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Magnetic permeability

Stan Zurek, Magnetic permeability, Encyclopedia Magnetica,

Magnetic permeability, μ (also written as mu) - a property of a given material or medium, quantifying its magnetic response of flux density B when subjected to magnetic field strength H.1)2) Magnetic permeability is proportional to the ratio of B and H changes: μ = ΔB/ΔH. Several different kinds of permeability can be defined, depending on the character of changes in B and H.

Permeability $μ$ of magnetic materials (ferromagnets $μ_{ferro}$) are much higher than those of non-magnetic materials (vacuum $μ_0 = μ_{vac}$, diamagnets $μ_{dia}$, paramagnets $μ_{para}$, superconductors $μ_{sup}$)

Permeability of free space (or vacuum) is a universal physical constant defined in the international system of units (SI) denoted as $μ_0$ and its value defined3)4) as $μ_0 = 4 · π · 10^{-7}$ henry per metre which is approximately $μ_0 \approx 1.2566 · 10^{-6}$ H/m.

The definition of relative magnetic permeability used widely in engineering is linked to magnetic susceptibility which is more useful in theoretical physics and chemistry, and the relationship is such that:5)

Permeability μr and susceptibility χvol
$$μ_r = χ_\text{vol} + 1$$ (unitless)
where: $χ_\text{vol}$ - volume magnetic susceptibility (unitless)

The name “permeability” was proposed in 1885 by Olivier Heaviside.

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Practical significance of permeability

Permeability of ferromagnetic materials is a non-linear function of the applied excitation and reduces when approaching saturation

From a practical viewpoint permeability is one of the basic parameters which allows categorising all materials as “magnetic” (permeability much greater than that of vacuum $μ >> μ_0$) and “non-magnetic” (permeability similar to that of vacuum $μ \approx μ_0$). In reality all materials are magnetic because they always exhibit some magnetic response to the magnetic excitation so that:

B and H relationship used in engineering
$$B = μ · H $$ (T)
where: B - magnetic flux density (T), μ - absolute magnetic permeability (H/m), H - magnetic field strength (A/m)

Ferromagnetic and ferrimagnetic materials have their permeabilities significantly greater than vacuum and are used for construction of magnetic cores in magnetic circuits.

However, at sufficiently high excitation all magnetic materials exhibit magnetic saturation, above which the permeability reduces to that of vacuum, so its relative value reduces to 1 (rather than zero).

High permeability is significant as a figure of merit only for soft magnetic materials, and permeability is inversely correlated with coercivity. Semi-hard and hard magnetic materials exhibit large coercivity and thus the permeability can be much lower, for example μr = 1.05 but for these materials the coercivity and energy storing capabilities are much more important parameters.

Superconductors have an apparent permeability of zero, because they expel magnetic field from their inside by the surface currents.

Overview of properties of soft magnetic materials: saturation, permeability, frequency range, material cost6)


Family of B-H loops of a typical grain-oriented electrical steel

Inductance is related to permeability of the medium. The following relationship holds for a simple geometry such as toroid, with closed magnetic circuit (no air gap) and with the wire wound tightly on the magnetic core:

Inductance L
$$L = \frac{μ·N^2·A}{l} = \frac{μ_r · μ_0·N^2·A}{l} $$ (H)
where: $μ$ - absolute permeability (H/m) of magnetic core, $μ_r$ - relative permeability (unitless), $μ_0 = 4·π·10^{-7}$ (H/m) - absolute permeability of vacuum, $N$ - number of turns of the winding (unitless), $A$ - cross-sectional area of the core (m2), $l$ - magnetic path length (m)

This equation also holds for non-magnetic materials, for example for an air-cored inductor (provided that the magnetised medium is uniform, i.e. completely surrounds the winding).

However, if the magnetic core has an air gap, or is composed of several materials with different permeabilities, then the demagnetising effect arises due to the shape anisotropy and the effective permeability of such gapped core can be much lower than the permeability of the material.

For a magnetic core made of high-permeability material, with a single small air gap, the magnetic path length can be assumed to be equal to that of the gap length, and the permeability of vacuum can be used in the above equation to estimate the value of the inductance (see also: Calculator of inductance with a gapped core).

The value of measured permeability depends on the slope of the B/H ratio. For a magnetically closed core, under small-signal excitation the location of the working point on a hysteresis loop will strongly depend on the magnetic history of the material. If there was previous saturation the material will remain in the remanence Br point, with much smaller reversible permeability as compared to lower offset values (see the illustration with the family of B-H curves, and the slopes indicating the permeability at local points for B=0). Therefore, inductance of a winding wound on a magnetically closed core can vary in a wide range.

Air gap reduces the effective permeability
Even a very small air gap can have a significant impact on effective permeability and thus on inductance

The effective permeability is reduced significantly if an air gap is introduced to the core. This linearises the B-H loop and thus also the slope at all the points below saturation becomes similar. Inductors with an air gap can hold much better tolerances, but they still have typically values varying by a much larger margin than it is the case for typical capacitors or resistors.

For example, inductance can be specified to have the margins as wide as ±50% for some inductors,7) whereas ±10% specification is typical for capacitors.

Calculator of effective magnetic permeability from air gap

This section is an interactive calculator.

For a magnetic circuit with uniform cross-section the value of effective permeability μeff can be calculated if the lengths and permeabilities of both parts are known.

$$ \mu_{eff} = \frac{\mu_{core}}{ {\frac{l_{gap}}{l_{core}} ⋅ \mu_{core} + 1 } } $$

Relative μcore = (unitless)       core length lcore =       gap length lgap =

       Effective permeability = (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 lcore » lgap, and if µcore » 1.

Types of permeability

See also the main article: Types of magnetic permeability.
Illustration of several types of permeability, calculated as a ratio (slope) of change of B to change of H: μa or μampl - amplitude, μmax - maximum, μi - initial, μΔ - incremental, μrev - reversible

In general, magnetic permeability is defined as the ratio of some change of B to some change of H, namely: μ = ΔB/ΔH.

However, the respective values ΔB and ΔH can be defined in various ways, at different points or locations in a B-H loop, and therefore there can be several “types” of permeability, as illustrated, and as listed at the end of this article.

For example, relative amplitude permeability μampl or simply μr is calculated from using the maximum values at a given level of excitation as Bpeak / Hpeak.

The maximum permeability μmax is simply the point at which the amplitude permeability is at its maximum, namely where the slope of the straight line is the steepest, as drawn between the origin of the axes and the Bpeak / Hpeak.

At higher excitation frequencies the B-H loop becomes rounded, and because the B vector lags behind H by some angle due to losses, their peaks no longer coincide. This makes the calculation of amplitude permeability ill-conditioned and less useful at very high frequencies (e.g. when the eddy current loss is much greater than the hysteresis loss).

Amplitude permeability is a ratio of Bpeak/Hpeak, but at high frequencies the B-H loop is rounded and the ratio becomes ill-defined; instead complex permeability can be used

Therefore, at high frequencies it is customary to make use of complex permeability, whose real part μ' (marked with “prime”) represents the lossless inductive component, and the imaginary part μ“ (marked with “double prime”) represents the lossy behaviour. This is because if B is completely in phase with H (zero phase shift) then the material is lossless and only the real component of the complex permeability is present.

Incremental permeability μΔ is measured with an excitation composed of a DC offset and small AC amplitude. Under such conditions a minor loop is traced in the main B-H loop, and the incremental changes in B and H are used for calculating the permeability.8)

Reversible permeability is measured in the same way as the incremental value, but with the amplitude of the AC excitation as small as practically possible (H → 0).9) The name reversible implies that the process is lossless, as it is the case also for the initial permeability.

Initial permeability is related to the slope of the curve when the material is first demagnetised (by demagnetisation or annealing),10) and then excited with a very small amplitude, as small as measurable, for example at Hpeak = 0.4 A/m.

With very small amplitude of AC excitation the domain walls do not change their positions, but remain pinned and undergo bowing (so that there is no Barkhausen jumps). Therefore, there is no hysteresis loss and only very small (even negligible) losses related to local eddy currents.

For sufficiently small excitation frequency, around power frequencies, the initial permeability becomes constant and independent of frequency.

Relative and absolute permeability

Absolute permeability has the unit of henry per metre (H/m), and relative permeability is unitless, referred to the permeability of vacuum.11)

The value of absolute permeability μ expresses the direct ratio of B (T) to H (A/m), and therefore the resulting SI unit is (H/m).

Absolute permeability μ
$$ μ = \frac{B}{H} $$ (T) / (A/m) ≡ (H/m)

The value of relative permeability μr expresses the ratio of absolute permeability μ of a given material to the absolute permeability of vacuum μ0. Both values have the same units, to the result is unitless.

Relative permeability μr
$$ μ_r = \frac{μ}{μ_0} = \frac{B}{μ_0·H} $$ (H/m) / (H/m) ≡ (unitless)
(T) / ( (H/m)·(A/m) ) ≡ (unitless)

For most materials the values of absolute permeability are rather low, and thus not very intuitive to use. For example, permeability of vacuum μ0 = 1.26 × 10-6 (H/m) and the absolute permeability μ of grain-oriented electrical steel can be 5.03 × 10-2 (H/m).

The relative values allow obtaining an easy-to-follow figure of merit. In a common language - the relative permeability expresses how much the given material is “better” than vacuum at “concentrating” the magnetic field. Following the example given above, the relative permeability of grain-oriented electrical steel can be expressed as μr = 40 000 which gives a much more intuitive information about its magnetic properties (in this case: 40 thousand times “better” than vacuum or other non-magnetic materials).

In the CGS system of units there is no absolute permeability, only relative.

Calculator of relative and absolute permeability

This section is an interactive calculator.

This calculator of permeability relies on the following equations (in SI units):

Relative Absolute
$$ μ_r = \frac{ΔB}{μ_0·ΔH} $$ (unitless) $$ μ = \frac{ΔB}{ΔH} $$ (H/m)

Flux density B =            Magnetic field strength H =


Relative magnetic permeability μr = (unitless)            Absolute magnetic permeability μ = (H/m)

See also


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magnetic_permeability.txt · Last modified: 2023/09/09 23:34 by stan_zurek

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