Electric Power Generation, Transmission, and Distribution

13.5.3 Capacitance of a Three-Phase Line

Consider a three-phase line with the same voltage magnitude between phases, and assuming a balanced
system with abc (positive) sequence such thatqAþqBþqC¼0. The conductors have radiirA,rB, andrC,
and the space between conductors areDAB,DBC, andDAC(whereDAB,DBC, andDAC>rA,rB, andrC).
Also, the effect of earth and neutral conductors is neglected.
The expression for voltages between two conductors in a single-phase system can be extended to
obtain the voltages between conductors in a three-phase system. The expressions forVABandVACare

VAB¼

1

2 p« 0

qAln

DAB

rA



þqBln

rB

DAB



þqCln

DBC

DAC



ðÞV (13:72)

VAC¼

1

2 p« 0

qAln

DCA

rA



þqBln

DBC

DAB



þqCln

rC

DAC



ðÞV (13:73)

If the three-phase system has triangular arrangement with equidistant conductors such
thatDAB¼DBC¼DAC¼D, with the same radii for the conductors such thatrA¼rB¼rC¼r(where
D>r), the expressions forVABandVACare

VAB¼

1

2 p« 0

qAln

"

D

r

#

þqBln

"

r

D

#

þqCln

"

D

D

"##

¼

1

2 p« 0

qAln

"

D

r

#

þqBln

"

r

D

"##

ðÞV (13:74)

VAC¼

1

2 p« 0

qAln

"

D

r

#

þqBln

"

D

D

#

þqCln

"

r

D

"##

¼

1

2 p« 0

qAln

"

D

r

#

þqCln

"

r

D

"##

ðÞV (13:75)

Balanced line-to-line voltages with sequence abc, expressed in terms of the line-to-neutral voltage are

VAB¼

ffiffiffi

3

p

VANff 30 andVAC¼VCA¼

ffiffiffi

3

p

VANff 30 ;

whereVANis the line-to-neutral voltage. Therefore,VANcan be expressed in terms ofVABandVACas

VAN¼

VABþVAC

3

(13:76)

and thus, substitutingVABandVACfromEqs. (13.67)and (13.68)we have

VAN¼

1

6 p« 0

qAln

"

D

r

#

þqBln

"

r

D

"##

þ qAln

"

D

r

#

þqCln

"

r

D

"#"##

¼

1

6 p« 0

2 qAln

"

D

r

#

þ qBþqC

!

ln

"

r

D

##

ðÞV

"

(13:77)

Under balanced conditionsqAþqBþqC¼0, orqA¼(qBþqC) then, the final expression for the line-
to-neutral voltage is

VAN¼

1

2 p« 0

qAln

D

r



ðÞV (13:78)