Analytical analysis of a rectangular shielded multilayer coupled coplanar waveguide

Abstract

The closed-form expressions for calculating the quasi-static parameters of a rectangular shielded multilayer coupled coplanar waveguide (RSMCCPW) are presented. Expressions of the odd- and even-mode characteristic impedances, which can provide accurate and fast calculations, are derived by using conformal-mapping techniques. The numerical results are obtained and compared with those available in the literature for similar unshielded structures. Good agreement between the results is observed. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 41: 392–395, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20149

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Analytical Analysis of a Rectangular Shielded Multilayer
Coupled Coplanar Waveguide
Zhiyong Shan, Yanhua Zhang, Xilang Zhou, and Ji Yao
Department of Electronic Engineering
Shanghai Jiaotong University
Shanghai 200030, P. R. China
E-mail: xlzhou@sjtu.edu.cn
Abstract：： The closed-form expressions are presented for calculating the quasi-static
parameters of a rectangular shielded multilayer coupled coplanar waveguide (RSMCCPW).
The expressions of the odd- and even-mode characteristic impedances which can provide
accurate and fast calculations are derived by using conformal mapping techniques. The
numerical results are obtained and compared with those available in the literature for
unshielded similar structures. Good agreement is observed between the results.
1．． Introduction
Advances in substrate technologies and progress in computer-aided design (CAD) have led
to the development of multilayer radio frequency/microwave integrated circuits (RF/MIC's).
Coplanar waveguides (CPW's) and coupled coplanar waveguides (CCPW's) are widely used
in modern RF/MIC's and high- speed integrated circuits [1-3]. Many authors have investigated
extensively unshielded/partial shielded CPW's and CCPW's by using various methods, such as
spectral-domain full-wave analysis, finite-difference time-domain method, finite-element
method, and conformal mapping techniques etc. To the best of our knowledge, the quasistatic
characteristics of the RSMCCPW has not been analyzed using conformal mapping techniques.
In this paper, an analytical analysis of the RSMCCPW structure is presented using conformal
mapping techniques. The numerical results for the present structure are presented for a
comparison with those previously obtained by [4] and [5].
2．． Theoretical Analysis
To simplify its calculation, we consider a RSMCCPW structure with three layers (region I,
II, and III) of isotropic and lossless dielectrics shown in Fig.1, where the two conductor strips
1

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of width () and three-ground strips of width , (
c b
−
a
a d
−
), (w d
−
) of CCPW on
multilayer substrates is located symmetrically inside a enclosure of height () and is
assumed to be infinitely thin and perfectly conducting. We also assume
1hh
+
3
3
r
2
rεε>
and
, so that the field components parallel to the dielectric/dielectric interface are much
larger in comparison with the normal components, and all dielectric/dielectric interfaces may
be approximated as magnetic walls.
3hh
?
2
The odd- and even-mode are analyzed by assuming that a magnetic wall (MW) and an
electric wall (EW) are present at the central symmetric line
AA
′
−
. The odd- and even-mode
capacitances per unit length,
and , can be obtained by summing up the capacitances
per unit length in region I, II and III, as shown in Fig.2. That is,
. The odd- and even-mode characteristic
impedances can be derived by means of the capacitances per unit length
and . Due to
symmetrical nature of the RSMCCPW structure, we can only consider the half of the cross
section of each region. Fig.2 shows the analytic configuration of the odd mode for the half of
the region I, II, and III, where the relative dielectric constants of the region I, II, and III are
o
C
e
C
12312
,
ooooeee
CCCCCCCC
=++=++
3
e
3
o
C
e
C
123
, (),
rrrr
εεεε−
, respectively.
2.1 Odd-mode parameters
Fig.3 gives the conformal transformations for the odd-mode configuration shown in Fig.2
(a). First, the configuration of Fig.3(a) is transformed onto the upper half of the
- plane in
Fig.3(b) using the following mapping function
T
()
2
11
o
11
o
sn,
z
t K k
W
⎛
⎜
⎝
⎞
⎟
⎠
=
k
(1)
where is determined by
11
ok
11
o
11
o
′
1
()()
K kK kW h
=
, and
( )
K k is the complete
elliptical integral of the first kind with the module and
k
( )
K k′ its complement. Then, the
upper half of
-plane in Fig.3(b) is transformed onto the upper half of
- plane in Fig.3(c)
using the mapping function
TT
2

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13
t
11
t
t
12
t
t t
−
14
t
111312
t
w
−
−
−
=
(2)
where
(
(
)
)
(
(
)
)
()
()
()
()
222
11
t
t
11
o
11
o
12
t
11
o
11
o
-plane in Fig.3(c) is transformed
13
t
11
o
11
o
sn
sn
,
,
, sn,
W
, sn,,
aK k
dK k
W k
W k
bK k W kcK k W k
=
=
==
2
14 11
o
11
o
. Last, the upper half of
into the internal region of a rectangular using the known Schwarz-Christoffel transformation,
and the module can be determined from correspondence of vertices in T -plane and
-plane. It can be expressed as
1
ok
W

(
(
)
)
(
(
)
)
14
t
t
11
t
t
13
t
t
12
t
t
1
14121311
14
1
w
o
k
−
−
−
−
==
(3)
Thus, the odd-mode capacitance per unit length of the region I is

(
(
)
)
1
101
1
2
o
′
or
o
K k
K k
C
ε ε=
(4)
To obtain the capacitance per unit length of the region II in Fig.2(a), the right half of the
region II in Fig.4(a) is transformed onto the lower half of the
-plane in Fig.4(b) using the
following mapping function
T

x
x
t
μ β
γ β
γ
μ
−
−
−
−
=
(5)
where
22
21 2121 21
sn ((), ), sn ((), ),
oooo
x zK kW k aK kW k
α==

222
21
(
21 2121 2121
sn (
(
K k
(),
=
), sn ((),),
T
sn ((),),
oooooo
bK k
)
o
W k
)
cK kW kdK k W k
βγμ===
and
21212
o
′
K kW h
.Then, the lower half of -plane in Fig.4(b) is transformed
into the rectangular region of W-plane in Fig.4(c) using the Schwarz-Christoffel transformation
0
2322 28 27
d
()()()(
t
t
w
tttttttt
=
−−−−
∫
)
(6)
where
2328
(
ν
)() ((
γ
)()),( )( ) ((
γ
)( )),
tt
μ βνγβμαμ βαγβμ=−−−−=−−−−

2
o
222721
() (), 1,1
tt
βγβμν=−−==
k
, and dimensional parameters can be determined
by
3

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0
02
(
(
)
)
o
′
o
W
H
K k
K k
=
22
2
(7)

1 22
1
022
(arcsin
K k
,
)
)
(
o
o
′
FA k
W
W
=
(8)

2 22
2
0 22
(arcsin
K k
,
)
)
(
o
o
′
F A k
W
W
=
(9)
where
1
(
ν
)() (( )() (
μ
)()) ,
A
μ γμν γ βν μ βγ=−−−−−−−


2
(
ν
)() (( )( ) (
μ
)()) ,
A
γ γμνγβνμ βγ=−−−−−−−

022
(()() (
μ
)()) ((
γ
)() (
γ
)()) ,
k
νγβν μ βαμ ναγ νμ=−−−−−−−−−−

and
),(
kF φ
is the incomplete elliptical integral of the first kind.
In order to evaluate the capacitance per unit length of the rectangular region in Fig.4(c),
we can assume a magnetic wall present at the center of the slot
) 2)( (
21
WW +
in the
W-plane. Thus, the odd-mode capacitance per unit length in the region II is obtained as

()
(
(
)
)
(
(
)
)
2324
2023
2324
2
o
′
o
′
orr
oo
K k
K k
K k
K k
C
εεε
⎛
⎜⎜
⎝
⎞
⎟⎟
=−+
⎠
(10)
where and can be obtained by the formulas proposed by [6,7 ].
23
ok
24
ok
Following the same analysis as the configuration of the Fig.3(a), is given as
3
o
C


(
(
)
)
3
303
3
2
o
′
or
o
K k
K k
C
ε ε
=
(11)
where is similar to (3), provided that is displaced by .
3
ok
1h
3h
Therefore, the total odd-mode capacitance per unit length is

(
(
)
)
()
(
(
)
)
(
(
)
)
(
(
)
)
1 2324
01233
12324
2
o
′
o
′
o
′
orrrr
ooo
K k
K k
K k
K k
K k
K k
K k
K k
C
εε
⎨
⎪
⎩
εεε
⎧
⎪
⎫
⎪
⎬
⎪
⎭
⎡
⎢
⎣
⎤
⎥
⎦
=+−++
3
3
o
o′
(12)
and odd-mode characteristic impedance is
4

Page 5

(
(
)
)
(
(
)
)
1
23 24
2324
60
ε
o
′
o
′
o
oo
eo
K k
K k
K k
K k
Z
π
−
⎡
⎢
⎣
⎤
⎥
=+
⎦
(13)
where

(
(
)
)
()
(
(
)
)
)
)
(
(
)
)
)
)
(
(
)
)
(
(
(
(
1 2324
1233
123
K k
K k
24
23 24
2324
o
′
o
′
o
′
rrrr
ooo
eo
o
′
o
′
oo
K k
K k
K k
K k
K k
K k
K k
K k
K k
K k
εεεε
ε
⎡
⎢
⎣
⎤
⎥
⎦
+−++
=
⎡
⎢
⎣
⎤
⎥
⎦
+
3
3
o
o′
(14)
2.2 Even-mode parameters
For even-mode case, the EW along the imaginary axis in Fig.2 should be changed into the
correspondence MW. Similar to the analysis for odd-mode case, through a sequence of the
conformal mapping, the total even-mode capacitance per unit length can be derived as
(
(
)
)
(
(
)
)
()
(
(
)
)
(
(
)
)
(
(
)
)
(
(
)
)
11
′
12
′
21
′
22
′
31
′
32
′
01233
11 12 21 2231 32
2
eeeeee
errrr
eeeeee
K kK kK kK kK kK k
C
K kK kK kK kK kK k
εεεεε=++−+++
⎧
⎨
⎩
⎡
⎢
⎣
⎤
⎥
⎦
⎡
⎢
⎣
⎤
⎥
⎦
⎡
⎢
⎣
⎤⎫
⎬
⎦⎭
⎥
(15)
where the modulus and can also be determined by the similar
formulas derived by the odd-mode case.
11
e
12
e
212231
,,,,
eee
kkkkk
32
ek
According to (15), the even-mode characteristic impedance can easily be derived as
(
(
)
)
(
(
)
)
(
(
)
)
(
(
)
)
1
11
e
′
12
e
′
31
′
32
′
11
e
12
e
3132
60
ε
ee
e
ee
ee
K k
K k
K k
K k
K k
K k
K k
K k
Z
π
−
⎧
⎪
⎨
⎪
⎩
⎫
⎪
⎬
⎪
⎭
⎡
⎢
⎣
⎤
⎥
⎦
⎡
⎢
⎣
=+++
⎤
⎥
⎦
(16)
where
(
(
)
)
(
(
)
)
K k
()
(
(
⎤ ⎡
⎥ ⎢
⎦ ⎣
)
)
K k
(
(
)
)
K k
(
(
)
)
(
(
)
)
(
(
)
)
(
(
)
)
(
(
)
)
(
(
)
)
11
′
12
′
⎡
⎢
⎣
21
′
22
′
31
′
32
′
1233
11 1221223132
11
′
12
′
31
′
32
′
11123132
eeeeee
rrrr
eeeeee
ee
eeee
eeee
K kK k K kK kK k K k
K kK k K k K k K kK k
K k
K kK kK kK k
εεεε
ε
=
++−+++
+++
⎡
⎢
⎣
⎤
⎥
⎦
⎡
⎢
⎣
⎤
⎥
⎦
⎡
⎢
⎣
⎤
⎥
⎦
⎤
⎥⎦
(17)
3．． Results and discussions
In order to verify the correctness of the expressions derived using the conformal mapping
techniques, the quasistatic parameters of the present structure are compared with those obtained
for the unshielded CCPW's proposed in [4]and [5]. Table 1 gives the comparison of the odd-
5