Electric Power Generation, Transmission, and Distribution

The differential flux dfenclosed in a ring of thickness

dy, from pointD 1 to pointD 2 , for a 1-m length of

conductor is

df¼Bydy¼

m 0

2 p

I

y

dyðÞWb=m (13:20)

As the total currentIflows in the surface conductor,

then the differential flux linkage dlhas the same

magnitude as the differential flux df.

dl¼df¼

m 0

2 p

I

y

dyðÞWb=m (13:21)

The total external flux linkage enclosed by the ring is

obtained by integrating fromD 1 toD 2

l 1  2 ¼

ðD 2

D 1

dl¼

m 0

2 p

I

ðD 2

D 1

dy

y

¼

m 0

2 p

Iln

D 1

D 2



ðÞWb=m (13:22)

In general, the total external flux linkage from the surface of the conductor to any pointD, per unit
length, is

lext¼

ðD

r

dl¼

m 0

2 p

Iln

D

r



ðÞWb=m (13:23)

The summation of the internal and external flux linkage at any pointDpermits evaluation of the total
inductance of the conductorLtot, per unit length, as follows:

lintlþlext¼

m 0

2 p

I

1

4

þln

D

r



¼

m 0

2 p

I ln

D

e^1 =^4 r



ðÞWb=m (13:24)

Ltot¼

lintþlext

I

¼

m 0

2 p

ln

D

GMR



ðÞH=m (13:25)

where GMR (geometric mean radius)¼e^1 =^4 r¼0.7788r
GMR can be considered as the radius of a fictitious conductor assumed to have no internal flux but
with the same inductance as the actual conductor with radiusr.

13.4.4 Inductance of a Two-Wire Single-Phase Line

Now, consider a two-wire single-phase line with solid cylindrical conductors A and B with the same
radiusr, same lengthl, and separated by a distanceD, whereD>r, and conducting the same currentI,as
shown in Fig. 13.9. The current flows from the source to the load in conductor A and returns in
conductor B (IA¼IB).
The magnetic flux generated by one conductor links the other conductor. The total flux linking
conductor A, for instance, has two components: (a) the flux generated by conductor A and (b) the flux
generated by conductor B which links conductor A.
As shown in Fig. 13.10, the total flux linkage from conductors A and B at pointPis

lAP¼lAAPþlABP (13:26)

lBP¼lBBPþlBAP (13:27)

I

r

dy

y

D 1 D 2

x

FIGURE 13.8 External magnetic field.