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mw-ui-icon-wikimedia-expand"></span> <span>Toggle Resistance–reluctance model subsection</span> </button> <ul id="toc-Resistance–reluctance_model-sublist" class="vector-toc-list"> <li id="toc-Hopkinson's_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hopkinson's_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Hopkinson's law</span> </div> </a> <ul id="toc-Hopkinson's_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reluctance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reluctance"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Reluctance</span> </div> </a> <ul id="toc-Reluctance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Permeability_and_conductivity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Permeability_and_conductivity"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Permeability and conductivity</span> </div> </a> <ul id="toc-Permeability_and_conductivity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Summary_of_analogy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Summary_of_analogy"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Summary of analogy</span> </div> </a> <ul id="toc-Summary_of_analogy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limitations_of_the_analogy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limitations_of_the_analogy"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Limitations of the analogy</span> </div> </a> <ul id="toc-Limitations_of_the_analogy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Circuit_laws" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Circuit_laws"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Circuit laws</span> </div> </a> <ul id="toc-Circuit_laws-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Magnetic circuit</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" 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Available in 25 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-25" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">25 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D8%A6%D8%B1%D8%A9_%D9%85%D8%BA%D9%86%D8%A7%D8%B7%D9%8A%D8%B3%D9%8A%D8%A9" title="دائرة مغناطيسية – Arabic" lang="ar" hreflang="ar" data-title="دائرة مغناطيسية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9A%E0%A7%8C%E0%A6%AE%E0%A7%8D%E0%A6%AC%E0%A6%95_%E0%A6%AC%E0%A6%B0%E0%A7%8D%E0%A6%A4%E0%A6%A8%E0%A7%80" title="চৌম্বক বর্তনী – Bangla" lang="bn" hreflang="bn" data-title="চৌম্বক বর্তনী" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D0%B8%D1%82%D0%BD%D0%B0_%D0%B2%D0%B5%D1%80%D0%B8%D0%B3%D0%B0" title="Магнитна верига – Bulgarian" lang="bg" hreflang="bg" data-title="Магнитна верига" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Circuit_magn%C3%A8tic" title="Circuit magnètic – Catalan" lang="ca" hreflang="ca" data-title="Circuit magnètic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D0%B8%D1%82%D0%BB%D0%B0_%D0%B2%C4%83%D1%87%C4%83%D1%80%D0%B0" title="Магнитла вăчăра – Chuvash" lang="cv" hreflang="cv" data-title="Магнитла вăчăра" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Magnetischer_Kreis" title="Magnetischer Kreis – German" lang="de" hreflang="de" data-title="Magnetischer Kreis" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Magnetahel" title="Magnetahel – Estonian" lang="et" hreflang="et" data-title="Magnetahel" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B1%CE%B3%CE%BD%CE%B7%CF%84%CE%B9%CE%BA%CF%8C_%CE%BA%CF%8D%CE%BA%CE%BB%CF%89%CE%BC%CE%B1" title="Μαγνητικό κύκλωμα – Greek" lang="el" hreflang="el" data-title="Μαγνητικό κύκλωμα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Circuito_magn%C3%A9tico" title="Circuito magnético – Spanish" lang="es" hreflang="es" data-title="Circuito magnético" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Zirkuitu_magnetiko" title="Zirkuitu magnetiko – Basque" lang="eu" hreflang="eu" data-title="Zirkuitu magnetiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AF%D8%A7%D8%B1_%D9%85%D8%BA%D9%86%D8%A7%D8%B7%DB%8C%D8%B3%DB%8C" title="مدار مغناطیسی – Persian" lang="fa" hreflang="fa" data-title="مدار مغناطیسی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Circuit_magn%C3%A9tique" title="Circuit magnétique – French" lang="fr" hreflang="fr" data-title="Circuit magnétique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%A3%D5%B6%D5%AB%D5%BD%D5%A1%D5%AF%D5%A1%D5%B6_%D5%B7%D5%B2%D5%A9%D5%A1" title="Մագնիսական շղթա – Armenian" lang="hy" hreflang="hy" data-title="Մագնիսական շղթա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%9A%E0%A5%81%E0%A4%AE%E0%A5%8D%E0%A4%AC%E0%A4%95%E0%A5%80%E0%A4%AF_%E0%A4%AA%E0%A4%B0%E0%A4%BF%E0%A4%AA%E0%A4%A5" title="चुम्बकीय परिपथ – Hindi" lang="hi" hreflang="hi" data-title="चुम्बकीय परिपथ" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Circuito_magnetico" title="Circuito magnetico – Italian" lang="it" hreflang="it" data-title="Circuito magnetico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%95%E0%B2%BE%E0%B2%82%E0%B2%A4_%E0%B2%AE%E0%B2%82%E0%B2%A1%E0%B2%B2" title="ಕಾಂತ ಮಂಡಲ – Kannada" lang="kn" hreflang="kn" data-title="ಕಾಂತ ಮಂಡಲ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Magn%C4%93tisk%C4%81_%C4%B7%C4%93de" title="Magnētiskā ķēde – Latvian" lang="lv" hreflang="lv" data-title="Magnētiskā ķēde" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%A3%81%E6%B0%97%E5%9B%9E%E8%B7%AF" title="磁気回路 – Japanese" lang="ja" hreflang="ja" data-title="磁気回路" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Magnetisk_krets" title="Magnetisk krets – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Magnetisk krets" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Magnetow%C3%B3d" title="Magnetowód – Polish" lang="pl" hreflang="pl" data-title="Magnetowód" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D0%B8%D1%82%D0%BD%D0%B0%D1%8F_%D1%86%D0%B5%D0%BF%D1%8C" title="Магнитная цепь – Russian" lang="ru" hreflang="ru" data-title="Магнитная цепь" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Magnetisk_krets" title="Magnetisk krets – Swedish" lang="sv" hreflang="sv" data-title="Magnetisk krets" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D1%96%D1%82%D0%BD%D0%B5_%D0%BA%D0%BE%D0%BB%D0%BE" title="Магнітне коло – Ukrainian" lang="uk" hreflang="uk" data-title="Магнітне коло" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D9%82%D9%86%D8%A7%D8%B7%DB%8C%D8%B3%DB%8C_%D8%AF%D9%88%D8%B1" title="مقناطیسی دور – Urdu" lang="ur" hreflang="ur" data-title="مقناطیسی دور" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%A3%81%E8%B7%AF" title="磁路 – Chinese" lang="zh" hreflang="zh" data-title="磁路" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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href="/wiki/File:Electromagnet_with_gap.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Electromagnet_with_gap.svg/370px-Electromagnet_with_gap.svg.png" decoding="async" width="370" height="406" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Electromagnet_with_gap.svg/555px-Electromagnet_with_gap.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/Electromagnet_with_gap.svg/740px-Electromagnet_with_gap.svg.png 2x" data-file-width="674" data-file-height="739" /></a><figcaption>Magnetic field <i>(<span style="color:green;">green</span>)</i> induced by a current-carrying wire winding <i>(<span style="color:red;">red</span>)</i> in a magnetic circuit consisting of an iron core <i>C</i> forming a closed loop with two air gaps <i>G</i> in it. In an analogy to an electric circuit, the winding acts analogously to an <a href="/wiki/Electric_battery" title="Electric battery">electric battery</a>, providing the magnetizing field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span>, the core pieces act like wires, and the gaps <i>G</i> act like <a href="/wiki/Resistor" title="Resistor">resistors</a>.<br /> <i>B</i> – magnetic field in the core<br /> <i>B<sub>F</sub></i> – "fringing fields". In the gaps <i>G</i> the electric field lines "bulge" out, so the field strength is less than in the core: <i>B<sub>F</sub></i> < <i>B</i><br /> <i>B<sub>L</sub></i> – <a href="/wiki/Leakage_flux" class="mw-redirect" title="Leakage flux">leakage flux</a>; magnetic field lines which don't follow complete magnetic circuit<br /> <i>L</i> – average length of the magnetic circuit. It is the sum of the length <i>L<sub>core</sub></i> in the iron core pieces and the length <i>L<sub>gap</sub></i> in the air gaps <i>G</i>.<br /> </figcaption></figure> <p>A <b>magnetic circuit</b> is made up of one or more closed loop paths containing a <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a>. The flux is usually generated by <a href="/wiki/Permanent_magnet" class="mw-redirect" title="Permanent magnet">permanent magnets</a> or <a href="/wiki/Electromagnet" title="Electromagnet">electromagnets</a> and confined to the path by <a href="/wiki/Magnetic_core" title="Magnetic core">magnetic cores</a> consisting of <a href="/wiki/Ferromagnetic_material" class="mw-redirect" title="Ferromagnetic material">ferromagnetic materials</a> like iron, although there may be air gaps or other materials in the path. Magnetic circuits are employed to efficiently channel <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic fields</a> in many devices such as <a href="/wiki/Electric_motor" title="Electric motor">electric motors</a>, <a href="/wiki/Electric_generator" title="Electric generator">generators</a>, <a href="/wiki/Transformer" title="Transformer">transformers</a>, <a href="/wiki/Relay" title="Relay">relays</a>, lifting <a href="/wiki/Electromagnet" title="Electromagnet">electromagnets</a>, <a href="/wiki/SQUID" title="SQUID">SQUIDs</a>, <a href="/wiki/Galvanometer" title="Galvanometer">galvanometers</a>, and magnetic <a href="/wiki/Recording_head" title="Recording head">recording heads</a>. </p><p>The relation between <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a>, <a href="/wiki/Magnetomotive_force" title="Magnetomotive force">magnetomotive force</a>, and <a href="/wiki/Magnetic_reluctance" title="Magnetic reluctance">magnetic reluctance</a> in an <a href="/wiki/Magnetic_saturation" class="mw-redirect" title="Magnetic saturation">unsaturated</a> magnetic circuit can be described by <a href="/wiki/Hopkinson%27s_law" class="mw-redirect" title="Hopkinson's law">Hopkinson's law</a>, which bears a superficial resemblance to <a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm's law</a> in electrical circuits, resulting in a one-to-one correspondence between properties of a magnetic circuit and an analogous electric circuit. Using this concept the magnetic fields of complex devices such as <a href="/wiki/Transformer" title="Transformer">transformers</a> can be quickly solved using the methods and techniques developed for electrical circuits. </p><p>Some examples of magnetic circuits are: </p> <ul><li><a href="/wiki/Horseshoe" title="Horseshoe">horseshoe</a> <a href="/wiki/Magnet" title="Magnet">magnet</a> with iron <a href="/wiki/Magnet_keeper" title="Magnet keeper">keeper</a> (low-<a href="/wiki/Reluctance" class="mw-redirect" title="Reluctance">reluctance</a> circuit)</li> <li>horseshoe magnet with no keeper (high-reluctance circuit)</li> <li><a href="/wiki/Electric_motor" title="Electric motor">electric motor</a> (variable-reluctance circuit)</li> <li>some types of <a href="/wiki/Magnetic_cartridge" title="Magnetic cartridge">pickup cartridge</a> (variable-reluctance circuits)</li></ul> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Magnetomotive_force_(MMF)"><span id="Magnetomotive_force_.28MMF.29"></span>Magnetomotive force (MMF)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Magnetic_circuit&action=edit&section=1" title="Edit section: Magnetomotive force (MMF)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Magnetomotive_force" title="Magnetomotive force">magnetomotive force</a></div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output 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title="Gauss's law for magnetism">Gauss magnetic law</a></li> <li><a href="/wiki/Magnetic_moment" title="Magnetic moment">Magnetic dipole</a></li> <li><a href="/wiki/Magnetic_field" title="Magnetic field">Magnetic field</a></li> <li><a href="/wiki/Magnetic_flux" title="Magnetic flux">Magnetic flux</a></li> <li><a href="/wiki/Magnetic_scalar_potential" title="Magnetic scalar potential">Magnetic scalar potential</a></li> <li><a href="/wiki/Magnetic_vector_potential" title="Magnetic vector potential">Magnetic vector potential</a></li> <li><a href="/wiki/Magnetization" title="Magnetization">Magnetization</a></li> <li><a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">Permeability</a></li> <li><a href="/wiki/Right-hand_rule#Electromagnetism" title="Right-hand rule">Right-hand rule</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" 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title="Electromagnetic radiation">Electromagnetic radiation</a></li> <li><a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday law</a></li> <li><a href="/wiki/Jefimenko%27s_equations" title="Jefimenko's equations">Jefimenko equations</a></li> <li><a href="/wiki/Larmor_formula" title="Larmor formula">Larmor formula</a></li> <li><a href="/wiki/Lenz%27s_law" title="Lenz's law">Lenz law</a></li> <li><a href="/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential" title="Liénard–Wiechert potential">Liénard–Wiechert potential</a></li> <li><a href="/wiki/London_equations" title="London equations">London equations</a></li> <li><a href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell equations</a></li> <li><a href="/wiki/Maxwell_stress_tensor" title="Maxwell stress tensor">Maxwell tensor</a></li> <li><a href="/wiki/Poynting_vector" title="Poynting vector">Poynting vector</a></li> <li><a href="/wiki/Synchrotron_radiation" title="Synchrotron radiation">Synchrotron radiation</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Electrical_network" title="Electrical network">Electrical network</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Alternating_current" title="Alternating current">Alternating current</a></li> <li><a href="/wiki/Capacitance" title="Capacitance">Capacitance</a></li> <li><a href="/wiki/Current_density" title="Current density">Current density</a></li> <li><a href="/wiki/Direct_current" title="Direct current">Direct current</a></li> <li><a href="/wiki/Electric_current" title="Electric current">Electric current</a></li> <li><a href="/wiki/Electric_power" title="Electric power">Electric power</a></li> <li><a href="/wiki/Electrolysis" title="Electrolysis">Electrolysis</a></li> <li><a href="/wiki/Electromotive_force" title="Electromotive force">Electromotive force</a></li> <li><a href="/wiki/Electrical_impedance" title="Electrical impedance">Impedance</a></li> <li><a href="/wiki/Inductance" title="Inductance">Inductance</a></li> <li><a href="/wiki/Joule_heating" title="Joule heating">Joule heating</a></li> <li><a href="/wiki/Kirchhoff%27s_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff laws</a></li> <li><a href="/wiki/Network_analysis_(electrical_circuits)" title="Network analysis (electrical circuits)">Network analysis</a></li> <li><a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm law</a></li> <li><a href="/wiki/Series_and_parallel_circuits#Parallel_circuits" title="Series and parallel circuits">Parallel circuit</a></li> <li><a href="/wiki/Electrical_resistance_and_conductance" title="Electrical resistance and conductance">Resistance</a></li> <li><a href="/wiki/Resonator#Electromagnetics" title="Resonator">Resonant cavities</a></li> <li><a href="/wiki/Series_and_parallel_circuits#Series_circuits" title="Series and parallel circuits">Series circuit</a></li> <li><a href="/wiki/Voltage" title="Voltage">Voltage</a></li> <li><a href="/wiki/Watt" title="Watt">Watt</a></li> <li><a href="/wiki/Waveguide_(radio_frequency)" title="Waveguide (radio frequency)">Waveguides</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a class="mw-selflink selflink">Magnetic circuit</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/AC_motor" title="AC motor">AC motor</a></li> <li><a href="/wiki/DC_motor" title="DC motor">DC motor</a></li> <li><a href="/wiki/Electric_machine" title="Electric machine">Electric machine</a></li> <li><a href="/wiki/Electric_motor" title="Electric motor">Electric motor</a></li> <li><a href="/wiki/Gyrator%E2%80%93capacitor_model" title="Gyrator–capacitor model">Gyrator–capacitor</a></li> <li><a href="/wiki/Induction_motor" title="Induction motor">Induction motor</a></li> <li><a href="/wiki/Linear_motor" title="Linear motor">Linear motor</a></li> <li><a href="/wiki/Magnetomotive_force" title="Magnetomotive force">Magnetomotive force</a></li> <li><a href="/wiki/Permeance" title="Permeance">Permeance</a></li> <li><a href="/wiki/Magnetic_complex_reluctance" title="Magnetic complex reluctance">Reluctance (complex)</a></li> <li><a href="/wiki/Magnetic_reluctance" title="Magnetic reluctance">Reluctance (real)</a></li> <li><a href="/wiki/Rotor_(electric)" title="Rotor (electric)">Rotor</a></li> <li><a href="/wiki/Stator" title="Stator">Stator</a></li> <li><a href="/wiki/Transformer" title="Transformer">Transformer</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Covariant_formulation_of_classical_electromagnetism" title="Covariant formulation of classical electromagnetism">Covariant formulation</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">Electromagnetic tensor</a></li> <li><a href="/wiki/Classical_electromagnetism_and_special_relativity" title="Classical electromagnetism and special relativity">Electromagnetism and special relativity</a></li> <li><a href="/wiki/Four-current" title="Four-current">Four-current</a></li> <li><a href="/wiki/Electromagnetic_four-potential" title="Electromagnetic four-potential">Four-potential</a></li> <li><a href="/wiki/Mathematical_descriptions_of_the_electromagnetic_field" title="Mathematical descriptions of the electromagnetic field">Mathematical descriptions</a></li> <li><a href="/wiki/Maxwell%27s_equations_in_curved_spacetime" title="Maxwell's equations in curved spacetime">Maxwell equations in curved spacetime</a></li> <li><a href="/wiki/Relativistic_electromagnetism" title="Relativistic electromagnetism">Relativistic electromagnetism</a></li> <li><a href="/wiki/Electromagnetic_stress%E2%80%93energy_tensor" title="Electromagnetic stress–energy tensor">Stress–energy tensor</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampère</a></li> <li><a href="/wiki/Jean-Baptiste_Biot" title="Jean-Baptiste Biot">Biot</a></li> <li><a href="/wiki/Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Coulomb</a></li> <li><a href="/wiki/Humphry_Davy" title="Humphry Davy">Davy</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Michael_Faraday" title="Michael Faraday">Faraday</a></li> <li><a href="/wiki/Hippolyte_Fizeau" title="Hippolyte Fizeau">Fizeau</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Heaviside</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz</a></li> <li><a href="/wiki/Joseph_Henry" title="Joseph Henry">Henry</a></li> <li><a href="/wiki/Heinrich_Hertz" title="Heinrich Hertz">Hertz</a></li> <li><a href="/wiki/John_Hopkinson" title="John Hopkinson">Hopkinson</a></li> <li><a href="/wiki/Oleg_D._Jefimenko" title="Oleg D. Jefimenko">Jefimenko</a></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li> <li><a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Kirchhoff</a></li> <li><a href="/wiki/Joseph_Larmor" title="Joseph Larmor">Larmor</a></li> <li><a href="/wiki/Emil_Lenz" title="Emil Lenz">Lenz</a></li> <li><a href="/wiki/Alfred-Marie_Li%C3%A9nard" title="Alfred-Marie Liénard">Liénard</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Franz_Ernst_Neumann" title="Franz Ernst Neumann">Neumann</a></li> <li><a href="/wiki/Georg_Ohm" title="Georg Ohm">Ohm</a></li> <li><a href="/wiki/Hans_Christian_%C3%98rsted" title="Hans Christian Ørsted">Ørsted</a></li> <li><a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li> <li><a href="/wiki/John_Henry_Poynting" title="John Henry Poynting">Poynting</a></li> <li><a href="/wiki/William_Ritchie_(physicist)" title="William Ritchie (physicist)">Ritchie</a></li> <li><a href="/wiki/F%C3%A9lix_Savart" title="Félix Savart">Savart</a></li> <li><a href="/wiki/George_Singer" title="George Singer">Singer</a></li> <li><a href="/wiki/Charles_Proteus_Steinmetz" title="Charles Proteus Steinmetz">Steinmetz</a></li> <li><a href="/wiki/Nikola_Tesla" title="Nikola Tesla">Tesla</a></li> <li><a href="/wiki/J._J._Thomson" title="J. J. Thomson">Thomson</a></li> <li><a href="/wiki/Alessandro_Volta" title="Alessandro Volta">Volta</a></li> <li><a href="/wiki/Wilhelm_Eduard_Weber" title="Wilhelm Eduard Weber">Weber</a></li> <li><a href="/wiki/Emil_Wiechert" title="Emil Wiechert">Wiechert</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Electromagnetism" title="Template:Electromagnetism"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Electromagnetism" title="Template talk:Electromagnetism"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Electromagnetism" title="Special:EditPage/Template:Electromagnetism"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>Similar to the way that electromotive force (<a href="/wiki/Electromotive_force" title="Electromotive force">EMF</a>) drives a current of electrical charge in electrical circuits, <a href="/wiki/Magnetomotive_force" title="Magnetomotive force">magnetomotive force</a> (MMF) 'drives' magnetic flux through magnetic circuits. The term 'magnetomotive force', though, is a misnomer since it is not a force nor is anything moving. It is perhaps better to call it simply MMF. In analogy to the definition of <a href="/wiki/Electromotive_force" title="Electromotive force">EMF</a>, the magnetomotive force <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"></span> around a closed loop is defined as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}=\oint \mathbf {H} \cdot \mathrm {d} \mathbf {l} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>=</mo> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">l</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}=\oint \mathbf {H} \cdot \mathrm {d} \mathbf {l} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aeb3302906799dd15c6dc22a5ba96a66c167484d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.058ex; height:5.676ex;" alt="{\displaystyle {\mathcal {F}}=\oint \mathbf {H} \cdot \mathrm {d} \mathbf {l} .}"></span></dd></dl> <p>The MMF represents the potential that a hypothetical <a href="/wiki/Magnetic_monopole" title="Magnetic monopole">magnetic charge</a> would gain by completing the loop. The magnetic flux that is driven is <b>not</b> a <a href="/wiki/Magnetic_current" title="Magnetic current">current of magnetic charge</a>; it merely has the same relationship to MMF that electric current has to EMF. (See microscopic origins of reluctance below for a further description.) </p><p>The unit of magnetomotive force is the <a href="/wiki/Ampere-turn" title="Ampere-turn">ampere-turn</a> (At), represented by a steady, direct <a href="/wiki/Electric_current" title="Electric current">electric current</a> of one <a href="/wiki/Ampere" title="Ampere">ampere</a> flowing in a single-turn loop of electrically conducting material in a <a href="/wiki/Vacuum" title="Vacuum">vacuum</a>. The gilbert (Gb), established by the <a href="/wiki/International_Electrotechnical_Commission" title="International Electrotechnical Commission">IEC</a> in 1930,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> is the <a href="/wiki/CGS" class="mw-redirect" title="CGS">CGS</a> unit of magnetomotive force and is a slightly smaller unit than the ampere-turn. The unit is named after <a href="/wiki/William_Gilbert_(astronomer)" class="mw-redirect" title="William Gilbert (astronomer)">William Gilbert</a> (1544–1603) English physician and natural philosopher. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}1\;{\text{Gb}}&={\frac {10}{4\pi }}\;{\text{At}}\\[2pt]&\approx 0.795775\;{\text{At}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.5em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>1</mn> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>Gb</mtext> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>10</mn> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>At</mtext> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≈</mo> <mn>0.795775</mn> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>At</mtext> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}1\;{\text{Gb}}&={\frac {10}{4\pi }}\;{\text{At}}\\[2pt]&\approx 0.795775\;{\text{At}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/878d429a721eebfdbb12e72667b77d46b5f2ca6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:20.851ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}1\;{\text{Gb}}&={\frac {10}{4\pi }}\;{\text{At}}\\[2pt]&\approx 0.795775\;{\text{At}}\end{aligned}}}"></span><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></dd></dl> <p>The magnetomotive force can often be quickly calculated using <a href="/wiki/Amp%C3%A8re%27s_circuital_law" title="Ampère's circuital law">Ampère's law</a>. For example, the magnetomotive force <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"></span> of a long coil is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}=NI}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>=</mo> <mi>N</mi> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}=NI}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7005f7c37d60a86a05e84668c699420e3dbd4e3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.26ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}=NI}"></span></dd></dl> <p>where <i>N</i> is the number of <a href="/wiki/Turn_(geometry)" class="mw-redirect" title="Turn (geometry)">turns</a> and <i>I</i> is the current in the coil. In practice this equation is used for the MMF of real <a href="/wiki/Inductor" title="Inductor">inductors</a> with <i>N</i> being the <a href="/wiki/Winding_number" title="Winding number">winding number</a> of the inducting coil. </p> <div class="mw-heading mw-heading2"><h2 id="Magnetic_flux">Magnetic flux</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Magnetic_circuit&action=edit&section=2" title="Edit section: Magnetic flux"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Magnetic_flux" title="Magnetic flux">Magnetic flux</a></div> <p>An applied MMF 'drives' <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a> through the magnetic components of the system. The magnetic flux through a magnetic component is proportional to the number of <a href="/wiki/Magnetic_field#Magnetic_field_lines" title="Magnetic field">magnetic field lines</a> that pass through the cross sectional area of that component. This is the <i>net</i> number, i.e. the number passing through in one direction, minus the number passing through in the other direction. The direction of the magnetic field vector <i><b>B</b></i> is by definition from the south to the north pole of a magnet inside the magnet; outside the field lines go from north to south. </p><p>The <a href="/wiki/Flux" title="Flux">flux</a> through an element of <a href="/wiki/Area" title="Area">area</a> <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> to the direction of magnetic field is given by the product of the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> and the <a href="/wiki/Area" title="Area">area</a> element. More generally, magnetic flux Φ is defined by a <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar product</a> of the magnetic field and the area element vector. Quantitatively, the magnetic flux through a surface <i>S</i> is defined as the <a href="/wiki/Integral" title="Integral">integral</a> of the magnetic field over the area of the surface </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{m}=\iint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{m}=\iint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80fd0c6b25d2a7e459a345cc1b5d0e2f74cf062a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.655ex; height:5.676ex;" alt="{\displaystyle \Phi _{m}=\iint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {S} .}"></span></dd></dl> <p>For a magnetic component the area <i><b>S</b></i> used to calculate the magnetic flux Φ is usually chosen to be the cross-sectional area of the component. </p><p>The <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> <a href="/wiki/Unit_of_measurement" title="Unit of measurement">unit</a> of magnetic flux is the <a href="/wiki/Weber_(unit)" title="Weber (unit)">weber</a> (in derived units: volt-seconds), and the unit of magnetic flux density (or "magnetic induction", <span class="texhtml mvar" style="font-style:italic;">B</span>) is the weber per square meter, or <a href="/wiki/Tesla_(unit)" title="Tesla (unit)">tesla</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Circuit_models">Circuit models</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Magnetic_circuit&action=edit&section=3" title="Edit section: Circuit models"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The most common way of representing a magnetic circuit is the resistance–reluctance model, which draws an analogy between electrical and magnetic circuits. This model is good for systems that contain only magnetic components, but for modelling a system that contains both electrical and magnetic parts it has serious drawbacks. It does not properly model power and energy flow between the electrical and magnetic domains. This is because electrical resistance will dissipate energy whereas magnetic reluctance stores it and returns it later. An alternative model that correctly models energy flow is the <a href="/wiki/Gyrator%E2%80%93capacitor_model" title="Gyrator–capacitor model">gyrator–capacitor model</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Resistance–reluctance_model"><span id="Resistance.E2.80.93reluctance_model"></span>Resistance–reluctance model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Magnetic_circuit&action=edit&section=4" title="Edit section: Resistance–reluctance model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The resistance–reluctance model for magnetic circuits is a <a href="/wiki/Lumped-element_model" title="Lumped-element model">lumped-element model</a> that makes electrical resistance analogous to magnetic <a href="/wiki/Reluctance" class="mw-redirect" title="Reluctance">reluctance</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Hopkinson's_law"><span id="Hopkinson.27s_law"></span>Hopkinson's law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Magnetic_circuit&action=edit&section=5" title="Edit section: Hopkinson's law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In electrical circuits, <a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm's law</a> is an empirical relation between the <a href="/wiki/Electromotive_force" title="Electromotive force">EMF</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}"></span> applied across an element and the <a href="/wiki/Current_(electricity)" class="mw-redirect" title="Current (electricity)">current</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span> it generates through that element. It is written as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}=IR.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mo>=</mo> <mi>I</mi> <mi>R</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}=IR.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac148123ce562ae4faf354489b26d261a671168d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.992ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}=IR.}"></span> where <i>R</i> is the <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">electrical resistance</a> of that material. There is a counterpart to <a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm's law</a> used in magnetic circuits. This law is often called <b>Hopkinson's law</b>, after <a href="/wiki/John_Hopkinson" title="John Hopkinson">John Hopkinson</a>, but was actually formulated earlier by <a href="/wiki/Henry_Augustus_Rowland" title="Henry Augustus Rowland">Henry Augustus Rowland</a> in 1873.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> It states that<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}=\Phi {\mathcal {R}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}=\Phi {\mathcal {R}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21bc4c017bb5bf2a1a3373ba6cbb2eddbf3336e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.321ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}=\Phi {\mathcal {R}}.}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"></span> is the magnetomotive force (MMF) across a magnetic element, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span> is the <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a> through the magnetic element, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}"></span> is the <a href="/wiki/Magnetic_reluctance" title="Magnetic reluctance">magnetic reluctance</a> of that element. (It will be shown later that this relationship is due to the empirical relationship between the <i><b>H</b></i>-field and the magnetic field <i><b>B</b></i>, <span class="nowrap"><i><b>B</b></i> = <i>μ<b>H</b></i></span>, where <i>μ</i> is the <a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">permeability</a> of the material). Like Ohm's law, Hopkinson's law can be interpreted either as an empirical equation that works for some materials, or it may serve as a definition of reluctance. </p><p>Hopkinson's law is not a correct analogy with Ohm's law in terms of modelling power and energy flow. In particular, there is no power dissipation associated with a magnetic reluctance in the same way as there is a dissipation in an electrical resistance. The magnetic resistance that is a true analogy of electrical resistance in this respect is defined as the ratio of magnetomotive force and the rate of change of magnetic flux. Here rate of change of magnetic flux is standing in for electric current and the Ohm's law analogy becomes, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}={\frac {d\Phi }{dt}}R_{\mathrm {m} },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}={\frac {d\Phi }{dt}}R_{\mathrm {m} },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f5ca3a4feb158ae7d770b232f2426d7f3f41600" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.767ex; height:5.509ex;" alt="{\displaystyle {\mathcal {F}}={\frac {d\Phi }{dt}}R_{\mathrm {m} },}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathrm {m} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathrm {m} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3a48d48e165dd949439c8193c6da9686fe9c41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.365ex; height:2.509ex;" alt="{\displaystyle R_{\mathrm {m} }}"></span> is the magnetic resistance. This relationship is part of an electrical-magnetic analogy called the <a href="/wiki/Gyrator-capacitor_model" class="mw-redirect" title="Gyrator-capacitor model">gyrator-capacitor model</a> and is intended to overcome the drawbacks of the reluctance model. The gyrator-capacitor model is, in turn, part of a <a href="/wiki/Mechanical%E2%80%93electrical_analogies#Other_energy_domains" title="Mechanical–electrical analogies">wider group of compatible analogies</a> used to model systems across multiple energy domains. </p> <div class="mw-heading mw-heading3"><h3 id="Reluctance">Reluctance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Magnetic_circuit&action=edit&section=6" title="Edit section: Reluctance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Reluctance" class="mw-redirect" title="Reluctance">Reluctance</a></div> <p><b>Magnetic reluctance</b>, or <b>magnetic resistance</b>, is analogous to <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a> in an <a href="/wiki/Electrical" class="mw-redirect" title="Electrical">electrical</a> <a href="/wiki/Electrical_network" title="Electrical network">circuit</a> (although it does not dissipate magnetic energy). In likeness to the way an <a href="/wiki/Electric_field" title="Electric field">electric field</a> causes an <a href="/wiki/Electric_current" title="Electric current">electric current</a> to follow the <a href="/wiki/Path_of_least_resistance" title="Path of least resistance">path of least resistance</a>, a <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> causes <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a> to follow the path of least magnetic reluctance. It is a <a href="/wiki/Scalar_(physics)" title="Scalar (physics)">scalar</a>, <a href="/wiki/Intensive_and_extensive_properties#Extensive_properties" title="Intensive and extensive properties">extensive quantity</a>, akin to electrical resistance. </p><p>The total reluctance is equal to the ratio of the MMF in a passive magnetic circuit and the <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a> in this circuit. In an AC field, the reluctance is the ratio of the amplitude values for a <a href="/wiki/Sine_wave" title="Sine wave">sinusoidal</a> MMF and magnetic flux. (see <a href="/wiki/Phasor_(sine_waves)" class="mw-redirect" title="Phasor (sine waves)">phasors</a>) </p><p>The definition can be expressed as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}={\frac {\mathcal {F}}{\Phi }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> <mi mathvariant="normal">Φ</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}={\frac {\mathcal {F}}{\Phi }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd4902c1d55edd0f54f3841f3cff74e2f9b0be8b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.479ex; height:5.176ex;" alt="{\displaystyle {\mathcal {R}}={\frac {\mathcal {F}}{\Phi }},}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74532dc308c806964b832df0d0d73352195c2f2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.971ex; height:2.176ex;" alt="{\displaystyle {\mathcal {R}}}"></span> is the reluctance in <a href="/wiki/Ampere-turn" title="Ampere-turn">ampere-turns</a> per <a href="/wiki/Weber_(unit)" title="Weber (unit)">weber</a> (a unit that is equivalent to turns per <a href="/wiki/Henry_(unit)" title="Henry (unit)">henry</a>). </p><p>Magnetic flux always forms a closed loop, as described by <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>, but the path of the loop depends on the reluctance of the surrounding materials. It is concentrated around the path of least reluctance. Air and vacuum have high reluctance, while easily magnetized materials such as <a href="/wiki/Soft_iron" class="mw-redirect" title="Soft iron">soft iron</a> have low reluctance. The concentration of flux in low-reluctance materials forms strong temporary poles and causes mechanical forces that tend to move the materials towards regions of higher flux so it is always an attractive force(pull). </p><p>The inverse of reluctance is called <i><a href="/wiki/Permeance" title="Permeance">permeance</a></i>. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}={\frac {1}{\mathcal {R}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}={\frac {1}{\mathcal {R}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e81244bace92b3993c2475fd2e6fc81ed3bb9f6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.256ex; height:5.343ex;" alt="{\displaystyle {\mathcal {P}}={\frac {1}{\mathcal {R}}}.}"></span> </p><p>Its <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> derived unit is the <a href="/wiki/Henry_(unit)" title="Henry (unit)">henry</a> (the same as the unit of <a href="/wiki/Inductance" title="Inductance">inductance</a>, although the two concepts are distinct). </p> <div class="mw-heading mw-heading3"><h3 id="Permeability_and_conductivity">Permeability and conductivity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Magnetic_circuit&action=edit&section=7" title="Edit section: Permeability and conductivity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The reluctance of a magnetically uniform magnetic circuit element can be calculated as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}={\frac {l}{\mu A}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>l</mi> <mrow> <mi>μ</mi> <mi>A</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}={\frac {l}{\mu A}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f339f598764a9886819ae3c1bba087fbcc52e6a9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.697ex; height:6.009ex;" alt="{\displaystyle {\mathcal {R}}={\frac {l}{\mu A}}.}"></span> where </p> <ul><li><span class="texhtml mvar" style="font-style:italic;">l</span> is the length of the element,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\mu _{r}\mu _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\mu _{r}\mu _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f0ed9f700ac993e9d7a771740a1c35690469d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.332ex; height:2.176ex;" alt="{\displaystyle \mu =\mu _{r}\mu _{0}}"></span> is the <a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">permeability</a> of the material (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{\mathrm {r} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\mathrm {r} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc6a7458faaa5ff5912633d5305ae3c086041a21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.279ex; height:2.176ex;" alt="{\displaystyle \mu _{\mathrm {r} }}"></span> is the relative permeability of the material (dimensionless), and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2fd9b8decb38a3cd158e7b6c0c6e2d987fefcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{0}}"></span> is the permeability of free space), and</li> <li><span class="texhtml mvar" style="font-style:italic;">A</span> is the cross-sectional area of the circuit.</li></ul> <p>This is similar to the equation for electrical resistance in materials, with permeability being analogous to conductivity; the reciprocal of the permeability is known as magnetic reluctivity and is analogous to resistivity. Longer, thinner geometries with low permeabilities lead to higher reluctance. Low reluctance, like low resistance in electric circuits, is generally preferred.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2009)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Summary_of_analogy">Summary of analogy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Magnetic_circuit&action=edit&section=8" title="Edit section: Summary of analogy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following table summarizes the mathematical analogy between electrical circuit theory and magnetic circuit theory. This is mathematical analogy and not a physical one. Objects in the same row have the same mathematical role; the physics of the two theories are very different. For example, current is the flow of electrical charge, while magnetic flux is <b>not</b> the flow of any quantity. </p> <table class="wikitable"> <caption>Analogy between 'magnetic circuits' and electrical circuits </caption> <tbody><tr> <th colspan="3">Magnetic </th> <th rowspan="2"> </th> <th colspan="3">Electric </th></tr> <tr> <th>Name</th> <th>Symbol</th> <th>Units </th> <th>Name</th> <th>Symbol</th> <th>Units </th></tr> <tr> <td><a href="/wiki/Magnetomotive_force" title="Magnetomotive force">Magnetomotive force</a> (MMF)</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}=\int \mathbf {H} \cdot \mathrm {d} \mathbf {l} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">l</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}=\int \mathbf {H} \cdot \mathrm {d} \mathbf {l} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34666c5480d31922dc7ef3c28578bbe5366cdcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.411ex; height:5.676ex;" alt="{\displaystyle {\mathcal {F}}=\int \mathbf {H} \cdot \mathrm {d} \mathbf {l} }"></span></td> <td><a href="/wiki/Ampere-turn" title="Ampere-turn">ampere-turn</a></td> <td></td> <td><a href="/wiki/Electromotive_force" title="Electromotive force">Electromotive force</a> (EMF)</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}=\int \mathbf {E} \cdot \mathrm {d} \mathbf {l} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">l</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}=\int \mathbf {E} \cdot \mathrm {d} \mathbf {l} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cb492bc091f70acf2fc40d25c45cb40cbcb76ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.461ex; height:5.676ex;" alt="{\displaystyle {\mathcal {E}}=\int \mathbf {E} \cdot \mathrm {d} \mathbf {l} }"></span></td> <td><a href="/wiki/Volt" title="Volt">volt</a> </td></tr> <tr> <td><a href="/wiki/Magnetic_field" title="Magnetic field">Magnetic field</a></td> <td><i><b>H</b></i></td> <td><a href="/wiki/Ampere" title="Ampere">ampere</a>/<a href="/wiki/Meter" class="mw-redirect" title="Meter">meter</a></td> <td></td> <td><a href="/wiki/Electric_field" title="Electric field">Electric field</a></td> <td><i><b>E</b></i></td> <td><a href="/wiki/Volt" title="Volt">volt</a>/<a href="/wiki/Meter" class="mw-redirect" title="Meter">meter</a> = <a href="/wiki/Newton_(unit)" title="Newton (unit)">newton</a>/<a href="/wiki/Coulomb" title="Coulomb">coulomb</a> </td></tr> <tr> <td><a href="/wiki/Magnetic_flux" title="Magnetic flux">Magnetic flux</a></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"></span></td> <td><a href="/wiki/Weber_(unit)" title="Weber (unit)">weber</a></td> <td></td> <td><a href="/wiki/Electric_current" title="Electric current">Electric current</a></td> <td><i>I</i></td> <td><a href="/wiki/Ampere" title="Ampere">ampere</a> </td></tr> <tr> <td>Hopkinson's law or Rowland's law</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}=\Phi {\mathcal {R}}_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}=\Phi {\mathcal {R}}_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7176884ed2e075c4b02767ca2605b7e8c73338ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.349ex; height:2.509ex;" alt="{\displaystyle {\mathcal {F}}=\Phi {\mathcal {R}}_{m}}"></span></td> <td><a href="/wiki/Ampere-turn" title="Ampere-turn">ampere-turn</a></td> <td></td> <td><a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm's law</a></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}=IR}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">E</mi> </mrow> </mrow> <mo>=</mo> <mi>I</mi> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}=IR}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1fc2846ab84bf43d432b4f40237167161108341" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.345ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}=IR}"></span></td> <td> </td></tr> <tr> <td><a href="/wiki/Reluctance" class="mw-redirect" title="Reluctance">Reluctance</a></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}_{\mathrm {m} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}_{\mathrm {m} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5103f693f3e98192b6bfd42fa444e7a9d29275c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.572ex; height:2.509ex;" alt="{\displaystyle {\mathcal {R}}_{\mathrm {m} }}"></span></td> <td>1/<a href="/wiki/Henry_(unit)" title="Henry (unit)">henry</a></td> <td></td> <td><a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">Electrical resistance</a></td> <td><i>R</i></td> <td><a href="/wiki/Ohm" title="Ohm">ohm</a> </td></tr> <tr> <td><a href="/wiki/Permeance" title="Permeance">Permeance</a></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}={\frac {1}{{\mathcal {R}}_{\mathrm {m} }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}={\frac {1}{{\mathcal {R}}_{\mathrm {m} }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b68ffae8cd37b201b887da6942a5d34f274b3ffa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.21ex; height:5.509ex;" alt="{\displaystyle {\mathcal {P}}={\frac {1}{{\mathcal {R}}_{\mathrm {m} }}}}"></span></td> <td><a href="/wiki/Henry_(unit)" title="Henry (unit)">henry</a></td> <td></td> <td><a href="/wiki/Electric_conductance" class="mw-redirect" title="Electric conductance">Electric conductance</a></td> <td><i>G</i> = 1/<i>R</i></td> <td>1/<a href="/wiki/Ohm" title="Ohm">ohm</a> = <a href="/wiki/Mho" class="mw-redirect" title="Mho">mho</a> = <a href="/wiki/Siemens_(unit)" title="Siemens (unit)">siemens</a> </td></tr> <tr> <td>Relation between <i><b>B</b></i> and <i><b>H</b></i></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} =\mu \mathbf {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =\mu \mathbf {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a39244f3c9bf55730932029179ce5972a26d72e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.493ex; height:2.676ex;" alt="{\displaystyle \mathbf {B} =\mu \mathbf {H} }"></span></td> <td></td> <td></td> <td>Microscopic Ohm's law</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} =\sigma \mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>=</mo> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} =\sigma \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/292075ba422a0b35e4db48b3f7baaca71e1fdd63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.566ex; height:2.176ex;" alt="{\displaystyle \mathbf {J} =\sigma \mathbf {E} }"></span></td> <td> </td></tr> <tr> <td><a href="/wiki/Magnetic_flux_density" class="mw-redirect" title="Magnetic flux density">Magnetic flux density</a> <i><b>B</b></i></td> <td><i><b>B</b></i></td> <td><a href="/wiki/Tesla_(unit)" title="Tesla (unit)">tesla</a></td> <td></td> <td><a href="/wiki/Current_density" title="Current density">Current density</a></td> <td><i><b>J</b></i></td> <td><a href="/wiki/Ampere" title="Ampere">ampere</a>/<a href="/wiki/Square_meter" class="mw-redirect" title="Square meter">square meter</a> </td></tr> <tr> <td><a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">Permeability</a></td> <td><i>μ</i></td> <td><a href="/wiki/Henry_(unit)" title="Henry (unit)">henry</a>/<a href="/wiki/Meter" class="mw-redirect" title="Meter">meter</a></td> <td></td> <td><a href="/wiki/Electrical_conductivity" class="mw-redirect" title="Electrical conductivity">Electrical conductivity</a></td> <td><i>σ</i></td> <td><a href="/wiki/Siemens_(unit)" title="Siemens (unit)">siemens</a>/<a href="/wiki/Meter" class="mw-redirect" title="Meter">meter</a> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Limitations_of_the_analogy">Limitations of the analogy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Magnetic_circuit&action=edit&section=9" title="Edit section: Limitations of the analogy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The resistance–reluctance model has limitations. Electric and magnetic circuits are only superficially similar because of the similarity between Hopkinson's law and Ohm's law. Magnetic circuits have significant differences that need to be taken into account in their construction: </p> <ul><li>Electric currents represent the flow of particles (electrons) and carry <a href="/wiki/Power_(physics)" title="Power (physics)">power</a>, part or all of which is dissipated as heat in resistances. Magnetic fields don't represent a "flow" of anything, and no power is dissipated in reluctances.</li> <li>The current in typical electric circuits is confined to the circuit, with very little "leakage". In typical magnetic circuits not all of the magnetic field is confined to the magnetic circuit because magnetic permeability also exists outside materials (see <a href="/wiki/Vacuum_permeability" title="Vacuum permeability">vacuum permeability</a>). Thus, there may be significant "<a href="/wiki/Leakage_flux" class="mw-redirect" title="Leakage flux">leakage flux</a>" in the space outside the magnetic cores, which must be taken into account but is often difficult to calculate.</li> <li>Most importantly, magnetic circuits are <a href="/wiki/Nonlinear_element" class="mw-redirect" title="Nonlinear element">nonlinear</a>; the reluctance in a magnetic circuit is not constant, as resistance is, but varies depending on the magnetic field. At high magnetic fluxes the <a href="/wiki/Ferromagnetic_materials" class="mw-redirect" title="Ferromagnetic materials">ferromagnetic materials</a> used for the cores of magnetic circuits <a href="/wiki/Saturation_(magnetic)" title="Saturation (magnetic)">saturate</a>, limiting further increase of the magnetic flux through, so above this level the reluctance increases rapidly. In addition, ferromagnetic materials suffer from <a href="/wiki/Hysteresis" title="Hysteresis">hysteresis</a> so the flux in them depends not just on the instantaneous MMF but also on the history of MMF. After the source of the magnetic flux is turned off, <a href="/wiki/Remanent_magnetism" class="mw-redirect" title="Remanent magnetism">remanent magnetism</a> is left in ferromagnetic materials, creating flux with no MMF.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Circuit_laws">Circuit laws</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Magnetic_circuit&action=edit&section=10" title="Edit section: Circuit laws"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Magnetischer_Kreis.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Magnetischer_Kreis.svg/220px-Magnetischer_Kreis.svg.png" decoding="async" width="220" height="172" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Magnetischer_Kreis.svg/330px-Magnetischer_Kreis.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Magnetischer_Kreis.svg/440px-Magnetischer_Kreis.svg.png 2x" data-file-width="273" data-file-height="213" /></a><figcaption>Magnetic circuit</figcaption></figure> <p>Magnetic circuits obey other laws that are similar to electrical circuit laws. For example, the total reluctance <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}_{\mathrm {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}_{\mathrm {T} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7ad65e14151ca5eac12a32cf6c0b3378289288a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.39ex; height:2.509ex;" alt="{\displaystyle {\mathcal {R}}_{\mathrm {T} }}"></span> of reluctances <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}_{1},\ {\mathcal {R}}_{2},\ \ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}_{1},\ {\mathcal {R}}_{2},\ \ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3b965f71cb356e83e8604e3f5cab95f367123a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.389ex; height:2.509ex;" alt="{\displaystyle {\mathcal {R}}_{1},\ {\mathcal {R}}_{2},\ \ldots }"></span> in series is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}_{\mathrm {T} }={\mathcal {R}}_{1}+{\mathcal {R}}_{2}+\dotsm }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}_{\mathrm {T} }={\mathcal {R}}_{1}+{\mathcal {R}}_{2}+\dotsm }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc94e88e3ed715c399227c9a8420d47f36cd554f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.942ex; height:2.509ex;" alt="{\displaystyle {\mathcal {R}}_{\mathrm {T} }={\mathcal {R}}_{1}+{\mathcal {R}}_{2}+\dotsm }"></span> </p><p>This also follows from <a href="/wiki/Amp%C3%A8re%27s_law" class="mw-redirect" title="Ampère's law">Ampère's law</a> and is analogous to <a href="/wiki/Kirchhoff%27s_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff's voltage law</a> for adding resistances in series. Also, the sum of magnetic fluxes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{1},\ \Phi _{2},\ \ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{1},\ \Phi _{2},\ \ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c33661258175924b04c71d7a941013eb48e48f16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.804ex; height:2.509ex;" alt="{\displaystyle \Phi _{1},\ \Phi _{2},\ \ldots }"></span> into any node is always zero: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{1}+\Phi _{2}+\dotsm =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{1}+\Phi _{2}+\dotsm =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d578977512a783824f391ea915ddf8d9193e6a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.776ex; height:2.509ex;" alt="{\displaystyle \Phi _{1}+\Phi _{2}+\dotsm =0.}"></span> </p><p>This follows from <a href="/wiki/Gauss%27s_law_for_magnetism" title="Gauss's law for magnetism">Gauss's law</a> and is analogous to <a href="/wiki/Kirchhoff%27s_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff's current law</a> for analyzing electrical circuits. </p><p>Together, the three laws above form a complete system for analysing magnetic circuits, in a manner similar to electric circuits. Comparing the two types of circuits shows that: </p> <ul><li>The equivalent to resistance <i>R</i> is the <i>reluctance</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {R}}_{\mathrm {m} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {R}}_{\mathrm {m} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5103f693f3e98192b6bfd42fa444e7a9d29275c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.572ex; height:2.509ex;" alt="{\displaystyle {\mathcal {R}}_{\mathrm {m} }}"></span></li> <li>The equivalent to current <i>I</i> is the <i>magnetic flux</i> Φ</li> <li>The equivalent to voltage <i>V</i> is the <i>magnetomotive Force</i> <i>F</i></li></ul> <p>Magnetic circuits can be solved for the flux in each branch by application of the magnetic equivalent of <a href="/wiki/Kirchhoff%27s_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff's voltage law</a> (<a href="/wiki/KVL" class="mw-redirect" title="KVL">KVL</a>) for pure source/resistance circuits. Specifically, whereas KVL states that the voltage excitation applied to a loop is equal to the sum of the voltage drops (resistance times current) around the loop, the magnetic analogue states that the magnetomotive force (achieved from ampere-turn excitation) is equal to the sum of MMF drops (product of flux and reluctance) across the rest of the loop. (If there are multiple loops, the current in each branch can be solved through a matrix equation—much as a matrix solution for mesh circuit branch currents is obtained in loop analysis—after which the individual branch currents are obtained by adding and/or subtracting the constituent <a href="/wiki/Loop_current" class="mw-redirect" title="Loop current">loop currents</a> as indicated by the adopted sign convention and loop orientations.) Per <a href="/wiki/Amp%C3%A8re%27s_law" class="mw-redirect" title="Ampère's law">Ampère's law</a>, the excitation is the product of the current and the number of complete loops made and is measured in ampere-turns. Stated more generally: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=NI=\oint \mathbf {H} \cdot \mathrm {d} \mathbf {l} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>N</mi> <mi>I</mi> <mo>=</mo> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">l</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=NI=\oint \mathbf {H} \cdot \mathrm {d} \mathbf {l} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd5ad08fdb036678a3e170faa2f9d22428db1d27" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.206ex; height:5.676ex;" alt="{\displaystyle F=NI=\oint \mathbf {H} \cdot \mathrm {d} \mathbf {l} .}"></span> </p><p>By Stokes's theorem, the closed <a href="/wiki/Line_integral" title="Line integral">line integral</a> of <span class="texhtml"><i>H</i>·d<i>l</i></span> around a contour is equal to the open <a href="/wiki/Surface_integral" title="Surface integral">surface integral</a> of curl <span class="texhtml"><b>H</b>·d<b>A</b></span> across the surface bounded by the closed contour. Since, from <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>, <span class="texhtml">curl <b>H</b> = <b>J</b></span>, the closed line integral of <span class="texhtml"><b>H</b>·d<b>l</b></span> evaluates to the total current passing through the surface. This is equal to the excitation, <span class="texhtml"><i>NI</i></span>, which also measures current passing through the surface, thereby verifying that the net current flow through a surface is zero ampere-turns in a closed system that conserves energy. </p><p>More complex magnetic systems, where the flux is not confined to a simple loop, must be analysed from first principles by using <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Magnetic_circuit&action=edit&section=11" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Air gaps can be created in the cores of certain transformers to reduce the effects of <a href="/wiki/Saturation_(magnetic)" title="Saturation (magnetic)">saturation</a>. This increases the reluctance of the magnetic circuit, and enables it to store more <a href="/wiki/Energy" title="Energy">energy</a> before core saturation. This effect is used in the <a href="/wiki/Flyback_transformer" title="Flyback transformer">flyback transformers</a> of cathode-ray tube video displays and in some types of <a href="/wiki/Switched-mode_power_supply" title="Switched-mode power supply">switch-mode power supply</a>.</li> <li>Variation of reluctance is the principle behind the <a href="/wiki/Reluctance_motor" title="Reluctance motor">reluctance motor</a> (or the variable reluctance generator) and the <a href="/wiki/Alexanderson_alternator" title="Alexanderson alternator">Alexanderson alternator</a>.</li> <li><a href="/wiki/Multimedia" title="Multimedia">Multimedia</a> <a href="/wiki/Loudspeaker" title="Loudspeaker">loudspeakers</a> are typically shielded magnetically, in order to reduce magnetic interference caused to <a href="/wiki/Television" title="Television">televisions</a> and other <a href="/wiki/Cathode-ray_tube" title="Cathode-ray tube">CRTs</a>. The speaker magnet is covered with a material such as <a href="/wiki/Soft_iron" class="mw-redirect" title="Soft iron">soft iron</a> to minimize the stray magnetic field.</li></ul> <p>Reluctance can also be applied to variable reluctance (magnetic) <a href="/wiki/Pick_up_(music_technology)" class="mw-redirect" title="Pick up (music technology)">pickups</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Magnetic_circuit&action=edit&section=12" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Magnetic_capacitance" class="mw-redirect" title="Magnetic capacitance">Magnetic capacitance</a></li> <li><a href="/wiki/Magnetic_complex_reluctance" title="Magnetic complex reluctance">Magnetic complex reluctance</a></li> <li><a href="/wiki/Tokamak" title="Tokamak">Tokamak</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Magnetic_circuit&action=edit&section=13" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.iec.ch/about/history/overview/">"International Electrotechnical Commission"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=International+Electrotechnical+Commission&rft_id=http%3A%2F%2Fwww.iec.ch%2Fabout%2Fhistory%2Foverview%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+circuit" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Matthew M. Radmanesh, <i>The Gateway to Understanding: Electrons to Waves and Beyond</i>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NANN_b5hc_EC&dq=gilbert&pg=PA539">p. 539</a>, AuthorHouse, 2005 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1418487406" title="Special:BookSources/1418487406">1418487406</a>.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Rowland H., Phil. Mag. (4), vol. 46, 1873, p. 140.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.ginerdelosrios.org/pizarra/electronica/nemesio/pizarra_neme/simuladores/parametros_magneticos.swf">"Magnetism (flash)"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Magnetism+%28flash%29&rft_id=http%3A%2F%2Fwww.ginerdelosrios.org%2Fpizarra%2Felectronica%2Fnemesio%2Fpizarra_neme%2Fsimuladores%2Fparametros_magneticos.swf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+circuit" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTescheMichel_IanozTorbjörn_Karlsson1997" class="citation book cs1">Tesche, Fredrick; Michel Ianoz; Torbjörn Karlsson (1997). <i>EMC Analysis Methods and Computational Models</i>. Wiley-IEEE. p. 513. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-15573-X" title="Special:BookSources/0-471-15573-X"><bdi>0-471-15573-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=EMC+Analysis+Methods+and+Computational+Models&rft.pages=513&rft.pub=Wiley-IEEE&rft.date=1997&rft.isbn=0-471-15573-X&rft.aulast=Tesche&rft.aufirst=Fredrick&rft.au=Michel+Ianoz&rft.au=Torbj%C3%B6rn+Karlsson&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+circuit" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Magnetic_circuit&action=edit&section=14" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060309173902/http://www.analogzone.com/col_0909.pdf"><i>Magnetic–Electric Analogs</i></a> by Dennis L. Feucht, Innovatia Laboratories (PDF) <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120717003534/http://www.analogzone.com/col_0909.pdf">Archived</a> July 17, 2012, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20080804012619/http://www.magnet.fsu.edu/education/tutorials/java/magneticshunt/">Interactive Java Tutorial on Magnetic Shunts</a> National High Magnetic Field Laboratory</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output 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