Table of Contents
Magnetic flux
Stan Zurek, Magnetic flux, Encyclopedia Magnetica, http://www.e-magnetica.pl/doku.php/magnetic_flux |
Magnetic flux, Φ - a physical quantity that expresses the amount of magnetic flux density B which penetrates the given cross-sectional area A of interest.^{1)}^{2)}^{3)}^{4)}
If there is some magnetic field in a given volume, there is always some magnetic flux Φ associated with it,^{5)} provided that the area of interest is not defined as a closed three-dimensional surface. Φ is always a scalar value, even if B and A are treated as vectors in the calculations.
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The magnetic flux density B is the fundamental quantity which is used for defining the existence of magnetic field. And the magnetic flux Φ quantifies the amount of B that penetrates surface A. The names are related, because one quantity is calculated from the other.
Magnetic flux is measured in the SI unit of weber (Wb),^{7)} (in the CGS system the unit was “maxwell”, Mx).
Relation of magnetic flux Φ and its unit weber (Wb) to other values | |
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(1a) (quantities) | $$ Φ = \vec{B_{avg}}·\vec{a}·A ≡ V·t $$ |
(1b) (units) | $ \mathrm{ Wb ≡ T·m^2 ≡ V·s ≡ \frac{kg·m^2}{s^2·A} } $ |
where: $\vec{B_{avg}}$ - vector of magnetic flux density (T) averaged over area $A$, $\vec{a}$ - normal vector (unitless) to the surface $A$, $A$ - area (m^{2}), $V$ - voltage (V), $t$ - time (s) |
Changes in magnetic flux Φ induce voltage in the related electric circuit. Therefore, Φ is related to B and A, as well as to the induced voltage and time (through the Faraday's law of induction).
If the given coil has N number of turns, then the quantities such as induced voltage are scaled accordingly, as explained in the following sections.
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Magnetic flux through surface
Flux of field
For any surface, flux of field is defined as the net “flow” of that field through that surface. This is applicable to any vector field, such as velocity of molecules in liquid, or electromagnetic quantities.^{8)}
Mathematically flux is calculated as the vector dot product of the vector of field and the normal vector to the surface at that point. This operation extracts the component of the field which is perpendicular (normal) to the surface, and thus summation (or rather integration) over the whole surface computes the amount of flux penetrating the surface.
A field which is completely parallel to the surface produces zero flux through that surface, because there is no perpendicular component of the field crossing such surface.
According to the accepted convention, in a right-handed system of coordinates, positive flux is defined as such that flows out of the given surface.^{9)}
Magnetic flux
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However, magnetic field in the form of magnetic flux density B is solenoidal, which means that the imaginary field lines have no beginnings and no ends, but always loop back on themselves. For such a field it can be shown that for any completely closed three-dimensional surface (such as the surface of a sphere) the magnetic flux is zero. This is because any line which exits the bubble of the surface must also enter it, so that the net value for that specific field line is zero (and therefore it is zero for any such line).
On the other hand, any field line which closes completely within the volume of the bubble does not cross the surface so it does not contribute to the total flux calculation. Similarly, any field line which misses the shape does not contribute at all. As a result, flux of B through any closed 3D surface is always zero.
This is synonymous with the magnetic field not being produced by magnetic monopoles and therefore the divergence (“sourceness”) of magnetic flux density is always zero, which is one of the Maxwell's equation, namely the Gauss's law for magnetism.
Gauss's law for magnetism^{10)}^{11)} | |
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(2a) (differential form) | $$ \text{div} \vec{B} ≡ \text{div} \mathbf{B} ≡ ∇·\mathbf{B} = 0 $$ |
(2b) (integral form) | $ \int_{C} \vec{B}·d \vec{A} = 0 $ |
where: B - flux density (T), A - area (m^{2}), C - closed 3D surface | |
Note: various equivalent notations are used in the literature. |
However, if the magnetic flux is calculated through some open area (2D or 3D), then the result in most cases will not be zero.
Note that Φ is always a scalar quantity, and it is only illustrated here as the red arrow to show that it is positive because B points away from the penetrated surface A
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It is important to note that the magnetic flux is based on the magnetic flux density B. Theoretically, it would be also possible to calculate flux of other vector field quantities, such as magnetic polarisation J, magnetic field strength H, or magnetisation M, but these are not used, mostly because their practical usefulness in physical calculations is very limited. This is because only flux of B is guaranteed to be zero for a closed 3D surface.
It is of course possible to calculate the flux of electric field Φ_{E} with similar rules as described above. And because the electric field is produced by electric charges such as electrons and protons (which are by definition “electric monopoles”), then the value of Φ_{E} for a given closed 3D surface quantifies the amount of net electric charge closed by such surface.^{12)}
Flux in ideal closed magnetic circuit
In a closed magnetic circuit made of high-permeability magnetic material it can be typically assumed that all the flux lines remain confined inside the magnetic core.
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In such a circuit, magnetic flux is directed parallel to the surface, and therefore for a given perpendicular cross-sectional area only the perpendicular component of flux is present, so that the average of the flux density B can be used to calculate the magnetic flux. Thus, the calculations can be simplified to an all-scalar representation.
This is also true for the parts of the core which have different cross-sectional area, such as in the corners. This is because the average flux density reduces (area is larger so the flux lines spread out), but the area is increased so their product remains the same.
As illustrated, near the inner corners of the frame core (pink arrow) the flux density B can be increased, whereas near the outer corners (green arrow) it can be reduced. But it is the average B that is applicable for calculation of magnetic flux Φ and that remains unchanged throughout the core (unless magnetic saturation and flux leakage takes place).
However, the fact that only average flux density matters has also an impact on the measurements of magnetic quantities. Namely, the voltage induced in a coil can be used to calculate only the average value penetrating the given coil, and thus local variations such as saturation cannot be detected.^{13)}
Faraday's law - flux rule
Faraday's law of induction relates the changes in magnetic field penetrating given area with the electric field or voltage induced due to such changes. This law is presented in the literature in multiple ways. The minus sign arises because of the Lenz's law (the Nature opposes change).
^{S. Zurek, E-Magnetica.pl, CC-BY-4.0}
The Faraday's law of induction (also called the flux rule) states that changes in time of the value of magnetic flux Φ (equivalent to average magnetic flux density B) penetrating the given coil with number of turns N and area A generate electromotive force (EMF). Therefore, the induced voltage is proportional to the time derivative of Φ or B.
Faraday's law of induction - the flux rule | ||
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(3a) (expressed by Φ) | $$ EMF = - N ⋅ \frac{dΦ}{dt} $$ | (V) |
(3b) (expressed by B) | $$ EMF = - N ⋅ A ⋅ \frac{dB_{avg}}{dt} $$ | (V) |
(3c) (expressed by λ) | $$ EMF = - \frac{dλ}{dt} $$ | (V) |
where: $EMF$ - electromotive force (V) measurable as voltage (V), $N$ - number of turns in the coil (unitless), $Φ$ - magnetic flux (Wb), $t$ - time (s), $A$ - area of the coil (m^{2}), $B_{avg}$ - spatial average of flux density in the coil (T), $λ = N·Φ$ - flux linkage (Wb) |
The variation (d/dt) can arise because of the changing amplitude, direction (due to the cosine of the angle), or position of the conductor (due to change in the area).^{14)}
Typically, in books related to electromagnetic machines and devices the flux rule is expressed by using equation (3a). This is because the flux linkage λ is related to the calculation of inductance L, which is useful in computations related to electromagnetic performance.^{15)}^{16)}^{17)}
Some authors follow the name “flux rule” literally, and teach that the magnetic flux is the only basis of the Faraday's law.^{18)} However, the magnetic flux is defined as the flux of B so all such equations are mathematically and physically equivalent.
If the coil has N turns which are closely packed and connected in series, then in practice can be often assumed that the same flux penetrates all the turns, and therefore the flux linkage is increased by multiplicity of the turns (compare equations (3a) and (3c)). The voltages generated in each turn add up so that the total voltage is N times greater.
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The Faraday's law relates the changes in B or Φ to the induced voltage by such changes. This induced voltage “circulates”^{20)} around the area penetrated by the magnetic field, and this can be calculated directly by using the curl of electric field E generated by the temporal changes of magnetic flux density B. It should be noted that in this equation it is the flux density B that is the fundamental quantity, not flux Φ. Equations such as (3a, 3b, 3c) are only a convenient way for calculating specific simplified cases.
Faraday's law in a differential form ^{21)} | |
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(4) | $$ \vec{∇} × \vec{E} = - \frac{∂ \vec{B}}{∂t} $$ |
where: circulation of the vector of electric field E equals the changes of flux density B in time t |
For a given open surface (2D or 3D), if a wire was to be placed around its boundary, then the circulating electric field will give rise to the induced voltage, and if the circuit was completed - to the current circulating around the perpendicular changes in magnetic field. This is the reason why eddy currents are generated in all electrically conductive materials (whether they are magnetic or not).
Measurement of Φ and B
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The Faraday's law of induction can be employed for measuring magnetic flux density B or magnetic flux Φ. ^{23)}^{24)}
The value of B or Φ can be learned by calculating an integral (in an analogue electronic circuit or numerically) of the voltage induced in a search coil. Such processing is conveniently executed by a measuring device called fluxmeter, whose name implies that the signal integration is performed. This is in contrast to a gaussmeter which typically measures the signal directly, without calculating the integral (e.g. from a Hall-effect sensor).
It should be noted that such measurements can only detect the net magnetic flux (total flux) Φ or the average magnetic flux density B_{avg} penetrating the coil.
Therefore, even if the magnetic material saturates locally this is “hidden” in the signal which is proportional to the average quantity, perhaps with the exception of harmonic distortions in sinusoidal signals. Local variation can be only detected with more precision with appropriately smaller coils placed in the regions of interest.^{25)}
It should be noted that the minus sign in equations (5a, 5c) (or their equivalents) is applicable only to the EMF signal, because the measured voltage at the output of the coil has reversed polarity due to the second Kirchhoff's law.^{26)}^{/p.516} These equations are sometimes erroneously stated with the minus even for the measured voltage, for example as in the international standard EN 60404-6:2003^{27)}, or for the EMF without the minus as in some books.^{28)}
Measurement of magnetic flux Φ or magnetic flux density B with a search coil | ||
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(5a) (measuring flux) | $$ Φ = - \frac{1}{N} ⋅ \int_0^T (EMF) dt + Φ_0 $$ | (Wb) |
(5b) (measuring flux) | $$ Φ ≈ \frac{1}{N} ⋅ \int_0^T (V_{out}) dt + Φ_0 $$ | (Wb) |
(5c) (measuring flux density) | $$ B_{avg} = - \frac{1}{N⋅A} ⋅ \int_0^T (EMF) dt + B_0 $$ | (T) |
(5d) (measuring flux density) | $$ B_{avg} ≈ \frac{1}{N⋅A} ⋅ \int_0^T (V_{out}) dt + B_0 $$ | (T) |
where: $EMF$ - induced electromotive force (V), $V_{out}$ - output voltage (V) of the coil measured on the terminals of the coil induced due to EMF, assuming very high impedance of the voltmeter such that $V_{out} ≈ - EMF$, $N$ - number of turns of the coil (unitless), $A$ - cross-sectional area (m^{2}) of the coil, $T$ - time interval (s), $t$ - time (s), $Φ_0$ - magnetic flux at the start of the integral (Wb), $B_0$ - magnetic flux density at the start of the integral (T) | ||
Important note: the EMF inside the coil has an opposite sign to the voltage measured at the output of such coil.^{29)}^{/p.516} This results directly from the second Kirchhoff's law: the sum of voltages around a single loop must remain zero. |
Inductance
Inductance L is the parameter which quantifies the “inertia” to changes in the amount of energy stored in the magnetic field.^{30)}
Self-inductance
Assuming linear behaviour of magnetic material, such that its relative permeability $μ_r$ is constant (e.g. absence of magnetic saturation), the inductance for a given coil can be calculated by any of the equivalent expressions in equation (6).
If there are no other magnetic or electric circuits to be considered, and all the flux lines penetrate only the single coil of interest, then the inductance of equation (6) is referred to as self-inductance. This can be also illustrated (see the next section), as the flux which penetrates only the coil with its own current (rather than some additional flux from any other coil).
Depending on the type of computational analysis the calculations can be performed in the way which is most useful or accessible due to the physical quantities of the given magnetic circuit.
Each of these equations has several underlying assumptions (such as linearity and lack of saturation), even though they might not be explicitly stated. For example, there is a certain amount of self-inductance which arises from the amount of magnetic field which is present inside of the wire or conductor. In most cases this is negligibly small as compared to the “global” inductance of the coil, but this is not always the case.^{31)}^{32)}
Self-inductance of a simple coil ^{33)}^{34)} | ||
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(6) | $$L = \frac{N⋅Φ}{I} = \frac{λ}{I} = \frac{N⋅B⋅A}{I} = \frac{N^2⋅B⋅A}{N·I} = \frac{N^2⋅B⋅A}{H·l} = \frac{N^2⋅μ_r⋅μ_0⋅A}{l} = \frac{N^2}{R} $$ | (H) |
where: $N$ - number of turns (unitless), $Φ$ - magnetic flux (Wb), $I$ - electric current (A), $λ$ - flux linkage (Wb), $B$ - magnetic flux density (T), $A$ - cross-sectional area (m^{2}), $H$ - magnetic field strength (A/m), $l$ - magnetic path length (m), $μ_r$ - relative permeability (unitless), $μ_0$ - permeability of vacuum (H/m), $R$ - reluctance factor (1/H) |
Self-inductance also arises for a straight conductor (any conductor), rather than a loop or coil - for more details see inductance of a straight conductor.
Mutual inductance
If the magnetic field generated by one coil penetrates another coil, then there is some amount of “mutual flux”, which gives rise to mutual inductance L_{M} or simply M, as illustrated by the fluxes $Φ_{12}$ and $Φ_{21}$. The indices denote that the flux penetrating the second coil is generated by the current in the first coil, and vice versa.
There can be magnetic coupling between more than two coils, and the calculations are carried out in a similar manner.
Mutual inductance M ^{35)} | ||
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(7a) | $$ M_{12} = M_{21} = M $$ | (H) |
(7b) | $$ M_{12} = \frac{N_1⋅Φ_1}{I_2} = \frac{λ_1}{I_2} $$ | (H) |
(7c) | $$ M_{21} = \frac{N_2⋅Φ_2}{I_1} = \frac{λ_2}{I_1} $$ | (H) |
where: $M$, $M_{12}$ and $M_{21}$ - mutual inductance (Wb), $N_1$ and $N_2$ - number of turns (unitless) of coil 1 and 2 respectively, $Φ_{1}$ and $Φ_{2}$ - magnetic flux (Wb) of each coil, $I_1$ and $I_2$ - electric current (A) of each coil, $λ_1$ and $λ_2$ - flux linkage (Wb) of each coil |
The mutual inductance is sometimes expressed by the magnetic coupling coefficient k, which can take a maximum value of k = 1 for all the flux being coupled, k = 0 if there is no magnetic coupling between the coils, or k = -1, if the coupling is negative but complete (with reversed polarity of one of the coils). In a typical well-coupled transformer |k| > 0.95, but it can be even higher than 0.99, meaning that almost all the flux of the first winding penetrates the second winding.
Magnetic coupling coefficient k ^{36)} | ||
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(8) | $$ k_{12} = \frac{M}{\sqrt{L_1 ⋅ L_2}} $$ | (unitless) |
where: $k_{12}$ - magnetic coupling coefficient (unitless) between coils 1 and 2, $M$ - mutual inductance (H), $L_1$ - inductance (H) of the first coil, $L_2$ - inductance (H) of the second coil |
The concept of coupling coefficient k is widely used in SPICE software (such as PSpice, LTspice, Qspice) for simulating electronic and electric circuits. Coupling between multiple coils can be defined in the same k statement.^{37)}^{38)}^{39)}