Analysis
o
leakage
in
mul ilaye ed mic os ip lines
using complex images
J.
Be nal,
E
Mesa
,
F.
Medina
GN~O de Mic oondas. Dep . Elec mica y Elec omagne ismo
Facul ad de Fsica
A da. Reina Me cedes s/n,
41012
Se illa (SPAIN)
Fax:
+
34
5
4239434
email: [email p o ec ed]
Abs ac
The Mixed Po en ial
In eg al
Equa ion (MPIE)
is
combined
wi h
he
complcx
iinagc
me hod
Lo
analyze he comple e spec um o mul ilaye ed p in ed ansmission lines. A ele an con ibu-
ion
o
his me hod
IS
i s
abili y
o
s udy
bo h
he
bound
and leaky
egimes
in
a
e y simple and
e icien way. Since he analysis is ca ied ou
in
he spa ial domain, his wo k also makes
i
possible
o
analyze
he
leakage phenomenon o s uc u es wi h nonze o hickness conduc o s.
1
In oduc ion
The p opaga ion cha ac e is ics o guiding s uc u es in laye ed media a e e y e i-
cien ly compu ed by means o in eg al equa ion me hods. Fo he laye ed geome y,
he equi ed G een’s unc ions a e only known in closed- o m in he spec al domain.
I
a
spa ial domain o mula ion we e used, he spa ial-domain G een’s unc ions can be
ob ained om hei spec al e sions h ough Fou ie ans o m in e sion
[I].
S ic ly
ze o- hickness p in ed ansmission lines can be sol ed bo h in he spec al and he spa-
ial domains wi h a he low compu a ional e o . Non-plana conduc o s a e be e ac-
coun ed o by means o he spa ial-domain e sion o he in eg al equa ion
[Z].
Du ing
he
las
decade,
a
powe ul ool - he Disc e e Complex Images Technique (DCIT-
was de eloped o ob ain he spa ial G een’s unc ions o laye ed s uc u es
[3].
This
echnique has been p edominan ly applied o
3D
plana ci cui s and an ennas p oblems.
Recen ly his me hod has been adap ed o deal wi h
2D
guiding p oblems, whe e he
ele ance o
a
as gene a ion o he G een’s unc ions
is
e en mo e signi ican han in
he 3D con ex . Thus, plana s ip-like
[4]
and slo -like
[5]
s uc u es ha e been sol ed
ollowing ha p ocedu e. Howe e , he impo an ques ion posed by he exis ence o
bo h su ace and space leaky-wa e solu ions has no been conside ed in hose wo ks
whe e only bound modes we e ea ed. Leaky-wa es ha e been comp ehensi ely s ud-
ied in he ame o he Spec al Domain Analysis (SDA)
[6].
Wo king in SDA implies
o pe o m nume ical in eg a ion along
a
p ope ly chosen pa h
in
he spec al a iable
complex plane. Va ious physical and ma hema ical a gumen s ha e been employed o
de e mine he ea u es o his pa h. In his pape we will show how o adap he me hod
in
[4]
o include he leaky egime. This new app oach has he ad an age o being e y
simple and e icien and he capabili y o dealing wi h non-plana conduc ing s uc u es.
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2 Analysis
The p oblem
o
he de e mina ion
o
he p opaga ion cons an s
o
a mul is ip sys em
embedded
in
a
laye ed subs a e can be posed in e ms
o
a
Mixed Po en ial In eg al
Equa ion
iis
shown in
[4].
The
G een' unc ions o
he
scala and ec o po en ials,
Ji"'
and
K,;
,
inus be ob ained oin hei spec al e bions. This Implies
o
pe o m
Fou ie
~iiis o m in e sions o he ype:
whe e
A,J
=
m,
A.,
b ing
lie
ans e w wa enuinbe and
I;:
=
siiined p opaga ion cons an .
-
j.u
he as-
The DClT can be ad an ageously
uaed
in his p oble n
o
a oid he edious nume -
iciil
in eg a ion
in ol ed
in
(I
).
The unde lying idea
ia
o expand he spec al unc ion
ii i
a
sum
o complex exponen ial unc ions whose spa ial-domain coun e pa
is
known
hanks o
a
2D
Som nc eld iden i y
[4].
Howe e , his echnique was
only
applied o
cha x e ize bound modes. Nex we will explain he guidelines o he me hod along wi h
lie
modi ica ions ha mus be in oduced o gene alize he app oach o accoun o leaky
modes.
-
I
is el1 known ha he spec al unc ions
G(b:,)
show wo ypes
o
singula i ies:
b anch
poin a
a
b.,
=
and poles. They a e espec i ely ela ed
o
he adia-
i~oii
iii o
i e
space
and o he inodcs
o
he backg ound wa eguide. Since he complex
im,igcs
cxlmision
ciiii
only
i
analy ical unc ions, he abo e singula i ies
mus
be p op-
e ly haiidled.
This
IS
done by i s in oducing a change
o
a iable
(ILO
= m)
ha elimina es b anch poin s and second ex ac ing
ou
in analy ical o m he signi -
icm
polcs
(in
p x ice,
hobe
poles associa ed wi h abo e cu o wa eguide modes).
. a ic con ibu ion
IS
also explici ly ex ac ed ou . Thus, he spec al domain
G een' 'unc lons
a e
spli in o h ee con ibu ions:
-
--
whe e
G,,
ih
he quasi-s a ic c in
(Go
=
Ii~iia,+~
G),
Gp
ep esen s he con ibu ion
o he u hce wa e e ms and
LG,
is he emaining ha is sui able o complex expo-
nen~i;iI expansion.
/!
"
The spa ial doinain e sion
o
Gp
u ns ou
o
be
[4]:
whe e
No
i
he numbe
o
signi ican poles,
k,
is
he loca ion o hep- h pole,
4
i s
eqidue
and
=
k::
-
k,$
In
he bound egime
(k:
=
l
>
k,,)
his exp ession yields
exponen ially decaying ields
in
he anb e se di ec ion. I we a e in e es ed in he leaky
egime
asocia ed
o
ii
pa icula su ace-wa e pole,
i
can be p o ed ha
i
su ices
o
chenge
lie sign
o
6,>
o ha pa icula
pole
in
(3).
58
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-
The
G,
unc ion can be app op ia ely expanded as a
siini
o coinplcx exponen ial
unc ions:
! I,:,
G;,
c
u*c-y~I(o
(4)
1=I
The space domain con ibu ion o he e ms
in
he abo e expansion can be analy ically
ob ained by using he ollowing iden i y:
whe e
(
=
,/(:I;
-
:I:')~
+
y2,
A'"
ib
he ze o o de modi ied Bessel unc ion o he sec-
ond kind and
Hi2)
hc ze o o de Haenkel unc ion
o
he wcond kind. The i s op ion
gi e5
place
o
g owing
ields
o
accoun ing
o leaky egime whe eas lic second
op-
ion p o ides he bound solu ions. In
[6,
71
he space-wa leaky modes a c ob ained
by imposing an al e na i e in eg a ion pa h in he
I;,
complex plane o he nume ical
e alua ion o :he spec al in eg als
(I).
This ciilcula ion is supe seded by he p esen
app oach. The e m in
(I)
can be conside ed
a
pa icula case
o (5).
To summa ize, he p oposed me hod allow us o analy ically ob ain he spacc doinai
ke nel o he in eg al equa ion in bound and lcaky egimes. Once he space domain
ke nels ha e been ob ained, he in eg al equa ion is sol ed by using he Gale kin me hod
wi h adequa e basis unc ions leading o quasi-analy ical e illua ion
o
ma ix en ies
141.
3
Nume ical
esul s
Ou me hod has been alida ed by compa ing ou esul s wi h he ones ob ained by
means o a well es ablished
SDA
code
[7]
de eloped o ze o- hickiie~h mip-like s iic-
u es. The ag eemen in bo h bound and leaky egimes is excellen
b -
id1
he cascs
conside ed.
As
an example, Figu e
1
shows p opaga ion cons an da a o
a
space-wa e
leaky mode suppo ed by
a
mic os ip line. Fo he ze o- hickness case, his igu e coin-
pa es
SDA
esul s e sus ou esul s o
B/k0
and
o/kU.
An excellen ag eeinen is
obse ed. This s uc u e was also analyzed in
[I],
showing ou esul s
good
ag eemen
wi h he da a epo ed in
[I].
Al hough he ex ension o ou app oach o nonze o hick-
ness conduc o s equi es u he ma hema ical s eps han hose p esen ed in his pape ,
his ask has been done and some nume ical esul s ha e been include in he igu e.
4
Conclusions
The
MPIE
app oach combined wi h he complex iinage me hod has been adap ed o
analyzing mic os ip ansmission lines in leaky egime hy he i s ime. Bo h su ace-
wa e and space-wa e leaky modes a e inco po a ed o he analysis in a e y simple way.
The complex image echnique p o ides high accu acy and e iciency o he analysis.
Wo king in he spa ial domain allows o an ex ension o nonplana conduc o s.
Al-
hough his gene aliza ion is no s aigh o wa d, he heo y epo ed in his pape o e s
he necessa y backg ound.
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I2l
1
.o
06
A-
CO
,
4
6 8
io
12 i4
F equency
(GHz)
Figu e
1
:
No malized p opaga ion
(P/ko)
and a enua ion
(culko)
cons an s o
a
space-
wa e
leaky
mode in a mic os ip line:
E,
=
9.8,
w
=
3
mm,
h
=
0.635
mm,
=
s ip
hickness.
Re e ences
[I]
K.
A. Michal ky and
D.
Zheng, "Rigu ous analysis
o
open mic os ip lines o a bi a y
c oss sec ion in hound and leaky egimes,"
IEEE
T ans. Mic owa e Theo y Tech.,
ol. 37,
pp. 2005-2010, Dec. 1989.
[2]
C-I.G.Hsu,
R.F.Ha ing on, K.A.Michalski and D.Zheng "Analysis
o
mul iconduc o
ansmission lines o a bi a y c oss sec ion in mul ilaye ed uniaxial media,"
IEEE T ans.
Mic owa e Theo y Tech.,
ol. 41, pp.
10-18,
Jan. 1993.
[3]
D.G. Fang,
J.J.
Yang and
G.Y.
Delisle, "Disc e e image heo y o ho izon al elec ic
dipoles in a mul ilaye ed medium,''
P oc. Ins . Eh. Eng.,
ol.
135,
pp, 297-302, Oc .
1988.
[4] J.Bc nal, EMedina, R.R.Boix and M.Ho no, "Fas ull wa e analysis
o
mul is ip ans-
mission lines based
on
MPIE
and complex images,''
IEEE T ans. Mic owa e Theo y Tech.,
ol.
48, pp.445452, Ma .
2000.
[51
E.
A. Soliman,
P.
Pie e s,
E.
Beyne and
G.A.E.
Vandenbosch, "Nume ically e icien
spa ial-domain momen me hod o mul islo ansmission lines in laye ed media
-
aplica ion o mul islo in MCM-D echnology,"
IEEE
T ans. Mic owa e Theo y Tech.,
ol.
47, pp. 1782-1787, Sep. 1999.
[61
N.K. Das and D.M.
Poza ,
"Full-wa e spec al-domain compu a ion
o
ma e ial, adia ion
and guided wa e losses in in ini e mul ilaye ed p in ed ansmission lines,"
IEEE T ans.
Mic owa e Theo y Tech.,
ol. 39, pp.
54-63,
Jan. 2000.
[71
R.
Ma quCs, F. Mesa, "Spec al domain analysis
o
highe o de leaky modes in mic os ip
lines: a new spec al-gap e ec ,"
Jou nal
o
Elec omagne ic Wa es and Applica ions,
ol.
II,
pp. 1367-1384.1997.
583
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