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Ring Opening Polymerisation of ϵ-Caprolactone Using MicrowaveElectric and Magnetic Heating

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UMI
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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M EA SU R EM EN T O F
R E F L E C T IO N
TH E MICROWAVE
C O E F F IC IE N T O F LAMINAR D IE L E C T R IC S
by
F o u a d H. F a n a k i
S u b m itte d in p a r t i a l fu lfillm e n t
of th e r e q u ir e m e n ts fo r th e d e g re e of
M a s te r of S c ie n c e
F a c u lty of G ra d u a te S tu d ies
The U n iv e rs ity of W e s te rn O n ta rio
London, C anada
1962
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UMI Number: EC45214
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UMI
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A p p ro v ed fo r the
A p p ro v e d fo r th e
D e p a rtm e n t of P h y s ic s
D e p a rtm e n t of P h y s ic s
by th e E x a m in in g C o m m itte e
May 8, 1962
by th e A d v is o ry C o m m itte e
r
d
- T
U
11
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A B STRA C T
It h a s b een found th a t r e fle c tio n s of c o n s id e ra b le im p o rta n c e to
th e p ro p a g a tio n of u l t r a h ig h fre q u e n c y r a d io w av es o c c u r co m m o n ly
w ith in th e tr o p o s p h e r e .
If is b e lie v e d th a t th e s e tr o p o s p h e r ic r e f le c tio n s
a r e c a u s e d by w ea k i r r e g u l a r i t i e s in th e d ie l e c tr ic c o n s ta n t of th e a i r .
F ew a tte m p ts h av e b e e n m a d e to u n d e r s ta n d th e p h y s ic a l p r o c e s s e s th a t
le a d to th e s e i r r e g u l a r i t i e s .
O ne of th e s e is th e s tu d y of m ic ro w a v e
r a d a r " a n g e ls " (i. e . th e r e f le c tio n of r a d io w av es f r o m a s e n s ib ly c le a r
r e g io n of th e tro p o s p h e r e ).
O ne a s p e c t of th e s tu d y of m ic ro w a v e a n g e ls is th e a n a ly s is of th e
z
) of d ie le c tr ic la m in a .
r
S u ch la m in a w ould r e p r e s e n t la y e r - ty p e i r r e g u l a r i t i e s in r e f r a c t i v i t y of
th e tr o p o s p h e r e .
A m ic ro w a v e r a d a r a t 10, 000 M c /s e c . w as c o n s tr u c te d fo r th is
p u rp o se .
By u s in g s e p a r a te t r a n s m i t t e r and r e c e i v e r a n te n n a s , m e a s u r e ­
m e n ts w e r e ta k e n in th e n e a r fie ld to p ro v id e a h ig h d e g re e of s e n s itiv ity
r—*2
in
. F r o m th e o ry , i t is found th a t th e d iffe re n c e b e tw e e n th e s ig n a ls
s c a t te r e d f r o m f la t d i e l e c tr ic and co n d u ctin g la m in a e of lik e g e o m e tr y is
th e d iffe re n c e in
of th e F r e s n e l z o n e .
2 w hen th e w id th of th e la m in a is l e s s th a n th e d ia m e te r
M e a s u re m e n ts w e re m a d e on s ig n a ls s c a t t e r e d by
g la s s an d b r a s s d is c s a s a t e s t of th is p r in c ip le .
F r o m th is in fo rm a tio n
i t w as found th a t J""12 fo r a g la s s d is c of th ic k n e s s 0 .9 4 c m . and d i e le c tr ic
c o n s ta n t 6 .7 1 is e q u a l to 0 .4 3 + .0 5 , in c lo s e a g r e e m e n t w ith th e th e o r e tic a l
v a lu e .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ACKNOW LEDGE MEN TS
The a u th o r w is h e s to th a n k D r . A. D . M is e n e r , F . R . S . C . ,
D r . R . J . U ffen, P r o f e s s o r R . L . A lien and D r . P . A. F o r s y th , F . R . S . C.
f o r th e u s e o f th e f a c ilitie s o f th e D e p a rtm e n t of P h y s ic s .
/
H e is g r e a tly in d e b te d to th e m e m b e r s of h is A d v is o ry C o m m itte e
an d in p a r t ic u la r to D r . D . R . H ay w ho s u p e r v is e d th is w o rk and w h o se
a d v ic e an d a s s i s ta n c e w e r e of g r e a t v a lu e .
A p p re c ia tio n is e x p r e s s e d to
th e T r o p o s p h e ric P h y s ic s G ro u p a s a w hole fo r th e ir f rie n d ly a d v ic e
th ro u g h o u t th e w o rk .
The a u th o r w is h e s to e x p r e s s h is s p e c ia l g r a titu d e to th e g o v ern p aen t
o f th e U n ited A ra b R e p u b lic f o r th e s c h o la r s h ip h e ld by h im d u rin g th e
p e r io d o f th is in v e s tig a tio n .
He is g r a te f u l f o r th e h e lp g iv en in te c h n ic a l m a tte r s b y M r. D . G.
R um boJd, M r. E . T h o m p so n , M r. i i . J e f f r i e s , M r. T . D av id so n and
M r. F . S e ltx e r .
T h is in v e s tig a tio n w a s s u p p o rte d b y a g r a n t f r o m th e D efence
R e s e a r c h B o a rd of C an a d a, fo r w h ich th e a u th o r w is h e s to e x p r e s s h is
g r a titu d e .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE O F CONTENTS
A B STR A C T
--------
iii
A C K N O W LE D G E M E N TS--------------------------------------------------L IS T O F IL L U S T R A T IO N S
L IST O F TA B LES
iv
-------------
v iii
-----------
C h a p te r I
I n tr o d u c tio n ...............
C h a p te r II
T he R a d a r E q u a tio n
**
l
....................
5
..................................
2.1 In tro d u c tio n
2* 2 T he R a d a r E q u a tio n
5
............................
. ....
2*3 S c a tte r in g C ro s s ^S ection fo r a C o n d u c to r. • . .
•
C h a p te r III E le c tr o m a g n e tic S c a tte rin g . • • ........................
5
6
9
3*1 In tro d u c tio n • • • .......................
9
3* 2 E , M* S c a tte r in g byAC o n d u c to r.....................
9
3* 2*1 G eom etry* • • . . . *.................................
9
3* 2. 2 The F ie ld I n te n s ity a t P
3* 3 E , M* S c a tte r in g by a D i e l e c t r i c
•••
...........
ID
13
3* 3* 1 R e fle c tio n C o e ffic ie n t............................
13
3* 3. 2 The D iffra c tio n F ie ld - .......................
13
3* 4 S c a tte r in g C r o s s - S e c tio n fo r D i e l e c t r i c s ................
14
C h a p te r IV M ethods of T h e o r e tic a l D e riv a tio n of R e fle c tio n . . . .
C o e ffic ie n t
16
4* 1 I n tr o d u c tio n ..........................................................................
16
4* 2 R e fle c tio n f r o m a S e m i-In fin ite S u r f a c e .................
16
4*3 R e fle c tio n f r o m Two P a r a l l e l P la n e I n t e r f a c e s . .
18
4*3* 1 R e fle c tio n C o e ffic ie n t of a G ra d u a l T r a n s ­
itio n of D ie le c tr ic C o n s ta n t..............
18
4* 3.* 2 R e fle c tio n C o e ffic ie n t of an A b ru p t T r a n s ­
itio n AD ie le c tr ic C o n s ta n t
..............
21
4* 4 R e fle c tio n f r o m a C u rv ed S u r f a c e ..............................
24
4. 5 S u m m a r y ................
24
v
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C h a p te r V
M ethods of M e a su rin g R e fle c tio n C o e f f ic ie n t,.................
26
5 .1
I n tro d u c tio n
26
5. 2
W aveguide R e f l e c t o m e t e r .............................
26
5 .3
W aveguide B rid g e N e tw o r k ...............
28
5 .4
M icro w av e I n t e r f e r o m e t e r ..........................
28
5.
5
29
5 .6
P u ls e d R a d a r
M icro w av e M ir r o r
5, 7 D o p p ler R a d a r
5 .8
.....................
.
................
29
................
F re q u e n c y M o d u lated R
31
a
d
a
r , ...............
31
5 .9 S u m m a r y ...................
32
C h a p te r VI T he N e a r F ie ld M e th o d ..............................
6 .1
I n tr o d u c tio n ......................................................................
33
33
...........................................................
35
6. 2 ,1 T r a n s m ittin g and R e c e iv in g A n te n n a s .. . . . .
35
6* 2. 2 A u to m a tic F re q u e n c y C o n tro l S y s te m .. . . . .
35
6. 2. 3 B a c k g ro u n d S u p p r e s s i o n .................. ...................
37
6*3 T he T a r g e t R e g io n ...............................................................
37
6* 4 T a r g e t M ount
43
6* 2 L a b o ra to r y R a d a r
6 .5
.............
M e a s u re m e n t P r o c e d u r e ............................................
C h a p te r VII P r e c is io n of M e a s u r e m e n t.
7 .1
I n tr o d u c tio n
48
....................................................
..............................
7. 2 S y s te m S ta b ility *
..............................
49
49
49
7. 2 .1
T r a n s m itte r P o w e r .......................
49
1.2,2
C a lib r a te d A tte n u a to r S e t t i n g . . . . . . . . . . . . .
50
7. 2. 3 S y s te m N o ise B e v e l .....................
50
7. 2, 4 R e p e a ta b ility of M e a s u re m e n t . . . . . . . . . . . .
50
7. 3 M e a s u re m e n ts upon S ta n d a rd T a rg e ts • • • • • • • • • r ••
50
7. 3 .1
R e fle c tio n F r o m C onducting S p h e re s • r > • •• 51
7. 3. 2 R e fle c tio n F r o m C onducting D is c s . . . . . . . . 56
7* 4 S u m m a ry
................
57
C h a p te r VIII R e fle c tio n C o e ffic ie n t of G la s s B a m i n a ............................
8 . 1 I n tr o d u c tio n
..............................
vi
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60
60
8. 2 T h e o ry of R e fle c tio n
60
8. 3 Me a s u r m e a ts
61
8 .4
62
R e m a rk s
C h a p te r IX C o n clu sio n s
64
A ppendix I D ie le c tr ic L a y e r R e fle c tio n
66
A ppendix II C ir c u it D ia g ra m s f o r C. W. R a d a r
74
REFERENCES
79
v ii
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LIST O F ILLUSTRATIONS
F ig u r e 1
On th e C u r r e n t D is trib u tio n M e th o d . ........................................11
F ig u r e 2
D ie le c tr ic L a y e r R e f le c tio n ............................................. . .............. 20
F ig u re 3
R e fle c tio n of a P la n e D ie le c tr ic S h e e t a s a F u n c tio n of
T h ic k n e s s ...............................................
22
F ig u re 4
M ethods o f M e a su rin g th e R e fle c tio n C o e ffic ie n t.................. 27
F ig u r e 5
M ethods o f M e a su rin g th e R e fle c tio n C o e f f ic ie n t.................30
F ig u r e 6
D iffra c tio n of W aves a t S m a ll A p e r t u r e ............. ...... ..............34
F ig u r e 7
L a b o r a to r y R a d a r F o r D ie le c tr ic R e fle c tio n M e a s u r e ­
m ent
36
F ig u r e 8
F ie ld I n te n s ity M e a su rin g E q u ip m e n t...................................... 38
F ig u r e 9
A x ial R a d ia tio n P a t t e r n ................................................................. 39
F ig u re 10
T r a n s v e r s e R a d ia tio n P a tte r n ( T r a n s m itte r A n te n n a ) .. . 40
F ig u r e 11
T r a n s v e r s e R a d ia tio n P a t t e r n (R e c e iv e r A n ten n a)................. 41
F ig u r e 12
P h a s e M e a su rin g E q u ip m e n t........................................................ 42
F ig u r e 13
T r a n s v e r s e P h a s e C hange ( T r a n s m itte r A n te n n a )
F ig u r e 14
T r a n s v e r s e P h a s e C hange (R e c e iv e r A ntenna).................... 45
F ig u r e 15
T a r g e t M ount ( S p h e r e s ) ......................................................................46
F ig u r e 16
T a r g e t M ount ( D is c s ) ....................................................................... 47
F ig u re 17
S c a tte r in g F r o m M eta l S p h e r e ................................................. 54
F ig u r e 18
S c a tte r in g F r o m M etal and G la s s D i s c s ............. .................. 58
F ig u r e 19
I. F . A m p lifie r (F o u r s ta g e s ) ....................................................... 75
F ig u r e 20
I. F . A m p lifie r (n in e s t a g e s ) ....................................................... 76
F ig u re 21
A* F . C. S y s te m
F ig u r e 22
2K25 K ly s tro n P o w e r S u p p ly ....................................................... 78
44
............................................................................... 77
v iii
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LIST O F TABLES
T a b le I
F o r m u la s fo r R a d a r G r o s s - S e c t i o n ............................
T a b le II
P o s s ib le E r r o r s
T a b le III
C h a r a c te r is tic s of G la s s D i s c s .....................................................
...........................................................
ix
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8
55
62
CHAPTER I
INTRODUCTION
The s u b je c t of e le c tr o m a g n e tic p ro p a g a tio n th ro u g h th e d e n s e lo w e r
re g io n s of th e a tm o s p h e r e ( tr o p o s p h e re )* h a s r e c e iv e d c o n s id e ra b le
th e o r e tic a l an d e x p e r im e n ta l s tu d y .
One of th e r e a s o n s fo r th is stu d y
h a s b ee n th e e n o rm o u s im p o rta n c e of c o m m u n ic a tio n s by s c a t te r ■•pro­
p a g a tio n .
B e fo re th e d e v e lo p m e n t of a c c u r a te in s tr u m e n ts f o r th e m e a s u r e m e n t
o f th e r e f r a c t i v e in d e x of th e a tm o s p h e r e , l i ttl e w as know n a b o u t d is c r e te
r e f le c tin g r e g io n s in th e tr o p o s p h e r e .
R e c e n tly , h o w e v e r, C ra in (1955)
and B irn b a u m (1955) h av e p ro v id e d m o re c o m p le te in fo rm a tio n a b o u t the
r e f r a c t i v e in d e x of th e tr o p o s p h e r e th ro u g h th e u s e of th e m ic ro w a v e r e fra c to m e te r.
S tu d ie s w ith th is in s tr u m e n t, h o w e v e r, a r e e x p e n s iv e to
co n d u ct on a l a r g e s c a le an d a r e lim ite d to th e e x a m in a tio n of only one
p o in t in th e tr o p o s p h e r e a t any one m o m e n t.
The u s e of r a d a r , on the
o th e r h an d , e n a b le s th e e x p lo ra tio n of la r g e v o lu m e s of s p a c e in a v e r y
s h o r t tim e .
This a lte r n a tiv e a p p ro a c h , w h ich p r o v id e s in fo rm a tio n on th e fin e
s tr u c t u r e of th e tr o p o s p h e r e , w as s u g g e s te d by the a c c id e n ta l o b s e rv a tio n s
of M itra (1936), W atson - W att e t a l (1936) and C olw ell e t a l (1939), w o rk in g
in d e p e n d e n tly .
T hey d is c o v e re d th a t in a d d itio n to th e e x p e c te d ion osp h eric
ec h o e s in th e u p p e r a tm o s p h e r e , th e r e a r e m a n y w eak r e f le c tio n s f ro m
* The tr o p o s p h e r e is th a t p a r t of th e e a r t h 's a tm o s p h e re w h ich lie s b e ­
tw een th e g ro u n d an d th e tro p o p a u s e , w h e re clouds f o r m and co n v e c tio n
is a p p r e c ia b le . Its d e p th is ab o u t th ir te e n k ilo m e te r s .
1
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2
le v e ls w e ll belo w th e E - la y e r o£ th e io n o s p h e re .
S u b se q u e n tly s e v e r a l
l e t t e r s and p a p e r s (P la n k , 1956) a p p e a r e d on th e s u b je c t of r e f le c tio n s .
T h e se r e fle c tio n s w e re r e c e iv e d f r o m r e g io n s in th e a tm o s p h e r e w h e re
no a p p a r e n t r e f le c tin g o r s c a tte r in g s o u r c e s e x is te d .
e c h o e s h av e b e e n te r m e d " a n g e ls ”.
w ith in th e tr o p o s p h e r e .
T h e se c l e a r - s k y
T he r e f le c tio n m a y o c c u r a n y w h e re
Io n iz a tio n do es n o t p la y any p a r t in m ic ro w a v e
an g e l r e f le c tio n s in c e o b s e rv a tio n s m a d e w ith m e te o r o lo g ic a l sou n d in g
b a llo o n s b y G ish and B o o k er (1939) in d ic a te th a t the n e c e s s a r y io n iz a tio n
is n o t p r e s e n t in th e tr o p o s p h e r e .
P id d in g to n (1939) an d G ordon (1949) s u g g e s te d th a t th e a n g e l r e f le c tio n s
a r e c a u s e d by w eak i r r e g u l a r i t i e s in th e d ie l e c tr ic c o n s ta n t of th e a i r in th e
tr o p o s p h e r e an d th is s u g g e s tio n is s u p p o rte d by m ic ro w a v e r e f r a c to m e te r
m e a s u r e m e n ts .
L ittle in fo rm a tio n is a v a ila b le on th e d im e n sio n s of th e s e
i r r e g u l a r i t i e s , an d th e p r o c e s s e s by w hich th e y o c c u r w ith in th e tr o p o s ­
p h ere.
P id d in g to n (1939) in h is e a r ly r e p o r t show ed th a t th e m a x im u m
p o w er r e f le c tio n c o e ffic ie n t of tro p o s p h e r ic a n g e ls is of th e o r d e r of 10
-10
.
M o re r e c e n t e x p e rim e n ts by C h m ela and A r m s tr o n g (1955) and S to u t
an d S pock (1955) h av e show n th a t a n g e l e c h o e s m ay a ls o o c c u r n e a r th e
b o tto m , s id e s and to p of c lo u d s.
H o w ev er, m a n y o th e r e x p e r im e n ta l
o b s e rv a tio n s ( L e a s u r e e t a l 1957; A tlas 1959) of tr o p o s p h e r ic r e fle c tio n s
h av e b ee n o b ta in e d f r o m e s s e n tia lly c le a r s k ie s .
H ay an d R e id (1962) u s e d a 6770 M c /s e c . fre q u e n c y -m o d u la te d
r a d a r to o b ta in f u r th e r in fo rm a tio n on th e a n g e l s tr u c t u r e of th e tr o p o s p h e r e .
T hey found th a t r e f le c tin g r e g io n s in th e tr o p o s p h e r e co m m o n ly h av e p o w er
• 16
*11
r e f le c tio n c o e ffic ie n ts ly in g betw een 10
an d 10
, if th e y are h o riz o n ta l
f la t s t r a t a .
The d e d u c e d r e f le c tio n c o e ffic ie n ts r e q u ir e th a t th e r e ­
fle c tin g la y e r b e only a few c e n tim e te r s in d e p th .
M icro w av e r e f r a c t ­
o m e te r so u n d in g te c h n iq u e s h av e n o t p e r m itte d s p a tia l r e s o lu tio n of th is
o r d e r , an d h e n c e no d i r e c t r e f r a c tiv ity m e a s u re m e n ts a r e a v a ila b le to
s u b s ta n tia te th is c la im .
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3
F r o m th is b r i e f r e s u m e . I t is e v id e n t th a t m o re c o m p le te in f o r m ­
a tio n a n th e m i c r o s t r u c tu r e of th e tr o p o s p h e r e is n e e d e d .
An e x a m in ­
a tio n of th e r e la tio n b e tw e e n th e p o w e r r e f le c tio n c o e ffic ie n t and th e
g r a d ie n t in d i e l e c t r ic c o n s ta n t is an im p o r ta n t s te p in th e s tu d y of th e s e
m ic ro s tru c tu re s .
It i s n o te d th a t th e o r e tic a l a n a ly s e s o f p o w er r e f le c tio n
c o e ffic ie n ts h av e b e e n w o rk e d o u t in a v e r y lim ite d n u m b e r of c a s e s .
H e n c e , a s tu d y o f Hie e ffe c ts of a n o n - u n if o r m illu m in a tio n of th e t a r g e t
an d o f v a r io u s d is tr ib u tio n s of th e d ie le c tr ic c o n s ta n t a t th e t a r g e t upon
th e p o w er r e f le c tio n c o e ffic ie n t, is n e e d e d .
T he p r e s e n t a c c o u n t d e s c r ib e s a la b o r a to r y m e th o d o f m e a s u r in g
th e w eak p o w e r r e f le c tio n c o e ffic ie n ts o f f la t la m in a r d i e l e c t r ic s .
T h is
m e th o d m u s t p ro v id e f o r m e a s u r e m e n ts upon la m in a th a t a r e m a n y w a v e ­
le n g th s in l a t e r a l e x te n t s o th a t a l a t e r s tu d y m a y be c a r r i e d o u t on th e
e f fe c t of n o n -u n ifo rm ity o f illu m in a tio n upon th e p o w er r e f le c tio n c o ­
e ffic ie n t o f th e la m in a .
A b r ie f c o m m e n ta ry on th e r a d a r e q u a tio n and th e s c a tte r in g c r o s s s e c tio n a r e p r e s e n te d in C h a p te r U.
The th e o r y of e le c tr o m a g n e tic
s c a t te r i n g by c o n d u c tin g an d d i e l e c t r ic b o d ie s is s ta te d b r ie f ly in C h a p te r
111, to g e th e r w ith th e d e fin itio n o f th e p o w er r e f le c tio n c o e ffic ie n t and its
r e la tio n to Hie r a d a r c r o s s - s e c t i o n .
A s u m m a r y o f th e e x is tin g th e o r e tic a l
a n a ly s e s o f r e f le c tio n c o e ffic ie n ts is in c lu d e d in C h a p te r IV .
C h a p te r V
o u tlin e s th e v a r io u s m e th o d s u s e d in m e a s u r in g r e f le c tio n c o e f fic ie n ts .
T he p r in c ip le o f th e n e a r - f i e l d m e th o d em p lo y e d by th e w r i te r an d a
d e s c r ip tio n o f th e e q u ip m e n t a r e g iv en in C h a p te r VI.
C h a p te r VXX
p r e s e n ts an a n a ly s is o f th e p r e c is io n of m e a s u r e m e n ts upon th e c o n d u ctin g
s p h e r e s an d co n d u ctin g d is c s ( s ta n d a r d ta r g e ts ) .
T he s a m e C h a p te r
c o n ta in s , in a d d itio n , th e th e o r e tic a l a n a ly s is of r e f le c tio n f r o m the
s ta n d a r d t a r g e t s .
The m e a s u r e m e n t o f th e p o w er r e f le c tio n c o e ffic ie n t
o f a g la s s la m in a is p r e s e n te d in C h a p te r V III.
C h a p te r IX c o n ta in s th e
c o n c lu s io n s d ra w n f r o m th e e x p e r im e n ta l and th e o r e tic a l w o rk *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
T he m a th e m a tic a l a n a ly s is of th e r e f le c tio n c o e ffic ie n t of a th in
d i e l e c t r ic la y e r is d is c u s s e d in A ppendix 1.
C ir c u it d ia g r a m s of th e
C-W r a d a r a r e g iv e n in A ppendix II.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHA PTER II
TH E RADAR EQUATION
2 .1
In tro d u c tio n
The m e th o d o f d ed u c in g th e r e f le c tio n c o e ffic ie n t of a la m in a , as
d e s c r ib e d in th is th e s i s , m a k e s u s e of th e r a d a r e q u a tio n and th e s c a tte r in g
c r o s s - s e c t i o n of th e r e f le c tin g ta r g e t.
T his C h a p te r h,*gina w ith a b r ie f
c o m m e n ta ry on th e r a d a r e q u a tio n , fo llo w ed by th e d e fin itio n of the
s c a tte r in g c r o s s - s e c t i o n .
T he f o rm u la s fo r th e r a d a r c r o s s - s e c tio n s
of r e p r e s e n ta tiv e ta r g e ts a r e p r e s e n te d in a ta b le .
2. 2 The R a d a r E q u a tio n
L#et u s c o n s id e r a r a d i a to r of e le c tr o m a g n e tic e n e rg y w ith
a s the
p o w e r r a d ia te d u n ifo rm ly in a ll d ir e c tio n s (i. e . an is o tr o p ic r a d ia t o r ) .
At a r a n g e R th e p o w e r d e n s ity f r o m th a t r a d ia to r is
P t /4irR 2
(1)
If th is r a d i a t o r is n o n - is o tr o p ic
a n g le
and h a s a g a in G^(0,
<j>), w h e re 6 is the
of e le v a tio n an d <j> is th e a z im u th , th e n th e p o w er d e n s ity
a tra n g e
R is g iv en by
P .G J 9 , <j>)
( 2)
4irR
T h is fo rm u la a s s u m e s th a t th e fie ld of th e a n te n n a is d iv e r g e n t r a d ia lly
and h e n c e id e n tifie s it a s th e F ra u n h o fe r (o r d is ta n t) fie ld .
If an o b je c t
is lo c a te d a t d is ta n c e R f r o m th is a n te n n a , its r e a c tio n to th e in c id e n t
f ie ld is d e s c r ib e d by its s c a tte r in g c r o s s - s e c t i o n (<f), w h ich w ill be
d is c u s s e d in a l a t e r s e c tio n .
T he d e fin itio n of th e s c a tte r in g c r o s s -
s e c tio n a s s u m e s th a t th e o b je c t e ffe c tiv e ly in te r c e p ts p o w er f r o m an
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6
a r e a $ of th e in c id e n t w av e and th e n r e r a d i a t e s i t is o tr o p ic a lly .
C on­
s e q u e n tly , th e r e - r a d i a t e d p o w er d e n s ity a t d is ta n c e R f r o m th e i n t e r ­
ce p tin g o b je c t is
P t Gt
2
4trR
*
4uR
(3)
2
A gain, th is im p lie s th a t th e r e - r a d i a t e d fie ld is d iv e r g e n t r a d ia lly
f r o m th e o b je c t, a s in its F ra u n h o fe r re g io n ,
A s e c o n d a n te n n a is lo c a te d
a t d is ta n c e R f r o m th e s c a tte r in g o b je c t and h a s a g ain G ^ S 1, $ ().
T his
a n te n n a w ill h a v e an e ffe c tiv e r e c e iv in g c r o s s - s e c t i o n A^ w h ich is r e la te d
to G (0*, 4>*) by
r
,
4ir A
G <0, $) « — 5 - i
r
x*
w h e re
X
b y th e
r a d a r a n te n n a ,
(4)
is th e s ig n a l w ave le n g th (S ilv e r 1949),
w ill th e n b e g iv en by
<j>)
P
.
The p o w er in te r c e p te d
<p
X2G (e*,4>‘)
----- 2----------------- 2
4irR
4irR
£-----4ir
<5>
the
T h is is^ w ell know n r a d a r e q u a tio n ( s e e fo r ex a m p le K e r r 1951),
2. 3 S c a tte r in g C r o s s - S e c tio n f o r a C onductor
The s c a tte r in g c r o s s - s e c t i o n (T o f a r e f le c to r is d e fin e d a s th e a r e a
in te rc e p tin g th a t a m o u n t of p o w er w h ich , if s c a tte r e d is o tr o p ic a lly , w ould
r e t u r n to a p o in t P an am o u n t of p o w er e q u a l to th a t a c tu a lly r e c e iv e d a t
P.
M a th e m a tic a lly it is d e fin e d as fo llo w s (M en tx er 1955):
CT* « lim
R -► 00
w h e re
an d
2 I W I
4irR | |
i
(6)
is th e s c a t te r e d p o w er d e n s ity a t d is ta n c e R f r o m th e s c a t t e r e r ,
is th e p o w er d e n s ity in an in c id e n t p la n e w ave f ie ld .
F o r a p e r f e c tly
co n d u ctin g s c a t t e r e r a l l of th e in c id e n t E . M. fie ld is r e - r a d i a t e d and none
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7
is l o s t by t r a n s m is s io n th ro u g h th e s c a t t e r e r o r d is s ip a tio n w ith in i t .
A
s p e c ia l k in d of s c a tte r in g c r o s s - s e c t i o n is th a t o b s e rv e d w hen th e r e ­
c e iv in g and tr a n s m ittin g a n te n n a s a r e lo c a te d a t th e s a m e p la c e .
H e re cr
is th e r a d a r c r o s s - s e c t i o n .
The r a d a r c r o s s - s e c t i o n fo r p e r f e c tly co n d u ctin g r e f l e c t o r s of
v a r io u s s h a p e s , a ll b ein g u n ifo rm ly illu m in a te d , hav e b e e n c a lc u la te d by
v a rio u s a u th o rs u s in g th e above d e fin itio n .
A few e x a m p le s o f th e c a l ­
c u la te d (P a r e q u o ted in th e follow ing ta b le to show th e d e p e n d e n c e of 6"
on th e g e o m e try , illu m in a tio n , and th e p h y s ic a l p r o p e r tie s of th e ta r g e t.
T he m eth o d o f c a lc u la tin g < p fo r co n d u ctin g s p h e r e s and f la t co n d u ctin g
d is c s is i l l u s t r a t e d in l a t e r C h a p te rs .
F r o m ta b le 1 i t can be s e e n th a t (p is a fu n ctio n of g e o m e try (the
f i r s t fo u r e x a m p le s ) an d of illu m in a tio n (the l a s t tw o e x a m p le s ).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE I
FORM ULAS F O R RADAR CROSS-SECTIONS
S c a tte re r
A spect
1) p la te , s e m i- in f in ite
c o n d u c to r
N o rm a l
in c id e n c e
2) la r g e , f la t, co n d u ctin g
p la te
N o rm a l
in c id e n c e
3)
N o rm a l
in c id e n c e
4)
la r g e , co n d u ctin g s p h e re
la r g e , co n d u ctin g
c ir c u l a r c y lin d e r
b ro ad s id e on
5) la r g e , f la t, r e c ta n g u la r
p la te
in c id e n c e
a t a n g le 0
6) la r g e , co n d u ctin g
c ir c u la r c y lin d e r
in c id e n c e
a t an an g le
6 to b r o a d ­
s id e
D efin itio n of S y m b o ls
vR
4irA ^/X ^
ua
2tt a 1
R e fe re n c e
R = R ange to p la te
K e r r (1951)
A = P la te a r e a
X * w a v e le n g th
K e r r (1951)
a * R ad iu s
M e n tz e r (1955)
1 * c y lin d e r le n g th
K e r r (1951)
4vA s in (k a s in 6)
k * 2tr
2 k a s in 9
X
X 2
A ssu m in g th e p la te is in th e
(co s 6).
x - y p la n e th en a is th e s id e
p a r a l l e l to th e x - a x is
2
aX co s 8 s in (k 1 s in 6)
2
2ir
s in 9
M e n tz e r (1955)
CD
C H A P T E R HI
E L E C T R O M A G N E T IC SC A T TE R IN G
1 In tro d u c tio n
The n a tu r e of th e s c a tte r in g c r o s s - s e c t i o n ((P) of co n d u ctin g and
d i e l e c t r ic ta r g e t s w ill be e x a m in e d to show how th e s c a tte r e d s ig n a l
d ep en d s upon th e p h y s ic a l p r o p e r tie s of th e ta r g e t.
A th e o r e tic a l d i s ­
c u s s io n on th e s c a t t e r e d fie ld of co n d u ctin g an d d ie le c tr ic o b s ta c le s is
c a r r i e d o u t.
T h is is fo llo w ed b y a d e fin itio n of th e r e f le c tio n c o e ffic ie n t.
T he r e la tio n s h ip b e tw e en th e s c a tte r in g c r o s s - s e c t i o n and th e r e f le c tio n
c o e ffic ie n t of d ie l e c t r ic an d co n d u ctin g o b je c ts is d e d u ced f r o m th is
th e o r y .
3. 2 E . M. S c a tte r in g b y a c o n d u c to r
The s c a t te r e d f ie ld in th e p r e s e n c e of a co n d u ctin g o b je c t m a y be
d e te r m in e d by th e " c u r r e n t d is tr ib u tio n " m e th o d .
T his te c h n iq u e r e p la c e s
th e o b je c t b y th e in d u c e d c u r r e n t d is tr ib u tio n o v e r th e s c a tte r e r * s s u r f a c e .
Its u s e a s s u m e s th a t:
(i) th e r e is no c u r r e n t o v e r th e shadow a r e a of th e s c a t t e r e r ,
(ii) th e c u r r e n t d is tr ib u tio n of th e illu m in a te d r e g io n is th e s a m e a t
e a c h p o in t.
The " c u r r e n t d is tr ib u tio n " m e th o d d e s c r ib e d by S ilv e r (1949) w ill
b e o u tlin e d b r ie f ly h e r e .
3. 2. 1 G e o m e try
C o n s id e r a s c a tte r in g o b je c t
w h ich is lo c a te d in a h o m o g en eo u s,
s o u r c e - f r e e , is o tr o p ic an d n o n -c o n d u c tin g r e g io n V (F ig . l , a ) .
9
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
is an
10
a r b i t r a r y s u r f a c e e n c lo s in g th e v o lu m e o f s p a c e V.
V h a s a g iv en
s in g u la r ity P w h ic h i s th e p o in t o f o b s e rv a tio n o f th e s c a t te r e d f ie ld .
R i s th e d is ta n c e f r o m P to an y o th e r p o in t on th e s c a t t e r e r .
a is a
u n it v e c to r n o r m a l to a bou nding s u r f a c e and d ir e c te d in to th e r e g io n V.
1 .2 .2
T he F ie ld in te n s ity a t P
F o r th e lin e a r m e d iu m in V th e v e c to r H e lm h o its e q u a tio n s m a y be
d e r iv e d f r o m M ax w ell’s e le c tr o m a g n e tic f ie ld e q u a tio n s a s :
V
*V x 1
2
-
—
m
-
V s V * H * k H +V* J
w ith e q u a tio n s o f c o n tin u ity :
V . d + j w^ » o
*7
- 4 j s <)I
^ • K
U)
* 0
w h e re k 2 « w2 ^i £ * ( ^ h k * w av elen g th ).
E q u a tio n s U ) an d (2) a r e in te g r a te d u s in g G r e e n 's th e o r e m .
and H
P
If &
a r e th e s c a t t e r e d f ie ld in te n s itie s a t P th en
% * " 4v
♦
f
/
W wJ 1 *
[ - 4
♦ (a x
V
dv
<n * I ) x V f
s as
1 2
+ ( n .E ) V ^ ] d S
(3)
w h e re 4* ** •
•
1
T he f ie ld s a t th e o b s e rv a tio n p o in t P h a v e th u s b e e n e x p r e s s e d a s th e s u m
o f c o n trib u tio n s d u e to J an d j h i th e v o lu m e V* a n d E an d H on th e
s u r f a c e s 8^ a n d S ^ .
1
P
A s i m i l a r a n a ly s is fo llo w s f o r th e m a g n e tic v e c to r
.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
11
LLUMINATED
REGION
PLANE
INCID EN T
WAVE
SHADOW
REGION
J DIRECTION
OF
PROPAGATION
(b)
FI0.1. ON THE
CURRENT
STRI BUT 10 N
METHOD
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
12
L e t th e bounding s u r f a c e
in fin ite .
( a r b itr a r y ) expand u n til its r a d iu s is
T hen E q . (3) b e c o m e s
*
■
b-
J
0
" / +7
/ [ - i w jx
-
(n x H) + (n x E) x V t|i
si
+ <n.E)V >j/j
dS
(4)
Now a s s u m e th a t th e s c a t t e r e r S j is a p e r f e c t c o n d u c to r.
b o u n d a ry c o n d itio n s a t
T hen th e
a re :
n x E * 0
'O
n. E * —
n. H - 0
_
n x H ■K
(5)
A ssu m e a ls o th a t th e s c a t t e r e r is in th e f a r f ie ld of a t r a n s m i t t e r , s o th a t
th e w ave in c id e n t (E^, IT ) upon th e t a r g e t is a p la n e wave*
T hen th e
a p p ro x im a te illu m in a tio n is d e d u c e d f r o m g e o m e tr ic a l o p tic s ( r a y o p tic s)
b y a s s u m in g th a t its to ta l s u r f a c e S j is d iv id e d as fo llo w s (F ig . l , b )
S
w h e re
l
*S
o
+ S. + S
b
s
(6)
S
= illu m in a te d a r e a
o
Sb - boundary betw een illu m in a t e and .h a d .d a r .a
S
* shaded a re a ,
s
T h e re is n o E . M . f ie ld o v e r Sg, an d h e n c e n o in d u c e d c u r r e n ts o r c h a r g e s .
O v e r SQ th e r e a r e in d u c e d a c u r r e n t K an d a c h a r g e ^ * F r o m th e e q u a tio n s
of c o n tin u ity an d s in c e th e c u r r e n t on
c a n n o t s to p a b ru p tly a t S^> th e r e
i . ovar Sb an in d u e d lin e d i.tr ib u tio n of ch arge
.
F or g e o m e tr ic a l
r e f le c tio n o f a p la n e w av e f r o m a p e r f e c t c o n d u c to r, th e s u r f a c e c u r r e n t
d e n s ity a n th e r e f l e c t o r is
K * 2(n x H.) * 2(n
i
w h e re
r
x H )
* r e f le c te d m a g n e tic fie ld .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(7)
13
T he s u r f a c e c h a rg e d e n s ity is found to be
2 € (n . E ) * 2 6 (n « E )
i
r
(8)
T he in d u c e d lin e d is tr ib u tio n of c h a rg e $ is o b tain ed f r o m th e c o n tin u ity
e q u a tio n s and g iv en by
(9)
w h e re T is a u n it v e c to r alo n g S^*
T h . f l.I d
to a s c a t te r e d .ig n a l f r o m Sj i . found fc o m E q. H )
by th e s u b s titu tio n o f n x H * 2{n x H ^), (n x E) * 0 and (n . E) * 2(n • E^)
f o r th e s u r f a c e in te g r a l o v e r S ; and J * 0,
° —
i
(r
v o lu m e in te g r a l alo n g S, . T hen, E = - -— J |
b
E q , (10) r e p r e s e n t s th e s c a t te r e d f ie ld a t th e o b s e rv a tio n p o in t P in th e
p r e s e n c e of a p e r f e c t c o n d u c to r,
3 . 3 E , M, S c a tte r in g b y a D ie le c tr ic
T he s c a t te r e d fie ld in th e p r e s e n c e of a p e r f e c t d ie l e c tr ic o b je c t
m a y b e d e te r m in e d by th e a p p lic a tio n of th e d e fin itio n of th e r e f le c tio n
c o e ffic ie n t and th e u s e of E q , (10).
3 ,3 .1
R e fle c tio n C o e ffic ie n t
T he c o m p le x p la n e E , M. w ave r e f le c tio n c o e ffic ie n t ( |
a s ( s e e fo r e x a m p le R am P
r
) is d efin e d
and W h in n ery 1946):
-s ’Fke ta n g e n tia l c o m p o n en t of th e e l e c t r i c f ie ld in th e r e f le c te d w ave
T he ta n g e n tia l c o m p o n en t of th e e le c t r i c f ie ld in th e in c id e n t w ave
O r , w ith r e s p e c t to th e m a g n e tic fie ld :
( 11)
n x Jrli
T he s tu d y of I sh o w ed th a t th e r e f le c tio n c o e ffic ie n ts of m e ta ls
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
14
a r e p r a c tic a lly in d e p e n d e n t of th e an g le of In c id e n c e and d iffe r b y a
n e g lig ib le a m o u n t f r o m u n ity .
T hen f r o m e q u a tio n (11)
n x H. * n x H
i
r
(12)
fo r a m e ta l.
F o r a d i e l e c t r ic r e f l e c to r E q , (11) ia a p p lie d and its v a lu e lie s
b e tw e e n 1 an d -1.
In g e n e r a l |""*varies w ith th e an g le of in c id e n c e and
p la n e of p o la r iz a tio n . *
3 .3 . 2 The D iffra c tio n F ie ld
F o r a d ie l e c t r ic p la n e of a r b i t r a r y m a t e r ia l th e s c a t te r e d s ig n a l
E g iv en by E q . (10) m a y b e d e d u ced f r o m th e above a n a ly s is n o tin g th e
P
fo llo w in g :
~
wn x n
r
* - .r - z 1
nn xv H
Wi
(ii)
E q . (10) a p p lie s fo r |^= - 1, w h e re th e e le m e n t of s u r f a c e dS is
a p la n e an d th e in c id e n t w ave is p la n e .
W ith th e ab o v e s ta te m e n ts , E q . (10) m a y be w r itte n , fo r a d i e le c tr ic p la n e
s c a t t e r e r ( j" " ^ !) , a s :
p
2-nr jw e
J
S
[ f i n x H ,). V ( V * ) + k 2 [ " k x H.) +]
o
If j"1m a y b© a s s u m e d c o n s ta n t o v e r th e w hole s u r f a c e S ^, th en
o
E p (fo r a d i e l e c t r ic r e f le c to r ) =|""\
(fo r a co n d u ctin g r e f le c to r ) (14)
3. 4 S c a tte r in g C r o s s - S e c tio n f o r D ie le c tr ic s
C h a p te r II p r e s e n te d th e d e fin itio n of th e s c a tte r in g c r o s s - s e c t i o n
a s th e r a t i o of th e s c a t te r e d p o w er d e n s ity to th e p o w er d e n s ity in an
in c id e n t p la n e w av e .
T h is r a t i o is p r o p o rtio n a l to th e s q u a r e of th e r a t io
of th e s c a t te r e d to in c id e n t f ie ld s .
It c a n be s e e n th en f r o m E q . (14)
* F o r f u r th e r d is c u s s io n s e e S tra tto n (1941).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
15
th a t <T fo r a d i e le c tr ic r e f le c to r is given as
<P ( d ie le c tr ic ) * p
. (T (co n d u ctin g r e f l e c to r hav in g
th e s a m e g e o m e try ),
p ro v id e d th a t
c o n s ta n t
o v e r th e s u r f a c e of the s c a t t e r e r .
M e a s u re m e n ts of th e d iffra c tio n p a tte r n s in f r o n t of s m a ll m e ta llic
d is c s w e r e c o m p a re d w ith th o s e in f ro n t of s m a ll d ie le c tr ic d is c s by
S e v e rin a n d B ae c k m a n n (1951),
T he r e s u l t s c o n firm e d th e th e o r e tic a l
a n a ly s is g iv en a b o v e, a t l e a s t f o r s m a ll u n ifo rm ly illu m in a te d s c a t t e r e r s .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C H A PT E R IV
M ETHODS O F T H E O R E T IC A L DERIVATION O F R E F L E C T IO N
C O E F F IC IE N T
4 .1 In tro d u c tio n
The p r o b le m of d e te rm in in g th e r e f le c tio n c o e ffic ie n t a t a change
in d i e l e c t r ic c o n s ta n t h a s b e e n g iv en c o n s id e ra b le a tte n tio n by v a rio u s
a u th o r s .
The w o rk done on th e r e f le c tio n c o e ffic ie n t is q u ite lim ite d
an d i t c o v e r s o nly th e s im p le s t ty p e s of d is tr ib u tio n of d i e le c tr ic c o n ­
s ta n t.
The p u rp o s e of th is C h a p te r is to p r e s e n t a s u r v e y on th e e x is tin g
d e r iv a tio n s of th e r e f le c tio n c o e ffic ie n t.
in fin ite s u r f a c e is d is c u s s e d in S e c . 4. 2.
T he r e f le c tio n f r o m a s e m i ­
In S e c . 4. 3 f o rm u la s a r e g iven
f o r th e r e f le c tio n c o e ffic ie n t of tw o p a r a l l e l p la n e i n te r f a c e s , fo r a
g r a d u a l lin e a r an d a b r u p t tr a n s itio n in th e d ie le c tr ic c o n s ta n t.
R e fle c tio n
f r o m a c u rv e d s u r f a c e is d is c u s s e d in S e c . 4 .4 .
4 . 2 R e fle c tio n f r o m a S e m i-In fin ite S u rfa c e
S n e ll's law s an d F r e s n e l 's e q u a tio n s to g e th e r d e te r m in e th e d e ­
p e n d e n c e o f th e r e f le c tio n c o e ffic ie n t upon th e e le c tr o m a g n e tic p r o p e r tie s
of th e r e f l e c t o r , a n g le of in c id e n c e and p o la r is a tio n .
S in ce th e fo re g o in g
m a t e r ia l is w r itte n in d e ta il in m a n y te x t books on E . M. th e o r y , (fo r
in s ta n c e , s e e S tra tto n 1941), th is d is c u s s io n w ill be co n fin ed to th e r e s u l t
of th e ir a n a ly s is .
If Z 1 an d Z_ a r e th e in tr in s ic im p e d a n c e s of m e d ia (1) and (2), th e
1
2
r e f le c tio n c o e ffic ie n t
16
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
w h e re
0^ * an g le of in c id e n c e .
an g le of r e f r a c ti o n w ith in m e d iu m (2).
The s u b s c r ip t v and h d en o te v e r t ic a l an d h o riz o n ta l p o la r iz a tio n r e s p e c t ­
iv e ly . * i t is s e e n f r o m £ q s . (1) and (2) th a t p 1 d ep en d s upon th e
e le c tr o m a g n e tic p r o p e r t i e s , th e a n g le of in c id e n c e and th e p o la r iz a tio n .
F o r n o r m a l in c id e n c e (0^ * 0^ * 0) th e d is tin c tio n b e tw e e n th e v an d h -c o m p o n e n ts d is a p p e a r s , and th e r e f le c tio n c o e ffic ie n t s im p lif ie s to
r.p r !ll*i
'
I v “
Ih *
Z , + Z,
2
1
(3)
Now if m e d iu m (2) is a m e ta l, th e n b e c a u s e of th e h ig h c o n d u c tiv ity
( i . e . Z^ < < Z^), E q . (3) r e d u c e s to
P « - 1
.
(4)
A lso if b o th m e d ia a r e n o n -c o n d u c tin g and n o n -m a g n e tic , E q . (3) c a n b e r e w r itte n
w h e re
r
«-i
/-i
(5)
F z *F\
« a p e r m ittiv ity of th e m e d iu m .
F r o m S n e ll's law s
“ "21
<6>
* " H o riz o n ta l" an d " v e r ti c a l" p o la r iz a tio n a s u s e d h e r e m e a n th a t th e
e l e c t r i c v e c to r is p e r p e n d ic u la r to o r lie s in th e p la n e of in c id e n c e .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
18
w h e re n
is th e r e la tiv e in d e x of r e f r a c t io n of th e tw o m e d ia .
bl
of n 2l* E q . (5) r e d u c e s to
r
" a ' 1
“a
S in c e
« *« « ,
r o
w h e re
«
and
e
r
In te r m s
(7)
+ 1
(8)
= r e la tiv e d ie l e c tr ic c o n s ta n t
of f r e e s p a c e .
* --perm ittivity...
E q . (5) b e c o m e s
r
if
2 - K
(9)
j K z + J K i
« « i
r
th 6 “
J 7 ] 2 + - i% i *
a“ d
•K z
-f c l
K
2
^
2' < r l
T hus E q. (9) m a y be s im p lif ie d ( F r ie n d 1949) to
r2
“
rl
(10)
4
4 . 3 R e fle c tio n f r o m Two P a r a l le l P la n e I n te rf a c e s
M ed iu m (2) of th e p re c e d in g s e c tio n m a y n o t r e p r e s e n t a s e m i ­
in fin ite s u r f a c e b u t m a y c o n s is t of a d ie l e c tr ic la y e r of a g iv en th ic k n e s s
fo llo w ed by a th ir d m e d iu m (3).
4 .3 .1 R e fle c tio n C o e ffic ie n t of a G ra d u a l T r a n s itio n of D ie le c tr ic
C o n sta n t
(a)
T he r e f le c tio n f r o m a la y e r w h e re in th e d ie le c tr ic c o n s ta n t
ch a n g es g r a d u a lly , m e d iu m (3) b ein g th e s a m e as m e d iu m (1). and fo r
n o r m a l in c id e n c e , is d is c u s s e d by F r ie n d (1949).
H is a n a ly s is is o u tlin e d
in A ppendix I in m o r e d e ta il, a s h is p u b lish e d a n a ly s is is n o t a d e q u a te to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
19
g iv e a, fu ll u n d e rs ta n d in g o f th e p r o b le m .
H it a n a ly s is in d ic a te d th a t:
f o r a m o n o to n ic tr a n s itio n (F ig . 2, a) th e m a g n itu d e o f r e f le c tio n c o ­
e f fic ie n t ie g iv e n by
In 1
II |
eifth. [ 5 * (V T T l?
-
I f£. S (^ 4
+
f-i
s la te
w h e re
s
♦ Ml
o
.~K )1
J~T+ Mo
,u)
- N ' )1
I
S * l a y e r th ic k n e e e in w av elen g th e
■ an in c r e m e n t o f « ab o v e u n i t y
o
r
3
N « t r a n s itio n d eviation ( « , £ * * f j).
M
As a c h e c k o n th is r e s u lt i t c a n be s e e n th a t f o r $ a p p ro a c h in g sera* th e
eq u a tio n f o r
J-1 b e c o m e s
n
y r n ?
-
j
i t
y
F
♦j i ' tl T' . ' S1*
‘•Vi ♦
us
T h is is th e F r e s n e l e q u a tio n for r e fle c tio n b etw een tw o s e m i- in f in ite
m e d ia (S e c . 4 .2 ) .
r,
F o r a s y m m e tr ic a l la y e r (F ig . 2* b)
f>os/rnr>
■ f H
-----
r jr r c y
f <i - v
r
* f ( i * d 2 - JC*y i ♦
"s
+ d i>
-I
cos 2 ( d ? + j d )
1 -------- L
r—■
^
|
w h e re d^ a n d d ^ a r e th e im a g in a ry and r e a l p arts o f
1♦ 4 3 M
- dj)
(13)
r esp e ctiv ely * and M is the la y e r d ev ia tio a i P = G a m m a f u n c t i o n ) .
(b)
F or a lin e a r grad ien t o f the d ie le c tr ic constant* the r e fle c tio n
is d is c u s s e d b y F la n k (1956) w h ere
w h e re
and
r
^
is g iv e n a s
t.> n 1
4 Iv a
(14)
n * r e f r a c t i v e in d e x
d « th ic k n e s s of th e l a y e r .
T h is c o n fig u ra tio n i s a ls o d is c u s s e d b y K e r r (1951) in t e r m s o f the
s c a t te r i n g c r o s s - s e c t io n ( (T ).
If the la y e r is at a d ista n ce R fr o m a
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
o
<3
Ll I
__1
Ll .
I- LU
z
o. < CH
- *(0
z
o
o (X
LU
in
o or >*
•- <
o
111 — J
-I
111
O 5 O
in
cr
H
O
in
o
LU
-J
UJ
1HOI3H
Q
2-
C\J
<
h
*
o
o
©
Ll.
* 2
O* “
o
bl
J
bl
in
o
in
1H0 I jIH
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
21
r a d ia tio n s o u r c e an d e x te n d s to E + A R o v e r an a r e a A p e r p e n d ic u la r
to th e d ir e c tio n of th e b e a m , th e s c a tte r in g c r o s s - s e c t i o n is
/T»
A 2 f d(n - 1) I 2
“
4 ,3 .
I P L“5r ------J
r 2 ,2ir A R 4
8in ( T
/ir4
“ )•
<15)
2 R e fle c tio n C o e ffic ie n t of an A b ru p t T r a n s itio n in
D ie le c tr ic C o n sta n t
F o r an a b ru p t tr a n s itio n th e r e la tio n s h ip b etw een th e r e f le c tio n
c o e ffic ie n t and th e th ic k n e s s d o f a p la n e h o m o g en eo u s d ie le c t r ic s h e e t
h a s b e e n d is c u s s e d by m an y a u th o rs (fo r e x a m p le , s e e S tra tto n 1941),
It is a s s u m e d th a t th e s h e e t (m e d iu m 2) is l o s s l e s s , p la n e and n o r m a l
to th e d ir e c tio n of in c id e n c e of th e p la n e £ . M. w av e.
H e re th e r e f le c tio n
c o e ffic ie n t is
P
P
\
s
12 +
‘23 6
..—
2jk V
d
1+
w h e re
2jk ,d
P
r
12
U
r
(16)
'2 3 *
1*
P
Z 2 " Z1
| ^ * r e f le c tio n c o e ffic ie n t a t th e b o u n d a ry 1, 2 = - —
r.
(S ec. 4. 2)
2 . 7X
3 * 2
= r e f le c tio n c o e ffic ie n t a t th e b o u n d a ry 2, 3 = - — -——
23
3 +
2
7
If th e m e d iu m on e ith e r s id e of th e s h e e t is th e s a m e ( n , - -
n .,.
E q . (16) r e d u c e s to
r-
~
2 jk ,d
12 (1 " e
IT z
)
7Jk2d *
(1? )
e
T he p o w er r e f le c tio n c o e ffic ie n t (|~~1 ) is e q u a l to th e s q u a r e of
th e a b s o lu te v a lu e o f th e s e r a t i o s . E q . (17) in t e r m s of P 2 b e c o m e s
H 2
■
4 P 2 s i n 2 (k ,d )
12
r
(18)
< - r > >
+ 4 r i 2 ^
<k2d^
T he r e la tio n s h ip b etw ee n j”""* and d is p lo tte d in F ig . (3) fo r a p la n e s h e e t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
22
<
8
IU
LU
I
o_.
y
a:
CO
CO
tu
E
H
O
_l
UJ
0
1
Q
LU
Z
THICKNESS
CO
-I
CL
OF
<
Lu
O
Z
o
o
UJ
_l
Ll
LU
QC
ID
in
1N3I0IJJ3O0
in
N 0 I 1 0 3 “ld3M
O
ro
•
O
u.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
FUNCTION
<
23
o f C o rn in g 0080 g la s s ( d ie le c tr ic c o n s ta n t * 6 . 7 1 (Von H ip p el 1954) a t
2
10*000 M c /s e c . )• I t ca n b e s e e n th a t |~ ' is an o s c illa tin g fu n c tio n of
th e th ic k n e s s d w ith c o n s ta n t a m p litu d e .
T he f i r s t m a x im u m in
is a t
d * . 75 c m (i. e . d « ~ fo r K = 3 c m . ).
The c a s e o f r e f le c tio n o b liq u e ly f r o m a d ie le c tr ic s h e e t is d is c u s s e d
by B a c h y n sk i (1958) w h e re th e m e d iu m s u rro u n d in g th e s h e e t is l o s s l e s s .
A d d itio n al r e f le c tio n s w ill th e r e f o r e o c c u r fet th e b o u n d a ry s u r f a c e s and
p a r t of th is e n e rg y w ill r e t u r n to th e s u rr o u n d in g m e d iu m .
w ave th u s is co m p o se d of s e v e r a l c o m p o n e n ts.
H ence
The r e f le c te d
|~~' w ill v a r y w ith
th e illu m in a te d a r e a s in c e th e in c id e n t and r e f r a c t e d a n g le s a r e n o t c o n ­
s ta n t o v e r th e t a r g e t s u r f a c e .
H e re E q . (17) b e c o m e s
p
P
i1 -
r ?
7>
e j t >
( 1 -----------= n r - ......71—
u -
rf*
w h e re
(19)
is th e p h a s e s h ift and
5
w h e re
)
k2
2 d (c o a T 2 - k l ta n ° 2 a in 81>
0^ = an g le of in c id e n c e
» 2 = a n g le of r e f r a c tio n .
If m e d iu m (3) is a m e ta l <
at d.
id
-1) to ta l r e f le c tio n ta k e s p la c e
In th is c a s e E q . (16) c a n b e r e w r itte n
n
r
- e 2* 2*
“ ---- — - p
2jir,d
XIl2 e
2
<20>
As a f u r th e r exam ple* fo r th e b o u n d a ry b e tw e e n m e d ia 1-3 to be in v is ib le
(i. e .
|~* ^ « 0). E q . (16) b e c o m e s
2 jk 2d
f i
2
+
f »
‘
'
* °
•
F o r a non f e r r o - m a g n e tic m e d ia th e p h a s e r e la tio n
e 2Jk3d - - 1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
<2 1 >
24
i. e .
d = an odd muSfc ip le of ~ ,
P
an d
”
P
I 23 *
«
In te r m s of th e r e l a t iv e d ie le c tr ic c o n s ta n t
* r2
* r3
rl
r2
.
( 22 )
H en ce th e b o u n d a ry b e c o m e s in v is ib le w hen th e th ic k n e s s and d ie le c tr ic
c o n s ta n t of th e i n s e r t e d la y e r 2 s a tis f y E q . (22).
4 . 4 R e fle c tio n f r o m a C u rv ed S u rfa c e
F o r r e f le c tio n f r o m a c u rv e d s u r f a c e , th e r e f le c te d w ave d iv e r g e s .
To a c c o u n t fo r th is d iv e rg e n c e in m e a s u rin g th e r e f le c tio n c o e ffic ie n t, J""1
f o r a p la n e s u r f a c e is m u ltip lie d b y a d iv e rg e n c e f a c to r D ( K e r r 1951).
O is th e s q u a r e r o o t of th e r a t i o of th e c r o s s - s e c t i o n of th e co n e of r a y s
a f te r r e f le c tio n by th e c u rv e d s u r f a c e to th a t a f te r r e f le c tio n by a p la n e .
P id d in g to n (1939) sh o w ed th a t if a s u r f a c e of d is c o n tin u ity in th e
d i e l e c t r ic c o n s ta n t is n o t an in fin ite p la n e b u t a co n cav e r e f le c tin g s u r f a c e ,
2
m a y b e m a n y tim e s g r e a t e r th a n th a t o b ta in e d f r o m th e p la n e .
P
4 . 5 S u m m a ry
T he th r e e fo llo w in g ty p e s of r e f le c tio n c o e ffic ie n ts w e re d is c u s s e d :
(i)
R e fle c tio n f r o m a s e m i- in f in ite s u r f a c e .
(ii) R e fle c tio n f r o m tw o p a r a l l e l p la n e in te r f a c e s .
(iii) R e fle c tio n f r o m a c u rv e d s u r f a c e .
The c a s e of o b liq u e r e f le c tio n f r o m a m e d iu m w ith a m o n o tb n ic o r
s y m m e tr ic a l tran sition in the d ie le c tr ic constant has apparently not been
d is c u s s e d in th e l i t e r a t u r e .
T his re v ie w a ls o sh o w s th a t la y e r s w h ich a r e la r g e w ith r e s p e c t to
X., w ith la r g e r a d i i of c u r v a tu r e , a r e th e only c o n fig u ra tio n s th a t h av e b een
a n a ly s e d .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
25
In s u b se q u e n t s e c tio n s of th is th e s is " re fle c tio n c o e ffic ie n t" w ill
r e f e r to p o w er r e f le c tio n c o e ffic ie n t, u n le s s o th e rw is e in d ic a te d .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C H A PT E R V
M ETHODS O F M EASURING R E F L E C T IO N C O E F F IC IE N T
5.1 In tro d u c tio n
A s u r v e y of e x is tin g m e th o d s of m e a s u rin g r e f le c tio n c o e ffic ie n t
is p r e s e n te d in th is C h a p te r.
T h is in fo rm a tio n w ill p ro v id e a b a s is fo r
th e d e s ig n of th e la b o r a to r y r a d a r th a t is d e s c r ib e d in C h a p te r V I.
A
f e a tu r e th a t is co m m o n to a ll m e th o d s is th e ir a b ility to d is c r im in a te
b etw ee n th e E . M. w ave tr a v e llin g to w a rd s th e r e f le c tin g o b je c t and th e
w av e r e f le c te d f r o m th e s a m e o b je c t.
P r a c t ic a l m e th o d s of o b tain in g
th is d is c r im in a tio n a r e i l l u s tr a te d by s ix ty p e s of a p p a r a tu s .
5. 2 W aveguide R e fle c to m e te r
D ir e c tio n a l c o u p le rs th a t a r e u s e d fo r th e m e a s u r e m e n t of r e ­
fle c tio n c o e ffic ie n ts a r e te r m e d r e f l e c to m e t e r s .
The d ir e c tio n a l c o u p le r
is s e n s itiv e e s s e n tia lly on ly to a w ave tr a v e llin g in one d ir e c tio n .
T hen
f o r th e m e a s u r e m e n t of th e r e f le c tio n c o e ffic ie n t (|“ ' ^), tw o d ir e c tio n a l
c o u p le rs a r e n e e d e d , one to m e a s u r e th e in c id e n t p o w er P^ and th e o th e r
th e r e f le c te d p o w er P . By c a r e fu l d e s ig n of d ir e c tio n a l c o u p le rs (F ig .
^
__ 2
4 , a ), E n g e n and B e a tty (1951) w e re a b le to m e a s u r e M a s s m a ll as
-10
*
10
. The tr a n s m is s io n lin e u s e d by th e m w as a w av eg u id e w ith a
n e g lig ib le a tte n u a tio n f a c to r , and th e r e f le c tin g o b je c t w as lo c a te d w ith in
th e w a v eg u id e. H ^ is th en o b ta in e d by th e r e la tio n
In th is m eth o d th e m a x im u m s iz e of th e r e f le c tin g o b je c t is lim ite d by
th e g u id e c r o s s - s e c t i o n .
It sh o u ld b e n o te d a ls o th a t w av eg u id e m o d es
26
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REFLECTOMETER
BRIDGE
NETWORK
■TR ANSM ITTER
RECEIVER
RECEIVER
RESISTIVE
TER MI NA TI ON
A
z
*2
■TARGET
HYBRID
T
JUNCTION
TARGET
DIRECTIONAL
COUPLER
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MICROWAVE
INTERFEROMETER
TARGET
Re c e i v e r
PHASE
SHIFTER
TRANSM ITTER
<
ATTENUATOR
LOCAL
O SC ILLA TO R
MATCHING
TERMINATION
<*-TRANSNITTER
(O)
F IG .4-.
METHODS
OF M E A S U R IN G THE REFLECTION
COEFFICIENT
tVJ
28
w ill p r e s e n t n o n -u n ifo rm illu m in a tio n of th e o b je c t w ith in th e g u id e.
5. 3 W aveguide B rid g e N etw o rk
In F ig . (4, b) Z j, Z 2 »
g u id e b r id g e .
an d
a r e fo u r im p e d a n c e s in a w a v e ­
A s ig n a l s o u r c e ( tr a n s m itte r ) is c o n n e c te d a t T and a
s ig n a l a m p litu d e d e te c to r ( r e c e iv e r ) a t R .
For
Z.
Z
—L =
Z2
—
(1)
Z4
n o n e of th e s ig n a l f r o m T a r r i v e s a t R .
T his r e p r e s e n t s th e co n d itio n
w hen th e t a r g e t h a s th e s a m e im p e d a n c e (Z =
w ave) a s th e s u rr o u n d in g m e d iu m .
p o w e r (P ) is e q u a l to z e r o .
X
fo r a p la n e in c id e n t
T hen |"""^ * 0, b e c a u s e th e r e c e iv e d
If Z . ( ta r g e t im p ed an ce) d o es n o t s a tis f y
^
E q . (1), th e b r id g e b a la n c e is d e s tr o y e d and p o w er f r o m T r e a c h e s R .
T he am o u n t of T th a t r e a c h e s R is an in d ic a tio n of th e t a r g e t im p e d a n c e
2
o f th e ta r g e t .
r
M e a s u re m e n ts of
c a n th e n be m a d e by th e b rid g e n e tw o rk u sin g
a w av eg u id e T ju n c tio n (F ig . 4, b ).
im p e d a n c e s w ith in th e w a v eg u id e.
is th e t a r g e t im p e d a n c e .
Z^ an d Z ^ a r e e q u iv a le n t to ju n c tio n
Z^ is a b a la n c in g te r m in a tio n and
L a b ru m (1952) m e a s u r e d [”1^ by f i r s t b a la n c in g
th e b r id g e fo r z e r o t r a n s m is s io n b etw een th e E - and H - a r m s b y a d ­
ju s tin g Z^ an d Z^» an d th en m e a s u rin g th e r e c e iv e d s ig n a l a f te r i n t r o ­
d u cin g th e t a r g e t .
The lim ita tio n on |"""* ^ is due to th e d iffic u lty in o b tain in g m e c h a n ic a l
s ta b ility of th e w av eg u id e b rid g e im p e d a n c e s Z and Z T h e
n z
m e a s u r e s m a ll 1
Z
r
a b ility to
d ep en d s on th e fre q u e n c y s ta b ility of th e g e n e r a to r .
is found to b e 10
-10
in p r a c tic e .
5. 4 M icro w av e I n te r f e r o m e te r
The m ic ro w a v e i n t e r f e r o m e te r , in p r in c ip le , is a n o th e r f o r m of
th e n u ll m e th o d ,
P h illip s (1959) h a s m e a s u r e d th e s ig n a l s c a tte r e d by
tu rb u le n t a i r by th is m e th o d .
H is m e a s u r e m e n ts in v o lv e tr a n s m is s io n s
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29
th ro u g h th e ta r g e t , b u t th e m eth o d a ls o is a p p lic a b le to r e f le c tio n f r o m
th e t a r g e t .
a n te n n a s .
He em p lo y e d a s e p a r a te t r a n s m i t t e r and dou b le r e c e iv e r
W ith th is e q u ip m e n t, show n in F ig . 4, c, and w ith no t a r g e t
in th e f ie ld , th e r e la tiv e p h a s e s and a m p litu d e s w e r e a d ju s te d f o r a
n u ll ou tp u t
P, * -
p 3
.
W hen th e t a r g e t is in tro d u c e d
The e q u ip m e n t w as
c a lib r a te d by m e a n s of s p h e r e s of know n s c a tte r in g c r o s s - s e c t i o n and
th ro u g h th e u s e of th e r a d a r e q u a tio n .
T his m eth o d is u s a b le
w ith a w ide r a n g e of t a r g e t s iz e .
5. 5 M ic ro w a v e M ir r o r
A s e m i t r a n s p a r e n t g la s s p la te is u s e d a s a m i r r o r , in th is m e th o d ,
to s e p a r a t e th e s ig n a l f r o m th e t r a n s m i t t e r (P^) and the s ig n a l r e f le c te d
f r o m th e t a r g e t (P ^ ).
t a r g e t lo c a tio n .
in F ig . (5, a).
is m e a s u r e d by p la c in g th e r e c e i v e r a t the
F o r P , th e r e c e iv e r is r o ta te d to th e p o s itio n show n
T
T his s y s te m is c a lib r a te d b y u sin g ta r g e ts of know n <P
M e a s u re m e n ts of (p fo r c ir c u la r m e ta llic d is c s w e re o b ta in e d
by S c h m itt (1959) by ap p ly in g th e s a m e m e th o d .
The s y s te m is s u rr o u n d e d
by a b s o rb in g m a t e r ia l c o n s is tin g of e n m e s h e d g ra p h itiz e d a n im a l h a ir .
T h is m ic ro w a v e a b s o r b e r is p o s itio n e d to p r e v e n t r e f le c tio n f r o m u n ­
w an ted o b je c ts .
The s e n s itiv ity is lim ite d by s p u rio u s r e f le c tio n s f r o m
th e b a c k g ro u n d - m ic ro w a v e a b s o r b e r , w a lls , e tc .
T his m e th o d is
u s a b le w ith a w id e r a n g e of t a r g e t s iz e s .
5 .6
P u ls e d R a d a r
M icro w av e a n g e l r e s e a r c h in r e c e n t y e a r s h av e b e e n fre q u e n tly
co n d u cted w ith a p u ls e d r a d a r .
By tra n s m ittin g th e s ig n a l in s h o r t p u ls e s ,
th e tr a n s m itte d s ig n a l (P^) is s e p a r a te d f r o m the r e c e iv e d s ig n a l
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MIRROR
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MICROWAVE
TIME
Id)
(C)
PIG.5 .
METHODS
OF
M EAS UR IN G
THE
REFLECTION
CO EFFIC IENT
a fin ite tim e in te r v a l (F ig . 5, b ), due to th e v e lo c ity of p ro p a g a tio n of
th e e le c tr o m a g n e tic w av e b e tw e e n th e a n te n n a and th e ta r g e t.
d e r iv e d f r o m th e r a d a r e q u a tio n (C h a p te r II) and th e s c a tte r in g c r o s s s e c tio n ( 6^ ) of th e t a r g e t . A c a lib r a tio n is o b ta in e d f r o m ta r g e t s of
|2
know n
an d
The s e n s itiv ity of th is m eth o d is lim ite d b y the
d e te c to r n o is e an d by th e m in im u m r a n g e to th e t a r g e t a s d e p e n d e n t on
th e tim e in te r v a l b etw een th e a d ja c e n t ed g es of th e tr a n s m itte d an d r e ­
c e iv e d p u ls e s .
5. 7 D o p p le r R a d a r
T he b a s ic p r in c ip le of th e d o p p le r r a d a r is th e p r e s e n ta tio n of th e
r e c e iv e d s ig n a ls in s u c h a w ay th a t only m oving ta r g e ts can b e d e te c te d .
A m o v in g t a r g e t g iv es r i s e to a r e f le c te d s ig n a l w h ich d iff e r s s lig h tly in
fre q u e n c y f r o m th a t of th e outgoing s ig n a l.
T he fre q u e n c y d iff e r e n c e is
2v
a s m a ll f r a c tio n of th e o r ig in a l fre q u e n c y f and is e q u a l to (“ ) f, w h e re
v is th e v e lo c ity of th e t a r g e t alo n g th e lin e of s ig h t and c is th e v e lo c ity
of lig h t.
The r e c e iv e d s ig n a l (F ) f r o m th e m oving t a r g e t is m e a s u r e d
r
a n d th en c o m p a re d w ith th e t r a n s m itte d s ig n a l P (F ig . 5, b ). C a lib ra tio n
t
of th e r a d a r w ith s ta n d a r d ta r g e ts an d u s in g th e r a d a r eq u a tio n , y ie ld s
f o r th e t e s t ta r g e t.
T he s c a tte r in g of E . M. w av es by s p h e r e s w as
m e a s u r e d by H ey e t a l (1956) u s in g th is p r in c ip le .
The s c a tte r in g o b je c ts
w e r e m o u n te d on a lo w - re f le c tio n w ooden tr o lle y (F ig . 5, c).
T he t a r g e t
w as allo w e d to m o v e w ith a c o n s ta n t v e lo c ity to w a rd s th e a n te n n a to give
10 c / s e c . D o p p ler s h if t b etw een th e tr a n s m itte d an d r e c e iv e d s ig n a l
2
-10
m e a s u r e d by H ey is found to be 10
r
5. 8 F re q u e n c y M o d u lated R a d a r
T h is m e th o d is u s e d fo r s ta tio n a r y t a r g e ts .
m e n t a s in S e c . 5. 7.
I t h a s th e s a m e a r r a n g e ­
The r e c e iv e d s ig n a l d iff e r s f r o m th e tr a n s m itte d
s ig n a l by an a m o u n t A f (F ig . 5, d) due to th e fin ite s p e e d of p ro p a g a tio n
of th e E . M. w av e to th e t a r g e t an d b a c k to the r e c e i v e r .
The r e c e i v e r
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32
is s e n s itiv e on ly to A f an d th is w ill d is c r im in a te b etw een
and P ^ .
^ of t e s t ta r g e ts is found th ro u g h th e u s e of th e r a d a r e q u atio n and
(P of th e s ta n d a r d t a r g e ts .
F o r th e stu d y of tro p o s p h e r ic r e f le c tio n s , a fre q u e n c y m o d u la te d
r a d a r w as u s e d by H ay and R eid (1962),
S e p a r a te t r a n s m i t t e r and r e -
c e iv e r a n te n n a s w e re d ir e c te d v e r tic a lly u p w a rd s .
.
to be of th e o r d e r 10
mm.
w as found
5. 9 S u m m a ry
T he fo llo w in g p o in ts sh o u ld b e b o rn e in m in d in th e d e s ig n of
e q u ip m e n t fo r stu d y in g
1.
The w av eg u id e r e f le c to m e te r is co n fin ed to s m a ll t a r g e t s i z e s .
The t a r g e t is n o t u n ifo rm ly illu m in a te d .
2.
The b rid g e n e tw o rk an d th e m ic ro w a v e in te r f e r o m e te r r e q u ir e
fre q u e n c y s ta b iliz a tio n .
3.
T he m ic ro w a v e m i r r o r s e n s itiv ity is lim ite d b y in te r - a n te n n a
co u p lin g , f o r 9 0 ° b etw een a n te n n a a x e s .
4.
P u ls e d r a d a r in v o lv e s c o m p le x e q u ip m e n t fo r v e r y s h o r t p u ls e -
le n g th s .
5.
D o p p ler r a d a r h a s r e la tiv e ly s im p le te c h n iq u e s .
6.
F . M. r a d a r in v o lv e s c o m p le x eq u ip m e n t.
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C H A PT E R VI
TH E N E A R -F IE E D M ETHOD
6 .1 In tro d u c tio n
The s u rv e y of th e e x is tin g m e th o d s of m e a s u rin g th e r e fle c tio n
c o e ffic ie n t in th e p re v io u s C h a p te r p ro v id e s in fo rm a tio n fo r th e d e s ig n
of e q u ip m e n t to b e d e s c r ib e d below .
The s u b s titu tio n m eth o d is e m p lo y ed .
T h is m eth o d p e r m its th e s tu d y of r e f le c tio n c o e ffic ie n ts fo r a w ide r a n g e
of t a r g e t s iz e s an d a n g le s of in c id e n c e .
F u r th e r , th e d e s ig n of the
s y s te m is f le x ib le , s o th a t if th e n e e d a r i s e s a high d e g r e e of s e n s itiv ity
m a y be a c h ie v e d th ro u g h p r o g r e s s iv e m o d ific a tio n of th e s y s te m c o m p o n e n ts.
The r a d a r e q u atio n d is c u s s e d in C h a p te r II in d ic a te d th a t
P
r
OC
R
4
w h e re
P y a r e c e iv e d p o w er
and
R
a r a d a r ra n g e .
T h is in d ic a te s th a t, in p r in c ip le , R m a y be re d u c e d a s m u c h as d e s ir a b le
to p e r m it s y s te m s e n s itiv ity f o r s m a ll r e f le c tio n c o e ffic ie n ts .
If a p la n e
w av e p a s s e s th ro u g h a n a r r o w a p e r tu r e an a p p r e c ia b le b ro a d e n in g is
o b s e rv e d an d its b r e a d th i n c r e a s e s a s th e a p e r t u r e is f u r th e r n a r ro w e d
(Je n k in s an d W hite 1957).
is th e s a m e .
F ig . (6) in d ic a te s th e in te n s ity a t A, B and C
T h is m e th o d u s e s s e p a r a te t r a n s m i t t e r and r e c e iv e r a n te n n a s
to is o la te th e tr a n s m itte d p o w er
fro m P .
The e le c tr o n ic e q u ip m e n t
is of c o n v e n tio n a l d e s ig n , b u t s p e c ia l a tte n tio n h a s b e e n p a id to th e m ounting
of th e r e fle c tin g o b je c t.
T his C h a p te r b eg in s w ith a g e n e r a l d e s c rip tio n
33
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34
>
>
>
DIFFRACTED
INCIDENT
WAVEFRONT
WAVEFRONTS
DIFFRACTING
SCREEN
F IG .6 .
D IF F R A C T IO N
OF W A V ES
AT
SMALL
APERTURE
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
of th e e q u ip m e n t an d is fo llo w ed by s u b s e c tio n s giving b r ie f d e ta ils on the
co m p o n e n ts, w ith s p e c ia l e m p h a s is on th e t a r g e t re g io n , t a r g e t m o u n t,
a n d th e m e a s u r e m e n t p ro c e d u re
6.
2 L a b o ra to r y R a d a r
A co n tin u o u s w av e r a d a r h a s b een d e s ig n e d fo r la b o r a to r y m e a s u r e ­
m e n ts , a s show n in F ig . (7).
10, 000 M c /s e c . .
A m ic ro w a v e t r a n s m i t t e r is o p e r a te d a t
S e p a r a te t r a n s m i t t e r an d r e c e iv e r a n te n n a s w ith
p a r a b o lo id a l r e f l e c t o r s a r e d ir e c te d h o r iz o n ta lly a t th e ta r g e t.
t a r g e t is in th e n e a r f ie ld of th e a n te n n a s ,
The
A lo w - re f le c tio n b a c k g ro u n d
is o b tain ed by s u rro u n d in g th e a n te n n a fie ld w ith m ic ro w a v e a b s o r b e r .
A s u p e rh e te ro d y n e m ic ro w a v e r e c e iv e r w ith a co n v e n tio n a l A. F . C. s y s te m
fo r c o n tro l of lo c a l o s c illa to r fre q u e n c y is u s e d .
The r e c e iv e d s ig n a l
p o w e r is in d ic a te d by a m e te r .
D e ta ils on co m p o n e n ts of th is s y s te m a r e given in th e follow ing
s e c tio n s .
6.
2.1
T r a n s m ittin g an d R e c e iv in g A ntennas
Two p a r a b o lic r e f l e c to r s w e re u s e d fo r tr a n s m ittin g and re c e iv in g
a n te n n a s .
T h is ty p e of r e f le c t o r w as ch o se n to p ro d u c e a u n ifo rm p la n e
w ave o v e r th e r e g io n of th e t a r g e t lo c a tio n (K e lle h e r 1950).
The a p e r tu r e s
of th e tr a n s m ittin g and r e c e iv in g a n te n n a s a r e 53 and 46. 5 c m . r e s p e c t ­
iv e ly .
T he tr a n s m ittin g a n te n n a is s u p p o rte d by a m e ta l s ta n d c la m p e d
to a ta b le m e a s u rin g 2. 5 by 1. 2 m e t e r s .
The re c e iv in g a n te n n a is
c la m p e d on a tu rn ta b le b e s id e th e tr a n s m i t t e r a n te n n a .
6.
2. 2 A u to m a tic F re q u e n c y C o n tro l (A. F . C . ) S y s te m
T he lo c a l o s c illa to r k ly s tr o n w as fre q u e n c y s ta b iliz e d by a 32 M c /s e c .
c o n v e n tio n a l A. F . C. s y s te m in o r d e r to m a in ta in th e in te r m e d ia te
fre q u e n c y a t a fix e d p o in t in th e r e c e iv e r p a s s b an d .
The r e f e r e n c e s ig n a l
fo r th e A. F . C. u n it w as d e r iv e d f r o m th e t r a n s m it te r c ir c u it th ro u g h a
3 c m la b o r a to r y w av eg u id e to th e lo c a l o s c illa to r c ir c u it.
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36
MICRO WAVE
ABSORBER
TARGET
PARA BO L 1C
RECEIVER
ANTENNA
PA RABOLIC
TRANSMITTER
ANTENNA
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ATTENUATOR
COUPLER
ADJUSTABLE
TER M IN AT IO N
magic
ATTENUATOR
t
COUPLE R
T R A N S M IT TE R
OSCILLATOR
LOCAL
OS C I L L A T O R
I CRYSTAL
DETECTOR
,
FREQUENCY
METER
A M M ET E R
7=9
c rystal
, DETECTOR
■V C O U P L E R
POWER
DISCRIMINATOR
SU PPL Y
A M P L I F IER
TUNED
POWER
AUDIO
AMMETER
SUPPLY
FIG. 7.
L ABO RA TO RY
A M PLIFIER
RADAR
FOR
A M PL IFIER
DIELECTRIC
REF LBCTIO N
MEASUREMENT
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
37
6.
2. 3 B a c k g ro u n d S u p p re s s io n
In o r d e r to m in im iz e e x tra n e o u s r e fle c tio n s f r o m th e s u rro u n d in g
f ra m e w o r k an d w a lls a m ic ro w a v e a b s o r b e r (e n m e sh e d g r a p h itiz e d
a n im a l h a ir ) # h a s b e en m o u n ted on a w ooden fra m e w o r k s u rro u n d in g
th e t a r g e t a r e a .
The m ic ro w a v e a b s o r b e r r e f le c ts a s ig n a l below the
n o is e le v e l of th e re c e iv in g s y s te m .
6 *3
The T a r g e t R egion
It is n e c e s s a r y to m e a s u r e th e fie ld p o w er and p h a s e d is tr ib u tio n
in f r o n t of e a c h a n te n n a to in d ic a te th e p e r m itte d t a r g e t lo c a tio n s .
m e th o d s u s e d h av e b ee n d e s c r ib e d by M o n tg o m ery (1947).
The
F ig . ( 8 )
i l l u s t r a t e s th e e x p e r im e n ta l e q u ip m e n t.
A 2K 25 r e f le x k ly s tr o n tu n ed to a w a v e le n g th of 3. 2 c m . w as th e
s o u r c e o f th e e n e rg y .
In f r o n t of th e a n te n n a a p e r tu r e a p ro b e d e v ic e w as
allo w e d to m o v e in a s tr a ig h t lin e alo n g a c a lib r a te d s te e l o p tic a l b e n c h .
M e a s u re m e n ts of th e p o w er d is tr ib u tio n along th e d ip o le a x is w e re
ta k e n a t eq u a l i n te r v a ls .
T h e o re tic a l a n a ly s is (H an sen e t a l 1959) show s th a t th e p o w er is
p ro p o rtio n a l to
in th e d is ta n t f ie ld (R is th e d is ta n c e to th e a p e r tu r e ) .
R
I t h a s a m a x im u m v a lu e a t a c e r ta in r a n g e .
It m a y b e n o te d f r o m F ig . (9) th a t th e p e a k p o w er d e n s ity o c c u rs
a t a b o u t 72 c m . f r o m th e a n te n n a a p e r tu r e .
T his w as th en ch o se n to
b e th e t a r g e t lo c a tio n .
The t r a n s v e r s e p o w er d is tr ib u tio n m e a s u re m e n ts w e re ta k e n a t
s e v e r a l r a n g e s f r o m th e a n te n n a s .
and (11).
T h e se a r e r e p r e s e n te d in F ig s . (10)
T he f ie ld p a tte r n n e a r th e t a r g e t lo c a tio n h a s th e c h a r a c t e r i s t i c s
of a F r e s n e l d iff r a c tio n r e g io n .
F o r th e m e a s u r e m e n t of t r a n s v e r s e
p h a s e ch a n g e, th e a r r a n g e m e n t show n in F ig . (12) is u s e d .
A s a m p le of
# B . F . G o o d ric h C om pany, Sponge P r o d u c ts D iv isio n . S h elto n ,
C o n n ec tic u t.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
AMMETER
AMPLIFIE R '
PARABOLIC
MI CROWAVE
ABSORBER
TRANSMITTER
FREQUENCY
METER
ANTENNA
TRANSMITTER
CALIBRATED
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40
T R A N S M IT T E R
A N T E N N A
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field
a n t e n n a
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PATTERN
ANTENNA)
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ANTENNA)
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41
Reproduced
with permission
of the copyright ow ner.
w a v e g u i d e
A M M ETER
- i— T r a n s f o r m e r
AMPLIFIER
PARABOLIC
CRYSTAL
MICROWAVE
TRANSMITTER
DETECTOR^
ABSORBER
COAXIAL
ANTENNA
Further reproduction
FREQUENCYrL.
M ETE R —
1
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CABLE
WAVEGUIDE
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HYBRI D
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RECEIVE R
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^TRA NSM ITTER
CRYSTAL
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without p e r m i s s i o n .
CALIBRATED
OPTICAL
BENCH
POW ER
TUNED
AUDIO
AMPLIFIE R
SUPPLY
FIG. 12.
PHASE
MEASURI NG
EQUI PMENT
AMME TER
43
r a d ia tio n is p ic k e d up by th e p ro b e in th e a n te n n a fie ld .
A p h a s e s h if te r
c o m p a re s th e p h a s e of th is s ig n a l w ith a r e f e r e n c e s ig n a l w h ich co m es
d ir e c tly f r o m th e s o u r c e .
The p ro b e is allo w ed to m ove on th e o p tic a l
b e n c h in f r o n t of th e a n te n n a .
M e a s u re m e n ts of th e ch an g e in p h a s e
w e re r e c o r d e d by m ak in g in c r e m e n ta l ch an g es in th e p ro b e p o s itio n .
The r e s u l t s a r e p lo tte d in F ig s . (13) and (14).
It is s e e n th a t the p h a s e
ch an g e a t th e t a r g e t lo c a tio n o v e r a 6 c m . d is ta n c e c e n te r e d a b o u t th e
d ip o le a x is is 4 ° .
T he th e o r e tic a l a n a ly s is (C h a p te r III) s u g g e s ts th a t n o n -u n ifo rm
illu m in a tio n of th e t a r g e t sh o u ld n o t in flu e n c e th e m e a s u r e m e n t of
n 2
2
r e f le c tio n c o e ffic ie n t ( j ) by th is te c h n iq u e a s long as |“ ' is e v e ry w h e re
c o n s ta n t o v e r th e t a r g e t s u r f a c e .
H o w ev er, th e v a lid ity of th is a s s u m p tio n
w ill be te s te d by m ak in g u s e of th e in fo rm a tio n on t a r g e t illu m in a tio n
g iv en h e r e .
6.
4 T a r g e t M ount
In o r d e r to m a in ta in an o b je c t in a d e s ir e d a s p e c t fo r o b s e rv a tio n s ,
s o m e m eth o d of c o n tro lle d s u p p o rt is r e q u ir e d .
It is n e c e s s a r y to e n s u r e
th a t r e fle c tio n s f r o m th e s u p p o rtin g s tr u c t u r e do n o t p ro d u c e an a p p r e c i­
a b le in te r f e r in g s ig n a l a t th e r e c e i v e r .
A s a tis f a c to r y la b o r a to r y m eth o d
is d e s c r ib e d h e r e , in w h ic h th e o b je c t is p la c e d on a s p e c ia lly d e s ig n e d
lo w - r e f le c tio n s u p p o rt.
The a p p a ra tu s c o n s is ts of a w ooden s ta n d
h av in g tw o u p rig h ts and th e b a s e (F ig . 15).
T h e re a r e th r e e p u lle y s
a ro u n d w h ic h m a y b e r o ta te d a n y lo n lin e ( ra d iu s 0 . 0 1 c m . ) c a r r y in g
ta r g e ts in to th e n e a r fie ld of th e a n te n n a s a t th e p o in t of in te r s e c tio n of
th e ir a x e s .
The p u lle y s y s te m f a c ilita te s o p tim a l p la c in g of th e t a r g e t
in r e la tio n to th e t r a n s m i t t e r .
If th e t a r g e t is in th e f o r m of a c ir c u la r
d is c , a n y lo n lin e fa s te n e d to th e d is c is hoo k ed a t p o in ts a , b , c and d
(F ig . 16).
T his allo w s th e d is c to be m o v ed f r e e ly in th e v e r t ic a l p la n e .
A w ooden a r m F ^ F ^ is f r e e to m ove in b o th c i r c u la r and t r a n s v e r s e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
44
TRA N SM ITTER
A N TENN A
200R *35 cm
100
15
10
5
0
5
10
15
D (cm)
9 (cleg.)
200
R* 5 5 c m
100
15
10
0
5
5
10
15
c m
0 (deg.)
200
R =75 cm
100
15
10
0
5
5
10
15
Q(cm
6 (cl eg. )
200
R =95 cm
100
10
0
5
5
10
15
D (cm)
9 (deg.)
200
R =115 c m
100
15
R* axial
distance
D= d i s t a n c e
0 * relative
FIG.13.
fro m
p h a se
from
10
5
0
5
10
15
D (c
an tenna
an ten n a
axis
of a n t e n n a
field
TRA N S V ER SE
PHASE
CHANG E (TRANSMITTER ANTENNA)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45
RECE IVE R
ANTENNA
200
100
15
10
R * 5 5 .2
0
5
5
10
cm
15
0 (deg.)
R = 75.2 c m
15
10
5
0
5
10
15
D (cm )
9 (deg.)
200
R= 9 5 . 2 c m
100
10
D( c m )
9(deg.)
200
R = 115 2 c m
100
15
10
0
5
5
10
15
D(cm)
200
R= 1 3 5 . 2
100
15
R=
axial
D=
d is ta n c e
fro m
0=
re la tiv e
phase
F I0 .I4 .
d ista n c e
fr o m
10
TRANSVERSE
5
10
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a n t e n n a
of
0
5
cm
a x is
an ten n a
field
PHASE
CHANGE
( R E C E IV E R
ANTENNA)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
46
PULLEY
f NYL ON
\
LI NE
TARGET
MICROWAVE
A B SQ R B E R
FIG 15. TARGET
MOUNT (SPHERES}
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CLAMP
microwave
absorber
'RANSMITTER
Fl° - , e -
target
M 0 U NT
Reproduced
« h P e m s s to n of ,he coPy rlghl
(OlSCS)
48
d ire c tio n s to allo w th e d is c to be in th e r e g io n of m a x im u m f ie ld in te n s ity .
6.
5 M e a s u re m e n t P r o c e d u r e
T he fo llo w in g s te p s d e s c r ib e th e p r o c e d u re of m e a s u r e m e n t w h ich
is a p p lie d in th e s u b s titu tio n m eth o d :
1.
The t e s t t a r g e t is in s e r te d a t th e p o s itio n of m a x im u m fie ld
in te n s ity to give th e m a x im u m s ig n a l a t th e r e c e i v e r .
2.
T he c a lib r a te d w av eg u id e a tte n u a to r is a d ju s te d to re d u c e th e
r e c e iv e d s ig n a l to th e s e le c te d r e f e r e n c e le v e l and th e a tte n u a to r re a d in g
is n o te d .
3.
The t e s t t a r g e t is r e p la c e d by a m e ta l t a r g e t of th e s a m e s h a p e ,
an d th e c a lib r a te d w av eg u id e a tte n u a to r is r e a d ju s te d to give a r e c e iv e d
s ig n a l a t th e s e le c te d r e f e r e n c e le v e l.
4.
E a c h of th e ab o ve m e a s u r e m e n ts is r e p e a te d s e v e r a l tim e s and
th e a v e r a g e r e s u l t is ta k e n to re d u c e th e m e a s u r e m e n t e r r o r .
5.
T he r e f le c tio n c o e ffic ie n t is o b ta in e d by applying th e a n a ly s is
g iv e n p r e v io u s ly .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C H A PT E R VH
PR E C ISIO N O F M EASUREM ENT
7.1 In tro d u c tio n
The p u rp o s e of th is C h a p te r is to p r e s e n t an a n a ly s is of th e p r e ­
c is io n of r e f le c tio n c o e ffic ie n t m e a s u r e m e n ts .
T h e re a r e v a r io u s s o u r c e s
o f e r r o r in th is m e a s u re m e n t:
1.
S ta b ility of th e p o w er o u tp u t f r o m th e t r a n s m i t t e r .
2.
P r e c i s i o n of th e c a lib r a tio n of th e w av eg u id e a tte n u a to r .
3.
S ig n a l-to - n o is e r a t io fo r r e c e iv e d s ig n a l (th is in c lu d e s th e
m a g n itu d e of th e b a c k g ro u n d r e f le c tio n r e la tiv e to th e r e f le c tio n f r o m a
ta r g e t) .
4.
R e p e a ta b ility of m e a s u r e m e n ts on a given t a r g e t, due to
p o s itio n in g in th e a n te n n a f ie ld s .
5,
S a tu ra tio n of th e r a d a r r e c e i v e r by th e r e c e iv e d p o w e r.
T he p r e s e n t s tu d y w as c a r r i e d o u t th ro u g h th e follow ing s te p s :
1.
T he p r e c is io n of th e m e a s u r e m e n t of th e s c a tte r e d p o w er f r o m
ta r g e ts of know n s c a tte r in g c r o s s - s e c t i o n and r e f le c tio n c o e ffic ie n t w as
a s c e r ta in e d .
2.
T he n o is e le v e l of th e r e c e iv e r v/as m e a s u r e d .
3.
The e ffe c t *of n o n -u n ifo rm illu m in a tio n of th e t a r g e t upon
w as e x a m in e d .
7. 2 S y s te m S ta b ility
7. 2.1
T r a n s m itte r P o w er
T he s ta b ility of th e s ig n a l f r o m th e t r a n s m it te r w as in d ic a te d by
co u p lin g th e t r a n s m i t t e r th ro u g h a la r g e a tte n u a to r to th e r e c e iv e r and
49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
r e c o r d in g th e r e c e iv e d s ig n a l.
T he am o u n t o£ in s ta b ility w as o b ta in e d
f r o m th e s e r e c o r d s by c a lib r a tin g th e r e c e iv e r o u tp u t w ith a p r e c is io n
a tte n u a to r .
T he r e s u l t s sh o w ed th a t th e tr a n s m itte d s ig n a l v a r ie d by
l e s s th a n t 0 .2 d e c ib e ls o v e r a 115 m in u te in te r v a l.
7. 2. 2 C a lib r a te d A tte n u a to r S ettin g
P ro c e e d in g a s in th e p re v io u s s u b s e c tio n (7. 2.1), r e p e a te d s e ttin g s
w e re m a d e of th e c a lib r a te d a tte n u a to r a t th r e e s e le c te d s ig n a l le v e ls .
T h e se show how p r e c i s e l y th e a tte n u a to r s e ttin g m ay b e r e a d .
I t w as
found th a t th e a tte n u a to r co uld b e r e s e t p r e c i s e l y to + 0. 3 d e c ib e ls .
7. 2. 3 S y s te m N o ise L e v e l
The r e c e i v e r n o is e le v e l w as d e d u ced by p ro c e e d in g a g a in as in the
p re v io u s s u b s e c tio n .
A d d itio n al a tte n u a tio n w as in s e r te d u n til th e r e ­
c e iv e d s ig n a l d e c r e a s e d to th e n o is e le v e l of th e r e c e i v e r .
The t r a n s ­
m itte d p o w er and th e a m o u n t of a tte n u a tio n w e re a s c e r ta in e d .
The e x ­
p e r im e n ta l r e s u l t s sh o w ed th a t th e a tte n u a tio n b etw een tr a n s m i t t e r and
r e c e i v e r w as 77 d e c ib e ls and th e tr a n s m itte d pow er w as 113 m illiw a tts .
T his in d ic a te s th a t th e r e c e i v e r 's n o is e le v e l w as - 5 7 .5 db. m .w .
7. 2. 4 R e p e a ta b ility of M e a s u re m e n ts on T a rg e ts
T his d e s c r ip tio n a p p lie s to m e a s u r e m e n ts w ith th e t r a n s m i t t e r and
r e c e i v e r c o n n e c te d to th e ir a n te n n a s and th e t a r g e t in p o s itio n .
The
s ig n a l m e a s u r e m e n ts w e re r e p e a te d w ith a g iven t a r g e t s e v e r a l tim e s and
th e m e a n v a lu e w as ta k e n .
F r o m th e s e , th e s ta n d a rd d e v ia tio n ( )
of
th e s e m e a s u re m e n ts w as c a lc u la te d .
7. 3 M e a s u re m e n ts Upon S ta n d a rd T a rg e ts
The o v e r a ll p r e c is io n of th e m e a s u r e m e n t w as a s s e s s e d by m e a s u rin g
th e r e c e iv e d p o w er f r o m ta r g e ts w ith u n ity r e f le c tio n c o e ffic ie n ts and
know n s c a tte r in g c r o s s - s e c t i o n s .
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51
7 .3 .1
(a)
R e fle c tio n F r o m C onducting S p h e re s
T h e o re tic a l S c a tte r in g G ro s s -S e c tio n
The d iffra c tio n o f a p la n e w ave by s p h e r e s w as f i r s t in v e s tig a te d
by M ie (1908) an d h a s b e e n c a r r i e d on by S tra tto n (1941) w hose m eth o d
of tr e a tm e n t w ill be o u tlin e d b r ie f ly h e r e .
If th e in c id e n t fie ld is a p la n e w ave tr a v e llin g in th e Z - d ir e c tio n
w ith an e l e c t r i c v e c to r
E. = E
l x
w h e re
e^k Z “ **Wt
(1)
'
= a m p litu d e v e c to r in th e X - d ire c tio n ,
an d
k
= p ro p a g a tio n c o n s ta n t,
w
= a n g u la r fre q u e n c y ,
t
= tim e ,
.
in th e p r e .e n c e of a e p h e re of x a d iu . a, th e in c id e n t fie ld can w r itte n ae
~
jwt
n 2n + 1 f
.,
E = e
h (j)
7 — TiT
m
- j b n , )■ . (2)
i
, w
n (n + l)
V. n
oln * n
e ln j
n=i
and b^ a r e th e s c a tte r in g a m p litu d e s a s s o c ia te d w ith th e m a g n e tic
an d e l e c t r i c 2 n - p o le s in d u c e d in th e s p h e r e by the in c id e n t e le c tr o m a g n e tic
f ie ld ,
a ^ an d b ^ d ep en d on th e d i e le c tr ic c o n s ta n t and r a d iu s of th e s p h e r e ,
and on th e w a v e le n g th .
The s u b s c r ip ts e and o d e s c r ib e s o lu tio n s th a t
a r e ev en o r odd.
an d n , a r e s p h e r ic a l v e c to r w ave fu n c tio n s d e eln
F r o m E q . (2) th e s c a t te r e d fie ld cam th e n be
m
oln
fin e d by S tra tto n (1941).
w r itte n a s
—
-jw t ~ , , 4n
E =e
2 (j)
s
,
n=l
2n
+ 1
7 -7
f
4a m
n + 1 \ n
oln
- jb
* n
n . I
elnf
J
A pplying th e b o u n d a ry c o n d itio n s to E q . (3) th e c o e ffic ie n ts a
found to be
(3)
n
, . p.
jn ( 5 )
a
“
h (1) ( 5 )
n
J
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and b
n
a re
J
d5
w h e re
^ * K a,
n
' '
' '
____
j ( ? ) = /-rr- J
*n J 7“ 2 ^ n + f
, i( ? ) ( J ( ? ) is a B e s s e l fu n ctio n of
J
n J
th e f i r s t k in d of o r d e r n),
an d
h (1) ( ? ) = / —
n
J /y 2 5
th e f i r s t k in d of o r d e r n ).
n+ f
(H(1\ % ) is a H an k el fu n ctio n of
n
J
R e c a llin g th e d e fin itio n of th e s c a tte r in g c r o s s - s e c t io n p , g iv en
in C h a p te r II, as E q . ( 6 ) and a s s u m in g th a t th e in c id e n t e le c tr o m a g n e tic
w ave is p la n e and th e s p h e r e is u n ifo rm ly illu m in a te d and im m e r s e d in
a h o m o g en eo u s m e d iu m , th e s c a t te r e d p o w er d e n s ity W ca n be w r itte n
r
as
g2
(-1)“ (n + | ) (an - b j I2
2 <o )
n=1
1
an d th e in c id e n t p o w er d e n s ity W is
2
E
w. = °
i
2
*lo
w h e re
E ^ * a m p litu d e of th e in c id e n t p la n e w ave
an d
^
T hus
£* <•• h rr*m II ^ . (-I)” (2n + 1) (a n - b XX) II 2,
m
ITa 1 L „ 'n - l
1
2 2
* i n tr in s ic im p e d a n c e of f r e e s p a c e = 1 2 0 tt o h m s.
or
4ir
w h e re f is d e fin e d a s
f, *
5Z, |( - i]n
p
1) ( 2 n + l) —rrr—
n»i
V X)( j )
* $ jn(
$ ** I
771-------- ^d f---------
- i - ( ; h J j N j ) r1
The v a lu e s of th e fu n c tio n f w e re c o m p u ted by H ey and c o - w o rk e r s (1956)
fo r p e r f e c tly co n d u ctin g s p h e r e s of r a d iu s ra n g in g f r o m 0 to 1. 5 w a v e ­
le n g th s .
O th e r c o m p u te d v a lu e s of (P ( p riv a te C o m m u n icatio n , by
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
53
W. P h illip M elling) sh o w ed th a t th e c o m p u ted v a lu e s by H ey sh o u ld be
m u ltip lie d by a f a c to r 4 to o b ta in th e c o r r e c t v a lu e s of (P .
(b) C o m p a riso n w ith E x p e rim e n ta l M e a s u re m e n ts
S m o o th p e r f e c tly co n d u ctin g s p h e r e s of ra d iu s ra n g in g f r o m 0.17
to 1. 32 c m . w e r e s u p p o rte d by p a s s in g th ro u g h th e m th e nylon lin e of
th e t a r g e t m o u n t (F ig . 15).
The t r a n s m i t t e r and lo c a l o s c il l a t o r s , p o w er s u p p lie s , and I. F .
and tu n e d a m p lif ie r s w e r e o p e r a te d fo r s o m e h o u rs b e fo re m e a s u r e m e n ts
w e re ta k e n in o r d e r to o b ta in o p e ra tio n s ta b ility .
F in a l tu n in g and
a d ju s tm e n ts w e r e m a d e fo r m a x im u m p o w er o u tput.
The p o w er r e f le c te d f r o m e a c h s p h e r e , P ^ , w as m e a s u r e d by th e
s u b s titu tio n m e th o d .
R an d o m e r r o r s w e r e re d u c e d b y r e p e a tin g e a c h
m e a s u r e m e n t s ix tim e s and a v e ra g in g th e r e s u l t s .
c ro s s -s e c tio n
The s c a tte r in g
(p w as c a lc u la te d by th e u s e of th e r a d a r eq u a tio n
(C h a p te r U) w h ich can be w r itte n in th is c a s e as
P
= C (p
r
w h e re C is a c o n s ta n t in te r m s of r a d a r r a n g e , w a v e le n g th , a n te n n a g ain
and tr a n s m itte d p o w e r.
T he r e s u l t s of th e m e a s u r e m e n ts a r e p lo tte d in F ig . (17) fo r a
w a v e le n g th (X.) * 3. 2 c m .
T he f u ll- lin e c u rv e w hich is o b ta in e d f r o m th e
c o r r e c t e d th e o r e tic a l c o m p u ta tio n b y H ey h a s b een f itte d to th e o b s e rv e d
p o in ts .
In a d d itio n to th e e r r o r due to th e in s ta b ility of th e tr a n s m itte d
p o w er and th e r e s e ttin g of th e a tte n u a to r (S ec. 7. 2), e r r o r s a r i s e in
th e m e a s u r e m e n t of th e s ig n a l w a v e le n g th ( \ ) and th e r a d iu s of th e s p h e r e
(a ).
T he p o s s ib le e r r o r s in th e m e a s u r e m e n ts a r e g iv en in T a b le II.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3*5
SPHERES
54
>j0
>0 vO
oO
METAL
FROM
2.5
2.0
FIG. 17.
1,5
(K0)
T H E O R E T IC A L
SCATTERING
*0
3*0
lO * tO
O.
o
o
o
o
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
55
T A B L E II
P O SS IB L E ERRORS
P a r a m e t e r _____________________
P e r c e n ta g e e r r o r
a
0 .6
X
0 .3
I n s ta b ility of
th e tr a n s m itte d s ig n a l
5 .0
R e s e ttin g th e a tte n u a to r
7 .0
O'
1 2 .0
(T'
Ka
0 .9
In F ig . (17), th e to ta l p o s s ib le e r r o r in th e a b s c is s a e is 0 .9 p e r c e n t
a n d in th e o r d in a te s i s 1 3 .0 p e r c e n t.
It is n o te d th a t th e e x p e r im e n ta l
an d th e o r e tic a l r e s u l t s a r e w ith in th e c a lc u la te d p o s s ib le e r r o r fo r a ll
(S''
p o in ts n o t ly in g n e a r th e m a x im u m an d m in im u m v a lu e s of “^£2. •
(c)
T he M in im u m D e te c ta b le S ig n al
The l i m i t o f s e n s itiv ity o f th e r a d a r r e c e iv e r is r e a c h e d w hen th e
r e c e iv e d s ig n a l le v e l f a lls below th e n o is e le v e l.
Any i n c r e a s e in
a m p lific a tio n d o e s n o t im p ro v e th e s e n s itiv ity b e c a u s e s ig n a ls and n o is e
a r e a m p lifie d to g e th e r .
F o r a n o is e le v e l e q u a l to * 5 7 .5 db. m . w . in
th e s u p e rh e te ro d y n e r e c e i v e r , th e m in im u m m e a s u r a b le s c a tte r in g
c r o s s - s e c t i o n of a c o n d u c tin g s p h e r e is
^
,
■ 0.017 c m * ,
m in .
T h en b y ap p ly in g R a y le ig h 's s c a tte r in g law ( K e r r 1951)
1 .403 ( | ) 4 . 104 .
va
When X ■ 3. 2 c m , th e r a d iu s of to e s m a l l e s t d e te c ta b le s p h e r e is
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
56
a * 0 . 085 c m .
7 .3 .2
(a)
R e fle c tio n F r o m C onducting D is c s
T h e o re tic a l S c a tte r in g C r o s s - S e c tio n
The s o lu tio n o f d iffra c tio n of e le c tr o m a g n e tic w aves by a p e r f e c tly
co n d u ctin g d is c w as found f i r s t by M eix n e r (1948).
The s o lu tio n w as
o b ta in e d by th e m eth o d of s e r i e s e x p a n sio n in te r m s of s e p a r a te s o lu tio n s
of th e w av e e q u a tio n .
The n u m e r ic a l c o m p u ta tio n is v e r y te d io u s b e ­
c a u s e of th e p o o r c o n v e rg e n c e of th e s e r i e s .
A nother s o lu tio n fo r the
p r o b le m w as g iv en by M eix n e r and A n d re je w sk i (1950) and A n d re je w sk i
(1953).
T h e ir s o lu tio n c an be p r a c tic a lly e v a lu a te d fo r any w a v e le n g th .
T his e x a c t s o lu tio n m a y b e o b ta in e d by expanding th e H e r tz v e c to r s of
th e d iff r a c te d fie ld in a s e r i e s of o b la te s p h e ro id a l fu n c tio n s a p p r o p ria te
to th e fo llo w in g c o n d itio n s:
1.
F o r a p la n e in c id e n t E . M. w ave w ith fie ld s tr e n g th E . and H.
iwt
—
—
an d w ith tim e d e p en d en c e e
, th e s c a t te r e d fie ld E and H f r o m a
s
s
co n d u ctin g c ir c u l a r d is c m u s t s a tis f y M a x w e ll's e q u a tio n s.
2.
On th e s u r f a c e of th e d is c th e ta n g e n tia l c o m p o n en t of th e
to ta l e l e c t r i c fie ld E . + E
v a n is h e s .
s
A t a la r g e d is ta n c e f r o m th e d is c th e d iff r a c te d w ave b e h a v e s
1
3.
lik e an outgoing e le c tr o m a g n e tic s p h e r ic a l w ave w ith d ire c tio n - d e p e n d e n t
a m p litu d e .
4.
E a c h f in ite v o lu m e c o n ta in s a fin ite am o u n t of th e fie ld e n e rg y
(edge c o n d itio n ).
F o r a d is c , r a d iu s a , o r ie n te d in th e p la n e Z = 0 and e x p o se d to
p la n e e le c tr o m a g n e tic w av es of w a v e le n g th X a t n o r m a l in c id e n c e ,
A n d re je w s k i's s o lu tio n y ie ld s
<T
| Vq ( 1, k a , 0 ) - U(ka) +
(1, ka) 12 ,
k
1
fo r th e b a c k - s c a tte r in g c r o s s - s e c tio n ; w h e re th e fu n c tio n s V (1, k a , 0),
o
U (ka), and $ (1* ka) a r e d efin ed by A n d re je w sk i (1953). S c h m itt (1959),
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
57
fo r h is m e a s u r e m e n t of th e b a c k s c a tte r in g f r o m a c ir c u la r d is c , co m p u ted
f o u rte e n v a lu e s of th e fu n c tio n s V (1, k a , a), U(ka) and $ (1, ka) in th e
o
o
ra n g e 0 < k a < 10 in a d d itio n to th e v a lu e g iv en n u m e r ic a lly by
A n d re je w sk i fo r k a = 10.
(b)
C o m p a riso n w ith E x p e rim e n ta l M e a s u re m e n ts
A n u m b e r of s a m p le s in th e f o r m of c ir c u la r m e ta llic d is c s , ra d iu s
ra n g in g f r o m 0. 7 to 13.9 c m . , w e r e c u t o u t of fu ll b r a s s on a la th e .
G reat
c a r e w as ta k e n to o b ta in f la t s m o o th s u r f a c e s .
The d is c s w e re s u p p o rte d
fo r m e a s u r e m e n t a s in F ig . 16 ( s e e S e c . 6 .4 ) .
The r e f le c te d p o w er
w as m e a s u r e d , f r o m w h ich th e s c a tte r in g c r o s s - s e c t i o n w as d e d u c e d as
in th e c a s e of s p h e r e s .
A g ra p h of th e s c a tte r in g c r o s s - s e c t i o n ( <P ) as
a fu n c tio n of th e d is c r a d iu s (a) is show n in (F ig . 18).
h a s b e e n o b ta in e d f r o m th e th e o r e tic a l c o m p u ta tio n .
The s o lid c u rv e
T he o b s e rv e d p o in ts
h av e b ee n f itte d to th e th e o r e tic a l c u rv e by th e l e a s t s q u a r e s m e th o d .
It is s e e n th a t th e e x p e r im e n ta l and th e o r e tic a l r e s u l t s a r e w ith in
th e c a lc u la te d p o s s ib le e r r o r of 12. . 96 *o r
a = 3 . 2 cm .
p o in ts f r o m a = 0. 7 to
T he l a r g e s t e r r o r o c c u rs a t a > 3 . 8 c m .
T h is d e p a r tu r e
is due to th e f a c t th a t th e a m p litu d e and p h a s e of th e f ie ld in th e re g io n
of th e d is c is n o t c o n s ta n t (n o n -u n ifo rm illu m in a tio n S e c . 6 .3 ) .
7. 4 S u m m a ry
An e x p e r im e n ta l in v e s tig a tio n of th e s ta b ility of th e tr a n s m itte d
p o w e r an d th e r e s e ttin g of th e a tte n u a to r in d ic a te d th a t th e p o s s ib le e r r o r
is 1 2 %
T he th e o r e tic a l a n a ly s is of th e d iffra c tio n p ro b le m show ed th a t fo r
a p e r f e c tly co n d u ctin g s p h e r e and f la t d is c
and
f
£ ( - 1)
I n= l
( 2 n + l) U
-b
V
(1, k a , 0) - U(ka) <(*(!, ka)
n
)
n
E x p e rim e n ts to a s s e s s th e o v e r a ll p r e c is io n of th e m e a s u r e m e n t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
58
IN D IC A T E D
0.43
0.41
R E F L E C T IO N
C O E F F IC IE N T
041 Q32 Q32 0.27 0.24
0.22 0.21
5
10
SCATTERING
CROSS-SECTION
(cm
T H E O R E T IC A L
E X P E R IM E N TA L
4
10
METALLIC
10
0
3
GLASS
DISC
DISC
2
10
DISC
FlG.18.
S C A T T E R IN G
FRO M
RADIUS
M ETAL
(cm.)
AND
GLASS
DISCS
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
59
w e re c a r r i e d o u t by m e a s u rin g
s p h e r e s an d f la t d is c s ) .
th e th e o r e tic a l fo rm u la e .
f r o m s ta n d a rd ta r g e ts (co n d u ctin g
T he e x p e r im e n ta l r e s u l t s w e r e c o m p a re d w ith
T his c o m p a ris o n le d to th e follow ing:
(a) In th e c a s e of th e s p h e r e s , th e e x p e r im e n ta l and th e o r e tic a l
r e s u l t s a r e w ith in th e c a lc u la te d p o s s ib le e r r o r e x c e p t a t th e m a x im u m
—
tra
(b) In th e c a s e of th e d is c s , th e e x p e r im e n ta l and th e o r e tic a l
r e s u l t s a r e w ith in th e p o s s ib le e r r o r fo r a < 3. 2 c m .
non u n ifo rm illu m in a tio n of th e d is c fo r a > 3. 8 c m upon
T he e ffe c t of
a c c o u n ts
fo r th e d e v ia tio n f r o m th e th e o r e tic a l c u rv e .
F o r a n o is e le v e l e q u a l to -5 7 . 5 db. m . w . , th e r a d iu s of th e
s m a l l e s t d e te c ta b le s p h e r e is
a = 0. 085 c m .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER VIII
R E F L E C T IO N C O E F F IC IE N T O F GLASS LAMINAS
8 .1 In tro d u c tio n
The r e f le c tio n c o e ffic ie n t ( |
) of a la m in a r d ie le c tr ic h a s b een
m e a s u r e d as a p r a c t i c a l t e s t of th e s u b s titu tio n m e th o d .
T he m e a s u r e m e n t
w a s a p p lie d to a body of s im p le g e o m e tr ic a l c o n fig u ra tio n , n a m e ly , a
g la s s la m in a in w h ich th e f r o n t and b a c k s u r f a c e s a r e p a r a ll e l p la n e s .
In th is C h a p te r th e o r e tic a l a n a ly s is of r e f le c tio n f r o m s u c h a c o n z
r
.
R e m a rk s p e r ta in in g to th e e x p e r im e n ta l r e s u l t s a r e a ls o p r e s e n te d .
8.
2 T h e o ry of R e fle c tio n
z
of a g la s s la m in a , i t is n e c e s s a r y to r e c a l l th e
r
lo u o w m g p r in c ip le s :
a fu n c 1.
tio n Tof
e t agl eaonma ly
e trsyis, gofiv th
t a rCgheat pillu
a tio n ed
and
he ththe etoarregtic
ene in
te r mIIIin show
th aoft $its is
2.
In C h a p te r VII i t w as s e e n th a t <P is p r o p o rtio n a l to th e s q u a r e
of th e r a t i o of th e s c a t te r e d to in c id e n t fie ld s a t th e ta r g e t.
The t h e o r ­
e tic a l a n a ly s is g iv e n in C h a p te r HI in d ic a te d th a t th e s c a t te r e d fie ld in
f r o n t of a d ie l e c t r ic p la n e is e q u a l to th a t in f r o n t of a co n d u ctin g p la n e
o f th e s a m e s h a p e m u ltip lie d by
th e
p la n e ).
H e re
<S'' ( d ie le c tr ic ) = P
2
*. (P (co n d u ctin g ).
S e v e rin and B ae c k m a n n (1951) h av e c o n firm e d th is a n a ly s is fo r
s m a ll d i s c s .
60
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
61
3.
T he th e o r y of r e f le c tio n f r o m a la y e r w h e re in th e ch an g e in
d ie l e c t r ic c o n s ta n t ( € ) is a b ru p t, is d is c u s s e d by S tra tto n (1941) (s e e
—.2
S e c , 4 .3 ) . In h is a n a ly s is th e r e la tio n b e tw e e n
and th e th ic k n e s s
of th e la y e r (d) fo r a n o r m a lly in c id e n t w ave is
4 f- 1 2
r 2- ( i- -
s in (kd)
ner
1
(k d )
w h e re
I—12
|
= r e f le c tio n c o e ffic ie n t a t th e s u r f a c e of th e la y e r
cixid
k -
^ •
F ig . (3) sh o w s th a t fo r a p la n e s h e e t of c o rn in g 0080 g la s s (« = 6 . 71
z
o ccu rs at
r
d = . 75 c m and th a t fo r d * 0 .9 4 c m
= 0 .5 .
One im p o r ta n t r e s u l t of th e above a n a ly s is is th a t th e s ig n ific a n c e
of p 2 ca n be t r e a t e d se p a ra te ly f r o m th e e ffe c ts of b o th t a r g e t illu m in ­
a tio n and g e o m e try by c o m p a rin g th e s c a tte r e d s ig n a ls f r o m d ie le c tr ic
z
fo r a d i e le c tr ic
r
s h e e t can b e found by c o m p a rin g th e s c a t te r e d s ig n a l f r o m a d ie le c tr ic
d is c of th e s a m e th ic k n e s s and e , w ith th a t f r o m a m e ta l d is c of th e
2
s a m e s iz e , p ro v id e d th a t n
is a c o n s ta n t o v e r th e t a r g e t. It follow s
I
i—j 2
f r o m th e th e o r y of C h a p te r III th a t
fo r a d ie l e c tr ic d is c w ill be
id e n tic a l w ith th a t f o r an in fin ite p la n e of th e s a m e d ie le c tr ic c o n s ta n t
and th ic k n e s s .
8.
a mmine ne tsth e p r in c ip le s m e n tio n e d ab o v e, g la s s d is c s (c o rn in g
3 MTo
e a seuxre
0800) of th ic k n e s s 0 .9 4 c m . w e re c u t w ith r a d i i a p p ro x im a te ly e q u a l to
th o s e of th e b r a s s d is c s (C h a p te r V II).
T h e se d is c s w e re c la m p e d in th e
s a m e m a n n e r a s th e m e ta llic d is c s F ig . (18).
The m e a s u r e m e n ts of <p of th e d i e le c tr ic d is c s w e re ta k e n p r in c ip a lly
z
a s in d ic a te d in th e p re v io u s s e c tio n . The m e a s u r e m e n t
r
p r o c e d u r e is th e s a m e as th a t d e s c r ib e d e a r l i e r in C h a p te r VII (S ec. 7. 2).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
62
r
T he a v e r a g e e x p e r im e n ta l v a lu e s of (P to g e th e r w ith th e th e o r e tic a l
d e r iv e d f r o m S tra tto n a r e ta b u la te d fo r e a c h d is c in T a b le III
T A B L E III
CH A RA C TERISTICS O F GLASS DISCS
D is c
No
D isc r a d iu s
a cm
T h ic k n e ss
d (cm )
A v e ra g e E x p .
(T (c m ) 1 0 “
.....
1
0 .9 4 9
0 .4 9 8
1. 20
0 .1 4 + 0 .0 2
2
0 .9 4 5
0. 500
1.
0 . 21
3
0 .9 4 6
0. 500
1.95
0 .5 2 t 0 .0 6
4
0 .9 4 6
0. 500
2 . 11
0 .8 3 t 0. 10
5
0 .9 4 8
0 .4 9 9
2.
60
1. 48 + 0. 18
6
0 .9 4 7
0 .4 9 9
3 .4 0
3 .3 1 + 0 .4 0
7
0 .9 4 9
0 .4 9 8
3 .8 0
5 .0 1 + 0 .6 0
8
0 .9 4 6
0 .5 0 0
4 .2 1
5 .5 0 + 0 .6 5
9
0 .9 4 5
0 .5 0 0
6 .3 0
3 .3 2 t 1 .0 0
10
0 .9 4 9
0 .4 9 8
10. 21
8 .5 1 t 1 .0 1
60
+ 0 .0 2
F ig . (18) (C h a p te r VII) show s th e e x p e r im e n ta l r e s u l t s , w ith ( P a s a
fu n c tio n of th e d is c r a d iu s a .
T he c r o s s e s r e p r e s e n t <TN of th e o onducting
d is c s an d th e e n c ir c le d p o in ts ( p of th e g la s s d is c s .
r e p r e s e n t s th e th e o r e tic a l( P
8.
T he s o lid c u rv e
of u n ifo rm ly illu m in a te d co n d u ctin g d is c s .
4 R e m a rk s
I t fo llo w s f r o m th e th e o r e tic a l a n a ly s is (S ec. 8 . 2) th a t th e r a t io of
*s r e p r e s e n t e d by th e tw o b r o k e n - lin e c u r v e s in F ig . (18) sh o u ld r e ­
p r e s e n t J”" o f th e g la s s la m in a .
j ~ d e d u c e d in th is w ay is in d ic a te d
by th e n u m b e rs ra n g in g f r o m 0. 43 + . 05 to 0. 21 + 0. 02 a c r o s s th e top of
th e f ig u r e .
It is n o te d th a t th e e x p e r im e n ta l an d th e o r e tic a l v a lu e s of J * ^
(T a b le II) a r e w ith in th e p o s s ib le e r r o r (12.
% ) fo r 1. 2 < a < 3 . 4 .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
For
63
j2
g r e a t e r th a n th e e x p e r im e n ta l e r r o r .
r
f r o m th e th e o r e tic a l v a lu e is
T he th e o r e tic a l a n a ly s is above
s u g g e s ts th a t th is d e p a r tu r e is due to th e f a c t th a t
.
.
2
is n o t c o n s ta n t
f o r ch an g in g a n g le o f in c id e n c e o v e r th e s u r f a c e of th e la m in a (S ec. 4 .3 . 2).
It is b e lie v e d th a t th e n o n -u n ifo rm illu m in a tio n of th e la m in a d o es n o t
in flu e n c e th e m e a s u r e m e n t b e c a u s e of th e s u b s titu tio n m e th o d .
w ill b e e x a m in e d in fu tu r e s tu d ie s .
s m a ll d is c s ,
B u t th is
T he e x p e r im e n ta l r e s u l t s fo r th e
a < 3 . 8 c m . in d ic a te d th a t th is m eth o d c a n b e a p p lie d fo r
m o n o s ta tic o r b is ta tic m e a s u r e m e n ts of (p .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER IX
CONCLUSION
E x p e rim e n ts w ith o u td o o r s y s te m s lo r m e a s u rin g r e f le c tio n c o ­
e ffic ie n ts d u e to i r r e g u l a r i t i e s in d ie le c tr ic c o n s ta n t w ith in th e t r o p o s ­
p h e r e h a v e show n th a t, b e c a u s e of w in d , r a i n and snow , good m e a s u r e ­
m e n ts co u ld b e o b ta in e d only a few d ay s p e r m o n th .
The in d o o r m e th o d
h a s b e e n found to b e c o m p le te ly f r e e of th e s e lim ita tio n s and to be
c a p a b le of o p e ra tio n a t its u ltim a te s e n s itiv ity w h e n e v e r r e q u ir e d .
T h is th e s is is c o n c e rn e d w ith th e p r o b le m of s e ttin g up e q u ip m e n t
to m e a s u r e th e r e f le c tio n c o e ffic ie n t of a g la s s la m in a , th is b ein g a p a r t
o f a l a r g e r p r o b le m c o n c e rn in g r a d a r r e f le c tio n s f r o m d ie le c tr ic in h o m o ­
g e n e itie s in th e tr o p o s p h e r e .
A s u r v e y of e x is tin g m e th o d s and t h e o r ­
e tic a l a n a ly s e s is fo llo w e d b y a n e x p e r im e n ta l in v e s tig a tio n of r e f le c tio n
c o e ffic ie n t m e a s u r e m e n ts .
T he p r o b le m o f co n tin u o u s w av e r a d a r d e s ig n is d is c u s s e d .
A
s u rv e y o f th e e x is tin g m e th o d s of m e a s u rin g r e f le c tio n c o e ffic ie n t p r o ­
v id e s th e b a s is f o r th e d e s ig n .
T he p r o b le m o f s u p p o rtin g th e t a r g e t
w ith o u t p ro d u c in g s ig n if ic a n t in te r f e r in g s ig n a l a t th e r e c e iv e r w as s o lv e d
b y a n o v e l w ooden s u p p o rt.
T he f ie ld a m p litu d e an d p h a s e d is tr ib u tio n s in f r o n t of e a c h a n te n n a
in d ic a te d th a t th e t a r g e t sh o u ld b e lo c a te d a t 72 c m . f r o m th e a n te n n a
a p e r t u r e , th is b e in g th e p o in t of m a x im u m f ie ld in te n s ity .
C onducting s p h e r e s an d c i r c u la r d is c s w e r e c h o s e n to e x a m in e
th e p r e c is io n o f th e m e a s u r e m e n t.
T he p o s s ib le e r r o r due to th e i n ­
s ta b ility o f th e p o w e r o u tp u t an d th e r e s e ttin g of th e a tte n u a to r w as found
to b e 12
.
T he e x p e r im e n ta l r e s u l t s in d ic a te d th a t;
64
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
65
1.
-*kc
in ^ c ase of th e c o n d u ctin g s p h e r e s , th e e x p e r im e n ta l and t h e o r ­
e tic a l r e s u l t s a r e w ith in th e p o s s ib le e r r o r e x c e p t in th e r e g io n of th e
m a x im a an d m in im a of th e flu c tu a tin g s c a tte r in g c r o s s - s e c t i o n .
ikC
2. In c a s e of th e co n d u ctin g d is c s th e e x p e r im e n ta l and A n d re je w s k i's
th e o r e tic a l r e s u l t s a r e w ith in th e p o s s ib le e r r o r fo r d is c s of d ia m e te r s
l e s s th a n th e b e a m w id th .
S e v e r a l s ig n ific a n t r e s u l t s in th is th e s is sh o u ld be e m p h a s iz e d :
a) A th e o r e tic a l stu d y of th e r e la tio n s h ip b etw een th e s c a tte r in g
c r o s s - s e c t i o n s an d r e f le c tio n c o e ffic ie n ts , w hich h a s p r e v io u s ly n o t b een
d is c u s s e d in th e l i t e r a t u r e , r e v e a le d th a t th e s c a tte r in g c r o s s - s e c t i o n of
a d i e le c tr ic body d iff e r s f r o m th a t of a co n d u ctin g body of th e s a m e sh a p e
by th e r e f le c tio n c o e ffic ie n ts o n ly .
T he e x p e r im e n ta l r e s u l t s in d ic a te d
th a t th is is tr u e u n d e r th e c o n d itio n th a t th e r e f le c tio n c o e ffic ie n t is c o n s ta n t
o v e r th e s u r f a c e of th e r e f l e c t o r .
T his a p p lie s fo r m o n o s ta tic o r b is ta tic
m e a s u re m e n ts of th e s c a tte r in g c r o s s - s e c t i o n .
b)
The f a c t th a t th e r e f le c tio n c o e ffic ie n t can b e g iv en b y th e r a tio
o f th e s c a tte r in g c r o s s - s e c t i o n o f a g la s3 d is c to th a t of a p e r f e c tly c o n ­
d u ctin g d is c o f lik e g e o m e try , m a k e s it p o s s ib le to s e p a r a te th e c o n trib u tio n
of t a r g e t r e f le c tio n c o e ffic ie n t f r o m th e c o n trib u tio n s of t a r g e t illu m in a tio n
and t a r g e t g e o m e try .
T h is is im p o r ta n t s in c e i t p e r m its th e d e te r m in a tio n
of th e p la n e w ave r e f le c tio n c o e ffic ie n t fo r a s e m i- in f in ite body, th ro u g h
th e u s e of n o n -p la n e -w a v e m e a s u r e m e n ts upon a fin ite body.
c)
The e x p e rim e n ta lly d e te r m in e d r e f le c tio n c o e ffic ie n t of th e g la s s
la m in a ra n g e s f r o m 0 .4 3 + .0 5 to 0. 21 + .0 2 .
T his v a r ia tio n in the
r e f le c tio n c o e ffic ie n t is due to th e change in th e r e f le c tio n c o e ffic ie n t
w ith th e an g le of in c id e n c e .
H o w ev er, th e e x p e r im e n ta l r e s u l t s c o n firm e d
th e v a lid ity of th e s u b s titu tio n m e th o d , w h ich a p p a re n tly h a s n o t b een
r e p o r te d e ls e w h e re in th e l i t e r a t u r e .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A P P E N D IX I
D IE L E C T R IC LA Y ER R E F L E C T IO N S
The p r o b le m of r e f le c tio n f r o m a la y e r w h e re in th e d ie l e c tr ic
c o n s ta n t is a fu n c tio n of p o s itio n ( F r ie n d 1949) is d is c u s s e d u n d e r th e
fo llo w in g c o n d itio n s:
1.
P la n e w av e in c id e n t n o r m a lly on a h o riz o n ta l l a y e r .
2. The d i e l e c t r ic c o n s ta n t is a fu n c tio n of h e ig h t.
3. The a b s e n c e of io n iz a tio n o r f r e e c h a rg e .
F o r th e above a s s u m p tio n s M axw ell w ave e q u a tio n , w ith th e tim e d e p e n d j ^t
ence e
, can b e w r itte n as
( 1)
w h e re th e e l e c t r i c f ie ld of th e in c id e n t w ave is given by
( 2)
k =
and
#
c
s i m i l a r l y fo r th e m a g n e tic f ie ld .
By tr a n s f o r m in g th e h y p e r g e o m e tric d if f e r e n tia l eq u atio n (H g. D . E . )
a p p r o p r ia te ly , i t m a y be c o n v e rte d to th e f o r m of E q . (1).
H en ce, the
s o lu tio n of th e w ave e q u a tio n can be o b ta in e d f r o m th e know n s o lu tio n s of
th e Hg. D. E .
The Hg. D. E . h a s th e f o rm
<*?y.
+
or
J . + ,*>*
x (l - x)
^ 2
+
v = 0
dx
x (l - x)
y = 0
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3)
67
w h e re a , |3 an d
y
a r e a r b i t r a r y c o n s ta n ts .
By th e tr a n s f o r m a tio n
x = x( f )
(4)
and
th e Hg. D . E . b e c o m e s
dx .
.e . d 2 X
d 2r
^
dr
d*l .
d 2x
dr
J
+ y - (a+ ft+ l)x
1_
x (l-x )
* dx
d^
... d ^ .
/c
(5)\
+
d<P
d^
^
aP
. r (f) = 0
x (l-x )
E q u a tin g th e c o e ffic ie n t of ( d ^ / d ^ ) to z e r o
_dr
dA Z x
. * .
dx
dx 2
- P (x) . ( 7 s )
** 0
( 6)
/d x a
• (d ' '
(7)
dV
In te g ra tin g E q . ( 6 ) one g ets
r (^ ) * r 0 .x
w h e re r
2
q+ (3 - y+1
2
- x)
(1
is th e in te g ra tio n c o n s ta n t.
W ith th is v a lu e o f r ( ? ) E q . (5) r e d u c e s to
a2?
w h e re
3
d x
g(
v
or
. dx
y( y~^)
(8)
3_
d$
= i -
.d x .2
" d?
0
=
' 4 '
2
,d x
. dx . 2
d * 2
dr
(9)
4 2 (2 a p - Y(<*+ft-l)x + ((a-|3 ) - i) x '
*
4 2 (1-x)*2
4x
.e v_ i d
d^2
i x)\ 2 ^
- /d
( j £ - In
dx
df
1
" 4
. d
dJ
dx . 2
\
(10)
Kl + K 2 1 T 7 + K 3 ( H - x ) 2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
68
4K^ * y i y - 2 ),
w h e re
4K ^ = l - ( a - P ) 2 + y(Y“ 2 ),
4K 3 = ( a + p - y ) 2 - 1 ,
or
a = ^ (1 + 7 1 + 4 ^
+ /l+ 4 K 3
- ,/+ 4 ( K ^ K ^ ) ) .
p = i ( l + 0/l+ 4 K ^
+^1+4K 3
+ , / 1+4(K1-K 12) ) .
V= V1+4k| + 1
.
(11)
If P( ^ ) = In ( -x ), g ( ^ ) r e d u c e s to
,« v
1
d 3P
, dP
3 ,d 2P
- ( ? 5Y “ ) 2 \| '(K
+ 44 ') “ K 2-» 1? ?+ ’*e x^p .
1
- K
e x ^ -..a l
t
3 (1 + e x p .
P(
( dP,
, _ ,
(1 2 )
P( y
(,,K te r m " )
t )
)2
W ith th e tr a n s f o r m a tio n s g iv en ab o v e , th e Hg. D. E . is tr a n s f o r m e d in to
a w ave e q u a tio n w h o se f o r m a g r e e s
« ( ^ ) v a ry in g w ith p o s itio n .
H en ce,
w ith E q s . (9, 10, 11, 12)and
w ith
f r o m th e know n s o lu tio n s of th e
Hg. D. E . th e so lu tio n of ( 8 ) can b e o b ta in e d .
T he s o lu tio n o f th e H g. D . E . h a s a lr e a d y b een g iv en by G a u s s .
h a s th e s o lu tio n s
F (a , p , y, x) ,
an d
x^“ Y F ( a - y+ 1»
forJxl<^l
i.e .
P~Y+1»
2 -y , x)
- 1 < ^ x <^1
w h e re F (a , p , y, x) d e n o te s th e Hg. s e r i e s
i 4.
* 1*
+ a ( a + l) P ( P + l) x 2
1. Y
1. 2 . y( y + D
. a ( a + l) ( a + 2 ) p ( p + l) ( p + 2 )x 3
1. 2 . 3 y( y + 1 ) ( V + 2 )
B o th s o lu tio n s a r e l in e a r ly in d e p e n d e n t an d , th e r e f o r e , in d ic a te a
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
It
69
fu n d a m e n ta l sy stem *
The H g. D . E . h a s th r e e s in g u la r p o in ts 0, 1, oo .
F o r e a c h p o in t
th e r e is a fu n d a m e n ta l s y s te m w h ich c o n v e rg e s in th o s e r e g io n s .
F o r x * 0 th e H g. D. E . h a s th e o r ig in a l s o lu tio n s , n a m e ly ,
F j = F (a , (3,
F^
y*
*)j
- Xl - Y F ( a - y+1,
P
-
Y + 1,
2-y,
x )
•
F o r x = 1, th e s u b s titu tio n x = (1-Z) tr a n s f o r m s th e H g. D. E . in to
Z (l-Z ) ~~~\ + a + p + l - y -(a + P + l)Z
dZ
^
- a(3y = 0
w h ic h y ie ld s th e so lu tio n s in te r m s of x
F 3 = F (a , p, a+ p - y+1, 1-x);
F
(14)
ss ( l - x ) Y“a "’^ . F( Y -P, Y-a » 1+ Y“a ~P» l* x ) •
F o r x = 00 , th e s u b s titu tio n x = ^ t r a n s f o r m s th e Hg. D. E . in to
2
Z 2 (1-Z) ^
dZ
+ Z (1 -a - p ) -
( 2 - y)
z
+ apy =
0 ,
w h ic h y ie ld s a so lu tio n in te r m s of x
F 5 = x~a . F (a , a-y + 1 , a-p + 1 , ^ )
;
(1 5 )
F 6 = x " P . F ( p , P -y + 1 , p - a + 1, ~ ) •
T hen in g e n e r a l th e o r ig in a l eq u a tio n h a s th e a d d itio n a l so lu tio n s F ^ , F ^ ,
F_ and F , .
5
6
Up t i l l now th e s o lu tio n h a s b e e n g iv en in a s e r i e s f o r m w h ich
c o n v e rg e s o n ly in a p a r t ic u la r re g io n , though th is s e r i e s d e fin e s an
a n a ly tic fu n c tio n w h ic h is a s o lu tio n of th e Hg. D. E . fo r th e c o n v e rg e n t
r e g io n of th e s e r i e s .
C o n sid e r now F ^ a n a ly tic a lly c o n tin u ed to the
c o n v e rg e n c e r e g io n o f th e g iv en s e r i e s F^ and F ^ .
r e p r e s e n t th e s e r i e s an d its a n a ly tic a l co n tin u a tio n .
F^ s h a ll h e n c e fo rth
T hen w e h av e
b e s id e s th e fu n ctio n F and F _ a t th a t lo c a tio n , th e s o lu tio n F _ .
1
Z
D
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
70
G au ss s ta te d th e r e la tio n b etw een F , F
I d
f
5
= < -irQ
' '
P (l-P ).
w h e re
F
r ( l + a-Y)
3
in th e f o rm
1
( 16 )
P ( a . - J 3 - H ) . P (y -1 )
"P (Y -P )....p l'a ) ------ 2 *
, ,
+ (-1)
and F
'
P is th e G a u s sia n g a m m a fu n c tio n .
F o r e v e r y s o lu tio n of y of th e Hg. D . E . o b ta in e d th ro u g h th e
tr a n s f o r m a tio n , a s o lu tio n ^ of th e w ave eq u a tio n m a y a ls o b e o b ta in e d
fo r e a c h F^.
H en ce, f r o m E q s . (4) and (7) fo r e v e ry F^, th e r e e x is ts ^
s u c h th a t
rn
1
(.
a j& lX t1
« T 1 x 2 ( 1 -x)
x
(f| ) " 2 F i
2
U7)
W ith th e tr a n s f o r m a tio n
K-i
x = - A e
1C>0
and fo r la r g e p o s itiv e v a lu e of ^
one can g e t
Y -l- 2 a
V)
-1
= rQ
( 5
-2
VC.
(18)
2
• x
q + g -y + l
.
.
• d - x)
2
• F (
1 .
-x ) •
Y lL - a
fc li
1 y e “ 2 (-l) 2
a
2 .. • x p .n
A
i pil?
j f:i* V C ^)
*
(1 9 )
c o n s id e rin g o nly th e f i r s t t e r m of th e Hg. s e r i e s F ( ................ ~ ).
S in u ld .r ly ^
v*l
=
(<1
|C 2 (-A )
2
t
ill.
^
. exp (
(20)
(-A ) 2 . exp ( ~ 1
l z =ro
|C ^ );
IC ^ ) .
C hoosing th e c o n s ta n ts of th e H g. D . E . s u c h th a t
and
J
3
a r e r e a l an d p o s itiv e , th e above a s y m p to tic a p p ro x im a tio n fo r ( &
r e p r e s e n t s th e p o s itio n d e p e n d en ce of a p la n e w ave tr a v e llin g in the
- d ir e c tio n .
If
j
r 2K A
1
0
(K = c o n sta n t)
j
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.
71
an d
—7 ^- * 2K
j
2
V
0
(K
( j * r^ 1
= c o n s ta n t),
a 6*
- , ; 1
S im ila r ly f o r \
2
and
^
2j
lC ^ (-A )
and
« p . m k 2 K $,.
2 . ex p . ( - j
\C ^ )
1- y
7 2=r Ql
(“A>2 * eXP* (J ^1 ^ V ‘
F r o m E q . (2) th e e l e c t r i c fie ld s can be given as
5
i
”2
-1
(-1) 2
ro hr
A
. ex p . ( j » t - j K ^ lfC ^ ),
JdL
1
"2
E = r " 1 K r ‘ ( “A) 2 .
1
o
( 21)
Izl
E = r _1 1 C ' “2 ( - A ) 2
2
o
1
and
F r o m th e c o m b in a tio n (16), i t can be s e e n th a t th e p r im a r y p la n e
w av e E j is s p lit in to tw o p a r ts on p a s s in g th ro u g h th e la y e r , a tr a n s m itte d
w ave E_ and a r e f le c te d w ave E _.
5
2
F r o m E q . (16) and b y d e fin itio n , th e r e f le c tio n c o e ffic ie n t \
p
fo r
A=1 is g iv en by
r
■
P ( v - 1). r q - p ) .
f(n - a - Y )
[ (l-Y). ('(y-P). p ( a )
(??.
*
T he V a ria tio n of th e D ie le c tr ic C o n sta n t
B y th e tr a n s f o r m a tio n
th e D t e r m o f E q . (12) v a n is h e s , and w ith A=1 E q . (12) tr a n s f o r m s in to
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F o r la r g e n e g a tiv e v a lu e s of
VC2
>—
-
,
1
( Ki n
» •
( 24)
F o r la r g e p o s itiv e v a lu e s of
)
(K i + L _
k
2) .
(25)
k
B etw e en th e v a lu e s g iv en b y E q s . (24) and (25), lie s th e r e g io n in
w h ic h «
r
v a rie s .
Its b r e a d th is g iv en by 1 / yd .
If c
is s lig h tly g r e a te r th an u n ity by M , a s in th e tr o p o s p h e r e ,
r
o
« f ( ^ ) is g iv en by
M 4 e x p v VC? _
« ( f ) = l + M - N g £ j £ ...5
-. M
r ^
o
1 + e x p .^
(1 + e x p .vC J ) 2
w h e re N an d M a r e r e l a t e d to th e c o n s ta n ts K^,
J~1+4K ^
by th e follow ing:
i
js . yiTk? =
jS . J l+ M ^ -N 1
and
(26)
« J 1+4(K 1 -K 2) 1 ;
(27)
= J 1+4 S 2M '
= 2 (d 2 + jd x)
.
H e re d an d d_ a r e th e im a g in a ry and r e a l p a r ts of th e r o o t,
1
r e s p e c tiv e ly , and S is th e r e la tiv e la y e r th ic k n e s s g iv en by
e
3
2k
VC
25
4ir
i-
^
\K L
In E q . (26), th e s e c o n d t e r m r e p r e s e n t s a "m o n o to n ic tr a n s itio n "
of th e d i e l e c t r ic c o n s ta n t « ( f ) f r o m 1 + M to 1 + M - N. T he th ir d
r \
o
o
t e r m r e p r e s e n t s a " s y m m e tr ic a l l a y e r " w h ich is of a lo w e r v a lu e th an
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73
(1 + Mq ) w hen M is p o s itiv e , and of a h ig h e r v a lu e w hen M is n e g a tiv e .
It is n o te d th a t w hen M = M , e
..
= (1 + M - N) r e a c h e s a
o
r( m in im u m )
o
lim it of 1. H ence N and M a r e ta k e n a s th e tr a n s itio n d e v ia tio n and th e
la y e r d e v ia tio n r e s p e c tiv e ly Fig* (2).
E q . (27) can b e w r itte n in te r m s of th e Hg. c o n s ta n ts a s :
a = i + d2 + j |
(J T T i?
P = i + d 2 + j | (j 1 + Mo'
an d
- J l +Mo - N 1+
jd j ) .
+J 1 + MQ - N +
jd x) , (28)
v = 1 + j s jT T k P
F r o m E q . (22) an d E q s . (28), th e r e f le c tio n c o e ffic ie n t can be
w r itte n a s :
f u s j w
) . r j i - d 2 - j ( f ( J m r + J i + m o - 5 T + d i) |
. [ \ i + d 2 - j ( | ( j m r + J1+ M ^-N ) *
[ ( - j s / ! + ! ? ) . [ 1 1 - d 2 + j ( |( f l + h ? - J l + M o - N )
- d j ) ]
• f | i + d 2+ j ( | ( j r r s r . j i+ m o - n » + d1>]
F o r a m o n o to n ic tr a n s itio n , (M=0, d =0 and d_=|-), th e r e f le c tio n
X
«
c o e ffic ie n t is g iv en by
|
n ,
s in h
\ f Sl/TuT - J l t M o -N |
(30)
'
*
s in h 1 1 S (JT + m P
+ J l+ M Q-N )j
F o r a s y m m e tr ic a l l a y e r , (N=0)
i
p
i (jS J T + U
\\ \ -
P
. |
(i
)
+ d2 -
■|,( i - d 2 -j(s /in p + 0,))
COS v ( d
+
jd
)
jf s J l+ M ^ - dj) ) . ---------- £ ------- i-
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(31)
A PPEN D IX II
T h is A ppendix c o n ta in s th e c ir c u i t d ia g ra m s fo r th e C-W r a d a r .
D ia g ra m s a r e p r e s e n te d fo r th e follow ing c ir c u its :
(19) I. F , a m p lifie r (fo u r s ta g e s )
(20) I, F . a m p lifie r (n in e s ta g e s )
(21) A. F , C, s y s te m
(22) 2K25 k ly s tr o n p o w er su p p ly ,
74
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75
flL
STAGES)
vw
N
U
(FOUR
<
(0
Cvj
I. F. AMPLIFIER
VW
o -LO
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FIG. 19.
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8
T
fs.
Z Q
<0
ynr-
Ow
z
LU111
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-Jfc*
A
lio v
12A T7
XTAL
2 -6 A H 6
6A H 6
6AH6
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ft
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1 0 DB
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BIAS VOLTS
6A S7
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1 1 0 v AC
1 1 0 v AC
B IA S VOLTS
R
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150
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§560
ffl 1000
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(N IN E STAGES)
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FIG. 2 I.
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c
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REFERENCES
A n d re je w sk i, W ., Z . an g ew . P h y s ik , 5, 157 (1953).
B a c h y n sk i, M. P . , C an. J o u r . P h y s . , 36, 456 (1958).
B a u e r, J . R . and M e y e r, J . H . , T r a n s a c tio n A m e ric a n G e o p h y sic a l
U nion, 39, No. 4, (1958).
B irn b a u m , G. an d B u s s e y , H. E . , P r o c . I n s t. R ad io E n g r s , 43, 1412 (1955).
C h m e la , A. C. an d A rm s tro n g , G. M ., P r o c . F ifth W eath er R a d a r
C o n fe re n c e , 63 (1955).
C o lw ell, R . C . , F r ie n d , A. W ., H a ll, N. I. and H ill, L . E . , N a tu re , 138,
245 (1939).
C ra in , C, M ., P r o c . I n s t. R ad io E n g r s ., 43, 1405 (1955).
E n g en , G .E . and B e a tty , R . W ., T r a n s . I n s t. R adio E n g r s ., M T T -7 , 351
(1951).
F r ie n d , A. W ., P r o c . I n s t. R ad io E n g r s ., 37, 116 (1949).
G ish , O. H. and B o o k e r, H. G . , P r o c . In s t. R adio E n g r s .,
27,117 (1939).
G o rd o n , W. E . , P r o c . I n s t. R ad io E n g r s . , 37, 41 (1949).
H a n se n , R . C . , B a ilin , E . E . , T r a n s . I n s t. R adio E n g r s .,
A P -7 , 8458 (1959),
H ay, D . R . an d R e id , W. M ., Can. J o u r. P h y s ., 40, 128 (1962).
H ey, J . S . , S te w a rt, G. S . , P in s o n , J . T . and v . P r in c e , P . E . , P r o c .
P h y s . S o c. B , 69, 1038 (1956).
Yon H ip p el, A. R . , " D ie le c tr ic M a te ria ls and A p p lic a tio n s" , W iley and
S o n s, N . Y ., (1958).
J o rd a n , E . C , , " E le c tr o m a g n e tic W aves and R a d ia tin g S y s te m " P r e n ti c e H a ll, I n c ., E nglw ood C liffs, N . J . , (1950).
J e n k in s , F , A. an d W hite, H. E . , " F u n d a m e n ta ls of O p tic s " , M cG raw H ill
(1957).
K ay, A, F . , T r a n s . In s t. R adio E n g r s ., A P - 8 , 586 (1960).
79
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
K e lle h e r , K . S ., J o u r . A ppl. P h y s . , 21, 573 (1950).
K e r r , D . E . , " P ro p a g a tio n of S h o rt R ad io W av es" M c G ra w -H ill, (1951).
L a b ru m , N. R . , J o u r. A ppl. P h y s ., 23, 1320 (1952).
L e a s u r e , R . B . , D u rh a m , K. S . , T o b ia s , J . J . and D u d raw , R . A . , P r o c .
S ix th W ea th er R a d a r C o n fe re n c e s , 261 (1957).
M e n tz e r, J . R . , " S c a tte rin g and D iffra c tio n of R ad io W av es" P e rg a m o n
P r e s s , N. Y. (1955).
M itra , S . K . , N a tu re , 135_, 953 (1936).
M ix n e r, J . and A n d re je w s id , W ., A nn. d e r P h y s ik 7, 156 (1950).
M o n tg o m ery , C. G . , "T ec h n iq u e of M icro w av e M e a s u r e m e n ts " M cG raw H ill, N . Y . , (1947).
P id d in g to n , J . H . , P r o c . In s t. R ad io E n g r s . 27, 753 (1939).
P id d in g to n , J . H . , P r o c . P h y s . S o c ., J31, 129 (1939).
P h illip s , C. E . , T r a n s . I n s t. R ad io E n g r s . A P -7 , 245 (1959).
P la n k , V . G ., G eo p h y s. R e s . P a p e r s , N o. 50, A .F . C .R . C. (1956).
S c h m itt, H . J . , T r a n s . I n s t. R adio E n g r s . , A P -7 , 15 (1959).
S e v e rln , H . , and B ae c k m a n n , W ., A new . P h y s ., 3, 22 (1951).
S ilv e r , S . , "M icro w av e A ntenna T h e o ry and D e sig n " M c G ra w -H ill, N. Y .,
(1949).
S to u t, G .E . and S p ack , M. J r . , P r o c . F ifth W eath er R a d a r C o n fe re n c e ,
67 (1955).
S tra tto n , J . A ., " E le c tr o m a g n e tic T h e o ry " M c G ra w -H ill.
R a m o , S . and W h im n e ry , J. R . , " F ie ld s and W aves in M o d ern R a d io " .,
John W iley and S o n s, I n c ., N. Y. (1944).
W atson^ R , A . , e t a l . , P r o c . R o y . S o c . , 161, 81 (1 9 3 6 ).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
VITA
NAME:
F o u a d K w am S aid H u s se in F a n a k i
BORN:
A le x a n d ria , E g y p t, 1931
EDUCATION:
P rim a ry -
E l- M a a r e f
B a c c o s , A le x a n d ria , 1944-1948
S e c o n d a ry
E l- E r w a E l-W s k a
A le x a n d ria 1948-1953
U n iv e rs ity
U n iv e rs ity of A le x a n d ria
A le x a n d ria 1953-1957
P h y s ic s and M a th e m a tic s
U. W. O. F a c u lty of G ra d u a te S tu d ie s (1959R a d io P h y s ic s
D e g re e
B .S c . 1957
S c h o la rs h ip
T he G o v e rn m e n t of th e U nited A ra b R ep u b lic
E X P E R IE N C E :
R e s e a r c h A s s is ta n t, D e p a rtm e n t of P h y s ic s ,
U n iv e rs ity of W e s te rn O n ta rio , 1959 - 1961
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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