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 was created by analogy with the concept of resistance.
The reverse of the reluctance is called perméance magnetic, but this term and this concept are used rather little.
Analogy of Hopkinson
Principles
This analogy consists in making a parallel between the electrical circuits and the magnetic circuits.
Electrical circuits Magnetic circuits
Intensity of electric current I Flow of the magnetic field in the circuit φ
Resistance R Reluctance ℜ
Conductivity σ Magnetic permeability µ
Electromotive force E Magnetomotive force F ou ∑ nI
Law of ohm E = R * I Law of Hopkinson F = ℜ * σ
where µ = µ0µr where µris the relative permeability.
Note : saturation
This analogy can make it possible to envisage the behavior of a circuit, either in statics, or into alternate, in the quite precise case of the linearity constant permeability, or total linearity between field and magnetic induction. Whenever these conditions are not completely met, it can be possible to extend the concept of reluctance.
Indeed, the value of permeability is likely to vary according to various parameters of the problem considered, even in a simple case.
The variation of the value of permeability can find its origin in saturation for the strong magnetic fields, which corresponds to the saturation of the curves of hysteresis B-H.
In statics, this effect of saturation can be taken into account by introduction of zones having a lower permeability to the places where saturation occurs. However, in alternating signal, saturation takes place only for certain fractions of the cycle. The concept of reluctance then becomes difficult to handle.
In addition, the permeability is likely to vary with the high frequencies. The variation of permeability according to the frequency depends on materials and their geometries lamination on conducting magnetic materials by insulators in order to minimize the eddy currents.
Determination of the reluctance of a homogeneous magnetic circuit
General case
For a homogeneous magnetic circuit, thay is to say 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:
ℜ = 1 ⁄ µ * l ⁄ s en H-1
µ being the magnetic permeability in kg·m.A-2·s-2
l the length in meters
s the section in m2
Equivalent reluctance of an air-gap
The reluctance of an air-gap low thickness is given by
ℜ = e ⁄ µ0 * S
with
e : thickness of the air-gap
µ0 : permeability of the vacuum
S : 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 thay is to say 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, thay is to say of the same section and consisted of same material.
Association in series : When two homogeneous sections having respectively for reluctance ℜ1 and ℜ2 follow one another, the reluctance of the unit is ℜeq.serie = ℜ1 + ℜ2
Association in parallel : When two homogeneous sections having respectively for reluctance ℜ1 and ℜ2 are placed side by side, the reluctance of the unit is ℜeq.⁄⁄such as 1 ⁄ ℜeq.⁄⁄ = 1 ⁄ ℜ1 + 1 ⁄ ℜ2, that is to say still ℜeq.⁄⁄ = ℜ1 * ℜ2 ⁄ ℜ1 + ℜ2
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 mains transformers 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:
ℜ = ℜ’ * ℜ’ ⁄ 2ℜ’ = ℜ’ ⁄ 2
The magnetic circuit of reluctance ℜ’ itself is consisted of association in series of two homogeneous sections: the part out of ferromagnetic material and the air-gap. One thus has : ℜ’ = ℜfer + ℜe
The required reluctance ℜ is thus equal to : ℜ = 1 ⁄ 2S * (e ⁄ µ0 + lfer ⁄ µfer
Frequently, the term e ⁄ µ0 is very large in front of the term lfer ⁄ µfer. The reluctance of the circuit is then practically equal to the reluctance of the air-gap.
Representation of a permanent magnet
In any rigor, a permanent magnet is a strongly nonlinear element, prohibiting the use of the model of reluctance. It is however possible to extend it by introducing a primary circuit, subject to the knowledge of the curve of hysteresis of the magnet in the direction considered, and of the appreciation of the excursion of field considered invalid if the excursion approaches the coercitive force.
A magnet length L, section S, slope of right-hand side of retreat µ and remanent induction B is represented by:
a material of reluctance ℜ = : ⁄ µ * S
a primary circuit N * i = l * B ⁄ µ
Dielectric reluctance
It is the reverse of the logical size of study of the electric phenomena and which is named permeance dielectric space
Equation of dimensions of the dielectric réluctace: L-4 * M-1 * T4 * I2
Symbol of size: R
Unity S.I. + : F ⁄ m²
R = C ⁄ S
R = σ’ x t
R = b’ * ω ⁄ S
R = g* ⁄ I
R (F ⁄ m²-sr) = dielectric reluctance
C (F) = capacity
b’(F ⁄ sr) = permittance
σ’(S ⁄ m) = electric conductivity
ω (sr) = solid angle
g*(F-m²) = polarizability
Î(m4) = moment of inertia
Specific dielectric reluctance
It is the dielectric reluctance brought back to the solid angle
Equation of dimensions : L-4 * M-1 * T4 * I2 * A-1
Symbol of size : r’
Unity S.I. + : F ⁄ m²-sr
Relation between specific reluctance and perméance dielectric
r’ = 1 ⁄ e’
r’ = R ⁄ ω
r’(F ⁄ m²-sr) = Specific dielectric reluctance
e’(m²-sr ⁄ F) = permeance dielectric corresponding
Magnetic reluctance
Equation of dimensions of the magnetic reluctance : L-2 * M-1 * T2 * I2
Symbol : W*
Unity S.I. + : H-1
W* = ω * H ⁄ T
ω(sr) = solid angle in which is exerted the magnetic effect
T(Wb ⁄ m) = magnetic potential of induction with a field of excitation H (mOe)
Specific magnetic reluctance
It is the magnetic reluctance brought back to the solid angle
This size is the reverse of the logical size of study of these phenomena and which is named permeance magnetic (Λ)
Equation of dimensions : L-2 * M-1 * T2 * I2 * A-1
Symbol size : w*
Unity S.I + = H-1 sr-1
w* = H ⁄ T
w* = i.dl ⁄ µ * S
w* = I’ ⁄ Φ (formulate Hopkinson)
w* (H-1 sr-1) = specific magnetic reluctance of a tube of induction of permanent polarization negligible
T (Wb ⁄ m) = magnetic potential of induction creating a field of excitation H (mOe)
µ (H-sr ⁄ m) = magnetic permeability
dl(m) = element of circuit, S (m²) the section and I (A) intensity
I (A ⁄ sr) = magnetomotive force or potential difference of excitation magnetic
Φ (Wb) = magnetic flow of induction
specific magnetic reluctance for core of a reel
w* = n * i ⁄ Φ * ω
w*(H-1 * sr-1)= specific magnetic reluctance of a core of a reel of N whorls traversed by a current i(A)
ΦWb = magnetic flow of induction through the core
ω(sr)= solid angle in which is exerted the magnetic effect
reluctance for a transformer, the formula above becomes
w* = ]n1 * i1 + n2 * i2_ ⁄ Φ * ω
n1 et n2 = many whorls of 2 rollings up where the intensities are i1, i2(A)
The part of flow of the 1° winds crossing second is named useful flow and the part of flow lost names parasitic flow
Relation between reluctance and inductance
A formula connects the inductance of a rolling up carried out around a magnetic circuit, the reluctance of this magnetic circuit and the number of whorls of rolling up :
L = N² ⁄ ℜ
L : inductance of rolling up in H
N : many whorls of rolling up, without unit

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