IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 48, NO. 3, MARCH 2000 445
Fas Full-Wa e Analysis o Mul is ip T ansmission
Lines Based on MPIE and Complex Image Theo y
Joaquin Be nal, F ancisco Medina, Membe , IEEE, Ra ael R. Boix, Membe , IEEE, and Manuel Ho no, Membe , IEEE
Abs ac —The mixed-po en ial elec ic- ield in eg al equa ion
is used in conjunc ion wi h he Gale kin’s me hod and complex
image heo y o analyzing a ansmission line wi h mul iple
s ips embedded in di e en laye s o a mul ilaye ed uniaxially
aniso opic dielec ic subs a e. The wo-dimensional G een’s
unc ions o he scala and ec o po en ials a e analy ically
ob ained in he space domain due o he app oxima ion o i s
spec al-domain e sion wi h complex images, hus a oiding
leng hy nume ical e alua ions. Double in eg als in ol ed in he
compu a ion o Gale kin’s ma ix en ies a e quasi-analy ically
ca ied ou o he chosen basis unc ions, which a e well sui ed o
he p oblem.
Index Te ms—Complex image me hod, in eg al equa ions, lay-
e ed media, plana ansmission lines.
I. INTRODUCTION
THE analysis o a plana mul is ip sys em such as ha
shown in Fig. 1 has been ca ied ou by using a a ie y
o echniques du ing he pas h ee decades, including bo h
quasi-TEM and ull-wa e o mula ions. Achie ing high nu-
me ical e iciency has been he goal o many ecen pape s. A
sample o his ype o wo k in he ame o he quasi-TEM anal-
ysis can be ound in [1] and e e ences he ein. In his pape ,
emphasis is placed on he ull-wa e app oach. Ve y e icien
algo i hms dealing wi h he ull-wa e analysis o plana lines
ha e been also epo ed, including he singula in eg al-equa-
ion me hod [2], [3] and he eigen alue app oach [4] o boxed
s uc u es, he Wiene –Hop me hod [5], and a ious enhanced
implemen a ions o he spec al-domain analysis (SDA)
[6]–[9]. In his pape , he au ho s p opose a e y as analysis
o he s uc u e in Fig. 1 based on he mixed-po en ial in eg al
equa ion (MPIE) [10]–[12]. The nume ical pe o mance o his
app oach is d as ically imp o ed by using a sui able wo-di-
mensional (2-D) space-domain ep esen a ion o he po en ial
G een’s unc ions and quasi-analy ical compu a ion o he
eac ion in eg als appea ing when a Gale kin scheme is used
o sol ing he MPIE o ind he su ace cu en s. This as and
accu a e compu a ion o Gale kin’s ma ix en ies is he key
Manusc ip ecei ed July 27, 1999; e ised Decembe 10, 1999. This wo k
was suppo ed by he Comisión In e minis e ial de Ciencia y Tecnología, Spain
unde P ojec TIC95-0447.
J. Be nal is wi h he Depa men o Applied Physics, Uni e si y o Se ille,
41092 Se ille, Spain.
F. Medina and R. R. Boix a e wi h he Mic owa e G oup, 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]).
M. Ho no, deceased, was wi h he Mic owa e G oup, Depa men o Elec-
onics and Elec omagne ism, Uni e si y o Se ille, 41012 Se ille, Spain.
Publishe I em Iden i ie S 0018-9480(00)02050-0.
Fig. 1. C oss sec ion o he mul iconduc o ansmission line unde analysis.
poin o ge e y high e iciency. In his way, he de e mina ion
o space-domain G een’s unc ions is ca ied ou ia he com-
plex images echnique [13]–[15], hus a oidingcommonly used
nume ical spec al 2-D Somme eld- ype in eg a ion [16], [17].
This me hod, o iginally in ended o he analysis o a adia ing
dipole in a h ee-dimensional (3-D) s a i ied medium, has been
adap ed he e o ou 2-D p oblem. A his poin , i should be
men ioned ha a co ec ion o he o mula ion o [15], in o-
duced by Kipp and Chan in [18], mus be also applied in he
2-D case. As i is well known, i s - and second-kind Chebyshe
polynomials weighed by he p ope s ip edge condi ion a e
e y sui able basis unc ions o he cu en expansion [9]. The
eac ion in eg als in ol ing hese unc ions and he closed- o m
exp ession o he G een’s unc ions ob ained wi h he complex
images me hod a e quasi-analy ically compu ed. The e o e,
he mos ime-consuming s ep in sea ching o he p opaga ion
cons an s, which is he compu a ion o Gale kin’s ma ix, is
d as ically accele a ed.
II. FORMULATION OF THE INTEGRAL EQUATION
Le us conside a ansmission line consis ing o in in-
i ely hin s ips embedded in he a ious laye s o a mul ilay-
e ed subs a e (see Fig. 1). Each laye is a uniaxial aniso opic
dielec ic, wi h i s op ical axis pe pendicula o he in e aces
be ween laye s. Since we a e in e es ed in modes ha p opaga e
in he -di ec ion, we assume a common phase ac o o
ields and cu en s, whe e is he unknown p opaga ion con-
s an . By en o cing he bounda y condi ion o he angen ial
elec ic ield a he su ace o he conduc o s, we ob ain an elec-
ic- ield in eg al equa ion (EFIE). The ke nel o his EFIE has
a se e e singula i y ha makes i unsui able o a di ec appli-
ca ion o he me hod o momen s [10]. To o e come his di -
icul y, we can ans o m his in eg al equa ion in o an MPIE
o m, whose ke nel has a weake singula i y [10], [11], [19].
0018–9480/00$10.00 © 2000 IEEE
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446 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 48, NO. 3, MARCH 2000
Since he sou ces in ou s uc u e a e pe pendicula o , he
MPIE has he ollowing o m:
(1)
on he conduc ing s ips. and in (1) a e he magne ic
ec o po en ial and elec ic scala po en ial due o he su ace
cu en on he h conduc o
(2)
(3)
In (1)–(3), s ands o he su ace o he h conduc o
placed a he plane . I is well known
ha o a ho izon ally di ec ed dipole, wo componen s o
he ec o po en ial a e necessa y o sa is y he bounda y
condi ions a he in e aces [20]. We ha e used he adi ional
Somme eld’s o mula ion o he ec o po en ial [21] so
he componen o he ec o po en ial is chosen oge he
wi h he componen pa allel o he sou ce. This o mula ion
is con enien o he analysis o plana s uc u es because
. Mo eo e , he e olu ion symme y o ou
p oblem subs a e a ound he -axis leads o .
The e o e, only one spec al in eg al is necessa y o ob aining
he 2-D G een’s unc ion o he magne ic ec o po en ial.
III. KERNEL OF THE INTEGRAL EQUATION
I is easible o ob ain a closed- o m exp ession o
and in he spec al domain
[21], [22], namely, and , being
( and a e he Ca esian
spec al a iables and is he adial pola spec al a iable).
I should be poin ed ou ha i he s uc u e has conduc o s
placed a di e en le els, and
mus be e alua ed o sou ce and obse a ion poin s a any o
he le els. Taking in o accoun he ecip oci y o he G een’s
unc ions, his lead o combina ions o sou ce
and obse a ion planes. Once he spec al e sion o he ke nel
o ou in eg al equa ion is known, i s 2-D spa ial coun e pa
can be ob ained om he ollowing spec al in eg al:
(4)
In (4) and s and o he spa ial (2-D) and spec al ep e-
sen a ions o any o he and unc ions.
The -dependence is no explici ly shown since i will no play
any ole in he de elopmen he ea e .
The in eg and in (4) may ha e se e al poles in he eal axis
o he -plane, which depend on he s uc u e and equency.
Thesepolescanbeeasily emo ed,as will beexplainedla e on.
Ano he impo an opological ea u e o he spec al-domain
G een’s unc ions is heexis enceo b anchpoin s a .
These b anch poin s a e ela ed o he ee-space unbounded
uppe laye o he s uc u e, and hey will play an impo an ole
in he de elopmen o he nume ical app oach.
The compu a ion o (4) akes a signi ican pa o he o e all
compu a ion ime since he in eg ands a e ypically oscilla o y
and slowly decaying. Since he in eg ands depend on , ha in-
eg al mus be ecalcula ed o e e y alue o he p opaga ion
cons an in he oo sea ch p ocess. The e o e, a as me hod o
e alua e (4) is o pa amoun impo ance. The complex image
me hod al eady used in he analysis o plana ci cui s, an ennas,
and sca e ing p oblems [13]–[15], [18] can be adap ed o ac-
complish ha goal. The basic idea o his me hod is o ex ac
om he spec al ke nel i s quasi-s a ic and su ace-wa e con i-
bu ions, and o app oxima e he emaining unc ion by a sum o
complex exponen ials. In he 3-D case, he Somme eld iden i y
can henbeemployed oe alua einclosed o m heSomme eld
in eg als. This leads o a e y e icien algo i hm p o ided ha
we ha e a mean o e alua e he quasi-s a ic and su ace-wa e
con ibu ions in closed o m. In he 2-D case, as a as he au-
ho s know, he spec al in eg als in (4) a e usually nume ically
calcula ed [16], [17]. Al hough e icien nume ical in eg a ion
algo i hms a e used, he p ocedu e is no as e icien as hose
epo ed in [13]–[15], [18]. Wha we p opose in his pape is o
adap he complex image me hod o ou p oblem. In o de o do
his, i is ins uc i e o examine he spec al-domain e sion o
bo h he ec o - and scala -po en ial G een’s unc ion o a a -
eling-wa e line sou ce in he ee space a a heigh abo e a
g ound plane. These o mulas can be w i en in he ollowing
o m:
(5)
whe e ,( ), o he
componen o he dyadic spec al G een’s unc ion o he
ec o po en ial and o he scala po en ial. The
i s e m in (5) co esponds o he e ec o he sou ce i sel ,
whe eas he second e m is he image con ibu ion. Hence, i
seems easonable o hink ha i he sou ce is embedded in a
s a i ied medium, he spec al G een’s unc ion is sui able o
be exp essed as a quasi-s a ic e m (which accoun s
o he nea - ield con ibu ion o he sou ce and has a singula
space-domain coun e pa ) plus a numbe o images o he
o m . Howe e , i is well known ha a s a i ied
medium is also capable o p opaga ing su ace wa es, which a e
independen o he sou ce. The in luence o hese p opaga ing
modes in he spec al-domain G een’s unc ion is he exis ence
o a ini e numbe o poles ha do no appea in he ee-space
p oblem. These poles modi y he spec al-domain G een’s
unc ion beha io ha can no longe be exclusi ely exp essed
as a sum o exponen ial unc ions. As a mean o e alua ing (4)
in a e icien way, we hen w i e he spec al G een’s unc ion
in he ollowing app oxima ing o m:
(6)
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BERNAL e al.: FAST FULL-WAVE ANALYSIS OF MULTISTRIP TRANSMISSION LINES 447
The e m is he quasi-s a ic con ibu ion, ep esen s
he su ace wa e e m, and is he emaining e m, which
is o be expanded by a ini e se ies o complex exponen ial unc-
ions.
A. Quasi-S a ic Te m
The quasi-s a ic ields a e dominan when he dis ance
be ween he sou ce and ield poin s is small compa ed o he
ee-space wa eleng h. In such a case, he complex exponen-
ial appea ing in he in eg and o (4) oscilla es wi h a la ge
pe iod. The e o e, he beha io o he spa ial G een’s unc ion
is s ongly a ec ed by he asymp o ic alues o he spec al
G een’s unc ion ( ). Since he in eg ands dec ease e y
slowly, he ollowing asymp o ic beha io o
and mus be ex ac ed ou :
o he wise (7)
o he wise (8)
whe e is he posi ion o he h in e ace (see Fig. 1). On
he o he hand, and a e he ela i e pe mi i i ies o he
h laye in di ec ions pe pendicula and pa allel o he -axis,
espec i ely. No e ha we will ha e nonze o asymp o ic e ms
only when sou ce and obse a ion poin s a e a he same le el.
F om (7) and (8), we can, in gene al, w i e
(9)
whe e κis a cons an ha is ze o i and whose alue
depends on he cases ea ed in (7) and (8) i sou ce and obse -
a ion poin s a e a he same le el ( ). The cons an
was de ined in (4).
I should be poin ed ou ha he b anch poin s appea ing in
in he spec al G een’s unc ions a e also p esen in
(9). The e o e, his e m does no in oduce any new b anch
cu in he -plane opology [18]. In o de o calcula e he 2-D
space-domain e sion o (9), he ollowing spec al in eg al
mus be ca ied ou :
(10)
whe e .αis supposed o be posi i e since
we a e in e es ed only in he bound egime (as opposi e o he
leaky egime). The in eg al (10) can be analy ically calcula ed
by using he ollowing esul [23]:
(11)
whe e is he ze o h-o de modi ied Bessel unc ion o he
second kind. The in eg al in (10) can be seen as he limi o (11)
when , hence, he con ibu ion o he quasi-s a ic e m in
he space domain is
(12)
No e ha since o small , he 2-D space-
domain G een’s unc ions ha e a loga i hmic singula i y when
and he ield poin app oaches he sou ce poin ( ).
B. Su ace-Wa e Poles Con ibu ion
The complex image scheme can be applied o e a complex
pa h, hus a oiding p oblems ela ed o he p esence o poles on
he eal axis o he complex plane [24]. Howe e , he e a e
heo e ical and nume ical easons ha make i ad isable o e-
mo e hepolecon ibu ions om hespec alG een’s unc ions.
Complex exponen ial unc ions canno ep oduce accu a ely in
hespec aldomain hebeha io associa ed o hesepoles. Since
he spec al unc ions a e e en unc ions o , he poles always
appea in pai s. Consequen ly, we can w i e [14], [15]
(13)
whe e is he numbe o poles, is he loca ion o he h
pole in he -plane, and is i s esidue
The e o e, he space-domain con ibu ion o he su ace wa e
poles is
(14)
whe e is supposed o be posi i e (bound egime).
Ananaly icalexp ession o hein eg alin(14)isa ailable om
[23], in such a way ha
(15)
No e ha , in con as wi h he 3-D case [13], [14], he con-
ibu ion om he su ace-wa e poles in ou 2-D si ua ion does
no in oduce any singula i y. Thus, he ea men s epo ed in
[25] o [26] o deal wi h his p oblem is no equi ed inou case.
The e o e, we can di ec ly ex ac he su ace-wa e con ibu ion
om he complex image expansion, ob aining a well-beha ed
app oxima ion o any alue o he spa ial a iable. This makes
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448 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 48, NO. 3, MARCH 2000
an impo an di e ence be ween he ansmission and adia ion
p oblems.
C. Applica ion o he Complex Image Me hod
We ha e ound closed- o m exp essions ha allow us o ex-
ac he asymp o ic and su ace wa e e ms o he spec al-do-
main G een’s unc ion and o eco e hem in he 2-D space do-
main.The emainingspec al—domain unc ionisnowsui able
o be expanded as a ini e sum o complex exponen ial unc ions
by using, o ins ance, he gene al pencil o unc ion (GPOF)
me hod [27]
(16)
whe e is he numbe o employed complex images. I is
expec ed ha a sho numbe o images is enough o p ac ical
pu poses because o he sui abili y o he expanding unc ions:
hey a e he spec al e sion o cylind ical wa es.
In o de o ob ain he unknown coe icien s and
appea ing in (16), we ha e sampled ou spec al unc ions on a
pa h in he -plane, which a oids he poles and b anch poin
singula i y. To ensu e an op imum esul , we ha e applied
a wo-s ep p ocedu e ha makes i possible o ake mo e
samples nea he o igin, whe e, owing o he p oximi y o he
b anch poin , he spec al unc ion p esen s as a ia ions.
This wo-s ep app oach has been p oposed by Aksun in [24]
o app oxima ing 3-D G een’s unc ions (al hough his au ho
does no ex ac he su ace-wa e con ibu ion). The use o he
wo-s ep algo i hm is also ad an ageous o deal wi h s uc u es
ha ing e y hin laye s. The easons o ha a e explained
in [28] in he con ex o he quasi-s a ic analysis o coplana
wa eguide (CPW) s uc u es. On he o he hand, i is wo h
no ing ha al hough we could also expand he e m as
a sum o complex exponen ials in he
o a iables, his would
in oduce a new b anch poin in he plane opology and,
he e o e, he expansion would ail o app oxima e he ac ual
unc ion in a co ec way [18].
Now, he las e m in (6) can be ans o med in o he 2-D spa-
ial domain by using (11). The e o e, wecan w i e he ollowing
exp ession o he whole 2-D space domain G een’s unc ion:
(17)
The 2-D spa ial-domain G een’s unc ions a e ob ained as a
sum o adial wa es (plus he su ace-wa e con ibu ion) in he
samewayassphe icalwa esa eob ained o he spa ial-domain
3-D G een’s unc ions [13]. No e ha (17) has he impo an ad-
an ageo beinganexplici unc iono hep opaga ioncons an
h ough and ( ). In he oo sea ching
p ocess in ol ed in he esolu ion o he eigen alue p oblem,
he p opaga ion cons an is changed many imes, bu hose
changes a e au oma ically aken in o accoun by (17). I a nu-
me ical in eg a ion scheme is applied o sol e o (4), he in e-
g als ha e o be ecompu ed o each new guess alue o . The
use o ou app oach ob iously implies impo an cen al p o-
cessing uni (CPU) ime sa ings.
IV. APPLICATION OF GALERKIN’S METHOD
Once he space-domain ke nel o he in eg al equa ion has
been e icien ly ob ained, we can apply he Gale kin’s me hod.
A well-es ablished se o basis unc ions o plana - ype s uc-
u es has been chosen. In ac , o a plana s ip o wid h and
whose cen al poin coo dina e is , he basis unc ions o he
componen s o he cu en densi y ha e been aken o be
(18)
(19)
whe e is he numbe o basis unc ions employed o he
ans e se componen o he cu en densi y (one mo e basis
unc ion mus be used o he axial componen o he cu en
densi y o ensu e ha he o al cu en ul ills he con inui y
equa ion). and s and o i s - and second-kind
Chebyshe polynomials, espec i ely. These unc ions mimic
he eal beha io o he cu en s nea he edge o he conduc ing
s ips and a e quasi- o hogonal o he space ke nel we a e
using. This allows us o a ain accu a e esul s while using e y
ew basis unc ions.
Thenex s epis ocalcula e hecon olu ionandinne p oduc
in eg als by using he basis unc ions in (18) and (19) and he
ke nel in (17). Thanks o he ela ionship be ween i s - and
second-kindChebyshe polynomials[29],weonlyneed ocom-
pu ein eg alsin ol ing i s -kindpolynomials.The unc ion
p esen s a loga i hmic singula i y ha migh cause p oblems
in he con olu ion in eg al. Fo una ely, he con ibu ion o he
con olu ionin eg also his singula i ycan behandled in closed
o m, such as explained in [1]. The es o he ke nel is egula
and does no gene a e in eg a ion p oblems. Due o he ype o
singula i y p esen in he basis unc ions, low-o de Chebyshe
quad a u es a e sui able o accu a ely ca y ou he in eg a ions
in ol ing he egula pa o he ke nel. In his way, he elemen s
o Gale kin’s ma ix a e gene a ed bo h e y accu a ely and e -
icien ly.
V. NUMERICAL RESULTS
The i s s ep o checking he pe o mance o he p oposed
app oach is o e i y ha he app oxima ion o he 2-D
space-domain G een’s unc ions is co ec . These unc ions
show an exponen ial decay wi h he dis ance be ween sou ce
and ield poin s, which is qui e di e en om ha ob ained
in he 3-D case. This decaying is as e o la ge alues o .
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BERNAL e al.: FAST FULL-WAVE ANALYSIS OF MULTISTRIP TRANSMISSION LINES 449
We ha e compa ed in his pape he space-domain G een’s
unc ions compu ed by di ec in eg a ion a combina ion o he
Rombe g’s me hod, and he weigh ed a e ages me hod e-
po ed in [11] has been used agains hose ob ained by using
he echnique in his pape . In his way, Fig. 2 shows o
he wo-laye s uc u e depic ed in he igu e. The ela i e
di e ence be ween nume ical and GPOF esul s is plo ed.
The GPOF has been applied wi h and wi hou pole ex ac-
ion (bu always wi h ex ac ion o he quasi-s a ic e m). Fo
he 9-GHz case [see Fig. 2(a)], we can see ha he GPOF
app oxima ion is e y accu a e (bu ai ly be e i pole ex-
ac ion is applied) in he whole ange o in e es . Rela i e
e o is la ge only o hose egions whe e he alues o
he app oxima ed unc ion is negligible. In Fig. 2(b), simila
da a a e plo ed o a equency o 33 GHz ( wo poles a e
in ol ed in his case). No e ha i poles a e no emo ed,
la ge e o s a e ob ained, while a e y good app oxima ion is
achie ed a e emo ing hem. The e o e, emo ing he poles
is s ongly ad ised, a e all, i is nei he di icul no ime
consuming o ind hem [30], while nume ical bene i s a e
impo an . I is wo h men ioning ha he example conside ed
in Fig. 2 co esponds o a con igu a ion ha ing a e y hin
dielec ic laye . This could cause se ious nume ical p oblems,
which ha e been o e come hanks o he applica ion o he
wo-s ep scheme used in ou s udy. Al hough we ha e concen-
a ed ou a en ion on , simila conclusions a e alid o
. This s udy has been ca ied ou o many combina ions
o subs a es and sou ce and ield poin loca ions (coplana
and noncoplana ). The o e all conclusion is ha he wo-s ep
app oach in conjunc ion wi h he quasi-s a ic e m and pole
ex ac ion p o ide an excellen space-domain ep esen a ion o
he equi ed 2-D G een’s unc ions.
Once we a e ce ain abou he accu acy o he 2-D space-do-
main G een’s unc ions compu ed ia(6), we ha e e alua ed he
global pe o mance o ou me hod. Fi s o all, we ha e checked
he accu acy and con e gence p ope ies o he eac ion in e-
g als de ining he en ies o Gale kin’s ma ix. We ha e con-
i med ha hese en ies a e compu ed wi h ex eme accu acy
(mo e han six co ec igu es) using e y low-o de Chebyshe
quad a u es and closed- o m e alua ion o he loga i hmic sin-
gula i y con ibu ion. On he o he hand, we ha e ca ied ou
exhaus i e compa isons wi h p opaga ion cons an s compu ed
using nume ical e alua ion o he G een’s unc ion and using
enhanced e sions o he SDA [31]. The ag eemen be ween
he a ious esul s is o al and we only de ec di e ences in he
compu a ional e o (CPU ime). I has been e i ied ha pole
ex ac ion is necessa y o many cases because o he wise he
e o in he space G een’s unc ions meaning ully a ec s he
inal esul o he p opaga ion cons an s. As an example, some
nume ical esul s o he undamen al and i s wo highe o de
modes o a simple mic os ip line a e included in Table I. Those
esul s ha e been ob ained by using ou basis unc ions o he
longi udinal cu en and h ee unc ions o he ans e se one
( h ee and wo a e enough o he undamen al mode) wi h and
wi hou pole ex ac ion. Ex ac ion o poles is clea ly necessa y
o equencies abo e 35 GHz. O he wise esul s a e no eliable
because hey a e s ongly dependen on he numbe o images,
sample poin s, and quad a u e poin s. Mo eo e , spu ious solu-
(a)
(b)
Fig. 2. Magni ude o
K
(solid line) and ela i e di e ence be ween
nume ical in eg a ion compu a ion and complex images compu a ion wi h
(black do s) and wi hou (whi e do s) pole ex ac ion o : (a) 9 GHz and (b) 33
GHz. Da a:
d
=
1
mm,
d
=0
:
01
mm,
"
=
"
=10
,
"
=2
:
25
, and
"
=1
:
5
.
TABLE I
=k
FOR THE FUNDAMENTAL AND TWO
FIRST HIGHER MODES OF THE MICROSTRIP IN THE TOP FIGURE.
w
=3
:
0
mm,
h
=0
:
635
mm, AND
"
=9
:
8
.LEFT-HAND-SIDE COLUMN:WITH SURFACE
POLE EXTRACTION.RIGHT-HAND-SIDE COLUMN:NOPOLE EXTRACTION
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450 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 48, NO. 3, MARCH 2000
ions may appea . Eigh complex images in he app oxima ion
o he G een’s unc ions and ou quad a u e poin s in he e al-
ua ion o he eac ion in eg als ha e been used in his able.
In o de o illus a e CPU ime sa ing, we ha e compa ed
he echnique p oposed in his pape wi h nume ical e al-
ua ion o he space-domain G een’s unc ion. E en hough
he in eg a ion scheme we ha e used o gene a e he nu-
me ical samples o he G een’s unc ion is e y e icien ,
he applica ion o he me hod p oposed he e s ill yields an
impo an educ ion o CPU ime. Mo eo e , complex im-
ages ha e no o be ecompu ed when is changed in he
oo sea ching p ocess, whe eas new nume ical in eg a ions
would be equi ed o new alues o . Hence, he ela i e
impac o using his app oach in he analysis o ansmission
lines (eigen alue 2-D p oblem) is p obably s onge han he
impac o using a simila echnique in a 3-D plana p oblem
since, in he la e case, he gene a ion o he G een’s unc-
ions is a small ac ion o he o al nume ical e o . As
an example, Fig. 3 shows he a io o CPU imes using nu-
me ical in eg a ion agains he me hod epo ed he e as a
unc ion o he numbe o s ips ( he same numbe o basis
unc ions has been used in each s ip so as o keep he
same accu acy le el). CPU ime educ ion is signi ican o
any case, becoming mo e impo an as he complexi y o he
mul is ip sys em inc eases.
As a inal example, Fig. 4 shows he dispe sion cu es o
he undamen al modes o he i e conduc o mic os ip ans-
mission line depic ed in he igu e. Two cases a e conside ed:
in Case A, he s ips a e in he ai –dielec ic in e ace; in Case
B, he cen e conduc o esides on he op in e ace o a e y
hin co e laye . Dispe sion cu es o he con igu a ion (A)
we e published by Ki azawa in [32] and la e ep oduced by
Hsu in [33]. Resul s o he con igu a ion B a e gi en in [33].
Ki azawa uses a a ia ional me hod, whe eas Hsu employs
an MPIE scheme sol ed in he space domain by using he
me hod o momen s wi h piecewise linea basis unc ions and
nume ical compu a ion o he spec al in eg als. The ag eemen
be ween ou esul s and hose p esen ed in [32] and [33] is
e y good, as can been seen in he g aphical ep esen a ion. In
o de o ep oduce hose da a, we ha e used h ee longi udinal
and wo ans e se basis unc ions along wi h ou quad a u e
poin s and eigh complex images o app oxima ing he egula
pa o he spec al-domain G een’s unc ions. Many o he
esul s epo ed in he li e a u e ha e been ep oduced wi h ou
me hod, bu hey a e no included he e o he sake o b e i y.
As a inal commen on he accu acy and obus ness o he
p oposed me hod, we ha e o say ha e y accu a e esul s a e
also ob ained o he cu en dis ibu ion. A sys ema ic inc ease
o he numbe o basis unc ions does no in oduce nume -
ical ins abili ies and all he coe icien s o he cu en expan-
sion a e compu ed wi h e y good accu acy ( i e co ec igu es
a e easily ob ained o he expansion coe icien s). This is mo e
signi ican ega ding he quali y o he employed echnique han
p opaga ioncons an esul s[9].Tosumup,ou manynume ical
expe imen s con i m ha he de eloped me hod wo ks p ope ly,
p o iding e yaccu a e esul s and impo an compu a ional-e -
o sa ings.
Fig. 3. CPU ime a io o a mic os ip analysis using nume ical gene a ion
o he G een’s unc ion and he echnique in his pape as a unc ion o he
numbe o s ips. Longi udinal and ans e se cu en s ha e been app oxima ed
by means o h ee and wo basis unc ions, espec i ely. Subs a e: hickness
=
0
:
635
mm,

=9
:
8
. S ip wid h
=3
mm. S ip sepa a ion: 1.5 mm.
Fig. 4. Dispe sion cu es o he i e undamen al modes o he s uc u e o
he igu e. Dielec ic da a as in Fig. 2.
w
=1
mm,
s
=0
:
2
mm. Case (A):
d
=0
. Black squa es: esul s in [32] and [33], solid line: ou esul s. Case (B):
d
=0
:
01
d
. Whi e squa es: esul s in [33], dash line: ou esul s.
VI. CONCLUSIONS
A new me hod has been p oposed o he compu a ion o
he dispe sion cu es o mul ile el mul iconduc o plana ans-
mission lines embedded in a uniaxially aniso opic s a i ied
medium.Theapp oachisbasedon hecompleximage echnique
and MPIE o mula ion. We ake ad an age o a closed- o m
de i a ion o he 2-D space-domain G een’s unc ion and o he
use o a sui able se o basis unc ions o ob ain a as and ac-
cu a e compu e code. The e alua ion o Gale kin’s ma ix en-
iesis pe o medin a e ye icien way. Nume ical esul sha e
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BERNAL e al.: FAST FULL-WAVE ANALYSIS OF MULTISTRIP TRANSMISSION LINES 451
been p esen ed and compa ed wi h da a a ailable in he li e a-
u e and supplied by o he me hods. Ve y good ag eemen has
been ound in all cases by using e y modes compu a ional e-
sou ces.
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[33] C. G. Hsu, “Analysis o a mul iconduc o ansmission line embedded
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Joaquin Be nal was bo n in Se ille, Spain, in 1971.
He ecei ed he Licenciado deg ee in physics om
he Uni e si y o Se ille, Se ille, Spain, in 1994, and
is cu en ly wo king owa d he Ph.D. deg ee a he
Uni e si y o Se ille.
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 De-
pa men o Applied Physics, Uni e si y o Se ille.
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.
M . Be nal was he ecipien o a 1995 schola ship p esen ed by Jun a de
Andalucía.
F ancisco Medina (M’90) was bo n in Pue o Real,
Cádiz, Spain, in No embe 1960. He ecei ed he Li-
cenciado and he 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
a he Labo a oi e de Mic oondes de l’ENSEEIHT,
Toulouse, F ance.
F om 1985 o 1989, he was a P o eso Ayudan e
wi h he Depa men o Elec onics and Elec omag-
ne ism, Uni e si y o Se ille, and since 1990, he has
been P o eso Ti ula o elec omagne ism. He is cu en ly he Head o he Mi-
c owa es G oup, Uni e si y o Se ille. His esea ch in e es s include analy ical
and nume ical me hods o plana s uc u es and ci cui s and he in luence on
hese ci cui s o aniso opic ma e ials.
D .Medinawasamembe o heTechnicalP og ammeCommi eeo he23 d
Eu opean Mic owa e Con e ence, Mad id, Spain (1993). He is on he edi o ial
boa d o he IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES.
He was he ecipien o a Minis e io de Educacion y Ciencia/Minis e e de la
Reche che e la Technologie Schola ship.
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452 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 48, NO. 3, MARCH 2000
Ra ael R. Boix (M’97) was bo n in Melilla, Spain,
in 1962. He ecei ed he Licenciado and doc o
deg ees in physics om he Uni e si y o Se ille,
Se ille, Spain, in 1985 and 1990, espec i ely.
Since 1985, he has been wi h he Elec onics
and Elec omagne ics Depa men , Uni e si y o
Se ille, whe e he became an Associa e P o esso
in 1994. Du ing he summe s o 1991 and 1992, he
was wi h he Elec ical Enginee ing Depa men ,
Uni e si y o Cali o nia a Los Angeles, as a Visi ing
Schola . Du ing he summe o 1996, he was wi h
he Elec ical and Compu e Enginee ing Depa men , Sy acuse Uni e si y,
Sy acuse, NY, as a Visi ing Schola . His cu en esea ch in e es is ocused on
he analysis o he e ec s o complex subs a es on he pe o mance o plana
ansmission-line discon inui ies, plana passi e mic owa e ci cui s, plana
esona o s, and p in ed ci cui an ennas.
ManuelHo no(M’75)wasbo n inTo edel Campo,
Jaén, Spain, and died in Sep embe 1998, in Se ille,
Spain. He ecei ed he Licenciado and he doc o de-
g ees om he Uni e si y o Se ille, Spain, in 1969
and 1972, espec i ely, bo h in physics.
In Oc obe 1969, he joined he Depa men o
Elec onics and Elec omagne ism, Uni e si y o
Se ille, whe e he became an Assis an P o esso in
1970, Associa e P o esso in 1975, and P o esso in
1986. His main ields o in e es included bounda y
alue p oblems in elec omagne ic heo y, wa e
p opaga ion h ough aniso opic media, and mic owa e in eg a ed ci cui s.
Du ing his inal yea s, he was engaged in he analysis o plana ansmission
lines embedded in aniso opic ma e ials, mul iconduc o ansmission lines,
and plana an ennas. He was a membe o he Elec omagne ism Academy,
Massachuse s Ins i u e o Technology (MIT), Camb idge.
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