1002 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-35, NO. 11, NOVEMBER 1987
Capaci ance and Induc ance Ma ices o
Mul is ip S uc u es in Mul ilaye ed
Aniso opic Dielec ics
FRANCISCO MEDINA AND MANUEL HORNO, MEMBER, IEEE
Ab. ac —In his pape we p esen auni ied a ia ional app oach o
de e mine he capaci ance and induc ance ma ices o gene alized mul i-
s ip sys ems embedded in amul ilaye ed iso/aniso opic dielec ic lossless
medium. The analysis is ea ned ou in he spec a domain o ake
ad an age o p e iously ob ained ecu ence ela ions which calcula e he
G een’s unc ion in he spec al domain. The me hod leads o alow-o de
sys em o linea equa ions, which is shown explici ly. Examples and
compa ison wi h p e iously published esul s ha e been included.
I. INTRODUCTION
T
HE PHYSICAL beha io o uni o m mul iconduc o
ansmission lines when hey a e used in he design o
high- equency elec ical il e s and couple s is well unde -
s ood [1]. I is also well known ha a he low- equency
end o he spec um, aquasi-TEM app oxima ion can be
assumed e en when inhomogeneous and/o aniso opic
dielec ics a e in ol ed [2, 3]. Unde his assump ion, he
p oblem educes o he de e mina ion o he Maxwell
capaci ance ma ix o he sys em. Pa icula in e es has
been ocused on he plana s uc u es used in MIC ech-
nology. Nume ous pape s dealing wi h quasi-s a ic p opa-
ga ion in wo- o h ee-conduc o s ip sys ems can be
ound in he li e a u e [4–8]. The mo e gene al p oblem o
de e mining he Maxwell capaci ance ma ix o agene -
alized mul is ip sys em has been ea ed o homogeneous
[9] and nonhomogeneous [10] media. Howe e , o plana
s uc u es embedded in amul ilaye ed dielec ic medium,
he spec al-domain analysis seems o be especially sui -
able [11, 12]. Recen ly an e icien ecu ence algo i hm o
ob ain he G een’s unc ion in he spec al domain associ-
a ed wi h an aniso opic mul ilaye ed dielec ic medium
has been used o analyze se e al coplana [13] and non-
coplana [14] s uc u es. Simila wo k dealing wi h iso-
opic dielec ics has also been ecen ly published [15]. The
aim o he p esen pape is o apply a ia ional analysis in
he spec al domain in o de o ob ain he capaci ance and
induc ance ma ices ha cha ac e ize amul is ip sys em
o he ype shown in Fig. 1. The me hod leads o a
low-o de sys em o linea equa ions whose coe icien s
and independen e ms a e explici ly shown in he wo k.
Manusc ip ecewed Ap il 20, 1987; e ised July 2, 1987. This wo k
was suppo ed by he Comisin Aseso a de In es igaci6n Clen i ica y
T4cnica (P ey. 3028/83), Spain.
The au ho s a e wi h he Depa amen o de Elec ncldad yElec onics,
Facul ad de Fisica, U u e sidad de Se illa, 41012, Se illa, Spain.
IEEE Log Numbe 8716592,
@
~——––— ~–———=
I%%1
I
ig H, I
LO_______ ———______L ________— d
63
(a)
@
~= ——————
—– ——– ———–––– ——– ——— ——–~
,E% H% I
JL
.“ .
‘ ..
]q‘
HI
L—— ———_J_______________________J
IbI
(b)
Fig. 1. (a) C oss sec ion o gene ic mul ilaye ed coplana s ip lines. (b)
C oss sec ion o agene al noncoplana mul is ip sys em. Uand B
deno e elec ic wall, magne ic wall, o o~en bounda : L and R
deno e elec ic o magne ;c wall. ‘J
The choice o basis unc ions is discussed and se e al
examples a e in oduced o show he me hod’s s eng h.
II. STATEMENT OF THE PROBLEM:
SPECTRAL ANALYSIS
Conside asys em o s ip lines in as a i ied lossless
dielec ic egion enclosed in ase o ec angula bounda y
condi ions. Conduc o s can be coplana (Fig. l(a)) o
0018-9480/87/1100-1002$01.00 01987 IEEE
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MEDINA AND HORNO; CAPACITANCE AND INDUCTANCE MATRICES
non-coplana (Fig. l(b)) s ips. The op and bo om shields
can be conside ed elec ic walls, magne ic walls, o open
bounda ies. So, open geome ies can be simula ed by
choosing an open bounda y in he op shield and aking
“b“ in Fig. 1la ge enough. The pe mi i i y o he j h
dielec ic laye is adiagonal enso ~j. (I is impo an o
ake in o accoun he dielec ic aniso opy om ap ac ical
and heo e ical poin o iew [16].) I nonmagne ic ma e i-
als (p, =p~)a e assumed, he p oblem will be educed o,
he de e mina ion o he capaci ance ma ix o he s uc-
u es unde conside a ion wi h ([ C,j]) and wi hou ([C:])
subs a es [3]. These wo-dimensional elec os a ic p ob-
lems can be eadily o mula ed in he spec al domain
ins ead o wo king in he space domain [13] –[15]. The i s
s ep in he solu ion p ocess is o de e mine G een’s unc-
ion in he spec al domain associa ed wi h he s uc u e.
When se e al dielec ic laye s a e in ol ed, his ask may
become e y edious. Ne e heless, e y simple ecu ence
exp essions o de e mine G een’s scala unc ion o
coplana s uc u es [13] (Fig. l(a)) and G een’s unc ion
ma ix o noncoplana ones [14] (Fig. l(b)) ha e been
published. In his way, he Fou ie ans o ms o he
po en ial unc ion ~, and he su ace cha ge ~, on he
in e aces whe e he s ips a e loca ed a e ela ed ia
~( z)=G.(lZ) -Fs( z) (coplana s ips) (la)
[~(n)] =[G,.(n)] [p,(n)] (noncoplana s ips).
(lb)
These ela ions will be used in he ollowing pa ag aphs
o de i e he induc ance and capaci ance ma ices.
HI. VARIATIONAL ANALYSIS
In amul iconduc o sys em such as he one desc ibed in
he p e ious sec ion, cha ges and po en ials on he s ips
a e ela ed by means o he ma ices [C] and [P] in he
ollowing way:
[Q,]=[c,, ][L]
[K] =[~,,][Q,] [8,] -1=[c,,]. (2)
The i, j h elemen o he [C] ma ix is he ee cha ge pe
uni leng h on he i h conduc o when all conduc o s
excep he j h one a e g ounded and he j h conduc o is
cha ged o apo en ial o 1V. Hence, he elemen s o [C]
can be de e mined by ela ing he cha ge on he conduc-
o s o hei po en ials. Howe e , in ou wo k, we will ake
adi e en app oach, one based on ene gy calcula ions. In
his way, we can ake ad an age o he a ia ional na u e
o he ene gy exp essions. Fo an a bi a y dis ibu ion o
cha ges on he s ips, he elec os a ic ene gy s o ed in he
sys em pe uni leng h can be exp essed as
u=+[Q,][~,J][Qj] =+i[Izl[c ,][TJl (3)
No e ha [P] can be exp essed in e ms o he elec o-
s a ic ene gy o di e en si ua ions in he ollowing way:
P,, =2U’’/Q:
P,, =(UiJ– U“- UjJ)/Q,Q,, i+j (4)
1003
whe e U‘J is he elec os a ic ene gy s o ed in he sys em
pe uni leng h when all conduc o s excep he i h and j h
ones a e isola ed and discha ged and he i h and j h
conduc o s suppo cha ges Q, and Qj, espec i ely. Now,
()
ou p oblem is he e alua ion o he ene gy o he N+“c
di e en dis ibu ions o cha ges necessa y o compu e ill
he elemen s o he [P] ma ix (and consequen ly he [C]
ma ix).
The elec ic ene gy can be exp essed in he Fou ie
domain as ollows:
u=: g’[F+’z)]’[ i$ (n)][ i (n)]
n—1
(5)
M=numbe o in e aces wi h conduc ing s ips
(M= 1 o coplana s ip5)
s, =l,. ... M
whe e P,(n) is he Fou ie ans o m o he su ace cha ge
densi y on he j h in e ace ( he exp essions (1) ha e been
used in (5)). Aknowledge o hese dis ibu ions o cha ges
is only possible in e y limi ed cases. Howe e , he s a-
iona y na u e o (5) allows us o apply Ri z’s minimiza-
ion p ocedu e o ob ain a e y accu a e es ima ion o U.
Le n~be he numbe o conduc ing s ips lying on he
s h in e ace. The su ace cha ge densi y can be expanded
in ase o basis unc ions:
(0 elsewhe e
(6)
whe e
b
J()
p:. x.U!x=l
and
b
~()
P:p x.dx=O.
The a ia ional coe icien s a~P a e ob ained by minimi-
za ion o he elec os a ic ene gy in (5) (Ri z’s p ocedu e).
This p ocess leads o he ollowing sys em o linea equa-
ions:
Mn, #
whe e he coe icien ma ix is exp essed in e ms o he
Fou ie ans o ms o he basis unc ions in he ollowing
way: m
and he independen e ms column is ela ed o he pa -
icula dis ibu ion o cha ged and isola ed s ips o be
conside ed o each ene gy calcula ion. To de e mine he
in e se o he capaci ance ma ix om (4), i is clea ha
we mus only conside cases wi h one o wo s ips cha ged
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1004 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-35, NO. 11, NOVEMBER 1987
I“s
~ss —II
.’ 1(
‘~... ––—— b1
I1
Fig, 2. De ail o he s h in e ace o he s uc u es in Fig. 1.
and he es isola ed and wi hou cha ge. The e o e, i he
l h s ip o he u h in e ace and he m h s ip o he u h
in e ace a e cha ged, he independen e ms o he equa-
ion sys em will be gi en by
(9)
n=l
Once (7) is sol ed, he elec ic ene gy pe uni leng h can
be compu ed om he ollowing exp ession:
Exp ession (10) is e alua ed o all possible alues o
~,~=l,... ,Mand 1,m=1,. . . . n,; he [P] coe icien s
can hen be ob ained by using (4). This p ocess mus be
ca ied ou o he s uc u e wi h and wi hou dielec ic
laye s o ob ain [Ci,] and [C;]. In his way, he capaci ance
and induc ance ma ices a e ob ained and he sys em is
comple ely cha ac e ized unde aquasi-TEM ope a ion.
IV. TRIAL FUNCTIONS
Be o e gene a ing nume ical esul s, a good choice o
ial unc ions is necessa y. Di e en se s o basis unc-
ions ha e been conside ed in his wo k. F om his s udy,
we conclude ha an adequa e choice o ial unc ions
mus ake in o accoun he singula i ies o he cha ge
dis ibu ions a he edges o he s ips.
A e y simple se o unc ions is o conside acons an
e m co e ing he o al cha ge on he s ip and wo e ms
o he singula i ies a he edges (see Fig. 2):
[1
h(x) =; FL(x)= 1-2 “j)
sT
~—
-&i 3
w:
,2(+=(A-k’ (11)
This choice is good enough o yield e y accu a e esul s
o coplana s ip s uc u es. This ac has been concluded
om sys ema ic compa isons wi h p e iously published
esul s and wi h he da a gene a ed by using amo e
comple e se o basis unc ions such as he ollowing one
(Fig. 2): {1 1/2
2
p:o(x)=— W$’
[1
~_ (XL) 2
(Wy;)
Hp (-L))
P:p(x) =;P:o(x)” Cos —’ x x :
-COS(;)JO(;)). (12)
In agene al case in ol ing b oadside couplings, he
cha ge dis ibu ion is mo e complica ed and one mus use
he expansion in (12) o ob ain good esul s. The con e -
gence o he Fou ie se ies is slowe wi h hese unc ions,
bu he accu acy in ce ain cases is imp o ed subs an ially.
In o de o imp o e he con e gence, he asymp o ic pe -
o mance o he se ies appea ing in (8) and (9) has been
conside ed in he compu e p og ams. In his way, he
numbe o Fou ie e ms necessa y o ob ain he desi ed
accu acy is d as ically educed and bo h choices o ial
unc ions a e sui able o quick calcula ions. CPU ime is
less han one second pe conduc o s ip o mul is ip
s uc u es on a VAX1 1/780 compu e .
V. NUMERICAL RESULTS
The heo y p esen ed in his pape was used o w i e a
compu e p og am which p o ides he capaci ance and
induc ance ma ices o asys em o coupled s ips em-
bedded in amul ilaye ed iso opic o aniso opic medium.
F om hese ma ices, mode impedances and phase eloci-
ies a e eadily ob ained [3]. In o de o check he esul s,
we analyzed se e al pa icula s uc u es p e iously s udied
by o he au ho s. In he ollowing pa ag aphs, we show
hese compa isons and some new esul s.
In o de o show he e ec o a good choice o basis
unc ions we compa e ou esul s wi h he ones epo ed
by Koul and Bha [12]. (These au ho s p o ide ‘a use ul
me hod o analyze awide a ie y o symme ical s iplike
s uc u es on aniso opic subs a es.) In Fig. 3we compa e
he esul s ob ained in [12] o he in e elec ode capaci-
ance o acoplana s uc u e (elec o-op ic modula o )
wi h he ones calcula ed by means o he me hod in his
pape . We show wo cu es by using he ial unc ions in
(12) wi h n=Oand n=5. In bo h cases we ound a e y
good ag eemen o na ow s ips. Howe e , asigni ican
disc epancy is obse ed o wide ones. In ou opinion, is
disc epancy is due o he ac ha he ial unc ion used in
[12] canno con enien ly ep esen he cha ge dis ibu ion
on wide s ips. These cu es we e also compu ed by using
(11) and he disc epancy was less han 0.5 pe cen o all
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MSDINA AND HORNO: CAPACITANCE AND INDUCTANCE MATRICES
w(}D)
1005
Lo 80 120 160 . .. .
...
S.mn
~“’””4
..”, --
‘oo .7
,Ooc
40 80 120 160 20
s(pm)
Fig. 3. In e elec ode capaci ances o elec o-op ic modula o as a
unc ion o he wid h o he s ips and hei sepa a ion. ---- and —
a e he esul s epo ed in [12]. –.– .– ep esen ou compu a ions wi h
one basis unc ion, and . . . . ep esen ou compu a ions wi h i e
basis unc ions.
dimensions. This ac con i ms ha (11) is a good se o
basis unc ions o coplana s ips.
The s uc u es shown in Fig. 4(a) and (b) we e analyzed
by Ki azawa and Mi a [7] and by Ki azawa and Hayashi
[8]. The e ec i e dielec ic cons an s o Cand n modes o
hese s uc u es a e ep esen ed in Figs. 5 and 6. As we can
see, he esul s in [7] and [8] a e in good ag eemen wi h
ou da a.
Table Icompa es ou da a wi h hose ob ained by Wei
e al. [10] o he case o wo coupled mic os ip lines
be ween wo g ound planes in ahomogeneous and iso-
opic medium. Table II shows he esul s ob ained o a
s uc u e wi h h ee s ips embedded in a h ee-laye ed
dielec ic medium be ween wo g ound planes. This s uc-
u e was also s udied by Wei e al. in [10]. We compa e
bo h esul s in Table II. Signi ican di e ences a e de-
ec ed, bu hese a e wi hin he ma gin o e o gi en in
[10].
Mo e exac calcula ions we e p e iously epo ed by
Kammle on mul iconduc o s uc u es in ahomogeneous
and iso opic medium [9]. In Table III, we show he esul s
ob ained in he analysis o apai o asymme ical coupled
s ips be ween wo g ound planes. An excellen ag eemen
(wi hin +0.002 in all cases) was ound. (I mus be em-
phasized ha he Kammle esul s a e exac o wi hin
~0.001.) These da a we e gene a ed wi h he ial unc-
ions in (12). A ypical disc epancy o 1pe cen o 5
pe cen was obse ed by using (11). Howe e , he se o
I
(a)
I)’
h
Zd
h
i
(b)
Fig. 4. (a) C oss-sec iona iew o asymme ical coupled s ip lines. (b)
C oss-sec ionaJ iew o b oadside-coupled s ip lines o unequa wid h.
12.0 ,
n
----- ------- --------- --------- --------- —--..-=
P...-....
10.0 COUPLED STRIPS WITH OVERLAY
L—C-mode p esen wo k
8.0 ‘--’~’/
+!
~b---
u
,0 /cOu’’’:_::::::::>>>~##=#=”z”==z”==
-------------
---------------
------------
---------
---.” -_
4.0 .-.. ......... ............ ~
ICOUPLED SUSPENDED STRIPS
2.0 L..—dL_LL_LL—L—_J
0.1 1.0 4.0
Fig. 5. Quasi-s a ic cha ac e is ics e sus W2 WI, (The dimensions and
dielec ic cons an s ha e been akcn mn [7j:)
basis unc ions in (11) allows us o ob ain e y good esul s
in he case o coplana s uc u es. In his way, Table IV
compa es ou esul s o he case o se e al mul iconduc-
o , coplana s uc u es wi h hose epo ed in [9]. The
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1006 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-35, NO. 11, NOVEMBER 1987
8.0, TABLE II
VALUES OF THE CAPACITANCE AND INDUCTANCE COEFFICIENTS
FOR THREE COUPLED STRIP LINES IN ATHREE-LAYERED DIELECTRIC
BETWEEN GROUND PLANES AND COMPARISON WITH THE DATA
REPORTED IN [10]
h
6.0
l
—P esen wo k
~e , ll .Re . [81
o2 4 6 8
SI!J ~
!- .6 ~.15
J’s
Ej. 4.2 E. ‘-. 2~ ~
1—.6< .2
I4x
(a)
Re . (10 )
0.4900 x10 -9
4.5737 x10 -12
-0.6457 x10-10
0.2459 x10
-0.6138 x10::0
0.2865 x10 -9
0.7773 x10-’0
-0.1036 x10 -12
-0.7193 x10-”
0.5212 x10-10
-0.9766 x10 -11
0.3676 x10-’0
P esen win-k Di e ence
140 h‘cl -9
0.5115 x10
-12
–0.5929 x10
-0.6972 x. 10-10
0.2572 X10-9
–0.6659 x10 –lo
0.2977 x10 -9
0.8110 x10-10
-0.1075 x10-’2
-0.7812 x,0-’1
-10
0.5445 x10
-0.1C158 x10-10
0.4040 x10-10
4.3 %
3.3 %
7.7 %
4.5 %
8.1 %
3.8 %
c11
c12
c13
c22
C23
c33
c011
c012
c013
c022
c023
c033
L11
L12
L13
’22
L23
L33
c
60
SINGLE SIJSPENDED
sTRIPLINES
. — W=W1
s
— —— w~= W2
4.2 %
3.7 %
6.2 %
4.4 %
8.9 %
4.1 %
04 6 e
S/wl
(b)
Fig. 6. (a) E ec i e dielec ic cons an e sus S/ W1. (b) Cha ac e is ic
impedances e sus ,S/ WI. (Dimensions and dielec ic cons an s ha e
been aken om [8].)
-6
0.1456 x10
-8
0.5630 x10
0.2s44 x10-7
0.2240 x10 *
0.5762 X10 -7
-6
0.3065 x10
-6
0.1403 x10
o.58a7 xlC 4
-7
0.2862 X10
0.2157 X10+
-7
0.5%15 x10
0.2963 x10 -6
3.9 %
4.5 %
0.6 %
3.8 %
1.0 %
3.4 %
TABLE I
VALUES OF THE CAPACITANCE AND INDUCTANCE
COEFFICIENTS FOR Two COUPLED MICROSTRIP
LINES BETWEEN Two GROUND PLANES IN ATABLE III
ASYMMETRICAL BROADSIDE-COUPLED STRSPSBETWEEN GROUND
PLANES: COMPA USON WITH THE CAPACITANCE COEFFICIENTS
REPORTED BY IQMLER [9]
HOMOGENEOUS MEDIUM AND COMPABJSON WITH
THE DATA REPORTED IN [10].
1
-3 +’+ ,’:;3
1.5
Re . ~10~ P esen wo k
–9 -9
Cll 0.5356 X10 0.5320 x10
c-0.9250 x10-11 -0.1008 x10 -1(
12
c0.7834 X10 -9 0.7790 x10 -9
22
c0.5466 x10-10 0.5600 x10-1(
011
c-0.9439 . 10-12 -0.1062 x10-’”
012
c022 0.7994 x10-’0 0.8200 x10-1[
L0.2033 X10-6 0.1987 X10-6
11 -8 -8
’12 0.2401 X10 0.2570 X10
-6 -6
L22 0.1390 x10 0.1357 x10
3
)
)
1
3
‘2
1.0
0.8
0.6
0.5
0.4
0,3
0.2
0.1 I
0.05
–4
C1l/.o
9.136
8.862
8.434
8.179
7.904
7.611
7.300
6.956
6.746
c12/ o
-5.355
-4.743
-3.953
-3.518
–3.064
-2,592
-2.100
-1.561
-1.234
C2240
9.136
7.760
6.301
5.558
4,807
4.048
3.268
2.424
1.916
.
c/.
11 ~
‘1
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
I
*I●
c12/e oC22]’0
I
–5.352 9.133
-4.742 7.759
–3.953 6.301
–3.518 5.557
-3.063 4.807
–2.592 4.046
-2.099 3.264
-1.561 2.423
-1.235 1.915
9.133
8.860
8.434
8.179
7.903
7.610
7.299
6.956
6.746
w~e ence ~9] P esen wo k
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MEDINA AND HORNO: CAPACITANCE AND INDUCTANCE MATMCES 1007
TABLE IV
CAPACITANCE COEFFICIENTS COMPUTED FOR SEVERAL COPLANAR
CONFIGURATIONS AND COMPARISON WITH RESULTS IN
KAWLER [9]
—. —
— — —
P esen wo k (Re . [$
cll/s O=2.4617 (2.4618)
C1lAO =2.8878 (2.8!388)
c12/c ~=-1 .0372 (-1 .0379;
cll/Fo =2.8903, (2.8914)
C22/E ~=3.2908 (3.2915)
C121E ~=-1 .0060 (-1 .0064)
C13/E ~=-0.0834 (-0.0841)
C1l/FO =2. E!904 (2.8914)
C22/C0 =3.2921 (3.2938)
CJFO =-1 .0057 (-1 .0061)
C231F0 =-0.9766 (-0.9767)
C131C0 =-0.0788 (-0.0795)
C141E0 =-0.0124 (-0.0125)
C1l/sO =2.8904 (2.8914)
C22/E0 =3:2921 (3,2939)
c33/co =3.2943 (3.2961)
CIZ/CO =-1 .0057 (-1 .0061)
C2J=0 :-0.9763 (-0.9764)
c13/Eo =-0.0789 (-0.0794)
c24/F ~=-0.0745 (-0.0751)
c14/co =-0.0117 (-0.0117)
c15/Eo .-0.0020 (.0.0020)
di e ence is less han 0.05 pe cen o he wo s o he
cases s udied.
Finally, ano he con igu a ion was analyzed wi h ou
me hod, one consis ing o a h ee-line symme ical couple
wi h a wo-laye ed aniso opic subs a e: sapphi e (h z)
and py oli ic bo on ni ide (P.B.N.) (h J. Special couple
s uc u es such as his one a e o en equi ed in communi-
ca ion sys ems and o he mic owa e applica ions. In hese
s uc u es, he quasi-TEM modes (A, B, C) can be
p opaga ed [4]–[6]. Fig. 7 ep esen s he dependence o he
mode cha ac e is ics on he a io hi/h. I can be no iced
ha he e a e wo alues o hi/h ha equalize he mode
phase eloci ies. This in e es ing esul is aconsequence o
he combined e ec o he geome y o he s uc u e and
he use o aniso opic subs a es [17].
VI. CONCLUSIONS
In his pape he au ho s ha e discussed he analysis o
shielded mul iconduc o s ip lines embedded in amul i-
laye ed aniso opic medium by employing a a ia ional
echnique in Fou ie ’s disc e e domain. The calcula ion o
8.0
7.0
6.0
5.0
4.0
3.0
l E=5.12; E=
xl 3.40 (P. B.N
yl
mode AE== 9.40 ;EY2 .11-60
(sapphi e)
.2 .d .6 .8 hi/h
Fig. 7. A, B, and Cmode e ec i e dielec ic cons an s o h ee s ips
on wo aniso opic dielec ic laye s as a unc ion o he ela i e
hickness o each-laye .
he capaci ance and induc ance ma ices o lossless con-
igu a ions is achie ed by compu ing he elec ic ene gy
pe uni leng h o he s uc u es. The Rayleigh-Ri z p oce-
du e has been applied o op imize he solu ion using
adequa e ial unc ions. The me hod is nume ically e y
e icien and can be easily implemen ed in acompu e
p og am.
The numbe o aniso opic dielec ic laye s and conduc-
o s ips is no longe adi icul y because he G een’s
unc ion ma ix is e alua ed by means o a e y simple
ecu ence algo i hm. Some examples ha e’ been included
o illus a e he s eng h o he me hod and i s accu acy.
The p opaga ion modes o h ee lines wi h wo aniso opic
laye s ha e been s udied as apa icula applica ion.
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[1]
[2]
[3]
[4]
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ac e is ic e mina ions o mul iconduc o ansmission lines wi h
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ol. MTT-21, pp. 450–457, July 1973.
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conduc o ansmission linesj’ IEEE T ans. Mic owa e Theo y
Tech., ol. MTT-29, pp. 812-817, Aug. 1981.
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aniso opic ansmission lines: Applica ion o MIC lines,” IEEE
T ans Mzc owaue Theo~ Tech., ol. MTT33, pp. 927-932, Oc .
1985.
D. Pa lidis and H. L. Ha nagel, “The design and pe o mance o
h ee-hne mic os lp couple s,” IEEE T ans. Mic owa e Theosy
Tech., ol. MTT-24, pp. 631-640, Oc . 1976.
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side-coupled s iplines wi h aniso opic subs a es,” IEEE T ans.
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Ap . 1984.
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F ancisco Medina was bo n in Pue o Rea , Spain,
on No embe 9, 1960. He ecei ed he
Licencia u a deg ee in physics om he Uni e -
si y o Se ille, Spain, in 1983.
He is cu en ly Assis an P o esso o Elec-
ici y and Magne ism in he Depa men o
Elec ici y and Elec onics, Uni e si y o Se ille,
whe e he is s udying o he Ph.D. deg ee. His
cu en in e es is in mul iconduc o plana
ansmission lines and MIC design.
*
Manuel Homo (M75) was bo n in Te e del
Campo (Ja+n), Sps in, on June 29, 1947. He
ecei ed he deg ee o Licenciado in physics in
1969 and he deg ee o Doc o en Ciencias in
Physics in 1972, bo h om he Uni e si y o
Se illa, Spain.
Since Oc obe 1969 he has been wi h he De-
pa 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, Associa e P o esso in
1975, and P o esso in 1986. His main ields o
in e es include bounda y a ue 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 ia s, mul iconduc o ansmission lines,
and plana slow-wa e s uc u es.
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