Design of travelling wave tubes

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S u b m itte d i a p a r t i a l f u l f i l l m e n t o f th e re q u ire m e n ts f o r th e d e g re e o f MASSEB 01' SCIENCE

u a iV E R s m o f m o T m t On t a r i o LONDON 1950

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The a u t h o r i s i n d e b t e d t o D r. fi. C. D e a r i e f o r s u g g e s t i n g t h i s p r o b le m , f o r h i s e n c o u ra g e m e n t and f o r h is c o n stru c tiv e c ritic is m

o f th e m a n u s c rip t.

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them es a r e a l s o e x te n d e d t o M r. S . E . T u l l en d Mr. J . H. B l& e x w e li f o r t h e i r a s s i s t a n c e and many h e l p f u l s u g g e s t ­ io n s.

F i n a l l y , h e w is h e s t o e x p r e s s h i s g r a t i t u d e t o h i s

w if e , C y n th ia , f o r ty p in g th e m a n u sc rip t and f o r h e r c o n s t a n t e n c o u r a g e m e n t.

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i i l

LIST OF ILLUSTRATIONS....................................................................

v

.ABSTRACT.

vi

...............................................

SECTION I II

INTRODUCTION........................ DESCRIPTION OF A TYPICAL TRAVELLING WAVE TUBE

III IV

1

THEORY

........................................... ..................................

3 7

DESIGN AND CONSTRUCTION OF TUBE.........................

36

V REVIEW OF EXPERIMENTAL PAPERS..............................

46

APPENDIX........................................................................................................... 54 BIBLIOGRAPHY.

......................................

63

V I T A .................................................................................................................... 66

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LIST OF ILLUSXRATIONS F i g u r e No. 1.

Page

S c h e m a tic d ia g r a m o f t h e t r a v e l l i n g wave a m p lifie r.

............................

-4

2.

E q u i v a l e n t c i r c u i t f o r t h e h e l i x and beam .

. 10

3.

H e m i s p h e r i c a l s h e l l e n c l o s i n g e l e c t r o n beam

. 11

4*

G ra p h o f c i r c u i t im p e d a n c e p a r a m e t e r m u l t i p l y i n g f a c t o r f o r a beam o f r a d i u s r .

. . 25

5.

G rap h o f c i r c u i t im p e d a n c e p a r a m e t e r . . . . .

6.

G raph o f r e l a t i v e p h a s e v e l o c i t y o f a wave on a c h a r g e f r e e h e l i x

27

..................................... 33

7.

P h o t o g r a p h o f gun a sse m b ly an d s p a c i n g d i s c s

. 43

6.

P h o t o g r a p h o f o u t g a s s l n g a p p a r a t u s . ...................... 4 6

9*

P h o t o g r a p h o f t u b e i n wave g u i d e ..............................* 4 7

10.

P h o t o g r a p h o f vacuum s t a t i o n ......................................... 55

11.

p h o t o g r a p h o f McLeod g a u g e ........................................... 56

12.

C i r c u i t d i a g r a m ................................

13.

C a lib ra tio n cu rv e.

. . . . .

59 ......................................

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62

isJbS'XltttCT

A b r i e f rev ie w o f th e c i r c u i t ana f i e i u t h e o r i e s o f t h e t r a v e l l i n g wave t u b e i s p r e s e n t e d w i t h e m p h a s is on t h e f a c t o r s w h ic h i n f l u e n c e t h e g a i n and b a n d w i d th .

A

d e s i g n p r o c e d u r e i s g i v e n f o r a t u b e w h ich w i l l o p e r a t e a t t h r e e th o u s a n d m e g a c y c l e s .

An a p p e n d i x d e s c r i b e s t h e

c o n s t r u c t i o n o f a h i g h vacuum s y s te m u s e d t o e v a c u a t e t h i s tu b e.

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-1 -

I INTRQDUCTIQH Man h a s l o n g r e c o g n i z e d t h e i m p o r t a n c e o f c o m m u n ic a tio n s i n h i s d a i l y l i f e a n d h e h a s w orked d i l i g e n t l y

to d ev elo p

d e v i c e s and t o d i s c o v e r t e c h n i q u e s w h ic h w ould e n h a n c e h i s a b ility

to re la y I n t e l l i g e n c e ,

P r i m i t i v e man u s e d t h e m ed ia

o f v i s i b l e l i g h t an d low f r e q u e n c y so un d t o t r a n s m i t i n f o r m ­ a t i o n b u t t h e l i m i t a t i o n s o f h i s m e th o d s r e s t r i c t e d t h e s ig n a ls to th e c r u d e s t m essages.

T h e r e was t h e a d d i t i o n a l

d is a d v a n ta g e t h a t th e tr a n s m is s io n d i s t a n c e , u n d er th e m ost f a v o u r a b l e c o n d i t i o n s , was l i m i t e d by t h e r a n g e o f t h e human e a r o r eye and by t h e c u r v a t u r e o f t h e e a r t h .

M odern c o m m u n ic a tio n d e v i c e s a r e b a s e d on t h e d e t e c t i o n o f a n e l e c t r i c c u r r e n t i n a c i r c u i t an d i n t e l l i g e n c e c a n now be conveyed o v e r g r e a t d i s t a n c e s a lm o s t i n s t a n t a n e o u s l y .

The

n a rro w b a n d s c o m p risin g th e a u d io and v i s i b l e l i g h t f r e q u e n ­ c i e s o r i g i n a l l y em p lo y ed a r e now o n ly a s m a ll p a r t o f t h e s p e c t r u m o f f r e q u e n c i e s w h ic h c a n b e u s e d f o r c o m m u n i c a ti o n s . The p r o g r e s s made i n t h e r a d i o f r e q u e n c y b a n d w h ic h e x t e n d s fro m a b o u t one t h o u s a n d c y c l e s t o one h u n d r e d t h o u s a n d m eg a-

R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.

-2 -

c y c l e s i s m ost n o ta b le *

The d e v e lo p m e n t o f e l e c t r o n t u b e s w h ich c a n b e u s e d t o g e n e r a t e f r e q u e n c i e s i n t h e r e g i o n fro m t h r e e t h o u s a n d t o t e n t h o u s a n d m e g a c y c le s h a s a d v a n c e d r a p i d l y , i n t h e p a s t tw e n ty f i v e y e a r s b u t an a m p l i f i e r w ith a band w id th p r o p o r t i o n a l to th e freq u en cy space a v a i la b l e i n t h i s re g io n d id n o t e x i s t u n til re c e n tly .

I n th e o ld s t y l e r a d io frequency a m p lif ie r th e re s o n a n t c i r c u i t an d t h e i n p u t c a p a c i t a n c e w h ic h w ere a s s o c i a t e d w i t h t h e t u b e , p l a c e d a r e s t r i c t i o n on t h e b a n d w id th o f t h e a m p l i ­ fie r,

and t h e e l e c t r o n t r a n s i t tim e r e s t r i c t e d t h e h i g h e s t

f r e q u e n c y w h ic h c o u l d b e a m p l i f i e d . was p a r t l y

I1he l a t t e r d i f f i c u l t y

overcom e when t h e c o n v e n t i o n a l ty p e o f t u b e was

r e p l a c e d by t h e K l y s t r o n .

The b a n d w i d th p ro b le m r e m a in e d

u n t i l 1943 when R u d o lp h K o m p f n e r ^ l i n g wave t u b e .

H i s d e v i c e was t h e r e s u l t o f a s e a r c h f o r a

m ore e f f i c i e n t m e th o d o f re m o v in g m o d u la t e d b e d s i n

pro d u ced th e f i r s t t r a v e l ­

a K ly stro n .

t h e e n e rg y fro m t h e v e l o c i t y

D u r in g t h e p a s t few y e a r s ,

s e v e r a l g r o u p s , b o t h i n E u ro p e and t h e U n i t e d S t a t e s , h a v e b e e n c o n d u c t i n g I n v e s t i g a t i o n s i n t o t h e t h e o r e t i c a l and p r a c t ic a l a sp e c ts

The p u r p o s e

o f th ese tu b e s .

of th is paper i s

to g iv e a re v ie w o f th e

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two

m a in p r e s e n t a t i o n s o f t h e t h e o r y arid t o n o t e t h e f a c t o r s i n t h e t h e o r y w h ic h a r e i m p o r t a n t i n t h e d e s i g n o f a t u b e .

The

p r o g r e s s made i n t h e d e v e lo p m e n t o f a t u b e to o p e r a t e i n t h e t e n c e n tim e te r band w i l l a l s o be d e s c r i b e d .

II

(a)

DESCRIPTION OF A TRAVELLING WAVE TUBE.

F i g . 1 show s a s c h e m a t i c d ia g ra m o f a t r a v e l l i n g wave tub© and t h i s may b e d i v i d e d i n t o t h r e e p rim a ry

p a rts.

1.

E l e c t r o n S o u r c e a n a beam f o r m in g e l e c t r o d e s .

2.

Slow wave s t r u c t u r e .

3.

A c o lle c tin g anode.

The e l e c t r o n s o u r c e c o n s i s t s o f a n o x i d e c o a t e d c a t h o d e an d i t s a s s o c i a t e d f i l a m e n t .

The e m i t t e d e l e c t r o n s a r e c o n ­

c e n t r a t e d i n t o a beam by t h e e s t a b l i s h m e n t o f a p o t e n t i a l g r a d i e n t b e tw e e n t h e e l e c t r o d e s an d any s t r o n g l y d i v e r g e n t e l e c t r o n s a r e rem oved by c a u s i n g t h e beam t o p a s s t h r o u g h s m a l l a p e r t u r e s i n th e e l e c t r o d e s . is

A s t r u c t u r e o f t h i s ty p e

u se d i n c a t h o d e r a y t u b e s an d i s c o m m e r c ia lly a v a i l a b l e b u t

s u p p l i e s a maximum e m i s s i o n c u r r e n t o f t h e o r d e r o f one m i l l i arp ere.

I t w i l l b e shown l a t e r , i n t h e t h e o r y ,

th a t i t is

d e s i r a b l e t o h a v e & l a r g e beam c u r r e n t d e n s i t y an d s p e c i a l (15) h i g h c u r r e n t d e n s i t y guns h a v e b e e n d e s i g n e d by P i e r c e ' ' f o r use i n th ese tu b e s.

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The slo w wave s t r u c t u r e shown I n t h e f i g u r e c o n s i s t s o f a h e l i x o f c o n d u c tin g w ire .

The f u n c t i o n o f t h e h e l i x may

a l s o b e p e r f o r m e d by t h e u s e o f c o i l e d wave g u id e o r a. s e r i e s of d isc s.

T h i s p a p e r i s c o n c e r n e d o n ly w i t h t h e h e l i c a l t y p e

of stru c tu re .

I& ch e n d o f t h e h e l i x h a s a s m a ll p r o b e a t t a c h e d

w h ic h i n t u r n i s f a s t e n e d t o a m e t a l c o l l a r f o r m e c h a n i c a l su p p o rt.

The p r o b e s a r e u s e d t o c o u p l e th e s i g n a l t o t h e i n p u t

end o f t h e h e l i x and t o rem ove t h e a m p l i f i e d wave a t t h e o t h e r end.

The c o m p le te s lo w wave s t r u c t u r e t o g e t h e r w i t h t h e e l e c ­ t r o n gun an d c o l l e c t o r a r e a r r a n g e d i n l i n e i n & g l a s s t u b e w h ic h i s e v a c u a t e d .

The t u b e i s m o u n te d so t h a t i t p a s s e s

t h r o u g h two p i e c e s o f wave g u id e ( F i g . 1) and t h e p r o b e s , w h ich a r e p a r a l l e l t o t h e a x i s o f t h e t u b e , b e h a v e a s s m a l l a n te n n a e i n s i d e th e g u ia e .

The c o l l e c t i n g a n o d e i s a d i s c o f c o n d u c t i n g m a t e r i a l s u c h a s n i c k e l w h ich h a s weak m a g n e ti c p r o p e r t i e s .

S tro n g ly

f e r r o m a g n e t i c m a t e r i a l s a r e .a v o id e d t h r o u g h o u t t h e t u b e a s t h e i n t r o d u c t i o n o f u n c o n t r o l l e d m a g n e t i c f i e l d s may c a u s e £ d e v i a ­ t i o n o f tiie e l e c t r o n beam from i t s a x i a l p a t h .

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(b )

PRINCIPLE OF kCTIOH.

E l e c t r o n s a r e e m i t t e d t h e r m i o n i c a l l y by t h e c a t h o d e and a r e f o c u s s e d i n t o & beam w h ic h i s d i r e c t e d a l o n g t h e a x i s o f th e h e l i x .

The a x i a l v e l o c i t y

o f t h e e l e c t r o n s i s g o v e rn e d

by t h e m a g n i t u d e o f t h e a c c e l e r a t i n g v o l t a g e .

The s i g n a l i s c o u p l e d t o t h e h e l i x a t t h e c a t h o d e end and i s p r o p a g a t e d i n t h e fo rm o f an e l e c t r o m a g n e t i c wave. T h i s wave f o l l o w s t h e s p i r a l p a t h o f t h e w i r e w i t h a n a x i a l p h a s e v e l o c i t y w h ic h i s g o v e r n e d by t h e d i m e n s i o n s o f t h e h e lix .

E l e c t r i c and m a g n e t i c f i e l d s a r e a s s o c i a t e d w i t h t h e

e l e c t r o m a g n e t i c wave a n a t h e e f f e c t o f t h e a x i a l co m ponent o f t h e e l e c t r i c f i e l d w h ic h p e r i o d i c a l l y

changes i t s a m p litu d e

and d i r e c t i o n i s o f p a r t i c u l a r i n t e r e s t .

Those e l e c t r o n s w h ich a r e r e t a r d e d by t h e a x i a l e l e c t r i c fie ld

slow down w h e r e a s t h o s e w h ic h a r e a c c e l e r a t e d by t h e

f i e l d w i l l s p e e d up and o v e r t a k e t h e s lo w e r e l e c t r o n s .

The

b u n c h e s o f e l e c t r o n s t h u s fo rm e d w i l l a p p e a r i n th e r e g i o n s o f th e r e t a r d i n g e l e c t r i c f i e l d i f i s p ro p erly a d ju s te d .

t h e e l e c t r o n beam v e l o c i t y

T h i s r e s u l t s i n a t r a n s f e r o f e n e rg y

fro m t h e beam t o t h e e l e c t r o m a g n e t i c wave t h e r b y c a u s i n g a m p l i ­ f i c a t i o n o f th e l a t t e r .

The a m p l i f i c a t i o n i s f o u n d t o t a k e p l a c e when t h e e l e c t r o n

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beam h a s I t s v e l o c i t y a x i a l phase v e lo c ity

a d ju ste d , to e q u a l a p p ro x im a te ly

th e

o f t h e e l e c t r o m a g n e t i c wave.

Ill IHEOKf Q£ TBAVbLLIKC iYAVE TUBbd

(a)

I n t r o d u c t i o n to T heory.

The th e o r y o l t r a v e l l i n g wave t u b e s i s n o t s u f f i c i e n t l y c o m p l e t e t o g i v e p r e c i s e r e l a t i o n s b e tw e e n a l l t h e p a r a m e t e r s w h ic h a r e i n v o l v e d .

Many a s s u m p t i o n s a r e made i n o r d e r t o

f o r m u l a t e t h e p r o b le m and t o s i m p l i f y q u e n tly ,

th e m a th e m a tic s , c o n s e ­

t h e t h e o r y g i v e s a good q u a l i t a t i v e , b u t o n ly a n

a p p r o x i m a t e q u a n t i t a t i v e p i c t u r e o f t h e phenomena w h ic h t a k e pi*-* c e .

The s u b j e c t h a s b e e n d i s c u s s e d by s e v e r a l a u t h o r s and i n m ost c a s e s ,

t h e p ro b le m h a s b e e n i n v e s t i g a t e d by one o r o t h e r

o f t h e f o l l o w i n g m e th o d s . 1.

C i r c u i t m e th o d .

The e x c i t a t i o n o f a one d i m e n s i o n a l

t r a n s m i s s i o n l i n e by a n e l e c t r o n beam i s c o n s i d e r e d . 2.

f i e l d m e th o d .

M ax w ell e q u a t i o n s a r e u s e d t o d e s c r i b e

c o n d itio n s b o th i n s id e ana o u ts id e th e h e l i x .

The b e s t known p a p e r o f t h e f i r s t t y p e i s t h a t by P i e r c e ^ a n d t h e se c o n d m eth o d h a s b e e n p r e s e n t e d by Chu an d J a c k s o n ^ ) an d K y d b e e k . ' ^ )

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A p r e s e n t a t i o n o f t h e th e o r y its

o f t h e t r a v e l l i n g wave t u b e ,

a s s o c ia te d c i r c u i t s ana o th e r r e l e t e d

to p ic s, has re c e n tly

a p p e a r e d a s a s e r i e s o f a r t i c l e s i n B e l l S y stem t e c h n i c a l j o u r n a l by P i e r c e . t h e s e

a r t i c l e s a r e so o n t o b e

p u b l i s h e d i n b o o n fo rm . (b ) 1.

P ie r c e T h eo ry .

A s s u m p ti o n s .

T h is a n a l y s i s i s b a s e d on th e f o llo w in g a s s u m p tio n s ; (a)

S m a ll s i g n a l .

(b )

E l e c t r o n s h a v e a x i a l b u t no t r a n s v e r s e v e lo c ity .

(c)

A i l e l e c t r o n s a r e a c t e d on by t h e seme e x te rn a tin g f i e ld .

(d )

E le c tro n v e lo c ity

i s s m a l l com pared w i t h

th e v e lo c it y o f l i g h t .

I n common w i t h m o s t a u t h o r s w r i t i n g on t r a v e l l i n g wave tu b e s , P ie r c e d e v e lo p s a sm a ll s i g n a l th e o r y .

T h is a ssu m p tio n

i m p l i e s t h a t t h e a l t e r n a t i n g com pon ent o f t h e v e l o c i t y o f a n e l e c t r o n i n t h e beam i s v e r y v e lo c ity of th e e le c tro n .

s m a l l co m pared w i t h t h e o v e r a g e

The p u r p o s e o f t h i s s u p p o s i t i o n i s

t o m a le a s i m p l i f i c a t i o n i n t h e m a t h e m a ti c s -which would o t h e r ­ w is e c o n t a i n n o n - l i n e a r d i f f e r e n t i a l e q u a t i o n s .

An i d e a l i s e d t u b e i s c o n s i d e r e d i n w h ich t h e e l e c t r o n s a r e c o n s tr a in e d to fo llo w a s t r a i g h t p a th p a r a l l e l to th e a x is

R e p ro d u c e d with perm ission of the copyright owner. F urther reproduction prohibited without perm ission.

j -

o f th e h e li x ,

T h i s c o n d i t i o n i s a p p r o a c h e d i n p r a c t i c e by

t h e u s e o f a s t r o n g m a g n e ti c f o c u s s i n g f i e l d w h ic h i s p r o ­ d u c e d by a n e x t e r n a l s o l e n o i d s u r r o u n d i n g t h e h e l i x .

The t h i r d a s s u m p t i o n i m p l i e s e i t h e r t h a t t h e e l e c t r o n beam h a s a v e r y

s m a l l d i a m e t e r o r t h a t t h e beam h a s a h o l lo w

c y li n d r ic a l shape.

The f o u r t h a s s u m p t i o n i s made so t h a t any c h a n g e i n t h e m ass o f a n e l e c t r o n due t o i t s v e l o c i t y may b e n e g l e c t e d . The o r d e r o f m a g n itu d e o f t h e e l e c t r o n v e l o c i t y i s a b o u t one t e n t h t h e v e l o c i t y

2.

i n m ost tu b e s

of lig h t.

F o r m u l a t i n g t h e P ro b le m .

The a m p l i f i c a t i o n w h ic h o c c u r s i n a t r a v e l l i n g wave tu b e i s du e t o a n e x c h a n g e o f e n erg y b e tw e e n t h e e l e c t r o n beam and t h e e l e c t r o m a g n e t i c f i e l d o f t h e s i g n a l on th e h e l i x .

V e lo c ity

m o d u l a t i o n o f a n e l e c t r o n beam o c c u r s when t h e beam t r a v e r s e s a r e g i o n i n w h ich a n e l e c t r i c f i e l d a l t e r n a t e s i n t h e d i r e c t i o n of e le c tro n tra v e l. m a tic a l e q u a tio n .

T h i s a c t i o n c a n b e e x p r e s s e d by a m a th e ­ L i k e w i s e , when a v e l o c i t y m o d u la t e d beam

p a s s e s th ro u g h an e l e c t r i c f i e l d , a l t e r n a t i n g a s p r e v io u s ly , t h e r e i s a n e f f e c t on t h e f i e l d w h ic h may a l s o b e e x p r e s s e d m a t h e m a t i c a l l y by a n e q u a t i o n .

I n th e f a m i li a r k ly s tr o n tu b e , th e f i r s t o p e ra tio n ta k e s p l a c e i n t h e b u n c h e r c a v i t y w h e re a s t h e se c o n d o c c u r s i n t h e

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c a tc h e r c a v ity .

I n an a n a lo g o u s m anner, b o th o p e r a t io n s p ro ­

c eed sim u lta n e o u s ly a lo n g d i f f e r e n t s e c t i o n s o f th e h e l i x i n t h e t r a v e l l i n g wave t u b e .

H ence, i f

th e e q u a tio n s r e p r e s e n t ­

i n g e n e r g y e x c h a n g e o u t l i n e d a b o v e , a r e c o m b in e d , t h e n t h e c o n d itio n s n e c e s s a ry f o r s u c c e s s f u l o p e r a tio n o f th e tu b e as an a m p l i f i e r sh o u ld a p p e a r.

(a)

F i e l d p r o d u c e d by a v e l o c i t y m o d u la te d e l e c t r o n beam ,

C o n s i d e r a s e c t i o n o f an i n f i n i t e , one d i m e n s i o n a l , l o s s ­ l e s s t r a n s m i s s i o n l i n e w i t h a c o n v e c t i o n c u r r e n t I 0+-i f l o w i n g p a r a l l e l to th e l i n e ( F ig . 2 ) . ic -

i 0 *- i

n n m n n n rw w

± F ig. 2

I h e d i s t r i b u t e d i n d u c t a n c e and c a p a c i t a n c e o f t h e t r a n s ­ m i s s i o n l i n e a r e so c h o s e n t h a t i / r n c = Vp w h e re Vp i s

th e p h ase v e lo c ity

(i) o f t h e wave on t h e h e l i x a c t i n g

R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.

-li­ on t h e e l e c t r o n s . Let

v

= IO+ 1

- e le c t r o n c o n v e c tio n c u r r e n t.

I0

- fi.vere.ge c o n v e c t i o n c u r r e n t ,

i

- a l t e r n a t i n g com p o n en t o f c o n v e c t i o n c u rre n t,

( s i n u s o i d a l w i t h tim e )

Z

= 0 +- jX

- s e r i e s im p e d a n c e p e r u n i t l e n g t h .

1

~ 0 + jB

- s h u n t a d m itta n c e p e r u n i t l e n g t h ,

d

- d isp lac em e n t c u r r e n t ( s in u s o id a l w i t h tim e ) .

E n c l o s e a s m a l l vo lu m e o f t h e e l e c t r o n c o n v e c t i o n c u r r e n t by a h e m i s p h e r i c a l s u r f a c e a s shown i n F i g . 3*

I c ~

1

't~ 1

F ig . 3 S i n c e t h e e l e c t r o n beam i s v e l o c i t y m o d u l a t e d , a d i s p l a c e m e n t c u r r e n t e x i s t s and t h e r e l a t i o n b e tw e e n t h i s and t h e e l e c t r o n c o n v e c t i o n c u r r e n t i s now t o b e e s t a b l i s h e d .

From e le m e n t a r y t h e o r y , t h e e l e c t r i c f l u x t h r o u g h t h e h e m i­ s p h e r ic a l s u rfa c e i s r e l a t e d to th e ch arg e en clo sed a s fo llo w s ; ^ D .d s

= q

w here D -

(2) e l e c t r i c f l u x d e n sity

R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.

-1 2 -

q - e l e c t r i c c h a r g e e n c l o s e d by t h e s u r f a c e d s - e le m e n t o f s u r f a c e . A lso th e c o n v e c tio n c u r r e n t f lo w in g o u t th ro u g h th e s u r f a c e m ust e q u a l th e tim e r a t e o f d e c r e a s e o f ch arg e, e n c lo s e d , th u s I e = - V a ft C om bining e q u a t i o n s (2) and (3 ) we g e t At

b lc

-

0

U)

Let le w h ere

= f J c - as

J c - c o n v e c tio n c u r r e n t d e n sity

Then fro m e q u a t i o n (4 ) jF ( J d *ds + J c *ds)

= 0

(5 )

T h i s e q u a t i o n s t a t e s t h a t t h e sum o f t h e d i s p l a c e m e n t c u r r e n t and t h e c o n d u c t i o n c u r r e n t t h r o u g h t h e s u r f a c e i s z e r o .

P ie rc e

a ss u m e s t h a t t h e d i s p l a c e m e n t c u r r e n t a l o n g t h e s tr e a m i s n e g ­ l i g i b l e com pared w i t h t h a t fro m In th is case

J

=

t h e stream , t o

th e c i r c u i t .

~ j jl c As

= if

(6)

The e q u a t i o n s f o r t h e l o s s l e s s l i n e may now b e w r i t t e n = - jLI

(7 )

= -jBV - hJ

( 6)

Az

L e t a l l q u a n t i t i e s v a ry w i t h d i s t a n c e a s e x p ( - r z ) th e p ro p a g a tio n c o n sca n t o f th e E q u a t i o n s (6 )

com b in ed beam

w here

T is

ana l in e .

(7 ) an d (d) becom e r e s p e c t i v e l y

j

=

ri

R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.

(*)

-1 3 -

-rv «

-jxi

( iq )

-n = -jBv-hri

(u)

E l i m i n a t e I fro m t h e s e e q u a t i o n s t o o b t a i n

v ( r V bx)

= -jrxi

I n th e a b se n c e o f th e e l e c t r o n stre am ( i = o )

(12) we h a v e fro m

e q u a t i o n (1 2 ) t h e p r o p a g a t i o n c o n s t a n t T, o f a n o r d i n a r y t r a n s ­ m issio n l i n e . w h e re

T, =- j/B X

(1 3 )

I t s c h a r a c t e r i s t i c im p e d a n c e i s

2 = IT °

From e q u a t i o n s (1 2 )

Jb

(M)

(1 3 ) and (1 4 )

y -

-ij l z a

r - r,7-

( 15)

T h i s e q u a t i o n e x p r e s s e s t h e r e l a t i o n s h i p b e tw e e n t h e v o l t a g e ¥ w h ich w i l l a p p e a r on t h e t r a n s m i s s i o n l i n e and t h e e l t e r n & t i n g com ponent o f t h e e l e c t r o n c o n v e c t i o n c u r r e n t .

(b )

E f f e c t o f e l e c t r i c f i e l d on t h e e l e c t r o n beam .

L e t uQ - a v erag e v e lo c ity o f th e e le c tr o n s VQ - D. C. a c c e l e r a t i n g v o l t a g e I Q - av erag e e le c tr o n c o n v e c tio n c u r r e n t PQ - a v e r a g e c h a r g e p e r u n i t l e n g t h v

- a l t e r n a t i n g com p onen t o f v e l o c i t y

i

- a l t e r n a t i n g com p o n en t o f e l e c t r o n C o n v e c t io n c u r ­ ren t

P - a l t e r n a t i n g com ponent o f l i n e a r c h a rg e d e n s ity e /m - c h a r g e t o m ass r a t i o o f a n e l e c t r o n a

- a c c e le ra tio n

R e p ro d u c e d with p erm ission of the copyright owner. F urther reproduction prohibited without perm ission.

- H -

The f o r c e e q u a t i o n f o r &n e l e c t r o n due t o t h e l o n g i t u d ­ in a l e le c tric fie ld is ma - - e£ and s i n c e

- - V\[ bz

£

th en

( 16 )

ma =

A c c e le ra tio n i s

t h e tim e r a t e o f c h a n g e o f th e v e l o c i t y and

t h e r e f o r e e q u a t i o n (1 6 ) may b e w r i t t e n d ( u 0 +- v ) _