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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

Page 2

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

tK 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

Page 3

13

t

11

t

t

12

t

t t

−

14

t

1113 12

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 kW kcK kW k

=

=

==

2

1411

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

141213 11

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

212121 21

sn ((), ),sn ((),),

oooo

x zK kW kaK k W k

α==

222

21

(

21 212121 21

sn (

(

K k

(),

=

),sn ((), ),

T

sn ((), ),

oooooo

bK k

)

o

W k

)

cK k W 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 2827

d

()( )()(

t

t

w

tttttttt

=

−−−−

∫

)

(6)

where

23 28

(

ν

)() ((

γ

)()),( )( ) ((

γ

)( )),

tt

μ βνγβμα μ βαγβμ=−−−−=−−−−

2

o

22 2721

() (),1,1

tt

βγβμν=−−==

k

, and dimensional parameters can be determined

by

3

Page 4

0

02

(

(

)

)

o

′

o

W

H

K k

K k

=

22

2

(7)

122

1

0 22

(arcsin

K k

,

)

)

(

o

o

′

F A 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 γμνγβν μ βγ=−−−−−−−

022

(()( ) (

μ

)()) ((

γ

)() (

γ

)()) ,

k

νγβνμ βα μ να γ 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

()

(

(

)

)

(

(

)

)

23 24

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 23 24

1233

1 23

K k

K k

24

2324

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 1221 223132

2

eeeeee

errrr

eeeeee

K k K k K kK k K kK k

C

K kK k K kK k K 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

2122 31

,,,,

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

31 32

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 1221 22 3132

11

′

12

′

31

′

32

′

11123132

eeeeee

rrrr

eeeeee

ee

eeee

eeee

K kK kK kK k K kK k

K kK k K k K kK k K k

K k

K kK k K 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-

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