Page 283 - Engineering Electromagnetics, 8th Edition

CHAPTER 8 Magnetic Forces, Materials, and Inductance 265

Figure 8.14 A portion of a coil showing partial flux
linkages. The total flux linkages are obtained by adding
the fluxes linking each turn.

flux at the mean radius times the total number of turns. In order to obtain the total
flux linkages we must look at the coil on a turn-by-turn basis.

(N ) total = 1 + 2 + ··· + i + ··· + N
N

= i
i=1
where i is the flux linking the ith turn. Rather than doing this, we usually rely on
experience and empirical quantities called winding factors and pitch factors to adjust
the basic formula to apply to the real physical world.
An equivalent definition for inductance may be made using an energy point
of view,

2W H
L = (52)
I 2
where I is the total current flowing in the closed path and W H is the energy in the
magnetic field produced by the current. After using (52) to obtain several other general
expressions for inductance, we will show that it is equivalent to (49). We first express
the potential energy W H in terms of the magnetic fields,

B · H dν
vol (53)
L =
I 2
and then replace B by ∇× A,
1
L = H · (∇× A)dν
I 2 vol