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Robust development of hybrid finite element methods for antennas and microwave circuits

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R ob u st D evelopm ent o f H ybrid F inite
E lem ent M eth ods For A ntennas and
M icrowave Circuits
by
Jian G ong
A dissertation su bm itted in partial fulfillment
o f th e requirem ents for th e degree o f
D octor o f Philosophy
(E lectrical Engineering)
in T he U niversity o f M ichigan
1996
D octoral C om m ittee:
Professor
Professor
Professor
Professor
Professor
John L. V olakis. Chairperson
Linda P.B . K atehi
T hom as B .A . Senior
K am al Sarabandi
G regory M. H ulbert
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
UMI Number: 9711974
UMI Microform 9711974
Copyright 1997, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized
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R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
/-v
^
Jian Gong
1996
All R ights Reserved
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
To Wit) p a ren ts
mt) memort) ant) mt) encouragement
To D ongti arift Cfrertcfyen
mt) inspiration ant) mt) fulfillment
11
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ACKNOW LEDGEM ENTS
M any have helped m e during m y academ ic life and along m y P h .D . journey and
th ey definitely deserve m y deep appreciations.
First of all, I am grateful to Prof. John L. Volakis who has been m y advisor since
1991 when he ‘grabbed’ m e from RIT in R ochester. N ew York. I would not have
stayed here w ithout his persuasive effort to the adm ission ad m in istration , where
tedious and ineffective processes were usually required.
His advisory, assistance,
encouragem ent and con tin u ou s support m ade the d issertation research possible. My
sincere gratitude also goes to m y co m m itte e m em bers, Profs. T h om as Senior. Linda
K atehi, K am al Sarabandi and G regory H ulbert. for their evaluations and tim e spent
to review the m anuscript to im prove th e quality of th e dissertation.
Having stayed in th e R ad iation Laboratory for alm ost five years. I am full of
pleasant m em ories o f various even ts and a ctiv ities, acad em ically and socially. Many
colleagues o f m ine during th is period, who m ay have already left or are still hanging
around, have helped m e here and there in variety of m eans. We all enjoy discussions
of diverse and m u tu ally in terestin g (cu ltural, political and techn ical) issues and w e’ve
learned from each other a great deal in m any aspects (at least from m y point o f view ).
T his atm osphere and environ m en t is w hat I always prefer and w hat I will m iss in
th e future. A m ong m an y o f th em in th e R adiation Laboratory, th e D epartm ent of
EEC’S, or even the U n iversity, are (I chose to only list th e m ost current JLV's group
in alphabetic order) H ristos A n astassiu . Lars A ndersen. Brown Arik. Yunus Erdem li.
iii
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
Sunil B indigavanale, Youssrv Botros. Mark C’asciato. Stephane L egault.1 Zhifang Li.
M ike Nurnberger and Tayfun O zdem ir.
I'd like to m en tion another fellow, Prof.
T .S .M . M aclean, who was w ith the
U n iversity o f B irm ingham . England and is now retired. During 19SS-L990, I had the
chance to work w ith him to in vestigate certain interesting radiowave propagation
and scatterin g problem s. W hat I was im pressed is his excellent scientific research
a ttitu d e and his personal m odel. His trust and instrum ental recom m endations on
m any occasions provided m e w ith th e im portant opportunities. W ithou t his effort.
I susp ect that I would not have achieved w hat I have today.
1 am deeply indebted to m y fam ily.
M y father died when I was little during
th e Great Culture Revolution (G C R ) in C hina.
(H e was physically tortured and
beaten to death w ithin five days by th e Red Guards.) T h e custod y burden o f th e
children was then en tirely loaded to m y m o th er’s shoulders. In spite o f the incredible
am ount of spiritual and physical pressures, she took good care of her children and
paid particular a tten tio n to their education. It was her effort and high e x p e c ta tio n of
us that encouraged m e to pursue excellen ce in m y school and college stu d y and later
in m y academ ic career. U nfortunately sh e passed away at her early fifties because
of th e long term em otion al depression. M y sisters and I experienced this deplorable
incident occurring in our family, and th is in cid en t, however, established an unusually
close relation am ong us. I am so proud o f all m y five elder sisters!
Last, but not th e least, my wife D ongli deserves m y sincere gratitude and appre­
ciation. Our lovely son Davin (C henchen) was two years old when th is dissertation
started being prepared. He often shows his clear preference betw een his m oth er and
m e because of the m uch less tim e I spent w ith him . M y in -law parents have helped
O ffic ia lly S teph ane does not belong to the JLV's group. He is however alw ays included as an
active m em ber o f the group.
iv
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us a great deal, esp ecially in the later stage o f this draft preparation. For years and
years their patience, understanding, support and love have m otivated and inspired
m e in m y research work. W ithou t th em , this dissertation would not be so successful.
v
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
TABLE OF CONTENTS
D E D I C A T I O N ...........................................................................................................................
A C K N O W L E D G E M E N T S ................................................................................................
ii
iii
L IS T O F
F I G U R E S ..............................................................................................................
ix
L IS T O F
T A B L E S ..................................................................................................................
xv
L IS T O F
A P P E N D I C E S ...................................................................................................... x v i
CHAPTER
I . I n t r o d u c t io n
............................................................................................................
1
O v e r v i e w .......................................................................................................
Fundam entals o f E lectrom agn etic T h e o r y .....................................
1.2.1 M axw ell E q u a t i o n s ................................................................
1.2.2 B oundary C onditions and Boundary V alue Problem s
1.2.3 U niqueness T heorem and Equivalence P rinciple . . .
1.2.4 Integral E quation and D yadic Green's F unction . .
1
5
6
7
10
13
F i n i t e E l e m e n t A n a ly s i s in E l e c t r o m a g n e t i c s .................................
20
1.1
1.2
II.
2.1
2.2
2.3
2.4
Functional F o r m u la t io n .........................................................................
2.1.1 P ertinent Functional For L ossy/A n isotrop ic M edia
— I ................................................. . ' ............................................
2.1.2 P ertinent Functional For L ossv/A n isotrop ic M edia
— I I ............................................. . ' ...........................................
G alerkin F o r m u la t i o n .............................................................................
T otal F ield and Scattered F ield F o r m u la tio n s .............................
2.3.1 S ca ttered /In cid en t Fields and Boundary C onditions
2.3.2 G alerkin's M e t h o d ....................................................................
2.3.3 V ariational M e t h o d ................................................................
Param eter E x t r a c t i o n .............................................................................
2.4.1 R adiation and RCS P attern ..............................................
2.4.2 G ain and A xial R a t i o ............................................................
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21
25
27
2S
30
31
34
36
38
3S
40
III. Edge—Based FE—BI T e c h n iq u e ........................................................
3.1
3 .2
43
I n t r o d u c t i o n ..................................................................................................
Hybrid S ystem F u n c tio n a l........................................................................
3.2.1
FEM S u b s y s t e m .......................................................................
3.2.2 Boundary Integral S u b sy stem .........................................
3.2.3 C om bined F E -B I S y s t e m .....................................................
N um erical I m p le m e n t a t io n ...................
Selected N um erical R e s u l t s ....................................................................
43
44
45
48
50
51
54
IV. Efficient Boundary IntegralSubsystem — I ...............................
61
3.3
3.4
4.1
4.2
4.3
4.4
I n t r o d u c t i o n ..................................................................................................
A pplication o f C onjugate G radient A l g o r i t h m s ...........................
4.2.1 BiCG A lgorithm W ith P r e c o n d it io n in g .......................
4.2.2 B iC G -F F T A lgorithm For Linear S y s t e m .....................
4.2.3 C onvolutional Form o f B oundary I n t e g r a l.....................
M esh O verlay S c h e m e ...............................................................................
4.3.1 F ield T r a n sfo r m a tio n s............................................................
R e s u l t s ............................................................................................................
V. Efficient Finite Element Subsystem — II
5.1
5.2
5.3
5.4
5.5
..................................
61
62
62
64
65
71
71
76
SO
Introduction .................................................................................................
Hybrid F E -B I F o r m u la t io n ....................................................................
E d ge-B ased P rism atic E lem ents .........................................................
A p p l i c a t i o n s .................................................................................................
C oncluding R e m a r k s ...................................................................................
80
81
85
S7
95
VI. A ntenna Feed M o d e lin g ...................................................................
98
6.1
6.2
6.3
6.4
Probe F e e d .....................................................................................................
98
6.1.1 Sim ple Probe Feed ................................................................ 98
6.1.2 V oltage Gap F e e d ...................................................................
99
A p erture-coupled M icrostrip M odel .................................................
99
Coax Cable F e e d ..............................................................................................102
6.3.1 M o tiv a t io n ...................................................................................... 102
6.3.2
Hybrid F E -B I S y s t e m ............................................................103
6.3.3
Proposed Coax Feed M o d e l ....................................................103
6.3.4 R esults and C o n c lu s io n ............................................................108
C o n c l u s i o n ........................................................................................................ 109
VII. Circuit M odeling
.................................................................................. 115
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7.1
7.2
7.3
7.4
I n t r o d u c t i o n ..................................................................................................... 115
N um erical D e - e m b e d d in g ............................................................................117
Truncation U sing D M T ............................................................................... 120
Truncation U sing PM L ............................................................................... 121
7.4.1
T h e o r y ............................................................................................122
7.4.2
R e s u l t s ............................................................................................ 124
VIII. AWE:
8.1
5.2
8.3
A sym ptotic Waveform E v a lu a tio n .............................133
B rief O verview o f A W E ............................................................................... 133
T h e o r y ................................................................................................................ 134
8.2.1
FEM System R e c a s t .................................................................. 134
8.2.2
A sym p totic W aveform E v a l u a t i o n ...................................... 136
N um erical Im plem entation ........................................................................ 138
EX. C o n c lu sio n s............................................................................................... 142
9.1
9.2
9.3
D iscussion on th e Research W o r k .............................................................. 142
Suggestions for Future T a s k s .................................................................... 145
M odular D e v e lo p m e n t...................................................................................145
A P P E N D I C E S ....................................................................................................... 147
B IB L IO G R A P H Y ................................................................................................. 159
viii
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LIST OF FIGURES
Figure
1.1
(a) R ecessed cavity in a PE C ground plane, (b) R ecessed cavity in
a d ielectrically coated PEC ground p lan e.......................................................
12
Illustrations of equivalence principle w hen applied to th e structure
show n in fig. 1.1a........................................................................................................
14
Illustrations o f equivalence principle w hen applied to th e structure
show n in fig. 1.1b.......................................................................................................
15
E xam ples o f protruding configurations (a) on a planar platform (b)
on a curved platform in consideration o f th e equivalence principle. .
16
2.1
Illustration of a typical conform al an ten n a configuration.........................
22
2.2
Illustration of a scattering problem setu p for scattered field form u­
la tio n ................................................................................................................................
30
3.1
Illustration of a typical radiation and scatterin g problem ........................
45
3.2
A tetrahedron and its local n o d e /e d g e num bering s c h e m e ....................
46
3.3
Pair o f triangles sharing the zth e d g e .............................................................
49
3.4
A typ ical g eo m etry /m esh for a c a v ity -b a ck ed circular patch antenna.
51
3.5
A flow chart describes the m ajor im p lem en tation procedures from
th e m esh generation, a few data preprocessors, the F E -B I kernel, to
th e BiC G solution and finally th e d ata o u tp u t............................................
53
C om parison of the com puted and m easured age backscatter RCS as
a function o f frequency for th e show n circular patch. T h e incidence
angle was 30° off the ground p lane....................................................................
56
1.2
1.3
1.4
3.6
ix
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3.7
Com parison o f the com p uted and m easured input im pedance for th e
circular patch shown in fig. 3.6. T h e feed was placed 0.8 cm from th e
center o f the patch and th e frequency was sw ept from 3 to 3.8 G H z.
56
Illustration o f the configuration and m esh o f the one-arm conical
spiral used for th e com p u tation o f fig. 3 .9 ......................................................
58
Com parison of th e calculated radiation pattern ( E#), taken in the
6 = 90°-p lan e. w ith d ata in reference for th e one-arm conical spiral
shown in fig. 3 . 8 ........................................................................................................
58
Com parison o f input im pedan ce ca lcu la tio n s for the illustrated cavitybacked slo t.....................................................................................................................
59
V isualization o f th e near field d istrib u tion at the lower layer o f a
stacked circular patch an ten n a.............................................................................
60
4.1
Structured m esh consists of equal right t r i a n g l e s .......................................
66
4.2
Illustration of two triangles w ith th e corresponding indices to help
to prove the convolutional property o f th e boundary integral...............
70
Printed circular patch antenna is m od eled using the recessed schem e
to incorporate th e B iC G -F F T algorith m , (a) Illustration o f the
configuration; (b) C om parisons o f th e B iC G -F F T result w ith that
o f th e ordinary F E -B I techn iqu e presen ted in chapter 3 ..........................
72
O verlay of a structured triangular apertu re m esh over an u n struc­
tured m esh, shown here to conform to a circular patch............................
73
Illustration of the param eters and g eo m etry used in con stru ctin g the
transform ation m atrix elem en ts betw een th e structured and unstru c­
tured m esh .....................................................................................................................
74
4.6
Illustration of the cavitv-b ack ed triangular patch array............................
77
4.7
Com parisons of th e m on ostatic radar cross section scattering by a 2 x
2 triangular patch array shown in fig. 4.6. T h e reults were com p u ted
using th e regular BiCG F E -B I tech n iq u e described in chapter 3 and
using th e B iC G -F F T proposed in th is ch ap ter............................................
7S
3.8
3.9
3.10
3.11
4.3
4.4
4.5
x
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4.S
Com parisons o f the m on ostatic radar cross sectio n scattering by an
em p ty aperture w ith th e sam e ca v ity size and dielectric filling (er= 1)
as th e structure shown in fig. 4.6. A gain, th e resu lts were calculated
using th e regular BiC G F E -B I techn iqu e described in chapter 3 and
using the B iC G -F F T proposed in th is chapter............................................
IS
B istatic RCS scatterin g by a crcular patch an tenna m odeled using
th e regular BiCG F E -B I and th e B iC G -F F T algorithm w ith over­
laying transform ..........................................................................................................
79
o .l
G eom etry of ca vity-b ack ed m icrostrip antennas
....................................
SI
5.2
Illustration o f tessellation using p r i s m s .........................................................
84
5.3
Right angled p r i s m ..................................................................................................
84
5.4
G eom etry of the annular slot an tenna backed by a cavity 23.7 cm in
diam eter and 3 cm deep .....................................................................................
88
Scattering: B ista tic (c o -p o l) RCS patterns com p u ted using th e tetra­
hedral F E -B I code and th e prism atic F E -B I code. T he norm ally
incident plane wave is polarized along th e © = 0 plane and th e ob­
servation cut is perpendicular to th at plane. Radiation: X -p o l and
C o-p ol radiation patterns in th e © = 0 plane from the annular slot
antenna shown in fig. 5.4. T h e solid lines are com puted using the
tetrahedral F E -B I cod e whereas th e d otted lines are com puted us­
ing the prism atic F E -B I code. T h e ex cita tio n probe is placed at the
point (y = 0 ) m arked in fig. 5 .4 ............................................................................
89
Illustration of th e setup for com p u tin g the FSS transm ission co­
efficient Upper figure: periodic elem en t (top view ): Lower figure:
periodic elem en t in cavity (cross-sectional v i e w ) .....................................
90
C alculations and com parisons o f transm ission through the FSS struc­
ture shown in fig. 5 . 6 .............................................................................................
92
4.9
•5.5
5.6
5.7
5.8
U pper figure: geom etry o f the m ultilayer frequency selective sur­
face (F SS) used for m odeling: lower figure: m easured and calculated
transm ission coefficient through th e FSS s t r u c t u r e ....................... 93
5.9
Illustration of a typical 2 -a rm slo t-sp ira l d e s i g n ...........................
5.10
R adiation P attern at f = I .lG H z (center frequency design). A good
axial ratio is achieved up to 60° degree.................................................
96
xi
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
96
5.11
5.12
R adiation Pattern at f= 0.944G H z (low er end o f frequency range).
It can be seen that th e axial ratio o f th e pattern becom es larger
com pared to that at th e center frequency, but still rem ains w ithin
3dB for a wide angle range. T h is indicates that th e num ber o f the
outer turns in the spiral contour design is m ost likely sufficient. . . .
97
R adiation Pattern at f= 1 .2 5 6 G H z (higher end o f frequency range).
It can be seen that th e axial ratio of th e pattern is deteriorated
com pared to those at the center frequency and lower frequency. This
certain ly shows that th e num ber o f inner loops still needs to be
increased to insure a good q u ality p attern .....................................................
97
6.1
C ross-section of an aperture coupled patch antenna, show ing the
c a v ity region I and th e m icrostrip line region II for two different
FEM com putation d om ain s.......................................................................................100
6.2
slot and its discretization (a) slot aperture: (b) typical m esh from
c avity region; (c) uniform m esh from m icrostrip line region...................... 100
6.3
Illustration of a cavity-backed patch antenna w ith a coax cable feed. 110
6.4
(a) Side view of a cavity-backed antenna w ith a coax cable feed; (b)
Illustration of the FEM m esh at th e cavity-cab le ju n ction (th e field
is set to zero at the center conductor su rface)..................................................110
6.5
Field distribution in a shorted coax cable as com p uted by th e finite
elem en t m ethod using th e expansion ( 6 .IS). — : analytical: xxx:
num erical, (a) Field coefficient eo along the length o f th e cable (left­
m ost point is the location o f th e short); (b) Field along the radial
coordinate calculated at a d istan ce A /4 from th e shorted te r m in a tio n .I ll
6.6
M easured and calculated input im pedan ce for a cavity-backed circu­
lar patch antenna having th e follow ing specifications: patch radius
r= 13m m ; cavity radius R = 2 1 .1 m m ; substrate thickness t= 4 .1 m m ;
er= 2 .4 ; and feed location x /= 0 .S cm distan ce from center. R esults
based on the sim ple probe m odel are also shown for com parison.
Our m odeling retains th e vertical wire connection to th e patch and
uses th e incom ing coaxial m ode field for excitation , (a) Real part:
(b) Im aginary part........................................................................................................ 112
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
6.7
M easured and calcu lated input im p ed an ce for a circular patch an­
ten n a having the follow ing specifications: patch radius r=2cm : sub­
stra te thickness d = 0.21844cm ; feed location from center j /= 0 .7 c m ;
er= 2 .3 3 ; fa n 6 = 0 .0 0 1 2 . — : m easurem ent; xxx: th is m ethod; o o o :
probe m odel, (a) R eal part; (b) Im aginary p art..............................................113
7.1
Illustration of a shielded m icrostrip lin e............................................................... 118
7.2
Illustration of th e cross section of a sh ield ed m icrostrip line...................... 121
7.3
A rectangular w aveguide (a) and a m icrostrip line (b) truncated
using th e perfectly m atched uniaxial absorbing layer............................122
7.4
P lane wave incidence on an interface betw een tw o diagonally anisotropic
half-spaces..................................................................................................................123
7.5
T yp ical field values o f T E to m ode inside a rectangular waveguide
term in ated by a perfectly m atched uniaxial layer................................... 128
7.6
F ield values o f th e T E io m ode inside a w aveguide term inated by a
p erfectly m atched uniaxial layer. T h e absorber is 10 elem en ts thick
and each elem ent was 0.5 cm which tran slates to about 13 sam ples
per w avelength at 4.5 G H z.................................................................................128
7.7
R eflection coefficient vs 2 f 3 t / \ g ( a = 1) for th e perfectly m atched
uniaxial layer used to term in ate the w aveguide shown in fig. 7.6. . . 129
7.8
R eflection coefficient vs 2 3 t / \ g, w ith a = ,3, for th e perfectly m atched
uniaxial layer used to term in ate the w aveguide shown in fig. 7.6. . . 129
7.9
R eflection coefficient vs 2 3 t / X g w ith a = l . for th e shielded m icrostrip
line term in ated by th e perfectly m atch ed uniaxial layer...................... 130
7.10
R eflection coefficient vs 2 3 t / X g w ith a = 3 , for th e shielded m i­
crostrip line term in ated by th e perfectly m atch ed uniaxial layer. . . 130
7.11
Input im pedance calculations for th e PM L term in ated m icrostrip as
com pared to the theoretical reference d a ta ................................................. 131
7.12
Illustration of a m eander line geom etry used for com parison with
m easurem en t.............................................................................................................132
7.13
C om parison of calculated and m easured results for th e m eander line
shown in fig.7.12...................................................................................................... 132
xiii
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
8.1
Illustration o f the shielded m icrostrip stu b ex cited w ith a current
probe....................................................................................................................................L40
8.2
Im pedance calculations using trad ition al FEM frequency analysis for
a shielded m icrostrip stu b show n in figure 8.1. Solid line is th e real
part and th e dashed line d enotes th e im aginary part of th e solutions.
T h ese com p utations are used as reference for com parisons........................ 140
8.3
4th order and 8th order AW E im p lem en tation s using one point ex ­
pansion at 1.78 G Hz are shown to com pare w ith th e reference data.
W ith the 4th order AW E solu tion s. 56% and 33% bandw idth agree­
m ent can be achieved for th e real (a) and im aginary (b) parts of
im pedan ce com p utations, resp ectively. It is also show n that th e 8th
order solu tions agree e x c elle n tly w ith th e reference data over th e
en tire band, (a) Real Part (b) Im aginary Part com p utations . . . .
141
9.1
M u lti-m od u lar FEM e n v ir o n m e n t ........................................................................146
A .l
(a) A tetrahedron, (b) its local n o d e /e d g e num bering schem e . . . .
XIV
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
148
LIST OF TABLES
Table
5. 1
Com parisons o f gain and axial ratio at different operatin g frequencies 95
6.1
T h e correspondence betw een th e edge numbers and the n od e pairs
for each coord in ate(r. or r) along w ith th e definition o f th e tilded
param eters in (6 .1 8 )......................................................................................................108
xv
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
LIST OF APPENDICES
Appendix
A.
E valuation o f M atrix E lem ents for T etrahedrals
...............................................14S
B.
E valuation o f the Boundary Integral S y stem M a t r i x ......................................... 152
C.
Form ulation for Right A ngle P r i s m s .........................................................................154
D.
S y stem D erivation From A F u n c t io n a l.....................................................................157
xvi
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
CHAPTER I
Introduction
O ne o f the primary goals in th is d issertation is concerned w ith th e d evelop m ent
o f robust hybrid finite elem en t-b o u n d a ry integral (F E -B I) techniques for m odeling
and design o f conformal antennas o f arbitrary shape. B oth the fin ite elem en t and
integral equation m ethods will be first overview ed in this chapter w ith an em p hasis
on recen tly developed hybrid F E -B I m ethodologies for antennas, m icrow ave and
m illim eter wave applications. T h e stru ctu re o f the dissertation is th en o u tlin ed . We
con clu d e th e chapter with discu ssions o f certain fundam ental con cep ts and m ethods
in electrom agn etics, which are im p ortant to this study.
1.1
Overview
T h e developm ent of sim ulation tech n iq u es for conform al antennas ty p ica lly m ounted
on veh icles is a challenging task . B y and large, existin g analysis and design m eth ­
ods are restricted to planar and m o stly rectangular patch antennas.
T h ese tech­
niques have difficulty in being ex te n d ed to non-rectangular/non-planar configurations
loaded w ith dielectrics and com prised o f intricate shapes to attain larger bandw idth
and gain perform ance [1-4]. M oreover, practical antenna designs m ay also require
a sop h isticated feeding structure, such as coaxial cable, m icrostrip line, stripline.
1
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
proxim ity or aperture coupled circuit netw ork, e tc . and integral equation m ethods
are not ea sily adaptable to m odeling th ese structures, esp ecially in th e presence
o f finitely sized dielectric loadings.
Partial differential equation (P D E ) techniques
(e.g. finite elem en t and finite difference m eth od s) m ay also experience difficulties in
m odeling unbounded field problem s, such as th ose found in antenna radiation and
scatterin g. T h e m otivation of this d issertation is therefore based on th e need to de­
velop gen eral-p u rp ose analysis techniques which can accurately sim u late conform al
antennas o f arbitrary shape with diverse feeding schem es. W ith the rapid grow th of
personal cellular G PS and other com m u n ication sy stem s, there is an increasing need
for such techniques since even traditional and protruding low frequency antennas
are being re-designed for conform ality and to m eet requirem ents for a host o f new
ap p lications [5.6]. B esides, m ost develop m en t of com p u tation al electrom agn etics in
th is su b ject can be applied to m edical diagnosis and treatm ent which have shown a
trem endous research and application p oten tial [7.8].
T h e co m p lex ity o f new antennas dem ands that analysis and design softw are be de­
veloped based on m ethodologies that are robust, versatile, and geom etrically ad ap t­
able. R ecen tly it has been dem onstrated that th e finite elem ent m eth od w hen cou­
pled w ith th e m ore traditional integral equation approach becom es q u ite attractive
for m odeling a w ide variety of existin g and em ergin g antenna configurations [9]. The
finite elem en t m eth od is indeed ideal for m od elin g the interior volum e o f th e an­
ten n a structure (m ulti-layer substrate, finite size dielectric loading, stacked elem ent
design, feed network and cavity volu m e, e tc .)
and is one of the m ost celebrated
analysis m ethods in engineering. On th e other hand, th e boundary integral offers
th e m ost accurate representation of th e fields exterior to th e antenna. Thus th e com ­
bination o f th e finite elem ent and th e boundary integral (F E -B I) m eth od s provides
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
3
for the handling of th e geom etrical com p lexity w ith ou t com prom ising accuracy. This
hybrid m eth od ology appears to be very a ttractive for conform al antenna m odeling.
H owever, its d evelop m ent and application to m ore practical and em erging antennas
presents us w ith m any theoretical and num erical challenges, w hich will be exten sively
investigated in the work.
Specifically, m esh term in ation plays an im portant role in FEM sim ulations and.
in m any cases, the accuracy is subject to th e perform ance o f the dom ain truncation
schem e.
For conform al antenna m odeling, a boundary integral (B I) equation has
been em ployed in this dissertation for term in ating th e anten n a’s radiating surface
and this m ethod is th eoretically free o f approxim ation. T h us, a desired accuracy can
be achieved w ithout fundam ental lim itations. A n ten n a configurations o f arbitrary
shape can be readily tessellated using m esh generation packages in the context of
th e F E -B I technique. In m odeling the interior region or the feed network, a superior
artificial absorbing m aterial — perfectly m atched layer (P M L ) — has been used to
ensure a m inim um im pact due to truncation walls. An in tensive stu d y of the PML's
perform ance has been carried out and the op tim al selection o f PML param eters has
been designed and em p loyed herewith in shielded structure m odeling.
Frequency dom ain m ethods provide th e necessary inform ation for engineering
design.
However, when wideband responses are needed, th ey can quickly becom e
exp en sive com pared to tim e dom ain techniques. A m eth od , referred to as th e asym p­
to tic waveform evaluation (A W E ), can be used to a llev ia te this issue. It has already
been successfully used in VLSI and circuit analysis. In the con text of the FEM . we
shall in vestigate the su ita b ility and validity of AW E for sim u latin g MMIC devices.
One of the im portant issues in antenna analysis is th e feed design. M odeling a
feed using the finite elem en t m ethod is indeed a challenging problem , and a sim ­
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
4
plified probe feed m odel fails to accu rately predict th e input im p ed an ce.
On the
oth er hand, the num erical sy stem can b ecom e ill-cond itioned w hen a feed network
is m od eled w ithout careful consid eration s. In this dissertation various feed m odels
will be in vestigated in consideration o f accuracy and efficiency. T h ey in clu d e current
and vo lta g e gap generators, strip lin e. m icrostrip line, coaxial cable, apertu re coupled
m icrostrip , etc.
In regards to the develop m ent and applications o f the sim u lation techn iqu es, the
test and design benchm ark m odels o f particular interest are m icrostrip (rectangular
and circular) patch antennas, d u al-stack ed patch antenna, ring slot an tenna, and
cone an ten n a, etc.
It is noted th at som e o f th em are not necessarily planar or
conform al.
R eferring to the dissertation stru ctu re, we begin with a description o f electro­
m agn etic fundam entals and then proceed to discuss the boundary con d ition s, equiv­
alence principle. Dyadic G reen's functions and the related theorem s.
T h e finite
elem en t m eth od as applied to tim e-h a rm o n ic electrom agn etic fields and waves is
su b seq u en tly described and th e basic FEM equations are derived from b oth varia­
tional and G alerkin techniques. T h e derivation is given in algebraic form allow ing
th e in clusion o f general anisotropy. T h e em phasis of th e discussion is on th e gen­
eralization of th e variational functional and G alerkin techniques w hen anisotropic
and lossy m aterials are present.
C hapter 3.4.5 and 6 discuss th e d evelop m en t of
e d g e -b a se d F E -B I techniques w ith significant efficiency im provem ent for antennas
and feed network m odeling. T h e em p h asis in th ese chapters is on d evelop in g novel
m eth od ologies to m inim ize th e required com p u tin g resources.
C h apter 7 is devoted to circuit m odeling w here specialized tru ncations suited
for gu id e w ave structures are presented.
T he perfectly m atched layer (P M L ). an
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
anisotropic artificial absorber used for m esh truncation, is investigated in term s of
perform ance and ap p lications.
W ideband system responses prom pt us to look at m ore efficient analysis tools
to replace th e current brute force frequency dom ain analysis approaches. C hapter S
discusses a prelim inary develop m en t o f th e FEM in connection w ith th e AW E.
In th e last chapter, we su m m arize and discuss th e anticipated future research
work to ex ten d the ca p ab ility and ap p lications o f the robust FEM d evelop m en t. A
list of suggested topics is included w ith specific recom m endations.
1.2
Fundam entals o f E lectrom agnetic T heory
Since m any fundam ental con cep ts and theorem s o f electrom agn etics w ill be em ­
ployed. w e will describe th e pertinent ones in this section for reference purpose.
This will also ensure con sisten cy in nom enclature and conventions throughout the
dissertation.
T he vector wave equ ation — th e on ly partial differential equation (P D E ) con­
sidered in this research — w ill be first derived from M axw ell equation s.
Various
boundary conditions w ill be stu d ied to estab lish the general m ath em atical m odels of
boundary value problem s (B V P ). T h e equivalence principle, uniqueness theorem and
the half-space dyadic G reen's functions are then briefly discussed for EM solutions
in radiation and scatterin g problem s.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
6
1 .2 .1
M a x w e ll E q u a t io n s
T im e-h arm on ic M axw ell eq u ation s o f differential form in a linear, an isotrop ic and
uniform m edium are given by [10]
V x E
-ju p
V x H
JLJC
H - M,
• E -T J;
(1.1)
( 1. 2 )
V-f-E
9e
(1.3)
V -^ -H
Pm
(1.4)
where E and H are th e electric and m agn etic field intensity, respectively, u; is th e
radian frequency and the factor eJ'wt is assum ed and suppressed throughout this
dissertation: M , and J , are th e im pressed m agnetic and electric current, resp ectively,
to serve as possible sources in th e m ed iu m under consideration: finally p e and p m
d en ote th e electric and m agnetic charge density. B oth M , and pm are fictitiou s and
non-physical quantities, which fa cilita te th e form ulation of physical problem s when
th e equivalence principle is em p loyed . T h e m aterial tensors c and p represent the
p erm ittiv ity and perm eability, resp ectively, and m ay be w ritten, in general, as
c=
p =
e0£r =
p op r =
c0
p0
^11
£12
^13
£21
^22
£23
£31
£32
£33
P ll
Pl2 P l3
P21
P22 P23
P3 1
P32 P33
w ith c0 and p 0 being the free sp ace p erm ittiv ity and perm eability.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(1.5)
( 1.6 )
T h e procedure to derive the vector wave equation begins by elim in atin g one of
th e two field q u an tities from (1.1) and (1.2). To do so. we first take a dot product
of (1.1) w ith th e ten sor Jlt 1 and then take th e curl on both sid es o f (1.1) to obtain
V x Jlt 1 • V x E = —ju!fi0V x H — V x /Ir 1 • M ,
(1.7)
S u b stitu tion o f (1.2) into (1.7) yields
V x ( j l r 1 • V x E j = uj2fioe0I r - E — jujpoJi — V x ^FjT1 • M , j
or
V x ( ji~ l • V x E^ - k l t r • E = - ju,y/0J i - V x ^ T 1 • M ,j
( l.S)
w here Ar0 = u-’^/^oCo is th e free space wave num ber. T h e dual o f ( l.S ) is given by
v x (f;1 •v x h) - ki%
h = - iwcoM,- + v x ( f ; 1 ■j,)
(1.9)
and can be sim ilarly derived startin g with (1.2). E quations ( l.S ) and (1.9) are the
vector wave equation s o f th e desired form.
1.2.2
Boundary Conditions and Boundary Value Problem s
T hree types of boundary conditions are typically encou n tered , and in th e context
of th e finite elem en t m eth od , th ese boundary conditions m u st be considered and
carefully treated. In w hat follows we shall discuss these con d ition s.
Dirichlet Boundary Condition
Consider two m ed ia separated by a surface T w hose unit norm al fi points from
m edium 1 to m edium 2. T h e fields on two sides of the interface satisfy th e relation
h x ( E 2 - E i) = - M s
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(1-10)
s
where
Ms is a fictitious m agnetic surface current and E i and E 2 are the electric field
inside m edium 1 and m ed ium 2. respectively. If m ed ium 1 is a perfectly m agnetic
conductor (P M C ), then E i vanishes and (1.10) becom es f i x E 2 = —M s. T h e surface
m agnetic current
M s can eith er be an im pressed source (ex cita tio n ) or m ay represent
a secondary (induced) current. If m ed iu m I is a perfectly electric conductor (P E C ).
n x
E2 also vanishes and thus Ma = 0 on the PE C surface.
Sim ilarly, for the m agnetic field.
h x (H 2 — H ,) =
Js
(1.11)
where J s denotes an electric surface current. T h e PE C surface can support electric
currents, given by h x H 2 = J*. sin ce H i is zero w ith in the conductor. B y duality,
th e PM C surface does not support electric currents, i.e. V x H 2 .
T h e relations (1.10) and (1.11) are inhom ogeneous D irichlet boundary conditions.
T h ey becom e hom ogeneous when
Ms = 0 and J s = 0. and in th ose cases th ey im ply
the tangential field continuity across th e dielectric interfaces. O ften, J s and
Ms are
introduced as fictitious currents w hen applying the equivalence principle (excep t in
special cases where they are specified a priori). T h e im p lication of this issue will be
discussed later in the develop m ent.
Neum ann and M ixed Boundary Condition
In form ulating a physical problem u sing hybrid finite elem en t m eth od s, we usually
work w ith eith er
E or H field. If, for in stan ce, we choose E as th e working quantity,
then (1.11) m ust be rewritten as
h x (^ ~ l • V x E )
where ( j i
1•V x E j
-
( F 1• V x E) J =
i = 1 .2 are evalu ated just inside
-ju ;J s
(1.12)
the /th m edium approaching
the boundary (from the ith m ed ium ). If m edium 1 is a PE C . th en V x E t = 0. and
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(1.12) reduces to a stan dard N eum ann boundary con d ition , by which a con straint
on the derivative o f E at th e interface is defined. T h e dual o f (1.12) is given by
h x [ ( I -1 - V x
- ( r l •V x H )
J
= -jw M s
(1.13)
and this condition is used when working w ith th e H field. In m any ap p lication s, th e
single field form ulation is often desired sin ce th e sy stem size m ay be kept m inim um
in this m anner. H owever, it is already seen th at the sin gle field form ulation im p lies
use o f the second order conditions referred to as natural conditions. F ortunately, it
is rather straightforward to im pose th is typ e o f conditions in regard to finite elem en t
sim ulations.
As for m ixed boundary conditions, an exam p le is th e resistive surface w here th e
electric and m agn etic fields satisfy th e condition
h x n x E + Rii x [H]
= 0
w ith R being th e effective resistivity of th e surface and [H]
(1.14)
= H + - H ~ th e field
difference above and below the surface. T his is a typical m ixed (third ty p e) hom o­
geneous boundary con d ition . A nother exam p le of a m ixed condition occurs in tran s­
m ission line problem (e.g . a coax cable, or other guided w ave structures), w here the
electric and m agn etic fields at a cross-section of th e line are given by
E
=
E ‘e - ^' + T E 'e7'
(1.15)
H
=
H 'e - 7 ' - TFTe*'"
(1.16)
and
n x E ‘ = —Z H ‘
(1.17)
where h = —z and ( E ‘. H 1) are th e incom ing fields before encountering a d iscon tin u ity
or load along the transm ission line. A lso. Z is th e wave im pedance associated w ith
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
10
th e transm ission line m ode o f the guide w ave structure. E lim inating T from (1.15)
and (1.16) in view o f (1.17) yield s th e relation
h x E - Z H = 2n x E 'e " 7-'
(1.18)
w hich is an exam p le of inhom ogeneous m ixed (third type) boundary condition. T h is
becom es apparent when H is expressed in term s o f th e derivative (curl) of E ,1 and
therefore the left hand sid e contains b oth differentiated and undifferentiated quan­
titie s. (In this case, the right hand sid e is usu ally considered as a known function.)
T h e m ixed boundary condition (1.18) is found very useful when applying the FEM to
guided wave structures for truncation and e x c ita tio n sim ultaneously. It is basically
a form o f absorbing boundary condition (A B C ).
1 .2 .3
U n iq u e n e s s T h e o r e m a n d E q u iv a le n c e P r in c ip le
T h e uniqueness theorem and the equivalence principle will be exp licitly or im p lic­
itly applied to this work w hen dealing w ith integral equations to term in ate the FEM
m esh and when evaluating th e far-held pattern. T ogether w ith dyadic Green's func­
tions, it becom es convenient to apply th ese con cep ts to construct integral equations
associated w ith various geom etries in radiation and scattering problem s. It is our
intent to discuss the theorem and th e principle (w ith ou t proof) for later applications.
U n iq u e n e s s T h e o r e m
P artial differential equations (P D E ) can be solved using various approaches and
th e corresponding results can also be represented in num erous forms given certain
boundary conditions.
ation , e tc .)
M oreover, m any (boundary, initial, natural, essential, radi­
conditions o f P D E m odels can b e extracted from the m athem atical
^ a r e m ust be taken when a curl operation is perform ed at a boundary discontinuity. It should
be appropriate to evaluate th e field derivative at a d istan ce from a discontinuity and then let th e
d istan ce tend to zero.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
11
sp ecification s o f well defined physical problem s. T h e q u estion then arises as to how
to relate th e solutions and how m any con d ition s are sufficient to achieve th e 'cor­
rect' solu tion . Uniqueness theorem s offer th e answer to th is question. Specifically in
electrom a gn etics, the E M solutions are uniquely d e te rm in e d by the sources in a given
region plus the tangential com ponents o f the electric field on boundaries, o r plus the
tangential com ponents o f the m agnetic field on boundaries. 2
E q u iv a l e n c e P r in c ip le
From th e uniqueness theorem , an EM problem can be uniquely solved if th e
tan gen tial (eith er the electric or m agn etic) field com p onent at th e boundary is pre­
scribed. In th is work, of interest is an EM problem where a dielectric inhom ogen eou s
region e x ists in th e presence o f a large P E C platform , probably coated w ith a d ielec­
tric slab. T h e typical geom etries are show n in fig. 1.1. w here we consider th e upper
half space to be the exterior region and th e ca v ity th e interior region.
In EM analysis, the fields in th e exterior region can be represented in integral
form con taining the equivalent current sources.
From (1.10) or (1.11) th e tan gen ­
tial electric or m agnetic field near th e apertu re (or th e discon tin u ity region) m ay be
eq u ivalen tly expressed in term s o f th e surface currents M , an d /or J,-. B y 'equiva­
len ce', we dem and the field d istribution rem ain th e sam e when the fictitiou s surface
currents are used to replace the interior region (cavity volu m e). It can be show n
through th e uniqueness theorem th at th is su b stitu tio n indeed ensures an identical
EM field distribution in th e exterior region.
W hen th e interior region is exclu d ed from consid eration , th e current sources in
(1.10) and (1.11) m ay be arbitrarily chosen leading to an infinite num ber o f choices
for th e equivalent currents. H owever, in our work th e field behavior in th e interior
2See the proof in reference [10].
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
L2
equivalent currents
cavity
ground plane
(a)
equivalent currents
\
cavity
coated ground plane
(b)
Figure 1.1: (a) R ecessed cavity in a PEC ground plane, (b) Recessed c a v ity in a
dielectrically coated P E C ground plane.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
L3
region is also needed, and m ost specifically, th e coupling of th e fields in th e inner and
outer cavity regions is desired. It is therefore convenient to select th e to ta l tan gential
electric or m agnetic field to sp ecify th e equivalent currents as
M, = E x h
and
J s = h x H.
(1-19)
T h is choice im plies the assu m p tion o f zero interior fields when th e exterior region
is considered, and zero exterior fields w hen th e interior region is needed. Fig. (1.2)
and (1.3) illustrate the details o f ap p lyin g th e principle, where th e fictitiou s currents
afFect th e region of interest (R O I) on ly w ith zero EM fields outsid e o f th e ROI. It is
observed that this choice of equivalent currents perm its a convenient in terior/exterior
sy stem couplin g for the utotal field form ulation" in hybrid FEM ap p lication s.
1 .2 .4
I n t e g r a l E q u a tio n a n d D y a d ic G r e e n ’s F u n c t io n
T h e D yadic Green's functions are particularly convenient for con stru ctin g integral
eq u ation s in th e presence of certain canonical platform s. For a planar structure, the
platform of particular interest is th e PE C infinite ground plane in w hich a cavity is
recessed w ith dielectric loading or absorption depending on applications.
T h e choice of the dyadic G reen's function varies depending on th e FEM form u­
lations.
For the electric field form ulation, we are seeking an appropriate integral
representation to find the m agn etic field in th e exterior region using th e inform ation
on or near th e region of the aperture. To this end. let us start w ith th e structure
con taining a possible protrusion as show n in fig. 1.4. where th e equivalen ce princi­
ple has been used on the outer contours o f th e structures to obtain th e equivalent
currents.
C onsider th e wave equation
V x V x G (r . r') - ^ 2//0eoG (r. r') = - l S ( r - r')
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(1.20)
14
equivalent currents
(E,H)=(0,0)
ground plane
Region of Interest (ROI): Exterior
(a)
equivalent currents
(E,H)=(0,0)
C roD
cavity
Region of Interest (ROI): Interior
(b)
F igure 1.2: Illustrations of equivalence principle when applied to the structure shown
in fig. 1.1a.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
15
equivalent currents
(5 D
(E,H)=(0,0)
ground plane
Region of Interest (ROI): Exterior
(a)
equivalent currents
(E,H)=(0,0)
cavity
C Ro O
Region of Interest (ROI): Interior
(b)
Figure 1.3: Illustrations of equivalence principle when applied to th e structure show n
in fig. 1.1b.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(b)
Figure 1.4: E xam ples of protruding configurations (a) on a planar platform (b) on a
curved platform in consideration o f th e equivalence principle.
w here G
is th e dyadic Green's function G in association with (1.9) (assum ing M , =
0). and I is th e idem factor defined as I = x x + y y + z z . A lso, note th e id en tity
• (V x
y
X
Q) -
(V
X
V
= - J J h • [P x V x Q + ( V
X
X
P ) • Q } dV
P ) x Q] dS
(1.21)
and upon se ttin g P = H and Q = G , we get
III{H
•(y x y x G) - (y x y x H) • g } dv
= - J f s h - [ K x V x G + ( y x H ) x G ] dS
From (1.9) and (1.20), th e left hand side (LH S) of (1-22) reduces to
LHS = —H (r') - J J J y x J - G ( r |r ') dV
and th e right hand side can be rearranged as
RHS = J J ^H • [h
X
y
X
G]
+ (y
X
H ) • [h
X
G] dS
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(1.22)
E q u atin g th e LHS and RHS y ield s
H(r)
=
- JJJ
V ' x J • G(r'|r) d V
• [n x V ' x G ] + (V ' x
w here
V
H) • [n x G ] } d S '
(1.23)
r and r' have been interchanged w ithout loss o f generality. As can b e realized.
is th e volum e containing th e d istrib u ted electric current source and S ' is the
surface enclosing the entire upper h a lf space.
To elim in a te the curl on J , we use th e dyadic identity.
V - ( J x G ) = ( V x J ) - G —J - ( V x G )
and th e divergence theorem to get
/ n t volume = - J J J v J • ( V ' x
G) d V ' - J J ^ h - { J x G) dS'
(1.24)
w here th e Som m erfeld radiation con d itio n was invoked to elim in ate th e integral at
infinity. T herefore. S' is only over th e ou ter surface of the body.
It rem ains to represent th e surface integrals in term s of th e electric field near the
c a v ity sin ce this field is typ ically th e com p u tab le quantity. T his is carried out by
insertin g (1.24) into (1.23). y ield in g
H(r) =
- JJJ^ J ’( V ' x G ) d V
- J J
=
-J J J ^
- J J
{H • (h x V ' x G ) + ( V' x H - J) • {h x G ) } d S '
J-(V'xG)dV'
{H • (n x V ' x G ) + j u c 0E ■{h x G ) } d S ’
(1.25)
w here th e M axwell equation (1.2) has been used. It should be rem arked th at the
above field representation is general, i.e. not restrictive to planar or conform al cases.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
For instance, in the presence o f a P E C platform . (1.25) is valid for protruding con­
figurations as shown in fig. 1.4. In th ese cases, th e surface integrations are carried
ou t over th e platform plus th e ou ter contours of th e structures.
T he field representation (1.25) shall be exam in ed and com pared for conform al
and protruding structures. To this end, we rewrite th e surface integral as
I n t surface
=
- J J
=V x
[f
H) dS' + V x ( V x G ) r - ( h x E ) j
j ( V x G ) T - (h x
| h
J J S’ I
- (« x G ) d 5 ' - —
(n
x E ) - ( V ' x G ) } dS'
J
dS'
(1.26)
w here T denotes a transpose operation of th e dyadic and th e integral in th e last
step is proportional to th e electric field.
If G (r |r ') is th e electric dyadic Green's
function of th e first kind defined as h x G = 0. th e first term in the integrand
o f (1.26) vanishes on the platform , provided S' is coin cident w ith th e platform . For
dielectric protrusion, this term reduces to th e integration on ly over th e outer contour
not conform al to the platform . A n altern ative is to define an electric dyadic Green's
function which satisfies th e con d ition h x (V ' x G ) r = 0.
A s can be seen, this
definition of th e Green's function equivalently leads to the sam e vanishing term in
(1.26). G is referred to as th e dyadic Green's function o f first kind. T he equivalence
o f both definitions can be proved from th e sym m etry properties o f th e dyadic Green's
functions [11].
For a planar PEC platform , G reduces to
G (r |r ') = G „ (r |r ') - G 0(r |r ') + 2 f r G 0(r |r ')
w here G q is the free space G reen's function given by
-ytoir-r'i
6'o(r|r') =
4 - r - r'
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(1.27)
Inserting (1.26) and (1.27) into (1.25), we obtain
H (r )
=
H 'BC(r) + H re/(r) + 2 jk Y 0 J ^ G 0(r|r') • (h x E ) d S '
(1.28)
T h is is th e desired form o f the m agn etic field representation used to establish th e
boundary integral equation for a planar platform .
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
C H A P T E R II
F inite Elem ent A nalysis in Electrom agnetics
T h e finite elem en t m eth od (F E M ) has been applied to electrom agnetics ( EM)
sin ce several decades ago [12].
E specially in th e late eighties and early n in eties,
it is observed th at the p ub lication volum e associated w ith the FEM in electrical
engineering grew in a fairly rapid pace [13]. T h is is prim arily because electro m a g ­
n etic problem s in engineering designs becom e increasingly com plex and a n a lytical
approaches or other num erical techniques no longer m eet practical needs. W ith its
num erous a ttractive features over other num erical techniques, th e FEM has been
ex ten siv ely investigated and exp loited for various EM applications [13].
T his chapter is organized as follows.
S ection 1 and 2 describe th e th eoretical
form ulations to construct th e FEM equation s.
T h ese are usually considered th e
indisp en sab le fundam entals o f th e technique, even though som e interesting issues
associated w ith these basics are still in develop m en t stage, especially in term s o f nu­
m erical im plem entation. O f interest in this co n tex t is the discussion of the variational
functional and G alerkin's techniques when ap plied to general anisotropic and lossy
electrom agn etic problem s. T h is topic is one o f th e least studied and d ocu m en ted
in the literature related to com p utational electrom agn etics. A nisotropic m aterials
have been used for dom ain truncations (refer to C hapter 7) and therefore th e general
20
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
21
vector or tensor form w ill be used w henever possib le in th e necessary derivations
for th e F E M . W ith th is typ e o f form ulations, isotropic sy ste m s m ay be regarded as
special cases.
T h e chapter is concluded w ith th e discussion o f th e physical quan tities for antenna
analysis in association w ith th e com p u tation o f electrom agn etic fields. The formulas
given in th is con text require m inim um am ount o f effort for com p utations.
2.1
Functional Form ulation
T h e FEM was first developed w ith th e aid o f functional analysis.
Tradition­
ally, m any standard boundary value problem s (B V P ) encountered in practice can be
eq u ivalen tly related to th e extrem ization o f a certain variational functional. W ith
th e R a y leig h -R itz procedure to project a continuous function space onto a discrete
finite exp ansion space, th e variational functional m eth od can be used to solve those
physical problem s and therefore becom es one o f th e two im portant approaches to
form ulate th e FEM . A functional version o f th e FEM for th e vector wave equations
( l.S ) or (1.9) is discu ssed in this section , which can readily incorporate boundary
con d ition s, sources, resistive cards and other constraints into th e form ulation. It is
regarded as a natural, convenient and som etim es ph ysically m eaningful approach.
Furtherm ore, the functional m ay represent a true physical q u an tity (e.g.
in low
frequency, power transm ission applications) and hence th is form ulation provides a
feature o f m erit for its evaluation. A lso, as can be seen in th is chapter, the varia­
tional m eth od in general non-self-adjoint cases m ay be rigorously treated to result
in a final sym m etric sy ste m , a subset o f w hich is id entical to that obtained from
G alerkin's technique. Last, but not the least, th e variational functional form ulation
can be used to validate th e expressions based on G alerkin's m ethod.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
C onsider a typical radiation or scatterin g problem shown in fig. 2.1. where th e
Z
4
Radiating element
G round plane
Aperture
Cavity
Feed
x
Figure 2.1: Illustration o f a typical conform al antenna configuration.
radiating elem en ts (or array) are enclosed in a region Q. T h e platform surrounding
th e rad ia tio n /sca tterin g geom etry can be a planar ground plane, certain canonical
shape (cylin d er/sp h ere), or even a doubly curved surface in which case the Green's
function is not available. In Q, electrom agn etic fields satisfy th e wave equation ( l . S)
or (1 .9 ). which can be concisely described using a linear operator C given by
C $=K i
(2.1)
where $ denotes th e field E or H , and
C
=
x iTr ' - V x ) — (fc^e,.-)
for electric field
(2.2)
C
=
x t r ‘ - V x ) - (ATq/v)
for m agn etic field
(2-3)
K , is th e source term associated w ith th e im pressed electric and m agnetic currents
and m ay be e x p lic itly given by
K,
= —juinoJi — V x ( j l r 1 • M , j
for electric field
(2.4)
K,-
= —
for m agnetic field
(2.5)
+ V x
)
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
As already m en tioned. £ is a linear operator and
sym m etric dielectric tensors Ji and e. th e
one can readily
show that for
pertinent functional o f th e original
PD E
has th e form
^ ($ ) = ^ < $ . £ $ > - <
$.K, >
(2.6)
where th e inner product < . > is defined as
< A .B > =
I A • B “ dV
Jn
(2.7)
(w ith B “ being th e com plex conjugate o f B ) for lossless m edia, or m ore generally as
< A .B > =
j A-BdV
JQ
(2.8)
for both lossless and lossy m edia.
T h e equivalent variational problem can now be stated as th e extrem ization o f the
functional (2.6) in conjunction w ith th e essential boundary conditions (e.g. Dirichlet B C rs).
Specifically, the boundary value problem is equivalent to th e following
variational m odel
& F ($) = 0
(2.9)
Essential Boundary C onditions
B ecause th e effect of com plex m aterials on th e resultant system is o f prim ary
interest to us. we restrict m ost of our discussions in this chapter to hom ogeneous
Dirichlet boundary conditions unless otherw ise specified. Therefore, the variational
approach o f (2.9) ensures a sym m etric num erical sy stem . T his is significant since
m any physical problem s retain a certain sym m etry property and th e corresponding
m ath em atical m odels should therefore reflect this property. M oreover, the sym m etry
of a num erical system is always desirable since it leads to m ore efficient solu tion and
less storage requirem ents.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
24
T he ex iste n c e o f th e functional (2.6) requires th e operator C be self-adjoin t, a
property usually defined to satisfy
< £$ ,'!> > = < $ . £ $ >
where $ and $ represent any two ad m issible fu n ction s.
(2.10)
If (2.10) holds, not only
does the num erical sy ste m derived from th e functional (2.6) rem ain sym m etric, the
m in im iza tio n /m a x im iza tio n becom es p h ysically m ean ingful.
O f m ost concern is th e situation where th e partial differential operator o f a system
is no longer se lf-a d jo in t. M ath em atically there ex ists no such a natural functional in
th e case sim ilar to (2 .6 ). A typical exam p le is th e p resen ce o f a lossy and anisotropic
m edium , w hose d ielectric m aterial tensors are not sy m m e tr ic or H erm itian. and this
type o f problem s is m ore often seen now adays. T h e develop m en t o f finite elem ent
m ethods for th o se problem s is still at an early sta g e becau se it involves num erous
challenges.
T raditionally, th ese physical problem s were fictitio u sly sim plified and dealt with
using available num erical approaches. K onrad [14] first tried to form ulate a 3-D FEM
with three vector com p on en ts to represent electro m a g n etic fields in anisotropic but
loss-free m edia. T h e tensors were therefore assum ed to b e H erm itian in his study.
A few years later in 1980's, th e num ber o f p u b lication s in this subject increased
typically w ith a p p lication s to waveguide structures. U nfortunately, the variational
approaches reported by different authors during th at period con sisten tly led to non­
standard and non-H erm itian eigenvalue sy stem s (even w ith the aid of an adjoint
system [15,16]).
Even worse, the num erical sy stem s derived in this m anner were
usually doubled in size. A s indicated in [17]. when a non-standard eigenvalue system
was m anipulated to reduce to the standard form , th e size o f the system was doubled
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
again. Sim ilar reports were also seen in later papers [18-23] for th e problem s o f wave
propagation inside anisotropic m edia. No radiation and scattering analysis has been
reported in this c o n tex t.
In what follow s, we generalize th e FEM form ulation to lossy and anisotropic
electrom agnetic problem s. Specifically, we show th at tw o different m ethods, one with
th e aid of an adjoint au xiliary system and the o th e r w ith th e Lagrange m ultiplier, can
be used to con stru ct th e pertinent functional. G alerk in ’s m eth od is then com pared
to th ese two variational techniques.
2.1.1
Pertinent Functional For L ossy/A nisotropic Media — I
As is known, th e natural variational functional no longer exists for non-H erm itian
operators since no m a tter what definition is g iven for inner products (see (2.7) or
(2 .8 )), one cannot ob tain a self-ad joint operator necessary for natural functional
design. In th ese cases, we consider a generalized functional
T = < C $ , ' $ > - < $,Ka > - <
'F .K >
(2-11)
where $ is th e unknow n solution function of th e original PD E problem and K is
th e rig h t-h a n d -sid e function as in (2.1). Sim ilarly, 'F is th e solution function of the
adjoint PD E such that
£ at f = K a
(2.12)
where Ca can be derived from
<£$,'?>=<
>
(2.13)
w ith C ^ Ca.
It should be rem arked that the functional (2. 11) reduces to (2.6) (excep t for a
possible constant coefficient) if C is self-ad join t.
A lso, th e original P D E and its
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
26
adjoint counterpart (2.12) can be recovered through the variational process w ith
resp ect to th e functions $ and
resp ectiv ely (th e sim ple derivation is o m itted
here). A fter discretization is carried o u t. th e final num erical system is sym m etric.
T h is can be shown as follows. Let
$ =
* = 5 > jV j
(2.14)
w here V , is th e basis function used for b oth unknown functions and x,-. t/, are th e
corresponding expansion coefficients. Inserting (2.14) into (2.11) yields
T = Y , Z
‘ j
x 'Hi < £ V - V > > - Y , Xi < V ” K a > - ^
<
j
</, < V , , K >
(2.15)
L<pon perform ing the differentiation w ith respect to x,- and y j individually, we get
th e two decoupled system s o f linear equation s
^ 0 Qx\ fx\
Qy
0
(kA
(2.16)
W
w here th e m atrices Q x. Q y and th e colu m n vectors K x. K y are given by
<CVj.Vt >
Q*
y
Qr-
_
< C V i.V j >
Kf
< V .-.K >
K yf
< V,.Ka >
In general. Q x # <2X. Q ytJ ^ Q y, and Q x ^ Q yy
However, Q x = Q y, = { Q yJ)T.
T h ese relations indicate a loss o f sym m etry o f th e original problem , but th e sy m m etry
holds for the overall system !
T h e storage requirement is a function of .Y /2. where N is the dim ension of (2.16).
E ven though there is an auxiliary system needed to com p lete the analysis, in practice
th is system does not require storage.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
2.1.2
Pertinent Functional For L ossy/A nisotropic Media — II
An altern ative to using an adjoint sy stem is to em p loy the Lagrange m ultiplier
technique in con stru ctin g th e pertinent functional. T h e Lagrange m ultiplier is usually
used to incorporate additional constraints to a sy ste m . To illustrate th e technique,
consider th e sa m e P D E m odel as described in (2 .1 ). and we first rewrite it as
£ $ - K :=0
(2.17)
N ext, we assum e an expansion space w here the so lu tio n is defined and solved. T h e
unknown function $ and th e m ultiplier function A are expanded in th e sam e space.
If (2.17) is regarded as a ^constraint", we try to add th e constraint to a "null" system
and get
:F(<f>.A) = < A , £ $ - K , >
( 2 .IS)
T his functional is now used to form ulate th e F E M . A s described above, on applying
the R a y leig h -R itz procedure to both $ and A u sin g th e sam e set of basis functions,
viz.
$ = 5 1 x-v" A= E
yjVj
(2-19)
j
•
we obtain
n*. A) = E E xw < v ;-CVi ~ K‘ >
■
Carrying out differentiation with respect to
/„
o
(2-20)
j
/
Q
and yj . individually, yields
Kr
( 2 .2 1 )
0
)
\yj
\ o
where Qfj = Q j { as in (2.16). We observe that (2 .2 1 ) is sim ilar to (2.16) with two
decoupled su b system s of the sam e size. T he properties of the subsystem s are also
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
28
sim ilar to those in (2.16). W e further observe th at th e Lagrange m ultiplier technique
m ay be regarded as a special case to (2.11) w here a hom ogeneous adjoint P D E is
now virtually assum ed (i.e. K a = 0). A gain, it should be remarked that when a
self-ad join t problem is considered, th e above form ulations (b oth the adjoint system
approach and th e Lagrange m ultiplier techn iqu e) will result in a sym m etric system
and th e auxiliary su b system becom es eith er redundant (for adjoint approach) or un­
necessary (for th e m ultiplier tech n iq u e).
In th e later case, th e m ultiplier can be
considered identical to th e unknown function $ exp an d ed using th e sam e basis func­
tions. T h is results in an interestin g coin cidence w ith G alerkin technique described
next.
2.2
Galerkin Form ulation
G alerkin's m eth od is now considered to form ulate th e finite elem en t m ethod. Tra­
ditionally. G alerkin's technique used in con junction w ith integral equation em ploys
the sam e testin g and expansion functions to ob tain a sym m etric dense num erical
system . However, in th e case o f the FE M . G alerkin's m eth od does not always lead
to a sym m etric system . Apart from boundary con d ition s, th e linear operator of a
PD E problem determ ines th e sy m m etry feature o f th e resu ltin g system .
A lso, unlike th e variational approach. G alerkin's m eth od solves th e weighted PD E
by a testin g process as
< '& .£ $ > = < 'F .K i >
(2.22)
where $ and 'F are both defined in th e sam e function space. Specifically, in Galerkin's
m ethod one seeks the solu tion for th e unknown function $ which satisfies certain
prescribed constraints and (2 .2 2 ). with th e aid o f another arbitrarily chosen function
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
29
Sim ilarly to the variational approach, th e R a y leig h -R itz procedure m ay also be
used to project th e continuous space onto a finite discrete separable H ilbert space.
T h e m ath em atical problem is then rephrased to seek a discrete solu tion set whose
entries are the coefficients of th e expansion. T h e testin g function $ m ust, o f course,
be defined in th e sam e discrete space to ensure that th e original P D E is solved with
proper boundary conditions.
It is obvious that if th e linear operator C is self-ad join t, the choice o f th e testin g
function $ = $ results in a sy m m etric num erical system . O therw ise, no m a tte r how
the inner product is defined, th e final discrete system in general does not exh ib it
sym m etry property.
We observe that G alerkin's m eth od can usually be applied to any linear operators
even when
th e corresponding
natural functional
does not exist. A lso in th e general
cases (as considered when describing th e functional approach). G alerkin's m eth od
leads to th e sam e num erical sy ste m as th e desired portion in (2.16) and (2 .2 1 ). T his
can be dem onstrated as follows. Inserting (2.14) into (2.22). we readily ob tain
£ £ nsj < Vj. rv, >= £ > < Vj, k, >
«
J
j
or
Y ,y j
j
< V ,,£ V , > - < V , . K , > | = 0
(2.23)
As assum ed, ^ is an arbitrary function. T hus th e term in th e curly bracket should
vanish, yielding
Q Tx =
Kx
(2.24)
which is exactly th e sam e as th e su b system derived from the variational approach.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
30
It is noted that if one chose th e Lagrange m ultiplier (in the variational approach)
as th e testin g function
th e sam e num erical sy stem would result.
A nalytically,
though, th e entire sy stem is tw ice th e size of that derived via G alerkin's m ethod.
T h is is due to the fundam ental difference o f th e two techniques.
R egardless, as
e x p e c te d , the obtain ed num erical system s o f interest are virtually identical!
2.3
Total Field and Scattered Field Form ulations
In th is section, we focus on a general scattering problem as illustrated in fig. 2.2,
w here a perfectly conductin g electric (P E C ) body is coated w ith a d ielectric layer
w hose relative perm eability and p erm ittiv ity are jld and I d, respectively. (N o te that
Q. j:
Qa:
Tp:
Vd :
Tj :
r0:
dielectric coated region ( c d , f i d )
free space ( I f = /J/ = 1)
absorbing layer (I a ,JLa )
boundary of the PEC body
boundary of the dielectric coating and free space region
boundary of the absorber and free space region
PEC boundary of the outer absorber
Figure 2.2: Illustration of a scatterin g problem setup for scattered field form ulation.
for the purpose of generality, th e m edium is assum ed anisotropic.)
T h e situ ation w ith absorbing boundary conditions for truncating the FEM do­
m ain has been analyzed before (see e.g. [24] or [25]). However, two issues associated
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
w ith this typ e of problem s have not been carefully addressed in th e literature. O ne
o f th em is the equivalence betw een th e variational and the G alerkin’s m eth od when
th e scattered field is used as th e working variable. Proof of th is equivalen ce can be
ted ious and cum bersom e but it is nevertheless an im portant issue. U nfortunately,
one is used to assum ing th at th ese two form ulations are equivalent w ith ou t proof.
A nother issue relates to th e recen tly introduced perfectly m atched absorbing m ate­
rial. A s it turns out, there are several advantages to use artificial absorbing m aterials,
including accuracy control, conform ality, ease o f boundary treatm en t, e tc . However,
their inclusion introduces ad d ition al artificial conditions
inside th e absorber layer
and care m ust be taken w hen th ose conditions are enforced in th e FE M form ulation.
M oreover, although a m eta l-b a ck ed absorber layer sim plifies the FE M im plem enta­
tion , th e m ultilayered FEM region contains high inhom ogeneity, w hich again requires
a careful presentation of th e form ulation. To this en d , we exten d our th eoretical dis­
cussions on th e FEM to sca ttered field representations, where the treatm en t of the
boundary and transition con d ition s w ill also be described.
2 .3 .1
S c a t t e r e d / I n c i d e n t F i e ld s a n d B o u n d a r y C o n d it io n s
Referring to fig. 2.2. w e begin w ith the wave equation in term s of H (th e E
form ulation m ay be readily hand led by du ality). To proceed w ith G alerkin’s m ethod,
we first w rite H f0‘ as
H to t =
H scat + H in c
(2 /2 5 )
where H iC“ and H 'nc are th e scattered and incident field, respectively. N e x t, w eight­
ing th e source free wave eq u ation w ith the testin g function V yields
J V •{V
X
t;1•V
X
H - k l f r • h } d9. = 0
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(2.26)
32
w here fl = fid + f l / + fl* encom passes th e entire com p u tation al region. T o proceed
with th e derivation o f the weak form wave equation , it is necessary to introduce
certain constraints on th e scattered and incident fields w ithin fl and on th e boundary.
First, sin ce th e incident field is not allow ed to pass through the absorber layer as
well as th e m etal back wall T0, we note that
jja c a i
H
jjtn c
r G fld + f l /
(r) =
(2.27)
jjscai
r G otherw ise
with th e incident field satisfying
= 0
r G f la + f l /
~ ^ o / v• j H '
(2.28)
7
^0
r G otherw ise
It is thus evid en t that the scattered field satisfies th e hom ogeneous w ave equation in
regions f2a -f- f l / and th e inhom ogeneous wave equation in fidT h e boundary conditions on H 3Ca* can be readily derived by c o n sisten tly applying
the field decom p osition (2.25). N ote that in accordance with (2 .2 7 ). the electric field
is likew ise decom p osed as
(2.29)
H owever, one should be cautioned that E ,nc and H mc do not satisfy M axw ell equa­
tions in th e dielectric region. T hat is,
M o E inc ± V x H '
r G fid
(2.30)
which conflicts w ith what one would in tu itiv ely assum e. C onventionally, th e incident
field is assum ed to exist in the d ielectrically coated region fid as if th ere was no
d ielectric there. A fter a quick glance, one would im m ed iately arrive at a conclusion
that (2.30) is against M axwell theory. In reality, it can be proved th a t if E ,nc and
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
33
H'"1' indeed satisfied M axw ell equations in f ij . a contradictory boundary condition
would im m ed iately resu lt. T h is can be seen by taking a curl operation on both sides
of (2.25).
In view o f (2.29) and M axwell equations in dielectric m edia, we would have
? - V x
=
rfl ■V
X
H ,nc +
rdl - V
X
H 3cat
r e 9 .d
Im posing the con d ition o f to ta l field tangential con tin u ity at the boundary
(2.31)
would
yield
h x | f j l • V x H scat|r+ - f j l • V x H 3Cai|r- } = 0
(2.32)
which is a hom ogeneous N eum ann boundary condition. H owever, if we start w ith
(2.27), upon tak in g th e curl operation and im posing th e condition o f total field
tangential con tin u ity at th e boundary Tj, we get
h x | f ^ 1 • V x H sca‘|r+ - I } 1 • V x H 5cat|r - | = - n x { f j l - f j 1} - V x H ,rlc (2.33)
which is an inhom ogen eou s N eum ann boundary condition.
This in con sisten cy is because (2.32) was derived on th e basis o f th e d ecom p osi­
tion (2.29) and th e assu m p tion that the incident field w ith in dielectric regions also
satisfies M axwell eq u ation s.
H owever, the decom p osition (2.29) is artificial and it
is therefore necessary to keep in m ind that only (2.27) holds true when deriving
boundary con d itions.
As a rule of th u m b , an appropriate interpretation o f the phenom enon should read:
the incident field inside dielectric media existed in the sa m e fashion as in free space
as if the free space was replaced with the media.
M ath em atically, this im plies the
condition
M o E ,nc = Td l ■V x H*nc
r € 9.d
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(2.34)
34
C on sisten tly one can derive th e oth er boundary conditions required for th e FEM
form ulation following th e sam e procedure. T h ey are classified as D irichlet and N eu­
m ann conditions as follows.
• D irichlet C onditions
Boundaries
TP
C onditions
h x H 3C“* = K “ — h x H mc ( K , unknown current)
h x { H 5C“J|+ — H sca£|_ } = 0
r/
h x { H sca‘|+ - H icaf|_ } = - f i x H ‘nc
To
h x H scat = K* ( K a unknown current)
• N eum ann C onditions
Boundaries
C onditions
rP
h x l ^ 1 • V x H sca‘ = -- h x Cj l • V x H ,nc
+
+
= -i
h X { t ; 1 • V X H 5C“*|
= —n x er
•V x H in<:
T/
h x [ e r~ l • V x H seaf |
To
r i x ^ - V x H 3C“ = 0
+
= h x tj
7 x H 'nc
where { } |* denotes { } | — { } |_ and er and H ac“* should take the values at
positive and n egative sides o f a specific boundary.
2.3.2
Galerkin’s M ethod
R eturning to (2.26) and in view o f th e vector identity
A • V x B = ( V x A) • B — V • (A x B)
(2.35)
and th e divergence theorem , we ob tain th e corresponding weak form w ave equation
Int^ + In t/ + Inta = 0
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(2.36)
35
where
f
=
Intrf
[ V x V - ^ - V x
Jud '■
+ J
+
•V x h ) dS + j
V • (no x
J
( H iCa£ + H ,nc) -
tfV
■% ■ ( H sc“£ + H ,nc)
V - ^ x ^ ' - V x
H*'nc) d S
H sco£) d S
V - (» ,X £ ;l - V x
d9.
(2.37)
fr d
Int /
a
=
J
+
f V ■ ( h 2 x t ~ l - V x H sca£) d S
JVf
=
f
J
+
| v - ( «
V x v - £ ; l - v x H sca'( f n + |
V
X
V •f;
3
1
•V
X
H sca£
dn +
v -^ -h tx f}1
I V - (-h 2
* Ff
x n scat)
ds
(2.5
X
t~l • V
X
H sea£)
X ^ - V x H sca£) d S
dS
(2.39)
w ith no, hi, h2 and h2 being th e unit norm als at th e boundaries r p. T j. F/ and Torespectively. T h ey all point aw ay from th e center of the PEC b ody (i.e. outw ards).
Invoking the boundary conditions as tab u lated above, we observe that th e surface
integrals on Tp in (2.37) and on To in (2.39) vanish. Also, the sum o f th e surface
integrals on T / in (2.3S) and (2.39) reduces to
- [
J r,
V • h2 x
■V x H.incd S
(2.40)
Sim ilarly, the sum o f th e surface integrals (in volving H sca£) on Tj in (2.37) and (2.3S)
becom es
£
V-hi
X
( r f l - ! J l ) • V x K ,ncd S
(2.41)
N ote that both integrals (2.40) and (2.41) will not contribute to th e sy ste m , but to
th e excitation on th e right hand side. O f course, the excitation term (2.41) would
disappear if the perm itivity is continuous across the boundary Tj. B y d u ality, for
an electric field form ulation, a term sim ilar to (2.41) will appear for d iscon tinuous
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
36
perm eability. T h is observation does not hold for (2 .4 0 ). w hich arises from th e inher­
ent incident field definition on the two sides o f th e boundary T /. as m ath em atically
show n in (2.27). Furtherm ore, the second term in (2.41) will be cancelled ou t with
th e integral on Tj involving H ,nc in (2.37). A fter a sim p le m anipulation, th e final
sv stem reduces to
i(
V x V - ^ - V x H scat - k * V - JIr • H Jca‘)
=
-
f ( v X V •f ;
Jsid
1
•V
X
H 'nc - A-qV •
- [ V • hi x f j l • V X H ,nc dS +
Jrd
f
Jrf
V
dn
• H ”1^
dQ.
• h2 X Tjl ■V
X H ‘nc dS
(2.42)
where er is again the relative p erm ittivity w ith respect to th e specific region. This
is th e desired weak form o f th e wave equation , and a sim ilar equation was obtained
in
[24]
2.3.3
using th e functional form ulation for isotropic m ed ia.
Variational M ethod
It is now o f interest to em ploy the functional form ulation to obtain the equation
corresponding to
(2.4 2).
To this end. it is in tu itiv e to begin with the total field
representation and use th e functional
.F (H )
= \
f
(V x H • I]?1 • V
- jQd+nf +na v
x
H
- AqH • ^7r • h ) dQ.
(2.43)
7
where, as before. er and Jlt denote the corresponding relative p erm ittivity and perm ebility. respectively. A lso sim ilar to G alerkin's m eth o d , one would now logically
proceed with the field decom postion H = H 5cai + H ‘nc and express the functional
in term s of th e scattered field. It is unfortunate that th is approach will result in a
different form of linear system than
(2.42)
and th e d etailed m athem atical proof is
presented in A p pendix D.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
T h e discrepancy arises from th e assum ption o f th e functional (2 .4 3 ), w hich is not
a valid expression w hen th e corresponding P D E operator is no longer self-ad joint. Re­
call in section
2
. 1 . 1 that th e presence o f general anisotropic and lossy m ed ia requires
th e application o f an auxilarv adjoint sy stem , together w ith w hich th e pertinent
functional is given by a generalized form in (2.11). It is therefore necessary to begin
w ith this generalized functional rather than (2.43). In a source-free region. (2.11)
exp licitly takes th e form
^■(H„, H ) = J
Ha •|v
x qTl - V x H — k & r - H } d V
(2.44)
and the variation is im posed on the adjoint variable H a to get
= 0
£ F ( H a.H )
(2.45)
*H =o
T h e variational functional m eth od of this version proves valid and identical to Galerkin's
technique for any linear operators in electrom agnetic problem s. It is also interesting
to note that on ce com pared to G alerkin's m ethod, th e adjoint field qu an tity H a in
(2.44) seem s to take place o f th e testin g function V in (2 .1 1 ). H owever. H a theo­
retically differs from V in that the former is defined as a solu tion function for the
adjoint system o f th e original P D E problem , whereas th e latter is ju st an arbitrary
adm issible testin g function that does not have to be a solu tion to th e adjoint system .
Apart from th e concept difference as m entioned above, th e m a th em atical proce­
dure required to derive th e FEM system is sim ilar to that presented in th e previous
subsection, and w ill not be discussed here to avoid rep etition. O ne can be assured
that the final sy ste m obtained from (2.44) and (2.45) is o f course identical to (2.42).
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
38
2.4
Param eter E xtraction
A n accurate full w ave an alysis can on ly predict th e near field (for P D E type
m ethods) or current (for integral based techniques) distributions, w hich can be used
to obtain certain practical param eters d ep ending on applications. For in tricate sys­
tem s. involved num erical m od els m ay b e needed for output d a ta ex tra ctio n , including
far field evaluation in th e presence o f non-canonical platform s and a d e-em b ed d in g
process for antenna feed netw ork or circuit sim ulations.
A ntenna param eters can be readily evaluated after th e near field distribution
is achieved via full w ave an alysis.
T h e d e-em b ed d in g process is required for feed
network or circuit m od elin g and will be discussed in chapter 7. For a non-planar
platform the far field evalu ation can be obtain ed from th e general discussion in
chapter
1
. where the form ulation in term s o f th e dyadic G reen s function m ust be
used to consider the eq uivalent currents and th e free space G reen's function.
2.4.1
Radiation and RCS Pattern
In the case o f antennas, we are m o stly interested in their radiation and scattering
patterns and other related param eters such as gain and axial ratio. (T h e near field
quan tities such as input im p ed an ce, feedline S-param eters. e tc .. will be disccussed
in later chapters).
B oth radiation and radar cross section (R C S) p atterns can be
readily characterized w ith respect to th e 3 -D spherical coordin ates 6 and o.
Consider the planar ca v ity -b a c k e d antenna as shown in fig.
2
. 1 . O nce th e field
distribution on th e aperture S o f th e conform al antenna is ob tain ed from th e full wave
analysis, we can then proceed to evalu ate the far field pattern. T h e m ost straightfor­
ward approach for this co m p u ta tio n in th e presence of an infinite con d u ctin g ground
plane is use of th e equivalence principle. To do so. we define th e m agn etic current
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
39
as M = —z x E 5 . where E 5 is th e electric field evaluated on th e apertu re and z is
norm al to th e aperture surface. T h e e le ctric vector potential is then g iven by
F(0.d)
=
e°e lkT J[Jfs M ejk r ' d S '
=
f° e
^ JfJ[S (E s
x ~)ejk r'd 5 '
(2.46)
where th e electric field is typ ically expan d ed in term s of the surface vector basis
function S '. Introducing this exp an sion . (2.46) becom es
F(0.<
-t~ r
e° e Jhr- V
4~r
(2.47)
where V’ denotes th e sum of th e surface integral over each o f th e d iscretization
elem en ts on the aperture. T h e far zon e m agn etic field H becom es
H r = —j u i F j
(2.4S)
where H r represents the transverse com p on en t of the far zone m agnetic field, whose
6 and 6 projections are given by
Hg = —j cz
e0 e - -,tr
4 ~r
{cos 0 cos o \ x + cos 6 sin 6Vy — sin 0V-}
Pe
4 "£r e - Jt,
;-------{ —sin &VX + cos o V y }
4a r
Ho =
^
in which
(2 -4 9 >
- ,tr p
4 -r
*
Pg = —j {•} and P0 = —j { •}. and {•} stands for the corresponding term s
in the curly
brackets.
The RCS of th e t (t = 9 or o) com ponent can be represented
in term s o f Pg and Pc to yield
5^1*1’
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
<2-50»
40
where A and Z 0 = \Jfio/co are th e free space wavelength and th e intrinsic im pedance,
respectively. In (2.50). the incident w ave \H'\ was set to unity w ith ou t loss o f general­
ity. In practice, it is custom ary to express RCS in dB , where a t is usu ally norm alized
to A2 or to squared m eter first.
T h e radiation pattern m ay also be represented in the sam e m anner as m entioned
above. T h e difference in procedure here is the norm alization w ith respect to a m ax­
im um radiation field value. T h e reason is that in radiation m od e, th e antenna is
ex cited by an interior source rather than the incident plane w ave H l as in the scat­
tering case. O f interest therefore is the relative field intensity in th e far zone. To get
th is, we represent the far field in ten sity in term s o f the above calcu lated quantity
crrcs to avoid a repetition of post processing. Specifically, th e form ula
=
/TrCS
i
( C ? n m ax
(2.51)
1
is used for radiation analysis.
2.4.2
Gain and Axial R atio
Gain ( G ) and A xial R atio (A R ) are two im portant param eters w hich indicate the
antenna's performance. It is also noted that these two qu an tities typ ically charac­
terize th e far zone features o f th e antenna.
By definition, the gain G for a lossless antenna (w ith 100% efficiency) is given by
o-r/y
G = “ n
P rad
(2.52)
for a cavity-backed structure, where t ’max is the m axim um pow er d en sity and Prac\
denotes th e total radiated pow er from th e antenna in the upper h alf space. (It is
noted that this definition of G is identical to that of directivity for lossless antennas.)
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
41
Since the m axim um power d en sity Umax can also be expressed as
t max =
7 3
4n
(tffi + <To)
(2.53)
th e antenna gain becom es
G =
(2.54)
—■* rad
T h u s, th e com p utation o f G is rather trivially done once cr is found as given in the
previous subsection.
In reality, Praa m ay be evalu ated on an assum ption that all
input at the feed is transferred to antenna e lem en t(s) and radiated. In th is case, one
has Praa = / 2/?inr w here I is th e known current source on th e feed, and R m is the
input resistance of th e an tenna m easured at th e reference plane.
H owever, this schem e m ay not alw ays work sin ce th e gain (m ore precisely direc­
tivity) reflects th e far field behavior, w hile th e input resistance com p u tation relies
on an accurate full w ave m odel for near field prediction. It is well known that th e
far field and th e near field com p u tation s offer different accuracy. T his accuracy in­
con sisten cy arises if R-in is used to determ in e th e gain G , a far zone pattern w hose
accuracy is directly governed by th e near field com p u tation w ithout averaging effect.
To avoid this accuracy inconsisten cy, one m ay calcu late th e gain or the d irectivity by
evalu ating Prad from th e far field radiation p attern. Specifically. Prad can be obtained
by integrating th e radiation in ten sity
=HLuja
=
+°o)<lP-
(2.55)
over th e half space. It is obvious that if a certain sym m etry of the pattern rem ains
such as circular about th e vertical Z axis, then U { 9 . o ) reduces to i'(9) and th e above
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
42
integration can be effectively evalu ated . O therw ise, a num erical
2
-D integration is
required.
A xial ratio ( A R ) is another im p ortan t antenna param eter, esp ecia lly when a
circular polarization (C P ) of a n ten n a ’s perform ance is o f the primary concern. Since
A R also features the far zone effect o f th e antenna, it is desirable to d eterm in e A R
w ith a m inim um com p utational load. It is noted that given the above p re-ca lcu la ted
eg and a#, one is again able to d eterm in e A R uniquely by
(2.56)
w ith
w here 3 = 2(3>//s —$ h0 ) i the tw ice o f th e phase difference betw een the tw o m agnetic
field com p onents.
(2 .5 0 ).
It can be o b ta in ed from the quan tities Pg and P# defined in
Since these two q u an tities are com p lex num bers, the phase difference is
readily represented as
(2.5S)
w ith Im {•} being the im aginary part.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
C H A P T E R III
Edge—Based F E -B I Technique
3.1
Introduction
N um erical m ethods have been serving th e engineers and researchers for m any
years in antenna analysis and design.
A m on g th em , th e m om ent m ethod in con­
ju n ction w ith various integral equation (IE ) form ulations played a m ajor role [1-3].
H owever, IE m ethods are associated w ith field representations in which the appro­
priate G reen's function for th e specific geom etry m ust be em ployed and this lim its
their versatility. M oreover. IE techniques are u su ally form ulated on th e assum ption
of an infinite layered (not inhom ogeneous) su b strate, a m odel which deviates from
th e practical configuration and leads to inaccuracies for larger bandw idth antennas.
Furtherm ore, in th e context o f IE m eth od s, an ten n a ex cita tio n s are represented us­
ing sim plified m odels that differ m ore or less from th e actu al configurations. Also,
due to th e singularity of th e current distrib u tion near th e feed ju n ctio n (s), special
m easures m ust be taken [26] for proper m odeling.
In contrast, the hybrid F inite
E lem en t-B ou n d ary Integral (F E -B I) techn iqu e a lleviates th ese difficulties and this
was recently dem onstrated when th e m ethod was applied to inhom ogeneous objects
of canonical shape scattering [27.28] and rectangular patch antennas [9].
Based on the past success of F E -B I m eth od s for an tenna analysis, it is desirable
43
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
44
to e x te n d th e m eth od to antennas o f arbitrary sh ap e. In this chapter, an edge-based
hybrid fin ite elem ent-boundary integral form ulation is presented for th e character­
ization o f arbitrarily shaped cavity-backed an ten n as [29].
An exam p le of such a
configuration is shown in fig. 3.1. where a c a v ity is recessed in a m etallic ground
plane en closin g th e FEM volum e. T he an ten n a elem en ts on th e aperture m ay be
ex cited by different schem es, such as a sim p le probe, a m agnetic frill generator, a
practical coaxial cable, m icrostrip line, slot or a C P W line. In th e con text o f the
FE M , th e c a v ity is first discretized into a n um ber o f tetrahedral elem en ts that natu­
rally reduce to triangles on th e c a v ity ’s ap ertu re. For non-rectangular patches this
triangular gridding is, in general, non-uniform and th e exact boundary integral for­
m ulation based upon this m esh applies to any patch shape. As a result, th e hybrid
FE -B I tech n iq u e is capable o f m odeling arbitrarily shaped cavitv-backed antenna
configurations, different substrate in h om ogen eities, anisotropies, as well as various
practical e x c ita tio n schem es.
3.2
H ybrid System Functional
In th is section , th e edge-based hybrid F E -B I m eth od will be form ulated using
the variational principle, where m atrix algeb ra n otation is em ployed so that one can
readily e x te n d the form ulae to the general anisotropic case.
As presented in [9],
the c o m p le te functional pertinent to the sca tterin g and radiation by a cavity-backed
configuration shown in fig. 3.1 m ay be w ritten as
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
45
Z
i
Patch
Ground plane
Aperture
Cavity
Coax cable opening
(surface C)
Coax cable
Figure 3.1: Illustration o f a typical radiation and sc a tte rin g problem .
F(E) = ^ / /X { (VxE)- V x E)- ^ rE.E}d,
+ 2 j k 0 Z0
+
J
JJJ E' X
- 2kl j j {
w here
j { E x H ‘) • = d S
i + V x — M tj dv
E x z) • |
J J jE
x~)-(i+
(3.1)
G'0 (r . r') r fS 'j d S
J,- and M, represent interior electric and m agnetic current sources within
the cavity V: H 1 is th e incident field, if any. from the exterior region: th e surface
S encom passes the cavity aperture exclu d in g the portion o ccu p ied by th e antenna
elem ents: er and
d en o te, resp ectively, the relative p e r m ittiv ity and perm eability:
ko is the free space wave num ber, I the unit dyad, and
6
'o(r, r') th e free space Green's
function w ith r and r' d en otin g th e observation and integration points.
3.2.1
FEM Subsystem
In proceeding w ith th e d iscretization of (3.1). it is convenient to decom p ose it as
F = Fv + Fs
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(3.2)
46
where
Fy
d enotes th e volum e integral contributions and sim ilarly
Fs
accounts for th e
surface integral contributions. T h e cavity volum e is subdivided into N tetrahedral
elem en ts Ve (e = 1 .2 ....N ). and w ithin each tetrahedron as shown in fig. 3.2 th e field
1
2
nodes/vertices
Figure 3.2: A tetrahedron and its local n o d e/ed g e num bering schem e
is expanded using edge-based elem en ts as
E = [V]J{E}e
(3.3)
with
( V
‘'ul N
'u2
{Vu } =
u = x.y.
V
v’u6
/
( El N
V E* J
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(3.4)
47
in w hich V'u, is the u [u = x. y or z) com p on en t o f th e volum e vector basis functions
along th e 1th edge. T h e unknown vector { £ } e has six entries, one for each tetrah e­
dron edge. (H ere we use square brackets for m atrices and curly brackets for vectors).
Inserting (3.3) into (3.1), and taking th e first variation of Fv w ith respect to { £ } e,
vie Ids
SFV = £
{[.-!]«{£}« + {A'K>
(3.5)
where
/
w. =
Jffn
j k QZo
Jiy
M lx
\
M iy
> dv
(3-7)
M, ,
\ Jiz )
T y { X z ] - T = {V>]
[ D V ] eT
(3.S)
=
f j v
-fa
n
To carry out th e above integrations, it rem ains to introduce the volum e expansion
or shape functions V e. For our im p lem en tation we em ployed the linear edge-based
shape functions for tetrahedral elem en ts given in [30.31]. T he explicit finite elem ent
m atrix entries associated with a typ ical tetrahedron (as shown in fig.
. ) are given
3 2
in A p pendix A for reference.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
48
3 .2 .2
B o u n d a r y In teg r a l S u b sy ste m
To d iscretize th e surface integrals in (3 .1 ), th e aperture is su bd ivided into trian­
gular elem en ts sin ce these correspond to the faces o f th e tetrahedrals. W ithin each
triangle, th e field is represented as
(3.9)
where
' ul
{S u }
=
u = x.y
■u2
V
5 “3
(3.10)
/
( n-si
E N
{Esh
=
Js2
V
E S3
^ / e
in w hich 5 UI is th e u(u = x . y ) com ponent of th e surface vector basis functions along
th e ith edge. On su b stitu tin g (3.9) into th e surface integrals o f (3.1) and taking the
first variation o f Fs w ith respect to { E s } e. we obtain
SFs = £
{ [ B M M , + { £ } .}
(3.11)
where
[ B } e
=
- 2 k 2Q[Se][Se]r + 2
•G'0 (r. r') d S dS'
(3.12)
and
HI
{ L } e = j 2 k 0Z0 J J ^ [ S e}
dS
-H I
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(3.13)
49
N ote that in (3.12) the elem en ts o f th e array [Se] are functions o f the observation
vector r. w hereas the elem ents o f [5e]r are w ith respect to the integration point r'.
A su itab le set o f linear edge-based surface basis functions is
.
x ( r ~ r«)e(r)
? € Se
S i(r ) =
(3.14)
otherw ise
0
In this expression (referring to fig. 3 .3 ). /,• d en otes th e length o f the 1th edge and r, is
\ th edge
Figure 3.3: Pair o f trian gles sharing th e ith edge
the position vector o f the vertex o p p osite to th e ith edge. Since each edge shares tw o
triangles, on e is defined as the plus and th e oth er as th e m inus triangle. T herefore.
e(r) is given by
/
r € 5+
1
e(r) = <
(3.15)
r € S~
- 1
where Se = S * + S ~ . The constant .4e in (3.14) d enotes the area of th e plus or
m inus triangle depending on w hether r
6
5 e+ or r G 5 " . We note that S ,(r ) x i
yields th e basis functions used by R ao. et al. [32] in their m om ent m ethod solu tion
of boundary integral equations. T he e x p lic it expressions for the boundary integral
equation su b system is given in A p pendix B for reference.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
50
3.2.3
Combined FE—B I System
To construct th e final system for th e solu tion o f th e electric field com ponents we
com bine (3.5) and (3 .1 1 ). and after assem bly w e obtain th e system
{ { A \ { E } + {A '}} + { [ £ ] { £ , } + { L } } = 0
(3.16)
In this. {A*} and { L } are the ex citation vectors due to the interior current sources
and the exterior ex cita tio n , respectively.
T h e unknown electric field vector { E }
consists o f all field expansion coefficients w ith respect to th e elem en t edges except
those coin ciding with perfectly electrically con d u ctin g (P E C ) w alls. PEC antenna
elem en t(s) or PEC pins inside the cavity. F inally, th e vector { E s } represents the
unknown surface fields w hose entries are part o f th o se in { A } w ith their corresponding
edges on th e aperture. T h e explicit expressions for th e m atrices and vectors in (3.16)
can be readily extracted from (3.6), (3.7) and (3.12) (see also [33]). It is evident that
[A] and [B] are sym m etric as a result o f the assum ed isotropic m ed ium and reciprocity.
In addition. [A] exh ib its high sparsity due to th e FEM form ulation whereas [B] is
fully populated.
Two approaches m ay be followed in carrying out th e solu tion o f th e com bined
subsystem s w hen an iterative solver is em p loyed such as th e biconjugate gradient
(BiC’G) m eth o d [34]. T h ese approaches differ in th e m anner used for the evaluation of
m atrix-vector products called for in th e iteration step s. O ne could sum th e coefficient
m atrices [A] and [B] by adding up the corresponding m atrix entries prior to the
execution o f the BiCG algorithm , or instead th e resulting vectors m ay be sum m ed
after carrying out th e individual m atrix-vector products. W e observe that the first
approach is m ore efficient in term s of com p u tation tim e after reordering the com bined
m atrix and storing only th e non-zero elem en ts. T h is is because, in the context of
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
•51
this schem e, the com bination of the two m atrices is perform ed on ly once o u tsid e the
iteration. However, th e second approach is com p atib le w ith th e B iC G -F F T sch em e,
where th e F F T algorithm is em ployed to exp loit the convolutional property o f the
integral operator, thus elim inatin g a need to ex p licitly store the entire B I m atrix.
T h e B iC G -F F T technique will be discussed in chapter 4.
3.3
N um erical Im plem entation
B ased on the above presented F E -B I form ulation, the hybrid m eth od was im ­
plem ented and a com p uter program was develop ed for the analysis of radiation and
scatterin g by cavitv-backed patch antennas o f arbitrary shape. T he antenna geom e tr v /m e sh is first generated as shown in fig. 3.4 and supplied to this program in an
Figure 3.4: A typ ical g e o m e tr v /m esh for a cavity-b ack ed circular patch an tenna.
input file which, as a m inim um , contains
(a) the nodes and their (x . y . z ) coordinates:
(b) the tetrahedral elem en ts and th e corresponding nodes form ing each elem en t:
(c) the nodes identifying the cavity aperture:
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(d) the nodes identifying m etallic boundaries, anten n a p a tch (es), feed(s). and pos­
sible vertical posts.
For arbitrary antenna geom etries, it is necessary to em p loy a sop h isticated volu m e
m esh generation package and a num ber of th ese are available com m ercially. T y p ica lly
each of th ese packages generates a ^universal file" that can be readily preprocessed to
extract th e aforem entioned input list. G iven th e above list of d ata, an interpretation
routine is used to convert the inform ation from n o d e-elem en ts to e d g e-elem en ts. We
usually refer this procedure as data preprocessing. T h e flow chart shown in fig. 3.5
describes th e m ajor im plem entation procedures from the m esh generation, a few d ata
preprocessors, th e F E -B I kernel, to the B iC G solu tion and finally the data o u tp u t.
O ne of th e prim ary com plications in th e hybrid techn iqu e im plem entation is the
efficient treatm en t o f com bining th e two separate su b system s. It is noted th at th e
FEM sparse m atrix is large in dim ension but requires less storage space, w hile th e
boundary integral sub system is always sm all in size but can be dom inant in term s
o f m em ory dem and. T h is is particularly true w hen th e n o n -m eta llic portion on th e
cavity aperture predom inates over the anten n a radiating elem en ts.
Furtherm ore,
the boundary integral su bsystem in a general purpose hybrid F E -B I im p lem entation
is entirely independent from that of the FEM and even th e basis functions can
be independently developed. T his also accounts for th e two arbitrary num bering
system s and com b ining them is a relatively com p lex task . O ne m ajor advantage of
the m ethod how ever is that these two su b system s can b e developed and validated
individually.
O nce b oth of the subsystem s are verified, the couplin g of th e sub system s is a c­
com plished by enforcing the boundary conditions im p licitly on tangential H fields via
the integral representation and ex p licitly on tangential E fields over the interior and
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
IMPLEMENTATION FLOW CHART
z:
MESHER
z_
T
Universal File
.Z
!
T
Pre-Processors j
T
FEM System
r
T
FOR
BI System
Feed Models
Combinations
Data File Set
— FEM System
Core Program
Bl System
Feed Models
BiCG Solver
Combinations
OUTPUT
Figure 3.5: A flow chart describes th e m ajor im plem entation procedures from the
m esh generation, a few d a ta preprocessors, the F E -B I kernel, to the
BiCG solution and finally th e data ou tp u t.
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
•54
ex terio r regions. A significant effort w as d evoted to developing a program in such a
m anner so that both the storage and com p u ta tio n a l requirem ents can be m inim ized.
Specifically, th e boundary con d ition s on th e m etallic surfaces are enforced a prior i
to con dense th e system which in volves o n ly nonzero field com p onents. T o th is point,
th e sparse finite elem ent m atrix was stored as a single array of length N v N m , where
iVt. is th e total number o f unknow ns w ith in th e cavity volum e and iVnz d en otes the
m axim u m num ber of nonzero row en tries.
T h e B l m atrix was stored in different
w ays for the evaluation of the m a trix -v ecto r products. If the BiCG so lu tio n was to
be carried out w ithout special tre a tm e n ts (such as incorporating th e F F T ). then the
N s x N s BI integral m atrix is added to th e FE array resulting in a
1
-D array about
A v V nz + iVj long. For slot anten n as, in clu d in g cavity-backed spirals, and m oderately
sized sy stem s, it was found preferable to use this schem e. In that case th e generation
o f a sin gle com bined FE-BI m atrix before execu tion of the BiCG algorith m reduces
th e com p u tation al requirem ents. T h is is because a number of op erations associated
w ith th e repeated com binations o f th e F E and BI subsystem s w ithin th e BiC G iter­
ation is avoided. T he alternative is to carry out the m atrix-vector prod u cts for the
FE M and BI subsystem s separately. T h e advantage of the schem e b ecom es appar­
ent w hen a special treatm ent is perform ed on to the num erical sy stem for efficiency
consid eration and this will be in v estig a ted at certain depth in chapter 5.
3.4
Selected Num erical R esu lts
We present below som e rep resen tative num erical results for th e purpose o f vali­
d a tin g and dem onstrating the robustness of the tetrahedral form ulation for scattering
and radiation by different configurations o f cavity-backed antennas. In each case the
com p u ted results via the FE -B I m eth o d are com pared with reference m easured or
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
calculated data.
Scattering and radiation by a circular patch:
Fig. 3.6 illustrates a circular patch residing on th e surface of a 0.406 cm thick
substrate having a relative dielectric constan t o f t T = 2.9. T he patch's diam eter is
2.6 cm and th e substrate is enclosed in a circular c a v ity 6.292 cm wide. T h is ca v ity
and the patch were recessed in a low cross se ctio n b ody for m easuring its RCS. A
com parison of th e m easured and calcu lated b ack scatter agg RCS as a function o f
frequency is also shown in fig. 3.6. For th is co m p u ta tio n th e direction of the incident
plane wave was 60° from norm al, and as seen th e agreem ent between m easurem en ts
and calculations is very good throughout th e 4 -9 G H z band. Input im pedan ce m ea­
surem ents and calculations for the sam e patch are displayed in fig. 3.7. T h e probe
feed in this case was placed 0.S cm from th e p atch 's center and it is again seen th at
the m easurem ents and calculations are in good agreem ent.
Radiation by a one-arm conical spiral:
We considered th e m odeling o f this radiator to dem on strate the geom etrical ver­
sa tility of the FE -B I m ethod. Tw o projections o f th e spiral radiator and surface m esh
are illustrated in fig. 3 .S. T h e top and b o tto m ed ges o f the strip form ing th e spiral
follow the lines p = 0 .0 5 0 3 A e x p [0 .2 2 1 (d ± 2 .6 6 )], z = a± ex p (0.221o). where {p. o . z)
denote the standard cylindrical coordinates, a± are equal to O.OS32A and 0.0257A.
respectively, and 0 < d < 2tt. T his spiral arm resides on an inverted cone (9.24 cm
tall) whose b ottom cross section has a diam eter o f
1 .6 8
cm and the top cross section
has a diam eter o f 21.78 cm . For our calcu lation s A = 30 cm ( / = 1 G Hz) and th e
spiral was situ ated in a circular cavity 10.01 cm d eep . T h e com puted E0 principal
plane radiation pattern taken in th e 6 = 9 0 °-p la n e, using a probe feed at th e ca v ity
base, is given in fig. 3.9. It is seen that this p a ttern is in good agreem ent with th e
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
56
-10.0
e
CD
■
o
co
U
Q£
ea
c
FH-Bl Method
1/3
-
20.0
-
30.0
-
40.0
-
50.0
•c
H
u
d * 2.6 cm
CO
V)
0.406 cm
C
J
CO
pa
2 * 3.146 cm
-
60.0
4.00
6.00
5.00
8.00
7.00
f r e q u e n c y (G H z)
Figure 3.6: Com parison o f th e com p uted and m easured agg backscatter RCS as a
function of frequency for th e show n circular patch. T h e incidence angle
was 30° off the ground plane.
1.0
0.5
•x
.2.0
>5.0
• measured
X calculated
1.0
5.0
'5.0
2.0
1.0
Figure 3.7: Com parison of the com puted and m easured input im pedan ce for the
circular patch shown in fig. 3.6. T h e feed was placed O.S cm from the
center o f the patch and th e frequency was sw ept from 3 to 3.8 GHz.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
data given in [35]. As can be exp ected , the Eg pattern (not show n) differed from
th e m easured d a ta near th e horizon because o f interference from th e finite circular
cavity housing th e spiral which was included in the an a ly tica l m od el. T h e latter was
not part o f th e m easurem ent configuration which con sisted o f th e spiral antenna on
a large circular plate.
Annular slot impedance:
Fig. 3.10 show s a narrow circular (0.75 cm wide) annular slot situ ated in a cir­
cular c a v ity 24.7 cm wide and 3 cm deep. Because th e annular slot is narrow, the
im p lem en tation o f the BI su b system is very sm all for this app lication and as a result
there is no need to invoke th e F F T in the BiCG algorithm . T h e F E -B I m ethod is
basically q u ite effective in m odeling sm all aperture configurations w ithout a need for
special co m p u tation al considerations. Input im pedance calcu lation s as a function of
frequency for th is radiator, excited by a probe placed across th e slo t, are shown in
fig. 3.10, and agree well w ith th e values calculated via a m odal-boundary integral
m ethod [14]. For these calculations, the frequency was sw ept from 700-1000 MHz.
T h e d ielectric constan t o f th e m aterial filling th e cavity was set to er = 1.35 as in [36]
and this is an effective value to account for the presence o f a dielectric slot cover used
as part o f th e m easurem ent m odel for holding the plate.
Stacked circular patch antenna:
To d em on strate th e capability of the developed hybrid techn iqu e, we now present
a q u alitative stu d y and visualization of the near field d istrib u tion inside a ca v ity backed, stacked circular patch antenna as shown in fig. 3.11. N ote that the sim ilar
configuration w ith rectangular patch shape has been in vestigated and found a sig­
nificant band w idth increase. T his is because o f the dual frequency resonance due
to the two stacked patches. T he circular patches are m ore attra ctiv e than stacked
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
58
Figure 3.8: Illustration o f th e configuration and m esh o f th e one-arm conical spiral
used for the c o m p u ta tio n o f fig. 3.9.
s
- 10.
E
u
- 20 .
-30.
E$ (FE-BI)
E-, measured
-40.
-50.
-90.
-60.
-30.
0
.
30.
60.
90.
9
Figure 3.9: Com parison of th e calcu lated radiation pattern ( E $), taken in th e o =
9 0 °-p la n e, w ith d a ta in reference [35] for the one-arm conical spiral shown
in fig. 3.8
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
59
a = 12.35 cm
b = 0.75 cm
p0=
7.7 cm
probe
0.7 < f < 1 GHz
d = 3 cm^ |
.
£reff = 1.35
|
1.0
.
2.0
5.0
— Calculated
• FE-BI Method
1.0
2.0
5.0
■5.0
2.0
0.5
1 .0
Figure 3.10: C om parison o f input im pedance calcu lation s for the illustrated cavitybacked slot.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
60
rectangular patches because th ey occupy a sm all area when operated at the sam e fre­
q u en cy [37]. U nfortunately, no sufficient research on this geom etry has been reported
in th e literature due to a lack o f analysis tool.
It turns ou t that th e above presented hybrid FEM technique is well su ited for
th is stu d y. To show this, we chose a c a v ity -b a c k e d , stacked circular patch an tenna
fed w ith an offset single vertical post und erneath th e lower patch to link th e cavity
base (view ed as a ground plane). N o direct electric contact between th e upper and
th e lower patches exists and thus the power transfer has to rely com p letely on th e
electro m a g n etic coupling from th e lower to th e upper patch. This can be clearly
verified from th e near field visu alization, w hich is available and com plete on ly via th e
P D E related techniques. (T he laboratory m easu rem en t m ay provide the im age o f th e
field d istrib u tion above and near a m icrostrip [38].) A nother interesting point is that
th ough th e an tenna was fed w ith a single offset probe, the energy is concentrated at
two d istin ct regions. O ne is around the probe feed, and th e other is near th e o p p o site
location w ith respect to the center o f th e p atch. T h e two regions act as out-of-phase
electric pole to effectively ex cite th e antenna. A lth ou gh the patches are circular in
sh ap e, th e offset excitation ensures th e linear p olarization in radiated fields.
F igure 3.11: V isualization o f the near field d istrib u tion at the lower layer o f a stacked
circular patch antenna.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
C H A P T E R IV
Efficient Boundary Integral Subsystem — I
4.1
Introduction
As is know n, th e hybrid finite elem en t-b o u n d a ry integral m ethod is accurate and
capable o f hand ling a variety o f conform al antennas.
However, th e drawback as­
sociated w ith th is technique and any oth er global truncation approaches can m ake
it less attra ctiv e. T h is is especially true if one is interested in m odeling large an­
ten na sy stem s (arrays). A lthough the F E -B I m eth od is particularly su ited for the
configurations w ith relatively sm all aperture size and possibly com plex cavity design
(including feedlines, isotropy/anisotropy, oth er layers of m etals, e tc .). it would be
much m ore useful to accelerate the speed and reduce the C P U requirem ent for the
hybrid approach. O ne possible solution is th e C G -F F T technique discussed in this
chapter.
T h e boundary integral (BI) equation su b sy stem leads to a fully populated m atrix
whose size is d eterm in ed by th e num ber o f aperture m esh edges. For large apertures,
th is analysis b ecom es im practical in term s o f storage and com putation tim e require­
m ents, and to overcom e this inefficiency, a uniform zoning of th e aperture is required.
B y resorting to th e structured m esh, the boundary integral m atrix can be cast into
a discrete convolutional form, thus p erm ittin g th e com p utation of the m atrix-vector
61
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
62
products via th e discrete Fourier transform (D F T ) and avoiding a need to store the
full B I m atrix. This m em ory saving sch em e has already been applied to IE solu tions
in volvin g rectangular grids [39.34], and a sim ilar im plem entation was also reported
for triangular surface grids [40] involving inheren t approxim ations. In th is chapter,
we first show that the B iC G -F F T solver can be precisely im plem ented on uniform
triangular m eshes. T he differences betw een th e rectangular and triangular m eshes
are also described. For non -rectan gu lar a n te n n a geom etries, a special treatm en t re­
ferred to as th e overlaying scheme is proposed and discussed in section 4.3. A few
results are presented which dem on strate th e m eth od 's validity.
4.2
A pplication o f C onjugate G radient A lgorithm s
T h e C onjugate Gradient (C G ) iterative so lu tio n of linear system s o f equations has
been ex te n siv e ly in vestigated and the rep resen tative references are collected in [41].
A lth ou gh th e state-of-the-art CG algorith m s do not lend th em selves as a robust
in p u t/o u tp u t "black-box“ [42]. they are in d eed capable of handling large-scale com ­
p u tation al problem s which m ay be im p o ssib le for direct system solvers. It is esp e­
cially desirable to em ploy th e algorithm s w h en one seeks the solu tions of large-scale
num erical system s w ithout resorting to c o stly com p u tin g resources.
4.2.1
BiCG Algorithm W ith Preconditioning
C on jugate gradient (C G ) algorithm s h ave been developed for over forty years
[43.44] and one o f th e prim ary applications now adays is to solve large scale linear
sy stem s, as aforem entioned.
It is n otew orth y that there exist various versions of
the CG algorithm s, taking advantage o f different properties o f th e m atrix such as
sym m etric and sparsity. A lso, preconditioning is often used to speed up convergence.
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63
As su ggested in [45.46], th e algorithm used in this work is as follows:
G iven
= rt = b —A • xi
Pi = ? i
For A: = 1 ,2 ,3 ,...
fl-
Tfc
Pfc • A • pfc
rfc+i = rt - Qt A - p k
rjfc+i = rfc - a'kA ' T ■ p k
,3
^"fc+l ’ ^k+1
Jk = ——--------rJk • Vk
Pfc+l = Vk + $ kP k
P it+ i =
r t+ i +
3 kp k
X/fc+1 = i t + QtPt
where * d en otes th e com p lex conjugate and T is th e transpose. T h is version of the
iterative algorithm is quite general in term s of th e sy stem m atrix to be solved and
is usually referred to as th e B iconjugate gradient (B iC G ) m eth od for unsym m etric
system s.
If th e m atrix
A is sym m etric and the initial value is chosen as iq =
r j, th e algorith m can then be shortened to require on ly one m a tr ix -v e c to r product
per iteration , sin ce in each step r k and p fc are com p lex conjugate o f rjt and p fc.
respectively.
T h e ordinary conjugate gradient algorithm can be considered as a special case of
the BiC G when
A is H erm itian (i.e. A = A _r). A gain in this case, th e algorithm can
be shortened to have about 50% less com p utational effort. T h e CG algorithm is also
am enable to a straightforward interpretation of its convergence principle. Basically.
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64
th e algorithm m inim izes the function
/(x ) =
•A •x —b •x
H ence x o b tain ed from the CG algorith m after th e k iteration steps becom es th e
solu tion o f th e equation
A /(x ) = A • x —b = 0
T he CG typ e of algorithm s did not b ecom e c o m p etitiv e until preconditioning
was introduced to improve the original sy ste m con d ition and significantly reduce th e
convergence rate. One sim ple preconditioner is th e inverse of the original diagonal
m atrix. In our applications this precon d ition in g has been q u ite successful and is used
in conjunction with the BiCG algorithm .
4 .2 .2
BiC G r—F F T A lg o r it h m F o r L in e a r S y s t e m
In our work, the linear system o f eq u ation s is usually large in size, partially full
and partially sparse. T he conjugate gradien t (C G ) typ e of algorithm s can be used
to allev ia te th e m em ory requirem ent sin ce th e LU decom position requires excessive
m em ory and C P U tim e. However, th e p artially dense m atrix due to the boundary
integral equ ation may still dom in ate th e C P U d em and. T his is because th e m eth od
o f m om ent (M oM ) always leads to a d en se sy ste m by its nature. Solving th e dense
sy stem in a traditional m anner requires 0 ( N 2) order o f operations per iteration ,
where ;V is th e boundary integral sy ste m d im en sion .
R eduction of the operation
counts w ill o f course significantly decrease th e solu tion tim e and this can be accom ­
plished by recasting the BI system onto a few T oep litz subm atrices and m aking use
o f the fast Fourier transform (F F T ) to carry out th e m atrix-vector products in th e
BiCG solver.
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65
As described La th e next section , th e boundary integral equ ation can be cast into
a convolutional form if a uniform grid is applied for discretization . T h is is not sur­
prising sin ce th e G reen's function is involved in th e integration. To solve th e equation
using th e CG algorithm , it rem ains to carry out the convolution at each iteration.
To this en d , one m ay calcu late th e convolution by taking th e Fourier Transform of
tw o spatial d ata sequences (arrays) in w hich th e convolution becom es th e product of
th e two 'frequency' sequences. A n inverse transform of th e product y ield s th e result.
In contrast to th e order o f 0 { N 2) C P U requirem ent for a m a tr ix -v e c to r product in
a traditional fashion, the sch em e needs 0 ( /V lo g 2 N) operation counts if th e F F T
algorithm is em p loyed. T h e operation reduction is indeed significant and the tech­
nique has th e lowest C P U dem and am ong integral equation solvers, includ ing the
fast m u ltipole m ethod (F M M ) [47], thus is always preferred.
4.2.3
Convolutional Form of Boundary Integral
T h e boundary integral equation is discretized using th e structured triangular grid,
and the relation betw een th e unstructured and structured m esh is described in the
next section. We recognize that th e triangular grid consists o f equal right triangles
as shown in fig. 4.1 and thus involves three different classes o f edges (class 1. 2 and
3). T hese include th e x-d irected , {/-directed and the diagonal edges, all of which are
uniform ly spaced. For the F F T im p lem en tation each class o f edges is in dependently
num bered in accordance w ith their geom etric location. Specifically, th e Lth class will
carry the num bering ( m ,n ) if th e edge is th e m th along th e x direction and th e nth
along the y direction. T h e indices ( m ,n ) take the values
m
=
0 , 1. 2 . . . . , \r
n
=
0 . 1 . 2 ......... :V‘
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66
CM
O
•■
c
m: 0
1
2
3
M
Figure 4.1: Structured m esh consists o f equal right triangles
w ith i =
1
for th e (/-directed edges, i = 2 for th e diagonal edges and i = 3 for the
x-d irected edges. C onsequently, we find that
M - 2
Ml =
i = 1
M -
1 i' = 2
M -
1
i = 3
-V* =
'V — 1
1= 1
N -
i
N -2
1
= 2
(4.1)
i = 3
where M and N denote th e num bers of elem en ts along th e x and y directions,
respectively.
To perform th e integrations for the evalu ation o f th e boundary integral m atrix
e lem en ts, it is now convenient to rewrite th e basis functions (3.14) in term s of the
new indices ( m ,n ) . We readily find that th e ed ge-b ased basis functions associated
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67
w ith each o f th e aforem entioned class o f edges can be rew ritten as
( n A y - y ) x + (x - m A x ) y
slmn(x-y) =
(x.y)
( y — {n + l ) A y ) x + ( ( m + 2 ) A x — x ) y
Ax
S e+
6
{x,y)eS~
(4-*2)
otherw ise
0
( n A i / - i / ) i + ( r - ( m + l ) A i ) y £ S*
V(Ax)2 + (Ay)2
Smnfau)
=
( y — (n + l ) A y ) x + { m A x — x ) y
AxAy
otherw ise
0
s i n ( x -y)
=
Ay
G S e' M )
{{n + 2 )A y — y ) x + (x - (m + l ) A x ) y
{x.y) £ 5+
(y - n A y )x + (m A x - x )y
( x .y )
0
otherw ise
6
S~
(-4--*=)
w here th e superscripts refer to the edge class. After th e d iscretization and assem bly
processes, one ob tain s a discretized version o f the BI sy stem , from which each entry
of th e boundary m atrix-vector product can now be calculated as
vp
ap
{B I su b system } = [B ] { £ s} =
(4.5)
j= i
m'=0 n '= 0
in w hich (m . n ) are th e geom etric location indices for the ith class observation edges
whereas (m ' . n ') are the sam e for th e _/th class edges belonging to integration el­
em en ts.
T hus, the specification of th e indices i , m and n co m p letely defines the
entry fc, = n M ‘ + M of the colum n resu ltin g after the execu tion of th e boundary
m atrix-vector product. It is readily found that
=
- 2 k l JJ^ j j ^ . S ^ C ^ T . ^ i x i y d x ' d y ’
+
JJS g
(4.6)
£'(r ) £j ( r ) lib Go(r. r') d x dy dx' dy'
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68
w ith
Ay
li =
i = l
x /(A x ) 2 + { A y ) 2
Ax
i= 2
('I-'')
i = 3
More im portantly, it can be shown that th e BI su b system (4.6) ex h ib its th e convo­
lutional property B l^ nm,n, = B^m_ m, n_ n<) and thus we can rewrite (4.5) as
3
[ B ] { £ .} = £
B '1 » EJ
(4.S)
j=1
where th e * denotes convolution.
T h e proof will be presented in th e end of this
section to ensure th e sm ooth discussion o f th e rem aining procedure. It is now seen
that th e com p utation o f th e boundary m atrix-vector product can be perform ed by
em p loyin g the 2-D discrete Fourier transform (D F T ). thus avoiding a need to store
the BI m atrix other than those entries w hich are unique.
property o f B^m_m, n_ n
W hen th e sym m etry
is also invoked, im plying
B(L-m',n-n') =
m'-m.n'-n)
it is concluded that the total non-redundant entries in th e BI m atrix are
3
3
•v » ■= E E v ‘‘ u ‘ ■=i j =i
T his should be com pared to th e (Xw=i -V/‘iV‘ )
2
- I:'
( 4 -l 0 >
entries w hose storage would norm ally
be required if the BI system was not cast in convolutional form and it is notew or­
thy th at Np is much sm aller in num ber.
B'ijn—m' n-n') =
cast 'n a
To avoid aliasing, it is necessary that
array which has th e usual periodic form,
and zero padding m ay also be required to m ake use of the standard F F T routines.
Specifically, the m atrix-vector product (4.5) is execu ted by using th e M F T x X F T
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69
arrav
0 < m < M'
0 < n < Nj
MFT - A/' +
0 < n < Nj
B i}( m . n -
1
- NFT).
1
< m < MFT
0 < m < A/'
NFT - -VJ +
B ij( m - 1 - M FT .n - 1 - NFT).
0
( 4 .1 1 )
1
< n < NFT
MFT - AP +
NFT - N j +
1
1
< m < MFT
< n < NFT
otherwise
w ith th e corresponding field vector given by
EUfh.n) =
(4.12)
0
otherw ise
and M FT and N F T m ust be powers o f 2 if a radix 2 F F T algorith m is used.
In th e B iC G -F F T algorithm th e BI su b system vector is sym b o lica lly com puted
as
{B I su b system } = ^ 5 { D F T
i=i
" 1
{ D F T { S ‘J} • D F T { £ £ } } }
(4.13)
T h e presence of the operator 5 indicates the necessary reordering of th e 2-D array
which results after the inverse F F T operation into a single colu m n w ith th e proper
indexing for addition to th e FEM su b system . It should be rem arked that in contrast
to [40] th e integrals (4.6) are evaluated w ithout introducing any approxim ation. This
is necessary to preserve th e sym m etry feature o f th e global com b ined sy stem .
As prom
ised, we now show v(4.S).
r
7 or th e relation B ml*n . m ' n ' =
B)J
( m —m , . n —n') to
conclude this section. To sim p lify th e proof, we refer to fig. 4.2 and consider only
the first integral in (4.6). T h e sam e proof can be appied to th e second integral in
(4.6).
In addition, w ith no loss of generality for th e proof, th e / =
1
class edges
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Figure 4.2: Illustration of tw o triangles w ith the corresponding indices to help to
prove the convolutional property of the boundary integral.
(v -d irected ) in S * for the trial function and the i = 2 class edges (diagon al) in S~
for th e testin g function will be used. To this point, su b stitu tn g S ^ x . y ) in S + and
5 ^ /n/(x .f/) in S ~ into the first term in (4.6) yields
Intm„,m/„< = C J J
J J ' [ ( n A y - y ) { n ' A y - y') + (x - m A x ) { m A x + 2 A x - x')]
G'o [(x - x')x + { y - y')y] d x d y d x ' d y '
w here
C is a constant
coefficient and its detailed form
th is proof.N ote that the integration lim its should be set
(4.14)
is not of our concern for
as [m A x . (m + l) A x ] and
[n A )/,(n + l)A y ] for the unprim ed coordinates and sim ilarly for th e prim ed ones.
Therefore, (4.14) will be sim plified if th e following transform s are introduced, viz.
x = mAx + £
x' — m A x +
(4.15)
y = n A y + rj
y = nAy + y
Indeed, on su b stitu tin g the transform s, one obtains
/* f i r A j
Intjnn.m»„» = C J j
/• r A x A y
JJ
[yy' + £(2 A x - £')] G’o {x [(m - m ' ) A x +
( f - £')] + « / [ ( « - n')-^y + ( n ~ 'Z)]} d^dy d g d r f
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(4.16)
71
A lthough this integral m ay not b e expressible in term s o f an elem en tary fu n ction , one
is how ever assured th at th e resu ltin g expression m ust be a function o f [(m — m'). (rc —
n')] no m atter what form th e G reen's function Go w ill take. M athem atically, it m eans
Intm„.m'n' — Intm_ m'_n_ n'
( - l .li )
T his is th e desired relation. From th e proof, we can conclude that the convolution
in (4.S) holds.
4.3
M esh Overlay Schem e
A s described above, th e B iC G -F F T solver requires uniform aperture gridding
so th at the BI subsystem can b e put in block circulant form. T his can be alw ays
achieved during m esh generation w henever th e patches are rectangular in sh ap e or in
case o f radiators which are placed at som e d istan ce (usually sm all) below the apertu re
as show n in fig. 4 .3 (a ).
In th e latter case, one m ight need to add an appropriate
absorbing m aterial around th e e d g e/co rn er o f th e cavity near th e aperture to avoid
possible ed ge/corn er effects, esp ecia lly when th e aperture size is not large enough.
Fig. 4.3(b ) shows th e exam p le o f th is im plem entation.
4.3.1
Field Transformations
H owever, for circular, triangular, or other non-rectangular patches on th e aper­
ture. it is not natural to con stru ct a uniform m esh using the m esh generator. T y p ­
ically. the aperture m esh is necessary to conform to the patch shape, leading to an
unstructured free surface grid. In this case, to m ake use of th e efficient, low m em ­
ory B iC G -F F T algorithm , an approach is proposed to overlay on th e unstructured
aperture grid another coincident structured grid as shown in fig. 4.4. T he boundary
integral subsystem is then con stru cted by using the overlaid uniform grid w hose edge
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uniform grid
cavity
recessed patch
(a)
Circular Patch Recessed Underneath the Ground Plane
30.
20 .
10 .
CQ
-3
a.
oa
U
C£
o
-
10 .
-
20 .
permittivity = 4
-30.
permittivity = I
-40.
permittivity = 4 (with FFT)
-50.
permittivity = 1 (with FFT)
-60.
0
.
10.
20.
30.
40.
50.
60.
70.
80.
90.
0
(b)
Figure 4.3: P rin ted circular patch antenna is m odeled using th e recessed schem e to
incorporate th e B iC G -F F T algorithm , (a) Illustration o f th e configura­
tion: (b) C om parisons o f the B iC G -F F T result w ith that of th e ordinary
F E -B I technique presented in chapter 3.
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73
I
0
n
m
0
1
2
?
Figure 4.4: O verlay o f a structured triangular aperture m esh over an unstructured
m esh, shown here to conform to a circular patch.
fields can be related to th ose on th e unstructured grid via two sparse transform ation
m atrices. T hat is. it is necessary to append to th e system (3.11) th e relations
{£.}» =
{ £ .} „ ,
=
P>] { £ ,}„,
P y { £ .} „
(4.18)
where th e subscripts u and nu refer to th e field coefficients o f the uniform and
non-uniform aperture grids, respectively. A lso. [TV] and [7 s] refer to th e forward
and backward transform ation m atrices, respectively, with N u and N nu den otin g the
num bers of the uniform and non-uniform m esh edges on th e cavity aperture.
To derive the elem en ts o f [7V]. we begin w ith th e expansion (3.14) and enforce it
at three points on each ed ge belonging to th e uniform grid. We con ven ien tly place
these three points at th e center and ends of th e ed ge (see fig. 4.5).
G iven the fields at th e se p oints, we can interpolate the field along th e ( m . n) edge
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overlaid unstructured
m esh triangles
end2
mid
(m,n) uniform grid edge
O
endl
\
/
/
Figure 4.5: Illustration o f the param eters and geom etry used in constructin g the
transform ation m atrix elem ents betw een th e structured and unstructured
m esh.
o f the uniform grid using the weighted average
j
♦’enai
S
^
*wmia
+
^ 3 E nu(rniid)
- V m id
+
E ‘ “(r “ d,)
fc = I
Y
end2
53
k= i
^ n u ( r end2 )
(4.19)
in which eu d en otes the unit vector along x. y or th e diagonal, depending on the class
o f edge being considered. T he quantities E *u represent th e fields in the non-uniform
grid triangles w ith th e superscript k being a sum variable in case rendj. renj 2 or
specify a point shared by more than one triangle. O bviously. .Vendl. A ' ^ and A’end2
denote th e num ber o f non-uniform grid triangles sharing th e node at rend!- Fmid and
rend2- resp ectively, and will typically be equal to unity.
A fter assem bling (4.19) into (4.18) we find that th e elem en ts of the forward
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transform ation m atrix are given by
9v
j
5Z X^^S/CrendJ
endl k=i (=i
^
*^mid 3
(Tmid)
. » rr~ '
^
k= 1 (=l
:^endj3 ^
3
1
I
+ 9 y d 5 Z y~! £iJ>S/(rend2) f
jt=i *=i
in which
(d-20)
/
1 j = J(
eiji =
0
otherw ise
and th e global indices i and j correspond to the ith uniform grid edge and the j t h nonuniform grid edge. T h e subscript jr is the global index used in num bering the nonuniform grid edges, whereas the subscript f. ( = 1. 2 or 3) is the local edge index used
in th e definition of the basis functions S*. W e remark th at the exp licit com p utation
and storage of the transform ation m atrix elem en ts results in a substantial increase in
efficiency because it avoids th e usual assem bly process during each iteration step and
that th e proposed overlay sch em e allow s the analysis o f large non-rectangular patch
arrays because storage of fully populated BI sy stem m atrix is avoided.
T he user
needs to on ly provide an additional data file which flags th e uniform grid edges lying
on a PE C elem ent and this is an im portant user-oriented feature o f the form ulation.
Follow ing the sam e procedure, we can obtain the expression for the entries of the
backward transform ation m atrix.
It should be noted th at assum ing each uniform
grid edge traverses three or less non-uniform grid triangles, the non-zero entries in
each row o f [TV] will be 9 or less. However, th ey can reach a m axim um of IS if the
m idpoint and endpoints reside on an edge of th e non-uniform grid. T he m axim um
non-zero entries in each row of [7g] will be 15. but the typ ical num ber is much less.
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76
4.4
R esu lts
Figure 4.6 show s a cavity-b ack ed 2 x 2 p atch array, where each patch is a rig h tangled triangle.
S in ce this geom etry is a d ap tab le to a uniform m esh w ith rig h t-
angled triangles, it is used to verify our proposed F E -B I technique incorporated
w ith th e B iC G -F F T system solver. T h e d evelop ed F E -B I code with the B iC G -F F T
is first com pared w ith th e original version o f th e hybrid F E -B I technique described
in chapter 3. A s show n in fig. 4.7, the m o n o sta tic radar cross section (RCS) pattern
over the space 0 < 0 < 90° at th e 6 = 0 plane agrees very well with that com p uted
using the regular BiC G solver.
It is also in form ative to com pare the sca ttered fields by the sam e cavity-backed
structure w ithout th e patch array to find th e con trib u tion o f th e array to scattering.
Figure 4.S show s th e m on ostatic RCS patterns by th e aperture with the absence of
the patch array. A gain, th e com p utations were ob ta in ed using both the regular BiCG
and B iC G -F F T versions of th e F E -B I m eth o d s. As can be seen, the level o f th e
scattered field at th e norm al incidence reaches ab ove zero in d B /A 2 with the presence
of the patch array, w hereas the scattering by th e aperture at the sam e incidence is
about 23 d B /A 2 lower.
To varifv th e overlaying schem e for nonrectangular geom etry, we evaluated a
bistatic RCS sca tterin g as shown in fig. 4.9 by a cavity-backed circular patch an­
tenna. In this case, th e dielectric fillings o f er = 4 and er = 10 inside the cavity were
used, respectively, and th e results obtained using th e regular BiCG F E -B I m ethod
are com pared w ith th ose com puted using th e B iC G -F F T w ith the overlaying trans­
form ations.
It is observed that th e agreem ent is q u ite satisfactory in scattering
analysis.
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Patch Array
Figure 4.6: Illustration o f th e cavity-b ack ed triangular patch array.
For radiation analysis (e.g . input im pedan ce) where an accurate prediction of
th e n ea r-zo n e fields m ay be required, th e accuracy of the overlaying sch em e can be
sign ifican tly enhanced by considering th e tria l-testin g elem en t's interactions in the
boundary integral. Specifically, it is suggested to separate the interactions betw een
th e closed -region elem en ts from th e far zone weak couplings.
T h e strong c lo se -
region couplings are treated using th e norm al m ethod of m om ents, whereas th e weak
coulings are com puted using the fast algorithm . T his approach has been reported
(see e .g . [47]). and once com bined w ith the overlaying schem e, it can be used to
control th e accuracy of th e F E -B I technique in an adaptive m anner.
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7S
10.
5a
s
E
S
£
V
at
■■■■I............ r l........ i............. i
i .........
a
-10.
J
-20.
-30.
---------- Sigm aJIheti: with no FFT
-40,
----------S lpni P h r with no FFT
B0
•ee -5a
a.
S ljn a .lh e t* : with FFT
▼
Slgnn_Phi; with FFT
•60.
£
S3
-70.
-80.
•...........i—
10.
... i ............1............. 1............. 1............. 1- .......1....... ...1............
20.
30.
40.
50.
60.
70.
80.
90.
F ig u r e -1.7: Com parisons o f th e m on ostatic radar cross section scattering by a 2 x 2
triangular patch array show n in fig. 4.6. T he reults were com p uted using
th e regular B iC G F E -B I tech n iq u e described in chapter 3 and using th e
B iC G -F F T proposed in th is chapter.
-
10.0
-
20.0
-30.0
CQ
TJ
e
-«0 .0
this method (H-pol)
this method (E-poO
O
-50.0
reference (H-pol)
reference (H-pol)
-600
-700
-800
0 00
1000 20.00 30.00 40.00 50 00 60.00 70.00 8 0 (X) “imki
Figure 4.S: Com parisons o f the m on ostatic radar cross section scattering by an em p ty
aperture with the sam e cavity size and dielectric filling (er= l ) as the
structure shown in fig. 4.6. A gain, th e results were calculated using
th e regular BiCG F E -B I techn iqu e described in chapter 3 and using the
B iC G -F F T proposed in th is chapter.
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79
RCS Comparison of Transform With No Transform
o.
7TTTTr" 'l
............i............ i............ i........... 1......■ " " |'" rTTTTT
-8.
<<
3
09
rz
c
iu
3
cu
00
U
05
a
V!
-16.
ii'i
......
-24.
v
i'i
t
i
|
A*
-
7 v ’
* V*7
-32.
AX
:
-40. -48. -
---------
With No Transform (permittivity=10)
.......... -
With No Transform (permittivity=4)
^ \\ ■
irV
-56.
-64.
S
*
With Transform (permittivity=10)
7
With Transform (permittivity=4)
-72.
........... I............ I............ I............I............ I........... 1........... 1.
-80.
0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
6
Figure 4.9: B ista tic RCS scattering by a crcular patch antenna m o d eled using the
regular BiCG F E -B I and th e B iC G -F F T algorithm w ith overlaying
transform .
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CHAPTER V
Efficient F inite Elem ent Subsystem — II
5.1
Introduction
As dem onstrated in th e previous two chapters (also reported in [1 3 ,2 9 .4 8 ]), a
hybrid finite elem en t-b ou n d ary integral technique [48.13] can be em p loyed for char­
acterizing conform al antennas o f arbitrary shape [29]. Indeed, p lan ar/n on -p lan ar.
rectan gu lar/n on -rectan gu lar design s, ring slot or spiral slot antennas w ith probe,
coax cable or m icrostrip line feeds can be sim ulated w ith th is form ulation.
This
is because of th e geom etrical ad ap tab ility o f tetrahedral elem en ts used for the im ­
plem en tation . However, in practice, certain configurations require extrem ely high
sam p ling rates due to the presence of fine geom etrical d e ta ils. A m on g them are a
variety o f slot antennas (spirals, rings, slot spirals, cross slo ts, log-p eriod ic slots,
e tc .), w here the slot w idth is m uch sm aller than the o th er dim en sion s (cavity diam ­
eter or in ter-d istan ce o f slo ts).
In these cases, the m esh is e x tr e m ely dense (with
over 50. 100 or even higher sam p les per w avelength), w hereas typ ical discretizations
involve only 10-20 elem en ts per w avelength. This den se sam p lin g rate is especially
severe for 3 -D tetrahedral m esh es, where the geom etrical d etails usu ally distort the
tetrahedrals. T h e num erical sy ste m assem bled from this ty p e o f m esh often leads to
a large system condition due to th e degraded mesh quality. A lso, m esh generation is
SO
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81
Ground Plane
xl)in s,ot
Figure 5.1: G eom etry of c a v ity -b a ck ed m icrostrip antennas
tedious and th e solu tion C PU tim e is u n accep tab lv large.
In this chapter, we propose a finite elem en t-b o u n d a ry integral form ulation using
ed ge-b ased triangular prism elem en ts. It can be show n that this elem en t choice is
ideally suited for planar antenna configurations con tain in g spirals, circular and trian­
gular slots. A m ong m any advantages o f th e p rism atic elem en ts, th e m ost im portant
is the sim p licity o f m esh generation.
A lso, m uch sm aller num ber of unknow ns is
required for an accurate and efficient m odeling o f com p lex geom etries. B elow , we
begin by first ou tlin in g the finite elem en t-b o u n d a ry integral (F E -B I) form ulation for
slot antenna m odeling. A new, p h ysically m ean ingful, set of ed ge-b ased functions
for prism s is then presented to gen erate th e discrete system of equation s. Finally,
th e applications o f th e proposed hybrid F E -B I m eth od to various antenna radiation
and scattering problem s are given to conclu de th e chapter.
5.2
H ybrid F E -B I Form ulation
Consider th e cavity-backed slot antenna show n in fig. 5.1 where th e c a v ity is
recessed in a ground plane. To solve for th e E -field inside and on the aperture o f the
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cavity, a standard approach is to e x trem ize th e functional (3 .1 ) which, for radiation
and scattering problem s, m ay be generalized as
F (E )
=
+ JJf E• (jkaZ„J‘
+
dV
j k QZo 1 1
E • (H x h ) d S
J J So+Sf
(5.1)
where er and Jir denote th e rela tiv e tensor con stitu tive param eters of th e cavity
m ed ium , Z q and ko are th e free sp ace im pedance and propagation constan t, respec­
tively. So represents th e aperture (or slots) excluding th e m etallic portions and 5'/
denotes the junction op ening to th e guided feeding structures. A lso. Vj is th e volum e
occupied by the source(s) ( J ‘ or M ') and H is the corresponding m agnetic field on
So and S j whose outer norm al is given by h. As before, th e exp licit know ledge o f H
in (5.1) is required over th e surface So and 5 / (also referred to as m esh truncation
surfaces) for a unique solu tion o f E . Specifically, the m agn etic field H over So m ay be
replaced in term s o f E via a boundary integral (B I) or absorbing boundary condition
(A B C ), whereas H on S f is d eterm in ed on the basis of th e given feeding structure.
T h is version of th e functional as com pared to (3.1) allows an easy inclusion o f various
feed m odels, such as aperture cou p led slo t, coax cable, etc. (see chapter
6
for d etails).
In this chapter since we con cen trate on im proving the FEM efficiency, therefore the
boundary integral m ethod w ill be em p loyed as in chapter 3. It will be seen that this
im plem entation indeed m eets th e accuracy need w ithout ex tra C PU burden. In the
con text of the F E -B I. H is represented by the integral
H = H s° + 2 j k 0Yo J J s G „(r, r') • (£ x E (r ')) d s ’
(5.2)
where G is the electric dyadic G reen's function o f the first kind such that h x G = 0 is
satisfied on the (planar, spherical or cylindrical) m etallic platform (refer to chapter
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1
).
S3
For th e antenna problem shown in fig. -5.1 where th e platform is a planar ground
plane. G becom es the half space dyadic G reen's function
1
/
\
e - J > o |r - r '|
G=(i+r v) 3 ^ -
,5 -3 >
w ith r and r' being the observation and integration points, resp ectively, and I =
x x + y y -f ££ is the unit dyad. In con n ection w ith our problem , i.e. th a t o f a cavity
recessed in a ground plane,
is equal to th e sum o f the incident and reflected
H 5°
fields for scattering com p utations, or zero for antenna param eter evalu ation s.
To d iscretize the functional (5 .1 ). we choose to su b d ivid e th e volu m e region using
prism atic elem en ts as
shown in fig. 5.2 and fig. 5.3. T h e field in each o f th e prism s
can be approxim ated using the linear ed g e-b a sed expansion [49-51]
9
e
‘ = ^ £ ;
1=1
v
; = [v i J -{E '} .
(5.4)
where [V ]e = [{V^.}. {V],}, {V I}], and { E e} = { £ { . E \ , . . . . £ ’| } r . T h e vectors { Vu}. u x . y . z . are o f dim ension m = 9 and th ey sim p ly represent the x . y . z com p on en ts of
V j associated with the j t h edge o f th e eth elem en t. Since V e- are chosen to be e d g e based functions, the unknown coefficients E
represent the average field along the
Jth edge of th e eth elem ent. A corresponding representation for th e ap ertu re fields
is
E(r) = ^ £ ? s ; ( r ) = [ 5 1 f { £ * } .
(5.5)
1=1
where [S']s = [5r . 5 y], and V ( r ) reduces to S 5 (r) when th e position vector is on the
slot.
To generate the discrete sy stem for EJ, (5.4) and (5.5) are su b stitu ted into (5.1)
and subsequently F ( E ) is differentiated w ith respect to each unknown E j . W ith the
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Figure 5.2: Illustration of tessellation using prisms
A
Z
©
Z=Zc
i=1,2,3
6
j=4,5,6
k=7,8,9
Figure 5.3: Right angled prism
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So
understanding that th e surface field coefficients E* are a subset o f E j , we obtain
{If ! =
+E ib*h£'}+!>''}+E{i’}=o
e=l
e=l
e=l
s
(s-6)
where th e sum s are over th e to ta l num ber o f volum e or surface e le m e n ts. In this, the
m atrix elem ents are given by
A tJ
=
/ / / v { V x V i . r - V x V J- l f c g V . f P . v } *
(5.7)
A'f
=
jkoZ o J J J
(5.8)
Bh
=
Vi -[j' + V x ^ - V x
M ‘] d v
-J ^ J J ^ k Z S 'iW -S W G o ir.^ d sd s'
+ 2 J J s J J s> [V x S?(r)]_.[V' x S ‘ (r')].-C 0 (r. r') ds ds'
L\
=
2 j k 0Z0 J f s S \ ■( H 1 x =) d s
where th e subscript z in
(5.9)
(5.10)
d en otes to take the z com p onent. It is noted that L\
is rem oved in case of radiation problem s and that the sam e holds for K f w hen the
scattering problem is considered.
5.3
Edge—B ased P rism atic Elem ents
Consider th e right angled prism shown in fig. 5.3 whose vertical (z-d ir e c te d ) sides
are parallel (righ t-an gled prism ). We now design two geom etric q u a n tities as
hi = i t ■z x (r - r,-).
S ■=
(5.11)
where r, denotes th e location o f th e ith node, e, is th e unit vector along the ith
triangular edge op p osite to th e ith node. /; denotes th e length o f this ed ge and r
is any position vector term in ated inside the triangle. One way to obtain an ed g ebased field representation for th e prism is to utilize th e nodal basis functions [52]
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86
and then apply the procedure discussed in [4 9 .5 3 ,5 4 ]. However, an alternative and
m ore p h y sically m eaningful approach can b e em p loyed for the construction o f the
edge elem en ts. Referring to fig. 5.2, it is ev id en t th a t if r is in th e x - y plane, then
5 f e in (5.11) gives th e area of another triangle 12'3' such that th e len gths of edges
joining th e nodes 2 — 3 and 2' — 3' are equal. W ith this definition o f r. we define a
vector a vector
S. = - ^ 7 ^ x ( r - r.)
where e,- • S, =
S‘
(5.12)
T hat is. the vector com p on en t along e,- has a m agnitude which
is equal to th e ratio of the areas of th e triangle 12'3' to that o f 123.
We observe
that (5.12) is sim p ly the ed ge-b ased exp an sion for the triangular elem en ts [32] and
is the appropriate expansion to be used in (5 .5 ). T h e corresponding volum etric basis
functions can be obtained by in sp ection, viz.
V‘
= ~r ^
Vj
= —
V fc
==Q
-S .
~ ~ 1 S,
i = 1, 2. 3
j = 4. 5. 6
(5.13)
A = 7. S. 9
where Q is the triangle sim plex coordinate a ssociated w ith the Arth prism vertex at
{xic-Uk)- A s illustrated in fig. 5.3. rc and h = A z represent the offset coordinate and
the prism h eight, respectively. W hen (5.13) are su b stitu ted into (5 .7 ). the resulting
integrals can be evaluated in closed form as given in th e A p p en d ix C. However,
the integrals resulting from the su b stitu tion o f (5.12) into (5.9) m ust be carried out
num erically, excep t the self-cells which can be perform ed an alytically as discussed
by W ilton [55].
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87
5.4
A pplications
T h in slot an ten n a structures have been treated using th e above form ulated F E -B I
techn iqu e and certain m odeling results w ill be presented in this section to dem on­
strate th e valid ity and capability o f th e approach.
Radiation and scattering by an Annular Slot:
To evalu ate th e accuracy and efficiency of th e prism atic m esh and th e afore­
m en tioned im p lem en tation , we first consider th e analysis o f the narrow annular slot
(0.75cm w ide) show n in fig. 5.4. T h e slot is backed by a m etallic circular cavity 24.7
cm in d iam eter and 3 cm deep. T h e F E -B I m eth od is q u ite attractive for this ge­
om etry becau se th e slot is very narrow and m ost of th e com p utational requirem ents
are sh ifted on th e finite elem ent portion of th e sy stem . T h e calculation shown in
fig. 5.5 were carried out using th e prism atic and tetrahedral elem en ts [29]. As seen,
th ey overlay each other. However, o n ly 1024 prism s were needed for m odeling the
cavity, w hereas th e num ber of th e tetrahedral elem en ts for this hom ogeneously filled
cavity were 2898 for acceptable elem en t d istortion. If a m u lti-layered structure was
considered, or a sim ilar system condition was used as a criterion for m esh generat ion,
then m uch m ore tetrahedrals than prism s would be needed for m odeling such a struc­
ture. M oreover, th e prism atic m esh is trivially generated given the slot ou tlin e. In
contrast, su b stan tial tim e investm ent is required for generating and p ost-p rocessin g
th e tetrahedral m esh.
Frequency Selective Surfaces:
Frequency selectiv e surfaces (F SS) structures [56.57] are arrays o f tigh tly packed
periodic e lem en ts which are typ ically sandw iched betw een dielectric layers.
T he
periodic elem en ts m ay be of printed form or slot configurations designed to resonate
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88
prbbe
a=12.35 cm
b=0.75 cm
p0= 7.7 cm
0. 7 <f <1 GHz
d=3 cm 11
e^r1-35
Figure 5.4: G eom etry of th e annular slot an ten n a backed by a cavity 23.7 cm in
diam eter and 3 cm deep
at specific frequencies. As such, they are p en etrab le around the elem ent resonances
and b ecom e com p letely reflecting at other frequencies. To m eet bandw idth design
sp ecification s, stacked elem ent arrays m ay be used in conjunction w ith dielectric
layer loading.
Here we shall consider the analysis o f FSS structures (w ith slot elem en ts) via
th e F E -B I m ethod. B ecause of the fine geom etrical detail associated w ith th e FSS
surface, th e finite elem ent m ethod has yet to be applied for th e characterization of
FSS structures, but use of prism atic elem en ts m akes this a much easier task.
Of
particular interest in FSS design is th e d eterm in ation of th e transm ission coefficient
as a function o f frequency, and since th e array is periodic, it suffices to consider a
sin gle cell o f th e FSS. For com puting th e tran sm ission coefficient T. th e periodic cell
is placed in a cavity as shown in fig. 5.6 and th e structure is excited by a plane wave
im pinging at norm al incidence. A ssum ing th at near resonance the wave tran sm itted
through th e FSS screen will retain its TE M character, th e transm ission line concept
can be used to find the scattered field
Es =
aT2
1 — ct R
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89
-1 0
-20
o
Prism Elements
Tetrahedral Elements
a: -40
-60
-70
-80
40
100
theta (degree)
-1 0
-6 0
-7 0
-8 0 '
-1 0 0
'
-80
'
-6 0
'
-4 0
l-2 0
40
100
theta (degree)
Figure 5.5: Scattering: B istatic (co -p o l) RCS patterns com puted using th e tetrah e­
dral F E -B I code and the prism atic F E -B I code. T he norm ally incident
plane wave is polarized along th e 6 = 0 plane and the observation cut
is perpendicular to that plane. R adiation: X -p o l and C o -p o l radiation
patterns in the o = 0 plane from th e annular slot antenna show n in
fig. 5.4. T he solid lines are com p u ted using the tetrahedral F E -B I code
whereas the dotted lines are co m p u ted using th e prism atic F E -B I code.
T h e excitation probe is placed at th e point (y = 0 ) marked in fig. 5.4.
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90
f
^
^ \0 .5 c m
0.391cm
0.061cm
i
e8
radome cover
(on 8=4.5 substrate)
0.0762cm
0.0762cm
metal backwall
absorber
Figure 5.6: Illustration o f the setup for com p u tin g th e FSS transm ission coefficient
U pper figure: periodic elem en t (top view ): Lower figure: periodic elem ent
in cavity (cross-sectional view )
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91
w here T is th e transm ission coefficient o f th e FSS, R = I — T and a is th e reflec­
tion coefficient associated w ith the c a v ity base. To reduce th e m ultiple interactions
w ith in th e cavity, it is appropriate to term in ate th e cavity w ith som e absorber, thus
reducing the value of a to less than 0.1. S in ce R is also sm all near resonance, a good
approxim ation for T is
Es
Tjb =
10
log
a
and upon considering the next higher order cavity interactions, we have
T<ib ~ T j g +
10
log [ l - a ( l - r (0)) | .
A m ore direct and traditional com p u tation o f T,i b would involve th e placem ent of
th e FSS elem en t in a thick slot [58].
H owever, this requires enforcem ent of the
boundary integral over the entire lower surface of the slot, leading to a m uch m ore
com p u ta tio n a lly intensive im p lem entation.
T h e above FSS m odeling approach was applied for a characterization of single
layer and m u lti-layer FSS structures. In both cases, the periodic elem en t was a slot
configuration. T h e geom etry o f the sin gle layer periodic elem ent is show n in fig. 5.6
and con sists o f a planar slot array on a d ielectric layer 0.0762 cm thick and having
er = 4.5. T h e F E -B I calculation using p rism atic elem en ts is given in fig. 5.7. Clearly,
our calculations are in good agreem ent w ith th e m easurem ents and d a ta based on
th e m ore traditional m ethod of m om ents [59,60].
T h e geom etry of the m ultilayer radom e considered in our stu d y is given in fig. 5.8.
T h e total thickness of the FSS was 6.3072 cm and is com prised o f tw o slot arrays (of
th e sam e geom etry) sandw iched w ithin th e dielectric layers. For m od elin g purpose,
a 1.54cm thick absorber is placed below th e FSS as shown in fig 5.8.
From the
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92
-5
-10
\
0 061cn
.
-2 0
-2 5
Frequency (GHz)
Figure 5.7: C alcu lation s and com parisons of transm ission through th e FSS structure
show n in fig. 5.6
calculated results, it is seen that th e results generated b y th e F E -B I m ethod are in
good agreem ent w ith th e m easurem ents.
Radiation P roperty study of Conformal Slot Spiral Antenna:
Consider a ty p ica l A rchem idean slot-sp iral an tenna show n in fig. 5.9. T his an­
tenna is built on a d o u b le-sid ed PC B with its tw o arm s follow ing th e expression:
r = a d + i3. w here a = 0.1333cm and j3 = 2.S595cm . O ne arm can be determ ined
from th e other by rotating 180° counterclockw isely. It is noted that this structure
differs from th e con ven tion al design in that th e central portion o f th e spiral is not
fabricated. T h e reasoning for it relies on the facts that th e antenna is designed with
a bandw idth less th an 30%. and that the central portion usually requires a careful
fabrication becau se o f the geom etric details, and still th a t th e central space m ay be
used for possibly com p lex feed network. One o f our goal is to stu d y th e effect of this
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93
slot array placed
60m ils below top surface o f 90m ils layer
slot array at 30mils
below iod o f 90mils
•90mils
85cm
90m ils
absorber, E
6 .4 5 c m
-10
Ul I rjn
—15 -
*
-20
m easured
calculated
-25
-3 0 '
0.4
0.6
0.8
1.4
1.8
Figure 5.S: U pper figure: geom etry o f the m ultilayer frequency se le c tiv e surface
(F SS) used for m odeling: lower figure: m easured and calcu lated trans­
m ission coefficient through th e FSS structure
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94
spiral shape on its perform ance.
A benchm ark test m odel is design ated to operate from 1 IS M Hz to 157 MHz.
to replace th e conventional protruding blade antenna.
T h e siz e however is much
com pact w ith its conform ality property. Our sim ulation m odel is scaled by
1
/S to
operate at 944M H z to 1256M Hz w ith th e center frequency 1100M H z. T h e values of
q and ,3 above were determ ined based on this frequency band and also th e num ber
of turns (4.5). T h e cavity is filled w ith a dielectric slab (er = 2.2) o f 0.3 cm depth,
corresponding to approxim ately
0 .0 1 1
free space w avelength at th e center frequency.
T he antenna's d irectivity is analyzed from the radiated pattern at lower, center and
higher frequencies and the results are tab u lated in Table 5.1.
Figure 5.10 - 5.12 show th e radiation patterns for frequency 944. 1100 and 1256
M Hz. respectively. T h e Eg and E# at th e principle plane o = 90° are p lotted. It
is understood th at when th e frequency varies, the active region travels along the
slot spiral. T hus the principle plane m ay not be coincident w ith the E -p lan e. (In
fact, the E -plane is not clearly defined in this case.) T he op tim u m axial ratio for
the three cases are tabulated also in T able 5.1. and it shows th at the spiral shape
design really plays an im portant role to insure a good quality radiation pattern. At
both center and lower frequencies, less than 3 dB A R has been achieved. W hen the
frequency increases, th e active region m oves inwards to the center and becom es closer
to the feeds where the EM fields exh ib it a com paratively strong profile. T h e radiated
pattern therefore is m ost likely affected and this explains why th e AR increases at
the high frequency. T h e A R deterioration can be avoided by adding a couple of spiral
turns inside. It is seen, nevertheless, th at a CP m ode can be achieved w ithin the
entire design ated bandw idth and w ith a w ide azim uthal angle (as wide as 60° in the
optim um case). In practice, we n otice that absorbing m aterials m ay be needed to
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95
regulate th e m agnetic currents at th e b eginning or ends o f the slot spiral, especially
when th e num ber o f turns is m inim ized.
Frequency (G H z)
0.944
7.22
1 .1 0 0
1.256
G ain (dB )
6 .6 6
5.23
A xial R atio (dB )
2.7
1 .0
3
Table 5.1: Com parisons o f gain and axial ratio at different op eratin g frequencies
5.5
Concluding Rem arks
A hybrid finite elem en t-b ou n d ary integral (F E -B I) form ulation was presented
for m odeling narrow slots in m etal backed cavities. P rism atic elem en ts were used
in connection with th e F E -B I im p lem en tation , and in contrast to th e tetrahedral
elem en ts, these offer several advantages. A m ong them , low sam p ling rates are needed
for generating m eshes and th e m esh generation process is su b stan tially sim plified.
O ther advantages o f the prism atic elem en ts over the tetrahedral elem en ts include
b etter system conditions and faster p r e /p o st data processing.
T h e exp licit expressions for F E -B I im plem entation of prism atic elem en ts were
tab ulated and num erical results for slot antennas and frequency selective surfaces
were presented to dem onstrate the validity and capability of th e technique.
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96
Figure 5.9: Illustration o f a typical 2-arm slot-sp iral design
frequency=l.1G H z
- - 0 - - - ..
-1 0
/
-2 0
/
i
-30
L
solid line: E_phi, d ashed line: E J h e ta
Figure 5.10: R adiation P attern at f = l.l G H z (center frequency d esign ). A good axial
ratio is achieved up to 60° degree.
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97
frequency=0.944GHz
-1 0
solid Gne: E_phi. dashed line: E Jh e ta
Figure 5.11: R ad iation P attern at f= 0.944G H z (lower end o f frequency range). It
can be seen that th e axial ratio o f the pattern becom es larger compared
to th a t at th e center frequency, but still rem ains w ithin 3dB for a wide
angle range. T h is indicates that the num ber o f th e outer turns in the
spiral contour design is m ost likely sufficient.
frequency=1.256 GHz
-to
/
-2 0
/
i
solid line: E_phi, dashed tine: E.theta
Figure 5.12: R ad iation P attern at f= 1.256G H z (higher en d of frequency range). It
can be seen that th e axial ratio of the pattern is deteriorated compared
to th o se at th e center frequency and lower frequency. T his certainly
show s th at th e num ber of inner loops still needs to be increased to
insure a good q u ality pattern.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
CHAPTER VI
A ntenna Feed M odeling
For scatterin g problem s where th e plane w ave incidence is usually considered as
the 'source', th e rig h t-h a n d -sid e excitation has been exp licitly expressed in (3.7) and
(5.10) and w ill not be discussed further. H owever, for antenna im pedance evalu ation s,
we have proposed and reported several feeding schem es [6 L] associated w ith various
practical feed designs for m icrostrip antennas. S om e o f these are discussed below .
6.1
6
.1 .1
P rob e Feed
S i m p le P r o b e F e e d
For thin substrates th e coaxial cable feed m ay be sim plified as a thin current
filament of len gth I carrying an electric current I I .
Since this filam ent is located
inside the cavity, th e first term o f th e integral in (3.7) needs to be considered for this
m odel. Specifically, the ith (global) entry o f th e excitation vector
I<i = j k o Z 0I I - V,-(r).
becom es
i = j \ . J 2 • ••-.jm
where r is th e location o f th e filam ent, m is th e num ber of (n o n -m eta llic) elem en t
edges and j m is th e global edge num bering in d ex.
In general, m such entries are
associated w ith m elem ent edges, and thus th e index i goes from j \ up to j m. T h is
9S
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
99
expression can be further reduced to A',- = j k o Z o I I. provided that th e ith edge is
coincident w ith th e current filam ent.
6 .1 .2
V o lt a g e G a p F e e d
T his ex cita tio n is also referred to as a gap generator and am ounts to sp ecifyin g a
prior i th e electric voltage V across the op en in g o f th e coax cable or any other gap.
Since V = E • d, where d is a vector w hose m agn itu d e is th e gap w idth, and E the
V
electric field across th e gap. we have that A, = - -----—. where cosOi is equal to
dcosVi
1
if the
fth edge is parallel to d. Num erically, th is gap voltage m odel can be realized by first
settin g th e diagonal term .4,-,- equal to u n ity and th e off-diagonal term s Aij {i ^ j )
to zero.
For the righ t-h an d -sid e vector, on ly th e entry corresponding to th e ith
(global) edge across the gap is specified and set equal to the value A, whereas all
other entries associated w ith edges not in th e gap are set to zero.
6.2
A p erture-coup led M icrostrip M odel
As show n in fig. 6.1. when the m icrostrip antenna is fed w ith a m icrostrip line
network underneath the ground plane (c a v ity ’s base) via a coupling aperture, special
treatm ent o f th e feed structure m ust be considered in th e FEM form ulation. T his
is because th e m icrostrip line is usually designed to have different size and shape as
com pared to th e c a v ity ’s geom etries. H ence, th e conventional sim ulation of treatin g
the entire 3-D dom ain using a single ty p e o f elem en ts is not efficient or appropriate
for this feed.
Referring to fig. 6.1. it is appropriate to separate th e com p utational dom ains
because o f th e sm all elem ent size required in m odeling th e guided feed structure.
O ne difficulty encountered when this d ecom p osition is im plem ented is how to m odel
the coupling through the aperture.
As an exam p le let us consider a rectangular
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
100
aperture which has been e x ten siv ely em ployed in practice. T h e ca v ity fields m ay be
discretized using tetrahedral elem en ts, whereas in th e m icrostrip line region rectan­
gular bricks are th e b est can d id ate since th e feed stru ctu re is rectangular in shape
and the substrate is o f con stan t thickness. A lthough b oth typ es o f elem en ts em p loy
Antenna Elments
Truncation Plane
Coupling Aperture
Figure 6 . 1 : C ro ss-sectio n o f an aperture coupled patch an ten n a, show ing the cavity
region I and th e m icrostrip line region II for tw o different FEM com p u­
tation d om ain s.
(a)
(b)
(c)
Figure 6.2: slot and its d iscretization (a) slot aperture; (b ) typ ical m esh from cavity
region; (c) uniform m esh from m icrostrip line region.
edge-b ased field exp an sion s, th e m eshes across th e com m on area (coupling aperture)
are different, and th is causes difficulty in enforcing field con tin u ity across the slot
aperture.
H owever, sin ce th e aperture is very narrow, a 'static' field distribution
m ay be assum ed at any given frequency. Therefore, the potential concept m ay be
again applied to relate th e fields below and above th e aperture.
Specifically, the
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
101
'eq u i-p o ten tia F continu ity condition is enforced, and to proceed to do so. let us first
classify th e slot edges as follows
T e t r a h e d r a l M e s h ( C a v it y R e g io n I ):
• Ejl
j = 1 .2 .3
vertical edges
• Ej2
j=
diagonal edges
1 .2 .3 ....
B r ic k M e s h ( F e e d R e g io n I I ):
• Ej
j = 1 .2 .3 _____
vertical edges
T h en th e 'eq u i-p o ten tia l' continu ity condition requires that
( 6 . 1)
( 6 .2 )
in which
whereas t and d are the lengths of th e vertical and diagonal ed ges, respectively.
That is. t is sim p ly th e w idth of th e narrow rectangular aperture. T h e coefficient
is equal to
± 1
depending on th e sign conventions associated w ith th e m eshes above
and below th e coupling aperture.
T h e co n n ectiv ity schem e for entirely different com p u tation al dom ains m ay be
exten d ed by generalizing this concept.
It is apparent that th is approach makes
the FEM im p lem en tation straightforward for different g e o m e tr y /siz e dom ains that
would be sign ificantly inefficient if only one ty p e of elem en ts w ere used for m odeling
the structure. In addition, the technique ensures a good sy stem condition since the
num ber of distorted elem en ts in the m esh are m inim ized.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
102
6.3
C oax Cable Feed
6.3.1
M otivation
T h e coax cable is w idely used as a feeding structure for m icrostrip or cavitybacked patch antennas because o f its sim p licity and low spurious radiation. Indeed,
abundant literature ex ists on th e theoretical and exp erim en tal in vestigation o f coax
cable feeds [62-64].
M ost of th ese papers present integral equation techn iqu es in
conjunction w ith th e pertinent G reen's functions. H owever, the G reen's function is
only available for a certain class o f geom etries, and this lim its th e app lication of
th e integral techniques to th ose geom etry designs.
A lso, th e form ulation m ust be
m odified and recoded for different antenna configurations corresponding to G reen's
function variations. To avoid th e com p lexity o f the G reen's function, we recently
proposed a hybrid finite elem ent - boundary integral approach [29] which is described
in chapter 3 and 4. For antenna radiation, it is observed that a sim ple probe m odel
w ith a constant current along th e inner conductor linking th e grounded base to the
antenna elem ent is straightforward and efficient. B ut th e probe feed is on ly valid
for thin substrates and this is consistent w ith th e M om ent M ethod (M M ) results.
To m odel an electrically thick su b strate, in this section a more so p h isticated feed
m odeling schem e is proposed in th e con text o f the finite elem ent m eth od (F E M )
using linear edge-based tetrahedral elem en ts. T h e form ulation of the entire hybrid
num erical system will be first described in th e presence o f the necessary functional
term for feedline. T h e proposed feed m odel is then presented on the basis o f a TEM
m ode ex cita tio n . M odel im provem ents are also discussed for th e case w hen th e TEM
assum ption at the cavity-cab le ju n ction does not hold.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
103
6 .3 .2
H y b r id F E —B I S y s t e m
T h e functional pertinent to th e radiation by a cavity-backed antenna w ith a coax
cable feed (as shown in fig.
F(E)
=
6.3) is given by
i///((V x E ).i(V x E )-^ E .E j
-
2
$ J J { E x z) - y
j j E x z) -
6 'o ( r .r ') c /5 ' | d S
- j k 0Zo J J (E x H ) - £ dS.
(6.3)
where V refers to th e c a v ity volum e and the surface S encom passes th e ca v ity aperture
exclud ing the portion o ccu p ied by the m etallic an ten n a elem ents: er and f.iT d enote,
respectively, th e relative p erm ittiv ity and perm eability; kQ is the free space wave
num ber, Z q is the free space intrinsic wave im p ed an ce, I is th e unit dyad, and
G'o(r. r') is the free sp ace G reen's function w ith r and r' den otin g th e observation
and integration points: th e surface C is the cross section of th e coax cab le at th e
ca v ity -ca b le ju n ction .
Follow ing th e standard discretization procedure [29]. we obtain a sy stem o f equa­
tions of th e form
£ {(*,]{£;}} +£ {[ey{£;}}+£ dFcg f H)
e=l
eSS
egC
=
i6--*)
1
where th e exp licit expressions for Aij and Bij m ay be found in [29]and th e functional
term Fc is th e surface integral on C in (6.3).
6 .3 .3
P ro p o sed C o a x F eed M o d el
To proceed w ith th e evalu ation of
Fc = - j k 0Z0 J J {E x H ) - £ dS.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(6.5)
104
a boundary constraint relating E to H is needed. To this en d . we assum e a TEM
m ode on C and consequently the fields w ith in th e c a v ity m ay be expressed as (see
fig. 6.4)
E = - ^ = ( l + r )ir ,
H = f(l-r )U
r
In
r
(6.6)
where t rc is th e relative p erm ittiv ity inside th e coax cable; T denotes th e reflection
coefficient m easured at c =
location.
0
and I0 is th e given input current source at the sam e
A lso, (r, 6. r) are th e polar coordin ates o f a point in th e cable with the
center at r = 0. To sim plify the analysis, w e introduce th e quantities
eo= 4 2 ^ ( 1 + r ,
A0 = A ( 1 _ D .
2
2 -y /tr c
(6.7)
"
H ence.
E = — r.
r
H = — 6.
r
( 6 .S)
and from (6.7) it follows
hQ = - ^ ^ - e
Zo
0
+
3
'i
.
(6.9)
which is the desired constraint at the cab le ju n ction in term s of th e new quantities
hQ and e0. N ote that e 0 and h0 are field coefficients as new unknowns in place of the
fields E and H . and it is therefore appropriate to rew rite Fc in term s o f th ese new
coefficients. To do so, we su b stitu te (6.7) and (6.9) into (6.5) and upon m aking use
o f the axisym m etric field property we ob tain
Fc = - 2 - j k 0Zoe0hQCl n { - ) .
a
( 6 . 10 )
where a and b are the radii o f th e inner and outer cable conductors. T he superscript
src stan d s to indicate that h0 is treated as a source term in th e extrem ization of the
functional.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
105
We choose th e linear ed ge-b ased tetrahedral elem en ts to d iscretize th e cavity
and th e corresponding m esh on th e cross section C is show n in fig. 6 .4 (b ). In this
form ulation, the field across th e p th edge. p = 1 . 2 ,..., N c { N c is the num ber o f cavity
m esh edges on C ). is set to a constan t as d ictated by the linear edge-based expansion
function inside the cavity.
1
H owever, th e cable TEM m odal fields ( 6 . 6 ) b ehave as
/ r and this m odeling in con sisten cy m akes it difficult to apply the tan gen tial field
con tinu ity condition at th e cable ju n ction ( i.e. over the aperture C ). To overcom e
th ese difficulties, we can relate th e fields across th e cable ju n ction by recognizing that
th e potential difference betw een th e inner and outer conductors m ust be th e sam e
as com puted by th e fields o f th e cavity or th ose in th e cable region. Specifically,
if th e Nph edge of th e cavity region borders and is also across the coax cable, from
( 6 . 6 )—( 6 .S) it then follows th a t the appropriate eq u i-p o ten tia l condition is
A V = Ei{b - a) = e0l n ~ .
a
i = N p(p = 1 . 2
Nc)-
(6.11)
w here A V denotes th e p oten tia l difference betw een the inner and outer surface o f the
cable. T his condition sim p ly provides a relation betw een the constant c a v ity edge
field and the coax cable m odal field. W hen used into the functional Fc- it introduces
th e excitation into th e hybrid finite elem en t system w ithout a need to ex te n d the
m esh inside th e cable or to em p loy a fictitious current probe. It rem ains to rew rite
Fc in term s o f
i.e. th e field value o f th e edges bordering th e cable and to do so
we su b stitu te (6.9) and (6.11) into (6.10). T h en upon taking into account all N c
cavity m esh edges on the cab le ju n ction , we obtain
* =
t
v.
a
)
E-
<*•*»
i= :V p
In this expression, rather th an representing the functional Fc in term s o f a sin gle edge
field, we m ade use of the average field across th e cable as com puted by th e to ta lity
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
106
o f the equal elem ent fields on th e cable's aperture (b ecau se of th e axisym m etric
property, all elem ents fields at th e ca b le?s aperture are eq u al).
T h e factor inside
th e curly brackets o f ( 6 . 1 2 ). w ith th e superscript src, functions as a source in the
extrem ization process. H ence, th e extrem ization o f (6.12) yields
dFc
m
1
•/
”7
/1
\
f
\ Z £ rc
b —a
=
=
UiEi-Vt,
i = iYp( p =
1 .2. ....Yc ).
(6.13)
where
(6-14)
K
=
j ^ k QZ0{b - a) I0.
(6-15)
We observe that the ‘con stan t ca v ity field' along each m esh edge at th e cable ju n c­
tion is just a fictitious field representation and its m eaningful physical interpretation
is governed by the e q u i-p o te n tia l constraint (6.11).
To proceed, we assem ble the
FEM system together w ith (6 .1 3 ). Specifically, each (7, is added to th e A c diagonal
entries of the finite elem ent m a trix which is associated w ith th e A c edges bordering
th e coax cable. A lso, th e e x c ita tio n colum n of the hybrid system is nullified every­
where except for th e A c entries w hich are set to VJ. O nce th e hybrid FE -B I system
is solved [29], the input a d m itta n c e at z = 0 is calculated from
H • r rclo
'2I q
1
-
( 6 - 16)
where Z c is the characteristic im p ed an ce o f the coax cable.
In the above feed m odel we assum ed the presence of o n ly th e d om in an t(T E M )
m od e at th e cavity-cable ju n ctio n , an assum ption which m ay not b e su itab le for
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
LOT
certain applications.
To overcom e this lim ita tio n , one approach is to ex te n d th e
m esh (say. a distance d) into th e cable. T h e eq u i-p o ten tia l condition will then be
applied at z = -d . where all higher order m odes vanish. T his schem e requires a m ore
su itab le expansion for the fields in the —d < z <
0
section to avoid th e com p lication
o f exten d in g th e tetrahedral m esh into th e cable and. thus, retain th e efficiency o f th e
eq u i-p o ten tia l feed m odel. Since in m ost cases th e antenna is operated in a frequency
range far below the c u t-o ff o f the first higher order m ode o f th e coax cable, the field
distribution near the ju n ction C will still be dom inated by th e fundam ental TE M
m ode. W ith this understanding, a possib le su itab le expansion for th e field in th e
coax cable (using shell elem en ts rather th an tetrahedrals) is
^
.N * (r .< k r )
e= E
Ei e;9
-r " ■
(6'17)
where q = r . 6 or z. i= 1 .2 ,3 or 4 and N ^ (r. o , ~) is th e shape function for each o f th e
12 edges (3 d irectio n sx 4 edges per direction ). T h ey are given by
N ’° =
A qiAq ^ ~
w ith qa .qb and qc representing r .
6
^
9c ~~ ^
°
^6 ' 1 8 ^
and z in cyclic rotation and correspondingly
qa.qb and qc represent th e param eters r. 6 and
A lso, i denotes the ed ge num ber
along each coordinate, and \ q a is the w idth of th e edge along the qa d irection . T h e
correspondence betw een the edge num bers and th e node pairs for each coord in ate(r. o
or r) is given in Table 6.1 along with th e definition o f the tilded param eters in ( 6 . IS).
W hen an axisym m etric field property is assum ed, th e num erator o f the expansion in
(6.17) reduces to the standard brick elem en t form at for the radial and z com p on en ts,
independent o f th e 6 variable. N ote also that th e particular property of th is exp an ­
sion is the introduction of the 1 /r factor, sim ulating the coaxial cable m ode. T h e
accuracy of (6.17) is dem onstrated in fig. 6.5. where we show that only 2-3 elem en ts
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
10S
T A B L E l______________________
o
I
Z
l
1
2
3
I
(1-2)
(1-3)
(8-7)
(5-6)
(1-1)
(5-8)
(6-7)
(2-3)
(1-5)
(2-6)
(3-7)
(1-8)
e
T
+
J.
-
r
O
o, + A o
01
01
01 + A O
rj
r,
+
+
Ar
Ar
+
r i
-
rI
ri + A r
ri
ri
r, + A r
-1-
4-
-
z
z\ *f A r
+ Ar
-I
•1
+ Ar
-1
-1
+
,
r
i
I
2
3
I
I
j 2
3
i
z
param eters
01
Ar
-1-1
q
node
pairs
o
coordinates
+
A i5
01
Ol
X
Table 6.1: T h e correspondence betw een th e ed ge num bers and the node pairs for each
coord in ate(r. 6 or z) along w ith th e defin ition of the tilded param eters in
(6 .1 8 ).
are needed along the radial direction for the accu rate prediction of th e dom inant
field d istrib u tion . W hen com pared to th e conven tion al linear tetrahedral elem en ts,
the efficiency o f this expansion is apparent (i.e .. m any m ore tetrahedrals are needed
to m odel th e sam e cable region).
6.3.4
R esults and Conclusion
To valid ate our proposed feed sim ulation, tw o circular patch antenna configura­
tions were used for calculation. O ne patch an tenna was of radius 1.3 cm printed
atop o f a circular cavity (rad ius=2.1 cm ) filled w ith a dielectric (er= 2 .9 ) m aterial
0.41 cm deep. For this patch, th e feed was placed 0.S cm from the cen ter and the
input im p ed an ce was m easured over the band 2 - 5 G H z. In fig.
6 .6
we com pare the
m easured input im pedance w ith data com p uted on th e basis o f the proposed equip otential feed m odel. Clearly, the results from m easurem en ts and the e q u i-p o ten tia l
m odel are in excellen t agreem ent whereas th e probe m odel yields su b stan tially inac­
curate results near resonance.
Figure 6.7 show s the com parison betw een m easurem en ts and calcu lation s for an­
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
109
oth er patch antenna whose input im p ed a n ce was m easured by A berle and Pozar [65].
T h is patch had a radius o f 2.0 cm and th e 0.21S cm thick substrate had er= 2 .3 3 and
a loss tangent tan<S=0.0012. T h e feed was located 0.7 cm from th e cen ter, and for
our F E -B I calculation the patch was placed in a circular cavity o f 2.44 cm in radius.
A s show n in fig. 6.7. the eq u i-p oten tial m odel is again in excellent agreem ent w ith
m easu rem en ts, as opposed to th e resu lts by th e probe m odel in [65].
In conclu sion, the presented e q u i-p o te n tia l feed m odel has been show n to be
e x tr e m ely accurate in m odeling coax feed structures. M oreover, its im p lem en tation
in th e con text o f a finite elem ent form ulation is very sim ple and as easy to im plem ent
as th e probe feed. It was also d em on strated how the proposed feed m od el can be
gen eralized to th e case of asym m etric feed structures where evanescent m odes m ay
be present.
6 .4
Conclusion
In d evelop ing numerical techniques for antennas, the feed network is on e o f chal­
len gin g problem s to solve in consideration o f accuracy, efficiency and sim p licity . T h is
is prim arily because the antenna feed in fabrication has certain in stru m en tal uncer­
ta in ties on one side. On the other, as aforem entioned, the accuracy o f num erical
resu lts is usually extrem ely sen sitive to th e feed m odel, th e feed lo ca tio n , sam p ling
rate around th e feed point, and so on.
T h e proposed numerical feeds in th is chapter resem ble th e practical sy stem s as
clo sely as possib le, and w ith a thorough consideration of their num erical im p lem en ­
ta tio n s. we realize that they can be used for m ostly encountered an ten n a problem s.
A s an addition to the group of feed m od els, we also developed a circuit m odal feed
w hich coincides with dom ain tru ncations. Since this m odel has to do w ith m icrow ave
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
110
z
i
Ground plane
Aperture
'
^
7
— - —-------
>
X
-*T"
_ Coax cable opening
(surface C)
'
' ------------
-« - -- "
Coax cable
Figure 6.3: Illustration o f a cavity-backed patch antenna with a coax cable feed.
° aVity
---------‘
patch
r4
«P
T
-
2b
2a
cavity-cable junction
(a)
T
(b)
F ig u r e 6.4: (a) Side view o f a cavity-backed an ten n a with a coax cable feed: (b)
Illustration of th e FEM m esh at th e cavity-cab le junction (th e field is set
to zero at th e center conductor surface).
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
1L1
Comparison of £-FiefcJ Along (1-Wavetengtti) Coax Cable
150
100
c
o
nc3
01
2
2
£o
til
*50
•100
-150
se g m e n t nu m b er
(a)
Comparison of E-Field Distribution Along R
6000
5500
34 500
5 3500
2000
1500
0.04
0.06
distance in cm from the center
(b)
0.08
Figure 6.5: Field distribution in a shorted coax cable as com puted by the finite ele­
m ent m ethod using the exp an sion ( 6 . IS). — : analytical: xxx: num erical,
(a) Field coefficient e 0 along th e len gth o f th e cable (leftm ost point is the
location of the short): (b) Field along th e radial coordinate calculated at
a distan ce A/4 from th e shorted term in ation .
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
112
180.
140.
[• a
this method
probe model
u
100.
3
60.0
w
C.
20.0
-
20.0
1.00
3.00
2.00
4.00
5.00
frequency (GHz)
(a)
180.
y
140.
100
this method
.
3
c.
60.0
20.0
- 20.0
-60.0
1.00
2.00
3.00
4.00
5.00
frequency (GHz)
(b)
Figure 6 .6 : Measured and calcu lated input im pedance for a cavity-backed circular
patch antenna having th e follow ing specifications: patch radius r= 13m m :
cavity radius R = 2 1 .1 m m : su b strate thickness t= 4 .1 m m : er= 2.4: and feed
location x /= 0 .S cm d ista n ce from center. R esults based on th e sim p le
probe m odel are also show n for com parison. Our m odeling retains th e
vertical wire connection to th e patch and uses the incom ing coaxial m od e
field for excitation , (a) R eal part: (b) Imaginary part.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
113
120
100
a 40
2.6
2.7
2.65
2.75
2.8
2.85
frequency (GHz)
(a)
100
•20
5 -40
2.55
Z6
2.65
2.7
2.75
2.85
2.9
frequency (GHz)
(b)
Figure 6.7: M easured and calcu lated input im pedance for a circular patch an­
tenna having th e follow ing specifications: patch radius r = 2 cm : su b strate
thickness d=0.'21S44cm : feed location from center j /= 0 .7 c m : er= 2.33:
fam $=0.00r2. [65]. — : m easurem ent: xxx: this m ethod: o o o: probe
m odel [65] (a) Real part: (b) Im aginary part.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
114
circuit, we shall leave th e top ic to the next chapter in conjunction w ith other related
su b jects.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
C H A P T E R V II
Circuit M odeling
M any designs o f m icrostrip antennas require certain feedlines to carry e lectrom ag­
netic signals from the source. F in ite elem en t m ethods are also su ited and applicab le
to these wave propagation problem s. T h is chapter is devoted to circuit m odeling.
A fter an overview of recent m esh term in ation techniques, we discuss a num erical
d e-em bedding process appropriate for th e finite elem en t analysis, and then turn to
th e topic o f m esh truncations for circuit sim u lation .
7.1
Introduction
One o f th e m ost im portant asp ects o f finite elem en t im plem entations is th e trun­
cation of the com putational volum e. An ideal truncation schem e m ust ensure that
outgoin g waves are not reflected backwards at th e m esh term ination surface, i.e. the
m esh truncation schem e m ust sim u late a surface which actually does not e x ist. To
d ate, a variety of non-reflecting or absorbing boundary conditions ( A B C s) have been
em ployed for truncating the com p utational volum e at som e distance from th e radi­
atin g or scattering surface, and applications to m icrow ave circuits and devices have
also been reported. T he A BCs are typ ically second or higher order boundary condi­
tions and are applied at the m esh term in ation surface to truncate the com p u tation al
115
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
116
volu m e as required by any P D E so lu tio n . A m ong them , a class o f A B C s is based on
th e o n e -w a y wave equation m eth od [66,67] and another is derived startin g with the
W ilcox Expansion [68.31]. A lso, higher order A B C s using H igdon’s [6 9 ,70]formulation and problem specific num erical A B C s have been successfully used, particularly
for tru ncating meshes in guided stru ctures [71]. There are several difficulties with
trad ition al A B C s. A m ong them is accuracy control, conform ality. ease o f parallelization and im plem entation difficulties w hen dealing with higher order A B C 's. Also, the
ap p lication s o f A B C ’s in m icrow ave circuit m odeling requires a p r io r i knowledge of
th e propagation constants which are ty p ica lly not available for high d en sity packages.
A n alternative to traditional A B C s is to em ploy an artificial absorber for mesh
tru n cation . Basically, instead o f an A B C , a thin layer o f absorbing m aterial is used
to tru n ca te the m esh, and the perform ance for a variety o f such absorbers have been
considered [72.73].
N evertheless, th e se lossy artificial absorbers (hom ogeneous or
not) still exhibit a non-zero reflection at incidence angles away from norm al. Re­
cently, though. Berenger [74] in trodu ced a new approach for m od elin g an artificial
absorber that is reflectionless at its interface for all incidence angles. In two dim en­
sion s, his approach requires th e sp littin g o f the field com p onents involving normal
(to th e boundary) derivatives and assign in g to each com ponent a different conduc­
tivity. In this manner an additional degree o f freedom is introduced that is chosen
to sim u la te a reflectionless m ed iu m at all incidence angles. Provided th e m edium
is lossy, th is property is m aintained for a finite thickness layer. B erenger refers to
th e la tter as a perfectly m atched layer(P M L ) and generalization of his idea to three
d im en sion s have already been considered [75.76]. Also, im p lem en tation s o f the ab­
sorber for truncating finite differen ce-tim e d om ain (F D T D ) solu tions has so far been
found highly successful. N everth eless, it should be noted that Berenger's PML does
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117
not satisfy M axwell equations and cannot be ea sily im plem ented in finite elem en t
(F E M ) solu tion.
A new anisotropic(un iaxial) artificial absorber [77] was introduced recently for
tru ncating FEM m eshes. T h is artificial absorber is also reflectionless at all in cidence
angles. B asically, by m aking appropriate choices for th e con stitu tive param eter ten ­
sors. th e m edium im pedance can be m ade independent of frequency, polarization,
and wave incidence angle. A PM L layer can then be constructed by introducing suf­
ficient loss in th e m aterial properties. T h e im p lem entation of this artificial absorber
for tru ncating finite elem ent m eshes is straightforw ard and. m oreover, th e absorber
is M axw ellian.
7.2
N um erical D e—em bedding
D e-em bedd ing presented here is a num erical process used to extract certain circuit
q u an tities. Specifically, we are interested in S-p aram eters for a uniform transm ission
line term in ated w ith any loads den otin g th e possible discontinuities, w hich m ay arise
from line-to-line or line-to-antenna couplings. T h e dom inant transm ission line m ode
is assum ed at and near a reference plane S ref in this discussion.
C onsider a transm ission lin e o f certain length as shown in fig. 7.1.
W ith an
appropriate shielding schem e, th e line is included in th e com putational dom ain. T he
full wave analysis provides th e E field d istribution anyw here including th e region
along th e line. O ne is therefore able to represent E field along the transm ission line
w ith respect to the locations to get
V{ ~) = Vie- ''' + Vre'l=
(7.1)
where V is proportional to th e m agn itu d e of E w ith V| being the incom ing and Vr
the reflected wave am plitude, c is m easured from th e reference plane S ref •
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7
is the
118
Microstrip Line
Shielded FEM Region
Substrate
Probes
Figure 7.1: Illustration o f a shielded m icrostrip line.
propagation constan t to be determ in ed and R = Vr/Vj is th e reflection coefficient,
or S n .
Since in (7.1)
7
. Vj and Vr are three independent quan tities that characterize the
wave propagation and reflection, to d eterm in e them one needs to specify th e field
values for V’(r ) at three points c _ . ~o and z + o f equal inter-distances
V(z.)
z+ —z0
along
V ( z 0)
V(z+ )
=
c 0 — c_
th e line. To sim plify the problem , we choose th e reference plane
center point such that
(7.2)
right at th e
r0 = 0 and z + = —c_ = d. G iven th e three field values from
FEM com p u tation s, it follows
To solve for
7
V( d)
= v;e - ^ + VTe ,d
(7.3)
V (0)
= Vi + Vr
(7.4)
V(-d)
= \'}e>d +
VTe~^d
(7.5)
. we first add (7.3) to (7.5) to get
( i; + Vr ) [ e ld + e -" rf) = V( d) + V ( - d )
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(7.6)
119
T h en e lim in a tin g V] + VT from (7.4) and (7 .6 ). we obtain
. L,
V{d) + V {- d )
W m —
cosh M = —
from w h ich
7
can be determ ined. T h e effective guided w avelength \ g and effective
d ielectric constan t ee/ / m ay then be calcu lated by
w ith (3 = I m { 7 } and A0 being the free space w avelength. From (7.3) and (7 .4 ). Vi­
and Vr are expressed as
v;
V (0 1 e * - V[ d)
2
vr
sin h ( 7 <f)
V'(0) - V)
(7.9)
T herefore, the reflection coefficient becom es
T h is d e-em b ed d in g process is su ited for one port network analysis.
However,
th e tech n iq u e m ay be readily exten d ed to tw o -p o rt networks. For instance, on the
assu m p tion of a perfect term ination or m atch at port
2
, once V; is d eterm in ed .
5^1
can be ob tain ed by V0f\'}. where V0 is th e ou tgoin g wave at port 2 predicted by FEM .
As m en tion ed before, S-param eter evalu ations depend on term ination m eth od s.
Low q u ality term inations result in prediction errors and m ake th e analysis less reli­
able. T herefore, high perform ance term in ation m ethods are always desirable and we
next discu ss this issue.
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120
7.3
Truncation U sing D M T
As already indicated, S-param eters from a p ossib le discon tinuity region along a
transm ission line may be extracted at a distant reference plane, w here there ex ists
on ly a dom inant m ode. For shielded m icrostrip lines at the input port ( # 1 ) and
o u tp u t port ( # 2 ) , (sim ilar to that in fig. 7.1), th e m odes underneath th e lines are
given by
|
£ * (* )= S
+ R e n:)
z € Sin
(7.11)
TEo(x)e~r2Z
z e S out
w here E q{x ) denotes the field distribution of incident wave at the incident plane
(port
1
) and R is the reflection coefficient at th e sa m e plane. T represents the trans­
m ission coefficient m easured at th e plane S out (port
propagation constants at port
1
and port
2
2
). and
7 1
.
72
are th e effective
, resp ectively.
For tru ncating the FEM m esh at a specific port, it is necessary to first determ ine
the E field pattern across the shielded structure.
T h is can be accom plished by
assum ing a sta tic m odel shown in fig. 7.2, where th e sta tic potential satisfies
V 2o
=
0
p
=
Vo
on m eta llic line
(7.12)
w here E = —V p . Sove this standard P D E m od el, and w ith a tedious m athem atical
derivation, it is finally found that
cos ( 7 - * ) sinh ( ^ y )
T , n=i.odd
o(x.y) = <
l^n=x.odd
_
-1
m
An
= ^ d
/
V
sinh ( ^ d )
( n ~ \ . , / n ~ ,,
\
;-----co s ( — x l s i n h f — (6 —y))
" s in h ( ^ ( 6 — ^ ) )
\a ) \aKy,J
z< d
where
1
—
1 ■*
. fn~w\
sin -----v
2 a J
1
1
« 2/ n
F
-•in — ‘ 0 -------- 7 7 ----------Z
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( 7- 1 3 )
121
i
4
0 -a/2
a/2
Figure 7.2: Illustration of the cross section o f a sh ield ed m icrostrip line.
w ith
F
fn
A com p lete FEM system m ay now be constructed by introducing th e EM fields
at Sin to tru n cate th e com p utational dom ain. T h is tru n cation sim ultan eou sly in­
troduces an e x c ita tio n to th e num erical system and th e S-param eters m ay then be
extracted by m easuring th e field distributions at th e in p u t and output ports as m en­
tioned before.
7.4
Truncation U sing PM L
Below , we begin with a brief presentation o f th e artificial absorber, and this is
followed by an exam in ation of the absorber's perform ance in term inating guided
structures and volu m e m eshes in scattering problem s. R esu lts are presented which
show the absorber's perform ance as a function o f th ick n ess/freq u en cy and for differ­
ent loss factors.
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7.4.1
T heory
C onsider th e waveguide, shielded m icrostrip line and scatterer show n in fig.
7.3.
O f interest is to m odel th e wave propagation in these structures using th e finite
Electric
Probe
£x = e/ = ^ = a -jP
^x = M-y= £ = < H P
d+t=40cm t=5 cm
cross-section: 4.755x2.215 cm
H t
(a). waveguide
Microstrip line
Absorbing layer
\
\
d + 1 = 12.0 cm
—*
hK
♦
w
H
h = 0.21 cm
L
H = 1.06 cm
j
t ♦-
L = 2.38 cm
E
= 3.2
£r =
w - 0.548 cm
(b). Microstrip Line
Figure 7.3: A rectangular w aveguide (a) and a m icrostrip line (b) tru ncated using
the perfectly m atched uniaxial absorbing layer.
elem en t m eth od . For a general uniaxial m ed iu m , the functional to be m inim ized is
F =
f
Jv
V x E • (^T l - V x E) - ^ f r • E • E d V
-f
E x (^T 1 • V x E) • dS .
(7.14)
* «^»n’h«^out
in which JIT and Ir denote the perm eab ility and p erm ittivity tensors whereas E is the
total electric field in the m ed ium . T h e surface integrals over 5,-„ and S out m ust be
evaluated by introducing an independent boundary condition and th e A B C serves
for this purpose but alternatively an absorbing layer m ay be used. An approach to
evalu ate the perform ance of an absorbing layer for term inating th e FE m esh is to
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
extract the reflection coefficient com p u ted in the presence of th e absorbing layer used
to term in ate the com p u tation al d om ain . In this stu d y vve consider th e perform ance
o f a thin uniaxial layer for term in atin g the FE m esh in a rectangular waveguide
and a m icrostrip line. Such a uniaxial layer was proposed by Sacks et.al. [77] who
considered the plane wave reflection from an anisotropic interface (see fig. 7.4 ). If
Region 2
Transmitted wave
Incident wave
X
Figure 7.4: Plane w ave incidence on an interface betw een two d iagonally anisotropic
half-spaces.
l*r and er are th e relative co n stitu tiv e param eter tensors o f th e form
/
J1r = « r =
a2
0
0
o
6-2
0
0
0
c2
V
\
( “-15)
/
th e TE and TM reflection coefficients at the interface (assum ing free space as the
background m aterial) becom e
cosOi — sJ^ cosB t
r te
c°s6i + ^ c o s S t
(7.16)
~ c o s O t — COsQi
R™
=
a2
1
1
c os0t* +' x\ I /^-cos6t
■?-N
I
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
124
and bv choosing a? = b2 and c> = — it follows that R TE = R Txl = 0 for all incidence
02
angles, im plying a perfectly m atched m aterial interface. If we set a 2 = a — j ‘3 . the
reflected field for a m etal-backed uniaxial layer is
\R{Ql )\ = e - 23ktcos6'
(7.17)
where t is the thickness of th e layer and 0, is the plane wave in cidence angle. The
param eter a k is sim p ly th e w avenum ber in the absorber.
B asically, th e proposed
m etal backed uniaxial layer has a reflectivity o f -30 dB if 3 tcos9{ = 0.275A or oodB if 3tcos6i = 0.5A. w here A is th e w avelength o f th e background m aterial.
T he reflection coefficient (7.17) can be reduced further by backing th e layer with an
A B C rather than a PE C . H owever, th e PE C backing is m ore a ttr a c tiv e because it
elim in ates altogether th e integrals over th e surfaces. Clearly, alth ou gh th e interface
is reflectionless. th e finite thickn ess layer is not and this is also true for Barenger's
PML absorber.
Below we present a num ber o f results which show the perform ance o f th e proposed
uniaxial absorbing layer as a function o f th e param eter 3 . th e layer thickness t and
frequency for the guided structures show n in Figure 7.3. We rem ark that for the
m icrostrip line it is necessary to let a2 = er6 (a — j,3) for th e p erm ittiv ity tensor
and a 2 = firb{oc — j 3 ) for th e perm eability tensor, where er(, and fj.rb are th e relative
con stitu tive param eters o f th e background m aterial (i.e. the su b strate).
7.4.2
Results
Rectangular Waveguide
Let us first consider th e rectangular w aveguide show n in fig. 7.3. T h e guide's crosssection has dim ensions 4.755 cm x 2.215 cm and is chosen to propagate only th e T £ \o
m ode. It is excited by an electric probe at the left, and fig. 7.6 show s the m ode field
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
strength inside th e w aveguide which has been term in ated by a perfectly m atched
uniaxial layer. As exp ected , th e field decay inside th e absorber is exponential and
for 3 values less than unity th e w ave does not have sufficient decay to suppress
reflections from th e m etal backing o f this 5cm layer.
C onsequ en tly, a VSYVR of
about 1.1 is observed for 3 = 0.5. However, as 3 is increased to unity, the V SW R
is nearly
1 .0
and th e wave decay is precisely given by e _l3t£cos6, = e ~V^cos9ip %where
t is th e wave travel distance m easured from the absorber interface. P = 2 3 t / X g and
here 0, = 44.5°.
It is noted that when 3 is increased to larger values, th e rapid
decay is seen to cause unacceptable V S W R ’s. O ne is therefore prom pted to look
for an o p tim u m decay factor for a given absorber thickn ess and fig. 7.7 provides a
plot o f the T E i o m ode reflection coefficient as a function o f 2 3 t / X g. where we chose
to norm alize w ith respect to th e guided w avelength Xg. Figure 7.7 is typical o f the
absorber perform ance and dem onstrates its broadband nature and the existence of
an optim um value o f 3 for m inim izing th e reflection coefficien t. Basically, the results
suggest that 3 m ust be chosen for a given absorber thickn ess to provide the slowest
decay w ithout causing reflections from th e absorber backing.
T h at is. th e lowest
reflection m ay be achieved when th e entire absorber w id th is used to reduce the
wave a m p litu d e before it reaches the absorber's backing. A s ex p e c te d , this optim um
value o f 3 changes w ith frequency but th e broadband properties o f th e absorber are
still m aintained sin ce accep table low reflections can still be achieved for unoptim ized
3 values. For ex am p le, in th e case o f / = A .oG H z (dashed lin e) th e optim um value of
3 =
1
gives a reflection coefficient o f -45dB whereas th e value o f 3 = 3 (corresponding
to 2 3 t / X g = 2.3) gives a reflection coefficient of -37 dB w hich is still acceptable for
many ap plications. It should be noted though that se ttin g 3 = 3 allows use of an
absorber which is about 2cm or 1/3 free space w avelengths. A lso, as can be realized
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
126
th e d iscretization rate plays a role in finding th e op tim u m value of 3 t / X g and thus
th e presented curves refer to a sam pling rate o f around 1S/A 3 for th e w aveguide
exam p le.
Not surprisingly (see ( 6 . 6 )). for this ex a m p le , th e value of a does not play an
im portant role in th e perform ance of the absorber and this is dem onstrated in fig. 7.8.
As seen, se ttin g alw ays a = 3 gives the sam e perform ance as the case of a = 1 shown
in fig. 7.7.
Our tests also show that other choices o f a give th e sim ilar absorber
perform ance.
H owever, it is exp ected th at a w ill play a role in the presence of
atten u a tin g m odes and it is therefore recom m en ded to choose a = 3 to ensure that
all m odes are absorbed.
M icrostrip line
T h e perform ance of the perfectly m atched uniaxial layer in absorbing th e shielded
m icrostrip lin e m od e is illustrated in fig. 7.9 w here th e reflection coefficient is plotted
as a function of 2 3 t [ X g, where Xg = A0 / y/ e7ff and e?/ / is the effective dielectric
constan t.
In th is case, the m icrostrip line is term in ated with a 1.87 cm thick. 5 -
layered absorber and the line is extended up to 4 layers inside th e absorber to avoid
an electric con tact w ith the m etallic wall.
S im ilarly to the w aveguide, we again
observe th at an optim um 3 value exists and it was verified th at in th e absorber
the wave e x h ib its the sam e attenuation behavior as show n in fig. 7.6. T h e reflection
coefficient at th e op tim u m 3 = 1 is now —42d B and if better perform ance is required,
a thicker absorbing layer m ay be required. A gain as in the case of the w aveguide
exam p le, th e value of a plays little role in th e perform ance of th e absorber and this
is illustrated in fig. 7.10. However, o f im p ortan ce is the behavior of th e reflection
coefficient as a function of 2 3 t / X g. For th e w aveguide and m icrostrip ex a m p les, we
observe th at th e absorption is m axim ized for ap p roxim ately the sam e value of 2 3 t / X g
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(about O.S). T hus these curves can be used for other applications as w ell, althou gh
it should be noted that the discretization rate plays an equally im portant role and
this needs further investigation.
T h e accuracy and validity o f th e PM L applications for circuit param eter com p u­
tation s can also b e seen from th e result illustrated in fig. 7.11. It is seen th a t use
o f th e o p tim ized 4.5 cm PML layer, w ith a = 1 and , 3 = 1 , yield s very accurate
input im p ed an ce values. The show n m icrostrip line im pedan ces were co m p u ted by
m easuring th e vertical field at the probe's location w ithout a need to e x tr a c t the
V S W R w hich is often difficult w ith unstructured finite elem en t m eshes. N o te that
th e shielded m icrostrip line dim ensions for th e data are given in fig. 7.11.
M e a n d e r li n e
A nother ex a m p le is the m eander line show n in fig. 7.12. For the FEM sim u lation ,
the structure was placed in a rectangular ca v ity o f size 5.8m m x IS.Omm x 3.175m m .
T h e cavity was tessellated using 29 x 150 x 5 edges and on ly 150 edges w ere used
along th e y-a x is. T h e dom ain was term in ated with a 10 layer PM L. each layer being
of thickness t = 0.12m m .
T h e 5 n resu lts are shown in fig 7.13 and are in good
agreem ent w ith th e m easured d ata [7S].
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
12S
E le c tric
P ro b e
Real Part
Imaginary Part
e '»
Magnitude
4
w
-
= b
------ M t H
£=
ex = eY=
^ 1=
p , = py =
« -M
1 - j '3
1 -jP
d + t= 4 0 c m
t=1 c m
Samples Along Waveguide (14samplesA.)
Figure 7.5: T ypical field values o f T E 1 0 m ode inside a rectangular w aveguide term i­
nated by a perfectly m atched uniaxial layer.
—
beta=0.5
10 .
120
64
66
68
70
74
Segments number along waveguide (alpha=1.0,
:1 f=4.5GHz)
Figure 7.6: Field values of th e T E io m ode inside a waveguide term in ated by a per­
fectly m atched uniaxial layer. T h e absorber is 10 elem ents thick and each
elem ent was 0.5 cm which translates to about 13 sam ples per w avelength
at 4.5 GHz.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
129
—
f= 4.0G H z
f= 4.5G H z
-10
f= 5.0G H z
— -3 0
-3 5 -4 0
-4 5
-5 0
Figure 7.7: R eflection coefficient vs 2 f i t / \ g (a = 1 ) for th e perfectly m atched uniaxial
layer used to term in ate the waveguide show n in fig. 7.6.
-5
-1 0
—
f= 4.0G H z
f= 4.5G H z
f= 5.0G H z
2 -1 5
3
0)
> -20
— -3 0
-3 5
-4 0
-4 5
-5 0
2|3t in X.g (a=(3)
Figure 7.S: R eflection coefficient vs 2 3 t j \ g. with a = 3. for the perfectly m atched
uniaxial layer used to term inate the w aveguide shown in fig. 7.6.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
130
f= 4.0G H z
f= 4.5G H z
-10
f= 5.0G H z
0 .-1 5
-20
-2 5
— -3 0
-3 5
-4 0
-4 5
-5 0
0 .5
1.5
2
2.5
3.5
2(Jt in Xg (a=1.0)
Figure 7.9: R eflection coefficient vs 2 3 t / \ g w ith a = l . for the shielded m icrostrip line
term in ated by the p erfectly m atched uniaxial layer.
-5
—
f=4.0G H z
f=4.5G H z
-10
f=5.0G H z
Q. -1 5
S’- 2 5
r -30
-3 5
-4 0
-4 5
-5 0
0.5
1.5
3.5
Figure 7.10: Reflection coefficient vs 2 l 3 t / \ g w ith a = 3 . for the shielded m icrostrip
line term in ated by th e perfectly m atched uniaxial layer.
R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
131
—
Theory
FEM
ttsotngtyer
width of m icrostrip (cm)
Figure 7.11: Input im pedan ce calculations for th e PM L term inated m icrostrip as
com pared to the theoretical reference data.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
132
.305
.61
.305
1.525
.61
Figure 7.12: Illustration of a m eander line geom etry used for com parison with m ea­
surem ent.
0.9
0.8
0.7
0.6
0.4
0.3
Measured
- -Calculated
0.1
frequency (GHz)
Figure 7.13: Com parison o f calcu lated and m easured results for th e m eander line
shown in fig.7.12.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
C H A P T E R V III
AWE: A sym ptotic Waveform Evaluation
8.1
B rief Overview o f AW E
A lthough full wave electrom agnetic sy stem s are large and cum bersom e to solve,
ty p ically o n ly a few param eters are needed by th e designer or an alyst. A reduced or­
der m odeling o f these param eters (input im pedance. S param eters, far field pattern,
e tc) is therefore an im portant consid eration in m inim izing th e C P U requirem ents
needed for generating the frequency response o f th e param eter.
T h e A sym p totic
W aveform E valuation!A W E) m eth od is one approach to construct a reduced order­
ing m odel o f th e input im pedance or other useful electrom agn etic param eters. AW E
relies on a Pade approxim ation of th e given param eters to avoid th e repeated solution
of th e sy stem at each frequency value. It has already been applied to problem s in cir­
cu it analysis and in this paper we d em on strate its application and validity when used
in conjunction with the finite elem en t m eth o d to sim u late full wave electrom agnetic
problem s.
T h e m eth od of A sym p totic W aveform Evaluation (A W E ) is a reduced-order m od­
eling o f a linear system and has already been successfully used in VLSI and cir­
cuit analysis to approxim ate the transfer function associated w ith a given set of
p orts/variab les in circuit networks [79-S2]. T he basic idea o f th e m ethod is to de-
133
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
134
velop an approxim ate transfer function of a given linear sy stem from a lim ited set of
spectral solu tions. T ypically, a P ade exp ansion o f th e transfer function is postulated
w hose coefficients are then determ ined by m atch in g th e Pade representation to the
available spectral solu tions of th e co m p lete sy ste m .
In th is chapter we in vestigate the a p p licab ility o f th e AW E m eth od for approx­
im atin g the response o f a given param eter in full wave sim ulation of radiation or
scatterin g problem s in electrom agnetics. O f particular interest is to use AW E for
evalu atin g th e input im pedan ce o f the an ten n a over an entire bandw idth from a
know ledge o f th e full wave solu tion at a few (even a sin gle) frequency points. Also,
th e m eth o d can be used to fill-in a b ackseattering pattern w ith respect to frequency
from a p r io r i know ledge o f th e sim ulation sy ste m w ith a few d ata sam p les o f that pat­
tern. G iven that practical partial differential e q u a tio n (P D E ) system s involve several
thousand unknow ns. AW E can indeed have a dram atic reduction o f C P U require­
m en ts in generating a response for a given sy ste m param eter (sta te variable) w ithout
a need to resolve th e sy stem for th e fields in th e entire com p u tation al grid. Below we
first describe th e recasting of th e FEM sy ste m for application o f the AW E. We then
proceed to describe th e AW E m eth od and d em on strate its ap p lication, accuracy and
efficiency for com p u tin g the input im p ed an ce o f a shielded m icrostrip stu b.
8.2
8
.2
.1
T heory
F E M S y stem R eca st
T h e application o f th e finite elem en t m eth od to full wave electrom agn etic solu­
tions am ounts to generating a linear sy stem o f equations by extrem izin g the func­
tional [S3]
T = < V x E .! f - V x E > - A
2
< E . 3 ■E > + k b . t .
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(S .l)
w here < . > denotes an inner-product and b .t . is a possible boundary term w h ose
specific form is not required for th is discu ssion . A lso, th e dyadics a and l3 are m a te2tt
*js
rial related coefficients, k = — = — is th e w avenum ber and
A
c
operatin g frequency with c being th e speed o f light.
lj
is the corresponding
A discretized form o f (8 .1 )
incorporating th e appropriate boun dary conditions is [29]
^Ao + k A i -f k 2A -^j {A '} = { / }
w here A , d en ote th e
(8 .2 )
usual square (sparse) m atrices and { / } is a colum n m a trix
describing the specific excitation .
C learly (S.2 ) can be solved using direct or iterative m ethods for a given value of
th e w avenum ber. Even though A i s sparse, th e solu tion of th e system (8.2) is c o m ­
p u tation ally intensive and m ust be further repeated for each k to obtain a frequency
response.
A lso, certain analyses and designs m ay require both tem poral and fre­
quency responses placing additional com p u tation al burdens and a repeated so lu tio n
o f (8.2) is not an efficient approach in generatin g these responses. An ap p lication of
AW E to achieve an approxim ation to th e se responses is an attractive altern ative and
below we form ulate AWE in co n n ectio n w ith th e FEM system (8.2). w hose im p le­
m en tation is considered in con n ection w ith antenna and m icrowave circuit p roblem .
For these problem s it turns out th at th e ex citation colum n { / } is alinear fu n ction
o f th e w avenum ber and can therefore be sta ted as
{ /} = H /i}
(S.3)
w ith { / i } being independent o f frequency. T h is observation will be specifically used
in our subsequent presentation.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
136
8.2.2
A sym ptotic Waveform Evaluation
To describe th e basic idea o f AW E in conjunction w ith th e F E M , we begin by
first expanding th e solution {A '} in a Taylor series ab ou t k0 as
{A"}
=
{ A 'o } +
( k — k0) { X
i}
-f {k — kQ)2 { X
2
}
+ ...
+ ( b ~ ko)‘ { X ,} + O {(Ar - M ' +1}
where
{X o}
(8.4)
is th e solution o f ( 8 . 2 ) corresponding to th e w avenum ber ko. B y intro­
ducing this expansion into
( 8 . 2 ) and eq u atin g equal powers o f k in conjunction with
(8 .3 ), after som e m anipulations we find that
{X o}
=
fc o A - ^ /J
}
=
A 0 1
{ X '2 }
=
—A
{ A 'i
[{ / 1 } —
A i {Xo} —
0 1 [A i { X t }
+
2/l0A 2 { X o } ]
A 2({X o}
+ 2k0
{ X j} )]
(8 .5 )
{X,} = - V [ A i { X t_l} + A 2({X,_2} + 2fco{X,_i})]
w ith
Ao — Ao + k0A i + A:qA2
(8.6)
E xpressions (8.-5) are referred to as the system m om ents whereas ( 8 . 6 ) is the
system at th e prescribed w avenum ber (&o). A lthough an exp licit inversion of Aq
1
m ay be needed as indicated in (8 .5 ), th is inversion is used repeatedly and can thus
be stored o u t-o f-c o r e for th e im p lem entation of AW E. A lso, given that for input
im pedan ce com p u tation s we are typ ically interested in th e field value at one location
of th e com p u tation al dom ain, only a sin gle entry o f { AT/( A:)} needs be considered, say
(th e pth entry) X [ ( k ) . T he above m om en ts can then be reduced to scalar form and
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
the expansions (8.5) becom e a scalar representation o f .V? [k) about th e corresponding
solu tion at ko. To yield a m ore convergent expression, we can in stead revert the
m om ents to a Pade expansion, w hich is a conventional rational function in form. A
special case of the qth order o f such an exp ansion is given by
y p i i
‘
\ _
“
a ° +
1
+
a i(k
6
~
fro) +
a 2 (k — ^ o )2 +
- • • +
o-qjk — k 0 )q
x(/c - ko) + b2(k - k 0 )2 + . . . + bq(k - fco) 9
_
'
where a, and 6 , (i = 0 .1 ..... q) are referred to as th e Pade coefficients.
For transient analysis, it is observed th at th e Pade expansion can be reform ulated
by partial fraction decom p osition [82.84] as
.V J W - A +
( 8 .8 )
1=1
where X qo is th e lim itin g value w hen k ten d s to infinity. Clearly, th is is th e represen­
tation suitable for tim e/freq u en cy dom ain transform ation. T h e residues and poles
(rt and kQ + ki) in (8.7) or ( 8 . 8 ) correspond to th ose of th e original physical system
and play im portant roles in d eterm in in g th e accuracy of the ap p roxim ation. In gen­
eral. higher order expansion contains m ore system residues and poles and usually
provides a better approxim ation. Since th e accuracy of AWE relies on th e dom inant
residues and poles located in a com p lex plane closest to the point on the real axis
ko from the origin, in practice th e num ber o f poles (and residues) needed to obtain
a sufficiently accurate expansion can be m uch sm aller than that o f th e original nu­
m erical system , which is the b eau ty o f AW E m eth od . (D etailed an alysis and theory
o f Pade expansion can be found for in stan ce in [85].)
For hybrid finite elem ent - boundary integral sy stem , th e im p lem en tation of
AW E is more involved because th e fully p opu lated subm atrix o f th e overall system
m ay be associated w ith a m ore com p lex d ep en d en ce on frequency. In this case it is
attractive to instead generate th e full su b m atrix by introducing a spectral expansion
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
of th e exp on en tial boundary integral kernel to facilitate th e extraction of th e system
m om en ts. T h is approach does increase th e com p lications for im p lem en tin g AW E.
It how ever rem ains far m ore efficient in term s of C P U requirem ents when com pared
to th e conventional approach to continu ously repeating th e solu tion o f th e entire
sy stem .
8.3
N um erical Im plem entation
A s an application of AW E to a full w ave electrom agnetic sim u lation , w e consider
the evalu ation of the input im p ed an ce for m icrostrip stu b shielded in a m etallic
rectangular cavity as shown in fig.
8.1. A s exp ected , th e stu b ’s input im p ed an ce is
a stron g function o f frequency from 1-3 GHz and this exam p le is therefore a good
d em on stration of AVVE’s capability.
T h e sh ielded cavity is 2.38cm x 6.00cm x 1.06cm in size and the m icrostrip stub
resides on a 0.35cm thick substrate having a dielectric constant o f 3.2. T h e stub is
0.79cm w ide and A/ 2 long at about 1.8 G H z. We n ote that the cavity is term inated
at th e p erfectly electric conductor (P E C ) back wall by an artificial absorber having
relative con stan ts of er = (3.2. —3.2) and
= (1.0. —1.0). In th is stu d y th e artificial
absorber was used for settin g up an appropriate forced problem rather than to es­
tablish a perfectly m atched interface. N evertheless th e num erical FEM sy stem was
already d em onstrated valid and accurate for m icrowave circuit analysis [8 6 ].
T h e frequency response of th e sh ielded stub was first com p uted using a full wave
finite elem en t code from
1
to 3 G Hz at 40M Hz intervals (50 points) to serve as the
reference solu tion. We then chose th e single input im pedan ce solution at 1.78GHz
in conjunction w ith the 4th order and Sth order AW E representation given in (S. 8 )
to ap p roxim ate the reference response.
As seen in fig. 8.3. the 4th order AW E
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
139
representation is in agreem ent w ith th e real and reactive parts o f th e reference input
im p ed an ce solution over about -56% and 33% bandw idth, respectively. T h is clearly
show s th at th e contributions o f th e sy ste m poles in the com p lex k plane lead to an
accuracy difference to th e real and rea ctiv e com ponents. Surprisingly, th e Sth order
AW E representation recovers the reference solution over th e entire l-3G H z band for
both im pedance com ponents. W e also observed that th e C P U requirem ents for 4th
and Sth order com putations are nearly th e sam e excep t for a few m ore tim es of
m a tr ix -v e c to r products. T h e num ber of th ese product operations is in th e order of
th e AW E approxim ation order q and therefore much sm aller than th e size of the
original num erical system .
It is also apparent that to d em o n stra te th e AWE efficiency w e on ly solved the
sy ste m once at one frequency p oin t. T h e save of C PU tim e can be easily estim ated
w hen com pared to solve the sy ste m con ven tion ally for each frequency over th e entire
band.
T hus, the AWE representation is an extrem ely useful ad d ition to electro­
m agn etic sim ulation codes and packages w hen a wideband frequency response of the
sy ste m is required. T h e develop m en t and u tility of th e m ethod for m ore com plex
num erical system s and m ultiple p aram eter sim ulation can be readily exten d ed and
will be considered in th e future.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
140
Figure S .1: Illustration of the shielded m icrostrip stub excited with a current probe.
035
-
■o 30
1.2
1.4
1.6
2-2
.S
frequency in GHz
1
2.4
Z6
2.8
Figure 8.2: Im pedance calculations using trad ition al FEM frequency analysis for a
shielded m icrostrip stu b show n in figure 8 . 1 . Solid line is th e real part
and th e dashed line d en otes th e im aginary part of the solu tion s. T h ese
com p utations are used as reference for com parisons.
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
141
40
Exact
20
- - AWE-arn order
- - AWE-4th order
1.4
1.6
2J2
frequency in GHz
1.8
2.4
2.6
2.8
2.4
2.6
2.8
(a)
- - AWE-8th Order
- - AWE-4th Order
a 30
£25
1.2
1.4
1.6
2-2
1.8
2
frequency in GHz
(b)
Figure S.3: 4th order and Sth order AW E im p lem en tation s using one point exp an ­
sion at 1.7S GHz are shown to com pare w ith the reference d ata. W ith
the 4th order AW E solu tions, 56% and 33% bandw idth agreem ent can be
achieved for the real (a) and im aginary (b) parts o f im p ed an ce com p u­
tations, respectively. It is also shown that th e Sth order solu tion s agree
excellen tly w ith the reference data over th e entire band, (a) Real Part
(b) Im aginary Part com p utations
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
C H A P T E R IX
Conclusions
9.1
D iscu ssion on the Research Work
During th e period of developing th e hybrid finite elem en t m ethods, m any e x ­
pected and u n exp ected issues were frequently en countered.
A m ong them are the
understanding o f physical system s, developm ent o f m a th em a tica l m odels, interpre­
tation o f results, lack of m easurem ent d ata for com p arison, and increased com pu­
tational dem an d s, e tc . We can com fortably sta te th at significant progress has been
m ade during th e course of this work. Som e of our accom p lish m en ts are sum m arized
below.
• G e n e r a l p u r p o s e h y b r id F E - B I m e t h o d d e v e l o p m e n t
O nce the FE and BI subsystem s and th e hybrid m eth od were m ath em ati­
cally form ulated, a m ajor effort was then devoted to th e integration of the
tw o su b sy stem s. T he interface betw een th e F E -B I program and a com m ercial
(SDRC’-ID E A S ) m esh generator was d evelop ed w ith m inim um but sufficient
geom etry and m eshing data. T he latter task was im p ortan t in perm itting the
geom etrical m odeling and m eshing o f printed an tenna configurations of arbi­
trary shape. It is this general version of th e F E -B I cod e th at can (in theory)
be used to sim u late any planar conform al antenna.
142
R e p r o d u c e d with p e r m issio n o f th e co p y r ig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
143
• I t e r a t iv e s y s t e m s o l v e r
A m em ory saving algorithm ITPA C K was intertw ined w ith th e hybrid FEM
subsystem to register on ly th e non-zero FEM entries.
T h e B iC G iterative
solver was in d ep en d en tly d evelop ed for partially sparse and partially full m a­
trices in conjunction w ith th e ITPA C K algorithm .
•
U n if o r m g r id B I s u b s y s t e m —
B iC G - F F T
To facilitate the efficient storage and evaluation of the BI su b -sy stem , a uni­
form right triangular zoning sch em e for discretization of th e boundary integral
equation was introdu ced by re-num bering th e triangle edges as d ic ta te d by their
geom etrical location s. T his approach leads to a BI su b -system w hich could be
cast as a 2-D discrete con volution, thus allow ing use of F F T for fast execution
o f the iterative solver. T h is tru n cation /term in ation th e "exact" evaluation of
rectangular and right-triangular patches.
•
N o n - u n if o r m B I s u b s y s t e m —
O v e r la y -B iC G -F F T
For non-rectangular p atches, an interpolation schem e was proposed to m ake use
of the efficient B iC G -F F T techn iqu e by overlaying a fictitious uniform grid with
th e original arbitrary m esh. T h e forw ard/backward tran sform ation m atrices
to account for field in terp olation s using localized basis functions were derived
and th ey were indeed highly sparse.
• F e e d m o d e l in g
Feed m odeling is one of th e m ost im portant and challenging tasks in th e context
of the general purpose FE M . To this end. a series o f co m m o n ly used feed
structures were m odeled using th e hybrid technique, esp ecially in consideration
of efficiency and accuracy. T h ese include p rob es/generators, apertu re coupled
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144
slotlin e. m icrostrip line, coax cab le, e tc .
• P r is m a t ic F E M E l e m e n t s I n c o r p o r a t io n
A m ajor problem in any hybrid F E M analysis is th e ted ious pre-processing
for m esh generation. Thin layer substrates in the presence of thick spacer(s)
are often found in practical conform al antenna design s. However, this typical
configuration leads to large num erical system s w hen tetrahedral elem en ts are
used. To alleviate these difficulties, th e prism atic edge-based elem en ts were
d evelop ed and incorporated in th e hybrid system . T h is form ulation exhibits
certain features/advantages th at tetrahedral FEM does not. It can therefore
be used to com p en sate the tetrahedral FEM as a su b system m odule.
• M e s h T r u n c a t io n s W i t h D M T a n d P M L
T h e uniaxial or other anisotropic m ed iu m sim ulation m ay be readily accom ­
plished using the proposed hybrid FE M technique due to the geom etrical adapt­
ab ility of th e tetrahedral elem en ts. H ence the PM L was first introduced into
the 3-D FE M . Various perform ance stu d ies were carried out to o p tim ize the
application of the PM L to m icrow ave circuit sim u lation s. In the m ean tim e, an
an alytical approach, dom inant m od e tru ncation (D M T ), was proposed and im ­
p lem ented as an alternative m esh tru ncation of the FEM dom ain for m icrostrip
lines and sim ilar structures.
•
R e d u c e d o r d e r a p p r o x im a t io n — A W E
AW E has been reported useful in RLC and VLSI ap p lications. For wideband
and highly varying frequency resp onses, this techn iqu e is particularly efficient.
G iven th e prom ise of the m ethod for broadband sim ulations of VLSI circuits,
we consider its application to electrom agn etic system .
In particular. AWE
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145
was incorporated into th e finite elem ent m ethod. It was indeed observed that
the attractive features of AW E are m aintained when used in electrom agnetic
problem s.
9.2
Suggestions for Future Tasks
T h e following is a list o f su ggested tasks for further develop m en t o f th e finite
elem ent m ethods
• H ig h e r o r d e r e d g e - b a s e d F E M d e v e l o p m e n t
• A d a p t iv e e l e m e n t s
• M ix e d E l e m e n t s a n d in t e r f a c e
• A n is o t r o p y ( w it h l o s s ) F E M I n v e s t ig a t io n s / a p p l ic a t io n s
• I n c o r p o r a t io n o f M o r e R o b u s t t r u n c a t io n s
• M o d u l a r d e v e l o p m e n t a n d in t e g r a t io n ( w it h u s e r in t e r f a c e )
9.3
M odular D evelopm ent
Hybrid finite elem ent m eth od s for the analysis of various electrom agn etic prob­
lem s encountered in practice are still on the way to reach its m aturity. A s well known,
any general purpose techn iqu e (such as th e com m ercially available softw are in electro­
m agnetics) either loses its efficiency or becom es incapable when sim u la tin g intricate
problem s. It is anticipated th at at th e current stage of the FEM d evelop m en t with
the lim ited capacity of co m p u tin g resources, more and m ore specialized techniques
will be desired, particularly when efficiency and speed becom e a key consideration
in large scale com p utations and in engineering design.
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146
W ith a w hole set of sp ecially d evelop ed techniques and m eth od ologies, one should
then consider to create an integration environm ent. As shown in fig. 9.1. we propose
this FEM m odular environm ent for future com putational electro m a g n etic applica­
tions. A well designed m odular finite elem en t m ethods will be th e m ost capable and
robust in th e future!
FEM Multi-Module Environment
User Interface
User's Modules
Kernel
Mesh
Truncation
Modules
FEM Modules
bricks
prisms
tetras
FeedLine
Modules
Postprocessor
Figure 9.1: M u lti-m od u lar FEM environm ent
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APPENDICES
147
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L4S
A P P E N D IX A
Evaluation of M atrix Elem ents for Tetrahedrals
Referring to Fig. A .l and the associated table, th e fields in th e eth tetrahedron
1
2
nodes/vertices
(a)
Table
Edge
N um bering
(D
©
(3)
<3>
<§>
V ertex N um bering
1
1
1
2
*2
2
3
4
4
3
2
3
4
(b)
Figure A .l: (a) A tetrahedron, (b) its local n o d e /e d g e num bering schem e
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149
are expanded as
6
E= £
£ 'w :
i=i
where th e basis functions W ' are given by
w 7_,-(r)
=
f7_i + g 7_, x r
r € V;
0
outsid e elem en t
;
fr-i =
777 - ru x
6V'e
b jb ,.,
g7_,
r,2
r,,. r,2 : p osition vectors o f vertices
and 12 (see Table)
e,
6V
6v;
(r,-a - r.-J
e, =
bi
bi
= |r,-2 — r t l| = len gth of the ith ed ge (see Table)
Ve
= elem en t's volum e
We note that
V
indicating that
W “
•W ? =
0
V x
W ? =
are divergenceless. Furtherm ore.
6tJ = If
W ' ( r ’ ) - e J = 6,J:
1 i=j
{ 0
where r 7
2g,
i? j
has its tip on th e j th edge of th e tetrahedron. T his last property
that th e coefficients E ‘
ensures
= E • e, represent th e average field value at the ith edge of
th e tetrahedron.
Using th e above basis functions, we now proceed w ith th e derivation of the m atrix
elem en ts A*-. We have
fff
J J J \ \
- ( V
I* r
x W ') •
(v
x W
') =
- i g , . ■g
v;
Hr
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150
Also.
Er [
V n - W ' j]d v
=
Jv*
—
c
rIIIv
{ (f,"
fj) +
( r
D )
+
( g ‘' x
r ) ' ( g j
x
dv
M A + I2 + £})
where
D = (fi x g j ) + (fj x g j
and
=
h
=
Since f is a con stan t vector.
• f , dv
I I I / - ' -
//Xr
dv
D
J J J ( g £ X r ) ’ (gj X r ) dv
reduces to
h = fi • fj-v;
To evalu ate I2 w e first set
x —^
i= l
y — ^ ' Liyi.
_ — ^ ^ Li~i
i= l
i= l
where x,-. r/,-. x, (f = 1___ .4 ) d en ote th e (x . y . z ) coordinates o f th e tetrahedron's
vertices and £ , are th e sim p lex coordinates or shape functions for th e sam e elem en t.
That is. Li is th e norm alized volum e of th e tetrahedron form ed by its three corners
other than th e fth . and th e point ( x . y . z ) located w ithin th e tetrahedron. U sing the
standard form ula for volum e integration w ithin a tetrahedral elem en t and sim plifying,
we have
12
—
—
Dx ^
i= t
x, + D y ^
t—1
t/i + D z ^
zt
i= l
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
L5L
where D m is th e m th com ponent of
D. T h e evalu ation of / 3 can be sim plified by the
use o f basic vector id en tities. We have
h
|r|2 d v - £ (g , • r ) ( g J • r) d v
=
g, • g j
=
(9iy9jy + 9izgjz)
J \\
-r2 d v + {glxgJX + gi:gj : ) I y 2 d v + {gixgj r + giygj y ) I z 2 d v
Jv.
Jv,
(9ix9jy d" gjxgiy) J
Xy dv
[gix9jz T 9jx9iz ) J
ZX
dv
(giygjz T gjygiz) J" y z d v
where y,-m represents th e m th com ponent o f th e vector g ,. Each o f th e above integrals
can be easily evalu ated an alytically and th e result can be expressed in th e general
form
r
I
1w
.
for /, m = 1____ .3 .
f
4
Y
. 1=1
4
au
4
+ ^ ^ 0>li ^ ^ @>TT
1=1
1=1
T h e param eters a[ or a m can represent any o f th e rectilinear
variables x . y , z.
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
APPEN D IX B
Evaluation of the Boundary Integral System
M atrix
T h e exp licit representation o f the boundary integral subsystem m atrix is given
by (3.12) and can be rew ritten as
Bpqtj = 2
JJpJJ
q ( ~ koSi • S j + V x Si • V x S j ) G 0(r.
r') dS' d S
( B .l)
where G'o(r, r') is th e free space G reen's function, and T f is the pth triangle of th e
triangle pair S p as show n in fig. 3.3. Sim ilar to th e finite elem en t assem bly procedure,
it should be recognized that the definition o f ( B .l ) virtu ally involves an assem bling
over the triangles.
To proceed. (3.14) is used to discritize th e field region and thus its curl is given
by
V x S ,(r ) = £ ( r ) i i - i
Ap
(B .2 )
w here e(r) is defined by (3.15). N ote that when deriving (B .2). the fact that
located inside the pth triangle in a planar surface is considered and therefore
r is
V-r = 2.
G iven the Green's function, it is straightforward to express the m atrix entries as
rr
Z.2 / . / .
=
+
^
A
J
J
rr
II
I I
S
T:S
U
L
/
L
r\
^
t
'\
f \
i \
P-jk0R
T'') ( i ~ T'i ) t i ( T ) c A T ) t ^ ~
dS' dS
‘- ^ r € M t ‘ ( T ) d S ' d S
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
m )
in which R = |r — r'|.
T h ese integrals can be readily evalu ated for n o n -se lf-c e ll
term s by num erical in tegrations.
It is also observed that once T f coincides with
TJ . the integrands b ecom e singular because o f the G reen's function. In th is case,
the singularity should be rem oved. For th e second integral, this is accom plished by
subtracting and adding an additional term . T hat is
ff ff
J J t ,p J J t ?
, - j k 0R
e- ^ - c t(r)c J( v ) d S ' d S
“
[[ j[[
r r
=
J J t?
M
r r
g -jfc0 /J _
J t?
•<?
^
~~~E>— "€‘(r )ei( r ) dS' d S
K
t
^ i l r W r )d S 'd S
(B .4)
T h e first integral in (B .4 ) is evalu ated using num erical integrations and the second
one is carried out a n a lytically [55]. Sim ilarly, th e first integral in (B .3) is rearranged
as
J I Tp J J T^ V ~
' (r “ rj)e'(r)£i(r) - ^ —dS ' d S
I It”ILir~^ ^~
=
R-- dS'dS
+
(B .5)
in which the first integral on th e right hand side is num erically integratable with
singularity rem oved and th e second one again m ay be expressed in a sim p le analytical
form [55].
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
154
A P P E N D IX C
Form ulation for R ight Angle Prism s
For FEM im p lem en tation , th e follow ing quan tities are required
e
V
P.mn
-
L
-
L
X
Vm•V
X
V ndV
(C -l)
V m - V n dV
Q
(C.2)
where the curls are given by
V x V,
=
li
[(x - X |)x + {y - yi)y - z(z - cc)]
2S ' A z
li
VxV^ =
+ ( y ~ yj )y +
~
+ ^ - ;)i j = 4.5.6
1
V x Vfc
i = 1 .2 .3
= 5 ^ 7 K'Ik2 ~ Xkl ^ + (yk2 ~ ykl
(c -3)
k = 7 .S .9
To this end. we follow th e n otation defined in (4.13) and (4 .1 4 ). where i . i ' = 1.2.3
represent the top triangle edges, j ,j ' = 4 .5 .6 denote the b o tto m triangle edges and
k. k ' = l.S .9 stand for th e vertical three edges. It is found th at (C .2) and (C’.3) can
be analvticallv evalu ated and we tab u late th e results as follows
Pa> =
Pjj'
=
Co, D i i 'A z + - S e( A z f
C
jj
'
D ^Az+^-S^Azf
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(C .4)
(C.5)
155
(C .6)
P
r ‘j
=
rPj ‘ = —C ij D i j ± z ~ ^ " ( A c ) 3
(C .7)
(C .8 )
Pik
=
Pjk
Pki =
li
[x ■P { S X - x tS e ) + y ■Ik( S Y - <,tS'e)]
4(5«)s
Pkl = 4 ( #
^ ‘ J k [ S X ~ X j S ' ] + * ' J k [ S Y ~ yj5'e)l
(C .9)
(C .10)
( C - ll )
Qn'
(A c )3
— ^— Cn'Dii'
=
(C .12)
(A c )3
Qn' =
— ^— Cjj> Djj>
(C .13)
(C .14)
Qw
=
A z S eTkk>
(A c )3
Qu =
Qji =
Qik
Qh = Qjk = Qkj
=
(C .16)
6
= o
(C-17)
where
Tkk'
= 1 /6
r
=
,J
Dij
f o r k = k'\
1 /1 2
fo rk ^ k '
hlj
4 ( S eA c ) 2
(CMS)
= S X X - (xi + x j ) S X + X iX jSe + S Y V - (y, + y j ) S Y + y i y j S e
T h e rem aining q u an tities in th e above list of the expressions are defined as
Se
=
[
Js«
d xd y
x d xd y
Js*
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
T h ese integrals can be expressed in term s o f th e global coordinates o f the th ree nodes
(Xi. Vi). (Xj . Yj). (.Ym. V*m). Specifically, assum ing that the three nodes i.j and m of
a triangle are in counterclockw ise rotation , w e then have.
= J
SX
dxdy = |
1
-r.
Hi
1
x j
Uj
1
m
x dxdy = ^
{Xi + Xj + x m )
y dxdy = ^
( V- + Yj + Ym )
l/m
- L
SY
■/,
S'
SXX
= J
Se
i 2 d x d y = ^ {(.V,- + A"j + Xm) 2 + (.V? + X ; + A 'i)}
SYY
= J
y 2d x d y - ^
SXY
=
f xydxdy =
Js=
{(V , + Vj + v m ) - + ( V , 2 + \ 2 + V,‘ ) }
{(.V t + X j + .Vm) (Vj- + Yj + Ym)
+ ( .v tv; + . v , v ' + .v mv m)}
R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
A PPEN D IX D
System D erivation From A Functional
Referring to fig. 2.2. we begin w ith th e functional
jc-(H ) = I /
( V x H - 1 7 1 • V x H - A-^H -
- Jnd+n} +na v
-H )
'
dP.
( D .l)
to derive the sy stem in term s of th e scattered field. On inserting th e field decom po­
sition H = H sca* + H ‘nc. th e functional becom es
H sc“£. H ,nc)
=
(H scal. H scat) | Qd+Q{ +Qa + ( H SC“‘. H ' nc) | n<<+ ( H nc. H 3cat) \^
+
( H sca(. H ,nc) |n / + ( H ,nc. H sca£) |n/
(D .2)
w here th e fact that H ‘nc does not exist in Ra has been considered, and ( .) represents
th e integral o f the sam e form as
in ( D .l ) . O nce a self-adjoint system operator is
assum ed, it then follows that
( H ,nc. H 5ca£) |ntf= ( H sca£. H ,nc) |nrf
(D .3)
( H mc. H sca£) |n / = ( H sca£. H ,nc) |n/
(D .4)
A lso, in free space 0 / .
U pon invoking the divergence theory, we have
2 ( H sca£. H mc) |n/
=
j f K scat • ( h i x f ^ 1 • V x H mc) d S
-
J
f H scat - ( n 2 x Ts ' • V x H inc) d S
rt
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
(D .o)
158
Evidently, for a self-adjoint operator, one readily recovers the sy ste m (2.42) obtained
via G alerkin's m ethod.
It should be noted th at besides the b ou n d ary/tran sition
conditions, the self-adjoint property o f a sy stem operator sim ply requires
V
1 =
( ? 7 l)
%
-
=
{ f r ) T
( D .6 )
In the case of a non-self-adjoint operator, it is generally not possib le to recover the
system given by (2.42) in th e sam e m anner. T h is is because th e functional (D .2) in
term s o f th e scattered field is o f th e form
.F (H )
(V x H 3cat • f ; 1 • V
=
- /
2 JQd+Qf +Qa y
+
-f
+
- f
2 Jod K
-
( v X H sca£ • f ; 1 • V X H ,BC - k * H scat ■% ■H ,nc) d9.
(V x H inc ■f ; 1 • V
+
x H scat - ^ H sca£ • % ■H sca£) rffl
J
x H 3Cat - k%H ,ne • jid ■H aca£) dQ
'
H scaf • ( n x x Tf l ■V x H ,nc) d S
f
H ii:a£ • ( n 2 x f ] 1 ■V x H ,nc) d S
(D .7)
JVf
It is observed that the first integral shows th e sam e form of the FEM sy ste m as that in
(2.42). A ll other integrals in (D .7) con tribute to th e system ex cita tio n . A pparently,
the two integrals over dom ain 9d are not id en tical, leading to a different FEM system
than (2.42).
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
B IB L IO G R A P H Y
159
R e p r o d u c e d with p e r m issio n o f th e co p y rig h t o w n e r . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .
160
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