AC power - Wikipedia

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>AC power - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"f1db2e22-a47b-49c0-8691-dd514926aef1","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"AC_power","wgTitle":"AC power","wgCurRevisionId":1258338867,"wgRevisionId":1258338867,"wgArticleId":1780823,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Webarchive template wayback links","Articles with short description","Short description matches Wikidata","AC power","Electric power"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"AC_power","wgRelevantArticleId":1780823,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{ "status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":30000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q111954952","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready", "user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming", "ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.5"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/4/42/City_lights_in_motion.jpg/1200px-City_lights_in_motion.jpg"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="900"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/4/42/City_lights_in_motion.jpg/800px-City_lights_in_motion.jpg"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="600"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/4/42/City_lights_in_motion.jpg/640px-City_lights_in_motion.jpg"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="480"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="AC power - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/AC_power"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=AC_power&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/AC_power"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-AC_power rootpage-AC_power skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=AC+power" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=AC+power" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=AC+power" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=AC+power" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Active,_reactive,_apparent,_and_complex_power_in_sinusoidal_steady-state" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Active,_reactive,_apparent,_and_complex_power_in_sinusoidal_steady-state"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Active, reactive, apparent, and complex power in sinusoidal steady-state</span> </div> </a> <ul id="toc-Active,_reactive,_apparent,_and_complex_power_in_sinusoidal_steady-state-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calculations_and_equations_in_sinusoidal_steady-state" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Calculations_and_equations_in_sinusoidal_steady-state"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Calculations and equations in sinusoidal steady-state</span> </div> </a> <ul id="toc-Calculations_and_equations_in_sinusoidal_steady-state-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Power_factor" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Power_factor"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Power factor</span> </div> </a> <ul id="toc-Power_factor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reactive_power" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reactive_power"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Reactive power</span> </div> </a> <button aria-controls="toc-Reactive_power-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 Reactive power subsection</span> </button> <ul id="toc-Reactive_power-sublist" class="vector-toc-list"> <li id="toc-Capacitive_vs._inductive_loads" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Capacitive_vs._inductive_loads"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Capacitive vs. inductive loads</span> </div> </a> <ul id="toc-Capacitive_vs._inductive_loads-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reactive_power_control" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reactive_power_control"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Reactive power control</span> </div> </a> <ul id="toc-Reactive_power_control-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Unbalanced_sinusoidal_polyphase_systems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Unbalanced_sinusoidal_polyphase_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Unbalanced sinusoidal polyphase systems</span> </div> </a> <ul id="toc-Unbalanced_sinusoidal_polyphase_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real_number_formulas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Real_number_formulas"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Real number formulas</span> </div> </a> <ul id="toc-Real_number_formulas-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiple_frequency_systems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Multiple_frequency_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Multiple frequency systems</span> </div> </a> <ul id="toc-Multiple_frequency_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</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">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">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">AC power</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 2 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-2" 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">2 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%90%EB%A5%98_%EC%A0%84%EB%A0%A5" title="교류 전력 – Korean" lang="ko" hreflang="ko" data-title="교류 전력" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</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%E7%94%B5%E5%8A%9F%E7%8E%87" title="交流电功率 – Chinese" lang="zh" hreflang="zh" data-title="交流电功率" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q111954952#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/AC_power" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:AC_power" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/AC_power"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=AC_power&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=AC_power&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/AC_power"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=AC_power&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=AC_power&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/AC_power" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/AC_power" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=AC_power&oldid=1258338867" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=AC_power&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=AC_power&id=1258338867&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FAC_power"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FAC_power"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=AC_power&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=AC_power&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q111954952" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Power in alternating current systems</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">This article is about power in AC systems. For information on utility-supplied AC power, see <a href="/wiki/Mains_electricity" title="Mains electricity">Mains electricity</a>.</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:City_lights_in_motion.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/City_lights_in_motion.jpg/250px-City_lights_in_motion.jpg" decoding="async" width="250" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/City_lights_in_motion.jpg/375px-City_lights_in_motion.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/City_lights_in_motion.jpg/500px-City_lights_in_motion.jpg 2x" data-file-width="2048" data-file-height="1536" /></a><figcaption>The blinking of non-incandescent city lights is shown in this motion-blurred long exposure. The AC nature of the mains power is revealed by the dashed appearance of the traces of moving lights.</figcaption></figure> <p>In an electric circuit, <a href="/wiki/Instantaneous_power" class="mw-redirect" title="Instantaneous power">instantaneous power</a> is the time rate of flow of energy past a given point of the circuit. In <a href="/wiki/Alternating_current" title="Alternating current">alternating current</a> circuits, energy storage elements such as <a href="/wiki/Inductor" title="Inductor">inductors</a> and <a href="/wiki/Capacitor" title="Capacitor">capacitors</a> may result in periodic reversals of the direction of energy flow. Its SI unit is the <a href="/wiki/Watt" title="Watt">watt</a>. </p><p>The portion of instantaneous power that, averaged over a complete cycle of the <a href="/wiki/AC_waveform" class="mw-redirect" title="AC waveform">AC waveform</a>, results in net transfer of energy in one direction is known as instantaneous active power, and its time average is known as <b>active power</b> or <b>real power</b>.<sup id="cite_ref-IEEE_1459_1-0" class="reference"><a href="#cite_note-IEEE_1459-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 3">: 3 </span></sup> The portion of instantaneous power that results in no net transfer of energy but instead oscillates between the source and load in each cycle due to stored energy is known as instantaneous reactive power, and its amplitude is the absolute value of <b>reactive power</b>.<sup id="cite_ref-ThomasRosaToussaint_2016_2-0" class="reference"><a href="#cite_note-ThomasRosaToussaint_2016-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-IEEE_1459_1-1" class="reference"><a href="#cite_note-IEEE_1459-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 4">: 4 </span></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Active,_reactive,_apparent,_and_complex_power_in_sinusoidal_steady-state"><span id="Active.2C_reactive.2C_apparent.2C_and_complex_power_in_sinusoidal_steady-state"></span><span class="anchor" id="Active_power"></span><span class="anchor" id="Reactive_power"></span><span class="anchor" id="Apparent_power"></span><span class="anchor" id="Complex_power"></span><span class="anchor" id="Real_power"></span>Active, reactive, apparent, and complex power in sinusoidal steady-state</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AC_power&action=edit&section=1" title="Edit section: Active, reactive, apparent, and complex power in sinusoidal steady-state"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a simple alternating current (AC) circuit consisting of a source and a <a href="/wiki/Linear_circuit" title="Linear circuit">linear</a> <a href="/wiki/Time-invariant_system" title="Time-invariant system">time-invariant</a> load, both the current and voltage are <a href="/wiki/Sine_wave" title="Sine wave">sinusoidal</a> at the same frequency.<sup id="cite_ref-Das_2015_3-0" class="reference"><a href="#cite_note-Das_2015-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> If the load is purely <a href="/wiki/Resistive" class="mw-redirect" title="Resistive">resistive</a>, the two quantities reverse their <a href="/wiki/Electrical_polarity" class="mw-redirect" title="Electrical polarity">polarity</a> at the same time. Hence, the instantaneous power, given by the product of voltage and current, is always positive, such that the direction of energy flow does not reverse and always is toward the resistor. In this case, only active power is transferred. </p><p>If the load is purely <a href="/wiki/Electrical_reactance" title="Electrical reactance"><i>reactive</i></a>, then the voltage and current are 90 degrees out of phase. For two quarters of each cycle, the product of voltage and current is positive, but for the other two quarters, the product is negative, indicating that on average, exactly as much energy flows into the load as flows back out. There is no net energy flow over each half cycle. In this case, only reactive power flows: There is no net transfer of energy to the load; however, electrical power does flow along the wires and returns by flowing in reverse along the same wires. The current required for this reactive power flow dissipates energy in the line resistance, even if the ideal load device consumes no energy itself. Practical loads have resistance as well as inductance, or capacitance, so both active and reactive powers will flow to normal loads. </p><p>Apparent power is the product of the <a href="/wiki/Root-mean-square" class="mw-redirect" title="Root-mean-square">RMS</a> values of voltage and current. Apparent power is taken into account when designing and operating power systems, because although the current associated with reactive power does no work at the load, it still must be supplied by the power source. Conductors, transformers and generators must be sized to carry the total current, not just the current that does useful work. <a href="/wiki/Voltage_control_and_reactive_power_management" title="Voltage control and reactive power management">Insufficient reactive power can depress voltage levels</a> on an electrical grid and, under certain operating conditions, collapse the network (a <a href="/wiki/Power_outage" title="Power outage">blackout</a>). Another consequence is that adding the apparent power for two loads will not accurately give the total power unless they have the same phase difference between current and voltage (the same <a href="/wiki/Power_factor" title="Power factor">power factor</a>). </p><p>Conventionally, capacitors are treated as if they generate reactive power, and inductors are treated as if they consume it. If a capacitor and an inductor are placed in parallel, then the currents flowing through the capacitor and the inductor tend to cancel rather than add. This is the fundamental mechanism for controlling the power factor in electric power transmission; capacitors (or inductors) are inserted in a circuit to partially compensate for reactive power 'consumed' ('generated') by the load. Purely capacitive circuits supply reactive power with the current waveform leading the voltage waveform by 90 degrees, while purely inductive circuits absorb reactive power with the current waveform lagging the voltage waveform by 90 degrees. The result of this is that capacitive and inductive circuit elements tend to cancel each other out.<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> </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Cmplxpower.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Cmplxpower.svg/280px-Cmplxpower.svg.png" decoding="async" width="280" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Cmplxpower.svg/420px-Cmplxpower.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Cmplxpower.svg/560px-Cmplxpower.svg.png 2x" data-file-width="793" data-file-height="499" /></a><figcaption><b><u>The Power Triangle</u></b><br />The complex power is the vector sum of active and reactive power. The apparent power is the magnitude of the complex power.<br />  <b>Active power</b>, <i>P</i><br />  <b>Reactive power</b>, <i>Q</i><br />  <b>Complex power</b>, <i>S</i><br />  <b>Apparent power</b>, <i>|S|</i><br />  <b>Phase of voltage relative to current</b>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span></figcaption></figure> <p>Engineers use the following terms to describe energy flow in a system (and assign each of them a different unit to differentiate between them): </p> <ul><li><b>Active power</b>,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <i>P</i>, or <b>real power</b>:<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> <a href="/wiki/Watt" title="Watt">watt</a> (W);</li> <li><b>Reactive power</b>, <i>Q</i>: <a href="/wiki/Volt-ampere_reactive" class="mw-redirect" title="Volt-ampere reactive">volt-ampere reactive</a> (var);</li> <li><b>Complex power</b>, <i>S</i>: <a href="/wiki/Volt-ampere" title="Volt-ampere">volt-ampere</a> (VA);</li> <li><b>Apparent power</b>, |<i>S</i>|: the <a href="/wiki/Magnitude_(vector)" class="mw-redirect" title="Magnitude (vector)">magnitude</a> of complex power <i>S</i>: volt-ampere (VA);</li> <li><b>Phase of voltage relative to current</b>, <i>φ</i>: the angle of difference (in degrees) between current and voltage; <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi =\arg(V)-\arg(I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> <mo>=</mo> <mi>arg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>arg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi =\arg(V)-\arg(I)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c675d101cd33c462d67d58e0d1e7233274a9692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.51ex; height:2.843ex;" alt="{\displaystyle \varphi =\arg(V)-\arg(I)}"></span>. Current lagging voltage (<a href="/wiki/Quadrant_(plane_geometry)" title="Quadrant (plane geometry)">quadrant</a> I vector), current leading voltage (quadrant IV vector).</li></ul> <p>These are all denoted in the adjacent diagram (called a power triangle). </p><p>In the diagram, <i>P</i> is the active power, <i>Q</i> is the reactive power (in this case positive), <i>S</i> is the complex power and the length of <i>S</i> is the apparent power. Reactive power does not do any work, so it is represented as the <b>imaginary axis</b> of the vector diagram. Active power does do work, so it is the real axis. </p><p>The unit for power is the <a href="/wiki/Watt" title="Watt">watt</a> (symbol: W). Apparent power is often expressed in <a href="/wiki/Volt-ampere" title="Volt-ampere">volt-amperes</a> (VA) since it is the product of <a href="/wiki/Root_mean_square" title="Root mean square">RMS</a> <a href="/wiki/Voltage" title="Voltage">voltage</a> and RMS <a href="/wiki/Electric_current" title="Electric current">current</a>. The unit for reactive power is var, which stands for <a href="/wiki/Volt-ampere_reactive" class="mw-redirect" title="Volt-ampere reactive">volt-ampere reactive</a>. Since reactive power transfers no net energy to the load, it is sometimes called "wattless" power. It does, however, serve an important function in <a href="/wiki/Electrical_grid" title="Electrical grid">electrical grids</a> and its lack has been cited as a significant factor in the <a href="/wiki/Northeast_blackout_of_2003" title="Northeast blackout of 2003">Northeast blackout of 2003</a>.<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> Understanding the relationship among these three quantities lies at the heart of understanding power engineering. The mathematical relationship among them can be represented by vectors or expressed using <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, <i>S</i> = <i>P</i> + <i>j Q</i> (where <i>j</i> is the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>). </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Active-and-reactive-power-064pf-en.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Active-and-reactive-power-064pf-en.svg/500px-Active-and-reactive-power-064pf-en.svg.png" decoding="async" width="500" height="325" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Active-and-reactive-power-064pf-en.svg/750px-Active-and-reactive-power-064pf-en.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Active-and-reactive-power-064pf-en.svg/1000px-Active-and-reactive-power-064pf-en.svg.png 2x" data-file-width="800" data-file-height="520" /></a><figcaption>Instantaneous power in AC systems when the current lags behind the voltage by 50 degrees.</figcaption></figure> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Calculations_and_equations_in_sinusoidal_steady-state">Calculations and equations in sinusoidal steady-state</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AC_power&action=edit&section=2" title="Edit section: Calculations and equations in sinusoidal steady-state"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The formula for complex power (units: VA) in <a href="/wiki/Phasor" title="Phasor">phasor</a> form 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 S=VI^{*}=|S|\angle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>V</mi> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">∠</mi> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=VI^{*}=|S|\angle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627819074a22c3fa9ba90295cfd00ea705f63175" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.745ex; height:2.843ex;" alt="{\displaystyle S=VI^{*}=|S|\angle \varphi }"></span>,</dd></dl> <p>where <i>V</i> denotes voltage in phasor form, with the amplitude as <a href="/wiki/Root_mean_square" title="Root mean square">RMS</a>, and <i>I</i> denotes current in phasor form, with the amplitude as RMS. Also by convention, the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of <i>I</i> is used, which is denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9682fb2bdb0d34a1894bad921a50c2be9c787f75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.271ex; height:2.343ex;" alt="{\displaystyle I^{*}}"></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 {\overline {I}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>I</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {I}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43b8e66ae8bc148a0ef25e32292c223ee9ae24d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.383ex; height:3.009ex;" alt="{\displaystyle {\overline {I}}}"></span>), rather than <i>I</i> itself. This is done because otherwise using the product V I to define S would result in a quantity that depends on the reference angle chosen for V or I, but defining S as V I* results in a quantity that doesn't depend on the reference angle and allows to relate S to P and Q.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Other forms of complex power (units in volt-amps, VA) are derived from <i>Z</i>, the load <a href="/wiki/Electrical_impedance" title="Electrical impedance">impedance</a> (units in ohms, Ω). </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 S=|I|^{2}Z={\frac {|V|^{2}}{Z^{*}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>I</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>Z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=|I|^{2}Z={\frac {|V|^{2}}{Z^{*}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1647305cc61e38a85fcb193c98227d817c4f23fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:17.868ex; height:6.009ex;" alt="{\displaystyle S=|I|^{2}Z={\frac {|V|^{2}}{Z^{*}}}}"></span>.</dd></dl> <p>Consequentially, with reference to the power triangle, real power (units in watts, W) is derived as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=|S|\cos {\varphi }=|I|^{2}R={\frac {|V|^{2}}{|Z|^{2}}}\times {R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>I</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=|S|\cos {\varphi }=|I|^{2}R={\frac {|V|^{2}}{|Z|^{2}}}\times {R}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2014caff6f2b875213c15644951f0e4eac597f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.099ex; height:7.176ex;" alt="{\displaystyle P=|S|\cos {\varphi }=|I|^{2}R={\frac {|V|^{2}}{|Z|^{2}}}\times {R}}"></span>.</dd></dl> <p>For a purely resistive load, real power can be simplified 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 P={\frac {|V|^{2}}{R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>R</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\frac {|V|^{2}}{R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8c7c78d1abf08cb79ece5ebc1c696cf55f2033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.815ex; height:6.176ex;" alt="{\displaystyle P={\frac {|V|^{2}}{R}}}"></span>.</dd></dl> <p><i>R</i> denotes resistance (units in ohms, Ω) of the load. </p><p>Reactive power (units in volts-amps-reactive, var) is derived as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=|S|\sin {\varphi }=|I|^{2}X={\frac {|V|^{2}}{|Z|^{2}}}\times {X}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>I</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=|S|\sin {\varphi }=|I|^{2}X={\frac {|V|^{2}}{|Z|^{2}}}\times {X}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1f4c85b7ae886059dd9adf952c687ec7e20d907" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.368ex; height:7.176ex;" alt="{\displaystyle Q=|S|\sin {\varphi }=|I|^{2}X={\frac {|V|^{2}}{|Z|^{2}}}\times {X}}"></span>.</dd></dl> <p>For a purely reactive load, reactive power can be simplified 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 Q={\frac {|V|^{2}}{X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>V</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>X</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q={\frac {|V|^{2}}{X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44a65bfe6f7a1f9be242aee225b3253976681db4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.908ex; height:6.009ex;" alt="{\displaystyle Q={\frac {|V|^{2}}{X}}}"></span>,</dd></dl> <p>where <i>X</i> denotes <a href="/wiki/Electrical_reactance" title="Electrical reactance">reactance</a> (units in ohms, Ω) of the load. </p><p>Combining, the complex power (units in volt-amps, VA) is back-derived as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=P+jQ}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>P</mi> <mo>+</mo> <mi>j</mi> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=P+jQ}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2344a6b29d8b81f396b8904fe49114810f5cdaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.98ex; height:2.509ex;" alt="{\displaystyle S=P+jQ}"></span>,</dd></dl> <p>and the apparent power (units in volt-amps, VA) as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |S|={\sqrt {P^{2}+Q^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |S|={\sqrt {P^{2}+Q^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4c6dd10b231e2b1e5836b65b9784a41165e8d32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:16.824ex; height:4.843ex;" alt="{\displaystyle |S|={\sqrt {P^{2}+Q^{2}}}}"></span>.</dd></dl> <p>These are simplified diagrammatically by the power triangle. </p> <div class="mw-heading mw-heading2"><h2 id="Power_factor">Power factor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AC_power&action=edit&section=3" title="Edit section: Power factor"><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/Power_factor" title="Power factor">Power factor</a></div> <p>The ratio of active power to apparent power in a circuit is called the <a href="/wiki/Power_factor" title="Power factor">power factor</a>. For two systems transmitting the same amount of active power, the system with the lower power factor will have higher circulating currents due to energy that returns to the source from energy storage in the load. These higher currents produce higher losses and reduce overall transmission efficiency. A lower power factor circuit will have a higher apparent power and higher losses for the same amount of active power. The power factor is 1.0 when the voltage and current are in <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a>. It is zero when the current leads or lags the voltage by 90 degrees. When the voltage and current are 180 degrees out of phase, the power factor is negative one, and the load is feeding energy into the source (an example would be a home with solar cells on the roof that feed power into the power grid when the sun is shining). Power factors are usually stated as "leading" or "lagging" to show the sign of the phase angle of current with respect to voltage. Voltage is designated as the base to which current angle is compared, meaning that current is thought of as either "leading" or "lagging" voltage. Where the waveforms are purely sinusoidal, the power factor is the cosine of the phase angle (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }"></span>) between the current and voltage sinusoidal waveforms. Equipment data sheets and nameplates will often abbreviate power factor 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 \cos \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98a7ed057494fc925837ca7d6aa0ccf2cc948d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.884ex; height:2.509ex;" alt="{\displaystyle \cos \phi }"></span>" for this reason. </p><p>Example: The active power is <span class="nowrap">700 W</span> and the phase angle between voltage and current is 45.6°. The power factor is <span class="nowrap">cos(45.6°) = 0.700</span>. The apparent power is then: <span class="nowrap">700 W / cos(45.6°) = 1000 VA</span>. The concept of power dissipation in AC circuit is explained and illustrated with the example. </p><p>For instance, a power factor of 0.68 means that only 68 percent of the total current supplied (in magnitude) is actually doing work; the remaining current does no work at the load. Power Factor is very important in Power sector substations. Form the national grid the sub sectors are required to have minimum amount of power factor. Otherwise there are many loss. Mainly the required vary around 0.90 to 0.96 or more. Better the power factor less the loss. </p> <div class="mw-heading mw-heading2"><h2 id="Reactive_power">Reactive power</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AC_power&action=edit&section=4" title="Edit section: Reactive power"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a direct current circuit, the power flowing to the load is proportional to the product of the current through the load and the potential drop across the load. The power that happens because of a capacitor or inductor is called reactive power. It happens because of the AC nature of elements like inductors and capacitors. Energy flows in one direction from the source to the load. In AC power, the voltage and current both vary approximately sinusoidally. When there is inductance or capacitance in the circuit, the voltage and current waveforms do not line up perfectly. The power flow has two components – one component flows from source to load and can perform work at the load; the other portion, known as "reactive power", is due to the delay between voltage and current, known as phase angle, and cannot do useful work at the load. It can be thought of as current that is arriving at the wrong time (too late or too early). To distinguish reactive power from active power, it is measured in units of "<a href="/wiki/Volt-amperes_reactive" class="mw-redirect" title="Volt-amperes reactive">volt-amperes reactive</a>", or var. These units can simplify to watts but are left as var to denote that they represent no actual work output. </p><p>Energy stored in capacitive or inductive elements of the network gives rise to reactive power flow. Reactive power flow strongly influences the voltage levels across the network. Voltage levels and reactive power flow must be carefully controlled to allow a power system to be operated within acceptable limits. A technique known as <a href="/wiki/Flexible_AC_transmission_system" title="Flexible AC transmission system">reactive compensation</a> is used to reduce apparent power flow to a load by reducing reactive power supplied from transmission lines and providing it locally. For example, to compensate an inductive load, a shunt capacitor is installed close to the load itself. This allows all reactive power needed by the load to be supplied by the capacitor and not have to be transferred over the transmission lines. This practice saves energy because it reduces the amount of energy that is required to be produced by the utility to do the same amount of work. Additionally, it allows for more efficient transmission line designs using smaller conductors or fewer bundled conductors and optimizing the design of transmission towers. </p> <div class="mw-heading mw-heading3"><h3 id="Capacitive_vs._inductive_loads">Capacitive vs. inductive loads</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AC_power&action=edit&section=5" title="Edit section: Capacitive vs. inductive loads"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Stored energy in the magnetic or electric field of a load device, such as a motor or capacitor, causes an offset between the current and the voltage waveforms. A capacitor is a device that stores energy in the form of an electric field. As current is driven through the capacitor, charge build-up causes an opposing voltage to develop across the capacitor. This voltage increases until some maximum dictated by the capacitor structure. In an AC network, the voltage across a capacitor is constantly changing. The capacitor opposes this change, causing the current to lead the voltage in phase. Capacitors are said to "source" reactive power, and thus to cause a leading power factor. </p><p>Induction machines are some of the most common types of loads in the electric power system today. These machines use <a href="/wiki/Inductors" class="mw-redirect" title="Inductors">inductors</a>, or large coils of wire to store energy in the form of a magnetic field. When a voltage is initially placed across the coil, the inductor strongly resists this change in a current and magnetic field, which causes a time delay for the current to reach its maximum value. This causes the current to lag behind the voltage in phase. Inductors are said to "sink" reactive power, and thus to cause a lagging power factor. <a href="/wiki/Induction_generator" title="Induction generator">Induction generators</a> can source or sink reactive power, and provide a measure of control to system operators over reactive power flow and thus voltage.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Because these devices have opposite effects on the phase angle between voltage and current, they can be used to "cancel out" each other's effects. This usually takes the form of capacitor banks being used to counteract the lagging power factor caused by induction motors. </p> <div class="mw-heading mw-heading3"><h3 id="Reactive_power_control">Reactive power control</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AC_power&action=edit&section=6" title="Edit section: Reactive power control"><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/Voltage_control_and_reactive_power_management" title="Voltage control and reactive power management">Voltage control and reactive power management</a></div> <p>Transmission connected generators are generally required to support reactive power flow. For example, on the United Kingdom transmission system, generators are required by the Grid Code Requirements to supply their rated power between the limits of 0.85 power factor lagging and 0.90 power factor leading at the designated terminals. The system operator will perform switching actions to maintain a secure and economical voltage profile while maintaining a reactive power balance 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 \mathrm {Generator\ MVARs+System\ gain+Shunt\ capacitors=MVAR\ Demand+Reactive\ losses+Shunt\ reactors} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">G</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mtext> </mtext> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">s</mi> <mo>+</mo> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> <mtext> </mtext> <mi mathvariant="normal">g</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mo>+</mo> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mtext> </mtext> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">s</mi> <mo>=</mo> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">R</mi> <mtext> </mtext> <mi mathvariant="normal">D</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">d</mi> <mo>+</mo> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">v</mi> <mi mathvariant="normal">e</mi> <mtext> </mtext> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">s</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">s</mi> <mo>+</mo> <mi mathvariant="normal">S</mi> <mi mathvariant="normal">h</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> <mtext> </mtext> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Generator\ MVARs+System\ gain+Shunt\ capacitors=MVAR\ Demand+Reactive\ losses+Shunt\ reactors} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/985c99dbddcdfbaef7fe2c683f85364220e421f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:108.562ex; height:2.509ex;" alt="{\displaystyle \mathrm {Generator\ MVARs+System\ gain+Shunt\ capacitors=MVAR\ Demand+Reactive\ losses+Shunt\ reactors} }"></span></dd></dl> <p>The "<a href="/wiki/Ferranti_effect" title="Ferranti effect">system gain</a>" is an important source of reactive power in the above power balance equation, which is generated by the capacitative nature of the transmission network itself. By making decisive switching actions in the early morning before the demand increases, the system gain can be maximized early on, helping to secure the system for the whole day. To balance the equation some pre-fault reactive generator use will be required. Other sources of reactive power that will also be used include shunt capacitors, shunt reactors, <a href="/wiki/Static_VAR_compensator" title="Static VAR compensator">static VAR compensators</a> and voltage control circuits. </p> <div class="mw-heading mw-heading2"><h2 id="Unbalanced_sinusoidal_polyphase_systems">Unbalanced sinusoidal polyphase systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AC_power&action=edit&section=7" title="Edit section: Unbalanced sinusoidal polyphase systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While active power and reactive power are well defined in any system, the definition of apparent power for unbalanced polyphase systems is considered to be one of the most controversial topics in power engineering. Originally, apparent power arose merely as a figure of merit. Major delineations of the concept are attributed to <a href="/wiki/William_Stanley_Jr." title="William Stanley Jr.">Stanley</a>'s <i>Phenomena of Retardation in the Induction Coil</i> (1888) and <a href="/wiki/Charles_Proteus_Steinmetz" title="Charles Proteus Steinmetz">Steinmetz</a>'s <i>Theoretical Elements of Engineering</i> (1915). However, with the development of <a href="/wiki/Three_phase" class="mw-redirect" title="Three phase">three phase</a> power distribution, it became clear that the definition of apparent power and the power factor could not be applied to unbalanced <a href="/wiki/Polyphase_systems" class="mw-redirect" title="Polyphase systems">polyphase systems</a>. In 1920, a "Special Joint Committee of the AIEE and the National Electric Light Association" met to resolve the issue. They considered two definitions. </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 S_{A}=|S_{\mathrm {a} }|+|S_{\mathrm {b} }|+|S_{\mathrm {c} }|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{A}=|S_{\mathrm {a} }|+|S_{\mathrm {b} }|+|S_{\mathrm {c} }|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7da801a997db583dfea74eea2009a2e2f4de261" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.987ex; height:2.843ex;" alt="{\displaystyle S_{A}=|S_{\mathrm {a} }|+|S_{\mathrm {b} }|+|S_{\mathrm {c} }|}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {pf} _{A}={P_{\mathrm {a} }+P_{\mathrm {b} }+P_{\mathrm {c} } \over S_{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {pf} _{A}={P_{\mathrm {a} }+P_{\mathrm {b} }+P_{\mathrm {c} } \over S_{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2187e1bc7220431d2711f42636d21c272648aada" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.724ex; height:5.676ex;" alt="{\displaystyle \mathrm {pf} _{A}={P_{\mathrm {a} }+P_{\mathrm {b} }+P_{\mathrm {c} } \over S_{A}}}"></span>,</dd></dl> <p>that is, the arithmetic sum of the phase apparent powers; and </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 S_{V}=|P_{\mathrm {a} }+P_{\mathrm {b} }+P_{\mathrm {c} }+j(Q_{\mathrm {a} }+Q_{\mathrm {b} }+Q_{\mathrm {c} })|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi>j</mi> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{V}=|P_{\mathrm {a} }+P_{\mathrm {b} }+P_{\mathrm {c} }+j(Q_{\mathrm {a} }+Q_{\mathrm {b} }+Q_{\mathrm {c} })|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5eef8f5be6a72bcd01533fb9b1f9afa80185d62c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.6ex; height:2.843ex;" alt="{\displaystyle S_{V}=|P_{\mathrm {a} }+P_{\mathrm {b} }+P_{\mathrm {c} }+j(Q_{\mathrm {a} }+Q_{\mathrm {b} }+Q_{\mathrm {c} })|}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {pf} _{V}={P_{\mathrm {a} }+P_{\mathrm {b} }+P_{\mathrm {c} } \over S_{V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">f</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {pf} _{V}={P_{\mathrm {a} }+P_{\mathrm {b} }+P_{\mathrm {c} } \over S_{V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc9896c99c7bea6a76af433dea5ba8d13eca1c07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.755ex; height:5.676ex;" alt="{\displaystyle \mathrm {pf} _{V}={P_{\mathrm {a} }+P_{\mathrm {b} }+P_{\mathrm {c} } \over S_{V}}}"></span>,</dd></dl> <p>that is, the magnitude of total three-phase complex power. </p><p>The 1920 committee found no consensus and the topic continued to dominate discussions. In 1930, another committee formed and once again failed to resolve the question. The transcripts of their discussions are the lengthiest and most controversial ever published by the AIEE.<sup id="cite_ref-Emanuel_1993_10-0" class="reference"><a href="#cite_note-Emanuel_1993-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Further resolution of this debate did not come until the late 1990s. </p><p>A new definition based on <a href="/wiki/Symmetrical_components" title="Symmetrical components">symmetrical components</a> theory was proposed in 1993 by Alexander Emanuel for unbalanced linear load supplied with asymmetrical sinusoidal voltages: </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 S={\sqrt {\left(|V_{\mathrm {a} }^{2}|+|V_{\mathrm {b} }^{2}|+|V_{\mathrm {c} }^{2}|\right)\left(|I_{\mathrm {a} }^{2}|+|I_{\mathrm {b} }^{2}|+|I_{\mathrm {c} }^{2}|\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\sqrt {\left(|V_{\mathrm {a} }^{2}|+|V_{\mathrm {b} }^{2}|+|V_{\mathrm {c} }^{2}|\right)\left(|I_{\mathrm {a} }^{2}|+|I_{\mathrm {b} }^{2}|+|I_{\mathrm {c} }^{2}|\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb2d35fc262c4d47fb6c5a55b81d5cc0ebb414f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:46.418ex; height:4.843ex;" alt="{\displaystyle S={\sqrt {\left(|V_{\mathrm {a} }^{2}|+|V_{\mathrm {b} }^{2}|+|V_{\mathrm {c} }^{2}|\right)\left(|I_{\mathrm {a} }^{2}|+|I_{\mathrm {b} }^{2}|+|I_{\mathrm {c} }^{2}|\right)}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {pf} ={P^{+} \over S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> <mi mathvariant="normal">f</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mi>S</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {pf} ={P^{+} \over S}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96f217fc509077e49fb49dcaf299ed6af3cf0b84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.425ex; height:5.676ex;" alt="{\displaystyle \mathrm {pf} ={P^{+} \over S}}"></span>,</dd></dl> <p>that is, the root of squared sums of line voltages multiplied by the root of squared sums of line currents. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7161b1309d204da1254c6fdeeec8f54d5e8c71c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.332ex; height:2.509ex;" alt="{\displaystyle P^{+}}"></span> denotes the positive sequence power: </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^{+}=3|V^{+}||I^{+}|\cos {(\arg {(V^{+})}-\arg {(I^{+})})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>=</mo> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>arg</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>−</mo> <mi>arg</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P^{+}=3|V^{+}||I^{+}|\cos {(\arg {(V^{+})}-\arg {(I^{+})})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e78c53deebf3c29491099ebab2b18af83ccd3e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.891ex; height:3.009ex;" alt="{\displaystyle P^{+}=3|V^{+}||I^{+}|\cos {(\arg {(V^{+})}-\arg {(I^{+})})}}"></span></dd></dl> <p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b2aafc830b7719ab820d87c8a8eec52e8756b9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.428ex; height:2.509ex;" alt="{\displaystyle V^{+}}"></span> denotes the positive sequence voltage phasor, 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 I^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c7e3c9c9ec6be223e286f8a0cf03c2cb0c57e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.727ex; height:2.509ex;" alt="{\displaystyle I^{+}}"></span> denotes the positive sequence current phasor.<sup id="cite_ref-Emanuel_1993_10-1" class="reference"><a href="#cite_note-Emanuel_1993-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Real_number_formulas">Real number formulas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AC_power&action=edit&section=8" title="Edit section: Real number formulas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A perfect resistor stores no energy; so current and voltage are in phase. Therefore, there is no reactive power 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 P=S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acbfb568b54edf1db2e826688e8a352ec64abb37" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.343ex; height:2.176ex;" alt="{\displaystyle P=S}"></span> (using the <a href="/wiki/Passive_sign_convention" title="Passive sign convention">passive sign convention</a>). Therefore, for a perfect resistor </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=S=V_{\mathrm {RMS} }I_{\mathrm {RMS} }=I_{\mathrm {RMS} }^{2}R={\frac {V_{\mathrm {RMS} }^{2}}{R}}\,\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>S</mi> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo>=</mo> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>R</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=S=V_{\mathrm {RMS} }I_{\mathrm {RMS} }=I_{\mathrm {RMS} }^{2}R={\frac {V_{\mathrm {RMS} }^{2}}{R}}\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bae339fa2c890783ae2269c9ff996ed4d695fdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-right: -0.387ex; width:38.833ex; height:6.176ex;" alt="{\displaystyle P=S=V_{\mathrm {RMS} }I_{\mathrm {RMS} }=I_{\mathrm {RMS} }^{2}R={\frac {V_{\mathrm {RMS} }^{2}}{R}}\,\!}"></span>.</dd></dl> <p>For a perfect capacitor or inductor, there is no net power transfer; so all power is reactive. Therefore, for a perfect capacitor or inductor: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P&=0\\Q&=|S|=V_{\mathrm {RMS} }I_{\mathrm {RMS} }=I_{\mathrm {RMS} }^{2}|X|={\frac {V_{\mathrm {RMS} }^{2}}{|X|}}\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>P</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>Q</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">S</mi> </mrow> </mrow> </msub> <mo>=</mo> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P&=0\\Q&=|S|=V_{\mathrm {RMS} }I_{\mathrm {RMS} }=I_{\mathrm {RMS} }^{2}|X|={\frac {V_{\mathrm {RMS} }^{2}}{|X|}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/049e4d3a2c81503646525f6e234983d850652646" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:42.093ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}P&=0\\Q&=|S|=V_{\mathrm {RMS} }I_{\mathrm {RMS} }=I_{\mathrm {RMS} }^{2}|X|={\frac {V_{\mathrm {RMS} }^{2}}{|X|}}\end{aligned}}}"></span>.</dd></dl> <p>where <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span></i> is the <a href="/wiki/Electrical_reactance" title="Electrical reactance">reactance</a> of the capacitor or inductor. </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> is defined as being positive for an inductor and negative for a capacitor, then the <a href="/wiki/Absolute_value" title="Absolute value">modulus</a> signs can be removed from S and X and get </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 Q=I_{\mathrm {RMS} }^{2}X={\frac {V_{\mathrm {RMS} }^{2}}{X}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">M</mi> <mi mathvariant="normal">S</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>X</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=I_{\mathrm {RMS} }^{2}X={\frac {V_{\mathrm {RMS} }^{2}}{X}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c86c12c52e35f8695555b4b6a98ddadc146330f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.955ex; height:6.009ex;" alt="{\displaystyle Q=I_{\mathrm {RMS} }^{2}X={\frac {V_{\mathrm {RMS} }^{2}}{X}}}"></span>.</dd></dl> <p>Instantaneous power is defined as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(t)=v(t)\,i(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>i</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(t)=v(t)\,i(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8307cb88907b06d4c42ea1b93b8479663f008339" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:14.621ex; height:2.843ex;" alt="{\displaystyle p(t)=v(t)\,i(t)}"></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 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> 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 i(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06931e7bcffd32010f83c3d5dbef2d9bcbdcb670" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.451ex; height:2.843ex;" alt="{\displaystyle i(t)}"></span> are the time-varying voltage and current waveforms. </p><p>This definition is useful because it applies to all waveforms, whether they are sinusoidal or not. This is particularly useful in power electronics, where non-sinusoidal waveforms are common. </p><p>In general, engineers are interested in the active power averaged over a period of time, whether it is a low frequency line cycle or a high frequency power converter switching period. The simplest way to get that result is to take the integral of the instantaneous calculation over the desired period: </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{avg}}={\frac {1}{t_{2}-t_{1}}}\int _{t_{1}}^{t_{2}}v(t)\,i(t)\,\mathrm {d} t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>avg</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>i</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{avg}}={\frac {1}{t_{2}-t_{1}}}\int _{t_{1}}^{t_{2}}v(t)\,i(t)\,\mathrm {d} t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b761bf50d58cb3e29734922511efa0ddc8ffbc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.829ex; height:6.509ex;" alt="{\displaystyle P_{\text{avg}}={\frac {1}{t_{2}-t_{1}}}\int _{t_{1}}^{t_{2}}v(t)\,i(t)\,\mathrm {d} t}"></span>.</dd></dl> <p>This method of calculating the average power gives the active power regardless of <a href="/w/index.php?title=Harmonic_content&action=edit&redlink=1" class="new" title="Harmonic content (page does not exist)">harmonic content</a> of the waveform. In practical applications, this would be done in the digital domain, where the calculation becomes trivial when compared to the use of rms and phase to determine active power: </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{avg}}={\frac {1}{n}}\sum _{k=1}^{n}V[k]I[k]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>avg</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>V</mi> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> <mi>I</mi> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\text{avg}}={\frac {1}{n}}\sum _{k=1}^{n}V[k]I[k]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea58eff331b5263dbf27b58073ba0a5f0a0e68bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.664ex; height:6.843ex;" alt="{\displaystyle P_{\text{avg}}={\frac {1}{n}}\sum _{k=1}^{n}V[k]I[k]}"></span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Multiple_frequency_systems">Multiple frequency systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AC_power&action=edit&section=9" title="Edit section: Multiple frequency systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since an RMS value can be calculated for any waveform, apparent power can be calculated from this. For active power it would at first appear that it would be necessary to calculate many product terms and average all of them. However, looking at one of these product terms in more detail produces a very interesting result. </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&A\cos(\omega _{1}t+k_{1})\cos(\omega _{2}t+k_{2})\\={}&{\frac {A}{2}}\cos \left[\left(\omega _{1}t+k_{1}\right)+\left(\omega _{2}t+k_{2}\right)\right]+{\frac {A}{2}}\cos \left[\left(\omega _{1}t+k_{1}\right)-\left(\omega _{2}t+k_{2}\right)\right]\\={}&{\frac {A}{2}}\cos \left[\left(\omega _{1}+\omega _{2}\right)t+k_{1}+k_{2}\right]+{\frac {A}{2}}\cos \left[\left(\omega _{1}-\omega _{2}\right)t+k_{1}-k_{2}\right]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>t</mi> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mn>2</mn> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mi>t</mi> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&A\cos(\omega _{1}t+k_{1})\cos(\omega _{2}t+k_{2})\\={}&{\frac {A}{2}}\cos \left[\left(\omega _{1}t+k_{1}\right)+\left(\omega _{2}t+k_{2}\right)\right]+{\frac {A}{2}}\cos \left[\left(\omega _{1}t+k_{1}\right)-\left(\omega _{2}t+k_{2}\right)\right]\\={}&{\frac {A}{2}}\cos \left[\left(\omega _{1}+\omega _{2}\right)t+k_{1}+k_{2}\right]+{\frac {A}{2}}\cos \left[\left(\omega _{1}-\omega _{2}\right)t+k_{1}-k_{2}\right]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c773be852c41c009df2d70fc4d2e5d7ef7a36ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:67.488ex; height:14.009ex;" alt="{\displaystyle {\begin{aligned}&A\cos(\omega _{1}t+k_{1})\cos(\omega _{2}t+k_{2})\\={}&{\frac {A}{2}}\cos \left[\left(\omega _{1}t+k_{1}\right)+\left(\omega _{2}t+k_{2}\right)\right]+{\frac {A}{2}}\cos \left[\left(\omega _{1}t+k_{1}\right)-\left(\omega _{2}t+k_{2}\right)\right]\\={}&{\frac {A}{2}}\cos \left[\left(\omega _{1}+\omega _{2}\right)t+k_{1}+k_{2}\right]+{\frac {A}{2}}\cos \left[\left(\omega _{1}-\omega _{2}\right)t+k_{1}-k_{2}\right]\end{aligned}}}"></span></dd></dl> <p>However, the time average of a function of the form <span class="nowrap">cos(<i>ωt</i> + <i>k</i>)</span> is zero provided that <i>ω</i> is nonzero. Therefore, the only product terms that have a nonzero average are those where the frequency of voltage and current match. In other words, it is possible to calculate active (average) power by simply treating each frequency separately and adding up the answers. Furthermore, if voltage of the mains supply is assumed to be a single frequency (which it usually is), this shows that <a href="/wiki/Harmonics_(electrical_power)" title="Harmonics (electrical power)">harmonic currents</a> are a bad thing. They will increase the RMS current (since there will be non-zero terms added) and therefore apparent power, but they will have no effect on the active power transferred. Hence, harmonic currents will reduce the power factor. Harmonic currents can be reduced by a filter placed at the input of the device. Typically this will consist of either just a capacitor (relying on parasitic resistance and inductance in the supply) or a capacitor-inductor network. An active <a href="/wiki/Power_factor_correction" class="mw-redirect" title="Power factor correction">power factor correction</a> circuit at the input would generally reduce the harmonic currents further and maintain the power factor closer to unity. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AC_power&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1259569809">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{clear:left;float:left;margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Crystal_energy.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Crystal_energy.svg/29px-Crystal_energy.svg.png" decoding="async" width="29" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Crystal_energy.svg/44px-Crystal_energy.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Crystal_energy.svg/59px-Crystal_energy.svg.png 2x" data-file-width="130" data-file-height="124" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Energy" title="Portal:Energy">Energy portal</a></span></li></ul> <ul><li><a href="/wiki/War_of_the_currents" title="War of the currents">War of the currents</a></li> <li><a href="/wiki/Electric_power_transmission" title="Electric power transmission">Electric power transmission</a></li> <li><a href="/wiki/Transformer" title="Transformer">Transformer</a></li> <li><a href="/wiki/Mains_electricity" title="Mains electricity">Mains electricity</a></li> <li><a href="/wiki/Deformed_power" title="Deformed power">Deformed power</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AC_power&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-IEEE_1459-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-IEEE_1459_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-IEEE_1459_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation book cs1"><i>IEEE Standard Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions</i>. IEEE. 2010. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FIEEESTD.2010.5439063">10.1109/IEEESTD.2010.5439063</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7381-6058-0" title="Special:BookSources/978-0-7381-6058-0"><bdi>978-0-7381-6058-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=IEEE+Standard+Definitions+for+the+Measurement+of+Electric+Power+Quantities+Under+Sinusoidal%2C+Nonsinusoidal%2C+Balanced%2C+or+Unbalanced+Conditions&rft.pub=IEEE&rft.date=2010&rft_id=info%3Adoi%2F10.1109%2FIEEESTD.2010.5439063&rft.isbn=978-0-7381-6058-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAC+power" class="Z3988"></span></span> </li> <li id="cite_note-ThomasRosaToussaint_2016-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-ThomasRosaToussaint_2016_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThomasRosaToussaint2016" class="citation book cs1">Thomas, Roland E.; Rosa, Albert J.; Toussaint, Gregory J. (2016). <i>The Analysis and Design of Linear Circuits</i> (8 ed.). Wiley. pp. 812–813. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-119-23538-5" title="Special:BookSources/978-1-119-23538-5"><bdi>978-1-119-23538-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Analysis+and+Design+of+Linear+Circuits&rft.pages=812-813&rft.edition=8&rft.pub=Wiley&rft.date=2016&rft.isbn=978-1-119-23538-5&rft.aulast=Thomas&rft.aufirst=Roland+E.&rft.au=Rosa%2C+Albert+J.&rft.au=Toussaint%2C+Gregory+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAC+power" class="Z3988"></span></span> </li> <li id="cite_note-Das_2015-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Das_2015_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDas2015" class="citation book cs1">Das, J. C. (2015). <i>Power System Harmonics and Passive Filter Design</i>. Wiley, IEEE Press. p. 2. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-118-86162-2" title="Special:BookSources/978-1-118-86162-2"><bdi>978-1-118-86162-2</bdi></a>. <q>To distinguish between linear and nonlinear loads, we may say that linear time-invariant loads are characterized so that an application of a sinusoidal voltage results in a sinusoidal flow of current.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Power+System+Harmonics+and+Passive+Filter+Design&rft.pages=2&rft.pub=Wiley%2C+IEEE+Press&rft.date=2015&rft.isbn=978-1-118-86162-2&rft.aulast=Das&rft.aufirst=J.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAC+power" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://electricalnotes.wordpress.com/2011/03/21/importance-of-reactive-power-for-system/">"Importance of Reactive Power for System"</a>. 21 March 2011. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150512200158/https://electricalnotes.wordpress.com/2011/03/21/importance-of-reactive-power-for-system/">Archived</a> from the original on 2015-05-12<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-04-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Importance+of+Reactive+Power+for+System&rft.date=2011-03-21&rft_id=https%3A%2F%2Felectricalnotes.wordpress.com%2F2011%2F03%2F21%2Fimportance-of-reactive-power-for-system%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAC+power" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><i><a rel="nofollow" class="external text" href="http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-11-42">Definition of Active Power in the International Electrotechnical Vocabulary</a></i> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150423120137/http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-11-42">Archived</a> April 23, 2015, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><i>IEEE 100 : the authoritative dictionary of IEEE standards terms.-7th ed.</i> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7381-2601-2" title="Special:BookSources/0-7381-2601-2">0-7381-2601-2</a>, page 23</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20071020070028/http://www.ferc.gov/industries/electric/indus-act/blackout/09-12-03-blackout-sum.pdf">"August 14, 2003 Outage – Sequence of Events"</a> <span class="cs1-format">(PDF)</span>. <a href="/wiki/FERC" class="mw-redirect" title="FERC">FERC</a>. 2003-09-12. Archived from <a rel="nofollow" class="external text" href="https://www.ferc.gov/industries/electric/indus-act/blackout/09-12-03-blackout-sum.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2007-10-20<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-02-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=August+14%2C+2003+Outage+%E2%80%93+Sequence+of+Events&rft.pub=FERC&rft.date=2003-09-12&rft_id=http%3A%2F%2Fwww.ferc.gov%2Findustries%2Felectric%2Findus-act%2Fblackout%2F09-12-03-blackout-sum.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAC+power" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClose" class="citation book cs1">Close, Charles M. <i>The Analysis of Linear Circuits</i>. pp. 398 (section 8.3).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Analysis+of+Linear+Circuits&rft.pages=398+%28section+8.3%29&rft.aulast=Close&rft.aufirst=Charles+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAC+power" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20151025015726/http://web.media.mit.edu/~dolguin/mas961/loads.html">"Load differentiation"</a>. Archived from <a rel="nofollow" class="external text" href="http://web.media.mit.edu/~dolguin/mas961/loads.html">the original</a> on 2015-10-25<span class="reference-accessdate">. Retrieved <span class="nowrap">2015-04-29</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Load+differentiation&rft_id=http%3A%2F%2Fweb.media.mit.edu%2F~dolguin%2Fmas961%2Floads.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAC+power" class="Z3988"></span></span> </li> <li id="cite_note-Emanuel_1993-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Emanuel_1993_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Emanuel_1993_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="Emanuel_1993" class="citation journal cs1">Emanuel, Alexander (July 1993). "On The Definition of Power Factor and Apparent Power in Unbalanced Polyphase Circuits with Sinusoidal Voltage and Currents". <i>IEEE Transactions on Power Delivery</i>. <b>8</b> (3): 841–852. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F61.252612">10.1109/61.252612</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Power+Delivery&rft.atitle=On+The+Definition+of+Power+Factor+and+Apparent+Power+in+Unbalanced+Polyphase+Circuits+with+Sinusoidal+Voltage+and+Currents&rft.volume=8&rft.issue=3&rft.pages=841-852&rft.date=1993-07&rft_id=info%3Adoi%2F10.1109%2F61.252612&rft.aulast=Emanuel&rft.aufirst=Alexander&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAC+power" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=AC_power&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.circuit-magic.com/acpower.htm">"AC Power Java Applet"</a></li></ul> <div class="navbox-styles"><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 li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Electricity_delivery" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><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:Electricity_delivery" title="Template:Electricity delivery"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Electricity_delivery" title="Template talk:Electricity delivery"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Electricity_delivery" title="Special:EditPage/Template:Electricity delivery"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Electricity_delivery" style="font-size:114%;margin:0 4em"><a href="/wiki/Electricity_delivery" title="Electricity delivery">Electricity delivery</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Automatic_generation_control" title="Automatic generation control">Automatic generation control</a></li> <li><a href="/wiki/Backfeeding" title="Backfeeding">Backfeeding</a></li> <li><a href="/wiki/Base_load" title="Base load">Base load</a></li> <li><a href="/wiki/Demand_factor" title="Demand factor">Demand factor</a></li> <li><a href="/wiki/Droop_speed_control" title="Droop speed control">Droop speed control</a></li> <li><a href="/wiki/Electric_power" title="Electric power">Electric power</a></li> <li><a href="/wiki/Electric_power_quality" title="Electric power quality">Electric power quality</a></li> <li><a href="/wiki/Electrical_fault" title="Electrical fault">Electrical fault</a></li> <li><a href="/wiki/Energy_demand_management" title="Energy demand management">Energy demand management</a></li> <li><a href="/wiki/Energy_return_on_investment" title="Energy return on investment">Energy return on investment</a></li> <li><a href="/wiki/Grid_code" title="Grid code">Grid code</a></li> <li><a href="/wiki/Grid_energy_storage" title="Grid energy storage">Grid energy storage</a></li> <li><a href="/wiki/Grid_strength" class="mw-redirect" title="Grid strength">Grid strength</a></li> <li><a href="/wiki/Home_energy_storage" title="Home energy storage">Home energy storage</a></li> <li><a href="/wiki/Load-following_power_plant" title="Load-following power plant">Load-following</a></li> <li><a href="/wiki/Merit_order" title="Merit order">Merit order</a></li> <li><a href="/wiki/Nameplate_capacity" title="Nameplate capacity">Nameplate capacity</a></li> <li><a href="/wiki/Peak_demand" title="Peak demand">Peak demand</a></li> <li><a href="/wiki/Power_factor" title="Power factor">Power factor</a></li> <li><a href="/wiki/Power-flow_study" title="Power-flow study">Power-flow study</a></li> <li><a href="/wiki/Repowering" title="Repowering">Repowering</a></li> <li><a href="/wiki/Utility_frequency" title="Utility frequency">Utility frequency</a></li> <li><a href="/wiki/Variable_renewable_energy" title="Variable renewable energy">Variability</a></li> <li><a href="/wiki/Vehicle-to-grid" title="Vehicle-to-grid">Vehicle-to-grid</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="8" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/File:Abspannportal.jpg" class="mw-file-description" title="Portal pylons of Kriftel substation near Frankfurt"><img alt="Portal pylons of Kriftel substation near Frankfurt" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Abspannportal.jpg/120px-Abspannportal.jpg" decoding="async" width="120" height="90" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Abspannportal.jpg/180px-Abspannportal.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Abspannportal.jpg/240px-Abspannportal.jpg 2x" data-file-width="2048" data-file-height="1536" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sources</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:7em"><a href="/wiki/Non-renewable_resource" title="Non-renewable resource">Non-renewable</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Fossil_fuel_power_station" title="Fossil fuel power station">Fossil fuel power station</a> <ul><li><a href="/wiki/Coal" title="Coal">Coal</a></li> <li><a href="/wiki/Natural_gas" title="Natural gas">Natural gas</a></li> <li><a href="/wiki/Oil_shale" title="Oil shale">Oil shale</a></li> <li><a href="/wiki/Petroleum" title="Petroleum">Petroleum</a></li></ul></li> <li><a href="/wiki/Nuclear_power" title="Nuclear power">Nuclear</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:7em"><a href="/wiki/Renewable_energy" title="Renewable energy">Renewable</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Biofuel" title="Biofuel">Biofuel</a></li> <li><a href="/wiki/Biogas" title="Biogas">Biogas</a></li> <li><a href="/wiki/Biomass" title="Biomass">Biomass</a></li> <li><a href="/wiki/Geothermal_power" title="Geothermal power">Geothermal</a></li> <li><a href="/wiki/Hydroelectricity" title="Hydroelectricity">Hydro</a></li> <li><a href="/wiki/Marine_energy" title="Marine energy">Marine</a> <ul><li><a href="/wiki/Marine_current_power" title="Marine current power">Current</a></li> <li><a href="/wiki/Osmotic_power" title="Osmotic power">Osmotic</a></li> <li><a href="/wiki/Ocean_thermal_energy_conversion" title="Ocean thermal energy conversion">Thermal</a></li> <li><a href="/wiki/Tidal_power" title="Tidal power">Tidal</a></li> <li><a href="/wiki/Wave_power" title="Wave power">Wave</a></li></ul></li> <li><a href="/wiki/Solar_power" title="Solar power">Solar</a></li> <li><a href="/wiki/Sustainable_biofuel" title="Sustainable biofuel">Sustainable biofuel</a></li> <li><a href="/wiki/Wind_power" title="Wind power">Wind</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Power_station_technology" title="Category:Power station technology">Generation</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">AC power</a></li> <li><a href="/wiki/Cogeneration" title="Cogeneration">Cogeneration</a></li> <li><a href="/wiki/Combined_cycle_power_plant" title="Combined cycle power plant">Combined cycle</a></li> <li><a href="/wiki/Cooling_tower" title="Cooling tower">Cooling tower</a></li> <li><a href="/wiki/Induction_generator" title="Induction generator">Induction generator</a></li> <li><a href="/wiki/Micro_combined_heat_and_power" title="Micro combined heat and power">Micro CHP</a></li> <li><a href="/wiki/Microgeneration" title="Microgeneration">Microgeneration</a></li> <li><a href="/wiki/Rankine_cycle" title="Rankine cycle">Rankine cycle</a></li> <li><a href="/wiki/Three-phase_electric_power" title="Three-phase electric power">Three-phase electric power</a></li> <li><a href="/wiki/Virtual_power_plant" title="Virtual power plant">Virtual power plant</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;"><a href="/wiki/Electric_power_transmission" title="Electric power transmission">Transmission</a><br />and <a href="/wiki/Electric_power_distribution" title="Electric power distribution">distribution</a></div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Demand_response" title="Demand response">Demand response</a></li> <li><a href="/wiki/Distributed_generation" title="Distributed generation">Distributed generation</a></li> <li><a href="/wiki/Dynamic_demand_(electric_power)" title="Dynamic demand (electric power)">Dynamic demand</a></li> <li><a href="/wiki/Electric_power_distribution" title="Electric power distribution">Electric power distribution</a></li> <li><a href="/wiki/Electric_power_system" title="Electric power system">Electric power system</a></li> <li><a href="/wiki/Electric_power_transmission" title="Electric power transmission">Electric power transmission</a></li> <li><a href="/wiki/Electrical_busbar_system" title="Electrical busbar system">Electrical busbar system</a></li> <li><a href="/wiki/Electrical_grid" title="Electrical grid">Electrical grid</a></li> <li><a href="/wiki/Electrical_substation" class="mw-redirect" title="Electrical substation">Electrical substation</a></li> <li><a href="/wiki/Electricity_retailing" title="Electricity retailing">Electricity retailing</a></li> <li><a href="/wiki/High-voltage_direct_current" title="High-voltage direct current">High-voltage direct current</a></li> <li><a href="/wiki/High-voltage_shore_connection" title="High-voltage shore connection">High-voltage shore connection</a></li> <li><a href="/wiki/Interconnector" title="Interconnector">Interconnector</a></li> <li><a href="/wiki/Load_management" title="Load management">Load management</a></li> <li><a href="/wiki/Mains_electricity_by_country" title="Mains electricity by country">Mains electricity by country</a></li> <li><a href="/wiki/Overhead_power_line" title="Overhead power line">Overhead power line</a></li> <li><a href="/wiki/Power_station" title="Power station">Power station</a></li> <li><a href="/wiki/Pumped-storage_hydroelectricity" title="Pumped-storage hydroelectricity">Pumped hydro</a></li> <li><a href="/wiki/Single-wire_earth_return" title="Single-wire earth return">Single-wire earth return</a></li> <li><a href="/wiki/Smart_grid" title="Smart grid">Smart grid</a></li> <li><a href="/wiki/Super_grid" title="Super grid">Super grid</a></li> <li><a href="/wiki/Transformer" title="Transformer">Transformer</a></li> <li><a href="/wiki/Transmission_system_operator" title="Transmission system operator">Transmission system operator</a> (TSO)</li> <li><a href="/wiki/Transmission_tower" title="Transmission tower">Transmission tower</a></li> <li><a href="/wiki/Utility_pole" title="Utility pole">Utility pole</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Failure modes</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Black_start" title="Black start">Black start</a></li> <li><a href="/wiki/Brownout_(electricity)" title="Brownout (electricity)">Brownout</a></li> <li><a href="/wiki/Cascading_failure" title="Cascading failure">Cascading failure</a></li> <li><a href="/wiki/Power_outage" title="Power outage">Power outage</a> <ul><li><a href="/wiki/Rolling_blackout" title="Rolling blackout">Rolling blackout</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Protective<br />devices</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arc-fault_circuit_interrupter" title="Arc-fault circuit interrupter">Arc-fault circuit interrupter</a></li> <li><a href="/wiki/Circuit_breaker" title="Circuit breaker">Circuit breaker</a> <ul><li><a href="/wiki/Earth-leakage_circuit_breaker" title="Earth-leakage circuit breaker">Earth-leakage</a></li> <li><a href="/wiki/Sulfur_hexafluoride_circuit_breaker" title="Sulfur hexafluoride circuit breaker">Sulfur hexafluoride</a></li></ul></li> <li><a href="/wiki/Generator_interlock_kit" title="Generator interlock kit">Generator interlock kit</a></li> <li><a href="/wiki/Numerical_relay" title="Numerical relay">Numerical relay</a></li> <li><a href="/wiki/Power_system_protection" title="Power system protection">Power system protection</a></li> <li><a href="/wiki/Protective_relay" title="Protective relay">Protective relay</a></li> <li><a href="/wiki/Residual-current_device" title="Residual-current device">Residual-current device</a> (GFI)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Economics<br />and policies</div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Availability_factor" title="Availability factor">Availability factor</a></li> <li><a href="/wiki/Capacity_factor" title="Capacity factor">Capacity factor</a></li> <li><a href="/wiki/Carbon_offsets_and_credits" title="Carbon offsets and credits">Carbon offsets and credits</a></li> <li><a href="/wiki/Cost_of_electricity_by_source" title="Cost of electricity by source">Cost of electricity by source</a></li> <li><a href="/wiki/Energy_subsidy" title="Energy subsidy">Energy subsidies</a></li> <li><a href="/wiki/Environmental_tax" title="Environmental tax">Environmental tax</a></li> <li><a href="/wiki/Feed-in_tariff" title="Feed-in tariff">Feed-in tariff</a></li> <li><a href="/wiki/Fossil_fuel_phase-out" title="Fossil fuel phase-out">Fossil fuel phase-out</a></li> <li><a href="/wiki/Load_factor_(electrical)" title="Load factor (electrical)">Load factor</a></li> <li><a href="/wiki/Net_metering" title="Net metering">Net metering</a></li> <li><a href="/wiki/Pigouvian_tax" title="Pigouvian tax">Pigouvian tax</a></li> <li><a href="/wiki/Renewable_Energy_Certificate_(United_States)" title="Renewable Energy Certificate (United States)">Renewable Energy Certificates</a></li> <li><a href="/wiki/Renewable_energy_commercialization" title="Renewable energy commercialization">Renewable energy commercialization</a></li> <li><a href="/wiki/Renewable_Energy_Payments" title="Renewable Energy Payments">Renewable Energy Payments</a></li> <li><a href="/wiki/Spark_spread" title="Spark spread">Spark/Dark/Quark/Bark spread</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Statistics and<br />production</div></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Electric_energy_consumption" title="Electric energy consumption">Electric energy consumption</a></li> <li><a href="/wiki/List_of_electricity_sectors" title="List of electricity sectors">List of electricity sectors</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Electric_power_distribution" title="Category:Electric power distribution">Category</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=AC_power&oldid=1258338867">https://en.wikipedia.org/w/index.php?title=AC_power&oldid=1258338867</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:AC_power" title="Category:AC power">AC power</a></li><li><a href="/wiki/Category:Electric_power" title="Category:Electric power">Electric power</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Webarchive_template_wayback_links" title="Category:Webarchive template wayback links">Webarchive template wayback links</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 19 November 2024, at 04:12<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=AC_power&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5857dfdcd6-ngp8f","wgBackendResponseTime":880,"wgPageParseReport":{"limitreport":{"cputime":"0.523","walltime":"0.716","ppvisitednodes":{"value":1884,"limit":1000000},"postexpandincludesize":{"value":43300,"limit":2097152},"templateargumentsize":{"value":1679,"limit":2097152},"expansiondepth":{"value":16,"limit":100},"expensivefunctioncount":{"value":5,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":44048,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 528.773 1 -total"," 35.28% 186.574 1 Template:Reflist"," 21.14% 111.794 4 Template:Cite_book"," 20.18% 106.700 1 Template:Electricity_generation"," 19.77% 104.542 2 Template:Navbox"," 14.70% 77.734 1 Template:Short_description"," 9.34% 49.388 2 Template:Pagetype"," 7.05% 37.304 1 Template:About"," 6.78% 35.854 2 Template:Rp"," 5.61% 29.643 2 Template:R/superscript"]},"scribunto":{"limitreport-timeusage":{"value":"0.296","limit":"10.000"},"limitreport-memusage":{"value":6601442,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-5857dfdcd6-ngp8f","timestamp":"20241203070253","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"AC power","url":"https:\/\/en.wikipedia.org\/wiki\/AC_power","sameAs":"http:\/\/www.wikidata.org\/entity\/Q111954952","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q111954952","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2005-04-22T17:52:30Z","dateModified":"2024-11-19T04:12:39Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/4\/42\/City_lights_in_motion.jpg","headline":"power in alternating current systems"}</script> </body> </html>

CINXE.COM

AC power - Wikipedia