Presented By:

Anisur Rahman,ID: 1402EEE00033

Department of EEEManarat International University (MIU)

Ashulia, Khagan, Savar, Dhaka, Bangladesh

Definition Of Inductance

Flux Linkages of Conductors

1. Flux linkages inside the conductor

2. Flux linkages outside the conductor

Flux linkages of one conductor in a group of

conductors

Inductance of a single phase two wire line

Inductance of 3-phase overhead line

Bundled conductors

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4

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Consider a conductor of radius r carrying a

current I. At a distance x from the center of

this conductor, the magnetic field intensity Hx

The internal inductance per meter is

intint

8H ml

I

7

70int

4 10

8 8H ml

If the relative permeability of the conductor is 1 (non-ferromagnetic

materials, such as copper and aluminum), the inductance per meter

reduces to

The external inductance per meter is :

The total external flux linkages per meter can be found via integration

To find the flux linkages external to a

conductor, we will consider the portion of

flux between two points P1 and P2 that lie

at distances D1 and D2 from the center of

the conductor.

Theoretically , the flux due to a conductor extends from

the center of the conductor to right unto infinity.

Assuming that the flux linkages will extend unto a point P

very far from the group of the conductors , and the

distances are as shown in fig.

Consider a group of conductors 1, 2, 3,…… n such that the

sum of currents in all these conductors is zero.

If the currents carried by respective strands are I1, I2, I3, …In

we have I1+I2+I3+…+In = 0

ψ1p1 = All flux linkages of conductor 1 due to tis own current

I1 , internal and external , unto point P ….

Ψ1p1 = 2 × 10 I1 ln D1p/r1‘ Wb. T/Mt

Ψ1p2 = Flux linkages with conductor 1 due to current in

conductor-2

Ψ1p2 = 2 × 10 × I2 ln D2p/D12

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Similarly ψ1p3…….ψ1pn

Ψ1p=

2 × 10 { I1 ln D1p/r'1 + I2 ln D2p/D12 +...+ In ln Dnp/D1n}

Flux linkages with conductor 1 due to I1 , I2…..In

Net flux linkages ψ1p

Ψ1p = 2 × 10 { I1 ln 1/r1'+ I2 ln 1/D12 +…+ In ln 1/D1n }

Wb-turns/m

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The inductance of a single-phase line

consisting of two conductors of radii r

spaced by a distance D and both carrying

currents of magnitude I flowing into the

page in “A” conductor and out of the page

in the “B“ conductor.

x xH dl I Ñ

Since the path of radius x2 encloses both

conductors and the currents are equal and

opposite, the net current enclosed is 0 and,

therefore, there are no contributions to the

total inductance from the magnetic fields

at distances greater than D.

A B

The total inductance of a line per unit length in this transmission line is a sum of

the internal inductance and the external inductance between the conductor surface

(r) and the separation distance (D):

int

1ln

2 4ext H m

Dl l l

r

By symmetry, the total inductance of the other line is the same, therefore, the total

inductance of a two-wire transmission line is

1

ln4

H mD

lr

Where r’=is GMR (Geometric mean radius)

For a solid conductor G.M.R = 0.7788 times the radius of conductor.

D is the distance between conductors

‘

‘

‘

In a 3-phase transmission line, the inductance of each

conductor is considered instead of loop inductance.

The conductor of a 3-phase overhead line may be placed

symmetrically or unsymmetrically on the towers.

With Symmetrical Spacing :

A 3-phase line in which the space between any two

conductor is the same as shown in fig. the line is called

symmetrical line.

Fig. shows the conductor of a 3-phase line conductor has

a radius r meters and spacing between the conductors is

D meters.

Under balanced three-phase phasor currents, the algebraic sum of the currents in the conductors is zero.

Hence, Ia + Ib + Ic = 0

The flux linkages of the conductor ‘a’ are

ψa = 2 × 10 [ Ia ln 1/Daa + Ib ln 1/Dab + Ic ln 1/Dac ] Wb-T/m

Inductance of conductor a,

La = ψa/Ia

= 2 × 10 ln D/r‘ H/m

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A 3-phase line in which the space between the conductors is

different as shown in fig., the line is called unsymmetrical line.

Consider 3-phase line with conductors a, b and c each of

radius r meters.

Let, the spacing between them be Dab, Dbc and Dca and the

currents flowing through them be Ia, Ib and Ic respectively as

shown in fig.

From fig, the flux linkages of the conductor a

Ψa = 2 × 10 [ Ia ln 1/r' + Ib ln 1/D12 + Ic ln 1/D31 ] Wb-T/m-7

A bundle conductor is a conductor made up of two or

more sub-conductors and is used as one phase

conductor.

Lines of 400kv and higher voltages invariably use

bundled conductors.

Sub-conductors of a bundled conductor are separated

from each other by a constant distance varying from 0.2

m to 0.6 m depending upon designed voltage and

surrounding conditions throughout the length of the line

with the help of spacers.

It reduces corona loss.

It reduces radio interference

The bundled conductor lines transmit bulk power with

reduced losses, thereby giving increased transmission

efficiency

Bundle conductor lines have a higher capacitance to

neutral so they have higher charging current, which helps

in improving power factor

By bundling, the GMR is increased, the inductance per

phase is reduced. As a result reactance per phase is

reduced.