Application of Conformal Mapping for Electromagnetic

Applications of Conformal Mappings for Electromagnetics

Yuya SaitoElectrical and Computer Engineering

Introduction

Modern applications of conformal mapping

•Heat Transfer

•Fluid FlowHydrodynamics and Aerodynamics

•ElectromagneticsStatic field in electricity and magnetism, Transmission

line and Waveguide, and Smith Chart etc

Transient Heat Conduction

Conformal Mappings for Electromagnetics

　　 u=constant （ blue line ） ⇔　 Electrical Flux 　

　　 v=constant （ red line) ⇔ 　 Magnetic Field (or electrical

potential)

Conformal Mapping ： z=f(w) 　　 z, w: complex values 　

z-plane 　　　　　　　　　　　　　　 w-plane

⇔

iy 　　 z=x+jy v1

x

v2

iv 　　 w=u+jvv1

uv2

Mapping a region in one complex plane onto another complex plane

For Electromagnetics

Capacitance

a b1

a1

d1

εr

V

Electrical Flux

1

11

dbaC r

1

1

daC r

Two dimensional problem if b1=1

Coaxial Cable

E field

H field

I

abC r

/ln2

r

VQC

the capacitance per unit length

dvsdEV vs

)( 0

Gauss’s Lawa b

１

２3

4

5

6 7

8

E field

H field

x

yZ-plane

123

45

6 7 8

a b

LogZW Mapping Function

iLogr

rθ

W-plane

1

2

3

4

5

6

7

8

π

2π

Logau

0

Logbu

0ar br

u

v

abC r

/ln2

1

1

daC r

a1

d1

Conformal Mapping for the Coaxial Cable

・

= u + iv

Transmission Lines for Microwave Circuits

Transistor Resistor Substrate

Ground Plane

Center conductor

Air Bridge

εr

Coplanar Waveguide Slot lineMicrostrip Line

Center conductor

Ground

Coplanar Waveguide (CPW)

Center ConductorGround Plane

Substrate

εr

Current

How can we derive the capacitance of unit per length?

x

yair

εrCross section

Unit length

Schwarz-Christoffel Transformation

Schwarz-Christoffel Transformation

w-planeZ-plane

x

y

P1P2

P3P4

P5

α1

α2

α3α4

α5

u

v

・X’1 ・ ・・ ・

X’2 X’3 X’4 X’5

(P1) (P2) (P3) (P4) (P5)

1)/('1)/('2

1)/('1 )()()( 21 n

nxwxwxwAdwdz

∞+∞-

Assumption•Ground plane is long enough

•Substrate thickness is large enough

•The thickness of the metal is small enough

+∞x

y

-∞

air

εr

-i∞

SC transformation for CPWs

SC transformation

Metal thickness is small enough

Substrate

+∞x

y

-∞

air

εr

-i∞

SC Transformation for CPWs

Symmetry

E-field

Parallel plate capacitor!!

u

v π/2 radπ/2 rad

air

① ③②④

①

②③

④

⑤

⑤

⑥

⑥

π/2 radπ/2 rad

①④

③∞∞

⑤

⑥

②

Z-planeSC transfrom

u

ivW-plane

SC Transformation for CPWs

u1=K(k)

au

bzaz

Adzdw0 22220 ))((

1

)(1 kKu

))(( 2222 bzaz

Adzdw

where A :constant, k=a/b

First kind complete elliptic function

+∞x

y

-∞air

a-a b-b

Z-plane

0

)(kK

u

vW-plane

SC Transformation for CPWs

u1+iv1=K(k)+iK(k’)

K’(k)

)'(1 kKv +∞x

y

-∞air

a-a b-b

Z-plane

)'()(2

1

1

kKkK

vuC rr

where A :constant, k’2=1-k2

The substrate case is the same as the air region case

b

a

ivu

u bzaz

Adzdw))(( 2222

11

1

Assumption•Ground plane is long enough •Substrate thickness is large enough

•The thickness of the metal is small enough+∞

x

y

-∞

air

εr

-i∞

Metal thickness is small enough

Substrate

Can we still use Conformal Mapping???

Consideration of the assumption

Finite length of the ground plane

y+∞

x-∞

air

a-a b-b c-c

Z-plane y+∞

xa b cSymmetry

i∞

+∞t1 t2 t3

Mapping Function2zt

0

21 at 2

2 bt 23 ct

-∞

T-plane

① ② ③ ④ ⑤⑥

u

v π/2 radπ/2 rad

②

③④

⑤⑥

π/2 radπ/2 rad

① ②③

④

⑤⑥

①

SC Transformation

-i∞Substrate Substrate

Substrate

air

Z-plane

Finite thickness of the substrate

y+∞

x-∞

air

a-a b-bh

ih

hat

2sinh1

hbt

2sinh2

Mapping Function

hzt

2sinh

-i∞

t1 t2 +∞-∞ -t1-t2

T plane W plane

SC Transformation

u

v

Substrate

Air region is the same as previous way

Finite thickness of the metal

+∞x

y-∞

airZ-plane

Substrate

z1

z2 z3

z4z5

z6z7

z8

+∞-∞air

W-plane

w1・・・・・・・・ w ２ w ３w ４w ５w6w7w8

η1・ ・

・・・・

・・ η2

η3η4η5η6

η7 η8

η-plane

SC Transformation

SC Transformation

Summary

•Show the derivation of the capacitance for the EM (RF) devicesex: phase velocity, characteristic impedance, and attenuation loss

•Conformal mapping is powerful way to get the analytical solutions!!

constrain•Only 2 dimensional problem

•Some assumptions are needed

•Limitation of mapping functions

Mapping Function

yZ-plane

0x

vＷ -plane

0u

nZW

n/

yZ-plane

0x

vＷ -plane

0u2/

2ZW n=2

Mapping Function

W-plane

0 ・ ・・・

・・

D ECB

I

HG

・

uvyZ-plane

x0 ・ ・・・

・ ・・

D ECB

I HG

Mapping Function

hZW

2sinh

ih

i∞

a1 a2

d1

a

Must be Uniform

1daC r

1

3

1

2

1

1

da

da

da

rrr

a

d1

Uniform E field Non Uniform E field

1da

r

321 CCCC

Non Uniform E field in the capacitor

a3

Strong field Strong fieldWeek field