(Definition of Inductance, Flux linkages of current carrying conductor, Inductance of a single phase two wire line)

CONTENTS Defination Of Inductance

Flux Linkages of Conductors1. Flux linkages inside the conductor2. Flux linkages outside the conductor

Inductance of a single phase two wire line

General value of Inductance & Capacitance in Transmission Lines

WHAT IS INDUCTANCE ??

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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 can be found from Ampere’s law:

x xH dl I

1. Flux linkages inside the conductor

Where Hx is the magnetic field intensity at each point along a closed path, dl is a unit vector along that path and Ix is the net current enclosed in the path.

For the homogeneous materials and a circular path of radius x, the magnitude of Hx is constant, and dl is always parallel to Hx. Therefore:

22

xx x x

IxH I Hx

We had already assumed uniform current density ;

2

2xxI Ir

Thus, the magnetic intensity at radius x inside the conductor is

22x H mxH Ir

The flux density at a distance x from the center of the conductor is

2 [ ]2x x T

xIB Hr

The differential magnetic flux contained in a circular tube of thickness dx and at a distance x from the center of the conductor is

2 [ ]2

xId dx Wb mr

The flux linkages per meter of length due to flux in the tube is the product of the differential flux and the fraction of current linked:

2 3

2 4 [ ]2

Wb turns mx x Id d dxr r

d = Bx * (dx * 1)

The total internal flux linkages per meter can be found via integration…3

int 40

[ ]2 8

r

Wb turns mx I Id dxr

Therefore, the internal inductance per meter is

intint 8

H mlI

7

70int

4 108 8

H ml

If the relative permeability of the conductor is 1 (non-ferromagnetic materials, such as copper and aluminum), the inductance per meter reduces to

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.

In the external to the conductor region, the magnetic intensity at a distance x from the center of conductor is

2 2x

xI IH

x x

The flux density at a distance x from the center of conductor is

2x xIB Hx

The differential magnetic flux contained in a circular tube of thickness dx and at a distance x from the center of the conductor is :

[ ]2

Id dx Wb mx

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

2 2

1 1

1

2

[ ]ln2 2

D D

extD D

Wb turns mDI Id dx

x D

The external inductance per meter is :

t 2t

1

ln2

exex H m

DlI D

Inductance of a single-phase 2-wire line

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

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

Inductance of a single-phase 2-wire transmission lineThe 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):

int1 ln

2 4ext H mDl l lr

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 mDlr

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

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Example based on Inductance of Single phase 2 wire tranmission line

Questions ..??1) When the flux increases , inductance will -------- ??

2) GMR is abbreviated as _________ and it is equals to ______ times radius of conductor ??

3) If current is 2A and inductance is 4H , what will be the value of flux ??

Answers A1) Increase

A2) GEOMETRIC MEAN RADIUS & GMR = 0.7788 times radius of conductor

A3) 8 Wb

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