ring9 Ring10 Ring11 DesignL15W13P21N2 Ring12 0Geom Ring13 Ring14 Ring15 Ring16 Ring17 Ring18 Ring19 Ring20 Ring21 Ring22 Ring23 Ring24

316

Ring25 Ring26 Ring27 Ring28 Ring29 Ring30 Ring31 Ring32 Ring33 Ring34 Ring35 Ring36 Ring37 Ring38 Ring39 Ring40 TL1 TL2 TL3 TL4 TL5 TL6 TL7 TL8 TL9 TL10 TL11 TL12 TL13 TL14 TL15 TL16 TL17 TL18 TL19 TL20 TL21 TL22 TL23 TL24 TL25 TL26 TL27 TL28 TL29

317

TL30 TL31 TL32 TL33 TL34 TL35 TL36 TL37 TL38 TL39 TL40 TL41 TL42 TL43 TL44 TL45 TL46 TL47 TL48 TL49 TL50 TL51 TL52 TL53 TL54 TL55 TL56 TL57 TL58 TL59 TL60 TL61 TL62 TL63 TL64 TL65 TL66 TL67 TL68 TL69 TL70 TL71 TL72 TL73 TL74

318

TL75 TL76 TL77 TL78 TL79 TL80 lowerXBoundary upperXBoundary lowerYBoundary Grids Grid upperYBoundary lowerZBoundary upperZBoundary Electric Field Magnetic Field kind = External Magnetic Field Fields ExternalB0: b field specification Bx = BSTATIC By = 0.0 Bz = 0.0 inputSignal: kind = Coaxial Waveguide inner radius = RWIRE_COAX Field outer radius = BOXRADIUS_COAX Dynamics frequency = FREQ voltage = 1.5 relative permittivity = 1.0 FieldBoundaryConditio start time = 1.0e-6 ns stop time = 3.5e-9 turn on time = 3.5e-9 coaxial waveguide surface = upper-y waveguide X-center coordinate = X_COAX_IN Z-center coordinate = 0.0 Current Distributions KineticParticles: Beam emission: electron kind = Settable Flux Nominal density = emitter type = beam emitter 74981074818806.33 start time = 0.0 Particle weights = stop time = 3.40282e+038 Particle variable weights mean velocity = 4.1e+07, 0, 0 Dynamics Weight setting = thermal velocity = 0, 0, 0 computed weights flux setting = emission current density macroparticles per cell current density = JDENS = 48.88806242676887 surface = shape emitter particle dynamics = emission offset = 0.01 relativistic mask function = maskJ

319

object name = emitterDisk localize velocity directions to shape = False

electronabsSaveCutCell: kind = Cut-Cell Absorb and Save: tube Ring1 Ring2 Ring3 Ring4 Ring5 Ring6 Ring7 Ring8 Ring9 Ring10 Ring11 Ring12 Ring13 Ring14 Ring15 Ring16 Ring17 Ring18 Ring19 Ring20 Ring21 Ring22 Ring23 Ring24 Ring25 Ring26 Ring27 Ring28 Ring29 Ring30 Ring31 Ring32 Ring33 Ring34 Ring35 Ring36 Ring37 Ring38 Ring39 Ring40

320

TL1 TL2 TL3 TL4 TL5 TL6 TL7 TL8 TL9 TL10 TL11 TL12 TL13 TL14 TL15 TL16 TL17 TL18 TL19 TL20 TL21 TL22 TL23 TL24 TL25 TL26 TL27 TL28 TL29 TL30 TL31 TL32 TL33 TL34 TL35 TL36 TL37 TL38 TL39 TL40 TL41 TL42 TL43 TL44 TL45 TL46

321

TL47 TL48 TL49 TL50 TL51 TL52 TL53 TL54 TL55 TL56 TL57 TL58 TL59 TL60 TL61 TL62 TL63 TL64 TL65 TL66 TL67 TL68 TL69 TL70 TL71 TL72 TL73 TL74 TL75 TL76 TL77 TL78 TL79 TL80 AbsorbedCurrent kind = Absorbed Particle Current numberOfMacroParticl kind = Number of Macroparticles es currEmit kind = Emitted Current inputVoltage kind = Pseudo-potential at Coordinates outputVoltage kind = Pseudo-potential at Coordinates Histories inputPower kind = Poynting Flux outputPower kind = Poynting Flux poyntingA kind = Poynting Flux poyntingB kind = Poynting Flux poyntingC kind = Poynting Flux poyntingBGN kind = Poynting Flux

322

poyntingEND kind = Poynting Flux poyntingXY kind = Poynting Flux poyntingXZ kind = Poynting Flux poyntingD kind = Poynting Flux

323