### Table of Contents

# Effective magnetic permeability

Stan Zurek, Effective magnetic permeability, Encyclopedia Magnetica, http://e-magnetica.pl/doku.php/effective_magnetic_permeability |

**Effective magnetic permeability** (also **apparent magnetic permeability**^{1)}), often denoted as ** μ_{e}**,

**or**

*μ*_{eff}**- a term used in analysis of magnetic performance of gapped cores. For a non-homogeneous core (e.g. gapped or composed of powder-like particles) this would be the value of magnetic permeability of a hypothetical homogeneous material which would exhibit the same permeability.**

*μ*_{a}^{2)}

^{S. Zurek, E-Magnetica.pl, CC-BY-4.0}

## Effect of air gap

^{3)}

^{S. Zurek, E-Magnetica.pl, CC-BY-4.0}

Magnetic permeability of a magnetic material is linked to the slope of a B-H curve or (or B-H loop). With increasing air gap the slope is reduced, and changes caused by non-linearity of the material (due to variations in flux density, temperature, bias, time, etc.) are reduced.^{4)}

With the gap present, higher magnetomotive force (excitation) is required to reach the same flux density. Similar behaviour could be obtained if the magnetic circuit was made not from a gapped core but from a non-gapped core made from material with proportionally lower permeability. A value of permeability required to obtain equivalent *B-H* performance is therefore the value of effective permeability.

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## Equations and calculations

^{S. Zurek, E-Magnetica.pl, CC-BY-4.0}

It is possible to analytically calculate the value of relative effective permeability for simple magnetic circuits, with a uniform gap.^{5)}

There are several assumptions:

- the cross section area of the magnetic circuit is constant at every point of the circuit, and is the same for the core and for the gap
- the length of the air gap is much shorter than the path length of the magnetic core
- the magnetisation is uniform and fringing effect is negligible
- permeability of the core material is much greater than the permeability of air gap
- relative permeability of the medium in the air gap is assumed to be unity (i.e. the same as vacuum).

The equation is derived by using the concept of magnetic reluctance^{6)} and with the assumptions listed above. All values of permeability (input and output) are given as relative permeability (so the value of “1” means permeability of the air gap itself). The length of the core and the gap must be given in the same units. For instance, if the core length is given in millimetres, then also the air gap length must be given in millimetres. But the equation holds for any other length units: inches/inches, metres/metres, etc.

Equations can also be derived for multi-path or non-uniform magnetic circuits, but these are obviously configuration-dependent and must be calculated for each specific structure.^{7)} For example, in a three-leg core (as shown in the image) the path length is considered to be only around one window (e.g. just the orange path).

### Calculator of effective magnetic permeability from air gap

This section is an interactive calculator. |
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For a magnetic circuit with uniform cross-section the value of effective permeability *μ _{eff}* can be calculated if the lengths and permeabilities of both parts are known.

$$ \mu_{eff} = \frac{\mu_{core}}{ {\frac{l_{gap}}{l_{core}} ⋅ \mu_{core} + 1 } } $$ | (unitless) |

*Note: The equation is valid only for a simple magnetic circuit, with cross-sectional area of the core and gap being equal, for relative permeability if l _{core} » l_{gap}, and if µ_{core} » 1.*

## Composite materials

^{S. Zurek, E-Magnetica.pl, CC-BY-4.0}

The value of effective permeability is important for composite materials, which may contain significant volumetric percentage of non-magnetic material. The small particles (as in powder cores) have rather high permeability, but the bulk of the core made out of such material exhibits effective permeability whose value is tailored for specific applications.^{8)}

For instance, Ferrotron 119 used for flux concentrators in induction heating has a maximum relative permeability of 8.0 (despite being made from ferromagnetic particles), because it is designed to work at high frequency (up to 5 MHz) and high excitation (20 kA/m) without saturation.

However, because such a magnetic core does not have a concentrated air gap then the simple equation given above cannot be used. Depending on the complexity of given material the calculations can become very difficult to solve or even formulate.^{9)}

Hence, the end users of composite cores can rely on the effective permeability values given by the manufacturers of the materials or magnetic cores. If the product is a magnetic core, then the AL value (inductance per turn) is often more useful than the value effective permeability as such.^{10)}

Nevertheless, in order to easier distinguish the type of material from which a given core is made the name of the material often refers to the value of effective permeability, for instance Ferroxcube uses notation Sendust 75, where 75 is the value of effective relative permeability at room temperature, so that^{11)} *μ _{e}* = 75.