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class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#AC_power_supply_frequencies"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>AC power supply frequencies</span> </div> </a> <button aria-controls="toc-AC_power_supply_frequencies-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 AC power supply frequencies subsection</span> </button> <ul id="toc-AC_power_supply_frequencies-sublist" class="vector-toc-list"> <li id="toc-Low_frequency" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Low_frequency"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Low frequency</span> </div> </a> <ul id="toc-Low_frequency-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-High_frequency" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#High_frequency"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>High frequency</span> </div> </a> <ul id="toc-High_frequency-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Effects_at_high_frequencies" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Effects_at_high_frequencies"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Effects at high frequencies</span> </div> </a> <button aria-controls="toc-Effects_at_high_frequencies-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 Effects at high frequencies subsection</span> </button> <ul id="toc-Effects_at_high_frequencies-sublist" class="vector-toc-list"> <li id="toc-Techniques_for_reducing_AC_resistance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Techniques_for_reducing_AC_resistance"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Techniques for reducing AC resistance</span> </div> </a> <ul id="toc-Techniques_for_reducing_AC_resistance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Techniques_for_reducing_radiation_loss" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Techniques_for_reducing_radiation_loss"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Techniques for reducing radiation loss</span> </div> </a> <ul id="toc-Techniques_for_reducing_radiation_loss-sublist" class="vector-toc-list"> <li id="toc-Twisted_pairs" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Twisted_pairs"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Twisted pairs</span> </div> </a> <ul id="toc-Twisted_pairs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coaxial_cables" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Coaxial_cables"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Coaxial cables</span> </div> </a> <ul id="toc-Coaxial_cables-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Waveguides" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Waveguides"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.3</span> <span>Waveguides</span> </div> </a> <ul id="toc-Waveguides-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fiber_optics" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Fiber_optics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.4</span> <span>Fiber optics</span> </div> </a> <ul id="toc-Fiber_optics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Formulation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formulation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Formulation</span> </div> </a> <button aria-controls="toc-Formulation-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 Formulation subsection</span> </button> <ul id="toc-Formulation-sublist" class="vector-toc-list"> <li id="toc-Root_mean_square_voltage" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Root_mean_square_voltage"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Root mean square voltage</span> </div> </a> <ul id="toc-Root_mean_square_voltage-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Power" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Power"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Power</span> </div> </a> <ul id="toc-Power-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples_of_alternating_current" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples_of_alternating_current"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Examples of alternating current</span> </div> </a> <ul id="toc-Examples_of_alternating_current-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Information_transmission" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Information_transmission"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Information transmission</span> </div> </a> <ul id="toc-Information_transmission-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>History</span> </div> </a> <button aria-controls="toc-History-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 History subsection</span> </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Transformers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transformers"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Transformers</span> </div> </a> <ul id="toc-Transformers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pioneers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pioneers"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Pioneers</span> </div> </a> <ul id="toc-Pioneers-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</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">Alternating current</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 91 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-91" 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">91 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/Wisselstroom" title="Wisselstroom – Afrikaans" lang="af" hreflang="af" data-title="Wisselstroom" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Wechselstrom" title="Wechselstrom – Alemannic" lang="gsw" hreflang="gsw" data-title="Wechselstrom" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A0%E1%88%98%E1%8A%95%E1%89%BA_%E1%8C%85%E1%88%A8%E1%89%B5" title="አመንቺ ጅረት – Amharic" lang="am" hreflang="am" data-title="አመንቺ ጅረት" 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/%D8%AA%D9%8A%D8%A7%D8%B1_%D9%85%D8%AA%D8%B1%D8%AF%D8%AF" title="تيار متردد – Arabic" lang="ar" hreflang="ar" data-title="تيار متردد" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Corrient_alterna" title="Corrient alterna – Aragonese" lang="an" hreflang="an" data-title="Corrient alterna" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target"><span>Aragonés</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Corriente_alterno" title="Corriente alterno – Asturian" lang="ast" hreflang="ast" data-title="Corriente alterno" 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/D%C9%99yi%C5%9F%C9%99n_c%C9%99r%C9%99yan" title="Dəyişən cərəyan – Azerbaijani" lang="az" hreflang="az" data-title="Dəyişən cərəyan" 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%AA%E0%A6%B0%E0%A6%BF%E0%A6%AC%E0%A6%B0%E0%A7%8D%E0%A6%A4%E0%A7%80_%E0%A6%A4%E0%A6%A1%E0%A6%BC%E0%A6%BF%E0%A7%8E_%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%AC%E0%A6%BE%E0%A6%B9" title="পরিবর্তী তড়িৎ প্রবাহ – Bangla" lang="bn" hreflang="bn" data-title="পরিবর্তী তড়িৎ প্রবাহ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B5%D1%80%D0%B0%D0%BC%D0%B5%D0%BD%D0%BD%D1%8B_%D1%82%D0%BE%D0%BA" title="Пераменны ток – Belarusian" lang="be" hreflang="be" data-title="Пераменны ток" 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%97%D1%8C%D0%BC%D0%B5%D0%BD%D0%BD%D1%8B_%D1%82%D0%BE%D0%BA" title="Зьменны ток – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Зьменны ток" 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%9F%D1%80%D0%BE%D0%BC%D0%B5%D0%BD%D0%BB%D0%B8%D0%B2_%D1%82%D0%BE%D0%BA" title="Променлив ток – Bulgarian" lang="bg" hreflang="bg" data-title="Променлив ток" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bo mw-list-item"><a href="https://bo.wikipedia.org/wiki/%E0%BD%A6%E0%BE%A4%E0%BD%BA%E0%BD%A3%E0%BC%8B%E0%BD%A2%E0%BE%92%E0%BE%B1%E0%BD%B4%E0%BD%82%E0%BC%8B%E0%BD%82%E0%BE%B3%E0%BD%BC%E0%BD%82%E0%BC%8B%E0%BD%A2%E0%BE%92%E0%BE%B1%E0%BD%B4%E0%BD%93%E0%BC%8D" title="སྤེལ་རྒྱུག་གློག་རྒྱུན། – Tibetan" lang="bo" hreflang="bo" data-title="སྤེལ་རྒྱུག་གློག་རྒྱུན།" data-language-autonym="བོད་ཡིག" data-language-local-name="Tibetan" class="interlanguage-link-target"><span>བོད་ཡིག</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Naizmjeni%C4%8Dna_struja" title="Naizmjenična struja – Bosnian" lang="bs" hreflang="bs" data-title="Naizmjenična struja" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Corrent_altern" title="Corrent altern – Catalan" lang="ca" hreflang="ca" data-title="Corrent altern" 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%A3%D0%BB%D1%88%C4%83%D0%BD%D1%83%D0%BB%D0%BB%C4%83_%D1%82%D0%BE%D0%BA" title="Улшăнуллă ток – Chuvash" lang="cv" hreflang="cv" data-title="Улшăнуллă ток" 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/St%C5%99%C3%ADdav%C3%BD_proud" title="Střídavý proud – Czech" lang="cs" hreflang="cs" data-title="Střídavý proud" 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/Vekselstr%C3%B8m" title="Vekselstrøm – Danish" lang="da" hreflang="da" data-title="Vekselstrøm" 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/Wechselstrom" title="Wechselstrom – German" lang="de" hreflang="de" data-title="Wechselstrom" 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/Vahelduvvool" title="Vahelduvvool – Estonian" lang="et" hreflang="et" data-title="Vahelduvvool" 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%95%CE%BD%CE%B1%CE%BB%CE%BB%CE%B1%CF%83%CF%83%CF%8C%CE%BC%CE%B5%CE%BD%CE%BF_%CF%81%CE%B5%CF%8D%CE%BC%CE%B1" title="Εναλλασσόμενο ρεύμα – Greek" lang="el" hreflang="el" data-title="Εναλλασσόμενο ρεύμα" 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/Corriente_alterna" title="Corriente alterna – Spanish" lang="es" hreflang="es" data-title="Corriente alterna" 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/Alterna_kurento" title="Alterna kurento – Esperanto" lang="eo" hreflang="eo" data-title="Alterna kurento" 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/Korronte_alterno" title="Korronte alterno – Basque" lang="eu" hreflang="eu" data-title="Korronte alterno" 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%AC%D8%B1%DB%8C%D8%A7%D9%86_%D9%85%D8%AA%D9%86%D8%A7%D9%88%D8%A8" title="جریان متناوب – Persian" lang="fa" hreflang="fa" data-title="جریان متناوب" 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/Courant_alternatif" title="Courant alternatif – French" lang="fr" hreflang="fr" data-title="Courant alternatif" 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/Sruth_ailt%C3%A9arnach" title="Sruth ailtéarnach – Irish" lang="ga" hreflang="ga" data-title="Sruth ailtéarnach" 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/Corrente_alterna" title="Corrente alterna – Galician" lang="gl" hreflang="gl" data-title="Corrente alterna" 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/%EA%B5%90%EB%A5%98" title="교류 – Korean" lang="ko" hreflang="ko" data-title="교류" 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/%D5%93%D5%B8%D6%83%D5%B8%D5%AD%D5%A1%D5%AF%D5%A1%D5%B6_%D5%B0%D5%B8%D5%BD%D5%A1%D5%B6%D6%84" title="Փոփոխական հոսանք – Armenian" lang="hy" hreflang="hy" data-title="Փոփոխական հոսանք" 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%A4%A4%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A5%80_%E0%A4%A7%E0%A4%BE%E0%A4%B0%E0%A4%BE" title="प्रत्यावर्ती धारा – Hindi" lang="hi" hreflang="hi" data-title="प्रत्यावर्ती धारा" 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/Izmjeni%C4%8Dna_elektri%C4%8Dna_struja" title="Izmjenična električna struja – Croatian" lang="hr" hreflang="hr" data-title="Izmjenična električna struja" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-gor mw-list-item"><a href="https://gor.wikipedia.org/wiki/Arus_bolak-balik" title="Arus bolak-balik – Gorontalo" lang="gor" hreflang="gor" data-title="Arus bolak-balik" data-language-autonym="Bahasa Hulontalo" data-language-local-name="Gorontalo" class="interlanguage-link-target"><span>Bahasa Hulontalo</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Korento_alternanta" title="Korento alternanta – Ido" lang="io" hreflang="io" data-title="Korento alternanta" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Arus_bolak-balik" title="Arus bolak-balik – Indonesian" lang="id" hreflang="id" data-title="Arus bolak-balik" 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/Ri%C3%B0straumur" title="Riðstraumur – Icelandic" lang="is" hreflang="is" data-title="Riðstraumur" 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/Corrente_alternata" title="Corrente alternata – Italian" lang="it" hreflang="it" data-title="Corrente alternata" 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%96%D7%A8%D7%9D_%D7%97%D7%99%D7%9C%D7%95%D7%A4%D7%99%D7%9F" title="זרם חילופין – Hebrew" lang="he" hreflang="he" data-title="זרם חילופין" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AA%E0%B2%B0%E0%B3%8D%E0%B2%AF%E0%B2%BE%E0%B2%AF(%E0%B2%B8%E0%B2%B0%E0%B2%A6%E0%B2%BF%E0%B2%AF%E0%B2%82%E0%B2%A4%E0%B3%86_%E0%B2%AC%E0%B2%B0%E0%B3%81%E0%B2%B5)_%E0%B2%B5%E0%B2%BF%E0%B2%A6%E0%B3%8D%E0%B2%AF%E0%B3%81%E0%B2%A4%E0%B3%8D_%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%B5%E0%B2%BE%E0%B2%B9" title="ಪರ್ಯಾಯ(ಸರದಿಯಂತೆ ಬರುವ) ವಿದ್ಯುತ್ ಪ್ರವಾಹ – Kannada" lang="kn" hreflang="kn" data-title="ಪರ್ಯಾಯ(ಸರದಿಯಂತೆ ಬರುವ) ವಿದ್ಯುತ್ ಪ್ರವಾಹ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%AA%E1%83%95%E1%83%9A%E1%83%90%E1%83%93%E1%83%98_%E1%83%93%E1%83%94%E1%83%9C%E1%83%98" title="ცვლადი დენი – Georgian" lang="ka" hreflang="ka" data-title="ცვლადი დენი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D0%B9%D0%BD%D1%8B%D0%BC%D0%B0%D0%BB%D1%8B_%D1%82%D0%BE%D0%BA" title="Айнымалы ток – Kazakh" lang="kk" hreflang="kk" data-title="Айнымалы ток" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Kouran_alt%C3%A8natif_(AC)" title="Kouran altènatif (AC) – Haitian Creole" lang="ht" hreflang="ht" data-title="Kouran altènatif (AC)" 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-ku mw-list-item"><a href="https://ku.wikipedia.org/wiki/Rewteka_p%C3%AAlp%C3%AAlok%C3%AE" title="Rewteka pêlpêlokî – Kurdish" lang="ku" hreflang="ku" data-title="Rewteka pêlpêlokî" data-language-autonym="Kurdî" data-language-local-name="Kurdish" class="interlanguage-link-target"><span>Kurdî</span></a></li><li class="interlanguage-link interwiki-lld mw-list-item"><a href="https://lld.wikipedia.org/wiki/Strom_mud%C3%A0_ju" title="Strom mudà ju – Ladin" lang="lld" hreflang="lld" data-title="Strom mudà ju" data-language-autonym="Ladin" data-language-local-name="Ladin" class="interlanguage-link-target"><span>Ladin</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Mai%C5%86str%C4%81va" title="Maiņstrāva – Latvian" lang="lv" hreflang="lv" data-title="Maiņstrāva" 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/Kintamoji_elektros_srov%C4%97" title="Kintamoji elektros srovė – Lithuanian" lang="lt" hreflang="lt" data-title="Kintamoji elektros srovė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Corrent_alternada" title="Corrent alternada – Lombard" lang="lmo" hreflang="lmo" data-title="Corrent alternada" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/V%C3%A1ltakoz%C3%B3_%C3%A1ram" title="Váltakozó áram – Hungarian" lang="hu" hreflang="hu" data-title="Váltakozó áram" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9D%D0%B0%D0%B8%D0%B7%D0%BC%D0%B5%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D1%81%D1%82%D1%80%D1%83%D1%98%D0%B0" title="Наизменична струја – Macedonian" lang="mk" hreflang="mk" data-title="Наизменична струја" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Zitina_miherina" title="Zitina miherina – Malagasy" lang="mg" hreflang="mg" data-title="Zitina miherina" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%A4%E0%B5%8D%E0%B4%AF%E0%B4%BE%E0%B4%B5%E0%B5%BC%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%BF%E0%B4%A7%E0%B4%BE%E0%B4%B0%E0%B4%BE_%E0%B4%B5%E0%B5%88%E0%B4%A6%E0%B5%8D%E0%B4%AF%E0%B5%81%E0%B4%A4%E0%B4%BF" title="പ്രത്യാവർത്തിധാരാ വൈദ്യുതി – Malayalam" lang="ml" hreflang="ml" data-title="പ്രത്യാവർത്തിധാരാ വൈദ്യുതി" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Arus_ulang-alik" title="Arus ulang-alik – Malay" lang="ms" hreflang="ms" data-title="Arus ulang-alik" 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/Wisselstroom" title="Wisselstroom – Dutch" lang="nl" hreflang="nl" data-title="Wisselstroom" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%8F%E0%A4%B8%E0%A5%80_%E0%A4%95%E0%A4%B0%E0%A5%87%E0%A4%A8%E0%A5%8D%E0%A4%9F" title="एसी करेन्ट – Nepali" lang="ne" hreflang="ne" data-title="एसी करेन्ट" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BA%A4%E6%B5%81" title="交流 – Japanese" lang="ja" hreflang="ja" data-title="交流" 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/Wakselstruum" title="Wakselstruum – Northern Frisian" lang="frr" hreflang="frr" data-title="Wakselstruum" 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/Vekselstr%C3%B8m" title="Vekselstrøm – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Vekselstrøm" 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/Vekselstraum" title="Vekselstraum – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Vekselstraum" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Corrent_alternatiu" title="Corrent alternatiu – Occitan" lang="oc" hreflang="oc" data-title="Corrent alternatiu" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/O%CA%BBzgaruvchan_elektr_toki" title="Oʻzgaruvchan elektr toki – Uzbek" lang="uz" hreflang="uz" data-title="Oʻzgaruvchan elektr toki" 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-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AC%E0%A8%BF%E0%A8%9C%E0%A8%B2%E0%A9%80_%E0%A8%A6%E0%A9%80_%E0%A8%AC%E0%A8%A6%E0%A8%B2%E0%A8%B5%E0%A9%80%E0%A8%82_%E0%A8%A7%E0%A8%BE%E0%A8%B0%E0%A8%BE" title="ਬਿਜਲੀ ਦੀ ਬਦਲਵੀਂ ਧਾਰਾ – Punjabi" lang="pa" hreflang="pa" data-title="ਬਿਜਲੀ ਦੀ ਬਦਲਵੀਂ ਧਾਰਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A2%D9%84%D9%B9%D8%B1%D9%86%DB%8C%D9%B9%D9%86%DA%AF_%DA%A9%D8%B1%D9%86%D9%B9" title="آلٹرنیٹنگ کرنٹ – Western Punjabi" lang="pnb" hreflang="pnb" data-title="آلٹرنیٹنگ کرنٹ" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pr%C4%85d_przemienny" title="Prąd przemienny – Polish" lang="pl" hreflang="pl" data-title="Prąd przemienny" 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/Corrente_alternada" title="Corrente alternada – Portuguese" lang="pt" hreflang="pt" data-title="Corrente alternada" 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/Curent_alternativ" title="Curent alternativ – Romanian" lang="ro" hreflang="ro" data-title="Curent alternativ" 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%9F%D0%B5%D1%80%D0%B5%D0%BC%D0%B5%D0%BD%D0%BD%D1%8B%D0%B9_%D1%82%D0%BE%D0%BA" title="Переменный ток – Russian" lang="ru" hreflang="ru" data-title="Переменный ток" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Alternatin_current" title="Alternatin current – Scots" lang="sco" hreflang="sco" data-title="Alternatin current" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-stq mw-list-item"><a href="https://stq.wikipedia.org/wiki/Wikselsponnenge" title="Wikselsponnenge – Saterland Frisian" lang="stq" hreflang="stq" data-title="Wikselsponnenge" data-language-autonym="Seeltersk" data-language-local-name="Saterland Frisian" class="interlanguage-link-target"><span>Seeltersk</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Rryma_e_ndryshueshme" title="Rryma e ndryshueshme – Albanian" lang="sq" hreflang="sq" data-title="Rryma e ndryshueshme" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Alternating_current" title="Alternating current – Simple English" lang="en-simple" hreflang="en-simple" data-title="Alternating current" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Striedav%C3%BD_pr%C3%BAd" title="Striedavý prúd – Slovak" lang="sk" hreflang="sk" data-title="Striedavý prúd" 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/Izmeni%C4%8Dni_elektri%C4%8Dni_tok" title="Izmenični električni tok – Slovenian" lang="sl" hreflang="sl" data-title="Izmenični električni tok" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%AA%DB%95%D8%B2%D9%88%D9%88%DB%8C_%DA%AF%DB%86%DA%95%D8%A7%D9%88" title="تەزووی گۆڕاو – Central Kurdish" lang="ckb" hreflang="ckb" data-title="تەزووی گۆڕاو" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9D%D0%B0%D0%B8%D0%B7%D0%BC%D0%B5%D0%BD%D0%B8%D1%87%D0%BD%D0%B0_%D1%81%D1%82%D1%80%D1%83%D1%98%D0%B0" title="Наизменична струја – Serbian" lang="sr" hreflang="sr" data-title="Наизменична струја" 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/Izmjeni%C4%8Dna_struja" title="Izmjenična struja – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Izmjenična struja" 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-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Arus_listrik_bulak-balik" title="Arus listrik bulak-balik – Sundanese" lang="su" hreflang="su" data-title="Arus listrik bulak-balik" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Vaihtovirta" title="Vaihtovirta – Finnish" lang="fi" hreflang="fi" data-title="Vaihtovirta" 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/V%C3%A4xelstr%C3%B6m" title="Växelström – Swedish" lang="sv" hreflang="sv" data-title="Växelström" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Alternating_current" title="Alternating current – Tagalog" lang="tl" hreflang="tl" data-title="Alternating current" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AE%BE%E0%AE%B1%E0%AF%81%E0%AE%A4%E0%AE%BF%E0%AE%9A%E0%AF%88_%E0%AE%AE%E0%AE%BF%E0%AE%A9%E0%AF%8D%E0%AE%A9%E0%AF%8B%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%AE%E0%AF%8D" title="மாறுதிசை மின்னோட்டம் – Tamil" lang="ta" hreflang="ta" data-title="மாறுதிசை மின்னோட்டம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%A6%E0%B1%8D%E0%B0%B5%E0%B0%BF%E0%B0%AE%E0%B1%81%E0%B0%96_%E0%B0%B5%E0%B0%BF%E0%B0%A6%E0%B1%8D%E0%B0%AF%E0%B1%81%E0%B0%A4%E0%B1%8D_%E0%B0%AA%E0%B1%8D%E0%B0%B0%E0%B0%B5%E0%B0%BE%E0%B0%B9%E0%B0%82" title="ద్విముఖ విద్యుత్ ప్రవాహం – Telugu" lang="te" hreflang="te" data-title="ద్విముఖ విద్యుత్ ప్రవాహం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%84%E0%B8%9F%E0%B8%9F%E0%B9%89%E0%B8%B2%E0%B8%81%E0%B8%A3%E0%B8%B0%E0%B9%81%E0%B8%AA%E0%B8%AA%E0%B8%A5%E0%B8%B1%E0%B8%9A" title="ไฟฟ้ากระแสสลับ – Thai" lang="th" hreflang="th" data-title="ไฟฟ้ากระแสสลับ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Alternatif_ak%C4%B1m" title="Alternatif akım – Turkish" lang="tr" hreflang="tr" data-title="Alternatif akım" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a 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class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Alternating_current" title="Alternating current – Waray" lang="war" hreflang="war" data-title="Alternating current" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wo mw-list-item"><a href="https://wo.wikipedia.org/wiki/Dawaan_bu_safaanu" title="Dawaan bu safaanu – Wolof" lang="wo" hreflang="wo" data-title="Dawaan bu safaanu" 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/%E4%BA%A4%E6%B5%81%E7%94%B5" title="交流电 – Wu" lang="wuu" hreflang="wuu" data-title="交流电" 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-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%95%D7%95%D7%A2%D7%A7%D7%A1%D7%9C%D7%A9%D7%98%D7%A8%D7%90%D7%9D" title="וועקסלשטראם – Yiddish" lang="yi" hreflang="yi" data-title="וועקסלשטראם" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%BF%95%E9%9B%BB" title="濕電 – Cantonese" lang="yue" hreflang="yue" data-title="濕電" 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/%E4%BA%A4%E6%B5%81%E9%9B%BB" title="交流電 – Chinese" lang="zh" hreflang="zh" data-title="交流電" data-language-autonym="中文" data-language-local-name="Chinese" 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.ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Alternating_current" title="Special:EditPage/Alternating current">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Alternating+current%22">"Alternating current"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Alternating+current%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Alternating+current%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Alternating+current%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Alternating+current%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Alternating+current%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">March 2023</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Types_of_current.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Types_of_current.svg/220px-Types_of_current.svg.png" decoding="async" width="220" height="119" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Types_of_current.svg/330px-Types_of_current.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/38/Types_of_current.svg/440px-Types_of_current.svg.png 2x" data-file-width="738" data-file-height="400" /></a><figcaption>Alternating current (green curve). The horizontal axis measures time (it also represents zero voltage/current); the vertical, current or voltage.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist 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about</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File/Frameless"><a href="/wiki/File:VFPt_Solenoid_correct2.svg" class="mw-file-description" title="Solenoid"><img alt="Solenoid" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/190px-VFPt_Solenoid_correct2.svg.png" decoding="async" width="190" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/285px-VFPt_Solenoid_correct2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/380px-VFPt_Solenoid_correct2.svg.png 2x" data-file-width="490" data-file-height="200" /></a></span></td></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Electricity" title="Electricity">Electricity</a></li> <li><a href="/wiki/Magnetism" title="Magnetism">Magnetism</a></li> <li><a href="/wiki/Optics" title="Optics">Optics</a></li> <li><a href="/wiki/History_of_electromagnetic_theory" title="History of electromagnetic theory">History</a></li> <li><a href="/wiki/Computational_electromagnetics" title="Computational electromagnetics">Computational</a></li> <li><a href="/wiki/List_of_textbooks_in_electromagnetism" title="List of textbooks in electromagnetism">Textbooks</a></li> <li><a href="/wiki/List_of_electrical_phenomena" title="List of electrical phenomena">Phenomena</a></li></ul></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/Electrostatics" title="Electrostatics">Electrostatics</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Charge_density" title="Charge density">Charge density</a></li> <li><a href="/wiki/Electrical_conductor" title="Electrical conductor">Conductor</a></li> <li><a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb law</a></li> <li><a href="/wiki/Electret" title="Electret">Electret</a></li> <li><a href="/wiki/Electric_charge" title="Electric charge">Electric charge</a></li> <li><a href="/wiki/Electric_dipole_moment" title="Electric dipole moment">Electric dipole</a></li> <li><a href="/wiki/Electric_field" title="Electric field">Electric field</a></li> <li><a href="/wiki/Electric_flux" title="Electric flux">Electric flux</a></li> <li><a href="/wiki/Electric_potential" title="Electric potential">Electric potential</a></li> <li><a href="/wiki/Electrostatic_discharge" title="Electrostatic discharge">Electrostatic discharge</a></li> <li><a href="/wiki/Electrostatic_induction" title="Electrostatic induction">Electrostatic induction</a></li> <li><a href="/wiki/Gauss%27s_law" title="Gauss's law">Gauss law</a></li> <li><a href="/wiki/Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li> <li><a href="/wiki/Permittivity" title="Permittivity">Permittivity</a></li> <li><a href="/wiki/Polarization_density" title="Polarization density">Polarization</a></li> <li><a href="/wiki/Electric_potential_energy" title="Electric potential energy">Potential energy</a></li> <li><a href="/wiki/Static_electricity" title="Static electricity">Static electricity</a></li> <li><a href="/wiki/Triboelectric_effect" title="Triboelectric effect">Triboelectricity</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/Magnetostatics" title="Magnetostatics">Magnetostatics</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Amp%C3%A8re%27s_circuital_law" 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 class="mw-selflink selflink">Alternating current</a></li> <li><a href="/wiki/Capacitance" title="Capacitance">Capacitance</a></li> <li><a href="/wiki/Current_density" title="Current density">Current density</a></li> <li><a href="/wiki/Direct_current" title="Direct current">Direct current</a></li> <li><a href="/wiki/Electric_current" title="Electric current">Electric current</a></li> <li><a href="/wiki/Electric_power" title="Electric power">Electric power</a></li> <li><a href="/wiki/Electrolysis" title="Electrolysis">Electrolysis</a></li> <li><a href="/wiki/Electromotive_force" title="Electromotive force">Electromotive force</a></li> <li><a href="/wiki/Electrical_impedance" title="Electrical impedance">Impedance</a></li> <li><a href="/wiki/Inductance" title="Inductance">Inductance</a></li> <li><a href="/wiki/Joule_heating" title="Joule heating">Joule heating</a></li> <li><a href="/wiki/Kirchhoff%27s_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff laws</a></li> <li><a href="/wiki/Network_analysis_(electrical_circuits)" title="Network analysis (electrical circuits)">Network analysis</a></li> <li><a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm law</a></li> <li><a href="/wiki/Series_and_parallel_circuits#Parallel_circuits" title="Series and parallel circuits">Parallel circuit</a></li> <li><a href="/wiki/Electrical_resistance_and_conductance" title="Electrical resistance and conductance">Resistance</a></li> <li><a href="/wiki/Resonator#Electromagnetics" title="Resonator">Resonant cavities</a></li> <li><a href="/wiki/Series_and_parallel_circuits#Series_circuits" title="Series and parallel circuits">Series circuit</a></li> <li><a href="/wiki/Voltage" title="Voltage">Voltage</a></li> <li><a href="/wiki/Watt" title="Watt">Watt</a></li> <li><a href="/wiki/Waveguide_(radio_frequency)" title="Waveguide (radio frequency)">Waveguides</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible 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>Alternating current</b> (<b>AC</b>) is an <a href="/wiki/Electric_current" title="Electric current">electric current</a> that periodically reverses direction and changes its magnitude continuously with time ranging between some maximum and minimum values, in contrast to <i><a href="/wiki/Direct_current" title="Direct current">direct current</a></i> (DC), which flows only in one direction. Alternating current is the type of electric current through which <a href="/wiki/Electric_power" title="Electric power">electric power</a> is delivered to businesses and residences, and it is the type of <a href="/wiki/Electrical_energy" title="Electrical energy">electric current</a> that consumers typically use when they plug <a href="/wiki/Kitchen_appliance" class="mw-redirect" title="Kitchen appliance">kitchen appliances</a>, <a href="/wiki/Television" title="Television">televisions</a>, <a href="/wiki/Fan_(machine)" title="Fan (machine)">fans</a> and <a href="/wiki/Electric_lamp" class="mw-redirect" title="Electric lamp">electric lamps</a> into a <a href="/wiki/Wall_socket" class="mw-redirect" title="Wall socket">wall socket</a>. The abbreviations <i>AC</i> and <i>DC</i> are often used to mean simply <i>Alternating Current</i> and <i>Direct Current</i>, respectively, as when they modify <i><a href="/wiki/Electric_current" title="Electric current">current</a></i> or <i><a href="/wiki/Voltage" title="Voltage">voltage</a></i>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><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> </p><p>The usual <a href="/wiki/Waveform" title="Waveform">waveform</a> of alternating current in most electric power circuits is a <a href="/wiki/Sine_wave" title="Sine wave">sine wave</a>, whose positive half-period corresponds with positive direction of the current and vice versa (the full period is called a <i><a href="/wiki/Wave_cycle" class="mw-redirect" title="Wave cycle">cycle</a></i>). In certain applications, like <a href="/wiki/Guitar_amplifier" title="Guitar amplifier">guitar amplifiers</a>, different waveforms are used, such as <a href="/wiki/Triangle_wave" title="Triangle wave">triangular waves</a> or <a href="/wiki/Square_wave" title="Square wave">square waves</a>. <a href="/wiki/Audio_frequency" title="Audio frequency">Audio</a> and <a href="/wiki/Radio_frequency" title="Radio frequency">radio</a> signals carried on electrical wires are also examples of alternating current. These types of alternating current carry information such as sound (audio) or images (video) sometimes carried by <a href="/wiki/Modulation" title="Modulation">modulation</a> of an AC carrier signal. These currents typically alternate at higher frequencies than those used in power transmission. </p> <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-3"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="Transmission,_distribution,_and_domestic_power_supply"><span id="Transmission.2C_distribution.2C_and_domestic_power_supply"></span>Transmission, distribution, and domestic power supply</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=1" title="Edit section: Transmission, distribution, and domestic power supply"><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 articles: <a href="/wiki/Electric_power_transmission" title="Electric power transmission">Electric power transmission</a> and <a href="/wiki/Electric_power_distribution" title="Electric power distribution">Electric power distribution</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Electric_Transmission.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Electric_Transmission.png/220px-Electric_Transmission.png" decoding="async" width="220" height="113" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Electric_Transmission.png/330px-Electric_Transmission.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Electric_Transmission.png/440px-Electric_Transmission.png 2x" data-file-width="3225" data-file-height="1650" /></a><figcaption>A schematic representation of long distance electric power transmission. From left to right: G=generator, U=step-up transformer, V=voltage at beginning of transmission line, Pt=power entering transmission line, I=current in wires, R=total resistance in wires, Pw=power lost in transmission line, Pe=power reaching the end of the transmission line, D=step-down transformer, C=consumers.</figcaption></figure> <p>Electrical energy is distributed as alternating current because AC <a href="/wiki/Voltage" title="Voltage">voltage</a> may be increased or decreased with a <a href="/wiki/Transformer" title="Transformer">transformer</a>. This allows the power to be transmitted through <a href="/wiki/Power_line" class="mw-redirect" title="Power line">power lines</a> efficiently at <a href="/wiki/High_voltage" title="High voltage">high voltage</a>, which reduces the energy lost as heat due to <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a> of the wire, and transformed to a lower, safer voltage for use. Use of a higher voltage leads to significantly more efficient transmission of power. The power losses (<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_{\rm {w}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">w</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\rm {w}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abc657948b2077aa4e0c1c2f3de7ec851c1078ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.911ex; height:2.509ex;" alt="{\displaystyle P_{\rm {w}}}"></span>) in the wire are a product of the square of the current ( I ) and the <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a> (R) of the wire, described by the formula: </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 P_{\rm {w}}=I^{2}R\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">w</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\rm {w}}=I^{2}R\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8efda57cea8199be3ce8d9667896d8963d4b2395" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.078ex; height:3.009ex;" alt="{\displaystyle P_{\rm {w}}=I^{2}R\,.}"></span></dd></dl> <p>This means that when transmitting a fixed power on a given wire, if the current is halved (i.e. the voltage is doubled), the power loss due to the wire's resistance will be reduced to one quarter. </p><p>The power transmitted is equal to the product of the current and the voltage (assuming no phase difference); that is, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\rm {t}}=IV\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>I</mi> <mi>V</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\rm {t}}=IV\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fb217fe2922665f20d2077f6c2a9df94994ec37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.456ex; height:2.509ex;" alt="{\displaystyle P_{\rm {t}}=IV\,.}"></span></dd></dl> <p>Consequently, power transmitted at a higher voltage requires less loss-producing current than for the same power at a lower voltage. Power is often transmitted at hundreds of kilovolts on <a href="/wiki/Transmission_tower" title="Transmission tower">pylons</a>, and transformed down to tens of kilovolts to be transmitted on lower level lines, and finally transformed down to 100 V – 240 V for domestic use. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Highvoltagetransmissionlines.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Highvoltagetransmissionlines.jpg/220px-Highvoltagetransmissionlines.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Highvoltagetransmissionlines.jpg/330px-Highvoltagetransmissionlines.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Highvoltagetransmissionlines.jpg/440px-Highvoltagetransmissionlines.jpg 2x" data-file-width="3264" data-file-height="2448" /></a><figcaption>Three-phase high-voltage <a href="/wiki/Transmission_line" title="Transmission line">transmission lines</a> use alternating currents to distribute power over long distances between <a href="/wiki/Electric_generation" class="mw-redirect" title="Electric generation">electric generation</a> plants and consumers. The lines in the picture are located in eastern <a href="/wiki/Utah" title="Utah">Utah</a>.</figcaption></figure> <p>High voltages have disadvantages, such as the increased insulation required, and generally increased difficulty in their safe handling. In a <a href="/wiki/Power_plant" class="mw-redirect" title="Power plant">power plant</a>, energy is generated at a convenient voltage for the design of a <a href="/wiki/Electric_generator" title="Electric generator">generator</a>, and then stepped up to a high voltage for transmission. Near the loads, the transmission voltage is stepped down to the voltages used by equipment. Consumer voltages vary somewhat depending on the country and size of load, but generally motors and lighting are built to use up to a few hundred volts between phases. The voltage delivered to equipment such as lighting and motor loads is standardized, with an allowable range of voltage over which equipment is expected to operate. Standard power utilization voltages and percentage tolerance vary in the different <a href="/wiki/Mains_power_systems" class="mw-redirect" title="Mains power systems">mains power systems</a> found in the world. </p><p><a href="/wiki/High-voltage_direct_current" title="High-voltage direct current">High-voltage direct-current</a> (HVDC) electric power transmission systems have become more viable as technology has provided efficient means of changing the voltage of DC power. Transmission with high voltage direct current was not feasible in the early days of <a href="/wiki/Electric_power_transmission" title="Electric power transmission">electric power transmission</a>, as there was then no economically viable way to step the voltage of DC down for end user applications such as lighting incandescent bulbs. </p><p><a href="/wiki/Three-phase_electric_power" title="Three-phase electric power">Three-phase</a> electrical generation is very common. The simplest way is to use three separate coils in the generator <a href="/wiki/Stator" title="Stator">stator</a>, physically offset by an angle of 120° (one-third of a complete 360° phase) to each other. Three current waveforms are produced that are equal in magnitude and 120° <a href="/wiki/Out_of_phase" class="mw-redirect" title="Out of phase">out of phase</a> to each other. If coils are added opposite to these (60° spacing), they generate the same phases with reverse <a href="/wiki/Electric_polarity" class="mw-redirect" title="Electric polarity">polarity</a> and so can be simply wired together. In practice, higher "pole orders" are commonly used. For example, a 12-pole machine would have 36 coils (10° spacing). The advantage is that lower rotational speeds can be used to generate the same frequency. For example, a 2-pole machine running at 3600 rpm and a 12-pole machine running at 600 rpm produce the same frequency; the lower speed is preferable for larger machines. If the load on a three-phase system is balanced equally among the phases, no current flows through the <a href="/wiki/Neutral_point" class="mw-redirect" title="Neutral point">neutral point</a>. Even in the worst-case unbalanced (linear) load, the neutral current will not exceed the highest of the phase currents. Non-linear loads (e.g. the switch-mode power supplies widely used) may require an oversized neutral bus and neutral conductor in the upstream distribution panel to handle <a href="/wiki/Harmonic_(electrical_power)" class="mw-redirect" title="Harmonic (electrical power)">harmonics</a>. Harmonics can cause neutral conductor current levels to exceed that of one or all phase conductors. </p><p>For three-phase at utilization voltages a four-wire system is often used. When stepping down three-phase, a transformer with a Delta (3-wire) primary and a Star (4-wire, center-earthed) secondary is often used so there is no need for a neutral on the supply side. For smaller customers (just how small varies by country and age of the installation) only a <a href="/wiki/Single-phase_electric_power" title="Single-phase electric power">single phase</a> and neutral, or two phases and neutral, are taken to the property. For larger installations all three phases and neutral are taken to the main distribution panel. From the three-phase main panel, both single and three-phase circuits may lead off. <a href="/wiki/Split-phase_electric_power" title="Split-phase electric power">Three-wire single-phase</a> systems, with a single center-tapped transformer giving two live conductors, is a common distribution scheme for residential and small commercial buildings in North America. This arrangement is sometimes incorrectly referred to as "two phase". A similar method is used for a different reason on construction sites in the UK. Small power tools and lighting are supposed to be supplied by a local center-tapped transformer with a voltage of 55 V between each power conductor and earth. This significantly reduces the risk of <a href="/wiki/Electric_shock" class="mw-redirect" title="Electric shock">electric shock</a> in the event that one of the live conductors becomes exposed through an equipment fault whilst still allowing a reasonable voltage of 110 V between the two conductors for running the tools. </p><p>A <a href="/wiki/Ground_and_neutral" title="Ground and neutral">third wire</a>, called the bond (or earth) wire, is often connected between non-current-carrying metal enclosures and earth ground. This conductor provides protection from electric shock due to accidental contact of circuit conductors with the metal chassis of portable appliances and tools. Bonding all non-current-carrying metal parts into one complete system ensures there is always a low <a href="/wiki/Electrical_impedance" title="Electrical impedance">electrical impedance</a> path to ground sufficient to carry any <a href="/wiki/Fault_(power_engineering)" class="mw-redirect" title="Fault (power engineering)">fault</a> current for as long as it takes for the system to clear the fault. This low impedance path allows the maximum amount of fault current, causing the overcurrent protection device (breakers, fuses) to trip or burn out as quickly as possible, bringing the electrical system to a safe state. All bond wires are bonded to ground at the main service panel, as is the neutral/identified conductor if present. </p> <div class="mw-heading mw-heading2"><h2 id="AC_power_supply_frequencies">AC power supply frequencies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=2" title="Edit section: AC power supply frequencies"><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/Mains_electricity_by_country" title="Mains electricity by country">Mains electricity by country</a></div> <p>The <a href="/wiki/Utility_frequency" title="Utility frequency">frequency of the electrical system</a> varies by country and sometimes within a country; most electric power is generated at either 50 or 60 <a href="/wiki/Hertz" title="Hertz">Hertz</a>. Some countries have a mixture of 50 Hz and 60 Hz supplies, notably <a href="/wiki/Electricity_sector_in_Japan#Transmission" title="Electricity sector in Japan">electricity power transmission in Japan</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Low_frequency">Low frequency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=3" title="Edit section: Low frequency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A low frequency eases the design of electric motors, particularly for hoisting, crushing and rolling applications, and commutator-type <a href="/wiki/Traction_motor" title="Traction motor">traction motors</a> for applications such as <a href="/wiki/Railway" class="mw-redirect" title="Railway">railways</a>. However, low frequency also causes noticeable flicker in <a href="/wiki/Arc_lamp" title="Arc lamp">arc lamps</a> and <a href="/wiki/Incandescent_light_bulb" title="Incandescent light bulb">incandescent light bulbs</a>. The use of lower frequencies also provided the advantage of lower transmission losses, which are proportional to frequency. </p><p>The original Niagara Falls generators were built to produce 25 Hz power, as a compromise between low frequency for traction and heavy induction motors, while still allowing incandescent lighting to operate (although with noticeable flicker). Most of the 25 Hz residential and commercial customers for Niagara Falls power were converted to 60 Hz by the late 1950s, although some<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Avoid_weasel_words" class="mw-redirect" title="Wikipedia:Avoid weasel words"><span title="The material near this tag possibly uses too vague attribution or weasel words. (December 2011)">which?</span></a></i>]</sup> 25 Hz industrial customers still existed as of the start of the 21st century. 16.7 Hz power (formerly 16 2/3 Hz) is still used in some European rail systems, such as in <a href="/wiki/Austria" title="Austria">Austria</a>, <a href="/wiki/Germany" title="Germany">Germany</a>, <a href="/wiki/Norway" title="Norway">Norway</a>, <a href="/wiki/Sweden" title="Sweden">Sweden</a> and <a href="/wiki/Switzerland" title="Switzerland">Switzerland</a>. </p> <div class="mw-heading mw-heading3"><h3 id="High_frequency">High frequency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=4" title="Edit section: High frequency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Off-shore, military, textile industry, marine, aircraft, and spacecraft applications sometimes use 400 Hz, for benefits of reduced weight of apparatus or higher motor speeds. Computer <a href="/wiki/Mainframe_computer" title="Mainframe computer">mainframe</a> systems were often powered by 400 Hz or 415 Hz for benefits of <a href="/wiki/Ripple_(electrical)" title="Ripple (electrical)">ripple</a> reduction while using smaller internal AC to DC conversion units.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (August 2022)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Effects_at_high_frequencies">Effects at high frequencies</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=5" title="Edit section: Effects at high frequencies"><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/Skin_effect" title="Skin effect">Skin effect</a></div> <p>A direct current flows uniformly throughout the cross-section of a homogeneous <a href="/wiki/Electrical_conductivity" class="mw-redirect" title="Electrical conductivity">electrically conducting</a> wire. An alternating current of any frequency is forced away from the wire's center, toward its outer surface. This is because an alternating current (which is the result of the acceleration of <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a>) creates <a href="/wiki/Electromagnetic_waves" class="mw-redirect" title="Electromagnetic waves">electromagnetic waves</a> (a phenomenon known as <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a>). Electric conductors are not conducive to electromagnetic waves (a <a href="/wiki/Perfect_conductor" title="Perfect conductor">perfect electric conductor</a> prohibits all electromagnetic waves within its boundary), so a wire that is made of a non-perfect conductor (a conductor with finite, rather than infinite, electrical conductivity) pushes the alternating current, along with their associated electromagnetic fields, away from the wire's center. The phenomenon of alternating current being pushed away from the center of the conductor is called <a href="/wiki/Skin_effect" title="Skin effect">skin effect</a>, and a direct current does not exhibit this effect, since a direct current does not create electromagnetic waves. </p><p>At very high frequencies, the current no longer flows <i>in</i> the wire, but effectively flows <i>on</i> the surface of the wire, within a thickness of a few <a href="/wiki/Skin_depth" class="mw-redirect" title="Skin depth">skin depths</a>. The skin depth is the thickness at which the current density is reduced by 63%. Even at relatively low frequencies used for power transmission (50 Hz – 60 Hz), non-uniform distribution of current still occurs in sufficiently thick <a href="/wiki/Electrical_conductor" title="Electrical conductor">conductors</a>. For example, the skin depth of a copper conductor is approximately 8.57 mm at 60 Hz, so high current conductors are usually hollow to reduce their mass and cost. This tendency of alternating current to flow predominantly in the periphery of conductors reduces the effective cross-section of the conductor. This increases the effective AC <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a> of the conductor, since resistance is inversely proportional to the cross-sectional area. A conductor's AC resistance is higher than its DC resistance, causing a higher energy loss due to <a href="/wiki/Ohmic_heating" class="mw-redirect" title="Ohmic heating">ohmic heating</a> (also called I<sup>2</sup>R loss). </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Techniques_for_reducing_AC_resistance">Techniques for reducing AC resistance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=6" title="Edit section: Techniques for reducing AC resistance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For low to medium frequencies, conductors can be divided into stranded wires, each insulated from the others, with the relative positions of individual strands specially arranged within the conductor bundle. Wire constructed using this technique is called <a href="/wiki/Litz_wire" title="Litz wire">Litz wire</a>. This measure helps to partially mitigate skin effect by forcing more equal current throughout the total cross section of the stranded conductors. Litz wire is used for making <a href="/wiki/Quality_factor" class="mw-redirect" title="Quality factor">high-Q</a> <a href="/wiki/Inductor" title="Inductor">inductors</a>, reducing losses in flexible conductors carrying very high currents at lower frequencies, and in the windings of devices carrying higher <a href="/wiki/Radio_frequency" title="Radio frequency">radio frequency</a> current (up to hundreds of kilohertz), such as switch-mode <a href="/wiki/Power_supply" title="Power supply">power supplies</a> and <a href="/wiki/Radio_frequency" title="Radio frequency">radio frequency</a> <a href="/wiki/Transformer" title="Transformer">transformers</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Techniques_for_reducing_radiation_loss">Techniques for reducing radiation loss</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=7" title="Edit section: Techniques for reducing radiation loss"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As written above, an alternating current is made of <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> under periodic <a href="/wiki/Acceleration" title="Acceleration">acceleration</a>, which causes <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">radiation</a> of <a href="/wiki/Electromagnetic_waves" class="mw-redirect" title="Electromagnetic waves">electromagnetic waves</a>. Energy that is radiated is lost. Depending on the frequency, different techniques are used to minimize the loss due to radiation. </p> <div class="mw-heading mw-heading4"><h4 id="Twisted_pairs">Twisted pairs</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=8" title="Edit section: Twisted pairs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>At frequencies up to about 1 GHz, pairs of wires are twisted together in a cable, forming a <a href="/wiki/Twisted_pair" title="Twisted pair">twisted pair</a>. This reduces losses from <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a> and <a href="/wiki/Inductive_coupling" title="Inductive coupling">inductive coupling</a>. A twisted pair must be used with a <a href="/wiki/Balanced_line" title="Balanced line">balanced</a> signaling system, so that the two wires carry equal but opposite currents. Each wire in a twisted pair radiates a signal, but it is effectively cancelled by radiation from the other wire, resulting in almost no radiation loss. </p> <div class="mw-heading mw-heading4"><h4 id="Coaxial_cables">Coaxial cables</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=9" title="Edit section: Coaxial cables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Coaxial_cable" title="Coaxial cable">Coaxial cables</a> are commonly used at <a href="/wiki/Audio_frequency" title="Audio frequency">audio frequencies</a> and above for convenience. A coaxial cable has a conductive wire inside a conductive tube, separated by a <a href="/wiki/Dielectric" title="Dielectric">dielectric</a> layer. The current flowing on the surface of the inner conductor is equal and opposite to the current flowing on the inner surface of the outer tube. The electromagnetic field is thus completely contained within the tube, and (ideally) no energy is lost to radiation or coupling outside the tube. Coaxial cables have acceptably small losses for frequencies up to about 5 GHz. For <a href="/wiki/Microwave" title="Microwave">microwave</a> frequencies greater than 5 GHz, the losses (due mainly to the dielectric separating the inner and outer tubes being a non-ideal insulator) become too large, making <a href="/wiki/Waveguide_(electromagnetism)" class="mw-redirect" title="Waveguide (electromagnetism)">waveguides</a> a more efficient medium for transmitting energy. Coaxial cables often use a perforated dielectric layer to separate the inner and outer conductors in order to minimize the power dissipated by the dielectric. </p> <div class="mw-heading mw-heading4"><h4 id="Waveguides">Waveguides</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=10" title="Edit section: Waveguides"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Waveguide_(electromagnetism)" class="mw-redirect" title="Waveguide (electromagnetism)">Waveguides</a> are similar to coaxial cables, as both consist of tubes, with the biggest difference being that waveguides have no inner conductor. Waveguides can have any arbitrary cross section, but rectangular cross sections are the most common. Because waveguides do not have an inner conductor to carry a return current, waveguides cannot deliver energy by means of an <a href="/wiki/Electric_current" title="Electric current">electric current</a>, but rather by means of a <i>guided</i> <a href="/wiki/Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a>. Although <a href="/wiki/Current_density" title="Current density">surface currents</a> do flow on the inner walls of the waveguides, those surface currents do not carry power. Power is carried by the guided electromagnetic fields. The surface currents are set up by the guided electromagnetic fields and have the effect of keeping the fields inside the waveguide and preventing leakage of the fields to the space outside the waveguide. Waveguides have dimensions comparable to the <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> of the alternating current to be transmitted, so they are feasible only at microwave frequencies. In addition to this mechanical feasibility, <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">electrical resistance</a> of the non-ideal metals forming the walls of the waveguide causes <a href="/wiki/Dissipation" title="Dissipation">dissipation</a> of power (surface currents flowing on lossy <a href="/wiki/Electrical_conductor" title="Electrical conductor">conductors</a> dissipate power). At higher frequencies, the power lost to this dissipation becomes unacceptably large. </p> <div class="mw-heading mw-heading4"><h4 id="Fiber_optics">Fiber optics</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=11" title="Edit section: Fiber optics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>At frequencies greater than 200 GHz, waveguide dimensions become impractically small, and the <a href="/wiki/Ohmic_heating" class="mw-redirect" title="Ohmic heating">ohmic losses</a> in the waveguide walls become large. Instead, <a href="/wiki/Fibre_optics" class="mw-redirect" title="Fibre optics">fiber optics</a>, which are a form of dielectric waveguides, can be used. For such frequencies, the concepts of voltages and currents are no longer used.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (October 2024)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading2"><h2 id="Formulation">Formulation<span class="anchor" id="Mathematics"></span><span class="anchor" id="Mathematics_of_AC_voltages"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=12" title="Edit section: Formulation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Sine_voltage.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Sine_voltage.svg/220px-Sine_voltage.svg.png" decoding="async" width="220" height="229" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Sine_voltage.svg/330px-Sine_voltage.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Sine_voltage.svg/440px-Sine_voltage.svg.png 2x" data-file-width="816" data-file-height="850" /></a><figcaption>A sinusoidal alternating voltage.<div><ol><li>Peak,</li><li>Peak-to-peak amplitude,</li><li>Effective value,</li><li>Period</li></ol></div></figcaption></figure> <p>Alternating currents are accompanied (or caused) by alternating voltages. An AC voltage <i>v</i> can be described mathematically as a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of time by the following equation: </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 v(t)=V_{\text{peak}}\sin(\omega 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> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v(t)=V_{\text{peak}}\sin(\omega t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c05731005b2d245f908da771c57de4da5b89bcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.134ex; height:3.009ex;" alt="{\displaystyle v(t)=V_{\text{peak}}\sin(\omega t)}"></span>,</dd></dl> <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 V_{\text{peak}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{peak}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57804b7a3ccbc7cf3a67e709854cdd7919aab732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.921ex; height:2.843ex;" alt="{\displaystyle V_{\text{peak}}}"></span> is the peak voltage (unit: <a href="/wiki/Volt" title="Volt">volt</a>),</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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> is the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> (unit: <a href="/wiki/Radians_per_second" class="mw-redirect" title="Radians per second">radians per second</a>). <div class="paragraphbreak" style="margin-top:0.5em"></div>The angular frequency is related to the physical frequency, <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> (unit: <a href="/wiki/Hertz" title="Hertz">hertz</a>), which represents the number of cycles per second, by the equation <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>.</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 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> is the time (unit: <a href="/wiki/Second" title="Second">second</a>).</li></ul> <p>The peak-to-peak value of an AC voltage is defined as the difference between its positive peak and its negative peak. Since the maximum value 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 \sin(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a990a5545cac26c1c6821dca95d898bc80fe3a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.995ex; height:2.843ex;" alt="{\displaystyle \sin(x)}"></span> is +1 and the minimum value is −1, an AC voltage swings between <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{peak}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +V_{\text{peak}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de3811465bc7eedda488bc20e052c6734d4a47d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.73ex; height:2.843ex;" alt="{\displaystyle +V_{\text{peak}}}"></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 -V_{\text{peak}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -V_{\text{peak}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2094fdab248b0420bbdd07e30eb603aec5bc45a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.73ex; height:2.843ex;" alt="{\displaystyle -V_{\text{peak}}}"></span>. The peak-to-peak voltage, usually written as <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{pp}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>pp</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{pp}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0adf4eef1890e2ae65472447581c6e57464a6b9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.415ex; height:2.843ex;" alt="{\displaystyle V_{\text{pp}}}"></span> or <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-P}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>P-P</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{P-P}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c980c2388818f19bfc8f4a7c16fd38ecf870584" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.374ex; height:2.509ex;" alt="{\displaystyle V_{\text{P-P}}}"></span>, is therefore <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{peak}}-(-V_{\text{peak}})=2V_{\text{peak}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{peak}}-(-V_{\text{peak}})=2V_{\text{peak}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f592666e37d1b4e6f0c516d24bbb8dc26726eb3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.483ex; height:3.009ex;" alt="{\displaystyle V_{\text{peak}}-(-V_{\text{peak}})=2V_{\text{peak}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Root_mean_square_voltage">Root mean square voltage<span class="anchor" id="Root_mean_square"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=13" title="Edit section: Root mean square voltage"><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/RMS_amplitude" class="mw-redirect" title="RMS amplitude">RMS amplitude</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">For broader coverage of this topic, see <a href="/wiki/Root_mean_square_voltage" class="mw-redirect" title="Root mean square voltage">Root mean square voltage</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sine_wave_2.svg" class="mw-file-description"><img alt="A graph of sin(x) with a dashed line at y=sin(45)" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Sine_wave_2.svg/220px-Sine_wave_2.svg.png" decoding="async" width="220" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Sine_wave_2.svg/330px-Sine_wave_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Sine_wave_2.svg/440px-Sine_wave_2.svg.png 2x" data-file-width="520" data-file-height="332" /></a><figcaption>A sine wave, over one cycle (360°). The dashed line represents the <a href="/wiki/Root_mean_square" title="Root mean square">root mean square</a> (RMS) value at <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 {\sqrt {0.5}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>0.5</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {0.5}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/906816e723e5f427bd12c033f1b9087eeca7e6d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.908ex; height:2.843ex;" alt="{\displaystyle {\sqrt {0.5}}}"></span> (about 0.707).</figcaption></figure> <p>Below an AC waveform (with no <a href="/wiki/DC_component" class="mw-redirect" title="DC component">DC component</a>) is assumed. </p><p>The RMS voltage is the square root of the <a href="/wiki/Mean_of_a_function" title="Mean of a function">mean</a> over one cycle of the square of the instantaneous voltage. </p> <div><ul><li>For an arbitrary periodic waveform <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> of period <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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>: <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 V_{\text{rms}}={\sqrt {{\frac {1}{T}}\int _{0}^{T}{[v(t)]^{2}dt}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rms</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mrow> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{rms}}={\sqrt {{\frac {1}{T}}\int _{0}^{T}{[v(t)]^{2}dt}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef708d0af8135ef76f3d2398403e698f8e7b98ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.598ex; height:7.509ex;" alt="{\displaystyle V_{\text{rms}}={\sqrt {{\frac {1}{T}}\int _{0}^{T}{[v(t)]^{2}dt}}}.}"></span></dd></dl></li><li>For a sinusoidal voltage: <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}V_{\text{rms}}&={\sqrt {{\frac {1}{T}}\int _{0}^{T}[{V_{\text{peak}}\sin(\omega t+\phi )]^{2}dt}}}\\&=V_{\text{peak}}{\sqrt {{\frac {1}{2T}}\int _{0}^{T}[{1-\cos(2\omega t+2\phi )]dt}}}\\&=V_{\text{peak}}{\sqrt {{\frac {1}{2T}}\int _{0}^{T}{dt}}}\\&={\frac {V_{\text{peak}}}{\sqrt {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="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rms</mtext> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mrow> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>T</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mn>2</mn> <mi>ϕ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mi>d</mi> <mi>t</mi> </mrow> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>T</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>t</mi> </mrow> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V_{\text{rms}}&={\sqrt {{\frac {1}{T}}\int _{0}^{T}[{V_{\text{peak}}\sin(\omega t+\phi )]^{2}dt}}}\\&=V_{\text{peak}}{\sqrt {{\frac {1}{2T}}\int _{0}^{T}[{1-\cos(2\omega t+2\phi )]dt}}}\\&=V_{\text{peak}}{\sqrt {{\frac {1}{2T}}\int _{0}^{T}{dt}}}\\&={\frac {V_{\text{peak}}}{\sqrt {2}}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d702e731c7dae922caedc8378d246e417f156f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.338ex; width:44.328ex; height:29.843ex;" alt="{\displaystyle {\begin{aligned}V_{\text{rms}}&={\sqrt {{\frac {1}{T}}\int _{0}^{T}[{V_{\text{peak}}\sin(\omega t+\phi )]^{2}dt}}}\\&=V_{\text{peak}}{\sqrt {{\frac {1}{2T}}\int _{0}^{T}[{1-\cos(2\omega t+2\phi )]dt}}}\\&=V_{\text{peak}}{\sqrt {{\frac {1}{2T}}\int _{0}^{T}{dt}}}\\&={\frac {V_{\text{peak}}}{\sqrt {2}}}\end{aligned}}}"></span></dd> where the <a href="/wiki/Trigonometric_identity" class="mw-redirect" title="Trigonometric identity">trigonometric identity</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 \sin ^{2}(x)={\frac {1-\cos(2x)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sin ^{2}(x)={\frac {1-\cos(2x)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2babf93bbda4c46a82cf91fa319b7ee98b5ef77b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.399ex; height:5.676ex;" alt="{\displaystyle \sin ^{2}(x)={\frac {1-\cos(2x)}{2}}}"></span> has been used and the factor <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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}"></span> is called the <a href="/wiki/Crest_factor" title="Crest factor">crest factor</a>, which varies for different waveforms.</li><li>For a <a href="/wiki/Triangle_wave" title="Triangle wave">triangle waveform</a> centered about zero <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 V_{\text{rms}}={\frac {V_{\text{peak}}}{\sqrt {3}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rms</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{rms}}={\frac {V_{\text{peak}}}{\sqrt {3}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/393328da39ebe29095e07a03c32dcbbb30f6c9f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:13.752ex; height:6.509ex;" alt="{\displaystyle V_{\text{rms}}={\frac {V_{\text{peak}}}{\sqrt {3}}}.}"></span></dd></dl></li><li>For a <a href="/wiki/Square_wave" title="Square wave">square waveform</a> centered about zero <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 V_{\text{rms}}=V_{\text{peak}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rms</mtext> </mrow> </msub> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{rms}}=V_{\text{peak}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34a5df4cb1a345c1e8b1f2d3667323af9bcc2925" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.916ex; height:2.843ex;" alt="{\displaystyle V_{\text{rms}}=V_{\text{peak}}.}"></span></dd></li></ul></div> <div class="mw-heading mw-heading3"><h3 id="Power">Power</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=14" title="Edit section: Power"><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/AC_power" title="AC power">AC power</a></div> <p>The relationship between voltage and the power delivered is: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(t)={\frac {v^{2}(t)}{R}}}"> <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> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>R</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(t)={\frac {v^{2}(t)}{R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f16c2878c3406e79780c1e537a42025577028bba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:12.673ex; height:6.009ex;" alt="{\displaystyle p(t)={\frac {v^{2}(t)}{R}}}"></span>,</dd></dl> <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 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/4b0bfb3769bf24d80e15374dc37b0441e2616e33" 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 R}"></span> represents a load resistance. </p><p>Rather than using instantaneous 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>, it is more practical to use a time-averaged power (where the averaging is performed over any integer number of cycles). Therefore, AC voltage is often expressed as a <a href="/wiki/Root_mean_square" title="Root mean square">root mean square</a> (RMS) value, written as <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{rms}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rms</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{rms}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af4009701e886869febe6552f9a97b08cdc08066" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.249ex; height:2.509ex;" alt="{\displaystyle V_{\text{rms}}}"></span>, because </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 P_{\text{average}}={\frac {{V_{\text{rms}}}^{2}}{R}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>average</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rms</mtext> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{average}}={\frac {{V_{\text{rms}}}^{2}}{R}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45882396ab9fbb65d1e248e685b47c90b913353d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.048ex; height:5.843ex;" alt="{\displaystyle P_{\text{average}}={\frac {{V_{\text{rms}}}^{2}}{R}}.}"></span></dd></dl> <dl><dt>Power oscillation</dt> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}v(t)&=V_{\text{peak}}\sin(\omega t)\\i(t)&={\frac {v(t)}{R}}={\frac {V_{\text{peak}}}{R}}\sin(\omega t)\\p(t)&=v(t)i(t)={\frac {(V_{\text{peak}})^{2}}{R}}\sin ^{2}(\omega t)={\frac {(V_{\text{peak}})^{2}}{2R}}\ (1-\cos(2\omega t))\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> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>i</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>R</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mi>R</mi> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>p</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>i</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>R</mi> </mfrac> </mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>R</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}v(t)&=V_{\text{peak}}\sin(\omega t)\\i(t)&={\frac {v(t)}{R}}={\frac {V_{\text{peak}}}{R}}\sin(\omega t)\\p(t)&=v(t)i(t)={\frac {(V_{\text{peak}})^{2}}{R}}\sin ^{2}(\omega t)={\frac {(V_{\text{peak}})^{2}}{2R}}\ (1-\cos(2\omega t))\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/970bc0c19b0ab8da811293b6fcdcc7f755ac6eac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:61.488ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}v(t)&=V_{\text{peak}}\sin(\omega t)\\i(t)&={\frac {v(t)}{R}}={\frac {V_{\text{peak}}}{R}}\sin(\omega t)\\p(t)&=v(t)i(t)={\frac {(V_{\text{peak}})^{2}}{R}}\sin ^{2}(\omega t)={\frac {(V_{\text{peak}})^{2}}{2R}}\ (1-\cos(2\omega t))\end{aligned}}}"></span></dd></dl> <p>For this reason, AC power's waveform becomes <a href="/wiki/Rectifier#Full-wave_rectification" title="Rectifier">Full-wave rectified</a> sine, and its fundamental frequency is double of the one of the voltage's. </p><p><br /> </p> <div class="mw-heading mw-heading3"><h3 id="Examples_of_alternating_current">Examples of alternating current</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=15" title="Edit section: Examples of alternating current"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To illustrate these concepts, consider a 230 V AC <a href="/wiki/Mains_power_systems" class="mw-redirect" title="Mains power systems">mains</a> supply used in <a href="/wiki/Mains_power_systems#Table_of_mains_voltages,_frequencies,_and_plugs" class="mw-redirect" title="Mains power systems">many countries</a> around the world. It is so called because its <a href="/wiki/Root_mean_square" title="Root mean square">root mean square</a> value is 230 V. This means that the time-averaged power delivered <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_{\text{average}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>average</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{average}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e36ebffc9c21b32af4d29e954ad6c146290ea14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.163ex; height:2.843ex;" alt="{\displaystyle P_{\text{average}}}"></span> is equivalent to the power delivered by a DC voltage of 230 V. To determine the peak voltage (amplitude), we can rearrange the above equation 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 V_{\text{peak}}={\sqrt {2}}\ V_{\text{rms}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mtext> </mtext> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rms</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{peak}}={\sqrt {2}}\ V_{\text{rms}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27beea29b4e53ae71e956ef7b0cc8161bd2fd9cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.948ex; height:3.343ex;" alt="{\displaystyle V_{\text{peak}}={\sqrt {2}}\ V_{\text{rms}}}"></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 P_{\text{peak}}={\frac {(V_{\text{rms}})^{2}}{R}}{\frac {(V_{\text{peak}})^{2}}{(V_{\text{rms}})^{2}}}={\text{P}}_{\text{average}}{\sqrt {2}}^{2}={\text{2}}P_{\text{average}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rms</mtext> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>R</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rms</mtext> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>P</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>average</mtext> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>2</mtext> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>average</mtext> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{peak}}={\frac {(V_{\text{rms}})^{2}}{R}}{\frac {(V_{\text{peak}})^{2}}{(V_{\text{rms}})^{2}}}={\text{P}}_{\text{average}}{\sqrt {2}}^{2}={\text{2}}P_{\text{average}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d9eb63ac85af7b13059dba1d83d43d778ecffff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.302ex; height:6.676ex;" alt="{\displaystyle P_{\text{peak}}={\frac {(V_{\text{rms}})^{2}}{R}}{\frac {(V_{\text{peak}})^{2}}{(V_{\text{rms}})^{2}}}={\text{P}}_{\text{average}}{\sqrt {2}}^{2}={\text{2}}P_{\text{average}}.}"></span></dd></dl> <p>For 230 V AC, the peak 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_{\text{peak}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\text{peak}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57804b7a3ccbc7cf3a67e709854cdd7919aab732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.921ex; height:2.843ex;" alt="{\displaystyle V_{\text{peak}}}"></span> is therefore <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 230{\text{ V}}\times {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>230</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> V</mtext> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 230{\text{ V}}\times {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a62b346a4d3e2423775bceec63970163ac5e7f59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.75ex; height:3.009ex;" alt="{\displaystyle 230{\text{ V}}\times {\sqrt {2}}}"></span>, which is about 325 V, and the peak 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_{\text{peak}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>peak</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{peak}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13fd01a515155de886a12fef814897c6766a7a58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.058ex; height:2.843ex;" alt="{\displaystyle P_{\text{peak}}}"></span> 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 230\times R\times W\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>230</mn> <mo>×</mo> <mi>R</mi> <mo>×</mo> <mi>W</mi> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 230\times R\times W\times 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d916ce83cdc4853d4b343368e019dc5754321a69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.37ex; height:2.176ex;" alt="{\displaystyle 230\times R\times W\times 2}"></span>, that is 460 RW. During the course of one cycle (two cycle as the power) the voltage rises from zero to 325 V, the power from zero to 460 RW, and both falls through zero. Next, the voltage descends to reverse direction, -325 V, but the power ascends again to 460 RW, and both returns to zero. </p> <div class="mw-heading mw-heading2"><h2 id="Information_transmission">Information transmission</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=16" title="Edit section: Information transmission"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Alternating current is used to transmit <a href="/wiki/Information" title="Information">information</a>, as in the cases of <a href="/wiki/Telephone" title="Telephone">telephone</a> and <a href="/wiki/Cable_television" title="Cable television">cable television</a>. Information signals are carried over a wide range of AC frequencies. <a href="/wiki/Plain_old_telephone_service" title="Plain old telephone service">POTS</a> telephone signals have a frequency of about 3 kHz, close to the <a href="/wiki/Baseband" title="Baseband">baseband</a> audio frequency. Cable television and other cable-transmitted information currents may alternate at frequencies of tens to thousands of megahertz. These frequencies are similar to the electromagnetic wave frequencies often used to transmit the same types of information <a href="/wiki/Wireless" title="Wireless">over the air</a>. </p> <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=Alternating_current&action=edit&section=17" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first <a href="/wiki/Alternator" title="Alternator">alternator</a> to produce alternating current was an electric generator based on <a href="/wiki/Michael_Faraday" title="Michael Faraday">Michael Faraday</a>'s principles constructed by the French instrument maker <a href="/wiki/Hippolyte_Pixii" title="Hippolyte Pixii">Hippolyte Pixii</a> in 1832.<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> Pixii later added a <a href="/wiki/Commutator_(electric)" title="Commutator (electric)">commutator</a> to his device to produce the (then) more commonly used direct current. The earliest recorded practical application of alternating current is probably by <a href="/wiki/Guillaume_Duchenne" class="mw-redirect" title="Guillaume Duchenne">Guillaume Duchenne</a>, inventor and developer of <a href="/wiki/Electrotherapy" title="Electrotherapy">electrotherapy</a>. In 1855, he announced that AC was superior to <a href="/wiki/Direct_current" title="Direct current">direct current</a> for electrotherapeutic triggering of muscle contractions.<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> Alternating current technology was developed further by the Hungarian <a href="/wiki/Ganz_Works" title="Ganz Works">Ganz Works</a> company (1870s), and in the 1880s: <a href="/wiki/Sebastian_Ziani_de_Ferranti" title="Sebastian Ziani de Ferranti">Sebastian Ziani de Ferranti</a>, <a href="/wiki/Lucien_Gaulard" title="Lucien Gaulard">Lucien Gaulard</a>, and <a href="/wiki/Galileo_Ferraris" title="Galileo Ferraris">Galileo Ferraris</a>. </p><p>In 1876, Russian engineer <a href="/wiki/Pavel_Yablochkov" title="Pavel Yablochkov">Pavel Yablochkov</a> invented a lighting system where sets of induction coils were installed along a high voltage AC line. Instead of changing voltage, the primary windings transferred power to the secondary windings which were connected to one or several <a href="/wiki/Yablochkov_candle" title="Yablochkov candle">'electric candles'</a> (arc lamps) of his own design,<sup id="cite_ref-maglab_5-0" class="reference"><a href="#cite_note-maglab-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><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> used to keep the failure of one lamp from disabling the entire circuit.<sup id="cite_ref-maglab_5-1" class="reference"><a href="#cite_note-maglab-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In 1878, the <a href="/wiki/Ganz_Works" title="Ganz Works">Ganz factory</a>, Budapest, Hungary, began manufacturing equipment for electric lighting and, by 1883, had installed over fifty systems in <a href="/wiki/Austria-Hungary" title="Austria-Hungary">Austria-Hungary</a>. Their AC systems used arc and incandescent lamps, generators, and other equipment.<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> <div class="mw-heading mw-heading3"><h3 id="Transformers">Transformers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=18" title="Edit section: Transformers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Alternating current systems can use <a href="/wiki/Transformer" title="Transformer">transformers</a> to change voltage from low to high level and back, allowing generation and consumption at low voltages but transmission, possibly over great distances, at high voltage, with savings in the cost of conductors and energy losses. A bipolar open-core <a href="/wiki/Transformer" title="Transformer">power transformer</a> developed by <a href="/wiki/Lucien_Gaulard" title="Lucien Gaulard">Lucien Gaulard</a> and <a href="/wiki/John_Dixon_Gibbs" title="John Dixon Gibbs">John Dixon Gibbs</a> was demonstrated in London in 1881, and attracted the interest of <a href="/wiki/Westinghouse_Electric_(1886)" class="mw-redirect" title="Westinghouse Electric (1886)">Westinghouse</a>. They also exhibited the invention in <a href="/wiki/Turin" title="Turin">Turin</a> in 1884. However, these early induction coils with open magnetic circuits are inefficient at transferring power to <a href="/wiki/Electrical_load" title="Electrical load">loads</a>. Until about 1880, the paradigm for AC power transmission from a high voltage supply to a low voltage load was a series circuit. Open-core transformers with a ratio near 1:1 were connected with their primaries in series to allow use of a high voltage for transmission while presenting a low voltage to the lamps. The inherent flaw in this method was that turning off a single lamp (or other electric device) affected the voltage supplied to all others on the same circuit. Many adjustable transformer designs were introduced to compensate for this problematic characteristic of the series circuit, including those employing methods of adjusting the core or bypassing the magnetic flux around part of a coil.<sup id="cite_ref-FJU1889_8-0" class="reference"><a href="#cite_note-FJU1889-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The direct current systems did not have these drawbacks, giving it significant advantages over early AC systems. </p><p>In the UK, <a href="/wiki/Sebastian_Ziani_de_Ferranti" title="Sebastian Ziani de Ferranti">Sebastian de Ferranti</a>, who had been developing AC generators and transformers in London since 1882, redesigned the AC system at the <a href="/wiki/Grosvenor_Gallery#Generating_station" title="Grosvenor Gallery">Grosvenor Gallery power station</a> in 1886 for the London Electric Supply Corporation (LESCo) including alternators of his own design and open core transformer designs with serial connections for utilization loads - similar to Gaulard and Gibbs.<sup id="cite_ref-FOOTNOTEHughes199398_9-0" class="reference"><a href="#cite_note-FOOTNOTEHughes199398-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> In 1890, he designed <a href="/wiki/Deptford_Power_Station" title="Deptford Power Station">their power station at Deptford</a><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> and converted the Grosvenor Gallery station across the Thames into an <a href="/wiki/Electrical_substation" class="mw-redirect" title="Electrical substation">electrical substation</a>, showing the way to integrate older plants into a universal AC supply system.<sup id="cite_ref-FOOTNOTEHughes1993208_11-0" class="reference"><a href="#cite_note-FOOTNOTEHughes1993208-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Pioneers">Pioneers</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=19" title="Edit section: Pioneers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:ZBD_team.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/ZBD_team.jpg/220px-ZBD_team.jpg" decoding="async" width="220" height="135" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/ZBD_team.jpg/330px-ZBD_team.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/ZBD_team.jpg/440px-ZBD_team.jpg 2x" data-file-width="800" data-file-height="491" /></a><figcaption>The Hungarian "ZBD" Team (<a href="/wiki/K%C3%A1roly_Zipernowsky" title="Károly Zipernowsky">Károly Zipernowsky</a>, <a href="/wiki/Ott%C3%B3_Bl%C3%A1thy" title="Ottó Bláthy">Ottó Bláthy</a>, <a href="/wiki/Miksa_D%C3%A9ri" title="Miksa Déri">Miksa Déri</a>), inventors of the first high efficiency, closed-core shunt connection <a href="/wiki/Transformer" title="Transformer">transformer</a></figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:DBZ_trafo.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/DBZ_trafo.jpg/220px-DBZ_trafo.jpg" decoding="async" width="220" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/DBZ_trafo.jpg/330px-DBZ_trafo.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/DBZ_trafo.jpg/440px-DBZ_trafo.jpg 2x" data-file-width="1712" data-file-height="1368" /></a><figcaption>The prototype of the ZBD transformer on display at the Széchenyi István Memorial Exhibition, <a href="/wiki/Nagycenk" title="Nagycenk">Nagycenk</a> in <a href="/wiki/Hungary" title="Hungary">Hungary</a></figcaption></figure> <p>In the autumn<sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify"><span title="The text near this tag is ambiguous, and needs clarification. (January 2023)">ambiguous</span></a></i>]</sup> of 1884, <a href="/wiki/K%C3%A1roly_Zipernowsky" title="Károly Zipernowsky">Károly Zipernowsky</a>, <a href="/wiki/Ott%C3%B3_Bl%C3%A1thy" title="Ottó Bláthy">Ottó Bláthy</a> and <a href="/wiki/Miksa_D%C3%A9ri" title="Miksa Déri">Miksa Déri</a> (ZBD), three engineers associated with the <a href="/wiki/Ganz_Works" title="Ganz Works">Ganz Works</a> of Budapest, determined that open-core devices were impractical, as they were incapable of reliably regulating voltage.<sup id="cite_ref-FOOTNOTEHughes199395_12-0" class="reference"><a href="#cite_note-FOOTNOTEHughes199395-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Bláthy had suggested the use of closed cores, Zipernowsky had suggested the use of <a href="/wiki/Shunt_(electrical)" title="Shunt (electrical)">parallel shunt connections</a>, and Déri had performed the experiments;<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> In their joint 1885 patent applications for novel transformers (later called ZBD transformers), they described two designs with closed magnetic circuits where copper windings were either wound around a ring core of iron wires or else surrounded by a core of iron wires.<sup id="cite_ref-FJU1889_8-1" class="reference"><a href="#cite_note-FJU1889-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> In both designs, the magnetic flux linking the primary and secondary windings traveled almost entirely within the confines of the iron core, with no intentional path through air (see <a href="/wiki/Transformer#Toroidal_cores" title="Transformer">toroidal cores</a>). The new transformers were 3.4 times more efficient than the open-core bipolar devices of Gaulard and Gibbs.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> The Ganz factory in 1884 shipped the world's first five high-efficiency AC transformers.<sup id="cite_ref-Halacsy_(1961)_15-0" class="reference"><a href="#cite_note-Halacsy_(1961)-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> This first unit had been manufactured to the following specifications: 1,400 W, 40 Hz, 120:72 V, 11.6:19.4 A, ratio 1.67:1, one-phase, shell form.<sup id="cite_ref-Halacsy_(1961)_15-1" class="reference"><a href="#cite_note-Halacsy_(1961)-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>The ZBD patents included two other major interrelated innovations: one concerning the use of parallel connected, instead of series connected, utilization loads, the other concerning the ability to have high turns ratio transformers such that the supply network voltage could be much higher (initially 1400 V to 2000 V) than the voltage of utilization loads (100 V initially preferred).<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> When employed in parallel connected electric distribution systems, closed-core transformers finally made it technically and economically feasible to provide electric power for lighting in homes, businesses and public spaces.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> The other essential milestone was the introduction of 'voltage source, voltage intensive' (VSVI) systems'<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> by the invention of constant voltage generators in 1885.<sup id="cite_ref-FOOTNOTEHughes199396_21-0" class="reference"><a href="#cite_note-FOOTNOTEHughes199396-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> In early 1885, the three engineers also eliminated the problem of <a href="/wiki/Eddy_current" title="Eddy current">eddy current</a> losses with the invention of the lamination of electromagnetic cores.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> Ottó Bláthy also invented the first AC <a href="/wiki/Electricity_meter" title="Electricity meter">electricity meter</a>.<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><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><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>The AC power system was developed and adopted rapidly after 1886 due to its ability to distribute electricity efficiently over long distances, overcoming the limitations of the <a href="/wiki/Direct_current" title="Direct current">direct current</a> system. In 1886, the ZBD engineers designed the world's first <a href="/wiki/Power_station" title="Power station">power station</a> that used AC generators to power a parallel-connected common electrical network, the steam-powered Rome-Cerchi power plant.<sup id="cite_ref-IEC_Techline_27-0" class="reference"><a href="#cite_note-IEC_Techline-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> The reliability of the AC technology received impetus after the Ganz Works electrified a large European metropolis: <a href="/wiki/Rome" title="Rome">Rome</a> in 1886.<sup id="cite_ref-IEC_Techline_27-1" class="reference"><a href="#cite_note-IEC_Techline-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:WestinghouseEarlyACSystem1887-USP373035.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/WestinghouseEarlyACSystem1887-USP373035.png/220px-WestinghouseEarlyACSystem1887-USP373035.png" decoding="async" width="220" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/WestinghouseEarlyACSystem1887-USP373035.png/330px-WestinghouseEarlyACSystem1887-USP373035.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3b/WestinghouseEarlyACSystem1887-USP373035.png/440px-WestinghouseEarlyACSystem1887-USP373035.png 2x" data-file-width="840" data-file-height="618" /></a><figcaption>Westinghouse Early AC System 1887<br /> (<a rel="nofollow" class="external text" href="https://web.archive.org/web/20090325121254/http://www.pat2pdf.org/patents/pat373035.pdf">US patent 373035</a>)</figcaption></figure> <p>Building on the advancement of AC technology in Europe,<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> <a href="/wiki/George_Westinghouse" title="George Westinghouse">George Westinghouse</a> founded the <a href="/wiki/Westinghouse_Electric_Corporation" title="Westinghouse Electric Corporation">Westinghouse Electric</a> in Pittsburgh, Pennsylvania, on January 8, 1886.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> The new firm became active in developing alternating current (AC) electric infrastructure throughout the United States. The <a href="/wiki/Edison_Electric_Light_Company" class="mw-redirect" title="Edison Electric Light Company">Edison Electric Light Company</a> held an option on the US rights for the Ganz ZBD transformers, requiring Westinghouse to pursue alternative designs on the same principles. George Westinghouse had bought Gaulard and Gibbs' patents for $50,000 in February 1886.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> He assigned to <a href="/wiki/William_Stanley,_Jr." class="mw-redirect" title="William Stanley, Jr.">William Stanley</a> the task of redesigning the Gaulard and Gibbs transformer for commercial use in United States.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> On March 20, 1886, Stanley conducted a demonstrative experiment in <a href="/wiki/Great_Barrington,_Massachusetts" title="Great Barrington, Massachusetts">Great Barrington</a>: A <a href="/wiki/Siemens" title="Siemens">Siemens</a> generator's voltage of 500 volts was converted into 3000 volts, and then the voltage was stepped down to 500 volts by six Westinghouse transformers. With this setup, the Westinghouse company successfully powered thirty 100-volt incandescent bulbs in twenty shops along the main street of Great Barrington.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> The spread of Westinghouse and other AC systems triggered a push back in late 1887 by <a href="/wiki/Thomas_Edison" title="Thomas Edison">Thomas Edison</a> (a proponent of direct current), who attempted to discredit alternating current as too dangerous in a public campaign called the "<a href="/wiki/War_of_the_currents" title="War of the currents">war of the currents</a>". In 1888, alternating current systems gained further viability with introduction of a functional <a href="/wiki/AC_motor" title="AC motor">AC motor</a>, something these systems had lacked up till then. The design, an <a href="/wiki/Induction_motor" title="Induction motor">induction motor</a>, was independently invented by <a href="/wiki/Galileo_Ferraris" title="Galileo Ferraris">Galileo Ferraris</a> and <a href="/wiki/Nikola_Tesla" title="Nikola Tesla">Nikola Tesla</a> (with Tesla's design being licensed by Westinghouse in the US). This design was independently further developed into the modern practical <a href="/wiki/Three-phase" class="mw-redirect" title="Three-phase">three-phase</a> form by <a href="/wiki/Mikhail_Dolivo-Dobrovolsky" title="Mikhail Dolivo-Dobrovolsky">Mikhail Dolivo-Dobrovolsky</a> and <a href="/wiki/Charles_Eugene_Lancelot_Brown" title="Charles Eugene Lancelot Brown">Charles Eugene Lancelot Brown</a> in Germany on one side,<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Jonas_Wenstr%C3%B6m" title="Jonas Wenström">Jonas Wenström</a> in Sweden on the other, though Brown favoured the two-phase system. </p><p>A long-distance alternating current transmission was used in 1883 for the Metropolitan Railway station lighting in <a href="/wiki/London" title="London">London</a>, while the single-phase 1884 system in <a href="/wiki/Turin" title="Turin">Turin</a>, Italy, was the first multiple-user AC distribution system in the world.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Ames_Hydroelectric_Generating_Plant" title="Ames Hydroelectric Generating Plant">Ames Hydroelectric Generating Plant</a>, constructed in 1890, was among the first hydroelectric alternating current power plants. A long distance transmission of single-phase electricity from a hydroelectric generating plant in Oregon at Willamette Falls sent power fourteen miles downriver to downtown Portland for street lighting in 1890.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> In 1891, another transmission system was installed in Telluride Colorado.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> The first <a href="/wiki/Three-phase_electric_power" title="Three-phase electric power">three-phase system</a> was established in 1891 in <a href="/wiki/Frankfurt" title="Frankfurt">Frankfurt</a>, Germany. The <a href="/wiki/Tivoli,_Lazio" title="Tivoli, Lazio">Tivoli</a>–<a href="/wiki/Rome" title="Rome">Rome</a> transmission was completed in 1892.<sup id="cite_ref-Holjevac_37-0" class="reference"><a href="#cite_note-Holjevac-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> The San Antonio Canyon Generator was the third commercial single-phase hydroelectric AC power plant in the United States to provide long-distance electricity. It was completed on December 31, 1892, by <a href="/wiki/Almarian_Decker" title="Almarian Decker">Almarian William Decker</a> to provide power to the city of <a href="/wiki/Pomona,_California" title="Pomona, California">Pomona, California</a>, which was 14 miles away. Meanwhile, the possibility of transferring electrical power from a waterfall at a distance was explored at the <a href="/wiki/Gr%C3%A4ngesberg" title="Grängesberg">Grängesberg</a> mine in Sweden. A <span class="nowrap"><span data-sort-value="7001450000000000000♠"></span>45 <a href="/wiki/Metre" title="Metre">m</a></span> fall at Hällsjön, Smedjebackens kommun, where a small iron work had been located, was selected. In 1893, a three-phase <span class="nowrap"><span data-sort-value="7000950000000000000♠"></span>9.5 <a href="/wiki/Kilovolt" class="mw-redirect" title="Kilovolt">kv</a></span> system was used to transfer 400 <a href="/wiki/Horsepower" title="Horsepower">horsepower</a> a distance of <span class="nowrap"><span data-sort-value="7004150000000000000♠"></span>15 <a href="/wiki/Kilometre" title="Kilometre">km</a></span>, becoming the first commercial application.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup>In 1893, Westinghouse built an alternating current system for the <a href="/wiki/Chicago_World_Exposition" class="mw-redirect" title="Chicago World Exposition">Chicago World Exposition</a>.<sup id="cite_ref-Holjevac_37-1" class="reference"><a href="#cite_note-Holjevac-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> In 1893, Decker designed the first American commercial <a href="/wiki/Three-phase" class="mw-redirect" title="Three-phase">three-phase</a> power plant using alternating current—the hydroelectric <a href="/wiki/Mill_Creek_No._1_Hydroelectric_Plant" class="mw-redirect" title="Mill Creek No. 1 Hydroelectric Plant">Mill Creek No. 1 Hydroelectric Plant</a> near <a href="/wiki/Redlands,_California" title="Redlands, California">Redlands, California</a>. Decker's design incorporated 10 kV three-phase transmission and established the standards for the complete system of generation, transmission and motors used in USA today. The original Niagara Falls <a href="/wiki/Adams_Power_Plant" class="mw-redirect" title="Adams Power Plant">Adams Power Plant</a> with three two-phase generators was put into operation in August 1895, but was connected to the remote transmission system only in 1896. The <a href="/wiki/Jaruga_Hydroelectric_Power_Plant" title="Jaruga Hydroelectric Power Plant">Jaruga Hydroelectric Power Plant</a> in Croatia was set in operation two days later, on 28 August 1895. Its <a href="/wiki/Electric_generator" title="Electric generator">generator</a> (42 Hz, 240 kW) was made and installed by the Hungarian company <a href="/wiki/Ganz" class="mw-redirect" title="Ganz">Ganz</a>, while the transmission line from the power plant to the City of <a href="/wiki/%C5%A0ibenik" title="Šibenik">Šibenik</a> was 11.5 kilometers (7.1 mi) long, and the municipal distribution grid 3000 V/110 V included six transforming stations.<sup id="cite_ref-Holjevac_37-2" class="reference"><a href="#cite_note-Holjevac-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> </p><p>Alternating current circuit theory developed rapidly in the latter part of the 19th and early 20th century. Notable contributors to the theoretical basis of alternating current calculations include <a href="/wiki/Charles_Steinmetz" class="mw-redirect" title="Charles Steinmetz">Charles Steinmetz</a>, <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a>, and many others.<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> Calculations in unbalanced three-phase systems were simplified by the <a href="/wiki/Symmetrical_components" title="Symmetrical components">symmetrical components</a> methods discussed by <a href="/wiki/Charles_LeGeyt_Fortescue" title="Charles LeGeyt Fortescue">Charles LeGeyt Fortescue</a> in 1918. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span 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title="Utility frequency">Utility frequency</a></li> <li><a href="/wiki/War_of_the_currents" title="War of the currents">War of the currents</a></li> <li><a href="/wiki/AC/DC_receiver_design" title="AC/DC receiver design">AC/DC receiver design</a></li></ul> </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=Alternating_current&action=edit&section=21" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite 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Kuzle, Igor (2019). <a rel="nofollow" class="external text" href="https://hrcak.srce.hr/238713">"Prvi cjeloviti višefazni elektroenergetski sustav na svijetu – Krka Šibenik"</a>. <i>Godišnjak Akademije tehničkih znanosti Hrvatske</i> (in Croatian). <b>2019</b> (1): 162–174. <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/2975-657X">2975-657X</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Godi%C5%A1njak+Akademije+tehni%C4%8Dkih+znanosti+Hrvatske&rft.atitle=Prvi+cjeloviti+vi%C5%A1efazni+elektroenergetski+sustav+na+svijetu+%E2%80%93+Krka+%C5%A0ibenik&rft.volume=2019&rft.issue=1&rft.pages=162-174&rft.date=2019&rft.issn=2975-657X&rft.aulast=Holjevac&rft.aufirst=Ninoslav&rft.au=Kuzle%2C+Igor&rft_id=https%3A%2F%2Fhrcak.srce.hr%2F238713&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlternating+current" 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="CITEREFHjulström1940" class="citation book cs1">Hjulström, Filip (1940). <a rel="nofollow" class="external text" href="https://www.antikvariat.net/sv/rod151504-elektrifieringens-utveckling-i-sverige-en-ekonomisk-geografisk-oversikt-hjulstrom-filip"><i>Elektrifieringens utveckling i Sverige, en ekonomisk-geografisk översikt</i></a>. [Excerpt taken from YMER 1941, häfte 2.Utgiven av Sällskapet för antropologi och geografi: Meddelande från Upsala univeristets geografiska institution, N:o 29, published by Esselte ab, Stockholm 1941 no. 135205]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elektrifieringens+utveckling+i+Sverige%2C+en+ekonomisk-geografisk+%C3%B6versikt&rft.date=1940&rft.aulast=Hjulstr%C3%B6m&rft.aufirst=Filip&rft_id=https%3A%2F%2Fwww.antikvariat.net%2Fsv%2Frod151504-elektrifieringens-utveckling-i-sverige-en-ekonomisk-geografisk-oversikt-hjulstrom-filip&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlternating+current" 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="CITEREFGrattan-Guinness2003" class="citation book cs1">Grattan-Guinness, I. (September 19, 2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=f5FqsDPVQ2MC&q=theoretical++alternating+current++Oliver+Heaviside&pg=PA1229"><i>Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences</i></a>. JHU Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8018-7397-3" title="Special:BookSources/978-0-8018-7397-3"><bdi>978-0-8018-7397-3</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Companion+Encyclopedia+of+the+History+and+Philosophy+of+the+Mathematical+Sciences&rft.pub=JHU+Press&rft.date=2003-09-19&rft.isbn=978-0-8018-7397-3&rft.aulast=Grattan-Guinness&rft.aufirst=I.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Df5FqsDPVQ2MC%26q%3Dtheoretical%2B%2Balternating%2Bcurrent%2B%2BOliver%2BHeaviside%26pg%3DPA1229&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlternating+current" 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="CITEREFSuzuki2009" class="citation book cs1">Suzuki, Jeff (August 27, 2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lew5IC5piCwC&q=theoretical++alternating+current++Charles+Steinmetz&pg=PA329"><i>Mathematics in Historical Context</i></a>. MAA. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88385-570-6" title="Special:BookSources/978-0-88385-570-6"><bdi>978-0-88385-570-6</bdi></a> – via Google Books.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+in+Historical+Context&rft.pub=MAA&rft.date=2009-08-27&rft.isbn=978-0-88385-570-6&rft.aulast=Suzuki&rft.aufirst=Jeff&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dlew5IC5piCwC%26q%3Dtheoretical%2B%2Balternating%2Bcurrent%2B%2BCharles%2BSteinmetz%26pg%3DPA329&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlternating+current" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Alternating_current&action=edit&section=22" title="Edit section: Further reading"><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>Willam A. Meyers, <i>History and Reflections on the Way Things Were: Mill Creek Power Plant – Making History with AC</i>, IEEE Power Engineering Review, February 1997, pp. 22–24</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=Alternating_current&action=edit&section=23" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><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="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Alternating_current" class="extiw" title="commons:Category:Alternating current">Alternating current</a></span>.</div></div> </div> <ul><li>"<i>AC/DC: <a rel="nofollow" class="external text" href="https://www.pbs.org/wgbh/amex/edison/sfeature/acdc.html">What's the Difference</a>?</i>". Edison's Miracle of Light, <a rel="nofollow" class="external text" href="https://www.pbs.org/wgbh/amex/index.html">American Experience</a>. (<a href="/wiki/Public_Broadcasting_Service" class="mw-redirect" title="Public Broadcasting Service">PBS</a>)</li> <li>"<i>AC/DC: <a rel="nofollow" class="external text" href="https://www.pbs.org/wgbh/amex/edison/sfeature/acdc_insideacgenerator.html">Inside the AC Generator</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141228182024/http://www.pbs.org/wgbh/amex/edison/sfeature/acdc_insideacgenerator.html">Archived</a> 2014-12-28 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></i>". Edison's Miracle of Light, American Experience. (PBS)</li> <li><a rel="nofollow" class="external text" href="http://www.technology.niagarac.on.ca/people/mcsele/Rankine.html">Professor Mark Csele's tour of the 25 Hz Rankine generating station</a></li> <li>Blalock, Thomas J., "<i><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070607042254/http://www.ieee.org/organizations/pes/public/2003/sep/peshistory.html">The Frequency Changer Era: Interconnecting Systems of Varying Cycles</a></i>". The history of various frequencies and interconversion schemes in the US at the beginning of the 20th century</li> <li><a rel="nofollow" class="external text" href="http://edisontechcenter.org/AC-PowerHistory.html">AC Power History and Timeline</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 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