====== Magnetic flux ====== |< 100% >| | //[[user/Stan Zurek]], Magnetic flux, Encyclopedia Magnetica//, \\ @PAGEL@ | **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.[(Feynman>[[https://www.feynmanlectures.caltech.edu/II_36.html|Richard Feynman, Robert Leighton, Matthew Sands, Ferromagnetism, The Feynman Lectures on Physics, Vol. II, Basic Books, ISBN: 9780465079988]])][(Jiles>[[http://books.google.com/books?isbn=9780412798603|David C. Jiles, Introduction to Magnetism and Magnetic Materials, Second Edition, Chapman & Hall, CRC, 1998, ISBN 9780412798603]])][(Hyperphysics>[[http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/fluxmg.html|R. Nave, Magnetic Flux, Electricity and Magnetism, HyperPhysics]], {accessed 2022-11-12})][(LibreTexts>[[https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_Physics_(Boundless)/22%3A_Induction_AC_Circuits_and_Electrical_Technologies/22.1%3A_Magnetic_Flux_Induction_and_Faradays_Law|LibreTexts, Physics, 22.1: Magnetic Flux, Induction, and Faraday’s Law]], {accessed 2022-11-12})] If there is some [[magnetic field]] in a given volume, there is always some magnetic flux //Φ// associated with it,[(Jiles)] 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. **Magnetic flux //Φ//** is the product of average normal component of [[magnetic flux density]] //B// and the [[cross-sectional area|surface area]] //A// [(Feynman)] [[file/magnetic_flux_magnetica_png|{{magnetic_flux_magnetica.png}}]] {{page>insert/by_SZ}} 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),[(BIMP_2019>[[https://www.bipm.org/en/publications/si-brochure|Bureau international des poids et mesures, Le Système international d’unités (SI) 9e édition 2019, The International System of Units (SI), 9th edition, 2019, ISBN 978-92-822-2272-0]], {accessed 2021-06-29})] (in the [[CGS system]] the unit was "maxwell", Mx). ^ Relation of magnetic flux //Φ// and its unit weber (Wb) to other values ^^ | (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 (m2), $V$ - 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. {{page>insert/paypal}} ===== Magnetic flux through surface ===== ==== Flux of field ==== Normal vectors (blue, $\hat{n}$) for each infinitesimal fragment of surface $S$ can be used to extract the perpendicular component of the vector field (red, $F$) - this allows calculation of the **flux of vector field**; summation needs to be carried out through all infinitesimal areas $dS$ [[file/surface_integral_definition_chetvorno_png|{{surface_integral_definition_chetvorno.png}}]] //by Chetvorno, [[https://en.wikipedia.org/wiki/File:Surface_integral_-_definition.svg|Wikimedia Commons]], [[https://creativecommons.org/licenses/by/4.0/|CC-BY-4.0]]// 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.[(Feynman)] Mathematically flux is calculated as the [[vector dot product]] of the [[vector field|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 hand rule|right-handed system of coordinates]], positive flux is defined as such that flows out of the given surface.[(Feynman)] ==== Magnetic flux ==== **Magnetic flux //Φ//** due to [[magnetic flux density]] //B// through a closed 3D surface //C// is always zero [[file/magnetic_flux_through_closed_surface_magnetica_png|{{magnetic_flux_through_closed_surface_magnetica.png}}]] {{page>insert/by_SZ}} However, magnetic field in the form of [[magnetic flux density]] //B// is [[solenoidal field|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 monopole|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]][(Feynman)][(Jiles)] ^^ | (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 (m2), //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. **Magnetic flux //Φ//** due to [[magnetic flux density]] //B// through a flat 2D surface with area //A// is related to //B// normal to the surface //A//: a) //B// at some arbitrary angle $α(\vec{B},\hat{a})$, b) parallel to surface so $α(\vec{B},\hat{a})$=90°, c) perpendicular to the surface so $α(\vec{B},\hat{a})$=0°). \\ 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// [[file/magnetic_flux_at_angle_magnetica_png|{{magnetic_flux_at_angle_magnetica.png}}]] {{page>insert/by_SZ}} 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]] //ΦE// with similar rules as described above. And because the electric field is produced by [[electric charge|electric charges]] such as [[electron|electrons]] and [[proton|protons]] (which are by definition "[[electric monopole|electric monopoles]]"), then the value of //ΦE// for a given closed 3D surface quantifies the amount of net electric charge closed by such surface.[(Purcell>[[https://isbnsearch.org/isbn/9781107014022|E.M. Purcell, D.J. Morin, Electricity and magnetism, 3rd edition, Cambridge University Press, 2013, ISBN 9781107014022]])] ==== Flux in ideal closed magnetic circuit ==== In a closed [[magnetic circuit]] made of [[high permeability|high-permeability magnetic material]] it can be typically assumed that all the [[flux lines]] remain confined inside the magnetic core. In a closed [[magnetic circuit]], [[magnetic flux]] can be equated to the product of the average [[magnetic flux density]] //B// and effective [[cross-sectional area]] //A//; note that all the flux lines remain inside the [[magnetic core]] [[file/magnetic_flux_in_magnetic_circuit_magnetica_png|{{magnetic_flux_in_magnetic_circuit_magnetica.png}}]] {{page>insert/by_SZ}} 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.[(Zurek)] ===== 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). [[Faraday's law of induction]] - changing [[magnetic field]] (flux density //B// or flux //Φ//) by moving the [[magnet]] induces [[electromotive force]] EMF (measurable as voltage //V//) in a loop or coil of a conductor; the voltmeter needle deflects in opposite directions if the magnetic field is increasing or decreasing [[file/faraday_law_magnetica_png|{{faraday_law_magnetica.png}}]] {{page>insert/by_SZ}} 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 ^^^ | (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 (m2), $B_{avg}$ - 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).[(Fleisch_Maxwell>[[https://isbnsearch.org/isbn/9780521877619|Daniel Fleisch, A Student’s Guide to Maxwell’s Equations, Cambridge University Press, Cambridge, 2008, ISBN 9780521877619]])] 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.[(Kothari>[[https://isbnsearch.org/isbn/9780070699670|D.P. Kothari, I.J. Nagrath, Electric Machines, 4th edition, The McGraw-Hill Companies, New Delhi, 2010, ISBN 9780070699670]])][(Guru>[[https://isbnsearch.org/isbn/9780195138900|Bhag S. Guru, Huseyin R. Hiziroglu, Electric Machinery and Transformers, 3d edition, Oxford University Press, New York, 2001, ISBN 9780195138900]])][(Hurley>[[https://isbnsearch.org/isbn/9781119950578|W.G. Hurley, W.H. Wölfle, Transformers and Inductors for Power Electronics: Theory, Design and Applications, Wiley, 2013, ISBN 9781119950578]])] Some authors follow the name "flux rule" literally, and teach that the magnetic flux is the only basis of the Faraday's law.[(Sahay>[[https://isbnsearch.org/isbn/9788122418361|Kuldeep Sahay, Basic Concepts of Electrical Engineering, New Age International, 2006, ISBN 9788122418361]])] 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 [[series connection|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. Intuitive illustration of the [[vector_calculus#stokes_curl_theorem|Stokes' curl theorem]],[(Griffiths)] which shows how the vector can "circulate" around the area of interest [[file/stokes_curl_theorem_magnetica_png|{{stokes_curl_theorem_magnetica.png}}]] {{page>insert/by_SZ}} The Faraday's law relates the changes in //B// or //Φ// to the induced voltage by such changes. This induced voltage "circulates"[(Griffiths>[[http://books.google.com/books?isbn=0321856562|David J. Griffiths, Introduction to electrodynamics, 4th ed., Pearson, Boston, 2013, ISBN 0321856562]])] 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 [(Fleisch_Maxwell)] ^^ | (4) | $$ \vec{∇} × \vec{E} = - \frac{∂ \vec{B}}{∂t} $$ | | where: [[curl|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 [[conductor|electrically conductive materials]] (whether they are [[magnetic materials|magnetic or not]]). ==== Measurement of Φ and B ==== Examples of a single-turn [[search coil|search coils]] perpendicular to each other which allow measuring the //Bx// and //By// components; the wire is wrapped through the small holes drilled in the sample ([[lamination]]) under test[(Zurek)] [[file/2_orthogonal_b-coils_jpg|{{2dmch3/2_orthogonal_b-coils.jpg}}]] {{page>insert/by_SZ}} The Faraday's law of induction can be employed for measuring magnetic flux density //B// or magnetic flux //Φ//. [(Tumanski>[[https://isbnsearch.org/isbn/9780367864958|Sławomir Tumański, Handbook of magnetic measurements, CRC Press / Taylor & Francis, Boca Raton, FL, 2011, ISBN 9780367864958]])][(Zurek>[[https://isbnsearch.org/isbn/9780367891572|S. Zurek, Characterisation of Soft Magnetic Materials Under Rotational Magnetisation, CRC Press, 2019, ISBN 9780367891572]])] 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 //Bavg// penetrating the coil. Therefore, even if the magnetic material [[magnetic saturation|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.[(Zurek)] 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]].[(Zurek)]/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[(IEC60404_6_2003>[[https://webstore.iec.ch/publication/2078|IEC 60404-6:2003, Magnetic materials - Part 6: Methods of measurement of the magnetic properties of magnetically soft metallic and powder materials at frequencies in the range 20 Hz to 200 kHz by the use of ring specimens]], {accessed 2022-12-30})], or for the EMF without the minus as in some books.[(Kothari)] ^ Measurement of magnetic flux //Φ// or magnetic flux density //B// with a search coil ^^^ | (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]] (m2) 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.[(Zurek)]/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]].[(Hurley)] ==== Self-inductance ==== Field lines in a solenoid (cross-section view) [[file/h_around_solenoid_section_png|{{2dm/h_around_solenoid_section.png}}]] {{page>insert/by_SZ}} [[Magnetic field lines]] around a [[magnetic dipole]] [[file/dipole_2_png|{{dipole_2.png}}]] \\ //by Geek3 and S. Zurek, [[https://creativecommons.org/licenses/by-sa/3.0|CC-BY-SA-3.0]]// 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.[(Paul>[[https://isbnsearch.org/isbn/9780470461884|Clayton R. Paul, Inductance, Loop and Partial, John Wiley & Sons, IEEE Press, 2009, ISBN 9780470461884]])][(Furlani>[[https://isbnsearch.org/isbn/0122699513|Edward P. Furlani, Permanent magnet and electromechanical devices, Academic Press, London, 2021, ISBN 0122699513]])] ^ [[Self-inductance]] of a simple [[coil]] [(Hurley)][(Furlani)] ^^^ | (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]] (m2), $H$ - [[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 ==== Illustration of self-flux ($Φ_{11}$ and $Φ_{22}$) and mutual flux ($Φ_{12}$ and $Φ_{21}$), giving rise to **self-inductance** and **mutual inductance**, respectively; if the coils are identical then the mutual fluxes between them are equal [[file/self_and_mutual_inductance_magnetica_png|{{self_and_mutual_inductance_magnetica.png}}]] {{page>insert/by_SZ}} 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]] //LM// 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// [(Hurley)] ^^^ | (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// [(Kothari)] ^^^ | (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.[(Analog>[[https://www.analog.com/media/en/technical-documentation/lt-journal-article/LTJournal-V24N1-04-di-LTspice-GabinoAlonso.pdf|Gabino Alonso, What’s New with LTspice IV?, LT Journal of Analog Innovation, April 2014, p. 20]], {accessed 2023-09-30})][(PSpice>[[https://www.pspice.com/resources/application-notes/using-inductor-coupling-symbols|Using the Inductor Coupling Symbols, Application Note, Cadence Design Systems, 2016]], {accessed 2023-09-30})][(Qorvo>[[https://forum.qorvo.com/t/magnetic-coupling-element/15516|Magnetic coupling element - QSPICE - Qorvo Tech Forum]], {accessed 2023-09-30})] ===== See also ===== * [[Magnetic flux density]] * [[Magnetic field]] * [[Flux of field]] ===== References ===== ~~REFNOTES~~ {{tag> Magnetic_flux Magnetic_flux_density Magnetic_field Counter}}