applications of conformal mappings for electromagnetics - saito
TRANSCRIPT
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