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Student ： Jhih Liang Shiao Advisor ： Yi Sheng Yeh

[ NTHU ]

Stability analysis of a gyrotron backward-wave oscillation with an external injection signal

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Outline

I. Introduction

II. Stability analysis of a gyrotron backward-

wave oscillation with an external injection

signal

III. Summary

IV. References

[ NTHU ]

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Applications of Millimeter Wave

• * 朱國瑞、張存續、陳仕宏 “電子迴旋脈射 - 原理及應用 ”物理雙月刊（廿八卷二期） 2006年 4 月

m ateria ls processing

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gyromonotron

gyroklystron

gyro-BWO

gyro-TWT

forward wave interaction

zk

2222cz ωckω

γ/vkω ezz

0zk

ω

Types of Gyrotron Tubes Four types of gyrotron tubes ：

(1) Gyromonotron

High power oscillator

(2) Gyroklystron

High power amplifier

(3) Gyro-TWT

Broad bandwidth amplifier

(4) Gyro-BWO

Oscillator (frequency

tuning)

forward wave interaction

zk

2222cz ωckω

γ/vkω ezz

ωwaveguide cavity modes

backward wave interaction

zk

2222cz ωckω

γ/vkω ezz

0zk

ω

5

E0 cosω tB0

Electron 2

Electron 1

Basic Mechanism of Gyro-TWA※Δω=ω-Ωc 0≧

where Ωc = eB0/γm, γ=(1-v2/c2)-1/2

※Electron #1 initially gaining energy→γ increases →Δω increases→out of synchronism→weaker interaction

※Electron #2 initially losing energy→γ decreases →Δω decreases→approaching synchronism →strong interaction

[ UCD ]

output waveinput wave electron beam

oB

rcrw

F

F

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Cycltron Resonance Maser Interaction Mechanism

Time = 0.0, Efficiency = 0.0 %

Time = 4.0, Efficiency = -5.0 %

Time = 8.0, Efficiency = 8.2 %

Time = 12.0, Efficiency = 26.8 %

R

＊ V. L. Granatstein, and I. Alexeff, High-power Microwave Source, Artech House, 1985.

eB

mvR

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Stability Analysis of an Injection-Locking Gyro-BWO

(a ) p h y sic a l co n f ig u ra tio n

0 .00 .10 .20 .30 .40 .5

r w(z

) (c

m)

(b ) m ag n e tic f ie ld

1 3 .4

1 3 .6

1 3 .8

Bz(

z) (

kG)

(c ) f ie ld p ro f ile

0 2 4 6 8 1 0 1 20 .00 .30 .50 .81 .0

| f(z

)|z (c m )z 1 z 2

0 .3 3 cm 0 .2 6 8 cm 0 .3 3 cm

-4 -2 0 2 4kz (cm-1)

0

20

40

60

80

100

f (

GH

z)

s=1 TE11

B0=13.8 kG

Fig. (a) Profile of the interaction structure . (b) Magnetic field . (c) Normalized field profile versus z in a gyro-BWO. The oscillation frequency on free-running operation is 32.8525 GHz in the gyro-BWO. Parameters are Vb=100 kV, B0 =13.8 kG, Ib=5 A, α=1.1, and rc=0.09 cm.

＊ Y. S. Yeh, T. H. Chang, and Y. C. Yu, “Stability analysis of a gyrotron backward-wave oscillation with an external injection signal,” IEEE. Trams. Plasma Sci., vol. 34, no. 4, 2006.

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Stability Analysis of an Injection-Locking Gyro-BWO

= 32.84987 GHz [ 00)( / 5108 ]

(a )

0 .00 .20 .40 .60 .81 .0

| f(z

)|

(b )

0 .00 .20 .40 .60 .81 .0

| f(z

)|

(c )

0 2 4 6 8 1 0 1 20 .00 .20 .40 .60 .81 .0

| f(z

)|

z (c m )

)( 01

msin

cos

dt

dm

22

，

outin0minmax )2(2)( P/PQ//m

0dt

d

Stable solution

Steady-state solution

where

a

b

c

c

a

b

0 .0 1 0 .1 1 1 0

P in (k W )

-2 0 0

-1 5 0

-1 0 0

-5 0

0

(a )

(b )

0 .0 1 0 .1 1 1 0

P in (k W )

0

5

1 0

1 5

2 0

2 5

3 0

3 5

s tab le

u n s ta b le

s tab le

u n s ta b le

th eo re tic a l

0.05 7.5

-15.5°

1.5

-186°

-180 °

31.44

21.96

4.14

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Numerical Method and Simulation Model

mtimnmmnz erkJzfkB )(2

Fields of the circularly polarized TEmn mode

*2

22 2 *

1

, , ,8 Nb

z jjmn mn

j j jz r t zIdk f z i W

dz x K z f z

j

z j

v E

v

Field equation

Relativistic equation of motion

ext

d ee

dt c P E v B B

Boundary conditions (injection-locking regime)

)()(' 22 zfikzf z

1 11

z zik z ik zf z f e f e

1 11' ( )z zik z ik z

zf z ik f e f e

1 1' zf z ik f zBoundary conditions (free-running regime)

2 2' zf z ik f z

1zik zef

waveguide

1z z 2z z

z

1zik zef

10

- 8 - 4 0 4 8

(-)1 0 4 /

-7 0

-6 0

-5 0

-4 0

-3 0

-2 0

-1 0

0

Pin

/Pou

t (dB

)

s im u la tedth eo re tica l

0 2 4 6 8 1 0 1 20 .0

0 .2

0 .4

0 .6

0 .8

1 .0

| f(z

)|

z (cm )

(-) / = 81 0

81 0

B0=13.8 kG

Pin=4 kW

Pin=5.1 kW

1)(2 0inout0 /P/PQ

Adler Curve

Stability Analysis of an Injection-Locking Gyro-BWO

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B o= 1 3 .9 k G

0 2 4 6 8 1 0 1 20 .0

0 .2

0 .4

0 .6

0 .8

1 .0

| f(z

)|

z (c m )

in j e c tio n -lo ck in g o p e ra tio n

f re e -ru n n in g o p e ra tio n

z = 1 .2 7 c m

B o= 1 3 .7 8 k G

0 2 4 6 8 1 0 1 20 .0

0 .2

0 .4

0 .6

0 .8

1 .0

| f(z

)|

z (c m )

in j ec tio n -lo ck in g o p e ra tio n

f re e -ru n n in g o p e ra tio n

z = 1 .3 7 c m

Stability Analysis of an Injection-Locking Gyro-BWO

B o= 1 4 .2 k G

0 2 4 6 8 1 0 1 20 .0

0 .2

0 .4

0 .6

0 .8

1 .0

| f(z

)|

z (c m )

in j ec tio n -lo ck in g o p e ra tio n

f re e -ru n n in g o p e ra tio n

z = 1 .0 3 c m

1 3 .6 1 3 .8 1 4 .0 1 4 .2 1 4 .4 1 4 .6 1 4 .8 1 5 .0

B 0 (k G )

0

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

(%

)

3 2 .0

3 2 .4

3 2 .8

3 3 .2

3 3 .6

3 4 .0

Freq

uenc

y (G

Hz)

in j ec tio n -lo ck in g re g im ef re e -ru n n in g reg im e

13.78 13.914.2

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Summary

• This study presents that the perturbation of a stable solution must decay in time to determine the stability of a steady-state solution, and the phase difference of the stable solution is restricted in the range between -90 ° and 90°

• The simulated locking bandwidth curve is slightly asymmetrical at frequencies below the oscillation frequency, the reason may be that the field concentrates toward the upstream end at frequencies below the oscillation frequency, and show that the locking bandwidth curve is about 28 MHz at a relative power ratio of -20 dB.

• The simulation result of the peak efficiency on the injection-locking operation increases to 34% at the magnetic field of 13.78 kG whereas the efficiency on the free-running operation is 30% at a magnetic field of 13.9 kG.

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References

[1] 物理雙週刊 p.426~433 (2006.4)[2] G. S. Nusinovich and O. Dumbrajs, IEEE Trans. Plasma Sci. 24, 620 (1996).[3] V. K. Yulpatov, Radiophys. Quantum Electron. 10, 471 (1967).[4] V. L. Granatstein, and I. Alexeff, High-power Microwave Source, Artech House,

1985.[5] R. A. York and T. Itoh, “Injection- and phase-locking techniques for beam

control,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 11, pp. 1920–1929, Nov. 1998.

[6] Y. S. Yeh, T. H. Chang, and Y. C. Yu, “Stability analysis of a gyrotron backward-wave oscillation with an external injection signal,” IEEE. Trams. Plasma Sci., vol. 34, no. 4, pp.1523-1528 , Aug. 2006.

[7] 羅智陽， “ A Gyrotron Backward-Wave Oscillator Driven by an External

Signal” [8] 廖志偉， “ A Gyrotron Backward-Wave Oscillator Driven by an External

Signal”