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1
Prof. David R. JacksonDept. of ECE
Fall 2013
Notes 8
ECE 6340 Intermediate EM Waves
2
Signal Propagation on Line
Introduce Fourier transform:
( ) ( )
1( ) ( )
2
j ti i
j ti i
v v t e dt
v t v e d
vi (t)+ vo (t)-
+ -
Z0 () , g ()
Semi-infinite line z = lz = 0
3
Signal Propagation on Line
1( ) ( )
2
j ti iv t v e d
Goal: To get this into a form that looks like
( ) Re , 0nj ti n n
n
v t A e
The transform variable can then be interpreted as (radian) frequency.
4
Signal Propagation on Line (cont.)
( ) ( )
j ti iv v t e dt
Since ( ) Real ( )iv t t
*( ) ( )i iv v it follows that
0
0
1 1( ) ( ) ( )
2 2
j t j ti i iv t v e d v e d
Next, use
We start by considering a useful property of the transform.
Transform definition:
(assuming that is real)
5
Signal Propagation on Line (cont.)0
0
1 1( ) ( ) ( )
2 2
j t j ti i iv t v e d v e d
Use = - 0 0
0
*
0
*
0
1 1( ) ( )
2 2
1( )
2
1( )
2
1( )
2
j t j ti i
j ti
j ti
j ti
v e d v e d
v e d
v e d
v e d
6
Signal Propagation on Line (cont.)
Hence
*
0
*
0
0
1( ) ( ) ( )
2
1Re ( ) ( )
2
1Re ( )
j t j ti i i
j t j ti i
j ti
v t v e v e d
v e v e d
v e d
0
1( ) Re ( ) j t
i iv t v d e
or
The transform variable is now interpreted as radian frequency.
2 f
7
Signal Propagation on Line (cont.)
0
1( ) Re ( ) j t
i iv t v d e
t
iv t
Any signal can be expressed as a collection of infinite sinusoidal signals.
8
Signal Propagation on Line (cont.)
1( )n i nA v d
Denote
( ) Re nj ti n
n
v t A e Then
d
nA
n
0
1( ) Re ( ) j t
i iv t v d e
The integral is approximated as a sum.
9
Signal Propagation on Line (cont.)
( ) Re nj ti n
n
v t A e
Taking the limit,
o
0
1( ) Re ( ) l j t
iv t v e e d
( ) where
Therefore,
o ( ) Re
1Re ( )
n n
n n
l j tn
n
l j ti n
n
v t A e e
v d e e
( )n n
10
Signal Propagation on Line (cont.)
o
0
1( ) Re ( ) l j t
iv t v e e d
( ) where
Final result:
( ) ( ) j ti iv v t e dt
11
Signal Propagation on Line (cont.)
o
0
1( ) Re ( ) l j t
iv t v e e d
Compare:
o o
0
1( ) Re ( ) j tv t v e d
Output voltage expression
Output voltage expressed in inverse transform form
o ( ) ( )iv v e
Conclusion:
12
Signal Propagation on Line (cont.)
The most general scenario:
T()Vi () Vo ()
T() is the frequency-domain transfer function.
o ( )i
VT
V
the ratio of the phasor amplitudes
o
0
1( ) Re ( ) j t
iv t v T e d
o iv v T
Then we have
13
Lossless Line
Denote
/
( )p
p
j j LC
j v
v
constant
/
o
0
( / )
0
1( ) Re ( )
1Re ( )
p
p
j v l j ti
j t l v
i
v t v e e d
v e d
p
lt t
v
Then o
0
1( ) Re ( ) j t
iv t v e d
o
0
1( ) Re ( ) l j t
iv t v e e d
14
Compare with
o
0
1( ) Re ( ) j t
iv t v e d
0
1( ) Re ( )
j ti iv t v e d
Hence o ( )iv t v t
o ( / )i pv t v t l v or
Lossless Line (cont.)
15
Lossless Line (cont.)
o ( / )i pv t v t l v
pv
The pulse moves at the phase velocity without distortion.
16
Lossless Line (cont.)
Note that the shape of the pulse as a function of z is the mirror image of the pulse shape as a function of t.
pv
iv t
z = 0
, ( / )i pv z t v t z v
z > 0
pz v t
t
iv tSawtooth pulse
t
Note the time delay in the trace!
17
Lossless Line (cont.)
The sawtooth pulse is shown emerging from the source end of the line.
iv t
z = 0
t=0t = t1 t = t2 t = t3
t = t4
t
iv tSawtooth pulse
, ( / )i pv z t v t z v
18
Signal Propagation with Dispersion
(dispersion)
Assume 0( ) ( )cosiv t S t t
( )p pv v
S(t) = slowly varying envelope function
vi (t)+ vo (t) -
+ -
Low-loss line
z = lz = 0
19
Signal Propagation with Dispersion (cont.)
S t
t
iv t
This could be called a “wave packet”:
The signal consists of not a single frequency, but a group of closely-spaced frequencies, centered near the carrier frequency 0.
20
The spectrum of S(t) is very localized near zero frequency:
( ) ( ) j tS S t e dt
( )S
The envelope function is narrow in the (transform) domain.
Signal Propagation with Dispersion (cont.)
21
Use
o
0
1( ) Re ( ) l j t
iv t v e e d
0( ) ( )cosiv t S t t
0 0
0 0
1 1( )
2 2
1 1( ) ( )
2 2
j t j tiv S t e e
S S
F
Signal Propagation with Dispersion (cont.)
Hence
22
0 0
1 1( ) ( ) ( )
2 2iv S S
Peak near 0 Peak near -0
0
1( ) ( )
2iv S
Hence, we can write
Signal Propagation with Dispersion (cont.)
0
0
( )iv o
0
1( ) Re ( ) l j t
iv t v e e d
23
Hence
Neglect (second order)
o 0
0
1( ) Re ( )
2l j tv t S e e d
0
0
0 0
0 0
( ) ( )
( ) ( )
d
d
d
d
Next, use
with
Signal Propagation with Dispersion (cont.)
j
0( )S
0
24
o 0
0
1( ) Re ( )
2l j tv t S e e d
0 0
00
o 0
0
1( ) Re ( )
2l j l
dj l
d j t
v t S e e
e e d
0
0 0
0
( )d
d
Signal Propagation with Dispersion (cont.)
0 0
0 0
Hence
Denote
25
0 0
00
o 0
0
1( ) Re ( )
2l j l
dj l
d j t
v t S e e
e e d
Signal Propagation with Dispersion (cont.)
0 0
000 0
o 0
0
1( ) Re ( )
2l j l
dj l
d j tj t
v t S e e
e e e d
Multiply and divide by exp (j0t):
26
Let
0 0 0 0
0
o
1( ) Re ( )
2
dj l
dl j l j t j tv t e e e S e e d
0
Signal Propagation with Dispersion (cont.)
0 0
000 0
o 0
0
1( ) Re ( )
2l j l
dj l
d j tj t
v t S e e
e e e d
27
Extend the lower limit to minus infinity:
(since the spectrum of the envelope function is concentrated near zero frequency).
00 0 0o
1( ) Re ( )
2
dj t l
dl j l j tv t e e e S e d
Signal Propagation with Dispersion (cont.)
0 0 0 0
0
o
1( ) Re ( )
2
dj l
dl j l j t j tv t e e e S e e d
( )S
0
Hence
28
Hence
00 0 0o
1( ) Re ( )
2
dj t l
dl j l j tv t e e e S e d
0 0 0
0
o( ) Re l j l j t dv t e e e S t l
d
Signal Propagation with Dispersion (cont.)
29
or
Define group velocity:
0
0
o 0 0( ) cosldv t S t l e t l
d
0
g
dv
d
0o 0( ) / cos / l
g pv t S t l v t l v e
Signal Propagation with Dispersion (cont.)
30
Summary
0o 0( , ) / cos / l
g pv t z S t z v t z v e
0( ) ( )cosiv t S t t
z
o ,v t z vg
vp
Signal Propagation with Dispersion (cont.)
31
z
o ,v t z
z
o ,v t z
z
o ,v t z
t = 0
t =
t = 2
Example: vg = 0, vp > 0Phase velocityGroup velocity
32
Example
Example from Wikipedia (view in full-screen mode with pptx)
This shows a wave with the group velocity and phase velocity going in different directions. (The group velocity is positive and the phase velocity is negative.)
http://en.wikipedia.org/wiki/Group_velocity
“Backward wave”
(The phase and group velocities are in opposite directions.)
33
Example
Example from Wikipedia (view in full-screen mode with pptx)
Frequency dispersion in groups of gravity waves on the surface of deep water. The red dot moves with the phase velocity, and the green dots propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The red dot overtakes two green dots when moving from the left to the right of the figure.
New waves seem to emerge at the back of a wave group, grow in amplitude until they are at the center of the group, and vanish at the wave group front.
For surface gravity waves, the water particle velocities are much smaller than the phase velocity, in most cases.
http://en.wikipedia.org/wiki/Group_velocity
34
Notes on Group Velocity
In many cases, the group velocity represents the velocity of information flow (the velocity of the baseband signal).
This is true when the dispersion is sufficiently small over the frequency spectrum of the signal, and the group velocity is less than the speed of light.
Example: a narrow-band signal propagating in a rectangular waveguide.
In some cases the group velocity exceeds the speed of light. In such cases, the waveform always distorts sufficiently as it propagates so that the signal never arrives fast than light.
35
Notes on Group Velocity (cont.)
Sometimes vg > c (e.g., low-loss TL filled with air)
t = 0 t > 0
lossless
low-loss
The first non-zero part of the signal does not arrive faster than light.
36
Signal Velocity
0s
lv
t
Relativity:
sv ct
o ( )v t
0t
V0
+ vo (t) -
+ -
TL
z = l
t = 0 l
z = 0
37
Energy Velocity
Definition of energy velocity vE :
zE
l
v P
W
<Wl > = time-average energy stored per unit length in the z direction.
lz l E
zv
t t
W W
P W
vE
z
V
<W>
38
Energy Velocity
zE
l
v P
W
Note: In many systems the energy velocity is equal to the group velocity.
2 2 2 21 1 1 1 1
4 4 4 4l c c c c
V S
E H dV E H dSz
W
vE
z
V
<W>
*1ˆRe
2z
S
P E H z dS where
39
Example
Rectangular Waveguide
22
0
22
0 0
ka
a 22
0 0
pv
a
b
a
TE10
2
0
1
p
cv
k a
Multiply top and bottom by c / = 1 / k0 :
40
Example (cont.)
To calculate the group velocity, use
22 2
0 0
0 0
2
0 0
2 2 ( )
1 1( )
g
p
a
d d
dv c
d v
2g pv v c
Hence
Note: This property holds for all lossless waveguides.
41
Example (cont.)
2
0
1
p
cv
k a
pv
c
gv
( 0)c
v
2
0
1gv ck a
42
Example (cont.)
Graphical representation ( - diagram):
22
0 0 a
0
g
dv
d
pv
slope pv
slope gv
( 0)
c ca
c
c
slope
vs.
g pv v
43
Dispersion
Theorem: If there is no dispersion, then vp = vg .
1
1
1
1
a
a
d a d
da
d
g pv v
( )pv constant (from the definition of dispersion)
Proof:
Hence:
An example is a lossless transmission line (no dispersion).
Note: For a lossless TL we have
1 1p gv v
LC
44
Dispersion (cont.)
Theorem: If vp = vg for all frequencies, then there is no dispersion.
ln lnC
d
d
d d
C
e
Proof:
pv
constant
45
Dispersion (cont.)
pv constant (no dispersion)
p gv v
EXAMPLES:
- Plane wave in free space
- Lossless TL
- Distortionless (lossy)TL
46
Dispersion (cont.)
No dispersion Attenuation is frequency independent
No distortionand
Note: Loss on a TL causes dispersion and it also causes the attenuation to be frequency dependent.
(The phase velocity is frequency independent.)
47
Dispersion (cont.)
“Normal” Dispersion:g pv v
“Anomalous” Dispersion: g pv v
Example: waveguide
Example: low-loss transmission line
This is equivalent to 0/
0d k
d
This is equivalent to 0/
0d k
d
48
Backward Wave
Definition of backward wave: 0g pv v
The group velocity has the opposite sign as the phase velocity.
This type of wave will never exist on a TEM transmission line filled with usual dielectric materials, but may exist on a periodic artificial transmission line.
Note: Do not confuse “backward wave" with "a wave traveling in the backward direction."
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