Concept of Magnetic Circuit

Magnetic Circuit

Magnetism that depends on a flow of electric current is called electromagnetism. Fig. 7-1 shows a typical magnetic pattern produced by a loosely wound coil or solenoid with air core.

Fig. 7-1. Solenoid With Air Core

While most of the magnetic flux links all the turns of the solenoid, some of it leaks through, as at a, and does not link the end turns. This is called flux leakage with respect to the end turns. It may be shown by calculation¹ that, if a solenoid whose length is many times its diameter be wound with N turns uniformly distributed over its length l, the field intensity at its center will be

and

where

I = current flowing, in amperes;
l = length of solenoid, in centimeters;
N = number of turns in solenoid;
H = field intensity, in gausses.

Fig. 7-2. Closed-Loop Solenoid

If the solenoid were bent so that its axis formed a closed loop, as in Fig. 7-2, and there were no flux leakage, the amperes required to produce a field intensity H with N turns would be

The flux leakage from such a solenoid or coil may be reduced to an amount negligible for most purposes by substituting for the air core a core of some other material such as iron in which a magnetic field may be established more easily than in air.

Thus, for a magnetic circuit all or most of which is in iron, equation (7-1) may be used except as modified in equation (7-5). See the solution of Example 7-1. It is much easier to magnetize iron than a vacuum, or air.

The ease with which a material may be made to carry magnetic lines of induction, compared to the ease with which they may be established in a vacuum, is called its permeability.

The following comparison may be made between the electric circuit and the magnetic circuit.

Electric Circuit	Magnetic Circuit
Electromotive Force, emf or E Produced by any one of several means. Unit is the volt.	Magnetomotive Force, mmf or M Produced only by ampere-turns, Unit is the gilbert, which equals 0.4πNI.
Conductivity, K Varies for different materials, from practically zero to high values. Has its highest values for a few pure metals, notably silver and copper. Is independent of current density, but varies with temperature, the change being different for different metals.	Permeability, μ Is unity for a vacuum, air, and many other materials. Varies from very slightly below 1 for a few diamagnetic materials to more than 1000 for iron and some alloys. Varies greatly with flux density B, but is not materially affected by moderate temperature. The amount of change with B varies greatly for different alloys.
Resistivity, Ρ Reciprocal of conductivity.	Reluctivity Reciprocal of permeability.

If equation (7-1) were used to find the flux density in the core of a solenoid like that in Fig. 7-1, which is provided with an iron core of uniform-cross-section and of permeability μ, then

Also,

or

where

B = flux density in the iron, in lines per square centimeter;
ϕ = total flux in the iron;
A = cross-sectional area of the iron, in square centimeters;
I = length of the iron, in centimeters.

The term

is called reluctance, and its symbol is Ρ (rho). Reluctance in a magnetic circuit corresponds to resistance in an electric circuit. Thus,

[7-6]

or

[7-7]

in which M = magnetomotive force.

Equation (7-6) or equation (7-7) is called Ohm's Law of the magnetic circuit because of its similarity to the equation

of the electric circuit.

The resistance of air to low electrical potentials is infinite, and so it is possible to "open" an electric circuit. The reluctivity of air, however, is 1; so it is not possible to have a magnetic path of infinite reluctance and hence, it may be said that a magnetic circuit cannot be opened.

Reference

1	Principles of Direct-Current Machines, by A, S. Langsdorf, McGraw-Hill Book Co.