====== Effective magnetic permeability ======

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| //[[user/Stan Zurek]], Effective magnetic permeability, Encyclopedia Magnetica//, \\ @PAGEL@   |


**Effective magnetic permeability** (also **apparent magnetic permeability**[([[http://google.com/books?isbn=9781566775601|M. Bedenbecker, Z. Celinski, H.H. Gatzen, Directional permeability dependence in electroplated patterned permalloy layers, The Electrochemical Society Transactions, 3 (25), 2007, ISBN 9781566775601, p. 123]])]), often denoted as **//μ<sub>e</sub>//**, **//μ<sub>eff</sub>//** or **//μ<sub>a</sub>//** - a term used in analysis of magnetic performance of [[gapped core|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.
[([[http://www.intl-magnetics.org/pdfs/SFG-98.pdf|Soft Ferrites, A User's Guide, Magnetic Materials Producers Association, 1998, p. 37]])]

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Even a very small [[air gap]] can have a significant impact on effective permeability
[[/file/effective_permeability_vs_air_gap_in_rm4_epcos_e-m_png|{{effective_permeability_vs_air_gap_in_rm4_epcos_e-m.png}}]] 
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An [[air gap]] in a [[magnetic circuit]]
[[file/Air gap diagram_png|{{Air gap diagram.png}}]]
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===== Effect of air gap =====

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Changes of effective permeability and linearisation of the [[B-H loop]] caused by increasing air gap[(Finke)]
[[file/influence_of_air_gap_magnetica_png|{{influence_of_air_gap_magnetica.png}}]]
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Air gap reduces the effective permeability
[[file/Air gap in BH loop_png|{{Air gap in BH loop.png}}]]
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[[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.[([[http://www.ferronics.com/files/Toroid.pdf|Toroids, Design considerations, Ferronics Inc., {accessed 24 Jun 2013}]])]

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  =====

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[[Gapped core]] with: total [[magnetic path length]] $l$ (orange), length of core $l_{core}$ (blue) and length of gap $l_g$ (red)
[[file/gapped_core_magnetica_jpg|{{gapped_core_magnetica.jpg}}]]
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It is possible to [[analytical calculations|analytically calculate]] the value of relative effective permeability for simple magnetic circuits, with a uniform gap.[(Finke>[[http://www.magmet.com/pdf/GappedCoreStruct.pdf|G.B. Finke, Gapped magnetic core structures, Magnetic Metals Corporation, {accessed 17 Jun 2013}]])]

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 [[flux fringing|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 [[permeability of free space|vacuum]]).

The equation is derived by using the concept of [[magnetic reluctance]][(Finke)] 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.[(Finke)] 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).

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===== Composite materials =====

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The ratio of air gap and the particles in a [[powder core]] dictate the value of effective permeability. The black lines illustrate distribution of [[magnetic flux]].
[[file/fem_of_powder_core_magnetica_png|{{fem_of_powder_core_magnetica.png}}]]
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The value of effective permeability is important for [[composite material|composite materials]], which may contain significant volumetric percentage of [[non-magnetic material]]. The small particles (as in [[powder core|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.[([[http://www.ti.com/lit/ml/slup124/slup124.pdf|L. Dixon, Magnetic core characteristics, Texas Instruments, p, 2-3, {accessed 22 May 2013}]])]

For instance, [[Ferrotron 119]] used for [[flux concentrator]]s 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|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.[([[http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1381502|Patrick Quéffélec, David Bariou,  Philippe Gelin, A Predictive Model for the Permeability Tensor of Magnetized Heterogeneous Materials, IEEE Transactions on Magnetics, Vol. 41, (1), p. 17]])]

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.[([[http://www.mag-inc.com/File%20Library/Product%20Datasheets/Powder%20Core/New%20Powder%20Cores/Toroids/140%20SIze/0077140A7.pdf|Data sheet, KoolMu toroid 140 (0077140A7), 125μ, Magnetics, {accessed 24 Jun 2013}]])]

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[([[http://www.ferroxcube.com/prod/assets/sendust.pdf|Sendust, Material specification, Ferroxcube, {accessed 22 Jul 2013}]])] //μ<sub>e</sub>// = 75.

===== See also =====
  * [[Magnetic permeability]]
  * [[Air gap]]
  * [[Magnetic core]]

===== References =====
~~REFNOTES~~

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