Reluctance

The reluctance makes it possible to quantify a physical property: the aptitude of a Magnetic circuit to be opposed to its penetration by a Magnetic field.

This size at summer created by analogy with the concept of resistance.

The reverse of the reluctance is called perméance , but this term and this concept are used rather little.

Determination of the reluctance of a homogeneous magnetic circuit

General case

For a homogeneous magnetic circuit, i.e. made up of only one material and homogeneous section, there exists a relation making it possible to calculate its reluctance according to the material which constitutes it and of its dimensions:

$\backslash \; mathcal\; \{R\}\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; driven\}\; \backslash \; cdot\; \backslash \; frac\; L\; S\; \backslash ,$ in H ^-1

• $\backslash \; driven\; \backslash ,$ being the magnetic Permeability in kg·m.A^-2·s^-2,

• $l\; \backslash ,$ the length in Meter S,
• $s\; \backslash ,$ the section in m².

Equivalent reluctance of an air-gap

The reluctance of an air-gap low thickness is given by:

$\backslash \; mathcal\; \{R\}\; =\; \backslash \; frac\; \{E\}\; \{\backslash \; mu\_0\; \backslash \; cdot\; S\}\; \backslash ,$

With:

• $e\; \backslash ,$ thickness of the air-gap,
• $\backslash \; mu\_0\; \backslash ,$ permeability of the vacuum,
• $S\; \backslash ,$ section of the air-gap.

If the thickness of the air-gap is large, there are more possible to consider only the lines of magnetic field remain perpendicular to the air-gap. One must then take account of the blooming of the magnetic field i.e. to consider that the section S is larger than that of the metal parts on both sides of the air-gap.

Reluctance of a magnetic circuit of complex form

Principle of calculation

The laws of association of the reluctances make it possible to calculate that of a magnetic circuit of form complex or composed of materials to the different magnetic characteristics. One breaks up this circuit into homogeneous sections, i.e. of the same section and consisted of same material.

• Association in series: When two homogeneous sections having respectively for reluctance $\backslash \; mathcal\; \{R\}\; \_1$ and $\backslash \; mathcal\; \{R\}\; \_2$ follow one another, the reluctance of the unit is $\backslash \; mathcal\; \{R\}\; \_\; \{eq.\; series\}\; =\; \backslash \; mathcal\; \{R\}\; \_1\; +\; \backslash \; mathcal\; \{R\}\; \_2$

• Association in parallel: When two homogeneous sections having respectively for reluctance $\backslash \; mathcal\; \{R\}\; \_1$ and $\backslash \; mathcal\; \{R\}\; \_2$ are placed side by side, the reluctance of the unit is $\backslash \; mathcal\; \{R\}\; \_\; \{eq.\; //\}$ such as $\backslash \; frac\; \{1\}\; \{\backslash \; mathcal\; \{R\}\; \_\; \{eq.\; //\}\}\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; mathcal\; \{R\}\; \_1\}\; +\; \backslash \; frac\; \{1\}\; \{\backslash \; mathcal\; \{R\}\; \_2\}$, is still $\backslash \; mathcal\; \{R\}\; \_\; \{eq.\; //\}\; =\; \backslash \; frac\; \{\backslash \; mathcal\; \{R\}\; \_1.\; \backslash \; mathcal\; \{R\}\; \_2\}\; \{\backslash \; mathcal\; \{R\}\; \_1\; +\; \backslash \; mathcal\; \{R\}\; \_2\}$.

Using these laws one can calculate the reluctance of the magnetic circuit complexes in his entirety.

Example

Magnetic circuits of the same form that represented opposite are frequently used to produce transformers of Alimentation to cutting. Winding is carried out in the window and surrounds the core. To calculate his reluctance, one starts by considering that it consists of two magnetic circuits of form simple coupled one against the other, therefore in parallel . One can then write:

$\backslash \; mathcal\; \{R\}\; =\; \backslash \; frac\; \{\backslash \; mathcal\; \{R\}\; “.\; \backslash \; mathcal\; \{R\}”\}\; \{2\; \backslash \; mathcal\; \{R\}\; “\}\; =\; \backslash \; mathcal\; \{R\}”\; /2$.

The magnetic circuit of reluctance $\backslash \; mathcal\; \{R\}\; “\backslash ,$ itself is consisted of association in series of two homogeneous sections: the part out of ferromagnetic material and the air-gap. One thus has:

$\backslash \; mathcal\; \{R\}”\; =\; \backslash \; mathcal\; \{R\}\; \_\; \{iron\}\; +\; \backslash \; mathcal\; \{R\}\; \_e\; \backslash ,$

The reluctance $\backslash \; mathcal\; \{R\}\; \backslash ,$ required is thus equal to:

$\backslash \; mathcal\; \{R\}\; =\; \backslash \; frac\; \{1\}\; \{2S\}\; \backslash \; cdot\; (\backslash \; frac\; \{E\}\; \{\backslash \; mu\_0\}\; +\; \backslash \; frac\; \{l\_\; \{iron\}\}\; \{\backslash \; mu\_\; \{iron\}\})\; \backslash ,$

Frequently, the term $\backslash \; frac\; \{E\}\; \{\backslash \; mu\_0\}$ is very large in front of the term $\backslash \; frac\; \{l\_\; \{iron\}\}\; \{\backslash \; mu\_\; \{iron\}\}\; \backslash ,$. The reluctance of the circuit is then practically equal to the reluctance of the air-gap.

Representation of a permanent magnet

A magnet length $l$, section $S$, slope of right-hand side of retreat $\backslash \; mu$ and of remanent induction $B$ is represented by:

• a material of reluctance $R\; \backslash \; =\; L\; (\backslash \; driven\; \backslash \; cdot\; S)$
• a primary circuit $N\; \backslash \; cdot\; I\; \backslash \; =\; (L\; \backslash \; cdot\; B)/\backslash \; mu$

Relation between reluctance and Inductance

A formula connects the inductance of a Enroulement realized around a magnetic circuit, the reluctance of this magnetic circuit and the number of whorls of rolling up:

$L\; =\; \backslash \; frac\; \{N^2\}\; \{\backslash \; mathcal\; \{R\}\}\; \backslash ,$

• $L\; \backslash ,$ inductance of rolling up in H

• $N\; \backslash ,$ many whorls of rolling up, without unit

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