Table of Contents
Magnetic field strength
Stan Zurek, Magnetic field strength, Encyclopedia Magnetica, http://e-magnetica.pl/doku.php/magnetic_field_strength |
Magnetic field strength H - a physical quantity used as one of the basic measures of the intensity of magnetic field.^{1)}^{2)} The unit of magnetic field strength^{3)} is ampere per metre or A/m.
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From the engineering viewpoint, magnetic field strength $H$ can be thought of as excitation and the magnetic flux density $B$ as the response of the medium.^{4)}^{5)} This naming convention is defined in the SI system of units.
From theoretical physics viewpoint, the field $H$ is defined as the vectorial difference between flux density $B$ and magnetisation $M$. The H field is sometimes referred to as “auxiliary” or simply “field H”.^{6)}^{7)}^{8)}
These two approaches are identical in the sense of the physical quantities in question (with the same physical units of A/m), but are referred to by different names, and different emphasis put on their meaning and use in derivation of some equations.
Magnetic field is a vector field in space, and is a kind of energy whose full quantification requires the knowledge of the vector fields of both magnetic field strength $H$ and flux density $B$ (or other values correlated with them, such as magnetisation M or polarisation J). In vacuum, at each point the $H$ and $B$ vectors are oriented along the same direction and are directly proportional through permeability of free space, but in other media they can be misaligned (especially in non-uniform or anisotropic materials).
The requirement of two quantities is analogous for example to electricity. Both electric voltage $V$ and electric current $I$ are required to fully quantify the effects of electricity, e.g. the amount of transferred energy.^{9)}
The name magnetic field strength and the symbol $H$ are defined by International Bureau of Weights and Measures (BIPM) as a one of the coherent derived physical units.^{10)} Therefore, strictly speaking, other names like magnetic field intensity or magnetic field (or even just field) which can be encountered in everyday technical jargon^{11)} are incorrect if used when referring to a specific value of H in A/m.
There are many other names which are used in the literature, all denoting the same quantity:
- magnetic field intensity H ^{12)}
- magnetic field H ^{13)}^{14)}^{15)}
- field H ^{16)}
- field H' ^{17)}
- H-field strength ^{18)}
- H-field ^{19)}
- magnetization field strength H ^{20)}
- magnetizing field strength H ^{21)}
- magnetising force H ^{22)}
- magnetic force H ^{23)}
- intensity of magnetic force H ^{24)}
- auxiliary field H ^{25)}
- and probably several others.
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Magnetic flux density B
See separate article on: Magnetic flux density |
See separate article on: Confusion between B and H |
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The magnetic flux density B is a separate physical quantity, with different physical units in the SI system. The H and B are interlinked such that:^{26)}
$$\vec{B} = \vec{J} + μ_0 · \vec{H} = μ_0 · (\vec{H} + \vec{M}) $$ | (T) |
where: $μ_0$ - absolute permeability of vacuum (H/m), $μ_r$ - relative permeability of material (unitless), $μ = μ_0 · μ_r$ - absolute permeability of material (H/m), $J$ - magnetic polarisation (T), $M$ - magnetisation (A/m) |
Magnetisation M represents orientation of subatomic magnetic dipole moments per unit volume, and magnetic polarisation J is M scaled by the permeability of vacuum.
In a general case, all the three vectors B, H and J (or B, H and M) can point in different directions (as shown in the illustration for an anisotropic material), but always such that the vector sum in the equation above is fulfilled.
For uniaxial magnetisation the equation can be simplified to the scalar form, which is used widely in engineering applications:
$$B = μ_0 · μ_r · H = μ · H $$ | (T) |
Relative permeability $μ_r$ is a figure of merit of soft magnetic materials and has values significantly greater than unity.
For hard magnetic materials $μ_r \approx$ 1, and it is a much less important parameter.
For non-magnetic materials also $μ_r \approx$ 1, but such that paramagnets are weakly attracted to any polarity of magnetic field ($μ_r$ slightly greater than unity), and diamagnets are always weakly repelled it ($μ_r$ slightly less than unity). Depending on the viewpoint, superconductors can be classified as ideal diamagnets for which $μ_r$ = 0, and thus they are quite strongly repelled from magnetic field, sufficiently for magnetic levitation.^{27)}
Difficulty with definition
It is difficult to give a concise definition of such a basic quantity like magnetic field, but various authors give at least a descriptive version. The same applies to magnetic field strength, as well as the other basic quantity - magnetic flux density.
The table below shows some examples of definitions of $H$ given in the literature (exact quotations are shown).
Publication | Definition of magnetic field | Definition of magnetic field strength $H$ | Definition of magnetic flux density $B$ |
---|---|---|---|
R. Feynman, R. Leighton, M. Sands The Feynman Lectures on Physics^{28)} | First, we must extend, somewhat, our ideas of the electric and magnetic vectors, E and B. We have defined them in terms of the forces that are felt by a charge. We wish now to speak of electric and magnetic fields at a point even when there is no charge present. We are saying, in effect, that since there are forces “acting on” the charge, there is still “something” there when the charge is removed. | We choose to define a new vector field H by $$\mathbf{H} = \mathbf{B} − \frac{\mathbf{M}}{ε_0 c^2} $$ […] Most people who use the mks units have chosen to use a different definition of H. Calling their field H' (of course, they still call it H without the prime), it is defined by $$\mathbf{H'} = ε_0 c^2\mathbf{B} − \mathbf{M}$$ Also, they usually write $ε_0 c^2$ as a new number 1/μ_{0} | We can write the force F on a charge q moving with a velocity v as $$\mathbf{F} = q(\mathbf{E} + \mathbf{v} × \mathbf{B})$$ We call E the electric field and B the magnetic field at the location of the charge. |
Richard M. Bozorth Ferromagnetism^{29)} | A magnet will attract a piece of iron even though the two are not in contact, and this action-at-a-distance is said to be caused by the magnetic field, or field of force. | The strength of the field of force, the magnetic field strength, or magnetizing force H, may be defined in terms of magnetic poles: one centimeter from a unit pole the field strength is one oersted. | Faraday showed that some of the properties of magnetism may be likened to a flow and conceived endless lines of induction that represent the direction and, by their concentration, the flow at any point. […] The total number of lines crossing a given area at right angles is the flux in that area. The flux per unit area is the flux density, or magnetic induction, and is represented by the symbol B. |
David C. Jiles Introduction to Magnetism and Magnetic Materials^{30)} | One of the most fundamental ideas in magnetism is the concept of the magnetic field. When a field is generated in a volume of space it means that there is a change of energy of that volume, and furthermore that there is an energy gradient so that a force is produced which can be detected by the acceleration of an electric charge moving in the field, by the force on a current-carrying conductor, by the torque on a magnetic dipole such as a bar magnet or even by a reorientation of spins of electrons within certain types of atoms. | There are a number of ways in which the magnetic field strength H can be defined. In accordance with the ideas developed here we wish to emphasize the connection between the magnetic field H and the generating electric current. […] The simplest definition is as follows. The ampere per meter is the field strength produced by an infinitely long solenoid containing n turns per metre of coil and carrying a current of 1/n amperes. | When a magnetic field H has been generated in a medium by a current, in accordance with Ampere's law, the response of the medium is its magnetic induction B, also sometimes called the flux density. |
Magnetic field, Encyclopaedia Britannica^{31)} | Magnetic field, region in the neighbourhood of a magnetic, electric current, or changing electric field, in which magnetic forces are observable. | The magnetic field H might be thought of as the magnetic field produced by the flow of current in wires […]^{32)} | […] the magnetic field B [might be thought of] as the total magnetic field including also the contribution made by the magnetic properties of the materials in the field.^{33)} |
E.M. Purcell, D.J. Morin, Electricity and magnetism^{34)} | This interaction of currents and other moving charges can be described by introducing a magnetic field. […] We propose to keep on calling $\mathbf{B}$ the magnetic field. | If we now define a vector function $\mathbf{H}(x, y, z)$ at every point in space by the relation $$ \mathbf{H} \equiv \frac{\mathbf{B}}{μ_0} - \mathbf{M} $$ […] As for $\mathbf{H}$, although other names have been invented for it, we shall call it the field $\mathbf{H}$, or even the magnetic field $\mathbf{H}$. | […] any moving charged particle that finds itself in this field, experiences a force […] given by $$ \mathbf{F} = q·\mathbf{E} + q·\mathbf{v} × \mathbf{B} $$ […] We shall take the equation as the definition of $\mathbf{B}$. |
Analogy to electric circuits
Electric circuit
At the fundamental level, all the electricity is linked to the presence and movement of electric charges, so knowing their positions would be sufficient to fully quantify all electric effects, including electric field. However, in practice, it is much simpler to operate with directly measurable quantities such as current $I$ and voltage $V$.
From a macroscopic viewpoint, values of $I$ and $V$ are both required to fully quantify the effects of electricity in electric circuits. In direct current circuits the proportionality between $V$ and $I$ is dictated by electrical resistance $R$ of a given medium (according to Ohm's law), such that $V = R·I$.
The product of $V$ and $I$ is proportional to power $P$ and energy $E$ in a given electric circuit.
Magnetic circuit
By analogy both magnetic field strength $H$ and magnetic flux density $B$ (or their representations by other related variables) are required for quantifying the effects of magnetism in magnetic circuits. The proportionality between $H$ and $B$ is dictated by magnetic permeability $μ$ of a given medium.^{35)}
All magnetic field effects are also linked to the movement and intrinsic properties of electric charges. Knowing these properties (such as spin magnetic moment) and the details of movement of the charges (taking into account relativistic effects) it would be possible to completely describe the magnetic field. However, in practice it is much simpler, especially from engineering viewpoint, to utilise the directly measurable quantities such as $H$ and $B$ to quantify power and energy in a given magnetic circuit.
Under steady state conditions, the product of $H$ and $B$ is a measure of specific energy in J/m^{3}, stored in the magnetic field contained in the given medium. The $B·H$ product (the amount of stored energy) is used for example for classification of permanent magnets.^{36)}
H due to electric current
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From a macroscopic viewpoint, the fields can be treated as averaged over some volume of material, and their magnitudes can be linked to measurable signals such as current or voltage. Therefore, this approach is used extensively in engineering.^{37)}^{38)}^{39)}
$H$ is always generated around electrical current $I$, which can be a solid conductor with current or just a moving electric charge (also in free space). The direction of the $H$ vector is perpendicular to the direction of the current $I$ generating it, and the senses of the vectors are assumed to follow the right-hand rule.^{40)} It can be said that H “circulates” around the current I.
Without other sources of magnetic field and in a uniform and isotropic medium the generated magnetic field strength $H$ depends only on the magnitude and direction of the electric current $I$ and the physical sizes involved (e.g. length and diameter of the conductor, etc.) so according to the Ampere's law the proportionality is dictated by the magnetic path length $l$:
$$ \int_C \vec{H} · d \vec{l} = I $$ | (A) |
where: C - closed path over which the integral is calculated, $dl$ - infinitesimal fragment of magnetic path length (m), $I$ - current (A) |
In a linear isotropic medium the values from various sources combine and can be calculated from the superposition of the sources. For simple geometrical cases the value of $H$ can be calculated analytically, but for very complex systems it is possible to perform computation for example with finite-element modelling.
The relationship between $H$ and $I$ is often shown by employing the Biot-Savart's law^{41)} or Ampere's law.^{42)} Often (but not always^{43)}) both of these are stated with the variable of flux density $B$ so that the permeability of the medium is automatically taken into account.
In many examples given in the literature there is an implicit assumption (typically not stated) that the derivation is carried out for vacuum and not for an arbitrary medium with a different permeability^{44)}. When the $μ_0$ permeability is reduced in the equations on both sides then $H$ is proportional only to $I$ and this is true for any uniform isotropic medium with any permeability, even non-linear.^{45)}
The situation is slightly different for anisotropic or discontinuous medium. They can give rise to additional sources of magnetic field because new magnetic poles can be generated by the excited medium, and these poles must be taken into account in order to accurately describe distribution of $H$. For instance, pole pieces in an electromagnet affect $H$, whose distribution is no longer dictated by just the coils with electric current.
H due to M and B
The microscopic viewpoint is used often in theoretical physics.^{46)}^{47)}
Each atom responds to the externally applied magnetic field B with some magnetisation M, which is defined as the vector sum of magnetic moments per given volume. The “auxiliary” magnetic field H is then defined as the vector difference between the applied magnetic field B and the magnetisation M:^{48)}
$$\vec{H} = \frac{\vec{B}}{μ_0} - \vec{M}$$ | (A/m) |
where: $μ_0$ - absolute permeability of vacuum (H/m) |
For DC excitation, in non-magnetic or magnetic but isotropic materials B and H vectors are parallel. For ferromagnetic (and other ordered structures) the crystal or shape anisotropy can introduce significant angle between the two vectors.
Maxwell's equations
Maxwell's equations are typically given with respect to magnetic flux density B because in that form they are valid under more general conditions.^{49)}
However, under certain conditions it is also possible to express them with respect to H. This approach is extensively used in numerical calculations such as finite-element modelling (FEM), where the direct link between the electric current (expressed by current density J) and H is exploited, through the Ampere's law, both for solutions and formulations of boundary conditions.^{50)}^{51)}^{52)}^{53)}^{54)}
Example of notation used in FEM documentation (after reference ^{55)} ) | |
---|---|
$$ \nabla · \mathbf{D} = ρ$$ | $$ \nabla · \mathbf{B} = 0$$ |
$$ \nabla \times \mathbf{E} = - \frac {\partial \mathbf{B}}{\partial t}$$ | $$ \nabla \times \mathbf{H} = \mathbf{J} + \frac {\partial \mathbf{D}}{\partial t} $$ |
H in electromagnetic waves
In vacuum, in the absence of charges and currents, the Maxwell's equations simplify, and they can be written either with respect to magnetic flux density B (as shown in the table below), or magnetic field strength H. This can be done because of the linearity of vacuum (or other non-magnetic medium), which has no free charges, so there are no additional electric currents which have to be taken into account.^{56)} The format with B is valid under more general conditions.^{57)}
Maxwell's equations in vacuum (in a differential form)^{58)}^{59)} | |||
---|---|---|---|
magnetic field represented by H | magnetic field represented by B | ||
$$ \text{div } \mathbf{E} = 0$$ | $$ \text{div } \mathbf{H} = 0$$ | $$ \text{div } \mathbf{E} = 0$$ | $$ \text{div } \mathbf{B} = 0$$ |
$$ \text{curl } \mathbf{E} = - \mu_0 · \frac {\partial \mathbf{H}}{\partial t}$$ | $$ \text{curl } \mathbf{H} = \epsilon_0 · \frac {\partial \mathbf{E}}{\partial t} $$ | $$ \text{curl } \mathbf{E} = - \frac {\partial \mathbf{B}}{\partial t}$$ | $$ \text{curl } \mathbf{B} = \mu_0 · \epsilon_0 · \frac {\partial \mathbf{E}}{\partial t} $$ |
In vacuum the two notations, with B or H are exactly equivalent, with the latter quite popular for analysing radiation from antennas.^{60)} For example, using the Poynting vector which represents power, as a product of electric field E in V/m and magnetic field H in A/m, the result is V·A/m^{2} or W/m^{2} (power density).
Defining H with force
It is shown in the literature that magnetic field strength at a given point in space can be defined as the mechanical force acting on unit pole at the given point.^{61)} However, calculation of force requires $B$, which depends on the properties of medium. Indeed, the original experiment performed by Biot and Savart involved physical forces acting on wires.^{62)}
The forces acting on two magnetised bodies will be different if they are placed in oxygen (which is paramagnetic) or in water (which is diamagnetic). This difference will be directly proportional to the relative permeabilities of the involved media. However, the $H$ produced around the wire will be the same (as long as the medium is uniform and isotropic).
The magnitude of magnetic force (Lorentz force) is always proportional flux density $B$.^{63)}
Generation of H
Known values of H are generated by utilising the Ampere or Biot-Savart laws mentioned above. If relativistic effects can be ignored, then the proportionality is exactly direct such that instantaneous values of magnetic field strength $H$ correspond to instantaneous values of the applied current $I$:
$$ H(t) = c · I(t) $$ | (A/m) |
where: $c$ - proportionality constant of a given circuit (1/m) |
Under certain conditions the generated magnetic field can be calculated so precisely that it can be used for calibration of other sensors or definition of values, as recommended by BIPM.^{64)}
Two typical devices which can be used for generating known values of H are the solenoid and the Helmholtz coil.^{65)}^{66)} They can be even used in a combined setup, in which the external Helmholtz coils compensate for Earth's magnetic field (or other unwanted sources) and the internal solenoid for generation of the precisely known magnetic field.^{67)}
Solenoid
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See also the main article: Solenoid |
In an infinitely long uniform solenoid, the value of H at its axis depends only on the value of current in the coil and the number of turns per unit length.
For a solenoid with a finite length, the magnetic field at its geometrical centre can be calculated as in the equation below. For “thin” (wire diameter much smaller than coil diameter) and “long” (coil diameter much smaller than its length $d \ll l$) solenoid, the equation simplifies:^{68)}
$$ H_{centre} = \frac{N·I}{\sqrt{l^2 + d^2}} \approx \frac{N·I}{l} $$ | (A/m) |
where: $N$ - total number of turns in the solenoid (unitless), $I$ - current (A), $l$ - solenoid length (m), $d$ - solenoid diameter (m) |
If the thickness of the wire in the solenoid is significant, or there are many layers of the coil, some additional correction terms are required in the equation.^{69)}
Helmholtz coil
See also the main article: Helmholtz coil |
Another widely used source of H is the Helmholtz coil. The device comprises two identical coils resembling circular current loops, positioned parallel on the same axis, and separated precisely by the radius of the circle.
For two coils, each with radius r and each comprising number of turns N_{each}, the value of magnetic field at the geometrical centre can be calculated as:^{70)}
(turns per coil) | $$ H_{centre} = \frac{N_{each}·I·\sqrt{0.8^3}}{r} \approx \frac{0.71554·N_{each}·I}{r} $$ | (A/m) |
(turns total) | $$ H_{centre} = \frac{N_{total}·I·\sqrt{0.8^3}}{2·r} \approx \frac{0.35777·N_{total}·I}{r} $$ | (A/m) |
where: $N_{each}$ - number of turns of each coil (unitless), $N_{total}$ - total number of turns of both coils (unitless), such that $N_{total}=2·N_{each}$, $I$ - current (A), $r$ - radius of each coil and spacing between them (m) |
Shapes other than circular are also used (e.g. square) but at the expense of uniformity of the obtained field distribution.
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Magnetic circuit with a small gap
The Ampere's law relates the integral around a closed path, to the current enclosed by such path.
This relationship is used extensively in engineering by employing the concept of magnetomotive force (product of current and turns of the coil, expressed in ampere-turns). For a simple magnetic circuit with one air gap it can be written that:
$$ N·I = H_{core}·l_{core} + H_{gap}·l_{gap} $$ | (A-turns) ≡ (A) |
where: $N$ - number of turns of the winding (unitless), $I$ - current (A), $H_{core}$ - H in the core (A/m), $l_{core}$ - length of the core (m), $H_{gap}$ - H in the air gap (A/m), $l_{gap}$ - length of the air gap (m) |
In a magnetic circuit with a relatively small air gap the value of magnetic flux density is such that $B_{gap} \approx B_{core}$. However, the value of H required to support some value of B is scaled by the inverse of relative permeability. Hence, for a magnetic material with large permeability, leads to the condition of $H_{core} \ll H_{gap}$ and also to $H_{core}·l_{core} \ll H_{gap}·l_{gap}$, and therefore only the terms related to the air gap are significant. This allows simplifying the equation as:^{71)}
$$ H_{gap} \approx \frac{N·I}{l_{gap}} $$ | (A/m) |
where: $N$ - number of turns of the winding (unitless), $I$ - current (A), $l_{gap}$ - length of the air gap (m) |
However, for more complex magnetic circuits, effects such as flux fringing or magnetic energy stored in the material must be taken into account, and this can be done by numerical methods such as finite-element modelling.^{72)}
Addition of air gap allows storing energy in it. The B-H loop is “sheared”, extending the operation to higher H. These effects are widely used for example in flyback transformers.
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Demagnetising field
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Magnetic field is generated not only by the electric currents, but also by the magnetic moments which can store magnetic energy, for example by alignment due to previously applied process of magnetisation. The collections of such intrinsic magnetic moments amounts to magnetisation $M$ and becomes a source of magnetic field, as it is the case in permanent magnets. If magnetic poles are created then magnetic field lines (of $H$) are by convention assumed to point from the N to the S pole.
The magnetic field lines will close through the medium surrounding the magnet, but also through the magnet itself, in the direction opposite to the magnetisation $M$, thus lowering the effective magnetisation of the body, which is the reason why this effect is called the demagnetising field $H_d$.
The effect can be quantified with a unitless demagnetising factor $N_d$, which is proportional to the magnetisation and it is a function of dimensions of the body, such that for for very long structures or for magnetically closed circuits $N_d=0$ and for thin flat structures of infinite dimensions magnetised perpendicularly to the surface $N_d=1$.
The value of demagnetising factor can be calculated analytically for ellipsoids and other very simple geometric shapes, and for a sphere $N_d=1/3$.^{74)}
$$H_d = - N_d·M $$ | (A/m) |
where: $N_d$ - demagnetising factor (unitless), $M$ - magnetisation (A/m) |
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The illustration shows a permanent magnet magnetised with uniform M (or J), surrounded by vacuum. The magnetic field lines are shown separately for each field: M, H and B, which is the vector sum of M and H.
The demagnetising field H_{d} points in the opposite direction but it is non-uniform because some magnetic flux closes also through the outside of the body (where M = 0 and J = 0).
As a result, inside the body B is also non-uniform and in terms of magnitude B < J.
Outside the body, the field lines of B and H have the same shape, because in vacuum the two vector quantities differ only by a scalar constant of the vacuum permeability μ_{0} (both M and J are zero).
This illustration also shows that at the boundary between the two media with different permeability values, for H the tangential component H_{t} is preserved, and for B the normal component B_{t} is preserved.
Measurement of H
The value of H cannot be measured directly, but it is derived by other means.
In some magnetic measurement systems the proportionality to current is used explicitly, as for example in such devices as Epstein frame, single-sheet tester or toroidal sample. The measured quantity is current (e.g. by means of a shunt resistor), and H is calculated from it.^{76)}^{77)}^{78)}
In some other applications, it is possible to utilise the principle that the tangential component of H does not change at the interface between two materials. Therefore, by measuring the tangential component of magnetic field right at the surface of the sample it is possible to learn about the field immediately below the surface. However, such measurement relies on the assumption that the B-H relationship inside the sensor is linear, because the sensing coil operation is based on the Faraday's law of induction, in which the measured value is the magnetic flux density B. This is typically achieved by using a non-magnetic material as the former on which the H-coil is wound. The signal in the coil is induced proportionally to B in the H-coil, but due to linearity it can be recalculated to extract the information about H.^{79)} Other detectors such us Rogowski-Chattock potentiometer or Hall-effect sensor can be also used to detect the tangential component of H, but they too measure the quantity of B which can be then re-calculated and expressed as H.^{80)}
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Energy and energy density
Energy density $E_d$ of the energy stored in magnetic field, in a given material, can be calculated as:^{84)}
$$E_d = \int H · dB $$ | (J/m^{3}) |
which for a material with linear characteristics, including high-energy permanent magnets, can be simplified to:
$$E_d = \frac{H·B}{2} $$ | (J/m^{3}) |
It should be noted that the last equation above encompasses both the field which is applied as well as the response of the material to being magnetised (regardless which quantity is assumed to be “fundamental”, B or H).
However, in non-magnetic materials for which $μ_r$ ≈ 1 it can be written that:
$$B = μ_0 · H $$ | (T) | and | $$\frac{B}{μ_0} = H $$ | (A/m) |
Therefore, substitution can be made such that eliminates one of the variables, making the energy density proportional to the square of either just B or just H. Depending on the publication, both forms are used,^{85)}^{86)} often not stating the implicit assumption of $μ_r$ ≈ 1. These two forms are equivalent, although expression with B appears to be more popular. If the assumption $μ_r$ ≈ 1 cannot be made then energy is proportional to the product of $B·H$, or the equation has to include also the relative permeability $μ_r$.^{87)}
for $μ_r \approx$ 1 | $$E_d = \frac{B^2}{2·μ_0} = \frac{μ_0·H^2}{2} $$ | (J/m^{3}) |
for $μ_r \neq 1$ | $$E_d = \frac{B^2}{2·μ_r·μ_0} = \frac{μ_r·μ_0·H^2}{2} $$ | (J/m^{3}) |
Hysteresis loop and power loss
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Soft magnetic materials are used for energy transformation under alternating or pulsed magnetisation regimes. Energy efficiency of a magnetic circuit depends on the power lost in the given magnetic material.
For one cycle of magnetisation (for time from 0 to T), the total energy lost in the material is proportional to the area of the traced B-H loop (hysteresis loop). The numerical value of loss can be calculated as:^{88)}
$$P = \frac{f}{D}·\int_0^T \left(\frac{dB}{dt} · H \right) dt $$ | (W/kg) |
where: $f$ - frequency of magnetisation (Hz), $D$ - density of material (kg/m^{3}) |
The specific power loss is an important figure of merit for soft magnetic materials, and for example it is the basis of categorisation of electrical steels.^{89)}
Because of the operating conditions such B-H loops are measured under conditions of sinusoidal voltage, which also enforces sinusoidal B. The waveform of H can become severely distorted especially when material operates close to saturation. This is effect is responsible for example for the inrush current in transformers.
Models of B-H loop
There are several analytical, statistical and numerical models which are used for mathematical description of the B-H loop trajectories, for the purpose of “prediction” of material behaviour under pre-defined or arbitrary magnetisation conditions. Total magnetic losses can be calculated because real B-H loops represent such total losses, and the models attempt to represent the non-linear trajectories of such loops.
Models such as Jiles-Atherton or Preisach use H as the independent variable representing the applied excitation, as dictated by the Ampere's circuital law.^{90)}
Coercivity
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Coercivity H_{c} is defined as the point at which the hysteresis loop crosses the horizontal axis (B or J or M = 0), as measured for a given material. The values of coercivity are used for the broad classification of magnetic materials into: soft (H_{c} < 1 kA/m), hard (H_{c} > 100 kA/m) and semi-hard (1 kA < H_{c} < 100 kA/m).^{92)}
The of coercivity is linked to the amount of energy which is required to magnetise (and demagnetise) a given magnetic material. Soft magnetic materials have narrow B-H loop, they are easy to magnetise and therefore they have low values of coercivity.
In high-energy permanent magnets the coercivity values are very high (wide hysteresis loop), and because of the significant differences between the values of magnetic flux density B and magnetic polarisation J two values of coercivity can be distinguished: _{J}H_{c} and _{B}H_{c}. For such magnets the almost straight line extending from the point of coercivity _{B}H_{c} to the point of remanence B_{r} denotes the operating conditions of a magnetic in a given magnetic circuit.^{93)} Application of magnetic field greater than coercivity can permanently demagnetise even a high-energy magnet. For lower energy magnets the demagnetisation can happen even at fields lower than coercivity.
In soft magnetic materials under normal operating conditions (significantly below saturation) $B \approx J$ and therefore just single value of coercivity H_{c} is measured (because _{J}H_{c} ≈ _{B}H_{c}).