# Magnetic Reluctance

A description of the magnetic reluctance, also discussing a way to calculate it

## Overview

**Key facts**The magnetic reluctance is defined as: where is the magnetomotive force, and the magnetic flux. For a magnetic circuit of length , cross-sectional area , and relative magnetic permeability , the magnetic reluctance can be calculated with: where is the magnetic permeability of free space. <br/>

**Constants**

The magnetic reluctance of a magnetic circuit can be regarded as the formal analog of the resistance in an electrical circuit. The magnetic reluctance can be expressed as:
where is the magnetomotive force (mmf), and is the magnetic flux.
In order to calculate the magnetic reluctance, consider a magnetic circuit of length and cross-sectional area , as diagramed in Figure 1.
We know that the magnetic field strength can be written as:
where is the current in the coil, and is the number of turns (for a more detailed discussion on the magnetic field strength see Field Strength ). Furthermore, can be related to the magnetic flux density with the equation:
where is the magnetic permeability of free space, and the relative magnetic permeability of the material.
As the magnetic flux is defined as:
equation (3) can also be written as:
from which the magnetic field strength becomes:
Considering that is uniform, equation (2) becomes:
Using the expression form of from (7) in (6), we get that:
which leads to:
As the magnetomotive force of a coil is given by:
equation (9) becomes:
or:

Taking into account the definition of the magnetic reluctance from (1), we get that can be calculated as:

Example:

[metric]

##### Example - Magnetic flux and flux density of a toroid

Problem

Consider a toroid with the mean length of , the cross section of , and the relative magnetic permeability of . What is the magnetic flux and the magnetic flux density if the coil has 10 turns and the current is 2 amperes ?

Workings

As the magnetic reluctance is given by:
and, in our case, (), and (), we get that:
from which we obtain:
The magnetic flux can be written as:
where , the magnetomotive force, is given by:
As, in our case, , , and also considering (3), we obtain the magnetic flux:
Taking into account that the cross-sectional area is (), the magnetic flux density becomes:
As a side note, if the toroid has an air gap of length , then its total magnetic reluctance, , would be the magnetic reluctance of the toroid plus the magnetic reluctance of the air gap:
Thus, in this case, the total magnetic flux, , would be given by:

Solution