# Encyclopedia Magnetica™

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# Relative magnetic permeability

 Stan Zurek, Relative magnetic permeability, Encyclopedia Magnetica, http://www.e-magnetica.pl/doku.php/relative_magnetic_permeability

Relative magnetic permeability typically denoted by μr - magnetic permeability expressed as the ratio of the absolute magnetic permeability to the value of magnetic permeability of vacuum μ0.1)2) Relative magnetic permeability gives an intuitive figure of merit expressing the magnetic performance of a given material. For example, relative permeability μr = 100 means that the material has permeability 100 times greater than that of vacuum. Also, for vacuum μr = 1, by definition.

Relative permeabilities of magnetic materials (e.g. ferromagnets) are much higher than non-magnetic materials (e.g. vacuum, diamagnets and paramagnets)

Magnetic permeability of vacuum is a universal physical constant3)4) $μ_0 = 4 · π · 10^{-7}$ (H/m) which is approximately $μ_0 \approx 1.2566 · 10^{-6}$ H/m.

By definition, the relative value is the ratio of the absolute value to the value in vacuum:5)

Relative permeability
$$\mu_r = \frac{\mu_{material}}{\mu_0} = \frac{\mu}{\mu_0}$$ (unitless)
where: $\mu_{material} = \mu$ - absolute permeability of material (H/m), $\mu_0$ - absolute permeability of vacuum (H/m)

Materials commonly referred to as “magnetic” (ferromagnetic and ferrimagnetic) have permeability much greater than unity. The highest permeability can be encountered in materials such as NiFe alloy or Co-based amorphous ribbon which with careful preparation can reach μr = 1,000,000. However, widely used electrical steels and soft ferrites have typically permeability between 1,000 - 50,000.

Paramagnets have μr slightly greater than 1 (e.g. for rather “strongly” paramagnetic oxygen 1.00000037), and diamagnets slightly lower than 1 (e.g. for water 0.999991). However, μr = 0 for superconductors because they expel magnetic field from their inside.6)

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

 → → → Helpful page? Support us! → → → PayPal ← ← ← Help us with just $0.10 per month? Come on… ← ← ← ## Types of relative permeability  See also the main article: Types of magnetic permeability. There are many types of relative permeability, because there can be many definitions of magnetic permeability. The main definition of absolute magnetic permeability is based on the change of the flux density ΔB to the change of magnetic field strength ΔH. Therefore, depending on how the change Δ is measured the definition can be related to the peak values (amplitude permeability), difference to some previous state (differential permeability), expressed in complex numbers (complex permeability), and so on. Absolute permeability $$μ = \frac{ΔB}{ΔH}$$ (H/m) where:$μ$- absolute permeability (H/m),$ΔB$- change of magnetic flux density (T),$ΔH$- change of magnetic field strength (A/m) All such values can be calculated as absolute or relative. And if expressed as relative, then, for the example of the absolute permeability being 1000 times greater, it can be simply stated that$\mu_r$= 1000 (unitless), and typically the equation of relationship between B and H is written as:7) B-H relation expressed with absolute and relative permeability $$B = \mu_{material} · H = \mu_r · \mu_0 · H$$ (T) For simpler magnetic problems, relative permeability can be expressed as a scalar value and it is useful for analysing magnetic circuits which can be represented by a one-dimensional problem, i.e. such that the anisotropy of the material or shape can be neglected. However, in certain cases also the permeability can be expressed with two orthogonal values, e.g. μr,x and μr,y, which approximate anisotropy with an elliptical function. This approach is used for example in some finite-element modelling software.8) For full vector analysis, apart from B and H, also either the magnetisation M or magnetic polarisation J need to be taken into account.9) In magnetic structures such as laminated magnetic cores there is a large 3D anisotropy due to the the material properties (in X and Y directions), and also due to the shape anisotropy (in the Z direction) because of the thin laminations. Current state of the art FEM solvers are still unable to model fully laminated cores in 3D.10) ## 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) ## Permeability and susceptibility 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:11) Permeability μr and susceptibility χvol $$μ_r = χ_\text{vol} + 1$$ (unitless) where:$χ_\text{vol}\$ - volume magnetic susceptibility (unitless)

It should be noted that susceptibility has different values in SI and CGS systems of units, even though it is dimensionless in both cases. This difference is not always clear from the context of a given publication. Also, the susceptibility can be defined with respect to mass or molar quantity, which then results with different units and numerical values. Again, special attention must be paid when converting between SI and CGS systems.

## Permeability of materials

Overview of magnetic properties of soft magnetic materials: saturation, relative permeability (denoted as μ in this figure), frequency range, material cost12)