Encyclopedia Magnetica™

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====== Magnetic saturation ======

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| //[[user/Stan Zurek]], Magnetic saturation, Encyclopedia Magnetica//, \\ @PAGEL@ {updated: ~~LASTMOD~~, accessed: @YEAR@-@MONTH@-@DAY@}  |

**Magnetic saturation** - a phenomenon of non-linearity of [[magnetic properties]] of [[magnetic materials]] at high amplitude of magnetic excitation.[(Feynman>[[https://www.feynmanlectures.caltech.edu/II_36.html|Richard Feynman, Robert Leighton, Matthew Sands, Ferromagnetism, The Feynman Lectures on Physics, Vol. II, Basic Books, ISBN: 9780465079988]])][(Cullity>[[https://books.google.co.uk/books?isbn=9780471477419|B.D. Cullity, C.D. Graham, Introduction to Magnetic Materials, 2nd edition, Wiley, IEEE Press, 2009, ISBN 9780471477419]])][(Purcell>[[https://isbnsearch.org/isbn/9781107014022|E.M. Purcell, D.J. Morin, Electricity and magnetism, 3rd edition, Cambridge University Press, 2013, ISBN 9781107014022]])][(Tumanski>[[https://isbnsearch.org/isbn/9780367864958|Sławomir Tumański, Handbook of magnetic measurements, CRC Press / Taylor & Francis, Boca Raton, FL, 2011, ISBN 9780367864958]])] At complete magnetic saturation, the values of [[magnetisation]] //M// (A/m) or [[magnetic polarisation]] //J// (T) reach the highest possible values of **saturation magnetisation** //M<sub>sat</sub>// or **saturation polarisation** //J<sub>sat</sub>//, respectively, which are also a function of the chemical composition of the material and temperature.[(Fiorillo>[[https://isbnsearch.org/isbn/9780122572517|Fausto Fiorillo, Measurement and Characterization of Magnetic Materials, Academic Press, 2005, ISBN 9780122572517]])]

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[[B-H loop|B-H loops]] (hysteresis loops) showing the onset of magnetic saturation at higher amplitudes - //[[magnetic flux density|B]]// increases very slowly even though the applied //[[magnetic field strength|H]]// increases significantly
[[file/m4_b-h_loop_50hz_png|{{m4_b-h_loop_50hz.png}}]]
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However, the values of [[magnetic flux density]] //B// (T) or [[magnetic field strength]] //H// (A/m) do not reach saturation, so there are no corresponding values of //B<sub>sat</sub>// or //H<sub>sat</sub>//, even though such denotations can be encountered in the literature. //H// can be always increased and thus the associated //μ//<sub>0</sub>·//H// component of //B// will be also increased with increasing excitation. More explanation is provided below.

[[Ferromagnetic]] (and [[ferrimagnetic]]) materials respond with increased magnetisation to the applied external magnetic field. 

At lower amplitudes, magnetic response of [[soft magnetic materials]] can be treated as [[quasi-linear]] (with certain simplifying assumptions), so that the excitation can be applied and response to be utilised over some amplitude useful in practical applications, such as [[magnetic core|magnetic cores]] of [[transformer|transformers]], [[inductor|inductors]], [[motor|motors]], and [[generator|generators]]. 

One of the practically useful properties of a magnetic core is its high [[permeability]] so that the [[magnetic flux]] can be guided in a controlled way in a [[magnetic circuit]].[(Fitzgerald>[[https://isbnsearch.org/isbn/0071121935|A.E. Fitzgerald, Charles Kingsley Jr., Stephen D. Umans, Electric Machinery, 6th edition, Boston, 2003, ISBN 0-07-112193-5]])]


However, at higher excitation the onset of saturation decreases the value of permeability, eventually (at sufficiently high amplitude) making the [[effective permeability]] similar to that of non-magnetic materials (i.e. //μ<sub>r</sub>// = 1). The [[inductance]] (and hence [[impedance]]) of the associated [[winding|windings]] decreases significantly, leading to [[inrush current|inrush]] and [[fault current|fault currents]],[(Fitzgerald)] which can be dangerous to the reliability or operation of the [[electromagnetic device]].

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Magnetic saturation (//J<sub>sat</sub>// = 2.15 T) of [[pure iron]] (linear plot)
[[file/saturation_of_armco_pure_iron_magnetica_png|{{/saturation_of_armco_pure_iron_magnetica.png}}]]
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Magnetic saturation (//J<sub>sat</sub>// = 2.15 T) of [[pure iron]] (semi-log plot)
[[file/saturation_of_armco_pure_iron_log_magnetica_png|{{/saturation_of_armco_pure_iron_log_magnetica.png}}]]
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Therefore, magnetic saturation limits the practical operating range of electromagnetic devices (transformers, inductors, motors, generators) so that correct functioning outside of their [[nominal]] voltage, current, or frequency range might not be possible. 

Depending on the type of analysis (theoretical or practical), there can be several types of saturation (explained in more detail in the following sections):
  * complete theoretical **magnetisation saturation** at absolute zero temperature
  * **forced saturation** (of precessing moments)
  * theoretical **magnetisation saturation** or **polarisation saturation** at operating temperature
  * **technical saturation** of [[magnetic domain]]
  * operation above the "**knee point**" (above the quasi-linear region)

In [[semi-hard magnetic materials]] and [[hard magnetic materials]], the saturation limits the amount of energy which can be stored in the magnetised material, but also [[coercivity]] is the other limiting factor. 

Additionally, in all magnetic materials the level of saturation is related to the temperature of operation in an inverse way. The closer the temperature to the [[Curie temperature]] the lower the saturation.[(Cullity)]

[[Non-magnetic materials]] also react to magnetic field, but with much weaker response. [[Paramagnets]] (which have positive [[magnetic susceptibility]] due to unpaired [[magnetic dipole moment|magnetic moments]]) can become magnetically saturated, but only at very low temperatures and very high magnetic fields. [[Diamagnets]] (negative susceptibility, no unpaired magnetic moments) slightly oppose and weaken the applied magnetic field, and cannot be saturated.[(Cullity)]

[[Superconductor|Superconductors]] lose their superconducting state at sufficiently high magnetic fields. However, the physics of the superconductivity is different from the ordinary magnetic saturation of magnetic materials.[(Cullity)]

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===== Typical saturation values =====

There are many magnetic materials ([[alloy|alloys]], [[ferrite|ferrites]], and [[composite|composites]]) widely used in industry, for which well-established values of magnetic saturation are known.[(Cullity)][(Fiorillo)] Significant research was carried out in the past in order to identify the material with the highest saturation magnetisation, and it was discovered that the Co-Fe alloys resulted with the highest values. However, cobalt is an expensive material, and thus it would be beneficial if alternative materials were identified. Recent research effort suggests that further optimisation can be achieved for new alloys, which push the boundaries to as high as 2.43 T.[(YahaoLi>[[https://doi.org/10.1016/j.scriptamat.2024.116485|Yahao Li, Errui Jiang, Kaijia Hu, Yongqian Peng, Ziqi Ni, Fengqi Liu, Yicong Ye, Shun Li, Shuxin Bai, Active learning-enabled the discovery of ultra-high saturation magnetization soft magnetic alloys, Scripta Materialia,
Vol. 257, 2025, 116485, https://doi.org/10.1016/j.scriptamat.2024.116485]])]

^  Typical saturation polarisation //J<sub>sat</sub>// values [(Fiorillo)][(Cullity)][(YahaoLi)]    ^^
|  Fe70-Co25-Ni4-Si1    |  2.43 T  |
|  Co49-Fe49-V2         |  2.35 T  |
|  Fe                   |  2.15 T  |
|  Co                   |  2.15 T  |
|  Ni50-Fe50            |  1.60 T  |
|  Ni36-Fe64            |  1.30 T  |
|  Ni80-Fe15-Mo5        |  0.80 T  |
|  Ni77-Fe14-Cu5-Mo4    |  0.78 T  |
|  Ni                   |  0.608 T  |
|  FeFe<sub>2</sub>O<sub>4</sub>  |  0.603 T  | 
|  CoFe<sub>2</sub>O<sub>4</sub>  |  0.534 T  | 
|  MnFe<sub>2</sub>O<sub>4</sub>  |  0.503 T  |
|  NiFe<sub>2</sub>O<sub>4</sub>  |  0.340 T  |
|  MgFe<sub>2</sub>O<sub>4</sub>  |  0.151 T  |

Nevertheless, magnetic saturation is only one parameter important for design of [[electromagnetic device|electromagnetic devices]], and for optimum performance other values are also critical, such as [[power loss]], or [[magnetic permeability]]. 





===== Process of magnetisation =====

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[[Magnetisation]] //M// is the sum of [[magnetic dipole moment|magnetic dipole moments]] //m<sub>i</sub>// per unit of volume //V//
[[/file/volume_magnetisation_magnetica_png|{{volume_magnetisation_magnetica.png}}]]
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Unpaired (uncompensated) [[magnetic dipole moment|magnetic dipole moments]] of [[atom|atoms]] respond to the magnetic field by aligning to it. 

The vector sum of all the individual moments averaged over a given volume represents [[magnetisation]] $M$ (A/m), or the equivalent quantity of [[magnetic polarisation]] $J$ (T) (which is the magnetisation scaled by the [[permeability of vacuum]]).[(Griffiths>[[http://books.google.com/books?isbn=0321856562|David J. Griffiths, Introduction to electrodynamics, 4th ed., Pearson, Boston, 2013, ISBN 0321856562]])][(Zurek>[[https://isbnsearch.org/isbn/9780367891572|S. Zurek, Characterisation of Soft Magnetic Materials Under Rotational Magnetisation, CRC Press, 2019, ISBN 9780367891572]])][(Jiles>[[http://books.google.com/books?isbn=9780412798603|David C. Jiles, Introduction to Magnetism and Magnetic Materials, Second Edition, Chapman & Hall, CRC, 1998, ISBN 9780412798603]])]


|  [[magnetisation]]  |  $$ \vec{M} = \frac{\sum \vec{m_i}}{V} $$  |  (A/m)  |
|  [[magnetic polarisation]]  |  $$ \vec{J} = μ_0 · \vec{M}$$  |  (T)  |
| where: $M$ - [[magnetisation]] (A/m), $\sum m_i$ - sum of [[magnetic moment|magnetic moments]] of individual atoms (A·m<sup>2</sup>), $V$ - unit [[volume]] (m<sup>3</sup>), $J$ - [[magnetic polarisation]] (T), $μ_0$ - [[permeability of vacuum]]   |||

If the interactions between the neighbouring moments are strong enough they can spontaneously align over macroscopic distances, thus forming [[magnetic domain|magnetic domains]]. Domains are separated by [[domain wall|domain walls]].[(Cullity)] (In paramagnets the interactions between the neighbours are too weak to form domains.)

In each domain, all the available moments are aligned, which is synonymous with local saturation, because no further alignment is possible at the given temperature.


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[[magnetic domain|Magnetic domains]] in a single [[grain]] (outlined with a black line) of [[non-oriented electrical steel]]. The image shows an area 0.1 mm wide, and the arrows show orientation of the domains. The domains are separated by [[domain wall|domain walls]].
[[file/magnetic_domain_with_arrows_magnetica_png|{{magnetic_domain_with_arrows_magnetica.png}}]]
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Magnetic domain structure ([[lancet domain|lancet]] combs switching during magnetisation) in [[high-permeability grain-oriented electrical steel]]. [[Bar domain|Bar domains]] are visible in the upper part.
[[file/lancet_comb_domains_in_go_steel_magnetica_gif|{{lancet_comb_domains_in_go_steel_magnetica.gif}}]]
//<sup>Copyright (c) Oles Hostanar</sup>//
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[[Magnetisation process]] involves changes in the configuration of magnetic domains; //B// is not limited to some "//B<sub>sat</sub>//" value
[[file/b-h_and_j-h_curve_with_domains_png|{{2dm/b-h_and_j-h_curve_with_domains.png}}]]
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However, the domains within a magnetic material can point in opposing directions, so that the net magnetisation over the whole body of a ferromagnetic (or ferrimagnetic) material can be much smaller, even zero if the material is completely [[demagnetisation|demagnetised]]. 

If some magnetic field is applied to a demagnetised material, then the magnetic domains whose direction coincides with the applied field grow, at the expense of anti-aligned domains. The domain walls move accordingly to accommodate such changes.[(Cullity)][(Jiles)][(Zurek)]

During this process, a greater proportion of the whole volume will have domains aligned with the applied field, so the net magnetisation will increase proportionally. This is the range of operation in which the soft magnetic materials respond in a quasi-linear way to the applied excitation.

At even higher excitation, most of the volume become consumed by the domains aligned to the applied field. If their directions differ slightly, then eventually the vector will be rotated to be parallel with the field, but this requires much stronger amplitude to be achieved. 

Once full alignment is achieved, then the internal magnetisation //M// or polarisation //J// cannot increase any more, and thus the full magnetic saturation values of [[magnetisation saturation]] //M<sub>sat</sub>// or [[polarisation saturation]] //J<sub>sat</sub>// were achieved. 

Further increase of the applied field cannot increase //M// or //J//. However, total magnetic response of the material is quantified in the value of magnetic flux density //B//, which always increases by some amount if //H// is increased.

^   [[Magnetic flux density]] //B//  ^^^
|  (expressed \\ with polarisation //J//)  |  $$ \vec{B} = \vec{J} + μ_0 · \vec{H} $$  |  (T)  | 
|  \\ (expressed \\ with magnetisation //M//)  |  $$ \vec{B} = μ_0 · \vec{M} + μ_0 · \vec{H} $$  \\  $$ \vec{B} = μ_0 · ( \vec{M} + \vec{H} )$$ |  (T)  | 

The [[hysteresis component of power loss|hysteresis]] and [[excess component of power loss|excess components]] of [[power loss]] in a magnetic material are related to the movements of [[domain walls]].[(Jiles)][(Tumanski)] Therefore, if the amplitude of [[AC excitation]] is increased the loss will also increase, even though the material might be reaching (and exceeding) the state of full saturation at the peak of the excitation. This is because, with alternating magnetisation process the excitation reverses cyclically, and thus the resulting magnetisation, polarisation and flux density also reverse, with large domain movements as they cross through their corresponding zero values.

==== Saturation values: Hsat, Bsat, Msat, Jsat ====

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Illustration of [[magnetic field strength]] //H//, [[flux density]] //B//, [[magnetisation]] //M//, and [[magnetic polarisation|polarisation]] //J// in a **[[ferromagnet]]**; all three vector quantities //H//, //M// and //B// can be at different angles due to energy of [[anisotropy]]; in the shown state the material is close to saturation, but not yet fully saturated because the direction of //M//, //J//, and //B// vectors is not yet parallel to the applied //H//
[[file/magnetic_field_flux_density_magnetisation_ferromagnet_e-m_png|{{magnetic_field_flux_density_magnetisation_ferromagnet_e-m.png}}]]
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The name "saturation" implies that no further increase of some quantity is possible. Full magnetic saturation is extremely difficult to achieve, and can happen only under very high excitation, not available in ordinary engineering applications.[(Jiles)]

The four main values which quantify the magnetic field in materials can be characterised as follows:
  * [[magnetic field strength]] **//H//** - Theoretically, magnetic excitation can be always increased with increasing amount of applied electric current, therefore there is no limited value of "//H<sub>sat</sub>//" arising from the magnetic properties of the magnetised material. There is a practical limitation to the amount of current which can be sustained in a conductor due to thermal and/or superconducting [[critical field]] limits, but these are not related to the magnetic saturation as understood from the viewpoint of aligning magnetic dipole moments.
  * [[magnetic flux density]] **//B//** - If //H// is present, then the $μ_0·H$ component is always present, and therefore increasing the excitation in a form of //H// will always produce an increase in the corresponding magnetic flux density //B// (when discussing amplitudes near saturation). Hence, it is also not possible to achieve some limiting value of "//B<sub>sat</sub>//" (and consequently there is no "saturation flux density" nor "saturation induction"). However, in practical cases it is sometimes useful to refer to the state of a material which is near saturation with some specific value of "//B<sub>sat</sub>//". Nevertheless, there is an implicit assumption that only some "technical saturation" is attained rather than full saturation. For example, in analysis of hysteresis loops of permanent magnets the "//B<sub>sat</sub>//" (or "//B<sub>s</sub>//") point could mean a state when just //J<sub>sat</sub>// was reached, even though //B// would continue to increase linearly beyond that point.[(Arnold>[[https://www.arnoldmagnetics.com/wp-content/uploads/2017/10/Vocabulary-of-Magnetism.pdf|Arnold Magnetic Technologies, TECHNotes, The Vocabulary of Magnetism, 2016]], {accessed 2021-05-02})]
    * in the CGS system the [[magnetic polarisation]] is referred to as [[intrinsic induction]], and therefore it is possible to have **saturation intrinsic induction** //B<sub>i,sat</sub>// which is typically just referred to as "saturation induction" //B<sub>sat</sub>// or //B<sub>s</sub>// even though it represents a different quantity from the ordinary induction[(Bozorth)]
  * [[magnetisation]] **//M//** - The magnetisation //M// in (A/m) is the vector average of the individual magnetic moment over a given volume. Once all the individual moments are completely aligned with the applied excitation then full **saturation magnetisation //M<sub>sat</sub>//** is achieved. Further increase of magnetic excitation will //not// produce further increase of //M//, because full alignment was already achieved.
  * [[magnetic polarisation]] **//J//** - The value of polarisation //J// in (T) is the same quantity as //M// but scaled by the [[permeability of vacuum]] so that $J = μ_0 · M$, and therefore $J_{sat} = μ_0 · M_{sat}$. Therefore, there is also the limiting value of **saturation polarisation //J<sub>s</sub>//**.



==== Soft magnetic materials ====

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[[Magnetisation process]] involves changes in the configuration of domains, at full saturation all domains are aligned in the same direction
[[file/magnetic_domains_vs._hysteresis_loop_png|{{2dmch3/magnetic_domains_vs._hysteresis_loop.png}}]]
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Soft magnetic materials are typically used within a range of excitation in which the difference between the flux density //B// and the polarisation //J// is negligible in practice. For example, an excitation of 10 kA/m would contribute only $μ_0 · H$ = 0.01 T, whereas saturation of various soft magnetic materials ranges between 0.3-2.4 T.

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[[Fluxgate magnetometer]] FLC 100[([[http://www.stefan-mayer.com/Lcsing.htm|Miniature Fluxgate Magnetic Field Sensor FLC 100, Stefan Mayer Instruments {accessed 2012-10-23}]])]
[[file/fluxgate_magnetometer_commercial_jpg|{{fluxgate_magnetometer_commercial.jpg}}]]
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Above the "knee" of the non-linear magnetisation curve (see also explanation in the next sections), the [[permeability]] begins to decrease, and for ordinary devices such as transformers and motors this should be avoided.

However, certain applications make use of the saturation. For example, in a [[fluxgate sensor]], some part of a [[magnetic core]] is cyclically saturated magnetically, with a symmetrical excitation. An additional magnetic field makes the saturation asymmetrical, which can be detected and correlated with the additional field, thus producing a mechanism for accurate measurement of that field.[(Tumanski)]

Another example of saturation used as a wanted feature is the [[saturable fault current limiter]] technology. An appropriately shaped magnetic core is pre-saturated with superconducting windings or permanent magnets so that another winding placed on such core has very low impedance. Upon the occurrence of a fault current, the arising magnetomotive force opposes the saturating one, so that the core is taken instantly out of saturation, its permeability increases, inductance and reactance of the winding increases, and thus the fault current is limited accordingly.[(FCL>[[https://www.entsoe.eu/Technopedia/techsheets/fault-current-limiter|Fault Current Limiter, ENTSO-E Technopedia]], {accessed 2021-12-30})]

==== Semi-hard and hard magnetic materials ====

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Optimum operating point //P// for a [[permanent magnet]] is where the product //B// and //H// reaches maximum //BH<sub>max</sub>//; in a magnet //B// and //J// differ significantly[(Leonowicz)]
[[/file/permanent_magnet_bhmax_curve_magnetica_png|{{permanent_magnet_bhmax_curve_magnetica.png}}]]
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In [[semi-hard magnetic materials|semi-hard]], and especially in [[hard magnetic materials]] the range of applied magnetic field strength can reach high and very high values, where the difference between //J// and //B// is so significant that a distinction has to be made between the [[B-H loop|B-H]] and [[J-H loop|J-H loops]].  

When a sufficiently high magnetic field is applied to a permanent magnet then a saturation state is achieved (//J<sub>sat</sub>//). Then, after removing the applied field a significant amount of energy is stored in the internal magnetic field of the material. This energy can be stored for infinitely long time (for all practical purposes) if the material is not subjected to negative field or high temperature.

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Magnetic domain structure in a [[bonded magnet]] based on Nd14.8Fe76Co4.8B4.2Nb0.2[(Dospial)]
[[/file/domain_structure_of_bonded_magnet_nd14.8fe76co4.8b4.2nb0.21_dospial_mdpi_e-m_jpg|{{domain_structure_of_bonded_magnet_nd14.8fe76co4.8b4.2nb0.21_dospial_mdpi_e-m.jpg}}]] \\
//<sup>M. Dośpiał, [[https://www.mdpi.com|MDPI]], [[https://creativecommons.org/licenses/by/4.0/|CC-BY-4.0]]</sup>//
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For strong permanent magnets such as [[NdFeB]] the application of a magnetic field in an opposing direction, even such that the resulting //B// reduces to zero (//<sub>B</sub>H<sub>C</sub>// point) the magnet remains magnetised. This inherent energy storage is a basis for information storage (hard drives) and for generation of strong magnetic fields by permanent magnets.

In permanent magnets the size of grains and domains is very small, which helps in strong [[domain wall pinning]] thus increasing the amount of energy which can be stored in the magnetic field.[(Leonowicz>[[http://books.google.com/books?isbn=8320430496|Marcin Leonowicz, Jerzy J. Wysłocki, Modern magnets, Technologies, Coercivity mechanisms, Applications (in Polish: Współczesne magnesy, Technologie, Mechanizmy koercji, Zastosowania), Wydawnictwa Naukowo-Techniczne, Warsaw, 2005, ISBN 8320430496]])][(Dospial>[[https://doi.org/10.3390/ma14112849|Marcin J. Dośpiał, The Domain and Microstructure of Resin-Bonded Magnets. Materials 2021, 14, 2849. https://doi.org/10.3390/ma14112849]])]

Permanent magnets are typically in a [[magnetic circuit]] with an air gap, which produces [[demagnetising effect]] and thus makes the magnet to operate in the second quadrant of the hysteresis loop, in any case away from saturation. However, a magnet must be first exposed to very high (saturating, or almost saturating field) before it can become magnetised.

===== Anisotropy =====

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**Saturation polarisation** //J<sub>sat</sub>// of [[single-crystal iron-silicon]] alloy is the same regardless of direction of excitation (designed here by the values of [[Miller indices]])[(Bozorth)]
[[file/j-h_curves_for_three_basic_crystallographic_directions_of_single_crystal_png|{{2dm/
j-h_curves_for_three_basic_crystallographic_directions_of_single_crystal.png}}]]
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Ordinary magnetic materials can be saturated in a "positive" or "negative" direction (or in any other direction in 3D), with the same value of saturation, but perhaps with just negative sign (if "negative" excitation was applied).[(Cullity)] 

Highly anisotropic materials such as [[grain-oriented electrical steel]] have some privileged directions ("[[easy magnetisation direction]]") in which the material responds with higher //M//, //J// or //B// at lower applied field //H//. It is therefore easier to saturate the material when the magnetic excitation is applied in that direction.

However, ultimately, the limiting value of saturation //J<sub>sat</sub>// or //M<sub>sat</sub>// is the same regardless of the direction of applied field, because it is a property related to the chemical composition of the material, not the [[crystallographic structure]] as such. This is because a saturation is a state in which all the magnetic moments are aligned, regardless the direction.[(Bozorth>[[https://isbnsearch.org/isbn/0780310322|Richard M. Bozorth, Ferromagnetism, Wiley-IEEE Press, 1993, ISBN 0780310322]])]

Similar saturation behaviour can be observed if the anisotropy is caused by the [[shape anisotropy]] of the sample, rather than the intrinsic properties of the material of the sample.[(Cullity)]

===== Temperature =====

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Schematic of alignment of moments in a domain vs. Curie temperature //T<sub>C</sub>//:[(Jiles)] a) above //T<sub>C</sub>// there is no alignment (random), b) just below //T<sub>C</sub>// there is alignment but thermal agitation disturbs it, c) at temperature much lower than //T<sub>C</sub>// the moments are aligned but continue to precess so //M// is lower than //M<sub>sat</sub>// at //T//=0, d) full alignment at absolute zero temperature so //M// = //M<sub>sat</sub>//
[[file/magnetisation_vs_temperature_jiles_magnetica_png|{{magnetisation_vs_temperature_jiles_magnetica.png}}]]
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The phenomenon of ferromagnetism is linked to the spontaneous alignment of [[magnetic moment|magnetic moments]] due to the neighbour-neighbour interaction. These interactions are counteracted by thermal agitation of the constituting [[atom|atoms]]. If the temperature increases then there is more thermal energy and the magnetic interactions are impacted by a greater amount. At sufficiently high temperature, beyond [[Curie temperature]], ferromagnets become paramagnetic because the spontaneous alignment of magnetic moments cannot be sustained in the overpowering thermal agitation. 

On the other hand, if the temperature is lower, then the magnetic interactions are impacted to a lesser degree. For instance, [[gadolinium]] is paramagnetic at room temperature, but becomes ferromagnetic upon cooling to lower values.

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Normalised [[spontaneous magnetisation]] //M/M<sub>sat</sub>// as a function of temperature for Fe, Co, Ni and Gd;[(Jiles)][(Nigh)] dashed lines show room temperature relative to the Curie temperature //T/T<sub>C</sub>//, //J<sub>e</sub>// denotes the [[exchange integral]] coefficient
[[file/magnetisation_of_ferromagnets_vs_temperature_e-m_png|{{magnetisation_of_ferromagnets_vs_temperature_e-m.png}}]]
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At the temperatures near [[absolute zero temperature|absolute zero]] the highest possible spontaneous alignment can be achieved, with the highest possible values of **saturation magnetisation** //M<sub>sat</sub>// or the equivalent **saturation polarisation** //J<sub>sat</sub>// for the given material. These thermal effects have implications on "technical saturation" (described in more detail below).

<WRAP right 35%>
|< 95% >|
^  [[Curie temperature]] \\ for ferromagnetic elements[(Jiles)]  ^^^
^   ^   (°C)  ^  (K)  ^
|  [[cobalt]] Co  |  1131  |  1404  |
|  [[iron]] Fe  |  770  |  1043  |
|  [[nickel]] Ni  |  358  |  631  |
|  [[gadolinium]] Gd  |  20  |  293  |
|  [[terbium]] Tb  |  -54  |  219  |
|  [[dysprosium]] Dy  |  -188  |  85  |
|  [[thulium]] Tm  |  -241  |  32  | 
|  [[erbium]] Eb  |  -253.5  |  19.5  |
|  [[holmium]]  Ho  |  -254  |  19  |
</WRAP>

If the saturation magnetisation at absolute zero temperature is taken as the reference then, as shown in the illustration, the spontaneous magnetisation //M// decreases the more rapidly the closer the material is to its Curie temperature. The shape of the //M/M<sub>sat</sub>// curve is related to the configuration of unpaired magnetic moments in the atomic structure of the given element.[(Nigh>[[https://doi.org/10.31274/rtd-180813-4593|Harold Eugene Nigh, Magnetization and electrical resistivity of gadolinium single crystals, PhD thesis, Iowa State University, USA, 1963, https://doi.org/10.31274/rtd-180813-4593]])] 

If an electromagnetic device has to operate at elevated temperatures these effects become important. For example, electric generators inside jet engines can be exposed to high temperatures (even exceeding 200 °C) and therefore the required magnetic properties must be ensured for the built-in electric generator.[(Andersson>[[https://lup.lub.lu.se/luur/download?fileOId=8942189&func=downloadFile&recordOId=8942183|Fredrik Andersson, Integrated generator for use in aircraft engines, Pre-study, Lunds Universitet, Lund, 2018]], {accessed 2022-01-01})] This applies to the magnetic cores as well as permanent magnets.

Also, in induction heating of steel for hardening, the metal might be heated above its Curie temperature, which has to be taken into account during the heating process, because it changes the amount of energy that can be delivered to the heated element by electromagnetic means.[(Rudnev>[[https://www.routledgehandbooks.com/doi/10.1201/9781315117485-3|Valery Rudnev, Don Loveless, Raymond L. Cook. Handbook of Induction Heating, CRC Press, ISBN 9781466553958]])]

Information storage on semi-hard magnetic materials (e.g. hard drives) relies on retention of local magnetisation over long periods of time. This process is limited by [[superparamagnetic limit]], which dictates the thermal stability of the magnetic storing structure.[(Li>[[https://eng.libretexts.org/@go/page/337|Tina Li, Superparamagnetism. (2021, September 8). https://eng.libretexts.org/@go/page/337]], {accessed 2022-01-01})]

===== Technical saturation - knee point =====

The [[magnetisation curve]] of soft magnetic materials has a characteristic shape, which can be split into two simplified regions.[(MDPI_Lee>[[https://doi.org/10.3390/en7074571|Lee, B.E.; Lee, J.; Won, S.H.; Lee, B.; Crossley, P.A.; Kang, Y.C. Saturation Detection-Based Blocking Scheme for Transformer Differential Protection. Energies 2014, 7, 4571-4587. https://doi.org/10.3390/en7074571]])] 

At lower excitation, the material has relatively high permeability and often it can be approximated with a constant [[magnetic permeability]] so that the material can be thought of as responding in a quasi-linear way to the applied excitation. Operation of a magnetic core in this region is predictable and generally safe.[(Furlani>[[https://isbnsearch.org/isbn/9780122699511|Edward P. Furlani, Permanent Magnet and Electromechanical Devices: Materials, Analysis, and Applications, Academic Press, 2001, ISBN]])]

At high amplitude of excitation the material is operating close to saturation, and its permeability reduces significantly. If a device was designed to operate in the quasi-linear region then entering the saturation (e.g. as caused by some fault condition) can be dangerous and can lead to a permanent failure. 

<box 50% right #f0f0f0>
The "knee" point is above the quasi-linear region
[[file/saturation_knee_quasi-linear_region_magnetica_png|{{/saturation_knee_quasi-linear_region_magnetica.png}}]]
{{page>insert/by_SZ}}
</box>

Therefore, in some practical application it is useful to distinguish between these two regions, marking the "knee" point which separates them. Therefore, the colloquial meaning of the "knee point" is the location on a magnetisation curve which lies just above the quasi-linear region, and anything above the knee is referred to as "saturation",[(EV>[[https://www.electricalvolt.com/2018/09/how-to-calculate-knee-point-voltage-of-current-transformer/|Electrical Volt, How to Calculate Knee Point Voltage of Current Transformer?]], {accessed 2021-12-31})] which strictly speaking is incorrect (because full magnetic saturation requires much higher excitation). However the **technical saturation** is a useful concept, both from a practical and theoretical viewpoint.

From a theoretical viewpoint, the state of technical saturation can be also referred to in the [[process of magnetisation]]. When a sufficiently high magnetic excitation is applied to a given magnetic material, eventually a single-domain state will be reached. This single-domain state represents the theoretical **technical saturation**, because all the magnetic moments are parallel and thus the material is saturated to **//M<sub>sat</sub>//** in the direction parallel to that domain. However, if the exciting field is applied as some other angle and even higher value will be applied (so that the [[energy of anisotropy]] can be overcome), then the domain will eventually rotate to become parallel to the applied field. Moreover, even the precession of dipoles can be forced to lower angles, and then the **complete saturation //M//<sub>0</sub>** will be reached. This slow increase of magnetisation due to rotation in high applied field is known as [[forced magnetisation]].[(Jiles)]

In some specific practical or engineering cases the [[knee point]] is defined quite precisely, as it is for instance for [[current transformer|current transformers]] (CT) used in electricity distribution industry.[(EV>[[https://www.electricalvolt.com/2018/09/how-to-calculate-knee-point-voltage-of-current-transformer/|Electrical Volt, How to Calculate Knee Point Voltage of Current Transformer?]], {access 2021-12-31})][(Milner>[[https://megger.com/support/technical-library/et-magazine-july-2021|Stan Zurek, Dave Milner, What is a knee point?, Electrical Tester, 2021, Issue 003, p. 71]], {accessed 2021-12-30})] For example, the international standard IEC 61869 defines the knee point as such at which an incremental increase of AC excitation voltage (at nominal operating frequency) by 10% causes the increase of resulting current by 50%.[(Milner)] Therefore, an increase in one quantity causes a disproportional increase in the related quantity, and thus the magnetic non-linearity (the location of the "knee") can be detected simply by measuring the associated electrical signals.

Another approach is described in the standard IEEE C57.13 which gives two other definitions, referred to [[ANSI 45]]° and [[ANSI 30]]°. With the applied AC voltage plotted against the resulting current with the same scaling  (e.g. [[log-log]]) the knee point is where the line tangential to the plotted V-I curve lies at an angle of 45° or 30°, respectively.[(Milner)] 

In a CT it is important to know where the knee point lies, because the accuracy of transducing the primary current can be guaranteed only below this point. Additionally, the voltage across the winding of a CT is related to the amount of power that can be delivered to the [[CT burden]] (and for example to safely operate a [[protection relay]]). Therefore, the measurement of the knee point (saturation point, or [[saturation voltage]]) is a tool for verification of the capabilities of a given CT. There are commercial products for such measurements.[(Megger_MVCT>[[https://embed.widencdn.net/pdf/plus/megger/hofbpc6rcr/MVCT_DS_EN.pdf|MVCT, Megger VT & CT Analyzer, Megger, Datasheet, 2021]], {accessed 2021-12-30})]

Yet another definition of practical magnetic saturation is given in the standard IEC 60404-7:2019.[(IEC>[[https://webstore.iec.ch/publication/33513|International Standard, IEC 60404-7:2019, Magnetic materials - Part 7: Method of measurement of the coercivity (up to 160 kA/m) of magnetic materials in an open magnetic circuit]])] Practically sufficient level of saturation is obtained if increasing the saturating field by 50% does not increase the measured [[coercivity]] by more than 1%. The saturating field can be applied by an [[electromagnet]] (solenoid) or a [[permanent magnet]].[(IEC)] In commercial [[coercimeter|coercimeters]] saturation of soft magnetic materials is achieved typically with a field of 140-450 kA/m, which can be obtained by DC current from a variable DC power source, or by pulse methods from a LC circuit.[(Brockhaus>[[https://www.brockhaus.com/measurements/products/softmagnetic/est-series/cft-200|Brockhaus Group, Coercimeter CFT 200, Product brochure, 2023]], {accessed 2023-09-14})][(LE>[[https://www.laboratorio.elettrofisico.com/wp-content/uploads/2021/05/coercimeter-14.pdf|Laboratorio Elettrofisico, CR/03 - Coercimeter, Measuring system made for irregularly shaped samples, Product brochure, 2023]], {accessed 2023-09-14})]
===== Inrush current =====

If an ordinary AC transformer (50/60 Hz) is energised at a random instant of time with a sinusoidal voltage, then the [[magnetic flux]] can build up and add to the [[remanence]], and thus push the magnetic core into magnetic saturation (i.e. deep above the [[knee of magnetisation curve]]).[(Fitzgerald)][(Wen>[[https://doi.org/10.3390/en10040480|Wen, X.; Zhang, J.; Lu, H. Automatic J–A Model Parameter Tuning Algorithm for High Accuracy Inrush Current Simulation. Energies 2017, 10, 480. https://doi.org/10.3390/en10040480]])][(Komarzyniec>[[https://doi.org/10.3390/en14206714|Komarzyniec, G. Calculating the Inrush Current of Superconducting Transformers. Energies 2021, 14, 6714. https://doi.org/10.3390/en14206714]])]

<box 70% right #f0f0f0>
Example of an [[inrush current]] in a transformer, due to asymmetric magnetic saturation of the [[magnetic core]][(Wen)]: left - measured waveform of the [[inrush current]] with the first cycle having the highest peak, right - corresponding asymmetric [[B-H loop]] traced during the inrush event
[[file/wen_et_al_inrush_current_mdpi_2017_e-m_png|{{/wen_et_al_inrush_current_mdpi_2017_e-m.png}}]]
//<sup>X. Wen, J. Zhang, H. Lu, [[https://www.mdpi.com|MDPI]], [[https://creativecommons.org/licenses/by/4.0/|CC-BY-4.0]]</sup>//
</box>

Under such condition, the impedance of the windings can drop significantly because of technical saturation. The AC transformers are typically energised from a relatively "stiff" voltage source ([[electricity grid]]) and therefore fixed voltage amplitude and low winding impedance results in elevated amplitude of resultant current, at the level similar to a fault (such as [[short circuit]]). In the illustration, in the measured data for a small transformer the worst-case peak current reaches up to 1.1 A, even though the nominal current is 0.15 A, thus there was an [[order of magnitude]] increase,[(Ekwue>[[http://dx.doi.org/10.4314/njt.v37i4.27|A. O. Ekwue, B. Rawn, Investigations into the transformer inrush current problem, Nigerian Journal of Technology (NIJOTECH), Vol. 37 (4), 2018, pp. 1058-1064, http://dx.doi.org/10.4314/njt.v37i4.27]])] and it can last for relatively long time of up to several seconds.[(Ge>[[https://doi.org/10.3390/en14248244|Ge, W.; Zhang, C.; Xie, Y.; Yu, M.; Wang, Y. Analysis of the Electromechanical Characteristics of Power Transformer under Different Residual Fluxes. Energies 2021, 14, 8244. https://doi.org/10.3390/en14248244]])]

Similar inrush events can happen in electric motors.[(Faraday_Predictive>[[https://www.faradaypredictive.com/technical-education/diagnostic-outputs/464-motor-start-up-or-inrush-currents/|4.6.4 Motor Start-up or Inrush Currents, MBVI Diagnostics – Motor Start-up or inrush currents, Faraday Predictive, https://www.faradaypredictive.com]], {accessed 2022-01-01})]

There are multiple implications of inrush current events, including **thermal**, **mechanical**, and **protection**:[(Fitzgerald)][(Ge)] 
  * **Thermal** - Power loss dissipated in the windings is proportional to the square of current. Therefore, if the the current increases by a factor of 10 then the instantaneous power loss would increase by a factor of around 100. This can cause overheating of electrical insulation and thus catastrophic electrical failure of such machine.
  * **Mechanical** - Fault-like current levels in the windings can exert mechanical forces large enough that they can distort and bend the windings, to the point of destruction,[(Ge)] especially if high-voltage insulation was damaged in such process. Methods such as [[frequency-sweep analysis]] can detect physical distortions of the windings in a non-destructive way by a low-energy electrical measurements.
  * **Protection** - high-power electrical machines such as transformers and motors are protected against fault currents (such as [[short-circuit]] or [[locked rotor]] conditions) by suitable [[circuit breaker|circuit breakers]] or [[fuse|fuses]]. The activation level of such protection has to be low enough to protect the devices, and the lower the threshold the better level of protection. However, if the tripping level is set too low then every time the device is powered and an inrush current event happens then the protection would activate every time, resulting in [[nuisance tripping]]. In order to cater for inrush currents, there are circuit breakers which have "delayed" action so that a relatively short overload in a form of inrush current would not activate it. This is also one of the main reasons why there are "slow" and "fast" fuses, because a fast fuse would be activated every time by an inrush current caused by magnetic saturation of a the magnetic core.

===== Demagnetisation =====

The process of [[demagnetisation]] means that a material which was previously magnetised or saturated and it remains in some //B// ≠ 0 (at //H// = 0) state is brought to the state //B// = 0 (at //H// = 0). This minimises the amount of energy stored in the magnetic field outside and inside the volume of the magnetic material.

A magnetic material can be demagnetised in two basic ways: **reducing AC excitation** (most popular) and **thermal** (rarely used, but theoretically effective).

In the **reducing AC excitation** method the excitation is applied in the alternating way (e.g. sinusoidal or switched DC). For correct demagnetisation, the applied AC magnetisation needs to have the amplitude at least equal, or higher than the excitation which was previously applied to the material. The AC amplitude is then slowly reduced, so that that the material is exposed to multiple cycles of excitation in a "positive" and "negative" direction. Such excitation makes the B-H curve to trace a reducing spiral. If the demagnetisation is successful, the material ends at the state //H// = 0 and //B// = 0. 

If the amplitude is reduces too quickly, in too few steps, then the demagnetisation process might not be successful, such that //B// ≠ 0 at //H// = 0. Similar applies if the applied excitation is lower than the previous level of magnetic field to which the material was exposed - because then it might not be possible to fully "erase" the [[magnetic history]] of the material.[(Zurek)]

<box 100% left #f0f0f0>
[[Demagnetisation]] through a slow reduction of [[AC]] excitation: a) waveforms of current (proportional to //H//) and voltage (proportional to //dB/dt//) applied during AC demagnetisation, b) resulting "spiral" of B-H characteristics with the final point after demagnetisation (black dot) ideally ending close to the demagnetised state //H//=0, //B//=0 (the demagnetisation as shown in the image was not perfect because //B// ≠ 0 at //H// = 0)
[[file/ac_demagnetisation_example_png|{{/2dmch4/ac_demagnetisation_example.png}}]]
{{page>insert/by_SZ}}
</box>


In the **thermal method** the magnetised material has to be heated to a temperature above [[Curie temperature]] //T<sub>c</sub>// of the given material. The thermal agitation of magnetic moments can destroy their spontaneous alignment, and upon cooling the material will become demagnetised, due to randomised orientation of all the contributing domains. However, in practice this method is used very rarely, because for most magnetic alloys //T<sub>c</sub>// is several hundred degrees Celsius, which would damage other associated materials such as varnish used for protection of the [[magnetic core]] or the [[winding|windings]]. 


===== Remanence =====

<box 50% right #f0f0f0>
[[Remanence]] //B<sub>r</sub>// and [[coercivity]] //H<sub>c</sub>// are measured after saturating, and then reversing the applied H into the [[second quadrant]]
[[file/coercivity_remanence_soft_magnetic_materials_magnetica_png|{{/coercivity_remanence_soft_magnetic_materials_magnetica.png}}]]
{{page>insert/by_SZ}}
</box>

It should be noted that $ \vec{B} = \vec{J} + μ_0 · \vec{H} $, and thus at //H//=0 it is true that $B_{H=0} = J_{H=0}$. If the material was first saturated, and then the external excitation was brought back to zero then the material will be put in the state of //H//=0, and it will retain some level of magnetised state. This "remaining" amount of //B// or //J// is called a **[[remanence]]**, and can be denoted by **//B<sub>r</sub>//** or **//J<sub>r</sub>//** (both values have the same unit of tesla, T). 

The equality //B<sub>r</sub>// = //J<sub>r</sub>// holds also for permanent magnets (see illustration above).

The remanence point can be also quantified as //M<sub>r</sub>//. The value can be determined simply by first measuring the value of //J<sub>r</sub>// (T) and dividing it by //μ<sub>0</sub>// (H/m), thus obtaining //M<sub>r</sub>// (A/m).

In some literature, the definition of remanence requires the material to be saturated first, so the remanence value will be always the same for the same material under the same conditions. 

However, in some analyses the "full" saturation is not required so that the value of remanence become a function of the magnetic history, so that the "remanence" can vary from its maximum value all the way to zero (if the material is nearly demagnetised). This is for instance the studies of [[paleomagnetism]] in which the term "remanence" can refer to the amount of remnant flux density of some mineral due to the [[Earth's magnetic field]] (without magnetic saturation).[(Tumanski)][(Kodama>[[https://isbnsearch.org/isbn/9781118384152|Kenneth P. Kodama, Paleomagnetism of Sedimentary Rocks, John Wiley & Sons, 2012, ISBN 9781118384152]])]
===== Coercivity =====

After saturation, if the excitation is first reduced to zero (//H// = 0), and then slowly applied in the negative direction through the second quadrant, until //B// = 0, then a point called **[[coercivity]]** **//H<sub>c</sub>//** will be reached. And because //H// has to be applied in a negative direction, the value of //H<sub>c</sub>// is negative.

In [[soft magnetic materials]], the value of //H<sub>c</sub>// is relatively low, the lower the better. For that reason the component $μ_0 · \vec{H}$ in the equation $\vec{B} = \vec{J} + μ_0 · \vec{H}$ is negligibly smaller than $J$ and therefore $J \approx B$ = 0 at the coercivity point of $H_c$.

<box 30% right #f0f0f0>
In permanent magnets | //<sub>B</sub>H<sub>c</sub>// | < | //<sub>J</sub>H<sub>c</sub>// |
[[/file/permanent_magnet_bhmax_curve_magnetica_png|{{permanent_magnet_bhmax_curve_magnetica.png}}]]
{{page>insert/by_SZ}}
</box>

In [[hard magnetic materials]] the value of //H<sub>c</sub>// is relatively high, the higher the better. The contribution of $μ_0 · \vec{H}$ is no longer negligible and thus two points have to be distinguished (see also illustration in previous sections):[(Leonowicz)]
  * at //B// = 0 the coercivity value is //<sub>B</sub>H<sub>c</sub>//
  * at //J// = 0 the coercivity value is //<sub>J</sub>H<sub>c</sub>//
  * and such that: | //<sub>B</sub>H<sub>c</sub>// | < | //<sub>J</sub>H<sub>c</sub>// |

As with remanence, coercivity can be defined as measured after saturation so that it has always the same value, related to material properties. 

But in other types of analyses it can be useful to use the "coercivity" as measured after other amplitudes or modes of excitation (e.g. varying frequency), for any B-H loop crossing the //B//=0 level.[(Najgebauer>[[http://doi.org/10.12693/APhysPolA.131.633|Mariusz Najgebauer, Fractional Scaling of Magnetic Coercivity in Electrical Steels, Acta Physica Polonica A, Vol. 131 (4), 2017, pp. 633-635]])]
===== Rotational magnetisation =====

In magnetic cores of [[rotating machine|rotating]] and [[polyphase machine|polyphase machines]] there are regions which are magnetised in a rotating way. 

As mentioned above, the hysteresis and excess component of power loss is related to the movement of [[domain wall|domain walls]]. As the material is excited with high amplitude close to saturation, then if the magnetising field rotates then there are fewer domain walls, and at full saturation the material has just one [[magnetic domain]] (no domain walls) rotating together with the applied field.[(Zurek)]

The eddy currents are still induced in the the magnetised material and thus they generate loss, which increases proportionally to the square of frequency. But at very slow angular frequency of rotation the power loss in the material can decrease to very small values, much lower than the losses under AC magnetisation, as shown below for some typical measured data. The loss under rotational magnetisation reaches a characteristic peak at about //B<sub>rot</sub>// between 1.4 - 1.7 T for [[electrical steel|electrical steels]] and begins to decrease for higher excitation.[(Zurek)] 

If the frequency of rotation is increased then the eddy currents become the dominant mechanism for loss generation and thus the peak in the power loss characteristics is less pronounced, and does not occur for even higher frequencies.[(Zurek_IEEE>[[https://doi.org/10.1109/INTMAG.2006.374884|S. Zurek et al., Rotational power losses under controlled flux density or magnetic field in electrical steel sheets, 2006 IEEE International Magnetics Conference (INTERMAG), 2006, pp. 853-853, doi: 10.1109/INTMAG.2006.374884]])] 

<box 45% left #f0f0f0>
Animation of //B// and //H// vectors measured during rotational magnetisation of [[grain-oriented electrical steel]]
{{youtube>jCgZO4MbLRY?medium}} {{page>/insert/by SZ}}
</box>

<box 45% left #f0f0f0>
Comparison of power loss in [[electrical steel]] under AC and rotational (ROT) magnetisation[(Zurek)]
[[file/comparison_of_alternating_and_rotational_loss_by_sz_png|{{/2dmch7/
comparison_of_alternating_and_rotational_loss_by_sz.png}}]]
{{page>insert/by_SZ}}
</box>
===== See also =====
  * [[Magnetic dipole moment]]
  * [[Magnetic domain]]
  * [[Magnetic materials]]

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

{{tag> Magnetic_saturation Magnetic_domains Magnetic_domain_walls Knee_point B-H_curve J-H_curve M-H_curve Magnetisation_curve Magnetic_properties  Counter}}