Electric Power Generation, Transmission, and Distribution

Ampere’s law determines the magnetic field intensityHx, constant at any point along the circle
contour as

Hx¼

Ix

2 px

¼

I

2 pr^2

xðÞA=m (13:12)

The magnetic flux densityBxis obtained by

Bx¼mHx¼

m 0

2 p

Ix

r^2



ðÞT (13:13)

wherem¼m 0 ¼ 4 p 10 ^7 H=m for a nonmagnetic material.
The differential flux dfenclosed in a ring of thickness dxfor a 1-m length of conductor and the
differential flux linkage dlin the respective area are

df¼Bxdx¼

m 0

2 p

Ix

r^2



dxðÞWb=m (13:14)

dl¼

px^2

pr^2

df¼

m 0

2 p

Ix^3

r^4



dxðÞWb=m (13:15)

The internal flux linkage is obtained by integrating the differential flux linkage fromx¼0tox¼r

lint¼

ðr

0

dl¼

m 0

8 p

IðÞWb=m (13:16)

Therefore, the conductor inductance due to internal flux linkage, per unit length, becomes

Lint¼

lint

I

¼

m 0

8 p

ðÞH=m (13:17)

13.4.3 External Inductance

The external inductance is evaluated assuming that the total currentIis concentrated at the conductor
surface (maximum skin effect). At any point on an external magnetic field circle of radiusy(Fig. 13.8),
the magnetic field intensityHyand the magnetic field densityBy, per unit length, are

Hy¼

I

2 py

ðÞA=m (13:18)

By¼mHy¼

m 0

2 p

I

y

ðÞT (13:19)

df

x

I

r

Hx

Ix

dx

FIGURE 13.7 Internal magnetic flux.