2328
IEEE TRANSACTIONS
ON
MICROWAVE THEORY AND TECHNIQUES. VOL.
42,
NO.
12.
DECEMBER
1994
Quick Compu a ion
o
[C]
and [L]
Ma ices
o
Gene alized Mul iconduc o
Coplana Wa eguide T ansmission Lines
En ique
D ake,
F ancisco
Medina, and Manuel
Homo,
Membe ,
ZEEE
Abs ac -An enhanced spec al domain quasi-TEM analysis o
gene alized coplana wa eguide ansmission lines (GCPWTL)
is p esen ed. The analysis s a s om he o mula ion o a
con olu ion- ype in eg al equa ion o he elec ic ield a he
slo s.
Chebyshe polynomials including Maxwell singula i ies a e
used as basis unc ions o sol e he in eg al equa ion by he
Gale kin me hod. Fas and accu a e quasi-analy ical o mulas
a e used o calcula e he Gale kin’s ma ix en ies, he eby
signi ican ly educing he in ol ed CPU ime and inc easing
eliabili y and accu acy. These ea u es make his echnique
use ul and compe i i e as
CAD
ool o coplana wa eguide
designs.
I.
INTRODUCTION
OPLANAR wa eguide (CPW) ansmission lines a e
C
becoming a compe i i e al e na i e o mic os ip in many
applica ions (including bo h hyb id and monoli hic echnolo-
gies). A numbe o a ac i e ea u es-loca ion o he signal
g ounds on he same subs a e su ace as he signal line,
low pa asi ic induc ances, easy shun and se ies connec ions,
a oidance o he need o ia holes, good isola ion in di ec-
ional couple s, e c.
[
I]-[3]-makes his ansmission medium
pa icula ly in e es ing. Due o his ac oge he wi h he
ela i e lack o design da a ii ailable o CPW s uc u es (in
compa ison wi h he mic os ip line), esea ch on many aspec s
ela ed o he cha ac e iza ion o CPW s uc u es is s ill going
on [4]-[7].
The compu a ion o he p opaga ion cha ac e is ics o CPWs
has ecei ed some a en ion in old and ecen li e a u e (see,
o example,
[8]-[
1
11,
which include exhaus i e bibliog aphy
sweeping a wide a ie y
o
analy ical and nume ical ech-
niques). As i is well known, he e alua ion o dispe sion,
adia ion, highe -o de modes o leakage phenomena equi es
igo ous hyb id-mode app oaches. Howe e , he quasi-TEM
app oxima ion can be expec ed o yield use ul esul s in he
equency band whe eon MIC’s usually ope a e oday, a leas
o hose s uc u es and componen s which a e no pa icula ly
equency-sensi i e [8], and e en
up
o 40 GHz in he design o
coplana MMIC’s
[9].
Since he quasi-TEM analysis equi es
Manusc ip ecei ed No embe
17,
1993; e ised Janua y 27, 1994.
This wo k was suppo ed
in
pa by he DGICYT, Spain, unde con ac
TIC91-1018.
The au ho s a e wi h he Mic oy es G oup. Depa men
o
Elec onics
and Elec omagne ism. Facul ad de Fisica A da. Reina Me cedes
s/n,
41012
Se illa, Spain.
IEEE Log Numbe 9405369.
much less compu a ional e o , i is mo e adequa e
o
design
pu poses.
In
addi ion, quasi-TEM da a could e en ually be
used as ini ial guesses in ull-wa e algo i hms, hus imp o ing
hei e iciency.
The quasi-TEM analysis o ce ain pa icula CPW
geome ies has been al eady ca ied ou in a e y e icien
way (sui able o CAD applica ions). Fo ins ance, a quasi-
analy ical me hod o deal wi h a single CPW embedded in
a s a i ied medium is epo ed in
[l
11.
In ha pape he
eade can ind a lis o e e ences epo ing o he quasi-
analy ical me hods o analyze a a ie y o symme ical and
asymme ical single CPW geome ies (mos o hem based
on con o mal mapping app oaches). One o he mos ecen
con ibu ions based on con o mal mapping can be ound
in [6]. Howe e , mo e complex CPW s uc u es in ol ing
mul iple dielec ic laye s and coupled conduc o s ha e
ecei ed less a en ion in spi e o i s ob ious in e es in
p ac ical applica ions ( il e , couple s, e c.). Some analy ical o
app oxima e solu ions o symme ical coupled s uc u es ha e
been epo ed in he li e a u e
[l],
[12]. Mo e sophis ica ed
geome ies ha e been conside ed in [13], [14]. A ecen
wo k
[
151 p oposes an e icien Wiene -Hopi solu ion o
a mul iconduc o CPW sys em ( o applica ion as in e digi al
ansduce ), bu i is es ic ed o geome ies symme ically
placed be ween wo g ound planes wi h homogeneous
medium. The a ailabili y o quick and e sa ile mul iconduc o
sol e s is impo an om he designe ’s pe spec i e, since
op imiza ion p ocesses in ol e he i e a i e e alua ion o
he pa ame e s o a s uc u e o a wide ange o design
a iables. In his sense, ex emely e icien algo i hms ha e
been al eady de eloped o he quasi-TEM analysis
o
gene al mul is ip geome ies
[
161-[
181.
Howe e , as a as
we know, a sys ema ic and quasi-analy ical ea men o a
gene alized coplana wa eguide ansmission line (GCPWTL)
sys em-including an a bi a y numbe o me allic s ips
be ween wo g ound planes embedded in a mul ilaye
medium-has no been explici ly gi en ye . The cu en
pape con ibu es o he compu e -aided design
o
coplana -
ype ci cui s by o e ing a quasi-analy ical p ocedu e o
compu e he quasi-TEM p opaga ion pa ame e s o he
GCPWTL sys em in Fig.
1.
The mul islo geome y o he
GCPWTL sys em makes sui able o s a e he analysis in
e ms
o
he ape u e elec ic ield. The e o e, a con olu ion
in eg al equa ion is p oposed o connec he elec ic ield
a he slo s wi h he ee cha ge on he me alliza ions. The
0018-9480/94$04.00
0
1994 IEEE
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[C] AND
[L]
MATRICES
OF
GENERALIZED MULTICONDUCTOR COPLANAR WAVEGUIDE TRANSMISSION LINES
2329
e.w.,
m.w.
o
0.b.
-------I
,///////,///,////,
e.w.,
m.w.
o
0.b.
x=a
2-
X=O
Fig.
1,
ansmission
line
(GCPWTL).
C oss-sec ion
o
he
gene alized mul iconduc o coplana wa eguide
me hod used o sol e he elec ic ield in eg al equa ion is an
enhanced Gale kin spec al domain analysis (SDA). Quasi-
analy ical o mulas a e p o ided o compu e he Gale kin’s
ma ix en ies. A dual ea men o mul is ip geome ies
in e ms
o
he ee cha ge densi y on he me alliza ions
was success ully used and epo ed in [17]. High speed
o compu a ion and ex eme accu acy could make he use
o his me hod use ul
in
he CAD
o
coplana wa eguide
ci cui s.
11.
OUTLINE
OF
THE
PROBLEM
Fig. 1 shows he c oss-sec ion o he GCPWTL analyzed
in his wo k. T ansla ional symme y in he p opaga ion
di ec ion (z-axis) is assumed. The whole s uc u e is
enclosed in o a ec angula box de ined by he planes
s
=
0,
J;
=
a,
y
=
0,
and
y
=
b.
The la e al
planes
.I‘
=
0
and
z
=
a
a e elec ic walls (e.w.). The
planes
y
=
0
and
y
=
b
can be chosen o be elec ic
walls, magne ic walls (m.w.) o open bounda ies (0.b.).
The subs a e is a Ni-laye ed 1osslessAossy iso/aniso opic
linea medium. An a bi a y numbe
(N)
o conduc o
s ips-alloca ed be ween wo g ounded me al ins-a e
p in ed on he M h in e ace. Le us cha ac e ize each o
he
N
+
1
slo s be ween hese conduc o s ips by bo h
i s wid h
(3%;
z
=
1..
. .
,N
+
1)
and he posi ion
(zcz;
1
=
1,.
.
.
,
N
+
1)
o i s middle poin (wi h espec o he le
la e al wall). This s uc u e includes a la ge g oup o CPW
geome ies as pa icula cases. Mo eo e , he con igu a ion in
Fig.
1
accoun s o possible echnological cons ain s (uppe
and la e al shielding, conduc o backing, and line- o-line
coupling).
As is well known, all he quasi-TEM p opaga ion pa ame e s
o
a N-conduc o ansmission line may
be
ob ained om
i s capaci ance,
[C],
and induc ance,
[L],
pe uni leng h
(p.u.1.) ma ices. The de e mina ion o
[C]
implies o sol e
an elec os a ic p oblem.
[L]
may be compu ed om he
capaci ance,
[C’],
o a ela ed s uc u e
[19].
I he elec os a ic
p oblem is s a ed in e ms o he su ace ee cha ge densi y
on he conduc o s, each coe icien ,
C2,,
o
[C]
o
[C’]
is
iden i ied as he ee cha ge on he z- h conduc o when he j- h
conduc o is se
o
ol age uni y and he es o he conduc o s
a e g ounded (canonical ol age exci a ion)
[
171. Howe e , he
mul islo geome y is mo e e icien ly cha ac e ized in e ms
o
he ape u e elec ic ields han
in
e ms o he cha ge
dis ibu ion. Owing o his, i is mo e di ec o calcula e i s
coe icien s o po en ial ma ix
[PI,
i.e., he se o coe icien s,
Pi,,
which linea ly ela es he po en ial
V,
o any conduc o
o he ee cha ges
Q,
on
all
o
he conduc o s, including
i sel :
N
,
=
Pi&),
(i
=
1,.
.
.
,N).
J=l
No e ha
[PI
=
[GI-’.
Each coe icien
P;j
may be de ined
as he ol age
o
he i h conduc ing s ip when he j- h s ip
is cha ged wi h cha ge uni y, and he es
o
he s ips a e
discha ged (canonical cha ge exci a ion). The ol age o each
s ip is compu ed in eg a ing he elec ic ield 2-componen
along he slo s exis ing be ween one o he g ounded pla es
and ha s ip. The e o e,
[PI
(and i s in e se,
[C])
will be
ob ained i we compu e he ape u e ields o
N
independen
canonical cha ge exci a ions.
111.
THE
SLOT
ELECTRIC FIELD
EQUATION
F om he G een’s heo em, he elec os a ic po en ial
@(z)
on he me allized in e ace o he s uc u e in Fig.
1
is ela ed
o he ee cha ge densi y
o(2)
by
whe e
G(z,
z’)
is he po en ial G een’s unc ion associa ed o
he hl h in e ace
o
he s uc u e. This is he ee cha ge den-
si y in eg al equa ion usually sol ed when mul is ip geome-
ies a e analyzed. Howe e , we a e now in e es ed in using
an in eg al equa ion o he slo elec ic ield 2-componen .
Owing o he exis ence o elec ic walls in bo h
z
=
0
and
z
=
a,
he Fou ie se ies expansion o
G(z,z’)
yields
(3)
whe e- om Fou ie ans o m heo y
[20]--G,(a)
is he
spec al domain G een’s unc ion (SDGF) associa ed o he
la e ally open s uc u e, i.e.,
a
+
30.
De ining he ollowing
pai s o sine(cosine)-Fou ie ans o ms:
and using
(3),
i is s aigh o wa d o con e
(2)
in o
(4)
by sine-Fou ie ans o m. Simple manipula ions le us o
ew i e
(5)
in he ollowing way:
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2330
lEEE
TRANSACTIONS
ON
MICROWAVE
THEORY
AND
TECHNIQUES,
VOL.
42,
NO.
I?,
DECEMBER
1994
whe e
(7)
No e ha
(6)
may be iden i ied as he cosine-Fou ie ans o m
o
q(x)
=
.I”
L(J,L’)E,(T’)
ds’
(8)
whe e
Ez( ’)
is he elec ic ield 2-componen along he
me allized in e ace.
q(~)
in
(8)
is he amoun o ee cha ge
alloca ed om
d
=
0
o
T’
=
L,
i.e.,
x
q(x)
=
4,
O(d)
dxl
and
L(2.z’)
is
(9)
a e used as basis unc ions in he expansion o he elec ic
ield x-componen
2
-7
E+,
(L’)
=
{
2
[I
-
(*)
]
Tq(
*);
. ‘
E
S,
(15)
el sew he e.
The choice o his se o unc ions is sugges ed by he na u e
o he in eg al ke nel. The applica ion o he Gale kin me hod
con e s (12) and (13) in o a sys em o algeb aic linea
equa ions o he coe icien s
aq,,,
A7+1
I
IS
ob ious ha
q(x)
s ands cons an along each slo
(s
E
U0,J
=
0
S,
=.
[L,,
-
s,/2
5
.
5
. ea
s,/2]),
and i s alue is
J=1
i-1
q(: :€Si)=CQj;
i=1
....,
N+l
(11)
j=O
whe e
Qj
(.j
=
1,.
. . ,
N)
is he o al ee cha ge on he
j- h s ip, and
Qo,
he o al ee cha ge on he le coplana
g ounded in. Since
QO
is
no known, he o al alue o
q(x)
along a slo can no be compu ed when
a
cha ge exci a ion
is imposed. Howe e , when
z
skips om
a
gi en slo o
he nex one, he inc ease in
q(x)
is
equal o he amoun o
ee cha ge suppo ed by he s ip alloca ed be ween he wo
slo s. The e o e, he elec ic ield 2-componen o he k h
(k
=
1:
. . .
,
N)
canonical cha ge exci a ion (cha ge uni y on
he k h s ip keeping he es o he s ips discha ged) ul ils
he ollowing condi ion
1“
L(z
E
,Si+l,x’)Ez(x’)
dd
whe e he en ies
A;:$
O ,
=
0,
. .
.
,
n i
-
1;
y
=
(I,
.
.
.
, i, ;
-
1;
i,j
=
1,.
. .
,
N
+
1)
o he sys em a e:
A;;{
=
J’
dz
Exp,,(x)
J’
d.c’
zc2
+s,
/2
x,,+. ,!2
s,,-s,/2
a,,-s,/2
x
EZq,,
(d)L(X,
d).
(17)
Howe e
L(x,z’)
in (17) is no known (excep o special
cases) in closed o m. Fo una ely, (7) shows ha he spec al
ans o m_,
L(c ),
o
he
ke nel o (17) is ela ed
o
he
SDGF,
Go(a),
associa ed o he la e ally open e sion
o
he
mul ilaye ed con igu a ion. The e o e,
i
is mo e sui able o
ob ain a spec al domain exp ession o he en ies
A;;:,
and,
hen, ake ad an age om he e icien algo i hm epo ed
in
[19], [21] o compu ing he SDGF o an a bi a y laye ed
con igu a ion. The spec al e sion o
(17)
may be deduced
om Pa se al and con olu ion heo ems
(12) whe e
kz0
a e he cosine-Fou ie ans o ms o
II
Ezq,,
(d),
Le.,
whe e
b,k
is
he K onecke del a
(1
i
i
=
k,
0
i
z
#
k).
In
J~(?)(-I):
cos(a,.~,~)
Jq( )(-l)w
sin(a,s,,])
i
y
is
e en
i
q
is odd
(19)
addi ion, he g ounding o he la e al ins makes he elec ic
-
ield x-componen ul ils
E%,,(-)
=
1‘
Ez(2-’)dT’
=
0.
(13) wi h
Jq(-)
being he i s kind Bessel unc ion o o de
q.
Once he sys em (16) has been sol ed, he coe icien s o
po en ial a e di ec ly compu ed om he expansion ze o h-
IV.
METHOD
OF
ANALYSIS o de coe icien s
P obably, one o he bes known echniques o sol e in eg al
equa ions as (12) and (1
3)
is he Gale kin me hod. In his wo k,
a
,=1
pzk
=
-
i
=
1,
.
.
. ,
L
k h
exci a ion
his me hod has been also chosen. The i s kind Chebyshe
polynomials,
Tq(.),
weighed by he Maxwell edge singula i y (20)
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[CI AND [L] MATRICES
OF
GENERALIZED MULTICONDUCTOR COPLANAR WAVEGUIDE TRANSMISSION
LINES
2331
0.1
0.0
1
I
o-~
2
3
4
5
6
7
8
9
1011
a
/I
Fig.
2. Rela i e di e ences be ween he capaci ance coe icien s
o
a la -
e ally closed h ee-conduc o CPW- ype ansmission line
(Ccl)
and i s
co esponding
open
e sion
(C‘,,p),
Da a:
h
=
0.635
mm,
1
=
2.4
mm,
.sl
=
s2
=
sa
=
sq
=
0.2
mm,
. l
=
u/2-1.1
mm,
.cc2
=
a/2-0.5
mm,
. p:<
=
a/S
+
0.5
mm,
z,.4
=
~/2
+
1.1
mm,
cZ*.
=
13 o,
Fyy
=
loco.
since only he ze o h-o de basis unc ions in
(15)
ha e non-
anishing in eg als along hei de ini ion in e als.
The compu a ional s ep o he de e mina ion o
[PI
in ol -
ing mos o he CPU ime is p ecisely he sum o he spec al
se ies in
(18).
I s di ec sum is no ad isable because o i s
e y slow con e gence. In his wo k,
he
Kumme ’s me hod
(ex ac ion o an asymp o ic ail) is used o imp o e he se ies
con e gence. The se ies o
(18)
a e hen spli as ollows:
whe e
<k
(k
=
M,
+
1)
is he pe mi i i y (o he equi -
alen pe mi i i y
[E],
[21] in he aniso opic case)
o
he
k- h kye . Since
La,
has he same asymp o ic beha io
han
C
o la ge
an,
he emainde se ies ( i s e m a he
igh hand o (21)) con e ges e y quickly. The asymp o ic
ails
S;;:
a e ex emely slow con e gen se ies, bu quasi-
analy ical exp essions o hem a e p o ided in Appendix.
The me hods employed
o
ob ain he o mulas in Appen-
dix ha e been al eady desc ibed in
[17];
hus, we ha e
jus included in his Appendix he inal o mulas which ap-
plies o he se ies in ol ed in he analysis o GCPWTL
s uc u es.
V.
NUMERICAL
RESULTS
A FORTRAN p og am (MULTISLOT) has been de el-
oped implemen ing he heo y in his pape . The compu e
code uns on a PC/486/33 MHz. In o de o alida e ou
me hod, we ha e ep oduced analy ical da a ob ained by
means o exac con o mal mappings (which a e a ailable o
some pa icula geome ies). Good ag eemen has also been
ound wi h da a epo ed o mo e complica ed geome ies
which we e ob ained by nume ical p ocedu es. An in e es ing
compa ison has been ca ied ou wi h he esul s epo ed
in
1151.
The me hod used in ha pape is inhe en ly e y
accu a e (al hough, in p inciple, i is limi ed o homogeneous
o symme ical geome ies). The esul s epo ed in
11.51
a e
he Fou ie ans o ms
o
he su ace cha ge dis ibu ions
a he “ac i e” me alliza ion plane o se e al SAW (su -
ace acous ic wa e) s uc u es o se e al elec ode exci a-
ion condi ions. We ha e ep oduced hei esul s wi h e y
good ag eemen and e y ew basis unc ions and sho
CPU
ime ( ypically less han one second on he a o e-men ioned
compu e
pla o m),
Mino disc epancies we e de ec ed o
la ge alues o he Fou ie a iable, due o he di e en
na u e o he basis unc ions used in he expansions
o
he
unknown unc ions (su ace cha ge densi y o slo elec ic
ield). We belie e ou esul s a e e en mo e accu a e since
ou basis unc ions inco po a e he singula beha iou a
he me allic edges and he esul s do no modi y when he
numbe o unc ions inc ease abo e a ce ain alue (nume ical
s abili y).
In
addi ion, exhaus i e con e gence es s ha e been pe -
o med o iden i y he geome ical dimensionless a ios go -
eming he con e gence o ou codes. This kind o s udy
inc eases ou con idence in ou esul s.
The
conclusions om
hese es s a e analogous o he ones epo ed in
[I71
o
he mul is ip case. We can summa ize he e he main poin s
highligh ed by his s udy:
The esidual spec al se ies ( i s e m a he igh hand
in (21)) show exponen ial con e gence. The main ge-
ome ical pa ame e go e ning his con e gence is he
a io
h/a
(h
being he hickness o he hinnes laye
adjoining he me allized in e ace and
a
being he wid h o
he enclosu e). The e o e, he pa ame e
a.
should no be
chosen unnecessa ily la ge when la e ally open s uc u es
ha e o be simula ed, since his could inc ease he numbe
o Fou ie e ms o be added. Fo judicious alues o
a
jus a ew spec al e ms a e ypically equi ed. An
example o he in luence o he
box
wid h
on
he capac-
i ance coe icien s
is
shown in Fig. 2. In his igu e, he
ela i e di e ence be ween he capaci ance coe icien s o
a closed s uc u e and i s la e ally open e sion is plo ed.
No e ha om a p ac ical poin o iew he box wid h
does no need o be e y la ge o simula e he open
s uc u e.
I is impo an o highligh ha he wid h o he slo s/s ips
egion (dis ance be ween he wo coplana g ound planes)
has no in luence on he con e gence o he esidual spec-
al se ies, hus a oiding
he
ypical con e gence p oblems
a ising when o ce b u e summa ion is used o analyze
na ow slo s/s ips egions. These s a emen s a e illus a ed
wi h he example in Table
I.
No e he impo an
CPU
ime
sa ings o all o he geome ies.
The ob aining o he asymp o ic ails,
Si.:,
in ol es a neg-
ligible compu a ional cos in compa ison wi h s aigh o -
wa d Fou ie se ies summa ion. Ne e heless, i is use ul
o know which ac o (s) may a ec o he con e gence o
he powe se ies o Gauss-Chebyshe quad a u es shown
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2332
IEEE TRANSACTIONS
ON
MICROWAVE THEORY AND TECHNIQUES. VOL.
42,
YO.
I?,
DECRMHER
1994
TABLE
1
NLHBEK
OF
TPRMS
OF
TH~
SPECTRAL SERIES
WITH
()I,
)
AND
WITHOUT
OBTAIV
TIE
CAPACITAYCE
OF
A
CPW
WITH
0.1%
ACCURACY
(ns)
ASYMFTO~IC
EXTRACTlOlV WHICH SHOULD
BE
ADDED
TO
(hi
=
>?
=
,
=
9
9 o)
RATIO
OF
CPU
TIMES
(
/ a)
[T/TqT-
0.01 0.2
0
28000
0.05 0.2
1
0.10
0.2
3
10.25
1
0.2
1
3
E
0.25
0.10
0.01
3
0
3
7
15
3
7
__
__
5500
2900
1500
700
1500
1500
1500
1500
1500
0.025
0.046
0.088
0.086
0.086
0.086
0.100
in
he Appendix. The el iciency o he compu a ion o
he ails is essen ially con olled by he ela i e p oximi y
be ween he slo s, in such a way ha when e y igh ly
coupled slo s ( e y na ow s ips) a e p esen mo e e ms
mus be e ained o add up he powe se ies
(24)
and
mo e quad a u e poin s a e necessa y in Gauss-Chebyshe
in eg a ions
(28).
Ne e heless, e y ew powe senes
e ms
a e
equi ed excep o imp ac icable s ip wid h,
and he numbe o quad a u e poin s do no need o
be la ge han he numbe o basis unc ions (al hough
i can e en ually be inc eased
o
accoun o ex emely
na ow s ips). In any case, asymp o ic ex ac ion is
al~ nys
ad isable, since di ec summa ion has
always
much wo se
nume ical pe o mance.
3)
The numbe o basis unc ions,
n ,
;
i
=
1,.
. .
,
N+
1.
o
be e ained o e each slo
is
ela ed o i s wid h. Thanks
o he app op ia e ea u es o he basis unc ions,
no
mo e
han wo
o
h ee o hem ha e o be used on each slo
in mos cases.
A
ypical con e gence pa e n is shown
in Table 11. An in e es ing poin o be emphasized he e
is
ha when he numbe o basis unc ions
is
inc eased,
no nume ical p oblems a ise. On he con a y we ha e
ound nume ical ins abili ies when di ec summa ion o
Fou ie se ies is applied. As an addi ional ad an age o
ou p ocedu e, we can say ha he expansion coe icien s in
(14)
a e compu ed wi h ex eme accu acy. The slo elec ic
ield is hen ob ained in addi ion o he elec ical pa am-
e e s. This is e y di icul wi h o ce b u e summa ion
unless a p ohibi i ely la ge numbe o Fou ie e ms is
e ained.
In o de o check he esul s o ou compu e p og am when
applied o a bi a y mul islo geome ies- o which we ha e
no ound published da a-we ha e compa ed wi h esul s
gene a ed wi h a p og am (MULTISTRIP) w i en o e icien ly
analyze mul is ip s uc u es,
[
171. When his code is used,
TABLE
11
CONVERGENCE
OF
THE
CAPACITANCE COEFFICIENTS
(NORMALIZED
TO
el))
WITH
THE
NUMBER
OF
BASIS
FUNCTIONS
AT
EACH
SLOT.
DATA:
?’HE STKUCTURE IN
FIG.
2
WITH
a
=
40
mm,
11
=
0.635
mm,
1
=
5.2
mm,
.~i,
=
.s.
=
1
.0
mm,
.s2
=
.s3
=
0.1
mm,
: <.I
=
17.9
mm,
. C2
=
18.95
min,
. ,3
=
21.05;
=
1360,
cy?,
=
1060,
n :i
=
j ~,
n 4
=
) i
mm,
. 4
=
22.1
mm,
TABLE
111
CAPACITANCE COEFFICIENTS FOR THE
EQUIVALENT
MI
I.TIS
TRIP
TRANSMISSION
LINE (MSTL)
AND COPLANAR WAVEGUIDE
TRANSMISSION
LI~E
(CPWTL)
GEOMETRIES
SHOWN
IN (A)
AND
(B).
DATA
(I
=
20
mm,
11’1
=
w?
=
0
5
mm,
i 2
=
1.0;
mm,
5
=
0
2
mm,
h
=
0
635
mm.
=
9
G o
EO
MSTL
(A)
a
Canaci ance coe icien s
o
s uc u
A
!
Capaci ance coe icien s
o
s uc u e
B
DC,
I
16.6210 1-7.7423 1-0.2990
I
18.8537
he GCPW geome y is simula ed by using wide g ounded
s ips
o
app oxima ely accoun o la e al g ound planes.
Table
I11
shows he capaci ance coe icien s o a h ee s ips
CPW s uc u e when compu ed wi h MULTISLOT and when
compu ed wi h MULTISTRIP. In he las case, he o iginal
s uc u e is simula ed wi h a i e s ip con igu a ion wi h
g ounded ex eme s ips. The esul s o his simula ion o
se e al alues o he wid hs o he ex eme s ips a e shown
in Table 111. Consis en esul s o he capaci ance pa ame e s
ha e been ound wi h bo h compu e codes. Howe e , he
CPU ime used by MULTISTRIP is en imes he CPU ime
used by MULTISLOT
(0.12
seconds in
a
PC/486/33
MHz
compu e , including he compu a ions o he s uc u e in
acuum). This di e ence is due o he ollowing ac : o
his ype o geome ies he numbe o basis unc ions e-
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e
a/.:
QUICK COMPUTATION
OF
[C]
AND
[L]
MATRICES
OF
GENERALIZED MULTICONDUCTOR COPLANAR WAVEGUIDE TRANSMISSION LINES
2333
TABLE
IV
CAPACII
ANC~
COE~FICIENTS
OF
THI:
S x
STRIPS
COPLANAR
WAVEGUIDE
TYPE
GEOMETRY
SHOWN
ui
THE
FIGURE.
THE
PLANE
AA'
Is
w:!
=
0.6
mm,
w3
=
0.8
mm,
51
=
"4
=
0.3
mm,
s2
=
0.2
mm,
s1
=
0.1
mm,
hl
=
0.2
mm,
hl
=
0.2
mm,
FI
=
9.6~0,
FZ
=
4.0~0
A
SYMMETRY
PLANE.
DATA:
u
=
20
mm,
u'l
=
0.4
mm,
.-.-
A
-.-
A'
qui ed o app oxima e he su ace cha ge densi y on he
s ips is much highe han he numbe o basis unc ions
equi ed o accu a ely app oxima e he 2-componen o he
slo elec ic ield.
As a inal nume ical example, we ha e compu ed he
pa ame e s o he six conduc o s geome y shown in Table
IV.
The ho izon al symme y plane (AA') is conside ed i s
an
elec ic wall and hen a magne ic wall in o de o exploi
he symme y. Accu acy is se o ou decimal igu es o he
no malized capaci ance coe icien s. This accu acy is achie ed
by using ou basis unc ions a he 0.3 mm slo s and h ee
basis unc ions a he 0.1 mm and 0.2 mm slo s. To al
CPU
ime was
0.7
seconds on a PC/486/33 MHz.
VI.
CONCLUSION
In his pape we ha e p esen ed a echnique o deal wi h
he quasi-s a ic analysis o mul iconduc o plana s uc u es
belonging o he amily o coplana wa eguides. The me hod
is based on he e icien solu ion o an in eg al equa ion o
he elec ic ield exis ing a he slo s. E iciency
is
achie ed
by means o analy ical p ep ocessing o nume ical se ies. The
compu e p og ams de eloped on he basis o his me hod a e
ex emely accu a e and nume ically s able. In addi ion,
CPU
imes a e sho enough o conside hese p og ams use ul in
he ame o a compu e aided design sys em. The elec ic
ield and su ace cha ge densi y can be also compu ed wi h
ex eme accu acy.
APPENDIX
Two al e na i e o mula ion-which we ha e called spec-
al and spa ial domain o mula ions espec i ely-ha e been
used o he quasi-analy ical de e mina ion o he asymp o ic
ails
S,"::.
A.
Spec al Domain Compu a ion
The compu a ion o
Si;:
in
(22)
implies he addi ion o
slowly con e gen igonome ical se ies o he ollowing ype:
whe e
di
=
%
and
e$
=
I(xcj
2,i).
These se ies ha e al eady appea ed in he analysis o a
mul is ip con igu a ion
[
17,
(S)].
The esidui calculus ech-
nique may be used o con e (23) in o a much mo e quickly
con e gen powe se ies (see
[17]
o mo e de ails)
23
F[-k,
-p
-
IC;
q
+'l;
(dj/ &)2]
(iE
+
i) (p
+
IC
+
1)
X
00
yp+4+2k-1
1
dy
sinh(7 y)
.
(24)
whe e
F
is he hype geome ic unc ion, and
,
he gamma
unc ion. The hype geome ic unc ion
F[-k,
-y
-
k;
q
+
1;
(dj/d;)2]
is a k-deg ee polynomial in
(dj/di)2
cosh[(n
-
&)y]
;
p
+
q
e en
;
p+q
odd
sinh[(n
-
cij)y]
x
F[-k,
-p
-
k;
q
+
1;
(dj/di)2]
-
k
(k
+
i) (p
+
IC
+
i) (q
+
i)(dj/di)2m
-
n=O
(25)
Two
al e na i e closed o m exp essions a e known o he
in eg als appea ing in (24)
whe e
p
=
p
+
q
+
2k,
p
=
7
-
e$,
and
<
is
he Riemann's
ze a unc ion. The i s exp ession in (26) is used o he i s
ew e ms o he k-se ies. This su ices o mos cases, bu i
la ge alues o
k
a e needed, he second exp ession in
(26)
p o ides an al e na i e quick solu ion.
The case
p
=
q
=
0
equi es a sligh ly di e en ea men
(2k
+
1)F[-k, -k;
1;
(d3/d2)']
di
03
(S,":,j)'
=
k=l
~IC~~(IC
+
1)
(
Jk
x
{+:$I
+<[2k,L$]}
-
In [sin
($)I
-
111(2)
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IEEE TRANSACTIONS
ON
MICROWAVE THEORY AND TECHNIQUES,
VOL.
42,
NO.
12,
DECEMBER
1994
No e he necessa y p esence o he in eg a ion cons an , ln(2),
which was no calcula ed in
[17]
because in ha case i
canceled ou when he de ini i e
S,”:!
was compu ed.
B.
Spa ial Domain Compu a ion
Pa se al and con olu ion heo ems p o ide a second e -
icien al e na i e o compu e
S;::
om he quasi-analy ical
in eg a ion o i s spa ial coun e pa s
x,,+s,/z
Tp(-)
:Si;:
=/
dx--
xLL-s*12
X
(28)
27
7l
=
--In
4
sin
-15
-
.’I]
sin
[-(x
+
x0]}.
7
{
1:2: 2n
(29)
The squa e oo in he denomina o o he in eg ands in
(28)
makes he Gauss-Chebyshe quad a u e o mula o be spe-
cially sui able o he compu a ion o hese in eg als. Howe e ,
di ec Gauss-Chebyshe summa ion o he con olu ions is no
ad isable because o he loga i hmic singula i y o
&(x,
x’)
in
5
=
2’.
The e o e, a p e ious ex ac ion and sepa a e
in eg a ion
o
his singula i y is e y use ul o inc ease he
e iciency o he quad a u es. When mul is ip con igu a ions
a e analyzed
[17],
he e en ual p oximi y be ween he s ips
and he la e al walls in oduces quasi-singula beha io o he
in eg and o
z
+
x’
-+
0
o
x
+
x’
+
2n. In coplana
wa eguide- ype con igu a ions he e is no such possibili y as
a consequence o he p esence o he wo la e al g ounded
ins. Consequen ly, he “singula pa ,”
S(x,
XI),
o
Lus(x,
2’)
should be de ined as:
(30)
26
S(z,x’)
=
--In
Ix
-
J?I.
7T
I
he ke nel o
(28)
is spli in o he wo ollowing pa s:
&s(x
2.’)
=
[&(z,
x’)
-
S(X,
d)]
+
S(Z,
z’)
(31)
he con ibu ion o he i s e m ( e y smoo h unc ion) o
he con olu ion in eg als is compu ed wi h a low o de Gauss-
Chebyshe quad a u e, and he con ibu ion o he second e m
(singula pa ) can be analy ically e alua ed. Le
[q;
x]
be
he con olu ion in eg al o he “singula pa ” excep a cons an
ac o
hen, i can be demons a ed ha o
i
#
j
(33)
whe e sgn(-) is he sign unc ion, and o
i
=
j
The las s ep o he compu a ion o
(28)
is
o ca y ou he
inne p oduc s. Closed o m exp essions ha e been ound only
o he case
i
=
j
The es o he inne p oduc s ha e been nume ically e alua ed
by
low
o de Gauss-Chebyshe quad a u es.
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DRAKE
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al.:
QUICK
COMPUTATION
OF
[C
I
AND
[L]
MATRICES
OF
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R.
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F.
Medina and M. Homo, “Uppe and lowe hounds on mode ca-
paci ances
o
a la ge class o aniso opic mul ilaye ed mic os ip-
like
ansmission lines,”
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Ins .
Elec.
Eng.
(Mic owa es, Op ics
An ennas),
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3,
pp. 157-163, June 1985.
En ique D ake
was bo n Sep embe
4,
1966, in
Mon illa, Chdoba, Spain. He ecei ed he Licenci-
ado deg ee in physics om he Uni e si y
o
Se ille,
Spain, in 1990.
He
is cu en ly ollowing
a
Ph.D.
p og am in Mic owa es.
He
is
Assis an
P o esso
a he Depa men
o
Applied Physics a he Uni e si y o Se ille since
1992. His esea ch in e es ocus on
he
analysis o
plana s uc u es and mul iconduc o lines.
F ancisco Medina,
o
a
pho og aph and biog aphy,
see
page 1631 o
he
Sep embe issue o his TRANSACTIONS.
Manuel Ho no
(M75),
o
a pho og aph and biog aphy,
see
page 432
o
he
Ma ch issue o his TRANSACTIONS.
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