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8/8/2019 Slide07 Chapter04

1/89

BEE3133

Electrical Power SystemsChapter 3

Transmission Line Parameters

Rahmatul Hidayah Salimin


2/89


3/89

RESISTANCE

Important in transmission

efficiency evaluation and

economic studies.

Significant effect Generation ofI2R loss in

transmission line.

ProducesIR-type voltage dropwhich affect voltage regulation.


4/89

RESISTANCE

The dc resistance of a solid roundconductorat a specified temperatureis

Where :

= conductor resistivity (-m),

l = conductor length (m) ; and

A = conductor cross-sectional area (m2)

dc

lR

AV!


5/89

RESISTANCE

Conductor resistance isaffected by three factors:-

Frequency (skin effect)

Spiraling

Temperature


6/89

RESISTANCE

Frequency Skin Effect

When ac flows in a conductor, the

current distribution is not uniform over

the conductor cross-sectional area andthe current density is greatest at the

surface of the conductor.

This causes the ac resistance to be

somewhat higher than the dcresistance. The behavior is known as

skin effect.


7/89

RESISTANCE

The skin effect is where alternating

current tends to avoid travel through

the center of a solid conductor, limiting

itself to conduction near the surface.

This effectively limits the cross-

sectional conductor area available to

carry alternating electron flow,

increasing the resistance of thatconductor above what it would

normally be for direct current


8/89

RESISTANCE


9/89

RESISTANCE

Skin effect correction factoraredefined as

Where

R = AC resistance ; and

Ro= DC resistance.

O

R

R


10/89

RESISTANCE

Spiraling

For stranded conductors, alternatelayers of strands are spiraled inopposite directions to hold the strands

together. Spiraling makes the strands 1 2%

longer than the actual conductorlength.

DC resistance of a stranded conductoris 1 2% larger than the calculatedvalue.


11/89

RESISTANCE

Temperature

The conductor resistance increasesas temperature increases. Thischange can be considered linear overthe range of temperature normally

encountered and may be calculatedfrom :

Where:R1 = conductor resistances at t1 in C

R2 = conductor resistances at t2 in C

T = temperature constant (depends on

the conductor material)

22 1

1

T tR R

T t

!


12/89

RESISTANCE

The conductor resistance is best

determined from manufacturers

data.

Some conversion used incalculating line resistance:-

1 cmil = 5.067x10-4 mm2

=5.067x10

-6

cm

2

= 5.067x10-10 m2


13/89


14/89

RESISTANCE

Example:-

A solid cylindrical aluminum

conductor 25km long has an area

of 336,400 circular mils. Obtain theconductor resistance at

(a) 20C and

(b) 50C.

The resistivity of aluminum at 20C is

= 2.8x10-8-m.


15/89

RESISTANCE

Answer (a)

25

8 3

4

6

2.8 10 25 10

336, 400 5.076 10

4.0994 10

l km

l

R A

V!

!

v v v!

v v

! v ;


16/89

RESISTANCE

Answer (b)

5050 20

20

6

6

228 504.0994 10

228 204.5953 10

C

C C

C

T tR R T t

rr r

r

!

! v

! v ;


17/89

RESISTANCE

Exercise 1

A transmission-line cable consists of

12 identical strands of aluminum,

each 3mm in diameter. Theresistivity of aluminum strand at

20C is 2.8x10-8-m. Find the 50C

ac resistance per km of the cable.

Assume a skin-effect correctionfactor of 1.02 at 50Hz.


18/89

RESISTANCE

Exercise 2:-A solid cylindrical aluminum

conductor 115km long has an area

of 336,400 circular mils. Obtain the

conductor resistance at:

(a) 20C

(b) 40C

(c) 70C

The resistivity of aluminum at 20C is

= 2.8x10-8-m.


19/89

RESISTANCE

Exercise 3

A transmission-line cable consists of

15 identical strands of aluminum,

each 2.5mm in diameter. Theresistivity of aluminum strand at

20C is 2.8x10-8-m. Find the 50C

ac resistance per km of the cable.

Assume a skin-effect correctionfactor of 1.015 at 50Hz.


20/89

INDUCTANCE :

A SINGLE CONDUCTOR

A current-carrying conductor producesa magnetic field around the conductor.

The magnetic flux can be determined

by using the right hand rule.

For nonmagnetic material, the

inductance L is the ratio of its total

magnetic flux linkage to the currentI,

given by

where=flux linkages, in Weber turns.

LI

P!


21/89

INDUCTANCE :

A SINGLE CONDUCTOR

For illustrativeexample, considera long roundconductor with

radius r, carryinga currentIasshown.

The magnetic

field intensityHx,around a circle ofradiusx, isconstant andtangent to thecircle.

2

xx

IH

xT!


22/89

INDUCTANCE :

A SINGLE CONDUCTOR

The inductance of the conductorcan be defined as the sum of

contributions from flux linkages

internal and external to theconductor.


23/89

Flux Linkage


24/89

INDUCTANCE :

A SINGLE CONDUCTOR


25/89

INDUCTANCE :

A SINGLE PHASE LINES


26/89

INDUCTANCE :

3-PHASE TRANSMISSION LINES


27/89

INDUCTANCE :



28/89

What and How to Calculate:-

Lint , Lext @ L?

L1 , L2 @ L?

L11 , L12 @ L22? GMR?

GMD?


29/89

INDUCTANCE :

A SINGLE CONDUCTOR

INTERNAL INDUCTANCE

Internal inductance can be express as

follows:-

Where

o = permeability of air (4 x 10-7 H/m)

The internal inductance is independent of

the conductor radius r

70int 1 10 /

8 2L H mQ

T! ! v


30/89

INDUCTANCE :

A SINGLE CONDUCTOR

INDUCTANCE DUETO EXTERNAL

FLUX LINKAGE

External

inductancebetween to point

D2 and D1 can be

express as

follows:

7 2

1

2 10 ln /extD

L H mD

! v


31/89

INDUCTANCE :


A single phase lines consist of asingle current carrying line with a

return line which is in opposite

direction. This can be illustrated as:


32/89

INDUCTANCE :


Inductance of a single-phaselines can be expressed as

below with an assumption

that the radius of r1=r2=r.7 7 2

int

1

7 7 7

1

7 741

4

7

0.25

110 2 10 ln /

2

1 110 2 10 ln / 2 10 ln /

2 4

12 10 ln ln / 2 10 ln ln /

2 10 ln /

ext

D L L L H m

D

D DH m H m

r r

D De H m H m

r re

DH m

re

! ! v v

! v v ! v

! v ! v

! v


33/89

SELF AND MUTUAL

INDUCTANCES

The series inductance per phase canbe express in terms of self-inductanceof each conductor and their mutualinductance.

Consider the one meter length single-phase circuit in figure below:-

Where L11 and L22 are self-inductanceand the mutual inductance L

12


34/89

SELF AND MUTUAL

INDUCTANCES

D

xD

xL

DxL

erxL

ILLID

xer

xIL

ILL

ILL

mHD

xer

xL

mH

D

xerxL

1ln102

1

ln102

1ln102

1

ln102

1ln102

1ln102

/1

ln1021

ln102

/1ln102

1

ln102

77

12

7

12

25.01

7

11

112111

7

25.0

1

7

111

222212

112111

7

25.0

2

7

2

7

25.0

1

7

1

!

!

!

!

!

!!

!

!

!

!

P

P

P


35/89

SELF AND MUTUAL

INDUCTANCES

L11, L22 and L12 can be expressed asbelow:-

7

11 0.25

1

7

22 0.25

2

7

12 21

1

2 10 ln

12 10 ln

12 10 ln

L r e

L

r e

L LD

! v

! v

! ! v


36/89

SELF AND MUTUAL

INDUCTANCES

Flux linkage of conductor i

ijDIerIx

n

j ij

j

i

ii {

! !

1

ln1

ln1021

25.0

7

P


37/89

INDUCTANCE :


Symmetrical Spacing Consider 1 meter length of a three-phase

line with three conductors, each radius r,

symmetrically spaced in a triangular

configuration.


38/89

INDUCTANCE :


Assume balance 3-phase current

Ia+ Ib+ Ic = 0

The total flux linkage of phase a

conductor

Substitute for Ib + Ic=-Ia

!

DI

DI

erIx

cb

a

aa

1ln

1ln

1ln102

25.0

7P

25.0

7

25.0

7 ln1021

ln1

ln102

!

!

er

DIx

DI

erIx

a

aa

a

aaP


39/89

INDUCTANCE :


Because of symmetry, a=b=c

The inductance per phase per

kilometer length

kmmH

re

Dx

I

L /ln10225.0

7

!!P


40/89

INDUCTANCE :


Asymmetrical Spacing Practical transmission lines cannot maintain

symmetrical spacing of conductors because of

construction considerations.

Consider one meter length of three-phase line with

three conductors, each with radius r. Theconductor are asymmetrically spaced with

distances as shown.


41/89

INDUCTANCE :


The flux linkages are:-

v!

v!

v!

2313

25.0

7

2312

25.0

7

1312

25.0

7

1ln

1ln

1ln102

1ln

1ln

1ln102

1ln

1ln

1ln102

DI

DI

reI

DI

DI

reI

DI

DI

reI

bacc

cabb

cbaa

P

P

P


42/89

INDUCTANCE :


For balanced three-phase currentwithI

aas reference, we have:-

a

o

ac

a

o

ab

aIIII

aII

!!!!

120240

2


43/89

INDUCTANCE :


Thus La,Lb and Lc can be foundusing the following equation:-

v!!

23

2

25.0

12

7 1ln1

ln1

ln102

D

a

reD

a

I

L

b

b

b

P

v!!

1312

2

25.0

7 1ln1

ln1

ln102D

aD

areI

La

a

a

P

v!!

25.0

2313

27 1ln1

ln1

ln102reD

aD

aI

Lc

c

c

P


44/89

INDUCTANCE :


Transpose Line Transposition is used to regain symmetry

in good measures and obtain a per-phaseanalysis.


45/89

INDUCTANCE :


This consists of interchanging the phaseconfiguration every one-third the length sothat each conductor is moved to occupy thenext physical position in a regular sequence.

Transposition arrangement are shown in the

figure


46/89

INDUCTANCE :


Since in a transposed line eachphase takes all three positions,

the inductance per phase can be

obtained by finding the averagevalue.


47/89

0.25

12 13

7

0.25

23 12

0.25

13 23

7

0.25

12

3

1 1 1ln 1 240 ln 1 120 ln

2 10 1 1 1ln 1 240 ln 1 120 ln

3

1 1 1ln 1 240 ln 1 120 ln

2 10 1 1 13ln ln ln

3

a b c L L L

L

re D D

re D D

re D D

re D D

!

r r

v ! r r

r r

-

v!

R R

R R

R R

23 13

312 23 137

0.25

1ln

2 10 ln

D

D D D

re

! v


48/89

Since in a transposed line each phasetakes all three positions, the

inductance per phase can be obtained

by finding the average value.

3

cba

a

LLL

L

!


49/89

Noting a + a2 = -1

Inductance per phase per kilometer

length

25.0

3

1

1323127

3

1

132312

25.0

7

132312

25.0

7

ln102

1ln1ln102

1ln

1ln

1ln

1ln3

3

102

v!

v!

v!

re

DDD

DDDre

DDDreL

kmmH

re

DDDL /ln2.0

25.0

3

1

132312

!


50/89

What and How to Calculate:-

Lint , Lext @ L?

L1 , L2 @ L?

L11 , L12 @ L22? GMR?

GMD?


51/89

Inductance of Composite

Conductors

In evaluationofinductance, solid round

conductors were considered.However,in

practicaltransmissionlines, stranded

conductors are used.

Consider a single-phase line consistingof

twocomposite conductorsx and y as shown

in Figure 1. The currentinx isIreferenced

intothe page, andthe returnin y is I.


52/89


Conductors

Conductorx consistofn identical strands or

subconductors, each with radius rx.

Conductory consistofm identical strands or

subconductors, each with radius ry

.

The currentis assumedtobe equally divided

amonthe subconductors. The current per

strands isI/n inx andI/m iny.


53/89


Conductors

x y


54/89

nncnbnax

mnmncnbnan

n

nanacabx

mamacabaaa

a

nanacabx

mamacabaa

a

amacabaa

anacabx

a

DDDr

DDDDn

nIL

DDDr

DDDD

nnIL

DDDr

DDDDI

or

DDDDm

I

DDDrn

I

...'

...ln102

/

...'

...

ln102/

...'

...ln102

1ln...

1ln

1ln

1ln102

1ln...

1ln

1ln

'

1ln102

'''7

'''7

'''7

'''

7

7

v!!

v!!

v!

v

v!

P

P

P

P


55/89

'...

)...)...(...(

)...)...(...(

/ln102

2

''''

7

xnnbbaa

nnnnbnaanabaax

mnnmnbnaamabaa

x

x

rDDD

where

DDDDDDGMR

DDDDDDGMD

where

mHGMR

GMDL

!!!

!

!

v!


56/89

GMR of Bundled Conductors

Extra high voltage transmissionlines are

usually constructed with bundledconductors.

Bundling reduces the line reactance, which

improves the line performance andincreases

the powercapability ofthe line.


57/89

GMR of Bundled Conductors

4 316 42/1

3 29 3

09.1)2(

)(

dDdddDD

bundleorsubconductfourthefor

dDddDD

bundleorsubconductthreethefor

ss

b

s

ss

b

s

v!vvvv!

v!vv!

dD

dDD

bundleorsubconducttwothefor

DDDDDDGMR

ss

b

s

nnnnbnaanabaax

v!v!

!

4 2

)(

)...)...(...(2

I d t f Th h


58/89

Inductance of Three-phase

Double Circuit Lines

Athree-phase double-circuittransmission

line consists oftwoidenticalthree-phase

circuits. To achievebalance, each phase

conductormustbe transposed withinitgroup

and with respecttothe parallelthree-phase

line.

Consider a three-phase double-circuitline

with relative phase positions a1

b1

c1

-c2

b2

a2

.

I d t f Th h


59/89



c

GMDbetween each phase group

422122111

422122111

422122111

cacacacaC

cbcbcbcbBC

babababaB

DDDDD

DDDDD

DDDDD

!

!

!

I d t f Th h


60/89



The equivalentGMD per phase is then

3ACBCAB

DDDGMD !

Similarly,GMRofeach phase group is

214 2

21

214 2

21

214 2

21

)(

)(

)(

cc

b

cc

b

SC

bb

b

bb

b

SB

aa

b

aa

b

SA

DDDDD

DDDDD

DDDDD

ss

ss

ss

!!

!!

!!

where is the geometricmean radius of

bundledconductors.

b

sD

I d t f Th h


61/89



The equivalentGMR per phase is then

3SCSSAL DDDGMR !

The inductance per-phase is

mHGMR

GMDL

L

x /ln1027v!

INDUCTANCE :


62/89

INDUCTANCE :


Question 4

A three-phase, 50 Hz transmission line has a

reactance 0.5 per kilometer. The conductor

geometric mean radius is 2 cm. Determine thephase spacing D in meter.

INDUCTANCE :


63/89

INDUCTANCE :


Question 4

A three-phase, 60 Hz transmission line has a

reactance 0.25 per kilometer. The conductor

geometric mean radius is 5 cm. Determine thephase spacing D in meter.

INDUCTANCE :


64/89

INDUCTANCE :


Question 4

A three-phase, 50 Hz transmission line has

Xc = 0.5 per kilometer. The conductor geometric

mean radius is 2 cm. Determine the phase spacingD in meter.


65/89

CAPACITANCE

Transmission line conductorsexhibit capacitance with respect

to each other due to the potential

difference between them.

The amount of capacitancebetween conductors is a function

ofconductor size, spacing, and

height above ground.

Capacitance C is:-

qC

V!


66/89

LINE CAPACITANCE

Consider a longround conductor

with radius r,

carrying a

charge ofqcoulombs per

meter length as

shown.

The electricalflux density at a

cylinder of radius

x is given by: 2

q qD

A xT! !


67/89

LINE CAPACITANCE

The electric field intensity E is:-

Where permittivity of free space, 0 = 8.85x10-12 F/m.

The potential difference between cylinders

from position D1 to D2 is defined as:-

The notation V12 implies the voltage drop from 1

relative to 2.

0 02

D qE

xI TI! !

212

0 1

ln2

q DVDTI

!

CAPACITANCE OF SINGLE


68/89

CAPACITANCE OF SINGLE-

PHASE LINES

Consider one meter length of a single-phase line consisting of two long solid

round conductors each having a

radius r as shown.

For a single phase, voltage betweenconductor 1 and 2 is:-

12

0

ln /q D

V F mrTI

!



69/89


PHASE LINES

The capacitance between theconductors:-

012 /

ln

C F mD

r

TI!



70/89


PHASE LINES

The equation gives the line-to-line capacitance between the

conductors

For the purpose of transmission

line modeling, we find itconvenient to define a

capacitance Cbetween each

conductor and a neutral line as

illustrated.



71/89


PHASE LINES

Voltage to neutral is half ofV12 and the capacitance to

neutral is C=2C12 or:-

02 /

ln

C F mD

r

TI!

Potential Difference in a


72/89

Potential Difference in a

Multiconductor configuration

Consider n parallel long conductorswith charges q1, q2,,qncoulombs/meter as shown below.

ki

kj

n

k

kijD

DqV ln

2

1

10!! TI

Potential difference between conductori and j due to the presence of all

charges is

CAPACITANCE OF THREE-


73/89


PHASE LINES

Consider one meter length of 3-phaseline with three long conductors, each

with radius r, with conductor spacing

as shown below:

CAPACITANCE OF THREE


74/89


PHASE LINES

For balanced 3-phase system, the

capacitance per phase to neutral is:

1/3

12 23 13

2F/m

ln

a o

an

qC

VD D D

r

TI! !



75/89


PHASE LINES

1/312 23 130.0556 F/km

ln

C

D D D

r

Q!

The capacitance to neutral in F perkilometer is:


76/89

Effect of bundling

mF

r

GMDCb

/ln

2 0TI

!

The effect of bundling is introduce an

equivalent radius rb. The radius rb issimilar to GMRcalculate earlier for the

inductance with the exception that

radius rof each subconductor is used

instead ofDs.


77/89

Effect of bundling

Ifd is the bundle spacing, we obtain for

the two-subconductor bundle

drrb v!

For the three-subconductor bundle

3 2drrb v!

For the four-subconductor bundle4 309.1 drrb v!

Capacitance of Three-phase


78/89

Capacitance of Three phase


mF

GMRGMD

C

c

/

ln

2 0TI!

The per-phase equivalent capacitanceto neutral is obtained to

GMD is the same as was found for

inductance calculation

4

22122111

422122111

422122111

cacacacaAC

cbcbcbcbC

babababaA

DDDDD

DDDDD

DDDDD

!

!

!

Capacitance of Three-phase


79/89

Capacitance of Three phase


The equivalentGMD per phase is then

3ACCA

DDDGMD !

The GMRC of each phase is similar to

the GMRL, with the exception that rb isused instead of b

sD

This will results in the following equ

21

21

21

cc

b

C

bb

b

B

aab

A

Drr

Drr

Drr

!

!

! 3CBAC rrrGMR !

EFFECT OF EARTH ON THE


80/89


CAPACITANCE

For isolated charged conductor theelectric flux lines are radial andorthogonal to cylindrical equipotentialsurfaces, which will change theeffective capacitance of the line.

The earth level is an equipotentialsurface. Therefore flux lines are forcedto cut the surface of the earthorthogonally.

The effect of the earth is to increasethe capacitance.



81/89


CAPACITANCE

But, normally, the height of theconductor is large compared to thedistance between the conductors, andthe earth effect is negligible.

Therefore, for all line models used forbalanced steady-state analysis, theeffect of earth on the capacitance canbe negligible.

However, for unbalance analysis suchas unbalance faults, the earths effectand shield wires should be considered.

MAGNETIC FIELD


82/89

MAGNETIC FIELD

INDUCTION

Transmission line magnetic fieldsaffect objects in the proximity of

the line.

Produced by the currents in theline.

It induces voltage in objects that

have a considerable length

parallel to the line (Ex: telephone

wires, pipelines etc.).

MAGNETIC FIELD


83/89

MAGNETIC FIELD

INDUCTION

The magnetic field is effected bythe presence of earth return

currents.

There are general concernsregarding the biological effects of

electromagnetic and electrostatic

fields on people.

ELECTROSTATIC


84/89

ELECTROSTATIC

INDUCTION

Transmission line electric fields affectobjects in the proximity of the line.

It produced by high voltage in thelines.

Electric field induces current inobjects which are in the area of theelectric fields.

The effect of electric fields becomes

more concern at higher voltages.

ELECTROSTATIC


85/89

ELECTROSTATIC

INDUCTION

Primary cause of induction to vehicles,buildings, and object of comparable

size.

Human body is effected to electric

discharges from charged objects in thefield of the line.

The current densities in human cause

by electric fields of transmission lines

are much higher than those induced by

magnetic fields!

CORONA


86/89

CORONA

When surface potential gradientexceeds the dielectric strength ofsurrounding air, ionization occursin the area close to conductor

surface. This partial ionization is known as

corona.

Corona generate by atmosphericconditions (i.e. air density,humidity, wind)

CORONA


87/89

CORONA

Corona produces power loss andaudible noise (Ex: radio

interference).

Corona can be reduced by: Increase the conductor size.

Use of conductor bundling.

Re ie


88/89

Review

Transmission Line Parameters: Resistance

Skin effect

Inductance Single phase line

3 phase line equal & unequal spacing

Capacitance Single phase line

3 phase line equal & unequal spacing

Conductance Neglected Corona

Review


89/89

Review

Effect of Earth on theCapacitance

Magnetic Field Induction

Electrostatic Induction Corona

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