Parallel Plate Transmission Line
y
c
a
Assume e  jz
er
b
x
Partially loaded parallel
Plate waveguide
variation , no variation with x
2
2

k

in the air region
 0
2
2
2
TM modes :  t ez  k c ez  0 , k c   2
2
k

in the dielectric region

for air region
p
Let k c  
 for the dielectric region
 2  p 2  (e r  1)k 02
 je r k 0Y0 e z
  2
y
hx ( y )  
jk Y ez
 02
 p y
e z ( y )  C1 sin y
e z ( y )  C 2 sin p (b  y )
For dielectric region
for air region
0 ya
a yb
C1 sin a  C 2 sin pc
er
1
C1 cos a   C 2 cos pc

p
tan a  e r p tan pc Transceden tal equation must be solved
simultaneo usly with  2  p 2  (e r  1) k 02
 , p
  k 02  p 2  k 2  l 2
Most of the modes will be nonpropaga ting if  is imaginary
The variation is e
 z
and the field decays exponentia lly.
The value of  between k 0 and k can occur if p  jp0
Let  0 to be the correspond ing value of  then :
 0 tan  0 a  e r p0 tan p0 c
 20  p02  (e r  1)k 02
Low Frequency Solution
When the frequency is low,
k 02 is very small number ,  0 and p0 are very small 
 0 a  e r p02 c
2
(e r  1) k 02  p02 
e r p02 c
a
2
(
e

1
)
k
2
r
0a
or p0 
The solution for  is
a  e rc
  k  p0
2
0
2
e rb

k0  e e k0
a  e rc
e e is the effective dielectric constant
y
c
b
er
a
x
-W
e e   LC
W
L, C are the static inductance and capacitanc e
per meter.
The time average stored magnetic energy is Wm
0
0
1 2
Wm 
H dxdy 
WbJ  LI z


0

w
4
2
4
b
I z  2WJ z
L 0
2W
b
w
2
x
2
z
C2  C1 sin  0 a / j sinh p0 c   jC1  0 a / p0 c
0 ya
ez  C1 0 y
j
b
ey  
C1   jC1
0
(e r  1)c
je r k 0Y0
(e r c  a )e r
hx 
C1  jY0C1
0
(e r  1)c
Ca Cd
e 0 2W
e 0e r 2W
The capacitanc e C 
, Ca 
, Cd 
Ca  Cd
c
a
e 0e r 2W
C
e rc  a
e 0e r  0 b
LC 
e rc  a
   LC
The capacitanc e C 
e 0e r 2W
C
e rc  a
e 0e r  0 b
LC 
e rc  a
Ca Cd
Ca  Cd
, Ca 
e 0 2W
c
   LC
Field expression s :
C2  C1 sin  0 a / j sinh p0 c   jC1  0 a / p0 c
0 ya
ez  C1 0 y
ey  
j
b
C1   jC1
0
(e r  1)c
, Cd 
e 0e r 2W
a
je r k 0Y0
(e r c  a )e r
hx 
C1  jY0C1
0
(e r  1)c
In the air region :
(b  y )
e z  C1 0 a
c
j 0 a
b
e y   2 C1   jC1e r
(e r  1)c
p0 c
hx 
j k 0Y0
0
C1  jY0C1
b
V    e y dy  jC1
0
I z  2WJ z  2WH x
(e r c  a )e r
(e r  1)c
b
(a  e r c)
(e r  1)c
The characteri stic impedance is :
Z0
V
Zc  
I 2W
( a  e r c )b
er
L

C
In the low frequency limit, the dominant mode of
propagatio n becomed a TEM mode (quasi - TEM mode)
At hight frequency the mode of propagatio n is an E mod
High Frequency Solution:
At high frequency k 0 and  0 , p0 are large.
tanh p0 c  1
 0 tanh  0 a  e r p0  e r (e r  1)k 02   20
 20  p02  (e r  1)k 02 The solution is independen t of b
e z ( y )  C1 sin  0 y
for 0  y  a
sinh p0 (b  y )
e z ( y )  C 2 j sinh p0 (b  y )  C1 sin  0 a
sinh p0 (b  a )
e p0 ( b y )
 C1 sin  0 a p0 ( ba )  C1 sin  0 ae p0 ( y a ) for a  y  b
e
The field decays exponentia lly away from the air - dielectric surface
and does not depend on b as long as p0 c  p0 (b  a ) is large. The field
is guided by the dielectric sheet. This type of mode is called surface
wave mode.
Microstrip Transmission Line
y
w
H
x
r ( x, y , z )  r s ( x, z ) ( y  H )
J ( x, y , z )  J s ( x, z ) ( y  H )
B    A,
  E   jB   j  A
E   jA  

  H  jD  J
For anisotropi c dielectric :
D  e 0e r ( E x aˆ x  E z aˆ z )  e 0e y E y aˆ y
    A    A   2 A  j 0 D   0 J


D  e 0e r [ j ( Ax aˆ x  Az aˆ z )  aˆ x
 aˆ z
]
x
z

 e 0e y ( jAy aˆ y  aˆ y
)
y
let
  A   je 0e r  0  (Lorentz condition)
-  2 A  j 0 [  je 0e r A  e 0  e r

 
 e 0 (e y  e r ) jaˆ y Ay  aˆ y
]   0 J
y 

J does not have a y component 
 2 Ax  e r ( y )k 02 Ax    0 J x
 2 Az  e r ( y )k 02 Az    0 J z
 e r
 
 Ay  e y ( y )k Ay  j 0e 0 
 (e y  e r )


r

y


2
2
0



 j 0e 0 (e y  e r )
  ( H )e r  1) ( y  H ) 
y


 D  r

  2   2      
e r  2  2    e y
  e r2 k 02 
y  y  y 
 x
Ay
r
   j (e y  1) Ay ( H ) ( y  H )  j (e y  e r )
e0
y
Boundary conditions:
lim  0 
H 
H 
 Ax
Ax
dy 
2
y
y
2
H
H
 lim  0 
H 
H 
  0 J sx ( x, z ) ( y  H )dy
   0 J sx
Ax
y
H
   0 J sx
y

y
,
H
Ax
 0 H z
y
Ay
Az
y
,
H
   0 J sz
H
Az
 0 H x
y
H
 j 0e 0 (e r  1) ( H )
H
H

ey
y
H
rs

 j (e y  1) Ay ( H )
e0
In the substrate region away from the interface we have :
( 2  e r k 02 ) Ax  0
( 2  e r k 02 ) Az  0
( 2  e y k 02 ) Ay  j 0e 0 (e y  e r )

y
Ay
 2  2 e y  2
2
 2 
 e r k 0    j (e y  e r )
2
2
x
z
e r y
y
In the air region e r  e y  1
For an isotropic substrate e r  e y
d 2 ez
2


ez  0
2
dy
for 0  y  a
d 2 ez
2

p
ez  0
for a  y  b
2
dy
ez ( y )  0
at y  0, b , e z ( y ) is continuous at y  a
H x is continuous at y  a 


ey ( y)  


j
2
j
p2
ez
y
e z
y
e r ez
 2 y
y a
1 ez
 2
p y
For dielectric region
for air region
y a