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, also known simply as magnetic resistance, opposes the production of magnetic flux in a magnetic circuit. It is a characteristic of materials that resist the formation of magnetic flux.

In an electrical circuit, the resistance opposes the flow of current in the circuit and it dissipates the electric energy. The magnetic reluctance in a magnetic circuit is analogous to the resistance in an electric circuit as it opposes the production of magnetic flux in a magnetic circuit but it does not give rise to the dissipation of energy rather it stores magnetic energy.

Reluctance increases with the length of the magnetic circuit and decreases with a larger cross-sectional area of the magnetic path. It is a scalar quantity, denoted by ( S ), meaning it is defined only by its magnitude and not by direction.

Mathematically it can be expressed as

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

where, l = length of the magnetic path in meters

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

$\mu_r$ = relative permeability of a magnetic material

$A$ = Cross sectional area in square meters ( $m^2$ )

In both AC and DC magnetic fields, reluctance is defined as the ratio of magnetomotive force (m.m.f.) to magnetic flux. This value fluctuates in pulsating AC or DC fields.

Thus it can be expressed as

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

Reluctance in a Series Magnetic Circuit

Like in a series electrical circuit, the total resistance is equal to the sum of the individual resistances,

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

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

Similarly, in a series of magnetic circuits, the total reluctance equals the sum of the individual reluctances encountered around the closed flux path.

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

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

What is Permeability?

Permeability, or magnetic permeability, measures a material’s ability to support the formation of a magnetic field by allowing magnetic lines of force to pass through it.

The SI unit of permeability is Henry/meter (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 defined as the degree to which the material is a better conductor of magnetic flux as compared to free space.

It is denoted by $\mu_r$ .

What is Reluctivity?

Reluctivity or a specific reluctance is defined as the reluctance offered by a magnetic circuit of a unit length and a unit cross-section.

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

When l = 1 m and A = 1 m² then, we have

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

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

Its unit is meter/Henry.

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

Permeance vs Reluctance

Permeance is defined as the reciprocal of reluctance. It is denoted by P.

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

Reluctance Units

The unit of reluctance is ampere-turns per Weber (AT/Wb) or 1/Henry or H^-1.

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 $\frac {NI}{l} = H$

put this into equation (3) we get,

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

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

Magneto Motive Force (M.M.F)

M.M.F is defined as the force that tends to establish the flux through a magnetic circuit.

It is equal to the product of the current flowing through the coil and the number of turns of the coil.

Hence, $m.m.f = NI$

Its unit is ampere-turns (AT).

Thus, $AT = NI$

Magnetomotive force (m.m.f) is the effort required to move a unit magnetic pole (1 Wb) through a magnetic circuit, similar to electromotive force (e.m.f) in electrical circuits.

Applications of Reluctance

Some of the applications of reluctance include:

In the transformer, reluctance is mainly used to reduce the effect of magnetic saturation. The constant air gaps in a transformer increase the reluctance of the circuit and hence store more magnetic energy before saturation.
Reluctance motor is used for many constant speed applications such as electric clock timer, signaling devices, recording instruments, etc, which is works on the principle of variable reluctance.
One of the main characteristics of the magnetically hard materials is that it has a strong magnetic reluctance which is used to create permanent magnets. Example: Tungsten steel, cobalt steel, chromium steel, alnico, etc….
The speaker magnet is covered with a soft magnetic material such as soft iron to minimize the effect of the stray magnetic field.
Multimedia loudspeakers are magnetically shielded in order to reduce magnetic interference caused to TV (televisions) and CRTs (Cathode Ray Tube).

Permeance	Reluctance
Permeance is a measure of the ease with flux can be set up in the magnetic circuit.	Reluctance opposes the production of magnetic 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 or 1/Henry or H^-1.
It is analogous to conductance in an electric circuit.	It is analogous to resistance in an electric circuit.