Coercivity, magnetic coercivity, coercive field, or coercive force, typically denoted by H_C - such a value of the externally applied unidirectional magnetic field strength H that reduces the instantaneous state of magnetisation of a material to zero (B = 0, J = 0, or M = 0), typically from the initial point of remanence B_R.^{1)2)3)4)5)6)7)} Because of its position on the hysteresis loop (second quadrant) the value of coercivity is typically stated as a negative number in A/m in the SI units (or oersted in the CGS). For hard magnetic materials there are two distinctly different coercivities _BH_C and _JH_C, but for soft magnetic materials these two values are negligibly close so only one value of H_C is used. Coercivity is an example of a magnetic property, which quantifies the resistance of a material to being demagnetised.

Position of induction coercivity _BH_C on the B-H loop and polarisation coercivity _JH_C on the J-H loop for hard magnetic materials⁸⁾

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Coercivity H_c in soft magnetic materials is measured after saturating, switching the excitation off (H = 0), and then reversing the applied H into the second quadrant

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The value of coercivity is an important parameter and it is used for classification of magnetic materials into three broad groups:⁹⁾

• soft magnetic materials (H_C < 1 kA/m),
• semi-hard magnetic materials (1 kA/m < H_C < 100 kA/m)
• hard magnetic materials (H_C > 100 kA/m)

But this is not a strict classification because other publications might give different limits or not even include the class of semi-hard materials.¹⁰⁾ Also, the function, performance, or application of the given material can dictate the actual “type”.¹¹⁾ Semi-hard materials are typically used for storage of information (hard drives, magnetic tapes), rather than in applications for energy generation or transformation.

Reaching the coercivity point, at which the state of magnetisation as measured by B, J, or M = 0 does not mean that the material was fully or even partially demagnetised, and much more elaborate procedures are needed to ensure that demagnetisation is achieved.

The “state of magnetisation” can be expressed in different ways and hence there can be several types of coercivity, which can be used interchangeably in the literature, depending on the context. Some typical examples are as follows:

• for magnetic flux density B (also known as magnetic induction), so for the B-H loop it is the induction coercivity _BH_C (also written as H_cB)¹²⁾
• for magnetic polarisation J (J-H loop) it is the polarisation coercivity _JH_C (also written as H_cJ)¹³⁾
• for magnetisation M (M-H loop) it is the magnetisation coercivity, which could be marked as _MH_C or H_CM ¹⁴⁾¹⁵⁾
• in CGS units the polarisation can be referred to as “intrinsic induction” and hence there is intrinsic coercivity H_Ci, which can also be denoted by _iH_C¹⁶⁾ - this is synonymous with _JH_C but typically expressed in different units^17)18)19)20) (SI system uses the name “magnetic polarisation” J in tesla, but CGS uses “intrinsic induction” B_i in gauss, or magnetisation M which can be expressed either in gauss or in oersted)
• the name intrinsic coercivity _iH_c can be also applied to data expressed by magnetisation ²¹⁾²²⁾
• in soft magnetic materials the difference between B and J is negligibly small when measured experimentally away from magnetic saturation (e.g. below the knee of magnetisation), so when μ₀·H « J it is _BH_C ≈ _JH_C and thus typically no distinction is made between the two values and just H_C is used
• if clear from the context then only H_C is used as a symbol for any type of coercivity²³⁾
• different units of a coercivity value can be used in various publications: A/m, T, G, Oe, as dictated by the differences in defining the “state of magnetisation” ^24)25)26)
  • for example in some cases the value of H_C is quoted as μ₀·H_C so that the unit of T can be used instead of A/m ²⁷⁾
• remanence coercivity H_C,r is such such a value of H that after its application and reduction to zero the remnant flux density becomes zero (see explanation and illustration below in the text)²⁸⁾²⁹⁾
• another name for coercive field used in the past was starting field H_s because it was “the field necessary to start the change in magnetization along the steep part of the loop” ³⁰⁾
• this list is not exhaustive and other names and definitions can be employed in the literature

Progress of the range of coercivities in magnetic materials ³¹⁾

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Coercivity is not a directly important parameter for soft ferromagnets as such, because for them permeability is typically a more pertinent figure of merit. However, there is an inverse relationship between the two quantities, because higher coercivity increases the width of the hysteresis loop and therefore reduces permeability and increases power loss.³²⁾

Therefore, a low coercivity is a pre-requisite to obtaining high permeability, and vice versa. The materials which have the highest permeability μ_r also have necessarily very low coercivity H_C, as listed in the table below.^33)34)35) Especially for the initial permeability, there is a reciprocal relationship between the permeability and coercivity μ_i ~ 1/H_C.³⁶⁾

The value of coercivity is also closely and directly linked with hysteresis loss especially under alternating excitation. It is therefore critical that for a low loss material its coercivity has to be a low value.

Yet, for power transformation it is the power loss that is typically specified, not coercivity. Only in some special signal application (such as fluxgate sensors or impulse transformers) coercivity might be a directly relevant quantity. This parameter is also important for semi-hard magnetic materials used in magnetic data storage, because it dictates the immunity to external magnetic fields so that the data can be safely retained once recorded in the magnetic structure.³⁷⁾ Obviously, coercivity dictates the effort required to magnetise the material, so that the information could be recorded in the first place (and erased or re-recorded as needed).

For hard ferromagnets the value of coercivity is one of the critical parameters, and thus development of new magnets is often focused on increasing its value.³⁸⁾³⁹⁾

Indeed, development of modern magnetic materials is directly related to the “mastery of coercivity”,⁴⁰⁾ because appropriate control of the relevant coercivity mechanisms allows obtaining better magnetic materials, whether they are magnetically soft, semi-hard, or hard. Typically used magnetic materials have their saturation magnetisation spanning only around just one order of magnitude, but currently coercivity spans across eight orders of magnitude.

Properties of soft ferromagnets: saturation, permeability, frequency range, material cost⁴⁸⁾

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Properties of semi-hard ferromagnets: remanence, coercivity, and energy density⁴⁹⁾

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Properties of hard ferromagnets: remanence, coercivity, temperature range, material cost⁵⁰⁾

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Coercivity H_c is expressed in the same units as magnetic field strength H. In engineering applications, in the SI system this is typically in A/m (ampere per metre), and in the CGS system it is in Oe (oersted).^51)52)53)

However, in some physics experiments the magnetic field is sometimes expressed in the same units as magnetic flux density, namely T (tesla), by scaling the values by the permeability of vacuum.⁵⁴⁾

The CGS units are still widely used in the physics experiments and thus they are still frequently used in scientific papers as well as in industry. Especially in the permanent magnet nomenclature, CGS units remain very popular.⁵⁵⁾⁵⁶⁾ For example, the classification of permanent magnets is based on their energy level, which is expressed typically as a grade of the magnet “N52”, where the value “52” denotes the amount of maximum magnetic energy in MGOe (mega-gauss-oersted), and this energy is proportional to the coercivity. However, in the CGS system attention needs to be paid to the 4π constant, because magnetisation can be expressed with our without it, thus changing the values by an order of magnitude. In the SI system there is no such caveat.

Coercivity depends not only the chemical composition of the material, but also on the physical state of that material. For example for iron, its very pure monocrystal can have coercivity around 1 A/m, in the polycrystalline state it could be 100 A/m, and for single-domain iron particles of an appropriate size it could be even beyond 10 kA/m ⁵⁷⁾, so spanning over 5 orders of magnitude, even though all these materials are made of pure iron.

Changes in magnetisation state of a material can occur as a result of domain wall motion (including domain nucleation) or domain rotation. Therefore, any factors which can impact on these processes can also affect the coercivity. Domain wall movement can require less energy than rotation, and therefore as a general rule for high-coercivity materials the state of a single domain is preferable, whereas for low coercivity (high permeability) free domain wall motion is required.⁵⁸⁾

The underlying physics of coercivity is extremely complex, with various magnetic energy components competing, depending on the volume of particles, presence of domain walls, crystallographic defects, non-magnetic inclusions, etc. Because of this complexity there is no single mathematical model of coercivity valid for all materials, and some simplifications are applied so that mathematical functions can be defined for simplified systems.⁵⁹⁾

There are several coercivity mechanisms which are widely discussed in the literature, with the major ones related to:⁶⁰⁾

• domain wall motion (impeded by various effects)
• anisotropy (due to various terms), and magnetisation reversal
• domain nucleation (due to small particle limit)

Many complex mathematical models of coercivity were presented in the literature over the years. The physics of the phenomena are complex, and especially for permanent magnets it is difficult to be certain about all the components that contribute to coercivity. Magnets produced industrially (or even at a laboratory scale) attain only about 10-20 % of the values calculated from the theoretical considerations, whether for well-established technologies⁶¹⁾ or for new types of materials,⁶²⁾ and many researchers and scientists internationally work on improving the understanding of the physics of coercivity.

Domain wall pinning is one of the mechanisms that impacts domain wall movement.⁶³⁾ Any non-uniformities in the magnetic materials such non-magnetic voids, crystal defects, grain boundaries, inclusions, or precipitates can give rise to a local energy increase which must be overcome before the domain wall can move through such an entity. A large entity will create “strong pinning”, but a group of small entities (e.g. each of which is significantly smaller than the domain wall width) may give rise to “weak pinning”.⁶⁴⁾

Under a static condition a domain wall can be “pinned” to such location, or that such defects are “pinning sites”.⁶⁵⁾⁶⁶⁾ There is actually a physical evidence of such pinning, because the domain wall movements happen in a jerky motion (rather than smooth and continuous) and this can be detected for example through the Barkhausen noise phenomenon.

In the absence of other factors a domain wall will rest in some position of a local energy minimum, such as being “stuck” at one of the pinning sites. And for example when some external magnetic field is applied there will be magnetic pressure exerted on the wall, which when increased sufficiently high will make the wall overcome the pinning force and the domain can rapidly accelerate towards the next pinning position (Barkhausen jump event). The process is then repeated, with the next local jump, and so on. If there is a lower local energy minimum just after a larger one then the wall may jump over it.

Simplified animation of a domain wall crossing a non-magnetic void, ⁶⁷⁾ at the non-magnetic defect the domain wall will experience some pinning force

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Evolution of micro-hysteresis J-H loop due to local energy of domain wall pinning ⁶⁸⁾

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However, when the direction of the applied magnetic field is reversed, then different pinning sites might be involved in the process, so that effectively a different path is generated, as shown in the image below. Each such Barkhausen jump represents rapid change in magnetisation which generates local micro eddy currents, which dissipate energy, so the process is lossy in its nature (hysteresis loss).

Magnetic domain structure (lancet combs switching during magnetisation) in high-permeability grain-oriented electrical steel. Wide bar domains are visible in the upper part. The different domain structures are separated by a ground boundary. ^{Copyright © Oles Hostanar}

These individual energy losses add up, and if the magnetisation process is cyclic in one direction then a loop is formed, called hysteresis loop. As illustrated in the drawing of the evolution of micro-histeresis, the position of the coercivity and remanence points on such a loop can be easily identified by extracting the relevant zero crossings J = 0 and H = 0, respectively.⁶⁹⁾ Thus coercivity is related to the “strength” of the energy minima dictating the domain wall movements in a given material, especially applicable to the soft magnetic materials.⁷⁰⁾

In most soft magnetic materials the magnetic domain structure is extremely complex, for example due to angular misalignment of local grains in the polycrystalline structure, or when the applied magnetic field is at some angle to the easy magnetisation direction.⁷¹⁾

The re-organisation of the domains occurs by domain wall motion, and this can be impeded also by the position of grain boundaries, which by definition involve local atomic “dislocations” in the crystalline structure. Such defects give rise to additional magnetostatic energy which has to be minimised, and which contributes to coercivity.

However, the grain size itself dictates some energy conditions on the domain walls, and it was shown experimentally that the domain structure is affected by the grain size or sample thickness.^72)73)74)

Therefore, several energy terms play a significant part and can behave in an “additive” way. This approach has limited validity only for soft magnetic materials⁷⁵⁾, and the “additivity” property of the energy due to H contributions is the basis of the widely used Jiles-Atherton model for the hysteresis loop.⁷⁶⁾

Coercivity H_C vs. grain size D for various soft magnetic materials⁷⁸⁾

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Because the different energy terms interact, the changes of the physical state (e.g. the size of grains) of the material can significantly influence the resulting coercivity. A well-known graph (illustrated here) collating results of several materials shows that below certain grain size (around 20 nm) the local domain wall movement is no longer impacted significantly and thus very low coercivity (and very high permeability) can be obtained for the nanocrystalline and the amorphous magnetic materials.⁷⁹⁾

As the grain diameter D increases, the coercivity becomes its very strong function so that H_C ~ D⁶, reaching the maximum at around 100 nm size. This property is exploited in high-energy permanent magnets which are assembled from fine-particle powders, for example by the sintering process,⁸⁰⁾ or by growing grains of appropriate size in alloyed magnets such as Alnico.⁸¹⁾

For large grains, their boundaries occupy proportionally less volume of the material, but inside of each grain behaves akin to a monocrystal. Therefore, large grains can also beneficial for lowering coercivity and this property is exploited in grain-oriented electrical steel, in which a single grain can be as large as 20 mm. However, even though static coercivity can be low, too large domains eventually lead to excessive additional loss due to fast domain wall movements, and for this reason the domain wall refinement techniques are employed, in order to reduce the domain widths in large-grain grain-oriented electrical steels. This allows achieving even higher permeabilities (and thus lower AC coercivities).⁸²⁾

Magnetic anisotropy means different magnetic properties as measured in different directions. Such directional difference can arise due to many factors, related both to intrinsic material property as well as due to the sample shape property.⁸³⁾ Therefore, there can be magnetostatic terms of energy due to crystal anisotropy, as well as due to sample anisotropy (including the shape of grains).

For example, if there is an easy axis of magnetisation, then the material or sample with the magnetisation M aligned with the easy axis will have a minimum energy stored. But if M is deflected by some angle (e.g. due to external H applied at some angle) then the anisotropy energy density E_a will depend on the angle of deflection, and can be expressed as:

In soft ferromagnets anisotropy increases coercivity, but in hard ferromagnets various anisotropy terms can reduce the maximum attainable coercivity, and mathematical models show that the limit is at the order of:

An anisotropic particle in the Stoner-Wohlfarth model ⁸⁶⁾

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The simplest analytical model of an anisotropic particle is the Stoner-Wohlfarth model, which describes an ellipsoidal particle with uniform magnetisation (i.e. with coherent rotation). The total energy density of such a system can be described with the equation as below, with the first term being similar to the anisotropy energy density mentioned above.

Normalised magnetisation loops due to the Stoner-Wohlfarth model, for several angles of applied field for a single particle, and for a large array of randomly oriented particles ⁸⁸⁾

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Minimising this equation with respect to θ can give rise to either one or two minima, depending also on other factors (such as particle size and the amount of applied field).⁸⁹⁾⁹⁰⁾ Magnetic hysteresis (and thus coercivity) arises under the condition with two minima, and the direction of magnetisation switches abruptly (reverses by 180°) at the points where $d^2E/dθ^2 = 0$, with the strength of the applied “switching” field H representing the resulting coercivity H_C of such system - and it is equal to the coercivity limit as expressed by the equation above.⁹¹⁾

With H applied at α = 0° the switching is abrupt and the hysteresis loop is rectangular. But if the α angle is different then the loop shape becomes rounded, and the resulting coercivity is reduced, due to the effect of the shape anisotropy, which can be also expressed by the demagnetising factor N, which leads to the following equation as shown below.

In a hypothetical arrangement of a large array of such particles oriented randomly (but neglecting particle-particle interactions) the resulting hysteresis loop is a mixture of all the contributions, thus resulting with an “average” shape of the loop for such an array. The resulting coercivity of such collection is H_C = 0.482·H_C,limit.⁹²⁾

The Stoner-Wohlfarth model describes only coherent rotation of magnetisation, i.e. such in which all magnetic moments of all atoms are always parallel to each other. In reality this is rarely the case and other modes are possible, in which the angles of the individual moments can change gradually, even ending up in a circularly closed domains, or at least with some amount of “curling”.⁹⁴⁾ Such models are typically investigated computationally (micromagnetics), on large clusters of individual magnetic moments, because the moment-to-moment interactions cannot be neglected and they strongly influence the outcome.⁹⁵⁾ Micromagnetics-type calculations cannot be performed for large-volume samples because due to the number of nodes the computation becomes prohibitively too expensive even for supercomputers. For this reason simplified “global” analytical approaches are often employed, such as Jiles-Atherton model or various homogenisation procedures (especially in finite-element modelling).

Coercivity is linked to reversal of the state of magnetisation, which can occur by reversal of the magnetic domains, or by domain wall movement. If there are multiple domains then initially the movement of domain walls can be the process which involves the least amount of energy, as compared to other mechanisms. Domain rotation typically requires larger effort.

However, in a hypothetical state in which the whole material volume is magnetised to a single-domain state there are no domain walls which can be moved. Therefore, the energy minimisation must occur either by rotation, or by first nucleation of an opposing domain, so that a new domain wall is generated and can move in order to reduce the magnetostatic energy.⁹⁶⁾

Energy consideration show that sufficiently small particles will prefer to remain in a single-domain state, because creation of a domain wall (due to nucleation of another domain) will expend more energy, for example as compared to “curling”, “buckling” and other mechanisms. For a hypothetical spherically-shaped particle with demagnetising factor N = 1/3 and cubic anisotropy the maximum radius for the single-domain particle can be estimated as in the equation below. So if the particles are kept small enough they can remain in a single-domain state and thus the coercivity can be increased by prohibiting nucleation of smaller additional domains. This mechanism is exploited at least partially in high-energy magnets such as sintered NdFeB, in which the grains have size typically of the order of 10 μm, but in nanocomposite magnets the grains can be as small as 25 nm.⁹⁷⁾

In a larger structures, large non-magnetic inclusions or surface defects can act as the domain nucleation sites, because the non-magnetic discontinuity will introduce certain amount of “curling” or “reversing” in the local magnetic moments, due to local field enhancement. Once a sufficiently large volume of a reversed magnetisation is created (of the order of $V ≈ d^3_w$, where d is domain wall width), then effectively a domain wall has been created locally, and can begin to travel outwards, thus growing the new domain at the expense of the original domain, if the magnetic pressure allows and if there are no other sufficiently strong pinning sites to stop it.

Proximity sensor with an elongated Alnico magnet

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Magnetisation curve is affected by temperature, and all ferromagnetic materials become paramagnetic at sufficiently high temperatures (above their Curie temperature). Under quasi-static excitation (DC) for all paramagnets coercivity is zero, but remanence is also zero (i.e. there magnetisation curve passes through B = 0 H = 0), and therefore no magnetostatic energy is stored inside such material (even though at lower temperature it could be stored if some ferromagnetism is developed).

For ferromagnets below their Curie temperature, as the temperature is elevated typically the coercivity reduces. In hard magnetic materials this means reduction of the amount of stored magnetic energy and easier self-demagnetisation, not only due to thermal excitation, but also due to the magnetic field existing in a given magnetic circuit. This is why weaker magnets such as Alnico cannot be used as very short or flat items (magnetised through the short axis), because the self-demagnetising field would be strong enough to partially demagnetise the magnet (irreversibly), due to moderate value of the magnet's coercivity.

This is not the case for high-coercivity magnets such as NdFeB which can be made as flat disks without the danger of self-demagnetising under normal operating temperatures. However, when the temperature is elevated the coercivity drops significantly.^99)100)

In soft ferromagnets coercivity depends less strongly on temperature, but it also reduces.¹⁰¹⁾ This leads to an interesting phenomenon that the initial permeability increases when approaching the Curie temperature point, even though the saturation magnetisation decreases.¹⁰²⁾

In high-energy magnets coercivity reduces significantly more rapidly with temperature than remanence, but in magnetically soft materials the changes of a comparable order of magnitude for both of these parameters.

Changing H_C for soft ferrite 3C90 at 25°C and 100°C ¹⁰³⁾

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Changing _JH_C and _BH_C for permanent magnet VACODYM 745 HR between 20°C and 120°C ¹⁰⁴⁾

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In soft ferromagnets, coercivity is not a directly important parameter, so typically it is not quantified over temperature. But for hard magnetic materials coercivity is critical, and therefore the changes vs. temperature are quite well detailed in the relevant datasheets, and these are an important characteristics that need to be taken into account during magnetic designs.¹⁰⁵⁾ The table below summarises typical values of temperature coefficients TK for B_R and _JH_C, for most frequently used permanent magnets.

It should be noted that a coefficient as high as -0.6 %/°C means that increasing the temperature by 100 °C would reduce the coercivity by as much as -60 %, which could render the given magnet completely unsuitable for its application, either because the operation of the device could be no longer maintained, or that the magnet could be irreversibly demagnetised (as described below). Therefore, hard magnets have typically their maximum operating temperature specified, and this value is much lower than their Curie temperature.

Effect of reversible (nominal, nom) and irreversible (irr) operation of a permanent magnet on coercivity _BH_C and remanence B_r due to temperature (T₁, T₂) and demagnetising field caused by different magnetic load (a,b) ¹⁰⁹⁾

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With slight increase in temperature the reduction of coercivity can be reversible, so that upon reduction of temperature the coercivity can increase back to its nominal or previous value. However, with excessive changes, such that the knee of demagnetisation in the B-H or J-H curve is exceeded, either directly by the thermal changes or by the demagnetising field then irreversible demagnetisation can occur.¹¹⁰⁾ As illustrated, at lower temperature T₁ the magnet can safely operate over the entire range 0 > H > _BH_C, and all the changes of the operating point will be completely reversible, regardless the position of the magnetic load line (a,b).

However, at elevated temperature T₂ (or for lower-energy magnets with non-linear demagnetisation curve), only operation above the load line “a” is safely reversible (above point ₂P_a). But if at this temperature the load line “b” is applied, then the operating point is pushed beyond the knee of demagnetisation and the material will demagnetise proportionally to the depth of that incursion, thus shifting to a new curve, parallel to the nominal, but proportionally lower. This will irreversibly reduce the remanence from the nominal B_r,nom to the new B_r,irr. If the temperature is lowered from T₂ back to T₁, then also the original curve will be modified, and the coercivity will also be impacted irreversibly. Such changes effectively mean that less energy is stored in the magnet, so the value of (BH)_max is lowered accordingly.

If no chemical and physical changes occurred in the magnet then it is theoretically possible to magnetise it back to the nominal state, but this is impractical as the magnetic circuit would have to be disassembled, so typically this is not carried out for industrial devices.

Structural changes will typically lead to irreversible changes and reduction of coercivity which cannot be restored by simple re-magnetisation.

For uniform materials coercivity is symmetrical because the hysteresis loop is symmetrical. However, asymmetry of coercivity values can be induced, especially in multi-layer materials such as spin valves.^111)112) The asymmetry arises because the magnetically harder phase can introduce an additional biasing field which must be compensated for by the externally applied field. In a single-layer structures additional anisotropy terms can be introduced by annealing in a magnetic field.¹¹³⁾

In most magnetic materials used for energy transformation (rather than information processing or storage) the values of coercivity are symmetrical.

Asymmetric hysteresis M-H loops (with asymmetric coercivities) for multi-layer materials (magnetoresistive sensors) ^{114) by P.P. Sharma, E. Albisetti, M. Monticelli, R. Bertacco, D. Petti, CC-BY-4.0}

M-H loops of Co particles cooled with and without H ¹¹⁵⁾

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M-H loops of an Ni-Mn alloy cooled with and without H ¹¹⁶⁾

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General procedure for measuring coercivity ¹¹⁷⁾

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Coercivity of magnetic materials is measured by using a measurement system appropriate for the type of material magnetic (soft, hard, semi-hard). The particular procedures vary greatly, as dictated by the value of coercivity and the shape and size of the samples under test. However, from the conceptual magnetic viewpoint the procedure is similar in all cases, and can be broadly described as follows:

• Before the procedure, the material can be in any state of magnetisation (demagnetised, or magnetised, to any level, or in its remanence state).
• The first step (step 1 in the illustration) is to apply magnetic field which is high enough so that the state of magnetic saturation is obtained. The level of excitation is dictated by the type of material under test, and its expected value of coercivity. In any case, the applied saturating H must be much greater than the H_C to be measured, so at least H_sat > H_C but more accurately H_sat » H_C.
• After saturation, the excitation is reduced (step 2) to H = 0 (thus returning the material to the remanence point B_R = J_R, or M_R).
• Then a negative field is applied (step 3) progressively, so that the state of magnetisation begins to approach zero. The quantity of magnetisation (flux density B, polarisation J, or magnetisation M) is measured continuously during this phase. The value of H at which B, J, or M becomes zero represents the coercive field H_C for that value, namely _BH_C, _JH_C, or _MH_C.
• Measurement of remanence coercivity H_C,R is more complicated, because it can only be quantified after reducing the applied negative field back to zero (step 4) and checking if the remnant flux density, polarisation, or magnetisation is zero. If the curve does not return to (0,0) then the saturating procedure should be repeated and a different level of negative field needs to be applied, etc.
• Coercivity can be measured in both directions, for positive or negative applied field. The procedure is identical in both cases, but simply performed with the opposing magnetic polarity, or reversed position of the sample in the measurement system (and the value of H_C,R may be then averaged from both directions, for symmetrical materials).¹¹⁸⁾

Influence of air gap (l_g = 0.07 mm) on the shape of B-H loop for a cut core (l_c = 250 mm), so ratio l_g / l_c = 0.00028

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Permeability values of typical soft magnetic materials are high and for lower fields the difference between the flux density B and the polarisation J is negligible in practice. Therefore, there is no need to define or measure two separate values, because around the coercivity values it is also true that B ≈ J, and consequently H_C ≈ _BH_C ≈ _JH_C.

Furthermore, the presence of air gap in a magnetic circuit of the sample leads to significantly lower effective permeability so that the B-H loop becomes “sheared” (slanted). However, the sheared loop (B-H, J-H, or M-H) still crosses the horizontal axis at the point which is in practice negligibly close to the loop for the closed magnetic circuit (i.e. without the air gap present). This allows measurement of coercivity of soft magnetic materials on open samples, as described for instance by the international standard IEC 60404-7 Method of measurement of the coercivity (up to 160 kA/m) of magnetic materials in an open magnetic circuit.¹¹⁹⁾

Coercivity H_c in soft magnetic materials is measured after saturating, switching the excitation off (H = 0), and then reversing the applied H into the second quadrant

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The excitation is applied in a typical way as described above, by first saturating the sample of material to some suitable value (e.g. 200 kA/m for highly permeable materials)¹²⁰⁾, then reducing the field to zero, and then applying negative field until the measured B reduces to zero.

IEC 60404-7 ¹²¹⁾ specifies the relationship between the induction coercivity _BH_C and the polarisation coercivity _JH_C by the equation below, which for high-permeability materials reduces to negligible difference between the two values.

For example, if the slope of differential permeability around the zero cross is μ_diff = 100 then the difference between the two values will be 1 %. So for more permeable materials this difference is indeed negligible in practice.

According to IEC 604040-7, sample under test should be ideally of elongated shape (ratio of length to width 5:1 or more) and it should be placed in the axis of the demagnetising solenoid.¹²³⁾

Setup for measurement of coercivity in soft and semi-hard magnetic materials:¹²⁴⁾ 1) sample, 2) saturating and demagnetising solenoid, 3) inner radial off-centre Hall-effect sensor, 4) external radial differential fluxgate sensors, 5) vibrating coil (orange lines - magnetic field due to solenoid, red lines - magnetic field due to magnetised sample)

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It should be noted that the absolute accuracy of the B measurement is not important, because the important information is the location of the zero crossing. Therefore, the B measurement can be optimised as a zero-detector. The zero crossing can be detected by several means as illustrated for sample (item 1) in a solenoid (item 2):

• Hall-effect sensor (item 3) can be placed near one end of the sample, but in such a way that it is off the sample axis and configured to detect the radial component (indicated by the blue arrow).
• Pair of fluxgate sensors (item 4) can be placed outside of the solenoid. The sensors should be configured to measure the radial field (green arrows), making them a differential pair which will suppress effects of some unwanted external far field. For best results the sensor should be placed near the centre of the solenoid, and the sample end should coincide with the location of the sensors.¹²⁵⁾
• Vibrating coil (item 5) can be placed near or around one end of the sample, to measure the axial component (purple arrow).

Practically sufficient level of magnetic 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 with appropriate closure yokes if necessary.¹²⁶⁾

In commercial systems saturating soft magnetic materials is achieved typically with a field 140-450 kA/m, which can be obtained by DC current from a variable DC power source, or by pulse methods from an LC circuit.^127)128) The saturating solenoid can be enclosed in a shielding box.¹²⁹⁾

For physically large and electrically conductive samples there can be significant eddy currents and the saturating field should be applied for sufficient long time for these currents to die away and for the field to penetrate and to saturate also the inside of the sample. The standard IEC 60404-7 suggests that in some cases the saturation dwell time might be up to 20 seconds, especially for thick or large samples.¹³⁰⁾ The standard allows using permanent magnets as the source of magnetic field for saturation.

Hysteresisgraph measurement system for measuring magnetically closed samples, capable of measuring coercivity and remanence

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For magnetically closed samples the measurement requires the least amount of excitation in the sense of the magnetising current that needs to be supplied to the sample under test.

An example of such measurement system is shown in the photograph. The specimen under test is not shown, and can be in the form of a toroidal sample, or a bar enclosed in a suitable magnetising yoke.

At the top there is the display with the measured hysteresis loop, from which the coercivity and remanence points (zero crossings) can be extracted.

On the first shelf from the top are the Lakeshore fluxmeters, which are used for performing analogue integration of the induced secondary voltage (in the secondary winding), so that the accurate instantaneous flux density can be measured. The other fluxmeter can be used as a gaussmeter if a Hall-effect sensor is used.¹³¹⁾

On the second shelf is an Agilent digitising voltmeter, which is used for measuring the voltage drop across a precision shunt resistor, connected in series with the primary winding, so that the instantaneous primary current can be measured. For magnetically closed samples the magnetic field strength H can be calculated from the primary current.

On the third shelf is a Kepco DC digital power amplifier, which varies the applied DC current as dictated by the digital signal communicated by the controller in the system.

On the bottom shelf is the controlling computer, which dictates the changes in the primary current, collects all the measurement data, processes and displays the measured values, and interacts with the operator.

For high-energy permanent magnets the required measurement (and saturation) field might be even higher than H > 1 MA/m, so it is technically more difficult to apply such fields, but otherwise the procedure is similar to soft ferromagnets because sufficiently large current has to be passed through the magnetising/demagnetising coil. Appropriate values of H, B, J, M have to be measured accordingly to establish the zero crossings.

In permanent magnets | _BH_c | < | _JH_c |

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In soft magnetic materials, the value of H_c 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 $H_c$.

In hard magnetic materials the value of H_C is relatively high, the higher the better. The contribution of $μ_0 · \vec{H}$ is no longer negligible and thus two H_C points have to be distinguished:¹³²⁾

• at B = 0 the coercivity value is _BH_c
• at J = 0 the coercivity value is _JH_c
• and such that: | _BH_c | < | _JH_c |

It should be noted that for a high-energy material such as sintered NdFeB magnet at the point _BH_c there is no significant magnetisation reversal, so the energy stored in such magnet is not impaired. This is because only B reduces due to the increasing μ₀·H component, but J or M remain practically at their saturation level (this might not be the case for elevated temperatures, for which the immunity to demagnetisation is compromised).

However, the point _JH_c is by definition when J or M reduce to zero, so significant irreversible demagnetisation will occur, comparable almost to a fully demagnetised state. If not damaged (thermally, chemically, or physically) then such magnet can be still re-magnetised back to the full energy.^133)134)

Concept of pulsed measurement ¹³⁵⁾

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It is possible but difficult in practice to obtain full saturation of magnets with solenoids driven by a DC current, because of the resistive losses in the conductor of the solenoid. If the slowly varying current is used then the measurement is similar to as described above for the soft ferromagnets.¹³⁶⁾

Less expensive methods used in production and testing employ pulsed currents. A capacitor is charged to a controlled level of voltage and then discharged through the magnetising coil (via a diode to stop oscillations). This produces a uni-directional pulse of very high current which is capable of magnetising, saturating, or demagnetising a magnet.¹³⁷⁾ Magnetising yokes can be used to focus and direct the magnetising/demagnetising field.

Widening of the B-H loop caused by eddy currents in pulsed measurement can be utilised in the extrapolation-to-DC method ¹³⁸⁾

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Pulses can be applied by successive demagnetisation, or by always returning to the same saturation point, synonymous with first-order reversal curves (FORC).¹³⁹⁾ Varying the level of impulse current (for example by changing the voltage to which the impulse capacitors are charged) allows for repetitive impulses to be applied, and for the relevant parameters to be extracted either by a directly measured data, or by extrapolation of the curves.

Extrapolation is required because a shorter or faster pulse represents higher frequency and the measured loop is widened due to the global eddy currents induced in the body of the magnet. If it is not possible to test under quasi-static conditions, then pulses of different duration can be used, and the extrapolation to “zero frequency” can be performed from the data measured at higher effective frequencies (due to limited pulse width). ¹⁴⁰⁾

Coercivity discussed in the papers of theoretical physics is often implicitly assumed to be measured under DC-like (quasi-static) conditions. However, the same definition of coercivity can be applied to measurements performed under AC excitation.

B-H loop for conducting non-magnetic material under AC excitation, with wider loop at higher frequency ¹⁴¹⁾

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B-H loop for conducting but non-magnetic material under quasi-DC excitation ¹⁴²⁾

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Any conducting material exposed to varying magnetic field will have some eddy currents induced in it. These eddy currents will contribute to power loss, which for sinusoidal excitation will be manifested as an elliptically shaped B-H loop, which will be wider at higher frequency. The loop area will be directly proportional to the dissipated loss. Such loop will exhibit a different rising an falling branches, for which the “coercivity” value can be interpreted, using the same definition as for the static excitation, from the zero crossings. However, for such a material there is no inherent energy storing (only energy dissipation), and this can be easily discerned if a quasi-static measurement is performed, because the lower the excitation frequency the lower the value of apparent coercivity that will be detected, all the way to zero when extrapolated to zero frequency - if the material does not exhibit any ferromagnetism.

The value of apparent coercivity can be easily extracted from such AC magnetisation loops, and this is frequently done in some applications, for example as used for non-destructive testing of materials, both magnetic and “non-magnetic”. However, measurement at DC-like excitation (or extrapolated to such conditions) is required to detect if the material possesses some energy-storing capabilities.

Most of the discussion of coercivity in the literature is implicitly carried out with the assumption that the excitation or analysis is uni-axial, or that the saturation is applied at some specific direction, and demagnetisation is applied in the same (or similar) direction, but with opposite sense.

By definition, the coercivity point is such a value of H at which B = 0, J = 0 or M = 0, so the problem simplifies in practice to detect the zero crossing of these curves: B-H, J-H, or M-H, respectively. But there are certain modes of magnetic excitation in which the vectors can be held at an always-positive amplitude, such us under the rotational magnetisation.¹⁴³⁾

Rotational magnetisation with the B vector loci controlled to follow a circular path (of a constant amplitude) ¹⁴⁴⁾

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Corresponding B_x-H_x loops decomposed from rotational magnetisation ¹⁴⁵⁾

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Corresponding B_y-H_y loops decomposed from rotational magnetisation ¹⁴⁶⁾

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Example of 3D magnetisation curves ¹⁴⁷⁾

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Therefore, if there is no point at which the amplitude of B = 0 then proper “coercivity” point cannot be established. It is possible to decompose such rotational magnetisation into orthogonal X and Y components (as shown in the graphs), for which such zero crossings could be identified. However, these would only represent some “apparent coercivity”, rather than a value that is indicative of energy storing capability.

Similar limitation applies to other complex modes of magnetisation, with arbitrary 2D or 3D paths. For example, under 3D magnetisation an arbitrary virtual “plane” can be established and the vector of a constant amplitude can be rotated within that plane. This would still not produce a “zero crossing” event which could be used in the sense of the proper definition of the coercivity. Again, decomposition into directional loops could be carried out, but any such apparent coercivity will represent a completely different value from the “proper” uni-directional measurement, even under quasi-static conditions.

However, the 2D and 3D measurement systems can be run in a mode in which alternating magnetisation (uni-directional) is generated at a single specified direction. And for such case it is then possibly to apply the conventional definition and meaning of coercivity (i.e. zero crossings).

Coercive materials can be modelled in electromagnetic simulation software, for example by employing the finite-element modelling approach.

The internal stored magnetisation can be mathematically approximated as a current-carrying solenoid, whose outer envelope of the shape is synonymous with the shape of the magnet. By utilising several simplifying assumptions about the shape and position of the magnetisation curve, in combination with the coercivity value, an iterative solution can be calculated, which approximates the behaviour of a permanent magnet in a magnetic circuit (e.g. FEMM software).¹⁴⁸⁾ With more complex definitions in more capable software (e.g. COMSOL software) it might also possible to address non-linearities pertinent to lower-energy magnets or additional effects such as eddy-current heating.¹⁴⁹⁾

The area of a hysteresis loop represents power loss, and is related to coercivity

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The power loss dissipated in a given magnetic material is proportional to the area of hysteresis enclosed by the B-H loop. Therefore, the width of the loop is almost directly proportional to the value of coercivity of a given material, and with all other parameters constant (such as B_R) the power loss P is almost directly proportional to the value of coercivity H_C so: P ∝ H_C.

But for the more flexible definition of coercivity, if any amplitude is taken into account, then the changes of loop area are proportional to the changes of H_C and B_R, namely to their product. But increasing amplitude of excitation tends to increase proportionally both of these values (for a family of B-H loops at the same frequency), and therefore the power loss is roughly proportional to the square of coercivity, so: P ∝ H_C². ¹⁵⁰⁾

This property is more important for soft ferromagnets, for which the power loss is a directly more applicable parameter than the coercivity as such.

In hard ferromagnets coercivity is critical for storing large amount of magnetic energy (BH)_max. For high-energy magnets their relative permeability is close to unity, and when simplifications and linearisation is assumed then a maximum theoretical value of stored energy can be estimated, by using the practically achievable coercivity and saturation magnetisation. ¹⁵¹⁾

In high-energy permanent magnets the maximum energy density E_max = BH_max is at the point P denoting half of the straight line in the second quadrant

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For costs and efficiency reasons, volume of magnets is typically minimised in magnetic devices. It follows that the system should be designed to operate at the most efficient point, from the magnet's viewpoint (which is typically the most expensive part). This point is where the (B·H) product becomes a maximum, typically referred to in the literature as: (BH)_max.

For high-energy magnets such as NdFeB, the demagnetisation curve in the second quadrant is linear. Therefore, the maximum value of the (B·H) product is the product of B_r / 2 and _BH_C / 2 because this results the largest area (green rectangle in the illustration), as given in the equation above (scaled by the permeability of vacuum). For lower energy magnets (Alnico, ferrites) the demagnetisation curve is more non-linear, but the procedure is analogous.

Real or apparent coercivity H_C and remanence B_R can be defined for each hysteresis loop in the family of different amplitudes (even without prior saturation)

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As with remanence, coercivity can be defined as measured after saturation so that it has always the same value for the same sample, related only to the material properties, rather than the state of temporary magnetisation.

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.¹⁵⁵⁾ In such cases the value of “coercivity” is a function of the amplitude of excitation as well as the frequency, or even waveshape of the applied signals.

Such values can be then trended and used as an additional parameter which characterises the behaviour of the sample, giving further insights to the physics of the phenomena occurring in the investigated magnetic material.^156)157)

Coercivity is related to microstructure and this is exploited for non-destructive testing (NDT). Hardened surfaces exhibit increased level of hardness due to the specific physiochemical state, dissolved carbon structures, atomic dislocations, etc. This state gives rise to coercivity, which can be detected by electromagnetic means.

In some materials the relationship between hardness and coercivity is almost linear,¹⁵⁸⁾ which lends itself into very straightforward detection of the level of damage of the material.

Relation between Vickers hardness and coercivity in the P91 heat-resistant martensitic steel with different thermal ageing durations ^{159) by Y. Li, C. Sun, K. Liu, T. Xu, B. He, CC-BY-4.0}

However, due to significant difficulties with mathematical modelling of the underlying physics of the coercivity mechanisms, the changes of coercivity cannot be defined by means of absolute values, and therefore most such NDT techniques rely on detecting relative changes, as compared to some reference sample. If reference data is not available then it might not be possible to define the level of damage of the investigated material.

Also, the given investigated material might be undergoing various modes of being “damaged”. For example, for the steel as shown in the graph, the steel has the highest coercivity (and the highest hardness) in its nominal state, and exposure to prolonged high temperature causes reduction of coercivity.¹⁶⁰⁾

But a magnetic steel exposed to mechanical loading (inducing strain) will undergo changes to microstructure involving increased amount of crystal dislocations, which may increase coercivity.¹⁶¹⁾ This is why the reference data is required for NDT based on coercivity (as well as on other effects such as Barkhausen noise).

• Remanence
• Magnetic field strength
• Magnetic properties