application of conformal mapping for electromagnetic
DESCRIPTION
Schwarz-Christoffel Transformation applied to calculate capacitance of Coplanar WaveguideTRANSCRIPT
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Applications of Conformal Mappings for Electromagnetics
Yuya SaitoElectrical and Computer Engineering
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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
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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
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Capacitance
a b1
a1
d1
εr
V
Electrical Flux
1
11
dbaC r
1
1
daC r
Two dimensional problem if b1=1
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Coaxial Cable
E field
H field
I
abC r
/ln2
r
VQC
the capacitance per unit length
dvsdEV vs
)( 0
Gauss’s Lawa b
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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
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Transmission Lines for Microwave Circuits
Transistor Resistor Substrate
Ground Plane
Center conductor
Air Bridge
εr
Coplanar Waveguide Slot lineMicrostrip Line
Center conductor
Ground
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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
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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
∞+∞-
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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
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+∞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
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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
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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
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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
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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
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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
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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
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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
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Mapping Function
yZ-plane
0x
vW -plane
0u
nZW
n/
yZ-plane
0x
vW -plane
0u2/
2ZW n=2
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Mapping Function
W-plane
0 ・ ・・・
・・
D ECB
I
HG
・
uvyZ-plane
x0 ・ ・・・
・ ・・
D ECB
I HG
Mapping Function
hZW
2sinh
ih
i∞
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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