chapter – 3 conformal mapping technique – an...

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Chapter – 3

Conformal Mapping Technique – An Overview

3.1 Introduction

The conformal mapping method is invaluable for solving problems in engineering and

physics. By choosing an appropriate mapping function, the analyst can transform the

inconvenient geometry into a much more convenient one. This conformal mapping

technique (CMT) is equivalent to a coordinate transformation and its application to

planar and non-planar transmission lines is described in this chapter.

Coplanar lines and their modifications are widely used in modern MIC and MMIC [105],

and high-speed integrated circuits [43]. Rigorous analysis of waveguiding properties of

these lines - characteristic impedance (Z0), effective relative permittivity (effε ) and losses

(α ), is carried out by using one of the well-known full-wave analysis techniques [119].

Calculations of Z0 and effε of coplanar lines by CMT were first presented by Wen [27].

Although his analysis leads to simple analytical expressions for the line parameters, they

are valid only for the case of infinite substrate thickness and infinite dimensions for the

two side ground strips. Several numerical methods such as the relaxation method, method

of moment etc are developed to calculate line parameters taking into account the effect of

finite substrate thickness and finite line dimensions. These methods do not provide

closed-form expressions for line parameters, useful for the CAD applications [23].

However the refined CMT methods [107] lead to compact closed-form expressions for

impedance, effective relative permittivity and microwave losses [48, 74,104] in terms of

finite line dimensions and substrate thickness. The CMT has also been used to get the

analytical expressions for electric-field distributions in simple CPW structures used in the

Conformal Mapping Technique: An Overview

46

integrated optical (IO) modulators [96]. The CMT is further extended for modeling of

characteristic impedance, effective relative permittivity [19-20,66-67,86,91], conductor

and dielectric [47-48,84-85] losses and also for charge and field [33] distributions in

multilayered planar and non-planar transmission lines. The analysis is based on the

partial capacitance technique [33] that assumes the magnetic-wall approximation at the

dielectric/dielectric interfaces. The good agreement of results for the line parameters

(Z0, effε ,α ), obtained from the CMT based analytical expressions, has been observed

against the results of measurements, simulations and field-analysis [74,107].

In this thesis, the application of CMT is extended to obtain closed-form analytical

solutions of effε and Z0 for single-layer and multilayer planar and non-planar CPW, CPS

and slotline structures. We present below a summary of the CMT.

3.2 CMT of Complex Variable

The determination of the distributed capacitance and inductance of a transmission line

requires a solution of Laplace’s equation in two dimensions. For the planar transmission-

line structures, it is difficult to construct solutions for Laplace’s equation. The conformal

mapping method maps the boundaries of the complex geometry into a simpler

configuration for which solutions to Laplace’s equation are easily found.

3.2.1 Mapping

The analytic function F(z), describing the geometrical shape in the complex z-plane,

transform the shape to other shape in the w-plane. The process is shown in Fig.(3.1). The

transformation is described by

y)jv(x,+y)u(x,=F(z)=w (a) (3.1)

jy+x=z (b)


47

where, z is defined in the complex z-plane (x-y plane), shown in Fig. (3.1a) and w is

defined in the complex w-plane (u-v plane), shown in Fig. (3.1b).

3.2.2 Properties of Conformal Mapping

The conformal mapping, or transformation of two intersecting curves from the z-plane to

the w-plane, preserves the angles between every pair of curves; that is, if two curves in

the z-plane intersect at a particular angle, the corresponding curves will also intersect at

the same angle, although the transformed curves in w-plane may not have any

resemblance to the original curves. Any conformal transformation would, therefore,

transform parallel or orthogonal curves in the z-plane into parallel or orthogonal curves in

the w-plane respectively. This is the most important property of the conformal

transformation that is achieved only by the analytical function, as mentioned above.

Some more properties of the conformal mapping transformations are summarized below

[38, 138].

Our z-plane i.e. the (x, y) coordinate system is an orthogonal one. For an analytical

function, satisfying Cauchy-Reimann condition, the w-plane i.e. the (u, v) coordinate

system is also orthogonal. We have the following useful and interesting consequences of

orthogonality of (u, v) coordinate system:

• Laplace’s equation remains invariant on the transformation from the (x, y) to the

(u, v) coordinate systems.

• The electrostatic energy in both (x, y) and (u, v) coordinate systems remains invariant.

Consequentially, the capacitance of a system of conductors remains the unchanged on

the transformation of arrangement of conductors from the (x, y) to the (u, v)

coordinate systems. Thus, under the conformal mapping transformation from the (x,

y) to (u, v) coordinate system there is a change in the geometrical shape of the


48

conductors arrangement without any change in the capacitance. This is a very

important property for the computation of the transmission line parameters.

• The CMT is a geometrical process not a physical process involving changes in the

material properties like electrical, thermal etc. Thus the physical nature of boundaries

of any region formed by curves or lines remain unchanged on their transformation

from (x, y) to (u, v) coordinate system. However, the electrical boundaries, i.e. the

electric wall with its boundary condition, as well as the magnetic wall with it

boundary condition, remain unchanged in the conformal mapping transformation,

even if their geometrical shapes are changed. The property of a medium, described by

the permittivity, permeability and conductivity remains unchanged in such

transformation.

• In an electrostatic system, the coordinate u represents the electrostatic potential (φ )

and the coordinate v represents electric flux (Ψ ) that corresponds to the charge on a

conductor. The coordinates u and v can equally represent the electric flux and

potential respectively.

3.2.3 Applications of Conformal Mapping

Conformal transformation has been employed frequently to determine the characteristics

of transmission lines and discontinuities in them. The advantage of this approach is that

the analytical solution based on conformal transformation leads to design equations.

Also, an open geometry like that of planar lines is transformed into a closed geometry.

The limitation of this method is that it provides solution for the static fields only, which

may be used at best for the quasi-static cases. A series of conformal mappings are

performed to obtain the characteristics for a range of different geometric parameters [27].

Conformal mapping method has also been used in the field of medical physics such as

brain surface mapping [140]. The CMT is used in scattering and diffraction problems too.


49

However the conformal mapping approach is limited to problems that can be reduced to

two dimensions and to problems with high degrees of symmetry.

3.3 Schwarz-Christoffel Transformation

The Schwarz-Christoffel (SC) – transformation helps to solve an important class of

boundary value problems that involve regions with polygonal boundaries. This

transformation is useful technique for the analysis of planar transmission lines.

(a) (b)

Fig.(3.1): Schwarz-Christoffel transformation: (a) z-plane; and (b) w-plane.

SC-transformation:

( ) ( ) ( ) ( )∫ +−−−== −−−

z

z

kN

k2

k1

0

N21 B'dzx'z.........x'zx'zAzFw

(3.2a)

where 2k.........kkk N....321 =++ , and B is an arbitrary constant that determines the

position of the polygon. The size and orientation of the polygon can be controlled

through the magnitude and angle of A, respectively. The inverse SC-transformation

corresponding to equation-(3.2a) is given by

( ) ( ) ( ) ( )∫ +−−−== −−−

w

w

KN

K2

K1

0

N21 d'dww'w.........w'ww'wCwFz

(3.2b)


50

where 2K.........KKK N....321 =++ (3.2c)

The SC – transformation given above maps a given polygon in the w–plane, Fig. (3.1b)

into the line segments along the real axis in the z-plane, Fig. (3.1a) [38,138]. In the

process of the geometrical transformation, the interior space of the polygon in w- plane is

transformed to the upper half space in the z-plane. The role of the z- plane and the w-

plane can be interchanged. Therefore, the SC- transformation also transforms the line

segments in the w-plane along the real u- axis into the polygon in the z-plane.

3.3.1 Elliptic Sine Function

Determination of the analytic functions from the SC- transformation involves definite

integrals that may not have solutions in term of the standard algebraic and trigonometric

functions. Often the integrals involved in the SC- transformation are treated through a

function known as the elliptic sine function [37-38]. The transformation

);k,w(snz = jyxz += (3.3a)

is an important mapping function in the study of planar transmission lines [38]. The

elliptic sine function )k,w(sn transforms the boundary of a rectangle in the w-plane onto

the real axis of the z-plane, and the interior of the rectangle is mapped onto the upper half

of the z-plane. The inverse transformation is given by

)k,z(snw 1−= (3.3b)

It transforms the x-axis into a rectangle in the w-plane. The rectangle may represent a

parallel plate capacitor.

The elliptic sine function is a generalization of the sine function to the complex domain

and is periodic along both the axis. The periodicity of the elliptic sine function is 4K(k)


51

along the u-axis and 2K′(k) along the v-axis. The parameter k is called the modulus. The

function K(k) defines the inverse elliptic sine function over the range 10 ≤≤ z :

( )( ) )k,1(snzk1z1

dz)k(K 1

1

0222

−=−−

= ∫

(3.4)

The complex function ( ) ( )[ ]kjKkK ′+

defines the inverse elliptic sine function over the range kz 10 ≤≤ :

( )( ) ( )( ) ( )( )∫∫∫−−

+−−

=−−

=+k/k/

zkz

dz

zkz

dz

zkz

dz)k('jK)k(K

1

1222

1

0222

1

0222 111111

(3.5a)

where

( )( )∫−−

=k

zkz

dz)k('K

1

1222 11

(3.5b)

The integral in equation - (3.4) may be expressed in a standard form by substituting

θsinz = ,

=−

= ∫ 2,kF

sink1

d)k(K

2

022

πθ

θπ

(3.6)

It is called the complete elliptic integral of the first kind. Similarly, substituting

( ) ( ) θ2222 cosk11zk −=− in equation-(3.5b) gives

)'k(K

2,'kF

sin'k1

d)k('K

2

022

=

=−

= ∫π

θθπ

(3.7)

where 2k1'k −= . The approximate expressions can be used to evaluate the elliptic

integral [38,104,136]. The ratio of complete integrals ( ) ( )kKkK ′ occurs very commonly


52

in the design of planar lines. Fortunately, this ratio can be determined accurately to 1 part

in 105 using the following simple expression [136]:

( )( )

≤≤

−+

≤≤

′−′+

=′

−

1k7.0for,k1

k12ln

1

7.0k0for,k1

k12ln

1

kK

kK

1

π

π (3.8)

3.4 Quasi-Static Analysis of Planar/Non-Planar Transmission Lines

In this section, we obtain the closed – form expressions of the characteristics of different

configurations of single-layer and multilayer CPW, CPS and slotline with the use of

CMT, in which quasi-static solutions of the effective relative permittivity and

characteristic impedance of the structures are derived and calculated for their use in the

CAD applications.

3.4.1 Coplanar Waveguide

CPW on Finite Thickness Substrate and Infinite Ground Plane

A CPW with finite thickness substrate h and relative permittivity rε is shown in

Fig.(3.2a). Both the ground planes are infinite in width that is not a practical situation.

This will be discussed in the next section.

An exact conformal mapping of the structure shown in Fig.(3.2a) involves mapping of

the dielectric and air interface in the lower half-space. Such transformation is difficult.

However, based on following assumption we can obtain an approximate conformal

mapping [23].


53

The total line capacitance p.u.l. of a CPW structure, shown in Fig.(3.2a) on a substrate of

thickness h is equal to the line capacitance p.u.l. of the CPW line on the air substrate

(both upper and lower half- spaces) + the line capacitance p.u.l. of the lower half- space

of a CPW on the infinitely thick substrate of relative permittivity ( rε - 1). The

conformal mapping is achieved in two steps. The first transformation, shown in Fig.(3.2c)

converts half of the original structure shown in Fig.(3.2b) to a conventional CPW on the

infinitely thick substrate having relative permittivity ( rε - 1). It is achieved by

following mapping function that transforms the structure Fig.(3.2b) from the z-plane to

the t-plane, shown in Fig.(3.2c)

=

h2

zsinht

π (3.9)

(a) (b)

(c) (d)

Fig. (3.2): Conformal mapping of CPW with finite substrate thickness (a) Original structure, (b) z-plane,

(c) t-plane, (d) w-plane


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The second CMT, shown in Fig. (3.2d), is achieved by using the SC- transformation that

transforms the strip located on the real x-axis of the t-plane, Fig. (3.2c) into a rectangle in

the w-plane with the aspect ratio

( )h2/bsinh

)h2/asinh(

t

tk

2

11 π

π== 211 k1k −=′ (3.10)

If C1 is the line capacitance p.u.l. of one half of the structure on the substrate with relative

permittivity ( rε -1) then

( ) ( ))k(K

kK12C

1

1r01 ′

−= εε (3.11)

The line capacitance p.u.l. of upper half of the structure is

( ))k(K

kK2C

0

000 ′

= ε where 2000 k1k,

b

ak −=′= (3.12)

Based on our initial assumption, the total line capacitance p.u.l. of a CPW on the finite

thickness substrate is

( ) 1L0U0rT CCCC ++=ε (3.13)

where U0C is the line capacitance p.u.l. of upper half of the CPW on the air- substrate

and L0C is the line capacitance p.u.l. of lower half of the CPW on the air- substrate. In

the present case U0C = L0C = 0C . The total line capacitance of CPW p.u.l. on the air

substrate is

0rT0 C2)1(C ==ε (3.14)


55

The effε and Z0 of the present form of the CPW is obtained from

( )( )

( ) ( ))k(K

kK

)k(K

kK

2

)1(1

C

C

2

11

C2

CC2

1C

C

0

0

1

1r

0

1

0

10

rT0

rTeff

′′

−+=

+=

+=

==

εεεε (3.15)

( ))k(K

kKZ

eff 0

00

30 ′=

επ

(3.16)

CPW with Finite Ground Planes

In previous section, the ground planes have been considered to be of infinite width.

However, in practice the ground planes have to be of finite widths as shown in Fig (3.3a)

for the CPW on a finitely thick substrate. The sequence of CMT is shown in Fig.(3.3).

The finite extent of the ground planes modifies the aspect ratio. Thus after modification,

)k( 0 and )k( 1 become )k( G0 and )k( G

1 respectively, for the infinitely and finitely thick

substrates with finite width ground planes

110G0 F

b

aFkk == (a) where,

2

1

2

2

1)c/a(1

)c/b(1F

−−= (b) (3.17)

( ) 221G1 F

h2/bsinh

)h2/asinh(Fkk

ππ== (a) where,

2

1

2

2

2

)h2/csinh(

)h2/asinh(1

)h2/csinh(

)h2/bsinh(1

F

−

−

=

ππ

ππ

(b) (3.18)

In both cases for c ∞→ (infinitely wide ground planes) F1→ 1 and F2→ 1.

Firstly the conductor strips are located on the infinite air substrate in the upper half- plane

and secondly they are also located at the finite thickness substrate on the lower - half


56

plane [47]. Both the cases should be treated separately. The sequence of conformal

mapping of the CPW structure of finite ground planes on the finite substrate thickness is

shown in Fig. (3.3). Fig. (3.3b) shows the first quadrant of the original structure. Its upper

part on the infinite air substrate is mapped into the upper half of the t-plane shown in Fig.

(3.3c) by using following function

t = z2 (3.19)

Fig. (3.3d) shows that the conducting strip (electric wall, EW) and the dielectric interface

(magnetic wall, MW) are mapped as a rectangle in the w-plane by following SC-

transformation

( )( )( )∫ −−−==+

t

0t 321 ttttttt

dt)t(wvju

(3.20)

The mapping starts from location – (4) to (5-6) to (1) such that a parallel plate capacitor

is formed. The above integral can be transformed to a standard elliptic integral of the first

kind by change of variables which results in the aspect ratio of the CPW on the infinite

substrate with finite extent coplanar ground planes:

( )( )

( )( )22

22

222

222G02

ac

bc

b

a

bac

abckkk

−−=

−

−=== (3.21)

The aspect ratio, k0 = a/b is for the infinite width ground planes and F1

= ( ) ( )2222 ac/bc −− is the correction factor due to the finite width of the ground

planes.


57

(a) (b)

(c) (d)

Fig.(3.3): Conformal mapping of CPW with finite ground width (a) Original structure, (b) z-plane, (c) t-

plane and (d) w-plane The line capacitance p.u.l. of a CPW on the air substrate is

)k(K

)k(K2C

2

200 ′

= ε (3.22)

To compute the line capacitance of lower half of the plane shown in Fig (3.4a), the

dielectric and air interface are replaced by a magnetic wall and the relative permittivity

by ( rε -1) of the infinite extent half-space. The transformation of region from the z-plane,

Fig.(3.3b) to the x-plane, shown in Fig. (3.4a) is achieved by following mapping function

=h2

zcoshx 2 π

(3.23)


58

(a) (b)

Fig.(3.4): (a) Transformation of region to x-plane, (b) Mapping to rectangle forming parallel plate

capacitance

The mapped structure is shown in Fig (3.4a). It can be further mapped as a rectangle

forming the parallel plate capacitance shown in Fig (3.4b) in the w-plane. The SC-

transformation is

( )∫ −−−−

==+x

1x 4321 )xx)(xx)(xx(xx

dxwvju (3.24)

The integral given above can be transformed to a standard elliptic integral of the first

kind, giving us aspect ratios:

−−×

−−==

12

12

22

2213

)h/b(cosh

)h/a(cosh

)h/a(cosh)h/c(cosh

)h/b(cosh)h/c(coshkk

2

2

22

22G

ππ

ππππ

×

−−=

h)b/2sinh(

h)a/2sinh(

)h/a(sinh)h/c(sinh

)h/b(sinh)h/c(sinh22

22

ππ

ππππ 2

1

22

22 (3.25)

It gives us the correction factor F2 of equation-(3.18b). The line capacitance p.u.l. of

lower half of the CPW with dielectric layer is

( ) ( ))k(K

kK12C

3

3r01 ′

−= εε (3.26)


59

The total line capacitance p.u.l. of a CPW is

( ) ( ))k(K

kK)1(2

)k(K

kK4CC2C

3

3r0

2

2010T ′

−+′

=+= εεε (3.27)

The effε and Z0 of the present form the CPW is

( )( )

( ) ( ))k(K

kK

)k(K

kK)(

C

Cr

rT

rTeff

2

2

3

3

01

2

11

1

′′

−+==

= εεεε (3.28)

( ))k(K

kK30Z

2

2

eff0

′=

επ

(3.29)

Asymmetrical CPW

The conformal mapping of the asymmetrical CPW (ACPW), shown in Fig. (3.5a), is

similar to the that given in [47]. The total capacitance p.u.l. of ACPW is equal to the sum

of the air-filled line capacitance in the absence of substrate C1 and the capacitance of the

substrate C2, assumed to have dielectric ( 1−rε ).

21rT CC2)(C +=ε (3.30)

First in order to obtain the air capacitance C1, assuming the ACPW without the substrate,

the upper half plane in Fig.(3.5b) is transformed in to the rectangular region in w-plane

by means of the mapping

∫ −−−−=

t

0t 4321 )tt)(tt)(tt)(tt(

dtw (3.31)


60

(a) (b)

(c) (d)

Fig.(3.5):Conformal mapping of ACPW of finite thickness substrate (a) Original structure, (b) z-plane,

(c) t-plane, (d) w-plane. In this case, the air capacitance p.u.l. of the line is given by

)k(K

)k(KC

'4

401 ε= (3.32)

where [ ]

[ ] [ ])sw()sw(

s/)ww(k

21

214 11

1

++++

= (a) 244 1 kk' −= (b) (3.33)

Second, in order to compute the dielectric capacitance C2, the dielectric region in

Fig.(3.5b) is transformed into the lower half region in Fig.(3.5c) by using equation-

(3.9).Then, the lower half plane in Fig.(3.5c) is transformed into the rectangular region in

w –plane for which t-plane transformation is not needed . The locations z1, z2, z3, z4 in the

z-plane can be directly transformed to w-plane by using the above mentioned SC-

transformation to form the rectangle in the w-plane, i.e.


61

44332211 ,,, ztztztzt →→→→

where, 243211 w2

sz,

2

sz,

2

sz,

2

swz +==−=

+−= . (3.34)

The line capacitance of the ACPW for the case of lower half space filled with infinite

extent ( 1−rε ) dielectric medium is

( ))k(K

)k(KC

'r5

502 1−= εε (3.35)

where

++

+

+

++

+

=)ws(

hsinh

h

ssinh

h

ssinh)ws(

hsinh

)ws(h

sinh)ws(h

sinhh

ssinh

k

21

21

52

4442

4

24

244

2

ππππ

πππ

(a) (3.36)

55 1 kk' −=

(b)

The total capacitances of ACPW with finite extent substrate in the lower half space and

air medium in the upper half space are

)k(K

)k(K)(

)k(K

)k(K)(C

'r'rT5

50

4

40 12 −+= εεεε (a)

)k(K

)k(K)(C

'rT4

400 21 εε == (b) (3.37)


62

The effε and Z0 of the ACPW on the finite extent dielectric substrate is

)k(K

)k(K

)k(K

)k(K)(

)(C

)(C'

'

rrT

rTeff

5

5

4

4

01

2

11

1−+=

== ε

εεε (3.38)

)k(K

)k(KZ

'

eff 4

40

60

επ= (3.39)

Non-Planar CPW

We present four CPW structures on non-planar surfaces, shown in Fig. (3.6) :

i. Elliptical CPW (ECPW)

ii. Circular Cylindrical CPW (CCPW)

iii. Semi-Ellipsoidal CPW (SECPW)

iv. Semi-Circular Cylindrical CPW (SCCPW)

Fig. (3.6a) shows cross section of the elliptical CPW (ECPW). The structure is formed by

two confocal ellipses. Their semi major axis and semi minor axis are a1, a2, a3, b1, b2 and

b3, respectively. The 2nd ellipse is the inner side of the strip conductor and 3rd ellipse is

the outer side of the strip conductor, not shown in Fig. (3.6a). The dimensions a3 and b3

accounts for the finite strip conductor thickness. The central arc strip width is 2ψ and the

gap between the strip and the ground plane is( )ψθ − . For ECPW and circular cylindrical

CPW (CCPW), ground plane width is ( )θπ 22 − .In Fig. (3.6a), the conductor thickness

for semi-major and semi-minor axes are ( )23 aa − and ( )23 bb − , respectively. Moreover,

the dielectric substance thickness for the semi-major and semi-minor axes are ( )12 aa −

and( )12 bb − , respectively. The relative dielectric constant is rε in both the cases.


63

(a) (b)

(c) (d) Fig.(3.6): CPW on the curved surfaces: (a) Elliptical CPW (ECPW) ,(b) Circular Cylindrical CPW

(CCPW), (c) Semi-ellipsoidal CPW (SECPW) and (d) Semi- circular cylindrical CPW (SCCPW).

With the help of conformal mapping method [85], ECPW is first transformed to the

CCPW. The CCPW is then transformed to the planar CPW line on the finite thickness

substrate. Finally the planar CPW, with the help of SC- transformation, is transformed to

the parallel plate capacitor. Due to the invariance of the electrostatic energy, the

capacitance of the parallel plate capacitor gives the line capacitance of the ECPW [38, 59,

106]. Based on the partial capacitance approximation, the overall capacitances per unit

length of the both structures are considered as the sum of the capacitance in free-space

(upper region) and dielectric layer. C0 and C1 capacitances stand for the free-space

capacitance and the dielectric layer capacitance, respectively. For the calculation of total

unit length capacitance, the partial capacitances C0 and C1 can be obtained using

successive conformal transformation steps [47,131]. So, the overall capacitances per unit

length of both the structures are given by,


64

1L0U0T CCCC ++= (3.40)

where the air-region capacitances can be expressed as:

)k(K

)k(K2CC

'0

00L0U0 ε== (3.41)

where

( )( ) )ii(k1'k)i(k 2

0022

22

2

2

0 −=−−=ψπθπ

θψ (3.42)

In the above equation, K(k) is the complete elliptic integral of the first kind. Similarly, the

capacitance of the dielectric layer is,

( ))k(K

)k(K12C

'1

1r01 −= εε (3.43)

where the related modulus can be expressed as;

( )( ) )ii(k1'k)i(

xx

xx

x

xk 2

1122

22

2

2

1 −=−

−=

ψπ

θπ

θ

ψ (3.44)

also,

( )( )

( )( ) )ii(1

H0,

2Hexp

2Hexp1x

)i(H

1,2Hexp

2Hexpx

4

2

2

2

<<

+−−=

∞<<

+−

=

π

πππ

π

π (3.45)

The parameter H is dependent on dimensions of ECPW and CCPW. It is expressed by,


65

++

=

=

11

22

1

2

ba

baln

r

rlnH (3.46)

xψ and xθ are found using the relations given below:

(3.47)

where F (ϕ, k) is the incomplete elliptic integral of the first kind in Jacobi’s notation.

Equation – (3.47) is evaluated by means of using the approximate formulations given in

[76].

Finally, the effε and 0Z of ECPW and CCPW, using the conformal mapping method, are

written as:

)'k(K

)k(K

)k(K

)'k(K

2

11

1

1

0

0reff

−+=

εε (3.48)

)k(K

)'k(K30Z

0

0

eff0 ε

π= (3.49)

The cross-section of the original elliptical cylinder, shown in Fig.(3.6a), is located in the

ζ -plane. The focal distance between three confocal ellipses is

23

23

22

22

21

21 bababac −=−=−= (3.50)

( )( ) ( )

( )( ) ( ) )ii(xKx,x/xarcsinF

)i(xKx,x/xarcsinF

πππθ

πππψ

πθπψ

=

=


66

The confocal elliptical cylinders are transformed into the circular cylinders in the z-plane

on using the following mapping function [85,144]

[ ]22(1

cc

z −= ζζ m (3.51)

The above mapping function provides us the concentric circular cylinders of radius r1, r2

and r3:

)iii(

ba

bar)ii(

ba

bar)i(

ba

bar

33

333

22

222

11

111 −

+=−+=

−+= (3.52)

The CPW on the concentric circular cylinders is transformed into the planar CPW using

the following mapping function [85,144]

2r

zlnju

2

π+

= (3.53)

(a) (b)

Fig.(3.7): Cross sectional view of non-planar CPW with finite ground plane width: (a) (2π-2θ) and (b) (π-2θ).


67

The above mapping function provides us the following conducing strip width (s), slot

width (w), substrate thickness (h) and strip conductor thickness (t) of the transformed

ECPW into the corresponding planar CPW.

)iv(r

rlnt)iii(

r

rlnh)ii(w)i(s

2

3

1

22 ==−== ψθΨ (3.54)

When a1 = b1 = a, a2 = b2 = b, a3 = b3 = d i.e., c = 0, h becomes h = ln (b/a) and t

becomes t = ln (d/b), then equation- (3.52) gives the structural parameters for the CCPW,

as shown in Fig. (3.6b).For the ground plane width ( )θπ 2− , the CPW structure on the

semi- ellipsoidal (SECPW) and semi-circular cylindrical surfaces (SCCPW) are obtained.

It is shown in Fig.(3.6c) and (3.6d) respectively.

On applying conformal mapping, the modulus k0 and k1 and their complementary

modulus '0k and '

1k involved in the complete elliptic integral of the first kind are defined,

in terms of the structural parameters, as follows:

( )( )( )( ) )ii(k1'k)i(2

2k 2

0022

22

2

2

0 −=−

−=ψπθπ

θψ

(3.55)

( )( ) )ii(k1'k)i(

xx

xx

x

xk 2

11222

222

2

2

1 −=−

−=

ψπ

θπ

θ

ψ (3.56)

also,

( )( )

( )( ) )ii(

H,

Hexp

Hexpx

)i(H

,Hexp

Hexpx

12

022

221

21

22

22

4

2

2

2

2

2

<<

+−−=

∞<<

+

−=

π

πππ

π

π (3.57)

xψ and xθ can be found using the relations given below;


68

(3.58)

On substituting the above equations to equation-(3.48) and (3.49),effε and 0Z of SECPW

and SCCPW are obtained.

Multilayer CPW a. Planar CPW

The conformal mapping of the multilayered CPW structure is based on the partial

capacitance technique [110]. The partial capacitance technique is based on splitting the

multilayered substrate into several single-layer “substrates” with modified dielectric

constants, as shown in Fig.(3.8), assuming magnetic walls at the dielectric/dielectric

interfaces. Although this is an approximation, its general validity may be explained by

the fact that the total electrostatic energy in the CPW, Fig.(3.8a), is equal to the sum of

electrostatic energies in the partial capacitances. The capacitance of each of the single

layered “substrates” is evaluated using conformal mapping technique [23,47] and the

total capacitance is the sum of the partial capacitances. However, capacitance is

computed w.r.t. the height of the layer from the central capacitance. The p.u.l.

capacitance of any of the single-layer substrate CPW’s in Fig.(3.8) can be written as

3,2,1iwhere)k(K

)k(K2C

'i

i)i(r0i == εε

(3.59)

)iii(1- )ii( )i(- r3r2r2r1 rrrεεεεεεε =−== 321 1

(3.60)

while in the case of free space, p.u.l. capacitance is obtained from equation-(3.12).

( )( ) ( )( )( ) ( ) )ii(xK

2x,x/xarcsinF

)i(xK2

x,x/xarcsinF

222

222

πππθ

πππψ

πθπψ

=

=


69

Fig.(3.8): Splitting of a three-layered substrate CPW according to the partial capacitance technique.

The total p.u.l. capacitance is the sum of all partial capacitances and can be written as

3210T CCCCC +++=

(3.61)

As a result, effε and 0Z of multilayer CPW can be written as:

)'k(K

)k(K

)k(K

)'k(K

2

1

)'k(K

)k(K

)k(K

)'k(K

2

1

)'k(K

)k(K

)k(K

)'k(K

21

3

3

0

03r

2

2

0

02r

1

1

0

02r1reff

−+

−+

−+=

εεεεε

(3.62)

)k(K

)'k(K30Z

0

0

eff0 ε

π= (3.63)

where modulus ki (i=1,2,3) and 0k along with their complements can be computed using

equation-(3.10) and (3.12) respectively. The substrate thickness h in equation – (3.10)

will be replaced by hi (i=1,2,3) accordingly.


70

b. Non-planar CPW

The cross sections of multilayer non-planar CPW on elliptical, circular cylindrical, semi-

ellipsoidal and semi- circular cylindrical surfaces are shown in Fig.(3.9). The structures

encircle two dielectric substrates with dielectric constants 1rε and 2rε respectively. At

the upper side there is another dielectric layer with dielectric constant 3rε . The arc of the

central elliptical strip of CPW is 2ψ, the gap between the strip and the ground plane

is( )ψθ − and the ground plane width is ( )θπ 22 − . The transformation of MECPW into a

multilayer cylindrical CPW (MCCPW), in the z-plane is done using equation-(3.51).

Thus, five confocal ellipses in the ζ-plane are mapped into circles in the z-plane with their

radii obtained using equation-(3.52). Then MCCPW is transformed into multilayer CPW

with a finite ground plane using equation-(3.53) which gives the structural parameters:

)vi(r

rlnh)v(

r

rlnh)iv(

r

rlnh

)iii(r

rlnt)ii(w)i(s

3

53

1

32

2

31

3

42

===

=−== ψθΨ

(3.64)

The total capacitance CT p.u.l. of the multilayer CPW is the sum of the capacitances of

three CPWs and the free space capacitance. It can be written as

3210T CCCCC +++=

(3.65)

where C0 is the free space capacitance after removing the dielectric layers and C1 is the

capacitance of dielectric material of thickness H1=h1 and equivalent dielectric

constant( )2r1r εε − . Also C2 and C3 are the capacitances of dielectric layers of thickness

H2=h1+h2 with dielectric constant( )12r −ε , and of thickness H3=h3 with dielectric

constant( )13r −ε , respectively, as explained in [4]. Based on this method, the relation of


71

the free space and dielectric capacitance is given by equation-(3.41) and (3.43)

respectively.

As a result, effε and 0Z of MECPW and MCCPW using the conformal mapping method

can be obtained using equation-(3.62) and (3.63) respectively. When ground plane width

is ( )θπ 2− , the multilayer CPW on the semi- ellipsoidal (MSECPW) and semi-circular

cylindrical surfaces (MSCCPW) is obtained as shown in Fig.(3.9c) and (3.9d). Using

equation-(3.53) to (3.58), the modulus and their complementary can be obtained which

can be substituted in equation (3.62) and (3.63) to obtain effε and Z0 of MSECPW and

MSCCPW.

(a) (b)

(c) (d)

Fig.(3.9): Multilayer CPW on the curved surfaces: (a) Elliptical CPW (MECPW) ,(b) Circular Cylindrical CPW (MCCPW) ,(c) Semi-ellipsoidal CPW (MSECPW) and (d) Semi- circular cylindrical CPW (MSCCPW).


72

3.4.2 Coplanar Strip Line

CPS on Finite Thickness Substrate

A CPS with finite thickness substrate h and relative permittivity rε , shown in

Fig.(3.10a), is a complementary structure of the CPW structure i.e. the slots and

conductor strips of the CPW are replaced by the conductor strips and slot to get the CPS

structure. The product of CPSZ0 and CPWZ0 is a constant and is expressed as [106]

CPWeff

CPSeff

CPWCPS ZZεη

εη

44

22

00 === (3.66)

where η is the intrinsic impedance of the medium and is equal to 120π. However, the

complementary model of CPS fails w.r.t. the CPW structure on the finitely thick substrate

[105].

The conformal mapping is achieved in two steps. The first transformation, shown in

Fig.(3.10b) converts one half of the CPS with electric wall (1)-(6), that is treated as the

perfect conductor; to the coplanar strip on the infinitely thick substrate having relative

permittivity ( 1−rε ) by using equation – (3.23). The mapping function transforms one-

half of lower part of the CPS with the electric wall into conducting strip in the t-plane.

The second CMT, shown in Fig.(3.10c), is achieved by transforming coplanar strip

structures located in the t-plane into a rectangle in the w-plane using the SC-

transformation

)k(Kj)k(K)tt)(tt)(tt)(tt(

jvuw 't

t55

6 3216

1 +=−−−−

=+= ∫ (3.67)


73

The line capacitance C0 p.u.l for the upper half of the CPS structure with air medium

( 1r =ε ) and line capacitance C1 p.u.l of the CPS on the ( 1r −ε ) substrate are

)b()k(K

)k(K)(C)a(

)k(K

)k(KC

'r

'

6

601

0

000 2

1

2

−==

εεε (3.68)

The modulus k0 and complementary modulus '0k are given by equation – (3.12), whereas

the modulus k6 and complementary modulus 'k6 are given by

)h/btanh(

)h/atanh(k

2

26 π

π= , 26

26 1 k'k −= (3.69)

(a) (b)

(c)

Fig. (3.10): Lower half space conformal mapping of CPS with finite substrate thickness (a) z-plane, (b) t-plane, (c) w-plane.


74

The total line capacitance of the CPS on finite thick substrate is

CT ( rε ) = )k(K

)k(K)(

)k(K

)k(KCC

'r

'

6

60

0

0010 2

12

−+=+

εεε (3.70)

The total line capacitance of CPS on infinite air substrate is

C0T ( rε =1) = )k(K

)k(K

0

'00ε

(3.71)

Finally the effε and 0Z of the CPS on finite thick substrate are

( )

)k(K

)k(K

)k(K

)k(K

)(C

)(C '

'r

rT

rTeff

6

6

0

0

0 2

11

1

−+=

==

εεεε (3.72)

)k(K

)k(KZ

'eff 0

00

120

επ= (3.73)

Asymmetrical CPS

The asymmetrical CPS (ACPS) structure is shown in Fig. (3.11a). It has two strip

conductors of width w1 and w2 separated by the slot gap s. The conducting strips and slot

edges in the z-plane, shown in Fig. (3.11b) are marked by points δγβα ,,, such that

.δγβα >>> The total line capacitance C0T p.u.l. for ACPS in the free space is :

)k(K

)k(KC

'

T7

700 2ε= (3.74)


75

The modulus k7 and complementary modulus 'k7 are given by

( )

( )( ) )b(kk)a(wsws

swwsk ' 2

7721

217 1−=

++++= (3.75)

The line capacitance of the ACPS for the case of lower half space filled with infinite

extent ( 1−rε ) dielectric medium is

( ))k(K

)k(KC

'

r8

801 1−= εε (3.76)

The total line capacitance of the ACPS line is

( ))k(K

)k(K

)k(K

)k(K)(C

'

r

'

rT8

80

7

70 12 −+= εεεε (3.77)

(a) (b)

(c)

Fig.(3.11): Conformal mapping of ACPS of finite thickness substrate (a) Original structure, (b) z-plane, (c) w-plane.


76

The effε and Z0 of the ACPS on the finitely thick substrate is

)k(K

)k(K

)k(K

)k(K)(

)(C

)(C '

'rrT

rTeff

8

8

7

7

01

2

11

1−+=

== ε

εεε

(3.78)

)k(K

)k(KZ

'eff 7

70

60

επ= (3.79)

where

( ) ( )

( ) ( )

−

++

−

+

−

+

−

++=

h

wexpsww

hexpsw

hexp

h

wexpsw

hexpsww

hexp

k1

211

1121

8 221

2

221

2

πππ

πππ

(a) (3.80)

88 1 kk' −=

(b)

Non- Planar CPS

Fig.(3.12) shows the following four CPS structures on the non-planar surfaces:

i. Elliptical CPS (ECPS)

ii. Circular Cylindrical CPS (CCPS)

iii. Semi-Ellipsoidal CPS (SECPS)

iv. Semi-Circular Cylindrical CPS (SCCPS)

In Fig. (3.12a), the angle subtended by the arc of strip conductor and the gap between the

two strips at center are θ and 2ψ, respectively. For elliptical CPS (ECPS) and circular

cylindrical CPS (CCPS), ground plane width is ( )ψθπ 222 −− . Again, based on the


77

partial capacitance approximation, the overall capacitances per unit length of structures

are considered as the sum of the capacitance in free-space C0 and dielectric layer C1. For

the calculation of total unit length capacitance, the partial capacitances C0 and C1 can be

obtained using successive conformal transformation steps [47,131]. So, the overall

capacitances per unit length of both the structures are given by,

10 CCCT += (3.81)

where the air-region and dielectric layer capacitances are obtained from equation - (3.68).

The related modulus can be obtained from equation-(3.42) - (3.47) after replacing θ with

ψθ + .

(a) (b)

(c) (d)

Fig.(3.12): CPS on the curved surfaces: (a) Elliptical CPS (ECPS) ,(b) Circular Cylindrical CPS (CCPS) ,

(c) Semi-ellipsoidal CPS (SECPS) and (d) Semi- circular cylindrical CPS ( SCCPS ).

As a result, effε and 0Z of ECPS and CCPS using the conformal mapping method can be

written as:


78

)k(K

)'k(K

)'k(K

)k(Kreff

1

1

0

0

2

11

−+=

εε (3.82)

)'k(K

)k(KZ

eff 0

00

120

επ= (3.83)

The confocal elliptical cylinders are transformed into the circular cylinders in the z-plane

and then into the planar CPS using the equations-(3.50) – (3.53) [85,144]. This provides

the following conducing strip width (w), slot width (s), substrate thickness (h) and strip

conductor thickness (t) of the transformed ECPS into the corresponding planar CPS:

)iv(ba

baln

r

rlnt)iii(

ba

baln

r

rlnh)ii(s)i(w

22

33

2

3

11

22

1

22++

==++==== Ψθ (3.84)

When c = 0 equation- (3.52) gives the structural parameters for the CCPS, as shown in

Fig. (3.12b).When ground plane width is (π-2θ-2ψ), the CPS on the semi- ellipsoidal

(SECPS) and semi-circular cylindrical surfaces (SCCPS) is obtained as shown in

Fig.(3.12c) and (3.12d). On applying conformal mapping, the modulus k0 and k1 and their

complementary modulus '0k and '1k involved in the complete elliptic integral of the first

kind can be obtained using equations-(3.55) and (3.56) after replacing θ with

ψθ + [130]. On their substitution to the equation - (3.82) and (3.83), the effε and 0Z of

SECPS and SCCPS are obtained.

Multilayer CPS a. Planar CPS

A CPS sandwiched between multilayer dielectric substrates is shown in Fig. (3.13a). The

method of superposition of partial capacitances is used to determine the capacitance p.u.l.


79

of the structure. Hence the total p.u.l. capacitance is the sum of all partial capacitances

and can be written as

3210T CCCCC +++=

(3.85)

The configuration of these capacitances are similar to that shown in Fig.(3.8). The p.u.l.

capacitance of free space can be obtained from equation –(3.68a) and capacitance of any

of the single-layer substrate CPS’s can be written as [130]:

3212

0,,iwhere

)k(K

)k(KC

i

'i

)i(r

i ==εε

(3.86)

(a) (b)

Fig.(3.13): Multilayered CPS according to the partial capacitance technique.

The expressions for ( )irε

(i =1, 2, 3)

can be obtained from equation – (3.60). As a result,

effε and 0Z of multilayer CPS can be written as:

)k(K

)'k(K

)'k(K

)k(K

)k(K

)'k(K

)'k(K

)k(K

)k(K

)'k(K

)'k(K

)k(K rrrreff

3

3

0

03

2

2

0

02

1

1

0

021

2

1

2

1

21

−+

−+

−+=

εεεεε

(3.87)


80

)'k(K

)k(KZ

eff 0

00

120

επ= (3.88)

where modulus k0 and ki (i=1,2,3) along with their complements can be computed using

equation-(3.12) and (3.69) respectively. The substrate thickness h in equation – (3.69)

will be replaced by hi (i=1,2,3) accordingly.

b. Non-planar CPS

The cross sections of multilayer non-planar CPS is shown in Fig.(3.14) which encircle

two dielectric substrates with dielectric constants 1rε and 2rε respectively, and at the

upper side there is another dielectric layer with dielectric constant 3rε . The

transformation of MECPS into a MCCPS, in the z-plane is done using equation-(3.51).

Thus, five confocal ellipses in the ζ-plane are mapped into circles in the z-plane with their

radii obtained using equation-(3.52). Then MCCPS is transformed into multilayer CPS

using equation-(3.53) which gives structural parameters as:

)vi(r

rlnh)v(

r

rlnh)iv(

r

rlnh

)iii(r

rlnt)ii(w)i(s

3

53

1

32

2

31

3

42

===

=== θΨ

(3.89)

The total capacitance CT p.u.l. of the multilayer CPS is the sum of the capacitances of

three CPSs and the free space capacitance. It can be written as

3210T CCCCC +++=

(3.90)

The expressions of the free space and dielectric capacitance are given by equation-(3.68a)

and (3.43) respectively. As a result, effε and 0Z of MECPS and MCCPS using the


81

conformal mapping method can be obtained using equation-(3.87) and (3.88)

respectively. Using equation-(3.55) to (3.56) after replacing θ with ψθ + , the modulus

and their complementary for MSECPS and MSCCPS can be obtained which can be

substituted in equation (3.87) and (3.88) to obtain effε and 0Z of the structure.

(a) (b)

(c) (d)

Fig.(3.14): Multilayer CPS on the curved surfaces: (a) Elliptical CPS (MECPS) ,(b) Circular Cylindrical

CPS (MCCPS),(c) Semi-ellipsoidal CPS (MSECPS) and (d) Semi- circular cylindrical (MSCCPS).

3.4.3 Slotline

Standard Slotline The slotline, shown in Fig.(3.15a), of the finite width conductors could be viewed as the

CPS that supports the quasi-static TEM mode. The quasi-static nature of the slotline is


82

mentioned by Cohn and others [32,67,112]. The structure is analyzed for the computation

of the effε and 0Z with the help of conformal mapping method combined with some

semi-empirical modeling [67]. The dielectric layer is assumed to be homogenous and

isotropic, and the strips are assumed to have negligible thickness.

Firstly, the properties of slotline on air substrate are computed and then on a dielectric

substrate. In accordance with basic slotline theory [74,112], the analysis is made on the

assumption that the main part of electromagnetic field with significant influence on

slotline parameters is concentrated only on the “virtual” distance h0 below and above the

slot. It is achieved by following mapping function that transforms the structure from the

z-plane to the t-plane [67]:

)a(

h

ztanht

=2

π

)b(

)sk)(s()k(Kds

dw' 22

02

0 11

11

−−⋅−=

where the complex variable

)c(t

ts

0

02 1

2

1

αα

−−⋅

+=

and the parameters on air and dielectric substrate are

)d(h

wtanh

=

00 2

πα )e(h

wtanh

=21πα

)f(

h

.hh

r

++⋅=

20

0 1

013301

λε

(3.91)

where, λ0 is the free space signal wavelength. The structure is transformed from the z-

plane in Fig.(3.15b) and Fig. (3.15d) via the t-plane into the final form in the w-plane in

Fig. (3.15e).


83

(a) (b) (c)

(d) (e)

Fig. (3.15): Conformal mapping of slotline with finite substrate thickness (a) Original structure, (b) z-plane

without dielectric substrate, (c) t-plane, (d) z-plane with dielectric substrate, (e) w-plane.

K(k0) and K( 0k′ ) are complete elliptic integrals of the first kind and can be computed by

the approximate expression from equation-(3.8). Their modulus k0 and complementary

modulus 0k′ are given by

0

020 1

2α

α+

⋅=k (a) 20

20 1 kk −=′ (b) (3.92)

On dielectric substrate, modulus k1 and complementary modulus 1k′ are given by

1

121 1

2α

α+

⋅=k (a) 21

21 1 kk +=′ (b) (3.93)


84

The effε and 0Z of the standard slot line structure shown in Fig.(3.16) due to conformal

mapping is given by [67]:

)'k(K

)k(K

)k(K

)k(K 'r

eff0

0

1

1

2

11

−+=

εε (3.94)

)(

)(60

0

00 kK

kKZ

eff′

=ε

π (3.95)

The expressions are applicable for range: h/λ ≤ 0.01, 0.02 ≤ w/h ≤ 1 and 2.22 ≤ εr ≤ 20

with accuracy within 2%.

Non-Planar Slotline

Fig.(3.16) shows the following four slotline structures on the non-planar surfaces, located

in the ζ -plane:

i. Elliptical Slotline (ES)

ii. Circular Cylindrical Slotline (CS)

iii. Semi-Ellipsoidal Slotline (SES)

iv. Semi-Circular Cylindrical Slotline (SCS)

The angle subtended by its slot gap at the center is 2ψ and rε is the relative dielectric

constant of the substrate. For ES and CS ground plane width is ( )ψπ 22 − and for SES

and SCS ground plane width is( )ψπ 2− . For this transmission line, the same method, as

mentioned above, can be used. Assuming that only TEM modes exist, the confocal

elliptical cylinders are transformed into the circular cylinders in the z-plane and then into

the planar slotline using the equations-(3.50)-(3.53) [83].This provides the following


85

center slot gap (w), substrate thickness (h) and strip conductor thickness (t) of the

transformed ES into the corresponding planar slotline:

)iii(ba

baln

r

rlnt)ii(

ba

baln

r

rlnh)i(w

22

33

2

3

11

22

1

22++

==++

=== ψ (3.96)

After applying CMT and substituting above equation in equations – (3.91) - (3.95),

related modulus, their complementary, effε and 0Z of the non-planar slotline structure

can be obtained.

(a) (b)

(c) (d)

Fig.(3.16): Slotline on the curved surfaces: (a) Elliptical Slotline (ES) ,(b) Circular Cylindrical Slotline

(CS),(c) Semi-Ellipsoidal Slotline (SES) and (d) Semi- Circular Cylindrical Slotline (SCS).


86

Multilayer Slotline

Two multilayered forms of slotline had been investigated by Svačina et. al. [67] using

CMT: a double-layer sandwich slotline and a slotline on a double-layer composite

substrate. However, only composite substrate slotline is discussed here as it has been

used for further study of characteristics of multilayered slotline in this thesis.

a. Planar Slotline

A slotline on a double-layer composite substrate is shown in Fig.(3.17). For the virtual

distance h0, the following expression has been found by modification of equation-(3.91f)

and empirical modeling:

+−++⋅=

2

2

0

2

1212

20

2)(

0133.01

hh

hhh

rrr

λ

εεε (3.97)

Fig.(3.17): Multilayered Slotline: composite substrate.

By this treatment, similar to Fig. (3.15e), effε of a composite substrate slotline is

computed from


87

⋅−

+

⋅−

+=)'k(K

)k(K

)k(K

)k(K

)'k(K

)k(K

)k(K

)k(K 'r

'rr

eff0

0

2

22

0

0

1

121

2

1

21

εεεε (3.98)

where, i

iik

αα+

⋅=1

22 (a) 22 1 ii kk +=′ (b)

=

ii h

stanh

2

πα [i =1,2] (c) (3.99)

k0 and 0k′ are computed from equation –(3.92). However, for computing α0, we use

equation – (3.97) in equation (3.91d). The 0Z of the multilayered planar slotline is given

by equation-(3.95) with equation – (3.91) taken into consideration.

b. Non-planar Slotline

The cross sections of multilayer non-planar slotline are shown in Fig. (3.18).

(a) (b)

(c) (d)

Fig.(3.18): Multilayer Slotline on the curved surfaces: (a) Elliptical Slotline (MES) ,(b) Circular

Cylindrical Slotline (MCS),(c) Semi-Ellipsoidal Slotline (MSES) and (d) Semi- Circular Cylindrical Slotline (MSCS).


88

In these structures, non-planar slotline encircles two dielectric substrates with dielectric

constants 1rε and 2rε respectively. The ground plane width is ( )ψπ 22 − for multilayer

ES (MES) and multilayer CS (MCS) and is( )ψπ 2− for multilayer SES (MSES) and

multilayer SCS (MSCS). The transformation of MES in the ζ-plane into a MCS in the z-

plane and then into the planar slotline is done using equation - (3.50) - (3.53) [83], which

give structural parameters as:

)iv(r

rlnh)iii(

r

rlnh)ii(

r

rlnt)i(w

1

32

2

31

3

42 ==== Ψ

(3.100)

After applying CMT and substituting above equation in equations – (3.97) - (3.99),

related modulus, their complementary, effε and 0Z of the multilayered non-planar

slotline structure can be obtained.

chapter – 3 conformal mapping technique – an...

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