As overhead conductors are identical, thenr¼rA¼rB¼rC. Also, as the conductors have equilateral
arrangement,D¼DAB¼DBC¼DCA
VAB¼
1
2 p« 0
qA ln"
D
r#
ln"
DAB^0
DAA 0!#
þqB
ln"
r
D#
ln"
DBB^0
DAB 0#!
qCln"
DBC^0
DAC 0"##
VÞð (13:93)Similarly, expressions forVBCandVACare
VBC¼1
2 p« 0"
qAln"
DCA 0
DBA 0#
þqB
lnD
r#
ln"
DCB 0
DBB 0#!
þqC
ln"
r
D#
ln"
DCC 0
DBC 0"##!
VÞð (13:94)VAC¼1
2 p« 0"
qA ln"
D
r#
ln"
DCA 0
DAA 0#!
qBln"
DCB 0
DAB 0#
þqC
ln"
r
D#
ln"
DCC 0
DAC 0#!#
VÞð (13:95)The phase voltageVANbecomes, through algebraic reduction,
VAN¼VABþVAC
3¼1
2 p« 0
qA lnD
r
ln" ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(^3) DAB 0 DBC 0 DCA 0
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(^3) DAA 0 DBB 0 DCC 0
p
!#
VÞð (13:96)
Therefore, the phase capacitanceCAN, per unit length, is
CAN¼
qA
VAN
¼
2 p« 0
ln
h
D
r
i
ln
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(^3) DAB (^0) DBC (^0) DCA 0
p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(^3) DAA 0 DBB 0 DCC 0
p
ðÞF=m (13:97)
Equations (13.79)and (13.97) have similar expressions, except for the term ln ((DAB^0 DBC^0 DCA^0 )^1 =^3 =(DAA^0
DBB^0 DCC^0 )^1 =^3 ) included in Eq. (13.97). That term represents the effect of the earth on phase
capacitance, increasing its total value. However, the capacitance increment is really small, and is usually
A
B
C
B
A C
DA DC
qB
qA qC
−qA −qC
−qB
Image Conductors
Overhead Conductors
DCC = 2DC
DB
DAA = 2DA Earth’s SurfaceDBB = 2DB
FIGURE 13.19 Arrangement of image conductors in a three-phase transmission line.