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class="vector-toc-numb">2</span> <span>Classification by technology</span> </div> </a> <button aria-controls="toc-Classification_by_technology-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 Classification by technology subsection</span> </button> <ul id="toc-Classification_by_technology-sublist" class="vector-toc-list"> <li id="toc-Passive_filters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Passive_filters"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Passive filters</span> </div> </a> <ul id="toc-Passive_filters-sublist" class="vector-toc-list"> <li id="toc-Single_element_types" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Single_element_types"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Single element types</span> </div> </a> <ul id="toc-Single_element_types-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-L_filter" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#L_filter"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>L filter</span> </div> </a> <ul id="toc-L_filter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-T_and_π_filters" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#T_and_π_filters"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>T and π filters</span> </div> </a> <ul id="toc-T_and_π_filters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiple-element_types" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Multiple-element_types"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.4</span> <span>Multiple-element types</span> </div> </a> <ul id="toc-Multiple-element_types-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Active_filters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Active_filters"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Active filters</span> </div> </a> <ul id="toc-Active_filters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_filter_technologies" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_filter_technologies"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Other filter technologies</span> </div> </a> <ul id="toc-Other_filter_technologies-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-The_transfer_function" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_transfer_function"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>The transfer function</span> </div> </a> <ul id="toc-The_transfer_function-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classification_by_topology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Classification_by_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Classification by topology</span> </div> </a> <ul id="toc-Classification_by_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classification_by_design_methodology" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Classification_by_design_methodology"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Classification by design methodology</span> </div> </a> <button aria-controls="toc-Classification_by_design_methodology-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 Classification by design methodology subsection</span> </button> <ul id="toc-Classification_by_design_methodology-sublist" class="vector-toc-list"> <li id="toc-Direct_circuit_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Direct_circuit_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Direct circuit analysis</span> </div> </a> <ul id="toc-Direct_circuit_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Image_impedance_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Image_impedance_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Image impedance analysis</span> </div> </a> <ul id="toc-Image_impedance_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Network_synthesis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Network_synthesis"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Network synthesis</span> </div> </a> <ul id="toc-Network_synthesis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes_and_references" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes_and_references"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes and references</span> </div> </a> <ul id="toc-Notes_and_references-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" 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Available in 30 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-30" 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">30 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B1%D8%B4%D8%AD_%D8%A5%D9%84%D9%83%D8%AA%D8%B1%D9%88%D9%86%D9%8A" title="مرشح إلكتروني – Arabic" lang="ar" hreflang="ar" data-title="مرشح إلكتروني" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Filtre_electr%C3%B2nic" title="Filtre electrònic – Catalan" lang="ca" hreflang="ca" data-title="Filtre electrònic" data-language-autonym="Català" data-language-local-name="Catalan" 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/Elektronick%C3%BD_filtr" title="Elektronický filtr – Czech" lang="cs" hreflang="cs" data-title="Elektronický filtr" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Elektronisk_filter" title="Elektronisk filter – Danish" lang="da" hreflang="da" data-title="Elektronisk filter" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Filter_(Elektrotechnik)" title="Filter (Elektrotechnik) – German" lang="de" hreflang="de" data-title="Filter (Elektrotechnik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Sagedusfilter" title="Sagedusfilter – Estonian" lang="et" hreflang="et" data-title="Sagedusfilter" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%97%CE%BB%CE%B5%CE%BA%CF%84%CF%81%CE%BF%CE%BD%CE%B9%CE%BA%CF%8C_%CF%86%CE%AF%CE%BB%CF%84%CF%81%CE%BF" title="Ηλεκτρονικό φίλτρο – Greek" lang="el" hreflang="el" data-title="Ηλεκτρονικό φίλτρο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Filtro_electr%C3%B3nico" title="Filtro electrónico – Spanish" lang="es" hreflang="es" data-title="Filtro electrónico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Iragazki_elektroniko" title="Iragazki elektroniko – Basque" lang="eu" hreflang="eu" data-title="Iragazki elektroniko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%DB%8C%D9%84%D8%AA%D8%B1_%D8%A7%D9%84%DA%A9%D8%AA%D8%B1%D9%88%D9%86%DB%8C%DA%A9%DB%8C" title="فیلتر الکترونیکی – Persian" lang="fa" hreflang="fa" data-title="فیلتر الکترونیکی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Filtre_(%C3%A9lectronique)" title="Filtre (électronique) – French" lang="fr" hreflang="fr" data-title="Filtre (électronique)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%87%E0%A4%B2%E0%A5%87%E0%A4%95%E0%A5%8D%E0%A4%9F%E0%A5%8D%E0%A4%B0%E0%A5%89%E0%A4%A8%E0%A4%BF%E0%A4%95_%E0%A4%AB%E0%A4%BF%E0%A4%B2%E0%A5%8D%E0%A4%9F%E0%A4%B0" title="इलेक्ट्रॉनिक फिल्टर – Hindi" lang="hi" hreflang="hi" data-title="इलेक्ट्रॉनिक फिल्टर" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Elektri%C4%8Dni_filtri" title="Električni filtri – Croatian" lang="hr" hreflang="hr" data-title="Električni filtri" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Filtro_(elettronica)" title="Filtro (elettronica) – Italian" lang="it" hreflang="it" data-title="Filtro (elettronica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A1%D7%A0%D7%9F_(%D7%90%D7%9C%D7%A7%D7%98%D7%A8%D7%95%D7%A0%D7%99%D7%A7%D7%94)" title="מסנן (אלקטרוניקה) – Hebrew" lang="he" hreflang="he" data-title="מסנן (אלקטרוניקה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B6%E0%B3%8B%E0%B2%A7%E0%B2%95" title="ಶೋಧಕ – Kannada" lang="kn" hreflang="kn" data-title="ಶೋಧಕ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Filter_(elektronica)" title="Filter (elektronica) – Dutch" lang="nl" hreflang="nl" data-title="Filter (elektronica)" data-language-autonym="Nederlands" data-language-local-name="Dutch" 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/%E3%83%95%E3%82%A3%E3%83%AB%E3%82%BF%E5%9B%9E%E8%B7%AF" title="フィルタ回路 – Japanese" lang="ja" hreflang="ja" data-title="フィルタ回路" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Elektrisk_filter" title="Elektrisk filter – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Elektrisk filter" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Filtr_(elektronika)" title="Filtr (elektronika) – Polish" lang="pl" hreflang="pl" data-title="Filtr (elektronika)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Filtro_eletr%C3%B4nico" title="Filtro eletrônico – Portuguese" lang="pt" hreflang="pt" data-title="Filtro eletrônico" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Filtru_electronic" title="Filtru electronic – Romanian" lang="ro" hreflang="ro" data-title="Filtru electronic" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D0%B8%D0%BB%D1%8C%D1%82%D1%80_(%D1%8D%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BE%D0%BD%D0%B8%D0%BA%D0%B0)" title="Фильтр (электроника) – Russian" lang="ru" hreflang="ru" data-title="Фильтр (электроника)" data-language-autonym="Русский" data-language-local-name="Russian" 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/Filter_(electronics)" title="Filter (electronics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Filter (electronics)" 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-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Elektronski_filter" title="Elektronski filter – Serbian" lang="sr" hreflang="sr" data-title="Elektronski filter" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Elektri%C4%8Dni_filter" title="Električni filter – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Električni filter" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Elektroniikan_suodattimet" title="Elektroniikan suodattimet – Finnish" lang="fi" hreflang="fi" data-title="Elektroniikan suodattimet" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Elektronik_filtre" title="Elektronik filtre – Turkish" lang="tr" hreflang="tr" data-title="Elektronik filtre" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BE%D0%BD%D0%BD%D0%B8%D0%B9_%D1%84%D1%96%D0%BB%D1%8C%D1%82%D1%80" title="Електронний фільтр – Ukrainian" lang="uk" hreflang="uk" data-title="Електронний фільтр" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%94%B5%E5%AD%90%E6%BB%A4%E6%B3%A2%E5%99%A8" title="电子滤波器 – Chinese" lang="zh" hreflang="zh" data-title="电子滤波器" 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/Q327754#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> 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filters</a>)</span></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">Electronic device</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 lumped-element electrical filters. For distributed-element types, see <a href="/wiki/Distributed-element_filter" title="Distributed-element filter">Distributed-element filter</a>. For other filter technologies, see <a href="/wiki/Filter_(signal_processing)#Technologies" title="Filter (signal processing)">Filter (signal processing) § Technologies</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_footnotes_needed plainlinks metadata ambox ambox-style ambox-More_footnotes_needed" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article includes a list of <a href="/wiki/Wikipedia:Citing_sources#General_references" title="Wikipedia:Citing sources">general references</a>, but <b>it lacks sufficient corresponding <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help to <a href="/wiki/Wikipedia:WikiProject_Reliability" title="Wikipedia:WikiProject Reliability">improve</a> this article by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">March 2013</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Hilofilter.agr.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Hilofilter.agr.jpg/200px-Hilofilter.agr.jpg" decoding="async" width="200" height="126" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/73/Hilofilter.agr.jpg/300px-Hilofilter.agr.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/73/Hilofilter.agr.jpg/400px-Hilofilter.agr.jpg 2x" data-file-width="1493" data-file-height="941" /></a><figcaption>Television signal splitter consisting of a high-pass filter (left) and a low-pass filter (right). The antenna is connected to the screw terminals to the left of center.</figcaption></figure> <p><b>Electronic filters</b> are a type of <a href="/wiki/Signal_processing_filter" class="mw-redirect" title="Signal processing filter">signal processing filter</a> in the form of electrical circuits. This article covers those filters consisting of <a href="/wiki/Lumped-element_model" title="Lumped-element model">lumped</a> electronic components, as opposed to <a href="/wiki/Distributed-element_filter" title="Distributed-element filter">distributed-element filters</a>. That is, using components and interconnections that, in analysis, can be considered to exist at a single point. These components can be in discrete packages or part of an <a href="/wiki/Integrated_circuit" title="Integrated circuit">integrated circuit</a>. </p><p>Electronic filters remove unwanted frequency components from the applied signal, enhance wanted ones, or both. They can be: </p> <ul><li><a href="/wiki/Passive_filter" class="mw-redirect" title="Passive filter">passive</a> or <a href="/wiki/Active_filter" title="Active filter">active</a></li> <li><a href="/wiki/Analogue_filter" title="Analogue filter">analog</a> or <a href="/wiki/Digital_filter" title="Digital filter">digital</a></li> <li><a href="/wiki/High-pass_filter" title="High-pass filter">high-pass</a>, <a href="/wiki/Low-pass_filter" title="Low-pass filter">low-pass</a>, <a href="/wiki/Band-pass_filter" title="Band-pass filter">band-pass</a>, <a href="/wiki/Band-stop_filter" title="Band-stop filter">band-stop</a> (band-rejection; notch), or <a href="/wiki/All-pass_filter" title="All-pass filter">all-pass</a>.</li> <li><a href="/wiki/Discrete-time" class="mw-redirect" title="Discrete-time">discrete-time</a> (sampled) or <a href="/wiki/Continuous_signal" class="mw-redirect" title="Continuous signal">continuous-time</a></li> <li><a href="/wiki/Linear_filter" title="Linear filter">linear</a> or <a href="/wiki/Non-linear_filter" class="mw-redirect" title="Non-linear filter">non-linear</a></li> <li><a href="/wiki/Infinite_impulse_response" title="Infinite impulse response">infinite impulse response</a> (IIR type) or <a href="/wiki/Finite_impulse_response" title="Finite impulse response">finite impulse response</a> (FIR type)</li></ul> <p>The most common types of electronic filters are <a href="/wiki/Linear_filters" class="mw-redirect" title="Linear filters">linear filters</a>, regardless of other aspects of their design. See the article on linear filters for details on their design and analysis. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=1" title="Edit section: History"><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/Analogue_filter" title="Analogue filter">Analogue filter</a></div> <p>The oldest forms of electronic filters are passive analog linear filters, constructed using only <a href="/wiki/Resistors" class="mw-redirect" title="Resistors">resistors</a> and <a href="/wiki/Capacitor" title="Capacitor">capacitors</a> or resistors and <a href="/wiki/Inductor" title="Inductor">inductors</a>. These are known as RC and RL single-<a href="/wiki/Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">pole</a> filters respectively. However, these simple filters have very limited uses. Multipole LC filters provide greater control of response form, bandwidth and <a href="/wiki/Transition_band" title="Transition band">transition bands</a>. The first of these filters was the <a href="/wiki/Constant_k_filter" title="Constant k filter">constant k filter</a>, invented by <a href="/wiki/George_Ashley_Campbell" title="George Ashley Campbell">George Campbell</a> in 1910. Campbell's filter was a ladder network based on <a href="/wiki/Transmission_line" title="Transmission line">transmission line</a> theory. Together with improved filters by <a href="/wiki/Otto_Zobel" class="mw-redirect" title="Otto Zobel">Otto Zobel</a> and others, these filters are known as <a href="/wiki/Image_parameter_filter" class="mw-redirect" title="Image parameter filter">image parameter filters</a>. A major step forward was taken by <a href="/wiki/Wilhelm_Cauer" title="Wilhelm Cauer">Wilhelm Cauer</a> who founded the field of <a href="/wiki/Network_synthesis_filter" class="mw-redirect" title="Network synthesis filter">network synthesis</a> around the time of <a href="/wiki/World_War_II" title="World War II">World War II</a>. Cauer's theory allowed filters to be constructed that precisely followed some prescribed frequency function. </p> <div class="mw-heading mw-heading2"><h2 id="Classification_by_technology">Classification by technology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=2" title="Edit section: Classification by technology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Passive_filters">Passive filters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=3" title="Edit section: Passive filters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Passive implementations of linear filters are based on combinations of <a href="/wiki/Resistor" title="Resistor">resistors</a> (R), <a href="/wiki/Inductor" title="Inductor">inductors</a> (L) and <a href="/wiki/Capacitor" title="Capacitor">capacitors</a> (C). These types are collectively known as <i>passive filters</i>, because they do not depend upon an external power supply and they do not contain active components such as <a href="/wiki/Transistor" title="Transistor">transistors</a>. </p><p>Inductors block high-frequency signals and conduct low-frequency signals, while <a href="/wiki/Capacitor" title="Capacitor">capacitors</a> do the reverse. A filter in which the signal passes through an <a href="/wiki/Inductor" title="Inductor">inductor</a>, or in which a capacitor provides a path to ground, presents less <a href="/wiki/Attenuation" title="Attenuation">attenuation</a> to low-frequency signals than high-frequency signals and is therefore a <i><a href="/wiki/Low-pass_filter" title="Low-pass filter">low-pass filter</a></i>. If the signal passes through a capacitor, or has a path to ground through an inductor, then the filter presents less attenuation to high-frequency signals than low-frequency signals and therefore is a <i><a href="/wiki/High-pass_filter" title="High-pass filter">high-pass filter</a></i>. <a href="/wiki/Resistor" title="Resistor">Resistors</a> on their own have no frequency-selective properties, but are added to inductors and capacitors to determine the <i>time-constants</i> of the circuit, and therefore the frequencies to which it responds. </p><p>The inductors and capacitors are the <a href="/wiki/Reactance_(electronics)" class="mw-redirect" title="Reactance (electronics)">reactive</a> elements of the filter. The number of elements determines the order of the filter. In this context, an <a href="/wiki/LC_circuit" title="LC circuit">LC tuned circuit</a> being used in a band-pass or band-stop filter is considered a single element even though it consists of two components. </p><p>At high frequencies (above about 100 <a href="/wiki/Hertz" title="Hertz">megahertz</a>), sometimes the inductors consist of single loops or strips of sheet metal, and the capacitors consist of adjacent strips of metal. These inductive or capacitive pieces of metal are called <a href="/wiki/Stub_(electronics)" title="Stub (electronics)">stubs</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Single_element_types">Single element types</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=4" title="Edit section: Single element types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Frame"><a href="/wiki/File:1st_Order_Lowpass_Filter_RC.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/1st_Order_Lowpass_Filter_RC.svg/250px-1st_Order_Lowpass_Filter_RC.svg.png" decoding="async" width="250" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/1st_Order_Lowpass_Filter_RC.svg/375px-1st_Order_Lowpass_Filter_RC.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/1st_Order_Lowpass_Filter_RC.svg/500px-1st_Order_Lowpass_Filter_RC.svg.png 2x" data-file-width="250" data-file-height="140" /></a><figcaption>A low-pass electronic filter realised by an <a href="/wiki/RC_circuit" title="RC circuit">RC circuit</a></figcaption></figure> <p>The simplest passive filters, <a href="/wiki/RC_circuit" title="RC circuit">RC</a> and <a href="/wiki/RL_circuit" title="RL circuit">RL</a> filters, include only one reactive element, except for the <a href="/wiki/Hybrid_LC_Filter" title="Hybrid LC Filter">hybrid LC filter</a>, which is characterized by inductance and capacitance integrated in one element.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="L_filter">L filter</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=5" title="Edit section: L filter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An L filter consists of two reactive elements, one in series and one in parallel. </p> <div class="mw-heading mw-heading4"><h4 id="T_and_π_filters"><span id="T_and_.CF.80_filters"></span>T and π filters</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=6" title="Edit section: T and π filters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Lowpass_Pi_Filter.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Lowpass_Pi_Filter.svg/250px-Lowpass_Pi_Filter.svg.png" decoding="async" width="250" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Lowpass_Pi_Filter.svg/375px-Lowpass_Pi_Filter.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Lowpass_Pi_Filter.svg/500px-Lowpass_Pi_Filter.svg.png 2x" data-file-width="250" data-file-height="140" /></a><figcaption>Low-pass π filter</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Highpass_T_Filter.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Highpass_T_Filter.svg/250px-Highpass_T_Filter.svg.png" decoding="async" width="250" height="140" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/Highpass_T_Filter.svg/375px-Highpass_T_Filter.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/Highpass_T_Filter.svg/500px-Highpass_T_Filter.svg.png 2x" data-file-width="250" data-file-height="140" /></a><figcaption>High-pass T filter</figcaption></figure> <p>Three-element filters can have a 'T' or 'π' topology and in either geometries, a <a href="/wiki/Low-pass" class="mw-redirect" title="Low-pass">low-pass</a>, <a href="/wiki/High-pass" class="mw-redirect" title="High-pass">high-pass</a>, <a href="/wiki/Band-pass" class="mw-redirect" title="Band-pass">band-pass</a>, or <a href="/wiki/Band-stop" class="mw-redirect" title="Band-stop">band-stop</a> characteristic is possible. The components can be chosen symmetric or not, depending on the required frequency characteristics. The high-pass T filter in the illustration, has a very low impedance at high frequencies, and a very high impedance at low frequencies. That means that it can be inserted in a transmission line, resulting in the high frequencies being passed and low frequencies being reflected. Likewise, for the illustrated low-pass π filter, the circuit can be connected to a transmission line, transmitting low frequencies and reflecting high frequencies. Using <a href="/wiki/M-derived_filter" title="M-derived filter">m-derived filter</a> sections with correct termination impedances, the input impedance can be reasonably constant in the pass band.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Multiple-element_types">Multiple-element types</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=7" title="Edit section: Multiple-element types"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Multiple-element filtration are usually constructed as a <a href="/wiki/Ladder_network" class="mw-redirect" title="Ladder network">ladder network</a>. These can be seen as a continuation of the L,T and π designs of filters. More elements are needed when it is desired to improve some parameter of the filter such as stop-band rejection or slope of transition from pass-band to stop-band. </p> <div class="mw-heading mw-heading3"><h3 id="Active_filters">Active filters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=8" title="Edit section: Active filters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Active_filter" title="Active filter">Active filters</a> are implemented using a combination of passive and active (amplifying) components, and require an outside power source. <a href="/wiki/Operational_amplifier" title="Operational amplifier">Operational amplifiers</a> are frequently used in active filter designs. These can have high <a href="/wiki/Q_factor" title="Q factor">Q factor</a>, and can achieve <a href="/wiki/Electrical_resonance" title="Electrical resonance">resonance</a> without the use of inductors. However, their upper frequency limit is limited by the bandwidth of the amplifiers. </p> <div class="mw-heading mw-heading3"><h3 id="Other_filter_technologies">Other filter technologies</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=9" title="Edit section: Other filter technologies"><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/Filter_(signal_processing)" title="Filter (signal processing)">Filter (signal processing)</a></div> <p>There are many filter technologies other than lumped component electronics. These include <a href="/wiki/Digital_filter" title="Digital filter">digital filters</a>, <a href="/wiki/Crystal_filter" title="Crystal filter">crystal filters</a>, <a href="/wiki/Mechanical_filter" title="Mechanical filter">mechanical filters</a>, <a href="/wiki/Surface_acoustic_wave" title="Surface acoustic wave">surface acoustic wave</a> (SAW) filters, <a href="/wiki/Thin-film_bulk_acoustic_resonator" title="Thin-film bulk acoustic resonator">thin-film bulk acoustic resonator</a> (TFBAR, FBAR) based filters, <a href="/wiki/YIG_sphere" title="YIG sphere">garnet filters</a>, and atomic filters (used in <a href="/wiki/Atomic_clock" title="Atomic clock">atomic clocks</a>). </p> <div class="mw-heading mw-heading2"><h2 id="The_transfer_function">The transfer function</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=10" title="Edit section: The transfer function"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><i>see also <a href="/wiki/Filter_(signal_processing)" title="Filter (signal processing)">Filter (signal processing)</a> for further analysis</i></dd></dl> <p>The <a href="/wiki/Transfer_function" title="Transfer function">transfer function</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 H(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b91390324fc9c33ec00fe57e3924ad7118cc1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.963ex; height:2.843ex;" alt="{\displaystyle H(s)}"></span> of a filter is the ratio of the output signal <span class="mwe-math-element"><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(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd2fc8c3dee5ab3d3e58d95991c2ade1f680858e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.673ex; height:2.843ex;" alt="{\displaystyle Y(s)}"></span> to that of the input signal <span class="mwe-math-element"><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(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f01b165ffc7afda3f07e31b39ca4dc7d2284a899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle X(s)}"></span> as a function of the complex frequency <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>: </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 H(s)={\frac {Y(s)}{X(s)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>X</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)={\frac {Y(s)}{X(s)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc803bc360262f1c4c4ae127bbda43f1a8834f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.778ex; height:6.509ex;" alt="{\displaystyle H(s)={\frac {Y(s)}{X(s)}}}"></span>.</dd></dl> <p>The transfer function of all linear time-invariant filters, when constructed of <a href="/wiki/Lumped-element_model" title="Lumped-element model">lumped</a> components (as opposed to <a href="/wiki/Distributed-element_model" title="Distributed-element model">distributed</a> components such as transmission lines), will be the ratio of two polynomials in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span>, i.e. a <a href="/wiki/Rational_function" title="Rational function">rational function</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/949f49e8db56fbce881a915af8a9f4f9b0ecce2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.671ex; height:1.676ex;" alt="{\displaystyle \ s}"></span>. The order of the transfer function will be the highest power of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ s}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>s</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ s}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/949f49e8db56fbce881a915af8a9f4f9b0ecce2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.671ex; height:1.676ex;" alt="{\displaystyle \ s}"></span> encountered in either the numerator or the denominator. </p> <div class="mw-heading mw-heading2"><h2 id="Classification_by_topology">Classification by topology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=11" title="Edit section: Classification by topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Electronic filters can be classified by the technology used to implement them. Filters using <a href="/wiki/Passive_filter" class="mw-redirect" title="Passive filter">passive filter</a> and <a href="/wiki/Active_filter" title="Active filter">active filter</a> technology can be further classified by the particular <a href="/wiki/Electronic_filter_topology" title="Electronic filter topology">electronic filter topology</a> used to implement them. </p><p>Any given filter transfer function may be implemented in any <a href="/wiki/Electronic_filter_topology" title="Electronic filter topology">electronic filter topology</a>. </p><p>Some common circuit topologies are: </p> <ul><li><a href="/wiki/Cauer_topology" class="mw-redirect" title="Cauer topology">Cauer topology</a> – passive</li> <li><a href="/wiki/Sallen%E2%80%93Key_topology" title="Sallen–Key topology">Sallen–Key topology</a> – active</li> <li><a href="/wiki/Multiple_feedback_topology" class="mw-redirect" title="Multiple feedback topology">Multiple feedback topology</a> – active</li> <li><a href="/wiki/State_variable_filter" title="State variable filter">State variable topology</a> – active</li> <li><a href="/wiki/Biquad_filter" class="mw-redirect" title="Biquad filter">Biquadratic topology</a> – active</li></ul> <div class="mw-heading mw-heading2"><h2 id="Classification_by_design_methodology">Classification by design methodology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=12" title="Edit section: Classification by design methodology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 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.mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title" style="font-size:130%"><a class="mw-selflink selflink">Linear analog<br />electronic filters</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:lavender; padding:2px 5px;;color: var(--color-base)"><a href="/wiki/Network_synthesis_filters" title="Network synthesis filters">Network synthesis filters</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align:left;"> <ul><li><a href="/wiki/Butterworth_filter" title="Butterworth filter">Butterworth filter</a></li> <li><a href="/wiki/Chebyshev_filter" title="Chebyshev filter">Chebyshev filter</a></li> <li><a href="/wiki/Elliptic_filter" title="Elliptic filter">Elliptic (Cauer) filter</a></li> <li><a href="/wiki/Bessel_filter" title="Bessel filter">Bessel filter</a></li> <li><a href="/wiki/Gaussian_filter" title="Gaussian filter">Gaussian filter</a></li> <li><a href="/wiki/Optimum_%22L%22_filter" title="Optimum "L" filter">Optimum "L" (Legendre) filter</a></li> <li><a href="/wiki/Linkwitz%E2%80%93Riley_filter" title="Linkwitz–Riley filter">Linkwitz–Riley filter</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:lavender; padding:2px 5px;;color: var(--color-base)"><a href="/wiki/Composite_image_filter" title="Composite image filter">Image impedance filters</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align:left;"> <ul><li><a href="/wiki/Constant_k_filter" title="Constant k filter">Constant k filter</a></li> <li><a href="/wiki/M-derived_filter" title="M-derived filter">m-derived filter</a></li> <li><a href="/wiki/General_mn-type_image_filter" title="General mn-type image filter">General image filters</a></li> <li><a href="/wiki/Zobel_network" title="Zobel network">Zobel network</a> (constant R) filter</li> <li><a href="/wiki/Lattice_phase_equaliser" title="Lattice phase equaliser">Lattice filter</a> (all-pass)</li> <li><a href="/wiki/Bridged_T_delay_equaliser" title="Bridged T delay equaliser">Bridged T delay equaliser</a> (all-pass)</li> <li><a href="/wiki/Composite_image_filter" title="Composite image filter">Composite image filter</a></li> <li><a href="/wiki/Mm%27-type_filter" title="Mm'-type filter">mm'-type filter</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:lavender; padding:2px 5px;;color: var(--color-base)">Simple filters</div><div class="sidebar-list-content mw-collapsible-content" style="text-align:left;"> <ul><li><a href="/wiki/RC_circuit" title="RC circuit">RC filter</a></li> <li><a href="/wiki/RL_circuit" title="RL circuit">RL filter</a></li> <li><a href="/wiki/LC_circuit" title="LC circuit">LC filter</a></li> <li><a href="/wiki/RLC_circuit" title="RLC circuit">RLC filter</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar" style="padding-top:0;"><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:Linear_analog_electronic_filter" title="Template:Linear analog electronic filter"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Linear_analog_electronic_filter" title="Template talk:Linear analog electronic filter"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Linear_analog_electronic_filter" title="Special:EditPage/Template:Linear analog electronic filter"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>Historically, linear analog filter design has evolved through three major approaches. The oldest designs are simple circuits where the main design criterion was the <a href="/wiki/Q_factor" title="Q factor">Q factor</a> of the circuit. This reflected the radio receiver application of filtering as Q was a measure of the frequency selectivity of a tuning circuit. From the 1920s filters began to be designed from the <a href="/wiki/Image_impedance" title="Image impedance">image</a> point of view, mostly being driven by the requirements of telecommunications. After <a href="/wiki/World_War_II" title="World War II">World War II</a> the dominant methodology was <a href="/wiki/Network_synthesis_filters" title="Network synthesis filters">network synthesis</a>. The higher mathematics used originally required extensive tables of polynomial coefficient values to be published but modern computer resources have made that unnecessary.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Direct_circuit_analysis">Direct circuit analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=13" title="Edit section: Direct circuit analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Low order filters can be designed by directly applying basic circuit laws such as <a href="/wiki/Kirchhoff%27s_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff's laws</a> to obtain the transfer function. This kind of analysis is usually only carried out for simple filters of 1st or 2nd order. </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:RL_filter_response.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/RL_filter_response.png/250px-RL_filter_response.png" decoding="async" width="250" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/RL_filter_response.png/375px-RL_filter_response.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/RL_filter_response.png/500px-RL_filter_response.png 2x" data-file-width="766" data-file-height="602" /></a><figcaption>RL filter frequency response</figcaption></figure> <div style="clear:left;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Image_impedance_analysis">Image impedance analysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=14" title="Edit section: Image impedance analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Composite_image_filters" class="mw-redirect" title="Composite image filters">Composite image filters</a></div></dd></dl> <p>This approach analyses the filter sections from the point of view of the filter being in an infinite chain of identical sections. It has the advantages of simplicity of approach and the ability to easily extend to higher orders. It has the disadvantage that accuracy of predicted responses relies on filter terminations in the image impedance, which is usually not the case.<sup id="cite_ref-MYJ_4-0" class="reference"><a href="#cite_note-MYJ-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Constant_k_order_5.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Constant_k_order_5.png/250px-Constant_k_order_5.png" decoding="async" width="250" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Constant_k_order_5.png/375px-Constant_k_order_5.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fd/Constant_k_order_5.png/500px-Constant_k_order_5.png 2x" data-file-width="768" data-file-height="603" /></a><figcaption>Constant k filter response with 5 elements</figcaption></figure> <figure class="mw-halign-none" typeof="mw:File/Thumb"><a href="/wiki/File:Zobel_5_sections.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Zobel_5_sections.png/250px-Zobel_5_sections.png" decoding="async" width="250" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Zobel_5_sections.png/375px-Zobel_5_sections.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Zobel_5_sections.png/500px-Zobel_5_sections.png 2x" data-file-width="768" data-file-height="603" /></a><figcaption>Zobel network (constant R) filter, 5 sections</figcaption></figure> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:M-derived_order_1.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/M-derived_order_1.png/250px-M-derived_order_1.png" decoding="async" width="250" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f5/M-derived_order_1.png/375px-M-derived_order_1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f5/M-derived_order_1.png/500px-M-derived_order_1.png 2x" data-file-width="766" data-file-height="602" /></a><figcaption>m-derived filter response, m=0.5, 2 elements</figcaption></figure> <figure class="mw-halign-none" typeof="mw:File/Thumb"><a href="/wiki/File:M-derived_order_5.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/M-derived_order_5.png/250px-M-derived_order_5.png" decoding="async" width="250" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/M-derived_order_5.png/375px-M-derived_order_5.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/M-derived_order_5.png/500px-M-derived_order_5.png 2x" data-file-width="766" data-file-height="602" /></a><figcaption>m-derived filter response, m=0.5, 5 elements</figcaption></figure> <div style="clear:left;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Network_synthesis">Network synthesis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=15" title="Edit section: Network synthesis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Network_synthesis_filters" title="Network synthesis filters">Network synthesis filters</a></div></dd></dl> <p>The <a href="/wiki/Network_synthesis" title="Network synthesis">network synthesis</a> approach starts with a required transfer function and then expresses that as a polynomial equation of the input impedance of the filter. The actual element values of the filter are obtained by continued-fraction or partial-fraction expansions of this polynomial. Unlike the image method, there is no need for impedance matching networks at the terminations as the effects of the terminating resistors are included in the analysis from the start.<sup id="cite_ref-MYJ_4-1" class="reference"><a href="#cite_note-MYJ-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Here is an image comparing Butterworth, Chebyshev, and elliptic filters. The filters in this illustration are all fifth-order low-pass filters. The particular implementation – analog or digital, passive or active – makes no difference; their output would be the same. </p> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Filters_order5.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Filters_order5.svg/500px-Filters_order5.svg.png" decoding="async" width="500" height="462" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Filters_order5.svg/750px-Filters_order5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Filters_order5.svg/1000px-Filters_order5.svg.png 2x" data-file-width="702" data-file-height="648" /></a><figcaption></figcaption></figure> <p>As is clear from the image, elliptic filters are sharper than all the others, but they show ripples on the whole bandwidth. </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=Electronic_filter&action=edit&section=16" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Analog_filter" class="mw-redirect" title="Analog filter">Analog filter</a></li> <li><a href="/wiki/Audio_crossover" title="Audio crossover">Audio crossover</a></li> <li><a href="/wiki/Audio_filter" title="Audio filter">Audio filter</a></li> <li><a href="/wiki/Cascaded_integrator-comb_filter" class="mw-redirect" title="Cascaded integrator-comb filter">Cascaded integrator-comb filter</a></li> <li><a href="/wiki/Comb_filter" title="Comb filter">Comb filter</a></li> <li><a href="/wiki/DSL_filter" title="DSL filter">DSL filter</a></li> <li><a href="/wiki/Electronic_low-pass_filter" class="mw-redirect" title="Electronic low-pass filter">Electronic low-pass filter</a></li> <li><a href="/wiki/Nyquist_filter" title="Nyquist filter">Nyquist filter</a></li> <li><a href="/wiki/RF_and_microwave_filter" title="RF and microwave filter">RF and microwave filter</a></li> <li><a href="/wiki/Switched-capacitor_filter" class="mw-redirect" title="Switched-capacitor filter">Switched-capacitor filter</a></li> <li><a href="/wiki/Tone_control_circuits" class="mw-redirect" title="Tone control circuits">Tone control circuits</a></li> <li><a href="/wiki/Voltage-controlled_filter" title="Voltage-controlled filter">Voltage-controlled filter</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes_and_references">Notes and references</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electronic_filter&action=edit&section=17" title="Edit section: Notes and references"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Dzhankhotov V., <i>Hybrid LC filter for power electronic drives: Theory and Implementation</i>, 2009</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">The American Radio Relay League, Inc.: "The ARRL Handbook, 1968" page 50</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Bray, J, <i>Innovation and the Communications Revolution</i>, Institute of Electrical Engineers<i></i></span> </li> <li id="cite_note-MYJ-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-MYJ_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-MYJ_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Matthaei, Young, Jones <i>Microwave Filters, Impedance-Matching Networks, and Coupling Structures</i> McGraw-Hill 1964</span> </li> </ol></div></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <dl><dd><ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFZverev,_Anatol,_I1969" class="citation book cs1">Zverev, Anatol, I (1969). <i>Handbook of Filter Synthesis</i>. John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-98680-1" title="Special:BookSources/0-471-98680-1"><bdi>0-471-98680-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Filter+Synthesis&rft.pub=John+Wiley+%26+Sons&rft.date=1969&rft.isbn=0-471-98680-1&rft.au=Zverev%2C+Anatol%2C+I&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectronic+filter" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span> Catalog of passive filter types and component values. The Bible for practical electronic filter design.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliams,_Arthur_BTaylor,_Fred_J1995" class="citation book cs1">Williams, Arthur B; Taylor, Fred J (1995). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/electronicfilter00will_0"><i>Electronic Filter Design Handbook</i></a></span>. McGraw-Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-070441-4" title="Special:BookSources/0-07-070441-4"><bdi>0-07-070441-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electronic+Filter+Design+Handbook&rft.pub=McGraw-Hill&rft.date=1995&rft.isbn=0-07-070441-4&rft.au=Williams%2C+Arthur+B&rft.au=Taylor%2C+Fred+J&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felectronicfilter00will_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectronic+filter" class="Z3988"></span></li></ul></dd></dl> </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=Electronic_filter&action=edit&section=18" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> 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