Dispersion for a Rectangular WG
Consider  2  k 2  kc2 .
i)
If f<fc,    j and  2  k 2  kc2
  kc2  k 2
2
 2   2 
 
 

 c    
2
2
 1  1
 2     
 c    

2
c

ii)
k 
1  
 kc 
2
c
2
2
 f 
1  
 fc 
2
If f>fc,  is real and k 2   2  kc2 .
DEGENERATE MODES
In WG’s, several modes with different configurations have the cutoff frequency. These are called
degenerate modes. It can be seen that in a waveguide the possible TEmn and TMmn modes are always
degenerate. The waveguide dimensions are always selected in a way that only the desired mode
(generally TE10 or TM11) propagate and higher modes are not supported.
TEH DOMINANT MODE (TE10 MODE)
With m=1 and n=0 , we get the dominant mode with the lowest cutoff frequency.
fc
10
 m 2  n 2 

 
 
  
 a   b   m1 a
n 0
f c10 
1
2
1 
1

 a 2a 
c 
10
2
2

 2a
kc10  a
g 
10

1    2a 
2
Note that the cutoff wavenumber, cutoff frequency, cut off wavelength all depends on the geometry.
The Field Components of the Dominant Mode
  x   j10 z
H z  A10 cos 
e
 a 
Ex  0
 j a 
  x   j10 z
Ey   
 A10 sin 
e
 

 a 
j a
  x   j10 z
Hx 
A10 sin 
e

 a 
Ez  0
Hy  0
 

a
2
With,   k 2  
We see that the field components are independent of y. Only the y-directed component of the electric
field exists and has a simple sinusoidal variation of strength across the width of the WG. The magnetic
field exits in the closed loops in the x-z plane and the whole field pattern moves along the WG at the
phase velocity v p 

 f g .

The electric field lines terminate in electric charges in the walls of the WG. As the wave travels along hte
WG, the currents in the WG walls redistribute these charges so that the electric field always correctly
terminated.
Field lines corresponding to TE10 MODE
The Surface Currents on the walls of the WG (TE10 Mode)
The surface current density on a perfect conductor is:
J s  nˆ X H
On x=0 wall,

nˆ  aˆ x , J s  aˆ x X A10e  j10 z aˆ z
J s  aˆ y A10e

 j 10 z
On x=a wall,

nˆ  aˆ x , J s  aˆ x X  A10e j10 z aˆ z

J s  aˆ y A10e j10 z
On y=0 wall,
 j a
x 
  x    j10 z
nˆ  aˆ y , J s  aˆ y X 
A10 sin 
 aˆ x  A10 cos 
 aˆ z  e

a
a



 


j  a   x    j10 z
x  ˆ
J s  A10  cos 
sin 
 ax 
 aˆ z  e

 a 
 a  

On y=b wall,
 j a
x 
  x    j10 z
nˆ  aˆ y , J s  aˆ y X 
A10 sin 
 aˆ x  A10 cos 
 aˆ z  e
 a 
 a  
 

j  a   x    j10 z
x  ˆ
J s   A10  cos 
sin 
 ax 
 aˆ z  e

 a 
 a  

The surface charge density  s can be determined from s  nˆ. D ,
On x=0 wall,
nˆ  aˆ x , nˆ. D  aˆ x . 0 E y aˆ y   0
s  0
On x=a wall,
nˆ  aˆ x , nˆ. D  aˆ x . 0 E y aˆ y   0
s  0
On y=0 wall,
nˆ  aˆ y , nˆ. D   0 E y
s  
j 0  a


y 0
A10 sin
x
a
j 0 a

  x   j z
A10 sin 
e
 a 
e j10 z
On y=b wall,
nˆ  aˆ y , nˆ. D   0 E y
s 
j 0  a

A10 sin
y 0
x
a

j 0 a

  x   j z
A10 sin 
e
 a 
e j10 z
Surface current on waveguide walls for TE10 mode.
The current is redistributing the charge in order to support the electric field intensity one quarter of a
cycle later.
Power Flow down the Waveguide (TE10 Mode)
The time- average power per unit area is:
1
Pav  Re  EXH * 
2
 x  j10 z   j  a *
 j a
x  ˆ
  x  ˆ  j10 z
*
EXH *   
A10 sin
e
aˆ y  X  
A10 sin 
 ax  A10co s 
 az  e

a

  
 a 
 a  
j a  j  a  2   x 
2

 sin 
 A10 (aˆ z ).aˆ z
   
 a 
2
a
2
x 
A10 sin 2 
 EXH * .aˆz  

2

 a 
 EXH .aˆ
*
z

For a propagating mode  is real, so
Pav .aˆ z 
 a 2
2
x 
A10 sin 2 

2
2
 a 
The power crossing the WG cross-section:
P10,av 
 a 2
 a3b
2 a
2
A
b

A10
10
2
2
2
2
4
P10,av 
 a3b
2
A10
2
4
Characteristic Impedance of a Waveguide
In order to apply the transmission line theory to WG’s, we must be able to determine  or g as well as
the characteristic impedance Z 0 of the particular guide configuration. We have,  
g 

2
 f

1    c  
f


2
g
and
.
Define:
V
,where V  is the voltage between the conductors and I  is the conduction current in the

I
propagation direction for the traveling wave. For TEM lines Z 0 is uniquely defined since the value of
Z0 
V    E.dl is independent of the integration path. In the WG case, V  is a function of the integration
path.
For the dominant TE10 mode
Z 0  60
1
r 
2
2
ab 1   f c f  


r
ATTENUATION in a WAVEGUIDE
Attenuation in a WG can be caused by either dielectric loss and/or conductor loss. If  d is the
attenuation constant due to dielectric loss and  c is the attenuation constant due to the conductor loss,
then the total attenuation constant is:
  c  d
If the WG is completely filled with a homogeneous dielectric the attenuation can be calculated from the
propagation constant and this result applies to any guide with a homogeneous dielectric filling the
guide.   kc2  k *2
k * : Complex wavenumber.   kc2   2 0 0 r 1  j tan   2 .
1
In practical applications. So, with
a x
2
2
2
Then,   kc    0 0 r 1  j tan  
Since

1
2
2
1  x2 
a    for x  a
2 a 
kc2  k 2 
kc2  k 2  j  , k    0 r 0
k 2 tan 
 j
2
Then,  d 
k 2 tan 
( Np / m) For TE and TM waves.
2
1 jk 2 tan  k 2 tan 

 j
2 kc2  k 2
2
For TEM waves, kc  0,   k ,  d 
k tan 
( Np / m)
2
PERTURBATION METHOD FOR CALCULATING ATTENUATION DUE TO CONDUCTOR LOSS
The method uses the fields of the lossess line, with the assumption that the fields of the lossy line are
not greatly different from the fields of the lossless line.
The power flow along a lossy TL, in the absence of reflections:
P( z )  P0e2 z
Where P0 is the power at z=0 plane and  is the attenuation constant to be determined. Now, define
the power loss per unit length along the line as:
Pl  z   
P
 2 P0e2 z  2 P  z 
z
Where the (-) sign on the derivative was chosen so that Pl would be a positive quantity. Then,

Pl  z  Pl  z  2 z

e
2 P  z  2 P0
The power loss per unit length due to finite wall conductivity is:
Pl 
Rs
2
J
2
s
dl
c
Rs=the wall surface resistance.
C=integration contour which encloses the perimeter of the guide walls.
TM MODES
  2ez  2ez

 2  kc2ez   0 ,
2
y
 x

We have H z  0 and ez  x, y  must satisfy 
The general solution is,
ez  x, y    A cos k x x  B sin k x x  C cos k y y  D sin k y y 
The boundary conditions are:
kc2  k 2   2
ez  x, y   0 for x  0 and x  a
ez  x, y   0 for y  0 and y  b
Using these B.C.’s we obtain:
A=0 and k x 
m
, m  1,2,3,...
a
C=0, and k y 
n
, n  1,2,3,...
b
So,
 m
Ez  x, y, z   Bmn sin 
 a
  n
x  sin 
  b

y  e  j z

The other field components are:
Ex  x, y, z   
j  m
 m   n   j z
Bmnco s 
x  sin 
y e
2
akc
 a   b 
E y  x, y, z   
j  n
 m
Bmn sin 
2
bkc
 a
H x  x, y, z  
j n
 m
Bmn sin 
2
bkc
 a
H y  x, y , z   
  n   j z
x  cos 
y e
  b 
  n
x  cos 
  b
j m
 m
Bmnco s 
2
akc
 a
  n   j z
x  sin 
y e
  b 
 2  m 2  n 2 
  k  k  k  
 
 
 a   b  

2

y  e  j z

1
2
2
c
If either m or n is zero, the fields vanish identically. So there are no TM00, TM01 or TM01 modes.
The lowest order TM mode is the TM11 mode with:
   2    2 
f c11 
     
2   a   b  
1
1
2
 f cTE10
Since f c of the lowest order TM mode is greater than the f c of the lowest order TE mode, TE10 is the
lowest among all modes.
The wave-impedance
ZTM 
E
Ex

 y  
Hy
Hx k