This document discusses transmission line propagation coefficients including reflection coefficient and transmission coefficient. It defines the reflection coefficient as the ratio of reflected to incident voltage or current. Reflection and transmission coefficients are derived for a transmission line terminated by a load impedance. Standing wave patterns on transmission lines are also analyzed. Key properties of standing waves include maximum and minimum voltages occurring at intervals of half wavelength and voltages/currents being 90 degrees out of phase.

1. Course: Electromagnetic Theory
paper code: EI 503
Course Coordinator: Arpan Deyasi
Department of Electronics and Communication Engineering
RCC Institute of Information Technology
Kolkata, India
Topic: Transmission Line – Propagation Coefficients
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2. Reflection Coefficient
It is the ratio of reflected voltage(/current) with the incident voltage (/current)
for a given transmission line
ref ref
inc inc
V I
V I
 = = −
z
z 0 inc
inc 0
0 0
V e V
I I e
Z Z
+ −
+ −
= = =
z
z 0 ref
ref 0
0 0
V e V
I I e
Z Z
− 
− 
= = − = −
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3. Reflection Coefficient
z z
0 0
V(z) V e V e
+ −
− 
= +
Now
z z
0 0
V(z) V e V e
+ +
− −
= + 
( )
z
0
V(z) V e 1
+ −
= + 
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4. Reflection Coefficient
Similarly
z z
0 0
I(z) I e I e
+ −
− 
= −
z z
0 0
I(z) I e I e
+ +
− −
= − 
( )
z
0
I(z) I e 1
+ −
= − 
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5. Reflection Coefficient
If the transmission line has a length ‘l’
( )
l
0 l
V(l) V e 1
+ −
= + 
( )
l
0 l
I(l) I e 1
+ −
= − 
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6. Reflection Coefficient
If the transmission line is terminated by load ‘ZL’
( )
( )
l
0 l
l
0 l
V e 1
V(l)
Z(l)
I(l) I e 1
+ −
+ −
+ 
= =
− 
( )
( )
l
0
l
1
Z(l) Z
1
+ 
=
− 
L 0
l
L 0
Z Z
Z Z
−
 =
+
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7. Tranmission Coefficient
Transmission coefficient
tr tr
inc inc
V I
T
V I
= =
L 0
L 0
Z Z
T 1 1
Z Z
−
= +  = +
+
L
L 0
2Z
T
Z Z
=
+
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8. Transmission Coefficient
From power conservation principle
in ref tr
P P P
= +
tr in ref
P P P
= −
( ) ( ) ( )
2 2 2
l l l
tr 0 0
L 0 0
V e V e V e
2Z 2Z 2Z
+ + −
− − −
= −
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9. Transmission Coefficient
( ) ( ) ( )
( )
2 2 2
l l l
tr 0 0
2
l
L 0
0
V e V e V e
1
2Z 2Z V e
+ + −
− − −
+ −
 
 
= −
 
 
 
2
2 L
l
0
Z
T 1
Z
 
= − 
 
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10. Problem 1
A certain transmission line has characteristic impedance (75+j0.01) Ω and is
terminated by load impedance of (70+j50) Ω. Calculate reflection coefficient
and transmission coefficient.
Soln
L 0
l
L 0
Z Z
Z Z
−
 =
+
l
(70 j50) (75 j0.01)
(70 j50) (75 j0.01)
+ − +
 =
+ + +
l 0.08 j0.32
 = +
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11. T 1
= + 
T 1 0.08 j0.32
= + +
T 1.08 j0.32
= +
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12. Problem 2
A lossless transmission line has characteristic impedance 50 Ω and is terminated by
load impedance of 75 Ω. If the line is energized by a generator with output impedance
of 50 Ω and open circuit output voltage of 30 V [rms). Find magnitude of instantaneous
load voltage, instantaneous power delivered to the load. Consider length of the line is
2.25λ.
Soln
2
l .
4 2
  
 = =

L 0
l
L 0
Z Z 75 50
0.2
Z Z 75 50
− −
 = = =
+ +
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13. Instantaneous voltage at load
( )
l
0 l
V(l) V e 1
+ −
= + 
( )
j l
0 l
V(l) V e 1
+ − 
= + 
( )
V(l) 30 1 0.2
= +
V(l) 36 V
=
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14. Instantaneous power delivered
2
l
L
V
P(l)
Z
=
2
36
P(l)
75
=
P(l) 17.28 W
=
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15. Standing wave
General solution of transmission line equation
z z
0 0
V(z) V e V e
+ −
− 
= +
z z
0 0
I(z) I e I e
+ −
− 
= −
j
 =  + 
Propagation constant
z z
0 0
V(z) V e V e
+ −
− 
= +
z j z z j z
0 0
I(z) I e e I e e
+ −
− −   
= −
z j z z j z
0 0
V(z) V e e V e e
+ −
− −   
 = +
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16. z j z z j z
0 0
V(z) V e e V e e
+ −
− −   
= +
Standing wave
Now
( ) ( ) ( ) ( )
z z
0 0
V e cos z jsin z V e cos z jsin z
+ −
− 
=  −  +  + 
   
   
( ) ( )
z z z z
0 0 0 0
V e V e cos z V e V e sin z
+ − + −
−  − 
   
= +  + − 
   
j
0
V e− 
=
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17. Standing wave
where
( ) ( )
1/ 2
2 2
z z 2 z z 2
0 0 0 0 0
V V e V e cos z V e V e sin z
+ − + −
−  − 
 
   
= +  + − 
   
 
 
( )
( )
( )
z z
0 0
1
z z
0 0
V e V e
tan tan z
V e V e
+ −
− 
−
+ −
− 
 
−
 
 = 
 
+
 
Voltage for
standing-wave pattern
Phase for standing-wave pattern
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18. Standing wave: properties
Maximum voltage amplitude
z z
max 0 0
V V e V e
+ −
− 
= +
 
z
max 0
V V e 1
+ −
= + 
Minimum voltage amplitude
z z
min 0 0
V V e V e
+ −
− 
= −
 
z
min 0
V V e 1
+ −
= − 
occurs at z n
 = 
occurs at z (2n 1)
2

 = −
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19. Standing wave: properties
The distance between two successive minima or maxima is one-half wavelength
z n
 = 
n n
z
2
 
= =



n
z
2

=
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20. Standing wave
z j z z j z
0 0
I(z) I e e I e e
+ −
− −   
= −
Again
( ) ( ) ( ) ( )
z z
0 0
I e cos z jsin z I e cos z jsin z
+ −
− 
=  −  −  + 
   
   
( ) ( )
z z z z
0 0 0 0
I e I e cos z I e I e sin z
+ − + −
−  − 
   
= +  − − 
   
j
0
I e− 
=
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21. Standing wave
where
( ) ( )
1/ 2
2 2
z z 2 z z 2
0 0 0 0 0
I I e I e cos z I e I e sin z
+ − + −
−  − 
 
   
= −  + + 
   
 
 
( )
( )
( )
z z
0 0
1
z z
0 0
I e I e
tan tan z
I e I e
+ −
− 
−
+ −
− 
 
+
 
 = 
 
−
 
Current for
standing-wave pattern
Phase for standing-wave pattern
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22. Standing wave: properties
Maximum voltage amplitude
Minimum voltage amplitude
occurs at
occurs at
z z
max 0 0
I I e I e
+ −
− 
= −
 
z
max 0
I I e 1
+ −
= +  z n
 = 
z z
min 0 0
I I e I e
+ −
− 
= +
 
z
min 0
I I e 1
+ −
= −  z (2n 1)
2

 = −
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23. Standing wave: properties
When
z z
0 0
V e V e
+ −
− 
=
then standing wave pattern with zero phase is given by
( )
z
S 0
V 2V e cos z
+ −
= 
This is pure standing wave
Vmax
Vmin
λ/2
λ/2
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24. When
z z
0 0
I e I e
+ −
− 
=
then standing wave pattern with zero phase is given by
( )
z
0
S
0
V
I 2j e sin z
Z
+
−
= − 
This is pure standing wave
Standing wave: properties
Vmax
Vmin
λ/2
λ/2
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25. Voltage and current standing waves are 90° out of phase along any time
Standing wave: properties
Voltage nodes and current nodes are interlaced a quarter-wavelength apart
V I
λ/4
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26. Standing Wave Ratio
It is the ratio of maximum standing wave pattern (voltage/ current) to the minimum of
that value (voltage/ current)
max max
min min
V I
SWR
V I
= =
1
SWR
1
+ 
=
− 
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27. 24-11-2021 Arpan Deyasi, EM Theory 27
Problem 3
A transmission line has characteristic impedance (50+j0.01) Ω and is terminated by
load impedance of (73-j42.5) Ω. Calculate SWR.
Soln
L 0
l
L 0
Z Z
Z Z
−
 =
+
l
(73 j42.5) (50 j0.01)
(73 j42.5) (50 j0.01)
− − +
 =
− + +
l 0.377 42.7
 =  
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28. 24-11-2021 Arpan Deyasi, EM Theory 28
1
SWR
1
+ 
=
− 
1 0.377
SWR
1 0.377
+
=
−
SWR 2.21
=
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29. 24-11-2021 Arpan Deyasi, EM Theory 29
Problem 4
Soln
A transmission line of characteristics impedance of Z0 = 50 Ω is terminated by a load
RL = ZL = 100 Ω. Find V.S.W.R, ZMIN and ZMAX
L 0
L
L 0
Z Z
Z Z
−
 =
+
L
100 50 50
100 50 150
−
 = =
+
L 0.33
 =
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30. 24-11-2021 Arpan Deyasi, EM Theory 30
L
L
1
V.S.W.R
1
+ 
=
− 
1
1
3
V.S.W.R
1
1
3
+
=
−
V.S.W.R 2
=
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31. 24-11-2021 Arpan Deyasi, EM Theory 31
L
MAX 0
L
1
Z Z
1
 
+ 
=  
− 
 
L
MIN 0
L
1
Z Z
1
 
− 
=  
+ 
 
MAX
Z 100
= 
MIN
Z 25
= 
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32. 24-11-2021 Arpan Deyasi, EM Theory 32
Problem 5
Soln
A lossless transmission line with a characteristics impedance of 75 Ω is terminated
by a load of impedance 120 Ω. If the magnitude of incident voltage is 10 V. Calculate
the minimum and maximum values of voltages on the line.
L 0
L
L 0
Z Z
Z Z
−
 =
+
L
45
195
 =
L
120 75
120 75
−
 =
+
L 0.243
 =
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33. 24-11-2021 Arpan Deyasi, EM Theory 33
( )
Max 0 L
V V 1
= + 
( )
Min 0 L
V V 1
= − 
( )
Max
V 10 1 0.243
= +
Max
V 12.43V
=
( )
Min
V 10 1 0.243
= −
Min
V 7.57V
=
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