260
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TRANSACTIONS
ON
MICROWAVE
THEORY
AND
TECHNIQUES, VOL.
41.
NO.
2,
FEBRUARY
1993
Imp o ed Quasi-TEM Spec al Domain Analysis
O
Boxed Coplana Mul iconduc o Mic os ip Lines
En ique
D ake, F ancisco Medina,
Membe ..
IEEE,
and
Manuel
Homo,
Membe -,
IEEE
Abs ac -
This pape p esen s a e y e icien quasi-TEM
analysis
o
mul is ip ansmission sys ems embedded in a laye ed
medium. The numbe o conduc o s and subs a es is a bi a y,
and he whole s uc u e is assumed o be enclosed in a ec an-
gula se o bounda y condi ions. The analysis makes use o he
Gale kin me hod in he spec al domain. Chebyshe polynomials
wi h edge condi ions a e used as basis and es unc ions o
he s ips ee cha ge dis ibu ion. This s anda d echnique is
conside ably enhanced by means o wo al e na i e p ocedu es o
accele a e he compu a ion
o
he en ies o he Gale kin ma ix.
Ex emely accu a e esul s o a mul is ip sys em, including he
cha ge dis ibu ion, can hen be ob ained on a
PC
compu e in
a sho
CPU
ime.
I.
INTRODUCTION
ULTICONDUCTOR TRANSMISSION LINES (MTL)
M
a e widely used in (monoli hic) mic owa e in eg a ed
ci cui s, high speed in e connec ing buses and o he applica-
ions. Once he p opaga ion cha ac e is ics o
a
MTL sys em
a e
known, i s equency domain o ime domain elec ical
esponses can be ob ained by means o well known me hods.
The p opaga ion pa ame e s ha e been compu ed by means o
bo h quasi-TEM and ull-wa e app oaches. In many p ac ical
si ua ions he quasi-TEM analysis p o ides esul s which a e
accu a e enough, and in hese cases
i
is p e e ed
o
he
much mo e compu a ionally in ol ed ull-wa e app oach. In
addi ion, quasi-TEM da a can be used
as
an ini ial guess in
ull-wa e algo i hms, hus imp o ing hei e iciency.
I quasi-TEM ope a ion is assumed. he p opaga ion pa am-
e e s a e compu ed om he capaci ance,
IC],
and induc ance,
[L],
pe uni leng h (p.u.1.) ma ices o he
MTL.
Powe ul
me hods ha e been epo ed
in
he li e a u e o compu e
[C]
and [L] o mul iconduc o sys ems ha ing a bi a y geome y
[1]-[3]
o
plana geome y
[4],
[SI.
Speci ic echniques ha e
also
been de eloped o mic os ip geome ies, which esul
in
pa icula ly e icien compu e algo i hms. Fo ins ance,
some mul iconduc o s uc u es can
be
exac ly sol ed by using
con o mal mapping [6],
[7]
o
e y e icien ly handled by
means o he in eg al equa ion echnique
[8].
Fo
he gene al
mic os ip-like geome y embedded
in
a
laye ed linea medium
(see Fig.1) he spec al domain app oach (SDA)
-
combined
wi h he Gale kin me hod
[9,
IO],
a ia ional o mula ion
[I I].
[I21
o i e a i e echniques
1131,
[I41
-
is p obably he mos
Manusc ip ecei ed Feb ua y
24.
1992:
e ised May
26.
1992.
This
wo k
was
suppo ed
by
he DGICYT. Spain (P ojec No. TIC91-1018).
The au ho qa e wi h he Depa amen o de Elec onica
y
Elec omagne ismo.
Facul ad
de
Fisica,
Uni e sidad
de Se illa.
A da.
Reinci
Me ccdcc
s/n,
41
01
2
Se illa, Spain.
IEEE Log
Numbe
9204482.
Y
Elec ic wall, magne ic wall
1
o
open bounda y
I
=o
1-
x=O
Elec ic wall, magne ic wall
x=a
o
open bounda y
Fig.
1,
C o $ sec ion
o
he gene alized
boxed
coplana mul is ip line unde
s udy.
simple and widely used
ool.
Al hough he di ec applica ion
o hese echniques gi es place o accu a e, eliable and quick
compu e codes, p ope analy ical p ep ocessing d as ically
imp o es hei pe o mance. A a ie y o echniques in ol ing
hea y analy ical wo k has been applied
o
he solu ion o he
single mic os ip p oblem (see
[
151 and he e e ences he ein).
The p esen pape is
a
meaning ul ex ension o he wo k in
[
151 which deals wi h mul is ip geome ies ha ing a bi a y
s ip wid hs. The echnique is essen ially an enhanced spec al
domain analysis. Two e icien schemes a e p o ided o ac-
cele a e he compu a ion o he spec al se ies in ol ed in he
Gale kin ma ix in a d as ical way. The applica ion
o
hese
echniques makes i possible
o
compu e he cha ac e is ic
pa ame e s o he mic os ip MTL in Fig.
1
wi h high accu acy
in
a
sho CPU ime. The cha ge dis ibu ion is simul aneously
ob ained wi h ex eme accu acy. The analysis o
a
ypical
mul is ip sys em can be ca ied
ou
on a PC/AT compu e
wi h ma h cop ocesso in no mo e han one o wo seconds.
The de eloped p og ams can be used o CAD applica ions
on a wo ks a ion. This so wa e could be use ul o enginee s
dealing wi h mul is ip geome ies.
11.
STATEMENT
OF
THE
PROBLEM
The c oss sec ion o he mic os ip-like sys em conside ed
in
his wo k is shown in Fig.
1.
T ansla ional symme y
in
he di ec ion is assumed. An a bi a y numbe ,
N.
o
ze o- hickness pe ec ly conduc ing s ips a e placed on he
31 h
in e ace o
a
N,-laye ed medium. The i h s ip is
0~J18-9480/93$03.00
0
1993
IEEE
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DRAKE
c
ul.:
IMPROVED
QUASI-TEM
SPECTRAL
DOMAIN
ANALYSIS
26
1
cha ac e ized by i s wid h,
w;.
and he posi ion,
si.
o i s
middle poin . The laye ed subs a e is composed o
NE
slabs
o
lossless/lossy/iso/aniso opic
linea ma e ials. The j h laye
is cha ac e ized by i s complex dielec ic pe mi i i y enso ,
Ai::
(I
c;,,(an)
.
G(a,)
.
aq,j(an)
(4)
ij,
(o equi alen pe mi i i y enso
[lo]).
The s uc u e is
enclosed in o a ec angula ame bounded by he planes
=
b
(see Fig.
1).
A
wide amily
o
coplana mic os ip-like ansmission lines can
be
conside ed
o be a pa icula case o his gene ic s uc u e.
As
i is well known, all he quasi-TEM pa ame e s o he
MTL sys em can be ob ained om
i s
capaci ance,
[C].
and
induc ance,
[L],
p.u.1. ma ices. Usually
[L]
is compu ed om
0q3J(”71)={
J4
(
y)
(-l)(q-1)/2
cos(cy,,sj)
i
q
is odd
he capaci ance p.u.1. ma ix,
[C’],
o
a p ope ela ed s uc u e
a e he ollowing spec al se ies:
2m
n=l
=
0,
=
a,
=
0,
whe e
(1,
=
nn/a
is he Fou ie a iable,
G
is he SDGF, and
cq,,
(Q71)
a e he sine-Fou ie ans o ms o he basis unc ions
in
(3):
Ly
W’
-
J~
(y)
(--1)q/2
sin(a,,sj) i
q
is e en
(<
[IO].
Then, he quasi-TEM analysis educes o sol ing wo
elec os a ic- ype bidimensional p oblems. Each coe icien
C,,
(i.J
=
1,.
.
.
.
N)
o
[C]
o
[C’]
can be de ined as he ee
cha ge
on
he i h s ip when he j h s ip is se o ol age uni y
and he es o he s ips a e g ounded (canonical exci a ion).
The e o e he compu a ion o
[C]
(o
[C’])
equi es o sol e he
ee cha ge densi y in eg al equa ion
N
imes ( o
N
canonical
exci a ions):
N
nn
whe e
G(z.a:’)
is
he s a ic G een’s unc ion,
@(x)
is he
ol age (one o ze o
on
each s ip depending
on
he pa icula
exci a ion), and
j(z’)
is he ee cha ge densi y on he j h
s ip.
No
gene al closed o m exp essions a e known o he spa-
ial domain G een’s unc ion o ou p oblem, bu he spec al
domain G een’s unc ion (SDGF) can
be
easily compu ed (see,
o ins ance,
[
1
11,
[IO])
o
an
a bi a y laye ed con igu a ion.
Acco ding o his,
i
is mo e con enien o wo k wi h
(1)
in
he spec al domain. A use ul echnique o sol e he spec al
domain e sion o (1)
is
he Gale kin me hod. As
i
has
been es ablished in he li e a u e
on
his subjec , Chebyshe
polynomials weighed by he Maxwell edge singula i y a e
pa icula ly sui able es and basis unc ions o he s ips ee
cha ge densi y. When hese unc ions a e used
aj(: ’)
can be
w i en as ollows:
JI
whe e
.Iq
is he i s kind o Bessel unc ion
o
o de
q.
The sum o he se ies in
(4)
is he compu a ional s ep in-
ol ing meaning ul CPU ime cos . The e o e, he cons uc ion
o a highly e icien compu e code equi es he analy ical
p ep ocessing o hose se ies. Kumme ’s me hod (ex ac ion
o an asymp o ic ail) is used in o de o accele a e he
con e gence o he se ies. Acco ding o his me hod, he se ies
(4)
a e spli as ollows:
71=1
(7)
k
(k
=
h . h
+
1)
being he pe mi i i y (o he equi alen
pe mi i i y
[
111,
[
101
in he aniso opic case) o he k h laye .
Since
Gas
is chosen o be he asymp o ic beha io o
G
o
la ge
an.
he emainde se ies ( i s e m o
(6))
con e ges
e y quickly. The asymp o ic ails
Si::
a e ex emely slow
con e gen se ies, bu hey can be educed o quasi-analy ical
exp essions by means o he wo p ocedu es desc ibed in he
wo ollowing sec ions.
qn nx,
111.
TRANSFORMATION
OF
THE
TAILS
INTO
POWER
SERIES
dz’)
=
%.,%(J’)
(2)
I can be seen om
(5)
and
(7)
ha he compu a ion o
S;::
in ol es he addi ion
o
slowly con e gen igonome ical
se ies o he ollowing kind:
q=O
whe e
w
71
71x1
mq
,
(
.E’)
=
(3)
The applica ion o he Gale kin me hod leads o a sys em
o
algeb aic linea equa ions o
uq
,.
The en ies
24;
:
(JJ
=
0. . .
.
.
p,,,,,,
:
q
=
0.
.
.
.
.
qlIlaXl
:
1.
j
=
1.
. . .
.
N)
o his sys em
whe e
d,
=
nw,/2u
and
:
=
(7 /a)(sJ
S~).
The esidues calculus echnique makes i possible o ans-
o m
(8)
in o much mo e quickly con e gen powe se ies. The
i s s ep is o iden i y
(8)
as he addi ion o he in ini e esidues
--T
T-
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262
IEEE
TRANSACTIONS
ON
MICROWAVE
THEORY AND
TECHNIQUES,
VOL.
41,
NO.
2.
FEBRUARY
1993
o he ollowing p ope ly chosen complex-plane unc ion:
When (9) is in eg a ed along he closed pa h shown in
Fig.
2,
he esidues, Cauchy heo em p o ides an al e na i e
exp ession
o
(S,;:),’:
cos11
[(T
-
c*
)y]:
p
+
q
e en
x
(10)
.
{
si ih
[(T
-
cL3)y]:
p
+
y
odd
Now,
he p oduc o modi ied Bessel unc ions
IJ,
is ex-
panded as a se ies o powe s
[[16],
p.
9601
shown
in
(11)
below.
F
being he hype geome ic unc ion, and
being he
gamma unc ion. The hype geome ic unc ion
F[-k.
-p
-
k;
y+l;
(d3/d,)2]
is a k-deg ee polynomial in (d,/dO2 shown
in (12) below.
In addi ion, he in eg als appea ing in
(1
1)
a e known in
closed o m. Two al ema i e exp essions
[ [
161,
pp. 349-3501
o hem a e:
whe e
p
=
p
+
q
+
2k,
[j
=
T
-
cG1
and
C
is he Riemann’s
ze a unc ion. The i s exp ession in (13) is used o he i s
ew e ms o he k-se ies. This su ices o mos cases, bu i
1
jy
z-plane
Fig.
2.
In eg a ion pa h in he complex plane
o
he compu a ion
o
he
spec al se ies
by
means
o
he esidues calculus echnique.
la ge alues o
k
a e needed, he second exp ession in (13)
p o ides an al e na i e quick solu ion.
This p ocedu e is s ill alid in he case
p+q
=
1
because he
new pole o
(z)
in
z
=
0
p esen s a pu ely imagina y esidue.
Howe e , he case
p
=
q
=
0
equi es a sepa a e ea men
because o he double pole o
(z)
in
z
=
0.
Le us conside :
This auxilia y se ies can be compu ed by means o he
echnique explained abo e when
(z)
in
(9)
is eplaced by
g(z)
=
z
.
(z).
I he powe se ies esul ing om his
p ocedu e is in eg a ed wi h espec o
c:
exp ession
(13,
cos11
[(n
-
c*)yl:
siiih
[(T
-
c,,)y]:
p
+
y
e en
p
+
q
odd
2
!Jp+q+2k-l
’
/I’
d‘y sin11
(ny)
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DRAKE
e
a/.:
IMPROVED
QUASI-TEM
SPECTRAL
DOMAIN
ANALYSIS
263
which is shown a he bo om o his page, is ob ained. In his
exp ession,
h(d,,
d,)
is he in eg a ion cons an , which does
no need o be calcula ed because i cancels ou when
S,”::
is
compu ed. I should be no iced ha he gene al exp essions
p esen ed in his sec ion o he compu a ion o
5’;;:
educes
o he mo e simple exp essions appea ing in
[
151
when he
case
I
=
j
is conside ed (basis and es ing unc ions on he
same s ip).
The powe se ies in his sec ion p o ide ex eme accu acy
when jus a ew e ms a e e ained. In some pa icula cases
(which co espond o a he heo e ical han p ac ical si ua-
ions, such as ex emely high coupling le els o e y close
p oximi y o he s ips o he side walls) mo e e ms need o
be conside ed. Anyway, he Shanks ans o ma ion
[
17, pp.
369-3741 p o ides a use ul ool o accele a e he powe se ies
in such
a
way ha he echnique is e en use ul in hese
c i ical
cases.
I .
SPATIAL DOMAIN COMPUTATION
OF
THE
TAILS
The second echnique conside ed o he compu a ion o
(7) is based on he quasi-analy ical in eg a ion o he spa ial
coun e pa o he se ies
5’;;;.
By applying Pa se al and
con olu ion heo ems, we can w i e
(1
6), which is shown a he
bo om
o
his page, wi h
Gas(x.
d)
being he spa ial domain
G een’s unc ion whose Fou ie ans o m is
Gas(n),
ha is:
No e ha
Gas
may be physically in e p e ed as he G een’s
unc ion o he “asymp o ic s uc u e” which esul s by p o-
longing he M h laye o
y
=
-m
and he
(A4
+
1) h laye
o
y
=
+w.
The squa e oo in he denomina o o he in eg ands
in
(16)
makes hese in eg als specially sui able o be compu ed by
means o he Gauss-Chebyshe quad a u e o mula. Howe e ,
he di ec applica ion
o
his quad a u e o he compu a ion
o he con olu ion in eg als is no e icien enough because
o he loga i hmic singula i y o
GaS(x..d)
in
.c
=
.E’.
In addi ion, i he s ips #1 and/o
#N
a e e y close o
he la e al elec ic walls,
Gas
(.I:,
s’)
exhibi s a quasi-singula
beha io when
LC
+
IC’
+
0
o
x
+
:c‘
-+
2a.
In o de o
o e come hese compu a ional d awbacks, he singula and
quasi-singula con ibu ions o
Gas
(x,
d)
mus be ex ac ed
ou and con enien ly ea ed. The h ee e ms causing he
nume ical p oblems can be joined o gi e he ollowing
“singula pa ” o
Gas(x,
d):
I should be no iced ha om a physical poin o iew,
S(a,
2’)
accoun s o he con ibu ions o he eal cha ge line
and he i s wo image lines e lec ed by he conduc ing walls.
The con olu ion in eg als shown in (16) a he bo om
o
he page, can be e y e icien ly e alua ed by spli ing he
kemel in o wo pa s:
Gas(z.z’)
=
S(:E.:I;’)
+
[G,,(z.s’)
-
S(x,
d)].
The second e m in his exp ession is a e y smoo h
unc ion. The e o e, i s con ibu ion o he con olu ion is
ob ained wi h a low o de Chebyshe quad a u e. On he o he
hand, he con olu ion in ol ing
S(: .
d)
has been analy ically
e alua ed. Le
us
de ine
I[;
[q;
z]:
111
lx
-
2’1
i =O
In
(x
+
d)
i =-1
(19)
In
(2a
-
:E
-
d)
i
=
$1
s,
-
WL/2
5
s
5
s,
+
u1,/2
I complex plane in eg a ion echniques a e used, he closed
o m exp essions shown in
(20)
a he bo om o his page a e
ob ained o (19), whe e
’WZ
‘W
2-
2
s,
-
-
<
:I:
5
s,
+
2:
I1
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264
IEEE
TRANSACTIONS
ON
MICROWAVE
THEORY
AND
TECHNIQUES.
VOL
41.
NO
2.
FEBKLARY
1993
A
2.00
1.00
0.50
0.10
0.05
0.01
and sgn(.) is he sign unc ion. Fo he pa icula case
i
=
j:
u
=
0:
L2
S:;:
S ,',"
S,",':
($,':)*
0.43
0
0
1
O.GO
1
14
3
0.75
1
2
8
5
0.94
2
2
35 16
0.!)7
3
3
79
3F
0.99
3
0
229
94
The nex s ep o he compu a ion o
(16)
is
o
ca y ou he
inne p oduc s. Only he pa in ol ing
I:!/
has been ound o
ha e a closed o m:
-l/2p
ln(w /4)
i
p
=
y
#
0
i
p
=
q
=
0
={()
i
p
#
q
(22)
The es o he inne p oduc s ha e been nume ically e al-
ua ed by low o de Gauss-Chebyshe quad a u es.
In mos cases, he numbe o poin s employed
in
bo h
con olu ion and inne p oduc quad a u es has been wo
o
h ee mo e han he numbe o basis unc ions used on each
s ip. This is su icien o ensu e mo e han eigh signi ican
igu es in he calcula ions. Ne e heless, he compu e p og am
which implemen s his echnique makes
i
possible o in oduce
a la ge numbe o quad a u e poin s o he inne p oduc
in eg als in ol ing
I<:.
12:,11 .
and
(1
=
1:'.
.
N
-
1).
This possibili y can be use ul because hese inne p oduc s
equi e a ew mo e quad a u e poin s
in
he
c i ic,a/
cases
men ioned a he end o Sec ion
111
(i simila accu acy is
equi ed o all he inne p oduc s).
V.
NUMERICAL RESULTS
Two double p ecision FORTRAN codes ha e been w i en
o implemen he echniques discussed abo e. Exhaus i e
nume ical wo k has been ca ied ou o alida e he compu e
codes on bo h a
PC/386
compu e (wi h ma h cop ocesso )
and a VAX/6410. This wo k has been use ul o es ablish
he pa ame e s which ha e an in luence on he accu acy and
e iciency o he wo al e na i e me hods p oposed
in
his
pape .
Fi s ly, we ha e checked
he
con e gcnce
o
he
se ies
(1
I)
and
(IS).
The con e gence o
(S;::)'
has been ound o be
mainly a ec ed by he a ios
z
=
111,
+
,wJ/'2(.~,,
+
.s,)
and
7.;
=
'iu;
+
1,/2(.s,
-
s ).
In mos p ac ical si ua ions. he
con e gence is eached in ew e ms. Ne e heless. he con-
e gence becomes slowe when
z
and/o
, ,j
a e e y close
o one (uni y). This
occu s
i some s ip is e y close o he
la e al me allic walls
(7.;
E
1)
and/o i e y igh ly coupled
s ips a e p esen
( 6
E
1).
In hese cases, mo e e ms mus be
e ained o add up he co esponding se ies.
In
Table
I
we show
a ypical con e gence pa e n o he se ies
5':::;
associa ed
o
a pai
o
asymme ical s ips o di e en coupling le els. The
numbe
o
e ms equi ed
in
he compu a ion o
S::,':
inc eases
when
y2
app oaches o
uni .
Fo his s udy i e signi ican
co ec igu es a e always imposed
in
he compu a ion o
('l,l.
TABLE
I
TO
BE
RETAINED
FOR
FIVE
SK~NIFICAN~
Flm
RES
Acci
RACY
lh
THE
MAXIMUM
NUMBER
OF
TERMS
A,,,,,,
OFl-H SERIES
.$:,:
I.,/
=
1.2
C4PAClTANCE.
THE
As1
FRISKED
COLL IlN INCLUDES
I
HE
RESLLTS
OBTAINED
BY
USING
THE
TWICE
ITERATED
SHANKS
TRANSFORMATION.
(DATA:
(I
=
10.111
=
1.h'
=
2.
(I',
=
1.
2
=
2.il
=
3).
€0
4WlA
w2
I.-
No e ha he use o he wice i e a ed Shanks ans o ma ion
(as e isked column) in oduces a signi ican accele a ion (o e
SO%)
in he con e gence o his se ies. I mus be emphasized
ha he cases in ol ing a e y la ge
A:,,,,,
a e no ealis ic.
Anyway, he di ec summa ion o he o iginal Fou ie se ies
would equi e much mo e compu a ional e o (p ohibi i e
i
e y high accu acy is desi ed). The e o e he applica ion o
his echnique is always p e e able o di ec summa ion.
The spa ial domain echnique p esen s simila di icul ies
in he same cases, bu
i
has been ound o be less sensi i e
o hose p oblems.
In
gene al, he con olu ion in eg al and
he inne p oduc s a e e y accu a ely compu ed by using
a numbe o quad a u e poin s equal
o
o
sligh ly la ge
han he numbe o basis unc ions employed on each s ip.
Typical compu a ions a e
so
accu a e ha mo e han eigh
meaning ul digi s can be ob ained o he coe icien s o
he expansion
o
he cha ge dis ibu ion. Howe e , when
I.:
and/o
I,;
ake ( heo e ical a he han p ac ical) alues
e y close o
1.
he numbe
o '
quad a u e poin s
IV~T
used
o he compu a ion o he c i ical inne p oduc s in ol ing
1,;.
I.:,'.,
and/o
(i
=
I..
. . .
.V
-
1)
mus be inc eased
o keep he accu acy pa e n.
In
Table
11.
we display he alues
o
lV,y
needed o ge ho h i e and en signi ican igu es
Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:45:33 UTC om IEEE Xplo e. Res ic ions apply.

DRAKE
e[
a/
:
IMPROVED QUASI-TEM SPECTRAL DOMAIN ANALYSIS
Ihn
I
2
3
1
5
7
8
9
Ill
II
I
2
3
1
5
G
7
S
9
6
265
A
=
0.
I
mi
CII
Cl2
25.7199 -7.73533
30.8053 -9.00472
31.3117 -9.01270
31.1235 -9.1175G
31.1139
-9.13304
31.4455
-9.13GG9
31.1190
-9.13687
31.1191
-9.l3602
31.1191
-9.13604
31.4191
-9.13694
5.70529 -4.73082
11.9197 -7.53461
13.5927
-9.20794
13.9190
-9.19782
13.9535
-9.56750
13.9914
-9.57426
13.9020
-9.57489
l3.0920
-0.57191
1:1.9920
-9.57491
31.1179
-0.13600
TABLE
I1
NUMBER
OF
GAUSS-
CHEBYSHEV
QUADRATURE
POINTS
. -,:
USED IN THE
CRITICAL INNER PRODUCT INVOLVING TO GET BOTH
FIVE
AND
TEN
SIGYIFICANT
FIGURES
CAPACITANCES.
(DATA:
EQUAL
TO
THOSE
IN
TABLE
I).
A
2.00
1
.oo
0.50
0.10
0.05
0.01
-
1'1
2
0.43
0.60
0.75
0.94
0.97
0.99
-
iV;
(5
digi s)
2
2
3
7
11
Nd
(10
digi s)
13
20
35
I
o
CI,~
o di e en coupling le els. In ha example, he
con olu ion in eg als and he non-c i ical inne p oduc s ha e
been compu ed wi h 4 Gauss-Chebyshe quad a u e poin s.
The aspec s discussed in he p e ious pa ag aphs deal wi h
he quali y o he compu a ions o he Gale kin ma ix en ies.
Howe e , he accu acy o he capaci ance and induc ance
coe icien s also depends on he numbe o basis unc ions.
The ial unc ions in
(3)
a e e y sui able o he mul is ip
p oblem, since ex eme accu acy o bo h capaci ance coe i-
cien s and cha ge dis ibu ion can
be
achie ed in all p ac ical
si ua ions. Typical esul s ob ained wi h ou p og ams (bo h
p og ams p o ide exac ly he same esul s wi hin he compu e
accu acy) can be seen in Tables
111
and IV. F om Table
111,
i can be seen ha mo e ial unc ions a e equi ed when
he s ips a e e y close o when hey a e adjacen o hin
dielec ic laye s ( he s ips a e in his case ela i ely wide in
compa ison wi h he hicknesses
o
he laye s). Anyway, om
a p ac ical poin o iew, no mo e han h ee o ou basis
unc ions a e necessa y o ob ain use ul highly accu a e esul s.
Table IV shows an example o he expansion coe icien s
ob ained o he cha ge dis ibu ion when di e en numbe o
basis unc ions a e used. The coe icien s o he expansion a e
no sensi i e o he addi ion o new e ms when con e gence
has been achie ed ( his ac sugges s ha
(2,
3) is a quasi-
o hogonal expansion). As
i
is expec ed om he s a iona y
na u e o he i s coe icien , his is much mo e accu a ely
compu ed han he cha ge dis ibu ion.
In ou opinion, an impo an ea u e o he me hods epo ed
in his pape is ha when hey a e used, no nume ical p oblems
a ise i he numbe o basis unc ions is inc eased. In he pas ,
he au ho s ha e used o he asymp o ic ails-in ol ing la ge
a gumen app oxima ion
o
he Bessel unc ions- o accele a e
he con e gence o he spec al se ies
[
181.
Fo mos p ac ical
pu poses, ha app oach wo ks p ope ly (al hough mo e slowly
han hose p esen ed in he p esen pape ), bu some p oblems
can be obse ed when a ela i ely la ge numbe o basis
unc ions a e used. This is a consequence o cumula i e
nume ical e o s when se ies in ol ing la ge o de Bessel
unc ions need
o
be compu ed. This d awback has no been
de ec ed wi h he me hods de eloped
in
he p esen wo k.
The nume ical da a gene a ed wi h
ou
p og ams ha e been
compa ed wi h highly accu a e da a (o exac solu ions) e-
,'
TABLE
I11
TO
6")
VERSUS
THE
NUMBER
OF
BASIS
FUNCTIONS
(33111.1).
DATA:
6,-,.2
13.
cyY2
=
10.
~,.~:3
=
yy:i
=
2.51.
(1
=
20
mm,
.s1
E
9.15
mm,
u1
=
1
mm,
(172
=
1.5
mm,
CASE
I
(RI
=
0.h2
=
0.635 mm,
hg
=
O.h4
=
10
mm),
CASE
I1
(hl
=
0.51
mm,
112
=
0.125
mm,
h:j
=
0.125
mm,
hd
=
9.875
mm).
CONVERGENCE
OF
THE
CAPACITANCE
MATRIX
(NORMALIZED
Cases
1
~
c22
-
37.1587
39.1877
39.4832
39.5505
39.5639
39.5663
39.5GG3
39.5671
39.5671
39.5671
39.5671
10.1251
13.OSG9
14.7678
15.0354
15.1115
15.1
170
15.1
1
77
15.1177
15.1 177
__
__
~
C1
I
~
27.1524
27.1841
27.2127
27.2132
27.2132
27.2132
~
6.57581
7.06826
7.18892
7.
ISGO2
7.19835
7.19849
7.19850
7.19850
-
3
=
lmn
c12
-1.22612
-
1.22926
-1.19820
-1.19824
-
1.19829
-
1.19829
-1.74512
-
1.83526
-1.94558
-1.94596
-1.94900
-1.94901
-1.94901
-
1
.9ma
1
C12
35.1310
35.1583
35.2954
35.2955
35.2956
35.2956
-
-
7.99884
8.14886
8.28897
8.29218
8.29925
8.29927
8.29928
8.29928
po ed in he li e a u e o pa icula s uc u es. The ag eemen
has been always ound o be excellen (all he signi ican
igu es epo ed ha e been in a iably ob ained). Fo example,
nume ical da a o he capaci ance coe icien s
o
a i e s ips
s iplike con igu a ion in homogeneous medium a e epo ed in
[[8],
Table
VII]
and
[[7],
Fig. 61. The o me uses an enhanced
in eg al equa ion echnique and ex apola ion p ocedu es and
he la e gi es a con o mal mapping solu ion (al hough nume -
ical compu a ion o he hipe ellip ic unc ions
is
equi ed). Fi e
igu es a e co ec ly gi en in
[8]
and six igu es a e gi en in
[7]
o he no malized capaci ance coe icien s.
Ou
p og ams
ep oduce all he signi ican igu es epo ed in hose wo ks.
To
ob ain i e igu es accu acy, 4 basis unc ions ha e been
e ained on each s ip, and he CPU ime was abou
I
.5
seconds
on a PC/386 compu e (abou
0.15
seconds on a VAXJ6410).
Six igu es we e ob ained wi h
5
basis unc ions
(2.2
seconds
on a PC/386 compu e , abou
0.2
seconds on a VAXJ6410).
These CPU imes e e o he spa ial domain echnique. The
same esul s we e ob ained by means o he o he echnique
desc ibed in his pape wi h sligh ly highe CPU imes. As
a inal example, in Table
V
we compa e ou esul s wi h
7
T
Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on July 17,2020 a 15:45:33 UTC om IEEE Xplo e. Res ic ions apply.
266
IEEE
TRANSACTIONS
ON
MICROWAVE
THEORY
AND
TECHNIQUES,
VOL.
41,
NO.
2.
FEBRUARY
1993
3
1
2
4
1
2
4
5
1
2
4
5
G
3
3
3
TABLE IV
CHARGE DISTRIBUTION COEFFICIENTS
OF
A
PAIR
OF
ASYMMETRICAL COUPLED
STRIPS
FOR
DIFFERENT VALUES
OF
THE
NUMBER
OF
BASIS FUNCTIONS
~n ey.
DATA: EQUAL
TO
THOSE
IN
TABLE 111, CASE
I,
1
=
1
mm.
-1.95733
27.2132
0.10152
0.01094
27.2132
0.10156
0.01094
0.04654
27.2132
0.10156
0.01094
0.04654
0.00033
-1.95706
-1.95705
-1.95705
Qmax
1
2
3
4
5
G
Dics cl
[5]
4.633
7.857
5.723
-2.547
-2.338
-0.oso
-0.553
-0.064
-0.013
Exci .
(1,O)
This
wo k
4.642
7.871
5.738
-2.555
-2.346
-0.080
-0.553
-0.064
-0.013
a92
-1.22612
-1.22926
1.70030
1.68933
-1.19820
-0.82322
-
1
A9824
1.68674
0.31610
1.68675
0.31610
-0.82325
-
1.19829
-0.82309
-0.09211
-
1.19829
1.68676
0.31609
-0.0921
1
0.02020
-0.82309
Exci ,
%,I
-
1.226 12
-1.22926
-1.51918
-
1.19820
-
1.48469
-0.55950
-1.19824
-1.48439
-0.55952
-0.15337
-1.19829
-1.48446
-0.55955
-0.15338
-0.03038
-1
A9829
-1.48446
-0.55955
-0.15338
-0.03038
-0.00439
:oJ)
a9,2
35.1310
35.1583
35.2954
-0.11748
-0.11830
-4.65142
35.2955
-0.11827
-4.65132
-0.02298
35.2956
-0.11828
-4.65161
-0.02298
0.16107
35.2956
-0.11829
-4.65161
-0.02298
0.16107
-0.00
150
hose ob ained by he me hod o lines (wi h nonequidis an
disc e iza ion) in
[5]
o a i e conduc o mic os ip s uc u e
wi h di e en wid hs and inhomogeneous subs a e. To ob ain
4
digi s accu acy we ha e used
5
basis unc ions, and he
CPU
ime was less han
1.5
seconds
(PC/386).
A ac ion o a
second was necessa y o ge he same le el o accu acy o he
example in
[5].
In gene al, e y eliable and accu a e esul s
can be ob ained o he capaci ance and induc ance ma ices o
mic os ip s uc u es on a
PC/386
compu e wi h
CPU
imes
anging om a ac ion o a second o wo o h ee seconds
(depending on he numbe o s ips and basis unc ions). The
su ace cha ge dis ibu ions a e also p o ided wi h e y good
accu acy.
VI.
CONCLUSIONS
In he p esen wo k, a nume ically imp o ed spec al domain
app oach is employed o he e icien and accu a e quasi-
TEM analysis o a wide class o mul is ip ansmission
lines.
A
p ope analy ical p ep ocessing is inco po a ed in
he compu a ion o he en ies o he Gale kin equa ions
sys em in o de o achie e ex eme accu acy in a sho
CPU
ime. To each his goal, wo di e en echniques ha e been
p oposed and compa ed. The double p ecision FORTRAN
p og ams implemen ing hese echniques a e able o analyze
mul is ip con igu a ions embedded in mul ilaye ed subs a es
on a
PC/386
compu e in less han a ew seconds. To al
ag eemen has been ound wi h exac esul s (up o he
accu acy epo ed in he li e a u e) o simple pa icula con ig-
.
i
-~
TABLE V
COMPARISON BETWEEN
OUR
RESULTS
AND
THOSE REPORTED
IN
[S]
FOR
THE
CAPACITANCE MATRIX (NORMALIZED
TO
F~,
)
OF
A
FIVE-
CONDUCTOR MICROSTRIP CONFIGURATION. DATA:
12
=
0.8
mm,
=
2.5~0.
(1’1
=
0.1C
mm,
=
0.47
mm
A
=
0.07
mm,
I(
=
3.7
mm.
€0
I
4
w1
kkW24i
w1
H--w2-l-l
w1
k-
I
I
€0
i
w1
k-k-wzii
w1
H--w244
w1
k-
4k-
n
€
h
Ik-
-
n
€
Tl
I
u a ions. Good ag eemen has been also ound wi h many o he
esul s epo ed o mo e gene al s uc u es. The su ace cha ge
dis ibu ion can be ob ained wi h accu acy and eliabili y i
his quan i y is equi ed.
REFERENCES
[I] C. Wei,
R.
F. Ha ing on,
J.
R. Mau z, T. K.
Sa ka ,
“Mul iconduc-
o ansmission lines in mul ilaye ed dielec ic media,”
/€E€
T ans.
Mic owa e Theo y Tech..
ol. MTT-32, pp, 439450, Ap . 1984.
[2]
Z.
Pan ic and R. Mi a, “Quasi-TEM analysis o mic owa e ansmis-
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IEEE T ans.
Mic mu e
Th o y
Tech..
ol. MTT-34, pp. 1096-1103, No . 1986.
[3] F. Olyslage , N. Fach6, and D. de Zu e , “New as and accu a e line
pa ame e calcula ion o gene al mul iconduc o ansmission lines
in
mul ilaye ed media,”
IEEE T ans. Mic owm Theo y Tech..
ol. 39, pp.
901-909, June 1991.
[4] V. K. T ipa hi and R. J. Bucolo,
“A
simple ne wo k analog app oach
o he quasi-s a ic cha ac e is ics
o
gene al lossy, aniso opic, laye ed
s uc u es,”
IEEE T ans.
Mic owwe
Theo y Tech..
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1458-1464, Dec. 1985.
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AEU,
ol.
41, pp. 169-175, 1987.
[6]
L.
J.
P. Linne ,
“A
me hod o he compu a ion o he cha ac e is ic
immi ance ma ix
o
mul iconduc o s iplines wi h a bi a y wid hs,”
IEEE T ans. Mic, owa e Theo y Tech..
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1974.
[7] D. Homen co schi, A. Manolescu, A. M. Manolescu, and L. K eindle ,
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IEEE T ans. Mic owa e Theo y Tech..
ol. MTT-36, pp.
1002-1007, June 1988.
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o
cha ac e is ic admi ances and coupling
coe icien s o s ip ansmission lines,”
IEEE T ans.
Mic owai
Theo y
and
Tec,h..
ol. MTT-16, pp. 925-937, No . 1968.
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a/.:
IMPROVED QUASI-TEM SPECTRAL DOMAIN ANALYSIS
261
[9]
T
Ki azawa and
Y.
Hayashi, “Asymme ical h ee-line coupled s iplines
wi h aniso opic subs a es,” IEEE
T,-ans
M c m~ai
Theo y Tech
,
ol.
MTT-34, pp 767-772, July 1986
dnaly is o mul ilaye ed, mul iconduc o coplana s uc u es wi h dielec-
ic and magne ic aniso opy including subs a e
losses,”
IEEE
Tiuns
M ciowaie
Theo y Tech,
ol. 38, pp. 1059-1068. Aug. 1990.
[I
I]
F
Medina and M Homo, “Uppe and lowe bounds on mode ca-
paci ances o a la ge class o aniso opic mul ilaye ed mic os np-
like ansmission lines,”
P oc
Ins
Elec Eiig
(M c ou~u es. Op i s
&
An ennuhi,
ol. 132,
no
3, pp 157-163, June 1985 lines.
1121
A
Sawicki and K Sachse, “Lowe and uppe bound calcula ions
on
he capaci ance o mul iconduc o p in ed ansmission line using
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Ti
ans
Mi ou~a
Theo y Tech,
ol. MTT-34, pp 236244, Feb. 1986.
[I71
C H Chdn and R. Mi a, “Analysis o MMIC s uc u es using an
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(M’75) was bom in
To e
del
e icien i e a i e app odch,” IEEE
T ans
Micinna e Theo b Tech
.
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76,
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96-105, Janua y 1988. deg ee in Physics in June 1969, and he Doc o
[
141 E. D ake,
F
Medina, and M. Homo, “An imp o ed i e a i e echnique deg ee in physics in Janua y 1972, bo h om he
o
he quasi-TEM analysis
o
gene alized plana lines,” IEEE
T ans
Uni e si y o Se ille, Spain.
Mi nwm3e
Theo Tc h,
ol. 40, Ap 1992 Since Oc obe 1969 he has been wi h he
[IS]
F.
Medina and M. Homo, “Quasi-analy ical s a ic solu ion o he boxed Depa men o Elec onics and Elec omagne ism
mic os ip line embedded in a laye ed medium,” IEEE
T ans
Mic
I
ouwe
a he Uni e si y o Se ille, whe e he became an
Theo Tech
,
ol 40, Sep . 1992. Assis an P o esso in 1970, Associa e P o esso
[16]
I.
S.
G adsh eyn and
I.
M Ry hik,
Tuhle
oj
I eq als.
Se ies
and
in 1975 and P o esso in 1986. He is a membe
P ndicc s New Yo k. Academic P ess, 1980 o he Elec omagne ism Academy o M.1
T.,
[I71
c
M Bende and
s
A O szag.
Ad anced
Mo hema cal Me hodc oi
Camb idge. HIS main ields o in e es include bounda y alue p oblems
S
ie ilim
and
Enginee s
New
Yo k
McG dw-Hill, 1978. in elec omagne ic heo y, wa e p opaga ion h ough dniso opic medid, and
Medim
and M. Homo, “SFc al and a ld lonal analYsl5
o
gene - mic owa e in eg a ed ci cui s He
is
p esen ly engaged in he analysis o
dized
cYllnd lcal and elllP lcal ip and mlc o5 lP Ilnes,” [EEE T ans plana ansmission lines embedded in ani o opic ma e ials, mul iconduc o
M ciouai
Theo
Te l
.
ol 38, pp 1287-1293. Sep 1990 ansmission lines, and plana slow-wa e 5 uc u es
F ancisco Medina
(M’91) was bom in Pue o Real,
CBdiz, Spain,
on
No embe , 1960. He ecei ed
he Licenciado deg ee in Sep embe 1983 and he
[IO] M Homo,
F
L Mesa,
F
Medina, and
R.
Ma quis, “Quasi-TEM Doc o deg ee in 1987, bo h in physics, om he
Uni e si y
o
Se ille, Spain.
He is cu en ly Associa e P o esso o Elec ici y
and Magne ism in he Depa men o Elec onics
and Elec omagne ics, Uni e i y o Se ille. His
esea ch deals mainly wi h analy ical and nume ical
me hods o plana s uc u es dnd mul iconduc o
En ique D ake
was bom in Mon illa, Co doba,
Spain,
on
Sep embe 4, 1966. He ecei ed he
Licenciado 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 e5 wi h a schola ship
o he Spanish Go e nmen . His esea ch in e es
ocus on he analysis o plana s uc u es and mul-
iconduc o line5
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