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Design and modeling of microwave and millimeter -wave vacuum electronic devices

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Design and Modeling of Microwave and
Millimeter-Wave Vacuum Electronic Devices
By
HsinLu Hsu
B.S. (Physics Department, National Tsing Hua University, Taiwan) 1999
M.S. (Physics Department, National Tsing Hua University, Taiwan) 2001
DISSERTATION
Submitted in partial satisfaction of the requirements for the degree of
DOCTOR OF PHILOSOPHY
in
Engineering - Applied Science
in the
OFFICE OF GRADUATE STUDIES
of the
UNIVERSITY OF CALFORNIA
DAVIS
Approved:
David Hwang
-/y\ sa (CK-a r^j
p.
Jonathan P. Herii
Neville CUuhmann, Jr. (committee chair)
Committee in Charge
2006
i
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ACKNOWLEDGEMENTS
Many individuals have contributed to this dissertation, helping with a satisfying end
to my Ph.D. work. There are numerous people I would like to thank, and I am blessed to
have encountered every one of them.
First, I would like to give enormous thanks to my research advisor at U.C. Davis,
Professor N. C. Luhmann.
Professor Luhmann’s extremely broad acknowledge of
physics and engineering earned my deepest respect.
I enjoyed his style and attitude
towards teaching very much, serious but relaxed, giving students the freedom to develop
their own ideas. Although he has a very busy schedule, he always makes time for his
students. His concern and assistance on academic, research, career, and other issues have
been tremendous. I am very grateful to have been one o f his students.
I would also like to thank Dr. George Caryotakis for having me at SLAC.
His
lifetime o f great contributions to klystrons and his extreme dedication to the field is
indeed amazing and earned my highest respect.
Dr. Glenn P. Scheitrum, my advisor in SLAC, has my highly sincere appreciation.
He is an ideal supervisor. My decision to move to SLAC two years ago has resulted in a
significant improvement in my knowledge o f vacuum electronics, and in large part this is
due to Dr. Scheitrum.
His experience and knowledge, both within and outside the
vacuum electronics field, is too extensive to describe. I am also impressed by his attitude
toward students and others, and I highly appreciate his endless guidance, patience, and
caring.
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My thesis committee members, Professor John S. De Groot, Professor David Hwang,
and Professor Jonathan P. Heritage, have my special thanks for their warm support.
I would also like to thank Professor Kwo-Ray Chu, my research advisor while I was
earning my masters degree at NTHU. Professor Chu is a great scientist and his serious
attitude toward research had a significant impact on me. He is always willing to help his
students. I really appreciate that I had such a distinguished advisor when I started my
graduate program.
I also would like to thank Dr. Larry Ludeking and Dr. David Smithe for their
patience, effort and help with the MAGIC program.
I am indebted to the members of the Microwave and Millimeter Wave Technology
Group at UCD and to many individuals in the Klystron Department at SLAC for their
friendly help and support while I earned my Ph.D. I would like to thank Bill Coombe,
Arcot Prakash, Jay Sung, Kolbot By, and Tom Ninnis for their friendship and
tremendous help over the past years while residing in California, with special thanks to
Tom Ninnis for reviewing my writing.
Finally I would like to express my deepest gratitude to my parents, my sister and my
little brother.
Their caring has always supported me greatly and gave me infinite
confidence. No matter what happens, I know they will be always there and on my side.
This work has been supported by AFOSR MURI program.
HsinLu Hsu
Menlo Park, California
June 2006
iii
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Design and Modeling of Microwave and
Millimeter-Wave Vacuum Electronic Devices
Abstract
by
HsinLu Hsu
Doctor o f Philosophy - Applied Science Engineering
University o f California, Davis, 2006
Professor Neville C. Luhmann, Jr., Chair
Precise design and modeling of microwave tubes is becoming increasingly
important as applications move to ever increasing power levels, frequencies, and
bandwidths. Here, traditional approaches of frequency scaling of existing devices and
the use o f a number o f experimental iterations are unacceptable, both from an economic
and time viewpoint.
In addition, the added dimensional constraints and material
demands associated with the move to high power millimeter wave applications make it
imperative to realize optimal designs with maximum tolerance to dimensional and
material property variations, effects that demand extensive, fully realistic computer
simulations and parameter space searches.
iv
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In this thesis, the above approach is applied to the design o f an Extended
Interaction Klystron (EIK), whose frequency is chosen to be Ka-Band, both for
specificity as well as the fact that there are increasing numbers of applications aimed at
this frequency region. This thesis demonstrates the design and modeling of a unique
geometry which allows more flexible tuning of each cavity in the EIK to provide higher
power and broader bandwidth, as well as accounting for the inevitable errors in
fabricated dimensions.
In addition to the EIK, several other types of vacuum electronic devices have been
extensively demonstrated with 3D or 2D modeling using a Particle-In-Cell (PIC) code,
including a Ka-Band TunneLadder TWT and a W-Band Reflex Klystron.
This dissertation also demonstrates the optimization and modeling of the
Magnetron Injection Gun (MIG) and the input coupler for the UCD W-band TEoi gyroTWT. The MIG is optimized using a 2D trajectory code and the coupler is optimized
using a 3D electromagnetic-field simulation code. Both devices have been built and are
currently employed in the TWT which has produced more than 100 kW with gain in
excess o f 60 dB.
v
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Contents
Acknowledgements ................................................................................................. ii
Abstract ...........................................................................................................................iv
Introduction
A. Overview o f Microwave Vacuum Electronic Devices......................................... 1
A.1 Solid Sate versus Vacuum Electronics.................................................... 2
A.2 High Frequency Applications................................................................... 7
A.3 The Future of Vacuum Electronics...........................................................9
B. Motivation for Computer Modeling of Vacuum Electronics................................11
C. Scope o f the Dissertation........................................................................................ 12
Chapter 1
Introduction of Design and 3D Computer Modeling of a Ka-Band EIK
1.1 Klystron Overview.................................................................................................. 24
1.2 Motivation for 3D Modeling of Ka-Band EEK and Desired Goals......................28
1.3 State-of-The-Art EIK.............................................................................................. 32
Charter 2
EIK Modeling Tools and Circuit Design
2.1 Modeling Tools......................................................................................................33
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2.1.1 ID MathCAD Calculation Worksheet.................................................... 34
2.1.2. ID AJ Disk................................................................................................35
2.1.3. 3D PIC code - MAGIC 3 D .................................................................... 38
2.2 Circuit Design L oop............................................................................................... 41
2.3 Coupled Cavity D esign.......................................................................................... 46
2.3.1. M2(R/Q)....................................................................................................49
2.4 Design Parameters...................................................................................................53
2.5 Final Design and Modeling Results....................................................................... 55
2.6 Focusing Fields........................................................................................................61
Charter 3
EIK MAGIC 3D Modeling Techniques and Analysis
3.1 Convergence T e st....................................................................................................64
3.2 Reduced Plasma Wavelength..................................................................................66
3.3 Resonant Frequency and Q Measurements............................................................68
3.3.1 Frequency Domain Approach................................................................. 68
3.3.2 Time Domain Approach......................................................................... 69
3.4 R/Q M odeling.......................................................................................................... 74
3.5 Coupling Coefficient Modeling..............................................................................77
3.5.1 “Cold Test” M Simulations...................................................................... 78
3.5.2 “Hot Test” M Simulations....................................................................... 80
3.6 Input and Output Cavity M odeling........................................................................81
3.7 The Input Gap V oltage........................................................................................... 86
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3.8 Define Input Power..................................................................................................91
3.8.1 Input Power Simulation using M AGIC.................................................... 92
3.8.2 Measuring Input Power in MAGIC using S n ........................................... 95
3.8.3 Input Power in AJDISK..............................................................................96
3.8.4 Conclusions..................................................................................................97
3.9 Over Drive Approach.............................................................................................. 98
3.9.1 Driving a Single Cavity At The Resonant Frequency............................. 98
3.9.2 Driving a Single Cavity Off The Resonant Frequency........................... 100
3.9.3 Driving the Input Coupled Cavity............................................................. 101
3.9.4 Conclusions..................................................................................................103
3.10 Beam Modeling and Investigations in M A G IC ..................................................104
3.10.1 Built-in LAMINAR BEAM command in MAGIC................................104
3.10.2 Import Files for MAGIC 2D .....................................................................107
3.10.3 Summary.................................................................................................... 115
3.10.4 Additional Information.............................................................................117
3.11 Stability A nalysis..................................................................................................118
3.11.1 Stability calculation using MathCAD W orksheet...................................118
3.11.2 Cavity M aterial...........................................................................................125
3.12 Beam Loading in MAGIC 3 D ..............................................................................126
3.12.1 Beam Loaded Resonant Frequency and Q Simulations.......................... 126
3.12.2 Input Cavity Beam Loading Measurements............................................ 131
3.12.3 Result Comparison.................................................................................... 132
3.12.4 Penultimate Cavity Beam Loading........................................................... 135
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3.12.5 Mode Contamination...............................................................................137
3.13 Noise Investigations...............................................................................................138
3.13.1 Fourier Analyses........................................................................................ 142
3.13.2 Numerical Effect and Grid Jiggling..........................................................144
3.13.3 Algorithm....................................................................................................146
3.13.4 Discussion of the Physical Mode.............................................................. 146
3.13.5 Summary.....................................................................................................150
3.14 Cavity Tuning Using Fake T u n e r........................................................................151
3.15 Simulation Mesh Size and Time Step.................................................................. 152
Charter 4
Summaries of Design and Computer Modeling of EIK.........................154
Chapter 5
TunneLadder TWT Modeling Using MAGIC 3D
5.1 Intro, and Motivation for 3D Modeling of Ka-Band TunneLadder TW T.......... 158
5.2 Overview o f TunneLadder TW T............................................................................161
5.3 Goal and Purpose.....................................................................................................163
5.4 Circuit Modeling Using MAGIC 3D ..................................................................... 164
5.5 Cold Test Simulations..............................................................................................165
5.5.1 Dispersion Simulations................................................................................ 165
5.5.2 Impedance Simulations and Calculations.................................................. 166
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5.5.3 Attenuation Simulations and Calculations..................................................169
5.5.4 Waveguide Transition and coupler M odeling........................................... 171
5.5.5 Optimized Coupler....................................................................................... 175
5.6 Hot Test Simulations...............................................................................................176
5.6.1 Non-Symmetry of Hot Test Simulations....................................................176
5.6.2 Synchronous Hot Test Simulations............................................................. 177
5.6.3 Experimental Hot Test Simulation ............................................................ 179
5.7 Discussion and Summaries..................................................................................... 184
5.8 Additional Information............................................................................................186
Chapter 6
Reflex klystron Modeling Using MAGIC 2D ........................................ 187
6.1 Introduction o f Reflex Klystron M odeling........................................................... 187
6.2 Operating Principles o f the Reflex Klystron.......................................................... 188
6.3 Circuit Modeling Using MAGIC 2D ......................................................................190
6.4 Electronic Tuning Simulation Results....................................................................195
6.5 THz Reflex Klystron Modeling.............................................................................. 197
6.6 Additional Modeling Information...........................................................................198
6.7 Summaries and Future W ork.................................................................................. 199
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Chapter 7
UCD 94 GHz Gyro-TWT Input Coupler & MIG Optimization
7.1 Input Coupler Optimization.................................................................................... 200
7.1.1 Introduction and Overview........................................................................ 200
7.1.2 Optimization of the Coupler for Broad Bandwidth Operation............... 202
7.1.3 Conclusions..................................................................................................205
7.2 Magnetron Injection Gun Optimization................................................................. 205
7.2.1 Introduction and Overview........................................................................ 205
7.2.2 Magnetic Field Simulation......................................................................... 207
7.2.3 MIG Optimization using EGUN................................................................ 209
7.2.4 Discussion, Summary and Future work.................................................... 212
Appendix
I. MathCAD Worksheet for EIK Small Signal Calculation.......................................214
II. MathCAD Worksheet for Input Cavity Gap Voltage...........................................222
III. MathCAD Worksheet for Stability Analysis.........................................................224
IV. PIC Code Introduction............................................................................................ 229
V. Reproduction of MTHU M IG .................................................................................237
References .................................................................................................................... 238
xi
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Introduction of
Design and Computer Modeling of
Microwave Vacuum Electronics
A. Overview of Microwave Vacuum Electronic Devices
Microwave power electronics is critically for numerous aspects of our daily life in the
21st century. There is no doubt that our society has been increasingly dependent on
microwave devices over the last few decades.
Without them, we would not have
microwave ovens [21], wireless communication [22], satellite communication [4], radar
[23], GPS (global positioning system) [24], medical accelerators [25], electronic
countermeasures [26], broadcasting media (TV, radio) and many more products that are a
ubiquitous part of modem society.
It should be noted that the frequency regimes of the electromagnetic spectrum are not
precisely determined. Very often, “RF” represents frequencies between -3 0 kHz to 0.3
GHz; “microwave” represents -0.3 GHz to 30 GHz; “millimeter wave” represents - 30
GHz to 300 GHz, and “sub-millimeter wave” represents roughly -100 GHz to 3000 GHz
(also called the terahertz regime). Nevertheless, “RF” and “microwave” very often refer
to this entire frequency range.
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2
A.1 Solid Sate versus Vacuum Electronics
All microwave power electronic devices operate on the principle of converting the
kinetic energy o f an electron stream (electron current) to coherent electromagnetic
radiation [4]. In a solid state amplifier, a microwave frequency AC (alternating-current)
voltage at low current is applied to the electrodes at the two extreme ends of the device.
The corresponding microwave frequency modulation o f the large current across the
device equates to a net current and power gain for the microwave signal [4] [9]. The
electron stream diffusively drifts through a solid medium (semiconductor) and
experiences numerous collisions with atoms, converting a significant fraction o f its
kinetic energy into heat inside the device. In a vacuum electronic amplifier (Fig. 1), a
low power microwave signal is fed into one end of an interaction circuit, where the
electron stream interacts with the electric field of the microwave radiation.
This
interaction converts electron kinetic energy into more intense microwave power, which is
extracted from the other end of the interaction circuit.
The electron stream travels
collisionlessly through an evacuated region. The fact that the solid-state electron stream
travels collisionally inside the semiconductor, and thus with lower mobility than the
electron stream traveling through a vacuum, gives an intrinsic physical advantage to
microwave vacuum electronics for producing significantly higher power at higher
frequencies compared to solid state devices (Fig. 2).
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3
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C a th o d e
C oolant
(air or liquid)
C ollector su rfa c e h e a t
(h o tte st te m p e ra tu re zo n e)
power
supply
Fig. 1: Vacuum Electronic Amplifier Configuration [4] © 2005 IEEE
10 cm
0.1 m
o.i mm
1.0 mm
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Electronic
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Fig. 2: Average RF output power versus frequency for various electronic devices [6]
©2002 IEEE
Over the last few decades, both the technology and performance has continued to
increase in both solid-state and vacuum electronic devices. Each has advantages and
disadvantages when compared to the other, and as a result the choice of the device
depends on the application requirements.
In some devices, both technologies are
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combined, such as the MPM (microwave power module) and MMPM (millimeter wave
power module), to take advantages of both technologies [5].
Solid state devices
generally have advantages when comparing to vacuum electronics devices in the
following areas:
1. Low Noise
2. Low cost (easily-manufactured in large quantity)
3. Compact size (this characteristic and the next apply when comparing to
conventionally-manufactured vacuum electronics for low frequency applications)
4. Lightweight
Because o f the reasons above, solid state devices are increasingly replacing vacuum
electronic device in many applications. However, to date, for high frequency and high
power driven devices, vacuum electronics are superior over solid state devices due to the
following fundamental physical differences:
1. Higher electron mobility
2. Better heat dissipation
3. Ability to operate at higher temperature
4. Ability to reuse the unspent energy from the spent beam to increase the efficiency
It is generally agreed that for high power (>100 watts average) and high frequency (>30
GHz or so) applications, vacuum electronic devices are considered to be smaller, lighter,
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5
less expensive, more efficient, and can handle more extreme environmental conditions
when compared to solid state devices. For low power and low frequency applications,
solid state devices are considered to be more reliable, easier to maintain, safer (no high
voltage), lower cost for high volume products, and have some performance advantages,
especially in communications (because of linearity). Nevertheless, in many applications,
vacuum electronics devices are not anticipated to be replaced by solid-state devices in the
near future. One particularly good example is the case of commercial geosynchronous
satellites (Fig. 3). Over the years of 1992-2002, the market overwhelmingly preferred a
vacuum electronic device (in this case traveling wave tubes amplifiers or TWT As) over
solid state devices (solid state power amplifiers or SSPAs) due to price, performance (the
need to provide high microwave power at very high efficiency), long-life, and the
capability to operate at high temperatures [4]. Up to date, the vacuum electronics remain
being the essential role in many places, especially at the higher frequencies and higher
power area.
OHPAs PER S/C
5> 50
I SSPAs PER S/C
■ TWTAs PER S/C
tu 20
1992
1993
1994
199S
1996
1997
1998
1999
2000
2001
2002
CALENDAR YEAR
Fig. 3: Plot o f average number of high-power amplifier (HP As)
on commercial geosynchrous satellites [4] © 2005 IEEE
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6
As we move into the 21st century, it is increasingly apparent that more and more
applications are moving to even increasing power level, frequencies, and bandwidths.
For example, broadband wireless communication is expected to migrate to increasingly
higher frequencies following the November, 2003, FCC decision to make available the
71-76 GHz, 81-86 GHz, and 92-95 GHz bands (commonly referred to as the "E-Band")
for commercial wireless broadband communications (Fig. 4). Advancements in vacuum
electronics devices will lead to numerous applications in this millimeter-wave spectrum.
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Fig. 4: Average atmospheric attenuation versus frequency
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7
A.2 High Frequency Applications
Vacuum electronic devices are also expected to play a vital role in the terahertz
frequency region (also called sub-millimeter wave region), where the electromagnetic
spectrum remains one of the least explored regions. In addition to the possibility that
sub-millimeter wave technology might be useful for detection of planets containing
atmospheric conditions suitable for extraterrestrial life [7], it would also contribute
tremendously to the needs of the medical community as well as homeland security.
Since the photon energy o f terahertz radiation is a million times less than that of X-rays
and produces no biological tissue damage, a security system using terahertz radiation
could be placed in highly populated areas to detect hidden weapons. Terahertz radiation
can easily penetrate most dielectrics, including paper, cloths, many building materials,
and can provide not only spectroscopic information, but also high-resolution images of
metallic objects, without concern that the system could harm animals or plants [8] [3].
Besides defense and homeland security applications, the fact that terahertz radiation is
quickly absorbed by water and similar fluids makes it useful for distinguishing healthy
cells from cancerous cells because of the difference in blood flow, and therefore could
possibly be used to detect the first stages of demineralization [3] [8].
The critical roadblock to full exploitation of the sub-millimeter-wave bands has been
lack of coherent radiation sources that are powerful, efficient, frequency agile (sufficient
bandwidth), reliable, compact, and affordable. The terahertz region is the area where
electrons mobility in semiconductors is limited and leads to device heating problems and
reduced efficiency, but electrons traveling in vacuum would not have such limitations,
assuming the electric and magnetic fields can be shaped to modulate the beam and
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convert their energy to RF power [3]. Furthermore, vacuum devices with depressed
collectors can reuse the beam energy to increase the efficiency, while solid state devices
cannot.
However, when the conventional vacuum electronic device interaction space
diminishes at short wavelengths, the structure complexity and surface field concentration
present fabrication difficulties as well as power handling problems. By comparison, the
interaction space for optical radiation by lasers is not restricted in size. However, since
an atom upon each excitation emits only a single photon of the desired energy, the
efficiency is limited. Furthermore, this accounts for the decreasing power output of
lasers with decreasing photon energy, so that lasers are most often employed in the
infrared and shorter-wavelength region.
Such opposing limitations of classical and
quantum-mechanical sources result in the millimeter and sub-millimeter “gap” (Fig. 5)
[15].
ECM-based
devices -
S10
S’10
'
Lasers
C o n v e n tio n a l
m ic ro w a v e t u b e s
W a v e le n g th (m m )
Fig. 5: Average power capabilities of lasers, conventional microwave tubes,
and ECM-based devices (such as gyrotrons) [87]
Reprinted figure with permission from Reference [87], © 2004 The American Physical Society
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9
As indicated in Fig. 5 the electron cyclotron interactions provide one possible
solution for the “gap”.
This kind of vacuum electron device is called an Electron
Cyclotron Maser (ECM). One good example is the gyro-TWT, to be discussed further in
later sections [4] [17].
ECM-based devices indeed have delivered unprecedented power levels in the
millimeter and sub-millimeter wave regions; unfortunately, they have been bulky and
very expensive. Although traditional vacuum electronic power sources have suffered
from simple physical scaling problems, advances in micro-fabrication technology for
microstructures are allowing new opportunities for vacuum electron devices producing
RF radiation at higher frequencies. Micro-fabricated vacuum electron devices, being
powerful, efficient, and affordable, show promise as sub-millimeter wave power sources.
A.3 The Future of Vacuum Electronics
Vacuum electronic devices are playing, and will continue to play, a major role in a
variety o f commercial and defense applications. In the 21st century, we are looking for
the power device to be smaller, lighter, cheaper, and with improved performance. A few
o f the technologies and approaches that will contribute to this movement are as follows
[1]:
1. Advancements in computer hardware and software as well as the modeling skills
to support faster and more accurate simulations.
2. Development of improved and more sophisticated high-voltage power electronics
technology and design, resulting in an increase in reliability and performance and
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10
a reduction in the volume and weight of power supplies for vacuum electronic
devices.
3. New materials, electron sources, and manufacturing technologies to improve
vacuum electronic device performance and reduce costs.
The approach and success of MPMs is a good example of sophisticated technology
innovations. A low-gain vacuum electronic amplifier (TWTA) driven by a solid-state
power amplifier (SSPA) gives superior performance over both technologies alone in
certain circumstances (Fig. 6). This approach can be used with other vacuum electronic
devices, assuming help from advanced computer modeling tools and design skills.
M odem numerical techniques and advanced simulation software will provide the
capability to design and optimize for specified design goals and will be used to exploit
the full potential o f existing devices, as well as innovations and venturing into new or
previously unexplored regimes of amplifier operation. Furthermore, advancements in
Typical MPM Advantages
compared to
Traditional TWTA s and SSPA s
(100 watt Ku band example)
MPMs vs. TWTAs
Size*
5:1 reduction
MPW* V* s s e a s
8:1 reduction
Weight*
5:1 reduction
8:1 reduction
Noise
100:1 reduction
- 2 dB degradation
Efficiency
50% improvement
3:1 improvement
"Includes Power Supply and Coding
Fig. 6: MPM advantages versus TWTAs and SSPAs (Ku band: 12-18 GHz) [11]
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11
micro-fabrication technology provide new opportunities for vacuum electronic devices
producing power at higher frequencies (millimeter frequency and sub-millimeter
frequency range). Techniques such as those being developed for MEMS (micro-electro­
mechanical systems) and including techniques such as LIGA, SU-8, deep reactive ion
etching, and electrical discharge machining and some others [3], will be adapted in the
drive towards increasing sophistication in vacuum electronic devices [1]. With MEMS
technology, it becomes possible to bond tube-like structures onto a semiconductor
substrate.
These new technologies will lead to continuing advancements in vacuum
electronic devices.
B. Motivation for Computer Modeling of Vacuum Electronics
As mentioned above, one of the keys to future vacuum electronic device performance
optimization is the development of faster and more mature numerical modeling and
simulation capability, including hardware, software, and design skills. Advancements in
powerful computational tools and computer hardware aids have contributed significantly
to the improvement of vacuum electronic devices.
Using the conventional design
approach, extension o f the VPB (vacuum power booster) efficiency from 15% to 33%
has required the design, fabrication, modification, and testing of fourteen prototypes over
two years. However when the first of the advanced computer modeling codes became
available, the electronic efficiency was extended to 42% in a single step requiring less
than two months [1] [2]. Clearly, the motivation behind the use of computer design is to
reduce development cost by streamlining the overall design process - the design
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12
iterations and optimizations are made on computers instead o f by producing hardware, so
that prototypes are realized with fewer test cycles [4].
With the development o f high power, high frequency vacuum electronic devices,
operation becomes more sensitive to device dimensions and material properties. The
traditional approach o f scaling existing devices is therefore unacceptable and the added
sensitivity increases the importance of computation to obtain optimized designs. The
dynamic optimization of a vacuum electronic device to achieve better performance can
be made with the aid of highly sophisticated modeling and simulation capabilities.
Computation is used in all areas of microwave vacuum electronics design, including
the interaction circuit, the electron gun, the collector, the focusing system, and the
calculation o f thermal and physical stress on the structure.
This dissertation
demonstrates device modeling of several subassemblies. A more detailed decription of
the motivation behind each modeling effort will be discussed in the relevant section.
C. Scope of the Dissertation
The purpose of this dissertation is to develop computer design and modeling
capability for microwave vacuum electronic devices using simulation tools.
The
important members of the existing family of microwave vacuum electronic devices can
be classified into three categories according to the interaction between the
electromagnetic wave and the electron beam in each device (Fig. 7) [4]: linear-beam,
cross-field, and fast-wave devices.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
In linear-beam devices, also referred to as “O-type” devices [10], the electron beam
essentially flows in a straight line and an axial magnetic field (aligned with the axis of
the beam) is primarily employed. The general direction of beam motion is aligned with
both electric and magnetic fields as illustrated below [10]:
►B
E
— -
' - I electron beam
In contrast to linear beam devices, in crossed-field tubes the DC electric field is
perpendicular to the magnetic field.
The general motion of the electron beam is
perpendicular to both fields as illustrated below [10]:
* electron beam
In the slow-wave devices, one major category in the linear-beam device family, the
circuit structure is configured so that the phase velocity of the electromagnetic field is
slowed to the electron beam velocity. However, in fast-wave tube devices, the circuit is
usually a smooth waveguide or a large resonator in which no attempt is made to reduce
the velocity of the wave. Instead, the electron beam is injected in such a manner that
interaction can take place [10].
In the thesis, an extended interaction klystron (EIK) interaction circuit, a
TunneLadder TWT interaction circuit, a reflex klystron interaction circuit, and the RF
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14
input coupler, as well as the electron gun of a gyrontron traveling-wave amplifier (gyroTWT), are selected to demonstrate the design and modeling work.
Transit Short-Gap
Klystrons
Klystrons
Reflex Klystrons
Extended Interaction
Klystrons
Linear-Beam
Devices
Klystrons with
TW Output Section/
TWystrons
Traveling-Wave
Tubes (TWTs)
Slow-wave
Devices
Crossed-Field
Devices
Magnetrons
Backward-Wave
Oscillators (BWOs)
h Orotrons / Ledatrons
Crossed-Field
Amplifiers (CFAs)
Aroplltron
M-type TWA
l-type BWA
Gyrotrons
Fast-Wave
Devices
Monotron
08cillatore
Klystron
Configurations
\
..
FELs / Ubitrons
TWT
Configurations
BWO
Configurations
Fig. 7: Microwave vacuum electronic device family tree [4] © 2005 IEEE
Chapter 1 ~ 4 are the major part of this dissertation and describe the development of
the capability to design a complete interaction circuit for a 35 GHz EIK using computer
modeling tools. A unique, non-cylindrically-symmetric cavity geometry was chosen and
the complete interaction circuit structure, including the input/output waveguide along
with the electron beam and the RF, are fully modeled using a three-dimension particle-incell (PIC) computation code, as well as with two one-dimension simulation tools.
The klystron is one of two major classes of devices categorized as linear-beam tubes
and is a common type of microwave source based on coherent transition radiation from
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15
electrons. Transition radiation occurs when electrons pass through a boundary between
two media with different refractive indices, or through some perturbations in the medium
such as plates or grids [4]. In microwave wave tube devices, such as klystrons, these
perturbations are short-gap cavities within which the microwave fields are localized.
Figure 8 below contains the basic two-cavity klystron amplifier configuration. The
electron beam from the cathode at the left side of the figure is velocity modulated by the
RF signal in the 1st, or the “buncher” cavity. The modulation results in the electron
bunching which occurs as the beam drifts towards the 2nd, or the “catcher” cavity. The
resulting AC current in the beam induces current in the catcher cavity and results in RF
power out of the klystron which is substantially lager than the RF input power. To
improve the efficiency and the device gain, a number of intermediate cavities can be
inserted.
Input signal
Buncher
cavity
Electron
gun
Cathode
B e a m D r i f t space
Output signal
Catcher
cavity
Anode
Fig. 8: Schematic of two-cavity klystron amplifier [4] © 2005 IEEE
The extended interaction klystron (EIK) replaces one or more of the single-gap
cavities o f the conventional klystron by a structure containing two or more interaction
gaps. Compared to conventional klystrons, the EIK has increased bandwidth due to the
combined use o f the slow-wave structure (here a coupled-cavity TWT). Furthermore, the
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16
power-handling capability is also increased because the voltage at each interaction gap is
reduced. The configuration comparison between the klystron and EIK is as shown below:
out
Interaction
Structure
Gun
C o lle c to r
Interaction
Intensity
D ista n c e
Fig. 9: Klystron [86] © 1973 IEEE
r
1 H
‘C U L J
r~ n
r H
^
I I 1 L-LJ
Fig. 10: Extended Interaction klystron [86] © 1973 IEEE
Because of high efficiency and the capability of providing high peak and average
powers, klystrons have been used in many applications such as radar, communications,
and accelerators (medical and for particle physics) [4]. However, the combined needs for
relatively wider bandwidth, higher frequencies and higher power have led to a number of
developments using EIKs, such as the 94-GHz radar for the CloudSat project, NASA’s
Earth Space Science Pathfinder mission [12].
This dissertation is concerned not only with developing the skills and techniques for
designing an EIK interaction circuit using computer modeling tools, it also focused on
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17
the validation and improvement of a particular 3D PIC code by comparing the results to
theory and results from other simulations. Many software bugs were found and fixed
during the modeling process and this contributed to the maturation and advancement of
this PIC code [Appendix IV].
Chapter 5 presents the computer modeling work of a 29 GHz TunneLadder TWT
(traveling wave tube) with a very unique geometry that can be constructed using modem
fabrication technologies.
This work was initially performed during the design and
modeling of a mini vacuum power booster for Ka band MPM applications [70].
The two primary elements of a TWT are an electron beam and a guided, traveling
electromagnetic wave. TWTs amplify microwave energy by continuous deceleration of
electrons moving synchronously with an electromagnetic wave. There are two types of
TWTs. One type uses a slow-wave-structure (SWS) where the electrons interact with
longitudinal electric field components in the wave; the other uses a fast-wave-structure
(FWS) where the electrons interact with transverse electric field components in the
guided wave. Currently, most o f the TWTs used in defense and commercial applications
are of the SWS type, as is the TunneLadder TWT.
Along with the klystron, the SWS TWT is one of two major categories of devices
know as linear-beam, or O-type tubes based on Cherenkov radiation [4]. Cherenkov
radiation requires the electromagnetic wave phase velocity (vPh=oi/kz) to be less than the
electron velocity (ve).
The electrons propagate near a periodic structure (the SWS),
whose fields can be represented as the superposition of spatial harmonics. The phase
velocity o f the synchronous spatial harmonic of the RF wave is close to the electron
beam velocity, as illustrated in Figure. 11. The beam line (co =kzve), characterizing the
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electron beam, intersects the dispersion curve of the SWS and therefore continuous
interaction between the electrons and the RF wave takes place.
The group velocity
(vg=d<j)/dkz) is positive for a TWT (a forward wave device), and negative for a BWO (a
backward wave oscillator).
Although collaboration between the parties was suspended and the design work was
discontinued, this work has resulted in the development of modeling capability for this
uniquely shaped TWT. This capability can contribute to the future design work for this
type o f microwave amplifier, to fulfill the need for a compact, low cost, and high power,
high frequency microwave amplifier.
Uniform Axial Magnetic Raid
---------------------------
Electron Beam
forward wave
Output Waveguide
Cutoff Waveguide
(0
JW T
.Interactlor
BWO /
Interaction
Dispersion Curve
' for Slow-Wave
| Structure |
Fig. 11: Schematic diagram and dispersion diagram for slow wave structure [4]
©2005 IEEE
Chapter 6 demonstrates the validation work o f a millimeter wave reflex klystron from
CPI in Canada. The validation work of the interaction circuit is not only to demonstrate
the modeling ability using this particular PIC modeling tool, but also to develop the
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19
design skills for reflex klystrons in higher frequency regions (in particular, the THz
region).
A reflex klystron is a single-cavity klystron oscillator. As shown in Figure 12, an
electron beam passes through a cavity gap into a region where the negative voltage on a
repeller electrode causes the electron to reverse direction. Assuming that a signal exists
across the cavity grids, the electron beam is velocity modulated, similar to the
modulation in klystron. As the electron beam drifts into the repeller region, bunching
occurs and if the bunched electrons return to the modulating gap at the correct time, they
induce a current that reinforces the current in the cavity.
Thus, the electron beam
provides the feedback look that causes the device to oscillate [10] [68].
^
E lectron B eam
ficflcetor
Colliodc
Electrode Structure
Fig. 12: Reflex klystron Scheme
Reflex klystrons are low power and low noise devices that have historically been
used as local oscillators. While most of the low frequency reflex klystrons have been
replaced by solid state devices due to lower cost, new micro-fabrication technologies
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20
provide a possibility o f reflex klystron applications in millimeter wave and sub
millimeter wave regions.
As mentioned in the previous section (high frequency applications), the sources of
radiation generally used for Terahertz (or sub-millimeter-wave) science and technology
have so far been expensive or bulky [19]. The reflex klystron is a promising candidate
for use as a source of Terahertz power at the milli-watt level [20]. Because only one
cavity is used and no RF driver or bulky magnetic focusing system is required, the reflex
klystron has good chance to fulfill the need for Terahertz power sources for smallfootprint, cost-effective and efficient fundamental sources for medical imaging and
remote sensing applications [20].
Chapter 7 demonstrates the design optimization and computer simulation o f the RF
input coupler and the electron gun for the UCD (University of California, Davis) 94 GHz
Gyro-TWT.
Both o f these two devices have been built, tested, and demonstrated
superior performance over the previous subassemblies [17].
By employing a non-resonant interaction circuit structure (basically a waveguide, see
Fig. 14), the gyro-TWT can provide significantly higher peak power over the state-of-art
linear-beam amplifiers while retaining the broadband capability that makes it attractive
for advanced radar system and communication applications [15].
As a particular
example, the 92-94 GHz atmospheric window at W-band offers superior performance
through cloud, fog and smoke in radar applications.
Therefore, the University of
California, Davis, has developed a state-of-the-art W-band TEoi gyro-TWT (Fig. 13).
This UCD W-band TEoi gyro-TWT has undergone numerous modifications, with each
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21
version extensively hot tested, and the experimental results have been encouraging [14]
[17] [72]-[83].
Gun Vacuum Pump
Beam Tunnel
Interaction Circuit
s—
Superconducting
Magnet
MIG
RF Input
Collector
RF Output
d
Main
Vacuum Pump
Fig. 13: Schematic of the TEOI gyro-TWT amplifier
E.M. WAVE
EM . WAVE
V j *i__________ i*iJ /
w
e beam
H
v w
y w v w t v w T w v tn n n n r
,1UUU*QJUULajlJUUL&SLJkJUULSUlM .~
/
N
Fig. 14: Gyrotron traveling-wave tube amplifier
The RF input coupler and output coupler (Fig. 13) are employed in the W-band
TEoi gyro-TWT system to efficiently couple the RF power into and out of the interaction
circuit. For a high performance gyro-TWT amplifier, a well-matched input coupler is
essential (return loss ~ 20 dB). It is also very important that the wave is coupled into the
desired waveguide mode of the interaction circuit [18]. The input coupler of the UCD
gyro-TWT serves as a mode converter that converts the fundamental rectangular TEio
mode to the circular TEoi operating mode [17].
Important issues to consider in the
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22
design o f a high order mode coupler include: the coupling strength and bandwidth, the
mode purity, and reflection coefficient.
This dissertation demonstrates the optimization of the previous input coupler using a
3D electromagnetic-field simulation code. The optimized coupler has been built and
tested using a vector network analyzer and showed superior performance compared to the
previous design.
Efficient operation of the gyro-TWT requires a high quality electron beam with lowvelocity spread and uniform current density around the circumference of the beam. The
magnetron injection gun (called MIG) (Fig. 13) has been utilized most extensively for
gyro applications [16]. In MIGs, electrons are drawn in a generally radial direction from
the cathode surface and then formed into a hollow beam by electric and magnetic fields
(Fig. 15 & 16) [10].
Gun Anode
C athode
Cathode
Focus
Electrode
Accelerating
Anode
E lectron B eam
Fig. 15: Magnetron Injection Gun; double anode design
(MIG can also be single anode design)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Axial D ista nce
Fig. 16: Electron trajectories in gun region of gyrotron [52] © 1985 IEEE
Advanced computer modeling allows a design providing electron velocity spreads of
less than 8%, which is adequate for efficient operation [13]. This dissertation employed
a computer trajectory code to optimize and redesign the previous MIG for the UCD Wband gyro-TWT and has reduced the velocity spread significantly, as well as the RF
break down or arcing possibility. The optimized MIG has been built and contributed to
the improvement of the UCD W-band gyro-TWT experimental performance [14].
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24
Chapter 1
Introduction of Design and
3D Computer Modeling of
A Ka-Band EIK
1.1 Klystron Overview
Klystrons are the most efficient linear beam devices and are capable of the highest
peak and average power. The principles can be simply described by reference to Fig.
1.1.1 where:
1 The electron gun produces a flow of electrons (DC).
2 The bunching cavities regulate the speed of the electrons so that they arrive in
bunches at the output cavity. The electron velocities are modulated by the first
cavity gap and the bunches are formed around the electrons crossing the first cavity
gap where the sinusoidal voltage transitions from negative to positive (from
decelerating to accelerating). Bunches arrive at the output cavity (the second cavity
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25
in the figure) with a period of T0 which corresponds to the period of the sinusoidal
power input to the first cavity [25].
3 The electron bunches excite microwaves in the output cavity of the klystron.
4 The microwaves flow into the waveguide.
5 The electrons are absorbed in the beam collector.
Russell Varian, one of the klystron inventors, provided the following explanation of
velocity modulation [27]: “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 the 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 electron
beam is produced by this principle of bunching, rather than by the direct control of the
grid of atriode...”
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26
-—I
RF
OUTPUT
RF — *1
INPUT *
, MMMfttMCOWt
Fig. 1.1.1: The Applegate diagram [28]
The bunching figure shown in the Applegate diagram is ballistic and does not
account for electron space charge.
In reality, the space charge will modify these
trajectories and the interaction between the cavities and the beam can be better described
using space charge wave theory [25].
In order to maximize the efficiency, the best
location for the second cavity is at one-fourth of the reduced plasma wave length where
the maximum electron bunching occurs [10]. More detailed about plasma wave length
will be discussed in the plasma wave length modeling session.
The diagram above is a basic geometry of a two-cavity klystron.
Intermediate
cavities can be added to increase the output power and the bandwidth. The gain of multi­
cavity klystrons can be 60 dB or higher, and efficiencies of 70 % are not unusual. On the
other hand, klystrons are narrow-band devices compared to other linear beam devices,
such as traveling wave tubes [4].
A klystron cavity can be usually treated as an equivalent parallel RLC circuit driven
by an RF current source, (Fig. 1.1.2). This current comes from the fundamental current
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27
component in the beam that has been bunched by preceding cavities. Such a circuit is
characterized by its resonant frequency coo, the total cavity factor Qt, and the cavity
structure characteristic R/Q:
11
1
Fig. 1.1.2: A parallel RLC circuit representing a klystron cavity
_L-J_ _L J_
Q t ~ Q b + Qo + Qe
R _
Q
1 _ l~L _
co0C
V2
VC 2co0W
V - Voltage across the gap
w - Energy stored in the cavity
The three cavity factors above correspond to the beam loading losses (Qb), the cavity
ohmic losses (Qo), and the power loss due to the external load (Qe, applied to the input
and the output cavities). The ratio of shunt impedance to the cavity quality factor, R/Q,
can be physically described by the ratio of the square of the voltage across the interaction
gap of a klystron cavity and the energy stored in the cavity.
Klystron performance depends on the requirements of the application. Klystrons can
be used to power the electron accelerators that are used as both a source of X-rays for
medical applications and high energy electron beams for high energy physics. Klystrons
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28
can also be used for communications, such as high power amplifiers in satellite ground
stations and mobile troposcatter communication terminals [50].
1.2 Motivation for 3D Modeling of 35 GHz EIK and
Desired Goal
The traditional way o f making vacuum electron tubes is trial and error. To save time,
expense, and human resources, accurate design and modeling is necessary as the
applications move to higher power, frequency, and bandwidth. Modeling of vacuum
electronic tubes is a complex task and the simulation software is not always accurate.
This work is to develop the 3D design and modeling capability for a 35 GHz extended
interaction klystron (EIK). The reason for selecting a frequency of 35 GHz is because it
is in a frequency window o f relatively low atmospheric attenuation and therefore there
are an increasing number of applications in this frequency range, such as radar, plasma
heating, radio astronomy, and communications [4] [51]. The motivation for selecting an
EIK device is to increase the output power and bandwidth of klystrons.
Since the
maximum peak power in klystrons is limited by RF breakdown, the ability to handle
higher output power can be obtained by adding more gaps in a cavity (Fig. 1.2.1). A
geometry of a traveling wave tube (TWT) is employed (shown here is a coupled cavity
TWT geometry) and the electrons can interact with more phase velocities of the RF, thus
the operational bandwidth increases.
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29
M ultiple G a p
S in g le G a p
Coupling Slot
S in g le K lystron C av ity
EIK c o u p le d c avity
Fig. 1.2.1: The changes from a single gap cavity to a multiple cap cavity
The conventional klystron cavity is cylindrically symmetric and, once the cavity is
constructed, it is not simple to tune the resonant frequency of each cavity. However, the
flexibility o f tuning each cavity frequency is necessary due to fabrication imperfections
or design and modeling inaccuracies. The design illustrated in Fig. 1.2.2 provides a
sufficiently flat and deformable surface so that the cavity frequency can be easily tuned
to optimize output power and bandwidth. This two-gap racetrack cavity design also
increases the R/Q value from 105 (single racetrack cavity) to 175, resulting in higher
output power as well as broader operating bandwidth.
Beam
*
Coupling Slots
Adjustable
surfaces
Flat, deformable
wall surfaces
permit easy tuning
Fig. 1.2.2: Tunable single racetrack cavity and coupled racetrack cavity
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30
Note that for the geometry shown above, a 10 micron movement in the wall results in
a resonance shift of about 60 MHz, and therefore a tunable wall is very important.
Due to the lack o f verified, accurate, efficient and mature PIC codes, people did not
start using PIC codes to fully model vacuum electron tubes until very recently [29] [30].
2D modeling using PIC codes was usually performed on tubes with a cylindrically
symmetric geometry or by use of an equivalent circuit for a non-cylindrically-symmetric
tube to simply the modeling [31].
Because o f the unique non-cylindrically-symmetry geometry of the racetrack coupled
cavity (Fig. 1.2.3), 3D modeling is required for accurate results. The purpose of this
dissertation work is to develop the 3D design and modeling capability for noncylindrically-symmetric circuits.
The specifications for the EIK design study are as
follows:
♦
Output Power: 1 kW at 35 GHz (CW)
♦ G ain : 35 dB
♦ Bandwidth : 200 MHz
♦ Efficiency: 18 %
♦ Beam Parameters: 14 kV, 400 mA
♦ Beam Tunnel Radius: 0.5 mm
In an EIK, the beam power is determined based upon the required output power and
the electronic efficiency of the circuit. There is a tradeoff between the beam voltage and
the beam current. Lower perveance (P=I/V15) requires lower magnetic fields because of
lower currents; thus, there is a weight reduction of the focusing system. However, lower
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31
current density results in a longer tube length because of the longer reduced plasma wave
length. On the other hand, higher perveance (lower voltage) reduces the RF breakdown
possibility. Normally, the beam voltage and the beam current are determined based on
the beam power and the selected perveance.
In this dissertation case, the beam
parameters and the beam tunnel radius are fixed by existing gun parameters. The output
power can be increased by adding more cavities; however, that increases the total length
and the weight (also longer focusing system) of the device.
A 3D PIC code - MAGIC 3D (more detail in the design tools session and Appendix
IV) is chosen to demonstrate this work, as well as two other codes that perform ID
simulations using equivalent circuits.
3D PIC modeling requires considerably more
simulation time and computational memory compared to ID simulations. Due to the
non-cylindrically-symmetric characteristics, validation of the ID equivalent circuits
using 3D modeling is necessary.
This work also helps to advance this particular PIC code as many bugs o f the
software were found and fixed during the simulation process. A mature PIC code will
support the accurate design and modeling of vacuum electronics devices.
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32
1.3 State-of-the-Art EIK
The state-of-the-art EIK as demonstrated by CPI Canada in the year of 2005 is 2 kW
peak pulsed power with 1.4% duty cycle (average power 28 watts) at 94 GHz with a 1 dB
bandwidth of 250 MHz [12]. Electronic efficiency over 20% has been achieved and with
a depressed collector the EIK overall efficiency was increased to 32%. The weight is 6.2
kg.
The best commercially available CW Ka-Band EIK (CPI Canada) is 1 kW output
power with 250 MHz bandwidth at the frequency range from 27 GHz to 31 GHz. This
device serves as a source for satellite communication uplink applications [51].
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33
Chapter 2
EIK Modeling Tools and
Circuit Design
2.1 Modeling Tools
There are three computational tools used for this EIK design: the one dimensional
small signal MathCAD calculation worksheet (Appendix I), the one dimensional large
signal code AJDISK [25], and the three dimensional particle-in-cell (PIC) code MAGIC
3D [Appendix IV] [34]. The first two ID codes were developed by klystron experts and
are basically restricted to use in klystron design. In contrast, MAGIC is a more general
application PIC code and can be employed in the simulation of many other electronic
devices, such as TWT, klystron, and EIK etc.
The ID small signal code is written in MathCad format and is primarily intended for
tuning the individual cavity parameters and determining the spacing between the cavities
to achieve the desired small signal gain and the bandwidth in the linear region of
operation. Since the klystron usually operates in the nonlinear region, the ID MathCAD
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34
is only suitable for an initial design. The ID large signal code AJDISK models the
interaction between the electron beam and the RF field to predict the klystron
performance in the nonlinear region [25]. However, since only the axial component of
the electron motion is considered in ID AJDISK, the design needs to be validated using
MAGIC 3D, which fully models the interaction between the electrons and RF in all
dimensions.
2.1.1 ID MathCAD Calculation Worksheet
The ID MathCAD code (Appendix I) [25] is written based on kinematic theory and
space charge theory; which is specifically used to predict the klystron gain-bandwidth
performance in the linear region of operation. This calculation uses a small signal space
charge wave and the equivalent circuit model of the cavities to predict the gain,
bandwidth, and other parameters. A few assumptions are listed below:
1. The beam velocity is assumed to be uniform throughout the circuit (the faster
electron bunch does not pass the slower electron bunches).
2. No electron motion or current is assumed to exist along any dimension, other than
the axial dimension (this requires the presence of a strong magnetic field).
3. The analysis assumes small-signal RF variables. Products of RF quantities are
neglected (the ratio o f the each calculation output; such as the gain; is presented
and no input or output power quantities are calculated).
4. This particular template only considers velocity modulation.
The level of
inaccuracy increases when the cavities are spaced by less then a quarter of the
reduced plasma wave length. When the cavities are placed close to each other,
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35
the current modulations need to be considered to obtain more accurate
calculations [36].
The calculation is based on:
1. Kinematic Theory - Calculations of coupling coefficient M and beam loading
conductance.
2. Space Charge theory - Consideration of depressed potential and the plasma
reduction factor.
3. Small-Signal Stagger-Tuning Theory - Covering a wide bandwidth by separating
the resonances of individual narrow band resonators on gain-bandwidth
calculations.
The coupling coefficient is measured at the beam tunnel radius in each cavity
modeling using MAGIC 3D and averaged through the beam tunnel in MathCAD
(Appendix I).
2.1.2 ID AJDISK
The ID AJDISK code slices the beam into a set of charged disks that can move
separately (Fig.2.1.1). This was first written by Hiroshi Yonezawa [35] and later
modified by A. Jason [25]. This code is used to predict klystron performance in the
saturated region of operation because it can model the space charge in the large signal
region (a faster electron bunch can pass a slower electron bunch).
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36
Fig. 2.1.1: An electron beam is sliced into a set of charged disks in AJDISK
In AJDISK, each disk can move freely along the axial direction at a different velocity
- the motion is governed by the space charge from other disks and the electric field
associated with each cavity. The algorithm is as listed below:
1. Evaluate the equation o f motion based on the field.
2. Calculate the induced current for each cavity.
3. Calculate the fundamental component of the induced current using a Fourier
expansion and the induced current from the previous step.
4. Calculate the induced voltage in each cavity, where the induced voltage
is
impedance times the fundamental component of the induced current.
5. Repeat the above steps until the calculated voltages of two successive iterations
are equal to within some predefined percentage.
6. Calculate the gain and the efficiency.
Similar to ID MathCAD, ID AJDISK also assumes that the beam is sufficiently well
focused so that no electron or current can move in any direction other than the axial
direction; therefore, all disks have the same diameter. This requires a strong magnetic
field.
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the gap
37
Note that in AJDISK, the coupling coefficient M is represented by the parameter “k”,
which determines the shape of the electric field distribution across the klystron gap
according to the following Gaussian equation [25]:
E2{ z ) - - ^ = e *2<z w )2
Assuming that Zcenter is located at the origin, the coefficient M is defined as
J Ez(z)eM2dz
J Ez(z)dz
The denominator integration is normalized to one with its substitution into M:
£ ,(z ) = 4 . « - * v
sjn
Therefore, k can be calculated by
k(M ) = ■
Pt
2 ALn
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
38
This function is used to transfer coefficient M to k in AJDISK. In practice, k has been
found by using a program called SUPERFISH [25] to simulate the field at 0.707 times
the beam radius (one method of averaging - the radius where is half current located) [25].
In this paper, M is measured at the 0.707 times the beam radius in each cavity using
MAGIC 3D, and then transferred to k using the function listed above into AJDISK.
2.1.3 3D PIC Code - MAGIC 3D
MAGIC 3D is a three dimensional particle-in-cell code [Appendix IV], developed by
Mission Research Corporation (MRC), now ATK Mission Research; which can be
generally used to model any device involving RF and particles. Both MathCAD and
AJDISK are 1-D simulation tools with some restrictions and assumptions, while MAGIC
3D can model a complete 3D structure, including the particles (beam) and the
interactions with the RF in all the directions.
Below is a flow chart of the simulation setup with MAGIC (Fig. 2.1.2)
Imttf OuitntNt
m iSs__________ ___ _________________
Fig. 2.1.2: Design flow chart to create a MAGIC simulation
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
39
Coordinates can be Cartesian, cylindrical, polar, or spherical. The object includes
points, lines, areas, and volumes of diverse geometries. Gridding must be specified
correctly. The spatial grid is the primary determinant of the simulation result accuracy
and all spatial phenomena of interest must be appropriately resolved. Mesh size can be
set differently at difference places to save the simulation time. A finer mesh may be used
on the area where the geometry is more complicated and where the particle trajectories
are. Time is specified with the duration to be covered by the simulation. About the field
algorithm, one may replace the default electromagnetic field algorithm with another if
needed.
The default field algorithms are suitable for mildly relativistic particle
simulations, e.g., y up to 1.5 (beam voltage approximately 255.5 kV).
Physical
properties are determined by conductivity, permittivity, and dielectric constant etc. The
boundaries must be appropriately setup.
Regarding to the initial conditions, most
simulations start from trivial initial conditions without fields or particles. The particles
have to be defined through emission processes and the particle algorithms can be altered
with the kinematics time step to help save the computation time in non-relativistic
simulations. Finally the output needs to be explicitly specified. Choosing what should
be output is important to obtain the results needed for analyses while remaining an
efficient usage of the computer memory and simulation time.
It has been shown by using these three tools, that a cylindrically symmetric klystron
can be reasonably designed (Fig. 2.1.3) [25]. In the figure below, only MAGIC 2D was
used because the demonstrated circuit is a cylindrically symmetric device.
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40
1 MW CW B-Factory Klystron (Small Signal)
00
■o
c
■to«
O
45
-> " - T
t
WCT%jL
—
—
M ea su red (Pin=4.1 W)
AJ DISK (Pin=4.1 W)
—
MAGIC 2D (P in = 4 .1 W )
MathCAD
-.wa-amn
35
470
472
474
476
478
480
482
Frequency (MHz)
Fig. 2.1.3: 1MW CW B-Factory Klystron (Small Signal) [25]
This dissertation chose a non-cylindrically-symmetric coupled cavity Ka-band EIK
(which is one o f the most general circuit and has numerous applications within this
frequency range) to demonstrate the design and modeling with MAGIC 3D code and to
compare the results with ID calculation and simulations.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
41
2.2 Circuit Design Loop
The center frequency of this EIK was selected to be 35 GHz. The beam voltage,
beam current, tunnel radius, and beam radius are fixed due to the use of an existing gun
design.
Since MAGIC 3D requires tremendous calculation time and computational
resources, we designed the initial EIK equivalent circuit using ID MathCAD and
AJDISK, including the cavity R/Q, external Qe, Qo, cavity frequency, and the gap-to-gap
distance (the spacing o f the center of a coupled cavity to the center of the adjacent
coupled cavity, see Fig. 2.2.1). These variables are highlighted in pink and yellow colors
in the MathCAD template (Fig. 2.2.2) and the fixed parameters are highlighted in green
color. We first use MAGIC 3D to model the initial coupled cavity with a gap length (the
distance between noses in each cavity, see Fig. 2.2.3) of 0.4 mm to obtain the R/Q, Qo,
and coupling coefficient M at the tunnel radius, then input these initial values to
MathCAD and adjust the Qe, Qo, cavity frequency, and gap-to-gap distance to optimize
the design in the small signal region. Then these variables are used as inputs in AJDISK
to calculate the large signal results (Fig. 2.2.4). The design loop is as shown below (Fig.
2.2.5):
G ap-to-G ap D istance
Fig. 2.2.1: Complete modeling with beam and RF using MAGIC 3D
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
42
Ka-Band EIK
B eam Voltage (V)
Num ber o f D ata Points
B eam C u rre n t (A)
Calculation Bandwidth (GHz) B W a .3 1
C enter Freqency (GHz)
Gain (Max)
T unnel Radius (m)
npoints a 30
Gain a t C enter Frequency
P lot M a rk e rs
B eam Radius (m)
fl . 35.083
fh - 35.145
Rem em ber to update the shape factor
d2 i f the fill-factor changes.
ld B Bandwidth (GHz)
pe = 3208.421
Bndwthdb = 0.041
3dB Bandwidth (GHz)
|lq = 157.904
% B W (ldB )
fbwl =0.118
% BW (3dB)
fbw3 = 0.177
f a = 1.562
Em pirical expression fo r Qe o f the output
cavity- not used in the g ain calculation:
46.5
( A-10 )
Ilf a 1.5
39.5
Vo
Io
Q'N
f
TT
2
"1
25.5
Q eN = 434.786
N um ber o f K lystron Cavities:
Na 7
Gun Microperveance (uA/VAl 5 )
K=
B rillouin Field (gauss)
18.5
1.241
34.8 34.84 34.88 34.92 34.96
Bbr = 1928.113
Jbeam = 203.718
Cathode C u rre n t D ensity (A/cm2)
Jcathode := 5
_
Gun Convergence
Jbeam
Set "Sw itch" equal to 1 to use the JO knife edge
approxim ation fo r Ma. Set “ Sw itch" equal to 0 to
use the M a defined below.
= 41
Switch a 0
J cathode
Cavity Rs/Q
( Ohms )
Gap Length
(m)
Q o j b 1837
f j * 34.858
Lj b 0
d t a 0.00045
Q e 2 = oo
Q o 2 a 650
f2 b 34.895
L j 3 0.0065
6^ a
0.00045
R Q 3 * 154
Q e3 b oo
Q o3 b 655
f3 b 35.051
L j a 0.0065
dj
0.00045
RQ4 b 154.5
Q »4 ‘ ®
o
O’
f4 s 35 .111
s
155
Q e5 s o o
R Q 0O 99
Q a6 - o o
RQ ? b 162
Q «7
b
191
II
Q e j b 355
00
R Q j a 158.4
R Q 2 b 152
RQ 5
Q o5 b 1821
f5 b 35.141
Q o6 s 2242
f6 =
Q o?
f? b 35.00
b
2089
35.04 35.08 35.12 35.16 35.2
fos
Beam C u rre n t D ensity (A/cm2)
„
35
a
C oipling Coef.
( @ r= a )
M t|
M a2
M aj
Coupling Coef.
(T o ta l)
a
0.9318
a
0.9312
b
0.9296
0.5869
0.5865
( 0.5883^
0.5879
L4 b 0.0065
d4 a 0.00045
M a 4 a 0.9290
L j = 0.0065
d5 a 0.00045
M a^ a 0.9286
d6 a 0.0005
M a fi a 0.9318
dj
M a?
0.5863
0.5883
35.20
a 0.0045
L,
b
0.0045
a
0.00045
b
0.9302
Fig. 2.2.2: MathCAD template for EIK
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1,0.5873
Fig. 2.2.3: Coupled Cavity Modeling Using MAGIC 3D
k a b a n d EIK
■ 1 4 .0 0 0 kV
■ 0 .400 A
- 3 6 .0 0 0 GHz
0 .6 0 0 m m
« 0 .2 6 0 m m
0 .2 0 0 W
■ 1 .0 0 7 kW
■ 3 7 .3 9 0 dB
Vo
lo
f
a
b
P in
Pout
G ain
z(m )
f(MHz)
Qe
Qo
R/Q
k
d(m )
V(kV)
0.0 0 0 0
3 4 8 6 8 .0 0
3 6 6 .0 0 0
1 8 3 7 .0 0 0
1 6 8 .4 0 0
23 0 4 .0 0
0.0 0 0 4 6
0.0 9 1 8
0.0066
3 4 8 9 6 .0 0
9 9 9 9 9 .0 0 0
6 6 0 .0 0 0
1 6 2 .0 0 0
2 301.00
0 .00046
0.2208
0.0 1 3 0
3 6 0 6 1 .0 0
9 9 9 9 9 .0 0 0
6 6 6 .0 0 0
1 6 4 .0 0 0
2 3 0 2 .0 0
0 .0 0 0 4 6
0.3 6 3 8
0.0 1 9 6
3 6 1 1 1 .0 0
9 9 9 9 9 .0 0 0
1 8 4 6 .0 0 0
1 6 4 .6 0 0
23 0 2 .0 0
0 .0 0 0 4 6
1.2 0 0 7
0 .0 2 6 0
3 6 1 4 1 .0 0
9 9 9 9 9 .0 0 0
1 8 2 1 .0 0 0
1 6 6 .0 0 0
2 3 0 0 .0 0
0 .0 0 0 4 6
2 .8 9 4 6
0.0 3 0 6
3 6 2 0 0 .0 0
9 9 9 9 9 .0 0 0
2 2 4 2 .0 0 0
9 9 .0 0 0
2 3 0 6 .0 0
0 .0 0 0 6 0
2 .6 9 6 0
0.0 3 6 0
3 6 0 0 0 .0 0
1 9 1 .0 0 0
2 0 8 9 .0 0 0
1 6 2 .0 0 0
2 3 0 6 .0 0
0 .0 0 0 4 6
8.2381
0.6
=
H
i* S I5 rF r~ ~
E
0.5
V
§ |l |g
\
: !-:
^ -: / |
i i . ! ■i '
!' i i! i
'
0.4
0.3
||S
~n u a n c e k. /
0.2
0.1
Distance (mm) x1/Step
38
1.S
(II / l o ) * 5
(12/10J*5
B
0>
t 1.0
3
O
0.S
•1
1
II
II
I I ..
- -
Energy/Eo
2.0
Lamda Q ■0.04229m
M=
0.01738
B eta*
1.00000
Theta(V7) 4.10140
EfflC. K s
0.24659
Emc. E = 0.24546
o 2.0
|
■
Distance (mm) x l/step
38
1.0
1.8
*
0o
SEE
0.8
o 0.4
0.2
Distance (mm) xl/Step
38
1.0
Fig. 2.2.4: AJDISK output
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Electric Field
Mm
m
R e so n a n c e Frequency,
R/Q, Q
Coupling C oefficient M
Fig. 2.2.5: Design Loop
Note that the MathCAD code is only suitable in the small signal region, and since the
EIK operates in the large signal region, one needs to iterate between MathCAD and
AJDISK to ensure that the large signal results are satisfied. MathCAD calculations take
about a second to perform, while AJDISK may take a few minutes, depending on the
number of calculation points and the CPU speed. When a satisfactory design is obtained
from AJDISK, a complete simulation, including the 3-D geometry, RF, and beam, are
modeled using MAGIC 3D (Fig. 2.2.1). The MAGIC 3D modeling takes approximately
41 to 55 hours on a PC with 3 GHz CPU depending upon the driving frequency (more
detail in the mesh size and time step section). The results are then compared to the 1-D
simulation results.
After tuning, the R/Q, Qo, and M values of each cavity will vary slightly because the
geometry is adjusted to tune each cavity to the correct resonant frequency. Each cavity is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45
then modeled again using MAGIC 3D and the new values of each parameter are used for
the MathCAD and AJDISK inputs.
It is important to verify the ID equivalent circuit with a complete 3D simulation. If
the results are comparable, the ID equivalent circuit is verified and therefore can be used
to save a significant amount of simulation time and computational resources in the design
process. However, a 3D modeling is necessary beforehand to validate these equivalent
circuit models. In addition, 3D simulations allow further analysis into issues such as
beam scalloping, beam transport and focusing using different magnetic fields, noise and
stability from other undesired modes, and RF breakdown possibilities in real devices, all
items which can not be explored in ID.
Note that the coupling coefficient M in MathCAD is measured at the beam tunnel
radius in each cavity modeling by MAGIC 3D and times an averaging coefficient across
the beam, giving a value around 0.59 (Appendix I). M in AJDISK is measured at 0.707
times the beam radius in each cavity modeling by MAGIC 3D, giving a value about
0.616. This number is then transferred to k using the function listed in section 2.1.2 into
AJDISK. Although the M input in AJDISK is slightly higher then M input in MathCAD,
the simulations gave more reasonable results comparing to those when the same M is
used. The reason could be that the beam radius remains almost constant in the small
signal region, but varies significantly in large signal region. In the large signal region,
the electron bunches come closer to the gap and result in higher M. The modeling results
will be provided in a later section of this chapter.
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46
2.3 Coupled Cavity Design
The initial cavity geometry was derived from an existing industrial design (Fig. 2.3.1).
The primary modifications are changing the slot at the side to a rectangular shape in
order to simply fabrication (Fig. 2.3.2), and adding an additional slot to increase the
bandwidth.
Larger slots increase the coupling and the bandwidth; however, excess
coupling causes the operation to switch from the cavity mode to the slot mode (Fig.
2.3.3).
To avoid having energy stored only in the slots, and therefore having no
interaction between the gap fields and the electrons, this device needs to be operated in
the cavity mode. To operate the device in the cavity mode and to achieve the maximum
bandwidth, an optimum coupling slot size can be determined by performing a series of
dispersion simulations (Fig. 2.3.4).
The optimum coupling slot size maximizes the
coupling while operating in the cavity mode. In addition to increasing the bandwidth,
higher coupling also increases the frequency separation between the n mode and 2k
mode. As will be discussed later in the stability analysis session, the highest n mode
frequency o f the final design (33.68 GHz; occurs in the 5th coupled cavity) is about 1.2
GHz lower then the operation frequency edge (about 34.87 GHz). Larger separation
between these two modes ensures that a lower-frequency n mode will not be excited, and
that operation remains in the desired 2n mode.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
47
Beam
Will be replaced by
a rectangular coupling
slot and put at the side
to make th e fabrication
easier.
S in g le elliptic cavity
EIK coupled cavity
Fig. 2.3.1: Initial single gap cavity and coupled cavity design
C o u p lin g S lo ts
Coupling Slots
Fig. 2.3.2: Coupled cavity with modified coupling slots
Operating Frequency
Mode switch by larger slots
_L
SLOT
MODE
MERGED
MODES
CAVITY
>
CAVITY
MODE
MODE
0
0
PHASESHIFT/CAVITY
Fig. 2.3.3: Dispersion diagram of the coupled cavity [47] © 1988 IEEE
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
48
33.16 T
33.14 N
33.12
X
vO
^-*'
o
33.1
§
33.08
Increase
the number of slots
or the size of slots
33.06
33.04
O p tim u m c o u p lin g slo t for b ro a d b a n d
33.02
0
17t
271
P h a s e S h ift P e r C avity
Fig. 2.3.4: Dispersion diagram for different slot sizes
For the two-gap coupled cavity, there are two standing wave modes, as illustrated in the
pictures below. The field pattern is TMoio-like mode.
0 - mode
Ez on axis
j i ■ mode
Ez on axis
#4
5^
N
°
T
e
•4
-2
0
Z (l)
2
4
•I
(E-3)
0
Z (rn)
I
(E-3)
Fig. 2.3.5: Axial direction electric field pattern of zero (27i) mode and 7i mode of the
coupled cavity
For our I n mode EIK, the spacing between the gaps in the coupled cavity is designed to
be 2 mm, corresponding to a transit angle of 2n for a 14 kV beam.
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49
2.3.1 M2(R/Q)
Since the gain is proportional to the value of M2(R/Q), and the values of M and R/Q
vary due to the gap size, it is important to use a gap size that maximizes the value of
M2(R/Q). Larger gap results in higher R/Q, but lower M. A series of simulations can be
performed to find the optimum gap size. As the gap size increases, the capacitance of the
cavity increases, resulting in a lower resonant frequency. Therefore, the geometry of the
cavity needs to be adjusted to maintain the same resonant frequency. For larger gap, the
cavity length needs to be decreased.
Initially, a rough grid model was used to determine the local maximum M2(R/Q)
value (Fig. 2.3.6 & 7). A rough grid model was used in order to save computation time
compared to a finer grid model. A rough grid affects the quantity (accuracy) of each
result, but it remains the quality consistency - thus, a geometry gives a higher value of
M2(R/Q) then the other with a rough grid model, will also have higher value with a finer
grid model.
Fig. 2.3.6: MAGIC 3D single cavity modeling
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
W
31
30
0.2
0.3
0.4
0.5
0.7
0.6
Gap Size (mm)
Fig. 2.3.7: M2R/Q versus gap size with fixed cavity length
The results above show that the maximum M2(R/Q) occurs at a gap size of
approximately 0.5 mm. A cavity with a 0.5 mm gap size is then retuned to 35 GHz by
changing the cavity length. At this point, we can proceed with finer grid modeling as
shown below (Fig. 2.3.8).
F=36.60 GHz
35.6
35.5
o
CN
2
!i«v *' , -r
35.4
35.3
* * ~ rt t h
rf% 0 0 r f '
35.2
3.5
3.6
3.7
3.8
3.9
4
4.1
Cavity Length (mm)
Fig. 2.3.8: M2R/Q versus cavity length with fixed gap size
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
51
From the results above, we can see that, at the resonant frequency of 35 GHz, the
M2(R/Q) value of a cavity with a 0.5 mm gap size is no better than the original design
using a 0.4 mm gap size. Further simulations were performed with decreased cavity
lengths to increase the M2R/Q value, with the results shown below (Fig. 2.3.98c 10)
Current cavity design
gap 0.5 mm
gap 0.4 mm
gap 0.38 mm
gap 0.36 mm
gap 0.35 mm
35.9
..
35.7
O
35.5
£
CN
35.3
35.1
mmamm
34.9
3.1
3.3
3.5
3.7
3.9
4.1
Cavity Length (mm)
t
j
Fig. 2.3.9: M R/Q versus cavity length for various gap sizes
C u rre n t D esig n
35.0 GHz
35.2 GHz
O
t2
CN
j bi(<
0.35
0.40
0.45
0.50
Gap Size (mm)
Fig. 2.3.10: M2R/Q versus gap size for different frequencies
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
52
The results show that the optimum gap size for 35 GHz is 0.38 mm. The electrical
length for this single gap cavity with a 0.38 mm gap size is approximately 1.215 radians
at 35 GHz and 1.222 radians at 35.2 GHz.
The electrical length is the electron
propagation constant multiplied by the gap length (i.e. pe d). The normal range for the
electrical length for a klystron gain cavity is between 1.0 and 1.5 radians. An electrical
length close to 2 n may result in un-modulated electrons after passing the gap.
The gap size o f the existing coupled cavity is 0.4 mm, which is about 0.3% lower
then the optimum value. The optimization space is limited by the existing electron gun,
as the coupling coefficient is a function of the beam tunnel diameter and the beam
diameter. A higher filling factor (the ratio of beam diameter to beam tunnel diameter)
and a smaller tunnel diameter result in a higher coupling coefficient, as the interaction
between the electrons and the gap takes place where the electric field is higher. R/Q is
determined by the cavity geometry. Additional information on these subjects can be
found in the R/Q and coupling coefficient modeling sections in Chapter 3.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
2.4 Design Parameters
To increase the output power of an electron tube, higher beam power is required.
Beam power is a function of the required output power and the electronic efficiency [10].
The gain is usually determined by the interaction circuit length, depending upon RF input
power level (related to the required output power), considering the applications, noise
level o f the driver, and the noise in interaction circuit. Higher gain can be obtained with
a higher value o f M2(R/Q) of each cavity, as detailed in the previous section. The gain
can also be increased by adding more cavities. However more cavities result in longer
circuit length.
Higher efficiency can be obtained with lower perveance (lower current density hence
more efficient electron bunching) [48] and higher coupling coefficient. The coupling
coefficient can be increased by a higher filling factor, and smaller drift tube radius. In
addition, a higher value of R/Q allows a lower external Q of the output cavity, hence
higher circuit efficiency [49]. A multiple gap output cavity is usually used to increase
R/Q and reduce RF breakdown possibility.
Wide bandwidth in a klystron requires low beam impedance (i.e. high perveance)
[25]. Increasing the number o f cavities also allows for wider bandwidth.
In this design, a previously-designed electron gun determines the beam power, the
beam perveance, and optimization limitation of the coupling coefficient. The circuit is
based on seven cavities, considering the required gain and the approximate gain
contributed from each cavity.
The first five and the output cavity are coupled-gap
cavities in order to increase R/Q. The penultimate cavity is a single-gap cavity so that it
can be located close to the output cavity, as the electron bunching reaches a maximum
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
54
quickly after the penultimate cavity (the resonant frequency in which is tuned upward to
maximize output power) [10]. The distances between cavities are optimized using ID
simulations.
The optimum location of the output cavity is where the maximum RF
current occurs.
However, the output cavity in the final design was located slightly
further due to the minimum wall thickness required.
The initial distances between the penultimate cavity to the previous cavity and to the
output cavity were optimized to 4 mm. Unfortunately, this resulted in a maximum cavity
wall thickness o f 0.325 mm; which was too thin for thermal strength. The distances were
then adjusted to 4.5 mm, allowing for a wall thickness of 0.825 mm. More details are
provided in the following section.
The ideal external Q for the input cavity maximizes the power transfer and is equal to:
Q
=
—
- —
00 ^ 0 6
The ideal external Q for the output cavity maximizes the output efficiency and is equal to:
where Irf is the ratio of RF current in the output cavity to the DC current; Vo/Io is the
beam impedance and M is the coupling coefficient.
As will be mentioned in the
following section, both external Qs were lowered to approximately half of the ideal
values to increase the bandwidth.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2.5 Final Design and Modeling Results
The design parameters shown in Table 2.5.1 are the initial design parameters
predicted through ID small signal calculation and ID large signal simulation.
Table 2.5.1: Initial Design Parameters
Cavity Number
1
3
4
175
175
175
6
7
175
105
175
oo
oo
oo
oo
oo
190
1900
665
665
1900
1900
1900
1900
34.86
0
34.90
35.05
35.11
35.14
35.20
35.00
6.5
6.5
6.5
6.5
4.5
4.5
Gap Length (mm)
0.4
0.4
0.4
0.4
0.4
0.4
0.4
M
0.582
0.582
0.582
0.582
0.582
0.583
0.582
R/Q
175
Q.
350
Qo
f (GHz)
Gap-Gap (mm)
2
5
In the 3D modeling, the solution accuracy improves with smaller mesh size,
especially for our special "racetrack" cavity; consequently, fine meshes are required to
model the geometry appropriately. However, reducing the mesh size in 3D modeling
significantly increases the simulation time and the computing memory load. Cutting the
mesh length in half along three axes increases the number of cells (computational
memory required) by a factor of 8. In addition, the Courant condition requires that the
time step be sufficiently short that the electromagnetic fields cannot propagate across the
shortest cell dimension in one time step (more detail in Appendix IV). Therefore, the
total simulation time would increase approximately by a factor o f 16.
This huge
computational load makes the 3D modeling extremely challenging for our seven-cavity
EIK (six o f them are coupled cavities). An ideal 3D laminar beam modeling in Cartesian
coordinates was also an issue (Chapter 3) in MAGIC 3D.
The new version of MAGIC
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56
3D (2005.0201.1602) was able to model the immersed flow in Cartesian coordinates
appropriately (the previous version had some internal bugs and not able to model the
beam correctly when the IMPORT LAMINARBEAM was used). More discussion of
the beam modeling will be provided in the next chapter. The individual cavity R/Q,
coupling coefficient M, Q, beam transportation, coupling iris/reduced-height waveguide,
have been modeled/designed with MAGIC 3D using a PC with a 3 GHz CPU. The
complete 3D EIK hot test simulations were planed to use MAGIC 3D parallel version on
a Linux cluster at SLAC. However, since the Linux parallel version of MAGIC 3D had
problems to finish a complete EIK simulation due to the geometry complexity, the mesh
size has been increased in order to complete the simulations on single PC. The total
number o f cells is reduced to 1/6 by increasing the smallest cell size from 0.05 mm to 0.1
mm. To tune each cavity resonant frequency accurately in the complete seven-cavity
EIK modeling without using smaller mesh size is also challenging.
Thus, a few “fake”
tuners have been added to tune the cavity resonant frequency accurately without reducing
the mesh size; this will be described in more detail in the fake tuner session. Because the
mesh size was increased, the resonant frequency was reduced in the simulations
(described in more detail in the convergence session), in order to tune the cavity back to
the original frequency and not to change the deformable wall significantly (this avoids
having the flat side come too close to the gap), the cavity was truncated at the racetrack
end.
Another consequence of the coarser mesh is that the M 2-R/Q factor which
dominates the gain is slightly changed (~ 1%). The number of particles imported to the
simulation was also reduced from 100 to 36.
The final parameters modeled with
MAGIC 3D using the coarser mesh are listed below:
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57
Table 2.5.2: Final Parameters in MAGIC 3D modeling with bigger mesh
C avity N um ber
1
2
3
4
5
6
7
R/Q
158.4
152
154
154.5
155
99
162
Q«
355
00
00
00
00
00
191
Qo
1837
650
655
1846
1821
2242
2089
f (G H z)
34.858
34.895
35.051
35.111
35.141
35.20
35.00
G ap -G ap (m m )
0
6.5
6.5
6.5
6.5
4.5
4.5
G ap Length (m m )
0.45
0.45
0.45
0.45
0.45
0.5
0.45
M
0.588
0.588
0.587
0.587
0.586
0.588
0.587
This design is verified to yield 1.1 kW output power and 240 MHz bandwidth (see
Figs. 2.5.1 ~ 4). All cavities are two-gap coupled cavities, except for the penultimate
cavity which is a single gap cavity in order for it to be located closer to the output cavity.
The wall thickness between the adjacent cavities, however, is 0.825 mm in order for it to
have sufficient thermal strength. The input/output cavity external Q is lowered by a
coupling iris connected to a reduced-height waveguide. The output cavity external Q is
lowered to half o f the optimized value in order to have wider bandwidth coverage. In
addition, the 2nd and the 3rd cavities are coated by TiNitride loss material (Titanium
evaporated in Nitrogen gas and coated on the cavity surface) to increase the bandwidth.
The coated area and the thickness are determined by the desired Qo.
material is copper and the Qo is 1900.
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Other cavity
58
Output Power
Input Power
3.815 cm
Side View
Bottom View
Coupling slots at
the 1st, 2"«, 3'“, 4th, 5th, 7th Cavities
penultimate cavity
(single gap) ^
_L
Fig. 2.5.1: Seven Cavity Structure Modeled by MAGIC 3D
Saturation = 1.2 kW, 37 dB at 35 GHz
1400
*s
g
1200
1000
dj
800
£
600
MAGIC 3D
§.
400
1D Disk Modeling
o
200
0
0
0.1
0.2
0.3
0.4
0.5
Input Power (Watts)
Fig. 2.5.2: Gain Simulation Results
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59
m 35
MAGIC 3D
1D Disk Modeling
1D Math cad
8g*a/t8tans*«ttBg
34.8
34.9
35
35.2
35.1
Frequency (GHz)
Fig. 2.5.3: Small Signal Simulation Results
O u tp u t P o w e r= 1.1 kW, 38 dB
MAGIC 3D
1D Disk Modeling
34.8
34.9
35
35.1
35.2
Frequency (GHz)
Fig. 2.5.4: Large Signal Simulation Results
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60
The seven-cavity EIK was designed with a ID equivalent circuit and verified with
MAGIC 3D using an existing gun (Table 2.5.3). The results met the requirements (Table
2.5.4) with output power of 1.1 kW and 240 MHz bandwidth.
Table 2.5.3: Beam Parameters (existing gun)
Beam Voltage (kV)
Beam Current (A)
Tunnel Radius (mm)
Beam Radius (mm)
14
0.4
0.5
0.25
Table 2.5.4: EIK Requirements and Simulation Performance
Parameter
Frequency
Output Power
Gain
Bandwidth (3dB)
Electronic Efficiency
Requirement
35 GHz
1 kW
35 dB
200 MHz
17.9 %
Performance
35 GHz
1.1 kW
38 dB
240 MHz
19.6 %
Magic 3D resulted in higher gain than AJDISK in saturation region (Fig. 2.5.2),
perhaps because the electron bunching comes very close to the gaps in the large signal
region thereby resulting in higher modulations, while AJDISK assumes that the beam
radius is constant along the tube regardless of the focusing.
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61
2.6 Focusing Fields
The minimum magnetic field (axial component) required to prevent beam divergence
due to space-charge forces is called Brillouin flow:
a v jtt]£qu
where I is the beam current, uz is the axial beam velocity, rj is the electron charge to mass
ratio, and 60 is the dielectric constant of free space [10].
The focusing field needs to be higher than the Brillouin flow value because of the
radial component o f the electron velocity and various beam radii (electron bunching and
scalloping). Since the Brillouin flow is proportional to the square root of the current and
the theoretical maximum bunching ratio is 1.84, the minimum of v l.84
is required
when maximum bunching occurs. AJDISK predicts the maximum RF current ratio of
the EIK to be 1.6 and, with a Brillouin flow of 0.2 Tesla, a minimum B field of 0.25
Tesla in the large signal region is required. A B-field of 0.3 Tesla was used in the
MAGIC 3D simulation results shown in the previous section.
EIK modeling results with different magnetic fields are shown below using MAGIC
3D (ID simulation codes do not account for magnetic fields and assume a constant beam
radius).
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Input Power (Watts)
Fig. 2.6.1: Gain simulation resuls using different magnetic fields
34.8
34.9
35
35.1
35.2
Frequency (GHz)
Fig. 2.6.2: Bandwidth simulation results using different magnetic fields
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63
Higher magnetic fields give better beam transport; however, they result in electron
bunching further away from the gap and therefore reduce the modulation and output
power. Higher magnetic fields also require more magnets, thereby resulting in heavier
devices.
The best magnetic field is the minimum value required to give good beam
transport in the saturation region.
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64
Chapter 3
EIK MAGIC 3D Modeling Techniques
And Analysis
This chapter begins with a convergence test of MAGIC 3D, followed by a validation
of the plasma reduced wavelength modeling using MAGIC. After this, several other 3D
modeling techniques associated with the EIK are presented, such as cavity resonant
frequency and quality factor simulations, R/Q and M modeling, cavity tuning, input and
output cavity modeling (including the waveguide and iris), an overdrive method, laminar
beam modeling, and beam loaded resonant frequency and Q measurements. In addition,
analyses o f the input gap voltage, input power, circuit stability and noise are also
discussed.
3.1 Convergence Test
A pillbox with a radius of 1.265 cm and a height of 4.0 cm was modeled using
MAGIC to test the resonance simulation convergence.
The results of TMoio are
compared to theory (Fig. 3.1.1)
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65
MAGIC 2D
...
■
-p 'I.
MAGIC 3D
Theory
Fig. 3.1.1: The resonant frequency versus mesh size (Mesh Size = Radius/ 10/N)
The error is about 0.1 % for N = 2, which represents a mesh size of approximately 0.1
cm. An error o f less then 0.5 % results for N=1 at approximately 9 GHz (Fig. 3.1.1).
Therefore, a mesh size of 0.03 cm should provide very accurate results at approximately
30 GHz. A mesh size o f 0.05 mm was initially used in the EIK modeling to better
describe the gap geometry.
However, due to insufficient computer memory and
tremendous computation time, the mesh size was increased to 0.1 mm and the resonant
frequency of each cavity is decreased approximately 1.3 %.
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66
3.2 Reduced Plasma Wavelength
An accurate reduced plasma wavelength simulation demonstrates a simulation
program is capable to model the electron bunching and space charge accurately, which is
very important for precise hot test simulation results. The reduced plasma wavelength
[25] [Appendix I] was calculated using MAGIC 3D by modulating the beam with a
cavity gap and allowing the electrons to bunch along the drift tube (Fig. 3.2.1). The
distance between the RF current peaks was measured and compared to the theory (Fig.
3.2.2) [25] [Appendix I]. For this frequency, beam tunnel, and beam current of our EIK,
the half reduced plasma wavelength is calculated to be 20.098 mm, compared to 20.073
mm in MAGIC 3D. A magnetic field of 0.4 Tesla was used in this modeling. Higher
magnetic fields result in higher space charge, hence higher plasma frequency and shorter
plasma reduced wavelength [10]. For a two-cavity klystron, the best location of the
output cavity is at the quarter of the reduced plasma wavelength where the RF current
maximum occurs.
Pi
A RF signal at the gap to modulate the beam
Fig. 3.2.1: Electron bunching along the drift tube
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67
u
tw
i
( 8 - 1)
14 reduced
plasma
wavelength
§s
0
10
10
* <■)
30
(1-3)
Fig. 3.2.2: The particle energy and the RF current versus z
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68
3.3 Resonant Frequency and Q Measurement
There are two approaches to measuring the resonant frequencies of a cavity using
MAGIC - using the EIGENMODE solver in the frequency domain or using a Gaussian
signal in the time domain.
3.3.1 Frequency Domain Approach
The built-in EIGENMODE command in MAGIC specifies an eigenvalue solution of
the fully time-dependent Maxwell’s equations. By initializing the starting electric fields
using the PRESET command, MAGIC has the appropriate model structure that we desire,
as illustrated as below:
Total L en g th
------ »)
Fig. 3.3.1: Coupled cavity geometry
The proper setup for the MAGIC input file is:
B E TA = l _ P I * M O D E _ N U M B E R / T o t a l _ L e n g t h ;
- A s s ig n
d e s ire d
d iffe re n t
RF p h a s e a c r o s s
th e
c ir c u it
fo r
e ig en m o d e
FUNCTION E Z _ S E T ( X , Y , Z )
= C O S(B E T A *(Z -Z B G N JT U N N E L )) ;
P R E S E T E 3 FUNCTION E Z _ S E T ;
EIGENMODE;
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each
69
By assigning the “Mode Number” in MAGIC (Magic will calculate the phase difference
between the coupled-cavities for each eigenmode), we can obtain the desired eigenmode
results. The results of the resonant frequency and the Q for each eigenmode can be found
in the MAGIC Log file. Below are the Ez profiles of each eigenmode of the coupled
cavity.
(a)
(b)
Fig. 3.3.2: Ez profile of (a) In mode and (b) n mode of the coupled cavity
The EIGENMODE solver only solves problems with closed boundaries. If the object has
an open boundary, such as an input or output cavity connected to a waveguide, then we
need to use the time domain approach to measure the resonant frequencies and Q as
demonstrated in the following section.
3.3.2 Time Domain Approach
We can also measure the resonant frequencies and Q of each cavity in the time
domain. The approach is to place an antenna (signal Driver) at each gap (where the field
is approximately the strongest, as shown in Fig. 3.3.3 below), and then apply a Gaussian
signal (Fig. 3.3.4) on each antenna. After driving the cavity for a short time and then
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70
turning off the signal, the field will fall into the cavity eigenmode. This approach can be
used on an object with either closed or open boundaries.
Driver
Fig. 3.3.3: Locate the drivers on each gap
S
g
G aussian signal is
turned off after a while
Fourier Transform
o
Gaussian Signal
8
©
8
e
N
0
O
300
400
<00
•00
T im e (P ic o s e c o n d s )
»
4f
frequency (GHz)
Fig. 3.3.4: Gaussian signal and its Fourier transform
Distinct from the frequency domain, where we assign a different “Mode Number” for
each eigenmode, in the time domain we setup a different antenna signal for each desired
eigenmode in order to properly initialize each eigenmode electric field. For the 2 7t mode
(where the electric fields in both gaps have the same phase, or where the phase of the
electrons passing the second gap is the same as when they pass the first gap if the beam is
present, if the beam is present), the antenna signal setup can be:
P erio d
= 1 /F re q ;
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71
A n te n n a J T r = 5 * P e rio d ;
PI
=
2 .
- The w i d t h o f t h e G a u s s ia n s i g n a l
* a co s(0 .);
FU NCTION A n t _ D r i v e ( T )
= A n t e n n a _ J * S I N (2 . 0*P I*F R E Q *T ) *
E X P ( - 6 * ( ( T - A n t e n n a _ T r ) / A n t e n n a _ T r ) **2) ;
D R IV E R J 3 A n t _ D r i v e LOWER_ANTENNA ;
D R IV E R J 3 A n t _ D r i v e
UPPER_ANTENNA ;
O BSERVE F IE L D _ IN T E G R A L E . D L A X I S AM PLITU D E F r e q 4 ;
For the 7t mode (where the electric fields in the gaps have opposite phase, or where the
phase of the electrons passing the second gap is 180 degrees different from when they
pass the first gap, if the beam is present), the antenna signal setup can be:
FUNCTION A n t _ D r i v e _ P O S ( T )
=
A n t e n n a _ J * S I N ( 2 . 0 * P I*F R E Q *T ) *
EXP( - 6 * ( (T -A n te n n a _ T r )/A n te n n a _ T r )**2);
FUNCTION A n t _ D r i v e _ N E G ( T )
=
- A n t e n n a _ J * S I N ( 2 . 0*P I*F R E Q *T )*
EXP ( - 6 * ( ( T - A n t e n n a _ T r ) / A n t e n n a J T r ) * * 2 ) ;
D R IV E R J 3 A n t _ D r i v e _ P O S LOWER_ANTENNA ;
D R IV E R J 3 A n t _ D r i v e _ N E G
UPPER_ANTENNA;
OBSERVE F IE L D _ IN T E G R A L E . D L L O W E R _ H A L F _ A X IS AM PLITU D E F r e q 4 ;
OBSERVE F IE L D _ IN T E G R A L E . D L U P P E R _ H A L F _ A X IS AM PLITU D E F r e q 4 ;
The driving frequency is set close to the eigenmode frequency (the Gaussian signal we
used in Fig. 3.3.4 covers about ± 5 GHz, approximately centered at the resonant
frequency). The Gaussian signal width determines the precision of the driving frequency.
With a wider Gaussian signal width, the driving frequency is more precise.
It should be noted that:
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72
1. The antenna signals have the same direction (phase) for each antenna to initiate
the 271 mode, but the opposite direction to initiate the 7t mode. If we apply signals
with the same phase to both antennas, the field will fall into the 2 n mode
regardless of the driving frequency.
2. The OBSERVE FIELDINTEGRAL AMPLITUDE can calculate the cavity
resonant frequency and the cavity Q for each eigenmode as shown in Fig. 3.3.5
below. Since the direction of the n mode electric fields are opposite across the
coupled cavity, the OBSERVE FIELD_INTEGRAL AMPLITUDE analysis needs
to be measured at the upper and lower part separately, otherwise the voltage will
be integrated to zero across the circuit.
M
/
tr
Frequency
0
1
Frequency falls to the
dominate cavity resonance
after the driver turned off
“ .....
Oi
1
s>
200
400
Time (p s)
COO
000
Magic c alculates the
cponentiiil decay ftaramete r,
b__ IQ, basec on an a*ssumed -(O./2 0 —
_
arnplitude decay go ing as &
------ (a fter the d river is tijm ed ofT ------
©
Time (ns)
Fig. 3.3.5: Resonant frequency and Q obtained from AMPLITUDE analysis
3. When we use OBSERVE FIELD INTEGRAL AMPLITUDE, the Q will not be
analyzed until the fields begin decaying; therefore, we need to use the
SURFACE LOSS command to enable the loss calculation due to the material in
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73
the time domain. If there is no loss input, MAGIC can still measure the resonant
frequency with the voltage/field oscillation (see Fig. 3.3.6 below) from
AMPLITUDE analysis, but the Q will not be calculated.
F IELD INTEGRAL E.DL at A XIS
1 p e rio d
0
lOO
200
400
500
600
(E— 12)
Fig. 3.3.6: MAGIC measures the resonant frequency from the field oscillation
The time domain approach is especially useful when we need to measure the resonant
frequency on an open boundary object such as an input or output cavity.
A similar
approach will be used later when we measure the beam loaded resonant frequency and Q
of each cavity eigenmode.
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74
3.4 R/Q Modeling
R/Q is the ratio of the shunt impedance R of a cavity (defined as, R=VV*/W=|V|2/W,
i.e. the ratio of the square of the voltage across the gap to the power lost in the resonator
walls) and the cavity Q (defined as g=co U/W, i.e. 2 r f times the energy stored in the
cavity divided by the power lost in the resonator walls) [25].
If the cavity is assumed to be gridless and the field Ez is assumed to extend beyond
the actual gap, combing the two equations above gives:
Q
2coU
2 coU
where R/Q is a function of geometry, but not frequency or wall losses in the cavity.
Consequently, a 35 GHz cavity that is a scaled copy o f a 10 GHz cavity will have exactly
the same R/Q. Determination of R/Q by modeling can be done with any electromagnetic
simulation software that solves for cavity fields given a cavity geometry. Below we list
the steps to determine the R/Q of the EIK racetrack coupled cavity modeling using
MAGIC 3D.
1. Model the cavity for a given specific geometry and drive the cavity using the
antenna located on the gap (see the section on determining the resonant frequency
using the time domain approach for more detail).
2. Measure the voltage along the axis of the cavity using the MAGIC command:
O BSERVE F I E L D _ I N T E G R A L E . D L a x i s
SU F FIX v a x i s
(F ig .
3.4.1)
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75
Voltage
on axis
...
~ §
n
a
i§
o
8
0 ,9
9 .2
9 .4
0 .«
9 .9
1 .9
Time (ns)
Fig. 3.4.1: Measure voltage on axis
3. Measure the total electromagnetic energy stored in the cavity using the MAGIC
command:
O BSERVE
F IE L D _ E N E R G Y EM OSYS$VOLUME F I L T E R S T E P p e r i o d
SU F FIX U
(The alternate STEP filter provides time averaging of the variable over the time
period.)
EM
energy
§
o
0 .4
0 .4
1 .0
Time (ns)
Fig. 3.4.2: Measure EM energy in the cavity
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76
4. Calculate the R/Q using the MAGIC command:
FU N C T IO N
RQjCAVITY =
O BS$VAXIS*O BS$VAXIS/( 1 . e24* (O B S $U .E Q .0 ) + (4P I*F req*O B S$U ))
OBS ERVE TRANSFORM R Q J C A V I T Y WINDOW T IM E 2 0 * p e r i o d
sim ulation _tim e
The first term (OBS$U.EQ.O) in the FUNCTION denominator equals 1 if the
energy in the cavity is zero. This ensures that R/Q will not go to infinity if there
is no energy in the cavity. Instead, the R/Q value goes to zero if there is no
energy stored in the cavity.
ill
0 .»
0 .7
Time (ns)
Fig. 3.4.3: Calculate the R/Q
The R/Q measured in each cavity is listed as below:
Table 3.4.1: R/Q Modeling Results o f each cavity
Cavity
R /Q
1st
158.4
2nd
152.0
3rd
154.0
4th
154.5
5th
155.0
6th
99.0
yttl
162.0
The penultimate cavity is a single gap cavity so that the R/Q is lower than others.
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77
3.5 Coupling Coefficient Modeling
The coupling coefficient (M) is arguably the most important parameter used in the
design o f a klystron. It describes the interaction between the beam and the cavity fields,
where a higher M value means better interaction, hence higher efficiency and gain (the
klystron gain is a function of M2", where n is the number of cavities).
The definition o f M is:
V
f* £ (z)dz
>where Pe =o>lvz (vz is the electron velocity)
This is a function o f the electron propagation constant, and therefore accounts for finite
transit-time effects. The coupling coefficient is the ratio of the effective voltage applied
on the beam modulation (Veff), normalized to the RF voltage across the interaction region
(V). Both the effective voltage and the normalized voltage can be calculated along the
beam tunnel radius multiplied by an averaging coefficient across the beam (Appendix I),
or, along 0.707 o f the beam radius, where the half current is.
Because the coupling coefficient is very sensitive to the beam diameter, which is
never known precisely; discrepancies in the calculated or simulated gain of a klystron can
usually be traced to inaccuracies in the coupling coefficient calculation.
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78
3.5.1 “Cold Test” M Simulations
From the definition above, the electric fields can be simulated and the coupling
coefficient can be calculated using MAGIC, without the presence of the electrons (i.e. RF
only; or “cold test” simulations). The frequency domain solver or the time domain solver
can be used to obtain the Ez field on the beam tunnel radius followed by integration along
the axial direction to obtain the normalized voltage in MAGIC (Fig. 3.5.1 ~ 3):
FUNCTION EABS (Z, E) = A BS (E)
RANGE FIELD E3 tunnel
TSYSSEIGENMODE
TRANSFORM EABS
INTEGRATE POSITIVE
Since the Ez field is symmetric with respect to the center of the coupled cavity, only the
cosine portion o f the effective electric fields need to be integrated (Fig. 3.5.4):
FUNCTION M (Z, E) = E*Cos (2pi*FREQ*(Z- Zcenter)/Vz)
RANGE FIELD E3 tunnel
TSYSSEIGENMODE
TRANSFORM M
INTEGRATE POSITIVE
The pictures below show the coupled cavity simulation and the coupling coefficient
calculation using MAGIC 3D. Note that the EIK and the coupled cavity are designed to
be operated in the 2n mode, which means the electric fields in both cavities have the
same phase.
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79
1
I/
4
-2
0
•
Z (mm)
2
V
H
(B«)
1
*
\
¥
\
\n
\
■a
Na
$
’
\
-i
Z (mm)
Fig. 3.5.3: Voltage at the tunnel radius
1
Fig. 3.5.2: Ez versus. Z
Fig. 3.5.1: Ez field pattern on the Y-Z plane
Integral (MS)*n
.J
-i
»
»
Z (mm)
1
r
*
Fig. 3.5.4: Effective Voltage at the tunnel
radius
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80
3.5.2 “Hot Test” M Simulations
The coupling coefficient can also be calculated in “hot test” simulations (with the
presence of the electrons and RF) with MAGIC 3D modeling (Fig. 3.5.5) using:
V - V ini. .
M = - max
v
In this case, the coupling coefficient is calculated from the difference between the initial
electron energy and the maximum energy after passing the gap (where the cavity electric
fields vanishes) normalized to the voltage across the gap at 0.707 of the beam radius.
Maximum Beam
Voltage
(Right After The Gap)
Initial B eam
V oltage At
------
r = 0.707-B eam R ad iu s
Gap Ending Z (mm)
Fig. 3.5.5: Electron energy modulation along the axis due to the gap fields
The coupling coefficient for the intermediate cavity is approximately 0.58 using cold test
simulations and 0.582 using hot test simulations with MAGIC 3D.
This topic was
addressed further in the design tool and modeling result sections (Chapter 2).
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81
3.6 Input and Output Cavity Modeling
Three dimensional modeling allows us to model the input and output cavities with the
coupling irises and connecting waveguides as per the actual structure (Fig. 3.6.1). This
cannot be done using standard ID or 2D simulation tools.
Output W avegu id e
Coupling
^ Iris
Input W avegu id e
Input Cavity
Output C avity
Coupling Iris
Fig. 3.6.1: 3D EIK modeling
In ID calculations, the opening effect of the input cavity is included in the input gap
voltage calculations and the output cavity Qe is included in the gain calculations
(Appendix I). In 2D simulations, the circuit is usually modeled in cylindrical coordinates
(using cylindrically-symmetric circuit geometry or non-cylindrically-symmetric circuits
transferred to cylindrically-symmetric equivalent circuits), and no waveguide or opening
iris can be added [31]. Consequently, “fake” loss material is added to the input and
output cavity to simulate the actual Qe (Fig. 3.6.2).
Since the waveguide cannot be
incorporated into cylindrically symmetric modeling, a Driver (Fig. 3.6.2) is added in the
input cavity to excite the input power. The power in the loss material in the output cavity
is calculated to obtain the output power.
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82
Driver
Symmetry Axis
Fig. 3.6.2: The input/output cavity in 2D simulations
In 3D modeling, the external Q is included by adding the real coupling iris connected
to a reduced-height waveguide.
The RF input is at the input waveguide which is the
actual situation (please refer to the input power modeling section), and there is no need to
introduce a current using a Driver as in 2D modeling. The external Q is given by:
Q
_
®
‘
U stored
load
In order to model the input/output cavity with the opening iris and waveguide
appropriately with the desired Qe and the resonant frequency, the following steps were
adopted:
1. Model the loaded cavity with the opening iris and waveguide without the other
half of the coupled cavity. (Fig. 3.6.3)
2. Adjust the iris dimensions until half the Qe is obtained. (Because only half of the
coupled cavity is modeled, the stored energy is only half of the total energy so
that Qe is only half o f its actual value.)
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83
3. Due to the opening iris, the resonant frequency of the loaded cavity is lowered.
The cavity dimensions must then be reduced in order for its resonant frequency to
be at the desired frequency.
4. Both the iris dimensions and the cavity dimensions would cause Qe and the
resonant frequency to shift. Repeat the 2nd and the 3rd steps until the desired Qe
and resonant frequency are achieved.
5. Open the coupling slot and model the other half of the coupled cavity (Fig. 3.6.4).
The coupled cavity should achieve the desired Qe and resonant frequency (the
other half of the unloaded cavity should have been tuned to the desired resonant
frequency previously).
ris Length
C o u p lin g
S lo ts
Fig. 3.6.3: The loaded cavity
Fig. 3.6.4: The complete input/output cavity
The reason we tune the unloaded cavity and loaded cavity separately is because the
resonant frequency measurement (as well as Q measurement) on the axis shows
fluctuations if both cavities are present at the same time and have different resonant
frequencies (Fig. 3.6.6), and that makes it difficult to measure the resonance and Q
accurately, as well as tune the iris size. This situation happens when only one cavity is
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loaded and whenever the cavity dimension or the iris dimension are changed, since
structure modifications change the cavity resonance. Therefore, it is easier and more
efficient to tune both cavities separately. To increase the accuracy of the tuning, half
portions of the slots were included in the loaded cavity; the others were added when the
other cavity was added.
0.0
0.2
0.4
0.6
Time
0. 6
1.0
0.0
1.2
0.2
0.4
0.6
Time
(sec)
Fig. 3.6.5: Stable resonance measured
0. 1
1.0
1.2
(sec)
Fig. 3.6.6: Non-stable resonance measured
(Resonances at both sides are similar)
(Resonances at both sides are .off)
The total Q is equal to
J_ = J_
_1_ J _
Q r ~ Q b + Qo + Qe
If there is no beam present, the total Q is a function of Qb and Qo only. In the time
domain, the SURFACE_LOSS command can be used to calculate the loss in the material.
Therefore, we can use this command to calculate the total Q without the beam. Without
using this command, the total Q would be equal to Qe in the simulations.
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85
The input/output cavity results and the coupling iris dimension after tuning are listed
below:
Table 3.6.1: The Input/Output Cavity Modeling Results
Input Cavity
Output Cavity
fo (GHz)
34.858
35.000
Qe
355
191
Iris Height
0.60 mm
1.25 mm
Iris Width
3.0 mm
3.2 mm
Waveguide Height
0.60 mm
1.25 mm
Both the input and output coupling iris length are 0.8 mm in order to have sufficient
physical strength for machining between the cavity and the waveguide. Both reducedheight waveguides are the same height as the connected iris and have a width of 0.28
inches (WR28).
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86
3.7 The Input Gap Voltage
After the input cavity is modeled successfully and the beam loaded Q is measured, we
can compare the voltage at the input cavity gap to the theoretical value derived by J.
Vaughan [37]. This theoretical value includes the effects o f cavity mismatch on or off
resonance and o f beam loading and detuning, and correctly divides the available drive
power between the cavity losses, the real power transferred to the beam, and the reflected
power. In this derivation, it is assumed that the input line is matched at the generator end,
and that the generator itself is unaffected by reflected power.
The known parameters are assumed to be the cavity cold resonant frequency fo, the
cold unloaded Qo and external Qe, the shunt resistance R/Q, the beam admittance Gb+jBb
(see Appendix I for Gb and Bb calculation), and the available drive power Pin at frequency
/ [ 37].
The fractional detuning is
s=£zA
fo
The beam-loading Q is defined as:
Qb
4
=
—
- —
Gb(R /Q )
The hot unloaded Q is given by
QuH = \/Q 0 + l/Q b
The conductance is
C
1
r
Q.AR/Q)
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87
The terminal susceptance B t includes the cavity and beam susceptances, and is:
BT = ^ - + Bb
T R /Q
b
The total admittance is
Yt = Gt + jB r
The equivalent conductance of the drive line is
Ge = ----- -----Qe { R ! Q )
The complex reflection coefficient is
Ge - Y t
g e + yt
The input VSWR is
a =
H 4
i-H
Now the rms input gap voltage (the peak voltage divided by the square root of two)
can be defined in accordance with conservation o f energy:
y , = T ^ ^ Q . H(.R/Q)
l + (J
More detailed calculations can be found in Appendix II.
Below is the list o f the unloaded resonant frequency fo , Qo and loaded resonant
frequency f , and Qb for the input cavity, measured from MAGIC 3D simulations:
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88
Table 3.7.1 Unloaded and loaded resonant frequency and Q
Cavity #
1st
2 7t mode fo
34.858
2 7t mode Qo
1837
2 7t mode f ,
34.836
2 7t mode Qb 2 7t mode Qb
1072
1388*
*Calculated from J. Vaughan’s equation. See Appendix I & II.
With the fo , Qo, Qb values above, along with the Qe and R/Q values we listed earlier (see
the sections of the input cavity modeling and R/Q modeling respectively), we can
calculate the input gap voltage using J. Vaughan’s equations (Appendix II). The Gb, Bb,
and Qb values are calculated from MathCAD using a ID equivalent circuit configuration
(Appendix I). We can also measure the input gap voltage directly with MAGIC 3D
simulations. The theoretical value and directly measured value from MAGIC 3D are
compared and plotted in Fig. 3.7.1.
Figure 3.7.2 is the comparison between the theoretical results using the Qb value
measured from MAGIC 3D and the measured voltage results directly from MAGIC 3D
modeling with a 19.5 MHz frequency shift. These two plots indicate that with the use of
the measured Qb from the MAGIC 3D modeling, the theoretical input gap voltage value
and the voltage measured directly in the simulations are very similar, except with a 19.5
MHz frequency shift. There is further discussion about this frequency shift in the section
o f beam loading in MAGIC 3D.
The simulations and modeling results below use a constant input power level of
0.1507 watts - which is the input power at 35 GHz that gives the output power 1 kW.
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34.8
3 4.9
35
35.1
35.2
Frequency (GHz)
Fig. 3.7.1: Theoretical input gap voltage compared to the measured results from MAGIC
3D modeling
M A G IC
T h e o ry (u sin g M a g ic Q b )
M agic S h ifte d
34.8
34.9
35
35.1
35.2
Frequency (GHz)
Fig. 3.7.2: Theoretical input gap voltage compared to the frequency-shifted MAGIC 3D
results
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90
Since this trend appears in all the bandwidth results when comparing MAGIC to
AJDISK (see the section o f beam loading in MAGIC 3D; note that the input gap voltage
in AJDISK is calculated based on J. Vaughan’s theory), it would be very beneficial to
have experimental data to help determine whether MAGIC 3D handles beam loading
appropriately. With the aid of the experimental results, MAGIC can be advanced to a
more accurate and matured PIC code.
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91
3.8 Define Input Power
It is necessary to define the input power in simulations appropriately in order to
accurately calculate the gain (Gain=l0-Log [P0ut/Pm])It is extremely important to distinguish between “input power” and “net input power”.
The input power we normally refer to in gain calculations is the total power supplied,
regardless o f the reflection level. Net input power, including what we measure directly
from MAGIC, refers to the net power that actually goes into or goes out of the
measurement area after reflection. Different driving frequencies result in different net
input power levels due to the reflection caused primarily by the detiming of the cavity
resonance (when the driving frequency is shifted from the input cavity resonance) and the
mismatch o f the Qe (when the Qe is not a perfect match to the input cavity). When the
cavity is driven at its beam loaded resonant frequency and the Qe is perfectly matched,
the reflection should theoretically be zero. To transfer the most net input power from the
waveguide into the input cavity, the optimum value for Qe is given by
®e = J _ + J _
Qo + Qt
The current value of Qe has been lowered to approximately half of the optimum value in
order to broaden the bandwidth.
In order to compare the results between AJDISK and MAGIC, we need to understand
and define the input power correctly in both simulations.
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92
3.8.1 Input Power Simulation Using MAGIC
Since MAGIC does not provide the input power directly, we need to set up an
appropriate signal at the input port in order to obtain the desired input power, as
illustrated as below:
PORT
I
a v e g u id e
Coupled
C a v it y '
Fig. 3.8.1: Input Cavity Modeling with Waveguide
Proper setup for the MAGIC input file (Fig. 3.8.1):
FREQ_W G_CUTOFF
=
PORT_BETA_PHASE
0 . 5 _ C /W G _ W ID T H ;
=
1 / S Q R T ( 1 - ( F R E Q _W G _C U T O F F /F R E Q )* * 2 )
F U N C T IO N
W G _SIG N A L T
F U N C T IO N
E Z _ P R O F I L E (X , Y , Z )
=
S I N ( 1 _ P I * ( X - 0 . 5 * W G _W ID T H )/W G _W ID T H )
F U N C T IO N
E X _ P R O F IL E (X ,Y , Z)
=
0
PORT
IN PU T
PORT
=
;
W G_VOLTS*SM O OTH _RAM P( 0 . 2 * F R E Q * T ) * S I N ( 2 _ P I * F R E Q * T ) ;
N E G A T IV E
PHASE
IN C O M IN G
WG S I G N A L
N O R M A L IZ A T IO N
OBSERVE
F IE L D
;
POWER
S .D A
IN PU T
V E L O C IT Y
F U N C T IO N
VOLTAGE
PORT
PORT
IN PU T
F IL T E R
BETA
PHASE
E l
EX
P R O F IL E
E3
EZ
PR O FILE
VOLT
STEP
L IN E ;
TWO
<3.8.1>
P E R IO D S ;
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93
In order to generate a proper TEio RF signal into the waveguide, the signal needs to be set
up correctly by assigning the Ez and Ex profiles [41] at the port (x-z plane in Fig. 3.8.1).
We can then measure the input power using the OBSERVE FIELD_POWER command.
Detailed descriptions o f each command can be found in the MAGIC User Manual [34].
The above is a proper setup of the input waveguide/input cavity modeling in “hot
test” simulations - when both the RF signal and the electron beam are present in the tube.
However, we need to be aware that the power we measure above would be the “net input
power”. It is found that when the iris, cavity, and the beam are present, it is difficult to
measure the input power accurately because there is also power being reflected back from
the iris/cavity. This will be further detailed in the next section. In order to conveniently
define the input power, we simply model the input waveguide itself (without the
iris/cavity transient and the beam) and setup the input port at one side and an output port
at the other side by matching the phase velocity of the RF signal (as with the input port)
and measure the power at either of these two ports (both of them should give the same
power with different directions). This approach is adopted to ensure that the net input
power we measure at the port is also the total input power, since there is no reflection.
Proper setup for the output port in the MAGIC input file:
PORT
OUTPUT_PORT
P O S IT IV E
PH A SE _V E L O C IT Y
y n r-7 "7
/ V/ /
—
/
7—
PORT_BETA_PHASE;
- 7— ^ O
u tp u t PORT
/ / '
Input PORT
Fig. 3.8.2: Input waveguide modeling
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<3.8.2>
94
By assigning a voltage V j cross the waveguide (Fig.
3 .8 .2 ) ,
we can use the power-voltage
definition to express the impedance o f the TEoi mode in a rectangular guide [44]:
Z „=S- = 3 7 7 ~ E A
P
a is, A
where — = ,
A
1
— , f c=—
2a
(3.8.1)
Therefore, with the fixed waveguide dimensions and the voltage we can calculate the
input power P;n at different frequencies using Eqn. (3.8.1). The input waveguide for our
EIK is a reduced height WR-28 (Ka band standard rectangular) waveguide with a width
of 7.112 mm and height o f 0.6 mm. Table 3.8.1 below lists the input power versus
frequency with a constant voltage across the waveguide.
Table 3.8.1: Theoretical input power versus driving frequency
/(G H z)
V i (volts)
Pin (watts)
34.85
4.9
0.1503
34.90
4.9
0.1504
34.95
4.9
0.1505
35.00
4.9
0.1507
35.05
4.9
0.1508
35.10
4.9
0.1509
35.15
4.9
0.1510
The calculation indicates that the input power changes only slightly across the operating
frequency band. With a proper setup, we are able to obtain the input power from MAGIC
3D to within 0.001 % accuracy using a 0.1 mm mesh size at 35 GHz. We can then input
any desired input power by timing the voltage across the waveguide using Eqn. (3.8.1).
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95
3.8.2 Measuring Input Power in MAGIC Using Sn
Our initial approach for measuring the input power is to measure the input power at
the port directly and the reflection coefficient in the waveguide in “hot test” simulations
(while the cavity, RF, and beam are all present). We import the beam to the input cavity
(Fig. 3.8.1) and input the RF from the port and measure the power at the port (here, the
power we measure is the “net input power” after the reflection). A proper setup for
measuring the reflection in MAGIC:
RANGE
F IE L D
E3
V S W R _L IN E
R a n g e _ T im e r
S T A N D IN G _W A V E _R A T IO
FREQ;
Note that in order to correctly perform the VSWR measurement, the measurement must
extend at least a quarter wavelength along the measurement axis. (Note that extending
the waveguide longer than the required length will not change the results and will result
in a longer simulation time, although the time required in the simulation setup is the same
due to the same cavity Q.) The results are listed in Table 3.8.2 below:
Table 3.8.2: Reflection coefficient and net input power measurements
when the cavity is loaded
/(G H z)
34.861
35.00
35.15
Vi (volt)
4.9
4.9
4.9
Sn
0.505
0.845
0.620
Net Pin (w)
0.12277
0.02495
0.00856
Pin (w)
0.1648
0.0873
0.0139
From the results, we see that the closer the driving frequency is to the beam loading
resonance o f the input cavity (which is 34.8564 GHz), the more the net input power. By
measuring Sn and the net input power, we should be able to obtain the same input power
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96
results as Table 3.8.1 from the previous section.
The input power should remain
essentially constant regardless of the driving frequency. However the results show a very
different trend (Table 3.8.2). The reason may be because when the cavity transient and
the beam are added, there is increased noise which makes it difficult to measure and
determine the input power. Therefore, we should always use the theoretical input power
value in the gain calculations (the same as measuring the net input power in MAGIC
when only the waveguide is modeled).
Note that when there is no beam present, the reflection is much higher because the
drift tube has a cutoff frequency of 175.7 GHz. The Ka-band RF signal either gives
power to the beam, is attenuated due to the wall loss (the input cavity material is chosen
as copper, which results in a limited amount o f loss), or is reflected from the iris/cavity.
3.8.3 Input Power in AJDISK
Table 3.8.1 indicates that the total input power is almost constant across the
frequency band (changes less than 0.5% from 34.85 GHz to 35.15 GHz).
AJDISK
employs the total input power (not the net input power) as a constant value across the
operating frequencies when calculating the bandwidth. The gap voltage of the input
cavity and the gain calculations account for the effect of the reflection due to frequency
detuning and Qe mismatch [37].
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97
3.8.4 Conclusions
To summarize this section:
1. When a constant voltage is applied across the rectangular waveguide, the total
input power is determined primarily by the waveguide dimensions.
2. In order to obtain accurate input power levels from MAGIC 3D, a simple
waveguide model with the proper signal setup has been completed and the results
match the theory.
3. When the cavity transient and the beam are present in MAGIC simulations, it is
difficult to calculate the input power accurately using the net input power and the Si i
measurement.
Therefore, we should always use the theoretical input power or the
measurement from the waveguide modeling (without cavity/beam etc.) in the gain
calculations.
4. AJDISK employs the total input power as a constant when calculating the
bandwidth and does take into account the reflection due to the frequency detiming
and Qe mismatch.
After the input power definition is clear in both AJDISK and MAGIC, the gain can be
correctly calculated and the results can be compared between MAGIC and AJDISK.
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98
3.9 Over Drive Approach
Since complex Magic 3D simulations take a tremendous amount of time for problems
such as for our seven-cavity-EIK (six of them are coupled cavities), it is important to use
an approach that minimizes the simulation time. One technique is to shorten the time
required for the RF signal to fill the cavities.
The time required for the RF signal to fill each cavity depends on the cavity quality
factor Q. The cavity response to the RF signal is as follows:
E{t) = E0{ \- e - * l2Q)
Using the OBSERVE FIELD INTEGRAL AMPLITUDE command in MAGIC, we can
obtain 1/Q by calculating - (2/co)*(d/dt) log (amplitude) versus time.
3.9.1 Driving a Single Cavity At The Resonant Frequency
In order to reduce the simulation time, we can shorten the RF filling time by over
driving the cavity. Figure 3.9.1 below shows how the RF signal fills the cavity using a
normal driving signal; Figure 3.9.2 shows how the RF signal fills the cavity using an over
driven signal. The results show that by using an over driven signal we can fill the cavity
faster, thereby saving simulation time.
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99
Signal A nalysis Amplitude o f PIELD_IKTEGRAL E.DL a t AXIS
S ignal A nalysis Amplitude of FIELD INTEGRAL E.DL a t AXIS
| Drive Signal | = V0
D rive S ig n a l | = O v e rD riv e _ R a tio *V 0
| Drive Signal | = V0
n---------- r
4
Time
0 toD
6
(secj
2
4
6
Time
(E-9)
(sec)
IB-9)
Fig. 3.9.1: Driving the cavity with
Fig. 3.9.2: Over driving the cavity
the typical method
for a period of time
Typically, we drive the cavity with a signal as shown below, providing the results in
Figure 3.9.1:
Drive _signal{t) = V0 •Sin{cot)
The over-drive signal shown below provides the results shown in Figure 3.9.2:
Drive _ signal(t) = V0 ■Sin(cot) •(OverDrive _ Ratio •Step(tOD, t) + Step(t, tOD))
=
OverDrive _ Ratio •V0 •Sin(cot) ,
if
t < t0D
V0 •Sin(cot) ,
if
t > tOD
(Vo is the driving voltage corresponding to V in Fig. 3.9.1 and Fig. 3.9.2)
It indicates that if we drive the cavity harder at the beginning and then drive it normally,
the cavity can be filled faster. The relationship between the over-drive time and the over­
drive ratio (over drive voltage divided by normal driving voltage) is:
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100
OverDrive _ Ratio •V0(tOD) = V0
OverDrive _ Ratio •V(l - e~<ot°D/2Q) = V
Once we determine how hard to drive the cavity (the over-drive ratio), the over-drive
time can be calculated. Figure 3.9.1 and 3.9.2 are the results for driving a single cavity at
its resonant frequency o f 34.861 GHz.
The over-drive ratio used in Figure 3.9.2 is
5.92666 for ten periods (one period is 1/34.861 GHz). The total Q here is equal to the
external Q o f 170 when there is no wall (material) loss.
3.9.2 Driving a Single Cavity Off The Resonant Frequency
From the results above, it is apparent that the over-drive method reduces the
simulation time significantly if we drive the cavity at the resonant frequency. However,
it is not clear if the over-drive method would be useful when driving the cavity at a
frequency other than the resonant frequency. This is investigated in the following.
Figures 3.9.3 and 3.9.4 are the results when we drive and over-drive a cavity with a
resonant frequency o f 34.861 GHz with a driving signal of 35 GHz. Figures 3.9.5 and
3.9.6 are the results when we drive and over-drive a cavity with a resonant frequency of
35.182 GHz with a driving signal of 35 GHz. These results indicate that the over-drive
method is not assured o f filling the cavity to the stable point faster if the cavity is driven
at a frequency other than the resonant frequency.
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101
S ig n a l A n a ly sis A n p litu d e o f FIELD_INTEGRAL E.DL a t AXIS
S ig n a l A n a ly sis A nplitude o f FIELD_INTEGRAL E.DL a t AXIS
•" " '
■' • ' “
:
S
!
I
/
/
/
..../L.......
/
/
/
0
-■
1•• 6
2
Time
1
8
(se c)
0
6
2
8
(E-9)
(E-9)
Fig. 3.9.3: Driving the cavity at a
Fig. 3.9.4: Driving the cavity at a
higher frequency; typical method
higher frequency; over-drive method
S ig n a l A n a ly sis A m plitude o f FIELD_INTEGRAL E.DL a t AXIS
S ig n a l A n a ly sis A m plitude o f FIELD_INTEGRAL E.DL a t AXIS
..............................
! \
1 A\ \
\
1
v
1
I...........
,
:
/
- ..........-
-
/
!
:
;
...............................................................................................
/
.. . . ... . . . . . . . . . . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0
Time
(sec )
(E-9)
2
4
6
Time
(se c )
.
.
8
(E-9)
Fig. 3.9.5: Driving the cavity at a
Fig. 3.9.6 Driving the cavity at a
lower frequency; typical method
lower frequency; over-drive method
3.9.3 Driving the Input Coupled Cavity
The single cavity example above indicates that we should save a significant amount
of simulation time if we over drive the input coupled cavity at its resonant frequency
(note that we cannot over drive the following cavities because those cavities need to be
fed naturally from the RF signal carried by the beam). The resonant frequency of the
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102
input cavity is 34.86 GHz. The results shown below in Figure 3.9.7 and Figure 3.9.8
indicate that the over-drive method is effective for our input coupled cavity. However,
when we drive it off the resonant frequency (in this case at the center frequency of 35
GHz), the results in Fig. 3.9.9 and Fig. 3.9.10 show that the over-drive method does not
result in the RF signal filling the cavity to the stable point faster.
S ig n a l A n a ly s is A m plitude o f FIELD_INTEGRAL E.DL a t AXIS
Z
S ig n a l A n a ly s is A m plitude o f FIELD_INTEGRAL E.DL a t AXIS
i
'
/
..........
/
/
/
/
;
""
/
i
i
Time
(se c )
Time
(E-9)
(sec)
(E-9)
Fig. 3.9.7: Driving the input coupled
Fig. 3.9.8: Driving the input coupled cavity
cavity at its resonance; typical method
at its resonance; over-drive method
S ig n a l A n a ly s is A m plitude o f FIELD_INTEGRAL E.DL a t AXIS
S ig n a l A n a ly s is A m plitude o f FIELD_INTEGRAL E.DL a t AXIS
-— - •
/i
/
f
1
+ V
;
..... -..H... -.. ..............
—..i......
0
"-
2
4
6
8
Time
10
12
0
14
(s e c )
2
8
4
Time
10
(se c )
12
14
(E-9)
(E-9)
Fig. 3.9.9: Driving the input coupled
Fig. 3.9.10 Driving the input coupled cavity
cavity at off its resonance; typical method
at off its resonance; over-drive method
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103
The total Q o f the input coupled cavity is about 297 and the over-drive time used was
ten periods. One period is equal to l/(driving frequency). The over-drive ratio used was
9.9785. The results above seem there is mismatch between the over-drive ratio and the
over-drive time which is probably due to a small error in the Q calculation. The resonant
frequency and Q results have more fluctuations in input coupled cavity modeling because
the two half-cavities have different resonant frequencies. The first-half cavity geometry
has been changed due to the opening iris and it is difficult to exactly tune the cavity back
to its original resonant frequency (without loading).
3.9.4 Conclusions
To summarize above:
1. The over drive method reduces the simulation time significantly when we over drive
the cavity at its resonant frequency with an appropriate signal for an appropriate time.
2. The over-drive method is not very useful when we drive the cavity at off-its-resonant
frequency, either higher or lower then the resonant frequency.
3. The over-drive method is effective with our input coupled cavity. However, the over­
drive time and the over-drive ratio may not exactly match to each other due to the
calculation error of the total Q and the resonant frequencies of the two half-cavities
with an opening iris at one side.
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104
3.10 Beam Modeling and Investigations in MAGIC
Precise beam modeling is required to obtain accurate hot test simulation results.
Initially, a built-in beam modeling command in MAGIC was used to model the electron
beam. However, it was found to perform poorly to model an ideal laminar beam.
Consequently, the beam was subsequently modeled using manually edited electron
profiles. This work helped ATK identify a software error associated with the default
beam modeling in MAGIC 3D (Cartesian coordinates).
3.10.1 Built-in LAMINARBEAM Command in MAGIC
The LAMINAR BEAM option in MAGIC [34], together with a magnetic field
profile, is designed to create an ideal laminar flow (electron beam) [10] with a given
beam voltage, current, and radius.
This built-in command was first tested with a
Brillouin flow [10] in 3D Cartesian coordinates, as our EIK is modeled in 3D Cartesian
coordinates. The results, showing the electron momentum and the energy along the axis
using this MAGIC command, are shown in Fig. 3.10.1. The results indicate that the
built-in command does not result in a perfectly rigid rotating laminar beam. The electron
momentum, Px, Py and Pz, should be constant along the Z axis, but instead have
fluctuations that lead to a non-consistent total energy.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
105
I
■2
Time 2 .9 9 5 n s :
^
2
(E-3)
PHASESPACE f o r a l l p a r t i c l e s
r-r-- -
» § 18S ll8SL,
r
0
(tn)
(E -3)
3
h?';
■2
0
2
(E-3)
Fig. 3.10.1: The electron momentum (Px, Py, Pz) and energy (eV) versus axis using
MAGIC 3D
In order to avoid long simulation time and to simplify calculations when later
manually editing the beam profile, an initial investigation and simulations were
performed using 2D cylindrical coordinates. When using IMPORT LAMINAR_BEAM
with Brillouin flux density [10] in MAGIC 2D, the results (Fig. 3.10.2) do not correspond
to theory.
The momentum Pr should be zero; however, it is not.
Furthermore, the
electron energy, Pz and Pe, should be constant along the Z axis, but they are not. Clearly,
the built-in “IMPORT LAMINAR BEAM” command does not accurately model an ideal
laminar beam with Brillouin fields. Overcoming this required manually editing the beam
profile and then importing the edited files into MAGIC simulations.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
106
Is
Beam Tunnel
r
- - J a r ticle-Trafeetories:
^ — Initial energy value
!
5
20
(E-3)
(M)
20
0
5
10
15
2 mi
15
20
Fig. 3.10.2: The electron momentum (Pz, PT, Pe) and energy (eV) versus axis using
MAGIC 2D
Note that the initial value of the particle energy generated from LAMINAR_BEAM
in MAGIC 2D (version 2004.0728.1246) was not correct - it is slightly higher than the
theoretical depressed potential, but MAGIC 3D does provide correct results.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
107
3.10.2 Import Files for MAGIC 2D
To model the beam appropriately, a proper import file is necessary. In MAGIC 2D,
there are two files associated with the IMPORT command: *.grd contains the electric
field and *.par contains the particle information. Both files need to be manually edited
and imported to the simulation.
a. Electric Field (*.grd file)
With 2D cylindrical coordinates, the axial and radial electric fields
(Er
and Ee) are
setup in the *.grd file.
Er - According to Gauss’s law, for a perfect laminar beam (assuming the tube is
infinitely long), the radial electric field in the beam is:
r - Radius (r<b or r=b)
I - Beam current
Er(r) = -
— —2
2 Jibe0u0
b - Beam radius
Co - Dielectric constant of free space
uo - Beam axial velocity
The electric field in the radial direction
(Er)
reaches a maximum at the beam
radius and then decreases following a 1/r2 dependence outside the beam radius
(Fig. 3.10.3).
Ee - Ee equals zero in a uniform laminar beam.
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108
Er and Ee can then be assigned at the mesh points in the electric field file,
corresponding to the imported plane coordinates.
M
8
<
I
B
>
s
I
100
300
200
th e
400
<E-«)
B eam radius
Fig. 3.10.3: The radial electric field versus the beam tunnel radius
b. Particle Profile (*.par filel
In the particle file, the electrons are divided into several rays to represent the
macro beam behavior. Each ray represents: J (the current of each ray), Z (the
particle axial position relative to the IMPORT plane), R (the particle radial
position), and Pz, Pr, and Pe (the particle momentum in the z, r, and 0 directions).
Note that in this file, the momentum represents the particle velocity multiplied by
the relativity factor, in units of meters/second.
i.
J (amps), Z (m), R (m)
The beam is assumed to be uniformly distributed throughout the radial direction.
The electron ray is located at the center radius of the representative area and the
represented current is proportional to each represented annular area.
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109
ii. Pz, Pr, Pe (m/s)
The total momentum and energy of each ray can be calculated from the
depressed potential. The potential beam depression, V<j can be calculated from
the formula below:
r - Radius position
v - Beam voltage
b - Beam radius
a - Tunnel radius
K - Microperveance
P z - At the Brillouin flow, all electrons move at the same axial velocity as the
velocity o f the electrons on axis, which have no rotation. Therefore, we can
assign P z of all the beams with the value calculated from the depressed beam
potential o f the beam on axis.
Pe - According to the principle of conservation of energy, the additional energy
o f the beam off axis would contribute to the angular momentum.
At the
Brillouin flow, the angular frequency equals the Larmor frequency, which is
half o f the cyclotron frequency:
..
(y- the relativity factor)
2 ym
Therefore, we can assign Pe, with the radius multiplied by the Larmor frequency
and y . Due to the depressed beam potential, gamma (y) varies with the radius.
However, since the change with the radius is not significant, we can use the
value at 0.707 o f the beam radius as an average value through the radial
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110
direction. Using this simplification, it appears that the results are very close to
those obtained using the actual gamma value.
Pr - Pr is assigned to be zero with Brillouin flow because the space charge force
and the magnetic field force are equal with opposite directions.
Using the theory above, three different approaches are detailed in the following
section. The first approach (previously used at SLAC, with the depressed beam potential
added by the author) initially imports a partially edited beam profile to MAGIC and then
obtains the final import files from MAGIC exported files. However, this method did not
work well. The second and third approaches involved manually editing all the beam
parameters to eventually achieve the desired laminar beam.
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Ill
Import file I
In this approach, the initial particle file was edited to set P r and Pe to zero, with all of
the energy assigned to P z. The modeling results are shown below. The total electron
energy is not constant and electrons move inward and then outward periodically due to
the imbalance between the space charge force and the magnetic force. Therefore, the
field and particle files are exported at the balanced point (Fig. 3.10.4), where Pe is at half
o f the maximum, in order to obtain the particle files for the subsequent simulations.
T
i
Beam Tunnel
§
i
|
s
d
Particle Trajectories
t
§{
s
d
0
2
2
6
4
(E-3)
(M)
8
Z
12
10
14
(E-3)
(K)
Export files where the half
maximum value occurs
A ll depressed beam potential
contributes to Pz initially
5
\
£
7
2
0
2
6
8
Z IN)
10
12
It
A
0
2
(E-3)
1
6
8
10
12
It
i
e
2 IN)
10 12
it
IE-3)
Fig. 3.10.4: The electron momentum (Pz, Pr, Pe) and energy (eV) versus axial position
using MAGIC 2D with import file I
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The exported particle file is modified with Pr set to zero and then imported again. The
results below reflect a laminar beam, with Pz and Pe remaining constant along the axis
and the electron trajectories not intersecting (except for very few rays at the edge, for the
drift tube length modeled). However, the beam radius is larger than the desired value and
Pe is incorrect. If we decrease the beam radius in the initial imported particle file, the
ideal beam radius may be reached, but this would likely require several iterations.
I
Beam Tunnel
1
g
Desired Beam Radius
-8
I
s
o
0
2
4
6
8
Z
(M)
10
12
0
14
(E -3)
2
4
6
8
Z
i IB
10
12
(M)
14
(E -3)
2
n)
Fig. 3.10.5: The electron momentum (Pz, Pr, Pe) and energy (eV) versus axial position
using MAGIC 2D with final import file I
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113
Import file II
Since Import file I did not provide the desired beam, all the parameters in the initial
beam profile were manually edited, with Pz set to the on axis value and Pe set for
Brillouin flow. The results are shown below:
Edge Beam
8
0
5
10
IS
20
£ r at beam edge
j
s
-
I I-
-i
Sz
i
200
300
Fig. 3.10.6: The electron momentum (Pz, Pr, Pe) and energy (eV) versus axial position
using MAGIC 2D with import file II
The results show a laminar beam with the desired beam radius, except for a few rays at
the beam edge. The cause o f these few rays was suspected to be due to the electric field
error at the beam edge because of insufficient resolution. Therefore, a finer mesh model
with more rays is employed.
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114
Import file III
To correct the problem at the beam edge seen with Import file II, a finer mesh model with
more rays was used (otherwise, this model was the same as Import File II). Using this
technique, the edge beam effect was reduced, with Pr at the edge beam reduced
significantly. The non-laminar portion of the beam is then reduced, as can be seen in the
figures below.
Edge Beam
3
S
s
I
s
12
2
2
2D
0
IB-3)
5
10
Z INI
15
20
(E-3)
£
at beam edge
Fig. 3.10.7: The electron momentum (Pz, Pr, Pe) and energy (eV) versus axial position
using MAGIC 2D with import file III
These results indicate that the previous simulations did not have sufficient resolution to
accurately model Er, and that resulted in lower Er at the beam edge, leading to the
outermost beam going inward. As one ray starts becomes non-laminar, it affects the
other rays and the beam quality decreases, especially with a longer drift tube. Modeling
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
115
with higher resolution reduces this effect. Another possibly appropriate method, if a
larger mesh needs to be used, is to reduce P q of the outermost beam in the imported beam
file to obtain similar results.
3.10.3 Summary
An ideal laminar beam is critical for accurate hot test simulations. Since the MAGIC
(version 2004.0728.1246) command IMPORT LAM INARBEAM did not result in a
perfectly laminar beam, the manually edited beam profiles were investigated.
The
imported files were manually edited under Brillouin flow in cylindrically coordinates
because the parameters are easier to calculate with Brillouin flux.
With any other
focusing field, the beam profile would be much more complicated. By manually editing
the imported electric field and particle profiles, with all parameters assigned to the
theoretical value under Brillouin flow, a high-quality laminar beam can be simulated.
Only a small portion o f the beam at the beam edge is non-laminar, and this effect can be
reduced by increasing the electric field resolution at the edge. However with a finer mesh,
the beam quality could degrade with a longer drift tube. In addition, importing a laminar
beam in 2D cylindrical coordinates to 3D Cartesian coordinates in MAGIC remains an
issue (the EIK is modeled in 3D Cartesian coordinates). Manually edited beam profiles
in 3D Cartesian coordinates with Brillouin flow has not yet been tested. However, from
these simulation results, a perfect Brillouin flow is demonstrated to be very difficult to
model precisely in MAGIC because Brillouin flow is very sensitive to every parameter
involved. Therefore, confined flow should be always used in simulations, which is the
case with practical devices.
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116
As a result o f this investigation, ATK was able to locate a software error associated
with the emission process in MAGIC 3D Cartesian coordinates. The LAMINAR BEAM
command was fixed and the new version (2005.0201.1602) provides more stable radial
electron beam momentum (Px and Py) along the z axis (the previous built-in LAMINAR
command had higher fluctuations in Px and Py). In comparison with the previous version
o f MAGIC (2004.0728.1246), the new version resulted in a higher voltage cross the
individual cavity gap but appears to provide similar gain results. The hot test results on
the final EIK design uses the revised version of MAGIC 3D.
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117
3.10.4 Additional Information
The author would like to clarify confusion concerning beam modeling that arose in
discussions about this software problem with ATK. If there is no magnetic field present
before the IMPORT plane, the Brillouin field is the only magnetic field that can generate
the beam at a certain radius without scalloping. If the magnetic field at the cathode is
zero and the magnetic field is higher or lower than the Brillouin field, then the beam
would either go inward or outward. However, if a magnetic field is present at the cathode,
then the magnetic field can be adjusted higher than the Brillouin field to have the beam at
the same beam radius without scalloping (Pe cross Bz balances with the space charge
force and the centrifugal force). In such a case, the beam would rotate slower than with
Brillouin flow (hence lower values of Pe). This situation only happens when there is a
magnetic field threading the cathode and increasing appropriately along the axis.
It is concluded that the LAMINAR BEAM command in MAGIC 3D can model the
confined (immersed) flow appropriately, but not the Brillouin flow. Using this command,
it is possible to model a confined flow beam with magnetic fields higher than Brillouin
field, assuming that there is a magnetic field present at the cathode and this field
increases appropriately along the beam tunnel. Using the LAMINAR_BEAM command
with magnetic fields higher than Brillouin flux results in the electron Px, Py being lower
than it is with Brillouin flow because the confined beam rotates slower.
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3.11 Stability Analysis
It is important to perform a complete stability analysis by calculating the beam loaded
Q-value o f every eigenmode in each cavity. If the beam loaded Q of an eigenmode is
negative and it results in the total Q assuming a negative value, the circuit may turn into
an oscillator. Here, an RF input is not required and the noise will start to grow (i.e.,
oscillating) at that eigenmode frequency and destroy the beam [43].
3.11.1 Stability Calculation Using MathCAD Worksheet
A MathCAD program in Appendix III is used to calculate the beam loaded Q using
the electric fields o f each cavity eigenmode and the Wessel-Berg M+/M. coupling
coefficient approach [25]. Each cavity is modeled using MAGIC 3D and the electric
fields on the axis and the drift tube radius of each eigenmode are obtained and
appropriately inserted into the MathCAD worksheet.
The MathCAD program then
employs these electric fields to calculate the coupling coefficients for the slow and fast
space-charge waves as well as the beam loaded conductance, thereby making it possible
to obtain the beam loaded Q. Here, the electric fields are the Ez profile exported from
MAGIC simulations.
Below are the electric fields of the 2nd coupled cavity (intermediate cavity):
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119
—— Ez on A x is ------ Ez on Drift Tube Radius
n
N
LU
■□mi
•D as
□
am
dus
am a
Z(m)
Fig. 3.11.1: Electric Field Profile of the 2 n Mode
—
Ez on A xis
•DJS1
Ez on Drift Tube Radius
a
omz
aan
Z*n)
Fig. 3.11.2: Electric Field Profile of the n Mode
In order to ensure that there is no noise start oscillating during the beam voltage turn-on
and turn-off period (for pulsed beam; for DC beam, there will be only the turn-on
process). The plot extends from below 5% up to 120% (in case of overshooting) of the
operating beam voltage. Although there is not much power while the beam voltage is
below a few percent o f the normal operating voltage, we would like to ensure that the
total Q remains positive at all the times. The beam loaded Q values o f each intermediate
cavity are shown below (Fig. 3.11.3 ~ 7).
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120
Beam Loaded Q
3000 -*+-+»-
2 PI Mode
PI Mode
2000
-
16.7
Voltage (kV)
Fig. 3.11.3: The Beam Loaded Q of the 2nd coupled cavity (Qo=650)
n
*
V
l
\
*
* s'
111;;
k
m
i
it* r . S -
Beam
Loaded Q
VifaVnW
»
2 PI Mode
PI Mode
I
6.7
8.7
10.7
16.7
Voltage (kV)
Fig. 3.11.4: The Beam Loaded Q of the 3rd coupled cavity (Q0=655)
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121
3000
Beam
Loaded Q
2000
1000
0
2 PI Mode
-1000
PI Mode
-2000
-3000
Voltage (kV)
Fig. 3.11.5: The Beam Loaded Q of the 4th coupled cavity (Q0=1846)
Beam
Loaded Q
3000
2000
X lur fi hiii/iiVmif (
1000
-/ /
o
[ill hrjail
\ . i»* '1 ^ <’•^'■0^
* 9-■S
v ** ’4 3
2 PI Mode
-1000
PI Mode
-2000
-3000
0.7
2.7
4.7
6.7
8.7
10.7
12.7
14.7
V o lta g e (kV)
Fig. 3.11.6: The Beam Loaded Q of the 5th coupled cavity (Q0=1821)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
16.7
122
7500 ■
5000 -
o-
' .
*
„ i ' j,i* i/ia.
*>»' ,
tfiffT
■%
.*
-2500 - ■;•';■»■
■ ‘f
.v
tftw *'
*5000 - ......»■»•... —
$
V-"
’7500 ■7'^i»i'^ *,N|,»^ ■M
y,i*aIr<^
0.7
2.7
4.7
6.7
8.7
10.7
12.7
14.7
16.7
Voltage (kV)
Fig. 3.11.7: The Beam Loaded Q of the 6 th coupled cavity (Q0=2242, single gap)
The mode frequencies and Q-values of the intermediate cavities at beam voltage (14 kV)
are as follows:
Table 1 3.11.1: The resonant frequency and beam loaded Q of each cavity
Intermediate Cavity #
2 PI mode f (GHz)
PI mode f (GHz)
2 PI mode Qb
PI mode Qb
2 PI Q0
PI Qo
Conductivity (mhos/m)
2 nd
34.895
33.469
3170
46547
650
710
3 x l0 5
3rd
35.051
33.602
1775
-78935
655
719
3xl0 5
4 th
35.111
33.654
1737
-38540
1847
1607
5.99x10'
5th
35.141
33.680
1717.4
-30834
1845
1606
5.99x10'
6
35.200
7552
2273
5.99x10'
The input and output cavities have open irises which are connected to the waveguides.
The stability analyses of both cavities were skipped because the loaded waveguide results
in low external Q. Therefore, it is very unlikely to have oscillation from both cavities.
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123
It is not necessary to perform the stability analysis of the input/output cavities unless
they have higher R/Q (different geometry) or more gaps (more eigenmodes) than the
intermediate cavities.
Note that the resonant frequencies of the % mode frequency o f each cavity are all
below the bandwidth edge (34.85 GHz ~ 35.15 GHz).
The 6 th cavity has only a single gap; therefore, there is only one eigenmode.
Although the table above shows good stability at the beam voltage (14 kV), the total Q
becomes negative when the voltage was raised in the 4th and the 5th cavities (see Figs.
3.11.5 & 6 ). The n mode Qb is about -890 around 9.3 kV and the 2 n mode Qb is about 1362 around 5.1 kV.
The light loading in these two cavities (the wall material was
chosen to be copper which has a conductivity of 5.99x107 mhos/m) may result in
oscillation when the voltage is raised and continue to oscillate at higher voltages. To
resolve this issue, one possibility is to shorten the cavity along the z-direction so that the
slower electrons would leave the cavity sooner; however, that requires redesign of the
cavity. The other way is to load the cavity which is easier and that is our preferred path.
We would like to load the cavities so that Qo is about 90% o f the absolute minimum
Qb value (890). (Therefore Qo is around 800.) The results from AJDISK and MathCAD
are shown in the following pages (Fig. 3.11.8 ~ 10). Since these two are intermediate
cavities, the reduction o f Qo did not change the gain or bandwidth significantly.
At later sections o f this chapter (3.12 & 3.13), the beam loading and noise analysis
using MAGIC indicate that our original EIK design does not result in any physical
oscillations; therefore, we do not need to coat these two extra cavities. The simulation
results o f coating the 4th and the 5th cavities are for reference.
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■ A J D IS K sim u ltaio n with n e w Q o
A JD IS K sim u latio n with p re v io u s Q o
Input Power (w)
Fig. 3.11.8: Gain simulations with Qo reduction
1D MathCAD with new Qo
1D AJDISK with new Q o
34.8
34.9
35
F re q u e n c y (QHz)
35.1
35.2
Fig. 3.11.9: Small signal simulations with Qo reduction
AJDISK input power 0.17 w (new Cb)
■A J D S K input p o w er 0 .1 5 w (n e w Qo)
AJDSK input power 0.15w(pnevous Qo)
34.8
34.9
35
F req u en cy (QHz)
35.1
352
Fig. 3.11.10: Bandwidth simulations with Qo reduction
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125
In Fig. 3.11.10, the input power for the previous bandwidth simulation before Qo
reduction was 0.15 watts which gave output power 1.02 kW at the center frequency and 3
dB bandwidth 234 MHz. After Qo reduction the output power reduced to 958 watts with
the same input power. In order to obtain the output power over 1 kW, 0.17 watts were
put in (with the Qo reduction) and 1 kW and 3 dB bandwidth 220 MHz were obtained.
3.11.2 Cavity Material
When we have a negative total Q , we can coat the cavity using loss material to lower
the Qo and thereby suppress the oscillation (i.e., we make the total Q positive again). The
Qb value will not be changed by different materials of the cavity.
Specifically, Qb is
determined by the fast and slow wave coupling coefficients which are primarily
determined by the cavity geometry and the tube radius, etc.
A new conductance value o f 4.3 xlO5 mhos/meter is used for the 4th and 5th cavities in
the results shown above and this gave values of Qb 800 and 715 for the n mode and the
27t mode, respectively for both cavities. This change ensures that the total Q is always
positive when the voltage is raised.
One coated loss material possibility could be “TiNitride” (Ti - Titanium; Nitride Nitrogen). The Titanium can be evaporated in Nitrogen gas and be coated on the desired
cavity surfaces. We can choose how much portion of the surfaces and what thickness we
would like to depends on the desired Qo.
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126
3.12 Beam Loading in MAGIC 3D
In the previous section (section 3.11), we employed a MathCAD calculation
worksheet (Appendix III) to calculate the beam loaded Q using the electric fields of each
cavity eigenmode (from MAGIC 3D) and the Wessel-Berg M+/M. (coupling coefficient)
method. In this section, we will use MAGIC 3D to directly measure the beam loaded Q
and resonant frequency with both the beam and RF signal present (often referred to as
“hot test” simulations).
3.12.1 Beam Loaded Resonant Frequency and Q Simulations
The gain and the bandwidth simulations in both small and large signal operation were
completed when we applied RF continuously to the input PORT and concurrently
imported the DC beam to the circuit. However, we cannot measure the beam loaded
resonant frequency when the cavity is continuously driven with an RF signal. In order to
measure the beam loaded resonant frequency and the beam loaded Q of each cavity in
MAGIC 3D, the following approach was adopted. This approach is similar to how we
measure the unloaded resonant frequency and Q of a cavity (the referenced figures are
from 2 7i mode simulations):
1. Place a Driver on each gap. (Fig. 3.12.1)
2. Apply a Gaussian signal to each Driver (Fig. 3.12.2). Setup the signal to properly
initialize the electric field for each eigenmode (please refer to the section
describing the measurement of the resonant frequency o f an unloaded cavity).
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127
3. When there is no loss input, the gap voltage will stabilize (Fig. 3.12.3). Note that
in a previous section we inputted the material loss and measured the unloaded
resonant frequency and Q of the cavity.
4. Instead o f inputting the material loss, we import the beam (Fig. 3.12.4) after the
voltage stabilizes. Sufficient Driver current is required to build up a significant
voltage across the gap (the gap voltage is proportional to the RF current applied)
compared to the beam voltage (Fig. 3.12.5); otherwise it may be difficult to see
the RF fields when the DC signal is present.
o
i-i
Gaussian Signal
o
* Driver *
o
D R IV E R J 3 D r iv e _ s ig n a l A N T E N N A
Fig. 3.12.1: Locate Driver at each gap
0
600
400
200
TiM
800
(*C )
Fig. 3.12.2: Apply Gaussian Signal
to each Driver
hVUtwfc
§
Gap Voltage
Without Beam
0
400
200
Tia*
600
(m c )
Fig. 3.12.3: Temporal growth
Import Beam at Appropriate Timing
Fig. 3.12.4: Load the beam into the circuit
o f voltage on the circuit axis
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
128
RF
8
■-
o
From Boam
i
i
Gap Voltage Decay
Due to Beam Loading
o
0
-4
4
2
(■)
o
oO
I
0.0
I
I
I
I
I
0.0
1.0
1.0
2 .0
2.0
Tim
Fig. 3.12.5: Electric field on the axis
(me)
(E-9|
Fig. 3.12.6: RF loss due to beam loading
(2 tc mode)
s
£'
QO
©
Beam Loading
—
Resonant Frequency
1/Beam Loaded Q -
o
IM
©
©
o.o
©
1.0
1.0
0.0
Tim
2 .0
2 .0
(sec)
0.0
0.0
1.0
1.0
(E-91
Tim
2.0
2.0
((-91
|s*c)
Joule
c
\
1
Energy Decay Due
to Beam Loading
8
o
0
1
2
3
ided Q ......—..
_
_
r
o
1
3
1
1
8
D
O _=..
A
1
(E-9)
Fig. 3.12.7: Loaded resonant frequency and loaded Q of each eigenmode
♦
j
f
4
2
Tim
3
(» c)
Fig. 3.12.8 Loaded Q of each eigenmode calculated from the RF energy decay
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(1-91
129
5. With OBERVE FIELDJNTEGRAL AMPLITUDE, we see that the fields decay
due to the beam loading (Fig. 3.12.6). This command also calculates the resonant
frequency and the Q from the field oscillations (Fig. 3.12.7 & 8 ). Since the beam
is imported and no other loss is present, the results obtained are the beam loaded
resonant frequency and the beam loaded Q of each eigenmode.
6 . We
can
also
use OBSERVE FIELD ENERGY
EM
OSYSSVOLUME
Q_OF_SIGNAL FREQ to determine the beam loaded Q (Fig. 3.12.8).
This
command is designed to measure the decay o f electromagnetic energy as a
function o f time. The “FREQ” is the reference frequency for analysis (here we
put in the eigenmode frequency o f each cavity). This method is useful for 7i mode
simulations since OBSERVE FIELD INTEGRAL AMPLITUDE needs to be
measured at the upper and lower gap separately for n mode.
However, with
Q_OF_SIGNAL we can conveniently measure the Q from the decay o f the total
energy in the coupled cavity.
Table 3.12.1 below summarizes the results of the MAGIC 3D simulations. The sixth
cavity is a single cavity, so there is only one eigenmode. Each Qb is calculated at its
eigenmode resonant frequency.
One can see that the results of Qb are very different from the results we obtained in
the stability analysis section using the MathCAD worksheet (Table 3.12.2) using
kinematic and space charge theory.
The discrepancy may be due to the coupling
coefficients calculations. However, the trend is similar. The results also showed that at
the operating voltage there is no possibility of oscillation from undesired modes. The
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130
complete interaction circuit simulation, including the beam, using MAGIC 3D also
displayed no noise from n modes (please see the section of noise discussion).
The
unloaded resonant frequency and the theoretical beam loaded resonant frequency are
displayed below for references (Table 3.12.3 & 4).
Table 3.12.1: Beam loaded resonant frequency fb and Qb results from MAGIC 3D
Intermediate Cavity
2 7t mode fb (GHz)
n mode fb (GHz)
2 n mode Qb
n mode Qb
2 7t mode Qo
7i mode Qo
2 nd
34.874
3rd
4th
5th
35.030
35.090
35.120
33.495
33.626
33.678
33.074
1062
1019
1006
1000
8113
650
710
18680
655
719
16307
1846
1607
42905
1821
1606
6 th
35.201
2299
2242
Table 3.12.2: Beam loaded Qb results from stability analysis calculation (Appendix III)
2 nd
Intermediate Cavity #
2 7i mode Qb
n mode Qb
3170
46547
3rd
1775
-78935
4th
1737
-38540
5*
1717.4
-30834
6 th
7552
Table 3.12.3: Unloaded resonant frequency of each cavity (MAGIC 3D)
2 nd
Intermediate Cavity #
2 7i mode fo (GHz)
7i mode fo (GHz)
34.895
33.469
3rd
35.051
33.602
4 th
35.111
33.654
5tn
35.141
33.680
6 th
35.200
Table 3.12.4: Beam loaded resonant frequency fb and Qb results from MathCAD using
kinematic theory (Appendix I)
Intermediate Cavity #
2 71 mode fb (GHz)
2 7t mode Qb
2 nd
34.8935
1447
3rd
35.0495
1428
4 th
35.1095
1423
5tn
35.1394
1419
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6th
35.199
2226
131
3.12.2 Input Cavity Beam Loading Measurements
For the first cavity beam loading modeling (input cavity), we use the same technique
as described above, with the addition of setting up the input PORT (Fig. 3.12.9). As
opposed to the setup of the input PORT in the section on input power, the input PORT of
the input cavity now needs to be setup as an output PORT in the beam loaded Q
measurement because the drivers are located internally in this simulation. The PORT
setup is as following:
FREQ _W G _CU TO FF
=
PO R T_B ETA _PH A SE
PO RT
O U TPU T_PO R T
0 . 5 _ C /W G _ W ID T H ;
=
l / S Q R T ( 1 - ( F R E Q _ W G _ C U T O F F /F R E Q )* * 2 ) ;
N E G A T IV E
P H A S E _ V E L O C IT Y
PO R T_B ETA _PH A SE;
PORT
Waveguide
Cavity
Fig. 3.12.9: Input Cavity Configuration
Note that only the 2n mode was modeled for the input cavity. Because of the low loaded
Qe, oscillation in the input cavity is unlikely. The results are presented and discussed in
the section about input gap voltage (3.7) [45].
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132
3.12.3 Result Comparison
Except for the penultimate cavity (the sixth cavity, to be discussed later), all other
intermediate cavity beam loaded resonant frequencies from MAGIC 3D (Table 3.12.1)
are about 21 MHz lower than the unloaded resonant frequencies (Table 3.12.3), and about
19.5 MHz lower than the beam loaded resonant frequencies from MathCAD calculations
(Table 3.12.4) using kinematic theory (Appendix I).
In previous bandwidth simulations, it appears that the MAGIC 3D small and large
signal results have a small frequency shift downward when compared to AJ Disk results
(Fig. 3.12.9 and Fig. 3.12.10). AJ Disk uses kinematic theory to handle the beam loaded
resonant frequency - the same as MathCAD. If we shift the MAGIC 3D results by the
19.5 MHz - which is the difference between the MAGIC 3D and AJ Disk beam loaded
resonant frequencies (Tables 3.12.1 and 3.12.3) (assuming the MAGIC 3D calculate the
beam loaded resonant frequencies the same as AJDISK), the match between these two
modeling tools is improved significantly (Fig. 3.12.11 and Fig. 3.12.12).
This frequency offset indicates that either MAGIC or AJ Disk may calculate the beam
loaded resonant frequency to be lower or higher than the actual value. However, it is
difficult to recognize which tool is more accurate before performing actual measurements
on the EIK.
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34.8
34.9
35
35.1
35.2
Frequency (GHz)
34.8
34.9
35
35.1
35.2
F req u en cy (GHz)
Fig. 3.12.9: Small signal simulation
Fig. 3.12.10: Large signal simulation
result comparison
result comparison
MAGIC 3D
1D Disk Modelinfll
MAGIC 3D
1D Disk Modeling
Frequency (GHz)
F requency (GHz)
Fig. 3.12.11: Small signal simulation
Fig. 3.12.12: Large signal simulation
result after shifting
result after shifting
Fortunately, the BFK in SLAC already has the measurement data required for
comparison with simulations (Fig. 3.12.13).
It can be clearly seen that there is an
approximately 0.4 MHz downward shift of the entire MAGIC results. Previously, the
reason for this effect was unknown. However, after the beam loaded resonant frequency
modeling and the analysis above, it is very possible that this downward frequency shift is
due to how MAGIC handles the beam loading. Since a MAGIC simulation using a larger
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134
mesh size resulted in a lower unloaded resonant frequency, a coarser mesh size was
initially suspected to result in a larger beam loading frequency shift. Therefore, the beam
loaded resonant frequency was measured with a finer mesh size using MAGIC 3D.
However, use of a finer mesh did not change the amount of this downward frequency
shift. This may need to be investigated further, both using AJDISK and discussing the
MAGIC results with ATK.
1 MW CW B-Factory Klystron (Small Signal)
55 [
I
1
<5 45
(
'
35
/
I M
,
'L- i ^A—1 W 1
. *
470
472
S
1
«
■«*
474
476
—
M easu red (P,n=4.1 W)
—
—
AJ DISK <p in=4-1 w >
MAGIC 2D (P,„=4.1W )
478
480
482
Frequency ( MHz)
Fig. 3.12.13: BFK measurements and simulation result comparisons
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135
3.12.4 Penultimate Cavity Beam Loading
In contrast to the MathCAD worksheet calculation (Table 3.12.4), the penultimate
cavity beam loaded resonant frequency measured with MAGIC 3D (Table 3.12.1) is
slightly higher than the unloaded resonant frequency (Table 3.12.3). The reason for this
is as following.
The transit angle is the change of the phase when electrons pass the gap and is equal
to the electron propagation constant (3e (2tcIX) times the gap length d. The susceptance Bb
o f the beam loading conductance Gb is positive between the transit angle of zero and tc
and becomes inductive (negative) after n (Fig. 3.12.14) [46]. The beam loaded resonant
frequency is
r
____ ____
fo
0.25
0.15
0.10
0.05
0
-0.05
- 0.10
•0.15
0
n
271
31C
47t
P«rf
Fig. 3.12.14: Normalized beam loading conductance and susceptance
for a gridded gap, as a function of transit angle [46]
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136
Therefore, if the susceptance is positive, the beam loaded resonant frequency is lower
then the unloaded resonant frequency, and vice versa.
■2
•1
f f i
0
1
z M
Fig. 3.12.15: The penultimate cavity
2
«-3|
Fig. 3.12.16: Ez profile on axis from
MAGIC 3D
The MathCAD small signal worksheet calculates the transit angle using the physical
gap distance of the cavity (Fig. 3.12.15), thereby giving a transit angle of 0.5 lu when d =
0.5 mm. Hence, the beam loaded resonant frequency calculated using MathCAD is lower
than the unloaded resonant frequency. However, the electric fields extend into the drift
tube so that the length over which electrons are actually modulated is longer than the gap
length (Fig. 3.12.16).
The penultimate cavity modeling using MAGIC 3D gives an
“effective gap length” (Fig. 3.12.16, L = 3.35 mm) that is much longer than the gap
length d and results in a transit angle of 3.427t. Therefore, the beam loaded resonant
frequency measured using MAGIC 3D is slightly higher than the unloaded resonant
frequency.
Normally, the klystron cavity (single gap) is operated between transit angles between
1 to 1.5 radians (about 60 to 90 degree).
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137
3.12.5 Mode Contamination
The results o f the rc mode Qb modeling contains significant ripple at the beginning of
the simulation (Fig. 3.12.17). That is due to competition from the 2n mode (the Gaussian
signal excites both modes at the same time, although with much a stronger signal at the n
mode frequency). However, since the 2n mode has a lower Qb (Table 3.12.1), it decays
faster than the n mode and the ripple decreases with time. Conversely, with Qb modeling
in the 2n mode (Fig. 3.12.18), the n mode contamination will eventually show up through
the faster 2n mode decay. Hence, in this case the ripple gets worse with time and the Qb
measurement exhibits an upward tilt due to the n mode contamination. Using a more
precise Gaussian signal to more smoothly build up the RF and to turn the beam on slower
may reduce the mode contamination.
®
o
..............
o
<4-1
<4-1
O
o
.... i----------- i-----------
®
.. ... ..
10
20
Time
30
(sec)
40
(E-9)
20
Time
(sec)
Fig. 3.12.17: The 7t-mode Qb
Fig. 3.12.18: The 27i-mode Qb
modeling of the 5th cavity
modeling of the 2 nd cavity
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(E-9)
138
3.13 Noise Investigations
When the RF is applied (Fig. 3.13.1) and the beam is simultaneously imported into
the circuit concurrently (often referred to “hot test” simulations), unexpected noise was
discovered when driving the circuit with higher frequencies. The figures (Fig. 3.13.2 & 3)
shown below are the results of OBERVE FIELD INTEGRAL AMPLITUDE from the
axis of the 1st cavity. The AMPLITUDE option analyzes the amplitude envelope of an
oscillation with a given reference frequency. With the cavity driven at the frequency of
34.85 GHz (Fig. 3.13.2), the voltage oscillation along the axis (al) is analyzed and the
amplitude envelope is obtained (a2).
The results show a very clean signal and are
consistent in both figures. However, when the cavity is driven at the frequency of 35.10
GHz (Fig. 3.13.3), there is a high level signal that develops in the voltage oscillation plot
(bl) that does not appear in the amplitude envelope plot (b2). This usually indicates that
the signal frequency is much higher than the given reference frequency.
oo
Input PORT
0
Output PORT
10
20
30
40
Z (m)
Fig. 3.13.1: The 7-cavity EIK geometry
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(E-3)
139
iVvvw^.. -------
10
Tim e
~ r
15
(se c )
20
10
(E-9)
15
Tiae
(al)
(sec)
(a2 )
Fig. 3.13.2: The voltage oscillation (al) and the amplitude envelope (a2) along the 1st
cavity axis when the cavity is driven at 34.85 GHz
0
5
10
Tim e
15
(se c )
20
0
(bl)
5
15
10
Tiae
20
25
(sec)
(b2 )
Fig. 3.13.3: The voltage oscillation (bl) and the amplitude envelope (b2) along the 1st
cavity axis when the cavity is driven at 35.10 GHz
This noise phenomenon has been observed in many simulations at higher driving
frequencies. The operating frequency range is approximately 34.85 GHz to 35.15 GHz,
and the frequency where the phenomenon starts appearing increases when the input
power increases. For example, when the input power is 0.0063 W (the voltage across the
waveguide is 1 volt), the noise starts appearing at a driving frequency around 34.05 GHz
and above. When the input power is 0.15 W (the voltage across the waveguide is 4.9
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140
volts), the noise starts appearing at a driving frequency around 35.10 GHz and above.
Whenever this phenomenon appears, the results exhibit unwanted noise in all plots, such
as the Ez profile on axis (Fig. 3.13.4) and the PHASESPACE plots of the electrons (Fig.
3.13.5).
..
■\ \ 1
1
\
J
\ t\ '
,
lilt
pi
|
ll
1
..... ; .. }
t fi'
L
i
./n
#
.....f 4 ..u... i .. %
l v
hi
1*
\
7
I f'li
iffl
------------------ J
-I------------0
10
20
30
z (m)
40
0
10
(E-31
(Cl)
20
z (m)
30
40
(E-3)
(c2)
Fig. 3.13.4: The Ez profile along the EIK
I
is with no noise (cl) and with noise (c2 )
I
§
o
s
o
^ 0
10
20
Z (n)
30
40
(E-3)
(dl)
Fig. 3.13.5: The gamma vs. z plots
10
20
30
2 (ml
(d2 )
ith no noise (dl) and with noise (d2 )
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40
(E-3)
141
The eigenmode with the lowest cutoff frequency of the circular waveguide is TEn
[41] and the cutoff frequency is equal to
f , = — •1 ^ 1
2n
a
(a - drift tube radius)
The drift tube radius is 0.5mm. That provides a cutoff frequency o f 176 GHz, which
ensures that our Ka-band EIK will not have any RE traveling forward or backward along
the tube. However, from the CONTOUR plots below we see that the wave is traveling
along the tube when the noise is present.
(e2 )
Fig. 3.13.6: The CONTOUR plot of the EIK modeling (y-z plane) without noise (el)
and with noise (e2) (Brighter color, higher magnitude o f Ez).
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142
3.13.1 Fourier Analyses
In order to determine the frequency components of the noise, we performed a Fourier
analysis on the axis and on the beam (just before the electrons exit the circuit). The
“FFT” command in MAGIC allows us the option o f performing the Fourier analysis on a
line or a plane o f our choice. Fig. 3.13.7 below is the Fourier analysis o f the axis and
beam when the noise is not present, indicating that the only signals appearing are
harmonic frequencies. However, the FFT plot when noise is present shows frequency
components other than harmonic frequencies (Fig. 3.13.8). After reviewing several FFT
plots when noise is present, it is concluded that the frequencies of the noise are above 200
GHz and they vary depending on the driving frequency with a separation (between each
noise frequency components) equal to the driving frequency (£2).
It is important to determine the source of this noise oscillation in order to eliminate it.
The cause can be physical or numerical. Physically, the unexpected oscillation can be
due to the cavity stability (when the total Q is negative because of the beam loaded Q) or
the slot mode (when most o f the energy stays in the slots instead o f the cavities).
However, if the noise results from physical modes, the frequencies will not vary due to
the driving frequency as shown in the FFT results in the previous session.
Further investigation, to be discussed later in this section, indicates that the noise is
unlikely to be physical oscillations. When we looked at the PHASESPACE plots of the
beam (Fig. 3.13.9), we found that the beam without noise looks normal (gl) while the
plot o f the beam with noise (g2 ) shows the electrons to lined up in small layers, with
spacing between 1 to 2 “cells” [Appendix IV]. According to Dr. David Smithe in ATK,
this usually indicates a numerical instability.
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143
5
1
Is
1 1- L.J
100
200
F re q u e n c y (GHz)
0
300
34 90
100
LO-
-i
u
300
200
Frequency (GHz)
(el)
(e2 )
Fig. 3.13.7: The Fourier analysis of the axis (el) and the beam (e2) with a driving
frequency o f 34.90 GHz
f
35.15 Gl Iz ap a rt
............... J..A...
\
.. i
... I — 1--J
100
200
F re q u e n cy (GHz)
(fl)
300
Frequency
(GHz)
(f2 )
Fig. 3.13.8: The Fourier analysis of the axis (fl) and the beam (f2) with a driving
frequency o f 3 5.15 GHz
(gl)
(g2 )
Fig. 3.13.9: The PHASEPACE of the beam without noise (gl) with noise (g2)
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144
3.13.2 Numerical Effect and Grid Jiggling
In finite difference simulations, light does not always travel at the speed of light. The
numerical speed o f light depends upon the cell resolution. In other words, at any given
frequency the numerical dispersion of the speed of light varies.
In the case of a
relativistic beam, the speed o f the particles may cross the light line, resulting in the
artificial Cerenkov radiation effect.
Cerenkov radiation occurs when a fast particle
moves through a medium at a constant velocity v, which is greater than the velocity of
light in that medium [39]. Due to the cell size in simulations, when the numerical speed
o f light crosses the speed o f the particles, they may interact like two circuit elements with
nearly identical resonant frequencies. When this numerical coherent effect occurs, the
interaction may increase at particular frequencies, especially after a long period of
running time. In MAGIC, this effect usually happens when the beam voltage is greater
than 150 kV, resulting in the numerical instability such as shown in the pictures above.
However, at the beam voltage we use here (14 kV), this kind o f effect should not take
place and the reason for it is unknown at this time.
In order to prevent the artificial Cerenkov radiation effect, we can systematically
detune the numerical grid (grid jiggling) to prevent any long range growth of instability.
This causes the speed o f light to vary slightly as the particles pass from cell to cell,
thereby preventing a coherent effect. It provides little or no time for the interactions
between the numerical speed of light and the particles to gain strength at particular
frequencies.
Although the reason for the numerical noise at this low beam voltage has not been
determined yet, we found that grid jiggling can remove noise with characteristics similar
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145
to those from the artificial Cerenkov radiation effect. The picture below on the left side
(hi) is the initial grid pattern that results in the noise with higher driving frequencies.
When we change the uniform grid to a non-uniform grid on the right side (h2), the
numerical noise is eliminated. This suggests that the noise does not come from a physical
mode.
It should be noted that the non-uniform grid should not physically change the
resonant frequencies o f the cavity or other similar characteristics. The cell size variation
between adjacent cells should not exceed 20 %.
0.1 m m Uniform Grid
0.12 m m
(hi)
0 .1 m m
0 .1 1 m m
0 .1 m m
0.12 m m
(h2 )
Fig. 3.13.10: The grid pattern in the cavity using a uniform grid (fl) and a non-uniform
grid (£2)
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146
3.13.3 Algorithm
In addition to grid jiggling, there is another approach to remove this numerical noise.
All previous simulations were done by using the default electromagnetic algorithm
“MAWELL CENTERED” [34] which is suitable for use without particles or with nonrelativistic particles (beam voltage below 150 kV).
The “MAXWELL H IG H Q ”
algorithm artificially damps electromagnetic fields and it is especially suitable for cases
involving relativistic particles and cavities [34].
It was found that when the
electromagnetic algorithm “MAXWELL HIGH Q” is assigned instead of the default
setting, the noise was eliminated without the need for grid jigging.
However, the
MAXWELL HIGH Q algorithm results in much longer simulation time compared to the
MAWELL CENTERED algorithm.
3.13.4 Discussion of the Physical Mode
Whenever there is an undesired oscillation in the circuit, we first need to perform the
Fourier analysis to determine the frequency components and in which cavity it appears
first, then we can proceed to determine if the noise results from a physical or
computational mode.
Although the noise eventually proved to be numerical, the
following physical modes were investigated initially.
a. Slot Mode
The circuit is designed to be operated in the cavity mode.
If the slots are
sufficiently large, the RF may stay in the slots (slot mode) instead of the cavities
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147
(cavity mode). The electrons then would not be modulated when passing the gaps
in the cavities. The resonant frequency of a rectangular cavity is given by [38]
fm n l
2
V
a
b
According to the dimensions of the slots in the intermediate cavities (2.8mm x
0.3mm x 0.75mm), the dominant-resonant mode (lowest resonant frequency) is
TEioi which is 207 GHz. If the dominant resonant mode does not appear, there will
not be higher modes present. This is because as the frequency increases at the
higher modes, the wavelength is shorter and the Q is lower; consequently, the
dominant mode usually has the strongest signal. Since there is no 207 GHz signal
shown in the FFT plots, it is very unlikely that the noise results from the slots. The
CONTOUR [38] plot from EIK modeling also shows that most o f the RF energy is
stored in the cavities, not in the slots, when there are no particles present (Fig.
3.13.11).
S lots
Fig. 3.13.11: The CONTOUR plot of the EIK modeling (x-z plane) when there are
no particles (brighter color, higher magnitude o f Ez).
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148
b. 7t Mode
Our EIK is designed to be operated at the 2n mode (cavity mode). The operating
frequency range is approximately 34.85 GHz to 35.15 GHz.
The n mode
frequencies o f the intermediate cavities are within the 33.47 GHz to 33.68 GHz
range. The FFT plots do not show noise signals with these frequency components.
c. Stability
The measurements o f the beam loaded Qb in MAGIC 3D “hot test” simulations
demonstrate that the Qb values of each eigenmode are all positive in the
intermediate cavities and consequently there should not be any oscillation arising
with a beam voltage o f 14 kV. The 2nd and the 3rd cavities are coated with loss
material, reducing the possibility of oscillations arising from these two cavities. We
also coated the 4th and the 5th cavities with loss material for testing, but the results
showed there were stronger noise signals present. This also indicates that the noise
is not due to the cavity beam loaded instability.
Assuming the noise arises from a certain cavity, in addition to the beam loaded Qb
measurement; there is another approach to figure out which cavity is the source.
We can model the 1st cavity only; the 1st plus the 2nd; the 1st, the 2nd, plus the 3rd; the
1st, the 2nd, the 3rd plus the 4th; and so on, with RF driven continuously and the beam
imported concurrently, to purify the starting point of these noises. When this was
performed on the EIK, we found there was no noise present up to the 4th cavity;
however, the noise appeared when the 5th cavity was added. In order to determine
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149
tli
tli
if the 5 cavity is where the noise originates, we changed the 5 cavity geometry to
be the same as the 4th cavity, but the noise did not disappear. This indicates that
neither the 4th cavity nor the 5th cavity is the reason that physically results the noise.
d. Intermodulation Products
When any amplifier is driven by a single tone input signal, the output spectrum
consists in general o f a superposition of the input signal frequency and its
harmonics.
When the amplifier is driven by a multiple-tone input signal, the
nonlinearities that generate harmonics of the single-tone act to create beats among
the multiple tones; signals at these beat frequencies are usually known as
intermodulation products [40] [42]. However, since our EIK is driven by a pure
single frequency in all simulations that we have conducted so far, there should not
be any intermodulation products present.
To summary, the FFT figures do not indicate any noise at frequencies located within
any physical mode regions.
We eventually proved that the noise is computational.
Nevertheless, the discussion above is useful for determining the origin o f any physical
mode noise and should be referenced whenever we encounter noise in simulations.
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150
3.13.5 Summary
To summarize:
1. There is high noise that appears when the EIK was driven at higher frequencies.
The frequency components of the noise depend on the driving frequency.
2 . Significant investigations into physical noise modes were conducted and it was
determined that they are unlikely to be the origins of the noise oscillations.
3. The grid jiggling method and the Maxwell H i g h Q algorithm were used and both
approaches eliminated the noise successfully, demonstrating that the noise results
from a numerical source and not a physical mode.
4. Usually, this kind o f numerical noise occurs in relativistic beams (beam voltage
above 150 kV). The reason why it appears in this low beam voltage (14 kV) is
not yet determined.
5. The grid jiggling method is preferable for removing the numerical noise because
Maxwell High Q [34] requires more simulation time.
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151
3.14 Cavity Tuning Using Fake Tuner
As illustrated in the first chapter, the resonant frequency o f each racetrack cavity is
tuned by adjusting the position of the flat surface of the cavity wall. In the simulation, as
with actual hardware, each cavity is initially tuned by changing the cavity length (Fig.
3.14.1). However, since the resonant frequency of each cavity can differ by only 30 MHz,
and since a 10 micron movement in the cavity wall results in approximately a 60 MHz
shift in the resonant frequency, precise tuning of the cavity length requires simulations to
be performed using a very small mesh size. As previously mentioned, using a very small
mesh requires a very large amount o f computer memory and extensive computation time.
Therefore, the mesh size was increased from 0.05 mm to 0.1 mm for the complete hot test
simulations, making accurate tuning of cavities extremely difficult, especially with this
seven-cavity-EHC (with a total of 13 signal gap cavities, and with each cavity length
located at different positions along the Y axis). Therefore, instead o f tuning the cavity
along the Y axis, a “fake tuner” was created along the Z axis o f each cavity (Fig. 3.14.1).
Each tuner can be adjusted in the X and Z axes (and Y, if necessary). The timer can be as
small as one “cell” [Appendix IV]. The main advantage of this tuning technique is that
the cavities differ primarily in the Z axis instead of the Y axis, and this allows this
complete EIK to be modeled using a larger mesh size while retaining the accuracy of
each cavity resonant frequency.
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152
C a v ity
\ * ~ L e n g th
m
Y
U
I Tuner
(a)
t
I
(b)
Fig. 3.14.1: 3D Cavity Modeling in MAGIC (a) 3D view (b) Side view
3.15 Simulation Mesh Size and Time Step
The PIC code performed in the design and analysis of this EIK is MAGIC3D/Double
precision, version 2005.0201.1602. A mesh size of 0.05mm was used initially. However,
the simulation was aborted due to insufficient computer memory, operating on a single
PC with 3GB o f RAM.
There are various potential reasons can cause simulation
abortions, such as boundary setup errors or improper use o f commands etc.
The
indication of the abortion due to insufficient memory can be found at the end of LOG file,
as shown below:
***
E rro r
in
***
F a ilu re
***
E rro r
ro u tin e
u sin g
is
The
i n s u f f ic ie n t
***
ABORT
OKAY.
***
a llo c a tin g
dynam ic
J1 ,J 2 ,J3 ,Q 0
a re a
can n o t
be
re s e rv e d
b ecau se
of
a re a .
CODE
E x ecu tio n
ALLOCAT
ERROR
AT:
te rm in a te d
ALLOCAT:
by
code
e x i t ! ;
a b o rt.
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CON DITIO N:
153
This memory usage for the simulation can be viewed on the “Memory Usage” in the
“Process” of the Window Task Manager during the initialization stage o f the simulation.
To complete the simulation, the mesh size was increased to 0.1mm, which also
significantly reduced the computation time. The minimum memory required for this EIK
modeling using a 0.1mm mesh size is about 800MB (480MB for single precision
execution).
The total number of cells simulated is 2,180,364 (comparing with
approximately 13,167,000 using a 0.05mm mesh size). The number o f particles imported
using “IMPORT LAMINARBEAM ” is 36 (comparing to previous 100).
The total
number o f particles created during the simulation period o f 26ns is 5,890,320.
The default electromagnetic time step was originally 1.54><10' 13 seconds. To speed
the simulation, the time step was increased manually to 1.59xl0 ' 13 seconds, which still
meets the Courant condition. (The default value in MAGIC equals to 80% o f the Courant
criterion) [34].
Duration time for this modeling is between 26ns to 32ns, depending on
the driving frequency and stabilization. With a single PC, Pentium(R) 4 CPU 3.00 GHz
(3GB RAM), the run time is approximately 41 to 55 hours.
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154
Chapter 4
Summaries of Design and
Computer Modeling of EIK
The purpose o f this work is to develop 3D design and modeling capability for a noncylindrically-syrnmetric microwave vacuum electronic circuit.
Precise design and
modeling o f microwave tubes is becoming increasingly important as applications move
to higher power levels, frequencies, and bandwidths. The conventional approaches of
making vacuum electronic tubes, such as frequency scaling o f existing devices and the
use of a number o f experimental iterations are unacceptable, both from an economic and
time viewpoint. In addition, the added dimensional constraints and material demands
associated with the move to high power millimeter wave applications make it imperative
to realize optimal designs with maximum tolerance to dimensional and material property
variations, effects that demand extensive, fully realistic computer simulations and
parameter space searches.
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155
In this section o f the dissertation, the above approach is applied to the design of an
extended interaction klystron (EIK), whose frequency is chosen to be Ka-Band, both for
specificity as well as the fact that there are increasing numbers of applications aimed at
this frequency region. This work demonstrates the design and modeling of a unique
geometry which allows more flexible timing of each cavity in the EIK to provide higher
power and broader bandwidth, as well as accounting for the inevitable errors in
fabricated dimensions.
To increase the output power and the bandwidth, seven cavities were used.
To
increase the tuning range, a “racetrack” coupled cavity was created, the analysis of which
requires 3D modeling. The flexibility of tuning each cavity frequency is necessary due
to fabrication imperfections or design and modeling inaccuracies. The design illustrated
in Chapter 1 provides a sufficiently flat and deformable surface so that the cavity
frequency can be easily tuned to optimize output power and bandwidth. This two-gap
racetrack cavity design also increases the R/Q value from 105 (with a single racetrack
cavity) to 175, contributing to higher output power and increased bandwidth. A ID
small signal calculation worksheet written in MathCAD and a large signal computer code
AJDISK as well as a 3D PIC code MAGIC 3D were used for the cavity and the circuit
design; these codes are detailed described in Chapter 2.
The design parameters predicted through ID small signal and large signal simulations
yield 1 kW output power and 220 MHz bandwidth. All cavities are two-gap coupled
cavities except for the penultimate cavity, which is a single gap cavity in order for it to be
located closer to the output cavity. The input/output cavity external Q is lowered by a
coupling iris connected to a reduced-height waveguide. The second and third cavities are
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156
coated by loss material to increase the bandwidth.
The final design, with 3D PIC
modeling, yields an output power of 1.1 kW and bandwidth o f 240 MHz.
The ID
calculation results using equivalent circuits and the 3D simulation of a complete circuit,
including the iris and waveguide, are in good agreement in both small and large signal
regions. The results are presented in Chapter 3.
Because o f the unique non-cylindrical-symmetry geometry of the racetrack coupled
cavity, 3D modeling is required for accurate results.
A 3D PIC code was chosen -
MAGIC 3D - which requires considerably more simulation time and computational
memory compared to ID simulation. The complete 3D EIK hot test simulations were
planned to be performed on a Linux cluster at SLAC using the parallel version of
MAGIC 3D. However, since the Linux parallel version of MAGIC 3D had problems
completing the EIK simulation due to the geometry complexity, the mesh size was
increased in order to complete the simulations on single PC. In addition to developing
modeling capability using 3D PIC code, this work also contributed to the advance of this
particular code, as there were few internal bugs were found and fixed throughout the
process. One o f an internal bugs associated with the laminar beam import is extensively
investigated and presented in Chapter 4. Chapter 4 also demonstrates some very useful
modeling techniques and few important issues associated with this EIK analysis,
including the techniques to save a significant amount of the simulation time and circuit
stability analysis etc.
Due to the extensive computation time, ID simulation codes were used for the initial
design, with the assistance o f the 3D code on constructing the ID equivalent circuits.
However, since ID simulation codes do not allow for a complicated circuit geometry
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input and have many assumptions, such as perfect beam transport, no focusing field input,
no undesired noise or oscillations, a fixed beam diameter, and symmetric simplicity, 3D
modeling is required to validate the final design.
This work has demonstrated a
successful design using ID equivalent circuits (using MATHCAD and AJDISK) and
developed the 3D PIC modeling capability (using MAGIC 3D) on a non-cylindricallysymmetric microwave tube (a Ka band coupled cavity EIK). This will allow us to pursue
more ambitious goals with more complex structures, and significantly reduce time and
expense of the microwave vacuum electronic tube development process [71].
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158
Chapter 5
TunneLadder TWT Modeling
Using MAGIC 3D
As discussed in the Introduction of this dissertation, vacuum electronic devices can
generate higher power at higher frequencies compared to solid state device due to
intrinsic physical differences. However, the limitations associated with conventional
manufacturing processes result in higher-ffequency vacuum electronics devices being
very difficult to fabricate and quite expensive. This chapter presents a device with a
unique geometry which allows small dimensions at high frequencies to be achieved with
the use o f modem fabrication technologies.
5.1 Introduction and Motivation for 3D modeling of KaBand TunneLadder TWT
For wider bandwidth applications, TWTs [4] [10] are widely used. The two most
common types are the helix TWT and the coupled cavity TWT [4] [54]. The practical
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159
upper operating frequency of the helix TWT is limited to approximately 50 GHz due to
difficulties in manufacturing, heat dissipation, and fragility associated with the small size
of the structure (Fig. 5.1.1). For higher frequency operation, the coupled cavity TWT has
therefore predominated. However, there are a number of problems in their construction
that result in extremely low yield and attendant high cost. Basically, each of the cavities
must be individually matched (ferrule), then stacked and finally brazed to each other (Fig.
5.1.2).
Tolerances must be kept extremely tight throughout the whole process.
To
reduce the manufacturing cost, a device with fewer pieces and simpler assembly
procedures is required.
Magnetic Field
RF Input
Attenuator
Helix
Delay Une
RF Output
Collector
Beam
Fig. 5.1.1: Helix TWT Structure [64] © 1973 IEEE
(a)
(b)
Fig. 5.1.2: Coupled-Cavity TWT Structure (a) End view (b) Top view [63] © 1995 IEEE
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160
The TunneLadder TWT (Fig. 5.1.3) offers the potential for cost reduction because
the ladder and the outer shell, or enclosure, are fabricated separately, permitting a variety
of techniques (i.e. micro-fabrication) to be employed, techniques that are not
conventionally used in tube manufacture. The ladder portion of the circuit is fabricated
by milling slots in a solid piece of copper. Not only is machining the ladder portion a
relatively simple operation, but the tolerances can be checked prior to further assembly of
the circuit. The remaining parts of the circuit are two cover plates, into which channels
are machined to provide coupling. Following brazing of the two so-called mating combs
together, the circuit is completed [54].
This obviously represents considerably less
assembly time than that required for the standard coupled cavity TWT.
Fig. 5.1.3: Assembly procedure of TunneLadder TWT [54]
Used by permission o f NASA from Reference [54]. Permission by NASA does not constitute an official
endorsement, either expressed or implied, by the National Aeronautics and Space Administration.
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161
5.2 Overview of TunneLadder TWT
The TunneLadder TWT was derived from the Karp circuit (Fig. 5.2.1), which is a
ridged waveguide whose broad wall contains an array of transverse slots. In the base
band o f this circuit, the electrons alternately see a strong field and a weak field as they
pass, respectively, over a slot and over the metal between the slots, making spatial
harmonic operation possible [55]. The Karp circuit inventor, Arthur Karp, provided the
following explanation of the operation principle. Consider one slowly rotating disc as an
analog o f the electron beam, and a second, faster disc as the analog of the RF wave. At
any one instant, the angular positions of the discs will correspond to the phase positions
o f an electron, and o f the wave, respectively. To synchronize the RF and electrons,
reducing the velocity of the second disc (representing RF) to match that of the first
(representing the electrons), is analogous to what is done in a conventional travelingwave tube. However, the two original discs can be made to appear synchronous by
observing the second disc under stroboscopic light. By adjusting the rate at which the
brief flashes o f light illuminate the second disc, it can appear to be in synchronism with
the first regardless o f its original speed or direction. This situation is analogous to that in
a space-harmonic traveling-wave tube, where an electron beam “thinks” it “sees” an RF
wave in synchronism with itself - even if the wave is actually traveling faster or in the
opposite direction - because it views the wave intermittently.
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162
Fig. 5.2.1: Basic form of Karp circuit structure [56] © 1957 IEEE
The ladder is then formed and doubled to allow the passage of a pencil beam (Fig. 5.2.2).
This modification provides a thermal path through the dielectric posts to the ridges for
the beam-interception heating localized where the rungs (ladders) are closet to the beam.
Also, the field is large near the diamond dielectric chips (Fig. 5.2.2), which contributes to
the high axial electric field in the center of the beam hole, and thus to the high impedance
o f the circuit. The bandwidth became greater yet when a ridge was introduced within the
waveguide.
LADDER
R ID G K
DKLXCTRIC
SU PPO R T R O D
Fig. 5.2.2: Flat single ladder to “TunneLadder” design [54]
Used by permission o f NASA from Reference [54]. Permission by NASA does not constitute an official
endorsement, either expressed or implied, by the National Aeronautics and Space Administration.
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163
Early experiments were performed using Karp circuit structures for millimeter wave
TWTs [55] and backward wave oscillators [56].
Recent theoretical analysis of the
TunneLadder TWT, a modified Karp circuit, was reported by Kosmahl and Palmer with
references to earlier theoretical work [58], followed with several fabricated devices and
measurements reported by Varian Associates, Inc. [54] [57] [59] [60].
Advances in micro-fabrication technology for microstructures are allowing new
opportunities for vacuum electron devices that produce RF power at higher frequencies
[3].
Because o f the special structure of TunneLadder TWT, modem fabrication
manufacturing can be used, showing promise for efficient, lower-cost and compact
microwave and millimeter wave power amplifiers.
5.3 Goal and Purpose
The original goal o f this work was to develop a vacuum power booster for Ka-Band
MPM applications [1] [2] [11] using the TunneLadder TWT circuit.
Due to the
discontinuation o f collaborations between SLAC and the collaborating company, the
design work was discontinued.
However, as a result of this work, 3D modeling
capability for a TunneLadder TWT for millimeter wave applications has been developed.
This work started from the modeling of a Varian 29 GHz TunneLadder TWT (contracted
with NASA), including cold test and hot test simulations and associated calculations.
The cold test simulation results are compared to the measurements and previous
simulation results reported by others. The hot test simulations results are also presented
and compared to measurements. There are no previous hot test simulation results of this
circuit available for comparison.
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164
5.4 Circuit Modeling Using MAGIC 3D
To develop computer modeling capabilities using a 3D PIC program, the 29 GHz
TunneLadder TWT (Fig. 5.4.1), fabricated and tested by Varian Associates, Inc., under
contract with NASA Lewis Research Center (NAS3-22466), was selected for the initial
simulation work.
Laddergap—i
Baam tunnel—•
-OMactric chip
v km -, r ° * 'Mo,Mp
y
Ladder rung -
VJ
■
w .
2.66
cm
___
WwjssS///,
m
(a)
(b)
Fig. 5.4.1: 29 GHz TunneLadder TWT Circuit Structure (a) Top View (b) End View [61]
Used by permission o f NASA from Reference [61]. Permission by NASA does not constitute an official
endorsement, either expressed or implied, by the National Aeronautics and Space Administration.
The cold test simulation results, including the dispersion diagram, impedance
calculations, attenuation calculations, as well as the coupler and waveguide transition
modeling are presented using MAGIC 3D.
The results are compared to previous
simulation results using MAFIA and to measurements [61]. The hot-test simulations
with electrons and the focusing fields will also be presented using MAGIC 3D. The
results will be discussed and compared to measurements.
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165
5.5 Cold Test Simulations
This section presents the frequency versus phase dispersion, interaction impedance,
and circuit attenuation simulation results of the 29 GHz TunneLadder TWT, as well as
the VSWR modeling of the RF coupler and the waveguide transition.
5.5.1 Dispersion Simulations
The dispersion diagram describes the relationship between the resonant frequencies
and the phase shifts o f each cavity (or per period) in the circuit corresponding to different
standing wave modes. The phase shift per period is the period length times the axial
phase constant of the nth RF space harmonic. The phase constant is defined as:
_ _ </>+ 2nn
P n~
L
where cp is the phase shift (in radians) per period o f the fundamental space harmonic, n is
the RF space harmonic order, and L represents the length of one period. Here n equals
zero because the TunneLadder TWT is operated at the fundamental forward wave mode.
The simulation results using MAGIC 3D using a 24 period circuit model are shown
below (Fig. 5.5.1), along with the experimental data and previous simulation results
using MAFIA as reported by Kory [61].
The maximum discrepancy between the
measurements and the MAGIC simulation is approximately 0.7 %.
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Phase Shift Per Period (Radians)
Fig. 5.5.1: The frequency-phase dispersion characteristics of the TunneLadder TWT
(a) End View
(b) 3D View
Fig. 5.5.2: TunneLadder TWT modeling using MAGIC 3D (a) End View (b) 3D View
5.5.2 Impedance Simulations and Calculations
High circuit impedance is necessary to ensure high gain per unit length to yield
sufficient total gain in the allowable length, and for high electronic efficiency to provide
sufficient RF output power at the available beam voltage and current [53]. Therefore, the
impedance analysis is very important.
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167
The beam on-axis interaction impedance for the nth RF space harmonic is defined as
[61]:
, where
v
E„\ The nth Fourier component of the total on-axis axial electric field
Pgp: Time-averaged RF power flow
N: Number o f total periods simulated
vg: RF traveling wave group velocity
W: Time-averaged stored electromagnetic energy per unit length
WT: The total energy in N periods
In order to calculate the beam on-axis interaction impedance, the axial field profile, Ez(Z),
acquired from MAGIC 3D simulation, is directly substituted for the Fourier-transform of
the space harmonics. For this TunneLadder TWT operation, we only need to consider
the situation when n equals zero. The electric field on axis can be presented as:
where /?n is the axial phase constant of the nth space harmonic defined by the phase shift
per period divided by the length per period and En is calculated from Fourier analysis:
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168
The time-averaged power flow is the total electromagnetic energy multiplied by the RF
wave group velocity. The group velocity can be calculated from the dispersion diagram
(Fig. 5.5.1). For the total energy stored in the circuit, power loss, and the electric field
calculations, it should be noted that when using the frequency domain solver (i.e.
EIGENMODE command in MAGIC 3D) to perform the dispersion modeling, a standing
wave is simulated within the device to calculate the resonances, hence the total
electromagnetic energy and other associated parameters such as power loss and the
electric fields are twice the value of the traveling wave (as a standing wave is formed by
a forward-traveling-wave and a backward-traveling-wave).
The impedance result using MAGIC 3D modeling is shown below, and can be
compared to previous MAFIA simulation results (Fig. 5.5.3):
140
140
? 120
G 120
f
100
a , d 8 /c a v < ty
160
100
20
40
60
80
100
120
140
160
P h a se s h ttt p a r c a v ity , B t. d a g
Phase Shift P er Period
(a)
(b)
Fig. 5.5.3: Impedance simulation results (a) using MAGIC 3D (b) using MAFIA
Figure (b): Used by permission o f NASA from Reference [61J. Permission by NASA does not constitute an
official endorsement, either expressed or implied, by the National Aeronautics and Space Administration.
It is suspected that the group velocity discrepancy between MAFIA and MAGIC shown
on the dispersion diagram (Fig. 5.5.1) is the primary reason for the impedance calculation
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169
discrepancy. Nevertheless, the calculations using MAGIC and MAFIA show reasonable
agreement. The impedance versus the corresponding resonance is shown below:
Impedance,
f (GHz)
Fig. 5.5.4: Impedance versus the corresponding resonance using MAGIC 3D
From the results shown in Fig. 5.5.4, the highest impedance is approximately at 29 GHz,
where the highest gain can result. More periods can be modeled to accurately calculate
the group velocity and more precisely locate the frequency corresponding to the
maximum impedance.
5.5.3 Attenuation Simulations and Calculations
Although the circuit is constructed from copper (conductivity 5.99xlO 7 S/m), an
effective conductivity o fl.9 x l0 7 S/m is used to account for the actual losses in the
circuit.
These circuit losses are consistently greater then the theoretically predicted
values because of surface irregularities [61].
The dielectric constant, 5.5, and the
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170
dielectric support loss tangent, 0.0003 [61], were input into the attenuation simulations
using MAGIC 3D. The attenuation per period, in decibels, is defined as:
a =
4.343 P,Loss
prf -n
where PLoss is the total power loss of the traveling wave, N is the number of total
simulated periods, and PRF is the time-averaged RF power flow. The calculation results
using MAGIC 3D are shown below and can be compared to the previous simulation
results using MAFIA (Fig. 5.5.5).
1.6
160
1.4
140
1.0
C 130
§ ‘-0
100
'S' 0.8
1
0.6
C
0.4
02
0.0
20
40
60
80
100
120
140
160
Phaaa ahKt par cavity. 0L, dag
Phase Shift P er Period
(a)
(b)
Fig. 5.5.5: Attenuation calculation results (a) using MAGIC 3D (b) using MAFIA
Figure (b): Used by permission o f NASA from Reference [61]. Permission by NASA does not constitute an
official endorsement, either expressed or implied, by the National Aeronautics and Space Administration.
The MAGIC 3D attenuation results are plotted versus the corresponding resonances
below:
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171
Attemu.ition
f (GHz)
Fig. 5.5.6 Attenuation versus the corresponding resonance using MAGIC 3D modeling
The attenuation results from MAGIC 3D appear much higher than the MAFIA results
at higher phase shifts per period (i.e. higher resonant frequencies). It is suspected to be
due to higher group velocity calculation discrepancy at higher resonant frequencies.
Nevertheless, they closely agree near the center frequency of 29 GHz. A smaller mesh
size and more periods can be used in the modeling to increase the calculation accuracy.
5.5.4 Waveguide Transition and Coupler Modeling
One method for making a good broadband transition between two waveguides is to
slowly taper the geometry of one so that it gradually turns into the other, but such
transitions have the disadvantages of requiring long length, resulting in an increase of the
device size, weight and power dissipation. It is obviously preferable to have a short
length o f slow-wave structure along the electron beam, but this requires an abrupt
geometry change in the coupler and thus an abrupt change in impedance and field
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172
configuration, resulting in reflections. This sudden change can be cancelled by a shunt
susceptance of suitable magnitude placed appropriately in the feed waveguide [62].
The waveguide transition and coupler modeling was based on the structure described
in “A millimeter-Wave TunneLadder TWT” [54].
The geometry of this structure is
shown below (Fig. 5.5.7 & 8 ).
- Shorting plunger
*
y , f
J/ »
# M.// //
Capacitive post—<•
inductive iris
y / f
JJ ? 7 -7 r
1
MM
%
■M x
Fig. 5.5.7: TunneLadder TWT circuit to waveguide coupler structure [54]
Used by permission o f NASA from Reference [54]. Permission by NASA does not constitute an official
endorsement, either expressed or implied, by the National Aeronautics and Space Administration.
Input port
Inductive in s
Slow -w ave
circuit se c tio n
C apacitive
p o st
"
Output port-
Fig. 5.5.8: Geometry o f TunneLadder TWT circuit to waveguide transition [62]
Used by permission o f NASA from Reference [62]. Permission by NASA does not constitute an official
endorsement, either expressed or implied, by the National Aeronautics and Space Administration.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
173
Both input and output couplers have a reduced height waveguide short which is tapered
from standard waveguide height. An inductive iris and capacitive post are placed at the
point o f transition to the circuit, having the beam axis as the axis about which they are
symmetrical.
Because o f the Y-axis symmetry, and to reduce the simulation time and required
computer memory, the coupler and circuit were cut in half to perform the RF simulations
(Fig. 5.5.9). In order to reduce the simulation time, the smallest mesh size of the ladder
thickness was used.
RF
In p u t
RF
O u tp u t
►
Fig. 5.5.9: Circuit to waveguide modeling using MAGIC
The VWSR simulation results versus frequency are shown below.
Note that the
measurements and the MAFIA simulation used a 10:1 scaled model of the 29 GHz
coupler, while the MAGIC 3D simulations were done on a 29 GHz model.
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174
5
4
MAFIA
Measured
■♦— MAGIC 3D
3
2-
2
□ □ □
1
o
28.3
28.8
29.3
29.8
2.93
2.83
2.98
Frequency (GHz)
(a)
(b)
Fig. 5.5.10: VSWR results (a) using MAGIC 3D (b) using MAFIA, and measurements
Figure (b): Used by permission o f NASA from Reference [62]. Permission by NASA does not constitute an
official endorsement, either expressed or implied, by the National Aeronautics and Space Administration
Varying the thickness (t) and aperture size (a) of the inductive iris, and the outer
diameter (D) and gap size (g) of the capacitive post (Fig. 5.5.7) resulted in different
impedance matches and hence different VSWR. The calculation results using MAGIC
3D are shown below and can be compared to the previous simulation results using
MAFIA [62] (Fig. 5.5.11). There is no measurement data available for comparison.
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175
s
5 T
- ■ — Post lap, £
-A—
-A —
O ■■—
Iris£ap, a
- A — Iris width, t
Iris width, t
Iris gap, g
Post diamatar, 0
Post gap, g
Postdiaraeer, D
-0.06
-0 .0 6
-0 .0 4
-0 .0 2
0
0.02
0.04
-0.04
0.06
-0.02
0.00
0X12
Change In dimension. In.
0.04
0.06
C h a n g e in d im ension, in.
(a)
(b)
Fig. 5.5.11: VSWR results as a function of dimensional variations on original structure
(a) using MAGIC 3D (b) using MAFIA (for the 10X scale model)
Figure (b): Used by permission o f NASA from Reference [62], Permission by NASA does not constitute an
official endorsement, either expressed or implied, by the National Aeronautics and Space Administration
The VSWR results from MAGIC 3D appear to be slightly higher then the
measurements (Fig. 5.5.10). This m aybe due to the geometry discrepancy resulting from
the use of a bigger mesh, assuming the fabricated geometry is as designed. In addition,
for complex geometries, improper use of the MARK command [34] can lead to
dimensional errors in the MAGIC simulations. This is discussed further at the end of this
chapter, under Additional Information.
5.5.5 Optimized Coupler
A coupler with an inner poster radius increased from 0.3048 mm to 0.5388 mm was
discovered to result in very little reflection, with a VSWR of 1.02 (Sn of 0.01). This
coupler was originally selected for continuing with the hot test simulations; however, in
order to allow comparison to the experiment data, the original coupler was used in the
experimental hot test simulations, presented in a later section.
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176
5.6 Hot Test Simulations
5.6.1 Non-Symmetry of Hot Test Simulations
To avoid the enormous amount of time and computer memory required for hot test
simulations o f the entire device, it was hoped that the device could be accurately
simulated with cutting the device in half because of the Y axis symmetry, like previous
cold test simulations. Unfortunately, since the electron motion is not symmetric to any
axis, the entire device needs to be modeled with the electrons present. The figures below
show that MAGIC 3D encountered difficulties in simulating the electron beam when
only half o f the symmetrical device is modeled.
W /////////////////>
»'
.
/////////////////A
(a)
(b)
Fig. 5.6.1: (a) Top view of the simulation when only half circuit is modeled
(b) End view (top picture) and 3D view without the outer shells (bottom picture)
when entire circuit is modeled
With the same dimension, the same beam parameters, and appropriate beam focusing
fields, the results indicate that if only half the device is modeled (Fig. 5.6.1 (a)), the beam
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ill
trajectories reflect a very different motion from when the entire device is modeled (Fig.
5.6.1 (b)). This proves that the RF circuit symmetry property does not work in hot test
simulations.
5.6.2 Synchronous Hot Test Simulations
The synchronous case is when the electron beam velocity is the same as the velocity
of the signal on the circuit in the absence of the beam. Theoretically, the gain of a
traveling wave tube, for the synchronous case and without losses and space charge, can
generally be given by [10] [53]:
Gain = -9.54 + 47.3 C N (dB)
where C is the Pierce’s gain parameter and N is the distance along the RF structure in
wavelengths and the quantity C is defined by:
C3= K/4Ro
where K is the circuit impedance and Ro is the beam impedance.
The RF wavelength in the TunneLadder TWT is measured to be approximately 1.905
mm at 29 GHz with MAGIC 3D, leading to an RF phase velocity of 0.55 * 108 m/s (Fig
5.6.2),.
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178
2
i
3
> {»
(8-3)
Fig. 5.6.2: Axial electric field verse z of the TunneLadder TWT
To synchronize the electrons with the RF phase velocity, a gamma of 1.0175 and a
beam voltage of 8.944 kV were calculated. Since a microperveance of 0.2 was used in
the experiments [59], and given the beam voltage, the beam current is calculated to be
0.169 amps. Further, assuming a beam radius of 0.2 mm, a Brillouin flow of 1767 Gauss
was calculated. Focusing fields of 0.2 Tesla and the above beam parameters were used
for the synchronous simulation and the results are compared to theory (Fig. 5.6.3).
14 T
12
“
i
o
Lossless Synchronous
SlmUatlon Results
-
Theory
10
8
•
4
2
0.2
0 .5
0 .3
CN
Fig. 5.6.3: Lossless synchronous simulation results compared to theory
(without space charge consideration)
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179
Fig. 5.6.4: Hot test simulation including the coupler, interaction circuit
(12 periods in this picture) and electrons
In the results above, the circuit impedance of 140.2 ohms at 29 GHz was used to
calculate the Pierce gain parameter, which is 0.87 for a beam voltage of 8.944 kV and
current o f 169 mA. The simulation results were obtained with a circuit length o f 18, 24,
and 30 periods, corresponding to 3, 4, and 5 RF wave lengths respectively. The result
show excellent agreement with theory.
All the simulations were in the small signal
region. The optimized coupler from the previous section was used in these simulations
(Fig. 5.6.4).
5.6.3 Experimental Hot Test Simulations
In order to compare to the measurements [54], the experimental parameters of 10 kV
beam voltage, 215 mA beam current, 0.47 mm beam diameter, 0.32 Tesla focusing fields
and the original coupler were used for simulation inputs. A circuit with a total of 70
periods was used in the experiments, consisted of an output section of 42 periods and an
input section of 28 periods. The circuit was manufactured using two sections, most
likely to help prevent oscillations.
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Since the saturation output power would be the same in a shorter circuit with a lower
gain, a circuit o f 18 periods is initially modeled for the saturation output power
measurement to reduce the simulation time. However, because the 18 period circuit only
resulted in an approximate 4.5 dB gain, it was difficult to recognize where saturation
occurs by looking at the plot of the output power versus input power, although 500 watts
o f saturated output power could be estimated by looking at the electron momentum plot.
A longer circuit o f 30 periods was then simulated and the saturation output power was
measured to be 537 watts with a 12 dB gain at the center frequency of 29 GHz (Fig.
5.6.5). The electron momentum plots show the electrons are slowed down and give
energy to the RF signal (Fig. 5.6.6)
537 watts
60
30
15
20
25
30
35
40
45
50
Input Power (dBm)
Fig. 5.6.5: Output power versus input power plot using 30 period circuit model
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181
CNl
99999924
>Q
■)
M
0
2
6
4
Z
8
10
(E -3)
<m)
(a2 )
971873^694
a
&
UJ
u
0
2
4
6
8
10
6
8
10
Z (a)
(bl)
(b2 )
u
<D
U
U
O.
1
0
(cl)
2
4
(c2 )
Fig. 5.6.6: Top view of electron bunching with a 30 period circuit model (left figure)
and electron energy versus z (right figure)
(al) (a2 ) small signal; (bl) (b2 ) saturation; (cl) (c2 ) overdrive.
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182
In the above small signal plots (al & a2), the electrons are velocity modulated but do not
bunch into groups. In the saturation region (bl & b2), the faster electrons catch up the
slower electrons and they bunch into groups. More slower electrons then faster electrons
(compared to the initial velocity) indicated that the electrons are giving up energy to the
RF signal. In the overdrive region, the faster electrons pass the slower electrons, and
therefore they are over bunched and the gain decreases.
The bandwidth simulation used an 18 period circuit, was performed for both small
signal and large signal operation. A 2 dB bandwidth was calculated to be approximately
4.1% o f the center frequency o f 29 GHz (Fig. 5.6.7).
25
55
23
53
E
E
g
00
21
51
19
5o
a
49
■5
o
17
15
47
45
28
28.5
29
29.5
30
28
28.5
29
F requency (GHz)
F requency (GHz)
(a)
(b)
29.5
30
Fig. 5.6.7: Bandwidth simulations using an 18 period circuit model
(a) small signal region (b) rated output power o f 200 W
A circuit with 70 periods without separation was simulated but unfortunately it
resulted in noisy oscillations. The origin of the noise needs to be further investigated.
The experiment divided the entire circuit into two sections and did not experience
excessive noise. However, since the gain linearly increases with the circuit length, and
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183
from the simulation results with 18, 24, and 30 period circuits, the gain is estimated to be
58 dB per inch in small signal region and 56 dB per inch at the rated output power of 200
watts (Fig. 5.6.8).
Small Signal
Rated Output Power o f200 W
40
Estimated
C
O
TS
c
75
©
20
10
2
3
4
5
6
7
8
9
10
11
12
C ircuit L e n g th (w a v e le n g th )
Fig. 5.6.8: Gain versus circuit length in the number o f wavelengths
In “A millimeter-Wave TunneLadder TWT” [54], the saturated output power was
reported to be measured at 400 watts at 28.2 GHz, compared to the prediction of 480
watts at 29 GHz. This report indicates that the center frequency shifted from 29 GHz to
28.2 GHz and that this could be easily compensated for with minor adjustments to the
circuit geometry (note that the previous cold test measurements indicating a center
frequency of 29 GHz shown in previous sections were performed on a 10:1 scaled model
at that time). The comparison between the predicted performance, measurements on the
device, and MAGIC 3D simulation results are listed below (Table 5.6.1):
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184
Table 5.6.1: Circuit performance comparison between the predicted,
actual and MAGIC results.
Predicted
Actual
MAGIC 3D
Operating Voltage
Mlcroperveance
10 kV
10 kV
10 kV
0.2
0.215
0.215
Center Frequency
29 GHz
28.2 GHz
29 GHz
Gain
Saturation
Electronic Efficiency
Beam Diameter
58 dB/in
> 52 dB/in
~ 56 dB/in
480 watts
400 watts
537 watts
24 %
>21%
25%
0.47 mm
N/A
0.47 mm
N/A
96%
99.8%
1+%*
2.27%
4.1 %
200 watts
200 watts
200 watts
Beam Transmission
2 dB bandwidth
Rated Operating Power
*The predicted bandwidth is the ldB bandwidth.
The MAGIC simulation results indicated better agreements with the predicted
performance then the actual measurements. Potential reasons include fabrication error on
the device and better beam transmission in the simulations.
5.7 Discussion and Summaries
After several simulations and calculations as listed above, the following items are
noted:
a. The beam tunnel shape affects the dispersion result significantly.
Since the
actual TWT may not be fabricated exactly as modeled, minor geometry
adjustments may result in better agreement between the simulations and
measurements.
b. From the simulation plots, the maximum Ez appears between the beam tunnel
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185
ladders. Thus, if we minimize the beam tunnel radius, it may contribute to
higher interaction impedance because of a higher Ez on axis. However, this
modification may result in problems, such as heat dissipation difficulties or RF
breakdown. The optimization process must consider the thermal analysis.
c. A coupler with a very low VSWR was discovered, with a slightly higher inner
post radius then the original structure. However, in order to obtain results that
can be compared to the measurements, the original coupler and circuit design
will be used in the experimental hot test simulations. This RF model, with a
bigger mesh size, indicated that the cold-test simulation results are in a
reasonable range.
d. The hot test result showed very good agreement with theory for synchronous
operation. The simulation results showed some discrepancies compared to the
measurement, but better agreement with predicted performance. This could be
due to the center frequency of the fabricated circuit shifting and a worse beam
transmission in the experiments. Nevertheless, the gain per inch and the output
power results are very comparable.
This work has developed the capability of 3D modeling of TunneLadder TWTs and will
contribute to the future design work for this type of micro fabricated microwave power
amplifier [69].
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186
5.8 Additional Information
The simulation program demonstrated in the cold test simulations and the
synchronous
hot
test
simulations
is
MAGIC
3D,
single
precision,
version
2003.0220.1416; and version 2005.0201.1602 for the experimental hot test simulations.
Due to the manual mesh in MAGIC simulations, often a modeled object appears to
have different dimensions than what was input. All gridding must be properly done
manually in order to simulate a device with accurate dimensions, while considering the
efficiency o f the computation time and computer memory. The MARK command [34]
must be used as necessary, with an appropriate mesh size, especially when modeling a
complex geometry such as TunneLadder TWT. If not, the dimensions of each object can
easily shift. This is especially important when a larger mesh is used. The only way to
check if the device is accurately simulated is to check each dimension in the model using
VIEWER [34]. The latest discovery of improper usage was a missing MARK on the
dielectric support width in the X direction, and that caused the dielectric support width to
slightly increase by 1.6% and the center frequency to slightly increase by 0.4% when a
mesh size o f one ladder thickness was used. This MARK was not added to previous cold
test simulation results presented above, but it has been added for the experimental hot
test simulations.
However, the synchronous hot test simulations (Fig. 5.6.3) used a
model without the support width MARK and with the optimized coupler. When the
MARK and original coupler are input, the gain is slightly higher then the results
presented in Figure 5.6.3 at 29 GHz.
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187
Chapter 6
Reflex Klystron Modeling
Using MAGIC 2D
6.1 Introduction of Reflex Klystron Modeling
As mentioned in the Introduction of this dissertation, the reflex klystron is a
promising candidate for use as a local oscillator in THz heterodyne receiver applications
[7] [19] [20] [65] [66 ] [67]. This has been made possible through advances in micro­
fabrication techniques. For example, in the case of the “nanoklystron” concept conceived
by JPL researchers, the electron beam is produced by a carbon nanotube based field
emitter cold cathode and both the emitter grid and cavity are micro-machined. Given the
potential importance o f this source for the advancement of the THz field, it is thus crucial
to have good design and modeling capability. To provide confidence and a foundation,
the THz device design and modeling was started by benchmarking a documented 100
GHz CPI Canada reflex klystron, model VRB-2113 A (the exact dimensions and some
specific parameters can not be shown here since they were provided under a Non-
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188
Disclosure Agreement signed by UC Davis and CPI Canada). Since the cavity geometry
is cylindrically symmetric, MAGIC 2D modeling was initially employed to save
simulation time. The output coupling has been replaced by an artificial loss material in
the 2D modeling. Although a few parameters were not available and have been assigned
by the author, the modeling results with MAGIC 2D are found to agree reasonably well
with measurement data.
6.2 Operating Principles of the Reflex Klystron
As described in the Introduction of this dissertation, in a reflex klystron, electrons
emitted from a cathode pass through a cavity gap into a region where the negative voltage
on a repeller electrode causes the electrons to reverse direction (Fig. 6.2.1).
If the
bunched electrons return to the modulating gap at the correct time, they induce an RF
current that reinforces the current in the cavity, resulting in a feedback loop that causes
the device to oscillate. The principal features can be illustrated in Fig. 6.2.2 below [68 ]
[ 10].
The phase at which the electron bunch returns to cause oscillations is (n+3/4) cycles
after the beam initially passed through the bunching cavity. By changing the repeller
voltage, the phasing of the electrons returned to the cavity can be changed (Fig. 6.2.3).
Also, the oscillation frequency in any mode of operation is a function of the time during
which the electrons return in that mode [68].
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189
Infraction Cop, VmowC,
y
H
C’-41
Electrode Structure
Potential Distribution.
Fig. 6.2.1: Electrode structure and potential distribution in reflex klystrons [6 8 ]
© 1958 J. J. Hamilton
Petition of BefUctei
Mid-plane of
Ocoonotor G ridt
1 frW
U
J %1!«
Fig. 6.2.2: Applegate diagram for reflex klystrons [6 8 ]
© 1958 J. J. Hamilton
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190
Frequency
%
4-
o
CL
■s
t
o
Mode U
5-
6-
Reflector Voltage Increasing Negatively
Fig. 6.2.3: Output power versus reflector voltage for reflex klystrons [10]
6.3 Circuit Modeling Using MAGIC 2D
One o f the most important aspects of reflex klystron modeling is the repeller. The
approach shown below with PORT command [34] can also be used in 3D modeling when
a non-cylindrically symmetric geometry is present.
Tim e 1 4 . 9 9 2 n s :
X 2 - I n t e g r a l o f E rho
(V/m) a t 0SYS$AREA
0.000
PORT
- 5.660
- 14.150
- 16.960
- 19.810
■
- 22.640
25.470
- 28.300
Z
M
Fig. 6.3.1: Repeller modeling with the equi-potential line shown in color
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191
MAGIC 2D input:
F u n c tio n
P o te n tia l
F u n c tio n
G rh o (Z ,
PORT
R)
R e p e lle r_ P O R T
In c o m in g
F IE L D
=
=
1 /R ;
E2
IN T E G R A T E
0 . 5 * R e p e lle r_ V o lta g e * S m o o th _ R a m p (T /T ris e );
N E G A T IV E
P o te n tia l
N o rm a liz a tio n
CONTOUR
(t)
F u n c tio n
V o lta g e
OSYS$AREA
E2
G rho
R e p e lle r_ P O R T ;
TSYS$LAST
Y N E G A T IV E
A X IS
Z
SHADE
R e p e lle r_ V o lta g e
0
DV;
Note that in the first function, because the circuit is terminated with an open circuit (i.e.
infinite impedance, as there is no output PORT with a perfect match), the signal is
completely reflected and it doubles the voltage at the input on the injection end; thus, the
driving voltage is reduced by half.
The second function above assigns the voltage
difference between two conductors. The last CONTOUR command shows the potential
difference between the cavity and the repeller (Fig. 6.3.1).
The modeling only includes the beginning of the beam tunnel through the repeller and
ignores the part between the cathode and the beginning o f the beam tunnel, as the
reflector voltage is confined to the area shown in Fig. 6.3.1. Therefore, this modeling
approach provides reasonable results while also resulting in a significant reduction in
simulation time. The beam is assumed to only have a z-component of velocity when it
enters the beam tunnel. Several simulation plots are shown below. The electrons are first
velocity modulated when passing the gap and then bunch into groups in the returned path
(Fig. 6.3.3).
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192
Time 14.982 ns:
-1 .0
PHASESPACE f o r a l l p a r t i c l e s
-0 .5
0 .0
Z
0 .5
1.0
(M)
(E-3)
Fig. 6.3.2: Electrons reflected by repeller
Time 1 4 .9 8 2 n s :
PHASESPACE f o r a l l p a r t i c l e s
Time 1 4 .9 8 2 n s :
PHASESPACE f o r a l l p a r t i c l e s
CD
E n e r g y /p a r tic le
(eV)
03
- 1.0
- 0 .8
-0.6
- 0 .4
Z
- 0 .2
(M)
(a)
0.0
0.2
0 .4
(E -3)
- 1 .0
- 0 .B
■0 . 6
-0 .4
Z
- 0 .2
0.0
(M)
0.2
0 .4
(E -3)
(b)
Fig. 6.3.3: Electron bunching picture (a) energy versus z (b) momentum versus z
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193
Mesh Size
The smallest mesh size initially selected was 0.03 mm in order to reduce the
simulation time, as 0.1 mm mesh size gave reasonable results for the 35 GHz EIK
modeling. However, this mesh size was not able to model the radial electric field inside
or around the beam properly, with only three cells across the smallest beam tunnel radius.
The radial electric field inside the beam should be proportional to R and 1/R outside the
beam. Without adequate resolution, these fields can not be modeled accurately and the
results will have a high noise level as shown below.
Tim e 1 9 .9 8 0 n s :
PHASESPACE f o r a l l p a r t i c l e s
Tim e 1 9 .9 8 0 n s :
»
a p ll
PHASESPACE f o r a l l p a r t i c l e s
C-J
CM
o
- 1.0
:-3)
(a)
•0 . 8
- 0 .6
-0 .4
- 0 .2
Z (M)
0. 0
0. 2
0 .4
(E -3)
(b)
Fig. 6.3.4: Noise on (a) electron trajectories (b) in electron momentum
In order to have proper resolution across the beam tunnel and the difference between the
radius o f the left gap and the right gap, the smallest mesh size was changed to 0.012 mm.
At the voltage where the strongest oscillation occurs for each mode (the center of each
mode), it takes about 15 nanoseconds in MAGIC 2D to give stable results (corresponding
to about 8 hours simulation time on a 3 GHz CPU PC). However, at the voltages where
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194
the oscillations are weaker (further away from the center of each mode), it can take 30
nanoseconds or even longer (up to 26 hours!).
External O
In this 2D modeling, an artificial loss material was added to model the external Q.
Since there was no external Q recorded from the company documents, an optimized
external Q has been found with a series of MAGIC simulations shown below (Fig 6.3.5).
Lower external Q means a larger coupling iris and more output power is extracted from
the circuit; however, this lowers the electric field at the gap and reduces the oscillations,
and vice visa. The optimized external Q was found to be approximately 304, and this
number was used for the remainder of the modeling.
The cavity material is assigned to
be copper and gives an unloaded Q of 1208.
,____________
50
0
200
400
600
BOO
1000
1200
1400
1600
ExternalQ
Fig. 6.3.5: Output power versus external Q with a repeller voltage of negative 500 V
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195
R /0 and Coupling Coefficient M
The R/Q and the coupling coefficient M are modeled using the approach that was
described in the EIK chapter (Chapter 3). Because there is no beam focusing, the radius
o f the beam that passes and subsequently returns to the gap varies depending upon the
repeller voltage value. We calculate the radius of the average radius of the gaps on both
sides multiplied by a coefficient of 0.707 as the beams come very close to the gap
(sometimes the beam touches the gaps). The coupling coefficient M is determined to be
approximately 0.6 and the R/Q value is about 87.
6.4 Electronic Tuning Simulation Results
While both the cavity dimension timing and the repeller position tuning data are not
available from company documents, we proceeded with electronic tuning simulations by
changing the repeller voltage with the repeller position fixed.
The cavity resonant
frequency was tuned to be 100.38 GHz (by the author). While specific parameters can
not be described due to the previously mentioned NDA, the modeling results below are
very comparable with the measurements available (Fig. 6.4.1 & 2).
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196
0
100
2 0 0 3 0 0 4 0 0 6 0 0 6 0 0
700
8 0 0 9 0 0
R epeller V oltage (Volts)
Fig. 6.4.1: Output power versus repeller voltage
0
100
200
300
400
600
600
700
800
900
Repeller Voltage (Volts)
Fig. 6.4.2: Oscillation frequency versus repeller voltage
The operating modes were determined by the transition time between the electrons
passing and returning to the modulation gap, calculated from the electron momentum
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197
plots. The first three modes are 2.75, 3.75 and 4.75 cycles, corresponding to repeller
voltages o f approximately 825, 500, and 330 volts, respectively. The last two modes
were not easily determined from the electron momentum plots, nor from the first order
theory provided by J. Hamilton [6 8 ].
If the momentum plots were taken at the desired
timing of the exact period from MAGIC simulations, this might improve the electron
velocity calculation.
The available power output and electronic tuning have a contrary variation with
respect to the number o f drift cycles in the repeller space.
As the repeller voltage
decreases (mode number increases), the electrons take longer to return to the gap after the
first transit.
Therefore, optimum bunching is obtained at a lower RF gap voltage,
resulting in less output power but wider bandwidth. The cavity resonant frequency can
also be varied by changing the dimensions for different frequency operations.
Note that the repeller voltage that people commonly refer to is the difference between
the negative beam voltage and the repeller voltage in the simulation setup. Consequently,
if the beam voltage is 2500 volts and the repeller voltage is documented as 500 volts, it is
negative 3000 volts in the repeller modeling.
6.5 THz Reflex Klystron Modeling
Because the W-band reflex klystron modeling showed very convincing results when
compared to the measurements, it was scaled to 600 GHz for initial THz design and
modeling. The dimensions, mesh size, and time steps were all reduced by a factor of six.
The beam voltage and the beam current were reduced to 500 V and 4 mA (final values
still under investigation). There were no oscillations observed initially with a variety of
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198
repeller voltages.
After extending the simulation time, a very small oscillation was
observed. It is suspected that the repeller needs to be redesigned in order to be placed
closer to the cavity and reflect the beam properly at the same time to have stronger
oscillations. Several different repeller positions and geometries have been attempted;
however to date, none of them have worked very well.
Another issue is the long
simulation time. Each test run takes about 24 hours on a 3 GHz CPU PC just to see if
there is an oscillation starting in the circuit.
As noted earlier, due to the small dimensions, the THz reflex klystron is planned to be
micro-fabricated. Therefore, a rectangular shape is expected and it requires 3D modeling
after the initial simulations. Compared to 2D modeling, MAGIC 3D takes significantly
more simulation time and computational resources. Use of a reasonably equivalent 2D
model for the initial design iterations will make the design process much more efficient.
6.6 Additional Modeling Information
To model the repeller properly in the reflex klystron, several methods were attempted
in MAGIC, such as POISSON (which can only be used in 2D), DRIVER, CIRCUIT, and
PORT commands [34]. POISSON specifies an electrostatic solution and is usually used
when the standard PORT is not suitable. With POISSON, all relative voltages must be
specified on all conductors. Multiple POISSON solutions are necessary when there are
multiple conductors with different relative voltage offsets.
preferable dynamic approach compared with POISSON.
PORT or DRIVER is a
With PORT, a signal is
introduced through a boundary. With DRIVER, a signal can be modeled to introduce a
relative potential between floating conductors. With CIRCUIT, a floating conductor such
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199
as the cathode can be assigned a particular voltage, but a RESISTOR or INDUCTOR
must be used to allow a leakage current path. For the geometry demonstrated here, with
PORT it was easier to properly model the repeller.
Another issue is that the reflex klystron simulations with MAGIC 2D double
precision have always failed on the SLAC server (WTS01) when they run overnight. The
RAM and free space on the server have been determined to be sufficient for the runs. It
is suspected that the automatic back up in SLAC is related to the failure; however, the
real reason has not yet been conclusively determined.
This has made the modeling
difficult as only one PC was available to provide complete simulation results.
The results shown herein were obtained using MAGIC 2D double precision, version
2005.0201.1602. Note that the double precision execution is necessary for the modeling
o f this W band reflex klystron. The single precision execution option was initially used,
but it did not give reasonable results in many cases.
6.7 Summaries and Future Work
A W-band reflex klystron was successfully benchmarked using MAGIC 2D.
Although some detail parameters were not available, the modeling results showed
reasonable agreement with available measurement data.
This work has resulted in
development o f the modeling capability necessary for THz reflex klystron design.
Although 3D modeling must be performed on a micro-fabricated shape (such as
rectangular geometry), use o f a 2D equivalent circuit of a THz reflex klystron for the
initial design iterations will significantly reduce the simulation time and speed the overall
design process.
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200
Chapter 7
UCD 94 GHz Gyro-TWT
Input Coupler and MIG Optimization
This chapter briefly summaries the input coupler and magnetron injection gun
optimization work for the UCD 94 GHz Gyro-TWT experiments [17] [72]-[83]
7.1 Input Coupler Optimization
7.1.1 Introduction and Overview
The RF input coupler is used to efficiently couple the RF input power into the device.
For the UCD 94 GHz Gyro-TWT, the input coupler also serves as a mode converter,
converting the fundamental rectangular TEi0 mode to the circular TEoi operating mode
(Fig. 7.1.1).
The design of the coupler has been performed using High-Frequency
Structure Simulator (Ansoft HFSS 7.0).
TF,,
Fig. 7.1.1: UCD 94 GHz Gyro-TWT input coupler
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
201
The structure consists o f a TEio rectangular port, a TE5i coaxial cavity, five coupling
slots, and the circular waveguide piece that connects to the TEoi gyro-TWT circuit (Fig.
7.1.2). The RF input power coupled through the WR-10 fundamental TEio rectangular
port is converted to the TE 51 mode in the coaxial cavity. Then, through five coupling
slots spaced equally 72 degree azimuthally, the T E 51 mode couples into the circuit T E 01
mode.
(a)
(b)
Fig. 7.1.2: (a) HFSS cross-sectional view o f electromagnetic field intensity in the coupler,
(b) Electric Field pattern o f TEjo (rectangular waveguide), T E 5 1 and TE01 (circular
waveguide)
Previous C oupler M easurem ent
Previous C oupler S im ulation Results
0
10
1?
a"
20
30
Frequency (GHz)
35
90
92
94
96
98
100
Frequency (GHz)
Fig. 7.1.3: Previous coupler performance
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
202
In the previous coupler simulation result, the bandwidth was limited due to the short
cutoff frequency near 96.23 GHz. The 3 dB bandwidth was approximate 4 GHz.
7.1.2 Optimization of the Coupler for Broad Bandwidth Operation
In order to optimize the coupler, several parameters were optimized (Fig. 7.1.4).
First, the circular short radius was changed from 1.90 mm to 1.80 mm; thus resulting in
the cutoff frequency of the circular short radius changing from 96.23 GHz to 101.58 GHz.
This increased the frequency where the sharp decrease in coupling occurs, thereby
improving the bandwidth.
Second, the resonant frequency of the coaxial cavity was increased by approximate 1
GHz. The coaxial cavity length was reduced from 2.87 mm to 2.70 mm, and the coaxial
cavity inner radius increased from 2.51 mm to 2.61 mm; resulting the resonance
increasing from 94 GHz to 94.93 GHz. It therefore improved the coupling at higher
frequency. In addition, the dimensions of the rectangular input guide were changed to
2.70 mm x 1.35 mm.
Coaxial .Cjavj!
Cavity Length
Coupling
Slot “
D e lta
Fig. 7.1.4 Coupler optimization
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203
Then, different cases were examined, with different parameters optimized in each of
the three cases (Fig. 7.1.4 & 5). In the case of optimization I, the dimensions of the
coupling slots were optimized to be 2.70 mm x 0.40 mm, and the distance between the
coaxial cavity and circular short (Delta shown in Fig. 7.1.4) was decreased from 0.365
mm to 0.265 mm. In the case of optimization II, because the results of a model with
coupling slot dimensions of 2.70 mm x 0.5 mm with zero Delta indicated that the
coupling is lower near 91.50 GHz and better at higher frequencies; the model dimensions
were enlarged by a factor of 1.0 1 , and connected to the original circular waveguide.
Optimization III is the same as optimization II except a taper instead of a step is used
between the coaxial cavity and the circular waveguide.
The model was split in half and the center-plane assigned to be electric field
symmetric to save computation time. In addition, calculations were performed assuming
both an infinite and finite length of short. If the circular short is infinitely long, there
would be no EM wave with a frequency lower than the short cutoff frequency can be
transmitted through the short. However, in reality, there is some attenuation loss from
the short due to the finite length of the short, even if the frequency is under the cutoff.
Both cases showed very similar results.
Since the model was split in half to save
simulation time, a complete model needs to be calculated to ensure that unwanted modes
that did not appear in the half model will not appear in a complete model.
Two identical couplers have been fabricated to enhance measurement accuracy by
permitting a “back-to-back” connection of input couplers (Fig. 7.1.6). The material was
assumed to be perfect conductor in the simulations and this resulted in less insertion loss
than measured.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
204
Fabricated
1
o p tim iza tio n 3
c
/
Coupling (dB)
-3'
r
>tiinization 1
Cross Section O f The Fabricated
Broadband RF Input Coupler
-
r V
origii al
zatil
R e ctan g u la r
Input w aveguide
r
-9-
■II.M )
V
x
-
(Him )
oavial C avity
V
12-
hi(cnu(i<m ( ii ui(
( I f'ull )
-15-
1
90
1
92
1
94
” '1
96
i
98
1
100
Dime
102 /(G H z )
Fig. 7.1.5: Comparison of simulation results of the original and optimized couplers
Input
coupler
Back-To-Back Measurement
1
I
S
-10
-15
-20
-25
-30
90
W -Build v ecto r n etw o rk an aly zer em ployed
in the cold test o f the T E 01 R F couplers
92
96
98
100
Frequency (OHj J
Comparison between sim ulations and measurements
Fig. 7.1.6: Comparison o f simulation results and measurements of the optimized coupler
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
205
7.1.3 Conclusions
To optimize the coupler, the short radius was reduced to increase the cutoff frequency,
the resonant cavity geometry was changed to increase the coupling at higher frequency,
and the coupling slot geometry was enlarged to increase the TEoi coupling and
suppressed other mode coupling. The optimization resulted in the 3 dB bandwidth o f the
RF input coupler improving from approximate 4 GHz to 8 GHz.
7.2 Magnetron Injection Gun Optimization
7.2.1 Introduction and Overview
The magnetron injection gun (MIG) has been utilized most extensively for gyro
applications (Fig. 7.2.1). In MIGs, electrons are drawn in a generally radial direction
from the cathode surface and then formed into a hollow beam by electric and magnetic
fields. Electrons make many cycles in this region of gentle tapering o f the external
magnetic field, transforming their axial energy into rotational energy adiabatically [4]
[10]. As electrons pass further through the interaction region, a portion of their kinetic
energy is converted into electromagnetic radiation energy.
This interaction occurs
mainly due to the orbital component of the electron velocity and the transverse
component o f the electric field of the RF wave. Efficient operation of the Gyro-TWT
requires a high quality electron beam with low-velocity spread and uniform current
density around the circumference of the beam. Since the output power, efficiency, gain
and operating bandwidth decreases as the velocity spread increases, a good MIG with a
proper magnetic field design are very crucial to successful operational results.
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206
Electron
sTt«|ec(ory
A ccelerating
Anode
[Cgthodc,
Axial Distance
Q: Peipendicular-To-Parallel Velocity Ratio
Axial Velocity Spread: AvJ vz
(v2 - The Average Axial Velocity, Avz - The Standard Deviation)
Fig. 7.2.1: Magnetron injection gun schematic diagram
The original design o f the MIG for the UCD 94 GHz Gyro-TWT is shown in Table
7.2.1.
The simulation result comparison between the original designer (using
FINELGUN) and author (using EGUN) are listed in Table 7.2.2 [84] [85].
calculation discrepancy is primarily due to the difference in magnetic field inputs.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The
207
Table 7.2.1: Original Design Parameters
Length
Cathode Radius(rc)
Emitting Strip Length(ls)
Cathode to Anode Distance(d)
Cathode Angle
Cathode Upper Angle
Cathode Lower Angle
Anode Voltage
Beam current
Magnetic Field(Bo)
Magnetic compression Ratio
Cathode position
40cm
5.10 mm
1.88 mm
2.381 cm
74°
0°
0°
101 kV
5A
35.60 kG
35.058
3.05cm
Table 7.2.2: Simulation Results
Computation Code
Cathode Loading
Beam Voltage
Gamma(Y)
Perpendicular velocity Spread
Axial velocity Spread
Velocity Ratio(a)
Guiding Center
Guiding center spread
Maximal Electrical Field
FINELGUN
8.8 A/cm 2
99.6 kV
1.1949
5.911%
6.064%
1.008
0.882 mm
10 .2 2 %
N/A
EGUN
8.8 A/cm 2
98.2 kV
1.1887
8.163%
10.82%
1.183
0.897 mm
10 .2 1 %
180kV/cm
7.2.2 Magnetic Field Simulation
In order to produce such high magnetic fields (i.e.>35 kG), a refrigerated
superconducting magnet is used in the experiment.
The magnet consists of four
independent solenoid coils: an interaction coil, a gun coil, an interaction taper coil and a
gun taper coil. Each o f these coils is made using thousands of turns of wires. In order to
model the MIG precisely, the magnetic fields need to be calculated accurately. The
magnetic induction on the axis due to each turn can be represented by (Fig. 7.2.2):
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208
2\n
c
Bzo (z)‘
2 n 3 /2
+ ( Z-Zln
2b
a l : inner radius of coil
a2 : outer radius of coil
J
2 b: total length of coil
B zO
Zc: center position of coil
Zc
Fig. 7.2.2: Solenoid cross section view (symmetric wrt to the axis)
The current density J in the solenoid is assumed to be uniform:
NI
2 b(a 2 - a,)
J=-
N is the total turns of the coil. Based on the theory above, a calculation written with
FORTRAN gave the results below:
Bz (Q au ss)
70000
Interaction coil: 72.51 (Amps)
Gun coil: 48.87 (Amps)
Interaction taper: 44.75 (Amps)
Gun Taper: 52.66 (Amps)
60000
50000
40000
My sim ulation
S u p e r F ish
M easured d a ta (by cyro)
30000
20000
10000
-40
0
40
80
120
160
200
Z (cm)
Fig. 7.2.3: Magnetic field simulation result and measurement comparison
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
209
The maximum discrepancy between the calculation and the measurements is 1.9 %. This
will be discussed further in a later section.
7.2.3 MIG Optimization Using EGUN
EGUN is a trajectory code that computes trajectories of charged particles in
electrostatic and magnetostatic fields, including the effects of space charge and selfmagnetic fields. This program is a 2.5 dimensional code, meaning 2D for all EM fields
and 3D for all particle motion. Magnetic fields are to be specified externally by the user.
Poisson’s equation is solved by finite difference equations using boundary conditions
defined by specifying the type and position of the boundary.
determined by differentiation of the potential distribution.
Electric fields are
The electron trajectory
equations are fully relativistic and account for all possible electric and magnetic field
components. Space charge forces are realized through appropriate deposition of charge
on one cycle followed by another solution of Poisson’s equation, which is in turn
followed by another cycle o f trajectory calculations [85].
Here, the magnetic fields are input with the calculations from the previous section.
The electron profiles are specified externally, including initial electron position, emitting
angle, energy and current o f each macro particle. This approach was used to predict the
performance o f the MIG used at National Ting Hua University, Taiwan (Appendix V).
Several optimizations o f the UCD MIG are summarized and the results are shown
below (Fig. 7.2.4 & 5).
The nose and the upper part of the cathode are modified;
reducing the maximum electric fields from 180 kV/cm to 115 kV/cm, reducing the
possibility o f arcing. This also modified the equal potential line around the emitter and
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210
reduced the beam edge spread. The magnetic field is timed to achieve the desired alpha
ratio o f 1 and to reduce the velocity spread to 2%.
A nearly flat magnetic field is
designed around the cathode to reduce the sensitivity of the cathode location and the
magnetic field difference across the emitter length.
The results showed significant
improvement compared to original design.
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211
M IG D esign
O rig in a l
C a th o d e p o sitio n (cm)
34.25
In te ra c tio n coil (A)
50.64
I n te ra c tio n ta p e r coil (A)
0
G u n t a p e r coil (A)
J~\
G u n coil (A)
O rig in al M IG C a th o d e A ssem bly
O ptim ized
-38.5
-1.4
- 1 .6
C o m p ressio n R a tio
32.99
34.85
A xial v elocity S p re a d (% )
5.96
2.17
0.93
0.96
169
115
V elocity R a tio
a
M a x im u m E -fleld (kV /cm )
E q u i p o t e n t i a l L in e M o d if ie d
T o R e d u c e V e lo c ity S p r e a d
R e d u c e RF
B re a k d o w n
P o s s ib ility
E lectrons
Cathode
O ptim ized M IG C a th o d e Assem bly
Fig. 7.2.4: Comparison between the original and optimized MIG
Assembled MIG
Cathode
Stalk
Cathode
Emission
Ring
Designed MIG beam parameters
Beam voltage
100 kV
Beam current
5A
Velocity ratio (v±/
1.0
Velocity spread
2%
Cathode radius
5.1 mm
Guiding center radius 0.9 mm
Activated MIG
C a th o d e
G e o m e tr y
M o d ified
Glowing Cathode Emission
; '
:
j
M a g n e tic
F ie ld
R e d e sig n e d
S m o o th B F ield a ro u n d c a th o d e to re d u c e
v elocity s p r e a d a n d c a th o d e lo c atio n sen sitiv ity
Fig. 7.2.5: Fabricated MIG and the performance predicted by the simulation
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
212
7.2.4 Discussion, Summary and Future Work
The optimized MIG along with the optimized input coupler have been fabricated and
are currently used in this state-of-the-art Gyro-TWT (Fig. 7.2.6) [72]-[83].
□
to5
B
▼
O
A
•
■
♦
A
▼
i
03
1
a.
500
O
k
•
1979N RL
1979 Varian
1980 NRL
1989 NTHU
1990 NTHU
1995 NTHU
1998 NTHU
1995 UCLAfUC Davis
1998
2002
2003
2002
tAP/U Strathclyde
NRL
MIT
IAP
2004 UC Davis
10'
2005 CPI
P/X2=0 5 kW/cm2 0 5
+
2005 MIT
2005 UC Davis
10’
Frequency (GHz)
■ 1 1 0 kW , 71 dB g ain , 5 % BW , 2 2 .5 % e ffic ie n c y p re d ic te d fo r c u rre n t d e v ic e
V 1 4 0 kW , 6 0 dB g ain , 4 .5 % B W (4 .2 G H z), ri > 2 2 % a c h ie v e d in c u rre n t e x p e r im e n t
Fig. 7.2.6: UCD 94 GHz Gyro-TWT experiment
However, several issues associated with this MIG have been identified.
Due to
additional coils inside the magnet that are not shown on the manufacturer’s drawings,
and to potential errors in the manufacturer’s test data, significant discrepancies exist
between the simulations and the measurements. In addition, a boundary input error with
EGUN has been identified within the previous optimization simulations. Furthermore,
the cathode edge was coated to prevent current leakage; resulting in a higher current
density which was not included in the original design and optimization simulations.
With the updated coil data from the manufacturer and the correct boundary setup, the
optimized design was determined (100 kV, 5A, spread 2% and alpha of 1) with an
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
interaction coil current o f 51.25 A, a gun coil current of 0.27 A and a gun taper coil
current o f negative 40.81 A. By including the emitter edge coating in the model, the
velocity spread increased to 3.5%. However, current measurements with the Gyro-TWT
indicate a much higher velocity spread then the simulations. There are several possible
reasons for this. One is emission from the edges of the emitter. In the fabricated cathode,
the gaps next to the emitter were larger than expected, and those gaps can become virtual
cathodes. Although some coating on the edge could help this problem, electrons from
those gaps can cause serious instabilities and EGUN does not account for those effects.
The cathode surface roughness can also contribute to a much higher velocity spread. A
new MIG with lower compression has been proposed (by Dr. Barnett). With the updated
coil data and lower compression design, and hopefully smaller fabricated gaps next to the
emitters, the next generation MIG is expected to contribute to higher output power and
wider bandwidth operation of the UCD 94 GHz Gyro-TWT [83].
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214
Appendix I
MathCAD Worksheet for EIK Small Signal Calculation
SLAC's Small-Signal Code for 35 GHz EIK
ORIGIN = 1
Define C onstants:
c s 2.997925-108
Speed o f L ig h t
m e n 9.109-10" 31
-1 9
e = 1.602-10
E lectron R e st M ass
eo s 8.8541878-10
E lectron C harge
-
12
P erm ittivity o f F ree Space
Initialize V ariables:
n
—
11
E lectron C harge to M ass R atio
r) - 1.759 x 10
B eam C u rre n t D ensity (A/m2)
Jo = 2.037 x 10
A/m 2
B eam C u rre n t D ensity (A/cm2)
Jbeam - 203.718
A/cm 2
M icroperveance
K - 0.241
Vd := .0304-Vo K-l 0.25 + ln| -
P otential B eam D epression
( @ r = 0.707 * b )
Vd = 96.928
V o d :-V o -V d
D epressed B eam P otential
Vod - 1.3903 x 10
(D epressed) B eam Adm ittance
Go = 2.877 x 10
R elativistic M ass Factor
a -1 .0 2 7
v/c fo r a R elativistic P a rtic le
w ith P otential D epression
v o n = 0.229
E lectron propagation co n stan t
fie = 3208.421
R adial propagation constant
y
lo :=
me
Io
n-b
Jo
Jb e a m :
100
K : - - ^ - -106
AVf
1 <
Vo
Go :=
Io
Vod
a := 1 +
Vod
511000
1 -JVa
/( 2 - 1J
,)
v o n :- —
|Se :=
2-n-fD-lO
von-c
i peyl - v o n
- 3123.440
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
( A -l )
215
Io “
Bbr .-8.304-
j 2 _ l ° - 23
Ld +
(b-Vod0 2 5 )
B rillo u in F lux (G auss)
a )J
Bbr = 1928.113
( A-2 )
PLASMA REDUCTION FACTORS
Calculation o f the P lasm a Reduction Factor, R , for the P en cil Beam Geometry, Using the M ethod o f B ra n ch and M ihran.
yb := y-b
yb “ 0.781
7a :-
7a
y-a
-1 .5 6 2
P lasm a Frequency
rap
As a F irs t Estim ate o f R, Use:
1
Jo
:=
50- von- c
|
a
(*:=
1.5
rap
R1 := j l - 2 I l ( 7 b ) - K l ( 7 b ) - 2 ^ ^ - I l ( 7 b ) ^
Lp
2 ----
Pp
R1 = 0.454
(Initial G uess)
" T|? "
Solve the T ranscendental Equation & r the P lasm a Frequency Reduction Factor, R:
Given
t J 1 k ' b)
JO (fb)
my
KD(P *'tJ ,11 (P»'b ) + K l(pe b) I0 (p e a )
K0(pe-b)-I0(pe-a) - KD(|Se-a)-IO(|Se-b)
( A
-3
)
T := FindW
T - 5967.237
1
R:=
/vw
1+
e r
Reduced Plasm a Frequency, Propogation Constant, and W avelength
raq:=rap-R
pq :=
raq
von-c
w := 1 .. npoints i:=l..N
1
j := 1.. N - 1
m
AAA>
pq
Indices for Calculating Group Delay
Indices for C alculating the V arious D rift L engths
npoints is the Num ber o f Frequency Points to Calculate W ith in the B W
s :■ 1 .. npoints
n :- 1.. N
Xq:=^
1.. N - 1
N is the N um ber o f K lystron Cavities
Zg is the Position o f the Gap C enters w here L is the Gap-to-Gap Spacing
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
216
W ~ Zg* - Zgn
D ista n ce fro m a Given Gap to a ll th e O th e rs
Pqii ( B
Gap-to-Gap T r a n s it A n g le i n D e g re e s
T
36Q'\
BW
BW
fo := fD + s -----------------------s
n p o in ts
2
A s s ig n F re q u e n c ie s S y m m etrica lly A bout th e C e n te r F re q u e n c y
CALCULATE BEAM COUPLING COEFFICIENTS
Total Coupling Coefficient:
Mi>
I0 (y b )2 - I l ( y b ) 2
“( H
Ma.-
if (Switch = 1)
(A41
10(7-a)
IO(yb)2 - II(y b)2
-
’
T*1®Extra Factor is the Radial Coupling Coefficient,
Averaged A cross the Beam
otherwise
io(ya)
CALCULATE BEAM-LOADING CONDUCTANCES
K1 :■ IO(y-b)2 -
II (y b)2
M W
IO(ya)
Calculate Beam-Loading Conductance Using the Form ula to r a Knile-Edge Gap with a Hyperbolic Field D istribution:
Gb.:
1
< K 1 ) + — JO
2
•JO
1
2
VV
■J1
I 2J I 2J
to
ped.
T.a., “4dl4 y cico _ h(t b)
I0(ya)
I0(ya)2 .
Io
(a + 1) J
Vod
(A -5)
Calculate Q:
Q b . ----------1 RQ.Gb.
Q t:=
(—
V,
Qe.
1
--------------
—
—
Qb.
^ 1
Qo.
^ 1
Qur = ( —
Qe.
^
1
—
Qo.
1
CALCULATE BEAM-LOADING SUSCEPTANCES ------------
Conqiute beam-loading susceptance using W arnecke & G uenard,p.522
(
2.405 ^
5.520
Assum es th a tb /a ■ 1
C heck the Sum fo r Convergence
2
8.654
Zeroes o f JO(x):
p .s
11.792
14.931
1
\2
4
,3
n- 1
...
n[(M®)2 + 7&2]
Mu
18.071
V21.212 )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A- 6 )
217
Appty the Relativistic Correction:
Bb Go := Bb Go— ;------ r
M M W /V W
—
tJ-fo
+
1)
Beam-Loading Susceptance:
Bb := Bb Go-Go
Calculate Beam-Loaded Frequency:
fb :=
n ^1 + RQn-Bb_Go-Go
Unloaded Frequency:
( 34.8580 ^
r 34.85643 ^
34.8950
34.89349
35.0510
35.04946
f = 35.1110
Loaded Frequency:
f = 35.10945
35.1410
35.13945
35.2000
35.19901
V35.0000 j
<34.99838 >
Define the Impedance of the nft Gap:
RQ.n
”'s
i
Qt
n
2- m
fo
s
fb
2
(A-7)
n
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218
DEFINITION OF THE KLYSTRON SMALL SIGNAL POW ER GAIN
MM
m ,n
:= —
-M
M sinffiql
}
a .(a + i)
m «
m ,r j
Definition o f Transconductance: Ratio o f
Circuit Transconductance:
1 . Io
>n =
2 n fo l0
2 ° ’Vod
(MM
} (
'
m' n'
<Bq
A-8 )
H“ KF C u rren t a t th e n * Gap to the RF
Voltage a t the m * Gap w hich Produced
the Modulation.
G. := 1
l,s
q := 2.. N
q-1
Gq , is the T ran sfer Function i^/ij at
QeN = 191.000
Frequency f t,
RL:=RQNQeN
R g en :- R Q jQ e j
Calculate the Gain:
P :=(G n )2 4 - ^ N,S^
s
\ N,sJ
Rgen-RL
(A-9)
Gain := 10 log(|P s |)
60
Find the Maximum Gain, PPmax:
Ptest := P
s
s
20.tog(|G2
PPmax > 101og(max(Ptest))
201og(|G3
201og(|G4
Find the Gain a t the C enter Frequency:
2 0 b g (|G 3
PPfD :« Gain
. .
npoints
2 0 b g (|G 6
2
20 b g ( |G j
Find the Index, s, at W hich the Maximum Gain Occurs:
20
N..
0
2 0 b g (|G 8
sPmax :=
while Gaitij < PPmax
■20
34.8
34.9
35
fot
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
35.1
35.2
219
Fimd Hie 3dB Bandwidth and Fractional Bandwidth:
fl>
fh
sPl <— sPmax
while p a i n gpj > (PPmax - 3)J
a
(sPl > 1)
while |Gaingpj4 > (PPmax - 3)J
fl>
(sP h < npoints)
a
(sP h < npoints)
sPl
fh :=
0 if |fl s fojJ
fl otherwise
Bndwth :=
a
sPh i - sPh + 1
sPl < - sPl - 1
f l< - fo
sPh <r- sPmax
0 if ( f h s f o
. , )
\
npomtsf
fh otherwise
100
fbw3 := (Bndwth) ■
0 if |fh - fl| > BW
(fh - fl) otherwise
F indthe ldB Bandwidth and Fractional Bandwidth:
fldb:
sPl
fhdb :=
sPmax
while p a i n gpj 2 (PPmax - 1)J
a
sPh « - sPh + 1
sPl < - sPl - 1
fldb <- fo
fld b :-
AWMP
sPh <— sPmax
while p a i n gp^ > (PPmax - 1)J
(sP l > 1)
fhdb
sPl
0 if |fldb a fOjj
fhdb :=
fo m
sPh
0 if (fhdb = fonpointg)
fhdb otherwise
fldb otherwise
Bndwthdb := |0 if | fhdb - fldb | > BW
fbwl := (Bndwthdb)
| (fhdb - fldb) otherwise
100
fO
Calculate the M a rk e r Locations for the ldB and 3dB Bandwidth Pow er Levels:
P3 > PPmax - 3
PI := PPmax - 1
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220
Ka-Band EIK
Beam Voltage (V)
N um ber o f D ata P oints
Beam C u rre n t (A)
Calculation Bandwidth (GHz) BW a .31
C enter Freqency (GHz)
Gain (Max)
npoints a 30
Gain a t C enter Frequency
T unnel Radius (m)
P lot M a rk ers
Beam Radius (m)
fi =■ 35.083
f b - 35.145
Rem em ber to update the shape factor
d2 i f the fill-factor changes.
ldB Bandwidth (GHz)
(le = 3208.421
Bndwthdb = 0.041
3dB Bandwidth (GHz)
(iq = 157.904
<W>B W (ldB )
fbwl = 0.118
% BW (3dB)
fbw3 = 0.177
y a = 1.562
Em pirical expression fo r Qe o f the output
cavity - not used in the gain calculation:
( A-10 )
Irf a 1.5
39.5
Vo
Io
Qe N "
Gai»s 32.5
25.5
Q eN - 434.786
Num ber o f K lystron Cavities:
Na 7
18.5
Gun Microperveance (uA/VA 1 5 )
B riU ouin Field (gauss)
1.241
B b r= 1928.113
B eam C u rre n t D ensity (A/cm2)
Jbeam = 203.718
Cathode C u rre n t D ensity (A/cm2)
J c atho de := 5
Gun Convergence
34.8 34.84 34.88 34.92 34.96
Set "Sw itch" equal to 1 to u se the JO knife edge
approximation for Ma. Set "S w itch" equal to 0 to
use the M a defined below.
Jbeam = 41
J cathode
Switch a 0
!2 = 152
Gap Length
(m)
Qej a 355
Q oj a 1837
fj a 34.858
Qe2 a »
Q o2 a 650
f2 a 34.895
Qe3 a OO
Q o3 a 655
f3 a 35.051
Qe4 aoo
Qo4 a l8 4 6
f4 a 35.111
35.04 35.08 35.12 35.16 35.2
fos
Cavity Rs/Q
( O hm s)
RQ)j a 158.4
35
Lj a 0
a 0.0065
dt a 0.00045
a
0.00045
Coupling Coef.
(@ r-a)
Coupling Coef.
(T o ta l)
M aj a 0.9318
a
''0.5883'
0.9312
0.5879
RQ.>3 = 154
>4 = 154.5
>3 = 155
Q e j-c o
RQ,■6 * 99
Qe6 aoo
RQ.)? a 162
Qe? a 191
Lj a 0.0065
a 0.00045
L4 a 0.0065
d4 a 0.00045
M&j a 0.9296
Ma4 a 0.9290
Q o3 a 1821
f5 a 35.141
L5 a 0.0065
d3 a 0.00045
Ma5 a 0.9286
Q o6 a 2242
f6 a 35.20
Lg a 0.0045
dfi a 0.0005
M a g a 0 .9 3 1 8
Q o, a 2089
f7 a 35.00
L j a 0.0045
d j a 0.00045
Ma7 a 0.9302
0.5869
M -
0.5865
0.5863
0.5883
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
,0.5873,
221
4.548 x 10“
4.548 x 10" 6
4.548 x 10” 6
G b.
Qb =
4.548 x 10" 6
4.548 x 1 0 ' 6
( 0 .0 0 0 ^
f 0.5883''
' 0.000 s
■'1.444''
1446.627
0.007
0.5879
58.807
1.444
650.000
1427.840
0.013
0.5869
58.807
1.444
655.000
1423.219
0.020
1418.628
0.026
0.5863
58.807
1.444
1821.000
2225.537
0.031
0.5883
40.713
1.604
2242.000
^ 1357.329,
,0.035;
1,0.5873 j
1,40.713;
1,1.444,
, 175.000 >
4.539 x 10" 6
' 245.000 '
4.548 x lCf 6
448.486
M =
0.5865
ql-
58.807
(Sed -
1.444
449.019
Bb » 5.701 x 10 7
Qt =
' 297.507 '
'1388.177''
803.636
797.413
1116.869
. 155.014 }
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
Q u-
1846.000
222
Appendix II
MathCAD Worksheet for Input Cavity Gap Voltage
CALCULATION OF GAP VOLTAGES
This section has been included so that comparisons can be made with other simulation packages such as AJ Disk,
MAGIC, etc. The only variable declarations necessary are the Power delivered and the input operating frequency.
Finally, for further reference the input gap voltage will be calculated using methods suggested by b oth R. Symons
andR. Vaughan (R . Vaughan's results will be used to calculate the remaining gap voltages).
i-V^i
M
“
CALCULATING THE INPUT GAP'S PEAK VOLTAGE
R. Vaughan
R. Symons
fin - f,
fin - fb
Detuning (with Beam Loading)
delta S :
Detuning (without Beam Loading)
delta V :
fb,
RQj-Qtj-V^Pd
Q uh :=
Vgapl_Symons :=
Jl + i-2 Q tj delta_Sj ^RQ^ Qe^
Or instead o f using absolute value signs, the equivalent is:
(RQj Qtj
= 79.77
1
1
Qoj
Qbj
1
GT:
Unloaded Hot Q
Total Conductance
Quh-RQ
BT := ^ delta_V ^ ^
RQj
Total Susceptance
YT := GT + i-BT
Total Admittance
(itam Loading is belutkd U rt)
[ l + [2 Q tj ■delta_S)2j (^RQ j Q e J 2
GE :=
r>
1
QejRQj
G E -Y T
Drive Line Conductance
Reflection Coefficient
G E + YT
VSWR
1+ r
l- |r|
Vgapl Vaughan : = ------/8PdV S W R Q uhR Q .
e ~
1 + VSWR V
1
Vgapl_Symons - 79.77
Volts
Vgapl_Vaughan - 79.77
Volts
V_1 := Vgapl_Vaughan
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223
CALCULATING THE REMAINING GAP VOLTAGES
Z Cav
n
Cavity Impedance at Frequency fin
1
C«*02 - [fbj2
— + --------5
i
_
Qt
n
M M Cav
-
fin fb
n
y ~ — r-M
m>n := —
o ( a + l)
m
M
n
sinfpa l
V
m
1
>l J
Definition o f Transconductance: Ratio o f
1 . Io 2-n-fin-lO
— j2 Vod
<oq
g_cav
|M M _C avm J
the RF Current at the a * Gap to the RF
Voltage at the mf t Gap which Produced
the Modulation.
G_CaVl 1
q-1
G_Cw 4 is the T ransfer Function i^/ij at
G-CaV = Z [(^ cavj,q) z-Cavj G-Cavil
j-1
Frequency fin
Vj := V_1
Define and Calculate Cavity Voltages
Z Cav
V := G Cav
q
■V 1
q Z_CaVj
RESULTS & INPUTS FOR GAP VOLTAGE CALCULATIONS
| Power Delivered to the Input Cavity (Watts)
( 79.77
Gap Voltages
\ 139.21
Pout :=
Operating Frequency (GHz)
Z_CaVj-VN
4-Pd
Qy HQj-QVHQj,
Pout - 0.31
Ouqiut Power (Watts)
G a in := 1 0 .1 o g ( ^
Gain = 3.19
G ain a t F req u en cy fin
Acknowledgement
The author would like to thank Aaron Jensen for the help with the MathCAD worksheet
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
224
Appendix III
MathCAD Worksheet for Stability Analysis
Note:
1. The electric fields are the Ez profile exported from MAGIC simulations.
2. The mode frequency is the resonance of each eigenmode. Be careful to update this
parameter every time when a new field is input.
3. The “stored energy” is the total electromagnetic energy in the cavity which equals to 1
joule when the eigenmode solver is used. (The total electric energy plus the magnetic
energy is two joules in MAGIC when eigenmode solver (standing wave) is used,
therefore the total energy of the traveling wave is 1 joule.) When the time-domain solver
is used (for input or output cavity), we needs to put in the total EM energy simulated
corresponding to the time when the fields are exported.
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225
EIK Stability Calculations Using MAGIC
Define Constants:
c s 2.997925-108
S p eed of Light
me s 9.109-10” 31
19
e = 1.602 10
Electron Rest Mass
s o s 8.8541878-10
-12
Electron Charge
Permittivity of Free S p ace
S w eep P aram eters:
Io
Ko :=
Vo
P erv ean ce
1.5
Ko = 2.4147 x 10
numjpoints := npoints + mod(npoints,2)
Make npoints an Even N um ber
o := 0.5-num_points
Index W here th e V oltage Array, V, is Vo
i := 0.. num_points
Plotting Resolution
__ TT
1.99-Vo
1.99-Vo
V. := Vo + l-----------------------------num_points
2
Assign V oltages Sym m etrically
About Vo
Assign Currents A ssociated With
Each V oltage to Produce Constant P erv ean ce
Initialize V ariables:
me
Jo :=
it-b
11
Electron C harge to Mass Ratio
r) = 1.7587 x 10
Beam Current Density (A/m2)
Jo - 2.0372 x 10
Beam Current Density (A/cm2)
JbeamQ = 203.7183
M icroperveance
K = 0.2415
o
Jo
Jbeam :=
1002
K := Ko-10
6
(@r.0J07.
tt f 1
V d ,*non,,
0 4 .V
.K .^ + ,to^ j j
Beam Depression
Potential Beai
b)
Vd = 96.928
Vod:= V - V d
D epressed Beam Potential
Vod = 1.3903 x 10
(Depressed) Beam A dm ittance
Go = 2.8771 x 10
in
Go :=
Io
Vod
o
o
-5
o
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226
a := 1 +
Vod
von :=
(ie:=
Relativistic Mass Factor
c 0 = 1.0272
v/c for a Relativistic Particle
with Potential Depression
von = 0.2286
Electron propagation Constant
pe0 = 3.199 x 10
Radial P ropagation Constant
^o = 3.1142 x 10^
511000
2-n-fO-lO
o
von-c
:= (pe ^
- von^)
PLASMA REDUCTION FACTORS
Calculation of the Plasm a Reduction Factor, R, for th e Pencil Beam G eom etiy, Using the
Method of Branch and Mihran.
ya -=y. a
yao = 1.5571
■yb := y b
yb0 = 0.7786
P lasm a F requency
rap :
n-
Jo
rap0 = 2.3338 x 10
10
so-von-c I ^1.5
C alculate th e A pproxim ate Reduction Factor, R:
■-—
>
R .=
Jl
- 2-11 (yb)-Kl fob) - 2 - ^ - I l ( Tb )2
R educed P lasm a Frequency and Propogation Constant
coq := (mp-R)
H'=
raq
von-c
FILE DATA
R ead in data (on axis) from the MAGIC file listed later of this w orksheet.
0
< >
z axis := file axis
z (location on the z-axis)
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227
Coupling Coefficient A veraged Over the Beam Radius
fl0(vo-b)2 - I l ( ? 0 -b) 2
m (p )
:= Ma(p)-
------------------------ Qb C alculations-- ------------------------Gb/Go C alculated Using MAGIC and the M+ M- A pproach (from W esselBerg).
M_Min := M(pe - pq)
M_Plus := M(pe + pq)
Gb Go := ^ - [ ( |M _ M i n l ) 2 -( |M _ P lu S|) 2]8pq
0
2
(0 + 1J
Qb :=
Gb Go
RQ
“
Vod
Input Data and Results
Results
txt" is the input file associated with the certain m ode for th e EIK a t r=a.
■*.txt" Is the input file associated with the certain m ode on ax is (r=0).
To create an equivalent files for other output structures :
(1) Run MAGIC with RANGE FIELD E3 using the chosen geom etiy.
(2) MAGIC will c reate a *.grd file.
(3) Open th e c re a te d *.grd file and copy the Ez profile to *.txt file (this d ata should be
associated with th e field at r^O for R/Q calculations and r*=a for all o ther calculations).
(4) P lace th e files in the sam e directory a s this one.
(5) Enter th e files' nam e inside of the READPRN functions above.
Rem inder: Update the input file, ID, U, etc. for every n ew m ode or cavity.
Beam Voltage (V)
Beam Current (A)
H
M
Units P e r cm
(> d uses m eler)
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228
Mode Frequency (GHz)
0 - even sym m etry
1 - no sym m etry
Tunnel Radius (m)
Beam Radius (m)
Plotting Resolution
Stored Energy
(From MAGIC M O G file)
The following are variab les evaluated a t the operating frequency, current, and voltage.
|Se0 := 3208
Propagation Constant
Gb Go = 0.067
Gb/Go from MAGIC Using M+ M-
-
o
Coupling Coefficient @ r^a from MAGIC
Ma(pe0) - 0.9156
M(|ie0) = 0.5795
Coupling Coefficient (Averaged) from MAGIC
RQ - 163.6512
R/Q from MAGIC (Use the Field on Axis)
Plots & Figures
1.5 10
110 10
Ez axis
0
,9
-5 10'
-0.006
-0.004
-0.002
0.002
0.004
0.006
z,z_axis
Ez Field Distributions vs. Distance
2000
0
-2000
5000
1 104
1.5 IO4
Vi
Qb vs. Voltage
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229
Appendix IV
PIC Code Introduction
IV.l Overview
The basic electromagnetic computational processing cycle of MAGIC is shown in
Fig. IV.l. The fields in Maxwell’s equations are represented on a finite-difference grid.
From an alternative engineering viewpoint, the grid creates a circuit element at each
location in space, which is coupled to elements at neighboring locations. MAGIC can be
used in a variety o f field algorithms with a wide range o f speed, stability, and numerical
smoothness to provide special models that treat material properties and complex, finestructure geometric details.
Particles are represented using the particle-in-cell (PIC)
approach, in which a computational particle (hereafter referred to as a macro particle)
represents a large number a physical matter of the same properties (e.g., electrons). A
variety o f algorithms for emission processes, particle kinematics, and current density
allocation are available, all based upon the same mathematical foundation.
Material
models provide accurate treatment of the complexities of real systems can include both
field and particle effects. Boundary conditions connect the domain of the simulation with
the spatial regions and beyond. These algorithms are based on the fundamental response
o f Maxwell’s equations under the specification of Neumann, Dirichlet, and other
symmetries, as well as other, more complicated conditions.
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230
on
pfMMMkiBcyctopof tlmootap
Fig. IV.l: The processing cycle (Many algorithmic options are available for each step).
IV.2 Mathematical Foundations
Since most physical problems involve fields defined continuously over space and
astronomical number o f particles, the challenge is to achieve good solutions with finite
computational resources. Conditions for the overall validity o f the electromagnetic PIC
approach have been thoroughly documented [32]. The following is a brief introduction
of the mathematical foundation.
(a) Physical Basis
MAGIC performs a time integration of Faraday’s law, Ampere’s law, and particle
force equation:
— = -V x £
dt
QE = -----+
J —
1
Y7
—
Vx D5
dt
e ue
dp f .
1 ± = ^ , F i =qi[E(xi) + vi xB (xi)]
dt mi
(rv -1)
dx,
Pt
—L= v,,v, = —
dt
Yi
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231
subject to constraints provided by Gauss’s law and the corresponding rule for the
divergence o f B,
s?- e
=££
V -5 = 0
In above equations, E(x) and B(x) are the electric and magnetic fields, x/ and Pi are the
position and momentum of the zth charged particle, and J(x) and p(x) are the current
density and charge density resulting from those particles.
(b) Discrete Time
If we were to follow just the trajectory of one particle through given fields (so-called
trajectory calculations), then we probably would choose to keep velocity and position
information from several previous time steps and use a method o f time integration with a
high order o f accuracy. However, the minimum information needed for integration is the
particle velocity and position. We nearly always will choose to use the least information
(storage) and the fastest method we can. One commonly used integration is called leap­
frog method. The two first order differential equations to be integrated separately for
each particle are
dv ^ dx
m — = F ,— = v
dt
dt
where F is the force. These equations are replaced by the finite-difference equations
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The flow in time and notation is shown is shown in Fig. IV.2.
The computer will
advance v, and xt to v<+A<, xt+&t, even though v and jc are not known at the same time.
The time-integration scheme is based upon a fixed time interval, At, between variable
updates, such that when the time derivative of Eq. (IV.l) are approximated in finite
8E
(E - E )
difference from (e.g., — <------—-----— , they provide an equation for a new value of
dt
At
the variable at a time, At, later than the previous value). The leapfrog scheme of Eq.
(IV.l) is illustrated in Fig. IV.3.
VELOCITY
POSITION
ti me
Fig. IV.2: Sketch o f leap-frog integration method showing time-centering of force
F while advancing v, and of v while advancing x.
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233
faces F
cmenis
J
Fig. IV.3: Leapfrog time integration scheme
(c) Discrete space
All o f the spaces are divided into regular and orthogonal cells in one of four
coordinate systems (Cartesian, cylindrical, polar, and spherical). The continuous field
variables, E(x) and B(x), are stored in computer memory at specific “full-cell” and “halfcell” locations on the cell as shown in Fig. IV.4. The division of space into cells is
performed as two (or three) separate, one-dimensional discretizations, each specifying
full-grid intersections along axis. The full-grid intersections create the cells. The space
of full-grid locations may vary according to resolution requirements.
■■
Fig. IV.4: Spatial definition o f fields.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
234
(d) Divergence and Curl Operations
The finite-difference divergence and curl operations are derived by interpreting them
as averages over cell face or cell volume. For example, by Green’s theorem, the faceaveraged curl operation becomes— \dA•V x E = —— \dl-E = V —*-•.£ , which is a
dAk 3
dAk 3
dAk
loop sum o f the electric-field components and their edge-length elements, d l., around the
face-area, dAk, associated with a magnetic field component. The cell-averaged divergence
operation involves the same face area. The numerical equivalents o f the mathematical
identities, V •(Vx A) = 0 andV x(V ^) = 0, are also preserved exactly. This ensures that
transverse (curl) and longitudinal (divergence) fields remain mutually isolated. Thus,
time-step ping the fields does not create errors in the divergence.
(e) Macro Particle Representation
The number o f physical particles in a problem almost always exceeds the capacity of
any computer memory. Hence, a single macro particle typically must represent many
physical particles, which have the same mass and charge and are approximately co­
located in phase space. The representation is satisfactory when there are sufficient macro
particles to represent all phase space region, which contributes to the result. Such a
simulation is said to have good statistics. Particles and fields interact in two places in Eq.
(IV.l): the current density in Ampere’s law, J, and the Lorentz force, F l Since the fields
are defined at discrete locations on the grid, each particle is also mapped to the grid. The
particle charge is allocated to grid points surrounding the particle. The current density, J,
basically reflects the change in allocation of particle charge during a time step.
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235
The idea o f working with something like 1025 charges with as many calculations of
\ was dismissed, since E can be obtained from charge density p. Nearly all of the
r
plasma physics (or electron beams) require knowledge only down to some scale length,
with charge density (and current density) considered continuous; the finer grained
behavior is omitted. Hence, we are encouraged to take advantage of the simplifications
that come about in using a mathematical spatial grid, as shown in Fig. IV. 5, usually fine
enough to measure the charge density and calculate the electric field E. In Fig. IV.5, a
mathematical grid is set into plasma region in order to measure charge and current
densities p, J; from these we will obtain the electric and magnetic fields E, B on the grid.
A charged particle q at (x, y) will typically be counted in terms of p at nearby grid points
(0, 0), (1, 0), (1, 1), (0, 1) and in terms of J at the faces between these points. The force
on q will be obtained from the fields at these nearby points.
y
£
$
£
V
(0,1)
X.
•q
s
(1,1) V,
N
(O.O)
N ,
(1,0)
x x, '
%
—
SMA
1
Fig. IV.5: A mathematical grid is set into plasma region.
(f) Simulation Constraints
In general, large cells and time steps are desired to reduce simulation expense.
However the cell size and time step is constrained by spatial and temporal resolution
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236
requirements for the physical phenomenon of interest. These may be further constrained
by several numerical instabilities which can arise in plasma simulations. First numerical
instability occurs when the time step, At, is too large to resolve light waves of very short
wavelength (nearly equal to the grid spacing).
cAt < Axmin, where c is the speed of light and
The Courant stability condition is
is the minimum cell dimension.
Another numerical instability occurs when the time step is too large to resolve the
oscillation o f particles at the plasma frequency.
For stable plasma oscillations, the
plasma stability condition, copAt < 2 must be satisfied, where cop is the maximum plasma
frequency o f the charge distribution.
(g) Initial Conditions
The default initial condition is one in which all fields identically vanish and there are
no particles. However, arbitrary initial conditions can be established by presetting fields
and populating the space with particles. In this case, Gauss’s law must be satisfied by
solving Poisson’s equation and deriving the electric fields from the gradient of the scalar
potential.
IV.3 Conclusion
This introduction has demonstrated the basic mathematic foundation of the PIC
code - MAGIC. This concept continues to guide development. The implementation of
new models and algorithms is facilitated by a robust mathematical foundations and
software architecture [33] [34].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
237
Appendix V
Reproduction of the NTHU MIG
The NTHU MIG structure, the magnetic fields and the beam trajectories simulated by
EGUN:
B(Z) GAUSS
R (0.025cm )
0
600
m
1500
1200
1800
Z (0.025cm )
Reproduction of the NTHU Single Anode MIG:
NTHU results
Cathode Radius (rc)
Emitting Strip Length (ls)
Cathode to Anode Distance (d)
Cathode Angle
Beam Voltage
Beam Current
Magnetic Field (Bo)
Magnetic compression Ratio
Gamma
Perpendicular velocity Spread
Axial velocity Spread
Velocity Ratio
Maximal Electrical field
Maximal Electrical field position
Author’s Simulations
2.85 mm
1.65 mm
3.1 cm
67.1°
95 kv
2A
12.5 KG
17.2
1.1859
4.7579 %
3.8624 %
0.92656
114 kV/cm
(z= 151.87cm, r=5cm)
1.1809
4.547 %
3.866 %
0.9459
114.7 kV/cm
(z= 151.89cm, r=5cm)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
238
References
[1]
Vacuum electronics, Parker, R.K., Abrams, R.H., Jr.; Danly, B.G.; Levush, B.,
IEEE Transactions on Microwave Theory and Techniques, Volume 50, Issue
3, March 2002 Page(s): 835 - 845
[2] Vacuum electronics for the 21st century, Abrams, R.H.; Levush, B.; Mondelli,
A.A.; Parker, R.K.; IEEE Microwave Magazine, Volume 2, Issue 3, Sept. 2001
Page(s): 61 - 72
[3]
Microfabrication o f high-frequency vacuum electron devices, Ives, R.L., IEEE
Transactions on Plasma Science, Volume 32, Issue 3, Part 1, June 2004 Page(s):
1277-1291
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