Telegrapher's equation in transmission line

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 – Telegrapher’s Equation
17-11-2021 Arpan Deyasi, EM Theory 1
Arpan Deyasi
Electromagnetic
Theory

2. Transmission Line
It is a means of transferring information form one point to another
It’s primary use is to transmit electrical/optical signal from source to load,
where points are separated by a distance not less than quarter wavelength of
propagating signal
S L
17-11-2021 Arpan Deyasi, EM Theory 2
Arpan Deyasi
Electromagnetic
Theory

3. i. Resistance per unit length [R]
ii. Inductance per unit length [L]
iii. Conductance per unit length [G]
iv. Capacitance per unit length [C]
Transmission Line Parameters
It is customary and convenient to describe a transmission line in terms of line
parameters which have been listed below
17-11-2021 Arpan Deyasi, EM Theory 3
Arpan Deyasi
Electromagnetic
Theory

4. Transmission Line - classification
Lumped line ----- Time delay encountered by propagating wave in traversing
the fundamental elements (line parameters) is negligible compared with
frequency of the signal
Distributed line ----- Time delay encountered by propagating wave in
traversing the fundamental elements (line parameters) is significant
compared with frequency of the signal
17-11-2021 Arpan Deyasi, EM Theory 4
Arpan Deyasi
Electromagnetic
Theory

5. Line parameters when source is voltage
V
R L
17-11-2021 Arpan Deyasi, EM Theory 5
Arpan Deyasi
Electromagnetic
Theory

6. Line parameters when source is current
G C
I
17-11-2021 Arpan Deyasi, EM Theory 6
Arpan Deyasi
Electromagnetic
Theory

7. Equivalent Circuit of Transmission Line
GΔz CΔz
RΔz LΔz
I(z,t)
V(z,t) V(z+Δz,t)
I(z+Δz,t)
Towards generator Towards load
ΔI
17-11-2021 Arpan Deyasi, EM Theory 7
Arpan Deyasi
Electromagnetic
Theory

8. Equation of Transmission Line
Applying KVL
V(z,t) R zI(z,t) L z I(z,t) V(z z,t)
t

=  +  + + 

V(z,t) V(z z,t)
RI(z,t) L I(z,t)
z t
− +  
= +
 
V(z z,t) V(z,t)
RI(z,t) L I(z,t)
z t
+  − 
 
− = +
 
 
 
17-11-2021 Arpan Deyasi, EM Theory 8
Arpan Deyasi
Electromagnetic
Theory

9. Equation of Transmission Line
Under limiting condition
z 0
 →
z 0
V(z z,t) V(z,t)
Lt RI(z,t) L I(z,t)
z t
 →
+  − 
 
− = +
 
 
 
V(z,t) RI(z,t) L I(z,t)
z t
 
− = +
 
……………… (1)
17-11-2021 Arpan Deyasi, EM Theory 9
Arpan Deyasi
Electromagnetic
Theory

10. Equation of Transmission Line
Applying KCL
I(z,t) I(z z,t) I
= +  + 
   
I(z,t) I(z z,t) G z V(z z,t) C z V(z z,t)
t

− +  =  +  +  + 

   
I(z,t) I(z z,t)
G z V(z z,t) C z V(z z,t)
z t
− +  
=  +  +  + 
 
17-11-2021 Arpan Deyasi, EM Theory 10
Arpan Deyasi
Electromagnetic
Theory

11. Equation of Transmission Line
   
I(z z,t) I(z,t)
G z V(z z,t) C z V(z z,t)
z t
+  − 
 
− =  +  +  + 
 
 
 
Under limiting condition
z 0
 →
z 0
I(z z,t) I(z,t)
Lt GV(z,t) C V(z,t)
z t
 →
+  − 
 
− = +
 
 
 
I(z,t) GV(z,t) C V(z,t)
z t
 
− = +
 
……………… (2)
17-11-2021 Arpan Deyasi, EM Theory 11
Arpan Deyasi
Electromagnetic
Theory

12. Equation of Transmission Line
Differentiating (1) w.r.t ‘t’,
2 2
2
V(z,t) R I(z,t) L I(z,t)
t z t t
  
− = +
   
……………… (3)
Differentiating (2) w.r.t ‘z’,
2 2
2
I(z,t) G V(z,t) C V(z,t)
z z z t
  
− = +
   
……………… (4)
17-11-2021 Arpan Deyasi, EM Theory 12
Arpan Deyasi
Electromagnetic
Theory

13. Equation of Transmission Line
Substituting (3) and (1) in (4)
2 2
2 2
I(z,t) G RI(z,t) L I(z,t) C R I(z,t) L I(z,t)
z t t t
 
   
 
= + + +
 
 
   
   
( )
2 2
2 2
I(z,t) LC I(z,t) RC LG I(z,t) RGI(z,t)
z t t
  
= + + +
  
……………… (5)
17-11-2021 Arpan Deyasi, EM Theory 13
Arpan Deyasi
Electromagnetic
Theory

14. Equation of Transmission Line
Differentiating (1) w.r.t ‘z’,
2 2
2
V(z,t) R I(z,t) L I(z,t)
z z t z
  
− = +
   
……………… (6)
Differentiating (2) w.r.t ‘t’,
……………… (7)
2
2
I(z,t) G V(z,t) C V(z,t)
t z t t
  
− = +
   
17-11-2021 Arpan Deyasi, EM Theory 14
Arpan Deyasi
Electromagnetic
Theory

15. Equation of Transmission Line
Substituting (7) and (2) in (6)
2 2
2 2
V(z,t) R GV(z,t) C V(z,t) L G V(z,t) C V(z,t)
z t t t
 
   
 
= + + +
 
 
   
   
( )
2 2
2 2
V(z,t) LC V(z,t) RC LG V(z,t) RGV(z,t)
z t t
  
= + + +
  
……………… (8)
17-11-2021 Arpan Deyasi, EM Theory 15
Arpan Deyasi
Electromagnetic
Theory

16. Equation of Transmission Line
The coupled equations (5) and (8) can be described using the following general equation
( )
2 2
2 2
U(z,t) LC U(z,t) RC LG U(z,t) RGU(z,t)
z t t
  
= + + +
  
……………… (9)
Equation (9) is called Telegrapher’s equation
17-11-2021 Arpan Deyasi, EM Theory 16
Arpan Deyasi
Electromagnetic
Theory

17. Solution of Telegrapher’s Equation
j t
0
U(z,t) U (z)e 
=
Let trial solution for Telegraphist’s equation
……………… (10)
Substituting the trial solution
( )
2
2
0 0 0 0
2
d
U LCU j RC LG U RGU
dz
= − +  + +
( )( )
2
0 0
2
d
U R j L G j C U
dz
= +  + 
17-11-2021 Arpan Deyasi, EM Theory 17
Arpan Deyasi
Electromagnetic
Theory

18. Solution of Telegrapher’s Equation
2
2
0 0
2
d
U U
dz
= 
where ‘γ’ is propagation constant
( )( )
R j L G j C
 = +  + 
17-11-2021 Arpan Deyasi, EM Theory 18
Arpan Deyasi
Electromagnetic
Theory

19. ( )( )
j R j L G j C
 =  +  = +  + 
Let
( )( )
1/ 2
2 2 2 2 2 2 2
1
RG LC R L G C
2
 
  = −  + +  + 
 
 
( )( )
1/ 2
2 2 2 2 2 2 2
1
LC RG R L G C
2
 
 =  − + +  + 
 
 
Propagation constant
17-11-2021 Arpan Deyasi, EM Theory 19
Arpan Deyasi
Electromagnetic
Theory

20. Problem 1:
A lossy cable with R = 2.5 Ω/m, L = 10 μH/m, C = 10 pF/m, G = 0 operates at 1 GHz. Find
attenuation constant and phase constant of the line.
Soln
( )( )
j R j L G j C
 =  +  = +  + 
Propagation constant
( )( )
9 6 9 12
2.5 j 2 10 10 10 j 2 10 10 10
− −
= +        
0.00125 j0.0628
= +
17-11-2021 Arpan Deyasi, EM Theory 20
Arpan Deyasi
Electromagnetic
Theory

21. Attenuation constant
0.0628 rad/m
 =
1
0.00125 m−
 =
Phase constant
17-11-2021 Arpan Deyasi, EM Theory 21
Arpan Deyasi
Electromagnetic
Theory

22. Then transmission line equations can be rearranged as
2
2
2
d
V(z) V(z)
dz
= 
2
2
2
d
I(z) I(z)
dz
= 
&
Solution of Telegrapher’s Equation
17-11-2021 Arpan Deyasi, EM Theory 22
Arpan Deyasi
Electromagnetic
Theory

23. Solution of Telegrapher’s Equation
Solutions are
z z
0 0
V(z) V e V e
+ −
− 
= +
z z
0 0
I(z) I e I e
+ −
− 
= −
Current wave proceeding in
the +ve direction at z axis =
Incident wave
Voltage wave proceeding in
the -ve direction at z axis =
Reflected wave
Voltage wave proceeding in
the +ve direction at z axis =
Incident wave
Current wave proceeding in
the -ve direction at z axis =
Reflected wave
17-11-2021 Arpan Deyasi, EM Theory 23
Arpan Deyasi
Electromagnetic
Theory

24. ‘α’ is called attenuation constant
‘β’ is called phase constant
For D.C signal, ω=0
RG 0
  =  =
For very high frequency, ω →
0 LC
  =  = 

Propagation constant
17-11-2021 Arpan Deyasi, EM Theory 24
Arpan Deyasi
Electromagnetic
Theory

25. Problem 2:
Calculate phase constant of propagating wave in a transmission line having L = 0.5 mH/km and
C = 0.08 μF/km where operating frequency is 400 KHz. Assume R and G are negligible
Soln
LC
 = 
3 3 6
2 400 10 0.5 10 0.08 10
− −
 =      
15.9 rad/km
 =
17-11-2021 Arpan Deyasi, EM Theory 25
Arpan Deyasi
Electromagnetic
Theory

26. Problem 3:
Calculate attenuation constant of propagating wave in a transmission line having R = 1 Ω/km
And G = 0.4 mho/km when DC signal is subjected.
Soln
RG
 =
1 0.4
 = 
1
0.02 m−
 =
17-11-2021 Arpan Deyasi, EM Theory 26
Arpan Deyasi
Electromagnetic
Theory

27. Phase Velocity
Phase velocity is defined by
p
v

=

( )( )
p 1/ 2
2 2 2 2 2 2 2
v
1
LC RG R L G C
2

=
 
 − + +  + 
 
 
( )( )
p 1/ 2
2 2 2 2 2 2 2
2
v
LC RG R L G C

=
 
 − + +  + 
 
 
17-11-2021 Arpan Deyasi, EM Theory 27
Arpan Deyasi
Electromagnetic
Theory

28. Problem 4:
Calculate phase constant of propagating wave inside a transmission line operated at 100 MHz
and velocity of wave is 2.5⨯108 m/sec
Soln
Phase constant
p
v

 =
6
8
2 100 10
2.5 10
 
 =

2.51 rad/m
 =
17-11-2021 Arpan Deyasi, EM Theory 28
Arpan Deyasi
Electromagnetic
Theory

29. Problem 5:
Calculate phase velocity of propagating wave in a transmission line having L = 0.5 mH/km and
C = 0.08 μF/km
Soln
Phase velocity
p
v

=

p
1
v
LC LC

= =

p 3 6
1
v
0.5 10 0.08 10
− −
=
  
5
p
v 1.583 10 km/sec
= 
17-11-2021 Arpan Deyasi, EM Theory 29
Arpan Deyasi
Electromagnetic
Theory

30. Wavelength
Operating wavelength
2
 =

( )( )
1/ 2
2 2 2 2 2 2 2
2
1
LC RG R L G C
2

 =
 
 − + +  + 
 
 
( )( )
1/ 2
2 2 2 2 2 2 2
2 2
LC RG R L G C

 =
 
 − + +  + 
 
 
17-11-2021 Arpan Deyasi, EM Theory 30
Arpan Deyasi
Electromagnetic
Theory

31. Problem 6:
Calculate phase constant of propagating wave inside a transmission line operated at 0.5 m
wavelength
Soln
Phase constant
2
 =

2
0.5

 =
12.567 rad/m
 =
17-11-2021 Arpan Deyasi, EM Theory 31
Arpan Deyasi
Electromagnetic
Theory

32. If a signal of 30 MHz is transmitted through a coaxial line having C = 30 pF/m and L = 500
nH/m, then calculate [i] time delay, [ii] propagation velocity for 1 m long cable.
Soln
Problem 7:
Time delay
d
t LC
=
9 12
d
t 500 10 30 10
− −
=   
d
t 3.87 ns
=
17-11-2021 Arpan Deyasi, EM Theory 32
Arpan Deyasi
Electromagnetic
Theory

33. Velocity of wave propagation
p
d
L
v
t
=
p 9
1
v
3.87 10−
=

8
p
v 2.5839 10 m/sec
= 
17-11-2021 Arpan Deyasi, EM Theory 33
Arpan Deyasi
Electromagnetic
Theory