Unit: III Line Parameters computations

3.1 Introduction to line parameters: Resistance, Inductance and Capacitance

A transmission line is used for the transmission of electrical power from generating substation to the various
distribution units. It transmits the wave of voltage and current from one end to another. The transmission
line is made up of a conductor having a uniform cross-section along the line. Air act as an insulating or
dielectric medium between the conductors.
The performance of transmission line depends on the parameters of the line.
The transmission line has mainly four parameters, resistance, inductance, and capacitance and shunt
conductance.
These parameters are uniformly distributed along the line. Hence, it is also called the distributed parameter
of the transmission line.

The inductance and resistance form series impedance whereas the capacitance and conductance form the
shunt admittance.

Line inductance – The current flow in the transmission line induces the magnetic flux. When the current
in the transmission line changes, the magnetic flux also varies due to which emf induces in the circuit. The
magnitude of inducing emf depends on the rate of change of flux. Emf produces in the transmission line
resist the flow of current in the conductor, and this parameter is known as the inductance of the line.

Line capacitance – In the transmission lines, air acts as a dielectric medium. This dielectric medium
constitutes the capacitor between the conductors, which store the electrical energy, or increase the
capacitance of the line. The capacitance of the conductor is defined as the present of charge per unit of
potential difference.

Capacitance is negligible in short transmission lines whereas in long transmission; it is the most important
parameter. It affects the efficiency, voltage regulation, power factor and stability of the system.

Shunt conductance – Air act as a dielectric medium between the conductors. When the alternating voltage
applies in a conductor, some current flow in the dielectric medium because of dielectric imperfections. Such
current is called leakage current. Leakage current depends on the atmospheric condition and pollution like
moisture and surface deposits.

Shunt conductance is defined as the flow of leakage current between the conductors. It is distributed
uniformly along the whole length of the line. The symbol Y represented it, and it is measured in Siemens.
A transmission line has four parameters
Resistance (R) causes power loss in conductor

Inductance(L) magnetic field surrounding the conductor

depends upon the power transmission capacity of the line.

Capacitance (C) electric field between the conductors causes

charging current to flow in the line and assumes importance for medium
and long lines.

Shunt conductance(G) currents along insulators strings and

corona i.e caused by leakage current . Its effect is small and usually
neglected.
 Line Resistance:
Resistance to d.c current is given by

The variation of resistance of metallic conductors with temperature is

practically linear over the normal range of operation. Suppose R1 and R2
are the resistances of a conductor at t 1ºC and t2 ºC (t2 > t1) respectively.
If a1 is the temperature coefficient at t1°C, then,
Inductance
The flux linkage per current (in ampere) is called inductance.

Inductance of single conductor

Consider a long st. cylindrical conductor of radius r meter and carrying
current I ampere (r.m.s).
Current will set up magnetic field.
Magnetic lines of force will exist inside as well as outside the conductor.
Both these flux will contribute to the inductance of the conductor.
Total inductance of single conductor per meter length is
LT = Lint + Lext
external inductance
due to external flux

internal inductance The return path is

assumed to be
due to internal flux
situated so far
inductance due to internal flux linkage away that the
return current
does not affect the
magnetic field of
conductor.
As Current density is uniform
Current density (σ) = =
Current (IX) = current density × area enclosed
The differential flux in small region of thickness of length 1 M

For total flux linkage, we integrate 0 to r.

External inductance due to external flux linkage

Assumption : x>r
But we have to consider the flux linkage from conductor surface to any
external distance, i.e. r to D
Inductance of single phase two wire

Conductor current are equal in

magnitude but opposite in
direction

1 2
Go return
As a simplifying assumption we can assume that all the external flux set
up by current in conductor 1 links all the current up to the centre of
conductor 2 and that the flux beyond the centre of conductor 2 does
not link any current.
This assumption gives quite accurate results especially when D is much
greater than r1 and r2.
Thus, to obtain the inductance of conductor 1 due to the external flux
linkage, substituting D1 = r1 and D2 = D
This eqn gives the inductance of a two wire line in which one
conductor acts as a return conductor for the other. This is known
as loop inductance.
Example: A two conductor single phase line operates at 50 Hz. The
diameter of each conductor is 20 mm and the spacing between the
conductor is 3 m. calculate
(i) The inductance of each conductor per km.
(ii) the loop inductance of the line per km.
(iii) the inductive reactance per km.
Example: A single phase overhead line 25 km long is to be constructed
of conductor 1.5 cm diameter. Calculate the maximum spacing between
the conductor in order that the loop inductance of conductor is not
more than 0.08H.
Flux Linkages of One Conductor in a Group

➢composite conductor which is made up of

two or more strands which are in parallel.
➢all the strands are identical and share the
current equally.
➢The sum of the currents in all the
conductors is zero.

➢The conductors 1, 2, 3, ... n carry the currents I1,

I2, I3 ... In.
➢Let the distances of the conductors from a point
P be D1p, D2p , D3p ... Dnp respectively.
➢Let ψ1p1 be flux linkages of conductor due to its own current I1 due
to internal and external flux.
➢The flux beyond point P is excluded.

Now ψ1p2 is the flux linkages of conductor 1 due

to current is equal to flux produced by
I2 between the point P and conductor 1.

Similarly the flux linkages ψ1p with conductor 1 due to all the conductors in the group
By expanding the logarithmic terms and rearranging the terms

The sum of all currents is zero.

Substituting this value in above equation

If point P is at infinite distance so that ln (D1p/Dnp) ln (D2p/Dnp) etc will approach to
zero (since ln 1 = 0) then we have,
Inductance of composite conductor
Stranded conductor come under the general classification of composite
conductors which means conductors composed of 2 or more elements
or stranded electrically in parallel.
All strands are identical and share the current equally.

Consider the composite conductor ‘p’ and ‘Q’.

DM = mutual GMD (geometric mean distance)
DS = self GMD or GMR (Geometric mean radius)
Example : figure shows an arrangement of conductors for single
phase supply, the current being equally divided between conductors
X and X’ and between y and y ’ . If the diameter of each conductor is
8 mm. find the inductance per km of the line.
Inductance of three phase transmission line with symmetrical spacing

c
Inductance of three phase transmission line with unsymmetrical
spacing
Consider 3 –conductors a,b,c
having radius r.
From above expression
Individual phase inductance of line which unsymmetrical spaced is
a complex number.
Imaginary part in expression for inductance represents exchange
of energy between phases.
Inductance of 3 –Φ line with unsymmetrical spacing but transposed
➢asymmetrical spacing gives voltage drops even under balanced
current conditions.
➢ this leads unbalanced voltage at receiving end of the line.
➢However, one way to regain symmetry in good measure and obtain a
per phase model by exchanging the positions of the conductors at
regular intervals along the line such that each conductor occupies the
original position of every other conductor.
➢Such an exchange of conductor positions is called transposition.
➢The transposition is usually carried out at switching stations.
 = equivalent equilateral spacing

is also called geometric mean distance of line (DM).

Note : it is not present practice to transpose the power lines at regular

intervals. How ever an interchange in position of the conductors is
made at switching stations to be balance the inductance of the phase.
Example: A three phase transmission line has conductor diameter of 1.6
cm each, The conductors being spaced as shown in figure. The line is
carrying balanced load and it is transposed. Find the inductance of the
line per km per phase.
 Double circuit three phase line
➢A double circuit three phase line has two parallel conductors for each
phase.
➢ use of double circuit 3-phase line is to increase transmission
reliability and a higher transmission capacity.
Inductance of three phase double circuit with symmetrical spacing
Consider the three phase double circuit with the conductors placed at the
vertices of a regular hexagon.
Inductance of three phase double circuit with unsymmetrical
spacing but transposed
Example: figure shows the spacing of a double circuit 3 phase
overhead line. The conductor radius is 1.5 cm and line is
transposed. Find the inductance per phase per kilometer.
Bundled conductor
To transmit large amount of power over long distances by EHV and UHV
line are usually constructed with bundled conductors.
Bundled conductors increases self GMD and line inductance is reduced
which increase power capability of transmission line.

Fig: Configuration of bundled conductor

GMR of bundled conductor can be obtain with same as stranded
conductor.
Example:
Determine the geometric mean radius of a conductor in terms of the
radius r of an individual strand for (a) three equal strands and (b) four
equal strands.

Fig : (a) Fig : (b)

A single circuit three phase transposed transmission line is composed of
2 ACSR conductor per phase with horizontal configuration as shown in
figure. Find the inductive reactance per km at 50 Hz. Radius of each sub
conductor in the bundle is 1.725 cm.
Example:
Determine the inductance of a single phase transmission line consisting
of three conductors of 2 cm radii in the ‘go’ conductor and two
conductors of 4 cm radii in the return.
 Skin Effect
When an Alternating Current flows through a conductor, it is not distributed uniformly
throughout the conductor cross-section. AC current has a tendency to concentrate near
the surface of the conductor. This phenomenon in alternating currents is called as
the skin effect. Due to the skin effect, current is concentrated between the outer
surface of the conductor and a level called as the skin depth (skin depth is shown by ẟ
in the following figure). If the frequency of AC current is very high, the current is
restricted to a very thin layer near the conductor surface. Skin effect increases with
increase in the frequency.

Due to skin effect, the effective cross-section of the conductor through which the current
flows is reduced. Consequently, the effective resistance of the conductor is slightly
increased.
A solid conductor split into a large number of strands, each strand carrying a small part
of current. The inductance of each strand will vary according to its position. Strands
located at the center would be surrounded by a greater magnetic flux and, therefore, will
have a larger inductance than those near the surface. Higher inductance (and hence,
higher reactance) of the inner strands causes the alternating current to flow through the
strands having lower reactance, i.e. near the surface.

The skin effect depends upon the following factors:

Conductor material: Better conductors and ferromagnetic materials experience higher

skin effect

Cross-sectional area of the conductor: skin effect increases with increase in the cross-
sectional area

Frequency: increases with increase in the frequency

Shape of the conductor: skin effect is lesser for stranded conductors than solid
conductors
Proximity Effect

When two or more conductors carrying alternating current are close to each other, then
distribution of current in each conductor is affected due to the varying magnetic field of
each other. The varying magnetic field produced by alternating current induces eddy
currents in the adjacent conductors. Due to this, when the nearby conductors carrying
current in the same direction, the current is concentrated at the farthest side of the
conductors. When the nearby conductors are carrying current in opposite direction to
each other, the current is concentrated at the nearest parts of the conductors. This effect
is called as Proximity effect. The proximity effect also increases with increase in the
frequency. Effective resistance of the conductor is increased due to the proximity effect.

Skin effect and proximity effect both

are absent in case of DC currents, as
frequency of DC current is zero.
GMR(self GMD)
A conductor is composed of seven (7) identical copper each having
radius r as shown in the figure. Find the self GMD of the conductor
given D14 = 4r ,D12 = 2r ,D26 = 2√3r

Answer :2.177 r
Capacitance of transmission line

➢Capacitance is the ability of an object to hold an electric charge.

➢Capacitance in a transmission line results due to the potential

difference between the conductors.

➢The conductors get charged in the same way as the parallel plates
of a capacitor.
➢Capacitance between two parallel conductors depends on the size
and the spacing between the conductors.

➢Usually the capacitance is neglected for the transmission lines that

are less than 50 miles (80 km) long.
➢However the capacitance becomes significant for longer lines with
higher voltage.
Electric field and potential differences:
If a long st. cylindrical conductor carry a uniform charge(q)
throughout its length.
Electric flux density;

From gauss’s theorem the electric field intensity E at any point is

consider the long straight conductor of Fig. that
is carrying a positive charge q C/m.

Let two points P1 and P2 be located at

distances D1 and D2 respectively from the center of
the conductor.
The conductor is an equipotential surface in which
we can assume that the uniformly distributed
charge is concentrated at the center of the
conductor.
The potential difference V12 between the
points P1 and P2 is the work done in moving a unit
of charge from P2 to P1 .
Therefore the voltage drop between the two
points
Capacitance of a single phase two-wire transmission line

The voltage drop between

conductor a and b can be
obtained by adding voltage
drop by each charge alone.
The potential difference between each conductor and the ground
(or neutral) is one half of the potential difference between the two
conductors. Therefore, the capacitance to ground of this single-
phase transmission line will be

2
cn = can = cbn =
D
ln
r
Capacitive reactance between one conductor and neutral;

XC=

Line to neutral susceptance;

BC = =

Example: A two conductor single phase line operates at 50Hz. The diameter of each conductor
is 2 cm and are spaced 3 m apart. Calculate:
(a) Capacitance of each conductor to neutral per km.
(b) Line to line capacitance
(c) Capacitive susceptance to neutral per km.
Capacitance of a three phase line with equilateral spacing

qa,qb and qcare charges on respective conductors

The p.d between conductor a and b

& conductors a and c can be written
as

…………….(I)

……………..(ii)
For balanced load

……………..(iii)
Fig shows the phasor diagram of balanced
voltage of three phase line.

Line to neutral capacitance

Charging current per phase;

Capacitance of a three phase line with unsymmetrical spacing
It is assumed that the charge per unit length of the conductor remains
same in different position of transposition cycle.
Example: A 3 –phase 50 Hz , 200 km line consists of three conductors
each of diameter 21 mm. the spacing between the conductor are A-
B=3m, B-C= 5m , C-A=3.6m.
Find the capacitance and capacitive reactance per km of the line.
If the line operates at 132 kV. find the charging current per km and the
reactive voltamperes generated by the line per km.
Capacitance of 3 –phase double circuit
Normally two configuration of conductors are used in a double circuit
line.
(i) Hexagonal spacing
(ii) flat vertical spacing
Capacitance of 3 –phase double circuit
Flat vertical spacing (unsymmetrical spacing)

I ii iii
Example : A 3 phase double circuit line is shown in figure. Find the
capacitance per phase to neutral. Diameter of each conductor is 2cm.
Effect of earth on capacitance
It was assumed that the conductor are situated in free space so the
effect of earth is neglected.
But actually the conductors run parallel to the ground.
The earth is assumed to behave like an infinite, perfectly conducting
plane.

The effect of earth on capacitance of the line can be modeled by the

method of images suggested by Lord Kelvin.

Any one conductor are located at equal distances but in opposite

direction from ground surface.
Their charges and potential are equal but opposite sign.
The electric flux between the overhead
conductors and their images is perpendicular
to the plane i.e this plane forms an
equipotential surface.
Single phase line
Example:
Find the capacitance of a single phase line 40 km long consisting of
2 parallel wires each 4mm in diameter and 2m apart.
Determine the capacitance of the same line taking into account
effect of ground. The height of conductors above ground is 5m.
Comparing this equation with expression for capacitance of single
phase line without considering the effect of earth, we see that
earth tries to increase the capacitance of line by small amount.
But the effect is negligible if the conductors are high above ground
compared to distances between them.
Bundled conductors
Bundled conductors
Q. Find out the capacitance and charging current per unit length of the line when the
arrangement of the conductor is shown in figure below. The line is completely
transposed. The radius of the conductor is 0.75 cm and operating voltage is 220 kV.