Wave propagation over a finite ground plane
by Leo Raymond Spogen
A THESIS Submitted to the Graduate Committee in partial fulfillment of the requirements for the
degree of Master of Science in Electrical Engineering at Montana State College
Montana State University
© Copyright by Leo Raymond Spogen (1952)
Abstract:
The consideration of propagation over a finite ground plane is coming into prominence more each day
especially with the increasing use of line of sight transmission. Each time a television broadcasting
antenna is placed at the top of a building a finite ground plane or an irregular ground plane must be
considered. An excellent example of radiation over a finite ground plane is radiation from an antenna
on an airplane.
Although practically all problems pertaining to propagation over a finite ground plane will consider
three dimensional surfaces and not plane surfaces, it is feasible that a thesis of this nature may be the
stepping stone to future developments. The ground planes considered, in this thesis were circular
disks® The author has attempted to obtain the patterns ex= perimentally and then mathematically
verify the results.
In this thesis, the author has not only considered the effects of the discontinuity caused by the finiteness
of the ground plane but, also, the hole separating the antenna from the ground. The mathematical
results do not have close agreement with experimental results. The author has shown where errors can
exist and gives an explanation of how these errors affect results. WAVE PROPAGATION OVER
A FINITE GROUND PLANE
by
LEO RAYMOND SPOGEN9 JR0
A THESIS
S u b m itted t o th e G rad u ate Committee
in
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
M aster o f S c ie n c e i n E l e c t r i c a l E n g in e e rin g
at
Montana S t a t e C o lleg e
j a r D epartm ent
Bozieman9 Montana
Ju n e 9 19$2
2'
ACKNOWLEDGEMENT
T h is t h e s i s was u n d e rta k e n a t th e su g g e s tio n o f Montana S ta te
C ollege
P r o f e s s o r s R o b ert C. S e ib e l o f th e E l e c t r i c a l E n g in e e rin g
D epartm ent and Henry A ntosiew icz o f th e M athem atics D ep artm en t.
The
a u th o r w ish es to e x p re s s h i s a p p r e c ia tio n t o a l l th e E l e c t r i c a l E n g in eer­
in g and M athem atics s t a f f members f o r t h e i r v a lu a b le h e lp and s u g g e s tio n s
and p a r t i c u l a r l y to P r o fe s s o r S e ib e l, u n d er whose s u p e r v is io n th e work
was com pleted.
103061
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENT
ABSTRACT
0 * 0 0 * 0 * 0 * 0
0 * * 0 » 0 * * * » 0 0 0 » 0 * 0
INTRODUCTION
2
0.
0 *- » * * 0 0 0 » » » 0 0 * 0 » 0 » * 0 0 * 0 0 » » » 0 * 0 * * 0 » » 0 * 0 * » » *
* » » O 0 » 0 0 0 0 0 0 0 0 0
0^0
OOO 0 » » * 0
O O » 0 O *
»0*"0
0 * 0 0 » 0 0
0 » *
OO 0 » * * * 0 » »
5
OBTAINING EXPERIMENTAL PATTERNS
The Use o l Model A ntennas o*************************** * * * ** 0 6
T e s t Appax8a t u s o*** ******0 . 0 o**** ***** ** o** **o ** * ** ** ***o o***
6
TeSt P^OCedUre 0***0* 000 000 *000*0000 0'*000 0*0**00*0**0******* H '
MATHEMATICAL ANALYSIS
Preliminary C o n sid e r a tio n s and B a s ic A ssum ptions o********** 13
S o lu t io n f o r V ecto r M agnetic P o t e n t ia l in I n t e g r a l Form «*«» 16
Method o f I n te g r a t io n o****************** * * * *»****a*»*»»*oo* 20
S o lu t io n f o r th e M agnetic F i e l d I n t e n s i t y * * * «* *»«*»o.* * * * * 23
0
The. E l e c t r i c F i e l d I n t e n s i t y &** ***.* 0
DISCUSSION OF RESULTS
0
tk6 o o t t o o e e o
0
6
0
6
0 0 0 0 0
*0 ** 0 ** 0 *0 0 *** 0
0
0 0
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» e o t t o * o o o o o o o - c o o o o t t o o o o t t o . o a o » » e o
33
APPENDIX A
63?u n e n t Tt©Slil*bS ©o*o o » o o o o ©oo o« a o » o o oo c o a c o o oo Oo o o Oo Oo o o 36
APPfiMDIXB
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IsITfiRATUPihi CITED AMD CONSlHiTfiD ©o®»o.oooooo6»oo»6oft6o»oooa, 000» &&00900
ABSTRACT
The c o n s id e r a tio n o f p ro p a g a tio n o v e r a f i n i t e ground p la n e is .
coming i n t o prom inence more each day e s p e c i a l l y w ith th e in c r e a s in g u se o f
Iine=O f=-Sight tra n s m is sio n ,, Each tim e a t e l e v i s i o n b ro a d c a s tin g a n ten n a
i s p la c e d a t th e to p o f a b u ild in g a f i n i t e ground p la n e o r an i r r e g u l a r
ground p la n e must be considered® An e x c e lle n t example o f r a d i a t i o n o v er a
f i n i t e ground p la n e i s r a d i a t i o n from an a n te n n a on an airp lan e®
,
A lthough p r a c t i c a l l y a U problem s p e r ta in in g t o p ro p a g a tio n over a
f i n i t e ground p la n e w i l l c o n s id e r th r e e d im e n sio n a l s u r f a c e s and n o t p la n e .
s u r f a c e Sg i t i s f e a s i b l e t h a t a t h e s i s o f t h i s n a tu r e may be th e step p in g
s to n e t o f u t u r e developments® The ground p la n e s c o n s id e re d , in t h i s t h e s i s
w ere c i r c u l a r disks® The a u th o r has a tte m p te d t o o b ta in th e p a tte r n s ex=
p e r!m e n ta lly and th e n m a th e m a tic a lly v e r i f y th e re s u lts ®
I n t h i s t h e s i s 3 th e a u th o r h as n o t o n ly c o n sid e re d th e e f f e c t s o f
th e d i s c o n t in u ity c a u se d .b y the. f i n i t e n e s s o f th e ground p la n e b u t , a l s o ,
th e h o le s e p a r a tin g th e a n te n n a from th e ground®
The m a th e m atica l r e s u l t s
do n o t have c lo s e agreem ent w ith e x p e rim e n ta l re s u lts ® The a u th o r has
shown w here e r r o r s can e x i s t and g iv e s an e x p la n a tio n o f how th e s e e r r o r s
a f f e c t re s u lts ®
INTRODUCTION
A. r a d i a t i o n p a t t e r n i s a p o la r cu rv e form ed by p l o t t i n g th e
e l e c t r i c f i e l d i n t e n s i t y a t v a rio u s a n g le s , b u t a t a c o n s ta n t d is ta n c e
fro m a r a d i a t i n g s o u rc e 0
l i e i n a common p la n e 6
I n o rd e r to have, any m eaning, th e p o in ts must
P a tte r n s i n th e h o r iz o n ta l and v e r t i c a l p la n e s
c o n ta in in g th e o r i g i n a r e th e m ost common*
A lm ost e x c lu s iv e ly i n l i t e r a t u r e , r a d i a t i o n p a t t e r n s a re o b ta in e d
by c o n s id e rin g p ro p a g a tio n over an i n f i n i t e ground plane®
The o b je c t o f
t h i s r e s e a r c h was t o o b ta in r a d i a t i o n p a t t e r n s over a f i n i t e p la n e and
a tte m p t t o s u p p o rt th e e x p e rim e n ta l r e s u l t s m athem atically®
Because o f t h e i r symmetry, c i r c u l a r d is k s w ere u sed a s th e f i n i t e
planes®
The u se o f c i r c u l a r d is k s w i l l p ro d u ce c i r c u l a r h o r iz o n ta l p a t­
terns®
T h e re fo re , th e v e r t i c a l p a t t e r n i n o n ly one p la n e c o n ta in in g th e
o r i g i n need be c o n sid ered ^
-6 .
OBTAINING EXPERIMENTAL PATTERNS
The Use o f Model A ntennas
-
I n stu d y in g r a d i a t i o n p a t t e r n s e x p e rim e n ta lly <, i t i s o fte n d i f f i ­
c u l t t o o b ta in p a t t e r n s a t a r e q u ir e d fre q u e n c y s i n c e , i n m ost p r a c t i c a l
e a s e s 1, th e. wave le n g th w i l l be c o m p a ra tiv e ly I a r g e 0 T h is means t h a t f i e l d
m easurem ents sh o u ld be. ta k e n a t a g r e a t d i s t a n c e , which w ould r e q u ir e
e i t h e r e x tre m e ly s e n s i t i v e d e te c tio n d e v ic e s o r a r a t h e r la r g e power in p u t
t o th e t e s t a n te n n a 0 A nother a p p a re n t o b je c tio n i s f in d in g a n " o b s tru c tio n ■■
f r e e l o c a t io n f o r th e t e s t work*
The d i f f i c u l t y o f t e s t m easurem ents a t a. r e q u ir e d fre q u e n c y has
in tro d u c e d th e u se o f model a n te n n a s 0 A model t e s t a n te n n a i s a m in ia tu re
re p ro d u c tio n o f th e a c t u a l a n ten n a t o be te s te d *
a n te n n a i s
I f th e s iz e o f th e model
A tim e s th e s iz e o f th e a c t u a l a n te n n a , th e n th e fre q u e n c y t o
be u sed i n th e t e s t i n g o f th e model s h a l l be th e r e c ip r o c a l o f
A tim e s
th e o p e ra tin g fre q u e n c y 0 A l l dim ensions u n d er i n v e s t i g a t i o n s h a l l be
A
tim e s th e d im ensions a p p e a rin g un d er th e a c t u a l c o n d itio n s o The p a tte r n s
o b ta in e d in t h i s p a r t i c u l a r problem w ere o b ta in e d a t o n ly one fre q u e n c y
b u t , from model a n ten n a r e a s o n in g , a r e a p p lic a b le t o a l l r a d io fr e q u e n c ie s 6
The f u n c tio n o f a l l com ponents, o th e r th a n th e e l e c t r i c a l compon­
e n t s , a re assumed to be a p p a re n t a n d , t h e r e f o r e , s h a l l n o t b e d isc u sse d * .
A ll components w hich a r e n o t m entioned a s b e in g p a r t o f F ig u r e 2 o r F ig u re
3 w ill appear in fig u r e I 0
-
7-
FIGURE I
COMPLETE TEST APPARATUS
FIGURE 2
TEST ANTENNA AND ASSOCIATED COMPONENTS
-
9-
FIGURE 3
RECEIVING ANTENNA AND METERING COMPONENTS
The k ly stro n " tu b e i s u sed a s a n o s c i l l a t o r to p ro v id e th e . ra d io
frequency*
The p a r t i c u l a r k ly s tr o n u sed o p e ra te d a t a b o u t 3000 m egacycles
The k ly s tr o n r e c e iv e d i t s power from a, power su p p ly w hich c o n s is ts o f a
r e g u la te d and a n o n -r e g u la te d su p p ly i n s e r ie s *
The maximum v o lta g e
o b ta in a b le a t th e o u tp u t te rm in a ls o f th e power su p p ly i s a b o u t 11^0 v o l t s
The k ly s tr o n i s c o o led by a fan*
Power from th e k ly s tr o n i s f e d th ro u g h a
c o a x i a l l i n e t o th e t e s t a n te n n a and i t s a s s o c ia te d components*
..........F ig u r e 2 shows th e t e s t
a n ten n a and i t s a s s o c ia te d components*
The c o a x ia l l i n e from th e k ly s tr o n i s fe d t o a stub tu n e r and th e n to th e
t e s t a n te n n a o The purp o se of. th e s tu b tu n e r i s t o match th e a n ten n a t o
th e lin e *
A ll c o n n e c tio n s a re made in s id e th e c o l l a r w hich i s d e tac h a b le*
The a n te n n a i s con n ected t o th e c e n te r c o n d u c to r o f th e c o a x ia l, l i n e by a
s l i p f i t and th e ground p la n e t o th e o u te r co n d u cto r by a screw f i t t i n g *
The a n te n n a i s a p p ro x im a te ly o n e -q u a rte r wave le n g th *
The ground p la n e s
a r e made o f copper and t e s t i n g was done over p la n e s o f 2 9
10 c e n tim e te rs i n diam eter*
5s„ 6 , 8 , and
The c o l l a r 9 lo c a te d below th e ground p la n e s
i s a fr u s tr u m o f a r i g h t c i r c u l a r cone d e sig n e d t o r e f l e c t a l l waves
.
s trik in g it*
-These r e f l e c t e d w aves, t h e r e f o r e , cannot cau se in te r f e r e n c e
i n th e r e g io n i n w hich th e m easurem ents w ere taken*
The re c e iv in g a n te n n a and th e m easuring components a t e shown i n
F ig u re 3* ' T h e .re c e iv in g a n te n n a .is s u p p o rte d from th e a r c by a n o n - f le x i b l e c o a x ia l l i n e * T h is l i n e i s co nnected t o a q u a rte r-w a v e stu b and th e
s tu b t o a c r y s t a l h o ld e r*
The l i n e from th e c r y s t a l h o ld e r te rm in a te s a t
th e p a r a l l e l com bination o f a con d en ser and a d i r e c t c u r r e n t , 100
-IL
m icroam pere' am m eter»
The r e c e iv in g a n te n n a i s a h a lf-w a v e d ip o le t h a t
r e c e iv e s r a d i a t i o n from th e t e s t a n te n n a « The q u a rte r-w a v e s tu b i s in
p a r a l l e l w ith th e s e r i e s com bination composed o f th e c r y s t a l and th e p a r­
a l l e l com bination o f m eter and condenser®- The c r y s t a l r e c t i f i e s th e r a d io
frequency®
The s tu b , b e in g a s h o r t c i r c u i t t o d i r e c t c u r r e n t , com pletes
th e dAe, c i r c u i t b u t h as i n f i n i t e impedance t o th e r a d io frequency®
The
p u rp o se o f th e con d en ser i s t o b y -p a ss th e r a d io fre q u e n c y components
around th e meter®
" The f i r s t s te p ta k e n , to o b ta in d a ta f o r th e r a d i a t i o n p a t t e r n s ,
was th e a lig n m e n t o f th e arc*
T h is was done by u se o f a t r a n s i t , by su s­
pending a ;p o i n t e r by f i n e w ire from th e n in e ty d eg ree mark t o th e to p o f
th e t e s t a n te n n a , and by a d ju s tin g th e a d ju s t a b le le g s on th e a r c 9s
v e r t i c a l support®
.
Then th e k ly s tr o n was s e t i n t o o p e r a tio n and a d ju s te d f o r maximum
o u tp u t f o r th e power su p p ly v o lta g e a v a ila b le ®
The s tu b tu n e r was a d ju s te d
f o r maximum a n d , l a s t o f a l l , th e a n te n n a h e ig h t was v a r ie d f o r a maximum
reading®
E or each s te p th e a u th o r a t tem pted to keep below th e ground p la n e
i n o rd e r to p re v e n t e rro n e o u s p e ak in g due to r e f l e c t i o n s fro m h is body®
A..check on u n d e s ire d r e f l e c t i o n s was made by o b se rv in g changes in
m eter re a d in g s due t o th e p re s e n c e o f a l a r g e s h e e t o f copper, i n th e fie ld ®
R e f le c tio n s w ere checked a t a l l p o s s ib le lo c a tio n s®
The B allro o m o f th e
S tu d e n t U nion B u ild in g o f Montana S t a t e C o lleg e p re s e n te d no a p p a re n t
in te r f e r e n c e due t o r e f l e c t i o n s »
- The r e c e iv in g a n ten n a was th e n s e t i n th e d e s ir e d r a d i a l p o s itio n
and i t s p la n e o f p o l a r i s a t i o n made" p a r a l l e l t o th e v e r t i c a l l y p o la riz e d
s i g n a l e.
R eadings were th e n ta k e n a t f i v e d eg ree in c re m e n ts » To a s s u re t h a t
no e r r o r s o c c u rre d due t o f l u c t u a t i o n s o f l i n e v o lta g e , a re f e r e n c e a n ten n a
was p la c e d i n th e f i e l d and re a d in g s w ere ta k e n o n ly a t a c o n s ta n t v a lu e
.
.
o f r e f e r e n c e a n ten n a m easurem ents
was f i x e d th ro u g h o u t each t e s t r u n .
in g manners
angle®
The p o s i t i o n o f th e r e f e r e n c e a n ten n a
The re a d in g s were ta k e n i n th e fo llo w =
F i r s t th e a u th o r a d ju s te d th e r e c e iv in g a n te n n a a t th e d e s ir e d
-The a u th o r th e n low ered h im s e lf below th e ground p la n e .
When t h e .
r e f e r e n c e a n ten n a in d ic a te d i t s p re a s s ig n e d v a lu e , th e a u th o r s ig n a lle d
h i s a s s i s t a n t and th e a s s i s t a n t to o k th e r e a d in g .
The re a d in g s were ta k e n
a s c lo s e t o th e a n ten n a a s r e f l e c t i o n s would a llo w , . Even a t t h i s , th e
u se o f b in o c u la r s was fo u n d n e c e s s a ry ,
As soon a s th e m easurem ents were
ta k e n from z e ro t o n in e ty d e g re e s th e a u th o r and h is a s s i s t a n t in te rc h a n g e d
p o s itio n s and th e m easurem ents were r e p e a te d .
T h is p ro c e d u re was th e n
re p e a te d f o r a l l s i x p la n e s ,
’ S in c e th e r e c t i f i e d c u r r e n t th ro u g h a c r y s t a l v a r i e s ap p ro x im ately
a s th e sq u a re r o o t o f th e v o lta g e a c r o s s i t ( f o r low c u r r e n t v a lu e s ) th e
sq u a re r o o t o f a l l re a d in g s m ust be ta k e n b e fo re th e p a t t e r n can be p l o t ­
te d ,
The e x p e rim e n ta lly o b ta in e d p a t t e r n s a p p e a r i n A ppendix A5- alo n g
w ith c u r r e n t re a d in g s and sq u are r o o t v a lu e s f o r th e 6 and 10 c e n tim e te r
p la n e s'.
-13'
MATHEMATICAL AKALISIS
P re lim in a ry C o n s id e ra tio n s and B a sic A ssum ptions
I n a problem o f t h i s n a tu re many assu m p tio n s imist be made.
The
a ssu m p tio n s t h a t s h a l l be made a r e assu m p tio n s commonly made i n th e math­
e m a tic a l a n a ly s is o f r a d i a t i o n p a t t e r n s .
Among th e s e e d u ca te d g u esses
a re assu m p tio n s o f th e c u r r e n t d i s t r i b u t i o n alo n g th e a n te n n a 3 th e assump­
t i o n o f a n a n te n n a o f i n f i n i t e s i m a l d ia m e te rs and th e assu m p tio n t h a t th e
.
ground p la n e i s a p e r f e c t c o n d u c to r»
The e l e c t r i c f i e l d i n t e n s i t y o f a wave p a s s in g th ro u g h any medium
must be r e l a t e d t o th e m agnetic f i e l d i n t e n s i t y b y .th e i n t r i n s i c imped­
ance,
I f th e wave p a s s e s from one medium t o a n o th e r9 th e ta n g e n t i a l com­
p o n e n ts o f th e e l e c t r i c and th e m ag n etic f i e l d must be c o n tin u o u s a c ro s s
th e b oundary.
The norm al components a re n o t c o n s tr a in e d i n t h i s m anner, .
I n p a s s in g a c r o s s th e boundarys th e t a n g e n t i a l components m ust s t i l l be
r e l a t e d t o each O ther by th e i n t r i n s i c Im pedance3 b u t th e i n t r i n s i c imped­
ance o f th e two mediums a r e 3 i n g e n e r a l3 n o t e q u a l,
T h e re fo re 3 when a
wave p a s s e s from one medium t o a n o th e r3 i n o rd e r t h a t boundary c o n d itio n s
foe S a t i s f i e d 3 n o t a l l o f th e in c id e n t wave can c o n tin u e on i n t o th e second
medium.
T h is means t h a t r e f l e c t i o n m ust o c c u r .
I f a p e r f e c t l y conducting
ground p la n e i s c o n sid e re d ^ th e e l e c t r i c f i e l d i n t e n s i t y a t th e boundary
m ust e q u a l a e ro f o r a l l components s in c e a p e r f e c t l y c o n d u ctin g p la n e i s
a sh o rt c ir c u it.
S in c e th e v o lta g e and th e r e s i s t a n c e o f t h e ground i s
e q u a l t o a e r o 3 th e c u r r e n t i n th e p la n e i s o f in d e te rm in a n t fo rm .
I n o th e r
Words3 any v a lu e o f s u r f a c e m agn etic f i e l d i n t e n s i t y may e x i s t b u t3 s in c e
th e r e f l e c t e d wave cannot be g r e a te r th a n th e in c id e n t Wave9 th e ta n g e n t i a l
m ag n etic f i e l d i n t e n s i t y can n o t be l e s s th a n z e ro o r g r e a t e r th a n tw ice th e
in c id e n t m agnitude*
S in c e no power can b e tr a n s m itte d i n t o th e p e r f e c t l y
co n d u ctin g p la n e 9 th e r e f l e c t e d wave m ust c o n ta in th e same en erg y a s t h a t
e x i s t i n g i n t h e .in c id e n t waves i eeo p e r f e c t r e f le c tio n *
I n o rd e r f o r th e
energy on th e s u r f a c e o f th e r e f l e c t i n g p la n e to e q u a l z e r o s . th e in c id e n t
and r e f l e c t e d components o f en erg y m ust c an c e l*
T h is Im p lie s 9 by th e
P o y n tin g V e c to r9 t h a t t h e r e f l e c t e d m ag n etic i n t e n s i t y i s e q u a l t o th e
in c id e n t m agnetic i n t e n s i t y in d i r e c t i o n and m agnitude*
The d e r iv a tio n o f th e m ag n etic f i e l d I n t e n s i t y 9 a t a d i s t a n t p o in t
from a c u r r e n t c a r ry in g w ir e 9 may be o b ta in e d by f i r s t f in d in g an e x p re ss ­
io n f o r th e v e c to r m agnetic p o t e n t i a l a t any p o in t and th e n 9 by v e c to r
c a lc u lu s 9 ta k in g th e c u r l t o o b ta in a n e x p re s s io n f o r th e m ag n etic f i e l d
. /H
,
'
i n te n s ity * The d i r e c t i o n o f th e p o t e n t i a l v e c to r w i l l be i n th e same
d i r e c t i o n a s th e c u r r e n t flo w i n the' w ire*
The p o t e n t i a l V e c to r9 however9
w i l l la g th e c u r r e n t becau se o f th e f i n i t e tim e o f p ro p a g a tio n *
I f th is
c u r r e n t c a r ry in g w ire e x te n d s n o rm a lly fro m a p e r f e c t l y c o n d u ctin g p la n e s
th e n th e v e c to r p o t e n t i a l w il l be norm al t o th e p la n e and th e m agnetic
f i e l d i n t e h s i t y s h a l l be c o m p le tely ta n g e n tia l*
To s a t i s f y ^ th e boundary
c o n d itio n s o f a p e r f e c t l y c o n d u ctin g p la n e sim ply means t h a t th e r e f l e c t e d
v e c to r p o t e n t i a l be e q u a l t o th e in c id e n t v e c to r p o te n tia l*
I f th e ground
p la n e i s f i n i t e 9 th e n th e v e c to r m a g n e tic p o t e n t i a l s a t i s f i e s th e boundary
c o n d itio n s t o and a t th e d is c o n t i n u i t y and no r e f l e c t i o n o c c u rs p a s t t h i s
new boundary*
“ 1^="
The g e n e r a l e x p re s s io n f o r th e v e c to r m agnetic p o t e n t i a l i s
r id ?
w here
i
(I)'
I'
i s th e c u r r e n t d e n s ity and th e d i f f e r e n t i a l
volume i n t e g r a t i o n b
dV" in d ic a te s a
I f th e c u r r e n t c a r ry in g c o n d u cto r i s s t r a i g h t and o f
i n f i n i t e s i m a l d ia m e te rg th e n th e e q u a tio n f o r th e v e c to r p o t e n t i a l becomes
A - / w
?
'
(2)
w here th e c u r r e n t i s c o n sid e re d flo w in g i n th e
% d i r e c t i o n » The c u r r e n t
I
w i l l be a space f u n c tio n and i n th e e ase o f a tim e v a ry in g c u r r e n t
‘'
w i l l be a f u n c tio n o f b o th sp ace and tim e . I n th e d i s t a n t f i e l d s th e
"...
p o t e n t i a l i s tim e r e ta r d e d b ecau se o f th e f i n i t e tim e o f p ro p a g a tio n .
The
tim e la g a t a d i s t a n t p o in t w i l l be th e d is ta n c e t o t h a t p o in t d iv id e d by
th e V e lo c ity o f th e wave®
In f r e e s p a c e 9 th e v e l o c i t y o f r a d io waves i s
p r a c t i c a l l y 3 x 10® m eters p e r sec o n d .
The p h ase s h i f t caused by th e tim e
la g w i l l be th e p ro d u c t o f th e tim e la g and th e a n g u la r v e l o c i t y o f th e
w ave.
The fo re g o in g d is c u s s io n has shown how assu m p tio n s o f a p e r f e c t l y
c o n d u ctin g ground p la n e and a n a n te n n a o f i n f i n i t e s i m a l d ia m e te r s im p lif ie s
r a d i a t i o n problems®
A nother v e ry im p o rta n t c o n s id e ra tio n t h a t must be made
b e fo re a m a th e m atica l s o lu tio n f o r th e v e c to r m agnetic p o t e n t i a l can be
o b ta in e d i s th e c u r r e n t d i s t r i b u t i o n a lo n g th e a n te n n a .
The assu m p tio n
coMaonly made f o r a w ire a n te n n a i s t h a t t h e s ta n d in g c u r r e n t wave i s s in u ­
s o id a l,
The assu m p tio n was a r r iv e d a t fro m th e tra n s m is s io n l i n e th e o ry
o f a n open c i r c u i t e d line®
—16«»
S o lu tio n f o r V ecto r M agnetic P o t e n t i a l in I n t e g r a l Form
The c u r r e n t in th e a n ten n a v a r ie s s in u s o id a lly a lo n g th e an ten n a
and s h a l l be c o n sid e re d t o v a ry c o s in u s o id a lly w ith tim e .
The e x p re ssio n
f o r th e a n ten n a c u r r e n t o f an a n ten n a o f h e ig h t H a t any p o in t I from th e
ground p la n e s F ig u re k 9 i s
I ” I m s i n i l l (H -I) cos w t
A t a d i s t a n t p o in t
(3 )
P , th e v e c to r m ag n etic p o t e n t i a l i s tim e r e ta r d e d and
th e e x p re s s io n of c u r r e n t t h a t must be u sed i n s o lv in g f o r th e v e c to r po­
t e n t i a l becomes
(H -I) cos w ( t - I)
1 “ I n sin
F o r th e d i r e c t w ave, r
t o th e p o in t
P
i s th e d is ta n c e from th e p o in t
(U)
I
on th e a n ten n a
i n s p a c e , and f o r th e d i r e c t wave t h i s d is ta n c e s h a l l be
I f a c y l i n d r i c a l c o o rd in a te system i s chosen w ith th e a n te n n a on th e
z
a x is and th e b ase o f th e a n ten n a a s th e o r i g i n th e n th e d is ta n c e
by
r^
th e P y th ag o rean Theorm i s
rd ■
t/
( z - l ) ^ + (S2
(5 )
-1 7 -
F o r th e r e f l e c t e d wave, th e d is ta n c e
from
p o in t
I
r
i n E q u a tio n (Ii) i s th e d is ta n c e
t o th e ground p la n e p lu s th e d is ta n c e from th e ground p la n e to
P , rem em bering t h a t th e a n g le o f in c id e n c e must e q u a l th e a n g le o f
r e f l e c t i o n . F ig u re 5»
ig n a te d a s
F o r th e r e f l e c t e d wave t h i s d is ta n c e s h a l l be d e s­
r.
F ig u re 5
The r e f l e c t e d wave, a s seen from F ig u re 5» t r a v e l s th ro u g h th e same d i s t ­
ance a s though i t were p a s s in g th ro u g h th e ground p la n e and t o th e p o in t
( P p , - P2) « The e q u a tio n f o r
r^
is
rr - /( z + I)2 + p 2
(6 )
The e q u a tio n f o r th e d i f f e r e n t i a l m agnetic v e c to r p o t e n t i a l o f th e d i r e c t
w ave, from a d i f f e r e n t i a l le n g th , a lo n g th e a n te n n a , i s
d a
a 1Hi s i n •— (H -I) c o s W ( t - vI z" 1^
Ufr / ( Z- l ) 2 + p 2
* P2 ) d I
(? )
■=-18-
The p o t e n t i a l e q u a tio n f o r th e r e f l e c t e d wave i s
a n -y , s 1Hi 's' i 'n' "T
(H
-I)----------cos w---(t aa^ Az
A
'
......
r
To f in d th e t o t a l v e c to r p o t e n t i a l o f th e d i r e c t wave r e q u ir e s in te g r a tio n
o f th e d i f f e r e n t i a l v e c to r p o t e n t i a l betw een th e l i m i t s o f z e ro and H,
where H i s th e a n te n n a h e ig h t.
The r e f l e c t e d w ave, how ever, w i l l n o t
c o n ta in th e same l i m i t s o f i n t e g r a t i o n b ecau se o f th e
h o le in th e ground
p la n e around th e a n ten n a and th e o u te r p e rip h e ry o f th e ground p la n e .
These l i m i t s may be o b ta in e d from F ig u re 6 , where
h o le and
L
R
i s th e r a d iu s o f th e
th e d is ta n c e t o th e o u ts id e ed g e.
F ig u re 6
The c u r r e n t fla w in g from z ero to
th e c u r r e n t from
to
q
to
H w ill not r e f le c t.
th e low er l i m i t
q
q + z
q.
u
q
w i l l n o t r e f l e c t th e v e c to r p o t e n t i a l ,
w i l l r e f l e c t a v e c to r p o t e n t i a l , and from
The upper l i m i t o f i n t e g r a t i o n w i l l be
u
u
and
U sing s im ila r t r i a n g l e s , th e r e l a t i o n s h i p e x i s t t h a t
= £
9
®Lyc=»
The e q u a tio n f o r
q
becomes
Ra
p " R
( 9)
By e x a c tly th e same p ro ced u re^ th e u p p er l i m i t
u
i s fo u n d by th e e q u a tio n
u =
u
and
q
cm
c an n o t exceed
Ho -The .upper I i m i t s .Tshen
— p ~ ii
H
is
H
and th e low er l i m i t when
' f u n c tio n s o f
q .and
■.?.z. .
H i s a ls o H® T h is im p lie s t h a t th e
P" R
u / a r e c o n tin u o u s b u t t h a t t h e i r d e r iv a tiv e s a re
' i t
'
-
discontinuousc . These-derivatives^ therefbre^oidll not/exist .atothejpoint
:
o f : d is c o n t in u it y b u t -w illu e x i s t o n . e i t h e r i s i d e .. o f : t h a t point@ .; The Lbehayw
,
io r ' o f .the.; d e r iv a tiv e s .T s illL .la t e r .eap laih . jt h e . existen ce:;, o f a th e . plus/.and
m inus p o in ts e x i s t i n g i n th e c a lc u la te d p a t t e r n s o
The e x p re s s io n f o r th e d i r e c t v e c to r m ag n etic p o t e n t i a l w i l l b e ,
w ith th e s u b s t i t u t i o n t h a t
H
A
s in
TwrJ ^
— = ™L .
/*—---------(H -I) co s & F ( v t « y(z°-l)^ *
(U )
^ (z + l)^ + p 2
The r e f l e c t e d v e c to r m agnetic p o t e n t i a l w i l l be
Az r =
f
s in
(H -l.) cos
( v t - v(is+l)^ + p 2 ) d I
V (z + 1 )2 + p 2
The b e h a v io r o f
u
and
q
m ust be r e a l i z e d a t a l l tim e s .
( 12)
—20—
Method o f I n te g r a tio n
The i n t e g r a l s c o n ta in e d in th e e x p re s s io n s f o r th e v e c to r p o te n t­
i a l s o f th e d i r e c t and th e r e f l e c t e d w aves, r e q u ir e s im p lif ic a tio n b e fo re
an easy method o f i n t e g r a t i o n can be o b ta in e d .
F ig u re 7 w i l l show th e means try w hich in te g ra n d s can be changed
t o an in te g r a b le fo rm .
t h a t th e d is ta n c e
r
r
A re q u ire m e n t n e c e s s a ry f o r th e s im p lif ic a tio n i s
i s much g r e a te r th a n
H.
Then,
i n th e tim e v a ry in g f u n c tio n s
cannot be s u b s t i t u t e d f o r
b ecau se th e a d d itio n o r s u b tr a c ti o n o f
I
may cause c o n s id e ra b le phase
s h ift.
F ig u r e 7
The d is ta n c e r ^ and
rr
w ere d eterm in ed p re v io u s ly to be y ( z - l ^ +
re s p e c tiv e ly .
and
r^
i s r + dg.
Y
d^
From F ig u re 7 th e d is ta n c e
S in ce th e a n g le s ,
and
d^
a and
p
r^
is
and
r -
a re a c t u a l l y v e ry sm all
th e r e f o r e a r e e q u a l t o
I cos 0 .
The e q u a tio n s f o r th e d i r e c t and r e f l e c t e d v e c to r m agnetic p o te n t­
i a l s become
—
21—
Azd -
I
s in
(H -I ) co s
( v t - r + I cos 6 ) d I
(1 3 )
Az r =
/
s in
(H -I) cos
( v t - r - I cos 9 ) d I
(Iii)
and
By use o f th e trig o n o m e tr ic i d e n t i t y
sin A cos B = ^ Qsin (A+B) + sin (A-B)]
the equation for the reflected potential becomes
„u
s i n ^ r f y t - r + H - 1(1+ cos 9)]
+ s in
[ ( - v t + r ) + H -I ( i - cos 0 ) ] d I
(1 5 )
The in te g ra n d now i s e a s i l y in te g r a te d and th e v e c to r p o t e n t i a l becomes
J
A
- - lm>
cos 4 1' [ ( v t - r ) + H -I (I+ cos 9)]
zr
I S lT zT [
( I + cos 9)
+ co s
By f in d in g a common denom inator f o r
[ ( - v t + r ) + H -I ( 1 - c o s 9 ) ] I
( I - cos 9)
Jq
Az r
and making th e trig o n o m e tric
s u b s titu tio n s
cos (A+B) = cos A cos B - s in A s in B ,cos (-D ) = Cos D
and
s i n (-D ) = - s i n D
(16)
-2 2 -
th e e q u a tio n becomes
cos - ^ r ( v t - r ) r
---------- j ( 1 - C O S © )
s in 2 ©
Azr ’ '
COS
(1 + C O S 0 )1
I
+
*
(1 + C O S 0 )
COS - j H - l
( I - C O S0^
—s in —
£■ ( v t —r)
s in 2 0
T [1 -c o se ) s i n ■~T(h -1 (1+ cos0)J
C
— (l+ c o s 0 ) s in
(1—cosG^ji
(17)
S in c e
s in x
t s i n y - 2 s i n j (x -
y ) cos ^ (x - y )
cos x
+ cos y = 2 cos i (x +
y ) cos J (x - y )
cos x
- cos y - - 2 s in i (x
+ y ) s i n ^ (x - y )
and
Az r can be re d u c ed t o
cos
V =-
(v t-r)
|c o s
8 rr2r
(H- I) cos
( I cos©)
s in 2 8
- cos© s i n ~ (H- I) s i n
A
+ s in - r ( v t- r ) r
2
■ J cos
s in ^ 0
X
( I cos0)l
(H -I) s i n — ( I cos0)
+ cos© s i n
T
(H -I) cos
x
( I cos© )j|^l8)
The e q u a tio n f o r th e r e f l e c t e d m ag n etic v e c to r p o t e n t i a l , th e r e f o r e , i s
-2 3 -
A
^
cos 2 ? ( v t - r ) z
1In *
------------- - J cos
(H -u) cos
(u cos©)
(H -q) cos ~
(q cos6)
s in ^ Q
8
- cos
- cos© s in I l r (H -u) s i n
A
A
+ cos© s i n ~ f (H-q) s i n
+ s in
(u cos0)
(q c o s0 ^ |
(v t-r)
|c o s ^
(H -u) s in
(u cos0)
s in ^ 0
- cos
A
(H-q) s in
A
(q cos0)
+ COS0 s i n 2fr (H-u) cos
(u cos9)
A
^
- cos0 s in —^ (H -q) cos
(q c o s 0 ) ^ j (I? )
By p r a c t i c a l l y th e same p ro c e d u re th e e q u a tio n f o r th e d i r e c t
m agnetic p o t e n t i a l becomes
V L
cos ~
8 r r 2r
+ s in "
cos
(H c o s0 ) — cos -t^ H
A
A
( v t - r ) j~.
s in ^ 0
-s in
(H cos0) + cos0 s in i™’H
( v t - r )) f
s in ^ 0
The t o t a l m agnetic v e c to r p o t e n t i a l a t a p o in t i s th e v e c to r sum
o f th e r e f l e c t e d and th e d i r e c t w aves.
S in c e th e d i r e c t and r e f l e c t e d
v e c to r s have i n d e n t i c a l d i r e c t i o n th e a r ith m e tic sum i s u s e d .
S o lu tio n f o r th e M agnetic F i e l d I n t e n s i t y
The c u r l o f th e v e c to r m agnetic p o t e n t i a l i s th e m agnetic f i e l d
(Z)D
—2I4.—
in te n s ity .
The com plete v e c to r m agnetic p o t e n t i a l i s
cos - T r ( v t - r ) Z------- &-----------. J c o s zJT (H cos©)
s in 2 0
S f r 2F
- co s
^
H+co s ™ (H -u)
X
- cos 4™ (H -q) cos ^
co s ~
A
(u cos©)
(q co s6 )
- cos© s i n ~ (H -u) s i n ~ ( u cos©)
A
A
-x
+ cos© s i n £ £ (H -q) s in — (q cos©) I
A
A
2ff
+ s i n -r- ( v t - r ) f
ofT
--------A -------J - s i n ~ (H cos©)
SM 2 e
I
^
+ co s8 s in
+ cos©
A
~ + co s —
A
*
(H -u) s in - X (u co s8 )
s in ~ (H -u) cos “ (u cos©)
A
A
- co s 4 2 (H -q) s i n
(q cos©)
- cos© sin 2IT (H -q) cos
A
A
(q cos©)I
J
( 21 )
I n a s p h e r ic a l c o o rd in a te sy ste m , th e components o f th e c u r l o f a v e c to r
a re
C u rlr A
^
r s in © | j
C u rl0 A =
.
I
r s in 6
C urlv A
The v e c to r
Az
( 22)
( s i n 9 A *zO - ^ Ae
_ I
3 r A^
(23)
r
( 21+)
(r aS j - i r § ]
w i l l c o n ta in com ponents, i n th e
r
and th e
These components a re
i
0
d ir e c tio n .
Ar ”
(22)
co s 6
A g - - A g sin 0
S in c e
A ^ ' ® O and
Ar
( 26)
and
Ag
a re in d ep en d e n t o f
f i e l d i n t e n s i t y i s e n t i r e l y i n th e <f> d i r e c t i o n o
components o f
Aa
H
^ s th e magnetic,
I f th e
a r e s u b s t i t u t e d i n t o E q u a tio n (2U)a '
I
^
,
h K
- " 5 7 ( r a T K toe) -
r
and th e
©
becomes
COS ©
—g -g
(27)
w hich can be re d u c e ^ t o
r
^A z
in :+
cos 9
—
Az
( 28 )
-in n
A r a d i a t i o n p a t t e r n c o n s is ts o f t h r e e d i f f e r e n t f i e l d s i> They a r e
t h e e l e c t r o s t a t i c f I e l d s th e in d u c tio n f i e l d s and th e d i s t a n t f i e l d 0
E l e c t r o s t a t i c term s v a ry a s th e r e c i p r o c a l o f
a s th e r e c i p r o c a l o f
'r ^ 9 - th e in d u c tio n f i e l d
r ^ $ and th e d i s t a n t f i e l d a s th e r e c i p r o c a l o f
r„
S in c e p r a c t i c a l l y a l l p ro p a g a tio n problem s a r e concerned w ith th e f a r f ie ld s
o n ly te rm s d iv id e d by
r
s h a l l be r e t a i n e d i n th e space' d i f f e r e n t i a t i o n .
I n p o la r c o o rd in a te s ) when
u =
q - H and u ^ H<
L r cos ©
r s in © - L
(29)
R r cos 0 '
r s in ©
T h e ir d e r I v i a t i o n s s w ith p p s p e e t t o Qs a r e
(30)
-2 6 -
cta = -L r 2 +
r s in 9
HS
( r s i n 9 - L)^
-R
dc
f§
S in c e
r
R2 r s i n 9
( r s in 0 - r ) 2
q
or
(32)
i s c o n ta in e d in th e num erato r o f b o th
d e r i v a t i v e s , when
by
(3 1 )
u
and
q
and t h e i r
q ^ H and u ^ H, th e te rm s , in
, m u ltip lie d
o r t h e i r d e r iv a tiv e w i l l be p a r t o f th e d i s t a n t f i e l d .
d i s t a n t f i e l d term s o b ta in e d from cos©
it TTr^
u
^ z
a©
^ en
u ^ H
and
q
The
H a re
cos -Tp ( v t - r ) co-s ? / — s i n — ( Hm) cos — (u co s0 )
^ _
s i n m I d0_
A
A
+ ju s in 0 - cos©
cos ~ (Hm) s i n — (u co s9)
du s i n “ (H~q; cos ™ (q cos0)
d0
A
A
2 tt
(H-q) s i n — (q cos0)
-Jq sin© - cos© _ | c o s —
-
cos© cos ™ (H m ) s i n ~
- Ju sin© — cos©
+ ^
cos© cos ”
(u cos©)
cos© s i n
(H m ) cos
(u cos©)
(H~q) sin ™ (q cos©)
+ Jq sin© - cos© ^ |J c o s0 s in ~
(H -q) cos
(q c o s6 )j
+ s i n % ( v t - r ) £ 2 £ £ - { 0 i s i n ™ (H m ) s i n % (u cos© )
*
s i n m ( d©
A
a
-Iu sin© - cos©
co s S i (H"U) cos 2H (u cos©)
L
d©J
A
A
-
s i n ™ (H -q) s i n ~
(q
+ Jq sin© - cos© ^ J c o s ~
cob ©)
(H-q) cos ^
(q cos©)
- | | cos© cos ™ (H m ) c o s™ (u cos©)
+ |u sin© - cos© | | j c o s © s in ™ (Hm) s i n — (u cos©)
+ ^
cos© cos
(H -q) cos
(q cos©)
-2 7 -
“ [q sin© - cos© ^ J c o s © s i n — (H“q ) s i n ™ (q c o s © )j|
(3 3 )
H and u
The d is t a n t f i e l d , when
sin©
-
H, o b ta in ed from
d Az
m
4 tt r
co s
(v t-r )
s in ©
■f— s i n
(d r
(H-u) cos^H (u cos©)
A
- ™ c o s2© s i n ^
( Hm) cos^H (u cos©)
" ~
cos ~
(H -q) cos ^
(q cos©)
+ ^
c o s 2© s in — (H -q) cos ^
(q cos©)
- cos — (H m ) s i n S i (u cos©)
A
A
- cos© s i n ™ ( Hm) cos ~ (u cos©)
A
r\
+ cos ™ (H -q) s i n ■
— (q cos©)
+ cos© s i n ~
A
(H -q) cos
A
(q cos©)
+ s i n ™ (h cos©) - cos© s i n ~ h IA
A J
s in
(v t-r )
s in ©
- j ~ s in ^
- ^
( H m) s i n ^
(u cos©)
c o s 2© s i n ™ ( Hm) s in ^
_
s in ^
+
c o s2© s i n ~
+ cos ~
- cos ^
(H-q) s i n ~
(q cos©)
(u cos©)
( H m) s in ™ (u cos©)
(H -q) cos
+ cos© s i n ~
+ cos ”
(H -q) s i n ™ (q cos©)
( Hm) cos ^
- cos© s i n
(u cos©)
(q cos©)
(H -q) s in ~
(H cos©) - cos
(q cos©)
H
I
(3 4 )
-2 8 -
I n b o th p a r t s o f th e d i s t a n t f i e l d
# th e term s have n o t been c o lle c te d
so th e re a d e r may be a b le t o fo llo w th e d i f f e r e n t i a t i o n .
When
u -H
and q
= JT 7"
COS T
H,
( v t "r ) I - a C1 ) s i n ^
(H -u) cos ~
(u cos6)
( h - u ) s in ~
A
(u co s6)
+A(R) s i n “
(H-q) cos ™
(q co s6)
+B(R) cos ^
(H -q) s i n
(q co s0 )
-B (L ) cos
+
A
s i n ~ ( h c o s 8)
A
c o s6 s i n ™ h
A
s in 9
s in 8
+ s i n ™ ( v t - r ) j-C (L ) s i n ~
-D(L) cos
+C(R) s i n
(H -u) s i n
(H -u) cos ™ (u cos0)
A
(H -q) s i n ™ (q co s8)
%
+D(R) cos
4
(u cos0)
(H -q) cos ~
cos % (H co s8 )
A
s in 6
cos
(q co s6 )
A
s in 8
h
(35)
}
-2 9 -
When
q - H and
and ^
H, = —
P
4n r
^
v d .ll eq u a l z e r o .
(v t-r )
(H-q) cos ^
^A(R) sin ™
+ B(R) cos ^
(H -q) sin ~
(q cos6)
(q cos0) - cos© sin ~
h|
sin 0
+ sin ^
(v t-r )
(H -q) sin — (q cos0)
^C(R) sin
+ D(R) cos "
(H -q) cos ^
2 cos -gr (H COS0)
cos
sin 0
When
q = H and U = Hj
fi
(3 6 )
s in 0
> h! » I r 311(1 S i
C°S5f
4tt r
(q cos©)
f = 1” ^
s in — ( v t - r )
w il1 equal zero<
<H
' = = e =U1 f
2 tt
(3 ? )
The fu n c tio n s o f
L
and R, namely A(L), B(L) C(L), D(L), A(R), B(R),
C(R), and D(R) are o b ta in e d m erely by adding E q u ations ( 3 3 ) and ( 3 4 ) ,
c o l l e c t i n g terras, and making s im p lif y in g tr ig o n o m e tr ic s u b s t i t u t i o n s ,
The fu n c tio n s o f
e x c e p t th e
L
L
are e x a c t ly th e same as th e f u n c tio n s o f
has been s u b s t it u t e d fo r
s h a l l be r e f e r r e d t o a s fu n c tio n s o f
F
R.
R,
These f u n c t io n s , t h e r e f o r e ,
t o e lim in a te r e p e t i t i o n .
=30=
These c o e f f i c i e n t s a r e d e fin e d a s
A(F)
m2 ( r ~'r "s ^n ^ * Fr
( r sin©
n© - F r
* F& -s ^
B(F)
I
( r sin© - F )
(l=cds@)^ * 2 cot^ 9
I
2 j r
9
2
^
( s i n 2© - 2 cos© + I ,
(38)
C O tS v
san© - F r (2+cos© + -p g g )
+ F^
(1+3 cos©) I
sin©
J
I
r 2 sxn 2© +
O(F) = ( r sin© - F ) 2 |
g
+ F2
(39)
(g^2@ _ 2 Cos©)
sin© (cos© = I ) * cot© (cos© +1)1 (i|.0)
(cos© + I )
D(F)
r + F
( r sin© = F )' {■
(W .)
sin©
The E l e c t r i c F ie l d I n t e n s i t y
The e l e c t r i c f i e l d i n t e n s i t y m ust be o b ta in e d b e fo re th e r a d i a t i o n
p a t t e r n can be determ ined'^
The r e s u l t a n t e l e c t r i c f i e l d i n t e n s i t y i s th e
p ro d u c t o f th e i n t r i n s i c impedance and th e r e s u l t a n t m ag n etic f i e l d in te n ­
sity®
M axwell1s f i r s t e q u a tio n s t a t e s t h a t
6
O
C u rl H “ D “ . £ E
w here
D
(!^ )
i s th e e l e c t r i c d isp la c e m e n t d e n s it y and th e
th e f i r s t d e r iv a tiv e w ith r e s p e c t t o time®
dot
r e p r e s e n ts
From E q u a tio n s ( 2 2 ) , (23) and
( 2 l|) 3 i t i s e v id e n t t h a t th e e l e c t r i c f i e l d i n t e n s i t y can c o n ta in couponi
e a t s o n ly i n th e
.3
r
and th e ' © d i r e c t i o n s in c e th e m ag n etic f i e l d
i n t e n s i t y i s e n t i r e l y i n th e <f)
d ir e c tio n *
S in ce th e e q u a tio n f o r th e
m agnetic f i e l d i n t e n s i t y i s v e ry complex^ i t s h a l l be assumed t h a t th e
component i s n e g l i g i b l e „
r
T h is assu m p tio n i s b ased on th e f a c t t h a t f o r
an i n f i n i t e ground p la n e th e
r
component i s zero*
I f t h i s assu m p tio n
i s made and i s c o r r e c t th e n
" % Hjd
w here
q
(W)
i s th e i n t r i n s i c impedance w hich f o r f r e e sp ace c o n d itio n s i s
e q u a l t o IS O ire
The e q u a tio n f o r th e e l e c t r i c f i e l d i n t e n s i t y can now be
as
^ A cos ~
( v t - r ) + B s in
s in
(44)
(v t-r)
(4S)
(v t= r * ta n ^ g )
The e f f e c t i v e v a lu e of th e e l e c t r i c f i e l d i n t e n s i t y i s
jO E f f
—
5%
+ g2
(46)
The r a d i a t i o n p a t t e r n i s g iv e n by th e e x p re s s io n s / A.2' + B2 e
th e number o f term s c o n ta in e d i n
ic a l re s u lts fo r
A
and
A and
Because o f
B3 i t i s b e s t t o o b ta in numer­
B b e fo re s q u a r in g «
■■ The c a lc u la te d r a d i a t i o n p a tte r n s f o r th e 6 , and th e 10 c e n tim e te r
p la n e s a p p ea r i n Appendix B a lo n g w ith th e c a lc u la te d v a lu e s o f
B9 a n d * B^*
The p o i n t s 9 becau se o f th e co m p lex ity o f
A
q s u s A5
and
B9 -
w ere o n ly c a lc u la te d f p r t e n deg ree in c re m e n ts and a t th e a n g le a t which
=32”
d i s c o n t i n u i t i e s appear,, The a n g le s a t w hich th e d i s c o n t i n u i t i e s o ccu r
_
, ,
^ .
R r ‘cos©
L r cos©
a r e found by s e t t i n g ^ and —............ — e q u a l t o He At 0 ° th e
r sin© - R '
r sin© ” L
r a d i a t i o n p a t t e r n i s i n in d e te rm in a te , form b u t has a e ro value®
'i
■33-
DISCUSSION OF RESULTS
A com parison o f th e p a tte r n s co n ta in ed i n Appendix A t o th o se
co n ta in ed i n Appendix B i s f a i r l y good f o r th e 10 ce n tim e te r , p la n e e
The
p a tte r n s f o r th e 6 c e n tim e te r p la n e do h o t show a s good a com parison0
In
s e a r c h in g f o r th e re a so n f o r th e d isa g reem e n tg th e p o s s i b l e lo c a t io n o f
e r r o r s sh o u ld be r e v ie w e d » ■
The patterns o b ta in e d e x p e r im e n ta lly c o n ta in th e f o llo w in g e r r o r s g
w hich w ere n o t c o n t r o lla b le by th e au th ors
I0
The c o a x ia l lin e , r a d i a t e s . d i r e c t l y from i t s te r m in a tio n a t
th e ground p la n e a
2o
T h is sh o u ld g iv e e r r o r s a t sm a ll a n g le s o
B ecau se o f the- li m it e d v o lt a g e su p p ly 5 t h e d is t a n c e from
t h e t e s t antenna t o th e r e c e iv in g antenna was o n ly e ig h t
and one-=third wave I e n g t h s 0 When th e in d u c tio n f i e l d term s
a re o b ta in e d from th e v e c t o r m agn etic p o t e n t ia l^ some terms
w i l l h e d iv id e d by s in 3 ©»
A t sm a ll v a lu e s o f ©.s th e term s
c o n ta in in g s ! r3 © may be a p p r e c ia b le »
3a
A c e r t a in .amount o f u n d e sir a b le r e f l e c t i o n i s bound t o
occur©
I t i s d i f f i c u l t t o d eterm in e j u s t where th e s e
e r r o r s w i l l e x ist©
it©
"
The r e c e iv in g an ten n a3 t o p ic k up t h e - e l e c t r i c i n t e n s i t y at
a p o in t s sh o u ld be ex tr e m e ly sm a ll or th e d is t a n c e from th e
' t e s t antenna ex tre m e ly large©
case©
N eith er o f t h e s e were th e
The a p p r e c ia b le 'e r r o r s w i l l appear where th e change .
of electric field intensity with respect to
©
is large*
A suggestion for future work in obtaining experimental patterns is
to let the test antenna be the receiving antenna receiving radiation from
an antenna in front of a reflecting parabola*
reciprocal theorm*
This is an example of the
Most of the errors previously mentioned would be elim­
inated by this method*
Undesirable reflections would be practically elim­
inated because of the directivity of the reflector*
The distance from the
test antenna could be made greater, again because of the directivity®
Radiation from the coaxial line ,would not exist because the only power i n ,
the line would be power received by the antenna under test®
The mathematical results contain double value points at two angles
because of the discontinuous derivatives of
From the values of
u
and
u
and
q. with respect to
0*
q 9 included in Appendix B, it is clearly
evident how the slope varies and how and why the double values appear*
Because the curves were plotted at ten degree increments, maximums and mininrams may actually exist, but do not appear in the radiation patterns*
The
pattern for the 10 centimeter plane varies from the pattern obtained exper­
imentally in the range of smaller angles* If the radius of the hole is
allowed to approach zero, the pattern becomes a better comparison to the
experimental pattern*
This indicates that the diameter of the antenna is
appreciable or the effective size of the hole is smaller than the actual
size*
When the image antenna is considered, the. current distribution will
be sinusoidal from
u
to
q,
but the charge existing on the image should
"SS=
be the negative of that on. the antenna0 Since the area beneath the
current distribution curve is proportional to the effective charge* the
image charge is a better comparison to that of the antenna when the
ground plane is large» This explains why the results were better for the
10 centimeter plane than for the 6 centimeter plane«
Another assumption that was made was that the electric intensity
in the
r
direction was negligibleo The electric intensity in the
r
direction is known to exist and* therefore * may be appreciableo This
could produce a definite error because then the
0
component of electric
intensity would not be related to the magnetic field intensity by the
intrinsic impedance0
-3 6 - ,
APPENDIX A
EXPERIMENTAL RESULTS
Measured Current^ Square R oot V a iu ea and R a d ia tio n P a tte r n s
L ® 6 cm
0
Degrees
0
I
u .amps
.
L 53 10 cm
VT
o95
I
I
V T
u amps
0
I
0
5
Ie
I.
10
1.5
1.22
3o
1.73
15
1,6
■1.26
10.
3.16
20
2.
IoUl
21©
U.58
25
1 .5
1.22
38. .
6.16
30
Io
I.
53.
35
1.5
1.22
6U.
8.
UO
2.2
1.U8
8U.
9.17
U5
5 o'
. 2.2U
65.
8.06
5o
9*
3.
92.
9.59
55
11.9
3 .UU
9U.
9.70
60
12.
3.U6
80.
8.9U
65
15.
3.8?
60.
7.75
70
. 22.
UO69
. 50 .
7.07
75
19.
Uo36
5Uo
7.35
80
l6 .
Ue •
52.
7.21
85
15. '
3.87
32.
5.66
90
12 o
3oU6
13 0
3.61
.5
.71
'7.28
•
-3 7 0°
30"
W-..'-/
. w .
IX
\ -V, -L
F ig u r e 8
E xp erim en tal p a tte r n s o b ta in ed over c ir c u la r ground p la n e s o f (a ) 2 ,
(b ) it, ( c ) $ , (d ) 6 , ( e ) 8 , and ( f ) 10 c e n tim e te r s in d ia m e te r .
■33-
F ig u r e 3 (O c n t'd .)
-39
'
/
^x
F ig u r e 3 ( C o n V d .)
APPENDIX B
MAIHEMATICAL RESULTS
u 9 Q.,9 A.9 Bj9 V
D egrees
and R a d ia tio n P a tte r n s
W g v e Wave
L en gth s | L engths
L s IO cm
D egrees
Wave
L engths
Wave
Lengths'
O o O llO
—Itl-
M ath em atica lly d eriv ed p a tte r n s f o r r a d ia t io n over c ir c u la r ground p la n e s
o f (a ) 6 and (b ) 10 c e n tim e te r s in d ia m eter .
LITERATURE CITED AND CONSULTED
H ilderbrands F= B ., 19k9°
InCo5 New York
ADVANCED CALCULUS FQR ENGINEERS, P r e n tic e -H a ll
Jordan, Edward Co, 1950= ELECTROMAGNETIC WAVES AND RADIATING SYSTEMS, .
P r e n tic -H a ll Inc=, New York
M artin, Thomas L», J r 0, 1950«
H a ll, I n c 0 , New York
ULTRAHIGH FREQUENCY ENGINEERING,
P r e n tic e -
Ramo, Simon anc( Winnery, John R t5 19Wl« FIELDS AND WAVES IN MODERN RADIO,
John W iley and Sons, I n c 0, New York
R yder, John Do, 191*9o NETWORKS, LINES, AND FIELDS, P r e n tic e -H a ll, Inc =,
New York
S k i l l i n g , Hugh H ild r e th , 191*2«, FUNDAMENTALS OF ELECTRIC WAVES, John W iley
and S on s, In c * , New York..
■^ k i
103061
*1^I-I
MONTANA STATE UNTVFRSrrv i troaotcc
3
1762
1001 5 5 4 3
105061
;o78
3p65vr
9
c o p .2
AUTHOR
S p oken, Leo Raymond
______
Wave p r o p a g a tio n o v er a f i n i t e
vrm m d p la n e ,
t 1TLE
N3 78
Sjo (2>3yy
C O jD .2 ~
103061
^69