Short Notes: Transmission Lines | Short Notes for Electrical Engineering - Electrical Engineering (EE) PDF Download

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 Page 1


 
 
 
 
Skin Effect  
It is tendency of AC current to be concentrated on the surface of conductor. 
Cause:  Non-uniform distribution of magnitude flux linkages Due to skin effect, the effective 
area of cross section of conductor decrease and hence resistances increases. 
In case of DC, There is no skin effect so  
 
DC AC
RR ? 
? With increase in frequency, skin effect increases. 
? With increase in
r
? , skin effect increases. 
Inductance of a Transmission line 
? Single Conductor 
            Internal inductance 
0r
8
??
?
 
External inductance from distance ‘
1
d ’ to ‘
2
d ’ 
 
0r 2
ex
1
d
L ln
2d
??
??
??
??
??
?
?
 
Total inductance 
0 r 0 r 2
d
ln
8 2 r
? ? ? ?
??
??
??
??
??
 
  
0 r 0 r
1
4
dd
ln ln
2 2 r
re
?
??
? ? ? ? ??
?? ??
??
??
? ??
??
??
 
r 0.7788r ? ? = Geometric mean radius (GMR) 
 
? Single phase 2 – wire line 
Inductance of single wire 
0r
d
ln
2r
?? ??
?
??
? ?
??
 
Total inductance = 
12
LL ? 
0
sys
d
L ln
r
??
??
??
??
?
?
? ?
 
If radius of both wire is not same, assume radius of 1
st
 wire 
a
r & that of second wire is 
b
r
b
0
sys
a
d
L ln
rr
??
??
??
??
?
?
?
??
 
aa
r 0.7788r ? ?   &    
bb
r 0.7788r ? ? 
 
If instead of a single conductor per phase we use multiple conductor, then GMR is 
replaced by self GND (Geometric Mean Distance) and ‘d’ by mutual GMD. 
 
Page 2


 
 
 
 
Skin Effect  
It is tendency of AC current to be concentrated on the surface of conductor. 
Cause:  Non-uniform distribution of magnitude flux linkages Due to skin effect, the effective 
area of cross section of conductor decrease and hence resistances increases. 
In case of DC, There is no skin effect so  
 
DC AC
RR ? 
? With increase in frequency, skin effect increases. 
? With increase in
r
? , skin effect increases. 
Inductance of a Transmission line 
? Single Conductor 
            Internal inductance 
0r
8
??
?
 
External inductance from distance ‘
1
d ’ to ‘
2
d ’ 
 
0r 2
ex
1
d
L ln
2d
??
??
??
??
??
?
?
 
Total inductance 
0 r 0 r 2
d
ln
8 2 r
? ? ? ?
??
??
??
??
??
 
  
0 r 0 r
1
4
dd
ln ln
2 2 r
re
?
??
? ? ? ? ??
?? ??
??
??
? ??
??
??
 
r 0.7788r ? ? = Geometric mean radius (GMR) 
 
? Single phase 2 – wire line 
Inductance of single wire 
0r
d
ln
2r
?? ??
?
??
? ?
??
 
Total inductance = 
12
LL ? 
0
sys
d
L ln
r
??
??
??
??
?
?
? ?
 
If radius of both wire is not same, assume radius of 1
st
 wire 
a
r & that of second wire is 
b
r
b
0
sys
a
d
L ln
rr
??
??
??
??
?
?
?
??
 
aa
r 0.7788r ? ?   &    
bb
r 0.7788r ? ? 
 
If instead of a single conductor per phase we use multiple conductor, then GMR is 
replaced by self GND (Geometric Mean Distance) and ‘d’ by mutual GMD. 
 
 
 
 
 
 
Self GMD 
 
 
 
 
 
 
 
? ? ? ? ? ? ? ?
2
1
n
11 12 1n 21 22 2n n1 n2 nn
fwd
self GMD D D ........D D D .........D ....... D D ........D
??
??
? 
      Where 
ii i i
D r 0.7788r ?? ? 
? ? ? ? ? ?
2
' ' ' ' ' ' ' '
1
m
i'm'
i 1 i 2 ml mm
bwd
Self GMD D D ........D .............. D ................D
??
??
? 
    Where  
i i i i
D r 0.7788r
?
? ? ?
?? ? 
Mutual GMD 
Mutual GMD 
? ? ? ? ' ' ' ' ' '
1
mn
11 12 1m n1 n2 nm
D D ...........D ................. D D .................D
??
?
??
 
Now, with these terms all the inductance expressions change to  
            
0
d
Single wire : ln
2 self GMD
?
??
??
???
 
            
0
mutual GMD
1 , 2 wire: ln
2 Self GMD
???
??
??
?
??
 
Three – phase Transmission line 
Symmetrical configuration 
r 0
ph
D
L ln
2r
??
??
??
??
??
?
? ?
 
r 0.7788r ? ? 
 
Page 3


 
 
 
 
Skin Effect  
It is tendency of AC current to be concentrated on the surface of conductor. 
Cause:  Non-uniform distribution of magnitude flux linkages Due to skin effect, the effective 
area of cross section of conductor decrease and hence resistances increases. 
In case of DC, There is no skin effect so  
 
DC AC
RR ? 
? With increase in frequency, skin effect increases. 
? With increase in
r
? , skin effect increases. 
Inductance of a Transmission line 
? Single Conductor 
            Internal inductance 
0r
8
??
?
 
External inductance from distance ‘
1
d ’ to ‘
2
d ’ 
 
0r 2
ex
1
d
L ln
2d
??
??
??
??
??
?
?
 
Total inductance 
0 r 0 r 2
d
ln
8 2 r
? ? ? ?
??
??
??
??
??
 
  
0 r 0 r
1
4
dd
ln ln
2 2 r
re
?
??
? ? ? ? ??
?? ??
??
??
? ??
??
??
 
r 0.7788r ? ? = Geometric mean radius (GMR) 
 
? Single phase 2 – wire line 
Inductance of single wire 
0r
d
ln
2r
?? ??
?
??
? ?
??
 
Total inductance = 
12
LL ? 
0
sys
d
L ln
r
??
??
??
??
?
?
? ?
 
If radius of both wire is not same, assume radius of 1
st
 wire 
a
r & that of second wire is 
b
r
b
0
sys
a
d
L ln
rr
??
??
??
??
?
?
?
??
 
aa
r 0.7788r ? ?   &    
bb
r 0.7788r ? ? 
 
If instead of a single conductor per phase we use multiple conductor, then GMR is 
replaced by self GND (Geometric Mean Distance) and ‘d’ by mutual GMD. 
 
 
 
 
 
 
Self GMD 
 
 
 
 
 
 
 
? ? ? ? ? ? ? ?
2
1
n
11 12 1n 21 22 2n n1 n2 nn
fwd
self GMD D D ........D D D .........D ....... D D ........D
??
??
? 
      Where 
ii i i
D r 0.7788r ?? ? 
? ? ? ? ? ?
2
' ' ' ' ' ' ' '
1
m
i'm'
i 1 i 2 ml mm
bwd
Self GMD D D ........D .............. D ................D
??
??
? 
    Where  
i i i i
D r 0.7788r
?
? ? ?
?? ? 
Mutual GMD 
Mutual GMD 
? ? ? ? ' ' ' ' ' '
1
mn
11 12 1m n1 n2 nm
D D ...........D ................. D D .................D
??
?
??
 
Now, with these terms all the inductance expressions change to  
            
0
d
Single wire : ln
2 self GMD
?
??
??
???
 
            
0
mutual GMD
1 , 2 wire: ln
2 Self GMD
???
??
??
?
??
 
Three – phase Transmission line 
Symmetrical configuration 
r 0
ph
D
L ln
2r
??
??
??
??
??
?
? ?
 
r 0.7788r ? ? 
 
 
 
 
 
 
Asymmetrical configurations 
If conductors are placed horizontally or vertically. 
? ?
1
3
eq ab bc ca
D D D D ? ? ? 
eq
0r
ph
D
L ln
2r
??
??
??
??
??
?
? ?
 
In case of bundled conductor, more than one conductor per phase  
We replace   
? ? ab ab eq
DD ? = mutual GMD between a phase & b phase 
Similarly,       
? ?
bc bc eq
DD ? 
                      
? ? ca ca eq
DD ? 
In place of GMR, Self GMD is used 
? ? ? ? ? ?
1
3
a b c
Self GMD= Self GMD Self GMD Self GMD
??
??
 
Example:  Calculate inductance per phase of following circuit? 
 
 
 
Between successive conductors, distance = 3m , Radius of each conductor = 1m 
Solution 
? ? ? ?
1
4
ab a1b1 a1b2 a2b1 a2b2
eq
D D . D . D . D ? 
            ? ?
1
4
3 12 12 3 6m ? ? ? ? ? 
? ? ? ?
1
4
bc
eq
D 3 6 6 3 4.24m ? ? ? ? ? 
? ? ? ?
1
4
ca
eq
D 6 9 9 6 7.348m ? ? ? ? ? 
  
eq
D = mutual GMD 
         
1
3
ab.eq bc.eq ca.eq
 D D D ?? ? ? ?
??
 
         =   5.71m 
Page 4


 
 
 
 
Skin Effect  
It is tendency of AC current to be concentrated on the surface of conductor. 
Cause:  Non-uniform distribution of magnitude flux linkages Due to skin effect, the effective 
area of cross section of conductor decrease and hence resistances increases. 
In case of DC, There is no skin effect so  
 
DC AC
RR ? 
? With increase in frequency, skin effect increases. 
? With increase in
r
? , skin effect increases. 
Inductance of a Transmission line 
? Single Conductor 
            Internal inductance 
0r
8
??
?
 
External inductance from distance ‘
1
d ’ to ‘
2
d ’ 
 
0r 2
ex
1
d
L ln
2d
??
??
??
??
??
?
?
 
Total inductance 
0 r 0 r 2
d
ln
8 2 r
? ? ? ?
??
??
??
??
??
 
  
0 r 0 r
1
4
dd
ln ln
2 2 r
re
?
??
? ? ? ? ??
?? ??
??
??
? ??
??
??
 
r 0.7788r ? ? = Geometric mean radius (GMR) 
 
? Single phase 2 – wire line 
Inductance of single wire 
0r
d
ln
2r
?? ??
?
??
? ?
??
 
Total inductance = 
12
LL ? 
0
sys
d
L ln
r
??
??
??
??
?
?
? ?
 
If radius of both wire is not same, assume radius of 1
st
 wire 
a
r & that of second wire is 
b
r
b
0
sys
a
d
L ln
rr
??
??
??
??
?
?
?
??
 
aa
r 0.7788r ? ?   &    
bb
r 0.7788r ? ? 
 
If instead of a single conductor per phase we use multiple conductor, then GMR is 
replaced by self GND (Geometric Mean Distance) and ‘d’ by mutual GMD. 
 
 
 
 
 
 
Self GMD 
 
 
 
 
 
 
 
? ? ? ? ? ? ? ?
2
1
n
11 12 1n 21 22 2n n1 n2 nn
fwd
self GMD D D ........D D D .........D ....... D D ........D
??
??
? 
      Where 
ii i i
D r 0.7788r ?? ? 
? ? ? ? ? ?
2
' ' ' ' ' ' ' '
1
m
i'm'
i 1 i 2 ml mm
bwd
Self GMD D D ........D .............. D ................D
??
??
? 
    Where  
i i i i
D r 0.7788r
?
? ? ?
?? ? 
Mutual GMD 
Mutual GMD 
? ? ? ? ' ' ' ' ' '
1
mn
11 12 1m n1 n2 nm
D D ...........D ................. D D .................D
??
?
??
 
Now, with these terms all the inductance expressions change to  
            
0
d
Single wire : ln
2 self GMD
?
??
??
???
 
            
0
mutual GMD
1 , 2 wire: ln
2 Self GMD
???
??
??
?
??
 
Three – phase Transmission line 
Symmetrical configuration 
r 0
ph
D
L ln
2r
??
??
??
??
??
?
? ?
 
r 0.7788r ? ? 
 
 
 
 
 
 
Asymmetrical configurations 
If conductors are placed horizontally or vertically. 
? ?
1
3
eq ab bc ca
D D D D ? ? ? 
eq
0r
ph
D
L ln
2r
??
??
??
??
??
?
? ?
 
In case of bundled conductor, more than one conductor per phase  
We replace   
? ? ab ab eq
DD ? = mutual GMD between a phase & b phase 
Similarly,       
? ?
bc bc eq
DD ? 
                      
? ? ca ca eq
DD ? 
In place of GMR, Self GMD is used 
? ? ? ? ? ?
1
3
a b c
Self GMD= Self GMD Self GMD Self GMD
??
??
 
Example:  Calculate inductance per phase of following circuit? 
 
 
 
Between successive conductors, distance = 3m , Radius of each conductor = 1m 
Solution 
? ? ? ?
1
4
ab a1b1 a1b2 a2b1 a2b2
eq
D D . D . D . D ? 
            ? ?
1
4
3 12 12 3 6m ? ? ? ? ? 
? ? ? ?
1
4
bc
eq
D 3 6 6 3 4.24m ? ? ? ? ? 
? ? ? ?
1
4
ca
eq
D 6 9 9 6 7.348m ? ? ? ? ? 
  
eq
D = mutual GMD 
         
1
3
ab.eq bc.eq ca.eq
 D D D ?? ? ? ?
??
 
         =   5.71m 
 
 
 
 
? ? ? ?
1
4
a1a2 a2a1
a
Self GMD r D D r ?? ? ? ? ?  
                    ? ? ? ?
? ?
1
22 4
0.7788 0.01 15 ? ? ? 
                     =    0.341m 
? ? ? ?
1
4
b1b2 b2b1
b
Self GMD r D D r ?? ? ? ? ? 
                      =     0.2467m 
? ? ? ?
1
4
c1c2 c2c1
c
Self GMD r D D r ?? ? ? ? ? 
                      =     0.1528m 
? ? ? ? ? ? ? ?
1
3
ab
Self GMD Self GMD Self GMD Self GMD c ? 
                 =      0.2398m 
0
GMD
L ln
2 GMD Self
?
??
?
??
???
 
    
7
5.71
2 10 ln 0.634mH/km
0.2398
???
? ? ?
??
??
 
Remember, Inductance calculated using these formulas is per unit length. 
 
Transposition of Transmission line 
The position of different lines are changed after regular intervals to reduce radio interference in 
neighboring communication lines. 
 
 
 
 
Capacitance 
Single Phase 2 – Wire System 
                    
0r
ab
12
C
D
ln
rr
? ? ?
?
??
??
??
??
 
Line to neutral capacitance 
Page 5


 
 
 
 
Skin Effect  
It is tendency of AC current to be concentrated on the surface of conductor. 
Cause:  Non-uniform distribution of magnitude flux linkages Due to skin effect, the effective 
area of cross section of conductor decrease and hence resistances increases. 
In case of DC, There is no skin effect so  
 
DC AC
RR ? 
? With increase in frequency, skin effect increases. 
? With increase in
r
? , skin effect increases. 
Inductance of a Transmission line 
? Single Conductor 
            Internal inductance 
0r
8
??
?
 
External inductance from distance ‘
1
d ’ to ‘
2
d ’ 
 
0r 2
ex
1
d
L ln
2d
??
??
??
??
??
?
?
 
Total inductance 
0 r 0 r 2
d
ln
8 2 r
? ? ? ?
??
??
??
??
??
 
  
0 r 0 r
1
4
dd
ln ln
2 2 r
re
?
??
? ? ? ? ??
?? ??
??
??
? ??
??
??
 
r 0.7788r ? ? = Geometric mean radius (GMR) 
 
? Single phase 2 – wire line 
Inductance of single wire 
0r
d
ln
2r
?? ??
?
??
? ?
??
 
Total inductance = 
12
LL ? 
0
sys
d
L ln
r
??
??
??
??
?
?
? ?
 
If radius of both wire is not same, assume radius of 1
st
 wire 
a
r & that of second wire is 
b
r
b
0
sys
a
d
L ln
rr
??
??
??
??
?
?
?
??
 
aa
r 0.7788r ? ?   &    
bb
r 0.7788r ? ? 
 
If instead of a single conductor per phase we use multiple conductor, then GMR is 
replaced by self GND (Geometric Mean Distance) and ‘d’ by mutual GMD. 
 
 
 
 
 
 
Self GMD 
 
 
 
 
 
 
 
? ? ? ? ? ? ? ?
2
1
n
11 12 1n 21 22 2n n1 n2 nn
fwd
self GMD D D ........D D D .........D ....... D D ........D
??
??
? 
      Where 
ii i i
D r 0.7788r ?? ? 
? ? ? ? ? ?
2
' ' ' ' ' ' ' '
1
m
i'm'
i 1 i 2 ml mm
bwd
Self GMD D D ........D .............. D ................D
??
??
? 
    Where  
i i i i
D r 0.7788r
?
? ? ?
?? ? 
Mutual GMD 
Mutual GMD 
? ? ? ? ' ' ' ' ' '
1
mn
11 12 1m n1 n2 nm
D D ...........D ................. D D .................D
??
?
??
 
Now, with these terms all the inductance expressions change to  
            
0
d
Single wire : ln
2 self GMD
?
??
??
???
 
            
0
mutual GMD
1 , 2 wire: ln
2 Self GMD
???
??
??
?
??
 
Three – phase Transmission line 
Symmetrical configuration 
r 0
ph
D
L ln
2r
??
??
??
??
??
?
? ?
 
r 0.7788r ? ? 
 
 
 
 
 
 
Asymmetrical configurations 
If conductors are placed horizontally or vertically. 
? ?
1
3
eq ab bc ca
D D D D ? ? ? 
eq
0r
ph
D
L ln
2r
??
??
??
??
??
?
? ?
 
In case of bundled conductor, more than one conductor per phase  
We replace   
? ? ab ab eq
DD ? = mutual GMD between a phase & b phase 
Similarly,       
? ?
bc bc eq
DD ? 
                      
? ? ca ca eq
DD ? 
In place of GMR, Self GMD is used 
? ? ? ? ? ?
1
3
a b c
Self GMD= Self GMD Self GMD Self GMD
??
??
 
Example:  Calculate inductance per phase of following circuit? 
 
 
 
Between successive conductors, distance = 3m , Radius of each conductor = 1m 
Solution 
? ? ? ?
1
4
ab a1b1 a1b2 a2b1 a2b2
eq
D D . D . D . D ? 
            ? ?
1
4
3 12 12 3 6m ? ? ? ? ? 
? ? ? ?
1
4
bc
eq
D 3 6 6 3 4.24m ? ? ? ? ? 
? ? ? ?
1
4
ca
eq
D 6 9 9 6 7.348m ? ? ? ? ? 
  
eq
D = mutual GMD 
         
1
3
ab.eq bc.eq ca.eq
 D D D ?? ? ? ?
??
 
         =   5.71m 
 
 
 
 
? ? ? ?
1
4
a1a2 a2a1
a
Self GMD r D D r ?? ? ? ? ?  
                    ? ? ? ?
? ?
1
22 4
0.7788 0.01 15 ? ? ? 
                     =    0.341m 
? ? ? ?
1
4
b1b2 b2b1
b
Self GMD r D D r ?? ? ? ? ? 
                      =     0.2467m 
? ? ? ?
1
4
c1c2 c2c1
c
Self GMD r D D r ?? ? ? ? ? 
                      =     0.1528m 
? ? ? ? ? ? ? ?
1
3
ab
Self GMD Self GMD Self GMD Self GMD c ? 
                 =      0.2398m 
0
GMD
L ln
2 GMD Self
?
??
?
??
???
 
    
7
5.71
2 10 ln 0.634mH/km
0.2398
???
? ? ?
??
??
 
Remember, Inductance calculated using these formulas is per unit length. 
 
Transposition of Transmission line 
The position of different lines are changed after regular intervals to reduce radio interference in 
neighboring communication lines. 
 
 
 
 
Capacitance 
Single Phase 2 – Wire System 
                    
0r
ab
12
C
D
ln
rr
? ? ?
?
??
??
??
??
 
Line to neutral capacitance 
 
 
 
 
 
0r
an
1
2
C
D
ln
r
? ? ?
?
??
??
??
      ,      
0r
bn
2
2
C
D
ln
r
? ? ?
?
??
??
??
 
Three phase single conductor system 
0r
ph
2
C
GMD
ln
r
? ? ?
?
??
??
??
 
For bundled conductors 
00
ph
2
C
GMD
ln
Self GMD
? ? ?
?
??
??
??
 
In capacitance calculations, it must always be remembered that there is no concept of r, we 
simply use radius in calculating self GMD. 
Performance of Transmission line 
Classification of lines based on length  
1) Short Line 
          l < 80 km     or     l*f < 4000 ,          Where f = frequency 
 
2) Medium Line 
           80 km < l < 200 km 
           4000 < l*f < 10000 
 
3) Long Line 
          l    >  200 km 
          l*f > 10000 
Modeling of transmission lines  
Transmission lines are modeled as 2 – port network  
            
s R R
V AV BI ?? 
             
s R R
I CV DI ?? 
Under no load 
             
R
I0 ? , 
sR
V AV ? , 
s
R
V
V
A
? 
 
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FAQs on Short Notes: Transmission Lines - Short Notes for Electrical Engineering - Electrical Engineering (EE)

1. What is a transmission line in electrical engineering?
Ans. A transmission line is a specialized cable or conductor used to transfer electrical energy from one point to another. It is commonly used to transmit high-frequency signals, such as those used in telecommunications and power distribution systems.
2. What are the main types of transmission lines?
Ans. The main types of transmission lines are coaxial cables, twisted pair cables, and waveguides. Coaxial cables are commonly used in cable TV and internet connections, twisted pair cables are used in telephone systems, and waveguides are used in high-frequency applications.
3. What are the key parameters to consider in transmission line design?
Ans. The key parameters to consider in transmission line design are the characteristic impedance, attenuation, velocity of propagation, and electrical length. These parameters determine the performance and efficiency of the transmission line.
4. What is the significance of characteristic impedance in transmission lines?
Ans. Characteristic impedance is a fundamental parameter of a transmission line that determines how it responds to electrical signals. It is the ratio of voltage to current in a transmission line and plays a crucial role in impedance matching and signal transfer efficiency.
5. How can transmission line losses be minimized?
Ans. Transmission line losses can be minimized by using low-loss dielectric materials, reducing the length of the transmission line, increasing the conductor size, and properly terminating the line. Additionally, minimizing signal reflections and using efficient connectors can also help in reducing losses.
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