prof. david r. jackson dept. of ece notes 11 ece 5317-6351 microwave engineering fall 2011...
TRANSCRIPT
Prof. David R. JacksonDept. of ECE
Notes 11
ECE 5317-6351 Microwave Engineering
Fall 2011
Waveguides Part 8:Dispersion and Wave Velocities
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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)
Dispersion
2
1 2
1 1
1 21 2
1 21
2 2
21 1 2 2
cos cos
( ) ( )co
cos cos
cos ( ) c
s cos
os ( )
in a b
out a
a
a b
b
b
v A t B t
v A t L
A t B t
LB
B
LA t t
t L
0Zoutv+
-0 , ( )Z inv
+
-
L
Dispersion (cont.)
3
Consider two different frequencies applied at the input:
Matched load
2
1
a
a
b
b
jbin
j Lb
jain
j Lao
o
ut
ut
V Ae
V A
V Be
B
e
V e
1 21 2
1
1 1 2
1 2 2
2
( ) ( )cos co
cos ( ) cos ( )
s
out a
a b
bv A t L
L LA t
t
B
B
t
L
11( )p
Lt
v pv
0Zoutv+
-0 , ( )Z inv
+
-
L
Dispersion (cont.)
22( )p
Lt
v
4
Matched load
Recall:
No dispersion (dispersionless) ( )pv f
1 2( ) ( )p pv v
1 2t t
Dispersion ( )pv f
1 2( ) ( )p pv v
1 2t t
Phase relationship at end of the line is different than that at the beginning.
Dispersion (cont.)
5
( )iV o ( )V ( )Z
Consider the following system:
( ) j zZ A e
o ( ) ( ) ( )iV V Z
Signal Propagation
Amplitude Phase
2 f
The system will represent, for us, a waveguiding system.
6
Waveguiding system:
( )iS t o ( )S t( )Z
( ) ( )
1( ) ( )
2
j ti i
j ti i
S S t e dt
S t S e d
Fourier transform pair
*( ) ( )i iS S
Input signal
*
*
*
*
*
( )
1 1( ) ( )
2 2
1 1( ) ( )
2 2
1 1( ) ( )
2 2
( ) ( )
i i
j t j ti i
j t j ti i
j t j ti i
i i
S t S t
S e d S e d
S e d S e d
S e d S e d
S S
Proof:
Output signal
Property of real-valued signal:
Signal Propagation (cont.)
7
We can then show
0
1 1( ) ( ) ( ) Re ( )
2j t j t
i i i iS t S e d S t S e d
(See the derivation on the next slide.)
Signal Propagation (cont.)
8
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 . )
0
0
0
0
0 0
*
0 0
1( ) ( )
2
1 1( ) ( )
2 2
1 11 ( ) ( )
2 2
1 1( ) ( )
2 2
1 1( ) ( )
2 2
1
2
j ti i
j t j ti i
j t j ti i
j t j ti i
j t j ti i
S t S e d
S e d S e d
S e d S e d
S e d S e d
S e d S e d
*
0
0
( ) ( )
1Re ( )
j t j ti i
j ti
S e S e d
S e d
Signal Propagation (cont.)
9
o
0
0
1( ) Re (
1Re
)
)
(
(
)
j z
i
t
j t
jiA e
S t Z
S
e d
e
S
d
0
1( )( ) Re j
it
iS t e dS
Using the transfer function, we have
Interpreted as a phasor
Signal Propagation (cont.)
10
Hence, we have
(for a waveguiding structure)
o
0
1( ) Re ( ) ( ) j t
iS t Z S e d
Summary
Signal Propagation (cont.)
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( )iS t o ( )S t( )Z
A) Dispersionless System with Constant Attenuation
( ) j zZ A e
0
o
0
0
0
1( ) Re )
1Re )
(
( p
j z j
zj t
i
i
v
t
A S
S t A e S e d
e d
0A A constant
0pv
Constant phase velocity (not a function of frequency)
o 00
( ) ip
zS t A S t
v
The output is a delayed and scaled version of input.
The output has no distortion.
Dispersionless System
12
Now consider a narrow-band input signal of the form
Narrow band
0m
mm
( )E
B) Low-Loss System with Dispersion and Narrow-Band Signal
00( ) ( )cos( ) Re ( ) j t
iS t E t t E t e
(Physically, the envelope is slowing varying compared with the carrier.)
Narrow-Band Signal
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E t
t
iS t
0 0
0
0 0
( ) cos
1 1
2 21
( ) ( )2
i
j t j t
S F E t t
F E t e F E t e
E E
mm
( )E
00
iS
00( ) ( )cos( ) Re ( ) j t
iS t E t t E t e
Narrow-Band Signal (cont.)
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( )0
o
0
0
( )0
0
1 1Re ( )
2
1Re ( )
2
1( ) Re ( )
j t z
j z j t
t
i
j z
A E e d
A E e d
S t A S e e d
Hence, we have
Narrow-Band Signal (cont.)
15
Since the signal is narrow band, using a Taylor series expansion about 0 results in:
00
0
00
0 0 00
0 0 0
0 ( )( ) ( ) ( ) ...
( ) ( ) ...A
neglect
d
d
dAA A
dA
Low loss assumption
Narrow-Band Signal (cont.)
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Thus,
0 0 0
0 0 0 0 0
0 0 0
0
(
( )0 0
0
( ) ( )00
0
(
)o
)00
0
0
1Re ( )
2
Re (
1( ) Re ( )
2
)2
Re ( )2
s s
j z j zj t
j z j t j z j t
j t z j z j ts
j t
s
z
s
A E e e e d
Ae e E e e
S t A
d
Ae E e
E e d
e d
0 0 0( )0 Re ( )2
s sj t z j z j ts s
Ae E e e d
The spectrum of E is concentrated near = 0.
Narrow-Band Signal (cont.)
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0 0 0
0 0
( ) ( )0o
( )0
00 0 0
0
0
( )
cos (
( ) Re ( )2
)
Re
sj t z j t zs s
j t z E t z
AS t e E e d
A e
A t z E t z
Define
phase velocity @ 0
Define
00
1g
dv
d
group velocity @ 0
0
0pv
Narrow-Band Signal (cont.)
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Envelope travelswith group velocity
Carrier phase travels with phase velocity
p gv v
No dispersion1
1
1
1
p
g
cv
c
dc
d
v c
Proof :
constant
o 0 0( ) cosg p
z zS t A E t t
v v
Narrow-Band Signal (cont.)
19
z
o ,S t z vg
vp
/ gE t z v
o 0 0( ) cosg p
z zS t A E t t
v v
Narrow-Band Signal (cont.)
20
Recall2
2
a
Phase velocity:
Group velocity:1
g
d dv
d d
pv
22
pv
a
221
gva
21p g dv v c
Example: TE10 Mode of Rectangular Waveguide
After simple calculation:
Observation:
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Example (cont.)
y
z a
b
, x
o
o
PEC
1
10c
pv slope
Lossless Case c
cf f
(“Light line”)
gv slope
22
( )iS t o ( )S t( )Z
Filter ResponseInput signal
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.
Output signal
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o
0
1( ) Re ( )j j t
iS t A e S e d
From the previous results, we have
( ) jZ A e
( )iS t o ( )S t( )Z
Filter Response (cont.)Input signal
( ) jZ A e
Assume we have our modulated input signal:
Output signal
0o 0 0 0
0
( ) ( ) cosS t A E t t
0( ) ( )cos( )iS t E t t
0 0
0 0
d
d
whereThe output is:
0 0A A
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0o 0 0 0
0
( ) cos ( )S t A t z E t z
Let z -
( )iS t o ( )S t( )Z
Filter Response (cont.)Input signal Output signal
0o 0 0 0
0
( ) ( ) cosS t A E t t
Phase delay:
0
0p
Group delay:
0g
d
d
This motivates the following definitions: If the phase is a linear function of frequency, then
p g constant
In this case we have no signal distortion.
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o 0 0( ) ( ) cosg pS t A E t t
( )iS t o ( )S t( )Z
Linear-Phase Filter ResponseInput signal Output signal
o
0
1( ) Re ( )j j t
iS t A e S e d
( ) jZ A e
o 0
0
1( ) Re ( )j j t
iS t A e S e d
0A A Linear phase filter:
Hence
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Linear-Phase Filter Response (cont.)
o 0
0
0
0
0
1( ) Re ( )
1Re ( )
j j ti
j ti
i
S t A e S e d
A S e d
A S t
A linear-phase filter does not distort the signal.
We then have
o 0( ) iS t A S t
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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