IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 8, AUGUST 2002 1895
2-D Analysis o Leakage in P in ed-Ci cui Lines
Using Disc e e Complex-Images Technique
Joaquín Be nal, F ancisco Mesa, Membe , IEEE, and F ancisco Medina, Senio Membe , IEEE
Abs ac —The mixed-po en ial in eg al equa ion is combined
wi h he disc e e complex-images echnique o analyze he com-
ple e spec um o mul ilaye ed p in ed ansmission lines. A
ele an con ibu ion o he p esen wo-dimensional app oach
is i s abili y o s udy bo h he bound and leaky egimes in a e y
simple, sys ema ic, and e icien way. Since he analysis is ca ied
ou in he spa ial domain, his me hod makes i possible o analyze
he leakage phenomenon o s uc u es wi h nonze o- hickness
conduc o s. E icien quasi-analy ical echniques a e employed o
sol e he in eg al equa ion.
Index Te ms—Complex images, leaky modes, mixed-po en ial
in eg al equa ion, p in ed-ci cui lines.
I. INTRODUCTION
INTEGRAL-EQUATION me hods a e widely ecognized as
e y e icien ools o s udying he p opaga ion cha ac e -
is ics o p in ed-ci cui lines. O en, hese in eg al equa ions
a e posed in he spec al domain since he G een’s unc ions
o he laye ed medium a e only known in closed- o m in his
domain [1]. Thus, many wo ks ha e shown he e iciency
o he spec al-domain app oach (SDA) o compu e he
p opaga ion pa ame e s o bo h bound and/o leaky modes
[2]–[5]. Ne e heless, a well-known limi a ion o he SDA is
i s inadequacy o deal p ope ly wi h gene al nonze o hickness
conduc o s. A possible al e na i e o he SDA, able o ea
wi h nonze o- hickness conduc o s, is ound when he co -
esponding in eg al equa ion is di ec ly sol ed in he spa ial
domain [6]. This echnique equi es o compu e he spa-
ial-domain G een’s unc ions om hei spec al coun e pa s
h ough Fou ie - ans o m in e sion. Nume ically e icien
in eg a ion schemes ha e been used in he pas o pe o m
ha ask [7]. Howe e , a powe ul echnique p oposed a he
end o he 1980s, i.e., he disc e e complex-images echnique
(DCIT) [8], has become, in a a ie y o e sions [9]–[11], he
s anda d echnique o ob ain spa ial-domain G een’s unc ions
o laye ed s uc u es. Tha echnique, o iginally in ended o
he analysis o h ee-dimensional (3-D) plana ci cui s and an-
ennas, has been ecen ly adap ed o deal wi h wo-dimensional
(2-D) guiding s uc u es, including s ip-like [12], coplana
wa eguide [13], and a bi a y c oss-sec ional conduc o s [14].
The ele ance o a as gene a ion o he G een’s unc ion in
Manusc ip ecei ed Ap il 4, 2001. This wo k was suppo ed in pa by he
Comisión In e minis e ial de Cienciay Tecnología,Spain, unde P ojec TIC98-
0630, and by Jun a de Andalucía.
J. Be nal is wi h he Depa men o Applied Physics III, Uni e si y o Se ille,
41092 Se ille, Spain.
F. Mesa is wi h he Depa men o Applied Physics I, Uni e si y o Se ille,
41012 Se ille, Spain (e-mail: [email p o ec ed]).
F. Medina is wi h he Depa men o Elec onics and Elec omagne ism,
Uni e si y o Se ille, 41012 Se ille, Spain (e-mail: [email p o ec ed]).
Publishe I em Iden i ie 10.1109/TMTT.2002.801320.
2-D p oblems is ela i ely highe han in 3-D cases because he
CPU ime de o ed o sol e he inal sys em o linea equa ions
is usually negligible in he 2-D si ua ion and he G een’s
unc ions mus be gene a ed many imes o di e en alues o
he unknown p opaga ion cons an .
Al hough [12]–[14] could deal wi h a g ea a ie y o mul-
iconduc o and mul ilaye s uc u es, hey we e pu posely e-
s ic ed o s udy only he bound egime. The impo an ques ion
posed by he exis enceo bo h su ace and space leaky-wa e so-
lu ions has been o en ea ed in he ame o he SDA [3], [5] in
such a way ha only ze o- hickness s ip-like o slo -like s uc-
u es ha e been conside ed in dep h. Mo eo e , accoun ing o
leakage in he ame o SDA equi es a he sophis ica ed phys-
ical/ma hema ical easoning o p ope ly choose he in eg a ion
pa hs ha momen -me hod spec al in eg als ha e o un along.
Thesame easoninghas o beused i hespa ial-domain G een’s
unc ion is ob ained by means o a di ec in eg a ion in he spec-
al domain [7]. Thus, he aim o his pape is o show how o
ex end he me hod p oposed in [12] and [14] o also deal wi h
he leaky egime. This new app oach u ns ou o be e ysimple
and nume ically e y e icien o s ic ly plana s uc u es and
capable o dealing wi h nonplana conduc ing s uc u es. These
aluable ea u es can be also ad an ageously used o s udying
he p ac ical exci a ion o leaky modes [15], [16] since his 3-D
p oblem in ol es as one o i s c ucial s eps he compu a ion o
he eac ion in eg als appea ing in he 2-D case.
II. FORMULATION OF THE PROBLEM
Thespa ial-domainmixed-po en ialin eg alequa ion(MPIE)
will be applied o s udy he wa e p opaga ion in p in ed-ci cui
lines such as ha shown in Fig. 1. The equi ed ke nel o he
MPIE, namely, he spa ial-domain G een’s unc ions associa ed
wi h he scala and ec o po en ials [6], ha e o be ob ained
om hei co esponding spec al e sions. This calcula ion im-
plies o pe o m Fou ie ans o m in e sions o he ollowing
gene al ype:
(1)
whe e
(2)
wi h being he ans e se wa enumbe , being he ee-
space wa enumbe , and being he assumed p op-
aga ion cons an . (No e ha has been used ins ead o
wi h he pu pose o se ing all hese a iables as wa enumbe s;
0018-9480/02$17.00 © 2002 IEEE
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1896 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 8, AUGUST 2002
Fig. 1. C oss sec ion o he s uc u e unde s udy.
, whe e is he a iable used in [14] and o he pa-
pe s.) I can be checked ha he spec al-domain G een’s unc-
ions o he laye ed s uc u e a e ac ually unc ions o p o-
ided ha he media a e iso opic and/o uniaxial aniso opic
dielec ic. W i ing hese unc ions in his con enien way (in-
s ead o as unc ions o ) is e y ele an as i will become
appa en la e .
The DCIT can now be ad an ageously used o a oid he a -
duous compu a ional e o in ol ed in a di ec nume ical in e-
g a ion o (1). The e iciency o his app oach basically lies in
he ac ha he spa ial-domain coun e pa o an spec al ex-
ponen ial e m is known in closed o m hanks o a 2-D e -
sion o he Somme eld iden i y. This idea has been success-
ully applied in [12]–[14] o cha ac e ize bound modes. Ou
pu pose he e is o gene alize ha o mula ion o also accoun
o su ace- and space-wa e leaky modes. This gene aliza ion
equi es a p e ious ea men o he singula i ies (b anch poin s
and poles) o he spec al G een’s unc ions since hese singu-
la i ies a e di ec ly ela ed o adia ion in he o m o space and
su ace wa es [17]. Conside ing ha he unde lying idea o he
DCIT is o ind a sum o complex exponen ial unc ions ha i s
p ope ly and each exponen ial e m is a complex analy ical
unc ion, i is hen c ucial in his me hod o ind ha pa o he
G een’s unc ion ha is ac ually analy ic (i.e., ee o singula i-
ies) wi hin he ange o in e es . In his sense, i is impo an o
no e ha , as a unc ion o , is no a mul i alued unc-
ion because i does no show b anch poin s in he complex
plane. Ne e heless, is s ill me omo phic since i does
ha e poles, which should be ex ac ed ou in o de o isola e he
analy ic pa o . In ac , has an in ini e numbe
o poles a , whe e
(3)
wi h being he wa enumbe o he h mode o he g ounded
laye edsubs a e. Despi e he exis enceo in ini e poles,in mos
p ac ical cases, i is only necessa y o ex ac ou hose poles
appea ing on he eal axis, i.e., he poles associa ed wi h he
abo e-cu o su ace wa es o he g ounded laye ed subs a e
[14]. Fo con enience, he quasi-s a ic con ibu ion should be
also ex ac ed ou explici ly, as o iginally epo ed in [9] and
la e , in a 2-D con ex , in [12].
The combined MPIE–DCIT app oach was used in [14] o
s udy he bound egimeins uc u eswi h nonplana conduc o s.
This app oach will be b ie ly ou lined he e in o de o in oduce
he non i ial changes equi ed o ex end he me hod o dealing
wi h he leaky egime. When nonplana conduc o s a e consid-
e ed, he ke nel o he in eg al equa ion con ains se e al e ms
o he dyadic magne ic ec o -po en ial G een’s unc ion and
some de i a i es plus he e m associa ed o he scala -po en-
ial G een’s unc ion [14]. Thus, he G een’s unc ion p oblem
can be educed o ob aining he space-domain e sion o he ol-
lowing gene ic G een’s unc ion:
(4)
whe e is a equency-dependen cons an and
(5)
wi h being he spec al G een’s unc ion o
(apa om ).
The spec al unc ion can be spli in o he ollowing
h ee di e en con ibu ions:
(6)
whe e accoun s o he quasi-s a ic pa
(7)
(wi h beingaknowncons an ); ep esen s hecon ibu ion
o he signi ican su ace-wa e e ms
(8)
whe e is he pole associa ed wi h he h su ace wa e and
is i s esidue. is he emaining quasi-analy ical pa ha
can be accu a ely expanded as he ollowing sum o complex
exponen ial unc ions:
(9)
whe e and a e, espec i ely, heampli ude anda gumen o
each exponen ial e m ha expands he pa o no depending
o and .
A. T ea men o
The spa ial-domain e sion o can be w i en as
(10)
whe e is gi en by
(11)
wi h . Fo he nonplana case, he abo e in e-
g al and i s de i a i e wi h espec o can be e icien ly
compu ed using in eg a ion con ou echniques, as explained in
he Appendix. Howe e , o he plana case ( ),
(11), which will be deno ed as , can be ob ained in closed
o m.I ela ionship [18,(4.91)] is now adap edand analy ically
ex ended o make con inuous in he en i e complex
plane, can be con enien ly exp essed as
(12)
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BERNAL e al.: 2-D ANALYSIS OF LEAKAGE IN PRINTED-CIRCUIT LINES USING DCIT 1897
Fig. 2. Pa hs o in eg a ion in he complex
k
-plane. Fo simplici y, only a
pai o su ace-wa e poles a e shown (
222
).
in o de o emphasize he na u e o he associa ed ield. F om a
physical poin -o - iew, he case implies ha he
ields associa ed wi h he abo e-cu o h su ace wa e decay
exponen ially in he ans e se di ec ion. Taking all he su -
ace-wa e e ms in his way would be consis en wi h he bound
egime. The su ace-wa e leaky egime is accoun ed o by he
case, which gi es he cha ac e is ic exponen ially
g owing- ield beha io in he ans e se di ec ion [17]. Taking
he second choice o some o he abo e-cu o su ace wa es,
he esul ing leaky egime would accoun o a mode adia ing
in he o m o only hose su ace wa es hus conside ed.
The abo e esul s can be ela ed o he SDA by means o con-
ou in eg a ion echniques. I , o simplici y, only one su ace
wa e is assumed o be abo e cu o , he bound- egime esul
in (12) is equi alen o ha ing used he con en ional eal-axis
in eg a ion pa h when pe o ming he Fou ie - ans o m in e -
sion (pa h in Fig. 2). The su ace-wa e leaky- egime esul
can be iewed as a consequence o employing an in eg a ion
con ou de ou ing a ound he poles as pa h in Fig. 2. Taking
in o accoun ha he in eg and in (11) has no b anch poin s, in-
eg a ion along pa h is equi alen o in eg a ion along pa h
in Fig. 2, whe eas he in eg a ion pa h is equi alen o
he in eg a ion pa h . The e o e, swi ching be ween and
pa hs simply implies changing he pole ha is cap u ed by
he in eg a ion pa h. This swi ch is equi alen o choose he pole
acco ding o he sign o i s imagina y pa .
The abo e discussion has shown ha he selec ion o he
egime o be deal wi h (bound o su ace-wa e leakage) is
simply imposed by he p ope choice in (12). Following he
heo y gi en in [15], he momen -me hod de e minan unc ion
whose ze os a e he p opaga ion cons an s o he line de ines
a Riemann su ace in he longi udinal wa enumbe complex
-plane wi h mul iple b anch poin s. Thus, he choice made
in can be ela ed o he p ope /imp ope shee ha he
p opaga ion cons an will be loca ed on when sea ching o
he oo s o he dispe sion equa ion o he line.
B. T ea men o
Taking in o accoun he complex exponen ial expansion (9),
he space-domain con ibu ion associa ed wi h
can be exp essed as
(13)
Fig.3. In eg a ionpa hs
C
and
C
in hecomplex
k
-plane.In eg a ionpa hs
on he p ope /imp ope shee wi h espec o he
6
k
b anch poin s a e display
insolid/dashed lines.Fo simplici y,only apai o su ace-wa epoles a eshown
(
222
).
whe e
(14)
Followinga simila a ionale as ha p e iouslyused o , each
o hein eg al ep esen a ionsgi enby hein e seFou ie ans-
o m will be exp essed in an app op ia e closed o m. Thus,
ela ionship [18, (4.156)] is now adap ed and analy ically con-
inued in he en i e plane o w i e
(15)
whe e a e he b anch poin s
o he G een’s unc ion in he complex -plane,
, is he ze o h-o de
modi ied Bessel unc ion o he second kind, and is he
ze o h-o de Hankel unc ion o he second kind. Taking in o
accoun ha bo h he quasi-s a ic e m and he i s e m
in (4) can be conside ed as pa icula cases o he exponen ial
e ms in expansion (9) (when ), i s spa ial-domain
coun e pa can be eadily ob ained om (15).
Looking a (15), he i s op ion, i.e., ,gi es
place o exponen ially decaying ields in he no mal -di ec ion,
asi ua ion ha is compa iblewi h hebound egime.The second
op ionin(15)p o idesexponen iallyg owing ieldsin heuppe
hal -space, hus accoun ing o he space-wa e leaky egime.
The wo- alued (15) can be ela ed o he di e en in eg a ion
pa hs used in he SDA o space-wa e leaky modes [3], [16]. The
con en ional eal-axis in eg a ion pa h in Fig. 3 is ound o
be equi alen o he choice in (15), whe eas he
in eg a ion pa h in his igu e would e lec he
choice.
C. Sol ing he In eg al Equa ion
A ele an ea u e o he p esen app oach is ha e y ac-
cu a e closed- o m exp essions a e ob ained o he ke nels o
he in eg al equa ion o bo h he bound and leaky egimes. The
egime o be s udied (bound, su ace wa e, and space wa e) is
simply selec ed by he con enien choice in (12) and (15). This
p o ides a sys ema ic, e y e icien , and a he simple way o
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1898 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 8, AUGUST 2002
s udy he p opaga ion phenomena, supe seding he possible in-
con eniences o he SDA ela ed o he nume ical e alua ion
o he spec al in eg als along di e en in eg a ion pa hs in he
complex plane.
Once he space-domain ke nels a e ob ained, he in eg al
equa ion is sol ed by using he Gale kin momen me hod.
Chebyshe polynomials o he i s and second kinds weighed
by he edge condi ion a e used as basis unc ions o analyze
s uc u es wi h in ini ely hin conduc o s. S uc u es wi h
a bi a y c oss-sec ional conduc o s a e analyzed employing
nonuni o mly dis ibu ed piecewise linea unc ions as basis
unc ions. In bo h cases, a quasi-analy ical e alua ion o he
ma ix en ies is ca ied ou . Mo e de ails can be ound in [12]
and [14].
III. NUMERICAL RESULTS
The p esen me hod has been alida ed by compa ing ou e-
sul s wi h p e iously published esul s and wi h hose ob ained
by means o a well-es ablished SDA code [19] de eloped o
ze o- hickness s ip-like s uc u es. The ag eemen o bo h
bound and leaky egimes is excellen o all he cases consid-
e ed. As a i s example, Fig. 4(a) and (b) shows he no malized
phase and a enua ion cons an s o a space-wa e leaky mode
suppo ed by a mic os ip line p e iously s udied by Michalsky
and Zheng [7] only o he ze o- hickness case. The compa ison
be ween he da a epo ed in [7] wi h ou MPIE–DCIT esul s
shows an excellen ag eemen . Fig. 4(a) and (b) also includes
esul s o wo nonze o hickness cases, i.e., ec angula and
apezoidal c oss-sec ional conduc o s. These cu es show how
he shape o he conduc o can in luence he dispe sion cu es.
To gi e an idea o he o e all compu a ional e o equi ed
by ou me hod, 100 alues o he p opaga ion cons an o
he ze o- hickness case we e compu ed in 6 s wi h a Pen ium
II 450-MHz compu e . Rec angula o apezoidal s uc u es
equi e a ew seconds o ge a single alue o using he
same pla o m. Fig. 5 shows ano he compa ison wi h he
dispe sion cu es o a ci cula -wi e ansmission line epo ed
in [7, Fig. 7]. Again, he ag eemen ound bo h o he bound
() and he space-wa e leaky ( ) modes is e y good. I
is in e es ing o men ion ha , in he compu a ion o he abo e
esul s, because o he p esence o se e al slab wa eguide
modes ( , , and ), he explici con ibu ion o up
o h ee su ace-wa e poles had o be conside ed in he calculus.
Finally, and o show he capabili y o ou me hod o deal wi h
mul iconduc o and mul ilaye nonplana s uc u es, a wo-laye
LIGA mic omachined ansmission line has been analyzed. The
LIGA p ocess, desc ibed in [20], gi es ise o me alliza ions
wi h a high hickness/wid h a io. This makes i possible o use
he conduc o hickness as a new a iable in he design o il-
e s and couple s. The analyzed s uc u e consis s o a pai o
squa e c oss-sec ional conduc o s (200 200 m) sepa a ed
by a 120- m dis ance and placed on a laye ed subs a e. The
dispe sion cu es o he no malized p opaga ion and a enua-
ion cons an s o a su ace-wa e leaky mode appea ing in his
s uc u ea eshowninFig.6.Tos udy hee ec o heme alliza-
ion hickness, he dispe sion cu es o he same s uc u e as-
suming ze o- hickness conduc o s a e alsoshown. I can be seen
(a)
(b)
Fig. 4. No malized: (a) phase and (b) a enua ion cons an o a space-wa e
leaky mode in he mic os ip line shown in he inse :
"
=
9
:
8
,
w
=3
mm,
h
=0
:
635
mm.
Fig. 5. No malized phase and a enua ion cons an s o he undamen al mode
and he i s highe mode in a ci cula -wi e ansmission line. Lines: ou esul s.
Symbols: da a in [7]. S uc u al pa ame e s:
a=h
=0
:
25
,
"
=4
.
ha hespli ing poin ( he poin whe e acomplexleakysolu ion
me geswi hacomplexconjuga esolu ion,and he wosolu ions
hen spli apa as wo imp ope eal modes) appea s a lowe
equencies in he nonplana case. Resul s o he squa e-shape
conduc o s ha e been ob ained wi h 15 basis unc ion o e each
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BERNAL e al.: 2-D ANALYSIS OF LEAKAGE IN PRINTED-CIRCUIT LINES USING DCIT 1899
Fig. 6. No malized phase and a enua ion cons an s o a su ace-wa e leaky
mode in a wo-conduc o and wo-laye ansmission line wi h
w
=
200

m,
s
= 120

m,
h
=
h
= 210

m,
"
=3
, and
"
=4
:
6
.
conduc o . The CPU ime was much la ge o he nonplana
case because o he use o many basis unc ions (in compa -
ison wi h he ze o- hickness s ip case) and he need o com-
pu ing mo e ime-consuming in eg als. Rega dless, he me hod
p o ides he possibili y o e icien ly de e mining in a quan i a-
i e way he in luence o he hickness o he shape o he con-
duc o s in he leaky egime.
IV. CONCLUSIONS
A new app oach based on he combina ion o he MPIE and
DCIT has been de eloped o compu e in a simple and sys em-
a ic way he p opaga ion cha ac e is ics o bo h bound, su ace-,
and space-wa e leaky modes in p in ed-ci cui lines. One o he
main goals o his app oach has been he ob aining o accu a e
exp essions o he ke nels o he co esponding in eg al equa-
ioninaquasi-analy icalway.Thishasa oided hein ol ed ask
o in eg a ing along di e se in eg a ion pa hs in he ans e se
wa enumbe complex plane equi ed in he SDA. The use o
he complex images echnique has also p o ided high accu acy
and e iciency o he analysis. Wo king in he spa ial domain has
allowed o a di ec (al hough non i ial) ex ension o he anal-
ysis o nonplana conduc o s. Some esul s ha e been epo ed
o show he accu acy and e iciency o he compu e code de-
eloped.
APPENDIX
An e icien me hod o compu e in eg al (11) and i s de i a-
i e wi h espec o will be shown he e. To simpli y he
no a ion, he ollowing new a iables will be used:
and .
The in eg and in (11) has wo poles a and wo
b anch poin s a . I an in eg a ion pa h unning
om o along he eal axis is con enien ly closed in
he lowe hal -plane, he esul o his in eg al is de e mined by
he su ace-wa e pole enclosed by such a pa h. Conside ing ha
he closed pa h mus de ou a ound he b anch cu , (11) can be
exp essed as
(16)
whe e
(17)
The i s e m in (16) is a wo- alued unc ion accoun ing o he
su ace-wa e pole con ibu ion wi h —as in (12),
he sign o de e mines he bound/leaky egime unde
conside a ion. The second e m in (16) gi es he con ibu ion o
he in eg a ion along he b anch cu (assuming ha )
and, mo e speci ically, i comes om an in eg a ion along a e -
icalb anchcu wi h .Ashappensin(15),
he sign o in (17) se s he egime (bound/space-wa e
leaky) o be ea ed. The change o a iable
causes in eg al o be quickly calcula ed by using, o in-
s ance, Gauss quad a u es.
The de i a i e o can be eadily compu ed om he abo e
exp essions o gi e
(18)
whe e
(19)
No e ha , o la ge alues o , he in eg ands in (17) and
(19) become highly oscilla o y, hus gi ing ise o nume ical
d awbacks. In ha case, can be calcula ed by means o a di-
ec in eg a ion o (11). This nume ic in eg a ion mus be pe -
o med along pa h in Fig. 2 o accoun o he bound egime,
whe eas pa h / mus be chosen i we a e looking o a su -
ace-/space-wa e leaky egime.
The in eg al along he eal-axis pa h, deno ed as , can
be ew i en in he ollowing o m:
(20)
(), which can be e icien ly calcula ed by using
Gauss quad a u es. No e ha la ge alues o lead o a as
con e gence o .
The in eg a ion along pa h , i.e., , can be spli in o in e-
g a ion along he eal axis ( ) in addi ion o in eg a ion along
and pa hs in Fig. 2, namely,
(21)
whe e
(22)
wi h and .
Acco ding o [16], he in eg al along pa h , i.e., , can
be exp essed as in eg al (21) plus an in eg al unning along a
loop, as shown in [16]
(23)
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1900 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 50, NO. 8, AUGUST 2002
whe e
(24)
wi h and .
Thede i a i eo wi h espec o canbe eadily
ob ained om (20) and (22) yielding he ollowing exp essions:
(25)
(26)
(27)
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Joaquín Be nal was bo n in Se ille, Spain, in 1971.
He ecei ed he Licenciado and Doc o deg ees om
he Uni e si y o Se ille, Se ille, Spain, in 1994 and
2000, bo h in physics.
In 1995, he joined he Depa men o Elec onic
and Elec omagne ism, Uni e si y o Se ille. Since
1998, he has been an Assis an P o esso in he
Depa men o Applied Physics III a he same uni-
e si y. His esea ch in e es s ocus on he analysis
o plana s uc u es o in eg a ed mic owa e ci cui s
and high-speed e y la ge scale in eg a ion (VLSI)
in e connec s.
D . Be nal was he ecipien o a Jun a de Andalucía schola ship.
F anciscoMesa(M’94)wasbo ninCádiz,Spain,on
Ap il 1965. He ecei ed he Licenciado and Doc o
deg ees om heUni e si yo Se ille,Se ille,Spain,
in 1989 and 1991, espec i ely, bo h in physics.
He is cu en ly an Associa e P o esso in he De-
pa men o Applied Physics I, Uni e si y o Se ille.
His esea ch in e es ocuses on elec omagne ic
p opaga ion/ adia ion in plana lines wi h gene al
aniso opic ma e ials.
F ancisco Medina (M’90–SM’01) was bo n in
Pue o Real, Cádiz, Spain, in No embe 1960. He
ecei ed he Licenciado and Doc o deg ees om
he Uni e si y o Se ille, Se ille, Spain, in 1983 and
1987, espec i ely, bo h in physics.
F om 1986 o 1987, he spen he academic yea
wi h he Labo a oi ede Mic oondes de l’ENSEEIHT,
Toulouse, F ance. F om 1985 o 1989, he was a P o-
eso Ayudan e(Assis an P o esso )wi h heDepa -
men o Elec onics and Elec omagne ism, Uni e -
si y o Se ille, and since 1990, he has been a P o eso
Ti ula (Associa e P o esso ) o elec omagne ism. He is also cu en ly Head o
he Mic owa es G oup, Uni e si y o Se ille. His esea ch in e es includes an-
aly ical and nume ical me hods o guidance, esonan and adia ing s uc u es,
passi e plana ci cui s, and he in luence on hese ci cui s o aniso opic ma e-
ials.
D . Medina was a membe o bo h he Technical P og am Commi ee (TPC)
o he 23 d Eu opean Mic owa e Con e ence, Mad id, Spain, 1993, and he
TPC o ISRAMT’99, Malaga, Spain. He is on he Edi o ial Boa d o he IEEE
TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He has been a e-
iewe o o he IEEE and Ins i u ion o Elec ical Enginee s (IEE), U.K., pub-
lica ions.
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