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Generation of high power microwaves in a large orbit coaxial gyrotron

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Generation of High Power Microwaves in a
Large Orbit Coaxial Gyrotron
by
Reginald Lamar Jaynes
A dissertation subm itted in partial fu lfillm e n t
o f the requirements for the degree o f
Doctor of Philosophy
(Electrical Engineering)
in The University o f Michigan
2000
Doctoral Com m ittee:
Professor Ward D. Getty, Co-Chairman
Professor Ronald M. Gilgenbach, Co-Chairman
Professor Brian E. G ilchrist
Professor Yue Y. Lau
Dr. Thomas A. Spencer, Nuclear Engineer, USAF,
Air Force Research Lab,
Phillips Research Site
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UMI Num ber 9963819
UMI8
UMI Microform9963819
Copyright 2000 by Bell & Howell Information and Learning Company.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
Bell & Howell Information and Learning Company
300 North Zeeb Road
P.O. Box 1346
Ann Arbor, Ml 48106-1346
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reginald Lamar Jaynes 2000
All Rights Reserved
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I dedicate this work to Nikola Tesla,
the inventor o f alternating current electricity
and w ireless transmission of pow er and intelligence,
who’s name has inexplicably fallen into obscurity.
ii
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ACKNOWLEDGMENTS
i
would
like
to
thank
Professor
Ronald
Gilgenbach
fo r
accepting me into his research group so I could work on so many
d iffe re n t
and
interesting
experim ental intuition
projects.
W ithout
his
rem arkable
and guidance, I would not have been able to
complete my dissertation.
I am most gra te ful
for
his help and
patience w hile working with me to better my w riting skills.
I w ould
like
to thank Professor Y. Y. Lau fo r
his easy to
understand theories that gave so much insight into the operation o f
gyrotrons, m icrowave devices and charged p a rticle
beams.
He has
shown me how important it is fo r theorists and experim entalists
work together.
I would like
to thank Professor Ward Getty fo r
teaching me about plasma processing and to w ish
retirem ent.
to
Professor Brian G ilchrist,
him well
in h is
thank you fo r being on m y
co m m itte e .
Dr. Thomas Spencer, thanks
for
your
help
setting
up th e
Labview program your many helpful discussions about the g y ro tro n
and the countless trips you have made to Michigan to visit us.
A special thanks goes to Dr. Jonathan Hochman for helping to
build and rebuild the magnetic fields of the m agnetic cusp as well as
for teaching me to work in the lab and manage people.
I thank Dr.
Joshua R intam aki for helping me run the experim ent so I could g e t
the (thousands) of shots that my thesis required.
Thanks to Dr. Scott
Kovaleski fo r sharing his interesting philosophical views on life and
making sure the lab was never dull.
Bill Cohen, thank you for helping
me run the experim ent and fo r the numerous insightful conversations
iii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
concerning the gyrotron
experim ent and plasma generation.
Good
luck at GE Chris Peters, thanks for doing so much w ork on tim e frequency analysis.
I’m sure that com puter is going to miss you
when you graduate.
Mike Lopez, good luck on the magnetron p ro je c t.
Also
I want
sociology.
to
complement
him
fo r
his
amazing
theories
on
A special thanks to the people who helped by fillin g
in
and running the experim ent when the usual running crew was n ot
available:
Mark Johnson, Rex Anderson,
Scott
Anderson,
Nicholas Eidietis, Antwan Edson, and Mark Porter.
thanks for your help with
Bo Qi,
Dr. Joe G ettes,
computer problems and fo r
helping me
learn to program in C.
Ed Birdsall, thank you for keeping the computers running
fo r
saving my com puter when it crashed.
com puter problems, whether it is
w ith
CAEN did not know the answers, Ed did.
John
Luginsland,
Larry
Ludeking,
and
If there were any type o f
Macs or PC’s, when even
I would also like to thank Dr.
and
David
Smithe
for
th e ir
invaluable help with using the MAGIC code.
I would
electronics
like
to
thank
Ron Spears
for
his
and fo r keeping the experim ents running.
expertise
But most o f
all, thank you fo r all the discussions and shared interest
radio and electronic hobbies.
in
in ham
Mark Perrault, thank you fo r using th a t
Navy know-how to keep things running smoothly and efficiently.
Finally, I would like to thank my mom Peggy , my dad Gerry and
my sister Cindy fo r their love and support.
If it had not been fo r
them, I would not be where I am today.
This work was supported by the A ir Force O ffice of S c ie n tific
Research (AFOSR) in itia tiv e
through a Texas Tech subcontract, A ir
iv
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Force Research Lab, Northrup Grumman Corporation, and the AFOSRsponsored MAGIC Code User’s group.
v
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Table of Contents
DEDICATION
ACKNOWLEDGEMENTS
LIST OF FIGURES
LIST OF TABLES
LIST OF APPENDICIES
ii
iii
v ii
xvi
xv ii
CHAPTER
1. In tro du ctio n
2. Gyrotron Theory
1
5
2.1 Gyrotron dispersion relations.
2.2 Limiting Currents.
2.3 Magnetic Cusp dynamics
3.
Experim ental
C onfiguration
3.1 Gyrotron Configuration
3.2 a Measurement Experimental Configuration
3.3 Magnetic Cusp and Magnetic Field
3.4 Coaxial/Non-Coaxial Cavity Structure
4.
Experim ental
D iagnostics
4.1 Glass Plate e-Beam Diagnostic
4.2 Electron Beam Current Diagnostic
4.3 Electron Beam a Diagnostics.
4.4 Mixer Diagnostics
5.
Experim ental G yrotron
Results
5.1 MELBA Experimental Shot Signal
5.2 Slotted Aperture, Coaxial Cavity Gyrotron
5.3 Microwave Power Measurements, Open Aperture Gyrotron
5.4 Current Transport Measurements
a. Coaxial Cavity
b. Slotted Cavity Current
c. Non-coaxial Cavity Current Measurements
5.5 Microwave Frequency Measurements
5.6 Time Frequency Analysis
5.7 Microwave Pulse Shortening
6. Summary and Conclusions
6.1 Summary
6.2 Conclusions
5
11
20
26
26
31
32
37
42
42
46
49
56
58
58
59
63
70
70
78
79
81
84
89
92
92
94
96
APPENDICES
BIBLIOGRAPHY
158
vi
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List of Figures
F ig u re
Chapter
2 . 1.1
Dispersion relation for a waveguide mode (hyperbolic) and four e-beam
modes (straight lines) with a of 1 and magnetic fields of: 0.12. 0 .1 4 5 ,
0.17 and 0.2 Tesla from right to left. An e-beam voltage of 750 kV is
assumed.
2 . 1.2
Dispersion relation with e-beam mode with a of 1 and waveguide modes
TE„ andTEgt.
2 .1 .3
(a) Dispersion relation with e-beam mode, TE„ and TEj, waveguide modes
(hyperbolas) and the operating frequencies (straight lines) of the TEt11,
TE112, TE1ia, TE114, TE118, and TE11e cavity modes, (b) Chart of cavity
modes, their theoretical and cold test measured resonant frequencies.
Frequencies designated as NM are not measured.
2.3.1
Plot of theoretical velocity cutoff of electrons transmitted through a
magnetic cusp as a function of solenoid magnetic field for dude fields of
0.083, 0.107, 0.115, and 0.123 T.
Chapter 3
3.1.1
Experimental setup for gyrotron oscillator with magnetic field plot.
3.1.2
Three cavities used for gyrotron oscillator:
a) Unslotted coaxial cavity,
b) Slotted coaxial cavity,
c) Unslotted cavity with no center conductor.
v ii
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Axial magnetic fields vs. axial distance for four different diode and
solenoidal field combinations.
3.3.1
3 .3 .2
Radial magnetic field as a function of:
(a) theta, (b) axial distance
Chapter 4
4.1.1
Experimental Glass Witness Plate
4 .1 .2
Simulation Results
4.1 .3
Histogram plot of single panicle simulation of number of panicles vs. e beam a.
4.3.1
Setup for a measurement experiment.
4 .3 .2
Comparison of theoretically derived and experimentally measured Larmor
radius vs. magnetic field for glass plates.
Solid curves obtained from
theory.
4 .3 .3
Plot of e-Beam a vs. magnetic field for glass plates.
4.3 .4
Comparison of theory and experiment for Ar vs. solenoid magnetic field
for glass plates.
4 .3 .5
Measured outer e-beam radius vs. solenoidal magnetic field for glass plate
measurements.
4 .3 .6
Measured inner e-beam radius vs. solenoidal magnetic field.
Chapter 5
5.1.1
MELBA experimental shot, (a) dtode voltage (620kV/div),
aperture current (2190 A/div),
div),
(b) anode
(c) cavity entrance current (880 A /
(d) microwave power signal (25
MW peak),
mixer signal.
viii
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(e) heterodyne
5.2 .1:
Measured microwave power vs. solenoid magnetic field for the coaxial
gyrotron with the 3-sfotted aperture plate.
5 .2 .2 :
Experimental electronic efficiency vs. solenoid magnetic field for coaxial
gyrotron with 3-slotted aperture plate
5.2 .3:
Measured microwave frequency vs. solenoidal magnetic field for coaxial
gyrotron with 3-slotted aperture
5 .2 .4:
Experimental cavity entrance current vs. solenoidal magnetic field for
coaxial gyrotron with 3-slotted aperture
5 .3 .1 :
Measured microwave power vs. solenoidal magnetic field,
coaxial
gyrotron.
5.3 .2:
Measured microwave efficiency vs. solenoid magnetic field,
coaxial
gyrotron.
5 .3 .3 .
Measured microwave power vs. solenoid magnetic field, coaxial, slotted
gyrotron
5.3 .4:
Microwave power vs. solenoid magnetic field for non-coaxial, unslotted
gyrotron
5 .3 .5 :
Efficiency vs. solenoid magnetic field for non-coaxial, unslotted gyrotron
5.4.1
Currents from anode aperture, cavity entrance, and exit vs. solenoidal
magnetic field for coaxial gyrotron, with diode magnetic field of: (a) 0 .8 3
kG. (b) 1.07 kG. (c) 1.23 kG.
ix
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5.4.2:
Measured magnetic cusp and cavity current transmission efficiency vs.
solenoidal magnetic field for coaxial gyrotron, with ciode magnetic field
of: (a) 0.83 kG. (b) 1.07 kG. (c) 1.23 kG.
5.4.3:
Cavity entrance and exit currents vs. dfode magnetic field for coaxial
gyrotron, with solenoidal magnetic fields of 1.85-1.95 kG.
5 .4 .4
Diode Magnetic field vs. Microwave power, with
solenoidal magnetic
fields of 1.85-1.95 kG.
5.4.5:
Anode aperture and cavity exit current vs. solenoid magnetic field fo r
slotted coaxial gyrotron, with diode magnetic field of 0.83 kG.
5.4.6:
Cavity entrance and exit current vs. solenoid magnetic field for non­
coaxial gyrotron, with diode magnetic fields of 0.83 and 1.07 kG.
5.5.1:
Microwave frequency vs. solenoidal magnetic field for coaxial gyrotron,
with diode magnetic fields of 0.83, 1.07, 1.15, 1.23 kG.
5.5.2:
Microwave frequency vs. solenoid magnetic field for slotted, coaxial
gyrotron, with diode magnetic fields of 0.83 kG
5.5.3:
Microwave frequency vs. solenoid magnetic field
for
non-coaxial
gyrotron, with diode magnetic fields of 0.83 and 1.07 kG
5.6.1
Time frequency analysis of mode hopping in unslotted coaxial gyrotron.
Local oscillator is 2.2 GHz.
5 .6 .2
Time frequency analysis of mode competition
in
gyrotron. Local oscillator is 2.2 GHz
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
unslotted coaxial
5 .6 .3
Time frequency analysis of unslotted ooaxial gyrotron. (Top) frequency
fluctuations vs. time
and (Bottom) tfode voltage fluctuations vs. time.
Local oscillator set to 2.3 GHz.
5.6 .4
Time frequency analysis high power microwave spike for slotted coaxial
gyrotron. (Top) Microwave mixer signal, (Middle)
Time frequency
analyses, (Bottom) Diode Voltage. Local oscillator set to 2.3 GHz.
5.7.1:
Microwave power vs. pulse length for the three gyrotrons:
▲ is coaxial non-slotted gyrotron,
+ is coaxial slotted, gyrotron.
O is gyrotron with no center rod
Curve represents 1000/x[ns] [MW]
A ppendix A
A.1
Experimental Set-up
A.2
Alpha Measurement Model
A.3
Small Orbit Magnetic Field
A.4
Small Orbit Glass plate
A.5
Small Orbit Simulation Results
A.6
Small Orbit, Particle Number vs. Radius
A.7
Small Orbit, Alpha vs. Particle Distribution
A.8
Large Orbit Magnetic Field
A.10
Large Orbit Simulation Results
A.11
Large Orbit Alpha vs. Particle Number
xi
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A p p e n d ix D
D.1
Shot 6346, Dmax - 4.1 cm, Dmin - 3.0 cm, Dout - 6.0 cm, A r « 1.0 1.5, Bdiod - 1.1 KGauss, Bsoie -1 .4 9 kGauss
D.2
Shot #6589, Dmax - 3.5 cm., Dmin - 2.8 cm., Douter - 5.7 cm., A r 1.1 - 1.4 cm, Bdiod « 1.1 kGauss, Bsole - 1.86 kGauss
D.3
Shot #6610, Dmax - 3.7 cm, Dmin - 2.9 cm, Douter * 5.7 cm, A r « 1.0
•1.4 cm, Bdiod - 1.1 kGauss, Bsole - 1.71 kGauss
D.4
Shot #6631, Dmax « 2.7 cm, Dmin - 2.0 cm, Dout - 4.7 cm, Ar - 1.0 1.35. Bdiod - 1.1 kGauss, Bsole « 2.23 kGauss
D.5
Shot #6651, Dmax - 4.0, Dmin « 3.5. Dout ■ 6.0, Ar • 1.0 -1.25,
Bdiod ■ 1.1, kGauss, Bsole - 1.25 kGauss
D.6
Shot # 6671, Dmax • 4.6, Dmin - 3.1, Dout - 5.7, Ar « .55 -1.3,
Bdiod - 1 .2 kGauss. Bsole - 1.48 kGauss
D.7
Shot #6691 mislabeled as 6688, Dmax - 3.6, Dmin - 2.8, Dout - 5.9,
Ar - 1.15 -1 .5 5 , Bdiod - 1.2 kGauss. Bsole -1 .8 5 kGauss
D.8
Shot # 6711, Dmax - 2.8cm, Dmin - 2.4 cm, Dout « 5.1 cm, Ar - 1.15
>1.35 cm, Bdiod - 1.2 kGauss. Bsole - 2.22 kGauss
D.9
Shot # 6736, Dmax * 3.7, Dmin - 3.0, Dout - 5.8, Ar « 1.05 -1.4 cm,
D.10
Bdiod -1 .2 kGauss, Bsole - 1.7 kGauss
Shot 6762, Dmax - 2.8, Dmin - 2.4,Dout - 4.4,Ar - .8-1.0
- 1.2 kGauss, Bsole - 2.96 kGauss
D .11
Shot 6287 (three slotted aperture)
D.12
Shot 5219 (8-pinhole aperture) Witness plateshows incoherent offcentering due to azimuthal asymmetries in the cusp magnetic field.
cm, Bdiod
Appendix E
E.1
Measured S,, parameter of coaxial cavity with frequency range of 2 GHz to
4 GHz for: (a) loop antenna, (b) dipole antenna (5 dB/ div)
E.2
TE11t mode of coaxial cavity, resonant frequency is 2.35 GHz, Q is 94
xi i
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(frequency range: 2.0 GHz - 2.6 GHz. 2.125 dB/div)
E.3
TEm mode of coaxial cavity, resonant frequency is 2.53 GHz, Q is 505.
(frequency range: ZJ2. GHz - 3.0 GHz. 10 dB/ div)
E.4
Slotted, coaxial cavity.
Using dipole antenna 1.5 cm behind front
reflecting posts, (frequency range: 2.0 GHz - 4.0 GHz. 10 dB/div)
E.5
Slotted coaxial cavity. TE,„
mode, resonant frequency is 2.28 GHz and
cavity Q is 162. (frequency range 2.0 GHz • 4.0 GHz, 10 dB/div)
E.6
Slotted, coaxial cavity, TE1iamode, resonant frequency is 2.53,
cavity Q
is 632. (frequency range: 2.0 GHz - 4.0 GHz, 10 dB/div)
E.7
Non coaxial cavity, TE1t1 mode, resonant frequency is 2.29 GHz, cavity Q
is 458 using a loop antenna,
(frequency range: 2.0 Gt-te -
4.0 GHz. 1 0
dB/ div)
E.8
Non-coaxial cavity, TE„2 mode, resonant frequency is 2.5 GHz. cavity
Q
is 180 using a dipole antenna (frequency range: 2.0 GHz- 4.0 GHz, 1 0
dB /div)
E.9
Metal tank and S-band waveguide attenuation.
E.10
Power reflected in microwave horn
Appendix F
F.1
Timing Traces: (a) dkxJe magnetic field current, (b)solenoidal magnetic
field current, (c) gate pulse
x iii
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F.2
Schematic of timing circuit
Appendix G
G.1
Dispersion relation for TE„ waveguide mode, a of 1 and e-beam voltage of
750 kV solved for frequency as a function of magnetic field.
The upper
curve is the upper forward wave intersection and the lower curve is the
forward/backward wave.
G.2
Frequency vs. magnetic field for three slotted anode aperture.
Curves
represent a between 0.5 and 2.0.
G.3
Frequency vs. magnetic field for large open anode aperture.
Curves
represent simple theory with alpha between 0.5 and 2.0.
G.4
Frequency vs. solenoidal magnetic field for coaxial non-slotted gyrotron
compared to plasma shifted cutoff frequency for e-beam current of 0 A,
500A, 1000A, 1500A, and 2000A, for: (a) o - 0.5, (b) a - 1.0, and
(C) a - 2.0
G.5
Frequency vs. solenoidal magnetic field for coaxial non-slotted gyrotron
compared to plasma shifted cutoff frequency for e-beam current of 0A,
500A, 1000A, 1500A, and 2000A, for magnetic field dependent a.
Frequency vs. solenoidal magnetic field with 200 kV beam depression,
compared to plasma shifted cutoff frequency for e-beam current of 0A,
500A, 1000A, 1500A, and2000A, with magnetic field dependent alpha
for cases:(a) Coaxial non-slotted gyrotron. (b)
non-coaxial gyrotron,
and (c) slotted coaxial gyrotron
x iv
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
G.7
Plot of theoretically predicted a as a function of maser solenoid magnetic
field for an electron transmitted through a magnetic cusp for four dtode
B-field Of 0.83 T, 1.07 T. 1.15 T, 1.23 T.
Appendix H
H.1
Diagram of transmission line model for sc/oc and sc/sc
xv
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
List of Tables
T a b le
chapter 2
2.2.1
Solutions to limiting current calculations using Drobot’s theory for a
non-coaxial gyrotron with different e-beam thicknesses (Delta)
2 .2 .2
Solutions to limiting
current calculations using Drobot's theory fo r
planar gyrotron to approximate a coaxial gyrotron using different e-beam
thicknesses (Delta)
2 .2 .3
Summary of limiting
currents
in different sections of the gyrotron
experiment. For diode. r.« 1 9.0 cm, r, » 2.5 cm, and r, - 2.0 cm. For
pre-cusp drift tube, rw«4.9 cm, r0 - 2.5 cm, and r, - 2.0 cm. For post­
cusp drift tube. r„»4.9 cm, r, « 3.0 cm, and r, - 1.0 cm. For microwave
cavity, r„*3.6 cm, r, - 3.0 cm, and r, > 1.0 cm.
chapter 3
3.4.1
Calculated TE^, modes of the coaxial microwave waveguide
3 .4 .2
Calculated T M ^ modes of the coaxial microwave waveguide
3 .4 .3
Calculated short circuit/open circuit axial mods frequencies for coaxial
cavity
3 .4 .4
Calculated short circuit/short circuit axial modes for coaxial cavity
3 .4 .5
Calculated
TE ^ mode frequencies
for
the
non-coaxial
microwave
waveguide
3 .4 .6
Calculated short circuit/short
circuit axial mode frequencies for non­
coaxial cavity
xvi
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LIST OF APPENDICES
Appendix
A.
Small/Large orbit alngle particle almulation
97
B.
Single particle source code
108
C.
Glass plate compositions
12 2
D.
Pictures of experimental glass witness plates
123
E.
Microwave Cold Tests
129
F.
Trigger Sequencing
141
G.
Dispersion Relations with Analysis
144
H.
Axial Mode Calculation
156
x v ii
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1. Introduction
Large orbit coaxial gyrotrons u tilizin g
being investigated
a t The University
long e-beam pulses are
of Michigan for high pow er
microwave production in a compact m icrowave source. The purpose
of this
dissertation
is
to
operating
in
or
gyrotrons
advantages of higher
[COR93] and better
coaxial gyrotrons.
(0.5
ps)
experim ents
near
lim iting
the
current,
mode se le ctivity
large
orbit
fundam ental
low er
co a xia l
mode
voltage
have
depression
[BAR95, SHA95] over non-
It is also the goal of th is thesis to in v e s tig a te
the physics of m icrowave
long
assess w hether
pulse
pulse shortening
e-beams.
Previous
used short e-beam pulses (5-25
phenomenon u tiliz in g
large
orbit
g yro tro n
ns) [DES81, DES81b,
LAW85, DES88, DES89]. The long (0.5-1.0 ps) e-beam pulse produced
by MELBA (Michigan Electron Long Beam A ccelerator) introduces new
physics
and enables
the
study
of
m icrowave
pulse
shortening
phenomena in these devices. The present experim ents are the f i r s t
long pulse, smooth bore coaxial gyrotron u tilizin g
a large o rb it-a x is
encircling e-beam operating in or near the fundam ental mode. A lso
the addition of axial slots have been investigated to determ ine th e ir
effect
on polarization
control
and mode se le ctivity.
S lo tte d
gyrotrons have been shown to generate high power in a single mode
[GOL88]. Gyrotrons have possible application
of m aterials,
high resolution
in plasma processing
radar [GAP94, BEN92] and tokam ak
plasma heating [GIL80].
1
permission of the copyright owner. Further reproduction prohibited without permission.
Before the invention of the gyrotron, the m icrowave tube had
an upper power lim it at high frequencies. [EDG93]
r.f.output _ power
«=
------------------------ -------------------
(operating _ frequency)
This rf output power lim itation was due to the slow wave s tru c tu re .
Gyrotrons were able to overcome this lim ita tio n
by using a ro ta tin g
beam and a smooth walled interaction cavity.
The cyclotron
maser em ission was th e o re tica lly
in ve stig a te d
by Twiss and Schneider [TWI58, SCH59] using a quantum m echanical
approach and classically
by Gaponov [GAP59, GAP67].
H irs c h fie ld
and W achtel developed the ‘electron cyclotron maser* e x p e rim e n ta lly
[HIR64].
Gapovon used a Magnetron Injection
to produce a high power m icrowave
Gun (MIG) electron gun
device firs t
described as a
‘gyrotron’ (KIS74, GAP75].
The conventional gyrotrons
used sm all
o rb it
e-beams.
The
large o rb it e-beam has Larmor orbits comparable to the size o f th e
e-beam dim ensions which makes the e-beam more compact than th e
small orbit devices.
A large orb it e-beam is generated using a cusp
m agnetic field [CHR66, DES81, DES81b]. Cusp guns used to generate
large o rb it e-beam have higher current
region than Magnetron injection
the large o rb it
ca p a b ilitie s
in the diode
guns (MIG) [LAW87, LAW88] since
e-beam a is low er
in this
c ritic a l
diode region.
Electron beams generated by a sim ple cusp geom etry have co herent
o ff-cen te rin g
[RHE74].
due to the fin ite
distance of the fie ld reversal region
The coherent o ff-ce n te rin g
is undesirable since it makes
the e-beam expand in thickness and possibly decreases its ope ra tin g
efficiency [ZHA95].
2
permission of the copyright owner. Further reproduction prohibited without permission.
Previous
research
configurations
[DES89] has focused
because it offers
on the
the p o s s ib ility
large
of high harm onic
production th e o re tic a lly
[UHM78, LAU82a] and experim entally
magnetron like
ca vitie s
[DES81b, LAW85] and transverse
magnetic fields
[DES88].
investigated
[GAL96].
which
Large orbit peniotron cavities
operate
at the
fourth
harmonic
polarized
m icrowaves
Previous large o rb it experim ents have u tilize d
but in overmoded configurations
have been investigated
using
w ig g le r
have been
at
Large o rb it rectangular cross section cavities
used to generate linearly
o r b it
91
GHz
have been
[RAD93, HOC98].
coaxial g e o m e trie s,
[DES81] and cylind rica l
c a v itie s
operating in the fundamental mode [DES89].
V ariations of the large orbit cusp guns have been designed th at use
center magnetic pole pieces [SCG91] and Pierce gun focusing o f th e
e-beam prior to the cusp [GAL96] to decrease the spread in a and th e
coherent o ff-cen te rin g.
Coaxial gyrotrons
current,
lower
offer
voltage
the advantages of
depression
higher
[COR93] and
lim itin g
better
mode
selectivity [BAR95, SHA95] than non-coaxial gyrotrons.
Sm all o r b it
coaxial
fo r
use as
tubes
e x h ib it
gyrotrons
have been previously
investigated
gyroklystron am plifiers [FLA94, BLA99].
At high m icrowave
power levels,
microwave
pulse shortening or the turning off of m icrowave production b efo re
the end of the e-beam pulse. This phenomenon is undesirable sin ce
it lim its
produce.
the single-pulse energy a high power m icrowave tube can
Many high
power
m icrowave
devices
e xh ib it
pulse
shortening [AGE96, BEN97, GIL98, GRA96, HEN97, MIL96]. The coaxial
gyrotron exhibits the typical microwave power being proportional to
3
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1/(Pulse length) type of pulse shortening.
Tim e-frequency analyses
of the heterodyne mixer signal has been implem ented at UM to s tu d y
pulse shortening.
Plasma in the microwave tube is also considered
to be a contributing factor [COH95].
4
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2.
2.1. Gyrotron
Gyrotron Theory
Dispersion
Relation
The gyrotron o scillator is a resonant device th a t converts d.c.
beam current into high frequency electrom agnetic waves.
There are
two components essential for generating microwave radiation
gyrotron using the cyclotron maser effect, gain and feedback.
one needs a gain medium to am plify the rf radiation.
beam interacting
with the waveguide structure
in a
F irs t
The e le ctro n
is the gain medium
in the gyrotron. The gain is produced by the cyclotron maser e ffe c t,
also referred to as the negative mass instab ility
one needs a feedback mechanism.
resonant cavity.
Second,
The feedback is provided by the
The microwave cavity also acts like a filte r
selects the operation frequency.
phase-matching
[LAU82].
condition
losses are am plified.
Only frequencies with
and w ith
w hich
the c o rre c t
the highest gain and lo w e s t
From in itia l
random noise a narrow
band
resonant frequency can grow to a large level. When resonance occurs
in a gyrotron, the electron cyclotron wave and the e le ctro m a g n e tic
(E-M) wave in the waveguide are in synchronism.
The electron beam
gives up some o f its rotational energy to the electrom agnetic wave.
The E-M wave then grows at the expense of the electron
beam’s
rotational energy.
In the gyrotron the gain is produced by a resonant coupling o f
the electron beam and the E-M waves in the waveguide.
An e le ctro n
gyrates at the cyclotron frequency due to the axial magnetic fie ld .
The electrons
also have a stream ing
velocity
in the axial
5
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or z
direction.
The ratio of the perpendicular e-beam velocity, V ^ , to
the axial e-beam velocity, Vz, is defined as a. The e-beam s p ira ls
down the d rift tube.
We can w rite
a dispersion
relation
for th is
electron wave in the form of equation 2.1.1 [LAU82, SH087].
(O = k zv z + G)c
(2.1.1)
Where <d is the angular frequency in radians, kz is the propagation
vector in the axial direction, Vz is the velocity in the axial d ire c tio n
and a>c is the relativistic cyclotron frequency.
<Oc =
(2 .1.2)
ym
The cyclotron
frequency depends on q, the electron
electron mass, y =
charge, m, th e
— v 2/ c 2 and, B, m agnetic field.
Since the <0c
term of the dispersion relation is dependent upon B, it is possible to
tune the cyclotron frequency and change the dispersion by changing
the B -fie ld .
The
electrom agnetic
waves
in
the
waveguide
have
th e
follow ing dispersion relation [HAR61],
Io = -yl ( k zc ) 2 + © £ ,
(2.1.3)
where c is the speed of light
frequency.
in vacuum and ta*, is the c u to ff
Below the cu to ff frequency electrom agnetic waves can
not propagate.
This equation 2.1.3 applies to all uniform w aveguides
regardless of the cross-section
of the waveguide and to all modes.
The size and shape of the cross-section
the c u to ff frequencies.
and the mode only change
The phase ve lo city
in the axial direction
is
faster than the speed o f light and th is is w hy the gyrotron is c a lle d
a fast-w ave device.
6
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If we plot equations 2.1.1 and 2.1.3 on the same graph we can
get zero, one or two intersections.
d iffe re n t
modes of operation.
dispersion relation
equal to
The intersections correspond to
Figure 2.1.1
and four d iffe re n t
shows the waveguide
e-beam modes for e-beam a
1. The e-beam modes are at d iffe re n t
values of m agnetic
fie ld s and dem onstrate different modes o f operation.
beam mode, fa rth e st
shows no intersection.
to the right, w ith
magnetic field
o f 0.12 T
The next higher e-beam mode a t 0.145 T is
the grazing condition typically
solution.
The low est e-
used in gyrotrons and has only one
The e-beam mode above grazing at magnetic field o f 0.17 T
has two solutions, a forward wave at the low frequency in te rs e c tio n
and forward wave at the high frequency intersection.
magnetic field
intersection
o f 0.2 T has tw o
intersections,
The highest
a lower frequency
on the backward wave gyro-BWO and a forw ard
intersection at a higher frequency.
wave
For this case, the gyrotron m ost
like ly operates on the backward wave since it is less sensitive to
velocity spread in the e-beam than the forward wave is.
7
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f (GHz)
0.2 T 0.17 T
0.145 T
0.12 T
Figure 2.1.1
Dispersion relation for a waveguide mode (hyperbolic) and four e-beam
modes (straight lines) with a of 1 and magnetic fields of: 0.12. 0 .1 4 5 .
0.17 and 0.2 Tesla from right to left. An e-beam voltage of 750 kV is
assumed.
The waveguide used in the experim ents
has a coaxial c ro s s -
section with outer radius 3.6 and inner radius 0.3 cm.
The TEn is
the low est order non-TEM waveguide mode with a cutoff frequency
of 2.34 GHz. The e-beam a was experimentally measured to be about
1 (see Chapter 5).
If we use an e-beam energy of 750 kV. we get the
dispersion relation shown in Figure 2.1.2.
This dispersion re la tio n
shows a typical e-beam mode and the two low est waveguide modes,
the TEn and TEZi.
The next higher waveguide mode, TE2i , does not
provide mode com petition until at a relatively high magnetic field.
8
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f (GHz)
e-Beam
a=1.0
B = 0.2 T
Figure 2.1.2
Dispersion relation with e-beam mode with a of 1 and waveguide modes
TE„ and TEj,.
We have observed frequencies of operation between these waveguide
modes.
Due to the finite length of the waveguide, there are a se rie s
of cavity modes at interm ediate
TE2i cutoff frequencies.
frequencies between the TEn and
The axial modes’ frequencies are: TEm a t
2.34 GHz, TE112 at 2.5 GHz, are the main modes observed.
The modes
continue up to TE 1i6 which is nearly the same frequency as the TE21
waveguide cutoff.
Figure 2.1.3 shows the cavity modes.
9
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
f (GHz)
TE'116
T E'115
TE
T E'114
TE'113
T E'112
T E^111
e-Beam
a = 1 .0 ^
B = 0.2 T
k j [m l]
(a)
cavity mode frea.(G H z)
TE111
T E 11 2
T E 11 3
TE 114
T E 11 5
T E 11 6
M il frea. (GHz) c.t.
2 .3 5
2 .3 5
2 .4 9
2 .5 3
2 .7 4
NM
3 .0 9
NM
3 .4 9
NM
3 .9 4
NM
(b)
Figure 2.1.3 (a) Dispersion relation with e-beam mode, TE,, and TEj, waveguide modes
(hyperbolas) and the operating frequencies (straight lines) of the TE ,,,.
TE,12. TE113i TE114. TE115i and TE „, cavity modes, (b) Chart of cavity
modes, their theoretical and cold test measured resonant frequencies.
Frequencies designated as NM are not measured.
10
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2.2 Limiting
Currents
Limiting Current of MELBA Diode
The lim iting
e-beam current is the maximum current th a t can
be transported down a vacuum d rift tube. This current lim it
is due
to the potential energy needed to hold the e-beam together a g a in st
its own space charge. This energy is taken from the kinetic energy
of the electron beam.
the e-beam disrupts
When the current reaches the limiting c u rre n t,
and possibly causing reflection
Propagation of the e-beam no longer continues.
is dependent upon several factors
d rift
including
of e le ctro n s.
The lim iting
c u rre n t
the e-beam vo lta g e ,
tube dim ensions and geometry, e-beam d istrib u tio n ,
and e -
beam velocity ratio a .
The firs t
non-rotating
calculation of lim iting
current considered here is a
pencil beam in a cylindrical d rift
tube.
The fo llo w in g
is the equation for limiting current [REI94],
L
rv2/3 - n 3/2
—
l + 21n(£/a)
( 2 -2 1 )
An example corresponding to experimental param eters for the MELBA
diode is, y, the re la tiv is tic
factor, equals 2.46, b is the outer tube
radius of 19.0 cm, a is the outer e-beam radius of 2.5 cm. and
11
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/ 0 = Aite^mc1 /|e| which
is
approximately
17 kA.
This
gives
a
lim iting current of:
lL « 2.5 kA (non-coaxial, non-rotating pencil e-beam in MELBA diode)
The next calculation is for a non-rotating thin annular e le ctro n
beam in the MELBA diode. The annular beam gives a higher lim itin g
current than the pencil beam. [MIL82]
r v 2 /3
I L = 70 ^
- n 3/2
( 2- 2. 2)
0 21n(b/a)
L
lL= 3.4 kA (non-coaxial, non-rotating thin annular e-beam in MELBA
diode)
For finite beam thickness, the denominator in 2.2.2 is replaced
with the geometry factor for finite e-beam thickness to give
equation 2.2.3 [SPE98],
_ ix3/2
r v^2/3
z/3 —
IV
*7
V
Ir = Ic
r}
1 - 2 2 * 2 to (r0 tiri ) “ ln (rw / r0>
/o -n *
(2.2.3)
where r0 is the outer e-beam radius, r( is the inner e-beam radius,
and rw is the wall radius.
M icrowave
Cavity
The limiting current in the diode is 2.9 kA.
Lim iting
To calculate the lim iting
Current
current of the microwave cavity, i t
is necessary to include the rotation
low er the lim iting
of the e-beam since this can
current considerably.
It has been suggested by
12
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Spencer to replace 0 with
0,, to give a lower bound o f the lim itin g
current due to a rotating
e-beam.
Using an e-beam a o f 1. a w a ll
radius, rw, of 3.6 cm and an e-beam radius, rb,
of 3.0 cm, we get the
following lim iting current equation [SPE98],
1
A
r
,2/3
(2.2.4)
- — *
V l+ a :
2 \V3
l +a
\ 3/2
—1
y 2a 2 +1
I L = 1 7 kA
2 In
(2.2.5)
V
\ rb J
lL » 4.1 kA
(Non-coaxial, rotating annular e-beam in the cavity)
where rw is the outer wall radius and rb is the outer beam radius.
D robot's
c alc u la tio n
The next lim iting current calculation for the m icrowave c a v ity
is done using D robofs equations [D R 081]. This was done fo r ro ta tin g
beams in cylindrical and planar cavities.
For the lim itin g
current in
a cylindrical cavity with a rotating e-beam this theory gives:
= I ’G
<2-2-6)
/ * = [ ( r o / r x ) J / 3 - « 3'J
<2 -2 -7>
2 ne^mc* l\e\_____________________________
_
8 .546(fc4)_
_
IL
^ " g(rb + A / 2 ) + ln[rH>/(r6 + A /2 )] “ g(rb + A/2 ) + ln[rw /(r6 + A/2)]
(2.2.8)
where
A
is
the
e-beam
thickness.
For
thin
approximation in equation 2.2.9 can be used,
13
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e-beams,
th e
and where
1
(2 .2 .10 )
Yo
lYo + a 2 ] 1/
r±
[ l + a 2] l/2
(2.2.11)
Using the follow ing
MELBA param eters, conductor inner ra d iu s
rw«0.036 m, average beam radius rb« 0.02 m, beam thickness A « 0 .02
m, beam a«1. Vo-7 5 0 kV, y « 2.467, one obtains a limiting current for
a non-coaxial gyrotron o f
L « 4.76 kA.
(D robot’s non-coaxial, rotating annular e-beam)
Since this calculation
is only strictly
valid fo r thin beams, se ve ra l
calculations were done w ith thinner e-beams to see how good th is
approxim ation
results
is.
Table 2.2.1 shows lim itin g
for non-coaxial
gyrotrons
using various e-beam thicknesses.
the lim iting
w ith
current c a lc u la tio n
annular rotating
e-beam s
The thickness does not change
currents in drastic ways, so the use of a 2 cm th ic k
beam probably gives a reasonable calculation.
Delta (m)
I (kA)
0.02
0.002
0.0 0 0 2
0 .0 0 0 0 2
Table 2.2.1
4 .7 6
5.51
5 .5 3
5 .5 3
Solutions to limiting current calculations using Drobot’s theory for a
non-coaxial gyrotron with different e-beam thicknesses (Delta)
14
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For the lim iting
Drobot’s calculations
current of a coaxial gyrotron
of lim iting
approximate the coaxial case.
we can use
current for the planar case to
The planar case uses two p a ra lle l
plates with a width of L and separated by a distance xw. Between th e
two parallel plate there is a rotating
and a distance
xb from
the
lower
sheet e-beam of thickness A
plate.
This
is
used as an
approximation to a coaxial gyrotron geometry since if one ‘unrolls’ a
coaxial gyrotron, its geom etry is sim ilar
to that o f the p a ra lle l
plate case.
G ,
= _____________ (c0wc3 l\e \) L lx „ _____________
P
g (*b + & / 2 ) x b / x w + g ( x b) + x b( x w - x b - A / 2 )
= ______________ 1.360 ( k A ) L / x w_______________
“ g(x„ + A / 2 ) x b / x w + g ( x b ) + x b( x w - x b - A /2 )
(2.2.12)
(2.2.13)
For thin beams,
S U i + ^ ) S g ( x 4) = —
(2.2.14)
2
This approxim ation
requires the e-beam to be very thin
theory to give a reasonable answer (see Table 2.2.2).
for th is
The 2 cm beam
thickness does not give a reasonable result (not shown below).
15
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Delta (m)
I (kA)
0.0 0 0 0 2
0.0002
0.002
Table 2.2.2
6.71
6 .8 2
8 .1 7
Solutions to limiting
current calculations using Drobot's theory for
planar gyrotron to approximate a coaxial gyrotron using different e-beam
thicknesses (Delta)
w here:
xw * rw-rinn#r » .033
Xb *
t’b<*l'inn«r *
0 .0 1 7 .
The length L, is the width of planar gyrotron parallel plates.
It has
been assumed the length is the average circum ference of the coaxial
c a v ity .
L - 2*Pi*(xb) - .034*Pi-0.11 cm.
Coaxial Thin
Rotating e-Beam
An alternative geometric factor fo r a non-rotating, thin ebeam coaxial lim iting current is given by Krall [KRA92].
This factor
can be used in the rotating e-beam approximation by Spencer to
estim ate lim iting current of a coaxial rotating e-beam,
16
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
r
\
\3 /2
V3
1
1
-1
r
'
2 In 5 k-
V rt J
2 In
/w w y>
(2.2.15)
where rwo is the outer tube radius, r«, is the inner tube radius, and rb
is the e-beam ’s radius.
This calculation gives a limiting current of
4.4 kA in the cavity.
S tr iffle r ’s
C a lc u la tio n s
S triffle rs
calculations
emitted from an initia l
are
based
on the
annular
radius, Ri( and passing through a balanced o r
sym m etric magnetic cusp (Bmw* « BfinJ . The lim itin g
calculated by the initial and final e-beam radius.
determ ines the
non-coaxial
conductor,
and pz term s.
system
e-beam
Electron e-beam a
This is the lim itin g
even though
the
because no consideration
theory
current can be
current o f a
mentions
a c e n te r
was given to the change o f
potential energy of the e-beam due to the placing o f a center rod (i.e
nowhere in the equation does the re (center rod radius) appear.
equation fo r the limiting current is given in eq 2.2.16 [STR78].
-l
(2.2.16)
(2.2 .1 7)
I z = 2neRrtsvz = 17.05 * 103v fiz
17
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The
where r-R /R if rw - Rw/R„ p# - p ^ , Rt - in itia l
radius), Ri is the in itia l
e-beam radius (cathode
e-beam radius of 2.25 cm, R is the final e -
beam radius of 3 cm. and R» is the outer conductor wall o f 3.6 cm
radius.
Using the same parameters as above, gives the fo llo w in g
lim iting
currents o f
l(r-0.0255) -
14.3 kA.
This calculation
of lim itin g
current does not apply to the p re se n t
experim ent since the m agnetic cusp is not a balanced cusp.
C o rre a ’s
C alcu latio n s
Another calculation
of lim iting
current
uses the equations
derived by R. A. Correa and J. J. Barroso which are applied to c o a x ia l
and non-coaxial
cavitie s
with
thick
rotating
e-beams
[COR93].
These equations were solved for the coaxial gyrotron using MAPLE.
The equations for the lim iting current are as follows:
G (R " 0 )
L
(2.2.18)
G(Re,A R ') “
(1 - B ) 273
(2.2.19)
Q
e j V t - V Q \n ( R J R w)
(2.2.16)
m0c2y 0 IniRfo / Rw)
(2.2 .1 7)
18
permission of the copyright owner. Further reproduction prohibited without permission.
2
«'
(2 .2 .2 0 )
2Re
r
2R} . (
= l + 2 1 n (ft../« ,)-^ ? -ln
r
A
(2 .2 .21 )
-2-
(2 .2 .22)
Where the parameters used are:
Outer Radius o f conductor, Rw - 0.036 m
Inner Radius o f conductor, Rj„ - 0.003 m
Inner electron beam radius, R, - 0.01 m
outer electron beam radius, R2 - 0.03 m
center radius of electron beam, R. * (R1+R 2)/2 « 0.02 m
electron beam thickness, AR - 0.02 m
electron beam alpha, a - 1
Voltage of inner conductor, Vt - 0
Voltage of outer conductor, V2 « 0
For our param eters, electron beam radii are taken from pictures o f
radiation darkened glass plates, Figure D.11, Appendix D. We fin a lly
get a lim iting current for the coaxial case of:
lco«x - 4.41 kA
(Correa’s coaxial, rotating annular e-beam)
This equation can also be used to find the non-coaxial rotating
beam lim itin g
current.
The lim iting
current occurs at a vo lta g e
depression o f V1 - 1 8 6 kV and is:
Inoncou - 3.51 kA
e-
(non-coaxial, rotating annular e-beam)
19
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Summary of Limiting Currents
For a com plete summary o f the lim iting current calculations
see Table 2.2.3.
The e-beam is not rotating in the diode and pre-cusp
d rift tube so non-rotating e-beam calculations are used there.
The
outer and inner dimensions of the e-beam are re-2 .5 cm and r, - 2.0
cm for before the cusp and ro«3.0 cm and n -1 .0 cm for after the cusp.
The wall radii, rw, used for the calculations are: diode-19.0 cm, pre­
cusp and post-cusp d rift tube-4.9 cm, and cavity 3.6 cm.
ore-cuso d.t. post cuso d.t
(kA)
(kA)
Diode
(kA)
Non-coaxlal cavltv
Pencil e-beam TREI941
Thin annular e-beam ISPE981
Thick annular e-beam ISPE981
Thick annular e-beam IDR0931
Balanced cusp [STR78]
Thick annular e-beam [COR93]
Coaxial cavity
Planar TDR0931
Thin annular e-beam IKRA921
Thick annular e-beam TCOR931
Table 2.2.3
1C
1C
3 .4
ic
1C
tc
tc
Summary of limiting
experiment.
7.3 IC
9.4
8.1
8 .2
2 .5
3 .4
2 .9
3 .0
IC
1.5
0.9
3 .2
5.9
2.1
4.1
1.3
4 .7
14.3
3 .5
1.8
2.9
6.7
4.4
4.4
IC
11.2
10.7
cavity
(kA)
currents in different sections of the gyrotron
For dtode, r„«19.0 cm, re « 2.5 cm, and r, - 2.0 cm. For
pre-cusp drift tube. rw«4.9 cm, r. * 2.5 cm, and r, « 2.0 cm. For post­
cusp drift tube, r.-4 .9 cm, r0 - 3.0 cm, and r, - 1.0 cm. For microwave
cavity, rw«3.6 cm, r0 « 3.0 cm. and r, « 1.0 cm.
20
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2 .3
M agnetic Cusp Dynam ics.
Conventional gyrotrons have used MIG guns to produce s m a ll
orbit beams [KIS74, GAP75].
e-beam is d iffic u lt.
Generating a large orbit axis e n c irc lin g
One w ay to generate a large o rb it beam is by
using a m agnetic cusp fie ld .
gyrotron
devices
u tilizing
D estler has done extensive work on
large
orbit
e-beams
[DES81, DES81b,
DES88, DES89]. A cusp m agnetic field was used to generate an a x is
encircling
e-beam.
The beam has a c ritic a l
velocity
to enter th e
cusp for the e-beam to be transm itted through the cusp. It has been
shown th a t transport through the magnetic cusp can be more than
85% e ffic ie n t
[KAP74].
configurations
to
Others have used different
produce an axis
encircling
electron
large
orbit
gun
beam
[GAL96, SCG91]
The axis encircling beam used in these gyrotron experiments is
generated by a cusp m agnetic
field.
sym m etric cusp, in which the in itia l
The field
(diode) magnetic field
the same as the final (cavity) m agnetic field.
be controlled independently.
m agnetic fie ld
used is a nonis n o t
These two fields can
They are in opposite directions and th e
reversal region of the cusp occurs in about 10 cm.
Soft iron plates in the cusp region combined with
extra m agnetic
field co ils on either side of the cusp decrease the length £ of th e
field reversal transition. The cusp transition
possible
because this
decreases
the
should be as sharp as
amount
of
centering Ar of the beam from Rhee [RHE74],
21
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
coherent
o ff-
(2.3.1)
Ar = —sin 1—i —
2
f? * *
w here
(2.3.2)
The equations are derived for sym m etric cusp, (B i-B f) and where V0
is
in itia l
velocity,
r0 is
in itia l
radius
frequency in the solenoidal region.
and q>6 is the c y c lo tro n
The value of A r is the coherent
off-centering of the e-beam.
It scales with cusp transition
[RHE74].
tangent
The
hyperbolic
function
in
length £
equation
2.3.3
represents the z-component of the magnetic field.
(2.3.3)
The function
distance.
f is the m agnetic field
p rofile
and z is the a x ia l
The function f was curve fit to the experim ental m agnetic
field to determ ine the value of £ to be about 2.5 cm.
Coherent off-centering
is the off-centering
of the center o f
the Larmor orbits from the center line of the experiment.
centering
is dependent on the azim uthal
position
This o f f -
of the e le ctro n
beam entering the cusp as w ell as the m agnetic field profile.
The
end effe ct is that a thin annular electron beam entering a m agnetic
cusp may exit much thicker.
It
is
possible
to
extend
the
theory
to
account
fo r
an
asym m etrical cusp by observing that r\ - 1 is the cutoff condition for
a sym m etric
cusp.
Applying
conservation
of canonical
22
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
angular
momentum to a particle
in an axial
magnetic
fie ld
gives
th e
follow ing equations [REI94].
ym r2d + -^—w = const. •
2n
(2.3.4)
y/ = B r 2Jt‘
(2.3.5)
The boundary conditions require that the in itia l
q
, is zero since the e-beam is in itia lly
angular frequency.
not rotating
and that th e
final o be the cyclotron frequency.
Qx = 0
(2 3 .6 )
e .= -S h .
ym
(2.3.7)
Equations 2.3.4 through 2.3.7 can be combined to give equation 2.3.8.
This equation relates the initial
radius to initial
beam radius and fin a l
and final magnetic fields.
c y c lo tro n
(Note: if coherent o f f -
centering were zero then the final e-beam radius is the same as th e
final cyclotron radius., i.e. no e-beam thickness)
(2.3.8)
Equation 2.3.9
can bederived from
cyclotron equation.
a =
the definition o f a and the
This isa function of final cyclotron
1
radius.
(2.3.9)
Substituting eq 2.3.8 into eq 2.3.9 gives eq 2.3.10. This equation can
be used to predict the e-beam a based on the magnetic fie ld and th e
initial beam energy and radius.
23
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
We can define the cyclotron frequencies to be as follow s,
(2.3.11)
ym
(2.3.12)
a > ,= ^ -
'
ym
Using eq 2.3.11 and 2.3.12, equation 2.3.10 can be rew ritten
in a
more convenient form,
1
a =
r
-i
vo
(2.3.13)
\
-1
r?(QiG)r
where V 0 is the total in itia l
is when a approaches infinity.
velocity.
The e-beam cutoff c o n d itio n
This condition gives a cutoff v e lo c ity
for an asym m etric cusp:
Vcm S r ^ O ) ,® ,
(2.3.14)
We then define i\m for the coherent o ff-ce n te rin g
equation
in eq
2.3.15.
vo
*ln s «
(2.3.15)
COf
Equation 2.3.15 can be used in the coherent off-centering
equation
2.3.1 to extend this solution to a non-symmetric cusp.
The c ritic a l
cutoff
velocity
is plotted
function of p for differe nt diode B-fields.
as
the
m agnetic
field
is
increased,
in Figure 2.3.1 as a
The theory predicts th a t
the
amount
24
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
of
c u rre n t
transported
through the
sm aller fraction
magnetic
o f the e-beam w ill
cusp w ill
decrease because a
have high enough energy to g e t
through the cusp.
Diode
B-Field
0.123 [T]
0.115 [T]
0.107 m
0.083 m
08*
Solenoid B-Field [T]
Figure 2.3.1
Plot of theoretical velocity cutoff of electrons transmitted through a
magnetic cusp as a function of solenoid magnetic field for dtode fields of
0.083, 0.107, 0.115, and 0.123 T.
25
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3. Experimental Configuration
3 .1
G y ro tro n
C o n fig u ra tio n
The experim ental configuration
is depicted in Fig 3.1.1.
The
electron beam is generated by MELBA (Michigan Electron Long Beam
Accelerator) which is a Marx generator [PSI83] w ith
the fo llo w in g
parameters: beam energy of -0.7 to -1.0 MV, 1-10 kA diode c u rre n t,
0.2-2 kA extracted current and pulselength from 0.5-1 ps. MELBA i s
able to reach 1 pS fla t
top due to the reverse charged Abramyan
stage [ABR77]. The e-beam is generated by explosive emission fro m
an annular ring of scratched, bare aluminum on the cathode with
a
radius of 2.25 cm and thickness of 0.5 cm. The beam is generated in
a uniform diode m agnetic field of about 1 kG. The annular e le ctro n
beam is extracted through a 7.6 cm diameter aperture in the graphite
anode plate.
The coaxial rod (OD«6mm) begins a t the anode su rfa ce
and extends down the d rift tube to the m icrowave
output w indow .
This rod is shorted to the vacuum d rift tube which is at the same
ground potential as the anode. A soft iron plate 1.4 cm thick and
solenoidal windings create a magnetic cusp to generate a rotating e beam in which the electrons gyrate around the center conductor in a
large orbit configuration with some coherent o ff-centering
The coherent
o ff-cen te rin g
was
measured
to
give
[RHE74].
an e-beam
thickness of about 2.1 cm. The e-beam a was measured to be about
0.9-1.4 [JAY99], where a is defined as the ratio of the perpendicular
26
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
velocity
to
the
parallel
propagation direction.
ve lo city
with
respect
to
the
e-beam
The m icrowave cavity dim ensions are: length
of 26 cm and outer tube ID o f 7.4 cm. The microwaves are radiated
into a microwave-anechoic tank.
band w aveguides/directional
Microwave power is collected in S-
couplers and transported to a Faraday
cage where diode detector and heterodyne mixer are used to measure
power and tim e-frequency spectrum .
The first o f the three cases examined is the unslotted, co axia l
gyrotron [Fig. 3.1.2a]. The co a xia l gyrotron has two crossed copper
or stainless steel posts at the entrance of the interaction
region
that support the center rod at the upstream end of the cavity.
These
posts are 2 mm in diam eter
microwaves by shorting
and serve to reflect backward tra v e lin g
the upstream end of the
cavity.
They also
keep the center rod at ground potential w ith the outer cavity.
This
lowers the space charge depression of the e-beam and allow s for a
higher lim iting
gyrotrons.
current
At the exit
in the coaxial gyrotron than in standard
of the cavity
there
is a break
in th e
waveguide of about 1 cm and a reflecting washer 1.9 cm in d ia m e te r
by 3 mm thick mounted on the center conductor to give a d d itio n a l
reflection o f microwave power a t the exit of the cavity.
The c a v ity
measures about 26 cm from the beginning of the front shorting posts
to the break a t the back o f the cavity.
Both entrance and e x i t
currents
and
are measured
integrated d ig ita lly.
by Rogowski coils
th e ir
signals
are
The cavity is grounded to the inside of the 4 ”
d rift tube by a metal ring which is positioned between the entrance
Rogowski coil and the e xit Rogowski coil, 5.5 cm from the back o f
the cavity.
By the placement o f the metal ring there, a ll c u rre n t
27
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
striking
the center rod or inside of the cavity is measured by th e
entrance Rogowski coil but not the e xit coil.
The back end of th e
center rod is supported by a bolt in the center of the Lucite o u tp u t
w indow .
Slots in the top and bottom of the ca vity and output w aveguide
are introduced to p re fe re n tia lly
select
the T E „ waveguide mode.
The slots are axial to prevent azmuthal w all current flow ing
top and bottom of the waveguide.
in th e
Ideally the slots w ould have no
breaks in them, but to increase the mechanical strength
structure, some breaks in the slots are provided.
one break in the pair of slots respectively.
of th e
The cavity has only
This produces two s l o t s
on the top and two on the bottom of the cavity as shown in Figure
3.1.2b. The slots in the cavity are 10 cm long by 3 mm wide and have
a 1.7 cm axial space between them.
The output waveguide (also in
the vacuum d rift tube) has two breaks in the upper and low er s lo t s
respectively.
This produces three slots on the top of the wave guide
and three slots on the bottom.
The output waveguide is 58 cm long
and the slots are 13 cm long and also have a 1.7 cm axial space
between them .
These slots ensure that the polarization generated by
this gyrotron is the same as is detected.
The center conductor is
supported in the same way as in the coaxial, unslotted gyrotron.
slotted
cavity
The
also has the same 1 cm break between it and the
output waveguide and has the same re fle cto r
washer as described
above.
The non-coaxial cavity (Fig. 3.1.2c) s till
bars
at
the
front
of
the
cavity
to
has the copper cross
prevent
transm ission
of
m icrowaves out tow ard the diode, but there is no center conductor.
28
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Current is measured by Rogowski coils at the entrance and exit o f
the cavity.
Again, there is a break at the e xit
provide additional reflections
of the cavity
to
but in this case there is no r e fle c to r
w asher.
29
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B(kG) r
To Heterodyne
Mixer and
1 •
Crystal
Detectors
-1
W
o
CUSP
Graphite
Anode Plate
Maser Solenoidal Coils
Center
Rod
Cathode
Stalk
r» 2.25+/-025 cm
u
r
Interaction
.Region
Output Wtvefrtde
/
Entrance
Exit
Dumping megnets
Rogovski Coil Rogovaki
Output
Coil
Windov
S-Band RectangularWaveguide
Diode Magnetic
Field Coils
Figure 3.1.1
Experimental setup for gyrotronoscillator vith magnetic fiekl plot.
26 cm
d «7.4 cm
^aj
10 cm
(b)
Figure 3.1.2
Three cavities used for gyrotron oscillator:
a) Unslotted coaxial cavity,
b) Slotted coaxial cavity,
c) Unslotted cavity with no center conductor.
3.2
a
M easurem ent Experim ental
The electron
Configuration
beam is produced by MELBA as described above.
For the m easurement of a , the 7.6 cm anode aperture
with
an 8-pinhole
sym m etrically
graphite
anode [JAY97].
placed at a radius
diam eters are 1 mm.
The
is replaced
pinholes
of 2.25 cm and the
are
pinhole
The apertured graphite plate is 3 mm th ic k .
31
Reproduced with permission o f the copyright owner. Further re p ro d u c tio n prohibited without permission.
The 8 electron beam lets transverse down the stainless steel d r i f t
tube and through a m agnetic cusp field.
the axis encircling
The m agnetic cusp cre a te s
beam with an a ranging from 0.9 to 1.4 [JAY99],
for a diode B-field o f -1.1 kG and a maser solenoid B-field of 1.5 kG.
The cusp has a com plete
length.
For the
e-beam
field
a
reversal
over about 10 cm a x ia l
measurements,
the
normal
interaction cavity is replaced with a glass w itness plate.
copper
The g lass
plate was positioned about 60 cm downstream o f the anode a p e rtu re
which
is
15 cm downstream
interaction
cavity
would
be.
from
where
Since the
the
entrance
current
of th e
is so low
(~1
A/pinhole) the e-beam must be fired 20 tim es at the same B - f i e ld s
to appreciably darken the glass plates.
(anneal) with
tim e,
Since the glass plates fade
photographs of the patterns
are taken w ith
a
Polaroid camera d ire c tly follow ing the experim ental run. These are
then scanned into a personal computer for processing.
3 .3
Magnetic Cusp and Magnetic Fields
The rotating e-beam is generated by a m agnetic cusp.
The cusp
is composed of two separately controlled fie ld s, the diode fie ld and
the solenoidal field.
The diode field is a uniform field generated by
five large pancake magnets (52 cm ID, 84 cm OD, 4 cm thick) w i t h
equal spacing between them to give a total length of 21.8 cm.
The
diode magnetic coils generate 47 Gauss/Amp [HOC98] at the cathode
and are spaced 6 cm upstream from the so ft iron plate form ing th e
m agnetic field
reversals.
The solenoidal
fie ld
32
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
is generated
by
copper enameled magnet w ire wound around an 84 cm long, 10 cm
diam eter vacuum d rift tube. The windings have about 4.7 tu rn s /c m
and there are 2 layers to produce 11.6 Gauss/Amp in the uniform
cavity region [HOC98]. The electrons are accelerated across an A -K
gap of about 10.7 cm, then enter the solenoid d rift tube. There is a
field
reversal
in the magnetic field
from the cathode.
rotate
reversal the electron beam starts to
in a large orbit fashion.
centering
fields.
At the field
are determined
about 25.5 cm down stream
The exiting
beam's a and o f f -
by the diode and solenoidal
m agnetic
A soft iron plate a t 25.5 cm down stream from the cathode is
used to decrease the axial distance
o f the field
shorter the fie ld transition, the less off-centering
reversal.
The
the e-beam w i l l
have upon exiting the cusp [RHE74]. About 15 cm down stream fr o m
the m agnetic cusp the e-beam enters the microwave cavity.
There are two types of o ff-cen te rin g,
off-centering.
asym m etries
Incoherent
o ff-cen te rin g
in the magnetic fie ld .
corrected by winding axi-sym m etric
coherent off-centering
coherent and incoherent
is
to
a zim uthal
This type of off-centering
magnetic field
is an o ff-centering
windings.
is
The
of the Larmor orbits due
to the electrons entering the cusp at d iffere nt
Coherent off-centering
due
azim uthal p o sitio n s.
is undesirable since it increases the beam
thickness and thus leads to addition
beam scraping.
This type o f
off-centering can not be avoided (with this type of cusp gun). It can
be decreased by m inim izing
field reversal.
the tran sition s
length of the m agnetic
There is a lim it to how sharp this tran sition
which proportional to the diam eter of the field coil windings.
things were done to decrease this transition
length.
33
permission of the copyright owner. Further reproduction prohibited without permission.
can be
Two
First, a s o f t
iron piate was added to the cusp to intensify
the cusp region.
fit
tig h tly
radially.
the radial B -field
in
The iron plate consists of three iron sheets w h ic h
around
the vacuum
In additional to the
d rift
tube and extend out 45 cm
iron plates, extra field coils
wound on either side of the cusp.
w e re
These extra windings extend 3 cm
axially on either side of the magnetic cusp and serve to sharpen th e
cusp transition
regions.
This set of w indings is connected to th e
main solenoidal windings by jum per wires th at go around the o u ts id e
o f the iron
plates. This allow s
d rift
and it
tube
the plates to fit tig h tly against th e
also prevents
the fie ld s
from
the w ire
fro m
interfering with the critical cusp magnetic fields.
The
m agnetic
fields
connected to a digital scope.
various positions
were
measured
with
a Hall
probe
The magnetic fie ld s were measured a t
in the cavity and at d iffe re n t
charging vo lta g e s.
Magnetic fields were measured separately and later added to g e th e r
by superposition to get the total field.
measured with
Some te st fields were a lso
both diode and solenoidal
superposition of the fields.
fie ld s
pulsed to v e r ify
Figure 3.3.1 shows Bz as a function
axial position (measured by an axial Hall probe).
34
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
of
1.5
■ ■
1.0
;
0.5
i
o 0.0
f
: Plate
b 1 4 .4 *s 2 4
«
b lS « s l 8
b 1S +s24
a
B
T
b i4 .4 4 .s ia
a
•
i£ -0.5
0
•a
1.0
-
m ia o w we cavity
1
-
2.0
-2.5
Figure 3.3.1
a
a
2 -1.5
'fwmlf***1t « * N «
15
25
35
45
Distance from Anode (cm)
«
55
•
65
Axial magnetic fields vs. axial distance for four different cfiode and
solenoidal field combinations.
The radial magnetic fie ld s were also measured using a ra d ia l
probe in the cusp region.
Measurements as a function of both th e ta
and axial distance were taken.
35
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
DO
Oo
5 -10
-15
-15
10
5
-10
5
0
10*Bfield*sin(angular position) [kG]
15
a «*■
0 4S*»
4 •o*m
¥
m
* t> ii«
• <70 *a
oi
l -
a 11S««
ah
•a
-8
1U
12
-i_
14
_i_
16
P
18
20
Distance from Anode Plate
(b)
Figure 3.3.2
Radial magnetic field as a function of:
(a) theta, (b) axial distance
36
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 3.3.2 (a) shows a p lo t o f radial
theta.
These data shows that the fields
azimuthal direction.
m agnetic fields
are very uniform
vs.
in the
Figure 3.3.2 (b) shows the profile of the ra d ia l
magnetic field vs. axial distance.
The equations for the scaling
law of the diode and c a v ity
magnetic fields are shown in eq 3.3.3 for the cavity and eq 3.3.4 fo r
the diode, where X is the diode charging voltage in volts and Z is th e
solenoidal charging voltage in volts.
Cavity magnetic field:
B[kG] - -0.0800037 + 0.01057*Z - 7.9009e-5*X
(3.3.3)
Diode magnetic field:
B[kG] - -0.06199161 - 1.9421e-4*Z - 6.7201 e-4*X
3 .4
C oaxial/N on-C oaxial
The coaxial cavity
encircling
[HOC98J.
Structure.
was designed for
beam previously
section gyrotron
C avity
utilized
(3.3.4)
for
the large
the
The cross-section
orbit,
rectangular
a xis
c ro s s -
of the e-beam w as
measured on a glass witness plate and inner and outer diam eters o f
the coaxial cavity were determined by the size of the e-beam fo r
operation in the S-band. See Appendix D, Fig D.11, 3-slotted
glass
p la te .
The dimensions of the cavity are as follow s.
outer cavity is
The ID of th e
7.4 cm, the OD of the center rod is 6 mm and th e
37
permission of the copyright owner. Further reproduction prohibited without permission.
ca vity length from the front shorting posts to the rear break in th e
waveguide is about 26 cm. The ca lculation
of the waveguide modes
can be determined by equation 3.4.1 [POZ90],
Jn ( K a )
( k cb )
1 -0
( 3 .4 . 1 )
rn'(kca)
•'/(* < * )
where J n* is the derivative of the n,h order J Bessel function, Y„’ is
the derivative of the nth order Y Bessel function, ke is the propagation
num ber in the waveguide direction,
a is the inner conductor ra d iu s
of 0.003 m and b is the outer conductor radius of 0.037 m.
equation must be solved num erically
T his
to give the TE™ w aveguide
c u to ff frequencies shown in Table 3.4.1, where n is the radial mode
num ber and m is the azimuthal mode number.
T E nm (GHz)
Coaxial
CM
N
1c
CO
II
c
Table 3.4.1
E
o
II
c
n=1
m *1
5.0 4
2 .3 4
3.9 4
5.4 2
9.3 3
6.7
8.64
10 .34
Calculated TE^, modes of the coaxial microwave waveguide
The gyrotron
should
not couple
to
electrons bunch in the theta direction.
completeness.
the
TM modes because th e
These modes are included fo r
The equation that gives the cutoff frequencies of th e
TM modes is
38
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
J n ( k ca )
r.(tcb)
K { k cb)
Yn(kca)
0
(3 .4 .2 )
where J„ is the nlh order J Bessel function
Bessel function. Table 3.4.2 has a listing
and Y„ is the n1" order Y
of cu to ff frequencies fo r
the TM modes.
T M nm
(GHz)
Table 3.4.2
Coaxial
m =1
m =2
n=0
n=1
4 .1 6
5 .0 4
8 .6 4
n-2
6 .6 3
1 0 .8 8
9 .3 4
Calculated TM ^ modes of the coaxial microwave waveguide
This coaxial gyrotron
is known to
frequency range from experim ent.
operates
in the S-band
The only coaxial TE mode th a t
could couple to the beam mode in S-band is the TEt1 at 2.34 GHz.
Other modes at frequencies
between the T E „ and the next h ig h e r
waveguide mode, the TE21, were observed in experim ent.
These
modes can be calculated by taking the axial component of a re so n a n t
cavity into account.
The axial
part
of the cavity
transm ission line with reflection
short circuit/open
c irc u it
modes can be modeled as a
at the ends (see Appendix H). The
(sc/oc) transm ission
line
most c lo s e ly
matched the resonant frequencies observed and these values are a lso
compared to the short c irc u it/s h o rt
transm ission
line.
c irc u it
(sc/sc)
model of th e
The cavity should behave more like the (s c /o c )
39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
case becausethe cavity is shorted at the front by the shorting rods
and it is an open c irc u it
on the back side. Now the cavity resonant
frequencies including the axial mode components can be c a lc u la te d
from equation 3.4.3.
/
Jum p
<3-4 -3>
= J f p2 + Jf tu2n
The first five
calculated axial modes for (sc/oc) are listed in Table
3.4.3.
short
circuit/open
c irc u it
TE111
TE112
TE113
TE114
TE115
Table 3.4.3
(S.C./O .C.)
(GHz)
2 .3 5
2 .4 9
2 .7 4
3 .0 9
3 .4 9
Calculated short circuit/open circuit axial mode frequencies for coaxial
cavity
For completeness calculations
of the sc/sc axial mode fre q u e n cie s
are included in Table 3.4.4.
short
circuit/short
TE111
TE112
TE113
TE114
Table 3.4.4
circu it
(S.C ./S .C .)
(GHz)
2.41
2.61
2.91
3 .2 8
Calculated short circuit/short circuit axial modes for coaxial cavity
40
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The waveguide
modes fo r
the
non-coaxial
ca vity
can be
calculated using equation 3.4.4 [BAL89],
fiun
(3.4.4)
ln n -J jle
where a is the radius of the waveguide and
zero of the derivative
Results
of
represents the mlh
of the Bessel function
calculations
for
some
of
the
Jm o f the order n.
waveguide
c u to ff
frequencies are shown in Table 3.4.5.
TE nm
(GHz) Non-coaxial
m=1
4.94
2.37
3.94
n=0
n=1
n=2
Table 3.4.5
Calculated TE,*, mode frequencies
m=Z
9.04
6.87
8.64
for
the
non-coaxial
microwave
waveguide
In Table 3.4.6, axial
modes for the non-coaxial
cavity
are
included using the short circuit / open circuit approxim ation.
short
circuit/open
circuit
(GHz)
2 .3 9
2 .5 2
2 .7 7
3 .1 2
TE111
TE112
TE113
TE114
Table 3.4.6
(S.C ./O .C .)
Calculated short circuit/short circuit axial mode frequencies for non­
coaxial cavity
41
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4. Experimental Diagnostics
4 .1
Glass
Plats e-beam
Diagnostic
A typical glass w itness plate used to measure a is shown i n
Fig 4.1.1. The beam lets create eight circles of radiation darkening.
By measuring the Larmor radius o f the circles
and knowing
the
interaction B -field it is possible to determ ine the beam a using th e
equations in the fo llo w in g
paragraphs
[CH092].
creates an axis encircling beam, the circles overlap.
centering
Since the cusp
Coherent o f f -
through the magnetic cusp causes the circles
to
have
different center points depending upon the spatial position at w hich
the beamlets enter the cusp [RHE74].
In order to analyze the radiation
darkened pattern,
one can
compute a using equation 4.1.1:
a -
1
(4.1.1)
Here y and p are the usual re la tiv is tic
factors,
m is th e
electron rest mass, c is the speed of light, R is the Larm or radius, q
is the electron charge, B is the magnetic field.
range from 1.5 cm to 2.1 cm.
The measured radii R
This gives a measured a distribution of
42
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0.9 to 1.4.
The measured off-centering
o f Larmor orbits
gives an
increase in e-beam thickness of 2.1 cm.
Results of the com puter simulation (as described in appendices
A and B) o f the e-beam tra je cto rie s
are shown in Figure. 4.1.2. The
eight circle s were generated by superimposing eight s in g le -p in h o le
particle sim ulations with a 45° rotation about the axis.
is
in good agreement w ith
the experim ental
This p a tte rn
radiation
pattern shown in Figure 4.1.1. This program num erically
the
re la tiv is tic
particle
equations
of motion
is started at a position
o f a single
corresponding
darkened
in te g ra te s
particle.
to the anode and
stopped at an axial distance corresponding to the position
glass w itness plate.
then recorded.
p articles
The X-Y coordinates and the particles
w ith
of th e
a ‘s are
This procedure is repeated for 1000 particles.
have the same in itia l
(cylin d rica l coordinates).
direction
The
in velocity
experim ental
radius and theta
Each particle
position
is given a d iffe re n t
A ll
at z«0
in it ia l
such that v/c is set at 0.914 to correspond
param eters.
The
determ ines the final position and alpha.
particles
In itia l
in itia l
angle
angles are chosen
random ly to be less than 2.5° with respect to the center line a xis.
This
spread
in
in itia l
angles
gives
good
agreement
w ith
experim entally observed thickness of the radiation darkened p a tte rn .
Sim ulation a ‘s of the 1000 particles are histogram -plotted in Figure
4.1.3.
Most of the particles fa ll with in the a range of 1.0 and 1.5,
which agrees well w ith
the experim entally
measured a [Fig. 4.1.3].
The sim ulated radii are between 1.55 cm and 2.33 cm.
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The simulated
off-centering
also gives an increase in e-beam thickness of about
2.1 cm.
These sim ulations neglected space charge effects and e le c tric
fields. The equations of motion are derived from the single p a rtic le
re la tiv is tic
Lagrangian in cylind rica l
round off errors
necessary
to
in solving
introduce
the
coordinates.
these equations
follow ing
To avoid large
num erically,
dim ensionless
it
w as
va ria b le s,
denoted with a bar,
t = tco
,
c
Z = z /l
»
p = p /l
>
I = —
m
(4.1.2)
c
where t is tim e, z is axial distance, p is radial distance, eo is th e
re la tiv is tic
scale.
cyclotron frequency, and / is the ch a ra cte ristic
One obtains
a set of coupled d iffe re n tia l
length
equations
in
dim ensionless form :
e = _ 2 p + £ / (?/)+ ±7'(*/)*
P
P
2
t
(4.1.3)
(4 . 1 .4 )
P = p 0 2 -e p h z l)
F = - I p 20 f'( z /)
2H
,
(4.1.5)
These equations were solved num erically
order Runge-Kutta algorithm
with
variable
using a fo u r th - fifth
step size.
For more
details see Appendix A.
The sim ulated a of 1.0 to 1.5 compares well with the measured
a of 0.9 to 1.4 from radiation darkened Larmor radius measurements.
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 4.1.1: Experimental Glass Witness Plate
Figure 4.1.2: Simulation Results
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
120
100
»
v
80
o
60
20
O ' ------ —
0.0
1—
0.5
—
1.0
1.5
2.0
2.5
3.0
3.5
a
Figure 4.1.3
Histogram plot of single particle simulation of number of particles vs. e beam a.
4.2
Electron
Beam
The aperture,
Current Diagnostic
entrance
and exit
electron
beam
diagnostics measure the e-beam current using calibrated
coils.
c u rre n ts
Rogowski
The co ils are wound to pick up the theta component of th e
magnetic field generated by the pulsed e-beam. They give a voltage
proportional to the tim e changing flux through the Rogowski c o ils
[GRI89], where the flux O is proportional to the e-beam current.
(4.2.1)
46
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The signal is then integrated to give a current trace.
traces were integrated
problem with
using RC integrating
using an RC integration
In itia lly
circu its.
circuit.
these
There is a
A rapidly
varying
component of the signal can cause unphysical spikes in the c u rre n t
measurement.
v.
v in
*
^
This is demonstrated by the following [KN 089],
+ vo u ,t
(4.2.2)
where V*,* is the voltage out of the RC integrator,
V * is the signal
into the RC integrator, and x is the integration tim e constant.
If x i s
too large, the high frequency components of the signal are h ig h ly
attenuated since the RC circu it acts like a low pass filte r.
If x i s
too small, unphysical spikes in the current signal can occur since
Vout becomes comparable in size to the x(dV0lM/dt) term .
The more recent method of integrating
the Rogowski sig n a ls
involves direct sampling of the current signals and the num erical
integration
of
the
signals.
This
method
does
elim inate
the
unphysical spikes in the current resulting from the RC in te g ra tio n
methods. There are some lim itations
signals.
w ill
to num erically integrating the
If there is an average off-set, this must be subtracted or i t
cause the integrated signal to slope up or down.
Also due to
numerical errors, electrical noise, and frequency dependent response
of the detector, the signals w ill
not always go to a zero baseline
after the end of the pulse.
The aperture current Rogowski coil is at the entrance to the
d rift tube just downstream from the anode plate.
is housed in a stainless steel adapter flange.
The Rogowski c o il
The calibration
this coil is 3904 [A/V] using RC integrator # 3 w ith
47
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
fo r
an in te g ra tio n
time, x o f 20 pS.
For the num erical
integration,
the conversion
factor is 3.322 [A/V-nS], taking in to account the 2 ns AT sam pling
rate.
The entrance current Rogowski coil is ju s t
upstream to th e
entrance to the microwave cavity on the front side.
It has about 140
turns at 0.5 cm in diameter.
It is shielded on the front,
bottom from the e-beam by a metal ring.
top and
Note the ring does n o t
make a complete current path in the direction of the current loops o r
the Rogowski signal would be distorted.
The conversion factor fo r
this coil is 836 [A/V] using RC integrator # 8 w ith
time, x of 9.3 pS.
The numerical integration
an in te g ra tio n
conversion factor
is
0.2383 [A /V -ns].
The exit current Rogowski coil is 4-5 cm upstream of the re a r
break in the microwave cavity.
diameter.
It has about 35 turns at 0.5 cm
It is located just behind the brass ring which shorts th e
cavity to the d rift tube. The conversion factor fo r this coil is 9451
[A/V] using RC integrator # 21 with an integration
tim e, t o f 10 pS.
The numerical conversion factor is 1.441 [A/V-ns].
The signals
from
entrance
and e xit
transported to the back of the vacuum d rift
cables.
Rogowski
coils
are
tube via coaxial BNC
These cables are then fed through a vacuum feedthrough.
Then they are transported
to the screen room where they can be
measured.
48
permission of the copyright owner. Further reproduction prohibited without permission.
4.3
E lectron Beam a
D ia g n o s tic
The a measurements were made by using the set up as shown
in Figure 4.3.1.
A graphite apertured
anode with
eight holes w as
used to form eight beamlets. The holes w ere placed a radial d istan ce
of 2.25 cm from the center line o f the experim ent.
The pinholes
were 1 mm in diam eter and the graphite plate was 3 mm thick.
electron
beam then transversed
microwave
experim ent,
cavity was removed.
with
the d rift
the
The
tube as it would in a
exception
that
the
m icrow ave
The magnetic cusp starts the e-beam ro ta tin g .
A glass plate was placed in the d rift tube (about 15 cm dow nstream
of where the entrance of the cavity w ould be). Radiation darkening
of
the
glass
plate
was
used to
record
the
spatial
distribution of the beam.
e-beamlet
W
anode
glass
plate
Eight 1mm
pinhole apertures
Figure 4.3.1 Setup for a measurement experiment.
49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
e le ctro n
These radiation darkened patterns show the cross-section
the Larmor o rb its
patterns
w ill
[CH092].
fade with
Pictures
tim e.
must be taken since these
About 20
(repeated)
necessary to produce a dark enough pattern to discern.
such as the Larmor radii and o ff-cen te rin g
measured from these patterns.
of
shots
are
In fo rm a tio n
of the orbits, Ar, can be
(See Appendix D fo r the series o f
pictures of glass plates taken and their respective
Larmor o rb its .)
There is some spread in the size of the Larmor radii in the azim uthal
direction and also a spread in the off-centering
where maximum and
minimum measurements of respective values are plotted in Figures
4.3.2 and 4.3.4.
Some of the orbits are not com pletely circular.
is because d iffe re n t
parts of the radiation
darkened circles
produced by d iffe re n t
electrons which have s lig h tly
T his
are
d iffe re n t fin a l
values of a and Larmor radii. These differences are due to d iffe re n t
initial angles and different trajectories of the electrons through th e
magnetic fie ld s.
circle,
this
Even though there is some variation from an e xa ct
diagnostic
s till
can
be used to
get
measurement o f a. Using eq 4.3.1 and knowing the in itia l
an accurate
energy and
Larmor orbits, one can calculate a .
1
(4.3.1)
The data in Appendix D are compared to theory based on conservation
of canonical angular momentum (equation 4.3.2),
(4.3.2)
50
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where R<, the in itia l
radius of the e-beam a t emission, is 2.25 cm.
This theory is compared to experim ental data of measured Larm or
radii as a function of solenoidal magnetic fie ld in Figure 4.3.2. Two
different Larmor radii are measured correspond to the maximum and
minimum
possible
radii.
The variation
in
radii
are due to th e
radiation darkened circles having some thickness and some v a ria tio n
in size and shape. Two d ifferent diode fie ld s
are plotted.
The experim ental
of 0.11 T and 0.12 T
size of the Larmor orbits
are in
reasonable agreem ent with the theory.
0.028
u
ft m u Ek.11
B
rt mtn Bb.1 1
O itmvk.12
0.024
#
flm in B s.12
0.020
3 0.016
cc
B-t
§ 0.012
Diodte0.11 T
0.008
0.004
0 ooo*- ‘ 1 * i ■ i. ■ .4 « .i ■ 1 - » ■ ..................
0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27 0.29 0.31
Solenoid B-ReldT
Figure 4.3.2 Comparison of theoretically derived and experimentally measured Larmor
radius vs. magnetic field for glass plates.
Solid curves obtained from
theory.
51
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Next we compare the theoretical
a to the
e x p e rim e n ta lly
measured a (see Figure 4.3.3). The equation used to calculate a is eq
4.3.3 was derived previously in section 2.3.
1
a -
where
V0 is
(4.3.3)
the
in itia l
velocity
(V 0-p *c )
calculated
from
th e
accelerating voltage, o»i is the relativistic cyclotron frequency at the
diode, (Of is the relativistic cyclotron frequency in the solenoid d r i f t
region and r, is equal to 0.0225 (m), the radius of the e le c tro n
beam lets
transm itted
through
the
pinhole
aperture.
The
experim ental values of a were calculated from the measured Larm or
radii and using equation 4.3.1. The data show some spread betw een
the maximum and minimum values of a. Agreem ent is good at th e
lower
magnetic
fie ld s.
values of a, a starts
compare the two
Near e-beam
re fle c tio n /c u to ff,
to increase rapidly
since small experim ental
at
high
making it d iffic u lt
errors
lead to
to
la rg e
calculated errors in a. The 0.295 T data point had a radii such th a t
the alpha is between 2 and infinity.
52
permission of the copyright owner. Further reproduction prohibited without permission.
0iod»«0.12T
2.5
/
/
<►
2.0
*
Soda. 0.11 T
1.5
a
1.0
\
.
0.5
□
■
dpha mm bO .11
alpha min EW0.11 •
O alpha max B»0.12
• ripha min IW0.12
0.0
0 11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27 0.29 0.31
Solenoid B-Reld T
Figure 4.3.3
Plot of e-Beam a vs. magnetic field for glass plates.
To compare
theory
to
experim ent
for
the
centering o f the electron beam, we can substitute
coherent
the solution
o fffo r
the nonsym m etric cusp (eq 2.3.15) into Rhee and D estler’s [RHE74]
equation (eq 2.3.1).
We get the coherent off-centering equation fo r a
non-sym m etric cusp (equation 4.3.4),
r
Ar = —sin 1
2
\Y o
(4.3.4)
O)
y
where A r is defined as the radial off-centering
guiding centers.
magnetic fie ld .
o f the Larmor o r b it
Figure 4.3.4 shows a plot o f A r vs. so le n o id a l
In this plo t the theory is in reasonable agreem ent
w ith the experim ent, except very near the e-beam cu to ff c o n d itio n
(above 0.23 [T]).
This is expected since the assum ptions used by
53
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Rhee and Oestler to derive equation 2.3.1 are no longer valid (such as
r fr 0(Oc « 1 and r s r0)-
0.020
0.018
0.12 T
0.016
0.014
^
0012
^w
0.010
0.11 T
<3 0.008
0.006
0.004
0.002
a
D ftn in h 0 .1 1
■
0 R lM K h 0 .1 1
O
0 fM n h 0 .1 2
•
OR i m k h 0 .1 2
0.000
0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27 0.29 0.31
Solenoid B-Reld T
Figure 4.3.4
Comparison of theory and experiment for Ar vs. solenoid magnetic field
for glass plates.
The final param eters that can be interpreted
from the glass
plates are the outer and inner radii o f the annular e-beam.
This is o f
interest for e-beam transport through a microwave cavity.
Figure
4.3.5 is a plot of the measured outer e-beam radius as a function o f
solenoidal magnetic field.
outer
radius
decreases
This plot shows that the e-beam’s fin a l
as the
m agnetic
field
increases.
The
decrease in outer beam radius is due to the decrease in Larm or
radius as seen in eq 4.3.2. The outer e-beam
radius
is
not as
sensitive to changes in the magnetic fie ld (as inner) because as th e
54
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Larmor radii decrease, the off-centering,
effects
p a rtia lly
Ar, increases.
These tw o
cancel each other but the Larmor radius has a
stronger dependence, so the e-beam outer radius decreases o v e ra ll
with
increasing
magnetic fie ld .
This means the e-beam is le ss
likely to intersect the outer cavity wall a t higher magnetic fie ld s .
However e-beam scraping on the inner rod would increase at h igher
m agnetic fields.
0.030
0.025
2
0.020
§
0 0.015
CO
1
0.010
0.005
□
Dtato-0.11 T
• Died^0.12T
0.000
0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27 0.29 0.31
Solenoid B-Field T
Figure 4.3.5
Measured outer e-beam radius vs. solenoidal magnetic field for glass plate
measurements.
Figure 4.3.6 shows a plot of the inner radius of the annular e beam as a function
of solenoidal magnetic field.
Notice that th e
inner radius decreases as the magnetic field increases.
Since the e -
beam is axis encircling, both a decrease in the Larmor orbits
increase of off-centering
a t higher m agnetic fie ld s
55
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and
decrease th e
inner e-beam diam eter.
This im plies
that higher m agnetic fie ld s
w ill cause e-beam disruption by scraping on the center conductor.
0.010
O
•
0.009
Had* * 0.11 T
Ood»-0.12T
0.008
0.007
■3 0.006
0.005
0.004
0.003
0.11 0.13 0.15 0.17 0.19 0.21 0.23 0.25 0.27 0.29 0.31
Solenoidal B-Field T
Figure 4.3.6
4.4
Measured inner e-beam radius vs. solenoidal magnetic field.
M ixer
Diagnostics
The heterodyne m ixer diagnostic has been a valuable tool fo r
determ ining the frequency of the operation of the gyrotron.
from
the
d is trib u tio n
mixer
can
be analyzed
by
a
reduced
(RID) program developed by W. W illiam s
The data
in te rfe re n c e
[JEC92].
T his
gives tim e frequency analysis of the microwave signals.
The mixer used was a ZEM-4300MH with param eters of: lo c a l
o s c illa to r
(LO) and
radio
frequency
(RF)
interm ediate frequency (IF) of DC-1000 MHz.
of
300-4300
MHz,
This m ixer was given
56
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
a cw rf signal from a m icrowave source (HP8350B Sweep O s c illa to r
and HP83590A RF Plug In, 2-20 GHz) with a frequency between 2.0 to
4.0 GHz and a power lever of 13 dBm fo r the LO (Local O s c illa to r)
channel of the mixer.
The LO frequencies were varied depending on
the mode o f operation to keep the IF (interm ediate frequency) sig n a l
in the 500 MHz range of the d ig ita l sampling scope.
The LO s ig n a l
was run through an octave iso la to r (UTE M icrowave Inc. C T -3 0 4 0 -0 T
2-4
GHz isola to r)
power.
to save the source from
reflected
The LO and RF signals were transm itted
RG142 coaxial
cable at 75 Ohms.
The signal
m icrow ave
through M 1 7 /6 0 to
the RF (ra d io
frequency) channel of the m ixer was supplied by the experiment.
microwaves
were generated by the coaxial gyrotron,
down the output wave guide in vacuum and radiated
(Plexiglas)
microwave
window.
The m icrowaves
attenuated by 30dB in the anechoic chamber,
band microwave waveguide.
tra n sp o rte d
out the end
were
fu rth e r
then coupled into an S -
The signal is then converted to RG214/U
(Type N) coaxial cable and transm itted
room.
The
to the Faraday cage screen
Before the signal is sent to the RF channel o f the mixer, it is
attenuated another 30 dB by HP coaxial attenuators.
The IF signal i s
brought from the mixer through a BMC cable to the DSA602, a fa s t
dig itizin g
scope with a typical
sample rate o f 1 G iga-sam ple/sec.
After the signal is recorded, it can be digitally analyzed.
57
permission of the copyright owner. Further reproduction prohibited without permission.
5. Experimental Gyrotron Results
5 .1
MELBA Experimental Shot Signals
Figure 5.1.1 is a ty p ic a l set of signals from a MELBA shot fo r
the coaxial gyrotron using a diode m agnetic field
of 0.9 kG and a
solenoid m agnetic field of 1.94 kG. Signal (a) is the voltage in the
diode. The anode aperture current, shown in (b), is integrated using
an RC integrator.
Trace
(c)
is
the
cavity
entrance
current.
Microwave power is plotted in trace (d) with a peak power of 25 MW.
The heterodyne mixer signal is shown in trace (e).
58
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
'(a)
(c)
500
600
700
Figure 5.1.1
800
900
1000
TIM E
1100
1200
1300
1400
1500
MELBA experimental shot, (a) dfode voltage (620kV/div),
aperture current (2190 A/div),
div),
(b)
anode
(c) cavity entrance current (880 A /
(d) microwave power signal (25
MW peak),
(e)
heterodyne
mixer signal.
5.2
Slotted
A perture,
The firs t
gyrotron
Coaxial C avity
Gyrotron
experim ents were run with
an apertured
anode lim iting the am ount of current available to the d rift tube.
The
aperture has 3 slots in a graphite plate w ith inner radius of 2.0 cm
and outer radius of 2.4 cm. Only allow ing a small range o f radii o f
transm itted
e-beam to enter the m agnetic cusp decreases the fin a l
spread in a.
59
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A glass w itness
plate was taken w ith one shot o f the w hole
beam through the three slotted
aperture.
The radiation
darkened
pattern is shown in Appendix D, Figure D.11 (shot # 6287).
The gyrotron microwave power generated is up to 8 MW and has
a broad resonance of 1.8 kG to 2.2 kG.
Figure 5.2.1 is a plot o f
measured m icrowave power vs. solenoidal magnetic field.
a>
>
0u
O
1
O
1.5
1.7
1.9
2.1
2.3
Solenoidal B-Field (kG)
Figure 5.2.1: Measured microwave power vs. solenoid magnetic field for the coaxial
gyrotron with the 3-slotted aperture plate.
An im portant
parameter
in any microwave
electronic e fficie n cy, t v
P eak
M icrow ave
e —beam
Power
Pow er
60
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
device
is
the
The electronic efficien cy
Figure 5.2.2.
versus magnetic fie ld
is plotted
in
The peak electronic efficiency, i\m, is about 13%.
16
14
12
*
S
u
*
S
10
8
6
4
2
0
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
2.7
Solenoidal B-Field (kG)
Figure 5.2.2: Experimental electronic efficiency vs. solenoid magnetic field for coaxial
gyrotron with 3*slotted aperture plate
The frequency of the gyrotron microwave signal is plotted
in
Figure 5.2.3. As the magnetic field is increased in the solenoid, th e
operating frequency increases, as expected for the gyrotron.
This is
a sum of all the data taken with diode magnet coils set to 0.99, 1.07,
1.55 and 1.23 kG.
61
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.1
1.3
13
1.7
1.9
2.1
2.3
2.5
2.7
Solenoidal B-Field (kG)
Figure 5.2.3: Measured microwave frequency vs. solenoidal magnetic field for coaxial
gyrotron with 3-sk>tted aperture
Current entering
Rogowski coil
between
the cavity
(Fig 5.2.4).
50*250
A, which
was monitored
The figure
by an entrance
shows entrance c u rre n ts
means space charge effects
im portant.
62
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
are
not
300
275
250
^ 225
r
i
u
0
g
200
175
150
125
■■ ■
1 100
75
50
25
0
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
2.7
Solenoidal B-Field (IcGauss)
Figure 5.2.4: Experimental cavity entrance current vs. solenoidal magnetic field for
coaxial gyrotron with 3-sk>tted aperture
5 .3
Microwave
Pow er M easurem ents:
Open Aperture Gyrotron
The next set o f gyrotron data were taken with three d iffe r e n t
types of gyrotron
slotted
coaxial,
cavities:
(1) the
non-slotted
and (3) the non-slotted
coaxial,
non-coaxial.
(2) th e
An open
aperture of diam eter 7.6 cm made o f POCO graphite was used as th e
anode plate at the entrance to the d rift
aperture allows
tube.
The large
open
a much higher current through the aperture
and,
since it allows all radii through, the spread in a is much larger than
the slotted aperture.
63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Microwave power as a function o f solenoidal magnetic field for
the unslotted, coaxial gyrotron
is shown in Figure 5.3.1.
The TE112
mode peak power of 50 MW is generated a t a m agnetic field of about
1.9 kG.
The power of 28 MW corresponds to the TEm mode at a
magnetic field of 1.55 kG. The 4 MW microwave power peak at 1.3 kG
is probably a higher mode, TE113.
60
■s
T E i13
TE111
-i— 4t — ■
4
TE112
iT i
i
n
1— * --- '-----1-----*
i ' '
i
i'
50
S
0*
£
s,
4>
►
CQ
£
p
40
30
20
. '. . i
10
0
1.1
1 .3
1 .5
1 .7
1 .9
2 .1
2 .3
Solenoidal B-Field (kG)
Figure 5.3.1: Measured microwave
power vs. solenoidal
magnetic field,
gyrotron.
64
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
coaxial
The e fficie n cy
shown in Figure 5.3.2 is the peak e le c tro n ic
efficiency (defined below) vs. solenoid magnetic field.
A voltage o f
800 kV is used because that is the highest voltage on the fla tto p .
This
yields
a conservative
estim ate.
There
are
three
peaks
corresponding to the three cavity modes: TEt11 at 2.35 GHz, TE112 a t
2.5 GHz, and TE 113 at 2.74 GHz, w ith
respective
peak e le c tro n ic
efficiencies of 5%. 11%, and 1%.
TE,13 TE,t1
S'
su
S3
tel
■s
£
&
&
Solenoidal B-Field (kG)
Figure 5.3.2: Measured microwave efficiency vs. solenoid magnetic field,
coaxial
gyrotron.
Figure
5.3.3
magnetic fie ld
shows
a graph
fo r the slotted
of
gyrotron.
m icrowave
power
versus
This gyrotron exhibits
a
resonance at about 1.85 kG. This resonance is much narrower w ith
respect to the magnetic field
than either the unslotted
65
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
or non-
coaxial cavities.
This gyrotron only operated in a single mode, th e
T E 11Z with power levels in the 60-90 MW range.
this
gyrotron
operates
in
waveguide-mode interaction
slots
of
microwave
the
cavity
powers
favors
this
mode
due
of the tw o-fold
the
are attributed
TEm
to
to
e le c tro n -b e a m -
axial symmetry of th e
mode.
two
It is believed th a t
Higher
possible
m easured
m echanism s.
First, the slots in the cavity and output waveguide p re fe re n tia lly
oriented the output m icrowave power to the same polarization
the m icrowave
S-band detectors.
Second, these slots
as
could a lso
decreases mode com petition which would perm its higher m icrow ave
power production in a single mode [GOL88].
66
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
100
£
S
90
80
%
70
60
t
I
50
40
e
e
u
s
20
■a
il j J i r t a l 1 !
1.3
1.5
»»■ n
1.7
Ip
1.9
r* i
2.1
1
n
2.3
I
2.5
»■
2.7
Solenoidal B-Field (kG)
Figure 5.3.3: Measured microwave power vs. solenoid magnetic field, coaxial, slotted
gyrotron
E fficiency vs. magnetic fie ld
the entrance
Rogowski coil was
could not be calculated because
not functioning
properly
during
these experim ental runs.
A graph of microwave power as a function
of m agnetic fie ld
for the non-coaxial gyrotron is shown in Figure 5.3.4. This g yro tro n
has a wide resonance from 1.7 to 2.1 kG and m icrowave power was
generated up to 25 MW. There w ere numerous cases of zero pow er
shots corresponding to zero cu rre n t transport down the d rift
67
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
tube.
As w ill be discussed fu rthe r below,
the e-beam transport was le ss
reliable than for the coaxial tube.
i
30
4*
»
| —i —i
i
*
i
•
i
*
i
■
20
i
4*
>
C8
§
km
’
*
J
10
B
T B
0
1.1
1.3
■ ■
1" E B 1
1.5
1.7
■ i
I
1.9
2.1
11 ii
2.3
2.5
2.7
Solenoidal B-Field (kG)
Figure 5.3.4: Microwave power vs. solenoid magnetic field for non-coaxial, unslotted
gyrotron
Figure 5.3.5 graphs efficiency
coaxial gyrotron.
vs. magnetic field
The peak efficiency,
a little
for the non­
less than 10%, w as
slightly lower than the coaxial gyrotron.
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
tt
4 -
a
a
0 L-cKHH>rei i nnPRB?oir»
1.1
1.3
1.5
1.7
1.9
*—
2.1
2.3
2.5
2.7
Solenoidal B-field (kG)
Figure 5.3.5: Efficiency vs. solenoid magnetic field for non-coaxial, unslotted gyrotron
To summarize
the
efficiency
gyrotron gave the highest electronic
results,
efficiency
the
3 -s lo tte d -a n o d e
of 13%. the coaxial
unslotted gave 11% and the non-coaxial gave 10%. The slotted -an od e
aperture is expected to give the highest efficiency
in a is limited by the aperture slots.
give lowest efficiency
space-charge potential.
is consistent with
since the spread
The non-coaxial is expected to
since more of the beam’s energy is in the
The overall range of e fficie n cie s,
the BWO interaction
for which
10-13% ,
previous w o rk
[SPE91] showed insensitivity of efficency to e-beam a spread.
69
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5 .4
Current
Transport
Measurem ents
a. Coaxial Cavity
Figure 5.4.1(a) summarizes current
measurements
a t anode
aperture, cavity entrance, and cavity exit current as a function
solenoidal magnetic fie ld
current is 900-1900
with a diode fie ld
of 0.83 kG.
of
A p e rtu re
A and shows weak dependence on so le n o id a l
m agnetic field strength.
This is expected since the solenoid fie ld is
sm all near the aperture.
The entrance current is 700-1300
A and
tends to decrease with increasing solenoidal magnetic field, because
the c ritic a l
velocity
(v^,) to transm it
the e-beam
through
the
m agnetic cusp increases w ith the product o f the diode and solenoid
magnetic fields (see eq 2.3.14).
The part o f the e-beam which is not
transm itted is reflected, sim ilar to a m agnetic m irror.
is a large spread in e-beam energies,
the total,
Since th e re
e-beam current is
not reflected at a given critical magnetic fie ld combination.
Instead
the e-beam transport fraction decreases as the solenoidal and diode
magnetic fields are raised.
Another reason why e-beam tra n sp o rt is
dependent on m agnetic fie ld is due to changing e-beam dim ensions.
As the magnetic fie ld
increases, the Larmor orbits decrease in size
and coherent o ff-cen te rin g
of e-beam intersection
current
is
measured
increases.
w ith
at
This increases the lik e lih o o d
the center conductor.
300-1100
A and also
The ca vity e x it
decreases
increasing magnetic field, following the entrance current.
possible
that
m icrowave
fields
could
perturb
trajectories to cause additional scraping in the cavity.
70
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the
w ith
It is also
e-beam
Figure 5.4.1 (b) shows the currents in aperture, entrance, and
exit m onitors,
but with
a diode magnetic field of 1.07 kG.
aperture current is 1700-2900
2100 A and decreases with
is 200-1400
A and is
A.
The
The entrance current is 9 0 0 -
increasing magnetic field.
decreasing
as a function
Exit c u rre n t
of
so le n oid al
m agnetic fie ld .
Figure 5.4.1(c) depicts measurements of anode aperture, c a v ity
entrance and e xit currents w ith the diode magnetic fie ld set at 1.23
kG.
The anode aperture current ranges from
1400-2700
cavity entrance current is in the range of 900-2100
current is reduced to 300-1900
currents decrease with
A.
increasing
could s till
The
The e x it
Both cavity entrance and e x it
solenoidal magnetic field.
anode aperture current also decreases with
magnetic fie ld .
A.
A.
The
increase of so le n oid al
The solenoidal field is small at the aperture but i t
have an effect on the amount of beam focused into th e
d rift tube.
71
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2000
1500
e 1000
g3
+ ♦+
500
1.3
1.5
1.7
1.9
2.1
2.3
2.5
2.7
2.3
2.5
2.7
Solenoid B-Fiekl (lcG)
(a):
3000
2500
<
|
2000
1500
a
•••
1000
500
1.1
1.3
1.5
1.7
1.9
2.1
Solenoid B-Fiekl (kG)
(b):
72
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3000
2500
2000
<
'w'
c
1500
s9
1000
500
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
2.7
Solenoid B-Field (kG)
(C)
Figure 5.4.1 Currents from anode aperture, cavity entrance, and exit vs. solenoidal
magnetic field for coaxial gyrotron, with diode magnetic field of: (a) 0.8 3
kG, (b) 1.07 kG, (c) 1.23 kG.
Using current transport data for the coaxial gyrotron we can
calculate fractional transport of current through the m agnetic cusp
and the microwave cavity as a function of solenoidal m agnetic fie ld
for different diode field settings (Fig 5.4.2).
Tcu sp
=
Le n tra n c e
£
a p e ru tre
T
= * ca vu y
£
I .e x it
e n tra n c e
73
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.0
0.8
■■
■•
0.2
0.0
1.1
1.3
1.5
1.7
1.9
2.1
2.3
Solenoid B-Fieki (kG)
2.5
2.7
2.5
2.7
(a)
1.0
0.8
8
0.6
«■
0.4
0.2
0.0
1.1
1.3
1.5
1.7
1.9
2.1
2.3
Solenoid B-Field (kG)
(b )
74
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.0
0.8
‘
8
j
06
0.4
H
0.2
0.0
1.1
1.3
1.5
1.7
1.9
2.1
2.3
Solenoid B-Field (kG)
2.5
2.7
(C)
Figure 5.4.2: Measured magnetic cusp and cavity current transmission efficiency vs.
solenoidal magnetic field for coaxial gyrotron, with diode magnetic field
of: (a) 0.83 kG, (b) 1.07 kG, (c) 1.23 kG.
The diode magnetic field also has an effect on the transport o f
the cavity entrance and e xit current.
In Figure 5.4.3 entrance and
exit currents are plotted as a function
of diode magnetic field fo r
constant charging voltage on the (cavity) solenoid.
charging condition was used, the fie ld s
in the cavity
slig h t dependence on the charging voltage
fields.
Since a c o n s ta n t
region had a
of the diode m agnetic
Even though there is an iron plate between the diode and th e
cavity there is some leakage of the diode magnetic fields
cavity
region.
This gives a cavity fie ld
into th e
of 1.85 kG to 1.95 kG fo r
75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
diode magnetic fields of 0.65 to 1.38 kG respectively.
There is a
maximum current transport at the diode magnetic field
o f 0.83 kG.
At higher diode B -fields the e-beam transport efficiency decreases.
This is to be expected from the v ^ , condition of the m agnetic cusp.
The calculation
of the actual transm itted
because it depends on the d istrib u tio n
in itia l
angles.
current is u nattainable
of the particles energies and
The cut o ff energy of the magnetic cusp is e a sily
plotted for single particles as shown in Figure 2.3.1 and predicts a
higher cutoff velocity at higher magnetic fields.
magnetic fields
the transport
At the low er diode
is also decreasing.
magnetic field does not focus or contain the e-beam.
beam has several constraints due to geometry:
the
magnetic
field
both the
Larmor
radius
Too low of a
Also, the e -
With the changing o f
and coherent
o ff-
centering are changing. This means the outer and inner diam eters o f
the e-beam are also changing (as seen in section 4.3).
76
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1200
O
0
o
1000
o
800
o
O
o
o
600
Exit
8
I
♦
0
¥
*
+
o
o
8
+
i
O
4-
4-
0
0.
+•
8
400
200
E n trm a
o
o
O
O
4-
0.8
1.4
1.2
1.0
Diode B-Field (kG)
Figure 5.4.3: Cavity entrance and exit currents vs. dtode magnetic field for coaxial
gyrotron, with solenoidal magnetic fields of 1.85-1.95 kG.
We can
compare
the
previous
current
transport
to
th e
microwave power generated as a function of diode magnetic fie ld
in
Figure 5.4.4. The solenoidal m agnetic fields
to
used are restricted
the same values used above which is the tuning condition
T E 112 mode.
fo r th e
Notice there are two peaks in the microwave power.
The
first peak at 0.83 kG corresponds to the maximum current tra n s p o rt
shown above and gives a m icrowave power o f 43 MW.
The second
peak at 1.23 kG has a higher power of 51 MW. The higher power i s
probably due to the higher a a t the larger m agnetic fields.
77
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
60
50
e. 40
>
0u
O
30
1
>
o
i-l
o
2
□
a
a
a
8
a
a
1
a
20
a
10
g
B
a
o
□
B
0
—I p
■
0.8
■ ■!
a
a
1.0
1.4
1.2
Diode Magnetic Reid (kG)
Figure 5.4.4
Diode Magnetic field vs. Microwave power, with
solenoidal magnetic
fields of 1.85-1.95 kG.
b. S lo tte d
C a v ity
C u rre n t
Figure 5.4.5 is a plot of anode aperture and cavity exit c u rre n t
for a diode m agnetic field
of 0.83 kG for the slotted
measurements of entrance current are available
fo r
cavity.
No
the s lo tte d
cavity since the entrance Rogowski coil was not operating during the
experim ental
run.
Notice the aperture current
is constant w ith
respect to solenoidal magnetic fie ld with the exception of a s m a ll
peak at 1.8-1.9 kG. Cavity e x it current was measured to be 3 0 0 1900
A.
As in the case o f the
unslotted
coaxial
78
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
cavity,
th e
transm ission
of exit current of the slotted
cavity decreased w ith
increasing magnetic field.
4000r -»
» ri
■■—
—Ir" »
»
■ i' ■
■ i
•> i
■
Ti.
□
Apartuw
4- Brit
3500 ■•+• aw
3000 ■
a □
□
°
22500-
I 2000:
°
1500
1000 *
D
□
°
1.1
Qn
° ° a
a
a 0 ( Bf f l Q
0Q
° o
°
° °
°°
□ □
□
e
* a0
+
+
++ +
t£b.
md ?+ BB
c
*
+
+ +
+
13
1.7
:
+ + +
,
+
+
1----------------- -------- ------
-
-----------------
•
□
4-
13
•
=
+■ *
500‘
o'
?
15
21
23
25
2.7
Soleno idal B-Field (kG)
Figure 5.4.5: Anode aperture and cavity exit current vs. solenoid magnetic field fo r
slotted coaxial gyrotron. with diode magnetic field of 0.83 kG.
c. Non-coaxial Current Measurements
Figure 5.4.6 shows the anode aperture, cavity
cavity
e xit current transport
anode aperture
current
for the non-coaxial
ranges from
250-3200
Rogowski showed currents from 150-1200
entrance, and
gyrotron.
A, the
The
entrance
A and the exit m o n ito r
had currents from 150 to 950A in the normal transport range. There
were also numerous cases where the current did not transport down
79
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
the d rift tube.
In these cases no current was measured on the anode
aperture, the cavity entrance or cavity exit Rogowski coils.
These
shots did produce current in the diode but the e-beam was not
measured in the anode aperture Rogowski coil at the entrance to the
d rift
tube.
The diode is
operating
near its
calculated to be about 3kA (Table 2.2.3).
lim itin g
c u rre n t
Because the coaxial rod
extends up to the aperture plate it affects the electric fie ld s in the
diode.
It is believed that the coaxial rod modifies the e le c tric fie ld
in the diode, raising
the lim iting
Thus the presence of the center
current
in this
c ritic a l
region.
rod provides stabilized
e-beam
transport out of the diode in the coaxial gyrotron which
can not
occur in the non-coaxial gyrotron.
3500
a
+
•
3000
Aparui*
Enfrano*
Exit
2500
<
w
e
E
6
2000
1500
* "
1000
„•
500
*
o
1.1
iV
1.3
1.5
s ’a J S
1.7
1.9
; *
2.1
*
2.3
2.5
2.7
Soknoidal B-Field (kG)
Figure 5.4.6: Cavity entrance and exit current vs. solenoid magnetic field for non­
coaxial gyrotron, with diode magnetic fields of 0.83 and 1.07 kG.
80
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5 .5
Microwave Frequency Measurements
Presented in Figure 5.5.1 are the frequency-m agnetic
data for the unslotted coaxial cavity gyrotron.
operated
in two
main
dom inant mode with
The coaxial g y ro tro n
modes: TEn i and TE i 12. The TEn11 is
m agnetic fields
frequency of about 2.35 GHz.
fie ld
of 1.4-1.9
th e
kG and o p e ra tin g
The TE1t2 is the main mode in th e
magnetic field range of 1.9 to 2.4 kGauss at a frequency o f about 2.5
GHz.
Frequencies are tunable between the respective
modes by
changing the solenoidal m agnetic field.
N
as
S
u
e
Ol
3
er
4)
2.7
■ ■
2.6
mm m 1 ■
TEi 12
2.5
::
2.4
V
i . ' s
s *
■ w
M
2.3
2.2
1.1
1 ■ ■
1.3
1.5
1.7
W
■ ■
■ .
■
1.9
2.1
-
2.3
2.5
2.7
Solenoidal B-Field (kG)
Figure 5.5.1: Microwave frequency vs. solenoidal magnetic field for coaxial gyrotron,
with diode magnetic fields of 0.83, 1.07, 1.15. 1.23 kG
81
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 5.5.2 represents
fie ld
from the slotted,
data
o f frequency
coaxial gyrotron.
versus
m agnetic
This gyrotron
p rim a rily
operates in the TEn2 mode with an operating frequency of about 2.55
GHz. This mode had a much narrower resonance with respect to the
m agnetic field compared to the unslotted and non-coaxial c a v itie s
(see also Fig. 5.3.3).
Frequencies outside the main resonance show
som e tuning with m agnetic fields but very little power is associated
w ith these frequencies.
Higher m agnetic fields show some power in
the TE113 mode and the lower m agnetic fields show some power in
the TEt11 mode.
2.9
2.8
■
TE
N
33
2.7
s
2.6
u
■TE
2.5
o»
La
2.4
2.3
22
• *
*1.1
1.3
1
............................
1.5
1.7
1.9
2.1
2.3
i . -i
2.5
2.7
Solenoidal B-Field (kG)
Figure 5.5.2: Microwave frequency vs. solenoid magnetic field for slotted, coaxial
gyrotron, with diode magnetic fields of 0.83 kG
82
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Data are shown in Figure 5.5.3 of frequency as a function
magnetic field for the non-coaxial gyrotron.
The frequencies are
spread between 2.4 GHz and 2.65 GHz w ith
corresponding to the low er
d istinction
2.52
magnetic
of
the
fields.
lower frequencies
L ittle
evidence
of
between the two modes, T E ,,, at 2.39 GHz and TE112 a t
GHz are
observed
because the spread
in
frequencies
is
comparable to the mode spacing.
2.9
2.8
N
S3
s
TE
2.7
2.6
2.5
2.4
2.3
2.2
1.1
1.3
1.5
1.7
1.9
2.1
2.3
2.5
2.7
Solenoidal B-Field (kG)
Figure 5.5.3: Microwave frequency vs. solenoid magnetic field for non-coaxial gyrotron,
with diode magnetic fields of 0.83 and 1.07 kG
83
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5 .6
Time Frequency Analysis
Time frequency analysis [COH95, JEC92] perm its tim e resolved
measurements
of gyrotron
em ission
frequencies.
Figures 5.6.1 through 5.6.4 are the tim e
gyrotron-m ixer
[JEC92, PET98],
tim e
Presented
frequency analysis
signal using reduced interference
is on the horizontal
axis.
of a
d istrib u tio n
Frequency is represented on the ve rtica l
in
(RID)
axis and
The darker areas designate th e
highest power at a given frequency in tim e, and the lighter areas
designate low er intensities.
Figure 5.6.1 is a typical
mode hopping in the coaxial gyrotron.
example o f
The power is in itia lly
in th e
lower TE1in mode from 960 ns to 1100 ns then the power jumps to
the higher TE112 mode from 1110 to 1230 ns.
The local o s c illa to r
was set to 2.2 GHz so this yields frequencies for the lower and upper
modes of 2.46 GHz and 2.60 GHz.
shown in Figure 5.6.2.
same time.
In this
A case of mode com petition
case both modes oscillated
is
at th e
From 880 to 940 ns more power is generated in th e
upper mode (T E 112) and from 950 to 1030 the low er mode (T E ,,,)
becomes dom inant.
After 1030 ns, until the end of the pulse both
modes’ power levels are comparable.
With a local o sc illa to r s e ttin g
of 2.2 GHz, th is gives a frequency of 2.37 GHz fo r the lower mode
and 2.52 GHz fo r the upper mode. These frequencies correspond to
the TEm and TE112 modes. In Figure 5.6.3 frequency fluctuations
compared to the diode voltage.
The local o s c illa to r
GHz which gives frequency flu ctu a tio n s
2.51 GHz.
The frequency tracks with
are
was set to 2.3
ranging from
2.39 GHz to
the voltage and causes th e
84
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
cavity
to detune from
[PET98].
resonance several tim es
during the
pulse
Presented in Figure 5.6.4 is an example of a pulse shortened
high power
(72
MW) microwave
coaxial gyrotron.
spike
The local o scillator
generated
by the s lo tte d
was set to 2.3 GHz w h ich
yields a m icrowave frequency of 2.52 GHz. The mixer has a g re a te r
dynamic range compared to the diode detector
microwave power.
used to
m easure
It shows that the m icrowaves did not c o m p le te ly
shut off (as the diode detector indicated) but continued to o s c illa te
at about the same frequency but at a very low power level fo r
alm ost 150 (ns) a fte r the power level dropped. The shut o ff of the
m icrowaves in th is case can be explained by plasma form ing due to
r.f. breakdown across the slots causing de-Qing of the m icrow ave
cavity.
Plasma has been shown to cause pulse shortening
device (coaxial gyrotron) and previously
[GIL98, COH98].
in other gyrotron devices
These mode com petition
voltage flu c tu a tio n s
in t h i s
and plasma form ation
mechanisms as w ell
as
have been shown to be
contributing factors in microwave pulse shortening.
85
permission of the copyright owner. Further reproduction prohibited without permission.
450
$ \
*
400
350
"KT
2 3001
g250
I 200
LU
150
100
50
§00
Figure 5.6.1
900
1000
1100
Tim e (ns)
1200
1300
1400
Time frequency analysis of mode hopping in unslotted coaxial gyrotron.
Local oscillator is 2.2 GHz.
86
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450
400
i ' s - =11
350
r
300
*
N
4
g250|
3
si - ! i t ;
Spool
L±I
..40*
150
■*
100
50
9oo
Figure 5.6.2
800
900
1000
Tim e (ns)
1100
Time frequency analysis of mode competition
1200
in
gyrotron. Local oscillator is 2.2 GHz
87
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1300
unslotted coaxial
(AM)*J|OA
11
Figure 5.6.3
Time frequency analysis of unslotted coaxial gyrotron. (Top) frequency
fluctuations vs. time
and (Bottom) dkxje voltage fluctuations vs. time.
Local oscillator set to 2.3 GHz.
88
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Tim * (re )
.y, \
a1*1
/
s,s
Tim* (re)
-200
|
-600
-1000
1000
1100
1150
1200
1250
1300
1350
1400
1450
1500
Time (re )
Figure 5.6.4
Time frequency analysis high power microwave spike for slotted coaxial
gyrotron. (Top) Microwave mixer
signal, (Middle)
Time frequency
analyses, (Bottom) Diode Voltage. Local oscillator set to 2.3 GHz.
89
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5.7
Microwave Pulse Shortening
Figure 5.7.1 represents data that show the typical pulse
shortening curve which at the outer-most edge has P~1/x behavior.
The pulse shortening curve represents an upper limit on the energy
output of the gyrotron.
The coaxial, unslotted gyrotron shows peak
power up to 50 MW and moderate power levels of 20-40 MW with
pulselengths of 10-40 ns.
points.
The triangles designate the coaxial data
The slotted coaxial gyrotron (+ symbols) shows the highest
power of 60-90 MW and extremely short pulselengths of 10-15 ns.
The non-coaxial gyrotron tends to produce lower microwave power.
100
S 20
100
125
False Length (ns)
150
175
Figure 5.7.1: Microwave power vs. pulse length for the three gyrotrons:
▲ is coaxial non-slotted gyrotron.
+ is coaxial slotted, gyrotron,
O is gyrotron with no center rod
Curve represents 1000/tfns] (MW]
90
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There are several
explain
[BEN92].
possible
why the microwaves
pulse shortening
mechanisms to
end before the voltage
pulse ends
Time frequency analysis showed that voltage flu c tu a tio n
caused the resonant frequency to detune [PET98].
Time frequency
analysis has also demonstrated mode hopping and mode c o m p e titio n
in this gyrotron device.
Spectroscopy measurements of H-alpha lin e
in hydrogen showed a correlation between turn on of plasma in th e
microwave tube and turn off of high power microwaves [GIL98].
91
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6. Summary and Conclusions
6 .1
S u m m ary
The goals of this thesis were (a) to assess whether coaxial
structures could enhance the performance of large orbit gyrotrons by
increasing the limiting current and improving mode selection and (b)
to study the physics of microwave pulse shortening utilizing
(0.5-1.0
ps) pulse e-beam.
Power and frequency
a long
of m icrowave
emission were examined for the coaxial non-slotted, coaxial s lo tte d
and the non-coaxial gyrotron.
power levels of 20-40
The unslotted, coaxial gyrotron gave
MW with
pulselengths of 10-40
ns.
T his
coaxial gyrotron operates in two main modes: TE t11 and TEm w ith
frequencies
of
2.34
and 2.5
GHz respectively.
The gyrotron
frequency is tunable between by the magnetic field.
coaxial
gyrotron
extremely
showed the
short pulselengths
highest
power
of 10-15
ns.
of
The s lo tte d
60-90
The slotted
MW and
coaxial
cavity operated mainly in the TE112 mode. The non-coaxial gyrotron
tended to produce lower microwave power with broader resonance in
the magnetic field tuning conditions.
The current transport out of the diode was more stable for the
coaxial
lim iting
gyrotron
currents
experiment.
because of the
are
compared
lim iting
to
Calculation of lim iting
current.
currents
Theories
measured
in
of
the
current for coaxial and non­
coaxial gyrotrons can be obtained from an analytical
92
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solution of a
rela tivistic
rotating e-beam in a coaxial cavity.
Since, in the non­
coaxial case, the electron beam was lost before it entered the d r i f t
tube, the lim iting
disruption.
finite
current in the diode was the reason for e-beam
The limiting current was calculated to be about 3 kA fo r
thickness
e-beam with
outer e-beam radius of 2.5 cm and
inner e-beam radius of 2.0 cm and a diode chamber wall
19.0 cm.
radius o f
This is consistent with the non-coaxial gyrotron
3.2 kA
maximum e-beam current level, where the e-beam was lost in the
diode on several shots.
Apertured anode currents with
the coaxial
center rod reached 3.6 kA which exceeds the upper aperture c u rre n t
limit observed in the non-coaxial gyrotron and the e-beam was never
lost for the coaxial gyrotrons.
The coaxial rod apparently raised the
limiting e-beam current in the diode, allowing higher currents to be
extracted.
The non-coaxial aperture behaves like
an e le c tro s ta tic
defocusing lens, where the coaxial aperture is focusing, since the
center rod is at ground potential.
The presence of the center rod
near the diode aperture produces a change in the e lectric
the critical diode region which in turn raises the lim iting
the diode.
fields in
current o f
Since the e-beam space charge depression was less in the
coaxial cavity case than in the non-coaxial case, more kinetic energy
is available for microwave production.
We would therefore expect
the microwave power generated to be greater in the coaxial case.
This agrees with our experimental observations.
Time-Frequency
analysis
[COH95,
JEC92]
was
used
on
heterodyne mixer signals to gain further insight into the operation
of the microwave
shortening.
tube and to find
Fluctuation
causes of microwave
pulse
in e-beam voltage was shown to produce
93
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fluctuation in microwave output frequency [PET98].
This detuning o f
the microwave driving frequency causes microwaves to fluctuate
power.
in
Mode hopping and mode competition
the
microwave
signals
shortening mechanisms.
in
have also been observed
and have been identified
Formation of plasma
as
pulse
may also co n trib u te
to pulse shortening.
Single
particle
simulations
glass plate diagnostics
are compared
to
experim ental
to measure e-beam a (Vp.jp/Vp***,).
Both
simulation and experiment gave a ’s on the order of 1 and both had
comparable off-centering of the Larmor orbits.
6 .2
C o n c lu s io n s
The coaxial gyrotron displayed superior performance compared
to the non-coaxial gyrotron.
The coaxial gyrotron operated at h igher
power levels and had more reliable transport of e-beam current o u t
of the diode. The higher power in the coaxial tube could be due to
less
mode competition
because of the
slightly
larger
spacing
between the T E „ and TE1Z waveguide
modes.
Since the c e n te r
conductor is so small, it is more likely
the higher power is due to
the more reliable transport of current and the lower e-beam spacecharge depression in the coaxial gyrotron.
The results of the slotted coaxial gyrotron were mixed.
this gyrotron did preferentially
fundamental
gyrotron.
select the TE112 cavity mode.
mode. TE111t was almost
completely
The
The
absent in th is
It is believed that the two fold axial symmetry of th e
94
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slots interacting
w ith
the e-beam strongly
The slots also did appear to pre fe re ntia lly
favors the TE112 mode.
orient the polarization
the vertical direction (the same polarization
did give higher measured powers.
disadvantage to using the slots.
in
as the detector) w h ic h
However, there was a m a jo r
It is believed that the slots w e re
responsible for the very short microwave pulses, due to suspected
plasma
breakdown
operates
with
across
a rotating
the
slots.
electron
rotating electrom agnetic wave.
The gyrotron
beam which
mechanism
couples
to
Because the electromagnetic
the
w ave
is rotating, excessively high electric fields are produced across th e
already “sharp”
waveguide.
which
slots
of the slotted
gyrotron’s cavity and o u tp u t
This in turn causes plasma breakdown across the s lo ts
imm ediately
changes
the
resonant
coupling
and
th u s
terminates the microwave production.
The coaxial gyrotron exhibits the typical
the
microwave
output
power (P~1/x).
pulse shortening o f
Mode competition,
mode
hopping and voltage fluctuations have been identified as contributing
factors.
Plasma has also contributed to pulse shortening
in many
microwave devices, [BEN97] and has been shown to contribute
pulse shortening in the large orbit coaxial gyrotron
[COH98].
The
slotted coaxial gyrotron exhibited the shortest microwave pulses.
is suggested that slots
decrease the e-beam to waveguide
coupling by preventing the waveguide mode from
which in turn leads to plasma form ation.
rotating
to
It
mode
[BEN92]
This could explain the
extremely pulse-shortened microwave spikes observed in the slotted
coaxial gyrotron.
95
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APPENDICES
96
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Appendix A.
Alpha Measurements D iagnostic for R elativistic Electron Beam in an
A xially Varying M agnetic F ield
Theoretical Discussion.
Single particle motion in an axially varying magnetic field was compare to experimentally
observed particle distributions. These simulations neglected space charge effects and
electric fields. I started with the relativistic Lagrangian in cylindrical coordinates.
L = —m0c 2
dL
dqt
p 2 + p 2e 2 + i
f
c2
v
d 3L
dt dqt
I
—epQAe
~'"~v
(A.1)
„
(A.2)
If a vector potential as in (eqA.3a) is chosen, we get a general form for the axial component
of the B-field (eqA.3b).
Ae =
(A.3a)
Bz (z ) = S j0/ ( 2 )
(A-31*)
Upon plugging Lagrangian in to equations of motions (eqA.2) and assuming a vector
potential of the form (eqA.3) we get three second order coupled differential equations.
(a.4)
r(z )i
p = p d 2 - cOcQ pf(z )
(A-5)
S=
(A.6)
P2Qf'(z)
To avoid large round off errors in solving these equation numerically it was necessary to
introduce dimensionless variables.
t = tCOc
(A.7a)
(A.7b)
z = zl
97
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(A.7c)
p = pl
(A .7d)
We finally get a set of coupled differential equations in dimensionless form.
(A .8)
p = pO 2 - O pf(zl )
(A .9)
? = ~ p 2W '(zl)
(A.10)
vv UV1V the
Uiv derivatives
UV&iVaUTVO W
v defined
UW1AI1VU »«>••
Where
are
as.
< //(z )
d f^ d z
tfW ) 1
dz
and... 0 = 0
dz
dz
dz
I
T'(zl)
(A.11)
I
f =f
-—•= z etc...
dt
These coupled differential equations were solved numerically using Runge-Kutta 45th order with variable step size. In each simulation, 1000 particles with the same initial
position and energy of 750 keV. The particles were initially given a uniform/random
distribution o f angles with the initial alpha (Vperp/Vparallel) < 1/6 (which is 1/2 the
maximum Alpha that can be transmitted through the pinholes).The pin holes were 3mm
deep and 1mm in diameter. The final alphas and positions for each of the particles were
recorded at die axial distance of 60 cm. from the anode plate. These results were then
compared to the experimentally measured distributions of particles.
Experimental Setup
The Experimental electron beams were generated by M .E X .B .A (Michigan's
Electron Long Beam Accelerator) with following operation parameter of : Voltage = 750
KV, Current = 1-10 KA, pulse length o f 0.5 pS.
The Experimental set up for the Large orbit beam is shown in (Fig A .1). The
magnetic field for the large orbit beam is cusped. The field goes from 1.11 KGauss on the
diode to -1.85 KGauss in the interaction region. A complete field reversal with in 10 cm.
98
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was achieved by adding several opposing winding on either side o f a 1.5 cm. thick soft
iron plate.
Experimental Set-up
glass plate & transport current
measurement
To
Diffusion
Pump
r=2.25+/-0.25 cm
Cusp
Iron pole piece
Maser Solenoidal Coils 4
Glass plate
Cathode
Stalk
Pearson
Carbon Anode( with 8 (1mm)
pinholes at r=2.25cm )
^Rogowski Coil
(Aperture Current)
toDSA
'Diode Magnetic
Field Coils
(fig A.1)
The Alpha Measurements were made by using the model in (Fig A .2). A graphite
aperture with eight holes in it was used to form eight beamlets. The holes were placed a
radial distance of 2.25 cm. from the central line of the experiment The pin holes were
1mm in diameter and the graphite plate was 3mm thick. The electron beam was then
passed down the drift tube as it would be in a microwave experiment, with the exception
that the wave guide was removed. A glass plate was placed in the drift tube (about 10 cm.
down steam of where the interaction region starts). The glass plate was used to record the
spatial electron distribution of the beam.
99
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Alpha Measurement Model
(X X 3 Q Q Q Q .P .
q q
GRAPHTTE
APERTURE
r-2.5
cm
BZ o2 .
T
e
s
I
a
/
0.1
i
i
45
60
Distance from anode (cm)
(fig A.2)
Comparison of Simulation to Experimental Data
* SMALL ORBIT
Small orbit and large orbit beams were simulated. The small orbit simulations were
compared to Melba shots 4804-4823. The B-field on the cathode was lKGauss which was
adiabatically ramped up to 2.1 KGauss in the interaction region.
100
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B-fiftld vs. Axial Distancs - Small Orbit
2.5
2
■
(9
1.5
*
1
N
m
Experimental
Simulated
0.5
*
0
- 20
20
40
60
80
100
120
Axial Distance (cm)
(fig A.3)
Radiation darkening of a glass plate makes a permanent record of the particle
distribution in the electron beam. The glass plate was placed 60cm downstream of die
graphite anode aperture. There were eight circles corresponding to the eight pinhole in the
aperture. The darkened circles on glass plate were offset from die centerline of the
experiment in a small orbit configuration. Their measured radii were 6mm-8mm. If one
assumes that the radius of the darkened pattern is the Larmor radius, it is possible to
calculated the measured alpha by using (eq A. 12).
a =
1
(A.12)
For Melba Shot 4804 the following values were used: B = .21 Tesla, y = 2.46, f£ =
0.914. This gave alphas ranging from 0.34 to 0.48.
101
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(fig A .4)
The particle simulation only modeled the particle distribution from one pinhole
aperture (fig A.S). The simulation predicted measured radius of 7-8mm which compares
well with experimentally measured particle distributions (fig A.4). Notice that there are
some particles in the center o f die circle but die highest density of particles is around the
outside edge (fig A.6).
The simulation also calculated an alpha distribution. The
maximum alpha particle distribution ranges from 0.2 to 0.4 with a peak at 0.3 (fig A .7).
The alpha particle distribution for small orbit beams extends down to zero.
102
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Sim ulated Sm all Orbit Particle
D is trib u tio n , M 4804
0 .0 1
Y (Metera)
0.005
-0.005
-
0.01
0 .0 0 5
0.01
X
0.015
0 .0 2
0.025
(Meters)
(fig A.5)
Number/Radius
Particle
35000 j
3 0 0 0 0 ■■
25000 20000
•
■
1 5 0 0 0 ■■
10000
5000 4
■
Particle
Density vs. Radius,
M 4904
■
o -M W W h lW lM W IlD l
CO
CO
IA
CM
rCM
m
o
Radius
in
<o
(Metsrs)
103
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(fig A.6)
Alpha
vs.
Particle D istribution,
M 4804
8 0 -r
70
o
.
o
o
o
o
o
o
o
*
o
o
o
o
*
o
*
o
’
o
o
*
Alpha
(fig A.7)
♦LARGE ORBIT
The Large Orbit beam was generated by a cusped magnetic field. This is a
nonadiabatic process and requires a strongly varying B-ficld. Comparison of measured
and simulated Bz fields are shown in (fig A.8). The field goes from 1.11 to -1.85
KGauss.
104
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Large O rbit
B-Field
Simulation
m
m
Experimental
3
o
*
0.5
^
-0.5
20
60
40
Z-Distance
80
(cm.)
(fig A.8)
The experimental large orbit particle distribution is much different than the small
orbit particle distribution. The large orbit circles overlap and form an axis encircling beam
with off centering of the circles. Instead of being complete uniform circles, the large orbit
patterns have higher particle density on one side of the circle (fig A .9). Alpha
measurements can be made experimentally using (eq A. 12 ). The measured radii ranges
from 1.45 cm to 1.7 cm. This gives an measured alpha distribution of 0.97 to 1.4.
The simulated results show a particle distribution with radii of 1.5 cm. to 2.1 cm.
(fig A. 10). The alpha data is plotted in histogram from. The maximum number of particles
in the alpha distribution is in die bin of alpha = 1 to 1.5.(see fig A .l 1)
105
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Simulated
Particle
Distrubtution,
M 5053
0.02
-
(Meters)
0 .0 1
0
<
•
5 ,V
e
Sm
* • •
-
0 . 01
-
0.02
%
>
• ep? T V
Xr-
r r j
* ■
s
i___
1
p >1
1■
- 0 .0 3
>
- 0 .0 4
- 0 . 0 3 -0.02 -0.01
X
0
0.01
0.02
0.03
(Meters)
(fig A . 10)
106
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Simulation Alpha vs. Particla Distribution,
M 5053
90 j
•
80
70--
*
50”
* *
A
E 60-3
o ^ N n ^ i n < D N ( o a i O ’' M n « i n < O N C B O > o
Alpha
(fig A . l l )
Conclusion
The small orbit beam has a measured alpha of 0.3-0.5. This is comparable to die
maximum simulation alpha of 0.3. The large orbit beam has a measured alpha of 0.9 -1.4.
This compares well with the simulated alpha o f 1.0 to 1.5. It is interesting to note that die
large orbit alpha distribution has a much different shape than the small orbit alpha
distribution. The small mbit has an alpha distribution that goes down to zero. The large
orbit simulation has a zero particle distribution below a = 0.5. The large and small orbit
spatial particle distributions were quite different from each other. The large orbit beam is
axis encircling with no particles in the center. The small orbit beam is off centered and
simulations show that some particle are in the center of the circle.
107
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Appendix B. Single Particle Simulation
programmed in C
Discussion of Program
C o m p ilin g
This program
lib ra rie s .
was compiles
using ACC compiler
with
the m ath
Start outer loop
This start a new particle at the initial position. This loop is s ta rte d
after the last particle reached the axial position corresponding to
the glass plate in the experiment. This program uses a total of 1000
p a rtic le s .
In itia l
C onditions
resets particle counter (this program uses 1000
p a rtic le s )
Initial particle Energy of .75 MeV
EnMev-0.75
Contrails Magnitude of Magnetic cusp (in Tesla)
BzTesla * 0 .1
Rmeters * .0225 Initial radius of electrons
initail axial position
Zmeters - 0.0
Zm axm eters-0.6 Axial distance particles are stopped
Chooses a random angle in the Theta direction
angleTh * ...
(angle<(.5*(1+6A2)A(1/2)
Chooses a random angle in R
angle R * ...
Usual relativistic factor
gamma v /c
beta*
betaVth «
initial beta in the theta direction
w c*
cyclotron frequency
Imeter characteristic length scale (to make equations
dim ensionless)
length scale for magnetic cusp transition
Ic - 0.025
NA: Vz/(R w«-j*coc) no longer used in this version
eata *
Dimensionless radial position
r*
i-0
108
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vr =
th - 0.0
vth *
z «
zmax vz =
t-0
tstep - 0.5
upmax «
eps - 1e-6
dim ensionless radial velocity
initial theta position
dim ensionless theta velocity
dim ensionless axial position
dimensionless axial end position (were glass plate
is located in experiment)
dim ensionless axial velocity
tim e
tim e s te p
NA: no longer used in this versions
Maximum allowed error in stepper.
S ta rt lo o p fo r rk4
This is the start of the loop that num erically integrates th e
equations of motion for an individual particle.
The n um erical
integration technique used is Runge-Kutta 4th-5th order (rk4).
Check to see if particle is reflected
If the particle reflects in the cusp it w ill never reach the zmax
point and the program w ill run forever.
This sets a flag and
condition to stop the particle loop if the axial velocity ever becomes
negative.
p r in tf( " to o
la rg e ")
The second part of this statem ent is a variable step size procedure.
This is the reason I had to w rite this program. Since the m agnetic
field is rapidly varying in the cusp region (exponentially varying) and
it is almost flat for the rest of the time, this is a hard d iffe r e n tia l
equation to solve. A variable step is needed because a very s m a ll
step size is required in the cusp, but this would take an
unreasonable amount of time to run for the rest of the flat region.
Standard program written to solve ODE’s that I tried were not s ta b le
in the variable step size chooser, (the rk4 is stable) The standard
procedure is to take 1 step, then take 2 half step. If they agree
within some accepted error then the particle is advanced the ste p
and the step size is increase.
If they don’t the step sized is
109
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decreased and the singe and double step part is repeated. The normal
method raises the step size to some power to converge rapidly on an
optimal step size. It turns out that this method of choosing a step
size depends on the last guess and become unstable in the cusp
region i.e first guess too large next guess way too small next guess
much larger than fist guess, next guess much sm aller that second
guess. This oscillation continues until o ve rflow /u n de rflo w e rro r
occurs.
My method simply increases/decreases step size by a
percentage of the last guess. This may not converge to the o ptim al
step size as quickly as the standard stepper, but it is stable in the
cusp region,
(multiply by 1.7 to increase step size or by 0.6 to
decrease step size).
end loop
This ends the particle loop.
or reached its destination.
w rite
The particle either reflected in the cusp
data
This writes the final positions and alpha (Vw/VpmnxM) to a file called
‘regdatl.d’. This data can later be plotted to get particle sp a tia l
distribution and alpha distributions.
Notice that I set all negative
alphas - -1. I don’t care about negative alpha since they are p a rtic le
that reflected. It is easier to deal with them (later) if they are a ll
the some number (i.e. -1).
RK4
C alculations
These are the functions that do the rk4 numerical integration of the
differential equations. They are called previously in the program.
The cusp profile information is contained in the functions b field and
dbfield . Several of the cusp parameters have to be changed here. I
did not make specific variable for them. Note d b fie ld must be the
derivative of b fie ld or the program will not work.
110
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C Source Code
#include <stdio.h>
#include <math.h>
void rk4(double.double,double,double,double,double,double,double
*,double *,double *,double ‘ .double *,double *,double);
double raccel(double,double,double,double,double,double,double);
double thaccef(double,double,double,double,double,double,double);
double zaccel(double,double.double,double,double,double,double);
double bfield(double);
double dbfield(double);
double Imeters.lc;
double ypc.ypl .yp2.yp3.xpc.xp1 ,xp2,xp3,xpcn,ypcn,pcd,drad;
m ain()
{
FILE *fp;
int
int
int
int
up « 0;
i * 0;
upmax;
flag;
double
double
double
double
double
double
double
double
double
double
double
double
xcoord.ycoord;
r.rnew.rn.rnh;
vr.vrnew.vrn.vrnh;
th.thnew.thn.thnh;
vth.vthnew.vthn.vthnh;
z,znew,zn,znh,zmax,Zmaxmeters;
vz,vznew,vzn,vznh;
tshalf,tstep,t,tm ax,tm in;
EnMev,gamma,beta;
BzTesla,Rmeters,Zmeters,wc;
points[7][60000];
d1 ,d2,d3,d4,d5.d6.delta.eps;
111
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double eata,angleR,angleTH.betaVth;
double Vperp.Vpar.alpha;
/*
Initial
conditions
*/
i » 0;
ran d(tim e());
while(i <1000)
{ /‘ start outer loop */
EnMev * 0.75;
BzTesla * 0.1;
Rmeters - 0.0225;
Zmeters « 0.0;
Zmaxmeters - .6;
angleTH - (0.5)*(((rand() % 100)/100.0)-0.5)/(sqrt(1+6A2));
angleR - (0.5)*(((rand() % 100)/100.0)-0.5)/(sqrt(1+6A2));
gamma - 1.0+(10*EnMev)/(5.11);
beta - sqrt((1-1/(gamma*gamma)));
betaVth - beta*angleTH;
wc « 175600000000‘ BzTesla/gamma;
Imeters » 300000000/wc;
Ic = 0.025;
eata * 300000000*beta/(Rm eters*wc);
r • Rmeters‘ wc/300000000;
vr - beta*angleR;
th = 0.0;
vth * betaVth/r;
z - Zmeters*wc/300000000;
zmax - Zmaxmeters‘ wc/300000000;
vz « beta*sqrt(1-(angleR*angleR)-(angleTH*angleTH));
t * 0.0;
tstep - 0.5;
upmax - 50000;
eps - 1e-6;
112
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r
start loop fo r rk4
*/
up * 0;
flag = 0;
w hile(z<zm ax)
{
rk 4 (r,v rlth ,v th ,z >v z >tstep,&rn© w I& vrnew ,& thn© w ,& vthnew ,& znew ,& vz
n e w .t);
tsh alf = tste p/2.0;
rk 4 (r,v r,th ,v th ,z fv z fts h a lff8irnhI& vrnhl & th n h f&vthnh,& znhI& vzn h It);
rk4(rn h ,vrn h >th n h fvth n h Iznhtvznh,tshalf,& rn.& vrn,& thnI& vthnI& zn,& vz
n .t+ ts h a lf);
d1
d2
d3
d4
d5
d6
*
=
*
fabs(rn-rnew);
fabs(vrn-vrnew);
fabs(thn-thnew);
fabs(vthn-vthnew);
fabs(zn-znew);
fabs(vzn-vznew);
if ((d1 >d2) && (d1 >d3) && (d1>d4) && (d1>d5) && (d1>d6))
{
delta « d1;
}
else
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if ((d2>d3) && (d2>d4) && (d2>d5) && (d2>d6))
{
delta - d2;
}
else
{
if ((d3>d4) && (d3>d5) && (d3>d6))
{
delta « d3;
}
else
{
if ((d4>d5) && (d4>d6))
{
delta - d4;
}
else
{
if (d5>d6)
{
delta - d5;
}
else
{
delta * d6;
}
}
}
}
if (delta < 1e-20)
{
delta -
1e-20;
}
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if (delta <eps)
{
tstep « tstep* 1.7;
/* printf("too sm all"); */
r « rn;
vr * vrn;
th * thn;
vth = vthn;
z = znew;
vz « vzn;
points[0][i]
points[1][i]
points[2][i]
points[3][i]
points[4][i]
points[5][i]
points[6][i]
m t;
«
«
»
»
»
-
r;
vr;
th;
vth;
z;
vz;
t - t + tstep;
/* check to see if reflected */
if (vz < 0.0)
{
z * zmax +0.5;
flag - 1;
}
}
else
{
tstep « tstep*0.60;
/* printf(”too large"); */
115
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}
if (tstep<1e-6) tstep * 1e-6;
++up;
}
I* end integration loop V
i++;
}
/* end outer loop */
write data
/*
*/
if ((fp = fopen("regdat1 .d","w")) »« NULL)
{
printf("\nCould not open file");
exit(1);
}
up * 0;
fprintf(fp," dat
[ ");
for (;up < i-1;)
{
xcoord * points[1][up] * cos(points[3][up]);
ycoord « points[1 ][up] * sin(points[3][up]);
Vperp ■
s q rt((p o in ts[1 ][u p ])*(p o in ts[1 ][u p ])*(p o in ts[4 ][u p ])*(p o in ts[4 ][u p ])+ (p o
in ts [2 ][u p ])*(p o in ts [2 ][u p ]));
Vpar - (points[6][up]);
alpha - Vperp/Vpar;
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if (alpha < 0)
{
alpha - -1;
}
fprintf(fp,"[ %f , %f , % f ] ,
\n ", xc o o rd *lm e te rs ,yc o o rd *lm e te rs , alpha);
++up;
}
xcoord * points[1][up] * cos(points[3][up]);
ycoord « points[1][up] * sin(points[3][up]);
Vperp «
sq rt((p o in ts[1 ][u p ])*(p o in ts(1 ][u p ])*(p o in ts[4 ][u p ])*(p o in ts[4 ][u p ])+ (p o
in ts [2 ][u p ])*(p o in ts [2 ][u p ]» ;
Vpar « (points[6][up]);
alpha « VperpA/par;
if (alpha < 0)
{
alpha - -1;
}
fprintf(fp,"[ %f , %f , % f ] ]:
V n ",xcoo rd *lm e te rs,ycoo rd *lm ete rs,alp ha );
/*
xp1
yp1
xp2
yp2
end
«
«
write data
*/
points[1][i-1] * cos(points[3][i-1]);
points[1][i-1] * sin(points[3)[i-1]);
points[1 ][i-20] * cos(points[3][i-20]);
points[1][i-20] * sin(points[3][i-20]);
117
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xp3 - points[1][i-40] * cos(points[3][i-40]);
yp3 * points[1][i-40] * sin(points[3][i-40]);
ypcn * (0.5)*(xp3*xp1*xp1 - xp3*xp2*xp2+xp1 *yp2*yp2xp1*yp3*yp3 xp1 *xp3*xp3+xp1 *x p 2 *xp 2 + xp 2 *yp 3 *yp 3 + xp 2 *xp 3 #xp3xp2*xp1 *xp1 -xp 3 *yp 2*yp 2 + xp 3 *yp1 *yp1 -xp2*yp1 *y p 1 );
xpcn - (0.5)*(-xp1*xp1*yp3+xp1*xp1*yp2-yp1*xp2*xp2y p 2 *x p 3 *x p 3 yp1 *yp2*yp2+yp1 *xp 3 *xp 3 + xp 2 *xp 2 *yp 3 + yp 2 *yp 2 *yp 3 yp2*yp3*yp3-yp1 *yp1 *yp3+yp1 *yp1 *yp2+yp1 *yp3*yp3);
pcd - (-xp1*yp3 + xp1*yp2 +xp2*yp3 +xp3*yp1-xp3*yp2-xp2*yp1);
xpc = xpcn/pcd;
ypc * ypcn/pcd;
drad « sqrt(xpc*xpc + ypc*ypc);
I* find alpha */
Vperp - sq rt((p o in ts[1 ][i-1 ])*(p o in ts[1 ][i-1 ])#(points[4][i1 ])*(p o in ts [4 ][i-1 ])+ (p o in ts [2 ][i-1 ])* (p o in ts [2 ][i-1 ]));
Vpar » (points[6][i-1 ]);
printf("\n Vperp %f *c, Vpar %f *c , alpha % f \n",
Vperp, Vpar, Vperp/Vpar);
}
I*
rk4 calculations */
void
rk 4 (rfv r,th Ivth fz,v2,tstep,rnew p,vrnew pfthnew pfvthnew p,znew p,vzne
w p ,t)
double r,vr,th,vth,z,vz,tstep,t;
double *rnew p,*vrnew p,*thnew p,*vthnew pt*znew p,*vznew p;
{
double
r1 ,vr1 ,r2,vr2,r3,vr3,r4,vr4;
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double th l ,vth1 fth2,vth2,th3,vth3,th4,vth4;
double z1 ,vz1 ,z2,vz2,z3,vz3,z4,vz4;
r1 * tstep*vr;
vr1 * tstepVacceK r.vr.th.vth.z.vz.t);
r2 « tstep*(vr+vr1/2);
vr2 «
ts te p *ra c c e l(r+ r1 /2 ,vr+ vr1 /2 , th+th 1/2 , v th + v th 1/2,z+z1 /2 , vz+ vz1 /2
,t+ ts te p /2 );
r3 « tstep*(vr+vr2/2);
vr3 «
ts te p *ra c c e l(r+ r2 /2 ,v r+ v r2 /2 , th+th 2/2, v th + v th 2 /2 ,z + z 2 /2 ,v z + v z 2 /2
,t+ ts te p /2 );
r4 « tstep*(vr+vr3);
vr4 *
ts te p *ra c c e l(r+ r3 /2 ,v r+ v r3 /2 ,th + th 3 /2 fv th + v th 3 /2 ,z + z 3 /2 ,v z + v z 3 /2
,t+ ts te p );
th1 « tstep*vth;
vth1 * tstep*thaccel(r,vr,th,vthfz fvz,t);
th2 - tstep*(vth+vth1/2);
vth2 ts te p *th a c c e l(r+ r1 /2, vr+vr1 /2,th+th1 /2 ,vth + vth 1 / 2 tz+z1 /2 , vz+vz1 /
2 ,t+ ts te p /2 );
th3 ■ tstep*(vth+vth2/2);
vth3 ts te p * th a c c e l(r+ r2 /2 tv r+ v r2 /2 ,th + th 2 /2 ,v th + v th 2 /2 ,z + z 2 /2 ,v z + v z 2 /
2 ,t+ ts te p /2 );
th4 « tstep*(vth+vth3);
vth4 »
ts te p *th a c c e l(r+ r3 /2 ,v r+ v r3 /2 ,th + th 3 /2 fv th + v th 3 /2 ,z + z 3 /2 ,v z + v z 3 /
2 ,t+ ts te p );
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z1 - tstep*vz;
vz1 « tstep*zaccel(r,vr,th,vth,z,vz,t);
z2 - tstep*(vz+vz1/2);
vz2 «
ts te p *z a c c e l(r+ r1 /2 ,v r+ v r1 /2 , th + th 1/2, vth + vth 1 /2 ,z + z 1 /2 ,v z + v z 1 /2
,t+ ts te p /2 );
z3 * tstep*(vz+vz2/2);
vz3 «
ts te p *z a c c e l(r+ r2 /2 ,v r+ v r2 /2 ,th + th 2 /2 ,v th + v th 2 /2 ,z + z 2 /2 ,v z + v z 2 /2
,t+ ts te p /2 );
z4 - tstep* (vz+vz3);
vz4 ts te p *z a c c e l(r+ r3 /2 ,v r+ v r3 /2 ,th + th 3 /2 ,v th + v th 3 /2 ,z + z 3 /2 ,v z + v z 3 /2
,t+ tste p );
‘ vrnewp - vr+(vr1 +2*vr2+2*vr3+vr4);
*rnewp - r+(r1+2*r2+2*r3+r4);
* vth newp - vth+(vth 1+2* vth2+2* vth3+vth4);
*thnewp - th+(th1+2*th2+2*th3+th4);
*vznewp - vz+(vz1+2*vz2+2*vz3+vz4);
*znewp - z+(z1+2*z2+2*z3+z4);
}
double raccel(double redouble vr,double th,double vth,double z,double
vz,double t)
{
return(r*vth*vth - vth*r*bfield(z));
}
double thaccel(double r,double vr,double th,double vth,double
z,double vz,double t)
{
return(-(2*vth*vr/r) + (vr*bfield(z)/r) + (vz*dbfield(z)/2)
120
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);
}
double zaccel(double r,double vr,double th.double vth .double z.double
vz,double t)
{
re tu rn (-r*r*v th *d b fie ld (z )/2 );
}
double bfield(double z)
{
return(-0.55+
(1 .65)*(-tanh(
(lm eters*z-.15)/lc))
);
}
double dbfield (double z)
{
re tu rn ((1 .6 5 )*(-1 + ta n h ((lm e te rs * z -0 .1 5 )/lc )* ta n h ((lm e te rs *z 0 .1 5 ) /lc ) )* (lm e te rs /lc ) ) ;
}
121
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Appendix C. The composition of glass witness plates.
A special thanks goes to Guardian glass for the donation of the glass samples used in
making the glass witness plates and for them providing the chemical compositions of die
glass. It is die impurities in the glass that form die color-centers for radiation damage
darkening.
results (W t.% )
Determination sample 1
clear
0.56
A 1A
9.24
CaO
Fe20 3
0.108
0.14
KjO
MgO
4.00
13.84
NajO
0.22
So3
71.89
Si02
sample 2
green tint
0.13
8.79
0.625
0.04
3.80
13.86
0.24
72.52
Test Method
XRF
XRF
XRF
XRF
XRF
XRF
XRF
XRF
BD
122
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Appendix D. Glass Plates
FigureD.l
Shot 6346, Dmax = 4.1 cm, Dmin = 3.0 cm, Dout = 6.0 cm, Ar = 1.0 -l.S ,
Bdiod = 1.1 kG, Bsole =1.49 kG
Figure D.2
Shot #6589, Dmax —3.5 cm., Dmin = 2.8 cm., Douter = 5.7 cm., Ar = 1.1
- 1.4 cm, Bdiod = 1.1 kG, Bsole = 1.86 kG
123
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Figure D.3
Shot #6610, Dmax = 3.7 cm, Dmin = 2.9 cm, Douter = 5.7 cm, Ar = 1.0 1.4 cm, Bdiod = 1.1 kG, Bsole = 1.71 kG
Figure D.4
Shot #6631, Dmax = 2.7 cm, Dmin = 2.0 cm, Dout = 4.7 cm, Ar = 1.0 1.35, Bdiod = 1.1 kG, Bsole = 2.23 kG
124
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Figure D.S
Shot #6651, Dmax = 4.0, Dmin = 3.5, Dout = 6.0, Ar = 1.0 -1.25, Bdiod
= 1.1, kG, Bsole = 1.25 kG
FigureD.6
Shot#6671,Dmax = 4.6, Dmin = 3.1,Dout = 5.7, Ar = .55 -1.3, Bdiod
= 1.2 kG, Bsole = 1.48 kG
125
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Figure D.7
Shot #6691 mislabeled as 6688, Dmax = 3.6, Dmin = 2.8, Dout = 5.9, Ar
- 1.15 - 1.55, Bdiod = 1.2 kG, Bsole = 1.85 kG
Figure D.8
Shot # 6711, Dmax = 2.8cm, Dmin = 2.4 cm, Dout = 5.1 cm, Ar = 1.15 1.35 cm, Bdiod = 1.2 kG, Bsole = 2.22 kG
126
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Figure D.9
Shot # 6736, Dmax = 3.7, Dmin = 3.0, Dout = 5.8, Ar = 1.05 -1.4 cm,
Bdiod = 1.2 kG, Bsole = 1.7 kG
Figure D. 10
Shot 6762, Dmax = 2.8, Dmin = 2.4, Dout = 4.4, Ar = .8-1.0 cm, Bdiod =
1.2 kG, Bsole = 2.96 kG
127
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Figure D. 11
Shot 6287 (three slotted aperture)
Figure D . 12
Shot 5219 (8-pinhole aperture) Witness plate shows incoherent offcentering due to azimuthal asymmetries in the cusp magnetic field.
128
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Appendix E. Microwave Cold Tests
E .l
Introduction
Microwave cold tests axe performed on the same cavities as used in die experiments
to determine resonant conditions and cavity Q = (energy stored*oVenergy lost per unit
cycle). Cavity Q can be measured approximately by using equation E.1 [SLA50].
Q s —— —
2 f t - f in
s —
(E.l)
AT
Where f0 is the resonance frequency, ft/2 is the half power frequency and A f is the full
width half max about the resonance frequency.
These measurements were taken using a Hewlett Packard 8722D 50MHz-40GHz 2port network analyzer. The Su parameter is die reflection coefficient and was measured
versus frequency. The lowest dB value corresponds to the resonance frequency. The
difference in frequency between the 3dB points is Af. The microwave power from the
network analyzer is transported down a coaxial cable. The network analyzer is calibrated to
the end of die cable. Then different antennas are connected to the end o f the cable. The
two types of antenna used were the electric dipole and the magnetic loop antenna.
The scattering matrix is defined as die ratio of die incident or reflected wave to die
transmitted wave [POZ90]. The Su parameter is a measure of the reflection coefficient, V*
is the voltage of the forward propagating wave and V is the backward propagating wave.
Y1
Sn = —
V+
(E.2)
The Su parameter is related to how well the cavity can store energy at a given frequency
since the energy stored in the cavity is proportional to die energy lost due to resistance.
This gives us a means of measuring Q by measuring die reflected power as a function of
frequency.
The magnetic loop has an advantage over the electric dipole in that it perturbs the
cavity modes less than die dipole antenna. The magnetic loop has to be oriented with die
opening facing forward and back with respect to the cavity to properly excite die IE
modes. The dipole antenna has die advantage in that it only excites the TE modes.
129
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E.2
Coaxial unslotted
Figure E.1 (a) shows the measured Sn parameter using a loop antenna at a position
18 cm from the front posts o f the cavity. Figure E.1 (b) shows the measured Sn parameter
using a dipole antenna at position 18 cm from front of die cavity. Both plots show the
TE1U mode at 2.35 GHz and the TEm mode at 2.5 GHz as well as the higher mode TE^
mode (calculated to be about 3.94 GHz).
at
•
t|
o-.iu at
uinaa
IB
a
ti at*
s
a
um »
M
I«M
M
4
<7
V
'j
tTAxr u»«M »oib
(a)
130
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f
ttosim
miMMt
an
s
(b)
Figure E.1
sm §
ttro«
U
s
o
aa
MeasuredSn parameterof coaxial cavity with frequency rangeof 2 GHz to4 GHz for,
(a) loop antenna, (b) dipoleantenna(5 dB/ div)
The Q and resonant frequency are obtained from Figure E.2 for the TE1U mode. A
dipole antenna was used at position 18 cm from the front. The Q was measured to be 94
and the resonant frequency was 2.35 GHz.
131
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P 4 » fl«
s
w a « 9
II
IB
TTART L0aoa»0003tt
Figure E2
s ro r 2 «B 3 » Q » O k
TE1U mode o f coaxial cavity, resoiunt frequency is 2.35 GHz, Q is 94
(frequency range: 2.0 GHz - 2.6 GHz, 2.125 dB/div)
The TEu2 mode was also measured with die same antenna and position, from
Figure E.3 the Q was measured to be SOS and the resonant frequency to be 2.S3 GHz.
132
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STMT UDMD)flDD3lk
Figure E3
IIW S<BD3»0»OI(
TEn2 mode of coaxial cavity, resonant frequency is 2.53 GHz, Q is 505.
(frequency range: 22 GHz - 3.0 GHz, 10 dB/ div)
E .3
Slotted Cavity
These Q measurements and frequency measurements were taken using die 1.25”
dipole antenna and 1 cm diameter loop antenna. In Figure E.4 are shown the two main
modes, the TEm and TE ju measured using a dipole antenna 1.5 cm from die front of die
cavity. The respective resonance frequencies are 2.27 and 2.66 GHz. The resonant
frequencies were slightly perturbed by the antenna in the cavity.
133
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S O a llN
m
crAaT
Figure E.4
I M
I i*
y,
*.
iror ««§M) tix a if
Slotted, coaxial cavity. Using dipole antenna 1.5 cm behind front reflecting posts,
(frequency range: 2.0 GHz - 4.0 GHz, 10 dB/div)
The Q of the TEU1 mode was measured by a loop antenna near the front (entrance)
of the cavity, 1 cm from die front posts. The Q is 162 and die resonance frequency is 2.28
GHz (see Figure E.5).
134
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S.<
Figure E.5
Slotted coaxial cavity, TEm mode, resonant frequency is 2.28 GHz and cavity Q is 162.
(fieqaencynnge2.0G Ib-4.0G H E, 10 dB/div)
The Q of the TE j^ mode is measured by a dipole antenna placed at 11 cm from the
front posts. In Figure E.6, Q is 632 and the resonant frequency is 2.53 GHz.
135
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I W
<a»
•
I
n
D
Figure E.6
Slotted, coaxial cavity, TEm mode, resonant frequency is 2.53, cavity Q is 632.
(frequency range: 2.0 GHz • 4.0 GHz, 10 dB/div)
E .4
Non-Coaxial Cavity
Figure E.7 shows die cold test measurements of die TEU] mode using a loop
antenna at 18 cm from die end of die cavity. The Q was measured to be 458 and the
resonant frequency was measured to be 2.29 GHz. The TE„2 mode is also shown here
with a resonant frequency o f 2.57 GHz.
136
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
it am iftt xaamok
CHI •
u
ta«MAO
lO 4MU
M TO tf
s
«
■
9J*u.ioa
Figure E.7
Non coaxial cavity, TEul mode, resonant frequency is 2.29 GHz, cavity Q is 458 using a
loop antenna, (frequency range: 2.0 GHz - 4.0 GHz, 10 dB/div)
The TE112mode was measured using a dipole antenna 18 cm from the front posts of
the cavity. Figure 3.7.8 shows a resonant frequency of 2.S GHz and a cavity Q of 180.
137
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
aJio0
1
Car
3.97 OHz
START XOOO ODDOOO 0 8 *
Figure E.8
Non-coaxial cavity, TE,U mode, resonant frequency is 2.5 GHz, cavity Q is 180 using a
dipole antenna, (frequency range: 2.0G H z -4 .0 GHz, lOdB/div)
E .5
Power Calibration
Power calibration of the microwave signal is based on accurately knowing die
amount of attenuation of the microwave signal before it is measured. Since there are no
available detectors that directly measured microwave powers on the order o f mega-watts,
the microwave power must be attenuated in several stages before it can be measured by
diode detectors (~0.1W). The signal starts out at several mega-watts when it is radiated out
the output window. It is radiated into a chamber with microwave absorber. A portion of
this signal is coupled to S-band wave guide. This signal is then attenuated by 30dB
directional couplers (S-band). Then it is transported to die Faraday screen room by
138
with permission of the copyright owner. Further reproduction prohibited without permission.
approximately 50ft of RG-214/u type E15402 ooaxial cable and is attenuated about 7dB.
Finally HP calibrated attenuators are added in line to adjust the total amount of attenuation.
The chamber with absorber and S-band waveguide attenuation was calibrated using
a HP8722D network analyzer. The total attenuation was measure to be a minimum of 63
dB was dependent on frequency. 63dB is used to give a conservative estimate of the
microwave power. The attenuation was measured by the S12parameter which measures the
power transmitted through the system (see Figure E.9). (NoteiS^ = S12 for reciprocal
systems). The 63 dB was also independently verified by separate measurements o f (a) the
tank with absorber and (b) die S-band waveguide with attenuators, each on die order of 30
dB of attenuation (not shown). The noise floor of the network analyzer was measured to
be about 80 dB.
6 Nov 1999 17:01127
CHI s
REF O dB
12
M A E K E R 5
3 .2 6 3 2 0 0 0 0 *
G H z
«B
3.090 OF I
Hid
sta r t
> non nnmno o h .
Figure E.9
s to p
4«n n m n n n o M
Metal tank and S-band waveguide attenuation.
139
permission of the copyright owner. Further reproduction prohibited without permission.
To ensure that die loss of attenuation was not due to reflections at die output hom,
the Su parameter was also measured, shown in Figure E.10. This shows that between 97
to 99% of the signal is transmitted through the output hom.
Figure E.10
Power reflected in microwave hom
The output hom had the same cross-sectional dimensions as the coaxial gyration at
the output window with ID of 6mm and OD of 7.2cm. The coaxial part was 24 cm long
and fed by an S-band mode converter. The input S-band signal was linearly polarized and
oriented to die same polarization as the S-band detector waveguide.
140
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix F. Trigger Sequencing
In order for the gyrotron to operate properly, there are fo u r
triggered events that must be timed with
They are the firing
bank, the firing
trigger.
respect to each other.
of the Marx bank, the firing
of the solenoid
field
of the diode fie ld
cap bank and the crow bar
These events all take place on different tim e scales and the
timing of these are controlled by a sequence of the Systran Donner
and BNC delay generators.
The Marx must be timed to trigger when
the diode and solenoid fields
are at their
maximum.
The diode
magnets are the slowest part of the system and m ust be fired f ir s t .
They take about 116 ms to reach their peak after the ignitron on the
bank is triggered.
start the tim ing
A signal from the Systran Donner Pulse generator
sequence for the experiment.
The maser solenoid
field is the next slowest part in the system and must be trig g e re d
with
a delay with
respect to the initial
pulse.
In order for the
solenoid to reach the maximum field at the same tim e as the diode
magnets the initia l
trigger
is delayed 108.3 ms by a BNC delay
generator (BNC 7050, Berkeley Nucleonics Corp.).
The Marx bank
must be triggered at the peak of the magnetic fields.
It is trig g e re d
116 ms after the diode magnetic field
bank.
The Marx bank takes
about 450 ns to erect. When the crowbar delay is set to 960 ns th is
produces a e-beam voltage pulse of approximately 500 ns.
141
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Tim e (20 ms/division)
Figure F.1
Timing Traces: (a) dtode magnetic field current, (b) solenoidal magnetic
field current, (c) gate pulse
142
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Systron-Donner 101 Pulse Generators
O
o
O
External in
External in
Bank
TriggeF
o o
O
t
+
SCR
Trigger
Trigger Generator (MELBA only)
5V input
300V output
SV input
300V output
4" t
I I M
IUUUT-1
LUUK'K'J
level
O
level
O
slope
O
-
M
L'J
input
slope
+
f
©
G
31
O
mag remote
scope timer
Figure F.2
-♦
initial
delay
per.
©
& —
out
0
©
ext.
initial delay
initial delay
Schematic of timing circuit
143
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
level
O
1®
input
I ' j ; i j L 'j in
Appendix Q. Dispersion Relations with Analysis
To compare theory with experim ental data it is necessary to
solve the two
dispersion
equations
2.1.1
and 2.1.3
in term s
of
frequency as a function o f magnetic field.
2c2nqBf ± 2 ^ -4 c 2x 4f 2nV2y2M 2 +4*V ,4x 4f 2ly 2U j + V 2x 2c2q2B2
f
f ~
( 4 c 2X 2
- 4 V 2X2)M .r
(G.1)
The maser solenoid B-field is designated by B,. This equation gives
two solutions, an upper forward wave solution fo r the positive sign
and a forward (or backward) gyro-wave for the minus sign.
In te rm s
of e-beam a and initial e-beam 0O>
Y =
*
ft> c
(G.2)
-G Z ?
The coupled dispersion relation can be plotted versus m agnetic
field (Figure G.1). The upper branch is the upper forw ard wave and
the lower branch is the forward o r backward gyro-wave.
144
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
f (GHz)
30'
•
25
20
IS :
Forward
wave
10
BWQ
S
Solenoidal B
B-Field
Field T
Figure G.1
Dispersion relation for TE,, waveguide mode, a of 1 and e-beam voltage of
750 kV solved for frequency as a function of magnetic field.
The upper
curve is the upper forward wave intersection and the lower curve is the
forward/backward wave.
In the experim ent, only the low er curve fo r the g y ro -fo rw a rd
and backward waves were observed.
The frequencies
curve were m ainly above measurements taken.
in the upper
It is expected th a t
the m icrowave powers of these frequencies are sm all because th e
upper intersection
on the dispersion relation, which operates on a
much higher value of kz, is suppressed by the e-beam ’s v e lo c ity
spread. The curves are plotted in Figure G.2 for several values of a
and compared to experim ental
data from
the 3 -slo tte d
145
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
aperture.
This gyrotron’s current was lim ited
to less than 250 A. The d a ta
compared well to these dispersion relation for a between 0.05 and 2.
2.9
2.8
2.0
2.7
as
O
2.6
os
2.5
2.4
2.3
1.3
1.5
1.7
1.9
2.1
2.3
2.5
2.7
Solenoidal B-Field (kG)
Figure G.2
Frequency vs. magnetic field for three slotted anode aperture.
Curves
represent a between 0.5 and 2.0.
The same dispersion relations are also compared to the c o a x ia l
gyrotron using the 7.6 cm open aperture which allows up to 3 kA in to
the drift tube (Fig G.3).
to this data.
These dispersion relations do not fit as w e ll
This is expected since the e-beam current is very high
and space charge effects must be included.
magnetic
fie ld s
there
measured frequency.
are
two
Notice that at the lo w e r
components
of
e x p e rim e n ta lly
The higher component is about 2.75 GHz and a
lower component about 2.35 GHz. The upper frequencies are probably
due to a forw ard wave.
oscillation
The lower frequencies are probably due to
at o r near the absolute in sta b ility
[LAU81] which
146
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
is a
stronger interaction
than the forward wave.
The upper frequencies
are due to the forward wave on the TEn dispersion relation
w h ich
oscillates at the TE1i3 cavity mode.
22
1.1
13
13
1.7
1.9
2.1
23
2.5
2.7
Solenoidal B-Field (kG)
Figure G.3
Frequency vs. magnetic field for large open anode aperture.
Curves
represent simple theory with alpha between 0.5 and 2.0.
For the high current beams used in the large open a p e rtu re
gyrotron, the entrance currents are on the order of 1-2 kA.
charge effects
become im portant
at these current
electron beam behaves like a plasma.
Space
levels.
The
An intense electron beam can
raise the effective waveguide cutoff by equation G.3.
(G.3)
The plasma frequency for a gyrating electron beam can be w ritten
equation G.4 [SPE91],
147
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
as
(G.4)
where lb is the e-beam current, e is the electron charge, po is th e
permeability o f free space, p0 is the re la tiv is tic
outer cavity
radius.
fa cto r and re is th e
This equation is derived fo r a non-coaxial
cavity but since the center conductor used in the present e xpe rim en t
is so thin, this equation can be used.
Plasma shifted cu to ff frequencies of the dispersion re la tio n s
are compared
to
experim entally
measured
frequencies
for
th e
different values of a, 0.5,1.0, and 2.0 in Figure G.4. Notice that in
plot G.4 (a), the lower
magnetic fie ld points.
a of 0.5 matches better
with
the lo w e r
The a equal to 1.0 (in plot G.4 (b)) m atches
with the mid range m agnetic fie ld points and the a equal to 2.0 (in
plot G.4 (c)) matches best with the higher
magnetic field points.
a =0.5
2.9
2000A
1500A
1000A
500A
0A
2.2
1.1
(a)
13
13 1.7 1.9 2.1 23
Solenoidal B-Fleld (kG)
23
2.7
148
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
a=10
2.9
oa
2000A
1500A
2.7
1000A
2.6
500A
OA
2.5
2.4
23
1.1
1.3
Solenoidal B-Field (kG )
(b )
a = 2 .0
2000A
1500A
2.9
1000A
2.8
"S'
s
o
I
*
500A
2.7
2.6
2.5
2.4
f
2.3
2.2
13
1.7
1.9
2.1
2.3
Solenoidal B-Fleld (kG)
(C )
Figure G.4
Frequency vs. solenoidal magnetic field for coaxial non-slotted gyrotron
compared to plasma shifted cutoff frequency for e-beam current of 0 A,
500A. 1000A, 1500A. and 2000A, for: (a) a - 0.5, (b) a - 1.0, and
(c) a - 2.0
149
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
This m agnetic
field
dependence of a is
predicted
in
the
a nalytical equations derived for electron beam transport through a
magnetic cusp.
(See equations 2.3.10 or 2.3.13). These equations o f
alpha as a function of B( and Bi (initial and final magnetic fields) can
be substituted
relations
in to the already extrem ely com plicated
d isp e rsio n
and solved using MAPLE [ELL92] to give plasma s h ifte d
cu toff dispersion equations with magnetic field dependent a. These
equations are plotted
m agnetic field
experim ental
in Figure G.5 for
and are compared to
data.
The theory
frequency
vs. so le n oid al
coaxial non-slotted
has five
plasma
shifted
g yro tro n
c u to ff
frequency curves fo r e-beam current of 0, 500A, 1000A, 1500A, and
2000A. The alpha is a function of magnetic field.
2.9
20 0 0 A
2.8
s
cs
£
I
*
1500A
10 00A
2.7
5 00A
2.6
2.5
2.4
2.3
2.2
15
L7
1.9
2.1
2.3
Solenoidal B-Field (kG)
Figure G.5
Frequency vs. solenoidal magnetic field for coaxial non-slotted gyrotron
compared to plasma shifted cutoff frequency for e-beam current of 0A,
500A, 1000A, 1500A, and 2000A, for magnetic field dependent a.
150
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The variable a model shown in Figure G.5 is a better model
than the constant a model shown in Figure G.4, but there is one m ore
correction that should be included.
With an intense e-beam, there is
a large space charge depression.
potential
A portion
must be used to set up the
potential o f the D. C. space charge.
e ffe ctive ly
of th e
n o n -trivia l
a cce le ra tin g
e le c tro s ta tic
This means th a t the e-beam i s
accelerated by a voltage sm aller than the diode voltage.
This voltage depression can be large for IREB (intense r e la tiv is tic
electron
beams).
Calculations
o f lim iting
current
in section
2.2
show an estim ate of 186 kV e-beam depression fo r the non-coaxial
gyrotron.
This should also be comparable to the e-beam depression
of the coaxial gyrotron since the lim iting
same.
for
currents
are about th e
In Figure G.6 (a) is a comparison of dispersion relation theory
the five
e-beam currents
(l-O ,
500,
1000,
1500,
2000
A)
including a space charge depression of 200 kV w ith a comparison to
experim ental data.
The 200 kV was chosen because it gave b e st
agreement between theory and experiment.
energy, equation G.5 gives the relation
From conservation
between in itia l
of
gamma y ,
beam depressed gamma y0 , beam current lb , and k is a co n sta n t
[MIL82].
Y = Yo + —
me
r
(G -5 )
= > Y = Y o + krb
These dispersion curves give good agreement with
through the entire range of m agnetic field.
the data
They even predict th e
forward wave interaction with the TE n3 mode at the low er m agnetic
fields.
The lower frequencies to the left
of the dispersion re la tio n
151
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
are probably due to the absolute
in s ta b ility ,
grow th rate than the forw ard wave interaction
which
has a h ig h e r
[LAU81]. The th e o ry
is consistent with alm ost all the frequencies from low to high and
even predicts the spread in frequencies observed.
The next plot (Figure G.6 (b)) is a comparison of theory to th e
non-coaxial,
unslotted
gyrotron.
The
theory
also
predicts th e
interactions observed in the experim ent.
This fin al plot (Fig G.6(c)) is com parison of the theory to th e
slotted
coaxial cavity.
low er
magnetic
The theory is in
fie ld s,
but
fields
good agreement w ith th e
above
2.2
kG give
frequencies than can be accounted for by e-beam shifting
waveguide cutoff.
perturbing
the
h ig h e r
of plasm a
The frequency shifts are probably due to the s lo ts
cavity
frequency the cavity
modes and thus changing
modes.
This
in turn
the
shifts
resonance
the o p e ra tin g
frequencies due to frequency pulling o f the cavity mode.
152
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2.9
2.8
£
2-7
2.6
2.5
2.4
23
22 ,
1.1
-
13
13
1.7
1.9
2.1
2.3
23
2.7
23
2.7
Solenoidal B-Field (kG)
(a)
2.9
2.8
s
o
2.7
2.6
I
2.5
*
2.4
2.3
2.2
13
1.7
1.9
2.1
2.3
Solenoidal B-Field (kG)
(b)
153
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2.9
2.8
2.7
2.6
2.4
23
22
1.1
13
13
1.7
Solenoidal B-Field (kG)
(c)
Figure Q.6
Frequency vs. solenoidal magnetic field with 200 kV beam depression,
compared to plasma shifted cutoff frequency for e-beam current of OA,
500A, 1000A, 1500A, and 2000A, with magnetic field dependent alpha
for cases:(a) Coaxial non-slotted gyrotron, (b)
non-coaxial gyrotron,
and (c) slotted coaxial gyrotron
The frequency
vs.
m agnetic
field
plots
show
two
differences between the slotted and unslotted gyrotrons.
main
The f ir s t
is that the slotted gyrotron m ainly operates in the TE n2 mode. The
reason why the gyrotron only m ainly operates in the T E n 2 mode is
probably due to the 2 fold sym m etry in the slots which favors th is
interaction
slotted
over the TE1t1 mode.
gyrotron
also
The second difference
exhibited
short
spiky
is that the
m icrowave
154
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
signals.
These signals can be explained by the breaking down o f the c a v ity
slots and the form ation of plasma in the cavity.
change the
resonance condition
o scillation.
A dditional
and could
plasma fillin g
Breakdown w ould
stop
the
m icrow ave
of the gyrotron would also
shift the operating frequency up as observed on Figure G.6 (c).
The range o f as deduced in this final
depression is from 0.7 to 1.3.
analysis
using e-beam
Figure G.7 shows a plot of the change
in a over the m agnetic field range for four diode fields.
Diode
B-Fidd
0.123 T
0.115 T
0.107 T
1211..
0.083 T
a
08-
t a — KB— KB— Cl— i
Solenoidal B-Fidd T
Figure G.7
Plot of theoretically predicted a as a function of maser solenoid magnetic
field for an electron transmitted through a magnetic cusp for four dcde
B-fieW of 0.83 T, 1.07 T, 1.15 T. 1.23 T.
155
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix H. Axial Mode Calculation
Figure H.1 shows the two d iffe re n t
transm ission
line m odels
used to calcualte the axial modes o f the microwave cavity.
]
-
Short Circuit/
Open Circuit
Short Circuit/
Short Circuit
Figure H.1
Diagram of transmission line model for sc/oc and sc/sc
The difference
sh ift
between the two cases is th a t a 180 degree phase
is present on the (sc/oc) case which
(sc/sc)
case.
is not present in th e
It is possible to use these boundary conditions
calculate the resonance frequencies of the transmission lines.
we w ill examine the sc/oc conditions.
to
F ir s t
We require that the wave be
unchanged upon one transit of the cavity o f length L.
where the extra n is due to a 180 degree phase sh ift due to th e
re fle ctio n .
Setting the phase factors o f equation H.1 equal we g e t
equation H.2
2
;
—“ * + 2 p x
(H*2 )
156
permission of the copyright owner. Further reproduction prohibited without permission.
where X is the guide wavelength and p is an integer fo r the a x ia l
mode number.
f = £.
J
k
(H.3)
S ubstitution of eq H.3 into eq H.2 gives the resonance condition fo r
(sc/oc) in terms of frequency,
fs c /o c = ^ ( ~
2
+ P )
( H '4 )
Equation H.5 gives the resonance conditions for the (sc/sc) case.
/
,
J sc/sc
(H.5)
<2 ^
157
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B IB LIO G R A PH Y
158
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B IB LIO G R A P H Y
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W. E. Cohen, R. M . Gilgenbach, J. M . Hochman, R. L. Jaynes, J. L
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C. J. Edgcombe, Gyrotron Oscillators: Their Principles and
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W. Ellis Jr., E. Johnson, E. Lodi, and D. Schwalbe, Maole V Flight
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