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University
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International
300 N. Z ee b R oad
Ann Arbor, Ml 48106
8225890
Chi, Shou-Hsu
MICROWAVE DIAGNOSTICS OF SEMICONDUCTORS
PH.D.
Northwestern University
University
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t
NORTHWESTERN UNIVERSITY
MICROWAVE DIAGNOSTICS OF SEMICONDUCTORS
A DISSERTATION
S u b m itte d t o t h e G r a d u a te S chool
I n P a r t i a l F u l f i l l m e n t o f t h e R e q u ire m e n ts
f o r th e degree
DOCTOR OF PHILOSOPHY
F ie ld o f E l e c t r i c a l E n g in e erin g
and Com puter S c i e n c e
By
SHOU-HSU CHI
E v a n sto n , I l l i n o i s
J u n e 1982
ABSTRACT
MICROWAVE DIAGNOSTICS OF SEMICONDUCTORS
By Shou-Hsu Chi
A t h e o r e t i c a l a n a l y s i s o f a m ic ro w a v e r e s o n a n t c a v i t y p a r t i a l l y
f i l l e d w i t h b o t h i s o t r o p i c and a n i s o t r o p i c m a t e r i a l i s c a r r i e d o u t .
The r e s o n a n t c o n d i t i o n o f a c a v i t y i s d e r i v e d u n d e r t h e f o l l o w i n g
c rite ria .
(1)
The i m a g i n a r y p a r t
i s equ al to z e ro .
(2)
c a v i t y i s a minimum.
o f t h e i n p u t im p edance o f t h e c a v i t y
The t o t a l E - f i e l d a t t h e c o u p l i n g w a l l i n t h e
(3)
The r e f l e c t i o n c o e f f i c i e n t a s o b s e r v e d i n t h e
c o n n e c t i n g w a v e g u id e i s a minimum.
We f i n d t h a t t h e e q u a t i o n o f t h e
r e s o n a n t c o n d i t i o n d e r i v e d by c r i t e r i o n
(1 ) i s e x a c t l y t h e same a s t h e
e q u a t i o n o f t h e r e s o n a n t c o n d i t i o n d e r i v e d by c r i t e r i o n
(2 ).
When t h e
c a v it y i s un d erco u p led , th e re s o n a n t c o n d itio n c a lc u la t e d in th e c a v ity
i s a lm o s t t h e same a s t h e r e s o n a n t c o n d i t i o n o b s e r v e d i n t h e c o n n e c t i n g
w av eg u id e.
The c o n d i t i o n o f c o u p l i n g i s d e r i v e d u n d e r an a s s u m p t i o n t h a t a
s h u n t s u s c e p t a n c e c a n b e u sed t o r e p r e s e n t t h e c o u p l i n g i r i s .
The con­
d i t i o n o f th e c r i t i c a l c o u p lin g is g iv e n .
The q u a l i t y f a c t o r o f a r e s o n a n t c a v i t y i s c a l c u l a t e d by a fundamen­
t a l d e f i n i t i o n w hich c o n s i d e r s a l l E - f i e l d s and H - f i e l d s i n t h e w hole
c a v it y w ithout ap p ro x im atio n .
A s o - c a l l e d common c r o s s p o i n t m ethod i s d e v e lo p e d t o m e a s u r e t h e
t r a n s p o r t p a r a m e t e r s o f a t h i c k a n i s o t r o p i c s a m p le .
ii
F or a t h i n a n i s o t r o p i c s a m p le , we u s e t h e q u a s i - p e r i o d i c c h a r a c ­
t e r i s t i c s o f t h e r e f l e c t i o n c o e f f i c i e n t a s a f u n c t i o n o f m a g n e t ic f i e l d
t o m e a s u r e t h e p r o p a g a t i o n c o n s t a n t and c a l c u l a t e t h e c a r r i e r con­
c e n tra tio n s.
The p h a s e a n g l e o f t h e r e f l e c t i o n c o e f f i c i e n t and t h e r e s o ­
n a n t f r e q u e n c y a r e a l s o q u a s i - p e r i o d i c f u n c t i o n s when t h e y v a r y w ith
m ag n etic f i e l d .
T hese phenomena a l s o c a n be use d t o d e t e r m i n e c a r r i e r
c o n c e n tra tio n .
iii
To t h e memory o f my f a t h e r
C hih-K an Chi
ACKNOWLEDGMENT
I w is h t o e x p r e s s my g r a t i t u d e t o my a d v i s o r , P r o f . M o r r i s E.
B ro d w in , f o r h i s i n s p i r a t i o n , g u i d a n c e , and e n c o u ra g e m e n t t h r o u g h o u t t h e
c o u r s e o f t h e i n v e s t i g a t i o n and t h e e n t i r e g r a d u a t e p r o g ra m .
S i n c e r e t h a n k s a r e e x t e n d e d t o Mr. J o s e A r a n e t a f o r a c a r e f u l
r e a d i n g o f t h e m a n u s c r i p t , t o D r . Tom Wong f o r some h e l p f u l d i s c u s s i o n s .
I w is h t o t h a n k my w i f e , L in -M ie n , f o r h e r u n d e r s t a n d i n g and
e n c o u ra g e m e n t d u r i n g t h e s e y e a r s .
v
TABLE OF CONTENTS
LIST OF TABLES
LIST OF FIGURES
CHAPTER
I.
II.
Page
INTRODUCTION...............................................................................................
ELECTROMAGNETIC WAVES IN AN ANISOTROPIC SOLIDSTATE PLASMA................................................................................................
III.
1
6
THE RESONANT CONDITION AND THE QUALITY FACTOR
OF A PARALLEL PLATE CAVITY PARTIALLY FILLED
WITH A SLAB ..................................................................................................
14
1 1 1 . 1 . I n t r o d u c t i o n ...............................................................................
14
1 1 1 . 2 . T he R e s o n a n t C o n d i t i o n ( I s o t r o p i c C a s e ) ...............
15
1 1 1 . 2 . 1 . T h e L o s s l e s s C ase And The Low L o s s
A p p r o x i m a t i o n .......................................................
16
1 1 1 . 2 . 2 . T h e Im p e d a n ce M ethod .......................................
18
1 1 1 . 2 . 3 . T he Minimum E - f i e l d M e t h o d ........................
19
1 1 1 . 2 . 4 . T h e R e s o n a n t C o n d i t i o n As O b se rv ed
I n T h e C o n n e c t in g W a v e g u i d e ......................
IV .
V.
28
1 1 1 . 3 . T he R e s o n a n t C o n d i t i o n ( A n i s o t r o p i c C a s e )
46
1 1 1 . 4 . D i s c u s s i o n .......................................................................................
53
1 1 1 . 5 . The Q u a l i t y F a c t o r .................................................................
55
1 1 1 . 5 . 1 . T he I s o t r o p i c C a s e ...........................................
56
1 1 1 . 5 . 2 . T he A n i s o t r o p i c C a s e .......................................
66
THE CCMMON CROSS POINT METHOD ....................
74
THE RELATIONSHIP OF MAXIMA AND MINIMA OF QUALITY
FACTOR AND REFLECTION COEFFICIENT WITH VARIATION
V I.
OF STATIC MAGNETIC FIELD ...................................................................
89
V . l . I n t r o d u c t i o n ....................................................................................
89
V .2 . The I s o t r o p i c C ase .....................................................................
94
V .3 . The A n i s o t r o p i c C ase
97
..........................................................
CONCLUSION....................................................................................................... 104
vi
APPENDIX
A.
Page
PLANE WAVE ANALYSIS OF THE ANISOTROPIC SOLIDSTATE PLASMA..................................................................................................
B.
THE RESONANT CONDITION OF A PARALLEL PLATE CAVITY
PARTIALLY FILLED WITH A LOSSLESS SLAB .......................................
C.
106
109
THE DERIVATION OF EQUATION ( 3 . 7 ) TO ( 3 . 1 ) WHEN THE
SLAB I S LOSSLESS .........................................................................................
112
D.
THE DERIVATION OF EQUATION ( 3 . 7 ) TO ( 3 . 1 5 )
............................
114
E.
THE CONDITIONS OF I (Z. )=0 AND V(1 ) BE A MINIMUM
m in
v o
OF A LOADED TRANSMISSION LINE ARE EQUIVALENT ......................
116
F.
THE INPUT IMPEDANCE OF A LOADED CAVITY AT THE LEFT
SIDE OF THE COUPLING WALL...................................................................
118
G.
A DISCUSSION OF THE DEFINITION OF THE QUALITY FACTOR . .
120
H.
E-FIELD AND H-FIELD IN THE CAVITY..................................................
137
I.
143
J.
PROOF THAT U „ ,+ U„ = U . + UTT AT RESONANCE DEFINED
Ef
Es
Hf
Hs
BY I ( Z 3 )=0 ..................................................................................................
m in
THE POYNTING THEOREM IN COMPLEX FORM ..........................................
K.
THE QUALITY FACTOR WHEN THE SLAB I S A GOOD CONDUCTOR . .
149
L.
THE QUALITY FACTOR WHEN OBSERVE THE PARALLEL
COMPONENT OF THE E-FIELD ......................................................................
M.
THE DERIVATION OF EQUATION ( 5 . 5 )
147
151
....................................................
154
BIBLIOGRAPHY .....................................................................................................................
157
vii
LIST OF TABLES
TABLE
Page
3 .1 .
The p a r a m e t e r s o f t h e s a m p le and t h e c a v i t y ..............................
25
3 .2 .
The p a r a m e t e r s o f e x a m p le s ......................................................................
34
3 .3 .
1Q/X and B f o r e a c h p o i n t ( e x a m p le 1) .............................................
37
3 .4 .
1 /X a n d B f o r e a c h p o i n t (e x a m p le 2) ............................................
°
2
2
P o i n t s on t h e X - R - R c i r c l e a n d t h e i r B .........................
c
The p a r a m e t e r s o f t h e t h i c k In S b s a m p l e .....................................
37
3 .5 .
4 .1 .
40
76
4 .4 .
The v a l u e s o f <*> a n d v a t H, = 0 . 1 KG, Q = 1392 ..................
p
dc
The v a l u e s o f u an d v a t R , = 1KG, 10KG, 2 0 K G ......................
p
dc
The p a r a m e t e r s o f t h e t h i c k InS b s a m p l e .....................................
83
4 .5 .
M e a s u re d Q a t d i f f e r e n t v a l u e s o f Bq ............................................
83
5 .1 .
The t r a n s p o r t p a r a m e t e r s o f t h e t h i n InS b s a m p l e ...................
98
4 .2 .
4 .3 .
viii
76
81
LIST OF FIGURES
FIGURE
Page
1 .1 .
The summary o f h i s t o r y r e v i e w ................................................................
2 .1 .
E - f i e l d o f t h e i n c i d e n t , r e f l e c t e d an d t r a n s m i t t e d
w aves a n d E - f i e l d i n t h e s a m p l e .........................................................
2 .2 .
..............................................
9
N o r m a liz e d d i s p e r s i o n c u r v e f o r t h e p o s i t i v e and
n e g a t i v e p o l a r i z e d w a v e s . When
2 .4 .
7
N o r m a liz e d d i s p e r s i o n c u r v e f o r t h e p o s i t i v e and
n e g a t i v e p o l a r i z e d w a v e s . When
2 .3 .
2
u>^= w ..............................................
10
N o r m a l i z e d d i s p e r s i o n c u r v e f o r t h e p o s i t i v e and
u>^< u ..............................................
11
3 .1 .
The g e o m e try o f a n i d e a l p a r a l l e l p l a t e c a v i t y .......................
17
3 .2 .
The im p e d a n ce o f t h e p a r a l l e l p l a t e c a v i t y .................................
17
3 .3 .
The E - f i e l d i n t h e p a r a l l e l p l a t e c a v i t y ......................................
21
3 .4 .
The c o m p a r is o n o f r e s o n a n t f r e q u e n c i e s o b t a i n e d by
-2
4
Eq. ( 3 . 2 ) , ( 3 . 3 ) a n d ( 3 . 1 4 ) f o r « = 2 0 « o , a= 10
t o 10 /Q -cm
n e g a t i v e p o l a r i z e d w a v e s . When
3 .5 .
d= 0 . 0 lcm , 1 = 1 . 6 6 7 c m .................................................................................
o
The c o m p a r is o n o f r e s o n a n t f r e q u e n c i e s o b t a i n e d by
26
Eq. ( 3 . 2 ) , ( 3 . 3 ) a n d ( 3 . 1 4 ) f o r « =20 « , * = 10- 5 t o l 0 4 /fi-cm
3 .6 .
d= ^ / 4 , 1 - 1 . 6 6 7 c m .....................................................................................
o
( a ) The i n p u t im p e d a n c e o f t h e l o a d e d c a v i t y
( b ) I t s e q u i v a l e n t c i r c u i t .....................................................................
27
33
3 .7 .
T he l o c i o f
a n d Re(Z,j,) o f s a m p le 1 ...........................................
35
3 .8 .
The l o c i o f Z^
a n d Re(Z,j,) o f s a m p le 2 ...........................................
36
3 .9 .
The l o c u s o f X
= R-
..............................................................................
39
3 . 1 0 . The e r r o r i n r e s o n a n t l e n g t h i n t r o d u c e d by E q . ( 3 . 3 7 ) and
( 3 . 5 1 ) , l a r g e B a p p r o x i m a t i o n , v e r s u s B ......................................
v e r s u s B f o r t h e t h i n and t h i c k InS b
r
sa m p le .....................................................................................................................
44
3 . 1 1 . The e r r o r o f 1
ix
45
LIST OF FIGURES ( c o n t i n u e d )
FIGURE
Page
3 .1 2 .
The c o n f i g u r a t i o n o f t h e
c a v i t y and E - f i e l d
..........................
3 .1 3 .
The l r v e r s u s t h e s t a t i c m a g n e t i c f i e l d f o r t h e t h i n
In S b s a m p l e .........................................................................................................
3 . 1 4 . The 1
47
52
v e r s u s t h e s t a t i c m ag n e tic f i e l d f o r t h e t h i c k
In S b s a m p le .........................................................................................................
54
3 . 1 5 . The d e g e n e r a c y o f t h e r e f l e c t i o n c o e f f i c i e n t w i t h
d i f f e r e n t B .............................................................
o
4 .1 .
Q v e r s u s v f o r d i f f e r e n t to when H , = 0 . 1 K G ........................
p
dc
4 .2 . Q v ersu s
v f o r d i f f e r e n t w when H, = 1 K G ..................................
p
dc
4 .3 . Q versu s
v f o r d i f f e r e n t w when H, = 10 K G .............................
p
dc
4 .4 . Q versu s
v f o r d i f f e r e n t w when H, = 20 K G .............................
p
dc
4 .5 .
u v e r s u s v f o r d i f f e r e n t H,
..............................................................
p
dc
4 . 6 . Q v e r s u s v f o r d i f f e r e n t w when H, = 6 . 7 K G ...........................
p
dc
4 .7 . Q v ersu s
v f o r d i f f e r e n t w when H, = 8 .3 7 5 K G .........................
p
dc
4 .8 . Q versu s
v f o r d i f f e r e n t w when H, = 10 K G ..............................
p
dc
4 .9 .
u v e r s u s v f o r d i f f e r e n t H,
..............................................................
p
dc
5 . 1 . a . Q v e r s u s /7d .......................................................................................................
91
5 . 1 . b . Power l o s s i n s l a b v e r s u s /7d ..............................................................
91
£ d ........................................................................
92
5 . 1 . e . E n e rg y s t o r e d v e r s u s
5 . 1 . d . T o ta l en erg y s t o r e d v e r s u s
5 .2 .
/3d .........................................................
72
77
78
79
80
82
84
85
86
87
92
Q v e r s u s /9d, a , <*= 0 .0 1 6 / 0 - c m , b , a= 0 .0 2 4 /fl-c m
c , o = 0 . 0 3 2 / 0 - c m .............................................................................................
93
J
5 . 3 . IP
v e r s u s B f o r t h i n In S b s a m p le , a , d = 6 m i ls ,
I m-i-|
o
b , d = 1 2 m i l s , c , d = l 8 m i l s ..........................................................................
5 .4 .
5 .5 .
99
Argum ent o f P , v e r s u s B f o r t h i n In S b s a m p le , a , d = 6 m i l s
m+
o
b , d = 1 2 m i l s , c , d = l 8 m i l s ..........................................................................
101
w v e r s u s B f o r t h i n In S b s a m p le a , d = 6 m i l s , b , d = 1 2 m ils
r
o
c , d = 1 8 m ils .........................................................................................................
103
x
1
CHAPTER I .
INTRODUCTION
When a l i n e a r l y p o l a r i z e d e l e c t r o m a g n e t i c wave i s n o r m a l ly i n c i d e n t
upon a s e m i c o n d u c t o r i n a s t a t i c m a g n e t ic f i e l d d i r e c t e d
o f p ro p ag a tio n , th e tra n s m itte d
fie ld
in th e d i r e c t i o n
is e ll i p t i c a l l y p o larized .
The
g e n e r a tio n o f e l l i p t i c a l p o l a r i z a t i o n i s c a ll e d th e Faraday E f f e c t .
r e f l e c t e d wave becomes e l l i p t i c a l l y
The
p o l a r i z e d w i t h t h e m a jo r a x i s t i l t e d
a t an a n g l e w i t h r e s p e c t t o t h e e l e c t r i c
f ie ld o f th e in c id e n t r a d ia tio n .
T h i s r e f l e c t i o n o b s e r v a t i o n i s c a l l e d t h e m agneto m ic ro w a v e K e r r
e f f e c t .56
I n t h i s d i s s e r t a t i o n , we s t u d y t h e s e m i c o n d u c t o r i n a c a v i t y by
o b s e r v i n g t h e m agneto m icrow ave K e r r e f f e c t .
The o b s e r v a t i o n dep e n d s
upon t h e t r a n s p o r t p a r a m e t e r s o f t h e s e m i c o n d u c t o r and i s t h e r e f o r e u s e ­
fu l in d e te rm in in g th e s e p a ra m e te rs .
T h is p r o c e s s i s o f t e n c a l l e d
"M icrow ave D i a g n o s t i c s " o r m icrow ave c h a r a c t e r i z a t i o n .
B e f o r e g o in g i n t o
t h e d e t a i l s o f o u r w ork, we s h a l l r e v i e w b r i e f l y a l l p r i o r work t h a t h a s
b e e n done i n t h e a r e a o f m ic ro w a v e d i a g n o s t i c s o f s e m i c o n d u c t o r s .
We c o l l e c t e d n i n e t y - t w o p a p e r s from 1953 t o 1981, and l i s t e d
a ll
p a p e r s a c c o r d i n g t o t h e d a t e o f p u b l i c a t i o n , t h e n g r o u p e d t h e s e p a p e r s as
shown i n F i g .
d isse rta tio n .
n u m b e rs .
1 .1 .
The num bers r e f e r t o t h e r e f e r e n c e a t t h e end o f t h i s
The e a r l i e r p a p e r s a r e d e s i g n a t e d by t h e s m a l l e r r e f e r e n c e
The s i x g r o u p s on t h e l e f t - h a n d s i d e i n F i g .
1.1 d e a l w ith t h e
r e f l e c t i o n m ethod w h i l e t h e o t h e r s i x g r o u p s on t h e r i g h t hand s i d e d e a l
w i t h t h e t r a n s m i s s i o n m eth o d .
The u p p e r e i g h t g r o u p s shown i n F i g .
1 .1 ,
d e a l w i t h e i t h e r o f t h e two m eth o d s i n c o n j u n c t i o n w i t h a s t a t i c m a g n e tic
fie ld ,
Hj)(j.
The f o u r g r o u p s a t t h e b o tto m i n F i g .
1.1 d e a l w ith e i t h e r
2
R e f l e c t i o n M ethod <-
(A)
(B)
3
A
3°
A
13
5 ,1 4 ,2 0 ,2 1 ,
2 2 ,2 6 ,3 0 ,3 2 ,
3 3 ,3 7 ,3 9 ,4 5
4 7 ,4 8 ,5 1 ,5 8 ,
5 9 ,6 4 ,6 6 ,8 6 ,
87
o
TJ
X
4 ,6 ,8 ,1 0 ,2 4 ,
4 0 ,4 1 ,4 3
(E)
(F )
I
- R e f l e c t i o n MethodI
V
o
2 3 ,3 6 ,3 8 ,6 9
7 5 ,7 6 ,7 7 ,8 1 ,
8 2 ,8 3 ,8 4
I
|
|
o
4 4 , 5 2 , 5 5 , 7 0 , •o
X
7 9 ,8 9
(H)
(I)
W aveguide <-
-> C a v i t y
(G)
CD)
2 ,1 9 ,5 0 ,5 7
(G)
T r a n s m i s s i o n Method1 ,3 ,7 ,1 1 ,1 2 ,
1 5 ,1 6 ,1 7 ,1 8 ,
2 5 ,2 7 ,2 8 ,2 9 ,
3 5 ,4 3
(J)
W aveguide <-
T h e o r e t ic a l o n ly 9 ,4 6 ,4 9 ,8 0 ,8 5 ,9 0 .
F ig u re 1 .1 .
3 4 ,5 3 ,5 4 ,6 0 ,
6 7 ,8 8 ,
uc ycj
8 ,6 5 ,7 1 ,7 8
C a v ity
i— >
5 6 ,6 1 ,6 2 ,6 8
73
W aveguide <-
(jc=to<—
-> C a v i t y
The summary o f h i s t o r y r e v i e w .
7 2 ,7 4 ,9 1 ,9 2
(K)
-» C a v i t y
wc= < K — I
W aveguide <-
T r a n s m i s s i o n Method
3
o f t h e two m eth o d s when t h e r e i s no m a g n e t ic f i e l d .
A lso n o t e t h a t b o th
m e th o d s a r e s u b d i v i d e d i n t o t h e w a veguide and c a v i t y m e th o d s .
F u r t h e r m o r e , t h e m ethods f o r t h e c a s e w here t h e r e i s a s t a t i c m a g n e tic
fie ld
w.
i s a l s o s u b d i v i d e d i n t o one where u)c = w and t h e o t h e r where
The symbol wc r e p r e s e n t s t h e c y c l o t r o n r e s o n a n t
t h e o r e t i c a l p ap ers a re l i s t e d
frequ en cy .
i n t h e b o tto m row o f F i g .
ojc
>
Some
1 .1 .
Some r e m a r k a b l e p a p e r s from each o f t h e s e g r o u p s a r e :
(1)
Group A:
M. E. Brodwin and R. J . Vernon^® f i r s t o b s e r v e d
m ic ro w a v e K e r r e f f e c t
ju n c tio n .
th e
i n Ge, S i and InSb by u s i n g a t u r n - s t i l e w a v eguide
An im proved m ethod was g i v e n by R. J . Vernon and T. A.
D o r s c h n e r . 7 3
v.
S. C h a m p lin , P . S. R ange, e t a l . ^ ^ , 6 8 m e a s u re d t h e
s e m i c o n d u c t o r c o n d u c t i v i t y and t h e com plex H a l l f a c t o r by a r e f l e c t i o n
t e c h n i q u e u s i n g a TEqi c i r c u l a r w a v e g u id e .
(2)
Group B:
e f f e c t i v e mass when
L
u
7 1 , 7 8
R. N. D e x t e r , H. J . Z e i g e r and B. Lax® m ea su red t h e
wc =s ui and when U)c > w.
M. E. Brodwin and P. S.
d e sig n e d a p r e c i s e c a v it y to m easure th e t r a n s p o r t p a ra m e te rs o f
sem ico n d u cto rs at c ry o g e n ic te m p e ra tu re .
(3 )
Group C:
Many i n v e s t i g a t o r s o b s e r v e d t h e F a r a d a y
e f f e e t ,5 » 1 4 ,2 2 ,3 0 ,3 3 ,4 5 ,6 6
F araday r o t a t i o n . 2 0 ,2 1 ,3 2
ancj
r a a n y
m e a s u re d t h e H a l l m o b i l i t y from t h e
-j^g h e l i c o n wave p r o p a g a t i o n was o b s e r v e d i n a
w a v e g u id e by P. A i g r a i n ,^ ® A. L i c h a b e r and R.
a l .4 7 ,4 8 ,5 8 ,6 4 ,8 6 ,8 7
V e i l e x ® 7 , 3 9
and jj P e r r i n e t
^ p e r t u r b a t i o n a n a l y s i s o f a r e c t a n g u l a r w aveguide
c o n t a i n i n g a t r a n s v e r s e l y m a g n e t iz e d s e m i c o n d u c t o r was g i v e n by G. J .
G a b r i e l and M. E. B r o d w in .59
4
(4)
Group D:
H a l l m o b i l i t y and
The t r a n s m i s s i o n c a v i t y t e c h n i q u e was u sed t o m e a s u re
m a g n e t o r e s i s t a n c e . 8 4 - j 5 3 , 6 0 , 6 7
n i q u e was u sed by F. S e r f e r t . ^
e le c tro d e le ss
tech ­
A b im o d a l c a v i t y t e c h n i q u e was use d by
N. P. Ong and a . M. D o r t i s . 88
(5)
G roups E, F and G:
W. S c h o c k le y ^ s u g g e s t e d t h a t t h e m icrow ave
c y c l o t r o n r e s o n a n c e a b s o r p t i o n i n Ge a t low t e m p e r a t u r e c o u ld be use d t o
d e t e r m i n e t h e s h a p e o f t h e e n e r g y s u r f a c e in t h e B r i l l o u i n z o n e .
S everal
i n v e s t i g a t o r s u sed c y c l o t r o n r e s o n a n c e t o d e t e r m i n e t h e e f f e c t i v e mass o f
e l e c t r o n s and h o l e s i n Ge and S
i > 8 > 8 ,1 0 ,4 0
Some i n v e s t i g a t o r s o b s e r v e d
o n l y c y c l o t r o n r e s o n a n c e p h e n o m e n a .1 8 , 4 1 , 5 0 , 5 7
(6)
Group H:
The com plex d i e l e c t r i c c o n s t a n t was m e a s u re d f o r d i f ­
f e r e n t k i n d s o f m a t e r i a l s , such a s b u l k m a t e r i a l s ,89 h i g h l o s s
m a te ria ls ,? 5
s h e e t m a t e r i a l s ? ? and t h i n f i l m s . 84
New t e c h n i q u e s such as
t h e lumped e le m e n t m e t h o d ,82 w a v e g u id e r e s o n a t o r m ethod83 w ere d e v e l o p e d .
A q u a r t e r wave t r a n s f o r m e r m ethod was u s e d by P. K. Roy and A. N.
D a t t a . 81
(7)
Group I :
I m p u r i t y c o n d u c t i o n i n s e m i c o n d u c t o r s was i n v e s t i ­
g a t e d by S. Tanaka and H. T. F a n ^ and by S. T anaka and M.
K o b a g a s h i . ^ 2
A p r e c i s e c a v i t y t e c h n i q u e was p r e s e n t e d by M. E. Brodwin and P. S.
L u .55
a
m ethod u s i n g an e v a n e s c e n t w a v e g u id e c o u p le d t o a c a v i t y r e s o n a ­
t o r was used by A. Kumer and D. G. S m it h .? 9
t h e com plex p e r m i t t i v i t y and i t s
W. E. C o u rtn e y 8 9 m e a s u re d
t e m p e r a t u r e c o e f f i c i e n t by t h e r e s o n a n t
f r e q u e n c i e s and Q - f a c t o r s o f a d i e l e c t r i c d i s k s h o r t e d a t b o t h e n d s .
(8)
Group J :
In 1953, T. S. B e n e d i c t and W.
f i e l d o f m icrow ave d i a g n o s t i c s o f s e m i c o n d u c t o r s .
S h o c k l y l > 3
s ta rte d
S i m i l a r m icrow ave
th e
5
t r a n s m i s s i o n m eth o d s w e re u s e d by F . A. D 'A l t r o y and H. Y. Tan7 and by
J . M. G o ld e y and S . C. B row nH »12
m e a s u re t h e e f f e c t i v e mass and
r e l a x a t i o n t im e o f c h a r g e c a r r i e r s .
The
o f t h e i m p u r i t i e s w e re o b s e r v e d , 1 5 - 1 7 ,2 8
p o l a r i z a t i o n and a b s o r p t i o n
t h e l i f e t i m e was
m ea su red .2 5 ,2 7 ,2 9 ,3 5
(9)
Group K.
A m icrow ave c a v i t y p e r t u r b a t i o n t e c h n i q u e was u s e d t o
m e a s u re t h e d i e l e c t r i c p a r a m e t e r s . 7 2»7^ » 9 1 ,9 2
There a r e s i x c h a p te r s i n t h i s d i s s e r t a t i o n :
C h ap ter I :
H e re a g e n e r a l d e s c r i p t i o n o f t h e e n t i r e work i s p r e ­
se n te d as w e ll as a h i s t o r i c a l rev iew .
C hap ter I I :
A p l a n e wave a n a l y s i s i s i n t r o d u c e d t o s t u d y
e le c tro ­
m a g n e t i c wave p r o p a g a t i o n i n a n i s o t r o p i c s e m i c o n d u c t o r s .
C hap ter I I I :
The r e s o n a n t c o n d i t i o n s o f a p a r a l l e l p l a t e c a v i t y
p a r t i a l l y f i l l e d w ith a lo s s y s la b a re d e riv e d under th e fo llo w in g
c rite ria :
(1 ) th e im a g in a ry
equal to z e ro ,
(2) th e t o t a l
p a r t o f t h e i n p u t im p edance o f t h e c a v i t y i s
E - f i e l d in th e c a v ity
n e a r th e c o u p lin g w a ll
i s a minimum and ( 3 ) t h e r e f l e c t i o n c o e f f i c i e n t a s o b s e r v e d i n t h e con­
n e c t i n g w a v e g u id e i s a minimum.
q u a l i t y f a c t o r of th e c a v ity
C h a p t e r IV:
The common
The c o n d i t i o n s o f t h e c o u p l i n g and t h e
a re d eriv ed .
c r o s s p o i n t m ethod i s p r e s e n t e d t o m e a s u re
t h e p a r a m e t e r s o f a t h i c k s a m p le .
C h a p t e r V:
The u s e o f t h e p e r i o d i c c h a r a c t e r i s t i c s o f t h e r e f l e c ­
t i o n c o e f f i c i e n t v e r s u s Hqc c u r v e t o d e t e r m i n e t h e p a r a m e t e r s o f a t h i n
sa m p le i s p r e s e n t e d .
C h a p t e r V I:
A summary and c o n c l u s i o n a r e g i v e n .
6
CHAPTER I I .
ELECTROMAGNETIC WAVES IN AN ANISOTROPIC SOLID-STATE PLASMA
When a l i n e a r l y p o l a r i z e d wave n o r m a l l y i n c i d e n t upon a s e m i c o n d u c t o r
i n a s t a t i c m ag n etic f i e l d d i r e c t e d i n th e d i r e c t i o n o f p r o p a g a tio n ,
F i g . 2 . 1 , b o t h r e f l e c t e d an d t r a n s m i t t e d w aves become e l l i p i t c a l l y
p o larize d .
The e l e c t r o m a g n e t i c w aves i n t h e s a m p le c a n be assum ed t o be
t h e sum o f two c o u n t e r r o t a t i n g c i r c u l a r l y p o l a r i z e d waves w i t h d i f f e r e n t
p ro p ag a tio n c o n s ta n ts .
The p l a n e wave a n a l y s i s o f t h e s o l i d - s t a t e p la s m a i s g i v e n i n
A p p e n d ix A.
The d i s p e r s i o n r e l a t i o n s o f t h e two c i r c u l a r l y p o l a r i z e d
waves a r e g i v e n by
K+2 = w2ji0 ( e + , - j e + " )
(2 . 1)
w here
•}
Ne2 1 /2
, t h e p la s m a f r e q u e n c y
u>p => (----- ^
esm
eB0
u>c = — jf, t h e c y c l o t r o n r e s o n a n c e f r e q u e n c y
m
v **— £ , t h e c o l l i s i o n f r e q u e n c y
om
N, t h e c a r r i e r c o n c e n t r a t i o n
es , th e s t a t i c p e r m i tt iv i ty
m*, t h e e f f e c t i v e mass
E .-ellipse
E+-c irc le
E_-circl
sa m p le
Er - ellipse
F i g . 2 . 1 . E - f i e l d s o f t h e i n c i d e n t , r e f l e c t e d and
t r a n s m i t t e d waves and E - f i e l d i n t h e s a m p le .
'j
a , th e s t a t i c c o n d u c tiv ity
B0 , t h e s t a t i c m a g n e t i c f i e l d
The a t t e n u a t i o n f a c t o r s and t h e p h a s e c o n s t a n t s o f t h e two c i r c u l a r l y
p o l a r i z e d w aves a r e g i v e n by*
K± = S± - j a ±
(2 .4 )
a± = u / - | a
[ - e ± * + (£+ *2 + e ± "2 ) 1 / 2 ] 1 /2
(2 .5 )
0± - w / - — ■ [ e + ' + ( e ± ' 2 + e ± " 2 ) 1 / 2 ] 1 /2
(2 .6 )
E q u a t i o n s ( 2 . 5 ) and ( 2 . 6 ) a r e u s e d u n d e r
th e assu m p tio n o fs p h e r i c a l
e n e r g y s u r f a c e s and e n e r g y i n d e p e n d e n t r e l a x a t i o n tim e o f t h e h o s t
m a t e r i a l . ^5
To s t u d y t h e d i s p e r s i v e n a t u r e o f t h e two c i r c u l a r l y p o l a r i z e d w a v e s ,
we assum e t h e s a m p le i s l o s s l e s s .
The p h a s e c o n s t a n t s c a n t h e n be
s im p lif ie d to
,____
0+ = ui/p0e s [ 1 -
____
3 - = u / p 0e s [ 1
(<jon /u))2 i / o
j _
]
( 2 . 7 . a)
(<*Jn /u))2 i / o
i'"+ a)c /o) J
( 2 . 7 ,b )
0)c
The n o r m a l i z e d p h a s e c o n s t a n t s , 0±/o)/uoe s v e r s u s - j j - a r e p l o t t e d i n
F i g . 2 . 2 , 2 .3 and 2 . 4 , u n d e r t h e c a s e s o f u)p >
ui r e s p e c t i v e l y w i t h
ioc /( d
id,
Up = w and Wp <
a s th e in d ep e n d e n t v a r i a b l e .
*The n o t a t i o n s a r e d e f i n e d t o be
jK± = y+ = a± + j 0 + , a r e t h e p r o p a g a t i o n c o n s t a n t s
<x+, a r e t h e a t t e n u a t i o n f a c t o r s
0+ , a r e t h e p h a s e c o n s t a n t s
Re
helicon
region
1.0
Im
Fig. 2.2. Normalized dispersion curve for the positive and negative polarized
waves. When wp> u
Re
1.0
1.0
Im
Fig. 2.3. Normlized dispersion curve for the positive and negative polarized
waves. When Wp= w
o
Re
1.0
lm.
Fig. 2.4. Normlized dispersion curve for the positive and negative polarized
waves. When «p<w
12
From F i g . 2 . 2 , we s e e t h a t when u)p > a>, t h e p o s i t i v e c i r c u l a r l y
p o l a r i z e d wave c a n p r o p a g a t e o n l y when u>c >
a ssu m e d a s ^ O
10
u>.
The h e l i c o n r e g i o n c a n be
n e Sa t *-ve c i r c u l a r l y p o l a r i z e d
WC
wave c a n p r o p a g a t e o n l y w h e n —
>
“
d
)
o
From F i g . 2 . 3 , we s e e t h a t when u>p =
“ 1]«
oj, t h e
p o l a r i z e d wave c a n p r o p a g a t e o n l y when mc >w.
p o sitiv e c irc u la rly
The n e g a t i v e c i r c u l a r l y
p o l a r i z e d wave c a n p r o p a g a t e i n a l l r e g i o n s .
From F i g . 2 . 4 , we s e e t h a t when u)p < m, t h e p o s i t i v e c i r c u l a r l y
Wn n
Wq
p o l a r i z e d wave w i l l be c u t o f f when 1 - ( j f '
^ ~ ^
neS a tiv e c i r ­
c u l a r l y p o l a r i z e d wave w i l l p r o p a g a t e i n a l l r e g i o n s .
When t h e s t a t i c m a g n e t i c f i e l d g o e s t o i n f i n i t y , b o t h p o s i t i v e and
n e g a tiv e c i r c u la r ly
p o l a r i z e d waves w i l l p r o p a g a t e w i t h t h e same
t i o n c o n s t a n t , 3+ =
u>/p0e s .
T h is c asei s
tru e
f o r a l l v a lu e s
I n t h e h e l i c o n r e g i o n , i f t h e low l o s s a s s u m p t i o n , u c v»
propaga­
o f u)p .
1, h o ld s,
t h e p h a s e c o n s t a n t s and t h e a t t e n u a t i o n f a c t o r s can be s i m p l i f i e d a s ^ 6
3+ -
ol
“ /U ^
cq. == 3 - -
u)pa,1 / 2 /a,c 1 /2
(2 .8 )
u)p «
(2 .9 )
1 / 2 / ( 2 wc3 / 2 t )
From Eq. ( 2 . 8 ) and ( 2 . 9 ) we s e e t h a t 3+ and a - a r e much l a r g e r t h a n
cq. a n d 3 - .
So t h e p o s i t i v e c i r c u l a r l y p o l a r i z e d wave w i l l p r o p a g a t e
w i t h low a t t e n u a t i o n , and t h e n e g a t i v e c i r c u l a r l y p o l a r i z e d wave w i l l be
h ig h ly a tte n u a te d .
The p o s i t i v e c i r c u l a r l y p o l a r i z e d wave i s c a l l e d t h e
e x t r a o r d i n a r y wave and t h e n e g a t i v e c i r c u l a r l y p o l a r i z e d wave i s c a l l e d
t h e o r d i n a r y w ave.
When we s u b s t i t u t e t h e p a r a m e t e r s o f t h e s o l i d - s t a t e p la s m a i n t o Eq.
jL
( 2 . 8 ) , we w i l l s e e t h a t 3+ i s i n d e p e n d e n t o f e s , ra and p , and 3+ becomes
13
/ u)Ney0
0+ = / —
?
°o
T h is i s an i n t e r e s t i n g r e s u l t .
(2 .1 0 )
When 0+ i s m e a s u re d u s i n g Eq. ( 2 . 1 0 ) ,
we c a n c a l c u l a t e t h e c a r r i e r c o n c e n t r a t i o n i m m e d i a t e l y .
o f t h i s c o n c l u s i o n w i l l be g i v e n i n C h a p t e r V.
The a p p l i c a t i o n
14
CHAPTER I I I .
THE RESONANT CONDITION AND THE QUALITY FACTOR OF A
PARALLEL PLATE CAVITY PARTIALLY FILLED WITH A SLAB
III.l.
In tro d u c tio n
I f t h e s l a b , w hich p a r t i a l l y f i l l s
th e c a v i t y , i s l o s s l e s s , th e
re s o n a n t c o n d itio n o f th e c a v it y has been o b t a i n e d .9 3 ,9 4
jju t i f f h e
s l a b i s l o s s y , i t w i l l be shown t h a t no s o l u t i o n s o f t h e wave e q u a t i o n
c a n be used t o s a t i s f y t h e t h r e e b o u n d a r i e s s i m u l t a n e o u s l y .
In t h i s
c h a p t e r , we u s e t h r e e m eth o d s t o d e f i n e t h e r e s o n a n t c o n d i t i o n .
( 1 ) We lo o k a t t h e c a v i t y a s a t r a n s m i s s i o n l i n e p r o b le m * -a n d assume
t h a t t h e i m a g i n a r y p a r t o f t h e i n p u t im pedance o f t h e c a v i t y i s e q u a l to
zero .
T h is a s s u m p t io n l e a d s t o a r e s o n a n t c o n d i t i o n .
( 2 ) We s o l v e t h e wave e q u a t i o n f o r t h e E - f i e l d s i n t h e c a v i t y , and
assum e t h a t t h e E - f i e l d a t one end o f t h e c a v i t y i s a minimum.
T h is
a s s u m p t io n l e a d s t o a n o t h e r r e s o n a n t c o n d i t i o n .
( 3 ) We assume t h a t t h e r e f l e c t i o n c o e f f i c i e n t o f t h e c a v i t y as
o b s e r v e d i n t h e c o n n e c t i n g w a v e g u id e i s a minimum a t r e s o n a n c e .
T h is
lea d s to a t h ir d reso n a n t c o n d itio n .
We f i n d t h a t t h e r e s o n a n t c o n d i t i o n o b t a i n e d by m ethod ( 1 ) i s
e x a c t l y t h e same a s o b t a i n e d by m ethod ( 2 ) .
When t h e c o u p l i n g i r i s
is
s m a l l , i . e . , t h e c a s e o f a v e r y u n d e r c o u p le d c a v i t y , t h e r e s o n a n t con­
d i t i o n o b t a i n e d by m eth o d s (1 ) and ( 2 ) w i l l a p p ro a c h t h e c o n d i t i o n
o b t a i n e d by m ethod ( 3 ) .
The c o n d i t i o n s f o r c r i t i c a l c o u p l i n g a r e
d i s c u s s e d when m ethod ( 3 ) i s p r e s e n t e d .
The q u a l i t y f a c t o r o f t h e p a r a l l e l p l a t e c a v i t y p a r t i a l l y
w ith a lo s s y sla b is d e riv e d .
fille d
In t h e d e r i v a t i o n , we assume t h a t a l l
l o s s e s a r e c o n t a i n e d i n t h e s a m p le .
15
I I I . 2.
The R e s o n a n t C o n d i t i o n
( I s o t r o p i c Ca se )
Horner e t a l . ^ 3 o r i g i n a l l y o b tain e d the re so n a n t c o n d itio n of a
cavity p a r t i a l ly
the fie ld s
f i l l e d w ith a lo s s le s s
s a t i s f y t h e b o u n d a r y c o n d i t i o n s i n 1946.
P a r i s ^ m entioned t h a t
if
state excitation,
mum a t a c e r t a i n
a criterio n
w i l l be c h a n g e d .
frequency.
for resonance.
Under t h e s i n u s o i d a l
Paris
suggested th a t t h i s
c a v i t y by t h e
w i t h a low l o s s
f i r s t determ ine the input
f o r t h e l o s s l e s s c a s e and
c r i t e r i o n based
upon i m p e d a n c e .
im pe da nce Z^n '> ^ i n '
r e p re s e n ts the input
s e c tio n of lin e co n tain in g the s la b .
This is
resonance.
A comparison of th e r e s u l t s
in section I I I . 2 .3 .
th e ca-vity
E - fie ld near the coupling w all is
We c a l l t h i s a minimum c r i t e r i o n .
o b t a i n e d by t h e above m e t h o d s i s g i v e n
We a l s o d i s c u s s P a r i s '
( 4 ) The r e s o n a n t c o n d i t i o n s
not p h y s ic a lly r e a l i z a b l e
Then we s a y
t h e im pe da nce c r i t e r i o n .
( 3 ) We assume t h a t when t h e t o t a l
minimum, t h e c a v i t y i s a t
We
i m p e d a n ce o f a t r a n s m i s s i o n l i n e o f l e n g t h
i f t h e i m a g i n a r y p a r t o f t h e i n p u t im pe da nce i s z e r o ,
i s at resonance.
for a
a s s u m p t i o n d e v e l o p e d by Mihran^® and by u s .
1 d e t e r m i n e d by an e q u i v a l e n t
that
resonant conditions
f o l l o w i n g way.
( 2 ) We i n v e s t i g a t e a d i f f e r e n t
imp ed an ce o f a s h o r t
e f f e c t be u s e d as
criterio n .
( 1 ) We ex a m in e t h e work o f H o r n e r e t a l . ,
compare i t
steady-
But he d i d n o t d e r i v e a r e s o n a n t c o n d i t i o n
s e c t i o n , we i n v e s t i g a t e d i f f e r e n t
filled
I n 1964,
th e r e s p o n s e of the r e s o n a t o r reach es a r e l a t i v e maxi­
equation using t h i s
In t h i s
that
t h e s l a b i s l o s s y , th e r e s o n a n t c o n d i t i o n as
o b ta in e d by Horner e t a l .
partially
slab using the c r i t e r i o n
suggestion in g re a te r d e t a i l .
as s u g g e s t e d by t h e ab ove c r i t e r i a
are
s i n c e no e n e r g y i s c o u p l e d o u t o f t h e c a v i t y
16
nor i s the c a v ity e x c ite d e x t e r n a l l y .
A more r e a l i s t i c
a p p r o a c h would
be t o d e t e r m i n e t h e r e s o n a n c e by c a l c u l a t i n g t h e r e f l e c t i o n c o e f f i c i e n t
i n t h e c o n n e c t i n g wav eg ui de c o u p l e d t o t h e c a v i t y and a p p l y i n g t h e c r i ­
t e r i o n o f minimum r e f l e c t i o n c o e f f i c i e n t as c o r r e s p o n d i n g t o r e s o n a n c e .
The c o n d i t i o n s f o r c r i t i c a l c o u p l i n g a r e d i s c u s s e d and we a l s o c a l c u l a t e
t h e d i f f e r e n c e s b e tw e e n t h i s method and t h e above m e n t i o n e d m e t h o d s .
III.2 .1 .
The L o s s l e s s
The i d e a l p a r a l l e l
Case and The Low Loss A p p r o x i m a t io n
p late cavity
i s shown
in
F ig . 3 .1 .I f the slab
i s l o s s l e s s , we ca n s o l v e t h e wave e q u a t i o n s i n t h e c a v i t y , and t h e
s o l u t i o n s o f t h e wave e q u a t i o n s ca n be matched a t t h e t h r e e b o u n d a r i e s ;
t h e l e f t and r i g h t p e r f e c t m e t a l p l a n e and t h e b o u n d a ry b e tw e e n f r e e
s p a c e and t h e s l a b .
H o r n e r e t a l . ^ 3 showed t h a t t h e r e s o n a n t c o n d i t i o n
is*
t a n K0 10 _
K0
“
tan
Kd
^
K
where Kq and K a r e t h e
p r o p a g a tio n c o n s ta n t s in the f re e space
and
the slab reg io n s, r e s p e c ti v e ly .
When t h e s l a b i s l o s s y , t h e p r o p a g a t i o n c o n s t a n t K, i n t h e r i g h t
hand s i d e o f Eq. ( 3 . 1 ) becomes a complex number, b u t t h e l e f t hand s i d e
o f Eq. ( 3 . 1 )
s t i l l real,
so Eq. ( 3 . 1 ) c a n n o t h o l d i n t h e l o s s y c a s e .
o t h e r w o r d s , t h e t h r e e b o u n d a r i e s c a n n o t be matched s i m u l t a n e o u s l y by
t h e s o l u t i o n s o f t h e wave e q u a t i o n s .
In t h e low l o s s c a s e , Mihran^® u se d t h e f o l l o w i n g e q u a t i o n as t h e
resonant condition.
k
See Appendix B f o r t h e d e r i v a t i o n .
In
17
/
/
/
/
free
/
/
/
s la b
space
^ o , €o
/
/
/
/
i risI / .
✓
/
/
/
/
/
Z
z=0
— Iq
F i g . 3 . 1 . The g e o m e t r y o f a n i d e a l p a r a l l e l p l a t e c a v i t y .
/
/
/
/
/
/
/
/
s
s
/
■in
■in. ✓
s
\
\
s.
/
/
/
/
/*
/
/
/
/
'cr
s
\
F i g . 3 . 2 . The imp e d a n ce o f t h e p a r a l l e l p l a t e c a v i t y
18
(3.2)
N o t e t h a t t h e i m a g i n a r y p a r t o f t h e r i g h t hand s i d e o f E q . ( 3 . 1 ) ,
-Im(--— s — ) , w i l l n o t be e q u a l t o z e r o f o r a l o s s y s l a b , b e c a u s e
Z i n ' ) , where Zin ' i s t h e in p u t impedance o f th e
s l a b b a c k e d by a p e r f e c t c o n d u c t o r ,
and Re(Z£n ' )
4= 0 when t h e s l a b i s
lossy.
A l t e r n a t i v e l y , we c a n c o n s i d e r K, (K = 8 - j a )
assuming a «
P, a p p r o x i m a t e K b y P.
a s c o m p l e x , and
This lea d s to th e fo llo w in g
condition
t a n Kq I j,
tan
Pd
(3.3)
P
Ko
L a t e r i n t h i s c h a p t e r , we s h a l l d e r i v e two a d d i t i o n a l r e s o n a n t con­
d i t i o n s b a s e d on t h e im p e d a n c e m e th o d and t h e minimum E - f i e l d r e q u i r e ­
m ent.
T h e n , we s h a l l c om pa r e t h e f o u r r e s o n a n t c o n d i t i o n s .
We commence
w ith t h e impedance m ethod.
I I I . 2 .2.
The I m p e d a n c e Method
The f i e l d e q u a t i o n s c a n n o t s a t i s f y t h e b o u n d a r y c o n d i t i o n s
partially
filled
cavity.
the resonant c o n d itio n .
in the
So, we i n v e s t i g a t e a n o t h e r met ho d t o d e f i n e
As shown i n F i g . 3 . 2 , we c o n s i d e r t h e c a v i t y a s
a t r a n s m i s s i o n l i n e e q u i v a l e n t and c a l c u l a t e t h e i n p u t im p e d a n c e a t t h e
l e f t hand p l a n e .
We i g n o r e c o u p l i n g t o t h e c o n n e c t i n g w a v e g u i d e .
The
r e s o n a n t c o n d i t i o n i s o b t a i n e d by r e q u i r i n g t h a t th e i m a g in a r y p a r t o f
th e in p u t impedance be z e r o .
19
Then we h a v e , 99
(3.4)
z i n ' = J 11 t a n Kc*
Z j n ' c o s Kpl0 + j n Q s i n Kq10
i n ~ n°
(3.5)
cos Kol0 + j Z i n ' s i n KqIq
rir + j n j i s t h e impedance o f t h e s l a b
/ ^o .
nQ = »-— i s t h e impedance o f t h e f r e e sp a ce
eo
S u b s t i t u t i n g Eq. ( 3 . 4 ) i n t o ( 3 . 5 ) ,
j h t a n Kd + j n Q t a n Kq Iq
nQ - h t a n Kd t a n Kq I q
Then we r e q u i r e t h a t ^ ( Z ^ )
(3.6)
= 0 and t h e r e s u l t i n g e q u a t i o n becomes t h e
resonant condition
tan 2 K o l=
2n0 ( n r s i n 23d + ni s i n h 2 txd)
------- 2 _ E ----------------------]-------------------- r---------------------------------( 3 . 7 )
( n r 2 + Hjz ) ( c o s h 2ad - c o s 2 Sd) - q0 z ( c o s h 2ad + c os 23d)
E q u a t i o n ( 3 . 7 ) i s d e f i n e d a s t h e r e s o n a n t c o n d i t i o n by means o f t h e
impeda nce m et h o d .
I t i s shown i n Appendix C, E q . (3 7) r e d u c e s t o
Eq. ( 3 . 1 ) as t h e l o s s e s d i s a p p e a r .
I I I . 2.3.
The Minimum E - F i e l d Method
As d i s c u s s e d i n C h a p t e r I I , a wave p r o p a g a t i n g i n an a n i s o t r o p i c
m a t e r i a l ca n be vi ew ed a s t h e sum o f two c o u n t e r r o t a t i n g c i r c u l a r l y
p o l a r i z e d waves w i t h d i f f e r e n t p r o p a g a t i o n c o n s t a n t s .
Since our goal i s
t o s t u d y a c a v i t y w i t h an a n i s o t r o p i c m a t e r i a l , we s e e k a r e s o n a n t con­
d i t i o n d e r i v e d from c o n c e p t s which a r e n a t u r a l
f o r wave p r o p a g a t i o n .
t h e b e g i n n i n g , we examine t h i s pr obl em f o r t h e i s o t r o p i c c a s e .
In
20
We s h a l l
i n c l u d e th e e f f e c t s o f c o u p lin g to th e m e a s u r in g waveguide
in Section I I I . 2 .4 .
The t o t a l E - f i e l d a r e l i s t e d
d a r y between r e g i o n s
1 and 2 i s a p e r f e c t m e t a l ,
coupling i r i s ,
o f which,
The b o u n ­
and i s c o n s i d e r e d w i t h a
P j 2 > P2 1 > T12 an<* T21 a r e d e f i n e d a s t h e r e f l e c ­
t i o n and t r a n s m i s s i o n c o e f f i c i e n t
tiv ely .
in Fig. 3 .3 .
from r e g i o n 1 t o 2 and .2 t o 1 , r e s p e c ­
The b o u n d a r y b e t w e e n r e g i o n s 2 and 3 i s t h e b o u n d a r y b e t w e e n
f r e e s p a c e and t h e s l a b , w he r e P23> P32> T23 anc* T32 a r e d e f i n e d .
4 is a p erfect m etal.
•
•
•
Now, t h e f o l l o w i n g e q u a t i o n s c a n b e w r i t t e n .
'ic
B = P l 2^ + T2 i E e j K o l °
( 3 . 8 . a)
C = T12A + P21Ee^ K° 1(>
( 3 . 8 .b)
D = ce-JM 0
(3.8.c)
„ -jKd
E = P23D + T32He
(3 .8 .d )
„ -jKd
+ p32He J
(3.8.e)
F =
t 23 d
G = F e " i Kd
(3 .8 . f )
H = -G
(3.8.g)
Eq. ( 3 .8 .c )
t o Eq.
_D_ _ e “ j Kq -*-0
C
( 3 . 8 . g ) , we h a v e
( 3 . 9 . a)
E
C
p e - ^ o
(JmK
(3.9.b)
F _
C
tme
(3.9.c)
- ^ o 1*
-£-=
*
Region
See r e f e r e n c e 1 0 3 , p . 5 6 3 .
(3 .9 .d)
21
* c
F i g . 3 . 3 . The E - f i e l d i n t h e p a r a l l e l p l a t e c a v i t y .
22
( 3 - , -e)
- % e - j K 0 l „ e- j K d
where
p9*s — g“ J 2Kd
= — ----- - — r^v,T
1 - p23 e“ j2Kd
t
.
.
.
rs the t o t a l r e f l e c t i o n c o e f f i c i e n t at
the 2/3 i n t e r f a c e
(3.10)
1 + P23
.
.
.
. .
T_ = —----------------------------, i s t h e t o t a l t r a n s m i s s i o n c o e f f i c i e n t a t
1 - p23e “ j 2Kd
t h e 2/ 3 i n t e r f a c e
p23 =
03 - h 0
h3 + hQ
h
-
h Q
h + hQ
Eq. ( 3 . 1 0 ) i s t h e t o t a l r e f l e c t i o n c o e f f i c i e n t when m u l t i p l e r e f l e c ­
tio n s are p r e s e n t.
large,
so t h a t d >
I f the th ic k n e ss of the s la b , d, i s s u f f i c i e n t l y
> 6 , where 6 i s t h e s k i n d e p t h , t h e t o t a l r e f l e c t i o n
c o e f f i c i e n t can be assumed as o r i g i n a t i n g e n t i r e l y from t h e f r o n t s u r f a c e
reflectio n .!^
The t o t a l E - f i e l d i n t h e r e g i o n 2, t h e f r e e s p a c e r e g i o n , ca n be
obtained,
E t o t a l = D e " jK° Z + EejK° Z
= C e " j K o l ° ( e " jK° Z + Pme jK° Z)
Le t pm = pmr + j p mj ,
(3.11)
then the t o t a l E - f i e ld at z = - l 0 is
I E t o t a l I z = - 10 = I C I [ ( 1 + PmrZ + Pmj2 ) + 2 pmr c o s 2K0 lo +
2pmj s i n 2K0 1 J 1 /2
This is the f i e l d to the r i g h t of the coupling w a ll.
the resonant
c o n d i t i o n as assuming from | Et o t a i | z _ ^
(3.12)
I f we d e f i n e
be a minimum,
we
23
s h a ll take the p a r t i a l d e r i v a ti v e of | E ^otal I z = - l o ^ t h
t h e o p e r a t i n g f r e q u e n c y and s e t i t e q u a l t o z e r o .
r e s p e c t t o w,
I t i s very d i f f i c u l t
t o g e t an a n s w e r , b e c a u s e a l m o s t a l l t e r m s , s uc h a s pm r , pmj , K0 , 0 , a ,
a r e f u n c t i o n s o f m.
So, we f i x w and d and c o n s i d e r lQa s t h e v a r i a b l e ,
and t a k e t h e p a r t i a l d e r i v a t i v e o f j Et o t a i | z = _ i Qw i t h r e s p e c t t o l 0 and
s e t i t equal to zero.
We hav e
8 1 Et o t a l 1 z = - l o _
91o
u .u ;
c | K ______________ -4Pmr s i n Kq10+ 4pmj c o s 2Kq10___________
E(1 + Pmr^ +
= q
+ ^Pmr c os 2Kol0 + 2 pmj s i n KqIJ
T h i s c o n d i t i o n i s an i m p o r t a n t o n e , s i n c e i t en c om pa s s es a l l m e a s u r e m e n ts
o f d i e l e c t r i c s i n TEgin c i r c u l a r c a v i t i e s w i t h a mov abl e end w a l l .
E xa pn di ng Eq. ( 3 . 1 3 ) , we o b t a i n t h e f o l l o w i n g e q u a t i o n .
t a n 2Knlr = —21.
■o r
p.mr
(3.14)
Eq. ( 3 . 1 4 ) i s d e f i n e d a s t h e r e s o n a n t c o n d i t i o n on l Qby means o f t h e
minimum E - f i e l d m et h od .
The f o l l o w i n g comments a r e i m p o r t a n t .
( 1 ) Eq. (14 ) i s e x a c l t y t h e same a s Eq. ( 3 . 7 ) , t h e r e s o n a n t e q u a t i o n
b y t h e impedance m et ho d.
The m a t h e m a t i c a l d e r i v a t i o n d e m o n s t r a t i n g t h i s
e q u a l i t y i s shown i n Appendix D.
t h a t the c o n d itio n s of ^ ( Z ^ )
F u r t h e r m o r e , i n Appendix E, we show
= 0 and V ( y be a minimum o f a l o a d e d
transm ission lin e are e q u iv a le n t.
The e q u a l i t y i s t r u e f o r any c o n d i t i o n
of loss.
(2)
I f t h e s l a b i s l o s s l e s s , t h e f o l l o w i n g can be s t a t e d .
Eq. ( 3 . 1 4 )
i s t h e same a s Eq. ( 3 . 7 ) and t h e Eq. ( 3 . 7 )
Since
i s t h e same as
24
Eq. ( 3 . 1 )
i n t h e l o s s l e s s c a s e , so E q . ( 3 . 1 4 ) ca n be used i n t h e
l o s s l e s s c a s e by s e t t i n g a = 0 and c o n s e q u e n t l y P23j = 0*
The impor­
t a n t r e s u l t i s t h a t t h e r e s o n a n t c o n d i t i o n , d e r i v e d on t h e b a s i s o f a
l o s s l e s s s l a b , on t h e b a s i s o f Iin(Zin) = 0 , and on t h e b a s i s t h a t
I ^to tal I
a minimum a r e e q u i v a l e n t .
(3)
P aris^
s a id t h a t in a l i n e a r system, th e n a t u r a l f re q u e n c ie s of
o s c i l l a t i o n are the poles of the t r a n s f e r fu n c tio n , or thezeros
denominator.
A c c o r d i n g l y i f D(jui) d e n o t e s t h i s d e n o m i n a t o r ,
of i t s
the n a tu r a l
fre q u e n c ie s of o s c i l l a t i o n are the ro o ts of the a lg e b ra ic equation
D(jw) = 0 and t h e r e s o n a n t f r e q u e n c i e s may be d e f i n e d by t h e r o o t s o f
the equation
- j £ - | DCj«) | - 0
'
(3.15)
T h e r e f o r e , h e s a i d t h a t by a n a l o g y , t h e r e s o n a n t f r e q u e n c i e s a r e
s o lu tio n s of the equation
- (u ^ + v^) = 0
dio
(3.16)
where u and v a r e d e f i n e d below
u + jv =
I\q
t a n Kol0 + - " ~ t a n Kd
(3.17)
Upon e x p a n d i n g t h e RHS o f Eq. ( 3 . 1 7 ) d e t e r m i n g u and v and s u b s t i t u t i n g
i n t o Eq. ( 3 . 1 6 ) we ha ve
d
[(
t a n KqIo
Kq
+
“ s i n h 2ad + 6 s i n 23d
( a2 + g2 ) ( COsti 2ad + cos 23d)
)
o
+
25
(
a i n 2 3d -__3 s i n h 2ad_______ ) 2 ] = 0
( a 2 + 82) ( c o s j1 2ad + c os 2(3d)
We can s e e t h a t i t
Eq. ( 3 . 1 8 ) .
(3.18)
i s d i f f i c u l t t o s o l v e t h e r e s o n a n t c o n d i t i o n from
So, we u s e t h e c o m p u te r t o s o l v e t h e r e s o n a n t f r e q u e n c y from
Eq. ( 3 . 1 8 ) and compare i t w i t h t h e r e s o n a n t f r e q u e n c i e s g i v e n by
Eq. ( 3 . 2 ) ,
( 3 . 3 ) and ( 3 . 1 4 ) .
The p a r a m e t e r s o f t h e s l a b s and t h e c a v i t y a r e shown i n t h e f o l l o w i n g
table:
e
o in/cm
*Hn m
\in m
1
20 eQ
10“ 2 t o 104
0 .0 001
0.0 16 667
2
20 e0
10- 4 t o 104
0.00 18 36
0.0 16 667
Sample
Table 3 .1 .
The p a r a m e t e r s o f t h e sa m pl e s and t h e c a v i t y .
The r e s u l t s a r e shown i n F i g . 3 . 4 and 3 . 5 .
1 i s much t h i n n e r t h a n a w a v e l e n g t h .
The t h i c k n e s s o f sample
As shown i n F i g . 3 . 4 , t h e r e s o n a n t
f r e q u e n c i e s g i v e n by M i h r a n ' s a p p r o x i m a t i o n and P a r i s '
approximation
h a v e v e r y s m a l l d i f f e r e n c e s w i t h t h e r e s o n a n t f r e q u e n c i e s g i v e n by t h e
impedance method and t h e minimum E - f i e l d m et h o d .
At t h e h i g h l o s s
r e g i o n , t h e r e s o n a n t f r e q u e n c y g i v e n by Eq. ( 3 . 3 ) , t h e a p p r o x i m a t i o n by
u s h a s an e m p h a t i c d i f f e r e n c e w i t h t h e r e s o n a n t f r e q u e n c i e s g i v e n by
o ther equations.
The t h i c k n e s s o f sample 2 i s a l m o s t e q u a l t o
In t h i s c a s e ,
the
s l a b w i l l a b s o r b more e n e r g y t h a n sample 1, so as shown i n F i g . 3 . 5 ,
t h e d i f f e r e n c e s a r e v e r y em ph a si ze d i n t h e h i g h l o s s r e g i o n .
(4)
Eq. ( 3 . 1 4 ) can be r e w r i t t e n as
L . GHz
lmp.& Min.E field app.
906
Mihran’s app.
Paris' app.
904
Our app.
9.02
9.0
8.88
8.86
-cm
F i g . 3 . 4 . The c o m p a r is o n o f r e s o n a n t f r e q u e n c i e s o b t a i n e d by E q . ( 3 . 2 ) ,
t
a n /1
t 'I
1 /. \
1 f> 4
H
H i ____ 1 -1
N3
On
fr.
GHz
Imp.& Min.E field App.
9.6
------------ Mihran's APP.
9.4
P aris' APP.
OurAPP.
9.2
8.8
8.6
8.4
“5
-4
“ 3
-2
“ 1
.
0
1
2
3
F i g . 3 . 5 . The c o m p a r is o n o f r e s o n a n t f r e q u e n c i e s o b t a i n e d by Eq. ( 3 . 2 ) ,
/ Q Q\ ry-rxA t Q 1/. \
c —on *.
/ ,- m - S n 1 I r \ __ ri = l //. 1=1 ft
28
2Kq1q"~* t a n CPj^j/ Pdj^)
mr
n
(3.18)
0 ) 1 j 2 } ■>«
We may t h e r e f o r e d e f i n e t h e r e s o n a n t c o n d i t i o n a s :
When t h e p h a s e s h i f t o f t h e wave t a k i n g a round t r i p
i n t h e empty
s p a c e o f t h e c a v i t y i s e q u a l t o t h e ph a s e a n g l e o f t h e t o t a l r e f l e c t i o n
c o e f f i c i e n t a t t h e i n t e r f a c e b e tw e e n t h e s l a b and empty s p a c e , t h e
c a v ity is at resonance.
(5)
Our c o n t r i b u t i o n i s t h a t we d e r i v e d two e q u a t i o n s , Eq. ( 3 . 7 )
and ( 3 . 1 4 ) , u n d e r d i f f e r e n t a s s u m p t i o n s , and t h e s e two e q u a t i o n s have
b e e n shown t o be e q u i v a l e n t and s i m p l e i n form.
I I I . 2.4.
The R e s on a nt C o n d i t i o n a s Obser ved i n t h e C o n n e c t i n g
Waveguide
T he re i s a d i f f e r e n c e be tw e e n t h e c a l c u l a t e d r e s o n a n t c o n d i t i o n i n
t h e c a v i t y and t h e r e s o n a n t c o n d i t i o n o b s e r v e d i n t h e c o n n e c t i n g wave­
guide.
We t h e r e f o r e s o l v e f o r a r e s o n a n c e as o b s e r v e d i n t h e c o n n e c t i n g
wa ve gui de and ca n s e e t h e e f f e c t o f t h e c o u p l i n g i r i s .
Th at c r i t e r i o n
f o r r e s o n a n c e i s t h e o b s e r v a t i o n o f a minimum i n t h e r e f l e c t i o n c o e f ­
ficien t.
We assume t h e s l a b i s a l o s s y i s o t r o p i c m a t e r i a l and t h a t
t h e r e a r e no w a l l l o s s e s .
The c o n f i g u r a t i o n i s t h e same a s i n F i g . 3 . 3 .
F o r c o n v e n i e n c e , we r e w r i t e Eq. ( 3 . 8 . a) t o Eq.
( 3 . 9 . e ) below.
B = p 1 2 A + T2 1 E e " j K o l o
( 3 . 8 . a)
C = t 1 2 A + p 2 1 E e _ ^K olo
(3.8.b)
D = c e - ^ 0 ^3
(3.8.c)
E = p 2 3 D + T3 2 H e- - ^ 0 ^t3
(3.8.d)
F = t 2 3 D + p3 2 He_ j K o l O
( 3 . 8 ,e)
29
G = Fe“ j Kd
( 3 . 8 . f)
H = -G
(3.8.g)
- 2 . = e " j Ko lo
( 3 . 9 . a)
"§■" Pme " j K o l °
( 3 . 9 .b)
» Tme " j Kolo
(3.9.c)
-£-=■ Tm e" j Kol<fe“ j Kd
-S .- - T ^ o V
(3.9.d)
^
(3.9 .e)
From Eq. ( 3 . 9 . b ) we ha v e
C - JL .
Pm
(3.19)
S u b s t i t u t i n g Eq. ( 3 . 1 9 ) i n t o Eq. ( 3 . 8 . b )
-1- e ^ o Pm
t 12A + p2 i E e " j K o l o
(3.20)
That i s
E - A -
. 1 PmXl2
e JK0 l 0 _ Pmp21e
S u b s t i t u t i n g Eq. ( 3 . 2 1 )
B = p 12A + A •
J Ko Lo
(3.21)
i n t o ( 3 . 8 . a)
^ 1 2 T2lP m^_ °
e JK0 l 0- pmp21e jK° o
(3>22)
B
The t o t a l r e f l e c t i o n c o e f f i c i e n t i s - r A
R
B
A
P l 2e
12K 1
° °+
Pm(T12T21 ~P12P21>
P m p 2 1
(3 23)
30
Assume t h e t h i c k n e s s o f t h e w a l l c o n t a i n i n g t h e i r i s
is very th in ,
so t h a t we may assume t h a t
1 + P12 = t 12
( 3 . 2 4 . a)
1 + P21 = t 21
(3.24.b)
p 12 = p21
(3.24.c)
Then
t 12t
21 ” p 12p21 = 1
+2 p i 2
S u b s t i t u t i n g Eq. ( 3 . 2 4 . c )
(3.25)
and ( 3 . 2 5 ) i n t o ( 3 . 2 3 ) we h a v e
^
p 12e j 2Kolo + (1 + 2P l 2 )Pm
R ---------- ^
e
............
(3.26)
~ pmp 12
L e t P 12 = Pr + J P j j Pm = pmr + JPmj> Ecl • ( 3 . 2 6 ) becomes
R = [ ( p r cos2Ko l 0 - P j s i n 2 K 0 l 0 + pmr + 2pmrpr - 2pmj p j )
+ j ( p j c o s 2 K 0 l0 + pr s i n 2K0 10 + pmj
[ (cos2K0 l 0 ~ Pr Pmr
Pj Pmj ^
+ 2 pr pmj
+2 p j p mr) ] /
j ( s i n 2 K o l 0 - PrPmj
—PjPmr^J
(3.27)
Eq. ( 3 . 2 7 ) i s t h e t o t a l r e f l e c t i o n c o e f f i c i e n t i n t h e wa ve gu id e a t
the w all of the i r i s .
The te r m p^2 i s t h e r e f l e c t i o n c o e f f i c i e n t o f t h e i r i s ,
it
is given
by*
p 12 - T T j B
(3 .2 8 .a)
t 12 = 1 + P1 2 = 2 J j ,
(3.28.b)
w h e r e , jB i s t h e n o r m a l i z e d s h u n t s u s c e p t a n c e i n t h e e q u i v a l e n t c i r c u i t
of the i r i s .
The v a l u e o f B de p e n d s upon t h e s i z e o f t h e i r i s ,
t h e w a v e g u i d e , and t h e g u i d e w a v e l e n g t h .
the dimension of
In o u r p r o b l e m , t h e unbounded
*See r e f e r e n c e 97, p p . 3 0 1 - 3 0 2 ; 104, p p . 1 8 5 -1 8 6 ; 106, p . 238.
31
c a s e , t h e c o n c e p t o f a s h u n t s u s c e p t a n c e can s t i l l be u s e d .
The v a l u e
o f B i n o ur p r o b le m t h e n c a n be c o n s i d e r e d as a f u n c t i o n o f t h e s i z e of
the i r i s
and t h e w a v e l e n g t h .
In e f f e c t , we a r e r e p l a c i n g t h e unbounded
s y s t e m by an e q u i v l a e n t t r a n s m i s s i o n l i n e .
The c o u p l i n g c o n d i t i o n s ,
( u n d e r c o u p l e d , c r i t i c a l c o u p l e d and o v e r c o u p l e d ) a r e r e l a t e d t o t h e s i z e
o f the i r i s .
We now p r e s e n t a d e v e lo p m en t which e s t a b l i s h e s t h i s
relation.
I f we wi de n t h e h o l e , we ca n f i n d a v a l u e o f B c o r r e s p o n d i n g t o c r i ­
t i c a l coupling.
The c o n d i t i o n s o f c r i t i c a l c o u p l i n g a r e t h a t b o t h th e
r e a l and i m a g i n a r y p a r t s o f R a r e e q u a l t o z e r o .
From E q . ( 3 . 2 7 ) ,
Rr and Rj a r e
Rr =
( p r " (Pmr2 + Pmj2 ^ 2 P r 2 + 2P j 2 + Pr)
+ COS2KQ1Q- [pmr( 2 P r + 1 - Pr 2 - P j 2 ) - 2pmj P j ]
+ s i n 2K0 10 • [pmj ( 2 p r + 1 - pr 2 - P j 2 ) + 2pm r p j ] } /
[ (c o s 2 K 0 l 0— pr pmr + p j p mj ) 2 + ( s i n 2 K 0 l Q- PrPmj
Rj =
{P j (1 + Pmr2
Pmj2 )
c o s 2K g y P m j ^ P r 2
Pj2
“P j P m r ) 2 J
2 Pr
(3.29)
■*■!)■*■ 2pmi-p j]
- s i n 2 K o ] J p mr( p r 2 + p j 2 + 2pr + 1) - 2pmj p j ] } /
[ ( c o s 2Kol0 - Pr Pmr
Pj Pm j ) 2
( s i n 2KQlQ— P^Pmj — PjPmr^2 !
(3.30)
I t w i l l be shown t h a t t h e i m a g i n a r y p a r t o f t h e i n p u t impedance o f
t h e c a v i t y a t t h e r i g h t s i d e o f t h e c o u p l i n g w a l l w i l l be an i n d u c t i v e
or a capacitive reactance.
When Im(Z£n ) i s an i n d u c t i v e r e a c t a n c e , a
c a p a c i t i v e i r i s ca n be used t o o b t a i n t h e c o n d i t i o n s o f Rr = 0 and Rj =
0.
I f Im( 2 i n )
used.
a c a p a c i t i v e r e a c t a n c e , an i n d u c t i v e i r i s must be
I f the c h a r a c t e r i s t i c s o f the i r i s ,
i.e .,
an i n d u c t i v e o r a c a pa ­
32
citiv e iris ,
i s g i v e n , we must ch oo se a s u i t a b l e l e n g t h o f l 0 t o c a u s e
Z i n ) be a c a p a c i t i v e o r an i n d u c t i v e r e a c t a n c e .
Then when we widen
t h e h o l e , we c a n o b s e r v e t h e c r i t i c a l c o u p l i n g .
The c o n d i t i o n s o f t h e c r i t i c a l c o u p l i n g can a l s o be d e f i n e d as
follows:
As shown i n F i g . 3 . 6 . a ,
i f we assume t h e n o r m a l i z e d i n p u t impedance
o f t h e l o s s y s l a b i s Z£n ' , t h e n o r m a l i z e d i n p u t impedance o f t h e c a v i t y
a t t h e r i g h t s i d e o f t h e c o u p l i n g w a l l i s Zin , and t h e n o r m a l i z e d i n p u t
impedance o f t h e c a v i t y a t t h e l e f t s i d e o f t h e c o u p l i n g w a l l i s Z-j, i t s
e q u i v a l e n t c i r c u i t can be shown i n F i g . 3 . 6 . b .
I f I^Z-jO = 0 and RqCZ^)
= 1, t h e s e a r e t h e c o n d i t i o n s f o r c r i t i c a l c o u p l i n g .
I t w i l l be shown
t h a t t h e s e c o n d i t i o n s a r e t h e same a s s e t t i n g Rj- = 0 , Rj = 0
from
Eq. ( 3 . 2 9 ) and ( 3 . 3 0 ) .
As shown i n Appendix F,
z - -■ » -» j t t - B x W )
(1-BX) 2 + (BR) 2
( 3 31)
where R + jX => Z jn , and
z i n ' + i t a nK0 l 0
in
1 + i zin'
zin ' =
t a n Ko1©
t a n KoU
From Eq. ( 3 . 3 1 ) , we e v a l u a t e Im(Zx) and Rg(Zx) and s e t ^ ( Z ^ ) = 0 Re (Zx)
= 1 , we ha ve
I ^ Z t) =
X ~
= 0
(1-BX) 2 + (BR) 2
( 3 . 3 2 . a)
/
/
J
/
ZT ,
Zin’
Zin
(a)
J'B
Zin
(b;
Fig. 3.6.
( a ) The i n p u t impeda nce o f t h e l o a d e d c a v i t y ,
(b) I t s e q u iv a l e n t c i r c u i t .
34
Re (ZT) = — -------- -s n ~ ~— T5------ = 1
(1-BX) 2 + (BR) 2
(3.32.b)
We ha ve t h e f o l l o w i n g c o n d i t i o n s f o r t h e c r i t i c a l c o u p l i n g ,
B = -'v X o
X2 + R2
( 3 . 3 3 . a)
R = (1 - BX) 2 + (BR) 2
(3.33.b)
Eq. ( 3 . 3 3 . a ) and ( 3 . 3 3 . b ) a r e t h e r e l a t i o n s t h a t B, X and R ha ve t o
s a t i s f y for c r i t i c a l coupling.
sen for c r i t i c a l c o u p lin g .
Note t h a t B c a n n o t be i n d e p e n d e n t l y cho­
We s h a l l p r e s e n t two e xa mp les t o show how t o
f i n d t h e v a l u e o f B which w i l l b r i n g t h e c a v i t y t o t h e c r i t i c a l c o u p l i n g
condition.
The p a r a m e t e r s o f t h e s e two ex amples a r e shown below
a in /S 2-cm
e
1
10
100
2
Table 3 .2 .
d in m
f i n GHZ
0 .0 01 86 2
9
variable
20 eQ
0 . 0 0 18 62
9
variable
The p a r a m e t e r s o f e x a m p l e s .
We c a l c u l a t e Z^n u s i n g d i f f e r e n t v a l u e s o f 1<>.
c i r c l e i n t h e Smith c h a r t ,
value of
l a t e B.
Using t h i s v a l u e o f B,
Note t h a t v a l u e s
f o r c e I ^ Z -p ) = 0 .
t o p o i n t A1 .
For each
We t h e n u s e Eq. ( 3 . 3 3 . a) and c a l c u ­
we t h e n c a l c u l a t e Re (Z-j) by means o f
o f B c a l c u l a t e d from E q . ( 3 . 3 3 . a) w i l l
So t h e l o c u s o f Re (Z'p) i s a l i n e i n t h e Smith c h a r t
corresponding to the r e s i s t a n c e a x is .
i n g t o 10= 0 .
The l o c u s o f Z£n i s a
and shown i n F i g s . 3 . 6 and 3 . 7 .
we c a l c u l a t e X and R.
Eq. ( 3 . 3 3 . b ) .
^0
o
0)
oCM
Sample
Here Z£n ' = Z£n ,
For e xa m pl e, p o i n t A c o r r e s p o n d ­
s e t t i n g B = 0.944, po in t A w ill t r a n s f e r
Then we r e p e a t f o r p o i n t B.
The f o l l o w i n g t a b l e g i v e s th e
F i g . 3 . 7 . The l o c i o f Zi n and Re (ZT> o f s a m p l e 1.
36
0
F i g . 3 . 8 . The l o c i o f Zi n and R£ (ZT ) o f samp le 2.
37
v a l u e o f \ j \ and B f o r e ach p o i n t
Points
A
B
R
yx
0
0.063
0.159
B
0.944
1.31
4.7
Remark
C
0 .2 3 9
D
S
0 .3 1 9
0 .4 15
4.7
1.31
E
0 .4 89
critical
coupling
critical
coupling
\ J \ and B f o r e ach p o i n t (exa m ple 1) .
T a b l e 3 .3
When t h e l o c u s
o f Re(Zx) m e e t s t h e c e n t e r ;
i.e .,
R^Z-p) = 1> and th e
c a v i t y i s a t t h e c r i t i c a l c o u p l i n g c a s e , we d e f i n e t h e s e v a l u e s o f B
and 1<j as Bc and l c .
The v a l u e s o f
and B f o r e ach p o i n t o f example 2
i s shown below
Po i n t s
A
R
B
yx
0
0 .0 2 8 4
0 .2 0 7
B
9.3
4.1
2 .2
Remark
C
0 .2 0 3 2
D
0 .2 7 9 4
0 .4 5 8
2 .2
4.1
critical
coupling
Table 3 .4 .
S
E
0. 4 9 3 2
critical
coupling
V A anc* B f ° r each p o i n t (exa m ple 2 ) .
The f o l l o w i n g d i s c u s s i o n can t e l l us more a b o u t t h e c r i t i c a l
coupling c o n d itio n :
(1)
S u b s t i t u t i n g v a l u e s o f Bq and 1q i n t o Eq. ( 3 . 2 9 ) and ( 3 . 3 0 ) , we
o b t a i n Rj. = 0, Rj = 0.
So t h e c o n d i t i o n s o f Re(Z-p) = 1.
e q u i v a l e n t t o t h e c o n d i t i o n s o f Rr = 0 , Rj = 0 .
im^T^ = ® are
38
(2)
From F i g . 3 . 7 and 3 . 8 we know t h a t Ini(Zin ) may be an i n d u c t i v e
or a cap acitiv e reactance.
The c o r r e s p o n d i n g c o u p l i n g i r i s which b r i n g s
the c a v ity to the c r i t i c a l coupling is c a p a c itiv e or in d u c tiv e ,
respectively.
(3)
I f we s u b s t i t u t e Eq. ( 3 . 3 3 . a) i n t o ( 3 . 3 3 . b ) and e l i m i n a t e B, we
have
X2
=
r
-
r
(3.34)
2
This e q u atio n p r e s e n ts a n e c e s s a ry c o n d itio n fo r ach iev in g c r i t i c a l
c o u p l i n g , and i s e q u i v a l e n t o f t h e y = 1 + j b c i r c l e .
shown i n F i g . 3 . 9 .
This c i r c l e i s
That means t h a t p o i n t s which a r e l o c a t e d on t h e
c i r c l e , ca n b e t r a n s f e r r e d t o t h e c e n t e r by a s u i t a b l e v a l u e o f B.
v a l u e o f B ca n be c a l c u l a t e d by Eq. ( 3 . 3 3 . a ) and ( 3 . 3 3 . b ) .
The
P o in ts not
l o c a t e d on t h e c i r c l e c a n n o t be t r a n s f e r r e d t o t h e c e n t e r .
(4)
From Eq. ( 3 . 3 3 . a ) and ( 3 . 3 4 ) t h e e q u a t i o n o f B ca n be r e w r i t t e n
as
(3.35)
Combing Eq. ( 3 . 3 4 ) and ( 3 . 3 5 ) we c a n c o n c l u d e .
l o c a t e d on t h e c i r c l e
shunt susceptance.
(5)
range,
= R -
r2 ,
Those p o i n t s , which a r e
ca n be t r a n s f e r r e d t o t h e c e n t e r by a
The v a l u e o f t h e s h u n t s u s c e p t a n c e i s
X
From Eq. ( 3 . 3 5 ) , we s e e t h a t t h e v a l u e s o f B h a v e a v e r y wide
f o r e x a m p l e , a s shown i n F i g . 3 . 8 .
39
0
F i g . 3 . 9 . The l o c u s o f X2 = R - R2 .
00
40
Point
R
X
Bc
A ( o r A*)
1 /2
1/2
( o r - 1/ 2 )
1
( o r - 1)
B ( o r B*)
0.01
0.0995
(o r -0.0995)
9.95
(or -9.95)
C ( o r C*)
0.95
0.218
0.229
(or -0.229)
Table 3 .5 .
P o i n t s on t h e x2 = R -
r2
c i r c l e and t h e i r Bq .
The r e s o n a n t c o n d i t i o n o c c u r s when t h e r e f l e c t i o n c o e f f i c i e n t
o b s e r v e d i n t h e c o n n e c t i n g wav eg ui de i s a minimum.
The m a g n i tu d e o f t h e
r e f l e c t i o n c o e f f i c i e n t , R, i s o b t a i n e d from Eq. ( 3 . 2 7 ) .
I t i s g i v e n as
| R | = [ ( p r c os 2Kq10 - pj s i n 2Kol0 + 2 Pm r Pr + Pmr “ 2pmj p j ) 2
i
+ ( pr s i n 2Kol0 + Pj c o s 2Kol0 + praj + 2 pmj p r + 2prarp j ) 2 ]
(3.36)
/
a
/
1/
*7
[ ( c o s 2Kol0 - Pr Pmr + P jP m j ) 2 + ( s ^n 2K0 10 - PrPmj “ P j P m r) 2 !
To f i n d t h e r e s o n a n t l e g n t h , l r , we t a k e t h e p a r t i a l d e r i v a t i v e o f
| R | w ith resp ect to
l 0 and s e t i t e q u a l
to zero.
We ha v e
A c os 2KQ1,. + B s i n 2Kolr + C = 0
(3.37)
where
A = Pmr^Pmr2 + Pmj2 ) ( 6 pj 3 + 6 pr 2 pj + 8 pr pj + 2 p j )
+ Pmj( pmr2 + Pmj2 ) ( 4 Pr 4 + 8 Pq2 P j 2 + 4 Pj 4 + 10 pr 3 + 10pr p j 2
+ 8 pr 2 + 2pr )
+ pm r ( 2 p j 3 +
2p j p r 2 -
+ Pmj ( 2 p r 3 + 4pr 2 + 4 p j 2 + 2pr +2pr p j2 )
2p j )
(3.38)
41
B = Pmj^Pmr2 + Pmj2 ) <fiPj3 6PjPr2 + 8 PrPj + 2Pj)
" Pmr^Prar2 + Pmj2 ^ 4 Pr4 + 8 pr 2 P j 2 + 4 p j 4 + 10 pr 3 + 10
+ 8 pr 2 + 2
”
C
+ pmj ( 2 p j 3 + 2pr 2 pj - 2 p j )
PY )
Pmr^2 Pr3 +
Pr P j 2
2pr p j 2 + 4pr 2 + 4 p j 2 + pr )
(3.39)
pmj 2 ) ( p r 2 + p j 2 + pr )
(3.40)
= 8 p j ( p mr2 +
S o l v i n g l r from Eq. ( 3 . 3 7 ) , we hav e
2K0 l r = - t a n ~ l -§■ - s i n “ l
c
— + qir,
B
/A2 + B2
Eq. ( 3 . 4 1 ) i s v e r y com pl ex .
be s im p lif ie d .
small i r i s ,
(3.41)
I f we make some a p p r o x i m a t i o n s , i t can
These a p p r o x i m a t i o n s a r i s e from t h e u s u a l c a s e o f a
l a r g e B, and u n d e r c o u p l e d c a v i t y .
0 1 2 - - 3&
= _(i
From Eq. ( 3 . 2 8 ) , we know
= (1
I f B i s very la r g e ,
p
q = 0,1,2,3, ...
( 3 ' 42>
we can make an e x p a n s i o n o f P i 2 «
jB
+-T-TO+
(jB ) 2
>
Th at i s
<3 -4 3 >
N e g l e c t i n g t h e h i g h e r t e r m s , we hav e
p12 = _ i + 2
12
J
-
4
(jB) 2
(3.44)
I n t e r m s o f r e a l and i m a g i n a r y p a r t s ,
P12 " Pr + JPj = ( “ I
Then
+ ]J~) +
(3.45)
42
T12 = 1 + P12 =~ 2 “
(3.46)
The n e c e s s a r y c o n d i t i o n s f o r t h e v a l i d i t y o f t h i s a p p r o x i m a t i o n a r e
that;
( 1 ) | P1 2 \ < l s (2 ) Upon s u b s t i t u t i o n , | r | s h o u l d be l e s s t h a n
unity.
When we u s e t h r e e te r m s o f t h e e x p a n s i o n i n Eq. ( 3 . 4 3 ) , b o t h two
c o n d i t i o n s , i . e . , | P12 I ^ 1 anc* I R I ^
o n ly use the f i r s t
2
- 1 + j —.
can
satisfied.
But i f we
two t e r m s o f t h e e x p a n s i o n i n Eq. ( 3 . 4 3 ) , t h e n p ^2 =
1
1
4
The v a l u e o f | p^2 I 1S 1 + —2 ’
When we s u b s t i t u t i e i t
*-s l a r Se r t h a n u n i t y .
in to Eq. ( 3 .3 6 ) , | R ) i s also l a r g e r than u n ity .
T h i s i s t h e r e a s o n t h a t we ch o o s e t h r e e t erm s o f t h e e x p a n s i o n o f p ^2 as
th e approxim ation eq u a tio n .
We s u b s t i t u t e Eq. ( 3 . 4 5 ) i n t o Eq. ( 3 . 3 8 ) , ( 3 . 3 9 ) and ( 3 . 4 0 ) , and
n e g l e c t t h e s e t er m s which c o n t a i n l /B ^ o r h i g h e r e x p o n e n t s on B.
We
t h e n hav e
A- = pm j ( pmr^ + Pmj^)
* ^“ ^ 2“ ^ +
^32” ^
(3.47)
B * Pmr <Pmr2 * Pmj2 > ' <-£§-> - % r < ~ p >
<3.4 8)
C = 0
(3.49)
S u b s t i t u t i n g Eq. ( 3 . 4 7 ) ,
( 3 . 4 8 ) and ( 3 . 4 9 ) i n t o Eq. ( 3 . 3 7 ) , we ha v e
pm:
Jmj c os 2Kolr - pmr s i n 2Kolr = 0
(3.50)
That i s
t a n Kolr = - ^ pmr
Eq. ( 3 . 5 1 ) i s t h e same as t h a t o b t a i n e d i n S e c t i o n I I I . 2 . 3 .
(3.51)
This
r e s u l t i s e q u i v a l e n t t o t h e v e r y u n d e r c o u p l e d c o n d i t i o n i n which v e r y
43
little
power i s d i s s i p a t e d i n t h e c o n n e c t i n g w a v e g u i d e .
view point,
i f the coupling i r i s
From o u r
i s small ( i n a n o th e r words, B i s l a r g e ) ,
t h e r e s o n a n t c o n d i t i o n which we m ea s u r e i n t h e c o n n e c t i n g wavegu ide i s
t h e same a s t h a t which we c a l c u l a t e i n t h e c a v i t y .
An e s t i m a t e was made o f t h e e r r o r i n i n t r o d u c i n g t h e u n d e r c o u p l e d
approxim ation.
We c a l c u l a t e l r from Eq. ( 3 . 4 1 ) , t h e g e n e r a l c a s e , and
Eq. ( 3 . 5 1 ) , t h e u n d e r c o u p l e d c a s e and compare them t o s e e t h e e r r o r s .
F i r s t , we assume pmr i s e q u a l t o - 0 . 2 t o - 0 . 9 9 , and pmj i s e q u a l t o
0.02 to 0 .1 .
For g i v e n
Pm r ,
Pm j
, we assume t h a t B = 0 . 2 5 , 0 . 5 ,
1, 2, 4,
8 , 16, 32, 64 , 128, o r t h a t we commence w i t h a l a r g e i r i s o r o v e r c o u p l e d
c a s e and p r o c e e d t o t h e u n d e r c o u p l e d c a s e .
the Fig. 3.10.
l r calculated
the i r i s ,
The r e s u l t s a r e p l o t t e d i n
From F i g . 3 . 1 0 , we know t h a t :
(1) th e e r r o r s of
from Eq. ( 3 . 4 1 ) and ( 3 . 5 1 ) depend o n l y upon t h e s i z e o f
i.e.,
t h e v a l u e s o f B.
I n t h i s e x m a p l e , we assume t h a t
Pmr c o v e r s a v e r y wide r a n g e , pmr = - 0 . 2 t o - 0 . 9 9 and i t
to note th a t th ere i s very l i t t l e
d ifferen ce in e rr o r .
is in te re stin g
In o t h e r w o r d s ,
th e e r r o r i s almost independent of the m a t e r i a l in th e c a v i t y .
(2)
When B > 30 , t h e e r r o r w i l l be l e s s t h a n 1%, t h e r e f o r e Eq. ( 3 . 5 1 ) can be
u s e d when B > 30.
We n e x t a p p l y t h i s e r r o r a n a l y s i s t o t h e c a s e s o f t h e t h i n and t h i c k
InSb s a m p l e . 95
The % e r r o r i n l r a r e c a l c u l a t e d from E q . ( 3 . 4 1 ) and
(3.51) are presented in Fig. 3.11.
The r e s u l t s a r e i d e n t i c a l t o t h o s e
p r e s e n t e d i n F i g . 3 . 1 0 and f o r t h e s e m a t e r i a l s we s e e e s s e n t i a l l y an
e x p a n s i o n o f t h e B = 0 t o B = 15.
Error,-4^"x10°
15%
mr
5%-
B,r24.42
B = 0.44
1%
20
30
40
60
80
100
120
Norm.Shunt
Susceptance
F i g . 3 . 1 0 . The e r r o r i n r e s o n a n t l e n g t h i n t r o d u c e by t h e E q . ( 3 . 3 7 ) and
( 3 . 5 1 ) , l a r g e B a p p r o x i m a t i o n , v e r s u s B.
■p-
-p~
Error*
Air
12 % -
10%
8%
12
F i g - 3 . 1 1 . The e r r o r o f 1
Norm.Shunt Sus.
v e r s u s B f o r t h e t h i n and t h i c k InS b sa m p le s
95
-o
Ul
46
I I I . 3.
The R e s o n a n t C o n d i t i o n ( A n i s o t r o p i c C a s e ) .
I f t h e s l a b i s c o n s i d e r e d a s an a n i s o t r o p i c m a t e r i a l , a s shown
i n F i g . 3 . 1 2 , t h e r e f l e c t e d wave i s an e l l i p i t c a l l y p o l a r i z e d wave, and
t h e wave i n t h e s l a b c a n b e c o n s i d e r e d a s t h e sum o f two c o u n t e r
r o t a t i n g c i r c u l a r l y p o l a r i z e d w aves.
The m o t i v a t i o n f o r t h i s p r o c e d u r e
o r i g i n a t e s in th e f a c t t h a t f o r a n is o tr o p ic m a te r ia l in t h i s o r ie n ­
t a t i o n , t h e n o rm a l modes a r e c o u n t e r r o t a t i n g c i r c u l a r l y p o l a r i z e d
fie ld s.
F or c o n v e n ie n c e i n s o l v i n g t h i s p r o b le m , we d i v i d e t h e i n c i d e n t
w ave, a l i n e a r l y p o l a r i z e d w ave, and t h e r e f l e c t e d w ave, i n t o two c i r ­
c u l a r l y p o l a r i z e d w a v e s.
T h e r e f o r e Eq. ( 3 . 8 . a ) t o ( 3 . 8 . g ) m ust be
r e w r i t t e n as
( 3 . 5 2 . a)
( 3 . 5 2 .b )
[(i+ jj)
+
(3 .5 2 .c )
(3.52.d)
/
/
/
/
/
/
/
/
/
/
\
\
\
\
\
\
\
\
Ei
,Et+
*
y
lo'
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
\
^dc
F i g . 3 . 1 2 . The c o n f i g u r a t i o n o f t h e c a v i t y and E - f i e l d .
■'j
48
F+
—
f\
A
A
= [ Y
+ - 2“ ( i - j j )
A
A
D_ A
A
( i + j i ) x 23+ + — C i - j J ) t 23_] +
bjT ( i +j j ) P 3 2 + e ^K+d + —
^i _ j j ^ P 3 2 - e ^K - d ]
(3 .5 2 .e )
*+
. a -jK + d
*\ -iK _ d
-T- ( l + j j ) + - ^ - ( i - j j ) = - r - ( i + j j ) e J + + -r— ( i - j j ) e J
(3 .5 2 .£ )
Hj.
a
a
H— a
a
G+
— (i+i i )
a
= ” 2”
a
G~ T
a
a
( i “j ^
( 3 . 5 2 ,g)
So, we d i v i d e t h i s p r o b le m i n t o two p r o b l e m s , one i s t h e p o s i t i v e c i r ­
c u l a r l y wave p r o b le m , t h e o t h e r i s t h e n e g a t i v e c i r c u l a r l y p o l a r i z e d
wave p r o b le m .
D±
~C± ~
E±
C±
S o l v i n g Eq. ( 3 . 5 2 . c )
t o ( 3 . 5 2 . g ) , we h a v e
e -X o '-o
( 3 . 5 3 . a)
a
(3.53.b)
e ^ o
F±
~c± = Tm±e
ii
+11 +1
0 |o
Tm±e
-jK0 l 0
(3.53.c)
-j K 0 l o _ j K+d
e
"
H±
~ C + ~ —xm±e
(3.53.d)
-j K 0 lo-jK±d
e
(3 . 5 3 . e )
w heree
- j 2K+d
P23± - e J ~
P m i * * " 1 ^ -------------------------------------------------------------------------------------- <3 -5 4 >
Pm± = 1 - P23±e - 32K±d
1 + PO'l +
T» ± = 1 = P2 3 ± e
-
We s h a l l p r o c e e d i n t h e f o l l o w i n g m a n n e r.
( 3 -55)
F i r s t we s h a l l u s e t h e
minimunm E - f i e l d method t o d e r i v e t h e r e s o n a n t c o n d i t i o n and d e r i v e t h e
49
r e s o n a n t c o n d i t i o n a s o b s e r v e d i n t h e c o n n e c t i n g w a v e g u id e .
Next we
s h a l l c a l c u l a t e t h e e r r o r s o f l r from t h e s e two r e s u l t s d e p e n d in g on t h e
v a lu e of the shunt su sc e p ta n c e ,
o r th e s iz e o f th e i r i s .
I n t h e b e g i n n i n g , we assum e
C+ = C_ ( s e e F i g . 3 . 3 ) , t h a t means we
assum e t h a t t h e i n c i d e n t wave i n t h e c a v i t y i s a l i n e a r l y p o l a r i z e d
w ave.
Hence D+ = D_ and t h e t o t a l E - f i e l d
re fle c tio n )
( b o t h i n c i d e n t and
in th e fre e space is
Et o t a l = De"^K° Z i + y e^K° Z [ ( i - j j ) p m+ + ( i - j j ) P m- ]
(3 .5 6 )
Now, d e f i n e t h e x -c o m p o n e n t o f t h e r e f l e c t e d wave a s E„ and t h e y com ponent a s EA.
is
T h is means t h a t t h e x-co m p o n e n t o f t h e r e f l e c t e d wave
i n t h e same d i r e c t i o n a s t h e p o l a r i z a t i o n o f t h e i n c i d e n t w ave.
i s shown i n F i g . 3 . 1 2 .
T h is
The t o t a l E - f i e l d i s
E t o t a l - ° ~ jK ° Z * * EejK° Z (^
4
-^
) I * Da i R» Zj
( 3. 57)
D e fin in g
The
1 /2
( p m+
+
p m_ )
=
p„ = p„r
+
jp „ j
( 3 . 5 8 . a)
1 /2
( p m+
-
p m_ )
=
Pj. = P j.r
+
jP j.j
(3 .5 8 .b )
m a g n i tu d e o f E t o t a l a t z = - I q , o r t h e l e f t hand w a l l , i s
I Et o t a l |
_ j ^ = I D I I d + Pur2 + P » j2
+pi r ^
+p , j 2 ) +
2 p „ r c o s 2K0 10 + 2 p „ j s i n 2Ko y 1^2
(3 .5 9 )
Take t h e p a r t i a l d e r i v a t i v e o f E q . ( 3 . 5 9 ) w i t h r e s p e c t t o l 0 , a n d s e t
i t equal to z e ro .
We t h e n have
t a n 2K0 l r = - ^ Pur
(3 .6 0 )
50
E q . ( 3 . 6 0 ) i s t h e r e s o n a n t c o n d i t i o n when t h e s l a b i s an a n i s o t r o p i c
m a te ria l.
An a s s u m p t io n i s b e i n g made t h a t i s n o t i m m e d i a t e l y o b v i o u s .
When we assum e t h a t E£ i s l i n e a r l y p o l a r i z e d , we a r e n e g l e c t i n g t h o s e
re fle c tio n s
from t h e c o u p l i n g w a l l which a r e o r t h o g o n a l l y o r i e n t e d o r i n
t h e y o r EA d i r e c t i o n .
I n e f f e c t , t h i s d e v e lo p m e n t assum es t h a t t h e
o r t h o g o n a l component o f t h e wave r e f l e c t e d
from t h e m a t e r i a l i s so s m a ll
com pared t o t h e p a r a l l e l o r X d i r e c t e d p o l a r i z a t i o n t h a t i t p r o d u c e s
o n l y a se co n d o r d e r e f f e c t and t h i s c a n be n e g l e c t e d .
b e t r u e f o r w e a k ly a n i s o t r o p i c m a t e r i a l .
T h is i s l i k e l y to
D e r i v i n g t h e r e s o n a n t con­
d i t i o n a s o b s e r v e d i n t h e c o n n e c t i n g w a veguide a u t o m a t i c a l l y t a k e s i n t o
a c c o u n t t h e r e f l e c t i o n s from t h e c o u p l i n g w a l l .
We now p r o c e e d w i t h
th is d e riv a tio n .
Assume t h a t we o n l y o b s e r v e t h e p a r a l l e l com ponents o f t h e e x c i t i n g
E -fie ld .
Then Eq. ( 3 . 5 2 . a ) must be r e w r i t t e n a s
(3 .6 1 )
where
S o l v i n g E q. ( 3 . 5 2 . b ) ,
( 3 . 5 3 . b ) and ( 3 . 6 1 ) we h a v e t h e f o l l o w i n g e q u a t i o n
f o r t h e r e f l e c t i o n c o e f f i c i e n t o f t h e c a v i t y a s o b s e r v e d i n t h e con­
n e c t i n g w a v e g u id e .
R„ -
n
i
Note t h a t i t i s s i m i l a r t o Eq. ( 3 . 2 6 ) .
P i 2e ^ 2K° l o + ( l +2Pl2)Pm+
■
+
P i 2 ^ 2Kol° + (1 + Pl2^Pm-\
—
)
(3.62)
51
The m a g n itu d e o f R,,, | R„ | i s much more c o m p l i c a t e d .
It
is d if f ic u lt
t o t a k e t h e p a r t i a l d e r i v a t i v e o f | R„ | w i t h r e s p e c t t o ^ a n d s o l v e f o r
lr .
C o n s e q u e n t l y , we c a l c u l a t e | R,, | t o o b t a i n l r and com pare i t w ith
th e v a lu e of l r o b ta in e d
The
The
from Eq. ( 3 . 6 0 ) .
two v a l u e s o f l r a r e p l o t t e d
a s a f u n c t i o n o f Hqc i-n F ig*
3 .1 3 .
c a l c u l a t i o n i s p e rf o r m e d f o r f o u r v a l u e s o f B:
(1)
The s l a b i s t h e t h i n InSb s a m p l e ^ a s we u s e d i n t h e l a s t
se ctio n .
(2)
The o p e r a t i n g f r e q u e n c y i s assum ed t o be c o n s t a n t and e q u a l t o
24 GHz.
(3)
For
f ie ld v aries
(4)
we
For
e a c h v a l u e o f B/Y0 , we assum e t h a t t h e s t a t i c m a g n e t ic
from 6 KG t o 10 KG.
th e g iven
s h u n t s u s c e p t a n c e and t h e s t a t i c m a g n e t i c f i e l d ,
c a l c u l a t e l r by t h e a p p r o x i m a t e v a l u e o f E q . ( 3 . 6 0 ) and by t h e e x a c t
v a l u e from E q . ( 3 . 6 2 ) .
I n t h e l a t t e r c a s e , we n u m e r i c a l l y e v a l u a t e
|
R„ | by c h o o s in g a d i f f e r e n t v a l u e o f lo .
|
R„ | i s c a l l e d l r .
(5)
The v a l u e o f l Qt h a t m in i m i z e s
The s h u n t s u s c e p t a n c e B/Y0 i s v a r i e d from 60 t o 180.
s e n ts a v e ry u n d erco u p led c a s e .
I t repre­
N ote t h a t t h e c u r v e c a l c u l a t e d by
E q. ( 3 . 6 0 ) w hich n e g l e c t s c o u p l i n g i s a l s o f o r a v e r y u n d e r c o u p l e d
c a v ity .
The d i s c r e p a n c y b e tw e e n t h e two c a l c u l a t i o n s i s q u i t e l a r g e i n
th e sen se t h a t th e d i f f e r e n c e in th e m ag n etic f i e l d r e q u ir e d to o b serv e
a maximum i n l r i s q u i t e l a r g e .
iso tro p ic case.
T h is r e s u l t i s d i f f e r e n t from t h e
T h e r e , we showed t h a t c o u p l i n g c a n b e n e g l e c t e d a s we
proceed to th e v e ry underco u p led l i m i t .
We presum e t h a t t h e i n a b i l i t y
1.29
•• • •
% •••
1.25
from.(Eq 3.62)
• « • • •—
1.23
from(Eq 3.6Q)
1.21
F i g . 3 . 1 3 . The 1 v e r s u s t h e s t a t i c m a g n e t i c f i e l d f o r t h e
t h i n JnSb s a m p le .
53
o f t h e two c a s e s t o a p p r o a c h ea ch o t h e r i n t h e l i m i t s o f v a n i s h i n g
c o u p l i n g i s due t o t h e n e g l e c t o f t h e o r t h o g o n a l r e f l e c t i o n s .
From F i g . 3 . 1 3 , we s e e t h a t t h e r e s u l t s a r e n o t g o o d .
v e ry underco u p led c a s e , th e e r r o r s a re s t i l l v e ry l a r g e
Even f o r t h e
Alr / l r = 2.5%,
t h e r e a s o n i s t h a t we d i d n o t c o n s i d e r t h e r e f l e c t i o n from t h e c o u p l i n g
w a l l when we d e r i v e d Eq. ( 3 . 6 0 ) .
The n e x t exam ple w i l l show t h a t i f t h e
s l a b i s more h e a v i l y doped and t h i c k e r , E q . ( 3 . 6 0 ) c a n s t i l l be u s e d .
I n t h i s e x a m p le , we u s e t h e . t h i c k InSb s a m p l e ^ a s we u s e d i n t h e l a s t
sectio n .
The r e s u l t s a r e p l o t t e d i n F i g . 3 . 1 4 .
b e tw e e n t h e v a l u e s o f l r c a l c u l a t e d
E q. ( 3 . 6 2 ) i s 0.3% .
H e r e , t h e maximum e r r o r
from Eq. ( 3 . 6 0 ) and e v a l u a t e d from
T h is v a l u e o f e r r o r i s a p p r o x i m a t e l y t h e same as
th e v a lu e o f e r r o r in S e c tio n I I I . 2, th e i s o t r o p ic c a se .
I t s h o u ld be n o t e d t h a t t h e t h i n sam p le e x h i b i t s m u l t i p l e r e f l e c ­
t i o n s and t h u s t h e r e i s a maximum i n l r .
The t h i c k sam ple h a s o n l y a
s i n g l e r e f l e c t i o n from t h e f r o n t s u r f a c e and t h u s i s much l e s s s e n s i t i v e
t o t h e m a g n e t ic f i e l d .
I I I . 4.
D iscu ssio n
We d e r i v e d t h r e e e q u a t i o n s w hich c o u ld be u sed t o d e f i n e t h e r e s o ­
n an t c o n d itio n o f a p a r a l l e l p l a t e c a v ity p a r t i a l l y f i l l e d w ith a lo s s y
s la b .
When t h e c a v i t y i s o v e r c o u p l e d , we h a v e t o c a l c u l a t e t h e r e s o n a n t
c o n d i t i o n a s o b s e r v e d i n t h e c o n n e c t i n g w a v e g u id e .
When t h e c a v i t y i s
u n d e r c o u p l e d , and t h e s l a b i s an i s o t r o p i c m a t e r i a l , t h e r e s o n a n t con­
d i t i o n o b t a i n e d by t h e im pedance m ethod and t h e minimum E - f i e l d method
c o u ld be u s e d .
I f t h e s l a b i s an a n i s o t r o p i c m a t e r i a l , we m ust ch o o se
th e e q u a tio n o f th e re s o n a n t c o n d itio n v e ry c a r e f u l l y .
The b e s t c h o ic e
54
lr .cm
1.29
1.27 ■
fromEq.3.60
^frB-140&180'
\0-ioo
from Eq.3-62
B*60
1.25-
1.23* ■
1.21
10
11
Hdc. KG
F i g . 3 . 1 4 . T he 1 v e r s u s t h e s t a t i c m a g n e t i c f i e l d f o r t h e t h i c k
In S b s a m p le .
55
i s c a l c u l a t i n g th e re s o n a n t c o n d itio n as o bserved in th e c o n n e c tin g
w a v e g u id e .
B e cause t h e e f f e c t o f t h e r e f l e c t i o n o f t h e c o u p l i n g w a l l i s
i m p o r t a n t f o r some c a s e s , t h e c o n d i t i o n o f t h e c r i t i c a l c o u p l i n g was
d iscu ssed .
T h is d i s c u s s i o n g i v e s us an i n d i c a t o r which shows t h e con­
d i t i o n o f th e c o u p lin g o f a c a v it y .
I I I . 5.
The Q u a l i t y F a c t o r
The q u a l i t y f a c t o r o f an e l e c t r o m a g n e t i c r e s o n a n t s y s te m can be
d e f i n e d as
0Jr *sum o f t h e tim e a v e r a g e d e l e c t r i c and m a g n e t ic e n e r g i e s
_ _ _______________ s t o r e d i n t h e e n t i r e volume__________________________
tim e a v e r a g e o f t h e power l o s s d u r i n g one c y c l e
^ ^3)
T h is d e f i n i t i o n i s a g e n e r a l d e f i n i t i o n o f t h e q u a l i t y f a c t o r o f a
r e s o n a n t s y s te m w hich i s e x c i t e d by a s t e a d y s t a t e s i n u s o i d a l e x c i t a t i o n
a t t h e r e s o n a n t f r e q u e n c y , ior .
As shown i n A ppendix G, t h i s d e f i n i t i o n
c a n n o t be u sed when a low-Q r e s o n a n t s y s te m i s e x c i t e d by a t r a n s i e n t
e x c ita tio n .
I n g e n e r a l , we a r e u s u a l l y i n t e r e s t e d i n a s t e a d y s t a t e
s i n u s o i d a l e x c i t a t i o n o f a r e s o n a n t s y s te m .
So, Eq. ( 3 . 6 3 ) c o u ld be
used.
The g e o m e try o f t h e p a r a l l e l p l a t e c a v i t y p a r t i a l l y f i l l e d w i t h a
s l a b i s shown i n F i g . 3 . 1 .
S in c e t h e p a r a l l e l p l a t e c a v i t y i s formed by
two i n f i n i t e p a r a l l e l m e t a l w a l l s , t h e e n e r g y s t o r e d p e r u n i t a r e a and
t h e a v e r a g e power l o s s p e r u n i t a r e a w i l l be u sed i n t h e d e v i a t i o n o f Q.
We f i r s t c a l c u l a t e t h e e n e r g y s t o r e d i n t h e c a v i t y by f i n d i n g t h e
e n ergy s to re d in th e e l e c t r i c
tic
fie ld .
f i e l d and t h e e n e r g y s t o r e d i n t h e magne­
We c a n t h e n d e t e r m i n e t h e tim e a v e r a g e s t o r e d e n e r g y .
In
t h i s m anner we need n o t c o n s i d e r an e q u i v a l e n t lumped c o n s t a n t c i r c u i t .
56
N e x t , we w i l l c a l c u l a t e t h e power l o s s by t h e P o g n t i n g th e o r e m .
We
assum e t h a t t h e p a r a l l e l m e t a l w a l l s a r e p e r f e c t c o n d u c t o r s , and t h e
s la b c o n ta in s a l l lo s s e s .
I I I . 5 .1 .
Hie I s o t r o p i c Case
In g e n e ra l, th e e l e c t r i c
f i e l d and t h e m a g n e t ic f i e l d w i l l be f u n c ­
t i o n s o f tim e and p o s i t i o n , i . e . , ¥ = E f ( x , y , z , t ) , H = H ( x , y , z , t ) .
S in c e
t im e d e p e n d e n c e i s s e p a r a b l e , t h e r e f o r e :
E ( x , y , z , t ) = E0 f ( x , y , z ) t g ( t )
( 3 . 6 4 . a)
H ( x , y , z , t ) = H0g ( x , y , z ) t H( t )
( 3 .6 4 .b )
w here f and g r e p r e s e n t t h e v e c t o r mode f u n c t i o n s , t E and t g r e p r e s e n t
t h e tim e f u n c t i o n s and w i l l be s i n u s o i d a l f u n c t i o n s when t h e s y s te m i s
e x c i t e d by a s i n u s o i d a l s t e a d y s t a t e e x c i t a t i o n .
The sum o f t h e e l e c t r i c and m a g n e t ic e n e r g i e s s t o r e d i n t h e e n t i r e
volum e a r e
U = UE + UH = f - ^ r
J J v I E ( x ,y ,z ,t ) I 2 dvdt
X
+ y°Y J J V I H K x , y , z , t )
| 2 dvdt
(3 .6 5 )
o r w ith v a r i a b le s e p a ra te
U =1
T
+ -jf Y
J qK
1 I Eo ! 2 *1 f ( x , y , z ) | 2 d v ] t E2 ( t ) d t
X
2
/ 0 / v 1 I Ho I ' | g ( x , y , z ) I 2 d v ] t H2 ( t ) d t
(3.66)
57
H e r e , we a r e a ssu m in g t h a t t h e c a v i t y i s u n i f o r m l y e x c i t e d .
t h e r e i s no l o s s i n t h e a i r s e c t i o n o f t h e c a v i t y .
T h a t means
I t w i l l be t r u e i f
t h e l e n g t h o f t h e a i r s e c t i o n i s n o t v e r y l a r g e when com pared w i t h t h e
w a v e l e n g t h o f o p e r a t i n g w ave.
wave a n a l y s i s ,
F o r a p a r a l l e l p l a t e c a v i t y , and p l a n e
f and g a r e n o t f u n c t i o n s o f x and y , t h e e n e r g y s t o r e d
p e r u n i t a r e a i s g i v e n by
T z2
U = U/(AxAy) = - | y
+
J0 f z I
I Eo I 2 • I
J 0 J 21 tl H0 I 2 *1
From A ppendix H, t h e e l e c t r i c
I 2 d z ] t E2( t )
dt
I 2d z ] t H2 ( t ) d t
(3 .6 7 )
f i e l d s and t h e m a g n e t ic f i e l d s i n t h e
f r e e s p a c e r e g i o n and i n t h e s l a b a r e
~
v
. . -jK 0 z
jK0 z j u t
E f ( z , t ) = RetE0 ( e
+ pme
)e
}
(3 .6 8 )
Es ( z , t )
B
s i n K ( z - d ) e ^ U)t }
(3 .6 9 )
. Eo / “ jK 0 z
jK0 z , j u t
H f ( z , t ) = Rg{
(e
- pme
)e
}
%
(3 .7 0 )
/)
„
~E0 (1+ Pm) -K
Hs ( z , t ) - Rgt
g in M
(3 .7 1 )
= Rq {-E0 — :—
c
u s i n Kd
ju t}
where t h e symbol (~ ) h a s b e e n p l a c e d o v e r t h e t i m e - v a r y i n g q u a n t i t y o r
in s ta n ta n e o u s v a lu e to d i s t i n g u i s h i t
from t h e p h a s o r q u a n t i t y .
The
s u b s c r i p t s f and s r e f e r t o t h e f r e e s p a c e and s l a b r e g i o n s ,
re sp e c tiv e ly .
E q . ( 3 . 6 8 ) t o ( 3 . 7 1 ) may be e x p r e s s e d as
E f(z ,t)
= | E0 ( e J °
+ pme J ° ) | c o s ( uit + <j>Ef )
(3 .7 2 )
58
1 + P
1ss ( z , t ) = | - E 0 — :—
. s i n i\u
H f(z ,t) = |
Hs ( z , t )
0 (e ^ °
ho
s i n K ( z - d ) | c o s (mt + <j>KS)
- Pme ^ K° Z) | c o s (cot + <J>nf)
= | " 'j ^ s i n ^ d
K C08 K (z" d ) I c o s (a)t + *Hs>
(3 .7 3 )
(3 .7 4 )
(3 .7 5 )
where
-jK 0 z
t(>Ef i s
th e
a r g u m e n t o f E0 ( e
^>gs
th e
a r g u m e n t o f ~ ^ o " 3 £n
is
■
.
.
4>Hf 1S t h e
jK 0 z
+ pme
1 + Pm
„
. Eo
argu m en t o f
.
(e
~ jK 0 z
)
s ^n K ( z - d )
-
prae
jK 0 z .
)
ho
- E ^ ( l + Pm)*K
<|>H
ns
is
th e
argu m en t o f
&
r------- :— =r:— c o s
jo ip s i n Kd
K (z -d )
N ote t h a t , a t any i n s t a n t , a t z = 0 , E f = Es , Hf = Hs a s p ro v e d i n
A p pendix H.
T hese e q u a l i t i e s r e p r e s e n t t h e c o n t i n u i t y b o u n d a ry
c o n d itio n s.
S u b s t i t u t i n g Eq. ( 3 . 7 2 ) t o ( 3 . 7 5 ) i n t o ( 3 . 6 7 ) , we c a n c a l c u l a t e :
(1)
t h e tim e a v e r a g e d e n e r g y s t o r e d i n t h e E - f i e l d i n t h e f r e e
s p a c e r e g i o n , U g f,
( 2 ) t h e tim e a v e r a g e d e n e r g y s t o r e d i n t h e E - f i e l d i n t h e s l a b , Ug8 ,
(3)
t h e tim e a v e r a g e d e n e r g y s t o r e d i n t h e H - f i e l d i n t h e f r e e
s p a c e r e g i o n , Ujjf,
(4)
t h e tim e a v e r a g e d e n e r g y s t o r e d i n t h e H - f i e l d i n t h e s l a b ,
uHs •
Then t h e t o t a l t im e a v e r a g e d e n e r g y s t o r e d i n t h e c a v i t y i s g i v e n by
u = UEf + UEa + UHf + UHg
(3 .7 6 )
59
T
uEf
= ' 2 ^r
o
*
J0 J - l J
I
Eo
I
2
( e
J
*
°
+
Pme ^ °
) | 2
dz
c o s 2 ( 0) t
+
< | > E f )d t
(3 .7 7 )
T
u Es = { {
d
Ja Jo
/
\
l " ^ s i n KdPm I 2 I s i n K ( z - d ) I 2 dz c o s 2 (uit + *Es> d t
(3 .7 8 )
M0 1
UHf “ T ' T
1 1
rT r° I E0 , 2 I, “ jK 0 z
jK 0 z , 2
n ,
/ n /_ i I
(e
" Pme
)
dz cos
( u t + <t>Hf^d t
°
le ti0
(3 .7 9 )
UHs = Y T
J o J o I ~ j ^ s i n ^ f < 2 I COS K (z_ d ) I 2 d t COs2 (a,t + ^
)dt
(3 .8 0 )
w h e re T i s t h e p e r i o d
6* i s t h e p e r m i t t i v i t y
A f t e r i n t e g r a t i n g , we h a v e
UEf m ^ f \ Eo | 2 [ (1 + | Pra I 2 ) • V -
Kq
( 1- c o s 2K o y + - ^ s i n
Kq
21^
(3 .8 1 )
,
uEs = ~
_
I Eo I 2 I d
%f = ~ 1 1
+ PmH
Eo I 2 f d
- i — s i n h 2ad - - i _ s i n
)
28d
c o s h 2ad - c o s 28d
+ l Pml 2) • l o - ^
d
- c o s 2K o t) -
(3 g2)
s i ^ K^
( 3 . 8 3 . a)
•
O
Po
*"^0
S i n c e 0o z = ---- , Ujjf becomes
£o
60
UHf = | 9 | E o l
2
[ d
+ l Pm I 2 )
‘ l o - * ® 3 ( 1_COS
2 K o t)
" ^
£ s in
2K° y
( 3 . 8 3 .b )
_ „2 + g2
Hs
2|
4 m2yQ
2 ^
°
m
^ sin h
s in l& d
**
c o s h 2ad - c o s 23d
^
34)
The t o t a l t im e a v e r a g e d e n e r g y s t o r e d i n t h e c a v i t y p e r u n i t a r e a i s
g i v e n by
e
D —r
I E<> 1 2 t o
* |pm 1 2 )
T a s i n h 2od - w
* - f I Eo 1 2 1 o
• y
s i n 2Sd
12
«2 .
„2 .
2
,
I
E0
|
Z
I
(
1 + Pm)
4 aj2p
c o s h 2ad - co s 23d
s i n h 2ctd +
+
-2
s in 2 3 d
(3 .8 5 )
c o s h 2ad - co s 23d
The second and t h i r d te r m s i n t h e r i g h t hand s i d e o f E q. ( 3 . 8 5 ) c a n be
c o m b in e d , and Eq. ( 3 . 8 5 ) becomes
T~\
E0 | 2
[(1
+|
Pm | 2 )
* y
+ - ^ " l
E 0 | 2 | (1
+
Pm)l 2
32 .
a2 .
— s m h 2ad + - r - s i n 23d
—--------------------- ®--------------c o s h 2ad - c o s 23d
*
(3 .8 6 )
I t i s i m p o r t a n t t o n o t e t h a t Ugf + Ujjf and Ugs $ Ujjs .
I f we d e f i n e
t h e r e s o n a n t c o n d i t i o n a s Im (Z in ) = 0 , ( S e c t i o n I I I . 2 . 2 ) , we c a n d e t e r ­
m ine a r e l a t i o n b e tw e e n t h e s e e n e r g i e s .
uE f + uEs = uHf +
h Hs
I t i s shown i n A ppendix I , t h a t
(3 .8 7 )
61
T h a t means t h e tim e a v e r a g e d e n e r g y s t o r e d i n t h e E - f i e l d i n t h e
c a v i t y i s eq u al to th e c o rre s p o n d in g en erg y s to r e d in th e H - f ie ld in th e
c a v i t y , under th e d e fin e d re s o n a n t c o n d itio n , ^ ( Z ^ )
a g r e e s w i t h t h e c o n c l u s i o n g i v e n i n A ppendix G.
= 0.
T h is r e s u l t
The t o t a l e n e r g i e s
s t o r e d i n t h e c a v i t y a r e t h e n g i v e n by
s i n 28d
(3 .8 8 )
c o s h 2ad - c o s 23d
When we d e f i n e t h e r e s o n a n t c o n d i t i o n b y o t h e r c r i t e r i a ( s u c h a s t h e
r e s o n a n c e a s o b s e r v e d i n t h e c o n n e c t i n g w a v e g u id e S e c t i o n I I I . 2 . 4 ) ,
E q . ( 3 . 8 7 ) c a n n o t b e u s e d , e x c e p t when t h e c a v i t y i s v e r y u n d e r c o u p l s d .
To d e t e r m i n e t h e tim e a v e r a g e power l o s s o v e r a c y c l e , we d e t e r m i n e
th e lo ss in th e s la b .
T h i s c a n b e a c c o m p li s h e d by e i t h e r d e t e r m i n i n g
t h e f i e l d s i n t h e s l a b and u s i n g
A v e ra g e power l o s s = Re [—■
E *J* dv]
(3 .8 9 )
d f by a p p l y i n g t h e P o y n t in g th e o r e m t o t h e volum e o f t h e s l a b and
show ing t h a t t h e l o s s c a n b e c a l c u l a t e d
from f i e l d s o v e r t h e s u r f a c e o f
t h e s l a b o r by
4 - Re I E x H* • d s =
Re / E • J * dv
2 ^ •'s
2 e Jv
(3 .9 0 )
D e t a i l s o f t h i s d e r i v a t i o n a r e i n A ppendix J .
The a v e r a g e power l o s s p e r u n i t a r e a i s
P = -|~ R e (E x
h* )
(3.91)
62
From Eq. ( 3 . 6 8 ) and ( 3 . 7 0 ) we know
JKq Z
(3 .9 2 )
)
jK0 z
(3 .9 3 )
)
So a t t h e i n t e r f a c e , z = 0 , t h e a v e r a g e power l o s s i s
i
E
Pz=0 “ I*"• R e [Eo (1 + pm> ~ Mq
(1 "
(3 .9 4 )
N o te t h a t i f (1 ) t h e s l a b i s l o s s l e s s , a = 0 , a = 0 , o r ( 2 ) t h e s l a b i s
t h e same a s a p e r f e c t m e t a l , a = °°, a = <*>, t h e m a g n i tu d e o f t h e t o t a l
r e f l e c t i o n c o e f f i c i e n t | pm | ^ = 1, so t h a t P = 0 .
S u b s t i t u t i n g Eq. ( 3 . 8 6 ) and ( 3 . 9 4 ) i n t o ( 3 . 6 3 ) , we h a v e t h e q u a l i t y
f a c t o r as
- £ ? s i n h 2ad + - ^ ? s i n 23d
_ a ______________ p___________■,
c o s h 2ad - c o s 28d
(3 .9 5 )
Eq. ( 3 . 9 5 ) i s a g e n e r a l e q u a t i o n o f t h e q u a l i t y f a c t o r o f a p a r a l l e l
p l a t e c a v it y p a r t i a l l y f i l l e d w ith a lo s s y s l a b .
We w i l l d i s c u s s t h i s
e q u a t i o n u n d e r two c a s e s :
( 1)
i.e ., d »
t h e t h i c k n e s s o f t h e s l a b i s much l a r g e r t h a n t h e s k i n d e p t h ,
6.
When d »
6 , t h e r e f l e c t i o n c o e f f i c i e n t o f t h e s l a b w i l l be dom ina­
t e d by t h e s u r f a c e r e f l e c t i o n ,
i.e .,
pm = p
63
/s: - / r
*
'1 ^ 7 7 ? ■ s
( 3 -96)
w h e re e = e ' - j s " , i s t h e com plex d i e l e c t r i c c o n s t a n t o f t h e s l a b .
The tim e a v e r a g e e n e r g y s t o r e d i n t h e f r e e s p a c e r e g i o n and i n t h e
s la b are
er
uf
- uE f + UHf “ Y I E° I
Us = UEs + UH s
[d + I ^ I ) * y
| E0 | 2 | (1 + pm) | 2 •
JL2s i n h 2ad + - g ? s i n 26d
a ______________ p__________
c o s h 2ad - c o s 20d
(3 .9 7 )
(3 .9 8 )
E q. ( 3 . 9 8 ) i s t h e sum o f E q . ( 3 . 8 2 ) and E q. ( 3 . 8 4 ) .
From Eq. ( 3 . 9 6 ) , t h e te r m s | 1 + pm | 2 a r e
I 1 * * 1* -
(3 - " >
w here | e | - / e ' 2 + s " 2
S in ce d »
2a
6 , t h e n ad »
(3 .1 0 0 )
1 , we h a v e
s i n h 2 ad ± - i ? s i n 28d
25
= 4 ta n h
c o s h 2ad - c o s 28d
2a
2od = 1 = 4 2a 2
(3 .1 0 1 )
S u b s t i t u t i n g E q. ( 3 . 1 0 0 ) i n t o ( 3 . 9 7 ) , s u b s t i t u t i n g Eq. ( 3 . 9 9 ) and
( 3 . 1 0 1 ) i n t o Eq. ( 3 . 8 2 ) and ( 3 . 8 4 ) , we h a v e
64
i
i 2
eo ^ e ’ + I e I )
Us=|E° i
T ^ T T r p -
5
'T
(3a03)
I t c a n be shown t h a t t h e e n e r g y s t o r e d i n t h e sa m p le i s s m a ll compared
to th e energy s to r e d in th e a i r .
eo + I e I
and i f l o »
F o r e x a m p le , n o t e t h a t
+I e I )
6 , we
(3 .1 0 4 )
h a v e Uf » Us .
The te r m 1 - | pm | 2 £s
l - k l » -
(3.105)
/e ^ + f e \ 2
The q u a l i t y f a c t o r i s o b t a i n e d by s u b s t i t u t i n g Eq. ( 3 . 1 0 2 ) and ( 3 . 1 0 5 ) i n t o
(3 .9 5 ), i t
Q -
is
u)r / p l ( e Q + / e ' 2 + e " 2 ) *10
r °
°
✓ 2( e ' + / e ' z + e"z)
(2)
(3 .1 0 6 )
The t h i c k n e s s o f t h e s l a b i s l e s s t h a n t h e s k i n d e p t h , i . e . ,
d } 6.
When d } 6 , i t
i s d i f f i c u l t to s im p lif y Eq. ( 3 .9 5 ) f o r a n o n -ze ro a .
When a = 0 , t h e power l o s s i n t h e s l a b i s z e r o .
s t o r e d i n t h e s l a b may n o t be n e g l e c t e d .
The tim e a v e r a g e d e n e r g y
The f o l l o w i n g d i s c u s s i o n i s g i v e n
under th e case o f a = 0 .
When
a = 0 , we h a v e
I 1 +
Pm
I
p ) 2 ( j ^ , ; ° s 2 6 d -) -
(3 .1 0 7 )
1 + | pm | 2 = - ^ - - P 2 ~ 2 p c o s2 3 d )
I 1 - p e " J 2Kd I 2
(3 .1 0 8 )
2 =
■
-
1 - p e - j 2Kd I 2
-
65
na2A
^
1 - c-o ds 2------3d
-j|-2s i n h 2otd + -£f?sin 23d
— --------------------c o s h 2ad ?------------------ c o s 23d
( 3 .1 0 9 )
S u b s t i t u t i n g Eq. ( 3 .1 0 8 ) i n t o ( 3 . 9 7 ) and s u b s t i t u t i n g Eq. ( 3 . 1 0 7 ) , ( 3 .1 0 9 )
i n t o ( 3 . 9 8 ) , we h a v e
Uf = | E j 2 •
s0 ' V
Us = | E0 | 2 •
e1 • d
1 * ^ - 2PCOs2gd| 1 - p e - j 2Kd | 2
( 3 .1 1 0 )
L + . .P2 + 2p
| 1 - p e - j 2Kd | 2
(3 .1 1 1 )
From Eq. ( 3 . 1 1 0 ) , we know t h a t
(a)
i f 23d =
Pf - 1*0 I 2 •
( 2 n + l ) i r , t h e n c o s 23d = - 1 , Uf becomes
• ! » • , 1 * "■ t
■■
| i _ pe-j2 K d | 2
The e n e r g y s t o r e d i n t h e s a m p le , Us , i s g i v e n b y Eq. ( 3 . 1 1 1 ) .
( 3 . 112)
Upon com­
p a r i n g Ug , Eq. ( 3 . 1 1 1 ) , and U f, E q . ( 3 . 1 1 2 ) , we s e e t h a t o n l y when s 0l 0 >>
e ' d , c a n we n e g l e c t t h e tim e a v e r a g e d e n e r g y s t o r e d i n t h e s l a b .
(b)
I f 23d = 2nir, t h e n c o s 2 3d = 1, Uf becomes
Pt - | E o | 2 - S o - V ^
^
]
T
“ •> »>
_
g 1
_
s i n c e 1 + p2 - 2p = — ( l + pz + 2 p ) , we c a n r e w r i t e Uf t o com pare w i t h
eo
Us »
66
C om paring Eq. ( 3 . 1 1 1 ) w i t h ( 3 . 1 1 4 ) , we s e e t h a t i f 1<j >> d , t h e tim e
av erag ed en erg y s to r e d in th e s la b can be n e g l e c t e d .
d iffe re n t
T h is r e s u l t i s
from t h a t i n c a s e ( a ) s i n c e t h e d i e l e c t r i c c o n s t a n t i s n o t
in v o lv e d in th e i n e q u a l i t y .
C a s e s ( a ) and (b ) a r e two e x tr e m e c a s e s .
d u c to r, s '
i s an o r d e r o f 10 eQ.
I f t h e s l a b i s a se m ic o n ­
In c o n c l u s i o n we c a n n o t n e g l e c t t h e
t im e a v e r a g e d e n e r g y s t o r e d i n t h e s l a b f o r c a l c u l a t i n g t h e Q when
m u ltip le r e f le c tio n s are p re s e n t.
I I I . 5 .2 .
The A n i s o t r o p i c C a s e .
I n t h i s s e c t i o n we s h a l l d e r i v e t h e e q u a t i o n f o r Q a s o b s e r v e d i n
th e c o u p lin g w aveguide.
We c o n t i n u e t o n e g l e c t e n e r g y
l o s t in th e
source.
S i n c e t h e e l e c t r o m a g n e t i c waves i n an a n i s o t r o p i c m a t e r i a l c a n be
assum ed a s t h e sum o f two c o u n t e r r o t a t i n g w a v e s , i t w i l l be c o n v e n i e n t
t o s e p a r a t e t h e i n c i d e n t wave and t h e r e f l e c t e d wave i n t o two c o u n t e r
r o t a t i n g w aves, r e s p e c t i v e l y .
As shown i n A ppendix J ,
t h e E - f i e l d s and
H - f i e l d s i n t h e f r e e s p a c e r e g i o n and i n t h e s l a b a r e
+ pm+E0+ ( i + j j ) e
E g(z,t)
— Rq { [ - E0 + ( l + p m+)
— ®o—^
+ Pm—^
s i n K _ ( z -d )
s i n K_d
( i - j j ) ] e J “t}
(3 .1 1 6 )
67
Hf ( z , t )
= Re {[ - ^ - E 0+ ( i + j j ) e “ jK° Z + 2 - E 0_ ( t - j j ) e " jK° Z
’•o
*o
+ 4 * Pra+Eo + ( i +j j ) e ^ K° Z "o
i E 0_pm_ ( i - j j ) e j K° Z] e j a)t}
r'o
(3 .1 1 7 )
• _
Hs ( z , t )
K + s in K + (z -d )
= R e { [ -E 0 + (1 + pm+)
my s i n K+d
a
* E° - < 1 *
a
(3 .1 1 8 )
Eq. ( 3 . 1 1 5 ) t o ( 3 . 1 1 8 ) c a n b e r e w r i t t e n a s
E ' f ( z . t ) = Re { [ [ E 0+ ( e " jK° Z + Pm+e j K° Z) + E0_ ( e " jK ° z + pm_ e jK° z ) ] i +
j [ E 0+ ( e - j K ° Z + P m + e ^ o 2) - E0_ ( e ' jK ° Z + pm_ e j K° Z) ] j ] e j “* }
(3 .1 1 9 )
^
s i n K + ( z -d )
, sin K -(z -d )
E3 ( 2*>t ) = Re { [ [ - E 0+ (1 + Pni+)
s inK+d
“ E0_ ( l + Pm- )
sinK_d
j [ - E 0_ ( l + Pm+)
s in K + ( z - d )
,
v sin K -(z -d ) a
.
+ E0_ ( l + Pm-)
.
",
]j]e
s m K +d
sm K _d
jmi.
}
(3 .1 2 0 )
H f ( z , t ) = R e { [ j [ - ^ - E 0+ ( e _jK ° Z - Pm+e jK ° Z) + j - E0_ ( e " jK° Z - pm_ e jK° Z) ] i +
n0
1>o
£ > ♦ (e-jK 0 z .
+i r
( e -jK „z _
jK0 z )]j5] , J » t }
(3.121)
68
^
,
K + s in K + (z -d )
K _ s in K _ ( z -d ) *
Hs ( z , t ) = Re ( [ j [ “ E0+ (1 + Pm +)jWM s i n
+ E0- ( l + Pm-> j wp s i n K_d ] l +
v K s i n K + (z -d )
K _ s in K _ ( z -d ) *
[E0+ ( l + Pm+> j u y g i n ^
+ E0- ( l + Pm-> j Uvi s i nK_d ] j J e
}
(3 .1 2 2 )
We c a n s e p a r a t e t h e
d id in s e c tio n
t im e f u n c t i o n s from Eq.
I I I . 5 . 1 , t h e n we
( 3 . 1 1 9 ) t o ( 3 . 1 2 2 ) a s we
s u b s t i t u t e the r e s u l t i n g e q u a tio n s in to
E q . ( 3 . 6 7 ) and i n t e g r a t e o v e r t h e d e f i n e d r a n g e o f z and t .
f i n a l l y , we
have,
UEf ~ Y
+ ^
I E°+ I
2 [ (1 + Ipm+ I 2 >
• 1° + - ^ ± (1 ” c o s 2Ko «
+ - | 2 | E0_ |
2 C(1 + | Pm- j 2 )
• lo + | ^ ~ (1 “ c o s 2K0 ]o) +
s i n 2Ko y
s in 2 K o y
( 3 .1 2 3 )
e+
UEs = ~2
,
.
+—
i
i 2 i /,
-j 2
' E°+ I I (1 + Pm+^l
x-L s i n h 2a+d - -jU- s i n 23+d
2a+
^ 2 3 +
c o s h 2a+d - co s 26+d
j
| 2 | ,
\ i 2
I Eo - I I (1 + Pm- ) |
4 — s i n h 2a_d - l d - s i n 23- d
2 a_
2 p_
c o s h 2ot_d - c o s 2 6 - d
UHf = ^ f - l E0+ | 2 [(1 + | Pm+ | 2 ) *
+ ^ - | E0_ | 2 [ ( 1 + |
pm_ | 2 ) • l o - ^
C1 - cos2KoV>
(1 - cos2K0 lo) “
-
(3 .1 2 4 )
sin2K o y
sin2K o y
( 3 .1 2 5 )
69
o L a ?
a+z + 3+* j
|
UHs =
2aj2p0
I E° + I
| ,,
v |
I (1 + pm+) I
2
4 — s i nh 2a+d + - iL - sin 26+ d
2a+
^
23+
c o s h 2a+d - cos23+d
2
a_2 + g_2
, .
o "2cL s in h 20ud + TBT S in 2 6 "d
1 E0_ | 2 ( 1 + pm_ ) 2 2a»2y0 1 0 - 1
m- i
c o s h 2 a_d - c o s 2 3 - d
(3 .1 2 6 )
The t o t a l e n e r g i e s s t o r e d i n t h e c a v i t y a r e
U - UE f + uEs + UHf + UHs
The a v e r a g e power l o s s i s
(3 .1 2 7 )
g i v e n by
P = - j R e ( E f x Hf * ) | z=Q
(3 .1 2 8 )
From Eq. ( 3 . 1 1 9 ) and ( 3 . 1 2 1 )
E f = [E0 + ( e
-jK 0 z
+ Pm+e
jK 0 z
) + E0_ ( e
-jK 0 z
+ pm_e
jK oz v ^
)]i
r
. ~j ^ o 2
jK oz v
_ , “ jK0 z
jR o z v-i.^
+ [E0+ ( e
+ Pm+e
) - E0_ ( e
+ Pm-e
)]jj
= Ex i + jE y j
r
Hf = t
(3 .1 2 9 )
1
/ “ j Ko z
J Ko z .
1 „ , “ i Ko z
j Ko z . , . ^
E0+ ( e
~ Pm+e
) + — E0_ ( e
- Pm§
)]ji
Ho
r,o
r 1 „
, “ i Ko z
+ [ — E0+ ( e
’'o
- Pm+e
J Ko z
1 _
, -jK o z
) + — E0_ ( e
Mo
= jHx i + Hyj
- Pm- e
jK0 z
A
)]j
(3 .1 3 0 )
I f x T f * = ExHy* + EyHx *
(3 .1 3 1 )
The a v e r a g e power l o s s p e r u n i t a r e a i s g i v e n by
* Re ( f x H*) | 2, 0 - - i- o [ | Eo t | 2 ( 1 - |
^
| 2) - | E0_ | 2 ( 1 - | % - | 2)
(3 .1 3 2 )
70
From A ppendix H, we know t h a t
I
Eq— | __ 1+ 1 P I
I Eo+ I ^
I Pm+ I
2 ( P f P m j + + PjPmr+^ s *’n
“ 2 ( P r P mr + “ P j P m j + ) C0S2K0 10 -
~ "l+ IP !Z I Pm- I ^ - 2 (P rPmr-"PjPmj-^c o s ^Eo^o“
2 ( p r pmj _+ pjp rar_ ) s i n Kq I q
-I c l 2
w here p = pr + j p r i s t h e r e f l e c t i o n c o e f f i c i e n t o f t h e c o u p l i n g i r i s .
The q u a l i t y f a c t o r i s
Q = T"i
j
( 1 - |pm+ |
5 1 2d
JL
|
I. 1
1 /,
(
|
) + | C | 2 <1 - | Pm- | z )
+ | pm_ | 2 ) • y
+ -1 ^
^ G°
" ^
I H - s i n h 2a- d " - f e . S in 23- d
2a)2y0
s i n h 2« , d + j j s - s i n 2 M
s i n h 2a - d
+ Pm
a +2 0+2
c o s h 2 a_d - c o s 23_d
c o s h 2 ot.d - c o s 2 B+d
^
* ■'•o +
| (1 + Pm+) I
s i n h 2a+d s i n 20+d
+
£- 1
l 2 i ,,
c o s h 2a+d - c o s 2 M
+ 2 ' ?
(
jl-
+ ^ Pra+ ^
a _2 ,
\ i
'
,
| (1 + Pm+)
S_2 ,
.
+ 2u^n0
*
+ j j f - s i n 2 3_d
c o s h 2a_d - cos 2(5_d
^
^
We w i l l d i s c u s s Eq. ( 3 . 1 3 4 ) by t h e f o l l o w i n g
134)
sta te m e n ts:
( 1 ) From S e c t i o n I I I . 5 . 1 , we know
(a)
if d »
can be n e g le c te d .
q _ a)r 11o eo t
5 and 10 >> 5, t h e tim e a v e r a g e e n e r g y s t o r e d i n t h e s l a b
Then t h e q u a l i t y f a c t o r becomes
+ I Pm+ I
(1
) * ^o* 1 £ I ( 1 +| Pm-
- | pm+ | 2 ) + | C| 2
(1 - | Pm- I 2 )
|
) * lp
(3 , 135)
71
(b)
the s la b .
if d «
6 , we c a n n o t n e g l e c t t h e tim e a v e r a g e e n e r g y s t o r e d i n
In many i n s t a n c e s , t h e n e g a t i v e l y p o l a r i z e d wave i s h i g h l y
a tt e n u a t e d (C h a p te r I I ) .
The t im e a v e r a g e e n e r g y i n t h e n e g a t i v e l y
p o l a r i z e d wave c a n b e n e g l e c t e d .
The q u a l i t y f a c t o r i s
Q --------- :------- ~ 2 ------- ^ 7 2 ---------:------ -- 2—
(1 - | pm+ |
) + | C|
(1 - | pm. |
| c | 2 ( i + 1 Pm_ | 2 ) • y
j L . s i n h 2a +d -
{eo [ d + I %+ I 2 ) * 1o+
+ - j 1 I Ci + Pm+) I 2 *
s i n 23+d
c o s h 2a+d - cos23+d
■2^-+ s i n h 2a +d -
)
a+2 + g+2
+
To
2aj2pc
1d
+ Pm+) I
s i n 20+d
^
c o s h 2a +d - c o s 20+d
( 2 ) I f t h e sam ple i s an i s o t r o p i c s e m i c o n d u c t o r w i t h an i n d u c e d a n i ­
s o t r o p y ( in d u c e d by a s t a t i c m a g n e t i c f i e l d ) t h e n o rm a l modes a r e
a f f e c t e d d i f f e r e n t l y by t h e m a g n e t i c f i e l d .
s i d e r a b l e d o u b t on
e x c ite d .
th e v a lu e o f
F o r e x a m p le , c o n s i d e r
T h i s r e s u l t c a s t s con­
a m ea su rem e n t o f Q when b o t h modes a r e
a c a v i t y w i t h l i n e a r p o l a r i z a t i o n and a
sam ple w i t h o u t any dc m a g n e t i c f i e l d .
The r e s o n a n c e c u r v e f o r a
r e f l e c t i o n c a v i t y would be a s shown i n F i g . 3 . 1 5 . a .
fie ld
i s in creased
mode i s d i f f e r e n t ,
th e reso n an t
When t h e m a g n e t ic
f r e q u e n c y f o r e a ch c i r c u l a r l y p o l a r i z e d
s i n c e 3+ + 3_ f o r Hqq 4= 0 .
As t h e m a g n e t i c f i e l d
is
i n c r e a s e d , we o b t a i n t h e s e q u e n c e o f o b s e r v a t i o n i n t h e c o n n e c t i n g wave­
g u i d e a s shown i n F i g . 3 . 1 5 . b t o 3 . 1 5 . d .
IPI
IPI
IPI
10
I.Oi
1.0
1.0
05
0.5
0.5
05
24-1
242
fr ,GHz
lb)
Et)=4KG
(1)
(2)
(3)
(4)
(5)
| P | i s t h e m a g n itu d e o f t h e r e f l e c t i o n c o e f f i c i e n t a s o b s e r v e d i n t h e c o n n e c t i n g
w a v e g u id e .
The n o r m a l i z e d s h u n t s u s c e p t a n c e i s assum ed t o b e B = 30.
The sam ple i s t h e t h i c k I n S b 9 5 } <}= 3 6 m ils and 3^ 1.25cm.
----------- r e p r e s e n t s a p o s i t i v e l y p o l a r i z e d i n c i d e n t wave.
-----------r e p r e s e n t s a n e g a t i v e l y p o l a r i z e d i n c i d e n t wave.
F i g . 3 . 1 5 . The d e g e n e r a c y o f t h e r e f l e c t i o n c o e f f i c i e n t w i t h d i f f e r e n t B0 .
N5
73
The i m p o r t a n t r e s u l t
is
th a t
th ere
i s no l o n g e r a s i m p l e r e l a ­
t i o n s h i p b e tw e e n t h e s h a p e o f t h e o b s e r v a t i o n and t h e c a l c u l a t i o n o f Q
from e n e r g y c o n s i d e r a t i o n s .
S o, i f we u s e a s i n g l e
fr
The e q u a t i o n Q = ^
no l o n g e r h o l d s .
c i r c u l a r l y p o l a r i z e d wave a s t h e i n c i d e n t
u)r
w a v e , we c a n o b s e r v e —— and c a l c u l a t e Q by t h e e n e r g y c o n s i d e r a t i o n .
More d e t a i l s o f t h e q u a l i t y
f a c t o r o f a r e s o n a n t c a v i t y w i l l be
d i s c u s s e d a t C h a p t e r IV , t h e t h i c k s a m p l e , and a t C h a p t e r V, t h e t h i n
sam p le.
74
CHAPTER IV.
THE COMMON CROSS POINT METHOD
I n t h i s c h a p t e r , we s u g g e s t a g r a p h i c a l m ethod w hich can be used t o
d e te rm in e th e p a ra m e te rs o f a t h i c k se m ico n d u cto r.
From E q . ( 3 . 1 0 6 ) , t h e q u a l i t y f a c t o r o f a p a r a l l e l p l a t e c a v i t y p a r ­
tia lly
f i l l e d w ith a t h i c k i s o t r o p i c s la b is *
i o _ / i n ( e 0 + / e '2 + e " z )
Q -
• 10
/,(« ■
. The q u a l i t y f a c t o r i s a f u n c t i o n o f u)r , l o , e '
and e " .
I f we f i x t h e
g e o m e t r y , and a p p l y a s t a t i c m a g n e t i c f i e l d , t h e s e m i c o n d u c t o r w i l l
become a n i s o t r o p i c and e + ' and e+" w i l l b e f u n c t i o n s o f Hjjq .
Then t h e
q u a l i t y f a c t o r i s a l s o a f u n c t i o n o f Hqq and i s d i f f e r e n t from Eq.
(4 .1 ).
As shown i n A p pendix L , i f we o n l y o b s e r v e t h e p a r a l l e l com­
ponent o f th e E - f i e ld , th e q u a lity f a c to r o f a p a r a l l e l p la te c a v ity
p a r t i a l l y f i l l e d w ith a t h i c k a n is o t r o p i c s la b i s
„ _ < V W 1 + I Rc I 2> * l o
Q "
,,
i - I Rc I 2
w here
P+ + CP-
* I f t h e s l a b i s s i m i l a r t o a good c o n d u c t o r , a s shown i n A ppendix K,
E q . ( 4 . 1 ) c a n be s i m p l i f i e d t o Q
nX
w here 6 i s t h e s k i n d e p t h .
T h is
e q u a t i o n i s t h e same a s t h e e q u a t i o n w hich i s shown i n r e f e r e n c e 102,
p . 176.
75
Eo? = ~ — i s t h e r a t i o o f t h e com plex a m p l i t u d e o f t h e n e g a t i v e l y
p o l a r i z e d wave t o t h e a m p l i t u d e o f t h e p o s i t i v e l y p o l a r i z e d
wave i n t h e f r e e s p a c e r e g i o n .
/e^ - /i+
P± =•
/e ^ + /e+
e+ = e+' - j e + "
We know t h a t , 1 ^ , ? , p+ a r e f u n c t i o n s o f u)p, v, HDC and t h e r e s o n a n t
f r e q u e n c y u^..
Q = Q(o)p,
So, Eq. ( 4 . 2 ) c a n be r e w r i t t e n as
v , Hd c , oir , t h e g e o m e try o f t h e c a v i t y )
(4 .3 )
F o r a f i x e d g e o m e try and a g i v e n s t a t i c m a g n e t ic f i e l d , we h a v e t h e
fo llo w in g r e l a t i o n
Q = Q(o>p,
v) | Hd c , ojj. a r e known
(4 .4 )
I f we m e a s u r e t h e Q 's a t d i f f e r e n t Hq^ , t h e n we h a v e
«i -
I % 01, M
Q2 - Q2 ( " p . v ) I „ DC2, ^
(4 .5 )
From t h e v ie w p o i n t o f t h e m a t h e m a t i c s , we c a n s o l v e (Dp and v from
Eq. ( 4 . 5 ) , b u t i t
is v ery c o m p lic a ted .
t o s o l v e t h i s p r o b le m .
So we s u g g e s t a g r a p h i c a l method
The f o l l o w i n g a r t i f i c i a l exam ple i s p r e s e n t e d t o
show t h i s g r a p h i c a l m eth o d .
Assume an InSb s a m p l e ^ and t h e r e s u l t s o f D.C. m e a s u r e m e n ts , t h e
p a r a m e t e r s o f t h i s sam ple and t h e c o r r e s p o n d i n g v a l u e s o f Up and v a r e :
76
m*
ss
0 .0 1 3 me
T a b le 4 . 1 .
p
1 8 .7 eQ
d
N
0 .0 0 0 6 5
0-cm
3 . 95x 10 ^
/ cm^
40 m i l s
V
“P
7 . 1 8 x l 0 13
rad /se c
5 . 5 6 x l 0 12
rad /se c
The p a r a m e t e r s o f t h e t h i c k InSb s a m p le .
(1)
U sin g Hj)c = 0 . 1 KG, we c a l c u l a t e Q v e r s u s v f o r d i f f e r e n t
v a l u e s o f (Op by Eq. ( 4 . 2 ) .
The r a n g e s o f u)p and v a r e c h o s e n n e a r th e
v a l u e s o f t h e r e s u l t s o f D.C. m ea u rem e n ts ( T a b le 4 . 1 ) .
v f o r d i f f e r e n t v a l u e s o f (Op i n F i g . 4 .1
We p l o t Q v e r s u s
I f t h e m e a s u re d Q i s 1392 a t
Hdc = 0 . 1 KG, t h a n from F i g . 4 . 1 , t h e p o s s i b l e v a l u e s o f oip and v a r e
shown i n T a b le 4 . 2 .
ojp x 101^
Q
hDC
0 .1 KG
v x 10l 2
8 .6 2 4
8 .9 8 4
9 .3 4 3
9 .7 0 3
5 .9 5
6 .4 5
6 .9 5 4
7 .5
8 .1
(2)
n
The v a l u e s o f oip and v a t HDC = 0 . 1
O'
T able 4 .2 .
1392
8 .2 6 4
1392.
F o l lo w in g t h e same p r o c e d u r e a s i n s t e p 1 f o r Hdc = 1 KG, 10 KG
and 20 KG r e s p e c t i v e l y , t h e r e s u l t a n t Q v e r s u s v c u r v e s a r e shown i n
F i g s . 4 . 2 , 4 . 3 and 4 . 4 .
HDc = 1 KG i s
1107.
We assumed t h a t t h e m ea su red v a l u e s o f Q f o r
1392; f o r HDC 10 KG, i t
i s 1288 and f o r HDC=20 KG, i t
U sing t h e same p r o c e d u r e s a s used i n s t e p 1, t h e r e s u l t a n t
c o r r e s p o n d i n g v a l u e s o f u)p and v a r e shown i n T a b le 4 . 3 .
is
77
1500
Q=1392
1300
9.703
9.344
1100
U)
= 8.984
X
10
8.624
8.264
900-
Fig. 4.1. Q versus V for different to
10
p
*10
when H, =0. 1 KG.
dc
78
1500
Q =1392
1300
9.703
9.344
1100
(J_=
8.984 X 10
8.624
8.264
900'
Fig. 4.2. Q versus V for different CJ
p
when H, = 1 KG.
dc
79
1500
1300-'
©©0
•Q=1288
9.703
9.344
0>_=8 584 X10
8.624
900
Fig. 4.3. Q versus V for different u
p
when H, = 10 KG.
dc
&0
81
Q
1 KG
1392
Q
hdc
10 KG
1288
Q
hdc
20 KG
T ab le 4 .3 .
(3)
8 .2 6 4
8 .6 2 4
8 .9 8 4
9 .3 4 3
9 .7 0 3
5 .9 5
6 .4 5
6 .9 5 4
7.5
8 .1
Up x 1013
.8.624
8 .9 8 4
9 .3 4 3
9 .7 0 3
v x 1012
6 .1 5
6 .9 5 4
7 .7 5
8 .5 5
Up x lO*3
8 .6 2 4
8 .9 8 4
9 .3 4 3
9 .7 0 3
v x 10^2
5 .7 5
6 .9 5 4
8 .3 3 4
9 .1 5 4
(Dp x 10*-3
HOC
1107
v X 1012
The v a l u e s o f Up and v a t Hqq = 1 KG, 10 KG and 20 KG
We p l o t uip v e r s u s v i n F i g . 4 . 5 , u s i n g t h e v a l u e s shown i n
T a b l e s 4 . 2 and 4 . 3 .
T hese c u r v e s h a v e one common c r o s s p o i n t .
oip = 8 .9 8 4 x 1013 r a d / s e c , v = 6 .9 4 5 x 1012 r a d / s e c .
It
is
I f th e v a lu e s of
m* and s g shown i n T a b le 4 . 1 a r e assumed c o r r e c t ,^5 t h e v a l u e s o f N and
pac w i l l b e : N = 4 . 4 2 x lO^-^/cm3 , pa c = 0 .0 0 0 7 2 6 J2-cm.
T h i s m ethod i s q u i t e s i m p l e .
S in c e t h e above exam ple i s an a r t i f i ­
c i a l p r o b le m , so a l l c u r v e s h a v e one common c r o s s p o i n t i n F i g . 4 . 5 .
In
a c t u a l c a s e , a l l c u r v e s may n o t have one common c r o s s p o i n t , b u t form a
sm all c i r c l e .
The c e n t e r o f t h e s m a l l c i r c l e g i v e s t h e v a l u e s o f Up and
v.
We use an a c t u a l exm aple t o exam ine t h i s m e th o d .
From R e f e r e n c e 9 5 ,
t h e p a r a m e t e r s o f DC m e a s u r e m e n t, T a b le 4 . 4 , and t h e q u a l i t y f a c t o r a t
d i f f e r e n t v a l u e s o f t h e s t a t i c m a g n e t ic f i e l d , T a b le 5 .5 a r e p r e s e n t e d
b e lo w :
82
10 . 0 '
-1 °
Hd c = 20 KG
9.0-
8.0
10 X10
F ig . 4 .5 .
oj
P
v e r s u s V f o r d i f f e r e n t H, .
QC
83
m
es
0 .0 1 3 mg
1 8 .7 eQ
T able 4 .4 .
p
N
8 x 10”6 f2-m
4 .9 5 x lO ^ /m S
The p a r a m e t e r s o f t h e t h i c k InSb s a m p l e . 95
Bo
6 . 7 KG
8 .3 7 5 KG
10 KG
Q
1755
1720
1685
T able 4 .5 .
M easured Q a t d i f f e r e n t v a l u e s o f B0 .
U sin g t h e same p r o c e d u r e s a s u s e d i n l a s t e x a m p le , we p l o t F i g . 4 . 6 ,
4 . 7 , 4 . 8 and 4 . 9 .
(1)
We s e e t h a t
From F i g . 4 . 9 , t h e s l o p e s o f t h e s e t h r e e Up v e r s u s v c u r v e s
have v ery l i t t l e
d iffe re n c e ,
so t h e common c r o s s p o i n t i s n o t c l e a r .
we can m e a s u re t h e q u a l i t y f a c t o r a t h i g h s t a t i c m a g n e t i c f i e l d s ,
If
(fo r
e x a m p le , a t l a s t e x a m p le , we u s e B0 = 20 KG), we w i l l h a v e some
Up v e r s u s v c u r v e s which s l o p e s w i l l h a v e l a r g e d i f f e r e n c e s .
Then t h e
common c r o s s p o i n t o r t h e a r e a form ed b y t h e c r o s s w i l l be c l e a r .
(2)
In t h i s e x a m p le , we h a v e Up = 9 .2 7 x 10 ^
rad /sec;
v = 1 2 .6 7 x
10^2 r a d / s e c , t h e c o r r e s p o n d i n g v a l u e s o f N and p a r e N = 6 .5 8 x
l O ^ / m ^ , p = 8 .6 9 x l O " ^ J2-m.
v a lu e has l i t t l e
The s c a t t e r i n g f a c t o r i s 1 . 3 3 .
T h is
d i f f e r e n c e when com pared t o t h e r e s u l t i n R e f e r e n c e
95.
(3)
I f t h e sam ple i s a t h i n s a m p le , i t w i l l be shown i n C h a p t e r V
t h a t t h e Q - f a c t o r v e r s u s Hqq i s a q u a s i - p e r i o d i c c u r v e .
S ince t h i s
&k
Q
Z\0°1
ed
Q
vr\©
©
©
- ©
- ©
9 .^3^
LAJ
a .tf
,©
s
9•
*A0'
8 .93
©
©
©
..n
►
•*1]
8-
viVve^ 3 o
itvt ^
■yeX S\iS
■giS*
.6.^
ion:
85
87
X 10
9.2
9-1
9.0
8-9
8.8
11.5
T i g . 4 .9 .
125
12
cj
P
v ersu s
v
for d iffe re n t B .
o
13.5 X 10
88
q u a s i - p e r i o d i c c h a r a c t e r i s t i c e x i s t e d , t h e ujp v e r s u s v c u r v e w i l l n o t be
a m o n o to n ic f u n c t i o n a t some c a s e s .
So, f o r a t h i n s a m p le , we do n o t
u s e t h i s m e th o d , t h e common c r o s s p o i n t m e th o d .
w i l l be d e v e lo p e d i n C h a p t e r V f o r a t h i n s a m p le .
A more p r e c i s e method
89
CHAPTER V.
THE RELATIONSHIP OF MAXIMA AND MINIMA OF QUALITY FACTOR AND
REFLECTION COEFFICIENT WITH VARIATION OF STATIC MAGNETIC
FIELD
V .l. In tro d u c tio n
F o r a f i x e d g e o m e t r y , i f we v a r y t h e s t a t i c m a g n e t ic f i e l d , Hpc» and
a d j u s t t h e o p e r a t i n g f r e q u e n c y t o r e s o n a n c e , we c a n o b t a i n a q u a l i t y
f a c t o r , Q, f o r e a c h m a g n e t ic f i e l d .
I f we p l o t t h e Q - v a lu e s v e r s u s Hq q ,
we w i l l s e e , i n some c a s e s , t h a t t h e Q - v a lu e s v e r s u s Hpc i s a m o n o to n ic
f u n c t i o n , and i n o t h e r c a s e s , i t
is a q u a si-p e rio d ic
fu n ctio n .
The
f o rm e r o c c u r s when t h e f r o n t s u r f a c e o f t h e sam p le d o m in a te s t h e r e f l e c ­
t i o n and t h e l a t t e r when m u l t i p l e r e f l e c t i o n s a r e p r e s e n t .
In C h a p t e r
IV, we s u g g e s t e d t h e common c r o s s p o i n t m ethod t o d e t e r m i n e t h e p a r a ­
m e t e r s o f t h e s e m i c o n d u c t o r f o r a t h i c k s a m p le .
In t h i s c h a p t e r , we w i l l
d i s c u s s t h e c a s e o f a t h i n s a m p l e , i . e . , when m u l t i p l e r e f l e c t i o n s a r e
p r e s e n t and when t h e Q - f a c t o r and t h e r e f l e c t i o n c o e f f i c i e n t v e r s u s
Hj)c i s a q u a s i - p e r i o d i c f u n c t i o n .
What i s t h e p e r i o d o f t h e Q - v a lu e s v e r s u s Hpc ( o r 0d) c u r v e ?
L e t us
u s e an a r t i f i c i a l exam ple t o exam ine t h i s p r o b le m .
(1)
III,
The g e o m e try o f t h e c a v i t y i s t h e same a s we u sed i n C h a p t e r
(F ig . 3 .1 ) .
Assume t h a t t h e c a v i t y i s u n d e r c o u p l e d , Eq. ( 3 . 1 4 )
h o l d s , and t a n 2X0 ^ =
Pmi / P mr c a n b e u s e d a s t h e r e s o n a n t c o n d i t i o n f o r
an und erco u p led c a v i t y .
(2)
Assume t h a t t h e d i e l e c t r i c c o n s t a n t o f t h e s l a b i s b e tw e e n
20 e 0 and 400 £0 and t h a t t h e c o n d u c t i v i t y i s 0.024/J2-cm , and t h e
90
t h i c k n e s s o f t h e sa m p le i s 0 .7 5 mm.
T hese v a l u e s i n s u r e a low l o s s
m u lt ip l y r e f l e c t i n g sam ple.
(3)
The o p e r a t i n g f r e q u e n c y i s 24 GHz, and we v a r y Iq t o c a u s e t h e
c a v ity to re s o n a te .
U s in g Eq. ( 3 . 9 5 ) , we c a l c u l a t e t h e q u a l i t y f a c t o r and p l o t i t v e r s u s
(3d i n F i g . 5 . 1 . a .
equal to
it.
We s e e t h a t t h e p e r i o d o f Q - v a lu e v e r s u s 0d i s a lm o s t
F u r t h e r m o r e , we p l o t t h e power l o s s i n t h e s l a b v e r s u s 0d,
( F i g . 5 . 1 . b ) , t h e e n e r g y s t o r e d i n t h e f r e e s p a c e r e g i o n and i n t h e s l a b
v e r s u s (3d ( F i g . 5 . 1 . c )
(3d, ( F i g . 5 . 1 . d ) .
and t h e t o t a l e n e r g y s t o r e d i n t h e c a v i t y v e r s u s
We s e e t h a t t h e p e r i o d i c c h a r a c t e r i s t i c s o f t h e Q v e r ­
s u s (3d c u r v e a r e d o m in a te d by t h e p e r i o d i c c h a r a c t e r i s t i c s o f t h e power
lo s s in the s la b .
We a l s o s e e t h a t t h e e n e r g y s t o r e d i n t h e s l a b c a n be
v e r y l a r g e when com pared w i t h t h e e n e r g y s t o r e d i n t h e f r e e s p a c e r e g i o n
f o r t h e c a s e when t h e t h i c k n e s s o f t h e s l a b i s an odd m u l t i p l e o f a
q u a r t e r w avelen g th .
I t a g re e s w ith th e d is c u s s io n in S e c tio n I I I . 5.
When we c h a n g e t h e v a l u e o f t h e c o n d u c t i v i t y , and c a l c u l a t e t h e
q u a l i t y f a c t o r , p l o t Q v e r s u s (3d, ( F i g . 5 . 2 . a , a = 0.016/J2-cm ,
F ig . 5 .2 .b ,
a
v a lu e s o f th e
= 0 .0 2 4 / f t - c m , F i g . 5 . 2 . c , a = 0 . 0 3 2 / f t - c m ) , we s e e t h a t t h e
q u a l i t y f a c t o r a r e c h a n g e d , b u t t h e v a l u e s o f 0d a t which
th e peaks o f Q o c c u rre d a r e n o t ch an g ed .
In t h e f o l l o w i n g s e c t i o n s , we w i l l exam ine t h i s p ro b le m a g a i n and
t r y to r e l a t e
t h e t r a n s p o r t p a r a m e t e r s o f t h e s e m i c o n d u c t o r to t h e p e r i o d
o f t h e q u a l i t y f a c t o r and t h e r e f l e c t i o n c o e f f i c i e n t v e r s u s (3d c u r v e .
I n S e c t i o n V .2 , we s t u d y t h e i s o t r o p i c c a s e , and t h e a n i s o t r o p i c c a s e i s
s t u d i e d i n S e c t i o n V .3 .
91
3000
2000-
1000
1
2
3
4
5
F ig . 5 .1 .a. Q v ersu s £d.
F i g . 5 . 1 . b . Power l o s s i n s l a b v e r s u s 0 d .
6
92
Energy
stored
in free space
in slab
c13
F i g . 5 . I . e . E n e rg y s t o r e d v e r s u s /3d.
Total
energy
stored
r13
F i g . 5 . 1 . d . T o t a l e n e r g y s t o r e d v e r s u s /9d.
93
3000
2000
10001
2000 +
1000 +
2000
1000
F i g . 5 . 2 . Q v e r s u s y3d, a ,
c,
= 0 .0 l6 /£ -c m , b ,
o = 0 .0 3 2 / 3 - c m .
o - 0 .0 2 4 /fi-c m ,
94
V .2 . The I s o t r o p i c Case
From t h e r e s u l t o f t h e a r t i f i c i a l exam ple i n t h e l a s t s e c t i o n , we
know t h a t t h e p e r i o d i c c h a r a c t e r i s t i c s o f t h e Q v e r s u s 3d c u r v e a r e
d o m in a te d by t h e p e r i o d i c c h a r a c t e r i s t i c s o f t h e power l o s s i n t h e s l a b ,
From Eq. ( 3 . 9 4 ) , we know t h a t t h e power l o s s i n t h e s l a b i s g i v e n by
Eo 1
2 n,
2
/,
|
| 2
(1 - I Pn I * )
(5 .1 )
The p e r i o d o f t h e Q v e r s u s 3d c u r v e i s t h e same a s t h e p e r i o d o f t h e
| Pm | ^ v e r s u s 3d c u r v e .
From Eq. ( 3 . 1 0 ) , we know t h a t
p23- e -j2 K d
(5 .2 )
Km
1 ~ P23e “ j ^ ^
p 23 * n + ^
-
pr +
K = 3 - jet
| Pm | ^ c a n b e o b t a i n e d from E q. ( 5 . 2 ) , and i s g i v e n by
2
P-m
pr 2 + pj2 + e-4 a d
2pr c o s2 3 d e ~ 2 ad + 2 p js in 2 3 d e -2 a d
1 + (Pj.2 + p j 2 ) e “ 4-ad - 2pr c o s 2 3 d e - 2a d _ 2 p j s i n 2 3 d e ~ 2 a d
(5 .3 )
T a k in g t h e p a r t i a l d e r i v a t i v e o f | pm | ^ w i t h r e s p e c t t o 3d , and s e t t i n g
i t eq u al to z e ro ;
S|P” | 2 - 0
83d
(5 .4 )
The d e t a i l e d d e r i v a t i o n i s shown i n Appendix M, and t h e r e s u l t i s t h a t
e x tr e m a o c c u r when
95
0 d = <(> + -^ir,
ra = 0 , 1 , 2 , • ' '
(5.5)
(5 .6 )
A = pr ( 1 B = Pj
pr 2 -
pj2)(i-e-4«<l)
( 1 + pr 2 + p j 2 ) ( !
( 5 . 7 . a)
+ e~4ad)
(5 .7 .b )
C = 4pr P je - 2 a d
(5 .7 .c )
From Eq. ( 5 . 5 ) , we s e e t h a t i f <j> i s v e r y s m a l l and c a n b e n e g l e c t e d ,
t h e e x tr e m e v a l u e s o f | pm | ^ w i l l o c c u r a t 0d =
ir, -^tt, • • • .
These
e x tr e m e v a l u e s o f | pm | ^ c o r r e s p o n d t o t h e e x tr e m e v a l u e s o f Q ( s e e
E q . ( 5 . 1 ) ) , and t h u s we n e e d c a l c u l a t e o n l y t h e p o s i t i o n o f e x tr e m a o f
I Pm I
t o l o c a t e e x tr e m a o f Q.
v a l u e f o r l a r g e 0d .
0d c u r v e i s it.
As shown i n a p p e n d ix M, <j> i s a v e r y s m a ll
We t h e r e f o r e assum e t h a t t h e p e r i o d o f t h e Q v e r s u s
T h a t m e a n s , i f we o b s e r v e Qmaxl a t ( 0d ) ^ and Qm ax2 a t
( 0d ) 2 » we h a v e
(5 .8 )
( 0d ) 2 “ ( 3d ) x = ir
The problem o f th e a p p l i c a t i o n o f E q . ( 5 . 8 ) i s th a t we have to fin d
a method to v a r y 8d in th e exp erim en t to o b se r v e Qmaxi ,
Qmax2»
etc*
I f we a p p l y a s t a t i c m a g n e t ic f i e l d t o t h e s a m p le , t h e n we ch an g e t h e
e l e c t r i c a l le n g th o f th e sam ple.
next se ctio n .
T h is m ethod w i l l be d i s c u s s e d i n t h e
H ere we s h a l l o n l y assum e t h a t d i s a v a r i a b l e .
One way o f p e r f o r m i n g t h e e x p e r i m e n t i s t o p h y s i c a l l y r e d u c e d by
g r i n d i n g o f f a c e r t a i n amount o f m a t e r i a l a f t e r e a c h m e a s u re m e n t.
P o s s ib ly l e s s te d io u s i s th e fo llo w in g te c h n iq u e .
c h a n g e t h e e l e c t r i c a l l e n g t h o f t h e s a m p le .
Use t h e f r e q u e n c y t o
In t h i s c a s e , l Qi s v a r i e d a t
96
e a c h f r e q u e n c y t o b r i n g t h e c a v i t y i n t o r e s o n a n c e and t h e Q i s m easu red
as a f u n c tio n o f r e s o n a n t fre q u e n c y .
However, we s h a l l c o n s i d e r d a s t h e
v a ria b le .
The complex p e r m i t t i v i t y o f a s e m i c o n d u c t o r sam ple i s ^ 9
U_2
e* -
es [ l ~ p
111'6 + v6
]
(5 . 9)
U 2V
e" = £s ' ~ uri( o?i6
+
( 5 - 10)
V 2T
6 )
We assume t h a t
(1)
u > Up, t h e n t h e EM wave c a n p r o p a g a t e i n t h e sam ple and
m u ltip le r e f l e c t io n s are p r e s e n t.
( 2)
u »
v , t h e low l o s s a s s u m p t io n i s n e c e s s a r y t o assume t h a t t h e
m u ltip le r e f l e c t io n s are p re s e n t.
Then E q s , ( 5 . 9 ) and ( 5 . 1 0 ) become
u 2
e ’ - e8 [ 1
V l
ui6
u_2 v
e" = e s - P
uJ
(5 .1 2 )
And t h e p h a s e c o n s t a n t w i l l be g i v e n by
3 » u /F l^
(5 .1 3 )
I f we o b s e r v e Qmaxi a t d = d ^ , t h e c o r r e s p o n d i n g r e s o n a n t f r e q u e n c y
i s ur i , and th e n 0^ x 2 a t d = d 2 > t h e c o r r e s p o n d i n g r e s o n a n t f r e q u e n c y
w i l l be ur 2 » t h e n we h a v e
2 1
01 = a)r l ' / lJo e s t 1
^U0 e s ( ^ i 2 - ojp2 ) 1 /2
“rl
(5 .1 4 )
97
e 2 = wr 2 ^
t1
= */ V*o^s ^“ r 2 2 ” “ p 2^
2
(5 .1 5 )
°*r2
We s u b s t i t u t e t h e s e v a l u e s o f 6 i n t o E q . ( 5 . 8 ) and t h e c o n d i t i o n f o r t h e
maximum v a l u e s o f Q becom es
/ y ^ e j ( u r 2 2 - Wp2 ) 1^ 2d2 - ^UQe s ( “ r l 2 “ “ p 2 ) 1^2 d l = u
(5 .1 6 )
Knowing ior j , ojr 2 , d ^ , d2 , we c a n s o l v e f o r (Dp from E q . ( 5 . 1 6 ) .
V .3 . The A n i s o t r o p i c Case
We s h a l l now d e s c r i b e p o s s i b l e ways o f d e t e r m i n i n g N w hich do n o t
r e l y on t h e m ea su rem e n t o f Q i n r e s o n a n t c a v i t y .
d e p e n d on t h e m a g n e t ic f i e l d
w av elen g th r e s o n a n c e s .
B a s i c a l l y t h e s e m ethods
f o r c i n g t h e sam ple t o p a s s t h r o u g h q u a r t e r
The m a g n i tu d e o f t h e r e f l e c t i o n c o e f f i c i e n t show
sh a rp v a r i a t i o n a t th e c r i t i c a l
len g th s.
The model c o n s i s t s o f a sam ple
i n a c i r c u l a r c y l i n d r i c a l w a v e g u id e o p e r a t i n g i n t h e TEj j mode b a c k e d by
a sh o rt c i r c u i t .
As shown i n C h a p t e r I I ,
in th e h e l i c o n r e g i o n , th e phase c o n s ta n t o f
t h e e x t r a o r d i n a r y wave c a n be s i m p l i f i e d t o
/ (DNep0
8+= / -------- Bo
(5 .1 7 )
w here N = t h e c a r r i e r c o n c e n t r a t i o n o f t h e s e m i c o n d u c t o r
B0 = t h e a p p l i e d s t a t i c m a g n e t i c f i e l d .
From E q . ( 5 . 1 7 ) we know t h a t :
( 1)
c<
, t h a t means we can use t h e s t r e n g t h o f t h e s t a t i c
m a g n e t ic f i e l d t o v a r y t h e p h a s e c o n s t a n t o f t h e e x t r a o r d i n a r y wave.
The
98
e l e c t r i c a l l e n g t h o f t h e sa m p le c a n t h e r e f o r e be v a r i e d , and we n e e d n o t
c h a n g e t h e p h y s i c a l l e n g t h o f t h e s a m p le .
(2)
Some t r a n s p o r t p a r a m e t e r s , su c h a s p , e s , ra*, a r e e l i m i n a t e d ,
and 8+ i s a f u n c t i o n o f N o n l y .
So, i f 0+ i s m e a s u r e d , N c a n be c a l c u ­
l a t e d im m ed ia te ly .
When a p o s i t i v e c i r c u l a r l y p o l a r i z e d wave i s i n c i d e n t upon a t h i n
a n i s o t r o p i c s a m p le , t h e m a g n i tu d e o f t h e t o t a l r e f l e c t i o n c o e f f i c i e n t ,
I pm+ I > w i l l a p p r o a c h a minimum when 8+d =
ir
( S e c tio n V .2 ).
T h is
phenomenon c a n be u s e d t o m e a s u r e 3+ , and from t h e m e a s u re d v a l u e o f 8+ ,
N can be c a l c u l a t e d .
F o r e x a m p le , we c a l c u l a t e | pm+ | v e r s u s B0 f o r a
t h i n InSb s a m p le and p l o t t h e r e s u l t s
F ig . 5 .3 .b ,
i n F i g . 5 . 3 . a , (d = 6 m i l s ) ,
(d = 12 m i l s ) and F i g . 5 . 3 . c , (d = 18 m i l s ) .
The t r a n s p o r t
p a r a m e t e r s o f t h e InSb s a m p le a r e shown i n T a b le 5 . 1 . ^
*
m
N
P
2 x 10^ / c m ^
0 .0 1 3 H-cm
T able 5 .1 .
0 .0 1 3 mg
es
1 8 .7 eQ
The t r a n s p o r t p a r a m e t e r s o f t h e t h i n InSb s a m p le .
We h a v e assum ed t h a t t h e s a m p le i s p u t i n a TE^j c i r c u l a r w a v e g u id e and
b a c k e d by a p e r f e c t c o n d u c t o r .
We assum ed t h a t t h e o p e r a t i n g f r e q u e n c y
i s 2 4 .7 8 GHz.
From F i g . 5 . 3 a , we s e e t h a t | pm+ | h a s a minimum v a l u e when BQ =
6 .1 2 5 KG, and 8+d = t t / 2 .
In a p r a c t i c a l m e a s u r e m e n t, we can h a v e B0 =
99
1.0
0.8
0. 6
-
0.4
0.2
12
20
24
0.8
0.6
04
0.2
12
16
20
0-8
0.6
0.2
12
F ig . 5 .3 .
pm+
v e r s u s Bq f o r t h i n InSb s a m p le .
a , d=6m i l s , b , d = 12m i l s , c , d = l 8m i l s .
20
24
100
2n+ 2
— tt.
6 .1 2 8 KG, b u t we do n o t know t h e v a l u e o f n i n 8+d =
6+d =
and c a l c u l a t e N.
So
we assume
I t is
m
/ 2n + l it. 2
_ /
N = ( - £ — - j ) x B0 /ajey0
and f o r B0 = 6 .1 2 5 KG
N = ( - ~ b 2 x 8*.3 x 1021
We c a n e s t i m a t e N b y c o m p a rin g t h e above r e s u l t t o t h e dc v a l u e .
F o r a r e a s o n a b l e dc v a l u e , n m ust be z e r o . The r e s u l t i n g
v a lu e
of N is
2 .0 7 x 102 V m 3 o r N = 2 .0 7 x 10^ / c m 2 .
From F i g . 5 . 3 . b , we s e e t h a t | p^+ | h a s two minima
and BQ2 ~ 2 8 .5 KG.
a t Ba l =2 .7 5 KG
The f o l l o w i n g r e l a t i o n c a n be used
( 8+d ) j - ( 8+d )2 3 v
(5 .1 8 )
The v a l u e o f N c a n b e c a l c u l a t e d a s
N = (-J-)2 ------------- \ --------- i-------- -- 1 .9 7 x 1 0 15/cm 3
d
a ie p o ^
-
)2
From F i g . 5 . 3 . c , we h a v e B0 i = 2 .2 5 KG, BQ2 = 6 .2 5 KG, N i s c a l c u ­
l a t e d a s N = 2 .1 x l O ^ / c m 3 .
The above c a l c u l a t i o n s show t h a t t h i s m ethod c a n be u s e d .
T h e re i s a n o t h e r m ethod w hich c a n b e u sed t o m e a s u r e t h e v a l u e o f
g+d .
When we c a l c u l a t e t h e p h a s e a n g l e o f P^*, we f i n d t h a t t h e ph a se
a n g le o f p ^ ,
IT
a r g pm+, w i l l be z e ro when B+d = —
3
Tr,-yn,
••*.
(F ig . 5 .4 ,
101
Arg
Pm+
16
20
16
20
A?
Km+
12
20
F i g . 5 . 4 . Argum ent o f Pm+ v e r s u s Bq f o r t h i n InSb s a m p le ,
a , d=6 m i l s , b , d= 12 m i l s , c , d= 18 m i l s .
2n+
a ,b ,c )
When B+d = ---- 2 ~lt> a r § Pra+
d r o p from p o s i t i v e t o n e g a t i v e
v e r y r a p i d l y and can be e a s i l y o b s e r v e d .
From Eq. ( 3 . 1 8 )
a r g pm+ = 2^ ^ =
Eq. ( 5 . 1 9 )
i n c h a p t e r IV, we know t h a t
t a n " 1 - ^ ! + nx
Pmr
(5 .1 9 )
i s th e re s o n a n t c o n d itio n o f th e p a r t i a l l y f i l l e d c a v it y a t
t h e r i g h t hand s i d e o f t h e c o u p l i n g w a l l .
So t h e se c o n d m ethod which i s
s u g g e s t e d by m e a s u r in g a r g pm+ i s t h e same a s m e a s u r in g t h e r e s o n a n t f r e
quency f o r a v e r y u n d e r c o u p l e d c a v i t y .
In F i g . 5 .5 we p l o t t h e r e s o n a n t
f r e q u e n c y , a ^ , v e r s u s B0 .
I t i s h ard to d e te rm in e th e ex a ct v a lu e o f
2n+ ^
B0 w hich c o r r e s p o n d s t o B+d
But tBie Pe r *-0<*
“ r v e r s u s 3+d
curve i s u.
F i g . 5 . 5 shows t h a t we c a n s t i l l c a l c u l a t e N from t h e
v a r i a t i o n o f th e re s o n a n t fre q u e n c y .
T h e r e a r e t h r e e m eth o d s w hich a r e m e n tio n e d i n t h i s s e c t i o n .
When | pm+ | a p p r o a c h e s a minimum, 8+d = ■
w i l l be z e ro when B+(j = nir.
f r e q u e n c y v e r s u s B0 i s it.
(3)
vm
( 2)
(1)
The p h a s e a n g l e o f
The p e r i o d o f 8+d o f t h e r e s o n a n t
The se c o n d and t h i r d m ethods c a n be u sed as
t h e a u x i l i a r y m ethod o f t h e f i r s t m ethod b e c a u s e t h e m easurem ent | pra+ | i
e a s y and a c c u r a t e .
103
n o
1.6
1.5
1.4
12
20
1.6
1.5
1.4
16
20
16
20
1.6
1.5
1.4
F ig . 5 .5 .
0)
versu s B
f o r t h i n In S b s a m p le .
a , d= 6 m i l s , b , d= 12 m i l s , c , d= 18 m i l s .
104
CHAPTER VI
(1)
CONCLUSION
A v ery d e ta ile d
a n a l y s i s o f a r e s o n a n t c a v i t y was g i v e n i n
C h ap ter I I I .
(a)
The r e s o n a n c e c o n d i t i o n o f a p a r a l l e l p l a t e c a v i t y p a r t i a l l y
f i l l e d w ith a lo s s y sla b i s d e riv e d under d i f f e r e n t c r i t e r i a .
When t h e c a v i t y i s u n d e r c o u p l e d , a l l e q u a t i o n s o f t h e r e s o n a n t
c o n d i t i o n w hich a r e d e r i v e d by d i f f e r e n t c r i t e r i a a p p ro a c h th e
same r e s u l t .
(b )
The c o u p l i n g c o n d i t i o n i s d i s c u s s e d u n d e r t h e a s s u m p t io n t h a t
th e c o u p lin g i r i s
(c)
is a shunt s u s c e p ta n c e .
The q u a l i t y f a c t o r o f a r e s o n a n t c a v i t y was d e r i v e d by c a l c u ­
l a t i n g t h e e n t i r e E - f i e l d and H - f i e l d i n t h e whole c a v i t y .
When t h e r e s o n a n t c o n d i t i o n o f t h e c a v i t y i s d e f i n e d as
^ ( Z i n ) = 0 , t h e tim e a v e r a g e d e n e r g y s t o r e d i n E - f i e l d , i s
e q u a l t o t h e tim e a v e r a g e d e n e r g y s t o r e d i n H - f i e l d .
re s o n a n t c o n d itio n o f th e c a v it y i s
I f th e
d e f i n e d by o t h e r c r i t e r i o n ,
t h e above e q u a l i t y no l o n g e r h o l d s .
(2 )
F o r a t h i c k a n i s o t r o p i c s a m p le , we s u g g e s t a m e th o d , t h e common
c r o s s p o i n t m e th o d , t o m e a s u re
As t h e
th e t r a n s p o r t
a p p l i e d s t a t i c m a g n e t ic f i e l d
p a r a m e t e r s o f t h e s a m p le .
is in cre ase d , th is
method w i l l show
b e tte r reso lu tio n .
(3)
F o r a t h i n a n i s o t r o p i c s a m p le , we c a n c a l c u l a t e t h e t r a n s p o r t
p a r a m e t e r s by m e a s u r in g ( a ) t h e m a g n itu d e o f t h e r e f l e c t i o n c o e f f i c i e n t
o r (b ) t h e p h a s e a n g le o f t h e r e f l e c t i o n c o e f f i c i e n t o r ( c ) t h e r e s o n a n t
f r e q u e n c y o f a c a v i t y p a r t i a l l y f i l l e d w ith t h i s a n i s o t r o p i c s a m p le .
In
105
th e m e a s u r e m e n t, t h e i n c i d e n t wave i s a p o s i t i v e c i r c u l a r l y p o l a r i z e d
wave.
106
APPENDIX A
PLANE WAVE ANALYSIS OF THE ANISOTROPIC SOLID-STATE PLASMA
Assume a p l a n e wave p r o p a g a t i n g i n t h e z - d i r e c t i o n w i t h a s t a t i c
m a g n e t i c f i e l d d i r e c t e d i n t h e same d i r e c t i o n , w hich i s n o r m a l l y t o t h e
s u r f a c e o f t h e s a m p le , ( F i g . 2 . 1 ) .
The e q u a t i o n o f m o ti o n o f e l e c t r o n s is-*-®®
m*
+ m*vv = - e [ E + v x "B0 ]
(A .l)
w here m : t h e e f f e c t i v e m ass
v:
th e v e lo c ity o f e le c tr o n s
v:
th e c o l l i s i o n freq u en cy
The c u r r e n t p e r u n i t a r e a J i s
J = -eNv
S u b s t i t u t i n g Eq. ( A . 2) i n t o
.
( A . 2)
( A . l ) , t a k i n g t h e t h r e e c o m ponents i n a
C a r t e s i a n c o o r d i n a t e s y s t e m , we h a v e
( v + jo i)J x - u)c J y = e s o>p2Ex
( A .3 . a )
( v + j o j ) j y + (0CJ X = e gwp2Ey
( A .3 .b )
( v + j(D)Jz = e gwp 2Ez
(A .3 .c )
eB0
where 0)c = ---------- , t h e c y c l o t r o n r e s o n a n t f r e q u e n c y
m
jje 2 1/2
wD = (.——t - )
, t h e p la s m a f r e q u e n c y
H
e gm
eg:
th e s t a t i c p e r m i tt iv i ty
107
R e w r i t i n g Eq. ( A .3) i n t o a m a t r i x fo rm , we h a v e
rx
=
y
0i
a2
0
-o2
01
0
0
03
0
-
Ex
Ey
Ez
wp2 e s ( v + ju>)
w here
°1 =
( v + jm ) 2 + mc 2
~ (0p 2 gs a>c
( v + j to) 2 + ojc 2
o3 =
( A .4)
^ p 2
V + jrn
( A .5 . a )
(A .5 .b )
( A .5 .c )
The Maxwell e q u a t i o n s a r e a p p l i e d t o o b t a i n a wave v e c t o r
E 3 - ja )p 0H
7 x 1
= joJEgE + T
( A .6 )
( A . 7)
The I T - f i e l d and H - f i e l d a r e assum ed a s
E = E0 ( x , y , z ) t e ( t ) = E0 e “ J K ' g e i 101-
( A .8 )
H = H g C x .y j z J t ^ C t ) = H^e- j K' z e i ^ 1-
( A .9)
S u b s t i t u t i n g Eq. (A .4 ) i n t o ( A .6 ) and ( A . 7 ) , t h e s i x com ponents o f
Eq. ( A . 6 ) and ( A .7 ) t h e n a r e
jKHy
0 l Ex + ° 2Ey + j “ es Ex
( A . 1 0 . a)
jKHx
a 2^
( A .I O .b )
0
jKEy
• OlEy - jWEgEy
(jOJEg + 03 )E Z
(A .IO .c )
-i^ o
( A .I O .d )
108
jKEx = jiop0 Hy
( A .I O .e )
0 = j 0JlioHz
(A .IO .f)
E q . ( A . 10) a r e f o u r hom ogeneous e q u a t i o n s w i t h f o u r v a r i a b l e s , t h e
f i e l d c o m p o n e n ts .
F o r a n o n t r i v i a l s o l u t i o n , t h e d e t e r m i n a n t m ust v a n i s h
and t h e d i s p e r s i o n r e l a t i o n s a r e found b y
K+2 = a)2 p e (1 - j —
—
S '
j
uies
± —
M
p.
o)eg
)
( A . 11)
7
S u b s t i t u t i n g E q . ( A .5 ) i n t o ( A . 1 1 ) , we h a v e
K+2 = aj2jj0 ( e + ' - j e + " )
(A. 12)
w here
e ,
=
{ , a
*
“
.
v
[ ( £ ) 2 - ( $ 2 ♦ l]2 * < ^ )2
( A . 13)
2
[(iHfi)2
+
(JA)2
+
u
±
i ^
“o
v
v
v2
e+" = e _ —S- {---------------------------------- -----------------------}
-
8 “V
[ ^ , 2
_ ( » )2 * 1 ,2 + (2H)2
(A. 14)
109
APPENDIX B
THE RESONANT CONDITION OF A PARALLEL PLATE CAVITY PARTIALLY
FILLED WITH A LOSSLESS SLAB
The g e o m e try o f a p a r a l l e l p l a t e c a v i t y p a r t i a l l y f i l l e d w i t h a
l o s s l e s s s l a b i s shown i n F i g . B . l .
Assum ing E - f i e l d o n l y h a s x-coraponent and t h e n H - f i e l d o n l y h a s
y - c o m p o n e n t, E - F i e l d s i n t h e f r e e s p a c e r e g i o n c a n b e w r i t t e n a s :
Ex = Ae^K° Z + Be“ ^K° Z
(B .l)
a t t h e b o u n d a r y , z = 0 , Ex = 0 , t h e n B = -A , Ex becomes
Ex = 2 A jsinK 0Z = CsinKQZ
( B .2 )
0H
U s in g Maxwell e q u a t i o n ,
3EX
E =
3H,
= cKqCosKqZ = - p 0 - ^ =
we ^ ave
- j u y 0Hy
( B .3 )
So,
Hy = j c -J—
cosK0 z
(B .4 )
The E - f i e l d i n t h e s l a b r e g i o n c a n be w r i t t e n a s :
Ex ' = e - j K Z ' + Fej k Z '
( B .5 )
w here Z1 = lo+ d - Z
At t h e r i g h t w a l l , Z' = 0 ,
Ex' = 0 ,
D = - F , Ex' becomes
Ex ' = 2jD s in K Z ' = C ' s i n K ( l 0+ d - Z)
(B .6)
110
fre e
space
slab
lo
F i g . B . l . The g e o m e tr y o f t h e c a v i t y .
Ill
The H - f i e l d c a n b e o b t a i n e d by a p p l y i n g Maxwell e q u a t i o n t o Eq. ( B . 6 ) ,
th a t is
V
= " j "Sir-
G' co sK ^ 1o +
d “ z)
( B .7 )
At t h e i n t e r f a c e o f t h e f r e e s p a c e and s l a b r e g i o n s , E - f i e l d and H - f i e l d
m ust b e c o n t i n u e d .
That i s ,
Z = l 0 ,E x = Ex ' ,
Hy = H y ' .
So,
CsinKo l 0 = C 's in K d
(B .8 )
CcosKo l 0 = -KC'cosKd
( B .9 )
D i v i d i n g Eq. ( B . 8 ) by Eq. ( B . 9 ) , we h a v e
t a n K oU
t a n Kd
—
(Bao)
E q . ( B .1 0 ) i s t h e r e s o n a n t c o n d i t i o n o f a p a r a l l e l p l a t e c a v i t y p a r t i a l l y
f i l l e d w ith a l o s s l e s s s la b .
112
APPENDIX C
THE DERIVATION OF EQUATION ( 3 . 7 ) t o ( 3 . 1 ) WHEN THE SLAB
IS LOSSLESS
Eq. ( 3 . 7 ) i s
2nri( n T. s i n 2Pd + ru s i n h 2nd)
(C .l)
t a n 2Kq10 = ------------- -----------------------------=!-------------------- ------------------------------------( Tlr + Uj ) ( c o s h 2ad - c o s 2(3d) - n0 ( c o s h 2ad + c o s 2Sd)
I f t h e s l a b i s l o s s l e s s , t h e n nj = 0 , a = 0 , Eq. ( C . l ) becomes
t a n 2KolQ=
2r|f)nr s i n 2 Sd
r
— ---------------nr 2 ( l - c o s 2 8 d ) - n02 ( l + c o s 2 gd)
( C .2 )
The f o l l o w i n g m a t h e m a t i c a l d e r i v a t o n c a n b e u s e d ,
t a n 2Kq 1 =
2TUhr c o s 6 d s i n 8 d
x-------------z------z-----nr 2 s i n 2 8d - no 2c o s 2 0d
ZtanK0 l 0
l-ta n ^ K g lg
(C .3 )
2q0 Ur t a n 8 d
(C .4 )
nr 2t a n 2 f3d - riQ2
tanK 0 l 0 Hr 2t a n 2 8d - n ^ t r a n K o ^ - n0 nr t a n 8d + no n2t a n 8d t a n 2Kol0 = 0
(C .5 )
( n 0tanK0 l0 + nr t a n B d ) ( n 0
T h e re a r e two s o l u t i o n s i n Eq.
- T ir ta n K o ^ a n g d ) = 0
(C .6 ) .
(C .6 )
We n e e d c h e c k e a c h one o f t h e two
s o lu tio n s.
(1)
n0 tanK0 lQ+ rir t a n 3 d
=0
( C .7 )
That is
ta n K P-19 = _ ta n 3 d _
Kq
p
(c>g)
113
Eq. ( C . 8 ) Ls t h e same a s E q . ( 3 . 1 ) , so i t i s d e r i v e d
(2)
Eq. ( 3 . 6 )
1o " nr tanKo ^ a n 0 d = 0
(C .9 )
Ln t e x t i s
jn ta n K d + jriotanK olj,
n°
h0 - ntanKdtanKolQ
( C .1 0 )
We s e e t h a t E q . ( C .9 ) w i l l c a u s e t h e d e n o m i n a to r o f Eq. ( C .1 0 ) g o in g t o
zero .
T h at means Z^n =
So t h i s s o l u t i o n i s n o t c o r r e c t .
114
APPENDIX D.
THE DERIVATION OF EQUATION ( 3 . 7 ) TO ( 3 . 1 5 )
Eq. ( 3 . 7 ) i s
(D .l)
2 r u ( r tr s i n 23d + n; s i n h 2 ctd)
t a n 2Ko l 0 = — ------ — ^
]---------- ----------- =---------------( n r z + nj M c o s h 2 ad - c o s 23d) - n0 ( c o s h 2ad + co s 23d)
Eq. ( 3 . 1 5 ) i s
(D .2 )
where
p_e j 2Kd
Pm = Pmr + JPmj = i _ pe-j2 K d
p
n - nQ
----------n + n0
(D .3 )
nr + j n ; - nQ
j ---------nr + j n j + nQ
(D .4 )
(D .5 )
e -j2 K d _ e “ 2 a d ('cos23d - j s i n 2 0 d )
S u b s t i t u t i n g Eq. ( D .4 ) and (D .5 ) i n t o
( D . 3 ) , we have
* ^ n i ~ n° - e“ 2 a d ( c o s 2 3 d - j s i n 2 3 d )
nr + j n -
*
nQ_^
__________________
(D .6 )
1 ~ h r + j nj - y —^ e ~ 2 a d ( c o s 2 gd - j s i n 20d )
R e a l i z i n g t h e d e n o m i n a to r o f Eq. ( D . 6 ) , we h a v e
MP + NQ + j(NP-MQ)
Pm "
P2 + Q2
where
(D.7)
M = nr “ h0 - e -2cid( n r c o s 2 3 d + n j s i n 2 3 d + qQc o s 2 3 d )
N _ n
+ e- 2 a d ( n r s in 2 0 d
- q jc o s 2 0 d +
n0 s in 2 3 d )
P * nr + % - e “ 2 o d ( r,rCOS2 3d + n j s i n 2 3 d - n0c o s 2 3 d )
Q = nj + e “ 2 a d ( nr s in 2 3 d
- n jC os23d -
n0 s in 2 3 d )
115
T hen, E q. (D .2 ) becomes
ta
NP +
~ M
Q
t a nn 2K
21^ 1 = M
Pmr -“ up
NQ
2~n0 (
+ 2e ~ 2otdrir s i n 2 gd - r ij e - ^ o d )
( n r 2 + Hj2 ) ( l + e“ ^ a<*-2e- 2 a t *cos28d) - ri02 ( l + e ” ^ 0^ + 2e“ 2a<*cos28d)
(D .8 )
D i v i d i n g Eq. ( D .8 ) b y e“ 2ot{*,
pmj
Pmr
2 r)0 [ n j ( e 2 ad- e - 2 a d ) + 2qr s in 2 3 d ]
(r)r 2 + r ij2 ) ( e 2 a ^ + e“ 2 ot^ -2 c o s 2 |3 d ) - Ti02 ( e 2ad + e -2ctd + 2cos2f3d)
(D .9 )
pm;
2h0 ( n ; s i n h 2 oid + nr s in 2 8 d )
tan 2 K 0 ln s= — ^ = ---------------- —— ^-------------------- --—----------------------------------------------pmr
( n r 2 + n j 2 ) ( c o s h 2ad - c o s 2 Pd) - n0 2 ( c o s h 2 ad + c o s 2 gd)
( D .1 0 )
E q . (D .1 0 ) i s e x a c t l y t h e same a s Eq. ( D . l ) .
Q .E .D .
116
APPENDIX E.
THE CONDITIONS OF Im(Zi n ) = 0 AND V ( y
BE A MINIMUM OF A
LOADED TRANSMISSION LINE ARE EQUIVALENT
C o n s i d e r a l o s s l e s s t r a n s m i s s i o n l i n e t e r m i n a t e d by a l o a d Z.
le n g th o f the tra n s m is s io n l in e i s y .
The
Then t h e i n p u t im pedance o f t h i s
lo ad ed tra n s m is s io n l i n e i s
7
in
V( IP
Z COSKQ1Q+ jZnSinK plo
u i o ) = ZOCOSXQJLQ+ jZ s in K 0 I 0
Zr cosK0 l Q+ j(Z ^ c o s K 0 l 0 + Z0 sinK 0 ^
(Z q COsKq Iq - Z js in K o y + j ( Z r sinK o y
(E .l)
w h e re Z = Zr + j Z j
The c o n d i t i o n o f I[n(Zin ) = 0 i s
Zr cosK0 l ( - Z r sinK o y
+ (ZjCosK0 l 0 + Z0 sinK o y ( Z q C o s K q L j - Z js in K o y
= 0
(E .2 )
It is
2z o z i
t a n 2Kol0 = — 5
S
5z r + z j 2 - zo 2
( E .3 )
The v o l t a g e a t l < j l e f t t o t h e l o a d i s
V(lo) = VcosK0 l<>+ jZ 0 I s i n K 0 lo
(E .4 )
= l[Z cosK 0 l0 + jZ 0 sinK o y
The m a g n itu d e o f V(l(j) i s
| V(lo) I = I [ ( Z r c o s K o y 2 + (Z0 s inK 0 lo +
ZjCosKo y 2 ] *^2
(E .5 )
w here Z = Zr + j Z j
To f i n d t h e v a l u e o f I q c a u s i n g | V ( y
| b e a minimum, l e t us t a k e t h e
p a r t i a l d e r i v a t i v e o f | V ( y | w ith r e s p e c t
t o lo and s e t i t e q u a l t o z e r o .
117
It
is
- i l *>»>..L . Q
(E .6 )
We h a v e
Z K o Z r ^ c o s K o ^ - s i n K o y + 2Ko (Zo sinK o l0 + ZjcosKo^CZoCOsKolo - Z j s i n K o y = 0
(E .7 )
That i s
ta r^ K ^
2Z • Z
------------Zr 2 + Zj2 - Zo2
E q. ( E . 8 ) i s t h e same a s Eq. ( E . 3 ) .
(E .8 )
T h at means a t I o j w h i c h t h e
i m a g i n a r y p a r t o f t h e i n p u t im pedance i s e q u a l t o z e r o i s t h e same as
| V(ij) |
b e a minimum.
118
APPENDIX F .
THE INPUT IMPEDANCE OF A LOADED CAVITY AT THE LEFT SIDE
OF THE COUPLING WALL.
■•in
Z in'
d
F ig . F . l .
ZQ = 1
zT
jB
'in
J
The i n p u t im pedance o f a lo a d e d c a v i t y and t h e
e q u iv a le n t c i r c u i t .
From above f i g u r e , we c a n w r i t e down t h e f o l l o w i n g e q u a t i o n s
Z£n ' = jn ta n K d = Zr + j Z j
(F .l)
w here n i s t h e n o r m a l i z e d c h a r a c t e r i s t i c im pedance o f t h e s l a b .
K i s th e p r o p a g a tio n c o n s ta n t o f th e s la b
Zin' + jta n K o 1©
= R + jX
i n " 1 + jZ iA ta n K Ql 0
(F .2 )
E v a l u a t i n g Eq. ( 2 ) , we h a v e
R =■
Zr ( l + tanK Q2 y
( F . 3 . a)
(1 - Z j t a n K Q p 2 + (Zr tanK o y ‘
Z j ( l ~ t a n 2Kpk) + (1 - Zy2 - Z j 2 ) t a n KqIq
(F .3 .b )
(1 - Z jta n K 0 lo) 2 + (Zr tanK 0 lo) 2
Then we h a v e
YX = jB
Ji n
1
jBR - BX + 1
= jB + R + jX~
R + jX
(F .4 )
119
The i n p u t im p e d a n c e , Zhp =■
T
Yt
, is th a t
________ R + iX
f(1l - BX)
RY') + jBR
=
R
i[X - B(X2 + R2 ) ]
(1 - BX) 2 + (BR) 2
The c r i t i c a l c o u p l i n g c o n d i t i o n s c a n b e d e f i n e d a s ImCZj.)
(F .5 )
0 and
RgCZ-j) = 1 , i t i s ,
B -
X
X2 + R2
R = (1 - BX) 2 + (BR) 2
( F . 6 .a )
( F . 6 .b )
120
APPENDIX G.
A DISCUSSION OF THE DEFINITION OF THE QUALITY FACTOR
The o r i g i n a l d e f i n i t i o n o f t h e q u a l i t y f a c t o r o f a lumped c o n s t a n t
r e s o n a n t s y s te m was g i v e n by
0
^
2ir
Sum o f t h e e n e r g y s t o r e d p e r c y c l e i n b o t h L and C
energy d i s s i p a t e d p e r c y c le in R
^
L e t us u s e a s e r i e s RLC c i r c u i t , F i g . 1, t o exam ine t h i s d e f i n i t i o n .
We s h a l l f i r s t assum e t h a t t r a n s i e n t e x c i t a t i o n i s a p p l i e d t o t h i s c i r ­
c u it.
C o n s i d e r an i n i t i a l c h a r g e on t h e c a p a c i t o r l e a d i n g t o an i n i t i a l
v o l t a g e VQ a c r o s s t h e c a p a c i t o r .
x t= o
=J=!Vr
F ig . G .l.
L
The s e r i e s RLC c i r c u i t w i t h an i n i t i a l c o n d i t i o n VQ on C.
The e q u a t i o n o f t h e
cu rren t at t
> 0 is
L ~ + Ri + ■£ J i d t « VQ
( G .2 )
Upon t a k i n g t h e d e r i v a t i v e w i t h r e s p e c t t o tim e t o e l i m i n a t e t h e i n t e g r a ­
t i o n , we h a v e
- d2i ♦ R 4 j + i . o
d t2
dt
C
L e t us assum e a s o l u t i o n f o r i ,
i = AePt
(0.3)
i n an e x p o n e n t i a l form ,
121
S u b s t i t u t i n g ( G .4 ) i n (G .3 ) and s o l v i n g f o r p from t h e r e s u l t i n g
e q u a t i o n , we h a v e
R ± /(_ 1 ) 2 . 1
P = - 2L
2L
LC
R 9
1
I f (xr-) > "77T i
ZL
Lu
( G .5 )
"the s o l u t i o n f o r i w i l l h a v e a g e n e r a l form shown i n t h e
fo llo w in g f ig u r e .
F i g . G .2 .
The r e s p o n s e o f i f o r t h e overdam ped c o n d i t i o n .
T h i s i s known a s t h e overdam ped c a s e .
R 9
1
I f ( r t -) < Tr , p i s c o m p le x .
2.L
LC
We c a n assum e a com plex f r e q u e n c y ,
p = jw c , t h e n Eq. (G .5 ) i s
R
2L _ J v/LC
J“c
4L
(G .6 )
or
u)c = j a ±
w here a
0) =
/LC
(G .7 )
oj
—> i s t h e dam ping c o e f f i c i e n t and
ZL
/
r.1
r 2c
, xis t h e f r e q u e n c y o f t h e f r e e r e s p o n s e .
The m a g n itu d e o f t h e com plex f r e q u e n c y , U)c i s
(G .8 )
122
f 0JC | = / a 2 + u2 == -± = r-
(G .9 )
The c u r r e n t , i , c a n b e shown t o be
y
i =
e“ a t sin<ot
(G .1 0 )
and t h e v o l t a g e a c r o s s t h e c a p a c i t o r i s
Vc = V0 e ” a t cosiot
(G .ll)
At any i n s t a n t , t h e e n e r g y s t o r e d i n t h e i n d u c t o r and i n t h e c a p a c i ­
t o r are
Ul
u
2
=
2
- —§— e“ 2 a t s i n 2 <ot
2o)2L
Uq = 4 c v c 2 = —2
e“ 2 cttc o s 2<ot
( G .1 2 )
(G .1 3 )
The e n e r g y s t o r e d i n t h e i n d u c t o r and t h e c a p a c i t o r p e r c y c l e i s t h e
tim e a v erag e o v e r a c y c l e .
C o n se q u e n tly ,
2
= Ti
I
^
— e“ 2a t s i n 2 (0t d t
2<o2L
Uq = ™ / — «— e ~ 2a t c o s 2<ot <jt
(G .1 4 )
(G .1 5 )
,
.
1
2 it
w h e re T = -z = ---r
to
A f t e r i n t e g r a t i o n , and s u b s t i t u t i n g Eq. ( G . 9 ) , a 2 + ai2 = - y , i n t o t h e
XiO
r e s u l t i n g e q u a t i o n , we h a v e
123
V 20)LC
_ m
UL = - f ^ R — (1 “ e
>
( G .1 6 )
r V 2u>LC
V 2 (dRC2
" o - t - 8i R - * ^ 3 S - ] °
- e
>
CG- 17)
N o te t h a t Ul $ Uq , t h a t m e a n s , u n d e r a t r a n s i e n t e x c i t a t i o n , w i t h non­
z e ro R th e en erg y s to r e d in L i s n o t eq u a l to th e en erg y s to r e d in C fo r
t h e s e r i e s RLC c i r c u i t .
.______
•i
The power l o s s a t any i n s t a n t i s
2
P = R^2 = —?—x - e “ 2 oits i n 2 0Jt
(G .1 8 )
The e n e r g y d i s s i p a t e d p e r c y c l e i s t h e n
T
v 2R T
W = / P d t = 2 j e ~ 2a t sin ^ (o t d t
o
a)^LA o
"J
(G .1 9 )
A f t e r i n t e g r a t i o n , and s u b s t i t u t i n g a 2 + a)2 = - r 4 LC
in to th e r e s u l ti n g
e q u a t i o n , we h a v e
W
(1 “ e“ 2aT)
(G .2 0 )
The q u a l i t y f a c t o r f o r t h e t r a n s i e n t c a s e becomes
Q = 2„
+
W
K.
( 0 . 21)
o
I f a low l o s s a s s u m p t io n i s m ade, a «
a), t h e n from E q . ( G . 9 ) , u> -
o r b y i n s p e c t i o n o f Eq. ( G . 8 )
r2°
4L
«
1
U sing Eq. ( G . 2 2 ) , we s e e t h a t i n Eq. ( G . 2 1 ) ,
(G . 2 2 )
,
124
_UL . .
R
uRC
8
In c o n c lu s i o n ,
(G.23)
th e q u a li ty
f a c t o r for
low l o s s , t r a n s i e n t e x c i t a t i o n o f a
s e r i e s RLC c i r c u i t becom es
uL
(G .2 4 )
Q =- R
N ex t, c o n s id e r a s te a d y s t a t e
RLC c i r c u i t
s in u s o id a l so u rce in s e r i e s w ith th e
a s shown b e lo w
F i g . G .3 .
The s e r i e s RLC c i r c u i t w i t h a s e r i e s
sin u so id a l v o lta g e
so u rce.
The
current
L
dt
e q u a tio n
+ Ri
The s o l u t i o n
~prJ
+
0
Z
t|> =
=
ta n
id t
= V0 c o s m t
fo r th e c u r r e n t ,
cos(u t
w h e re
is
-
A.2
i,
o f E q . ( G .2 5 ) i s g i v e n by
(G .2 6 )
ij>)
+ ( uL — -—
_i
1
( G .2 5 )
UC
UL - - L “c
r
( G. 2 7 )
( G . 28)
We d e f i n e t h e r e s o n a n t c o n d i t i o n a s ImCZ) = 0 , and f i n d t h a t
“r V E T
( G ' 29 )
125
f Z | = R, i|> = 0
It
(G .3 0 )
i s i m p o r t a n t t o n o t e , t h a t t h e r e s o n a n t f r e q u e n c y d o e s n o t depend
o n R i n t h i s c i r c u i t r e g a r d l e s s o f t h e v a l u e o f R.
T h is r e s u l t i s d i f ­
f e r e n t from t h e t r a n s i e n t c a s e i n w hich t h e f r e e o s c i l l a t i n g
freq u en cy
d o e s depend on R ( s e e E q . ( G . 8 ) ) .
The d i f f e r e n c e o r i g i n a t e s i n t h e c h o i c e o f c i r c u i t .
C o n sid er th e
i d e a l i z e d s e r i e s and p a r a l l e l c i r c u i t s i n F i g . G .4 .
F i g . G .4 .
The s e r i e s and p a r a l l e l RLC c i r c u i t .
H e r e , R i s p h y s i c a l l y s e p a r a t e from L and
C, t h u s 0)r i s n o t a f u n c t i o n o f
R fo r s in u s o id a l ste a d y s t a t e e x c i t a t i o n .
H ow ever, i n t h e r e a l c a s e o f l o s s i n
c irc u it
a d i e l e c t r i c , th e e q u iv a le n t
f o r a s e r i e s c o n f i g u r a t i o n becom es
V cosw t
R
c
F i g . G .5 .
The s e r i e s c i r c u i t w i t h a d i e l e c t r i c
lo ss.
126
We a r e i g n o r i n g r e s i s t i v e
lo s s e s in the in d u c to r.
The im pedance o f t h i s c i r c u i t i s
Rc
CRg2
Z ■ *8 *
♦ j » (L - ! + „ZC/ Rc2
>
(G. 31)
Once a g a i n , d e f i n i n g I n / Z ) = 0 a s t h e r e s o n a n t c o n d i t i o n , we h a v e
ur =
^ 1 “ ■ o
( G .3 2 )
Rc 2 C
A n o th e r c a s e i s t h e one d e s c r i b i n g l o s s i n an i n d u c t o r .
The
p a ra lle l c irc u it is
I cosw t
F i g . G .6 .
The p a r a l l e l c i r c u i t w i t h m a g n e t ic l o s s .
H e r e , we i g n o r e t h e d i e l e c t r i c
lo ss.
The im pedance i s g i v e n by
Z = Rg +
R
t
(1 - u 2LC) 2 + (i)2C2Rl 2
L -
0)2L2C - Rl 2C
j “ u -A c> * + A V
<G' 33)
D e f in e Im(Z) = 0 a s t h e r e s o n a n t c o n d i t i o n , and we h a v e
“ r ” /LC
J
iI - r l -----2c
( G .3 4 )
127
Now, we d i s c u s s t h e q u a l i t y f a c t o r o f a l l t h r e e c a s e s :
(1)
S e r i e s RLC c i r c u i t ,
l• r "-
V°
( r e f e r t o F i g . G .3) w i t h s e p a r a t e R
coso^t
(G .3 5 )
The i n s t a n t a n e o u s e n e r g y s t o r e d i n U i s
1
LV ^
-J L i r 2 = —
%
c o s 2tor t
2 L lr
2r 2
(G .3 6 )
The v o l t a g e a c r o s s t h e c a p a c i t o r i s
vc
/ irdt = " O c
s in “r t
(G ' 37)
The i n s t a n t a n e o u s e n e r g y s t o r e d i n C i s
«C - 1CVC2 - J f r - 2-
<G- 3 8 >
S u b s t i t u t i n g E q . ( G . 2 9 ) , u>r =
» in to (G .3 8 ),
LV ^
Uo =
s i n 2 cor t
u
2R2
r
( G .3 9 )
N ote t h a t t h e a m p l i t u d e o f Ul i s e q u a l t o
in sta n t,
th e t o t a l energy
th e a m p litu d e
o f Ug.
s to r e d i s not a fu n c tio n o ftim e ,
At any
and i s g i v e n
by
U = UL + Uc =j
^
L i2 +
^ C v i2 =
*■
2R^
(G .4 0 )
The e n e r g y d i s s i p a t e d p e r c y c l e i s
T
v 2 T
W= /
i r 2Rdt - —
/
Jo r
R ■'o
c o s 2o)t
A f t e r i n t e g r a t i o n , E q . ( G .4 1 ) i s
(G .4 1 )
128
TT
Vo 2 *
W ---------
(a 4 9 )
The q u a l i t y f a c t o r f o r a s e r i e s
RLC c
i r c u i t w ith s e p a r a te
R
e x c i t e d by a
s i n u s o i d a l s t e a d y s t a t e s o u r c e i s t h e n g i v e n by
Ut + Ur
" “ "
’ n
r
ior L
-
T
-
(G -4 3 )
and i n g e n e r a l i s d i f f e r e n t
from t h e Qf o r t h e t r a n s i e n t c a s e ( G . 2 1 ) .
(2)
w i t h d i e l e c t r i c l o s s , ( F i g . 5)
The e q u i v a l e n t c i r c u i t
V RC
** = ~ RRgC + L
c o s (J)r t
<G.4 4 )
•
•
•
1 . 9
The i n s t a n t a n e o u s e n e r g y s t o r e d i n L i s ^ L i r ,
t
v 2r2p2t
ttt c o s 2wr t
U l = -r- --------L
2 (RcRgC + l ) 2
r
The v o l t a g e a c r o s s t h e c a p a c i t o r
( G .4 5 )
is
v c = V0 coswr t - v Rg - vL
= -
or
V° L— -— [costor t Rc g
(0r Rsinrn
t]
(G .4 6 )
v o L ( l + a)r 2R c2c 2)
‘ — i ^ R~c '<• L
+ +>
CG' 47 )
w here <() = t a n “ ^0Jr Rc
The i n s t a n t a n e o u s e n e r g y s t o r e d i n C i s
1
„
i
Uc = 4 C v i2 = 4
c
2
1
2
Vn2L2C (l + 0)r 2Rc 2C2 )
— -------------------- rx----------(Rc Rg C + U) 2
S u b s t i t u t i n g Eq. ( G . 3 2 ) , n>r =
. v 2n 2p2j
UC = ' T /“2~ ^
r ~ c o s 2 U r t + <j>)
2 (Rc R„C + L) 2
~
n
c o s 2 (u)r t + <(>)
,
(G .4 8 )
' > i n t o Eq * ^G-4 8 ^
(G .49)
129
The sum o f t h e i n s t a n t a n e o u s e n e r g y s t o r e d i n L and C i s t im e d e p e n d e n t .
It
is d iffe re n t
from t h e c a s e o f a RLC s e r i e s c i r c u i t .
a m p l i t u d e o f Ul i s e q u a l t o t h e a m p l i t u d e o f Uq .
Note t h a t t h e
The t im e a v e r a g e o f t h e
e n e r g y s t o r e d i n L and C a r e
!
V0 2r c 2C2l
( G .5 0 )
4 (Rc RgC + L) 2
UC
It
1 rT 1 „ . 9 , .
T L 2 Cvi dt
vo 2Rc2C2L
1
“T
(G .5 1 )
(R^RgC- + li)2
i s i m p o r t a n t t o n o t e t h a t t h e tim e a v e r a g e o f t h e e n e r g y s t o r e d i n L
and C o v e r a c y c l e a r e e q u a l t o e a c h o t h e r .
m ethod t o e x p l a i n i t .
We c a n u se t h e f o l l o w i n g
From Eq. ( G . 3 1 ) , we may h a v e t h e f o l l o w i n g e q u i v a ­
len t c ir c u it,
Rc
2 2
1+ w C Rc
R?C
F i g . G .7 .
The e q u i v a l e n t c i r c u i t o f F i g . G .5 .
T h i s e q u i v a l e n t c i r c u i t i s s i m i l a r t o a s e r i e s RLC c i r c u i t , b u t t h e
e q u i v a l e n t c a p a c i t o r and r e s i s t o r a r e f r e q u e n c y d e p e n d e n t .
From t h e
r e s u l t o f a s e r i e s RLC c i r c u i t , i f we d e f i n e I ^ Z ) = 0 a s t h e r e s o n a n t
c o n d i t i o n * , we know t h a t t h e sum o f t h e e n e r g y s t o r e d i n L and i n C p e r
*We may d e f i n e t h e r e s o n a n t c o n d i t i o n b y o t h e r
c r i t e r i o n s
n o t h a v e Ul = Uq e v e n i n t h e s im p l e RLC c i r c u i t .
, ^U5 t h e n we can
130
c y c le i s equal to 2 tim es th e energy s to r e d in L p e r c y c le .
- y U, b u t U = Ul +
•
T h e r e f o r e Ul = Uq m ust h o l d .
The e n e r g y d i s s i p a t e d i n
g ■ J > M
e
- S
°
So Ul =
p er c y c le i s
.
(G .5 3 )
“r
(R g R g C + L ) 2
The q u a l i t y f a c t o r i s
Q = 2,
Ut + Up
. - k —
id—L
C =_ s _
(e M
T h i s r e s u l t o f t h e q u a l i t y f a c t o r i s t h e same a s t h e q u a l i t y f a c t o r f o r a
s e r i e s RLC c i r c u i t ,
(it
s h o u ld b e n o t e d t h a t i n c o r p o r a t i n g Rg i n t o t h e Q
c a l c u l a t i o n s may a l t e r t h e s e c o n c l u s i o n s . )
(3)
The e q u i v a l e n t c i r c u i t o f a p a r a l l e l c i r c u i t w i t h m a g n e t ic l o s s
( r e f e r F ig .
(G .6 ) ) .
T h i s c a s e i s s i m i l a r t o t h e above c a s e .
We l i s t
th e im p o rta n t
r e s u l t s as fo llo w s:
Rt
L “ u2L2C - RL2C
Z = Rg +
JL —
+ jw -------- 5— ^ ------- T T ~ T ~
®
(1 - oj2LC) 2 + (02C2R2
(1 - tozLC) 2 + cd2C2R£
( g *55)
N ote t h a t a l l e q u i v a l e n t R' , C ' , L 1 a r e f r e q u e n c y d e p e n d e n t .
w_ = —1
RT 2 C
// , 1 - —
h,
/L C
UL = 4 I o 2
L
2
■cos2 mr t
(G .5 7 )
Rl 2 C
UC = 4" I o 2
2
(G .5 6 )
L
c o s 2 (oir t + <|>)
Rt 2 C
( G .5 8 )
131
N ote t h a t t h e a m p l i t u d e o f t h e i n s t a n t e n e r g y s t o r e d i n L i s e q u a l t o t h e
i n s t a n t e n e r g y s t o r e d i n C.
( g ' 60)
W = -“^r L- Io 2 -Rl
r V2C"
(G-61)
W-.L
Q
(G .6 2 )
kL
From above r e s u l t s we c o n c l u d e t h a t u n d e r a s t e a d y s t a t e s i n u s o i d a l
e x c i t a t i o n , i f we d e f i n e l^CZ) = 0 a s t h e r e s o n a n t c o n d i t i o n , t h e n we
have:
(1)
At any i n s t a n t , t h e sura o f t h e e n e r g y s t o r e d i n L and i n C may
n o t b e tim e i n d e p e n d e n t , b u t
(2)
The e n e r g y s t o r e d i n L p e r c y c l e i s e q u a l t o t h e e n e r g y s t o r e d
i n C p e r c y c le r e g a r d le s s o f th e type o f th e l o s s .
(3)
The q u a l i t y f a c t o r o f t h e s e t h r e e c i r c u i t s , ( F i g . G .3 , G .5 ,
oj-L
G .6 ) a r e t h e same and g i v e n b y Q = ——— a s l o n g a s Rg i s i g n o r e d .
We assume t h a t t h e o b s e r v e d f r e q u e n c y d e p e n d e n c e o f t h e
f o r a c a v i t y i s t h e same a s
th at of
im pedance
t h e lumped c i r c u i t s p r e v i o u s l y
d isc u sse d .
We u s e an a c t u a l c a v i t y t o exam ine t h i s p r o b le m .
C o n s i d e r o u r fa m i­
l i a r p a r a l l e l p l a n e c a v i t y e x c i t e d i n t h e l o w e s t o r d e r TEM mode and
f i l l e d w ith a lo s s y d i e l e c t r i c .
132
y ,e ,a
-in
Z = 0
F i g . G.8 .
Z = d
The c o n f i g u r a t i o n o f t h e c a v i t y .
The i n p u t im pedance o f t h i s c a v i t y i s
(G .6 3 )
Z^n = jn ta n K d
D e f i n e t h e r e s o n a n t c o n d i t i o n a s Im C Z ^ ) - 0 , and we h a v e
8 s in 2 8 d + a s i n h 2ad = 0
( G .6 4 )
w here
J£W_
K
(G .6 5 )
K = 8 - jot
( G .6 6 )
n =
8 - »
( / i
* - 2i _
0)2e 2
J >je [ . / T
°2
a " “ 7“ 2 ^ 1 + ^ 2
* i|
(G .6 7 )
1)
(G .68)
-
The E - f i e l d a t any i n s t a n t i n t h e c a v i t y i s
(G .6 9 )
E = E0 s i n K ( Z - d ) s i n ait
A p p ly in g M a x w e ll's e q u a t i o n
—
K
H = En
c o s K ( z - d ) c o s o it
u ojp
3H
E = - P -gjr , we h a v e
(G.70)
133
The e n e r g y d e n s i t y o f E - f i e l d and H - f i e l d a t any i n s t a n t i s
UE = y l
Eo I
UH -
2 I s in K ( Z - d ) |
2 s i n 2 u)t
(G .7 1 )
2 I Eo I 2 I c o s K (Z -d ) | 2 c o s 2(ot
(G .7 2 )
At any i n s t a n t , t h e e n e r g y s t o r e d i n E - f i e l d and H - f i e l d p e r u n i t a r e a
are
UE
e
=y l
UH =
f
. 9
Eo I
I
Eo I
d
s i n 2 tot J
| s in K ( Z - d ) |
2 c o s 2 tot
|~ |
2 dZ
2 f Q | cos
(G .7 3 )
K(Z-d) | 2 dZ( G .7 4 )
A f t e r i n t e g r a t i n g , we h a v e
Ug = ■— | E0 | 2 s i n 2 tot [ - ^ s i n h
% = -‘-" 2
2otd -
I Eo I 2 c o s 2<ot
s i n h 2ad
s in 2 0 d ]
( G .7 5 )
+ -i^ sin 2 3 d ]
( G .7 6 )
S u b s t i t u t i n g t h e r e s u l t o f t h e r e s o n a n c e c o n d i t i o n , Eq. ( G . 6 4 ) , a s i n h 2 a d +
g s in 2 0 d - 0 , i n t o Eq. ( G .7 5 ) and ( G . 7 6 ) ,
2
UE = y I Eo I 2 s i n 2 tot • ”
1*1
U =
2
H 2 to V
2
^2S
s i n h 2nd
| e q | 2 c o s 2 tot • ——
° '
4a32
s i n h 2ad
( G .7 7 )
( G .7 8 )
We e l i m i n a t e t h e e x p l i c i t a p p e a r a n c e o f t h e c o n s t i t u t i o n p a r a m e t e r s .
From Eq. ( G .6 7 ) , ( G .6 8 ) ,
02- a 2 = to2 p e , | K | 2 = a 2 + 82 , so t h a t
134
e i „ i 2
a 2 + g2
s i n h 2ad
UH = t ( e o I
c o s 2 u)t
2 ' “° '
4a02
(G .7 9 )
The t o t a l s t o r e d e n e r g y , t h e sum o f t h e E - f i e l d e n e r g y and H - f i e l d
e n e r g y , i s tim e i n d e p e n d e n t and g i v e n by
e
U = UE + UH = — ) E0
2 a 2 + 02
s i n h 2ad
4ag2
(G .8 0 )
o r by
U = - | | E0 | 2 [ J j- s i n h 2c<d - A
s i n 20d]
(G .8 1 )
The e n e r g y d i s s i p i a t e d p e r c y c l e p e r u n i t a r e a i s
T d
W = / / <J | E | 2 dZ d t
o o
=-^-a I Eo I 2 tL4Aa s i n h 2ad - 4 48
a s i n 23d]
(G .8 2 )
and t h u s t h e q u a l i t y f a c t o r i s
Q = 2ir •
UE + UH
W
ue
( G .8 3 )
E q . (G .8 5 ) i s t h e same a s t h e e q u a t i o n o f t h e q u a l i t y f a c t o r i n r e f e r e n c e
9 7 , p . 42 5 .
The lumped c i r c u i t e q u i v a l e n t w hich c o u l d be u s e d t o r e p r e s e n t t h i s
c a v i t y i n t h e r e g i o n o f r e s o n a n c e i s a p a r a l l e l RLC c i r c u i t w i t h s e p a r a t e
R, and shown i n t h e f o l l o w i n g f i g u r e .
F i g . G .9 .
The p a r a l l e l RLC c i r c u i t w i t h a p a r a l l e l s i n u s o i d a l
v o lta g e source.
135
The impedance o f t h i s p a r a l l e l RLC c i r c u i t i s
Z
, j
[R(1-u)2LC) ] 2 + ui2L2
Once a g a i n , d e f i n i n g ^ ( Z )
j^ L (l-A c )
[R( 1-(D2LC) ] 2 + U)2L2
(G 84)
= 0 a s t h e r e s o n a n t c o n d i t i o n , we h a v e
“ r =/LC
(G,85)
At r e s o n a n c e , Z = R, u s i n g t h e same p r o c e d u r e s a s used b e f o r e , we h a v e
2
tt,
L
=
2cor 2L
s i n 2ait
(G.86)
2
Ur, = —^275-COS2tOr t
c
2tor 2L
(G.87)
Then, U = UL + UC = 2 J -%-
<G-8 8 >
The e n e r g y d i s s i p a t e d i n R p e r c y c l e i s g i v e n by
w -
(« -
b w
Thus, th e q u a l i t y f a c t o r i s
x
2Trj_U = R = W
ur L
r
Note t h a t ,
i f we r e l a t e t h e v a l u e s
(G>90)
o f Rand
C t o o and e , we h a v e , f o r an
i d e a l i z e d p a r a l l e l p l a t e c a p a c i t o r w ith as e n s i t i v i t y ,
p,
(G-9 °
C = e 4d
(G. 92 )
136
S u b s t i t u t i n g E q . ( G . 9 1 ) , (G„92) i n t o
(G .90),
a) e
Q
Eq.
(G.93)
(G.93)
i s t h e same a s E q . ( G . 8 3 ) , w h i c h was o b t a i n e d by s o l v i n g f i e l d
equations in the c a v ity .
F i n a l l y , we c o n c l u d e t h a t
(1)
When a s e r i e s RLC c i r c u i t w i t h s e p a r a t e R i s e x c i t e d b y a t r a n s i e n t
ex citation,
the energy sto re d
s to re d in L per c y c le .
in C per cycle i s not equal to the energy
The q u a l i t y f a c t o r o f t h i s c a s e w i l l be d i f f e r e n t
from t h e q u a l i t y f a c t o r o f t h e same c i r c u i t when e x c i t e d by a s i n u s o i d a l
steady s ta t e
(2)
ex citatio n .
When a s e r i e s c i r c u i t w i t h a p a r a l l e l c a p a c i t o r l o s s ,
or a p arallel
c i r c u i t with a s e r i e s
by a s in u s o i d a l
inductor lo ss ,
steady s t a t e e x c it a t i o n ,
(Fig.
G.6 ) , i s e x c it e d
and when we d e f i n e
t h e r e s o n a n t c o n d i t i o n , th e time a v erage e n e rg y s t o r e d
w i l l be e q u a l t o t h e t i m e a v e r a g e e n e r g y s t o r e d
(F ig . G .5),
^(Z )
= 0 as
in L over a cycle
in C over a c y c le ,
r e g a r d l e s s o f th e type of th e l o s s .
(3)
I f we d e f i n e t h e r e s o n a n t c o n d i t i o n by o t h e r c r i t e r i a ,
equivalent to ^ ( z )
w h ic h a r e n o t
= 0 , then the conclusions in (2 ) a re in c o r r e c t
for a
high loss c i r c u i t .
(4)
An i d e a l p a r a l l e l p l a t e c a v i t y f i l l e d w i t h l o s s y d i e l e c t r i c
r e p r e s e n t e d by a p a r a l l e l
U)£
t h e c a v i t y i s Q = ——.
RLC e q u i v a l e n t c i r c u i t .
c o u l d be
The q u a l i t y f a c t o r o f
137
APPENDIX H.
E-FIELDS AND H-FIELDS IN THE CAVITY
I f a s l a b i s ba c ke d by a p e r f e c t m e t a l ,
i t was shown i n S e c t i o n
I I I . 2 . 3 , t h a t t h e t o t a l r e f l e c t i o n and t r a n s m i s s i o n c o e f f i c i e n t a t t h e
i n t e r f a c e a r e g i v e n a s pm and Tm.
In t h i s a p p e n d i x , we w i l l c a l c u l a t e
t h e E - f i e l d s and H - f i e l d s i n t h e whole c a v i t y .
The c o n f i g u r a t i o n o f t h e c a v i t y i s shown i n F i g . H . l and a l l
E -fie ld s are l i s t e d .
At h e r e we assume t h a t t h e s l a b i s an i s o t r o p i c m a t e r i a l , and:
T
= the tra n s m is s io n c o e f f i c i e n t o f the coupling wall
p = the r e f l e c t i o n c o e f f i c i e n t of the coupling wall
P23~e"’j2Kd
Pm = ------------- 7-—,- was g i v e n i n S e c t i o n I I I . 2 . 3 .
1-P23e _ 3
tm
1 + P23
-----------------TovT was g i v e n i n S e c t i o n I I I . 2 . 3 .
1 - p23e“ J
The t o t a l E - f i e l d s i n c i d e n t upon t h e s l a b a r e
Ej. = TA8 - j Ko V ^ o Z ♦ T P %A e - j 3 Ko
—
—
'^ L i
1 - ppme
° ~ 3K° Z
The t o t a l E - f i e l d s r e f l e c t e d
+
•••
(H- l )
from t h e s l a b a r e
Er = TpmA a - f Co ¥ KZ ♦ Tppn 2Ae-j 3K° V jK° Z ♦ Tp2pm3Ae - j 5Ko H 3K» Z
.
TAe~jK° U
-jK 0Z
- i 2 K 1„ m
1 - ppme
.
.
xAe""^0 ^0
D e f i n e E0 = ------ — ----------------------------------------------------------------------------------- (H .3 )
1 - ppme~2 0 0
(slab)
(free space)
» - jJK °n Z
xAe
tAe- j Ko ^ - j K° Z
>
tp,
>
Ae- j 2K0 l ^ K 0 Z
-» % TAe- j K° V iKZ
xppmA e " j 2Ko V j Ko Z
>
xppmA e - j 3Ko 1% - j Ko Z
>
xppm2Ae":i4Kol% j KoZ
xppm2A e " j 3K° 1<fejKoZ
<-------
<
™
2
2
a
TP Pm Ae
-j^K0lo-jK 0Z
e J °
- % TAe-jK« V j 2Kde jKZ
*
TpTmpmAe ^31t» 1%” '’KZ
2
2 .
“ j5K0 l o j K n Z
xpz pmzAe J ° e J °
» Tp2 % pm2A e - 35K« > V j KZ
Fig. H .l.
The c o n f i g u r a t i o n o f t h e c a v i t y w i t h a l l E - f i e l d s l i s t e d ,
The total E-field in the free space region is
Ef - E0 ( e “ j K° Z + pme jK° Z)
(H.4)
o r w i t h t i m e - v a r y i n g form
Ef = Re (E0 ( e " jKo Z + pme j Ko Z) e j a ) t }
= | E0 ( e j Ro Z + p j jj e ^ o 2) | cos (ui t + <|>Ef )
( H .5 )
iK Z
iK Z
i s t h e argument o f E0 ( e J ° + pmeJ ° )
wher e
A p p l y i n g Maxwell e q u a t i o n
E = -jujiH t o Eq. ( H . 4 ) , Hf i s g i v e n by
/ “ jK o 2
jKn Z»
Hf = — ( e J ° _ Pme J ° )
^o
-
(H.6 )
E °
o r w i t h t i m e - v a r y i n g form
Hf = R
(
E
_j K° Z - Pme jK° Z) e jai,:}
= | —2 (e ^K° Z - pme^ K° Z) | c o s (ait + <t>nf)
where
( H .7 )
i s t h e argument o f —2 ( e j R o z _ pme ^ o Z)
ho
The t o t a l E - f i e l d s i n t h e s l a b a r e
Es = TmTAe ^KZe ^K° l o + r ppmTmAe ^Kze j 3Ko lo + Tp2 pm2 Tme ^Kze j 3Ko 1° +
•
- Tmxe - j K o V j 2 Kde jKZ _ TppmTmAe- j 3 K o V j 2 K d e- j K Z + . . .
TAe~jK o l °
- ppme - jJ2 K Qlo
-jK Z
m
"
TAe~jK o l °
-j2 K d jKZ
,1 - ppme - jJ2 K 0 ]o m
= E0 Tm( e - i KZ- e j KZe - j 2Kd)
Eq. ( H . 8 ) can be r e w r i t t e n as t h e form o f t h e s t a n d i n g wave
(H.8 )
140
Eg = TmE0 e " j Kd[ - 2 j s i n K ( z - d ) ]
(H .9)
The t r a n s m i s s i o n c o e f f i c i e n t xm can be w r i t t e n i n t e r m s o f 1 + pm,
(1 + P2 3 X I - e J ZKd)
-iK d
. .
1 + Pm - ----------------------- 3T2Kd-------------J % ( +2 j s i n K d )
(H .10)
1 - p23e J
So, Es can be r e p r e s e n t e d by
Es = “ Eo 4 - - U m s i n K ( Z - d )
(H .ll)
o r w i t h t h e t im e v a r y i n g from
Es = Re [ - V
g i nKdm- s i n K ( Z - d ) e ^ <1)t]
“ I “ Eo _sinKdm‘ s i n K ( Z - d ) | c o s ( w t + <t>ES^
(H.12 )
1 + pm
where <i>ES i s t h e argument o f - E 0 ~ s i n K ( Z - d )
A p p l y i n g Maxwell e q u a t i o n
Hs
s
E = -jrnyH t o Eq. ( H . l l ) , Hs i s g i v e n by
E n d + Pm) *K
----- :------ ------------ c o s K ( Z -d )
jwpsinKd
(H . 1 3 )
o r w i t h t i m e v a r y i n g form
**
nr
Eq (1 + Pm) *K
jut.
]
H» - ■ e t - jtiipsinKd ------- c o s K ( Z - d ) e
' I “ jSpoinKd")
co s K ( Z -d ) | c o e ( « * 4>h S>
where <j>HS i s t h e argument o f
(H . 1 4 )
E0 ( i + Pm) • K
------ cosK<z - d >
I f t h e s l a b i s an a n i s o t r o p i c m a t e r i a l , a l l E - f i e l d s a r e p l o t t e d i n t h e
follow ing f ig u r e .
. - j K 0 ZCtAe
° 1
Ar , ?
.'J.
. ' i . . - j K n Z - j K n l0
+ (l-jj)je
° e J ° °
-------- >
Ar ,C .'l.
“ jK.Z
,C
- jK _ Z . - j K 0 l_
-------- >
Tf [ ( i + j j ) P m + + ( i _ J 3 ^ Pm-^e ^K° Ze ^ 2Kol°
Tf t ( i +jj) Pm + + ( i - j j ) P m- ] e +^K° Ze " ^ Ko1°
<----------
Ar /C1 .Cv
+jK+Z - j K +2d
* *
jK_Z - j 2 K _ d . - i K - L ,
'C2 [ ( l + J J )Tm+e
e
+ ( 1- J J ) Tm-e
e J
]e J 0 0
<--------
Tp f ^ i + j j ) P m + + ( i - j j ) P m - ] e _ ^KoZe _^ 2Kol°
----------->
T p | t ( i + j j ) P m + + ( i - j j ) P m - ] e 3K° Ze
--------->
T p | [ ( i + j j ) p m+tm+e~3K+Z + ( i - j j ) T m_pm_e“ 3K- Z] e “ 3Kolo
---------- >
Ae j K° Zi
>
t p ^ [ ( i + j j ) p m+2 + ( i - j j ) p m_ 2 ]e ^K° Ze j 3Ko l0
<--------i
1
1
,
•
|
1
!
i
•
i
i
i
Us---------------------------------------- i ----------------------------------------- jl «------------------------------------------------ a --------------------------------------------------- 9
F i g . H .2 .
The E - f i e l d i n t h e c a v i t y p a r t i a l l y f i l l e d by an a n i s o t r o p i c s l a b ,
142
F o l l o w i n g t h e same p r o c e d u r e s a s we u s e d , we c a n o b t a i n t h e E - f i e l d s
and H - f i e l d s i n t h e f r e e s p a c e r e g i o n and i n t h e s l a b , and shown b e l o w .
E f ( Z , t ) = Rg{ [ [E0+ + ( e " jK° Z + Pm+e jK° Z) + E0_ ( e ‘ jKo Z + pm_ e +j Ko Z) ] i +
[E0+ ( e " jK° Z + pm+e jK° Z) - E0_ ( e ~ jK° Z + pm. e ^ ° Z) ] j j ] e j a ) t }
( H . 15)
~
s in K + ( Z -d )
Es ( Z , t ) = Re { [ [ - E 0 + (1 + pm+)
sinK+d
- E0_ ( l + pm_)
[ - E 0+( l + Pm+)
oinU ( 7-rO A
sinK_d ^ +
sinK+CZ-d)
sinK_(Z-d)
a
ginK+d
+ E0_ ( l + Pm_)
sin K_d l j j l e
>
(H .1 6 )
Hf ( Z , t ) = R g t l
MO
E0+ ( e " iKo Z- p m+ej Ko Z) + L ^ ( e ^ Ko Z- Pm. e i 'Ko Z) ] j i +
'o
r 1 _ , -jKnZ
jKn Z.
1 „ , -jK 0Z
jK0 Z. . a ju it .
[ - E0+( e J o
- pm+e J o ) + —-E0_ ( e
° - pm_ e J ° ) ] j ] e J
}
MO
nO
(H .1 7)
K+s i n K . ( Z - d )
K_s in K_( Z- d)
a
Hg ( Z , t ) = Re { [ [ - E 0+ ( l + p m+) j uyginK+d
+ Ep - ( 1+ Pm-)jajy s i nK_ d
li1 +
K+sinK+CZ-d)
[ Eo+ ( 1 +pm+ ) jojys inK+d
,
K_si nK_(Z-d)
+ Eo - ( 1+pm-) ja)uginK_d
a
i wt
}
( H . 18)
.
where
„
Ec+ - 2 ( 1_
^ a- i ^
—A e ~ ^ ° ^. °
En_ = ------ T
2 ( l - p p m_ e _ j 2Ko 1°)
( H . 19)
(H .2 0)
143
A PPE N D IX I .
PROOF THAT UE f + UE s
= UH f + UHg AT RESONANCE D EFINED
BY T j Z i n ) = 0
F rom
UEf
text,
UE f ,
UE g ,
UE f
I Eo I 2 t ( l + I Pm | 2 )
and
UEg
are
• lo + -T23- ( 1 - c o s 2Kot) + ^
Ko
sin 2 1 ^
(3.76)
,
uEs
%f =
- I — s i n h 2ad - - I — s i n 20d
I E° I 2 I (1 + Pm) I
*
“T - ! E° I 2 t (1 + l Pml 2)
c o s h 2ad - c os 20d
* ]o ‘ ^
(3 ??)
( 1 “ c o s 2Ko t> " ^ s ^ K o y
(3.78.b)
o - o
TT . _ a 2 + 02 i -
Hs
4oi2 p0
„
I 2 | n
° '
s i n h 2 ad + - | - r s i n 20d
p ) I 2 . _ 2a ______________ 20________
m 1
c o s h 2otd “ c o s 2f3d
( 3 79)
I f we d e f i n e t h e r e s o n a n t c o n d i t i o n a s I ^ Z i n ^ = 0 , from S e c t i o n I I I . 2 . 2 ,
we ha ve f o r t h e un c o u p le d c a v i t y ,
pmj c os 2 ^ 1 0 - pmr s i n 2KQ10
(1.1)
Then Eq. ( 3 . 7 6 ) and ( 3 . 7 8 . b ) become
«E£ - T - l
*ol
%f = x ! ^
2
[ <1 + 1 % l
2)
’
2 1 ( 1 + | p» i 2 ) ' ■“ " T ^ r 1
( I ’2)
( I -3)
144
Upon s u b t r a c t i o n , we h a v e
UEf - % f = - ^ 1
Eo l
2
*
(1.4)
2 ^ -
From S e c t i o n I I . 2 . 3 ,
"j2Kd
„ _ P e J
Pm “
1 - pe“ J 2Kd
Kq - K
where p = T T k 0
(1.5)
( 1 .6)
= pr + J pj
t h e n pmj i s g i v e n b y
e- 2 ad
Pmj = ------------- r n r j — r — [ 2 p j s i n h 2 a d + ( l - p r z - p j z ) s i n 2 0 d ]
I 1 - Pe
|
(1.7)
From E q . ( I . 6 )
Kp - g - jo t
_
P
3 “ j “ + Kq
.
2
Pr
Pi =
J
=
^
Kq2 + g2 + a 2 + 2Kq3
.2 - ________ 4Ko ^____________
K0 2 + a 2 + g2 + 2KQ0
2Ko«
(1 .8)
(1.9)
( 1 . 10)
K0 2 + a 2 + 02 + 2Kq0
1 + pr 2 + p j 2 + 2 pr =
s u b s titu tin g Eq. (1 .9 ),
Pm;
J
Kq2 - g2 - a 2 + 2jctKo
4K0 2
K0 2 + ot2 + g2 + 2Kq0
( 1 . 11)
( I . 10) i n t o ( 1 . 7 )
e _2ad
4Kq
----------------- ;--------7 * — «---- 5------- 7,---------- [ a s i n h 2 a d + 0s i n 20d]
| 1 - p e - 2JKd | 2
K0 2 + a 2 + g2 + 2Kog
( 1 . 12)
S u b s t i t u t i n g E q . ( 1 . 1 2 ) i n t o ( 1 . 4 ) , we hav e
145
2
Eq | fEg
u Ef ” uHf ” "j---------------. n„* , 2------ 1— “— „
1 - pe“ i 2Kd | 2 (Ko 2+ a 2+02 + 2Kq3)
[ a s i n h 2 a d + 8 si n 2 0 d ]
(1.13)
S u b s t r a c t i n g Eq. ( 3 . 7 9 ) by ( 3 . 7 7 ) , we hav e
-
<%8 - j \
S o I 2 I C l+ P i.) I 2 •c o s h 2 a d 1-
2
2
+ 4 s s i n 2 3 d ( - - v* 3
28
0)2 yo
- e' =
..?2 .t . E 2 + e ' = / e *2 + e "2
w2 y0
+e ' =
l
2
pm
=
'
+e ' ) ]
- — a* g2 - e ' = / e ' 2 + e"2
0)z iio
1 +
co , 2 M
e,)
(1.14)
(1.15)
w yQ
( 1 . 16 )
e“ 2 a d ( l + pr 2 + p i 2 + 2pr )
-------------------------------— ----------- 2( c o s h 2 ad - c o s 20d)
| i _ pe- j 2Kd | 2
(1.17)
S u b s t i t u t i n g Eq. ( 1 . 11) i n t o ( 1 . 1 7 )
r
1 + Pm I
2
2e “ 2 a d ( c o s h 2 ad - c o s 20d)
j i _ pe-j2 K d | 2
4Ko"
Kg2 + a 2 + 02 + 2Kg0
(1.18)
S u b s t i t u t i n g Eq. ( 1 . 1 5 ) ,
2
( 1 . 1 6 ) and ( 1 . 1 8 ) i n t o Eq. ( 1 . 1 4 ) we hav e
I Eg
2
K,a .2
uHs “ ^Es =
[ a s i n h 2 a d + 8sin2(3d]
| i _ pe-j2 K d | 2(Kq2
a 2 + $2 + 2Kg3)
(1.19)
146
S i n c e Kq2 = u>2 ji0 e0 , so
2 I Eo I 2 eo
Uug “ UEs = ----------------------------------------------------------------- [ a s i n h 2 a d + 3s in 2 6 d ]
| 1 - o e - J 2Kd | 2(Ko2 t a2 + s2 + 2K<)S,
( 1 . 20 )
From Eq. ( 1 . 1 3 ) and ( 1 . 2 0 ) we know
«
uE f “ uHf = UHs “ UEs
(1.21)
So we h a v e
uEf + uEs = uHf + uHs
Q.E.D.
(1.22)
147
APPENDIX J .
THE POYNTING THEOREM IN COMPLEX FORM
M a x w e l l 's c u r l e q u a t i o n s c a n be e x p r e s s e d i n p h a s o r form as
J7 x h
= ( a + jaie)E + T
(J.l)
7 X E = -jwp ¥
(J .2)
i n which T r e p r e s e n t s nonohmic c u r r e n t ,
such a s c o n n e c t i o n c u r r e n t o r
s p e c i f ie d source c u r r e n t .
S u b s t i t u t i n g Eq. ( J . l )
7
*I
x I*
and ( J . 2 )
= H* • V * E -
in to the follow ing v e c to r i d e n t i t y ,
f
• V * H*
( J .3)
(a
- j m e ) ¥ •I * - E•3*
(J.4)
we ha ve
7 •
e
x H* = -juiviH • H* -
I n t e g r a t i n g o v e r t h e volume V and a p p l y i n g t h e d i v e r g e n c e th eo re m t o t h e
l e f t hand s i d e , we ha ve
/ E x I*
*s
• dS = - i w J (pH • H* - eE • E*)dV - / 0E • E* dv
V
v
-
/ E
Jv
• J * dv
(J.5)
Ta ki ng t h e r e a l p a r t o f Eq. ( J . 5 ) and d i v i d i n g b y 2, we h a v e
^2 Re e /j E x g* • dS
=-Jr f a l • E* dv2 - 4e 'v
R e
2 Jv
/ E• J* dv
( J .6)
The l e f t t e r m r e p r e s e n t s t h e a v e r a g e power f l o w o u t w a rd t h e s u r f a c e which
i s t h e i n t e r f a c e o f t h e f r e e s p a c e and t h e s l a b i n o u r pr o b le m .
first
The
term o f t h e r i g h t hand s i d e r e p r e s e n t s t h e a v e r a g e power l o s s due
148
to the non-zero c o n d u c tiv ity .
face i t
Note t h a t , a t t h e l e f t s i d e o f t h e i n t e r ­
i s t h e f r e e s p a c e r e g i o n , so a = 0 , and t h e f i r s t t e r m o f t h e
r i g h t hand s i d e v a n i s h e d .
The se ocn d te r m o f t h e r i g h t hand s i d e r e p r e ­
s e n t s t h e a v e r a g e power l o s s due t o t h e c u r r e n t i n t h e s l a b ,
t i v e s i g n r e p r e s e n t s t h e power flo w i n t o t h e s l a b .
the nega­
So, we ca n c a l c u l a t e
t h e a v e r a g e power l o s s by
P = |
Re / E x H* • ds
(J.7)
o r by
P = i - R e Jy E • ' J * dv
(J .8)
149
APPENDIX K.
THE QUALITY FACTOR WHEN THE SLAB IS A GOOD CONDUCTOR
From Eq. ( 4 . 1 ) , t h e q u a l i t y f a c t o r i s
(V /^*o<eo + / e '2 + e " 2 ) • ^
Q '
h ''i
/ 2< C
The a t t e n u a t i o n f a c t o r a and t h e p h a s e c o n s t a n t 0 o f a l o s s y m a t e r i a l
w i t h a complex p e r m i t t i v i t y e = e 1 - j e " i s
/— / - e ' + / e '2 + e " 7
a = ai/iv, / ------------
r
e =
/ s '
+ /
(K. 2)
. e] g .
(K. 2)
I f t h e s l a b h a s a h i g h c o n d u c t i v i t y , e" =-^- , t h e n ,
( 1)
e" »
( 2)
0 - a
e ',
e" »
e0
So t h e q u a l i t y f a c t o r becomes
0
^r 2 P0 * ( e 0 + ^
+ e"2)
• lp
cor /ii^ V 2 ( e ' + / e ' + e "2
„ <7
e„ + / e ' 2 + e"^
2“ r z M0 ‘
—o
—
. i0
i— / s ' + / e ' z + e"2
2<V
r ^ _0
o /• -------- 2; --------------= alo
(K. 4)
The s k i n d e p t h 6 i s
6 =—
a
(K.5)
150
I f t h e s l a b i s s i m i l a r t o a good c o n d u c t o r , t h e r e s o n a n c e c o n d i t i o n i s
l 0 =-f*-
(K.6 )
S u b s t i t u t i n g Eq. ( K .5 ) and ( K . 6 ) , we h a v e
Q — §“
(K .7 )
151
APPENDIX L.
THE QUALITY FACTOR WHEN WE OBSERVE THE PARALLEL COMPONENT
OF THE E-FIELD (FIG. 3 . 1 2 )
From E q . ( 3 . 1 1 9 ) t o ( 3 . 1 2 2 ) , t h e p a r a l l e l component ( x -c o m p o n e n t ) o f
t h e E - f i e l d and y - c o m p o n e n t , H - f i e l d ,
i n t h e c a v i t y f o r an a r b i t r a r y
sa mp le a r e
Ef 3 Eo + ( e~jK° Z + Pm+e ^K° Z) + E0_ ( e ~ ^ K° Z + pm- e ^ K° Z)
( L .1)
s inKj. ( Z-d )
s inK_ ( Z-d )
Es 3 _Eo+ (1 + Pm+> ginK+d
" E° “ (1 + p®-* sinK_d
( L *2)
Hf 3
(L.3)
( e _iK° Z - P m + e ^ o 2)
no
Hs
=
Eo + d
+
l'o
( e ~ j Ko Z - pm_ e jK° Z)
v K+s inK+ ( Z-d )
K_si nK_ (Z-d )
jmysinK+d
+ E° - (1 + Pm_) jwysinK_d
Pm+ >
(L,4)
The f s u b s c r i p t r e f e r s t o t h e f r e e s p a c e r e g i o n and t h e s s u b s c r i p t
r e f e r s t o t h e sample r e g i o n .
S i n c e t h e s l a b i s a s s i g n e d t o be a t h i c k
s a m p l e , t h e e n e r g y s t o r e d i n t h e s l a b c a n be n e g l e c t e d ( S e c t i o n I I I . 5 ) .
The E - f i e l d and H - f i e l d i n t h e f r e e s p a c e r e g i o n ca n t h e n be w r i t t e n as
Ef -
e 0+
(.-* •* ♦
„ J
V
+ C(^ J V >
- Eo+
U+O [e -* ° z + f s i j ^ s r e3 V ]
= I*.
( 1 + 0 [ e ' iK ° Z + Rc e jK° Z]
(L .5 )
( 1 + ; ) [ e ' iK° Z - Rc e iK° Z]
(L .6)
Hf — ^
"o
E° where ?r = ——
Eo+
152
v - Pm+ + ^pmKc “
1 + 5
Usi ng t h e r e s u l t o f Eq. ( 3 . 8 1 ) ,
(3.83.b),
the energy s to re d in the
E - f i e l d and H - f i e l d a r e
UEf = - ^ - 1 E0+ | 2 | 1+5 | 2 [ ( 1 + | Rc | 2 ) * lo
p
•
+
R
(1 - c o s 2 K o l ) +
s in 2 K o l]
(L«7)
UHf “ TT | E0+ | 2 | 1+? | 2 [ (1 + | Rc I 2) ’ lo
- —
(1 - c o s 2 K o l ) -
s in 2 K o l]
(L .8 )
The t o t a l e n e r g y s t o r e d i n t h e c a v i t y i s
U - 4 s ! E0 + | 2 | 1+5 | 2 [ ( 1 + | Rc | 2 ) • U
(L.9)
The a v e r a g e power l o s s i n t h e s l a b i s
P —£ ^
(Ef x Hf * )
= 2 ^ ! Eo + l 2 i 1+5 I 2 (1 " I Rd
2)
( L ’ 10)
The q u a l i t y f a c t o r o f t h e c a v i t y i s t h e r e f o r e
«
' P
P
0>r no eo ( l + |
rc
| 2 ) • !<,
1 - I Rc I 2
For c h e c k i n g t h i s e q u a t i o n , we c a l c u l a t e Q v e r s u s H^c f ° r a t h i c k
InS b 95 a t room t e m p e r a t u r e and compose t h e r e s u l t w i t h t h e m ea s ur ed
v a l u e s i n R e f . 95.
We p l o t i t
in Fig. L .l .
153
2500
ca ic u le d
2000
m easured’
1500
500
Bq ,KG
F i g . L . l . Q v e r s u s H,
f o r t h i c k InSb
95
APPENDIX M.
THE DERIVATION OF EQ. ( 5 . 5 ) .
Eq. ( 5 . 4 ) i s
* | p j 2
= 0
30d
( M .l )
From Eq. ( 5 . 3 ) , | % | 2 i s
p
m
. 2
Pr 2 + P i 2 + e
“ 2pr c o s 2 gde 2ad + 2 p 4s i n 20d e 2otd
= — £_____ J_________________ z___________________ J----------------------- (M.2)
1 + ( p r 2 + P j 2 ) e “ 4oK* - 2 pr c o s 20d e “ 2od _ 2 p j s i n 2 0 d e - 2a<*
S u b s t i t u t i n g E q . (M.2) i n t o ( M . l ) , we hav e
s i n 2 g d [ P r ( l - P r 2- P j 2 ) ( l - e 4 a d )]
+ c o s 20d [ p j ( l + p r 2+ p j 2 ) ( i +e~4 a d ) ] _ 4pr p j e- 2 a d - q
Let
(M.3)
A = Pr ( l - P r 2 - p j 2 ) ( 1 _ e -4otd)
B = P j ( l + P r 2+ P j 2 ) ( l + e - 4 a d )
C = 4 p r P j e “ 2ad
Eq. (M.3) becomes
A si n2g d + Bcos23d - C = 0
(M.4)
D i v i d i n g e a ch t e r m o f Eq. (4 ) by / A2 + B , we ha ve
- 7=s====rr s i n 2 0 d +
/A 2 + B2
/A 2 + B2
D e f i n e 0 = t a n ” * — + nir,
c o s 2 gd -
n = 0,1,2,
>* C f
vA2 + B2
•••
= 0
(M.5)
(M.6 )
Eq. (M.5) becomes
c os9sin20d + sin0cos20d -
,
Ar
/A2 + B2
= 0
(M.7)
155
That i s
s i n ( 2 0 d + 0) = —= = = = =
/A2 + B2
(M.8 )
Then we hav e
20d + 0 = s i n - 1
(-=========-)+ 2pir, p = 0 , 1 , 2 *
/Az + Bz
(M.9)
S u b s t i t u t i n g Eq. (M.6 ) i n t o (M.9) we h a v e
20d = s i n “ l - = = = =
/A2 + B2
- t a n - ! -r- -mr + 2pir
A
0d = - | s i n - 1 - ^ 2 = 5
" 7 ta n -1 |
(M.10)
or,
_ 1 nlT +
p7r
(M. n )
n = 0 , 1 , 2 , ■*•
p = 0 , 1 , 2 , ***
P _ T n > 0
Eq. ( M . l l ) can be r e w r i t t e n as
0d = < | > + - | i r
where * = - -
1
_i B
1 . _i
tan 1 T * 1 a m
m = 0 , 1 , 2 , *••
(M.12)
C
Using t h e same p a r a m e t e r s which were used t o p l o t F i g . 5 . 1 , we p l o t
cf> v e r s u s 0d i n t h e f o l l o w i n g F i g u r e .
156
0.02
0-01
F i g . M . l . <f>v e r s u s fid.
157
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165
VITA
Name:
Shou-Hsu Chi
Born:
Dec. 2 5 ,
E ducation:
1943
Chung Cheng I n s t i t u t e o f T e c h n o l o g y , Ta iw a n ,
R epublic o f China.
1969, B . S . E . E .
1973, M . S . E . E .
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