Magnetic reluctance

Magnetic reluctance, or magnetic resistance, is a concept used in the analysis of magnetic circuits. It is analogous to resistance in an electrical circuit, but rather than dissipating electric energy it stores magnetic energy. In likeness to the way an electric field causes an electric current to follow the path of least resistance, a magnetic field causes magnetic flux to follow the path of least magnetic reluctance. It is a scalar, extensive quantity, akin to electrical resistance. The unit for magnetic reluctance is inverse henry, H⁻¹.

History

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 for a magnetic flux law, similar to Ohm's law for closed electric circuits, is attributed to H. Rowland.^[4]

Reluctance is usually represented by a cursive capital $\scriptstyle \mathcal R$.

Definition

In a DC field, the reluctance is the ratio of the "magnetomotive force” (MMF) in a magnetic circuit to the magnetic flux in this circuit. In a pulsating DC or AC field, the reluctance is the ratio of the amplitude of the "magnetomotive force” (MMF) in a magnetic circuit to the amplitude of the magnetic flux in this circuit (see phasors).

The definition can be expressed as follows:

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

where

$\scriptstyle \mathcal R$ ("R") is the reluctance in ampere-turns per weber (a unit that is equivalent to turns per henry). "Turns" refers to the winding number of an electrical conductor comprising an inductor.

$\scriptstyle \mathcal F$ ("F") is the magnetomotive force (MMF) in ampere-turns

Φ ("Phi") 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, while 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:

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

or

$\mathcal R = \frac{l}{\mu A}$

where

l is the length of the circuit in metres

$\scriptstyle \mu_0$ is the permeability of vacuum, equal to 4π × 10⁻⁷ henry per metre

$\scriptstyle \mu_r$ is the relative magnetic permeability of the material (dimensionless)

$\scriptstyle \mu$ is the permeability of the material ($\scriptstyle \mu \;=\; \mu_0 \mu_r$)

A is the cross-sectional area of the circuit in square metres

The inverse of reluctance is called permeance.

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

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

Applications

• Constant air gaps can be created in the core 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.
• Variable air gaps can be created in the cores by a movable keeper to create a flux switch that alters the amount of magnetic flux in a magnetic circuit without varying the constant magnetomotive force in that circuit.
• Variation of reluctance is the principle behind the reluctance motor (or the variable reluctance generator) and the Alexanderson alternator. Another way of saying this is that the reluctance forces strive for a maximally aligned magnetic circuit and a minimal air gap distance.
• Multimedia loudspeakers are typically shielded magnetically, in order to reduce magnetic interference caused to televisions and other CRTs. The speaker magnet is covered with a material such as soft iron to minimize the stray magnetic field.

Reluctance can also be applied to:

• Reluctance motors
• Variable reluctance (magnetic) pickups

See also

• Dielectric complex reluctance
• Magnetic capacitivity
• Magnetic capacitance
• Magnetic circuit
• Magnetic complex reluctance
• Reluctance motor

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

External links