MAGNETIC RELUCTANCE: UNITS, FORMULAS, CALCULATION, EXAMPLES - PHYSICAL - 2025

The magnetic reluctance or magnetic resistance is opposition means presents the passage of the magnetic flux: a greater reluctance more difficult to establish the magnetic flux. In a magnetic circuit, reluctance has the same role as electrical resistance in an electric circuit.

A coil carried by an electric current is an example of a very simple magnetic circuit. Thanks to the current, a magnetic flux is generated that depends on the geometric arrangement of the coil and also on the intensity of the current flowing through it.

Figure 1. Magnetic reluctance is a characteristic of magnetic circuits like the transformer. Source: Pixabay.

Formulas and Units

Denoting the magnetic flux as Φ _m, we have:

Where:

-N is the number of turns of the coil.

-The intensity of the current is i.

-ℓ _c represents the length of the circuit.

- A _c is the cross-sectional area.

-μ is the permeability of the medium.

The factor in the denominator that combines the geometry plus the influence of the medium is precisely the magnetic reluctance of the circuit, a scalar quantity which is denoted by the letter ℜ, to distinguish it from electrical resistance. So:

In the International System of Units (SI) ℜ is measured as the inverse of henry (multiplied by the number of turns N). In turn, the Henry is the unit for magnetic inductance, equivalent to 1 tesla (T) x square meter / ampere. Thus:

1 H ^-1 = 1 A / Tm ²

Since 1 Tm ² = 1 weber (Wb), the reluctance is also expressed in A / Wb (ampere / weber or more frequently ampere-turn / weber).

How is magnetic reluctance calculated?

Since magnetic reluctance has the same role as electrical resistance in a magnetic circuit, it is possible to extend the analogy by an equivalent of Ohm's law V = IR for these circuits.

Although it does not circulate properly, the magnetic flux Φ _m takes the place of the current, while instead of the voltage V, the magnetic voltage or magnetomotive force is defined, analogous to the electromotive force or emf in electrical circuits.

The magnetomotive force is responsible for maintaining the magnetic flux. It is abbreviated fmm and is denoted as ℱ. With it, we finally have an equation that relates the three quantities:

And comparing with the equation Φ _m = Ni / (ℓ _c / μA _c), it is concluded that:

In this way, the reluctance can be calculated knowing the geometry of the circuit and the permeability of the medium, or also knowing the magnetic flux and the magnetic tension, thanks to this last equation, called Hopkinson's law.

Difference with electrical resistance

The equation for magnetic reluctance ℜ = ℓ _c / μA _c is similar to R = L / σA for electrical resistance. In the latter, σ represents the conductivity of the material, L is the length of the wire and A is the area of ​​its cross section.

These three quantities: σ, L and A are constant. However, the permeability of the medium μ, in general, is not constant, so that the magnetic reluctance of a circuit is not constant either, unlike its electrical simile.

If there is a change in the medium, for example when going from air to iron or vice versa, there is a change in permeability, with the consequent variation in reluctance. And also magnetic materials go through hysteresis cycles.

This means that the application of an external field causes the material to retain some of the magnetism, even after the field is removed.

For this reason, every time the magnetic reluctance is calculated, it is necessary to carefully specify where the material is in the cycle and thus know its magnetization.

Examples

Although reluctance is highly dependent on the geometry of the circuit, it also depends on the permeability of the medium. The higher this value, the lower the reluctance; such is the case of ferromagnetic materials. Air, on the other hand, has low permeability, therefore its magnetic reluctance is higher.

Solenoids

A solenoid is a winding of length ℓ made with N turns, through which an electric current I is passed. The turns are generally wound in a circular fashion.

Inside it, an intense and uniform magnetic field is generated, while outside the field becomes approximately zero.

Figure 2. Magnetic field inside a solenoid. Source: Wikimedia Commons. Rajiv1840478.

If the winding is given a circular shape, it has a torus. Inside there may be air, but if an iron core is placed, the magnetic flux is much higher, thanks to the high permeability of this mineral.

Coil wound on a rectangular iron core

A magnetic circuit can be built by winding the coil on a rectangular iron core. In this way, when a current is passed through the wire, it is possible to establish an intense field flux confined within the iron core, as shown in figure 3.

The reluctance depends on the length of the circuit and the cross-sectional area indicated in the figure. The circuit shown is homogeneous, since the core is made of a single material and the cross section remains uniform.

Figure 3. A simple magnetic circuit consisting of a coil wound on an iron core in a rectangular shape. Source of the left figure: Wikimedia Commons. Frequently

Solved exercises

- Exercise 1

Find the magnetic reluctance of a rectilinear solenoid with 2000 turns, knowing that when a current of 5 A flows through it, a magnetic flux of 8 mWb is generated.

Solution

The equation ℱ = Ni is used to calculate the magnetic voltage, since the intensity of the current and the number of turns in the coil are available. It just multiplies:

Then use is made of ℱ = Φ _m. ℜ, taking care to express the magnetic flux in weber (the prefix "m" means "milli", so it is multiplied by 10 ^-3:

Now the reluctance is cleared and the values ​​are substituted:

- Exercise 2

Calculate the magnetic reluctance of the circuit shown in the figure with the dimensions shown, which are in centimeters. The permeability of the core is μ = 0.005655 T · m / A and the cross-sectional area is constant, 25 cm ².

Figure 4. Magnetic circuit of example 2. Source: F. Zapata.

Solution

We will apply the formula:

Permeability and cross-sectional area are available as data in the statement. It remains to find the length of the circuit, which is the perimeter of the red rectangle in the figure.

To do this, the length of a horizontal side is averaged, adding greater length and shorter length: (55 +25 cm) / 2 = 40 cm. Then proceed in the same way for the vertical side: (60 +30 cm) / 2 = 45 cm.

Finally the average lengths of the four sides are added:

Subtract substituting values ​​in the reluctance formula, not without first expressing the length and area of ​​the cross-section - given in the statement - in SI units:

References

1. Alemán, M. Ferromagnetic core. Recovered from: youtube.com.
2. Magnetic circuit and reluctance. Recovered from: mse.ndhu.edu.tw.
3. Spinadel, E. 1982. Electric and magnetic circuits. New Library.
4. Wikipedia. Magnetomotive force. Recovered from: es.wikipedia.org.
5. Wikipedia. Magnetic Reluctance. Recovered from: es.wikipedia.org.