High Power Microwave Tubes Basics and Trends, Volume 1

High Power Microwave Tubes: Basics and Trends Volume 1

High Power Microwave Tubes: Basics and Trends Volume 1 Vishal Kesari Microwave Tube Research and Development Centre, Bangalore, India

B N Basu Sir J C Bose School of Engineering, Mankundu, India

Morgan & Claypool Publishers

Copyright ª 2018 Morgan & Claypool Publishers All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher, or as expressly permitted by law or under terms agreed with the appropriate rights organization. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency, the Copyright Clearance Centre and other reproduction rights organisations. Rights & Permissions To obtain permission to re-use copyrighted material from Morgan & Claypool Publishers, please contact [email protected]. ISBN ISBN ISBN

978-1-6817-4561-9 (ebook) 978-1-6817-4560-2 (print) 978-1-6817-4563-3 (mobi)

DOI 10.1088/978-1-6817-4561-9 Version: 20180101 IOP Concise Physics ISSN 2053-2571 (online) ISSN 2054-7307 (print) A Morgan & Claypool publication as part of IOP Concise Physics Published by Morgan & Claypool Publishers, 1210 Fifth Avenue, Suite 250, San Rafael, CA, 94901, USA IOP Publishing, Temple Circus, Temple Way, Bristol BS1 6HG, UK

Dedicated to the Late Professor N C Vaidya, who established the Centre of Research in Microwave Tubes at Banaras Hindu University

Contents Preface

ix

Foreword

xi

Acknowledgments

xii

Author biographies

xiii

1

1-1

Introduction

1-11

References

2

Microwave tubes: classification, applications and trends

2-1

2.1 2.2 2.3

Classification Applications Trends in research and development References

2-1 2-1 2-5 2-9

3

Basic enabling concepts

3-1

3.1 3.2

Cathode Space-charge-limited and temperature-limited emission 3.2.1 The Child–Langmuir relation under the space-charge-limited condition of emission 3.2.2 The Richardson–Dushman relation under a temperature-limited condition of emission Space-charge waves and cyclotron waves 3.3.1 Space-charge waves 3.3.2 Cyclotron waves Electron bunching mechanism Induced current due to electron beam flow Space-charge-limiting current 3.6.1 Space-charge limiting current for an infinitesimally thin hollow electron beam in a metal envelope 3.6.2 Space-charge limiting current for a thick solid electron beam in a metal envelope Conservation of kinetic energy in M-type tubes References

3-1 3-3 3-4

3.3

3.4 3.5 3.6

3.7

3-5 3-5 3-5 3-6 3-8 3-11 3-13 3-13 3-15 3-17 3-18

4

Formation, confinement and collection of an electron beam

4-1

4.1

Electron gun

4-1 vii

High Power Microwave Tubes: Basics and Trends

4.2

4.3

4.1.1 Pierce gun derived from a flat cathode 4.1.2 Pierce gun derived from a curved cathode 4.1.3 Magnetron injection gun for the formation of a gyrating electron beam Magnetic focusing structure 4.2.1 Busch’s theorem 4.2.2 Brillouin focusing 4.2.3 Confined-flow focusing 4.2.4 Periodic permanent magnet focusing Multistage depressed collector References

5

Analytical aspects of beam-absent and beam-present slow-wave and fast-wave interaction structures

5.1

Analysis of helical slow-wave interaction structures 5.1.1 Sheath-helix model 5.1.2 Tape-helix model 5.1.3 Interaction impedance 5.1.4 Dispersion and interaction impedance characteristics Analysis of fast-wave disc-loaded waveguide interaction structures 5.2.1 Steps for obtaining dispersion relation/characteristics 5.2.2 Steps for obtaining interaction impedance characteristics 5.2.3 Models of axially periodic structures 5.2.4 Field intensities in structure regions 5.2.5 Relevant boundary conditions 5.2.6 Dispersion relation 5.2.7 Azimuthal interaction impedance 5.2.8 Structure characteristics Growing-wave interactions in slow-wave TWTs and fast-wave gyro-TWTs 5.3.1 Beam-present dispersion relations 5.3.2 Gain-frequency response References

5.2

5.3

viii

4-1 4-3 4-11 4-13 4-13 4-14 4-16 4-17 4-20 4-22 5-1 5-3 5-4 5-15 5-17 5-18 5-22 5-23 5-25 5-25 5-27 5-30 5-30 5-32 5-33 5-49 5-49 5-51 5-56

Preface This book grew out of our interaction with Professor Akhlesh Lakhtakia, of Pennsylvania State University, who motivated us to write it. The memoir was supposed to be of short length—a difficult task though—best captured in spirit while quoting Blaise Pascal: ‘I would have written a shorter letter, but I did not have the time,’ which was also echoed by many other thinkers: Saint Augustine, Pearl Buck, Pliny the Elder, Ezra Pound, Mark Twain and Oscar Wilde, to mention a few. Our aim in this book is to present an overview of microwave tubes (MWTs), which continue to be important despite competitive incursions from solid-state devices (SSDs). We have presented a broad and introductory survey, which we hope readers will be encouraged to read, rather than going through lengthier books for exploring the field of MWTs further in the selected areas of relevance to their respective interests. We hope that the present book will motivate newcomers to pursue research in MWTs and that it will apprise them, as well as the decision makers of the salient features and prospects of, as well as the trends of progress in, MWTs. In writing this book we have received help from High Power Microwaves by J Benford, J A Swegle and E Schamiloglu; Applications of High-Power Microwaves by A V GaponovGrekhov and V L Granatstein (eds); Microwave Tubes by A S Gilmour, Jr; Power Traveling-Wave Tubes by J F Gittins; Vacuum Tubes by K Spangenberg, and Electromagnetic Theory and Applications in Beam-wave Electronics by B N Basu (all listed in the references of this book). We have divided the book into volume 1 and volume 2, each comprising five chapters. The historical timeline starting from the beginning of the twentieth century on the evolution of traditional electron tubes followed by transit-time MWTs described in chapter 1 is intended to arouse the interest of readers in the developments made so far. In this chapter, we have also discussed the high-frequency limitations of electron tubes and how to overcome them; we have also pointed out in what respect MWTs are superior to SSDs. In chapter 2 we have outlined the classification of MWTs from various viewpoints (such as O and M types; Cerenkov, transition and bremsstrahlung radiation types, etc), and pointed out the applications of MWTs in defense/military, medical, scientific, civilian/domestic, industrial sectors, along with the trends in research and development in MWTs. In order to help newcomers in understanding the behavior of MWTs we have introduced some judiciously-selected enabling concepts in chapter 3. In chapter 4 we have discussed the basic concepts of electron guns—the Pierce gun derived from a flat or a curved cathode for O-type tubes, and the MIG and cusp gun for small-orbit and large-orbit gyro-tubes. In this chapter, we have also discussed magnetic focusing structures for the confinement of an electron beam under Brillouin and confined-flow focusing conditions. We have also discussed the salient features of a multistage depressed collector. In chapter 5 we have presented the analysis of a helical slow-wave structure and a disc-loaded waveguide to epitomize the analysis of beam-absent interaction structures. The analysis suggests how we can optimize the structure parameters to obtain the desired shape of the dispersion characteristics of the

ix


structure at a high value of interaction impedance leading to wideband performance of a TWT or a gyro-TWT. The analysis of the structure in the presence of an electron beam has also been outlined, which leads to the derivation of the dispersion relation of the beam’s present structure. The solution of the same can be interpreted to obtain the Pierce-type gain relation of a TWT or a gyro-TWT. Besides these five chapters in the present volume 1 of the book, there are five more chapters which are included in volume 2 of the book, in which we have discussed conventional tubes (chapter 6), fast-wave tubes (chapter 7), vacuum microelectronic tubes (chapter 8); provided handy information about the frequency and power ranges of common MWTs (chapter 9); and summed up the authors’ attempt to elucidate the various aspects of the basics of, and trends in, high power MWTs (chapter 10). We hope that readers following volume 1 (chapters 1 through 5) and subsequently volume 2 (chapters 6 through 10), will appreciate the basics and trends of MWTs, and be aware of the scope of the sustenance and development of MWTs in view of their ever-expanding applications in the high power and high frequency regime.

x

Foreword Vacuum electron devices (VEDs) have played a central role in electrical engineering almost since the birth of the profession near the end of the nineteenth century1. However, despite all the successes of VEDs, including the Voyager twin spacecraft, which are still chugging along, logging 35 000 miles an hour as they zoom farther and farther into the cosmos, forty years after their launch2, VEDs are still cast in a negative light. Recall Senator Lloyd Bentsen’s comments on NBC’s program Meet the Press during the 1988 United States Presidential Campaign as Michael Dukakis’s candidate for Vice President3: ‘You can’t compete if you build vacuum tubes in a solid-state world.’ Of course, nothing can be further from the truth. Although solid-state microwave devices are making progress in achieving higher output power levels, they have important limitations (electrons transport in a solid-state medium in solid-state devices, whereas electrons are ‘free’ in vacuum in VEDs) that will prevent them from overtaking VEDs4. VEDs play essential roles in communications, manufacturing, healthcare, homeland security, defense, manufacturing, the food industry, and in many other areas. This new book by Drs Kesari and Basu is targeted at students just entering the field. It is a welcome contribution since it provides a historical context to the pedagogical development of the subject. The material is accessible by undergraduates and easily grasped by graduate students. At a juncture where the practitioners in the field are aging, this book will help to bring a new generation of students into this vibrant area that promises to continue contributing to science and humanity. Edl Schamiloglu Distinguished Professor of Electrical and Computer Engineering IEEE Fellow Associate Dean for Research, School of Engineering University of New Mexico October 2017 1 IEEE Electron Devices Society, ‘50 Years of Electron Devices: The IEEE Electron Devices Society and Its Technologies 1952–2002’ (IEEE, Piscataway, NJ, 2002), available at http://ethw.org/w/images/f/ff/ 50_Years_of_Electron_Devices.pdf 2 ‘The Voyagers eventually will go quiet. The spacecrafts’ electric power, supplied by radioisotope thermoelectric generators, weakens each day.’ Dodd (Suzanne Dodd, the Voyager project manager at NASA’s Jet Propulsion Laboratory) said that scientists and engineers will likely begin shutting off instruments in 2020, a debate that she says is already underway. ‘These scientists have had their instruments on for 40 years,’ she said. ‘Nobody wants to be the first one turned off.’ The spacecrafts’ transmitters will be the last to go. They will die on their own, in the late 2020s or perhaps in the 2030s. ‘One day we’ll be looking for the signal and we won’t hear it anymore,’ Dodd said. From https://www.theatlantic.com/science/archive/2017/09/voyager-interstellarspace/538881/ 3 R S Symons 1998 Tubes: Still Vital after all these Years IEEE Spectrum p 52. 4 Refer to the article and accompanying text in: J H Booske 2008 Plasma physics and related challenges of millimeter-Wave-to-terahertz and high power microwave generation Phys. Plasmas 15 055502.

xi

Acknowledgments We profusely express our gratitude to Professor Akhlesh Laktakia for stimulating us to write this book. Mr Wayne Yuhasz has provided us with constant guidance in liaison with Professor Laktakia in the development of the manuscript. We sincerely thank Mr Joel Claypool of Morgan & Claypool Publishers for his suggestions to improve the presentation of the book. We sincerely acknowledge the support we have received from Mr B GuhaMallick, Chairman, SKFGI, and Dr S Kamath, Director, MTRDC. We sincerely thank Mr Amit Varshney for drawing the figures, and Mr Raktim Guha, Dr Udit Narayan Pal, Dr S Maurya, Dr Ranjan Kumar Barik, Dr S K Datta, and Dr Vishnu Srivastava for helping us to develop the reference section of the book. We wish to record our sincere thanks to Ms Sreelatha Menon for editing some portions of the manuscript of this book. We would also like to acknowledge Ms M Jayalaxmi, Librarian MTRDC, for her support in providing us with the published literature available at the Technical Information Centre of MTRDC. Vishal Kesari and B N Basu

xii

Author biographies Vishal Kesari Vishal Kesari received an MSc (Physics) degree from Purvanchal University, India, and a PhD (Electronics Engineering) degree from the Institute of Technology, Banaras Hindu University (IT-BHU) (now known as IIT-BHU), India, in 2001 and 2006, respectively. He has worked as a Research Fellow at the Centre of Advanced Study, Electronics Engineering Department, IT-BHU, and significantly contributed to sponsored research projects at the Centre of Research in Microwave Tubes, IT-BHU. He served as a lecturer at the Birla Institute of Technology, Ranchi, India, and the Indian Institute of Information Technology, Allahabad, India, before joining as a scientist at the Microwave Tube Research and Development Centre, Defence Research and Development Organisation, Bengaluru, India. His research interests include microwave and millimeter-wave vacuum electronic devices. He has authored two books: (i) Analysis of Disc-loaded Circular Waveguides for Wideband Gyro-TWTs (Lambert Academic Publishing AG & Co., Germany, 2009), and (ii) High Power Microwave Tubes: Basics and Trends (IOP Concise Physics; Morgan & Claypool Publishers, London, 2018) (with B N Basu as the co-author), and numerous research papers in peer-reviewed journals and conference proceedings. He has, to his credit, a number of international and departmental level awards including the DRDO Young Scientist Award, 2012. He has acted as a reviewer for various peer-reviewed journals and conference proceedings. He is a life member of the Vacuum Electronic Devices and Application Society, India.

B N Basu B N Basu received B.Tech, M.Tech and PhD degrees from the Institute of Radiophysics and Electronics, Calcutta University in 1965, 1966 and 1976, respectively. He served several organizations in India: RIT, Jamshedpur; CSIR-CEERI, Pilani; DRDO-DLRL, Hyderabad; IT-BHU, Varanasi; IFTM University, Moradabad; and SKFGI, Mankundu. He took visiting assignments abroad at Lancaster University, UK; Seoul National University, Korea; and KIT, Karlsruhe, Germany. He was the CSIR Distinguished Visiting Scientist at CSIR-CEERI, Pilani, and Consultant at DRDO-MTRDC, Bengaluru. He played a pivotal role in establishing MOUs (i) between the Department of Electronics Engineering, BHU and CSIR-CEERI, Pilani; (ii) between Seoul National University and CSIR-CEERI, Pilani; and (iii) between SKFGI, Mankundu and CSIR-CEERI, Pilani. He was President of the Vacuum Electron Devices and Application Society, Bengaluru. He has authored (or co-authored) more than a xiii


hundred research papers in journals of international repute (including 37 in IEEE Transactions) and six monograph chapters in the area of microwave tubes. He has authored four books: (i) Electromagnetic Theory and Applications in Beam-Wave Electronics (World Scientific, Singapore/New Jersey/London/Hong Kong, 1996) (ii) Technical Writing (Prentice-Hall of India, New Delhi, 2007), (iii) Engineering Electromagnetics Essentials (Universities Press, Hyderabad, 2015) (distributed by Orient Blackswan, India) and (iv) High Power Microwave Tubes: Basics and Trends (IOP Concise Physics; Morgan & Claypool Publishers, London, 2018) (with Vishal Kesari as the co-author). He is on the Editorial Board of the Journal of Electromagnetic Waves and Applications and he guest-edited a Special Issue on Microwave Tubes and Applications: Issue 17, Vol 31, 2017 of the Journal of Electromagnetic Waves and Applications (Taylor and Francis publication). He served on the Technical Committee on Vacuum Electronic Devices of the IEEE Electron Devices Society. He is a recipient of the SVC Aiya Memorial Award of IETE, Lifetime Achievement Award of the Vacuum Electronic Devices and Application Society, India and ISM Microwave Pioneer Award, Bengaluru.

xiv

IOP Concise Physics

High Power Microwave Tubes: Basics and Trends Volume 1 Vishal Kesari and B N Basu

Chapter 1 Introduction

The simplest electron tube is a vacuum diode, also known as a Flemming valve, which was invented by John Ambrose Flemming in 1904. In 1906 Lee DeForest invented the vacuum triode valve. In fact, the first two decades of the 19th century (1901–1920), besides the invention of the diode and the triode, saw the manufacturing of electron tubes by the Radio Corporation of America (RCA). In the second two decades of the same century (1921–1940), the invention of vacuum microwave tubes (MWTs), namely, the klystron, the travelling-wave tube (TWT) and the magnetron took place (table 1.1). Following the invention of an early form of magnetron by H Gerdien in 1910 and a split-anode magnetron by Albert Hull in 1920 and the subsequent experimentation on such magnetrons in the 1920s and 1930s, which had, however, operated at lower frequencies, the first magnetron of multiple-cavity type was developed independently by K Posthumas and H E Hollmann in 1935 and improved by John Randall and Harry Boot in 1940 in the centimeter-wave frequencies for radar. The invention and development of the klystron by George F Metcalf and William C Hahn in 1936 and by Russel Varian and Siguard Varian in 1937 was a significant event in the historical timeline. The TWT was independently invented by A V Haeff in 1933, N E Lindenblad in 1940 and Rudolf Kompfner in 1942 (table 1.1). In the third two decades of the 19th century (1941–1960), study in the area of TWTs intensified and the basic concept of electron cyclotron maser interaction, relevant to understanding the principle of the gyrotron, was developed (table 1.1). We had to wait until the fourth two decades of the 19th century (1961–1980) to see the development of the earliest versions of the gyrotron. The Joint European Tokamak (JET) and International Thermonuclear Experimental Reactor (ITER) programmes considered the gyrotron as the RF source for fusion plasma in the ninth decade of the same century (1981–1990), while during the period beyond 1990 various manufacturing companies, namely the Institute of Applied Physics (IAP) in Russia; Gycom in Russia; Forschungszentrum Karlsruhe (FZK) in Germany; Japan

doi:10.1088/978-1-6817-4561-9ch1

1-1

ª Morgan & Claypool Publishers 2018


Atomic Energy Research Institute. (JAERI) and Toshiba in Japan; Communications & Power Industries (CPI) in the USA; and Thomson Tubes Electroniques (TTE) in France, developed the technology of developing gyrotrons (table 1.1). The vacuum diode consists of two metallic electrodes: namely the cathode, which emits electrons, and the anode, also known as plate, raised to a higher electric potential than the cathode, which attracts electrons emitted from the cathode to

Table 1.1. Historical timeline.

Fleming valve (vacuum tube diode) First rudimentary radar Audion or triode valve Physics of electric oscillation and radio telegraphy Magnetron in early form Commercial electron tube Smooth-wall, split-anode magnetrons Tube scanning system for television Iconoscope or cathode-ray tube and kinescope Tetrode valve Beam diffraction oscillogram (beam and helix-wave interaction) Travelling-wave tube Multi-cavity magnetron Linear beam MWT theory Klystron Klystron Improved cavity magnetron for radar Travelling-wave tube

Travelling-wave tube Travelling-wave tube Travelling-wave tube Generation of microwaves by rotational energy of helical electron beam Maser Electron cyclotron maser interaction theory

1901–1920 John Ambrose Fleming C Hülsmeyer Lee DeForest G Marconi and K F Braun (Nobel prize)

1904 1904 1906 1909

H Gerdien Radio Corporation of America (RCA) 1921–1940 A W Hull Philo T Farnsworth Vladimir K Zworykin

1910 1920

Albert Hull and N H Williams at General Electric and Bernard Tellegen at Phillips A V Haeff

1926

A V Haeff K Posthumas, H E Hollmann Oskar Heil George F Metcalf and William C Hahn Russel Varian and Siguard Varian J T Randall and H A H Boot N E Lindenblad (US patent 2,300,052 filed on May 4, 1940 issued on October 27, 1942) 1941–1960 Rudolf Kompfner Lester M Field (US Patent 2,575,383) J R Pierce (US Patent 2,602,148) H Kleinwachter

1933 1935 1935 1936 1937 1939 1940

James P Gordon J Schneider R Twiss A Gaponov

1954 1957 1958 1959

1-2

1921 1922 1923

1933

1942 1946 1946 1950


1961–1980 1965 1965 1981–1990 Gyrotron in JET and ITER 1990 onwards Modern gyrotron technology IAP, Russia; Gycom, Russia; FZK, Germany; JAERI, Japan; Toshiba, Japan; CPI, USA; TTE, France; Centre de Recherches en Physique des Plasmas (CRPP), France, Multidisciplinary University Research Initiative (MURI), USA, and so on.

Gyrotrons (earliest version) in Russia

Figure 1.1. Vacuum triode.

form an electron beam—a flow of electrons—from the cathode to the anode. In 1906 Lee DeForest added another electrode in the tube called the grid consisting of a screen of wires through which the electrons can pass, and thus he invented the vacuum triode (figure 1.1). The word ‘triode’ is derived from the Greek τρίοδος, tríodos, from tri- (three) and hodós (road, way), originally meaning the place where three roads meet. The electric potential of the grid of the triode controls the flow of electrons in the tube. In fact, a new era of telephony, sound recording and reproduction, radio, television and computer in the beginning of the 20th century began after the advent of vacuum electron tubes. A directly or indirectly heated cathode, called the thermionic cathode, serves the purpose of an electron emitter in an electron tube (chapter 4). The potential on the grid of a triode can be changed to control the beam current that can be experienced in an external circuit connected to the tube. More and more electrodes can be added to an electron tube for additional functions. Thus, the fourth and the fifth grids can be added to make the so-called vacuum tetrode and vacuum pentode, respectively, in order to realize additional control of the flow of electrons [1–5] The present book deals with a particular type of vacuum electron tube, namely the MWT, in which the electrons in flow are bunched and the electron bunch is made to transfer its kinetic or

1-3


potential energy to electromagnetic waves supported by an interaction structure provided in the device [5, 6]. There are also other types of electron tubes such as the photo tube, in which the photoelectric effect is used for electron emission, and the gas-filled tube such as the thyratron, which contains a gas at a relatively low pressure that makes the device capable of handling much higher currents than the conventional vacuum tubes, thereby making it suitable as a high power electrical switch or a controlled rectifier. In a vacuum tube the accumulation of electrons or space charge in the path of electron beam flow exerts a repelling force on such flow of electrons thereby limiting the value of the current. High-current limitation of electron tubes due to the accumulation of such space charge can be alleviated by assisting MWTs, such as TWT, gyrotron, etc, by plasma. In some tubes, such as the virtual cathode oscillator (VIRCATOR), the space charge is used as an advantage to form the so-called ‘virtual cathode’. In the VIRCATOR, the electrons execute oscillatory motion across a wall of a resonant cavity between the actual cathode situated outside the cavity and the virtual cathode inside the cavity to generate microwaves [7–9]. In this book we intend to outline the basics of, and trends in, MWTs, addressing the various issues related to their high power, high efficiency, wideband and high frequency performances. The phrase ‘high power’ in the title of the book has to be judged vis-à-vis the application of the tube. What is usually ‘low power’, obtainable by a tube developed by vacuum microelectronics technology, can be considered as ‘high power’ in the terahertz frequency regime of application. Similarly, what is usually ‘high power’, for example in a radar system, becomes ‘low power’ for directed energy weapons (DEWs) [7–9]. Order of vacuum The vacuum is needed in a MWT to prevent the electrons emitted from the cathode (electron emitter) from colliding with the atoms thereby losing their energy before crossing or passing through the anode of the tube. Besides, the vacuum prevents ionization inside the tube caused by electrons colliding with atoms that produces positive ions, which can strike the cathode and damage it. A high order of vacuum prevents high power tubes from high voltage breakdown and arcing. The vacuum in MWTs depending on their applications (chapter 2) is created in the range of high vacuum (10−5–10−7 Torr) to ultra high vacuum (<10−7 Torr) (where 1 Torr = 1.333 22 millibar = 1 mm Hg = 133 Pascal). High-frequency limitations of electron tubes The factors responsible for setting a high-frequency limit of electron tubes are mainly (i) power loss due to skin effect (ii) I 2R loss caused by the capacitor charging currents, (iii) radiation losses, (iii) issues related to the thermal management of tiny tubes, (iv) interelectrode capacitance and lead inductance effects, (v) finite transit time of electron flight between electrodes and (vi) constancy of gain-bandwidth product [2–5]. Highly conducting materials should be used to make the tube parts to reduce power loss caused by the skin effect. I 2R loss caused by the capacitor charging

1-4


currents and associated losses can be reduced by reducing the interelectrode capacitances and by increasing the number of shunt paths along which the charging current flows. At high frequencies when the dimensions of the tube become comparable with wavelengths, electromagnetic waves may radiate out from the tube (interaction structure). In order to reduce such radiation losses the spacing between the electrodes needs to be reduced to the order of 1/100 of wavelengths, although at the cost of RF resistance of the conductors. Shielding the tube using a highly conducting shield is very effective in reducing radiation losses. At high frequencies, when the tube uses tiny parts, thermal management should be performed to cool the parts [4]. At high frequencies, the interelectrode capacitances and lead inductances of the tube become comparable with the capacitance and inductance, respectively, of the circuit connected to the tube, a resonant circuit for example, the dimensions of which are reduced at high frequencies [1, 2]. Thus, at such high frequencies, the reactance of the grid and cathode lead each increase and the reactance of the interelectrode capacitance between the grid and the cathode and that between the grid and the plate (anode) each decrease. Also, there can be a resonance between the lead inductance and the interelectrode capacitance of the tube at such frequencies. These high-frequency effects have been studied by the equivalent-circuit representation of an electron tube. For instance, a triode can be replaced by a constant source of current gmeg between the anode (plate) and the cathode, in parallel with the plate resistance rp. Here, gm is the transconductance of the tube and eg the incremental grid-cathode voltage Egk . We can find the input resistance Rg of the triode by analyzing the equivalent circuit of the triode connected to a load impedance Zl . For this purpose, we can consider the effect of only the cathode lead inductance Lk and ignore the effect of the grid lead inductance. We can also make the approximation that the potential across the cathode lead inductance is much less than Egk and rp >>Zl + jωLk . The grid-cathode capacitance Cgk can also be taken much larger than the grid-plate capacitance. Thus, such equivalent circuit analysis leads to the following expression for the input resistance Rg of the triode:

Rg ≅

1 . ω gmLkCgk 2

(1.1)

We can then appreciate from (1.1) that when the effect of the cathode lead inductance is much more significant than that of the grid lead inductance, the input resistance Rg of the triode is inversely proportional to the square of the operating frequency. Therefore, at high frequencies, energy is drawn from the signal source because of the coupling between the grid and cathode circuits caused by the cathode lead inductance [4]. Similarly, we can easily obtain the following approximate expression for the input admittance Yg if we set Lk = 0 and consider only the effect of Lg ≠ 0 [4]:

Yg ≅

jωCgk 1 − ω 2LgCgk

(Lk = 0, Lg ≠ 0).

1-5

(1.2)


Interestingly, it follows from (1.2) that at the frequency ω = 1/(LgCgk )1/2 , Yg → ∞, which corresponds to the occurrence of the resonance of the input circuit caused by the grid lead inductance Lg coupled to the grid-to-cathode capacitance Cgk . In other words, at this frequency of resonance, the signal input to the triode is short-circuited thereby making the input fail to cause any effect in the plate circuit [3]. The physical dimensions of the tube should be therefore reduced to minimize the effect of the electrode lead inductances and interelectrode capacitances. The reduction of these tube inductances and capacitances will also increase the maximum resonance frequency of a resonator circuit connected to the tube. Furthermore, at high frequencies, the transit time of the electrons between the cathode and grid becomes comparable with the time period of the modulating electric field in the cathode-grid space. As a result, the field may reverse its phase before electrons traverse this space, thereby causing the electrons to oscillate between the cathode and the grid or return to the cathode. The phenomenon can be easily understood considering the flight of an electron carrying a negative charge accelerated between a large, planar electrode to another similar electrode at a higher potential and studying the induced charges on these two electrodes while the electron is in transit between these electrodes. During the flight of the electron, the positive charge induced on the approaching electrode increases with time and that on the receding electrode decreases at the same time such that the sum of the two induced charges at any instant of time is equal to the magnitude of the electron charge. We can find the induced charge on the approaching electrode at any instant of time by equating the work done in transferring the induced charge to the approaching electrode, raised to a given potential with respect to the receding electrode, to the work done by the electron to move through a distance from the receding electrode at that instant of time. The induced charge so found becomes directly proportional to the distance of the electron from the receding electrode at that instant and, consequently, the induced current obtained by differentiating the induced charge with respect to time becomes proportional to the electron velocity at that instant. However, this electron velocity varies linearly with time since the electron has a constant acceleration, subject to the constant electric field between the electrodes. As a result, the induced current, which is proportional to the electron velocity, also varies linearly with time. Corresponding to this induced current, there will be a current flowing in the external circuit connected to the triode while the electron is in flight between the electrodes, contrary to the notion that some might have that the current would flow when the electron strikes the positive electrode and completes the path through the external circuit. The current ceasing to flow as the electron strikes the positive electrode is essentially a triangular pulse. For an electron beam, the total induced current is the addition of such triangular pulses of current associated with the motion of all the electrons in flight between the electrodes. Interestingly, current may even be induced in an electrode to which no flows of electrons are collected (for instance, the grid of a triode), if the number or velocity of electrons approaching the grid is greater than the number or velocity of electrons receding from it or vice versa depending on the grid bias voltage. From the concept of the induced current due to a finite transit time of electrons between the electrodes 1-6


developed here, it can be appreciated by simple analytical reasoning that the grid conductance Gg is jointly proportional to the square of signal frequency f and the transit time τ of electrons in the tube [2–5]:

Gg = Cgmτ 2f 2 .

(1.3)

A finite value Gg , due to the transit-time effect given by (1.3), is responsible for the power loss to the grid. The grid power loss can be reduced by increasing the plate voltage to reduce the value of τ , however, at the cost of the plate dissipation and/or by decreasing the interelectrode spacing, which, however, causes an undesirable increase of the interelectrode capacitance. This calls for the simultaneous decrease of the interelectrode spacing and electrode areas to avoid an increase of capacitance with allowable plate dissipation. The gain-bandwidth product limitation of an electron tube can be appreciated by studying the output of an electron tube in the form of a tuned resonator circuit comprising a tuning inductance L for the stray capacitance C of the tube. With the increase of the operating frequency, in the limit, the terminating leads form a short loop or a quarter-wave line terminated within the tube by the interelectrode capacitances [2–5]. The circuit analysis of such a tuned amplifier replacing the electron tube by a constant current source, supplying a current gmeg , in parallel with the plate resistance rp , yields the following expression for the gain-bandwidth product in terms of the transconductance gm of the tube and the stray capacitance C : g gain-bandwidth product = m . (1.4) C It follows from (1.4) that the gain-bandwidth product of an electron tube amplifier is a constant, being independent of the operating frequency and depending only on gm and C of the tube, suggesting that the gain of the amplifier can be increased only at the cost of its gain [4]. Tiny electron tubes to alleviate high-frequency limitation The lead inductance and interelectrode capacitance effects, as well as the transit-time effect, which limit the high-frequency performance of electron tubes, have been alleviated in tubes such as the acorn, doorknob and lighthouse tubes [2, 10]. The physical dimensions of these tubes are reduced in the same proportion as the highfrequency limiting effects are reduced without reducing the amplification capability of the tube. Although the operating frequencies of these tubes can be increased to UHF, the reduction of their size entails the reduction of their power handling capability as well. (The acorn tube is so named due to its glass cap resembling the cap of an acorn and the doorknob tube is an enlarged version of the acorn tube that enables the former to deliver higher power than the latter.) The limiting factor of this tube is the power dissipating ability of the grid in the proximity of the cathode [2]. The grid and plate of some of the acorn and doorknob tubes are each provided with two leads so that, if required, a section of parallel-wire line may be connected between each pair of grid and plate leads. Such an arrangement makes it possible to

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make the lead impedance high, for instance, if a quarter-wave is connected to the lead and is short-circuited at its load end [2]. The lighthouse tube has a planar construction—made of the cathode, grid and plate discs—to reduce interelectrode capacitance and lead inductance, which makes it resemble a lighthouse tower. The interelectrode distances of the tube are made a fraction of a mm and the terminals are made of flat discs welded to the end faces of glass cylinders; the edges of these discs projecting outside the vacuum tube envelope so that they could be connected to sections of coaxial lines of an oscillatory system [2]. The resonant circuit load of the lighthouse tube is constructed as the integral part of the tube (unlike the acorn and doorknob tubes) so that the undesirable effects of the lead inductance and interelectrode capacitance resulting from the tube and the resonant circuit load of the tube being separate units could be alleviated. Advent of transit-time microwave tubes The adverse effect of electron transit time in conventional electron tubes, such as the triode, which imparts a finite value of the grid conductance responsible for the power loss to the grid of the tube, can be used to advantage in MWTs. Thus, the concept of the induced current in an electrode of such a tube when the number or velocity of electrons approaching the electrode is different from the number or velocity of electrons receding from it can be used in a MWT such as the multi-cavity klystron. However, as the operating frequency is increased to the millimeter-wave regime, the sizes of the interaction structures of conventional MWTs need to be reduced limiting the device RF output powers. This has led to the development of fast-wave MWTs such as the gyrotron, which can deliver high powers even in the millimeter-wave regime since the sizes of the interaction structures of these devices do not reduce as much as those of conventional MWTs. Further, with the advent of vacuum microelectronic technology, the high-frequency capability of MWTs has been extended to the terahertz regime. Solid-state devices versus microwave tubes MWTs continue to be important despite competitive incursion from solid-state devices (SSDs) (figure 1.2). MWTs enjoy superiority over their solid-state counterparts with respect to having a lesser heat generated due to collision in the bulk of the device, a higher breakdown limit on maximum electric field inside the device, a smaller base-plate size (determined by the cooling efficiency), higher peak pulsedpower operability, ultra-bandwidth (three-plus octave) performance above a gigahertz, and so on (table 1.2). Further, unlike SSDs, MWTs—being fabricated out of metals and ceramics—are inherently hardened against radiation and fairly resistant to temperature and mechanical extremes (table 1.2). In fact, attempts were made to replace space-TWTs with SSDs, however, with limited success in view of the required ∼5 × 106 h MTBF (mean time between failures) in satellite qualified devices. Thus, although SSDs were tried in satellite communication systems during the last decade of the 20th century, for instance, replacing ∼50% TWTs with SSDs in 1995, such replacements declined beyond 1998 to only ∼10% making space TWTs more relevant than their SSD counterparts.

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Figure 1.2. Solid-state and vacuum device average power capabilities [11].

Table 1.2. Solid-state devices versus microwave tubes [10].

Issue Collisional heat produced by electron stream Operating temperature

Breakdown limit on maximum electric field inside the device Base plate size determined by cooling efficiency increasing with (i) the temperature difference between the hot surface and the cool environment and (ii) the surface area of the hot surface

Solid-state devices

Microwave tubes

Throughout volume

Only at the collector

Lower temperature operation for a longer life (lower mobility—a greater drag or inertial forces due to collision) Degradation at a higher temperature due to dopant migrating excessively, lattice becoming imperfect, mobility becoming reduced impairing high frequency performance Wide-band-gap semiconductors such as SiC and GaN to be used for high temperature operation Lower

Higher temperature operation

Larger

Higher Smaller (higher collector temperature)

(Continued)

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Peak pulsed power

Ultra-bandwidth performance (three-plus-octaves)

Hardening against radiation and tolerance to temperature and mechanical extremes

Lower (calls for power combining by multiple transistors and proportionate increase in package size) Possible below 1 GHz (corresponding to longer wavelengths ensuring negligible phase difference in the voltage between the emitter and base) Not possible

Direct cooling of heat zone Energy recovery out of waste beam Ionization

Not possible No recovery out of waste beam

Permissible operating temperature Handling power in interaction volume Noise figure Efficiency Process cost Performance Warm-up delay Periodic maintenance High voltage power Supply requirement

Lower (mobility of electrons is less at elevated temperature) Less power in smaller interaction volume Lower Lesser Lesser Linear Short Not required Not required

Ionization of lattice

Higher Beam may be pulsed in the region separated from the interaction region Usually not possible (controlling the structure dispersion is a challenging problem) Can be hardened and is fairly resistant to temperature and mechanical extremes Possible Significant recovery of spent beam energy Ionization of residual gasses (much less) Higher More power in smaller interaction volume Higher Higher Higher Nonlinear Long Required Required

Organization of the book The book is divided into two volumes comprising of ten chapters. Chapters 1 through 5 are contained in volume 1, and chapters 6 through 10 in volume 2. The present introductory chapter has presented the historical timeline of the development of MWTs (chapter 1, volume 1). Moreover, in this chapter, the order of vacuum required in conventional electron tubes and the high frequency limitations of these tubes have been discussed. How the development of tiny electron tubes and then the advent of transit-time tubes alleviated the high-frequency limitation of conventional electron tubes have also been discussed. An explanation for the sustenance of MWTs despite competitive incursions from solid-state devices has also been given. In the subsequent chapters, the classification and applications of

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MWTs and trends in their research and development (chapter 2, volume 1), the enabling concepts involved in understanding the principles of MWTs (chapter 3, volume 1), and the formation, confinement and collection of an electron beam in MWTs (chapter 4, volume 1) have been discussed. We have also analytically appreciated the various aspects of beam-absent and beam-present slow-wave and fast-wave interaction structures—the former typically with respect to a helical slowwave structure and disc-loaded cylindrical waveguide, respectively, and the latter typically with reference to the conventional TWT and the gyro-TWT, respectively (chapter 5, volume 1). A qualitative description has been presented for conventional and familiar microwave tubes, namely, TWTs, klystrons including multi-cavity and multi-beam klystrons, klystron variants, which include reflex klystron, inductive output tube, extended interaction klystron (EIK), extended interaction oscillator (EIO) and twystron, and also crossed-field tubes, namely, magnetron, crossed-field amplifier (CFA) and carcinotron (chapter 6, volume 2). Fast-wave tubes have also received attention encompassing the gyrotron, gyro-backward-wave oscillator, gyroklystron, gyro-travelling-wave tube, cyclotron auto-resonance maser (CARM), slow-wave cyclotron amplifier (SWCA), hybrid gyro-tubes and peniotron (chapter 7, volume 2). The book has further brought within its purview vacuum microelectronic, plasma-filled and high power microwave (HPM) tubes (chapter 8, volume 2). Handy information about the frequency and power ranges of common microwave tubes has also been given (chapter 9, volume 2) though more such information has been provided at relevant places in the rest of the book as and where necessary. An epilogue at the end summarizes the authors’ attempt to elucidate the various aspects of the basics of, and trends in, high power microwave tubes (chapter 10, volume 2).

References [1] Terman F E 1947 Radio Engineering (New York: McGraw Hill) [2] Spangenberg K 1948 Vacuum Tubes (New York: McGraw Hill) [3] Reich H J, Skalnik J G, Ordung F F and Krauss H L 1957 Microwave Principles (New York: Van Nostrand Reinhold Co) [4] Soohoo S F 1971 Microwave Electronics (Reading, MA: Addison-Wesley) [5] Hutter R G E 1960 Beam and Wave Electronics in Microwave Tubes (Princeton: D Van Nostrand) [6] Gilmour A S Jr 1986 Microwave Tubes (Norwood: Artech House) [7] Benford J and Swegle J A 1991 High Power Microwaves (Boston: Artech House) [8] Gaponov-Grekhov A V and Granatstein V L (ed) 1994 Applications of High-Power Microwaves (Boston: Artech House) [9] Benford J, Swegle J A and Schamiloglu E 2015 High Power Microwaves 3rd edn (New York: CRC Press) [10] Barker R J, Luhmann N C, Booske J H and Nusinovich G S (ed) 2005 Modern Microwave and Millimeter-wave Power Electronics (Piscataway: Wiley-IEEE Press) [11] Gilmour A S 2011 Klystrons, Traveling Wave Tubes, Magnetrons Crossed-Field Amplifiers, and Gyrotrons (Boston: Artech House)

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Chapter 2 Microwave tubes: classification, applications and trends

In a microwave tube (MWT) a beam of electrons is bunched in an interaction structure supporting electromagnetic waves, and subsequently the kinetic or potential energy of the electron bunch is converted into electromagnetic energy. In this chapter, we discuss the classification (section 2.1), applications (section 2.2) and trends in research and development of MWTs (section 2.3).

2.1 Classification MWTs have been classified from various angles. For example, the conventional TWT is classified from various angles such as TPO (tubes à propagation des ondes) (or simply O) type, slow-wave, non-relativistic bunching, axial bunching, propagating waveguide interaction, space-charge-wave interaction, distributed interaction, kinetic energy conversion, Cerenkov radiation, and so on (table 2.1). Similarly, the conventional magnetron and the crossed-field amplifier (CFA) can be classified as: tubes à propagation des ondes à champs magnetique (TPOM) (or simply M) type, slow-wave, non-relativistic bunching, cavity interaction, potential energy conversion, and so on. The magnetic field takes part in the interaction in a magnetron of M-type, unlike in a TWT of O-type where the magnetic field is used to confine or focus the electron beam. Further, relativistic types of MWTs also exist, such as the relativistic TWT and the relativistic magnetron to deliver high powers. Similarly, the gyrotron can be classified as fast-wave, relativistic bunching, azimuthal bunching, cavity interaction, kinetic energy conversion, bremsstrahlung radiation type; and so on (table 2.1).

2.2 Applications MWTs have well known applications in communication, radar and electronic warfare (EW). The use of MWTs makes it possible to establish a point-to-point doi:10.1088/978-1-6817-4561-9ch2

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Table 2.1. Classification of MWTs from various angles.

Angle of view Role of dc magnetic field in beam flow

Mechanism of the bunching of electrons

Classification TPO or O-type: dc magnetic field along beam flow M-type: beam flow perpendicular to crossed dc electric and magnetic fields Relativistic and non-relativistic bunching

Axial and azimuthal bunching

Interaction structure

Propagating waveguide (slow or fast) and cavity resonator types

Nature of wave supported by the beam

Space-charge and cyclotron-wave interaction

Nature of interaction

Localized and distributed interaction

Mechanism of energy transfer from the beam to electromagnetic waves Wave phase velocity vph of electromagnetic waves in the interaction structure Instability

Kinetic energy and potential energy conversion types

Slow-wave type: vph < c Fast-wave type: vph > c

CRM and Weibel instability types

Typical examples TWT, klystron, etc (O-type) Magnetron, CFA (M-type)

Gyrotron, gyro-TWT, etc (relativistic bunching) TWT, klystron, magnetron, SWCA, etc (non-relativistic bunching) TWT, klystron, SWCA, etc (axial bunching) Gyrotron, gyro-TWT, gyro-klystron, etc (azimuthal bunching) TWT, gyro-TWT, etc (propagating waveguide type) Klystron, gyrotron, gyro-klystron, etc (cavity resonator type) TWT, klystron, magnetron, etc (space-charge wave interaction) Gyrotron, gyro-TWT, etc (cyclotron-wave interaction) TWT, gyro-TWT, etc (distributed interaction) Klystron (localized interaction) TWT, klystron, gyrotron, gyro-TWT, gyro-klystron (kinetic energy conversion), etc Magnetron, CFA (potential energy conversion), etc TWT, magnetron, CFA, etc (slow-wave type) Gyrotron, gyro-TWT, gyro-klystron (fast-wave type)

Gyrotron (CRM instability), SWCA (Weibel instability) (Continued)

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Table 2.1. (Continued )

Angle of view

Nature of radiation

Classification

Electrons moving in the interaction structure with a DC velocity >vph : Cerenkov radiation type Electron beam passing through the boundary between two media with different refractive indexes or through perturbation in a medium such as conducting grids, or gaps between conducting surfaces Electron beam accelerated in electric and/or magnetic field: bremsstrahlung radiation

Typical examples CARM (combined CRM and Weibel instabilities) TWT (Cherenkov radiation)

Klystron, monotron (transition radiation type)

Gyrotron (bremsstrahlung in magnetic field) VIRCATOR (bremsstrahlung in electric field)

O stands for TPO—tubes à propagation des ondes, and M stands for TPOM—tubes à propagation des ondes à champs magnetique. CFA: crossed-field amplifier; SWCA: slow-wave cyclotron amplifier; CRM: cyclotron resonance maser; CARM: cyclotron auto-resonance maser; VIRCATOR: virtual cathode oscillator

communication link with more channel capacity. The application in communication also includes satellite-to-home communication and, in the millimeter-wave regime, high information density communication and deep-space and specialized satellite communication and high resolution radar and extension of radio range. The application of MWTs in civilian radar includes weather detection, highway collision avoidance, air-traffic control, burglar alarms, garage door openers, speed detectors (law enforcement), air-traffic control, mapping of ground terrain, and ground probing (for the detection of underground materials like gun emplacements, bunkers, mines, geological strata, pipes, voids, etc), remote sensing, imaging in atmospheric and planetary science, space debris phased-array mapping, analysis of cloud (as a sensor in environmental research), etc [1]. In the millimeter-wave regime, the applications are in long-distance radar, high resolution radar imaging and precision tracking which have been made possible by the advent of gyrotrons. In the military sector, the applications are in EW and electronic counter countermeasure (ECCM), encompassing electronic support measure (ESM) to detect, intercept, identify, locate, record, and/or analyse sources of radiated electromagnetic energy for the purposes of immediate threat recognition as required for making decisions involving electronic protection (EP), electronic attack (EA), avoidance, targeting, and other tactical employment of forces. The application in military radar includes missile tracking and guidance and information warfare (IW) involving a directed energy weapon (DEW) with the advent of high power microwave (HPM) tubes 2-3


Table 2.2. Application of some commonly used MWTs.

MWT

Application

Wideband helix TWT ESM, ECM, ECCM Narrow-band helix TWT Satellite and ground-based communication; telemetry and telecommand; radio astronomy; radar; missile seeker CC-TWT Radar Klystron Accelerator; broadcasting; radar; radio astronomy Magnetron Radar; microwave heating; broadcasting Gyrotron Plasma heating; industrial heating; powder metallurgy; material processing; low intensity conflict; active denial system Gyro-TWT Millimeter-wave radar Gyro-klystron Millimeter-wave radar VIRCATOR HPM/DEW MILO HPM/DEW Relativistic magnetron HPM/DEW Relativistic klystron HPM/DEW

(section 2.3) (table 2.2). In the terahertz regime, the applications are in imaging, security inspection, enhanced sensitivity spectroscopy and dynamic nuclear polarization enhanced nuclear magnetic resonance [2, 3]. Pulsed MWTs are extensively used in pulsed radars for military applications. Major applications include target detection, target recognition in surveillance radars, weapon control in fire-control radars, weapon guidance in missile systems, identifying enemy locations using imaging radars etc. Pulsed MWTs also find applications in other pulsed radars for air traffic control, weather observation (precipitation radar), satellite-based remote sensing, etc. The applications of continuous-wave (CW) MWTs are in unmodulated CW radars for traffic control, speed gauges, Doppler motion sensors, motion monitoring, etc, and also in frequencymodulated-continuous-wave (FM-CW) radars for imaging and non-imaging applications such as high-resolution imaging, navigation, radar altimeters, aircraft radio altimeters, etc. For satellite communication, we can use CW TWTs for multi-carrier communication downlinks and pulsed TWTs for remote sensing and imaging. For deep-space applications, we use pulsed tubes for imaging and both CW and pulsed tubes for the transfer of data to Earth through telemetry. In the EW system (while for ECCM a pulsed TWT can be used) we can—in order to implement deceptive anti-jamming by mimicking a radar echo in an ESM system—use the same TWT to operate in both CW and pulsed modes depending on the threat scenario [2, 3]. Peaceful applications of MWTs include industrial heating, material processing, waste remediation, civil, mining and public health engineering—including breaking of rock, breaking of concrete, tunnel boring and soil treatment, plasma heating for a controlled thermonuclear reactor (electron cyclotron resonance heating of fusion plasmas) involving heating of hydrogen isotopes typically at an ignition temperature

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of 108 K at 200 GHz. Scientific applications include RF linear accelerators, plasma diagnostics and chemistry, nonlinear spectroscopy, etc [1]. Material processing applications get a boost with the advent of high power MWTs at millimeter-wave frequencies, such as gyrotrons that can implement volumetric and selective heating. This, for instance, has made it possible to develop ceramic sintering and joining and production of new composite ceramics—stronger and less brittle—that retain their high strength under high temperature and corrosive conditions. Consequently, this has made it possible to develop lightweight ceramic engines for aircraft and automobiles as well as strong, long-lived ceramic walls for thermonuclear power reactors. Industrial heating applications include the drying of leather, paper, pharmaceuticals, tea, coffee, tobacco, textiles, etc. They also widely encompass the food industry: precooking, cooking, pasteurizing, sterilizing, dough proofing, thawing, tempering, pasta drying, roasting of food grains/beans, etc; the plastic industry: sealing/bonding, bulk heating, moulding plastic foam, plastic laminate production, drying, etc; the forest industry: hardwood drying, plywood-veneer drying, pulp/ wood-chip drying, destruction of fungi and insects in wood, etc; the rubber industry: vulcanization, curing sponge rubber tubing, curing and foaming polyurethane bulk heating, etc; and the chemical industry: drying paint and varnish, refractory processes, polymerizing, etc. Medical applications include medical diagnosis and treatment such as hyperthermia, that provides selective heating up of tissues without harming healthy ones, thus enabling such warmed-up tissues to receive more nutrients and antibodies thereby speeding up the healing process. Besides, there are applications in orthopaedics: arthritis, sciatica, rheumatism, etc; internal medicine: asthma, bronchitis, urology, etc; dermatology: boils, carbuncles, sores, chilblains, etc; oto-rhynolaryngology: abscesses, laryngitis, etc; dental care and ophthalmology. Further, other unconventional applications can be named such as satellite power stations, artificially created ionized layers for the extension of radio range, city lighting, nitrogen fertilizer raining on the Earth, and environmental control by both ozone generation and atmospheric purification of admixtures that destroy the ozone layer, and so on [1].

2.3 Trends in research and development The various types of MWTs developed (section 2.1) and their applications (section 2.2) have set a trend in their research and development. According to this trend we can categorize MWTs into five groups (figure 2.1) as follows. • Group 1: Improved-performance conventional tubes (multi-beam klystron, high-efficiency space TWT, multi-octave electronic warfare TWT, etc). • Group 2: Tubes accruing the advantages of both vacuum electronic and solidstate devices (micro-fabricated or vacuum microelectronic tube and miniature TWT co-existing with a solid-state power amplifier in a microwave power module (MPM)).

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Figure 2.1. Groups into which the trends in MWT research and development can be categorized.

• Group 3: IREB-driven virtual cathode oscillator (VIRCATOR), magnetically insulated line oscillator (MILO), relativistic backward-wave oscillator, relativistic klystron, electromagnetic bomb using a magnetic flux compression generator in conjunction with a VIRCATOR, etc. • Group 4: Fast-wave devices (gyrotron, gyro-TWT, etc) which fill up the technology gap in the high power, millimeter-wave frequency domain. • Group 5: Plasma-filled tubes in which the space charge is neutralized for enhanced beam current transport (plasma-assisted coupled-cavity TWT, backward-wave oscillator—PASOTRON (plasma-assisted slow-wave oscillator), gyrotron, etc). Group 1 MWTs are being continuously improved with respect to their performance characteristics by innovative design and technology (figure 2.1 and table 2.3). Thus, there has been global competition to enhance the life and efficiency of space TWTs, say, by tapering the pitch of the helix of the TWT for re-synchronization of the RF phase velocity with the beam velocity for efficiency improvement. Also, innovative helix loading techniques are being used to develop wideband (multi-octave) EW helix TWTs, such as by loading the metal envelope of the tube by metal, tapering the cross section of the dielectric helix supports, and by using multisection, multi-dispersion helical structures. Similarly, multi-beam technology, using multiple beam channels but a common interaction structure and a beam dump (collector), is being employed to develop compact, high power klystrons (figure 2.1). Group 2 MWTs—TWTs in microwave power modules (MPMs) and microfabricated vacuum microelectronic tubes—have removed the age-old rivalry between solid-state and vacuum electron devices (section 1.6) by accruing the advantages of solid-state devices and their relevant technology. The MPM is a synergic combination of a solid-state amplifier, a TWT power booster, sharing gains equally between them, and a built-in electronic power conditioner (EPC) to make a

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Table 2.3. Tubes and their envisaged features vis-à-vis trends.

Trend Improved performance conventional MWTs (Group 1)

Tubes accruing the advantages of both vacuum and solidstate devices/ electronics (Group 2)

Tubes

Features

Wideband electronic warfare TWTs High efficiency, long-life, lightweight space-TWTs High power compact multibeam klystrons, etc Micro-fabricated (vacuum microelectronic)—foldedwaveguide TWT, reflex klystron, etc

Innovative tube- envelope/tapered dielectric helix-support/pitch profiling/depressed collection Multi-beam electron gun

Helix TWT in MPM IREB driven HPM tubes (Group 3)

VIRCATOR, MILO, relativistic tubes: magnetron, TWT, BWO, klystron (RELTRON), OROTRON, multi-wave Cerenkov generators (MWCG), etc

Fast-wave tubes (Group 4)

Gyrotron, gyro-TWT, gyroklystron, etc

Plasma-filled tubes (Group 5)

Plasma-assisted TWT, PASOTRON, gyrotron, etc

Terahertz generation/batch production with microfabrication Coexistence of solid-state amplifier and a MWT (TWT) in MPM MPM in radar, EW and communication Bremsstrahlung of electrons in electrostatic field (VIRCATOR) E-bomb using a magnetic flux compression generator (FCG) in conjunction with a VIRCATOR Self-focusing (MILO) Information warfare Filling of mm-wave technology gap in the high power domain —Bremsstrahlung of electrons in magnetic field Periodic beam structures: vaneloaded, coaxial corrugated, photonic band gap (PBG) for mode selection/rarefaction Space-charge neutralization for enhanced beam current transport Relaxed beam focusing

compact module of an amplifier. Vacuum-microelectronic tubes have the fundamental advantages of (i) electron velocity in vacuum about a thousand times greater than that in semiconductor solids, and higher signal processing speeds, (ii) less collisions of moving electrons with atoms and less associated energy loss as heat, (iii) precision dimensioning of parts/electromagnetic structures in the millimeter-wave and terahertz regimes. Such tubes use cold field emission arrays cathodes (such as 2-7


carbon nanotubes) etc, and can be developed using micro-fabrication solid-stateelectronic technology such as EDM (electric discharge machining), DRIE (deep reactive ion etching) and LIGA (lithographie, galvanoformung, abformung) (x-ray lithography). These tubes have made it possible to extend the frequency range of high power MWTs (chapter 9, volume 2) to millimeter and terahertz frequency regimes and opened up the possibility of their batch production (figure 2.1 and table 2.3). Under Group 3, IREB-driven tubes are realized either from conventional tubes, such as the MILO, which is a modified crossed-field amplifier (CFA), or from unconventional tubes, such as the VIRCATOR, which is based on bremsstrahlung of electrons in an electrostatic field. They can be used for weapons based on HPM, the latter characterized by (i) long pulse duration, high-PRF, or CW and (ii) highpeak power, short-pulse duration, low-PRF, or single-shot operation (section 1.3). These weapons may be hard-kill or soft-kill types, the former for large-scale physical destruction of targets and the latter for disabling mission-critical equipment of the enemy. Such weapons require a power supply rather than an explosive as ammunition; can operate in all weather; can spread their effect by diffraction and thus permit coarse pointing for attack unlike laser weapons; remain operative even if the enemy system is switched off. Similarly, an intense electromagnetic pulse (EMP) of peak powers ∼10’s of TW of very short duration ∼100’s of ns (shock-wave) can be used for a directed energy weapon (DEW) (section 1.3). Typically, a flux compression generator (FCG) in conjunction with a VIRCATOR may be used for attacking a wide range of vulnerable equipment using front-door coupling through transmitting/receiving antennas associated with radar and communication equipment and back-door coupling through power connecting wires and cables, grills/holes in enclosure, display screens of computers, etc (figure 2.1 and table 2.3). Group 4 MWTs have filled the technology gap in the millimeter-wave frequency regime arising on one hand from the low-frequency limitation of quantum-optical devices such as the energy of each quantum decreasing with frequency and the difficulty of retaining popular inversion and, on the other hand, from the highfrequency limitation of conventional MWTs. Constant effort is being made to develop high-efficiency, mode-selective gyrotrons by designing innovative interaction structures such as a vane-loaded waveguide, a coaxial cavity with corrugated tapered cross-section coaxial insert for a larger beam current transport as well as for mode rarefaction. Similarly, wideband coalescence between the beam-mode and waveguide-mode dispersion characteristics has been achieved by dielectric lining of the waveguide interaction structure or using a tapered cross-section waveguide or a metal/dielectric disc-loaded waveguide to widen the bandwidth of a gyro-TWT. The cyclotron auto-resonance masers (CARMs), which use both CRM and Weibel instabilities, have been developed for higher power and wider bandwidth. Innovative harmonic multiplying tubes, requiring lesser magnetic fields due to cyclotron harmonic operation and a less expensive input drive due to low-frequency input, have been developed, such as the gyro-TWT and the inverted gyro-twystron, also known as PHIGTRON—twystron being a combination of the TWT and klystron [3]. 2-8


Under Group 5, due to plasma assistance that neutralizes the space charge, MWTs such as the coupled-cavity TWT, PASOTRON (plasma-assisted slow-wave oscillator (BWO-mode)) and gyrotron allow a greater beam current transport as well as relaxing the magnetic field requirement (figure 2.1 and table 2.2). Obviously, with respect to some of these tubes, the groups in which they have been categorized vis-à-vis the trends of their development may overlap. Thus, for instance, the gyrotron belonging to Group 4, if it is plasma-assisted, would come under group 5 as well. Similarly, the PASOTRON belonging to Group 5, if it is IREB-driven, would also come under the purview of Group 3.

References [1] Gaponov-Grekhov A V and Granatstein V L (ed) 1994 Applications of High-Power Microwaves (Boston: Artech House) [2] Gilmour A S Jr 1986 Microwave Tubes (Norwood: Artech House) [3] Gilmour A S 2011 Klystrons, Traveling Wave Tubes, Magnetrons Crossed-Field Amplifiers, and Gyrotrons (Boston: Artech House)

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Chapter 3 Basic enabling concepts

In a microwave tube (MWT) a bunch of electrons transfers their kinetic or potential energy to RF waves. Some of the basic concepts that help understanding the various aspects of the operation of MWTs are discussed in this chapter.

3.1 Cathode The cathode, an emitter of electrons in an electron tube, is said to be the ‘life’ of a MWT. A directly heated cathode is made in the form of a filament of an emitting material such as tungsten, thoriated-tungsten, each of the melting point 3410 °C, and tantalum of work functions 4.52, 2.63, and 4.1 eV, respectively [1, 2]. Tantalum is used in the cathodes of high voltage transmitting and diode rectifier tubes. In the magnetron of a microwave oven, thoriated tungsten is used. In crossed-field tubes such as the magnetron, the secondary electron emission adds to the mechanism of electron emission from the cathode. The majority of today’s MWTs, however, use indirectly-heated cathodes using a filament heater insulated from the cathode; the heater often potted with alumina to ensure high temperature cycling stability, stability with time, high rigidity to withstand rigors of environment, less warm-up time, etc. Such a cathode uses an emitting surface made of a metal cylinder usually of nickel coated with a mixture of barium, strontium and calcium carbonates, typically, in the percentage proportion by weight of 57.3:42.2:0.5. The nickel is doped with a small fraction of zinc, tungsten/zirconium or magnesium that functions as an activator. The cathode is heated or activated to reduce the carbonates into oxides. The work function of an oxide-coated cathode is lowered by the combination of barium oxide and free barium of a smaller work function ∼1.8 eV that enables it to emit hundreds of mA cm−2 under pulsed or CW operation and tens of A cm−2 under pulsed operation, at relatively low temperatures ∼650–700 °C. However, at low cathode temperatures, the residual gases in the device are likely to poison the cathode. But at high temperatures, the active cathode material, namely barium, gets evaporated from the cathode surface. This limitation of the oxide-coated cathode is doi:10.1088/978-1-6817-4561-9ch3

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overcome in the dispenser cathode that takes care in continually replenishing/ dispensing the active barium from the interior of the cathode, which eventually evaporates from the emitting surface. Such a cathode of L, A, B, Mormixed metal matrix (MM) types provides a higher emission current density, a higher pulse-length operation, a longer life, a lower heater power requirement, a reduced susceptibility to damage by residual gases in the tube, a potential for reactivation if exposed to leakage, etc. The L-type cathode consists of barium–strontium carbonates in a cavity behind a porous tungsten plug. The A-type cathode is first formed from porous tungsten and then impregnated with barium aluminate. In the B-type cathode, the impregnates used are barium aluminate, barium oxide and calcium oxide, which reduces the barium sublimation rate, while calcium oxide, which is otherwise hygroscopic, is stabilized by barium aluminate. The following typical chemical reaction shows barium as a product [3]:

W + 3Ba3Al2O6 + 6CaO = 3Ba2CaAl2O6 + Ca3WO6 + 3Ba. The B-type cathode can give an emission density of several A cm−2 at 1100 °C. In the M-type cathode, the porous tungsten impregnated with barium–calcium aluminate is coated (∼2000–10 000 AU) with a thin layer of osmium–iridium or osmium–ruthenium. The M-type cathode yields an emission current density comparable with or greater than the B-type cathode, with a longer life than the latter. In the MM cathode, an enhancing metal is put into the tungsten matrix itself. The present-day dispenser cathode technology has been making continuous progress aiming at high emission density, low operating temperature, less susceptibility to damage and surface degradation due to residual gases and diffusion of film coating, ruggedness, reduction of the electrical breakdown, RF losses, grid emission caused by sublimed materials, less warm-up time, etc. Thus, the coated particle cathode (CPC) has been developed which is made of specially coated particles bounded to a nickel surface. The deposition of a tungsten–osmium alloy on the surface of the porous tungsten matrix has also been tried out. In another version of dispenser cathodes, the pores have been provided on a thin foil of tungsten by laser drilling or ion-etching. The scandate cathode is prepared by adding scandium oxide to a dispenser cathode. Vacuum-microelectronic (micro-fabricated) tubes (section 2.3) use cold fieldemission array cathodes such as carbon nanotubes. Field-emission cathodes are also used in HPM tubes (section 2.3). In an explosive field emission cathode, the emission densities ∼ kA cm−2 have been obtained due to ∼100 kV cm−1 electric field from naturally occurring micro-points of the cathode materials such as graphite, aluminium and stainless steel. In such a cathode, field emission current heats micro-points, which rapidly heat and explode to form plasma flares within a few nanoseconds. Individual flares expand and merge within 5–20 ms to form a uniform emitter. Thus, in such an explosive field emission cathode, electrons are drawn from the plasma. Such a cathode is reusable from shot to shot in an HPM tube, though it poses the difficulty of gap closure shorting out the accelerating voltage. Such difficulty of gap closure is not encountered in the non-explosive field emission cathode in which the emission density of 100–1000 A cm−2 is obtained from arrays

3-2


of tungsten needle, grooved graphite, caesium iodide coated carbon fibre, common velvet, cloth, ferroelectrics such as PLZT (Pb, La, Zr, Ti), etc. A large local external electric field >5 × 109 GV m−1 pulls in free charges from the surroundings to the surface of the ferroelectric. The space charge that shields the PLZT surface from the external electric field is controlled by applying a rapidly changing electric field to cause emission from the surplus charge thus forming pulsed electron beams. The operating voltage of the non-explosive field emission cathode is less than that of the explosive field emission cathode. The cathodes for high power microwave (HPM) tubes are driven by an intensive relativistic electron beam (IREB). The IREB-driven cathodes for HPM tubes are discussed later (see section 8.3 in chapter 8, volume 2).

3.2 Space-charge-limited and temperature-limited emission Most MWTs, such as the TWT and the klystron, operate under the space-chargelimited condition of emission. The phenomenon could be simply understood by taking a diode operating under the space-charge limited condition. In a spacecharge limited diode, the cathode temperature is raised to a relatively high value, and the anode potential, which is positive with respect to the cathode, is raised to such a value that the number of emitted electrons exceeds the number of electrons reaching the anode. Under this condition, the anode current can be increased by increasing the anode potential and cannot be increase by increasing the cathode temperature, until the anode current reaches a saturation value. Beyond this value of the anode saturation current, however, the anode current can be increased by increasing the cathode temperature. The operating regions below and above this saturation value of the anode current obtained by increasing the anode potential are the space-charge-limited and the temperature-limited regions, respectively. In the space-charge limited region, there are a larger number of electrons available in the diode than those reaching the anode, such that there still remains scope to increase the anode current by increasing the anode potential. However, in the temperature-limited region, the anode potential is such that all the available electrons in the diode reach the anode and by increasing the anode potential one cannot increase the anode current. The anode current in this temperature-limited region can be increased by increasing the number of electrons available in the diode, which can be implemented by increasing the cathode temperature and not by increasing the anode potential. From an alternative point of view, we can look upon the temperature-limited region as the region where the anode current at a given anode potential increases with the cathode temperature until it reaches a saturation value such that the anode current beyond this value can no longer be increased by increasing the cathode temperature and could be increased only by increasing the anode potential. Accordingly, in this alternative viewpoint, the regions below and above the anode saturation current obtained by increasing the cathode temperature are the temperature-limited and the space-charge-limited regions, respectively [3–5].

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3.2.1 The Child–Langmuir relation under the space-charge-limited condition of emission For a planar diode, the anode current I0 and the anode voltage V0 can be related by the Child–Langmuir relation [3–5]:

I0 4 V 3/2 2 η ε0 0 2 (Child–Langmuir relation) = A 9 d

(3.1)

where d is the distance between the planar anode and cathode in parallel each of area A and η is the magnitude of the charge-to-mass ratio of an electron. In the derivation of (3.1) it is assumed that d is much smaller than the dimensions of the planar anode and cathode dimensions. The Child–Langmuir relation (3.1) is also known as the 3/2-power law since I0 , as can be seen from (3.1), is directly proportional to V03/2 , the index of power of V0 being 3/2. In order to derive (3.1) we consider the potential in the region between the anode and the cathode to vary only along z (∂/∂z ≠ 0; ∂/∂x = ∂/∂x = 0) taking the distance between them to be much smaller than their planar dimensions. Hence, we can write the onedimensional Poisson equation as [3, 5]:

d 2V ρ =− , 2 dz ε0

(3.2)

where ρ is the volume charge density of the space-charge constituted by the electrons in the cathode–anode region of the diode. The potential V in this region (0 ⩽ z ⩽ d ) can be obtained by solving (3.2) subject to the boundary conditions at the cathode (z = 0):

V = 0 (a) ⎫ ⎪ ⎬(z = 0). dV = 0 (b)⎪ ⎭ dz

(3.3)

The boundary condition (3.3(b)) implies that the electric field (=−dV /dz ) at the cathode (z = 0) is zero. This is tantamount to assuming that the negative spacecharge field at the cathode due to the electrons in flight between the cathode and the anode is neutralized by the positive electrostatic field due to the applied anode potential. If the slope dV /dz of potential distribution in a V versus z plot at the cathode (z = 0) were positive, the electric field (=−dV /dz ) would be in the negative z direction, that is, directed towards the cathode, and more electrons that carry negative charges would leave the vicinity of the cathode thereby increasing the negative space charge in the cathode region and hence depressing the V versus z plot towards the zero slope (dV /dz = 0) thereby making the electric field (=−dV /dz ) zero at the cathode. However, if the slope were negative, the electric field (=−dV /dz ) would be in the positive z direction, that is, directed away from the cathode, as a result of which the emitted electrons that carry negative charges would be forced back to the cathode which would reduce the negative space charge in the region and consequently lift the V versus z plot towards the zero slope (dV /dz = 0) and hence

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the electric field (=−dV /dz ) again zero at the cathode [3, 5]. We can then solve the differential equation (3.2) subject to the condition (3.3) making use of the expression for the current density equation J = ρ v of the electron flow and the equation balancing the kinetic and potential energies (1/2)mv 2 = e V , where v is the velocity of electrons assumed to be the same for all the electrons at a distance z from the cathode. The solution so obtained, interpreting the current density J = I0 /A in the cathode–anode region and V = V0 at the anode z = d , gives us the Child– Langmuir’s relation (3.1) for a space-charge limited diode. 3.2.2 The Richardson–Dushman relation under a temperature-limited condition of emission The electron emission current density J under a temperature-limited condition depends on the work function W of the material of the emitter—the latter defined as the minimum kinetic energy required to liberate an electron from the surface of the material. This dependence is expressed by the Richardson–Dushman relation [3, 4, 6, 7]:

⎛−W ⎞ ⎟, J = AT 2 exp ⎜ ⎝ kT ⎠

(3.4)

where W = eϕ is the work function of the emitter material in joules and ϕ is the same expressed in electron volt. k ( = 1.38 × 10−23 J K−1) is Boltzmann’s constant and T is the absolute temperature of the emitter. A = χA0 is a constant, in which A0 (=4πmek 2 /h3 ≈ 1202 mA mm−2 K−2) is a universal constant called Richardson’s constant, m and h ( =6.626 × 10−34 ) Js 1 being the electronic mass and Planck’s constant, respectively, and χ is a correction factor depending on the cathode material and is of the order of 0.5, typically, for materials such as molybdenum, nickel, tantalum, tungsten and barium. This value of χ ∼ 0.5 renders the value A ≈ 600 mA mm−2 K−2 for these materials.

3.3 Space-charge waves and cyclotron waves Two space-charge waves are generated when an electron beam is perturbed longitudinally in its motion and similarly two cyclotron waves are generated when an electron beam is perturbed transversely to its motion. 3.3.1 Space-charge waves Though a microwave tube is a vacuum electron device it does not operate in perfect vacuum and therefore both electrons and ions are present in the tube providing a charge-neutralized background. But for this neutral background the flow of electrons in the tube would not be possible due to the repulsive forces between them. Two space-charge waves are generated when such a flow of electrons is perturbed longitudinally, along z, with their respective propagation constants β and corresponding phase velocities vp(=ω /β ) given by the following alternative forms of dispersion relation [5, 8, 9]: 3-5


β = βe ∓ βp ; ω − βv0 ∓ ωp = ω ∓ ωp ω β= ; vp = v0 v0 ω ∓ ωp

0⎫ ⎪ ⎬ ⎪ ⎭

(3.5)

where v0 is the dc beam velocity of electrons, supposedly uniform for all the electrons of the beam. βe(=ω /v0 ) and βp(=ωp /v0 ) are the beam and the plasma propagation constants, respectively; ω is the wave frequency; and ωp is the angular plasma frequency of an electron which is related to the dc beam volume charge density ρ0 as [5, 8, 9]: 1/2

ωp = (ηρ0 / ε0)1/2 = ( η ∣ ρ0 ∣ / ε0) ,

(3.6)

η(=e /m ) being the charge-to-mass ratio of an electron. The dispersion relation (3.6) has been derived treating the electron beam as a charge fluid and assuming the perturbed quantities to vary as exp j (ωt − βz ) and taking help from the following expressions [5, 8, 9]:

J = ρv Dv1 = ηEs ρ ∂Es = 1 ∂z ε0 ∂ρ1 ∂J1 =0 + ∂t ∂z

(a)⎫ ⎪ (b) ⎪ ⎪ (c) ⎬ , ⎪ ⎪ (d) ⎪ ⎭

(3.7)

where J (=J0 + J1), ρ(=ρ0 + ρ1) and v(=v0 + v1) represent the current density, volume charge density and velocity of the perturbed electron beam, respectively, the subscripts 0 and 1 referring, respectively, to the unperturbed (dc) and perturbed (RF) parts of these quantities, it being further assumed that J1 ≪ J0, ρ1 ≪ ρ0 and v1 ≪ v0 . Es is the space-charge field created in the beam treated as a charge fluid of a large cross-sectional area when the electrons are displaced longitudinally along z from their mean position with respect to the positive ions. D(=∂/∂t + v0∂/∂z ) is a differential operator, which, in view of the dependence exp j (ωt − βz ) of perturbed quantities assumed, may be put as D = jω− jβv0 = j (ω − βv0 ). Further, here (3.7(a)) is the current density equation; (3.7(b)) is the one-dimensional force equation of an electron subject to the space-charge field Es ; (3.7(c)) is the Poisson equation; and (3.7(d)) is the one-dimensional continuity equation. The upper and lower signs of (3.5) correspond to the slow and fast space-charge waves of phase velocities vp(=ω /β ) less and greater than v0 , respectively (figure 3.1). 3.3.2 Cyclotron waves If an electron beam of a large cross-sectional area and uniform dc velocity v0 , immersed in a uniform dc magnetic field of flux density B along z in the axial direction of the beam, is perturbed in motion to a small extent in the transverse

3-6


Figure 3.1. Dispersion plots for the fast and the slow space-charge and cyclotron waves.

plane, then two waves are set up on the beam, called cyclotron waves. One of these waves is the fast cyclotron wave of phase velocity vp > v0 , while the other is the slow cyclotron wave of vp < v0, given by the following alternative forms of dispersion relation [3]:

β = βe ∓ βc ; ω − βv0 ∓ ωc = 0 ⎫ ⎪ ω ∓ ωc ω ⎬ β= ; vp = v0 ⎪ ω ∓ ωc v0 ⎭

(3.8)

where ωc = η B is the angular electron cyclotron frequency of the beam, the upper and lower signs referring to the fast and slow waves, respectively. The expression (3.8) can be derived starting from the Lorentz force expressions for electronic motion [3]:

Dv1x = ηBv1y = −ωcv1y Dv1y = −ηBv1x = ωcv1x

(a) ⎫ ⎬, (b)⎭

(3.9)

where v1x and v1y represent the perturbed electron velocities along x and y, respectively. Squaring (3.9(a)) and making use of (3.9(b)) we can then obtain

D 2 = −ωc 2 which, in view of the interpretation of the operator D given following (3.7), yields the dispersion relation (3.8) for cyclotron waves that is identical with (3.5) except that now ωc and the cyclotron propagation constant βc (=ωc /v0 ) have replaced ωp and βp(=ωp /v0 ), respectively (figure 3.1).

3-7


3.4 Electron bunching mechanism In a microwave tube the electrons in a beam of electrons are bunched in an interaction region of the device where their kinetic energy, for instance in a TWT, or their potential energy, for instance in a magnetron, is transferred to RF waves in an interaction region of the device. The bunching mechanism may be non-relativistic, for instance, in a TWT or relativistic, for instance, in a gyrotron. Russell Varian (who along with Siguard Varian invented the klystron), while explaining the electron bunching mechanism in the klystron, mentioned: ‘Just picture a steady stream of cars from San Francisco to Palo Alto; if the cars left San Francisco at equal increments and at the same velocity, then even in Palo Alto they would be evenly spaced and you would call this a direct flow of cars. But suppose somehow the speed of some cars, as they left San Francisco, was increased a bit and others retarded. Then, with time, the fast cars would tend to catch up with the slow ones and they would bunch into groups. Thus, if the velocity of cars was sufficiently different or the time long enough, the steady stream of cars would be broken and, under ideal conditions, would arrive in Palo Alto in clearly defined groups. In the same way an electron tube can be built in which the control of the e-beam is produced by the principle of bunching, rather than the direct control of a grid in a triode…’ In the Applegate diagram (figure 3.2) for a two-cavity klystron, a bunch of straight lines of slopes proportional to the electron velocities (figure 3.2) explains the arrival of electron bunches at the location of the crossing of these lines, at the catcher cavity of the klystron that consists of the input buncher and the output catcher cavities in its simplest two-cavity configuration (see section 6.2 in chapter 6, volume 2). This arrival of electron bunches at the output catcher cavity takes place at the interval of the time period T0 (=1/f0), e.g. of the sinusoidal input voltage of frequency f0, say, of the buncher cavity around the electrons that had crossed this

Figure 3.2. Applegate diagram showing the electron bunching in a klystron [10].

3-8


cavity when the input voltage crossed from its negative (decelerating) to positive (accelerating) value (figure 3.2). The bunching mechanism in the TWT can be explained considering, typically, two electrons, one at ‘A’ and the other at ‘D’, around a reference electron at ‘R’, all moving in the same direction from left to right along the interaction length of the device (figure 3.3); it being assumed that the DC electron beam velocity v0 is synchronous with the RF phase velocity vp(v0 = vp). In this interaction length, the electron at A is subjected to the accelerating RF electric field directed from right to left, and that at D subjected to the decelerating RF electric field directed from left to right, while the reference electron at R experiences no such fields (figure 3.3). As the time passes, the electron at A is accelerated and thus gains kinetic energy and, similarly, the electron at D is decelerated and thus loses kinetic energy, while the reference electron at R is neither accelerated nor decelerated and thus neither gains nor loses kinetic energy. Therefore, as the time passes the electrons both at A and D get closer to the reference electron at R and cluster or ‘bunch’ around the latter though the bunch of electrons taken together does not undergo any net change in its kinetic energy (figure 3.3). In fact, for the transfer of kinetic energy to take place from the bunch of electrons to the RF wave, the bunch must move to the decelerating electric field (figure 3.3) which is realized by slightly offsetting the synchronous condition to v0 ≳ vph (near-synchronous condition). Similarly, we can explain the bunching mechanism in a gyrotron (see section 7.2 in chapter 7, volume 2) considering at any instant of time, typically, two electrons, one at ‘A’ and the other at ‘D’, around a reference electron at ‘R’ in a uniform

Figure 3.3. Bunching of typically two electrons ‘A’ and ‘D’ subjected to the accelerating and the decelerating RF electric fields in the interaction region of a TWT around a reference electron ‘R’ that experiences no such fields.

3-9


magnetic field, all rotating in the same sense, say, anticlockwise in the interaction region of the device in which the RF electric field exists in a vertically upward direction (figure 3.4); it being assumed that the angular wave frequency ω is made synchronous with the angular cyclotron frequency ωc (ω = ωc ). The angular cyclotron frequency ωc of the rotating electrons can be written considering relativistic effects with the help of its expression given immediately following (3.8), however, replacing therein the electron mass m by the relativistic electron mass γ m as follows:

ωc = e B / γ m ,

(3.10)

where γ is the relativistic mass factor showing the dependence of the angular cyclotron frequency ωc of an electron on the value of γ , which in turn depends on the exchange of energy between the RF electric field and the electron. Thus, the electron at A (figure 3.4), which moves vertically downward and which is subjected to the accelerating force due to the electric field directed vertically upward, will receive energy from the RF electric field that will enhance the value γ of the electron and hence reduce the value of its angular cyclotron frequency ωc given by (3.10) and hence increase the value of the time period of its rotation Tc(=2π /ωc )(figure 3.4). Similarly, the electron at D (figure 3.4), which moves vertically upward and is

Figure 3.4. Bunching of typically two electrons ‘A’ and ‘D’ subjected to the accelerating and the decelerating RF electric fields, respectively, in the interaction region of a gyrotron around a reference electron ‘R’ that experiences no such fields.

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subjected to the decelerating force due to the electric field directed also vertically upward, will lose its energy to the RF electric field causing a decrease in the value of γ and in turn an increase in the value of its angular cyclotron frequency ωc given by (3.10) and hence a decrease in the value of the time period of its rotation Tc(=2π /ωc ) (figure 3.4). However, the electron at R (figure 3.4) moving transverse to the RF electric field experiences (figure 3.4) no force due to the RF electric field and will neither gain energy from nor lose energy to the RF field that would keep the value γ of this electron unchanged which in turn would not cause any change in the value of ωc and that of Tc(=2π /ωc ). As a result, with the passage of time, the electron at A will move slower and that at D faster than the electron at R around the circular orbit thereby causing the bunching of electrons at A and D around the electron at R (figure 3.4). However, the bunch of electrons at A, D and R taken together does not undergo any net change in its kinetic energy. For the transfer of energy to take place from the bunch of electrons to the RF wave, the bunch must move to the decelerating electric field (figure 3.4) which can be realized by slightly offsetting the synchronous condition to Tc ∼ > ωc (near-synchronous condition). > T , that is ω ∼

3.5 Induced current due to electron beam flow The concept of induced current on the electrodes such as grids due to the electron beam flow between them and realizing the current in an external circuit is of significance in a microwave tube, for instance, a klystron (see section 6.2 in chapter 6, volume 2). The concept can be developed considering an electron moving with acceleration from an electrode A to another electrode B raised to an electric potential V0, say, with respect to the potential of A. Such an electron motion will induce current in the electrodes A and B as long the electron is in flight between the electrodes. Further, such current can be made to flow through an external circuit since, with this movement of the electron, the electric lines of force emanating from the electron and terminating on the electrode B increase with time while those terminating on the electrode A correspondingly decrease (figure 3.5). If we equate (i) the potential energy q′V0 stored due to the charge q′, say, induced on the electrode B raised to the potential V0 with (ii) the work done by the electron as it moves through a distance z from the electrode A, we obtain (figure 3.5): q′V0 = e (V0 /d )z , where d is the distance between the electrodes (figure 3.5), the work done being obtained by multiplying the force e (V0 /d ) on the electron by the distance z , V0 /d being the electric field between the electrodes, whence q′ and the induced current i′ = dq′/dt can be obtained as

⎫ ⎪ ⎬, dz 1 e v⎪ dq′ i′ = = e = dt d d ⎭ dt q′ = e

z d

(3.11)

where v(=dz /dt ) is the electron velocity. The induced charge q on the electrode A is related to q′ as

3-11


Figure 3.5. Electric lines of force in the region between the electrodes A and B (a) and charges induced on them (b) at a typical position of the electron in the flight in the region, and the induced triangular current pulse at the electrode B (c) due to the electron flow between the electrodes.

q + q′ = e .

(3.12)

We can then find the induced charge q and current i (=dq /dt ) on the electrode A, with the help of (3.11) and (3.12), as

⎫ ⎛ z ⎞⎟ ⎜1 − ⎪ ⎪ ⎝ d⎠ ⎬. dz 1 e v⎪ dq i= =−e =− ⎪ dt d d ⎭ dt q= e

(3.13)

For a uniform acceleration of the electron, the electron velocity v increases linearly with time t and, correspondingly, as can be appreciated from (3.11) and (3.13), we get a triangular current pulse induced at the electrodes as long as the current is in the flight between the electrodes, the current ceasing to exist as v becomes nil when the electron reaches the electrode B. For an electron beam, the total induced current on an electrode is found by adding such triangular pulses of current associated with each electron. Further, current may even be induced in an electrode to which no

3-12


electrons flow, if the number or velocity of electrons approaching the electrode is different from the number or velocity of electrons receding from it.

3.6 Space-charge-limiting current The presence of the negative space-charge of the electrons in an electron tube restricts the upper limit of the beam current in the tube called the space-chargelimiting (SCL) current. This limit is imposed by the potential depression caused by the negative space-charge in the tube that retards the flow of electrons or reflects the flow back to form what is known as the virtual cathode. This phenomenon in turn restricts the power deliverable by the tube by increasing the beam current transport in the tube to its SCL value. In fact, there cannot be a flow of electrons or the existence of an electron beam in an electron tube unless there is a neutralizing background of positive ions which are always present due to the lack of ideal vacuum in the tube. The value of the SCL current and beam current transport in the tube and hence its power can be increased by a plasma-assisted tube (see section 8.2 in chapter 8, volume 2) in which the number of space-charge neutralizing positive ions are increased by reducing the vacuum of the tube (provided in the tube (chapter 1, volume 1) to prevent the electrons from losing their energy to the atoms by collision and to protect the cathode from the positive ions striking it). Further, the value of the SCL current is deliberately reduced to form a virtual cathode in a tube called the virtual cathode oscillator (VIRCATOR) that belongs to a class of high power microwave (HPM) tubes (see section 8.3 in chapter 8, volume 2). Beam current transport in a high power tube requiring a relativistic electron beam can be made possible when the relativistic kinetic energy of a beam electron exceeds its potential energy. The SCL current in the tube in its maximum limit therefore can be found by equating these two energies. 3.6.1 Space-charge limiting current for an infinitesimally thin hollow electron beam in a metal envelope Let us consider a drift tube—a circular metal envelope or wall of radius r0 representing an interaction region of a microwave tube, with a coaxial infinitesimally thin hollow electron beam of radius rb inside (figure 3.6). The radial electric field E at a point in the region between the electron beam and the surrounding metal envelope may be written, with the help of Gauss’s law, as ρl E= (rb ⩽ r ⩽ r0), (3.14) 2πε0r where r is the radial distance of the point in the region, and ρl is the beam line charge density. The potential energy PE of an electron of the beam can be found as the work done Wthin beam in moving the electron from the metal envelope to the position of the beam:

PE = Wthin beam =

∫r

b

3-13

r0

eE dr .

(3.15)


Figure 3.6. An infinitesimally thin hollow electron at the axis of a metal drift tube.

The relevant relations required to find the expression for PE by evaluating the integral in (3.15), which involves E in the integrand, are:

ρl = ρα ; ρ = ρb − ρi ρb = nee ; ρi = ni e J = ρ vb ; J = −Ib / α

⎫ ⎪ ⎬, ⎪ ⎭

(3.16)

where ρ is the volume charge density of the beam taking into account the presence of positively charged ions besides electrons. ρb and ρi are the volume charge densities of the electrons of the and ions, respectively, and ne and ni are their respective number densities. α is the beam’s cross-sectional area. vb is the electron beam velocity. J is the beam current density and Ib is the beam current interpreted as positive. Thus, we obtain PE after evaluating the integral in (3.15) and using the relations (3.16) as follows:

⎛ r0 ⎞⎛ n ⎞ ln ⎟⎜1 − i ⎟ ⎜ 1/2 ⎡ ⎤ ne ⎠ 2πε0⎢⎣ γb2 − 1 / γb⎥⎦c ⎝ rb ⎠⎝ (infinitesimally thin hollow beam),

PE = e

Ib

(

)

(3.17)

where γb is the relativistic mass factor given by

(

γb = 1 − vb2 / c 2

−1/2

)

.

(3.18)

The transport of an electron beam of high current is possible in a drift tube when the relativistic kinetic energy KE of the beam in the limit exceeds its potential energy PE ; the latter is given by (3.17). The expression for kinetic energy needs to be found giving due consideration to its reduction caused by the space-charge depression in the beam resulting from the depression of the effective accelerating potential in the drift tube. As the beam current increases the electrons are increasingly slowed down due to the space-charge depression until, at some point, the electron velocity is retarded from vb to nil and the beam current Ib increased to its limiting value—the so-called space-charge limiting current ISCL . This in turn corresponds to the

3-14


reduction of the relativistic kinetic energy of the electron by an amount mc 2(γb − 1) from its value mc 2(γc − 1), the latter corresponding to the potential Vc of the drift tube relative to the cathode, it being assumed that the drift space is grounded and that the electron beam is launched from a cathode at a negative voltage, Vc < 0, with respect to the wall such that the potential within the drift tube will be the potential of the wall, where γc is the relativistic mass factor corresponding to the potential Vc :

γc = 1 +

eVc . mc 2

(3.19)

Thus, under the influence of the space-charge depression described as above, the relativistic kinetic energy of the beam electron KE may be written as

KE = mc 2(γc − 1) − mc 2(γb − 1) = mc 2(γc − γb).

(3.20)

Since the transport of beam current Ib is possible in the drift tube if KE > PE, we can, by equating (3.17) with (3.20) (that is, putting PE = KE), obtain after simplification:

Ib =

IA

(

2 r γb − 1 2 ln 0 rb

1/2 ⎛ γc

)

⎜ ⎝

− γb ⎞ 1 , ⎟ γb ⎠ 1 − ni ne

(3.21)

where IA = 4πε0mc 3 / e = 17.1 kA. Further, the condition γb = γc1/3 put in (3.21) gives Ib max , the maximum of Ib, interpreted as ISCL as follows:

ISCL = Ib

1/3 γb=γc

3/2 1 2/3 r0 γc − 1 n 1− i 2 ln rb ne (infinitesimally thin hollow beam).

= Ib

max

=

IA

(

)

(3.22)

3.6.2 Space-charge limiting current for a thick solid electron beam in a metal envelope The expression for PE similar to (3.15), now for a thick solid beam in a metal envelope (figure 3.7), can be written as

PE = Wthick

beam

=

∫0

r0

eE dr =

∫0

rb

eE dr +

∫r

r0

eE dr .

(3.23)

b

Substituting the following expressions obtainable using Gauss’s law

rρ ⎫ (0 ⩽ r ⩽ rb) ⎪ ⎪ 2ε0 ⎬ ρl ρα (rb ⩽ r ⩽ r0)⎪ E= = ⎪ 2πε0r 2πε0r ⎭

E=

3-15

(3.24)


Figure 3.7. A thick electron at the axis of a metal drift tube.

in (3.24) and evaluating the integrals therein we obtain

PE =

∫0

rb

erρ dr + 2ε0

∫r

r0

b

eρ rb2 eρα r eρα ln 0 . dr = + 2πε0r 2ε0 2 2πε0 rb

(3.25)

In view of (3.16), (3.25) may be expressed as

PE =

eρα ⎛ r0 ⎞ ⎜1 + 2 ln ⎟ 4πε0 ⎝ rb ⎠

= e

⎛ Ib r0 ⎞⎛ ni ⎞ ⎜1 + 2 ln ⎟⎜1 − ⎟ 1/2 ⎡ ⎤ 2 rb ⎠⎝ ne ⎠ 4πε0⎣ (γb − 1) / γb⎦c ⎝

(3.26)

(thick solid beam). The expression (3.17) for the potential for an infinitesimally thin hollow beam is thus found to be identical with the expression (3.26) for a thick solid beam, except that the factor ln(r0 /rb ) in (3.17) has been replaced by (1 + 2 ln(r0 /rb ))/2 in (3.26). Interestingly, the expression (3.20) for the relativistic kinetic energy for an infinitesimally thin hollow beam continues to be valid for a thick solid beam as well. Therefore, if we follow the same procedure as described following (3.20) we can obtain the following expression for the space-charge limiting current by replacing the factor ln(r0 /rb ) in (3.23) by (1 + 2 ln(r0 /rb ))/2:

ISCL = Ib

γb=γc1/3

= Ib

max

=

IA

(

2/3 r0 γc − 1 1 + 2 ln rb (thick solid beam).

3/2

)

1 1−

ni ne

(3.27)

We can appreciate from (3.23) and (3.27) that the space-charge-limiting current is larger if the electron beam is closer to the drift tube. In high power tubes such as the gyrotron (see section 7.2 in chapter 7, volume 2), the beam current should be made larger though below this space-charge-limiting current. On the other hand, in some HPM tubes such as the virtual cathode oscillator (see section 8.3 in chapter 8, 3-16


volume 2), the space-charge limiting current is made smaller to encourage the formation of a virtual cathode such that the electrons oscillate between the actual and the virtual cathodes repeatedly passing back and forth in a resonant cavity for the transfer of energy from the beam to RF waves.

3.7 Conservation of kinetic energy in M-type tubes Let us introduce, in an interaction structure supporting RF waves, an electron with an initial dc velocity u 0 along positive z in a crossed dc electric field E0 along negative y and dc magnetic field B0 along positive y. Thus, the electron will be subjected to a z-directed RF electric field Ez besides the Lorentz force due to the magnetic field B0 , though the corresponding field along y is not significant compared to E0 . We can therefore write the force equation for the electronic motion as

⎫ d 2x 0 = ⎪ dt 2 ⎪ ⎪ d 2y m 2 = e( −E 0 + B0vz )⎬ , dt ⎪ ⎪ d 2z m 2 = −eB0vy + eEz ⎪ ⎭ dt

m

(3.28)

where vz = dz /dt and vy = dy /dt represent the electron velocities along z and y, respectively. Subject to the initial conditions at t = 0: x = vx(=dx /dt ) = 0; z = 0, vz (=dz /dt ) = u 0; and y = vy(=dy /dt ) = 0, we can obtain the solution of (3.28) as

⎫ ⎪ 1⎛ sin ωct ⎞⎪ E ⎛ E ⎞ y = − ⎜u 0 − 0 ⎟(1 − cos ωct ) + z ⎜t − ⎟⎪ B0 ⎝ B0 ⎠ ωc ⎠⎬ , ωc ⎝ ⎪ ⎛ E ⎞ sin ωct Ez E (1 − cos ωct ) ⎪ z = 0 t + ⎜u 0 − 0 ⎟ − ⎪ ⎝ B0 ⎠ ωc B0 ωcB0 ⎭

x=0

(3.29)

where ωc = ( −e /m )B0 is the electron cyclotron frequency (figure 3.8). Choosing u 0 = E0 /B0 we can find from (3.29)

dz E E sin ω t = 0 + z 2 c , dt B0 ωc B0 which gives the average z-directed velocity as E0 /B0, which is devoid of any timeperiodic component. From this z-directed velocity we can find z as

z=

E0 t. B0

(3.30)

The electron moving along z in time t subject to the electric field Ez along z is decelerated and thus loses its average kinetic energy equal to the amount of work done (force × distance) W z by the electron (by transferring energy to the field) given by 3-17


Figure 3.8. Electron trajectory in a crossed field system [11].

W

z

= e Ezz

which, on substituting z from (3.30) becomes

W

z

= e Ezz =

e E0 t. B0

(3.31)

At the same time, the electron moving along y in time t subject to the electric field E0 directed along the negative y direction becomes accelerated and gains an average kinetic energy equal to the amount of work done on it W y by the electric field given by

W

y

= e E 0y.

(3.32)

Substituting y = Ezt /B0—the latter obtained following the same approach as used in obtaining (3.30)—in (3.32) we obtain

W

y

= e E 0y =

e E 0Ez t. B0

(3.33)

Comparing (3.33) with (3.31) we notice that W y = W z , which means that in crossed electric and magnetic fields, on average the kinetic energy gained by an electron in motion along y is lost in its motion along z. This simple analysis helps one to appreciate that in crossed-field MWTs, on average, the kinetic energy of electrons remains unchanged and that it is their potential energy that is converted into RF energy (see section 6.4 in chapter 6, volume 2).

References [1] Harbaugh W E 1962 Tungsten, thoriated-tungsten, and thoria emitters Electron Tube Des. 16 90–8 [2] Cronin J L 1981 Modern dispenser cathodes IEEE Proc. 128 19–32

3-18


[3] Basu B N 1996 Electromagnetic Theory and Applications in Beam-wave Electronics (Singapore: World Scientific) [4] Spangenberg K 1948 Vacuum Tubes (New York: McGraw Hill) [5] Basu B N 2015 Engineering Electromagnetics Essentials (Hyderabad, India: Universities Press) [6] Hutter R G E 1960 Beam and Wave Electronics in Microwave Tubes (Princeton: D Van Nostrand) [7] Gilmour A S Jr 1986 Microwave Tubes (Norwood: Artech House) [8] Collin R E 1992 Foundations for Microwave Engineering 2nd edn (New York: Wiley-IEEE Press) [9] Ramo S 1939 Space-charge and field waves in an electron beam Phys. Rev. 56 276–83 [10] Gandhi O P 1981 Microwave Engineering and Applications (New York: Pergamon Press) [11] Gittins J F 1965 Power Traveling-Wave Tubes (New York: Elsevier)

3-19

IOP Concise Physics


Chapter 4 Formation, confinement and collection of an electron beam

In microwave tubes such as the TWT and the klystron, the electron beam of the desired current and cross-sectional area is formed by a system of electrodes, called the ‘electron gun’ (section 4.1). The electron beam is confined in the interaction structure by a ‘focusing structure’ (section 4.2), and, after the beam delivers part of its energy to RF waves supported by the interaction structure, the spent beam is collected by a system of electrodes called the ‘collector’ (section 4.3).

4.1 Electron gun If we had increased the distance between the cathode and the accelerating anode of a diode and accommodated an interaction structure between them, then the beam current would reduce to an insignificant value (see equation (3.1)). This calls for a system of electrodes called the ‘electron gun’ with a smaller distance between the cathode and an accelerating anode to throw the electron beam of the desired value of beam current and cross-sectional area beyond the anode into the interaction structure of a device. 4.1.1 Pierce gun derived from a flat cathode In the simplest form of an electron gun (called the Pierce gun), a parallel-flow, rectangular- strip electron beam, derived from a flat cathode located at z = 0 on the XY plane, is formed by a system of electrodes comprising, besides the cathode, an accelerating anode and an electrode, called the beam forming electrode (BFE), which is held at the potential of the cathode though kept thermally insulated from the latter so that it does not emit electrons (figure 4.1). The geometrical shapes of the BFE and the anode must ensure the condition that (i) at the beam edge (y = 0), the potential is given by

doi:10.1088/978-1-6817-4561-9ch4

4-1



Figure 4.1. Cross section of the parallel-flow Pierce gun derived from a flat cathode [1].

V (z )

y=0

⎛ ⎞2/3 9 J =⎜ ⎟ z 4/3, ⎝ 4(2 η )1/2 ε0 ⎠

(4.1)

the latter obtainable from the Child–Langmuir relation (3.1); and that (ii) there is no electrostatic force transverse to the rectilinear flow on the beam-edge electrons (y = 0) exerted on them by the electric field component −∂V /∂y , that is

∂V ∂y

= 0,

(4.2)

y=0

where J is the beam current density and η is the charge-to-mass ratio of an electron. Subject to the above conditions (i) and (ii) stated by (4.1) and (4.2), respectively, one can solve the two-dimensional Laplace equation valid outside the beam:

∂ 2V (z , y ) ∂ 2V (z , y ) =0 + ∂z 2 ∂y 2 to obtain the following expression for the potential outside the beam:

⎛ ⎞2/3 9 J V (z , y ) = Re f (z + jy ) = Re ⎜ ⎟ (z + jy )4/3 (y ⩾ 0), ⎝ 4 (2 η )1/2 ε0 ⎠

(4.3)

recalling the relevant mathematical concept that the real part of a complex function f (z + jy ), known as the analytic function, satisfies the two-dimensional Laplace equation. Further, putting

4-2


z + jy = (z 2 + y 2 )1/2 (cos θ + j sin θ )⎫ ⎪ ⎬ −1 y θ = tan ⎪ ⎭ x

(4.4)

we can express (4.3) as

⎛ ⎞2/3 4θ ⎞ 4θ 9 J 2/3⎛ ⎟ (y ⩾ 0). V (z , y ) = Re ⎜ + j sin ⎟ (z 2 + y 2 ) ⎜cos 1/2 ⎝ 3⎠ 3 ⎝ 4 (2 η ) ε0 ⎠

(4.5)

Corresponding to the BFE potential V = 0, which is the same as the cathode potential, we can put the right-hand side of (4.5) equal to zero giving cos(4θ /3) = 0 that corresponds to 4θ /3 = π /2, which may be written with the help of (4.4) as the BFE angle

θ = tan−1

3π y = = 67.5° (BFE angle). 8 x

(4.6)

It follows from (4.6) that the BFE will take a ‘planar-hat’ shape making an angle of 67.5° with the beam edge (figure 4.1). Similarly, in order to find the shape of the anode we have to find the potential distribution outside the beam edge with the help of (4.5), equating the latter to the anode potential V = V0 , and then identify the equipotential corresponding to V = V0 that gives the shape of the anode that starts from the beam edge at a right angle, deviating from a right angle away from the beam edge (figure 4.1). 4.1.2 Pierce gun derived from a curved cathode Further, one can enhance the beam current density with the help of the Pierce gun derived from a curved cathode of a relatively large area that forms a convergent-flow electron beam (figure 4.2). The principle of such a convergent-flow Pierce gun can be derived starting from a one-dimensional Poisson equation in spherical-polar coordinates for the potential V depending only on the radial coordinate r :

d 2V 2 dV + 2 dr r dr

= −

ρ ε0

and its Langmuir–Blodgett solution inside a diode consisting of a spherical cathode outside a concentric spherical anode, such that the solution passes on to (3.1) or (4.1) as a special case of the distance z ≪ rc of a point measured from and perpendicular to the cathode, rc being the radius of curvature of the cathode, as follows [1]:

⎛ ⎞2/3 9I0 V=⎜ ⎟ G 4/3(u ), ⎝ 16π (2 η )1/2 ε0 ⎠ where I0 is the beam current and u = ln(r /rc ); and

4-3

(4.7)


Figure 4.2. Convergent-flow Pierce-gun derived from a spherical-cup cathode, showing the anode aperture/ grid inside the beam; the BFE and the anode, outside the beam; the beam edge and the beam waist (or throat); and the concentric cathode and anode spheres, indicating the relevant parameters, namely the cathode and anode radii of curvature rc, ra , respectively, cathode-disc and anode-aperture radii rK , rA, respectively; cathodeto-anode distance d = (rc − ra ); beam-waist radius rM ; and beam-waist distance dm [1].

⎛ r⎞ 63 3 3 rc 3 2 rc r r ln4 c . ln ln G ⎜ln ⎟ = ln c + + + ⎝ rc ⎠ 4400 40 10 r r r r

(4.8)

The beam current I0 obtainable from (4.7), considering a spherical-cup emitting portion of the complete cathode sphere of radius rc , is reduced by a factor equal to the ratio of the area of the spherical-cup emitting portion, 2πrc2(1 − cos θ0 ), to the area of the cathode sphere, 4π rc2 (figure 4.2), which enables one to write with the help of (4.7) [1]:

k2 =

4 ⎛ 1 − cos θ 0 ⎞ ⎜ ⎟, 9 ⎝ G02(u ) ⎠

(4.9)

where G0(u ) is the value of G (u ) corresponding to r = ra , the anode radius of curvature; θ0 is the half cone angle subtended by the spherical cup at the common centre of curvature of the cathode and the anode spheres (figure 4.2); and k is defined in terms of the beam perveance perv(=I0 /V03/2 ) as

4-4


⎛ ⎞1/2 perv k=⎜ ⎟ ; ⎝ 2π ε0(2 η )1/2 ⎠

(4.10)

a quantity that also proves to be of relevance in studying the spread of the convergent beam in the field-free region beyond the aperture of the gun anode. Beyond the anode aperture, the electron beam will spread in the field-free region, which can be analyzed starting from the following expression for the force on an electron at the edge of the beam supposedly of a circular cross section of radius r :

d 2r ηρ r, = η Es = dt 2 2ε0

(4.11)

where Es (=ρ r /(2 ε0 )) is the space-charge electric field inside the beam obtainable with the help of Gauss’s law. Further, assuming dz /dt = v , the velocity of the beamedge electron, to be constant beyond the anode aperture in the field-free region; interpreting d 2r /dt 2 = v 2d 2r /dz 2 ; and making use of the relation J = ρv , where J is the beam current density and ρ is the volume charge density, and remembering that η and ρ each carry a negative sign, one may express the above force equation (4.11) as

d 2r η J r. = dz 2 2 v 3 ε0

(4.12)

Furthermore, making use of (i) the expression J = I0 /(πr 2 ), I0 being the beam current and (ii) the relation v = (2ηV0 )1/2 between the beam electron velocity v and the accelerating beam voltage V0, which is obtained by equating the kinetic and potential energies of the electron, we can express (4.12) as

1 d 2R = R 2 2 dZ

(4.13)

where R = r /r0 and Z = k (z /r0 ), k being given by (4.10) and r0 representing the beam radius r at a reference plane thereby also making R = 1 at such a reference plane. We can multiply (4.13) by 2(dR /dZ ) and integrate to obtain

dR = ±(ln R )1/2 dZ

(4.14)

by setting the integration constant to zero by taking the reference such that, at r = r0, it is at R = 1, dr /dz = dR /dZ = 0. Further, the electron beam flow beyond the anode aperture, instead of converging to a single point, would diverge out due to Coulomb repulsion between electrons and takes on a minimum value of radius rM , also referred to as the beam-waist radius. The plus and the minus signs in the right hand side of (4.14) refer to the positive and negative slopes of the beam trajectory, corresponding to the positive and negative values of dR /dZ , and those of dr /dz , respectively, and hence to the regions to the right and to the left of the beam waist of radius rM , respectively (figure 4.2). The distance dM of the beam waist from the anode, called the throw of the gun, can be found by integrating (4.14) taking the 4-5


negative sign in its right-hand side corresponding to the region to the left of the beam waist. For this purpose, we have to take the limits of integration between the limits Z = 0 (corresponding to z = 0 at the beam waist), R = 1 (corresponding to r = r0 = rM ), at the beam waist taken as the reference plane, and Z = −kdm /rM (corresponding to z = −dm ), R = r /r0 = rA /rM (corresponding to the anode-aperture radius r = rA) giving (figure 4.2)

∫0

d − krm M

dZ =

∫1

rA rM

−dR , (ln R )1/2

which when evaluated gives [1]

r ⎡ ln3/2 A 2rM ⎢ 1/2 rA rM dm = + ⎢ln k ⎢ rM 3(1 ! ) ⎣ (throw of the gun).

rA rM 5(2 ! )

ln5/2 +

rA rM 7(3 ! )

ln7/2 +

⎤ ⎥ + ⋯⎥ ⎥ ⎦

(4.15)

The beam transit angle θ0 at the anode, being equal to the half-cone angle subtended by the spherical-cup cathode at the centre of the spherical cathode (figure 4.2), may be expressed as (dr /dz )A = −tan θ0 which, with the help of (4.14) with the negative sign in its right hand side and the definitions of R and Z given following (4.13) with r0 interpreted as rM , may be written as k (log RA)1/2 = k (log(rA /rM )1/2 = tan θ0 whence we get

⎛ tan2 θ 0 ⎞ rA (=RA) = exp ⎜ ⎟. ⎝ k2 ⎠ rM

(4.16)

However, the expression (4.16) gets modified due to the anode aperture behaving as an electrostatic lens of second focal length f2 = f , say, causing a deviation δ = rA / f of the electron path, and in turn a change in the value of beam transit angle from θ0 to θ′ (figure 4.3), which enables one to write [1]

θ0′ = θ 0 − δ = θ 0 − rA/ f ⎫ ⎪ ⎬. tan2 θ0′ rA = exp ⎪ 2 rM k ⎭

(4.17)

The focal length f of the anode-aperture length depends on the anode voltage V0 and the electric fields E1 and E2 along the gun axis in the vicinity of the anode aperture in the regions to the left and the right of the anode, respectively. However, here we can take E2 = 0, the electron beam being launched beyond the anode aperture into the field-free region, and treat the electric field E1(=−∂V /∂z ) as negative as it is directed from the anode aperture to the cathode, that is, from the

4-6


Figure 4.3. Electron path showing the modified beam transit angle, θ′0 , caused by the deviation of the electron path, δ , at the anode-aperture lens (a) and the ‘paraxial-ray’ diagram to indicate the second focus of the lens, F2, and relate the magnitude of the second focal length, f2 = f , and the anode-aperture radius, rA (b) [1].

region to the right of the anode to the region to its left. In this case we can take the expression for f as [1]:

f=

4V0 4V0 = E1 −∂V / ∂z

= 0

4V0 ∂V / ∂r

(4.18) 0

interpreting the axis of the gun to be along z and the radial coordinate r of a point on the gun axis increasing from the right to the left, that is, along negative z giving ∂V /∂r = −∂V /∂z (figure 4.2), where the subscript ‘0’ refers to the derivative taken at the anode. Further, with the help of (4.7), we can express (4.18) as [1]:

f=

⎛ rc ⎞ ⎜ 2⎟ ⎝ ra ⎠

−6 G02 dG 2 d (rc / r )

.

(4.19)

0

With the help of (4.17) and (4.19) and using the relation θ0 ≅ rA /ra = rK /rc (figure 4.2), giving rK /rA = rc /ra , and taking tan θ′ = θ′ for smaller values of θ′, we can then write the expression for the beam convergence rK /rM as follows:

θ′2 rK rK r r = × A = K exp 20 rM rA rM rA k 2⎤ ⎡ 2⎛ θ′20 θ0 1 ⎛ rc ⎞⎛ 1 ⎞⎛ dG 2 ⎞ ⎞ ⎥ rc rc ⎢ = exp 2 = exp 2 ⎜⎜1 − ⎜ ⎟⎜ 2 ⎟⎜ ⎟⎟ , ⎢k ⎝ 6 ⎝ ra ⎠⎝ G0 ⎠⎝ d (rc / r ) ⎠0⎟⎠ ⎥⎦ ra k ra ⎣ which putting (dG 2 /d (rc /r ))0 = 2G0(dG /d (rc /r ))0 and θ02 /k 2 = (9/2)G02 , the latter obtainable from (4.9) taking cos θ0 ≈ 1 − θ0 2 /2 for small values of θ0 , becomes 2⎤ ⎡ 1 ⎛ r ⎞⎛ 1 ⎞⎛ 2 ⎞⎛ dG ⎞ ⎞ ⎥ 9 ⎛ rK r ⎟ = c exp ⎢ G02⎜⎜1 − ⎜ c ⎟⎜ 2 ⎟⎜ ⎟⎜ ⎟⎟ . ⎢2 ⎝ ⎝ ⎠ ⎝ ⎠ 6 ( / ) r G d r r rM ra G ⎝ ⎠0⎠ ⎥⎦ ⎝ ⎠ a 0 c 0 ⎣

4-7

(4.20)


Finally, with the help of (4.7) and its derivative, both evaluated at r = ra , we can obtain from (4.20) the following explicit expression for the beam convergence rK /rM :

⎡1 ⎛ ⎞2 ⎤ 369 3 rc 27 2 rc 12 rK rc rc ⎢ ln ln ln + = exp ⎜ −1 + + + …⎟ ⎥ . ⎢⎣ 2 ⎝ ⎠ ⎥⎦ 2200 40 5 rM ra ra ra ra

(4.21)

Taking the beam voltage V0, the beam current I0 , the beam radius, which is also equal to the beam-waist radius rM , and the cathode operating current density as the four input design parameters, we may proceed to find the output design parameters, namely, the cathode radius of curvature rc , the anode radius of curvature ra , the cathode-disc radius rK , the anode-aperture radius rA, the interelectrode spacing d (=(rc − ra )) and the throw of the gun dm through the following steps [1]: (i) The cathode disc area is equal to 2π rc2(1 − cos θ0 ) which may be approximated as π rk2 in view of the relations cos θ0 ≈ 1 − θ0 2 /2 and θ0 ≈ rK /rc , for small values of θ0. Multiplying this area, πrk2 , by the cathode operating current density JK,operating one gets the beam current I0 giving rK as: rK = (I0 /πJK ,operating )1/2 . (ii) Since rK is known from step (i), the beam convergence rK /rM is also known; rM being one of the known input beam parameters, which is essentially the beam radius to be maintained beyond the throw of the gun with the help of a separate focusing structure. (iii) Now that rK /rM is known, one may find rK /rM by inverting the series (4.21). (iv) Since rc /ra is known from step (iii), one may find G0 with the help of (4.8) interpreted at r = ra . Using the value of G0 thus found and that of the beam perveance prev(=I0 /V03/2 ) which is known from the given values of I0 and V0, we can then find θ0 with the help of (4.9). (v) One may use the relation θ0 ≈ rK /rc to find rc as rc = rK /θ0, where rK is known from step (i) and θ0 from step (iv). (vi) That rc /ra is known from step (iii) and rc from step (v) we can find ra as ra = rc /(rc /ra ). (vii) θ0 being known from step (iv) and ra from step (vi), rA can be found from the relation θ0 ≈ rA /ra as rA = θ0ra . (viii) rc and ra being known from steps (v) and (vi), respectively, the interelectrode spacing d can be found as d = rc − ra . (ix) The throw dm of the gun, measured as the distance of the beam waist from the anode, can be found from (4.15), now that rA is known from step (vii) and rM is known as one of the input parameters. Finding the design output parameters of the convergent-flow electron gun derived from a curved cathode as above, which is, in fact, known as the synthesis of the electron gun, becomes complete by predicting the shape of the electrodes, as has been done for a parallel-flow gun derived from a flat cathode. For this purpose, the configuration of the convergent conical beam problem is conformally mapped to the configuration of the strip beam of the parallel-flow Pierce gun derived from a flat

4-8


cathode. For this purpose, we can resort typically to the logarithmic transformation in which we take a suitable analytic function, here, the logarithmic function

W = f (Z ) = ln Z = u r + jui ⎫ ⎬ Z = z + jy = r exp( jθ ) ⎭

(4.22)

to transform the cross-sectional geometry of the problem from one plane, say, Z-plane to another, say, W-plane such that, corresponding to a point (z, y) on the Z-plane with z and y as the real and the imaginary axes, respectively, we have a point (ur, ui ) on the W-plane with ur and ui as the real and the imaginary axes, respectively (figure 4.4(a) and (b)). Through such transformation, called the conformal transformation, from one plane to another, the relative angles between the lines are preserved and the corresponding incremental areas remain similar in shape though they differ in scale. With reference to the present context, the beam edge of the conical beam after transformation from the Z- to W-plane coincides with the real axis of the W-plane, the solution for the potential function above the

Figure 4.4. Cross-sectional geometry of the conical beam configuration showing the half-cone angle θ = θ 0 , representing half the emitting portion of the cathode, the beam edge, the cathode and the anode spheres and the gun axis on the Z-plane (a); the first conformal transformation showing the cathode at ur = ur0 and the beam edge at ui = ui0 on the W-plane (b); and the second conformal transformation showing the cathode at u′r (=ur − ur0 ) = 0 and the beam edge at u′i (=ui − ui0 ) = 0 on the W′-plane (c) [1].

4-9


W-plane being presumably known. We look here for the potential function V (r, θ ) for the conical beam configuration outside the beam θ > θ0 which in the limit passes on to the potential function

⎛ ⎞2/3 9I0 V (r , θ 0 ) = ⎜ ⎟ G 4/3(u ) ⎝ 8πε0(2 η )1/2 (1 − cos θ 0) ⎠

(4.23)

at a radial coordinate r on the beam edge θ = θ0. The potential function (4.23) is essentially the generalized form of the potential function (4.9) at the specific radial coordinate r = ra of the anode which was obtained interpreting the Langmuir– Blodgett general solution (4.7). Now, with the help of (4.22) we obtain

u r = ln r ⎫ ⎬, ui = θ ⎭

(4.24)

which we can read at the cathode (r = rc ) on the beam edge (θ = θ0 )

u r 0 = ln rc ⎫ ⎬ ui 0 = θ 0 ⎭

(4.25)

as the reference coordinates ur = ur0 , ui = ui0 on the W-plane (figure 4.4(b)). In the second step, let us transform the cathode on the beam edge from the plane W (=ur + jui ) to the plane W ′(=u′r + ju′i ) so that the origin (u′r = 0, u′i = 0) of the plane W ′ is located at a point corresponding to the point (ur = ur0 , ui = ui0) of the plane W , and the cathode coincides with the imaginary axis u′r = 0 and the beam edge with the real axis u′i = 0 of the plane W ′ (figure 4.4(c)). Clearly, then u′r = ur − ur0 and u′i = ui − ui0 , which enables one to write

W ′ = u r′ + ju i′ = (u r − u r 0) + j (ui − ui 0) = (u r + jui ) − (u r 0 + jui 0), which can be read with the help of (4.24) and (4.25) as

W ′(=u r′ + ju i′) = In r + jθ − (In rc + jθ 0) = In(r / rc ) + j (θ − θ 0).

(4.26)

Now that by the second stage of transformation the beam edge and the cathode have been made to coincide with the real and the imaginary axes of the plane W ′, respectively, one has now actually changed the configuration of a conical beam to that of a rectangular strip beam. Therefore, the potential function V (r, θ ) at a point outside the beam (θ > θ0) can be written using the same method as that outlined following (4.3) for the formation of a rectangular strip beam derived from a flat cathode as follows:

V (r , θ ) = ReW ′ = Re f (u′r + ju′i ), which can be read with the help of (4.26) as

V (r , θ ) = ReW ′ = Re f (ln(r / rc ) + j (θ − θ 0)).

4-10

(4.27)


We need to find the potential function f in (4.27) subject to the conditions that (i) the potential V (r, θ ) satisfies the expression (4.23) for V (r, θ0 ) at a radial coordinate r on the beam edge θ = θ0 , and (ii) the derivative of potential at the beam edge (θ = θ0) becomes nil, that is, (∂V /∂θ )θ =θ0 = 0 to ensure that there exists no azimuthal component of electric field (=( −1/r )∂V /∂θ ) at the beam edge (θ = θ0 ) causing the deviation of the beam flow (figure 4.4(a)). At this stage, it is convenient to express G (u ) occurring in (4.23) with the help of (4.8) as

bn(In rc / r )n ⎫ ⎪ ⎪ n = 1,2,3, … n = 1,2,3, … ⎬. a1 = −1, a2 = 3/10, a3 = −3/40, a 4 = 63/4400, … ⎪ ⎪ bn = ( −1)nan (n = 1, 2, 3, … ⎭ G (u ) =

∑

an(In r / rc )n =

∑

(4.28)

Subject to the above conditions (i) and (ii) mentioned following (4.27) and in view of (4.28), one can then identify the function in (4.27) to write the following expression for the potential V (r, θ ) outside the beam (θ > θ0 ):

V (r , θ ) = ReW ′ = Re f (In(r / rc ) − j (θ − θ 0)) ⎛ = Re⎜ ⎝ 8πε(2

⎛ ⎞4/3 ⎞2/3 ⎜ n ⎟ ⎟ ⎜ ∑ an[In(r / rc ) + j (θ − θ 0)] ⎟ η 1/2 (1 − cos θ 0) ⎠ ⎝ n = 1,2,3, … ⎠ 9I0

⎞4/3 ⎛ ⎞2/3 ⎛ 9I0 = Re⎜ ⎟ ⎜ ∑ bn[In(rc / r ) + j (θ − θ 0)]n⎟⎟ ⎝ 8πε(2 η 1/2 (1 − cos θ 0) ⎠ ⎜⎝ n = 1,2,3, … ⎠

(4.29)

The potential function (4.29) thus obtained by conformal transformation in the conical beam case may be interpreted to find the equipotential lines on the crosssectional geometry of the conical beam configuration, as was done in the case of a rectangular strip beam configuration discussed following (4.5). From the equipotential one may find the shapes of the electrodes (here, those of the BFE and the anode); the method is referred to as the conformal mapping of electrode shapes. 4.1.3 Magnetron injection gun for the formation of a gyrating electron beam The magnetron injection gun (MIG), so named because of its cathode assembly resembling a magnetron—meant for fast-wave tubes like the small-orbit gyrotron (see section 7.2 in chapter 7, volume 2)—forms a hollow annular beam of gyrating electrons comprised of helical beamlets of small orbital radii compared to the transverse dimensions of the interaction structure of the device (figure 4.5) [2, 3]. A MIG consists of a convex (conical) thermionic dispenser cathode operating in the temperature-limited region to minimize the velocity spread in the beam (figure 4.5). Electrons are drawn off from the annular emitting portion of the lateral face of the cathode with a small angular velocity at an angle with the tube axis into a 4-11


Figure 4.5. Magnetron injection gun [2].

system of a crossed DC electric field established by the first anode (gun anode), and a magnetic field established by the gun solenoid to impart a small rotation of electrons. The non-emitting portion of the cathode of the MIG serves as a focusing electrode. At the same time, electrons in the gun acquire a large amount of axial velocity established by the second (accelerating) anode of the gun. Thus, electrons (i) move in cycloidal path near the cathode in a nearly axial magnetic field as in a magnetron and (ii) move at the same time axially away from the cathode to form an annular beam in the presence of an axial component of the electric field unlike in a magnetron [3]. A slowly increasing magnetic field or adiabatic compression region is provided to convert a large portion of the axial energy of the beam into its rotational energy in the beam tunnel of the MIG. The beam tunnel prevents the propagation of RF waves from the cavity of the gyrotron to its MIG region. Moreover, no beam-wave interaction that could degrade the beam quality is allowed to takes place in the beam tunnel where all parasitic oscillations generated are also arrested. The annular electron beam formed by the MIG comprises a number of helical beamlets each executing small orbits compared to the interaction cavity radius in gyrotron, which hence is referred to as a small-orbit gyrotron. However, in another version of the gyrotron gun, called the cusp gun, meant for a large-orbit gyrotron (LOG) (chapter 7, volume 2), the cusp near the cathode provides the reversals of magnetic field in which the axially moving electrons in a radial magnetic field acquire their azimuthal velocity from the Lorentz force (see section 4.1). A drift section following the gun provides an adiabatic magnetic field compression, that is, an increase in magnetic field from the gun to the interaction region to cause a reduction in the beam’s cross-sectional area and the formation of a beam of axis-circling electrons. The problem of lower efficiencies of LOG with higher beam-harmonic operation limiting the output power capability of such a gyrotron has been alleviated by the use of a Marx generator in combination with a pulse magnet providing an operating magnetic field of 12 T. This has opened up the possibility of a LOG to reach the border of the terahertz frequency range with an output power above 1–2 MW.

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4.2 Magnetic focusing structure The electrons beyond the throw of the gun, if left to them, would diverge out due to the Coulomb repulsive force between them. The function of the magnetic focusing structure is to provide the Lorentz force to counteract this Coulomb repulsive force, thereby constraining the electron beam to move parallel to the beam axis beyond the throw of the gun and transmit the beam into the interaction region of a linear-beam microwave tube such as the TWT. Though the magnetic flux density of the focusing structure is predominantly axial, it has adequate radial component, due to which the axially moving electrons of the beam will experience a Lorentz force to have an azimuthal velocity component. Consequently, the interaction between the azimuthal component of electron velocity and the axial component of the magnetic flux density provided by the structure would give rise to the required radial Lorentz force to counter-balance the space-charge force plus the centrifugal force of the circular electronic motion. For an electron in the focusing structure providing magnetic flux density B one can write: (i) the angular acceleration equation of an electron subject to the azimuthal component of Lorentz force and (ii) the radial acceleration equation of an electron subject to the force due to the radial component of the space-charge electric field Er together with the radial component of Lorentz force, respectively as follows [1]:

1 d ⎛ r 2 dθ ⎞ ⎜ ⎟ = η(v × B )θ r dt ⎝ dt ⎠

(angular acceleration)

⎛ dθ ⎞2 d 2r ⎜ ⎟ = η(E r + (v × B )r ) r − ⎝ dt ⎠ dt 2

(radial acceleration).

(4.30)

(4.31)

4.2.1 Busch’s theorem With the help of (4.30) we can find an expression for the angular frequency dθ /dt of the electron. For this purpose, it is convenient to express (4.30) as

⎛ dθ ⎞ d ⎜r 2 ⎟ = η r(Brvz − Bzvr ) dt ⎝ dt ⎠ which, in turn, may also be put as

⎛ r 2 dθ ⎞ η d⎜ ⎟ = − dϕB ⎝ dt ⎠ 2π

(4.32)

in terms of (dϕB (=−2πr(Br dz − Bz dr )), the element of magnetic flux through an element of beam strip generated by making a complete revolution of the element of an electron trajectory in the beam [1]; it being implied that the magnetic flux lines passing through such a beam strip also cut across the beam edge and that down the

4-13


focusing structure these flux lines become parallel to the axis of the beam. We can now integrate (4.32) to obtain

ηφ ηφ r 2dθ = − B + Bk , dt 2π 2π

(4.33)

the second term of the right-hand side of which being the integration constant, the latter obtained by putting the electron angular velocity dθ /dt = 0 at the cathode where from an electron is emitted with no angular motion, the subscript k referring to the cathode. Further, with the help of (4.33) we can write the following expression for the electron velocity known as Busch’s theorem [4]:

⎛ Bk ⎞⎛ rk ⎞2 ⎞ ⎛ Bk ⎞⎛ rk ⎞2 ⎞ dθ ηB ⎛ ωc ⎛ =− ⎜1 − ⎜ ⎟⎜⎝ ⎟⎠ ⎟ ⎜1 − ⎜ ⎟⎜⎝ ⎟⎠ ⎟ = ⎝B⎠ r ⎠ ⎝B⎠ r ⎠ dt 2 ⎝ 2 ⎝

(4.34)

(Busch’s theorem), where ϕB = πr 2B is the magnetic flux, treating the magnetic flux density B as predominantly axial and perpendicular to the circular cross section of the portion of the beam of radius r , that is, of area πr 2 , and ϕBk = πrk 2Bk is the magnetic flux at the cathode, rk being the beam radius at the cathode, and Bk the magnetic flux density at the cathode, and ωc(=−ηB ) is the electron cyclotron frequency. According to (4.34) the electron velocity in the beam in the magnetic focusing structure depends on its radial coordinate r . However, in the particular case of interest in which ϕBk = Bk = 0, say, for a case of no magnetic flux linked up with the cathode, we obtain from (4.34):

dθ ω = c (for a magnetically-shielded cathode; ϕBk = 0) , dt 2

(4.35)

which is independent of r meaning thereby that for such a case (ϕBk = 0) all the beam electrons rotate with the same angular velocity ωc /2, that is, the electron beam would rotate like a ‘rigid bar’. 4.2.2 Brillouin focusing Putting in (4.31) the space-charge electric field as Er = ρr/(2ε0 ), as was done in (4.11) (where the symbol Es was used to represent the space-charge electric field), we can express (4.31) as

⎛ ρr ⎞ ⎛ dθ ⎞2 d 2r ⎜ ⎟ , v B r = η + + ⎜ ⎟ θ ⎝ dt ⎠ ⎝ 2ε0 ⎠ dt 2

(4.36)

where ρ is the volume charge density of electrons. Using the expressions for (i) current density J = ρv already introduced following (4.11); (ii) electron angular velocity vθ = r(dθ /dt ); (iii) angular velocity given by (4.35), that is, dθ /dt = ωc /2 taking the cathode essentially as magnetically shielded; (iv) electron angular cyclotron frequency: ωc = −ηB = η B ; (v) current density: J = Ie /(πr 2 ), where Ie is the electron current through the cross-sectional area of radius r of the beam, we can 4-14


express (4.36). However, now taking the electrons as the beam-edge electrons and thus interpreting the electron current as Ie = −I0 , where I0 is the beam current constituted by the electron flow through the entire beam’s cross-sectional area, from (4.36) we can then obtain the following expression:

d 2r ⎛ η I0 ⎞ 1 ⎛ η 2 B 2 ⎞ =⎜ ⎟ −⎜ ⎟r ⎝ 4 ⎠ dt 2 ⎝ 2πε0v ⎠ r (beam-edge electron) (Brillouin focusing).

(4.37)

Interpreting d 2r /dt 2 = v 2d 2r /dz 2 , as done following (4.11), and making use of the expression v = (2ηV0 )1/2 for beam velocity v in terms of beam voltage V0 (introduced following (4.12)), we can then express (4.37) as

d 2r C = 1 − C2r (beam-edge electron) (Brillouin focusing) 2 dz r

(4.38)

⎫ η I0 3/2 ⎪ 2πε0(2 η V0) ⎪ ⎬. η 2B 2 ⎪ C2 = ⎪ 4(2 η V0) ⎭

(4.39)

where

C1 =

Let us now find the condition that the beam radius r down the focusing structure remains constant at a reference value r = a , at the beam-waist of a convergent Pierce electron gun, and beyond the beam-waist, and choose a = rM , the beam-waist radius defined following (4.14). Further, putting r = a + δ in (4.38), where δ ( <
δ ⎞−1 C1 ⎛ d 2δ C1 ⎜1 + ⎟ ( ) = − + δ = − C2a − C2δ , C a 2 a ⎝ a⎠ dz 2 a+δ which on binomial expansion of the first term of its right-hand side while ignoring higher powers of δ /a (since δ <
⎛C ⎞ ⎛C ⎞ d 2δ C ⎛ δ⎞ = 1 ⎜1 − ⎟ − C2a − C2δ = ⎜ 1 − C2a⎟ − ⎜ 21 + C2⎟δ . 2 ⎝ ⎠ ⎝a ⎠ ⎝a ⎠ dz a a

(4.40)

Now, if we choose C1 and C2 , each depending on the beam voltage V0 and current I0 according to (4.39), such that

C1 − C2a = 0 a we can write from (4.40)

⎛C ⎞ d 2δ = −⎜ 21 + C2⎟δ , 2 ⎝a ⎠ dz

4-15

(4.41)


which further putting C1/a 2 = C2 obtainable from (4.41) and m2 = 2C2 can be expressed

d 2δ = −2C2δ = −m 2δ , dz 2 which has the following solution:

δ = A sin mz + B cos mz

(4.42)

where A and B are constants. Further, we can appreciate from (4.42), under the conditions (i) z = 0 as the entry point of the electron beam with radius r(=a + δ ) = a corresponding to δ = 0 and (ii) dδ /dz = 0, the constants A and B each become nil (A = B = 0) and A = 0 which consequently make δ nil (δ = 0) independent of z , and, in turn, r = a + δ = a , also independent of z , thereby giving us an electron beam without scalloping in the focusing structure. Further, one can state the condition (4.41) with the help of (4.39) as

⎞1/2 ⎛ 2 I0 ⎟ , B = BB = ⎜ 1/2 ⎝ πε0 η 3/2 V0 a 2 ⎠

(4.43)

where BB is called the Brillouin magnetic flux density. We can appreciate from (4.43) that a larger magnetic flux density is required for an electron beam of a smaller radius, higher beam current and lower beam voltage. We can put together the conditions of no beam scalloping in the focusing structure, known as Brillouin conditions, as follows: (i) ϕBk = 0, that is, the cathode is shielded from the magnetic field, as stated preceding (4.35); (ii) the beam enters the focusing structure with a radius of r = a , which is desired to be maintained constantly throughout the beam transport, as stated preceding (4.43); (iii) dδ /dz = 0, that is, the beam should enter the magnetic field of the focusing structure parallel to the axis of the focusing structure or, in other words, there should be no radial component of beam velocity at the entry of the focusing structure, as stated preceding (4.43); and (iv) the magnetic flux density in the focusing structure should satisfy (4.43) relating it to the beam voltage, current and radius: V0, I0, a . 4.2.3 Confined-flow focusing In a magnetic focusing scheme, called confined-flow focusing, in which the magnetic flux lines are allowed to thread into the cathode (unlike in Brillouin focusing), the stringent conditions (i)–(iv) become relaxed and at the same time the beam radius becomes much less sensitive to the variation of beam current density. Presuming that the magnetic flux lines cutting through the beam-waist also extend over the entire cathode disc of a Pierce gun derived from a curved cathode (section 4.1), we can

4-16


equate the flux through the beam waist to the flux through the cathode disc (defined following (4.34)) to write BW π a 2 = Bkπ rk 2 giving

Bk = BW /(rk / a )2 ; BW = Bk(rk / a )2 ,

(4.44)

giving the flux density relation in terms of the area convergence of a convergent-flow gun, where BW is the magnetic flux density at the beam-waist of radius a(=rM ) (section 4.1). Further, since in this case ϕBk ≠ Bk ≠ 0, we need to take (4.34) instead of (4.35) for dθ /dt in the analysis which will modify (4.37) to be read as 4⎞ d 2r ⎛ η I0 ⎞ 1 η 2⎛ 2 2 rk B B = − − ⎜ ⎟r ⎜ ⎟ k dt 2 ⎝ 2πε0v ⎠ r 4 ⎝ r4 ⎠

(4.45)

(beam-edge electron) (confined-flow focusing). Comparing (4.37) for Brillouin focusing with (4.45) for confined-flow focusing interpreted at r = a for a beam-edge electron we can relate the confined-flow and Brillouin magnetic flux densities B and BB , respectively, in the following alternative forms while making use of (4.44): 4 rk4 2 2 2 rk ; B = B + B ; B 2 = BB2 + BW 2 k B a4 a4 B = pBB ; BW = (1 − 1/ p 2 )1/2 B = (p 2 − 1)1/2 BB

BB2 = B 2 − Bk 2

Bk = (1 − 1/ p 2 )1/2 (a / rk )2 B = (p 2 − 1)1/2 (a / rk )2 BB

⎫ ⎪ ⎪ ⎬, ⎪ ⎪ ⎭

(4.46)

where p is a factor defined as p = B /BB . Thus, it follows from (4.46) that, if, typically, we choose p = B /BB = 1.5 and the beam area convergence (πrk 2/πa 2(=(rk /a )2) = 50, we obtain BW ≈ 1.1BB and Bk ≈ 0.02BB , suggesting that the confined-flow focusing requires (i) a magnetic flux density BW at the beam-waist of the convergent gun, that is, at the entrance of the magnetic focusing structure greater—though not much greater—than the corresponding Brillouin value BB , and (ii) a small magnetic flux density Bk relative to BB threading into the cathode. 4.2.4 Periodic permanent magnet focusing A larger magnetic flux density is required for confining a beam of higher current as required for a higher power linear beam tube such as the TWT or for the operation of such a device at high frequencies for which the beam radius is required to be reduced commensurate with the reduced transverse dimension of the interaction structure of the device at such high frequencies. This can be appreciated from the relation (4.43) between the required magnetic flux density and the beam parameters, for instance, for Brillouin focusing. A solenoid focusing structure becomes rather heavy if it has to deliver a larger magnetic flux density; moreover, it also needs an external power supply to make it further heavier for practical devices, particularly

4-17


for air-borne and space applications. An alternative is the permanent magnet (PM) focusing structure which is, however, suitable for a microwave tube requiring a small interaction length such as the klystron and not for a distributed interaction device such as the TWT requiring a larger interaction length. However, if the length of a PM is increased by a factor of N , say, then all the other dimensions of the structure also have to be increased in the same proportion to ensure that the magnetic flux density in the interaction region does not fall, requiring an increase in the inner and outer radii, for instance, of a tubular PM each by the same factor N . This will amount to increasing the volume and hence the weight of the PM to N 3 times— attributable to the increase in magnetic field and magnetic energy stored outside the magnet which does not help in focusing the electron beam. In a periodic permanent magnet (PPM) focusing structure, instead of a single PM of a length increased by a factor of N , an array of N identical magnet cells are used (figure 4.6). The length of such a structure can be increased N times without requiring an increase to its transverse dimensions as is necessary in a PM. Thus, one realizes the advantages of a PPM over its PM counterpart in terms of weight by a factor of N 3 /N = N 2 . In such a PPM structure, the magnetic flux lines external to the magnet due to consecutive cells are directed oppositely causing a reduction in magnetic field and consequently a reduction in the loss of magnetic energy outside the magnet. Let us approximate the magnetic flux density in a PPM structure, supposedly varying periodically along the axial distance z while remaining uniform over its cross section, as [1]:

Figure 4.6. A periodic permanent magnet (PPM) providing a periodically varying magnetic field at the axis showing magnetic flux lines (dotted) and typically three magnet cells for the axial (a) and radial (b) magnetization [1, 6].

4-18


B = B0 cos

⎛ 1 + cos(4π z / L ) ⎞1/2 2π z ⎟ , = B0⎜ ⎝ ⎠ 2 L

(4.47)

where B0 is the peak magnetic flux density, L is the axial periodicity of variation in B , L/2 being the distance between two consecutive oppositely directed maxima (figure 4.6). We can express (4.38) taking the constants C1 and C2 from (4.39), however substituting B therein from (4.47) as the following trajectory equation for the PPM structure in terms of the normalized radius σ = r /a and the axial distance Z = 2π z /L of the beam-edge electron [1]:

d 2σ β + α(1 + cos 2Z )σ − = 0 2 dZ σ

(4.48)

where

η L2B02 64π 2V0 η L2I0 β= 8π 3a 2ε0(2 η V0)3/2

α=

=

⎫ ⎪ ⎪ ⎬. 2 η L (Perv ) ⎪ 8π 3a 2ε0(2 η )3/2 ⎪ ⎭

(4.49)

For a very small beam current I0 or beam perveance Perv that makes β = 0, as can be seen from (4.49), the trajectory equation (4.48) reduces to Maitheu’s equation

d 2σ + α(1 + cos 2Z )σ = 0, dZ 2

(4.50)

which has alternating pass and stop bands of solution on the scale of α that is proportional to B02 —between α = 0 and 0.66 (first pass band); between α = 0.66 and 1.72 (first stop band); α = 1.72 and 3.76 (second pass band); α = 3.76 and 6.1 (second stop band); and so on—the stop bands being wider than the pass bands and the widths of both the pass and stop bands increasing with the order of the band. In a pass band the solution of (4.50) for the normalized beam radius σ is stable and periodic with the normalized axial distance Z , the beam radius ripples depending on the value of α relative to that of β . This finding, arrived at for vanishingly small beam currents ( β = 0), continues to be valid even for higher beam currents and the detailed computational study shows that the minimum beam ripples if the condition α ≈ β is satisfied which, in view of (4.49), can be read as [1]:

⎛ ⎞1/2 ⎞1/2 ⎛ 2 2 I0 B0 2 I0 ⎟ ⎟ = BB , ⎜ Brms = = ⎜⎜ = ⎟ 1/2 1/2 2 ⎝ πε0 η 3/2 V0 a 2 ⎠ ⎝ πε0 η 3/2 V0 a 2 ⎠

(4.51)

where Brms(=B0 / 2 ) is the rms value of magnetic flux density of the PPM focusing structure. Further, one can choose the period L of the PPM vis-à-vis the first pass band corresponding to α < 0.66, as discussed following (4.50), which can be read with the help of (4.49) as

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⎛ 0.66 × 64 ⎞1/2 π V01/2 L<⎜ (first pass band). ⎟ ⎝ η ⎠ B0

(4.52)

Thus, Brms of the PPM magnetic flux density should be chosen as the Brillouin value BB that depends on the beam parameters according to (4.51) and the periodicity of the PPM L according to (4.52). Further, since confined-flow focusing demands more flux density than Brillouin focusing, in the former the PPM becomes of considerable relevance. Furthermore, the advent of light-weight magnetic materials such as samariumcobalt (SmCo5 and Sm2Co17) and ALNICO-5 has made it possible to design suitable light-weight PPM for air-borne and space applications.

4.3 Multistage depressed collector In a linear beam tube, the electron beam formed and confined by an electron gun and a focusing structure, respectively, and made to deliver part of its energy to RF waves in the interaction structure, becomes what is called the ‘spent beam’. Since the electrons of the spent beam no longer remain under the influence of the focusing structure they diverge out and get collected at the inner surfaces of a system of electrodes called the collector. The kinetic energy of the electrons of the spent beam striking the collector generates heat loss. However, in a collector called the depressed collector, the spent-beam electrons are slowed down by reducing or depressing the potential of the collector enabling them to ‘soft-land’ on the inner surface of the collector electrode thereby making it possible to partially recover the kinetic energy of the beam. Such a collector may be of two types: single-stage in its simplest form, and multistage (figure 4.7). Ideally, the collector would work with a hundred percent collector efficiency if the spent beams were retarded to a halt at the collector by applying on it a suitable depressed potential with respect to the interaction structure.

Figure 4.7. Schematic of a multi-stage depressed collector with six stages [3].

4-20


The electrons in the spent beam are distributed in various energy classes since they emerge out from the interaction region with a velocity spread due to the nonuniformity of electron emission from the cathode as well as magnetic focusing and RF modulation. Therefore, the collector needs to be segmented into various stages to make a multistage depressed collector (MDC) to collect the electrons of various energy classes. Each of these stages of MDC is biased at an appropriate potential to collect each energy class of electrons in the spent-beam. In a MDC, the highestpotential stage is situated nearest to the interaction structure that collects the slowest electrons, while the lowest potential stage, which could be depressed to be as low as the cathode potential, is located at the remotest point from the interaction structure that collects the fastest electrons. Further, by increasing the number of stages the collector efficiency can be increased; though with the improvement of incremental efficiency diminishing with an increase in the number of collector stages, more so beyond four stages. However, increasing the depressed-collector efficiency by increasing the number of stages invites more complexity in the design of the power supply calling for a design trade-off [1–3]. An alternative method of improving the performance of a MDC is to shape the electric and magnetic fields inside the collector in such a way that the electrons land on the electrodes perpendicularly thereby reducing their transverse energy and hence thermal loss [1–3]. Also, one should note that the optimum design of a MDC is a function of the input drive which controls the distribution of various collector currents. Over and above improving the device efficiency, a MDC relaxes the thermal management by reducing the collector heating and associated cooling problems and also reduces the hardness of x-rays due to reduced velocities of electrons impinging on the collector. However, in the design of a MDC another major issue is the backstreaming of electrons constituted by the slow spent-beam electrons reversing their flow, as well as by electrons reflected by the collector surface. Such electrons, likely to be collected by a higher potential than that just necessary for the purpose, off-set the power-saving purpose of a MDC; they also produce an excessive thermal load and degrade RF performance and cause instabilities. The problem is augmented also by the fast electrons in the spent beam that impinge on the collector to generate secondary electrons. A transverse magnetic field may be applied in the collector region to reduce back-streaming of reflected primary and secondary electrons [2, 3]. The introduction of the asymmetry in the geometry of the collector helps to recapture the secondary electrons. The material for the collector therefore, should have a low secondary electron emission coefficient [2, 3]. Besides, the material should have high emissivity and good thermal conductivity in a plane perpendicular to the deposition direction. It should also be of light weight and have at the same time good mechanical strength. Pyrolytic graphite consists of these properties and can be used as a material for the collector instead of the usual oxygen-free high conductivity (OFHC) copper. These collectors may be assembled simply by forcefitting without any brazing resulting in the least risk of out-gassing or long-term degradation problems. In the space TWT, the collector may also be projected beyond the satellite’s outer surface for cooling.

4-21


The depressed collector, which recovers the energy associated with the axial motion of the spent beam and which is suitable for a linear beam tube like the TWT, can also be employed for enhancing the efficiency of a gyro-device such as the gyrotron, which uses a gyrating electron beam having most of its energy in rotational motion. For this purpose, however, one has to adiabatically reduce the magnetic field, which is inherently large in the interaction region of the device, with the help of a magnetic cusp [3, 5].

References [1] Basu B N 1996 Electromagnetic Theory and Applications in Beam-wave Electronics (Singapore: World Scientific) [2] Gilmour A S Jr 1986 Microwave Tubes (Norwood: Artech House) [3] Gilmour A S 2011 Klystrons, Traveling Wave Tubes, Magnetrons Crossed-Field Amplifiers, and Gyrotrons (Boston: Artech House) [4] Jordan E C and Balmain K G 1968 Electromagnetic Waves and Radiating Systems (Englewood Cliffs: Prentice Hall) [5] Edgecombe C F (ed) 1993 Gyrotron Oscillators – Their Principles and Practice (London: Taylor and Francis) [6] Carter R G 1990 Electromagnetic Waves/Microwave Components and Devices (London: Chapman and Hall)

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IOP Concise Physics


Chapter 5 Analytical aspects of beam-absent and beam-present slow-wave and fast-wave interaction structures

The interaction structure of a microwave tube (MWT) where beam-wave interaction takes place for the transfer of energy from the electron beam to RF waves may be a propagating or a resonating type. For example, a metallic tape wound in the form of a helix that supports RF waves of phase velocity less than the speed of light is a propagating slow-wave structure (SWS) of a slow-wave traveling-wave tube (TWT). Similarly, a cylindrical waveguide is a fast-wave structure that supports RF waves of phase velocity greater than the speed of light. It is a propagating-type interaction structure which can be used as the interaction structure of a fast-wave gyro-TWT [1–4]. However, such a waveguide when short-ended or an open-ended and assigned a definite length can also be used as a resonator. For example, an open-ended waveguide resonator has been used as the interaction structure of a fast-wave gyrotron [1–3]. The three commonly used SWSs used in TWTs, besides the helix, are (i) the coupled-cavity structure, (ii) the ring-and-bar structure, which is essentially a helixderived structure with two inbuilt contra-wound helices and (iii) the folded waveguide (figure 5.1) [2]. A growing-wave beam-wave interaction takes place along the length of the SWS as the electron beam transmits through the structure when the electron beam velocity is made nearly synchronous with the RF phase velocity of the structure. The ring-and-bar SWS (figure 5.1(a)) is superior to a simple helix SWS in having a larger transverse dimension, a greater mass with more thermal contacts, and a higher interaction impedance, the latter measuring the electric field in the structure available for interaction with the electron beam for a given power propagating down the structure. The structure permits the use of such large transverse dimensions in the interaction with a higher voltage electron beam without entailing the risk of generating a backward-wave mode that can cause oscillations in

doi:10.1088/978-1-6817-4561-9ch5

5-1



Figure 5.1. Commonly used slow-wave structures of a TWT other than the helix: the coupled-cavity (a) ring-and-bar (b) and folded waveguide (c) structures [2].

a TWT. The structure being massive with thermal contacts with its support makes it better than a helix from the standpoint of thermal management and power handling capability. A higher interaction impedance of the structure providing a higher electric field for interaction with the electron beam yields a higher device gain and efficiency [1, 2, 5, 6]. The coupled-cavity SWS is a circular waveguide with obstacles in the form of washers placed at a regular axial interval to slow down the RF waves propagating through the waveguide [7]. It is constructed as a chain of cavities with successive cavities coupled with holes or slots, typically, in a staggered-slot combination, which is essentially a serpentine-line structure (figure 5.1(b)). The fields in the structure in two successive cavities are the same. However, they have a phase difference of 180°. The geometry of the structure also provides an additional phase difference of about 90° to make the total phase difference of about 270° between the fields of the successive cavities. Therefore, an electron beam leaving a cavity must arrive at the cavity next to it with a delay corresponding to a phase difference of about 270°. The ω−β dispersion plot of such a staggered-slot coupledcavity SWS exhibits a negative slope corresponding to a negative group velocity dω/dβ, corresponding to backward-wave propagation in the region between β = 0 and π, and hence the structure is called a fundamental backward-wave structure. The operating region of the TWT using such a structure is chosen, beyond this backward-wave region, between β = π and 2π in the ω−β dispersion plot where it exhibits a positive slope corresponding to a positive group velocity dω/dβ and a forward wave [7]. The coupled-cavity SWS also has a massive structure and thus has a better thermal capability and can deliver larger power outputs than a helix. However, the coupled-cavity SWS consisting of resonant cavities cannot give as wide a bandwidth as the helix, which is essentially a non-resonant structure. The folded-waveguide (figure 5.1(c)) slow-wave structure, which also belongs to the serpentine structure category, has the potential for being mini-fabricated combining vacuum electronics technology and silicon fabrication techniques. The fabrication of miniaturized folded-waveguide slow-wave structures has made it possible to develop terahertz TWTs. In this chapter we outline the analytical approaches to (i) a helix along with its dielectric supports and a vane-loaded envelope (section 5.1), and (ii) a disc-loaded loaded circular waveguide (section 5.2), to exemplify the analyses of two typical slow-wave and fast-wave interaction structures, respectively (figure 5.2). The control of the shape of the dispersion characteristics of both these structures at a high interaction impedance value is of relevance to widening the bandwidths of a conventional TWT and a gyro-TWT, respectively (section 5.3). Analysis of a helical structure is challenging in view of the intricacy due to the skew geometry of the helix

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Figure 5.2. A helix with dielectric helix-supports in a vane-loaded metal envelope (a) and a disc-loaded circular waveguide (b) [11].

and the periodicity of the structure—both axial periodicity due to helix turns and azimuthal periodicity due to the positioning of the dielectric helix support rods and vanes of the envelope at a regular angular interval. The analysis can be carried out as a boundary-value problem by suitably modeling the helix, finite helix thickness, dielectric helix-supports and metal envelope taking into account the space harmonics generated due to the axial and azimuthal periodicity of the structure. The rigor in the analysis can be added by considering the non-uniformity of the radial propagation constant over the structure’s cross section [8–11]. The analysis can be made still rigorous by taking into account the effect of asymmetry in discrete helix-support rods—in terms of their geometry, material properties and angular positioning—which can cause a stop-band in the dispersion characteristics and in turn band-edge oscillation in the device [11–13]. Similarly, the disc-loaded circular waveguide can be analyzed taking into account the effects of axial periodicity of the structure [4, 14–28]. Only the circular configuration of the helix has been considered in the analysis in this chapter (section 5.1). However, in the millimeter-wave/terahertz frequency regime, a planar helix is also suitable from the standpoint of the ease of fabricating the structure in tiny sizes using vacuum microelectronic, micro-fabrication technology (see chapter 8, volume 2). Further, a planar helix can be made compatible with a sheet electron beam in the device that can provide a larger beam current and relax the required magnetic field for the confinement of the beam. Analysis has been reported in literature for a planar helix modeled by a pair of conducting screens in different directions [29–31]. A planar helix with straight-edge connections has also been studied [32]. However, analysis of a planar helix has been kept outside the purview of the analysis in this chapter (section 5.1).

5.1 Analysis of helical slow-wave interaction structures The helix may be analyzed by the sheath-helix [5, 33–36] and the tape-helix [2, 5, 8, 33, 34, 37, 38] models. The sheath-helix model is valid for a large number of helix turns per guide wavelength. However, it cannot take into account the effects of space-harmonics generated due to the axial periodicity of helix turns, whereas the tape-helix model can do this. The helix can be studied by field and equivalent-circuit analyses. 5-3


5.1.1 Sheath-helix model Field analysis of a helix in free space In the sheath-helix model, the actual helix is replaced by a circular cylindrical sheath of infinitesimal thickness and radius equal to the mean radius of the actual helix and of infinite and zero conductivities parallel and perpendicular to the helix winding direction, respectively [2, 5, 36]. The structure supports both the TE (Ez = 0) and TM (Hz = 0) modes simultaneously, where the subscript z refers to the axial field component in the cylindrical system of coordinates (r, θ , z). For a helix in free space, and for slow waves: vph < c; β > k, where vph (=ω /β ) is the wave phase velocity, ω and β being the angular frequency and the axial phase propagation constant of the wave, respectively; by solving the wave equation we can obtain the following field solution [1, 38]:

Ezp = Ap I0(γr ) + BpK 0(γr ) Hzp = CpI0(γr ) + DpK 0(γr ) Eθp = −(jωμ0 / γ )[CpI1(γr ) − DpK1(γr )] Hθp = (jωε0 / γ )[Ap I1(γr ) − BpK1(γr )] Erp = (jβ / γ )[Ap I1(γr ) − BpK1(γr )] Hrp = (jβ / γ )[CpI1(γr ) − DpK1(γr )]

(a)⎫ ⎪ (b)⎪ ⎪ (c)⎪ ⎬, (d)⎪ (e)⎪ ⎪ (f)⎪ ⎭

(5.1)

where γ = (β 2 − k 2 )1/2 is the radial propagation constant, k (=ω(μ0 ε0 )1/2 = ω /c ) being the free-space propagation constant. The subscripts p = 1, 2 refer to the regions inside (0 ⩽ r ⩽ a ) (region 1) and outside (a ⩽ r ⩽ ∞) (region 2) the helix, respectively; a being the sheath-helix radius being equal to the mean radius of the actual helix. Iv(x ) and Kv(x )(v = 1, 2) are the modified Bessel functions of order v of the first and second kinds, respectively, and x(= γr ) is the argument of the modified Bessel functions. Ap , Bp, Cp and Dp(p = 1, 2) are the field constants. Further, in (5.1), the RF dependence exp j (ωt − βz ) is understood. However, it can be appreciated from (5.1) that in order to prevent the fields from blowing up to infinity inside the helix at the axis of the helix r = 0, one must put B1 = D1 = 0 since K 0(γr ) → ∞ as γr → 0. Similarly, one must put A2 = C2 = 0 in order to prevent the fields from blowing up to infinity outside the helix at r = ∞ since Io(γr ) → ∞ as γr → ∞. The remaining field constants, for the present case of a helix in free space, namely, A1,C1, B2 and D2 have non-zero values. The electromagnetic boundary conditions at the sheath-helix radius r = a , say, which is also the mean radius of the actual helix, are:

Ez1 − Ez 2 = 0 Eθ1 cos ψ + Ez1 sin ψ = 0 Eθ 2 cos ψ + Ez 2 sin ψ = 0 Hθ1 cos ψ + Hz1 sin ψ − (Hθ 2 cos ψ + Hz 2 sin ψ ) = 0

(a)⎫ ⎪ (b)⎪ ⎬ . (c)⎪ (d)⎪ ⎭(r=a )

(5.2)

The boundary condition (5.2(a)) represents the continuity of the axial component of electric field at the sheath helix (r = a ). The boundary conditions (5.2(b)) and (5.2(c)) each arise from the infinite conductivity of the sheath helix (r = a ) in the direction of

5-4


helix winding while the boundary condition (5.2(d)) arises from no current at the sheath helix (r = a ) perpendicular to the winding direction [2, 36]. Substituting the field expressions (5.1) involving the four constants A1,C1, B2 and D2 into four boundary conditions (5.2) we obtain four simultaneous equations in these constants. The condition for a non-trivial solution of these equations is that the 4 × 4 determinant formed by the coefficients of these constants becomes null. This condition invoking the relation Iν(γa )K ν+1(γa ) + K ν(γa )Iν+1(γa ) = 1/γa , where v is an integer (here, v = 0), leads to the following dispersion relation [9] of a helix in free space:

⎡ I (γa ) K 0(γa ) ⎤1/2 k cot ψ =⎢ 0 ⎥ γ ⎣ I1(γa ) K1(γa ) ⎦

(5.3)

(dispersion relation of a helix in free space obtained by field analysis in the sheathhelix model) where ψ (=tan−1(p′/(2πa )) is the helix pitch angle, p′ being the helix pitch. Equivalent-circuit analysis of a helix in free space Treating the helix as a transmission line and assuming it to be lossless, for the sake of simplicity, we can write its dispersion relation as

(5.4) β 2 = ω 2LC, where L and C are its series inductance per unit length and shunt capacitance per unit length, respectively [2, 8, 26]. The relevant boundary conditions at r = a required for the analysis are Ez1 = Ez 2

(5.5) Iza 2πa

Hθ 2 − Hθ1 = E θ1 = E θ 2 Hz1 − Hz 2 =

Iθa 2πa

(5.6) (5.7) (5.8)

Eθa cos ψ + Eza sin ψ = 0

(5.9)

−Iθa sin ψ + Iza cos ψ = 0

(5.10)

where Eza and Eθa are the axial and the azimuthal electric fields, respectively, and Iza and Iθa are the axial and the azimuthal currents, respectively, the subscript a referring to the quantities at the sheath-helix radius r = a . The boundary conditions (5.5) and (5.7) represent the continuity of the axial and azimuthal electric fields, respectively; (5.6) and (5.8) represent the discontinuity of the azimuthal and axial magnetic fields, respectively; (5.9) is (5.2(b)) written in an alternative form; and 5-5


(5.10) arises from zero conductivity perpendicular to the helix winding direction—all of them at the sheath-helix radius r = a . Further, we recall here the z-component of the relation between the electric field E ⃗ and the potential V in terms of the vector potential A⃗ [2, 39]:

Ez = −

∂Az ∂V − ∂t ∂z

(5.11)

and the z-component of Lorentz condition

∂V ∂Az = 0. + μ 0 ε0 ∂t ∂z

(5.12)

Besides, we also need to use the following telegrapher’s equations

∂V ∂Iza =0 +C ∂t ∂z ∂I ∂V + L za = 0 ∂t ∂z

⎫ (a)⎪ ⎪ ⎬. (b) ⎪ ⎪ ⎭

(5.13)

With the help of (5.5), (5.6), (5.11), (5.12) and (5.13(a)) and remembering the relation Iν(γa )K ν+1(γa ) + K ν(γa )Iν+1(γa ) = 1/γa (here, with ν = 0) we can obtain

C=

2πε0 . I0(γa )K 0(γa )

(5.14)

Similarly, with the help of (5.7)–(5.10) and (5.13(b)) we can obtain 2 μ0 ⎛ β ⎞ L= ⎜ ⎟ cot2 ψ I1(γa )K1(γa ). 2π ⎝ γ ⎠

(5.15)

The dispersion relation of the helix is then obtained by putting (5.14) and (5.15) in (5.4) as follows:

⎡ I (γa ) K 0(γa ) ⎤1/2 k cot ψ =⎢ 0 ⎥ γ ⎣ I1(γa ) K1(γa ) ⎦

(5.16)

(dispersion relation of a helix in free space obtained by equivalent-circuit analysis in the sheath-helix model). The relations (5.3) and (5.16) are identical establishing that the field and equivalent-circuit analyses yield one and the same dispersion relation of the helix. The equivalent circuit analysis also predicts, with the help of (5.14) and (5.15), the expression for characteristic impedance Z0 (=(L /C )1/2 of the helix—a parameter that is useful in the design of the input and output couplers for the helix. Modeling of dielectric helix-supports of a wedge cross section The identical wedge-shaped dielectric support rods of relative permittivity εr , symmetrically arranged around the helix in a metal envelope (figure 5.3), can be 5-6


Figure 5.3. Cross section of a helix supported by three identical wedge-shaped dielectric rods (N = 3) in a metal envelope showing the free-space region (1) inside the helix (0 ⩽ r ⩽ a ); the region (2) occupied by a support outside the helix (a ⩽ r ⩽ b; −ϕ /2 ⩽ θ ⩽ ϕ /2); and the free-space region (3) between two consecutive supports (a ⩽ r ⩽ b; ϕ /2 ⩽ θ ⩽ (2π /N ) − ϕ /2) [2, 8].

smoothed out into an equivalent continuous homogeneous dielectric tube of effective relative permittivity εr,eff which can be written, heuristically, considering the relative cross-sectional area occupied by the discrete dielectric supports in the structure, as follows [2, 8]:

εr,eff =

(εr )(As ) + (1)(A − As ) A ϕN = 1 + (εr − 1) s = 1 + (εr − 1) (As ) + (A − As ) A 2π

(5.17)

where As is the cross-sectional area of the dielectric support rods and A is the crosssectional area of the region between the helix and the envelope with the sub-regions occupied and unoccupied by the supports taken together. ϕ is the wedge angle for rods of the wedge cross section and N is the number of support rods, typically three (figures 5.3 and 5.4). The approach of deducing the dispersion relation for a helix in free space—(5.3) by field analysis and (5.16) by equivalent-circuit analysis—when extended to a helix surrounded by a dielectric tube of an effective relative permittivity εr,eff given by (5.17), while noting that though we can now continue to put B1 = D1 = 0 we can no longer put A2 = C2 = 0, yields the following dispersion relation:

⎡ I (γa ) K 0(γa ) ⎤1/2 k cot ψ =⎢ 0 ⎥ D lf γ ⎣ I1(γa ) K1(γa ) ⎦

(5.18)

where −1 ⎡⎧ I (γa ) K 0(γb) ⎫ ⎧ I (γa ) K1(γb) ⎫ ⎬ ⎨1 − 1 ⎬ D lf = ⎢⎨1 − 0 K 0(γa ) I0(γb) ⎭ ⎩ K1(γa ) I1(γb) ⎭ ⎢⎣⎩ −1/2 ⎧ ⎛ I1(γa ) K 0(γb) ⎞⎫⎤⎥ × ⎨1 + (εr,eff − 1) γa I0(γa ) K1(γa ) ⎜1 + ⎟⎬ K1(γa ) I0(γb) ⎠⎭⎥⎦ ⎝ ⎩

5-7

.

(5.19)


Figure 5.4. Cross section of a helix with dielectric support rods, typically of circular cross section, in a metal envelope (a) and its equivalent n-dielectric-tube model (b). a = 0 mean helix radius being equal to the sheathhelix radius, a0 = 0 outer helix radius of the helix and b = metal envelope radius [11].

The heuristically predicted expression (5.17) can be proved as a corollary to the deduction of the dispersion relation (5.18) of the structure by field analysis using the approach of Sinha reported by Basu et al [40] taking into account the azimuthal harmonics generated due to the angular periodicity of the dielectric wedges around the helix. Such an azimuthal-harmonic effect makes the RF quantities depend on the azimuthal coordinate θ as exp(jm(2π /Θ)θ ) = exp(jmNθ ), where m is the angularharmonic number; Θ = 2π /N is the azimuthal periodicity of the supports, N being the number of dielectric helix-supports which is also equal to the number of periods in the structure, each period consisting of a dielectric rod region and a free-space region between two consecutive dielectric support rods [40] (figure 5.4). Considering these angular harmonics we can write the following field expressions: ∞

∑

Ezs =

[As,m ImN (γr ) + Bs,mK mN (γr )]exp(jmNθ )

(5.20)

[Cs,mImN (γr ) + Ds,mK mN (γr )]exp(jmNθ )

(5.21)

m =−∞ ∞

Hzs =

∑ m =−∞

∞

∑

E θs =

′ (γr ) + Ds,mK mN ′ (γr ) − (mNβ /(γ 2r )] {( −jωμ0 / γ )[Cs,mImN

m =−∞

(5.22)

′ (γr ) + Bs,mK mN ′ (γr )]}exp(jmNθ ) × [As,m ImN ∞

Hθs =

∑

′ (γr ) + Bs,mK mN ′ (γr ) − (mNβ /(γ 2r )] {(jωεs / γ )[As,m ImN

m =−∞

(5.23)

′ (γr ) + Ds,mK mN ′ (γr )]}exp(jmNθ ), × [Cs,mImN where the subscripts s = 1, 2, 3 refer to the regions 1, 2, 3, respectively, as shown in figure 5.4. The boundary conditions at the interfaces between the three regions may be written as (figure 5.4) [40]:

5-8


Ez1 − Ez 2 = 0 (r = a ; −ϕ /2 ⩽ θ ⩽ ϕ /2)

(5.24)

Ez1 + Eθ1 cot ψ = 0 (r = a )

(5.25)

Ez 2(3) + Ez 2(3) cot ψ = 0 (r = a )

(5.26)

Hz1 tan ψ + Hθ1 = Hz 2 tan ψ + Hθ 2 (r = a ; −ϕ /2 ⩽ θ ⩽ ϕ /2)

(5.27)

Hz1 tan ψ + Hθ1 = Hz3 tan ψ + Hθ3 (r = a ; ϕ /2 ⩽ θ ⩽ (2π / N ) − ϕ /2).

(5.28)

Further, the boundary conditions at the metal envelope radius r = b of the structure may be written as Ez2(3) = 0 (r = b) (5.29)

Eθ 2(3) = 0 (r = b).

(5.30)

We can express the field constants in the field expressions (5.20)–(5.23) in terms of a single constant, namely, A1,m , with the help of the boundary conditions (5.24)–(5.26). Then we can multiply the boundary condition (5.27), in which the field expressions in terms of A1,m are put, by exp( −jqNθ ) and integrate it between the limits −ϕ /2 and ϕ/2. Next, we can then add the integration so obtained to what is similarly obtained by multiplying the boundary condition (5.28) also by exp( −jqNθ ) and integrating it between the limits ϕ/2 and (2π /N ) − ϕ /2 (figure 5.3). The subsequent result may be expressed as [40]: m(≠ q ) =∞

αqA1,q +

∑

δm,qA1,m

(5.31)

m(≠ q ) =−∞

put in two parts—one corresponding to m = q and the other to m ≠ q , where q is an integer. αq and δm.q are the functions of the structure parameters, the expressions for which can be found in [40]. Further, corresponding to the lower order values of the integers q and m: (i) q = 0, m = 1, m = −1 (ii) q = 1, m = 0, m = −1 and (iii) q = −1, m = 0, m = 1, the following set of three equations, respectively, can be obtained from (5.31):

α0A1,0 + δ1,0A1,1 + δ−1,0A1,−1 = 0

(5.32)

δ 0,1A1,0 + α1A1,1 + δ−1,1A1,−1 = 0

(5.33)

δ 0,−1A1,0 + δ1,−1A1,1 + α−1A1,−1 = 0.

(5.34)

Putting the condition for the non-trivial solutions of (5.32)–(5.34) that the determinant formed by the coefficients of field constants in their left-hand sides equals null yields, after algebraic simplification while using the relations δ1,0 = δ−1,1, δ0,1 = δ0,−1 and δ1,−1 = δ−1,1 [40], the following expression:

α0α1α−1 − α0δ1,2−1 + δ 0,1δ1,0(2δ1,−1 − α1 − α−1) = 0.

5-9

(5.35)


However, ignoring the last two terms of the left-hand side of (5.35), which each become insignificant with respect to its first term in typical practical situations [40], we obtain:

α0α1α−1 = 0.

(5.36)

The first of the three possible solutions (α0 = 0) appreciably differs in a wide range of frequencies from the remaining two and is of interest to TWT applications [40]. Putting α0 = 0, we obtain the dispersion that is identical with the heuristically obtained dispersion relation (5.18) with exactly the same interpretation of D lf and εr,eff as (5.19) and (5.17), respectively. Modeling of finite helix thickness Of the various approaches tested to take into account the effect of finite helix thickness in the model for analysis [55–44], the one by Swift-Hook [41] proves to be simple and useful in predicting the dispersion characteristics of the helix in agreement to experiment, for instance, in [56]. The approach is to take an additional freespace medium of half the helix thickness between the sheath at the mean helix radius and the beginning of the dielectric medium representing the helix supports [41]. This amounts to adding four field constants and correspondingly the same number of boundary conditions in the analysis of the structure. Consequently, this will modify the expression for D lf (5.19) which occurs in the dispersion relation (5.18) involving another helix parameter, namely, the outer radius of the helix a 0, such that a 0 − a is equal to half the helix thickness. Modeling of dielectric helix-supports deviating from a wedge cross section The helix becomes non-homogeneously loaded if the dielectric helix-supports deviate from the simple wedge geometry, for instance, of a circular, rectangular or T-shaped cross section. One can model such supports by smoothing them out into n homogeneous equivalent dielectric tube regions giving n + 2 regions in the equivalent analytical model including the free-space region inside the helix (0 ⩽ r ⩽ a ) and the free-space gap region between the sheath helix and the beginning of the dielectric (a ⩽ r ⩽ a 0 ) to account for the finite helix thickness [42] (figure 5.4). The expression for the effective relative permittivity of the pth dielectric tube (3 ⩽ p ⩽ n + 2)) is [30–21]

εr,eff,p =

(εr )(As,p ) + (1)(Ap − As,p ) As,p = 1 + (εr − 1) (3 ⩽ p ⩽ n + 2) (As,p ) + (Ap − As,p ) Ap

(5.37)

where Ap is the cross-sectional area of the pth equivalent dielectric tube and As,p is the area occupied by the discrete supports each of relative permittivity εr within the pth dielectric tube, the first dielectric tube being the third region of the model (figure 5.3(b)), the first two regions being the free-space regions for one has to interpret εr,eff,1 = εr,eff,2 = 1 corresponding to p = 1 and 2, respectively.

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The dispersion relation of the structure of the form (5.18) can be found, say, by field analysis following the procedure used in deducing (5.3). For the sake of simplicity, one can take the radial propagation constant to be the same as γ = (β 2 − k 2 )1/2 over all the structure regions of the model and use 4n + 6 boundary conditions—(i) four (4) boundary conditions at the sheath helix radius r = a(=( b1); (ii) two (2) boundary conditions at the conducting metal envelope radius r = b(=bn+2 ); (iii) four (4) boundary conditions arising from the continuity of the axial and azimuthal components of the electric field as well that of the axial and azimuthal components of the magnetic field at the inner radius of the first equivalent dielectric tube or the outer radius of the actual helix: r = b2 = a 0 ; and (iv) 4 × n − 1 boundary conditions at n − 1 interfaces between n equivalent dielectric tube regions of the structure [4 + 2 + 4 + 4 × (n − 1) = 4n + 6 boundary conditions]. The number of dielectric tube regions n can be increased until the converging results are obtained with respect to the value of the phase velocity from the dispersion relation of the structure. Effect of non-uniformity of radial propagation For the sake of rigor in the analysis, one needs to take into account the effect of nonuniform radial propagation constant over the structure’s cross section taking γp = (β 2 − k 2 )1/2 = γ (p = 1, 2) and γp = (β 2 − k 2εr,eff,p )1/2 (3 ⩽ p ⩽ n + 2). For this purpose, we can define the following wave impedance functions normalized with respect to the intrinsic impedance of the free space η0 = μ0 /ε01/2 [9, 10]:

ZE ,p =

⎫ ⎪ ⎪ ⎪ ⎬. ⎛ k ⎞ ⎛ I1(γ r ) − (Dp / Cp)K1(γ r ) ⎞⎪ p p ⎟⎟⎪ = −⎜⎜ ⎟⎟ ⎜⎜ ⎝ γp ⎠ ⎝ I0(γpr ) + (Dp / Cp)K 0(γpr ) ⎠⎪ ⎭

⎛ γp ⎞ ⎛ I0(γpr ) + (Bp / Ap )K 0(γpr ) ⎞ j Ez,p ⎟ ⎟⎜ =⎜ η0 Hθ,p ⎝ kεr,p ⎠ ⎜⎝ I1(γpr ) − (Bp / Ap )K1(γpr ) ⎟⎠

j Eθ,p ZH ,p = − η0 Hz,p

(5.38)

With the help of (5.38) and the boundary condition (5.2(d)) we can write

⎛ I (γa ) ⎞⎛ (B2 / A2 )−1 + K 0(γa )/ I0(γa ) ⎞1/2 k cot ψ = −⎜ 0 ⎟⎜ ⎟ , γ ⎝ I1(γa ) ⎠⎝ (D2 / C2 )−1 − K1(γa )/ I1(γa ) ⎠

(5.39)

in which the expressions for B2 /A2 and D2 /C2 can be obtained from the expressions for Bn+2 /An+2 and Dn+2 /Cn+2 . In order to find the expressions for Bn+2 /An+2 and Dn+2 /Cn+2 for this purpose we first write, with the help of the boundary conditions (5.29) and (5.30), interpreted here as Ez,n+2 = Eθ,n+2 = 0 at the metal envelope r = bn+2 = b, the following expression:

I0(γn+2b) ⎫ Bn+2 =− ⎪ An+2 K 0(γn+2b) ⎪ ⎬. I1(γn+2b) ⎪ Dn+2 = Cn +2 K1(γn+2b) ⎪ ⎭

5-11

(5.40)


We can then transfer (5.40) one step inward by using the boundary conditions that Ez,p, Eθ,p , Hz,p and Hθ,p are each continuous at the outermost dielectric–dielectric interface r = bn+1, interpreted with the help of the definition of impedance functions (5.38), as follows:

ZE ,(n+2) = ZE ,(n+1) ⎫ ⎬. ZH ,(n+2) = ZH ,(n+1) ⎭

(5.41)

The expression (5.41) gives the expressions for Bn+1/An+1 and Dn+1/Cn+1 in terms of Bn+2 /An+2 and Dn+2 /Cn+2 , respectively, the latter already obtained as (5.40). The procedure with another step inward at the interface r = bn similarly yields the expressions for Bn /An and Dn /Cn in terms of Bn+1/An+1 and Dn+1/Cn+1, respectively, obtained as above. Thus, we can carry out a similar procedure progressively inward until we finally obtain the explicit expressions for B2 /A2 and D2 /C2 required to be put in (5.39) to obtain the dispersion relation of the structure that takes into account the different radial propagation constants in the different structure regions. The dispersion characteristics so obtained considering this non-uniformity of radial propagation constant differ significantly from those obtained ignoring this effect, for a higher helix pitch angle ψ (=tan−1 p′/(2πa )) and for a wide range of values of εr,eff,3 and b /a [8]. Modeling of a vane-loaded metal envelope One can find the dispersion relation of a helix in a metal envelope provided with metal vanes (figure 5.2(a)) using the equivalent-circuit approach as has been done for a helix in free-space in the form of (5.16) with the help of the expressions for C and L given by (5.14) and (5.15), respectively. For a vane-loaded helix with a large number of vanes of infinitesimal thickness in a model, called the infinite number of vanes (INV) model [45], one can find the expression for C by assuming that the axial electric field gets shielded at the radius r = b′, say, of the vane tips. In the INV model, similarly, one can find the expression for L by assuming that the azimuthal electric field is unaffected by the presence of vanes and is shielded at the metal envelope radius r = b. This simple model gives the dispersion relation of the structure involving only the radial dimension of the vanes. A rigorous field analysis due to Kravchenko et al [45] that takes into account the azimuthal harmonics generated arising from the angular periodicity of metal vanes gave the dispersion relation of the structure revealing the effect of the number and angular and radial dimensions of vanes on the dispersion characteristics. However, since the analysis becomes highly involved and lengthy, a simpler empirical model, called the modified infinite number of vanes (MINV), was suggested [46]. The MINV model is based on the penetration of the axial electric field in the inter-vane region: Ezv(r ) relative to its value at the vane tip Ezv(b′) obtainable from [45], which can be expressed considering only the dominant first angularharmonic component as a first order of approximation as follows [46]:

Ezv(r ) K (γr )Iν(γb) − K ν(γb)Iν(γr ) ⎫ = ν ⎪ Ezv(b′) K ν(γb′)Iν(γb) − K ν(γb)Iν(γb′) ⎬ (b′ ⩽ r ⩽ b), ⎪ ν = 0.5nv /[1 − (nvϕ / π )] ⎭

5-12

(5.42)


where nv is the number of vanes and ϕv is half the angular thickness of each vane. One can thus estimate the order of penetration of the axial electric field in the intervane region with the help of (5.43). Thus, one can suggest with the help of (5.42) a new value r = b″ = b′ + Δ in the MINV model beyond which the axial electric field amplitude in the inter-vane region becomes insignificant relative to its value at the vane tips r = b′, where Δ is a function of nv and ϕ. Hence, in the MINV model, the dispersion relation can be obtained by replacing the vane-tip radius b′ of the INV model with b″. It has been found in [46] that by assigning Ezv(r )/Ezv(b′) ≈ 0.55 typically, in (5.42), we can obtain the value of b″, which will give the dispersion characteristics in the MINV model, reasonably agreeing with a prediction by a rigorous though highly involved analysis in [45]. Effect of structure losses The expression for the dielectric loading factor D lf , for instance, (5.19) appearing in the dispersion relation (5.18) of the structure, gets modified due to the loss in the helix and envelope materials as [10]:

⎫ ⎪ ⎪ β(material loss) = β [ [(1 + (R / ωL )2 )1/2 + 1]/2]1/2 ⎬ , ⎪ R = ωL[(1 + 2α 2 /(ω 2LC ))2 − 1]1/2 ⎪ ⎭

D lf (material loss) = D lf (β / β(material loss))

(5.43)

where β(material loss) is the axial propagation constant modified due to the loss [28]. L is obtainable by equivalent-circuit analysis (see, for example, (5.15)). R is the series resistance per unit length of the structure [10]. α —being the ratio of the power loss per unit length Ploss/length of the structure to twice the power transmitted through the structure—appearing in (5.43) is the attenuation constant of the structure due to the loss caused by the finite values of the resistivity of the materials making the helix and the envelope, the helix loss being more significant than the envelope loss [10]. Ploss/length results from the resistance offered to currents in directions parallel and perpendicular to the surface of the tape forming the helix, which results from the discontinuity of magnetic fields at the inner and outer tape surfaces as well as at the inner surface of the metal envelope [10]. Similarly, we can find the expression for D lf (atten coating), which is D lf modified in the presence of attenuator coating provided on the dielectric helix-support rods to prevent the TWT from oscillation caused by reflections of RF waves at the input and output ends of the structure [47]. The method for finding the desired expression for D lf (atten coating) involves an estimate of the surface current densities, in terms of the surface conductivity of coating, being the amounts of discontinuity of the axial and azimuthal magnetic field components [47]. Asymmetry of dielectric helix-support rods An analysis of the asymmetry of the dielectric helix-support rods with respect to their angular positioning around the helix and the rod-to-rod permittivity and/or cross-sectional area can be developed by the approach of Sinha reported by Joo et al [13], taking the wedge cross section of the supports. For this purpose, one can extend

5-13


Figure 5.5. Cross section of the region outside the helix divided into n number of wedge-shaped regions of different relative permittivity values and of different wedge angles [13].

Figure 5.6. Angular and permittivity off-sets in asymmetric structures [13].

the analysis considering angular harmonic effects for the supports enjoying symmetry as explained following (5.19) [40]. However, now we divide the cross section of the region outside the helix into n number of wedge-shaped regions of different relative permittivity values and of different wedge angles (figure 5.5) [13]. Further, we take n = 6 corresponding to three (3) dielectric regions and three (3) free-space regions over an angular extent of Θ = 2π and a single period (N = 1) (figure 5.6). Also, we take for angular offset: ∇θ ≠ 0 and ∇εr = 0 (εr1 = εr2 = εr3) and for support-permittivity offset: ∇θ = 0 and ∇εr ≠ 0 (εr1 = εr2 ≠ εr3), where εr1, εr2 and εr3 are the relative permittivities of the three helix-support rods (figure 5.6) [13]. The analysis predicts a stop-band in the ω − β dispersion plot of the structure at or near which the interaction impedance of the structure becomes large enough to result in a low starting beam current for oscillation, causing band-edge oscillations 5-14


thereby limiting the bandwidth of a helix TWT [2, 12, 13]. It is found that a distributed loss on the support rods can remove the stop band and stabilizes the device against such band-edge oscillations [2].

5.1.2 Tape-helix model The tape-helix model for the analysis of a helix due to Sensiper [37], unlike the sheath-helix model (section 5.1.1), is capable of taking into account the effect of space-harmonics generated due to the axial periodicity of helix turns and is valid even for a relatively small number of helix turns per guide wavelength. In this model, the actual helix is replaced by a tape of infinitesimal thickness that conducts in all directions [37]. It is usual to make the ‘narrow-tape’ approximation δ ≪ p′ in the analysis that is applicable to many practical situations [2, 5, 8, 34, 37, 38], where δ is the tape width and p′ is the helix pitch. We can apply Floquet’s theorem to write the expressions for RF fields in the helix while noting that the helix coincides with itself if either it is axially translated through its axial period p′ or axially translated through an arbitrary distance z′ < p′ and then rotated through an angle 2πz′/p′ (or, alternatively, it is rotated through an arbitrary angle θ < 2π and then translated through an axial distance p′θ /2π that makes the angular-harmonic number and the axial space-harmonic number n equal). Thus, we can write the expressions for the axial and azimuthal electric field components inside the helix as follows [11, 37]: ∞

Ez1(r , θ , z ) =

∑

A1n In(γn r ) exp ( −j 2πnz / p′)(exp − jβ0z )exp(jnθ )

(5.44)

⎡ − nβ ⎤ jωμ0 C1n In′(γn r )⎥ ⎢ 2 n A1n In(γn r ) − γn ⎣ γn r ⎦ n =−∞

(5.45)

n =−∞ ∞

Eθ1(r , θ , z ) =

∑

× exp ( −j 2πnz / p′)(exp − jβ0z )exp(jnθ ) × exp ( −j 2πnz / p′)(exp − jβ0z )exp (jnθ ), where

γn = ( βn2 − k 2 )1/2 ⎫ ⎬ βn = β0 + 2π n / p′⎭

(5.46)

and the subscript 1 represents the quantity inside the helix, and the prime over the modified Bessel function represents its derivative with respect to its argument. Dispersion relation of a helix in free space in the tape-helix model Taking δ ≪ p′ (narrow-tape approximation), let us then take the tape surface current density to be predominantly along the helix winding direction, presumably caused by the electric field along this direction. Hence, with the help of (5.44) and

5-15


(5.45), we can write the expression for the electric field component parallel to the helix winding direction E// (=Eθ cos ψ + Ez sin ψ ) at r = a as [2, 11]: ∞ ⎡⎛ ⎞ − nβ E1,//(a ) = ∑ ⎢⎜⎜ 2 n In{γna}cos ψ − In{γna}sin ψ ⎟⎟A1n ⎠ ⎣⎝ γn a n =−∞⎢

⎛ jωμ ⎞ ⎤ 0 −⎜ In′{γna}cos ψ ⎟C1n⎥ ⎝ γn ⎠ ⎥⎦

(5.47)

× exp( −jβ0z )exp − jn((2πz / p′) − θ ). where A1,n and C1n are the field constants that can be found with the help of the boundary conditions (5.5)–(5.11). These constants can be found in terms of the Fourier components of the axial and azimuthal tape surface current densities Jsz = Iza /(2πa ) and Jsθ = Iθa /(2πa ), respectively, as follows [14, 37]: ⎫ γ 2aK n(γna ) [Jˆs //,n sin ψ − (nβ / γn2a )Jˆs //,n cos ψ ]⎪ A1,n = − n jωε0 ⎬, (5.48) ⎪ C1n = −γna K n′(γna ) Jˆs //,n cos ψ ⎭ the hat over a quantity (here, surface current density Js ) representing its amplitudes. Jˆs //,n can be found as the nth Fourier component assuming a current distribution over the tape such that the amplitude of the tape surface current density remains constant at a value J over the tape width while its phase varies along the centerline of the tape winding as exp − (jβ0p′θ /(2π )), where β0 is the fundamental axial phase propagation constant using [2, 37]. Thus, we get ⎛ sin(β δ /2) ⎞⎛ δ ⎞ n Jˆs //,n = J ⎜ ⎟⎜ ⎟ . (5.49) ⎝ βnδ /2 ⎠⎝ p′ ⎠ Therefore, setting the boundary condition E// = 0 along the centerline of the tape surface r = a , which is expected to be valid over the entire tape surface as well, under the narrow-tape approximation, we obtain—with the help of (5.47) to be read in conjunction with (5.48) and (5.49)—the following dispersion relation of a helix in free-space in the tape-helix model: ∞

∑

[Mn(γna ) + Nn(γna )]

n =−∞

sin(βnδ /2) βnδ /2

=0

(5.50)

(dispersion relation of a helix in free-space obtained in the tape-helix model) where ⎛ nβ a ⎞2 n (5.51) cot ψ − γna⎟ In(γna )K n(γna ) Mn(γna ) = ⎜ ⎝ γna ⎠ (5.52) Nn(γna ) = k 02a 2 cot2 ψ In′(γna )K n′(γna ).

5-16


Dispersion relation of a loaded helix in the tape-helix model The analysis in the tape-helix model leading to the dispersion relation of a helix loaded with dielectric supports in a metal envelope is similar to that which has yielded the dispersion relation (5.50) of a helix in free space. However, the analysis now becomes somewhat involved in that one has to deal with 4n + 6 boundary conditions as discussed while modeling dielectric helix-supports deviating from of wedge cross section [8, 9]. Alternatively, one can use an elegant heuristic method in which to combine the dispersion relation of a ‘loaded helix’ in the ‘sheath-helix model’ with the dispersion relation of a ‘helix in free space’ in the ‘tape-helix model’ to obtain the dispersion relation of a ‘loaded helix’ in the ‘tape-helix model’ [48–50]. For this purpose, let us choose to express the dispersion relation (5.18) of a loaded helix in the sheath-helix model, putting β , γ and D lf as β0 , γ0 and Dlf0, respectively, as follows: 2 M0(γ0a )D lf0 (γ0a ) + N0(γ0a ) = 0

(5.53)

(dispersion relation of a loaded helix in the sheath-helix model), where γn , M0(γ0a ) and N0(γ0a ) are given by (5.46), (5.51) and (5.52), respectively, putting therein n = 0. Combining (5.50) with (5.53) one can intuitively write the following dispersion relation for a loaded helix in the tape-helix model [48–50]: ∞

∑

[Mn(γna )D lf2n(γna ) + Nn(γna )]

n =−∞

sin(βnδ /2) =0 βnδ /2

(5.54)

(dispersion relation of a loaded helix in the tape-helix model), where D lfn(γna ) is obtained by replacing 0 with n in the expression for the dielectric loading factor D lf0(γ0a ), which is the same as the expression for D lf in the dispersion relation of a loaded helix in the sheath-helix model in which γ is interpreted as γ0. Further, here, one has to take care to replace the modified Bessel functions and their derivatives with respect to arguments: I0(γ0a ), K 0(γ0a ), I0′(γ0a )(=I1(γa )) and K 0′(γ0a )(=−K1(γa )) with Im(γma ), Km(γma ), Im′ (γma ) and K m′ (γma ), respectively [48–50]. Interestingly, the heuristic dispersion relation (5.54) agrees with that obtained in [10, 24] by a rigorous analysis of a loaded helix in the tape-helix model. 5.1.3 Interaction impedance The interaction impedance K of a helical structure measuring the amplitude of the RF axial electric field Ez0 at a suitable position in the beam cross section, which may be typically taken at the axis of the beam, and is obtainable in the sheath-helix model, is given by [2]

K=

2 E z0 , 2β 2P

(5.55)

where P is the power propagating down the structure model comprising n + 2 regions as explained earlier preceding (5.37) while modeling discrete dielectric supports of finite helix thickness. P is given by [2]

5-17


1 P = Re 2 +

a(= b1)

∫

1 Re 2

(Er1Hθ∗1 − Eθ1Hr∗1)2πrdr

0 b 2 (= a 0 )

∫

(Er 2Hθ∗2 − Eθ 2Hr∗2 )2πrdr

(5.56)

b1(= a ) bp

n+2

1 + ∑ Re 2 p=3

∫

(ErpHθ∗p − EθpHrp∗ )2πrdr ,

bp −1

where the subscript p refers to the region of the structure model. p = 1 refers to the free-space region inside the helix and p = 2 to the free-space region of half the helix thickness outside the helix up to the beginning of the equivalent dielectric tube region. The subscripts p = 3 through p = n + 2 refer to n equivalent dielectric tube regions. r = b1 = a represents the sheath-helix radius (equal to the mean helix radius); r = b2 = a 0 represents the outer helix radius; and r = bn+2 = b represents the metal envelope radius. However, the expression (5.55) gets modified in the tape-helix model to be read as the nth space-harmonic interaction impedance Kn as follows [8]:

E z2,n

Kn =

,

∞

2βm2

∑

(5.57)

Pn

n =−∞

where Ez,n and Pn are the nth space-harmonic electric field amplitude and power transmitted through the structure, respectively. Further, in this nomenclature we can interpret the interaction impedance in sheath-helix model given by (5.55) as

K sheath−helix =

E z2,n 2β02P0

.

(5.58)

Therefore, with the help of (5.57) and (5.58) we can write the fundamental (n = 0) interaction impedance in the tape-helix model as [8]:

K 0 tape−helix =

E z2,0

=

∞

2β02

∑

Pn

E z2,0 2β02P0

n =−∞

P0

= K sheath−helixF ,

∞

∑

Pn

(5.59)

n =−∞

∞ P0 /∑n =−∞Pn

where F = is a reduction factor with respect to the interaction impedance obtainable by the analysis in the sheath-helix model. 5.1.4 Dispersion and interaction impedance characteristics The dispersion relation, for instance, (5.18) can be used to find k cot ψ /γ for a given value of γa . The former can be identified as the normalized phase velocity as follows: k cot ψ /γ = k cot ψ /( β 2 − k 2 )1/2 ≈ k cot ψ /β = ( ω /c ) cot ψ /(ω /vph ) = (vph /c ) cotψ. Further, we can multiply k cot ψ /γ thus obtained by γa to get the normalized 5-18


Figure 5.7. Comparison of theoretical dispersion characteristics of a helix supported by dielectric rods considering the effect of finite helix thickness (plots labeled as 1) and ignoring this effect (plots labeled as 2) with experiment [42], typically, for two structures labeled as I and II with all the relevant parameters given in [42].

frequency: (k cot ψ /γ ) × (γa ) = ka cot ψ = (ω /c )a cot ψ . Therefore, taking the different values of γa we can obtain the data to plot the normalized phase velocity k cot ψ /γ = (vph /c )cot ψ versus the normalized frequency ka cot ψ = (ω /c )a cot ψ (dispersion) characteristics. However, alternatively, if the helix dimensions are known we can interpret this plot as the normalized phase velocity vph /c versus frequency f (=ω /2π ) (dispersion) characteristics as well (figure 5.7). The tape-helix model revealed the appearance of allowed and forbidden zones in the dispersion characteristics caused by the periodicity of the helix. In the allowed zone, the phase velocity decreases with the tape width δ (figure 5.8(a)) and its value deviates from that predicted by the sheath-helix model and more so with the increase of the helix pitch (figure 5.8(b)). Furthermore, the dispersion characteristics predicted by the tape-helix model more closely agree with experiment than those by the sheath-helix model with respect to two typical helical structures in a metal envelope—one with rectangular (figure 5.9(a)) and the other with circular (figure 5.9(b)) dielectric helix-support rods [48]. Rao et al [53] developed a non-resonant perturbation theory of measurement to characterize a helical structure with respect to both the dispersion and interaction impedance-versus-frequency characteristics and hence also developed a computer-aided automated measurement setup based on this theory to obtain these characteristics experimentally. Their measurements [53] agreed with the theoretical prediction—more so in the tape-helix model than in the sheath-helix model [10] (figure 5.10).

5-19


Figure 5.8. Dispersion characteristics of a helix supported by dielectric wedge-shaped rods in a metal envelope in the tape-helix model taking (a) ψ = 10°, ϕ = 20°, N = 3, a = 1.0 mm, b /a = 2 , εr = 5.1 (APBN) taking the tape width as the parameter and (b) ψ = 20°, ϕ = 20°, N = 3, a = 0.75 mm, b = 1.6 mm, εr = 5.1 (APBN), the parameter p in the inset representing the helix pitch [2].

Figure 5.9. Dispersion characteristics of a helix in a metal envelope (a) using rectangular helix-supports with labels (i), (ii) and (iii) [55] and (b) using circular helix-supports with labels (iii), (ii) and (i) [52] to represent sheath-helix-model analysis, tape-helix-model analysis and experiment, respectively [48].

The INV model of a helical structure provided with a metal envelope has shown agreement with respect to dispersion characteristics with Scott’s experiment [54] using a relatively large number of thin vanes, typically 39 printed by photo-etching on quartz segments surrounding the helix between boron nitride helix-support rectangular rods (figure 5.11(a)). This agreement has been found in another study [48] to be closer in the tape-helix than in the sheath-helix model (figure 5.11(b)). However, for the dispersion characteristics of a helix with a vane-loaded metal envelope, one can go for the MNIV model [46] that has been found to be more 5-20


Figure 5.10. Comparison between the measurement by the non-resonant perturbation technique [53] and the analyses in the sheath-helix and tape-helix models [26] of a typical helical structure with respect to (a) dispersion and (b) interaction impedance-versus-frequency characteristics—solid lines with circles and rectangles in the latter representing the measured values including and excluding, respectively, the effect of space harmonics in the correction factor in the measurement formula [53]. The structure under measurement consists of a helix of mean radius 1.09 mm, outer radius 1.18 mm, tape width 0.76 mm and pitch angle 10.32°; three dielectric (APBN) helixsupport rods each of thickness 0.59 mm; and a metal envelope of radius 1.78 mm [53].

Figure 5.11. Agreement of the INV model with respect to the dispersion characteristics of a helix with rectangular supports: (a) in the sheath-helix model with Scott’s experiment, taking b /a = 2.52 , b′/a = 1.35, a 0 /a = 1.03, εr = 4 (boron nitride supports) [54], and (b) in the comparison study with Putz and Cascone’s experiment [51], labeling (i), (ii) and (iii) in the latter to represent the results of experiment [51], tape-helix model of analysis [48] and sheath-helix model of analysis [48], respectively, for a vane-loaded structure.

accurate than the INV model [2] in predicting results agreeing with those of the rigorous field analysis of Kravchenko et al [45] taking into account the finite number and dimensions of the vanes (figure 5.12). The interaction impedance versus frequency characteristics is as important as the dispersion characteristics since the value of the interaction impedance should not

5-21


Figure 5.12. The dispersion characteristics of a vane-loaded helical structure obtained by the INV and MINV models compared with those obtained by the rigorous analysis of Kravchenko et al [45] taking the parameters as nv = 39 , ϕv = π /24 , b /a = 1.9 , b′/a = 1.40 , a 0 /a = 1, εr = 1 [46].

deteriorate while designing a helical structure, say, for the desired nearly flat dispersion characteristics by vane loading the envelope or by tapering the cross section of the dielectric helix-supports for widening the bandwidth of a TWT. The analysis of a helix supported by dielectric rods in a metal envelope in a typical study [9] shows the need for adding the effect of non-uniformity of the radial propagation constant for the structure in which the discrete dielectric rods are smoothed out into a number of continuous homogeneous dielectric tubes of effective relative permittivity values (figure 5.13)—in particular for higher values of c /a (that is, for larger metal envelope radius relative to the mean helix radius) (figure 5.12(a) and (b)) and for lower values of cot ψ (that is, for higher values of helix pitch relative to the mean helix radius) (figure 5.12(c) and (d)). Furthermore, the analysis discussed at the end of section 5.1.1 shows the appearance of the stop band in the frequency versus phase shift dispersion characteristics caused by the asymmetry of the dielectric helix-support rods (figure 5.14). The width of the stop band is found to depend on the angular offset angle Δθ of the support rods taking Δεr = 0 (εr1 = εr2 = εr3) (figure 5.14(a)), as well as on the support-permittivity offset Δεr(εr1(=εr2 ) ≠ εr3) taking Δθ = 0 (figure 5.14(b)) (the parameters Δθ and Δεr having been explained in figure 5.7).

5.2 Analysis of fast-wave disc-loaded waveguide interaction structures In order to analyze a RF structure for its possibility to be a potential interaction structure for a MWT, it is very common to examine the interaction structure for wide bandwidth and/or higher gain. For this purpose, one may choose two

5-22


Figure 5.13. (a) Dispersion and (b) interaction impedance versus frequency characteristics taking c /a as the parameter; and (c) and (d) for the same two characteristics, respectively, taking cot ψ as the parameter, considering radial propagation constants to be non-uniform (γ3 ≠ γ4 ≠ ⋯ ≠γn+2 ) (solid line) and uniform (γ3 = γ4 = ⋯ =γn+2 ) (broken line) over the equivalent dielectric tube regions [9].

characteristics, namely dispersion and interaction impedance characteristics, to study the electron beam-absent RF behavior of the axially periodic structures. In the former characteristics, we look for a straight line portion of dispersion characteristics over a wide range of frequency in order to achieve a wideband coalescence of this dispersion characteristics with that of beam-mode for a wideband device performance. In the later characteristics, we may look for higher interaction impedance in the wideband frequency range that may lead to higher device gain. 5.2.1 Steps for obtaining dispersion relation/characteristics In this section, we summarize in general the various steps to obtain the dispersion relation/characteristics of an axially periodic structure. (i) Define the structure parameters and analytical regions for the analysis. 5-23


Figure 5.14. Stop-band characteristics of (a) the angularly asymmetric structure for offset angles Δθ = 0°, 10°, 20° labeled in the curves as A, B, C, respectively and (b) the dielectric asymmetric structure for relative permittivity offsets Δεr = 0% (εr = 6.5), Δεr = 10% (εr = 7.15), Δεr = 20% (εr = 7.8) (εr1(=εr2 ) ≠ εr3) labeled in the curves as A, B, C, respectively. (The other structure parameters are: a = 0.7 mm, b = 1.5 mm (envelope radius), εr = 6.5 mm, ϕ = 600 600 ) [53].

(ii) Define the phase propagation constants and type of harmonics (space or modal) in different regions based on the supporting waves (propagating or standing). (iii) Define the radial/transverse propagation constants for the slow-wave regime: phase propagation constant > free-space propagation constant; and for the fast-wave regime: free-space propagation constant > phase propagation constant. (iv) Consideration of modes (TE, TM or hybrid) based on the beam-wave interaction mechanism. (v) Write the relevant field intensity components. (vi) Define the relevant boundary conditions. (vii) Substitute the relevant field intensity components into the relevant boundary conditions. (viii) Do algebraic manipulation: multiply the resultant relations by trigonometric or exponential functions, integrate the relation within region of validity to get an infinite number of simultaneous equations in one of the field constants. (ix) The condition for a non-trivial solution of these simultaneous equations yields the determinantal form of dispersion relation of the structure. (x) Truncate the infinite order of the determinant and infinite summation in the equations to a sufficiently large number to find a converging solution (xi) Solve the dispersion relation and use the roots to plot the dispersion characteristics of the structure. (xii) Validate analytically in special cases of the structure and against the results obtained using some of the other analytical techniques and against that obtained using numerical simulation tools.

5-24


5.2.2 Steps for obtaining interaction impedance characteristics Similar to section 5.2.1, here we summarize the various steps to obtain the interaction impedance characteristics of an axially periodic structure. (i) Choose the structure parameters and corresponding roots of the dispersion relation. (ii) Choose the type of interaction impedance to be calculated: (i) axial interaction impedance Kz (for a linear beam device) or (ii) azimuthal interaction impedance K θ (for a gyrating beam device). (iii) Choose the space harmonic number for which the interaction impedance is to be calculated. (iv) Substitute the roots of the dispersion relation and space harmonic number into the expression for electric field intensity: axial Ez{r} (for a linear beam device) and azimuthal Eθ{r} (for a gyrating beam device), to get an expression for electric field intensity as a function of radial coordinate. (v) Choose r = 0 in the expression for Ez{r} (for a linear beam device); and plot Eθ{r} (for a gyrating beam device) against the radial coordinate and find out the radial position (r = rOpt ) for maximum azimuthal electric field intensity and substitute r = rOpt into the expression for Eθ{r}. (vi) Do algebraic manipulation to present all the field constants for different space harmonics in terms of the field constant for that space harmonic for which the interaction impedance is to be calculated. (vii) Calculate the total power transmitted through the structure by summing the contribution of each of the space harmonic component, obtained by integrating the average complex Poynting vector for the corresponding space harmonic over the cross-sectional area of the interaction structure. (viii) Substitute the relevant electric field intensity component, phase propagation constant and total power transmitted through the structure, contributed by all the space harmonic components, for a particular frequency into the expression for interaction impedance. (ix) Calculate the interaction impedance for various frequencies and plot the interaction impedance characteristics of the structure. (x) Validate analytically in special cases of the structure and against the results obtained using numerical simulation tools. 5.2.3 Models of axially periodic structures Here, we choose four variants of a disc-loaded circular waveguide for exploring their dispersion and interaction impedance characteristics. Infinitesimally thin metal disc-loaded circular waveguide: This interaction structure (figure 5.15) consists of a circular waveguide in which annular metal infinitesimally thin discs are arranged to maintain axial periodicity. The structure parameters are: the waveguide radius rW , the disc-hole radius rD , and the structure periodicity L (figure 5.15) [14, 15].

5-25


Figure 5.15. Infinitesimally thin metal disc-loaded circular waveguide [14, 15].

Figure 5.16. waveguide of finite disc-thickness [16–18].

Disc-loaded circular waveguide of finite disc-thickness: This interaction structure (figure 5.16) consists of a circular waveguide in which annular metal discs of finite thickness are arranged to maintain axial periodicity. The structure parameters are: the waveguide radius rW , the disc-hole radius rD , the disc-thickness T , and the structure periodicity L (figure 5.16) [16–18]. Interwoven-disc-loaded circular waveguide: The model of the structure considers a disc-loaded circular waveguide of varying inner radii of alternate discs. The discs are organized in the circular waveguide such that the disc of the bigger hole-radius is symmetrically in the middle of two alternate discs of the smaller hole-radii, and viceversa (figure 5.17). The structure parameters are: the waveguide radius rW , the holeradius of metal disc of smaller hole rSH , the hole-radius of metal disc of bigger hole rBH , the thickness of metal disc of smaller hole TSH , the thickness of metal disc of bigger hole TBH , and the structure periodicity L (figure 5.17). Thus, one may calculate the axial-gap between two consecutive discs (which is the distance between discs of smaller and bigger hole-radii) as (L − TSH − TBH )/2 [19]. Alternate dielectric and metal disc-loaded circular waveguide: The previous three variants of the disc-loaded circular waveguide are all-metal structures, and no consideration has been given to the presence of any dielectric region in the structure, which reportedly favors the wideband performance of a gyro-TWT. Also, the region between the two consecutive discs has been considered as a free-space region. However, in order to accrue the advantage of the dielectric loading in widening the device’s bandwidth, it would be worth examining the results of adding a dielectric region in the structure, for instance, in the form of dielectric discs interposed between 5-26


Figure 5.17. Interwoven-disc-loaded circular waveguide [19].

Figure 5.18. Alternate dielectric and metal disc-loaded circular waveguide [20].

the metal discs. This variant consists of a circular waveguide consisting of alternate dielectric and metal discs, such that the hole radius of the metal discs is less than that of the dielectric discs. That makes the disc-occupied region partly filled by dielectric. The structure parameters are: the waveguide radius rW , the hole radius of the metal disc rMD , the hole radius of the dielectric disc rDD , the thicknesses of the metal disc TMD , the thicknesses of the dielectric disc TDD , and the structure periodicity L(=TMD + TDD ) (figure 5.18) [20]. 5.2.4 Field intensities in structure regions For the sake of analysis, in general, the disc-loaded structures (figures 5.14–5.17) may be divided into two regions: (i) the central circular disc-free free-space region (region-I), and (ii) the disc-occupied region between two metal discs. The discoccupied free-space region may be referred to as region-II for the infinitesimally thin metal disc-loaded circular waveguide (figure 5.15) and the disc-loaded circular waveguide of finite disc-thickness (figure 5.16) [16–18]. On the other hand, for the interwoven-disc-loaded circular waveguide (figure 5.17) [19] and the alternate dielectric and metal disc-loaded circular waveguide [20] (figure 5.18), the discoccupied region may further be divided into two regions. For the interwoven-discloaded circular waveguide (figure 5.17), one may refer to the free-space region between two consecutive discs of smaller hole radius as region-II, and the free-space region between the discs of smaller hole radius and bigger hole radius as region-III. For the alternate dielectric and metal disc-loaded circular waveguide (figure 5.18), 5-27


one may refer to the free-space region between two consecutive discs as region-II and the dielectric region between two consecutive discs as region-III [14–20]. Disc-free region It is assumed that the disc-free region supports propagating waves and space harmonics. Due to axial periodicity, the structure needs to follow Floquet’s theorem that makes the phase propagation constant of the nth space harmonic as βnI = β0I +2π n /L(n = 0, ±1, ±2, …. ±∞), which appears in the expression for the transverse propagation constant for region-I as γnI = [k 2 − (βnI )2]1 2 for all of the four structures (figures 5.14–5.17). Therefore, the relevant TE mode (Ez = 0) field expressions considering propagating waves in the disc-free free-space region are given by [14–20]: +∞

HzI

∑

=

+∞

HzI,n

n =−∞

∑

1 I I I I An J0′{γn r}exp j (ωt − βn z ) γ n =−∞ n

(5.61)

+∞

E θI,n = jωμ0

n =−∞ +∞

HrI =

(5.60)

n =−∞

+∞

E θI =

AnI J0{γnI r}exp j (ωt − βnI z )

∑

=

∑ n =−∞

∑

+∞

HrI,n = −j

∑

βnI

γI n =−∞ n

AnI J0′{γnI r}exp j (ωt − βnI z )

(5.62)

Disc-occupied region It is assumed that the disc occupied region supports stationary wave and modal harmonics. Due to stationary wave formation between the metal discs, the phase propagation constant may be defined as βmII = mπ /(L − T ), (m = 1, 2, 3, …. ∞), (L − T ) being the distance between the metal discs for the disc-loaded circular waveguide of finite disc-thickness, which may be written as βmII = mπ /L for the infinitesimally thin metal disc-loaded circular waveguide [14–20], βmII= mπ /(L − TSH ) for the interwoven-disc-loaded circular waveguide [19], and βmII = mπ /(L − TMD ) for the alternate dielectric and metal disc-loaded circular waveguide [20]. Then, the transverse propagation constant in region-II may be defined as γmII = [k 2 − (βmII )2]1 2 . Further, for region-III, the phase propagation constant may be defined as βmIII = mπ /(L − TSH − TBH ), (m = 1, 2, 3, …. ∞) for the interwoven-disc-loaded circular waveguide (figure 5.17) [19], and βmIII = mπ /(L − TMD ) for the alternate dielectric and metal disc-loaded circular waveguide (figure 5.18) [20]. Then, the transverse propagation constant in region-III may be defined as γmIII = [εrIII k 2 − (βmIII )2]1 2 ; εrIII is the relative permittivity of the disc-occupied dielectric region, which takes unity (free-space) value εrIII = 1 for the interwovendisc-loaded circular waveguide (figure 5.17) and non-unity (dielectric) value εrIII ≠ 1 for the alternate dielectric and metal disc-loaded circular waveguide (figure 5.18). Therefore, the relevant TE mode (Ez = 0) field expressions considering stationary waves in disc-occupied region(s) are given by [14–20]: 5-28


Infinitesimally thin metal disc-loaded circular waveguide and a disc-loaded circular waveguide of finite disc-thickness [14–18]: In region-II: ∞

HzII =

∞

∑ HzII,m = ∑ AmII Z0{γmII r}exp(jωt )sin(βmII z ) m=1

∞

E θII =

(5.63)

m=1 ∞

1 II II II II Am Z0′{γm r}exp(jωt )sin(βm z ) γ m=1 m

∑ E θII,m = jωμ0 ∑ m=1

(5.64)

where Z0{γmII r} = J0{γmII r}Y0′{γmII rW } − J0′{γmII rW }Y0{γmII r} and Z0′ is obtained by taking the derivative of Z0 with respect to its argument. Interwoven-disc-loaded circular waveguide [19]: In region II: ∞

HzII =

∑ ⎡⎣AmII J0{γmII r} + BmII Y0{γmII r}⎤⎦exp(jωt )sin(βmz )

(5.65)

m=1 ∞

1 ⎡ II II II II ⎤ II ⎣Am J0′{γm r} + Bm Y0′{γm r}⎦exp(jωt )sin(βm z ) γ m=1 m

E θII = jωμ0 ∑

(5.66)

In region III: ∞

∑ AmIII Z0{γmIII r}exp(jωt )sin(βmIII z )

HzIII =

(5.67)

m=1 ∞

1 III III III III Am Z ′0 {γm r}exp(jωt )sin(βm z ) , γ m=1 m

E θIII = jωμ0 ∑

(5.68)

where Z0{γmIII r} = J0{γmIII r}Y0′{γmIII rW } − J0′{γmIII rW }Y0{γmIII r}. Alternate dielectric and metal disc-loaded circular waveguide [20]: In region II: ∞

HzII =

∑ ⎡⎣AmII J0{γmII r} + BmII Y0{γmII r}⎤⎦exp(jωt )sin(βmz )

(5.69)

m=1 ∞

1 ⎡ II II II II ⎤ II ⎣Am J0′{γm r} + Bm Y ′0 {γm r}⎦exp(jωt )sin(βm z ) . γ m=1 m

E θII = jωμ0 ∑

(5.70)

In region III: ∞

HzIII =

∑ AmIII Z0{γmIII r}exp(jωt )sin(βmz ) m=1

5-29

(5.71)


∞

E θIII

1

III III Am Z ′0 γ m=1 m

= jωμ0 ∑

{γmIII r}exp(jωt )sin(βmz),

(5.72)

where Z0{γmIII r} = J0{γmIII r}Y0′{γmIII rW } − J ′0 {γmIII rW }Y0{γmIII r}. 5.2.5 Relevant boundary conditions For the sake of analysis, one may write the boundary conditions: (i) arising from the vanishing tangential azimuthal component of the electric field at the conducting wall of the waveguide; (ii) arising from the continuity of the axial magnetic field between regions I and II; (iii) arising from the continuity of the azimuthal electric field between regions I and II; (iv) arising from the vanishing tangential azimuthal component of the electric field at the conducting inner circumferential surface of the metal discs; (v) arising from the continuity of the axial magnetic field between regions II and III; and (vi) arising from the continuity of the azimuthal electric field between regions II and III [14–20]. 5.2.6 Dispersion relation Further, one has to follow the steps (vii)–(ix) of section 5.2.1 to get the following determinantal dispersion relations of the structures: Infinitesimally thin metal disc-loaded circular waveguide [15]

det

I ⎡ J ′ γ Ir ⎤ 0 { n D} 1 J0{γn rD} ⎥ ⎢1 − II I 2 II II − (βnI ) ⎢⎣ γn Z ′0 {γm rD} γm Z0{γm rD} ⎥⎦

1

(βmII )2

=0

(5.73)

( −∞ < n < ∞ , 1 ⩽ m < ∞),

where Z0{γmII r} = J0{γmII r}Y0′{γmII rW } − J0′{γmII rW }Y0{γmII r}. Disc-loaded circular waveguide of finite disc-thickness [16–18]

det Mnm J0{γnI rD}Z0′{γmII rD} − Z0{γmII rD}J0′{γnI rD} = 0

(5.74)

where

Mnm =

γnI βmII 1 − ( −1)m exp⎡⎣ −jβnI (L − T )⎤⎦

(

)

γmII ⎡⎣βmII − exp( −jβ0I L ) βmII cos(βmII L ) + jβnI sin(βmII L ) ⎤⎦

(

)

.

Interwoven-disc-loaded circular waveguide [19]

det Mn m J0{γnI rSH } ⎡⎣J0′{γmII rSH } + ξ Y0′{γmII rSH }⎤⎦ −J0′{γnI rSH }⎡⎣J0{γmII rSH } + ξ Y0{γmII rSH }⎤⎦ = 0 ( −∞ < n < ∞ , 1 ⩽ m < ∞), (5.75)

5-30


where

Mnm =

ξ=

γnI βmII ⎡⎣1 − ( −1) m exp( −jβnI (L − TSH ))⎤⎦ γmII ⎡⎣βmII − exp( −jβ0I L )[βmII cos(βmII L ) + jβnI sin(βmII L )]⎤⎦

γpIII J0′{γmII rBH }Z0{γmIII rBH } − γmII J0{γmII rBH }Z0′{γmIII rBH } γmII Y0{γmII rBH }Z0′{γmIII rBH } − γmIII Y0′{γmII rBH }Z0{γmIII rBH }

,

,

Z0{γmIII r} = J0{γmIII r}Y0′{γmIII rW } − J0′{γmIII rW }Y0{γmIII r} . Alternate dielectric and metal disc-loaded circular waveguide [20]

det Mnm J0{γnI rMD}⎡⎣J0′{γmII rMD} + ξ Y0′{γmII rMD}⎤⎦

−J0′{γnI rMD}⎡⎣J0{γmII rMD} + ξ Y0{γmII rMD}⎤⎦ = 0 ( −∞ < n < ∞ , 1 ⩽ m < ∞) (5.76) where

Mnm =

ξ =

γnI βmII ⎡⎣1 − ( −1) m exp( −jβnI TDD)⎤⎦ γmII ⎡⎣βmII − exp( −jβ0I L )⎡⎣βmII cos(βmII L ) + jβnI sin(βmII L )⎤⎦⎤⎦ γmIII J0′{γmII rDD}Z0{γmIII rDD} − γmII J0{γmII rDD}Z0′{γmIII rDD} γmII Y0{γmII rDD}Z0′{γmIII rDD} − γmIII Y0′{γmII rDD}Z0{γmIII rDD}

Z0{γmIII r} = J0{γmIII r}Y0′{γmIII rW } − J0′{γmIII rW }Y0{γmIII r} . The dispersion relation of the infinitesimally thin metal disc-loaded circular waveguide [15], in special cases: (i) rD → rW , (ii) rW → rD , and (iii) L → 0 (densely populated discs, βmII (=mπ /L ) → ∞) passes on to that for smooth-wall circular waveguides of radii rW (special case I), rD (special case II), and rD (special case III), respectively. The dispersion relation of the disc-loaded circular waveguide of finite discthickness [16–18], in special cases: (i) rD → rW , (ii) rW → rD , and (iii) T → L (closely packed discs, βmII (=mπ /(L − T )) → ∞) passes on to that for smooth-wall circular waveguides of radii rW (special case I), rD (instead of rW ) (special case II), and rD (instead of rW ) (special case III), respectively; and in a special case: (iv) T → 0 passes on to that for the infinitesimally thin metal disc-loaded circular waveguide [15] (figure 5.15) (special case IV). The dispersion relation of the interwoven-disc-loaded circular waveguide [19] (figure 5.17), in special cases: (i) rSH = rBH → rW , (ii) rSH = rBH and TSH + TBH = L , (iii) rSH = rBH and TSH = TBH , (iv) rBH → rW , and (v) rSH → rW passes on to that for a

5-31


smooth-wall circular waveguide of radius rW (special case I), a smooth-wall circular waveguide of radius rBH (=rSH ) (special case II), the disc-loaded circular waveguide of finite disc-thickness of disc-hole radius rBH (=rSH ), disc-thickness TBH (=TSH ), and periodicity L/2 (special case III) [16–18], the disc-loaded circular waveguide of finite disc-thickness of disc-hole radius rSH , disc-thickness TSH , and periodicity L (special case IV) [16–18]; and the disc-loaded circular waveguide of finite disc-thickness of disc-hole radius rBH , disc-thickness TBH , and periodicity L (special case V) [16–18]. The special cases III–V with the further condition of TBH → 0 and TSH → 0 leads to the dispersion relation of the infinitesimally thin metal disc-loaded circular waveguide [14, 15], and which in the absence of the higher order harmonics passes to that published in [14]. The dispersion relation of the alternate dielectric and metal disc-loaded circular waveguide (figure 3.17), in special cases: (i) rMD = rDD → rW , (ii) TMD → L or TDD → 0, (iii) rDD → rW or εr = 1, and (iv) rDD → rW or εr = 1 (absence of dielectric) with TMD → 0 passes on to that for a smooth-wall circular waveguide of radius rW (special case I), a smooth-wall circular waveguide of radius rMD (special case II), a disc-loaded circular waveguide of finite disc-thickness of disc-hole radius rMD , discthickness TMD , and periodicity L (special case III) [16–18], and the infinitesimally thin metal disc-loaded circular waveguide of disc-hole radius rMD , and periodicity L (special case IV) [15]. In the absence of the higher order harmonics, the special cases IV further leads to the dispersion relation of the infinitesimally thin metal discloaded circular waveguide as published in [14]. 5.2.7 Azimuthal interaction impedance An estimate of the azimuthal electric field available for interaction with gyrating electrons in a gyro-TWT may be given by the azimuthal interaction impedance defined as [4, 16]:

K θ (r ) =

E θI 2(r ) , 2β 2Pt

(5.77)

where Pt is the total power transmitted through the structure. While calculating K θ (r ) of the infinitesimally thin metal disc-loaded circular waveguide, one has to calculate the power transmitted through the structure Pt(=∑∞ P ), contributed by all the n =−∞ n space harmonic components (n = 0, ±1, ±2, ..., ±∞), for which one may obtain an expression for Pn by integrating the average complex Poynting vector for the nth space harmonic over the cross-sectional area of the disc-free region I (0 ⩽ r < rD ) as:

Pn =

1 Re 2

rD

rD

∫r=0 (Εn × Η*n )z 2π r dr = π∫r=0 (ErI, nHθI,*n −EθI, nHrI,*n ) r dr.

(5.78)

The subscripts r and θ with the field intensity E (electric) and H (magnetic), respectively, represent the radial and azimuthal component of the corresponding field intensity. The superscript I represents the component of the field intensity in the 5-32


structure of region-I. It is assumed that there is no power flow in the disc-occupied region that presumably supports stationary waves, which, further, on the evaluation of the integration, yields the following expression for Pt : +∞

Pt =

∑

Pn = −πk η0

n =−∞

+∞ βnI I 2 2 I rD2 −1 ∑ An J1′ {γn rD} + ⎡⎣1 − (γnI rD ) ⎤⎦J12{γnI rD} , (5.79) 2 n =−∞ γnI 2

(

)

where η0 (=(μ0 /ε0 )1/2 ) is the intrinsic impedance of free-space. Thus, explicitly the azimuthal interaction impedance for the fundamental space harmonic component (n = 0) can be read as:

K θ,0{r} = η0

k R 012J0′2{γ0I r} βnI

+∞

π

β0I 2γ0I 2rD2

∑

( )

I n =−∞ γn

R 2 2 n1

(

−1 J1′ 2{γnI rD} + ⎡⎣1 − (γnI rD ) ⎤⎦J12{γnI rD}

)

,

(5.80)

where

R nm

II ⎡ I ⎤ II m I 1 γn Z0′{γm rD} βm ⎣( −1) exp(jβ0 L ) − 1⎦ = . ⎡ β I 2 − β II 2 ⎤ L γmII J0′{γnI rD} ( ) ( ) n m ⎣ ⎦

Interestingly, the expression for the azimuthal interaction impedance of the discloaded circular waveguide of finite disc-thickness becomes identical with that of the infinitesimally thin metal disc-loaded circular waveguide [4, 16]. The expressions for the azimuthal interaction impedance of the interwoven-disc-loaded circular waveguide and the alternate dielectric and metal disc-loaded circular waveguide can be obtained by replacing rD by rSH and rMD , respectively, in the expression for the azimuthal interaction impedance of the infinitesimally thin metal disc-loaded circular waveguide. 5.2.8 Structure characteristics Here we take two structure characteristics, namely, the dispersion and interaction impedance, to study the behavior of the structures. This is to explore the dependency of dispersion shaping with structure parameters. In order to widen the bandwidth of a gyro-TWT one has to widen the frequency range of the straight-line portion of the ω − β dispersion characteristics of the structure to ensure its grazing intersection or coalescence with the beam-mode dispersion line of the device [4, 14–28]. One has to optimize the structure parameters for widening the coalescence bandwidth, preferably near the waveguide cutoff, where the axial phase propagation constant of the structure tends to zero that would in turn make the Doppler shift rather small, in order to minimize the effect of beam velocity spread in the device [55]. While optimizing the structure parameters for widening the coalescence bandwidth due care has to be taken not to deteriorate the value of the azimuthal interaction impedance of the structure, as it may deteriorate the gain of the device. Realistically, from the standpoint of the application of the structure in a gyro-TWT, the gyrating 5-33


electron beam has to be placed at a location where the beam would experience maximum transverse RF field in the structure. Therefore, one should find the azimuthal interaction impedance of the structure at the radial position of the azimuthal electric field maxima. Dispersion characteristics The circular waveguide is inherently a high-pass filter with a single cutoff. The circular waveguide will exhibit alternate pass and stop bands with their respective higher and lower cutoffs when it is loaded periodically by the metal annular discs [39]. For a given mode (TE01, TE02 and TE03), the dispersion plots are found to be periodic, on the scale of normalized phase propagation constant, with the periodicity of β0I L = 2π . The extent of both the stop and pass bands, shown here on the krw scale, progressively decreases with the TE01, TE02 and TE03 modes. Further, while in the first pass band on the propagation scale β0I L , the RF group velocity (slopes of the krw versus β0I L dispersion plot) is positive for the TE01 or TE02 mode (fundamental forward wave mode), it is negative for the TE03 mode (fundamental backward wave mode) (figure 5.19). It is of interest to examine how the dependence of the structure dispersion characteristics, for typical mode TE01, on the disc-hole radius (figure 5.20) and the structure periodicity (figure 5.21) would change when the effect of the finite disc thickness (figure 5.16) is considered (T /rW ≠ 0) from when infinitesimally thin discs are considered. Further, with the decrease of either of the parameters, namely, the disc-hole radius and the structure periodicity, the lower and upper edge frequencies of the passband of the dispersion characteristics both increase, though to unequal extents, such that the passband decreases or increases according as the disc-hole radius decreases or the structure periodicity decreases, the mid-band frequency of the

Figure 5.19. Pass and stop band characteristics of the infinitesimally thin metal disc-loaded circular waveguide showing the pass and stop bands (including higher order harmonics n = 0, ±1, ±2, ±3, ±4, ±1; m = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11) [15].

5-34


Figure 5.20. Dispersion characteristics of the infinitesimally thin metal disc-loaded circular waveguide (T /rw = 0) (broken curve) and the disc-loaded circular waveguide of finite disc-thickness (T /rw = 0.1) (solid curve) taking the disc-hole radius as the parameter. For the parameter rD /rW = 1.0 , the broken curve with crosses refers to a smooth-wall circular waveguide [16–18].

passband however, shifting to a higher value for the decrease of both the parameters (figures 5.19 and 5.20). It is noted that although both the lower and the upper edge frequencies of the passband depend on the disc-hole radius for discs of finite thickness (figure 5.16), it is the lower edge frequency, and not the upper edge frequency, of the passband that depends on the disc-hole radius for infinitesimally thin discs (figure 5.20). Further, there exists an optimum value of the disc-hole radius corresponding to the maximum frequency range of the straight-line portion of the dispersion characteristics. The structure periodicity, however, is found to be more effective than the dischole radius in widening the frequency range of the straight-line portion of the dispersion characteristics, a decrease in the value of the structure periodicity causing an increase in the frequency range of the straight-line portion, however, accompanied by a shift of the range from the waveguide cutoff (figure 5.21). In the case of the infinitesimally thin metal disc-loaded circular waveguide, it is interesting to note that the upper-edge frequencies of the passband for different structure periodicity values relative to the waveguide-wall radius will all lie on the dispersion curve (hyperbola) of the smooth-wall circular waveguide (not shown) [15]. Thus, the dischole radius may be decreased (figure 5.21) and the structure periodicity increased (figure 5.22) for widening the device’s bandwidth. However, such broadbanding of coalescence is accompanied by the reduction of the bandwidth of the passband of the structure as well (figures 5.20 and 5.21). Similarly, with the decrease of the disc thickness, the lower and upper edge frequencies of the passband in the dispersion characteristics both decrease such that the passband first decreases and then increases passing through a minimum; and the mid-band frequency of the passband as well as the frequency corresponding to the beginning of the straight-line portion of the dispersion characteristics shifts to a 5-35


Figure 5.21. Dispersion characteristics of the infinitesimally thin metal disc-loaded circular waveguide (T /rW = 0) (broken curve) and the disc-loaded circular waveguide of finite disc-thickness (T /rW = 0.1) (solid curve) taking the structure periodicity as the parameter [15–18].

Figure 5.22. Dispersion characteristics of the disc-loaded circular waveguide of finite disc-thickness (T /rW = 0.1) (solid curve) taking the disc thickness as the parameter. For the parameter (T /rW = 0) (broken curve) refers to the infinitesimally thin metal disc-loaded circular waveguide [15–18].

lower value (figure 5.22). The dispersion characteristics taking the disc thickness as the parameter may be grouped for relatively thin and relatively thick discs (figure 5.22). The shape of the dispersion characteristics depends on the disc thickness, though not as much as it does on the disc-hole radius or the structure periodicity (figures 5.19–5.21) [15–18]. Similar to the disc-loaded circular waveguide of finite disc-thickness, both the holeradii (bigger and smaller) of the interwoven-disc-loaded circular waveguide are 5-36


Figure 5.23. Dispersion characteristics of the interwoven-disc-loaded circular waveguide taking the (a) bigger (rBH /rW ) and (b) smaller (rSH /rW ) hole-radii as the parameter [19].

responsive to dispersion shaping (figure 5.23). In general, the lower- and the uppercutoff frequencies increase with a decrease in hole-radii, however, in particular, the passband increases and decreases with a decrease of bigger and smaller hole-radii, respectively (figure 5.23). The disc-thickness of the bigger-hole-disc is neither very responsive for dispersion shaping, nor for the passband, however, the passband shifts to a higher frequency side (figure 5.24). One may use this nature for shifting the interaction band in order to achieve an optimum beam-wave interaction while designing a gyro-TWT with an interwoven-disc-loaded circular waveguide. The structure periodicity (figure 5.24) and the disc-thickness (figure 5.25) of the biggerhole-disc of the interwoven-disc-loaded circular waveguide are, respectively, the most and the least responsive for the passband as well as for dispersion shaping. In addition to dispersion shaping, which is required for designing a broadband gyro-TWT, the structure also holds the fascinating characteristic of an increase in passband with increase as well as with a decrease of disc-thickness of the smaller-hole-disc with reference to that of the bigger-hole-disc. Passband improvement can be achieved by varying the disc radii for a given beam-mode dispersion characteristic [19]. Similar to the disc-loaded circular waveguide of finite disc-thickness (figure 5.16), the structure periodicity of the interwoven-disc-loaded circular waveguide is the most responsive for the passband as well as for dispersion shaping (figure 5.24). While finding the straight-line portion of the dispersion characteristics of the structure, one may plot the slope of the dispersion (ω − β ) characteristics, which generally gives the group-velocity (vg /c ), versus frequency (figure 5.26). In the plot, for the broadband performance, one may look for region of constant group-velocity, which is basically a replica of the straight-line portion of the dispersion characteristics. The peak of the curve thus generated would typically correspond to the axial beam velocity required for beam-wave synchronism. While examining figure 5.26, one may find the results for the disc-loaded circular waveguides of constant disc-hole radii at two extreme ends. Here, it is necessary to point out that the disc-loaded circular waveguides presented at two extreme ends differ in periodicity. However, 5-37


Figure 5.24. Dispersion characteristics of the interwoven-disc-loaded circular waveguide taking the structure periodicity (L /rW ), as the parameter [19].

Figure 5.25. Dispersion characteristics of the interwoven-disc-loaded circular waveguide taking the discthickness of (a) bigger-hole-disc (TBH /rW ) and (b) smaller-hole-disc (TSH /rW ) as the parameter [19].

leaving the right-most broken curve apart, all other curves correspond to the same periodicity. One may observe the increase in the region of constant group-velocity with a decrease in the bigger hole-radius, however, with a shift of the frequency band (figure 5.26). The corresponding shift of the frequency band may be compensated with a change in either the waveguide radius or disc-thickness of the bigger-holedisc, or both. For examining the effect of the structure parameter of all-metal variants of the axially periodic structure, we choose only the lowest order azimuthally symmetric mode, however, while studying the alternate dielectric and metal disc-loaded circular 5-38


Figure 5.26. Normalized group-velocity versus normalized frequency characteristics of the interwoven-discloaded circular waveguide, taking a bigger hole-radius (rBH /rW ) as the parameter. Special cases leading to the disc-loaded circular waveguide of the constant disc-hole radius (figure 5.17) (broken curve) [19].

waveguide (dielectric loaded structure) we choose first three lowest order azimuthally symmetric modes. For the TE01 mode, with the increase of the relative permittivity of the dielectric discs, the lower and upper cutoff frequencies shift to a lower value, however, quantitatively the shift in the upper cutoff frequency is higher than that of the lower cutoff frequency, which in turn shortens the passband (figure 5.27(a)). For the TE02 mode, with the increase of the relative permittivity of the dielectric discs, the lower and upper cutoff frequencies shift to a lower value, however, quantitatively, the shift in the lower and upper cutoff frequencies are almost equal; effectively the passband does not change. It is interesting to note that in the absence of the dielectric discs, the group velocity takes zero value followed by negative values to again reach zero, and further positive over the propagation constant axis, whereas in the presence of the dielectric discs, it takes zero value followed by positive values to reach zero again and further negative (figure 5.27(b)), i.e., the inclusion of dielectric turns the negative dispersion into positive. For the TE03 mode, for both cases of absence and presence of the dielectric discs, the group velocity takes zero value followed by negative values to reach zero again and further positive, however, with the increase of the relative permittivity of the dielectric discs, quantitatively, the shift of the lower cutoff frequency is higher than that of the upper cutoff frequency, which in turn widens the passband. It has been observed that in the presence of dielectric discs the upper cutoff frequency shifts to a lower value, whereas the lower cutoff frequency remains unchanged with the increase of the relative permittivity of the dielectric discs and shortens the passband (figure 5.27(c)). For the TE01 mode, with the increase of the dielectric disc radius for a constant metal disc radius, the lower and upper cutoff frequencies shift up, however, quantitatively, the shift in the lower cutoff frequency is less than that of the upper 5-39


Figure 5.27. Dispersion characteristics of the alternate dielectric and metal disc-loaded circular waveguide for the modes (a) TE01, (b) TE02, and (c) TE03, taking the relative permittivity of the dielectric disc as the parameter. The broken curve represents the disc-loaded circular waveguide of finite disc-thickness [20].

cutoff frequency, which increases the passband (figure 5.28). Similarly, for the TE02 mode, the passband increases with the increase of dielectric disc radius. Here, it has been observed that the shift in frequency is specifically defined for the dielectric disc radius, for example, for the taken structure parameters (rMD /rW = 0.6, L /rW = 1.0, TDD /rW = 0.3, and εr = 5.0) the frequency shift is maximum for rDD /rW equal to 0.8 to 0.9, and minimum for 0.7 to 0.8, whereas for rDD /rW equal to 0.6 to 0.7 is found in between (figure 5.28(b)). For the TE03 mode, for the lower values of the inner dielectric disc radius, the group velocity takes zero value followed by positive values to reach zero again and further negative, whereas for higher values of the inner dielectric disc radius, it takes zero value followed by negative values to reach zero again and further positive over the propagation constant axis. Interestingly, in the process of changing the inner dielectric disc radius, the frequency corresponding to the point of β0L = vπ , where v = 1, 3, 5, …, remains almost unchanged (figure 5.28(c)). The change of group velocity close to β0 = 0 (operating point of a gyroTWT) due to rDD /rW may be utilized to avoid the negative group velocity close to β0 = 0, which may be the reason for oscillation in the device. 5-40


Figure 5.28. Dispersion characteristics of the alternate dielectric and metal disc-loaded circular waveguide the modes (a) TE01, (b) TE02, and (c) TE03, taking the inner radius of the dielectric disc (rDD /rW ) as the parameter [20].

In general, with the increase of periodicity of the alternate dielectric and metal disc-loaded circular waveguide, both the lower and upper cutoff frequencies shift down and shorten the passband with a higher relative shift in the upper cutoff frequency than that of the lower (figure 5.29). Clearly, the reflection of the change of the periodicity of the alternate dielectric and metal disc-loaded circular waveguide may be observed as the change in the period of the dispersion characteristics for all the three modes considered. For the TE01 and TE02 modes, the group velocity takes zero value followed by positive values to reach zero again and further negative, whereas for the TE03 mode, it takes zero value followed by negative values to reach zero again and further positive over the propagation constant axis [20]. Similar to all metal structures, for the disc-loaded circular waveguide of finite disc-thickness, the shape of the dispersion characteristics and the passband is not very sensitive to the change of dielectric disc thickness (figure 5.30), however, with

5-41


Figure 5.29. Dispersion characteristics of the alternate dielectric and metal disc-loaded circular waveguide for the modes (a) TE01, (b) TE02, and (c) TE03, taking structure periodicity (L /rW ) as the parameter [20].

a decrease of the dielectric disc thickness or with an increase of metal disc thickness, the passband shifts to the lower frequency side for the TE01 and TE02 modes (figure 5.29(a) and (b)). For the TE03 mode, the shape of the dispersion characteristics and the passband is least sensitive to the change of dielectric disc thickness, if very precisely seen, it has been observed that the lower cutoff frequency is insensitive and the upper cutoff frequency first decreases and then increases with a decrease of dielectric disc thickness or with an increase of metal disc thickness (figure 5.30). Further, for the constant difference between the radii of metal and dielectric discs, the lower and upper cutoff frequencies shift to a higher value and the passband increases with the increase of the metal disc radius for the TE01 and TE02 modes, whereas a major change of dispersion characteristics and the passband has been observed for the TE03 mode. It should also be noted here that for the TE03 mode the dispersion characteristics are very specific for each set of structure parameters considered (figure 5.31).

5-42


Figure 5.30. Dispersion characteristics of the alternate dielectric and metal disc-loaded circular waveguide for the modes (a) TE01, (b) TE02, and (c) TE03, taking the thickness (TDD /rW ) of the dielectric disc as the parameter [20].

Azimuthal interaction impedance characteristics In order to find the azimuthal interaction impedance of the structure at the radial position of the azimuthal electric field maxima, one has to take the value of the azimuthal electric field E θI,0 in region I , where the field would be maximum. Further, unlike the axial interaction impedance of a helical slow-wave structure of a conventional helix TWT [9], or the azimuthal interaction impedance of a smoothwall circular waveguide (non-periodic) of a gyro-TWT (figure 5.32) [55], which decreases with frequency approaching zero value at very high frequencies, the azimuthal interaction impedance K θ,0 of a disc-loaded waveguide, for the typical structure parameters and with reference to a typical waveguide mode TE01, decreases with frequency from a very high positive value (tending to infinity), at the lower cutoff frequency of the guide, via zero value, to a very high negative value (tending to negative infinity), at the higher cutoff frequency of the guide (figure 5.32). This nature of variation of K θ,0 with frequency may be correlated with the dispersion characteristics of the structure, obtained with the help of the dispersion relation, as follows (figure 5.32).

5-43


Figure 5.31. Dispersion characteristics of the alternate dielectric and metal disc-loaded circular waveguide for the modes (a) TE01, (b) TE02, and (c) TE03, taking the radius of the metal disc (rMD /rW ) as the parameter such that rMD /rW − rDD /rW = 0.1 [20].

The interaction impedance is inversely proportional to the power transmitted Pt , which in turn is equal to the group velocity of the RF waves multiplied by the energy stored per unit length of the guide. Thus, the interaction impedance becomes inversely proportional to the group velocity of RF waves, the latter being the slope of ω − β dispersion characteristics. At the lower and upper cutoff frequencies, the slopes of the dispersion characteristics (ω − β ) or group velocities of RF waves are each zero going positive and negative, respectively, with an increase of frequency (figure 5.33). The high positive and negative values of the interaction impedance at lower and upper cutoff frequencies, respectively, may be attributed to the aforesaid nature of variation of the slope of dispersion characteristics or group velocity. Similarly, the zero value of the interaction impedance at the point of intersection between the dispersion and interaction impedance characteristics observed corresponds to the large attainable group velocity at that point (figure 5.33). The azimuthal interaction impedance versus frequency characteristics of the infinitesimally thin metal disc-loaded circular waveguide, for the typical structure parameters, obtained by the present analysis, as a special case of rD = rW , pass on to 5-44


Figure 5.32. Azimuthal interaction impedance versus frequency characteristics of the infinitesimally thin metal disc-loaded circular waveguide passing on, as a special case [55, 56], to that of a smooth-wall waveguide [15].

Figure 5.33. Azimuthal interaction impedance versus frequency characteristics (solid line) vis-à-vis dispersion characteristics (broken line) of the infinitesimally thin metal disc-loaded circular waveguide [15].

those for a smooth-wall waveguide [55, 56] (figure 5.33). The value of the azimuthal interaction impedance increases with the introduction of discs in the waveguide except at higher frequencies, where it decreases rapidly to a value that is lower than that for a disc-free (smooth-wall) waveguide (figure 5.33). In a given passband, the value of the azimuthal interaction impedance K θ,0 decreases with the increase of the disc-hole radius (figure 5.34). Therefore, while adjusting the disc-hole radius for dispersion control, due consideration should be

5-45


Figure 5.34. Azimuthal interaction impedance versus frequency characteristics of the infinitesimally thin metal disc-loaded circular waveguide taking the normalized disc-hole radius rD /rW as the parameter [15].

Figure 5.35. Azimuthal interaction impedance versus frequency characteristics of the infinitesimally thin metal disc-loaded circular waveguide taking the normalized structure periodicity L /rW as the parameter [15].

made to see that the value of K θ,0 does not deteriorate. However, although the shape of the dispersion characteristics is strongly dependent on the structure periodicity (figure 5.21), the variation of K θ,0 with the structure periodicity cannot be significantly shown in a given passband since with the change of the structure periodicity the azimuthal interaction impedance versus frequency characteristics shifts from passband to passband (figure 5.35). The extent of the passband of the infinitesimally thin metal disc-loaded circular waveguide, as can be seen from the impedance versus frequency characteristics, for typical structure parameters and for the typical mode TE01, is found to be highly sensitive to the structure periodicity (figure 5.35). Therefore, within the range of a given passband, one cannot demonstrate the effect of the variation of structure periodicity on K θ,0 . 5-46


Figure 5.36. Azimuthal interaction impedance versus frequency characteristics (solid curve) vis-à-vis dispersion characteristics (broken curve) of the disc-loaded circular waveguide of finite disc-thickness [16–18].

Similar to the infinitesimally thin metal disc-loaded circular waveguide, in the discloaded circular waveguide with finite disc thickness, a high negative value of the azimuthal interaction impedance K θ,0 at the upper frequency edge of the passband is attributed to a negative value, close to zero, of the slope of the ω − β dispersion characteristics and hence a negative group velocity of RF waves just beyond the upper frequency edge of the passband (figure 5.36). (The dispersion characteristics have been shown here only up to the upper frequency edge of the passband in figure 5.36.) Further, the azimuthal interaction impedance K θ,0 versus frequency characteristics of the disc-loaded circular waveguide of finite disc-thickness, for the typical structure parameters and for the typical mode TE01, obtained by the present analysis, as a special case of rD = rW , pass on to those for a smooth-wall circular waveguide given, for instance, in Sangster [56] and Singh et al [55] (figure 5.37). The value of the azimuthal interaction impedance increases with the introduction of discs in the waveguide except at higher frequencies, where it decreases rapidly to a value lower than that for a disc-free (smooth-wall) circular waveguide (figure 5.37). The passband of frequencies of the disc-loaded circular waveguide is found to be much more sensitive to the structure periodicity of the structure (figure 5.38). Therefore, within the range of a given or specified passband, one cannot demonstrate the effect of the variation of structure periodicity on K θ,0 (figure 5.38). However, the value of the azimuthal interaction impedance K θ,0 decreases with the increase of the disc-hole radius (figure 5.37). Therefore, while adjusting the disc-hole radius for dispersion control, due consideration should be made to see that the value of K θ,0 does not deteriorate. For the special case of K θ,0 , the azimuthal interaction impedance characteristics pass on to that for a circular waveguide loaded with infinitesimally thin discs (figure 5.39). Interestingly, the variation of the disc thickness does not appreciably change the value of K θ,0 for a given passband; it merely shifts the frequency range over which the appreciable interaction impedance is obtained (figure 5.39). Furthermore, as the

5-47


Figure 5.37. Azimuthal interaction impedance versus frequency characteristics of the disc-loaded circular waveguide of finite disc-thickness taking the normalized disc-hole radius rD /rW as the parameter [16–18]. The solid curve with crosses refers to a smooth-wall circular waveguide (rD /rW = 1.0 ) [55, 56].

Figure 5.38. Azimuthal interaction impedance versus frequency characteristics of the disc-loaded circular waveguide of finite disc-thickness, taking the normalized structure periodicity L /rW as the parameter [16–18].

structure periodicity is increased, the passband of frequencies corresponding to the significant azimuthal interaction impedance shrinks and the centre frequency of the passband shifts towards lower frequencies. Also, the structure periodicity that yields a relatively large value of azimuthal interaction impedance (figure 5.38), in general, would be different from that providing the desired shape of the dispersion characteristics for wideband interaction in a gyro-TWT (see section 5.3), which suggests a suitable trade-off in the value of the structure periodicity chosen. 5-48


Figure 5.39. Azimuthal interaction impedance versus frequency characteristics of the disc-loaded circular waveguide of finite disc-thickness, taking the normalized disc thickness T /rW as the parameter [16–18]. The solid curve with circles refers to the infinitesimally thin metal disc-loaded circular waveguide (T /rW → 0 ) [15].

5.3 Growing-wave interactions in slow-wave TWTs and fast-wave gyro-TWTs The slow-wave conventional TWT operates at the point of intersection between the slow space-charge wave (see section 3.3 in chapter 3, volume 1) and the SWS-mode in the ω–β dispersion plot (see figure 6.2 in section 6.1 to follow in chapter 6, volume 2). 5.3.1 Beam-present dispersion relations Pierce [2, 5, 35, 36] has given an elegant approach to obtaining the dispersion relation of a TWT with the help of the two forms—known as the circuit and the electronic equations—of the same quantity, namely, the ratio of the circuit voltage to RF beam current. The circuit equation is obtained by treating the SWS as a distributed transmission line coupled to the electron beam of the device in a method in which the effect of an element of the modulated beam is simulated by an infinitesimal current generator sending two circuit waves in opposite directions. In this method deriving the circuit equation in terms of the interaction impedance of the SWS (see section 5.1.3), the electric field intensity at a circuit point is obtained by adding the integrated contributions from all such infinitesimal current generators to the left and to the right of the circuit point to the contribution from the injected input signal. The electronic equation is obtained basically from the force equation of an electron subjected to the circuit electric field together with space-charge fields. The two forms of the ratio of the circuit voltage to RF beam current, obtained as the circuit and the electric equations, respectively, when equated, yields the dispersion relation of the TWT as follows (see section 8.2.3 of [2] for the detailed steps of derivation):

5-49


( −β 2 + β02 )[( −(βe − β )2 + βp2 ] = −βe β 2β0 K

I0 (TWT dispersion relation), (5.81) 2V0

where β is the axial propagation constant, it being assumed that RF quantities in the beam-wave coupled system vary as exp j (ωt − βz ). β0 is the cold (beam-absent) propagation constant of the SWS circuit. βe = ω /v0 and βp = ω /v0 are the beam and the plasma propagation constants, respectively, v0 being the DC beam velocity. K is the interaction impedance of the SWS. V0 and I0 are the beam voltage and current, respectively. For weak coupling (I0 = 0), the right-hand side of (5.81) becomes null giving the circuit and beam space-charge waves decoupled yielding β = ±β0 corresponding to the forward and backward circuit waves, respectively, and β = βe ∓ βp corresponding to the fast and slow space-charge waves, respectively (see section 3.3 in chapter 3, volume 1). Similarly, the intersection between the beam-mode and waveguide-mode ω–β dispersion plots decides the operating point of the fast-wave gyro-TWT (see figure 7.1(e) in chapter 7, volume 2). The grazing intersection between the beam-mode and waveguide-mode ω–β dispersion characteristics of a gyro-TWT gives an expression for the required magnetic field in terms of the waveguide-mode cutoff frequency, beam-harmonic mode number, axial beam velocity and relativistic mass factor as has been outlined later (see (7.19) in chapter 7, volume 2). The derivation of the dispersion relation of a fast-wave gyro-TWT—similar to the dispersion relation (5.81) for a slow-wave TWT—is available in the literature [2, 4, 17–19, 21–24] (see, for instance, section 8.4.4 of [2] and chapter 6 of [4]). The analytical procedure starts from writing the wave equation in cylindrical coordinates (r, θ , z ) for the axial component of RF magnetic field in the presence of a thin hollow, annular, electron beam of radius rc , say, in helical trajectories of Larmor radius rL in a waveguide excited in the TEmn mode. The model of the beam considers mono-energetic electrons of the same Larmor radius and of the same thickness 2rL . Here, we make the tenuous-beam approximation according to which the spatial structure of the waveguide-mode remains unaffected by the presence of the beam in the analysis. Hence, one can write the wave equation substituting therein the beam-absent (cold) field expressions, however, interpreting the axial propagation constant as that corresponding to the beam-wave coupled system. The wave equation so written is then multiplied by the function rJm(ktr ) where kt(=Xmn /a ) is the transverse propagation constant of the waveguide, Xmn being the eigenvalue of the TEmn waveguide mode. The product of this multiplication is next integrated over the waveguide cross section between the centre (r = 0) and the wall (r = a ) of the waveguide to obtain the dispersion of the gyro-TWT. The foregoing procedure (detailed in section 8.4.4 of [2]), yields the following dispersion relation of a gyro-TWT:

(k

2 0

)

− β 2 − kt2 (ω − βvz − sωc / γ ) =

−μ0 e 2 N0 ηt2 (ω 2 − β 2c 2 )Hm,−s γ me 0 π a 2K mn

(gyro-TWTdispersionrelation),

5-50

(5.82)


where β is the axial propagation constant assuming RF quantities to vary as exp j (ωt − βz ). k 0 is the free-space propagation constant. vz is the axial component of electrons in the helical trajectory. s is the beam harmonic mode number. ωc is the non-relativistic angular cyclotron frequency. N0 is the number of electrons per axial length of the waveguide. ηt (=vt /c ) is the normalized transverse component of electron velocity vt . me0 and γ are the rest mass and relativistic mass factor of an electron, respectively (see chapter 7, volume 2). The functions Kmn and Hm,−s occurring in (5.82) are:

K mn = (1 − m 2 /(kt2a 2 ))Jm2 (kta )⎫ ⎬ ⎪ Hm,−s = Js2−m(ktrc )J ′s (ktrL ) ⎭ ⎪

the prime representative the derivative of Bessel function with respect to its argument. For weak coupling (N0 = 0), the right-hand side of (5.82) becomes null giving k 02 − β 2 − kt2 = 0, which is the waveguide-mode dispersion relation, and ω − βvz − sωc /γ , which is the beam-mode dispersion relation (see chapter 7, volume 2). 5.3.2 Gain-frequency response The beam-wave coupled system of a TWT supports a total of four waves—a forward wave and a backward wave supported by its slow-wave structure, and a forward slow space-charge wave and a forward fast space-charge wave supported by its electron beam. Similarly, four waves are supported by the beam-wave coupled system of a gyro-TWT comprising a forward wave and a backward wave supported by its waveguide, and a forward slow cyclotron wave and a forward fast cyclotron wave supported by its electron beam. The perception is correspondingly reflected in the dispersion relations (5.81) and (5.82) of the beam-wave coupled devices—the TWT and the gyro-TWT—each of which is essentially a fourth-degree equation with four solutions. It can be appreciated that with the help of the beam-wave coupled dispersion relations (5.81) and (5.82) that, in the TWT and the gyro-TWT, a real β corresponding to a complex ω and, at the same time, a real ω corresponding to a complex β refers to the convective instability and amplification in a TWT (table 5.1) [57]. In the analysis of beam-wave interaction, the contribution from only three forward-wave solutions of (5.81) or (5.82) is of relevance assuming that the interaction structure is so matched that the fourth wave, which is a backward wave, is not generated. Moreover, out of these three forward waves, the growingwave type with the propagation constant β having a positive imaginary part of β contributes to the device gain in view of RF dependence exp j (ωt − βz ) of the solutions of (5.81) or (5.82). The analysis (detailed in section 8.2.4 in [2]) leads to the following expression for the device gain in dB:

G = A + BCN

(TWT gain in dB)

5-51

(5.83)


Table 5.1. Interpretation of the dispersion relation for stability and instability.

where

⎛ ⎞ ⎫ 4QC ⎞ ⎛ 1 A = 20log10 ⎜1 + ⎟ ⎜ ⎟ ⎪ δ12 ⎠ ⎝ (1 − δ 2 / δ1) (1 − δ 3/ δ1) ⎠ ⎪ ⎝ ⎪ ⎬ (TWT), B = (40π log10e)x1 ≈ 54.6x1 ⎪ C = (KI0 /4V0)1/3 ⎪ ⎪ N = βe l /(2π ) = l / λe ⎭

(5.84)

in which QC = βp 2 /(4βe 2C 2 ) is called the Pierce’s space-charge parameter [2, 5, 35, 36]. δ1, δ2 and δ3 are the three values of the dimensionless parameter δ = j (βe − β )/(βeC ) measuring small departures of the three forward wave solutions of (5.81) for β from the beam propagation constant βe (corresponding to β ≅ βe(1 + jCδ )). x1 is the real part of δ1 supposedly having a positive imaginary part of β that contributes to the device gain. l is the interaction length and λe is the number of beam wavelengths. Similarly, the analysis (detailed in section 8.5 in [2]) leads to the following expression for the device gain in dB [2]:

G = A + BguideCgyroNgyro, guide (gyro − TWT gain in dB)

(5.85)

where ⎛ ⎞⎫ 4QC ⎞ ⎛ 1 A = 20log10 ⎜1 + ⎟ ⎟⎪ ⎜ δ12 ⎠ ⎝ (1 − δ 2 / δ1) (1 − δ 3/ δ1) ⎠ ⎪ ⎝ ⎪ ⎬(gyro − TWT). (5.86) Bguide = 40π (log10e)x1 ≅ 54.6x1 ⎪ 1/3 Cgyro = (K gyro I0 /4 V0 ) ⎪ ⎪ Ngyro, guide = βmnl /(2π ) ⎭

δ1, δ2 and δ3 are the three values of the dimensionless parameter δ = j (βmn − β )/(βmnCgyro ) measuring small departures of the three forward wave solutions of (5.82) for β from the beam-absent or cold TEmn mode waveguide-mode propagation constant βmn (corresponding to β ≅ βmn(1 + jCgyroδ )). The expression for Kgyro occurring in (5.86) is given by [2]:

5-52


K gyro =

(μ0 / ε0)1/2 ηt2kc2,mn(1 + α02 )Hm,−s 4 πK mna 2(vz / c )βmn

,

(5.87)

where α0(=vt /vz ) is the beam pitch factor being the ratio of the transverse velocity vt to axial velocity vz of beam electrons and kc,mn(=kt = Xmn /a ) is the TEmn waveguidemode cutoff propagation constant. However, Kgyro given by (5.87) does not appreciably differ from the azimuthal interaction impedance given by (5.77) (plotted in figures 5.32–5.39) (see sections 5.2.7 and 5.2.8); this has been appreciated typically for the TE01-mode over an approximately 10% bandwidth whose lower end is not close to the cut-off frequency of the waveguide [56]. Dimensional tapering for gyro-TWT broadbanding The optimum structure parameters (the disc-hole radius, the structure periodicity, and the thickness, besides the waveguide-wall radius) have been predicted for a wide bandwidth of coalescence between the beam-mode and the waveguide-mode dispersion characteristics that would in turn correspond to a wide bandwidth of a gyro-TWT. The optimization of the structure periodicity is found to be more effective than that of the other structure parameters, in maximizing the device bandwidth. The analysis for the azimuthal interaction impedance characteristics clearly shows its strong dependence on the structure parameters. Therefore, while optimizing the structure parameters, in particular, the structure periodicity, for the desired shape of the dispersion characteristics for wideband gyro-TWT performance, due consideration should be made to ensure that the value of the interaction impedance does not deteriorate causing a low device gain. There are two methods of broadbanding a gyro-TWT. In the first of these methods, the conventional waveguide interaction structure is loaded by a dielectric, or made to deviate from a simple smooth-wall geometry, and thus the dispersion characteristics of the waveguide are controlled for wideband coalescence between the beam-mode and waveguide-mode dispersion characteristics [2, 4, 14–24]. In this method, the structure’s cross-sectional dimensions are held uniform over the interaction length. In the second method of widening the device bandwidth, the waveguide cross-sectional dimensions of the structure are not held uniform and rather they are tapered, and, at the same time, the magnetic field and the beam parameters are synchronously profiled [4, 21–23, 58]. In the second method, although the bandwidth is increased due to different portions of the interaction length of the tapered-cross-section waveguide becoming effective for different frequency ranges, the gain of the device decreases due to the reduction in the effective interaction length of the device at each of the frequency ranges [58]. To compute the gain-frequency response of a gyro-TWT, one may substitute the phase propagation constant of an interaction structure from dispersion characteristics into the small-signal gain-equation of a gyro-TWT [2, 4]. A tapered disc-loaded waveguide (figure 5.40) accrues the advantages of the two methods of broadbanding a gyro-TWT, namely the method of tapering the waveguide cross section, to be accompanied by profiling the magnetic field, and the method of shaping the

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Figure 5.40. Tapered disc-loaded waveguide with finite disc thickness [21–23].

Figure 5.41. Gain-frequency response of a gyro-TWT in tapered and non-tapered disc-loaded and smoothwall circular waveguides, taking typically the interaction length as 162 mm, the mode as TE01, and the beam parameters as I0 = 9 A , V0 = 100 kV and α0 = 0.5. For the tapered device, the magnetic field and the beam parameters are typically B0,p /Bg,p = 1.0 (1 ⩽ p ⩽ nD ), rH,1 = 3.5 mm and rL,1/rW ,1 = 0.1. For the non-tapered device, the values of the corresponding quantities remain the same (B0 /Bg = 1.0 , rH = 3.5 mm and rL /rW = 0.1). The typical structure parameters are nD = 54 , Tp = 1.0 mm and Lp = 3.0 mm ( p = 1 to nD ) for the disc-loaded waveguide; and T = 1.0 mm and L = 3.0 mm, for the smooth-wall waveguide, with the start and end values rW ,start and rW ,end , respectively, indicated following the taper profile of the disc-hole radius of the corresponding tapered disc-loaded waveguide [21].

dispersion characteristics of the waveguide. In this model, the structure parameters, namely the disc thickness, the disc-to-disc distance or structure periodicity, and the disc-hole radius, are first optimized for the desired control of the structure dispersion and hence for wideband device performance. The bandwidth of the device would be

5-54


Table 5.2. Comparison of the tapered and non-tapered, smooth-wall and disc-loaded gyro-TWTs excited in the TE01 mode with structure parameters adjusted for the centre frequency ∼40 GHz (on the basis of the results presented in figure 5.41) [21].

Interaction structure

Bandwidth (3 dB)

Smooth-wall circular waveguide without a tapered cross section Smooth-wall circular waveguide with a tapered cross section Disc-loaded circular waveguide without a tapered cross section Disc-loaded circular waveguide with a cross section

4.8 6.5 5.0 7.3

GHz GHz GHz GHz

Gain 24.0 16.2 31.5 19.4

dB dB dB dB

further widened by tapering the structure cross section, more precisely by stepping the disc-hole and waveguide-wall radii [21–23]. It is expected that the reduction of the device gain due to the tapering or stepping of the structure’s cross section [58] would be compensated for by the enhancement of the interaction impedance of the structure caused by the introduction of the discs, and hence the device gain. This method requires that the magnetic and beam profiles are synchronous with the taper steps. It is found that the gain of the device would reduce in an attempt to widen its bandwidth, both for the smooth-wall and disc-loaded waveguides (figure 5.40, table 5.2). The double tapering scheme of a disc-loaded waveguide (with respect to both the waveguide-wall radius and disc-hole radius) has certainly predicted an enhancement of both the gain and bandwidth of a gyro-TWT as compared to the scheme of tapering the waveguide-wall radius of a smooth-wall waveguide, the mid-band frequency being adjusted, typically, ∼40 GHz, by an appropriate choice of the waveguide-wall radius, in the case of a smooth-wall waveguide; and the waveguidewall and disc-hole radii, in the case of a disc-loaded waveguide (figure 5.40, table 5.2). Thus, due to the tapering of the cross section, typically, the device bandwidth in a smooth-wall circular waveguide increases by 35% at the cost of 7.2 dB (33%) device gain. In comparison, due only to disc loading, both the device bandwidth and gain in a non-tapered circular waveguide increase by 4% and 7.5 dB (31%), respectively. However, one can combine the two methods of broadbanding, namely, tapering the waveguide cross section and disc loading, to increase the device bandwidth by 52% at a little cost of 4.6 dB (19%) device gain (figure 5.40, table 5.2). Thus, the relevant structure parameters being adjusted for the same mid-band frequency, (i) when its cross section is tapered, the device bandwidth for the smooth-wall waveguide increases and the gain decreases; (ii) when the waveguide is disc-loaded, the device bandwidth becomes more than for the smooth-wall waveguide without a tapered cross section and less than for the smooth-wall waveguide with a tapered cross section, and at the same time, the device gain becomes more than for the smooth-wall waveguide both with and without a tapered cross section; and (iii) when the discloaded waveguide is simultaneously tapered with respect to both the waveguide-wall radius and the disc-hole radius, the device bandwidth becomes more than for the smooth-wall waveguide both with and without a tapered cross section as well as for the disc-loaded waveguide without a tapered cross section, and at the same time, the

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device gain becomes more than for the smooth-wall waveguide with a tapered cross section and less than for both the smooth-wall and the disc-loaded waveguide each without a tapered cross section [21–23]. Interestingly, out of the structure parameters, namely, the disc-to-disc distance or structure periodicity in a disc-loaded circular waveguide of a non-tapered discloaded gyro-TWT, which is found to be the most effective optimizing parameter for controlling the dispersion characteristics of the structure and consequently, widening the device bandwidth, becomes the least significant taper parameter as compared to the other taper parameters, namely the waveguide-wall radius and the disc-hole radius, while implementing the scheme of double tapering. Also, while tapering the structure parameters for wide device bandwidth, in fact, care should be taken to synchronously profile the magnetic field and choose the beam parameters such that they obey the adiabatic beam-flow condition as well as the conservations of magnetic flux and that of the electron magnetic moment. In the present study, it is assumed that the electron beam is essentially monoenergetic with no velocity spread, an effect that would reduce the device efficiency, unless the operating point corresponds to a lower value of the phase propagation constant, towards the waveguide cutoff, which would however shift the operating point away from the grazing intersection or coalescence between the beam-mode and waveguide-mode dispersion characteristics. Therefore, future scope of the work should include the effect of beam velocity spread anticipating that, as a result of tapering the magnetic field, the electrons on the high end of the energy distribution tail would undergo less interaction than they would if the magnetic field were constant, leading to a larger velocity spread at the end of the interaction structure.

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[10] Ghosh S, Jain P K and Basu B N 1997 Rigorous tape analysis of inhomogeneously loaded helical slow-wave structures IEEE Trans. Electron Dev. 44 1158–68 [11] Kesari V and Basu B N 2017 Analysis of some periodic structures of microwave tubes: part 1: analysis of helical slow-wave structures of traveling-wave tubes J. Electromagn. Waves Appl. 37 1–37 [12] Lien E L 1979 Stop-bands produced by asymmetrical support rod systems in helix structures Tech. Dig. Int. Electron Dev. Meeting pp 412–5 [13] Joo Y D, Sinha A K and Park G S 2003 Electromagnetic wave propagation through an azimuthally asymmetric helix slow wave structure Jpn. J. Appl. Phys. 42 7585–93 [14] Choe J Y and Uhm H S 1981 Analysis of wide-band gyrotron amplifiers in a dielectric loaded waveguide J. Appl. Phys. 52 4506–16 [15] Kesari V, Jain P K and Basu B N 2004 Approaches to the analysis of a disc loaded cylindrical waveguide for potential application in wideband gyro-TWTs IEEE Trans. Plasma Sci. PS-32 2144–51 [16] Kesari V, Jain P K and Basu B N 2005 Analysis of a disc-loaded circular waveguide for interaction impedance of a gyrotron amplifier Int. J. Infrared Millim. Waves. 26 1093–110 [17] Kesari V, Jain P K and Basu B N 2005 Analysis of a circular waveguide loaded with thick annular metal discs for wideband gyro-TWTs IEEE Trans. Plasma Sci. 33 1358–65 [18] Kesari V and Basu B N 2013 Analysis of beam and magnetic field parameter sensitivity of a disc-loaded wideband gyro-TWT IEEE Trans. Plasma Sci. 41 1557–61 [19] Kesari V and Keshari J P 2013 Interwoven-disc-loaded circular waveguide for a wideband gyro-traveling-wave tube IEEE Trans. Plasma Sci. 41 456–60 [20] Kesari V and Keshari J P 2011 Analysis of a circular waveguide loaded with dielectric and metal discs Prog. Electromagn. Res. 111 253–69 [21] Kesari V, Jain P K and Basu B N 2006 Exploration of a double-tapered disc-loaded circular waveguide for a wideband gyro-TWT IEEE Electron Dev. Lett. 27 194–7 [22] Kesari V, Jain P K and Basu B N 2006 Analysis of a tapered disc-loaded waveguide for a wideband Gyro-TWT IEEE Trans. Plasma Sci. 34 541–6 [23] Kesari V, Jain P K and Basu B N 2007 Parameters to define the electron beam trajectory of a double-tapered disc-loaded wideband gyro-TWT in profiled magnetic field Int. J. Infrared Millim. Waves 28 443–9 [24] Kesari V 2010 Beam-present analysis of disc-loaded-coaxial waveguide for its application in gyro-TWT (Part-2) Prog. Electromagn. Res. 109 229–43 [25] Kesari V 2010 Beam-absent analysis of disc-loaded-coaxial waveguide for its application in gyro-TWT (Part-1) Prog. Electromagn. Res. 109 211–27 [26] Kesari V, Jain P K and Basu B N 2005 Modeling of axially periodic circular waveguide with combined dielectric and metal loading J. Phys. D: Appl. Phys. 38 3523–9 [27] Kesari V and Keshari J P 2012 Propagation characteristics of a variant of disc-loaded circular waveguide Prog. Electromagn. Res. M 26 23–37 [28] Kesari V 2014 Analysis of alternate dielectric and metal vane loaded circular waveguide for a wideband gyro-TWT IEEE Trans. Electron Dev. 61 915–20 [29] Sinha M P, Jha R K, Basu B N and Prasad R K 1985 Electromagnetic wave propagation in a planar helix with a metal shield J. Appl. Phys. 57 4480–1 [30] Chadha D, Aditya S and Arora R K 1983 Field theory of planar helix traveling-wave tube IEEE Trans. Microw. Theory Tech. 31 73–6

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High Power Microwave Tubes Basics and Trends, Volume 1

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