Inductance in Terms of Magnetic Reluctance and Magnetic Permeance

As mentioned previously, the magnetic circuit is similar in some respects to the electric circuit. In the case of the simple d-c circuit current, voltage and resistance are related to each other, under steady-state conditions, in accordance with Ohm's Law. The same kind of relationship is sometimes applied to magnetic circuits as follows

[4-27]

[4-28]

where

F = magnetomotive force (mmf) in ampere turns (Ni)
Φ = the equivalent flux linking all N turns of the exciting winding
R = magnetic reluctance in ampere turns per weber

The relationship expressed by Eq. 4-27 can be stated in terms of the magnetic permeance as follows

[4-29]

where

[4-30]

In the case of magnetic circuits in which there is no leakage, all the flux serves to link each turn of the winding and Φ represents the total flux. The magnetic circuit shown in Fig. 3-9 practically satisfies that condition. However, in magnetic circuits where there is magnetic leakage and the flux follows paths such that different amounts of flux link different numbers of turns in the exciting winding, the value of Φ in Eqs. 4-27,4-28, and 4-29 is less than the total flux. Such situations exist in air-core solenoids as well as in many circuits having ferromagnetic cores. Figure 4-3 shows a coil in which the current produces complete and partial flux linkages.

The symbol P stands for permeance, which is the reciprocal of reluctance. Magnetic reluctance in these relationships corresponds to resistance in the electric circuit; permeance corresponds to conductance. Both these quantities, reluctance and permeance, are functions of the magnetic permeabilities of the different parts of the magnetic circuit as well as the configurations of these parts.

Substitution of Eq. 4-28 in Eq. 4-27 yields

[4-31]

but F = Ni and λ/i = L, hence

from which

[4-32]

For the special case in which all the flux is confined to a path of constant cross-sectional area A and the mean length of flux path is l the expression for the reluctance is reduced to

[4-33]

Magnetic pull and magnetic torques can also be expressed as functions of permeance and reluctance from Eq. 4-32

[4-34]

and the developed force can be written as

[4-35]

and the developed torque as

[4-36]

The force and torque can be expressed in terms of reluctance by replacing P with 1/R in Eq. 4-36, thus

[4-37]

and

[4-38]

In Eqs. 4-35 to 4-38 the expressions for force and torque are those developed by the electromagnetic structure, in other words, motor action is involved when there is motion of translation or rotation in the direction of the developed forces.

Example 4-1:

The rotary electromagnetic device shown in Fig. 4-3 has 1,560 turns on each stator pole. The magnetic structure is of cast steel. The dimensions are as follows R1 = 0.75 in. radius of rounded part of rotor W = 1.00 in. width of rotor- and stator-pole face at right angles to page g =0.05 in. length of single air gap Θ = angle between stator-pole tip and adjacent rotor-pole tip Figure 4-3. Complete and partial flux linkages with a coil carrying current (a) Neglect the effect of the iron and of flux fringing and determine the self inductance in terms of Θ assuming a linear relationship between the area of the air gap and Θ(1) (b) Determine the maximum torque for a current of 1.2 amp. (c) Express the permeance as a function of the angle Θ. (d) Express the reluctance as a function of the angle Θ. (e) Express the energy stored in the air gap as a function of the angle Θ for a current of 1.2 amp.

Solution:

1	Actually this is a rather rough approximation. It is evident that the circuit has some inductance even for rotor positions in which the rotor iron does not overlap the stator iron, for example, when the magnetic axes of the rotor and stator are 90° apart. The effect of fringing, which is neglected here, also changes with the angle θ.