Electrical Power Systems

L = 0.4605
L
Resistance and Inductance of Transmission Lines 29
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
or balanced three-phase currents with I a as reference, we have
I b = a 2 I a
I c = a I a
where the complex operator a = 1 120° and a 2 =1 240° . Note that a 3 = 1.
Using eqns. (2.63) and (2.65), we get
La = la
Lb = l b
Lc = lc
HG
HG
HG
O
QP
mH/km ...(2.64)
...(2.65)
=
a a
I r¢ D D
+ +
1 2 1 1
0. 4605 log log log ...(2.66)
a ab ca
= a + a
I D r¢ D
+
1 1 2 1
0. 4605 log log log ...(2.67)
b ab bc
2 1 1 1
= 0. 4605 a log + a log + log ...(2.68)
I D D r
c ca bc
Equations (2.66), (2.67) and (2.68) show that the phase inductances are not equal and due
to mutual inductance they contain imaginary terms.
2.11 TRANSPOSE TRANSMISSION LINE
As mentioned in the previous section, asymmetrical spacing gives complex values of phase
inductances, which makes the study of power system difficult. However, one way to regain
symmetry in good measure and obtain a per phase model by exchanging the positions of the
conductors at regular intervals along the line such that each conductor occupies the original
position of every other conductor. Such an exchange of conductor positions is called transposition.
The transposition is usually carried out at switching stations. A complete transposition cycle is
shown in ig. 2.9. This arrangement causes each conductor to have the same average inductance
ig. 2.9: Transposition cycle of three-phase line.
I
KJ
I
KJ
I
¢ KJ