Applications of Conformal Mappings for Electromagnetics

1
Applications of Conformal Mappings for
Electromagnetics
Yuya SaitoElectrical and Computer Engineering
2
Introduction
Modern applications of conformal mapping

• Heat Transfer

Transient Heat Conduction

• Fluid Flow

Hydrodynamics and Aerodynamics

• Electromagnetics

Static field in electricity and magnetism,
Transmission line and Waveguide, and Smith Chart
etc
3
Conformal Mappings for Electromagnetics
For Electromagnetics
uconstant(blue line) ? Electrical Flux
vconstant(red line) ? Magnetic Field (or
electrical potential)
4
Capacitance
V
b1
a
a1
d1
er
Two dimensional problem if b11
Electrical Flux
5
Coaxial Cable
the capacitance per unit length
Gausss Law
a
b
I
E field
H field
6
Conformal Mapping for the Coaxial Cable
Z-plane
y
2
3

v
r
W-plane
1
4
x
2p
?
8
a
b
5
8
7
6
5
p
a1
6
7
4
3
2
u
E field
u iv
1
d1
H field
7
Transmission Lines for Microwave Circuits
Air Bridge
Center conductor
Ground Plane
er
Resistor
Transistor
Substrate
Center conductor
Ground
Microstrip Line
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Coplanar Waveguide (CPW)
Unit length
Schwarz-Christoffel Transformation
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Schwarz-Christoffel Transformation
w-plane
v
X1
X2
X3
X4
X5
8
8-





u
(P1)
(P2)
(P3)
(P4)
(P5)
10
SC transformation for CPWs
SC transformation
y
Metal thickness is small enough
air
-8
8
x
er
Substrate
-i8
11
SC Transformation for CPWs
Z-plane
SC transfrom
y
Symmetry
air
8
-8
x
?
?
?
?
?
?
v
p/2 rad
p/2 rad
er
?
?
?
?
-i8
Parallel plate capacitor!!
air
u
E-field
?
?
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SC Transformation for CPWs
Z-plane
W-plane
iv
where A constant, ka/b
u
First kind complete elliptic function
u1K(k)
0
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SC Transformation for CPWs
Z-plane
where A constant, k21-k2
W-plane
v
u1iv1K(k)iK(k)
K(k)
The substrate case is the same as the air region
case
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Consideration of the assumption
y
Metal thickness is small enough
air
-8
8
x
er
Substrate
-i8
Can we still use Conformal Mapping???
15
Finite length of the ground plane
i8
y
8
x
a
b
c
Symmetry
Substrate
Substrate
-i8
Mapping Function
v
p/2 rad
p/2 rad
T-plane
SC Transformation
air
?
?
?
?
?
?
?
?
?
?
8
t1
t2
-8
t3
0
u
Substrate
?
?
16
Finite thickness of the substrate
Z-plane
Air region is the same as previous way
Substrate
W plane
T plane
v
SC Transformation
u
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Finite thickness of the metal
SC Transformation
SC Transformation
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Summary
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Mapping Function
21
Mapping Function
ih
i8
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Non Uniform E field in the capacitor
a2
Must be Uniform
d1
Strong field
Strong field
Week field
Uniform E field
Non Uniform E field