Kvadratiskt medelvärde

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id="toc-Definition-sublist" class="vector-toc-list"> <li id="toc-Exempel" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Exempel"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Exempel</span> </div> </a> <ul id="toc-Exempel-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Kvadratiskt_medelvärde_för_olika_vågformer" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kvadratiskt_medelvärde_för_olika_vågformer"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Kvadratiskt medelvärde för olika vågformer</span> </div> </a> <ul id="toc-Kvadratiskt_medelvärde_för_olika_vågformer-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Summering_av_kvadratiska_medelvärden" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Summering_av_kvadratiska_medelvärden"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Summering av kvadratiska medelvärden</span> </div> </a> <ul id="toc-Summering_av_kvadratiska_medelvärden-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tillämpningar" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Tillämpningar"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Tillämpningar</span> </div> </a> <ul id="toc-Tillämpningar-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jämförelse_med_andra_medelvärden" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Jämförelse_med_andra_medelvärden"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Jämförelse med andra medelvärden</span> </div> </a> <ul id="toc-Jämförelse_med_andra_medelvärden-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Se_även" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Se_även"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Se även</span> </div> </a> <ul id="toc-Se_även-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referenser" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referenser"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Referenser</span> </div> </a> <button aria-controls="toc-Referenser-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>Växla underavsnittet Referenser</span> </button> <ul id="toc-Referenser-sublist" class="vector-toc-list"> <li id="toc-Noter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Noter"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Noter</span> </div> </a> <ul id="toc-Noter-sublist" class="vector-toc-list"> </ul> </li> </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="Innehåll" 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="Växla innehållsförteckningen" > <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">Växla innehållsförteckningen</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">Kvadratiskt medelvärde</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="Gå till en artikel på ett annat språk. Tillgänglig på 39 språk" > <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-39" 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">39 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AC%D8%B0%D8%B1_%D9%85%D8%AA%D9%88%D8%B3%D8%B7_%D9%85%D8%B1%D8%A8%D8%B9" title="جذر متوسط مربع – arabiska" lang="ar" hreflang="ar" data-title="جذر متوسط مربع" data-language-autonym="العربية" data-language-local-name="arabiska" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%B0%E0%A7%8D%E0%A6%97%E0%A6%AE%E0%A7%82%E0%A6%B2_%E0%A6%97%E0%A6%A1%E0%A6%BC_%E0%A6%AC%E0%A6%B0%E0%A7%8D%E0%A6%97" title="বর্গমূল গড় বর্গ – bengali" lang="bn" hreflang="bn" data-title="বর্গমূল গড় বর্গ" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A1%D1%8F%D1%80%D1%8D%D0%B4%D0%BD%D1%8F%D0%B5_%D0%BA%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D0%B5" title="Сярэдняе квадратычнае – belarusiska (tarasjkevitsa)" lang="be-tarask" hreflang="be-tarask" data-title="Сярэдняе квадратычнае" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="belarusiska (tarasjkevitsa)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D1%80%D0%B5%D0%B4%D0%BD%D0%BE_%D0%BA%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%B8%D1%87%D0%BD%D0%BE" title="Средно квадратично – bulgariska" lang="bg" hreflang="bg" data-title="Средно квадратично" data-language-autonym="Български" data-language-local-name="bulgariska" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Mitjana_quadr%C3%A0tica" title="Mitjana quadràtica – katalanska" lang="ca" hreflang="ca" data-title="Mitjana quadràtica" data-language-autonym="Català" data-language-local-name="katalanska" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kvadratick%C3%BD_pr%C5%AFm%C4%9Br" title="Kvadratický průměr – tjeckiska" lang="cs" hreflang="cs" data-title="Kvadratický průměr" data-language-autonym="Čeština" data-language-local-name="tjeckiska" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Effektiv_v%C3%A6rdi" title="Effektiv værdi – danska" lang="da" hreflang="da" data-title="Effektiv værdi" data-language-autonym="Dansk" data-language-local-name="danska" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Quadratisches_Mittel" title="Quadratisches Mittel – tyska" lang="de" hreflang="de" data-title="Quadratisches Mittel" data-language-autonym="Deutsch" data-language-local-name="tyska" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ruutkeskmine" title="Ruutkeskmine – estniska" lang="et" hreflang="et" data-title="Ruutkeskmine" data-language-autonym="Eesti" data-language-local-name="estniska" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Root_mean_square" title="Root mean square – engelska" lang="en" hreflang="en" data-title="Root mean square" data-language-autonym="English" data-language-local-name="engelska" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Media_cuadr%C3%A1tica" title="Media cuadrática – spanska" lang="es" hreflang="es" data-title="Media cuadrática" data-language-autonym="Español" data-language-local-name="spanska" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kvadrata_avera%C4%9Do" title="Kvadrata averaĝo – esperanto" lang="eo" hreflang="eo" data-title="Kvadrata averaĝo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Batezbesteko_koadratiko" title="Batezbesteko koadratiko – baskiska" lang="eu" hreflang="eu" data-title="Batezbesteko koadratiko" data-language-autonym="Euskara" data-language-local-name="baskiska" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AC%D8%B0%D8%B1_%D9%85%DB%8C%D8%A7%D9%86%DA%AF%DB%8C%D9%86_%D9%85%D8%B1%D8%A8%D8%B9%D8%A7%D8%AA" title="جذر میانگین مربعات – persiska" lang="fa" hreflang="fa" data-title="جذر میانگین مربعات" data-language-autonym="فارسی" data-language-local-name="persiska" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Moyenne_quadratique" title="Moyenne quadratique – franska" lang="fr" hreflang="fr" data-title="Moyenne quadratique" data-language-autonym="Français" data-language-local-name="franska" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%9C%EA%B3%B1%ED%8F%89%EA%B7%A0%EC%A0%9C%EA%B3%B1%EA%B7%BC" title="제곱평균제곱근 – koreanska" lang="ko" hreflang="ko" data-title="제곱평균제곱근" data-language-autonym="한국어" data-language-local-name="koreanska" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97_%E0%A4%AE%E0%A4%BE%E0%A4%A7%E0%A5%8D%E0%A4%AF_%E0%A4%AE%E0%A5%82%E0%A4%B2" title="वर्ग माध्य मूल – hindi" lang="hi" hreflang="hi" data-title="वर्ग माध्य मूल" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Rata-rata_kuadrat" title="Rata-rata kuadrat – indonesiska" lang="id" hreflang="id" data-title="Rata-rata kuadrat" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiska" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%95%D7%A8%D7%A9_%D7%9E%D7%9E%D7%95%D7%A6%D7%A2_%D7%94%D7%A8%D7%99%D7%91%D7%95%D7%A2%D7%99%D7%9D" title="שורש ממוצע הריבועים – hebreiska" lang="he" hreflang="he" data-title="שורש ממוצע הריבועים" data-language-autonym="עברית" data-language-local-name="hebreiska" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D1%82%D1%8B%D2%9B_%D0%BE%D1%80%D1%82%D0%B0%D1%88%D0%B0%D0%BB%D0%B0%D1%80" title="Квадраттық орташалар – kazakiska" lang="kk" hreflang="kk" data-title="Квадраттық орташалар" data-language-autonym="Қазақша" data-language-local-name="kazakiska" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/N%C3%A9gyzetes_k%C3%B6z%C3%A9p" title="Négyzetes közép – ungerska" lang="hu" hreflang="hu" data-title="Négyzetes közép" data-language-autonym="Magyar" data-language-local-name="ungerska" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Kwadratisch_gemiddelde" title="Kwadratisch gemiddelde – nederländska" lang="nl" hreflang="nl" data-title="Kwadratisch gemiddelde" data-language-autonym="Nederlands" data-language-local-name="nederländska" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BA%8C%E4%B9%97%E5%B9%B3%E5%9D%87%E5%B9%B3%E6%96%B9%E6%A0%B9" title="二乗平均平方根 – japanska" lang="ja" hreflang="ja" data-title="二乗平均平方根" data-language-autonym="日本語" data-language-local-name="japanska" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Kvadratisk_gjennomsnitt" title="Kvadratisk gjennomsnitt – norskt bokmål" lang="nb" hreflang="nb" data-title="Kvadratisk gjennomsnitt" data-language-autonym="Norsk bokmål" data-language-local-name="norskt bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Media_quadr%C3%A0tica" title="Media quadràtica – piemontesiska" lang="pms" hreflang="pms" data-title="Media quadràtica" data-language-autonym="Piemontèis" data-language-local-name="piemontesiska" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/%C5%9Arednia_kwadratowa" title="Średnia kwadratowa – polska" lang="pl" hreflang="pl" data-title="Średnia kwadratowa" data-language-autonym="Polski" data-language-local-name="polska" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Valor_eficaz" title="Valor eficaz – portugisiska" lang="pt" hreflang="pt" data-title="Valor eficaz" data-language-autonym="Português" data-language-local-name="portugisiska" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Medie_p%C4%83tratic%C4%83" title="Medie pătratică – rumänska" lang="ro" hreflang="ro" data-title="Medie pătratică" data-language-autonym="Română" data-language-local-name="rumänska" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D1%80%D0%B5%D0%B4%D0%BD%D0%B5%D0%B5_%D0%BA%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5" title="Среднее квадратическое – ryska" lang="ru" hreflang="ru" data-title="Среднее квадратическое" data-language-autonym="Русский" data-language-local-name="ryska" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Root_mean_square" title="Root mean square – Simple English" lang="en-simple" hreflang="en-simple" data-title="Root mean square" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kvadratick%C3%BD_priemer" title="Kvadratický priemer – slovakiska" lang="sk" hreflang="sk" data-title="Kvadratický priemer" data-language-autonym="Slovenčina" data-language-local-name="slovakiska" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%B2%D0%B0%D0%B4%D1%80%D0%B0%D1%82%D0%BD%D0%B0_%D1%81%D1%80%D0%B5%D0%B4%D0%B8%D0%BD%D0%B0" title="Квадратна средина – serbiska" lang="sr" hreflang="sr" data-title="Квадратна средина" data-language-autonym="Српски / srpski" data-language-local-name="serbiska" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Neli%C3%B6llinen_keskiarvo" title="Neliöllinen keskiarvo – finska" lang="fi" hreflang="fi" data-title="Neliöllinen keskiarvo" data-language-autonym="Suomi" data-language-local-name="finska" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%84%E0%B9%88%E0%B8%B2%E0%B9%80%E0%B8%89%E0%B8%A5%E0%B8%B5%E0%B9%88%E0%B8%A2%E0%B8%81%E0%B8%B3%E0%B8%A5%E0%B8%B1%E0%B8%87%E0%B8%AA%E0%B8%AD%E0%B8%87" title="ค่าเฉลี่ยกำลังสอง – thailändska" lang="th" hreflang="th" data-title="ค่าเฉลี่ยกำลังสอง" data-language-autonym="ไทย" data-language-local-name="thailändska" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Karek%C3%B6k_ortalama" title="Karekök ortalama – turkiska" lang="tr" hreflang="tr" data-title="Karekök ortalama" data-language-autonym="Türkçe" data-language-local-name="turkiska" 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id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Utseende"> <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">Utseende</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">flytta till sidofältet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">dölj</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">Från Wikipedia</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="sv" dir="ltr"><p><b>Kvadratiskt medelvärde</b> är ett <a href="/wiki/Statistik" title="Statistik">statistiskt</a> mätetal för variationerna hos en storhets belopp.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-reference-link-bracket">[</span>1<span class="cite-reference-link-bracket">]</span></a></sup> Kvadratiskt medelvärde är särskilt användbart om storhetens värden är både positiva och negativa, som till exempel för <a href="/wiki/Sinus" title="Sinus">sinusformade</a> förlopp. Det kvadratiska medelvärdet kan ses som ett <a href="/wiki/Generaliserat_medelv%C3%A4rde" title="Generaliserat medelvärde">generaliserat medelvärde</a> med <i>p</i> = 2. </p><p>Den engelska beteckningen för kvadratiskt medelvärde är <i>root mean square</i> eller RMS. Inom <a href="/wiki/Elektroteknik" title="Elektroteknik">elektrotekniken</a> kallas det kvadratiska medelvärdet av en <a href="/wiki/V%C3%A4xelstr%C3%B6m" title="Växelström">växelström</a> eller <a href="/wiki/V%C3%A4xelsp%C3%A4nning" title="Växelspänning">växelspänning</a> för växelstorhetens <a href="/wiki/Effektivv%C3%A4rde" title="Effektivvärde">effektivvärde</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&veaction=edit&section=1" title="Redigera avsnitt: Definition" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&action=edit&section=1" title="Redigera avsnitts källkod: Definition"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kvadratiska medelvärdet för en uppsättning värden (eller en tidskontinuerligt varierande vågform) är <a href="/wiki/Kvadratrot" title="Kvadratrot">kvadratroten</a> ur det aritmetiska medelvärdet av kvadraten på dessa värden (eller kvadraten på den funktion som definierar den kontinuerliga vågformen). </p><p>I fallet med en mängd av <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> diskreta värden <span class="mwe-math-element"><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_{1},x_{2},\dots ,x_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x_{1},x_{2},\dots ,x_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0416a78ac28f73940431b62ab4f8fd9ee84bbfde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.853ex; height:2.843ex;" alt="{\displaystyle \{x_{1},x_{2},\dots ,x_{n}\}}"></span> ges kvadratiska medelvärdet av </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 x_{\mathrm {rms} }={\sqrt {{\frac {1}{n}}\left(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</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> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\mathrm {rms} }={\sqrt {{\frac {1}{n}}\left(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc6aeb80ed31637cc820a796e530dd5c40db52a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.954ex; height:6.176ex;" alt="{\displaystyle x_{\mathrm {rms} }={\sqrt {{\frac {1}{n}}\left(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\right)}}}"></span></dd></dl> <p>I fallet när vågformen beskrivs av en kontinuerlig funktion <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}"></span> definierad på intervallet <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{1}\leq t\leq T_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≤</mo> <mi>t</mi> <mo>≤</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1}\leq t\leq T_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd4b90746157322251391e1a894a0438e405ca7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.86ex; height:2.509ex;" alt="{\displaystyle T_{1}\leq t\leq T_{2}}"></span> beräknas kvadratiska medelvärdet som </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 f_{\mathrm {rms} }={\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{[f(t)]}^{2}\,dt}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</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> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <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> <mrow class="MJX-TeXAtom-ORD"> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {rms} }={\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{[f(t)]}^{2}\,dt}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e1b23b04684325ebcec5c997cb037fcf4708351" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.195ex; height:7.676ex;" alt="{\displaystyle f_{\mathrm {rms} }={\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{[f(t)]}^{2}\,dt}}},}"></span></dd></dl> <p>och kvadratiska medelvärdet för en funktion över ett oändligt intervall beräknas som </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 f_{\mathrm {rms} }=\lim _{T\rightarrow \infty }{\sqrt {{1 \over {T}}{\int _{0}^{T}{[f(t)]}^{2}\,dt}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</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> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </mrow> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {rms} }=\lim _{T\rightarrow \infty }{\sqrt {{1 \over {T}}{\int _{0}^{T}{[f(t)]}^{2}\,dt}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e90e70a26cc03e83aefc05be23dd2da3a5e9e8a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.716ex; height:7.509ex;" alt="{\displaystyle f_{\mathrm {rms} }=\lim _{T\rightarrow \infty }{\sqrt {{1 \over {T}}{\int _{0}^{T}{[f(t)]}^{2}\,dt}}}}"></span></dd></dl> <p>Kvadratiska medelvärdet över ett oändligt intervall är för en periodisk funktion lika med kvadratiska medelvärdet för en period av funktionen. </p> <div class="mw-heading mw-heading3"><h3 id="Exempel">Exempel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&veaction=edit&section=2" title="Redigera avsnitt: Exempel" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&action=edit&section=2" title="Redigera avsnitts källkod: Exempel"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En sinusvåg beskrivs av </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 \ x(t)=A\sin(\omega t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x(t)=A\sin(\omega t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38c4a8d14f64f593ad1e30629b62623f751b9df4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.738ex; height:2.843ex;" alt="{\displaystyle \ x(t)=A\sin(\omega t)}"></span></dd></dl> <p>där <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a29b7d7604962065930f413ee72574d1f7a6e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:2.176ex;" alt="{\displaystyle \ A}"></span> är <a href="/wiki/Amplitud" title="Amplitud">amplituden</a> och <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b2efca4774f7032b3f7ff6aeb1e086457544bd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.009ex;" alt="{\displaystyle \ t}"></span> är tiden och <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abf2c1eb8ec34fbb75667b7bf5345c987c05ea33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.026ex; height:1.676ex;" alt="{\displaystyle \ \omega }"></span> vinkelfrekvensen i <a href="/wiki/Radian" title="Radian">radianer</a> per tidsenhet. </p><p>Då </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 \omega ={\frac {2\pi }{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\frac {2\pi }{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/689caf6d9f39dbc57a3d4ee99c82c98e31193e72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.875ex; height:5.176ex;" alt="{\displaystyle \omega ={\frac {2\pi }{T}}}"></span></dd></dl> <p>kan kvadratiska medelvärdet skrivas </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 x_{\text{rms}}={\sqrt {{1 \over T}\int _{0}^{T}\,A^{2}\sin ^{2}\left({\frac {2\pi }{T}}t\right)\,dt}}={\frac {A}{\sqrt {2\pi }}}{\sqrt {\int _{0}^{2\pi }\,\sin ^{2}(t)\,dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rms</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mspace width="thinmathspace" /> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{rms}}={\sqrt {{1 \over T}\int _{0}^{T}\,A^{2}\sin ^{2}\left({\frac {2\pi }{T}}t\right)\,dt}}={\frac {A}{\sqrt {2\pi }}}{\sqrt {\int _{0}^{2\pi }\,\sin ^{2}(t)\,dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e28df48018889dc67f926cedfe35925899104a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:59.185ex; height:7.509ex;" alt="{\displaystyle x_{\text{rms}}={\sqrt {{1 \over T}\int _{0}^{T}\,A^{2}\sin ^{2}\left({\frac {2\pi }{T}}t\right)\,dt}}={\frac {A}{\sqrt {2\pi }}}{\sqrt {\int _{0}^{2\pi }\,\sin ^{2}(t)\,dt}}}"></span></dd></dl> <p>Med hjälp av likheten </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 \int \,\sin ^{2}(t)\,dt={\frac {t}{2}}-{\frac {1}{4}}\sin(2t)+C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mspace width="thinmathspace" /> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \,\sin ^{2}(t)\,dt={\frac {t}{2}}-{\frac {1}{4}}\sin(2t)+C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/328304da6971177af372b3672e02aab4e2300f44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.566ex; height:5.676ex;" alt="{\displaystyle \int \,\sin ^{2}(t)\,dt={\frac {t}{2}}-{\frac {1}{4}}\sin(2t)+C}"></span></dd></dl> <p>kan kvadratiska medelvärdet beräknas till </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 x_{\text{rms}}={\frac {A}{\sqrt {2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rms</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{\text{rms}}={\frac {A}{\sqrt {2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e60b738b16f1781b81cb9c54a04e4f50b40c582" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.256ex; height:6.343ex;" alt="{\displaystyle x_{\text{rms}}={\frac {A}{\sqrt {2}}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Kvadratiskt_medelvärde_för_olika_vågformer"><span id="Kvadratiskt_medelv.C3.A4rde_f.C3.B6r_olika_v.C3.A5gformer"></span>Kvadratiskt medelvärde för olika vågformer</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&veaction=edit&section=3" title="Redigera avsnitt: Kvadratiskt medelvärde för olika vågformer" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&action=edit&section=3" title="Redigera avsnitts källkod: Kvadratiskt medelvärde för olika vågformer"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable" border="1"> <tbody><tr> <th>Vågform</th> <th>Ekvation</th> <th>Illustration</th> <th>Kvadratiskt<br />medelvärde </th></tr> <tr> <td><a href="/wiki/Konstant" title="Konstant">Konstant</a></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=a\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>a</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=a\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0889c2d47a260fa091061f65655f2b7bec3b57df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.871ex; height:2.009ex;" alt="{\displaystyle y=a\,}"></span></td> <td><span typeof="mw:File"><a href="/wiki/Fil:V%C3%A5gform_konstant.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/V%C3%A5gform_konstant.svg/280px-V%C3%A5gform_konstant.svg.png" decoding="async" width="280" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/V%C3%A5gform_konstant.svg/420px-V%C3%A5gform_konstant.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/V%C3%A5gform_konstant.svg/560px-V%C3%A5gform_konstant.svg.png 2x" data-file-width="800" data-file-height="200" /></a></span></td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d73aa5354c24942dab5316be466465a9d171510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.617ex; height:1.676ex;" alt="{\displaystyle a\,}"></span> </td></tr> <tr> <td><a href="/wiki/Sinus" title="Sinus">Sinus</a></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=a\sin(2\pi ft)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mi>a</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=a\sin(2\pi ft)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/437bd8b56b5d1a0751999aeb8f16f83a902bf0df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.536ex; height:2.843ex;" alt="{\displaystyle y=a\sin(2\pi ft)\,}"></span></td> <td><span typeof="mw:File"><a href="/wiki/Fil:V%C3%A5gform_sinus.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/V%C3%A5gform_sinus.svg/280px-V%C3%A5gform_sinus.svg.png" decoding="async" width="280" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6b/V%C3%A5gform_sinus.svg/420px-V%C3%A5gform_sinus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6b/V%C3%A5gform_sinus.svg/560px-V%C3%A5gform_sinus.svg.png 2x" data-file-width="800" data-file-height="200" /></a></span></td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{\sqrt {2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{\sqrt {2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dda608aff02ad5e622be07bf57b0390d88958b84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:3.934ex; height:5.676ex;" alt="{\displaystyle {\frac {a}{\sqrt {2}}}}"></span> </td></tr> <tr> <td><a href="/wiki/Fyrkantsv%C3%A5g" title="Fyrkantsvåg">Fyrkant</a></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\begin{cases}a&\{ft\}<0.5\\-a&\{ft\}>0.5\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> <mo><</mo> <mn>0.5</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>a</mi> </mtd> <mtd> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> <mo>></mo> <mn>0.5</mn> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\begin{cases}a&\{ft\}<0.5\\-a&\{ft\}>0.5\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd7b149e88fc28a567d7242cc1552f9dc8ff6480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.622ex; height:6.176ex;" alt="{\displaystyle y={\begin{cases}a&\{ft\}<0.5\\-a&\{ft\}>0.5\end{cases}}}"></span></td> <td><span typeof="mw:File"><a href="/wiki/Fil:V%C3%A5gform_fyrkant.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/V%C3%A5gform_fyrkant.svg/280px-V%C3%A5gform_fyrkant.svg.png" decoding="async" width="280" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/V%C3%A5gform_fyrkant.svg/420px-V%C3%A5gform_fyrkant.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/V%C3%A5gform_fyrkant.svg/560px-V%C3%A5gform_fyrkant.svg.png 2x" data-file-width="800" data-file-height="200" /></a></span></td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d73aa5354c24942dab5316be466465a9d171510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.617ex; height:1.676ex;" alt="{\displaystyle a\,}"></span> </td></tr> <tr> <td><a href="/wiki/Triangelv%C3%A5g" class="mw-redirect" title="Triangelvåg">Triangel</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=|2a\{ft\}-a\,|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>2</mn> <mi>a</mi> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> <mo>−</mo> <mi>a</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=|2a\{ft\}-a\,|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090d3c5fa26f6d4bd21d44df88383438c3404756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.84ex; height:2.843ex;" alt="{\displaystyle y=|2a\{ft\}-a\,|}"></span></td> <td><span typeof="mw:File"><a href="/wiki/Fil:V%C3%A5gform_triangel.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/V%C3%A5gform_triangel.svg/280px-V%C3%A5gform_triangel.svg.png" decoding="async" width="280" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/V%C3%A5gform_triangel.svg/420px-V%C3%A5gform_triangel.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b8/V%C3%A5gform_triangel.svg/560px-V%C3%A5gform_triangel.svg.png 2x" data-file-width="800" data-file-height="200" /></a></span></td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a \over {\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> </mrow> <annotation encoding="application/x-tex">{\displaystyle a \over {\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6df5978459e4f28c850c4e386055219fcf0d3de0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:3.934ex; height:5.676ex;" alt="{\displaystyle a \over {\sqrt {3}}}"></span> </td></tr> <tr> <td><a href="/wiki/S%C3%A5gtandskurva" title="Sågtandskurva">Sågtand</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=2a\{ft\}-a\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>2</mn> <mi>a</mi> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> <mo>−</mo> <mi>a</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=2a\{ft\}-a\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cffe2b658e53588e93adb915da9108899088ffe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.547ex; height:2.843ex;" alt="{\displaystyle y=2a\{ft\}-a\,}"></span></td> <td><span typeof="mw:File"><a href="/wiki/Fil:V%C3%A5gform_s%C3%A5gtand.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/V%C3%A5gform_s%C3%A5gtand.svg/280px-V%C3%A5gform_s%C3%A5gtand.svg.png" decoding="async" width="280" height="70" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/V%C3%A5gform_s%C3%A5gtand.svg/420px-V%C3%A5gform_s%C3%A5gtand.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/V%C3%A5gform_s%C3%A5gtand.svg/560px-V%C3%A5gform_s%C3%A5gtand.svg.png 2x" data-file-width="800" data-file-height="200" /></a></span></td> <td align="center"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a \over {\sqrt {3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> </mrow> <annotation encoding="application/x-tex">{\displaystyle a \over {\sqrt {3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6df5978459e4f28c850c4e386055219fcf0d3de0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:3.934ex; height:5.676ex;" alt="{\displaystyle a \over {\sqrt {3}}}"></span> </td></tr> <tr> <td colspan="4"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> är <a href="/wiki/Amplitud" title="Amplitud">amplituden</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> är <a href="/wiki/Frekvens" title="Frekvens">frekvensen</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span> är tiden och <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{ft\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{ft\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/477373f4d8fd59acc60c71f208ee95f97455a82d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.443ex; height:2.843ex;" alt="{\displaystyle \{ft\}}"></span> är decimaldelen av <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ft}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ft}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/715674977ba604775cd718b430eeb84c984ccc4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.118ex; height:2.509ex;" alt="{\displaystyle ft}"></span>, d.v.s <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq \{ft\}<1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mo fence="false" stretchy="false">{</mo> <mi>f</mi> <mi>t</mi> <mo fence="false" stretchy="false">}</mo> <mo><</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq \{ft\}<1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de6c9857afa96e71883a1b2af48d4cc46d5339a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.965ex; height:2.843ex;" alt="{\displaystyle 0\leq \{ft\}<1}"></span> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Summering_av_kvadratiska_medelvärden"><span id="Summering_av_kvadratiska_medelv.C3.A4rden"></span>Summering av kvadratiska medelvärden</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&veaction=edit&section=4" title="Redigera avsnitt: Summering av kvadratiska medelvärden" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&action=edit&section=4" title="Redigera avsnitts källkod: Summering av kvadratiska medelvärden"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vågformer som bildats genom summering av vågformer, har ett RMS-värde som är roten ur summan av kvadraterna av komponenternas RMS-värden om de ingående vågformerna är <a href="/wiki/Ortogonal" class="mw-redirect" title="Ortogonal">ortogonala</a> (det vill säga, om den genomsnittliga produkten av en av de ingående vågformerna med en annan är noll för alla par andra än en vågform multiplicerad med sig själv): </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 RMS_{Totalt}={\sqrt {{RMS_{1}}^{2}+{RMS_{2}}^{2}+\cdots +{RMS_{n}}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>M</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mi>o</mi> <mi>t</mi> <mi>a</mi> <mi>l</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>M</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>M</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mi>M</mi> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RMS_{Totalt}={\sqrt {{RMS_{1}}^{2}+{RMS_{2}}^{2}+\cdots +{RMS_{n}}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a085ba0bef8161d0a2622cc11a459f748c0bccd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:50.415ex; height:4.843ex;" alt="{\displaystyle RMS_{Totalt}={\sqrt {{RMS_{1}}^{2}+{RMS_{2}}^{2}+\cdots +{RMS_{n}}^{2}}}}"></span></dd></dl> <p>Exempelvis är vågformerna för sinus- och cosinusfunktionen ortogonala då </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 \int _{0}^{2\pi }\sin(x)\cos(x)dx=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>π</mi> </mrow> </msubsup> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{0}^{2\pi }\sin(x)\cos(x)dx=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82bc6a66aaf93cb9726ace058a938d807756d03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.286ex; height:6.176ex;" alt="{\displaystyle \int _{0}^{2\pi }\sin(x)\cos(x)dx=0}"></span></dd></dl> <p>och RMS-värdet för vågformernas summa blir </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 {\sqrt {\left({\frac {1}{\sqrt {2}}}\right)^{2}+\left({\frac {1}{\sqrt {2}}}\right)^{2}}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {\left({\frac {1}{\sqrt {2}}}\right)^{2}+\left({\frac {1}{\sqrt {2}}}\right)^{2}}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e6e0b1c90b40bfc50d2a0f70f4b08a49e53b255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.245ex; height:7.676ex;" alt="{\displaystyle {\sqrt {\left({\frac {1}{\sqrt {2}}}\right)^{2}+\left({\frac {1}{\sqrt {2}}}\right)^{2}}}=1}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Tillämpningar"><span id="Till.C3.A4mpningar"></span>Tillämpningar</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&veaction=edit&section=5" title="Redigera avsnitt: Tillämpningar" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&action=edit&section=5" title="Redigera avsnitts källkod: Tillämpningar"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Effektivv%C3%A4rde-png.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Effektivv%C3%A4rde-png.png/250px-Effektivv%C3%A4rde-png.png" decoding="async" width="250" height="265" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Effektivv%C3%A4rde-png.png/375px-Effektivv%C3%A4rde-png.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Effektivv%C3%A4rde-png.png/500px-Effektivv%C3%A4rde-png.png 2x" data-file-width="730" data-file-height="775" /></a><figcaption>Effektutvecklingen för en sinusformad ström och spänning i en <a href="/wiki/Resistor" title="Resistor">resistor</a>. Momentaneffektens medeleffekt kan beräknas som <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ u_{\text{rms}}i_{\text{rms}}={\frac {\hat {u}}{\sqrt {2}}}{\frac {\hat {i}}{\sqrt {2}}}={\frac {1}{2}}{\hat {u}}{\hat {i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rms</mtext> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>rms</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>i</mi> <mo stretchy="false">^</mo> </mover> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>u</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>i</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ u_{\text{rms}}i_{\text{rms}}={\frac {\hat {u}}{\sqrt {2}}}{\frac {\hat {i}}{\sqrt {2}}}={\frac {1}{2}}{\hat {u}}{\hat {i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82ac93759b6d85af9664a500840c7ff3a046fba0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.057ex; height:6.843ex;" alt="{\displaystyle \ u_{\text{rms}}i_{\text{rms}}={\frac {\hat {u}}{\sqrt {2}}}{\frac {\hat {i}}{\sqrt {2}}}={\frac {1}{2}}{\hat {u}}{\hat {i}}}"></span></figcaption></figure> <p>Inom <a href="/wiki/Fysik" title="Fysik">fysik</a> och <a href="/wiki/Elektroteknik" title="Elektroteknik">elektroteknik</a> används det kvadratiska medelvärdet för effektberäkningar av svängande system som till exempel elektriska svängningskretsar, akustiska vågor, ledningsresonatorer och <a href="/wiki/H%C3%A5lrumsresonator" title="Hålrumsresonator">hålrumsresonatorer</a>. Villkoret för det kvadratiska medelvärdets användbarhet för effektberäkningar är att det svängande systemets momentaneffekt är proportionell mot kvadraten på systemets momentanvärde. </p><p>För till exempel beräkning av vilken effektutveckling en periodisk växelström orsakar i en <a href="/wiki/Resistor" title="Resistor">resistor</a> med <a href="/wiki/Resistans" title="Resistans">resistansen</a> <i>R</i>, kan växelströmmen representeras av en konstant enligt </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=RI_{\text{RMS}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>R</mi> <msubsup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>RMS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=RI_{\text{RMS}}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fffed85e3a9be99ba79ed6ed4445e70d6f248ecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.494ex; height:3.176ex;" alt="{\displaystyle P=RI_{\text{RMS}}^{2}}"></span></dd></dl> <p>där <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\text{RMS}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>RMS</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\text{RMS}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b53295be04a218e69d29cdbc0a06d4b306e46b69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.886ex; height:2.509ex;" alt="{\displaystyle I_{\text{RMS}}}"></span> är växelströmmens RMS-värde vilket kan tolkas som värdet av den <a href="/wiki/Likstr%C3%B6m" title="Likström">likström</a> som i genomsnitt ger samma effektutveckling som växelströmmen. Genom att använda växelströmmens RMS-värde kan således effektproblemet behandlas som ett likströmsproblem. </p><p>Om det kvadratiska medelvärdet av den periodiskt varierande spänningen över <i>R</i> är <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{\text{RMS}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>RMS</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\text{RMS}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba6535926c3c4bb3a928355d116d4cef45e4af31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.45ex; height:2.509ex;" alt="{\displaystyle U_{\text{RMS}}}"></span>, vilket kan tolkas som värdet av den likspänning som ger samma genomsnittsliga effektutveckling i <i>R</i> som den periodiska växelspänningen, kan effekten också beräknas som </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 {U_{\text{RMS}}^{2}}{R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>RMS</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>R</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\frac {U_{\text{RMS}}^{2}}{R}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c57d9ae34183ac6ca311c5bdee9a42f2468ebb15" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.13ex; height:6.176ex;" alt="{\displaystyle P={\frac {U_{\text{RMS}}^{2}}{R}}}"></span></dd></dl> <p>eller </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=U_{\text{RMS}}I_{\text{RMS}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>RMS</mtext> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>RMS</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=U_{\text{RMS}}I_{\text{RMS}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/358e04f6e82dec40a1e08f3dd62f54ed20204a54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.18ex; height:2.509ex;" alt="{\displaystyle P=U_{\text{RMS}}I_{\text{RMS}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Jämförelse_med_andra_medelvärden"><span id="J.C3.A4mf.C3.B6relse_med_andra_medelv.C3.A4rden"></span>Jämförelse med andra medelvärden</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&veaction=edit&section=6" title="Redigera avsnitt: Jämförelse med andra medelvärden" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&action=edit&section=6" title="Redigera avsnitts källkod: Jämförelse med andra medelvärden"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Mean-values-2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Mean-values-2.png/300px-Mean-values-2.png" decoding="async" width="300" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Mean-values-2.png/450px-Mean-values-2.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Mean-values-2.png/600px-Mean-values-2.png 2x" data-file-width="807" data-file-height="483" /></a><figcaption>Geometrisk jämförelse av medelvärden</figcaption></figure><div style="clear:left;"></div> <p>Medelvärden av två tal, <i>a</i> och <i>b</i>, kan konstrueras geometriskt med hjälp av en halvcirkel med diametern <i>a</i> + <i>b</i>. </p> <dl><dd><i>A</i>: Aritmetiska medelvärdet</dd> <dd><i>Q</i>: Kvadratiska medelvärdet</dd> <dd><i>H</i>: Harmoniska medelvärdet</dd> <dd><i>G</i>: Geometriska medelvärdet</dd></dl> <p>Det framgår att </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 {\bar {a}}_{Q}\geq {\bar {a}}_{A}\geq {\bar {a}}_{G}\geq {\bar {a}}_{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msub> <mo>≥</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>≥</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mo>≥</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {a}}_{Q}\geq {\bar {a}}_{A}\geq {\bar {a}}_{G}\geq {\bar {a}}_{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a2443db625bbbcd4ea71aa6d4af7eee816212f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.427ex; height:2.676ex;" alt="{\displaystyle {\bar {a}}_{Q}\geq {\bar {a}}_{A}\geq {\bar {a}}_{G}\geq {\bar {a}}_{H}}"></span></dd></dl> <p>Denna ordning gäller även för ett godtyckligt antal tal. </p> <div class="mw-heading mw-heading2"><h2 id="Se_även"><span id="Se_.C3.A4ven"></span>Se även</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&veaction=edit&section=7" title="Redigera avsnitt: Se även" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&action=edit&section=7" title="Redigera avsnitts källkod: Se även"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Effektivv%C3%A4rde" title="Effektivvärde">Effektivvärde</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referenser">Referenser</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&veaction=edit&section=8" title="Redigera avsnitt: Referenser" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&action=edit&section=8" title="Redigera avsnitts källkod: Referenser"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Noter">Noter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&veaction=edit&section=9" title="Redigera avsnitt: Noter" class="mw-editsection-visualeditor"><span>redigera</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Kvadratiskt_medelv%C3%A4rde&action=edit&section=9" title="Redigera avsnitts källkod: Noter"><span>redigera wikitext</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <span class="reference-text"><cite style="font-style:normal" class="book"> <i><a rel="nofollow" class="external text" href="http://www.oxfordreference.com/view/10.1093/acref/9780199233991.001.0001/acref-9780199233991-e-2676">A Dictionary of Physics (6 ed.)</a></i>. Oxford University Press. 2009. <a href="/wiki/Special:Bokk%C3%A4llor/9780199233991" title="Special:Bokkällor/9780199233991">ISBN 9780199233991</a><span class="printonly">. <a rel="nofollow" class="external free" href="http://www.oxfordreference.com/view/10.1093/acref/9780199233991.001.0001/acref-9780199233991-e-2676">http://www.oxfordreference.com/view/10.1093/acref/9780199233991.001.0001/acref-9780199233991-e-2676</a></span></cite><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Dictionary+of+Physics+%286+ed.%29&rft.date=2009&rft.pub=Oxford+University+Press&rft.isbn=9780199233991&rft_id=http%3A%2F%2Fwww.oxfordreference.com%2Fview%2F10.1093%2Facref%2F9780199233991.001.0001%2Facref-9780199233991-e-2676&rfr_id=info:sid/en.wikipedia.org:Kvadratiskt_medelv%C3%A4rde"><span style="display: none;"> </span></span></span> </li> </ol></div> <style data-mw-deduplicate="TemplateStyles:r56287950">.mw-parser-output table.navbox{border:#aaa 1px solid;width:100%;margin:auto;margin-top:1em;clear:both;font-size:88%;text-align:center;padding:1px}.mw-parser-output link+table.navbox{margin-top:-1px}.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow,.mw-parser-output table.navbox th{text-align:center;padding-left:1em;padding-right:1em}.mw-parser-output .navbox-thlinkcolor .navbox-title button,.mw-parser-output .navbox-thlinkcolor .navbox-title .mw-collapsible-text,.mw-parser-output .navbox-thlinkcolor .navbox-title a{color:inherit}.mw-parser-output .nowraplinks a,.mw-parser-output .nowraplinks .selflink{white-space:nowrap}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right;font-weight:bold;padding-left:1em;padding-right:1em}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background:#fdfdfd}.mw-parser-output .navbox-list{border-color:#fdfdfd}.mw-parser-output .navbox-title,.mw-parser-output table.navbox th{background:#b0c4de}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background:#d0e0f5}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background:#deeafa}.mw-parser-output .navbox-even{background:#f7f7f7}.mw-parser-output .navbox-odd{background:transparent}</style><table class="navbox" style="border-spacing:0; 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