Power system fundamentals - Wind Power Integration: Connection and System Operational Aspects - page 45

found by integration. The inductance is then the flux linkage divided by the current,
as seen earlier.
Assume that the loop current is i . The magnetic field strength normal to the
elemental rectangle is obtained by applying (2.1) to both conductors:

i
2 p x þ

i
2 pð d x Þ

H ¼

The corresponding flux density is, from (2.2),

i
2 p x þ

i
2 p d x

B ¼ m 0

ð Þ

The flux enclosed by the rectangular element is the flux density multiplied by
the area normal to the field:

d x

m 0 li
2 p

1
x þ

1
d x

d F ¼

The total flux linked by the current is obtained by considering all rectangular
elements between the conductors:

ð

d r

dx

m 0 li
2 p

1
x þ

1
d x

F ¼

r

m 0 li
2 p

d r

¼ ½ ln ð x Þ ln ð d x Þ

r

m 0 li
p

d r
r

¼ ln

The loop inductance is then

m 0 l
p ln

d r
r

L l ¼ F= i ¼

It is more usual to work with conductor inductance, which will be half of the
loop inductance. Also, r is small in comparison with d , giving the following
approximate expression for the conductor inductance per unit length (Henry/m):

2 p ln d

m 0

L ¼ ð 2 : 20 Þ

r

This analysis ignores the flux linkage within the conductors, but any loss of
accuracy is of little importance here.
In the case of three-phase transmission, the expression for inductance per
phase is very similar. The only difference is that the distance between conductors d
is replaced by the geometric average of the distances between each pair of phase