ECE 5317-6351
Microwave Engineering
Fall 2011
Prof. David R. Jackson
Dept. of ECE
Notes 11
Waveguides Part 8:
Dispersion and Wave Velocities
1
Dispersion
Dispersion => Signal distortion due to “non-constant”
z phase velocity
=> Phase relationships in original signal spectrum are
changed as the signal propagates down the guide.
In waveguides, distortion is due to:
 Frequency-dependent phase velocity (frequency dispersion)
 Frequency-dependent attenuation => distorted amplitude
relationships
 Propagation of multiple modes that have different phase
velocities (modal dispersion)
2
Dispersion (cont.)
Consider two different frequencies applied at the input:
+
vin
-
Z 0 ,  ( )
+
vout
-
L
vin  A cos 1t  a   B cos 2t  b 
   
   
 A cos  1  t  a    B cos  2  t  b  
  1  
  2  

Z0
Matched load
Vina  Ae ja
Vinb  Be jb
j      L
Voaut  Ae  a 1 
j      L
Vobut  Be  b 2 
vout  A cos 1t   (1 ) L  a   B cos 2t   (2 ) L  b 
   (1 ) L a  
   (2 ) L b  
 A cos  1  t 
    B cos  2  t 
  
1
1  
2
2  
 
 
3
Dispersion (cont.)
+
Z 0 ,  ( )
vin
-
+
vout
Z0
-
Matched load
L
vout  A cos 1t   (1 ) L  a   B cos 2t   (2 ) L  b 
   (1 ) L a  
   (2 ) L b  
 A cos  1  t 
    B cos  2  t 
  
1
1  
2
2  
 
 
Recall:

vp 

t1 
L
v p (1 )
t2 
L
v p (2 )
4
Dispersion (cont.)
No dispersion (dispersionless)
 v p  f ( )
 v p (1 )  v p (2 )
 t1  t2
Dispersion  v p  f ( )
 v p (1 )  v p (2 )
 t1  t2
Phase relationship at end of the line is
different than that at the beginning.
5
Signal Propagation
Consider the following system:
  2 f
Z ( )
Vi ( )
Vo ( )
The system will
represent, for us, a
waveguiding system.
Vo ( )  Vi ( ) Z ( )
Waveguiding system:
Z ()  A   e
Amplitude
 j    z
Phase
6
Signal Propagation (cont.)
Input signal
Output signal
Z ( )
Si ( t )
Si ( ) 
So (t )

 jt
S
(
t
)
e
dt
i


1
Si (t ) 
2
Fourier transform pair

Proof:
jt
S
(

)
e
d
 i
Si (t )  Si*  t 


1

2
Property of real-valued signal:
Si (  )  Si ( )
*
1

2
1

2
1
S
(

)
e
d


 i
2



Si ( ) e
 j t


 S ( ) e
i


jt
 jt
 S ( ) e
*
i
 jt
d

1
d  
2
1
d 
2

*
 jt
S
(

)
e
d
i

  

 S ( ) e
*
i
 jt
d

 Si ( )  Si* ( )
7
Signal Propagation (cont.)
We can then show
1
Si (t ) 
2

 Si ()e d  Si (t ) 
jt

1


Re  Si ( )e jt d 
0
(See the derivation on the next slide.)
The form on the right is convenient, since it only involves positive values of .
(In this case,  has the nice interpretation of being radian frequency:  = 2 f . )
8
Signal Propagation (cont.)
1
Si ( t ) 
2
1

2


Si ( )e jt d 

0

1
jt
S
(

)
e
d


S
(

)
e
d
i
 i

2 0
jt

0
1
1
 j t
jt




1
S
(


)
e
d


S
(

)
e
d
  i
i

2
2 0


  

1
1
 j t
jt



S
(


)
e
d


S
(

)
e
d
i
i


2 0
2 0


1
1
*
 j t
jt



S
(

)
e
d


S
(

)
e
d
i
i


2 0
2 0

1 
jt
jt * 

S
(

)
e

S
(

)
e



  i
 d
i
2  0


1
  Re Si ( )e jt d 
 0
9
Signal Propagation (cont.)
Hence, we have

1

Si (t )   Re  Si ( ) e jt  d 


0
Interpreted as a phasor
Using the transfer function, we have



1

So (t )   Re  Z ( )  Si ( )  e jt  d




0

1



 Re A   e
 j    z

Si ( )e jt d
0
(for a waveguiding structure)
10
Signal Propagation (cont.)
Summary
So (t ) 
Si ( t )
1




Re  Z () Si () e jt d
0
Z ( )
So (t )
11
Dispersionless System
A) Dispersionless System with Constant Attenuation
Z ()  A   e
 So (t ) 

1



 Re A   e
A    constant  A0
 j    z
 j    z

Si ( )e jt d


v p0
0
1


Re  A0 Si ( )e

z 
j  t 
 v p 0 


d
Constant phase velocity (not
a function of frequency)
0

z 
So (t )  A0 Si  t 
 v 
p0 

The output is a delayed and
scaled version of input.
The output has no distortion.
12
Narrow-Band Signal
B) Low-Loss System with Dispersion and Narrow-Band Signal
E ( )
Now consider a narrow-band input signal of the form
Si (t )  E (t )cos(0t )  Re  E (t )e j0t 
Si  t 
m
m
E t 
Narrow band
t
 m  0
(Physically, the envelope is
slowing varying compared
with the carrier.)
13

Narrow-Band Signal (cont.)
Si (t )  E (t )cos(0t )  Re  E (t )e j0t 
E ( )
Si ( )  F E  t  cos 0t 



1
1
j0t
 F E t  e
 F E  t  e  j0t
2
2
1
  E (  0 )  E (  0 ) 
2

m
m

Si   
0
0

14
Narrow-Band Signal (cont.)
Hence, we have
So (t ) 
1


Re  A   Si ( )e
 j    z
e jt d
0

1
j (t     z )
d
 Re  A   E (  0 )e
2

0
1

1
j (t     z )
d
Re  A   E (  0 )e

2
0
15
Narrow-Band Signal (cont.)
Since the signal is narrow band, using a Taylor series expansion about
0 results in:
d
 ( )   (0 ) 
(  0 )  ...   0   0 (  0 )
d  0

0
 0
dA
A    A(0 ) 
(  0 )  ...  A0
d  0
A
0
neglect
Low loss assumption
16
Narrow-Band Signal (cont.)
Thus,

1
j ( t      z )
So (t ) 
Re  A   E (  0 ) e
d
2
0

1

Re  A0 E (  0 )e jt e  j0 z e  j0 ( 0 ) z d 
2
0
  j0 z j0t 

A0

Re  e e  E (  0 )e  j0 ( 0 ) z e j ( 0 ) t d  
2
0


 j ( t   z ) 

A0
 j  0s z jst
0
0

Re  e
E (s )e
e d s 

2


 0
s    0 
 j (0t  0 z ) 

A0
 j  0s z jst

Re  e
E
(

)
e
e
d

s
s

2



The spectrum of E is concentrated near  = 0.
17
Narrow-Band Signal (cont.)
 j (0t  0 z ) 

A0
js ( t  0 z )
So (t ) 
Re e
E (s ) e
d s 

2



 A0 Re e j (0t  0 z ) E (t  0 z ) 
  0
 A0 cos  0  t 
  0
Define
vp 
0
0

z   E (t   0 z )

phase velocity @ 0
Define
1  d 
vg 

 0  d   
0
group velocity @ 0
18
Narrow-Band Signal (cont.)

z  
z 
So (t )  A0 E  t   cos  0  t   
 v    v 
g 
p 

 
Carrier phase travels
with phase velocity
Envelope travels
with group velocity
No dispersion
 v p  vg
Proof :

 constant  c1

   c1
d

 c1
d
vp 
 vg  c1
19
Narrow-Band Signal (cont.)

z  
z 
So (t )  A0 E  t   cos  0  t   
 v    v 
g 
p 

 
So  t, z 
E  t  z / vg 
vg
z
vp
20
Example: TE10 Mode of Rectangular Waveguide
Recall
 
   2    
a

v

Phase velocity: p

Group velocity: vg 
2
After simple calculation:
vp 
d  d  


d   d 

 
2
  
2
 
a
1
vg 
1
 
2
 2    

a
Observation:
v p vg 
1

 cd2
21
Example (cont.)
Lossless Case
 c     
f  fc
PEC
y
b
, 
o

z
x
o
a
vg  slope
1
c10

(“Light line”)
v p  slope

22
Filter Response
Input signal
Output signal
Z ( )
Si ( t )
So (t )
What we have done also applies to a filter, but here we use the transfer
function phase directly, and do not introduce a phase constant.
Z ( )  A   e
j  
From the previous results, we have
So (t ) 
1



 Re A  e
j  

Si ()e jt d
0
23
Filter Response (cont.)
Input signal
Output signal
Si ( t )
Z ( )
Z ( )  A   e
So (t )
Let z -
j  
Assume we have our modulated input signal:
  
So (t )  A0 cos  0  t  0
  0

z   E (t   0 z )

Si (t )  E (t )cos(0t )
where
The output is:
  0  

So (t )  A0 E (t  0 ) cos  0  t   
  0  
A0  A 0 
0   0 
0 
d
0 
d
24
Filter Response (cont.)
Input signal
Output signal
Si ( t )
Z ( )
So (t )
  0  

So (t )  A0 E (t  0 ) cos  0  t   
  0  
If the phase is a
linear function of
frequency, then
This motivates the following definitions:
Phase delay:
Group delay:

p  0
0
d
g  
0 
d

So (t )  A0 E (t   g ) cos 0  t   p 
 p   g  constant
In this case we have
no signal distortion.

25
Linear-Phase Filter Response
Input signal
Output signal
Z ( )
Si ( t )
Z ()  A  e
So (t ) 



 Re A  e
j  
A    A0

Si ()e jt d
0
Hence
So (t ) 
j  
     
Linear phase filter:
1
So (t )
1




 j
jt
Re
A
e
S
(

)
e
d


0
i

0
26
Linear-Phase Filter Response (cont.)
We then have
So (t ) 
1




 j
jt
Re
A
e
S
(

)
e
d


0
i

0


j t 
 A0 Re   Si ( )e   d 

0

 A0 Si  t   
1
So (t )  A0Si t  
A linear-phase filter does not distort the signal.
It may be desirable to have a filter maintain a linear phase, at least over the
bandwidth of the filter. This will tend to minimize signal distortion.
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