harmonic generation in the beam current in a traveling...
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Peng Zhang, C. F. Dong1, D. Chernin, Y. Y. Lau, B. Hoff2, D. H. Simon, P. Wong, G. Greening, and R. M. Gilgenbach
Department of Nuclear Engineering and Radiological SciencesUniversity of Michigan, Ann Arbor
1Department of Atmospheric, Oceanic, and Space Sciences, University of Michigan2Air Force Research laboratory, Kirtland AFB, Albuquerque, NM
6th MIPSE Graduate Student SymposiumAnn Arbor, MIOctober 7, 2015
Harmonic generation in the beam current in a traveling wave tube
Work supported by AFOSR Award No. FA9550-15-1-0097, ONR Award No. N00014-13-1-0566, AFRL Award No. FA9451-14-1-0374, and L-3 Communications.
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Introduction
• Charge overtaking, or orbital crowding, can lead to harmonic generation, even in linear regime.
• This is well-known in klystron, but never studied in traveling wave tube (TWT).
• This paper extends the klystron theory of orbital bunching to a TWT to compute the harmonic content in the beam current [1].
[1] C. F. Dong, et al., IEEE Trans. Electron Devices (accepted, 2015).
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Harmonic generation due to orbital crowding
At t = t1, z = L1, current = infinite => significant harmonic content
Unperturbed orbit:
0 0 0 0( , ) ( )z z t t v t t= = −
With input signal:
0 0 1 0( , ) ( , )z z t t z t t= +
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Linear Theory of TWT
1 0 1 0( , ) Re[ ( , )]z t t Z t t=
Pierce’s 3-wave theory [2,3] :
[2] I. M. Rittersdorf, T. M. Antonsen, Jr., D. Chernin, and Y. Y. Lau, IEEE J. Electron Device Soc. 1, 117 (2013).
[3] J. R. Pierce, Traveling Wave Tubes, Van Nostrand (New York, 1950).
3 23 2 3 3 21 1 1
0 0 0 13 2( ) 4 [4 ( ) (1 ) ] 0d Z d Z dZj C b jd QC j C QC b jd Cb Zdt dt dt
ω ω ω+ − + + − + + =
C: gain parameterb: detuneQ: “space charge effect”d: cold-tube circuit loss
0 0
1 0
1 0
1 0 10
( ) 0,
( ) 0,
( ) .j t
Z t t
Z t teZ t t E em
ω
= =
= =
= =
Initial conditions
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Linear Theory of TWT (cont’d)
0 0 0 1 0 0 2 0 0 3 0( ) ( ) ( )101 0 1 2 32 2
0
( , ) ,j t C t t C t t C t teEZ t t e e e em C
ω ω δ ω δ ω δα α αω
− − − = + +
Pierce’s 3-wave dispersion relation [3]:
2 2( 4 )( ) (1 )QC jb d j Cbδ δ+ + + = − +
α1, α2, and α3 determined from (launching loss):
1 2 3
1 1 2 2 3 32 2 2
1 1 2 2 3 3
0,0,
1.
α α αα δ α δ α δ
α δ α δ α δ
+ + =
+ + =
+ + =
[3] J. R. Pierce, Traveling Wave Tubes, Van Nostrand (New York, 1950).
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TWT Bunching Parameter, X (new)
0 0 1 0( ) ( , )L v t t z t t= − +
yields relationship between the departure time (t0) and the arrival time (t1)
0 00 0 1 0 00 0 0
0 0 0
( , )( ) Re ( )i tL z t t Lt t X e R t tv v v
ωω ω ωω − = − ≡ − −
0 1 0 0 2 0 0 3 0( ) ( ) ( )0 1 2 3( ) .C t t C t t C t tR t t e e eω δ ω δ ω δα α α− − −− = + +
X= dimensionless “bunching parameter”
2 in
b
PXC P
=
Pin : input power Pb : DC beam power
10 0/wv eE mω=
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Relation between arrival time (t1) and departure time (t0)
(0) 00 0
0
( ) Lt tvωω − =
k-th order(k = 4 suffices)
0 0( ) ( 1)00 0 0
0
( ) Re ( ) , 1, 2,3,....i tk kLt t X e R t t kv
ωωω − − = − − =
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Zeroth order
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Harmonic Content of AC current on TWT [4]
charge conservation0 0( )LI t dt I dt=
0( ) jn tL n
nI t I e ω
∞
=−∞
= ∑
0 0
0 0 0 0 0 0
2 / 2 / 2( )0 0 0
0 0 0 00 0 0
( ) ( )2 2 2
in t in t in t in t tn L
II dtI t e I dt e d t eπ ω π ω π
ω ω ω ωω ω ωπ π π
− − − − −= = =∫ ∫ ∫
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contains n-th harmonic
[4] C. B. Wilsen, Y. Y. Lau, D. P. Chernin, and R. M. Gilgenbach, IEEE Trans. Plasma Sciences 30, 1176 (2002).
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Small signal electric field (E1) on CircuitLinearized force law
22 310 1 12 4 ( , )d Z eQC Z E z t
dt mω+ =
01 10(0, ) j tE t E e ω=
0 0 0 0
2 23 3( / ) ( / )21
1 11
( , ) 4(0, )
i iC z v C z vi i i
i i
E z t e QC eE t
δ ω δ ωα δ α= =
= +∑ ∑
This gives the rf power gain on circuit according to linear theory.
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Example: C-band TWT, Pin = 1mW
0 2 4.5 ,GHzω π= ×
Vb = 2.776 kV, I0 = 0.17 A, C = 0.1194, K = 111.2 ohm,
70 5.93 10 /v m s= ×
Pb = Vb I0 = 417.9 W3/ 2
0 / 1.16 micro-perveancebI V =
Excellent agreement for power gain on circuit,at low drive power
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Example: C-band TWT, Pin = 1mW
0 2 4.5 ,GHzω π= ×
Vb = 2.776 kV, I0 = 0.17 A, C = 0.1194, K = 111.2 ohm,
70 5.93 10 /v m s= ×
Pb = Vb I0 = 417.9 W3/ 2
0 / 1.16 micro-perveancebI V =
Excellent agreement for harmonic content,at low drive power
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Example: C-band TWT, Pin = 1mW
0 2 4.5 ,GHzω π= ×
Vb = 2.776 kV, I0 = 0.17 A, C = 0.1194, K = 111.2 ohm,
70 5.93 10 /v m s= ×
Pb = Vb I0 = 417.9 W3/ 2
0 / 1.16 micro-perveancebI V =
Excellent agreementfor harmonic content,at low drive power
Harmonic content at z = 4cm for 1 mW drive
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Example: C-band TWT, Pin = 54mW
0 2 4.5 ,GHzω π= ×
Vb = 2.776 kV, I0 = 0.17 A, C = 0.1194, K = 111.2 ohm,
70 5.93 10 /v m s= ×
Pb = Vb I0 = 417.9 W3/ 2
0 / 1.16 micro-perveancebI V =
Excellent agreement for power gain on circuit, at high drive power
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Example: C-band TWT, Pin = 54mW
0 2 4.5 ,GHzω π= ×
Vb = 2.776 kV, I0 = 0.17 A, C = 0.1194, K = 111.2 ohm,
70 5.93 10 /v m s= ×
Pb = Vb I0 = 417.9 W3/ 2
0 / 1.16 micro-perveancebI V =
Excellent agreement for harmonic content, at high drive power
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Example: C-band TWT, Pin = 54 mW
0 2 4.5 ,GHzω π= ×
Vb = 2.776 kV, I0 = 0.17 A, C = 0.1194, K = 111.2 ohm,
70 5.93 10 /v m s= ×
Pb = Vb I0 = 417.9 W3/ 2
0 / 1.16 micro-perveancebI V =
Excellent agreement for harmonic content, at high drive power
Harmonic content at z = 4cm for 54 mW drive
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Conclusion
• Crowding of the linearized electron orbits leads to significant harmonic content in the AC current. This is purely a kinematic effect (charge conservation).
• The harmonic currents calculated analytically agree with CHRISTINE simulation results.
• These harmonic AC currents may be used for frequency multiplication or TWT linearization (future work).
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