ECE 5317-6351
Microwave Engineering
Fall 2011
Prof. David R. Jackson
Dept. of ECE
Notes 5
Waveguides Part 2:
Parallel Plate Waveguide
1
Field Equations (from Notes 4)
Summary
Hx 
Ez
j 

 c
kc2 
y
kz
H z 

x 
These equations will be useful to us
in the present discussion.
Ez
H z 
j
H y  2   c
 kz

kc 
x
y 
Ex 
Ez
H z 
j

k


 z

kc2 
x
y 
k 2  2  c
kc   k  k
2
j 
Ey  2 
kc 
Ez
H z 
kz
 

y
x 

2 1/2
z
2
Parallel-Plate Waveguide
y
 Both plates assumed PEC
 w >> d, 


0
x
, , s
d
Neglect x variation,
edge effects
x
w
z
The parallel-plate stricture is a good 1ST order model for a
microstrip line.
w
,, 
d
3
TEM Mode
Parallel-plate waveguide
2 conductors  1 TEM mode
y
To solve for TEM mode:
for
t2  0
, , s
0 xw
0 yd
Boundary conditions:
( x,0)  0 ; ( x, d )  V0
 2 2 
     2  2   0
 x y 
2
t
d
x
z
w
k z    j  k    c  k   jk 
  k
  k 
c    j
s

4
TEM Mode (cont.)
2
0
2
y
where
  x,0  0 &   x, d   V0
 ( x, y )  A  By
V
  ( x, y )  0 y ; 0  x  w
d
0 yd
et  x, y   t    yˆ
@ y0
 A0
@yd
Vo

ˆ
  y
y
d
 E ( x, y, z )  et ( x, y )e
jkz
V0
  yˆ e
d
kz  k    c
B
V0
d
jkz
c    j
s

5
TEM Mode (cont.)
V
E ( x, y, z )   yˆ 0 e
d
y
jkz
Recall
H
1

( zˆ  E )
 H  x, y, z    xˆ
d
V0
e
d
, , s
x
jkz
z
w
For a wave prop. in + z direction
y
Time-ave. power flow in + z direction:
V0
E
H
, , s
x
P 
1 2  w   1  2 k z
V0   Re  *  e
2
 d   
*

1 
P   Re   ( E  H )  zˆ dS 
2 s

2
wd


1   V0
2 k z
 Re     * 2 zˆ  zˆ e dydx) 
2  0 0   d 


1
1 2 1 
 V0  2   wd  Re  *  e2 k z
2
d 
 
6
TEM Mode (cont.)
Transmission line voltage
y
0
V ( z )   E  yˆ dy
I
k z  k    c
d
V ( z )  V0 e
jkz
V
d
C
Transmission line current
w
I ( z )   H  x, d , z   xˆ dx
0
I0
Note:
PEC : J s  nˆ  H

sz
 J  Hz
, , s
-
x
w
z
V w
I ( z)   0 e
 d
I
+
Characteristic Impedance
V0 e jkz
Z0 
I 0 e jkz
(Assume + z wave)
d
Z0  
w
jkz
Phase Velocity (lossless case)
vp 


c


  
r  r
c = 2.99792458 108 m/s
7
TEM Mode (cont.)
For wave propagating in + z direction
Time-ave. power flow in +z direction:
(calculated using the voltage and current)
1
Re VI *
2
*

1
  V0 w  2 k z 

 Re V0 
e


2

d

 



P 
P 
Recall that we found from
the fields that:
1 2  w  1 
P  V0   Re  *  e2 k z
2
 d   

 1 
1
2 w
V0   Re  *  e 2 k z
2
 d   
same
This is expected, since a TEM mode is a
transmission-line type of mode, which is
described by voltage and current.
8
TEM Mode (cont.)
We can view the TEM mode in a parallel-plate waveguide as a
“piece” of a plane wave.
y
E
H
PEC
 ,  ,s
PMC
PMC
x
PEC
The PEC and PMS walls do not disturb the fields of the plane wave.
PEC :
nˆ  E  0
PMC :
nˆ  H  0
9
TMz Modes (Hz = 0)
Recall Ez ( x, y, z)  ez ( x, y) e
jkz z
y
where
 2 2
2


k
 2
c  ez  0,
2
 x y

d
1
2 2
z
, , s
x
kc  [ k 2  k ]
z
w
subject to B.C.’s Ez = 0 @ y = 0, d
ez  x, y   A sin(kc y )  B cos(kc y )
@y0 B0
@ y  d  kc d  n
n  0,1, 2,....  kc 
n
d
10
TMz Modes (cont.)
 n 
ez  x, y   A sin 
y
d


n  0,1, 2,...
 n 
 Ez  An sin 
y e
 d 
k z  k 2  kc2
jk z z
 n 
 k 

 d 
2
2
k 2  2  c
Recall:
j c Ez j c  n 
 n
Hx  2
 2 An 
 cos 
kc y
kc
 d 
 d
jk z Ez
jk z  n 
 n
Ey 

A
cos
n


kc2 y
kc2
 d 
 d
Ex  0
Hy  0
No x variation

ye


ye

Hz  0
jk z z
jk z z
11
TMz Modes (cont.)
Summary
 n 
Ez  An sin 
ye
 d 
y
jk z z
jk z
 n 
Ey 
An cos 
ye
kc
 d 
j c
 n 
Hx 
An cos 
ye
kc
 d 
Ex  H y  H z  0
kc 
n
;
d
n  0,1, 2,...
 n 
kz  k 2  

 d 
k 2   2  c
, , s
d
jk z z
x
w
z
jk z z
Each value of n corresponds to a unique
TM field solution or “mode.”
 TMn mode
2
Note:
n  0  kz  k
 TM 0  TEM
(In this case, we absorb the An coefficient
with the kc term.)
12
TMz Modes (cont.)
Lossless Case
kc


2

n



2

kz  k  


 d  




2
  k  k 
2
2
c
1
2
c     
1
2
y
n  0,1, 2,...
d
x
k   
2
2
for k 2  kc2
 k z    k 2  kc2
 propagating mode
, , s
z
w
for k 2  kc2
 k z   j kc2  k 2   j
 e  jk z z  e   z
Fields decay exponentially
 evanescent fields
 “cutoff” mode
13
TMz Modes (cont.)
Frequency that defines border between cutoff and propagation
(lossless case):
f  cutoff frequency
    
c
@ f  fcn
c
k  kc  cn  
n
f cn 
2d
1
cutoff frequency for TMn mode

TEM
prop.
single
mode
prop.
n
d
TM1
2
modes
prop
TM2
TM3
3
mode
prop.
….
f
cuttoff
0
f c1
fc2
fc3
14
TMz Modes (cont.)
Time average power flow in z direction (lossless case):

TMn
P
wd

1 
 Re     E  H *   zˆ dydx 
2 0 0

c     
wd

1 
  Re    E y H x*dydx 
2 0 0



PTMn

2

2  n
Re{
k
}
A
w
cos
y
z
n
0  d  dy
2kc2
d
y
d
, , s
x
z
w
d

;
n

0
2


  2 Re{k z } An w  2

2kc

d ; n 0 

n  0,1, 2,...

Real for f > fc
Imaginary for f < fc
15
TEz Modes
Recall
H z ( x, y, z)  hz ( x, y) e
y
jkz z
, , s
d
where
x
 2 2
2
 2  2  kc  hz  x, y   0,
 x y

subject to B.C.’s Ex = 0
1
2 2
z
kc  [ k 2  k ]
z
w
@ y=0, d
1  H z H y 
Ex 



j c  y
z 
 hz  A sin(kc y )  B cos(kc y )
PEC : H  nˆ  0
@y 0  A0
n
@ y  d  kc d  n , n  1, 2,3,...  kc 
d
16
TEz Modes (cont.)
 n 
hz  x, y   Bn cos 
y
 d 
 n 
 H z  Bn cos 
ye
d


n  1, 2,3, ...
k z  k 2  kc2
 n 
 k 

 d 
jk z z
Recall:
 j H z j  n   n  jkz z
Ex 
 2 Bn 
ye
 sin 
2
kc
y
kc
 d   d 
jk z H z
jk z  n   n  jkz z
Hy 


Bn 
y e
 sin 
2
2
kc y
kc
 d   d 
Hx  0
Ey  0
No x variation
2
2
k 2  2  c
Ez  0
17
TEz Modes (cont.)
Summary
 n 
H z  Bn cos 
ye
 d 
y
jk z z
j
 n  jk z z
Bn sin 
ye
kc
 d 
 jk z
 n  jk z z
Hy 
Bn sin 
ye
kc
 d 
d
Ex 
H x  E y  Ez  0
kc 
n
; n  1, 2,...
d
 n 
kz  k 2  

 d 
k 2   2  c
2
, , s
x
z
w
Each value of n corresponds to a
unique TE field solution or “mode.”
TEn mode
Cutoff frequency
n  1 
f cn 


2d   
18
All Modes
For all the modes of a parallel-plate waveguide, we have
n  1 
f cn 


2d   
TEM
prop.
single
mode
prop.
c     
TE1
TM1
TE2
TM2
TE3
TM3
3
modes
prop
5
mode
prop.
….
f
cuttoff
0
f c1
fc2
fc3
The mode with lowest cutoff frequency is called the “dominant”
mode of the wave guide.
19
Power in TEz Mode
Time average power flow in z direction (lossless case):

TEn
P
c     
wd

1 
*
 Re     E  H   zˆ dydx 
2 0 0

wd

1 
 Re    Ex H *y dydx 
2 0 0


d
P

d
2

2
c
4k
Re k z  Bn
, , s
x
 n 
  2 Re{k z } Bn W  sin 
y  dy
2kc
 d 
0

TEn
y
2
2
z
w
Wd 
n = 1,2,…..
Real for f > fc
Imaginary for f < fc
20
Field Plots
y
TEM
x
y
x
TM1
y
TE1
x
21