Coercivity, magnetic coercivity, coercive field, or coercive force, typically denoted by H_C - such a value of the externally applied magnetic field strength H that reduces the instantaneous state of magnetisation of a material to zero, typically from the initial point of remanence B_R.^1)2)3)4)5)6) Because of its position on the B-H loop (2nd quadrant) the value of coercivity is typically stated as a negative number.

Position of induction coercivity _BH_C on the B-H loop and polarisation coercivity _JH_C on the J-H loop for hard magnetic materials⁷⁾

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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: soft magnetic materials (H_C < 1 kA/m), hard magnetic materials (H_C > 100 kA/m), and semi-hard magnetic materials (1 kA/m < H_C < 100 kA/m),⁸⁾ but this is not a strict classification as the function, application, or application of the given material can dictate the actual “type”.⁹⁾

Reaching coercivity (i.e. where 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.¹⁰⁾¹¹⁾

The “state of magnetisation” can be expressed in different ways and hence there can be several types of coercivity, which can be used interchangeably depending on the context:

• for magnetic flux density B (also known as “magnetic induction”), so for the B-H loop it is the induction coercivity _BH_C
• for magnetic polarisation J (so J-H loop) it is the polarisation coercivity _JH_C
• for magnetisation M (so M-H loop) it is the magnetisation coercivity, which could be marked as _MH_C or H_CM ¹²⁾
• in CGS units the polarisation can be referred to as “intrinsic induction” and hence there is intrinsic coercivity H_Ci - this is synonymous with _JH_C but typically expressed in different units^13)14)15) (SI system uses the name “magnetic polarisation” J in tesla, but CGS uses “intrinsic induction” B_i in gauss)
• in soft magnetic materials the difference between B and J is negligibly small (hence _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 coercivity¹⁶⁾
• because of the differences in the magnetisation quantity different units can be used in various publications: A/m, T, G, Oe ^17)18)19)
• 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 kA/m ²⁰⁾
• 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 illustration below in the text)

Properties of soft ferromagnets: saturation, permeability, frequency range, material cost²¹⁾

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The value of coercivity is not directly important for soft ferromagnets, because for them permeability is a typically a better 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.²²⁾

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 very low coercivity H_C, as listed in the table below.²³⁾

Properties of hard ferromagnets: remanence, coercivity, temperature range, material cost²⁶⁾

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Properties of semi-hard ferromagnets: remanence, coercivity, and energy density²⁷⁾

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Coercivity depends not only the type of material, but also on the state of that material. For example, very pure monocrystal of iron can have coercivity around 1 A/m, in the polycrystalline state it could be 100 A/m, and for single-domain iron particles it could be even beyond 10 kA/m.²⁸⁾

Changes in magnetisation state of a material can occur as a result of domain wall motion, 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.²⁹⁾

Domain wall pinning.

Procedure for measuring coercivity³⁰⁾

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Coercivity of magnetic materials is measured by using a measurement system appropriate for the type of material (soft, hard, semi-hard). However, from the 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 (1) 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 much greater than the H_C to be measured, so H_sat » H_C.
• After saturation, the excitation is reduced (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 (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.
• Measurement of remanence coercivity H_C,R is more complicated, because it can only be quantified after switching the applied field back to zero (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 must 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).³¹⁾

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 $l_g / l_c$ = 0.00028

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The permeability of typical soft magnetic materials is 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 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) 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.³²⁾

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, by first saturating the sample of material to some suitable value (e.g. 200 kA/m for highly permeable materials)³³⁾, then reducing the field to zero, and then applying negative field until the measured B reduces to zero.

IEC 60404-7 ³⁴⁾ 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.

Setup for measurement of coercivity in soft and semi-hard magnetic materials:³⁵⁾ 1) sample, 2) saturating and demagnetising solenoid, 3) inner radial off-centre Hall-effect sensor, 4) external radial differential fluxgate sensors, 5) vibrating coil

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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.³⁶⁾

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 (1) in a solenoid (2):³⁷⁾

• Hall-effect sensor (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.
• Pair of fluxgate sensors (4) can be placed outside of the solenoid. The sensors should be configured to measure the radial field, making them a differential pair which will suppress effects of some external 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.³⁸⁾
• Vibrating coil can be placed near or around one end of the sample.

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.³⁹⁾

In commercial systems saturating soft magnetic materials is achieved typically with a field 140-450 kA/m,w hich can be obtained by DC current from a variable DC power source, or by pulse methods from a LC circuit.⁴⁰⁾⁴¹⁾ The saturating solenoid can be enclosed in a shielding box.⁴²⁾⁴³⁾

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.⁴⁴⁾ The standard allows using permanent magnets as the source of magnetic field for saturation.

In permanent magnets | _BH_c | < | _JH_c |

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For high-energy permanent magnets the required 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.

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:⁴⁵⁾

• 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 |

Concept of pulsed measurement⁴⁶⁾

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It is possible but difficult 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.⁴⁷⁾

Less expensive methods used in production and testing employ pulsed currents. A capacitor is charged to a controlled level of voltage and discharged through a coil, via diode. This produces a uni-directional pulse of very high current which is capable of magnetising, saturating, or demagnetising a magnet.

Widening of the B-H loop caused by eddy currents in pulsed measurement⁴⁸⁾

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a

b ⁴⁹⁾

Pulses can be applied by successive demagnetisation, or by always returning to the same saturation point, synonymous with first-order reversal curves (FORC).⁵⁰⁾

Furlani - overcome demagnetising field

a

b

Coercivity H_C and remanence B_R can be defined for each hysteresis loop (even without saturation)⁵¹⁾

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As with remanence, coercivity can be defined as measured after saturation so that it has always the same value, related to material properties (rather than the sample 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.⁵²⁾ 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.

a ⁵³⁾

bb⁵⁴⁾

a

Coercivity is related to microstructure and therefore it can be used for non-destructive testing.

• Remanence
• Magnetic field strength
• Magnetic properties