y
rectangular waveguides
TE10 mode
b
z
0
Ey
0
z
0
a
x
a
x
y
rectangular waveguides
TE10 mode
1
fc 
2
1
fc 
2
b
1

1

 
 
a
 
 
b
2
z
2
0
a
x
a
x
a
x
y
b
a = 2b
c
λc
c
a 

fc 
2fc
2
2a
0
z
z
0
b
0
0
0
y
y
rectangular waveguides
b
TE10 mode
1
fc 
2
1

 m 


 a 
2
z
0
a
0
x
a = 3 cm
2
1
fc 
2
1 m
3  10
  
  a 
2
3  10

2
8
8
m
 
a
2
 1   10  10  5 GHz


2
 0.03 
2
9
y
rectangular waveguides
b
TE10 mode

      z 0 0
     
 kx  ky
2
 
  
a
2
2
2
2
2
 
c    
a
x
a
2
 
    
a
 
2
2

2
2
c

y
rectangular waveguides
TE10 mode
 
2
0
a
x
1
 
    
1 2
 
a
c
c

2
2
 c 
  
1 
1  

 2fa 
 2a 
2

v  


 
    
a

b
2
2
2
z

0
c
 
 
a
y
rectangular waveguides
TE10 mode
 
    
a
 
v 
b
2
2
c
  
1  
 2a 
z
0
0
2

g 

f


2
  
1  
 2a 
v
2
a
x
y
rectangular waveguides
TE10 mode
b
z
0
a
0
Ey
z
0
a
g
x
x
y
rectangular waveguides
TE10 mode
b
2
 πx 

 c 
Hz  B cos
 
0
1  
x
a
c
 a 
  z 0
jB  πx 
jβ
 πx 
Ey  sin
Hx 
B sin


 
π
 a 
 a 
 
 
a
a
jωωμ
B
π
 
Ey
ωμ
a
Zc
ωμ

Z TE 




2
2
jβ
Hx
β
ω
B
 ωc 
 ωc 
1 
1 


π
c
 
ω
ω
a
Movie to illustrate phase mixing of
two propagating sinewaves in a
dispersive media.

c

Study of an amplitude modulated pulse
Movie to illustrate the propagation of an
amplitude modulated pulse in a
waveguide
dispersion
e
j t  z 
t1
z
t2
z
t3
z
dispersion
e
j t  z 
T
A( )   e
jo t
0
A( )  e
A( z, t ) 
e
 jt
  


  
e
dt
t1
z
t2
z
t3
z
2
j ( o t   ( o ) z )
2 
e
 ( t   '( o ) z 


2 


2
y
rectangular waveguides
TE10 mode
b
z
0
0
a
8.8.e. The transmission analogy can be applied to the
transverse field components, the ratios of which are
constants over guide cross sections and are given by
wave impedances. A rectangular waveguide of inside
dimensions [a = 4, b = 2 cm] is to propagate a TE10
mode of frequency 5 GHz. A dielectric of constant r=3
fills the guide for z>0 with an air dielectric for z<o.
x
y
rectangular waveguides
TE10 mode
Z TE 
Ey
Hx

Zc
 ωc 
1 

ω
2

b
Zc
z
0
2
0
a
x
 λ 
1  
 2a 
λ v ac
3  108
λdielectric 
 .035
λ v ac 

.06
εr
5  109
377
377
3  241
Zair 
 570 Z dielectric 
2
2
 3.5 
6
1 
1  

 8 
8
y
rectangular waveguides
TE10 mode
jB  πx 
Ey  sin

 
 a 
b
z
 
a
jβ
 πx 
Hx 
B sin

π
 a 
 
a
x  a, y  b
1
power 
Re
2
  E
x  0, y  0
power  EoHo ab
0
a
0
y

 Hx dxdy
*
x
y
rectangular waveguides
TE10 mode
b
z
0
0
a
lossy
dielectric
The wave will attenuate
as it propagates.
x
y
rectangular waveguides
TE10 mode
b
z
0
0
Loss in walls due to finite
conductivity of metal surfaces
Tangential H  surface current js
Ohmic power loss
Attenuation of the em wave
a
x
y
rectangular waveguides
TE10 mode
matching
b
z
0
Ey
0
z
0
a
x
a
x
y
rectangular waveguides
λc TE  2a b
8.8a. For f=3 GHz,
10
design a rectangular

λc TE 01  2b 0
x
waveguide with copper
a
z 0
10
conductor and air
λc
f
3  10

λ

10
cm
dielectric so that the
9
λ fc
3

10
TE10 wave will
propagate with a 30%
want λTE10  1.3 λ  13 cm
safety factor (f =
a = 6.5 cm
1.30fc) but also so that
wave type with next
λ
higher cutoff will be
want λ TE 01 
 7.7 cm
1.3
30% below its cutoff
frequency.
b = 3.85 cm
Cylindrical Waveguides
e
a
j t  z 

 

   H
t Ez      Ez
2
t Hz
2
2
2
2
2
z
z
j
Cylindrical Waveguides
a
j t  z 
e
2
2
2
t Ez      Ez  0


j
z


1   Ez  1  Ez
2
2
     Ez  0
r
 2
2
r r  r  r j
Ez  AJ0 kc r cosnj 
2
Bessel function
1
J0(x)
J1(x)
0
0
10
20
Cylindrical Waveguides
Ez  AJ0 kca  0
c a 2a
 zero of J0

kc a 
c
c
Ez  AJ0 kc r cosnj 
a
1
TMnl  TM01; TM02; TM11; ....
j
z
J0(x)
J1(x)
0
n is angular variation
l is radial variation
0
10
20
Cylindrical Waveguides
a
j t  z 
e
2
2
2
t Hz      Hz  0


j
z


1   Hz  1  Hz
2
2
     Hz  0
r
 2
2
r r  r  r j
2
Hz  BJn kc r cosnj 
Cylindrical Waveguides
a
j t  z 
e
j dJn kc r 
Ej 
B
cosnj 
kc
dr
z
j dJn kca 
Ej 
B
cosnj   0
1
kc
dr
J0(x)
TEnl  TE 01; TE02; TE11; ....
n is angular variation
l is radial variation
J1(x)
0
j
TE 01 mode
Loss decreases as frequency
increases
TE11 mode
Field distribution is similar to TE10
mode in rectangular waveguide
a
v
c
v
z
1
vg
c

j