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Synthesis of microwave structures by inversion of the TLM process

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S y n th e s is o f M ic r o w a v e S tr u c tu r e s
b y I n v e r s io n o f th e T L M P r o c e s s
by
M ic h e l F o re s t
A thesis presented to
T h e School o f G ra d u a te S tu d ie s a n d Research
o f th e U n iv e rs ity o f O tta w a
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fo r th e degree o f
M a s t e r o f A p p l i e d S c ie n c e in E le c t r i c a l E n g in e e r in g
O tta w a -C a rle to n In s titu te
fo r E le c tric a l E n g in e e rin g
©
Michel
Forest,
Ottawa,
Canada,
1992
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UNIVERSITE D’OTTAWA
e c o iE
des
Et u d e s
s u p Er ie u r e s e t d e la r e c h e r c h e
UNIVERSITY OF OTTAWA
SCHOOL OF GRADUATE STUDIES AND RESEARCH
Sm I
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♦ O r f DC I « / T T u « ‘ - * A V f O f A U T n O *
FOREST, MICHEL
M M C &S E PVS1KJE-HAJUHG A D D ftlZ l
h .1.
E a r
i V l ' l:
V audrnuil,
Ouebcc
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OkADC’O f O f f f f
M.Sc.A.
AMNEE D O e T T H T O i- rE A fl GRANTED
(Electrical Engineering)
19 9 2_____
TTTHC DC LA T H C S E - H rt/ O f THESIS
SYNTHESIS OF MICROWAVE STRUCTURES BY INVERSION OF THE TLM PROCESS
L'AITTEUR PERMET. PAR LA PRESENTE, LA CONSULTATION ET LE PR£ t
THE AUTHOR HEREBY PERMITS THE CONSULTATION AND THE LENDING OF
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F tU iN lN
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UNIVERS1TE D’OTTAWA
£c o l e
i
UNIVERSITY OF OTTAWA
d e s Et u d e s s u p E r ie u r e s
ET DE LA RECHERCHE
SCHOOL OF GRADUATE STUDIES
AND RESEARCH
FOREST, Michel
A L n T U ft 0 6 LA THS5 E ALffM0 * C * TH£$1$
M .Sc.A.
{Electrical Engineering)
GRA£>T0f<3»ff
ELECTRICAL ENGINEERING
ECOcEOEPASTEUEKT^ACULrr SCHOOL 0£PA*r*#E*r
TITHE D€ m THESE-TITTE O f THE THESIS
SYNTHESIS OF MICROWAVE STRUCTURES
BY INVERSION OF THE TLM PROCESS
W.J.R. Hoefer & M. Ney
o m e c r iu * m
l a t h e s e - r n fs is su p e r v is o r
EXAMINATEURS OE L> THESE-THESIS EXAWNEfiS
G. Costache
J . Wight
LE 0 0 f t * OC L-ECQLE DCS D U D 6S S U * »
H D ( LA RECHERCHE
SCHOOL O f
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UNIVERSITE D'OTTAWA
UNIVERSITY OF OTTAWA
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A b stra ct
T h is thesis presents a novel n u m e ric a l syn th e sis te c h n iq u e based on th e tim e reversal
p ro p e rty o f th e T ra n s m is s io n -L in e M a tr ix m e th o d ( T L M ) . I t a llo w s th e designer to
ge n erate th e g e o m e try o f a passive c ir c u it fro m its desired fre q u e n cy response using
a lte rn a te fo rw a rd and b ackw a rd tim e -d o m a in analyses.
T h is appro a ch opens new
doors in m ic ro w a v e synthesis and offers an advantageous a lte rn a tiv e to th e tr a d itio n a l
synthesis tech niq u e s usin g fre q u e n cy d o m a in analysis. To d e m o n s tra te and v a lid a te
th e p ro c e d u re , an in d u c tiv e ob stacle inside a w aveguide w ill be synthesized.
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S om m aire
C e tte these presente une n o u ve lle te ch n iq u e de synthese n u m e riq u e p e rm e tta n t d 'e ffe c tu o r
la c o n c e p tio n de s tru c tu re s e le c tro m a g n e tiq u e s a p a r t ir de s p e c ifica tio n s donnees sous
fo rm e de reponse en frequence. E lle est fo il dee sur la p ro p rie te d ’in ve rsio n du te m p s
de la m e th o d e T L M ( T r a n s m is s io n -L in e M a t r i x ) e t u tilis e en a lte rn a n c e des analyses
d ire c te s et inversees. La c o n ce p tio n d ’une s tru c tu re in d u c tiv e a I ’in te r ie u r d ’un g u id e
d ’onde est u tilis e e en guise d ’e xem ple p o u r illu s tr e r e t v a lid e r les d iffe re n te s etapes de
la p ro ce d u re . Les re s u lta ts d e m o n tre n t que ce tte n o u v e lle a p p ro ch e te m p o re lle o ffre
une a lte rn a tiv e avantageuse face au x te chniqu es tra d itio n e lle s q u i u tilis e n t p lu to t des
analyses fre q u e n tie lle s.
iii
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1 H e re b y declare t hat I a m th e sole a u th o r o f th is thesis. I a u th o riz e th e U n iv e rs ity
o f O tta w a to le n d th is thesis to o th e r in s titu tio n s fo r th e purpose o f sch o la rly research.
M ic h e fF o re s t
I fu r th e r a u th o riz e th e U n iv e rs ity o f O tta w a to reproduce; th is thesis by p h o to ­
c o p y in g o r by o th e r m eans, in to ta l o r in p a rt, a t th e request o f o th e r in s titu tio n s or
in d iv id u a ls fo r th e p u rp o se o f s c h o la rly research.
M ic h e l Forest.
iv
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A ck n o w led g em en ts
1 w ould lik e to express m y g r a titu d e to m y s u p e rv is o r D r. W o lfg a n g J .R . H oefer and
m y c o -s u p e rv is o r D r. M ic h e l N ey fo r th e ir co n tin u o u s encourag em ent th ro u g h o u t th is
w o rk .
Special th a n k s are due to M r.
P o m a n P .M . So, and to a ll th e m em bers o f th e
m icro w a ve g ro u p fo r m a ny h e lp fu l discussions.
1 w o u ld lik e to th a n k th e N a tu ra l Science and E n g in e e rin g Research C o u n cil o f
C a n a d a fo r g ra n tin g me a sch o la rsh ip fo r th e years 1990 to 1992.
Professors A lb e r t P a p ie rn ik and D o m in iq u e P om pei s h o u ld also be acknow ledged
fo r th e ir in te re s tin g discussions d u rin g m y sta y a t th e U n iv e rs ite de N ice - S op h ia
A n lip o lis .
V e ry special th a n k s are clue to .Jearn F rancois H u a rd fo r p ro o fre a d in g th is thesis.
1 w o u ld also lik e to th a n k th e stu d e n ts o f th e m icro w a ve g ro u p o f th e U n iv e rs ity
o f O tta w a and e sp e cia lly S y lv a in , R edouaue and M y k e fo r m a n y com m ents on m y
w ork.
F in a lly , 1 w o u ld lik e to express m y sincere a p p re c ia tio n to m y w ife C a ro lin e , fo r
s u p p o rtin g me d u rin g m y w o rk and s p e c ia lly d u rin g th e w r itin g o f th is thesis.
v
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
C on ten ts
1 Introdu ction
1
1.1
O v e r v ie w .......................................................................................
1
1.2
O rig in a l C o n trib u tio n s o f th is T h e s i s .............................
3
1.3
T hesis O u t l i n e ............................................................................
3
2 T h e T ransm ission-L ine M atrix M eth od
2.1
P ro p a g a tio n o f E le c tro m a g n e tic W a v e s .........................
5
2.2
L im ita tio n o f th e C o m p u ta tio n a l D o m a in ......................
10
2.3
E x c ita tio n and R e s p o n s e ......................................................
12
2.3.1
. . .
13
......................
16
2.4.1
In ve rsio n o f th e A l g o r i t h m ....................................
17
2.4.2
A p p lic a t io n s .................................................................
IS
C o n c lu s io n ...................................................................................
18
2.4
2.5
3
5
E v a lu a tio n o f th e S c a tte rin g P a ra m e te rs
T h e T im e Reversal o f th e T L M M e th o d
T he T echnique of Sh ape R eco n stru ctio n
21
3.1
S c a tte rin g o f an In c id e n t W ave b y a M e ta llic O b sta cle
21
3.2
P re s e n ta tio n o f th e P r o c e d u r e ............................................
25
3.3
P rocessing o f th e F in a l I m a g e ............................................
27
3.3.1
28
T h e T ransverse E le c tric F ie ld : E y ......................
v ii
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4
5
6
3.3.2
T ra n sve rsa l M a g n e tic F ie ld : I L ..............................................................
21
3.3.3
L o n g itu d in a l M a g n e tic F ie ld : / / . ...............................................................U
3.3.4
(C om b ina tion o( th e F ie ld C o m p o n e n t s ...........................................
31
3.4
M u ltip le O b s ta c le R e c o n s tr u c tio n ......................................................................
38
3.5
Shape re c o n s tru c tio n u sing th e 3 D - T L M condensed n o d e .........................
31)
T h e C om p lete S yn thesis T echnique
45
4.1
A p p ro x im a te A n a ly s is
........................................................................................
-Hi
4.2
U tiliz a tio n o f th e Shape R e c o n s tru c tio n T e c h n iq u e ..................................
46
4.3
D o m in a n t M o d e E x tr a c tio n
..............................................................................
47
4.4
M o d ific a tio n o f th e D o m in a n t
M o d e R e s p o n s e .............................................
51
4.5
C o n c lu s io n ...................................................................................................................
55
N u m erical E xam ple
58
5.1
S ynthesis o f an In d u c tiv e S c a t t e r e r ................................................................
58
5.2
C o n c lu s io n ...................................................................................................................
55
P arallelisation U sin g th e C on n ection M achine
66
6.1
T h e C o n n e c tio n M a c h in e ......................................................................................
66
6.2
T h e P a ra lle l T L M A lg o r ith m
...........................................................................
67
6.3
3 D -version o f th e T L M P r o g r a m ........................................................................
69
6.3.1
D a ta S t r u c t u r e ..........................................................................................
71
6.3.2
R e s u lt s .............................................................................................................
7!
C o n c lu s io n ...................................................................................................................
76
6.4
7 C onclusions
78
v iii
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List o f F igu res
1.1
Design process Ik ■lore th e a d ve n t o f d ig ita l c o m p u te rs ................
1
1.2
D esign process using c o m p u te r s im u la t io n s .........................................
2.1
T w o -d im e n s io n a l T L M sh u n t n o d e ......................................................
2.2
B a sic a lg o rith m o f th e T L M m e th o d
2.3
E x a m p le o f th e p ro p a g a tio n process in a tw o -d im e n s io n a l m esh a fte r
6
...............................................................
th e in je c tio n o f an im p u ls e .....................................................................
2.4
In s e rtio n o f a w all in to a 2 D - T L M mesh
2.5
(a ) R e c ta n g u la r w aveguide f ilt e r w ith o u tp u t p o in ts fo r
9
11
........................................................
13
S p a ra m e te rs
e x tra c tio n , (b ) S h o rt section o f th e same w a ve g u id e fo r th e reference
s ig n a l..............................................................................................................
2.6
E x a m p le d e m o n s tra tin g th e tim e re v e rs ib ility o f th e T L M
15
m e th o d . In
(a ) (b) and (c ). th e fie ld set up b y an im p u ls e in th e s tru c tu re u n d e r
s tu d y is disp la ye d at 1 = 1, t — 20 and t = 100 re s p e c tiv e ly . A fte r th e
in v e rs io n o f th e tim e sequence, th e source has been re c o n s tru c te d as
d isp la ye d in ( d ) ...........................................................................................
19
3.1
In c id e n t and s ca tte re d fie ld a ro u n d a m e ta llic o b s ta c le ...............
22
3.2
T h e e n tire s tru c tu re in c lu d in g th e sc.atterer y ie ld in g th e to ta l s o lu tio n
3.3
E m p ty s tru c tu re y ie ld in g th e hom ogeneous s o lu tio n ....................
24
3.4
S tru c tu re using a s u rro u n d in g sources................................................
24
ix
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23
3.0
l o p v ie w o f th e p a ra lle l p la tt' w aveguide c o n ta in in g th e m e ta llic scat ­
te rin '.................................................................................................................................
3.6
F ie ld c o m p o n e n ts used in the tw o -d im e n s io n a l T L M analysis o f a waveg­
u id e p ro b le m ................................................................................................................
3.7
26
37
D is tr ib u tio n o f th e m a x im u m values o f th e e le c tric fie ld o b ta in e d fro m
a reverse T L M a n a ly s is............................................................................................
29
3.8
R e s u lt o f th e e x tra c tio n o f th e lo ca l m a x im a fo r each lin e .......................
31)
3.9
D is tr ib u tio n o f th e m a x im u m value o f th e tra n sve rsa l m a g n e tic lie ld
o b ta in e d fro m a reverse T L M
a n a ly s is ...................................................
32
3.10 D is tr ib u tio n o f th e m a x im u m value o f th e lo n g itu d in a l m a g n e tic fie ld
o b ta in e d fro m a reverse T L M
a n a ly s is ...................................................
33
3.11 C o m p a ris o n betw een th e d is tr ib u tio n o f fie ld com ponen ts \ E V\ (a ) and
\ H r \ (b ). a n d th e d iffe re n ce \ H X\ — |i?y[ ( c ) ...........................................
3.r>
3.12 R e su lts o b ta in e d fro m th e d is p la y o f th e P o y n tin g ve cto r. M a g n itu d e
d is tr ib u tio n (a ) and e x tra c te d im age o f th e obsta cle ( b ) .................
3(i
3.13 Series o f im ages sh o w in g 3 in s ta n ts d u rin g th e an a lysis in (a ), (b) and
(c) an d th e re s u lt o f th e re c o n s tru c tio n in (d ). A ll im ages d is p la y th e
d is tr ib u tio n o l th e m a g n itu d e o f th e e le c tric fie ld co m p o n e n t n o rm a l
to th e p a p e r .......................................................................................................
37
3.14 Case o f th e re c o n s tru c tio n o f tw o obstacles h a v in g a large resonance
effe ct betw een th e m ........................................................................................
38
3.15 R e c o n s tru c tio n o f tw o id e n tic a l obstacles placed in sid e a p a ra lle l p la te
w aveguid e. T h e re c o n s tru c tio n o f th e in n e r w a lls is less a c c u ra te th a n
th e e x te rn a l ones.............................................................................................
3.16 R e p re s e n ta tio n o f th e s y m e tric a l condensed node a fte r Johns
40
. . . .
3.17 R e c ta n g u la r w aveguide w ith a m e ta llic ir is m o d e lle d w ith a th re e d im e n s io n a l T L M m esh................................................................................
x
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42
41
4.1
F lo w c h a rt o f th e c o m p le te synthesis p ro c e d u re u sin g th e in ve rsio n o f
th e T L M m e th o d ( D F T = D is c re te F o u rie r T ra n s fo rm ).............................
48
4.2
Idealized re p re s e n ta tio n o f a p a ra lle l p la te w a ve g u id e ..................................
50
4.3
D iffe re n t m odes o f p ro p a g a tio n s th a t can be seen in a tw o -d im e n s io n a l
analysis o f a p a ra lle l p la te w a ve g u id e .................................................................
4.4
P a ra lle l p la te w aveguide, (a ) w aveguide w ith an o b sta cle y ie ld in g th e
to ta l s o lu tio n (b ) e m p ty w aveguide y ie ld in g th e hom ogeneous s o lu tio n .
4.5
52
R e p re s e n ta tio n o f a T E M wave tra v e llin g to th e r ig h t (a) and to th e
le ft (b ). A is th e m a g n itu d e o f th e e le c tric fie ld o f th e w ave....................
4.6
51
N odes a d ja c e n t to th e le ft h a n d side a b s o rb in g b o u n d a ry .
56
V oltages
d e s c rib in g th e to ta l response (a) and volta g e s d e s c rib in g th e d ifference
s ig n a l ( b ) ........................................................................................................................
5.1
S h u n t in d u c ta n c e o f 0.9p H w h ich is used to generate th e s p e c ific a tio n s
fo r th e d e s ig n ................................................................................................................
5.2
59
S c a tte rin g p a ra m e te rs 5’n and S 21 o f an in d u c ta n c e o f 0.9p H . These
are th e design sp e c ific a tio n s fo r th is e x a m p le ..................................................
5.3
57
60
Several p o ssib le o b sta cle cross-sections w h ic h can a p p ro x im a te th e de­
sign c h a ra c te ris tic s o f th e sh u n t in d u c ta n c e .....................................................
61
5.4
A p p o x im a te s tr u c tu r e ...............................................................................................
61
5.5
S c a tte rin g p a ra m e te rs o b ta in e d fro m th e T L M a nalysis o f th e a p p ro x ­
im a te s tru c tu re . T h e y represent th e fre q u e n c y d o m a in response o f th e
d o m in a n t m o d e ............................................................................................................
62
5.6
N ew c o n fig u r a tio n ......................................................................................................
63
5.7
S c a tte rin g p a ra m e te rs o f th e new c o n fig u ra tio n .
T h e lo w fre q u e n c y
p a r t a lre a d y corresponds to th e design s p e c ific a tio n s ..................................
xi
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64
6.1
W a veguide s lr u c u ir e as s im u la te d using th e C o n n e ctio n M a ch in e . D i­
m ensions are given in A / .........................................................................................
6.2
C o m p a riso n betw een th e in fo rm a tio n to be stored in a 3D (a) and a.
2D (6) T L M synthesis p r o b le m ...........................................................................
6.3
E x a m p le o f re c o n s tru c tio n o f a th in iris in a re c ta n g u la r w aveguide.
T h e d is tr ib u tio n o f th e m a g n itu d e o f the ele ctic fie ld in th e cross section
o f th e g u id e is s h o w n ................................................................................................
6.4
E x a m p le o f re c o n s tru c tio n o f a th in iris in a re c ta n g u la r waveguide. In
th is case, th e ob sta cle is a c o n d u c tin g p la te w hich has tw o re c ta n g u la r
holes.
D is tr ib u tio n o f th e m a g n itu d e o f th e e le c tic fie ld in th e cross
section o f th e g u id e. L ig h t g re y in d ic a te s th e highest values o f th e field.
6.5
D ia g ra m o f th e m o d ific a tio n o f th e response o b ta in e d fro m a 3 D -T L M
a n a ly s is ...........................................................................................................................
xn
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List o f T ables
0.1
R e sults c o m p a rin g T L M s im u la tio n s on a D E C 5500 c o m p u te r and on
th e C o n n e ctio n M a c h in e ..........................................................................................
x iii
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C h ap ter 1
In tro d u ctio n
1.1
O v e r v ie w
T h e need fo r m o re e ffic ie n t design to o ls has alw ays been obvious. T h e epoch when
th e designe r had to b u ild a few c o s tly p ro to ty p e s b efore p ro v id in g a good product, is
lo n g past. T o d a y, c o m p u te r s im u la tio n s are an im p o r ta n t p a r t in th e design process.
W it h e ffic ie n t s im u la tio n techniqu es, i t is now po ssib le to p ro d u ce b e tte r re su lts (aster
a t a lo w e r p ric e because c o m p u te rs are able to p e rfo rm th e taslc fa ste r and wi t h
g re a te r fle x ib ility . O v e ra ll, w ith th e a d v e n t o f cheap m e m o ry and fast processors, it
is p o ssible to s im u la te v e ry large s tru c tu re s in a fa ir ly sm all a m o u n t o f tim e . These
Ser ie
of
P ro to ty p e s
F ig u re 1.1: D esign process b efore th e ad ve n t o f d ig ita l co m p u te rs.
1
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C o m p u te r
S im u la t io n :
F ig u re 1.2: D esign process u sing c o m p u te r s im u la tio n s
days, c o m p u te rs are be co m in g p o w e rfu l enough to suggest th a t th e y could n o t o n ly
p e rfo rm s im u la tio n s b u l also design o f s im p le p ro d u c ts b y them selves. P ro v id e d w ith
a p p ro p ria te s p e c ific a tio n s , c o m p u te rs are ab le to fin d th e o p tim u m values o f some
p a ra m e te rs th e device to be designed. M a n y te chniqu es can be used to achieve such a
goal. M o s t o f th e m are based on a fre q u e n c y d o m a in a n a lysis te c h n iq u e co m b in e d w ith
an o p tim iz a tio n s tra te g y . These techniqu es are c u r re n tly a p p lie d in m a n y fie ld s, such
as th e design o f th e rm a l s tru c tu re s in m e ch a n ica l e n g in e e rin g , th e design o f d ig ita l
c irc u its o r e le c tro m a g n e tic s tru c tu re s in e le c tric a l e n g in e e rin g , or flu id m echanics in
c iv il e n g ine e rin g . H ow e ver, tim e -d o m a in n u m e ric a l te chniqu es are n o t w ell s u ite d fo r
th is k in d o f design approach since every a n alysis re q u ire s th e tra n s ie n t b u ild u p o f th e
tim e response fro m an im p u ls iv e e x c ita tio n w h ich in vo lve s a la rg e c o m p u ta tio n tim e .
T h u s , th e repe ated analyses w ould be ve ry w a ste fu l.
Hence, a d iffe re n t a p proach
sh ou ld be ta ke n to p e rfo rm th e synthesis in th e tim e -d o m a in .
T h is thesis re p o rts a novel te c h n iq u e fo r th e syn th e sis o f c o n d u c tin g s c a tte re rs
based on a lte rn a te fo rw a rd and b a ckw a rd analyses w ith o u t re q u irin g tr a d itio n a l o p t i­
m iz a tio n proce d ure s. I t takes advantage o f th e tim e d im e n sio n since each tim e -d o m a in
a n a lysis p ro v id e s f u ll fre q u e n c y band in fo rm a tio n . I t is th e re s u lt o f a s tu d y to use
tim e reversal o f th e T L M m e th o d to p e rfo rm th e synthesis o f e le c tro m a g n e tic s tru c ­
tu re s, c o n d u c te d b y S o rre n tin o et al.
[ l ] in 1991.
R e su lts show th a t it offers an
9
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in te re s tin g a lte rn a tiv e to th e tr a d itio n a l synthesis techniques w h ich have heen used
so fa r.
1.2
O r ig in a l C o n tr ib u tio n s o f t h is T h e s is
T h e fo llo w in g o rig in a l c o n trib u tio n s w ere m ade w h ile in v e s tig a tin g th e tim e reversal
p ro p e rty o f th e T ra n s m is s io n -L in e M a tr ix m e th o d .
• T h e d e v e lo p m e n t oi an a c c u ra te te ch n iq u e fo r re c o n s tru c tin g th e im age ol a
m e ta llic, s c a tte re r fro m its T L M tim e response. Based on th e p rin c ip le s in tr o ­
duced in [1], several techniqu es fo r precise d e s c rip tio n o f a m e ta lic obstacle,
u s in g th e fie ld c o m p o n e n ts a va ila b le fro m a T L M a n a lysis, were developed,
• T h e d e v e lo p m e n t o f a new synthesis p ro c e d u re based on th e shape re co n stru c­
tio n te c h n iq u e u sing tim e reversal. T h is p ro ce d u re in clu d e s a novel te ch n iq u e
fo r m o d ify in g th e responses o f T L M analyses.
• T h e im p le m e n ta tio n o f th e new synthesis procedure s on a m assively p a ra lle l
c o m p u te r, th e C o n n e c tio n M a ch in e .
So fa r, no p re v io u s w o rk using a n u m e ric a l tim e reversal p ro ce d u re has been
p u b lis h e d . T h e syn th e sis te c h n iq u e presented here is c o m p le te ly new in th e field o f
n u m e ric a l m o d e lin g as a p p lie d to th e design o f m icro w a ve s tru c tu re s .
1.3
T h e s is O u tlin e
A fte r th is in tr o d u c tio n , C h a p te r 2 gives a s h o rt o v e rv ie w o f th e T ra n s m is s io n -L in e
M e th o d .
In C h a p te r 3. th e te c h n iq u e o f shape re c o n s tru c tio n is presented, and a
p ro g ra m m in g te c h n iq u e w ill be d e m o n s tra te d . Its p u rpose is to c le a rly d e m o n s tra te
how to re c o n s tru c t th e shape o f a p re v io u s ly analysed o b sta cle by reversing th e T L M
3
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process. C h a p te r 4 th e n presents th e co m p le te synthesis p ro c e d u re w h ich is based on
th e shape re c o n s tru c tio n te c h n iq u e . N u m e ric a l results are presented in C h a p te r 5, and
th e p a ra lle liz a tio n o f th e te c h n iq u e is e x p la in e d in C h a p te r 6. F in a lly , th e C on clu sio n
sum m arizes th e thesis and presents several a p p lic a tio n s o f th is novel p ro c e d u re in
some o th e r areas.
4
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C h ap ter 2
T h e T ran sm ission-L in e M a trix
M eth o d
T h e T ra n s m is s io n -L in e M a t r ix ( T L M ) m e th o d was firs t in tro d u c e d by Johns and
B e u rle [2] in th e e a rly seventies.
I t is a d iscre te tim e and d iscre te space s a m p lin g
m e th o d fo r m o d e lin g th e p ro p a g a tio n o f e le c tro m a g n e tic waves.
T h e p ro p a g a tio n
space is re p rese nte d b y a mesh o f in te rc o n n e c te d tra n s m is s io n lines.
p resents th e d iffe re n t aspects o f th is n u m e rica l te ch n iq u e .
T h is ch a p te r
For th e p urpose o f th is
p r o je c t, th e tw o -d im e n s io n a l version o f th e m e th o d w ill be em phasized.
A p p lic a ­
tio n to 3 D p ro b le m s is o n ly presented as an im p ro v e m e n t o f th e research p ro je c t in
S ection 3.5.
2.1
P r o p a g a t io n o f E le c tr o m a g n e t ic W a v e s
T h e T L M m e th o d is a d is c re tiz e d version o f H u yg e n s’ p rin c ip le . T h e la tte r specifies
th a t each wave fr o n t c rea te d by a r a d ia tin g source can be represented by a new series
o f sources, w h ich in tu r n , create a new series o f wave fro n ts .
T o m ode! th e wave
p ro p a g a tio n space, th e T L M m e th o d em ploys a g rid o f in te rc o n n e c te d tra n sm issio n
5
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F ig u re 2.1: T w o -d im e n s io n a l T L M s h u n t node.
lines w ith sp e cific lin e pa ra m e te rs. T h e p ro p a g a tio n o f e le c tro m a g n e tic waves is re p re ­
sented, in th e T L M m esh, b y th e m o vem ent o f vo lta g e im p ulses alo n g th e tra n sm issio n
lines. T h e d iffe re n tia l e qu a tio n s w h ich ru le th e voltages a n d th e c u rre n ts on th e lines
are o f th e same fo rm as M a x w e ll's e q u a tio n s. T h e re fo re , a re la tio n can be e stablishe d
betw een th e e le c tro m a g n e tic fie ld s in th e m e d iu m and th e vo lta g e s and c u rre n ts in
th e T L M m esh. T h e d iffe re n tia l e q u a tio n s re la tin g th e c u rre n ts a nd th e vo lta g e fo r
th e w ell kn o w n tw o -d im e n s io n a l sh u n t node show n in F ig u re 1.1 are:
dx
=
- 4 dtf .
P -D
^
dz
=
Ld
- ±
d t'
(2 0)
(
— + —
d z + dx
=
- 2 C ^-2 dt ’
(2 3)
1
j
w here L and C are th e in d u c ta n c e and th e ca p a cita n ce p e r u n ith le n g th o f th e ele­
m e n ta ry tra n s m is s io n lin e , w h ile Vy, I.x a n d I~ are th e v o lta g e a t th e T L M node and
th e c u rre n ts in th e mesh. S e ttin g ^
= 0, th e fo llo w in g e q u a tio n s are d e rive d fro m
th e M a x w e ll's e q u a tio n :
dEy
dHz
'
dx
=
§■
-
- /r‘ - dt
s f .’
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tn
.
(2-4)
<2-5>
w here E y. H r and H . are the o rth o g o n a l eleet.rie and m a gnetie Held com ponen ts in
th e .r, y and c d ire c tio n s . E q u a tio n s (1.1) to (1.3) can Ire com bined to o b ta in a Iwod im e n s io n a l wave e q u a tio n for th e voltages in th e n e tw o rk as a fu n c tio n o f tim e' and
space
<)x2
d=*
“
dr1
r*
d t- '
In the same m a n n e r we o b ta in fo r th e e le c tric fie ld
c)'2E,,
w here v n is th e v e lo c ity
o f th e
d 2E v
d 2E „
wave in th e m e d iu m .
spondence betw een th e fie ld co m ponen ts in
1 <)~EV
T h is yie ld s th e fo llo w in g c o rre ­
th e m e d iu m , and th e vo lta g e and c u rre n ts
in th e T L M mesh :
Eu
= l y,
Hr
=
I-L
= Jx ,
-L ,
( 2M)
C2.IU)
(2.11)
w ith th e c o n s titu tiv e p a ra m e te rs c and fi re la te d to lin e p a ra m e te rs L and C as
f
-
20,
(2.12)
//
-
L.
(2. Id)
T h e v e lo c ity o f th e w ave described by e q u a tio n 1.7 is th e n given by
=
[ ‘i
M
)
H ow ever, fo r th e e le m e n ta ry tra n sm issio n lin e o f p a ra m e te rs L and C , th e wave
v e lo c ity is
”' = 7 l c '
7
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i2 A r ,}
w hich is equal to th e speed o f lig h t c: in th e case o f a ir fille d tra n s m is s io n lines.
T h e re fo re th e v e lo c ity in th e T L M mesh is 1f y / 2 tim e s th e v e lo c ity on th e e le m e n ta ry
tra n sm issio n lines.
In a tw o -d im e n s io n a l hom ogeneous m e d iu m , th e c u rre n ts and th e v o lta g e a t each
node in th e mesh can be e va lua te d as
v»
=
(2.16)
| [ v v + v ; + v< + v ;].
\q
Vi
(2.17)
V
' 1' — V2
'3
(2.18)
Zo
w here th e V- represent th e vo lta g e im pulses in c id e n t on each node and Z 0 is th e
in tr in s ic im p e d a n c e o f th e m e d iu m .
T h u s , b y se le ctin g th e p ro p e r values o f lin e
p a ra m e te rs ( L an d (7), th e T L M mesh w ill m o d e l a re a lis tic m e d iu m o f p ro p a g a tio n .
A t each p o in t in th e m esh, th e im pulses are sca tte re d so as to re sp e ct th e co n ­
s e rv a tio n o f energy. T h e s c a tte rin g o f in c id e n t v o lta g e im pulses a t every tim e step k
y ie ld s th e re fle cte d im pulses at each node. F o r th e hom ogeneous m e d iu m w e have:
r
-1
’ V) ‘
V-2
V,
1
1 -1
1
—2
. v :> . fc+i
1
1
1
1
v2
1
V3
1
1 -1
1
1
1 -1
’
.
V,
'
(2.19)
V *.
O nce reflected fro m th e nodes, th e im pulses are in c id e n t on th e n e ig h b o u rin g nodes
and th e s c a tte rin g c o n tin u e s w ith th e new set o f in c id e n t im pulses.
(2 .20)
VI
T h e s c a tte rin g and exchange o f im pulses betw een th e nodes in th e mesh are th e basis
o f th e T L M a lg o rith m as shown in F ig u re 1.2. Since th e tra v e l tim e betw een e q u a lly
spaced nodes is c o n s ta n t, we say th a t th e process is synchronou s.
L e t d e n o te th e
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(o r) currents at nodes
as required
Loop for a
given num ber
o f iterations
C onvert the reflected
impulses in to
in cide nt impulses to
adjacent nodes
F ig u re 2.2: B asic a lg o rith m o f th e T L M m e th o d
9
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d is ta n c e between nudes as A / . YVe also set th e c o m p u ta tio n a l tim e step equal to A h
T h e tw o p a ra m e te rs are re la te d by th e wave v e lo c ity vi in th e e le m e n ta ry tra n sm issio n
lines
AI
At
(2 .21)
vi ’
F ig u re 1.3 illu s tra te s how a sin gle im p u ls e in c id e n t on a node can p ro p a g a te th ro u g h
th e mesh a t e v e ry tim e step (fo r th e tw o -d im e n s io n a l case). D ue to th e d is c re tiz a tio n
o f th e m e d iu m , some erro rs are in tro d u c e d in th e v e lo c ity o f th e wave p ro p a g a tin g in
th e mesh. These e rro rs and th e ir c o rre c tio n s are w ell described in [3. 4. 5].
F in a lly , th e s c a tte rin g m a tr ix e q u a tio n co rre s p o n d in g to th e m o re general case
w he re th e s h u n t nodes are loaded w ith p e r m itiv ity and loss s tu b s is:
r
V)
V'3
y
V.i
V's
/-*+1
2
2
2Vo
v,
2
2y 0
v2
f
t
1
2
2
2
-to - 2)
2
2t/0
Vh
2
2
2
2y 0
V'4
2
2
2
2
2
1
IT
2
1
ST
i
hO
-to-2)
(2jto ~
.
( 2 .2 2 )
.
w ith
y
w he re y0 and
= 4+
y0 + g0
(2.23)
are th e n o rm a liz e d c h a ra c te ris tic a d m itta n c e s o f th e p e r m itiv ity
s tu b and loss s tu b re sp e ctive ly, as illu s tr a te d in [2].
2 .2
L im ita tio n o f t h e C o m p u t a t io n a l D o m a in
Even i f la rg e c o m p u te r m e m o ry is a v a ila b le , th e e x te n t o f th e s tru c tu re m o d e le d by
th e T L M m esh m ust be lim ite d . B o u n d a rie s are used to r e s tric t th e c o m p u ta tio n a l
d o m a in .
E xce p t fo r closed s tru c tu re s , a b so rb in g b o u n d a ry c o n d itio n s are used to
10
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S c a tte rin g
- i■ t
J,
1~ «iT
1
k = 0
l
< j
i
\
i
S c a tt.e rin p -
,
— ►— —
J
T
l
4 ,
i
' - i
1
7
L
“ T
T
i
7
T
I
i
1
A
1
A t
i
■>
~ T
1
~T
-
L i.
1
h -1
F ig u re 2.3: E x a m p le o f th e p ro p a g a tio n process in a tw o -d im e n s io n a l mesh aftc
in je c tio n o f an im p u ls e .
11
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m odel th e p ro p a g a tio n o f th e e le c tro m a g n e tic waves o u t o f th e s tru c tu re . F o r e x a m ­
ple. to s im u la te a d is c o n tin u ity in a long w aveguide, a b so rb in g b o u n d a rie s are used
to re s tric t th e c o m p u ta tio n a l d o m a in to a sm a ll region a ro u n d th e d is c o n tin u ity . T h e
e quivalence betw een fie ld s a nd voltages o r c u rre n ts in th e mesh suggests th a t th is
cou ld be achieved b y u s in g a p ro p e r re fle c tio n co e ffic ie n t p w h ile depends on th e de­
sired ty p e o f b o u n d a ry .
In th e s h u n t connected tw o -d im e n s io n a l T L M node, ideal
e le c tric and m a g n e tic w a lls are represented b y a -1 and + 1 re fle c tio n co e ffic ie n t, re­
sp e ctive ly. T h e y are in se rte d h a lf-w a y betw een th e nodes to preserve th e sy n c h ro n is m
w ith o th e r im p ulse s in th e n e tw o rk . A n a b s o rb in g b o u n d a ry can be s im u la te d by a d ­
ju s tin g th e c o e ffic ie n t p to th e re fle ctio n
a general b o u n d a ry o f surface im p e d a n c e
011
Z c.
= zc- z_0
'
Zc + Z 0
1
>
H ow ever th is m o de l is v a lid o n ly fo r a n a rro w fre q u e n c y b a n d i f Z c is dispersive.
O th e r te ch n iq u e s fo r m o d e lin g a b s o rb in g b o u n d a rie s over a w id e fre q u e n c y b a n d
have been develop p ed [6] o r are u n d e r s tu d y . A m o n g o th e rs , th e Jo h n s m a tr ix te c h ­
n iq u e w h ic h is based
011
tim e -d o m a in d ia k o p tic s , p ro v id e s good a b s o rp tio n o ve r a
large fre q u e n cy b a n d . F ig u re 1.4 illu s tra te s how b o u n d a rie s are in c lu d e d in a T L M
m esh. N o te th e d is ta n c e o f A i / 2 betw een th e nodes and th e b o u n d a rie s.
2 .3
E x c it a t io n a n d R e s p o n s e
W h ile th e T L M m e th o d is a tim e -d o m a in n u m e ric a l te c h n iq u e , ty p ic a l a p p lic a tio n s
re q u ire re su lts in th e fre q u e n c y d o m a in .
A sin g le im p u ls e e x c ita tio n o f th e T L M
mesh p ro v id e s a f u ll b a n d w id th response in o n ly one a n a lysis. M a th e m a tic a lly , th e
im p u lses p ro p a g a tin g th ro u g h th e T L M mesh are fu n c tio n s w h ic h ta k e a f in ite va lu e
a t th e tim e k and are zero elsew here. T h u s , th e y do n o t co rre sp o n d e x a c tly to th e
D ira c d is tr ib u tio n 8. L e t J(1 ) d enote th e response o b ta in e d fro m a T L M a n a lysis a t
12
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A/
A/
2
•
•
F ig u re 2.4: In s e rtio n o f a w a ll in to a 2 D - T L M mesh
a s p e cific node. E q u a tio n (1 .2 5 ) illu s tra te s th e re su lts o f th e tim e -d o m a in a n alysis of
a s tru c tu re . A t a selected o u tp u t node, th e value o f th e node v o lta g e Vy is taken a t
each tim e step k. T h e d is c re te F o u rie r tra n s fo rm ( D F T ) o f /( /.) is then taken.
/ ( ' ) = E Vy6 ( u - k A l )
k= 0
12.25;
F{u>) = F ( f ( t ) )
(2.2(i)
F or a sin g le fre q u e n cy e x c ita tio n , one has th e p o s s ib ility to use a series o f im pulses
w h ic h sam p le a sine wave; a G aussian envelope can be selected fo r a pulse fo r a bandlim ite d e x c ita tio n .
I t is in te re s tin g to m e n tio n th a t th e e x c ita tio n o f th e s tru c tu re
can be done a t any node in th e m esh.
2 .3 .1
E v a lu a tio n o f t h e S c a tte r in g P a r a m e te r s
S c a tte rin g p a ra m e te rs - o r S p a ra m e te rs - are used to p ro v id e tra n s m is s io n and re­
fle c tio n c h a ra c te ris tic s o f s tru c tu re s u n d e r stu d y. A p ro c e d u re has been developed to
13
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c x tr a d those p a ra m e te rs iro m a T L M a n a lysis [7].
U su a lly , th e s tru c tu re u n d e r s tu d y
to he e v a lu a te d :
is s y m e tric a l and
o n ly tw o p a ra m e te rs need
th e re fle c tio n and th e tra n sm issio n coefficients S i j
and
Thus,
p ro v id e d th e in c id e n t, reflected and th e tra n s m itte d voltages are k n o w n , one can
c o m p u te th e va lu e o f h o th S’n and S-2\- T h e y are eva lu a te d a cco rd in g to th e fo llo w in g
expressions:
ri
,Sn
S»
=
I'oull
k o u fO
/cj
—------* cmiO
= ^ VWo
2
i" ) 7 \
(2-28)
w here VAnois th e reference o r in c id e n t v o lta g e o b ta in e d fro m a s im u lta n e o u s analysis
o f an e m p ty m a tch e d w aveguide section o f id e n tic a l cross-section, Voun is th e v o lta g e
m esured a t th e in p u t o f th e f ilte r (c o n ta in in g b o th th e in c id e n t and th e re fle cte d
waves) and \ 'out 2 F the vo lta g e ta k e n a t th e o u tp u t o f th e sam e f ilt e r (c o n ta in in g
th e tra n s m itte d wave). A ll those responses are in th e fre q u e n c y d o m a in . A F o u rie r
tra n s fo rm is used to get. th e m fro m th e tim e -d o m a in responses o b ta in e d by T L M
analysis.
T o e x p la in th is pro ced ure , le t us lo o k at th e fo llo w in g e xa m p le : suppose th a t we
w a n t to e xtract, th e S p aram e ters o f a w aveguide f ilt e r w h ic h co n ta in s tw o c irc u la r
posts. Its to p o lo g y is shown in F ig u re 1.5. T h e e x c ita tio n consists o f a set o f im pulses
w ith h a lf-s in e d is tr ib u tio n to la unch th e d o m in a n t T E \ o m ode. T h e re are tw o nodes
w h ere th e fie ld response w ill be p ic k e d -u p , a t th e in p u t and a t th e o u tp u t. T h e firs t
one w ill p ro v id e th e in fo rm a tio n on th e co m b in e d in c id e n t a n d th e re fle c te d voltages
w h ile th e second one w ill give th e tra n s m itte d vo lta g e .
T o e x tr a c t th e re fle cte d
voltages fro m th e to ta l vo lta g e m esured a t th e in p u t, a s h o rt se ctio n o f th e same
w aveguid e w ith o u t th e obstacles is used w ith th e same e x c ita tio n .
T h e value V0UtQ
p ick e d -u p in th is e m p ty s tru c tu re p ro vid e s th e in c id e n t v o lta g e since no re fle c tio n
occurs in th is w aveguide.
V
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' owfO
(b)
F ig u re 2.5:
(a) R e c ta n g u la r w aveguide filte r w ith o u tp u t p o in ts fo r S p aram eters
e x tra c tio n , (b ) S h o rt section o f th e sam e w aveguide fo r th e reference sig n a l.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A fte r F o u rie r tra n s fo rm o f those tim e d o m a in signals, th e fre q u e n cy d o m a in so lu ­
tio n s are o b ta in e d . U sing e q u a tio n s (1.27) and (1 .2 8 ), th e s c a tte rin g p a ra m e te rs are
c a lc u la te d .
E xp e rie n ce has show n th a t th e o u tp u t p o in ts m u s t be fa r enough fro m th e disco n ­
tin u itie s to g e t o n ly th e d o m in a n t m ode response. T h is c o n s tra in s us to e xte n d th e
c o m p u ta tio n a l d o m a in , th u s in cre a sin g th e c o m p u ta tio n a l tim e as w ell. T o im p ro v e
th is s itu a tio n , one can use a F o u rie r series expansio n o ve r th e tra n s v e rs a l d im ension
o f th e g u id e a t every tim e ste p near th e d is c o n tin u itie s , th u s re d u c in g th e size o f the
mesh, i t w ill re s u lt in a s e p a ra tio n o f th e a m p litu d e s o f every m odes. B y e x tra c tin g
th e F o u rie r serie c o e fficie n t c o rre sp o n d in g to th e m o d e , o f in te re s t, a t every tim e ste p ,
i t is possible to b u ild a tim e -d o m a in signal fo r a sin g le m o d e o f p ro p a g a tio n . U s u a lly,
th e m o d e o f in te re s t is th e d o m in a n t m ode. F o r m o re d e ta ils a b o u t th is te ch n iq u e ,
refer to S ection 4.8 w h ic h deals w ith th e e x tra c tio n o f th e d o m in a n t m o d e response
o f a s tru c tu re .
2 .4
T h e T im e R e v e r s a l o f t h e T L M M e t h o d
T h e T L M m e th o d , as w e ll as th e o th e r tim e -d o m a in n u m e ric a l te ch n iq u e s are re la ­
tiv e ly new in th e fie ld o f n u m e ric a l m o d e lin g o f e le c tro m a g n e tic s tru c tu re s . So fa r,
th e ir use is lim ite d to th e c o n v e n tio n a l tim e -d o m a in a n a lysis o f s tru c tu re s .
O th e r
avenues s till re m a in to be in v e s tig a te d . In th is se ctio n , an im p o r ta n t p ro p e rty o f th e
T L M m e th o d , th e tim e reversal p ro p e rty , is presented . T h is p ro p e rty p ro vid e s new
w ays o f d e sign in g m ic ro w a v e s tru c tu re s . In p a rtic u la r , i t is th e idea b e h in d th e new
T L M synthesis te c h n iq u e presented in th is thesis.
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2.4.1
In v ersio n o f th e A lg o r ith m
The T L M
a lg o rith m consists o f a series o f s c a tte rin g ite ra tio n s p e rfo rm e d at each
node. R e call th a t th e basic m a tr ix e q u a tio n fo r th e tw o -d im e n s io n a l T L M s c a tte rin g
is
I
Vi
v2
1
V-i
y
v4
A*+l
- { y - 2)
2
2
2
2g,
W
2
-(y-2)
2
2
2y
V2
2
2
-lit-2)
2
2y0
V:s
2
2
2
~ ( y - 2)
2y0
\/
• -i
2
2
2
2
(2y0 - y)
V'n
(2.29)
k
w h e re y a n d y 0 are v a ria b le s re la te d to th e d ie le c tric co n s ta n t and loss p a ra m e te r.
In th is syste m o f e q u a tio n s , th e im p u lse s in c id e n t on each nodes are used to c o m p u te
th e re fle c te d im p ulse s. T h is is done ite r a tiv e ly each tim e step a t every node.
A ls o , i t is easy to v e rify th a t th e s c a tte rin g m a tr ix is equal to its inverse1.
(2.:w )
.9 = ,9
T h is m eans th a t i t is p o s s ib le to o b ta in th e values o f th e in c id e n t im pulses Irorn th e
k n o w le d g e o f th e re fle cte d im p u lse s w ith o u t any m o d ific a tio n o f th e a lg o rith m since
th e basic m a tr ix e q u a tio n re m a in s th e same. T h is results in a tim e reversal of the
TLM
m e th o d .
In s te a d o f u s in g th e T L M
m e th o d to o b ta in th e fie ld d is tr ib u tio n
in tim e and space fro m a given source, it c o u ld be used b a ckw a rd s to o b ta in th e
e x c ita tio n c o rre s p o n d in g to a given fie ld d is trib u tio n .
T h is p r o p e r ty was f ir s t illu s tra te d by S o rre n tin o , So and H oefer [1] and was p re ­
sented fo r th e fir s t tim e a t th e E u ro p e a n M ic ro w a v e C onference in 1991.
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2 .4 .2
A p p lic a tio n s
A n im p o r ta n t advan ta ge o f tim e -d o m a in s im u la tio n s is th e p o s s ib ility to vis u a liz e
tin* a c tu a l p ro p a g a tio n o f waves in an y s tru c tu re . T h is makes th e T L M m e th o d an
in te re s tin g to o l in e d u c a tio n a l e le c tro m a g n e tic s . T im e -d o m a in s im u la tio n s h e lp th e
u n d e rs ta n d in g o f d y n a m ic p he n o m e n a d escribed by M a x w e ll’s e q u a tio n s u sin g vis u a l
a n im a tio n . W ith th e tim e reversal o f th e T L M m e th o d , it is now p o ssib le to s to p a
fo rw a rd s im u la tio n a t any g iven tim e step and proceed w ith th e b a ckw a rd s im u la tio n .
T h is helps in v is u a liz in g a spe cific effect d u rin g a s im u la tio n b y a lte rn a tin g between
fo rw a rd and b a ckw a rd tim e ste p in g .
A lth o u g h th e e d u c a tio n a l aspect o f th is fe a tu re is im p o r ta n t, th e m a in a p p lic a tio n
o f th is new p ro p e rty o f th e T L M m e th o d lies in th e a b ility to o b ta in a source fro m th e
held d is tr ib u tio n i t creates. T h e T L M m e th o d can be used to re c o n s tru c t th e source
fro m a held d is tr ib u tio n b y re v e rs in g the tim e sequence. A s an e xa m p le , F ig u re 1.6
shows a series o f im ages d e s c rib in g th is process.
In th e e x a m p le a s tru c tu re s u rro u n d e d b y m e ta lic w a lls is analysed , a n d th e process
is reversed a fte r several ite ra tio n s . In (a ), th e s tru c tu re is d isp la ye d w ith th e in itia l
e x c ita tio n w h ic h consist o f a sin g le im p u ls e a t a node in th e s tru c tu re .
F ig u re 1.6
(h) and (c) show th e s tru c tu re a fte r 20 and 100 ite ra tio n s . A fte r 200 ite ra tio n s , th e
process has been sto pp ed and s ta rte d in th e reverse tim e sequence to fin a lly o b ta in
th e re c o n s tru c te d source in (d ). T h is re c o n s tru c te d source is id e n tic a l to th e o rig in a l
one.
T h is e x a m p le shows th e s im p lic ity o f re c o n s tru c tin g a source u sin g th e tim e
re v e rs ib ility p ro p e rty o f th e T L M m e th o d .
2 .5
C o n c lu s io n
T h is ch a p te r has presented th e T ra n s m is s io n -L in e M a tr ix m e th o d w h ic h is a te c h n iq u e
based on H u y g e n s ’ p rin c ip le to s im u la te th e p ro p a g a tio n o f e le c tro m a g n e tic waves.
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F ig u re 2.6: E x a m p le d e m o n s tra tin g th e lim e re v e rs ib ility o f th e T L M m e th o d . In (« )
( b) and (c ), th e fie ld set u p by an im p u ls e in th e s tru c tu re u n d e r s tu d y is disp la ye d
a t t = 1, t = 20 and t = 100 re sp e ctive ly. A lte r th e in v e rs io n o f th e tim e sequence,
th e source has been re c o n s tru c te d as d isp la ye d in (d).
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T h e a p p lic a tio n o f th is tim e -d o m a in n u m e ric a l te ch n iq u e to th e a n a lysis o f general
.structures was show n.
F in a lly , th e p ro p e rty o f tim e reversal o f th e T L M
m e th o d
reveals p ro m is s in g avenues in th e fie ld o f tim e -d o m a in synthesis o f s tru c tu re s .
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C h ap ter 3
T h e T echnique o f S h a p e
R ec o n stru ctio n
T h e tim e reversal p ro p e rty o f th e T L M m e th o d can be used to develop a. new techniqu e
o f shape re c o n s tru c tio n . T h is ch a p te r shows how th e
TLM m e th o d can be used to
re c o n s tru c t th e shape o f a m e ta llic s c a tte re r th a t has p re v io u s ly been analysed. 'The
p ro c e d u re is e x p la in e d in d e ta il, and some exam ples o f re c o n s tru c tio n are presented.
3.1
S c a t t e r in g o f a n I n c id e n t W a v e b y a M e ta llic
O b s ta c le
W h e n a w a v e fro n t is in c id e n t upon a m e ta llic ob sta cle , a p e r tu r b a tio n o f th e field
q u a n titie s is cre ate d due to th e surface cu rre n ts in d u c e d on th e ob sta cle . T h e ta n ­
g e n tia l e le c tric fie ld E is re la te d to th e surface c u rre n t d e n s ity J by th e fo llo w in g
expression:
J = <t E ,
w here a is th e c o n d u c tiv ity o f th e obstacle.
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(3.1)
In n d e n t
Field
>
I—
r * \
Scattered
Field
Scatterer
F ig u re 3.1: Incident, and sca tte re d fie ld a ro u n d a m e ta llic obstacle.
A fte r th e w a v e fro n t has h it th e obstacle, i t becom es d iffic u lt to separate th e in ­
c id e n t and th e s c a tte re d electrom agn etic, fie ld s.
H ow ever, i t is possible to separate
these tw o fie lds i f we k n o w th e c u rre n t d is tr ib u tio n on th e surface o f th e obstacle
and th e value o f th e in c id e n t e le c tro m a g n e tic fie ld . T h is can be achieved e a sily using
n u m e ric a l te chn iqu e s such as th e m e th o d o f m o m e n ts. T h is m e th o d b u ild s a system
o f e q u a tio n s u sin g th e G re e n ’ s fu n c tio n approach w h ic h is th e n solved fo r th e c u r­
re n t d is tr ib u tio n . U s in g M a x w e ll’s eq u a tio n s, th e sca tte re d fie ld due to th is c u rre n t
d is tr ib u tio n on th e o b sta cle can be o b ta in e d .
T h e p o s s ib ility to se p arate th e in c id e n t a n d th e sca tte re d fie ld q u a n titie s suggests
th a t by a p p ly in g th e tim e reversal p ro p e rty to th e sca tte re d e le c tro m a g n e tic fie ld , it
w o u ld be p o s s ib le to o b ta in th e in d u ce d sources in th e same w a y as we can fin d th e
in itia l sources fro m a k n o w n e le c tro m a g n e tic fie ld d is tr ib u tio n (see C h a p te r 2 ). T h is
idea leads to th e te c h n iq u e o f shape re c o n s tru c tio n .
Because o f th e lin e a r ity o f th e T L M m e th o d , th e p rin c ip le o f s u p e rp o s itio n can
be a p p lie d .
T h e re fo re , th e im p u ls e response due to tw o d iffe re n t sources is equal
to th e sum o f th e im p u ls e reponses to each source, ta k e n se parately. L e t the. T L M
tim e -d o m a in reponse a t an o u tp u t node 0 be den o te d by
<j>i{i) and
L e t us also assign
to th e responses to tw o d iffe re n t e x c ita tio n s . T h u s , th e to ta l response
00
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O n tp m Point
o
•
•
St a tte re r
Source P o in t
A b s o rb in g b o u n d a ry
F ig u re 3.2: T h e e n tire s tru c tu re in c lu d in g th e s c a tte re r y ie ld in g th e to ta l so lu tio n
to th e c o m b in e d e x c ita tio n s is
M t) =
+ MO-
{*■'■!)
T h is illu s tra te s how th e p rin c ip le o f s u p e rp o s itio n can be a p p lie d to th e T L M
m e th o d .
N a tu ra ly , i t can also be used to separate th e in c id e n t and sca tte re d held
responses. In o th e r w o rd s, one can separate th e effects o f th e p rim a ry and the induced
sources in a lin e a r T L M s im u la tio n . In th e present case, we w a n t t.o e x tra c t th e field
response to th e in du ced sources fro m th e com bined response- due to b o th p rim a ry and
in d u c e d sources.
L e t us now in tro d u c e a new n o ta tio n to fa c ilita te th e u n d e rs ta n d in g o f th e p ro b le m .
I f <j>r(t) denotes th e to ta l h e ld s o lu tio n , 4>h {1) th e hom ogeneous s o lu tio n (in c id e n t field
w ith o u t th e s c a tte re r), th e n a c c o rd in g to th e p rin c ip le o f s u p e rp o s itio n , we can w rite
th e p a r tic u la r s o lu tio n as
(; u )
w h e re
represents th e fie ld s o lu tio n c o rre sp o n d in g to th e in d u ce d sources on th e
c o n to u r o f th e obstacle.
C o n se q u e n tly , th e to ta l response can be o b ta in e d a t a n y node o f th e T L M mode)
s im p ly b y a n a ly z in g the s c a tte rin g p ro b le m . T o o b ta in th e hom ogeneous s o lu tio n , th e
s c a tte re r is rem o ve d, w h ile a ll th e o th e r features o f th e p ro b le m re m a in th e same. T h e
d iffe re n c e betw een th e to ta l and th e hom ogeneous fie ld s o lu tio n s y ie ld s th e p a rtic u la r
s o lu tio n .
T h e o b je c tiv e o f th is se p a ra tio n o f the responses is to use th e p a rtic u la r
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•
Source P o in t
A b s o rb in g b o u n d a ry
F ig u re 3.3: E m p ty s tru c tu re y ie ld in g th e hom ogeneous s o lu tio n .
F ig u re 3.4: S tru c tu re u sin g a s u rro u n d in g sources.
s o lu tio n in a ba ckw a rd s im u la tio n to p ro v id e in fo rm a tio n a b o u t th e in d u c e d sources.
How ever, u sing a sin g le node as o u tput, p o in t does n o t p ro v id e enough in fo rm a tio n to
a c c u ra te ly re c o n s tru c t th e in d u ce d sources.
In o rd e r to fu lly describe th e lo c a tio n o f th e in d u c e d sources by re ve rsin g the
TLM
m e th o d , in fo rm a tio n a b o u t th e d is tr ib u tio n o f th e fie ld s in space is essential.
For exa m p le , if we s u rro u n d an obsta cle b y N nodes n u m b e re d fro m 1 to N , as seen
in F ig u re 3.4. and use th e m as o u tp u t p o in ts fo r th e a n a lysis, and as in p u t p o in ts fo r
the in je c tio n o f th e inverse sequence, th e re c o n s tru c tio n w o u ld be m o re e fficie n t.
T h e to ta l and th e hom ogeneous fie ld s o lu tio n s <£r(n, i ) and
are fu n c tio n s
o f space and tim e . T h e inverse s im u la tio n y ie ld s th e in d u ce d sources. These in duced
source's in d ic a te the c o n to u r o f th e m e ta llic obstacle. T h e re fo re , a re c o n s tru c tio n of
th e shape o f th e o bsta cle is possible.
In th e e xa m p le o f th e re c o n s tru c tio n o f an im p u ls iv e source {F ig u re 1.6) b y revers-
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ine, th e tim e sequence at te r an analysis, tile inverse process let! to a single source at
tim e t = 0. W h e n a n a ly s in g a s tru c tu re c o n ta in in g a m e ta llic sc a tte re r. th e sources
on th e c o n to u r o f th is o b sta cle are n o t in d u c e d sim u lta n e o u sly. In fa c t, th e w avefront
th a t h its th e o b sta cle creates a series o f asynchrono us ind u ce d sources.
T h e re fo re ,
w hen u sin g th e inverse process to o b ta in these ind u ce d sources, p a rts o f tin* co n ­
to u r w ill a p p e a r a t d iffe re n t tim e steps. For th a t reason, th e reverse process is not
able to re c o n s tru c t a ll th e in d u ce d sources sim u lta n e o u sly. M a n y s o lu tio n s have been
suggested to overcom e th is p ro b le m .
T h e n e x t se ctio n prese- ..s a te ch n iq u e o f shape re c o n s tru c tio n w hich is capable id
d e te rm in in g th e to p o lo g y o f th e sca tte re r.
3 .2
P r e s e n t a t io n o f th e P r o c e d u r e
T o e x p la in th e p ro c e d u re , a sim p le s tru c tu re is used so as to not obscure th e essence, oi
th e te ch n iq u e . L e t th e s tru c tu re be a p a ra lle l p la te w aveguide c o n ta n in g a m e ta llic o b ­
stacle. T h is s tru c tu re has been chosen because it can be tre a te d as a tw o -d im e n s io n a l
p ro b le m since th e re is no p h ysica l v a ria tio n in th e v e rtic a l d im e n sio n . A ls o , such a
s tru c tu re can p ro p a g a te a T E M m ode w h ic h co n s id e ra b ly s im p lifie s th e T L M a n a l­
ysis. A to p v ie w o f th is w aveguide is represented in F ig u re 3.5.
I t is com posed ol
tw o m a g n e tic w'alls and tw o a b so rb in g b o u n d a rie s to lim it th e c o m p u ta tio n a l d o m a in .
Those a b s o rb in g w alls are T E M a b so rb in g b o u n d a rie s represented by a single im p u lse
re fle c tio n c o e ffic ie n t.
T h e a n a lysis is p e rfo rm e d by p la c in g a lin e a r set of im p u ls iv e sources on th e le lt
side o f th e w aveguide. T h e d o m in a n t T E M m ode is generated by in je c tin g , a t every
node in th e e x c ita tio n zone, an im p u ls e h a v in g th e same a m p litu d e . T h e w a v e fro n t
th e n p ro p a g a te s th ro u g h th e g u id e to w a rd th e r ig h t u n til i t h its th e o bstacle.
At
th is m o m e n t, a re fle cte d wave is created. D u rin g th e w hole process, th e tim e -d o m a in
25
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o
Source node
. M agnetic wall
, A bsorbing B oundary
F ig u re '.I Jr. T o p v ie w o f th e p a ra lle l p la te w aveguide c o n ta in in g th e m e ta llic s c a tte re r.
response can be ta ke n at any p o in t in th e m esh. T h e p ro p o se d shape re c o n s tru c tio n
p ro ce d u re consists oi th e fo llo w in g th re e steps:
1. T h e c o m p le te s tru c tu re in c lu d in g th e o b sta cle is analysed. A t every ite ra tio n ,
th e values o f th e im p ulse s in c id e n t a t a ll nodes a d ja c e n t to th e tw o a b s o rb in g
b o u n d a rie s are saved. T h is yie ld s th e to ta l s o lu tio n ^ r ( r c , t) .
2. T h e same s tru c tu re is analysed w ith o u t th e o bstacle.
L ik e in th e firs t p a r t,
th e im pu lses a t b o th a b s o rb in g b o u n d a rie s are saved to y ie ld th e hom ogeneous
s o lu tio n </>//(??.,/).
3. T h e d iffe re n ce betw een th e to ta l and th e hom ogeneous s o lu tio n s , w h ic h is th e
p a r tic u la r s o lu tio n , is re -in je c te d in to th e e m p ty w aveguide in reverse tim e se­
quence. A fte r th e e n tire inverse process, th e in d u c e d sources on th e c irc u m fe r­
ence o f th e o b s ta c le are o b ta in e d .
A s we k n o w , th e fie ld s p ro p a g a tin g in th e s tru c tu re can be visu a lize d in th e tim e d o m a in using a m a p p in g o f th e fie ld q u a n titie s in to a c o lo r scale o r a 3D m esh. Since
th e scattered fie ld has a loca l m a x im u m o r m in im u m a t th e surface o f th e s c a tte re r,
th e value o f th e fie ld a t eve ry node is u p d a te d i f th e c u rre n t va lu e is h ig h e r th a n
any p re v io u s value. A t th e end o f th e process, a d is tr ib u tio n m a p o f th e m a x im u m
fie ld va lue is o b ta in e d .
e x tra c te d .
It is fro m th is m a p th a t th e shape o f th e s c a tte re r can be
D e p e n d in g on th e fie ld co m p o n e n t, a d iffe re n t im age is o b ta in e d and
26
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F ig u re 3.6: F ie ld c o m p o n e n ts used in th e tw o -d im e n s io n a l T L M analysis o f a waveg­
u id e p ro b le m .
d iffe re n t in fo rm a tio n can be d isp la ye d . T h e sources used in b o th analyses, th e first,
and th e second p a rts , m u st be th e sam e, and th e n u m b e r o f ite ra tio n s m u st be equal.
3 .3
P r o c e s s in g o f t h e F in a l I m a g e
In th e tw o -d im e n s io n a l T L M a n a lysis, a m a x im u m o f 3 fie ld co m p o n e n ts can be used.
In th e case o f a lo n g itu d in a l section o f a re c ta n g u la r w aveguide, th e com ponen ts
needed to describe th e p ro p a g a tio n o f a ll T E n0 m odes are E y , H x and I L .
As de­
scrib e d in c h a p te r 2, these fie ld co m p o n e n ts are represented by voltages and cu rre n ts
in th e T L M mesh.
F ro m th e inverse s im u la tio n , i t is possible to o b ta in m aps o f th e m a x im u m value,
o f these fie ld co m p on en ts o r a c o m b in a tio n o f th e m . N e x t we show th e k in d o f results
th a t can be e xp ect fro m these m aps and how to co m b in e th e m to enhance th e precision
o f th e shape re c o n s tru c tio n .
27
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3.3.1
T h e T ran sverse E le c tr ic F ield:
E y
B y d is p la y in g th e m ap o f th e m a x im u m a b so lu te value o f th e e le c tric fie ld d u rin g
th e inverse tim e sequence, one can g e t th e shape and th e p o s itio n o f th e p re v io u s ly
analysed o b sta cle fro m th e lo c a tio n o f th e m a x im u m values o f th e fie ld co m p o n e n t.
I f th e re -in je c te d signal o f th e b a ckw a rd s im u la tio n , w h ic h is th e p a r tic u la r s o lu tio n ,
is due to th e ind u ce d sources, th e n th e h ig h e s t values o f th e e le c tric fie ld should
c o rrespond to th e m ost im p o r ta n t in d u c e d sources on th e c o n to u r o f th e obstacle;
o b v io u s ly th e y w ill in d ic a te th e shape a n d th e lo c a tio n o f th e o bstacle.
F ig u re 3.7
shows as an e xa m p le , a m e ta llic s e p tu m in s id e a p a ra lle l p la te w aveguide. T h e value
o f th e fie ld on each node in th e g u id e is d iscre tize d a n d assigned an in te g e r value
between 0 and 255. T h e set o f in te g e r values has been chosen in o rd e r to use a color
m ap o f 256 colors o r S b its per p ix e l. In th is way, th e shape o f th e o b s ta c le is q u ite
well d e scribe d. In F ig u re 3.7, a c o n to u r re p re s e n ta tio n is used.
T o e x tr a c t th e dim en sion o f an obsta cle w ith b e tte r p re c is io n , an a lg o rith m is
used to lo c a te o f th e lo cal m a x im u m fo r every lin e o f th e m esh. A c rite r io n is used to
d e te rm in e a t w h a t level a m a x im u m va lu e is sto re d ; th is w ill be p a r t o f th e co n to u r.
For th e e le c tric fie ld re p re s e n ta tio n , a p o in t is considered a lo c a l m a x im u m o n ly if
th e a b s o lu te value o f th e slope on each side o f th e node in q u e s tio n is h ig h e r th a n a
given level. T h is level has been e va lu a te d e x p e rim e n ta lly to T h = 6 / A / . U sing th is
th re s h o ld va lu e w ith th e fo llo w in g a lg o rith m , th e lo c a l m a x im a are fo u n d .
A lg o r ith m fo r th e e x tra c tio n o f th e lo c a l m a x im a
• F or each lin e in th e s tru c tu re
— For every nodes on th e lin e
28
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70
0
46
i
F ig u re 3.7: D is tr ib u tio n o f th e m a x im u m values o f th e e le c tric fie ld o b ta in e d fro m a
reverse T L M a n alysis.
29
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F ig u re 3.S: R e s u lt o f th e e x tra c tio n o f th e lo ca l m a xim a , fo r each lin e .
30
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
* I f th e slope on th e left is hu-gei th a n 77; A N I) th e slope on th e rig h t
is s m a lle r th a n —77;. T h e n th e node is d e te cte d as a local m a x im u m
and is assigned a value “ a c tiv e ” .
T h e re s u lt is a m a p o f boolean values w here each a c tiv e no d e corresponds to a
d e te c te d lo c a l m a x im u m . T h i? m ap gives a good re p re s e n ta tio n o f th e to p o lo g y o f
th e s c a tte re r. U s in g th is a lg o rith m , th e exact shape o f th e se p tu m o f F ig u re 3.7 has
been o b ta in e d ; i t is d isp la ye d in F ig u re 3.8.
3 .3 .2
T ra n sv ersa l M a g n e tic F ield :
H x
T h e tra n s v e rs a l m agnetic, fie ld c o m p o n e n t can also be used to e x tra c t th e to p o lo g y
o f an o b stacle .
T h is fie ld c o m p o n e n t behaves in a s im ila r way as does th e e le c tric
fie ld e xcep t t h a t now th e p o s itio n s o f th e local m in im a describe th e shape ol th e
o bstacle .
T h u s , th e same processing can be done on th is im a g e e xce p t th a t th e
a lg o rith m e x tra c ts th e p o s itio n o f th e lo ca l m in im a in ste a d o f th e local m axim a,.
F or th e e x a m p le o f F ig u re 3.7, i t can be seen fro m F ig u re 3.9 th a t th e re su lt ol the
e x tra c tio n o f th e lo c a l m in im a y ie ld s th e same shape o f s e p tu m th a n F ig u re 3.8.
3 .3 .3
L o n g itu d in a l M a g n e tic F ield :
H z
T h e lo n g itu d in a l m a g n e tic fie ld c o m p o n e n t does n o t a p p e a r w hen a p u re T F M wave
p ro p a g a te s. H ow e ve r, it is g enerated in th e fo rm o f th e h ig h e r o rd e r m odes w hen th e
w ave h its an ob sta cle , and e sp e cia lly a t th e lo c a tio n o f th e sharp edges. I t m ig h t also
be g e n e ra te d b y th e coarseness effet o f th e T L M m esh a ro u n d those corners o r sharp
edges. W h e n re -in je c tin g th e p a r tic u la r s o lu tio n in to th e e m p ty w aveguide, th is field
c o m p o n e n t w ill m a in ly show th e p o s itio n o f th e corners o f th e ob sta cle . F ig u re 3.10
shows an e x a m p le o f th e fin a l lo n g itu d in a l fie ld im age o b ta in e d fro m th e inverse T L M
s im u la tio n o f th e m e ta llic s e p tu m .
31
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I2 “
11 2
I i2
O r£
k
F ig u re 3.9:
D is tr ib u tio n o f th e m a x im u m value o f th e tra n s v e rs a l m a g n e tic fie ld
o b ta in e d fro m a reverse T L M analysis.
32
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
F ig u re 3.10: D is tr ib u tio n o f th e m a x im u m value o f th e lo n g itu d in a l m a g n e tic fie ld
o b ta in e d fro m a reverse T L M analysis.
33
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3 .3 .4
C o m b in a tio n o f t h e F ie ld C o m p o n en ts
T h e o b s e rv a tio n th a t th e tra n s v e rs a l e le c tric and m a g n e tic fie ld co m p o n e n ts b o th
y ie ld th e c o n to u r o f o b je c ts in a s im ila r way, suggests th a t th e diffe re n ce between
b o th fie ld s yie ld s a m o re precise im a g e o f th a t o b je c t. Since th e p a r tic u la r e le c tric
fie ld s o lu tio n is m a x im u m a t th e in te rfa c e w h ile th e a c c o m p a n y in g m a g n e tic fie ld
has a lo c a l m in im u m a t th e same lo c a tio n , th e d ifference b etw een th e ir m a g n itu d e ,
j/7 x ( f, j ) j — \ E y ( i , j ) |, co m p u te d a t each node, w ill also d e lim it th e o b sta cle . R e p la cin g
by zero every n e g a tiv e re s u ltin g value, th e o b sta cle w ill be w e ll de scrib e d . T h e re s u lt­
in g fie ld value w ill th u s be m uch lo w e r in th e e m p ty region th a n close to th e obstacle.
F ig u re 3.11 illu s tra te s th e te c h n iq u e o f processing these tw o fie ld co m ponen ts.
A n o th e r ap proa ch is to use th e m a g n itu d e o f th e P o y n tin g v e c to r, w h ic h is a
c o m b in a tio n o f th e th re e fie ld c o m p o n e n ts. L e t | / 3( i, j ) j d e n o te th e m a g n itu d e o f th e
P o y n tin g v e c to r.
|r(i,j')| = |£,(i,j)|
j)|2
(3.4)
T a k in g in to a cco u n t w h a t has been said a b o u t th e fie ld d is tr ib u tio n o f th e d iffe re n t
co m p o n e n ts , it can be show n th a t th e lo c a tio n o f th e o b s ta c le w ill th e n be described
b y lo c a l m in im a o f P { i , j ) ■ F u rth e rm o re , th is app ro a ch gives a b e tte r re p re s e n ta tio n
o f th e o b s ta c le th a n th e lo c a l m in im a o b ta in e d b y th e tra n s v e rs a l m a g n e tic fie ld only.
F ig u re 3.12 d e m o n s tra te s th is . U sing th e a lg o r ith m th a t e x tra c ts th e p o s itio n o f th e
lo c a l m in im a fo r each h o riz o n ta l lin e o f nodes, th e shape and p o s itio n o f th e obsta cle is
re c o n s tru c te d w ith a good p re c isio n . A s a n o th e r exa m p le , F ig u re 3.13 shows a series
o f im ages in c lu d in g b o th th e a n a lysis and th e re c o n s tru c tio n o f a m e ta llic o b stacle.
O ne can see fro m th is e xa m p le th a t o n ly th e le ft side o f th e obsta cle is re c o n s tru c te d .
T h is is due to th e fa c t th a t th e in d u c e d sources have m a in ly be cre a te d on th is side.
34
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(a)
(b )
(c)
F ig u re 3.11: C o m p a ris o n bet.ween th e d is tr ib u tio n o f fie ld c o m p o n e n ts
\ H X\ (b ), and th e d iffe re n ce \ E y \ — \JJT\ (c).
35
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(a ) and
o
r fj
O
in
o
si
N
2
o
A©
■z^
So
(6 )
^'gUl-O 3.JO.
l: Results obtained from the display of the
distribution (a) and extracted imnef* i~r
Sc*o f i]
P
o v
0yiU iru r
e o b s t^
^
i
s
,,
e c tw - ^
3d
Repr^ c e , w m
Permis.sion
° f the
°°P yrig h t
^
FtJ1her
reProc]<Action
Pr°hibitecj
without
Perrrti;
®s/on.
„ ito d e
F ig u re 3.13: Series o f im ages s h e w in g 3 in s ta n ts d u rin g th e a n a lysis in (« ), (b) and
(c) a n d th e re s u lt o f th e re c o n s tru c tio n in (d ). A ll im ages d is p la y th e d is tr ib u tio n o f
th e m a g n itu d e o f th e e le c tric fie ld c o m p o n e n t n o rm a l to th e paper.
37
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T
INSIDE WALLS
CREATING RESONANCE
F ig u re 3.14:
Case u f th e re c o n s tru c tio n o f tw o obstacles h a v in g a la rg e resonance
e ffe ct bet,ween th e m .
3 .4
M u ltip le O b s ta c le R e c o n s t r u c t io n
So fa r. th e re c o n s tru c tio n o f sin gle obstacles does n o t present any m a jo r p ro b le m . T h e
e x a m p le s presented above illu s tr a te th a t it is easy to use th e T L M m e th o d to do so.
H o w e ve r, th e exte n sio n o f th e shape re c o n s tru c tio n te c h n iq u e to m u ltip le obstacles
is n o t s tra ig h t fo rw a rd . C o n sid e r fo r e x a m p le a w aveguide f ilt e r w h ic h in clu d e s tw o
m e ta llic posts. W ith the T L M m e th o d , fo rw a rd tim e -d o m a in a n a lysis is sim p le . T h e
resonance betw een th e tw o obstacles th a t creates th e filt e r c h a ra c te ris tic s is taken
in to account in th e T L M
a n alysis.
T h e tim e -d o m a in response o b ta in e d fro m th is
a n a lysis represents a c c u ra te ly th e repo rise o f th e filte r . H ow ever, th e a p p lic a tio n o f th e
re c o n s tru c tio n te c h n iq u e is n o t s im p le . T h e in fo rm a tio n o b ta in e d a t th e e x tre m itie s
o f th e w aveguide c o rre s p o n d in g to th e to ta l and th e hom ogeneous fie ld s o lu tio n s
c o n ta in s in d ire c t in fo rm a tio n a b o u t th e sources in d u ce d on th e in s id e w a ll o f th e
o bsta cle. F ig u re 3.14 illu s tra te s th is s itu a tio n .
The in d u c e d sources created on th e sides o f th e obstacles fa c in g th e w a ve fro n t are
38
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d ire c tly r a d ia tin g to w a rd th e a d so rb in g b o u d a ry .
T h e re s u ltin g in fo rm a tio n about
these sources is needed to reconstruct, th e c o n to u r o f th e m e ta llic s ca tte re r. H ow ever,
th e sources in d u ce d on th e in sid e w a lls o f th e obsta cle ra d ia te lirs t to w a rd th e w a ll oi
th e n e ig h b o u rin g o b sta c le , and before rea ch in g one o f th e a b so rb in g b o u n d a rie s , the
im p u lse s tra v e l a few tim e s between th e obstacles. T h is resonance elfect. causes the
a b s o rb in g b o u n d a rie s to receive o n ly second-hand in fo rm a tio n fro m th e in sid e w alls
o f th e obstacles.
I f th e p u rp o se o f th e s im u la tio n is to o b ta in o n ly th e frequency
response o f th e f ilte r , th is effect is o f no im p o rta n c e . T h e p ro b le m arises when th e
p a r tic u la r fie ld s o lu tio n Ov[ t ) (th e d ifference betw een th e to ta l and the hom ogeneous
fie ld s o lu tio n s o b ta in e d fro m th e tw o analyses) is used as a source in the inverse T L M
s im u la tio n w here it is exp ected to re c o n s tru c t th e shape o f th e obstacle. In th is case,
th e im p ulse s are n o t a b le to e n te r in to th e inverse resonance for th e reason that, th e
tw o obstacles are n o t th e re anym ore.
T h is re sults in a p o o r re c o n s tru c tio n o f th e
in s id e w a lls o f th e s c a tte re rs . A n exa m p le o f th is is given in F ig u re T i n ; it. is easy
to see th a t th e e x te rn a ! w ails are described w ith m uch b e tte r d e ta il th a n th e inside
w alls.
3 .5
S h a p e r e c o n s t r u c t io n u s in g t h e 3 D -T L M c o n ­
d en sed n od e
T h e te c h n iq u e de scrib ed in Section 3.4 can also be a p p lie d to th e th re e -d im e n sio n a l
T L M m e th o d [3, 4. o j.
In th is case, a m o re general p ro b le m can be analysed since
th e re is no lim ita tio n on th e n u m b e r o f fie ld co m p o n e n ts th a t can vary.
W ith th e
3 D - T L M m e th o d a ll six fie ld com ponen ts can be c o m p u te d , and th e co m p le te set
o f M a x w e ll’s e q u a tio n s can be s im u la te d .
T h e use o f th e s y m m e tric a l condensed
n od e p ro v id e s th e p o s s ib ility to im pose and c o lle c t a ll six field co m p o n e n ts a t tin ;
sam e lo c a tio n in spare. F ig u re 3.16 shows a sch e m a tic re p re se n ta tio n o f a 3 l ) - ' l L M
39
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F ig u re 3.15: R e c o n s tru c tio n o f tw o id e n tic a l obstacles placed in sid e a p a ra lle l p la te
w aveguide. T h e re c o n s tru c tio n o f th e in n e r w a lls is less a ccu ra te th a n th e e x te rn a l
ones.
condensed node.
As fo r the tw o -d im e n s io n a l case, th e s tu d y w ill be lim ite d to a w aveguide p ro b le m
w here th e re c o n s tru c tio n o f a s c a tte re r placed in th e m id d le o f a re c ta n g u la r w aveguid e
is p e rfo rm e d . In th is 3 D p ro b le m , th e o b sta cle is n o t re s tric te d to be u n ifo rm in y
d ire c tio n to preserve th e T E no m ode.
In fa c t, i t can be o f any shape.
T h e m a in
d ilf’erence w ith th e p re vio u s case resides in th e fa c t th a t in ste a d o f sa ving th e im p u lse s
on a lin e a t each o u tp u t, th e im p ulses e m e rg in g fr o m th e s tru c tu re are now p ic k e d
up in a plane on each side. T h u s, a la rg e r a m o u n t o f m e m o ry is needed to save th e
to ta l, the hom ogeneous and the p a rtic u la r s o lu tio n s .
T o illu s tr a te th e te c h n iq u e , a s im p le m e ta llic iris centered in a re c ta n g u la r w aveg­
uid e w ill be used. Since th is iris is m o d e lle d by an in fin ite ly th in m e ta llic w a ll, th e
le n g h t o f the w ave gu id e section can be r e la tiv e ly s m a ll co m pared to tw o o th e r d im e n ­
sions
and y. 'Phis re sults in a very dense mesh to d e scrib e th e tra n s v e rs a l p la n e o f
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V7
via
V4
V3
V6
V9
VS
r v i
V5
F ig u re 3.16: R e p re s e n ta tio n o f th e s y m e tric a ] condensed n o d e a fte r .Johns
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F ig u re 3 .]7 :
R e c ta n g u la r w aveguide w ith a m e ta llic iris m o d e lle d w ith a three-
d im e n s io n a l T L M mesh.
th e w aveguide. In th is way, a b e tte r d e s c rip tio n o f th e ir is w h ic h lies in a tra n sve rsa l
p lan e is o b ta in e d . T h e s tru c tu re tested c o n ta in s o n ly a few ‘'slice s’1 o f nodes. T h e y
each c o rre sp o n d to one o f th e x — y planes in th e w aveguide.
s im u la tio n , th e ir is is in s e rte d betw een tw o o f those slices.
F o r th e p u rp o se o f
F ig u re 3.17 illu s tra te s
th is m o d e llin g . F o u r e le c tric w a lls are used to s im u la te th e m e ta llic w a ve g u id e w alls.
F u rth e rm o re , t wo a b so rb in g b o u n d a rie s are needed on each sides to l i m i t th e c o m ­
p u ta tio n a l d o m a in . These b o u n d a rie s are lo ca te d a t z — 0 and z — Z (w h e re Z is
th e lo n g itu d in a l d im e n s io n o f th e w aveguid e) in th e T L M m esh h a lf-w a y betw een th e
nodes, as in th e tw o -d im e n s io n a l case. F o r each node connecte d to th e o u p u t planes,
th e re are tw o branches c o rre s p o n d in g to th e tw o p o la riz a tio n s o f th e p ro p a g a tin g
wave.
A s can be seen on F ig u re 3.16 re p re s e n tin g th e s y m e tric a l condensed node,
th e re are tw o o rth o g o n a l tra n s m is s io n -lin e s co n n e c tin g th e o th e r nodes in each o f th e
six d ire c tio n s . O n th e le ft-h a n d side o f th e w aveguide, th e tw o lin e s th a t co n n e ct to
th e a b s o rb in g b o u n d a ry are lines 2 and 4. O n th e rig h t-h a n d side, lin e s S and 9 are
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connected to th e o th e r a b s o rb in g b o u n d a ry .
I t is fro m those lines that- th e vo lta g e
im pulses are collected in each o f th e tw o o u tp u t planes. D ue to th e sm a ll n u m b e r o f
nodes in th e lo n g itu d in a l d ire c tio n , th e n u m b e r o f ite ra tio n s re q u ire d fo r th e fie ld to
vanish in th e s tru c tu re is re la tiv e ly s m a ll. F o r exam ple, in th e case o f th e e x c ita tio n
o f th e d o m in a n t m o de ( T E \ o). th e wave f-.unt w ill have reached th e rig h t o u tp u t a fte r
tw ic e th e n u m b e r o f nodes in th e Z d ire c tio n . O n ly som e rip p le s due to th e d ispersion
o f th e high fre q u e n c y c o m p o n e n ts w ill take m ore tim e . A fte r p ro p a g a tio n th ro u g h 10
tim e s th e n u m b e r o f nodes in th e lo n g itu d in a l d im e n s io n , the fie ld in th e w aveguide
vanishes. T h is c o n d itio n m u s t be sa tisfie d b efore re -in je c tin g th e p a rtic u la r s o lu tio n
in to th e e m p ty w aveguide. I t is im p o r ta n t to keep th e n u m b e r o f ite ra tio n s sm a ll since
a la rg e n u m b e r o f im p u lse s m u s t be saved a t each ite r a tio n . F u rth e rm o re , the o u tp u t
planes b e in g close to th e s c a tte re r, th e collected im p u lse s c o n ta in in fo rm a tio n on th e
h ig h e r o rd e r m odes o f p ro p a g a tio n .
In th is way, th e re c o n s tru c tio n o f the shape o f
th e s c a tte re r can be p e rfo rm e d w ith good re s o lu tio n and efficiency.
In th e case o f a th re e -d im e n s io n a l s tru c tu re , th e im a g in g o f th e field co m p o n e n ts
is n o t easy. T h e user m u s t select a tw o -d im e n s io n a l p la n e w here th e field values can
be co n ve rte d to a co lo r m a p and th e n displayed on th e screen. For th e exa m p le o f th e
iris , b y d is p la y in g th e fie ld in th e tra n sve rsa l p la n e ju s t in fro n t o f it , one can d e te c t
th e shape o f th e m e ta llic s c a tte re r a fte r re -in je c tio n o f th e p a r tic u la r response in tin reverse tim e sequence.
U s in g th e C o n n e c tio n M a c h in e (a p a ra lle l c o m p u te r), several s im u la tio n s have
been p e rfo rm e d to re c o n s tru c t a sim p le iris . A c o m p le te e xam ple is given in C h a p te r
6. T h is e x a m p le shows th a t th e te c h n iq u e o f shape re c o n s tru c tio n can be a p p lie d to
th e 3 D version o f th e T L M
m e th o d .
I t p ro vid e s g re a te r fle x ib ility to th e designer
th a n th e 2 D ve rsio n . T h e m e th o d co u ld also be a p p lie d to o th e r typ e s o f s tru c tu re s
such as ra d ia tin g ele m e n t, fo r exam ple.
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C o n c lu s io n
In th is C h a p te r, a new te c h n iq u e fo r re c o n s tru c tin g th e shape o f p re v io u s ly analysed
obstacles has been in tro d u c e d . T h e m e th o d is based on th e tim e reversal p ro p e rty o f
th e T L M m e th o d . N o te th a t th is te ch n iq u e o f shape re c o n s tru c tio n is n o t a synthesis
tech n iq u e .
H ow ever, th e e xam ples o f re c o n s tru c tio n suggest th a t by s p e c ify in g a
w a n te d response, one c o u ld use th e proposed te c h n iq u e to re c o n s tru c t th e shape
o f an ob sta cle th a t y ie ld th e w anted reponse. T h e fo llo w in g C h a p te r describes th e
c o m p le te synth esis te c h n iq u e in d e ta il.
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C h a p ter 4
T h e C o m p lete S y n th esis
T echnique
F ro m a p ra c tic a l p o in t o f v ie w , a genuin e synthesis does n o t start, w ith an a nalysis of
th e w a n te d s tru c tu re . U s u a lly it s ta rts w ith a fre q u e n cy response given as a design
s p e c ific a tio n . C o n s id e r an e xa m p le , th e s p e c ific a tio n s fo r a w aveguide bandpass filte r ,
w h e re lim it in g values fo r S n and S 2t. are given fo r a re la tiv e ly n a rro w fre q u e n cy band.
O u ts id e th is b a n d w id th , no s p e cifica tio n s are u s u a lly g ive n . F u rth e rm o re , in th e case
o f a w aveg uid e p ro b le m , o n ly a single m ode o f p ro p a g a tio n is specified.
H ow ever, th e shape re c o n s tru c tio n te ch n iq u e re q u ire s a re a lis tic tim e -d o m a in re­
sponse w h ic h c a n n o t be o b ta in e d fro m in c o m p le te s p e c tra l s p e c ific a tio n s . A ls o , th e
fie ld d is tr ib u tio n in th e tra n s v e rsa l p la n e o f th e w aveguid e filte r in tim e and space is
u n k n o w n since th e d is tr ib u tio n o f th e e nergy a m o n g th e d iffe re n t m odes o f p ro p a g a ­
tio n is also n o t specified.
H ence, th e shape re c o n s tru c tio n te c h n iq u e requires m ore
in fo r m a tio n th a n is u s u a lly given in th e s p e cifica tio n s.
T h e m is s in g in fo rm a tio n m u st be generate d som ehow to enable th e use o f th e T L M
m e th o d . S p e c ific a lly , we m u s t know th e S -param eters over a w id e r s p e c tru m and have
a good d e s c rip tio n o f h ig h e r o rd e r to re c o n s tru c t th e to p o lo g y o f th e sc a tte re r w ith an
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acce p ta b le accuracy. hi th e fo llo w in g section. a s o lu tio n to th is p ro b le m is proposed.
4 .1
A p p r o x im a te A n a ly s is
In th e tr a d itio n a l synthesis techniques w h ich in v o lv e an o p tim iz a tio n p ro ce d u re , th e
s ta r tin g p o in t is u s u a lly a guess o r an a p p ro x im a te to p o lo g y . T h is fir s t guess is based
on experie nce o r a p p ro x im a te fo rm u la s . In th e case o f c o n d u c tin g obstacles in waveg­
uides, a p p ro x im a te fo rm u la s e x ist in th e lite r a tu r e w h ich relates th e g e o m e try to th e
values o f an e q u iv a le n t lu m p e d e le m e n t c ir c u it fo r th e obstacles. T h e tim e -d o m a in
response o b ta in e d fro m th e T L M a n a lysis o f th e “ firs t guess g e o m e try ” is th e n m o d ­
ifie d in th e b a n d w id th o f in te re s t a c co rd in g to th e sp e cifica tio n s. T h u s , th e re q u ire d
w id e fre q u e n cy ba n d in fo rm a tio n can be generate d.
In o th e r w ords, th e d o m in a n t
m ode response o b ta in e d fro m th e a p p ro x im a te d an a lysis (see section 4.3) is replaced
by th e desired (spe cified ) d o m in a n t m o d e response.
T h u s , th e signal used to s y n ­
thesize th e o b sta cle is com posed o f th e specified response in th e o p e ra tin g fre q u e n cy
range and th e a p p ro x im a te d response a t th e o th e r frequencies. U su a lly, th e o rig in a l
s p e c ific a tio n s are given in th e frequency' d o m a in ; th e m o d ific a tio n o f th e a p p ro x i­
m a te d signal is also p e rfo rm e d in th e fre q u e n cy d o m a in a fte r a F o u rie r tra n s fo rm o f
th e tim e -d o m a in signals. O n ly th e d o m in a n t m ode tim e sig n a l is co n ve rte d in to th e
fre q u e n cy d o m a in . T h e te c h n iq u e used fo r m o d ify in g th e d o m in a n t m o d e response
w ill be presented la te r.
4 .2
U t iliz a t io n o f t h e S h a p e R e c o n s tr u c t io n T e c h ­
n iq u e
T h e m o d ifie d (o r h y b rid ) tim e -d o m a in sig n a l is used in th e shape re c o n s tru c tio n
te c h n iq u e as th e to ta l fie ld s o lu tio n . T h e hom ogeneous s o lu tio n is s u b tra c te d fro m
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it y ie ld in g th e h y b rid p a r tic u la r response. T h is response is re -in je c te d in th e reverse
tim e sequence in to th e e m p ty c o m p u ta tio n a l do m a in to y ie ld an im age o f a new o b ­
stacle. T h is new c o n fig u ra tio n is an im p ro v e d version of th e a p p ro x im a te d obstacle.
O b v io u s ly , a new ana lysis w o u ld v e rify i f th e ch a ra c te ris tic s co rrespond to th e spec­
ific a tio n s .
A few m ore such cycles can be p e rfo rm e d to achieve a good agreem ent
betw een th e s p e c ific a tio n s a n d th e response o f th e s y n lh e tiz e d o bstacle. T h is a lg o ­
r it h m converges q u ite fast.
E xp e rie n ce has shown th a t, d e p e n d in g on th e choice o f
th e a p p ro x im a te d to p o lo g y , o n ly 2 o r 3 ite ra tio n s are needed to achieve s a tis fa c to ry
re s u lts .
I t is im p o r ta n t to m e n tio n again th a t th e sp e cifica tio n s are com pared o n ly in the
b a u d w ith o f in te re s t a n d fo r th e d o m in a n t m ode. F ig u re -1.1 presents th e flow c h a rt
o f th e c o m p le te synth esis p ro ce d u re .
4 .3
D o m in a n t M o d e E x t r a c tio n
T h e use o f a tim e -d o m a in n u m e ric a l m e th o d fo r s im u la tin g th e p ro p a g a tio n of elec­
tro m a g n e tic waves in s id e a w aveguide is not, lim ite d to a single m ode of p ro p a g a ­
tio n .
E ven i f th e user has th e choice o f th e e x c ita tio n (one can e x c ite a g u id e in a
s p e cific m o d e ), h ig h e r o rd e r m odes are u s u a lly generated w hen a, w ave ir o n t h its a
d is c o n tin u ity in s id e th e w aveguide.
D e p e n d in g on th e ir c u t-o fr frequencies and th e
fre q u e n c y b a n d o f th e e x c ita tio n , these m odes can be e ith e r evanescent o r p ro p a g a t­
in g . F or e x a m p le , i f a W R .-90 re c ta n g u la r w aveguide is e x c ite d by a G aussian pulse
(a =
1 0 A I ) h a v in g a h a lf-s in e d is tr ib u tio n in th e tra n s v e rs a l d im e n s io n , th e 7'Ajci
m o de is e x c ite d . H ig h e r o rd e r m odes w h ich are generated by th e d is c o n tin u ity w ill be
evanescent. W h e n u sin g a tim e -d o m a in n u m e ric a l te c h n iq u e , th e to ta l tim e response
c o n ta in s a ll th e m odes, w h e th e r th e y are p ro p a g a tin g o r n o t. T h u s when lo o k in g at
th e tra n s v e rs a l d is tr ib u tio n o f th e e le c tric fie ld in th e w aveguid e, it is n o t possible
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Desired Frequency
Response
Approximated
Time Response
Approxim ated
B
Analysis
Geometn
Empty
Structure
C^) ^
Approximated
Freq. Response
Homogeneous
Time Response
I
Analysis
{—
Approximated
Freq. Response
Modified
rreq. Response
(M o d ifk a tio r)
Modified
Time Response
♦
r
Modified
Time Response
E
(^DdTerent^
Homogeneous
Time Response
Particular
Time Response
j
V
r
F
New
Geometry
V
New
Geometn1
■k
■§*
4 C sD k
Particular
Time Response
j
\
^ T L M r N
L s A n » ly s is y
New Time
Response
New Time
Response
F ig u re -1.1: F lo w c h a rt o f th e c o m p le te synthesis p ro c e d u re u s in g th e in v e rs io n o f th e
T L M m e th o d ( D F T —' D is c re te F o u rie r T ra n s fo rm ).
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)
to d is tin g u is li th e d iffe re n t mode's inside the w aveguide.
all mudes — th e to ta l fie ld s o lu tio n — is ava ila b le .
O n ly the s u p e rp o s itio n ol
Since th e tra n sve rsa l d is t r ib u ­
tio n o f th e fie ld com po ne n ts o f each p ro p a g a tin g m ode is represented by o rth o g o n a l
fu n c tio n s , one can e x tra c t th e m a g n itu d e o f these fu n c tio n s by p e rfo rm in g a F ourier
expansion in th e tra n s v e rs a l p lane o f th e w aveguide. T h e wave th a t propagates inside
th e w aveguid e can be decom posed in to an in fin ite sum o f m odes. For in sta n ce , the
e le c tric fie ld d is tr ib u tio n E y( x ) in th e cross section o f th e gu id e can be expanded in
an in fin ite F o u rie r series as
„
^
/?i7T.r\
£„(:*■) = «<_, +
cos
/H7T , r \
+ K s i"
1)
w here
jf
E y{ x ) d x ,
Each c o e ffic ie n t then corresponds to th e re la tiv e m a g n itu d e o f a m ode present in the
w aveguide.
For th e p a r tic u la r case o f th e p a ra lle l p la te w aveguide, th e dom inant, T E M m ode
corresponds to th e «o c o e ffic ie n t since th is m ode has a c o n ta n t d is tr ib u tio n a lo n g th e
x d im e n s io n . T h e re fo re , it corresponds to th e m ean value o f th e field in tra n sve rsa l
d ire c tio n o f th e w aveguide. H ig h e r o rd e r m ode have a m p litu d e coefficients an since
th e y w ill alw ays have a cosine d is trib u tio n a lo n g th e x axis.
T h e re fo re , fo r such a
w aveguide, all bn co e fficie n ts have a value o f 0.
In th e case o f a re c ta n g u la r w aveguide, a ll an are zero since th e c o n d itio n s a t x — 0
and x = a (th e w id th o f th e g u id e ) force E y to be zero. T h u s , th e field d is tr ib u tio n in
th e cross-section o f th e w aveguide is described by a sum o f sine fu n c tio n s , and every
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Mctallc
p la te
Magnetic
wall
Direction of
propagation
F ig u re -1.2: Id e a liz e d re p re s e n ta tio n o f a. p a ra lle l p la te w aveguide.
h„ co rresp on d to a m ode. T h e T
m ode can be e x tra c te d by c o m p u tin g th e value
o f h\ fro m E q u a tio n (4 .4 ).
H ow e ver, th e fa c t th a t th e responses are ta ke n ne a r th e T E M a b s o rb in g b o u n d a ry
can in tro d u c e an e rro r in th e e v a lu a tio n o f th e d o m in a n t m ode response.
U s u a lly ,
th is e rro r is n e g lig ib le in th e b a n d w id th o f in te rv a l.
le t us now s tu d y th e case o f th e p a ra lle l p la te w ave g u id e . T h is id e a lize d w a veguid e
consists o f tw o m e ta llic w a lls and tw o m a g n e tic w a lls has show n in F ig u re 4.2.
W h e n a n a ly s in g the d is tr ib u tio n o f th e e le c tro m a g n e tic fields in th e cross se ctio n
o f such a w aveguide w ith th e tw o -d im e n s io n a l T L M m e th o d , th e b o u n d a ry c o n d itio n s
im pose a set o f s o lu tio n s w h ich are o rth o g o n a l. F u rth e rm o re , when th e re is no d is tu r ­
bance in th e v o rtic a l d im e n s io n , a il th e e le c tro m a g n e tic fie ld co m p o n e n ts are c o n s ta n t
in th e y -d ire c tio n , so t h a t th e o n ly v a ria tio n s are a lo n g th e x axis. T h e w id th a, o f
th e w aveguide is an in te g e r m u ltip le o f th e h a lf-w a v e le n g th in tra n sve rse d ire c tio n .
U sin g th e a n alo g y w ith th e F o u rie r e xp a n sio n , i t is p o ssible to e x tra c t th e m a g n itu d e
o f e very m ode o f p ro p a g a tio n fro m th e coe fficie n ts o f th e series as m e n tio n e a rlie r.
F ig u re 4.3 shows th e firs t m odes o f th e series.
For th e case o f th e re c ta n g u la r w aveguide, d iffe re n t b o u n d a ry c o n d itio n s are ap-
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(b)
(c)
F ig u re 4.3: DifTerent- m odes o! p ro p a g a tio n s th a t can be seen in a t.w o -d iin n is io n a l
an alysis o f a p a ra lle l p la te w aveguide.
p lie d since th e cross-section o f th e g u ide is com posed o f fo u r e le c tric w alls.
The
d o m in a n t m od e o f p ro p a g a tio n is th e V’ /Mu m ode w hich has a h a lf-sin e d is tr ib u tio n
a lo n g th e a: a xis is c o n s ta n t along th e y axis. 'The constant v e rtic a l d is tr ib u tio n is
present fo r e ve ry T E , li} m ode o f p ro p a g a tio n . O n ly these T E 1ta modes exist, if th e re is
no p h y s ic a l v a ria tio n alo n g th e v e rtic a l axis inside th e guide. T h is means that, every
c u t a lo n g th e y axis w ill y ie ld an id e n tic a l x — z plane c o n te n t.
F in a lly , we have
show n how to e x tra c t the d o m in a n t m ode response o f a s tru c tu re . T h is response is
th e n m o d ifie d a c c o rd in g to th e s p e cifica tio n s, and used w ith th e shape re c o n s tru c ­
tio n te c h n iq u e to p ro v id e th e g e o m e try o f an obstacle c o rre s p o n d in g to th e given
s p e c ific a tio n s .
4 .4
M o d ific a tio n o f t h e D o m in a n t M o d e R e s p o n s e
A n im p o r ta n t p a rt o f th e T L M synthesis of m icro w a ve s tru c tu re s is the m o d ific a tio n
o f th e d o m in a n t m ode in fo rm a tio n p ro v id e d b y th e analysis o f a s tru c tu re w ith a p ­
p ro x im a te ly th e c o rre c t dim e n sio n s. T h e purpose o f th is section is to describe how
to m o d ify th e d o m in a n t m ode to achieve o u r synthesis.
In th e fo llo w in g , i t is shown how to e x tra c t th e d o m in a n t m ode response? o f a
p a ra lle l p la te w aveguide.
L e t th e tim e -d o m a in responses fo r th e e le c tric field o f a
s tru c tu re be denote d by o l j ^ - i . k ) and
/r).
T h e d o m in a n t m ode frequency
51
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Magnetic W all
Abs.
W all
Excitation
Zone pv-J
Abs.
W all
M agnetic W all
(a)
:t
M agnetic W all
Excitation
A bs.
W all
Abs.
WaH
Obstacle
M agnetic W all
z
(b )
f ig u r e 4.4:
P a ra lle l p la te w aveguide,
(a) w aveguide w ith an o b sta cle y ie ld in g th e
to ta l s o lu tio n (b ) e m p ty w aveguide y ie ld in g th e hom ogeneous s o lu tio n .
response is e x tra c te d b y p e rfo rm in g a F o u rie r e xp ansio n o f these signals over the
tra n sve rse d im e n s io n ol the w aveguide a t eve ry ite r a tio n , and by ta k in g th e F o u rie r
tra n s fo rm o f th e re s u ltin g tim e sequence 4>?ef t { k ) and
has a c o n s ta n t tra n sve rsa l d is tr ib u tio n (see F ig . 4.4).
T h e d o m in a n t m o d e
T h e re fo re , th e e x tra c tio n o f
th e d o m in a n t m ode is p e rfo rm e d fo r each ite ra tio n by c o m p u tin g th e average o f th e
e le c tric fie ld o ver th e .r axis, re s u ltin g in th e firs t te rm o f th e F o u rie r series. H ence,
1
ABA'
= A7B X
JrJ
52
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4-5)
and
>
MUX
fo r k — 0 , 1 , 2 . . . A b I 1 — 1. where' X B X a n ti N b lt a re re s p e ctive ly th e n u m b e r o f nodes
in th e tran sverse d ire c tio n and th e n u m b e r o f ite ra tio n s o f th e analysis.
Let
and
denote th e F o u rie r tra n s fo rm o f <j>jj.j,(k) and
Since th e s tru c tu re is e x c ite d fro m th e le ft,
T E M wave, w h ile
^'b
co n ta in s th e in c id e n t and th e re ile cfe d
^ 1(J tra n s m itte d T E M wave. T h is w ill be useful la te r.
L e t th e w a n te d fre q u e n c y responses be denote d by (I>) r / i( u;) and
((<*.’ )• These
are defined o n ly fo r a lim ite d fre q u e n cy band w here o n ly th e d o m in a n t m ode ra n
p ro p a g a te . T h e d iffe re n c e betw een th e w anted c h a ra c te ris tic s and th e response o f th e
d o m in a n t m o d e is o b ta in e d in th e fre q u e n cy d o m a in y ie ld in g
» ♦ £ /,(» )-♦ & .(« )
n-')
and
T h e re s u ltin g signals c o n ta in , re sp e ctive ly, a reflected and a tra n s m itte d T E M wave.
T h e re fo re ,
th e c o rre s p o n d in g tim e -d o m a in signals have a co n s ta n t
tr ib u tio n .
A s m e n tio n e d p re v io u s ly , i t is easy to represent
such
tra n s v e rs a l d is ­
a T E M tra v e llin g
wave in a 2 D - T L M n e tw o rk by using a lin e o f equal node voltages w ith th e p ro p e r
in c id e n t im pulses on each node. T h is is illu s tra te d in F ig u re 4.5. I f th e m a g n itu d e of
th e e le c tric fie ld , A , is set to u n ity , th e n fo r th e wave tra v e llin g to th e le ft, th e field
com p o n e n ts are c a lc u la te d as fo llo w :
= lE u
~
1rl
,
1
,
7^7, + 0 + 2 + ^
=
1.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.9)
Hr
=
4-(V<! - V2)
£(>
=
i(l-O)
/-Jo
b
il,
T h e re s u ltin g P o y n tin g v e c to r
=
~ ( V : - V 3)
=
J _ (2 —
Z0 2
=
0.
£o
I)
2
(4.11)
is in th e —zd ire c tio n ,
a g a tio n . F o r a T E M wave tr a v e llin g fro m th e
E,
=
i E
w
Hr
(4',0)
■
and so is th e d ire c tio n
le ft to th e rig h t,
we have :
l/<
i
=
^
+ 1 + ^ + 01
=
1,
=
U n ~ V 2)
£o
=
T (0 -1 )
(4.12)
_L
IE
o f p ro p ­
=
j ( V , - l / 3)
-
A
=
0.
(4.13)
z. 12 - - r)
(4 .1 4 )
N ow , th e P o y n tin g v e c to r is in th e p o s itiv e z d ire c tio n . T h e re fo re , fo r each tim e step
k\ th e difference signals can be represented by a lin e o f im pulses w e ig h te d by th e value
o f th e inverse F o u rie r tra n s fo rm o f i ^ e } ! i LJ) o r
F in a lly , th e m o d ifie d signals to be re in je c te d in to th e e m p ty c o m p u ta tio n a l d o m a in
54
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th a t yield th e shape of an obstacle according to specifications, are defined as:
{<*>)],
U sing E q u a tio n s (4 .7 ) and (4 .8) in ( 4 .In ) and (4.16) we
*)
=
< £ U . I U ')
=
< £ „« , « -
(•<■'■'>)
o b ta in
- ♦ 1 T /.M 1 -
*) -
~
,( “ ■'))■
H .IT I
d - lS )
F ig u re 4.6 (a) shows how th e tw o r ig h t han d side te rm s o f E q u a tio n s (4.17) and (4.18)
are rep re sen te d. T h e fir s t te rm s ,
and
k) , are th e to ta l voltages of
nodes i a t tim e k re s u ltin g fro m th e v o lta g e im pulses in c id e n t on th a t node.
It. is
th e to ta l s o lu tio n o b ta in e d fro m th e T L M
contain
analysis o f th e s tru c tu re . T h e y
th e d o m in a n t and a ll th e h ig h e r o rd e r m odes.
T h e second te rm s , w hich are the
inverse F o u rie r tra n s fo rm s o f the d iffe re n ce signals, c o n ta in o n ly a T E M wave (e ith e r
p ro p a g a tin g to w a rd th e r ig h t o r th e le ft). T h e re fo re , as m e n tio n e a rlie r, th e values of
th e in c id e n t im p u lse s d e s c rib in g th is signal are equal a lo n g th e tra n sve rsa l dim ension
show n in F ig u re 4.6 (b ) fo r th e le ft in p u t.
4 .5
C o n c lu s io n
In th is C h a p te r, a p a r tic u la r e xa m p le has been used to illu s tr a te how th e com plete
syn th e sis te c h n iq u e can be a p p lie d to o b ta in a sim p le m e ta llic s c a tte re r inside a
w aveg uid e. I t has also been shown how th e m issing in fo r m a tio n n o t in clu d e d in the
design s p e c ific a tio n can be ge nerate d by p e rfo rm in g an a nalysis o f an a p p ro x im a te d
s tru c tu re .
55
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TO W ARD THE R IG H T
TOW ARD THE LEFT
A
4 2' ■ 0
A
0 2' '
. ,.4
.4 2
4 2' ' 0
,.4
A 2
0 2' ’ 4
.,1
/i 2
4 2■ 0
i .A
A 2
0 2' ' 4
■A
2
A
4 21■ 0
.v4
A 2
0 21r 4
L.4
2
4
2
(b)
(a )
F ig u re 4.5: R e p re s e n ta tio n o f a T E M w ave tra v e llin g to th e rig h t (a ) and to th e le ft
(b ). .4 is th e m a g n itu d e o f th e e le c tric fie ld o f th e wave.
56
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A B S O R B IN G B O U N D A R Y ,
A
'2
A
\.A
2
a
(b)
F ig u re 4.6: N odes adjacent, to th e le ft hand aide a b s o rb in g b o u n d a ry . V oltages
s c rib in g th e to ta l response (a ) and voltages d e s c rib in g th e difference signal (b ).
57
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C h ap ter 5
N u m erica l E x a m p le
In th is c h a p te r, th e e n tire synthesis te c h n iq u e is presented step b y step b y d e sig n in g
an in d u c tiv e o bsta cle in s id e a p a ra lle l p la te w aveguide.
5 .1
S y n th e s is o f a n I n d u c tiv e S c a tte r e r
Suppose we w a n t to design an in d u c tiv e o b sta cle w ith an in d u c ta n c e o f 0.9p H , cen­
te re d in a p a ra lle l p la te w aveguide.
U sin g a lu m p e d e q u iv a le n t c ir c u it (show n in
F ig u re 5 .1 ), th e fre q u e n cy d o m a in c h a ra c te ris tic s are c o m p u te d . T h e s c a tte rin g p a ­
ra m e te rs £'n and .S'ai o f th e in d u c ta n c e are show n in F ig u re 5.2 vs th e n o rm a liz e d
freq ue n cy A / / A . R ecall th a t these curves are v a lid o n ly in th e sin g le -m o d e fre q u e n cy
ba n d and fo r th e d o m in a n t m o d e o f p ro p a g a tio n .
A t th is p o in t, th e desig n e r m u s t choose an a p p ro x im a te d s tru c tu re a m o n g several
p o ssible geom e trie s w h ic h c o u ld s a tis fy th e re q u ire m e n ts. T h e obsta cle c o u ld be, fo r
e x a m p le , a c irc u la r o r a re c ta n g u la r p o s t, a th in iris o r a th in tra n s v e rs a l se p tu m . W e
choose th e la tte r fo r th e s im p lic ity o f th e analysis.
As m e n tio n e d in th e p re v io u s c h a p te r, th is g e o m e try is used to gen e ra te an a p ­
p ro x im a te d w id e fre q u e n cy b a n d response w h ic h in clu d e s th e h ig h e r o rd e r m odes o f
58
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h
+
/■»
+
•
ju L
Vi
i
F ig u re 5.1: S h u n t in d u c ta n c e t;f 0.9p H w h ic h is used to generate th e sp e cifica tio n s
fo r th e design.
p ro p a g a tio n .
T o illu s tr a te th e convergence o f th e p ro ce d u re , we set th e firs t guess
w id th o f th e tra n s v e rs a l s e p tu m a r b itr a r ily to 41 A /. T h e 2 D - T L M mesh d e s c rib in g
th e g u id e c o n ta in s 29 b y 100 nodes. M a g n e tic w alls are used to s im u la te th e sides o f
th e p a ra lle l p la te w aveguide, and tw o T E M a b so rb in g b o u n d a rie s are used to lim it th e
c o m p u ta tio n a l d o m a in . T h e a b so rb in g b o u n d a rie s are m ade o f single re fle c tio n co e f­
fic ie n ts w h ic h can abso rb a T E M p la n e wave. F ig u re 5.4 show's th e e x a c t dim e n sio n s
o f th e a p p ro x im a te s tru c tu re .
T h e T L M a n a lysis o f th is s tru c tu re is p e rfo rm e d to o b ta in an a p p ro x im a te d re­
sponse. A s in th e shape re c o n s tru c tio n te ch n iq u e , a ll th e im p ulses in c id e n t up o n th e
a b s o rb in g b o u n d a rie s are saved to y ie ld to to ta l fie ld s o lu tio n
0 au<l'I'L jti'14 0 -
F ro m these, th e d o m in a n t m ode tim e -d o m a in response is e x tra c te d .
T h e n , u sing a.
d is c re te F o u rie r tra n s fo rm , th e fre q u e n c y d o m a in response is o b ta in e d . F in a lly , using
th e te c h n iq u e fo r th e e x tra c tio n o f th e s c a tte rin g p a ra m e te rs presented in tire ch a p te r
2, S \i
and
an> c o m p u te d .
T hese a p p ro x im a te d s c a tte rin g p a ra m e te rs are
show n in F ig u re 5.5.
T h e o b ta in e d response m u st be m o d ifie d a cc o rd in g to th e design sp e cifica tio n s.
In th e c u rre n t ban d w ith o f in te re s t, betw een 0 and 0.01 A / / A , th e a p p ro x im a te d
s c a tte rin g p a ra m e te rs are replaced by th e s c a tte rin g p a ra m e te rs o f th e 0 .9 /;// induc-
59
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Scattering P aram eters
S p e c ifi c a t i o n s f o r L = 0.9pF
0.9
0.8
0.7
0.6
a.
0.5
S21
0.3
0.2
0.0 ---0 .0 0 0
0.006
0.012
0.018
0.024
0.030
N o rm a liz e d F re q u en cy
F ig u re 5.2: S c a tte rin g p a ra m e te rs S n and S'21 o f an in d u c ta n c e o f 0.9p H . T hese are
th e design s p e c ific a tio n s fo r th is e xa m p le .
60
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F ig u re 5.3: Several possible ob sta cle cross-sections w h ich can a p p ro x im a te the design
c h a ra c te ris tic s o f th e shunt in d u cta n ce .
1
41 A l
F ig u re 5.4: A p p o x im a te s tru c tu re .
61
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Scattering P aram eters
A p p ro x im a te d S tru c tu re
0.8
S21
A m p litu d e
0 .7
0 .5
0 .4
0 .3
0.0
—
0 .0 0 0
0 .0 0 6
0.012
0 .0 1 8
0 .0 2 4
0 .0 3 0
N o rm a liz e d F requency
F igure15.5: S c a tte rin g p a ra m e te rs o b ta in e d fro m the T L M a nalysis o f th e a p p ro x im a te
s tru c tu re . T h e y represent th e fre q u e n cy d o m a in response o f th e d o m in a n t m ode.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
I
; 31 A /
F ig u re 5.6: N ew c o n fig u ra tio n .
tanc.e. T h is re s u lts in a h y b rid frequency response w h ich corresponds p a r tia lly to the
a p p ro x im a te d s tru c tu r e and to th e w anted in d u c ta n c e .
T h e h y b r id response is used as th e new to ta l fie ld s o lu tio n in th e shape recon­
s tru c tio n te c h n iq u e . P r io r to re -in je c tin g th e new to ta l field s o lu tio n in to th e e m p ty
c o m p u ta tio n a l d o m a in in reverse tim e sequence, th e d ifference between th is new re­
sponse and th e hom ogeneous response is co m p u te d .
F ro m th e b a c k w a rd s im u la tio n , a new c o n fig u ra tio n is e x tra c te d .
It can l»c seen
th a t it is an id e n tic a l s tru c tu re h a vin g a w id th o f 31A /. F ig u re 5.6 shows th is new
c o n fig u ra tio n . T o v e rify its v a lid ity , a new T L M a n alysis o f th is o b sta cle is pe rfo rm e d .
T h e re s u ltin g d o m in a n t m ode response is given in F ig u re 5.7.
F ro m these responses fo r ,S'n and .S'2J, we can see th a t th e design re q u ire m e n ts are
a lm o s t sa tis fie d . A n o th e r ite ra tio n w o u ld give m o re a c c u ra te re sults. D e p e n d in g on
th e re q u ire d a ccuracy, th e process m ay be sto p p e d here.
63
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Scattering P aram eters
F in a l S tr u c tu r e
0 .9
0.8
0 .7
-
0.6
a.
0.5
S21
0.4
0 .3
0.2
0.0
L —
0 .0 0 0
0 .0 0 6
0 .0 1 2
0 .0 1 8
0 .0 2 4
0 .0 3 0
N o rm a liz e d F requency
F ig u re 5.7: S c a tte rin g p a ra m e te rs o f th e new c o n fig u ra tio n . T h e lo w fre q u e n c y p a r t
a lre a d y corresponds to th e design sp e cifica tio n s.
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
5 .2
C o n c lu s io n
In th is C h a p te r, th e new tim e -d o m a iu synthesis te ch n iq u e has been c o m p le te ly illu s ­
tr a te d u s in g th e design o f an in d u c tiv e obstacle as a sim p le exam ple. T h e size o f a
th in s e p tu m has been co m p u te d in o n ly a few ite ra tio n s . R esults have shown th a t tor
th is case, a cce p ta b le re su lts w he re o b ta in e d in a single ite ra tio n .
65
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C h ap ter 6
P a ra llelisa tio n U sin g th e
C o n n ectio n M ach in e
A n im p o r ta n t pari, o f th e w o rk fo r th is thesis has been p e rfo rm e d in c o lla b o ra tio n
w ith Le L a b o ra to ire d ’E le c t run itp ie o f th e U n iv e rs ite c!e N ice — S o p h ia A n tip o lis in
France. P rofessor A lb e r t P a p ie rn ik , d ire c to r o f th e la b o r a to ry a u th o riz e d th e use o f
th e la b o r a to r y fa c ilitie s fo r fo u r m o n th s , in c lu d in g th e ac.ces to a C o n n e c tio n M achine,
a m a ssive ly p a ra lle l c o m p u te r.
D u rin g th is p e rio d , 1 was aide to te st th e synthesis te c h n iq u e on th is su p e rco m ­
p u te r since th e T ra n s m is s io n -L in e M a t r ix m e th o d is easy to im p le m e n t on it . I t has
re su lte d in a m a jo r re d u c tio n in c o m p u te r tim e fo r m ost o f th e s im u la tio n s . In th is
c h a p te r, th e essential results fro m th is w o rk in F rance are re p o rte d .
6 .1
T h e C o n n e c tio n M a c h in e
T h e C o n n e c tio n M a c h in e is a p a ra lle l c o m p u te r b u ilt by th e T h in k in g M a c h in e C o r­
p o ra tio n , w h ich has a la rg e n u m b e r o f v e ry sm a ll processors. A ll th e processors can
c o m m u n ic a te w ith each o th e r w ith h ig h speed.
H ow ever, th e c o n n e ctio n s betw een
66
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th e m is c o n tro lle d by th e softw are. T h is means that, th e processors can lie co n fig u re d
in such a w ay th a t i t tits tin* T L M mesh u n d e r s tu d y .
T h e re fo re , th e C o n n e ctio n
M a c h in e is a v e ry p o w e rfu l and fle x ib le to o l fo r T L M m o d e llin g .
T h e h a rd w a re o f th e C o n n e ctio n M a ch in e consists o f a few sets o f processors. Each
set is com posed o f 8192 processors. Each processor is o p e ra tin g o n ly on one h it. T h is
e x p la in s th e h ig h n u m b e r o f processors in a set. These can be com bined to o b ta in
a s u p e rc o m p u te r o f 64 th o u sa n d s processors. T h e m e m o ry is also d iv id e d in p a ra lle l
p a rts . Segments o f 256k B y te s are a tta c h e d to every processors. A lso, a p a ra lle l disk
that, p ro vid e s th e a b ility to sto re ve ry large p a ra lle l files w ith a. d ire c t acres fro m th e
processors, is connected to th e system .
F or th e user in te rfa c e , th e C o n n e ctio n M a c h in e is accessed fro m a U n ix w o rk s ta ­
tio n . T h e co m m a nd s to o p e ra te th e C o n n e c tio n M a c h in e are given fro m th is fro n t-e n d
c o m p u te r. F ro m th e user p o in t o f v ie w , th e C o n n e c tio n M a ch in e looks like a s ta n d a rd
U n ix system . O n ly a few new com m ands are needed to op e ra te it..
T o o p e ra te th e C o n n e c tio n M a c h in e , tw o m a jo r languages are a v a ila b le (in a d d i­
tio n to th e basic p a ra lle l assem bly code). T h e m ore o fte n used is tin* C M F o rtra n , a
special F o rtra n la n g ua g e w h ic h is a d a p te d to th e C o n n e ctio n M achine.
A lso , there
is th e C * (C s ta r), w h ic h is an im p ro v e d version o f C + + , th a t in clu d e s a series o f
fu n c tio n s and o p e ra to rs needed to create and use p a ra lle l variables. T h e (U language
has been chosen to do th e w o rk since th e T L M s im u la to r was w r itte n so fa r in C
langua ge. T h e re fo re , th e conversion to a p a ra lle l version was s im p lifie d .
6 .2
T h e P a r a lle l T L M A lg o r ith m
T h e T ra n s m is s io n -L in e M a t r ix m e th o d has been used so fa r a lm o s t e x c lu s iv e ly on
seq u en tia l c o m p u te rs .
T h e a lg o rith m was in it ia lly developed fo r such a c o m p u te r
s tru c tu re . T h e c o m p u ta tio n fo r u p d a tin g th e node vo lta g e is done a t each tim e step
67
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sequent ia lly lo r each node in I,Ik * s tru c tu re .
F rom one node to th e n e x t, o n ly th e
volta ge im p u lse values change, b u t th e c o m p u ta tio n is id e n tic a l.
T h e c o m p u ta tio n
has to be done a t every nodes one a t th e tim e . T h is re s u lts in a la rg e c o m p u ta tio n
tim e . B y p e rfo rm in g th e id e n tic a l c o m p u ta tio n on a ll nodes s im u lta n e o u s ly fo r each
tim e step, it is p ossible to reduce b y a large fa c to r (th e n u m b e r o f nodes) th e C P U
tim e re qu ire d to do a given T L M c o m p u ta tio n . T o th is end, each a v a ila b le processor
is associated w ith a T L M node, and th e basic s c a tte rin g is p e rfo rm e d a t every node
sim u lta n e o u s ly . T h e m e m o ry associated w ith each p rocessor co n ta in s th e re sp e ctive
values o f th e im pu lse s in c id e n t upon th e node.
T h e s im u la tio n s de scrib ed in th is thesis w here don e on th e C o n n e c tio n M a ch in e .
T h e c o m p u ta tio n tim e has been im p ro v e d b y a fa c to r o f a lm o s t 20, d e p e n d in g on th e
s itu a tio n .
A fte r e x e c u tin g th e p a ra lle l version o f th e T L M
synthesis p ro g ra m , th e
fo llo w in g p o in ts w here observed.
• T h e T L M c o m p u ta tio n tim e on th e C o n n e c tio n m a c h in e is im p ro v e d . H ow ever,
th e re d u c tio n in C P U tim e is low er th a n e xp e cte d .
T h e synthesis te c h n iq u e
requires w r itin g and re a d in g on th e d is k th e values o f th e im p u lse s in c id e n t
b o th a b s o rb in g b o u n d a rie s a t every ite ra tio n s .
011
T h is u s u a lly takes m o re tim e
th a n the T L M c o m p u ta tio n its e lf. T a b le 6.1 shows som e re sults c o m p a rin g th e
sequ e n tial and th e p a ra lle l e xe cu tio n tim e w ith and w ith o u t d isk w r itin g .
• T h e re d u c tio n in C P U tim e increases w ith th e n u m b e r o f nodes. In fa c t, th e re
is no m a jo r d iffe re n ce betw een th e se q u e n tia l and th e p a ra lle l c o m p u tio n a l tim e
if o n ly a few nodes are needed to describe th e s tru c tu re .
For e xa m p le , le t » d e no te th e n u m b e r o f o p e ra tio n s to be p e rfo rm e d on p
processors (o r n odes). I f each o f these o p e ra tio n s takes a C P U tim e t, th e n th e
idealized s e q u e n tia l c o m p u ta tio n tim e T s re q u ire d is g ive n by:
T s = n p i.
68
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(6 .1 )
H ow ever. th e p a ra lle l C F U lim e Tv required tu p e rfo rm tin - same task is:
(ti.'J)
1] , = u t .
H ence, th e im p ro v e m e n t given by th e ra tio r o f the sequen tial a m i the p a ra lle l
C P U tim e is:
7' =
= F1v
A s one can see. th e im p ro v e m e n t
011
th e c o m p u ta tio n a l tim e is d ire c tly pro
p o r tio n a l to th e n u m b e r o f nodes in th e T L M mesh.
T h e re fo re , th e use o f a
p a ra lle l c o m p u te r such as th e C o n n e ctio n M a ch in e is m ore efficient, lo r a large
n u m b e r o f nodes. U n fo rtu n a .te ly , th is im p ro v e m e n t is reduced by th e aspect of
th e c o m m u n ic a tio n s betw een th e nodes. T h e ra tio r depends also on th e c o m ­
m u n ic a tio n s w h ic h slow d ow n th e speed o f th e c o m p u ta tio n . T h e com parisons
m e n tio n e d above o n ly show th e effect o f th e n u m b e r o f nodes
011
th e increase' o f
c o m p u te r tim e .
• F in a lly , a ll th e scala r va ria b le s (as opposed to th e p a ra lle l va ria b le s) arc handled
by th e fro n t-e n d c o m p u te r. T h e use o f scalar variables reduces th e p a ra lle liza tio n fa c to r w h ic h , in tu r n , te n d s to reduce th e speed o f th e c o m p u ta tio n , '.['his is
due to th e slow c o m m u n ic a tio n s betw een th e C o n n e ctio n M a c h in e and its fro n tend c o m p u te r. E v e ry tim e th e p ro g ra m needs a scalar v a ria b le , th e c o n tro l is
tra n s fe rre d to th e fro n t-e n d c o m p u te r. T h is makes th e o v e ra ll c o m p u ta tio n a l
tim e la rg e r. B y re d u c in g th e access to the fro n t-e n d c o m p u te r, i f is possible to
reduce th is effect.
6 .3
3 D - v e r s io n o f t h e T L M P r o g r a m
Som e e x p e rim e n ts have been p e rfo rm e d on th e C o n n e ctio n M a c h in e using th e h i)
condensed T L M
node.
T h e p u rp o se was , in fa c t, to in v e s tig a te th e p o s s ib ility o f
69
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D e s c rip tio n o f
C o m p u ta tio n
C o m p u ta tio n
th e s im u la tio n
tim e : sequen tial
tim e : p a ra lle l
301 sec
195 sec
301 sec
18 sec.
600 sec.
20 sec
445 sec.
30 sec.
8192 nodes
1500 ite ra tio n s
8192 nodes
1500 ite ra tio n s
M in im iz a tio n o f th e
scala r variables
16384 nodes
1500 ite ra tio n s
M in im iz a tio n o f th e
sca la r variables
8192 nodes
1500 ite ra tio n s
S aving o f th e
im p u ls io n s a t th e
o u tp u ts
T a b le 6.1: R esults c o m p a rin g T L M s im u la tio n s on a D E C 5500 c o m p u te r a n d on th e
C o n n e c tio n M a ch in e .
70
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a p p l y i n g e f f i c i e n t l y th e 1 L M s y n th e s is te c h n iq u e t o a 31) p r o b le m .
6.3.1
D a t a S tr u c tu r e
T h e in v e s tig a tio n has been re s tric te d to th e p ro b le m s o f th in obstacles inside re c t­
a n g u la r w aveguides. F u rth e rm o re , due to th e re s tric tio n s on th e a rra n g e m e n t o f the
processors, w h ic h im p o se a n u m b e r o f nodes equal to a m u ltip le o f t i l 92, and each
d im e nsion to bo a m u ltip le o f 2. F ig u re 6.1 shows th e w aveguide g e o m e try chosen.
T h e d im en sion s used lo r th is p ro b le m were 126 - 6-1 ■ 8 = 65536. T h is data s tru c tu re
fo rm a t gives a ve ry good re s o lu tio n in th e ,r — y p lane. A lso , since th e g u id e lias a
s h o rt le n g th , th e n u m b e r o f ite ra tio n s re q u ire d u n til th e fie ld vanishes in th e g u id e is
reduced. M o re o ve r, th e p r o x im ity o f th e reference planes to th e o b sta cle provides a
g ood d e s c rip tio n o f th e h ig h e r o rd e r m odes.
T h e m o st im p o r ta n t d iffe re n ce between 2D and 3 D p ro b le m s concerns th e sa vin g o f
th e im p u lses a t b o th ends o f th e waveguide. In th e 3D case, em e rg in g im p u ls o s m u s l be
sto re d fo r a p la n e r a th e r th a n a lin e o f nodes. T h is increases co n sid e ra b ly th e a m o u n t
o f m e m o ry sto ra g e re q u ire d lo r the re c o n s tru c tio n o f th e obstacle.
th e re is th e p ro b le m o f th e a b s o rb in g b o und aries fo r th e 3D T L M
F u rth e rm o re ,
mesh.
In th is
e x p e rim e n t, a sin g le re fle c tio n co e fficie n t has been used to te rm in a te th e lines in th e
a b s o rb in g planes. T h is in tro d u c e s some errors in th e im p u ls e reponses e x tra c te d fro m
th e s tru c tu re b u t does n o t in fluence th e precision o f th e re c o n s tru c tio n . A w ideban d
d isp e rsive a b s o rb in g b o u n d a ry c o n d itio n w ould give m o re accu ra te results.
6.3.2
R e s u lt s
F ig u re 6.3 shows som e re s u lts o f th e re c o n s tru c tio n o f an in d u c tiv e iris inside a re c ta n ­
g u la r w aveguid e. T h e d is p la y shows th e d is trib u tio n o f th e m a g n itu d e o f th e e le c tric
fie ld in th e cross section o f th e g u id e .
F ig u re 6.3 (« ) shows th e d is tr ib u tio n of the
71
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64
Y
X
F ig u re 6.J: W a veguide st ru ct ure as s im u la te d using th e C o n n e c tio n M a c h in e . D im e n ­
sions are given in A / .
00
(b)
F ig u re 6.2: C o m p a ris o n betw een the in fo rm a tio n to be sto re d in a 3D (a ) and a 2 D
(b) T L M syn the sis p ro b le m
72
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F ig u re 6.3: E x a m p le o f re c o n s tru c tio n o f a th in iris in a re c ta n g u la r w avegim lo.
d is tr ib u tio n o f th e m a g n itu d e o f th e elec.tic fie ld in the cross section ol th e gui
show n.
73
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
e lectrir fie ld inside th e g u id e w ith o u t, th e m e ta llic obstacle.
I t is easy to recognize
the h a lf-s in e d is tr ib u tio n o f th e fie ld in th e g u id e . In (6), th e d is tr ib u tio n o f th e fie ld
for th e re c o n s tru c te d iris is show n. T h e edges o f th e iris are lo ca lize d by th e hig h e st
values o f th e field. These are in d ic a te d b y th e b rig h te s t red lines. T h e o b sta cle can
be e a sily re c o n s tru c te d w ith very goo d p re cisio n .
A lth o u g h ! th e c o m p u ta tio n is s tra ig h t fo rw a rd , th e p ro b le m o f v is u a liz in g th e
field com p o ne n ts in a th re e -d im e n s io n a l space is an im p o rta n t one. G o o d re su lts fo r
the re c o n s tru c tio n s were o b ta in e d o n ly fo r som e th in obstacles. F o r co m p le x o b je c ts ,
a c o m p lic a te d processing o f m u ltip le im ages s h o u ld be used. T h e re fo re , o n ly sim p le
s tru c tu re s were te ste d . On th e o th e r h a n d , since th e d is p la y used in th e re c o n s tru c tio n
lies in th e x —y p la n e , it gives a v e ry good re s o lu tio n in th a t p lane. T h is m akes possible
th e re c o n s tru c tio n o f irises o f any shape in th a t p la n e , as can be seen in F ig u re 6.4.
N u m e ric a l e x p e rim e n ts were p e rfo rm e d on th e synthesis o f th in irises fr o m th e
m o d ifie d responses o b ta in e d fro m th e T L M
m e n tio n e d here.
analyses.
Some o b se rva tio n s m u s t be
F ir s t, th e e x tra c tio n o f th e d o m in a n t m o d e response is d iffe re n t
fro m th e 21) case. N ow , th e d o m in a n t m ode is th e T E i o m o d e . I t corresponds in th e
F o u rie r exp an sio n to th e first, te rm o f th e sine series ra th e r th a n to th e co n s ta n t te rm
as in the case o f th e T E M m ode. Hence, th e e q u a tio n
w
)
=
* ( u , o ) si,.
,6.4)
is used to e x tra c t th e a m p litu d e o f th e e le c tric fie ld .
F rom th is lim e -d o m a in sig n a l, one can e x tra c t th e fre q u e n c y d o m a in sig n a l w h ich
m u s t be m o d ifie d a c c o rd in g to th e design s p e cifica tio n s. E x a c tly th e same te ch n iq u e
as fo r th e 2D case can be a p p lie d . T h e o n ly p ro b le m resides in th e C P U tim e needed to
e x tra c t th e d o m in an t, m ode in fo rm a tio n in th e tim e d o m a in . T h is o p e ra tio n requires
th e access to th e p a ra lle l d is k , w h ic h is r e la tiv e ly slow fro m th e fro n t-e n d c o m p u te r.
Since th e re s u ltin g tim e -d o m a in sig n a l is n o t a p a ra lle l va ria b le , i t consum es a la rg e
74
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F ig u re 6.4: E x a m p le o f re c o n s tru c tio n o f a th in iris in a re c ta n g u la r w aveguide. In th is
case, th e o b s ta c le is a c o n d u c tin g p la te w h ic h has tw o re c ta n g u la r holes. D is tr ib u tio n
o f th e m a g n itu d e o f th e e le c tic fie ld in th e cross section o f th e g u id e ,
in d ic a te s th e h ig h e s t values ol th e fie ld .
75
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b ig h t grey
a m o u n t o f tim e . F ig u re 0.5 illu s t r a t e th is c o n v e rs io n p rocess.
6 .4
C o n c lu s io n
T h e T L M syn th e sis te c h n iq u e has been im p le m e n te d on a m a ssively p a ra lle l c o m p u te r.
T h is has re su lte d in a large re d u c tio n o f th e c o m p u ta tio n tim e . H ow ever, even i f th e
conversion o f p a ra lle l to se qu e n tial d a ta is n o t as e ffic ie n t, th e p a ra lle liz a tio n s till
p ro vide s a re sp ecta ble im p ro v e m e n t fo r th e T L M
synthesis c o m p u ta tio n .
F u rth e r
im p ro v e m e n ts could be in tro d u c e d to accelerate th e o ve ra ll process. A m o n g o th e rs ,
th e p a ra lle l im p le m e n ta tio n o f the tim e and fre q u e n cy d o m a in response processing
w o u ld a cce lerate th e e xe cu tio n .
76
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t i . v . u . t)
lin :
1 III
11 1
I,,-,!
'I'
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1
F-
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0
F ig u re 6.5: D ia g ra m o f th e m o d ific a tio n o f th e response o b ta in e d iro m a 6 l> T L M
ana lysis.
(/
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C h ap ter 7
C on clu sion s
In th is thesis, a novel te r h u k jiie fu r th e tim e -d o m a in synthesis o f m e ta llic sca tte re rs
has been presented.
L in e M a tr ix m e th o d .
It is I.»ased on th e tim e reversal p ro p e rty o f th e T ra n s m is s io n It has been shown th a t b y a lte rn a tin g betw een fo rw a rd and
b a ckw a rd analyses, it is p o ssible to re c o n s tru c t th e shape o f a p re v io u s ly analysed
o bstacle.
I t has also been shown how th e high fre q u e n c y in fo rm a tio n m ission fro m
t he design s p e c ific a tio n s can be generated by a n a ly s in g an a p p ro x im a te d to p o lo g y o f
th e s tru c tu re .
A p p lic a tio n s o f th is te c h n iq u e are possible in m a n y fie ld s o f m icro w a ve technolo gy.
A m o n g these are th e design o f d is c o n tin u itite s , m a tc h in g c irc u its o f p la n a r s tru c tu re s
and re c ta n g u la r w aveguides. It m ay also be a p p lie d to th e design o f antennas in o rd e r
to o b ta in , fo r e xa m p le , th e shape o f a p rin te d a n te n n a co rre s p o n d in g to a specified
in p u t im p ed a nce .
F u rth e r research sh o u ld be c a rrie d o u t in th e syn th e sis o f m u ltip le obstacles since
most, p ra c tic a l m icro w a ve devices c o n ta in m ore th a n one d is c o n tin u ity .
T h e p o s s ib ility o f c o m b in in g im age processing w ith th is te c h n iq u e is also a to p ic
o f fu rth e r reaserch. l l m ig h t im p ro v e th e o ve ra ll e fficie n cy b y re d u c in g th e d e n s ity
o f nodes re q u ire d fo r an a c c u ra te re c o n s tru c tio n o f th e obstacle. Som e re ce n t devel-
78
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o p u le n t in sig na l processing co uld lie used
Vo
reduce th e n u m b e r o f T L M ite ra tio n .:
necessary to o b ta in a s a tis fa c to ry fre q u e n cy d o m a in responses.
A ls o , th e T L M a lg o rith m is a d va nta geously im p le m e n te d on a p a ra lle l c o m p u te r.
A la rg e re d u c tio n o f th e c o m p u ta tio n a l tim e can be achieved w hen using th e C onnec­
tio n M a ch in e .
F in a lly , th e te c h n iq u e is s till yo u n g and p ro m issiu g .
M u c h w o rk is needed to
e xte n d its a p p lic a tio n to o th e r types o f s tru c tu re and to reach a level of p rn tic a l
usefulness.
79
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B ib lio g ra p h y
[1] R. S o rre n tia o , P .P .M . So, VV.J.R. H oefer, “ N u m e rica l M ic ro w a v e S ynthesis My
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