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Finite-element analysis of microwave passive devices and ferrite-tuned antennas

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F I N I T E - E L E M E N T A N A L Y S IS O F M IC R O W A V E P A S S IV E D E V IC E S
A N D F E R R IT E - T U N E D A N T E N N A S
by
A n a sta sis C . P o iy c a rp o u
A D is s e rta tio n Presented in P a rtia l F u lfillm e n t
o f th e R e q u ire m e n ts fo r th e D egree
D o c to r o f P h ilo s o p h y
A R IZ O N A S T A T E U N I V E R S IT Y
M a y 1998
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UMI Number: 9817085
UMI Microform 9817085
Copyright 1998, by UMI Company. All rights reserved.
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UMI
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F IN I T E - E L E M E N T A N A L Y S IS O F M IC R O W A V E P A S S IV E D E V IC E S
A N D F E R R IT E - T U N E D A N T E N N A S
by
A nastasis C . P o ly c a rp o u
has been a p p ro v e d
J a n u a ry 1998
APPROVED:
C-
'
fi
. C h a irp e rso n
S u p e rviso ry C o m m itte e
PTED:
[e p a rtm e m C h a irp e rs o n
D ean, G ra d u a te C ollege
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ABSTRACT
T h is d is s e rta tio n deals w ith th e fo rm u la tio n o f a v e c to r fin ite e le m e n t m e th o d
for th e a na lysis o f m ic ro w a v e c irc u its , e le c tro n ic packages a n d fe rrite -tu n e d c a v ity backed slo t antennas.
A g e n e ra lize d eigenvalue p ro b le m is firs t fo rm u la te d to e x ­
a m in e th e d isp ersive p ro p a g a tio n c h a ra c te ris tic s o f tw o -d im e n s io n a l a n is o tro p ic and
lossy m icro w a ve s tru c tu re s . Q u a n titie s such as th e p ro p a g a tio n c o n s ta n t, a tte n u a ­
tio n c o n s ta n t, c h a ra c te ris tic im p e d a n c e and fie ld d is tr ib u tio n o f m e ta llic tra ces on
single and m u ltip le su b stra te s are e va lu a te d . In fo rm a tio n o b ta in e d fro m th e s o lu tio n
o f th e eigenvalue p ro b le m is s u b s e q u e n tly used fo r th e a n a lysis o f c o m p le x p la n a r
m icro w a ve c irc u its and e le c tro n ic packages. T h e re s u ltin g ^ -p a ra m e te rs are used to
evalua te th e e le c tric a l p e rfo rm a n c e o f th e s tru c tu re . A t m ic ro w a v e frequencies, p a r­
a s itic effects due to packa g in g, w ire b o n d in g and p o o r g ro u n d in g m a y s ig n ific a n tly
a lte r th e re sp e ctive .S'-param eters o f th e o rig in a l c ir c u it.
T h e fin ite e le m en t m e th o d is th e n h y b rid iz e d w ith a m ix e d s p e c tra l/s p a tia l d o ­
m a in m e th o d o f m o m e n ts and d iffr a c tio n th e o ry to c a lc u la te s c a tte rin g and ra d ia tio n
c h a ra c te ris tic s o f c a v ity -b a c k e d slo ts m o u n te d on in f in it e and fin ite g ro u n d planes
w ith a possible d ie le c tric o r m a g n e tic overlay.
T h e c a v ity m a y also be fille d w ith
m a g ne tize d fe rrite s , th u s p ro v id in g tu n in g c a p a b ilitie s th ro u g h a lte rin g th e e x te r­
n a lly bias fie ld . P re d ic tio n s o f th e ra d ia tio n c h a ra c te ris tic s o f te rrite -tu n e d ante nn a s
are co m p ared w ith m e a surem e n ts p e rfo rm e d in the anechoic ch a m b e r. F in a lly , th e
fin ite e le m e n t m e th o d is used to s im u la te wave p ro p a g a tio n th ro u g h a p e rfe c tly
m a tche d la ye r placed at th e b o u n d a ry o f th e c o m p u ta tio n a l d o m a in to abso rb p ro p ­
a g a tin g waves in th e o u tw a rd d ire c tio n . T h e effectiveness, e ffic ie n c y and a c c u ra c y of
th is tru n c a tio n te c h n iq u e is th o ro u g h ly in v e s tig a te d th ro u g h n u m e ric a l s im u la tio n s .
iii
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ACKNO W LED G M EN TS
S pecial th a n k s to m y a d v is o r D r. C o n s ta n tin e A . B a la n is fo r his guidance, en ­
co u ra g e m e n t a n d va lu a b le a d vice th r o u g h o u t th e c o m p le tio n o f th is d is s e rta tio n .
M y sincere a p p re c ia tio n is e xte n de d to th e o th e r fo u r m em bers o f th e c o m m itte e D r.
S a m ir E l-G h a z a ly . D r. G ua n g-W e n P an. D r. B ru n o D. YVelfert and D r. S u b ra m a n ia m
D. R a ja n fo r c a re fu lly re a d in g and e v a lu a tin g th e c o n te x t o f th is m a n u s c rip t. I w o u ld
also lik e to e xpress m y g ra titu d e to M ik e R. Lyons, an o ffice m a te a n d frie n d o f m in e ,
fo r th e co u n tle ss discussions and a rg u m e n ts we had re la te d to v a rio u s top ics fo u n d
in th is d is s e rta tio n . S pecial th a n k s are also d ire c te d to D r. Jam es T . A b e rle fo r his
sincere frie n d s h ip , professional h e lp , g u id a n c e and advice on several n u m e ric a l issues
raised d u r in g th e course o f th is s tu d y .
A t la st b u t not least. I w o u ld lik e to th a n k
C ra ig B ir tc h e r fo r p e rfo rm in g th e e x p e rim e n ts to va lid a te some o f th e th e o re tic a l
p re d ic tio n s p re se n te d in th is m a n u s c rip t.
I w o u ld a lso lik e express m y sin ce re a p p re c ia tio n to th e A d v a n c e d H e lic o p te r
E le c tro m a g n e tic s ( A H E ) p ro g ra m a n d th e U.S. A rm y Research O ffic e (A R O ) fo r
th e ir fin a n c ia l s u p p o rt o f th is research.
IV
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TABLE OF C O N TE X TS
Page
L IS T O F T A B L E S ..........................................................................................................................v iii
L IS T O F F I G U R E S ..................................................................................................................
ix
CHAPTER
1
I N T R O D U C T I O N ....................................................................................................
2
B R IE F H IS T O R Y O F T H E F I N IT E E L E M E N T M E T H O D
12
3
T W O - D IM E N S IO N A L M IC R O W A V E S T R U C T U R E S ..........................
20
3.1
I n t r o d u c t i o n ..................................................................................................
20
3.2
F o rm u la tio n o f the E igenvalue P r o b l e m .............................................
23
3.2.1
P rin c ip a l axis r o t a t i o n ............................................................
27
3.2.2
C h a ra c te ris tic i m p e d a n c e .....................................................
28
3.2.3
G en e ra lize d e igenvalue s o l v e r .............................................
31
-1
1
3.3
N u m e ric a l V a lid a tio n
..............................................................................
3-1
3.1
E le c tric A n is o tro p ie s in C 'o p la n a r W a v e g u id e s ...............................
37
3.0
S h ie ld in g E ffects
.........................................................................................
-Hi
3.6
C o n d u c to r L o s s e s .........................................................................................
10
3.7
F ie ld D i s t r i b u t i o n s .....................................................................................
07
3.S
C o n c lu s io n s .................................................................................................
71
T H R E E - D IM E N S IO N A L M IC R O W A V E S T R U C T U R E S
....................
73
-1.1
I n t r o d u c t i o n ..................................................................................................
73
1.2
F in ite E le m e n t F o r m u la t io n ...................................................................
7o
-1.3
A n O u tlin e o f th e F in ite -D ifF e re n ce T im e -D o m a in M e th o d
81
4.-1
N u m e ric a l V a lid a tio n and R e s u lt s ........................................................
. .
82
-1.4.1
M ic r o s trip in te rc o n n e c tio n th ro u g h a d ie le c tric b rid g e
4.4.2
B o n d -w ire in t e r c o n n e c t io n s .................................................
84
4.4.3
M ic ro s trip p atch a n te n n a
00
.....................................................
v
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83
CHAPTER
4.5
5
4.4.4
M icro w a ve low-pass filte r
.....................................................
92
4.4.5
S in g le -lo o p in d u c t o r s ................................................................
95
4.4.6
S p ira l in d u c t o r s .............................................................................. 100
4.4.7
D o u b le -v ia m ic ro s trip tra n s itio n p a c k a g e ........................... 106
4.4.8
T h e S O IC -8 p la s tic p a c k a g e ..................................................... 107
C o n c lu s io n s ........................................................................................................ 116
C A V IT Y - B A C K E D A P E R T U R E S ........................................................................118
5.1
I n t r o d u c t i o n ........................................................................................................ 118
5.2
S c a tte rin g fro m C a v ity -B a c k e d A p e r tu r e s .............................................. 119
5.5
5.2.1
F o rm u la tio n using F E M and a s y m p to tic s p e c tra l do­
m a in M o M .................................................................................... 121
5.2.2
A d m itta n c e m a tr ix using s p a tia l d o m a in M o M . . .
5.2.4
E x c ita tio n ve c to r using s p a tia l i n t e g r a t i o n ....................... 129
5.2.4
E le m e n ta l m a trice s using closed fo rm expressions . .
5.2.5
R a d a r cross section e v a lu a tio n ................................................. 159
5.2.6
V a lid a tio n o f raclar cross s e c tio n ......................................... 1 10
151
R a d ia tio n b y C a v ity -B a c k e d A p e r t u r e s ..................................................145
5.5.1
V a lid a tio n o f in p u t im p e d a n c e ................................................. 154
5.4
F re q u e n cy In te rp o la tio n o f A d m itta n c e M a t r i x ...................................160
5.5
D ir e c tiv ity . G a in a nd E f f ic ie n c y ................................................................ 164
5.5.1
6
126
V a lid a tio n o f g a in ...........................................................................167
5.6
H y b rid iz a tio n w ith th e U n ifo rm T h e o ry o f D if f r a c t io n ......................167
5.7
C o n c lu s io n s ........................................................................................................ 171
F E R R IT E - L O A D E D C A V IT Y - B A C 'K E D A P E R T U R E S ........................... 177
6.1
I n t r o d u c t i o n ........................................................................................................ 177
6.2
O r d in a r y and E x tra o rd in a ry W a v e s ......................................................... 179
6.5
D e m a g n e tiz a tio n E f f e c t s ...............................................................................182
6.4
S c a tte rin g fro m F e rrite -L o a d e d C a v ity -B a c k e d S l o t s ....................... 186
vi
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CHAPTER
Page
6.0
R a d ia tio n fro m F e rrite -L o a d e d C a v ity -B a c k e d S l o t s ......................... 195
6.6
7
C o n c lu s io n s ............................................................................................... -0 5
T H E A N IS O T R O P IC P E R F E C T L Y
M A T C H E D L A Y E R ......................... 20S
7.1
I n t r o d u c t io n ............................................................................................... 209
7.2
A 2 -D F in ite E le m e n t F o rm u la tio n o f th e
A n is o tro p ic P M L . . 211
7.3
A 3 -D F in ite E le m e n t F o rm u la tio n o f th e
A n is o tro p ic P M L . . 21-1
7.4
T e n so r R e p re s e n ta tio n o f th e A n is o tro p ic P M L ...................................216
7.5
N u m e ric a l R e su lts
7.6
.......................................................................................... 220
7.5.1
P a ra lle l-p la te w a v e g u id e ............................................................ 220
7.5.2
R e c ta n g u la r w a ve gu id e d i s c o n t in u it ie s .............................. 231
7.5.3
M ic ro w a v e c i r c u i t s .......................................................................235
D esign G u id e lin e s a n d C o n c lu s io n s ......................................................... 239
S
S U M M A R Y A N D G E N E R A L C O N C L U S I O N S .............................................. 213
9
R E C O M M E N D A T I O N S ...............................................................................................219
R E F E R E N C E S ................................................................................................................................. 251
A P P E N D IN
A
E L E M E N T A L M A T R IC E S O F T H E
E IG E N V A L U E P R O B L E M
vii
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. . 267
L IS T O F T A B L E S
T a b le
3.1
Page
P e rce n ta g e change in th e e ffe c tiv e d ie le c tric c o n s ta n t w ith in a range
100 G H z ...........................................................................................................................
42
P ercen tag e change in th e e ffe c tiv e d ie le c tric c o n s ta n t b y ro ta tin g the
c ry s ta l la ttic e 90° at a fre q u e n c y o f 50 G H z ( S -C P W ) ................................
44
3.3
C o n d u c tiv itie s o f few m e ta ls .....................................................................................
62
5.1
C o m p u ta tio n a l tim e re co rd e d on an IB M R IS C /6 0 0 0 w o rk s ta tio n fo r
th e square c a v ity-b a cke d a p e rtu re ..................................................................163
6.1
G e o m e try and m a te ria l s p e c ific a tio n s o f A n te n n a # 1...................................... 189
6.2
C o m p u ta tio n a l s ta tis tic s o f th e h y b r id F E M / M o M code ................................ 191
6.3
C o m p u ta tio n a l s ta tis tic s o f th e h y b r id F E M / M o M code w ith in te rp o ­
la t io n ............................................................................................................................191
6.4
G e o m e try and m a te ria l s p e c ific a tio n s o f A n te n n a # 2 ...................................... 197
7.1
M e sh in fo rm a tio n based on th re e d iffe re n t d is c re tiz a tio n s o f a p a ra lle lp la te w aveguide (.V = 10 and d = 20 m m ) .................................................221
3.2
v iii
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L IS T O F F IG U R E S
F ig u re
Page
3.1
T ria n g u la r ve cto r e le m e n t.......................................................................................
26
3.2
D e fin itio n o f p rin c ip a l axis ro ta tio n for a c ry s ta l la ttic e ............................
28
3.3
G e o m e try o f co u p le d m ic ro s trip lines on a b oron n itrid e su b stra te
( h i = 1.5 m m . h i = 3 m m . w = s = 1.5 m m . b = S.5 m m ) .........................
35
E ffe c tiv e d ie le c tric c o n s ta n t o f coupled m ic ro s trip lines on a boron
n itr id e su b stra te (exjr = 5.12. eyy = 3.4, e-- = 5.12). T h e m arkers
represent d a ta e x tra c te d fro m a p aper b y M o s ta fa et al. [1 5 7 ] ...............
35
G e o m e try o f a u n ila te ra l fin iin e on a d ie le c tric s u b s tra te w ith er = 3.8
(a. = 2b ~ 4.7752 m m . * = 0.127 m m . h -- 2.3876 m m , d = 0.47752
m m ) ..................................................................................................................................
36
C h a ra c te ris tic im p e d a n c e o f a u n ila te ra l fin iin e . T h e m a rke rs re p re ­
sent d a ta e x tra c te d fro m a paper by M a n so u r ct al. [ 1 5 8 ] ......................
36
G e o m e try o f a suspended c o p la n a r w aveguide (a = 7.112 m m . b =
3.556 m m . h i / a = 0.4.
h i / a — 0.1. h ^ / a = 0.5. w = .s = b / 5 )...........
38
D isp e rsio n curves fo r th e d o m in a n t m ode o f a suspended co p la n a r
waveguide. T h e m a rk e rs represent d a ta e x tra c te d fro m a p ap e r by
M a ze -M e rce u r t t al. [ 1-56].......................................................................................
38
C o n v e n tio n a l and suspended waveguides: (a ) C[ = 3.2 m m . c2 = 2.71
m m . d\ = dr, = 1.5 m m . d> = d:i = r/.t = 0.5 m m : (b ) ci = 2.27 m m .
c'l = 0.5 m m . c;5 - 2.71 m m . d L = c/ 5 = 1.5 m m . d 2 = f/3 = d { = 0.5 m m .
39
3.4
3.5
3.6
3.7
3.8
3.9
3.10 E ffe c tiv e d ie le c tric c o n s ta n t and c h a ra c te ris tic im p e d a n ce o f a conven­
tio n a l CPYV w ith a n is o tro p ic su b s tra te s ............................................................
3.11
E ffe c tiv e d ie le c tric c o n s ta n t and c h a ra c te ris tic im p e da n ce o f a sus­
pended C P W w ith a n is o tro p ic su b stra te s.........................................................
3.12
T h e effect o f c ry s ta l ro ta tio n on th e e ffe ctive d ie le c tric co n sta n t and
c h a ra c te ris tic im p e d a n c e o f a suspended CPW *......................................
15
3.13
L o n g itu d in a l fields fo r th e d o m in a n t and firs t H O M o f a c o n ve n tio n a l
C P W .................................................................................................................................
40
13
I
3.14 L o n g itu d in a l fields fo r th e d o m in a n t and firs t H O M o f a suspended
C P W .................................................................................................................................
47
3.15 M ic r o s trip lin e sh ie ld e d w ith a P E C box ( w = ! 0 0 // m . h = !0 0 ft m .
t = ! 0 / m i) ........................................................................................................................
48
3.16 E ffe c tiv e d ie le c tric c o n s ta n t o f a m ic ro s trip lin e enclosed by a s h ie ld in g
b ox o f various d im e n sio n s (<r = 13)...................................................................
ix
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5(1
F ig u re
Page
•'3.17 C h a ra c te ris tic im p edance o f a m ic ro s trip lin e enclosed by a sh ie ld in g
box o f various dim ensions (er
=
13)...................................................
50
•'3. IS E ffe c tiv e d ie le c tric co nsta nt o f a m ic ro s trip lin e enclosed by a s h ie ld in g
box o f various dim e nsion s (er
=
6 .2 )..................................................
51
3.19
C h a ra c te ris tic
b ox o f va rio us
im p edance o f a m ic ro s trip lin e enclosed by a sh ie ld in g
dim ensions (er
=
6 .2 )..................................................
51
•'3.20 E ffe c tiv e d ie le c tric co nsta nt o f a lossy w a veguide w ith er = 4 + _/100
(a = 26 ) .................................................................................................................................
51
3.21
A tte n u a tio n co n s ta n t o f a lossy w aveguide w ith er = -1 + _/100 ( a = 26).
54
3.22
G e o m e try o f a lossy m ic ro s trip lin e (er = 13. t = 3 /m i. h = 100/zm.
ic = 7 5 /m i. a c = 4.1 x 10‘ S /m . a 3 — 5.S x 10‘ S / m ) .................................
55
3.23
E ffe c tiv e d ie le c tric co nstant o f a lossy m ic r o s trip lin e ................................
56
3.24
A tte n u a tio n co n sta n t o f a lossy m ic ro s trip lin e .............................................
56
3.25 Real co m p o n e n t o f the c h a ra c te ris tic im p e d a n ce o f a lossy m ic ro s trip
lin e .....................................................................................................................................
58
3.26 Im a g in a ry c o m p o n e n t of th e c h a ra c te ris tic im p e d a n c e o f a lossy in ic ro s trip lin e ....................................................................................................................
58
3.27 G e o m e try o f a lossy co p la n a r w aveguide (er = 13. l a n d = 3.0 x 10_1.
a c = 3.0 x 10' S / m ) ...................................................................................................
60
3.28
E ffe c tiv e d ie le c tric co nstant o f a lossy c o p la n a r w a v e g u id e ......................
61
.'3.29
A tte n u a tio n co n sta n t o f a lossy co p la n a r w a ve g u id e ...................................
61
3.30
Real c o m p o n e n t o f the c h a ra c te ris tic im p e d a n ce o f a lossy co p la n a r
w a ve gu id e ........................................................................................................................
62
•'3.31 E ffe c tiv e d ie le c tric co nstant o f a lossy co p la n a r w a ve gu id e w ith d iffe r­
ent m e ta l c o n d u c tiv itie s ............................................................................................
63
3.32 A tte n u a tio n co n sta n t o f a lossy co p la n a r w a ve gu id e w ith d iffe re n t
m e ta l c o n d u c tiv itie s ...................................................................................................
63
3.33 Real c o m p o n e n t o f the c h a ra c te ris tic im p e d a n ce o f a lossy co p la n a r
w aveguide w ith d iffe re n t m e ta l c o n d u c tiv itie s .................................................
65
3.34 Im a g in a ry c o m p o n e n t o f th e c h a ra c te ris tic im p e d a n c e o f a lossy co p la ­
nar w aveguide w ith d iffe re n t m e ta l c o n d u c tiv itie s .........................................
65
3.35 G e o m e try o f a lossy m ic ro s trip ( t r = 13. t a n b = 3.0 x I 0_ l . crc =
3.0 x 107 S / m ) .............................................................................................................
66
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-3.36 E ffe c tiv e d ie le c tric co n sta n t o f a lossy m ic r o s trip lin e w ith d iffe re n t
m e ta l c o n d u c tiv itie s ....................................................................................................
68
3.37 A tte n u a tio n co n sta n t o f a lossy m ic r o s tr ip lin e w ith d iffe re n t m e ta l
c o n d u c tiv itie s .................................................................................................................
68
3.38 Real co m p on e nt o f th e c h a ra c te ris tic im p e d a n c e o f a lossy m ic ro s trip
lin e w ith d iffe re n t m e ta l c o n d u c tiv itie s ..............................................................
69
3.39 Im a g in a ry co m p o n e n t o f th e c h a ra c te ris tic im p e d a n ce o f a lossy m i­
c ro s trip lin e w ith d iffe re n t m e ta l c o n d u c tiv itie s .............................................
69
3.-10 E . (lo n g itu d in a l fie ld ) e va lu a te d a lo n g th e a ir-s u b s tra te in te rfa ce .
70
. .
3.-11
£ ( (tra n sverse fie ld ) e va lu a te d a lo n g th e a ir-s u b s tra te in te rfa ce .
. . .
70
3.42
H t (tra n sverse fie ld ) e v a lu a te d a lo n g th e a ir-s u b s tra te in te rfa ce .
. . .
71
4.1
T h re e -d im e n s io n a l re n d e rin g o f a ty p ic a l m ic ro s trip d is c o n tin u ity .
. .
76
4.2
T w o -p o rt g e o m e try w ith th e o u tp u t p o rt o rie n te d at an angle 0 w ith
respect to the in p u t p o r t ...........................................................................................
78
4.3
G e o m e try o f m ic ro s trip tra n s itio n tro u g h a d ie le c tric b rid g e .....................
8-3
4.4
.s'u o f a m ic ro s trip tra n s itio n th ro u g h a d ie le c tric b rid ge o f le n g th
d = 3.17« and d ie le c tric co n s ta n t er .....................................................................
8-3
>'n o f a m ic ro s trip tra n s itio n th ro u g h a d ie le c tric b rid ge o f le n g th
d = 6.35« and d ie le c tric co n s ta n t er .....................................................................
86
B o n d -w ire in te rc o n n e c tio n o f tw o m ic r o s tr ip lin e s .........................................
88
4.5
4.6
4.7
8'n
o f a b o n d -w ire in te rc o n n e c tio n o f tw o rn ic ro s trip lin e s...............
88
4.8
.8’j t
o f a b o n d -w ire in te rc o n n e c tio n o f tw o m ic ro s trip lin e s...............
89
Phase o f .s'u and 8'->i o f a b o n d -w ire in te rc o n n e c tio n o f tw o m ic ro s trip
lin e s ....................................................................................................................................
89
4.10 G e o m e try o f a re c ta n g u la r m ic r o s trip p a tc h a n te n n a on a R T /D u r o id
su b s tra te w ith er = 2 .2 ...............................................................................................
91
4.11 R e tu rn loss o f a re c ta n g u la r m ic r o s tr ip p a tch a n te n n a ................................
91
4.12 G e o m e try o f a low -pass filte r on a R T /D u r o id s u b s tra te w ith f r = 2.2.
93
4.13
R e tu rn loss o f a low -pass f ilt e r ...............................................................................
94
4.14
In s e rtio n loss o f a low-pass f ilt e r .............................................................................
94
4.15
Phase o f 8'n fo r a low -pass f ilt e r .............................................................................
95
4.16
Phase o f S>i fo r a low-pass f ilt e r .............................................................................
95
4.9
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S in g le -lo o p series in d u c to r p rin te d on a R T /D u r o id su b stra te w ith
d ie le c tric c o n s ta n t er = 2.2 and h e ig h t h — 0.794 m m (tv = 2.4 m m .
Ri = 1.5 m m . R a= 2.0 m m ) ....................................................................................
97
M a g n itu d e o f th e 5 -p a ra m e te rs o f a s in g le -lo o p series in d u c to r as a
fu n c tio n o f a ng le o ........................................................................................................
98
4.19 Phase o f th e 5 -p a ra m e te rs o f a sin g le -lo o p series in d u c to r as a fu n c tio n
o f a ng le o .........................................................................................................................
99
4 .IS
4.20 C o m p a ris o n o f |5’n | versus fre q u e n cy c a lc u la te d in d e p e n d e n tly u sin g a
fu ll-w a v e s im u la tio n and a lu m p e d e q u iv a le n t c ir c u it o f a series in d u c to r. 99
4.21 Series s p ira l in d u c to r p rin te d on an A lu m in a (er = 9.S) s u b s tra te o f
h e ig h t 0.635 m m . T h e e n d o f th e s p ira l is bon d ed w ith a m ic r o s trip
lin e th ro u g h a m e ta llic b rid g e o f h e ig h t 1.0 m m . T h e b rid g e has an
a rc shape defin e d b y th re e p o in ts (u q = 0.635 m m . iv2 =
— 0.2
m m . iL-.i = 2.3 m m . R \ = 1.9 m m . R 2 = 1.3 m m . R% = 0.7 m m ). . . .
4.22
101
M a g n itu d e o f 5’n a n d S22 fo r a series s p ira l in d u c to r w ith a b o n d -w ire
b rid g e ..................................................................................................................................... 102
4.23 M a g n itu d e o f S \ 2 a n d S21 fo r a series s p ira l in d u c to r w ith a b o n d -w ire
b rid g e ..................................................................................................................................... 102
1.24 S p ira l in d u c to r co n n e cte d in shunt w ith a m ic ro s trip line p rin te d on
an A lu m in a (er = 9 .8 ) s u b s tra te o f h e ig h t 0.635 m m (uq = 0.635 m m .
i r , = 0.2 m m . t c ( = 0.6 m m . R i = 1.9 m m . R 2 = 1.3 m m . R% = 0.7
m m ) ........................................................................................................................................ 103
1.25 M a g n itu d e o f 5 U and S21 fo r a s p ira l in d u c to r connected in sh u n t
across a rn ic ro s trip lin e ...................................................................................................104
1.26 L a y o u t o f a re c ta n g u la r s p ira l in d u c to r on an A lu m in a s u b s tra te w ith
cr = 9.8. (I = 0.635 m m . ic = 0.625 n u n . .s = 6 = 0.3125 m m .
h = 0.3175 n u n . / = 0.1 m m ........................................................................................ 105
4.27
M a g n itu d e o f 5 'n fo r a re c ta n g u la r s p ira l in d u c to r w ith an a ir b rid g e .
105
4.28
M a g n itu d e o f 5>i fo r a re c ta n g u la r s p ira l in d u c to r w ith an a ir b rid g e .
106
4.29
D o u b le -v ia tra n s itio n package w ith a flo a tin g g ro u n d . G e o m e try spec­
ific a tio n s : h\ = 0.8 m m . h 2 = 0.4 m m . /uj = 0.6 m m . ttq = 2.4 m m .
ir> = 0.8 m m . u’3 = 0.4 m m . uq = 5.2 m m .
= 6.8 m m . i/’h =
3.6 m m . t = 0.2 m m .......................................................................................................108
4.30
D o u b le -v ia tr a n s itio n package w ith a c o m m o n g ro u n d ................................... 108
4.31 5 -p a ra m e te rs versus fre q u e n cy fo r th e d o u b le -v ia tra n s itio n package. .
109
1.32 T h re e -d im e n s io n a l re n d e rin g o f a S O IC -8 p la s tic package. S p e c ific a ­
tio n s : L = 4.2 n u n . IF = 2.4 m m . p = 1.27 rnrn. «q = 0.4 m m . T h e
th ickn e ss o f a ll c o n d u c tin g surfaces (p a d d le , leads, e tc .) is 0.1 m m . . .
110
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4.33 T w o -d im e n s io n a l sid e vie w o f a SOIC'-S p la s tic package. S pe cifica ­
tio n s : \VP = 4.4 m m . L c = 0.7 m m . L g - 0.3 m m . d x = 0.535 m m .
d 2 = 0.1 m m . r/3 = 0.635 m m . c/4 = 0.2 m m . r/5 = 0.25 m m . T h e
th ickn e ss o f a ll c o n d u c tin g surfaces (p a dd le , leads, e tc .) is 0.1 r n r n . . .
110
4.34 D e ta il vie w o f g ro u n d in g th e lead to the m o th e rb o a rd ...................................... I l l
4.35 G e o m e try o f th e unpackaged M M IC : m ic r o s trip - th ro u g h co n n e ctio n
w ith a stu b . T h e th ickn e ss o f a ll c o n d u c tin g surfaces is 0.1 m m . A ll
d im e n sio n s are in m illim e te r s ..................................................................................... 112
4.36 S c a tte rin g p a ra m e te rs o f th e unpackaged M M IC : m ic ro s trip -th ro u g h
co n n e c tio n w ith a s tu b ...................................................................................................113
4.37 T w o -d im e n s io n a l to p vie w o f a SOIC'-S p la s tic package hou sin g a p la ­
n a r m ic ro s trip th ro u g h -c o n n e c tio n w ith a s tu b ................................................... 114
4.38 T h re e -d im e n s io n a l g e o m vie w re n d e rin g o f th e S O IC ’-S package...............114
4.39 S c a tte rin g p a ra m e te rs o f th e SO IC’-S p la s tic package.................................. 115
4.40 S'2 1 versus fre q u e n cy o f th e SOIC'-S p la s tic package u n d e r th e fo llo w in g
g ro u n d in g c o n d itio n s : (a) leads 2. 4. 5. S are g ro u n d e d , (b ) leads 2. S
a re g ro u n d e d , and (c ) leads 4. 5 are g ro u n d e d .....................................................116
5.1
G e n e ric shapes o f c a v ity -b a c k e d a p e rtu re a n te n n a s ............................................120
5.2
A 2-D view o f a c a v ity -b a c k e d p a tch a n te n n a m o u n te d on an in fin ite
g ro u n d p la n e .......................................................................................................................121
5.3
Eidge based te tra h e d ra l e le m e n t.................................................................................. 132
5.4
T h re e -s lo t a rra y backed by an a ir-fille d re c ta n g u la r c a v ity : L c = l l \ . =
0.75 cm . D c = 0.25 c m . L s - 0.5 cm . IF S = 0.05 cm . D s = 0.25 cm .
. 142
5.5
Frequency sweep o f an a rra y o f three slots backed by a re c ta n g u la r
c a v ity [ 0 l = 0 ° ).................................................................................................................. 143
5.6
A n g le sweep o f an a rra v o f th re e slots backed b v a re c ta n g u la r c a v itv
( / = 30 G E Iz).............. . ' ................................................ ‘
‘ .
143
5.7
C ir c u la r p atch backed by a c y lin d ric a l c a v ity . T h e c a v ity is fille d
w ith a d ie le c tric m a te ria l o f er = 2.2. t a n 8 e = 0.0009. /(,. = 1.0. and
t a n b m = 0 (/? i = 2.5 cm . R j = 3.0 cm . c = 0.5 c m ) ......................................14 1
5.S
M o n o s ta tic RCS versus fre q u e n cy o f a c ir c u la r p a tch backed b y a
c y lin d ric a l c a v ity (0, = 0 ° ) ........................................................................................... 144
5.9
A 2-D vie w o f a c a v ity -b a c k e d p atch a n te n n a m o u n te d on an in fin ite
g ro u n d plane and fed w ith a co a x ia l ca ble .............................................................147
5.10 A 3 -D view o f an a ir-fille d re c ta n g u la r c a v itv fed w ith a 50-Q co a xia l
ca b le o rie n te d in th e (/-d ire c tio n ................................................................................ 155
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5.11 A 2-D v ie w o f an a ir-fille d re c ta n g u la r c a v ity fed w ith a 50-Q co a xia l
cable o rie n te d in th e //-d ire c tio n ................................................................................155
5.12 Real p a rt o f th e re fle c tio n c o e ffic ie n t o f a closed a ir-fille d re c ta n g u la r
c a v ity fed w it h a 5 0 -fi co a xia l c a b le ........................................................................ 156
5.13 Im a g in a ry p a rt o f th e re fle c tio n c o e fficie n t o f a closed a ir-fille d re c t­
a n g u la r c a v ity fed w ith a 5 0 -fi c o a x ia l c a b le ....................................................... 156
5.11
In p u t re sista n ce o f an a ir-fille d c a v ity -b a c k e d s lo t a n te nn a fed w ith a
50-Q c o a x.............................................................................................................................157
5.15 In p u t re a cta n ce o f an a ir-fille d c a v ity -b a c k e d s lo t a n te nn a fed w ith a
50-Q co a x............................................................................................................................. 157
5.16 In p u t re sista n ce o f an a ir-fille d c a v ity -b a c k e d s lo t a nte nn a fed w ith a
sh o rt ca b le ...........................................................................................................................158
5.17 In p u t re a cta n ce o f an a ir-fille d c a v ity -b a c k e d s lo t a n te n n a fed w ith a
sh o rt c a b ie ...........................................................................................................................158
5.18 C irc u la r p a tc h backed by a c y lin d r ic a l c a v ity m o u n te d on an in fin ite
g ro u n d p la ne a n d fed w ith a v e r tic a lly o rie n te d c o a x ia l cable ( R x = 2.0
cm . /?2 = '-.44 c m . c = 0.218 c m . er = 2.33. t a nS e = 0.0012. /.ir - 1.0.
t a n b m = 0. R j = 0.7cm . a - 0 .0 4 5cm . b = 0 .1 5 cm . erc = 2.OS).................... 15!)
5.19
In p u t re sista n ce o f a c irc u la r c a v ity -b a c k e d p a tc h a n te n n a ...........................161
5.20 G e o m e try o f a square a p e rtu re backed by a square c a v ity w ith
sions A = B = C = 5 cm and a = b = 3 c m
d im e n ­
162
5.21
RCS c o m p u ta tio n s using lin e a r in te r p o la tio n fo r th e a d m itta n c e m a trix . 161
5.22
C o m p a riso n o f th e co-pol gain p a tte rn s o f an a ir-fille d ca vity-b a cke d
s lo t.......................................................................................................................................... 168
5.23 G e o m e try o f a c a v ity -b a c k e d a p e rtu re on a fin ite g ro u n d p la ne ..................16!)
5.24 C o m p a riso n o f th e E -p la ne g a in p a tte rn {Gg'j o f an a ir-fille d c a v ity backed s lo t a n te n n a m o u n te d on a fin ite g ro u n d p la n e ...................................173
5.25 D ir e c tiv ity p a tte rn s o f a c o a x ia l c a v itv at 5 G H z (h a rd p o la riz a tio n .
c .r-p la n e ).............................................................................................................................. 175
5.26 D ir e c tiv ity p a tte rn s o f a c o a x ia l c a v ity a t 5 G H z (h a rd p o la riz a tio n .
r//-p la n e ).............................................................................................................................. 175
6.1
(a) P e rp e n d ic u la r m a g n e tiz a tio n , (b ) P a ra lle l m a g n e tiz a tio n ......................183
6.2
G e o m e try o f a re c ta n g u la r fe r rite p ris m w ith a u n ifo rm DC m a g n e tic
fie ld
a p p lie d e x te rn a lly a lo n g th e v e rtic a l d ir e c tio n ...................................185
6.3
T h e b a llis tic d e m a g n e tiz in g fa c to r o f a u n ifo r m ly m a g ne tize d re c ta n ­
g u la r p ris m .......................................................................................................................... 187
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6.4
C o m p a ris o n betw een th e b a llis tic and m a g n e to rn e tric d e m a g n e tiz in g
fa c to rs o f a u n ifo rm ly m a g n e tiz e d re cta n g u la r p ris m w ith d im e n s io n
a —* oc ...................................................................................................................................1ST
6.5
G e o m e try o f a m u lti-la y e r fe rrite -lo a d e d CBS a n te n n a m o u n te d on an
in f in it e g ro u n d p la n e .......................................................................................................188
6.6
M o n o s ta tic RC'S {cr00) at n o rm a l in cid en ce o f a C B S a n te n n a loaded
w it h layers o f m a g n e tize d fe r r ite ( H 0 = 400 O e )..................................................190
6.7
F re q u e n cy tu n in g fo r soft p o la riz a tio n a t n o rm a l in cid e n ce (0 =
6 .S
E ffe c tiv e n o rm a liz e d p ro p a g a tio n co n sta n t o f th e e x tr a o rd in a ry wave
as a fu n c tio n o f fre q u e n cy............................................................................................. 193
6.9
F re q u e n cy tu n in g fo r h ard p o la riz a tio n a t n o rm a l in cid e n ce (0 = o — 0).194
o = 0). 195
6.10 T h e effect o f s a tu ra tio n m a g n e tiz a tio n ( 4 " . lf 3) on th e re so n a n t fre ­
q u e n c y o f th e m u lti-la y e r fe rrite -lo a d e d antenna (F/ 0 = 500 O e. A / / =
5 O e. 0 = o = 0 )............................................................................................................... 195
6.11 T h e effect o f lin e w id th on th e resonant frequency o f th e m u lti- la y e r
fe rrite -lo a d e d a n te n n a ( H a = 500 Oe. 4 ;r.\/a = 800 G . 0 = o = 0).
. .
196
6.12 G e o m e try o f a m u lti-la y e r fe rrite -lo a d e d CBS a n te n n a m o u n te d on
an in f in it e g ro u n d plane and fed w ith a 50-Q co a xia l ca b le a lo n g th e
/ /- d ir e c tio n ............................................................................................................................197
6.13 R e tu rn loss o f an a ir-fille d C B S a n te n n a m ounted on an in f in it e g ro u n d
p la n e and fed w ith a 50-H c o a x ia l cable along th e //- d ir e c tio n .......................198
6.14 M e a su re d m a g n e tic fie ld d is t r ib u t io n in sid e the c a v ity w h e n th e la tte r
is m a g n e tize d a lo ng the //-d ire c tio n using one p a ir o f m a g n e ts ................... 200
6.15 M e a sured m a g n e tic fie ld d is t r ib u t io n in side the c a v ity w h e n th e la tte r
is m a g n e tiz e d a long the //-d ire c tio n using tw o pairs o f m a g n e ts.
. . . 200
6.16
P re d ic te d and m easured in p u t im p e d a n ce versus fre q u e n cy o f a fe rrite lo a de d C B S a n te n n a using a s in g le p a ir o f magnets ( H 0 = 238 O e). . 202
6.17 P re d ic te d and m easured in p u t im p e d a n ce versus fre q u e n cy o f a fe rrite lo a de d C B S a n te n n a using tw o p a irs o f magnets ( H 0 = 445 O e ). . . . 202
6.18 P re d ic te d in p u t im p e d a n ce versus frequency o f a fe rrite -lo a d e d C B S
a n te n n a using various b ia s in g fie ld s .........................................................................203
6.19
P re d ic te d and m easured re tu r n loss versus frequency o f a fe rrite -lo a d e d
C B S a n te n n a using one and tw o p a irs o f m agnets ( H 0 = 238. 445 O e). 204
6.20
P re d ic te d and m easured g a in versus frequency o f a fe rrite -lo a d e d CBS
a n te n n a using va rious b ia s in g fie ld s (o -p o la riz a tio n )........................................205
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6 .2 L E- and H -p la n e a bso lu te g ain p a tte rn s o f a fe rrite -lo a d e d CBS antenna
biased a lo n g the //-d ire c tio n usin g tw o p a irs o f m agnets ( H a = 445 Oe
and / = S42.5 M H z ). ------- P r e d ic tio n s . ------ M e a surem e n ts.......................... 206
7.1
P a ra lle l-p la te waveguide te rm in a te d w ith a m u lti-la y e r P M L region
( b = 40 m m ) .................................................................................................................... 213
7.2
R e fle c tio n co efficie n t fro m a P M L te r m in a tio n as a fu n c tio n o f the
th e o re tic a l re fle ctio n co e fficie n t a t n o rm a l in c id e n c e ( T E M m ode. .V =
10. cl — 20 m m . m = 2 )..................................................................................................222
7.3
R e fle c tio n co efficie n t fro m a P M L te r m in a tio n as a fu n c tio n o f the
th e o re tic a l re fle ctio n co e ffic ie n t a t n o rm a l in c id e n c e ( T M t m ode. .V =
10. d = 20 m m . m = 2 )................................................................................................. 222
7.4
R e fle c tio n co efficie n t fro m a P M L te r m in a tio n as a fu n c tio n o f mesh
d e n s ity ( T E M m ode. .V = 10. d = 20 m m . m = 2,R =
1 0 ~ ') .................... 223
7.5
T h e e ffect o f s p a tia l p o ly n o m ia l o rd e r on th e re fle c tio n coefficient fro m
a P M L te rm in a tio n ( T E M m o d e, mesh # 3 . .V = 10. d = 20 m m .
R = lO " 4 ) ........................................................................................................................... 224
7.6
R e fle c tio n co efficient fro m a P M L te r m in a tio n as a fu n c tio n o f the
P M L d e p th (T E M m ode, mesh # 3 . .V = 5. R = 10 1)..................................225
7.7
R e fle c tio n co efficie n t fro m a P M L te r m in a tio n as a fu n c tio n o f the
P M L d e p th (T E M m ode, mesh # 3 . .V = 10. R = 10- '1)................................225
7.S
R e fle c tio n coefficient fro m a P M L te r m in a tio n as a fu n c tio n o f the
n u m b e r o f layers (T E M m o d e, mesh # 3 . m = 2. d = 20 m m . R = 10- '1).227
7.9
S in g le -la y e r P M L o p tim iz a tio n in te rm s o f 6 ( / = 100 M H z. average
ce ll size h = 0.5 m m ) ..................................................................................................... 22/
7.10 T h e e ffect o f d is c re tiz a tio n e rro r in o p tim iz in g a sin g le-la yer P M L
m e d iu m ............................................................................................................................... 229
7.11 N o rm a liz e d e le ctric fie ld d is tr ib u tio n in a p a ra lle l-p la te waveguide
( T E M m o d e . .V = 10. d = 20 m m . R = 1 0~‘ . m = 2 )..................................... 230
7.12 P a ra lle l-p la te waveguide loaded w ith a d ie le c tric d is c o n tin u ity w ith
er = 6 (b = 40 m m . ic = 30 m m . h = 20 m m ) .................................................... 230
7.13 C o n s e rv a tio n o f energy in th e presence o f a P M L in te rfa ce and a d i­
e le c tric d is c o n tin u ity a t close p r o x im ity ( T E M m o d e )................................... 232
7.14 C o n s e rv a tio n o f energy in th e presence o f a P M L in te rfa ce and a d i­
e le c tric d is c o n tin u ity at close p r o x im ity ( T M i m o d e ).....................................232
7.15 R e fle c tio n coefficient fro m a d ie le c tric d is c o n tin u ity o f er = 6 in sid e a
p a ra lle l-p la te w aveguide ( T E M m o d e )................................................................... 233
xvi
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F ig u re
Page
7.16 T ra n s m is s io n co efficie n t fro m a d ie le c tric d is c o n tin u ity o f er = 6 inside
a p a ra lle l-p la te waveguide (T E M m o d e )............................................................... 233
7.17 A d ie le c tric -lo a d e d waveguide te rm in a te d w it h a 5 -la ye r P M L (er = 6 .
6 = 1 c m ) ............................................................................................................................ 23-1
7.18 M a g n itu d e o f S'u for a d ie le c tric d is c o n tin u ity in sid e a re cta n g u la r
w a ve g u id e ........................................................................................................................... 234
7.19
A ir - fille d rig h t-a n g le bend te rm in a te d w ith a 5 -la y e r P M L ........................... 236
7.20
|S’2112 versus 2 a / X o f an a ir-fille d rig h t-a n g le b e n d .......................................... 236
7.21 G e o m e try o f a shielded m ic ro s trip lin e te r m in a te d w ith a sin g le-la yer
P M L . T h e s u b s tra te is R T /D u ro id o f er= 2 .2 .....................................................238
7.22 M a g n itu d e o f S’u versus th e o p tim iz a tio n p a ra m e te r 6 (average mesh
size h = 1 m m ) ................................................................................................................ 238
7.23 G e o m e try o f a low-pass filte r s u rro u n d e d b y a s in g le -la y e r P M L . The
s u b s tra te is R T /D u r o id o f heigh t 6 = 0.794 m m a nd er = 2 .2 .......................240
7.24
R e tu rn a nd in s e rtio n loss o f a low-pass f ilt e r ..................................................... 240
7.25
Phase o f 5’u and S->i o f a low-pass f ilt e r ...............................................................240
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CHAPTER 1
IN T R O D U C T IO N *
W ith recent advances in s e m ic o n d u c to r tech n olog y, w ireless c o m m u n ic a tio n s and
c o m p u te r in d u s try , th e d em an d fo r h ig h ly advanced a n d s o p h is tic a te d designs o f m i­
crow ave in te g ra te d c irc u its , e le c tro n ic packages and a n te n n a system s has increased.
T h e need fo r h igh p e rfo rm a n c e co m p o n e n ts and devices o fte n co -e xists w ith th e de­
sire o f low cost. H ow ever, to im p ro v e p e rfo rm a n ce and reduce cost, it is im p o rta n t
th a t th e process o f design, o p tim iz a tio n and te s tin g o f a d e vice be p e rfo rm e d using
c o m p u te r a id ed design (C A D ) to o ls ra th e r th a n be a tte m p te d in a fa b ric a tio n lab.
It is a co m m o n p ra c tic e in in d u s try th a t m ost o f th e design and s im u la tio n is done
e x p e rim e n ta lly , whereas o p tim iz a tio n is ra re ly a p p lie d .
re su lts in low p e rfo rm a n ce devices a t a h ig h cost.
Such m e th o d o lo g y u s u a lly
T h e a b ilit y to design, s im u la te
and o p tim iz e the o v e ra ll p e rfo rm a n c e o f a d evice u sin g n u m e ric a l techniques and
C A D tools is c e rta in ly an a dvantage. T h e m a in p ro b le m , how ever, is th a t n u m e ri­
cal techniques are not fu lly m a tu re d and developed. S u ffic ie n tly a ccu ra te m e tho d s
are u s u a lly lim ite d to sp ecific g e o m e trie s whereas th e m o re p o w e rfu l and v e rs a tile
m e tho d s are not as a ccu ra te o r c o m p u ta tio n a lly e ffic ie n t. It is th e re fo re im p o rta n t
th a t e ffic ie n t and h ig h ly a ccu ra te c o m p u ta tio n a l m e th o d s are developed to s im u la te
e le c tro m a g n e tic in te ra c tio n in p ra c tic a l a p p lic a tio n s in c lu d in g m icro w a ve c irc u its ,
in te rc o n n e c ts , e le c tro n ic packages and antennas.
In areas o f passive c irc u its and e le c tro n ic packaging it is d e sira b le n ot o n ly th e
device be ch a ra cte rize d in te rm s o f its 5 -p a ra m e te rs , p ro p a g a tio n c o n s ta n t, ch a ra c­
te r is tic im p e da n ce and c o n d u c to r losses, b u t also in te r m o f possible g ro u n d in g and
packaging effects.
T h e .S'-param eters o f a d evice d r a s tic a lly change once m o u n te d
in to a package. T h e package its e lf is u s u a lly m ade o f a s m a ll n u m b e r o f leads, an
elevated p a d d le w h ich is g ro u n d e d to th e m o th e rb o a rd g ro u n d , bond w ires, c o n d u c t­
in g vias and th e e n c a p tu la n t to p ro v id e p ro te c tio n to th e c ir c u it. W h e n the c irc u it
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is packaged, th e presence o f w ire b on d s, th e p o o r g ro u n d in g o f th e p a d d le , th e p ro x ­
im i t y o f th e leads, and th e e n c a p tu la n t its e lf cre a te p a ra s itic loads th a t a d ve rse ly
in flu e n c e th e p e rfo rm a n c e o f the o rig in a l c ir c u it. C o m p u ta tio n a l m e th o d s represent
u se fu l and p o w e rfu l to o ls th a t help to im p ro v e and o p tim iz e th e o v e ra ll p e rfo rm a n c e
o f th e package, a t least w ith in th e d e s ire d fre q u e n c y b a n d . O th e r p a ra s itic effects
such as tim e delays, m is m a tc h losses, r a d ia tio n losses and c o n d u c to r losses are also
im p o r ta n t q u a n titie s to ch a ra cte rize a n d a tte m p t to m in im iz e o r c o n tro l.
In a d d i­
tio n . effects due to m a te ria l a n is o tro p ie s in c o m m o n ly used su b stra te s sh o u ld also be
in v e s tig a te d .
A n te n n a s are essen tia l elem ents in c o m m u n ic a tio n system s since th e y p ro v id e
w ireless lin k s b etw een th e tr a n s m itte r a n d th e re ce ive r.
D e p e n d in g on th e specific
a p p lic a tio n and th e fre q u e n cy o f o p e ra tio n , th e a n te n n a c o n fig u ra tio n , d im e n s io n s
a n d ra d ia tio n c h a ra c te ris tic s m ay s ig n ific a n tly vary.
O f g re at in te re s t in n u m erou s
w ireless a p p lic a tio n s is th e c a v ity -b a c k e d s lo t a n te n n a w h ic h can be flu s h m o u n te d on
th e surface o f a v e h ic le w ith o u t a ffe c tin g its a e ro d y n a m ic p ro file . In a d d itio n to low
p ro file , it is o fte n d e s ira b le th a t th e a n te n n a is lig h t w e ig h t and o f s m a ll size, and it
e x h ib its high g ain, w id e b a n d w id th , a n d h ig h e ffic ie n c y . M o st o f these a ttr ib u te s can
be im p ro v e d a t th e expense o f o th e rs. F o r e x a m p le , an increase in g ain fo llo w s by a
decrease in b a n d w id th , and vice versa. A ls o , a decrease in size, and th e re fo re w e ig h t,
fo llo w s by an increase in th e q u a lity fa c to r w h ic h is in v e rs e ly p ro p o rtio n a l to the
b a n d w id th .
Design and o p tim iz a tio n o f a n te n n a p e rfo rm a n c e can be c o n v e n ie n tly
a chieve d th ro u g h th e im p le m e n ta tio n o f adva n ced n u m e ric a l tech n iq ue s. A lth o u g h
e x p e rim e n ts are u s u a lly th e m ost o b v io u s and d ire c t m eans to a n te n n a design and
a n a ly s is , these are o fte n tim e c o n s u m in g , e x p e n sive and prone to in s tru m e n ta tio n
e rro rs.
T h e in cre a sin g d e m a n d o f w ireless c o m m u n ic a tio n lin k s w ith in a w id e range o f
frequencies creates th e p ro b le m o f in te rfe re n c e and c o u p lin g am ong r a d ia tin g ele­
m e n ts th a t are lo c a te d a t close p r o x im ity .
To a v o id such p ro b le m s, th e n u m b e r
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o f a n te n n a s m o u n te d on a s in g le s tru c tu re , w h e th e r t h a t is m o b ile o r s ta tio n a ry ,
m u st be lim ite d . One way to re d uce th e n u m b e r o f a n te n n a s m o u n te d on a single
s tru c tu re is to use tu n a b le e le m e n ts. I f th e a n te n n a has th e c a p a b ility o f b eing elec­
t r o n ic a lly tu n e d to various fre qu e ncies, n o t o n ly in te rfe re n c e p ro b le m s are m in im iz e d
b u t also cost a nd c o m p le x ity . A n a n te n n a w ith h ig h p e rce n ta g e tu n in g c a p a b ilitie s
can c e r ta in ly replace th e fu n c tio n a lity o f o th e r a n te n n a s th a t o perate w ith in the
same b a n d w id th . T u n a b ility is u s u a lly achieved u sin g v a ra c to r diodes to c o n tro l the
loads o r th e e le c tric a l le n g th o f th e a n te n n a . F e rrite s can also be used fo r th e design
o f tu n a b le antennas because t h e ir e le c tro m a g n e tic p ro p e rtie s are s tro n g ly dependent
on th e s tre n g th and th e d ire c tio n o f an e x te rn a lly bias fie ld .
N u m e ric a l m e tho d s have been used fo r m a n y years to solve e le c tro m a g n e tic p ro b ­
lem s r e la tin g to m icrow ave c ir c u its and antennas.
T h e m o st p o p u la r m e th o d s are
th e m e th o d o f m o m e n ts ( M o M ) . th e fin ite -d iffe re n c e tim e -d o m a in (F D T D ) . th e fi­
n ite e le m e n t m e th o d ( F E M ) . th e m e th o d o f lines ( M o L ) and th e tra n sm issio n lin e
m e th o d ( T L M ) . T h e M o M . w h ic h is the m ost tr a d itio n a l a nd w id e ly used, was the
firs t to be a p p lie d fo r th e a n a ly s is o f p la n a r m icro w a ve c irc u its . T h ere e xist an e x te n ­
sive a m o u n t o f w o rk on th e a p p lic a tio n o f M o M o r s p e c tra l d o m a in M o M . o th e rw is e
ca lle d th e S D A . on tw o -d im e n s io n a l is o tro p ic and a n is o tro p ic su bstra te s [ l] - [ l. ‘)j. T h e
d is p e rs iv e c h a ra c te ris tic o f a tra n s m is s io n lin e , such as e ffe c tiv e d ie le c tric co n sta n t
and c h a ra c te ris tic im pedance, a re c a lc u la te d as a fu n c tio n o f frequency. T h e S D A
has also been e xte n de d to a n a ly z e th re e -d im e n s io n a l m ic ro w a v e s tru c tu re s in c lu d in g
n iic r o s tr ip p a tc h antennas [ 1 4 ]-[L6]. step d is c o n tin u itie s a n d s tu b s [17]. bends and Tju n c tio n s [ I S ] . [19]. gaps [20 ].[2 1 ]. filte rs and s p ira l in d u c to rs [16]. T h e S D A in vo lves
a G re e n 's fu n c tio n fo rm u la tio n w h ic h is sp ecific to a g e o m e try and a G a le r k in s d is ­
c re tiz a tio n based on th e chosen set o f basis fu n c tio n s .
For g e o m e tric a lly c o m p le x
s tru c tu re s , o b ta in in g a G re e n 's fu n c tio n can be e x tr e m e ly d iffic u lt if n ot im p o s s i­
ble. T h e re fo re , such an a p p ro a ch is lim ite d to m o s tly p la n a r s tru c tu re s w ith layered
m e d ia.
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O th e r n u m e ric a l m e th o d s have also been a pp lie d to s im ila r stru c tu re s . T h e fin ite d iffe re n ce fre q u e n c y -d o m a in ( F D F D ) m e th o d , w h ich is based on a ce n tra l diffe re n ce
d is c re tiz a tio n o f th e M a x w e ll's e q u a tio n s in frequency d o m a in , was a p p lie d in th e m id
SO's fo r th e a n a ly s is o f sh ie ld e d m ic ro w a v e s tru c tu re s [22] such as m ic ro s trip tr a n ­
sitio n s th ro u g h d ie le c tric w a lls and m ic ro s trip in te rc o n n e c tio n s th ro u g h w ire bonds.
T h e fin ite -d iffe re n c e tim e -d o m a in ( F D T D ) m e th o d , w h ic h has re ce n tly become one
o f th e m ost w id e ly used n u m e ric a l tech n iq u e s in e le c tro m a g n e tic s , was im p le m e n te d
in th e la te SCTs to c a lc u la te fre q u e n cy-d e p e n d e n t c h a ra c te ris tic s o f s im p le m ic ro s trip
d is c o n tin u itie s [23].[24]. T h e sam e a p p ro a ch has been su cce ssfu lly a pp lie d fo r m o re
c o m p lic a te d c ir c u its lik e p a tch a nte nn a s, low-pass filte rs and couplers [25].
The
fre qu e ncy d o m a in 5’-p a ra m e te rs are c o m p u te d using a F o u rie r tra n s fo rm o f th e tim e d o m a in response.
S ince th e e a rly 90's, num erous papers have appeared in th e l i t ­
e ra tu re on th e a p p lic a tio n o f th e F D T D m e tho d in th e area o f m icrow ave c irc u its
and packaging [26].[27].
A lth o u g h th e F D T D was p ro ve n to be a c o m p u ta tio n a lly
e ffic ie n t m e th o d fo r th e s o lu tio n o f h y p e rb o lic p a rtia l d iffe re n tia l e quations (P D F 's ),
e spe cia lly fo r ra d ia tio n and s c a tte rin g p ro ble m s in e le c tro m a g n e tic s , th e use ot re c t­
a n g u la r g rid s becom es a m a jo r d ra w b a c k . In a d d itio n , th e d iffic u lty in im p le m e n tin g
a n o n -u n ifo rm m esh, th e p o s s ib ility o f n u m e ric a l in s ta b ilitie s in th e s o lu tio n and the
in h e re n t slow convergence fo r h ig h -Q s tru c tu re s present even m o re problem s.
T h e M o L is a n o th e r n u m e ric a l te c h n iq u e o f in cre a sin g p o p u la rity in recent years.
It is based on s u b d iv id in g th e g u id in g s tru c tu re in to re c ta n g u la r hom ogeneous regions
and m a tc h in g th e fie ld c o m p o n e n ts a t th e interfaces.
T h is n u m e rica l concept was
successfully a p p lie d fo r th e a n a lysis o f m ic ro s trip d is c o n tin u itie s [2S].[29]. A lth o u g h
the a ccu ra cy o f th e m e th o d is q u ite p ro m is in g , its v e r s a tility is lim ite d to sp ecific
geom etries.
S im ila r lim ita tio n s are observed for th e m o d e -m a tc h in g approach [30]
w h ich becam e e x tre m e ly p o p u la r fo r th e analysis o f w a ve gu id e d is c o n tin u itie s .
A
m o re v e rs a tile m e th o d is th e T L M [31] w h ich was im p le m e n te d w ith great success fo r
the a na lysis o f m o re c o m p lic a te d g e o m e trie s in c lu d in g th re e -d im e n s io n a l e le c tro n ic
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packages [32]. In a d d itio n , lik e th e F D T D m e th o d , th e T L M allow s a n a lysis o f the
tra n s ie n t response. H o w e ve r, th e a c tu a l s im u la tio n is c o n stra in e d to o n ly re c ta n g u la r
shapes.
T h e F E M [3-3] is p ro b a b ly th e m ost v e rsa tile n u m e ric a l m e th o d fo r th e analysis
o f m ic ro w a v e in te g ra te d c irc u its and e le c tro n ic packages. T h e c o m p u ta tio n a l d om ain
is firs t d is c re tiz e d using u n s tru c tu re d g rid s, th e re fo re a llo w in g m o d e lin g o f a rb itra ry
g e o m e trie s a n d in h om o g en e o us regions.
M a x w e ll's e q u a tio n s in fre q u e n c y d om ain
are fo rm e d in to a lin e a r s y s te m o f equ a tion s by m in im iz in g th e w e ig h te d residual
u sin g C la le rk in 's a p p ro a ch. T h e syste m o f e quations is th e n solved to o b ta in th e field
d is tr ib u tio n in sid e th e c o m p u ta tio n a l d o m a in .
T h e a p p lic a tio n o f F E M in e le ctro m a g n e tic s began in th e la te 6 0 s a n d e a rly 70 s
w ith th e o b je c tiv e to solve fo r th e m odes o f hom ogeneous and in h om o g en e o us waveg­
uides [34]-[36]. T h e use o f n o d a l e le m en ts and th e presence o f m a te ria l in te rfa ce s in
th e d o m a in cre a te d th e p ro b le m o f spurious o r n o n -p h y s ic a l m odes.
T h e m e tho d
lo st p o p u la rity in e le c tro m a g n e tic s u n t il the e a rly 8 0 s when Xedelec in tro d u c e d the
v e c to r e le m en ts [37]. V e c to r e le m en ts were successfully a p p lie d fo r th e firs t tim e to
solve fo r th e p ro p a g a tio n c h a ra c te ris tic s o f p a rtia lly -lo a d e d d ie le c tric w aveguides [38].
It was show n th a t th e use o f v e c to r elem ents e lim in a te s sp u rio u s m odes fro m the
n u m e ric a l s o lu tio n . Since th e n , a n u m b e r o f v e c to r o r m ix e d v e c to r-n o d a l fo rm u la ­
tio n s have appeared in th e lite r a tu r e to tre a t inhom ogeneous. lossy, a n d a n is o tro p ic
tw o -d im e n s io n a l s tru c tu re s [3D]-[44].
T h e a ccuracy and v e rs a tility o f th e m e th o d
in e le c tro m a g n e tic s was e x te n s iv e ly d e m o n s tra te d fo r a v a rie ty o f tw o -d im e n s io n a l
w a veguide p ro b le m s. In th e m id 90 ’s, th e ve cto r F E M was a p p lie d fo r th e firs t tim e to
su cce ssfu lly solve c o m p le x m ic r o s trip d is c o n tin u itie s and in te rc o n n e c ts [2 7 ].[4 b ].[46].
In [27]. Y ook et al. used v e c to r te tra h e d ra l e lem ents in c o n ju n c tio n w it h th e e le c tric
fie ld fo r m u la tio n o f th e H e lm h o ltz 's e q u a tio n to ch a ra c te riz e h ig h -fre q u e n c y in te r­
connects. T h e fo rm u la tio n d id not ta ke in to account ra d ia tio n e ffects. In a d d itio n ,
th e e x c ita tio n was im p o se d using a sim p le d e lta source w h ich is n o t v e ry accurate.
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(i
T h e w o rk by W a n g and M it t r a [4-5] beca m e a tu r n in g p o in t fo r fu tu re a p p lic a tio n s o f
F E M in m ic ro w a v e c irc u its a n d p acka g in g. F irs t, th e y in tro d u c e d v e c to r a bso rb in g
b o u n d a ry c o n d itio n s (A B C 's ) to te r m in a te the p o rts a n d sid e w a lls o f th e s tru c tu re .
Second, th e in p u t p o rt was e x c ite d u s in g the g o v e rn in g m o d a l d is tr ib u tio n o f th e
tra n s m is s io n lin e a t a specific fre q u e n cy. T h e m a in d ra w b a c k o f th e ir fo rm u la tio n
stem s on th e a p p ro a c h used to c a lc u la te th e e x c ita tio n fie ld . T h e y s p e c ific a lly used a
3 -D e ig en va lue fo rm u la tio n to so lve fo r th e e x c ita tio n fie ld a t each fre qu e ncy, w h ic h is
c o m p u ta tio n a lly e xpensive. T h e y also assum ed th a t th e tra n s m is s io n lin e specifica ­
tio n s a t a ll p o rts a re id e n tic a l, th e re fo re th e re is no need to in c o rp o ra te c h a ra c te ris tic
im pedances in th e d e fin itio n o f 5’- p a ra m e te rs.
A n a ly s is o f m ic ro w a v e c irc u its and in te rco n n e cts s h o u ld n o t be re s tric te d o n ly
on the 5 -p a ra m e te rs o f th e s tru c tu re . C h a ra c te riz a tio n o f c o n d u c to r and ra d ia tio n
losses is a t least as im p o rta n t. A t c e rta in frequencies, tra n s m is s io n lines w ith fin ite
c o n d u c tiv ity m ig h t e x h ib it s k in d e p th s th a t are c o m p a ra b le to th e ir cross se ction
d im e n sio n s. In such cases, th e fie ld t o t a lly p enetrates in s id e th e c o n d u c to r th e re b y
re s u ltin g in h ig h losses.
P e rtu rb a tio n tech n iq ue s p ro b a b ly re present th e m ost t r a ­
d itio n a l a p p ro a ch to c h a ra c te riz in g c o n d u c to r losses in m o n o lith ic m icro w a ve and
m illim e te r w ave in te g ra te d c irc u its ( M M I C c ) [4 7 ]-[o lj.
These m e th o d s u tiliz e a
g e o m e try -in d e p e n d e n t surface im p e d a n c e to p e rtu rb th e su rface c u rre n ts o b ta in e d
fo r th e lossless case.
T h e e q u iv a le n t su rface im p e d a n ce re su lts in a good a p p ro x ­
im a tio n o n ly w h e n th e s trip th ic k n e s s is m uch la rg e r th a n th e s k in d e p th .
I f th e
s tr ip th ickn e ss is c o m p a ra b le to th e s k in d e p th , th e c o n d u c to r losses using p e rtu rb a ­
tio n te ch n iq u e s are u s u a lly u n d e re s tim a te d . M ore a c c u ra te n u m e ric a l tech n iq ue s o f
c h a ra c te riz in g c o n d u c to r losses in M .M lC s in clu d e th e m o d e m a tc h in g m e th o d [52].
th e M o L [53] and fo rm u la tio n s based on th e in te g ra l e q u a tio n ( I E ) [54].[55] o r th e
S D A [56]-[5S]. T h e F E M [59],[60]. h ow eve r, is p ro b a b ly th e m o st gen e ric and fle x ib le
o f a ll. a lth o u g h n o t necessarily th e m o st accurate.
In a d d itio n to M M IC 's . th e F E M was successfully a p p lie d to a v a rie ty o f s c a tte rin g
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ancl ra d ia tio n pro ble m s [33 ].[6 1 ]-[64 ]. O f great in te re s t is th e h y b rid iz a tio n o f F E M
w ith M o M to ana lyze 2-D and 3 -D s c a tte rin g and ra d ia tio n c h a ra cte ristics o f c a v ity backed a p e rtu re antennas m o u n te d on an in fin ite g ro u n d p la ne [65]-[70]. In a ll cases,
w ith th e e x c e p tio n o f G ong et al. [69], th e a p e rtu re had a lw a ys a re c ta n g u la r shape.
In o th e r w o rd s, th e ir fo rm u la tio n ca n n o t be a p p lie d fo r a pe rtu re s w ith a r b itr a r y
shapes. T h e reason is because th e u n d e rlin e d M o M a p p ro a ch involves ro o fto p basis
fu n c tio n s w it h re c ta n g u la r in ste a d o f tria n g u la r s u p p o rt. G ong et al. [69] were the
o n ly ones to use a s tru c tu re d tr ia n g u la r g rid fo r th e e x te r io r region to m ap th e fields
th a t co rre sp o n d to th e u n s tru c tu re d g rid in th e a p e rtu re .
Such process in volves
an in te r p o la tio n scheme betw een th e tw o g rid s th e re b y c o m p ro m is in g accuracy and
c o m p u ta tio n a l e ffo rt. A n o th e r im p o r ta n t issue is th e p la c e m e n t o f a d ie le c tric o ve rla y
on top o f th e g ro u n d plane to p ro v id e p ro te c tio n fo r th e c a v ity . T h e o n ly fo rm u la tio n
th a t co n side re d a d ie le c tric o v e rla y was the w o rk b y C h e ng et al. [70].
In th e ir
fo rm u la tio n , a pure sp e ctra l d o m a in M o M in c o n ju n c tio n w ith re cta n g u la r ro o fto p s
was im p le m e n te d . However, as it w ill be shown in th is d is s e rta tio n , th e use o f a pure
sp e c tra l d o m a in M o M to represent th e e x te rio r region o f th e c a v ity is c o m p u ta tio n a lly
e xpensive e s p e c ia lly fo r a large n u m b e r o f edges in th e a p e rtu re . D ie le c tric covered
c a v ity -b a c k e d a p e rtu re s were also a n a lyze d using th e s p e c tra l d o m a in M o M fo r b o th
th e in te r io r a n d e x te rio r regions o f th e c a v ity [71].
C’a v ity -b a c k e d a pe rtu re s o r slo ts have been a to p ic o f research and e x p e rim e n t for
th e last 40 years. C'alejs [72] was th e firs t to analyze th e ra d ia tio n c h a ra c te ris tic s o f a
ca v ity-b a cke cl slot using an in te g ra l e q u a tio n a p p ro a ch. He ca lcu la te d th e a d m itta n c e
o f the slo t a n d observed its dependence on th e c a v ity 's d im e n sio n s. He also re a lize d
th a t th e a n te n n a b a n d w id th decreases when the c a v ity is loaded w ith a d ie le c tric
m a te ria l.
A d a m s [73] used v a ria tio n a l m ethods to c a lc u la te a d m itta n c e , e fficien cy,
d ir e c tiv ity , b a n d w id th and resonant fre qu e ncy o f d ie le c tric and fe rrite -lo a d e d c a v ity backed slo ts. A lth o u g h lo a d in g th e c a v ity w ith m a te ria l reduces the b a n d w id th and
e ffic ie n c y o f th e a nte nn a , he illu s tr a te d th ro u g h a n a ly tic a l and e x p e rim e n ta l w o rk
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th a t th e resonant fre q u e n c y can be tun e d by v a ry in g th e e x te rn a lly D C m a g n e tic
fie ld . C ro sw e ll et al. [74] d e riv e d expressions fo r the a d m itta n c e o f a w aveguide-fed
re c ta n g u la r a p e rtu re u n d e r an inhom ogeneous plasm a. L o n g [75] p e rfo rm e d e x te n s iv e
e x p e rim e n ta l stud ie s on th e e ffect o f the c a v ity d im e n sio n s, th e lo a d in g m a te ria l o f
th e c a v ity a n d th e ty p e o f te r m in a tio n o f th e c a v ity on th e in p u t im p e da n ce o f the
slo t.
He th e n cam e up w ith a m a th e m a tic a l m o d el to re la te th e im p e d a n ce o f a
c a v ity -b a c k e d slo t a n te n n a to th a t o f an id e n tic a l slo t w h ic h is free to ra d ia te on
b o th sides o f th e g ro u n d p la n e [76]. A d d itio n a l p a ra m e tric s tu d ie s on the e ffect o f
th e c a v ity on th e in p u t im p e d a n c e o f the slo t were p e rfo rm e d by C o c k re ll [77]. He
illu s tra te d th a t as th e d e p th o f th e c a v ity increases th e re so na n t fre qu e ncy decreases
and th e b a n d w id th becom es n a rro w e r. M o re recent stud ie s on ca vity -b a c k e d slo ts o r
patch antennas have also a ppeared in the lite ra tu re [7S]-[S4],
As m e n tio n e d p re v io u s ly . A d a m s [73] was th e firs t one to use fe rrite s as lo a d ­
in g m a te ria ls in sid e a c a v ity -b a c k e d slot. E x te rn a l m a g n e tiz a tio n o f th e fe r rite can
p ro v id e fre q u e n cy tu n in g c a p a b ilitie s for th e ante nn a . A m o re recent s tu d y on fe rrite tu n e d c a v ity -b a c k e d s lo ts is th e w o rk by K o k o to ff [83]. He s p e c ific a lly used a M o M
a pproach to solve fo r th e fields in sid e and o u tsid e o f th e c a v ity .
T h is fo rm u la tio n
had num erou s lim ita tio n s in c lu d in g th e shape o f the c a v ity and th e d ire c tio n o f m a g ­
n e tiz a tio n .
In a d d itio n , th e fo rm u la tio n its e lf was prone to n u m e ric a l in s ta b ilitie s .
He a n a lyze d b o th s c a tte rin g and ra d ia tio n c h a ra c te ris tic s o f c a v ity -b a c k e d slo ts , a l­
th o u g h th e la tte r d id n o t c o m p a re fa v o ra b ly w ith m e a surem e n ts.
D u rin g th e last
decade, m a g n e tize d fe rrite s have also been used as su b stra te s fo r m ic ro s trip p atch
antennas [S5]-[92].
T h e d ire c tio n o f the e x te rn a lly bias fie ld can be chosen e ith e r
p a ra lle l o r n o rm a l to th e p la ne o f th e patch. I t was n u m e ric a lly illu s tra te d th a t v a ri­
ous s c a tte rin g and ra d ia tio n c h a ra c te ris tic s o f th e a nte nn a can be c o n tro lle d th ro u g h
a lte rin g th e s tre n g th o r th e d ire c tio n o f an e x te rn a lly bias m a g n e tic fie ld . Such c h a r­
a c te ris tic s in c lu d e re so na n t fre qu e ncy, ra d a r cross section, m a in beam , surface waves,
gain and e fficien cy.
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In th is d is s e rta tio n , a m ix e d e d g e -n o d a l F E M was firs t im p le m e n te d fo r th e a n a l­
ysis o f 2 -D m icro w a ve s tru c tu re s in th e presence o f e le c tric and m a g n e tic a n is o tro p ic
s u b s tra te s /s u p e rs tra te s and im p e rfe c t co n d u cto rs. T h e fu ll-w a v e fo r m u la tio n closely
fo llo w s th e w o rk by J in -F a Lee [42]; h o w e ve r it was e xte n d e d to a n is o tro p ic a n d lossy
m e d ia . T h e o b ta in e d g en e ra lize d e ig e n va lu e m a tr ix system is solved u s in g a pow erfo rw a rd ite ra tio n in c o n ju n c tio n w it h a G ra m -S c h m id t o rth o g o n a liz a tio n process.
T h e p ro p a g a tio n c o n s ta n t, a tte n u a tio n c o n s ta n t, c h a ra c te ris tic im p e d a n c e and field
d is tr ib u tio n are c o n v e n ie n tly c a lc u la te d versus frequency.
A close in v e s tig a tio n on
th e p ro p a g a tio n c h a ra c te ris tic s o f a m ic r o s trip and a c o p la n a r w a ve g u id e w ith fou r
c o m m o n ly used a n is o tro p ic s u b s tra te s is presented. T h e effect o f fin ite c o n d u c tiv ity
on th e p ro p a g a tio n c h a ra c te ris tic s o f m ic ro s trip s and co p la n a r w aveguides is also
e x a m in e d . T h is s u b je c t is p re sen ted in C h a p te r 3. In C h a p te r 2. a b r ie f h is to ric a l
b a c k g ro u n d o f th e F E M is g ive n .
A 3 -D v e c to r F E M is d e ve lo p e d a n d used fo r the a nalysis o f s h ie ld e d and open
M .M IC 's and e le c tro n ic packages.
T h e fo rm u la tio n was developed to tre a t e le c tric
and m a g n e tic a n iso tro p ie s, a lth o u g h a p p lic a tio n s in th is d is s e rta tio n a re lim ite d to
o n ly is o tro p ic m e d ia. T h e in p u t p o r t o f th e s tru c tu re is e x c ite d w ith th e g o v e rn in g
m o d e d is tr ib u tio n a t a sp ecific fre q u e n cy.
T h is e x c ita tio n fie ld is c a lc u la te d using
th e 2 -D eigenvalue fo rm u la tio n . T h e d is p e rs iv e p ro p a g a tio n c o n s ta n t o f th e tra n s m is ­
sion lin e a t each p o rt is u tiliz e d fo r an e ffic ie n t im p le m e n ta tio n o f th e A B C 's .
L he
c o rre s p o n d in g c h a ra c te ris tic im p e d a n c e is used in th e general d e fin itio n o f th e
p a ra m e te rs w h ich are c a lc u la te d based on th e to ta l and reference v o lta g e d iffe re n ce
b etw een th e tra n s m is s io n lin e a n d th e g ro u n d plane.
T h e e le c tric fie ld d is tr ib u ­
tio n in sid e th e s tru c tu re is o b ta in e d at each fre qu e ncy by s o lv in g a lin e a r system
o f e q u a tio n s . T h e code, w h ich was w r itt e n in F O R T R A N and in te rfa c e d w ith v a r­
ious g ra p h ic s packages, was e x te n s iv e ly v e rifie d and a p p lie d to m ic ro w a v e p la n a r
c irc u its , in te rc o n n e c ts , w ire b onds, a n d c o m p le x e le c tro n ic packages. T h e effects ot
p o o r g ro u n d in g and packaging are illu s tr a te d .
T h e c o rre sp o n d in g fo r m u la tio n aiu l
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10
n u m e ric a l re s u lts w ith co m p a ris o n s are presented in C h a p te r 4.
In a d d itio n to th e a n a lysis o f M M IC 's and e le c tro n ic packages, the F E M is also
used in th is d is s e rta tio n fo r th e a n a lysis and d e sign o f c a v ity -b a c k e d slot antennas
m o u n te d on in f in it e / f in it e g ro u n d planes. T h e c a v ity a nd a p e rtu re can be o f a r b i­
tr a r y shape sin ce th e d is c re tiz a tio n in vo lves te tra h e d ra s fo r th e c a v ity vo lu m e and
tria n g le s fo r th e a p e rtu re . In a d d itio n , th e c a v ity m ig h t be fille d w ith is o tro p ic or
fu lly a n is o tro p ic m a te ria ls in c iu d in g p la sm a a n d m a g n e tiz e d fe rrite s .
In th e case
o f an in f in ite g ro u n d plane, a sin g le la ye r o f d ie le c tr ic o r m a g n e tic o verlay can be
used to p ro te c t th e a n te n n a fro m e n v iro n m e n ta l c o n d itio n s . T h e m a in idea b e h in d
th e present a p p ro a ch is to use th e F E M to re p re s e n t th e fields inside th e c a v ity
and a h y b rid s p e c tra l-s p a tia l d o m a in M o M to re p re s e n t th e fields o utsid e th e c a v ity .
T h e h y b rid a p p ro a ch is a u g m e n te d w ith the use o f th e u n ifo rm th e o ry o f d iffra c tio n
( U T D ) to a c c o u n t fo r th e d iffr a c te d fields fro m th e edges o f a fin ite and u ncoated
g ro u n d p la n e . In a d d itio n , th e c o m p u ta tio n a l e ffic ie n c y o f th e h y b rid code is s u b s ta n ­
t ia lly im p ro v e d b y im p le m e n tin g a fre q u e n cy in te r p o la tio n o f th e a d m itta n c e m a trix .
R esults fo r ra d a r cross s e ctio n , in p u t im p e d a n ce , a b s o lu te g a in /d ir e c tiv ity p a tte rn s ,
re tu rn loss a n d e fficie n cy are c a lc u la te d fo r a v a rie ty o f a n te n n a co n fig u ra tio n s .
The
u n d e rlin e d fo rm u la tio n and n u m e ric a l v a lid a tio n o f th e h y b rid code is presented in
C h a p te r o.
In C h a p te r 6. th e h y b rid co de is a p p lie d fo r th e d esign and analysis o f a fe rrite tu n e d c a v ity -b a c k e c l a p e rtu re .
T h e c a v ity is fille d w it h layers o f solid fe rrite and
d ie le c tric m a te r ia l whereas th e c a v ity its e lf is m o u n te d on an in fin ite g ro u n d plane.
A n e x te rn a l m a g n e tic fie ld p a ra lle l to the a p e rtu re p la n e and alig ne d w ith th e d i­
re c tio n o f th e p ro b e is a p p lie d . T h e ra d a r cross s e c tio n o f th e a nte nn a is c a lc u la te d
fo r va rio us m a g n e tiz a tio n s . A d d itio n a l p a ra m e tric s tu d ie s are cond u cte d by a lte rin g
th e lin e w id th a n d th e s a tu ra tio n m a g n e tiz a tio n o f th e m a te ria l. T h e accuracy o f th e
s c a tte rin g co de is tested a g a in s t th e w o rk by K o k o to ff [S3], For th e ra d ia tio n case,
p re d ic tio n s are com p ared w it h m e asurem ents p e rfo rm e d in the anechoic c h a m b e r
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o f A S T . T h e a n a lysis presented in th is c h a p te r illu s tra te s th a t th e antenna can be
successfully tu n e d th ro u g h th e use o f m a g n e tize d fe rrite s .
T h e fin a l s u b je c t o f th is d is s e rta tio n , w hich is p re se n te d in C h a p te r 7. deals
w ith th e p ro p e r te rm in a tio n o f th e fin ite elem ent d o m a in fo r unb o un d ed ra d ia tio n
problem s.
S p e c ific a lly , th e a n is o tro p ic p e rfe c tly m a tc h e d la ye r ( P M L ) . w hich was
in tro d u c e d by Sacks et al. [93]. is fo rm u la te d and o p tim iz e d in th e co n te xt o f the
F E M . T h e P M L is a n on -p h ysical a r tific ia l absorber t h a t is p e rfe c tly m atched be­
tween tw o m e d ia to e lim in a te re fle ctio n s fro m the in te rfa c e . In a d d itio n , it decays the
p ro p a g a tin g wave in a d ire c tio n n o rm a l to th e in te rfa c e w h ic h makes it su itab le fo r
mesh tru n c a tio n in fin ite m e tho d s. T h e P M L concept was o rig in a lly in tro d u ce d and
im p le m e n te d in to th e F D T D by B erenger [94]. His im p le m e n ta tio n tho u g h in vo lved
a s p lit-fie ld fo rm u la tio n inside th e P M L region w hich is n o t ve ry s u ita b le for a fin ite
elem ent fo rm u la tio n . T h e idea o f the a n is o tro p ic P M L is c e rta in ly m ore su itab le fo r
F E M . H ow ever, it was shown by o th e r researchers [95 ].[9 6 j th a t its n u m e rica l im p le ­
m e n ta tio n m ig h t n ot be as a ccurate as in th e F D T D . In C h a p te r 7. th e n um e rica l
e rro r in tro d u c e d by th e a n is o tro p ic P M L u sin g the F E M is c[u a n tifie d and th o ro u g h ly
in v e stig a te d fo r va rious design p a ra m e te rs in c lu d in g d e p th , p ro file , m a xim u m co n ­
d u c tiv ity and mesh d en sity. It is illu s tra te d th a t the a n is o tro p ic P M L m ig h t be the
pre fe ra b le w ay o f te rm in a tin g th e c o m p u ta tio n a l d o m a in in F E M : however there a n '
s t ill a d d itio n a l p ro b le m s to be addressed.
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CHAPTER 2
B R IE F H IS T O R Y O F T H E F IN IT E E L E M E N T M E T H O D
D is c re tiz a tio n m ethods have becom e in cre a sin g ly p o p u la r in c o m p u ta tio n a l e le c tro ­
m a g n e tics because o f th e e n o rm o u s need to a c c u ra te ly and e ffic ie n tly s im u la te fie ld
in te ra c tio n a m o n g ra d ia tin g e lem ents in th e presence o f m a te ria l and g e o m e tric a l
c o m p le x itie s . A lth o u g h th e m o re tra d itio n a l in te g ra l m e th o d s, such as s p a tia l and
s p e c tra l d o m a in m e th o d o f m o m e n ts, are u s u a lly m o re a ccu ra te n u m e ric a l te c h ­
niq ue s. th e y are o fte n lim ite d to re la tiv e ly s im p le g eo m e trie s. Recent a dva n cem en ts
in c o m p u te r tech n o lo g y and c o m p u te r aided design becam e a tu rn in g p o in t fo r the
d e v e lo p m e n t and m a tu r ity o f w e ll-k n o w n d is c re tiz a tio n techniques such as th e fi­
n ite e le m e n t m e th o d ( F E M ) and th e fin ite -d iffe re n c e tim e -d o m a in ( F D T D ) m e th o d .
B o th m e th o d s can be e q u a lly used to solve b o u n d a ry -v a lu e p ro b le m s, e igenvalue
p ro b le m s , and in itia l-v a lu e p ro b le m s w h ich are c o m m o n ly fo u n d in areas o f flu id
m ech an ics, e le ctro m a g n e tics, s o lid m echanics, heat tra n s fe r, etc. A p p lic a tio n o f th e
F E M in e le ctro m a g n e tics becam e e x tre m e ly p o p u la r in th e e a rly 1980"s.
W h ile in
th e e a rly 1970's the n u m b e r o f p ub lish e d a rtic le s a b o u t fin ite elem ents in e le c tro ­
m a g n e tic p ro b le m s was less th a n a h a n d fu l, by th e e a rly 1990 s th is n u m b e r raised
to a few tho usands. A n e s tim a te o f a b o u t GOO new a rtic le s are p ub lish e d e ve ry year
in va rio u s professional jo u rn a ls and magazines.
T h e m a in p rin c ip le o f th e F E M is to replace th e e n tire c o m p u ta tio n a l d o m a in
b y a n u m b e r o f sm a lle r su b d o m a in s ca lle d t he f i n i t e elements. For tw o -d im e n s io n a l
(2 -D ) p ro b le m s these e le m en ts are u su a lly tria n g le s a n d rectangles, whereas fo r th re e d im e n s io n a l (d -D ) pro ble m s these are te tra h e d ro n s , tria n g u la r p rism s and re c ta n g u ­
la r b ricks. T h e u n kn o w n s o lu tio n is in te rp o la te d u s in g s im p le p o ly n o m ia ls o f degree
n w h ic h are u s u a lly associated w ith th e vertices, edges o r faces o f th e e le m e n t. A
set o f a lg e b ra ic equations is o b ta in e d by a p p ly in g th e G a le rk in s a p p ro a ch to th e
g o v e rn in g d iffe re n tia l e q u a tio n . S o lu tio n to th e syste m o f equ a tion s can be achieved
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13
using w e ll-k n o w n m a tr ix solvers such as c o n ju g a te g ra d ie n t, b i-c o n ju g a te g ra d ie n t,
e tc. In s u m m a ry , th e basic steps in a fin ite e le m e n t a n a lysis in v o lv e s th e fo llo w in g :
1. D is c re tiz a tio n o f th e c o m p u ta tio n a l d o m a in
2. S election o f in te rp o la tio n /b a s is fu n c tio n s
3. F o rm u la tio n o f th e system o f e q u a tio n s
4. S o lu tio n o f th e system o f e q u a tio n s
T h e re m a in in g p a rt o f th is c h a p te r is a s e le ctive h is to ric a l o v e rv ie w o f th e F E M as
a p p lie d to c o m p u ta tio n a l e le c tro m a g n e tic s . It b rie fly o u tlin e s m a jo r tu r n in g points
and stages o f th e m e th o d as w e ll as n u m e ric a l challenges associated w ith th e fo r­
m u la tio n at th e c u rre n t tim e . B y no m eans th e fo llo w in g h is to ric a l b a ckg ro u n d is a
co m p le te b ib lio g ra p h y o f th e F E M . O n th e c o n tra ry , it c o n ce n tra te s o n ly on m a jo r
and sp orad ic c o n trib u tio n s th a t are som ehow re la te d to th e c o n te x t o f th is d isse rta ­
tio n .
A lth o u g h th e F E M was o rig in a lly fo rm u la te d and o u tlin e d b y C’o u ra n t [97] in
th e e a rly I9 4 0 's. it re m a in e d to ta lly stra ng e to th e e le c tro m a g n e tic so c ie ty u n til the
la te I960's. S ilv e s te r [34] in tro d u c e d th e m e th o d to e le c tro m a g n e tic s b y p u b lis h in g a
classical p ap e r on th e fin ite e le m e n t s o lu tio n o f d o m in a n t and h ig h e r-o rd e r m odes for
hom ogeneous w aveguides. A h m e d and D a ly [3o] la te r used a s im ila r fin ite elem ent
fo rm u la tio n to a n a lyze inhom ogeneous w aveguides in s te a d .
A n a ly s is o f d ie le c tric -
loaded w aveguides using a fin ite e le m e n t d is c re tiz a tio n o f th e E z a n d H : scalar wave
e qu a tion s was also c a rrie d o u t b y C'endes and S ilv e s te r [36] in th e e a rly 1970's.
A d d itio n a l a rtic le s on to p ics such as m a g n e tics, n o n -lin e a r m a te ria ls and generic
w aveguide s tru c tu re s appeared th e re a fte r.
T h e w o rk done by th e pioneers o f th e F E M in th e la te 1960 s a nd e a rly 1970 s
becam e th e fo u n d a tio n fo r th e d e ve lo p m e n t o f one o f th e m ost p o w e rfu l n u m e rica l
techniques in e le c tro m a g n e tic s . T h e re was. h ow ever, one serious p ro b le m associated
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11
w ith th e o rig in a l fo rm u la tio n .
T h e p ro b le m evolved fro m th e use ot node-based
elem ents to re p re sen t ve cto r e le c tric a n d m a g n e tic fields. T h e o b ta in e d n u m e ric a l re­
sults were flo o d e d w ith n o n -p h ysica l s o lu tio n s re ferre d to as s p ur i ou s modes [36 ].[9S].
A n o th e r nam e som e tim e s used fo r n u m e ric a l sp u rio u s m odes is vector parasites. As it
was re p o rte d la te r in th e e a rly I9 9 0 's [99]-[ 103]. n o n -p h ysica l so lu tio n s are a ttr ib u te d
to the c o m b in e d effect o f lack o f e n fo rc e m e n t o f th e divergence c o n d itio n , th e in a b il­
it y to im p o se c o n tin u ity o f ta n g e n tia l fie ld s across m a te ria l in terfaces, and im p ro p e r
a p p ro x im a tio n o f th e n u ll space o f th e c u rl o p e ra to r. D e sp ite th e num erous e ffo rts by
researchers to p ro v id e a good and w id e ly a cce p ta b le e x p la n a tio n to the appearance
o f sp u rio u s m odes in e le c tro m a g n e tic p ro b le m s , th e search fo r th e o rig in o f these
modes is s t ill a to p ic o f c u rre n t research.
U n til th e e a rly I980's. the s p u rio u s -m o d e p ro b le m to t a lly p revented researchers
from u sin g th e F E M in e le c tro m a g n e tic a p p lic a tio n s . T h e in tro d u c tio n o f edge-based
elem ents by Neclelec [37] in L9S0 was a m a jo r b re a k th ro u g h in th e d eve lo p m e n t o f
the m e th o d .
F ir s t- and h ig h e r-o rd e r v e c to r e le m en ts fo r tria n g le s and te tra h e d ra s
follow ed th e re a fte r [42].[l04 ]-[1 0S ].
L 'n lik e n o d a l elem ents, edge-based e le m en ts a l­
lows c o n tin u ity o f th e ta n g e n tia l fie ld s across edges whereas the n o rm a l fields re m a in
d is co n tin u o u s.
modes.
In a d d itio n , edge-based e le m e n ts are know n to e lim in a te sp uriou s
W h ile th e reason is not w e ll-u n d e rs to o d y e t. it was shown th a t b y im p o s ­
ing th e d ive rg e n ce -fre e c o n d itio n , th e in f in it e n u m b e r o f s ta tic e ig en so lutio n s t o ta lly
disappear fro m th e co rre ct s o lu tio n . T h is d ive rg e n ce -fre e c o n d itio n can be e x p lic it ly
added to th e p a r tia l d iffe re n tia l e q u a tio n (p e n a lty a p p ro a ch ), o r o th e rw ise , in c o r­
porated in to th e v e c to r basis fu n c tio n s .
F o r e xa m p le , it can be e asily show n th a t
the v e c to r basis fu n c tio n s associated w ith lin e a r tria n g u la r and te tra h e d ra l e le m en ts
indeed s a tis fy th e divergence-free c o n d itio n : how ever, o th e r ty p e o f e le m ents, such
as lin e a r h exa h ed ra s. do not e x p lic it ly s a tis fy such c o n d itio n . To th is day. it is be­
lieved th a t th e u tiliz a tio n o f d ive rg e n c e -fre e v e c to r elem ents helps p ro vid e a u n iq u e
s o lu tio n th a t is free o f sp urious m o d e s, a lth o u g h the latest c la im states th a t vec­
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to r e le m en ts e lim in a te sp u rio u s m odes by a p p ro p ria te ly m o d e lin g th e n u ll space o f
the c u rl o p e ra to r [103]. A n a d d itio n a l a dvantage o f v e c to r elem ents is th a t th e y a l­
low e nfo rce m e nt o f D ir ic h le t b o u n d a ry c o n d itio n s in a very s tra ig h tfo rw a rd m a n n e r
regardless o f g e o m e tric a l c o m p le x itie s [109].[110].
F in a lly , edge-based elem ents are
kn ow n to e ffe c tiv e ly m o d e l th e a b ru p t change in th e d ire c tio n o f th e field at sharp
corners and edges [108]. H ow ever, th e m a g n itu d e o f th e field, w h ich becomes in fin ite
at those p o in ts , ca n n o t be p re cisely m odeled unless s in g u la r elem ents are used [1 1 1][113]. In co n tra st to edge-based e lem ents, n od a l e lem ents are know n to fa il in b o th
aspects.
B o th nodal and v e c to r fin ite e lem ents have been e xte n s iv e ly used to a n a lyze
p ro p a g a tio n c h a ra c te ris tic s o f g e o m e tric a lly and m a te r ia lly c o m p le x m icrow ave s tru c ­
tu re s. A ssu m in g a fie ld v a ria tio n e~~,: a long th e ^ -d ire c tio n , th e o rig in a l 3-D fin ite
ele m en t fo rm u la tio n can be reduced to a s im p le 2 -D p ro b le m . T h e nodal fo rm u la ­
tio n was th e firs t to be im p le m e n te d fo r th e a n a lysis o f hom ogeneously loaded d i­
e le c tric waveguides [34].[36]. g y ro m a g n e tic w aveguides [111], lossy waveguides [ll~>]
and o p tic a l fibers [116].
V e c to r fo rm u la tio n s have follow ed a fte rw a rd s w ith a p p li­
c a tio n s to a x ia lly m a g n e tiz e d fe r rite waveguides [43].[117]. m ic ro s trip s [118].[1 19].
fin lin e s [120]. and s trip lin c s [121].
P la n a r m ic ro w a v e c irc u its w ith d ie le c tric and
c o n d u c tin g losses [4 4 ].[o 9 ].[ 122] and m icro w a ve s tru c tu re s loaded w ith a n is o tro p ic
m a te ria ls [123].[124] have also been in v e s tig a te d u sin g ve cto r fo rm u la tio n s .
H a n o [38] was th e firs t to in tro d u c e re c ta n g u la r v e c to r elem ents to e xam ine p ro p a ­
g a tio n c h a ra c te ris tic s o f w a veguide s tru c tu re s th a t are p a r tia lly loaded w ith d ie le c tric
m a te ria ls . S pe cifica lly, he illu s tra te d th a t th e use o f edge-based elem ents c o m p le te ly
e lim in a te s spurious m odes fro m th e n u m e ric a l s o lu tio n . Since th e n , m any researchers
have used ve cto r e le m en ts to solve a n u m b e r o f e le c tro m a g n e tic problem s in c lu d in g
w aveguide p ro p a g a tio n [4 0 ].[4 2 ].[4 4 j.[1 25].[126]. s c a tte rin g [33].[61],[62].[67]. ra d ia tio n
[63].[67]. and m icro w a ve c ir c u it c h a ra c te riz a tio n [27],[45].[46].
E rro r analysis and a d a p tiv e mesh re fin e m e n t [127]-[ 133] has been a n o th e r to p ic
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16
o f g re at in te re s t in recent years. Som e o f th e m a jo r b e n e fits [134] o f th is tech n iq ue
in c lu d e h ig h degree o f accuracy, m in im u m m e m o ry a llo c a tio n , re d u c tio n in c o m p u ­
ta tio n a l tim e and co m p le te a u to m a tio n .
It is kn o w n to be e x tre m e ly useful in th e
analysis o f 3 -D p ro b le m s w here th e n u m b e r o f u n k n o w n s is o fte n in the o rd e r o f
thousands.
U tiliz a tio n o f e rro r ana lysis and a d a p tiv e m esh re fin e m e n t u su a lly re­
su lts in a c c u ra te so lu tio n s at a m in im u m cost o f m e m o ry space and c o m p u ta tio n a l
tim e . T h is can be achieved by s ta r tin g w ith a very coarse m esh, proceed w ith so lvin g
the fin ite e le m e n t system , c a lc u la te th e local e rro r in e v e ry e le m e n t, and then refine
by a d d in g new degrees o f freedom near th e regions w here th e lo ca l e rro r appears to
have exceeded th e th re s h o ld level. O nce th e mesh re fin e m e n t is accom plished, th e
m a trix se tu p a n d m a tr ix system s o lu tio n is repeated to o b ta in new estim a te s o f th e
local e rro r. T h is p ro ced u re keeps re p e a tin g its e lf u n til a ll e le m e n ts reach th e desired
e rro r level. F o llo w in g such an a p p ro a ch , th e mesh d is c re tiz a tio n is t ig h tly co n tro lle d
whereas th e a c cu ra cy o f th e s o lu tio n is s u b s ta n tia lly im p ro v e d .
Besides e rro r a nalysis and a d a p tiv e mesh re fin e m e n t, a b s o rb in g b o u n d a ry c o n d i­
tio n s ( A B C 's ) are also e x tre m e ly im p o rta n t in s im u la tin g u n b o u n d e d e le ctro m a g n e tic
pro ble m s. In m o st s itu a tio n s th e s tru c tu re is not sh ie lde d a n d . the re fo re, the fin ite
elem ent mesh ca n n o t be tru n c a te d w ith th e use o f an e le c tric o r m a gnetic w a ll.
How ever, w ith th e im p le m e n ta tio n o f A B C 's on an a r tific ia l b o u n da ry, th e fields
are allo w e d to p ro p a g a te u n d is tu rb e d in th e o u tw a rd d ire c tio n .
O nce the c o m p u ­
ta tio n a l d o m a in is e ffe c tiv e ly te rm in a te d , n o n -p h ysica l re fle c tio n s fro m the a rtific ia l
tru n c a tio n b o u n d a ry are m in im iz e d . A s ig n ific a n t c o n tr ib u tio n in th e developm ent
and im p le m e n ta tio n o f A B C 's in th e F E M is th a t o f B a yliss. G u n z b u rg e r and T u rk e l
[13-5]. P eterson [136].[137] and W ebb and Ixan ellopoulos [13S]-[140]. A com m on c h a r­
a c te ris tic o f a ll these a bso rb in g b o u n d a ry c o n d itio n s is t h a t th e y re ta in the s p a rs ity
o f th e m a tr ix .
T h is is a ve ry d e sira b le fea tu re , b o th in te rm s o f m e m o ry a llo c a ­
tio n and c o m p u ta tio n a l tim e .
T h e h ig h e r the s p a rs ity o f th e m a trix , th e s m a lle r
tlie n u m b e r o f n on -ze ro e n trie s: co n se q ue n tly, less m e m o ry re q u ire m e n ts are needed.
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17
T h e c o m p u ta tio n a l tim e re q u ire d to solve th e re s u ltin g lin e a r syste m is also less since
an ite ra tiv e so lve r, instead o f a d ire c t so lve r, is u sually im p le m e n te d .
In a d d itio n
to s p a rs ity , it is desirab le t h a t th e a b s o rb in g b ou n da ry c o n d itio n s do n o t d e s tro y
th e s y m m e try o f th e fin ite e le m e n t m a tr ix .
W ebb and K a n e llo p o u lo s [139] have
proposed firs t- and se co n d -o rd e r v e c to r A B C 's th a t re ta in th e s y m m e try o f th e d is ­
c re tiz e d e q u a tio n s. H ig h e r th a n se co n d -o rd e r a n a ly tic a l A B C 's are q u ite d if fic u lt to
d e riv e and im p le m e n t. H o w e ve r, n u m e ric a l A B C 's [141] is s t ill an a lte r n a tiv e choice.
A n o th e r w ay o f tru n c a tin g th e fin ite e le m e n t mesh is to im p le m e n t an in te g ra l
fo rm u la tio n usin g the free-space G re e n 's fu n c tio n . T h is a p p ro a ch is u s u a lly re fe rre d
to as th e fin ite e le m e n t-b o u n d a ry in te g ra l m e th o d ( F E - B I) . It was o r ig in a lly p ro ­
posed by M c D o n a ld and W e x le r [142] and la te r em ployed by J in and L ie p a [61] to
in v e s tig a te e le c tro m a g n e tic s c a tte rin g fro m 2-D cylin d ers. T h e F E M re p re sen ts th e
fie ld s th a t are in te r io r to th e a r t if ic ia l su rface whereas th e b o u n d a ry in te g ra l m e th o d
represents th e e x te rio r fie ld s. T h e tw o s o lu tio n s are co up le d th ro u g h th e c o n tin u ity
o f th e ta n g e n tia l fields across th e in te rfa c e . A lth o u g h such an a p p ro a ch is e x tre m e ly
a c cu ra te , it is n ot very p ra c tic a l fo r 3-D p ro b le m s because it d estro ys th e s p a rs ity of
th e m a tr ix system .
M esh tru n c a tio n o f th e c o m p u ta tio n a l d o m a in can also be achieved u s in g f ic t i­
tio u s absorbers [143]. o th e rw is e kn o w n as n on-physical lossy m a te ria ls .
H o w e ver,
th e m a jo r d ra w b a c k o f the se abso rb e rs is th a t th e y are designed to a b so rb in c i­
d e n t fields o n ly w ith in a r e la tiv e ly n a rro w frequency band. R e ce n tly. B e re n g e r [94]
has in tro d u c e d th e so-called p e r fe c tly m a tc h e d layer (P M L ) w h ic h , th e o re tic a lly , is
re fle ct ionless fo r a ll fre qu e ncies a n d angles o f incidence. He illu s tra te d th e e ffe c tiv e ­
ness o f th e P M L concept b y s o lv in g su ccessfu lly 2-D s c a tte rin g p ro b le m s u s in g th e
F D T D m e th o d .
H is im p le m e n ta tio n o f th e P M L in v o lv e d a s p lit-fie lc l fo r m u la tio n
in s id e tiie c o m p u ta tio n a l d o m a in . T h e a b so rb e r is s im p ly m a tch e d a t th e in te rfa c e
b etw een free space and P M L . w h ic h m eans th a t there are no re fle c tio n s caused by
th e in te rfa c e when an in c id e n t wave m akes a tra n s itio n fro m one m e d iu m to a n ­
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o th e r. S ubsequently, th e p ro p a g a tin g fie ld in sid e the lossy P M L m e d iu m b e g in s to
decay e x p o n e n tia lly . T h e a m o u n t o f loss can be chosen so th a t th e wave is t o t a lly
a tte n u a te d before it a c tu a lly reaches th e o u te r boundary.
Since th e in tro d u c tio n o f th e P M L . researchers have successfully im p le m e n te d
th e concept to solve a v a rie ty o f e le c tro m a g n e tic problem s. A tru n c a tio n e rro r w h ic h
is sm a lle r th a n —100 d B was o b ta in e d u sin g th e F D T D m e th o d [1 4 4 ].[L4-5]. T h is
tru n c a tio n e rro r is b y fa r s m a lle r th a n th e co rresponding e rro r observed w h e n using
a d iffe re n t ty p e o f a b so rb in g b o u n d a ry c o n d itio n to te rm in a te th e m esh.
T h e P M L co nce p t was la te r fo r m u la te d b y Pekel and M it t r a [146].[147] u s in g th e
F E M . T h e ir fo rm u la tio n o f th e P M L was so m e w ha t c o m p lica te d , whereas th e o b ­
ta in e d results were n ot as a c c u ra te as th e re su lts o btained using th e F D T D m e th o d .
A m ore e ffe ctive fin ite e le m e n t fo r m u la tio n o f th e P M L was d eveloped a n d im p le ­
m e n te d by Sacks et al. [93].
T h e p e r fe c tly m atched absorber was m o d e le d as a
u n ia x ia l lossy a n is o tro p ic m a te ria l w h ic h was la te r [148] found to be m a th e m a ti­
c a lly equ iva le nt to c o o rd in a te s tre tc h in g [149],[150]. The idea o f a p e rfe c tly m a tc h e d
a n is o tro p ic m e d iu m was also im p le m e n te d in th e F D T D m e th o d [148]-[153]. L n lik e
B erenger's o rig in a l fo rm u la tio n , th e use o f a u n ia x ia l a n iso tro p ic m e d iu m does n ot
re q u ire th e s p littin g o f th e fie ld s in s id e th e c o m p u ta tio n a l region.
E xcellen t re su lts re g a rd in g th e im p le m e n ta tio n o f p erfe ctly m a tc h e d la ye rs in th e
F E M were re c e n tly re p o rte d in [1 5 4 ].[1 5 5 ]. T h e p rim a ry focus o f th is w o rk was to
is o la te and q u a n tify th e n u m e ric a l e rro r caused by the presence o f th e a b s o rb e r.
N u m e ric a l results d e m o n s tra te cases w h e re th e tru n c a tio n e rro r reaches levels as low
as —80 d B in c lu d in g d is c re tiz a tio n e rro r. A lth o u g h the p e rfe c tly m a tch e d la y e r was
th e o re tic a lly proven to be an id e al a b s o rb e r fo r a ll p ro p a g a tin g waves, it is d e fin ite ly
n o n -id e a l w hen im p le m e n te d in to n u m e ric a l m e tho d s. I ’se o f th e P M L a b s o rb e r in
th e F E M . for e xa m p le , d estroys th e c o n d itio n n u m b e r of th e g lo b a l m a tr ix . C u rre n t
research stud ie s are p r im a r ily c o n c e n tra te d on im p ro v in g th e convergence ra te ot
m a tr ix solvers.
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W ith a ll these re ce nt advances in th e a re a o f c o m p u ta tio n a l e le ctro m a g n e tic s ,
th e F E M has reached a p o in t where it can now be successfully a p p lie d to a w id e
range o f e le c tro m a g n e tic p ro b le m s.
T h e m e th o d has a lre a d y been a p p lie d b y .Jin
and V o la kis [67] to c a v ity -b a c k e d m ic ro s trip p a tc h a n te n n a s m o u nte d on an in fin ite
g ro un d plane. T h e F E M was h y b rid iz e d w it h th e s p a tia l d o m a in m e th o d o f m o m e n ts
to solve ra d ia tio n a n d s c a tte rin g problem s. E dge-based elem ents in c o n ju n c tio n w ith
v e cto r A B C ’s were im p le m e n te d by C’h a tte rje e et al. [6-4] fo r the s o lu tio n o f 3 -D s c a t­
te rin g p ro b le m s. R e m a rk a b ly accurate re s u lts w ere o b ta in e d by p la cing th e mesh o n ly
a s m a ll fra c tio n o f a w a ve le n g th away fro m th e s c a tte re r. O th e r researchers have used
the F E M to c o m p u te th e 5 -p a ra m e te rs o f 3 -D w a ve g u id e d is c o n tin u itie s [1 -6 ].
An
im p re ssive c o n tr ib u tio n to th e m e tho d was th e w o rk b y W ang and M it t r a [46j.[46]
w ho su ccessfully im p le m e n te d a ve cto r F E M to e v a lu a te th e 5 -p a ra m e te rs o f va rio u s
m o n o lith ic m ic ro w a v e in te g ra te d c irc u it ( M M I C ) c o n fig u ra tio n s . N u m e ric a l m o d e l­
ing o f M M IC devices and ca vity-b a cke d a p e rtu re a nte n n a s is s till an o n -g o in g and
in te re s tin g research to p ic in th e area o f c o m p u ta tio n a l e le ctro m a g n e tics.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C H A P T E R :}
T W O - D IM E X S IO X A L M IC R O W A V E S T R U C T U R E S
T h is c h a p te r presents a fu ll-w a v e a n a lysis o f 2-D m ic ro w a v e c irc u its w hich m ig h t
e x h ib it e le c tric and m a g n e tic a n is o tro p ie s as w ell as d ie le c tr ic a n d c o n d u c to r losses.
D isp e rsive q u a n titie s such as th e p ro p a g a tio n co nsta nt a n d c h a ra c te ris tic im pedance
are e va lu a te d fo r each o f th e e x is tin g m odes. T h e fin ite -e le m e n t fo rm u la tio n y ie ld s a
g e n e ra lize d eigenvalue m a tr ix syste m w ith th e e ig e n ve cto r re p re s e n tin g the transverse
and lo n g itu d in a l e le c tric fie ld s in th e s tru c tu re and th e e ig e n va lu e representing th e
c o rre s p o n d in g p ro p a g a tio n c o n s ta n t.
T h e present fo r m u la tio n is to be in te rfa c e d
w ith a n o th e r fo rm u la tio n , w h ic h is o u tlin e d and discussed in th e fo llo w in g c h a p te r,
fo r th e a na lysis o f 3-D m ic ro w a v e c irc u its . T h e 2-D a n a ly s is solves fo r the g o ve rn in g
m o d a l d is tr ib u tio n at th e in p u t p o rt w h ich is needed to e x c ite th e s tru c tu re . In cases
w h e re m ore th a n one p ro p a g a tin g m o d e is su p p o rte d , th e e x c ita tio n field becomes
th e s u p e rp o s itio n o f a ll these m odes.
E v a lu a tio n o f th e c h a ra c te ris tic im pedances
and p ro p a g a tio n co nsta nts at a ll re m a in in g p o rts are also necessary fo r the a nalysis
o f 3 -D m icro w a ve c irc u its .
3.1
In tr o d u c tio n
A c c u ra te p re d ic tio n o f p ro p a g a tio n c h a ra c te ris tic s in p la n a r s tru c tu re s using is o tro p ic
and a n is o tro p ic su bstra te s is essen tia l in th e design o f m o n o lith ic m icrow ave in te ­
g ra te d c irc u its ( M M IC 's ) [9 ].[1 3 ].[lo 6 ]-[lo S ].
Since m a n y su b stra te s in m icrow ave
and m illim e te r wave a p p lic a tio n s e x h ib it d ie le c tric a n d / o r m a g n e tic anisotro p ies,
such as sa p p h ire s, ceram ics a n d fe rrite s , th e effects due to v a ria tio n s in the m a te ria l
p a ra m e te rs m u s t be fu lly a cco u n te d for. P rin c ip a l a xis ro ta tio n s o f a n is o tro p ic s u b ­
s tra te s in M M IC 's m ig h t also lead to s ig n ific a n t v a ria tio n s in th e e ffe ctive d ie le c tric
c o n s ta n t and c h a ra c te ris tic im p e d a n ce .
T h e d isp e rsive c h a ra c te ris tic s o f c o p la n a r
w aveguides (C 'P W 's ) and o th e r p la n a r s tru c tu re s , usin g e ith e r sin g le o r m u lti-la y e r
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is o tro p ic su bstrates, have been e x te n s iv e ly s tu d ie d in th e lite ra tu r e [ lo 9 j- [ 16-3]. I'o an
e x te n t, th e effects due to a n is o tro p y have been e x a m in e d using m e th o d s o th e r th a n
fin ite e le m e n t m e tho d ( F E M ) : h ow eve r p r im a r ily o n ly fo r u n ia x ia l a n d /o r b ia x ia l su b ­
s tra te s [9 ].[L56].[ 15S]. A x is ro ta tio n in va rio u s planes, w h ic h in tro d u c e s o ff-d ia g o n a l
e le m en ts in th e p e r m it t iv it y a n d p e r m e a b ility tensors, w ere also in v e s tig a te d in some
e x te n t u sin g th e tr a d itio n a l s p e c tra l d o m a in a p p ro a ch (S D A ) [13].[157].
In a d d itio n to m a te ria l a n is o tro p ie s , c o m m o n ly used su bstra te s in M M IC 's u s u a lly
e x h ib it d ie le c tric losses w h ic h have to be a cco u n te d fo r. M e ta llic traces w ith fin ite
c o n d u c tiv ity and s m a ll cross se ctio n also becom e e x tre m e ly lossy above a c e rta in
fre qu e ncy. T h e s m a lle r th e th ic k n e s s , th e h ig h e r th e loss. B o th typ e s o f losses are
d iffic u lt to e s tim a te u sin g q u a s i-s ta tic m e th o d s . A lth o u g h p e rtu rb a tio n m e th o d s [47][49] have been used fo r m a n y years to e s tim a te c o n d u c to r losses in M M IC 's . these
m e th o d s becom e in a c c u ra te w h e n th e th ickn e ss o f th e c o n d u c to r is on th e o rd e r o f the
s k in d e p th . T h e most s u ita b le m e th o d s to c o m p u te c o n d u c to r losses a t frequencies
w here th e thickness o f th e c o n d u c to r is on th e o rd e r o f th e skin d e p th in c lu d e the
m ode m a tc h in g te c h n iq u e [52]. th e m e th o d o f lines ( M o L ) [53]. th e s p e c tra l d o m a in
a p p ro a ch [56]-[5S] and th e in te g ra l e q u a tio n (IE ) m e th o d [54],[55]. H ow ever, none o f
these m e th o d s has th e p o te n tia l and v e r s a tility o f th e fin ite e le m en t m e th o d [59].[60].
T h e F E M can be used to m o d e l n ot o n ly d ie le c tric and c o n d u c to r losses b ut also
in h o n io ge n eo u s m edia a n d a r b it r a r y g e o m e trie s.
A lth o u g h q u a s i-s ta tic m e th o d s have been e m p lo y e d in the past to a n a lyze th e
d o m in a n t m ode c h a ra c te ris tic s o f C'PVV’s and o th e r p la n a r s tru c tu re s , such te c h ­
niques y ie ld a ccurate re s u lts o n ly at v e ry low frequencies [164]. M o re a ccu ra te fre ­
q ue n cy d ependent s o lu tio n s have been o b ta in e d u sin g fu ll-w a v e analyses such as the
S D A [13].[156].[163] and th e F E M [42]. W h ile th e S D A is a p o p u la r choice fo r a n a ly z ­
in g re g u la r p la n a r s tru c tu re s , th e F E M is th e m ost g e n e ra lly a p p lic a b le and v e rs a tile ,
since it is possible to m o d e l a r b it r a r y g e o m e tric and m a te ria l c o m p le x itie s .
In an
F E M im p le m e n ta tio n , th e c o m p u ta tio n a l d o m a in is d is c re tiz e d u sin g s im p le g eo m e t­
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11
ric shapes, such as tria n g le s ancl q u a d rila te ra ls , w here the fields are a p p ro x im a te d
u sin g lin e a r o r h ig h e r-o rd e r in te rp o la tio n fu n c tio n s . Because o f th is , it is also re la ­
t iv e ly s tra ig h tfo rw a rd to c o m p u te q u a n titie s o f in te re s t in M M IC tra n s m is s io n lines,
such as to ta l p ow er, vo lta g e d iffe re n ce a n d c h a ra c te ris tic im pedance. A m a jo r d ra w ­
back o f th e F E M is the presence o f n o n -p h y s ic a l o r sp uriou s modes. H o w e ver, these
n o n -p h ysica l s o lu tio n s to M a x w e ll's e q u a tio n s , w h ic h a ppear when u sin g n o d a l ele­
m e n ts . can be a voide d using v e c to r e le m en ts [42]. In a d d itio n to im p o s in g ta n g e n tia l
c o n tin u ity o f th e e le c tric and m a g n e tic fie ld s across ele m en t b o u n d a rie s, v e c to r ele­
m e n ts u su a lly s a tis fy th e d ivergence-free c o n d itio n . U sing th is ty p e o f e le m e n ts , the
o b ta in e d n u m e ric a l s o lu tio n s corre sp on d to th e tru e p hysica l so lu tio n s o f th e s tru c ­
tu re . A llo c a tio n o f c o m p u te r resources is also a m a jo r concern w hen u sin g th e F E M
since such a te c h n iq u e re q uires storage and m a n ip u la tio n o f large sparse system s. In
th is case, sparse lin e a r solvers are u s u a lly m o re s u ita b le th a n d ire c t solvers [123],
U n til now . m ost fin ite e le m en t fo rm u la tio n s have been used to a n a ly z e th e p ro p ­
a g a tio n c h a ra c te ris tic s o f is o tro p ic and b ia x ia lly a n is o tro p ic w aveguides [41].[123]
w ith e x p lic it a p p lic a tio n to o n ly is o tro p ic m ic r o s tr ip s tru c tu re s [42]. In th is c h a p te r,
an e xte n d e d edge-based fin ite ele m en t fo r m u la tio n fo r biaxinl and t ransversc- plari f
a n is o tro p ic m e d ia is presented and used to c h a ra c te riz e shielded 2-D m ic ro w a v e s tru c ­
tu re s . A n u m e ric a lly e ffic ie n t a lg o rith m fo r fin d in g th e largest e igenvalue a nd eigen­
v e c to r is presented based on a fo rw a rd ite r a tio n app ro a ch. H ig h e r e ig e n p a irs can be
fo u n d using a G ra m -S c h m id t o rth o g o n a liz a tio n process [165]. In a d d itio n , an e x p lic it
fo rm u la tio n fo r c a lc u la tin g c h a ra c te ris tic im p e d a n c e fo r s lo t-lik e M M IC s tru c tu re s
is presented fo r th e case o f usin g lin e a r tr ia n g u la r elem ents. N u m e ric a l re s u lts are
co m p a re d w ith e x is tin g p u b lish e d d a ta to v e r ify th e c u rre n t fo rm u la tio n .
T w o sp ecific g eo m e trie s, a co n v e n tio n a l (C -C P Y V ) and a suspended (S -C P W )
CPYV. are th o ro u g h ly in v e s tig a te d . T h e d o m in a n t and few h ig h e r-o rd e r m odes are
c lo se ly e x a m in e d in te rm s o f su b s tra te a n is o tro p y and p rin c ip a l axis ro ta tio n . C o n ­
to u r p lo ts fo r th e fie ld d is tr ib u tio n o f p ro p a g a tin g m odes are also p resented. \ isual-
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iz a tio n o f fie ld c o n c e n tra tio n p ro vid e s a d d itio n a l p h y s ic a l in s ig h t and u n d e rs ta n d in g
o f th e p ro p a g a tin g modes.
In a d d itio n to s u b s tra te a n is o tro p ie s , fin ite c o n d u c tiv ity has also an effect on th e
p ro p a g a tio n c h a ra c te ris tic s o f m ic ro w a v e c irc u its . T h is ty p e o f effect is in ve s tig a te d
fo r a v a rie ty o f m etals as the e ffe c tiv e p ro p a g a tio n c o n s ta n t, a tte n u a tio n co nsta nt
and c h a ra c te ris tic im p edance o f m ic ro s trip s and c o p la n a r w aveguides changes as a
fu n c tio n o f c o n d u c tiv ity . A t lo w e r fre qu e ncies, w h e re th e s tr ip thickness is on th e
o rd e r o f th e s k in d e p th , th e e le c tric fie ld t o t a lly p e n e tra te s th e c o n d u c to r surface
th e re b y in cre a sin g th e in d u cta n ce o f th e tra n s m is s io n lin e . T h is increase in in d u c ­
tance forces th e e ffe ctive d ie le c tric c o n s ta n t and c h a ra c te ris tic im pedance to also
increase.
A t these frequencies, it is e x tre m e ly im p o r ta n t th a t m e ta llic s trip s w ith
fin ite c o n d u c tiv ity and s m a ll cross se ctio n be m o d e le d u sin g a fu ll-w a v e a p p ro a ch.
Such a fo r m u la tio n is presented in th e fo llo w in g s e c tio n : it is based on a 2-D h y b rid
n o d a l-v e c to r fin ite elem ent m e th o d .
3.2
F o rm u la tio n o f th e E igenvalue P ro b le m
A fu ll-w a v e a nalysis o f shielded m ic ro w a v e c irc u its , w h ic h in c o rp o ra te s b o th e le c tric
and m a g n e tic a n iso tro p ie s, is d e scrib e d b y th e e le c tric -fie ld H e lm h o ltz 's e q u a tio n
given b y
V x (^ r
• V x E j - k-; T r E = 0.
( : u :
T h e p e r m it t iv it y and p e rm e a b ility ten so rs arc d e fin e d as
0
0
/'r
=
0
0
// rx
fljry
0
H y r
f l yy
0
0
0
(3 .2 )
c~
//,:
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
(3 .3)
21
whereas th e inverse p e rm e a b ility te n so r is d e fin e d as
K 7
/ Cy x"
0
K 7
o
r yy
^
0
f t 1™
(3-4)
T h e la tte r '« e valuated in closed fo rm using s y m b o lic m a n ip u la tio n s o f a 3 x 3 m a tr ix .
T h e fo rm u la tio n also assumes th a t th e re are no c u rre n t sources (J o r M ) in th e
c o m p u ta tio n a l d o m a in , and th a t th e c o rre s p o n d in g b o u n d a ry co n d itio n s a re given
bv
h x E
=
0
on a p e rfe c t e le c tric w a ll
(3 .5)
h x (V x E )
=
0
on a p e rfe ct m a g n e tic w a ll.
(3 .6)
T h e re p re se n ta tive fu n c tio n a l fo r th e H e lm h o ltz ’s e q u a tio n in a d o m a in f i can be
expressed as
1
; v X E ) /<r
( V x E ) ' - k l0E er E
(19. .
(3.7)
A s su m in g th a t th e dependence o f th e fields is e j k - z a lo n g th e r-d ire c tio n . th e fu n c ­
tio n a l / ’ ( E ) can be w ritte n in te rm s o f th e tra n s v e rs e and th e lo n g itu d in a l fie ld s:
F ( E ) = \ J J ^ [ ( V , x E () f t r ( V , x E , ) ‘ - k ; ( E ( 7 r E ” + E z t r E': )
+ ( V rE : + j k - E t )
k:Et
do.
(3.8)
w here V , is th e tra nsve rse del o p e ra to r. E t is th e tra n sve rse co m p o n e n t o f th e e le c tric
fie ld , and E z is the lo n g itu d in a l co m p o n e n t o f th e e le c tric fie ld .
T h e te n s o r / ' r -
re ferre d to as th e p s e u d o -p e rm e a b ility , has also been in tro d u c e d and is d e fin e d as
dyy
/'r
=
-/C
0
ini’
'J*
,nv
/ XX
0
0
0
/ / ‘JT
(3.9)
In case th e m e d iu m is is o tro p ic in stea d o f a n is o tro p ic , th e fu n c tio n a l in (3 .8 ) can
be s im p lifie d to the one re p o rte d in [42].
R e fe rrin g to Fig. 3.1. th e e le c tric field
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c o m p o n e n ts can be su b s e q u e n tly e x p a n d e d as a s u m m a tio n o f v e c to r and scalar
basis fu n c tio n s
et
=
= £ N ? e J ,.
;= i
(3-10)
e.
1=1
w here n represents th e n u m b e r o f degrees o f fre e d o m
fu n c tio n a l in (3 .8)
(D o F ) in each e le m e n t. T h e
is m in im iz e d , u s in g a s im ila r procedure
to [42 ]. to
o b ta in th e
fo llo w in g e le m e n ta l m a tric e s :
[-4“b
[B " b
[s;_.ju
=
JL l( v ' * N ' }r >r' {v‘ x N;i -
=
k m
=
/ / n { N ; } T /t{ N '}< /o
=
[ r ; t]0 .
=
J J jN
[fl:,],, =
w here / a nd j
u ( ND l
,, - * j p a #
< * • '«
w h e re t r => /7r
(3.13)
;(r
c u n
/ j f { v , . v n r ,?,{ n ;(,/ o
-
JJ
=
I*!],,
d9-
(3.1.51
[{v,.v;>T £{v,.v;} - «•»{.%•,<(T ?, {.v;}] r/f>
- * i [ 7 1 ],v
( 3 . Lb)
d e n o te th e row and th e c o lu m n , re sp e ctive ly, o f th e co rre s p o n d in g
e n try : th e size o f a ll e le m e n ta l m a tric e s is n x n.
Closed fo rm e xpressions for th e
above m a tric e s are g iv e n e x p lic itly , fo r th e case o f lin e a r tria n g u la r e le m e n ts , in A p ­
p e n d ix A . A lth o u g h th e d e riv a tio n o f these e q u a tio n s is ra th e r te d io u s and in v o lv e d .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
■26
Node 1
y
A
Edge 2
Edge 3
Node 2
Node 3
Edge 1
Fig. 3.1: T rian g ula r vector elem ent.
t h e ir c o m p u te r im p le m e n ta tio n is s tra ig h tfo rw a rd .
F o llo w in g the assem bly o f a ll
e le m e n ts in th e fin ite e le m e n t re g io n , a generalized e ig en va lue svstem is form e d
An
0
0
0
=
£
— k 'z
-
Bn
B t:
B zt
B ss
(3.17
e.
F ro m (3 .1 7 ). it is observed th a t th e m a tr ix on th e le ft h a n d side is sin g u la r: the re fo re,
th e re are X n eigenvectors th a t co rre sp on d to an e ig e n va lu e k : o f zero, w here X n is
th e n u m b e r o f nodes in th e fin ite ele m en t mesh. These so lu tion s are n on -p h ysical
since th e y do n ot s a tis fy th e H e lm h o ltz s e qu a tion . H o w e ver, th e y can be avoided by
re -w r itin g th e m a tr ix svste m in th e fo llo w in g fo rm :
'
B tt
B zt
B t;
B;Z
'
l
^
1
Bn +
- ki
B :t
rnux
B t:
B ::
'
£.
(3.LS)
w h e re k m.ix is th e m a x im u m p ro p a g a tio n constant in th e lo n g itu d in a l d ire c tio n . T h is
is defin e d as k'~l!ix -
k'*( m.i x [ i m,t x : cmrtx and
p e r m it t iv it y and p e rm e a b ility o f th e d om ain .
a re re sp e ctive ly th e m a x im u m
In m ic ro w a v e c irc u it a p p lic a tio n s ,
people are u s u a lly in te re s te d in th e few most d o m in a n t m odes o f th e s tru c tu re : i.e..
th e ones th a t correspond to th e m ost p o sitive F? a n d the re fo re, a la rg e r value o f
k m a r / { k m a r ~ k i ) - T h u s - fo r p o s itiv e values o f k . . th e eigenvalue ^ „ / ( ^ lax - k : )
ranges fro m 1 to " in f in it y " : 1 corresponds to k : = 0 and " in f in it y " corresponds to
k : = k rn,l x . As a re su lt o f th is tra n s fo rm a tio n , the zero eigenvalues are s h ifte d o u ts id e
th e range o f in te re st.
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I t is also im p o rta n t to re a lize here th a t a lth o u g h the g e n e ra lize d eigenvalue m a tr ix
syste m was fo rm u la te d a ssu m in g a p ro p a g a tin g wave o f th e fo rm
. th e re is no
re s tric tio n on w h e th e r k : is re a l o r c o m p le x . For e xam ple, one co u ld e asily assum e
th a t k . = 3 — j a . w h e re a is th e a tte n u a tio n co nsta nt and ,J is th e p ro p a g a tio n
c o n s ta n t in the lo n g itu d in a l d ire c tio n . T h u s , lossy m a te ria ls can also be considered
u sin g th e c u rre n t fo r m u la tio n .
T h e generalized e ig e n va lu e m a tr ix system can be solved u sin g e ith e r a sta n d a rd
d ire c t so lve r o r an ite r a tiv e so lve r. T h e fo rm e r u su a lly re su lts in th e c o m p u ta tio n
o f a ll th e eigenvalues a n d e ig en ve cto rs o f th e m a tr ix
system . H ow ever,
in p ra c tic e ,
o n ly th e firs t few d o m in a n t m odes are needed: the re fo re, an ite ra tiv e solver
is u s u a lly
m o re a p p ro p ria te .
3.2.1
P rin c ip a l axis r o ta tio n
T o a ccount fo r p rin c ip a l a xis ro ta tio n o f a n is o tro p ic su b stra te s, th e p e r m it t iv it y
te n so r. er . has to be m o d ifie d a c co rd in g to F ig . 3.2. A ssu m in g th a t
ci
0
0
'
0
Cy
0
0
0
c;i
fo r a b ia x ia l s u b s tra te , th e co rre sp o n d in g p e r m it t iv it y ten so r. c r . fo r any angle o f
r o ta tio n 0 is given by
C_ry
0
0
c;:
w here
Cj-j-
=
ci cos2(fl) 4- c >s in 2(0)
(3.19)
t,j,j
=
c i s in ’ (fl) + c> cos2(0)
(3.20)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
28
£j-y
—
£.3
—
£;/X
(3 .2 i;
(3.22)
(e -2 — C i) s in (0) cos(O).
In th is s tu d y . th e c ry s ta l la ttic e is ro ta te d betw een 0° and 90°.
As th e p rin c ip a l
axes o f th e c ry s ta l are ro ta te d , th e p ro p a g a tio n c h a ra c te ris tic s o f th e device are
m o d ifie d . T h e percen tag e change in th e e ffe ctive d ie le c tric c o n s ta n t a n d c h a ra c te r­
is tic im p e d a n c e o f th e tra n s m is s io n lin e as a fu n c tio n o f ro ta tio n a n g le becom es an
im p o rta n t q u a n tity d u rin g device design and m a n u fa c tu rin g .
.. z r
Crystal substrate
\
Fig. 3.2: D efinition o f p rin cip al axis ro tatio n for a crysta l la ttic e .
3.2.2
C h a ra c te ris tic im p e d a n ce
A fte r s o lv in g th e eig en va lue m a tr ix system at a specific frequency, th e p ro p a g a tio n
co n sta n t in th e c -d ire c tio n and th e co rre sp o n d in g n o rm a liz e d tra n s v e rs e and lo n g i­
tu d in a l fie ld s in s id e th e s tru c tu re can be o b ta in e d . B o th th e p ro p a g a tio n c o n sta n t
and g o v e rn in g fields are needed fo r th e c a lc u la tio n o f th e c h a ra c te ris tic im p e d a n ce .
A lth o u g h th e d e fin itio n o f th e c h a ra c te ris tic im p e d a n ce is n o t u n iq u e fo r in h o m o geneous w a ve g u id e s tru c tu re s [166]. th e vo lt age-pow er d e fin itio n was chosen for th e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
■2!)
c u rre n t analysis. T h e c o rre s p o n d in g expression is given by [lo S ]
V V
~T p
P = t
•
<*•*»>
p‘
1=1
w h e re V is th e vo lta g e d iffe re n c e between th e tra n s m is s io n lin e and th e g ro u n d plane.
Pi is th e p ow er flo w in g in th e r- d ir e c tio n th ro u g h th e cross s e c tio n o f th e i th elem ent.
P is th e to ta l pow er flo w in g in the r- d ir e c tio n . a n d .V is th e to ta l n u m b e r o f fin ite
e le m e n ts in the d o m a in o f in te re s t. T h e e le m e n ta l pow er P, is c o m p u te d using the
P o y n tin g v e cto r d efin e d as
y £
E , x H - - a s <fa] = ^ R e [ / £
( E r H"y -
E y / / ; ) ds
(3.24)
w h e re th e m a g n e tic fie ld co m p o n e n ts H r and FIV are c a lc u la te d d ir e c tly fro m M a x ­
w e ll's e q u a tion s: n ote th a t th e e le c tric fie ld c o m p o n e n ts are k n o w n in closed fo rm .
T h e in te g ra tio n in (3.24) is e valua ted fo r e v e ry sin g le e le m e n t in th e mesh.
Based on lin e a r tr ia n g u la r elem ents, w h e re th e edges re p re se n t transverse fields
a n d th e nodes represent lo n g itu d in a l fields, th e fin a l e xpressions fo r th e transverse
c o m p o n e n ts o f b o th th e e le c tric and th e m a g n e tic fields are e x p lic it ly g ive n below:
w h e re C \ - . \ and
C,
EA-r-u)
= C \ - C 2y
(3.23)
Ey{.r.ij)
= C:i. v - C A
(3.26)
P r { ' l' - y )
= D \ y T D 2 J-' + D j
(3 .2 1 )
H y(. v. y)
= D a ij + D-r, x +
(3.28)
are co n sta n ts d efined as fo llo w s:
=
d r
- / l i= i
c 2 = 2 - 1 ,1 , c-„
- • 1 i= i
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
(3.29)
30
—u‘nvL•
3
o. =
*
—
ninuk
3
=
-2rAf ^Lf i 0h f' rL[ ' “ «
£>3
=
J
2Au;fi0
- l‘r j
+ /C "
+ /C
yy
whore, j*,- a n d //, ( / =
i=1
i,‘nvL
—
0-
J
2Au.'[i0
=
•
3
D- = — V .
V l e'
-Au,fJ.o ,_2
E
W=1
* i £'(
E <>.• 4
>>1=1
E 'i‘i
i= 1
- j/C *=
E
« <.
W=l
- j/C /c -
E ' . - r . <.
i= i
- A # * .
E 'i'i'
i= i
- J K 7 k*
(3.33)
(3.3-1)
(E
1 .2 .3 ) d en o te th e co ordin a tes o f each tria n g le . /, ( i = 1 .2 .3 )
denote th e le n g th s o f the in d iv id u a l edges and A d e n o te s th e area o f th e tria n g u la r
elem ent w h ic h is equal to
.-I
T h e node n u m b e rin g
=
^ { r 16t + .v>b> + -r-.Ai} ■
(3.33)
o f each tria n g le is taken in a c o u n te r-c lo c k w is e (C C W ) sense.
A lso, th e 6,‘ s a nd c ,’ s are given by
hi =
/h — //3
Cj = .r.j — x i
b> =
1/.?- U\
ci
l>3 =
/ / 1-
C3 = x 2 -
>/2
= -r i ~ -!'.i
X i.
T h e in te g ra tio n in (3.2-1) o ver th e surface o f a tr ia n g u la r e le m e n t can be evaluated
very c o n v e n ie n tly u sin g the s im p le x coordinates.
In a d d itio n to c a lc u la tin g th e to ta l power flo w in g th ro u g h th e shielded s tru c tu re
at a g ive n fre q u e n cy, the vo lta g e difference b etw een th e s tr ip lin e and th e ground,
w hich is d e fin e d by
V = - j E - f l l .
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
(3.3(i)
31
m ust be c a lc u la te d . A s s u m in g th a t the slo t lies h o riz o n ta lly , th e above in te g ra l can
be s im p lifie d to
V = -
r
J X(
Er dx
(3.37)
where x / and .rr are th e le ft and rig h t x -c o o rd in a te s o f th e in te g ra tio n p a th . Since
th e slot is d is c re tiz e d in to fin ite lin e a r tr ia n g u la r e le m e n ts, th e lin e in te g ra l should
be evaluated fo r each e le m en t in th e slo t. T h e to ta l v o lta g e d ifference is g ive n by
•V.
=
(3.38)
1=1
w here .V3 is th e n u m b e r o f e le m en ts in th e s lo t and V'i is th e in d iv id u a l vo lta g e
difference. U sing lin e a r tr ia n g u la r elem ents, th is v o lta g e can be expressed as
Vi =
(x * - x f ) [(.(/! - h ) l i e tl + (i/2 - h ) lyCt -2 + (f/3 - h) l^ct3j
(3.39)
w here x* and .r' are th e lim its o f th e vo ltag e lin e in te g ra l e va lu a te d for each e le m e n t
in the slo t, and h is th e h e ig h t o f the slo t.
A s im ila r fo rm u la can be e q u iv a le n tly
o b ta in e d fo r a v e rtic a l slo t.
3.2.3
G en e ra lize d eigenvalue so lve r
T h e so lu tio n o f a g en e ra lize d eigenvalue p ro b le m d efin e d as
[A'lf.,-! = A [.U ]{.r}
(3.-10)
can be c o m p u ta tio n a lly in te n s iv e and tim e d e m a n d in g , e s p e c ia lly as the n u m b e r o f
unknow ns increases. T h e re are various m e th o d s o f s o lv in g fo r b o th the eigenvalues
and the co rre sp o n d in g eig en ve cto rs. The s im p le s t m e th o d is to store b o th m a tric e s
in a fu ll fo rm a t and th e n use a d ire c t so lve r, lik e th e ones a va ila b le in E IS P A C K .
Such a so lve r u s u a lly c o m p u te s a ll the eigenvalues and e ig envectors o f th e m a tr ix
system . T h is app ro a ch, how ever, is very in e ffic ie n t b o th in te rm s o f c o m p u ta tio n a l
tim e and m e m o ry re q u ire m e n ts .
O ne o f th e m ost s u ita b le m e tho d s fo r s o lv in g a
generalized eigenvalue p ro b le m is the subspace ite ra tio n , o th e rw is e kn ow n as fo r­
w ard and inverse p ow er ite ra tio n .
N ote th a t th e fo rw a rd pow er ite ra tio n is used
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
to e s tim a te th e largest eigenvalues o f th e m a tr ix syste m , whereas the inverse p ow er
ite ra tio n is used to e s tim a te th e low est eigenvalues. T h e m a jo r advantages o f u sin g a
p ow er ite ra tio n arc: firs t, spee d -u p in c o m p u ta tio n a l tim e : second, co m p le te u t iliz a ­
tio n o f th e s p a rs ity o f th e m a tric e s : t h ir d , c o m p u ta tio n o f o n ly a selected n u m b e r o f
e ig e n v a lu e /e ig e n ve cto r p airs. A s fa r as th e la tte r is co nce rn e d, o n ly th e m ost d o m i­
n ant m odes are im p o rta n t in th e a n a ly s is o f m ic ro w a v e s tru c tu re s : thu s, h ig h e r-o rd e r
eigenvalues and eigenvectors are n o t c a lc u la te d . In a d d itio n , th e a ccuracy in c a lc u ­
la tin g h ig h e r-o rd e r eigenvalues a n d eig en ve cto rs d e te rio ra te s w ith incre asing o rd e r,
w h ich is a n o th e r reason fo r n ot c a lc u la tin g m ore th a n a few eigenm odes.
T h e re su lts presented in th is c h a p te r were o b ta in e d u sin g a fo rw a rd pow er it e r ­
a tio n m e th o d . T h e a lg o rith m is q u ite s im p le b u t v e ry p o w e rfu l. T h e m a jo r steps
in v o lv e d in th e a lg o rith m are th e fo llo w in g :
Step # 1
• In itia liz e a s ta rtin g v e c to r u (0) (o th e r th a n th e zero v e c to r).
• Set th e ite ra tio n in d e x k = 0.
S te p 4 r -
• In cre m e n t th e ite ra tio n in d e x : k = k + I.
Step # 3
•
D e te rm in e a ve cto r v (K_l) = /\
For e fficien cy, th e m a trix -v e c to r m u ltip lic a tio n is p e rfo rm e d using a sparse storag e
fo rm a t.
Step # 1
• Solve th e lin e a r system M
= v (Ar-lb
A n e ffic ie n t way to solve th e a bo ve lin e a r system is to use a sparse L U solver. T h e
advantage o f using such a so lve r is th a t th e fa c to riz a tio n o f m a t r ix \ f takes place o n ly
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
once: th e re fo re , subsequent ite ra tio n s re q u ire o n ly a b a ckw ard s u b s titu tio n w h ic h
results in a s ig n ific a n t speed-up fo r th e a lg o rith m . O th e r solvers, such as th e ones
th a t use ite r a tiv e techniques, are also a p p ro p ria te sin ce th e sp a rsity o f th e m a tr ix
can be e ffic ie n tly u tiliz e d .
Step
• A ssig n v (t) = K u {k).
Step # 6
u -(* )v (*)
• E s tim a te th e largest eig en va lue usin g A**) = ——ttt— —rr.
U* I* Jyl »J
T h e s y m b o l * denotes co n ju g a te transpose.
Step # 7
...
• N o rm a liz e th e co rre sp o n d in g e ig e n ve cto r as fo llo w s : u l 1 =
1
°
°
u (*>
.
-(fc)v ( * - :)
Step # S
• C a lc u la te th e 2 -n o rm o f th e re s id u a l: ||/? ||(/ ' = ||/v u (t) — A .\/ u (t) ||2< toleran ce
=i- E x it th e a lg o rith m : o th e rw is e , go to Step # T .
As it was a lre a d y m e n tio n e d , th e above a lg o rith m converges to the largest e ig e n ­
value. p ro v id e d th a t the s ta r tin g v e c to r does n o t c o in c id e w ith one o f th e e ig en ­
vectors.
In o rd e r to c a lc u la te h ig h e r-o rd e r e igenvalues, th e s ta rtin g v e c to r has to
be chosen fro m a space \ I o rth o g o n a l to th e a lre a d y c a lcu la te d eigenvectors. Such
an o rth o g o n a liz a tio n is w e ll-k n o w n as th e G ra m -S c h m id t process [ L60 ] .
words, i f U [ . U 2
In o th e r
U m are co n side re d th e firs t m eigenvectors alre ad y c a lc u la te d
using th e p o w e r ite ra tio n , th e s ta r tin g ve cto r fo r each ite ra tio n , a ccordin g to th e
G ra m -S c h m id t o rth o g o n a liz a tio n . is g ive n by
u*-1 = u*"1 -
( u ' (A:_l,. \ / U i ) " U , -
( u " ( t _ 1). \ / U 2) " U 2 - . . . - ( u " ( t - 1, . U U m) " U „,
( d .- ii)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
U sing th is as a s ta rtin g vector in each ite ra tio n w ill re su lt in th e p re d ic tio n o f Am+1
and U m+1.
N o te th a t the accuracy o f th e a lg o rith m can be im p ro v e d by s e ttin g
th e co nve rg e nce tolerance to a lo w e r n u m b e r. It is also im p o rta n t to m e n tio n here
th a t th e fo rw a rd pow er ite ra tio n converges m uch fa ste r th a n th e inverse pow er ite ra ­
tio n [42 ]. a t least fo r the typ e o f p ro b le m s considered in th is s tu d y , w hich is a n o th e r
reason fo r im p le m e n tin g th is p a r tic u la r a lg o rith m .
3.3
N u m e ric a l V a lid a tio n
A c o m p le te F E M code based on th e a n a ly tic a l fo rm u la tio n presented in th e p re v i­
ous s e c tio n was w ritte n and tested fo r a v a rie ty o f geo m e trie s and m a te ria ls . T h e
F E M co de was in terfa ce d w ith I-D E A S . a softw are package fro m S tru c tu ra l D y n a m ­
ics R esearch C o rp o ra tio n (S D R C ) w ith preprocessing c a p a b ilitie s such as m e sh ing ,
m a te ria l d e fin itio n , and b o u n d a ry c o n d itio n s . It was also in te rfa c e d w ith o th e r w e llkn ow n packages such as P L O T M T V . G E O M Y IE W . T E C P L O T . and G X U P L O T
w h ich can be used fo r data v is u a liz a tio n and im p o rta n t g e o m e try checks.
T h e firs t g e o m e try considered to v a lid a te th e code was th e coupled m ic ro s trip
lines sh ow n in F ig . 3.3. T h is s tru c tu re was analyzed b y M o s ta la t t nl. [157] using
th e s p e c tra l d o m a in approach. T h e co u p le d m ic ro s trip lines rest on a u n ia x ia l b o ro n
n itrid e subst ra te w ith t XT = e-: = 3.12 and c,jy — 3.4. T h e e ffe ctive d ie le c tric co n s ta n t
( f rrf f ) versus th e c rysta l ro ta tio n angle (0 ). as defined in [157]. is d ep icte d in F ig . 3 .1
fo r tw o d iffe re n t frequencies: / = 10 G H z and / = 20 G H z. T h e com parison betw een
th e p re d ic te d re su lts and d a ta o b ta in e d fro m [157] illu s tra te s an e xcellent agreem ent
betw een th e tw o m ethods.
T h e second g e o m e try considered was a u n ila te ra l fin lin e . shown in Fig. 3.3. a n a ­
lyzed b y M a n s o u r el nl. [158]. T h e fre q u e n cy dependence o f th e e ffe ctive d ie le c tric
c o n s ta n t and c h a ra c te ris tic im p e d a n ce is illu s tra te d in F ig . 3.6.
C o m p a riso n d a ta
o b ta in e d fro m the corre sp on d ing fig u re in [158] are also show n in th is g ra p h .
a gree m e n t betw een the two sets o f d a ta is very good.
The
It is im p o rta n t to m e n tio n
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 3.3: G eom etry o f coupled m icrostrip lines on a boron n itrid e substrate ( h i
h 2 = 3 m m . tr = s = 1.5 mm. b - 8.5 m m ).
2.50
•
*
■
♦
Even Mode - 10 G H z (F E M )
Odd Mode - 10 G H z (F E M )
Even Mode - 20 G H z (F E M )
Odd Mode - 20 G H z (F E M )
Even Mode - 10 G H z
Odd Mode - 10 G H z
Even Mode - 20 G H z
Odd Mode - 20 G H z
^reff
2.25
2.00
1.751
1.50
20
30
40
50
60
70
90
0(deg)
Fig. 3.-1: Effective dielectric constant o f coupled m icrostrip lines on a boron n itride sub­
s tra te (erT = 5.L2. f vy = 3.-1. i z: = 5.12). The markers represent data extracted
from a paper by M ostafa et nl. [157]
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
H3--------------------------------------------------------------------------- H
a
Fig. 3.5: G eom etry o f a u n ila te ra l finline on a dielectric substrate w ith er = 3.8 (a = 2b
4.7752 mm. .s = 0.127 m m . h = 2.3876 mm. d = 0.47752 m m ).
1.6
500
450
1.4
400
1.2
Zc (Ohms)
350 [
1.0
300 i
250
200
0.6
150
100 I
—
Zc (FEM )
j
erefr(F E M ) j Q 2
Zc
J
^reff
0
10
20
30
40
50
60
70
!
0.0
80
Frequency (GHz)
Fig. 3.6: C haracteristic im pedance o f a u nilateral finline.
extracted from a paper by M ansour cl nl. [158]
The markers represent data
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
here th a t th e a c c u ra c y o f the c h a ra c te ris tic im p e da n ce depends on th e d e n s ity ot
th e mesh in th e c o m p u ta tio n a l d o m a in , e s p e c ia lly near th e s lo t. A n a ccurate repre­
s e n ta tio n o f th e fie ld s in th e v ic in ity o f th e s lo t, w h ich are k n o w n to e x h ib it ra p id
s p a tia l v a ria tio n s , re q u ire s e ith e r a fin e r m esh o r the use o f h ig h e r-o rd e r elem ents.
R e fe rrin g to F ig . 3.6. th e g e o m e try was d is c re tiz e d using 13 e le m e n ts across th e slot
and a to ta l o f 1.036 e le m en ts in th e e n tire s tru c tu re .
A suspended c o p la n a r waveguide w ith m a g n e tic a n is o tro p ic su bstra te s was also
s im u la te d u sin g th e present code.
T h is g e o m e try was in it ia lly a n a lyze d by Maze-
M e rc e u r et al. [156] u sin g th e sp e ctra l d o m a in a pproach. B y d e fin in g the n o rm a liz e d
p ro p a g a tio n c o n s ta n t squared as ere/ / - / 'r e / / = { k z / k o ) 2 [156]. th e effects due to both
e le c tric and m a g n e tic a niso tro p ie s can be a ccounted for. A co m p a riso n betw een the
tw o m e th o d s in c a lc u la tin g ere // • / 'r e / / versus fre q u e n cy is illu s tra te d in F ig . 3.8
fo r various is o tro p ic and m a g n e tic a lly a n is o tro p ic su bstra te s. A good agreem ent is
observed betw een th e tw o m ethods.
3.-1
E le c tric A n is o tro p ie s in C o p la n a r W aveguides
E le c tric a n is o tro p ie s are c o m m o n ly fo u n d in a v a rie ty o f su b stra te s used in in te g ra te d
m icro w a ve c irc u its and p rin te d -c irc u it ante nn a s.
A n is o tro p ie s o c c u r n a tu ra lly in
th e m a te ria l o r p u rp o s e ly im p la n te d d u r in g th e m a n u fa c tu rin g process to im p ro v e
c ir c u it p e rfo rm a n c e .
T h e effects o f a n is o tro p ie s on p ro p a g a tio n c h a ra c te ris tic s are
o fte n ig n ore d d u r in g device m o d e lin g and design.
T h is in tro d u c e s serious errors
in in te g ra te d -c irc u it design and reduces in te g ra te d -c irc u it re p e a ta b ility .
For these
reasons, m a te ria l a n iso tro p ie s should a lw a ys be accounted fo r using so p h is tic a te d
a n a ly tic a l o r n u m e ric a l m ethods.
T h e e ffe c tiv e d ie le c tric constant and c h a ra c te ris tic im p e d a n c e versus frequency
o f a c o n v e n tio n a l c o p la n a r w aveguide are firs t in v e s tig a te d .
T h e g e o m e try, w hich
is shown in Fig. 3 .9 (a ). is m odeled usin g fo u r c o m m o n ly used u n ia x ia l o r b ia x ia l
a n is o tro p ic s u b s tra te s :
sapphire ( t rx = t - : = 9.1. c,rj =
L 1.6). e p silam -1 0 ( i r j . =
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 3 . i: G eom etry o f a suspended coplanar waveguide (a = 7 .11'2 m m . b = 3.556 nun.
h \ / a -- 0.4. ho/a - 0.1. h ^ / a = 0.5. w = .s = 6 /5 ).
16
/ir= ( l . l . l )
^ = (1 ,1 .5 )
■ /V=(1.5.1)
♦ ;ir= ( 5 .l.l)
O ^ = (5 ,5 ,5 )
•
—
A
14
—
—
ftr= ( l . l . l ) (FE M )
^ = (1 .1 ,5 ) (FE M )
/ir=( 1.5.1) (FE M )
^•=(5,1,1) (FE M )
Mr= (5 .5 .5 )(F E M )
12
ereff ‘ /^reff
—
o'
10
s' s'
s '" °
8
♦
s'
6
♦
/
/
s'
/
/
jy '
4
♦
A
2
---------------------------■
--------------
-
------------------------------------------- 11
_
-mm
.^ .
" t
__
----------- -----------------------------*
"
. ~
a
• ........... -'-A—
O '------------------------------------------w
— m ---------------------■.- ............. 4
'
—
—
0
15
20
25
30
35
40
45
50
55
60
65
70
Frequency (G Hz)
Fig. 3.8: Dispersion curves for the dom inant mode o f a suspended coplanar waveguide.
The markers represent data extracted from a paper by M aze-M erceur ct al. [156]
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
39
dl
d2
d3
d4
d5
.... .. T ,
T
...
. (
.
c3
dl
T
d2 t
d4
d3
i
d5
l
c2
cl
(a)
(b)
Fig. 3.9: Conventional and suspended waveguides: (a) ct = 3.2 mm. c2 = 2.71 mm.
d \ = r/ 5 = l.o mm. do = d 3 = d., = 0.5 mm: (b) ci = 2.27 mm. c2 = 0.5 111111.
c3 = 2.71 mm. d x = d 3 = 1.5 mm. d> = d 3 = d 4 = 0.5 mm.
ezz =
13.0. eyy =
10.3). b o ro n n itr id e (e .^ = czz = 5.12. t yy = 3.-1). and P T F E
(P o ly te tra flu o ro e th y le n e ) c lo th (exx = 2.89. t yy = 2.45. t z: = 2 .95).
p e r m e a b ility o f a ll these c ry s ta ls is assum ed u n ity .
T h e re la tiv e
T h e c o rre sp o n d in g graphs are
illu s tr a te d in F ig . 3.10.
S ta r tin g w ith sapphire in F ig . 3.10. it is in te re s tin g to observe th a t th e d o m in a n t
m o d e is v e ry dispersive. T h e c o rre s p o n d in g erc/ / varies by as m u ch as 100C
X w ith in
th e 100
G H z range shown in th e fig u re .
m o d e occurs w ith in the range
T h e d isp e rsive b e h a vio r o f th e d o m in a n t
o f 10 to 30 G H z . w h ich lim its th e s in g le -m o d e o pe ra ­
tio n a l b a n d w id th . W ith in th e sam e fre q u e n cy range, th e co rre sp o n d in g c h a ra c te ris tic
im p e d a n c e o f th e d o m in a n t m o d e e x h ib its a ra p id decrease to w a rd zero. In a d d itio n ,
th e d o m in a n t m ode shows v e ry lit t le c o u p lin g w ith th e first fo u r H O M 's : how ever,
th e re is a s tro n g co u p lin g b etw een th e firs t and second H O M 's . As a re s u lt o f such a
s tro n g in te ra c tio n between these tw o m odes, th e c h a ra c te ris tic im p e d a n ce o f th e firs t
HO .M is o n ly shown up to a fre q u e n c y o f a b o u t 15 G H z . where s ig n ific a n t c o u p lin g is
o bse rve d . C o u p lin g between a n y tw o m odes is present when th e c o rre s p o n d in g phase
v e lo c itie s o r p ro p a g a tio n c o n s ta n ts are n e a rly th e same. T he closer these q u a n titie s
are. th e stro n g e r the c o u p lin g betw een th e tw o m odes.
W ith o u t discussing in d e ta il th e re m a in in g th re e sets o f g ra ph s in F ig . 3.10. it
is in te re s tin g to note th a t besides th e H O M c o u p lin g effects, th e re is an a d d itio n a l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10
100
Sapphire (9.4,11.6,9.4)
90 ■
D om inant Mode
F irst H O M
60:
U
50 r
Sapphire (9.4,11.6,9.4)
D om inant mode
H igher-order modes
30
40
50
70
60
90
10
30
20
100
35
40
50
Frequency (GHz)
Frequency (GHz)
Epsilam -10 (13.0.10.3,13.0)
12
•
D om inant Mode
First H O M
10 -
\
E psilam -10 (13.0,10.3,13.0)
20 •
D om inant mode
H igher-order modes
20
40
50
60
100
35
20
Frequency (GHz)
Frequency (GHz)
140 '
Boron N itrid e (5.12.3.4.5.12) ;
120
D o m in an t M ode
F irs t H O M
loos
JZ
O
Boron N itrid e (5.12,3.4.5.12)
D om inant mode
Higher-order modes
40
50
60
90
70
100
35
20
Frequency (GHz)
Frequency (GHz)
160 —
P TFE Cloth (2.89,2.45,2.95)
uo •
Dom inant Mode
F irs t H O M
120
g .o o
U‘
V4J
o
CS3
PTFE Cloth (2.89,2.45,2.95)
Dominant mode
Higher-order modes
10
20
30
40
50
60
70
Frequency (GHz)
60
80
40
20
90
100
15
20
25
30
35
40
Frequency (GHz)
Fig. 3.I0: Effective dielectric constant and characteristic impedance o f a conventional
CPW w ith anisotropic substrates.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
11
s tro n g in te ra c tio n between th e d o m in a n t mode and th e H O M s. B y e x a m in in g th e
fie ld d is tr ib u tio n o f th e d o m in a n t m ode in th e c o p la n a r w aveguide at va rio us fre ­
quencies. it was realized th a t th is m ode a c tu a lly crosses w it h a n u m b e r o f H O M s.
A lth o u g h such a process is cu m b e rso m e and tim e c o n s u m in g , it is th e o n ly w a y o f
tra c k in g dow n th e in d iv id u a l m odes w ith in a g ive n fre q u e n c y range.
T h is ty p e o f
d is p e rs io n e ffe ct, in a d d itio n to th e d o m in a n t m o d e d is p e rs io n c h a ra c te ris tic s , fu r ­
th e r lim its th e sin g le-m o de o p e ra tio n o f the s tru c tu re . A co m p a riso n o f th e percen t
change in th e e ffe c tiv e d ie le c tric c o n s ta n t o f the d o m in a n t m ode w ith in a fre q u e n c y
ra n ge o f 100 G H z is s u m m a riz e d fo r a ll fo u r su b stra te s in T a ble 3.1. S a p p h ire a p ­
pears to e x h ib it th e highest p e rc e n t change in ere //- w hereas P T F E c lo th e x h ib its
th e low est. T h e c h a ra c te ris tic im p e d a n ce o f th e d o m in a n t m ode rem ains re la tiv e ly
c o n s ta n t fo r a ll fo u r cases up to a fre q u e n cy where th e d is p e rs io n effect becomes q u ite
s ig n ific a n t. A t th a t frequency, th e c h a ra c te ris tic im p e d a n c e o f the d o m in a n t m o d e
begins to decay to w a rd zero. O n th e o th e r hand, th e firs t H O M corresponds to a
s lo t-lin e lik e m ode [163] and. th e re fo re , the c h a ra c te ris tic im p e d a n ce a s y m p to tic a lly
approaches in f in it y a t frequencies near cu to ff.
T h e d isp e rsio n curves c o n c e rn in g th e c o n ve n tio n a l c o p la n a r waveguide are c h a r­
a c te riz e d by a s tro n g in te ra c tio n between the d o m in a n t m o d e and H O M 's . w h ic h
c le a rly re stra in s th e sin g le -m o d e o p e ra tio n in M M IC ' s. F u rth e rm o re , th e re la tiv e ly
h ig h p ercen t change in the e ffe c tiv e d ie le c tric co n sta n t w it h in the 100 G H z fre q u e n c y
ra n ge in tro d u c e s a d d itio n a l u n w a n te d dispersion. T hese p ro b le m s can be o vercom e
b y in tro d u c in g a sin g le layer suspended co plan a r w a ve g u id e as shown in Fig. 3 .0 (b ).
T h e e ffe ctive d ie le c tric co nsta nt o f th e firs t five m ost d o m in a n t modes and th e c h a r­
a c te ris tic im p e d a n ce o f the firs t tw o m ost d o m in a n t m o d es were c o m p u te d and arc*
show n in Fig. 3.11. W ith o u t d iscu ssin g each g ra ph in d iv id u a lly , it is cle a r th a t th e re
is a s ig n ific a n t im p ro v e m e n t in th e e ffe ctive d ie le c tric c o n s ta n t and c h a ra c te ris tic
im p e d a n c e o f th e c u rre n t g e o m e try com pared to th e p re v io u s one.
F irs t, th e d is ­
p e rsive n a tu re o f th e d o m in a n t m o d e fo r all fo u r ty p e s o f su b stra te is s ig n ific a n tly
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
12
Table 3.1: Percentage change in the effective dielectric constant w ith in a range 100 GHz.
B oron N itr id e
P T F E c lo th
+7o%
+Sb%
+60%
+67%
+ 2 6%
+9%
S tru c tu re
S a p p h ire
E p sila m -1 0
C -C P W
+ 100%
S -C P W
+S 3%
reduced. T h e co rre sp o n d in g p ercent change in t rr j j is ta b u la te d in T a b le 3.1. T h e
d o m in a n t m ode when u sing sa p p h ire , e p sila m -1 0 . and b o ro n n itr id e s t ill e x h ib its a
s m a ll in te ra c tio n w ith H O M 's . b u t c le a rly th is m ode in te ra c tio n o ccurs a t frequen­
cies m u ch h ig h e r tha n those observed fo r th e c o n ve n tio n a l c o p la n a r waveguide. On
th e c o n tra ry , th e d o m in a n t m ode when u sin g P T F E c lo th , in a d d itio n to having the
less d isp e rs iv e c h a ra c te ris tic s , shows no in te ra c tio n w ith a n y H O M . a t least w ith in
a b a n d w id th o f 100 G H z. F u rth e rm o re , th e c u to ff frequencies fo r b o th th e first and
second H O M 's have s h ifte d to a h ig h e r fre qu e ncy. S p e cifica lly, th e c u to ff frequency ol
th e second H O M . in a ll fo u r cases e x a m in e d , s h ifts to a fre q u e n c y o f a b o u t 30 G H z.
B y in tro d u c in g resistive film s in th e s tru c tu re [167]. the firs t H O M can be s ig n ifi­
c a n tly a tte n u a te d , th e re b y in cre a sin g th e sin g le-m o de b a n d w id th to a p p ro x im a te ly
30 G H z w h ic h is a s ig n ific a n t im p ro v e m e n t com pared to th a t of th e co nve n tion a l
c o p la n a r w aveguide.
T h e d isp e rsio n c h a ra c te ris tic s o f th e suspended c o p la n a r w a ve gu id e can be fu r ­
th e r a lte re d b y ro ta tin g th e c ry s ta l la ttic e in th e transverse p la n e w h ile observing th e
v a ria tio n o f th e erP/ / and Z c a t a c e rta in frequency. C o n s id e rin g o n ly th e d o m in a n t
and firs t H O M at a fre q u e n cy o f bO G H z . th e c ry s ta l axes are ro ta te d according to
F ig . 3.2 fro m 0° to 90°. Based on o u r e xp e rie n ce , effects on th e p ro p a g a tio n c h a r­
a c te ris tic s . such as cre j j and Z... are m o st n o tice a b le at r e la tiv e ly h ig h frequencies.
T h e p re d ic te d results due to c ry s ta l ro ta tio n for a ll fo u r s u b s tra te s are shown in
F ig . 3.12.
W h e n using s a p p h ire , th e e ffe c tiv e d ie le c tric c o n s ta n t o f the d o m in a n t
m ode increases w ith th e ang le o f ro ta tio n 0 w h ic h , a cco rd in g to F ig . 3.10. w ill result
in a m o re disp ersive s tru c tu re . T h e increase in t rrf j as 0 changes fro m 0° to 90° oc-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
13
100
12.0
Sapphire (9.4,11.6,9.4)
105 •
Dom inant mode
Higher-order modes
fa
o
lm
vu
Sapphire (9.4,11.6,9.4)
20
30
40
so
80
90
o0
100
Frequency (GHz)
Dom inant Mode
F irst H O M
10
20
30
40
50
60
70
100
80
Epsilam -10 (13.0,10.3,13.0)
Dom inant mode
Higher-order modes
10F
fc
o
u
vu
Epsilam-10 (13.0,10.3,13.0)
Dom inant Mode
F irst H O M
20
30
40
50
60
100
7C
20
30
,0
50
60
70
60
90
100
Frequency (GHz)
Frequency (GHz)
1 10 -
Boron N itrid e (5.12,3.4,5.12)
120
D om inant mode
H igher-order modes
too -
s
o
fa
O'
Sm
vl>
-C
Boron N itrid e (5.12,3.4,5.12)
XJ
IN I
40 ■
Dom inant Mode
First H O M
20
10
20
30
40
50
60
70
AO
90
00
100
10
20
30
10
50
GO
70
80
90
100
Frequency (GHz)
Frequency (GHz)
1 6 0 ----
P T F E Cloth (2.89,2.45,2.95)
110
D o m in an t mode
H igher-order modes
120 ■
cn
g ,00
21
vu
PTFE Cloth (2.89,2.45,2.95) :
N)
Dom inant Mode
First H O M
,0
20
.10
»0
50
60
70
Frequency (GHz)
no
90
100
10
20
30
40
50
60
70
80
90
100
Frequency (GHz)
Fig. 3.11: Effective dielectric constant and characteristic impedance of a suspended CPU
w ith anisotropic substrates.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
44
Table 3.2: Percentage change in the effective dielectric constant by rotating the crystal
lattice 90° at a frequency o f 50 GHz (S-CPVY).
M ode
S a p p h ire
E p sila m -1 0
B o ro n N itr id e
P T F E c lo th
D o m in a n t
+ 4 . 47c
-3.2%
-7.0%
-2.3%
F ir-“ H O M
+ 1.1.7%
-9.2%,
-16.0%
-5.4%
cu rs because a) eyy > e ^ . a n d b) th e d o m in a n t fie ld c o m p o n e n t is o rie n te d along the
x - d ir e c tio n . In a d d itio n to a m o re d isp e rsive d o m in a n t m o d e, th e sp a cin g between
th e tw o m odes becomes s m a lle r w h ic h c le a rly re su lts in a s tro n g e r c o u p lin g . On th e
o th e r h a n d , also illu s tr a te d in F ig . 3.12. th e e ffe ct o f ro ta tio n on th e c h a ra c te ris tic
im p e d a n c e o f th e tw o m odes is re la tiv e ly s m a ll. T h u s , c ry s ta l r o ta tio n effects for th e
case o f s a p p h ire w ill n ot im p ro v e th e p ro p a g a tio n c h a ra c te ris tic s o f th e s tru c tu re , at
least as fa r as m ode in te r a c tio n a n d d isp e rsio n is concerned.
F o r th e re m a in in g th re e typ e s o f c ry s ta l, i.e .. e p s ila m -10. b o ro n n itr id e , a nd PTFF.
c lo th , th e effect o f r o ta tin g th e c ry s ta l la ttic e is q u ite o p p o s ite to th a t o f sa pp h ire . B y
r o ta tin g th e c ry s ta l, th e e ffe c tiv e d ie le c tric c o n s ta n t o f th e d o m in a n t m ode decrease's
s u b s ta n tia lly th e re b y im p ro v in g th e d isp e rsio n c h a ra c te ris tic s of th e s tru c tu re . This
decrease in cr? // as 0 increases o ccurs because a) cyy < cT r . and b ) th e d o m in a n t field
c o m p o n e n t is s t ill o rie n te d a lo n g th e x -d ire c tio n . In a d d itio n , a s u b s ta n tia l decrease
in th e c o u p lin g betw een th e tw o m ost d o m in a n t m odes is also obse rve d as th e angle
o f ro ta tio n increases fro m 0° to 90°.
T h e to t a l p e rce n t change in th e crr f f o f the
d o m in a n t m ode at 50 G H z . as th e c ry s ta l la ttic e is ro ta te d by 90°. is ta b u la te d in
T a b le 3.2.
A m o re c o m p le te u n d e rs ta n d in g o f th e e x is tin g m odes in a p a r tic u la r s tru c tu re
is u s u a lly achieved th ro u g h v is u a liz a tio n o f th e fie ld s.
A t a fre q u e n c y o f 15 G H z.
ty p ic a l c o n to u r p lo ts o f th e lo n g itu d in a l fie ld s fo r th e d o m in a n t a nd firs t highero rd e r m odes were g e n e ra te d fo r each C P \Y s tr u c tu r e u sin g a b o ro n n itr id e su bstra te .
T h e d o m in a n t and firs t h ig h e r-o rd e r modes o f th e c o n v e n tio n a l C P \V are d e p icte d in
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.')
6 4
62
60
fc 5*
O
u
VU
Sapphire (9.4,11.6,9.4)
60
56
D om inant Mode
F irs t H O M
54
Sapphire (9.4,11.6,9.4)
D om inant Mode
52
56
SO
0
10
20
55
30
40
50
60
70
60
90
F irs t HOM
0
10
20
30
40
50
60
70
60
90
R o ta tio n angle (degrees)
R o ta tio n a n g le (degrees)
60 i
58 •
Epsilam -10 (13.0,10.3,13.0)
Dom inant Mode
First H O M
Epsilam-10<13.0,10.3.13.01
Dominant Mode
First HOM
20
30
40
70
20
90
30
50
R o ta tio n angle (degrees)
R o ta tio n a n g le (degrees)
26
2I
Boron Nitride (5.12,3.4,5.12)
99
2 3
Boron N itride (5.12,3.4,5.12) ;
Dominant Mode
First H O M
t
2t
j
Dominant Mode
First H O M
90
j ISf 67
20
20
30
40
50
60
20
70
30
40
50
60
70
R o ta tio n angle (degrees)
R o ta tio n a n g le (degrees)
12 0 —
21
20
P T F E C loth (2.89.2.45,2.95) j
115
D o m in a n t Mode
F ir s t H O M
19
110 -
& '* 2 '- '
«
I I
io
16'
105
P T F E Cloth (2.89.2.45.2.95) !
Dom inant Mode
First H O M
100 ■
CS? »'
I 5
I 4 -
90 :
13
12—
JO
30
40
50
00
70
R o ta tio n a n g le (degrees)
R o ta tio n angle (degrees)
Fig. 3.12’ The effect o f crystal rotation on the effective dielectric constant and character­
istic impedance of a suspended C P W .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
46
F ig . 3.13. w h ile those o f th e suspended C 'P W are shown in F ig . 3.14. For b oth m odes,
th e c o n to u r p lo ts d e m o n s tra te th a t most o f th e lo n g itu d in a l fields are co n ce n trate d
w it h in th e s u b s tra te m a te ria l. T h e s y m m e tric n a tu re o f b o th CPYV s tru c tu re s over
th e v e rtic a l a xis is also illu s tra te d in these figures. A d is tin c tiv e c h a ra c te ris tic o f th e
d o m in a n t m ode is th a t o f a perfect m a g ne tic w a ll (P M C ) across th e lin e o f s y m m e try :
in c o n tra s t, th e firs t h ig h e r-o rd e r m ode is ch a ra cte rize d by a p erfe ct e le c tric w a ll
(P E C ). A s illu s tr a te d in F ig . 3.13. w h ich corresponds to th e c o n v e n tio n a l CPYV. th e
lo n g itu d in a l fie ld s are spread th ro u g h o u t the e n tire s u b s tra te , whereas in Fig. 3.14.
w h ic h corresponds to th e suspended CPYV. th e fields are com pressed tow ard th e
s u b s tra te in te rfa c e , th u s , re s u ltin g in a less d isp e rsive s tru c tu re .
3.0
S h ie ld in g E ffects
T h e eigenvalue fo r m u la tio n developed in th is c h a p te r is a p p lic a b le o n ly when th e
fin ite -e le m e n t mesh is te rm in a te d using a h a rd b o u n d a ry c o n d itio n .
T h is ty p e o f
b o u n d a ry c o n d itio n can be e ith e r a perfect e le c tric c o n d u c to r (P E C ) o r a p erfe ct
m a g n e tic c o n d u c to r ( P M C ) . However, in num erous occasions th e tra n sm iss io n lin e
(m ic ro s tr ip , c o p la n a r w aveguide, s lo tlin e ) is not shielded th e re b y th e fie ld d is tr ib u ­
tio n id e a lly e xte n d s to in fin ity .
In such cases, the c o m p u ta tio n a l d o m a in m ust be
tru n c a te d w ith an a b s o rb in g b o u n d a ry c o n d itio n as is u s u a lly done fo r ra d ia tio n
a n d s c a tte rin g p ro b le m s .
However, u n lik e ra d ia tio n o r s c a tte rin g p ro ble m s w here
th e fields are p ro p a g a tin g o u tw a rd s w ith o u t a tte n u a tio n , a 2 -D tra n sm iss io n lin e
does not ra d ia te . O n th e c o n tra ry , the fields in th e v ic in it y o f the tra n sm issio n lin e
are h ig h ly e%anescent in a d ire c tio n p a ra lle l to th e tra n sve rse plane. T h e respective
a tte n u a tio n c o n s ta n t is n o t o n ly a fu n c tio n o f fre qu e ncy b u t also dependent on th e
g e o m e tric a l and m a te ria l p ro p e rtie s o f th e c ir c u it. I f th e c ir c u it s u p p o rts a pure T E M
m o d e , lik e a s tr ip lin e [60] o r a co a xia l cable, the n th e ra te w ith w h ic h the fie ld is
d e c a y in g along th e tra n sve rse d ire c tio n is pre cisely kn o w . T h e re fo re , an a p p ro p ria te
a n d e fficie n t a b s o rb in g b o u n d a ry c o n d itio n can bo d e riv e d and successfully im p le -
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
17
0
1
2
3
4
Dom inant M ode
0
1
2
3
4
First HO M
Fig. 3.13: Longitudinal fields for the dominant and first HOM of a conventional CPYV.
Dominant M ode
First HOM
Fig. 3.14: Longitudinal fields for the dominant and first HOM of a suspended CPVW
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m e rite d in th e 2 -D eigenvalue p ro b le m . H ow ever, fo r s tru c tu re s th a t do n ot s u p p o rt
a p u re T E M m o d e, such as m ic ro s trip s and c o p la n a r w aveguides, a c c u ra te a b s o rb in g
b o u n d a ry c o n d itio n s cannot be d e rive d unless an a p p ro x im a te e m p iric a l fo rm u la fo r
th e a tte n u a tio n co nstant is firs t o b ta in e d . In th is d is s e rta tio n , an a b s o rb in g b o u n d ­
a ry c o n d itio n fo r evanescent waves in M M IC s tru c tu re s was n o t im p le m e n te d fo r
th is reason. T o s im u la te an open c ir c u it, th e tr u n c a tio n b o u n d a ry is u s u a lly placed
a t a re la tiv e ly large distance aw ay fro m th e tra n s m is s io n lin e (s ). A p e rfe c t e le c tric
c o n d u c to r is o fte n im posed on th e o u te r b o u n d a ry .
T h e effect o f s h ie ld in g on th e
e ffe c tiv e d ie le c tric constant and c h a ra c te ris tic im p e d a n c e o f th e tra n s m is s io n lin e is
in v e s tig a te d fo r various d im e nsion s o f th e e n closu re .
L
Fig. 3.15: M icrostrip line shielded w ith a PEC box (w =100 /zm. h=100 /m i. t = 10 /m i).
A m ic r o s trip lin e p rin te d on a s u b s tra te o f h e ig h t 100 /zm is d e p ic te d in E‘ ig. 3.15.
T h e w id th o f th e m ic ro s trip is 100 /zm and its th ic k n e s s is 10 /zm .
s u b s tra te m a te ria ls arc used: one w ith cr =
T w o d iffe re n t
13 a n d th e o th e r w it h er = 6.2. T h e
s h ie ld in g b o x is scptare in shape and varies fro m 500 /zm to 2000 /zm in steps o f 500 /zm.
T h e e ffe c tiv e d ie le c tric co n sta n t and c h a ra c te ris tic im p e d a n ce o f th e tra n s m is s io n
lin e , fo r th e case o f using a s u b s tra te w ith cr =
3.17. re sp e ctive ly.
13. are show n in Pigs. 3.16 and
It is in te re s tin g to observe th a t by in cre a sin g th e d im e n sio n s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
19
o f th e s h ie ld in g box. th e ePC/ / and Z c begin to increase.
T h e sh ielding effect is
m o re p ro m in e n t when th e d is ta n c e between th e m ic r o s trip and th e o u te r c o n d u c tin g
s u rfa ce is re la tiv e ly s h o rt. S ince th e fields away fro m th e s tr ip decay e x p o n e n tia lly ,
th e e ffe ct o f th e c o n d u c tin g b o x becomes in c re a s in g ly n e g lig ib le w ith distan ce .
It
is c le a r fro m Figs. 3.16 a n d 3.17 th a t the re la tiv e change in cre // and Z c w hen the
d im e n s io n s o f the s h ie ld in g b o x is increased fro m L =
2000// m is o n ly a fra c tio n o f 1%.
H =
1500/im to L = H =
E ven for th e case w here th e dim ensions o f the
s h ie ld in g b o x are L — H = 5 0 0 /im , th e d e v ia tio n in ePe/ / and Z c com pared to the
values o b ta in e d using th e la rg e s t c o n d u c tin g sh ie ld ( L = H = 2 00 0 /im ) is o n ly on
th e o rd e r o f ju s t 5%. S im ila r o bse rva tio ns are m ade fo r a su b stra te w ith er = 6.2.
T h e co rre sp o n d in g graphs a re show n in Figs. 3. IS and 3.19. A g a in , by c o n tin u o u s ly
in c re a s in g th e dim ensions o f th e c o n d u c tin g sh ie ld, th e e ffe c tiv e d ie le c tric co n s ta n t
and c h a ra c te ris tic im p e d a n ce o f th e m ic ro s trip lin e converge to values th a t represent
th e u n b o u n d e d case. In o th e r w ords, by using a c o n d u c tin g sh ie ld o f d im e nsion s 10
tim e s th e m ic ro s trip w id th , th e e rro r in erp/ / and Z c. w it h respect to the u nb o u n d e d
case, is o n ly 1-2 (7t.
3.6
W it h
C o n d u c to r Losses
recent advances in s e m ic o n d u c to r technology, m in ia tu r iz a tio n o f e le c tro n ic
packages and u tiliz a tio n o f h ig h e r clo ck speeds becom e th e p rim a ry focus o f to d a y 's
in d u s try .
For exam ple, c lo c k speeds fo r C M O S -b a sed ch ip s have increased fro m
2 M H z in th e e arly 80's to as h ig h as 500 M H z in th e m id 90"s. S e m ico n d u cto r chips
fo r w ireless c o m m u n ic a tio n s and m icrow ave a p p lic a tio n s are c u rre n tly o p e ra tin g at
c lo c k speeds in the low G ig a H e rtz range. In a d d itio n , m e ta llic traces on m u ltic h ip
m o d u le s (M C M 's ) e x h ib it thicknesses on the o rd e r o f a few / / m ’s w hich, a t lo w e r
fre q u e n cie s, are co m p a ra b le to th e s kin d e p th o f th e m e ta l. T h e skin d epth o f a good
c o n d u c to r is given by [168]
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
.")0
8.8
_ L=H,
L=H,
. L=H,
- L=H,
8.6
L /w = 5
L /w = 1 0
L /w = 1 5
L /w = 2 0
8 .4
£ 8.2
8.0
7.8
7.6
0
2
4
6
8
10
12
14
16
18
20
Frequency (GHz)
Fig. 3.16: Effective dielectric constant o f a m icrostrip line enclosed by a shielding box of
various dimensions (er = 13).
44
L=H,
_ L=H ,
_ L=H,
- • L=H,
L /w =5
L /w = 10
L /w =15
L /w = 20
42
S 41
39
38
37
0
2
4
6
8
10
12
14
16
18
20
Frequency (GHz)
Fig. 3.17: Characteristic impedance o f a m icrostrip line enclosed by a shielding box of
various dimensions (cr = 13).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
.->1
4.3
L=K,
L=H ,
_ L=H ,
L=H ,
4.25
4.2
L /w = 5
L /w = 1 0
L /w = 1 5
L /w = 2 0
4.15
4.1
“ 4.05
UJ
4.0
3.95
3.9
3.85
3.8
0
2
4
6
8
10
12
14
16
18
20
Frequency (GHz)
Fig. 3.18: Effective die lectric constant o f a m icrostrip line enclosed by a shielding box of
various dim ensions (cr = 6.2).
58
L=H ,
L=H,
L=H,
L=H,
L /w = 5
L /w = 1 0
L /w = 1 5
L /w = 2 0
N
54
0
2
4
6
8
10
12
14
16
18
20
Frequency (GHz)
Fig. 3.19: C h a racte ristic impedance o f a m icrostrip line enclosed by a shielding box of
various dim ensions (<r = 6.2).
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w here a.1 is th e a n g u la r fre q u e n cy. // is th e p e rm e a b ility o f th e m e d iu m and crc is
the c o rre s p o n d in g c o n d u c tiv ity .
As an e xa m p le , co nsider a m e ta llic s tr ip o f fin ite
th ickn e ss t. c o n d u c tiv ity crc = 5.S x 107 S /m . and an o p e ra tin g fre q u e n cy o f 1 G H z.
S u b s titu tin g these values in to (3.42). it can be show n th a t th e s k in d e p th is 6 =
2.09 f i m . A t lo w e r fre qu e ncies, th e skin d e p th becomes even la rg e r:
, a re s u lt, the
e le c tric fie ld , w h ic h p e n e tra te s th e c o n d u c to r surface, decays v e ry g ra d u a lly . In such
cases, th e c o n d u c to r losses m u s t be accounted fo r using a fu ll-w a v e ana lysis.
P e r tu rb a tio n te ch n iq u e s have been used fo r m a n y years to e s tim a te c o n d u c to r
losses in M M IC 's . These m e th o d s u tiliz e a g e o m e try -in d e p e n d e n t surface im p e d a n ce
to p e r tu r b th e su rfa ce c u rre n ts o b ta in e d fo r th e lossless case.
These su rface c u r­
rents are c a lc u la te d u sin g e ith e r q u a s i-T E M a p p ro x im a tio n s [49].[48] o r a fu ll-w a v e
a p p ro a ch [47].
In som e cases, e v a lu a tio n o f c o n d u c to r losses is achieved th ro u g h
the use o f an e q u iv a le n t su rface im pedance in c o n ju n c tio n w it h an in te g ra l equa­
tio n [50 ].[5 1 ]. T h e e q u iv a le n t surface im p e da n ce re sid ts in a good a p p ro x im a tio n o n ly
w hen th e s trip th ickn e ss is m u ch la rg er th a n th e skin d e p th .
I f th e s tr ip thickne ss
is c o m p a ra b le to th e s kin d e p th , the c o n d u c to r losses using p e r tu rb a tio n techniques
are u s u a lly u n d e re s tim a te d .
A n o th e r a p p ro a ch is th e use o f the in c re m e n ta l in d u c ta n c e ru le by Y\ heeler [169],
H ow ever, it recpiires e v a lu a tio n o f the tra n s m is s io n lin e in d u c ta n c e w h ic h is a v a ila b le
o n ly fo r T E M o r q u a s i-T E M modes. A v a ria tio n a l m e th o d u sin g th e v e c to r p o te n ­
tia l fo r m u la tio n [170] was also a p p lie d to skin -e ffe ct p ro ble m s. Such im p le m e n ta tio n
th o u g h , is m o re s u ita b le fo r re la tiv e ly s im p le g eom etries. For m o re c o m p le x geom e­
trie s. th e m o d e m a tc h in g te c h n iq u e [52] is one o f th e m ost p ro m is in g . In c o n tra s t to
the usual p e r tu rb a tio n m e th o d s , th e m e ta llic loss is a ccounted fo r w ith o u t th e use o f
a s k in -e ffe c t a p p ro x im a tio n . T h e m e ta llic regions are tre a te d th e same w a v as lossy
d ie le c tric layers. O th e r fu ll-w a v e techniques, in c lu d in g th e m e th o d o f lines ( M o L ) [53]
and fo rm u la tio n s based on th e in te g ra l e q u a tio n ( IE ) [54].[55] o r s p e c tra l d o m a in ap­
proach (S D A ) [56 ]-[5 8], are also e x tre m e ly a ccu ra te and p o w e rfu l. H ow ever, o f a ll the
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a v a ila b le n u m e ric a l tech n iq ue s, th e fin ite elem ent m e th o d ( F E M ) [59].[60] is p ro b a ­
b ly one o f th e m o st a p p ro p ria te and s u ita b le m ethods to be used fo r th e a na lysis o f
lossy M M IC s tru c tu re s . .Not o n ly it can handle in h om o g en e o us lossy m a te ria ls , b u t
also a r b itr a r y g e o m e trie s.
In th is se ctio n , th e fo rm u la tio n developed a t th e b e g in n in g o f th is c h a p te r is
a p p lie d to lossy p la n a r tra n s m is s io n lines. T h e o b je c tiv e is to in ve stig a te th e effect o f
fin ite c o n d u c tiv ity on th e p ro p a g a tio n ch a ra c te ris tic s o f th e tra n sm issio n lin e . These
in c lu d e e ffe ctive d ie le c tric co n s ta n t (eTr. j f ) o r p ro p a g a tio n co nsta nt ( J ) . a tte n u a tio n
co n sta n t ( a ) , and real and im a g in a ry p a rt o f th e c h a ra c te ris tic im p e da n ce ( Z L).
To v a lid a te th e a c c u ra c y o f th e code fo r lossy s tru c tu re s , consider th e same lossy
d ie le c tric w a ve gu id e o r ig in a lly e xa m in e d by Tan et nl. [60]. T h e d im e nsion s o f th e
w aveguide are a x 6 w here a =
26.
T h e loading m a te r ia l is ch a ra cte rize d by a
re la tiv e d ie le c tric c o n s ta n t er = -I + j l OO . B oth th e e ffe c tiv e d ie le c tric c o n sta n t and
a tte n u a tio n c o n s ta n t versus fre q u e n cy are ca lcu late d a n d co m p ared w ith th e exact
s o lu tio n . T h e e xa ct expression fo r th e co m p le x p ro p a g a tio n co nsta nt ("•) is g ive n by
2
r
>
T h e fin ite e le m e n t mesh consists o f 42 tria n gle s and a to ta l o f 67 u n k n o w n field
q u a n titie s . T h e co m p a riso n s between p re d ic tio n s and a n a ly tic a l so lu tio n are shown
in Figs. 3.20 and 3.21. A lth o u g h th e mesh is re la tiv e ly coarse, th e agreem ent betw een
the tw o d a ta sets is e x c e lle n t.
From F ig . 3.20. it is im p o r ta n t to realize th a t th e
e ffe ctive p ro p a g a tio n co n s ta n t depends not o n ly on th e real co m p on e nt o f cr but
also on th e im a g in a ry c o m p o n e n t.
P e rtu rb a tio n te c h n iq u e s , on the o th e r h an d ,
do n o t in c o rp o ra te th e lossy p a rt o f th e m e d iu m in th e c a lc u la tio n o f th e effective'
p ro p a g a tio n c o n s ta n t.
A n o th e r c o n fig u ra tio n th a t was ana lyze d by T a n et ul. [60] is the m ic r o s trip lin e
shown in F ig . 3.22.
and w id th ic =
T h e c o n d u c tin g m ic ro s trip has a fin ite thickness / =
7 5 //m .
3 //m
T h e su b stra te is o f G a lliu m A rs e n id e (CiaAs) w ith i r =
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
.">1
52.2
Exact
FEM
52.0
51.8
5=
a
su
51.6
51.4
51.2
51.0
50.8
14
Fig. 3.20: Effective dielectric constant o f a lossy waveguide w ith er = 4 + j 100 (n = 2b).
90
Exact
FEM
60
su
20
14
Fig. 3.21: A tte n u a tio n constant o f a lossy waveguide w ith cP = 4 + J 100 (a = 2b).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•)■)
13 and h e ig h t h =
100 / /m .
T h e c o n d u c tiv ity o f th e m e ta llic s tr ip is a c = 4.1 x
10' S /m whereas th a t o f th e g ro u n d p la n e is a 3 =
o.S x 10' S /n t.
A lth o u g h in
[60] th e g ro u n d p la n e was s im u la te d u sin g an im p e d a n c e b o u n d a ry c o n d itio n (m ix e d
b o u n d a ry c o n d itio n o f th e th ird k in d ), in th is s tu d y i t was tre a te d lik e th e ce n te r
c o n d u c to r. In d o in g so. th e g ro un d p la ne was m o d e le d as a re c ta n g u la r region w ith
a fin ite th ickn e ss o f 3 //m . T h e e ffe ctive p ro p a g a tio n co n sta n t is show n w ith o u t any
c o m p a riso n in F ig . 3.23. Its value begins to increase ra p id ly as th e fre q u e n c y drops
below o G H z . T h is effect is trig g e re d b y th e fin ite c o n d u c tiv ity o f th e s trip lin e . T h e
thickne ss o f th e c o n d u c tin g s trip , a t th e lo w e r fre q u e n c y end. becom es co m p a ra b le to
th e sk in d e p th and. th e re fo re , the fie ld in te n s ity p e n e tra te s th ro u g h th e m e ta l w ith o u t
s u b s ta n tia l a tte n u a tio n . T h is p henom enon in tro d u c e s an in te rn a l in d u c ta n c e w h ic h
causes th e e ffe c tiv e d ie le c tric co nsta nt to increase a t lo w e r fre quencies.
T h e a tte n u a tio n co n sta n t for th e sam e m ic r o s trip lin e is c o m p a re d in F ig. 3.24
w ith d a ta o b ta in e d fro m [60]. T h e c o m p a ris o n betw een th e tw o fin ite -e le m e n t fo r­
m u la tio n s is fa ir ly good. To o b ta in co n ve rg e n t re s u lts , e s p e c ia lly a t th e h ig h e r fre ­
q ue n cy end. th e m ic ro s trip lin e m u st be d is c re tiz e d s u ffic ie n tly . F or th is p a rtic u la r
s im u la tio n , th e th ickn e ss o f th e m ic r o s trip lin e was d is c re tiz e d u sing th re e tria n g u la r
elem ents. A coarser d is c re tiz a tio n re s u lts in p o o r p re d ic tio n s .
In a d d itio n , th e m i­
c ro s trip lin e was enclosed in to a re la tiv e ly large c o n d u c tin g b o x (20 tim e s the w id th
o f the m ic r o s trip ) w h ich closely resem bles th e open case.
w
1 1 ,
i
^
■ ■ ■ ■ ■ ■ §
h
Fig. 3.22: G eom etry o f a lossy m icrostrip line (er = 13. t = 3 /n n . h = 100//m . ir = 7-5//in .
er.. = 4.1 x 10' S /m . a.3 = o.S x 10' S /m ).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
,
9~
8
8
8
8
8,
8
8.
8.
8.u
0
5
10
15
20
25
30
35
40
45
50
Frequency (GHz)
Fig. 3.23: Effective dielectric constant o f a lossy rnicrostrip line.
0.09
P re s e n t fo rm u la tio n
Ref. [60]
0.08
0.07
0.06
0.05
0.04
0.03
0.02
001
0
5
10
15
20
25
30
35
40
Frequency (GHz)
Fig. 3.2-1: Attenuation constant o f a lossy rnicrostrip line.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
45
50
•
)
I
Besides th e p ro p a g a tio n and a tte n u a tio n constants, th e c h a ra c te ris tic im p e d a n ce
is also a ffe cte d by th e fin ite c o n d u c tiv itie s o f the rn ic ro s trip lin e and g ro u n d plane.
For th e lossless case, th e c h a ra c te ris tic im p e da n ce o f a tra n s m is s io n lin e is p u re ly
real: th e re fo re , th e expressions given in (3.23) and (3 .24 ) can be d ire c tly a p p lie d .
W h e n th e tra n s m is s io n line is lossy, th e pow er th ro u g h an e le m e n t. P,. is c o m p le x in
general: i.e..
O
p, = 5 [ f f E. XH- ■a,.*] = J [ f f (E, H; - E, Hi) ,U
T h e re fo re , th e c h a ra c te ris tic im p e da n ce, a ccordin g to (3 .2 3 ). is also c o m p le x . H o w ­
ever. th is q u a n tity is n ot u n iq u e ly d efined for d ispersive s tru c tu re s [I6 6 j. D e p e n d in g
on th e fo r m u la tio n , p ow er-voltage (P - Y ) . p o w e r-cu rre n t ( P - l) o r v o lta g e -c u rre n t ( \ I), th e e n d re s u lt is u su a lly d iffe re n t. A ll three fo rm u la tio n s converge to th e same
value o n ly at D C : at h ig he r frequencies, th e y begin to d iv e rg e fro m each o th e r [ L6G].
T h is p he n o m e n o n is illu s tra te d g ra p h ic a lly in Fig. 3.2o by c o m p a rin g th e c h a ra c te r­
is tic im p e d a n c e p re d ic te d using o u r fo rm u la tio n (P -Y d e fin itio n ) w ith d a ta o b ta in e d
fro m [60] ( P-1 d e fin itio n ). A lth o u g h th e agreem ent at lo w e r frequencies is e x c e lle n t,
the tw o fo rm u la tio n s begin to d ive rg e as the fre qu e ncy increases.
In a d d itio n , th e
im a g in a ry p a rt o f th e c h a ra c te ris tic im p e da n ce, w h ich is a ttr ib u te d to th e re a c tiv e
pow er, is fo u n d to be always p o s itiv e when the P -Y d e fin itio n is used: how ever,
when th e P -I d e fin itio n is used, th e im a g in a ry p a rt o f th e c h a ra c te ris tic im p e d a n ce
is a lw a ys n e g a tive . In o th e r words, th e c h a ra c te ris tic im p e d a n c e o b ta in e d based on
th e tw o fo rm u la tio n s are 180° o ut o f phase.
T h is is re la te d to th e fact th a t P in
one o f th e expressions occurs in th e d e n o m in a to r, whereas in th e o th e r expression
occurs in th e n u m e ra to r. To p ro vid e a fa ir com parison betw e e n o u r p re d ic tio n s and
th e d a ta e x tra c te d fro m [60]. the la tte r was in v e rte d . T h e co m p arison is show n in
F ig . 3.26. T h e tw o d a ta sets com pare favo ra b ly. X o te th a t th e im a g in a ry p a rt o f th e
c h a ra c te ris tic im p e d a n ce is s ig n ific a n tly sm a lle r th a n th e real p a rt.
T h e effects o f fin ite c o n d u c tiv ity on ro p la n a r w aveguides were also in v e s tig a te d .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•js
60
P re s e n t fo rm u la tio n (P -V d e fin itio n )
R ef. [60] ( P - I d e fin itio n )
58
56
w
s
o
54
-C 52
50
(S3
48
CD
46
44
t l
42
hi
40
5
10
15
20
25
30
35
40
45
50
Frequency (GHz)
F ie . 3.2o
Real com ponent o f the characteristic impedance o f a lossy rn icro strip line.
P re s e n t fo rm u la tio n (P -V d e fin itio n )
R ef. [60] (P - I d e fin itio n )
72
1.2
O
1.0
hS
0.4
15
20
25
30
35
Frequency (GHz)
Fig. 3.26: Im aginary com ponent o f the characteristic impedance o f a lossy rnicrostrip line
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T h e s tru c tu re , w h ich is show n in
F ig . 3.27. is th e sam e as th e one a n a ly z e d by
K e et al. [56] u sin g a m o d ifie d s p e c tra l d o m a in a p p ro a ch .
F n lik e th e rn ic ro s trip
c o n fig u ra tio n discussed p re v io u s ly , th is s tru c tu re is sh ie ld e d . T h e d e ta ile d d im e n s io n s
o f th e c o p la n a r w aveguide and s h ie ld in g b o x are shown in F ig . 3.27. T h e c o n d u c tiv ity
o f a ll m e ta llic regions is crc = 3.0 x 10' S /m . T h e s u b s tra te is G a A s w ith er = 12.9
a n d t an S = 3.0 x 10- 1 . T h e e ffe c tiv e d ie le c tric co n sta n t and a tte n u a tio n c o n s ta n t fo r
th is s tru c tu re are ca lc u la te d u sin g th e f in ite elem ent m e th o d and p lo tte d in F igs. 3.28
a n d 3.29. re sp e ctive ly, fo r a w id e ra n g e o f frequencies. T w o d iffe re n t d is c re tiz a tio n s
w ere considered:
a)
I e le m e n t across th e th ic k n e s s o f th e c o n d u c tin g regions
b ) 3 e le m en ts across th e th ic k n e s s o f th e c o n d u c tin g regions
A s e x p e c te d , th e fin e r th e mesh d e n s ity in sid e th e c o n d u c to rs , th e b e tte r th e accu ­
racy. T h e a greem ent betw een o u r p re d ic tio n s and th e referenced d a ta is e x c e lle n t.
T h e s lig h t d iscre p a n cy in cre/ / a n d a at h ig h e r frequencies is a ttr ib u te d to in s u ffic ie n t
d is c re tiz a tio n in sid e the c o n d u c to rs . U n lik e the rn ic ro s trip case, th e e ffe c tiv e d ie le c ­
t r ic co n sta n t fo r th e co p la n a r w a v e g u id e has a n eg a tive slope, w h ich has a lre a d y been
o bse rve d b o th th e o re tic a lly [52] a n d e x p e rim e n ta lly [171]. The real c o m p o n e n t o f th e
c h a ra c te ris tic im p e d a n ce is show n w ith o u t co m p arison in F ig . 3.30.
A s m e n tio n e d
b e fo re , th e c h a ra c te ris tic im p e d a n c e is n ot u n iq u e ly d e fin e d fo r d is p e rs iv e s tru c ­
tu re s . H ow ever, th e d iscre p a n cy a m o n g various fo rm u la tio n s is s m a ll. N e ve rth e le s s ,
a ll th re e fo rm u la tio n s can a c c u ra te ly p re d ic t th e ra p id increase in th e c h a ra c te ris ­
t ic im p e d a n ce as th e fre q u e n cy o f o p e ra tio n decreases.
As m e n tio n e d b e fo re , th is
p he n o m e n o n is a ttr ib u te d to an in c re a s in g in te rn a l in d u c ta n c e caused b y th e fin ite
c o n d u c tiv ity o f m e ta llic regions.
A d d itio n a l p a ra m e tric s tu d ie s u s in g th e same g e o m e try were focused on c h a n g ­
in g th e c o n d u c tiv ity value o f a ll m e ta llic layers w h ile o b s e rv in g th e v a ria tio n in th e
a tte n u a tio n c o n s ta n t, e ffe ctive d ie le c tr ic constant and c h a ra c te ris tic im p e d a n c e . S ix
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
()0
50
500
lim
M -m
50
600
(im
5
|r m
lin n
2 0 lim
Fig. 3.2 c G eom etry o f a lossy coplanar waveguide (er =
3.0 x 107 S /m ).
difFerent m e ta ls were used d u rin g these stud ie s.
50
lim
13. /n«(5 = 3.0 x 10 '. n,.
These in c lu d e s ilv e r, co pp e r, g old,
a lu m in u m , n icke l and p la tin u m . T h e associated c o n d u c tiv itie s are ta b u la te d in Ia b lc 3.3. T h e e ffe ctive d ie le c tric co n s ta n t as a fu n c tio n o f c o n d u c tiv ity is in ve s tig a te d
firs t. T h e co rre sp o n d in g g ra p h is show n in Fig. 3.31. As e xpe cted , th e e ffe ctive d i­
e le c tric c o n s ta n t fo r the co p la n a r w a ve gu id e decreases w ith in cre asing c o n d u c tiv ity .
As th e c o n d u c tiv ity approaches in fin ity , w h ich begins to create a b a rrie r fo r the fields
a ro u n d c o n d u c tin g surfaces, th e e ffe c tiv e d ie le c tric c o n s ta n t a t th e lo w e r frequencies
becom es fla t.
This flatness was a lre a d y observed in som e o f th e re su lts presented in
p re vio u s se ctions. Some in te re s tin g o b se rva tio n s arise fro m F ig . 3.31:
a) th e presence o f a n eg a tive slope in th e e ffe c tiv e d ie le c tric co n sta n t
b) th e ra p id increase in th e e ffe c tiv e d ie le c tric c o n s ta n t a t low er frequencies
N e ith e r phen om en o n can be observed u sin g co m m o n p e rtu rb a tio n techniques.
T h e c o rre s p o n d in g a tte n u a tio n c o n s ta n t, g ive n in d B / m m . is illu s tra te d fo r six
d iffe re n t m e ta l c o n d u c tiv itie s in f'ig . 3.32.
L ik e th e e ffe ctive d ie le c tric c o n s ta n t.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(il
8.2
3 -e le m en t m esh
1-elem ent m esh
SD A , Ref. [56]
8.0
7.8
ere(r
7.6
WO Mm
7.4
1. 5 Mm.
7.2
x
7.0
6.8
6.6
0
5
10
15
20
25
30
35
40
45
Frequency (GHz)
Fig. 3.28: Effective dielectric constant o f a lossy coplanar waveguide.
0.6
0.5
3 -e le m e n t m esh
1 -e le m e n t m esh
S D A , Ref. [56]
a(dB/mm)
0.4
0.2
0.0
20
30
45
Frequency (GHz)
Fig. 3.29: A tte n ua tio n constant o f a lossy coplanar waveguide.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
52.0
51.5
Real Zc (Ohms)
51.0
50.5
50.0
20 (itn
49.5
49.0
48.5
48.0
47.5
47.0
25
30
35
45
50
Frequency (GHz)
Fig. 3.30: Real component of the characteristic impedance o f a lossy coplanar waveguide.
Table 3.3: Conductivities of few metals.
M e ta l
C o n du ct iv it y. rr,. ( H - m ) - 1
Si 1vet-
6.2 X 10'
C 'oppcr
o.S X 10'
l . l X 10'
3.8 X 10'
C o ld
A lu m in u m
N ic k e l
P la tin u m
l.-l X 10'
9.o X 10°
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
7.6
S ilv e r
C op pe r
G o ld
A lu m in u m ~
_ N ic k e l
P la tin u m
7.4
7.2
7.0
NV
6.8
6.6
0
5
10
15
20
25
30
35
40
45
50
Frequency (GHz)
Fig. 3.31: Effective dielectric constant of a lossy coplanar waveguide w ith different metal
conductivities.
0.8
0.7
0.6
S ilv e r
C op pe r
G o ld
A lu m in u m
_ N ic k e l
P la tin u m
0.5
0.4
3 0.3
0.2
0.1
0.0
35
40
45
Frequency (GHz)
Fig. 3.32: A ttenuation constant o f a lossy coplanar waveguide w ith different metal conduc­
tivities.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(i-t
th e a tte n u a tio n co n sta n t also decreases w ith in cre a sin g c o n d u c tiv ity .
T h is is ex­
p e c te d since h ig h e r c o n d u c tiv ity re su lts in less fie ld c o n c e n tra tio n in s id e th e co nd u c­
to rs . N o te th a t s ilv e r a n d co p p e r have ve ry s im ila r c o n d u c to r losses. O n th e o th e r
h a n d , th e losses associated w ith g old and a lu m in u m are h ig h e r b y a p p ro x im a te ly
0.05 d B / m m .
T h is a m o u n t o f loss is re p re s e n ta tiv e o n ly fo r the s,,. c ific co plan a r
c o n fig u ra tio n analyzed in th is section. T h e re la tiv e ly sm a ll d iffe re n ce in co n d u c to r
losses betw een copper a n d a lu m in u m m ig h t have a s ig n ific a n t effect in th e overall
p e rfo rm a n c e o f an a c tu a l c ir c u it design.
T h e e ffect on th e real a n d im a g in a ry c o m p o n e n ts o f the c h a ra c te ris tic im pedance,
as th e m e ta l c o n d u c tiv ity changes, is illu s tr a te d in Figs. 3.33 and 3.34. As a lre ad y
o b se rve d fo r th e e ffe ctive d ie le c tric co n sta n t and a tte n u a tio n c o n s ta n t, b y increasing
th e c o n d u c tiv ity o f th e m e ta l, th e real and im a g in a ry com ponents o f th e ch a ra c te r­
is tic im p e d a n c e decrease. A t h ig h e r frequencies, how ever, th e effect o f fin ite m e tal
c o n d u c tiv ity begins to d im in is h .
In o th e r w o rd s, th e c h a ra c te ris tic im p e d a n c e ap­
proaches th a t o f a perfect e le c tric c o n d u c to r.
S im ila r p a ra m e tric s tu d ie s were p e rfo rm e d fo r a rn ic ro s trip lin e . T h e m a in ob ­
je c tiv e in these studies was to p ro v id e a fa ir co m p a ris o n betw een tw o o f the most
c o m m o n ly used c o n fig u ra tio n s in s e m ic o n d u c to r in d u s try : th e c o p la n a r waveguide'
and th e rn ic ro s trip .
T h e rn ic ro s trip lin e , w h ic h has th e same s p e c ific a tio n s as the
c o p la n a r waveguide e x a m in e d p re vio u sly, is show n in Fig. 3.35.
T h e o n ly d iffe r­
ence b etw een the tw o c o n fig u ra tio n s is the re m o va l o f th e u p p e r g ro u n d plane.
I he
e ffe c tiv e d ie le c tric c o n s ta n t versus fre q u e n cy is in ve stig a te d fo r d iffe re n t m e ta l con­
d u c tiv itie s . T h e c o rre s p o n d in g p lo ts are show n in F ig . 3.36. As was th e case w ith
th e c o p la n a r w aveguide, th e rn ic ro s trip lin e also e x h ib its a neg a tive slope. T h e slope
o f th e e ffe c tiv e d ie le c tric c o n s ta n t depends, a m o n g o th e r p a ra m e te rs, on th e w id th
o f th e s trip . B y in cre a sin g th e w id th o f th e s tr ip , th e slope o f th e e ffe c tiv e dielectricc o n s ta n t also increases, s o m e tim e s fro m a n e g a tiv e to a p o s itiv e value [52].
In g en e ra l, co plan a r w aveguides p ro v id e less d isp e rsive p ro p a g a tio n c h a ra c te ris tic s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
52
S ilv e r
C op pe r
G old
A lu m in u m
_ N ic k e l
P la tin u m
51
49
47
46
25
45
40
30
Frequency (GHz)
F ig. 3 .33 : Real com ponent o f the characteristic im pedance o f a lossy cop lan ar waveguide
w ith d ifferent m e ta l conductivities.
10
S ilv e r
C opper
G old
A lu m in u m ~
_ N ic k e l
P la tin u m
8
£
-C
9 s
tj
N
>>
CO .
C 4
•5b
ca
W
£
HH
2
0
0
o
10
15
20
25
30
35
40
45
50
Frequency (GHz)
F ig. 3 .34 : Im a g in a ry com ponent o f the c h aracteristic im pedance o f a lossy c op lan ar waveg­
uide w ith diffe re nt m etal con d u ctivities.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(id
500 M-rn
2 0 (dm
1.5 M-1L
I
600
i
5 nm .
t
Fig. 3.35: G eom etry o f a lossy rn icro strip (cr = 13. tarifi = 3.Ox
= 3.Ox 10' S/'rn).
th a n rn ic ro s trip lin e s: th e re fo re , th e y are fre q u e n tly p re fe rre d in h ig h -sp e e d c irc u it
a p p lic a tio n s . As a tra d e -o ff. c o p la n a r waveguides u s u a lly e x h ib it h ig h e r c o n d u c to r
losses th a n rn ic ro s trip lines. C o n c e rn in g the tw o g e o m e try c o n fig u ra tio n s discussed
in th is se ctio n , th e a tte n u a tio n c o n s ta n t associated w ith th e r n ic ro s trip lin e is by
a fa c to r o f 2 s m a lle r th a n th e a tte n u a tio n co n sta n t associated w ith th e c o p la n a r
w aveguide.
T h is is e v id e n t fro m c o m p a rin g F ig . 3.32 w ith
F ig . 3.37.
I t is clear
th a t th e c o n d u c to r losses associated w ith the rn ic ro s trip lin e a re b y fa r s m a lle r th a n
those associated w ith th e c o p la n a r w aveguide. T h e reason re la te s to th e fact th a t
co p la n a r w aveguides in v o lv e m o re c o n d u c tin g surfaces, e s p e c ia lly co rn e rs, near high
fie ld c o n c e n tra tio n , w h ic h leads to h ig h e r c o n d u c to r losses.
T h e real and im a g in a ry c o m p o n e n ts o f th e c h a ra c te ris tic im p e d a n c e fo r th e m i­
c ro s trip lin e are show n in Figs. 3.3S and 3.39. As is u s u a lly th e case, th e im a g in a ry
c o m p o n e n t is s ig n ific a n tly s m a lle r th a n the real c o m p o n e n t.
the zero value as th e fre q u e n c y o f o p e ra tio n increases.
A ls o , it approaches
In g e n e ra l, th e c h a ra c te r­
is tic im p e d a n ce , b o th real and im a g in a ry p a rts , alw ays decreases w ith in c re a s in g
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
c o n d u c tiv ity . A s im ila r o b se rva tio n was also re corded fo r th e co p la n a r w a ve gu id e.
3.7
F ie ld D is tr ib u tio n s
D u rin g a nalysis o f m ic ro w a v e c irc u its , it is o fte n useful to v is u a liz e th e fie ld d is t r i­
b u tio n in th e v ic in it y o f co n d u cto rs. F ie ld p e n e tra tio n in sid e th e co n d u cto rs can be
q u ite s u b s ta n tia l i f th e s tr ip thickne ss is on th e o rd e r o f th e s k in d e p th . T h e m a g ­
n itu d e o f lo n g itu d in a l e le c tric fie ld a lo n g th e a ir-s u b s tra te in te rfa c e o f th e c o p la n a r
w aveguide d e p icte d in F ig . 3.27 is show n in F ig . 3.40. T h e fre q u e n cy o f o p e ra tio n is
1 G H z. N o te th a t th e e le c tric fie ld c o m p o n e n t E z is re la te d to th e c u rre n t d e n s ity
J z a c co rd in g to
J : = < T CE ;
(3.1.'))
where a., is th e c o n d u c tiv ity o f th e m e ta l. In o th e r w o rd s. ./. is d ire c tly p ro p o rtio n a l
to E z.
As show n, th e lo n g itu d in a l c u rre n t d e n s ity J : is re la tiv e ly high near th e
corners o f th e ce n te r a n d g ro u n d c o n d u c to rs . U n lik e p erfe ct co n d u cto rs, h ow eve r. -Jz
is fin ite a t those co rn e rs.
T h e m a g n itu d e o f th e transverse co m p o n e n t o f th e e le c tric fie ld , co m prised o f E r
and E,r are shown in F ig . 3.41. T h e firs t o b s e rv a tio n fro m th is fig u re , as co m p a re d
to Fig. 3.40. is th a t th e transverse c o m p o n e n t is b y th re e orders o f m a g n itu d e la rg e r
th a n th e lo n g itu d in a l c o m p o n e n t. In a d d itio n , th e tra n sve rse co m p o n e n t, e s p e c ia lly
th e one a lo ng th e .r-d ire c tio n . is in te n se in sid e th e s lo t w h ic h is a ty p ic a l c h a ra c te ris tic
o f th e d o m in a n t m o d e o f th e s tru c tu re .
T h e e le c tric fie ld co m p o n e n t a long th e //-
d ire c tio n , a lth o u g h s t ill stro n g e r th a n th e lo n g itu d in a l c o m p o n e n t, is c le a rly s m a lle r
th a n th e one a long th e .r-d ire c tio n .
In F ig . 3.42. th e m a g n itu d e o f th e tra nsve rse c o m p o n e n t o f th e m a g n e tic fie ld
along th e a ir-s u b s tra te in te rfa c e o f th e co p la n a r w a ve gu id e are shown.
T h e s ta ir-
casing appearance o f these plots is due to th e use o f lin e a r basis fu n c tio n s and
o v e rs a m p lin g . It is in te re s tin g , h ow ever, to observe th a t th e h o riz o n ta l c o m p o n e n t
o f th e m a g n e tic fie ld is la rg e and a lm o s t u n ifo rm a lo n g th e le n g th o f the ce n te r con-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
7.6
S ilv e r
C opper
G old
. _ A lu m in u m
. _ N ic k e l
— P la tin u m
7.4
7.2
fa
vu
7.0
6.8
6.6
40
50
Frequency (GHz)
Fig. 3.36: Effective dielectric constant of a lossy rnicrostrip line w ith different metal con­
ductivities.
0.45
0.4
0.35
0.3
S ilv e r
C opper
_ _ _ Gold
A lu m in u m
_ _ N ic k e l
P la tin u m
c 0.25
-a
0.2
0.15
0.05
0.0
50
Frequency (GHz)
Fig. 3.37: Attenuation constant o f a lossy rnicrostrip line w ith different metal conductivi­
ties.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
69
17.0
S ilv e r
C opper
G old
A lu m in u m “
_ N ic k e l
P la tin u m
16.8
cn
s
j j 16.6
o
y
(SI
a 16.4
o
16.2
16.0
20
35
40
45
50
Frequency (GHz)
Fig. 3.38: Real component of the characteristic impedance o f a lossy rnicrostrip line w ith
different metal conductivities.
2.0
1.6
S ilv e r
C opper
G old
A lu m in u m
_ N ic k e l
P la tin u m
-
1.2
C 0.8
0.4
0.0
20
Frequency (GHz)
Fig. 3.39: Im aginary component of the characteristic impedance of a lossy rnicrostrip line
w ith different metal conductivities.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
70
3.0
f= l GHz
2.7
2.4
2.1
1.8
1.2
0.9
0.6
0.3
0.0
0
10
o
15
20
25
30
35
40
45
50
55
60
65
70
75
Frequency (GHz)
Fig. 3.40:
(longitudinal field) evaluated along the air-substrate interface.
E:
_
-
E x, ( f = l G H z)
E v, ( f = l G H z)
© 10
0
D
10
15
20
25
30
35
40
45
50
55
60
65
70
Frequency (GHz)
Fig. 3.41:
Et
(transverse field) evaluated along the air-substrate interface.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
75
71
d u c to r.
It becomes e x tre m e ly s m a ll in s id e th e slo t, and th e n , large a ga in near the
c o rn e r o f the g ro u n d plane. It a s y m p to tic a lly decays to zero as the d is ta n c e fro m the
s lo t increases. T h e v e rtic a l c o m p o n e n t o f th e m a g n e tic fie ld , on th e o th e r h a n d , is
s u b s ta n tia lly large in sid e th e slo t a n d n e a r c o n d u c to r edges: away fro m these regions,
it decays e x p o n e n tia lly .
45
H x, ( f = l G H z )
- - - H y, ( f = l G H z )
40
Frequency (GHz)
F ig. 3.42: I l t (tran sverse field) evalu ated along the air-su b stra te in te rfa c e .
3.S
C onclusions
A s u ita b le v e cto r-n o d a l fin ite e le m e n t fo rm u la tio n , w h ic h in c o rp o ra te s b o th elec­
t r ic and m a g ne tic b ia x ia l and tra n s v e rs e -p la n e a n is o tro p ic m a te ria ls , was developed
a nd a p p lie d to several M M IC s tru c tu re s .
In a d d itio n , a conve n ie nt c h a ra c te ris tic
im p e d a n ce fo rm u la tio n u sin g lin e a r tr ia n g u la r fin ite elem ents and a p o w e r-v o lta g e
d e fin itio n was presented. T h e re s u ltin g g en e ra lize d eigenvalue p ro b le m was solved
e ffic ie n tly using a fo rw a rd ite r a tio n a lg o rith m w h ile ta k in g fu ll a d v a n ta g e o f the
s p a rs ity o f the m a trice s.
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T h e sam e fo rm u la tio n is a p p lic a b le fo r lossy su b stra te s ancl co n d u c to rs w ith fi­
n ite c o n d u c tiv ity . In the presence o f loss, th e a tte n u a tio n c o n s ta n t fo r a wave p ro p ­
a g a tin g a lo n g th e lo n g itu d in a l d ire c tio n is non-zero. In a d d itio n , th e c h a ra c te ris tic
im p e d a n ce o f th e tra n sm issio n lin e e x h ib its not o n ly a real co m p o n e n t b u t also an
im a g in a ry c o m p o n e n t whereas th e co rre sp o n d in g p ro p a g a tio n co n sta n t increases w ith
in c re a sin g loss. In general, th e p ro p a g a tio n c h a ra c te ris tic s o f m icro w a ve c irc u its are
s u b s ta n tia lly affected by th e presence o f lossy co n d u cto rs. A c c u ra te c h a ra c te riz a tio n
o f p la n a r tra n s m is s io n lines w ith fin ite c o n d u c tiv ity becom es in c re a s in g ly im p o rta n t
in areas o f e le c tro n ic packaging and m icro w a ve c irc u its .
N u m e ric a l results and p a ra m e tric stud ie s co n ce rn in g th e effects o f a n iso tro p ie s
and c o n d u c to r losses on th e p ro p a g a tio n c h a ra c te ris tic s o f p la n a r c irc u its were p re ­
sented and discussed. C ode v e rific a tio n was achieved th ro u g h d a ta co m p a riso n w ith
e x is tin g jo u r n a l p u b lic a tio n s .
M o st o f th e analysis was p r im a r ily c o n c e n tra te d on
m ic ro s trip s and co planar w aveguides.
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CHAPTER 4
T H R E E - D IM E X S IO X A L M IC R O W A V E S T R U C T U R E S
A 3 -D ve cto r fin ite e le m en t m e th o d is fo rm u la te d in th is c h a p te r to ch a ra cte rize the
e le c tric a l p e rfo rm a n ce o f m icrow ave c irc u its and e le c tro n ic packages.
The fo rm u ­
la tio n is based on d is c re tiz in g th e M a x w e ll's e q u a tio n s in fre q u e n cy d o m a in .
The
c o m p u ta tio n a l space is s u b d iv id e d in to te tra h e d ra l elem ents w h ic h a p p ro x im a te the
s o lu tio n to M a x w e ll's e quations using lin e a r in te r p o la tio n fu n c tio n s .
H ig h e r-o rd e r
e le m en ts can a lw a ys be used to im p ro v e th e a c c u ra c y o f th e n u m e ric a l s o lu tio n . T he
s tru c tu re is e x c ite d using th e g o ve rn in g m o d e d is tr ib u tio n at th e in p u t p o rt w hich is
c a lc u la te d a p r i o r i using th e 2-D eigenvalue a n a ly s is d escribed in C h a p te r 3. T h e re­
m a in in g o u tp u t p o rts and s u rro u n d in g w alls are p ro p e rly te rm in a te d u sin g dispersive
a b so rb in g b o u n d a ry co n d itio n s. T h e a ccu ra cy o f th e fin ite e le m e n t code is verified
against a n u m b e r o f cases ra n gin g fro m p la n a r c irc u its to m o re c o m p le x e le ctro n ic
packages such as th e S -pin S O IC su rfa ce -m o u n t package. As show n in th e fo llo w in g
sections, the code is ve ry accurate, e ffic ie n t and v e rs a tile in a n a ly z in g m icrow ave
c irc u its and e le c tro n ic packages.
l.l
In tro d u c tio n
Recent advancem ents in m o n o lith ic m icro w a ve in te g ra te d c ir c u it (M .M IC ) te c h n o l­
ogy helped th e p ro d u c tio n o f e le c tro n ic packages w ith s ig n ific a n tly s m a lle r size and
a la rg e r n u m b e r o f p rin te d in te rco n n e cts on th e m o th e rb o a rd . A c c u ra te design and
o p tim iz a tio n o f high-speed h ig h-clen sity m ic ro w a v e c irc u its and packages becomes a
m a jo r challenge w hen it comes to h ig h p e rfo rm a n c e and lo w co st.
H ig h -fre q u e n c y
o p e ra tio n is u s u a lly th e m a in cause o f stro n g c o u p lin g and in te rfe re n c e a m o n g neigh­
b o rin g tra n sm issio n lines, th e re b y a ffe c tin g th e o v e ra ll e le c tric a l p e rfo rm a n c e o f the
package. T h e presence o f a b ru p t d is c o n tin u itie s , rn ic ro s trip bends, b o n d w ires, m e ta l­
lic bridges and v e rtic a l c o n d u c tin g vias re su lts in a d d itio n a l p a ra s itic effects such as
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71
ra d ia tio n and tim e delays. T h e m a jo r o b je c tiv e o f c u rre n t te c h n o lo g y is th e design
o f e le c tro n ic packages th a t are o p tim iz e d to m in im iz e severe p a ra s itic e ffe cts w ith o u t
necessarily in c re a s in g th e cost o r th e c o m p le x ity o f th e m a n u fa c tu rin g process.
T h e e x is tin g h ig h d e m a n d fo r th e d eve lo p m e n t o f m o re a ccu ra te , v e rs a tile and
e ffic ie n t n u m e ric a l m odels w h ic h can be used in th e design and c h a ra c te riz a tio n o f
m icro w a ve c irc u its m a n d a te s th e im p le m e n ta tio n o f fu ll-w a v e te ch n iq u e s such as the
fin ite-difF erence tim e -d o m a in ( F D T D ) m e th o d [172]. th e s p e ctra l d o m a in approach
(S D A ) [17]. and th e fin ite e le m e n t m e th o d (F E M ) [34]. T h e F D T D m e th o d is p ro b ­
a b ly th e m ost e x te n s iv e ly used te c h n iq u e fo r th e a nalysis o f g e o m e tric a lly complexp ackaging s tru c tu re s . It was in it ia lly a p p lie d fo r th e e v a lu a tio n o f fre q u e n c y d e p e n ­
d e n t p a ra m e te rs o f basic rn ic ro s trip d is c o n tin u itie s [23].[24]. It was la te r im p le m e n te d
su ccessfully in th e a na lysis o f m o re co m p le x s tru c tu re s such as filte rs , rn ic ro s trip
tra n s itio n s , bond w ires, b rid ge s, e tc. [22].[2-5]. H ow ever, th e m a in d ra w b a c k o f the
m e th o d is th a t c u rv e d surfaces and n o n -re c ta n g u la r vo lu m es are u s u a lly m o deled
using a s ta irc a s in g a p p ro a ch .
T h e S D A te ch n iq u e is also ve ry p o p u la r in th e area
o f m icro w a ve c ir c u it a na lysis arrd design.
Its m a in disa dva n ta g e th o u g h is th a t it
can o n ly h andle m e ta liz a tio n s in th e same plarte. A lth o u g h th e m e th o d can be e x­
ten d ed to tre a t c o n d u c tin g tra n s itio n s in the v e rtic a l p la ne [16]. it s t ill is re s tric te d
to sp ecific ty p e o f g eo m e trie s. O n th e c o n tra ry , the F E M is th e m o st v e rs a tile and
fle x ib le n u m e ric a l te c h n iq u e to be used in th e a nalysis o f g e o m e tric a lly c o m p lic a te d
e le c tro n ic packa g in g s tru c tu re s . T h e in tro d u c tio n o f v e c to r fin ite e le m e n ts [37]. the
va lu a b le c o n trib u tio n s on a b s o rb in g b o u n d a ry c o n d itio n s (A B C 's ) [136] and th e e f­
fectiveness o f sparse m a tr ix ite r a tiv e solvers created a co n d u cive e n v iro n m e n t for
tlie e v o lu tio n o f th e m e th o d in th e area o f c o m p u ta tio n a l e le c tro m a g n e tic s .
The
F E M has been e x te n s iv e ly used fo r s c a tte rin g and ra d ia tio n p ro b le m s [33 ]. w aveguide
p ro p a g a tio n p ro b le m s [I2 o j. and a na lysis o f tw o -d im e n s io n a l (2 -D ) M M I C s tru c tu re s
[42].[124]. R e ce n tly, th e m e th o d has been a p p lie d in th e .s’-p a ra m e te r e v a lu a tio n of
th re e -d im e n s io n a l (3 -D ) M .M IC s such as rn ic ro s trip tra n s itio n s , p la n a r d is c o n tin u ­
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Io
itie s and c o n d u c tin g vias [27].[45].[46].
T h is s tu d y fo rm u la te s a fu ll-w a v e analysis and im p le m e n ta tio n o f th e fin ite ele­
m e n t m e th o d to solve g e o m e tric a lly co m p le x and p ra c tic a l m icrow ave c irc u its . U n lik e
p re v io u s w o rk done on th e s u b je c t [27].[45].[46]. a 2 -D eigenvalue a n a lysis [124] is
n ow p e rfo rm e d at the in p u t p o rt to co m p u te th e fie ld d is trib u tio n o f th e d o m in a n t
o r h ig h e r-o rd e r modes: th e c ir c u it d is c o n tin u ity is th e n excited w ith th e g ove rn in g
m o d a l d is tr ib u tio n .
T h e eig en va lue analysis is also a p p lie d to th e o u tp u t p o rt in
o rd e r to c a lc u la te th e fre q u e n cy dependent p ro p a g a tio n constant and c h a ra c te ris ­
t ic im p e d a n c e o f the tra n s m is s io n line. T h e d is p e rs iv e p ro pa g atio n c o n s ta n t a t the
in p u t and o u tp u t p orts is used in th e im p le m e n ta tio n o f the A B C 's , whereas the
c h a ra c te ris tic im pedance is used in th e e v a lu a tio n o f th e ^'-pa ra m e ters. T h e c u rre n t
fo r m u la tio n is proven to be effi ci ent, flexible and e x tre m e ly accurat e in a n a ly z in g
c o m p le x 3 -D m icrow ave c irc u its . It is effici ent because o f the use o f a 2 -D eigenvalue
a n a ly s is to d e te rm in e th e e x c ita tio n fields and needed c irc u it p a ra m e te rs . It is ver­
sati l e because the in p u t and o u tp u t p o rts are n o t re s tric te d to a sin g le rn ic ro s trip
lin e : c o p la n a r waveguides, c o u p le d rn ic ro s trip lines and finlines can also be used.
It is accur at e because th e e x c ita tio n fie ld , p ro p a g a tio n constant and c h a ra c te ris tic
im p e d a n c e are co m p ute d at e ve ry frequency using a full-w a ve a p p ro a ch.
4.2
F in ite E lem ent F o rm u la tio n
A fu ll-w a v e fin ite elem ent m e th o d is used in th e a na lysis o f co m p lex e le c tro n ic p ack­
a g in g c irc u its p rin te d on sin g le o r m u lti-la y e r su b stra te s. A ty p ic a l rn ic ro s trip d is­
c o n tin u ity is illu s tra te d in F ig . 4.1. The in p u t p o rt o f th e s tru c tu re is e x c ite d using
th e d o m in a n t field d is tr ib u tio n a t a specific fre qu e ncy.
The g o v e rn in g fie ld d is t r i­
b u tio n a t th e in p u t p o rt is e va lu a te d a p r i o r i u sin g a 2-D eigenvalue a n a lysis [124].
B o th in p u t and o u tp u t p o rts are a p p ro p ria te ly te rm in a te d using a b s o rb in g b o u n d a ry
c o n d itio n s th a t are d ire c tly a p p lie d to the tra n sve rse e le c tric field co m p o n e n t at the
su rface . T h e same ty p e o f a b s o rb in g b o u n d a ry c o n d itio n s are also used to e ffe c tiv e ly
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Output Port
Side Walls
Discontinuity
Input Port
Fig. 4.1: Three-dim ensional rendering o f a typical rnicrostrip discontinuity.
te rm in a te th e side w alls o f open s tru c tu re s .
T h e 4 -D fin ite e le m en t a n a lysis begins w ith the d is c re tiz a tio n o f H e lm h o ltz 's
equ a tion in a source-free region
V x ( { f l r } - l . V x E ) - k i [ t r} E
=
(4 .1 )
0.
where [ r r ] and [// r ] are. re sp e ctive ly, th e re la tiv e p e r m it t iv it y and p e rm e a b ility tensors
o f th e d o m a in , l.’ sing th e w e ll-k n o w n C la le rkin 's te ch n iq u e . (4 .1) m a y be tra n s fo rm e d
in to a weak in te g ra l fo rm g ive n by
x E ) - ( ^ x N ) d \ - - k ; J^[ cr] - E - N d V +
x E ) - ( N x a n) d A = 0
(4 .2 )
where a n is th e n o rm a l to th e su rface u n it v e c to r p o in tin g o u ts id e th e fin ite e le m e n t
vo lu m e, a nd N denotes the v e c to r te s tin g fu n c tio n .
T h e closed surface in te g ra l in
(4.2) is n o n -ze ro o n ly on n o n -p e rfe c tly c o n d u c tin g surfaces. In gen e ra l. (4.2) can be
w ritte n as
[ ([//r]_ IV
J'.i
X
E ) • (V
X
N ) f l \ - - k - ; f \ c r] - E - N d V
Jn l
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+
/
( [ / 'r ] _ l V
+ fJ S2 ([,<r]-lV
+
f
JSi
( [ ^ r j _1V
X
E)-(N
X
(ln) d A
X
E) • (N
X
an) d A
x
E )-(N
X
an) d A = 0
(4.3)
where 5'i and i>’2 den o te th e in p u t and o u tp u t p o rts , re sp e ctive ly, whereas S 3 denotes
a ll open w a ll surfaces. T o e v a lu a te th e su rface in te g ra ls in (4 .3 ). a p p ro p ria te b o u n d ­
a ry co n d itio n s need to be deve lo p ed a t those surfaces [33].[45].
T h e d e ve lo p m e n t
o f such b o u n d a ry c o n d itio n s m a n da tes e xpressin g th e e le c tric fie ld o n th e surface in
term s o f the transverse and lo n g itu d in a l co m p o n e n ts, designated b y th e s u b s c rip ts t
and n. as follow s:
E = E , + an E n.
(4 .4)
T h e n o rm a l co m p o n e n t o f th e e le c tric fie ld E n is fu r th e r assum ed to be n e g lig ib le .
T h is a ssu m p tio n is im p le m e n te d o n ly in th e e v a lu a tio n o f the su rface in te g ra ls at th e
ports. As a re su lt, th e to ta l e le c tric fie ld at th e e x c ita tio n plane can be expressed as
a su p e rp o sitio n o f tw o tra n sve rse fie ld co m p o n e n ts
E ( . r . ;/. c ) ~ E t (-i'. //- - ) = E 7 lc(.r. >j. r ) + E ^ U - //• - )
(
where th e s u p e rscrip ts i nc and sea d e n o te in c id e n t and sca tte re d fie ld s, re sp e c tiv e ly .
T h e in c id e n t fie ld in (4.3) is o b ta in e d fro m th e 2-D eigenvalue a n a ly s is g ive n by
Er(.r.;,.c) =
E 0e t ( x . y ) t - Jk‘ :
(4.6)
where et {. v. y) is th e fie ld d is tr ib u tio n o f th e d o m in a n t m ode at th e in p u t p o rt and
is the co rre sp o n d in g p ro p a g a tio n c o n s ta n t. E xpressing th e in c id e n t fie ld in th e fo rm
shown in (4 .6). it was assum ed th a t th e in p u t p o rt lies on th e .r/y-plane. S u b s titu tin g
(4.6) in to
(4 .3 ). th e to ta l tra n sve rse fie ld a t th e in p u t p o rt is re p re sen ted as
E ( . r . ; / . r ) ~ E f (./-. //. - ) = E 0e t ( .r. //) r
+ R E 0e t( . r . y ) cj k : :
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(4.7)
7S
4 y
Input Port
O utput Port
z
Fig. -1.2: T w o -p o rt geometry w ith the o u tp u t p ort oriented at an angle 0 w ith respect to
the in p u t p ort.
w here R is th e re fle c tio n c o e ffic ie n t. Based on (1 .7). th e fo llo w in g firs t-o rd e r absorb­
in g b o u n d a ry c o n d itio n is v a lid [-13]:
- a . x ( V x E ) + j k ' . a . x (r i. x E ) = - 2 j k : E ,nc.
( l.S)
A s im ila r a b s o rb in g b o u n d a ry c o n d itio n can be d e rive d fo r th e o u tp u t p o rt as well.
T h e p la n a r surface o f the o u tp u t p o r t lies on th e r t j -p la n e w h e re a r x a y fo rm s a unit
ve cto r n o rm a l to th e p o rt plane: th is u n it ve cto r fo rm s an a ng le 0 w ith respect to
th e u n it v e c to r <7-. In o th e r w o rd s, th e o u tp u t p o rt p la n e does n o t necessary have
to be p a ra lle l to th e in p u t p o rt p la n e (see F ig . 1.2). As a re s u lt, g e o m e trie s such as
w aveguide and rn ic ro s trip bends m a y be s im u la te d .
R e fe rrin g to F ig . 1.2. th e to ta l fie ld at th e o u tp u t p o rt can be expressed in the
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fo llo w in g fo rm :
E ( x . y . z ) ~ E f ( x . y. z) = T E ae t ( r . u) e - j kn( r *'nd+'-cosd)
w h e re T
is
(4.9)
th e tra n s m is s io n co efficie n t ancl k n is th e p ro p a g a tio n c o n s ta n t in th e
d ire c tio n represented by
h = aT sin 9 + a z cos 0.
Based on the o rie n ta tio n
(4.10)
o f th e o u tp u t p o rt a nd th e co rre sp o n d in g fie ld d is tr ib u tio n
at th a t plane, a v a lid a b s o rb in g b o u n d a ry c o n d itio n is g ive n by [33]
a n x ( V x E ) + j k na n x ( a „ x E ) = 0.
(4 .1 1)
S u b s titu tin g (4.S) a nd (4.11) in to (4 .3). th e la t te r becomes
[ ( [//r ] —1V x E ) • ( V x N ) d V — k i [ [Cr] - E - N cIV
Jq
Jn
w h e re a ril =
+
j k z [ ( [ / 'r ] _ ‘ E
Js,
+
jkn
+
/ ( H - ' V x E ) - ( N x a j r f . 4
4s',
=
[ ( [ / 'r ] ' l E r X <
Js j
j
( [ / 'r ] " ‘ E
—</.. a n, =
x
a r n ) ■( N
x
ani)dA
X
« „,) • ( N
X
anj)dA
) • (N x ani)dA
f/r s in 0 + a .c o s f l. a nd
n o rm a l to th e open s id e w a ll surfaces.
(4.12)
is th e o u tw a rd u n it v e c to r
T h e a b s o rb in g b o u n d a ry c o n d itio n at th e
ope n sidew alls is ve ry s im ila r to the one im p le m e n te d at th e o u tp u t p o rt. T h e o n ly
d iffe re n c e between th e tw o is th a t kn is re p la c e d b y k 0 ^ / t r Ur w here er a nd / / r are.
re s p e ctive ly, th e (e ffe c tiv e ) p e r m it t iv it y a n d p e r m e a b ility o f the lo ca l m e d iu m . T h e
in te g ra l e q u a tio n in (4 .12 ) can be expressed as
f ([//r ] - l V
Jv.
x E)-(V
x
N )'l\--k ;
[ [f r ] - E - N r f V
Jv.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
so
+
j k ; [ ([/*>•]
JS1
lE
+
j k n [ ([/<r ] _ l E
Js2
+
j koyf t Ti Tr [ ( [ / / r ] - 'E x an3) ■ ( N x d „ , )
Js 3
=
-2jkzJsf j ([/ir]-1E r x a „ I) . ( N x a ni)</.4.
X
)
X
« „ ,)
• (N x
) dA
■( N x a „ 2) d A
(IA
(4.13)
In tro d u c in g te tra h e d ra l e le m e n ts, th e e le c tric fie ld E is e xp a n d e d in te rm s o f a set
o f ve cto r basis fu n c tio n s N to fin a lly o b ta in th e fo llo w in g e le m e n ta l m a trix system :
[.\r Hr B‘Si + B'jt +
= {//}
(4.14)
where
M '(iJ)
=
~
fJv.'{ V x N , - } r
-[/ i r] - l - { ^ x N J} r / r
H [ N ,r • [er] ■N j d V
Jv.'
(4.1-1)
B's,{i-j)
= jk. /
{N ,
X
#7„, } r - [/rr ] _1 • { N , x a ni } ( I A
( 1.10)
B s 2( i - J )
= jk n
{N , x
} r • [//r ]~ 1 - { N j x a „ , } d A
(4.17)
B sr A i . j )
= j k os/ e r f t r J ^ { N , x a „, } 7 • [//r] _ 1 • { N , x a n, } <7.4
6 '( f)
=
-2 jk :
f
Js:
{ N , x a,n } r • [ / ir]~ l • {E J nc x o n, } d A
(4. IS)
(4.10)
and i . j = 1.......6. T h e e le m e n ta l m a trice s are th e n assem bled in to the g lo b a l m a tr ix
using th e edge c o n n e c tiv ity in fo rm a tio n . T h e g lo b a l m a t r ix system is solved using
an e ffic ie n t C o n ju g a te G ra d ie n t Square (C G S ) so lve r w ith .Jacobi p re c o n d itio n in g .
For b e tte r a n d fa ste r convergence, th e ite ra tio n is done in d o u b le precision.
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81
O nce th e e le c tric fie ld d is tr ib u tio n is o b ta in e d e v e ry w h e re in th e s tru c tu re , the
next step is to e v a lu a te th e c o rre s p o n d in g voltages a t th e tw o p o rts .
X o te th a t
a lth o u g h th e th e o re tic a l fo rm u la tio n was based on a tw o -p o r t n e tw o rk , it can be
easily e xte n d e d fo r m u ltip le p o rts . T h e 5 -p a ra m e te rs o f th e s tru c tu re are e valuated
I • _ v re-f
S .i =
W
11
-
a a i’
where f j and V-i are th e voltages c a lc u la te d at p o rts
I a nd 2 (3 -D ana lysis), re­
sp e ctive ly. whereas V’1re^ is th e reference vo ltag e c a lc u la te d a t p o rt 1 (2-D ana lysis).
N o te also th a t Z c\ a nd Z c> are th e co rre sp o n d in g c h a ra c te ris tic im pedances o f the
tra n sm issio n lin e s at th e tw o p o rts . These are c a lc u la te d u sin g th e 2-D fin ite -e le m e n t
eigenvalue a na lysis.
4.3
A n O u tlin e o f th e F in ite -D iffe re n c e T im c -D o m a in M e th o d
T h e fin ite -d iffe re n c e tim e -d o m a in ( F D T D ) m e th o d is e x te n s iv e ly used in th is s tu d y
to v e rify som e o f th e p re d ic te d d a ta o b ta in e d using th e F F .M . In th is section, the
m e tho d is b rie fly in tro d u c e d by p re se n tin g some o f th e m o st im p o rta n t d e ta ils in
term s o f its im p le m e n ta tio n in p acka g in g and m ic ro w a v e c irc u its .
T h e F D T D m e th o d is one o f th e m o st p o p u la r n u m e ric a l tech n iq ue s fo r so lvin g
c o m p lex e le c tro m a g n e tic p ro b le m s. T h e F D T D m e th o d is fin d in g a p p lic a tio n s in a
w ide s p e c tru m o f s im u la tio n p ro b le m s in c lu d in g a nte n n a s fo r w ireless c o m m u n ic a ­
tio n s. b io m e d ic a l a p p lic a tio n s , m ic ro w a v e c irc u its , e le c tro n ic packa g in g and e le c tro ­
m a g ne tic s c a tte rin g and p e n e tra tio n .
T h e p o p u la rity o f th is m e th o d is a ttr ib u te d
to its s im p lic ity in im p le m e n ta tio n a n d c o m p u te r p ro g ra m m in g , its a b ility to handle
a rb itra ry and c o m p le x g eo m e trie s in c lu d in g d iffe re n t m a te ria ls , and th e fact th a t
it is a tim e -d o m a in m e th o d .
F re q ue n cy in fo rm a tio n is o b ta in e d th ro u g h a single
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
s im u la tio n over a b ro a d fre q u e n c y s p e c tru m .
A n F D T D code s u ita b le fo r h a n d lin g g e n e ra l m u lti-c o n d u c to r s tru c tu re s has been
d e v e lo p e d . T h e code is q u ite g e n e ra l in h a n d lin g d iffe re n t m a te ria ls and c o n d u c to r
d is c o n tin u itie s , such as th e ones fo u n d in e le c tro n ic packages. T h e d e ve lo p e d code
uses firs t-o rd e r M u r a b s o rb in g b o u n d a ry c o n d itio n s . These have been p ro ve n to w o rk
w e ll in a p p lic a tio n s in v o lv in g m ic ro w a v e c irc u its . T h e e le c tric w a ll source c o n d itio n
has been im p le m e n te d to e x c ite th e d o m in a n t m o d e o f s tru c tu re s in v e s tig a te d in th is
s tu d y . S in ce the F D T D m e th o d uses re c ta n g u la r b ric k s as the basic mesh e le m en ts,
it is p re d o m in a n tly s u ite d fo r p la n a r s tru c tu re s .
In o b ta in in g th e .S '-param eters o f a g iv e n s tru c tu re , a source p lane is im posed
a t th e in p u t p o rt.
T h e e x c ita tio n signal is a G aussian pulse in th e tim e d o m a in .
O n ce th e pulse is la u n ch e d , th e firs t-o rd e r M u r a b s o rb in g b o u n d a ry c o n d itio n s are
im m e d ia te ly tu rn e d o n.
T h e n u m e ric a l s im u la tio n is ca rrie d o u t tw ice .
s im u la tio n occurs in th e absence o f th e d is c o n tin u ity .
T h e firs t
T h is is re q u ire d in o rd e r
to e s ta b lis h a reference in c id e n t w a ve fo rm p ro p a g a tin g along th e m ic r o s tr ip lin e .
T h e re ference plane is d e fin e d .V cells away fro m th e b e g in n in g o f th e d is c o n tin u ity .
A second s im u la tio n is re p e a te d in the presence o f th e d is c o n tin u ity and th e tim e
s ig n a tu re o f the in c id e n t a n d re fle cte d vo ltag e s at th e reference p la ne is o b ta in e d .
U s in g th e tw o s im u la tio n s , th e in c id e n t a n d re fle cte d tim e -d o m a in w a ve fo rm s are
firs t c a lc u la te d and th e n used to e va lu a te th e a m p litu d e and phase o f th e re tu r n loss.
A s im ila r a rg u m e n t holds fo r th e tra n s m itte d v o lta g e used in th e e v a lu a tio n o f th e
in s e rtio n loss o f the s tru c tu re .
4.-1
N u m e ric a l V a lid a tio n a n d R esults
T h e f in ite elem ent fo r m u la tio n deve lo p ed in S e ctio n 4.2 was im p le m e n te d and a p p lie d
to a v a rie ty o f 3-D m ic ro w a v e c ir c u its and e le c tro n ic packages. Som e o f these c irc u its
in c lu d e p la n a r in te rc o n n e c ts , c o n d u c tin g via s. bon d w ires, bridges, single- and m u ltilo o p in d u c to rs , filte rs , d ie le c tric d is c o n tin u itie s as w ell as co m p le x packages such as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
th e 8 -p in S O IC p la s tic package. These g eom etries w ill be a n a lyze d using the fu ll-w a v e
a p p ro a ch discussed in th is c h a p te r.
For some cases, th e re s u lts o b ta in e d using th e
fin ite e le m e n t m e th o d are co m p a re d w ith s im u la tio n d a ta o b ta in e d using the fin ite d iffe re n ce tim e -d o m a in ( F D T D ) m e th o d , w h ich is b rie fly o u tlin e d in Section 4.3.
C o m p a riso n s w ith o th e r n u m e ric a l tech n iq ue s, such as in te g ra l m ethods and th e
fin ite -d iffe re n c e fre q u e n c y -d o m a in ( F D F D ) m e th o d , are also p ro v id e d .
4.4.1
M ic r o s trip in te rc o n n e c tio n th ro u g h a d ie le c tric b rid g e
fn th e design o f m u lti-c h ip m o d u le s (M C 'M ) engineers o fte n e n c o u n te r great d iffic u l­
ties in o p tim iz in g and im p ro v in g th e o v e ra ll e le c tric a l p e rfo rm a n c e o f a package. A
package u s u a lly houses m ore th a n a sin g le M .M IC . each one w it h a d is tin c t fu n c tio n ­
a lity . These iso la te d m odules are in te rc o n n e c te d th ro u g h th e use o f w ire b on d in g o r
m ic r o s trip tra n s itio n s and vias. These in te rc o n n e c ts o fte n in tro d u c e reflections th a t
d e te rio ra te th e p e rfo rm a n ce o f th e package. A c c u ra te design o f c h ip in te rco n n e cts
and bon d w ires necessitate th e use o f n u m e ric a l m e th o d s.
T h e in te rc o n n e c tio n o f tw o m ic r o s trip lines, w h ic h are s it t in g on separate s u b ­
s tra te s (C la A s). is analyzed u sin g th e fin ite elem ent code d e ve lo p ed d u rin g the course
o f th is s tu d y .
T h e m ic ro s trip tra n s itio n between th e tw o m o d u le s is encapsulated
b y a d ie le c tric region (b rid g e ) w ith re la tiv e d ie le c tric co n s ta n t cr and length cl. I he
g e o m e try o f th is s tru c tu re , w h ic h was analyzed p rio r to th is w o rk using the F D F D
m e th o d [22]. is shown in F ig . 4.3.
A ll c o n d u c tin g surfaces were tre ate d as p e r­
fect e le c tric co n d u cto rs. T h e sid e w a lls are also perfect e le c tric co n d u cto rs since th e
s tru c tu re is considered sh ie ld e d . T h e 5 'n versus th e n o rm a liz e d frequency ( k 0 - a) is
p lo tte d fo r va rio u s values o f er and cl. T h e d im e n sio n a used in th e n o rm a liz a tio n
was ta ke n to be 1 m m whereas th e fre q u e n cy o f o p e ra tio n ranges between 100 M H z
a nd 15 G H z. B y incre asing th e p e r m it t iv it y o f th e d ie le c tric b rid g e , it is e xpe cted
th a t th e m ic r o s trip tra n s itio n w ill p ro v id e a b e tte r m a tc h . T h e e ffe ctive d ie le c tric
c o n s ta n t
o f the m ic ro s trip lin e on a G aA s s u b s tra te (e r =
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12.0) was found
to be between 7.54 a n d 9.SL.
Thus, fo r a d ie le c tric b rid g e w ith d ie le c tric co nsta nt
close to t r ef f - th e m ic r o s trip tra n s itio n fro m one s u b s tra te to a n o th e r re su lts in sub­
s ta n tia lly less re fle c tio n s co m p ared to th e a ir b rid g e . T h e dependence o f |5’n | on the
d ie le c tric co n sta n t o f th e b rid g e is illu s tra te d in F ig . 4.4 fo r d = 3.17a. As show n in
th is fig u re , a d ie le c tric b rid g e w ith er = 9.S p ro v id e s th e best m a lc h m nong a ll fo u r
cases fo r a n o rm a liz e d fre q u e n cy o f k0 ■a < 0.2. A t h ig h e r frequencies, th e d ie le c tric
b rid g e , w h ich acts lik e a resonator, causes th e m a g n itu d e o f 5 'n to ra p id ly increase.
T h is resonant fre q u e n cy depends on th e tra n sve rse d im e n sio n s o f the w aveguide: in
o th e r words, it is n o t s tro n g ly in flu en ce d b y th e le n g th o f th e d ie le c tric b rid g e . A n ­
o th e r ty p e o f resonance, w h ich corresponds to |6’n | = 0. is a re su lt o f th e d e s tru c tiv e
c a n c e lla tio n o f th e in c id e n t and reflected fie ld s.
T h is ty p e o f resonance is s tro n g ly
d e p e n d e n t on th e le n g th o f th e d ie le c tric b rid g e ra th e r th a n th e transverse d im e n ­
sion o f th e w aveguide.
I t is evident fro m F ig . 4.5 th a t by increasing th e le n g th o f
th e d ie le c tric b rid g e , th e second ty p e o f resonance (|.h’n | = 0) alw ays s h ifts to low er
frequencies. It is also in te re s tin g to p o in t o u t t h a t, in b o th figures, the fin ite elem ent
p re d ic tio n s are co m p a re d w ith d a ta e x tra c te d fro m [22]. T h e agreem ent betw een the
tw o m e th o d s is e x c e lle n t.
4.4.2
B on d -w ire in te rc o n n e c tio n s
B o n d w ires are fre q u e n tly used in e le c tro n ic p acka g in g to p ro v id e a p h ysica l con­
n e c tio n between m ic ro s trip lines th a t are n o t o th e rw is e connected.
T h e ra d iu s ol
these w ires is o n ly a s m a ll fra c tio n o f the m ic r o s tr ip w id th u \ ty p ic a lly ra n g in g be­
tw een w /1 0 and w /5 . T h e bond w ire can be th o u g h t o f as a tra n s m is s io n lin e w ith
a c h a ra c te ris tic im p e d a n ce b eing p ro p o rtio n a l to
/
= V 0 + j^'C
< *■ *)
w h e re R. G. L and C a rc th e per u n it le n g th series resistance, shunt co nd u cta n ce ,
series in d u c ta n c e and sh un t capacitance, re s p e c tiv e ly . For a lossless lin e . (4.22) can
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
So
Fig. 4.3: G eom etry o f m ic ro s trip tra n s itio n trough a dielectric bridge.
1.0
er=1.0
0.9
—
er=2.32
- - er=3.78
er=9.8
C h ris t 8c H a rtn a g e l
= 0.7
m
' o 0.6
o
0.5
fin d
be U 4
cd
'k
0.3
0.2
0.1
0.0
0.0
-0.05
0.1
0.15
k0a
0.2
0.25
0.3
0.35
Fig. 4.4: S’n o f a m icrostrip tra n s itio n through a dielectric bridge o f length d = 3.17n and
dielectric constant f r .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
st>
1.0
—
0.9
0.8
er= 1 . 0
- er=2.32
- - er=3.78
er=9.8
C h r is t & H a rtn a g e l
o 0.6
'a 0.5
0.2
0.1
0.0
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
k0 a
Fig. 4.5: 5’n o f a m ic ro s trip tra n s itio n tliro u gh a d ie le ctric bridge o f length d = 6.35a and
dielectric constant cr -
be s im p lifie d to
B y decreasing th e w id th o f a m ic r o s trip lin e , th e p e r u n it le ngth series in d u c ta n c e L
increases th e re b y in c re a s in g th e c o rre sp o n d in g c h a ra c te ris tic im pedance. T h u s , it we
th in k o f a bon d w ire as b e in g s im ila r to a ve ry t h in m ic ro s trip lin e , its c h a ra c te ris tic
im p e da n ce sh o u ld be s ig n ific a n tly la rg er th a n th a t o f a w id e r m ic ro s trip lin e . As a
re su lt o f such m is m a tc h , s ig n ific a n t re fle ctio n s m ig h t be caused by th e presence ol
th e w ire bon d .
T h e g e o m e try sh ow n in F ig . 4.6. w h ich was p re v io u s ly analyzed by C h ris t and
H a rtn a g e l [22]. re p re sen ts a w ire bon d betw een tw o m ic ro s trip s p rin te d on separate
su bstra te s. T h e s u b s tra te m a te ria l is G aA s w ith t r = 12.9. The s tru c tu re is placed
in sid e a w a ve gu id e w ith tra n sve rse d im e nsion s 5 a x 3 .2 a. The bond w ire is re c ta n ­
g u la r in .shape h a v in g a cross se ctio n a l area 0.2« x 0.2« and an in n e r le n g th </. f o r
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
81
the s im u la tio n p a rt, a ll c o n d u c tin g surfaces were co nsidered as being p erfect e le c tric
c o n d u c to rs . T h e n o rm a liz a tio n c o n s ta n t a was set to 1 m m . whereas th e n o rm a liz e d
fre q u e n cy range is g iven b y 0 < k0a < 0.3a. T h e m a g n itu d e o f the 5 -p a ra m e te rs o f
th is s tru c tu re is p lo tte d in F ig s. 4.7 and 4.S fo r th re e d iffe re n t lengths d. C o m p a r­
isons w it h d a ta e x tra c te d fro m [22] are shown o n ly fo r th e m a g n itu d e o f 5 U . From
F ig. 4.7. it is e v id e n t th a t th e re is an excellen t agreem ent between results o b ta in e d
based on th is w o rk and re s u lts o b ta in e d using the F D F D m e th o d [22]. A n increase
in the le n g th o f th e bond w ire re s u lts in a la rg er o v e ra ll 5 U . therefore a s m a lle r S-i\.
A t h ig h e r frequencies, th e s tru c tu re w ill e v e n tu a lly resonate thu s forcin g 5'n to zero.
H ow ever, co m p a re d to th e m ic r o s tr ip in te rc o n n e c tio n th ro u g h an a ir b rid g e , w h ich
was presented in th e p re viou s s e c tio n , the bond w ire causes higher re fle ctio n s . For
e xa m p le , a t a frequency o f 4 ClPfz. a bond w ire w ith le n g th 3.17 m m and height
0.2 m m causes m o re th a n o0% re fle c tio n s . On th e c o n tra ry , th e m ic ro s trip in te rc o n ­
n e ctio n th ro u g h an a ir b rid g e , a t th e same frequency, causes o n ly 2o9? re fle ctio n s.
T h e re fle c tio n s caused by th e w ire b on d can be s ig n ific a n tly reduced if the bond wire'
its e lf is e n ca p su la te d w ith a d ie le c tric m a te ria l o f c e rta in d ie le c tric co nsta nt.
Besides m a g n itu d e , phase is also an im p o rta n t q u a n tity to consider when c a lc u la t­
ing th e 5 -p a ra m e te rs o f m ic ro w a v e c irc u its and packages. T h e phase is an in d ic a tio n
o f the tim e d elay it takes fo r th e sig n al to propagate th ro u g h the c irc u it. In m a n y
a p p lic a tio n s it is c ru c ia l th a t tim e delays are m in im iz e d o r a t least a c c u ra te ly pre­
d ic te d .
L in e a riz a tio n o f phase is u s u a lly one o f the m a in o b je ctive s in th e design
o f packages and m u lti-c h ip m o d u le s. Fig. 4.9 illu s tra te s th e phase o f th e b o n d -w ire
s tru c tu re fo r d iffe re n t values o f d. As shown in the fig u re , th e negative slope o f the
phase begins to increase w ith in c re a s in g d th e re b y re s u ltin g in a larger tim e delay.
It is also c le a r th a t fo r re la tiv e ly la rg e values o f d th e phase becomes n o n -lin e a r as a
fu n c tio n o f frequency.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
88
0.2a
0.2a ^
0.8a *
0.2a ^
4.8a
0.2a
0.2a
4.8a
Fig. -LG: Bond-wire interconnection of two m icrostrip lines.
1.0
*■
0.9
0.8
0.7
o 0.6
'a 0.5
03
0-4
S 0.3
_wl d=3.17a
.. _ d=6.35a
. . . d=12.7a
Christ & Hartnagel
0.2
0.1
0.0
0.0
0.05
0.1
0.15
kQa
0.2
0.25
Fig. -1.7: .6’u of a bond-wire interconnection of two m icrostrip lines.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.3
1.0
0.9
0.8
o 0.6
a>
" I 0.5
•
C n i
cc
'k
0.3
0.2
d=3.17a
d=6.35a
d=12.7a
0.0
0.05
0.1
0.15
koa
0.2
0.25
0.3
Fig. -1.8: 5 2i o f a bond-w ire interconnection o f two m icro strip lines.
200
160
120
40
?*
0
CO
'o
-40
-120
-160
-200
0.0
S21 (d=3.17a)
S.21 (d=6.35a)
S2l (d=12.7a)
Slt (d=3.17a)
Sn (d=6.35a)
S., (d=12.7a)
0.05
0.1
0.15
k0a
0.2
0.25
0.3
Fig. 4.9: Phase o f .S'n and S->i o f a bon d -w ire interconnection o f t\m m icro strip lines.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
90
4.4.3
M ic r o s trip p a tch a n te n n a
A n o th e r ty p e o f s tru c tu re considered in th is d is s e rta tio n , p rim a rily fo r v e rific a tio n
purposes, is th e re c ta n g u la r m ic ro s trip p a tc h a n te n n ^ cl'o \v n in Fig. 4.10. T h e p a tch
a n te n n a is p rin te d on a R T /D u ro id s u b s tra te w it h t r = 2.2 and height 0.794 m m . A
5 0 -fl m ic ro s trip lin e is used to e xcite th e p a tc h . T h e sam e g eo m e try was a na lyze d
in th e past by Sheen et al. [25] using th e F D T D m e th o d . In o rd e r to p ro v id e a fa ir
c o m p a riso n to th e FE.M . a s im ila r n u m e ric a l a p p ro a c h was im p le m e n te d d u rin g the
course o f th is s tu d y . T h e same mesh sizes suggested in [25] were also used here: i.e..
A .r = 0.3S9 m m . A y = 0.4 m m . A c = 0.265 m m and A t = 0.441 ps. These mesh
sizes re s u lt in an in te g ra l n u m b e r o f cells a lo n g th e w id th and le n g th o f th e p a tc h , b u t
not a lo ng th e w id th o f th e m ic ro s trip lin e fe e d in g th e p a tch . T h e re s u ltin g F D T D
mesh d im e n sio n s a re 61 x 100 x 17 cells.
T h e re tu rn loss (R L ) o b ta in e d using th e F E M a n d th e F D T D m e th o d is show n
in F ig . 4.11. A f a ir ly good agreem ent b etw een th e tw o m e th o d s is illu s tra te d .
For
frequencies low er th a n 10 G H z . where th e mesh d e n s ity is s u ffic ie n tly fine, th e agree­
m e n t betw een th e tw o n u m e ric a l techniques is e x c e lle n t. T w o d iffe re n t fin ite e le m e n t
d is c re tiz a tio n s w ere considered:
2S.8S3 te tra h e d ra s .
one w ith 2 2 .7 0 2 te lra h e d ra s and th e o th e r w ith
H ow ever, as shown in F ig . 4.11. o n ly a m in o r im p ro v e m e n t is
observed in th e p re d ic tio n s when using th e fin est d is c re tiz a tio n . A possible source
o f e rro r in th e c a lc u la tio n s is th e in a b ility o f th e F D T D m e th o d to p ro p e rly m a tc h
a ll m ic ro s trip su rface d im e nsion s. A n a rro w e r m ic r o s trip lin e , for e xa m p le , a lw ays
results in a la rg e r c h a ra c te ris tic im p e da n ce, th e re b y a ffe c tin g the re tu rn loss o f the
s tru c tu re , e s p e c ia lly at th e h ig h e r frequencies. O n th e o th e r hand, using th e F E M .
a ll g e o m e try d im e n s io n s are precisely m o d e le d .
In o rd e r to shecl in s ig h t in to th e c o m p u ta tio n a l e ffo rt re q u ire d by th e F E M . th e
fo llo w in g s ta tis tic s are re p o rte d . T h e o rig in a l m esh consisted o f 22.702 te tra h e d ra s
and a to ta l o f 2 5 .6 2 5 u nkn o w n s.
T h e c o m p u ta tio n a l tim e was a p p ro x im a te ly 30
m in u te s per fre q u e n c y p o in t in the low er fre q u e n c y range and 15 m in u te s p e r fre-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Fig. 4.10: Geometry o f a rectangular m icrostrip patch antenna on a RT/Duroid substrate
with cr = 2.2.
-10
3 -15 ;
CO -20 ;
a; -25 |
'O
3
I
-30 i
&£ o c
as -3 5
-40
-45 i
_ FEM (22,702)
_ FEM (28,883)
_ FDTD
Sheen’s results
-50
Frequency (GHz)
Fig. 1.11: Return loss of a rectangular m icrostrip patch antenna.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
cjuency p o in t in th e u p p e r fre q u e n c y range. T h is pro ble m was solved on a 370 IB M
R IS C /6 0 0 0 I ’ X I X w o rk s ta tio n .
T h e s o lu tio n toleran ce based on the re s id u a l n o rm
was set to 10- 6 . T h e re co rd e d c o m p u ta tio n a l tim e also accounts fo r th e c e n tra l p ro ­
cessing u n it (C P U ) tim e needed to e v a lu a te th e m o d a l field d is tr ib u tio n a t th e in p u t
p o rt.
O n th e o th e r hand, th e F D T D code to o k a p p ro x im a te ly 4o m in u te s fo r th e
o v e ra ll s im u la tio n : a to ta l o f S. 192 tim e steps were allowed fo r th e pulse to p ro p a g a te .
T h e s im u la tio n was done on a S ilic o n G ra p h ic s Power Ind ig o 2 w o rk s ta tio n w it h an
RSOOO processor. X o te th a t th e la t te r is a s ig n ific a n tly faster c o m p u te r th a n th e 370
IB M R IS C /6 0 0 0 .
4.4.4
M icro w a ve low-pass f ilt e r
T h e c irc u it a na lyzed in th is s e c tio n was also e x tra c te d fro m th e paper by Sheen f t
al. [2o]. T h is is th e low-pass f ilt e r illu s tr a te d in Fig. 4.12. T h e p e rfe c tly c o n d u c tin g
m ic ro s trip surfaces are p r in te d on a R T /D u r o id su bstra te w ith er = 2.2 a n d h e ig h t
0.794 m m . T h is g e o m e try was s im u la te d u sing b o th the F E M and th e F D T D codes
fo r a frequency range o f 20 G H z .
T h e m a g n itu d e o f S'n and Sbi versus fre q u e n c y
is illu s tra te d in Figs. 4.13 a n d 4.14. re s p e c tiv e ly , whereas th e c o rre s p o n d in g phases
are illu s tra te d in Figs. 4 .1 ”) a nd 4.16. T h e phase o f s’n was e va lu a te d at a d is ta n c e
4.233 m m aw ay fro m th e d is c o n tin u ity : th e phase o f
on th e o th e r h a n d , was
evaluated 3.3864 m m aw ay fro m th e d is c o n tin u ity . A ll fo u r figures show an e x c e lle n t
agreem ent between th e tw o m e th o d s . T h e fin ite elem ent mesh consisted o f 2 8.914
te tra h e d ra s and a to ta l o f 3 3 .0 3 2 u n k n o w n fie ld com ponents.
T h e c o rre s p o n d in g
C P U tim e fo r th is p ro b le m was a p p ro x im a te ly 20 m in u tes p e r fre q u e n cy p o in t in th e
low er frequency range and 10 m in u te s per fre qu e ncy p o in t in th e u p p e r fre q u e n c y
range.
X o te th a t a lth o u g h th e low -pass filte r is c o m p u ta tio n a lly a la rg e r p ro b le m
th a n th e m ic ro s trip patch a n te n n a , th e re q u ire d C P U tim e is s ig n ific a n tly less. T h e
reason is re la te d to th e c o n d itio n n u m b e r o f th e re su ltin g m a tr ix syste m .
A s far
as the F D T D m e th o d is co n c e rn e d , th e mesh dim ensions were th e fo llo w in g : A .r =
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
93
0.4064 m m . A y = 0.4233 m m . A c = 0.265 m m a nd A t = 0.441 ps. T h e o v e ra ll mesh
size was S i x 101 x IT cells. T h e re q u ire d c o m p u ta tio n a l tim e was a p p ro x im a te ly 50
m in u te s : a g a in , a to ta l o f S. 192 tim e steps w e re a llo w e d fo r the pulse to p ro p a g a te .
T h e s im u la tio n was ru n on a S ilic o n G ra p h ic s P ow er In d ig o 2 w o rk s ta tio n w ith an
RS000 processor.
20.32 mm
2.54 mm
0.794 mm
2.413 mm
Fig. 4.12: Geometry o f a low-pass filte r on a R T /D u ro id substrate w ith er = 2.2.
4.4.3
S in g le -lo o p in d u c to rs
L u m p e d e le m en ts are s y s te m a tic a lly used in th e design and m a n u fa c tu rin g of RL
and m ic ro w a v e c irc u its . These in c lu d e s tr a ig h t- lin e in d u c to rs , sin g le -lo o p and s p ira l
in d u c to rs , c h ip and in te r d ig ita te d ca p a c ito rs as w e ll as resistors. T h e ir d im e n s io n s are
u s u a lly m u c h sm a lle r th a n th e w a v e le n g th , t y p ic a lly / < A / 10 to a vo id u n d e s ira b le
effects such as p a ra s itic c a p a cita n ce a n d /o r in d u c ta n c e , resonances, fr in g in g fields
and o h m ic losses.
S m a ll values o f in d u c ta n c e can be re a lize d u sin g a sh o rt length o f s tra ig h t tra n s ­
m issio n lin e o r a sin g le-lo op in d u c to r , as sh ow n in F ig . 4.17. H a vin g a lo o p in d u c to r
o f le n g th I. an e s tim a te o f th e in d u c ta n c e can be o b ta in e d using th e tra n s m is s io n
lin e th e o ry w h ich states th a t th e c h a ra c te ris tic im p e d a n c e and p ro p a g a tio n c o n sta n t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Magnitude of S n (dB)
94
-10
-15
-20
-25
-30
-35
-40
-45
_ FEM j
- FDTD |
-50
14
20
Frequency (GHz)
Fig. 1.13: Return loss o f a low-pass filte r.
Magnitude of S2i (dB)
-10
-25
-30
-40
-45
_ FEM ;
- FDTD |
-50
20
Frequency (GHz)
Fig. 1.14: Insertion loss o f a low-pass filte r.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
180
FEM
FDTD
120
cn
01
0>
(h
he
0)
~a
co
<M
o
0
CO
CO
<-<
-120
-180
0
2
4
6
8
10
12
14
16
18
20
16
18
20
Frequency (GHz)
Fig. 4.15: Phase o f 5 U for a low-pass filte r.
180
_ FEM
- FDTD
120
-120
-180
0
2
4
6
8
10
12
14
Frequency (GHz)
Fig. 4.16: Phase o f S>\ for a low-pass filte r.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
96
o f a tra n sm issio n lin e are g iv e n , re sp e ctive ly, by
Z, =
3 =
{ I
(4.24)
*y/LC.
(4.25)
C o m b in in g the tw o e xpressions, it can be show n th a t
= 3 Z C => L = v ^ 7 7
w h e re v0 is the speed o f lig h t in free space.
(4.26)
For h ig h in d u c ta n c e , th e c h a ra c te ris ­
t ic
im p e d a n ce o f th e tra n s m is s io n lin e used to m a ke up th e
be
chosen large.
in d u c to r sh o u ld alw ays
As an e x a m p le o f show ing h ow these e qu a tion s
are
used, c o n ­
s id e r a single-loop in d u c to r d e fin e d b y an angle o = 280° and an average ra d iu s o f
R rlv = (R, + R0) / 2 = 1.75 m m . T h e in d u c to r is p rin te d on a su b stra te o f t r = 2.2
w ith an effective d ie le c tric c o n s ta n t o f a p p ro x im a te ly 1.8 a t low er frequencies. T h e
c h a ra c te ris tic im pedance o f th e m ic ro s trip lin e used to m ake up th e in d u c to r is 110 Q.
A c c o rd in g to (1.26). th e p e r -u n it-le n g th in d u c ta n c e o f th e lo o p is given by
/. = \ / l .8 ( — —— 77') = 0.192 n H /m m .
V3 x 10“ /
(4.27)
T h e le n g th o f the loop is c a lc u la te d using
1=
(iiS o) =
‘ •7 5 )
(
I
f
f
)=
s -3 6 n m '-
'
T h u s , th e q u a si-sta tic in d u c ta n c e o f th e sin g le -lo o p in d u c to r is a p p ro x im a te ly equal
to I ■ L = 8.56 • 0.-192 Ci -1.2 n i l . T h is value w ill be co m p a re d la te r to the in d u c ta n c e
va lu e o b ta in e d using a fu ll-w a v e s im u la tio n based on th e fin ite elem ent m e th o d .
T h e 5 -p a ra m e te rs o f a s in g le -lo o p series in d u c to r, show n in Fig. -1.17. are e x a m ­
in e d fo r d iffe re n t angles o . A s e xp e cte d , a c c o rd in g to th e tra n sm issio n lin e th e o ry ,
th e la rg e r the angle o. th e h ig h e r th e in d u c ta n c e o f th e lo o p : th e re fo re , th e la rg e r
th e slope o f |>’n |. T h is o b s e rv a tio n is d e p icte d in F ig . 1.18 w here th e m a g n itu d e o f
R eproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
!)7
♦
Fig. 4 .1 7 : Single-loop series in d u c to r p rin ted on a R T /D u ro id substrate w ith dielectric
constant er = '2.2 and height h = 0 .7 9 4 m m ( it = '2.4 m m . R, = t.o m m .
R0 = 2.0 m m ).
th e .S’-p a ra m e te rs is p lo tte d versus fre q u e n c y fo r tw o d iffe re n t values o f o . T h e c o r­
re s p o n d in g phase is show n in F ig . 4.19. B y in cre a sin g o. not o n ly th e slope o f |.S'n|
increases, w h ich is an in d ic a tio n o f an in creased in d u c ta n c e , b u t also th e slope ot th e
phase since th e tra n s m is s io n lin e p a th is n ow e le c tric a lly la rg er. T h e .S’-p a ra m e te rs
o b ta in e d fro m th e fu ll-w a v e a n a lysis were c u rv e -fitte d based on a s im p le e q u iv a le n t
c ir c u it m ade o f an in d u c to r in series w ith a c h a ra c te ris tic im p e d a n ce o f oO H .
The
o b ta in e d values fo r th e e ffe c tiv e lo o p in d u c ta n c e are L = I.So n H and L = 2.0o n il
fo r o = 240° and o = 2S0°. re s p e c tiv e ly .
T h e 5 -p a ra m e te rs p lo tte d based on th e
fu ll-w a v e fo rm u la tio n a n d th e e q u iv a le n t c ir c u it are show n in Fig. 4.20 fo r b o th cases.
For frequencies up to o G H z th e a g re e m e n t betw een th e tw o approaches is e x c e lle n t:
how eve r, as th e fre q u e n c y increases th e re e x is t a d d itio n a l p a ra s itic effects t h a t need
to be in c o rp o ra te d in to th e e q u iv a le n t c ir c u it to p ro v id e a good co m p a ris o n .
T h e s in g le -lo o p in d u c to r was p rin te d on a R T /D u r o id s u b s tra te w ith cr = 2.2 and
lin e w id th O.o m m . w h ic h corre sp on d s to a c h a ra c te ris tic im p e d a n ce o f a p p ro x im a te ly
1 10 O
T h e tra n s m is s io n lin e w id th at .h e in p u t and o u tp u t p o rts is 2.4 m m . w h ic h
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
98
is e q u iv a le n t to a c h a ra c te ris tic im pedance o f a p p ro x im a te ly 50 Q.
T h e height ot
the s u b s tra te is 0.794 m m . T h e n u m e rica l s im u la tio n was p e rfo rm e d using o n ly the
F E M . th e re fo re no co m p a ris o n d a ta is p ro v id e d .
X o w . b y c o m p a rin g th e in d u cta n ce o b ta in e d fro m th e e q u iv a le n t c irc u it when o =
280° w ith th e c o rre s p o n d in g value o b ta in e d using th e tra n s m is s io n lin e e q u a tio n s, it
is clear th a t th e la tte r ( L = 4.2 nPI) is tw ic e as large as th e fo rm e r ( L — 2.05 n H ). It
was n o te d in [173] th a t i f th e c h a ra c te ris tic im p e d a n ce o f th e tra n s m is s io n lin e used
to m ake u p th e in d u c to r is Z c. and the c h a ra c te ris tic im p e d a n c e o f th e tra n sm issio n
lin e at th e p o rts is Z 0. th e n th e effective in d u c ta n c e o f th e lo o p is reduced by a fa c to r
o f 1 — ( Z 0/ Z c )2 w h ic h in o u r case is equal to 0.79. In a d d itio n , th e tra n s m is s io n lin e
m odel does n o t ta k e in to accou n t any p a ra s itic effects lik e frin g in g fie ld s, d is trib u te r!
c a p a cita n ce and so on.
T h e exclusion o f these te rm s re s u lts in o v e re s tim a tin g the
e ffe ctive in d u c ta n c e o f th e loop.
1.0
0.9 p
21
o 0.8
2 0.6
0.5
S 0.4
0.1
0.0
Frequency (GHz)
Fig. -1.18: M a g n itu d e o f th e .S’-param eters o f a singledoop series in d u c to r as a function of
angle o.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
!)!)
180
150
120
90
|
60
2
30
cB
9"
m
o
*s
-30
£
-60
eB
£
21
-90
-120
-150
-180
0
1
3
2
4
5
6
7
9
8
10
Frequency (GHz)
F ig . 4.19: Phase o f the S -p a ra m e te rs o f a single-loop series in d u c to r as a function o f angle
O.
0.6
o 0.4
0.2
0.1
- 0=240
- 0=280°
L=1.85nH
L=2.05nH
1
2
3
4
5
Frequency (GHz)
F ig . 4.20: C om parison o f |.S’n | versus frequency calculated in d e p e n d e n tly using a full-w ave
sim u latio n and a lu m p ed equivalent circuit o f a series in d u c to r.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
100
4.4.6
S p ira l in d u c to rs
S in gle -loo p in d u c to rs are o fte n used in m ic ro w a v e c irc u it design when a s m a ll value
o f in d u cta n ce (u p to 2-3 n H ) is needed.
T h e re are num erou s design a p p lic a tio n s ,
how ever, w here a la rg e r value o f in d u c ta n c e (u p to 10-15 n H ) is desired. In such cases
m u ltip le tu r n s p ira l in d u c to rs are used in ste a d . T h e ir e q u iv a le n t in d u c ta n c e increases
w ith increasing th e n u m b e r o f tu rn s w h ic h also tra n sla te s to a la rg e r o h m ic loss and
d is trib u te d sh u n t ca pa cita nce .
T h e c o m b in a tio n o f a la rg e series in d u c ta n c e and
sh u n t capacitance leads to resonances w h ic h lim it th e m a x im u m o p e ra tin g fre qu e ncy.
A lth o u g h s p ira l in d u c to rs , as w e ll as in d u c to rs in general, are im p o rta n t e le m e n ts
in m icrow ave design, m ost o f th e re le v a n t a na lysis has been done using e ith e r q u a s i­
s ta tic m ethods [174] o r re c ta n g u la r-g rid m e th o d s such as th e F D T D . th e s p e c tra l
d o m a in approach (S D A ). th e tra n s m is s io n lin e m e th o d ( T L M ) and th e m e th o d o f
lines (M o L ) [29].
Q u a s i-s ta tic m e th o d s are k n o w n to be v a lid o n ly fo r frequencies
at w hich th e le n g th o f th e in d u c to r is a s m a ll fra c tio n o f th e o p e ra tin g w a v e le n g th ,
whereas th e accuracy o f re c ta n g u la r-g rid m e th o d s becomes q u e s tio n a b le w hen th e
s tru c tu re e x h ib its cu rve d m e ta fiz a tio n s . f sin g th e F E M on th e o th e r h an d , g e o m e tri­
cal and m a te ria l c o m p le x itie s can be h an d le d e ffe c tiv e ly w ith o u t s a c rific in g a ccuracy
o r c o m p u ta tio n a l resources.
A c irc u la r s p ira l in d u c to r c o n n e c te d in series w ith a m ic r o s trip lin e on an A lu m in a
su b stra te w ith er = 9.S is show n in F ig . 4.21. O ne end o f th e s p ira l is b on d ed w ith
th e m ic ro s trip lin e at p o rt 2 th ro u g h a c y lin d ric a l m e ta llic b rid g e .
T h e g e o m e try
o f th e b rid g e is defined by th re e p o in ts : one a t the ce n te r o f th e s p ira l, th e o th e r
one a t the edge o f the m ic r o s trip a n d th e th ir d one in th e m id d le o f th e gap (h e ig h t
o f 1.0 m m ). T h e s p ira l is m ade o u t o f 17 tu rn s w ith s tr ip w id th o f 0.2 m m . T h e
m ic ro s trip lin e at th e in p u t and o u tp u t p o rts is 0.635 m m w id e , and th e s u b s tra te
h e ig h t is also 0.635 m m . T h e m a g n itu d e o f .S’n and S>2 is illu s tra te d in F ig . 4.22
whereas th e m a g n itu d e o f .SY.> and .S’j i is illu s tra te d in F ig . 4.23.
A lth o u g h m e a ­
surem ents were not a va ila b le fo r d a ta c o m p a ris o n , the mesh d e n s ity was s u ffic ie n tly
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
101
fin e to ensure a c c u ra te s im u la tio n s . S p e c ific a lly , th e mesh co nsiste d o f 2 6.0 9 6 ele­
m e n ts and a to ta l o f 3 1 .0 9 3 u n kn o w n s. T h e c o rre s p o n d in g c o m p u ta tio n a l tim e was
a p p ro x im a te ly one h o u r p e r fre q u e n cy p o in t in th e low er range o f frequencies, and
20 m in u te s per fre q u e n c y p o in t in th e in te rm e d ia te -to -u p p e r range o f frequencies:
a g a in , a 370 IB M R IS C /6 0 0 0 w o rk s ta tio n was used to p e rfo rm th is s im u la tio n . M ost
o f th e c o m p u ta tio n a l tim e (9 0 f/c) was sp en t on s o lv in g th e lin e a r s y s te m o f e q u a tio n s.
C o m p a rin g th e m a g n itu d e s o f 5’n w ith S 2 2 an<J
$12
w ith .S'21. it is c le a r th a t those are
n o t id e n tic a l, a lth o u g h s im ila r . T h e m in o r d iffe re n ces are a ttr ib u te d to th e presence
o f th e m e ta llic b rid g e w h ic h m akes th e s tru c tu re n o n -re c ip ro c a l. It is also im p o rta n t
to re m in d the re a de r t h a t th e s p ira l in d u c to r acts lik e a lu m p e d ele m en t o n ly in
th e lo w e r range o f fre q u e n cie s, whereas a t h ig h e r frequencies the s tru c tu re begins to
resonate due to a d d itiv e c a p a c itiv e effects.
w.
Port #2
Port #1
w,
W
Fig. 4.21: Series spiral in d u c to r printed on an A lu m in a (er = 9.8) substrate o f height 0.635
mm. The end o f the spiral is bonded w ith a m icrostrip line through a m etallic
bridge o f height 1.0 m m . The bridge has an arc shape defined by three points
(«•, = 0.635 m m . ir> = «•:} = 0.2 m m . it\t = 2.3 mm. R\ = 1.9 m m . /?> = 1.3
mm.
= 0.7 m m ).
A s p ira l in d u c to r is u s u a lly co n n e cte d e ith e r in series o r in p a ra lle l.
T h e same
c o n fig u ra tio n as th e one used in th e p re v io u s e x a m p le is now co n n e cte d in sh u n t w ith
a m ic ro s trip lin e p rin te d 011 an A lu m in a s u b s tra te .
T h e g e o m e try and d im e n sio n s
o f th e s tru c tu re are sh ow n in F ig . 1.2 1. T h e c e n te r o f the s p ira l is g ro u n d e d using
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
102
1.0
0.9
a> 0.8
£ 0.7
5 0.6
m
0.5
-g 0.4
•S 0.3
be
J2 0.2
0.1
0.0
Frequency (GHz)
Fig. 4.22: Magnitude o f 5 I( and
S>>
for a series spiral inductor with a bond-wire bridge.
1.0
0.9
O
0.8
£ 0.7
S0.6
0.5
e 03
be
JS 0.2
|S2l|
0.1
0.0
Frequency (GHz)
Fig. 4.23; Magnitude o f .S’i^ and
S ji
for a series spiral inductor w ith a bond-wire bridge.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
103
a p la n a r c o n d u c tin g v ia . T h e m a g n itu d e p lo ts o f 5 n and S ji ca lcu la te d u sin g th e
F E M code are shown in F ig . 4.25.
A lth o u g h co m p a riso n s are not a va ila b le , it is
in te re s tin g to observe th a t at lo w e r frequencies th e s tru c tu re indeed behaves as a
lu m p e d in d u c to r co n n e cte d in s h u n t.
Such s tru c tu re th o u g h is h ig h ly re s o n a n t,
th e re fo re m u ltip le peaks a n d n u lls a p p e a r in th e h ig h e r fre qu e ncy range. T h e re fo re ,
th e re s u ltin g ^ '-p a ra m e te rs are p lo tte d o n ly up to 7 G H z . T h e fin ite e le m e n t mesh
fo r th is p ro b le m consisted o f 43.5SS te tra h e d ra s and a to ta l o f 51.270 u n kn o w n s .
Ground
Connection
In p u t P o rt
w
4
Fig. 4.21: Spiral in d u cto r connected in shunt w ith a m icro strip line printed on an A ltu n in a
(cr : 9.S) substrate o f height 0.035 turn ( »’i = 0.1535 mm . tc-j = 0.2 mm. t/\i = 0.0
mm. R i = 1.9 m m . R> = 1.3 m m . R^ = 0.7 nun).
A lth o u g h c irc u la r s p ira l in d u c to rs are c o m m o n ly fo u n d in M .M IC 's. re c ta n g u la r
s p ira l in d u c to rs are u s u a lly th e m ost p o p u la r. A ty p ic a l 1^ tu r n re c ta n g u la r s p ira l in ­
d u c to r w ith an a ir b rid g e is show n in F ig . 4.26. T h is s tru c tu re was in it ia lly a n a lyze d
by Becks f t al. [16] using th e s p e c tra l d o m a in approach a n d la te r by o th e rs u s in g th e
F D T D /m a tr ix - p e n c il (M P ) m e th o d [175] as w ell as in te g ra l m ethods [176].[177]. In
th is stu d y, th e s p ira l in d u c to r was d iscre tize d using te tra h e d ra l elem ents a n d s im ­
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
10-1
1.0
0.9
a> 0.8
a>
£ 0.7
S 0.6
CO
0.5
2 0.4
•5 0.3
2 0.2
0.1
0.0
0
1
2
3
4
5
6
7
Frequency (GHz)
Fig. 4.25: Magnitude o f 6'n and 5 ji for a spiral inductor connected in shunt across a
microstrip line.
u la te d using th e fin ite e le m en t m e th o d . Because o f the sharp co rn ers o f th e s p ira l,
i t is im p o rta n t th a t th e c o n d u c tin g surfaces be ade q ua tely d is c re tiz e d . S p e c ific a lly ,
th e m a x im u m size o f th e elem ent in s id e th e s u b s tra te was 0.15 m m whereas in sid e
th e a ir was 0.4 m m .
F irs t-o rd e r a b s o rb in g b o u n d a ry c o n d itio n s were used to te r ­
m in a te th e u n b o u n d e d space.
A lth o u g h in th e o rig in a l p aper b y Becks el al.
a ll
p la n a r s trip s were s im u la te d w ith o u t ta k in g in to co nsid e ra tio n th ic k n e s s , in o u r s im ­
u la tio n a ll c o n d u c tin g surfaces, in c lu d in g th e b o n d w ire, had th ic k n e s s o f 0.1 m m .
T h e m a g n itu d e o f S'n and $>i are sh ow n, re sp e ctive ly, in Figs. 4.27 and 4.28. T h e
fin ite ele m en t p re d ic tio n s are co m p a re d w ith th e measured d a ta e x tra c te d fro m [ I 6 j.
T h e agreem ent betw een the tw o d a ta sets is fa ir ly good.
P ossible sources o f e rro r
in c lu d e in s u ffic ie n t d is c re tiz a tio n and to le ra n ce s in the e x p e rim e n ts . T h e c o m p a ri­
son betw een o th e r n u m e ric a l tech n iq ue s and th e m easurem ents was also som ew hat
u n s a tis fa c to ry .
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
10-')
l b
b
Fig. 4.20: Layout o f a rectangular spiral inductor on an A lum ina substrate w ith cr =: 9.8.
d = 0.635 mm. w = 0.625 mm. .s = b = 0.3125 mm. h = 0.3175 mm. t = 0.1 m m .
1.0
FEM
M easurem ents
0.8
CO
o 0.6
S i 0.4
0.2
0.0
0
2
4
6
8
10
12
14
16
18
20
Frequency (GHz)
Fig. 1.27: M agnitude o f .S'n for a rectangular spiral inductor with an air bridge.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
106
1.0
FEM
M ea surem e nts ‘
0.8
o 0.6
£ ,0 .4
0.2
0.0
Frequency(GHz)
Fig. 4.28: M a g nitu d e o f .S'2i for a rectangular spiral in d u c to r w ith an a ir bridge.
1.4.7
D o u b lc -v ia m ic r o s trip tra n s itio n package
Besides p la n a r m ic ro w a v e c irc u its , a n u m b e r o f g e o m e tric a lly c o m p le x packages and
in te rc o n n e c ts were also analyzed.
T h e tw o -la y e r, d o u b le -v ia m ic ro s trip tra n s itio n
package show n in F ig . 4.29 is a re p re s e n ta tiv e s tru c tu re o f p ra c tic a l designs.
The
b o tto m la y e r is a d ie le c tric su b stra te w ith cr = 2.2 whereas th e to p la ye r is a n o th e r
ty p e o f s u b s tra te w ith er = 6.2. T h e tw o m ic ro s trip lines on th e m o th e rb o a rd , w hich
co rrespond to th e in p u t and o u tp u t p o rts , are connected to a m ic ro s trip lin e on the
to p la ye r th ro u g h v e rtic a l c o n d u c tin g vias. A flo a tin g g ro u n d , as shown in Fig. 4.29.
is placed on th e b o tto m face o f th e to p s u b s tra te , thus p ro v id in g p o te n tia l g ro u n d in g
fo r th e c ir c u it. T h e effectiveness o f the flo a tin g g ro u n d is in v e s tig a te d by s im u la tin g
th is package u sin g th e fin ite elem ent m e th o d as presented in th is ch a p te r.
d iffe re n t g ro u n d in g c o n d itio n s were considered:
a)
th e to p -la y e r g ro u n d plane was le ft flo a tin g
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Two
107
b)
th e to p -la y e r g ro u n d plane was sh o rte d to the m o th e rb o a rd u sin g fo u r c o n d u c t­
in g vias as show n in F ig . 4.30
T h e m a g n itu d e o f th e ^'-p a ra m e te rs fo r th e tw o cases a) and b) is illu s tra te d w ith o u t
co m p a riso n in F ig . 4.31. It is e v id e n t th a t w hen the to p -la y e r g ro u n d p la ne (p a d d le )
is s h o rte d to th e m o th e rb o a rd , th u s p ro v id in g a co m m o n g ro u n d , th e 5 - param eters
o f th e package s ig n ific a n tly im p ro v e . S pe cifica lly, th e |5 n | decreases and the |.Fj i |
increases, th e re fo re s h iftin g th e m a x im u m o p e ra tin g fre q u e n c y o f th e package to a
h ig h e r value. T h e reason for such an im p ro ve m e n t in th e 5’-p a ra m e te rs is tw o fo ld .
F irs t, th e cu rre n ts on the p a d d le have now a d ire ct p a th th ro u g h th e v e rtic a l vias
to th e m o th e rb o a rd g ro un d : th u s , w e akening p a ra sitic loads. Second, h ig h e r-o rd e r
modes betw een th e paddle and th e m o th e rb o a rd g ro u n d are e lim in a te d .
4.4.8
T h e S O IC -S p la s tic package
In p re v io u s sections, em phasis was pla ced m a in ly on th e a c c u ra te c h a ra c te riz a tio n
o f m o n o lith ic m ic ro w a v e /m illim e te r w ave in te g ra te d c irc u its ( M M I C ’ s) and in te rc o n ­
nects u sin g a fu ll-w a v e fin ite e le m e n t approach. In p ra c tic e , how ever, th e e le c tric a l
p e rfo rm a n c e o f the M M IC is d ra s tic a lly a lte re d when m o u n te d in to a package such as
the e ig h t-le a d s m a ll o u tlin e in te g ra te d c irc u it (S O IC -S ) s u rfa c c -m o u n t p la s tic pack­
age show n in Fig. 4.32. T h e presence o f th e package its e lf, in a d d itio n to w ire bonds,
leads and g ro u n d in g vias. re su lts in n u m e ro u s p a ra sitic effects such as spurious res­
onances caused by a d d itiv e in d u c ta n c e s and capacitances.
These effects should be
a c c u ra te lv a ccounted fo r using a fu ll-w a v e analvsis to s im u la te th e M M IC w ith and
w ith o u t th e presence o f the package.
A lth o u g h m ost o f c ir c u it design is based on
e q u iv a le n t-c irc u it m odels using lu m p e d elem ents, such m e th o d o lo g y m ig h t produce
erroneous re su lts w hen the o p e ra tin g fre q u e n cy is re la tiv e ly h ig h . E q u iv a le n t c irc u its
u s u a lly fa il to a c c u ra te ly a ccount fo r a ll com plex in te ra c tio n s , e.g.. d ispersion and
cro ssta lk, w h ic h ta ke place betw een th e package and th e M M IC .
P la s tic packages, such as th e one d e p ic te d in Fig. 4.32. have been used for years
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ms
w
Port #1
i
Port #2
h,
I Front View
Top View
w,
w
Er = 6.2
t—
Side View
Fig. 4.29: Double-via tra n s itio n package w ith a flo a tin g ground. Geometry specifications:
/*i = O.S mm . /i> = 0.4 m m . /i.-i = 0.6 m m . tri = 2.4 mm . t±%> = 0.S mm .
i/*3 = 0.4 mm . ir.i = 5.2 m m . tr5 = 6.8 mm . w6 - 8.6 mm. t = 0.2 mm.
i
!
■
■
Port #1
• i
t t
^
t i
t t
Ground via
i Front View!
Top View]
| Side View|
Fig. 4.80: D ouble-via tra n s itio n package w ith a common ground.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Port #2
10!)
1.0
0.9
0.8
0.7
0.6
0.5
Q. 0.4
CO
0.3
0.2
0.1
|Su |
|S 2i |
|Su |
- I S2i [
(flo a tin g p a d d le )
(flo a tin g p a d d le )
(grou nd ed p a d d le )
(grou nd ed p ad dle)
0.0
24
Frequency (GHz)
Fig. -1.31: .S’-paraineters versus frequency for the double-via transition package.
up to a fre q u e n c y o f a b o u t 2.o G H z. I t becomes e x tre m e ly d iffic u lt to u tiliz e these
packages fo r h ig h e r frequencies m a in ly because o f p o o r g ro u n d in g o f the paddle,
w h ic h is c le a rly show n in Fig. 1.32. T h e effects o f g ro u n d in g on th e .S'-parameters
o f th e o v e ra ll package, enclosing th e M M IC . are in v e s tig a te d in th is section using
th e fin ite e le m e n t m e th o d . R e fe rrin g to Fig. -1.32. th e S O IC -S su rfa ce -m o u n t p la s tic
package consists o f e ig h t leads, fo u r in each side o f th e s tr u c tu r e . T w o o f these leads
are c o n n e cte d th ro u g h w ire b o n d in g to th e in p u t and o u tp u t p o rts o f the M M IC
w h ic h , as illu s tr a te d in Fig. 1.33. is s lig h tly e le vate d fro m th e p ad d le . Some o f the
leads are d ir e c tly co nn e cte d to th e p a d d le and g ro u n d e d to th e m o th e rb o a rd th ro u g h
a v e rtic a l c o n d u c tin g v ia . as shown in F ig . -1.3-1. T h e re m a in in g leads are grounded
to th e m o th e rb o a rd b u t not o th e rw ise co n n e cte d to th e p a d d le . D iffe re n t g ro u n d in g
c o n d itio n s are in v e s tig a te d . A p oo r g ro u n d fo r th e M M IC m ig h t be proven disastrous
because o f th e d e ve lo p m e n t o f stro n g c u rre n ts on th e su rfa ce o f th e paddle.
A lso,
h ig h e r-o rd e r m odes betw een the p a d d le and th e m o th e rb o a rd g ro u n d m ig h t begin to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
HU
fo rm a t h ig he r frequencies th u s re in fo rc in g adverse p a ra s itic effects th a t m ig h t lim it
the m a x im u m o p e ra tin g fre qu e ncy o f the package.
Encaptulant
MMIC Paddle
Lead
5
Fig. -1.32: Three-dimensional rendering o f a SOIC-S plastic package. Specifications: L 4.2 mm. IF = 2.4 nun. p = 1.27 mm . nq = 0.4 mm. The thickness o f all
conducting surfaces (paddle, leads, etc.) is 0.1 mm.
M M IC
Wp
Fig. 4.33: Two-dim ensional side view o f a SOIC'-S plastic package. Specifications: U'p =
4.4 mm. L c = 0.7 mm . L fJ = 0.3 m m . d\ = 0.535 mm. d2 = 0.1 mm . d:i =
0.635 mm. d.\ = 0.2 mm . d$ = 0.25 mm . The thickness o f all conducting surfaces
(paddle, leads, etc.) is 0.1 nun.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ill
Grounding
Via
Motherboard
Ground plane
Fig. 4.3-1: Detail view o f grounding the lead to the m otherboard.
T h e M M IC shown in F ig . 4.35 is analyzed firs t in th e absence o f th e package.
T h is c ir c u it represents a basic th ro u g h -c o n n e c tio n w ith a s tu b .
W hen placed in to
th e package, th e c irc u it its e lf is elevated 0.1 m m above th e g ro u n d and e m bedded
in to a d ie le c tric m e d iu m w ith d ie le c tric constant er = 4. T h e w id th o f th e m ic ro s trip
lin e is 0.3 m m whereas a ll re m a in in g dim ensions are show n in F ig . 4.3-5. T h e same
M M IC s tru c tu re , in c lu d in g th e S O IC -8 su rfa ce -m o u n t package, was ana lyze d by
.Jackson [ 178j using the m e th o d o f m om ents and la te r by R ig h i ct al. [3*2] u sin g th e
T L M . In b o th studies, a ll c o n d u c tin g surfaces, e.g.. p a d d le , leads and m ic ro s trip s .
were s im u la te d w ith o u t ta k in g in to co nside ra tion a n y c o n d u c to r thickness.
In th is
stu d y , how ever, a c o n d u c to r th ickne ss o f 0.1 m m was considered w hich is a m ore
re a lis tic c o n fig u ra tio n . T h e 5 -p a ra m e te rs o f the th ro u g h -c o n n e c tio n were c a lc u la te d
using th e v e c to r fin ite e le m e n t m e th o d described in d e ta il in th is c h a p te r.
The
m a g n itu d e o f 5’u and 55 1 is show n versus fre qu e ncy in F ig . 4.36. It is e v id e n t th a t
th e c irc u it resonates at 10 G H z w h ic h corresponds to th e fre q u e n cy w here th e s tu b
becomes a q u a rte r o f a w a ve le n g th long.
A c c o rd in g to Jackson [ITS ], th e c irc u it
resonates at 0.5 G H z whereas a cco rd in g to R ighi f t al. [32]. th e c irc u it resonates at
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
112
Port #1
3.3
0.535
0.6
P o rt # 2
0 .7
8 r= 4
r 0 .3
Fig. -I..35: Geom etry o f the unpackaged M M IC ': m icro strip -th ro u g h connection w ith a stub.
The thickness o f all conducting surfaces is 0.1 mm. A ll dimensions are in m il­
limeters.
9 G H z : b o th o f these re s u lts are close to o u r s im u la tio n .
T h e e le c tric a l p e rfo rm a n c e o f th e M M IC is t o t a lly a lte re d when m o u n te d in to th e
S O IC -S p la s tic package w h ic h is shown in F ig . 4.32. T h e in p u t p o rt o f th e c ir c u it is
c o n n e cte d th ro u g h a b o n d w ire o f re c ta n g u la r cross section (0.1 n u n x 0.1 m m ) to
lead 1. and the o u tp u t p o rt o f th e c irc u it is c o n n e c te d th ro u g h an id e n tic a l b o n d w ire
to lead 7. as shown in F ig . 4.37. T h e g e o m e try in th is figure was d ra w n a c c o rd in g
to scale. T h e h eigh t o f th e bond w ire , fro m th e to p surface o f th e lead o r p a d d le to
th e in n e r surface o f th e h o riz o n ta l w ire , is 0.3-5 m m whereas th e b o n d -w ire le n g th ,
as m easured fro m th e in n e r surfaces o f th e v e rtic a l vias. is 0.6 m m .
Few of these
d im e n s io n s are s lig h tly d iffe re n t fro m those used b y .Jackson, w h ic h is a ttr ib u t e d to
th e use o f a fin ite th ickn e ss fo r a ll c o n d u c tin g surfaces. S p e cifica lly. .Jackson used a
b o n d w ire o f h eigh
O t 0.25 m m instead o f 0.35 m m . These are co n sid e re d o n lv
*■ m in o r
d iscre p a n cie s between th e tw o m odels. T h e re m a in in g g e o m e try s p e c ific a tio n s o f th e
package are shown in Figs. 4.32 and 4.33. Leads 2. 4. 5 and S are d ir e c t ly co n n e c te d to
th e p a d d le and also g ro u n d e d th ro u g h v e rtic a l c o n d u c tin g vias to th e m o th e rb o a rd , as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m
1.0
0.9
0.8
0.7
^
0.6
0.5
0 . 0.4
CO
0.3
0.2
0.1
0.0
20
Frequency (GHz)
Fig. 4.36: S catte rin g parameters o f the unpackaged M M IC : m ic ro s trip -th ro u g h connection
w ith a stub.
shown in F ig . 4.34. Leads 3 a nd 6 are gro un d ed to th e m o th e rb o a rd b u t n ot otherw ise
connected to th e paddle. T h e d is c re tiz e d c o n d u c tin g surfaces o f th e S O IC -S package
are show n in a th re e -d im e n s io n a l g e o m vie w fo rm a t in F ig . 4.38.
F ig . 4.39 shows
the c o rre s p o n d in g S’-p a ra m e te rs o f th e e n tire package. C o m p a re d to th e unpackaged
s tru c tu re , w hose .8 -p a ra m e te rs are illu s tra te d in F ig . 4.36. th e firs t resonance s h ifts
fro m 10 G H z to S G H z. w h ich agrees fa v o ra b ly w ith b o th .Jackson [178] and R ighi (/
al. [32]. T h e d o w n s h ift o f th e resonant fre q u e n cy o f th e M M IC is due to d is tra c tin g
in te rfe re n c e betw een th e c ir c u it and th e package. A d d itio n a l package resonances are
also o bserved in Fig. 4.39. These corre sp on d to 12 G H z . 15.5 G H z and 22.5 G H z. a ll
o f w h ich w ere also observed in [32].
It was p re v io u s ly m e n tio n e d th a t g ro u n d in g c o n d itio n s can have a sig n ifican t e f­
fect on th e o v e ra ll e le c tric a l p e rfo rm a n c e o f th e package.
A p o o r g ro u n d for the
M M IC p a d d le u s u a lly re sults in p a ra s itic effects th a t lim it th e m a x im u m o pe ra t-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ill
ilH H
v'szljitiyii&vZ-.
yyy///////y/yy/yy///.
T>yy///syys'>,V'
3
l l i
s s '/S
vM yyyyiiyii
'S
/s '//''/,
S'/s'S//.
ys/y/-y/‘''>
Fig. 4.37: Two-dimensional top view of a SOIC-8 plastic package housing a planar mi­
crostrip through-connection with a stub.
Fig. 4.38: Three-dimensional geomview rendering of the SOIC-8 package.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
11.')
1.0
0.9
0.8
0.7
0.6
0.5
& 0 .4
GO
0.3
0.2
0.1
0.0
IS
'21
0
5
10
15
20
25
Frequency (GHz)
Fig. -1.39: Scattering param eters o f the SOIC-8 plastic package.
ing fre q u e n cy o f th e package. T h re e d iffe re n t g ro u n d in g c o n fig u ra tio n s are used to
in v e s tig a te th is phenom enon:
a) leads 2. -1. 5 and 8 g ro u n d th e p a d d le to th e m o th e rb o a rd
b) leads 2 and 8 g ro u n d th e p a d d le to th e m o th e rb o a rd
c) leads -I and o g ro u n d th e p a d d le to th e m o th e rb o a rd
T h e re m a in in g leads are g ro u n d e d to th e m o th e rb o a rd b u t n ot o th e rw is e connected
to th e p addle. T h e m a g n itu d e o f b ’2 1 versus frequency fo r a ll th re e g ro u n d in g cases is
illu s tra te d in F ig . -1.40. W h e n th e p a d d le is p o o rly grounded to th e m o th e rb o a rd , th e
firs t resonance, w h ich corresponds to a zero tra n s m is s io n , is s h ifte d fa rth e r away fro m
the unpackagecl value o f 10 G H z. F or e x a m p le , case a) p rovides th e best g ro u n d in g
c o n d itio n s fo r th e paddle: th e re fo re , th e co rre sp o n d in g z e ro -tra n s m is s io n resonance
is th e closest to th e unpackaged va lu e. For th e o th e r tw o cases, i.e .. cases b) and c).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I Hi
_z -
-10
-15
-20
-25
-30
-35
_ Case (a)
_ Case(b)
-- Case (c)
-40
Frequency (GHz)
Fig. 4.40: S' 2 1 versus frequency o f the SO IC-8 plastic package under the follo w in g g ro un d ing
conditions: (a) leads 2. 4. 5. 8 are grounded, (b ) leads 2. 8 are grounded, and
(c) leads 4. o are grounded.
a lth o u g h b o th o f th e m co rre sp o n d to the sam e n u m b e r o f g ro u n d in g leads, case c)
p ro vid e s by fa r th e worse g ro u n d in g . For case c) leads 4 and o are th e ones g ro u n d e d :
h ow ever, th e ir lo c a tio n is on th e fa r side o f th e package w here the c u rre n t c o n c e n tra ­
tio n is r e la tiv e ly low . M o st o f th e cu rre n t c o n c e n tra tio n is u s u a lly u n d e rn e a th th e
i / p (le a d I) a nd o /p (le a d 7) leads o f the package.
4.o
C o n clu sio n s
In th is c h a p te r, a fu ll-w a v e v e c to r fin ite e le m e n t m e th o d was fo rm u la te d fo r th e
a n a lysis o f .‘3-D m icro w a ve c ir c u its and e le c tro n ic packages.
T h e in p u t p o rt o f th e
s tru c tu re is e x c ite d using th e g o v e rn in g m ode d is t r ib u t io n at a g ive n fre q u e n c y . 1 his
m ode d is tr ib u tio n , as w e ll as th e co rre sp o n d in g p ro p a g a tio n c o n s ta n t a nd c h a ra c ­
te r is tic im p e d a n c e o f th e tra n s m is s io n lin e , a rc a ll c a lc u la te d u sin g a 2-D a n a ly s is
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
LIT
described in C h a p te r 3. A ll p o rts and s u rro u n d in g w alls are te r m in a te d e ffe c tiv e ly
using a d isp ersive a b s o rb in g b o u n d a ry c o n d itio n .
A F O R T R A N code was developed based on th e fo r m u la tio n d escribe d in Sec­
tio n 4.2. T h is code was in te rfa c e d w ith a c o m m e rc ia l packaged c a lle d SDRC' I-D E A S
to p ro v id e the mesh in fo r m a tio n o f the d is c re tiz e d c o m p u ta tio n a l d o m a in . B o u n d ­
a ry c o n d itio n s such as P E C . P M C and A B C are also im p o sed u sin g th is g ra p h ic a l
package.
T h e code and fo r m u la tio n were e xte n s iv e ly v e rifie d by c o m p a rin g th e m a g n itu d e
and phase o f th e S’-p a ra m e te rs w ith o th e r c o m p u ta tio n a l te ch n iq u e s . In m ost cases,
th e fin ite e le m en t p re d ic tio n s were found to be in e xce lle n t a g re e m e n t w ith d a ta o b ­
ta in e d fro m o th e r in d e p e n d e n t sources. A v a rie ty o f c irc u its and in te rc o n n e c ts were
e x a m in e d ra n g in g fro m basic p la n a r c irc u its , such as m ic ro w a v e filte rs and patch
antennas, to sin g le -lo o p in d u c to rs and m u ltip le - tu r n s p ira l in d u c to rs , as w e ll as w ire
bonds and m ic ro s trip tra n s itio n s throu g h d ie le c tric w alls.
In a d d itio n , packaging
and g ro u n d in g effects o f p ra c tic a l packages, such as th e S O IC -S s u rfa c e -m o u n t p ack­
age. were also in v e s tig a te d using the fin ite e le m e n t m e th o d .
Based on num erous
stud ie s p e rfo rm e d d u r in g th e course o f th is w o rk , it was fo u n d th a t th e e le c tric a l
p e rfo rm a n ce o f a M M IC d e te rio ra te s w hen packaged. T h e package its e lf in tro d u ce s
p a ra s itic effects, such as a d d itio n a l resonances, w h ich are k n o w n to a dversely affect
th e 5 -p a ra m e te rs o f th e o rig in a l c irc u it. A lso , p o o r g ro u n d in g o f th e M M IC p addle
causes th e m a x im u m o p e ra tin g frequency o f th e package to s h ift to low er values.
T h u s , package o p tim iz a tio n becomes e x tre m e ly im p o rta n t w h e n e ffic ie n c y and high
p e rfo rm a n ce are d esirab le .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 5
C' A V I T V - B A C K E D A P E R T I ' R ES
In th is ch a p te r, a h y b r id fin ite e le m e n t m e th o d /m e th o d o f m o m e n ts ( F E M / M o M )
approach is used to a n a lyze c a v ity -b a c k e d a p e rtu re ante nn a s m o u n te d on an in fin ite
g ro u n d plane and c h a ra c te riz e d b y a rb itr a r y shapes and m a te ria l in h o m o g e n e itie s.
A d ie le c tric o r m a g n e tic o v e rla y m a y be d e p o site d on to p o f th e g ro un d p lane. A l­
th o u g h th e o v e rla y is is o tro p ic , th e lo a d in g m a te ria l in s id e th e c a v ity m a y be fu lly
a n is o tro p ic and fre q u e n cy d e p e n d e n t. T h e a n a lysis is c a rrie d o u t fo r b o th s c a tte rin g
and ra d ia tio n p ro b le m s. T h e ra d a r cross se ctio n , in p u t im p e d a n ce , re tu rn loss. g ain,
d ir e c t iv it y and e ffic ie n c y are c o m p u te d .
T h e p re d ic tio n s are co m p ared w ith o th e r
n u m e ric a l m e th o d s a nd m e a surem e n ts p e rfo rm e d at th e E le c tro M a g n e tic A n e ch o ic
C h a m b e r (E M A C ) o f A riz o n a S ta te U n iv e rs ity (A S U ).
T h e M o M p a rt is based on a pure s p e c tra l d o m a in a p p ro a ch using lin e a r basis
fu n c tio n s w ith tr ia n g u la r s u p p o rt. T h e c o m p u ta tio n a l e ffic ie n c y o f th e s p e c tra l do ­
m a in M o M degrades w ith in c re a s in g the n u m b e r o f u n kn o w n s in the a p e rtu re .
To
im p ro v e th e c o m p u ta tio n a l speed o f the m e th o d , the c o rre s p o n d in g in te g ra l is e val­
u ate d using an a s y m p to tic a p p ro a c h . F u rth e r im p ro v e m e n t is achieved th ro u g h the
use o f a fre qu e ncy in te r p o la tio n o f th e a d m itta n c e m a trix . F in a lly , an a d d itio n a l h y ­
b rid iz a tio n using th e u n ifo rm th e o ry o f d iffra c tio n ( U T D ) is im p le m e n te d to a ccount
fo r th e d iffra c tio n s a t th e edges o f a fin ite g ro u n d plane.
•5.1
In tro d u c tio n
C a v ity -b a c k e d a p e rtu re a n te n n a s, w hich have becom e th e focus o f in te re s t in re­
cent years, are w id e ly used in U H F and m ic ro w a v e frequencies. A p p lic a tio n s range
fro m ra d a r tra c k in g to m issile c o n tro l, n a v ig a tio n , s a te llite c o m m u n ic a tio n s , m o b ile
te le p h o n y, broadcast T V . a nd a irc ra ft c o m m u n ic a tio n s .
T y p ic a l c o n fig u ra tio n s o f
a p e rtu re s are re c ta n g u la r, c irc u la r, slits, slo ts, a n n u la r rin g s, and few o th e rs.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T he
1 1!)
placem ent o f a c a v ity u n d e rn e a th th e a p e rtu re helps im p ro v e ra d ia tio n c h a ra c te r­
istics and m in im iz e sp u rio u s r a d ia tio n to w a rd the back lobes.
O n e o f th e m ost
a ttra c tiv e features o f c a v ity -b a c k e d a p e rtu re antennas is th e ir lo w p ro file . T h e y are
o fte n flu s h m o u n te d on th e su rfa c e o f la rg e o b je c ts , lik e a irc ra ft, sp ace cra ft and m is ­
siles. th e re b y re ta in in g th e v e h ic le 's a e ro d y n a m ic p ro file .
T h e a n te n n a o p e n in g is
som etim es covered w ith a t h in d ie le c tric m a te ria l to p ro v id e a d d itio n a l p ro te c tio n
fro m e n v iro n m e n ta l c o n d itio n s .
T w o generic typ e s o f c a v ity -b a c k e d a p e rtu re antennas co nsidered in th is s tu d y
are illu s tra te d in F ig . 5.1. T h e firs t one d e p ic ts a c irc u la r m ic ro s trip p atch a n te n n a
backed by a c y lin d ric a l c a v ity .
T h e p a tc h is u su a lly e x c ite d w ith a co a x ia l cable
o rie n te d in th e v e rtic a l d ire c tio n . T h e in n e r c o n d u c to r is e xte n d e d fro m th e co axial
a p e rtu re to an offset p o in t on th e su rface o f th e p atch.
dep icts a n a rro w slo t backed b y a re c ta n g u la r ca vity.
is used to e x c ite th e s tru c tu re .
T h e second c o n fig u ra tio n
A h o riz o n ta l co a x ia l cable
A lth o u g h n o t shown in th is fig u re , b o th antennas
are flu s h m o u n te d on an in f in it e g ro u n d plane. T h e ra d ia tio n c h a ra c te ris tic s o f these
ante nn a c o n fig u ra tio n s can be c o n tro lle d th ro u g h the use o f m a te ria l lo a d in g s in sid e
the c a v ity , a lte rin g th e a p e rtu re shape, ch a n g in g the d im e nsion s o f th e c a v ity and
m o vin g th e lo c a tio n o f th e p ro b e w ith respect to the slo t.
5.2
S c a tte rin g fro m C a v ity -B a c k e d A p e rtu re s
A h y b rid iz a tio n o f F E M [33] a n d s p e c tra l/s p a tia l d o m a in M o M [S i]. [179] is u tiliz e d
in th e analysis o f c a v ity -b a c k e d a p e rtu re antennas m o u n te d on an in fin ite g ro u n d
plane. T h e p e rfe c tly c o n d u c tin g g ro u n d p la ne m ay be covered w ith a sin g le la ye r
o f d ie le c tric m a te ria l o f c o m p le x p e r m it t iv it y a n d /o r p e rm e a b ility .
A p la ne wave
in c id e n t at an angle 0 fro m th e n o rm a l to th e a p e rtu re axis is used to e x c ite the
antenna. A 2-D v ie w o f a m ic r o s tr ip p a tc h backed by a c a v ity o f a r b itr a r y shape is
shown in F ig . 5.2. T h e s p e c tra l d o m a in M o M represents th e fields in th e e x te rio r
o f th e c a v ity th ro u g h th e use o f th e h a lf-spa ce Cireen's fu n c tio n , whereas th e F E M
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
120
F ig . 5 .1 : G e n e ric shapes o f c a v ity -b a c k e d a p e r tu r e a n te n n a s .
represents th e fields in th e in te rio r of the c av ity th rough the use o f lin e a r edge-based
tetrahedral elem ents.
T h e tw o regions are coupled through th e c o n tin u ity o f the
tangential fields in th e a perture.
Th e M o M fo rm u la tio n can be to ta lly separated fro m the F E M fo rm u la tio n by
introducing, according to equivalence principle, a surface m agnetic current density
just above and below th e a p ertu re plane. T h e use o f te tra h e d ra l elem ents inside the
cavity volum e results in a tria n g u lar discretization o f b oth the a p e rtu re and patch
surfaces.
Since th e o b je c tiv e o f this study is th e analysis of a rb itra ry shape cavi­
ties and ap ertu res, th e most appropriate basis functions for the e x te rio r problem are
the ones w ith tria n g u la r support, otherwise know n as th e Rao. W ilto n and Glisson
(R W G ) basis functions [ISO ]. These are sim ilar to th e linear edge basis functions
frequently used in th e fin ite elem ent m ethod. T h e m a in difference betw een the two
sets is th at th e R W G basis functions, which are usually im plem ented in th e spectral
domain M o M . enforce th e con tin u ity of the norm al instead of the ta n g e n tia l com po­
nent of the field q u a n tity across edges. This p roperty o f the R W G basis functions is
needed to g uarantee th e c o n tin u ity of the norm al curren t across interfaces.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Incident '
Plane Wave
Infinite Half Free Space
Infinite Ground Plane
+M
+M
-M
-M
Perfectly
Conducting
Walls
Arbitrary Shaped
Cavity
Fig. 5.2: A 2-D view o f a cavity-backed patch antenna mounted on an in fin ite ground
plane.
5.2.1
F o rm u la tio n using F E M and a s y m p to tic s p e c tra l d o m a in M o M
T h e F E M fo rm u la tio n in a source-free space begins w ith th e d is c re tiz a tio n o f the
fre q u e n cy d o m a in H e lm h o ltz 's e q u a tio n given by
V x ( [ / / r ]-> - V x E ) - k-i [cr] E = 0
( o . l)
where [er ] and [/ / r] are re s p e c tiv e ly th e re la tiv e p e r m it t iv it y and p e rm e a b ility
o f th e m e d iu m , and E is th e u n kn o w n e le c tric fie ld .
tensors
T h e m a te ria l p a ra m e te rs are
m o deled as fu ll tensors to a llo w a n is o tro p ic and fre q u e n c y dependent m a te ria ls , lik e
fe rrite s a nd p la sm a , to be p a rt o f th e c o m p u ta tio n a l d o m a in .
D iric h le t b o u n d a ry
c o n d itio n s are im posed on a ll p e rfe c tly c o n d u c tin g surfaces im p ly in g th a t h x E = 0
on c a v ity w a lls.
S e ttin g th e w e ig hte d re sid u a l to zero and using th e w e ll-k n o w n
G a le rk in 's a p p ro a ch , th e H e lm h o ltz 's e q u a tio n becom es
[ ( [ f i r] - ' V x E ) - ( V x N ) d Q - k - 0
2 [ [er] ■E • N (IQ. = -
Jn
Jq
i ([ // r] " ‘
Js
V x E ) • ( N x a n) rIA
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
w h e re a n is th e n o rm a l to th e su rface u n it v e c to r d ire c te d o u tw a rd ly to the vo lu m e
o f th e ca vity. T h e surface in te g ra l on th e rig h t-h a n d side o f (5 .2 ) is nonzero o n ly in
th e a p e rtu re since th e c a v ity w a lls are p e rfe ct elect,-;'~ co n d u c to rs .
T h u s, w ith th e
use o f M a x w e ll's e qu a tion s th is in te g ra l can be s im p lifie d to
-
/ ( [ / ^ r ] ' 1v
X
E ) • (N
X
a n ) d A = j k 0Z 0
J S
f
H ■( N x a s) d A
(5 .3)
JAper
w here Z 0 and k Q are th e free-space in tr in s ic im p edance and p ro p a g a tio n co n s ta n t,
re sp ective ly.
B y im p o s in g th e c o n tin u ity o f th e ta n g e n tia l m a g n e tic fields in th e
a p e rtu re , one can w rite
H tan = H | ^ + H " ' + H “ ‘
w here the s u b s c rip t " t a n "
(5 ,1)
in d ic a te s ta n g e n tia l fields and th e s u p e rscrip ts " i n c " .
" r e f " and " e x t " in d ic a te in c id e n t, re fle cte d and e x te rn a l fie ld s, re sp ective ly.
im p o rta n t to c la rify here th a t
It is
is equal to 2H [ ^ o n ly in th e presence
o f an in fin ite p e rfe c tly c o n d u c tin g g ro u n d p lane. Using (5 .3 ) and (5 .4 ). th e in te g ra l
e q u a tio n (5.2) can be w ritte n as
f ([//r]-lV x E)-(V x N)dQ-ki Jf
JQ
jk0z0 [
J Aptr
[er]E • N
dQ. =
(h;:;- + h;;;{) • (N x <u)r/,i + jk0z0 [
h;^ • (N x dr)f/,i(5,v)
J Apf.r
T h e first in te g ra l on th e rig h t-h a n d side o f (5.5) represents th e e x c ita tio n v e c to r
w h ic h , as it w ill be shown la te r, is e v a lu a te d using a pure s p e c tra l d o m a in approach.
In case th a t th e in fin ite g ro u n d p la n e is n o t coated, the e x c ita tio n v e c to r m ay also be
e va lu a te d using a s p a tia l d o m a in a p p ro a c h . T h e second in te g ra l on th e rig h t-h a n d
side o f (5.5) represents th e a d m itta n c e m a t r ix fo r the e x te rio r o f th e c a v ity w h ich is
e va lu a te d using e ith e r a pure s p e c tra l d o m a in approach o r a m ix e d s p a tia l/s p e c tra l
d o m a in approach. U sing th e second m e th o d , th e e x p o n e n tia l b e h a v io r o f th e g o ve rn ­
in g G reen's fu n c tio n is n u m e ric a lly e x tra c te d to im p ro ve th e c o m p u ta tio n a l speed ol
th e sp ectra l in te g ra tio n . T h e a s y m p to tic p a rt is evaluated u sin g a s p a tia l in te g ra ­
tio n w hich is c o m p u ta tio n a lly m o re e ffic ie n t. In e va lu a tin g th e e x c ita tio n ve cto r and
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
a d m itta n c e m a tr ix , th e tr ia n g u la r su p p o rt v e c to r basis fu n c tio n s o r ig in a lly proposed
by Rao t t al. [ISO] were im p le m e n te d .
T h e F o u rie r tra n s fo rm o f these tria n g u la r
basis fu n c tio n s is kn o w n in closed fo rm [181]. T h u s , assum ing th e a p e rtu re lies on
th e x (/-p la n e ( r = 0 ). th e a d m itta n c e m a tr ix can be expressed in s p e c tra l d o m a in as
J h.
y
OZ*Q
4 ^
|G
f=
JZ IZ
•
=H
(^ -M - G
'
(AV .A-y ) |
• M j ( k T. k y ) d k r d k y
r
t /»
r
J —
M ,( ~ k r . - k y ) - G
( k x . ky ) ■M j ( k r . ky ) d k ^ l k y
(5.6 )
«/ —o c
where G
is th e d y a d ic G re e n 's fu n c tio n fo r a co a te d c o n d u c tin g g ro u n d plane in
= H
th e presence o f a m a g n e tic c u rre n t source. G
is th e d ya d ic G re en 's fu n c tio n o f
a hom ogeneous space w ith cfr l = er and f.ir/
= j.ir . w h e re er and / / r co rre sp o n d to
the re la tiv e p e r m it t iv it y and p e rm e a b ility o f th e c o a tin g m a te ria l, a n d M , is th e
F o u rie r tra n s fo rm o f th e i th basis fu n c tio n .
T h e firs t in te g ra l in (5 .6 ). den o te d as
Yij. is e v a lu a te d using a p u re sp e c tra l d o m a in a p p ro a c h a fte r c o n v e rtin g to p o la r
c o o rd in a te s . T h e second in te g ra l in (5 .6 ). d e n o te d as Y l ! . is e valua ted u sin g a s p a tia l
d o m a in M o M app ro a ch: i.e ..
v ;"
=
~ '2 k 2 J A M , ( r ) . [ J
+
2 f
J,\,
V • M t(r)
[
J
M j ( r ' ) G h( r . r ' ) d A '
G h( r . r ' ) ? ' - M j i r ^ d A '
a
dA
dA.
(5.7)
,
w here A t ( A j ) represents th e area o f th e tria n g le s u p p o rtin g th e i th ( j t h ) basis fu n c ­
tio n . r ( r ') is the p o s itio n v e c to r, and Gh is th e G re e n 's fu n c tio n fo r a hom ogeneous
m e d iu m w ith
= er and
= /rr . S im ila r to th e a d m itta n c e m a tr ix , th e fin ite e l­
e m e nt m a tric e s are e v a lu a te d b y first e x p a n d in g th e e le c tric fie ld E ( . r . / /. r ) in te rm s
o f a set o f lin e a r v e c to r basis fu n c tio n s fo r te tra h e d ro n s .
T h e re s u ltin g e le m e n ta l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
stifFness and mass m a tric e s . Ef- and F trJ. are g iv e n by
£?.
F
=
/
( V x N i l - W ' - I V x N ^ r
(5.8)
I = JfQ" N,- • [cr] • N y d V
(5.9)
w h e re Q e represents th e d o m a in o f a te tra h e d ro n a nd i . j
=
1
V e: .\ e being
th e n u m b e r o f degrees o f fre e d o m in each e le m e n t. T h ese are to be assembled in to
a g lo b a l m a tr ix based on th e e d g e -c o n n e c tiv ity in fo r m a tio n .
A ls o , depending on
th e p a r tic u la r choice o f basis fu n c tio n s , th e e le m e n ta l m a tric e s ZT'j and F^ m ay
be e v a lu a te d e ith e r a n a ly tic a lly o r n u m e ric a lly . E v a lu a tio n o f these m a trices is the
to p ic o f one o f th e fo llo w in g sections in th is c h a p te r. A s fa r as th e e x c ita tio n ve cto r
is co nce rn e d , w h ich is d e fin e d bv
F = jh z 0
(h ::;; + n z i ) • m ,
ha
(5.10)
.
its e v a lu a tio n begins b y e xp re ssin g th e m a g n e tic fie ld a t th e a p e rtu re plane in te rm s
o f th e g o ve rn in g G re e n 's fu n c tio n : i.e..
H ‘" + H rc / = ( ar sin O i - a y cos O i ) ^ - C - y ( k r s.k'ya) c o s 0 i €jkadn,se' t j k l ' T t j k ^
/. o
(5.11)
fo r h a rd p o la riz a tio n , and
H - + H r' / = ( (ir cos o, + rzv sin o , )
( kTS. k'ys) cos 6 t
ej k t , r t J k9ry ( - 12)
fo r soft p o la riz a tio n . A ll v a ria b le s in vo lve d in (5 .11 ) and (5 .1 2 ) are defined as
g-5u ( A v . , - M
=
* rr
c ; ” ( k r i .k-ya)
=
^
( 5 .1 1:
T„,
=
cr k-2 co s(kx<l) - f j k x s in ( k-X(l)
(5.15)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T,.
= k i cosi k' i d) + j f i r k-2 s\n{k-id)
(o. 16)
k,
= y/trflrk* ~ 32
(3.17)
k-2
=
y j k l - .P
(3. IS)
k rs
= k 0 sin 0t cos o,
(5.19)
k,JS
= k a sin 0, sin o,
(5.20)
w here d is th e th ickn e ss o f th e c o a tin g layer.
S u b s titu tin g (5.11) and (5.12) in to
(5.10) and u tiliz in g th e d e fin itio n o f th e F o u rie r tra n s fo rm , the e x c ita tio n ve cto r can
be w ritte n as
b ^ rd
=
^ - C f ( k TS.k,JS) cosOt e ^ d^ d'
O
( A’X5. kyS) sin Of
b ? Jt
=
.\/^ t (
AyS) cos o t ^
(o .2 l)
( k xs. kyS) sin o t ^
(■)— 1)
^ C ^ ( h r sA ^ ) c o s O t c ^ Uo^
/-o
( kTS, kys ) cos Oi “h
fo r h ard and soft p o la riz a tio n s , re sp e ctive ly, w here i denotes an edge in th e a p e rtu re
and x in d ic a te s c o m p le x co njug a te . T h e fin a l d is c re tiz e d e q u a tio n , in m a tr ix fo rm ,
is given by
[.!/ + > - ] { £ } = { b }
w here [.\/] = [£ ’] —
(5.2:1)
[/*'] is th e fin ite e le m e n t m a tr ix re p re sen tin g th e in te r io r o f the
c a v ity . [V ] is th e m e th o d o f m o m e n ts a d m itta n c e m a tr ix re p re sen tin g th e e x te rio r o f
th e c a v ity . { £ } is th e u n k n o w n e le c tric field ve cto r, and { b } is th e e x c ita tio n ve cto r
due to an in c id e n t p la n e wave. T h e tw o m a tric e s [,\/ ] and [} ] are not decoupled:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I2(i
on th e c o n tra ry , th e y p a r tia lly o v e rla p due to th e n o n ze ro ta n g e n tia l field in th e
a p e rtu re .
T h e g lo b a l fin ite e le m e n t m a tr ix has th e advantage o f b e in g sparse whereas th e
m e th o d o f m o m e n ts a d m itta n c e m a tr ix is always dense.
E ith e r an ite ra tiv e so lve r
o r a sparse L U solver can be u tiliz e d to solve th e re s u ltin g lin e a r system . Such a
lin e a r system u s u a lly co rre sp on d s to a large n u m b e r o f u n k n o w n s w h ich m ake it
a lm o s t c o m p u ta tio n a lly im p o s s ib le to im p le m e n t a d ire c t so lve r fo r its so lu tio n . A n
ite ra tiv e so lve r such as th e C o n ju g a te G ra d ie n t Squared (C 'G S ) a lg o rith m o r the B iC 'onjugate G ra d ie n t (B iC 'G ) a lg o rith m o r even th e Q uasi M in im u m R esidual (Q M R )
a lg o rith m are s u ita b le m o s tly fo r fre q u e n cy sweep c o m p u ta tio n s whereas a sparse
LC so lve r is s u ita b le fo r ang le sweep c o m p u ta tio n s .
o.'2.2
A d m itta n c e m a tr ix u sing s p a tia l d o m a in M o M
In th e fo rm u la tio n presented in th e p re vio u s section, it was show n th a t the e x te rio r
c a v ity p ro b le m is re p resented by a surface in te g ra l g ive n by
V -'
=
j k 0Zo [
H " ' ( r ) - ( N ( r ) x a n)d A
J A per
=
/
( [ / 'r ] ” 1^" x E ( r )) • ( N ( r ) x a n ) d A
(A M )
J Ape. r
w here
« „ is a
u n it v e c to r n o rm a l to th e a p e rtu re pla ne and d ire c te d o u tw a rd ly to
the in fin ite e x te rio r v o lu m e 1 ^ : i.e .. a n =
—a z.
Since th e te s tin g fu n c tio n N ( r )
is id e n tic a l to the basis fu n c tio n in te rp o la tin g the e le c tric fie ld E ( r ) (G a le rk in 's
a p p ro a ch ), e q u a tio n (5 .2 1 ) can s im p ly be w ritte n as
T 1' = -
f
([//r]-‘ V
X
E(r ) ) •
( E ( r ) x a : ) d , 1.
(5 .25 )
J Aprr
T h e c u r l o f th e e le c tric fie ld at an o b se rva tio n p o in t d e fin e d b y th e p o s itio n v e c to r r
is re la te d to th e free-space d y a d ic G re e n 's fu n c tio n G 0 as fo llo w s:
V x E ( r ) = —2A-" f
(5 ?
J A;.»»r
x
E ( r ') ) • G 0( r . r ') cl A '
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
(5.2(»i
121
w h e re
(•5.27)
G'0( r . r') = ( / + — V V ) G 'o ( r . r ')
h'Z
w ith
e - j k a\ r - r'|
6'0(r . r ) =
(5.28)
-l7r|r — r '|
and
/ = hr a T -f a ya y + d -d -.
(5-29)
S u b s titu tin g (5 .26 ) in to (5.2-1) a n d assum ing a free-space m e d iu m . Y ert becom es
Y cxt
=
- 2 k0
2 f
[d ; x E ( r ) ] •
J Apcr
=
- 2 kl [
[
J Apcr
dA
M ( r ' ) • G 0( r . r') d A 1 d A
M ( r ) • I" f
J Apcr
-2 [
( a z x E ( r ' ) ) • G 0(r . r') d A '
\,J A p c r
U A per
M (r) • [ [
L
iJJ A p c r
V V G 0( r . r ' ) • M ( r ' ) </ .- l '
dA.
(3.30)
U s in g th e G re en 's fu n c tio n id e n titv
(3.31 )
V G ’o = —V 'G ’o
th e second in te g ra l in (3.30) can be w ritte n as
2 /
M (r) •
J Apcr
f
V V ' G ' 0(r . r') • M ( r ' ) d A '
dA.
LJ A p c r
F u rth e rm o re , by im p le m e n tin g th e ve cto r id e n tity
V • (qB ) = o V • B + B • V o .
(3.33)
th e fo llo w in g expression holds:
M ( r ) • V ' G 0( r . r') = V
■[G-’0( r . r ' ) M ( r ' ) ] — G/0( r . r ' ) V ' • M ( r ' ) .
(5.31)
S u b s titu tin g (5.3-1) in to (3 .3 2 ). th e la tte r becomes
-
2 [
JApcr
2 [
J A r r
M (r) • [v f
L
M (r) • W
L
G’0( r . r ' ) V ' • M ( r ' ) d A 1 d A
JApcr
[
r - [ G ' 0( r . r ' ) M ( r ' ) ] f / . 4 '
JAp-r
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
d A.
(3 .3 3 i
U sing th is re su lt in to (5 .30 ). Y eIt can be expressed as
=
-2 k2
0 f
M (r) • [ f
LJ
JApcr
~
2 f
M (r) ■[v [
L
JApcr
+
2 f
J
L
dA
G 0( r . r ' ) V ' • M ( r ' ) d A '
Aper
M (r) • fv f
J Aper
dA
M ( r , ) - G ' 0( r . r / )rf/V/
Apcr
dA.
V ' . [ G 0( r . r ' ) M ( r ' ) ] d . 4 '
(5.36)
J A pp ec r
B y in tro d u c in g th e surface d ive rg e n ce th e o re m
jy - F d S
(5.37)
= jf hc -F d C \
where h c is a u n it v e c to r n o rm a l to th e p e rim e te r o f su rface S and p a ra lle l to th e
plane o f th a t su rface , th e last in te g ra l in (5.36) can be w r itt e n as
2 f
M (r) • fv
2 Apr r
L
C/0(r . r ' ) M ( r ' ) • h c d C
(5.38)
dA
JC A
where C’ a co rre sp on d s to the c o n to u r a lo ng the p e rim e te r o f th e a p e rtu re w h ic h , in
o u r case, lies in th e .ry-p la n e . X o te h ow ever th a t
M •
(5.3!))
= ( E x rt: ) • f>._. = E • ( « . x «,.) = E • a t
where d t is a u n it v e c to r ta n g e n tia l to C' I n
a d d itio n , th e ta n g e n tia l e le c tric fie ld
along th e p e rim e te r o f the a p e rtu re is zero: thu s, th e in te g ra l in (5.38) vanishes.
W ith th e use o f th e ve cto r id e n tity (5 .3 3 ). th e second in te g ra l in (5.36) can be
fu rth e r s im p lifie d to th e fo llo w in g :
- 2
f
J Aper
+
2 /
J Aper
V - ( \ f
\l«/.4p^r
G 0( r . r ' ) Y ' - M ( r ' ) r / . 4 ' ] M ( r ) )
[V • M ( r ) | [ f
J
/
G 0( r . r ' ) V ' • M ( r ' ) d A '
dA.
(5.-10)
L JJ A p ec r
A lso, w ith th e use o f th e surface d ive rg e n ce th e o re m , th e firs t in te g ra l in (5.-10)
vanishes fo r th e sam e reason th a t th e in te g ra l in (5 .3 8 ) d id .
T h u s , th e re s u ltin g
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expression fo r th e s p a tia l d o m a in m e th o d o f m o m e n ts a d m itta n c e p a rt is g iv e n by
=
2
f
V - ( |7
f
[V • M (r)] [
J Aper
- 2
G 0( r . r ' j X 7' - M ( r ' ) d A '
\ {.JA p e r
J Aper
f
M (r)
G'0( r . r') V ' - M ( r ' ) dA'
dA.
(5.41)
lJ A p ee r
T h is fin a l expression is d is c re tiz e d and e v a lu a te d fo llo w in g a procedure s im ila r to the
one used for th e fin ite e le m e n t m e th o d . S p e c ific a lly , th e unknow n surface magneticc u rre n t M ( r ) is e xp a n d e d in te rm s o f a set o f lin e a r ve cto r basis fu n c tio n s w ith
tria n g u la r s u p p o rt. T h e e v a lu a tio n o f th e a d m itta n c e surface in te g ra ls in (5.41) is
done n u m e ric a lly using a th irte e n -p o in t Gauss q u a d ra tu re for tria n g le s [1S2].
It is
im p o rta n t tho u gh to em p ha size th a t e x tra c a u tio n sh ou ld be used d u rin g n u m e ric a l
e v a lu a tio n o f these surface in te g ra ls because o f th e in h e re n t s in g u la rity ot th e freespace G reen's fu n c tio n .
5.2.3
E x c ita tio n v e c to r u sin g s p a tia l in te g ra tio n
T h e e x c ita tio n ve cto r fo r a c a v ity -b a c k e d a p e rtu re m o u n te d on an in fin ite g ro u n d
p lane and coated w ith a sin g le layer d ie le c tric m e d iu m was evaluated in S ectio n 5.2.1
using a pure s p e ctra l d o m a in approach.
In th e absence o f coating, th e e x c ita tio n
v e c to r can be e q u iv a le n tly e valua ted using a s p a tia l d o m a in in te g ra tio n , w h ic h is the
to p ic o f th is se ction .
In th e presence o f c o a tin g , th e s p a tia l d om ain a p p ro a ch can
becom e e x te n sive ly in v o lv e d , the re fo re, it is n ot u s u a lly recom m ended.
T h e e lem ental e x c ita tio n ve cto r based on a sin g le tria n g u la r elem ent in th e a p e r­
tu re is given by
b] = 2 j k 0Z 0 [
H;:;;
• (N,
x
b: )JA
(5.42)
J Ape r
w here th e surface in te g ra tio n is e valuated o v e r th e area o f the elem ent. T h e in c id e n t
m a g n e tic field is d efin e d as
H " 1" = { —<ir sin o, + a tJ cos o , ) H 0cj k *{™ n s'
si" * ' sin + : sin >
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(5.13)
1:J0
fo r h a rd p o la riz a tio n , a nd
H " ‘c = ( - a x cos 6i cos o , - a y cos 0, s in o , +«_- sin 0 , ) / / oejM x s in fl' “ * * .+ » « » * .
(5.4-1)
fo r soft p o la riz a tio n , w h e re o , and 0, are th e in c id e n t angles. H 0 is th e m a g n itu d e
o f th e in c id e n t m a g n e tic fie ld and k 0 is th e free-space p ro p a g a tio n co n sta n t.
The
a p e rtu re lies in the x i/-p la n e a t : = 0. th e re fo re d A = d x d y and H l,lc( x . y . : ) =
H mc(.r. y. 0). A lso , it is im p o r ta n t to em phasize here th a t, since th e c a v ity vo lu m e
is d is c re tiz e d using te tra h e d ra l e le m e n ts, o n ly three o f th e s ix edges c o n trib u te to
th e rig h t-h a n d side v e c to r: these are th e ones th a t lie in th e a p e rtu re plane.
The
in te g ra tio n in (5.42). a lth o u g h it c o u ld be c a rrie d out in closed fo rm , it was e va lu ­
ated n u m e ric a lly using G auss q u a d ra tu re . T h e in te g ra n d was firs t tra n s fo rm e d in to
s im p le x co o rd in a te s u sin g th e fo llo w in g tra n s fo rm a tio n :
.r
=
T n + '/•/•_> + ( ! - £ - r i ) x :i
!J
=
s//i +
m h.
+ (1 - s - '/)//••!
w here x t and //, (/ = 1 .2 .5 ) are th e x and
ij
(5.45a)
!’)b)
co ordinates o f a g ive n tria n g le in th e
a p e rtu re . T h e d iffe re n tia l area d A = d xdt j is tra n s fo rm e d , u sin g th e Jacobean, in to
(2.4)(/£(/// w here A is th e to ta l area o f th e tria n g le . A c c o rd in g to th is tra n s fo rm a tio n ,
th e e le m e n ta l rig h t-h a n d sid e v e c to r is w ritte n in term s o f s im p le x co ordin a tes as
•i
'>] = ( j ' 2 k 0Z 0 ) (2 .4 )
w here
r i-C
f f
Jo Jo
^"(s- ’I)dijd£.
(5.4(J)
is the tra n s fo rm e d in te g ra n d in sim p le x c o o rd in a te s. T h e expression
in (5.46) can be evalua ted n u m e ric a lly using Gauss q u a d ra tu re developed s p e c ific a lly
fo r tria n g le s [185]. H o w e ver. G aussian p o in ts and w eights are n o t u su a lly a v a ila b le
fo r o rd ers h ig h e r th a n th irte e n [185]: one m u st generate these G aussian p o in ts in
advance, if a h ig h e r-o rd e r in te g ra tio n is needed.
To avoid th e c o m p le x ity o f th is
p ro b le m , we decided to im p le m e n t a n o th e r va ria ble tra n s fo rm a tio n so th a t b o th
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
lim its o f th e in te g ra tio n m a p to 0 and 1. In such a case, a s im p le Gauss q u a d ra tu re
for re c ta n g u la r regions can be u tiliz e d . S pe cifica lly, if a = 1 — £ and r = q / ( \ — if),
the e x c ita tio n v e c to r s im p ly becom es
b] = j \ k oZ 0A
f f
Jo Jo
J -( u c . I — u ) u d v d u .
(5 .47 )
T h e o n ly d is a d v a n ta g e o f th is v a ria b le tra n s fo rm a tio n is th a t a large n u m b e r o f
G aussian p o in ts are c o n c e n tra te d in a s m a ll p o rtio n o f th e tria n g le w h ile le a v in g
the re m a in in g p o r tio n sp arsely sa m p le d . However, i f the to ta l n u m b e r o f in te g ra tio n
samples is chosen to be la rg e th e fin a l answ er is g ua ranteed to be q u ite a cc u ra te .
5.2.4
E le m e n ta l m a tric e s u sin g closed fo rm expressions
T he e le m e n ta l m a t r ix [ M €] is e v a lu a te d fo r each e le m en t based on th e d e fin itio n o f
the g o v e rn in g basis o r in te r p o la tio n fu n c tio n . In th e area o f e le c tro m a g n e tic s , w here
the u n k n o w n q u a n titie s represent v e c to r fields, th e m ost a p p ro p ria te in te r p o la n t is
u s u a lly a v e c to r fu n c tio n . Such e le m e n ts are often ca lled th e edge e le m en ts o r v e c to r
elem ents. Few e xa m p le s o f such e le m e n ts are the re c ta n g u la r b ric k , th e te tra h e d ro n ,
and th e h e x a h e d ro n . A m o n g th e th re e typ e s, the m ost a ttr a c tiv e ele m en t is p ro b a b ly
the te tra h e d ro n . T h e reason is th re e fo ld : firs t, the co rre sp o n d in g basis fu n c tio n s s a t­
isfy th e d ive rg e n ce -fre e c o n d itio n : second, a rb itra ry geom etries and cu rv e d surfaces
are m o re a c c u ra te ly m o d ele d : t h ir d , th e e le m en ta l m a tric e s can be e v a lu a te d a na ­
ly tic a lly . T h e re c ta n g u la r b ric k e le m e n t also satisfies th e d ive rge n ce-free c o n d itio n ,
how ever, it is c o n fin e d o n ly fo r th e a n a lysis o f s tru c tu re s w ith re c ta n g u la r shape. As
was th e case w it h th e te tra h e d ro n , th e co rre sp o n d in g e le m e n ta l m a tric e s can also be
e valua ted a n a ly tic a lly . O n th e o th e r h a n d , the h exahedron is not a d ive rg e n c e -fre e
e le m e n t: th e m o re d is to rte d th e e le m e n t is. the la rg e r th e d ivergence. In a d d itio n ,
e v a lu a tio n o f th e re s u ltin g e le m e n ta l m a tric e s requires th e use o f a n u m e ric a l in te ­
g ra tio n w h ic h is u s u a lly c o m p u ta tio n a lly expensive. T h u s , one sh o u ld t r y to avoid
using th is ty p e o f e le m e n t fo r p ra c tic a l a p p lica tio n s.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1T2
N2
E5
E4
z
N4
N3
E2
N1
Fig. a.3: Edge based te tra h e d ra l element.
h i th is s tu d y th e edge based te tra h e d ra l e le m e n t, shown in F ig . 5.3. was im p le ­
m e n te d in th e e va lu a tio n o f th e e le m e n ta l m a tric e s and e x c ita tio n ve cto r. T h e local
node and edge n u m b e rin g o f th is ty p e o f e le m e n t, in c lu d in g th e lo ca l edge d ire c tio n ,
is illu s tra te d in th is fig u re .
Each o f the six edges is represented by a v e c to r basis
fu n c tio n defined as
N,
= ( L t l V L l2
-
f.VL,,)/,
where /, (/' = 1...... 6) is th e le n g th o f th e edge a n d L , { j = I
in te r p o la tio n fu n c tio n associated w ith node j .
T h e s u b scrip ts
(a.lS)
-1) is th e scalar lin e ar
and /_> correspond
to th e tw o nodes associated w ith th e i th edge: n o te th a t th e lo ca l d ire c tio n o f th e
edge is defin e d fro m node /[ to i>. T h e scalar fu n c tio n L j is defined as
fo r j = 1...... 4: V r is th e v o lu m e o f th e te tra h e d ro n . T h e co n sta n ts a*. / / . c' . and
(t- are expressed in te rm s o f th e g lo b a l x . ij a n d r co ordin a tes o f th e fo u r nodes.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m
S p e c ific a lly , these are given by
=
=-l(x 2 !J.i — -i'3i/2) + - 3( 2*41/2— J-'iUl) *+*
a.% =
— .rxi/ 3 )
:
-1'2 .71 ) +
3
{ x i y A—
— x a!/ 3 )
(5.50a)
+ : i { x 4 t/ 3 — x 3 ij4)
(5.50b)
«3
=
- i ( - r i //2 -
~ JIZ/.|) + =i{-C’U\ - -r-iyi)
(5.50c)
«4
=
- 3 ( j'2 '/l — J-'lUl) + - 2 ( J’ l//3 — x zUl ) + —I ( J*3 i /‘2 — 3*27.3)
( 5 .5 0 c l)
- 2 ( 3*171
=
r 2(73 — 7 i) + -3 (//i — 72) + -.i (.(/■> _ !J:i)
(5.51a)
^2
—
~1(//-! — 7 .3 ) + -3(.{/1 ~ lJ i ) ~t~~ i ( 7.3 — !Ji)
(5.51b)
^3
=
- i (.'/2 — .Vi J + - 2( 7-1 — .Vi) + - i( '/ i — .72)
(5.51c )
b\
=
- 1( 7.3 — .7 2) + ->(.7i ~ ,7.i) + - 3(.72 — .7i)
(5 .5 Id )
=
—2( -2"t — .i';i) + - i (.2" 2 —3*.( )
(•)..)2ft)
c)
~I ( -2"3 — 3* >)
c% =
C[(.r.} — .r.|) + - 3(3*4 —.rt ) + ~_i ( .r 1 — 2 * 3
c-3
=
—1( .2*.| — .r 2 ) +
C,
=
Cl(.r-2 — 2*3) 4 - -2(2*3 — 3‘ \ ) + C;j(.l’ i
)
- 2(-f 1—.r.|) + —-1(-**•> — - i'i)
— X >)
(5.o2b)
(5.o2c)
(■>.•>2(1 )
d\
=
.72 (3.3 — -r i) + . 7 3 ( — 2 *2 ) -r //.((.i'j —
-c.i)
(5 . 5 - ia )
( ^2
=
.71(•*'•! — 3*3) -T .7 3 ( 3*1 — 3*.|) + .7 1 ( 3 *3 —
./*! )
(.)..).lb )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
134
4
= <J\{-1'2 ~ -i'-i) + t j i { .r.i - X ! ) + //.,(a-! -
x-i)
(5 .5 3 c)
4
= y lU ’3 - J-j) + '/•il-i-l - ^ 3) + !/3(-T-2 -
J-[ )•
(5..5:3cl)
T h e volum e o f a te tra h e d ro n can be c a lc u la te d using th e n o d a l co ordin a tes o f the
elem ent
Ve
=
^ { x . r ( L . 4 ) [ /y.!/ ( 2 . 4 ) c r ( 3 . 4 ) - y y ( : 3 . 4 ) c c ( 2 . 4 ) ]
b
+
tj!j( 1.4 ) [ r r ( 2 . 4 ) x x ( 3 . 4) — - - ( :3. 4 ) . r . r ( 2 . 4) J
+
” ( l.4 ) [.r .r ( 2 .4 ) ;y i/( 3 .4 ) - .r.r(:3.4 );yt/(2. 4 )]}
(5.54)
w here x x ( i . j ) . y y ( i . j ) and z z ( i . j ) are d e fin e d as
■c-i'('-j) =
-i-'-J-j
(5.55a)
!J!l{i'j) =
'A - !Jj
(5.55b)
r c ( /.y )
=
(5.55c)
T h e scalar in te r p o la tio n fu n c tio n L t has th e desirab le p ro p e rty o f b eing equal to
u n ity when e v a lu a te d a t th e i th e le m e n ta l node. In a d d itio n , it vanishes at a ll (jo in ts
on the surface o p p o s ite to th a t node. As a re s u lt, th e v e c to r basis fu n c tio n N , has a
co nsta nt ta n g e n tia l co m p o n e n t a long th e i lh edge and zero ta n g e n tia l co m p on e nt on
a ll o th e r edges. T h is p ro p e rty o f th e v e c to r basis fu n c tio n s becom es e x tre m e ly useful
w hen a zero ta n g e n tia l e le c tric fie ld m u s t be im posed on th e surface o f a p e rfe c tly
c o n d u c tin g w a ll.
A n o th e r im p o rta n t p ro p e rty o f these basis fu n c tio n s , in a d d itio n
to th e ones m e n tio n e d at th e b e g in n in g o f th is section, is th e a b ilit y to s a tis fy th e
c o n tin u ity
o f th e ta n g e n tia l fie ld across th e edges o f each e le m e n t. O n
th e o th e r
h an d , the n o rm a l com ponent o f th e fie ld is allow ed to be d is ro n tin u o u s across edges.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B y e m p lo y in g th e te tra h e d ra l ve cto r basis fu n c tio n s , w h ich we a lre a d y in tro d u c e d
a nd discussed in th is se ctio n , th e e le m e n ta l m a t r ix [ d / e] can be e v a lu a te d a n a ly tic a lly .
For s im p lic ity , it w o u ld be p ro b a b ly a good id e a to s p lit th e e le m e n ta l m a tr ix [ d / e]
in to tw o su b -e le m e n ta l m a tric e s d en o te d by [ E e] a n d [ F e] where
£7,
=
Fti
=
J V x N , ) - - ( V x
f
N>/1'
(5 .56 )
(5 .57 )
N , • [e,I • N j dV.
T h u s , th e o rig in a l e le m e n ta l m a t r ix [ . \E ] is s im p ly a s u p e rp o s itio n o f [ E f ] and [ E ' \
A/? = E] j - k i Ffy
(5 .58 )
T h e a n a ly tic a l e v a lu a tio n o f these tw o m a tric e s begins w ith the te n s o r d e fin itio n o f
p e r m it t iv it y and p e r m e a b ility g ive n by
^ j\r
9
=
^ zr
(
t
-
f yy
e V-'
f --.V
f..
I'r
=
/^ j- r
l Lx y
I 1r :
llyr
l ‘ «y
Vy=
1 ': :
.
T h e in ve rse p e rm e a b ility te n s o r, w h ich is e v a lu a te d s y m b o lic a lly u s in g basic lin e a r
a lg e b ra , is defined as
/ C
f‘r 7
*=
^
/ '- ' J
L /C
(5-59)
U s in g th e v e c to r in te r p o la tio n fu n c tio n s in tro d u c e d in (5.-18). th e c u r l of N , becomes
2 /,
^
x N ‘
=
+
w h ic h is c o n s ta n t.
}6 \
> [" r ( c i,
- </„ c „ ) + a y(d n bh - bn c/l2)
(>Al>nCl2 -
(5 .60 )
T h e re s u lt in (5.60) can be e q u iv a le n tly expressed in a m a tr ix
fo rm g iv e n by
Cf
V x N, =
C':
C;
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
( 5 .6 1)
156
where
Cf = cndi2 - dixa
(•5.62a)
Cf
=
dii^io
^*1^*2
(5.62b)
Ct
—
bXlci2
ct i bX2.
(5.62c)
S u b s titu tin g (5 .6 1 ) in to (5.56). th e e le m e n ta l m a tr ix [E ^ ] can be w ritte n as
i
inijv
r xin
\ cr
v
q
7,7
q r r yy
77 77 7=7 . LC;
r-e _
,J
-Mjlj
Cf Cf Cf
( 6 l " e )'* Jq'
■ /*£■
dV.
(5.63)
(5
.
A fte r p e rfo rm in g th e above m a tr ix m u ltip lic a tio n , a m o re s im p lifie d expression can
be o b ta in e d :
E*
,J
=
[ l‘ lj •
(6 \ "e )•'
1
c ' c; <l v+
[ r c;c<<n■+
,C
<7 / n. crc; <n' +
I
,
d V +
j a. c’q ^
1
+ / C / n.
c ’ c ' dV
c;c> ,iv
f . CCJ <«' + tfr Ja, c;c; </ij.
(■>.«i >
E very sin g le in te g ra n d in (5.64) is co n sta n t and. th e re fo re , th e fin a l expression for
[ E c] is s im p ly
r
=
[ V ' l ‘ lj
( 6 1 - ') '
{,c c fc ; + /-ircrc; + /c c fc ;
, C " C 'c ; + / < 7 t 7 c ; + i ^ c ; q
K?;cfc; + , c c c ;
+ ,.;rc/c;} ■
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
c>x.->>
l:E
M a tr ix [ F f \. w h ic h is defined in (5 .5 7 ). is evalua ted u sin g a s im ila r approach, h ir s t,
th e in te g ra l o ve r th e e le m e n ta l v o lu m e is w ritte n in a m o re c o n ve n ie n t m a trix fo rm
g ive n bv
a?
-zx
£yz
tzz
,4 '^
x!
=
f-xy
£yy
t-u
<-zy
. v
.
1
(
^xx
(IV
;
- V
( 0.6 6 )
.
w here X f . fo r e x a m p le , is the x c o m p o n e n t o f th e v e c to r basis fu n c tio n th a t c o r­
responds to th e lo c a l edge j .
T h e th re e co m p on e nts o f N are re la te d to the sca la r
in te rp o la tio n fu n c tio n s th a t co rre sp o n d to th e fo u r nodes o f th e e le m e n t. It can be
shown th a t these are
x? =
"( L i\ (-i>
61
-v ;
L"M'
[ -5.61 a )
L 12 Ct j )
(5.67b)
)
(5 .6 7 c )
=
w here L , t and L,, are th e scalar in te rp o la tio n fu n c tio n s associated w ith th e tw o node
o f edge i. M a t r ix m u ltip lic a tio n in (5 .6 6 ) results in
f
Jq-
X
?
J
\
+
X U X ' d V + ( :x f
Jc i'
x f x ; , i v + c „ J a_ . y .'.y ; ,i v +
/ fl,
X f X ; d V + e„
A'.'.V;" d V
JQ ’
a: v
,n ■
j a .Y».Y;,/r + t„ / q_ ,Y;.Y;rfr.
(5.68)
Each in te g ra n d in (5.6S) is a fu n c tio n o f p o s itio n , th e re fo re , it ca nn o t be factored o u t
o f th e in te g ra l. E v a lu a tio n o f these in te g ra ls w o u ld be easier and m o re u n d e rs ta n d a b le
i f th e fo llo w in g n o ta tio n is adopted:
'
"
=
i
V
' - V / - ' V
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(5.6!) 1
T iie rest
o f th e
in te g ra ls in
( 0 .6 8 )a re d e fin e d in a s im ila r way. S u b s titu tin g (5.67a)
in to (5 .6 9 ). th e in te g ra l I lcJi becomes
Iff
=
) ( £ „ * * - L „b h )4 V
=
{ m » X . L i <L j ‘ d V + 6" 4 j' L
-
b„bJ: [
j
L n L j , d V + bh b „
JQ'
J Q'-
£ i>L » d V
L „ L n <lv).
(5.70)
J
B y in tro d u c in g a n o th e r n o ta tio n , each o f th e in te g ra ls in (5.70) is w r itte n as
lim jn
=
[
(0.71 )
L > , n L Jn ( l V -
J n-
T h is v o lu m e in te g ra l can be e v a lu a te d a n a ly tic a lly using the fo llo w in g fo rm u la :
A -'/'/n 'n '
= 6 1 -— — — — - — - — .
(A: + / + rn + n + •>)!
r
/ ( L ^ iL ^ iU r iL .r d V
J n-
(5.72)
T h u s , th e fin a l expression o f m a tr ix [ F c] is g ive n by
F -J
=
t r x I f j r
+
+
+
C j - y f f f
+ <-z,, A '/ +
t r z f f j '
+
t y x t ? j T
+
+
(-u y f't?
< - y J ! j
/ rr
w here
A7
=
I lj
=
X;
=
Xj
^
( T v 77
bj J ' u i
l b ‘ 2 CJ2
( ( jV T 7
'<■
- M j. l -U2
+
"b
T
b J2
b ‘ l f Jl
F ijy
~
b i 2 CJ\ l ‘ \J2
l l 2 J2 ~~ b‘ 2 (h l
XlJl "bCn^Ji7!2 J 2
C‘2 bJ-.^:U2
~
b n b j2
[>2j>}
( r> - ~
~
b ‘ l C J2 f ' 2 J l }
( v )-<
~ bn C^J2 ^ 2 J\ }
(
‘ ! b J2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
^ ‘ 2J\
}
la
>
(5.71c)
(•)./ Id)
i:W
Aj
=
{ Cn C j2
A'j
=
{ C >2 d J 2 [ n n
A j
=
Aj
—
Aj
0.2.5
A 17. +
, f - \ V \ ’> { ( A 2 k j 2 A . J i
(61 * ) -
(6 V ’e)2
C I l CJ l E 2 J 2
~
C‘ l (hi I ‘ 2 J2
+
c'-'- A 'ji +
~
< A , A n / , ' 2J 2
cji A2j 2
~~ C'l
C l 2 CJ l ^ ‘ U 2
C ‘ 2 (b l ^ ‘ U 2
( l , 2 b Jl A iJ 2
~~ C,l ( ^ J 2
I ‘ 2J 1
A /2 I l 2 j l
fAjCj, / , U2
(g ^ -e )2 { d ‘2(tj2 A iji ■+■ fA[<Ai A 2J2
}
CJ 2
< M j, A U2
(5 .l4 e )
}
(-"j.T-lf)
}
( -I. I 4 g )
d i \ cj2 A 2 J1 }
fA ,fA i A 2J, } •
( 5 ./-th)
(5 .7 4 i)
R a d a r cross se ctio n e v a lu a tio n
O nce th e fin a l g lo b a l m a tr ix syste m is assem bled and solved to o b ta in th e e le c tric
fie ld d is tr ib u tio n in s id e th e c a v ity , th e far-zone sca tte re d fields can be ca lcu la te d
using th e surface m a g n e tic c u rre n t in th e a p e rtu re and th e s p e c tra l d o m a in Green s
fu n c tio n defined in ( 5 . ld )-(5 .2 0 ).
T h u s , th e sp h e rica l co m p o n e n ts o f th e far-zone
sc a tte re d e le c tric fie ld is g ive n by
ps
£ 9
_
~
j k o cos0 e~j k ° r - u
7,--------------- U 6 U - r s - H O '
r
Y f E j { - . U ^ f A v s . A-v.,) s in o 4je.\
El
=
A-y.,) cos 0 }
(5.75)
j k 0 cos Oc j k ° r ~
G " { k rs. k „ s )
•)-
y E j {.U ^ fA v .,. k J, S) cos o + M l j j ( k r s . kys) sin 0 }
jeA
w h e re A is th e a re a o f a tria n g le in th e a p e rtu re and j
=
(5.76)
I...A ..;: A .i being the
n u m b e r o f edges in th e a p e rtu re . It is in te re s tin g to realize th a t the se far-zone field
expressions can be w r itte n ve ry c o n v e n ie n tly in te rm s o f the e x c ita tio n vectors given
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
H)
in (3.21) and (5 .2 2 ). I t can be show n th a t these are e qu iva le nt to
El
j A'0 cos 0 t ~ j k ° r
=
hard
Z
(5 .7 i )
eA
J'6.4
Et
=
(5.78)
•£ E ,h ? ‘.
jeA
In o th e r w ords, th e far-zone sca tte re d fields are c a lcu la te d by a s im p le m u ltip lic a tio n
o f th e s o lu tio n v e c to r and th e e x c ita tio n ve cto r. T h erefo re , assum ing th e m a g n itu d e
o f th e in c id e n t e le c tric fie ld is u n ity , th e RCS o f a ca vity-b a cke d a p e rtu re is g ive n by
cree
ae0
=
-
phard^hard
I, 2 i (7512
(*o Z o )'
h m - l - r l E J = — -------
r — -x.
4
lim 1 - / - | E * |
=
tt
j€A
U-o Z 0 )2
s-jft I h a r d
Z
Er
Z
U J
jeA
( k 0Z 0 )2
(To0
r ' h a rd k sof^
j€--t
=
5.2.6
I-
i
2| r s i ' -
M
)
(5.79)
E * ° f t b*ojt
~
lim 4 - r \ E 0 \ = ---- ------r - :^
-1“
J
(5.80)
(5.81)
5.82)
j€A
V a lid a tio n o f ra d a r cross section
A 5 -D h y b rid F E M / M o M code was w ritte n using F O R T R A N 77 to c o m p u te th e RCS
o f c a v ity -b a c k e d a p e rtu re antennas m o u nte d on an in fin ite g ro u n d plane. T h e c a v ity
is d is c re tiz e d u s in g te tra h e d ra l elem ents. T h e m eshing is done w ith a c o m m e rc ia l
package called S DRC' I-D E A S [18-1].
Once th e mesh is c o m p le te d , th e b o u n d a ry
c o n d itio n s , such as D iric h le t and a bsorbing b o u n d a ry c o n d itio n s , are a p p lie d on p re ­
assigned surfaces. T h e mesh file is th e n e x p o rte d in to an A S C II fo rm a t: sp e c ific a lly ,
a C O S M IC X A S T R A X fo rm a t w h ich is la te r read and processed by th e m a in code.
W h ile ru n n in g , th e code p rin ts out valuable in fo rm a tio n such as m a te ria l d e fin itio n .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
to ta l n u m b e r o f e le m e n ts, types o f b o u n d a ry c o n d itio n s , and so on. T h is in fo rm a tio n
can be ve ry h e lp fu l d u rin g the d e b u g g in g process. I f a fu n d a m e n ta l e rro r occurs the
code a u to m a tic a lly te rm in a te s and p rin ts o u t an e rro r message id e n tify in g th e cause.
T h e code also w rite s th e g e o m e try in fo r m a tio n in to a d a ta file w h ic h can be read
and displayed b y G E O M V T E V V : th e la t te r is a g e o m e try v is u a liz a tio n package. T he
displayed c o lo r fo r surfaces depends on th e specified b o u n d a ry c o n d itio n , the re fo re
a llo w in g not o n ly th e g e o m e try to be v is u a liz e d b u t also to check th e v a lid ity o f the
im posed b o u n d a ry c o n d itio n s . F in a lly , as a post-p ro ce ssin g ste p , th e code is to ta lly
in terfa ce d w ith P L O T .M T Y and T E C 'P L O T fo r th e v is u a liz a tio n o f fie ld in te n s itie s
and cu rre n ts.
T h e code was v e rifie d for various s c a tte rin g p ro b le m s in c lu d in g RCS e va lu a tio n
o f a th re e -slo t a rra y backed by an a ir-fille d re c ta n g u la r c a v ity show n in F ig . 5.-1. A
frequency sweep o f th e a rra y at n o rm a l angle o f in cid e n ce is c o m p u te d and co m p ared
w ith d ata o b ta in e d u sing the sp e ctra l d o m a in m e th o d o f m o m e n ts. T h e co m p a riso n
between th e tw o d a ta sets is illu s tra te d in F ig . 5.5. T h e a greem ent is e x c e lle n t for
b o th p o la riz a tio n s .
H ow ever, as th e fre q u e n cy increases, th e d is c re tiz a tio n e rro r
m ig h t become large enough to affect th e a ccu ra cy o f th e re su lts.
T h e same g e o m e try was reconsidered to e va lu a te its RCS versus angle at a fre ­
quency o f 50 G H z . T h e co m parison betw een th e h y b rid a pp ro a ch a nd th e sp ectra l
d o m a in M o M is d e p ic te d in Fig. 5.6.
A g a in , an e xce lle n t a g ree m e n t betw een the
tw o m ethods is c le a rly shown. It is p ro b a b ly w o rth m e n tio n in g th a t an ang le sweep
s im u la tio n using an im p lic it n u m e ric a l te c h n iq u e , such as th e F E M o r th e M o M . is
c o m p u ta tio n a lly less d e m a n d in g th a n a fre q u e n cy sweep s im u la tio n . T h is is because
o f the fact th a t a change in the in c id e n t angle affects o n ly th e e x c ita tio n vector:
thu s, once th e L U fa c to riz a tio n is p e rfo rm e d , o n ly a b a c k -s u b s titu tio n is re q u ire d for
subsequent angles.
T h is is tru e o n ly i f an L U fa c to riz a tio n , in ste a d o f an ite ra tiv e
solver, has been u tiliz e d .
H ie code was also ve rifie d against n o n -re c ta n g u la r s tru c tu re s such as th e c irc u la r
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Lc
Wc
Ws
Ds
Dc
Fig. 5.4: Three-slot array backed by aa air-filled rectangular ca vity: L c — H'. = 0.75 cm.
D c — 0.25 cm . L s = 0.5 cm . lt s = 0.05 cm. D s = 0.25 cm.
patch , w h ich is backed b y a c y lin d ric a l c a v ity , d e p icte d in F ig . 5.7 . T h e c a v ity its e lf
is fille d w ith a lossy d ie le c tric m a te ria l o f t r — 2.2 and t a n S ,, = 0.0009. T h e c irc u la r
patch has a ra d ius o f 2.5 c m whereas the c a v ity has a ra d iu s o f 3 cm : the d e p th ol
the c a v ity is 0.5 cm . T h e m o n o s ta tic RCS p a tte rn o b ta in e d using the h y b rid code
is co m p a re d w ith a p u re s p e c tra l d o m a in M o M for a w id e range o f frequencies. T h e
sp e ctra l d o m a in M o M was im p le m e n te d using e n tire d o m a in basis fu n c tio n s in sid e
th e c a v ity [7 8 ].[SO]. A s illu s tr a te d in Fig. 5 .S. the tw o sets o f d a ta com pare w e ll w ith
each o th e r except near th e resonant frequency. T h e s h ift in th e resonant fre qu e ncy is
b a s ica lly a ttr ib u te d to th e in s u ffic ie n t d is c re tiz a tio n o f th e a n n u la r a p e rtu re . F u rth e r
in v e s tig a tio n o f th is e ffect p ro ve d th a t an increase in th e n u m b e r o f tria n g u la r facets
in th e a p e rtu re ten d s to s h ift th e resonant frequency o f th e a n te n n a to a s lig h tly
h ig h e r value.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1 l-'i
60
ode (FE M /M oM )
a68 (MoM)
a<txt> (FE M /M oM )
a oo (MoM)
50
40
30
Ls.
20
-10
-20
-30
-40 !
-50
24
20
28
32
40
36
Frequency (GHz)
Fig. 5.5: Frequency sweep o f an a rra y o f t hree slots backed by a rectangular ca vity ( 8t =
0° ) .
30
a60
aBB
cto$
aoo
20
lo t
(F E M A Io M )
(MoM)
(FE M /M oM )
(MoM)
-10
/L s ,
-30 •
Ws
-40
Dc
-50
-60
-70
-80
0
10
20
30
40
50
60
70
80
90
9 (Deg.)
Fig. 5.6: Angle sweep o f an a rra y o f three slots backed by a rectangular ca vity ( / =
30 GHz).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
11-
X
Fig. 5.7: Circular patch backed by a cylindrical cavity. The cavity is filled with a dielectric
material of cr = 2.2. t a n 6 e = 0.0009. //r = 1.0. and t a n S m = 0 ( R i = 2.5 cm.
R i = -1.0 cm. c = 0.5 cm).
0.0
H ybrid FEM /M oM j
M oM
(uisgp) QQo
-
10.0
-
20.0
-30.0
-40.0
-50.0
-60.0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Frequency (GHz)
Fig.
5 .S ;
Monostatic RCS versus frequency o f a circular patch backed by a cylindrical
cavity (Z), = 0 ° ) .
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5.3
R a d ia tio n b y C a v ity -B a c k e d A p e rtu re s
A lth o u g h ra d a r cross section e va lu a tio n p ro v id e s a g re a t deal o f in sig h t in to th e
p e rfo rm a n c e o f an a n te n n a , o th e r fig u re s -o f-m e rit such as in p u t im p e da n ce, e fficiency,
ra d ia tio n p a tte rn s , d ir e c tiv ity and gain are also im p o r ta n t and necessary, e spe cia lly
fo r design a n d o p tim iz a tio n purposes.
A b i lit y to n u m e ric a lly s im u la te ra d ia tio n
p ro b le m s a nd c o rre c tly p re d ic t a n te n n a q u a n titie s re q uires th e im p le m e n ta tio n o f an
a ccurate and e ffic ie n t feed m odel. V ario us feed m o d els have been im p le m e n te d in th e
past [33],[67]. how ever th e re is alw ays a tra d e -o ff betw een a ccu ra cy and efficiency.
For e x a m p le , th e p robe feed m odel (d e lta g a p ), w h ic h was firs t im p le m e n te d in FF.M
by J in and V o la k is [67]. a lth o u g h h ig h ly e ffic ie n t it is a ccu ra te o n ly when the gap
spacing is s m a ll. P a ra m e tric studies have show n th a t a c c u ra te re su lts can be o b ta in e d
w hen th e gap spacing is close to A /10 0 . For la rg e r spacings. th e c u rre n t across th e
gap begins to e x h ib it s p a tia l v a ria tio n th e re b y c re a tin g in a ccura cie s.
In a d d itio n ,
such an a p p ro a ch does not take in to a cco u n t th e fin ite ra d iu s o f th e in n e r c o a xia l
c o n d u c to r.
Ciong a n d V o la k is [IS o] re ce n tly proposed a n o th e r ty p e o f feed m odel w h ich is
based on th e m a g n e tic f r ill. The ra d iu s o f th e in n e r c o n d u c to r is im p lic it ly accounted
fo r by e n fo rc in g th e e q u ip o te u tia l c o n d itio n a t th e c o a x ia l o p e n in g . In o th e r w ords,
using th is ty p e o f feed m odel th e T E M -fie ld d is tr ib u tio n is im posed at the co axial
o pe n in g u n d e r th e c o n s tra in th a t th e vo ltag e betw e e n th e in n e r and o u te r co nd u cto rs,
w h ich do n o t p h y s ic a lly e x is t, m ust be c o n s ta n t as a fu n c tio n o f th e a z im u th a l angle.
T h is feed m o d e l, a lth o u g h not e x te n s iv e ly v e rifie d , it was fou n d in [185] to be m o re
a ccu ra te th a n th e probe feed m o d e l, h ow ever it is c e rta in ly m ore co m p lic a te d to
im p le m e n t.
In th is se ctio n , we present an a ccu ra te c o a x ia l feed m o d e l th a t overcom es m ost
o f th e lim ita tio n s fo u n d in o th e r p re v io u s ly proposed feed m odels,
[h is approach
is based on m o d e lin g th e coaxial cable as a c y lin d r ic a l w aveguide s u p p o rtin g o n ly
th e d o m in a n t T E M m ode, a lth o u g h h ig h e r-o rd e r m odes can also be in c o rp o ra te d . In
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
MG
o th e r w ords, ttie c o a x ia l ca b le is draw n and d is c re tiz e d using te tra h e d ra l elem ents
th e re b y tre a te d as p a rt o f th e fin ite -e le m e n t c o m p u ta tio n a l d o m a in .
A firs t-o rd e r
A B C is a p p lie d a t th e e x c ita tio n plane 10 absorb th e re fle cte d waves. T h e discon­
t in u it y a t th e c o a x -c a v ity in te rfa c e however, w ill m o st lik e ly g e n e ra te h ig h e r-o rd e r
evanescent m odes. I f th e A B C at the e x c ita tio n p la n e was n o t p ro p e rly fo rm u la te d
to a b so rb h ig h e r-o rd e r evanescent fields, th e e x c ita tio n p la ne has to be placed at a
c e rta in d ista n ce a w a y fro m th e d is c o n tin u ity to a llo w tho se fields to t o t a lly d ie out.
C o n s id e r th e g e o m e try show n in Fig. -5.9. A c o n d u c tin g p a tc h , w h ich resides in
th e a p e rtu re p la n e o f an a r b it r a r y ca vity, is e x c ite d w ith a c o a x ia l probe. A lth o u g h
in th is fig u re th e c o a x ia l ca ble is o rie n te d along th e v e rtic a l axis, o th e r d ire c tio n s o f
e x c ita tio n can be co n sid e re d .
In a d d itio n , a sin g le la y e r o f d ie le c tric m a te ria l may
be p la ced on to p o f th e g ro u n d plane. A h y b rid F E M / M o M a p p ro a ch is fo rm u la te d
to c o m p u te ra d ia tio n p a tte rn s and in p u t im p e d a n ce o f such a n te n n a s. As p re v io u s ly
discussed, th e F 'E M is a p p lie d in th e in te rio r o f th e c a v ity whereas th e s p e c tra l/s p a tia l
d o m a in M o M is a p p lie d in th e e x te rio r o f th e c a v ity . T h e tw o n u m e ric a l techniques
are c o u p le d th ro u g h th e c o n tin u ity o f the ta n g e n tia l fie ld s in th e a p e rtu re .
T h e a na lysis b e g in s w ith
th e d is c re tiz a tio n o f th e H e lm h o ltz ’s e q u a tio n in a
source-free region
V x ( [ / / , ] - ' • V x E ) - k -; [f r ] E = 0
(5.8:5)
w h ic h represents th e in te r io r o f th e ca vity. T h e p e r m it t iv it y and p e r m e a b ility o f the
m a te ria l in sid e th e c a v ity , w h ic h is denoted as d o m a in Q ( . are tre a te d as fu ll tensors.
T h e c o rre s p o n d in g w e ig h te d re s id u a l using th e G a ie r k in ’s te c h n iq u e is g ive n by
f
JSh
([^ J -'V x E M V x N M /n -^ [
Jci,
[er] - E - N r / 0 = - I
( W ' V x E l - l N x « n)r/,l
JSi
(5.8-1)
w h e re a n is th e n o rm a l to th e su rface u n it v e c to r d ire c te d o u tw a rd ly to th e v o lu m e of
th e c a v ity , and S\ is th e o u te r b o u n d a ry surface o f th a t v o lu m e . T h e surface in te g ra l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
i r
Infinite Half Free Space
Patch
Infinite Ground Plane
+M
+M
-M
-M
Perfectly
Conducting
Walls
Arbitrary Shaped
Cavity
Coaxial
Cable
Probe
Excitation Face
Fig. 5.9: A '2-D view o f a cavity-backed patch antenna mounted on an in fin ite ground
plane and fed w ith a coaxial cable.
shown to th e rig h t-h a n d side o f (5 .8 4 ). can he broken dow n in to
-
f
( [ / 'r ] _ I V x E ) • ( N
X
a n)d A
=
j k 0rh [
JS\
H m( • ( N
X
5 „) dA
J.Apfr
+
f.
( [/'r ]- 1 ^ 7 X
E ) • (N x a „ ) (IA.
(5.85)
J Coax
Im p o s in g th e c o n tin u ity o f th e ta n g e n tia l fields in the a p e rtu re p la n e , i.e .. H " * ' =
H e rt. th e surface in te g ra tio n o v e r th e a p e rtu re becomes
j k 0Z 0 [
H ‘n(
(N
x
a n )
dA
J A per
jk'0z0f
r h ( N x i „ ) (/,i
J Ape r
j k 0Z 0 [
H rrt
•
(N
X
aA dA
J Apr r
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1 is
=
j k 0Z 0 f
W rt
M s dA.
(5 .86 )
J Ape r
In d e riv in g (5.S6). we m ade use o f a n = a z and N x « . = M s. T h e in te g ra l in (5 .8 6 ).
w h ich represents th e e x te rio r to th e c a v ity p ro b le m , is e valua ted using a m ix e d
s p e c tra l/s p a tia l d o m a in M o M w ith tria n g u la r su p p o rt basis fu n c tio n s . T h is in te g ra l
results in th e same a d m itta n c e m a tr ix fo rm u la te d fo r th e RCS analysis.
W ith o u t
necessarily b eing re d u n d a n t, th e c o rre s p o n d in g a d m itta n c e m a tr ix is given by
Y ij
=
(Fr .Fy) j
• M j ( k r . ky ) dkr d k y
-
M , ( - C . -k,j)4 7T ~
where i . j =
G
( kr . k y ) ■M . j ( k r . k y ) dkr d k y (5.87)
J — yc J — yc
1....... \ A: ,\ Ab eing th e
in front o f (-5.87)is due to the
n u m b e r o f edges in the a p e rtu re . T h e m in u s sign
fa ct th a t th e co rre sp o n d in g in te g ra l has been
to the le ft-h a n d side o f th e weak fo rm expression o f H e lm h o ltz 's e q u a tio n .
m oved
Thus,
th e le ft-h a n d side o f th e lin e a r system o f e q u a tio n s is w ritte n as a su p e rp o s itio n o f
th e so-called globed Jin it t element m a t r i x , den o te d as [.!/]. and th e global met hod o f
moments admi t ta nce ma t r i x , d e n o te d as [»']•
T lie fo rm e r is o b ta in e d th ro u g h th e
assem bly o f a ll e le m e n ta l m a tric e s . T h e e le m e n ta l m a trix [ M ' } . w h ic h was e v a lu a te d
a n a ly tic a lly in S ection 5 .2 .-I. is g ive n by
M l = j
w here i . j = I
[(V
x
N.)
• ( V x N , ) - k ; N,- • [er] • N,] d V
•
(5 .88 )
Y c: .Ye being th e n u m b e r o f degrees o f freedom per ele m en t. For
th e rig h t-h a n d side ve cto r, on th e o th e r h a n d , the co rre sp on d ing in te g ra l is
RHS
=
f
( [ / / r ] " ‘ V X E ) ■( N x d n ) d A
J C'u'lT
=
[
( [ / 'r ] - 1 ^7 x E ) • ( h :
JC’o'i r
X
N ) (IA
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(5.8!))
assu m in g th a t the o rie n ta tio n o f th e co axial lin e is a lo n g th e p o s itiv e c -a x is .
I his
in te g ra l m u s t be e va lu a te d o ve r th e coaxial a p e rtu re w h ic h is part o f th e d is c re tiz e d
d o m a in . To e ffe c tiv e ly tru n c a te th e fin ite e le m e n t m esh in s id e the coax, an a b s o rb in g
b o u n d a ry c o n d itio n needs to be fo rm u la te d and e ffe c tiv e ly app lie d at th e co axial
a p e rtu re . A firs t-o rd e r A B C in c y lin d ric a l c o o rd in a te s is give n by
a z x { V x E(,o. o . c ) } — 7 a z x { « . x E (p. o. c ) } = 2 ' E " ^ [ p . o. z )
(o.90)
w here
E ( / j. o . c )
=
E !nc( p . o . z ) + E r e / ( p . o . z )
=
a p— r ~ ' : + a pR — t ' :
P
P
( o. 9 I )
and 7 = a z - f j k z. N o te th a t a . is the a tte n u a tio n c o n s ta n t, which is u s u a lly zero,
and A-. = A-0 y/TTc w here crc is th e d ie le c tric co nsta nt o f th e m a te ria l in sid e th e co axial
cable. T h e d e riv a tio n o f (b.91) begins w ith th e e v a lu a tio n o f V x E in c y lin d ric a l
c o o rd in a te s
—
v -.,
v
OPE
. 10 E P
V x E ( p . o. z) = u 0 — -------— —
az
p ao
,
(o.D'J)
to o b ta in
a. x { V x E ( p . o . z ) }
=
-
-
f
E.
{
P
- d p i - 7 —
E o
— c
c
e
P
E-o
' —H— t '
I P
=
n^O
+ J h - Zlt —
P
J
—~ E ( p . o. z) + 2 ^ E ‘ nc{ p . o . z).
(o .9 :l)
T h e d e riv a tio n co n tin u e s b y firs t re a liz in g th a t
7 <7.
x
{f/-
x
E (p . o. c ) } = —7- E ( /j.
o.
c).
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(o .D l)
I-=50
S u b tra c tin g (5 .9 4 ) fro m (5 .9 3 ) results in th e firs t-o rd e r A B C g ive n in (5.90). w h ich
can be im p le m e n te d in F E M to e ffe c tiv e ly a bso rb p la n e waves p ro p a g a tin g inside the
m esh. T h is e xp re ssio n can also be w ritte n as
( V x E ) t = (d . x E ) - 2~,{a: x E mc)
(5.95)
w here th e s u b s c rip t t in d ic a te s ta n g e n tia l c o m p o n e n ts . B y s u b s titu tin g (5.95) in to
(5.89) and a ss u m in g th a t th e p e rm e a b ility o f th e c o a x ia l cable is th a t o f free space,
th e rig h t-h a n d sid e in te g ra l becomes
RHS =
f
- ( d , x E ) • (d_- x N ) cl A -
J C'o'ix
[
J
C*o a x
2 - ( d , x E 'r,c) • (d - x N ) d A .
(5.96)
T h e firs t in te g ra l in (5 .96 ) can be moved to th e le ft side o f th e in te g ra l e q u a tio n to
be assem bled to th e g lo b a l m a tr ix . T h erefo re , th e le ft-h a n d side o f the o ve ra ll m a trix
e q u a tio n is w r itt e n as
[U + Y + B ] { E }
(5.97)
w here th e c o rre s p o n d in g e le m e n ta l m a trix [f? f ] is g ive n by
B'
fo r i . j = I
= —
f
' ( ( i : x N , ) - (d . x N j ) (IA.
(5.98)
V ' . T h e in te g ra l in (5.98) is e v a lu a te d o ver a ll tria n g le s in th e a p e rtu re
o f the coax. S im ila r ly , th e e le m e n ta l rig h t-h a n d side v e c to r is g ive n by
b] = - 2 -
f
(«_. x E '" 1') • ( d . x N , ) (IA
(5.99)
w here E 'nt: has th e fo llo w in g fo rm :
E '" c = a p — e ~ ' :
(5. LOO)
P
w h ich is th e fie ld re p re s e n ta tio n o f the T E M m o d e in a c y lin d ric a l co axial cable. T h e
in te g ra l in (5 .99 ) can be e v a lu a te d n u m e ric a lly by firs t r e w r itin g th e in c id e n t field
in re c ta n g u la r c o o rd in a te s : how ever, th e o rie n ta tio n o f th e coax m ust be taken in to
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a c co u n t. A ssu m in g th a t th e co a x ia l cable is o rie n te d in the p o s itiv e r- d ir e c tio n . the
in c id e n t fie ld can be expressed in re c ta n g u la r co o rd in a te s as
«j- cos o + <t, sin o
(. l01)
~ z j ) 2 + (i/ - H f ) 2
w here x j and y j are th e .r. y co o rd in a te s o f th e in n e r-p ro b e ce n te r and o is given by
o = a rc ta n ^
^
o G [ —ISO : ISO],
(5.102)
T h e su rface in te g ra tio n o v e r a tria n g le is e va lu a te d by tra n s fo rm in g th e re c ta n g u la r
c o o rd in a te s { x . y ) in to s im p le x c o o rd in a te s (£. //). T h e fin a l in te g ra tio n is ca rrie d o u t
by using Gauss q u a d ra tu re o v e r a u n it square region. Before e v a lu a tin g th e in te g ra ­
tio n . th e tria n g u la r region is m a p p e d in to a re c ta n g u la r region by im p le m e n tin g a
s im p le tra n s fo rm a tio n o f va ria b le s . A s im ila r tra n s fo rm a tio n was used to n u m e ric a lly
e v a lu a te th e e x c ita tio n v e c to r fo r th e RCS p ro b le m . T h is w ill n ot be repeated here.
A lte r n a tiv e ly , the e x c ita tio n v e c to r can be e va lu a te d using Gauss q u a d ra tu re fo r
tria n g le s [185]. T h e d ra w b a c k o f th is app ro a ch how ever, is the d iffic u lty to generate
th e a p p ro p ria te G aussian p o in ts . Based on th e a va ila b le lite ra tu re , these e xis t up to
an o rd e r o f th irte e n w h ich m ig h t n o t be a ccu ra te enough to c a rry o u t the in te g ra tio n
fo r th e e x c ita tio n ve cto r.
T h e re fle c tio n co e ffic ie n t fo r th e case w here th e probe is v e rtic a lly o rie n te d is
c a lc u la te d using th e to ta l fie ld expression a t th e e x c ita tio n plane.
E ( p . o . c = =c) = h p —
+ a pR — t ' z'
P
w here
(5.105)
P
is th e r c o o rd in a te o f th a t plane. E xpression (5.105) is firs t d o tte d w ith
= d p- f —
(5. 10-1)
P
w h ic h corresponds to th e T E M m ode o f th e coax and th e n in te g ra te d over th e co axial
a p e rtu re to o b ta in
/
/
J JC'xj'ir
E(/7. o .
■ e..or,j. ( I A
=
/ /
^
(i p — < ~ ' :r ■ e co,i r <IA
p
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
152
II (l
pR — ( ' :c ■ecotlT (/--I
J JCoax
r'i.~ rb
=
(5.105)
1
e_2‘'-'c /
/ E 0— p d p d o
Jo Ja
p*
R [
[ E 0— pdp do
Ja
p~
JO
=
£ V - " ln ( - ) e
a
+
R E 02~ l n ( - )
a
(5.100)
~'~c
(5.107)
w here a and 6 are. re sp e ctive ly, th e ra d ii o f th e in n e r and o u te r c o n d u c to rs o f the
c o a x ia l cable. T h e fin a l expression fo r th e re fle c tio n coefficient is
R =
r > -1| < 5' • [ [ E (p- • e <=oax d A - e_2‘ '''c.
n 0- " h H “ ) J JCuar
(5.108)
T h e re fle c tio n co efficient in (5.108) was d e riv e d based on the reference p lane at ; = 0.
A lth o u g h th e m a g n itu d e o f th e re fle c tio n co e fficie n t is not affected by th e lo ca tion
o f th e reference plane, th e phase does. H o w e ver, th e phase can be s h ifte d to a nother
reference p la ne using w e ll-k n o w n p ro p e rtie s o f th e .8-param eters. U sin g th e re flection
c o e ffic ie n t in (5 .10 8 ). the in p u t im p e d a n ce o f th e s tru c tu re is g ive n by
I
Zi* =
.
+ S u d2jkcL
T
v
Vl
{™
])
w here L is th e d ista n ce between reference planes. T h e value o f L m a y be p o s itiv e or
n e g a tiv e d e p e n d in g on th e d ire c tio n o f th e s h ift.
T h e far-zone ra d ia te d fields are s im ila r to th e far-zone sca tte re d fie ld s given in
S ectio n 5.2.5. These are ca lc u la te d u sin g th e m a g n e tic surface c u rre n t d is trib u tio n
in th e a p e rtu re . W ith o u t necessarily b e in g re d u n d a n t, the far-zo ne ra d ia te d fields o f
a c a v ity -b a c k e d a p e rtu re antenna are g iv e n by
_
j k o c o s 0 t - j k ° r ~,u
' r0 I ^
^ y.s) *
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Io 3
5 3 E j { - M ' J( k xs.k'ya) s m o + M y j i k ’rs.kys) cos o }
(5.110)
j€ A
_
—
j k 0 cos g e - j t ' f A V
.1
J tt
r
O l"rs- Avs/
E j { . \ r j j ( k xa. k ys) cos o + M y j { k r 3 . kyS) sin 0 }
j
(5 .1 1 L)
€A
w here .4 is a g ive n tria n g le in the a p e rtu re .
K n o w in g the far-zone fie ld s , a n te n n a
c h a ra c te ris tic s such as d ir e c tiv ity , g ain a nd e ffic ie n c y can be c o m p u te d .
•5.3.1
V a lid a tio n o f in p u t im p e da n ce
T h e c o a x ia l feed m o d e l th a t was fo rm u la te d in th e p re vio u s section was su ccessfu lly
im p le m e n te d in th e h y b r id F E M /M o M code.
can
T h e o rie n ta tio n o f th e c o a x ia l cable
be chosen a lo n g one o f th e th re e p r in c ip a l d ire c tio n s .
In o rd e r
to e ffe c tiv e ly
e v a lu a te th e a ccu ra cy o f th is feed m o d e l, it was decided th a t a closed e m p ty c a v ­
it y be a n a lyze d firs t. A 3-D vie w o f the c a v ity u n d e r co n sid e ra tio n is illu s tr a te d in
F ig . 5.10 whereas a m o re d e ta ile d g e o m e try d e s c rip tio n is shown in F ig . 5.11. T h is
sp ecific s tru c tu re was chosen fo r tw o reasons: fir s t, th e c a v ity is t o t a lly closed w h ic h
e lim in a te s possible n u m e ric a l o r e x p e rim e n ta l e rro r due to ra d ia tio n in to free space:
second, th e c a v ity has an in fin ite q u a lity fa c to r ( Q ) w h ich allow s us to e va lu a te th e
proposed feed m o d e l u n d e r th e w orst p ossible sce na rio . T h e m e a surem e n ts fo r th is
c a v ity were p e rfo rm e d using th e HPS510 n e tw o rk a n a lyze r at th e E le c tro .M a g n e tir
A n e ch o ic C h a m b e r ( E M A C ) o f A riz o n a S ta te I ’ n iv e rs ity . A co m p a riso n betw een th e
real and im a g in a ry p a rts o f th e c o rre s p o n d in g re fle c tio n coefficient fo r a fre q u e n c y
band o f 6 G H z is illu s tr a te d in Figs. 5.12 and 5.13. re sp ective ly. A lth o u g h th e c o a x ia l
cable was m o deled o n ly 1 cm long, w h ic h is e q u iv a le n t to A /6 at 5 G H z and A /3 at
10 G H z . th e c o m p a ris o n between the F E M and th e m easurem ents shows an e xcellen t
a gree m e n t.
D u rin g a n u m e ric a l s im u la tio n , one sh o u ld always m ake sure th a t the
le n g th o f the c o a x ia l ca ble is chosen lo n g e no u gh so th a t h ig h e r-o rd e r m odes, w h ich
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
are evanescent below 35 G H z . a tte n u a te b e fo re th e y reach th e e x c ita tio n piane. O th ­
erw ise . these h ig h e r-o rd e r m odes, w hich are n o t ta k e n in to a ccount w hen fo rm u la tin g
th e A B C a t th e e x c ita tio n plane, w ill d e s tro y th e T E M - lik e fields a t th e in p u t te r­
m in a l. A lo n ge r coax, p ro v id e d th a t the m esh d e n s ity re m a in s th e same, results in
m ore a ccu ra te p re d ic tio n s . H ow ever, the n u m b e r o f u n kn o w n s becomes s ig n ific a n tly
la rg e r th e re b y in cre a sin g th e c o m p u ta tio n a l tim e .
T h e same g e o m e try show n in Fig. 5.10 is re -e x a m in e d w ith one o f th e c a v ity
plates c o m p le te ly re m o ve d so th a t th e a n te n n a is now m o u n te d on an in fin ite ground
plane. S p e c ific a lly , th e face located at th e c = 0 p la ne , w h ic h is th e fa rth e s t surface
away fro m the co a xia l p ro b e , is rem oved.
T h u s , th e F E M is used to m odel the
fields in sid e th e c a v ity and th e coaxial ca b le , whereas th e s p e c tra l d o m a in M o M is
used to m odel th e fie ld s in th e e x te rio r re g ion o f th e c a v ity . T h e in p u t im pedance
o f th is c a v ity -b a c k e d s lo t a n te nn a is c o m p u te d w ith in a w id e fre q u e n cy band. T he
m e asurem ents were p e rfo rm e d using the HPS510 n e tw o rk a n a ly z e r at A riz o n a S tate
U n iv e rs ity . As fa r as th e e x p e rim e n t is co nce rn e d, th e a p e rtu re a n te n n a was m ounted
on a fin ite g ro u n d p la ne o f dim ensions 2-1 in x 21 in . T h e sh arp edges were covered w ith
a b s o rb in g m a te ria l to re d u ce d iffra c tio n s . In a d d itio n , th e a p e rtu re was ro ta te d by
an offset angle w ith respect to th e p rin c ip a l axes, th e re fo re d ire c tin g th e d iffra c tio n s
away fro m the a p e rtu re . A com parison o f in p u t im p e d a n c e betw een p re d ic tio n s and
m easurem ents is illu s tr a te d in Figs. 5.1 1 and 5.15. T w o d iffe re n t s im u la tio n s were
co nside re d: one w ith a coax o f length L - = 8 cm . and a n o th e r w ith a coax o f length
L c = 10 cm . B o th cases show an e xcellent a greem ent w ith th e m easurem ents. The
s lig h t d iscre p a n cy in th e resonance s h ift is m o st lik e ly a ttr ib u te d to a reference plane
m is m a tc h in th e m e a su re m e n ts.
However, i f th e coax is s h o rte r, e.g.
L c = I cm .
s ig n ific a n t d iscrepancies b eg in to appear at th e h ig h e r frequencies. Such o bservation
is illu s tra te d in Figs. 5.16 and 5.17.
As was p re v io u s ly n o te d , these discrepancies
betw een p re d ic tio n s and m easurem ents, w hen th e c o a x ia l cable is e le c tric a lly sh ort,
is a re s u lt o f h ig h e r-o rd e r modes th a t are n ot a cco u n te d fo r at th e e x c ita tio n plane.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
>
2
Fig. 5.10: A 3-D view o f an air-filled rectangular cavity fed w ith a 50-Q coaxial cable
oriented in the //-direction.
A
1.143 cm
I
I
2.286 cm I
iI
T
i
1
5.85089 cm
1.905 cm
Diameter = 0 .1 2 7 cm
1.016 cm
i 0.6 9 85 cm
JL
Diam eter = 0.4 1 02 1 cm
Fig. 5.11: A 2-D view o f an air-filled rectangular cavitv fed w ith a 50-Q coaxial cable
oriented in the //-direction.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reaj Part of Reflection Coefficient
1.0
0.8
M e a su re m e n ts
-• F E M (L c= l cm)
0.6
0.4
0.2
0.0
-
0.2
-0.4
-
0.6
-
0.8
-
1.0
o.
Frequency (GHz)
L2: Real part o f the reflection coefficient o f a closed air-filled rectangular cavity fed
w ith a 5 0-fi coaxial cable.
Imaginary Part Reflection Coefficient
1.0
0.8
__ M e a su re m e n ts
- F E M (L c= l cm)
0.6
0.4
0.2
0.0
-
0.2
-0.4
-
0.6
-
0.8
-
1.0
Frequency (GHz)
Fig. 5. Id: Im aginary part o f the reflection coefficient o f a closed a ir-fille d rectangular cavity
fed w ith a 50-f i coaxial cable.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
157
400
Measurements
.. FEM (Lc=8 cm)
FEM (Lc=10 cm)
Input resistance (Ohms)
360
320
280
240
200
160
80
40
o.
4
O
6
7
Frequency(GHz)
8
9
10
14: Input resistance of an air-filled cavity-backed slot antenna fed w ith a 50-Q coax.
300
M e a s u re m e n ts
Input reactance (Ohms)
250
FEM (Lc=8 cm)
FEM (Lc=10 cm)
200
150
100
50
-50
-150
-200
Frequency (GHz)
Fig. o .lo : Input reactance o f an air-filled cavity-backed slot antenna fed w ith a 50-Q coax.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
400
_ M e a su re m e n ts
-• F E M (L c= l cm )
Input resistance (Ohms)
360
320
280
240
200
160
120
80
40
4
o
6
7
8
9
10
p.
Frequency (GHz)
16: Input resistance of an air-fil!ed cavity-hacked slot antenna fed w ith a short cable.
300
Input reactance (Ohms)
250
—
M e a su re m e n ts
F E M ( L c= l cm )
200
150
100
50
-50
-150
-200
Frequency (GHz)
Fig. 5.17: Input reactance of an air-filled cavity-backed slot antenna fed w ith a short cable.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
150
z
A
Fig. 5.18: C ircu la r patch backed by a cylin d rica l ca vity mounted on an in fin ite ground
plane and fed with a ve rtica lly oriented coaxial cable (/?[ = 2.0 cm. /?•> = 2.4-1
cm. c = 0.218 cm. er = 2.34. tanb, = 0.0012. ftr = 1.0. t nn6m = 0 . R j = 0.7cm.
a = 0.045cm. b = 0.15cm. erc = 2.08).
Besides re c ta n g u la r shape a p e rtu re s , th e code was also va lid a te d fo r a c irc u la r
m ic ro s trip p a tch backed by a c y lin d ric a l c a v ity , as show n
in F ig . 5.18. T h e c y lin d r ic a l
c a v ity is fille d w ith a lossy m a te ria l o f cr = 2.33 and t an S r = 0.0012.
In a d d itio n ,
it
is flu s h m o u n tc d on an in fin ite p e rfe c tly c o n d u c tin g g ro u n d plane. T he c irc u la r p a tch
is e x c ite d w ith a 50-Q coax at an o ffse t d ista n ce o f 0.7 cm fro m the c e n te r ot th e
p atch . T h e re m a in in g geo m e try s p e c ific a tio n s are g iven in th e c a p tio n o f F ig . 5.18.
T h e in p u t resistance o f the patch a n te n n a versus fre q u e n cy is illu s tra te d in F ig . 5.10.
T w o d iffe re n t d is c re tiz a tio n s were co nside re d:
a) th e a p e rtu re consists o f 300 u n k n o w n s
b) th e a p e rtu re consists o f 600 u n kn o w n s
W h e n c o m p a rin g th e results o b ta in e d u sin g the h y b rid F E M / M o M code w ith re su lts
o b ta in e d u sing th e ent ire -d o m a in s p e c tra l M o M [80]. it is cle a r th a t th e a m p litu d e
o f th e in p u t resistance does not s tro n g ly depend on th e n u m b e r o f u n kn o w n s in th e
aperture’ , how ever t ho resonant fre q u e n cy does. T h e reason stem s fro m th e fact th a t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
160
th e a p e rtu re has a c irc u la r shape and c a n n o t be p re cise ly a p p ro x im a te d using a fin ite
n u m b e r o f lin e a r tria n g le s . In o th e r w o rd s, th e la rg e r the n u m b e r o f tria n g le s in the
a p e rtu re , th e m o re a ccu ra te the e x te rio r s p e c tra l/s p a tia l in te g ra tio n w ill be. From
F ig . 5.19. it is e v id e n t th a t by in cre asing th e n u m b e r o f u n kn o w n s in th e a p e rtu re ,
o u r p re d ic tio n s a p p ro a ch closer th e d a ta o b ta in e d using the e n tire -d o m a in M o M . U n ­
fo rtu n a te ly th o u g h , th e spectral d o m a in p a r t o f th e code becomes c o m p u ta tio n a lly
expensive as th e n u m b e r o f unknow ns in th e a p e rtu re increases. T h is is because o f
th e d ouble in fin ite in te g ra l in vo lve d in c a lc u la tin g th e a d m itta n c e m a tr ix elem ents.
A lth o u g h th e a s y m p to tic e va lu a tio n o f th e s p e c tra l in te g ra l im p ro ve s th e c o m p u ­
ta tio n a l speed b y a p p ro x im a te ly a fa c to r o f 5. it is s t ill re la tiv e ly c o m p u ta tio n a lly
expensive fo r fine d is c re tiz a tio n s . In such cases, a fre q u e n cy in te rp o la tio n o f the ad ­
m itta n c e m a trix e le m e n ts can be in tro d u c e d . In o th e r w ords, in ste a d o f e va lu a tin g
th e a d m itta n c e m a tr ix at each fre qu e ncy p o in t, it is e valua ted o n ly at few frequency
p o in ts and in te rp o la te d at in te rm e d ia te ones. T h is w ill be th e to p ic o f th e fo llo w in g
se ction .
5.4
Frequency In te rp o la tio n o f A d m itta n c e .M a trix
It was p re vio u sly m e n tio n e d th a t th e e x te r io r c a v ity p ro b le m is m o deled using a h y ­
b rid s p e c tra l/s p a tia l d o m a in m e tho d o f m o m e n ts . T h e s p a tia l in te g ra tio n is used to
e valua te the a s y m p to tic p a rt o f th e e x te rio r in te g ra l. A h y b rid s p e c tra l/s p a tia l do­
m a in approach, used to e valuate th e a d m itta n c e m a tr ix , is c e rta in ly c o m p u ta tio n a lly
m o re e fficien t th a n a p ure sp ectra l d o m a in a p p ro a ch . H ow ever, fo r cases where th e
n u m b e r o f edges in th e a p e rtu re are on th e o rd e r o f few h u n d re d s, even the h y b rid
s p e c tra l/s p a tia l d o m a in approach becom es in e ffic ie n t. S p e c ific a lly , the tim e re q uired
to f ill in the a d m itta n c e m a trix increases s u b s ta n tia lly w ith in cre a sin g th e n u m b e r o f
u nkn o w n s in th e a p e rtu re . To a lle v ia te th is p ro b le m , a fre q u e n cy in te rp o la tio n o f the
a d m itta n c e m a tr ix is recom m ended. T h e in d iv id u a l e n trie s o f th e a d m itta n c e m a trix
( Y , j ) are know n to change very g ra d u a lly w ith v a ry in g frequency. T h u s, kn o w in g the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
161
120
110
(F E M /M o M ) - NA=300
(F E M /M o M ) - NA=600
(M oM )
100
90
£
a
u
a
a
-m
•^
2.0
2.1
2.2
2.4
2.3
2.5
2.6
2.7
2.8
2.9
3.0
Frequency (GHz)
Fig. 5.19: In p u t resistance o f a circular cavity-backed patch antenna.
a d m itta n c e m a tr ix a t tw o fre qu e ncy p o in ts , e.g. f \ and /> . th e a d m itta n c e m a trix
a t an in te rm e d ia te fre q u e n c y p o in t / can be o b ta in e d using an in te r p o la tio n scheme.
A lth o u g h h ig h e r-o rd e r in te rp o la tio n schemes can be used, a lin e a r in te rp o la tio n was
fo u n d to be q u ite a c c u ra te and m ost e ffic ie n t in te rm s o f m e m o ry o r d is k savings.
T h e p ro ced u re b egins w ith firs t g e n e ra tin g th e a d m itta n c e m a tric e s V j 1 and Y'/.~
a t th e tw o frequencies f \ and f >. These are stored in tw o s e p a ra te s c ra tc h files on th e
d is k in o rd e r to save m e m o ry space. A t in te rm e d ia te fre q u e n cie s, in s te a d o f using
th e m e th o d o f m o m e n ts to fill in th e m a tr ix , in te rp o la tio n was a p p lie d using lin e a r
fu n c tio n s . In o th e r w o rd s.
Y/j = C \ Y /j 1 + C 2 Y / /
(o.l 12)
w here C\ and C'> are g iv e n by
/■ > C ,
/
=
( 0 .1 1 3 )
J
J —J l
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
162
Aperture
Infinite ground plane
Fig. 5.20: G eom etry o f a square aperture backed by a square cavity w ith dimensions .1
B = C' = 5 cm and a = b = 1 cm.
f - h
(6.111)
h - h
T o ensure a ccuracy in th e s o lu tio n . f 2 — f \ sh o u ld not be to o large.
For m ost ot
th e a n te n n a a p p lic a tio n s fo u n d in th is d is s e rta tio n , it is tru e th a t it f 2 — f i is equal
to o r s m a lle r th a n I G H z an e xcellen t a ccu ra cy in th e s o lu tio n is o b ta in e d .
I f the
tw o fre q u e n cy p o in ts f x and /_• are m oved fa rth e r a p a rt, the a ccu ra cy in th e s o lu tio n
begins to degrade.
T h e a ccuracy and o v e ra ll c o m p u ta tio n a l e ffic ie n c y o f th e lin e a r in te rp o la tio n
schem e is tested fo r th e e m p ty c a v ity -b a c k e d a p e rtu re shown in F ig . 6.20.
The
RCS (in d B s m ) o f th is a n te n n a is c o m p u te d versus frequency fo r a p lane wave at
n o rm a l in cidence.
T h e c a v ity was d is c re tiz e d using L.006 e le m e n ts, w h ich c o rre ­
sponds to 979 u n kn o w n s.
T h e n u m b e r o f u n kn o w n s in the a p e rtu re is 2-1.
I sing
a p u re sp e c tra l d o m a in a p p ro a ch , th e tim e needed to f ill the a d m itta n c e m a tr ix is
16 seconds (sec). T h is c o m p u ta tio n a l tim e increases e n o rm o u sly w ith in cre a sin g th e
n u m b e r o f u nkn o w n s in tiie a p e rtu re . X o t o n ly th a t, but this process is re peated at
each fre q u e n cy p o in t. T h e RCS o f th e c a v ity -b a c k e d a p e rtu re is show n to r I cases ot
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
in te r p o la tio n in F ig . 5.21. A to ta l o f 60 s a m p lin g p o in ts were used fo r each case. For
th e firs t s im u la tio n , no in te rp o la tio n was used. For th e re m a in in g th re e s im u la tio n s ,
e ith e r S. 4 o r 3 frequency p o in ts w ere used to c a rry o u t th e in te rp o la tio n at th e
in te rm e d ia te p o in ts .
In o th e r w ords, th e a d m itta n c e m a tr ix is evalua ted o n ly S. 4
o r 3 tim e s , re sp e ctive ly, w ith in a b a n d w id th o f 7 G H z . F ro m F ig . 5.21 it is e v id e n t
th a t th e in te r p o la tio n scheme w orks e x tre m e ly w e ll w h e n th e in te rp o la tio n p o in ts
are chosen 1 G H z a p a rt (f 2 — f i — 1 G H z ) w hich co rre sp on d s to case (6 0 /S ). W hen
those p o in ts are spread fa rth e r a p a rt, e.g. cases (6 0 /4 ) and (6 0 /3 ). th e a ccuracy o l
th e s o lu tio n , as shown in Fig. 5.21. becom es p ro g re ssive ly worse. T h e c o m p u ta tio n a l
tim e , re co rd e d on an IB M R IS C /6 0 0 0 w o rk s ta tio n , is ta b u la te d fo r each o f th e s im ­
u la tio n s in T a b le 5.1. A lth o u g h th is p ro b le m is co n side re d c o m p u ta tio n a lly s m a ll,
a c le a r in d ic a tio n on the ru n -tim e savings is o b ta in e d by c o m p a rin g th e in d iv id u a l
C'PL tim e s .
A n im p ro ve m e n t in c o m p u ta tio n a l speed by a fa c to r of 3 is
o b s e r v e d
when u sin g in te rp o la tio n , com pared to no in te r p o la tio n , w ith o u t necessarily s a c ri­
fic in g p re c is io n .
Im p ro vem en t by a la rg e r fa c to r is observed when th e n u m b e r o l
u n k n o w n s in th e a p e rtu re is increased.
Table 5.1: Com putational time recorded on an IBM RISC/6000 workstation for the square
cavity-backed aperture.
5.5
Scheme
C P C tim e (m im s e c )
No In te rp o la tio n
25:33
In te rp o la tio n 6 0 /8
9:58
In te rp o la tio n 6 0 /4
8:45
In te rp o la tio n 6 0 /3
8:22
D ir e c t iv ity . G a in and E fficie n cy
G a in p a tte rn s o f a ca vitv-b a cke d slot a n te n n a m o u n te d on an in fin ite g ro u n d plane,
w ith o r w ith o u t d ie le c tric overlay, are e va lu a te d based on th e far-zone ra d ia te d fields
in tht> presence o f a m agnetic surface c u rre n t M s in th e a p e rtu re .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A ssu m in g th a t
164
-10
-15
-20
-25
-40
-45
N o In te rp o la tio n
_ _ In te rp o la tio n (60/8)
.. _ In te rp o la tio n (60/4)
— In te rp o la tio n (60/3)
-50
-55
Frequency (GHz)
Fig. 5.21: RCS computations using linear interpolation for the admittance matrix.
th e a p e rtu re lies in th e .i-//-plane. th e sp he rica l c o m p o n e n ts o f th e e le c tric field in th e
fa r-fie k l zone are given b y (5.110) and (5.111) in S e c tio n 5.4.
K n o w in g the far-zone ra d ia te d fields o f th e c a v ity -b a c k e d s lo t (these fields are
v a lid c n ly in the u p p e r h a lf space), th e m a x im u m d ir e c t iv it y ot th e antenna is c a l­
c u la te d as follow s [186]:
D0 =
i - r r„.,
(5.115)
P rn*
w here V m.lx is th e m a x im u m value o f th e ra d ia tio n in te n s ity I ((.' ) . o ) given by
C (0.o) = ^
[ \ E 6( 0 . o )\-
+
\ E o ( 0 . o )\2]
.
(5.116!
T h e d ir e c t iv it y p a tte rn is. th e re fo re , defined as
D(0.o) =
l-C (O .o)
(5.117)
Pra-i
T h e ra d ia te d pow er P,.,,,/ is e valua ted by in te g ra tin g th e ra d ia tio n in te n s ity (' [( ) . o ]
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
16-3
o ver a h a lf sphere o f large ra d iu s : i.e..
Prad =
/
[ i ' ( 0 . o ) sin 0 dOdd.
Jo Jo
(-5.1 IS)
T h e d ir e c tiv ity a c co rd in g to th e new I E E E S t a n d a r d D e f i n i t i o n s o f Te rms f o r .-1ntennas is defined as "th e ra tio o f th e ra d ia tio n in te n s ity in a g iv e n d ire c tio n fro m the
a n te n n a to th e ra d ia tio n in te n s ity averaged over a ll d ire c tio n s ” . In o th e r w ords, the
d ir e c tiv ity does n o t a ccou n t fo r losses associated w ith th e a n te n n a e le m e n t. Such
losses m ig h t in c lu d e c o n d u c tio n a n d d ie le c tric losses.
A lth o u g h th e d ir e c t iv it y is a u se fu l m easure o f th e d ire c tio n a l p ro p e rtie s o f the
ante nn a , it p ro vide s no in fo r m a tio n on th e a n te nn a e ffic ie n c y . O n th e o th e r h an d , the
d e fin itio n o f gain accounts fo r b o th th e d ire c tio n a l p ro p e rtie s as w e ll as th e e fficien cy
o f th e antenna. T h e a n te n n a e ffic ie n c y , e x c lu d in g m is m a tc h losses, is d efin e d as
*
in
w here P,n is th e p ow er in p u t to th e a n te n n a , not the p o w e r i n p u t to the coax o r tin
t ransmi ssi on lined If Pul is ta k e n to be th e pow er in p u t to th e tra n s m is s io n lin e , then
m is m a tc h losses are p a rt o f th e r a d ia tio n e fficiency. In o th e r w o rd s.
e
w here
ec
is th e c o n d u c tio n e ffic ie n c y ,
(.3.120)
t j
is the d ie le c tric e ffic ie n c y , and T is the
re fle ctio n co efficie n t a t th e fe e d in g te r m in a l o f th e a n te n n a .
B ased on th e above
d e fin itio n s , th e a n te n n a g a in , a c c o rd in g to the new I E E E S t a n d a r d D e f i n i t i o n s o f
Terms f o r A n t e nn as is defin e d as
C ' ( O . o ) = ec t,i D ( 0 . o )
(-3.121)
o r o th e rw ise .
0 (0 . o) =
{ ~. l. y L o \
(-3.122)
*tn
For the ca vity -b a c k e d s lo t, th e p o w e r in p u t to the a n te n n a P,n is e q u a l to th e power
in p u t to the coax P f f m u ltip lie d by th e m is m a tc h loss 1 — 11 \2.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The pow er in p u t
100
to th e coax, a ssum ing fie ld p ro p a g a tio n a lo n g th e r-a x is . is ca lcu la te d based on the
e le c tric fie ld d is tr ib u tio n a t th e in p u t te r m in a l
E cx = — c~jkJr
15.122)
P
w h e re k c is th e p ro p a g a tio n co n sta n t in s id e th e coax, and l c is the a bsolute c co or­
d in a te o f th e e x c ita tio n plane. T h e c o rre s p o n d in g m a g n e tic fie ld , evaluated at the
sam e plane, is g ive n by
(o.L 2-1)
H cx =
rh P
w h e re r)c is th e in trin s ic im p e d a n ce o f th e m e d iu m in sid e th e co a xia l lin e . T h e pow er
flo w in g th ro u g h th e coax is ca lc u la te d b y in te g ra tin g th e P o y n tin g v e cto r over the
c ir c u la r surface o f th e c o a x ia l a p e rtu re
P"
=
J
{ E x H ' } p d p do
(o. 12'))
1 f 2~ rh 1
=
a- — /
2 Jo
/
-pdpdo
J i n.-p1
=
~ b
d - — ln ( - ) .
r/c
fl
(0 . 126)
( 0 . 12/ )
T h u s , th e pow er in p u t to th e a n te n n a is g iv e n by
Pin = — I n ( - ) ( L - | l f ) .
'h
o
(. 0 . 1 2 8 )
T h e above expression is used in (-5.122) to e v a lu a te th e gain p a tte rn s o f an
th a t
ise x c ite d w ith a co a xia l cable. It is im p o r ta n t to also em phasize
and
d ir e c t iv it y p a tte rn s
are u s u a lly d e fin e d a c c o rd in g
antenna
th a t b o th gain
to a given p o la riz a tio n in
a
g iv e n d ire c tio n . For e xa m p le , th e gain o f an a n te n n a m a y be w ritte n as
G
Go
+
G0
=
C iq
=
-i - r , ?
- j r 1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
( o . 12!))
(5.120)
167
G0
=
^
(5.131)
* in
w here i'g and U0 is th e ra d ia tio n in te n s ity a lo n g a c e rta in d ire c tio n c o n ta in in g th e 0
and o c o m p o n e n t o f th e e le c tric fie ld , re sp e c tiv e ly .
5.5.1
V a lid a tio n o f g a in
T h e fo rm u la tio n p resented in th is se ction was im p le m e n te d in th e h y b rid F E M / M o M
code to e valua te e fficie n cy, d ir e c tiv ity and g a in p a tte rn s fo r ca vity -b a c k e d s lo t an ­
ten n as. For v e rific a tio n purposes, co nside r th e C B S a nte nn a shown in Fig. 5 .1 1. T h e
a n te n n a is e x c ite d u sin g a s ta n d a rd 50-f i c o a x ia l cable a t a fre qu e ncy o f 7.5 G H z.
T h e g a in p a tte rn s Gg and G 0 w h ich , in th e absence o f losses are id e n tic a l to th e d i­
r e c t iv it y p a tte rn s , a rc e valua ted a long th e E- and H -planes. T h e p re d icte d p a tte rn s
are co m p a re d w ith M o M results p ro d u ce d by an in d e p e n d e n t source [ 1S7]. T h e c o m ­
p a riso n between th e tw o n u m e ric a l m e th o d s , fo r b o th 0 (E -p la n e ) and o (H -p la n e )
c o m p o n e n ts o f th e ra d ia te d fie ld , is illu s tra te d in Fig. 5.22. As show n, th e agreem ent
betw een th e tw o d a ta sets is e xce lle n t.
5.6
H y b rid iz a tio n w ith the F ilifo r m T h e o ry o f D iffra c tio n
C a v ity -b a c k e d slo t a nte nn a s are u s u a lly fla sh m o u n te d on re la tiv e ly large o b je c ts
such as a irc ra ft, m issile s and o th e r vehicles.
Based on th e fo rm u la tio n presented
iri th is c h a p te r, th e a n te n n a is alw ays m o u n te d on an in fin ite g ro u n d plane: th e re ­
fore. no d iffra c tio n s fro m s u rro u n d in g o b je c ts arc accounted for.
th e h y b r id F E M / M o M
In th is se ctio n ,
code is e xte n d e d to fin ite g ro un d planes b y in c o rp o ra tin g
sin g le and d o u b le d iffra c tio n s fro m edges to e va lu a te th e gain p a tte rn s on p rin c ip a l
planes. T h e d iffra c te d fields are c a lc u la te d based on th e u n ifo rm th e o ry o f d iffra c tio n
( F T D ) [1SS].[189]. w h ic h becomes e x tre m e ly a ccu ra te when th e d iffra c tio n p o in ts are
se pa ra ted by at teast 1 A. T h e h y b rid a p p ro a ch w hich in co rp o ra te s th e F T D m odel
in to th e o rig in a l code is s tra ig h tfo rw a rd [190]. T h e co m p le te ste p -b y-ste p p ro ce d u re
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
168
10
5
0
-5
CQ
3
C
-10
-15
'S3
O -20
-25
-30
E -p la n e
E -p la n e
. . . H -p la n e
H -p Ia n e
-35
-40
-80
-60
(F E M )
(M o M )
(F E M )
(M o M )
-40
-20
0
20
40
60
80
Elevation angle (degrees)
Fig. 5.22: C om parison o f the co-pol gain patterns o f an a ir-fille d cavity-backed .slot.
is d o c u m e n te d a nd c le a rly e x p la in e d in B a la n is's book " Advanced E n g i n t c r m g Elec­
t r oma gn et ic s " . C h a p te r 13. pp 811-814 [168]. fo r a A /1 m o n o p o le a n te n n a on top o f
a fin ite g ro u n d p la ne .
T h e firs t step begins w ith c a lc u la tin g th e ra d ia te d fields o f
an e q u iv a le n t A /2 d ip o le .
In o u r case, th e ra d ia te d fields fro m a c a v ity -b a c k e d slot
m o u n te d on an in fin ite g ro u n d plane arc kn o w n n u m e ric a lly in ste a d o f a n a ly tic a lly .
These ra d ia te d fields w ere ca lc u la te d based on th e surface m a g n e tic c u rre n t d e n sity
in th e a p e rtu re and are g ive n by (5.110) and (5.111).
In th e presence o f a fin ite
g ro u n d p la ne , th e ra d ia te d fields fro m th e a p e rtu re are d iffra c te d fro m th e edges o f
the g ro u n d p lane. In th e p rin c ip a l planes, th e d iffra c tio n s o c c u r at th e fo u r p o in ts
A . B. C'. and D sh ow n in F ig . 5.23. S p e c ific a lly , d iffra c tio n s fro m p o in ts A and B
w ill be added to th e o rig in a l fields ra d ia te d by th e slot in th e presence o f an in fin ite
g ro u n d p la ne to c a lc u la te th e to ta l far-zone p a tte rn a long th e c.r-p la n e . S im ila rly ,
d iffra c tio n s fro m p o in ts C and D w ill be a ccou n te d fo r to c a lc u la te th e to ta l far-zone
p a tte rn a lo ng th e c//-p la n e .
To c a lc u la te th e d iffra c te d fie ld s, th e in c id e n t field at
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
16!)
Aperture
Finite ground plane
Arbitrary
Cavity
Coaxial
C able
Fig. 5.2.'): G eom etry o f a cavity-backed a p e rtu re on a finite ground plane.
th e tw o d iffra c tio n p o in ts m u s t be e v a lu a te d firs t.
B y assum ing th a t th e ra d ia te d
fie ld em a na tes fro m th e c e n te r o f th e a p e rtu re , w h ic h is defined as th e o rig in ot
th e c o o rd in a te system , th e in c id e n t e le c tric fie ld at th e d iffra c tio n p o in ts , to r hard
p o la riz a tio n , is g ive n by
E 'M )
= - E ' OG( r = u-T/ 2 . 0 = - / 2 . o = Q)
E ‘0( B )
= j E rgG( r = U'r / 2 . 0 = - / - 2 . o = - )
m
= [- E rgG( r = icy/ 2 . 0 = - / 2 . o = x / 2 )
i
E'eiD)
= -l E r6G( r = u-,J/ 2 . ( ) = : r / 2 . o = : ] - / 2 )
(5.135)
i.1 3 1 )
(5.135)
w h e re EgG( r . 0 . o ) is th e far-zo ne fie ld ra d ia te d b y th e a p e rtu re in th e presence o f
an in fin ite g ro u n d plane.
K n o w in g th e in c id e n t fie ld a t th e d iffra c tio n p o in ts , th e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
d iffra c te d fields can be w r itt e n as [16S]
Ed
SQ{0) = E k‘ ( Q ) D k ( L ^ Q.J'0. n ) A ( s ' . s ) t - jks
(5.136)
w here Q in d ic a te s th e d iffr a c tio n p o in t. D h { L , ^ Q . 3'0. n ) is th e d iffra c tio n co efficie n t
fo r h a rd p o la riz a tio n and . 4 ( ^ '. s) is the s p a tia l a tte n u a tio n fa cto r. T h e re m a in in g
p a ra m e te rs are defined as fo llo w s :
s = d is ta n c e between th e o b s e rv a tio n and th e d iffr a c tio n p o in t
s' = d is ta n c e betw een th e so urce and the d iffr a c tio n p o in t
L = d ista n ce p a ra m e te r
fg = in c lu d e d angle betw een th e in c id e n t and d iffra c te d rays
J ' = o b liq u e in c id e n t angle
n = wedge fa c to r
For th e r j ’-p la ne . th e s p a tia l a tte n u a tio n fa c to r o f a sp he rica l wave w here obser­
vations are m ade in th e fa r zone is given by
J u'r / 2
-d (s ' = , r j 2 . s = r ?) = X —
rv
(5.137)
w here r 7 is th e d ista n ce b etw een th e d iffra c tio n p o in t Q and th e o b se rva tio n . O n the
o th e r h an d , fo r th e r/y-p la n e. th e co rre sp o n d in g s p a tia l a tte n u a tio n fa c to r is
j
U'y/2
, l( .s' = W yl 2. .S = r , ) =
(5.13$)
17
T h e value o f
fo r the d iffr a c tio n p o in ts A and B is g ive n by
£.4
=
^ + 0
f
SB
0 < 0 < ~
- 0
(5.13!))
0 < 0 < §
(5.110)
=
if _ 0 f < 0 < -
S im ila r expressions are used fo r
a ild £ d - re s p e c tiv e ly . T h e distance p a ra m e te r L is
equal to u'r / ' l a long th e r.r-p la n e and / r v/ 2 along th e r/y-p la n e. The o b liq u e in c id e n t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
171
a ngle 3'0 is ~ / 2 sin ce obse rva tio ns are m a d e o n ly on the p rin c ip a l planes. A lso , the
wedge fa c to r n is e q u a l to 2: th is corre sp on d s to a h a lf p lane. T h e re fo re , based on
these d e fin itio n s , th e single d iffra c te d fie ld s in the r.r-p la n e a re g ive n by
Ee
d A(0)
=
E'h( A ) D h ( L . ^ A . 3 f0. n ) ^ — d l e- ^
\/tc r /2
K- n )
n
(5.141)
,
e~J
S im ila rly , th e sin g le d iffra c te d fields in th e ry -p la n e are g ive n b y
E dc {9)
=
E i ( C ) D h( L . e c . 3 o . n ) ^ ^ - e - Jkr‘
r.-
(5.145)
E'eoiO)
=
E ‘h [ D ) D h { L . ^ D .3'0. n ) ^ l c - j k r E
I'd
(5.144
For fa r-fie ld o b se rva tio n s, th e phase v a ria tio n s are represented b y
r,,
=
r
- s in 0
rf,
=
/• -i— j - sin 0
(5 . 1
r.
=
r
(5.147)
r,{
=
r + -^-s\n9
2
(5.145)
sin 0
K i )
(5.148)
whereas th e a m p litu d e va ria tio n s are represented by
r „ = rt, = r c - r,{ = r.
(5.1 19)
T h e re fo re , th e sin g le d iffra c te d fields fro m th e edges o f th e f in it e g ro u n d plane can
be w ritte n as
. — ; A rr-
:5.150)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
172
£ & (*)
=
BUB)
e~jkr
—
(5 .W I)
fo r th e rx -p la n e . and
E « c (* )
=
E l ( C ) D h ( L ^ c . X ■ n ) ^ / ^ e ^ ' ^ 2)sin^ —
(5.152)
e - jk r
Ed
SD{6)
=
(5-153)
fo r th e r*/-p la n e . These d iffra c te d fields are added to th e ra d ia te d fields in th e pres­
ence o f an in fin ite g ro u n d p la ne , w h ic h are v a lid o n ly in th e u p p e r h a lf space, to
o b ta in th e to ta l fields e v e ry w h e re . In a d d itio n to sin g le d iffra c tio n s , d o u b le d iffr a c ­
tio n s were also ta ke n in to a ccou n t in th is study. These are c a lc u la te d in a s im ila r
m a n n e r.
[191].
For m o re in fo r m a tio n on m u ltip le d iffra c tio n s , th e reader is re fe rre d to
Slope d iffr a c tio n w ill n ot be in c o rp o ra te d here a lth o u g h th is te rm becomes
im p o rta n t fo r soft p o la riz a tio n .
T h e a ccu ra cy o f th e h y b rid F E M / M o M / F T D code is d e m o n s tra te d by e v a lu a tin g
th e E -p la n e g ain p a tte rn , fo r h a rd p o la riz a tio n , o f th e c a v ity -b a c k e d slo t a n te n n a
shown in F ig . 5.11. T h e a n te n n a its e lf is m o u n te d on a fin ite g ro u n d plane o f d im e n ­
sions 5 1 .7 c m x -10.57cm . T h e ce n te r o f th e g round p la n e co in cide s w ith th e c e n te r ot
th e a p e rtu re , whereas th e largest d im e n s io n o f th e g ro u n d p la ne is a lig n e d w ith th e
largest d im e n s io n o f th e a p e rtu re . T h e frequency o f o p e ra tio n is 7.5 G H z. T h e c o rre ­
sp o n d in g E -p la n e g a in p a tte rn (Gg) is c o m p u te d a n d co m p a re d w ith m ea surem e n ts
p e rfo rm e d at th e ane ch oic c h a m b e r o f A riz o n a S ta te U n iv e rs ity . C o n tra ry to th e new
d e fin itio n o f g a in , b o th d a ta sets in c lu d e the m is m a tc h loss ( I — |T ]~ ) betw een th e
a n te n n a and a 50-Q c o a x ia l cable. T h e co m p a riso n , illu s tra te d in F ig . 5.2-1. presents
an e xce lle n t a g ree m e n t betw een p re d ic tio n s and m ea surem e n ts.
T h e a ccu ra cy o f th e h y b rid code was fu rth e r in v e s tig a te d by c o n s id e rin g th e same
co a xia l c a v ity e x a m in e d by R e d d y ( t al. [190]. T h e c irc u la r c a v ity o f ra d iu s 1 in and
d e p tli 5 /S in is e x c ite d u sin g a c o a x ia l cable th a t is m o u n te d at th e b o tto m o f the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
90
90
120
120
FEM
Measurements
160
Fig. 5.24: Comparison o f the E-plane gain p atte rn (Go o f an a ir-filled cavitv-backed slot
antenna m ounted on a fin ite ground plane.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
c a v ity .
T h e ce n te r c o n d u c to r o f the coax w ith ra d iu s 0.0181 in is extended up to
th e a p e rtu re plane. T h e c h a ra c te ris tic im p e da n ce o f th e coax is 50 Cl. whereas th e
c a v ity is fille d w ith a ir. T h e a p e rtu re is fiu s h m o u n te d on a fin ite g ro un d p lane w ith
d im e n sio n s 24 in a lo n g th e .r-d ire c tio n . and 12 in along th e t/-d ire c tio n . The p re d ic te d
d ir e c t iv it y p a tte rn s fo r th e : x - and ri/-p la n e s are illu s tra te d in Figs. 5.25 and 5.26.
re s p e c tiv e ly . A lth o u g h co m p a riso n d a ta is n o t shown in these figures, o u r p re d ic tio n s
lo o k e x tre m e ly s im ila r to those p re d icte d and m easured by R e d d y et al. [190],
In
Figs. 5.25 and 5.26. th e d ir e c tiv ity p a tte rn s fo r an in fin ite g ro u n d plane are also
p resented. For th a t case, th e ra d ia te d fields below th e g ro u n d p lane are zero.
H y b r id iz a tio n o f lo w -fre q u e n c y n u m e ric a l m e tho d s w ith h ig h -fre q u e n cy a s y m p ­
to tic m e th o d s can be e xte n d e d , w ith a p p re cia b le e ffo rt, to m o re co m p lex g eom etries
such as a irfra m e s and g ro u n d vehicles. T h is can be a fu tu re to p ic fo r research.
5.7
C o n clu sio n s
In th is c h a p te r, a h y b rid fo rm u la tio n using th e fin ite e le m en t m e th o d and the m e th o d
o f m o m e n ts was deve lo p ed to analyze ca vity -b a c k e d slots m o u n te d on an in fin ite
g ro u n d p lane and co ated w ith a sin g le-la yer d ie le c tric m a te ria l. T h e a p e rtu re a n d /o r
c a v ity m a y have a r b itr a r y shape whereas th e c a v ity its e lf m ay be fille d w ith an in h o m ogeneous a n d /o r fre qu e ncy-de p en d en t m e d iu m . T h e c a v ity v o lu m e is d isc re tiz e d
w ith te tra h e d ra l e le m en ts and m odeled using th e F E M whereas th e e x te rio r v o lu m e
is m o d e le d using a h y b rid s p e c tra l/s p a tia l d o m a in M o M . T h e e x te rio r in te g ra l was
in it ia lly e valua ted u sin g a p ure sp ectra l d o m a in M o M . H ow ever, in o rd e r to speed up
th e c o m p u ta tio n a l tim e re q u ire d to e valua te th is in te g ra l, an a s y m p to tic approach
was in tro d u c e d w h ic h cre a te d tw o separate in te g ra ls.
O ne ot th e in te g ra ls is s till
e v a lu a te d using th e s p e c tra l approach: how ever it converges m u ch faster th a n th e
o rig in a l in te g ra l, and th e second one w h ich is th e a s y m p to tic p a rt is evaluated using
th e s p a tia l a p p ro a ch . T h is p rovides an im p ro v e m e n t in th e c o m p u ta tio n a l speed by
a p p ro x im a te ly a fa c to r o f 5. A d d itio n a l im p ro v e m e n t can be achieved by im p le m e n t-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
F in ite g ro u n d plane
In fin ite g ro u n d p lane '
■r -10
-25
-3 0
t
-3 5
-1 8 0
-1 3 5
-90
-4 5
0
45
90
135
180
Elevation angle (degrees)
Fig. 5.25: D ire c tiv ity patterns o f a coaxial ca vity at 5 GHz (hard p o la riza tio n . r.r-plane)
F in ite g ro u n d plane
In fin ite g ro u n d plane
S
-5
-4-)
•C -1 0
S -15
5 -20
-2 5
-3 0
-35
-1 8 0
-1 3 5
-90
-4 5
0
45
90
135
180
Elevation angle (degrees)
Fig. 5.26: D ire c tiv ity patterns o f a coaxial ca vity at 5 GHz (hard p o la riza tio n , r^-plane)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
in g a fre q u e n cy in te rp o la tio n schem e fo r th e a d m itta n c e m a trix .
T h e fo rm u la tio n was v e rifie d fo r a v a rie ty o f c a v ity -b a c k e d slo t a nte n n a s fo r b o th
s c a tte rin g and ra d ia tio n p ro b le m s .
For s c a tte rin g , th e RCS versus fre q u e n c y or
e le v a tio n angle is c a lc u la te d . C o m p a riso n s w ith d a ta o b ta in e d fro m an in d e p e n d e n t
s tu d y were presented and discussed. For ra d ia tio n , th e in p u t im p e da n ce, re fle c tio n
c o e ffic ie n t and a bso lu te g a in and d ir e c t iv it y p a tte rn s are ca lcu la te d .
C o m p a ris o n s
w ith m e a surem e n ts and o th e r sources were show n. T h e co m parisons were e x tre m e ly
fa v o ra b le .
T h e h y b rid F E M /M o M fo rm u la tio n was fu r th e r e xte n d e d to tre a t s im ila r ty p e o f
a nte n n a s m o u n te d on a fin ite , in ste a d o f an in fin ite , g ro u n d plane. T h e fin ite g ro u n d
plane is assum ed to be a p e rfe ct e le c tric c o n d u c to r w ith o u t co a tin g .
T h e ra d ia te d
fields fro m th e slot are d iffra c te d fro m th e edges o f th e g ro u n d plane. These d iffr a c ­
tio n s are a ccou n te d fo r u sin g th e u n ifo rm th e o ry o f d iffra c tio n . In o th e r w o rd s, the
to ta l fa r-zo n e ra d ia te d fields b y th e a n te n n a becom e th e su p e rp o s itio n o f th e o rig in a l
fie ld , w h ic h are v a lid o n ly above th e g ro u n d p la ne , and th e d iffra c te d fie ld s , w h ich
are present in the e n tire space s u rro u n d in g th e fin ite g ro u n d plane. T h e h y b r id iz a ­
tio n o f th re e m ethods was v e rifie d by c o m p a rin g p re d ic tio n s w ith m e a surem e n ts and
p u b lis h e d d a ta .
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CHAPTER 6
F E R R IT E - L O A D E D C A V IT Y - B A C K E D A P E R T U R E S
In th is c h a p te r, fe rrite s are used as lo a d in g m a te ria ls in side a ca vity-b a ck e d slot
(C B S ) m o u n te d on an in f in it e g ro u n d p la ne .
T h e in fin ite ground p lane m ay be
covered w it h a sin g le -la ye r d ie le c tric . T h e C B S is considered b o th as a s c a tte re r and
as a ra d ia to r. L: n lik e p re vio u s w o rk on m ic r o s trip antennas using fe rrite su b stra te s,
the c u rre n t s tu d y is p r im a r ily focussed on o p e ra tin g th e CBS in th e L’ H F band
ra th e r th a n th e m icro w a ve b a n d . A v e c to r fin ite e le m e n t m e th o d (F E M ) h y b rid iz e d
w ith th e m e th o d o f m o m e n ts ( M o M ) is used fo r th e analysis o f these antennas. T h e
v e rs a tility o f th e F E M a llo w s m a g n e tiz a tio n in a n y p rin c ip a l d ire c tio n . In a d d itio n ,
a r b itr a r y shapes o f c a vitie s a n d slots m a y be considered.
The e x c ita tio n is based
on a p lane w ave in cid en ce o r a c o a xia l feed m o d el im p le m e n te d using th e fin ite
elem ent m e th o d . R a d ar cross se ctio n , in p u t im p e d a n ce , re tu rn loss, and g a in versus
fre qu e ncy a re c a lc u la te d .
P a ra m e tric s tu d ie s in te rm s o f bias field, lin e w id th and
s a tu ra tio n m a g n e tiz a tio n are p e rfo rm e d .
N u m e ric a l re su lts are com pared w ith a
pure s p e c tra l d o m a in M o M fo rm u la te d in d e p e n d e n tly by K o k o to ff [S3] as well as
e x p e rim e n ta l d a ta .
A ll e x p e rim e n ts were p e rfo rm e d a t the anechoic ch a m b e r o f
A riz o n a S ta te U n iv e rs ity u s in g an HPS510 n e tw o rk a n a lyze r. C om parisons illu s tra te
e x ce lle n t a g re e m e n t between p re d ic tio n s a nd m easurem ents.
6.1
In tr o d u c tio n
Ferrites have been used fo r m a n y years in m icro w a ve and m illim e te r-w a v e devices
such as c irc u la to rs , iso la to rs, sw itch e s, and phase s h ifte rs [192]. T he m a te ria l p ro p ­
e rties o f fe rrite s are c o n tro lle d by th e d ire c tio n and s tre n g th o f an e x te rn a lly a p p lie d
m a g n e tic fie ld . T h is u n iq u e p ro p e rty o f fe rrim a g n e tic m a te ria ls has fou n d a p p lic a ­
tio ns n o t o n ly in m icro w a ve in te g ra te d c irc u its ( M IC 's ) b u t also in ante nn a te c h n o l­
ogy [S5]-[92].
T h e use o f m a g n e tiz e d fe rrite s in a n te n n a design provides d e sira b le
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features such as t u n a b ility . p o la riz a tio n d iv e rs ity , beam ste e rin g , ra d a r cross section
(R C S ) c o n tro l, surface wave re d u c tio n and gain e nh ancem ent.
A lth o u g h m a g n e tize d fe rrite s have been e x te n s iv e ly used to enhance a n te nn a c h a r­
a c te ris tic s th ro u g h a lte r in g th e bias fie ld , m ost o f previous w o rk was co n ce n tra te d
m a in ly on ra d a r cross se ctio n e v a lu a tio n o f p rin te d dipoles and m ic ro s trip patch
antennas. T h e resonant b e h a v io r o f th e a n te n n a e le m en t has been th o ro u g h ly inves­
tig a te d fo r va rious b ia s in g c o n d itio n s . A n im p o rta n t conclusion fro m these studies
is th a t th e bias fie ld s tro n g ly affects th e resonant fre qu e ncy o f a ll m odes w ith d o m i­
n an t e le c tric fie ld in th e d ire c tio n o f m a g n e tiz a tio n . A n e xce lle n t s tu d y on ra d ia tio n
c h a ra c te ris tic s o f p a tc h a ntennas p rin te d on n o rm a lly biased fe r rite su bstrates was
o rig in a lly p e rfo rm e d b y P ozar [87].[88]. N u m e ro us results were presented illu s tr a tin g
the effect o f b ia sing on a n te n n a e fficiency, resonant fre qu e ncy and in p u t im pedance.
U sing a s im ila r a p p ro a c h . V ang [89].[90] e xa m in e d th e effects o f a r b itr a r y m a g n e ti­
z a tio n on th e RCS response o f m ic ro s trip patches on fe rrite su bstra te s. He also used
fe rrite s u p e rs ta te s fo r th e design o f sw itc h a b ie antennas. A n o th e r s im ila r s tu d y was
re c e n tly p u b lish e d b y Lee t l al. [92] using in -p la n e biased fe r rite substrates.
RCS
com parisons betw een th e m a g ne tize d and u n m a g n e tize d cases were presented. A n a l­
ysis o f in fin ite d ip o le a rra y s p rin te d on fe rrite substrates was c a rrie d o u t by B u ris
et al. [91].
E m phasis was co n ce n tra te d p r im a r ily on scan p e rfo rm a n ce and in p u t
im p edance c a lc u la tio n s .
In th is d is s e rta tio n , th e a nalysis o f fe rrite -tu n e d CBS antennas is p e rfo rm e d using
a h y b rid iz a tio n o f F E M and M o M . T h e F E M based on lin e a r te tra h e d ra l elem ents
solves fo r th e e le c tric fie ld d is trib u tio n in sid e th e ca vity.
A s p e c tra l d o m a in M o M
is im p le m e n te d th ro u g h th e c o n tin u ity o f th e ta n g e n tia l m a g n e tic field in th e a p e r­
tu re to solve for th e fie ld d is tr ib u tio n in th e e x te rio r region: n o te th a t th e c a v ity is
m o u nte d on an in fin ite g ro u n d plane coated w ith a d ie le c tric layer. T h e m a in d ra w ­
back o f th e s p e ctra l d o m a in M o M is th a t it becomes e x tre m e ly slow w ith in creasing
the n u m b e r o f edges in th e a p e rtu re .
I his p ro b le m is overcom e by using an asym p-
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to ttc e x tra c tio n o f th e e x p o n e n tia l b e h a v io r o f th e G reen's fu n c tio n : th e a s y m p to tic
p a rt is e va lu a te d usin g a c o m p u ta tio n a lly e ffic ie n t sp a tia l d o m a in in te g ra tio n .
T h e c a v ity , w h ic h was chosen to be o f square shape, is e x c ite d u sin g a c y lin d ric a l
probe in a d ire c tio n p a ra lle l to th e g ro u n d plane.
A D C m a g n e tic fie ld is a p p lie d
in the d ire c tio n o f th e p ro be , a lth o u g h o th e r d ire c tio n s o f m a g n e tiz a tio n m ay be
considered. T h e firs t m o d e th a t is e x c ite d in sid e th e c a v ity is th e T E 10 w h ich e x h ib its
a d o m in a n t e le c tric fie ld in th e d ire c tio n o f the probe. Since th e m a g n e tiz a tio n is
o rie n te d in th e d ire c tio n o f th e d o m in a n t e le c tric fie ld , th e resonant frequency o f the
T E io m ode is e xp e cte d to s h ift if th e s tre n g th o f th e bias fie ld is a lte re d . T h is ty p e
o f m ode is so m e tim e s re ferre d to as a m a g n e to s ta tic m ode a ttr ib u te d to the presence
o f a stro n g e x tr a o rd in a ry wave inside th e fe rrite sam ple. In th e absence o f th e bias
fie ld , th e resonance due to th e m a g n e to s ta tic m ode to ta lly d isappears. T h is p ro p e rty
o f fe rrite s is o fte n u tiliz e d in th e design o f s w itc h a b le antennas [90].
N u m e ric a l s im u la tio n o f fe rrite -tu n e d antennas requires e s tim a tio n o f the dem ag­
n e tiz in g fie ld in sid e th e fe rrite layers. In th e case o f m ic ro s trip p a tc h antennas on
in fin ite ly long th in fe r rite su bstra te s, th e d e m a g n e tizin g fa c to r is e ith e r 0 o r 1 de­
p e n d in g w h e th e r th e bias fie ld is p a ra lle l o r n o rm a l to the endfaces o f th e substrate.
For the CBS a n te n n a , th e thickness o f each fe rrite layer in se rte d in th e c a v ity is com ­
parable to th e re m a in in g tw o dim e nsion s o f the sam ple.
T h u s , th e d em ag n etizin g
fa c to r m ust lie som ew here between 0 and 1. A good e s tim a te o f th e d em ag n etizin g
field in sid e th e fe rrite layers is e x tre m e ly im p o rta n t in p re d ic tin g a c c u ra te ly the ra d i­
a tio n c h a ra c te ris tic s o f C B S antennas. In th is d is s e rta tio n , th e d e m a g n e tiz in g fa c to r
is e s tim a te d based on a m a g n e to s ta tic m o d e l fo rm u la te d by .Joseph [193].
6.2
O rd in a ry and E x tra o rd in a ry Waves
T o shed in sid e and u n d e rs ta n d in g in to w ave p ro p a g a tio n in sid e fe rrite s , consider a
lin e a rly p o la rize d p la ne wave at n o rm a l in cid e n ce on a tra n s v e rs e ly bias fe rrite slab.
T w o d iffe re n t types o f wave are e x c ite d in sid e th e g y ro m a g n e tic m e d iu m : these are
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180
k n o w n as th e o rd in a ry and e x tra o rd in a ry waves [90j.[194j.
T h e o rd in a ry w ave is
s im p ly th e same as th e p la ne wave p ro p a g a tin g in sid e a d ie le c tric m e d iu m .
T h is
ty p e o f wave is t o ta lly u n a ffe cte d by th e m a g n e tiz a tio n o f th e fe rrite . O n th e o th e r
h a n d , th e e x tra o rd in a ry wave is p ro p a g a tin g a lo n g th e d ire c tio n o f th e bias m a g n e tic
fie ld , th e re b y a ffe ctin g its p ro p a g a tio n c h a ra c te ris tic s . T h e p ro p a g a tio n c o n s ta n t o f
th e e x tra o rd in a ry wave is g ive n by [19-1]
3f = a.- v'eJiTTf
I 6 - 1)
w ith
II2
w h e re a.-0 - f-‘ o~:(H0+ j
-
K 2
fi'fj
=
--------------
f*
=
11+
*
=
) and
( 6 .*-*)
.2 1
( G- i:
= ft0(
In these expressions, th e e x te rn a lly
a p p lie d m a g n e tic field is d e n o te d as H 0. th e lin e w id th o f the fe rrite m a te r ia l as A H .
a n d th e s a tu ra tio n m a g n e tiz a tio n as -[ ~. \I S. T h e o rd in a ry and e x tr a o rd in a ry waves
have fie ld co m p on e nts th a t are p e rp e n d ic u la r w ith each o th e r. In o th e r w ords, i f an
in c id e n t wave is p o la riz e d in th e d ire c tio n o f m a g n e tiz a tio n th e p ro p a g a tin g wave
in s id e th e tra n sve rse ly bias fe r rite w ill have a ll p ro p e rtie s o f th e e x tr a o r d in a r y wave.
O n th e o th e r hand, i f th e in c id e n t wave is p o la riz e d in the d ire c tio n p e rp e n d ic u la r
to th e d ire c tio n o f m a g n e tiz a tio n th e p ro p a g a tin g wave w ill behave lik e an o rd in a ry
wave. T h e tw o types o f wave are c o m p le te ly d e co u p le d o n ly fo r th e case o f n o rm a l
in c id e n c e .
I f an in c id e n t wave im p in g es th e fe r rite m a te ria l a t an a ng le , th e n th e
o r d in a r y and e x tra o rd in a ry waves are co u p le d . A lso , fro m (6 .2) it is a p p a re n t th a t
th e e ffe c tiv e p e rm e a b ility / i rj j m a y becom e n e g a tiv e fo r c e rta in values o f
wj0 and
w-,n . In such a case, th e p ro p a g a tio n co nsta nt also becomes n e g a tive , th e re fo re , th e
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wave a tte n ua tes ra p id ly (evanescent wave) as it p e n e tra te s the fe rrite slab.
T h is
phenom enon is u su a lly referred to as th e c u t- o ff s ta te o f the fe rrite m a te ria l.
An
in c id e n t wave p o la rize d along th e d ire c tio n o f m a g n e tiz a tio n w ill be to ta lly re fle cte d
i f /.ief f becomes negative. T h e fre qu e ncy ra n ge w h e re //e/ / is negative is g ive n by
\J^'o (~'o + -^rn ) ^
^'o +
(6 .0 )
Besides tra nsve rse -p la ne m a g n e tiz a tio n , th e fe r r ite slab m ay be o th e rw is e m a g ­
n etize d along th e d ire c tio n o f p ro p a g a tio n . T h e m a in o bservation is th a t th e fo rw a rd
tra v e lin g wave is p ro p a g a tin g w ith a d iffe re n t p ro p a g a tio n constant th a n th e b a ck­
w a rd tra v e lin g wave. T h e co rre sp o n d in g e xpressio n s are given by
3+
=
J-
=
+
(6.6)
(6.1
X o t o n ly the p ro p a g a tio n co nsta nt is d iffe re n t fo r th e fo rw a rd and backw ard waves,
b u t also the a tte n u a tio n co n sta n t, a ssum ing th a t th e fe rrite m a te ria l e x h ib its some
ty p e o f loss. T h is p ro p e rty o f fe rrite s is used fo r th e design o f isolators and phase
s h ifte rs . A lso, w hen one wave e x h ib its a rig h t-h a n d c irc u la r p o la riz a tio n , th e o th e r
wave always e x h ib its a le ft-h a n d c irc u la r p o la riz a tio n . T h e su pe rp o sition o f th e tw o
p ro p a g a tin g waves how ever, s till represents a lin e a r p o la riz a tio n .
T h e p e rm e a b ility o f m agnetized fe rrite s is re p re sen ted by a tensor n o ta tio n . D e­
p e n d in g on th e d ire c tio n o f the bias fie ld , th e s tru c tu re o f the tensor m ay vary. For
e x a m p le , w hen a fe rrite sam ple is m a g n e tize d a lo n g th e r-d ire c tio n . th e c o m p le x
p e rm e a b ility ten so r becomes
w here // and
k
//
- j K
0
jK
//
0
0
0
// o
are e x p lic itly give n in (6 .3 ) and (6.-1). respectively.
(6.S)
In case th e
d ire c tio n o f m a g n e tiz a tio n is along th e x - o r //-a x is , th e fe rrite p e rm e a b ility tensor
m u st be ro ta te d bv 00 degrees.
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6.3
D e m a g n e tiz a tio n Effects
W hen an
inside th e
a r b itr a r y piece
of
fe rrite is placed in a u n ifo rm m a g n e tic fie ld , the d ip o le s
m a te ria l te n d to a lig n them seives along th e d ire c tio n o f the fie ld .
Thus,
the net fie ld a t a n y p o in t in space is equal to the s u m m a tio n o f th e e x te rn a l fie ld
as w ell as th e fie ld due to th e a lig n m e n t o f a ll m a g n e tic d ip o le s.
In general, th e
field in sid e th e fe r rite is n o n -u n ifo rm and d iffe re n t fro m th e e x te rn a lly bias fie ld . To
d e m o n stra te th a t th e e x te rn a l and in te rn a l fie ld s are u s u a lly d iffe re n t, consider a th in
layer o f fe rrite , as illu s tra te d in F ig . 6 .1 (a ). th a t extends to in f in it y in the h o riz o n ta l
plane. T h e fe rrite sam ple is firs t m a g n e tize d p e rp e n d ic u la r to th e plane d efin e d by
the tw o in te rfa ce s. B y d e n o tin g th e in te r n a l fie ld as H 0 = a ~ H 0. th e m a g n e tic flu x
d ensity in sid e and o u tsid e th e fe rrite sheet is g ive n, re sp e ctive ly, by
B[ n =
=
H0 + - l - \ r 3
He
( 6 .9 )
(6.10)
where H e is th e e x te rn a lly a p p lie d m a g n e tic fie ld in th e r - d ir e c tio n . and -1".)/., is th e
sa tu ra tio n m a g n e tiz a tio n . E q u a tin g e q u a tio n s (6.9) and (6.10) leads to
H 0 = H e — Itt.IE ,.
(6.11)
Thus, th e in te rn a l fie ld is equal to th e e x te rn a l field m in u s th e s a tu ra tio n m a g n e ti­
za tion . Because o f th e in fin ite n a tu re o f th e fe rrite la ye r in th e la te ra l d ire c tio n , th e
in te rn a l fie ld is u n ifo rm : th is is not n ece ssa rily tru e fo r th e case o f a fe rrite la y e r o f
fin ite e x te n t. As show n in (6 .11 ). th e fie ld in sid e th e fe rrite , w h ic h is th e one th a t
c o rre c tly d e te rm in e s the elem ents o f th e p e rm e a b ility ten so r, is th e c o n trib u tio n ot
two fields:
th e e x te rn a lly bias fie ld and th e so-called "d e m a g n e tiz in g fie ld ” .
It is
referred to as th e d e m a g n e tiz in g fie ld because it acts in a d ire c tio n o pp o site to th e
bias field.
On th e o th e r h a n d , if th e a p p lie d fie ld is p a ra lle l to th e in te rfa ce s o f the s a m p le ,
as shown in F ig . 6 .1 (b ). th e fin a l result is t o t a lly d iffe re n t fro m (6.11). S p e c ific a lly .
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I S: )
Z
Z
A
A
H(
A ir
Ferrite
H,
A ir
H,
A ir
Ferrite
M
Ho
M
A ir
H,
H,
(a)
(b)
Fig'. 6.1: (a) Perpendicular m agnetization, (b ) Parallel m agnetization.
b y s a tis fy in g the c o n tin u ity o f th e ta n g e n tia l m a g n e tic fie ld at the tw o in terfaces, one
can show th a t
( 6 . 12 )
H, = Hr .
In o th e r words, the in te rn a l fie ld is id e n tic a l to th e bias fie ld . T h is re la tio n is va lid
even if th e fe rrite la ye r is fin ite , p ro v id e d th a t th e le n g th o f th e sam ple in th e d ire c tio n
o f m a g n e tiz a tio n is m uch la rg e r th a n th e thickness.
In general, th e in te rn a l field
depends on the in te n s ity o f th e bias fie ld , th e shape o f th e fe rrite , a n d its o rie n ta tio n
w ith respect to th e d ire c tio n o f th e bias fie ld .
T h u s, th e in te rn a l fie ld is u su a lly
w r itte n as
H 0 = H f -.V M
w h e re .V is a 3 x :) m a tr ix den o te d as th e d e m a g n e tiz in g fa c to r, and M
(6.13)
is the
m a g n e tiz a tio n ve cto r. In th is s tu d y , th e d e m a g n e tiz in g fa c to r is conside re d a sea la r
n u m b e r ra n gin g betw een 0 and I: its d e fin itio n assumes th a t th e d e m a g n e tiz in g field
is in th e o p p o site d ire c tio n o f th e bias fie ld : th a t is
ff0 =
H r -
(6 . 11)
A lth o u g h the d e riv a tio n o f th e d e m a g n e tiz in g fa c to r for a th in in fin ite fe rrite layer is
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s tra ig h tfo rw a rd , th e same a n a ly tic a l process becomes q u ite c u m b e rso m e to r g e n e ric
shapes. Joseph and S ch lo m a n n [195] d e riv e d clo se d -fo rm expressio n s fo r th e d e m a g ­
n e tiz in g fa c to r o f a re c ta n g u la r fe r r ite p ris m in a u n ifo rm D C m a g n e tic fie ld . T h e
a pproach is based on s o lv in g th e M a x w e ll's e q u a tio n s fo r a m a g n e to s ta tic p ro b le m .
T h e in te re s te d reader is re fe rre d to th e o rig in a l p ap e r [195].
A c c o rd in g to Joseph [196]. in m o st s itu a tio n s one is in te re s te d n ot in th e lo ca l
v a ria tio n s o f th e d e m a g n e tiz in g fa c to r in s id e the fe r rite v o lu m e b u t ra th e r in how
th e sam ple responds in som e average sense to the e x te rn a l b ia s fie ld .
T h u s , one
m a y suggest tw o d iffe re n t d e fin itio n s o f th e d e m a g n e tiz in g fa c to r: th e ballistic a n d
th e magne tomet ri c.
T h e b a llis tic d e m a g n e tiz in g fa c to r, d e n o te d by .Yj,. is d e fin e d ,
a cco rd in g to [196]. as th e average o f th e s p a tia lly v a ry in g d e m a g n e tiz in g fa c to r in
a plane p e rp e n d ic u la r to th e d ire c tio n o f th e a pp lie d fie ld and m id w a y betw een th e
endfaces o f th e sam ple. T h e m a g n e to m e tric d e m a g n e tiz in g fa c to r, denoted by ,Ym.
is d efin e d, a cco rd in g to [196], as th e average o f th e s p a tia lly v a ry in g d e m a g n e tiz in g
fa c to r o ve r th e v o lu m e o f th e s a m p le . In th is stu d y, th e b a llis tic d e m a g n e tiz in g fa c to r
is a d o p te d .
W ith reference F ig . 6.2. th e b a llis tic d e m a g n e tiz in g fa c to r is g ive n b y
[193]
-V,
=
—
{-l/;r/C O t- I [/? (p .r/)/(4 /;r/)] + q f [ { p . q ) + p H ( q . p )
2~pq >-
—
< ///(0 . r/) — p H (0. /;) -f- h { 0 . q ) + h ( 0 . p ) — h{p. q) — 1}
(6 .15 )
w here
li(u.c)
=
V l + l u 2 + 4 c-
(6.161
ff{u .c )
=
ln { [ / ) ( u . v) + 2 r ] / [ / t ( u . c) — 2c]
(6.17)
and
a
p = —
q =
b
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(6. K '')
f
c
__
a
Fig. 6.2: G eo m e try o f a rectangular fe rrite prism w ith a uniform DC magnetic field l l r
applied e x te rn a lly along the vertical direction.
T h e b a llis tic d e m a g n e tiz in g fa c to r is p lo tte d in F ig . 6.3 on a lo g a rith m ic scale fo r
various values o f p and q. For a co n sta n t value o f p. th e b a llis tic d e m a g n e tiz in g fa c to r
increases w it h q u n t il it e v e n tu a lly reaches a m a x im u m . T h is m a x im u m corresponds
to th e case w hen one o f th e h o riz o n ta l dim ensions e xte n d s to in fin ity . As q begins
to increase, w h ic h is e q u iv a le n t to in cre a sin g th e second h o riz o n ta l d im e n s io n , th e
e n tire g ra p h s h ifts u p w a rd .
E v e n tu a lly , when b o th h o riz o n ta l dim ensions increase
s ig n ific a n tly , w h ile m a in ta in in g th e v e rtic a l dim e nsion c o n s ta n t, th e b a llis tic d em a g ­
n e tiz in g fa c to r approaches I. W h e n one o f the h o riz o n ta l d im e nsion s, let us say a.
extends to in f in it y , e q u a tio n (6.15) becomes [193]
In o rd e r to p ro v id e a c o m p a riso n b etw een the b a llis tic and m a g n e to m e tric dem agn e tiz in g fa c to rs fo r a re c ta n g u la r p ris m , firs t it is necessary th a t the la tte r is defined
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
fo r th e case when p is in fin ity . T h e c o rre s p o n d in g e qu ation is g iv e n by [193]
Am = ~
1 4 ta n _1(<7) - 2q ln (r/) + ~ { q 2 - 1) In (1 + 2<y2) j .
(6.20)
B o th .Vi and .Ym are p lo tte d on th e sam e g ra p h , see F ig . 6.4. fo r various values o f
q = b/c. T h e tw o d e fin itio n s o f d e m a g n e tiz in g fa c to r result in s im ila r values p ro v id e d
q is la rg e r th a n 1: i.e .. b > c. T h e d e v ia tio n between th e tw o d e m a g n e tiz in g fa cto rs
increases as th e h o riz o n ta l d im e nsion becom es progressively s m a lle r th a n th e h eight
o f th e p ris m .
Based on th is m a g n e to s ta tic fo rm u la tio n , an a p p ro p ria te d e m a g n e tiz in g fa c to r
w ill be e s tim a te d in o rd e r to co m p u te , in an average sense, th e m a g n e tic field in s id e
a re c ta n g u la r fe r rite b lo ck. R e cta n g u la r fe r rite layers w ith fin ite thickne ss are used
in th is s tu d y as lo a d in g m a te ria ls fo r C B S antennas.
6.4
S c a tte rin g fro m F e rrite -L o a d e d C a v ity -B a c k e d Slots
T h e fin ite ele m en t m e th o d , w hich was f u lly h y b rid iz e d w ith th e s p e c tra l and s p a tia l
d o m a in m e th o d o f m o m e n ts, was im p le m e n te d fo r th e ana lysis o f CBS a ntennas
loaded w ith m a g n e tize d fe rrite s.
T h e a ccu ra cy and v a lid ity o f th e code was firs t
e v a lu a te d fo r n um erou s d ie le c tric -lo a d e d C B S antennas [197] a n d . la te r, fo r fe rrite loaded C B S antennas.
C o n sid e r th e m u lti- la y e r fe rrite -lo a d e d C B S shown in Fig. 6.5. T h is ante nn a was
o r ig in a lly designed and analyzed by K o k o to ff [S3] using b o th e x p e rim e n ts and m o ­
m e n t m e th o d s im u la tio n s .
T he c a v ity v o lu m e is p a rtitio n e d h o riz o n ta lly in to five
re c ta n g u la r sections. Each section is fille d w ith e ith e r d ie le c tric o r fe rrite m a te ria l.
T h e m a te ria l n u m b e rin g sta rts in ascen d in g o rd e r from b o tto m to to p . T h e d im e n ­
sions o f th e c a v ity a re a = 2 in . b = 2 in and c = 2 in . T h e in f in it e g ro u n d p la ne
is tre a te d as a p e rfe ct e le c tric c o n d u c to r w ith o u t overlay. M a te ria l p aram ete rs and
o th e r d im e n sio n s are ta b u la te d in T a b le 6.1. T h e m o n o sta tic RC’S is ca lc u la te d versus
fre q u e n cy fo r a p la ne wave at n o rm a l in cid e n ce . The fe rrite sam ples are m a g n e tize d
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ballistic Demagnetizing Factor
187
5
o
o
.o
10'
a/c=0.1
a/c=0.2
5
a /c= 0.3
a/c=0.5
_ a/c=1.0
a/c=10.0
9
b/c
J: The ballistic dem agnetizing facto r o f a u niform ly magnetized rectangular prism.
Demagnetizing Factors
O
n
5
2
_ B a llis tic (N b)
- • M a g n e to m e tric (N,
b/c
Fig. 0.4: Com parison between the ballistic and m agnetom etric dem agnetizing factors of
a u n ifo rm ly magnetized rectangular prism w ith dimension a — cc.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
188
z
t 3,
83, (i3
x l , e l , |nl
Fig. 6.5: G eom etry o f a m u lti-la ye r ferrite-loaded CBS antenna mounted on an in fin ite
ground plane.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 6.1: G eo m e try and m aterial specifications o f A ntenna # 1 .
M a te ria l # (i)
Thickness, r, (cm )
P e rm ittiv ity . er,
Perm eability. f.ir,
I
2
1.065
0.762
1
13.9
3
-I
0.737
1.790
2.2
13.9
5
0.726
2.2
1
Ferrite: f i T = 1. A / / = 5 Oe.
4 - \ I 3 = 800 C.
I
Ferrite: / / r = I. A H = 5 Oe.
4 ~ .\/s = 800 G
1
in th e //-d ire c tio n w ith an in te rn a l m a g n e tic fie ld o f H 0 = 400 Oe. T h e p re d icte d d a ta
(<x00) u sin g th e present fo r m u la tio n are co m p ared w ith d a ta o b ta in e d by K o k o to lf
u sin g p u re M o M [83].
A s d e p ic te d in F ig . 6.6. the tw o d a ta sets are in excellen t
a g re e m e n t. K o k o to ff's d a ta , show n as m a rke rs in this fig u re , are o n ly p lo tte d up to
a fre q u e n c y o f 850 M H z because th e a p p ro a ch im p le m e n te d in [83] becomes q u ite
u n s ta b le at h ig h e r frequencies, p o ss ib ly due to n um e rica l e rro rs. T h is can be th o u g h t
o f as a n o th e r advantage o f u sin g F E M in ste a d o f M o M to solve fo r th e fields inside
th e fe rrite -lo a d e d ca vity.
T h e fre qu e ncy range in w h ic h th e e x tra o rd in a ry wave begins to a tte n u a te is de­
p e n d e n t on th e a ctu a l fe r r ite p a ra m e te rs and the s tre n g th o f th e bias fie ld .
I his
fre q u e n cy range can be c o rre c tly e s tim a te d using the fo rm u la in (6 .5). R e fe rrin g to
th e fe rrite -lo a d e d a n te n n a show n in F ig . 6.5. the e x tra o rd in a ry wave w ill s ta rt de­
c a y in g a p p ro x im a te ly b etw e e n "2.0 G H z and 3.4 G H z. W ith in th is fre qu e ncy band, a
s ig n ific a n t d ro p o f th e ra d a r cross se ction m a y be observed d e p e n d in g on the p o la r­
iz a tio n o f th e in c id e n t fie ld . In such a case, it is possible th a t a d d itio n a l resonances
m ig h t a p p e a r in the lo w e r o r th e u p p e r range o f frequencies. T h is p ro p e rty o f fe rrite s
has been u tiliz e d in th e past to design s w itc h a b le m ic ro s trip p a tch antennas.
T h e fin ite elem ent d a ta show n in F ig . 6.6 were co m p u te d on a 370 1M B R IS C /6 0 0 0
w o rk s ta tio n . T h e th re e -d im e n s io n a l mesh was created using a co m m e rc ia l package
ca lle d S D R C I-D E A S . T h e to ta l n u m b e r o f te tra h e d ra l e le m en ts was 7.356. whereas
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
mo
o.o
T his M e th o d
M oM (K o k o to ff)
-
10.0
-
20.0
s
co
O -30.0
-40.0
-50.0
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Frequency (GHz)
Fig. 6.0: M onostatic RC'S {<J00) a t norm al incidence o f a CBS antenna loaded w ith layers
o f magnetized ferrite ( H 0 = 400 Oe).
th e n u m b e r o f unknow ns was 8.006. T h e re m a in in g setup param ete rs fo r th e p ro b le m
are given in T a b le 6.2. The m o n o s ta tic ra d a r cross se ction o f th e ante nn a was e v a lu ­
a te d a t 61 frequency points. T h e to ta l ru n tim e was 31 hours w hich is e q u iv a le n t to
a p p ro x im a te ly 30 m in u tes per fre q u e n c y p o in t. H ow ever, a closer lo o k at th e c o m ­
p u ta tio n a l s ta tis tic s in d icates th a t m ost o f the C P I tim e (25 m in u te s per p o in t) was
sp en t to f ill in th e .MoM a d m itta n c e m a tr ix , w hich represents th e e x te rio r p a rt o f the
p ro b le m .
It was then decided th a t a lin e a r in te rp o la tio n o f th e a d m itta n c e m a tr ix
be used across the frequency s p e c tru m . T h e same s im u la tio n was re p ea ted , b u t now
th e a d m itta n c e m a trix is e v a lu a te d o n ly at three fre q u e n cy p o in ts: at in -b e tw e e n
p o in ts , th e e n trie s o f th is m a t r ix are lin e a rly in te rp o la te d . T h e re m a in in g s e ttin g s
o f th e p ro b le m as well as th e c o rre s p o n d in g c o m p u ta tio n a l s ta tis tic s are illu s tra te d
in T a b le 6.3. S pe cifica lly, th e h y b r id cocle now takes o n ly a to ta l o f 5 h ou rs and 12
m in u te s to c o m p u te the m o n o s ta tic RCS o f the a nte nn a fo r 61 fre qu e ncy p o in ts . In
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
191
o th e r w ords, using lin e a r in te r p o la tio n fo r the a d m itta n c e m a trix , th e h y b rid code
re q uires on th e average o n ly 5 m in u te s p e r p o in t in s te a d o f 30 m in u te s p e r p o in t
observed w hen no in te rp o la tio n is a p p lie d . In te rm s o f accuracy, th e results in b o th
cases were id e n tic a l.
Table 6.2: C o m p u ta tio n a l statistics o f the h yb rid F E M /M o M code.
P ro b lem
P a ra m e ters
T otal
C P U T im e
E x te r n a l
In te g ra tio n T im e
C G S Solver
T im e
Elements = 7356
F E M Unknowns = 8006
M o M Unknowns = 209
S olution Tol. = L.0e-5
E valuation puts. = 6 1
IB M 370 R ISC/6000
31 hours
(30 m in /p n t.)
25 hours
(25 m in /p n t.)
3 hours
(3 m in /p n t)
Including both
polarizations
Table 6.3: C o m p uta tio na l s ta tistics o f the hybrid FE .M /M o.M code w ith in te rp o la tio n .
P ro b lem
P a ra m e ters
T otal
C P U T im e
E x te r n a l
In te g r a tio n T im e
C G S S olver
T im e
Elements = 7356
FE M Unknowns = 8006
M oM Unknowns = 209
Solution Tol. = l.0e-5
Evaluation pnts. = 61
In te rp o la tio n based on
3 frequency points
IB M 370 RISC/6000
5:12 hoursunin
(5 m in /p n t.)
1:15 hours:m in
3 hours
(3 m in /p n t)
Including both
polarizations
T h e a b ilit y to e ffe c tiv e ly tu n e th e fe rrite -lo a d e d C B S a n te n n a show n in F ig . 6."). is
illu s tr a te d b y v a ry in g the in te r n a l m a g n e tic fie ld H 0. T h e s tre n g th o f th e m a g n e tic
fie ld was c o n s ta n tly increased fro m 400 Oe to 700 O e.
A s shown in F ig . 6.7. th e
re so na n t fre q u e n cy o f the a n te n n a s h ifts to a h ig h e r fre q u e n c y as H 0 increases. T h is
fre q u e n cy tu n in g is a ttr ib u te d to th e e x t r a o r d i n a r y p ro p e rtie s o f th e fe rrite w h ic h
are c o n tro lle d by th e e ntries o f th e p e rm e a b ility te n s o r.
In o rd e r to gain a b e tte r
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
u n d e rs ta n d in g o f th e w ave b e h a v io r in s id e th e fe rrite , th e n o rm a liz e d p ro p a g a tio n
co nsta nt (sq u a re d ) o f th e e x tr a o rd in a ry wave inside a hom ogeneous fe rrite m e d iu m
o f er =
13. A H
=
0 O e and 4 ~ M S =
800 G is p lo tte d in F ig . 6.8 fo r various
values o f H 0. T h re e re g io n s o f in te re s t are to be id e n tifie d in th is g ra ph : th e lowfrequency region, th e r e so na nt f re qu e nc y re g io n and th e high-f requency region. In th e
lo w -fre q u e n cy re g io n , w h ic h is th e region o f in te re st in th is s tu d y , the n o rm a liz e d
p ro p a g a tio n c o n s ta n t s h ifts d o w n to lo w e r values as H 0 increases. T h is m eans th a t
th e e ffe ctive a p e rtu re o f th e a n te n n a becom es e le c tric a lly s m a lle r th e re b y s h iftin g th e
m a g n e to s ta tic resonance to h ig h e r frequencies. In th e h ig h -fre q u e n c y re g io n , w h e re
m ost m ic ro s trip d ip o le a nd p a tc h a nte nn a s o pe ra te, a s im ila r effect is o bserved.
B y in cre asing H 0. th e n o rm a liz e d p ro p a g a tio n co n sta n t fo r th e e x tra o rd in a ry wave
s ta rts to decrease: th e re fo re , th e m a g n e to s ta tic resonance s h ifts to a h ig h e r fre qu e ncy.
In th e resonant fre q u e n c y re g io n , th e n o rm a liz e d p ro p a g a tio n c o n s ta n t increases to
e x tre m e values befo re it a c tu a lly becomes im a g in a ry , th e re fo re suggesting a h ig h ly
lossy fe rrite . W h e n lossy, th e fe r rite can be used as an a b so rb e r. T h is p ro p e rty o f
fe rrite s finds n u m e ro u s a p p lic a tio n s in RC’S re d u c tio n and s w itc h a b le a ntennas.
The dependence o f th e resonant fre q u e n cy o f th e a n te n n a on th e bias fie ld fo r
h ard (0) p o la riz a tio n is illu s tr a te d in F ig . 6.9.
F n lik c soft ( o ) p o la riz a tio n , th e
m a g n e to s ta tic resonance o ccu rs a t re la tiv e ly h ig h e r frequencies,
rite reason fo r th is
s h ift is due to th e p o la riz a tio n o f th e in c id e n t fie ld w ith re sp ect to th e d ire c tio n o f
m a g n e tiz a tio n . Since th e d ire c tio n o f m a g n e tiz a tio n is set a lo n g th e //-axis, a n o rm a l
in c id e n t fie ld (0 = o = 0) th a t is p o la riz e d alo ng th e o -d ire c tio n is p ra c tic a lly p a ra lle l
to th e m a g n e tiz a tio n v e c to r. A c c o rd in g to Section 6.2. a s tro n g e x tra o rd in a ry wave
w ill appear in s id e th e fe r r ite sa m p le. O n th e o th e r h a n d , fo r an in c id e n t wave th a t
is cro ss-p o larize d w ith th e m a g n e tiz a tio n ve cto r, e.g.. h a rd p o la riz a tio n a t n o rm a l
in cid en ce (0 = o = 0 ). a s tro n g o rd in a ry wave w ill a p p e a r in s te a d . In g e n e ra l, th e
tw o waves co up le w ith each o th e r due to th e e xisten ce o f cro ss-p o la rize d fie ld s.
Besides v a ry in g th e bias m a g n e tic fie ld H 0. th e s a tu ra tio n m a g n e tiz a tio n 4 ~ . \ f s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m
-10
-20
-30
-50
-60
-70
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Frequency (GHz)
Fig. 6.7: Frequency tuning for soft polarization at normal incidence
(6
= o = 0).
150
125
100
^
50
~
25
-25
-50
0.0
1.0
2.0
3.0
Frequency (GHz)
Fig. 6.8: Effective norm alized propagation constant of the e x tra o rd in a ry wave as a func­
tion o f frequency.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
191
0.0
l?
-
10.0
-
20.0
-30.0
-40.0
-50.0 L0.70
0.75
0.80
0.85
0.90
0.95
1.00
Frequency (GHz)
Fig. 6.9: Frequency tuning for hard polarization at normal incidence
(0
= o = 0).
is also va rie d w h ile m o n ito rin g th e resonant fre q u e n cy o f the a nte nn a . T h e in te rn a l
m a g n e tic fie ld / / 0 is set to 500 Oe.
As - l~ . \ / 5 increases fro m 800 to 1200 Cl (see
F ig . 6 .1 0 ). th e firs t resonance s ta rts to sh ift to a lo w e r frequency. In cre a sin g 1~ / s
re s u lts in a la rg e r effe ctive p e rm e a b ility , w h ich means a lower resonant fre q u e n cy tor
th e m a g n e to s ta tic m ode. T h is is tru e o n ly in th e lo w -fre q u e n cy re g io n : in th e highfre q u e n c y region, the effect o f in cre a sin g l - . \ / s is reversed. A h ig h e r value fo r 1~.WS
re s id ts in a low er e ffe ctive p e rm e a b ility : how ever, th e a m o u n t o f s h ift in th e resonant
fre q u e n c y o f the antenna is a lm o s t n e g lig ib le since th e effe ctive / / r a s y m p to tic a lly
a pproaches 1.
T h e ra d a r cross section o f a m u lti-la y e r fe rrite -lo a d e d CBS a nte nn a also depends
on th e lin e w id th A / / o f th e fe rrite . T h e effect o f A H on the m a g n e to s ta tic -m o d e res­
o na n ce is illu s tra te d in F ig . 6.11 fo r o -p o la riz a tio n at n orrnal in cid e n ce (0 = o = 0).
T h e in te r n a l m agnetic fie ld was set to 500 Oe whereas the s a tu ra tio n m a g n e tiz a tio n
was set to 800 Cl. T h e lin e w id th is o fte n co n ce p tu a lize d as th e lossy te rm of the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I!).-)
-10
Ms=800 G
Ms=1000 G
Ms=1200 G
-20
-50
-60
-70 —
0.70
0.75
0.80
0.85
0.90
0.95
1 00
Frequency (GHz)
Fig. (5.10: The effect o f s a tu ra tio n m agnetization (4 ~ .\/s) on the resonant frequency o f the
m u lti-la ye r ferrite-loaded antenna ( f [ 0 = 500 Oe. _ \ / / = 5 Oe. 6 = o — 0).
fe rrite .
B y in cre a sin g th e lin e w id th . th e resonant peak o f th e m a g n e to s ta tic m o d e
is s ig n ific a n tly reduced d u e to e ne rg y d is s ip a tio n in s id e th e fe rrite .
H ow ever, th e
re so na n t fre q u e n cy re m a in s u n a ffe cte d .
6.5
R a d ia tio n fro m F e rrite -L o a d e d C a v ity -B a c k e d S lo ts
C o n s id e r a s im ila r CBS a n te n n a show n in Fig. 6.12. T h e a n te n n a is now tre a te d as
a ra d ia to r ra th e r th a n as a s c a tte re r.
It is e x c ite d h o r iz o n ta lly using a //-d ire c te d
c y lin d r ic a l p ro b e w ith d ia m e te r 0.0621 in and le n g th I.T o in . T h e probe is so ldered
to th e in n e r c o n d u c to r o f a oO-Q c o a x ia l cable centered at th e m id -p o in t o f th e c a v ­
it y 's s id e w a ll.
Before in tro d u c in g fe r rite and d ie le c tric layers in to th e c a v ity , th e
re tu r n loss o f an a ir-fille d C B S a n te n n a was p re d ic te d u sin g th e h y b rid F E M / M o M
code and co m p a re d w ith m e a su re m e n ts. T h e c o m p a riso n between p re d ic tio n s and
m e a su re m e n ts is illu s tra te d in F ig . 6.13: the tw o d a ta sets are in very good agree-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
196
-10
AH=5 Oe
AH=20 Oe
AH=40 Oe
-20
-30
-50
-60
-70
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Frequency (GHz)
Fig. 6.11: The effect o flin e w id th on the resonant frequency o f the multi-layer ferrite-loaded
antenna ( I I 0 = 500 Oe. -1/T.Us = 800 G . 9 = 0 = 0 ) .
m e rit. It is in te re s tin g to obse rve th a t th e a n te n n a p ro v id e s a fa ir ly good m a tc h at a
fre q u e n c y a ro u n d 1 G H z. H ow ever, th e o b je c tiv e o f th is s tu d y is to design a tu n a b le
C B S a n te n n a th a t operates w ith in th e F H F b a n d .
W ith such goal in m in d , fo u r
la ye rs o f fe r rite and d ie le c tric m a te ria l are placed h o r iz o n ta lly inside th e c a v ity . T h e
d im e n s io n s and m a te ria l s p e c ific a tio n s are d e p ic te d in T a b le 6.4.
T h e in s e rte d fe rrite layers are m a g n e tize d u sin g a p a ir o f p e rm a n e n t m a g n e ts:
one m a g n e t at each side o f th e C B S ante nn a . T h e e x te rn a lly bias fie ld is o rie n te d
a lo n g th e le n g th o f th e p ro b e . To be able to s im u la te th is a n te n n a using n u m e ric a l
te c h n iq u e s , th e s tre n g th o f th e a c tu a l m a g n e tic fie ld in s id e th e fe rrite slabs m u s t be
k n o w n pre cisely.
Its value is fo u n d by s u b tra c tin g th e d e m a g n e tiz in g fie ld , w h ic h
e x is ts in sid e th e re c ta n g u la r fe r rite sam ple, fro m th e e x te rn a lly bias fie ld . T h e a p ­
p ro p r ia te d e m a g n e tiz in g fa c to r can be e s tim a te d based on th e m a g n e to s ta tic m o d el
b r ie fly p resented in th e p re v io u s section.
T h e e x te r n a lly bias field was m e asured
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
197
Table 6.4: Geometry and m a te ria l specifications o f A nte n na # 2 .
M aterial # (i)
Thickness, r, (cm )
P e rm ittiv ity . er,
P erm eability. / i ri
1
2
0.635
0.762
2.2
13.9
3
4
1.974
1.790
2.2
13.9
1
Ferrite: f t r = 1. A / / = 9 Oe.
4 - . \ / s = 800 G
1
Ferrite: / / r = 1. A / / = 9 Oe.
4 - .U s = 800 G
z
a
Fig. 6.12: G eom etry o f a m ulti-layer ferrite-loaded CBS antenna m ounted on an
ground plane and fed w ith a oO-f? coaxial c a b l e along the (/-direction.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
in fin ite
198
GS -10
-12
FEM/MoM
Measurements
-14
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
Frequency (GHz)
4.0
4.4
4.8
5.2
Fig. 6.13: Return loss o f an air-filled CBS antenna mounted on an infinite ground plane
and fed w ith a 50-f2 coaxial cable along the (/-direction.
using a G au ssm e te r at va rio us d is c re te p o in ts inside th e c a v ity ,
[ lie d ie le c tric and
fe rrite layers had been rem oved fro m th e c a v ity befo re ha n d. T w o separate m easure­
m e n ts were p e rfo rm e d : th e firs t using one p a ir o f m agnets and th e second using tw o
p airs o f m agnets. T h re e -d im e n s io n a l p lo ts o f the m a g n e tic fie ld d is trib u tio n in sid e
th e e m p ty c a v ity are sh ow n in Figs. 6.14 and 6.15 for th e cases o f one and tw o p airs of
m agnets, re sp e ctive ly. T h e m e a su re m e n t d a ta were c u rv e -fitte d using cu b ic splines.
As e xpe cted , th e fie ld d is tr ib u tio n e x h ib its some frin g in g effects. T h is phenom enon
is p r im a r ily due to th e c a v ity w a lls and th e fin ite d im e n s io n s o f th e m agnets. N o te
th a t the m easured fie ld in sid e th e c a v ity is n o t the same as th e fie ld inside th e fe rrite .
To c o m p u te th e in te rn a l fie ld , th e d e m a g n e tiz in g field has to be e s tim a te d firs t based
on th e a c tu a l d im e n sio n s o f th e fe r rite sa m p le. How ever, in o rd e r fo r someone to use
th e re su lts d e p icte d in F ig . 6.3. th e e x te rn a lly bias fie ld has to be u n ifo rm .
I lie
a s su m p tio n o f a u n ifo rm bias fie ld was in tro d u c e d in th e n u m e ric a l m odel by aver-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
199
a ging th e fie ld d is tr ib u tio n shown in Figs. 6.14 a nd 6.1-5. A s im ila r ty p e o f averaging
was also in tro d u c e d b y Joseph [193].[196] in o rd e r to d e riv e th e expressions fo r th e
b a llis tic d e m a g n e tiz in g fa c to r used in th is s tu d y . For th e tw o cases o f m a g n e tiz a tio n ,
i.e.. u sin g one p a ir o r tw o pairs o f m agnets, th e averaged u n ifo rm m a g n e tic field was
fo u n d to be c ip p ro x im a te ly 375 Oe and 5S0 O e. re s p e c tiv e ly .
T h e d e m a g n e tiz in g fa c to r, as note d in th e p re v io u s se ctio n , is d e te rm in e d by
th e a c tu a l d im e n s io n s o f the fe rrite layers as w e ll as th e o rie n ta tio n o f the bias
fie ld .
S p e c ific a lly , th e re are tw o re c ta n g u la r fe r rite layers in sid e th e ca vity . T h e ir
d im e n s io n s are 5 .OS x 5.OS x 1.79 cm and 5 .OS x 5 .OS x 0.762 cm .
Due to th e ir
close p r o x im ity , th e tw o samples are tre a te d as a sin g le fe r r ite sa m p le w ith e ffe ctive
d im e n s io n s o f 5 .OS x 5.08 x 2.552 cm . T h e o r ie n ta tio n o f th e bias fie ld is along th e
(/-d ire c tio n .
U sin g F ig . 6.3. the co rre s p o n d in g d e m a g n e tiz in g fa c to r is found to be
a p p ro x im a te ly 0.17. T h u s , for the tw o cases o f m a g n e tiz a tio n , th e u n ifo rm m a g n e tic
fie ld (in an
average sense) inside th e fe r rite is g iv e n b y
H 0 ~ H f - -V • (1 ~ M S) = 375
- 0.17 • S00
= 239 Oe
(6.21)
— 0.17 • S00
= 444 Oe
(6.22
fo r one p a ir o f m a g n e ts, and
Ho ^ H ,
fo r tw o p a irs o f m a g ne ts.
- A'
• ( 4 - . U , ) = 5S0
i
It is im p o rta n t to e m p h a size here th a t these e s tim a te d
values o f th e in te r n a l m a g n e tic fie ld are s u b je c t to to le ra n ce s re la te d to e x p e rim e n ta l
e rro r a nd m in o r a ssu m p tio n s in tro d u c e d in th e m o d e l.
E x p e rim e n ta l errors were
in tro d u c e d in th e a n a lysis d u rin g th e process o f m e a s u rin g th e m a g n e tic field in sid e
th e c a v ity . Besides tolerances associated w ith th e C iaussm eter and o th e r in s tru m e n ts ,
it was o b se rve d d u r in g th e e x p e rim e n t th a t th e fie ld in s id e th e c a v ity was q u ite
s e n s itiv e to th e p o s itio n o f the m agnets. A s lig h t m is a lig n m e n t o f th e magnets w o u ld
re s u lt in a m in o r s h ift in th e resonant fre qu e ncy. T h u s , e x tr a p re ca u tio n s were ta ke n
d u rin g th e e x p e rim e n t to ensure th a t th e m a g n e ts w ere p e rfe c tly alig ne d and fixe d
to th e ir p o s itio n .
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
200
HOO
Fig. 6.1-1: Measured m agnetic field d istrib u tio n inside the ca vity when the latter is m ag­
netized along the (/-direction using one pair o f magnets.
Fig. 6.15: Measured magnetic field d istrib u tio n inside the cavity when the la tte r is m ag­
netized along the (/-direction using two pairs o f magnets.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
U sin g th e above estim ates fo r th e in te rn a l fie ld o f the fe rrite s , th e present fo r­
m u la tio n was im p le m e n te d to p re d ic t th e in p u t im p e da n ce o f th e fe rrite -lo a d e d CBS
a n te n n a show n in Fig. 6.12. To th e kn ow led g e o f th e a u th o r, th is is th e firs t com ­
p re he n sive s tu d y to present co m parisons betw een m easurem ents a nd p re d ic tio n s o f
fe rrite -tu n e antennas. A com parison d e p ic tin g th e in p u t im p e da n ce versus frequency,
w hen th e a n te n n a is m agnetized in th e .(/-direction using a sin g le p a ir o f m agnets,
is illu s tra te d in F ig . 6.16. A n in te rn a l m a g n e tic fie ld o f H 0 = 23S Oe was used for
th e s im u la tio n . T h e agreem ent betw een m easurem ents and p re d ic tio n is ve ry good.
N o te th a t th e existen ce o f th e in d ic a te d resonance is due to th e presence o f an ex­
te r n a lly bias fie ld . I f the m agnets are rem oved fro m th e sidew alls o f th e c a v ity , the
m a g n e to s ta tic resonance to ta lly disappears.
T h e m a in o b je c tiv e of th is p ro je c t, o f course, was to achieve fre q u e n c y -tu n in g
c a p a b ilitie s w ith in th e U H F band. T h u s , a n o th e r p a ir o f m agnets was placed on top
o f th e o ld ones. A n in te rn a l m a g n e tic fie ld o f H 0 = 44o Oe was used fo r th e s im u la ­
tio n . T h e co m p a riso n between m easurem ents and p re d ic tio n s is show n in Fig. 6.17.
T h e agreem ent between the tw o d a ta sets is e xce lle n t. T h e h y b rid F E M /.M o .M fo r­
m u la tio n p re d ic ts c o rre c tly not o n ly th e precise fre qu e ncy s h ift o f th e m a g n e to s ta tic
resonance b u t also the a m p litu d e and shape o f the in p u t im p e d a n ce . A lso , c o m p a r­
in g Figs. 6.16 and 6.17. it is in te re s tin g to observe th a t an increase o f 200 Oe in
th e e x te rn a lly bias fie ld causes th e resonant fre qu e ncy o f th e m a g n e to s ta tic m ode
to s h ift by as m uch as 120 M H z . W h e n th e in te rn a l m a g n e tic fie ld H 0 increases to
1.000 O e. th e resonant frequency s h ifts by 220 M H z . whereas w hen H 0 increases to
2 .0 0 0 Oe th e resonant frequency s h ifts b y n e a rly 300 M H z . T h e in p u t im pedance
versus fre q u e n cy fo r the last tw o cases o f m a g n e tiz a tio n is illu s tra te d in Fig. 6.18.
N o te th a t th e percentage change in fre q u e n cy tu n in g decreases s u b s ta n tia lly w ith
in c re a s in g H 0. T h e reason is re la te d to th e s e n s itiv ity o f th e e ffe c tiv e p e rm e a b ility
as a fu n c tio n o f H 0. From these figures, it is a pp a re n t th a t th e fe rrite -lo a d e d CBS
a n te n n a is tu n a b le between 700 to 1. 100 M H z .
A ssu m in g th e c e n te r fre q u e n cy ot
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202
200
150
M easurem ents (R e a l)
M easurem ents (Im a g .)
F E M /M o M (R eal)
F E M /M o M (Im a g .)
■“ ! -lo o
-200
-250 •—
0.70
0.71
0.72
0.73
0.74
0.75
0.76
Frequency (GHz)
Fig. 6.16: Predicted and measured in p u t impedance versus frequency o f a ferrite-loaded
CBS antenna using a single pair of magnets ( H 0 = 238 Oe).
400
300
M easurem ents (R e a l)
M easurem ents (Im a g .)
F E M /M o M (R eal)
F E M /M o M (Im a g .'
200
100
T3
O
CL
0
5 -loo
O'
-200
.300 ----------------------------------------------------------------- :---------------------------------------1
0.80
0.81
0.82
0.83
0.84
0.85
0.86
0.87
0.88
0.89
0.90
Frequency (GHz)
Fig. 6.17: Predicted and measured in p u t impedance versus frequency o f a ferrite-loaded
CBS antenna using two pairs o f magnets ( / / 0 = l-lo Oe).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
203
o p e ra tio n is 900 M H z . th e percentage tu n in g o f th e a n te n n a is a b o u t 45c/c.
The
tu n in g m e ch a n ism can be achieved q u ite e ffe c tiv e ly b y p la c in g an e le c tro m a g n e t in
th e d ire c tio n o f th e p ro b e .
500
H0= l,000 Oe (Real)
Ho=l,000 Oe (Imag.)
H0=2,000 Oe (Real.)
- - Ho=2,000 Oe (Imag.)
400
£ 300
o
'X
200
^
100
i—
a
a
a . -100
-200
-300 >—
0.90
0.93
0.96
0.99
1.02
1.05
1.08
1.11
1.14
Frequency (GHz)
F ig . fi. IS: Predicted in p u t im pedance versus frequency o f a ferrite-lo ad ed C B S a n te n n a
using various biasing fields.
T h e re tu rn loss o f th e a n te n n a fo r th e firs t tw o cases o f m a g n e tiz a tio n , i.e.. / / 0 =
23S Oe and / / 0 = 4-15 O e. is co m p ared w ith m e a su re m e n ts in F ig . 6.19. A g a in , th e
agreem ent betw een th e tw o d a ta sets is e x c e lle n t. N o te th a t fo r th e low er resonance,
th e re tu rn loss is —9 d B whereas fo r th e h ig h e r resonance, th e re tu rn loss im p ro v e s
to — 17 d B . T h e c o a x ia l ca ble was chosen to have a 50-Q c h a ra c te ris tic im p e d a n c e .
For b e tte r re tu rn loss, a cu s to m m ade co a xia l ca b le is p ro b a b ly a good choice. O f
course, a n te n n a design o p tim iz a tio n is alw ays a n o th e r o p tio n .
Besides in p u t im p e d a n c e and re tu rn loss, g a in is also an im p o rta n t fig u re -o fm e rit to c a lc u la te . In F ig . 6.20. th e gain (o -p o la r iz a tio n ) is shown fo r tw o d iffe re n t
m a g n e tiz a tio n s : / / 0 = 445 Oe and H 0 — 2 .000 O e. O ne set o f (dots represents th e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
204
-16 -___ ___ Measurements (1 magnet)
Measurements (2 magnets)
FEM/MoM (H0=238 Oe)
•
FEM/MoM (H„=445 Oe)
4
|
j
*
- 2 0 ---------------------------------------------------------------------------------------------------------------1
0.70
0.72
0.74
0.76
0.78 0.80
0.82
0.84
0.86
0.88
0.90
Frequency (GHz)
Fig. 6.19: Predicted and measured return loss versus frequency o f a ferrite-loaded CHS
antenna using one and two pairs o f magnets ( H 0 = 238. 4-15 Oe).
gain in c lu d in g th e m is m a tc h loss and th e o th e r set o f p lo ts represents th e g ain w ith o u t
th e m is m a tc h loss. O b v io u s ly , when th e m is m a tc h loss is a ccounted for. th e gain is
m a x im u m at th e resonant frequency o f th e a n te n n a . T h is m a x im u m was fo u n d to be
2.9 d B i at 842.5 M H z and 3.4 d B i at 1.010 M H z . B y n e g le ctin g th e m is m a tc h loss, the
e fficie n cy o f th e a n te n n a a t a frequency o f 842.5 M H z and I. 010 M H z is 60% and 90(/< .
re sp e ctive ly. T h is m eans th a t m ore energy is d is s ip a te d inside th e fe rrite w hen the
m a g n e tiz a tio n is set to 445 Oe ra th e r th a n 2 .0 0 0 Oe. M e a surem e n ts o f a b so lu te gain
for th e case o f u sin g tw o p a irs o f m agnets are also show n in Fig. 6.20. T h e agreem ent
betw een m e a surem e n ts and p re d ic tio n s is e x c e lle n t. T h e sm a ll discrepancies observed
are a ttr ib u te d to d iffr a c tio n s fro m th e edges o f th e fin ite g ro u n d p lane (9 ft long in
th e y -d ire c tio n ).
N o te also th a t gain im p ro v e m e n t can be achieved at th e cost ot
b a n d w id th : h o w e ve r, b a n d w id th is n ot a m a jo r issue fo r tu n a b le antennas.
One
way to im p ro v e g a in is by p la c in g d ie le c tric a n d /o r m a g n e tic o verlays on to p ot the
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
205
10
8
6
H „= 445 Oe (N o m is m a tc h )
H „= 44 5 Oe (In c l. m is m a tc h )
H o=2,000 Oe (N o m is m a tc h )
H o=2,000 Oe (In c l. m is m a tc h )
M e a su re m e n ts
4
£
2
‘
\
0-2
-4
-6
-8
-10
0.80
0.83
0.87
0.90
0.93
0.97
1.00
1.03
1.07
1.10
Frequency (GHz)
Fig. 6.20: Predicted and measured gain versus frequency o f a ferrite-loaded CBS antenna
using various biasing fields (o -p o la riz a tio n ).
g ro un d plane [198].[199]. A n o p tim iz a tio n code can be used to select o p tim u m values
for thickness and m a te ria l p a ra m e te rs .
T h e F- and H -p la n e a b s o lu te g ain p a tte rn s o f the fe rrite -tu n e d C B S antenna, at
a frequency o f 8-12.5 M H z a n d m a g n e tiz a tio n / / 0 = 4-15 O e. are sh ow n in Fig. 6.21.
T h e m easurem ents were p e rfo rm e d u sin g a g ro u n d plane o f d im e n s io n s 4 ft
x
9 It.
Besides the s m a ll rip p le caused by d iffra c tio n s fro m the edges o f th e g ro u n d plane,
th e agreem ent between s im u la tio n s a nd e x p e rim e n ts is ve ry good. T h e a bsolute gain
at broadside is close to 5 d B i.
6.6
C o n clusion s
A h y b rid F F .M /M o .M a p p ro a ch was fo rm u la te d to analyze fe rrite -tu n e d c a v ity-b a c k e d
slots m o u n te d on an in fin ite co ate d g ro u n d plane. T h e a ntenna u n d e r s tu d y is tu n a b le
w ith in th e C H F band th ro u g h a lte r in g th e e x te rn a lly bias m a g n e tic fie ld . S c a tte rin g
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0
30
30
60
60
90
90
10
0
-1 0
-20
-3 0
dB
-3 0
-2 0
-1 0
10
0
0
30
30
60
60
90
90
10
0
-1 0
-20
-3 0
dB
-3 0
-2 0
-1 0
0
10
Fig. 6.21: E- and H-pIane absolute gain patterns o f a ferrite-loaded CBS antenna b i­
ased along the //-direction using two pairs o f magnets ( H 0 = 44b Oe and
/ = 842.5 M H z). ------ P redictions. - — Measurements.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
a n d ra d ia tio n c h a ra c te ris tic s were c a lc u la te d a n d co m p a re d w ith m e a su re m e n ts and
d a ta fro m o th e r in d e p e n d e n t sources. T h e a g re e m e n t between p re d ic te d d a ta and
c o m p a ris o n d a ta was e x c e lle n t.
I t was also illu s tr a te d in th is c h a p te r th a t tu n in g
c a p a b ilitie s up to 45c/c can be achieved at a c e n te r fre qu e ncy o f 900 M H z .
a n te n n a has an a b so lu te g a in o f 3 d B i and a d ir e c t iv it y o f o d B i.
The
D e p e n d in g on
th e sp e cific a p p lic a tio n , d esign o p tim iz a tio n o f th is a n te n n a can be used to im p ro v e
c e rta in ra d ia tio n c h a ra c te ris tic s . F o r e x a m p le , a c o m b in a tio n o f layers can be used
in s id e o r o u ts id e th e c a v ity to im p ro v e th e a b s o lu te g ain o f th e a n te n n a .
For gain
im p ro v e m e n t, a lin e a r o r p la n a r a rra y o f slo ts m a y also be used. T h is is a to p ic o f
fu tu re in v e s tig a tio n .
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CHAPTER 7
T H E A N IS O T R O P IC ’ P E R F E C T L Y M A T C H E D L A Y E R
U n bounded e le c tro m a g n e tic p ro b le m s , such as s c a tte rin g a n d ra d ia tio n , are usually
solved using a n a ly tic a l o r n u m e ric a l m ethods.
A n a ly tic a l m e th o d s are m ore s u it­
able fo r c a n o n ic a l s tru c tu re s , whereas n u m e rica l m e th o d s are o fte n p re fe rred when
g e o m e trica l a nd m a te ria l c o m p le x itie s are present in th e d o m a in o f in te re s t.
An­
a ly tic a l m e th o d s in v o lv e re p re s e n ta tio n o f the fields in an u n b o u n d e d space using
a set o f c lo s e d -fo rm expressions th a t sa tisfy c e rta in b o u n d a ry c o n d itio n s .
On the
o th e r hand, n u m e ric a l m e th o d s, such as the fin ite e le m e n t m e th o d ( F E M ) and the
fin ite -d iffe re n c e tim e -d o m a in ( F D T D ) m e th o d , o p e ra te w ith in a fin ite d o m a in .
In
o th e r w ords, th e c o m p u ta tio n a l d o m a in needs to be n u m e r ic a lly tru n c a te d w ith o u t
d is tu rb in g wave p ro p a g a tio n in th e o u tw a rd d ire c tio n . T h u s , an e ffic ie n t absorbing
b o u n d a ry c o n d itio n (A B C ) m u st be im posed at th e o u te r b o u n d a ry so th a t spurious
backw ard re fle c tio n s are t o t a lly suppressed. S tu d y o f a b s o rb in g b o u n d a ry co n d itio n s
s u ita b le fo r im p le m e n ta tio n in fin ite m ethods has been an in te re s tin g research to p ic
for m a n y years [ 1AG]-[ 1-10].
A n a ccu ra te tru n c a tio n b o u n d a ry c o n d itio n , ca lle d th e p e rfe c tly m atched layer
(P .M L) [9-1]. was re c e n tly in tro d u c e d in the area o f c o m p u ta tio n a l e le ctro m a g n e tics.
T h is ty p e o f tru n c a tio n te c h n iq u e e x h ib its num erous advantages co m p a re d to m ore
tra d itio n a l A B C 's in c lu d in g b e tte r a b s o rp tio n , lo w e r b a c k w a rd re fle c tio n s , no fre­
quency o r angle dependence and ease o f im p le m e n ta tio n .
A lth o u g h th e P M L was
o rig in a lly d e ve lo p e d fo r th e F D T D m e th o d , it was la te r m o d ifie d and im p le m e n te d
in the F E M [93]. T h e present s tu d y is p r im a r ily c o n c e n tra te d on th e accuracy and
e fficie n t im p le m e n ta tio n o f P .M L in th e co n te xt o f th e F E M . P a ra m e tric studies are
co nducted n o t o n ly to e va lu a te th e accuracy o f P.M L in F E M . b u t also to shed insight
and u n d e rs ta n d in g to w a rd p ro p e r use o f the co nce p t.
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20!)
7.1
In tr o d u c tio n
Since the in tr o d u c tio n o f the p e rfe c tly m a tc h e d la ye r [94]. m a n y e ffo rts have concen­
tra te d on th e a c c u ra te and e ffic ie n t im p le m e n ta tio n o f such an a bso rb e r in th e fin ite
e le m en t m e th o d . Recent p u b lic a tio n s on th e to p ic d e m o n stra te th e effectiveness, ac­
c u ra cy and s im p lic ity o f the P M L in areas o f e le c tro m a g n e tic s c a tte rin g and antenna
ra d ia tio n [93 ].[2 0 0]. T h e P M L im p le m e n ta tio n in o th e r d is c ip lin e s , such as e le c tro n ic
packaging a n d m ic ro w a v e c irc u its , s t ill needs to be fu rth e r p ursue d . T h e P M L p e rfo r­
m ance c o n c e rn in g m icrow ave c irc u it s im u la tio n u sing the fin ite e le m e n t m e th o d was
b rie fly in v e s tig a te d by G ong f t al. [95]. T h e use o f the P M L in th e fin ite -d iffc re n c e
tim e -d o m a in m e th o d has a lre a d y show n s ig n ific a n t im p ro v e m e n t com pared to t r a d i­
tio n a l h ig h e r-o rd e r a bso rb in g b o u n d a ry c o n d itio n s . P ub lish e d d a ta [1 1 1],[14 5 ].[2 0 1]
d e m o n s tra te tru n c a tio n errors on the o rd e r o f — 100 dB o r even sm a lle r.
T h e o rig in a l im p le m e n ta tio n o f th e P .M L abso rb e r by B eren g er [94] is based on
a s p lit-fie ld fo rm u la tio n w h ich is kn o w n to be n o n -M a x w e llia n [202],
In a d d itio n ,
such an a p p ro a ch is s u ita b le o n ly in th e c o n te x t o f the F D T D m e th o d . R ecently, it
was show n by Sacks f t al. [93]. th a t th e P M L region can be e ffe c tiv e ly m odeled as a
u n ia x ia l a n is o tro p ic lossy m a te ria l. T h e P M L m e d iu m , w h ic h is classified as a ctive ,
causes no re fle c tio n s at the in te rfa ce and ra p id ly decays the in c id e n t fie ld in one o f
the p rin c ip a l d ire c tio n s . T he a n is o tro p ic P.M L was la te r im p le m e n te d in the F D T D
m e th o d by o th e rs [148].[152].[153] in o rd e r to a vo id s p littin g o f th e fields inside the
c o m p u ta tio n a l d o m a in . A lth o u g h a n o th e r a p p ro a ch to the d e ve lo p m e n t o f a p e rfe c tly
m a tche d la y e r, kn o w n as c o o rd in a te s tre tc h in g [149].[150]. a ppeared to be an a ccurate
and e ffic ie n t n u m e ric a l tru n c a tio n o f th e m esh, it d id not a ttr a c t p o p u la rity because o f
the in v o lv e d m a th e m a tic a l c o m p le x ity : h o w e ve r, it indeed in s p ire d th e developm ent
o f th e a n is o tro p ic p e rfe c tly m a tch e d la y e r.
T h is c h a p te r presents an e xte n sive in v e s tig a tio n on th e a ccu ra cy o f th e P.ML a b ­
so rb er m o d e le d as a u n ia x ia l a n is o tro p ic lossy m e d iu m . V a rio u s p a ra m e tric studies
a rc c o n d u c te d on b o th 2-D and 3-D p ro b le m s w ith p rim a ry o b je c tiv e to reach c e r­
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210
ta in co n c lu s iv e re m a rks on th e o v e ra ll p e rfo rm a n c e o f th e P M L region in th e c o n te x t
o f th e fin ite e le m e n t m e th o d . A n e xte n d e d s tu d y o f th e n u m e ric a l e rro r associated
w ith th e a n is o tro p ic P M L in th e c o n te x t o f F E M re c e n tly appeared in th e lite r a ­
tu re [1 5 4 ].[lo o ]. T h e m a in idea is to tre a t th e P M L re g io n as a n on -p h ysical u n ia x ia l
e le c tric a n d m a g n e tic a n is o tro p ic m a te ria l. In a d d itio n , th is m a te ria l is h ig h ly lossy
so th a t th e in c id e n t fie ld is s ig n ific a n tly a tte n u a te d before it a c tu a lly reaches th e
te r m in a tin g p erfe ct c o n d u c tin g w a ll. A lth o u g h the P M L te rm in a tio n p rovides a refle ction le ss b o u n d a ry c o n d itio n , th e b a ck-p la ce m e n t o f a P E C o r P M C surface re s u lts
in re fle c tio n s w ith in te n s ity b eing a fu n c tio n o f in c id e n t a ngle. H ow ever, by c o n tro l­
lin g th e e le c tric and m a g n e tic c o n d u c tiv itie s o f th e P M L m e d iu m , the in c id e n t fie ld
can be s ig n ific a n tly re d uce d before it a c tu a lly im p in g e s th e c o n d u c tin g w a ll.
It is
im p o rta n t to m e n tio n here th a t th e choice o f an A B C . ra th e r th a n a P E C o r P M C .
is also an a cce p ta b le te r m in a tin g w a ll fo r th e P M L re g io n [96].
A tw o -d im e n s io n a l p a ra lle l-p la te w a veguide was in it ia lly considered in th is s tu d y
to e va lu a te th e effectiveness and a ccu ra cy o f th e p e rfe c tly m a tch e d layer te rm in a tio n .
T w o -d im e n s io n a l p ro b le m s p ro v id e us w ith the f le x ib ilit y o f using re la tiv e ly dense
g rid s, th e re b y re d u cin g th e d is c re tiz a tio n e rro r, w ith o u t g e n e ra tin g m a trice s th a t art*
c o m p u ta tio n a lly in te n s iv e to solve.
Besides, a tw o -d im e n s io n a l p ro b le m is u s u a lly
s im p le r a n d m ore in t u itiv e th a n a th re e -d im e n s io n a l p ro b le m .
T h e p a ra lle l-p la te
w aveguide is e x c ite d w ith e ith e r a T E M o r T M i m o d e a n d operates w ith in a w id e
band o f frequencies. T h e a ccu ra cy o f th e P M L te r m in a tio n is th o ro u g h ly e x a m in e d
fo r various cases o f p a ra m e te riz a tio n . T h e idea o f u sin g th e P M L a n is o tro p ic m e d iu m
to te r m in a te a p a ra lle l-p la te w aveguide is th e n e xte n d e d to m ore p ra c tic a l th re e d im e n s io n a l p ro b le m s.
These in c lu d e b o th re c ta n g u la r w aveguide d is c o n tin u itie s
and m ic ro w a v e in te g ra te d c irc u its .
For re c ta n g u la r w aveguides, the in p u t p o rt is
e x c ite d u s in g th e d o m in a n t T E i0 m ode whereas th e o u tp u t p o rt is te rm in a te d w ith
an .Y -la ye r P M L region. Each la ye r has d iffe re n t e le c tric a n d m a g ne tic c o n d u c tiv itie s
s ta r tin g w it h ve ry low values near th e firs t in te rfa c e to in c re a s in g ly la rg e r values
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
211
near the te r m in a tin g w a ll. For m icro w a ve c irc u its , th e in p u t p o r t is e xcite d w ith the?
d o m in a n t m o d e o f th e m ic r o s trip o r c o p la n a r w aveguide lin e w hereas the o u tp u t p o rt
and re m a in in g s id e w a lls are te rm in a te d w ith a s in g le -la y e r1 P M L . As a re s u lt, th e
c o n d u c tiv ity o f th e m e d iu m m u st be chosen so th a t th e re fle c tio n e rro r is m in im iz e d .
T h e reason fo r im p le m e n tin g a s in g le -la y e r P M L . in ste a d o f a m u lti-la y e r P M L . in
th e area o f m ic ro w a v e c irc u its is to m a in ly s im p lify th e g e o m e tric a l m odel.
P re d icte d re s u lts on th e 5 -p a ra m e te rs o f w aveguide d is c o n tin u itie s and open m i­
crow ave in te g ra te d c ir c u its show a v e ry good agreem ent w ith d a ta o b ta in e d u sin g
o th e r c o m p u ta tio n a l te ch n iq u e s.
In g en e ra l, th e re su lts p re sen ted in the fo llo w in g
sections illu s tr a te th a t th e a n is o tro p ic P M L in th e c o n te x t o f th e fin ite e le m e n t
m e th o d can be used to tru n c a te th e c o m p u ta tio n a l d o m a in o f unbounded e le c tro ­
m a g n e tic p ro b le m s w ith d esirab le accuracy. T h e re are h ow eve r, a few pro ble m s to
s t ill overcom e w hen u sin g th e a n is o tro p ic P M L in th e fin ite e le m e n t m e tho d . T hese
in c lu d e d e g ra d a tio n o f th e m a tr ix c o n d itio n n u m b e r, slow convergence ra te fo r it e r ­
a tiv e solvers, a nd u n w a n te d c o m p le x itie s d u rin g m a te ria l d e fin itio n and g e o m e tric a l
m o d e lin g .
7.2
A 2-1) F in ite E le m e n t F o rm u la tio n o f th e A n is o tro p ic P .M L
In general, th e fin ite e le m e n t a nalysis o f e le c tro m a g n e tic w ave p ro ble m s begins w ith
th e d is c re tiz a tio n o f th e e le c tric held H e lm h o ltz 's e q u a tio n g iv e n by
7 x | / V ' T x E ) - [ - t r E = 0
(7 .1 )
w here / ' r and e r are th e re la tiv e p e rm e a b ility and p e r m it t iv it y tensors, re sp e c tiv e ly .
For p ro p e r im p le m e n ta tio n o f th e a n is o tro p ic P M L . th e p e r m e a b ility and p e r m it t iv it y
tensors m u st be d e fin e d as d ia g o n a l tensors o f th e fo llo w in g fo rm :
0
—
0
0
/ / yy
0
0
cr r
0
0
0
L
=
0
0
0
0
1A layer is d efin e d as a re c ta n g u la r region w h ere a ll m a te r ia l p a ra m e te r-
in c lu d in g ele ctric a n d
m a g n e tic c o n d u c tiv itie s , a re m a in ta in e d co n s ta n t.
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In tw o -d im e n s io n a l problem s, w here th e dependence o f th e e le c tric fie ld a lo ng th e cd ire c tio n is c o n s ta n t, the v e c to r wave e q u a tio n in (7 .1) is reduced u sin g th e sta n d a rd
G a le rk in 's app ro a ch to th e fo llo w in g tw o weak in te g ra l equations:
//,[
p ,(V,xE,)
J /ct
l
w here
(Vt
E,r
erE‘
(in
= 0
(7.2)
(V tE: ) H r ( v tE : ) - k 2
0E; t r E : dn = 0
(7.3)
Hr
X
- k;E t
refers to the p s e u d o -p e rm e a b ility te n so r defined in C h a p te r 3. N o te th a t
th e in te g ra l e q u a tio n in (7.2) in vo lves o n ly th e transverse co m p on e nt o f th e e le c tric
fie ld , whereas th e in te g ra l e q u a tio n in (7.3) in vo lve s o n ly th e lo n g itu d in a l co m p o n e n t
o f th e e le c tric fie ld . T he tw o e q u a tio n s are d ecoupled o n ly if th e p e r m it t iv it y and
p e rm e a b ility tensors e x h ib it a b lo ck d ia g o n a l s tru c tu re .
Based on a fin ite elem ent
d is c re tiz a tio n , th e transverse fields are e xpa n de d using lin e a r edge-based tria n g u la r
elem ents, whereas ih e lo n g itu d in a l fields are e xpa n de d using lin e a r node-based t r i ­
a n g u la r elem ents: th e d is c re tiz a tio n process o f th e tw o p a rtia l d iffe re n tia l e q u a tion s
(P D E 's ) is o u tlin e d in C h a p te r 3. In th e case o f a p a ra lle l-p la te w aveguide, th e in p u t
p o rt is e x c ite d w ith e ith e r a T E M o r T M t m o d e.
T h e field e x c ita tio n is im posed
using the fo llo w in g m ixed b o u n d a ry c o n d itio n
n x ( V x E ) -f j k p f i x (h x E ) = —2 j k pE""~
(7.1)
w here h is th e o u tw a rd u n it v e c to r n o rm a l to th e in p u t plane. kp is th e p ro p a g a tio n
co n sta n t
o f th e p ro p a g a tin g m o d e along th e d ire c tio n n o rm a l
is th e in c id e n t e le c tric
to th e p o r t, and E ‘ n'
fie ld . C o n sid e rin g th e g e o m e try and co o rd in a te syste m
shown
in F ig . 7.1. th e in c id e n t e le c tric fie ld fo r th e T E M and T M t m odes is re s p e c tiv e ly
d efined as
E " ,e
=
a y E 0c~j kzr
(7.5)
E ,m'
=
; i - E 0* i n ( ¥ - ) r - j k*r
b
(7 .fi)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
21:5
where kx is th e p ro p a g a tio n c o n s ta n t a lo n g th e .r-d ire c tio n . b is th e height o f th e
p a ra lle l-p la te w aveguide, and E 0 is th e a m p litu d e o f th e e x c ita tio n fie ld . Thus, the
th re e -d im e n sio n a l m ix e d b o u n d a ry c o n d itio n in (7.4) can be s ig n ific a n tly s im p lifie d
to th e fo llo w in g :
dEu
dx
- j k x E v = - 2 j k r E 0c~j k ‘ r
fo r T E M
dE- - j k x E z = - 2 j k j r E 0i > i n ( ~ ) e jklX
ox
b
(7.7)
fo r T M p
(7.8)
B oth e q u a tion s are v a lid o n ly a t th e e x c ita tio n plane. N o te th a t th is ty p e o f m ixe d
b o u n d a ry c o n d itio n , so m e tim e s re fe rre d to as an a bso rb in g o r ra d ia tio n b o u n d a ry
c o n d itio n , becomes exact i f th e fie ld d is tr ib u tio n at th e e x c ita tio n plane precisely
resembles th e assum ed m ode: th u s, som e people referred to th is ty p e o f b o u n d a ry
c o n d itio n as an e xa ct b o u n d a ry c o n d itio n .
U s u a lly th o u g h , th e presence o f a d is ­
c o n tin u ity in sid e th e w a veguide generates h ig h e r-o rd e r modes w h ic h are n ot to ta lly
a tte n u a te d by th e tim e th e y reach th e e x c ita tio n plane. T h e re fo re , th e use o f (7.1)
is considered o n ly an a p p ro x im a te tru n c a tio n o f the mesh a t th e in p u t p o rt.
T he
a ccuracy o f such an a b so rb in g o r ra d ia tio n b o u n d a ry c o n d itio n in c re a s in g ly becomes
b e tte r by m o v in g th e in p u t p o rt fa rth e r aw ay fro m the present d is c o n tin u ity .
T h e o u tp u t p o rt, on th e o th e r h an d . i.. te rm in a te d using th e a n is o tro p ic p e rfe c tly
m atched la ye r backed w ith a P E C o r P M C w a ll.
N ote th a t th e in p u t p o rt m ay
also be te rm in a te d using a p e rfe c tly m a tc h e d la ye r: how ever, th e e x c ita tio n has to
pEC
P M L R E G IO N
PEC
Fig. 7.1: P arallel-plate waveguide t e rm in a te d w ith a m ulti-layer P M L region (o = 10 m m ) .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
be im posed u sin g a s lig h tly d iffe re n t a p p ro a c h fro m th e one im p le m e n te d
in this
c h a p te r. Such a te c h n iq u e in volves im p o s in g m a g n e tic c u rre n ts on an a r tific ia l plane
betw een th e P M L in te rfa c e and th e fro n t d is c o n tin u ity .
T h e re fle c tio n and tra n s m is s io n c o e ffic ie n ts are ca lc u la te d based on th e p o rt equa­
tio n s at th e in p u t and o u tp u t planes. For th e T E M m ode, th e t o t a l neid a t th e in p u t
p o rt can be expressed as a s u p e rp o s itio n o f th e in c id e n t and re fle c te d waves
E y ( x = . r i. y ) = E 0e~jkzXx + Ft E 0ejkzXl
(7.9)
w here .iq is th e .r-c o o rd in a te o f th e in p u t p o r t. B y firs t in te g ra tin g b o th sides o f the
above expression and th e n solve fo r th e re fle c tio n co efficie n t Ft a t .r = .iq. th e final
re s u lt is
r
t
e
=
m
f h E j {
L 0b
r
=
Xl my ) d y
_
t - ^ «
.
(7.10)
Jo
In a s im ila r m a n n e r, th e re fle c tio n c o e ffic ie n t fo r th e T M j m ode is g iv e n by
R T 'U l
" V
=
l r
L 02b
I '
Jo
E : ( - r
=
. r l . v ) < i ! J - t ~ 2 j k l X l -
(7.11)
A t th e o u tp u t p o r t, th e co rre sp o n d in g tra n s m is s io n co efficie n t fo r each o f th e p ro p ­
a g a tin g m odes is g ive n by
t
t
e
m
=
f b
/.;v ( ,. =
s ^ u x h j
(7 .1 -J )
L 0b Jo
t t
"
x
= ~ F 02b
^ r Jo
t E^ J =
{hJ
w here ./•_> co rre sp on d s to th e .r-c o o rd in a te o f th e o u tp u t p o rt.
7.3
A 3-D F in ite E le m e n t F o rm u la tio n o f th e A n is o tro p ic P M L
In a th re e -d im e n s io n a l p ro b le m , a lth o u g h th e fin ite ele m en t fo r m u la tio n is u sua lly
m o re in vo lve d and cu m b ersom e , e s p e c ia lly fo r cases where th e a n a lysis involves
a n is o tro p ic m a te ria ls , th e approach is s t ill v e ry s im ila r to th e tw o -d im e n s io n a l case.
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T h e v e c to r wave e q u a tio n in ( 7 . 1) is firs t tra n s fo rm e d in to a weak in te g ra l fo rm given
by
[
E) •(V
X
Ju
X
N) d V -
L'Z [ [er] • E - N d V + /
Jq
Js
([//r]~lV x E ) • (N x h) d A
=
0
(7.14)
w here
h is th e
v o lu m e , and
n o rm a l to th e su rface u n it v e c to r p o in tin g o u tsid e th e fin ite elem ent
N
denotes th e v e c to r te s tin g fu n c tio n .
T h e closed surface in te g ra l in
(7 .14 ) is n on -ze ro o n ly on n o n -p e rfe c tly c o n d u c tin g surfaces. In th e a n a lysis o f m i­
crow ave c irc u its a n d g u id in g s tru c tu re s , a ll b o u n d a ry surfaces besides th e in p u t port
are te rm in a te d w it h a p e rfe c tly m a tch e d la y e r backed by e ith e r a P E C o r P M C . Thus,
th e surface in te g ra l in (7 .14 ) is e va lu a te d o n ly a t th e in p u t port w h ic h , as p re v io u s ly
n o te d , is te rm in a te d usin g th e S o m m c rfe ld -ty p e ra d ia tio n b o u n d a ry c o n d itio n given
in (7 .4).
T h e e x c ita tio n fie ld is o b ta in e d u sin g a tw o -d im e n s io n a l (2 -D ) fu ll-w a ve
eigenvalue a n a lysis [124] o f th e in p u t p o rt. In a d d itio n , th e 2-D e ig en va lue analysis
is used to e va lu a te th e d is p e rs iv e p ro p a g a tio n co n s ta n t and c h a ra c te ris tic im pedance
o f th e c o rre s p o n d in g tra n s m is s io n lin e . These q u a n titie s are essential in th e im p le ­
m e n ta tio n o f th e m ix e d b o u n d a ry c o n d itio n s a t th e p orts and th e e v a lu a tio n o f the
^'-p a ra m e te rs .
T h e re fle c tio n and tra n s m is s io n c o e fficie n ts at th e p o rts o f w a ve gu id e s tru c tu re s
are ca lc u la te d u sin g a s im ila r a p p ro a ch to th e tw o -d im e n s io n a l case. S p e c ific a lly , for
a re c ta n g u la r w a ve gu id e th a t is e x c ite d at th e in p u t p o rt using th e d o m in a n t T E io
m o d e, th e c o rre s p o n d in g re fle c tio n and tra n s m is s io n coefficients are g ive n by
R
=
T
=
f f
- -—
E ( . r . y. r t ) • e l0{ x . y ) d S - c ~ 2jkiaZl
a n t o J Jsl
~£
[
a nto J
f
E ( .r ./y .c 2) • e l0{ . r . y ) d S
(7.1.'))
(7.1fi)
w h e re S\ and S? are th e in p u t and o u tp u t p o rt surfaces, re sp e ctive ly. e to (.r./y ) =
dy sin( ^ ) c ~ jk'l0Z. a and b are th e h o riz o n ta l a n d v e rtic a l dim ensions o f th e re c ta n g u la r
cross section and Aq0 is th e p ro p a g a tio n co n s ta n t o f th e T E m m ode.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
216
In th e case o f m ic ro w a v e c irc u its , th e 5 -p aram e te rs are e va lu a te d based on th e
to ta l and reference vo ltag e s a t th e p o rts . M ore d eta ils on th e 5 -p a ra m e te r e va lu a tio n
are given in C h a p te r 4.
7.4
T e nso r R e p re s e n ta tio n o f th e A n is o tro p ic P M L
T he p e rfe c tly m a tch e d la y e r is m o deled as a u n ia x ia l a n is o tro p ic m a te ria l c h a ra c te r­
ized by th e fo llo w in g p e r m it t iv it y and p e rm e a b ility tensors [93].[152]:
[c]?m/
[/i]r '
=
=
k
0
0
0
At
0
0
0
1/At-
Cotr A.. = £0f r
//o /* r A : =
00 f i r
K'
0
0
0
At’
0
0
0
i
(7.17)
(7.18)
/ a.-’
where
a
(7.19)
- J-
=
r
r.20)
1-2-
For m ore d e ta ils on th e d e riv a tio n o f th e a n iso tro p ic absorber, the re a d e r is referred
to th e o rig in a l paper by Sacks f t al.
by Zhao f t al.
[151].
[93] as well as the re c e n tly p u b lish e d paper
T h e s tru c tu re o f the p e r m ittiv ity and p e rm e a b ility tensors
in (7.17) and (7.18) im poses s ig n ific a n t a tte n u a tio n for tra v e lin g waves along th e
c -d ire c tio n . For waves tra v e lin g in th e .r-d ire c tio n . th e e ntries
1 / a-
and I /
at'
located
in th e t h ir d row o f each te n so r m ust be interchanged w ith the co rre sp o n d in g e n trie s
k and k" lo c a te d in th e firs t row . T h e re s u ltin g m a trice s are d e n o te d as \ r and A ’
given bv
O
-
A,
=
1 / a-
0
0
0
a-
0
0
0
a-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(7.21)
217
a
:
1/ k ‘
0
0
=
0
K"
0
0
0
Km
(7.22)
B y e n fo rcin g th e B eren g er c o n d itio n [94]
a
a "
= --------.
e0er
/ t 0/ / r
(7.2:3)
th e c h a ra c te ris tic im p e da n ces o f th e tw o m e d ia at th e in te rfa ce are id e n tic a l, and
the re fo re, th e re are no re fle ctio n s caused by a plane wave in c id e n t on th a t in te rla c e .
Based on th is m a tc h in g c o n d itio n ,
k
is e qu a l to
k"
and A is id e n tic a l to A ‘ fo r a ll
three p rin c ip a l d ire c tio n s o f p ro p a g a tio n . A m o re generic expression fo r th e p e rfe c tly
m atched la ye r c o n d itio n is g ive n by [9-‘3]
[ p /
U
-
r wml
L/J
/*o/*r
(7.24)
Besides th e th re e p rin c ip a l d ire c tio n s , th e a n is o tro p ic P M L can be fo rm u la te d in
such a way as to absorb waves p ro p a g a tin g in o th e r d ire c tio n s as w e ll [1-51].
For
e xam ple, waves tra v e lin g in th e .r/y-d ire ctio n are b e tte r absorbed b y a P M L m e d iu m
th a t e x h ib its p ro p e rtie s o f b o th an .r-d ire c te d P.ML and a //-d ire c te d P M L [1 4 8 ].[ I b l]
ra th e r th a n ju s t one o f th e tw o : i.e..
[<]%'
=
to trA j- Ay
[/C r
=
/ w 'r A ;
a
7.2b i
;.
(7.26)
A ll re m a in in g c o m b in a tio n s are form ed in a s im ila r m anner. N o te also th a t th e ra te
w ith w h ich th e in c id e n t fie ld is a tte n u a te d in sid e th e P.ML region is d ir e c tly re la te d
to the a c tu a l values o f a and a ". In n u m erou s results presented by B erenger. it was
c le a rly p o in te d o u t th a t th e n u m e rica l re fle ctio n s fro m th e P M L in te rfa c e can be
s ig n ific a n tly reduced b y c a re fu lly sele cting th e values o f a and <r~. U s u a lly n u m e ric a l
re fle ction s o c c u r w hen th e tra n s itio n fro m one m e d iu m to a n o th e r becom es m o r:e
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
a b r u p t.
T h u s , fo r an a ccurate im p le m e n ta tio n o f the m u lti- la y e r P M L m o d e l, a
ju d ic io u s choice fo r cr and cr' is [94]
m
(7.27)
°{p)
V'(P)
o’,m a x
w h e re crm^r and cr^ax are. re s p e c tiv e ly , th e m a x im u m e le c tric and m a g n e tic c o n d u c ­
t iv it ie s o f th e a n is o tro p ic m e d iu m , d is th e e n tire d epth o f th e P M L re g ion , m is th e
o rd e r o f th e s p a tia l p o ly n o m ia l, and p a is th e p o sitio n o f th e firs t in te rfa c e in th e
d ire c tio n o f p ro p a g a tio n . T h e c o n tin u o u s d is trib u tio n o f a and cr' in sid e th e P M L
m e d iu m is sta irca se d based on th e n u m b e r o f abso rb in g layers in v o lv e d . H ow ever,
a lth o u g h a s ta irc a s in g approach is proven s im p le and s tra ig h tfo rw a rd to im p le m e n t.
it in tro d u c e s u n d e sira b le g e o m e tric a l and m a te ria l co m p le x itie s in F E M . T o avoid
such d iffic u ltie s , th e use o f a s in g le -la y e r P.M L m ay be an a lte r n a tiv e choice. T h e
s p a tia l decay o f th e field in sid e th e P.M L region is m a in ly c o n tro lle d by th e a c tu a l
values o f rrm,1T and cr"n,i r . A good choice fo r th e m a x im u m c o n d u c tiv ity value's is
given b y [94]
(7.29)
(7.40)
w h e re R is d e fin e d as the th e o re tic a l re fle c tio n coefficient a t n o rm a l in cid e n c e and c
is th e speed o f lig h t. S e ttin g th e re fle c tio n co e fficie n t to a desired value, e.g. 10- 1 .
a goo d e s tim a te fo r crmnT and cr'm,tJ. can be o b ta in e d . E q u a tion s (7 .29 ) and (7.20) do
n o t ta k e in to a ccou n t the d is c re tiz a tio n e rro r.
A lth o u g h th e im p le m e n ta tio n o f th e m u lti-la y e r P M L a bso rb e r g uarantees a n u ­
m e ric a l tru n c a tio n e rro r close to th e desired th e o re tic a l re fle c tio n c o e ffic ie n t /?. p ro ­
v id e d th e d is c re tiz a tio n e rro r is alw a ys s m a lle r, it was p u rp o s e ly a voide d in th e
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•219
a n a lysis o f m ic ro w a v e c irc u its because o f the associated g e o m e trica l and m a te ria l
c o m p le x itie s . As a lre a d y m e n tio n e d , each layer e x h ib its d is tin c t c o n d u c tiv ity values.
.Also, th e in te rs e c tio n o f these P M L layers form s "edges" and "c o rn e rs " w h ich also
e x h ib it d is tin c t c o n d u c tiv ity values. To avoid a ll these various c o m b in a tio n s o f m a­
te r ia l d e fin itio n , it was decide d th a t th e s in g le -la ye r P M L m odel be used. "th u s, the
s p a tia l v a ria tio n o f th e m a te ria l c o n d u c tiv itie s cr a n d cr" inside the P M L is co nstant:
i.e.. m = 0. As a re s u lt, th e e le c tric and m a g n e tic c o n d u c tiv itie s o f a single-layer
P M L region m a y be s im p ly w r itte n as
«
=
«r-
=
beat r c
—
( . ..ii)
(7.:|2)
w here 8 is an o p tim iz a tio n p a ra m e te r to be d e te rm in e d based on n u m e ric a l e x p e r­
im e n ta tio n .
A n a p p ro p ria te choice o f 6 is one th a t re su lts in m in im u m reflections
fro m th e P M L te r m in a tio n .
fle c tio n co e fficie n t
I. sing basic e le c tro m a g n e tic theory,
o f as in g le -la y e r P M L
region, backed by
th e th e o re tic a l re­
aperfect
c o n d u c to r, at
n o rm a l in cidence is g ive n by
R = t - uh-i
(T T 5 )
w here b = < j/(^ -t0cr ) and 3 is th e p ro p a g a tio n c o n s ta n t o f th e wave in sid e the m e d ium
p r io r to the P.ML in te rfa c e . N o te th a t 3 m ay be w r itte n as 3 0sJT^~f] where t r, j j is the
e ffe c tiv e d ie le c tric c o n s ta n t. S u b s titu tin g these p a ra m e te rs in to (7 .3 3 ). th e re s u ltin g
th e o re tic a l re fle c tio n c o e ffic ie n t is given by
R —- e ~2'd 17777)‘' ° y w 7 _
(7.31)
In case th a t th e tru n c a te d m e d iu m is free space, th e co rre sp o n d in g th e o re tic a l re­
fle c tio n co efficie n t s im p lifie s to
R = e~\
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(7.3-'))
220
In o th e r w ords. 8 is re la te d to th e th e o re tic a l re fle ctio n c o e ffic ie n t R as fo llo w s:
<5 = — L - l n f i ) .
\ / er e f f
\R J
(7 .36 )
Based on th e above d e fin itio n o f R a t th e P M L in te rfa ce , its value s ta rts decreasing
e x p o n e n tia lly as 6 increases. H ow ever, th e d is c re tiz a tio n e rro r becomes in c re a s in g ly
d o m in a n t, th e re b y the to ta l n u m e ric a l e rro r as a fu n c tio n o f 8 e x h ib its a m in im u m
value. A lso , u n lik e the m u lti- la y e r P M L m odel, the m a te ria l tra n s itio n a t th e P.M L
in te rfa c e is now m ore a b r u p t, th e re fo re , a d d itio n a l n u m e ric a l re fle ctio n s fro m th e
in te rfa c e are in tro d u c e d in th e s o lu tio n .
Thus, the value o f 8 is d e p e n d e n t not
o n ly on th e th e o re tic a l re fle c tio n co e fficie n t R b ut also on th e mesh d e n s ity and
n u m e ric a l re fle ctio n s fro m th e a b ru p t tra n s itio n at th e in te rfa c e . T h e la t te r can be
s ig n ific a n tly reduced by im p le m e n tin g th e m u lti-la y e r P M L m o d e l w ith m > 0.
It
is also im p o rta n t to em p ha size here th a t th e er and / / r o f th e m e d ia in v o lv e d at th e
P M L in te rfa c e be co n tin u o u s.
T.o
N u m e ric a l Results
T h e n u m e ric a l p erfo rm a n ce o f th e a n is o tro p ic P M L is tested fo r a v a rie ty o f p ro b le m s
ra n g in g fro m a tw o -d im e n s io n a l p a ra lle l-p la te waveguide to re c ta n g u la r w a ve gu id e
d is c o n tin u itie s and m icro w a ve c irc u its . A lth o u g h th e concept is a p p lic a b le to wave
p ro p a g a tio n as w ell as s c a tte rin g and ra d ia tio n , p a rtic u la r em p ha sis was p la ced on
wave p ro p a g a tio n inside g u id in g s tru c tu re s .
T .o .l
P a ra lle l-p la te w aveguide
T h e a ccu ra cy o f the a n is o tro p ic P M L absorber was firs t in v e s tig a te d in th e case o f a
p a ra lle l-p la te waveguide e x c ite d w ith e ith e r a T E M o r T M t m o d e. T h e p a ra lle l-p la te
w aveguide, w h ich is show n in F ig . 7.1. is a ir-fille d and te rm in a te d w ith a m u lti- la y e r
P.ML region backed w ith a p e rfe ct e le c tric co n d u cto r. T h e d e p th o f th e m u lti- la y e r
P.ML region is <1 = 20 m m a nd th e n u m b e r o f layers is .V = 10. A la ye r is d e fin e d as
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•221
a re c ta n g u la r re g io n w hich, d e p e n d in g on th e mesh size, m ig h t a cco m m o d a te m o re
th a n one e le m e n t in th e lo n g itu d in a l d ire r* io n . T h e o rd e r o f the s p a tia l p o ly n o m ia l
g ive n in (7.27) a n d (7.2S) is m =
2.
T h e effectiveness o f the m u lti-la y e r P M L
te rm in a tio n is in v e s tig a te d fo r va rio u s values o f R w h ich , as in d ic a te d by (7 .29 )
and (7.30). d ir e c tly relates to <rmrix a n d a'max- T h e re s u ltin g re fle c tio n co e ffic ie n t
as a fu n c tio n o f fre qu e ncy is d e p ic te d in Figs. 7.2 and
m o d e o f p ro p a g a tio n , respectively,
7.3 fo r th e T E M and T M j
ft is in te re s tin g to observe th a t th e o b ta in e d
n u m e ric a l re fle c tio n coefficie n t, fo r b o th cases o f e x c ita tio n , reduces c o n s id e ra b ly
w hen th e th e o re tic a l value o f R decreases fro m 0.1 to 0.0001.
N o te also th a t fo r
la rg e r values o f R. th e re fle ction co e ffic ie n t re m a in s co nsta nt versus fre q u e n cy w h ic h
is expected a c c o rd in g to the th e o ry o f th e P M L . However, fo r s m a lle r values o f
R. th e re fle ctio n co efficie n t is d o m in a te d b y th e d is c re tiz a tio n e rro r w h ich becom es
la rg e r w ith in c re a s in g frequency. It is also im p o rta n t to note here th a t th e re fle c tio n
co efficie n t fo r th e T M i inode increases ra p id ly as the fre qu e ncy approaches 4 G H z .
T h is is due to th e c u to ff frequency at f = 3.75 G H z. below w h ich the fields becom e
evanescent.
As illu s tra te d in Figs. 7.2 and
7.3. th e choice o f desired re fle ction co e ffic ie n t R
is ve ry im p o rta n t in designing an e ffe c tiv e m u lti-la y e r P M L region.
In a d d itio n to
/?. th e average m esh size is a n o th e r p a ra m e te r th a t affects th e n u m e ric a l re fle c tio n
c o e fficie n t. As a lre a d y m e n tion e d, th e n u m e ric a l re fle ctio n co efficie n t is a co m b in e d
effect due to a re fle c tio n from th e P M L m e d iu m as w ell as e rro r due to th e fin ite
e le m e n t d is c re tiz a tio n . Note th a t th e d is c re tiz a tio n e rro r is o f o rd e r I r w here h is th e
Table 7.1: Mesh inform ation based oil three different discretizations of a parallel-plate
waveguide (.V = 10 and d = 20 nun).
# o f Edges
# o f Nodes
5.175
7.872
2.698
11.339
17.169
5.831
22.571
34.084
11.514
M esh #
# o f T ria n g le s
I
2
3
|
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.0
-
10.0
— R=0.1
R=0.01
- R=0.0001
m -20.0
3
3 -30.0
a
0)
'o -40.0
g -50.0
u
g -60.0
g -70.0
I-S O .O
-90.0
-
100.0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
Frequency, GHz
Fig. 7.2: Reflection coefficient from a PM L term ination as a function o f the theoretical
reflection coefficient at normal incidence (T E M mode. .V = 10. d = '20 mm.
i n = 2).
0.0
-
R=0.1
R=0.01
j
- R=0.0001 !
10.0
'u -40.0 \
2
-50.0
g -60.0
H -70.0
q=
fg -80.0
-90.0
-
100.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
Frequency, GHz
Fig. 7.3: Reflection coefficient from a PM L term ination as a function of the theoretical
reflection coefficient at normal incidence (T M | mode. .V = 10. d — 20 mm.
i n = 2).
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
222
M e s h #1
M e s h #2 ~
M e s h #3
-10
• y -40
S -50
£ -60
-100
0
2
4
6
8
10
12
14
16
18
20
Frequency, GHz
Fig. 7.-1: Reflection coefficient from a P M L te rm in a tio n as a function o f mesh density
(T E M mode. .V = 10. d = 20 nun. m = 2. R — LO- '1)-
m a x im u m tria n g le edge. T h e e ffe ct o f d is c re tiz a tio n is illu s tra te d c le a rly in F ig. 7.1
fo r th re e d iffe re n t mesh d e n s itie s (see T a b le
7 .1 ).
B y in cre asing th e n u m b e r o f
tria n g le s in th e c o m p u ta tio n a l d o m a in , the d is c re tiz a tio n e rro r reduces to levels where
the n u m e ric a l re fle ctio n c o e ffic ie n t, at low er fre qu e ncies, is as low as —To d B w ith th e
tre n d o f re d u c in g it even fu r th e r b y a d d itio n a lly re fin in g th e mesh. A s th e fre qu e ncy
becomes la rg e r, the n u m e ric a l re fle c tio n c o e ffic ie n t seems to fo llo w a m o n o to n ic a lly
in cre a sin g slope.
T h is p h e n o m e n o n is so le ly a ttr ib u te d to the d is c re tiz a tio n e rro r
w h ich , as a lre a d y m e n tio n e d , is a fu n c tio n o f th e e le c tric a l size o f th e tria n g le edge.
A n o th e r p a ra m e te r th a t is c ru c ia l in th e design o f a m u lti-la y e r P M L m e d iu m
is th e o rd e r o f the c o n d u c tiv ity p ro file d en o te d b y m .
T h e o b je c tiv e here is to
c h a ra c te riz e th e s e n s itiv ity o f th e n u m e ric a l re fle c tio n co e fficie n t as th e o rd e r o f the
c o n d u c tiv ity p ro file increases. F ig . 7.5 d e m o n s tra te s th a t by in cre a sin g th e o rd e r o f
the s p a tia l p o ly n o m ia l in. th e n u m e ric a l re fle c tio n c o e ffic ie n t d e te rio ra te s a t th e low er
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
221
m=l
m=2 "
m=3
_ m=4
— m=5
-10
*3 -40
V/
-100
0
2
4
6
8
10
12
14
16
18
20
Frequency, GHz
Fig. 7.5: The effect o f spatial p o lyn o m ia l order on the reflection coefficient from a P M L
term in a tion (T E M mode, mesh #.'j. .V = 10. d = 20 m m . R = 10- 4 ).
frequencies, whereas at h ig h e r frequencies the fin ite d is c re tiz a tio n effect d o m in a te s .
In o th e r w ords, a lin e a r o r q u a d ra tic c o n d u c tiv ity p ro file in s id e th e P M L region seems
to p ro v id e a b e tte r a b so rp tio n a nd less sp uriou s re fle c tio n s co m p a re d to liig h e r-o rd c r
c o n d u c tiv ity profiles. T h e d iffe re n c e in th e re fle ctio n c o e ffic ie n t how ever between th e
tw o e x tre m e cases is less th a n 10 d B .
T h e same a ir-fille d p a ra lle l-p la te w a veguide was chosen to in v e s tig a te th e effect o f
chan g in g th e d e p th [d) o f th e P M L region as w ell as th e n u m b e r o f layers (.V ). T h e
re m a in in g p aram eters were m a in ta in e d th e same as in th e p re vio u s case. T h e re fle c­
tio n co e fficie n t versus fre q u e n cy fo r th re e d iffe re n t values o f d is shown in Figs. 7.6
and 7.7. fo r .V = .5 and .V = 10. re sp e ctive ly. B o th these figures correspond to th e
T E M m ode o f p ro p a g a tio n . T h e re s u lts illu s tra te th a t by in cre a sin g th e d e p th o f th e
P M L region, th e re fle ction c o e ffic ie n t a t th e low er fre q u e n c y range, w h ich is p r im a r ­
ily a ttr ib u te d to th e P.ML te r m in a tio n , becomes s m a lle r. S p e c ific a lly , in cre a sin g th e
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
o.o
-
d=10 mm
d=20 mm
d=30 mm
10.0
PQ -2 0 .0
Reflection coefficient
3
-30.0
-40.0
-50.0
-60.0
-70.0
-90.0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
Frequency, GHz
.6: Reflection coefficient from a P M L te rm in a tio n as a function o f the P M L depth
(T E M mode, mesh # 3 . .V = 5. R = 1 0 "1).
0.0
‘ ----- d=10 mm
d=20 mm
- • d=30 mm
Reflection coefficient (dB)
-10.0 r
-
20.0
-30.0
-40.0
-50.0
-60.0
-70.0
-80.0
-90.0
-
100.0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
Frequency, GHz
Fig. 7.7: Reflection coefficient from a P M L te rm in a tio n as a function o f the P M L depth
(T E M mode, mesh # 3 . .V = 10. R = 10- ')-
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
226
P M L d e p th , cl. fro m 10 m m to 30 m m th e re fle c tio n c o e fficie n t a t th e lo w e r end o f
th e fre q u e n cy range red uce s b y n e a rly as m uch as 20 d B . A lso , c o m p a rin g F ig . 7.6
w ith F ig . 7.7. it is c le a r t h a t th e re fle c tio n c o e ffic ie n t re m a in s a lm o s t u n a ffe c te d by
in cre a sin g th e n u m b e r o f la ye rs fro m A = 5 to .V =
10. T h is last re m a rk is b e tte r
illu s tra te d in Fig. 7.S w h e re th e re fle c tio n c o e ffic ie n t versus fre qu e ncy is p lo tte d fo r
various values o f .V. A c c o r d in g to th is figure, a decrease in th e n u m b e r o f layers fro m
16 to 2 re su lts in o n ly a s m a ll increase (o d B ) in th e re fle c tio n c o e ffic ie n t. In o th e r
w ords, incre asing th e n u m b e r o f P M L layers does n o t im p ro v e th e a c c u ra c y o f th e
results: on th e o th e r h a n d , it d e stro ys th e s im p lic ity o f th e g e o m e tric a l m o d e l. N o te
th a t a ll re m a in in g p a ra m e te rs such as m . cl and R are m a in ta in e d c o n s ta n t fo r a ll
cases considered. T h e m o s t in te re s tin g p h e n o m e n o n , h ow ever, is w hen a sin g le la ye r
o f P M L is used. It is o b v io u s fro m F ig . 7.S th a t th e re fle c tio n co efficie n t increases by
20 d B . T h e reason is re la te d to th e increase o f n u m e ric a l re fle ctio n s due to th e a b ru p t
d is c o n tin u ity at th e a ir-P .M L in te rfa c e .
T h e s m o o th e r th e m a te ria l tr a n s itio n , th e
s m a lle r th e b a ckw ard re fle c tio n s . In a d d itio n , th e d is c re tiz a tio n e rro r has increased
co m p ared to the m u lti- la y e r m o d e l since th e e n tire P M L region is now c h a ra c te r­
ized w ith a c o n d u c tiv ity v a lu e o f cr,n,l x . T h e la rg e r th e m a te ria l c o n d u c tiv ity , tin '
s m a lle r th e w a ve le n g th a n d . th e re fo re , the la rg e r th e d is c re tiz a tio n e rro r.
In such
a case, th e re is an o p tim u m va lu e o f a and <r" w h ic h p ro vid e s m in im u m re fle c tio n
c o e ffic ie n t. T h us, fo r a b e tte r P M L a b s o rp tio n th e associate d m a te ria l c o n d u c tiv itie s
should alw ays be o p tim iz e d to r a s in g le -la ye r te r m in a tio n .
O p tim iz a tio n o f a s in g le -la y e r P M L m e d iu m in te rm s o f b is show n in F ig . 7.9 fo r
th e case o f a p a ra lle l-p la te w a ve gu id e. T h e in p u t p o r t is a gain e x c ite d w ith a IT ..M
plane wave. T h e va lu e o f b increases fro m 0 to 15. Based on th e th e o re tic a l e xpressio n
o f th e re fle c tio n c o e ffic ie n t a t n o rm a l in cid e n ce , g iv e n b y (7 .3 5 ). th e n u m e ric a l re fle c ­
tio n co e fficie n t is e x p e c te d to decay e x p o n e n tia lly u n t il it reaches a p o in t w h e re th e
d is c re tiz a tio n e rro r a n d / o r th e sp u rio u s re fle c tio n s fro m th e m a te ria l d is c o n tin u ity
a t th e P.ML in te rfa ce b e g in s to d o m in a te . T h is is c le a tly illu s tra te d in F ig . 7.9 w here
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
N
N
N
N
N
-10
« -20
S
-30
C
0)
■ 3 -40
m
g -50
0
1 ' 6°
H -7 0 1
C£ -80 j~
/ X
V» w' / * ' % \V?.
!!
=
=
=
=
=
1
2
4
10
16
/
v?
-90
-100
6
8
10
12
Frequency, GHz
14
18
16
20
Fig. 7.S: Reflection coefficient from a P M L te rm in a tio n as a function o f the number o f
layers (T E M mode, mesh # 5 , m = 2. d = 20 mm. R = 1 0 " 1).
-10
-
N u m e ric a l ]
T h e o re tic a l ;
£ -5 0
£ -60 j
-100
6
Fig. 7.9: Single-layer P M L o p tim iza tio n in terms o f S ( / = 100 M H z. average cell size
h = 0.5 m m ).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
228
th e n u m e ric a l p re d ic tio n s closely fo llo w th e th e o re tic a l g ra p h fo r a ll values between
0 and —65 d B : th e co rre s p o n d in g value o f 6. w hen R = —65 d B . is a p p ro x im a te ly 9.
F u rth e r increase o f S re su lts in a m o n o to n ic a lly in cre a sin g n u m e ric a l re fle ctio n co effi­
c ie n t caused by b o th p o o r d is c re tiz a tio n and n o n -p h ysica l re fle ctio n s fro m the P M L
in te rfa ce . T h e la rg e r the value o f 8. th e la rg e r th e c o n d u c tiv ity o f th e lossy m e d iu m :
as a re s u lt, th e w a ve le n g th in s id e th e P M L region becom es in c re a s in g ly sm a lle r. In
o th e r w ords, a lth o u g h th e o p e ra tin g fre qu e ncy is re la tiv e ly lo w ( / = 100 M H z ) th e
d is c re tiz a tio n e rro r keeps in cre a sin g w ith incre asing 6. T h e average ce ll size used for
th is p a r tic u la r p ro b le m was 0.5 m m .
T h e effect o f th e d is c re tiz a tio n e rro r in o p tim iz in g a s in g le -la y e r P M L m e d iu m
is illu s tra te d in Fig. 7.10.
T h is fig u re p rovides a co m p a ris o n a m o n g th e o b ta in e d
n u m e ric a l re fle c tio n co e fficie n ts based on three d iffe re n t meshes: mesh # 1 has an
average cell size o f 0.5 m m : mesh # 2 has an average ce ll size o f 1.0 m m : mesh # 5
has an average cell size o f 2.0 m m .
From Fig. 7.10. it is cle a r th a t by increasing
th e average tria n g u la r edge le n g th (h ). th e d is c re tiz a tio n e rro r increases p ro p o rtio n ­
a lly a c co rd in g to / r . th e re b y s h iftin g th e local m in im u m to w a rd lo w e r values o f 8.
T h e re fo re , se le ctin g a va lu e fo r 8. w h ich m in im iz e s b a c k w a rd re fle ctio n s fro m the
P.ML te r m in a tio n , re q uires kn o w le d g e o f th e in h e re n t d is c re tiz a tio n e rro r. However,
th e d is c re tiz a tio n e rro r is b o th p ro b le m -d e p e n d e n t as w e ll as m e sh -dependent. Since
th e re is no clo se d-fo rm e xpressio n to p re cisely p re d ict th e level o f d is c re tiz a tio n e r­
ro r. it re m a in s up to the e xp e rie n ce and in tu itio n o f th e in d iv id u a l to p ro vid e a good
guess o f th e in v o lv e d d is c re tiz a tio n e rro r.
In a d d itio n to p lo ttin g th e re fle c tio n coefficie n t as a fu n c tio n o f frequency, it is also
in te re s tin g to observe th e n o rm a liz e d fie ld d is trib u tio n in s id e th e w aveguide when th e
o u tp u t p o rt is te rm in a te d w ith a P M L region. T h e c o rre s p o n d in g fie ld d is trib u tio n
fo r th e T E M case, when <I ~ 20. w = 2. and .V = 10. is illu s tr a te d in F ig . 7.11. T h e
m aO
g n itu d e o f th e tra nsve rse e le c tric fie ld b a sica llv
- re m a in s c o n sta n t e vervw he re in
th e w aveguide, whereas in s id e th e m u lti-la y e r P M L region it a tte n u a te s q u ite ra p id ly .
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
22!)
M e s h #1
M e s h #2
M e s h #3
T h e o re tic a l
-10
o -40
£ -5 0
£ -60
-100
6
Fig. 7.10: The effect o f discretization e rro r in o p tim iz in g a single-layer P M L medium.
A n o th e r im p o r ta n t p a ra m e te riz a tio n th a t was co n d u cte d usin g th e 2-D fin ite
e le m e n t fo r m u la tio n relates to the a ccuracy o f th e P M L te rm in a tio n w h e n th e la tte r
is placed close t o a d is c o n tin u ity . A re c ta n g u la r d ie le c tric d is c o n tin u ity ( w = 20 m m .
h =
20 m m . t r = 6) is in se rted in th e m id d le o f th e p a ra lle l-p la te w aveguide as
sh ow n in F ig . 7.12. T h e P M L in te rfa ce was placed at various d ista n ce s L fro m the
re a r end o f th e d is c o n tin u ity . T h e n o rm a liz e d e n e rg y ( | /?|2 + | T | 2) as a fu n c tio n ol
fre q u e n cy is sh ow n in Figs. 7.12 and
7.14 fo r d iffe re n t values o f L . I t is in te re s tin g
to observe, in b o th figures, th a t the n o rm a liz e d e n e rg y rem a in s v e ry close to u n ity
fo r a ll frequencies in th e low er end o f th e s p e c tru m . As th e fre q u e n c y o f o p e ra tio n
approaches th e c u to ff p o in t, p la cing th e P M L in te rfa c e near d is c o n tin u itie s creates
n u m e ric a l e rro rs . S p e c ific a lly , when th e P M L in te rfa c e is placed 10 m m away from
th e d is c o n tin u ity th e energy e rro r reaches values as high as 10c/r..
A s illu s tra te d
in Figs. 7.12 a n d 7.14. th is e rro r can be s ig n ific a n tly reduced by m o v in g th e P.ML
in te rfa c e fa r th e r aw ay fro m th e d is c o n tin u ity .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•2:50
AO
5 0
«
d ire ction
Fig. 7.11: Norm alized e le ctric field d is trib u tio n in a parallel-plate waveguide (T E M mode.
.V = 10. (I = 20 m m . R = 10~T m = 2).
P M L R E G IO N
PEC
N PEC
PEC
w
Fig. 7.12: P arallel-plate waveguide loaded w ith a dielectric d isco n tin u ity w ith f r — 0 (6 —
-10 nun. ir = 30 m m . h = 20 m m ).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T h e n u m e ric a l d is c re p a n c y in th e a c tu a l re fle c tio n and tra n sm issio n coefficients,
w hen p la cin g th e P M L re g ion a c e rta in d is ta n c e aw ay fro m th e d is c o n tin u ity , is
show n in Figs. 7.15 and 7.16 fo r th e T E M m o d e o f p ro p a g a tio n .
cases were co nside re d: one w ith L =
T w o d iffe re n t
10 m m a n d a n o th e r w ith L = 45 m m . T h e
re s u ltin g re tie c tio n and tra n s m is s io n co efficie n ts are co m p a re d w ith d a ta o b ta in e d by
te r m in a tin g th e o u tp u t p o rt u sin g th e "e x a c t” b o u n d a ry c o n d itio n (E B C ). In b oth
figures, it is show n th a t p la c in g a P M L te rm in a tio n o n ly a distan ce 10 m m away from
th e d ie le c tric d is c o n tin u ity re su lts in s ig n ific a n t e rro rs fo r frequencies above 5.0 G H z.
7.5.2
R e c ta n g u la r w a ve gu id e d is c o n tin u itie s
T h e n u m e ric a l a c c u ra c y o f th e a n is o tro p ic P M L a b s o rb e r in th re e -d im e n sio n a l p ro b ­
lem s was in v e s tig a te d in th e presence o f a d ie le c tric d is c o n tin u ity placed inside a
re c ta n g u la r w aveguide. T h e d ie le c tric co nsta nt o f th e d is c o n tin u ity is er = 6.
The
re c ta n g u la r w a ve gu id e is e x c ite d w ith the T E jo m o d e at th e in p u t p o rt whereas the
o u tp u t p o rt is te rm in a te d w ith a 5 -la ye r P M L m e d iu m backed w ith a p erfe ct e le c tric
c o n d u c to r.
T h e s p a tia l p ro file o f th e e le c tric and m a g n e tic loss in side th e m u lt i­
la ye r P M L region fo llo w s a p a ra b o lic shape.
T h e w aveguide region is discre tized
w ith te tra h e d ra l e le m en ts u sin g th e mesh g e n e ra to r o f SDRC' l-D E A S [ 1ST]. wheieas
th e fin a l m a tr ix syste m is solved using a sparse L I
d e co m p o s itio n . T h e n u m b e r of
u n kn o w n s in th e fin ite e le m e n t region is a p p ro x im a te ly 14.000.
The m a g n itu d e o f
>'u is c o m p u te d fo r values o f k Qb ra n g in g fro m 1.6 to 4.0. O u r re su lts are com pared
w ith d a ta o b ta in e d fro m a jo u r n a l paper w r itte n b y Ise f t nl. [125]. T h e c o m p a ri­
son betw een th e tw o d a ta sets is d e p icte d in F ig . 7.18. It is c le a rly illu s tra te d th a t
th e m a g n itu d e o f S'n c a lc u la te d using the p e rfe c tly m a tch e d la ye r as an absorber
in th e fin ite e le m e n t m e th o d is in excellen t a greem ent w ith the co rre sp o n d in g data
p u b lish e d by Ise.
T h e effectiveness and a c c u ra c y o f the a n is o tro p ic p e rfe c tly m a tch e d la ye r in a
th re e -d im e n s io n a l v e c to r fin ite elem ent code is also teste d for o th e r d ire c tio n s of
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
232
1.20
L=10
L=20
L=30
L=45
1.15
mm
mm
mm
mm
1.10
— 1.05
+ 1.00
PML REGION
PEC
TEM
0.90
PEC
TM
PEC
0.85
0.80
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Frequency, GHz
Fig. 7.13: Conservation of energy in the presence o f a P M L interface and a dielectric dis­
continuity at close proxim ity (T E M mode).
1 .20 i----------------------------------------------------------------------------------------------------------------------------------
j
1 15
^
I
L=10 mm
L=20 mm
L=30 mm
L=45 mm
i
1.10 I
I
— 1.05 i
+ 1-00
PML REGION
PEC
0.90 j
TEM
PEC
TM
PEC
0.85
0.80
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
Frequency, GHz
Fig. 7.1-1: Conservation of energy in the presence of a PM L interface and a dielectric dis­
continuity at close proxim ity (T .\I[ mode).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
f
i
0.0
L=10 mm
L=45 mm
EBC
Reflection coefficient (dB)
-4.0
-
-
8.0
12.0
-16.0
-
20.0
-24.0
P M L R EG IO N
-28.0
TEM
PEC
-32.0
TM,
PEC
-36.0
-40.0
0.5
0.0
1.0
1.5
2.5
2.0
3.0
3.5
Transmission coefficient (dB)
r.
Frequency, GHz
15: Reflection coefficient from a dielectric d isco n tin u ity o f cr = (j inside a parallelplate waveguide (T E M mode).
-4.0
-
8.0
PM L REGION
PEC
-
12.0
S PEC
TM
PEC
X.
-16.0
L=10 mm
L=45 mm
EBC
-
20.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Frequency, GHz
Fig. 7.16: Transmission coefficient from a dielectric d isco n tin u ity o f r r = 6 inside a parallelplate waveguide (T E M mode).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
231
PML
0.8 b
0.601 b
TE
0.399 b
0.556 b °-888 b 0.556 b
Fig. 7.17: A dielectric-loaded waveguide term inated w ith a o-layer P M L (cr = (i. b = 1 cm)
1.0
5-layer PML
ref. [125]
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
1.5
1.6
1.8
1.9
2.1
2.2
ko b
2.4
2.5
2.7
2.8
3.0
Fig. 7.18: M agnitude o f 8 'n for a dielectric d isco ntin u ity inside a rectangular waveguide.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
p ro p a g a tio n , besides th e r-a x is .
S p e c ific a lly , th e case o f an a ir-fille d re c ta n g u la r
w aveguide tu r n in g in to a rig h t-a n g le bend, as show n in F ig . 7.19. is in v e s tig a te d .
T h e d im e n sio n s o f th e w aveguide are chosen so th a t a = 2b. T h e o u tp u t p o rt, whose
n o rm a l ve c to r is p o in tin g a lo ng th e .r-a x is . is te rm in a te d w ith a 5-layer P M L m e d iu m
backed w ith a p e rfe ct e le c tric c o n d u c to r.
Because o f th e a pp a re nt change in the
o rie n ta tio n o f th e a bso rb e r, th e c o rre s p o n d in g p e r m it t iv it y and p e rm e a b ility tensors
m u st also be ro ta te d a c c o rd in g ly so th a t th e im p o sed a tte n u a tio n applies fo r in c id e n t
waves tra v e lin g in th e x- in stea d o f th e c -d ire c tio n . T h e size o f the c o m p u ta tio n a l
d o m a in corresponds to a p p ro x im a te ly 1-t.000 u n kn o w n s.
T h e p re d icte d |-S-2i | 2 is
co m p ared w ith d a ta e x tra c te d fro m th e p ap e r by Ise ct al. [125] for values o f 2 « /A
ra n g in g between 1.0 and 2.0. A v e ry good agreem ent betw een the tw o d a ta sets is
illu s tra te d in F ig . 7.20.
7.5..'J
M icro w a ve c irc u its
Based on num erous re su lts show n in p re vio u s sections, it is cle a r th a t th e m u lti-la y e r
P M L a bso rb e r can be used in th e fin ite e le m e n t m e th o d to e ffe c tiv e ly te rm in a te the
o u tp u t p o rts o f b o th 2-D and 3 -D g u id in g s tru c tu re s .
In a d d itio n , c e rta in design
p aram ete rs o f the P M L can be chosen a c c o rd in g ly to p ro v id e a n um erical tru n c a tio n
e rro r close to th e desired th e o re tic a l re fle c tio n co e fficie n t R. T h e p rim a ry o b je c tiv e
o f th is w o rk how ever, is the use o f th e a n is o tro p ic P M L in th e analysis o f m o re p ra c ­
tic a l e le c tro m a g n e tic p ro b le m s such as m icro w a ve in te g ra te d c irc u its ( M IC 's ) and
in te rco n n e cts. In such a case, th e c ir c u it m u st be t o ta lly enclosed by a re c ta n g u la r
P M L region th a t is designed to e ffe c tiv e ly a bsorb ra d ia te d fields caused by present
d is c o n tin u itie s .
Because o f th e in v o lv e d d iffic u ltie s d u rin g g e o m e try m o d e lin g and
m a te ria l d e fin itio n , it is u s u a lly easier to use th e s in g le -la ye r P M L m odel ra th e r th a n
th e m u lti-la y e r P M L m o d e l. T h u s , th e a n is o tro p ic m e d iu m m u st first be o p tim iz e d
based on the choice o f 6 to p ro v id e a m a x im u m wave a b s o rp tio n in th e d ire c tio n o f
in cid en ce . T h is was done fo r th e case o f a m ic ro s trip lin e o f w id th ir = 2.113 m m on
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
■m
PM L
b
a
Fig. 7.19: A ir-fille d right-angle bend term inated w ith a 5-layer P M L.
1.0
5-layer PML :
ref. [125]
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
1.0
1.1
1.2
1.3
1.4
1.5
2 a/A
1.6
1.7
1.8
1.9
Fig. 7.20: |S-2i | 2 versus 2n/ X o f an air-filled right-angle bend.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2.0
•237
a R T /D u ro ic I s u b s tra te o f h e ig h t h = 0.794 trim . A s in g le -la y e r P M L re g ion o f d e p th
d = 3 m m backed b y a m a g n e tic w a ll is chosen to te r m in a te th e o u tp u t p o r t. T h e
sid e w a lls are p erfe ct e le c tric c o n d u c to rs : th e re m a in in g m ic ro s trip d im e n sio n s are
show n in F ig . 7.21. T h e g e o m e try was d iscre tize d u s in g lin e a r te tra h e d ra l e le m en ts
o f average le n g th 1 m m . T h e re fle c tio n co e fficie n t as a fu n c tio n o f 8 is show n fo r fo u r
d iffe re n t frequencies in F ig . 7.22. B y in cre asing 8. th e re fle c tio n co efficie n t decreases
e x p o n e n tia lly .
H ow ever, w hen 8 increases above a c e rta in value, th e d is c re tiz a tio n
e rro r a n d n u m e ric a l re fle c tio n s fro m th e a b ru p t d is c o n tin u ity a t th e P M L in te rfa c e
becom e d o m in a n t: th e re fo re , a lo ca l m in im u m appears in th e g ra p h . T h is n u ll be­
comes d ee p er w ith d ecreasing fre q u e n cy because o f th e a p p a re n t re d u c tio n in th e
d is c re tiz a tio n e rro r. T h u s , th e value o f 8 should be ju d ic io u s ly chosen to m in im iz e
th e to ta l n u m e ric a l re fle c tio n e rro r caused by th e presence o f th e s in g le -la y e r P M L
re o
g ion . T h e th e o re tic a l re fle c tio n co efficie n t versus 8 is also show n in F igo. 7.22: an
cref f = 1.8 was used fo r th e c o m p u ta tio n .
O nce th e sin g le -la y e r P M L is o p tim iz e d based on th e a p p ro p ria te choice o f 8.
it can th e n be used as an a b s o rb e r to s im u la te open m ic ro w a v e c irc u its .
A ve c to r
fin ite e le m e n t approach was im p le m e n te d to c o m p u te th e .S'-param eters o f th e p la n a r
m ic ro w a v e low-pass filte r sh ow n in F ig . 7.23. T h is c ir c u it was o rig in a lly a n a lyze d by
Sheen f t at. [2o ] using th e F D T D m e th o d . T h e d e p th o f th e sin g le -la ye r P M L region
is 3 m m and th e d ista n ce to th e nearest d is c o n tin u ity is also 3 m m . T h e g e o m e try
was d is c re tiz e d using lin e a r te tra h e d ra l elem ents. T h e average size o f th e elem ent
used is a p p ro x im a te ly I m m . whereas th e to ta l n u m b e r o f e le m e n ts is 2 7 .8 3 4 . T h u s ,
th e e le c tric a l size o f a te tra h e d ro n in sid e th e s u b s tra te a t 20 G H z is a p p ro x im a te ly
A / 10.
A 2 -D fu ll-w a v e fin ite e le m e n t ana lysis is used to o b ta in th e fu n d a m e n ta l
m ode d is tr ib u tio n at th e in p u t p o rt, w hich is the n used as th e field e x c ita tio n for
th e th re e -d im e n s io n a l c ir c u it.
T h e m a g n itu d e o f th e c ir c u it s c a tte rin g p a ra m e te rs .
S'u and S 2 1 . versus fre q u e n cy are illu s tra te d in Fig. 7.24 whereas th e c o rre s p o n d in g
phase v a ria tio n is show n in F ig . 7.25.
These n u m e ric a l re su lts are co m p a re d w ith
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
IMS
3 mm
3 mm
0 .7 94 m m '
x
3 mm
2 .4 1 3 m m
Fig. 7.21: Geometry of a shielded m icrostrip line terminated with a single-layer PM L. The
substrate is R T /D tiroid o f er = 2.2.
f=5.00 GHz
f=3.00 GHz
f=1.00 GHz
f=0.75 GHz
Theoretical
CQ -10
-a
-15
-20
-35
-40
0.0
0.3
0.7
1.0
1.4
1.7
2.0
2.7
2.4
3.1
3.4
6
Fig. 7.22: M agnitude of .S’u versus the optim ization parameter
I mm).
b
(average mesh size
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
h =
2:59
d a ta o b ta in e d u sin g th e F D T D m e th o d . T h e c o m p a ris o n between th e tw o m e tho d s
illu s tra te s a ve ry g o o d agree m e n t, b o th in te rm s o f m a g n itu d e and phase. A t the
h ig h e r frequencies, how ever, th e d iscre p a n cy betw een th e tw o d a ta sets becomes
m ore v is ib le , w h ic h seems to be a ttr ib u te d to an in cre a sin g d is c re tiz a tio n e rro r in
b o th m e tho d s.
F o r th e phase c a lc u la tio n s , th e in p u t and o u tp u t port reference
planes were chosen a d ista n ce -1.233 m m and 3.3S64 m m . respectively, away fro m
th e d is c o n tin u ity . S p e c ific a lly , th e fin ite e le m e n t d a ta were sh ifte d to the predefined
reference planes used in th e F D T D code.
A lth o u g h it was show n here th a t the s in g le -la y e r P M L provides accurate re su lts
when im p le m e n te d in th e area o f m icrow ave c irc u its , it is in s tru c tiv e th a t we also
co m m e n t on th e convergence rate o f the ite r a tiv e so lve r used. S pe cifically, the re su lts
shown in the case o f th e m icrow ave low-pass f ilt e r were o b ta in e d using a C o n ju g a te
G ra d ie n t Square (C 'G S ) a lg o rith m w ith d ia g o n a l p re c o n d itio n in g . T h e ite ra tio n was
done in d ou b le p re c is io n .
H ow ever, the conve rg e nce ra te was s till e x tre m e ly slow
com pared to a case w h e re a firs t-o rd e r A B C was used in stea d o f a P M L . T h u s, the
convergence c rite rio n based on th e re la tiv e re s id u a l n o rm was s ig n ific a n tly re laxed in
o rd e r to speed up th e c o m p u ta tio n s . S p e c ific a lly , th e to le ra n c e was set to o.O/: — 2
w h ich is p ro b a b ly a n o th e r reason for some o f th e discrepancies shown in th e c o m ­
parisons. C u rre n t s tu d ie s are p r im a rily c o n c e n tra te d on im p ro v in g th e convergence
ra te o f th e ite ra tiv e so lve r.
7.6
Design G u id e lin e s and C onclusions
A p e rfe c tly m a tc h e d la ye r co m p rise d o f a lossy u n ia x ia l a n is o tro p ic m a te ria l was
successfully im p le m e n te d in to a 2-D and 3 -D fin ite e le m e n t fo rm u la tio n to c a lc u la te
th e S’-p a ra m e te rs o f various m icrow ave c ir c u its and g u id in g stru ctu re s.
N u m e ric a l
re su lts d e m o n s tra te th a t the a n is o tro p ic P.M L is a ccu ra te , s im p le to use. and s u ita b le
fo r im p le m e n ta tio n in th e fin ite elem ent m e th o d .
T h e a bso rb in g m a te ria l m a y be
m odeled e ith e r as a m u lti-la y e r o r as a s in g le -la y e r re g io n .
In th is stud y, it was
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•).\0
s'
A v(sub)
A
Geometry of a l o w - p a s s filter surrounded by a
!s RT/Duroid of height h = 0.794 mm and er
;
.... 1
0 I
—
-
tre 's'
vA P e ’
f u ( ^ eV
rftv5',s\°^
Auc®'
o ^ e
rep'
Cb0° pV°
fo u n d th a t th e m u lti- la y e r P M L p ro vid e s a b e tte r a b s o rp tio n th a n th e sin g le-la yer
P M L . S p e cifica lly, a n u m e ric a l e rro r close to th e th e o re tic a l re fle c tio n co efficie n t can
be achieved by ju d ic io u s ly choosing th e m a x im u m m a te ria l c o n d u c tiv itie s . crmar and
cr‘m u x
based on a d e s ira b le tru n c a tio n e rro r d e n o te d bv R. N o te t h a t th e o rd e r o f th e
s p a tia l p o ly n o m ia l has to be la rg e r th a n zero in o rd e r to m in im iz e s p u rio u s re flections
fro m the firs t P M L in te rfa c e . I f th e s p a tia l p o ly n o m ia l o rd e r is ze ro , th e n the P M L
te rm in a tio n corre sp on d s to a s in g le -la y e r m o d e l. In th a t case, th e tru n c a tio n e rro r
is aug m e n te d not o n ly by th e d is c re tiz a tio n e rro r b u t also b y a d d itio n a l n u m e rica l
re fle ctio n s fro m the firs t P M L in te rfa c e . T h e re fo re , a good ch o ice o f a and <r" has to
be d e te rm in e d by n u m e ric a l e x p e rim e n ta tio n . T h e la tte r is e q u iv a le n t to o p tim iz in g
th e n u m e rica l re fle c tio n e rro r based on th e choice o f 6.
A n o p tim u m value o f 6
re su lts in a n u m e ric a l re fle c tio n co e ffic ie n t th a t is re a son a bly close to th e re flection
c o e fficie n t o b ta in e d u s in g th e m u lti-la y e r P M L m o d el. S om e o f th e m ost im p o rta n t
fin d in g s o f th is s tu d y a re ta b u la te d below :
1. T h e P M L region can be designed e ith e r w ith m u ltip le la y e rs o f d is tin c t con­
d u c tiv itie s o r a sin g le la ye r o f u n ifo rm c o n d u c tiv ity .
G iv e n a specific mesh,
the m u lti-la y e r P M L m o d e l alw ays re su lts in a m in im u m re fle c tio n coefficient
whereas the s in g le -la y e r P M L m orh I m ust be o p tim iz e d .
2. T h e n u m e rica l re fle c tio n co e fficie n t fro m a P M L is a c o m b in e d effect ot the
d is c re tiz a tio n e rro r, re fle c tio n s fro m P M L in terfaces and th e b ack-p la ce m e n t ot
a hard b o u n d a ry c o n d itio n .
M esh re fin e m e n t in sid e th e P M L region reduces
b o th th e d is c re tiz a tio n e rro r and re fle c tio n s fro m in te rfa c e s .
■L T h e n u m e ric a l re fle c tio n co e ffic ie n t fro m a P M L decreases w ith
• In crea sin g th e P M L d e p th
•
D ecreasing th e o rd e r o f c o n d u c tiv ity p ro file ( m u lt i- la y e r m o d e l)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
212
-I. T h e n u m e ric a l re fle c tio n co efficie n t fro m a m u lti-la y e r P M L re m a in s in s e n s itiv e
to th e increase o f n u m b e r o f layers.
5. T h e p la ce m e n t o f a P M L re g ion in a close p r o x im ity to a m a te ria l d is c o n tin u ity
creates n u m e ric a l e rro rs.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER S
S U M M A R Y A N D G E N E R A L C O N C L U S IO N S
T h e focus o f th is d is s e rta tio n was th e d e ve lo p m e n t o f th e fin ite ele m en t m e th o d fo r
th e a na lysis o f co m p lex e le c tro m a g n e tic p ro b le m s ra n g in g fro m 2-D p la n a r m ic ro w a v e
c irc u its w ith a n is o tro p ic and lossy su bstra te s to 3 -D m icro w a ve d is c o n tin u itie s and
e le c tro n ic packages as w e ll as c a v ity -b a c k e d s lo t a nte n n a s w ith inhom ogeneous and
fre qu e ncy d e p e n d e n t m a te ria ls . For each o f these p ro je c ts a separate c o m p u te r code
was w r itte n , tested and v e rifie d a gainst o th e r n u m e ric a l m ethods and e x p e rim e n ts .
A lth o u g h a g re a t deal o f e ffo rt and em phasis was co n ce n tra te d on th e n u m e ric a l
issues in v o lv e d w ith th e p ro je c t, p ra c tic a l aspects and p ro b le m s related to th e a p p li­
ca tio n its e lf were seriously ta ke n in to c o n s id e ra tio n . E xte n sive discussions, personal
o p in io n s and ju d g m e n ts are p ro v id e d th ro u g h o u t th is m a n u s c rip t.
T h is d is s e rta tio n began w ith th e p re s e n ta tio n o f a fu ll-w a ve 2-D a n a ly s is o f
a n is o tro p ic a n d lossy m ic ro w a v e c irc u its .
A s s u m in g th a t an e x is tin g m o d e p ro p a ­
gates along th e tra n sm issio n lin e as a plane wave w ith a given p ro p a g a tio n c o n s ta n t.
M a x w e ll's e q u a tio n s can be fo rm u la te d th ro u g h th e use o f a fu n c tio n a l to o b ta in a
generalized eigenvalue p ro b le m . S o lu tio n o f th e e igenvalue p ro ble m at a g iv e n fre ­
q uency re s u lts in an e igenvalue th a t corresponds to th e dispersive p ro p a g a tio n co n ­
s ta n t and an e ig en ve cto r th a t corresponds to th e fie ld d is trib u tio n on th e tra n s v e rs e
plane. T h e c h a ra c te ris tic im p e d a n c e o f th e tra n s m is s io n lin e was s u b se q u e n tly c a lc u ­
la ted using th e p o w e r-vo lta g e d e fin itio n . T h e eig en va lue p ro b le m was solve d u sin g a
p ow er fo rw a rd ite ra tio n w ith G ra m -S c h m id t o rth o g o n a liz a tio n to o b ta in th e firs t few
d o m in a n t m odes. T he v a lid ity o f th e fo rm u la tio n a n d c o m p u te r code was illu s tr a te d
by a n a ly z in g p la n a r c irc u its w ith u n ia x ia l and b ia x ia l a n is o tro p ic su bstrates a n d c o m ­
p a rin g th e re s u lts w ith d a ta o b ta in e d using o th e r n u m e ric a l techniques. S u b s tra te
a n iso tro p ie s can a lte r th e e le c tric a l c h a ra c te ris tic s o f a tra n sm issio n lin e , th u s these*
should not be neglected.
In some cases, c ry s ta l ro ta tio n o f a n is o tro p ic s u b s tra te s
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
211
m ig h t be used to even im p ro v e th e e le c tric a l p e rfo rm a n c e o f c irc u its .
E xte n sive s tu d ie s on th e effect o f fin ite c o n d u c tiv ity o n th e p ro pa g atio n c h a ra c­
te ris tic s o f p la n a r c ir c u its w ere also show n. T w o d ifF c.w .t g e o m e trie s were considered:
the m ic ro s trip lin e and th e c o p la n a r w aveguide. In a d d itio n , d iffe re n t types o f m e ta l
were used: e.g.. a lu m in u m , co p p e r, n ic k e l, e tc. S tud ie s have show n th a t fin ite c o n d u c ­
t iv it y fo r m e ta llic traces has a great im p a c t on th e a tte n u a tio n co n sta n t, p ro p a g a tio n
co nsta nt and c h a ra c te ris tic im p e d a n ce o f th e tra n s m is s io n lin e . A t low er frequencies,
where th e cross s e ctio n o f th e c o n d u c to r is c o m p a ra b le to th e s k in d ep th, th e p ro p ­
agation c h a ra c te ris tic s are d ra s tic a lly a ffected. It was a lso show n th a t the c o p la n a r
w aveguide e x h ib its h ig h e r c o n d u c to r losses th a n th e m ic r o s tr ip lin e .
T h e fin ite e le m e n t a n a lysis o f 2-D m icro w a ve s tru c tu re s was extended to 2-1)
problem s. T y p ic a l s tru c tu re s in c lu d e p la n a r c irc u its and in te rc o n n e c ts , such as f il­
ters. in d u c to rs , w ire b onds, bridges and vias. as w e ll as c o m p le x packages such as
the 8 -p in S O IC s u rfa c e -m o u n t p la s tic package.
T h e in p u t p o rt o f the s tru c tu re is
e x cite d using th e d o m in a n t m ode d is tr ib u tio n whereas a ll o th e r p o rts and sidew alls
are te rm in a te d u sin g d is p e rs iv e a b so rb in g b o u n d a ry c o n d itio n s . S o lu tio n o f the lin ­
ear system o f e q u a tio n s was o b ta in e d th ro u g h th e use o f C o n ju g a te G ra d ie n t Square
a lg o rith m w ith d ia g o n a l p re c o n d itio n in g . T h e g lo b a l m a t r ix was stored in a sparse
fo rm a t to save m e m o ry space. T h e
p a ra m e te rs were c a lc u la te d based 011 the v o lt­
age d iffe re n ce a t th e p o rts and th e c o rre s p o n d in g c h a ra c te ris tic im pedances.
The
overall approach to th e p ro b le m was fou n d to be e x tre m e ly a ccu ra te and e ffic ie n t.
T he a ccu ra cy o f th e fo r m u la tio n and c o m p u te r code was v e rifie d by c o m p u tin g and
co m p a rin g th e S '-p a ra m cte rs o f n u m erou s s tru c tu re s w ith re s u lts o b ta in e d using o th e r
techniques in c lu d in g th e S D A and th e F D T D m e th o d . T h e agreem ent between th e
F E M p re d ic tio n s and co m p a riso n d a ta was fo u n d to be e x c e lle n t.
T h e 2-D F E M code was also used fo r th e analysis o f m ore com plex e le c tro n ic
packaging s tru c tu re s . .S pecifically, an 8 -p in S O IC s u rfa c e -m o u n t p la s tic package was
analyzed w ith in a fre q u e n c y range o f 1-20 G H z fo r d iffe re n t g ro u n d in g c o n d itio n s .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
245
T h e 5 -p a ra m e te rs o f an M M I C c ir c u it were firs t c o m p u te d in th e absence o f the
package. T h e n , it was m o u n te d o n to th e p a d d le o f th e S O IC package w ith c e rta in
leads grounded to th e m o th e rb o a rd g ro u n d .
As e xp e cte d , th e 5 - p a ra m e te rs o f the
c ir c u it have changed d r a s tic a lly once packaged, whereas th e package its e lf in tro d u c e d
a d d itio n a l resonances w h ic h lim it th e m a x im u m fre q u e n cy o f th e o v e ra ll s tru c tu re . It
was also found th a t g ro u n d in g c o n d itio n s can s h ift th e m a x im u m o p e ra tin g frequency
o f th e package to e ith e r lo w e r o r h ig h e r frequencies. T h u s, a good g ro u n d in g o f the
M M IC paddle can s u b s ta n tia lly im p ro v e th e e le c tric a l p e rfo rm a n ce o f th e package.
In a d d itio n to m ic ro w a v e c irc u its and e le c tro n ic packages, th e fin ite elem ent
m e th o d was h y b rid iz e d w it h a m ix e d s p e c tra l/s p a tia l d o m a in m e th o d o f m o m e n ts
to in ve stig a te s c a tte rin g and ra d ia tio n c h a ra c te ris tic s o f c a v ity -b a c k e d a pe rtu re s
m o u n te d on an in fin ite g ro u n d p la n e w ith d ie le c tric o r m a g n e tic overlay. T h e c a v ity
was tre a te d using th e v e c to r F E M whereas th e e x te rio r region was tre a te d using the
m ix e d s p e c tra l/s p a tia l d o m a in M o.M . T o im p ro v e th e e fficie n cy o f M o M . a frequency
in te rp o la tio n o f th e a d m itta n c e m a t r ix was in tro d u c e d . N ot o n ly th e a ccu ra cy o f the
m e th o d was re ta in e d , b u t also th e c o m p u ta tio n a l speed o f the code was s u b s ta n tia lly
im p ro v e d . T h e c a v ity -b a c k e d s lo t was tre a te d b o th as a sca tte re r and as a ra d ia to r.
For s c a tte rin g , a p la ne w ave in c id e n c e was co nsidered whereas fo r ra d ia tio n , a coaxial
feed m odel was fo rm u la te d . A lth o u g h th e feed co u ld have been e a sily m o d ele d as a
d e lta gap or a c u rre n t p ro b e , those are less a ccu ra te .
T h e a c cu ra cy o f th e h y b rid
FE.M /.M o.M approach was te s te d fo r a v a rie ty o f c a v ity -b a c k e d a p e rtu re s in c lu d in g
fin ite arrays o f a p e rtu re s . T h e p re d ic tio n s w ere co m p a re d w ith b o th m e asurem ents
and d a ta o b ta in e d u sin g a p u re s p e c tra l d o m a in M o M . T h e a greem ent was e x c e lle n t.
In m any p ra c tic a l a p p lic a tio n s , th e c a v ity -b a c k e d a p e rtu re is m o u n te d on a large
fin ite g round plane o r a la rg e su rfa ce such as an a irfra m e . T h e a na lysis o f th e e n tire
s tru c tu re using fu ll-w a v e c o m p u ta tio n a l te ch n iq u e s becomes e x tre m e ly e xpe n sive in
te rm s o f C P E tim e and m e m o ry space. For e le c tric a lly large s tru c tu re s a s y m p to tic
tech n iq ue s such as th e E T D becom e v e ry a ttr a c tiv e . T h e re fo re , th e h ig h -fre q u e n c y
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2-16
•
L T D m e th o d was in c o rp o ra te d in to th e a lre ad y h y b rid F E M / M o M code to o b ta in e d
d iffra c tio n s fro m the edges o f a fin ite g ro u n d plane on w h ich th e a n te n n a is m o u n te d
o n. T h e v a lid ity o f th e F E M / M o M / L 'T D h y b rid code was tested fo r a s m a ll n u m b e r
o f a n te n n a geom etries.
T h e p re d ic te d a bsolute gain p a tte rn s a lo n g th e p rin c ip a l
planes were com pared w ith m e a surem e n ts p e rfo rm e d a t th e a n e ch oic c h a m b e r o f
A S L '. T h e co m p arison illu s tra te s a v e ry good agreem ent a lth o u g h o n ly up to secondo rd e r d iffra c tio n term s w ere in c lu d e d in th e L T D fo rm u la tio n .
T h e h y b rid ante nn a code was e xte n d e d to analyze fe rrite -lo a d e d c a v ity -b a c k e d
slo ts m o u n te d on in fin ite o r fin ite g ro u n d planes. T h e loaded fe rrite is u s u a lly m a g ne ­
tiz e d in a given d ire c tio n using an e x te rn a l m a g n e tic fie ld . T h e e le c tric a l p ro p e rtie s
o f th e fe rrite m a te ria l a re s tro n g ly in flu en ce d by th e s tre n g th and d ire c tio n o f the
e x te rn a l fie ld .
By v a ry in g th e s tre n g th o f th e m a g n e tic fie ld , th e a n te n n a can be
tu n e d w ith in the designed b a n d w id th . T h is tu n in g effect was illu s tra te d in th is d is ­
s e rta tio n b o th n u m e ric a lly and e x p e rim e n ta lly . T h e p re d icte d d a ta co m p a re d very
w e ll w ith b o th a m o m e n t m e th o d s o lu tio n , fo r th e s c a tte rin g p ro b le m , and m easure­
m e n ts . fo r th e ra d ia tio n p ro b le m . A d d itio n a l p a ra m e tric stud ie s w ere p e rfo rm e d Inv a ry in g m a te ria l p a ra m e te rs as w e ll as th e in te n s ity o f th e bias fie ld . For s c a tte rin g
th e R C fj response o f a square c a v ity -b a c k e d a p e rtu re loaded w ith fe r rite /d ie le c t de­
layers was co m p ute d fo r b o th p o la riz a tio n s .
For ra d ia tio n , th e in p u t im pedance',
re tu r n loss, d ire c tiv ity , g a in and e ffic ie n c y were c o m p u te d and co m p a re d w ith m ea­
s u re m e n ts p e rfo rm ed in th e anechoic ch am be r o f A S F . A c c u ra c y in th e p re d ic tio n s ,
e s p e c ia lly fo r the resonant fre q u e n cy, is su bject to how a c c u ra te ly th e d e m a g n e tiz in g
fie ld in sid e th e fe rrite sam ples is e s tim a te d . To a c c u ra te ly p re d ic t th e s tre n g th o f
th e d e m a g n e tiz in g fie ld in s id e th e fe r rite sam ples, an a n a ly tic a l m a g n e to s ta tic m odel
fo r re c ta n g u la r geo m e trie s was in tro d u c e d .
O nce th e d e m a g n e tiz in g fie ld is d e te r­
m in e d . it is then s u b tra c te d fro m th e e x te rn a l fie ld to e s tim a te th e in te rn a l to the
fe r r ite fie ld . A n e rro r in e s tim a tin g th e d e m a g n e tiz in g fie ld can re su lt in a resonant
fre q u e n c y s h ift tow ard lo w e r o r h ig h e r frequencies.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
217
A square shape ca vity-b a cke d slo t fille d w ith la ye rs o f fe rrite and d ie le c tric was
designed, b u ilt and tested in th e anechoic c h a m b e r o f ASL*. T h e m a g n e tiz a tio n was
achieved th ro u g h th e use of p e rm a n e n t m agnets p la ce d a t tw o o f th e v e rtic a l sides
o f th e c a v it\r. T h e fie ld inside th e e m p ty c a v ity , w h ic h represents the e x te rn a l to th e
fe rrite fie ld , was m easured using a G aussm eter. T h e m a g n e to s ta tic m odel was used
to e s tim a te th e d e m a g n e tiz in g fie ld . T h is a n te n n a was s im u la te d using th e h y b rid
F E M / M o M code a nd th e p re d icte d in p u t im p e d a n ce , re tu rn loss and gain fo r va rio us
b ia sin g c o n d itio n s were com pared w ith m e a surem ents. T h e agreem ent is v e ry goo d .
T h e re tu rn loss ranges between —9 d B to —17 d B w ith in the desired b a n d w id th
whereas th e g a in was found to be on the average 3 d B i in c lu d in g m is m a tc h losses.
T u n in g c a p a b ilitie s close to 45 % can be achieved at a c e n te r frequency o f 900 M H z .
T h e fin ite e le m e n t m e tho d is classified as a d is c re tiz a tio n te ch n iq u e since th e
c o m p u ta tio n a l d o m a in is always d is c re tiz e d in to s m a ll elem ents.
L ik e a n y o th e r
d is c re tiz a tio n te c h n iq u e , the in fin ite d o m a in m ust be tru n c a te d to a fin ite d o m a in lo r
ra d ia tio n p ro b le m s : how ever, the tru n c a tio n b o u n d a ry m u st sim u la te o u tw a rd wave
p ro p a g a tio n w ith o u t a n y reflections. In o th e r w o rd s, th e p ro p a g a tin g wave m u st be
t o ta lly abso rb e d by th e presence o f th e b o u n d a ry. T h is can be achieved th ro u g h th e
use o f a b s o rb in g b o u n d a ry c o n d itio n s o r a r tific ia l a bsorbers.
In the last few years
an id e a l a b so rb e r was in tro d u ce d to s im u la te u n d is tu rb e d wave p ro p a g a tio n in th e
F D T D m esh.
T h e absorber is p e rfe c tly m a tc h e d , th u s no reflections o c c u r at th e
b o u n d a ry , a n d to t a lly decays th e fie ld as it p e n e tra te s th e m e d iu m . A lth o u g h th is
a bso rb e r had a g re a t success w ith th e F D T D m e th o d , it received m in im u m a tte n tio n
w ith th e F E M . In a d d itio n , m ost o f th e w o rk done on th e p e rfe c tly m a tc h e d la y e r
using th e F E M was som ew hat d is a p p o in tin g . N o t o n ly th e reflections were fo u n d to
be h ig h e r th a n those o b ta in e d by th e F D T D b u t also th e c o m p u ta tio n a l e ffic ie n c y
was m u ch worse. In th is d is s e rta tio n , a closer lo o k in to th e P M L as a p p lie d to th e
F E M was ta k e n . T h e concept is based on an a n is o tro p ic /lo s s y m e d iu m in tro d u c e d
by Sacks and J in -F a Lee. As was shown in th is d is s e rta tio n , the a n is o tro p ic P M L
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248
can be e x tre m e ly e ffe c tiv e in wave a b so rp tio n p ro v id e d th a t th e P M L is designed and
im p le m e n te d p ro p e rly . S p e c ific a lly , it was show n th a t th e P M L can be designed to
p ro d u ce a re fle ctio n c o e ffic ie n t low er th a n even —SO d P in c lu d in g d is c re tiz a tio n e rro r.
T h e re fle ctio n c o e ffic ie n t is s tro n g ly d ependent on va rio u s P M L design p aram ete rs,
such as d e p th , e le c tric a n d m a g n e tic c o n d u c tiv itie s , n u m b e r o f la ye r, s p a tia l p ro file ,
as w e ll as mesh d e n sity. N u m e ro u s p a ra m e tric s tu d ie s were c o n d u c te d and th o ro u g h ly
discussed in th is m a n u s c rip t. From these stud ie s, it is co n clu d e d th a t th e P M L has
a g re a t p o te n tia l in th e F E M : however, the re are several issues and obstacles th a t
need to be overcom e. F o r e xa m p le , use o f th e P M L o fte n d e stro ys th e c o n d itio n of
th e g lo b a l fin ite e le m e n t m a tr ix , thus th e c o m p u ta tio n a l e ffic ie n c y and a ccu ra cy o f
th e n u m e ric a l m e th o d g r e a tly d im in is h .
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CHAPTER 9
R E C O M M E X D A T IO XS
A lth o u g h m o s t o f the in it ia l o b je c tiv e s were successfully m e t d u rin g the course o f
th is w o rk , th e re are s t ill a d d itio n a l im p ro ve m e n ts to be m a d e . Som e useful recom ­
m e n d a tio n s on various issues d iscussed in th is m a n u s c rip t are g iv e n here.
C o n c e rn in g th e 2-D e ig e n v a lu e a n a lysis using th e F E M . th e fo rm u la tio n can be
e xte n d e d to a n a lyze a n is o tro p ic m a te ria ls defined by f u ll tensors instead o f b lo ck
d ia g o n a l tensors.
In a d d itio n , a b s o rb in g b o u n d a ry c o n d itio n s fo r evanescent waves
m u st be fo rm u la te d for a m o re e ffic ie n t analysis o f open m ic ro w a v e s tru c tu re s . T h a t
in v o lve s e s tim a tin g the a tte n u a tio n co n sta n t o f the evanescent waves in a d ire c tio n
p a ra lle l to th e tra nsve rse p la n e .
For th e a n a lysis o f 3 -D m ic ro w a v e c irc u its and e le c tro n ic packages, it is e x tre m e ly
im p o r ta n t th a t th e ite ra tiv e s o lv e r be im p ro ve d . C u rre n tly , a C o n ju g a te G ra d ie n t
S quare a lg o r ith m is used to s o lve th e lin e a r system o f e q u a tio n s .
Such a solver
g u a ra n te e s convergence: h o w e ve r th e convergence ra te is t e r r ib ly slow , especially tor
ill- c o n d itio n e d system s. A lth o u g h o th e r solvers have been te ste d by th e a u th o r, such
as th e B i-C o n ju g a te G ra d ie n t a n d th e Q u a s i-M in im u m R e s id u a l a lg o rith m s , none
o f th e m p e rfo rm e d in a s a tis fa c to ry m a n n e r. A d d itio n a l e ffo rt has to be placed to
im p ro v e th e c o m p u ta tio n a l speed o f th e solver. In te rm s o f a p p lic a tio n s , it w ould be
in te re s tin g a n d useful to c o n s id e r m o re p ra c tic a l and re a lis tic g eo m e trie s to analyze,
e s p e c ia lly e le c tro n ic packages lik e th e SO IC -S p la s tic package.
Im p le m e n ta tio n ot
h ig h e r-o rd e r edge e le m en ts is a lw a y s a n o th e r desirable im p ro v e m e n t for th e code.
As fa r as th e c a v ity -b a c k e d a p e rtu re s is concerned, s o lu tio n o f the m a tr ix sys­
te m is also slow .
T h e lin e a r s o lv e r has to be e ith e r im p ro v e d o r replaced w ith a
c o m p u ta tio n a lly faster a lg o r ith m .
A lth o u g h the a n a lysis a cco u n te d for a d ie le c tric
o r m a g n e tic o v e rla y on to p o f th e in fin ite ground p la ne , th e fo rm u la tio n was never
v e rifie d since no a va ila b le c o m p a ris o n d a ta was found in th e lite ra tu r e . The few jo u r ­
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n al papers a va ila b le on in p u t im p e d a n ce c a lc u la tio n s o f ca vity-b a cke d p a tch antennas
w it h d ie le c tric o verlays assum e an e x c ita tio n based on a d e lta gap betw een th e c a v ity
s id e w a ll and th e p a tch its e lf, [n o u r case, a co a x ia l cable was used in ste a d to e xcite
th e p a tch o r th e a p e rtu re . C o n c e rn in g the fe rrite -tu n e d c a v ity-b a cke d a p e rtu re , the
a n te n n a design m u st be im p ro v e d to p ro v id e a h ig h e r gain, low er re tu rn loss and
m o re a ttr a c tiv e ra d ia tio n c h a ra c te ris tic s in general.
Some o f these im p ro v e m e n ts
can be achieved th ro u g h th e use o f d ie le c tric layers placed e ith e r in sid e th e c a v ity
o r th e to p o f th e g ro u n d p lane.
A d d itio n a l o p tim iz a tio n can be achieved th ro u g h
g e o m e tric a l sh ap in g o f th e c a v ity a n d /o r th e a p e rtu re .
F in a lly , co n ce rn in g th e im p le m e n ta tio n o f th e P M L in the F E M . th e m a in d ra w ­
b ack o f th is te ch n iq u e is th e d e g ra d a tio n o f th e c o n d itio n n u m b e r o f th e g lo b a l m a trix
s y s te m . T h e reason stem s fro m th e use o f h ig h c o n d u c tiv itie s in sid e th e P M L region
th e re b y re s u ltin g in a fast d e ca yin g o f the p e n e tra tin g fieicl. A lth o u g h im p ro v e m e n t
o f th e m a tr ix c o n d itio n n u m b e r is h a rd to achieve, it is possible th a t m o re e ffe ctive
lin e a r solvers can be w r itte n to p ro v id e a s t ill a ccu ra te so lu tio n b u t w ith m in im u m
c o m p u ta tio n a l e ffo rt.
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26G
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX A
ELEMENTAL MATRICES OF THE EIGENVALUE PROBLEM
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
2(>S
T h e fo llo w in g co rre sp on d to a n a ly tic a l expressions o f th e e le m e n ta l m a tric e s fo r m u ­
la te d in S e c tio n 3.2 fo r th e e ig e n va lu e a n a lysis o f tw o -d im e n s io n a l m icro w a ve c irc u its
w ith a r b it r a r y cross se ctio n , a n is o tro p ic m a te ria ls a n u c o n d u c to r losses. T h e fin ite
e le m e n t a n a ly s is is based o n b lo c k -d ia g o n a l m a tric e s fo r th e p e r m it t iv it y and p e rm e ­
a b ilit y ten so rs a nd lin e a r n o d a l-e d g e tria n g u la r basis fu n c tio n s .
r a u
=
{ exx[&3 ~ ^ 6 3 + & »]
+
)[^.i c 3 — ( b >c3 + c 2 ^3 ) / - + ^2 C->]
~ C2C3 + C.]] }
( A . I)
+ c»«[c i - r i c'i + ci ] }
( A .2 )
fi
+ (fx.v +
£yr)[l>2C> ~
{ b ’ Ci +
)/2 +
^iCj]
d~cyy[c2 ~ C->C1 + Cl] }
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
( A . 3)
•26!)
P 7 ,],2
=
-4 - + 4.63)/2-M ,l
+ tx y [(^ 3 C l — 63C3 +
6 >C3 )/'2
+ £ >/x[( b\ Cz — 63C3 +
b 3 c > ) / 2 — b xc, }
- r t y , j [ { c 3 Ci -
r * ] 2l
[ T tt]
1:1
=
=
[ 7 7 f ] ia
where
C* +
I
^
<-yr
- /;>+
{f
- T ( r y [ { b 3 C1 ~
C > C ;s
b > C>
)/2
-
— b ) Ci \
C2 C , ] }
^
f try
(A .4 )
(A .h)
b>bx) / 2 -
b:xb x)
+
— A3C1]
+ f i/-r[ ( ^ - r 3 — ^ - C- “h b xc > ) / 2 — b xc3 ]
C'-iCi ~ C2
C- C 1 ) / -
"hero| Ix y
►t y r
" h f V !/[(
[r,']31 = [77,1,3
—
C 3 r
!
] }
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A .6
)
(A .7 )
270
C^7t]'»3
=
— b 2{ +
+
C ry [(^ lC >
l ) n /) 1 ) / 2
b xc x
—
+
63C i )/2 — b3c?j
+ C y j - [ ( ^ C [ — b [ C \ -+- b [ C
=
[T
u
:i)/'2
cf + C3C i )/2 -
-Kyy^Q O -
W A v.
— 6362]
— b n C ;i]
}
C 3C 2]
( A .S )
(A.!))
where •
]-23
1,1.
[ B \
J lj
[ B U ) 2j
=
Y 7 7 ^
-
Cj
[ b J
(/»:, -
=
=
(A. 10)
' = l ......... 3=2 = I.
=
b -i
-
b 2 ) ft £*'
[*,(*1 -
-p ^ Y
+
h i ) l l \jy
-
bj
( e.i -
M /C
CJ
(C l -
C2
)/< " “ '
c 2) //
7 = 1 ............ ::!
+ <•>’! - Cj)/'rT
1
(A. 12)
3
(A. 1:3)
- 61 )/<” ;; + c j( c 2 - ct )/
J =
[»:.],■> = [««%]„.
( A. 11)
" hH
[ „'«■
m"” -
L
A ry
/
t/j-
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(A. i n
271
= 1
:l :j = L
3
(A . 1-5)
i = I
P I - l, ,
=
ff(l+
3 : j = 1.......... :!
< « - { j
otherwise
* = 1......... 3 : j = 1.......... 3
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
( A . 16)
B IO G R A P H IC A L S K E T C H
A n a s ta s is C’ . P o lyca rp o u was b o rn in N ico sia . C y p ru s , in O c to b e r 1967. He c o m ­
p le te d h is e le m e n ta ry and h ig h sch oo l e d u c a tio n in C y p ru s , whereas in J u ly 1985 he
began a m ilit a r y service th a t lasted u n t il S e p te m b e r o f 19S7. In J a n u a ry o f 1990. he
cam e to th e L n ite d States to p ursu e a degree in E le c tr ic a l E ng in e ering . He received
th e B a c h e lo r o f Science (B .S .). surnm a cum laude. fro m A riz o n a S ta te U n iv e rs ity in
th e s u m m e r o f 1992. D u rin g his u n d e rg ra d u a te s tu d ie s he received sch olarsh ip s fro m
b o th th e g o v e rn m e n t o f C y p ru s and th e A riz o n a S ta te U n iv e rs ity . He la te r jo in e d
th e T e le c o m m u n ic a tio n s Research C e n te r ( T R C ) as a G ra d u a te S tu d e n t Research
A s s is ta n t. In A u g u s t 1994. he received th e M a s te r o f Science (M .S .) in E le c tric a l
E n g in e e rin g . H is research to p ic was h ig h -fre q u e n c y s c a tte rin g fro m trih e d r a l c o r­
ner re fle c to rs and c y lin d ric a l s tru c tu re s . He th e n p u rsu e d a D o c to r o f P h ilo s o p h y
(P h .D .) degree a t the same u n iv e rs ity . T h e m a in e m p h a sis o f his research was th e
d e v e lo p m e n t o f fin ite e le m en t m e th o d fo r th e s o lu tio n o f M a x w e ll’s e q u a tio n s. Re­
search was co n c e n tra te d on m ic ro w a v e c irc u its a n d e le c tro n ic packages as w e ll as slot
a nte n n a s in th e presence o f a c a v ity . D u rin g th e course o f his sta y in g at th e T R C .
he re ce ive d th e D e M u n d sch o la rsh ip fro m A riz o n a S ta te L 'n iv e rs ity . In a d d itio n , he
p u b lis h e d e ig h t papers in various a rc h iv a l jo u rn a ls and pre stigio u s m agazines, and
ten c o n fe re n ce papers, o f w h ich m o st o f th e m he p resented. He also w orked on a
v a rie ty o f p ro je c ts sponsored by p riv a te and g o v e rn m e n t agencies. He co -a u th o re d
n u m e ro u s re p o rts w ith his colleagues and a te c h n ic a l p ro p o sa l fo r th e A r m y Research
O ffice ( A R O ) . D u rin g his last sem ester a t A riz o n a S ta te U n iv e rs ity , he also had th e
o p p o r t u n it y to teach part o f a g ra d u a te course on A n te n n a A n a ly s is and D esign.
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