Electrical Power Systems

28 Electrical Power Systems
Using eqns. (2.56) and (2.57), we get
HG
1 1
log log
r¢
D
-
la = 0.4605 I a Ia
\ la = 0.4605 Ia log D
r¢
Therefore, L a = l a
I
a
= 0. 4605 log
I
KJ
I mWb–T/km HG KJ ...(2.58)
DI
mH/km HG KJ
...(2.59)
Because of symmetry, l a = l b = l c and hence three inductances are identical, i.e.,
L b = L c = L a.
2.10.1 Inductance of Three Phase Transmission Lines with
Asymmetrical Spacing
In actual practice, the conductors of a three phase
transmission line are not at the corners of an equilateral
triangle because of construction considerations. Therefore
with asymmetrical spacing, even with balanced currents,
the flux linkages and inductance of each phase are not the
same. A different inductance in each phase, resulting in
unbalanced receiving-end voltages even when sendingend
voltages and line currents are balanced. igure 2.8
shows the conductors of a three phase transmission line ig. 2.8: Three phase line with
with asymmetrical spacing.
asymmetrical spacing.
Using eqn. (2.46) will result in the following flux linkages.
or in matrix form
L
NM
l
l
l
a
b
c
l a = 0 4605
RST
r¢
1 1 1
. Ia
log I b log I c log
r¢
D
D
+ +
1 1 1
. I a log + I b log I c log
D r¢
D
+
l b = 0 4605
l c = 0 4605
O
QP
RST
RST
L
ab
1 1 1
. Ia
log + I b log + I c log
D D r¢
= 0.4605
ca
ab
bc
1 1 1
log log log
r¢ D D
1 1 1
log log log
D r¢ D
NM
ab ca
ab bc
1 1 1
log log log
D D r¢
ca bc
Therefore symmetrical inductance matrix L is given by
O L
QP
NM
I
I
a
I
b
c
ca
bc
O
QP
UVW
UVW
UVW
...(2.60)
...(2.61)
...(2.62)
...(2.63)