Remanence - a value of magnetic flux density remanence B_r, magnetic polarisation remanence J_r, or magnetisation remanence M_r expressing the state of magnetisation after the magnetic excitation is removed. Remanence is typically measured after first magnetically saturating the magnetic material under test, and then returning the external excitation of magnetic field strength to zero, H = 0, namely to the point at which the hysteresis loop crosses the H axis.^{1)2)3)4)5)6)7)8)}

Position of induction remanence B_r equal to polarisation remanence J_r on the B-H loop for hard magnetic materials⁹⁾

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Flux density remanence or induction remanence B_r equal to polarisation remanence J_r in soft magnetic materials is measured after saturating, and then switching the excitation off (H = 0)

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There are multiple names of remanence used in literature, such as:

• remanence ¹⁰⁾
• remanent flux density ¹¹⁾¹²⁾
• magnetic induction at remanence ¹³⁾
• remanent induction ¹⁴⁾
• remnant induction ¹⁵⁾
• remanent magnetic polarisation ¹⁶⁾
• remanent magnetisation ¹⁷⁾
• residual induction ¹⁸⁾
• retentivity ¹⁹⁾
• remanent magnetic field ²⁰⁾
• remanent state ²¹⁾
• remanent magnetism ²²⁾
• intrinsic remanent magnetization ²³⁾
• intrinsic remanent induction ²⁴⁾
• residual induction ²⁵⁾
• remanent induction flux ²⁶⁾
• residual flux ²⁷⁾

Also, symbols such as B_r and B_R are used interchangeably, and all of these values typically refer to the same point of H = 0 on the hysteresis loop.

However, there can be some differences in the exact definitions. For example, a distinction can be made between remanence which is measured after prior saturation, and remanent flux density / induction / polarisation / magnetisation which denotes the remaining state of magnetisation after reducing the H to zero, from any level (whether from prior magnetic saturation or not). Therefore, the remanence can be the an upper limit of the remanent value²⁸⁾, but this definition is not necessarily applied consistently in the literature.

By definition, the remanence point is measured for H = 0, and therefore the values of remanent flux density (or remanent induction) B_r and remanent polarisation J_r are synonymous, both as to their numerical value and the units, and thus B_r = J_r, because the component μ₀·H = 0.

In the CGS units the remanent induction and remanent intrinsic induction are also equivalent.

The remanent magnetisation M_r expresses exactly the same concept and the same value, but it is scaled by the permeability of vacuum μ₀, so that the units of (A/m) are obtained.

In other words, the remanence represent the same point on any hysteresis loop: B-H, J-H, or M-H, because they all cross the same H = 0 point, even though the loops can differ significantly in other places.

Domain wall pinning is one of the mechanisms that impacts domain wall movement.³⁰⁾ 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”.³¹⁾

Under a static condition a domain wall can be “pinned” to such location, or that such defects are “pinning sites”.³²⁾³³⁾ 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, ³⁴⁾ 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 ³⁵⁾

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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.³⁶⁾

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.³⁷⁾

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.³⁸⁾³⁹⁾

Therefore, several energy terms play a significant part and can behave in an “additive” way. This approach has limited validity only for soft magnetic materials, 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.⁴⁰⁾

An anisotropic particle in the Stoner-Wohlfarth model ⁴¹⁾

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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). Such particle exhibits shape anisotropy, and the total energy density of such a system can be described with the equation as below, with the first term being similar to the definition of anisotropy energy density (described in more detail in: coercivity).⁴²⁾

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 ⁴⁴⁾

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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).⁴⁵⁾ 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.

With H applied at α = 0° the switching is abrupt and the hysteresis loop is rectangular, making the remanence M_r = M_sat. But if the α angle is different then the loop shape becomes rounded, and the resulting remanence is reduced, due to the effect of the shape anisotropy, which can be also expressed by the demagnetising factor N.

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 remanence of such random collection is M_r = 0.5·M_sat, and for a random orientation of such non-interacting particles within one plane the result is M_r = 0.637·M_sat.⁴⁶⁾

Remanence and coercivity discussed in the literature 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 ⁴⁷⁾

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B-H loop for conducting but non-magnetic material under quasi-DC excitation ⁴⁸⁾

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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 power loss. Such loop will exhibit a different rising an falling branches, for which the “remanence” 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 remanence and 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 remanence 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.

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

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As with coercivity, remanence 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 “remanence” as measured after other amplitudes or modes of excitation (e.g. varying frequency), for any B-H loop crossing the H = 0 level.⁴⁹⁾ In such cases the value of “remanence” 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.⁵⁰⁾⁵¹⁾

Several magnetic properties, including remanence, are correlated to changes in the material, such as micro-hardness or microstructure, for example as caused by stress/strain applied to the samples.⁵²⁾ Such techniques are employed for example in non-destructive testing of magnetic structures and samples.

In permanent magnets typically J_r ≈ J_sat

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For high-energy permanent magnets magnetic saturation is required during production in order to achieve the maximum energy stored in the magnet. Because of high relative squareness of the hysteresis loop, the remanence value is typically very close to the saturation point.

However, the required 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.

If sufficiently high excitation is not applied first, then the magnet might be incompletely magnetised, thus reducing the amount of stored energy, and the practically useful “strength” of the magnet. Conversely, if the magnetic is demagnetised (by electromagnetic or thermal processes) this also leads to reducing remanence.

It should be noted that for a high-energy material such as sintered NdFeB magnet up to the point _BH_c there is no significant magnetisation reversal, so the energy stored in such magnet is not impaired, even though this point may mean B = 0, which is 100 % less than B_r. This is because only total B reduces due to the increasing μ₀·H component, but J or M remain practically at their remanence (≈ saturation) level. This might not be the case for elevated temperatures, for which the immunity to demagnetisation is compromised. Lower-energy magnets such as Alnico are more prone to demagnetisation and therefore reduction of B_r.

However, for any magnet with low or high energy, the point J = 0 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.⁵³⁾⁵⁴⁾

Properties of hard ferromagnets: remanence, coercivity, temperature range, material cost ⁵⁵⁾

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Properties of semi-hard ferromagnets: remanence, coercivity, and energy density ⁵⁶⁾

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Several magnetic properties relevant to hard magnetic materials are also relevant to semi-hard magnetic materials. However, semi-hard ferromagnets are typically used for temporary storage of magnetic energy, rather than permanent. This means that they have to be tailored for a given application, so that the state of magnetisation can be removed or reversed when required, e.g. for magnetic data storage.⁵⁷⁾ Large values of remanence are mostly sought for, and the variability of “magnetic hardness” is typically controlled by the changes in coercivity.⁵⁸⁾

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. And since in an ideal magnet the remanence point is close to saturation, then the remanence B_r can be used instead in such calculations.⁵⁹⁾

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.

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 both the remanence and coercivity reduce. 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. Magnetic poles generated by an open magnetic circuit contribute to the demagnetising field, which then becomes proportional to the operating magnetic polarisation, which in turn is proportional to the polarisation remanence.

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 remanence and coercivity drop significantly,⁶³⁾⁶⁴⁾ to eventually vanish above the Curie temperature.

In soft ferromagnets coercivity depends less strongly on temperature, but it also reduces.⁶⁵⁾ This leads to an interesting phenomenon that the initial permeability increases when approaching the Curie temperature point, even though the saturation magnetisation and also remancence decrease.⁶⁶⁾

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 ⁶⁷⁾

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Changing _JH_C and _BH_C for permanent magnet VACODYM 745 HR between 20°C and 120°C ⁶⁸⁾

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In soft ferromagnets, remanence is not necessarily a directly important parameter, so typically it is not quantified over temperature. But for hard magnetic materials both remanence and coercivity are 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.⁶⁹⁾ 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 -0.2 %/°C means that increasing the temperature by 100 °C would reduce the remanence by as much as -20 %, 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) ⁷³⁾

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With slight increase in temperature the reduction of remanence can be reversible, so that upon reduction of temperature the remanence can increase back to its nominal or previous value.⁷⁴⁾ 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.⁷⁵⁾

As illustrated for a high-energy magnet, 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.⁷⁶⁾

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 remanence which cannot be restored by simple re-magnetisation.

All ferromagnets, including soft magnetic materials exhibit some amount of remanence, and very soft magnetic materials (with very low coercivity) typically demonstrate relatively high remanence. As shown below, extremely soft magnetic materials can have an almost completely square hysteresis loop. By itself this is not detrimental to the operation or efficiency, because energy storage and loss is related to the product of remanence and coercivity, and by definition the latter parameter is reduced to a low value.

Some effects occurring in soft ferromagnets are qualitatively similar to those in hard ferromagnets, as described in the previous sections.

As mentioned above, simple and “ideal” magnetic structures can exhibit rectangular B-H loop. This is the case for example for some Co-based amorphous wires.⁷⁷⁾ The inner core of such wire can behave as a single-domain structure, and once sufficiently high external magnetic field is applied then a rapid switching of magnetic polarity will take place.

Typical Co-Fe amorphous microwires

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Co-Fe amorphous core (red arrow) in glass coating (blue arrows and translucent tip), 0.1 mm diameter

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Single Barkhausen jump in a microwire creates a rectangular B-H loop ^{78) by T. Charubin, M. Nowicki, R. Szewczyk, CC-BY-4.0}

Squareness ratio expressed as M_r / M_sat can be a used as a parameter for quantifying this type of behaviour of a given material, and obviously this ratio must be close to 1 to achieve large remanence.⁷⁹⁾

Soft magnetic materials with square hysteresis loop can be used for magnetic amplifiers, saturable reactors, bi-stable switching devices, or power inverter applications.⁸⁰⁾⁸¹⁾ Squareness ratio 0.8-0.95 can be achieved for such materials.

Hard magnetic materials typically exhibit high squareness ratio, due to their inherent energy storing capability.

Geological materials such as rocks can carry “natural magnetisation” caused or affected by various geological, physical, chemical processes, also influenced by the Earth's magnetic field.⁸²⁾⁸³⁾

Examples of some names used in geological literature are shown in the table below. Accurate measurement of remanent magnetisation in rock samples allows inferring some of the geological process that took place when the rock was formed.

General procedure for measuring remanence and coercivity ⁸⁵⁾

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Remanence 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). Even if the materials is in its remanence state, the next steps would be executed, to ensure reproducibility of the measured results.
  • In certain cases prior demagnetisation is required, especially if the measurements are performed according to international standards.⁸⁶⁾
• 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 (or at least the highest peak value, as required by the procedure⁸⁷⁾). 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). By continuous sampling of the values the point at H = 0 can be measured, and this point represents the measured value of remanence.
• Optionally, for measuring coercivity in the same setup:
  • 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 (and remanence) 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).⁸⁸⁾

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.⁸⁹⁾

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.

This particular hysteresisgraph can be used for performing measurements under low-frequency AC as well as quasi-DC excitation, and from such results the values can be extrapolated to zero frequency.

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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The value of measured remanence is affected by the performance of the magnetic circuit, rather than just the type of magnetic material. The addition of air gap to the magnetic circuit creates localised magnetic poles and the demagnetising field associated with them. This leads to reduction of effective permeability of the sample under test, and in consequence the shearing of hysteresis loop.

As a result, such “sheared” (slanted) hysteresis loop has a substantially different remanence point, because the H = 0 crossing points are different. The coercivity point is largely unaffected, but the apparent remanent point is changed significantly.

Therefore, for samples which are intended to be measured in magnetically closed circuits it is necessary to minimise the air gap between the sample and the pole pieces. This might require appropriate grinding, polishing, or lapping of the mating surfaces.⁹⁰⁾

For small air gaps magnetic force can be proportional to the square of magnetic flux density.⁹¹⁾ This may generate significant holding forces between parts, so that it is difficult to pry them away from each other, even though the total amount of magnetic energy stored in the system is low (e.g. when soft magnetic materials are used). Because the presence of air gap reduces the flux density then a small air gap may be beneficial in such cases, because the mechanical force would be reduced at the expense of effective permeability of the magnetic circuit, but the power loss would be unaffected in the first approximation.

• Coercivity
• Magnetic saturation