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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
?
Read More
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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Short Notes: Transmission Lines
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The content of Short Notes: Transmission Lines is prepared as per the latest Electrical Engineering (EE) syllabus.
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