An Investigation of Beam Loading in a Cavity

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Nov 15, 2013 (3 years and 6 months ago)

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An Investigation of
Beam Loading in a Cavity
Craig B.Wilsen, J.W.Luginsland
a
, Y.Y. Lau,
P. M.Tchou, L.Ludeking
b
, and R.M.Gilgenbach
Department of Nuclear Engineering and Radiological Sciences
University of Michigan, Ann Arbor, MI 48109-2104
a
Air Force Research Laboratory, Kirtland AFB, NM
b
Mission Research Corporation
2001 IEEE Pulsed Power Plasma Science Conference
MURI Review Poster Session: Friday, June 22, 2001
Las Vegas, NV
Supported by the AFOSR, and by the DUSD (S+T) under the Innovative Microwave Vacuum Electronics MURI Program,
managed by AFOSR under grant F49620-99-1-0297, and by the Northrop-Grumman Industrial Affiliates Program.
Introduction
• Beam loading is extremely important in high
power microwave sources (rf ~ KE ~ PE)
• Recently developed klystron intermodulation
theory requires accurate knowledge of the
beam-loaded Q and resonant frequency
• Beam loading is not conveniently measured
• Simulation with MAGIC2D, comparison
with several theoretical approaches
Definition of Q
• Power ratio definition, useful in simulations:
• Bandwidth definition, useful for transmission
measurement:
• Decay time definition:
Power Dissipated
Resonanceat Energy Stored Average
Q
dB
f
f
BW
Q
3
0
1


)/exp(
00
QtPP



MAGIC Simulation
• Eigenmode solver yields cold resonant
frequency
• Cold Q computed from energy ratio definition
• Loaded Q and frequency shift calculated from
frequency response of saturated gap voltage,
fitted to a Lorentzian:
2
0
0
max
21
)(











f
ff
Q
T
fT
Klystron Cavity Geometry
• Based on KlystronCavity.m2d example file
• I
RF
= 1.5 A, B
z
= 1.3 T
• f
0
= 5087 MHz, R/Q = 68, Q
cold
= 145
DC Beam Loading
V
b
= 40 kV
I
DC
= 3 A
• Constant perveance P = 0.375 microPerv
• Beam-loaded Q does not change much over
a broad range of beam current and -voltage
V
b
(kV) I
DC
(A)
Q
f
0
(MHz) V
g
(kV)
40 3 129.3 5087.4 13.28
150 21.8 129.9 5086.4 13.40
200 33.5 132.4 5086.2 13.61
250 46.9 133.4 5086.2 13.70
300 61.6 134.2 5086.2 13.78
350 77.6 135.0 5086.4 13.87
400 94.9 135.6 5086.4 13.87
450 113 136.3 5086.4 13.99
500 132 137.1 5086.4 14.05
Constant Perveance
Constant Perveance
Pervs 375.0
2/3

V
I
P
AC Beam Loading
AC Beam Loading
• DC beam current = 3A
• Beam-loaded Q is insensitive to AC beam
current over a broad range of modulation
depths:
I
AC
(A)
Q
f
0
(MHz) V
g
(kV)
0.1 129.4 5087.4 0.665
0.3 129.7 5087.2 2.02
0.6 129.3 5087.4 3.99
1 129.3 5087.4 6.64
1.5 128.8 5087.4 9.94
2 128.5 5087.4 13.2
Neutralized Beam
• Beam neutralization affects neither DC nor
AC beam loading
• Neutralizing background of infinitely
massive ions
• Also used an electron-positron beam: no
influence on beam loading
I
AC
(A)
Q
f
0
(MHz) V
g
(kV)
0 129.3 5087.4 13.28
0 129.6 5087.4 13.30
1 129.3 5087.4 6.64
1 129.2 5087.4 6.66
Neutral
Neutral
Intensely Relativistic
Electron Beam
Q = 81.3, f
0
= 5073 MHz
500 kV, 1000 A
B
z
= 2.6 T
I
RF
= 50 A
(Q
cold
=145, cold f
0
= 5087 MHz)
Detune at High Currents
V
b
= 240 kV
Changing the Cold Q
• Cavity wall conductivity varied
• DC beam loading at 240 kV and 188 A
• “Beam Q” computed from
Q
cold
Q
hot
f
0
(MHz) V
g
(kV) Q
beam
99 78.9 5079.7 8.0 389
145 102.6 5080.3 10.5 351
200 128.2 5080.5 13.1 357
300 160.9 5080.5 16.5 347
beamcoldhot
QQQ
111

Beam Loading Theory
• Various models:
A.Naïve theory (ignoring entrance conditions):
neutralized beam on a TM
010
mode predicts
detune, but no de-Q
B.1. As above, but orbits calculated and transformed
to Eulerian coordinates; detune and de-Q
2. Chodorow & Susskind: beam shunt admittance
3.R.G.E.Hutter: series beam impedance
[1,2,3 yield same results]
C.Antonsen’s latest model
Chodorowand Susskind
• Parallel RLC circuit for cavity, with shunt
admittance to model beam loading
BBB
jBGY 
0
0
cos
sinsin
2
1
V
I
G
B













 




0
0
cos
sincos
2
1
V
I
B
B













 




Fundamentals of Microwave Electronics, p. 80, McGraw-Hill, 1964.
Formulas from Chodorow
and Susskind
• Predict de-Q and frequency shift:
B
cold
loaded
RG
Q
Q


1
cold
QQRR )/(
B
BQR )/(
2
1



Beam Loading Calculations, using
formulas fromChodorow& Susskind
5087 MHz8510 MHz1850 MHz2700 MHzf
0
240 kV51 kV6 kV60 kVV
b
188 A11 A520 mA30 AI
b
–4.92 MHz
(–7.3 MHz)
–7.65 MHz–0.50 MHz
(– 0.9 MHz)
–5.49 MHz
(– 5 MHz)
f
0
145 to 137
(103)
3500 to 8441100 to 904
(934)
3000 to 432Q
MAGIC
simulations
JPL
VKX7864A
Wisconsin
4K3SL
Litton
L-5782
Conclusions
• MAGIC simulations indicate:
• de-Q perveance
• de-Q independent of I
AC
and of beam neutrality
• de-tune observed only at high current
These results are unexpected!
• Beam loading theory:
• Contradictory results (among theories, and with
simulations)