Magnetic Reluctance: What is it? (Formula, Units & Applications)

Contents

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Key learnings:

Magnetic Reluctance Defined: Magnetic reluctance is the opposition to magnetic flux within a magnetic circuit, functioning similarly to resistance in an electrical circuit.
Reluctance Formula: The reluctance is calculated by dividing the length of the magnetic path by the product of the permeability of free space, the relative permeability of the material, and the cross-sectional area of the magnetic path. This calculation illustrates how these factors collectively influence the magnetic resistance of a circuit.
Measurement Units: The standard unit of reluctance is expressed as ampere-turns per Weber (AT/Wb) or inverse Henry (H^-1).
Practical Applications: Applications of reluctance include its use in transformers to manage magnetic saturation, in reluctance motors, and in designing magnets for speakers.
Permeability and Reluctivity: Understanding permeability and reluctivity helps in analyzing how materials affect magnetic flux, crucial for designing efficient magnetic circuits.

What is Reluctance?

Magnetic reluctance is the ratio of magnetomotive force to magnetic flux in a lumped magnetic circuit. It depends on the path geometry and permeability, so it describes the complete magnetic path or one of its sections rather than a material alone.

In an electrical circuit, resistance relates voltage to current and dissipates power. Reluctance plays an analogous role by relating magnetomotive force to flux. Reluctance itself does not dissipate or store energy; the magnetic field stores magnetic energy, while hysteresis and eddy-current effects cause core loss.

For a uniform linear path, reluctance increases with path length and decreases with permeability and cross-sectional area. The page denotes it by S, though the script R symbol is also common. The lumped value is scalar for a one-dimensional path; anisotropic and nonlinear materials require more detailed field or incremental models.

For a uniform section with constant permeability and area, it is:

$\begin{align*} S = \frac {l}{\mu_0 \mu_r A} \end{align*}$

where, l = mean length of the magnetic path in metres

$\mu_0$ = permeability of free space (vacuum), approximately $4 \pi * 10^-^7$ henries per metre

$\mu_r$ = relative permeability of a magnetic material

$A$ = cross-sectional area in square metres ( $m^2$ )

For AC or DC excitation, reluctance is magnetomotive force divided by flux under the stated operating conditions. In ferromagnetic materials, permeability can change with flux density, frequency, temperature and magnetic history, so effective or incremental reluctance may also change.

This circuit-law definition is:

$\begin{align*} Relectance (S) = \frac {m.m.f}{flux} = \frac {F}{\phi} \end{align*}$

Reluctance in a Series Magnetic Circuit

In a series electrical circuit, individual resistances add:

$\begin{align*} R = R_1 + R_2 + R_3 +.............+R_n \end{align*}$

Where, $R = \frac {\rho l}{A} (\rho = Resistivity)$

Reluctances add when the same flux passes through consecutive magnetic sections and leakage and fringing are negligible.

$\begin{align*} S = S_1 + S_2 + S_3 +.............+S_n \end{align*}$

Where, $S = \frac {l}{\mu_0 \mu_r A}$

What is Permeability?

Magnetic permeability relates magnetic flux density to magnetic field strength. It may be a constant for a linear isotropic material, but ferromagnetic permeability depends on operating point and measurement conditions.

The SI unit of permeability is the henry per metre (H/m).

Mathematically, $\mu = \mu_0 \mu_r$ H/m

Where, $\mu_0$ = permeability of free space (vacuum) = $4 \pi * 10^-^7$ Henry/meter

$\mu_r$ = relative permeability of a magnetic material

It is the ratio of magnetic flux density (B) to magnetizing force (H).

$\begin{align*} \mu = \frac {B}{H} \end{align*}$

Relative Permeability

Relative permeability is the dimensionless ratio of a material’s permeability to the permeability of free space. Values can be greater than, close to or less than one, depending on the material and definition used.

It is denoted by $\mu_r$ .

What is Reluctivity?

Magnetic reluctivity, sometimes called specific reluctance, equals the reciprocal of absolute permeability and supplies the material factor in reluctance calculations.

We know the reluctance $S = \frac {l} {\mu_0 \mu_r A}$

When l = 1 m and A = 1 m², the numerical reluctance of the uniform sample equals its reluctivity:

$\begin{align*} S= \frac {1} {\mu_0 \mu_r (1)} = \frac {1} {\mu_0 \mu_r} =\frac {1} {\mu} \ ( \mu = \mu_0 \mu_r ) \end{align*}$

$\begin{align*} S (Specific \,\, Reluctance) = \frac {1} {Absolute \,\, Permeability (\mu)} \end{align*}$

Its SI unit is metre per henry (m/H).

It is analogous to resistivity (specific resistance) in an electric circuit.

Permeance vs Reluctance

Permeance is the reciprocal of reluctance. This page denotes it by P.

$Permeance (P) = \frac {1} {Reluctance(S)}$

Reluctance Units

The SI unit of reluctance is ampere-turns per weber (AT/Wb), equivalent to inverse henry or H^-1. A turn is dimensionless in SI, so some references write A/Wb.

Dimension of Magnetic Reluctance

$\begin{align*} S = \frac {l}{\mu A} \end{align*}$

$\begin{align*} \begin{split} \ S = \frac {M^0 L^1 T^0} {M^1 L^1 T^-^2 I^-^2 * M^0 L^2 T^0} \ \ = \frac {M^0 L^1 T^0} {M^1 L^3 T^-^2 I^-^2} \ \ = M^-^1 L^-^2 T^2 I^2 \ \end{split} \end{align*}$

Reluctance Formula

(1) $\begin{equation*} S = \frac {l}{\mu_0 \mu_r A} \end{equation*}$

Where, $\mu = \mu_0 \mu_r$ (In an electrical circuit $\epsilon = \epsilon_0 \epsilon_r$ )

Therefore, $S = \frac {l}{\mu A}$

Where, $\mu$ = permeability of the magnetic material

$\begin{align*} Reluctance (S) = \frac {m.m.f}{flux} \end{align*}$

(2) $\begin{equation*} S = \frac {NI}{\phi} \end{equation*}$

Comparing Equation (1) and (2), we get

$\begin{align*} \frac {l}{\mu_0 \mu_r A} = \frac {NI}{\phi} \end{align*}$

Rearranging terms, we get

(3) $\begin{equation*} \frac {\phi}{\mu_0 \mu_r A} = \frac {NI}{l} \end{equation*}$

But $\frac {\phi}{A} = B$ and $\frac {NI}{l} = H$

Substitution should give B divided by the complete permeability, not by free-space permeability alone. The first protected line below omits relative permeability; the second line gives the intended constitutive relation.

$\begin{align*} \frac {B}{\mu_0} = H \end{align*}$

$\begin{align*} B = \mu_0 \mu_r H = \mu H \ (where, \mu = \mu_0 \mu_r) \end{align*}$

Magnetomotive Force (M.M.F.)

Magnetomotive force is the magnetic-circuit excitation associated with the line integral of magnetic field strength around a closed path. Despite its name, it is not a mechanical force.

For an ideal N-turn winding carrying current I, magnetomotive force is the product NI.

Hence, $m.m.f = NI$

Its unit is the ampere-turn (AT).

Thus, $AT = NI$

Magnetomotive force is analogous to electromotive force in a lumped circuit model, but magnetic flux is not a flow of isolated magnetic poles. One weber is a unit of magnetic flux, not a unit magnetic pole.

Applications of Reluctance

Reluctance is used in these magnetic design and machine calculations:

In an energy-storage inductor or gapped flyback transformer, a designed gap raises reluctance, lowers effective permeability and helps manage magnetic saturation. Most air gaps are used to control inductance and allow more magnetic energy to be stored in the field; ordinary power transformers usually minimise unintended gaps.
Reluctance motors produce torque because a salient rotor tends to align its low-reluctance axis with the stator field. Applications include controlled industrial drives and positioning systems, not only clock timer mechanisms or constant-speed devices.
Designers select magnetically hard materials for high coercivity and useful remanence, which resist demagnetisation and support permanent-magnet operation. High reluctance alone does not define a permanent magnet.
A loudspeaker’s soft-magnetic pole pieces and return path guide flux through the voice-coil gap and can reduce stray field.
Magnetic shielding was especially useful near cathode-ray-tube televisions and monitors, where stray fields could distort the electron beam. Modern flat-panel displays do not have that CRT beam-deflection problem.

Permeance	Reluctance
Permeance relates flux to magnetomotive force in a magnetic circuit.	Reluctance relates magnetomotive force to flux in a magnetic circuit.
It is denoted by P.	It is denoted by S.
$Permeance = \frac{flux}{m.m.f}$	$Reluctance = \frac{m.m.f}{flux}$
Its unit is Wb/AT or henry.	Its unit is AT/Wb, inverse henry or H^-1.
It is analogous to conductance in an electric circuit.	It is analogous to resistance in an electric circuit.