Inductance - Wikipedia

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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Source_of_inductance" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Source_of_inductance"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Source of inductance</span> </div> </a> <ul id="toc-Source_of_inductance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Self-inductance_and_magnetic_energy" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Self-inductance_and_magnetic_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Self-inductance and magnetic energy</span> </div> </a> <ul id="toc-Self-inductance_and_magnetic_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inductive_reactance" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Inductive_reactance"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Inductive reactance</span> </div> </a> <ul id="toc-Inductive_reactance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calculating_inductance" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Calculating_inductance"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Calculating inductance</span> </div> </a> <button aria-controls="toc-Calculating_inductance-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Calculating inductance subsection</span> </button> <ul id="toc-Calculating_inductance-sublist" class="vector-toc-list"> <li id="toc-Inductance_of_a_straight_single_wire" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inductance_of_a_straight_single_wire"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Inductance of a straight single wire</span> </div> </a> <ul id="toc-Inductance_of_a_straight_single_wire-sublist" class="vector-toc-list"> <li id="toc-Practical_formulas" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Practical_formulas"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.1</span> <span>Practical formulas</span> </div> </a> <ul id="toc-Practical_formulas-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mutual_inductance_of_two_parallel_straight_wires" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mutual_inductance_of_two_parallel_straight_wires"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Mutual inductance of two parallel straight wires</span> </div> </a> <ul id="toc-Mutual_inductance_of_two_parallel_straight_wires-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mutual_inductance_of_two_wire_loops" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mutual_inductance_of_two_wire_loops"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Mutual inductance of two wire loops</span> </div> </a> <ul id="toc-Mutual_inductance_of_two_wire_loops-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Derivation</span> </div> </a> <ul id="toc-Derivation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Self-inductance_of_a_wire_loop" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Self-inductance_of_a_wire_loop"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Self-inductance of a wire loop</span> </div> </a> <ul id="toc-Self-inductance_of_a_wire_loop-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inductance_of_a_solenoid" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inductance_of_a_solenoid"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Inductance of a solenoid</span> </div> </a> <ul id="toc-Inductance_of_a_solenoid-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inductance_of_a_coaxial_cable" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inductance_of_a_coaxial_cable"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.7</span> <span>Inductance of a coaxial cable</span> </div> </a> <ul id="toc-Inductance_of_a_coaxial_cable-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inductance_of_multilayer_coils" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inductance_of_multilayer_coils"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.8</span> <span>Inductance of multilayer coils</span> </div> </a> <ul id="toc-Inductance_of_multilayer_coils-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnetic_cores" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Magnetic_cores"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.9</span> <span>Magnetic cores</span> </div> </a> <ul id="toc-Magnetic_cores-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Mutual_inductance" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mutual_inductance"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Mutual inductance</span> </div> </a> <button aria-controls="toc-Mutual_inductance-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Mutual inductance subsection</span> </button> <ul id="toc-Mutual_inductance-sublist" class="vector-toc-list"> <li id="toc-Derivation_of_mutual_inductance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivation_of_mutual_inductance"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Derivation of mutual inductance</span> </div> </a> <ul id="toc-Derivation_of_mutual_inductance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mutual_inductance_and_magnetic_field_energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mutual_inductance_and_magnetic_field_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Mutual inductance and magnetic field energy</span> </div> </a> <ul id="toc-Mutual_inductance_and_magnetic_field_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coupling_coefficient" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Coupling_coefficient"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Coupling coefficient</span> </div> </a> <ul id="toc-Coupling_coefficient-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrix_representation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matrix_representation"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Matrix representation</span> </div> </a> <ul id="toc-Matrix_representation-sublist" class="vector-toc-list"> <li id="toc-Multiple_Coupled_Inductors" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Multiple_Coupled_Inductors"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4.1</span> <span>Multiple Coupled Inductors</span> </div> </a> <ul id="toc-Multiple_Coupled_Inductors-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Equivalent_circuits" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equivalent_circuits"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Equivalent circuits</span> </div> </a> <ul id="toc-Equivalent_circuits-sublist" class="vector-toc-list"> <li id="toc-T-circuit" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#T-circuit"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5.1</span> <span>T-circuit</span> </div> </a> <ul id="toc-T-circuit-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-π-circuit" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#π-circuit"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5.2</span> <span>π-circuit</span> </div> </a> <ul id="toc-π-circuit-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Resonant_transformer" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Resonant_transformer"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.6</span> <span>Resonant transformer</span> </div> </a> <ul id="toc-Resonant_transformer-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ideal_transformers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ideal_transformers"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.7</span> <span>Ideal transformers</span> </div> </a> <ul id="toc-Ideal_transformers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Self-inductance_of_thin_wire_shapes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Self-inductance_of_thin_wire_shapes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Self-inductance of thin wire shapes</span> </div> </a> <ul id="toc-Self-inductance_of_thin_wire_shapes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_references" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#General_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>General references</span> </div> </a> <ul id="toc-General_references-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</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">Inductance</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" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 68 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-68" 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">68 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Induktansie" title="Induktansie – Afrikaans" lang="af" hreflang="af" data-title="Induktansie" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A5%E1%88%8D%E1%8A%A8%E1%8A%9D%E1%8A%90%E1%89%B5" title="እልከኝነት – Amharic" lang="am" hreflang="am" data-title="እልከኝነት" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AD%D8%A7%D8%AB%D8%A9" title="محاثة – Arabic" lang="ar" hreflang="ar" data-title="محاثة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Inductancia" title="Inductancia – Asturian" lang="ast" hreflang="ast" data-title="Inductancia" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0nduktivlik" title="İnduktivlik – Azerbaijani" lang="az" hreflang="az" data-title="İnduktivlik" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%86%E0%A6%AC%E0%A7%87%E0%A6%B6_%E0%A6%97%E0%A7%81%E0%A6%A3%E0%A6%BE%E0%A6%99%E0%A7%8D%E0%A6%95" title="আবেশ গুণাঙ্ক – Bangla" lang="bn" hreflang="bn" data-title="আবেশ গুণাঙ্ক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ti%C4%81n-k%C3%A1m" title="Tiān-kám – Minnan" lang="nan" hreflang="nan" data-title="Tiān-kám" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%86%D0%BD%D0%B4%D1%83%D0%BA%D1%82%D1%8B%D1%9E%D0%BD%D0%B0%D1%81%D1%86%D1%8C" title="Індуктыўнасць – Belarusian" lang="be" hreflang="be" data-title="Індуктыўнасць" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%86%D0%BD%D0%B4%D1%83%D0%BA%D1%82%D1%8B%D1%9E%D0%BD%D0%B0%D1%81%D1%8C%D1%86%D1%8C" title="Індуктыўнасьць – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Індуктыўнасьць" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D0%BD%D0%B4%D1%83%D0%BA%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82" title="Индуктивност – Bulgarian" lang="bg" hreflang="bg" data-title="Индуктивност" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Induktivit%C3%A4t" title="Induktivität – Bavarian" lang="bar" hreflang="bar" data-title="Induktivität" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Induct%C3%A0ncia" title="Inductància – Catalan" lang="ca" hreflang="ca" data-title="Inductància" 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%98%D0%BD%D0%B4%D1%83%D0%BA%D1%82%D0%B8%D0%B2%D0%BB%C4%83%D1%85" title="Индуктивлăх – Chuvash" lang="cv" hreflang="cv" data-title="Индуктивлăх" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Induk%C4%8Dnost" title="Indukčnost – Czech" lang="cs" hreflang="cs" data-title="Indukčnost" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Induktans" title="Induktans – Danish" lang="da" hreflang="da" data-title="Induktans" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Induktivit%C3%A4t" title="Induktivität – German" lang="de" hreflang="de" data-title="Induktivität" 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/Induktiivsus" title="Induktiivsus – Estonian" lang="et" hreflang="et" data-title="Induktiivsus" 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%91%CF%85%CF%84%CE%B5%CF%80%CE%B1%CE%B3%CF%89%CE%B3%CE%AE" title="Αυτεπαγωγή – Greek" lang="el" hreflang="el" data-title="Αυτεπαγωγή" 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/Inductancia" title="Inductancia – Spanish" lang="es" hreflang="es" data-title="Inductancia" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Induktanco" title="Induktanco – Esperanto" lang="eo" hreflang="eo" data-title="Induktanco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Induktantzia" title="Induktantzia – Basque" lang="eu" hreflang="eu" data-title="Induktantzia" 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/%D8%B6%D8%B1%DB%8C%D8%A8_%D8%AE%D9%88%D8%AF%D8%A7%D9%84%D9%82%D8%A7%DB%8C%DB%8C" title="ضریب خودالقایی – Persian" lang="fa" hreflang="fa" data-title="ضریب خودالقایی" 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/Inductance" title="Inductance – French" lang="fr" hreflang="fr" data-title="Inductance" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Ionduchtas" title="Ionduchtas – Irish" lang="ga" hreflang="ga" data-title="Ionduchtas" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Indutancia" title="Indutancia – Galician" lang="gl" hreflang="gl" data-title="Indutancia" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9C%A0%EB%8F%84%EA%B3%84%EC%88%98" title="유도계수 – Korean" lang="ko" hreflang="ko" data-title="유도계수" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%BB%D5%B6%D5%A4%D5%B8%D6%82%D5%AF%D5%BF%D5%AB%D5%BE%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Ինդուկտիվություն – Armenian" lang="hy" hreflang="hy" data-title="Ինդուկտիվություն" 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%AA%E0%A5%8D%E0%A4%B0%E0%A5%87%E0%A4%B0%E0%A4%95%E0%A4%A4%E0%A5%8D%E0%A4%B5" title="प्रेरकत्व – Hindi" lang="hi" hreflang="hi" data-title="प्रेरकत्व" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Elektri%C4%8Dni_induktivitet" title="Električni induktivitet – Croatian" lang="hr" hreflang="hr" data-title="Električni induktivitet" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Induktansi" title="Induktansi – Indonesian" lang="id" hreflang="id" data-title="Induktansi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Span" title="Span – Icelandic" lang="is" hreflang="is" data-title="Span" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Induttanza" title="Induttanza – Italian" lang="it" hreflang="it" data-title="Induttanza" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%94%D7%A9%D7%A8%D7%90%D7%95%D7%AA" title="השראות – Hebrew" lang="he" hreflang="he" data-title="השראות" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%98%E1%83%9C%E1%83%93%E1%83%A3%E1%83%A5%E1%83%AA%E1%83%98%E1%83%A3%E1%83%A0%E1%83%9D%E1%83%91%E1%83%90" title="ინდუქციურობა – Georgian" lang="ka" hreflang="ka" data-title="ინდუქციურობა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Endiktans" title="Endiktans – Haitian Creole" lang="ht" hreflang="ht" data-title="Endiktans" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%98%D0%BD%D0%B4%D1%83%D0%BA%D1%86%D0%B8%D1%8F%D0%BB%D1%83%D1%83%D0%BB%D1%83%D0%BA" title="Индукциялуулук – Kyrgyz" lang="ky" hreflang="ky" data-title="Индукциялуулук" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Induktivit%C4%81te" title="Induktivitāte – Latvian" lang="lv" hreflang="lv" data-title="Induktivitāte" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Induktyvumas" title="Induktyvumas – Lithuanian" lang="lt" hreflang="lt" data-title="Induktyvumas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%98%D0%BD%D0%B4%D1%83%D0%BA%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82" title="Индуктивност – Macedonian" lang="mk" hreflang="mk" data-title="Индуктивност" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kearuhan" title="Kearuhan – Malay" lang="ms" hreflang="ms" data-title="Kearuhan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Zelfinductie" title="Zelfinductie – Dutch" lang="nl" hreflang="nl" data-title="Zelfinductie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-new mw-list-item"><a href="https://new.wikipedia.org/wiki/%E0%A4%87%E0%A4%A8%E0%A5%8D%E0%A4%A1%E0%A4%95%E0%A5%8D%E0%A4%9F%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%B8" title="इन्डक्ट्यान्स – Newari" lang="new" hreflang="new" data-title="इन्डक्ट्यान्स" data-language-autonym="नेपाल भाषा" data-language-local-name="Newari" class="interlanguage-link-target"><span>नेपाल भाषा</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A4%E3%83%B3%E3%83%80%E3%82%AF%E3%82%BF%E3%83%B3%E3%82%B9" title="インダクタンス – Japanese" lang="ja" hreflang="ja" data-title="インダクタンス" 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-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Induktiwiteet" title="Induktiwiteet – Northern Frisian" lang="frr" hreflang="frr" data-title="Induktiwiteet" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Induktans" title="Induktans – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Induktans" 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-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Induktans" title="Induktans – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Induktans" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Induktivlik" title="Induktivlik – Uzbek" lang="uz" hreflang="uz" data-title="Induktivlik" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Indukcyjno%C5%9B%C4%87" title="Indukcyjność – Polish" lang="pl" hreflang="pl" data-title="Indukcyjność" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Indut%C3%A2ncia" title="Indutância – Portuguese" lang="pt" hreflang="pt" data-title="Indutância" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Inductan%C8%9B%C4%83" title="Inductanță – Romanian" lang="ro" hreflang="ro" data-title="Inductanță" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%98%D0%BD%D0%B4%D1%83%D0%BA%D1%82%D0%B8%D0%B2%D0%BD%D0%BE%D1%81%D1%82%D1%8C" title="Индуктивность – Russian" lang="ru" hreflang="ru" data-title="Индуктивность" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Induktiviteti" title="Induktiviteti – Albanian" lang="sq" hreflang="sq" data-title="Induktiviteti" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Induk%C4%8Dnos%C5%A5" title="Indukčnosť – Slovak" lang="sk" hreflang="sk" data-title="Indukčnosť" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Induktivnost" title="Induktivnost – Slovenian" lang="sl" hreflang="sl" data-title="Induktivnost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A1%D0%B0%D0%BC%D0%BE%D0%B8%D0%BD%D0%B4%D1%83%D0%BA%D1%86%D0%B8%D1%98%D0%B0" title="Самоиндукција – Serbian" lang="sr" hreflang="sr" data-title="Самоиндукција" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Induktivnost" title="Induktivnost – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Induktivnost" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Induktanssi" title="Induktanssi – Finnish" lang="fi" hreflang="fi" data-title="Induktanssi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Induktans" title="Induktans – Swedish" lang="sv" hreflang="sv" data-title="Induktans" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta badge-Q70893996 mw-list-item" title=""><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AF%82%E0%AE%A3%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%8D" title="தூண்டம் – Tamil" lang="ta" hreflang="ta" data-title="தூண்டம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%98%D0%BD%D0%B4%D1%83%D0%BA%D1%82%D0%B8%D0%B2%D0%BB%D1%8B%D0%BA" title="Индуктивлык – Tatar" lang="tt" hreflang="tt" data-title="Индуктивлык" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C4%B0nd%C3%BCktans" title="İndüktans – Turkish" lang="tr" hreflang="tr" data-title="İndüktans" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-tk mw-list-item"><a href="https://tk.wikipedia.org/wiki/Induktiwlik" title="Induktiwlik – Turkmen" lang="tk" hreflang="tk" data-title="Induktiwlik" data-language-autonym="Türkmençe" data-language-local-name="Turkmen" class="interlanguage-link-target"><span>Türkmençe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D0%BD%D0%B4%D1%83%D0%BA%D1%82%D0%B8%D0%B2%D0%BD%D1%96%D1%81%D1%82%D1%8C" title="Індуктивність – Ukrainian" lang="uk" hreflang="uk" data-title="Індуктивність" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%E1%BB%B1_c%E1%BA%A3m" title="Tự cảm – Vietnamese" lang="vi" hreflang="vi" data-title="Tự cảm" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wo mw-list-item"><a href="https://wo.wikipedia.org/wiki/Xiirtalu" title="Xiirtalu – Wolof" lang="wo" hreflang="wo" data-title="Xiirtalu" data-language-autonym="Wolof" data-language-local-name="Wolof" class="interlanguage-link-target"><span>Wolof</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%94%B5%E6%84%9F" title="电感 – Wu" lang="wuu" hreflang="wuu" data-title="电感" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%9B%BB%E6%84%9F" title="電感 – Cantonese" lang="yue" hreflang="yue" data-title="電感" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%94%B5%E6%84%9F" title="电感 – Chinese" lang="zh" hreflang="zh" data-title="电感" 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 href="https://www.wikidata.org/wiki/Special:EntityPage/Q177897#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div 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class="infobox-above">Inductance</th></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data"><span class="texhtml mvar" style="font-style:italic;">L</span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data"><a href="/wiki/Henry_(unit)" title="Henry (unit)">henry</a> (H)</td></tr><tr><th scope="row" class="infobox-label">In <a href="/wiki/SI_base_unit" title="SI base unit"><span class="wrap">SI base units</span></a></th><td class="infobox-data"><a href="/wiki/Kilogram" title="Kilogram">kg</a>⋅<a href="/wiki/Metre" title="Metre">m</a><sup>2</sup>⋅<a href="/wiki/Second" title="Second">s</a><sup>−2</sup>⋅<a href="/wiki/Ampere" title="Ampere">A</a><sup>−2</sup></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Derivations from<br />other quantities</div></th><td class="infobox-data"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"><ul><li><span class="texhtml"><i>L</i> = <i><a href="/wiki/Voltage" title="Voltage">V</a></i> / ( <i><a href="/wiki/Electrical_current" class="mw-redirect" title="Electrical current">I</a></i> / <i><a href="/wiki/Time" title="Time">t</a></i> )</span></li><li><span class="texhtml"><i>L</i> = <a href="/wiki/Magnetic_flux" title="Magnetic flux">Φ</a> / <i><a href="/wiki/Electrical_current" class="mw-redirect" title="Electrical current">I</a></i></span></li></ul></div></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td 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title="Ampère's circuital law">Ampère law</a></li> <li><a href="/wiki/Biot%E2%80%93Savart_law" title="Biot–Savart law">Biot–Savart law</a></li> <li><a href="/wiki/Gauss%27s_law_for_magnetism" 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" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">Electrodynamics</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Bremsstrahlung" title="Bremsstrahlung">Bremsstrahlung</a></li> <li><a href="/wiki/Cyclotron_radiation" title="Cyclotron radiation">Cyclotron radiation</a></li> <li><a href="/wiki/Displacement_current" title="Displacement current">Displacement current</a></li> <li><a href="/wiki/Eddy_current" title="Eddy current">Eddy current</a></li> <li><a href="/wiki/Electromagnetic_field" title="Electromagnetic field">Electromagnetic field</a></li> <li><a href="/wiki/Electromagnetic_induction" title="Electromagnetic induction">Electromagnetic induction</a></li> <li><a href="/wiki/Electromagnetic_pulse" title="Electromagnetic pulse">Electromagnetic pulse</a></li> <li><a href="/wiki/Electromagnetic_radiation" 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"><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 class="mw-selflink selflink">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 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/Magnetic_circuit" title="Magnetic circuit">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><b>Inductance</b> is the tendency of an <a href="/wiki/Electrical_conductor" title="Electrical conductor">electrical conductor</a> to oppose a change in the <a href="/wiki/Electric_current" title="Electric current">electric current</a> flowing through it. The electric current produces a <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> around the conductor. The magnetic field strength depends on the magnitude of the electric current, and follows any changes in the magnitude of the current. From <a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday's law of induction</a>, any change in magnetic field through a circuit induces an <a href="/wiki/Electromotive_force" title="Electromotive force">electromotive force</a> (EMF) (<a href="/wiki/Voltage" title="Voltage">voltage</a>) in the conductors, a process known as <a href="/wiki/Electromagnetic_induction" title="Electromagnetic induction">electromagnetic induction</a>. This induced voltage created by the changing current has the effect of opposing the change in current. This is stated by <a href="/wiki/Lenz%27s_law" title="Lenz's law">Lenz's law</a>, and the voltage is called <i><a href="/wiki/Back_EMF" class="mw-redirect" title="Back EMF">back EMF</a></i>. </p><p>Inductance is defined as the ratio of the induced voltage to the rate of change of current causing it.<sup id="cite_ref-Serway2017_1-0" class="reference"><a href="#cite_note-Serway2017-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It is a proportionality constant that depends on the geometry of circuit conductors (e.g., cross-section area and length) and the <a href="/wiki/Magnetic_permeability" class="mw-redirect" title="Magnetic permeability">magnetic permeability</a> of the conductor and nearby materials.<sup id="cite_ref-Serway2017_1-1" class="reference"><a href="#cite_note-Serway2017-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> An <a href="/wiki/Electronic_component" title="Electronic component">electronic component</a> designed to add inductance to a circuit is called an <a href="/wiki/Inductor" title="Inductor">inductor</a>. It typically consists of a <a href="/wiki/Electromagnetic_coil" title="Electromagnetic coil">coil</a> or helix of wire. </p><p>The term <i>inductance</i> was coined by <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a> in May 1884, as a convenient way to refer to "coefficient of self-induction".<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><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 is customary to use the symbol <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> for inductance, in honour of the physicist <a href="/wiki/Heinrich_Lenz" class="mw-redirect" title="Heinrich Lenz">Heinrich Lenz</a>.<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> In the <a href="/wiki/International_System_of_Units" title="International System of Units">SI</a> system, the unit of inductance is the <a href="/wiki/Henry_(unit)" title="Henry (unit)">henry</a> (H), which is the amount of inductance that causes a voltage of one <a href="/wiki/Volt" title="Volt">volt</a>, when the current is changing at a rate of one <a href="/wiki/Ampere_(unit)" class="mw-redirect" title="Ampere (unit)">ampere</a> per second.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The unit is named for <a href="/wiki/Joseph_Henry" title="Joseph Henry">Joseph Henry</a>, who discovered inductance independently of Faraday.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=1" title="Edit section: History"><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/History_of_electromagnetic_theory" title="History of electromagnetic theory">History of electromagnetic theory</a></div> <p>The history of electromagnetic induction, a facet of <a href="/wiki/Electromagnetism" title="Electromagnetism">electromagnetism</a>, began with observations of the ancients: electric charge or static electricity (rubbing silk on <a href="/wiki/Amber" title="Amber">amber</a>), electric current (<a href="/wiki/Lightning" title="Lightning">lightning</a>), and magnetic attraction (<a href="/wiki/Lodestone" title="Lodestone">lodestone</a>). Understanding the unity of these forces of nature, and the scientific theory of electromagnetism was initiated and achieved during the 19th century. </p><p>Electromagnetic induction was first described by <a href="/wiki/Michael_Faraday" title="Michael Faraday">Michael Faraday</a> in 1831.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> In Faraday's experiment, he wrapped two wires around opposite sides of an iron ring. He expected that, when current started to flow in one wire, a sort of wave would travel through the ring and cause some electrical effect on the opposite side. Using a <a href="/wiki/Galvanometer" title="Galvanometer">galvanometer</a>, he observed a transient current flow in the second coil of wire each time that a battery was connected or disconnected from the first coil.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This current was induced by the change in <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a> that occurred when the battery was connected and disconnected.<sup id="cite_ref-Giancoli_11-0" class="reference"><a href="#cite_note-Giancoli-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Faraday found several other manifestations of electromagnetic induction. For example, he saw transient currents when he quickly slid a bar magnet in and out of a coil of wires, and he generated a steady (<a href="/wiki/Direct_current" title="Direct current">DC</a>) current by rotating a copper disk near the bar magnet with a sliding electrical lead ("<a href="/wiki/Homopolar_generator" title="Homopolar generator">Faraday's disk</a>").<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Source_of_inductance">Source of inductance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=2" title="Edit section: Source of inductance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A current <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/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> flowing through a conductor generates a <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> around the conductor, which is described by <a href="/wiki/Ampere%27s_circuital_law" class="mw-redirect" title="Ampere's circuital law">Ampere's circuital law</a>. The total <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</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 \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> through a circuit is equal to the product of the perpendicular component of the magnetic flux density and the area of the surface spanning the current path. If the current varies, the <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</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 \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> through the circuit changes. By <a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday's law of induction</a>, any change in flux through a circuit induces an <a href="/wiki/Electromotive_force" title="Electromotive force">electromotive force</a> (EMF, <span class="nowrap"><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>)</span> in the circuit, proportional to the rate of change of flux </p><p><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}}(t)=-{\frac {\text{d}}{{\text{d}}t}}\,\Phi (t)}"> <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 stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>d</mtext> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}(t)=-{\frac {\text{d}}{{\text{d}}t}}\,\Phi (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe251817fc096040684579e785f3f7e721a3f59" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.549ex; height:5.509ex;" alt="{\displaystyle {\mathcal {E}}(t)=-{\frac {\text{d}}{{\text{d}}t}}\,\Phi (t)}"></span> </p><p>The negative sign in the equation indicates that the induced voltage is in a direction which opposes the change in current that created it; this is called <a href="/wiki/Lenz%27s_law" title="Lenz's law">Lenz's law</a>. The potential is therefore called a <a href="/wiki/Back_EMF" class="mw-redirect" title="Back EMF">back EMF</a>. If the current is increasing, the voltage is positive at the end of the conductor through which the current enters and negative at the end through which it leaves, tending to reduce the current. If the current is decreasing, the voltage is positive at the end through which the current leaves the conductor, tending to maintain the current. Self-inductance, usually just called inductance, <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is the ratio between the induced voltage and the rate of change of the current </p><p><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 v(t)=L\,{\frac {{\text{d}}i}{{\text{d}}t}}\qquad \qquad \qquad (1)\;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>L</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>i</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mspace width="2em" /> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)=L\,{\frac {{\text{d}}i}{{\text{d}}t}}\qquad \qquad \qquad (1)\;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/004860d12e24dbfab61c76a4f41a66403c9265b5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.366ex; height:5.509ex;" alt="{\displaystyle v(t)=L\,{\frac {{\text{d}}i}{{\text{d}}t}}\qquad \qquad \qquad (1)\;}"></span> </p><p>Thus, inductance is a property of a conductor or circuit, due to its magnetic field, which tends to oppose changes in current through the circuit. The unit of inductance in the <a href="/wiki/Systeme_International" class="mw-redirect" title="Systeme International">SI</a> system is the <a href="/wiki/Henry_(unit)" title="Henry (unit)">henry</a> (H), named after <a href="/wiki/Joseph_Henry" title="Joseph Henry">Joseph Henry</a>, which is the amount of inductance that generates a voltage of one <a href="/wiki/Volt_(unit)" class="mw-redirect" title="Volt (unit)">volt</a> when the current is changing at a rate of one <a href="/wiki/Ampere" title="Ampere">ampere</a> per second. </p><p>All conductors have some inductance, which may have either desirable or detrimental effects in practical electrical devices. The inductance of a circuit depends on the geometry of the current path, and on the <a href="/wiki/Magnetic_permeability" class="mw-redirect" title="Magnetic permeability">magnetic permeability</a> of nearby materials; <a href="/wiki/Ferromagnetic" class="mw-redirect" title="Ferromagnetic">ferromagnetic</a> materials with a higher permeability like <a href="/wiki/Iron" title="Iron">iron</a> near a conductor tend to increase the magnetic field and inductance. Any alteration to a circuit which increases the flux (total magnetic field) through the circuit produced by a given current increases the inductance, because inductance is also equal to the ratio of <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a> to current<sup id="cite_ref-Singh_13-0" class="reference"><a href="#cite_note-Singh-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Wadhwa_14-0" class="reference"><a href="#cite_note-Wadhwa-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Pelcovits_15-0" class="reference"><a href="#cite_note-Pelcovits-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Purcell_16-0" class="reference"><a href="#cite_note-Purcell-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p><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 L={\Phi (i) \over i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> <mi>i</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\Phi (i) \over i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0afdc96ea2467c757cfb63288bca8eab07920de0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.807ex; height:5.676ex;" alt="{\displaystyle L={\Phi (i) \over i}}"></span> </p><p>An <a href="/wiki/Inductor" title="Inductor">inductor</a> is an <a href="/wiki/Electrical_component" class="mw-redirect" title="Electrical component">electrical component</a> consisting of a conductor shaped to increase the magnetic flux, to add inductance to a circuit. Typically it consists of a wire wound into a <a href="/wiki/Electromagnetic_coil" title="Electromagnetic coil">coil</a> or <a href="/wiki/Helix" title="Helix">helix</a>. A coiled wire has a higher inductance than a straight wire of the same length, because the magnetic field lines pass through the circuit multiple times, it has multiple <a href="/wiki/Flux_linkage" title="Flux linkage">flux linkages</a>. The inductance is proportional to the square of the <a href="/wiki/Number_of_turns" class="mw-redirect" title="Number of turns">number of turns</a> in the coil, assuming full flux linkage. </p><p>The inductance of a coil can be increased by placing a <a href="/wiki/Magnetic_core" title="Magnetic core">magnetic core</a> of <a href="/wiki/Ferromagnetic" class="mw-redirect" title="Ferromagnetic">ferromagnetic</a> material in the hole in the center. The magnetic field of the coil magnetizes the material of the core, aligning its <a href="/wiki/Magnetic_domain" title="Magnetic domain">magnetic domains</a>, and the magnetic field of the core adds to that of the coil, increasing the flux through the coil. This is called a <a href="/wiki/Inductor#Ferromagnetic-core_inductor" title="Inductor">ferromagnetic core inductor</a>. A magnetic core can increase the inductance of a coil by thousands of times. </p><p>If multiple <a href="/wiki/Electric_circuit" class="mw-redirect" title="Electric circuit">electric circuits</a> are located close to each other, the magnetic field of one can pass through the other; in this case the circuits are said to be <i><a href="/wiki/Inductive_coupling" title="Inductive coupling">inductively coupled</a></i>. Due to <a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday's law of induction</a>, a change in current in one circuit can cause a change in magnetic flux in another circuit and thus induce a voltage in another circuit. The concept of inductance can be generalized in this case by defining the <a href="/wiki/Inductive_coupling" title="Inductive coupling">mutual inductance</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 M_{k,\ell }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>,</mo> <mi>ℓ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{k,\ell }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb496b799f3c8624fec6d354c66ec99f26abe978" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.486ex; height:2.843ex;" alt="{\displaystyle M_{k,\ell }}"></span> of circuit <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> and circuit <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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span> as the ratio of voltage induced in circuit <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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span> to the rate of change of current in circuit <span class="nowrap"><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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>.</span> This is the principle behind a <i><a href="/wiki/Transformer" title="Transformer">transformer</a></i>. <span class="anchor" id="self-inductance"></span> The property describing the effect of one conductor on itself is more precisely called <i>self-inductance</i>, and the properties describing the effects of one conductor with changing current on nearby conductors is called <i>mutual inductance</i>.<sup id="cite_ref-Sears_and_Zemansky_1964:743_17-0" class="reference"><a href="#cite_note-Sears_and_Zemansky_1964:743-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Self-inductance_and_magnetic_energy">Self-inductance and magnetic energy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=3" title="Edit section: Self-inductance and magnetic energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If the current through a conductor with inductance is increasing, a voltage <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 v(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/243a0bf98a12f48552ba6a70302122d81b237b3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.777ex; height:2.843ex;" alt="{\displaystyle v(t)}"></span> is induced across the conductor with a polarity that opposes the current—in addition to any voltage drop caused by the conductor's resistance. The charges flowing through the circuit lose potential energy. The energy from the external circuit required to overcome this "potential hill" is stored in the increased magnetic field around the conductor. Therefore, an inductor stores energy in its magnetic field. At any given time <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> the power <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 p(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b827c545ca1487214f0c498131228ef87718ece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:3.908ex; height:2.843ex;" alt="{\displaystyle p(t)}"></span> flowing into the magnetic field, which is equal to the rate of change of the stored energy <span class="nowrap"><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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>,</span> is the product of the current <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(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06931e7bcffd32010f83c3d5dbef2d9bcbdcb670" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.451ex; height:2.843ex;" alt="{\displaystyle i(t)}"></span> and voltage <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 v(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/243a0bf98a12f48552ba6a70302122d81b237b3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.777ex; height:2.843ex;" alt="{\displaystyle v(t)}"></span> across the conductor<sup id="cite_ref-Serway_18-0" class="reference"><a href="#cite_note-Serway-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Ida_19-0" class="reference"><a href="#cite_note-Ida-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Purcell2_20-0" class="reference"><a href="#cite_note-Purcell2-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p><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 p(t)={\frac {{\text{d}}U}{{\text{d}}t}}=v(t)\,i(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>U</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>i</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(t)={\frac {{\text{d}}U}{{\text{d}}t}}=v(t)\,i(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feae1412bd72ff2c75709c543c018b48c2b9039a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:21.631ex; height:5.509ex;" alt="{\displaystyle p(t)={\frac {{\text{d}}U}{{\text{d}}t}}=v(t)\,i(t)}"></span> </p><p>From (1) above </p><p><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 {\begin{aligned}{\frac {{\text{d}}U}{{\text{d}}t}}&=L(i)\,i\,{\frac {{\text{d}}i}{{\text{d}}t}}\\[3pt]{\text{d}}U&=L(i)\,i\,{\text{d}}i\,\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="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>U</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>i</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>i</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>U</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>i</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>i</mi> <mspace width="thinmathspace" /> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {{\text{d}}U}{{\text{d}}t}}&=L(i)\,i\,{\frac {{\text{d}}i}{{\text{d}}t}}\\[3pt]{\text{d}}U&=L(i)\,i\,{\text{d}}i\,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d65035bcbf7356315ad7c132bef205c2808d2823" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.775ex; margin-bottom: -0.229ex; width:16.501ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {{\text{d}}U}{{\text{d}}t}}&=L(i)\,i\,{\frac {{\text{d}}i}{{\text{d}}t}}\\[3pt]{\text{d}}U&=L(i)\,i\,{\text{d}}i\,\end{aligned}}}"></span> </p><p>When there is no current, there is no magnetic field and the stored energy is zero. Neglecting resistive losses, the <a href="/wiki/Energy" title="Energy">energy</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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> (measured in <a href="/wiki/Joule" title="Joule">joules</a>, in <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a>) stored by an inductance with a current <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> through it is equal to the amount of work required to establish the current through the inductance from zero, and therefore the magnetic field. This is given by: </p><p><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 U=\int _{0}^{I}L(i)\,i\,{\text{d}}i\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msubsup> <mi>L</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>i</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>i</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\int _{0}^{I}L(i)\,i\,{\text{d}}i\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0689c022dc512720c28f99c5a2ebf9094556a263" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.046ex; height:6.176ex;" alt="{\displaystyle U=\int _{0}^{I}L(i)\,i\,{\text{d}}i\,}"></span> </p><p>If the inductance <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 L(i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(i)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb6bda51bcf335bd6571acc37a0391eb5a7dbab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.195ex; height:2.843ex;" alt="{\displaystyle L(i)}"></span> is constant over the current range, the stored energy is<sup id="cite_ref-Serway_18-1" class="reference"><a href="#cite_note-Serway-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Ida_19-1" class="reference"><a href="#cite_note-Ida-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Purcell2_20-1" class="reference"><a href="#cite_note-Purcell2-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p><p><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 {\begin{aligned}U&=L\int _{0}^{I}\,i\,{\text{d}}i\\[3pt]&={\tfrac {1}{2}}L\,I^{2}\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="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>U</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>L</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <mi>i</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>i</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>L</mi> <mspace width="thinmathspace" /> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U&=L\int _{0}^{I}\,i\,{\text{d}}i\\[3pt]&={\tfrac {1}{2}}L\,I^{2}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f88094cd0bacb95c7052b70ccca2a1fbe5b417a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:15.186ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}U&=L\int _{0}^{I}\,i\,{\text{d}}i\\[3pt]&={\tfrac {1}{2}}L\,I^{2}\end{aligned}}}"></span> </p><p>Inductance is therefore also proportional to the energy stored in the magnetic field for a given current. This energy is stored as long as the current remains constant. If the current decreases, the magnetic field decreases, inducing a voltage in the conductor in the opposite direction, negative at the end through which current enters and positive at the end through which it leaves. This returns stored magnetic energy to the external circuit. </p><p>If <a href="/wiki/Ferromagnetic" class="mw-redirect" title="Ferromagnetic">ferromagnetic</a> materials are located near the conductor, such as in an inductor with a <a href="/wiki/Magnetic_core" title="Magnetic core">magnetic core</a>, the constant inductance equation above is only valid for <a href="/wiki/Linear_circuit" title="Linear circuit">linear</a> regions of the magnetic flux, at currents below the level at which the ferromagnetic material <a href="/wiki/Magnetic_saturation" class="mw-redirect" title="Magnetic saturation">saturates</a>, where the inductance is approximately constant. If the magnetic field in the inductor approaches the level at which the core saturates, the inductance begins to change with current, and the integral equation must be used. </p> <div class="mw-heading mw-heading2"><h2 id="Inductive_reactance">Inductive reactance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=4" title="Edit section: Inductive reactance"><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:Waveforms_-_inductive_reactance.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Waveforms_-_inductive_reactance.svg/220px-Waveforms_-_inductive_reactance.svg.png" decoding="async" width="220" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Waveforms_-_inductive_reactance.svg/330px-Waveforms_-_inductive_reactance.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/74/Waveforms_-_inductive_reactance.svg/440px-Waveforms_-_inductive_reactance.svg.png 2x" data-file-width="800" data-file-height="500" /></a><figcaption>The voltage <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 v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span>, blue)</i> and current <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 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/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>, red)</i> waveforms in an ideal inductor to which an alternating current has been applied. The current lags the voltage by 90°</figcaption></figure> <p>When a <a href="/wiki/Sinusoidal" class="mw-redirect" title="Sinusoidal">sinusoidal</a> <a href="/wiki/Alternating_current" title="Alternating current">alternating current</a> (AC) is passing through a linear inductance, the induced <a href="/wiki/Back-EMF" class="mw-redirect" title="Back-EMF">back-<abbr title="electromotive force">EMF</abbr></a> is also sinusoidal. If the current through the inductance is <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(t)=I_{\text{peak}}\sin \left(\omega t\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i(t)=I_{\text{peak}}\sin \left(\omega t\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07385ddc3d4f2413dfd5d1a40b7af8f1d58a548c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.477ex; height:3.009ex;" alt="{\displaystyle i(t)=I_{\text{peak}}\sin \left(\omega t\right)}"></span>, from (1) above the voltage across it 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 {\begin{aligned}v(t)&=L{\frac {{\text{d}}i}{{\text{d}}t}}=L\,{\frac {\text{d}}{{\text{d}}t}}\left[I_{\text{peak}}\sin \left(\omega t\right)\right]\\&=\omega L\,I_{\text{peak}}\,\cos \left(\omega t\right)=\omega L\,I_{\text{peak}}\,\sin \left(\omega t+{\pi \over 2}\right)\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="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>i</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>L</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mtext>d</mtext> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>ω</mi> <mi>L</mi> <mspace width="thinmathspace" /> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>ω</mi> <mi>L</mi> <mspace width="thinmathspace" /> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}v(t)&=L{\frac {{\text{d}}i}{{\text{d}}t}}=L\,{\frac {\text{d}}{{\text{d}}t}}\left[I_{\text{peak}}\sin \left(\omega t\right)\right]\\&=\omega L\,I_{\text{peak}}\,\cos \left(\omega t\right)=\omega L\,I_{\text{peak}}\,\sin \left(\omega t+{\pi \over 2}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dad77f4c1a6127e7e85ee68ac5bfbfc3fedd456d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.341ex; margin-bottom: -0.331ex; width:48.414ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}v(t)&=L{\frac {{\text{d}}i}{{\text{d}}t}}=L\,{\frac {\text{d}}{{\text{d}}t}}\left[I_{\text{peak}}\sin \left(\omega t\right)\right]\\&=\omega L\,I_{\text{peak}}\,\cos \left(\omega t\right)=\omega L\,I_{\text{peak}}\,\sin \left(\omega t+{\pi \over 2}\right)\end{aligned}}}"></span> </p><p>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 I_{\text{peak}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\text{peak}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b61c043a076d2e764d689c5f27f18c42c6868a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.589ex; height:2.843ex;" alt="{\displaystyle I_{\text{peak}}}"></span> is the <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> (peak value) of the sinusoidal current in amperes, <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 \omega =2\pi f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =2\pi f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1bf35d395c2d52391265e4bbda0aed14f52579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.317ex; height:2.509ex;" alt="{\displaystyle \omega =2\pi f}"></span> is the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> of the alternating current, with <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> being its <a href="/wiki/Frequency" title="Frequency">frequency</a> in <a href="/wiki/Hertz_(unit)" class="mw-redirect" title="Hertz (unit)">hertz</a>, 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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is the inductance. </p><p>Thus the amplitude (peak value) of the voltage across the inductance is </p><p><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 V_{p}=\omega L\,I_{p}=2\pi f\,L\,I_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mi>ω</mi> <mi>L</mi> <mspace width="thinmathspace" /> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mspace width="thinmathspace" /> <mi>L</mi> <mspace width="thinmathspace" /> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{p}=\omega L\,I_{p}=2\pi f\,L\,I_{p}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b1082f81eee13584b2a44fe8013ff1d3e6d74f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.322ex; height:2.843ex;" alt="{\displaystyle V_{p}=\omega L\,I_{p}=2\pi f\,L\,I_{p}}"></span> </p><p>Inductive <a href="/wiki/Reactance_(electronics)" class="mw-redirect" title="Reactance (electronics)">reactance</a> is the opposition of an inductor to an alternating current.<sup id="cite_ref-Gates_21-0" class="reference"><a href="#cite_note-Gates-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> It is defined analogously to <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">electrical resistance</a> in a resistor, as the ratio of the <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> (peak value) of the alternating voltage to current in the component </p><p><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 X_{L}={\frac {V_{p}}{I_{p}}}=2\pi f\,L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mspace width="thinmathspace" /> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{L}={\frac {V_{p}}{I_{p}}}=2\pi f\,L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea510d4c650038b2c7a1a2a5e41bdcd3a9cb18c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.466ex; height:6.176ex;" alt="{\displaystyle X_{L}={\frac {V_{p}}{I_{p}}}=2\pi f\,L}"></span> </p><p>Reactance has units of <a href="/wiki/Ohm_(unit)" class="mw-redirect" title="Ohm (unit)">ohms</a>. It can be seen that <a href="/wiki/Inductive_reactance" class="mw-redirect" title="Inductive reactance">inductive reactance</a> of an inductor increases proportionally with frequency <span class="nowrap"><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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span>,</span> so an inductor conducts less current for a given applied AC voltage as the frequency increases. Because the induced voltage is greatest when the current is increasing, the voltage and current waveforms are <a href="/wiki/Out_of_phase" class="mw-redirect" title="Out of phase">out of phase</a>; the voltage peaks occur earlier in each cycle than the current peaks. The phase difference between the current and the induced voltage is <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 ={\tfrac {1}{2}}\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi ={\tfrac {1}{2}}\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffbbc9cd5f5b9d6a0650d32c4ec8b291f04bc33a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.474ex; height:3.509ex;" alt="{\displaystyle \phi ={\tfrac {1}{2}}\pi }"></span> <a href="/wiki/Radian" title="Radian">radians</a> or 90 degrees, showing that in an ideal inductor <i>the current lags the voltage by 90°</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Calculating_inductance">Calculating inductance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=5" title="Edit section: Calculating inductance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the most general case, inductance can be calculated from Maxwell's equations. Many important cases can be solved using simplifications. Where high frequency currents are considered, with <a href="/wiki/Skin_effect" title="Skin effect">skin effect</a>, the surface current densities and magnetic field may be obtained by solving the <a href="/wiki/Laplace_equation" class="mw-redirect" title="Laplace equation">Laplace equation</a>. Where the conductors are thin wires, self-inductance still depends on the wire radius and the distribution of the current in the wire. This current distribution is approximately constant (on the surface or in the volume of the wire) for a wire radius much smaller than other length scales. </p> <div class="mw-heading mw-heading3"><h3 id="Inductance_of_a_straight_single_wire">Inductance of a straight single wire</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=6" title="Edit section: Inductance of a straight single wire"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As a practical matter, longer wires have more inductance, and thicker wires have less, analogous to their electrical resistance (although the relationships aren't linear, and are different in kind from the relationships that length and diameter bear to resistance). </p><p>Separating the wire from the other parts of the circuit introduces some unavoidable error in any formulas' results. These inductances are often referred to as “partial inductances”, in part to encourage consideration of the other contributions to whole-circuit inductance which are omitted. </p> <div class="mw-heading mw-heading4"><h4 id="Practical_formulas">Practical formulas</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=7" title="Edit section: Practical formulas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For derivation of the formulas below, see Rosa (1908).<sup id="cite_ref-Rosa1908_22-0" class="reference"><a href="#cite_note-Rosa1908-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> The total low frequency inductance (interior plus exterior) of a straight wire is: </p><p><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 L_{\text{DC}}=200{\text{ }}{\tfrac {\text{nH}}{\text{m}}}\,\ell \left[\ln \left({\frac {\,2\,\ell \,}{r}}\right)-0.75\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>DC</mtext> </mrow> </msub> <mo>=</mo> <mn>200</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mtext>nH</mtext> <mtext>m</mtext> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mi>ℓ</mi> <mrow> <mo>[</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mn>2</mn> <mspace width="thinmathspace" /> <mi>ℓ</mi> <mspace width="thinmathspace" /> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>0.75</mn> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\text{DC}}=200{\text{ }}{\tfrac {\text{nH}}{\text{m}}}\,\ell \left[\ln \left({\frac {\,2\,\ell \,}{r}}\right)-0.75\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39dc33d2f27647bf40a0b2261866624a63f7ecc5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.07ex; height:6.176ex;" alt="{\displaystyle L_{\text{DC}}=200{\text{ }}{\tfrac {\text{nH}}{\text{m}}}\,\ell \left[\ln \left({\frac {\,2\,\ell \,}{r}}\right)-0.75\right]}"></span> </p><p>where </p> <ul><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 L_{\text{DC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>DC</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\text{DC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c09438c413744c9e0e00918d928cd8e1fd0c1ee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.257ex; height:2.509ex;" alt="{\displaystyle L_{\text{DC}}}"></span> is the "low-frequency" or DC inductance in nanohenry (nH or 10<sup>−9</sup>H),</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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span> is the length of the wire in meters,</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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is the radius of the wire in meters (hence a very small decimal number),</li> <li>the constant <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 200{\text{ }}{\tfrac {\text{nH}}{\text{m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>200</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mtext>nH</mtext> <mtext>m</mtext> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 200{\text{ }}{\tfrac {\text{nH}}{\text{m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bad31e1d5ecc6bb11b6bfe1883f22193f8613217" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.051ex; height:3.343ex;" alt="{\displaystyle 200{\text{ }}{\tfrac {\text{nH}}{\text{m}}}}"></span> is the <a href="/wiki/Vacuum_permeability" title="Vacuum permeability">permeability of free space</a>, commonly called <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 _{\text{o}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\text{o}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f751f2baf43a6a82c6dd7b3cae6e05c590a1c8e9" 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 _{\text{o}}}"></span>, divided by <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 2\pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73efd1f6493490b058097060a572606d2c550a06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }"></span>; in the absence of magnetically reactive insulation the value 200 is exact when using the classical definition of <i>μ</i><sub>0</sub> = <span class="nowrap"><span data-sort-value="6993400000000000000♠"></span>4π<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−7</sup> H/m</span>, and correct to 7 decimal places when using the <a href="/wiki/2019_revision_of_the_SI" title="2019 revision of the SI">2019-redefined SI value</a> of <i>μ</i><sub>0</sub> = <span class="nowrap"><span data-sort-value="6994125663706212000♠"></span>1.256<span style="margin-left:.25em;">637</span><span style="margin-left:.25em;">062</span><span style="margin-left:.25em;">12</span>(19)<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−6</sup> <a href="/wiki/Henry_(unit)" title="Henry (unit)">H</a>/m</span>.</li></ul> <p>The constant 0.75 is just one parameter value among several; different frequency ranges, different shapes, or extremely long wire lengths require a slightly different constant (<a href="#current_distribution_parameter_Y">see below</a>). This result is based on the assumption that the radius <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is much less than the length <span class="nowrap"><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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span>,</span> which is the common case for wires and rods. Disks or thick cylinders have slightly different formulas. </p><p>For sufficiently high frequencies skin effects cause the interior currents to vanish, leaving only the currents on the surface of the conductor; the inductance for alternating current, <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 L_{\text{AC}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>AC</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\text{AC}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20e8d42edc75f9d47d88d381c0a005d139ab428b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.234ex; height:2.509ex;" alt="{\displaystyle L_{\text{AC}}}"></span> is then given by a very similar formula: </p><p><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 L_{\text{AC}}=200{\text{ }}{\tfrac {\text{nH}}{\text{m}}}\,\ell \left[\ln \left({\frac {\,2\,\ell \,}{r}}\right)-1\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>AC</mtext> </mrow> </msub> <mo>=</mo> <mn>200</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mtext>nH</mtext> <mtext>m</mtext> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mi>ℓ</mi> <mrow> <mo>[</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mn>2</mn> <mspace width="thinmathspace" /> <mi>ℓ</mi> <mspace width="thinmathspace" /> </mrow> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\text{AC}}=200{\text{ }}{\tfrac {\text{nH}}{\text{m}}}\,\ell \left[\ln \left({\frac {\,2\,\ell \,}{r}}\right)-1\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c01f6774dfc67af4b2f63877692be926882acc2e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.075ex; height:6.176ex;" alt="{\displaystyle L_{\text{AC}}=200{\text{ }}{\tfrac {\text{nH}}{\text{m}}}\,\ell \left[\ln \left({\frac {\,2\,\ell \,}{r}}\right)-1\right]}"></span> where the variables <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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span> 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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> are the same as above; note the changed constant term now 1, from 0.75 above. </p><p>In an example from everyday experience, just one of the conductors of a lamp cord <span class="nowrap"><span data-sort-value="7001100000000000000♠"></span>10 m</span> long, made of 18 <a href="/wiki/American_wire_gauge" title="American wire gauge">AWG</a> wire, would only have an inductance of about <span class="nowrap"><span data-sort-value="6995189999999999999♠"></span>19 μH</span> if stretched out straight. </p> <div class="mw-heading mw-heading3"><h3 id="Mutual_inductance_of_two_parallel_straight_wires">Mutual inductance of two parallel straight wires</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=8" title="Edit section: Mutual inductance of two parallel straight wires"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are two cases to consider: </p> <ol><li>Current travels in the same direction in each wire, and</li> <li>current travels in opposing directions in the wires.</li></ol> <p>Currents in the wires need not be equal, though they often are, as in the case of a complete circuit, where one wire is the source and the other the return. </p> <div class="mw-heading mw-heading3"><h3 id="Mutual_inductance_of_two_wire_loops">Mutual inductance of two wire loops</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=9" title="Edit section: Mutual inductance of two wire loops"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This is the generalized case of the paradigmatic two-loop cylindrical coil carrying a uniform low frequency current; the loops are independent closed circuits that can have different lengths, any orientation in space, and carry different currents. Nonetheless, the error terms, which are not included in the integral are only small if the geometries of the loops are mostly smooth and convex: They must not have too many kinks, sharp corners, coils, crossovers, parallel segments, concave cavities, or other topologically "close" deformations. A necessary predicate for the reduction of the 3-dimensional manifold integration formula to a double curve integral is that the current paths be filamentary circuits, i.e. thin wires where the radius of the wire is negligible compared to its length. </p><p>The mutual inductance by a filamentary circuit <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> on a filamentary circuit <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is given by the double integral <i><a href="/wiki/Franz_Ernst_Neumann" title="Franz Ernst Neumann">Neumann</a> formula</i><sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 L_{m,n}={\frac {\mu _{0}}{4\pi }}\oint _{C_{m}}\oint _{C_{n}}{\frac {\mathrm {d} \mathbf {x} _{m}\cdot \mathrm {d} \mathbf {x} _{n}}{\ \left|\mathbf {x} _{m}-\mathbf {x} _{n}\right|\ }}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </msub> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mtext> </mtext> <mrow> <mo>|</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{m,n}={\frac {\mu _{0}}{4\pi }}\oint _{C_{m}}\oint _{C_{n}}{\frac {\mathrm {d} \mathbf {x} _{m}\cdot \mathrm {d} \mathbf {x} _{n}}{\ \left|\mathbf {x} _{m}-\mathbf {x} _{n}\right|\ }}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c16aff97ada507d4481b0a4e06ad070423dfad6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.841ex; height:6.176ex;" alt="{\displaystyle L_{m,n}={\frac {\mu _{0}}{4\pi }}\oint _{C_{m}}\oint _{C_{n}}{\frac {\mathrm {d} \mathbf {x} _{m}\cdot \mathrm {d} \mathbf {x} _{n}}{\ \left|\mathbf {x} _{m}-\mathbf {x} _{n}\right|\ }}\ ,}"></span></dd></dl> <p>where </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 C_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/957e2166e45b11d83b27710666eb7aa38f5e4b89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.337ex; height:2.509ex;" alt="{\displaystyle C_{m}}"></span> 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 C_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0301812adb392070d834ca2df4ed97f1cf132f33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle C_{n}}"></span> are the curves followed by the wires.</dd> <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 \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 <a href="/wiki/Permeability_of_free_space" class="mw-redirect" title="Permeability of free space">permeability of free space</a> (<span class="nowrap">4<span class="texhtml mvar" style="font-style:italic;">π</span>×10<sup>−7</sup> H/m</span>)</dd> <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 \mathrm {d} \mathbf {x} _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mathbf {x} _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4350b23f6c494d91ca5432ab308852d3799ae6ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.379ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \mathbf {x} _{m}}"></span> is a small increment of the wire in circuit <span class="texhtml mvar" style="font-style:italic;">C</span><sub>m</sub></dd> <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 \mathbf {x} _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c877cff44ece60d864b32c5ad4b08c21acf7f27b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.086ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{m}}"></span> is the position of <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 \mathrm {d} \mathbf {x} _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mathbf {x} _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4350b23f6c494d91ca5432ab308852d3799ae6ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.379ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \mathbf {x} _{m}}"></span> in space</dd> <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 \mathrm {d} \mathbf {x} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mathbf {x} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47a827880e4a0863252d0e10cd47a3e064190fd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.922ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \mathbf {x} _{n}}"></span> is a small increment of the wire in circuit <span class="texhtml mvar" style="font-style:italic;">C</span><sub>n</sub></dd> <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 \mathbf {x} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19c4eb8112f7db1356461e380c1a7ff13a085688" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.629ex; height:2.009ex;" alt="{\displaystyle \mathbf {x} _{n}}"></span> is the position of <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 \mathrm {d} \mathbf {x} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} \mathbf {x} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47a827880e4a0863252d0e10cd47a3e064190fd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.922ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} \mathbf {x} _{n}}"></span> in space.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Derivation">Derivation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=10" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><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 M_{ij}\mathrel {\stackrel {\mathrm {def} }{=}} {\frac {\Phi _{ij}}{I_{j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-REL"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{ij}\mathrel {\stackrel {\mathrm {def} }{=}} {\frac {\Phi _{ij}}{I_{j}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba686177086289f8220e8fefb95cd4d50770021e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.269ex; height:6.176ex;" alt="{\displaystyle M_{ij}\mathrel {\stackrel {\mathrm {def} }{=}} {\frac {\Phi _{ij}}{I_{j}}}}"></span> </p><p>where </p> <ul><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 I_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7cb04680c2246f78082e43ec912f72afe7af266" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.933ex; height:2.843ex;" alt="{\displaystyle I_{j}}"></span> is the current through the <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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>th wire, this current creates the magnetic flux <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 _{ij}\ \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mtext> </mtext> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{ij}\ \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f1c21caa7785a1f4b3681e92eeb92d3a9d1bfd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.123ex; height:2.843ex;" alt="{\displaystyle \Phi _{ij}\ \,}"></span>through the <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/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>th surface</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 \Phi _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2dae28913f519171bf3a021c3cf392f4bc7b238" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.155ex; height:2.843ex;" alt="{\displaystyle \Phi _{ij}}"></span> is the <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a> through the <i>i</i>th surface due to the <a href="/wiki/Electrical_circuit" class="mw-redirect" title="Electrical circuit">electrical circuit</a> outlined by <span class="nowrap"><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 C_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4598eb9b4e1c079e0973de79383fc6f898744e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.572ex; height:2.843ex;" alt="{\displaystyle C_{j}}"></span>:</span><sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup></li></ul> <p><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 _{ij}=\int _{S_{i}}\mathbf {B} _{j}\cdot \mathrm {d} \mathbf {a} =\int _{S_{i}}(\nabla \times \mathbf {A_{j}} )\cdot \mathrm {d} \mathbf {a} =\oint _{C_{i}}\mathbf {A} _{j}\cdot \mathrm {d} \mathbf {s} _{i}=\oint _{C_{i}}\left({\frac {\mu _{0}I_{j}}{4\pi }}\oint _{C_{j}}{\frac {\mathrm {d} \mathbf {s} _{j}}{\left|\mathbf {s} _{i}-\mathbf {s} _{j}\right|}}\right)\cdot \mathrm {d} \mathbf {s} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> </msub> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mo>|</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>|</mo> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{ij}=\int _{S_{i}}\mathbf {B} _{j}\cdot \mathrm {d} \mathbf {a} =\int _{S_{i}}(\nabla \times \mathbf {A_{j}} )\cdot \mathrm {d} \mathbf {a} =\oint _{C_{i}}\mathbf {A} _{j}\cdot \mathrm {d} \mathbf {s} _{i}=\oint _{C_{i}}\left({\frac {\mu _{0}I_{j}}{4\pi }}\oint _{C_{j}}{\frac {\mathrm {d} \mathbf {s} _{j}}{\left|\mathbf {s} _{i}-\mathbf {s} _{j}\right|}}\right)\cdot \mathrm {d} \mathbf {s} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddde3e35569463a5ffd2a47e42e7de97adbbbe58" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:84.91ex; height:7.509ex;" alt="{\displaystyle \Phi _{ij}=\int _{S_{i}}\mathbf {B} _{j}\cdot \mathrm {d} \mathbf {a} =\int _{S_{i}}(\nabla \times \mathbf {A_{j}} )\cdot \mathrm {d} \mathbf {a} =\oint _{C_{i}}\mathbf {A} _{j}\cdot \mathrm {d} \mathbf {s} _{i}=\oint _{C_{i}}\left({\frac {\mu _{0}I_{j}}{4\pi }}\oint _{C_{j}}{\frac {\mathrm {d} \mathbf {s} _{j}}{\left|\mathbf {s} _{i}-\mathbf {s} _{j}\right|}}\right)\cdot \mathrm {d} \mathbf {s} _{i}}"></span> </p><p>where </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="margin-left: 1.6em;"> <ul><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 C_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc49dc02c0ec8c86b67e7d10518ac791eda0bf22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.461ex; height:2.509ex;" alt="{\displaystyle C_{i}}"></span> is the curve enclosing surface <span class="nowrap"><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 S_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de6e810a93f67802ecb603ee0e3324005c6e583e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.509ex;" alt="{\displaystyle S_{i}}"></span>;</span> 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 S_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de6e810a93f67802ecb603ee0e3324005c6e583e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.225ex; height:2.509ex;" alt="{\displaystyle S_{i}}"></span> is any arbitrary orientable area with edge <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 C_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc49dc02c0ec8c86b67e7d10518ac791eda0bf22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.461ex; height:2.509ex;" alt="{\displaystyle C_{i}}"></span></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 \mathbf {B} _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80a0200459f2e80ed4c3da665791a9fb100905e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.811ex; height:2.843ex;" alt="{\displaystyle \mathbf {B} _{j}}"></span> is the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> vector due to the <span class="nowrap"><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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>-th</span> current (of circuit <span class="nowrap"><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 C_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4598eb9b4e1c079e0973de79383fc6f898744e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.572ex; height:2.843ex;" alt="{\displaystyle C_{j}}"></span>).</span></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 \mathbf {A} _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65678ae27e9a7224fe6d7ae71e2b13fd3daf6894" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.929ex; height:2.843ex;" alt="{\displaystyle \mathbf {A} _{j}}"></span> is the <a href="/wiki/Vector_potential" title="Vector potential">vector potential</a> due to the <span class="nowrap"><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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>-th</span> current.</li></ul> </div> <p><a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes' theorem</a> has been used for the 3rd equality step. For the last equality step, we used the <a href="/wiki/Retarded_potential" title="Retarded potential">retarded potential</a> expression for <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 A_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6019bb70c912e59e9d5f442e9217517743ed4831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.653ex; height:2.843ex;" alt="{\displaystyle A_{j}}"></span> and we ignore the effect of the retarded time (assuming the geometry of the circuits is small enough compared to the wavelength of the current they carry). It is actually an approximation step, and is valid only for local circuits made of thin wires. </p> <div class="mw-heading mw-heading3"><h3 id="Self-inductance_of_a_wire_loop">Self-inductance of a wire loop</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=11" title="Edit section: Self-inductance of a wire loop"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Formally, the self-inductance of a wire loop would be given by the above equation with <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 \ m=n\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>m</mi> <mo>=</mo> <mi>n</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ m=n\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12f15e28fb29ef4843d17e174c4582735849c7aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.342ex; height:1.676ex;" alt="{\displaystyle \ m=n\ .}"></span> However, here <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 \ 1/\left|\mathbf {x} -\mathbf {x} '\right|\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>′</mo> </msup> </mrow> <mo>|</mo> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 1/\left|\mathbf {x} -\mathbf {x} '\right|\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5a9f5696cc4081ad12340c7224e1097c39623a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.901ex; height:3.009ex;" alt="{\displaystyle \ 1/\left|\mathbf {x} -\mathbf {x} '\right|\ }"></span> becomes infinite, leading to a logarithmically divergent integral.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> This necessitates taking the finite wire radius <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 \ a\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>a</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ a\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8124de742ae987fe73be9ca9d3d4ba8586e28b11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.391ex; height:1.676ex;" alt="{\displaystyle \ a\ }"></span> and the distribution of the current in the wire into account. There remains the contribution from the integral over all points and a correction term,<sup id="cite_ref-den12_26-0" class="reference"><a href="#cite_note-den12-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 L={\frac {\mu _{0}}{4\pi }}\left[\ \ell \ Y+\oint _{C}\oint _{C'}{\frac {\mathrm {d} \mathbf {x} \cdot \mathrm {d} \mathbf {x} '}{\ \left|\mathbf {x} -\mathbf {x} '\right|\ }}\ \right]+{\mathcal {O}}_{\mathsf {bend}}\quad {\text{ for }}\;\left|\mathbf {s} -\mathbf {s} '\right|>{\tfrac {1}{2}}a\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mi>ℓ</mi> <mtext> </mtext> <mi>Y</mi> <mo>+</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>C</mi> <mo>′</mo> </msup> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>′</mo> </msup> </mrow> <mrow> <mtext> </mtext> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>′</mo> </msup> </mrow> <mo>|</mo> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">b</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </msub> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mspace width="thickmathspace" /> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo>′</mo> </msup> </mrow> <mo>|</mo> </mrow> <mo>></mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mi>a</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\frac {\mu _{0}}{4\pi }}\left[\ \ell \ Y+\oint _{C}\oint _{C'}{\frac {\mathrm {d} \mathbf {x} \cdot \mathrm {d} \mathbf {x} '}{\ \left|\mathbf {x} -\mathbf {x} '\right|\ }}\ \right]+{\mathcal {O}}_{\mathsf {bend}}\quad {\text{ for }}\;\left|\mathbf {s} -\mathbf {s} '\right|>{\tfrac {1}{2}}a\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/894bc2bf45ec4caf3bb8bc823bfaa684943b698f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:64.469ex; height:6.343ex;" alt="{\displaystyle L={\frac {\mu _{0}}{4\pi }}\left[\ \ell \ Y+\oint _{C}\oint _{C'}{\frac {\mathrm {d} \mathbf {x} \cdot \mathrm {d} \mathbf {x} '}{\ \left|\mathbf {x} -\mathbf {x} '\right|\ }}\ \right]+{\mathcal {O}}_{\mathsf {bend}}\quad {\text{ for }}\;\left|\mathbf {s} -\mathbf {s} '\right|>{\tfrac {1}{2}}a\ }"></span></dd></dl> <p>where </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 \ \mathbf {s} \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mathbf {s} \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15110288d98d2ff3b63447739f1711d69b0d3e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:1.676ex;" alt="{\displaystyle \ \mathbf {s} \ }"></span> 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 \ \mathbf {s} '\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo>′</mo> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \mathbf {s} '\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07ce0c5b32b060ca7acfc3573205fe0680cb0162" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.902ex; height:2.509ex;" alt="{\displaystyle \ \mathbf {s} '\ }"></span> are distances along the curves <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 \ C\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>C</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ C\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b010f9498beac548bca41502f10621736d581b98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.928ex; height:2.176ex;" alt="{\displaystyle \ C\ }"></span> 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 \ C'\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>C</mi> <mo>′</mo> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ C'\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aab30cf3f746e86b199ec895add767c3c082a4fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.644ex; height:2.509ex;" alt="{\displaystyle \ C'\ }"></span> respectively</dd> <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 \ a\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>a</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ a\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8124de742ae987fe73be9ca9d3d4ba8586e28b11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.391ex; height:1.676ex;" alt="{\displaystyle \ a\ }"></span> is the radius of the wire</dd> <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 \ \ell \ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ℓ</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \ell \ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e027f732c772e69ea5a8e5d781c4517b35d3b88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.131ex; height:2.176ex;" alt="{\displaystyle \ \ell \ }"></span> is the length of the wire</dd> <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 \ Y\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>Y</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ Y\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9fae0bc05719eaaac8322fd4c65fa3851477fab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.935ex; height:2.009ex;" alt="{\displaystyle \ Y\ }"></span> is a constant that depends on the distribution of the current in the wire: <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 \ Y=0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>Y</mi> <mo>=</mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ Y=0\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d21b911ab60881ca2fb247e35faf6970db4b5410" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.196ex; height:2.176ex;" alt="{\displaystyle \ Y=0\ }"></span> when the current flows on the surface of the wire (total <a href="/wiki/Skin_effect" title="Skin effect">skin effect</a>),</dd> <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="{\textstyle \ Y={\tfrac {1}{2}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> </mtext> <mi>Y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \ Y={\tfrac {1}{2}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f049733dbcf0fe50fac1352c4bb9a3c5a52a3486" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.691ex; height:3.509ex;" alt="{\textstyle \ Y={\tfrac {1}{2}}\ }"></span> when the current is evenly over the cross-section of the wire.</dd></dl></dd> <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 {O}}_{\mathsf {bend}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">b</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {O}}_{\mathsf {bend}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aa10589db5dcdc7a0560f3eaa496d8496cba077" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.523ex; height:2.509ex;" alt="{\displaystyle \ {\mathcal {O}}_{\mathsf {bend}}\ }"></span> is an error term whose size depends on the curve of the loop: <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 {O}}_{\mathsf {bend}}={\mathcal {O}}(\mu _{0}a)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">b</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>a</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {O}}_{\mathsf {bend}}={\mathcal {O}}(\mu _{0}a)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aac6e71616edf333723dc4cdffed22dc8964a73c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.967ex; height:2.843ex;" alt="{\displaystyle \ {\mathcal {O}}_{\mathsf {bend}}={\mathcal {O}}(\mu _{0}a)\ }"></span> when the loop has sharp corners, and</dd> <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="{\textstyle \ {\mathcal {O}}_{\mathsf {bend}}={\mathcal {O}}{\mathord {\left({\mu _{0}a^{2}}/{\ell }\right)}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> </mtext> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">b</mi> <mi mathvariant="sans-serif">e</mi> <mi mathvariant="sans-serif">n</mi> <mi mathvariant="sans-serif">d</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>ℓ</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \ {\mathcal {O}}_{\mathsf {bend}}={\mathcal {O}}{\mathord {\left({\mu _{0}a^{2}}/{\ell }\right)}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6456505e5b0a6ca9a15b7be4526fc158a757fdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.474ex; height:3.176ex;" alt="{\textstyle \ {\mathcal {O}}_{\mathsf {bend}}={\mathcal {O}}{\mathord {\left({\mu _{0}a^{2}}/{\ell }\right)}}\ }"></span> when it is a smooth curve.</dd> <dd>Both are small when the wire is long compared to its radius.</dd></dl></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Inductance_of_a_solenoid">Inductance of a solenoid</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=12" title="Edit section: Inductance of a solenoid"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Solenoid" title="Solenoid">solenoid</a> is a long, thin coil; i.e., a coil whose length is much greater than its diameter. Under these conditions, and without any magnetic material used, the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic flux density</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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> within the coil is practically constant and is given by <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 B={\frac {\mu _{0}\,N\,i}{\ell }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> <mi>i</mi> </mrow> <mi>ℓ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B={\frac {\mu _{0}\,N\,i}{\ell }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f96348caa5755924bf1754fd6fd2ceb070f6876" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.795ex; height:5.509ex;" alt="{\displaystyle B={\frac {\mu _{0}\,N\,i}{\ell }}}"></span> </p><p>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 \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 <a href="/wiki/Magnetic_constant" class="mw-redirect" title="Magnetic constant">magnetic constant</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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" 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 N}"></span> the number of turns, <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/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> the current 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 l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"></span> the length of the coil. Ignoring end effects, the total magnetic flux through the coil is obtained by multiplying the flux density <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 B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> by the cross-section area <span class="nowrap"><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 A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>:</span> <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 ={\frac {\mu _{0}\,N\,i\,A}{\ell }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi>N</mi> <mspace width="thinmathspace" /> <mi>i</mi> <mspace width="thinmathspace" /> <mi>A</mi> </mrow> <mi>ℓ</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi ={\frac {\mu _{0}\,N\,i\,A}{\ell }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb70305cb61ca0cfee8a8b7894961aceec3fc21c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.486ex; height:5.676ex;" alt="{\displaystyle \Phi ={\frac {\mu _{0}\,N\,i\,A}{\ell }},}"></span> </p><p>When this is combined with the definition of inductance <span class="nowrap"><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 L={\frac {N\,\Phi }{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Φ</mi> </mrow> <mi>i</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\frac {N\,\Phi }{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dfdbd33e0a6dfb70025548ed8b46c3bf77eb12e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.646ex; height:5.176ex;" alt="{\displaystyle L={\frac {N\,\Phi }{i}}}"></span>,</span> it follows that the inductance of a solenoid is given by: <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 L={\frac {\mu _{0}\,N^{2}\,A}{\ell }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>A</mi> </mrow> <mi>ℓ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\frac {\mu _{0}\,N^{2}\,A}{\ell }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/340fb32942baac9aada07f51f2e865eb54307a8f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.315ex; height:5.843ex;" alt="{\displaystyle L={\frac {\mu _{0}\,N^{2}\,A}{\ell }}.}"></span> </p><p>Therefore, for air-core coils, inductance is a function of coil geometry and number of turns, and is independent of current. </p> <div class="mw-heading mw-heading3"><h3 id="Inductance_of_a_coaxial_cable">Inductance of a coaxial cable</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=13" title="Edit section: Inductance of a coaxial cable"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let the inner conductor have radius <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0b6d651eaf432dbf1f106021c8bb499ae83fd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.848ex; height:2.009ex;" alt="{\displaystyle r_{i}}"></span> and <a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">permeability</a> <span class="nowrap"><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 _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dea0a0293841cce9eef98b55e53a92b82ae59ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.201ex; height:2.176ex;" alt="{\displaystyle \mu _{i}}"></span>,</span> let the dielectric between the inner and outer conductor have permeability <span class="nowrap"><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 _{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f5a79bfd0a50678913082f322d849349359975c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle \mu _{d}}"></span>,</span> and let the outer conductor have inner radius <span class="nowrap"><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_{o1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{o1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50cfa3bd0be89e8b8fb64ba9b7cfaa593f348bc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.9ex; height:2.009ex;" alt="{\displaystyle r_{o1}}"></span>,</span> outer radius <span class="nowrap"><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_{o2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{o2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e58d9dd96489f1422fe7e8d856dd8c42f34a76d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.9ex; height:2.009ex;" alt="{\displaystyle r_{o2}}"></span>,</span> and permeability <span class="nowrap"><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>.</span> However, for a typical coaxial line application, we are interested in passing (non-DC) signals at frequencies for which the resistive <a href="/wiki/Skin_effect" title="Skin effect">skin effect</a> cannot be neglected. In most cases, the inner and outer conductor terms are negligible, in which case one may approximate </p><p><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 L'={\frac {{\text{d}}L}{{\text{d}}\ell }}\approx {\frac {\mu _{d}}{2\pi }}\ln {\frac {r_{o1}}{r_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>L</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>ℓ</mi> </mrow> </mfrac> </mrow> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L'={\frac {{\text{d}}L}{{\text{d}}\ell }}\approx {\frac {\mu _{d}}{2\pi }}\ln {\frac {r_{o1}}{r_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6d43ae982e76b7e1a5b24f57a9a82d2f31460d6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.957ex; height:5.676ex;" alt="{\displaystyle L'={\frac {{\text{d}}L}{{\text{d}}\ell }}\approx {\frac {\mu _{d}}{2\pi }}\ln {\frac {r_{o1}}{r_{i}}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Inductance_of_multilayer_coils">Inductance of multilayer coils</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=14" title="Edit section: Inductance of multilayer coils"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most practical air-core inductors are multilayer cylindrical coils with square cross-sections to minimize average distance between turns (circular cross -sections would be better but harder to form). </p> <div class="mw-heading mw-heading3"><h3 id="Magnetic_cores">Magnetic cores</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=15" title="Edit section: Magnetic cores"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many inductors include a <a href="/wiki/Magnetic_core" title="Magnetic core">magnetic core</a> at the center of or partly surrounding the winding. Over a large enough range these exhibit a nonlinear permeability with effects such as <a href="/wiki/Saturation_(magnetic)" title="Saturation (magnetic)">magnetic saturation</a>. Saturation makes the resulting inductance a function of the applied current. </p><p>The secant or large-signal inductance is used in flux calculations. It is defined as: </p><p><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 L_{s}(i)\mathrel {\overset {\underset {\mathrm {def} }{}}{=}} {\frac {N\ \Phi }{i}}={\frac {\Lambda }{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-REL"> <mover> <mo>=</mo> <munder> <mrow /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </munder> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>N</mi> <mtext> </mtext> <mi mathvariant="normal">Φ</mi> </mrow> <mi>i</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">Λ</mi> <mi>i</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{s}(i)\mathrel {\overset {\underset {\mathrm {def} }{}}{=}} {\frac {N\ \Phi }{i}}={\frac {\Lambda }{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a608172353d0f5abccab36923bfe09267a01fa0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.026ex; height:5.343ex;" alt="{\displaystyle L_{s}(i)\mathrel {\overset {\underset {\mathrm {def} }{}}{=}} {\frac {N\ \Phi }{i}}={\frac {\Lambda }{i}}}"></span> </p><p>The differential or small-signal inductance, on the other hand, is used in calculating voltage. It is defined as: </p><p><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 L_{d}(i)\mathrel {\overset {\underset {\mathrm {def} }{}}{=}} {\frac {{\text{d}}(N\Phi )}{{\text{d}}i}}={\frac {{\text{d}}\Lambda }{{\text{d}}i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-REL"> <mover> <mo>=</mo> <munder> <mrow /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </munder> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mo stretchy="false">(</mo> <mi>N</mi> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>i</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi mathvariant="normal">Λ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>i</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{d}(i)\mathrel {\overset {\underset {\mathrm {def} }{}}{=}} {\frac {{\text{d}}(N\Phi )}{{\text{d}}i}}={\frac {{\text{d}}\Lambda }{{\text{d}}i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56e92f426c2a65e29921c1f2c7c20a1d9e2fa4b7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.928ex; height:5.843ex;" alt="{\displaystyle L_{d}(i)\mathrel {\overset {\underset {\mathrm {def} }{}}{=}} {\frac {{\text{d}}(N\Phi )}{{\text{d}}i}}={\frac {{\text{d}}\Lambda }{{\text{d}}i}}}"></span> </p><p>The circuit voltage for a nonlinear inductor is obtained via the differential inductance as shown by Faraday's Law and the <a href="/wiki/Chain_rule" title="Chain rule">chain rule</a> of calculus. </p><p><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 v(t)={\frac {{\text{d}}\Lambda }{{\text{d}}t}}={\frac {{\text{d}}\Lambda }{{\text{d}}i}}{\frac {{\text{d}}i}{{\text{d}}t}}=L_{d}(i){\frac {{\text{d}}i}{{\text{d}}t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi mathvariant="normal">Λ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi mathvariant="normal">Λ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>i</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>i</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>i</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)={\frac {{\text{d}}\Lambda }{{\text{d}}t}}={\frac {{\text{d}}\Lambda }{{\text{d}}i}}{\frac {{\text{d}}i}{{\text{d}}t}}=L_{d}(i){\frac {{\text{d}}i}{{\text{d}}t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb1c392284e44a2022ba8755f53f0c2bab30688d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.778ex; height:5.509ex;" alt="{\displaystyle v(t)={\frac {{\text{d}}\Lambda }{{\text{d}}t}}={\frac {{\text{d}}\Lambda }{{\text{d}}i}}{\frac {{\text{d}}i}{{\text{d}}t}}=L_{d}(i){\frac {{\text{d}}i}{{\text{d}}t}}}"></span> </p><p>Similar definitions may be derived for nonlinear mutual inductance. </p> <div class="mw-heading mw-heading2"><h2 id="Mutual_inductance">Mutual inductance</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=16" title="Edit section: Mutual inductance"><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">Further information: <a href="/wiki/Inductive_coupling" title="Inductive coupling">Inductive coupling</a></div> <p>Mutual inductance is defined as the ratio between the EMF induced in one loop or coil by the rate of change of current in another loop or coil. Mutual inductance is given the symbol <span class="texhtml mvar" style="font-style:italic;">M</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Derivation_of_mutual_inductance">Derivation of mutual inductance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=17" title="Edit section: Derivation of mutual inductance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The inductance equations above are a consequence of <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>. For the important case of electrical circuits consisting of thin wires, the derivation is straightforward. </p><p>In a system of <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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}"></span> wire loops, each with one or several wire turns, the <a href="/wiki/Flux_linkage" title="Flux linkage">flux linkage</a> of loop <span class="nowrap"><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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>,</span> <span class="nowrap"><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 \lambda _{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f539901b6e206558d2a9e0ddfba3682cc56bab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.03ex; height:2.509ex;" alt="{\displaystyle \lambda _{m}}"></span>,</span> is given by <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 \displaystyle \lambda _{m}=N_{m}\Phi _{m}=\sum \limits _{n=1}^{K}L_{m,n}\ i_{n}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mtext> </mtext> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle \lambda _{m}=N_{m}\Phi _{m}=\sum \limits _{n=1}^{K}L_{m,n}\ i_{n}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d84d2f8cdb1487bb4f6a616498c3f8ef6e04ccb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.201ex; height:7.343ex;" alt="{\displaystyle \displaystyle \lambda _{m}=N_{m}\Phi _{m}=\sum \limits _{n=1}^{K}L_{m,n}\ i_{n}\,.}"></span> </p><p>Here <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 N_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c7e3465bc22700e96454f5f14ff3aaf63bea7fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.541ex; height:2.509ex;" alt="{\displaystyle N_{m}}"></span> denotes the number of turns in loop <span class="nowrap"><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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>;</span> <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e54271ff4d7f89332a69ec6cd1e004f057e671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.353ex; height:2.509ex;" alt="{\displaystyle \Phi _{m}}"></span> is the <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a> through loop <span class="nowrap"><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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>;</span> 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 L_{m,n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{m,n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/919bcf1275b14d15ab5a79ba31f53b1028acd772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.701ex; height:2.843ex;" alt="{\displaystyle L_{m,n}}"></span> are some constants described below. This equation follows from <a href="/wiki/Ampere%27s_law" class="mw-redirect" title="Ampere's law">Ampere's law</a>: <i>magnetic fields and fluxes are linear functions of the currents</i>. By <a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday's law of induction</a>, we have </p><p><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 \displaystyle v_{m}={\frac {{\text{d}}\lambda _{m}}{{\text{d}}t}}=N_{m}{\frac {{\text{d}}\Phi _{m}}{{\text{d}}t}}=\sum \limits _{n=1}^{K}L_{m,n}{\frac {{\text{d}}i_{n}}{{\text{d}}t}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle v_{m}={\frac {{\text{d}}\lambda _{m}}{{\text{d}}t}}=N_{m}{\frac {{\text{d}}\Phi _{m}}{{\text{d}}t}}=\sum \limits _{n=1}^{K}L_{m,n}{\frac {{\text{d}}i_{n}}{{\text{d}}t}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab74936d58a6887fd48b5a3029b7264e116266e4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.52ex; height:7.343ex;" alt="{\displaystyle \displaystyle v_{m}={\frac {{\text{d}}\lambda _{m}}{{\text{d}}t}}=N_{m}{\frac {{\text{d}}\Phi _{m}}{{\text{d}}t}}=\sum \limits _{n=1}^{K}L_{m,n}{\frac {{\text{d}}i_{n}}{{\text{d}}t}},}"></span> </p><p>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 v_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e39c47348c7acfafbf415f12afaaff6e0a0900ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.803ex; height:2.009ex;" alt="{\displaystyle v_{m}}"></span> denotes the voltage induced in circuit <span class="nowrap"><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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span>.</span> This agrees with the definition of inductance above if the coefficients <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 L_{m,n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{m,n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/919bcf1275b14d15ab5a79ba31f53b1028acd772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.701ex; height:2.843ex;" alt="{\displaystyle L_{m,n}}"></span> are identified with the coefficients of inductance. Because the total currents <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 N_{n}\ i_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mtext> </mtext> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{n}\ i_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b40fa6181339bb9f0de96987c7c04ea88df5f6f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.686ex; height:2.509ex;" alt="{\displaystyle N_{n}\ i_{n}}"></span> contribute to <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e54271ff4d7f89332a69ec6cd1e004f057e671" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.353ex; height:2.509ex;" alt="{\displaystyle \Phi _{m}}"></span> it also follows that <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 L_{m,n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{m,n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/919bcf1275b14d15ab5a79ba31f53b1028acd772" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.701ex; height:2.843ex;" alt="{\displaystyle L_{m,n}}"></span> is proportional to the product of turns <span class="nowrap"><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 N_{m}\ N_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mtext> </mtext> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{m}\ N_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d795ed0b52e10448a9cbc5baf6bcce04462790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.207ex; height:2.509ex;" alt="{\displaystyle N_{m}\ N_{n}}"></span>.</span> </p> <div class="mw-heading mw-heading3"><h3 id="Mutual_inductance_and_magnetic_field_energy">Mutual inductance and magnetic field energy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=18" title="Edit section: Mutual inductance and magnetic field energy"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Multiplying the equation for <i>v<sub>m</sub></i> above with <i>i<sub>m</sub>dt</i> and summing over <i>m</i> gives the energy transferred to the system in the time interval <i>dt</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 \sum \limits _{m}^{K}i_{m}v_{m}{\text{d}}t=\sum \limits _{m,n=1}^{K}i_{m}L_{m,n}{\text{d}}i_{n}\mathrel {\overset {!}{=}} \sum \limits _{n=1}^{K}{\frac {\partial W\left(i\right)}{\partial i_{n}}}{\text{d}}i_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> <mo>=</mo> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-REL"> <mover> <mo>=</mo> <mo>!</mo> </mover> </mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>W</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum \limits _{m}^{K}i_{m}v_{m}{\text{d}}t=\sum \limits _{m,n=1}^{K}i_{m}L_{m,n}{\text{d}}i_{n}\mathrel {\overset {!}{=}} \sum \limits _{n=1}^{K}{\frac {\partial W\left(i\right)}{\partial i_{n}}}{\text{d}}i_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acc33e234724de7a2f2bb641833e6e6ceae88c8c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:48.508ex; height:7.676ex;" alt="{\displaystyle \sum \limits _{m}^{K}i_{m}v_{m}{\text{d}}t=\sum \limits _{m,n=1}^{K}i_{m}L_{m,n}{\text{d}}i_{n}\mathrel {\overset {!}{=}} \sum \limits _{n=1}^{K}{\frac {\partial W\left(i\right)}{\partial i_{n}}}{\text{d}}i_{n}.}"></span> </p><p>This must agree with the change of the magnetic field energy, <i>W</i>, caused by the currents.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Symmetry_of_second_derivatives" title="Symmetry of second derivatives">integrability condition</a> </p><p><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 \displaystyle {\frac {\partial ^{2}W}{\partial i_{m}\partial i_{n}}}={\frac {\partial ^{2}W}{\partial i_{n}\partial i_{m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>W</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>W</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mi mathvariant="normal">∂</mi> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle {\frac {\partial ^{2}W}{\partial i_{m}\partial i_{n}}}={\frac {\partial ^{2}W}{\partial i_{n}\partial i_{m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f8c56ba74fe656a6e7334c88970d5225891431" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.04ex; height:6.176ex;" alt="{\displaystyle \displaystyle {\frac {\partial ^{2}W}{\partial i_{m}\partial i_{n}}}={\frac {\partial ^{2}W}{\partial i_{n}\partial i_{m}}}}"></span> </p><p>requires <i>L<sub>m,n</sub> = L<sub>n,m</sub></i>. The inductance matrix, <i>L<sub>m,n</sub></i>, thus is symmetric. The integral of the energy transfer is the magnetic field energy as a function of the currents, <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 \displaystyle W\left(i\right)={\frac {1}{2}}\sum \limits _{m,n=1}^{K}i_{m}L_{m,n}i_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mrow> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <munderover> <mo movablelimits="false">∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \displaystyle W\left(i\right)={\frac {1}{2}}\sum \limits _{m,n=1}^{K}i_{m}L_{m,n}i_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b59ac78841bd0c41d749606f2355a429e2e555e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:26.139ex; height:7.676ex;" alt="{\displaystyle \displaystyle W\left(i\right)={\frac {1}{2}}\sum \limits _{m,n=1}^{K}i_{m}L_{m,n}i_{n}.}"></span> </p><p>This equation also is a direct consequence of the linearity of Maxwell's equations. It is helpful to associate changing electric currents with a build-up or decrease of magnetic field energy. The corresponding energy transfer requires or generates a voltage. A <a href="/wiki/Impedance_analogy" title="Impedance analogy">mechanical analogy</a> in the <i>K</i> = 1 case with magnetic field energy (1/2)<i>Li</i><sup>2</sup> is a body with mass <i>M</i>, velocity <i>u</i> and kinetic energy (1/2)<i>Mu</i><sup>2</sup>. The rate of change of velocity (current) multiplied with mass (inductance) requires or generates a force (an electrical voltage). </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Mutually_inducting_inductors.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/9/95/Mutually_inducting_inductors.PNG" decoding="async" width="231" height="174" class="mw-file-element" data-file-width="231" data-file-height="174" /></a><figcaption>Circuit diagram of two mutually coupled inductors. The two vertical lines between the windings indicate that the transformer has a <a href="/wiki/Magnetic_core" title="Magnetic core">ferromagnetic core</a> . "n:m" shows the ratio between the number of windings of the left inductor to windings of the right inductor. This picture also shows the <a href="/wiki/Dot_convention" class="mw-redirect" title="Dot convention">dot convention</a>.</figcaption></figure> <p>Mutual inductance occurs when the change in current in one inductor induces a voltage in another nearby inductor. It is important as the mechanism by which <a href="/wiki/Transformer" title="Transformer">transformers</a> work, but it can also cause unwanted coupling between conductors in a circuit. </p><p>The mutual inductance, <span class="nowrap"><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 M_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6136d919b3f10cc58cc023e05a5c36b2f44e111f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.731ex; height:2.843ex;" alt="{\displaystyle M_{ij}}"></span>,</span> is also a measure of the coupling between two inductors. The mutual inductance by circuit <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/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> on circuit <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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span> is given by the double integral <i><a href="/wiki/Franz_Ernst_Neumann" title="Franz Ernst Neumann">Neumann</a> formula</i>, see <a href="#Calculating_inductance">calculation techniques</a> </p><p>The mutual inductance also has the relationship: <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 M_{21}=N_{1}\ N_{2}\ P_{21}\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mtext> </mtext> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{21}=N_{1}\ N_{2}\ P_{21}\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2c53291151121f7cdcb8061e796c866fc4264ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:17.599ex; height:2.509ex;" alt="{\displaystyle M_{21}=N_{1}\ N_{2}\ P_{21}\!}"></span> where </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="margin-left: 1.6em;"> <ul><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 M_{21}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{21}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d138a4939a763d5cf2b93dca0eecdd5a0b0d3c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.13ex; height:2.509ex;" alt="{\displaystyle M_{21}}"></span> is the mutual inductance, and the subscript specifies the relationship of the voltage induced in coil 2 due to the current in coil 1.</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 N_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b670ae74954b8aacd4d720d9b2b2081f6d3869" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{1}}"></span> is the number of turns in coil 1,</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 N_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/597ea9dac049261fdda77c5176b050e6588d6bb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{2}}"></span> is the number of turns in coil 2,</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 P_{21}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{21}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8d11e3ae15dde0a77bc9c9863bbbf5694d0b4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.368ex; height:2.509ex;" alt="{\displaystyle P_{21}}"></span> is the <a href="/wiki/Permeance" title="Permeance">permeance</a> of the space occupied by the flux.</li></ul> </div> <p>Once the mutual inductance <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is determined, it can be used to predict the behavior of a circuit: <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 v_{1}=L_{1}\ {\frac {{\text{d}}i_{1}}{{\text{d}}t}}-M\ {\frac {{\text{d}}i_{2}}{{\text{d}}t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mi>M</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1}=L_{1}\ {\frac {{\text{d}}i_{1}}{{\text{d}}t}}-M\ {\frac {{\text{d}}i_{2}}{{\text{d}}t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f560d1162e9c7d298900b3ec637979807e94bd9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.332ex; height:5.509ex;" alt="{\displaystyle v_{1}=L_{1}\ {\frac {{\text{d}}i_{1}}{{\text{d}}t}}-M\ {\frac {{\text{d}}i_{2}}{{\text{d}}t}}}"></span> where </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="margin-left: 1.6em;"> <ul><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 v_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98d33f5d498d528bd8c10edc8ac8c34347f32b3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{1}}"></span> is the voltage across the inductor of interest;</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 L_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e79dc1b001f8b923df475ed14de023cbc456013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{1}}"></span> is the inductance of the inductor of interest;</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 {\text{d}}i_{1}\,/\,{\text{d}}t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{d}}i_{1}\,/\,{\text{d}}t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e685baf26cec1bf6b20368c2d569808b5132a488" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.218ex; height:2.843ex;" alt="{\displaystyle {\text{d}}i_{1}\,/\,{\text{d}}t}"></span> is the derivative, with respect to time, of the current through the inductor of interest, labeled 1;</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 {\text{d}}i_{2}\,/\,{\text{d}}t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{d}}i_{2}\,/\,{\text{d}}t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ebd4a49a5d1e0c450a37e19df060da62e21c5d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.218ex; height:2.843ex;" alt="{\displaystyle {\text{d}}i_{2}\,/\,{\text{d}}t}"></span> is the derivative, with respect to time, of the current through the inductor, labeled 2, that is coupled to the first inductor; and</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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is the mutual inductance.</li></ul> </div> <p>The minus sign arises because of the sense the current <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_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14feff7997a635a64f7dfacfbd0374a24ab279bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.857ex; height:2.509ex;" alt="{\displaystyle i_{2}}"></span> has been defined in the diagram. With both currents defined going into the <a href="/wiki/Dot_convention" class="mw-redirect" title="Dot convention">dots</a> the sign of <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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> will be positive (the equation would read with a plus sign instead).<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Coupling_coefficient">Coupling coefficient</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=19" title="Edit section: Coupling coefficient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The coupling coefficient is the ratio of the open-circuit actual voltage ratio to the ratio that would be obtained if all the flux coupled from one <a href="/wiki/Magnetic_circuit" title="Magnetic circuit">magnetic circuit</a> to the other. The coupling coefficient is related to mutual inductance and self inductances in the following way. From the two simultaneous equations expressed in the two-port matrix the open-circuit voltage ratio is found to be: </p><p><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 {V_{2} \over V_{1}}_{\text{open circuit}}={M \over L_{1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>open circuit</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>M</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {V_{2} \over V_{1}}_{\text{open circuit}}={M \over L_{1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d09bbc663c577294a19c32903db9df7df5393863" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.583ex; height:6.009ex;" alt="{\displaystyle {V_{2} \over V_{1}}_{\text{open circuit}}={M \over L_{1}}}"></span> where </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="margin-left: 1.6em;"> <ul><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 M^{2}=M_{1}M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{2}=M_{1}M_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0619e4efeff7c69c9a1b570e5a2d0d39e4f3211f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.268ex; height:3.009ex;" alt="{\displaystyle M^{2}=M_{1}M_{2}}"></span></li></ul> </div> <p>while the ratio if all the flux is coupled is the ratio of the turns, hence the ratio of the square root of the inductances </p><p><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 {V_{2} \over V_{1}}_{\text{max coupling}}={\sqrt {L_{2} \over L_{1}\ }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>max coupling</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {V_{2} \over V_{1}}_{\text{max coupling}}={\sqrt {L_{2} \over L_{1}\ }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b30027cabc8a624ddd15d57394ed3b7ebcf84ecf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:22.624ex; height:7.509ex;" alt="{\displaystyle {V_{2} \over V_{1}}_{\text{max coupling}}={\sqrt {L_{2} \over L_{1}\ }}}"></span> </p><p>thus, </p><p><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 M=k{\sqrt {L_{1}\ L_{2}\ }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mtext> </mtext> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=k{\sqrt {L_{1}\ L_{2}\ }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ade306b1145bbafd4c16ca29b1d6d713f5a1ee10" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.511ex; height:3.343ex;" alt="{\displaystyle M=k{\sqrt {L_{1}\ L_{2}\ }}}"></span> where </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="margin-left: 1.6em;"> <ul><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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is the <i>coupling coefficient</i>,</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 L_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e79dc1b001f8b923df475ed14de023cbc456013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{1}}"></span> is the inductance of the first coil, and</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 L_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a952cfe42c86b7741f55a817da0e251793a358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{2}}"></span> is the inductance of the second coil.</li></ul> </div> <p>The coupling coefficient is a convenient way to specify the relationship between a certain orientation of inductors with arbitrary inductance. Most authors define the range as <span class="nowrap"><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 0\leq k<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>k</mi> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq k<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/907473f47f1507f50b023871549050cc2bcc5ac2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.733ex; height:2.343ex;" alt="{\displaystyle 0\leq k<1}"></span>,</span> but some<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> define it as <span class="nowrap"><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 -1<k<1\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mn>1</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1<k<1\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15a02353ca5a88fe10890c0e45e689b26f21ae67" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.928ex; height:2.343ex;" alt="{\displaystyle -1<k<1\,}"></span>.</span> Allowing negative values of <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> captures phase inversions of the coil connections and the direction of the windings.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Matrix_representation">Matrix representation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=20" title="Edit section: Matrix representation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mutually coupled inductors can be described by any of the <a href="/wiki/Two-port_network" title="Two-port network">two-port network</a> parameter matrix representations. The most direct are the <a href="/wiki/Z_parameters" class="mw-redirect" title="Z parameters">z parameters</a>, which are given by<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p><p><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 [\mathbf {z} ]=s{\begin{bmatrix}L_{1}\ M\\M\ L_{2}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <mi>M</mi> </mtd> </mtr> <mtr> <mtd> <mi>M</mi> <mtext> </mtext> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\mathbf {z} ]=s{\begin{bmatrix}L_{1}\ M\\M\ L_{2}\end{bmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/652ec55a09f63b383b59ce4fcb5527fd7abb66c6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.184ex; height:6.176ex;" alt="{\displaystyle [\mathbf {z} ]=s{\begin{bmatrix}L_{1}\ M\\M\ L_{2}\end{bmatrix}}.}"></span> </p><p>The <a href="/wiki/Admittance_parameters" title="Admittance parameters">y parameters</a> are given by </p><p><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 [\mathbf {y} ]={\frac {1}{s}}{\begin{bmatrix}L_{1}\ M\\M\ L_{2}\end{bmatrix}}^{-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>s</mi> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <mi>M</mi> </mtd> </mtr> <mtr> <mtd> <mi>M</mi> <mtext> </mtext> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [\mathbf {y} ]={\frac {1}{s}}{\begin{bmatrix}L_{1}\ M\\M\ L_{2}\end{bmatrix}}^{-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27053a9a0592616f59f1f2144ab57559ee6d4a4d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.648ex; height:6.509ex;" alt="{\displaystyle [\mathbf {y} ]={\frac {1}{s}}{\begin{bmatrix}L_{1}\ M\\M\ L_{2}\end{bmatrix}}^{-1}.}"></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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> is the <a href="/wiki/Complex_frequency" class="mw-redirect" title="Complex frequency">complex frequency</a> variable, <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 L_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e79dc1b001f8b923df475ed14de023cbc456013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{1}}"></span> 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 L_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6a952cfe42c86b7741f55a817da0e251793a358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:2.509ex;" alt="{\displaystyle L_{2}}"></span> are the inductances of the primary and secondary coil, respectively, 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 M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is the mutual inductance between the coils. </p> <div class="mw-heading mw-heading4"><h4 id="Multiple_Coupled_Inductors">Multiple Coupled Inductors</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=21" title="Edit section: Multiple Coupled Inductors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mutual inductance may be applied to multiple inductors simultaneously. The matrix representations for multiple mutually coupled inductors are given by<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>31<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 {\begin{aligned}&[\mathbf {z} ]=s{\begin{bmatrix}L_{1}&M_{12}&M_{13}&\dots &M_{1N}\\M_{12}&L_{2}&M_{23}&\dots &M_{2N}\\M_{13}&M_{23}&L_{3}&\dots &M_{3N}\\\vdots &\vdots &\vdots &\ddots \\M_{1N}&M_{2N}&M_{3N}&\dots &L_{N}\\\end{bmatrix}}\\\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="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo stretchy="false">]</mo> <mo>=</mo> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>N</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>N</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>N</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>N</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>N</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>N</mi> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&[\mathbf {z} ]=s{\begin{bmatrix}L_{1}&M_{12}&M_{13}&\dots &M_{1N}\\M_{12}&L_{2}&M_{23}&\dots &M_{2N}\\M_{13}&M_{23}&L_{3}&\dots &M_{3N}\\\vdots &\vdots &\vdots &\ddots \\M_{1N}&M_{2N}&M_{3N}&\dots &L_{N}\\\end{bmatrix}}\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67534d4088c19d6122bb82fabc11d9f0c04e5c86" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:42.613ex; height:17.509ex;" alt="{\displaystyle {\begin{aligned}&[\mathbf {z} ]=s{\begin{bmatrix}L_{1}&M_{12}&M_{13}&\dots &M_{1N}\\M_{12}&L_{2}&M_{23}&\dots &M_{2N}\\M_{13}&M_{23}&L_{3}&\dots &M_{3N}\\\vdots &\vdots &\vdots &\ddots \\M_{1N}&M_{2N}&M_{3N}&\dots &L_{N}\\\end{bmatrix}}\\\end{aligned}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Equivalent_circuits">Equivalent circuits</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=22" title="Edit section: Equivalent circuits"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="T-circuit">T-circuit</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=23" title="Edit section: T-circuit"><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:Mutual_inductance_equivalent_circuit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Mutual_inductance_equivalent_circuit.svg/220px-Mutual_inductance_equivalent_circuit.svg.png" decoding="async" width="220" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Mutual_inductance_equivalent_circuit.svg/330px-Mutual_inductance_equivalent_circuit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Mutual_inductance_equivalent_circuit.svg/440px-Mutual_inductance_equivalent_circuit.svg.png 2x" data-file-width="522" data-file-height="320" /></a><figcaption><i>T</i> equivalent circuit of mutually coupled inductors</figcaption></figure> <p>Mutually coupled inductors can equivalently be represented by a T-circuit of inductors as shown. If the coupling is strong and the inductors are of unequal values then the series inductor on the step-down side may take on a negative value.<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> </p><p>This can be analyzed as a two port network. With the output terminated with some arbitrary impedance <span class="nowrap"><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 Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>,</span> the voltage gain <span class="nowrap"><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 A_{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{v}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd85529417776f1ae43598d068b5ed468a4b704" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.773ex; height:2.509ex;" alt="{\displaystyle A_{v}}"></span>,</span> is given by, </p><p><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 A_{\mathrm {v} }={\frac {sMZ}{\,s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z\,}}={\frac {k}{\,s\left(1-k^{2}\right){\frac {\sqrt {L_{1}L_{2}}}{Z}}+{\sqrt {\frac {L_{1}}{L_{2}}}}\,}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <mi>M</mi> <mi>Z</mi> </mrow> <mrow> <mspace width="thinmathspace" /> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>s</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>Z</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mrow> <mspace width="thinmathspace" /> <mi>s</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </msqrt> <mi>Z</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mathrm {v} }={\frac {sMZ}{\,s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z\,}}={\frac {k}{\,s\left(1-k^{2}\right){\frac {\sqrt {L_{1}L_{2}}}{Z}}+{\sqrt {\frac {L_{1}}{L_{2}}}}\,}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba04108e0332262ecf8b05cf64ed811ae23f2c3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:60.74ex; height:8.176ex;" alt="{\displaystyle A_{\mathrm {v} }={\frac {sMZ}{\,s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z\,}}={\frac {k}{\,s\left(1-k^{2}\right){\frac {\sqrt {L_{1}L_{2}}}{Z}}+{\sqrt {\frac {L_{1}}{L_{2}}}}\,}}}"></span> </p><p>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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span> is the coupling constant 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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> is the <a href="/wiki/Complex_frequency" class="mw-redirect" title="Complex frequency">complex frequency</a> variable, as above. For tightly coupled inductors 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 k=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}"></span> this reduces to </p><p><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 A_{\mathrm {v} }={\sqrt {L_{2} \over L_{1}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">v</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mathrm {v} }={\sqrt {L_{2} \over L_{1}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7026f82f838c6acad6335dd521c66c3dbe963d0a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.739ex; height:7.509ex;" alt="{\displaystyle A_{\mathrm {v} }={\sqrt {L_{2} \over L_{1}}}}"></span> </p><p>which is independent of the load impedance. If the inductors are wound on the same core and with the same geometry, then this expression is equal to the turns ratio of the two inductors because inductance is proportional to the square of turns ratio. </p><p>The input impedance of the network is given by, </p><p><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 Z_{\text{in}}={\frac {s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z}{sL_{2}+Z}}={\frac {L_{1}}{L_{2}}}\,Z\,\left({\frac {1}{1+{\frac {Z}{\,sL_{2}\,}}}}\right)\left(1+{\frac {1-k^{2}}{\frac {Z}{\,sL_{2}\,}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>s</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>Z</mi> </mrow> <mrow> <mi>s</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>Z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>Z</mi> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Z</mi> <mrow> <mspace width="thinmathspace" /> <mi>s</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mfrac> <mi>Z</mi> <mrow> <mspace width="thinmathspace" /> <mi>s</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mrow> </mfrac> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{\text{in}}={\frac {s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z}{sL_{2}+Z}}={\frac {L_{1}}{L_{2}}}\,Z\,\left({\frac {1}{1+{\frac {Z}{\,sL_{2}\,}}}}\right)\left(1+{\frac {1-k^{2}}{\frac {Z}{\,sL_{2}\,}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f77e5f823cb1d4f76e6731eea6740fd62c073c6d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:69.544ex; height:8.843ex;" alt="{\displaystyle Z_{\text{in}}={\frac {s^{2}L_{1}L_{2}-s^{2}M^{2}+sL_{1}Z}{sL_{2}+Z}}={\frac {L_{1}}{L_{2}}}\,Z\,\left({\frac {1}{1+{\frac {Z}{\,sL_{2}\,}}}}\right)\left(1+{\frac {1-k^{2}}{\frac {Z}{\,sL_{2}\,}}}\right)}"></span> </p><p>For <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 k=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}"></span> this reduces to </p><p><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 Z_{\text{in}}={\frac {sL_{1}Z}{sL_{2}+Z}}={\frac {L_{1}}{L_{2}}}\,Z\,\left({\frac {1}{1+{\frac {Z}{\,sL_{2}\,}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>s</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>Z</mi> </mrow> <mrow> <mi>s</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mi>Z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>Z</mi> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Z</mi> <mrow> <mspace width="thinmathspace" /> <mi>s</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{\text{in}}={\frac {sL_{1}Z}{sL_{2}+Z}}={\frac {L_{1}}{L_{2}}}\,Z\,\left({\frac {1}{1+{\frac {Z}{\,sL_{2}\,}}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/189b2a8adaf9e00591672b8fc766501d28d1fc1f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:38.025ex; height:8.843ex;" alt="{\displaystyle Z_{\text{in}}={\frac {sL_{1}Z}{sL_{2}+Z}}={\frac {L_{1}}{L_{2}}}\,Z\,\left({\frac {1}{1+{\frac {Z}{\,sL_{2}\,}}}}\right)}"></span> </p><p>Thus, current gain <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 A_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}"></span> is <em>not</em> independent of load unless the further condition </p><p><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 |sL_{2}|\gg |Z|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>≫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |sL_{2}|\gg |Z|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a59e6d0e1fdbb1f2b7b837cc4131796224307061" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.609ex; height:2.843ex;" alt="{\displaystyle |sL_{2}|\gg |Z|}"></span> </p><p>is met, in which case, </p><p><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 Z_{\text{in}}\approx {L_{1} \over L_{2}}Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>in</mtext> </mrow> </msub> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{\text{in}}\approx {L_{1} \over L_{2}}Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25ec362576d9b6f10d097943ad5b9c1c3d57af8f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.443ex; height:5.509ex;" alt="{\displaystyle Z_{\text{in}}\approx {L_{1} \over L_{2}}Z}"></span> </p><p>and </p><p><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 A_{\text{i}}\approx {\sqrt {L_{1} \over L_{2}}}={1 \over A_{\text{v}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>i</mtext> </mrow> </msub> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>v</mtext> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\text{i}}\approx {\sqrt {L_{1} \over L_{2}}}={1 \over A_{\text{v}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0924068d74a729a19cc3a7d01a6648bf98786cfc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:18.106ex; height:7.509ex;" alt="{\displaystyle A_{\text{i}}\approx {\sqrt {L_{1} \over L_{2}}}={1 \over A_{\text{v}}}}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="π-circuit"><span id=".CF.80-circuit"></span>π-circuit</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=24" title="Edit section: π-circuit"><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:Mutual_inductance_pi_equivalent_circuit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Mutual_inductance_pi_equivalent_circuit.svg/220px-Mutual_inductance_pi_equivalent_circuit.svg.png" decoding="async" width="220" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Mutual_inductance_pi_equivalent_circuit.svg/330px-Mutual_inductance_pi_equivalent_circuit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Mutual_inductance_pi_equivalent_circuit.svg/440px-Mutual_inductance_pi_equivalent_circuit.svg.png 2x" data-file-width="906" data-file-height="406" /></a><figcaption><i>π</i> equivalent circuit of coupled inductors</figcaption></figure> <p>Alternatively, two coupled inductors can be modelled using a <i>π</i> equivalent circuit with optional ideal transformers at each port. While the circuit is more complicated than a T-circuit, it can be generalized<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> to circuits consisting of more than two coupled inductors. Equivalent circuit elements <span class="nowrap"><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 L_{\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/830b938e7494cb52b5217261f2a422fb386dc24e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.463ex; height:2.509ex;" alt="{\displaystyle L_{\text{s}}}"></span>,</span> <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 L_{\text{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\text{p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e91743414ca69036ee8bce4ae012c94b1224b165" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.729ex; height:2.843ex;" alt="{\displaystyle L_{\text{p}}}"></span> have physical meaning, modelling respectively <a href="/wiki/Magnetic_reluctance" title="Magnetic reluctance">magnetic reluctances</a> of coupling paths and <a href="/wiki/Magnetic_reluctance" title="Magnetic reluctance">magnetic reluctances</a> of <a href="/wiki/Leakage_inductance" title="Leakage inductance">leakage paths</a>. For example, electric currents flowing through these elements correspond to coupling and leakage <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic fluxes</a>. Ideal transformers normalize all self-inductances to 1 Henry to simplify mathematical formulas. </p><p>Equivalent circuit element values can be calculated from coupling coefficients with <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 {\begin{aligned}L_{S_{ij}}&={\frac {\det(\mathbf {K} )}{-\mathbf {C} _{ij}}}\\[3pt]L_{P_{i}}&={\frac {\det(\mathbf {K} )}{\sum _{j=1}^{N}\mathbf {C} _{ij}}}\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="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}L_{S_{ij}}&={\frac {\det(\mathbf {K} )}{-\mathbf {C} _{ij}}}\\[3pt]L_{P_{i}}&={\frac {\det(\mathbf {K} )}{\sum _{j=1}^{N}\mathbf {C} _{ij}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4739bf3c70b2d1403383f6f5d774cbda23a101d4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:17.943ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}L_{S_{ij}}&={\frac {\det(\mathbf {K} )}{-\mathbf {C} _{ij}}}\\[3pt]L_{P_{i}}&={\frac {\det(\mathbf {K} )}{\sum _{j=1}^{N}\mathbf {C} _{ij}}}\end{aligned}}}"></span> </p><p>where coupling coefficient matrix and its cofactors are 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 \mathbf {K} ={\begin{bmatrix}1&k_{12}&\cdots &k_{1N}\\k_{12}&1&\cdots &k_{2N}\\\vdots &\vdots &\ddots &\vdots \\k_{1N}&k_{2N}&\cdots &1\end{bmatrix}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>N</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>N</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>N</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>N</mi> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {K} ={\begin{bmatrix}1&k_{12}&\cdots &k_{1N}\\k_{12}&1&\cdots &k_{2N}\\\vdots &\vdots &\ddots &\vdots \\k_{1N}&k_{2N}&\cdots &1\end{bmatrix}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d8dfe3154eb0c32823d5b2e332d1f0c0c6d96e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:32.487ex; height:14.176ex;" alt="{\displaystyle \mathbf {K} ={\begin{bmatrix}1&k_{12}&\cdots &k_{1N}\\k_{12}&1&\cdots &k_{2N}\\\vdots &\vdots &\ddots &\vdots \\k_{1N}&k_{2N}&\cdots &1\end{bmatrix}}\quad }"></span> 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 \quad \mathbf {C} _{ij}=(-1)^{i+j}\,\mathbf {M} _{ij}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad \mathbf {C} _{ij}=(-1)^{i+j}\,\mathbf {M} _{ij}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfab064904caa30b301e7c51fe78531ffe3664d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.413ex; height:3.343ex;" alt="{\displaystyle \quad \mathbf {C} _{ij}=(-1)^{i+j}\,\mathbf {M} _{ij}.}"></span></dd></dl> <p>For two coupled inductors, these formulas simplify to </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 L_{S_{12}}={\frac {-k_{12}^{2}+1}{k_{12}}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>1</mn> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mfrac> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{S_{12}}={\frac {-k_{12}^{2}+1}{k_{12}}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991de47da49506114e819ff8a3a7fb2e088bba9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.477ex; height:6.343ex;" alt="{\displaystyle L_{S_{12}}={\frac {-k_{12}^{2}+1}{k_{12}}}\quad }"></span> 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 \quad L_{P_{1}}=L_{P_{2}}\!=\!k_{12}+1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mspace width="negativethinmathspace" /> <mo>=</mo> <mspace width="negativethinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad L_{P_{1}}=L_{P_{2}}\!=\!k_{12}+1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1beaed55431212bb217a912e176d3869a19ee3e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.886ex; height:2.843ex;" alt="{\displaystyle \quad L_{P_{1}}=L_{P_{2}}\!=\!k_{12}+1,}"></span></dd></dl> <p>and for three coupled inductors (for brevity shown only for <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 L_{\text{s12}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s12</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\text{s12}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cae1573cbe6d687927dfe668ccb292316b81207b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.107ex; height:2.509ex;" alt="{\displaystyle L_{\text{s12}}}"></span> 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 L_{\text{p1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>p1</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{\text{p1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bdb7139534b2fdd56763446e8deccf68b0f3a90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.551ex; height:2.843ex;" alt="{\displaystyle L_{\text{p1}}}"></span>) </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 L_{S_{12}}={\frac {2\,k_{12}\,k_{13}\,k_{23}-k_{12}^{2}-k_{13}^{2}-k_{23}^{2}+1}{k_{13}\,k_{23}-k_{12}}}\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>−</mo> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{S_{12}}={\frac {2\,k_{12}\,k_{13}\,k_{23}-k_{12}^{2}-k_{13}^{2}-k_{23}^{2}+1}{k_{13}\,k_{23}-k_{12}}}\quad }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c3712d150fbbaf57cb11948bb73a570bad7d91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.951ex; height:6.509ex;" alt="{\displaystyle L_{S_{12}}={\frac {2\,k_{12}\,k_{13}\,k_{23}-k_{12}^{2}-k_{13}^{2}-k_{23}^{2}+1}{k_{13}\,k_{23}-k_{12}}}\quad }"></span> 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 \quad L_{P_{1}}={\frac {2\,k_{12}\,k_{13}\,k_{23}-k_{12}^{2}-k_{13}^{2}-k_{23}^{2}+1}{k_{12}\,k_{23}+k_{13}\,k_{23}-k_{23}^{2}-k_{12}-k_{13}+1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="1em" /> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>−</mo> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>−</mo> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \quad L_{P_{1}}={\frac {2\,k_{12}\,k_{13}\,k_{23}-k_{12}^{2}-k_{13}^{2}-k_{23}^{2}+1}{k_{12}\,k_{23}+k_{13}\,k_{23}-k_{23}^{2}-k_{12}-k_{13}+1}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c988ab1657f6df72813dc35181e6f5d25505bd83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:48.357ex; height:7.176ex;" alt="{\displaystyle \quad L_{P_{1}}={\frac {2\,k_{12}\,k_{13}\,k_{23}-k_{12}^{2}-k_{13}^{2}-k_{23}^{2}+1}{k_{12}\,k_{23}+k_{13}\,k_{23}-k_{23}^{2}-k_{12}-k_{13}+1}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Resonant_transformer">Resonant transformer</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=25" title="Edit section: Resonant transformer"><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/Resonant_inductive_coupling" title="Resonant inductive coupling">Resonant inductive coupling</a></div> <p>When a capacitor is connected across one winding of a transformer, making the winding a <a href="/wiki/Tuned_circuit" class="mw-redirect" title="Tuned circuit">tuned circuit</a> (resonant circuit) it is called a single-tuned transformer. When a capacitor is connected across each winding, it is called a <a href="/wiki/Double_tuned" class="mw-redirect" title="Double tuned">double tuned transformer</a>. These <i><a href="/wiki/Transformer_types#Resonant_transformer" title="Transformer types">resonant transformers</a></i> can store oscillating electrical energy similar to a <a href="/wiki/Resonant_circuit" class="mw-redirect" title="Resonant circuit">resonant circuit</a> and thus function as a <a href="/wiki/Bandpass_filter" class="mw-redirect" title="Bandpass filter">bandpass filter</a>, allowing frequencies near their <a href="/wiki/Resonant_frequency" class="mw-redirect" title="Resonant frequency">resonant frequency</a> to pass from the primary to secondary winding, but blocking other frequencies. The amount of mutual inductance between the two windings, together with the <a href="/wiki/Q_factor" title="Q factor">Q factor</a> of the circuit, determine the shape of the frequency response curve. The advantage of the double tuned transformer is that it can have a wider bandwidth than a simple tuned circuit. The coupling of double-tuned circuits is described as loose-, critical-, or over-coupled depending on the value of the <a href="/wiki/Coupling_coefficient_(inductors)" class="mw-redirect" title="Coupling coefficient (inductors)">coupling coefficient</a> <span class="nowrap"><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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>.</span> When two tuned circuits are loosely coupled through mutual inductance, the bandwidth is narrow. As the amount of mutual inductance increases, the bandwidth continues to grow. When the mutual inductance is increased beyond the critical coupling, the peak in the frequency response curve splits into two peaks, and as the coupling is increased the two peaks move further apart. This is known as overcoupling. </p><p>Stongly-coupled self-resonant coils can be used for <a href="/wiki/Wireless_power_transfer" title="Wireless power transfer">wireless power transfer</a> between devices in the mid range distances (up to two metres).<sup id="cite_ref-Kurs_35-0" class="reference"><a href="#cite_note-Kurs-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> Strong coupling is required for a high percentage of power transferred, which results in peak splitting of the frequency response.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Ideal_transformers">Ideal transformers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=26" title="Edit section: Ideal transformers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When <span class="nowrap"><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 k=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c035ffa69b5bca8bf2d16c3da3aaad79a8bcbfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=1}"></span>,</span> the inductor is referred to as being closely coupled. If in addition, the self-inductances go to infinity, the inductor becomes an ideal <a href="/wiki/Transformer" title="Transformer">transformer</a>. In this case the voltages, currents, and number of turns can be related in the following way: </p><p><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 V_{\text{s}}={\frac {N_{\text{s}}}{N_{\text{p}}}}V_{\text{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> </msub> </mfrac> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{s}}={\frac {N_{\text{s}}}{N_{\text{p}}}}V_{\text{p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34b285fed80d10e867369a24335dc253a551f0cc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.684ex; height:5.843ex;" alt="{\displaystyle V_{\text{s}}={\frac {N_{\text{s}}}{N_{\text{p}}}}V_{\text{p}}}"></span> where </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="margin-left: 1.6em;"> <ul><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 V_{\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0622261e59ddd90ea5a01382cc5e12f756a45af2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.235ex; height:2.509ex;" alt="{\displaystyle V_{\text{s}}}"></span> is the voltage across the secondary inductor,</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 V_{\text{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b8b88f27718ebfa5b89eaaaf77c8a76e936972" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.501ex; height:2.843ex;" alt="{\displaystyle V_{\text{p}}}"></span> is the voltage across the primary inductor (the one connected to a power source),</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 N_{\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb67e0e9343fa1189fb928aac21766cd2272640" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.746ex; height:2.509ex;" alt="{\displaystyle N_{\text{s}}}"></span> is the number of turns in the secondary inductor, and</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 N_{\text{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{\text{p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ac247dc3071bb02df0785652cbb1f480443fd9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.012ex; height:2.843ex;" alt="{\displaystyle N_{\text{p}}}"></span> is the number of turns in the primary inductor.</li></ul> </div> <p>Conversely the current: </p><p><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 I_{\text{s}}={\frac {N_{\text{p}}}{N_{\text{s}}}}I_{\text{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> </msub> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mfrac> </mrow> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\text{s}}={\frac {N_{\text{p}}}{N_{\text{s}}}}I_{\text{p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d9dac1da5604f68adecbb04cf80ab4202b30c9b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.02ex; height:6.009ex;" alt="{\displaystyle I_{\text{s}}={\frac {N_{\text{p}}}{N_{\text{s}}}}I_{\text{p}}}"></span> where </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="margin-left: 1.6em;"> <ul><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 I_{\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa3ff2abfa47d060d611fe21c62e4f11111c6bc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.903ex; height:2.509ex;" alt="{\displaystyle I_{\text{s}}}"></span> is the current through the secondary inductor,</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 I_{\text{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\text{p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/061405f5fd9458cf15816b4f236ffa7d43a8889b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.169ex; height:2.843ex;" alt="{\displaystyle I_{\text{p}}}"></span> is the current through the primary inductor (the one connected to a power source),</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 N_{\text{s}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>s</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{\text{s}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb67e0e9343fa1189fb928aac21766cd2272640" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.746ex; height:2.509ex;" alt="{\displaystyle N_{\text{s}}}"></span> is the number of turns in the secondary inductor, and</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 N_{\text{p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>p</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{\text{p}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ac247dc3071bb02df0785652cbb1f480443fd9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.012ex; height:2.843ex;" alt="{\displaystyle N_{\text{p}}}"></span> is the number of turns in the primary inductor.</li></ul> </div> <p>The power through one inductor is the same as the power through the other. These equations neglect any forcing by current sources or voltage sources. </p> <div class="mw-heading mw-heading2"><h2 id="Self-inductance_of_thin_wire_shapes">Self-inductance of thin wire shapes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=27" title="Edit section: Self-inductance of thin wire shapes"><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">See also: <a href="/wiki/Inductor#Inductance_formulas" title="Inductor">Inductor § Inductance formulas</a></div> <p>The table below lists formulas for the self-inductance of various simple shapes made of thin cylindrical conductors (wires). In general these are only accurate if the wire radius <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> is much smaller than the dimensions of the shape, and if no ferromagnetic materials are nearby (no <a href="/wiki/Magnetic_core" title="Magnetic core">magnetic core</a>). </p> <table class="wikitable"> <caption>Self-inductance of thin wire shapes </caption> <tbody><tr> <th scope="col">Type </th> <th scope="col">Inductance </th> <th scope="col">Comment </th></tr> <tr> <th scope="row">Single layer <br />solenoid </th> <td> <p>Wheeler's approximation formula for current-sheet model air-core coil:<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> </p><p><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 {L}}={\frac {N^{2}D^{2}}{18D+40\ell }}}"> <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">L</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>18</mn> <mi>D</mi> <mo>+</mo> <mn>40</mn> <mi>ℓ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {N^{2}D^{2}}{18D+40\ell }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11bed0cff7a4d1888c8da0ed5a65e3ea53f7b1ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.923ex; height:6.009ex;" alt="{\displaystyle {\mathcal {L}}={\frac {N^{2}D^{2}}{18D+40\ell }}}"></span> (inches)        <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 {L}}={\frac {N^{2}D^{2}}{45.72D+101.6\ell }}}"> <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">L</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>45.72</mn> <mi>D</mi> <mo>+</mo> <mn>101.6</mn> <mi>ℓ</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {N^{2}D^{2}}{45.72D+101.6\ell }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b18be209a05ffc50dcbf70f61d4c19a54189d5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.866ex; height:6.009ex;" alt="{\displaystyle {\mathcal {L}}={\frac {N^{2}D^{2}}{45.72D+101.6\ell }}}"></span> (cm) </p><p>This formula gives an error no more than 1% <span class="nowrap">when <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 \ell >0.4\,D~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo>></mo> <mn>0.4</mn> <mspace width="thinmathspace" /> <mi>D</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell >0.4\,D~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b42b9d7bf7ead9cf841f7a9540540f8b9359227e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.579ex; height:2.176ex;" alt="{\displaystyle \ell >0.4\,D~.}"></span></span> </p> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"> <ul><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 {\mathcal {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">L</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9027196ecb178d598958555ea01c43157d83597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}"></span> inductance in μH (10<sup>−6</sup> henries)</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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" 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 N}"></span> number of turns</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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> diameter in (inches) (cm)</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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span> length in (inches) (cm)</li></ul> </div> </td></tr> <tr> <th scope="row">Coaxial <br />cable (HF) </th> <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 {L}}={\frac {\mu _{0}}{2\pi }}\ell \ln \left({\frac {b}{a}}\right)}"> <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">L</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mi>ℓ</mi> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {\mu _{0}}{2\pi }}\ell \ln \left({\frac {b}{a}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/245b0c425ebcdb65224a6d0c0fb7cb882d545536" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.816ex; height:6.176ex;" alt="{\displaystyle {\mathcal {L}}={\frac {\mu _{0}}{2\pi }}\ell \ln \left({\frac {b}{a}}\right)}"></span> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span>: Outer cond.'s inside radius</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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>: Inner conductor's radius</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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span>: Length</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 _{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>: see table footnote.</li></ul></div> </td></tr> <tr> <th scope="row">Circular loop<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </th> <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 {L}}=\mu _{0}\ r\ \left[\ln \left({\frac {8r}{a}}\right)-2+{\tfrac {1}{4}}Y+{\mathcal {O}}\left({\frac {a^{2}}{r^{2}}}\right)\right]}"> <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">L</mi> </mrow> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> <mi>r</mi> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>8</mn> <mi>r</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mi>Y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=\mu _{0}\ r\ \left[\ln \left({\frac {8r}{a}}\right)-2+{\tfrac {1}{4}}Y+{\mathcal {O}}\left({\frac {a^{2}}{r^{2}}}\right)\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/275873f317db17a666dc61f9d7890f7822266d6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.511ex; height:6.343ex;" alt="{\displaystyle {\mathcal {L}}=\mu _{0}\ r\ \left[\ln \left({\frac {8r}{a}}\right)-2+{\tfrac {1}{4}}Y+{\mathcal {O}}\left({\frac {a^{2}}{r^{2}}}\right)\right]}"></span> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span>: Loop radius</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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>: Wire radius</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 _{0},Y}"> <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> <mo>,</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0},Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b03a00a0db3c247e5feaaa861b82cb93f8c6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.263ex; height:2.676ex;" alt="{\displaystyle \mu _{0},Y}"></span>: see table footnotes.</li></ul></div> </td></tr> <tr> <th scope="row">Rectangle from <br />round wire<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> </th> <td> <p><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}{\mathcal {L}}={\frac {\mu _{0}}{\pi }}\ {\biggl [}\ &\ell _{1}\ln \left({\frac {2\ell _{1}}{a}}\right)+\ell _{2}\ \ln \left({\frac {2\ell _{2}}{a}}\right)+2{\sqrt {\ell _{1}^{2}+\ell _{2}^{2}\ }}\\&-\ell _{1}\ \sinh ^{-1}\left({\frac {\ell _{1}}{\ell _{2}}}\right)-\ell _{2}\sinh ^{-1}\left({\frac {\ell _{2}}{\ell _{1}}}\right)\\&-\left(2-{\tfrac {1}{4}}Y\ \right)\left(\ell _{1}+\ell _{2}\right)\ {\biggr ]}\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="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>π</mi> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mtext> </mtext> </mtd> <mtd> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mtext> </mtext> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mtext> </mtext> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>−</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mtext> </mtext> <msup> <mi>sinh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>sinh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mi>Y</mi> <mtext> </mtext> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {L}}={\frac {\mu _{0}}{\pi }}\ {\biggl [}\ &\ell _{1}\ln \left({\frac {2\ell _{1}}{a}}\right)+\ell _{2}\ \ln \left({\frac {2\ell _{2}}{a}}\right)+2{\sqrt {\ell _{1}^{2}+\ell _{2}^{2}\ }}\\&-\ell _{1}\ \sinh ^{-1}\left({\frac {\ell _{1}}{\ell _{2}}}\right)-\ell _{2}\sinh ^{-1}\left({\frac {\ell _{2}}{\ell _{1}}}\right)\\&-\left(2-{\tfrac {1}{4}}Y\ \right)\left(\ell _{1}+\ell _{2}\right)\ {\biggr ]}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/657fa5b4fe9cb240b275e22390f6f4d6df4c5f3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:51.939ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {L}}={\frac {\mu _{0}}{\pi }}\ {\biggl [}\ &\ell _{1}\ln \left({\frac {2\ell _{1}}{a}}\right)+\ell _{2}\ \ln \left({\frac {2\ell _{2}}{a}}\right)+2{\sqrt {\ell _{1}^{2}+\ell _{2}^{2}\ }}\\&-\ell _{1}\ \sinh ^{-1}\left({\frac {\ell _{1}}{\ell _{2}}}\right)-\ell _{2}\sinh ^{-1}\left({\frac {\ell _{2}}{\ell _{1}}}\right)\\&-\left(2-{\tfrac {1}{4}}Y\ \right)\left(\ell _{1}+\ell _{2}\right)\ {\biggr ]}\end{aligned}}}"></span> </p> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><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 \ell _{1},\ell _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell _{1},\ell _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c60295fa6a18a3bffa55120ba741b04726bc5e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.082ex; height:2.509ex;" alt="{\displaystyle \ell _{1},\ell _{2}}"></span>: Side lengths</li><li><span class="nowrap"><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 \ \ell _{1}\gg a,\ell _{2}\gg a\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≫</mo> <mi>a</mi> <mo>,</mo> <msub> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≫</mo> <mi>a</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \ell _{1}\gg a,\ell _{2}\gg a\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/303af9b7ddf16a272419bb539bfba94d93a70300" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.931ex; height:2.509ex;" alt="{\displaystyle \ \ell _{1}\gg a,\ell _{2}\gg a\ }"></span></span></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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>: Wire radius</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 _{0},Y}"> <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> <mo>,</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0},Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b03a00a0db3c247e5feaaa861b82cb93f8c6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.263ex; height:2.676ex;" alt="{\displaystyle \mu _{0},Y}"></span>: see table footnotes.</li></ul></div> </td></tr> <tr> <th scope="row">Pair of parallel<br /> wires </th> <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 {L}}={\frac {\ \mu _{0}}{\pi }}\ \ell \ \left[\ln \left({\frac {s}{a}}\right)+{\tfrac {1}{4}}Y\right]}"> <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">L</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mi>π</mi> </mfrac> </mrow> <mtext> </mtext> <mi>ℓ</mi> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mi>Y</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\frac {\ \mu _{0}}{\pi }}\ \ell \ \left[\ln \left({\frac {s}{a}}\right)+{\tfrac {1}{4}}Y\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89810637f022906cff6123b58e8ed787ec045386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.34ex; height:4.843ex;" alt="{\displaystyle {\mathcal {L}}={\frac {\ \mu _{0}}{\pi }}\ \ell \ \left[\ln \left({\frac {s}{a}}\right)+{\tfrac {1}{4}}Y\right]}"></span> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>: Wire radius</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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>: Separation distance, <span class="nowrap"><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 s\geq 2a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>≥</mo> <mn>2</mn> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\geq 2a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec6bf78f482942eff11ff9435ab23c5a9086f4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.581ex; height:2.343ex;" alt="{\displaystyle s\geq 2a}"></span></span></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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span>: Length of pair</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 _{0},Y}"> <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> <mo>,</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0},Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96b03a00a0db3c247e5feaaa861b82cb93f8c6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.263ex; height:2.676ex;" alt="{\displaystyle \mu _{0},Y}"></span>: see table footnotes.</li></ul></div> </td></tr> <tr> <th scope="row">Pair of parallel<br /> wires (HF) </th> <td> <p><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}{\mathcal {L}}&={\frac {\mu _{0}}{\pi }}\ \ell \ \cosh ^{-1}\left({\frac {s}{2a}}\right)\\&={\frac {\mu _{0}}{\pi }}\ \ell \ \ln \left({\frac {s}{2a}}+{\sqrt {{\frac {s^{2}}{4a^{2}}}-1}}\right)\\&\approx {\frac {\mu _{0}}{\pi }}\ \ell \ \ln \left({\frac {s}{a}}\right)\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="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>π</mi> </mfrac> </mrow> <mtext> </mtext> <mi>ℓ</mi> <mtext> </mtext> <msup> <mi>cosh</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>π</mi> </mfrac> </mrow> <mtext> </mtext> <mi>ℓ</mi> <mtext> </mtext> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>π</mi> </mfrac> </mrow> <mtext> </mtext> <mi>ℓ</mi> <mtext> </mtext> <mi>ln</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>a</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {L}}&={\frac {\mu _{0}}{\pi }}\ \ell \ \cosh ^{-1}\left({\frac {s}{2a}}\right)\\&={\frac {\mu _{0}}{\pi }}\ \ell \ \ln \left({\frac {s}{2a}}+{\sqrt {{\frac {s^{2}}{4a^{2}}}-1}}\right)\\&\approx {\frac {\mu _{0}}{\pi }}\ \ell \ \ln \left({\frac {s}{a}}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36bef7cfc428f7697516b1abb2ef1835d7b72ff5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:33.563ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {L}}&={\frac {\mu _{0}}{\pi }}\ \ell \ \cosh ^{-1}\left({\frac {s}{2a}}\right)\\&={\frac {\mu _{0}}{\pi }}\ \ell \ \ln \left({\frac {s}{2a}}+{\sqrt {{\frac {s^{2}}{4a^{2}}}-1}}\right)\\&\approx {\frac {\mu _{0}}{\pi }}\ \ell \ \ln \left({\frac {s}{a}}\right)\end{aligned}}}"></span> </p> </td> <td><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span>: Wire radius</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 s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>: Separation distance, <span class="nowrap"><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 s\geq 2a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>≥</mo> <mn>2</mn> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s\geq 2a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec6bf78f482942eff11ff9435ab23c5a9086f4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.581ex; height:2.343ex;" alt="{\displaystyle s\geq 2a}"></span></span></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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span>: Length (each) of pair</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 _{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>: see table footnote.</li></ul></div> </td></tr></tbody></table> <p><span class="anchor" id="current_distribution_parameter_Y"></span><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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span> is an approximately constant value between 0 and 1 that depends on the distribution of the current in the wire: <span class="nowrap"><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 Y=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56cd853e6606465d2259975da9d0a0bb08f612af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.034ex; height:2.176ex;" alt="{\displaystyle Y=0}"></span></span> when the current flows only on the surface of the wire (complete <a href="/wiki/Skin_effect" title="Skin effect">skin effect</a>), <span class="nowrap"><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 Y=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/867ae2de7c84119e258e68ca484e01e03b00bd73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.034ex; height:2.176ex;" alt="{\displaystyle Y=1}"></span></span> when the current is evenly spread over the cross-section of the wire (<a href="/wiki/Direct_current" title="Direct current">direct current</a>). For round wires, Rosa (1908) gives a formula equivalent to:<sup id="cite_ref-Rosa1908_22-1" class="reference"><a href="#cite_note-Rosa1908-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p><p><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 Y\approx {\frac {1}{\,1+a\ {\sqrt {{\tfrac {1}{8}}\mu \sigma \omega \,}}\,}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mn>1</mn> <mo>+</mo> <mi>a</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <mi>μ</mi> <mi>σ</mi> <mi>ω</mi> <mspace width="thinmathspace" /> </msqrt> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y\approx {\frac {1}{\,1+a\ {\sqrt {{\tfrac {1}{8}}\mu \sigma \omega \,}}\,}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b4e3bd13ed4b1dd3804d075a056771f5d89edcb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:20.842ex; height:8.009ex;" alt="{\displaystyle Y\approx {\frac {1}{\,1+a\ {\sqrt {{\tfrac {1}{8}}\mu \sigma \omega \,}}\,}}}"></span> </p><p>where </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist" style="margin-left: 1.6em;"> <ul><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 \omega =2\pi f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =2\pi f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1bf35d395c2d52391265e4bbda0aed14f52579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.317ex; height:2.509ex;" alt="{\displaystyle \omega =2\pi f}"></span> is the angular frequency, in radians per second;</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 _{0}\,\mu _{\text{r}}}"> <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"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>r</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\mu _{0}\,\mu _{\text{r}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829fcb31bf2740285ffdbfca1b0593495560dd44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.622ex; height:2.176ex;" alt="{\displaystyle \mu =\mu _{0}\,\mu _{\text{r}}}"></span> is the net <a href="/wiki/Magnetic_permeability" class="mw-redirect" title="Magnetic permeability">magnetic permeability</a> of the wire;</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 \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> is the wire's specific conductivity; and</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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> is the wire radius.</li></ul> </div> <p><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 {O}}(x)}"> <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">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/950828b454a9ba398992fb5ae9bdcbe1590e448e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.989ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(x)}"></span> is represents small term(s) that have been dropped from the formula, to make it simpler. Read the term <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 {O}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}+{\mathcal {O}}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b98f02b1113ec67d664bc372ae913bf5fd04ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.829ex; height:2.843ex;" alt="{\displaystyle {}+{\mathcal {O}}(x)}"></span> as "plus small corrections that vary on the order of <span class="nowrap"><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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>"</span> (see <a href="/wiki/Big_O_notation" title="Big O notation">big O notation</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=Inductance&action=edit&section=28" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Electromagnetic_induction" title="Electromagnetic induction">Electromagnetic induction</a></li> <li><a href="/wiki/Gyrator" title="Gyrator">Gyrator</a></li> <li><a href="/wiki/Hydraulic_analogy" title="Hydraulic analogy">Hydraulic analogy</a></li> <li><a href="/wiki/Leakage_inductance" title="Leakage inductance">Leakage inductance</a></li> <li><a href="/wiki/LC_circuit" title="LC circuit">LC circuit</a>, <a href="/wiki/RLC_circuit" title="RLC circuit">RLC circuit</a>, <a href="/wiki/RL_circuit" title="RL circuit">RL circuit</a></li> <li><a href="/wiki/Kinetic_inductance" title="Kinetic inductance">Kinetic inductance</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=29" title="Edit section: Footnotes"><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 reflist-columns references-column-width reflist-lower-alpha"> <ol class="references"> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"> The integral is called "logarithmically divergent" because <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 \ \int {\frac {1}{x}}\ \mathrm {d} x=\ln(x)\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>x</mi> </mfrac> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mo>=</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \int {\frac {1}{x}}\ \mathrm {d} x=\ln(x)\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5570746c9ec668c1cf1416c8886c0e117b9990" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.675ex; height:5.676ex;" alt="{\displaystyle \ \int {\frac {1}{x}}\ \mathrm {d} x=\ln(x)\ }"></span> for <span class="nowrap"><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 \ x>0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>x</mi> <mo>></mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x>0\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a8556b578a27826cd12ceff8f58bb323c843b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.752ex; height:2.176ex;" alt="{\displaystyle \ x>0\ }"></span>,</span> hence it approaches infinity like a logarithm whose argument approaches infinity.</span> </li> </ol></div> <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=Inductance&action=edit&section=30" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-Serway2017-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Serway2017_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Serway2017_1-1"><sup><i><b>b</b></i></sup></a></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 id="CITEREFSerwayJewettWilsonWilson2017" class="citation book cs1">Serway, A. Raymond; Jewett, John W.; Wilson, Jane; Wilson, Anna; Rowlands, Wayne (2017). "Inductance". <i>Physics for global scientists and engineers</i> (2 ed.). Cengage AU. p. 901. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780170355520" title="Special:BookSources/9780170355520"><bdi>9780170355520</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Inductance&rft.btitle=Physics+for+global+scientists+and+engineers&rft.pages=901&rft.edition=2&rft.pub=Cengage+AU&rft.date=2017&rft.isbn=9780170355520&rft.aulast=Serway&rft.aufirst=A.+Raymond&rft.au=Jewett%2C+John+W.&rft.au=Wilson%2C+Jane&rft.au=Wilson%2C+Anna&rft.au=Rowlands%2C+Wayne&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaker1976" class="citation book cs1">Baker, Edward Cecil (1976). <i>Sir William Preece, F.R.S.: Victorian Engineer Extraordinary</i>. Hutchinson. p. 204. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780091266103" title="Special:BookSources/9780091266103"><bdi>9780091266103</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sir+William+Preece%2C+F.R.S.%3A+Victorian+Engineer+Extraordinary&rft.pages=204&rft.pub=Hutchinson&rft.date=1976&rft.isbn=9780091266103&rft.aulast=Baker&rft.aufirst=Edward+Cecil&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span>.</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeaviside1894" class="citation book cs1">Heaviside, Oliver (1894). "The induction of currents in cores". <i>Electrical Papers, Vol. 1</i>. London: Macmillan. p. <a rel="nofollow" class="external text" href="https://archive.org/details/electricalpaper00heavgoog/page/353/mode/2up">354</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+induction+of+currents+in+cores&rft.btitle=Electrical+Papers%2C+Vol.+1&rft.place=London&rft.pages=354&rft.pub=Macmillan&rft.date=1894&rft.aulast=Heaviside&rft.aufirst=Oliver&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></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 id="CITEREFElert" class="citation web cs1">Elert, Glenn. <a rel="nofollow" class="external text" href="http://physics.info/inductance/">"The Physics Hypertextbook: Inductance"</a><span class="reference-accessdate">. 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Pearson / Prentice Hall. p. 255. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-241326-8" title="Special:BookSources/978-0-13-241326-8"><bdi>978-0-13-241326-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+applied+electromagnetics&rft.pages=255&rft.edition=5th&rft.pub=Pearson+%2F+Prentice+Hall&rft.date=2007&rft.isbn=978-0-13-241326-8&rft.aulast=Ulaby&rft.aufirst=Fawwaz&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-1GDkgEACAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</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="https://web.archive.org/web/20131213121232/http://www.nas.edu/history/members/henry.html">"Joseph Henry"</a>. <i>Distinguished Members Gallery, National Academy of Sciences</i>. 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Pearson Education India. p. 65. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-8131760611" title="Special:BookSources/978-8131760611"><bdi>978-8131760611</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electro+Magnetic+Field+Theory&rft.pages=65&rft.pub=Pearson+Education+India&rft.date=2011&rft.isbn=978-8131760611&rft.aulast=Singh&rft.aufirst=Yaduvir&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D0-PfbT49tJMC%26pg%3DPA65&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-Wadhwa-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wadhwa_14-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWadhwa2005" class="citation book cs1">Wadhwa, C.L. 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Cengage Learning. p. 153. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0766816982" title="Special:BookSources/0766816982"><bdi>0766816982</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Electronics&rft.pages=153&rft.pub=Cengage+Learning&rft.date=2001&rft.isbn=0766816982&rft.aulast=Gates&rft.aufirst=Earl+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DIwC5GIA0cREC%26pg%3DPA153&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-Rosa1908-22"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rosa1908_22-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rosa1908_22-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosa1908" class="citation journal cs1">Rosa, E.B. (1908). <a rel="nofollow" class="external text" href="https://doi.org/10.6028%2Fbulletin.088">"The self and mutual inductances of linear conductors"</a>. <i>Bulletin of the Bureau of Standards</i>. <b>4</b> (2). <a href="/wiki/U.S._Bureau_of_Standards" class="mw-redirect" title="U.S. Bureau of Standards">U.S. Bureau of Standards</a>: 301 ff. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.6028%2Fbulletin.088">10.6028/bulletin.088</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+Bureau+of+Standards&rft.atitle=The+self+and+mutual+inductances+of+linear+conductors&rft.volume=4&rft.issue=2&rft.pages=301+ff&rft.date=1908&rft_id=info%3Adoi%2F10.6028%2Fbulletin.088&rft.aulast=Rosa&rft.aufirst=E.B.&rft_id=https%3A%2F%2Fdoi.org%2F10.6028%252Fbulletin.088&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNeumann1846" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Franz_Ernst_Neumann" title="Franz Ernst Neumann">Neumann, F.E.</a> (1846). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1423608">"Allgemeine Gesetze der inducirten elektrischen Ströme"</a> [General rules for induced electric currents]. <i>Annalen der Physik und Chemie</i> (in German). <b>143</b> (1). Wiley: 31–44. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1846AnP...143...31N">1846AnP...143...31N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.18461430103">10.1002/andp.18461430103</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-3804">0003-3804</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annalen+der+Physik+und+Chemie&rft.atitle=Allgemeine+Gesetze+der+inducirten+elektrischen+Str%C3%B6me&rft.volume=143&rft.issue=1&rft.pages=31-44&rft.date=1846&rft.issn=0003-3804&rft_id=info%3Adoi%2F10.1002%2Fandp.18461430103&rft_id=info%3Abibcode%2F1846AnP...143...31N&rft.aulast=Neumann&rft.aufirst=F.E.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1423608&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJackson1975" class="citation book cs1">Jackson, J. D. (1975). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/classicalelectro00jack_0"><i>Classical Electrodynamics</i></a></span>. Wiley. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/classicalelectro00jack_0/page/176">176</a>, 263. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471431329" title="Special:BookSources/9780471431329"><bdi>9780471431329</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Electrodynamics&rft.pages=176%2C+263&rft.pub=Wiley&rft.date=1975&rft.isbn=9780471431329&rft.aulast=Jackson&rft.aufirst=J.+D.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fclassicalelectro00jack_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-den12-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-den12_26-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDengler2016" class="citation journal cs1">Dengler, R. (2016). "Self inductance of a wire loop as a curve integral". <i>Advanced Electromagnetics</i>. <b>5</b> (1): 1–8. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1204.1486">1204.1486</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2016AdEl....5....1D">2016AdEl....5....1D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.7716%2Faem.v5i1.331">10.7716/aem.v5i1.331</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:53583557">53583557</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Advanced+Electromagnetics&rft.atitle=Self+inductance+of+a+wire+loop+as+a+curve+integral&rft.volume=5&rft.issue=1&rft.pages=1-8&rft.date=2016&rft_id=info%3Aarxiv%2F1204.1486&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A53583557%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.7716%2Faem.v5i1.331&rft_id=info%3Abibcode%2F2016AdEl....5....1D&rft.aulast=Dengler&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text">The kinetic energy of the drifting electrons is many orders of magnitude smaller than W, except for nanowires.</span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNahviEdminister2002" class="citation book cs1">Nahvi, Mahmood; Edminister, Joseph (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nrxT9Qjguk8C&pg=PA338"><i>Schaum's outline of theory and problems of electric circuits</i></a>. McGraw-Hill Professional. p. 338. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-139307-2" title="Special:BookSources/0-07-139307-2"><bdi>0-07-139307-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Schaum%27s+outline+of+theory+and+problems+of+electric+circuits&rft.pages=338&rft.pub=McGraw-Hill+Professional&rft.date=2002&rft.isbn=0-07-139307-2&rft.aulast=Nahvi&rft.aufirst=Mahmood&rft.au=Edminister%2C+Joseph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnrxT9Qjguk8C%26pg%3DPA338&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThierauf2004" class="citation book cs1">Thierauf, Stephen C. (2004). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/highspeedcircuit00thie_269"><i>High-speed Circuit Board Signal Integrity</i></a></span>. Artech House. p. <a rel="nofollow" class="external text" href="https://archive.org/details/highspeedcircuit00thie_269/page/n70">56</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1580538460" title="Special:BookSources/1580538460"><bdi>1580538460</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=High-speed+Circuit+Board+Signal+Integrity&rft.pages=56&rft.pub=Artech+House&rft.date=2004&rft.isbn=1580538460&rft.aulast=Thierauf&rft.aufirst=Stephen+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhighspeedcircuit00thie_269&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKimKimJungKwon2009" class="citation journal cs1">Kim, Seok; Kim, Shin-Ae; Jung, Goeun; Kwon, Kee-Won; Chun, Jung-Hoon (2009). "Design of a Reliable Broadband I/O Employing T-coil". <i>JSTS:journal of Semiconductor Technology and Science</i>. <b>9</b> (4): 198–204. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5573%2FJSTS.2009.9.4.198">10.5573/JSTS.2009.9.4.198</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:56413251">56413251</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=JSTS%3Ajournal+of+Semiconductor+Technology+and+Science&rft.atitle=Design+of+a+Reliable+Broadband+I%2FO+Employing+T-coil&rft.volume=9&rft.issue=4&rft.pages=198-204&rft.date=2009&rft_id=info%3Adoi%2F10.5573%2FJSTS.2009.9.4.198&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A56413251%23id-name%3DS2CID&rft.aulast=Kim&rft.aufirst=Seok&rft.au=Kim%2C+Shin-Ae&rft.au=Jung%2C+Goeun&rft.au=Kwon%2C+Kee-Won&rft.au=Chun%2C+Jung-Hoon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAatre1981" class="citation book cs1">Aatre, Vasudev K. (1981). <a rel="nofollow" class="external text" href="https://archive.org/details/networktheoryfil0000aatr/page/n1/mode/2up"><i>Network Theory and Filter Design</i></a>. USA, Canada, Latin America, and Middle East: John Wiley & Sons. pp. 71, 72. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-470-26934-0" title="Special:BookSources/0-470-26934-0"><bdi>0-470-26934-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Network+Theory+and+Filter+Design&rft.place=USA%2C+Canada%2C+Latin+America%2C+and+Middle+East&rft.pages=71%2C+72&rft.pub=John+Wiley+%26+Sons&rft.date=1981&rft.isbn=0-470-26934-0&rft.aulast=Aatre&rft.aufirst=Vasudev+K.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fnetworktheoryfil0000aatr%2Fpage%2Fn1%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChuaDesoerKuh1987" class="citation book cs1">Chua, Leon O.; Desoer, Charles A.; Kuh, Ernest S. (1987). <a rel="nofollow" class="external text" href="https://archive.org/details/linearnonlinearc0000leon"><i>Linear and Nonlinear Circuits</i></a>. McGraw-Hill, Inc. p. 459. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-100685-0" title="Special:BookSources/0-07-100685-0"><bdi>0-07-100685-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+and+Nonlinear+Circuits&rft.pages=459&rft.pub=McGraw-Hill%2C+Inc.&rft.date=1987&rft.isbn=0-07-100685-0&rft.aulast=Chua&rft.aufirst=Leon+O.&rft.au=Desoer%2C+Charles+A.&rft.au=Kuh%2C+Ernest+S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flinearnonlinearc0000leon&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEslami2005" class="citation book cs1">Eslami, Mansour (May 24, 2005). <a rel="nofollow" class="external text" href="https://archive.org/details/circuitanalysisf0000esla/mode/2up"><i>Circuit Analysis Fundamentals</i></a>. Chicago, IL, US: Agile Press. p. 194. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-9718239-5-2" title="Special:BookSources/0-9718239-5-2"><bdi>0-9718239-5-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Circuit+Analysis+Fundamentals&rft.place=Chicago%2C+IL%2C+US&rft.pages=194&rft.pub=Agile+Press&rft.date=2005-05-24&rft.isbn=0-9718239-5-2&rft.aulast=Eslami&rft.aufirst=Mansour&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcircuitanalysisf0000esla%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRadeckiYuanMiuraAikawa2012" class="citation journal cs1">Radecki, Andrzej; Yuan, Yuxiang; Miura, Noriyuki; Aikawa, Iori; Take, Yasuhiro; Ishikuro, Hiroki; Kuroda, Tadahiro (2012). 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D.; Fisher, P.; Soljacic, M. (6 July 2007). "Wireless Power Transfer via Strongly Coupled Magnetic Resonances". <i>Science</i>. <b>317</b> (5834): 83–86. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007Sci...317...83K">2007Sci...317...83K</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.418.9645">10.1.1.418.9645</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.1143254">10.1126/science.1143254</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17556549">17556549</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17105396">17105396</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Wireless+Power+Transfer+via+Strongly+Coupled+Magnetic+Resonances&rft.volume=317&rft.issue=5834&rft.pages=83-86&rft.date=2007-07-06&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17105396%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2007Sci...317...83K&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.418.9645%23id-name%3DCiteSeerX&rft_id=info%3Apmid%2F17556549&rft_id=info%3Adoi%2F10.1126%2Fscience.1143254&rft.aulast=Kurs&rft.aufirst=A.&rft.au=Karalis%2C+A.&rft.au=Moffatt%2C+R.&rft.au=Joannopoulos%2C+J.+D.&rft.au=Fisher%2C+P.&rft.au=Soljacic%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-36"><span class="mw-cite-backlink"><b><a href="#cite_ref-36">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSampleMeyerSmith2011" class="citation journal cs1">Sample, Alanson P.; Meyer, D. A.; Smith, J. R. (2011). "Analysis, Experimental Results, and Range Adaptation of Magnetically Coupled Resonators for Wireless Power Transfer". <i>IEEE Transactions on Industrial Electronics</i>. <b>58</b> (2): 544–554. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTIE.2010.2046002">10.1109/TIE.2010.2046002</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14721">14721</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Industrial+Electronics&rft.atitle=Analysis%2C+Experimental+Results%2C+and+Range+Adaptation+of+Magnetically+Coupled+Resonators+for+Wireless+Power+Transfer&rft.volume=58&rft.issue=2&rft.pages=544-554&rft.date=2011&rft_id=info%3Adoi%2F10.1109%2FTIE.2010.2046002&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14721%23id-name%3DS2CID&rft.aulast=Sample&rft.aufirst=Alanson+P.&rft.au=Meyer%2C+D.+A.&rft.au=Smith%2C+J.+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRendon-HernandezHalimSmithArnold2022" class="citation book cs1">Rendon-Hernandez, Adrian A.; Halim, Miah A.; Smith, Spencer E.; Arnold, David P. (2022). "Magnetically Coupled Microelectromechanical Resonators for Low-Frequency Wireless Power Transfer". <i>2022 IEEE 35th International Conference on Micro Electro Mechanical Systems Conference (MEMS)</i>. pp. 648–651. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FMEMS51670.2022.9699458">10.1109/MEMS51670.2022.9699458</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-6654-0911-7" title="Special:BookSources/978-1-6654-0911-7"><bdi>978-1-6654-0911-7</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:246753151">246753151</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Magnetically+Coupled+Microelectromechanical+Resonators+for+Low-Frequency+Wireless+Power+Transfer&rft.btitle=2022+IEEE+35th+International+Conference+on+Micro+Electro+Mechanical+Systems+Conference+%28MEMS%29&rft.pages=648-651&rft.date=2022&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A246753151%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2FMEMS51670.2022.9699458&rft.isbn=978-1-6654-0911-7&rft.aulast=Rendon-Hernandez&rft.aufirst=Adrian+A.&rft.au=Halim%2C+Miah+A.&rft.au=Smith%2C+Spencer+E.&rft.au=Arnold%2C+David+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWheeler1942" class="citation journal cs1">Wheeler, H.A. (1942). "Formulas for the Skin Effect". <i>Proceedings of the IRE</i>. <b>30</b> (9): 412–424. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FJRPROC.1942.232015">10.1109/JRPROC.1942.232015</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:51630416">51630416</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+IRE&rft.atitle=Formulas+for+the+Skin+Effect&rft.volume=30&rft.issue=9&rft.pages=412-424&rft.date=1942&rft_id=info%3Adoi%2F10.1109%2FJRPROC.1942.232015&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A51630416%23id-name%3DS2CID&rft.aulast=Wheeler&rft.aufirst=H.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWheeler1928" class="citation journal cs1">Wheeler, H.A. (1928). "Simple Inductance Formulas for Radio Coils". <i>Proceedings of the IRE</i>. <b>16</b> (10): 1398–1400. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FJRPROC.1928.221309">10.1109/JRPROC.1928.221309</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:51638679">51638679</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+IRE&rft.atitle=Simple+Inductance+Formulas+for+Radio+Coils&rft.volume=16&rft.issue=10&rft.pages=1398-1400&rft.date=1928&rft_id=info%3Adoi%2F10.1109%2FJRPROC.1928.221309&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A51638679%23id-name%3DS2CID&rft.aulast=Wheeler&rft.aufirst=H.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> <li id="cite_note-40"><span class="mw-cite-backlink"><b><a href="#cite_ref-40">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFElliott1993" class="citation book cs1">Elliott, R.S. (1993). <i>Electromagnetics</i>. New York: IEEE Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electromagnetics&rft.place=New+York&rft.pub=IEEE+Press&rft.date=1993&rft.aulast=Elliott&rft.aufirst=R.S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span> Note: The published constant <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">−3</span>⁄<span class="den">2</span></span> in the result for a uniform current distribution is wrong.</span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrover1946" class="citation book cs1">Grover, Frederick W. (1946). <i>Inductance Calculations: Working formulas and tables</i>. New York: Dover Publications, Inc.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Inductance+Calculations%3A+Working+formulas+and+tables&rft.place=New+York&rft.pub=Dover+Publications%2C+Inc.&rft.date=1946&rft.aulast=Grover&rft.aufirst=Frederick+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="General_references">General references</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Inductance&action=edit&section=31" title="Edit section: General references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrederick_W._Grover1952" class="citation book cs1">Frederick W. Grover (1952). <i>Inductance Calculations</i>. Dover Publications, New York.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Inductance+Calculations&rft.pub=Dover+Publications%2C+New+York&rft.date=1952&rft.au=Frederick+W.+Grover&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths,_David_J.1998" class="citation book cs1">Griffiths, David J. (1998). <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoel00grif_0"><i>Introduction to Electrodynamics (3rd ed.)</i></a>. Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-805326-X" title="Special:BookSources/0-13-805326-X"><bdi>0-13-805326-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=Introduction+to+Electrodynamics+%283rd+ed.%29&rft.pub=Prentice+Hall&rft.date=1998&rft.isbn=0-13-805326-X&rft.au=Griffiths%2C+David+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoel00grif_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWangsness1986" class="citation book cs1">Wangsness, Roald K. (1986). <i>Electromagnetic Fields</i> (2nd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-81186-6" title="Special:BookSources/0-471-81186-6"><bdi>0-471-81186-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electromagnetic+Fields&rft.edition=2nd&rft.pub=Wiley&rft.date=1986&rft.isbn=0-471-81186-6&rft.aulast=Wangsness&rft.aufirst=Roald+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHughes,_Edward.2002" class="citation book cs1">Hughes, Edward. (2002). <i>Electrical & Electronic Technology (8th ed.)</i>. Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-582-40519-X" title="Special:BookSources/0-582-40519-X"><bdi>0-582-40519-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=Electrical+%26+Electronic+Technology+%288th+ed.%29&rft.pub=Prentice+Hall&rft.date=2002&rft.isbn=0-582-40519-X&rft.au=Hughes%2C+Edward.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AInductance" class="Z3988"></span></li> <li><a href="/wiki/Karl_K%C3%BCpfm%C3%BCller" title="Karl Küpfmüller">Küpfmüller K.</a>, <i>Einführung in die theoretische Elektrotechnik,</i> Springer-Verlag, 1959.</li> <li>Heaviside O., <i>Electrical Papers.</i> Vol.1. – L.; N.Y.: Macmillan, 1892, p. 429-560.</li> <li><a href="/wiki/Fritz_Langford-Smith" title="Fritz Langford-Smith">Fritz Langford-Smith</a>, editor (1953). <i><a rel="nofollow" class="external text" href="https://archive.org/stream/bitsavers_rcaRadiotr1954_94958503/Radiotron_Designers_Handbook_1954#page/n469/mode/2up">Radiotron Designer's Handbook</a></i>, 4th Edition, Amalgamated Wireless Valve Company Pty., Ltd. Chapter 10, "Calculation of Inductance" (pp. 429–448), includes a wealth of formulas and nomographs for coils, solenoids, and mutual inductance.</li> <li>F. W. Sears and M. W. Zemansky 1964 <i>University Physics: Third Edition (Complete Volume)</i>, Addison-Wesley Publishing Company, Inc. Reading MA, LCCC 63-15265 (no ISBN).</li></ul> </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=Inductance&action=edit&section=32" 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/20171115094017/http://www.cvel.clemson.edu/emc/calculators/Inductance_Calculator/index.html"><i>Clemson Vehicular Electronics Laboratory: Inductance Calculator</i></a></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 .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q177897#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4026770-2">Germany</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Inductance"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85065802">United States</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="http://olduli.nli.org.il/F/?func=find-b&local_base=NLX10&find_code=UID&request=987007548367805171">Israel</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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