IEEE
TRANSACTIONS
ON
MICROWAVE
THEORY
AND
TECHNIQUES,
VOL.
38,
NO.
8,
AUGUST
1990 1059
Quasi-TEM Analysis
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
MANUEL HORNO,
MEMBER,
IEEE,
FRANCISCO
L.
MESA, FRANCISCO MEDINA,
AND
RICARDO
MARQUES
Abs ac
--In his pape , a quasi-TEM analysis o mul iconduc o
plana lines embedded in a laye ed s uc u e in ol ing
lossy
iso/
aniso opic elec ic and/o magne ic ma e ials is achie ed. Condi ions
unde which a quasi-TEM assump ion is alid a e heo e ically de e -
mined.
An
e icien spec al-domain analysis is used o de e mine he
complex capaci ance,
[Cl,
and induc ance,
[Ll,
ma ices cha ac e izing
he ansmission sys em. The compu a ion o
[Ll,
when media cha ac-
e ized o a ully gene al pe meabili y enso a e p esen , is educed o
he compu a ion
o
an equi alen capaci ance ma ix. I is also shown
ha mos ac ual MMIC mic os ip- ype s uc u es (whe e semiconduc-
o subs a es a e p esen ) and possible u u e applica ions including
lossy magne ic ma e ials can be analyzed by using he simple quasi-TEM
model. The alidi y o he esul s has been e i ied by compa ison wi h
“ ull-wa e’’ heo e ical and expe imen al da a on mic os ip lines on
magne ic subs a es and slow-wa e s uc u es.
I.
INTRODUCTION
HE EVOLUTION in mic owa e sys ems poin s a
T
he comple e in eg a ion o passi e and ac i e ele-
men s (MMIC). Mic os ip-like ansmission lines on
semiconduc o subs a es used in hese kinds o ci cui s
should be con enien ly cha ac e ized. The high conduc i -
i y
o
hese subs a es p ecludes he use o pe u ba ional
echniques in o de o ake in o accoun he e ec o he
lossy ma e ial on he p opaga ion cha ac e is ics. In pa -
icula , he impo an “slow-wa e’’ modes suppo ed by
me al-insula o -semiconduc o (MIS) con igu a ions
mus be subjec ed o mo e de ailed analysis
[11-[31.
The
use o magne ic ma e ials in o de o imp o e he beha -
io o slow-wa e lines has been sugges ed in
[4],
whe e he
au ho s de elop a quali a i e discussion on he subjec .’
The c oss sec ion o he s uc u e conside ed in his
wo k is shown in Fig. 1, whe e he elemen s o he
pe mi i i y and pe meabili y enso s o e e y laye ,
[
eli
Manusc ip ecei ed No embe
15,
1989; e ised Ma ch 12, 1990. This
wo k was suppo ed by he DGICYT, Spain (P ojec PB87-0788-C03-01).
The au ho s a e wi h he Depa amen o de Elec 6nica
y
Elec omag-
ne ismo, Facul ad de Fisica, Uni e sidad de Se illa, 41012 Se ille, Spain.
EEE Log Numbe 9036419.
Du ing he e iewing pe iod, a pape dealing wi h plana ansmis-
sion lines wi h aniso opic magne ic media was published
[51.
Y=h,-,
in e ace
N
in e ace
M
in e ace
M-
1
in e ace
2
in e ace
1
C.12
[PI2
C.11
CPll
Fig.
1.
T ans e sal sec ion
o
a mul ilaye , mul iconduc o , coplana
s uc u e wi h dielec ic and magne ic aniso opy including losses.
and
[
p];,
espec i ely, a e in gene al complex quan i ies:
[=E
o
p
a
o
p=x
o
y.
The imagina y pa s o hese coe icien s display he e -
ec o subs a e losses.
Al hough mos pape s dealing wi h hese s uc u es
make use o he “ ull-wa e’’ app oach
[l],
[2],
he e is a
wide ange o si ua ions whe e he “quasi-TEM” assump-
ion is su icien ly co ec
[31.
The i s pu pose
o
he
p esen pape is o analyze he condi ions unde which
0018-9480/90/0800- 1059$01
.OO
0
1990 IEEE
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1060
IEEE
TRANSACTIONS
ON
MICROWAVE
THEORY
AND TECHNIQUES, VOL.
38,
NO.
8,
AUGUST
1990
he quasi-TEM app oach can be conside ed alid, consid-
e ing he p esence
o
lossy elec ic o /and magne ic ma-
e ials. A e ha , we p o ide an e icien me hod o
compu e he cha ac e is ics o mul is ip lines embedded
in a complex s a i ied lossy iso/aniso opic medium. Fo
his, we i s s a e he way o ob aining he [L] complex
induc ance ma ix pe uni leng h by sol ing an equi alen
elec ic p oblem and wo king ou an equi alen complex
capaci ance ma ix pe uni leng h,
[Ceq].
La e we apply
he Gale kin me hod in he spec al domain o ob ain
[Ceq]
and he complex capaci ance ma ix o he s uc-
u e,
[C].
P opaga ion cons an s and impedances a e eas-
ily compu ed om hese ma ices. The imagina y pa s o
he
[C]
and
[L]
ma ices will accoun o he shun
dielec ic and se ies magne ic losses, espec i ely.
A
ma hema ical appendix showing a echnique o imp o e
he con e gence o he in eg als in ol ed has been in-
cluded. In his way, a e y gene al and e icien algo i hm
o ea con igu a ions such as hose depic ed in Fig.
1
is
de eloped, making clea he ange o alidi y
o
such a
desc ip ion.
11. VALIDITY
OF
THE
QUASI-TEM
APPROXIMATION
As
was s a ed abo e, he analysis ca ied ou on he
s uc u e in Fig.
1
is based on he quasi-TEM app oxima-
ion. In [6] and [7], his app oxima ion is jus i ied o
lossless subs a es. In he con ex o his pape , i is
necessa y o know he limi a ions o he app oxima ion
when elec ic and/o magne ic losses canno be ne-
glec ed.
Fi s , in o de o a oid con usion in he no a ion o
aniso opic media, le
us
conside
wo
conduc o s,
cI
and
c2,
embedded in an inhomogenous, iso opic, and lossy
medium. Bo h dielec ic pe mi i i y,
E1( ),
and magne ic
pe meabili y,
i( ),
as well as he complex p opaga ion
cons an ,
y
=
p
-
ja,
will be complex quan i ies.
A
di-
mensional analysis o Maxwell’s equa ions will allow
us
o
es ablish he condi ions unde which he quasi-TEM ap-
p oach is easonable. By sepa a ing ans e sal and longi-
udinal componen s
o
ields and ope a o s (subsc ip
s ands o ans e sal componen and
1,
is he uni ec o
in he longi udinal di ec ion), Maxwell’s equa ions a e
w i en as
(1)
V,
x
E,
=
-
jw i( )Hz13
(2)
YE,
=
-
jyE,
-
jwi2( ) 3
X
H,
V,
X
H,
=
jw ( )E, ,
(3)
YH,
=
jyH,
+
jwi( ) ,
X
E,.
(4)
In o de o ca y ou he dimensional analysis,
(2)
is
in eg a ed along a pa h om any poin
a
o conduc o
c,
o any poin
b
o conduc o
c2,
and
(4)
is in eg a ed along
a pa h
C
su ounding a conduc o , namely
O=-jy/
h
E,.d +jw/h i( )H,.( ,Xd )
(5)
a
a
0
=
-
jy
$
H,.
d
-
jw
$
;(
)
E;
(
1,
X
d
)
.
(6)
C C
Fo ou pu pose, we conside
p,+
(llE,ll)d
(7)
(9)
@( )W3
x
d )
(l~( )l)(llE ll)d
(10)
whe e
x
N
y
deno es ha he o de o magni ude o
quan i ies
x
and
y
is he same,
(.)
deno es a e age
alue,
1.1
deno es modulus,
11.11
deno es ec o no m, and
d
is he ans e sal cha ac e is ic dimension o he line.
Combing
(5)
and (6) and aking in o accoun (7)-(10),
we ob ain
(I-4I)
l~ld(llE/ll)
(11)
IYI
-
wd(li2( )l)(IWl)
.
(12)
whe e
The e o e, we can conclude ha he ange o alidi y
o
he quasi-TEM app oxima ion,
(~Ez/)/(/~E,~/)
<<
1, is de-
e mined by
1
d
<<
(
13)
w
J<
I
i
(
1
I
) (
I
2
(
1
I)
Condi ion (13) is easily ex ended o he aniso opic case.
Fo
i ,
we ha e o subs i u e
i( )
o
C( )
in his exp es-
sion o he la ges elemen in he enso pe meabili y o
pe mi i i y.
Nex , we will conside sepa a ely he e ec s o dielec-
ic and magne ic losses on condi ion (13).
A.
Dielec ic
Lossy
Medium
In his case (13) becomes
1
d
<<
1/2
*
(14)
w
[
(
CL(
)
)(E(
1
)
{Z]
Two ypical limi si ua ions o en a ise in p ac ice. The
i s is
(a( ))2
<<
w2(E( ))*.
In his case (14) becomes
1
d
<<
(15)
wd(CL(
)
)
(4
)
)
Equa ion (15) is he usual exp ession o he case o
ossless medium.
The second limi si ua ion is
(a( ))2
>>
w2(E( ))2.
In his case,
1
d
<<
(16)
dw
(
CL(
1
)(a
(
1
)
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HVRNV
e
al.:
QUASI-TEM ANALYSIS 1061
+n
I
I
(b)
sec ion
o
a
mul iconduc o line.
Fig.
2.
(a) T ans e sal sec ion
o
a gene al in e ace.
(b)
T ans e sal
Thus i is ob ained ha
d
<<
(6( )),
whe e
6
is he skin
dep h.
The abo e si ua ions a e o a dielec ic wi h small
losses and a semiconduc o subs a e espec i ely. In he
i s case, condi ion
(15)
shows ha he quasi-TEM model
is alid when he ans e se dimensions a e much smalle
han he wa eleng h, such as in he lossless case. When
he subs a e is a semiconduc o , quasi-TEM p opaga ion
is in essence allowed i he skin e ec can be neglec ed.
B. Magne ic Lossy Medium
I he medium is a demagne ized homogeneous and
lossy e i e which is cha ac e ized by a scala pe meabil-
i y (see, o ins ance, [8] o an exp ession o his quan i y),
(13) becomes
eloci y o ligh
[6],
ia
As
is well known, i magne ic media a e in ol ed, his
simple ela ion does no hold. I is hen necessa y o sol e
a magne os a ic p oblem o compu ing he
[
Ll
ma ix.
Howe e , [9] and [lo] show he analogy be ween his
magne ic p oblem and o he equi alen dielec ic p ob-
lems when he magne ic medium is iso opic. The e o e,
hese wo ks a e no s ic ly applicable o he ea men o
impo an cases, such as pa ially magne ized o sa u a ed
e i es. In his sec ion, he equi alence is ex ended o
magne ic ma e ials cha ac e ized by a gene al pe meabil-
i y enso .
A.
Isomo phism Be ween Magne ic and Dielec ic P oblems
Le us conside a gene al in e ace be ween wo media
(see Fig.
2(a)),
cha ac e ized by he ela i e pe meabili y
and pe mi i i y enso s
[p Ii
and
(i
=
1,2), espec-
i ely, which can be complex quan i ies. Elec ic and
magne ic ields and po en ials obey he ollowing pa ial
di e en ial equa ions and bounda y condi ions:
1)
Elec ic Field:
4e
being he elec ic po en ial. Fo he dielec ic in e -
ace.
1
d
<<
whe e
y
is he gy omagne ic a io,
47M,
is he sa u a ion
magne iza ion, and
A
and
N
a e adjus able pa ame e s
in he imagina y pa o scala pe meabili y.
As
was expec ed, he accu acy
o
he quasi-TEM model
dec eases when equency inc eases o bo h kinds o
media. Ne e heless, in MMIC echnology he alue o
he ans e sal dis ances and he ange o use ul equen-
cies jus i y he use o he quasi-TEM app oxima ion o
mos p ac ical cases.
111.
EQUIVALENCE
BETWEEN MAGNETIC
AND
DIELECTRIC MEDIA
In
he ange o alidi y o he quasi-TEM assump ion, i
he medium is nonmagne ic, he induc ance ma ix is
ela ed o
[C,],
he acuum capaci ance ma ix, and
e,
he
Fo he conduc o in e ace,
2)
Magne ic Field:
Since ield
B
has only ans e sal
componen s,
B
can be w i en as
B
=
-
1,
X
V,A,,
whe e
A,
is he
z
componen o he ec o magne ic po en ial.
Wi h
[7Ii
=
[p l,:
',
he equa ion o his componen is
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1062 IEEE
TRANSACTIONS
ON
MICROWAVE THEORY
AND
TECHNIQUES, VOL.
38,
NO.
8,
AUGUST
1990
By means o ma hema ical manipula ion, he ollowing
iden i y can be p o ed:
Using (251, (24) can be ew i en as
.V,A,,,
=O
(i=1,2).
(26)
1
Bounda y condi ions o he magne ic ield a e, o he
magne ic in e ace,
B,,=
B,,-(I,Xn).V,A,,=(Z3Xn)~V,A,2
(27)
K,=
e2
-
[[T11.(~3
x
V,A,,)I
.
=
[[7712.(~3
x
YA,,)]
- .
(28)
Making use o (29, (28) becomes
Fo he conduc o in e ace,
B,,
=
0
=
(I,
X
n)
*V,A,,
=
0
(30)
n
x
[[TI~-(Z~
x
V,AZl)]
=
-
pOjs.
(31)
n
x
H,
=
j,
We can no e ha (31) is
Compa ing (19)-(23) wi h (26), (27), (29),
(30),
and (32),
we can no e ha hese se s o equa ions a e iden ical.
So,
we can s a e an isomo phism be ween
4e
and
A,
by
means o he ela ion
This isomo phism allows us o de e mine he magne ic
ield, when he magne ic medium is cha ac e ized by a
gene al pe meabili y enso , by sol ing an equi alen elec-
ic p oblem.
B.
Compu a ion
o
[L]
induc ance ma ix coe icien s a e de ined as
Le us conside he s uc u e in Fig.
2(b).
The complex
L..
=
-
(34)
ami
is he magne ic lux pe uni leng h associa ed wi h
conduc o
i,
which can be exp essed as
i
@mi=/V,A,i.dZ=
1
AZi.
(35)
Conduc o
1
has been conside ed he ze o magne ic
po-
en ial le el.
We can ew i e (34) as
whe e
$,
deno es in eg a ion a ound he j h conduc o
bounda y. On he o he hand,
i
he
[PI
ma ix is de ined
as he in e se o he complex capaci ance ma ix pe uni
leng h,
[PI
=
[Cl-',
we ha e
p..
=
-
These coe icien s can be w i en as
(37)
In subsec ion
III-A
i was p o ed ha he magne ic
p oblem can be educed o an elec ic equi alen one
simply by using he equi alen pe mi i i y gi en by (33).
So,
in o de o compu e
[Ll
and aking in o accoun he
o m o
(36)
and (381, we can conclude ha Li,
=
eopo
whe e
P,yq
is an elemen o he ma ix
[Peql
=
[Ceq1-',
[
Ceq]
being he complex capaci ance ma ix pe uni leng h
when he pe mi i i y enso s a e gi en by (33). We can
inally w i e
[
L]
=
Eopo[
Ceq]
-I.
(39)
The e o e, om he abo e exp ession, we can say ha
he calcula ion o he induc ance ma ix can be achie ed
by compu ing an equi alen capaci ance ma ix.
IV. COMPUTATION
OF
THE
COMPLEX
CAPACITANCE MATRICES
Unde he quasi-TEM app oxima ion, he s uc u e o
Fig.
1
is ully cha ac e ized by he complex capaci ance
and induc ance ma ices pe uni leng h,
[
C]
and
[
L].
The
physical meaning o he eal and imagina y pa s o he
[Cl
ma ix is discussed in [ll]. Simila ly, he eal pa o
[
L]
is he usual induc ance ma ix, and he imagina y pa
is a se ies esis ance ma ix ela ed o he magne ic losses.
Ne e heless, as s a ed abo e, he e alua ion o he [L]
ma ix is educed o he de e mina ion o an equi alen
capaci ance ma ix
[
Ceq].
In consequence, he analysis o
de e mine he cha ac e is ic pa ame e s o he line is
based en i ely upon he calcula ion o complex capaci-
ance ma ices. The e o e we will be in e es ed in he
complex capaci ance ma ix pe uni leng h o a s uc u e
such as he one shown in Fig. 1, bu wi hou magne ic
subs a es (since he magne ic pa o he p oblem is
eplaced by an elec ic equi alen p oblem). This ma ix
ela es he complex cha ge ec o
Q
o he ol age ec o
V
as ollows:
Q
=
[Cl-V.
I all he elemen s o he ol age ec o a e se o ze o
excep he j h, which is se o uni y, hen he j h column
o he capaci ance ma ix is equal o he cha ge ec o ,
Qi
=
Cpj
(p,j
=
1,.
. .
,
N,;
he supe sc ip
j
indica es which
conduc o is exci ed). The e o e, he e alua ion o [Cl is
he e alua ion o
N,
cha ge ec o s co esponding o
di e en exci a ions. To ind he cha ge dis ibu ion on
he M h in e ace and ob ain hese cha ge ec o s
Q',
we
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HORNO
e
01.:
QUASI-TEM
ANALYSlS
1063
ha e o sol e he ollowing ope a o equa ion:
&=4
p
=
0
ou side he su ace
o
he s ips (40)
whe e
p
=
p(x)
and
4
=
4(x)
a e, espec i ely, he cha ge
dis ibu ion and he po en ial on he M h in e ace ( he
po en ial is ze o excep in he exci ed s ip, which is se o
uni y), and
.9p
=
/mmG(x,
x’)p(x‘)
du‘,
G(x,
x’)
being he
G een’s unc ion o
y
=
h,.
The Gale kin me hod is used o sol e he abo e in e-
g al equa ion. I we conside
p=l
whe e
p,(x)+~
i
cp-~~xxcc,+-
W
WP
2
2
and
p,(x)
is expanded in he ini e base o unc ions
Tp
=
Ippo,ppl,.
*
.,pPN),
hen
p=l
n=O
(Subsc ip
p
o
q
will e e o he s ip and subsc ip
n
o
n
o he basis unc ion.)
Ca ying ou he Gale kin me hod, we ind he ollow-
ing sys em o linea equa ions:
b’
=
[
Tla’,
whe e
bJm
=
(pqm,4’)
a e he elemen s o he ec o
b’
and
qmpn
=
(pqm,9ppn)
he elemen s o he
[ l
ma ix
(q,p
=
1;.
.,
Nc; m,n
=
0;.
.,
N ).
The [ ] ma ix does no de-
pend on he exci a ion o he s ips. Acco ding o he
no a ion used,
N,
p=o
This equa ion
is
a gene al exp ession o he complex
capaci ance coe icien s in e ms o he complex G een’s
unc ion and he di e en basis unc ions. I he ollowing
well-known basis unc ions a e chosen:
(T,
being he Chebyshe polynomials) and he Pa se al
iden i y is used, we will ha e
and
whe e
J,(-)
s ands o he Bessel unc ion o o de
k.
In
o de o ’nd he complex G een’s unc ion in he spec al
domain,
G(a),
he algo i hms de eloped in [12] ha e been
sligh ly modi ied o include he e ec o a ully gene al
pe mi i i y enso (see Appendix I).
A his poin , i is e y impo an o conside he
nume ical e iciency in compu ing he
qmpn
pa ame e s.
I a di ec nume ical in eg a ion is ied ( o ins ance, he
Simpson o Rombe g me hod), he slow con e gence and
he oscilla ions
o
he in eg and in (45) make he compu-
a ion ime, in s anda d compu e s, oo long o become
p ac ical. The e o e, i is necessa y o apply some nume i-
cal in eg a ion scheme which p o ides a as compu a ion.
In his pape , an app op ia e asymp o ic ea men joined
wi h esidue calculus echniques (see Appendix 11) has
allowed us o educe he compu a ion ime. We ha e
ypically ob ained a a io be ween he ime employed by
di ec in eg a ion and by hese schemes o abou
50.
V.
RESULTS
In his sec ion, be o e gene a ing eliable da a, we a e
going o check he e iciency o he me hod epo ed in
his pape . Fi s , since ou analysis makes use o a nume -
ical scheme, we mus be su e ha he compu ed alues
con e ge when he numbe o basis unc ions inc eases.
Second, we need o e i y ha hese alues a e co ec .
To
ensu e he con e gence o he nume ical me hod, we
will s udy he in luence o he numbe o basis unc ions
on inal nume ical da a o a a ie y o signi ican s uc-
u es. The alidi y o he esul s will be checked by
compa ison wi h expe imen al and heo e ical ( ull-wa e)
da a ob ained by o he au ho s. Once all his has been
done, we can use ou p og ams wi h con idence o s udy
se e al s uc u es wi h semiconduc o and magne ic sub-
s a es.
A.
Con e gence Analysis
In o de o s udy he con e gence o he me hod, a
a ie y o con e gence pa e ns simila o he ones shown
in Fig. 3 ha e been gene a ed. The ela i e e o is
compu ed aking as e e ence alue he esul ob ained
wi h a la ge numbe o basis unc ions
(N
=
7).
Fo
p ac ical dimensions we ha e concluded ha no mo e
han h ee basis unc ions a e needed o ob ain
e y
accu a e esul s. Wo s cases always co espond o o -
diagonal elemen s o he capaci ance ma ices, and e en
o hese elemen s e o s a e below 3%. The la ge he
absolu e alues, he smalle he ela i e e o s; he e o e,
diagonal elemen s a e always compu ed wi h an e o o
less han 1%. On he o he hand, we ha e ound ha
odd-o de unc ions a e equi ed o ake in o accoun
s ongly coupled s uc u es, and e en-o de ones a e mo e
signi ican o la ge
W/h,
a ios ( o ins ance in Fig. 3).
B.
Nume ical Resul s and Discussion
LL
We now compa e ou alues wi h expe imen al da a,
./s;I.(
a?)G(a)~,(
aT)eja(cq-c
)
da
(45)
when hese a e a ailable, and wi h heo e ical ones ob-
ained by ull-wa e analysis
[2],
[13]. Ou pu pose is
o
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1064
IEEE
TRANSACTIONS
ON
MICROWAVE THEORY
AND
TECHNIQUES, VOL.
38,
NO.
8,
AUGUST
1990
show how nowadays in mos in e es ing p ac ical cases
o
MMIC echnology, a ull-wa e analysis is no necessa y o
ob ain accu a e alues. The ull-wa e analysis is much
mo e in ol ed han he quasi-TEM one, equi ing mo e
compu e capabili y. Wi h he asymp o ic beha io we
ha e in oduced, i is possible o use a PC, and he
scheme epo ed he e could be sui able o CAD pu -
poses. Ne e heless, a igo ous ull-wa e analysis mus be
ca ied ou in o de o es ablish he equency beha io
when quasi-TEM assump ions a e no alid. Nex , we
p esen examples o pa icula s uc u es. Fi s , mic os ip
on
demagne ized lossless e i e subs a es is analyzed in
Fig. 4. We can see ha in p oximi y o he esonance
equency, he quasi-TEM app oxima ion is no ye alid.
This ac was expec ed, because a ound his equency
condi ion (13) is no ul illed. Fo he es o he equen-
cies, he ag eemen is good.
In
Fig.
5
we ep esen he
c oss sec ion o a coupled slow-wa e MIS con igu a ion.
Fo his s uc u e, we plo he slow-wa e ac o and he
a enua ion cons an in Fig.
6.
In his igu e, some expe i-
men al esul s co esponding o he single s ip case
[2]
ha e been included o compa ison pu poses. E en- and
odd-mode cha ac e is ics end o hese esul s when he
s ips a e weakly coupled
(s
-
m).
The ag eemen wi h
bo h analy ical ull-wa e and expe imen al da a is
e y
good in he whole equency ange conside ed.
Second, we a e in e es ed in he applica ion o ou
p og ams o slow-wa e e omagne ic mic os ip lines such
as hose discussed in [4] (FM and FMS). In [4], hese
s uc u es a e s udied by means
o
a quali a i e pa allel-
pla e wa eguide model. Since he dimensions and cha ac-
e is ics o he s uc u es analyzed in ha pape obey
condi ion (131, we can pe o m a mo e accu a e s udy by
using he quasi-TEM app oxima ion. Fig.
7
shows he
slow-wa e ac o and he a enua ion cons an o mi-
c os ip lines in which he lowe laye in Fig.
5
is consid-
e ed o be ei he a demagne ized e omagne ic subs a e
(FM) o a demagne ized e omagne ic semiconduc o
subs a e (FMS). We ha e chosen se e al alues o he
cons an
A
in
(17)
( he la ge he alue o
A,
he bigge
he losses in he e omagne ic subs a e) o show how he
magne ic losses a ec he abo e ea u es. Ou esul s
exhibi he same quali a i e shape as hose in [4], bu we
can no ice ha o usual alues o
A
(in [8], A
-
lo-*),
hese s uc u es do no o e he impo an slow-wa e
e ec s epo ed in
[4].
Magne ic losses should be much
g ea e han he ac ual ones o ob ain ou s anding slow-
wa e beha io .
Also,
we ha e compu ed he slow-wa e
ac o and he a enua ion cons an o h ee di e en
mic os ip lines: MIS, demagne ized FMS, and sa u a ed
FMS. (Exp essions o he Polde enso when he sub-
s a e is magne ically sa u a ed pa allel o he
z
axis can
be ound in [14].) Sligh di e ences be ween he cu es
co esponding o he magne ic s uc u es and he MIS
one ha e been ound o mos geome ies. In Fig.
8,
his
can be checked o a pa icula geome ic con igu a ion.
We can no e how he shape o he cu es is due mainly o
he conduc i i y
o
he subs a e. The e o e, he me e
p esence o magne ic losses in he subs a es has li le
in luence
on
he imp o emen o slow-wa e e ec s wi h
espec o he p esence
o
elec ic ones. In p oximi y o
he esonance equency
(0.6
GHz o he demagne ized
FMS and
6
GHz o he sa u a ed FMS), he pe meabili y
enso elemen s ha e highe alues. This leads o he
disc epancies obse ed in his ange o equencies.
VI.
CONCLUSIONS
A sho e iew o he mos ele an poin s de eloped in
A
quali a i e heo e ical jus i ica ion o he quasi-
TEM model has been achie ed. This conce ns he
applica ion o his model o he mos p ac ical MMIC
ansmission lines.
The compu a ion
o
he complex induc ance ma ix
pe uni leng h, o s uc u es cha ac e ized by a
gene al pe meabili y enso , is educed o he compu-
a ion o an equi alen complex capaci ance ma ix.
A simple and nume ically e icien algo i hm o com-
pu e he complex capaci ance ma ix o e y gene al
mic os ip-like con igu a ions has been de eloped.
This algo i hm could be used as a CAD ool, since an
app op ia e nume ical ea men makes i possible o
achie e ole able CPU imes.
The e iciency o he algo i hm has been demon-
s a ed by he ag eemen wi h expe imen al da a and
wi h heo e ical esul s ob ained by means o much
mo e in ol ed ull-wa e compu a ions. Slow-wa e
s uc u es such as MIS, FM, and
FMS
con igu a ions
ha e been success ully analyzed wi h his algo i hm.
his pape ollows.
APPENDIX I
Following [12], he G een’s unc ion in he spec al
domain o he s uc u e shown in Fig.
1
is
whe e
j
=
2;.
.,
M
;(a)
=
2IdQ)
exp ess he con ibu ion o he laye s below he in e ace
M,
and
j=
N-2;..,M
Hh-l(a)
=
EN-1,N-La)
exp ess he con ibu ion o he laye s abo e he in e -
ace
M.
The meanings o he unc ions which appea abo e a e
explained in [151. The gene ali y o he pe mi i i y en-
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HORNO
e
al.;
OUASI-TEM ANALYSIS
hl
1065
E 1
*=*p
il
0.8
1
:;L.Ji
Imagina y pa s
I'
i;.,
0.1
0
-0.1
-0.2
-0.3
-0.4
I
I
I I
I
1
1
23456
Numbe o basis unc ions
Fig.
3.
Rela i e e o s in 1% o eal and imagina y pa s o capaci-
ance elemen s. F eq
=
1 GHz.
h,
=
250
ym.
h,
=
1
ym.
w,
=
160
ym.
w2
=
100
pm.
w3
=
120
ym.
s1
=
s2
=
190
ym.
E,,
=
12,
=
4,
)
0=8
(R.mm)-',
(-)
C
I,,
(-----)
C
12.
c,,,
(.
. . . . . .
. . .
.
.
.
.
.)
c
)
c23.
(
. . . . .-
22,
-
I
I
I
+
3
4
5
6 7
8
F eq.
(GHz)
Fig.
4.
Phase eloci y o a mic os ip o e e i e demagne ized
loss-
less subs a e G-1001.
w/h
=
0.431,
4 MS
=
1210 G.
E,
=
15.5.
(-)
P esen me hod,
(-
- -
-1
1131,
(+
+
+)
expe imen al alues.
16
b
,o
9
12
c.
0
0
I
0
i7i
8
c
0.1
:
E:
E-
CI-
a-
d
0.01
U
-
1
o-~
.
0.01
0.1
1
(a)
F eq
(GHz)
L..
,
.
a.....
1
I1
0.0
1
0.1
1
(b)
F eq
(GHz)
Fig.
6.
(a) Slow-wa e ac o and (b) a enua ion cons an o he cou-
pled mic os ip MIS con igu a ion shown in Fig. 5, wi h
w
=
160
pm,
h,
=
250
ym,
h2
=
1
ym,
(
=
5.10-3
(R.mm)-',
E,,
=
E,~
=
12,
s
=
800
ym
o (1) and
s
=
160
ym
o (2).
(-)
P esen me hod,
(-
- -
-)
[2],
(+
+
+)
o
(0 0
0)
expe imen al alues.
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1066
IEEE
TRANSACITONS ON
MICROWAVE
THEORY
AND
TECHNIQUES,
VOL.
38,
NO.
8,
AUGUST
1990
1
1
10
F eq
(GHz)
(a)
1
10
1
10
F eq
(GHz)
(b)
1
10
F eq
(GHz)
(d)
I
F eq
(GHz)
W)
Fig.
7.
Fo he con igu a ion shown in Fig.
5,
wi h
h,
=
100
pm,
h,
=
1
pm,
w
=
200 pm,
s
=
100
pm,
E,,
=
E,,
=
15.5,
~TM,
=
0.2
kG,
and
N
=
1.
(a) Slow-wa e ac o and
(b)
a enua ion cons an o a coupled mic os ip demagne ized FM.
(c)
Slow-wa e ac o and
(d)
a enua ion cons an o a coupled mic os ip demagne ized FMS:
(
=
1
(0
.mm)-
I.
(-
- -
-)
A
=
100,
(--.--.--.-)
A
=
1,
(-)
A
=
0.01.
so , whe e all he componen s can be di e en and com-
plex, leads o some di e ences wi h espec o he exp es-
sion o hese unc ions gi en in
[15].
We ha e now
wi h
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HORN0
e
al.:
QUASI-TEM
ANALYSlS
30
i
25
20
3
0 0
.c
'5
1
1
0
10
0
5
0.01
0.1
1
10
F eq
(GHz)
(a)
1067
whe e
1
Kas
=
jK
+
e;Sm
+
EK+~S~+~
and
L
This asymp o ic beha io o he G een's unc ion is used
in Appendix I1 in o de o accele a e he con e gence o
in eg als.
APPENDIX
I1
In o de o ob ain he alue o he coe icien s
qmpn,
i
is necessa y o calcula e he ollowing in eg als:
We choose as he asymp o ic beha io o he in eg and,
10-51
.
. .
.....I
.
.
.
.....I . . .
..A
0.01
0.1
1
10
F eq(CHz)
(b)
Fig.
8.
(a) Slow-wa e ac o and
(b)
a enua ion cons an o di e en
slow-wa e mic os ip con igu a ions o
wo
di e en alues
o
con-
duc i i y. In Fig.
5
h,
=
100
pm,
h,
=
1
p n,
w
=
200
pm,
s
em,
and
E,,
=
e z
=
15.5.
Fo demagne ized FMS, 4wMS
=
200
G.
A
=
100.
N=
1.
Fo
sa u a ed FMS,
H,=
2500
Oe.
AH
=
75
Oe,
(-)
MIS,
(-----I
demagne ized FMS,
(-.-.-.-
)
sa u-
a ed FMS.
and
1
--Si,;-
,(
a)
=
-
€;,asi
csch(
aSihi)e-jnR~h~.
EO
On he o he hand, i is easy o check ha he asymp-
o ic beha io o he G een's unc ion is
o
a
*m
6( )--
Kas
a
wi h
Kas
i
q=p
%aco h(a$)
G?,< )
=
i
q p.
1
a
(Kas
is de ined in Appendix I.) Now we will sepa a e he
compu a ion o
Zqmpn
in o
wo
pa s in such a way ha
Zqmpn
=
Zimpn
+
Zlmpn.
The i s e m will be
I will con e ge as because he in eg and is p ac ically
ze o when
lala4.
Then, by means o any nume ical
in eg a ion me hod, he compu a ion ime o his e m is
e y sho . The second e m is
Z;mpn
=
/~mJ,(a~p/2)Ga,(
a)J,(
awq /2)eja('q-'0) da,
and
wo
cases a e conside ed. The i s is whe e
q
=
p:
This in eg al can be no malized and hen i will no
depend on he dimension o he s uc u e,
so
in eg als o
his kind a e compu ed jus once and hen abula ed. The
second case
is
whe e
q
#
p:
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