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.

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).

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).

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=

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

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).

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

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

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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