IEEE
TRANSACTIONS
ON
MICROWAVE
THEORY
AND
TECHNIQUES,
VOL.
38,
NO.
9,
SEPTEMBER
1990 1287
Spec al and Va ia ional Analysis
o
Gene alized Cylind ical
and Ellip ical S ip and
Mic os ip Lines
Abs mc
-In his pape , he a ia ional echnique in he spec al
domain
(VTSD)
is shown o
be
an e icien me hod
o
compu ing he
quasi-TEM pa ame e s
o
a bi a y mul iconduc o and mul idielec ic
cylind ical
o
ellip ical s ip con igu a ions. Simple con o mal mappings
educe he cylind ical
o
ellip ical geome ies o an equi alen ec angu-
la one wi h pe iodic bounda y condi ions. Mino modi ica ions o
p e ious wo k on plana s uc u es allow
us
o analyze any cylind ical
o
ellip ical geome y, compu ing he capaci ance
[Cl
and induc ance
E1
ma ices, om which he e ec i e dielec ic cons an s and mode
impedances a e ob ained.
I. INTRODUCTION
N THE RECENT mic owa e li e a u e, se e al pape s
I
ha e been de o ed o he analysis o s iplike and
mic os iplike ansmission lines p in ed on lexible di-
elec ics w apped a ound cylind ical o ellip ical su aces.
These sys ems can be used o exci e con o mal a ays
moun ed on cylind ical o ellip ical objec s. Coaxial- o-
plana line ansi ions, slo ed lines, and wa ping due o
se e e en i onmen al changes can also be con enien ly
modeled wi h cylind ical s ips.
A numbe o au ho s ha e s udied cylind ical and ellip-
ical con igu a ions using he quasi-TEM model. Wang
uses a dual se ies ep esen a ion o analyze he homoge-
neous cylind ical s ipline and he inhomogeneous cylin-
d ical mic os ip [l]. The modi ied esidue calculus ech-
nique is used by Joshi
e
al.
o sol e cylind ical and
ellip ical s iplines [2]. Con o mal mappings ha e been
applied o se e al simple con igu a ions [31-[6] and, e y
ecen ly, he mo e in ol ed mul iconduc o s iplinelike
p oblem [7]. The G een’s unc ion o mula ion wi h a ia-
ional exp essions o he capaci ance is used o analyze
ellip ic single [81 and coupled [91 a c s ips and a nonsym-
me ical pai o coupled cylind ical s ips [lo]. Single [ll]
Manusc ip ecei ed Augus 29, 1989; e ised Ap il 6, 1990. This wo k
was suppo ed by DGICYT (P ojec PB87-0798-C03-01, 1988-91) and
by he Jun a de Andalucia (P ojec P5355-2, 1988-90).
The au ho s a e wi h he Depa men o de Elec 6nica
y
Elec omag-
ne ismo, Facul ad de Fisica, Uni e sidad de Se illa, 41012, Se illa,
Spain.
IEEE Log Numbe 9036755.
and b oad-side-coupled
[
121 cylind ical s ips ha e been
analyzed sol ing he Laplace equa ion subjec o he ap-
p op ia e bounda y condi ions in a h ee-dielec ic
medium. The spec al-domain echnique in cylind ical
coo dina es has been used o analyze single and symme -
ically coupled a c s ips [13]. A signi ican s ep in he
analysis o sys ems o his kind has been gi en in [141,
whe e a gene al class o mul iconduc o
cylind ical
lines
was s udied by aking ad an age o he pe iodici y o
hese s uc u es and using he FFT algo i hm in conjunc-
ion wi h an i e a i e scheme. In addi ion, a ull-wa e
analysis o single and coupled cylind ical s ips p in ed on
one-laye subs a e has been epo ed in [151 and [161.
The aim o he p esen pape is o show ha gene al-
ized cylind ical o con ocal ellip ical s ip geome ies (Fig.
l(a))
can be easily s udied by using he heo y and com-
pu e p og ams p e iously de eloped o plana s uc u es
wi h ec angula bounda y condi ions. No e ha mos o
he ansmission lines men ioned in he p e ious pa a-
g aph can be iewed as pa icula cases o his gene ic
mul iconduc o sys em. The me hod is based on he appli-
ca ion o con o mal mappings which ans o m he
cylind ical o ellip ical geome y in o a plana one wi h
ec angula pe iodic bounda y condi ions. This equi alen
s uc u e is analyzed by using he VTSD app oach de-
sc ibed in
[
171,
[
181 me ely in oducing sligh modi ica-
ions in he analy ical ea men and he compu e p o-
g ams. This analysis allows one o compu e he [,!,] and
[
C] ma ices cha ac e izing he sys em unde quasi-TEM
ope a ion and, om hese, he modal impedances and
e ec i e dielec ic cons an s. The o iginal p og ams based
on [17] and [18] ha e been imp o ed by using be e ial
unc ions and a con enien asymp o ic analysis in o de o
accele a e he con e gence o he se ies appea ing in he
nume ical compu a ion. In his way, we ha e a e y e i-
cien algo i hm o cha ac e ize, unde he quasi-TEM
assump ion, e y gene al cylind ical o con ocal ellip ical
ansmission sys ems in ol ing an a bi a y numbe o
dielec ic laye s and conduc o s dis ibu ed be ween hem,
e en i hey a e p in ed in di e en in e aces.
0018-9480/90/0900-1287$01.00
01990 IEEE
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1288
IEEE
TRANSACTIONS
ON
MICROWAVE
THEORY
AND
TECHNIQUES,
VOL.
38,
NO.
9,
SEPTEMBER
1990
ELLIPTIC
YULTXSTR PS
SYSTEM
w
-
plane
I'
I.'
2-
-
'1:.
I
N
Y
I
I
I
I I
1,!2
1
I
lh .
1
I
E2
I
I
I
I
U
(b)
-
I
CL
i
y1
0
I
'I
2n
I
uR
Fig.
1.
(a) C oss sec ion o gene alized ellip ical mul is ip and mul i-
dielec ic s uc u e. The no a ion
o
geome ical pa ame e s de ining
a c s ips and in e aces is included. When
c
=
0,
ai
=
bi
=
,,
we ha e
he cylind ical case.
(b)
Image s uc u e in
w
plane
o
he con igu a-
ion in (a). This s uc u e is pe iodic in he
U
di ec ion wi h pe iod
2iT.
11.
GENERAL CONFIGURATIONS
AND
CONFORMAL
MAPPINGS
The c oss sec ion o a gene alized
con ocal ellip ical
sys em
has been d awn in Fig. l(a). Ou pu pose is o
compu e he [L] and [C] ma ices o his mul iconduc o
ansmission line. Ins ead o wo king in an ellip ical ame,
we will exploi he heo y and he p og ams de eloped by
he au ho s o ea ing gene alized plana con igu a ions
[17], [NI. In o de o do his, we ans o m he o iginal
ellip ical (Fig. l(a)) geome y in o an equi alen one wi h
pe iodic ec angula bounda y condi ions (Fig. l(b)) ia
he ollowing con o mal mapping:
whe e he as e isk deno es complex conjuga e;
2c
=
ocal
dis ance;
z
=
x
+
jy;
w
=
U
+
ju; and
b,
and
a,
a e he
mino and majo semiaxes o inne g ound ellipse.
The o iginal ellip ical con igu a ion in he
z
plane is
mapped in o he ec angula one in he
w
plane. Wi h he
mapping gi en by (l), geome ical pa ame e s (a) and (b)
o Fig.
1
a e ela ed in he ollowing way:
Hi
=
anh-'
(
bi+'
/ai+')
-
anh-'
(
bi
/ai)
u
=
an-'
[(ai/bi)
an&]
uk= an-' [(ai/bi) ancpk]
(
2)
(using he no a ion in Fig. l(a) and (b)).
Fo a
cylind ical
con igu a ion in he
z
plane ( ha is,
c
=
0,
ai
=
b,
=
i
in Fig. l(a)) a sui able mapping unc ion
is
w(z)
=
jln(z*/a).
(3)
In his way he cylind ical sys em is ans o med in o he
ec angula one shown in Fig. l(b):
H~
=
In(
ip'
/ i)
u
=
'pi
U$
=
&.
'
(4)
The equi alen con igu a ion (Fig. l(b)) consis s o an
a bi a y numbe
(N,)
o conduc ing s ips lying in a
ce ain numbe
(M)
o plana in e aces be ween
Nd
lossless iso opic dielec ic laye s, he whole being en-
closed by a se o ec angula bounda y condi ions. The
cylind ical o ellip ical con igu a ion and hei co e-
sponding ans o med s uc u es ha e he same
[
C]
and
[
L] ma ices (nonmagne ic ma e ials a e assumed). A
gene ic con igu a ion simila o he one in Fig. l(b) has
been e icien ly analyzed by he au ho s using he a ia-
ional echnique in he spec al domain in [171 and [181.
Since he analysis is pa allel o he one epo ed in [17]
and [18], we e e he eade o hese pape s and o he
Appendix he ein o de ails. A b ie accoun will be gi en
in he ollowing sec ion.
111. NUMERICAL RESULTS
The
[C]
ma ix is compu ed om he elec ical ene gy
s o ed pe uni leng h in he s uc u e o an app op ia e
se o exci a ions o he conduc o s ips [18]. The un-
known su ace cha ge dis ibu ion on he s ips o each
exci a ion is expanded in o a se o basis unc ions and he
expansion coe icien s a e compu ed o minimize he elec-
ical ene gy (Rayleigh-Ri z me hod). In o de o do his,
we wo k in he Fou ie ans o m domain, since he
spec al G een's unc ion ma ix (SGFM) needed in he
analysis can be easily ob ained using he me hod epo ed
in [17]. As is well known, [L] is ob ained om he [C]
ma ix o he same s uc u e wi hou dielec ics.
The nume ical e iciency o he Rayleigh-Ri z algo-
i hm s ongly depends on he choice o ial unc ions o
app oxima ing he unknown su ace cha ge densi y on he
s ips. In he u esen wo k we use a se o basis unc ions
w(
z)
=
cos-l(
z*/c)
-
j
an-'
(b,
/a,)
(1) which has p b ed o be e y good o bo h quasi-TEM
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MEDINA AND HORNO: SPECTRAL AND VARIATIONAL ANALYSIS
1289
and ull-wa e analyses because hey inco po a e he sin-
gula beha io o he su ace cha ge densi y
(p)
a he
s ip edges and he p ope ies o Chebyshe polynomi-
nals:
n
p’(
U)
=
pi(
U)
+
aipi(
U)
(5)
m=l
S’
=
(U:.
+
4)/2 W’= uk
-
U:.
whe e
TJ )
is a Chebyshe polynominal o he i s kind
o o de
n.
A his poin , some a en ion mus be gi en o he
nume ical con e gence o Fou ie se ies appea ing in he
analysis (see
[18,
eqs.
(8)-(10)1).
The se ies a ising om
in e ac ions be ween noncoplana s ips a e quickly con-
e gen , because he o -diagonal elemen s o he SGFM
ela ing he Fou ie ans o m o he po en ial and he
su ace cha ge in di e en in e aces dec ease exponen-
ially. Howe e , in e ac ions be ween s ips lying a he
same in e ace yield poo ly con e gen se ies (i he basis
unc ions in
(5)
a e used, he con e gence is as
l/n2).
A
signi ican imp o emen is achie ed i he asymp o ic ail
ex ac ion echnique is applied (see he Appendix). In
Fig.
2,
we show
a
ypical con e gence pa e n wi h and
wi hou asymp o ic ex ac ion. Clea ly, asymp o ic ex ac-
ion will esul in an impo an educ ion
o
CPU ime.
This ac is e en mo e p onounced o mul iple-s ip
con igu a ions, and i has been e i ied o a wide a ie y
o simple and coupled s uc u es.
On he basis o his heo y we ha e w i en a compu e
p og am whose eliabili y and nume ical e iciency ha e
been con enien ly es ed. Table
I
shows he con e gence
o he no malized cha ac e is ic impedance
(E,“*z,)
wi h
he numbe o ial unc ions o a cylind ical homoge-
neous s ipline. We ha e included some esul s abula ed
in
[14,
able I,
N
=
20481
o compa ison. The pa icula
case
a
=
180,
which can be exac ly sol ed by con o mal
mapping (exac alue
=
5.361
O),
has also been included.
The ag eemen wi h da a epo ed in
[14]
is excellen , and
he exac alue is also ob ained. We can see om his
able ha bigge angles equi e mo e basis unc ions.
Ne e heless, CPU ime does no inc ease as quickly wi h
he a c wid h, since bigge angles equi e adding ewe
Fou ie e ms. Typically, ou signi ican digi s a e ob-
ained in
50-200
ms on a
VAX-11/785
compu e . No e
ha ou me hod does no impose es ic ions on he s ip
wid h, in con as o he me hod in
[14],
based in he
applica ion o he FFT.
The mul iconduc o case is also checked by compa -
isons wi h da a epo ed in
[14,
able
1111.
In Table I1 we
p esen esul s o he mode e ec i e dielec ic cons an s
o
six
coupled s ips. Sligh di e ences could be due o
he ela i ely small numbe o samples used in
I141
o
compu e hese da a. In gene al, e y good ag eemen wi h
da a published in
[14]
has been ound. Disc epancies wi h
‘0
70
20
30
Numbos
o
Founes
o-
Fig. 2. Pe cen de ia ion om he exac alue o he no malized
capaci ance o a cylind ical s ip as a unc ion
o
he numbe
o
e ained Fou ie e ms wi h (i) and wi hou (ii) asymp o ic ex ac ion.
g aphic da a published in
[l]
and
[ll]
poin ed ou in
[141
ha e also been de ec ed. Since his disag eemen has
been ound using wo comple ely di e en me hods, we
can also conclude ha he accu acy
o
he esul s in
111
and
[ll]
is ques ionable.
Edge-coupled symme ical a c s ips ha e been consid-
e ed o compa e wi h he esul s in
[5]
and
1131.
The
me hod in
[5]
is alid when he geome ical pa ame e s
a e wi hin a ce ain ange o alues. Fo hese, e y good
ag eemen is ound. Howe e , he au ho s in
[13]
use as
ial unc ion o he cha ge densi y jus he ze o h-o de
e m in
(5).
This choice esul s in se ious nume ical e o s
when s ips a e s ongly coupled. This is because he
cha ge dis ibu ion is no symme ical a ound he cen e
o he s ip, as is assumed in
[13].
When wide a c s ips
a e in ol ed, he cha ge dis ibu ion ends o be almos
uni o m. Thus, signi ican disc epancies wi h esul s in
[13]
also a ise in his case. Table I11 illus a es hese ac s.
Resul s in he i s column
(n
=
0)
co espond o he
me hod in
[13].
In o de o ob ain accu a e esul s, a ew
basis unc ions mus be e ained. No e ha odd ial
unc ions a e closely ela ed o he coupling e ec , since
hese ake in o accoun he asymme y o he cha ge
dis ibu ion on coupled s ips. E en unc ions mus be
added when wide s ips a e in ol ed. Simila accu acy
p oblems a e also expec ed o occu in
[8]-[101,
whe e
poo ial unc ions a e used.
Noncoplana s ips can be also ea ed wi h ou p o-
g am. We ha e also es ed his case. Fo ins ance, we
ha e conside ed he symme ical b oadside-coupled s ips
analyzed in
[6]
and
[12].
In hese pape s, he a io be-
ween adii
is
chosen in such a way ha e en and odd
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1290
IEEE
TRANSACTIONS ON
MICROWAVE
THEORY
AND
TECHNIQUES,
VOL.
38,
NO.
9,
SEPTEMBER
1990
a
Resul s
numbe
o
ial unc ions (n )
1.00
19.95
39.99
60.03
80.07
99.93
119.97
140.01
in
[141
N=2048
38.14
21.27
14.75
j
11.29
0
12
34
5 6
189.0
39.18 38.11
22.97 21.27 21.26
16.61 14.76 14.75
13.16 11.31 11.29
10.97 9. 180 9.162
9.437 7.718 7.698 7.6
8.303 6.659 6.638 6.6
180.00
TABLE
I1
EFFECTIVE
DIELECTRIC
CONSTANTS
OF
SIX
COUPLED
STRIPLINES
AS
A
FUNCTION
OF
SEPARATION
ANGLE
(a) Resul s epo ed in 114, able
1111.
(b)
Ou
esul s.
8.453 5.452 5.379 5.366 5.363 5.362 5.361
modes can be de ined, and he analysis is applied o he
single conduc o geome y co esponding o each mode.
We ha e analyzed hese s uc u es bo h by conside ing
hem as pa icula cases o he mo e gene al asymme ic
con igu a ion and by using he heo y
o
odd and e en
modes, in oducing he app op ia e elec ic o magne ic
wall
[12].
The nume ical esul s ob ained wi h bo h p oce-
du es a e indis inguishable. Ve y good ag eemen has
been ound wi h he con o mal mapping esul s in
[61
o
a wide ange o alues o
a.
Howe e , sligh disc epancies
wi h he a ia ional esul s in
[12]
we e ound o high
alues o
a.
This is due again o a lack o p ecision in he
es ima ion o cha ge dis ibu ion, as is e iden om Fig.
3.
In his igu e, we show he a io
o
phase eloci ies o
b oadside-coupled mic os ip lines using
n
=
0
and
n
=
6
in
(5).
The compu ed esul s coincide wi h g aphic da a
epo ed in
[12]
when
n
=
0, bu signi ican de ia ion is
ound
(10%
o wide s ips) when mo e e ms a e used o
app oxima e he cha ge dis ibu ion. In he limi ing case
a
=
180,
which can be exac ly sol ed, we ge i e signi i-
can digi s using ou basis unc ions in
(5).
In o de o check he alidi y o he p og am o
ellip ic
a c s ips, we ha e compa ed ou da a wi h he e y
accu a e con o mal mapping esul s epo ed in
[3].
These
TABLE
I11
EVEN-
AND
ODD-MODE
CHARACTERISTIC
IMPEDANCE
OF
EDGE-COUPLED
CYLINDRICAL
STRIPS
numbe
o
ial
unc ions
(n )
50.92 49.89 48.80 48.77 48.77 48.77
37.07 35.59 35.55 35.47 35.45 35.45
16.79 16.63 14.29 14.27 14.26 14.25 14.25
13.67 13.45 12.87 12.82 12.81 12.79 12.79
45.52 45.47 45.02 45.02
42.47 42.41 42.23 42.23
20
80
20
15.88 15.88 14.02 14.02
14.01
14.01
14.58 14.58 13.46 13.46 13.46 13.46
b/a=1.4; c/a=1.8;
c
=2
n
=
0
co esponds
o he
echnique
in
[131
1.S
.
1
1-4
b- -P-
-
- -
1.11
ow’
1
1
1
50
50
100
Hol --ongl
<d g >
Fig.
3.
Va ia ion
o
mode eloci y a ios
o
b oadside-coupled mi-
c os ip lines as a unc ion o hal -angle s ip a c. No a ion is he same
as ha in [12, ig.
61:
- - -
- -
esul s in
[12];
0
ou esul s wi h
n
=
0
in
(5);
-
ou
esul s wi h
n = 6.
(i)
p/a
=
0.1, (ii)
p/a
=
0.3,
(iii)
p/a
=
0.6,
(i )
p/a
=
1.0.
b/a
=
1.4,
E,
=
2.32.
esul s a e limi ed o single ellip ical s ipline in homoge-
neous medium, bu hey a e good enough o ou compa -
ison pu pose. We ha e made compa isons wi h all g aphic
da a epo ed in
[3],
and he ag eemen is excellen in all
cases. In Fig.
4,
o example, we show he cha ac e is ic
impedance o an ellip ical s ipline as a unc ion
o
he
a c wid h o di e en loca ions wi h espec o he majo
axis
(0,)
and
wo
di e en dis ances o he inne ellip ical
g ound
(h
/a).
As
expec ed, he cha ac e is ic impedance
depends on
Bo,
which en e s as a new a iable no exis ing
in he cylind ical case.
In conclusion, he compu e p og ams based on he
heo y in his pape can be used wi h con idence o
compu e he cha ac e is ic pa ame e s o a e y wide
a ie y o cylind ical o ellip ical ansmission sys ems.
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MEDINA AND HORNO:
SPECTRAL
AND VARIATIONAL ANALYSIS
Q
bo
i.
1
0
1291
c:
___-----__
_---
-
-1
..'
,
,
_....
~----
.................
_._._._____.___.___._._._._.
................
.:::;::
i
0
0.02
0.04
0.06
a,
.........'....
,
300
=
.5
I
A
,A,
::
1
10
1
Angle (deg ees)
Fig.
4.
Cha ac e is ic impedances o ellip ic s ipline in homogeneous
medium (ai ) e sus s ip angle
(A
4):
-
esul s epo ed in [31;
A
ou
esul s.
Wi h he aim o including an example no epo ed in
p e ious li e a u e, we show in Fig. 5 he e ec i e dielec-
ic cons an s and mode impedances
o
a pai o asymme -
ical coupled cylind ical s ips p in ed
on
a wo-laye
subs a e. In Fig. 5(a) we can see ha he c- and -mode
e ec i e dielec ic cons an s can be made almos equal by
a p ope choice o he hickness o he inne dielec ic
(which has he lowe dielec ic cons an ). The ange o
alues o
(b
-
a)/a
whe e he e ec i e dielec ic con-
s an s a e e y close is a ound a
c i ical poin
de ined by
he equaliza ion o he induc i e and capaci i e couplings.
In his zone, he mode line impedances exhibi he singu-
la beha io shown in Fig. 5(b). The c o
T
na u e
o
each
mode changes o each side o he c i ical poin ; ha is,
he co esponding mode numbe changes he sign. This
beha io is simila o ha epo ed in
[19]
o coupled
asymme ic mic os ips wi h o e lay.
A
de ailed analysis
o hese phenomena and an analysis o o he pa icula
s uc u es a e ou o he scope o his pape , which
ocuses mo e on he compu a ional me hod o sol e his
kind o s uc u e han on he analysis o he beha io o
pa icula con igu a ions.
IV.
CONCLUSIONS
In
his pape we ha e s a ed ha e y gene al single-
conduc o o mul iconduc o cylind ical/ellip ical s ip
con igu a ions embedded in a laye ed dielec ic medium
can be educed, ia con o mal mapping, o a ec angula
geome y wi h he same
[I,]
and
[C]
ma ices. The analy-
sis
o his equi alen s uc u e is achie ed by aking ad-
<b-a>/a
(b)
Fig.
5.
(a)
E ec i e dielec ic cons an s o c and
n-
modes o a pai o
asymme ic cylind ical s ips
on
wo-laye subs a e. (b) Modal line
impedances
o
he same s uc u e. No e he singula beha iou a ound
ZzT.
a1
=
lo",
az
=
20",
0
=
5",
c/a
=
1.2,
c1
=
2.32,
c2
=
10.
._._
he c i ical poin .
----
Zlc;
......
ZIT;
-
Zzc;
.-
an age
o
p e ious wo k on boxed plana s uc u es. The
nume ical con e gence o he p og ams is g ea ly accele -
a ed inco po a ing he asymp o ic beha io o he se ies
appea ing in he analysis, in such
a
way ha e icien
p og ams ha e been w i en. I has also been poin ed ou
ha excessi ely simple app oxima ions o he su ace
cha ge dis ibu ion yield meaning ul nume ical e o s,
mainly when s ong coupling o wide s ips a e in ol ed.
As
an applica ion example, he singula beha io o he
cha ac e is ic pa ame e s o asymme ic coupled s uc-
u es on mul ilaye subs a es has been shown.
APPENDIX
He e we include ce ain ema ks on h_e compu a ion o
he spec al G een's unc ion ma ix,
[Gij(n)],
wi h
i,
j
=
1,.
e,
M,
used in he
VTSD
[17],
[18]
and on nume ical
aspec s conce ning he addi ion o Fou ie se ies in ol ed
in he me hod.
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1292
IEEE TRANSACTIONS
ON
MICROWAVE
THEORY
AND
TECHNIQUES,
VOL.
38,
NO.
9,
SEPTEMBER
1990
The s uc u e analyzed in his pape (Fig. l(b)) di e s
om he con igu a ions s udied in [171 and [181 in he
na u e o he la e al bounda y condi ions. The s uc u e
in his wo k is pe iodic wi h pe iod 277 in he
U
di ec ion
ins ead o ha ing elec ic o magne ic sidewalls. Then, all
he quan i ies a e Fou ie ans o med using he de ini-
ion:
F(n)
=/2T (u)e~nudu.
0
(AI)
As
a consequence, [GJn)]-' =[L,,(n)] can be com-
pu ed om [17, eqs. (2)-(17)] making he ollowing subs i-
u ions:
k,-n
E,*:
=
Ey*:-E,
(A2)
whe e- he subsc ip
i
e e s o he i h laye in Fig. l(b),
and [L,,(n)] is a idiagonal symme ical ma ix wi h ele-
men s de ined by
whe e
Ljk
and
pk
a e he Fou ie ans o m o ee su ace
cha ge and he po en ial dis ibu ions on he k h in e -
ace (see Fig. l(b)). Tha is, he o iginal bounda y condi-
ion p oblem is sepa a ed in o simple p oblems de ined
by he bounda y condi ions in
(A3).
The solu ions o hese
p oblems a e gi en in [17, eqs. (12)-(17)] in e ms o
ce ain
gij(n)
unc ions de ined in [17, eq. (611. The physi-
cal meaning o hese unc ions is e iden om
[17,
eq.
(5)l. They a e he ac o s ela ing he Fou ie ans o m o
he ee cha ge densi y and he po en ial dis ibu ions in
he con igu a ions shown in Fig. 6.
On he o he hand, all he Fou ie se ies appea ing in
[171 and [181 mus be added om
n
=
--M
o
n
=
+-M
ins ead o om
n
=
1
o
n
=W.
In Sec ion I11 we e e ed
o he need o aking in o accoun he asymp o ic beha -
io o hese se ies in o de o achie e good con e gence.
F om [17, eqs. (6) o (1711 we can see ha [Lij(n)l
exponen ially con e ges o a diagonal ma ix whose ele-
men s a e
ik,k(n
+w)
=€o(El+E,+l)lnl=G~:(n
+w),
k=l,..*
,M
(A41
whe e
i
and
i
+
1
e e o he dielec ic laye s adjacen o
he k h in e ace
Fo he basis unc ions in
(5),
he Fou ie se ies appea -
ing when he me hod in [18] is applied has he ollowing
o m:
whe e
Jp,q,
Bessel unc ions o he i s kind o o de
p,q, a e he Fou ie ans o ms o he basis unc ions in
(5).
The con e gence o
S
is no sa is ac o y. Howe e , i
we conside he asymp o ic beha io o
A(n)
when
n
-
CO,
namely
A,(n),
we can ew i e
(A9
as
m
m
S=
C
[~(n)-~,(n)]+
C
AAn)
(~6)
n=
-m
n=
-m
whe e
cos
(nu
-
p. /2- ~/4) cos(
nb
-
q /2- ~/4)
n2
A,(
n)
=
C
wi h
C
=
2/
[
. (
ab)
1'2~o(
E:
+
E:'
')I
.
Now, a e some manipula ions, he second e m in (A6) is exp essed as a combina ion o igonome ic se ies. One
o hese se ies can be exp essed in closed o m; he o he is educed o an equi alen one con e ging in a ew e ms by
means o he esidue echnique. In ou p og ams we ha e used
T2
x(27-x)
,
0<X<2T
6 4
n=l
l-log(x)+x2
(
-+-
7:
1:4i0)],
4
9
sinh[m. (. -x)/2]
7
m,odd
m2sinh(m. 2/2)
[l- anh(n)]
-
-
sin(nx)
c
------"(
-(2. -x) l-log(2~-x)+(2. -x)2
n=l
n2
{
n=l
x
<
0.5
,
(27-X)
<OS
0.5
<
x
<
(2~ -0.5)
(eigh digi s being co ec in he wo s case using hese exp essions).
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1293
MEDINA
AND HORNO:
SPECTRAL
AND
VARIATIONAL
ANALYSIS
(b)
S uc u es de ining he elemen a y spec al unc ions needed o
comp e he global ma ix G een’s unc ion: (a) de ini ion o
2Jn)
7
c?~/Y;
(b) de ini ion
o
~ ,l~i(n)=C?l/~~i. The quan i ies
C?*
and
V,
a e he Fou ie ans o ms o he ee su ace cha ge densi y and he
po en ial dis ibu ion.
Fig. 6.
[14] C. H. Chan and R. Mi a, “Analysis o a class o cylind ical
mul iconduc o ansmission lines using an i e a i e app oach,”
IEEE T ans. Mic owa e Theo y Tech.,
ol. MTT-35, pp. 415-424,
Ap . 1987.
[15] N.
G.
Alexopoulos and A. Naka ani, “Cylind ical subs a e mi-
c os ip line cha ac e iza ion,”
IEEE T ans. Mic owaue Theo y
Tech.,
ol. MTT-35, pp. 843-849, Sep . 1988.
[16] A. Naka ani and N. G. Alexopoulos, “Coupled mic os ip lines on
a cylind ical subs a e,”
IEEE T ans. Mic owa e Theo y Tech.,
ol.
MTT-35, pp. 1392-1398, Dec. 1987.
1171
F.
Medina and M. Ho no, “De e mina ion o G een’s unc ion
ma ix
o
mul iconduc o and aniso opic mul idielec ic plana
ansmission lines: A a ia ional app oach,”
IEEE T ans.
Mi-
c owaue Theo y Tech.,
ol. MTT-33, pp. 933-940, Oc . 1985.
[18] F. Medina and M. Ho no, “Capi ance and induc ance ma ices o
mul is ip s uc u es in mul ilaye ed aniso opic dielec ics,”
ZEEE
T ans. Mic owa e Theo y Tech.,
ol. MTT-35, pp. 1002-1008, No .
1987.
[
191
K.
Sachse, “Analysis o asymme ical inhomogeneous coupled-line
di ec ional couple ,” in
P oc.
18 h
Eu opean Mic owa e
Con .,
S ockholm (Sweden), Sep . 1988, pp. 985-990.
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K.
Joshi and
B.
N. Das, “Analysis o ellip ic and cylind ical
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ol.
MTT-28,
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J.
S.
Rao, and
B.
N. Das, “Cha ac e is ic impedance
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287-291, Oc . 1980.
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cylind ical s iplines and mic os iplines,”
IEEE T ans. Mic owaue
Theo y Tech.,
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neous coupled cylind ical s iplines,”
Elec on. Le .,
ol. 23, no.
16, pp. 821-822, July 1987.
C.
J.
Reddy and
M.
D. Deshpande, “Cha ac e is ics o b oadside
coupled cylind ical s iplines,”
M c owaue and Op . Tech. Le .,
ol. 1, no. 4, pp. 133-136, June
1988.
D. Homen co schi, “A cylind ical mul iconduc o s ipline-like
mic os ip ansmission line,”
IEEE T ans. Mzc owaue Theo y
Tech.,
ol. 37, pp. 497-503, Ma . 1989.
B.
N. Das, A. Chak abo y, and
K. K.
Joshi, “Cha ac e is ic
impedance o ellip ic cylind ical s ip and mic os iplines illed
wi h laye ed subs a e,”
P oc.
Ins . Elec. Eng.,
p . H, ol. 130, pp.
245-250, June 1983.
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N. Das and
K.
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S.
V. R. P asad, “E en- and odd-mode
impedances o coupled ellip ic a c s ips,”
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Reddy and M. D. Deshpande, “Analysis o coupled cylind i-
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F ancisco Medina
was bo n in Pue o Real,
CQdiz, Spain, on No embe 9, 1960. He ecei ed
he Licenciado deg ee in Sep embe 1983 and
he Doc o deg ee in Sep embe 1987, bo h in
physics, om he Uni e si y o Se illa, Spain.
He is cu en ly Assis an P o esso o Elec-
ici y and Magne ism in he Depa men o
Elec onics and Elec omagne ism, Uni e si y o
Se illa. His esea ch ocuses mainly on nume i-
cal me hods o plana s uc u es and mul icon-
duc o lines.
Manuel
Ho no (M75) was bo n in To e del
Campo, Jah, Spain. He ecei ed he deg ee o
Licenciado in physics in June 1969 and he
deg ee o Doc o en Ciencias in physics in Jan-
ua y 1972, bo h om he Uni e si y o Se illa,
Spain.
Since Oc obe 1969 he has been wi h he
Depa men o Elec ici y and Elec onics a he
Uni e si y
o
Se illa, whe e he became an Assis-
an P o esso in 1970, an Associa e P o esso in
1975, and 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 aniso opic ma e ials, mul iconduc o ansmission
lines, and plana slow-wa e s uc u es.
8
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