IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 45, NO. 9, SEPTEMBER 1997 1619
Quick Quasi-TEM Analysis o Mul iconduc o
T ansmission Lines wi h Rec angula
C oss Sec ion
Joaqu´ın Be nal, F ancisco Medina, and Manuel Ho no, Membe , IEEE
Abs ac —This pape p esen s an e icien and accu a e p oce-
du e o compu ing he quasi-s a ic ma ix pa ame e s (
[C]
,
[L]
,
[G]
, and
[R]
) o ec angula -shaped conduc o s embedded in a
mul ilaye ed dielec ic medium o e an in ini e g ound plane. An
addi ional op g ound plane can also be conside ed. The p oblem
is o mula ed in e ms o he space-domain in eg al equa ion o
he ee-cha ge dis ibu ion on he slab conduc o su aces. The
spa ial G een’s unc ion is compu ed om i s spec al coun e -
pa using sys em iden i ica ion echniques [P ony’s me hod o
ma ix pencil me hod (MPM)]. The in eg al equa ion is sol ed
by means o a Gale kin scheme employing en i e domain basis
unc ions. This esul s in a small ma ix size. In addi ion, he
quasi-analy ical e alua ion o he en ies o he Gale kin ma ix
leads o a e y e icien and accu a e compu e code. A de ailed
s udy on he con e gence and accu acy o he me hod has been
included.
Index Te ms—Losses, mic os ip, mul iconduc o ansmission
lines, quasi-TEM analysis, hick conduc o s.
I. INTRODUCTION
ALARGE amoun o pape s ha e been de o ed o he
analysis o plana ansmission lines h oughou he las
h ee decades. Mos o he published wo k assumes negligible
me alliza ion hickness. This app oxima ion is good enough
o many p ac ical si ua ions, and pe mi s he simpli ica ion
and e icien use o he analysis ad hoc ma hema ical ech-
niques. Howe e , du ing he las ew yea s many au ho s
ha e paid a en ion o he p oblem o accoun ing o he
nonze o me alliza ion hickness. Apa om he aim o in-
c easing accu acy, his in e es comes om he necessi y o
analyzing he elec ical beha io o he ela i ely hick s ips
employed in monoli hic mic owa e ci cui s and high-speed
digi al ci cui s. In hese cases, me alliza ion hicknesses and
s ip wid hs a e in he same o de o magni ude. The e o e, he
o me can no longe be neglec ed. The s ip hickness mus
also be conside ed whene e igh edge coupling is p esen ,
e en hough ela i ely wide s ips a e unde conside a ion.
Mo eo e , he compu a ion o ohmic conduc o losses equi es
explici accoun abili y o he s ip hickness.
A numbe o au ho s ha e analyzed he e ec s o he
me alliza ion hickness by using a ull-wa e analysis, bu in
Manusc ip ecei ed Oc obe 4, 1996; e ised Ma ch 25, 1997. This wo k
was suppo ed by he DGICYT, Spain, unde P ojec TIC95-0447. The wo k
o J. Be nal was suppo ed by a g an om Jun a de Andalucia (Spain).
The au ho s a e wi h he G upo de Mic oondas, Depa amen o de
Elec ´
onica y Elec omagne ismo, Facul ad de F´
ısica, Uni e sidad de Se illa,
41012 Se ille, Spain.
Publishe I em Iden i ie S 0018-9480(97)06066-3.
his pape we a e only in e es ed in quasi-s a ic app oaches.
This is because o i s compa a i e simplici y, which makes i
use ul o de elop quick mic os ip sol e s. Full-wa e me h-
ods, excep when applied o ze o- hickness p in ed lines,
usually demand a lo o compu e ime. E en when a quasi-
TEM app oach is used, accoun ing o s ip- hickness e ec s
p ecludes, in p inciple, he use o some e y e icien analy -
ical echniques de eloped o ze o- hickness plana s uc u es
[1]–[3]. In p inciple, one migh use some ype o pu ely
nume ical app oach, such as he ini e-di e ence [4] o ini e-
elemen echniques [5]–[7]. Howe e , e en hough signi i-
can imp o emen s ha e been in oduced in he o mula ion
o hose me hods, hey a e mo e app op ia e when dealing
wi h complica ed geome ies which canno be analyzed wi h
lesse compu e esou ce-demanding echniques. A hyb id
nume ical/analy ical me hod— he me hod o lines—has been
ecen ly applied o he analysis o coupled mic os ips wi h
ini e hickness [8]. Ne e heless, solu ions based on in eg al-
equa ion o mula ions seem o be well sui ed o mos p ac ical
cases i he goal is o ge high accu acy wi h low compu a ional
cos . Thus, a bi a y-sec ion coupled conduc o s embedded in
a laye ed medium ha e been analyzed in [9] by means o an
in eg al-equa ion echnique based on he ee-space G een’s
unc ion. This me hod was gene alized in [10] o accoun
o nonlaye ed dielec ics. Impo an nume ical imp o emen s
on his echnique ha e ecen ly been epo ed in [11]. O he
au ho s p e e o use a dielec ic G een’s unc ion when
dealing wi h laye ed dielec ic subs a es, since he numbe
o unknowns is d as ically educed in his way [12]–[15].
The las i e pape s s essed he analy ical p ep ocessing
o he compu a ions so as o enhance bo h accu acy and
compu a ional speed. Some o he au ho s ha e epo ed di -
e en echniques o de elop quick compu a ional ools o he
quasi-TEM analysis o pa icula mic os ip s uc u es wi h
nonnegligible me alliza ion hickness [16]–[19]. This pape i s
in o his esea ch line.
In his pape , we p opose a new me hod ha combines
he ad an ages o di e en o mula ions in o de o build up
a quick quasi-s a ic compu e sol e o ec angula -shaped
conduc o s embedded in a mul ilaye ed dielec ic medium.
The me hod is based on sol ing he space-domain in eg al
equa ion o he ee-cha ge dis ibu ion on he conduc ing
slabs. The app op ia e space-domain G een’s unc ion ( he
ke nel) is con enien ly ob ained om he spec al one by
means o sys em iden i ica ion echniques (using he complex
0018–9480/97$10.00 1997 IEEE
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1620 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 45, NO. 9, SEPTEMBER 1997
Fig. 1. The mul iconduc o ansmission line s udied in his pape . I consis s
o
N
c
ec angula conduc ing slabs embedded in he
M
h laye o a
N
-laye s
dielec ic medium.
images concep [20]). En i e domain basis unc ions a e used
in a Gale kin scheme wi h he aim o keeping he size o he
inal Gale kin ma ix small. In addi ion, he compu a ion o
he elemen s o his ma ix is ca ied ou in a e y e icien
way by using sui able nume ical quad a u es and closed- o m
in eg a ion. Pu ing oge he all hese elemen s leads o an
e icien and accu a e compu e code which is sui able o
quick compu a ions e en on a PC pla o m.
II. STATEMENT OF THE PROBLEM
Conside a mul iconduc o ansmission line such as he
one shown in Fig. 1. An a bi a y numbe o ec angula -
shaped conduc o s a e embedded inside he h dielec ic
laye o a mul ilaye ed ( laye s) dielec ic medium. A bo om
g ound plane is always p esen , while he op g ound plane is
op ional. Ou main pu pose is o compu e, in an e icien and
accu a e way, he pe uni leng h complex capaci ance ma ix
o he mul iconduc o sys em (i.e., he capaci ance and
conduc ance ma ices). As is well known, he compu a ion
o he pe uni leng h induc ance ma ix educes o he
e alua ion o o he same s uc u e wi hou dielec ics.
Finally, by using he Wheele ’s inc emen al induc ance ule,
we can ob ain he esis ance ma ix o he mul iconduc o
sys em om , as shown in [21].
As s a ed in he Sec ion I, he e exis s a wide a ie y o ech-
niques o compu e he capaci ance ma ices o mul iconduc o
ansmission sys ems. We ha e chosen o sol e he in eg al
equa ion o he su ace cha ge dis ibu ion on he ec angula
conduc o s. This means ha an app op ia e G een’s unc ion
accoun ing o he mul iple bounda ies and he bo om (and
op) g ound pla es has o be compu ed. The spec al-domain
e sion o such a G een’s unc ion can be easily ob ained
owing o he laye ed geome y o he dielec ic egion (see,
o ins ance, he ans e se ansmission line (TTL) me hod
epo ed in [22] o he me hod in [23]). I sou ce and ield
poin s a e inside he h laye , he spec al G een’s unc ion
can be w i en in he ollowing o m:
(1)
in (1) s and o he e lec ion coe icien s seen om he
lowe and uppe su aces bounding he h laye ,
p o ided we a e using he equi alen ansmission-line model
o compu e he spec al G een’s unc ion [22]. O cou se,
a e known closed- o m unc ions o he Fou ie a iable and
o he pe mi i i ies and hicknesses o he dielec ic laye s
below o abo e he h one. In o de o eco e he
-dependence o he space-domain G een’s unc ion, a Fou ie
ans o m in e sion has o be ca ied ou :
(2)
Apa om some pa icula cases, he in e se Fou ie ans-
o m (2) canno be pe o med in closed o m. Howe e ,
he -dependen unc ions appea ing in (1) as mul iplica i e
ac o s o he exponen ial e ms could be w i en as a ini e sum
o complex exponen ials. In his way, we could apply he ideas
epo ed in [20] so as o ge a e y close app oxima ion o he
space-domain G een’s unc ion. This poin will be discussed in
Sec ion III. Once he G een’s unc ion is known, we can sol e
he in eg al equa ion o he ee-cha ge dis ibu ion by using,
o ins ance, he Gale kin me hod. In Sec ion IV, we will gi e
de ails on he ype o basis unc ions used in his pape and on
he echniques applied in o de o speed up compu a ions. F om
he cha ge on he conduc ing slabs, we ob ain he capaci ance
ma ix. I he elec ical pe mi i i ies o he dielec ic laye s
a e complex (dielec ic losses), he elemen s o his ma ix will
be complex. Thei imagina y pa s gi e he elemen s o .
On he o he hand, is he in e se o he capaci ance ma ix
o he s uc u e wi hou dielec ics o e ( is he speed o
ligh in acuo). Finally, unde s ong skin-e ec ope a ion, we
can compu e he pe -uni -leng h esis ance ma ix , om
by using he ex ension o a mul iconduc o ansmission-
line sys em o he Wheele ’s ule [21]. The e o e, wi h an
accu a e me hod o compu e complex capaci ance ma ices,
we will be able o ob ain all he quasi-s a ic ma ix pa ame e s
cha ac e izing ou lossy mul iconduc o sys em (including he
ma ix, p o ided he skin e ec on me als is s ong).
III. COMPUTATION OF THE SPACE-DOMAIN
GREEN’SFUNCTION
Following he guidelines in [20], he space-domain G een’s
unc ion can be usually exp essed as a sho summa ion o
e ms which can be easily compu ed om i s spec al-domain
e sion. The spec al G een’s unc ion is known in closed o m
o a laye ed s uc u e. The basic equa ion unde lying Chow’s
p ocedu e is he ollowing ela ionship be ween spec al and
spa ial unc ions:
(3)
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BERNAL e al.: QUICK QUASI-TEM ANALYSIS OF MULTICONDUCTOR TRANSMISSION LINES 1621
whe e s ands o Fou ie ans o m and is a gene alized
complex dis ance. I we can w i e (1) as a sum o e ms such as
he ones on he igh -hand side (RHS) o (3), i is clea ha we
will ha e he space-domain G een’s unc ion o ou p oblem.
This can be done almos in a s aigh o wa d way by expanding
he e ms by mul iplying he exponen ials in (1) as a sum o
complex exponen ials. The a gumen s o he exponen ials and
he coe icien s o he expansion can be compu ed by using
spec al es ima ion me hods. In his pape , we ha e used and
compa ed he P ony’s me hod employed in [20] and he MPM,
as epo ed in [24].
A cau ion should be exe ed when we a e conside ing
s uc u es ha ing a op g ound plane in addi ion o he bo om
one. In such a case, he spec al unc ions o be app oxima ed
a e singula a he poin . This beha io comes om he
ollowing ac o :
(4)
whe e and .A
homogeneous s uc u e wi h ela i e pe mi i i y ha ing
wo g ound planes sepa a ed by a dis ance has a spec al
G een’s unc ion ha can be exp essed as ollows:
(5)
The G een’s unc ion in (5) and he o iginal one ha e he
same beha io a ound he poin . In addi ion, he
space-domain G een’s unc ion o he homogeneous s uc u e
is known in closed o m:
(6)
Exp ession (6) can be conside ed as he i s con ibu ion o
he comple e G een’s unc ion o he laye ed s uc u e. The
addi ional e ms a e exp essed in he spec al domain as he
di e ence be ween (1) and (5). The e ms o his spec al
unc ion ha e no singula i ies a and can now be ea ed
wi hou p oblems.
In b ie and a e some algeb a, he spec al-domain G een’s
unc ion o he laye ed s uc u e can be w i en in he ollow-
ing use ul o m:
(7)
whe e
(8a)
(8b)
(8c)
being
In all he abo e exp essions in he p esence o a op
g ound plane, and i he e is no op g ound plane.
The i s wo e ms a he RHS in (7) co espond o he
sou ce poin and he i s eal image. The spec al unc ions
can be expanded as sums o complex
exponen ials. These e ms can be iewed as he con ibu ions
o ce ain complex images, ollowing he e minology in [20].
Usually jus a ew o hese complex images a e enough o
ge a e y accu a e ep esen a ion o he G een’s unc ion.
The e o e, a e app oxima ing he coe icien s in (8) as
sho se ies o complex exponen ial unc ions wi h a gumen s
and ampli udes , we ob ain om (3) and (7) he
ollowing space-domain G een’s unc ion:
(9)
whe e is he o al numbe o employed complex images.
Once again, he e m a ec ed by he ac o in (9) is p esen
only i we ha e a op g ound plane. This e m does no
p esen a loga i hmic singula i y in he de ini ion domain
and, he e o e, will no lead la e o in eg a ion p oblems.
Ne e heless, in he compu e implemen a ion o he code, we
ha e ex ac ed ou no only he singula e m, bu also he i s
wo eal images ( e lec ions on he op and bo om g ound
planes) in o de o a oid leng hy in eg a ions when he sou ce
and ield poin s a e e y close o he g ound pla es.
In o de o ge some insigh abou he ea u es o he
app oxima ion we a e using, we include in his sec ion some
nume ical examples. Thus, Fig. 2 shows he ela i e e o
when a ypical spec al unc ion—such as he ones ound
in ou p oblem—is app oxima ed by an inc easing numbe
o complex images. In his example, we ha e app oxima ed
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1622 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 45, NO. 9, SEPTEMBER 1997
(a)
(b)
Fig. 2. Rela i e e o in he app oxima ion o a ypical spec al unc ion
in ou p oblem by using (a) P ony’s me hod, and (b) MPM. We p esen
cu es o 2–5 complex images.
0
L
(

)
co esponds o a single dielec ic
laye
(
"
=2
;h
1
=1
mm) on a pe ec g ound plane.
, which is he sole unc ion o be app oxima ed
when dealing wi h he s anda d mic os ip s uc u e. The
complex images ha e been ound by applying P ony’s me hod
[Fig. 2(a)] and he MPM [Fig. 2(b)]. F om hese igu es we
conclude ha no mo e han ou o i e images a e necessa y o
ge a e y good ep esen a ion o he o iginal G een’s unc ion
in he spec al domain and, he e o e, in he space domain.
Fig. 2(b) also includes he unc ion o be app oxima ed. No ice
ha al hough he P ony app oxima ion is mo e accu a e when
inc eases, he use o he MPM seems o be ad isable
because his app oach yields be e esul s in he egion whe e
he app oxima ed unc ion is meaning ully di e en om
ze o. This ea u e has been con i med o a la ge numbe
o nume ical examples. In b ie , we conclude a e a lo o
nume ical es s ha jus a ew images compu ed wi h he
MPM will ensu e a high-quali y app oxima ion o he equi ed
space-domain G een’s unc ion.
IV. APPLICATION OF THE GALERKIN METHOD
In Sec ion III, we ha e desc ibed a simple me hod o ob ain
an app oxima e closed- o m exp ession o he space-domain
G een’s unc ion. Now we ha e o sol e he in eg al equa ion
o he su ace ee-cha ge dis ibu ion on he ec angula
conduc o s by using he Gale kin me hod. I is well known ha
he e iciency o his echnique depends o a la ge ex en on he
sui abili y o he chosen basis unc ions. In ou pa icula case
we should use unc ions accoun ing o he singula beha io a
he me allic co ne s. Ne e heless, we ha e chosen a di e en
c i e ion. We a e mo e in e es ed in using unc ions leading
o closed- o m o mulas o he elemen s o he Gale kin
ma ix in o de o speed up he illing o such a ma ix.
Bu , in addi ion, he numbe o unc ions needed o ge a
gi en accu acy should be kep as low as possible. These
wo goals can be achie ed by using each o he aces o he
ec angula conduc o s o he basis unc ions usually employed
o app oxima e he cha ge dis ibu ion on ze o hickness s ips:
i s -kind Chebyshe polynomials weighed by he Maxwell
dis ibu ion. In his way, o a gi en conduc ing ace o wid h
we w i e
(10)
whe e s ands o he o a iable ( o ho izon al and
e ical conduc ing aces, espec i ely), is he middle poin
o he in e al whe e he cha ge densi y is app oxima ed, and
is he numbe o e ained basis unc ions on ha ace.
An impo an poin is ha he unc ions in (10) only pa ially
accoun o he singula i ies a he co ne s. Howe e , hey
allow us a quick illing o he Gale kin ma ix. This is because
some o he equi ed in eg a ions can be pe o med in closed
o m. When we use he G een’s unc ion in he o m gi en in
(9) and he basis unc ions in (10), we ha e o ca y ou he
ollowing in eg a ions:
(11)
whe e s ands o o . The in eg als in (11) can be w i en
in e ms o he ollowing one:
(12)
This in eg al has a closed o m and can be exp essed as
ollows:
Case a) :
(13a)
Case b) :
(13b)
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BERNAL e al.: QUICK QUASI-TEM ANALYSIS OF MULTICONDUCTOR TRANSMISSION LINES 1623
whe e
(The sign be o e he squa e oo is chosen in such a way ha
).
We mus men ion he e ha he eal pa o he in eg al in
(12) was epo ed by Fikio is e al. in [1], and ha we ha e
also essen ially used he same echnique o ge he imagina y
pa o (12).
The emaining in eg a ions (inne p oduc s wi h he es
unc ions) can now be ca ied ou by using low-o de
Gauss–Chebyshe quad a u es. I a op g ound plane is p esen
we ha e o s ill accoun o he con ibu ion o he e m
in (9). This is easily done pe o ming a double low-o de
Gauss–Chebyshe quad a u e, owing o he ma hema ical
na u e o he in eg and. The inal esul is ha he Gale kin
ma ix is illed wi h low compu a ional e o . This could
also be done o simple subsec ional pulse o iangula basis
unc ions. The ad an age o using he unc ions in (10) is ha
we do no need oo many o hem o ge e y accu a e esul s,
as will be shown in he Sec ion V. The e o e, he Gale kin-
ma ix size will be small and he o e all compu a ion ime
will be low.
V. NUMERICAL RESULTS
As a i s s ep in he analysis o he nume ical beha io
o he p oposed echnique, we ha e iden i ied he di e en
ac o s a ec ing he accu acy o he inal esul s. In Sec ion III,
we said some wo ds abou he compu a ion o he space-
domain G een’s unc ion. Since his unc ion is nume ically
compu ed by means o an app oxima e me hod, i s alues
will be a ec ed by a ce ain e o . Howe e , his e o can be
sys ema ically educed by inc easing he numbe o complex
images. In ac , jus a ew o hese images ensu e a ela i e
e o well below one pa in 10 in he whole ange o
in e es . Ano he sou ce o e o o nume ical ype can be
ound in he e alua ion o he nume ical quad a u es needed o
compu e some in eg als. A e a lo o nume ical expe imen s,
we concluded ha e y good esul s will be ob ained by
using a numbe o quad a u e poin s exceeding in wo he
o de o he highe o de Chebyshe polynomial used in
he basis- unc ion’s expansion. In he case o inne p oduc s
co esponding o unc ions de ined on ouching s ip segmen s,
his numbe should be inc eased o 10. Wi h his choice, we
do no de ec e o s associa ed wi h e oneous compu a ion o
de ini e in eg als.
Howe e , he main ac o a ec ing he accu acy o he
inal esul s is he numbe o basis unc ions e ained in he
expansion o he ee-cha ge dis ibu ion. We a e in e es ed
in ge ing good enough esul s wi h ew basis unc ions so
as o keep he size o he Gale kin ma ix small. Ob i-
ously, ou compu e code pe mi s us o conside he case
o ze o- hickness s ips as a pa icula case. We ha e made
compa isons o he esul s p o ided by ou code wi h o he
p ac ically exac esul s epo ed in he li e a u e [1]–[3]. Ou
TABLE I
C
11
(
+)
AND
C
1
2
(
0
)
(pF/m) FOR THE STRUCTURE IN THE
FIGURE AGAINST THE NUMBER OF BASIS FUNCTIONS ON VERTICAL
(
N
)
AND HORIZONTAL
(
N
w
)
CONDUCTOR FACES.
w
=1
,
s
=
=0
:
2
,
h
1
=3
,
h
2
=2
(mm).
"
1
=2
:
5
,
"
2
=10
p og am also yields i ually exac esul s because o he ze o
hickness case, i is a quasi-analy ical mic os ip sol e (such
as he me hods epo ed in he ci ed pape s), wi h ex emely
low cen al p ocessing uni (CPU) ime consump ion. This
excellen pe o mance is ela ed o he na u e o he used basis
unc ions—which a e especially sui able o ze o hickness
s ips—and o he analy ical ea men o he compu a ion o
he Gale kin ma ix en ies. The e o e, as a subp oduc o ou
analysis, we ha e an ex emely e icien code o he analysis
o ze o hickness coupled s ips a ailable. Ne e heless, in
he con ex o his pape , we a e mo e in e es ed in he case
o nonze o hickness s ips. The e o e, we show in Table I
he con e gence o he capaci ance coe icien s o a pai o
hick coupled mic os ips. F om Table I, we can see ha
con e gence is no so good as he one we achie ed in he
analysis o ze o- hickness coupled s ips [2]. The eason o
his is ha he employed basis unc ions do no exac ly i he
co ne beha io . Howe e , hese unc ions a e e y good om
a p ac ical poin o iew: wo basis unc ions pe s ip side
yield esul s wi h an accu acy be e han 0.3%. O cou se,
he numbe o basis unc ions has o be inc eased when
he geome y is mo e c i ical ( o example, when coupling
be ween he s ips is ex emely s ong o in o he geome ically
complica ed si ua ions). Ne e heless, we usually ob ain e y
good esul s wi h jus a ew basis unc ions pe s ip side.
Ge ing simila accu acy wi h subsec ional- ype basis unc ions
would equi e many mo e basis unc ions, he e o e inc easing
he CPU ime.
A e analyzing he con e gence o he me hod, i is in-
e es ing o check he ypical a ainable accu acy using some
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1624 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 45, NO. 9, SEPTEMBER 1997
TABLE II
Z
0
OF A SYMMETRICAL SHIELDED SLAB LINE
N
w;
IS THE
NUMBER OF BASIS FUNCTIONS
W
=
SLAB WIDTH.
=
SLAB
THICKNESS
b
=
SEPARATION BETWEEN GROUND PLATES
app op ia e benchma k. We can use as a benchma k he
ec angula slab symme ically placed be ween wo in ini e
g ound planes, since o his s uc u e he e exis s an ana-
ly ical solu ion. Table II shows he compa ison be ween exac
solu ions aken om [25] (con o mal mapping app oach) and
nume ical alues compu ed wi h ou me hod. I is clea om
his able ha ou me hod can p o ide e y high accu acy i
we use enough basis unc ions. Howe e , e en using jus one
unc ion on each side o he slab, we ge esul s wi hin a 0.4%
e o excep o he wo s case ( ), whe e he e o
is a ound 1.5%. In his case, using jus one mo e unc ion on
he long sides o he slab esul s in a d as ic imp o emen
o he accu acy (less han 0.02% e o ). O he compa isons
ha e been made wi h some exac esul s epo ed in [26] wi h
simila conclusions. Con e gence o he co ec alue has hen
been demons a ed. No e ha his poin is impo an since
ou basis unc ions do no exhibi he igo ously co ec edge
beha io .
In [19], he au ho p esen s a echnique based on spec al-
domain analysis (SDA) o compu e he quasi-s a ic pa ame e s
o boxed coupled s ips o a bi a y hickness. We ha e made
compa isons wi h he esul s epo ed in [19] and we ha e
ound e y good ag eemen . In pa icula , we ha e compu ed
he modal pa ame e s o wo e y igh ly coupled s ips, which
canno be e icien ly ea ed by using he simple mul is ip
model epo ed in [15] (as was claimed in [19]). Ou esul s a e
e y accu a e o his case e en using only wo basis unc ions
on he aced sides o he s ips and one on he emaining sides
(0.2% e o ). This esul s in CPU imes much sho e han
he ones epo ed in [19] (ou s is below 1 s on a 66-MHz
pen ium-based PC).
Ou p og am can be used o calcula e he esis ance ma ix
o he mul iconduc o sys em jus applying he Wheele ’s
ule, such as discussed in [21]:
whe e is he su ace esis ance o he me al, is he
induc ance ma ix, and is he coo dina e a iable no mal o
he conduc o su ace. The de i a i e is nume ically compu ed,
so we only need o accu a ely compu e . Table III shows
he no malized esis ance o a ec angula slab on a g ound
TABLE III
NORMALIZED RESISTANCE OF RECTANGULAR SLAB OVER GROUND PLANE AS A
FUNCTION OF THE NUMBER OF BASIS FUNCTIONS
(
n
)
ON EACH STRIP SIDE
AND OF THE
1

INCREMENT USED IN THE NUMERICAL DERIVATION SLAB WIDTH
=2
a
,SLAB THICKNESS
=
a
,DISTANCE TO GROUND PLANE
=
a
Fig. 3. A enua ion ac o s o he undamen al quasi-TEM modes o he
ou mic os ips sys em analyzed in [27]. S ip wid hs:
w
1
=
w
4
=0
:
6
mm,
w
2
=
w
3
=0
:
3
mm, sepa a ion be ween s ips:
s
1
=
s
3
=0
:
3
mm,
s
2
=0
:
2
mm, s ip hickness:
=0
:
01
mm, subs a e hickness:
h
=0
:
635
mm, heigh o co e :
d
=6
:
0
mm,
"
=9
:
8
"
0
;
=5
:
1
1
10
7
S/m
plane as a unc ion o he numbe o basis unc ions used on
each s ip side. The esul is close o he one epo ed in [13]
. We ha e included se e al columns o
show how he inc emen used o nume ically pe o m he
de i a ion a ec s he inal esul . We can ypically choose
a ound 10 imes he smalles dimension o he conduc o .
The esul emains s able o lowe alues o . No e ha
he de i a i e is e y accu a ely compu ed e en hough he
o iginal unc ion is a ec ed by a ce ain e o , since his is
a sys ema ic e o ha a ec s in he same magni ude he
wo alues o he unc ion equi ed o nume ically pe o m
he de i a ion. Recen ly, Gen ili e al. [27] ha e epo ed a
echnique o compu e he modal a enua ion ac o s
o a mul iconduc o line. The alues o he
pa ame e s can be eadily ob ained om he cha ac e is ic ma-
ices compu ed in his pape . This way, we ha e compa ed ou
esul s wi h hose epo ed in [27], wi h e y good ag eemen .
Fo ins ance, we show in Fig. 3 he ou modal a enua ion
ac o s co esponding o he undamen al modes suppo ed by
a ou -coupled-s ip lossy sys em. Ou esul s a e e y close
o he ones compu ed by he me hod p oposed by Gen ili
e al. (sligh di e ences could be a ibu ed o he side walls
conside ed in ha pape ), bu some impo an disc epancies
ha e been ound when compa ing he esul s compu ed by
means o a ini e-elemen -based code.
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BERNAL e al.: QUICK QUASI-TEM ANALYSIS OF MULTICONDUCTOR TRANSMISSION LINES 1625
TABLE IV
COEFFICIENTS OF
[C]
,
[L]
,AND
[R]
FOR THE FIVE CONDUCTORS’ STRUCTURE IN THE FIGURE.
w
=3
,
s
=2
,
h
=1
,
=1
(mm),
"
1
=2
,
"
2
=1
.
Finally, Table IV shows he elemen s o and o
a i e-coupled-s ip sys em. All he igu es in Table IV a e
co ec . We ha e used 15 basis unc ions on each s ip side,
i.e., a o al o 300 basis unc ions, o ensu e he accu acy o
hese esul s. This esul s in a compu a ion ime o 75 s on
a Pen ium PC/66 MHz. Howe e , accu acy be e han 0.2%
o all he ma ix elemen s is achie ed by using i e unc ions
and 13 s o CPU ime. I an e o in he o de o 1% can be
ole a ed, we only ha e o use wo unc ions pe s ip side. We
ha e comple ed Table IV, including he no malized esis ance
ma ix elemen s.
VI. CONCLUSIONS
This pape desc ibes an e icien and accu a e echnique
o compu e he quasi-s a ic ma ix pa ame e s ( , , ,
) o a sys em o ec angula c oss-sec ion coupled con-
duc o s embedded in a laye ed dielec ic medium. Accu-
acy and nume ical e iciency a e achie ed by means o
wo main issues. Fi s , we use en i e domain basis unc-
ions o app oxima e he ee su ace-cha ge dis ibu ion on
he s ips. These unc ions allow us o keep he Gale kin
ma ix size small when compa ed wi h ypical ma ix size
associa ed wi h he use o subsec ional unc ions. Second,
we ge quasi-analy ical e alua ion o he in eg als de ining
he Gale kin-ma ix en ies. This has been done by aking
ad an age o he use o he complex image concep and
he ma hema ical p ope ies o he employed basis unc ions.
Any spec al es ima ion echnique can be used o compu e
he complex images, bu he MPM seems o pe o m be e
han P ony’s me hod. The inal p oduc is an accu a e and
quick compu e code ha pe mi s one o analyze unde quasi-
TEM assump ion a a ie y o ansmission lines consis ing o
coupled ec angula conduc ing slabs. The pe o mance o he
code has been exhaus i ely checked by making con e gence
es s and compa isons wi h o he echniques, some o which
ha e been included in Sec ion V. No e ha al hough in his
pape i has been assumed ha all he s ips we e embed-
ded in he same dielec ic egion, a mo e gene al si ua ion
can be ea ed by applying he same gene al ideas epo ed
he e.
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Joaqu´ın Be nal was bo n in Se illa, Spain, in 1971.
He ecei ed he Licenciado deg ee in physics om
he Uni e si y o Se ille, Spain, in 1994, and is
cu en ly wo king owa d he Ph.D. deg ee.
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.
F ancisco Medina was bo n in Pue o Real,
C´adiz, Spain, in No embe , 1960. He ecei ed he
Licenciado and he Doc o deg ees, bo h in physics,
om he Uni e si y o Se ille, Se ille, Spain, in
1983 and 1987, espec i ely. 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, on
schola ship om MEC-MRT.
F om 1985 o 1989, he was an Assis an
P o esso in he Depa men o Elec onics and
Elec omagne ics, 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 ics.
His esea ch deals mainly wi h analy ical and nume ical me hods o plana
s uc u es and ci cui applica ions o mul iconduc o lines.
D . Medina was a membe o he Technical P og amme Commi ee o he
23 d Eu opean Mic owa e Con e ence, Mad id, Spain, in 1993.
Manuel Ho no (M’75) was bo n in To e del
Campo, Ja´en, Spain. He ecei ed he Licenciado
and he Doc o deg ees, bo h in physics, om he
Uni e si y o Se ille, Se ille, Spain, in 1969, and
1972, espec i ely.
Since 1969, he has been wi h 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 Full
P o esso in 1986. His main ields o in e es include
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. He is p esen ly engaged in he analysis o plana ansmission
lines embedded in complex ma e ials, mul iconduc o ansmission lines, and
p in ed an ennas.
D . Ho no is a membe o he Elec omagne ism Academy, Massachuse s
Ins i u e o Technology (MIT), Camb idge.
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