electromagnetics lecture on transmission line theory

1. Applied Electromagnetics
1
ECEN 372
LECT. 1: The Transmission Line Theory
1
Dr. Ahmed Mahmoud
Assistant Professor, ECE Program
Email: amwmahmoud@nu.edu.eg

2. Telegrapher Equations
L= Series inductance per unit length for both conductors ( / )
H m
G= Shunt conductance per unit length ( / )
S m
C= Shunt capacitance per unit length ( / )
F m
( , )
( , ) ( , ) ( , ) 0
i z t
v z t R zi z t L z v z z t
t

       

( , )
( , ) ( , ) ( , ) 0
v z z t
i z t G zv z z t C z i z z t
t
  
         

Kirchoff’s Laws
R= Series resistance per unit length for both conductors ( / )
m

I
V L
t



V
I C
t



V IR

I VG


3. Telegrapher Equations (Cont’d)
( , )
( , ) ( , ) ( , ) 0
i z t
v z t R zi z t L z v z z t
t

       

( , ) ( , ) ( , )
( , ) 0
v z t i z t v z z t
Ri z t L
z t z
  
   
  
( , ) ( , ) ( , )
( , )
v z z t v z t i z t
Ri z t L
z t
   
  
 
( , ) ( , )
( , )
v z t i z t
Ri z t L
z t
 
  
 
0
( , ) ( , ) ( , )
lim ( , )
z
v z z t v z t i z t
Ri z t L
z t
 
   
  
 

4. Telegrapher Equations (Cont’d)
( , ) ( , )
( , )
v z t i z t
Ri z t L
z t
 
  
 
( , ) ( , )
( , )
i z t v z t
Gv z t C
z t
 
  
 
( )
( ) ( )
dV z
RI z j LI z
dz

  
( )
( ) ( )
dV z
R j L I z
dz

  
( )
( ) ( )
dI z
G j C V z
dz

  
( )
( ) ( )
dI z
GV z j CV z
dz

  
j
t





5. Phasors
Complex number that carries amplitude and phase information of a sinusoidal function
cos( ) sin( )
j
e i

 
 
cos( ) Re{ }
j
e 
  sin( ) Im{ }
j
e 
 
( )
cos( ) Re{ } Re{ }
j t j t j
m m m
v V t V e V e e
   
  
   
Re{ }
j j t
m
v V e e
 

{ cos( )}
j
m m
V V e P V t

 
   1 1 j
e 

 

6. Inverse Phasor Transform
1
{ } Re{ }
j j j t
m m
P V e V e e
  



7. Telegrapher Equations (Cont’d)
2
2
( )( ) ( )
d V
R j L G j C V z
dz
 
  
2
2
2
( ) 0
d V
V z
dz

 
2
2
( ) ( )
d V dI
R j L z
dz dz

  
( )
( ) ( )
dI z
G j C V z
dz

  
2


8. Telegrapher Equations (Cont’d)
2
2
2
( ) 0
d V
V z
dz

 
( ) z z
o o
V z V e V e
 
  
 
( ) ( )
dV
R j L I z
dz

  
( ) [ ]
z z
o o
I z V e V e
R j L
 


  

  

1
( )
( )
dV
I z
R j L dz

 

( ) [ ]
z z
o o
I z V e V e
R j L
 


  
 


9. Telegrapher Equations (Cont’d)
o
R j L R j L
Z
G j C
 
 
 
 

o o
o
o o
V V
Z
I I
 
 
  
( ) z z z z
o o
o o
o o
V V
I z I e I e e e
Z Z
   
 
   
   
( ) z z
o o
V z V e V e
 
  
 
( ) [ ]
z z
o o
I z V e V e
R j L
 


  
 

Characteristic Impedance

10. Telegrapher Equations (Cont’d)
( , ) | | cos( ) | | cos( )
z z
o o
v z t V t z e V t z e
 
     
    
     
( ) ( )
( ) | | | |
j j z j j z
o o
V z V e e V e e
     
 
    
 
( ) z z
o o
V z V e V e
 
  
 
( , ) Re{ ( ) }
j t
v z t V z e 

( ) ( )
( , ) Re{| | | | }
z j t z z j t z
o o
v z t V e e V e e
       
 
      
 

11. Telegrapher Equations (Cont’d)
( , ) | | cos( ) | | cos( )
z z
o o
v z t V t z e V t z e
 
     
    
     
2



ph
v f



 
( ) ( ( ) ) 2
t z t z
       
 
      
2
 

( )
t z Const
  
  
z t Const
  
  
t Const
z
 


 

dz
dt


