Transcript ENE 429 Transmission Lines

1

Transmission lines or T-lines are used to guide propagation
of EM waves at high frequencies.

Examples:
› Transmitter and antenna
› Connections between computers in a network
› Interconnects between components of a stereo system
› Connection between a cable service provider and aTV set.
› Connection between devices on circuit board

Distances between devices are separated by much
larger order of wavelength than those in the normal
electrical circuits causing time delay.
2

Properties to address:
› time delay
› reflections
› attenuation
› distortion
3

Types of transmission lines
4

The differential segment of the
transmission line
R’ = resistance per unit length
L’= inductance per unit length
C’= capacitor per unit length
G’= conductance per unit length
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
General transmission lines equations:
v( z, t )
i ( z, t )

 i ( z, t ) R ' L '
z
t
i ( z, t )
v( z , t )

 v( z, t )G ' C '
z
t
6

Time-harmonic waves on transmission
lines
dV ( z )
 ( R ' j L ') I ( z )
dz
dI ( z )
 (G ' jC ')V ( z )
dz
After arranging we have
where
d 2V ( z )
  2V ( z )  0
dz
  ( R ' j L ')(G ' jC ')    j  .
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
Instantaneous form
v( z, t )  V0 e z cos(t   z )  V0e z cos(t   z )
i( z, t )  I 0 e z cos(t   z )  I 0e z cos(t   z )

Phasor form
V ( z )  V0 e z  V0e z
I ( z )  I 0 e z  I 0e z
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
lossless when R’ = 0 and G’ = 0
 0
  j   j L ' C '
   L ' C '
and

1
up  

L 'C '
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
low loss when R’ << L’ and G’ << C’
1/ 2

    j    R ' j L ' (G ' jC ')1/ 2 


1/ 2
1/ 2

R'  
G'  
 j L ' C ' 1 
1


 
j

L
'
j

C
'
 
 

Expanding
1 x
in binomial series gives
2
x x
1  x  1    ......
2 8
for x << 1
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Therefore, we get
1
C'
L'
  (R '
G'
)
2
L'
C'
R' 
 1 G'
   LC 1  (

)
 8 C '  L ' 
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 Characteristic impedance Z0 is defined as
the the ratio of the traveling voltage wave
amplitude to the traveling current wave
amplitude.
V0
V0
Z0     
I0
I0
or
R ' j L '
Z0 
.
G ' jC '
For lossless line,
L'
Z0 
.
C'
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 Power transmitted over a specific distance is
calculated.
 The instantaneous power in the +z traveling wave
at any point along the transmission line can be
shown as
2
V
Pi  ( z, t )  v( z, t )i( z, t )  0 e2 z cos 2 (t   z ).
Z0
 The time-averaged power can be shown as
2
T
T
V
1


2 z 1
2
0
Pavg ( z )   Pi ( z, t )dt 
e
cos
(t   z )dt.

T0
Z0
T0
2
V

Pavg
( z )  0 e2 z
Z0
W.
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 A convenient way to measure power ratios
 Power gain (dB)
Pout
G (dB)  10 log(
)
Pin
dB
 Power loss (dB)
Pin
attenuation(dB)  10 log(
) dB
Pout
 1 Np = 8.686 dB
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 Representation of absolute power levels is the dBm
scale
P
G(dBm )  10log(
)
1mW
dBm
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a)
what fraction of the input power does it reach the
output?
b)
What fraction of the input power does it reach the
midpoint of the line?
c)
What is the attenuation constant?
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
To satisfy boundary conditions between two
dissimilar lines

If the line is lossy, Z0 will be complex.
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
The phasor voltage along the line can be shown as
Vi ( z )  V0i e z e j  z
Vr ( z )  V0 r e z e  j  z

The phasor voltage and current at the load is the
sum of incident and reflected values evaluated at
z = 0.
VL  V0i  V0 r
V0i  V0 r
I L  I 0i  I 0 r 
Z0
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
Reflection coefficient
V0 r Z L  Z 0
L 

  e jr
V0i Z L  Z 0

A reflected wave will experience a reduction in
amplitude and a phase shift.

Transmission coefficient
VL
2Z L
L 
 1  
  e jt
V0i
Z0  Z L
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Pavg ,i
1
1 V V 2 L  1 V0 2 L

 Re Vi I i  Re 
e

e

2
2  Z0
 2 Z0
Pavg ,r


(

V
)(

V
)
1
1

2 L 
0
0
 Re Vr I r  Re 
e

2
2 
Z0




1
 
2
2

0 0

W
2
2
V0 2 L
e
Z0
W
Pavg ,r

Pavg ,i
2
Pavg ,t
 1 
Pavg ,i
2
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W

The main objective in transmitting power to a
load is to configure line/load combination such
that there is no reflection, that means.
 0
 Z L  Z0 .
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

Incident and reflected waves create “Standing
wave”.
Knowing standing waves or the voltage amplitude
as a function of position helps determine load and
input impedances
Voltage standing wave ratio
Vmax
VSWR 
Vmin
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If a load is matched then no reflected wave
occurs, the voltage will be the same at every point.
 If the load is terminated in short or open circuit, the
total voltage form becomes a standing wave.
 If the reflected voltage is neither 0 nor 100 percent
of the incident voltage then the total voltage will
compose of both traveling and standing waves.

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
let a load be position at z = 0 and the input wave
amplitude is V0,
VT ( z )  V0e j z   V0e j  z
where
ZL  Z 0
 
  e j .
ZL  Z 0
VT ( z)  V0 (e j z   e j (  z  ) )
 V0e j / 2 (e j z e j / 2   e j z e j / 2 )
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we can show that
VT ( z)  V0 (1   )e
 j z
 2V0  e
traveling wave
j / 2

cos( z  ).
2
standing wave
 The maximum amplitude occurs when
VT ( z)  V0 (1   ).
 The minimum amplitude occurs when standing waves
become null,
VT ( z)  V0 (1   ).
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
The minimum voltage amplitude occurs when two
phase terms have a phase difference of odd
multiples of .
 z  ( z   )  (2m  1)
zmin

; m  0,1, 2,...


(  (2m  1) )
4
The maximum voltage amplitude occurs when two
phase terms are the same or have a phase
difference of even multiples of .
 z  ( z   )  2m
zmax
; m  0,1, 2,...


(  2m )
4
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
If  = 0,  is real and positive

and


zmin   (2m  1)
4
m
zmax  
.
2
Each zmin are separated by multiples of one-half
wavelength, the same applies to zmax. The distance
between zmin and zmax is a quarter wavelength.
We can show that
VT ,max 1  
VSWR 

.
VT ,min 1  
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