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I E E E T R A N S A C T I O N S O N N U C L E A R S C I E N C E

EFFECT O F I O N ENGINE EXHAUST O N THERADIATION FIELD OF A DIPOLE ANTENNA

P A R T I : I NF IN IT E S LA B MODEL FOR T HE E XH AU ST ION BEAMA . R . M . Rashad

SUMMARYA theory o f t h e effect o f an ionrocket e n g i n e ex hau st on t h e radiationpattern o f a d ip ol e a nt en na i s presented.T h e e l e ct r om a gne t i c e q u at i on s a r e c o m -bined with t h o s e describ ing t h e exhaustplasma beam t o calculate i t s equivalenteffective dielectric c o n s t a n t . In part Io f t h i s p a p e r , t h e beam i s represented b ya n i nf ini te slab o f a homogenous plasmam e d i u m , a s i s u s u a l l y considered i n s p a c ec h a r g e neutralization studies o f t h i s

t y p e o f e n g i n e . The equations o f prop-agation o f e l ec t ro m ag n et i c w av es throught h e beam medium, a r e used t o calculatet h e total dipole radiation f i e l d . T h emethod o f s te ep es t d es c en t i s applied f o rt h e evaluation o f t h e i n t e g r a l s . I t i sf o u n d that t h e dipole radiation patterndepends greatly on t h e beam c h a r a c t e r i s t i cp a r a m e t e r s .I . INTRODUCTION

T h i s paper considers t h e e f f e c t o ft h e e x h a u s t o f an ion engine rocket ont h e radiation field o f an e le ct ri c d ip ol eantenna installed in i t . I t i s wellknown from t h e design o f s uch a n enginethat t h e i o n beam provides t h e t h r u s t ,while t h e electron b ea m provides t h e nec-e s s a r y neutralization o f t h e space charge.T h i s neutralization i s necessary f o r ap r a c t i c a l operation o f t h e r o c k e t .Previous studies on t h e engine ionb e a m n eu t ra l iz a ti o n problems have consid-e r e d i t a s a h om o ge ne ou s p la s ma beam.Rosenbluth e t a l . ( 1 9 6 2 ) used a o n e -dimensional i n f i n i t e slab model t o rep-resent t h e beam. Other investigators e . g .S e i t z e t a l , 1960; Mirels, 1960;Baldwin,1 9 6 1 ; Halverson e t a l . , 1 9 b 0 a ls o basedt h e i r ion b e a m n e ut r al i za t io n s t u d i e s ons uch m o d e l .

T h i s pape r w i l l b e d i v i d e d i n t o twoparts: t h e first o n e c o n s i d e r s t h e i n -finite s l a b model to represent t h e engineion b e a m ; w h i l e the second part a s s ume s acy lind r ica l configuration for t h e beam.The effect o f these c on fi g ur at io ns ; r e p-resenting t h e engine ion beam exhaust; onthe radiation field o f a short e l e c t r i c* Electrical E n g i n e e r i n g D e p a r t m e n t ,U n i v e r s i t y o f Detroit, Detroit,Michigan

d i p o l e antenna i n s t a l l e d in i t will beanalyzed i n full d e t a i l s .I I . THE MATHEMATICAL M ODEL

In t h i s p a r t o f the p r e s e n t a t i o n , t h eengine i o n b e a m will be r e p r e s e n t e d as a ni n f i n i t e slab o f a h o mo ge ne o us p la sm am e d i u m , o f thickness I as shown i nf i g u r e ( 1 ) . The e le ct r ic d i p o l e will b eo r i e n t e d p a r a l l e l to t h e y a x i s , and a ta distance a above t h e b e a m . T h e b e a ms u r f a c e fluctuations wi l l be n e g l e c t e di n t h i s a n a l y s i s , h o w e v e r , these e f f e c t sw i l l be taken into consideration in afuture p a p e r .T h e d i p o l e s o ur ce current c a n b er e p r e s e n t e d as:j = (7z) g Y x -4) ____ ( 1 )

where: J i s t h e s o ur ce current d e n s i t y ,a i s a unit ve ct o r i n t h e a directiona n d i i s the usual Delta function.I I I . T H E E NGINE ION BE AM ALONE

The plasma:e r e d m o v i n g in 1v e l o c i t y - . If o r m o f e p [i n g e q u a t i o n s c xthe beam:- - tf r v'r

- B t - t- _ t - 7A("- )7X V

ion beam w i l l be consid-t h e z d i r e c t i o n with aF o r a d e p e n d e n c e i n t h eC g W f - K z ) 3 , t h e follow-D a n b e written t o describe----(2 )

=0 - _ _ - ( 3 )4 7 2 ___

* S 47Te4e,c2 IV

- 4fjc 2 j - CC - 4( 4 )

where: n i s t h e d e n s i t y , u - the v e l o c i t y ,.1

O r i g i n a l m a n u s c r i p t r e c e i v e d A u g . 5 , 1 9 6 4

1 2 A p r i l 1

46

e 4 -rn IV

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e t h e c h a r g e , /, t h e m a s s , ' , t h e un-perturbed d e n s i t y , a n d j . t h e c u r r e n to f t h e particles in t h e ion b e a m . Ina d d i t i o n , E r e p r e s e n t s t h e e l e c t r i cf i e l d , c t h e v e l o c i t y o f l i g h t i n t h ep l a s m a m e d i u m , w h i l e I ' i s t h e wavem n m b e r o Let c . b e the Maxwellian d i s -tribution o f t h e ion p a r t i c l e s w i t hr e s p e c t t o v e l o c i t i e s . U s i n g t h e per-t u r b a t i o n t e c h n i q u e t o s o l v e t h e a b o v enon linear e q u a t i o n s , t h e f o l l o w i n gr e l a t i o n s are o b t a i n e d :

L (C O 4, ,,4r=_ )e__- ( 6 )I n relations ( 5 ) a n d ( 6 ) ' i ' a n da r e c o n s i d e r e d a s s m a l l a e v i a t i o n sfrom t h e i r c o r r e s p o n d i n g s t a t i o n a r yv a l u e s respectively. From t h e s e e q u a -t i o n s , t h e f o l l o w i n g relation i so b t a i n e d :

J e 4 < = - ( ,)2 K 2 C 2 ( 7 )T h e d i e l e c t r i c constant o f t h e beam willtherefore b e :

T h e f u n c t i o n y ' w i l l b e a solution o f t h ef o l l o w i n g e q u a t i o n s t h a t describe t h ew a v e propagation through t h e plasma beamand t h e vacuum surrounding i t :

tX2 K 2 =_5 ( z ) 9 ( z - a )+ t

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I E E E T R A N S A C T I O N S O N N U C L E A R S C I E N C E

, = C e I ( + ) c s c \ g . s < \ ( x 7 g )for - - - - - ( 1 6 - c )where: X22 , , , 2

2 2 2

C , C 2 , C 3 a r d R a r e c o n s t a n t s t o b ed e t e r m i n e d . For o a n d "- t o b e c o n t i n -UOU8 a t , t h e r e f o r e :c Cr 3 + X 0 e s G - - - ( 1 7 )X = Reflection c o e f f i c i e n t a t t h e v a c u u m -beam i n t e r f a c e oA t - a,P i s c o n t i n u o u s , but a -i s d i s c o n t i r n o u s . I n t e g r a t i n g equation( 1 3 ) from x2 to g=ig i v e s :

Q

which s p e c i f i e s t h e d i s c o n t i n u i t y in t h ed e r i v a t i v e a t t h e s o u r c e . T h e b o u n d a r yc o n d i t i o n r a t x = e g i v e s n- c 2 J a

C - It(_ e2 2 t J

S u b s t i t u t i n g t h e s e v a l u e s r o r C , , C 2 ,C 3 and R in e q u a t i o n s ( 1 6 a , b & n d c ) ,t h e solution f o r 5 6 i s :r L J C x - - z ) -x4 )

< + = e t ~ t e1 2

2 + S-[sa ) -p ( I f W ) . cs c \ . s " , \ ( z o t2 , c lEquations ( 1 9 - a ) a n d ( 1 9 - b ) can b ecombined t o ge t h e r to giv e :

- 4 rI -a l0 = e2 - c s

for t ; i_ _ _ _ ( 1 9 - a )f o r O S t

( 1 9 - b )f _ o r ( 1 9 _ c ).Wwmm(19-c )

x v - a I+ w ? e f o r X a > o

_- (19-d)

By u s i n g t h e i n v e r s e t o t h e L a p l a c et r a n s f o r m a l r e a d y i n t r o d u c e d , t h e r e f o r e:

( a ( x , Z ) = -- I - c;K r I I-c*a I Y zw h e r e C i s a c o n t o u r t o b e d e t e r m i n e dwhich s a t i s f i e s t h e proper behavior o fthe function y ' a t i n f i n i t y . From equa-tion ( 2 0 ) , V ( x , z ) i s a f u n c t i o n o f ther e f l e c t i o n c o e f f i c i e n t R . From e q u a t i o n( 1 7 ) , t h e t e r m \ c O t ' x i s a n e v e nfunction o f \ . S i n c e 2 ) yt h e r e f o r e i t d o e s n o t matter which s i g no f t h e square root i s c h o s e n . H o w e v e r ,f o r t h e s e c o n d term in e q u a t i o n ( 1 7 ) ,t h e following relations s h o u l d b e s a t -i s f i e d i n o r d e r t o have o u t w a r d p r o p -a g a t i n g w a v e s :

W e ( j ) >o a n d J m ( S ) < T h e r e f o r e t h e f u n c t i o n y i ' b e c o m e s :

I. 2 .- e

7 / 2 , 4 AC/ / +

and

C Y d if z i d z i + ~ ' e i t l e z t e d_ _ - _ - ( 2 1 )2 2 1 t t 2C o g z 2 + t r 2 b (22;3+ & *

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o b t a i n e d . I f s u c h a p a t t e r n I s c o m p a r e dt o t h a t o f the d i p o l e r a d i a t i n g in a vac-u u m medium a l o n e , a big d i f f e r e n c e i n t h etwo p a t t e r n s i s n o t i c e d . T h i s d i f f e r e n c ei n d i c a t e s t h e e f f e c t o f e n g i n e e x h a u s tion beam on t h e radiation f i e l d o f thedipole a n t e n n a . I t c a n b e e a s i l y s h o w nfrom e q u a t i o n s ( 8 ) , ( 9 ) , ( 2 1 ) and ( 2 2 )t h a t t h i s deviation i n t h e radiationpattern i s a f u n c t i o n o f t h e o p e r a t i n gf r e q u e n c y a n d the beam p a r a m e t e r s .

V . C O N C L U S I O N S BT h i s study shows t h a t t h e e x h a u s tb e a m s from an ion r o c k e t e n g i n e h a s an o t i c e a b l e e f f e c t on t h e radiation p a t -tern o f a d i p o l e antenna attached to t h ee n g i n e . T h e radiation pattern o f t h i sa n t e n n a c a n b e c a l c u l a t e d f r o m t h e f i e l de x p r e s s i o n s g i v e n i n t h i s s t u d y .

T h e a ut h o r expresses his a p p r e c i a t i o nt o Professor R . W . A h l q u i s t , from t h eU n i v e r s i t y o f D e t r o i t , for t h e many h e l p -f u l d i s c u s s i o n s d u r i n g t h e p r e p a r a t i o n o ft h i s p a p e r .APPENDIX

L e t f ( i u ) = _ c /- C o s ( / 4 )= L e < S r , , ( I e g )

- a t , ' = or o = 6 , , o

Since a complex function ca n n o t have am a x i m m o r a m i n i m u m , t h e r e f o r e t h estationary p o i n t s are s a d d l e p o i n t s . Int h e vicinity o f a s a d d l e p o i n t , a T a y l o rexpansion can b e u s e d . T h i s g i v e s t h ef o l l o w i n g :o " > 0 , g /f --2?0~42

= _ 4 < f' + ' - 4 < , - ( r - 0 + 4

( / 4 - ) '

---- ( 2 4 )I t i s c l e a r from relation ( 2 4 ) t h a t t h ef i r s t d e r i v a t i v e v a n i s h e s , w h i l e thes e c o n d d e r i v a t i v e e q u a l s K at 6 eLet / o , J - r b e t h e r a d i a l a n d a n g u l a r c o -o r d i n a t e s with t h e o r i g i n a t 6 i n t h e , -p l a n e . T h e r e f o r e :T h e i n t e g r a l s f o r ( 4 d r e c t a n d4 e f l e c t e d shown in eq u a on ( 2 1 ) willb e e v a l u a t e d by t h e m e t h o d o f s t e e p e s tdescent .A: EVALUATION O F THE D I R E C T F I E L DI N T E G R A L

L e t y = < 1

Let J 4 LF1 2 /j&32

= 1 . /.4c i f / '= 'K 5>1 , # A

from which:

. . c < , = K C o s c r IJ ' 3 = K 3 , < , e r c a , X 2

T h e r e f o r e :2 .27 ~ K = - Kc o s

= z c ' A C o 7 s 9z = r s,

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taken a l o n g f o FF o r any value o f, / " : -2 =K > iB : E V A L U A T I O N O F 'F I E L D I N T E G R A L

e4 7 r l < r= . e/ 7 7 T - 1 - - _ - r ( 2 b )

T H E : R E F L E C T E DUsing the n o t a t i o n s g i v e n i n Section( b . l ) , t h e r e f l e c t e d f i e l d i n t e g r a lbecomes:

- KI . C o s ( j i - 6 )Along t h i s p a t h : /, = -z + et h e f i r s t quadrantand / 6 - /ei n t h e t h i r d q u a d r a n t

-4l

= e & / / i n t h eand - f fc / = . - e >46 / 4 i n t h eT h e r e f o r e , t h e i n t e g r a l o fb e c o m e s :Cf = - - C e

J c s D c( / < , , 7T ) 0 24 i -" 1 1

. : - - -4 C e C / / o 76

4 i nd 79 f )

f i r s t q u a d r a n tt h i r d q u a d r a n t .equation ( 2 3 )

_ /< ie C g ,I t i s c l e a r that w hen , < r i s very l a r g e ,e , ( - L I r { ) c a r s b e n e g l e c t e d for P > / -

_ C - I < r -_7r ) 6r fe d l { q = - 2t . eC.- eC

2- 2 { K re i

Due t o t h e r a p i d decay o f t h e e x p o n e n t i a l ,t h e i n t e g r a l f r o m o t o , , = i s n e a r l yequal t o t h e integralc - 0 2

2 / ; / I o 7 Te c " O - _ _/ J / 2 K < t0

-6107 ( U )

C

where:-c o s / - X cof/ I Co S / L , , - \ C . '

0 1 / - - - _ ( 2 7 )

x - 1 g _ _ _ _ - - ( 2 8 )The r a d i a l c o - o r d i n a t e w i l l have i t so r i g i n a t z= o and x = - . T h econtour C o f i n t e g r a t i o n g i v e n byequation ( 2 7 ) w i l l b e d e f o r m e d i n t o aSDC p a s s i n g t h r o u g h t h e s a d d l e p o i n t,a -= . I n v e s t i g a t i o n o f relation ( 2 8 )s h o w s t h a t ( / - ) m ay h a v e t h e f o l l o w i n gt y p e s o f p o l e s :C a s e a ) N o P o l e s f o r R ( g ) i n t h e v i c i n -i t y o f t h e S a d d l e P o i n t : -L e t R ( p ) h a ve i n g e n e r a l a s u r f a c ewave p o l e a n d a leaky wave p o l e a sshown i n f i g u r e ( 3 ) . T h e contour i sw a r p e d around t h e p o l e s as s h o w n . T h eresidue terms h a v e to b e c o n s i d e r e d i na d d i t i o n t o t h e i n t e g r a l along t h e S D C .L e t / ' s a n d / ' < b e t h e v a l u e s o f , ua tt h e s u r f a c e wave pole and t h e l e a k ywave p o l e r e s p e c t i v e l y . A l s o , G ( , " ) t or e p r e s e n t t h e r e s i d u e o f t(,) a t ap o l e . T h e r e f o r e , t h e i n t e g r a l o f e q u a -tion ( 2 7 ) be co me s :- z ? ( / J ) e c . _ 2 . Z G 6 / s ) e - i Z 0 E G ( 2 L )7T~~~~SL

S D CThe series s h o wn a b o v e arises from i n -t e g r a t i o n a b o u t the c i r c l e s s u r r o u n d i n gt h e p o l e s . E x p a n d i n g 2 i n a T a y l o rseries a b o u t 6 s

. . the i n t e g r a l o f equation ( 2 3 ) b e c o m e s :- ' 6 < - - 4 7 - ) f o r l a r g ee l

00 )7/ 4 ) _ R ( 4 ' ) + ' ? 6 /' 7 = /

_ ' ( K r _ - i f )

-i 1 . . - ^

A p r i l1 6

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R A S H A D : E F F E C T O F I O N ENGINE EXHAUST O N A D I P O L E , P A R T I

where: =-(&Al on g the SDC n e a r & * the followingrelations c a n be wr it te n:

c _ r4_ 1 - ]/ L 6 = f o e7 r

( -62 e 4= - e 4 c / /

T h e integral o f4 r ) -

'70

a n d( F i r s t q u a d r a n t )a n d( T h i r d q u a d r a n t )equation ( 2 7 ) b e c o m e s :

C ~ o o 2e-k ( /Q.-e

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H a l v e r s o n , W . D . ; D e G r o f f , H . M . a n dH o l m e s , R . A . ( 1 9 6 0 ) . An electro g a sdynamic approach to t h e ion j e t c h a r g eneutralization p r o b l e m , D i g e s t o fP a p e r s , American R o c k e t S o c i e t y S y m p o -s ium ( M o n t e r e y . , C a l i f . , N o v . 3 - 4 )M i r e l s , H . ( 1 9 b 0 ) , On i o n rocket neu-t r a l i z a t i o n , D i g e s t o f P a p e r s , AmericanR oc ke t S oc ie ty Symposium ( M o n t e r e y ,C a l i f . , N o v . 3 - 4 )O b e r h e t t i n g e r , F . ( J u l y - S e p t . 1 9 5 9 ) , O na m o d i f i c a t i o n o f W a t s o n ' s L e m m a J .R e s . N B S , 6 3 B , p p . 1 5 - 1 7R o s e n b l u t h , M . N . , P e a r l a t e i n , L . D . a n dS t u a r t , G . W . ( 1 9 6 2 ) . N a u t r a l i z a t i o no f ion b e a m s , D i g e s t o f P a p e r s ,A m e r i c a n Rocket Society Electric

P r opuls ion C o n f e r e n c e ( B e r k e l e y ,C a l i f . , March 1 4 - 1 6 )S e i t z , R . N . ; Shelton, R . and Stuhlinger,E . ( 1 9 6 0 ) , Present status o f t h e b e a mneutralization p r o b l e m . D i g e s t o f

Papers. A m e r i c a n Rocket So cie t y S y m p o -sium ( M o n t e r e y , C a l i f . , Nov. 3 - 4 )Stratton, J . A. ( 1 9 4 1 ) , Ele ct r o ma gne t icT h e o r y ( M c G r a w - H i l l C o . , New York)Van der Waerden, B . L. ( 1 9 5 0 ) , On themethod o f s a d d l e points, Appl. S c i .R e s e a r c h , 2 B , 3 3 - 4 3W a i t , J . R . ( D e c . 1 9 5 7 ) , Excitation o fsurface waves on c o n d u c t i n g , s t r a t i f i e d ,d i e l e c t r i c coated and corrugatedsurfaces, J . R e s . NBS 5 9 , p p . 3 6 5 - 3 7 7

D i p o l e

. Z

yF i g u r e 1 . E x h a u s t I o n B e a m a n dD i p o l e A n t e n n a C o n f i g u r a t i o n .

F i g u r e 2 . S t e e p e s t - d e s c e n t c o n t o u ro f i n t e g r a t i o n f o r t h e d i r e c t f i e l d . F i g u r e 3 . S t e e p e s t - d e s c e n t c o n t o u rf o r t h e R e f l e c t e d F i e l d .

1 8 A p r i l