Chapter 4_part 2

8/3/2019 Chapter 4_part 2

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Line Parameters

An electric transmission line is characterized by fourparameters:

series resistance

series inductance

shunt capacitance

shunt conductance

The inductance is the dominant series element due toits effect on power transmission capacity and voltage

drop It is associated with the induced voltage caused by the

change in flux that affect the conductivity and exchangecurrents in the circuit


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The capacitance is the dominating shunt element and itrepresents a source of reactive power.

Mvars VL2, the importance of the shunt capacitance

parameter increases with an increase in the magnitudeof the operating voltage.

The series resistance and shunt conductance are theleast important parameters as their effect on thetransmission capacity is relatively very less.

However, the series resistance completely determinesthe real power transmission loss and its presence mustbe considered.

Line Parameters


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The shunt conductance accounts for the resistiveleakage current

The leakage current:

flows along the insulator strings and ionised pathways

in the air varies appreciably with changes in the weather,

atmospheric humidity, pollution and salt content

The effect of the shunt conductance under normal

operating conditions is usually neglected Consider only three parameters for the development of

a transmission line model for use in power systemstudies

Line Parameters


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Line Resistance

Resistance causes power loss in a transmission line.

Direct current resistance Rdcis given as:

Where = conductor resistivity in -m

l = conductor length in m

A = conductor cross-sectional area in m2

A

lR

dc


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The conductor resistance affected by three factors:frequency, spiraling, and temperature.

When AC flows in a conductor, current distribution is notuniform over the conductor cross-sectional area

current density is greatest at the surface of the conductor(skin effects)

Thus, at 60 Hz, the ac resistance is about 2% higherthan DC resistance.

Stranded conductor is spiraled, each strand is longerthan the finished conductor this results a slightlyhigher resistance than the value calculated usingformula:

Line Resistance

A

lRdc


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Power losses cause a rise in temperature of the lineconductors.

The resistance of a conductor also depends on itstemperature.

Resistance Rt1 of a conductor at t10C is obtained fromthe relation:

where R0

= resistance of conductor at 00C

0 = temperature coefficient of the conductor at 00C

Line Resistance

11 001 tRRt


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The conductor resistance increase as temperatureincreases.

The resistance Rt2of a conductor at temperature t20C

can be found out from the expression:

Line Resistance

10

20

1

2

/1

/1

t

t

R

R

t

t

OR

1

2

1

2

tT

tT

R

R

t

t

1

2

12

tT

tTRR

tt

Where T = 1/0 temperature

constants


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Values of T for certain types of conductor:

Aluminium = 228

Annealed hard drawn copper = 234.5

(100% conductivity) Hard drawn copper = 241

(97.3% conductivity)

The conductor resistance is best determined from

manufacturers data.

Line Resistance


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Line Conductance

Under normal operating conditions, the effect of shuntconductance, G, is usually neglected because it isnegligible.


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Line Inductance

The inductance of a conductor is defined as the fluxlinkages per unit current in it, given by:

Flows of alternating current in conductor will change theflux which links the conductor

This flux consist of two: internal flux and external flux

Internal flux due to the current inside the conductor External fluxdue to conductors current and due to the

current of other conductors in the vicinity

Henry

I

NL


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The inductance due to internal flux internal inductance,Lin

The inductance due to external flux externalinductance, L

ex

The total inductance:

Line Inductance

exin LLL


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Line Inductance

Internal Inductance

Figure A shows the crosssection of a long cylindricalconductor with radius randcarrying a current Iamperes.

Assume the return path of thecurrent is far away does noteffect the magnetic field of theconductor

Therefore, we can assume thatthe magnetic flux lines to beconcentric with the conductor

dxxr

Figure A


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Line Inductance

The magnetic field intensity atthat distance x m is given by:

where IX = the current in dx2x = the length of the flux

path

Assuming N=1, then:

dxxr

Figure A

mATx

NIH XX /

2

x

IHx

X

2 (1)


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Line Inductance

IX = current density x areaenclosed

=

Substitute 2 in 1:

The flux density is given as:

Where =r0 = permeability

dxxr

Figure A

AIr

xx

r

I2

2

2

2

(2)

mATr

xI

xI

r

xHX /

224

12

2

2

2

2/

2mwb

r

xIHB Xx


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Line Inductance

The flux, d is given as:

Where Ax = the elemental area

normal to the flux path. Consider 1 m length of the

conductor, then the flux, dx isgiven as:

The flux linkages per meterlength of conductor is given as:

dxxr

Figure A

xxx ABd

wbdxr

xI

dxBd xx 22)1(

2

4

3

2

2

/

2

mWbTdx

r

Ixd

r

xd xx


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Line Inductance

The internal flux linkages isgiven as:

Thus, the internal inductancesis given as:

If relative permeability, r = 1and 0 = 4 x 10

-7 H/m, then:

dxxr

Figure A

mWbTI

dxr

Ixr

inx /82

0

4

3

Henry

I

L rinin

88

0

mHLin /10

2

1 7Internal inductance

independent of conductordimension


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External Inductance

Assume the flux lines betweenA1 and A2 lie within theconcentric cylindrical surfacepassing A1 and A2

The magnetic field intensity ata distance x meters from thecenter of the conductor is:

The flux density BX is given by:

Line Inductance

dx

x

D1

D2

A1

A2

Figure B

mATx

I

HX /2

2/

2

mWb

x

IHB XX


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The flux d in the tubularelement of thickness dx metersis (for an axial length 1 m):

The flux linkage for the solidconductor (assume N=1) is:

Line Inductance

dx

x

D1

D2

A1

A2

Figure B

mWbdx

x

IABd XXx /

2

1

212 ln

22

2

2

1DDIdx

xI

dxx

Idd

D

D

X

XX


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The external inductance isgiven as:

We take = 0 = 4 x 10-7 H/m

because outside theconductor, the medium is air(r = 1).

Line Inductance

dx

x

D1

D2

A1

A2

Figure B

1

27

1

20 ln102ln2 D

D

D

D

IL X

ex t


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r

D

Line Inductance

The total inductance for asingle conductor A is the sumof its internal and externalinductance which is given as:

'ln102ln102

ln1

ln102

lnln102

ln

4

1102

ln102102

1

ln102

102

1

7

4/1

7

4/1

7

4/17

7

77

7

7

r

D

re

D

r

D

e

r

De

r

D

r

DLLL

r

DL

L

ex tin

ex t

in

A


21/23

D

1 2

r1 r

2

mH

r

DLLL /

'

ln1047

21

Inductance for Single Phase 2 WireLine

kmmH

r

DL /

'

log921.0OR

The inductance in line 1 is given as:

The inductance in line 2:

The total inductance value:

mHr

DL /

'ln102

1

7

1

mHrDL /ln102

'

2

72


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Example 4.1

Inductance for Single Phase 2 WireLine


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Exercise (Single Phase Trans. Line)

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Chapter 4_part 2