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# Magnetic reluctance

Magnetic reluctance can be thought of as having an analogous function to resistance in an electrical circuit except that it cannot consume energy. The term was coined in May 1888 by Oliver Heaviside. [1] The notion of “magnetic resistance” was first mentioned by James Joule [2] and the term "magnetomotive force” (MMF) was first named by Bosanquet.[3] The idea about an application to a magnetic flux law, similar to Ohm's law for a closed electric circuit, is attributed to H. Rowland.[4]

It is equal to the ratio of the MMF in a passive magnetic circuit and the magnetic flux in this circuit or to the ratio of their amplitude values for a sinusoidal MMF and magnetic flux. It is a scalar value.

## Definition

The definition can be expressed as:

$\mathfrak R = \frac{\mathfrak F}{\Phi}$

where

$\mathfrak R$ is the reluctance in ampere-turns per weber (a unit that is equivalent to turns per henry)
$\mathfrak F$ is the magnetomotive force (MMF) in ampere-turns
Φ is the magnetic flux in webers.

It is sometimes known as Hopkinson's law and is analogous to Ohm's Law with resistance replaced by reluctance, voltage by MMF and current by magnetic flux.

Magnetic flux always forms a closed loop, as described by Maxwell's equations, but the path of the loop depends on the reluctance of the surrounding materials. It is concentrated around the path of least reluctance. Air and vacuum have high reluctance, whilst easily magnetized materials such as soft iron have low reluctance. The concentration of flux in low-reluctance materials forms strong temporary poles and causes mechanical forces that tend to move the materials towards regions of higher flux so it is always an attractive force(pull).

The reluctance of a uniform magnetic circuit can be calculated as:

$\mathfrak R = \frac{l}{\mu_0 \mu_r A}$

where

l is the length of the circuit in metres
μ0 is the permeability of free space, equal to $4 \pi \times 10^{-7}$ henry per metre
μr is the relative magnetic permeability of the material (dimensionless)
A is the cross-sectional area of the circuit in square metres

The inverse of reluctance is called permeance.

$\mathfrak P = \frac{1}{\mathfrak R}$

Its SI derived unit is the henry (the same as the unit of inductance, although the two concepts are distinct).

## Applications

Air gaps can be created in the cores of certain transformers to reduce the effects of saturation. This increases the reluctance of the magnetic circuit, and enables it to store more energy before core saturation. This effect is also used in the flyback transformer.

Variation of reluctance is the principle behind the variable reluctance generator and the Alexanderson alternator.

Reluctance can also be applied to:

• Reluctance motors
• Variable reluctance (magnetic) pickups

## References

1. ^ Heaviside O., Electrical Papers. Vol.2. – L.; N.Y.: Macmillan, 1892, p. 166.
2. ^ Joule J., Scientific Papers, vol. 1. – 1884, p. 36.
3. ^ Bosanquet, Phil. Mag., vol. 15, 1883, p. 205.
4. ^ Rowland H., Phil. Mag. (4), vol. 46, 1873, p. 140.