Applications of Conformal Mappings for Electromagnetics

Yuya Saito Electrical and Computer Engineering

IntroductionModern applications of conformal mapping

Heat TransferTransient Heat Conduction

Fluid FlowHydrodynamics and Aerodynamics

ElectromagneticsStatic field in electricity and magnetism, Transmission line and Waveguide, and Smith Chart etc

Conformal Mappings for Electromagnetics

Conformal Mapping values z-plane iyv1 x v2 z=x+jy

z=f(w)

z, w: complex w-plane w=u+jvu v2

iv v1

Mapping a region in one complex plane onto another complex plane

For Electromagnetics u=constant blue line v=constant red line) potential) Electrical Flux Magnetic Field (or electrical

Capacitance V

aa1 d1

b1

a1b1 C ! Ir d1 a1 C ! Ir d1rTwo dimensional problem if b1=1

Electrical Flux

Coaxial Cablethe capacitance per unit length

T T (I 0 E ) y ds ! V v dvs V

Q C! V

Gausss Lawa b

IrI E field H field

2TI r C! ln b / a

Conformal Mapping for the Coaxial Cable y3 4 5 6a

Z-plane

4 5

3 2

1

6 7 8

r8b

xMapping Function

v

28 7 6

W-planeC ! Ir a1 d1

7

W ! LogZ ! Logr iU

a1

5 4 3 2 1

C!

2TI r ln b / a

E field H field

= u + ivu ! Logar!a U !0

ur !b U !0

d1

u ! Logb

Transmission Lines for Microwave CircuitsAir Bridge

Center conductor Ground Planer

TransistorCenter conductor

Resistor

Substrate

Ground

Microstrip Line

Coplanar Waveguide

Slot line

Coplanar Waveguide (CPW)

Center Conductor

Ground Plane

y air

Currentr

xUnit length

Substrate

Cross section

r

How can we derive the capacitance of unit per length? Schwarz-Christoffel Transformation

Schwarz-Christoffel Transformation

yP44 3

Z-plane P3

v X1 X2(P1) (P2)

w-plane

P55 2 1

xP2

X3 X4 X5(P3) (P4) (P5)

P1

dz ' ' ! A( w x1' ) (E1 / T )1 ( w x2 ) (E 2 / T ) 1 ( w xn ) (E n / T ) 1 dw

+ u

SC transformation for CPWs SC transformation yMetal thickness is small enough

air + x

Assumption Ground plane is long enough Substrate thickness is large enough The thickness of the metal is small enough

-Substrater

-i

SC Transformation for CPWs

Z-plane y Symmetry -r

SC transfrom air/2 rad /2 rad

+ x/2 rad

vair

/2 rad

-i

Parallel plate capacitor!! E-field

u

SC Transformation for CPWs Z-plane y --b -a a b

air + x

dw A ! dz ( z 2 a 2 )( z 2 b 2 )

W-plane iv

u1

0

dw !

a

Adz ( z 2 a 2 )( z 2 b 2 )

0

u1 ! K (k )where A :constant, k=a/b

u0 u1=K(k)

K (k )First kind complete elliptic function

SC Transformation for CPWs Z-plane y --b -a a b

air + x

u1 iv1

u1

dw !

b

Adz ( z 2 a 2 )( z 2 b 2 )

a

v1 ! K (k ' )where A :constant, k2=1-k2

W-plane v

u1+iv1=K(k)+iK(k) K(k)

u1 2 K (k ) C ! Ir ! Ir v1 K (k ' )

u

The substrate case is the same as the air region case

Consideration of the assumption

Assumption yMetal thickness is small enough

Ground plane is long enough

air + x

Substrate thickness is large enough The thickness of the metal is small enough

-Substrater

-i

Can we still use Conformal Mapping???

Finite length of the ground planeZ-plane

ya b

i

air + c xSymmetry

ya b

--b -a -c Substrate

+ c xSubstrate

-iMapping Function

t ! z2T-planeair

/2 rad SC Transformation t3+

v

/2 rad

-

0 t1

t2

Substrate

u/2 rad /2 rad

t1 ! a 2

t2 ! b 2

t3 ! c 2

Finite thickness of the substrate Z-plane --b -a Substrate

ya b

air + x h

Air region is the same as previous way

ihMapping Function

T plane -

Tz t ! sinh 2ht1 t2

W plane v +SC Transformation

-t2 -t1

-i

Ta t1 ! sinh 2h Tb t 2 ! sinh 2h

u

Finite thickness of the metal Z-plane -SC Transformationz8 z5 z7 z6

yz1 z2

airz4 z3

+

x

Substrate

-plane W-plane air -w8 w7 w6 w w1 w w w7 8 1 2 6 5 4 3

SC Transformation

+

Summary

Conformal mapping is powerful way to get the analytical solutions!!constrain Only 2 dimensional problem Some assumptions are needed Limitation of mapping functions

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

Mapping Function

W !ZZ-plane

n

yT /n0

-plane

v x0

u-planeW ! Z2n=2

Z-plane

yT /20

v u0

x

Mapping FunctionMapping Function

TZ W ! sinh 2h

Z-planeB C

y0 D E

W-plane

vC 0 D E

x

B

u

G

ih

I

H G

I H

i

Non Uniform E field in the capacitor a a2

aMust be Uniform

a1 d1 d1

a3

Strong field Week field Strong field

Uniform E field

a C ! Ir d1

Non Uniform E field C ! C1 C2 C3a1 a2 a3 ! Ir Ir Ir d1 d1 d1

a ! Ir d1