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Magnetic reluctance

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Resistance to magnetic flux

Magnetic reluctance
Common symbols	${\mathcal {R}}$ , ${\mathcal {S}}$
SI unit	H⁻¹
Derivations from other quantities	${\frac {1}{\mathcal {P}}}$ , ${\frac {\mathcal {F}}{\Phi }}$ , ${\frac {l}{\mu _{0}\mu _{r}A}}$
Dimension	M^–1L^–2T²I²

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Magnetic reluctance, ormagnetic resistance, is a concept used in the analysis ofmagnetic circuits. It is defined as the ratio ofmagnetomotive force (mmf) tomagnetic flux. It represents the opposition to magnetic flux, and depends on the geometry and composition of an object.

Magnetic reluctance in a magnetic circuit is analogous toelectrical resistance in anelectrical circuit in that resistance is a measure of the opposition to theelectric current. The definition of magnetic reluctance is analogous toOhm's law in this respect. However, magnetic flux passing through a reluctance does not give rise to dissipation of heat as it does for current through a resistance. Thus, the analogy cannot be used for modelling energy flow in systems where energy crosses between the magnetic and electrical domains. An alternative analogy to the reluctance model which does correctly represent energy flows is thegyrator–capacitor model.

Magnetic reluctance is ascalar extensive quantity. The unit for magnetic reluctance is inversehenry, H⁻¹.

History

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The termreluctance was coined in May 1888 byOliver Heaviside.^[1] The notion of "magnetic resistance" was first mentioned byJames Joule in 1840.^[2] The idea for amagnetic flux law, similar toOhm's law for closedelectric circuits, is attributed toHenry Augustus Rowland in an 1873 paper.^[3] Rowland is also responsible for coining the termmagnetomotive force in 1880,^[4] also coined, apparently independently, a bit later in 1883 by Bosanquet.^[5]

Reluctance is usually represented by acursive capital ${\mathcal {R}}$ .

Definitions

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In both AC and DC fields, the reluctance is the ratio of themagnetomotive force (MMF) in amagnetic circuit to themagnetic flux in this circuit. In a pulsating DC or AC field, the reluctance also pulsates (seephasors).

The definition can be expressed as follows: ${\mathcal {R}}={\frac {\mathcal {F}}{\Phi }}$ where

${\mathcal {R}}$ ("R") is the reluctance inampere-turns perweber (a unit that is equivalent to turns perhenry). "Turns" refers to thewinding number of an electrical conductor comprising an inductor.
${\mathcal {F}}$ ("F") is themagnetomotive force (MMF) in ampere-turns
Φ ("Phi") is themagnetic flux in webers.

It is sometimes known asHopkinson's law and is analogous toOhm's law with resistance replaced by reluctance, voltage by MMF and current by magnetic flux.

Permeance is the inverse of reluctance: ${\mathcal {P}}={\frac {1}{\mathcal {R}}}$

ItsSI derived unit is thehenry (the same as the unit ofinductance, although the two concepts are distinct).

Magnetic flux always forms a closed loop, as described byMaxwell'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 assoft 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: ${\mathcal {R}}={\frac {l}{\mu _{0}\mu _{r}A}}={\frac {l}{\mu A}}$

where

l is the length of the circuit inmetres
$\mu _{0}$ is the permeability of vacuum, equal to ${\textstyle 4\pi \times 10^{-7}\mathrm {\frac {H}{m}} }$ (or, ${\textstyle \mathrm {\frac {kg\cdot m}{A^{2}\cdot s^{2}}} }$ = ${\textstyle \mathrm {\frac {s\cdot V}{A\cdot m}} }$ = ${\textstyle \mathrm {\frac {J}{A^{2}\cdot m}} }$ )
$\mu _{r}$ is the relativemagnetic permeability of the material (dimensionless)
$\mu$ is the permeability of the material ( $\mu =\mu _{0}\mu _{r}$ )
A is the cross-sectional area of the circuit insquare metres

Applications

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Constant air gaps can be created in the core of certain transformers to reduce the effects ofsaturation. This increases the reluctance of the magnetic circuit, and enables it to store moreenergy before core saturation. This effect is also used in theflyback 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 constantmagnetomotive force in that circuit.
Variation of reluctance is the principle behind thereluctance motor (or the variable reluctance generator) and theAlexanderson alternator. Another way of saying this is that thereluctance forces strive for a maximally aligned magnetic circuit and a minimal air gap distance.
Loudspeakers used in conjunction withcomputer monitors or other screens are typically shielded magnetically, in order to reduce magnetic interference caused to the screens such as intelevisions orCRTs.^[6] The speaker magnet is covered with a material such assoft iron to minimize the stray magnetic field.

Reluctance can also be applied to:

References

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^Heaviside O. (1892)Electrical Papers, Vol 2 – L.; N.Y.: Macmillan, p. 166
^Joule J. (1884)Scientific Papers, vol 1, p.36
^Rowland, Henry A. (1873). "XIV. On magnetic permeability, and the maximum of magnetism of iron, steel, and nickel".Philosophical Magazine. Series 4.46 (304):140–159.doi:10.1080/14786447308640912.
^Rowland, Henry A,"On the general equations of electro-magnetic action, with application to a new theory of magnetic attractions, and to the theory of the magnetic rotation of the plane of polarization of light" (part 2),American Journal of Mathematics, vol. 3, nos. 1–2, pp. 89–113, March 1880.
^Bosanquet, R.H.M. (1883)."XXVIII.On magnetomotive force".Philosophical Magazine. Series 5.15 (93):205–217.doi:10.1080/14786448308628457.
^Crawford, Walt (1995-04-01)."Multimedia madness: Notes along the way".Library Hi Tech.13 (4):101–119.doi:10.1108/eb047972.ISSN 0737-8831.

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