VAB transp¼
VAB firstþVAB secondþVAB third
3
¼1
6 p« 0
qAlnDABDACDBC
r^3
þqBlnr^3
DABDACDBC
þqClnDACDACDBC
DACDACDBC
ðÞV (13:86)VAC transp¼VAC firstþVAC secondþVAC third
3
¼
1
6 p« 0qAln
DABDACDBC
r^3
þqBln
DACDACDBC
DABDACDBC
þqCln
r^3
DACDACDBC
ðÞV (13:87)For a balanced system whereqA¼(qBþqC), the phase-to-neutral voltageVAN(phase voltage) is
VAN transp¼
VAB transpþVAC transp
3
¼1
18 p« 0
2 qAlnDABDACDBC
r^3
þðÞqBþqC lnr^3
DABDACDBC¼
1
6 p« 0qAln
DABDACDBC
r^3
¼
1
2 p« 0qAln
GMD
r
ðÞV (13:88)where GMD¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(^3) DABDBCDCA
p
¼geometrical mean distance for a three-phase line.
For bundle conductors, an equivalent radiusrereplaces the radiusrof a single conductor and is
determined by the number of conductors per bundle and the spacing of conductors. The expression ofre
is similar to GMRbundleused in the calculation of the inductance per phase, except that the actual outside
radius of the conductor is used instead of the GMRphase. Therefore, the expression forVANis
VAN transp¼
1
2 p« 0
qAln
GMD
re
ðÞV (13:89)
wherere¼(dn^1 r)^1 =n¼equivalent radius for up to three conductors per bundle (m)
re¼1.09 (d^3 r)^1 =^4 ¼equivalent radius for four conductors per bundle (m)
d¼distance between bundle conductors (m)
n¼number of conductors per bundle
Finally, the capacitance and capacitive reactance, per unit length, from phase to neutral can be
evaluated as
CAN transp¼
qA
VAN transp
¼
2 p« 0
ln
GMD
re
ðÞF=m (13:90)
XAN transp¼
1
2 pfCAN transp
¼
1
4 pf« 0
ln
GMD
re
ðÞV=m (13:91)
13.5.5 Capacitance Due to Earth’s Surface
Considering a single-overhead conductor with a return path through the earth, separated a distanceH
from earth’s surface, the charge of the earth would be equal in magnitude to that on the conductor but of
opposite sign. If the earth is assumed as a perfectly conductive horizontal plane with infinite length, then
the electric field lines will go from the conductor to the earth, perpendicular to the earth’s surface
(Fig. 13.17).