RLC circuit - Wikipedia

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class="vector-toc-link" href="#Bandwidth"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Bandwidth</span> </div> </a> <ul id="toc-Bandwidth-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Q_factor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Q_factor"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span><span>Q</span> factor</span> </div> </a> <ul id="toc-Q_factor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scaled_parameters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Scaled_parameters"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Scaled parameters</span> </div> </a> <ul id="toc-Scaled_parameters-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Series_circuit" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Series_circuit"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Series circuit</span> </div> </a> <button aria-controls="toc-Series_circuit-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 Series circuit subsection</span> </button> <ul id="toc-Series_circuit-sublist" class="vector-toc-list"> <li id="toc-Transient_response" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transient_response"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Transient response</span> </div> </a> <ul id="toc-Transient_response-sublist" class="vector-toc-list"> <li id="toc-Overdamped_response" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Overdamped_response"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Overdamped response</span> </div> </a> <ul id="toc-Overdamped_response-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Underdamped_response" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Underdamped_response"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.2</span> <span>Underdamped response</span> </div> </a> <ul id="toc-Underdamped_response-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Critically_damped_response" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Critically_damped_response"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.3</span> <span>Critically damped response</span> </div> </a> <ul id="toc-Critically_damped_response-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Laplace_domain" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Laplace_domain"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Laplace domain</span> </div> </a> <ul id="toc-Laplace_domain-sublist" class="vector-toc-list"> <li id="toc-Laplace_admittance" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Laplace_admittance"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Laplace admittance</span> </div> </a> <ul id="toc-Laplace_admittance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poles_and_zeros" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Poles_and_zeros"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Poles and zeros</span> </div> </a> <ul id="toc-Poles_and_zeros-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_solution" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#General_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.3</span> <span>General solution</span> </div> </a> <ul id="toc-General_solution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sinusoidal_steady_state" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Sinusoidal_steady_state"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.4</span> <span>Sinusoidal steady state</span> </div> </a> <ul id="toc-Sinusoidal_steady_state-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Parallel_circuit" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Parallel_circuit"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span><span>Parallel circuit</span></span> </div> </a> <button aria-controls="toc-Parallel_circuit-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 <span>Parallel circuit</span> subsection</span> </button> <ul id="toc-Parallel_circuit-sublist" class="vector-toc-list"> <li id="toc-Frequency_domain" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Frequency_domain"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Frequency domain</span> </div> </a> <ul id="toc-Frequency_domain-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_configurations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_configurations"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Other configurations</span> </div> </a> <ul id="toc-Other_configurations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-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 Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Variable_tuned_circuits" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Variable_tuned_circuits"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Variable tuned circuits</span> </div> </a> <ul id="toc-Variable_tuned_circuits-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Filters" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Filters"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Filters</span> </div> </a> <ul id="toc-Filters-sublist" class="vector-toc-list"> <li id="toc-Low-pass_filter" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Low-pass_filter"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.1</span> <span>Low-pass filter</span> </div> </a> <ul id="toc-Low-pass_filter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-High-pass_filter" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#High-pass_filter"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.2</span> <span>High-pass filter</span> </div> </a> <ul id="toc-High-pass_filter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Band-pass_filter" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Band-pass_filter"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.3</span> <span>Band-pass filter</span> </div> </a> <ul id="toc-Band-pass_filter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Band-stop_filter" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Band-stop_filter"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2.4</span> <span>Band-stop filter</span> </div> </a> <ul id="toc-Band-stop_filter-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Oscillators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Oscillators"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Oscillators</span> </div> </a> <ul id="toc-Oscillators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Voltage_multiplier" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Voltage_multiplier"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Voltage multiplier</span> </div> </a> <ul id="toc-Voltage_multiplier-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pulse_discharge_circuit" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pulse_discharge_circuit"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Pulse discharge circuit</span> </div> </a> <ul id="toc-Pulse_discharge_circuit-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">RLC circuit</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 34 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-34" 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">34 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A0%E1%88%AD_%E1%8A%A4%E1%88%8D_%E1%88%B2_%E1%8B%91%E1%8B%B0%E1%89%B5" title="አር ኤል ሲ ዑደት – Amharic" lang="am" hreflang="am" data-title="አር ኤል ሲ ዑደት" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AF%D8%A7%D8%A6%D8%B1%D8%A9_%D9%85%D9%82%D8%A7%D9%88%D9%85%D8%A9_%D9%88%D9%85%D9%84%D9%81_%D9%88%D9%85%D9%83%D8%AB%D9%81" title="دائرة مقاومة وملف ومكثف – Arabic" lang="ar" hreflang="ar" data-title="دائرة مقاومة وملف ومكثف" 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/Circuit_RLC" title="Circuit RLC – Catalan" lang="ca" hreflang="ca" data-title="Circuit RLC" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A1%D1%83%D0%BB%D0%BB%D0%B0%D0%BD%D1%83%D1%81%D0%B5%D0%BD_%C4%95%D0%BB%D0%BA%D0%B8" title="Сулланусен ĕлки – Chuvash" lang="cv" hreflang="cv" data-title="Сулланусен ĕлки" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Obvod_RLC" title="Obvod RLC – Czech" lang="cs" hreflang="cs" data-title="Obvod RLC" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/V%C3%B5nkering" title="Võnkering – Estonian" lang="et" hreflang="et" data-title="Võnkering" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Circuito_RLC" title="Circuito RLC – Spanish" lang="es" hreflang="es" data-title="Circuito RLC" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/RLC-cirkvito" title="RLC-cirkvito – Esperanto" lang="eo" hreflang="eo" data-title="RLC-cirkvito" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AF%D8%A7%D8%B1_%D8%A2%D8%B1%D8%A7%D9%84%E2%80%8C%D8%B3%DB%8C" title="مدار آرال‌سی – Persian" lang="fa" hreflang="fa" data-title="مدار آرال‌سی" 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/Circuit_RLC" title="Circuit RLC – French" lang="fr" hreflang="fr" data-title="Circuit RLC" data-language-autonym="Français" data-language-local-name="French" 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/RLC_%ED%9A%8C%EB%A1%9C" title="RLC 회로 – Korean" lang="ko" hreflang="ko" data-title="RLC 회로" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8F%D5%A1%D5%BF%D5%A1%D5%B6%D5%B8%D5%B2%D5%A1%D5%AF%D5%A1%D5%B6_%D5%AF%D5%B8%D5%B6%D5%BF%D5%B8%D6%82%D6%80" title="Տատանողական կոնտուր – Armenian" lang="hy" hreflang="hy" data-title="Տատանողական կոնտուր" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%86%E0%A4%B0_%E0%A4%8F%E0%A4%B2_%E0%A4%B8%E0%A5%80_%E0%A4%AA%E0%A4%B0%E0%A4%BF%E0%A4%AA%E0%A4%A5" title="आर एल सी परिपथ – Hindi" lang="hi" hreflang="hi" data-title="आर एल सी परिपथ" 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/Titrajni_krug" title="Titrajni krug – Croatian" lang="hr" hreflang="hr" data-title="Titrajni krug" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Sirkuit_RLC" title="Sirkuit RLC – Indonesian" lang="id" hreflang="id" data-title="Sirkuit RLC" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Circuito_RLC" title="Circuito RLC – Italian" lang="it" hreflang="it" data-title="Circuito RLC" 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%A2%D7%92%D7%9C_RLC" title="מעגל RLC – Hebrew" lang="he" hreflang="he" data-title="מעגל RLC" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Sirkuit_RLC" title="Sirkuit RLC – Javanese" lang="jv" hreflang="jv" data-title="Sirkuit RLC" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A2%D0%B5%D1%80%D0%B1%D0%B5%D0%BB%D0%BC%D0%B5%D0%BB%D1%96_%D0%BA%D0%BE%D0%BD%D1%82%D1%83%D1%80" title="Тербелмелі контур – Kazakh" lang="kk" hreflang="kk" data-title="Тербелмелі контур" data-language-autonym="Қазақша" data-language-local-name="Kazakh" 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/Rezg%C5%91k%C3%B6r" title="Rezgőkör – Hungarian" lang="hu" hreflang="hu" data-title="Rezgőkör" data-language-autonym="Magyar" data-language-local-name="Hungarian" 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/RLC-kring" title="RLC-kring – Dutch" lang="nl" hreflang="nl" data-title="RLC-kring" 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/RLC%E5%9B%9E%E8%B7%AF" title="RLC回路 – Japanese" lang="ja" hreflang="ja" data-title="RLC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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Obw%C3%B3d_RLC" title="Obwód RLC – Polish" lang="pl" hreflang="pl" data-title="Obwód RLC" 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/Circuito_RLC" title="Circuito RLC – Portuguese" lang="pt" hreflang="pt" data-title="Circuito RLC" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BB%D0%B5%D0%B1%D0%B0%D1%82%D0%B5%D0%BB%D1%8C%D0%BD%D1%8B%D0%B9_%D0%BA%D0%BE%D0%BD%D1%82%D1%83%D1%80" title="Колебательный контур – Russian" lang="ru" hreflang="ru" data-title="Колебательный контур" 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/RLC_circuit" title="RLC circuit – Simple English" lang="en-simple" hreflang="en-simple" data-title="RLC circuit" 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/%D0%A0%D0%9B%D0%A6_%D0%BA%D0%BE%D0%BB%D0%BE" title="РЛЦ коло – Serbian" lang="sr" hreflang="sr" data-title="РЛЦ коло" 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/RLC_kolo" title="RLC kolo – Serbo-Croatian" lang="sh" hreflang="sh" data-title="RLC kolo" 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/RLC-piiri" title="RLC-piiri – Finnish" lang="fi" hreflang="fi" data-title="RLC-piiri" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D1%82%D1%83%D1%80%D0%B8_%D0%BB%D0%B0%D0%BF%D0%BF%D0%B8%D1%88" title="Контури лаппиш – Tajik" lang="tg" hreflang="tg" data-title="Контури лаппиш" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/RLC_devresi" title="RLC devresi – Turkish" lang="tr" hreflang="tr" data-title="RLC devresi" 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%9A%D0%BE%D0%BB%D0%B8%D0%B2%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BA%D0%BE%D0%BD%D1%82%D1%83%D1%80" title="Коливальний контур – Ukrainian" lang="uk" hreflang="uk" data-title="Коливальний контур" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%E1%BA%A1ch_%C4%91i%E1%BB%87n_RLC" title="Mạch điện RLC – Vietnamese" lang="vi" hreflang="vi" data-title="Mạch điện RLC" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/RLC%E7%94%B5%E8%B7%AF" title="RLC电路 – Chinese" lang="zh" hreflang="zh" data-title="RLC电路" 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/Q323477#sitelinks-wikipedia" 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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 href="/wiki/Electronic_filter" title="Electronic filter">Linear analog<br />electronic filters</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><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 mw-collapsed"><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 class="mw-selflink selflink">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> <figure typeof="mw:File/Thumb"><a href="/wiki/File:RLC_series.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/RLC_series.png/350px-RLC_series.png" decoding="async" width="350" height="123" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/RLC_series.png/525px-RLC_series.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/RLC_series.png/700px-RLC_series.png 2x" data-file-width="1417" data-file-height="498" /></a><figcaption>A series RLC network (in order): a resistor, an inductor, and a capacitor</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Tuned_circuit_of_shortwave_radio_transmitter_from_1938.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Tuned_circuit_of_shortwave_radio_transmitter_from_1938.jpg/220px-Tuned_circuit_of_shortwave_radio_transmitter_from_1938.jpg" decoding="async" width="220" height="169" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Tuned_circuit_of_shortwave_radio_transmitter_from_1938.jpg/330px-Tuned_circuit_of_shortwave_radio_transmitter_from_1938.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/5/50/Tuned_circuit_of_shortwave_radio_transmitter_from_1938.jpg 2x" data-file-width="383" data-file-height="294" /></a><figcaption>Tuned circuit of a <a href="/wiki/Shortwave" class="mw-redirect" title="Shortwave">shortwave</a> <a href="/wiki/Radio_transmitter" class="mw-redirect" title="Radio transmitter">radio transmitter</a>. This circuit does not have a resistor like the above, but all tuned circuits have some resistance, causing them to function as an RLC circuit.</figcaption></figure> <p>An <b>RLC circuit</b> is an <a href="/wiki/Electrical_circuit" class="mw-redirect" title="Electrical circuit">electrical circuit</a> consisting of a <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistor</a> (R), an <a href="/wiki/Inductor" title="Inductor">inductor</a> (L), and a <a href="/wiki/Capacitor" title="Capacitor">capacitor</a> (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC. </p><p>The circuit forms a <a href="/wiki/Harmonic_oscillator" title="Harmonic oscillator">harmonic oscillator</a> for current, and <a href="/wiki/Resonance" title="Resonance">resonates</a> in a manner similar to an <a href="/wiki/LC_circuit" title="LC circuit">LC circuit</a>. Introducing the resistor increases the decay of these oscillations, which is also known as <a href="/wiki/Damping" title="Damping">damping</a>. The resistor also reduces the peak resonant frequency. Some resistance is unavoidable even if a resistor is not specifically included as a component. </p><p>RLC circuits have many applications as <a href="/wiki/Electronic_oscillator" title="Electronic oscillator">oscillator circuits</a>. <a href="/wiki/Receiver_(radio)" class="mw-redirect" title="Receiver (radio)">Radio receivers</a> and <a href="/wiki/Television_set" title="Television set">television sets</a> use them for <a href="/wiki/Tuner_(electronics)" class="mw-redirect" title="Tuner (electronics)">tuning</a> to select a narrow frequency range from ambient radio waves. In this role, the circuit is often referred to as a tuned circuit. An RLC circuit can be used as a <a href="/wiki/Band-pass_filter" title="Band-pass filter">band-pass filter</a>, <a href="/wiki/Band-stop_filter" title="Band-stop filter">band-stop filter</a>, <a href="/wiki/Low-pass_filter" title="Low-pass filter">low-pass filter</a> or <a href="/wiki/High-pass_filter" title="High-pass filter">high-pass filter</a>. The tuning application, for instance, is an example of band-pass filtering. The RLC filter is described as a <i>second-order</i> circuit, meaning that any voltage or current in the circuit can be described by a second-order <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> in circuit analysis. </p><p>The three circuit elements, R, L and C, can be combined in a number of different <a href="/wiki/Topology_(electronics)" class="mw-redirect" title="Topology (electronics)">topologies</a>. All three elements in series or all three elements in parallel are the simplest in concept and the most straightforward to analyse. There are, however, other arrangements, some with practical importance in real circuits. One issue often encountered is the need to take into account inductor resistance. Inductors are typically constructed from coils of wire, the resistance of which is not usually desirable, but it often has a significant effect on the circuit. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_concepts">Basic concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=1" title="Edit section: Basic concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Resonance">Resonance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=2" title="Edit section: Resonance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An important property of this circuit is its ability to resonate at a specific frequency, the <a href="/wiki/Electrical_resonance" title="Electrical resonance">resonance frequency</a>, <span class="texhtml"><i>f</i><sub>0</sub></span>. Frequencies are measured in units of <a href="/wiki/Hertz" title="Hertz">hertz</a>. In this article, <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a>, <span class="texhtml"><i>ω</i><sub>0</sub></span>, is used because it is more mathematically convenient. This is measured in <a href="/wiki/Radian" title="Radian">radians</a> per second. They are related to each other by a simple proportion, </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 _{0}=2\pi f_{0}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>π</mi> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}=2\pi f_{0}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/974c8a67af65caebe6b983d9b714e56f564d67b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.32ex; height:2.509ex;" alt="{\displaystyle \omega _{0}=2\pi f_{0}\,.}"></span></dd></dl> <p><a href="/wiki/Resonance" title="Resonance">Resonance</a> occurs because energy for this situation is stored in two different ways: in an electric field as the capacitor is charged and in a magnetic field as current flows through the inductor. Energy can be transferred from one to the other within the circuit and this can be oscillatory. A mechanical analogy is a weight suspended on a spring which will oscillate up and down when released. This is no passing metaphor; a weight on a spring is described by exactly the same second order differential equation as an RLC circuit and for all the properties of the one system there will be found an analogous property of the other. The mechanical property answering to the resistor in the circuit is friction in the spring–weight system. Friction will slowly bring any oscillation to a halt if there is no external force driving it. Likewise, the resistance in an RLC circuit will "damp" the oscillation, diminishing it with time if there is no driving AC power source in the circuit. </p><p>The resonant frequency is defined as the frequency at which the <a href="/wiki/Electrical_impedance" title="Electrical impedance">impedance</a> of the circuit is at a minimum. Equivalently, it can be defined as the frequency at which the impedance is purely real (that is, purely resistive). This occurs because the impedances of the inductor and capacitor at resonant are equal but of opposite sign and cancel out. Circuits where L and C are in parallel rather than series actually have a maximum impedance rather than a minimum impedance. For this reason they are often described as <a href="/wiki/Antiresonance" title="Antiresonance">antiresonators</a>; it is still usual, however, to name the frequency at which this occurs as the resonant frequency. </p> <div class="mw-heading mw-heading3"><h3 id="Natural_frequency">Natural frequency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=3" title="Edit section: Natural frequency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The resonance frequency is defined in terms of the impedance presented to a driving source. It is still possible for the circuit to carry on oscillating (for a time) after the driving source has been removed or it is subjected to a step in voltage (including a step down to zero). This is similar to the way that a tuning fork will carry on ringing after it has been struck, and the effect is often called ringing. This effect is the peak natural resonance frequency of the circuit and in general is not exactly the same as the driven resonance frequency, although the two will usually be quite close to each other. Various terms are used by different authors to distinguish the two, but resonance frequency unqualified usually means the driven resonance frequency. The driven frequency may be called the <a href="/wiki/Undamped" class="mw-redirect" title="Undamped">undamped</a> resonance frequency or undamped natural frequency and the peak frequency may be called the damped resonance frequency or the damped natural frequency. The reason for this terminology is that the driven resonance frequency in a series or parallel resonant circuit has the value.<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> <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 _{0}={\frac {1}{\sqrt {L\,C~}}}\,~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>L</mi> <mspace width="thinmathspace" /> <mi>C</mi> <mtext> </mtext> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}={\frac {1}{\sqrt {L\,C~}}}\,~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cd1e4ac873be7ab86315de1edfc76771d14ce49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.302ex; height:6.176ex;" alt="{\displaystyle \omega _{0}={\frac {1}{\sqrt {L\,C~}}}\,~.}"></span></dd></dl> <p>This is exactly the same as the resonance frequency of a lossless LC circuit – that is, one with no resistor present. The resonant frequency for a <i>driven</i> RLC circuit is the same as a circuit in which there is no damping, hence undamped resonant frequency. The resonant frequency peak amplitude, on the other hand, does depend on the value of the resistor and is described as the damped resonant frequency. A <i>highly damped</i> circuit will fail to resonate at all, when not driven. A circuit with a value of resistor that causes it to be just on the edge of ringing is called <a href="/wiki/Critically_damped" class="mw-redirect" title="Critically damped">critically damped</a>. Either side of critically damped are described as <a href="/wiki/Underdamped" class="mw-redirect" title="Underdamped">underdamped</a> (ringing happens) and <a href="/wiki/Overdamped" class="mw-redirect" title="Overdamped">overdamped</a> (ringing is suppressed). </p><p>Circuits with topologies more complex than straightforward series or parallel (some examples described later in the article) have a driven resonance frequency that deviates from <span class="mwe-math-element"><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 _{0}=1/{\sqrt {L\,C~}}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>L</mi> <mspace width="thinmathspace" /> <mi>C</mi> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \omega _{0}=1/{\sqrt {L\,C~}}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb2b3f48be08f9b42a40706773cc8b8d963b9b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.338ex; height:3.176ex;" alt="{\displaystyle \ \omega _{0}=1/{\sqrt {L\,C~}}\ }"></span>, and for those the undamped resonance frequency, damped resonance frequency and driven resonance frequency can all be different. </p> <div class="mw-heading mw-heading3"><h3 id="Damping">Damping</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=4" title="Edit section: Damping"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Damping" title="Damping">Damping</a> is caused by the resistance in the circuit. It determines whether or not the circuit will resonate naturally (that is, without a driving source). Circuits that will resonate in this way are described as underdamped and those that will not are overdamped. Damping attenuation (symbol <span class="texhtml mvar" style="font-style:italic;">α</span>) is measured in <a href="/wiki/Neper" title="Neper">nepers</a> per second. However, the unitless <a href="/wiki/Damping" title="Damping">damping factor</a> (symbol <span class="texhtml mvar" style="font-style:italic;">ζ</span>, zeta) is often a more useful measure, which is related to <span class="texhtml mvar" style="font-style:italic;">α</span> by </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 \ \zeta ={\frac {\alpha }{\ \omega _{0}\ }}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>ζ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mtext> </mtext> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \zeta ={\frac {\alpha }{\ \omega _{0}\ }}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/145faba4aeef8f0d9c94392caeb327a39bff8abf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.499ex; height:5.009ex;" alt="{\displaystyle \ \zeta ={\frac {\alpha }{\ \omega _{0}\ }}\ .}"></span></dd></dl> <p>The special case of <span class="texhtml"><i>ζ</i> = 1</span> is called <i>critical damping</i> and represents the case of a circuit that is just on the border of oscillation. It is the minimum damping that can be applied without causing oscillation. </p> <div class="mw-heading mw-heading3"><h3 id="Bandwidth">Bandwidth</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=5" title="Edit section: Bandwidth"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The resonance effect can be used for filtering, the rapid change in impedance near resonance can be used to pass or block signals close to the resonance frequency. Both band-pass and band-stop filters can be constructed and some filter circuits are shown later in the article. A key parameter in filter design is <a href="/wiki/Bandwidth_(signal_processing)" title="Bandwidth (signal processing)">bandwidth</a>. The bandwidth is measured between the <a href="/wiki/Cutoff_frequency" title="Cutoff frequency">cutoff frequencies</a>, most frequently defined as the frequencies at which the power passed through the circuit has fallen to half the value passed at resonance. There are two of these half-power frequencies, one above, and one below the resonance frequency </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 \Delta \omega =\omega _{2}-\omega _{1}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>ω</mi> <mo>=</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \omega =\omega _{2}-\omega _{1}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7455d7ae74006b7f8ed7e7101b615cb4e3fb2ab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.355ex; height:2.509ex;" alt="{\displaystyle \Delta \omega =\omega _{2}-\omega _{1}\,,}"></span></dd></dl> <p>where <span class="texhtml">Δ<i>ω</i></span> is the bandwidth, <span class="texhtml"><i>ω</i><sub>1</sub></span> is the lower half-power frequency and <span class="texhtml"><i>ω</i><sub>2</sub></span> is the upper half-power frequency. The bandwidth is related to attenuation by </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 \Delta \omega =2\alpha \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>ω</mi> <mo>=</mo> <mn>2</mn> <mi>α</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \omega =2\alpha \,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/826f4233fa3a85108531be4503fd7dadf122c6ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.164ex; height:2.509ex;" alt="{\displaystyle \Delta \omega =2\alpha \,,}"></span></dd></dl> <p>where the units are radians per second and <a href="/wiki/Neper" title="Neper">nepers</a> per second respectively.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="this is only an approximation for small attenuations, so please cite your sources (January 2012)">citation needed</span></a></i>]</sup> Other units may require a conversion factor. A more general measure of bandwidth is the fractional bandwidth, which expresses the bandwidth as a fraction of the resonance frequency and is given by </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 B_{\mathrm {f} }={\frac {\Delta \omega }{\omega _{0}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>ω</mi> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{\mathrm {f} }={\frac {\Delta \omega }{\omega _{0}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/257dfd0744a76438a1d8ddc34e4c72f7393b1eb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.958ex; height:5.676ex;" alt="{\displaystyle B_{\mathrm {f} }={\frac {\Delta \omega }{\omega _{0}}}\,.}"></span></dd></dl> <p>The fractional bandwidth is also often stated as a percentage. The damping of filter circuits is adjusted to result in the required bandwidth. A narrow band filter, such as a <a href="/wiki/Notch_filter" class="mw-redirect" title="Notch filter">notch filter</a>, requires low damping. A wide band filter requires high damping. </p> <div class="mw-heading mw-heading3"><h3 id="Q_factor"><span class="texhtml mvar" style="font-style:italic;">Q</span> factor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=6" title="Edit section: Q factor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Q_factor" title="Q factor"><span class="texhtml mvar" style="font-style:italic;">Q</span> factor</a> is a widespread measure used to characterise resonators. It is defined as the peak energy stored in the circuit divided by the average energy dissipated in it per radian at resonance. Low-<span class="texhtml mvar" style="font-style:italic;">Q</span> circuits are therefore damped and lossy and high-<span class="texhtml mvar" style="font-style:italic;">Q</span> circuits are underdamped and prone to amplitude extremes if driven at the resonant frequency.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> <span class="texhtml mvar" style="font-style:italic;">Q</span> is related to bandwidth; low-<span class="texhtml mvar" style="font-style:italic;">Q</span> circuits are wide-band and high-<span class="texhtml mvar" style="font-style:italic;">Q</span> circuits are narrow-band. In fact, it happens that <span class="texhtml mvar" style="font-style:italic;">Q</span> is the inverse of fractional bandwidth </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q={\frac {1}{\,B_{\mathrm {f} }\,}}={\frac {\omega _{\text{o}}}{\,\operatorname {\Delta } \omega \,}}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">Δ</mi> </mrow> <mo>⁡</mo> <mi>ω</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q={\frac {1}{\,B_{\mathrm {f} }\,}}={\frac {\omega _{\text{o}}}{\,\operatorname {\Delta } \omega \,}}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e45cf3d82b7e02cd9c1bcccf1f784af72339f5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.925ex; height:5.509ex;" alt="{\displaystyle Q={\frac {1}{\,B_{\mathrm {f} }\,}}={\frac {\omega _{\text{o}}}{\,\operatorname {\Delta } \omega \,}}\;.}"></span><sup id="cite_ref-Long-2004-04-15_3-0" class="reference"><a href="#cite_note-Long-2004-04-15-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></dd></dl> <p><span class="texhtml mvar" style="font-style:italic;">Q</span> factor is directly proportional to <a href="/wiki/Selectivity_(radio)" title="Selectivity (radio)">selectivity</a>, as the <span class="texhtml mvar" style="font-style:italic;">Q</span> factor depends inversely on bandwidth. </p><p>For a series resonant circuit (<a href="#Series_circuit">as shown below</a>), the <span class="texhtml mvar" style="font-style:italic;">Q</span> factor can be calculated as follows:<sup id="cite_ref-Long-2004-04-15_3-1" class="reference"><a href="#cite_note-Long-2004-04-15-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q={\frac {X}{\,R\;}}={\frac {1}{\,\omega _{\text{o}}R\,C\,}}={\frac {\,\omega _{\text{o}}L\,}{R}}={\frac {1}{\,R\;}}{\sqrt {{\frac {L}{\,C\,}}\,}}={\frac {\,Z_{\text{o}}\,}{R\;}}\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>X</mi> <mrow> <mspace width="thinmathspace" /> <mi>R</mi> <mspace width="thickmathspace" /> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mi>R</mi> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mi>L</mi> <mspace width="thinmathspace" /> </mrow> <mi>R</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mi>R</mi> <mspace width="thickmathspace" /> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mrow> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mspace width="thinmathspace" /> </mrow> <mrow> <mi>R</mi> <mspace width="thickmathspace" /> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q={\frac {X}{\,R\;}}={\frac {1}{\,\omega _{\text{o}}R\,C\,}}={\frac {\,\omega _{\text{o}}L\,}{R}}={\frac {1}{\,R\;}}{\sqrt {{\frac {L}{\,C\,}}\,}}={\frac {\,Z_{\text{o}}\,}{R\;}}\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0dd8708dfe615874755c02bcd4fd9508f13ee19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:49.948ex; height:6.176ex;" alt="{\displaystyle Q={\frac {X}{\,R\;}}={\frac {1}{\,\omega _{\text{o}}R\,C\,}}={\frac {\,\omega _{\text{o}}L\,}{R}}={\frac {1}{\,R\;}}{\sqrt {{\frac {L}{\,C\,}}\,}}={\frac {\,Z_{\text{o}}\,}{R\;}}\;,}"></span><sup id="cite_ref-Long-2004-04-15_3-2" class="reference"><a href="#cite_note-Long-2004-04-15-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,X\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>X</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,X\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8fae704edc6b604313b9df9a4b674c40a7334bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.754ex; height:2.176ex;" alt="{\displaystyle \,X\,}"></span> is the reactance <i>either</i> 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 \,L\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>L</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,L\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c9ee80f5d8f948f5995ad5cd6d9c4c20a956cac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.357ex; height:2.176ex;" alt="{\displaystyle \,L\,}"></span> or 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 \,C\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,C\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/163a8873285c2bf583c476bc5d2a1f7c3b3ecd17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.541ex; height:2.176ex;" alt="{\displaystyle \,C\,}"></span> at resonance, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,Z_{\text{o}}\equiv {\sqrt {{\frac {L}{\,C\,}}\,}}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>o</mtext> </mrow> </msub> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mrow> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </msqrt> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,Z_{\text{o}}\equiv {\sqrt {{\frac {L}{\,C\,}}\,}}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f006b1b37c38a78af247467e0714c370fbbd7be7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.507ex; height:6.176ex;" alt="{\displaystyle \,Z_{\text{o}}\equiv {\sqrt {{\frac {L}{\,C\,}}\,}}\;.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Scaled_parameters">Scaled parameters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=7" title="Edit section: Scaled parameters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The parameters <span class="texhtml mvar" style="font-style:italic;">ζ</span>, <span class="texhtml"><i>B</i><sub>f</sub></span>, and <span class="texhtml mvar" style="font-style:italic;">Q</span> are all scaled to <span class="texhtml"><i>ω</i><sub>0</sub></span>. This means that circuits which have similar parameters share similar characteristics regardless of whether or not they are operating in the same frequency band. </p><p>The article next gives the analysis for the series RLC circuit in detail. Other configurations are not described in such detail, but the key differences from the series case are given. The general form of the differential equations given in the series circuit section are applicable to all second order circuits and can be used to describe the voltage or current in any <a href="/wiki/Electrical_element" title="Electrical element">element</a> of each circuit. </p> <div class="mw-heading mw-heading2"><h2 id="Series_circuit">Series circuit</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=8" title="Edit section: Series circuit"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:RLC_series_circuit_v1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/RLC_series_circuit_v1.svg/250px-RLC_series_circuit_v1.svg.png" decoding="async" width="250" height="346" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fb/RLC_series_circuit_v1.svg/375px-RLC_series_circuit_v1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fb/RLC_series_circuit_v1.svg/500px-RLC_series_circuit_v1.svg.png 2x" data-file-width="135" data-file-height="187" /></a><figcaption><b>Figure 1:</b> RLC series circuit <style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"><ul><li><span class="texhtml mvar" style="font-style:italic;">V</span>, the voltage source powering the circuit</li><li><span class="texhtml mvar" style="font-style:italic;">I</span>, the current admitted through the circuit</li><li><span class="texhtml mvar" style="font-style:italic;">R</span>, the effective resistance of the combined load, source, and components</li><li><span class="texhtml mvar" style="font-style:italic;">L</span>, the inductance of the <a href="/wiki/Inductor" title="Inductor">inductor</a> component</li><li><span class="texhtml mvar" style="font-style:italic;">C</span>, the capacitance of the <a href="/wiki/Capacitor" title="Capacitor">capacitor</a> component</li></ul></div></figcaption></figure> <p>In this circuit, the three components are all in series with the <a href="/wiki/Voltage_source" title="Voltage source">voltage source</a>. The governing <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> can be found by substituting into <a href="/wiki/Kirchhoff%27s_voltage_law" class="mw-redirect" title="Kirchhoff's voltage law">Kirchhoff's voltage law</a> (KVL) the <a href="/wiki/Constitutive_equation" title="Constitutive equation">constitutive equation</a> for each of the three elements. From the KVL, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\mathrm {R} }+V_{\mathrm {L} }+V_{\mathrm {C} }=V(t)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathrm {R} }+V_{\mathrm {L} }+V_{\mathrm {C} }=V(t)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/334c49f2a0c29404bd1560ca0de2598152d4607d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.435ex; height:2.843ex;" alt="{\displaystyle V_{\mathrm {R} }+V_{\mathrm {L} }+V_{\mathrm {C} }=V(t)\,,}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">V<sub>R</sub></span>, <span class="texhtml"><i>V</i><sub>L</sub></span> and <span class="texhtml"><i>V</i><sub>C</sub></span> are the voltages across <span class="texhtml mvar" style="font-style:italic;">R</span>, <span class="texhtml mvar" style="font-style:italic;">L</span>, and <span class="texhtml mvar" style="font-style:italic;">C</span>, respectively, and <span class="texhtml"><i>V</i>(<i>t</i>)</span> is the time-varying voltage from the source. </p><p>Substituting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{R}=R\ I(t)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>=</mo> <mi>R</mi> <mtext> </mtext> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{R}=R\ I(t)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d73b9e8bba55d0f68b32cae99db5bd42609c1757" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.133ex; height:2.843ex;" alt="{\displaystyle V_{R}=R\ I(t)\,,}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,V_{\mathrm {L} }=L{\frac {\mathrm {d} I(t)}{\mathrm {d} t}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">L</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,V_{\mathrm {L} }=L{\frac {\mathrm {d} I(t)}{\mathrm {d} t}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12102511b3e85d2a68022392026dcdd1a97b8fb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.02ex; height:5.843ex;" alt="{\displaystyle \,V_{\mathrm {L} }=L{\frac {\mathrm {d} I(t)}{\mathrm {d} t}}\,}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,V_{\mathrm {C} }=V(0)+{\frac {1}{\,C\,}}\int _{0}^{t}I(\tau )\,\mathrm {d} \tau \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">C</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>I</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>τ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,V_{\mathrm {C} }=V(0)+{\frac {1}{\,C\,}}\int _{0}^{t}I(\tau )\,\mathrm {d} \tau \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94360ffb5d4af9f340bacb591df7e7c3da68741a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.752ex; height:6.176ex;" alt="{\displaystyle \,V_{\mathrm {C} }=V(0)+{\frac {1}{\,C\,}}\int _{0}^{t}I(\tau )\,\mathrm {d} \tau \,}"></span> into the equation above yields: </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 RI(t)+L{\frac {\,\mathrm {d} I(t)\,}{\mathrm {d} t}}+V(0)+{\frac {1}{\,C\,}}\int _{0}^{t}I(\tau )\,\mathrm {d} \tau =V(t)\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>I</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>τ</mi> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RI(t)+L{\frac {\,\mathrm {d} I(t)\,}{\mathrm {d} t}}+V(0)+{\frac {1}{\,C\,}}\int _{0}^{t}I(\tau )\,\mathrm {d} \tau =V(t)\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5926f68f3b982c19616601a46fbb4aa2a9bda719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:50.503ex; height:6.176ex;" alt="{\displaystyle RI(t)+L{\frac {\,\mathrm {d} I(t)\,}{\mathrm {d} t}}+V(0)+{\frac {1}{\,C\,}}\int _{0}^{t}I(\tau )\,\mathrm {d} \tau =V(t)\;.}"></span></dd></dl> <p>For the case where the source is an unchanging voltage, taking the time derivative and dividing by <span class="texhtml mvar" style="font-style:italic;">L</span> leads to the following second order differential equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}I(t)+{\frac {R}{\,L\,}}{\frac {\mathrm {d} }{\mathrm {d} t}}I(t)+{\frac {1}{\,L\,C\,}}I(t)=0\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mrow> <mspace width="thinmathspace" /> <mi>L</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mi>L</mi> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}I(t)+{\frac {R}{\,L\,}}{\frac {\mathrm {d} }{\mathrm {d} t}}I(t)+{\frac {1}{\,L\,C\,}}I(t)=0\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08462b05595b763aff0a5da2ff567f5eb8b6fc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:38.226ex; height:6.009ex;" alt="{\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}I(t)+{\frac {R}{\,L\,}}{\frac {\mathrm {d} }{\mathrm {d} t}}I(t)+{\frac {1}{\,L\,C\,}}I(t)=0\;.}"></span></dd></dl> <p>This can usefully be expressed in a more generally applicable form: </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 {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}I(t)+2\alpha {\frac {\mathrm {d} }{\mathrm {d} t}}I(t)+\omega _{0}^{2}I(t)=0\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}I(t)+2\alpha {\frac {\mathrm {d} }{\mathrm {d} t}}I(t)+\omega _{0}^{2}I(t)=0\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b39a6e98c130f832005e97d0e3bbc5bda10a95e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:34.837ex; height:6.009ex;" alt="{\displaystyle {\frac {\mathrm {d} ^{2}}{\mathrm {d} t^{2}}}I(t)+2\alpha {\frac {\mathrm {d} }{\mathrm {d} t}}I(t)+\omega _{0}^{2}I(t)=0\;.}"></span></dd></dl> <p><span class="texhtml mvar" style="font-style:italic;">α</span> and <span class="texhtml"><i>ω</i><sub>0</sub></span> are both in units of <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a>. <span class="texhtml mvar" style="font-style:italic;">α</span> is called the <i>neper frequency</i>, or <i>attenuation</i>, and is a measure of how fast the <a href="/wiki/Transient_response" title="Transient response">transient response</a> of the circuit will die away after the stimulus has been removed. Neper occurs in the name because the units can also be considered to be <a href="/wiki/Neper" title="Neper">nepers</a> per second, neper being a <a href="/wiki/Logarithmic_unit" class="mw-redirect" title="Logarithmic unit">logarithmic unit</a> of attenuation. <span class="texhtml"><i>ω</i><sub>0</sub></span> is the angular resonance frequency.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>For the case of the series RLC circuit these two parameters are given by:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\alpha &={\frac {R}{\,2L\,}}\\\omega _{0}&={\frac {1}{\,{\sqrt {L\,C\,}}\,}}\;.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>α</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mrow> <mspace width="thinmathspace" /> <mn>2</mn> <mi>L</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>L</mi> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </msqrt> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\alpha &={\frac {R}{\,2L\,}}\\\omega _{0}&={\frac {1}{\,{\sqrt {L\,C\,}}\,}}\;.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95ebcf1b7de32223f7ed36e3ea7e37350e2189d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:15.312ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}\alpha &={\frac {R}{\,2L\,}}\\\omega _{0}&={\frac {1}{\,{\sqrt {L\,C\,}}\,}}\;.\end{aligned}}}"></span></dd></dl> <p>A useful parameter is the <i>damping factor</i>, <span class="texhtml mvar" style="font-style:italic;">ζ</span>, which is defined as the ratio of these two; although, sometimes <span class="texhtml mvar" style="font-style:italic;">ζ</span> is not used, and <span class="texhtml mvar" style="font-style:italic;">α</span> is referred to as <i>damping factor</i> instead; hence requiring careful specification of one's use of that term.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </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 \zeta \equiv {\frac {\alpha }{\,\omega _{0}\,}}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mspace width="thinmathspace" /> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta \equiv {\frac {\alpha }{\,\omega _{0}\,}}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efda41fdb1da78cee7fef144d758a9840a0da012" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.596ex; height:5.009ex;" alt="{\displaystyle \zeta \equiv {\frac {\alpha }{\,\omega _{0}\,}}\;.}"></span></dd></dl> <p>In the case of the series RLC circuit, the damping factor is given by </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 \zeta ={\frac {\,R\,}{2}}{\sqrt {{\frac {C}{\,L\,}}\,}}={\frac {1}{\ 2Q\ }}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mi>R</mi> <mspace width="thinmathspace" /> </mrow> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mrow> <mspace width="thinmathspace" /> <mi>L</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mn>2</mn> <mi>Q</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta ={\frac {\,R\,}{2}}{\sqrt {{\frac {C}{\,L\,}}\,}}={\frac {1}{\ 2Q\ }}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a394c86b33315d27c97c1889b8d1f3afe20b28d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.796ex; height:6.176ex;" alt="{\displaystyle \zeta ={\frac {\,R\,}{2}}{\sqrt {{\frac {C}{\,L\,}}\,}}={\frac {1}{\ 2Q\ }}~.}"></span></dd></dl> <p>The value of the damping factor determines the type of transient that the circuit will exhibit.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Transient_response">Transient response</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=9" title="Edit section: Transient response"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:RLC_transient_plot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/RLC_transient_plot.svg/350px-RLC_transient_plot.svg.png" decoding="async" width="350" height="281" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/RLC_transient_plot.svg/525px-RLC_transient_plot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8e/RLC_transient_plot.svg/700px-RLC_transient_plot.svg.png 2x" data-file-width="1001" data-file-height="805" /></a><figcaption>Plot showing underdamped and overdamped responses of a series RLC circuit to a voltage input step of 1 V. The critical damping plot is the bold red curve. The plots are normalised for <span class="texhtml"><i>L</i> = 1</span>, <span class="texhtml"><i>C</i> = 1</span> and <span class="texhtml"><i>ω</i><sub>0</sub> = 1</span>.</figcaption></figure> <p>The differential equation has the <a href="/wiki/Linear_homogeneous_differential_equation" class="mw-redirect" title="Linear homogeneous differential equation">characteristic equation</a>,<sup id="cite_ref-Argawal656_8-0" class="reference"><a href="#cite_note-Argawal656-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}+2\alpha s+\omega _{0}^{2}=0\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>α</mi> <mi>s</mi> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{2}+2\alpha s+\omega _{0}^{2}=0\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/955f6d702e420613bbc28aaec1d69f9fc3c2e469" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.361ex; height:3.343ex;" alt="{\displaystyle s^{2}+2\alpha s+\omega _{0}^{2}=0\,.}"></span></dd></dl> <p>The roots of the equation in <span class="texhtml mvar" style="font-style:italic;">s</span>-domain are,<sup id="cite_ref-Argawal656_8-1" class="reference"><a href="#cite_note-Argawal656-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}s_{1}&=-\alpha +{\sqrt {\alpha ^{2}-\omega _{0}^{2}\,}}=-\omega _{0}\left(\zeta -{\sqrt {\zeta ^{2}-1\,}}\right)={\frac {-\omega _{0}}{\ \zeta +{\sqrt {\zeta ^{2}-1\ }}\ }}\ ,\\s_{2}&=-\alpha -{\sqrt {\alpha ^{2}-\omega _{0}^{2}\,}}=-\omega _{0}\left(\zeta +{\sqrt {\zeta ^{2}-1\,}}\right)\;.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>α</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="thinmathspace" /> </msqrt> </mrow> <mo>=</mo> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>ζ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mspace width="thinmathspace" /> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mtext> </mtext> <mi>ζ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>α</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="thinmathspace" /> </msqrt> </mrow> <mo>=</mo> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>ζ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mspace width="thinmathspace" /> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s_{1}&=-\alpha +{\sqrt {\alpha ^{2}-\omega _{0}^{2}\,}}=-\omega _{0}\left(\zeta -{\sqrt {\zeta ^{2}-1\,}}\right)={\frac {-\omega _{0}}{\ \zeta +{\sqrt {\zeta ^{2}-1\ }}\ }}\ ,\\s_{2}&=-\alpha -{\sqrt {\alpha ^{2}-\omega _{0}^{2}\,}}=-\omega _{0}\left(\zeta +{\sqrt {\zeta ^{2}-1\,}}\right)\;.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1cfeeff21eaf2abeca109069b7a926ffc8bcbc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:66.099ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}s_{1}&=-\alpha +{\sqrt {\alpha ^{2}-\omega _{0}^{2}\,}}=-\omega _{0}\left(\zeta -{\sqrt {\zeta ^{2}-1\,}}\right)={\frac {-\omega _{0}}{\ \zeta +{\sqrt {\zeta ^{2}-1\ }}\ }}\ ,\\s_{2}&=-\alpha -{\sqrt {\alpha ^{2}-\omega _{0}^{2}\,}}=-\omega _{0}\left(\zeta +{\sqrt {\zeta ^{2}-1\,}}\right)\;.\end{aligned}}}"></span></dd></dl> <p>The general solution of the differential equation is an exponential in either root or a linear superposition of both, </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 I(t)=A_{1}e^{s_{1}t}+A_{2}e^{s_{2}t}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>t</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>t</mi> </mrow> </msup> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(t)=A_{1}e^{s_{1}t}+A_{2}e^{s_{2}t}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2dec72ad1e8474f8930e4e36210af0a87640668" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.67ex; height:3.009ex;" alt="{\displaystyle I(t)=A_{1}e^{s_{1}t}+A_{2}e^{s_{2}t}\;.}"></span></dd></dl> <p>The coefficients <span class="texhtml"><i>A</i><sub>1</sub></span> and <span class="texhtml"><i>A</i><sub>2</sub></span> are determined by the <a href="/wiki/Boundary_condition" class="mw-redirect" title="Boundary condition">boundary conditions</a> of the specific problem being analysed. That is, they are set by the values of the currents and voltages in the circuit at the onset of the transient and the presumed value they will settle to after infinite time.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> The differential equation for the circuit solves in three different ways depending on the value of <span class="texhtml mvar" style="font-style:italic;">ζ</span>. These are overdamped (<span class="texhtml"><i>ζ</i> > 1</span>), underdamped (<span class="texhtml"><i>ζ</i> < 1</span>), and critically damped (<span class="texhtml"><i>ζ</i> = 1</span>). </p> <div class="mw-heading mw-heading4"><h4 id="Overdamped_response">Overdamped response</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=10" title="Edit section: Overdamped response"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The overdamped response (<span class="texhtml"><i>ζ</i> > 1</span>) is<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </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 I(t)=A_{1}e^{-\omega _{0}\left(\zeta +{\sqrt {\zeta ^{2}-1}}\right)t}+A_{2}e^{-\omega _{0}\left(\zeta -{\sqrt {\zeta ^{2}-1}}\right)t}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>ζ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mi>t</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mi>ζ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mi>t</mi> </mrow> </msup> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(t)=A_{1}e^{-\omega _{0}\left(\zeta +{\sqrt {\zeta ^{2}-1}}\right)t}+A_{2}e^{-\omega _{0}\left(\zeta -{\sqrt {\zeta ^{2}-1}}\right)t}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c30df6d45b03992a1c677987146dd194ec532197" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.406ex; height:5.343ex;" alt="{\displaystyle I(t)=A_{1}e^{-\omega _{0}\left(\zeta +{\sqrt {\zeta ^{2}-1}}\right)t}+A_{2}e^{-\omega _{0}\left(\zeta -{\sqrt {\zeta ^{2}-1}}\right)t}\;.}"></span></dd></dl> <p>The overdamped response is a decay of the transient current without oscillation.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Underdamped_response">Underdamped response</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=11" title="Edit section: Underdamped response"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The underdamped response (<span class="texhtml"><i>ζ</i> < 1</span>) is<sup id="cite_ref-Nilsson295_12-0" class="reference"><a href="#cite_note-Nilsson295-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </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 I(t)=B_{1}e^{-\alpha t}\cos(\omega _{\mathrm {d} }t)+B_{2}e^{-\alpha t}\sin(\omega _{\mathrm {d} }t)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>t</mi> </mrow> </msup> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>t</mi> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(t)=B_{1}e^{-\alpha t}\cos(\omega _{\mathrm {d} }t)+B_{2}e^{-\alpha t}\sin(\omega _{\mathrm {d} }t)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f6d9f477021c8247201c5a0db3efebe04c4e205" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.133ex; height:3.009ex;" alt="{\displaystyle I(t)=B_{1}e^{-\alpha t}\cos(\omega _{\mathrm {d} }t)+B_{2}e^{-\alpha t}\sin(\omega _{\mathrm {d} }t)\,.}"></span></dd></dl> <p>By applying standard <a href="/wiki/List_of_trigonometric_identities#Linear_combinations" title="List of trigonometric identities">trigonometric identities</a> the two trigonometric functions may be expressed as a single sinusoid with phase shift,<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </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 I(t)=B_{3}e^{-\alpha t}\sin(\omega _{\mathrm {d} }t+\varphi )\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>t</mi> </mrow> </msup> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(t)=B_{3}e^{-\alpha t}\sin(\omega _{\mathrm {d} }t+\varphi )\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/973d1e933c54af59a4ed1ce76f7673f88b02b790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.113ex; height:3.009ex;" alt="{\displaystyle I(t)=B_{3}e^{-\alpha t}\sin(\omega _{\mathrm {d} }t+\varphi )\;.}"></span></dd></dl> <p>The underdamped response is a decaying oscillation at frequency <span class="texhtml"><i>ω</i><sub>d</sub></span>. The oscillation decays at a rate determined by the attenuation <span class="texhtml">α</span>. The exponential in <span class="texhtml">α</span> describes the <a href="/wiki/Envelope_(waves)" title="Envelope (waves)">envelope</a> of the oscillation. <span class="texhtml"><i>B</i><sub>1</sub></span> and <span class="texhtml"><i>B</i><sub>2</sub></span> (or <span class="texhtml"><i>B</i><sub>3</sub></span> and the phase shift <span class="texhtml mvar" style="font-style:italic;">φ</span> in the second form) are arbitrary constants determined by boundary conditions. The frequency <span class="texhtml"><i>ω</i><sub>d</sub></span> is given by<sup id="cite_ref-Nilsson295_12-1" class="reference"><a href="#cite_note-Nilsson295-12"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </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 _{\mathrm {d} }={\sqrt {\omega _{0}^{2}-\alpha ^{2}}}=\omega _{0}{\sqrt {1-\zeta ^{2}\,}}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>=</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>ζ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </msqrt> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathrm {d} }={\sqrt {\omega _{0}^{2}-\alpha ^{2}}}=\omega _{0}{\sqrt {1-\zeta ^{2}\,}}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dc22f20162d14b34c780ca36fd018bd0597bdfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:31.673ex; height:4.843ex;" alt="{\displaystyle \omega _{\mathrm {d} }={\sqrt {\omega _{0}^{2}-\alpha ^{2}}}=\omega _{0}{\sqrt {1-\zeta ^{2}\,}}\;.}"></span></dd></dl> <p>This is called the damped resonance frequency or the damped natural frequency. It is the frequency the circuit will naturally oscillate at if not driven by an external source. The resonance frequency, <span class="texhtml"><i>ω</i><sub>0</sub></span>, which is the frequency at which the circuit will resonate when driven by an external oscillation, may often be referred to as the undamped resonance frequency to distinguish it.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Critically_damped_response">Critically damped response</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=12" title="Edit section: Critically damped response"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The critically damped response (<span class="texhtml"><i>ζ</i> = 1</span>) is<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </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 I(t)=D_{1}te^{-\alpha t}+D_{2}e^{-\alpha t}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>t</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>t</mi> </mrow> </msup> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(t)=D_{1}te^{-\alpha t}+D_{2}e^{-\alpha t}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a27728f303c7f6ac45a04571bdd2d9aed8f27c78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.328ex; height:3.009ex;" alt="{\displaystyle I(t)=D_{1}te^{-\alpha t}+D_{2}e^{-\alpha t}\;.}"></span></dd></dl> <p>The critically damped response represents the circuit response that decays in the fastest possible time without going into oscillation. This consideration is important in control systems where it is required to reach the desired state as quickly as possible without overshooting. <span class="texhtml"><i>D</i><sub>1</sub></span> and <span class="texhtml"><i>D</i><sub>2</sub></span> are arbitrary constants determined by boundary conditions.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Laplace_domain">Laplace domain</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=13" title="Edit section: Laplace domain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The series RLC can be analyzed for both transient and steady AC state behavior using the <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a>.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> If the voltage source above produces a waveform with Laplace-transformed <span class="texhtml"><i>V</i>(<i>s</i>)</span> (where <span class="texhtml mvar" style="font-style:italic;">s</span> is the <a href="/wiki/Complex_frequency" class="mw-redirect" title="Complex frequency">complex frequency</a> <span class="texhtml"><i>s</i> = <i>σ</i> + <i>jω</i></span>), the <a href="/wiki/KVL" class="mw-redirect" title="KVL">KVL</a> can be applied in the Laplace domain: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(s)=I(s)\left(R+L\,s+{\frac {1}{\,C\,s\,}}\right)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <mi>R</mi> <mo>+</mo> <mi>L</mi> <mspace width="thinmathspace" /> <mi>s</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> <mi>s</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(s)=I(s)\left(R+L\,s+{\frac {1}{\,C\,s\,}}\right)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/503370d0dbc1eb64662f3a24e15c944505b67c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.446ex; height:6.176ex;" alt="{\displaystyle V(s)=I(s)\left(R+L\,s+{\frac {1}{\,C\,s\,}}\right)\,,}"></span></dd></dl> <p>where <span class="texhtml"><i>I</i>(<i>s</i>)</span> is the Laplace-transformed current through all components. Solving for <span class="texhtml mvar" style="font-style:italic;"><i>I</i>(<i>s</i>)</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 I(s)={\frac {1}{\,R+L\,s+{\frac {1}{\,C\,s\,}}\,}}V(s)\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mi>R</mi> <mo>+</mo> <mi>L</mi> <mspace width="thinmathspace" /> <mi>s</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> <mi>s</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(s)={\frac {1}{\,R+L\,s+{\frac {1}{\,C\,s\,}}\,}}V(s)\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1052bc5807a72f42e0685c4bf2ab414928abbc49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:29.282ex; height:6.843ex;" alt="{\displaystyle I(s)={\frac {1}{\,R+L\,s+{\frac {1}{\,C\,s\,}}\,}}V(s)\;.}"></span></dd></dl> <p>And rearranging, we have </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 I(s)={\frac {s}{\,L\ \left(s^{2}+{\frac {R}{\,L\,}}s+{\frac {1}{\,L\,C\,}}\right)\,}}V(s)\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mrow> <mspace width="thinmathspace" /> <mi>L</mi> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mrow> <mspace width="thinmathspace" /> <mi>L</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mi>s</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mi>L</mi> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(s)={\frac {s}{\,L\ \left(s^{2}+{\frac {R}{\,L\,}}s+{\frac {1}{\,L\,C\,}}\right)\,}}V(s)\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1a1133109c5e387c71907dff60f9ffc7356c1c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:36.097ex; height:7.343ex;" alt="{\displaystyle I(s)={\frac {s}{\,L\ \left(s^{2}+{\frac {R}{\,L\,}}s+{\frac {1}{\,L\,C\,}}\right)\,}}V(s)\;.}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Laplace_admittance">Laplace admittance</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=14" title="Edit section: Laplace admittance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Solving for the Laplace <a href="/wiki/Admittance" title="Admittance">admittance</a> <span class="texhtml"><i>Y</i>(<i>s</i>)</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 Y(s)={\frac {I(s)}{\,V(s)\,}}={\frac {s}{\,L\ \left(s^{2}+{\frac {R}{\,L\,}}s+{\frac {1}{\,L\,C\,}}\right)\,}}\,.}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mspace width="thinmathspace" /> <mi>V</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mrow> <mspace width="thinmathspace" /> <mi>L</mi> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mrow> <mspace width="thinmathspace" /> <mi>L</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mi>s</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mi>L</mi> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y(s)={\frac {I(s)}{\,V(s)\,}}={\frac {s}{\,L\ \left(s^{2}+{\frac {R}{\,L\,}}s+{\frac {1}{\,L\,C\,}}\right)\,}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/645ef6e417eb19dcc54f80011849c483bc72026e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:41.149ex; height:8.343ex;" alt="{\displaystyle Y(s)={\frac {I(s)}{\,V(s)\,}}={\frac {s}{\,L\ \left(s^{2}+{\frac {R}{\,L\,}}s+{\frac {1}{\,L\,C\,}}\right)\,}}\,.}"></span></dd></dl> <p>Simplifying using parameters <span class="texhtml mvar" style="font-style:italic;">α</span> and <span class="texhtml"><i>ω</i><sub>0</sub></span> defined in the previous section, we have </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 Y(s)={\frac {I(s)}{\,V(s)\,}}={\frac {s}{\,L\ \left(s^{2}+2\alpha s+\omega _{0}^{2}\right)\,}}\;.}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mspace width="thinmathspace" /> <mi>V</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mrow> <mspace width="thinmathspace" /> <mi>L</mi> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>α</mi> <mi>s</mi> <mo>+</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y(s)={\frac {I(s)}{\,V(s)\,}}={\frac {s}{\,L\ \left(s^{2}+2\alpha s+\omega _{0}^{2}\right)\,}}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c449fa2f91695dbb8c25d0f0c001062f3577b1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.816ex; height:7.009ex;" alt="{\displaystyle Y(s)={\frac {I(s)}{\,V(s)\,}}={\frac {s}{\,L\ \left(s^{2}+2\alpha s+\omega _{0}^{2}\right)\,}}\;.}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Poles_and_zeros">Poles and zeros</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=15" title="Edit section: Poles and zeros"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Zero_(complex_analysis)" class="mw-redirect" title="Zero (complex analysis)">zeros</a> of <span class="texhtml"><i>Y</i>(<i>s</i>)</span> are those values of <span class="texhtml mvar" style="font-style:italic;">s</span> where <span class="texhtml"><i>Y</i>(<i>s</i>) = 0</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 s=0\quad {\mbox{and}}\quad |s|\rightarrow \infty \;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>and</mtext> </mstyle> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=0\quad {\mbox{and}}\quad |s|\rightarrow \infty \;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37fd3384ea98f0d4061b85125300016cf800611e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.358ex; height:2.843ex;" alt="{\displaystyle s=0\quad {\mbox{and}}\quad |s|\rightarrow \infty \;.}"></span></dd></dl> <p>The <a href="/wiki/Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">poles</a> of <span class="texhtml"><i>Y</i>(<i>s</i>)</span> are those values of <span class="texhtml">s</span> where <span class="texhtml"><i>Y</i>(<i>s</i>) → ∞</span>. By the <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic formula</a>, we find </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=-\alpha \pm {\sqrt {\alpha ^{2}-\omega _{0}^{2}\,}}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mo>−</mo> <mi>α</mi> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="thinmathspace" /> </msqrt> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=-\alpha \pm {\sqrt {\alpha ^{2}-\omega _{0}^{2}\,}}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11682fee474029a8b7e7ec750aba9a2fe9143d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:22.21ex; height:4.676ex;" alt="{\displaystyle s=-\alpha \pm {\sqrt {\alpha ^{2}-\omega _{0}^{2}\,}}\;.}"></span></dd></dl> <p>The poles of <span class="texhtml"><i>Y</i>(<i>s</i>)</span> are identical to the roots <span class="texhtml"><i>s</i><sub>1</sub></span> and <span class="texhtml"><i>s</i><sub>2</sub></span> of the characteristic polynomial of the differential equation in the section above. </p> <div class="mw-heading mw-heading4"><h4 id="General_solution">General solution</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=16" title="Edit section: General solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For an arbitrary <span class="texhtml"><i>V</i>(<i>t</i>)</span>, the solution obtained by inverse transform of <span class="texhtml"><i>I</i>(<i>s</i>)</span> is: </p> <ul><li>In the underdamped case, <span class="texhtml"><i>ω</i><sub>0</sub> > <i>α</i></span>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(t)={\frac {1}{\,L\,}}\ \int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\left[\cos(\omega _{\mathrm {d} }\tau )-{\frac {\alpha }{\ \omega _{\mathrm {d} }\ }}\sin(\omega _{\mathrm {d} }\tau )\right]\,d\tau \,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mi>L</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>V</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>τ</mi> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mtext> </mtext> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mtext> </mtext> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(t)={\frac {1}{\,L\,}}\ \int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\left[\cos(\omega _{\mathrm {d} }\tau )-{\frac {\alpha }{\ \omega _{\mathrm {d} }\ }}\sin(\omega _{\mathrm {d} }\tau )\right]\,d\tau \,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91b41e0e443a918d8e3841ce47450578d04636e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:59.79ex; height:6.343ex;" alt="{\displaystyle I(t)={\frac {1}{\,L\,}}\ \int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\left[\cos(\omega _{\mathrm {d} }\tau )-{\frac {\alpha }{\ \omega _{\mathrm {d} }\ }}\sin(\omega _{\mathrm {d} }\tau )\right]\,d\tau \,,}"></span></dd></dl></li> <li>In the critically damped case, <span class="texhtml"><i>ω</i><sub>0</sub> = <i>α</i></span>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ I(t)={\frac {1}{\ L\ }}\ \int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\ \left[\ 1-\alpha \tau \ \right]\ \mathrm {d} \tau \ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>L</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>V</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>τ</mi> </mrow> </msup> <mtext> </mtext> <mrow> <mo>[</mo> <mrow> <mtext> </mtext> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mi>τ</mi> <mtext> </mtext> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>τ</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ I(t)={\frac {1}{\ L\ }}\ \int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\ \left[\ 1-\alpha \tau \ \right]\ \mathrm {d} \tau \ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e7e9db9cb9752337607155caa71339efd24234d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.504ex; height:6.176ex;" alt="{\displaystyle \ I(t)={\frac {1}{\ L\ }}\ \int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\ \left[\ 1-\alpha \tau \ \right]\ \mathrm {d} \tau \ ,}"></span></dd></dl></li> <li>In the overdamped case, <span class="texhtml"><i>ω</i><sub>0</sub> < <i>α</i></span>: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ I(t)={\frac {1}{\ L\ }}\ \int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\left[\cosh(\omega _{\mathrm {r} }\tau )-{\frac {\alpha }{\ \omega _{\mathrm {r} }\ }}\sinh(\omega _{\mathrm {r} }\tau )\right]\,d\tau \ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>L</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>V</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mi>τ</mi> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>α</mi> <mrow> <mtext> </mtext> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mtext> </mtext> </mrow> </mfrac> </mrow> <mi>sinh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ I(t)={\frac {1}{\ L\ }}\ \int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\left[\cosh(\omega _{\mathrm {r} }\tau )-{\frac {\alpha }{\ \omega _{\mathrm {r} }\ }}\sinh(\omega _{\mathrm {r} }\tau )\right]\,d\tau \ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2527f90fc993b69c1f37efe033d27ab8df93db74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:62.728ex; height:6.343ex;" alt="{\displaystyle \ I(t)={\frac {1}{\ L\ }}\ \int _{0}^{t}V(t-\tau )e^{-\alpha \tau }\left[\cosh(\omega _{\mathrm {r} }\tau )-{\frac {\alpha }{\ \omega _{\mathrm {r} }\ }}\sinh(\omega _{\mathrm {r} }\tau )\right]\,d\tau \ ,}"></span></dd></dl></li></ul> <p>where <span class="texhtml"><i>ω</i><sub>r</sub> = <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>α</i><sup>2</sup> − <i>ω</i><sub>0</sub><sup>2</sup></span></span></span>, and <span class="texhtml">cosh</span> and <span class="texhtml">sinh</span> are the usual <a href="/wiki/Hyperbolic_function" class="mw-redirect" title="Hyperbolic function">hyperbolic functions</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Sinusoidal_steady_state">Sinusoidal steady state</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=17" title="Edit section: Sinusoidal steady state"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:RLC_Series_Circuit_Bode_Magnitude_Plot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/RLC_Series_Circuit_Bode_Magnitude_Plot.svg/220px-RLC_Series_Circuit_Bode_Magnitude_Plot.svg.png" decoding="async" width="220" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/RLC_Series_Circuit_Bode_Magnitude_Plot.svg/330px-RLC_Series_Circuit_Bode_Magnitude_Plot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/RLC_Series_Circuit_Bode_Magnitude_Plot.svg/440px-RLC_Series_Circuit_Bode_Magnitude_Plot.svg.png 2x" data-file-width="1971" data-file-height="937" /></a><figcaption>Bode magnitude plot for the voltages across the elements of an RLC series circuit. Natural frequency <span class="nowrap"><i>ω</i><sub>0</sub> = 1 rad/s</span>, damping ratio <span class="nowrap"><i>ζ</i> = 0.4</span>.</figcaption></figure> <p>Sinusoidal steady state is represented by letting <span class="texhtml"><i>s</i> = <i>jω</i></span>, where <span class="texhtml mvar" style="font-style:italic;">j</span> is the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>. Taking the magnitude of the above equation with this substitution: </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 {\big |}Y(j\omega ){\big |}={\frac {1}{{\sqrt {R^{2}+\left(\omega L-{\frac {1}{\,\omega C\,}}\right)^{2}}}\,}}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mi>ω</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>ω</mi> <mi>L</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mi>ω</mi> <mi>C</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big |}Y(j\omega ){\big |}={\frac {1}{{\sqrt {R^{2}+\left(\omega L-{\frac {1}{\,\omega C\,}}\right)^{2}}}\,}}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c6e5c7ffdb92df57c7ca9f739d273f4dba88c08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:34.457ex; height:9.343ex;" alt="{\displaystyle {\big |}Y(j\omega ){\big |}={\frac {1}{{\sqrt {R^{2}+\left(\omega L-{\frac {1}{\,\omega C\,}}\right)^{2}}}\,}}\;.}"></span></dd></dl> <p>and the current as a function of <span class="texhtml mvar" style="font-style:italic;">ω</span> can be found from </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 {\big |}I(j\omega ){\big |}={\big |}Y(j\omega ){\big |}\cdot {\bigl |}V(j\omega ){\bigr |}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mi>I</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mi>ω</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mi>ω</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mi>V</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mi>ω</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">|</mo> </mrow> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\big |}I(j\omega ){\big |}={\big |}Y(j\omega ){\big |}\cdot {\bigl |}V(j\omega ){\bigr |}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e22d2324b6ea291abd5052ef8ec43a5afd49091d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.322ex; height:3.176ex;" alt="{\displaystyle {\big |}I(j\omega ){\big |}={\big |}Y(j\omega ){\big |}\cdot {\bigl |}V(j\omega ){\bigr |}\;.}"></span></dd></dl> <p>There is a peak value of <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>I</i>(<i>jω</i>)</span>|</span>. The value of <span class="texhtml mvar" style="font-style:italic;">ω</span> at this peak is, in this particular case, equal to the undamped natural resonance frequency.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> This means that the maximum voltage across the resistor, and thus maximum heat dissipation, occurs at the natural frequency. </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 _{0}={\frac {1}{\,{\sqrt {L\ C\ }}\,}}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>L</mi> <mtext> </mtext> <mi>C</mi> <mtext> </mtext> </msqrt> </mrow> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}={\frac {1}{\,{\sqrt {L\ C\ }}\,}}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ac0b501defadea4a31ad5cd1ffefd940debfc52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.947ex; height:6.176ex;" alt="{\displaystyle \omega _{0}={\frac {1}{\,{\sqrt {L\ C\ }}\,}}\;.}"></span></dd></dl> <p>From the frequency response of the current, the frequency response of the voltages across the various circuit elements can also be determined (see figure). Moreover, the maximum voltage across the capacitor happens at a frequency </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 _{C}={\frac {\ {\sqrt {{\tfrac {L}{\ C\ }}-{\tfrac {1}{2}}R^{2}}}\ }{L}}\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>L</mi> <mrow> <mtext> </mtext> <mi>C</mi> <mtext> </mtext> </mrow> </mfrac> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mtext> </mtext> </mrow> <mi>L</mi> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{C}={\frac {\ {\sqrt {{\tfrac {L}{\ C\ }}-{\tfrac {1}{2}}R^{2}}}\ }{L}}\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/926f77bfe31beba4c5ae723cf02152081cc3ebae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.202ex; height:7.676ex;" alt="{\displaystyle \omega _{C}={\frac {\ {\sqrt {{\tfrac {L}{\ C\ }}-{\tfrac {1}{2}}R^{2}}}\ }{L}}\;,}"></span></dd></dl> <p>whereas the maximum voltage across the inductor occurs at </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 _{L}={\frac {1}{\ C{\sqrt {{\tfrac {L}{\ C\ }}-{\tfrac {1}{2}}R^{2}\ }}\ }}\;.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>L</mi> <mrow> <mtext> </mtext> <mi>C</mi> <mtext> </mtext> </mrow> </mfrac> </mstyle> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mspace width="thickmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{L}={\frac {1}{\ C{\sqrt {{\tfrac {L}{\ C\ }}-{\tfrac {1}{2}}R^{2}\ }}\ }}\;.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c161593a8eb980289870283be1504df8783cae88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:24.419ex; height:8.009ex;" alt="{\displaystyle \omega _{L}={\frac {1}{\ C{\sqrt {{\tfrac {L}{\ C\ }}-{\tfrac {1}{2}}R^{2}\ }}\ }}\;.}"></span></dd></dl> <p>It holds: <span class="mwe-math-element"><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 _{C}<\omega _{0}<\omega _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo><</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{C}<\omega _{0}<\omega _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce2dbd8e4792864135c5325b12909db157ca645a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.421ex; height:2.176ex;" alt="{\displaystyle \omega _{C}<\omega _{0}<\omega _{L}}"></span>. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Parallel_circuit"><span class="anchor" id="parallel_circuit_anchor">Parallel circuit</span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=18" title="Edit section: Parallel circuit"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:RLC_parallel_circuit_v1.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/RLC_parallel_circuit_v1.svg/200px-RLC_parallel_circuit_v1.svg.png" decoding="async" width="200" height="110" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d0/RLC_parallel_circuit_v1.svg/300px-RLC_parallel_circuit_v1.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d0/RLC_parallel_circuit_v1.svg/400px-RLC_parallel_circuit_v1.svg.png 2x" data-file-width="208" data-file-height="114" /></a><figcaption><b>Figure 2.</b> RLC parallel circuit<br /> <span class="texhtml mvar" style="font-style:italic;">V</span> – the voltage source powering the circuit<br /> <span class="texhtml mvar" style="font-style:italic;">I</span> – the current admitted through the circuit<br /> <span class="texhtml mvar" style="font-style:italic;">R</span> – the equivalent resistance of the combined source, load, and components<br /> <span class="texhtml mvar" style="font-style:italic;">L</span> – the inductance of the inductor component<br /> <span class="texhtml mvar" style="font-style:italic;">C</span> – the capacitance of the capacitor component</figcaption></figure> <p>The properties of the parallel RLC circuit can be obtained from the <a href="/wiki/Duality_(electrical_circuits)" title="Duality (electrical circuits)">duality relationship</a> of electrical circuits and considering that the parallel RLC is the <a href="/wiki/Dual_impedance" title="Dual impedance">dual impedance</a> of a series RLC. Considering this, it becomes clear that the differential equations describing this circuit are identical to the general form of those describing a series RLC. </p><p>For the parallel circuit, the attenuation <span class="texhtml mvar" style="font-style:italic;">α</span> is given by<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </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 \alpha ={\frac {1}{\,2\,R\,C\,}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mn>2</mn> <mspace width="thinmathspace" /> <mi>R</mi> <mspace width="thinmathspace" /> <mi>C</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha ={\frac {1}{\,2\,R\,C\,}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4efe8428133ede9d7ea204c1b0e3f9d0d956b07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.663ex; height:5.343ex;" alt="{\displaystyle \alpha ={\frac {1}{\,2\,R\,C\,}}}"></span></dd></dl> <p>and the damping factor is consequently </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 \zeta ={\frac {1}{\,2\,R\,}}{\sqrt {{\frac {L}{C}}~}}\,~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mn>2</mn> <mspace width="thinmathspace" /> <mi>R</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>C</mi> </mfrac> </mrow> <mtext> </mtext> </msqrt> </mrow> <mspace width="thinmathspace" /> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta ={\frac {1}{\,2\,R\,}}{\sqrt {{\frac {L}{C}}~}}\,~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/114d245358720839a7fe6081dc9cd541ff5b308b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.239ex; height:6.176ex;" alt="{\displaystyle \zeta ={\frac {1}{\,2\,R\,}}{\sqrt {{\frac {L}{C}}~}}\,~.}"></span></dd></dl> <p>Likewise, the other scaled parameters, fractional bandwidth and <span class="texhtml mvar" style="font-style:italic;">Q</span> are also reciprocals of each other. This means that a wide-band, low-<span class="texhtml mvar" style="font-style:italic;">Q</span> circuit in one topology will become a narrow-band, high-<span class="texhtml mvar" style="font-style:italic;">Q</span> circuit in the other topology when constructed from components with identical values. The fractional bandwidth and <span class="texhtml mvar" style="font-style:italic;">Q</span> of the parallel circuit are given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}B_{\mathrm {f} }&={\frac {1}{\,R\,}}{\sqrt {{\frac {L}{C}}~}}\\Q&=R{\sqrt {{\frac {C}{L}}~}}\,~.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mi>R</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>C</mi> </mfrac> </mrow> <mtext> </mtext> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>Q</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <mi>L</mi> </mfrac> </mrow> <mtext> </mtext> </msqrt> </mrow> <mspace width="thinmathspace" /> <mtext> </mtext> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}B_{\mathrm {f} }&={\frac {1}{\,R\,}}{\sqrt {{\frac {L}{C}}~}}\\Q&=R{\sqrt {{\frac {C}{L}}~}}\,~.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e48c33fc8811398d0904b80880a014383716a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:15.343ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}B_{\mathrm {f} }&={\frac {1}{\,R\,}}{\sqrt {{\frac {L}{C}}~}}\\Q&=R{\sqrt {{\frac {C}{L}}~}}\,~.\end{aligned}}}"></span></dd></dl> <p>Notice that the formulas here are the reciprocals of the formulas for the series circuit, given above. </p> <div class="mw-heading mw-heading3"><h3 id="Frequency_domain">Frequency domain</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=19" title="Edit section: Frequency domain"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:RLC_parallel_plot.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/RLC_parallel_plot.svg/350px-RLC_parallel_plot.svg.png" decoding="async" width="350" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/RLC_parallel_plot.svg/525px-RLC_parallel_plot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/RLC_parallel_plot.svg/700px-RLC_parallel_plot.svg.png 2x" data-file-width="1001" data-file-height="626" /></a><figcaption><b>Figure 3.</b> Sinusoidal steady-state analysis. Normalised to <span class="texhtml"><i>R</i> = 1 <a href="/wiki/Ohm" title="Ohm">Ω</a></span>, <span class="texhtml"><i>C</i> = 1 <a href="/wiki/Farad" title="Farad">F</a></span>, <span class="texhtml"><i>L</i> = 1 <a href="/wiki/Henry_(unit)" title="Henry (unit)">H</a></span>, and <span class="texhtml"><i>V</i> = 1 <a href="/wiki/Volt" title="Volt">V</a></span>.</figcaption></figure> <p>The complex admittance of this circuit is given by adding up the admittances of the components: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {1}{\,Z\,}}&={\frac {1}{\,Z_{L}\,}}+{\frac {1}{\,Z_{C}\,}}+{\frac {1}{\,Z_{R}\,}}\\&={\frac {1}{\,j\,\omega \,L\,}}+j\,\omega \,C+{\frac {1}{\,R\,}}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mi>Z</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mi>j</mi> <mspace width="thinmathspace" /> <mi>ω</mi> <mspace width="thinmathspace" /> <mi>L</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>j</mi> <mspace width="thinmathspace" /> <mi>ω</mi> <mspace width="thinmathspace" /> <mi>C</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thinmathspace" /> <mi>R</mi> <mspace width="thinmathspace" /> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {1}{\,Z\,}}&={\frac {1}{\,Z_{L}\,}}+{\frac {1}{\,Z_{C}\,}}+{\frac {1}{\,Z_{R}\,}}\\&={\frac {1}{\,j\,\omega \,L\,}}+j\,\omega \,C+{\frac {1}{\,R\,}}\,.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7c7a81c42b4f678de111baa6bbea377b5b8a0a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:28.545ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {1}{\,Z\,}}&={\frac {1}{\,Z_{L}\,}}+{\frac {1}{\,Z_{C}\,}}+{\frac {1}{\,Z_{R}\,}}\\&={\frac {1}{\,j\,\omega \,L\,}}+j\,\omega \,C+{\frac {1}{\,R\,}}\,.\end{aligned}}}"></span></dd></dl> <p>The change from a series arrangement to a parallel arrangement results in the circuit having a peak in impedance at resonance rather than a minimum, so the circuit is an anti-resonator. </p><p>The graph opposite shows that there is a minimum in the frequency response of the current at the resonance 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 ~\omega _{0}=1/{\sqrt {\,L\,C~}}~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mspace width="thinmathspace" /> <mi>L</mi> <mspace width="thinmathspace" /> <mi>C</mi> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~\omega _{0}=1/{\sqrt {\,L\,C~}}~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11ec32212c440ba039d213c03a0768c8b4a2ae9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.725ex; height:3.176ex;" alt="{\displaystyle ~\omega _{0}=1/{\sqrt {\,L\,C~}}~}"></span> when the circuit is driven by a constant voltage. On the other hand, if driven by a constant current, there would be a maximum in the voltage which would follow the same curve as the current in the series circuit. </p> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Other_configurations">Other configurations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=20" title="Edit section: Other configurations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:RL_series_C_parallel.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/RL_series_C_parallel.svg/160px-RL_series_C_parallel.svg.png" decoding="async" width="160" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/RL_series_C_parallel.svg/240px-RL_series_C_parallel.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/RL_series_C_parallel.svg/320px-RL_series_C_parallel.svg.png 2x" data-file-width="396" data-file-height="413" /></a><figcaption><b>Figure 4.</b> Series RL, parallel C circuit with resistance in series with the inductor is the standard model for a self-resonant inductor</figcaption></figure> <p>A series resistor with the inductor in a parallel LC circuit as shown in Figure 4 is a topology commonly encountered where there is a need to take into account the resistance of the coil winding and its self-capacitance. Parallel LC circuits are frequently used for <a href="/wiki/Bandpass_filter" class="mw-redirect" title="Bandpass filter">bandpass filtering</a> and the <span class="texhtml mvar" style="font-style:italic;">Q</span> is largely governed by this resistance. The resonant frequency of this circuit is<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </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 _{0}={\sqrt {{\frac {1}{\ LC\ }}-\left({\frac {R}{\ L\ }}\right)^{2}~}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>L</mi> <mi>C</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mrow> <mtext> </mtext> <mi>L</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \omega _{0}={\sqrt {{\frac {1}{\ LC\ }}-\left({\frac {R}{\ L\ }}\right)^{2}~}}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb588607b57d0905b9934ff2b718f4587afa3ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.554ex; height:7.676ex;" alt="{\displaystyle \ \omega _{0}={\sqrt {{\frac {1}{\ LC\ }}-\left({\frac {R}{\ L\ }}\right)^{2}~}}\ .}"></span></dd></dl> <p>This is the resonant frequency of the circuit defined as the frequency at which the admittance has zero imaginary part. The frequency that appears in the generalised form of the characteristic equation (which is the same for this circuit as previously) </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ s^{2}+2\alpha s+{\omega '_{0}}^{2}=0\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>α</mi> <mi>s</mi> <mo>+</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ s^{2}+2\alpha s+{\omega '_{0}}^{2}=0\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a39d473d2f941091d6703d82d71eb41717aed3e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.543ex; height:3.343ex;" alt="{\displaystyle \ s^{2}+2\alpha s+{\omega '_{0}}^{2}=0\ }"></span></dd></dl> <p>is not the same frequency. In this case it is the natural, <i>undamped</i> resonant frequency:<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </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 '_{0}\equiv {\frac {1}{\ {\sqrt {LC~}}\ }}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>L</mi> <mi>C</mi> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \omega '_{0}\equiv {\frac {1}{\ {\sqrt {LC~}}\ }}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e540b36f962b9dae081aa9e17109a80dafcb60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.27ex; height:6.176ex;" alt="{\displaystyle \ \omega '_{0}\equiv {\frac {1}{\ {\sqrt {LC~}}\ }}\ .}"></span></dd></dl> <p>The frequency <span class="texhtml"><i>ω</i><sub>max</sub></span>, at which the impedance magnitude is maximum, is given by<sup id="cite_ref-Cartwright_et_al._22-0" class="reference"><a href="#cite_note-Cartwright_et_al.-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </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 _{\mathrm {max} }=\omega '_{0}\ {\sqrt {-{\frac {1}{\ Q_{L}^{2}\ ~}}+{\sqrt {1+{\frac {2}{\ Q_{L}^{2}\ }}~}}~}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>=</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mtext> </mtext> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mtext> </mtext> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \omega _{\mathrm {max} }=\omega '_{0}\ {\sqrt {-{\frac {1}{\ Q_{L}^{2}\ ~}}+{\sqrt {1+{\frac {2}{\ Q_{L}^{2}\ }}~}}~}}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bbf1c40754e5fb6d206829ad8035de60d690c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:38.27ex; height:8.176ex;" alt="{\displaystyle \ \omega _{\mathrm {max} }=\omega '_{0}\ {\sqrt {-{\frac {1}{\ Q_{L}^{2}\ ~}}+{\sqrt {1+{\frac {2}{\ Q_{L}^{2}\ }}~}}~}}\ ,}"></span></dd></dl> <p>where <span class="texhtml"><i>Q<sub>L</sub></i> ≡ <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num"><i>ω′</i><sub>0</sub><i>L</i></span><span class="sr-only">/</span><span class="den"><i>R</i></span></span>⁠</span></span> is the <a href="/wiki/Q_factor" title="Q factor">quality factor</a> of the coil. This can be well approximated by<sup id="cite_ref-Cartwright_et_al._22-1" class="reference"><a href="#cite_note-Cartwright_et_al.-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </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 _{\mathrm {max} }\approx \omega '_{0}\ {\sqrt {1-{\frac {1}{\ 2Q_{L}^{4}\ }}~}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>≈</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mn>2</mn> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \omega _{\mathrm {max} }\approx \omega '_{0}\ {\sqrt {1-{\frac {1}{\ 2Q_{L}^{4}\ }}~}}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec46385349f696b6b9748e6a1ce4cc69986576f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:25.981ex; height:7.676ex;" alt="{\displaystyle \ \omega _{\mathrm {max} }\approx \omega '_{0}\ {\sqrt {1-{\frac {1}{\ 2Q_{L}^{4}\ }}~}}\ .}"></span></dd></dl> <p>Furthermore, the exact maximum impedance magnitude is given by<sup id="cite_ref-Cartwright_et_al._22-2" class="reference"><a href="#cite_note-Cartwright_et_al.-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </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 \ |Z|_{\mathrm {max} }={\frac {RQ_{L}^{2}}{\ {\sqrt {2Q_{L}{\sqrt {Q_{L}^{2}+2\ }}-\left(2Q_{L}^{2}+1\right)~}}\ }}={\frac {RQ_{L}}{\ {\sqrt {2{\sqrt {1+2/Q_{L}^{2}\ }}-\left(2+1/Q_{L}^{2}\right)~}}\ }}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Z</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mtext> </mtext> </msqrt> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mrow> <mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mtext> </mtext> </msqrt> </mrow> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ |Z|_{\mathrm {max} }={\frac {RQ_{L}^{2}}{\ {\sqrt {2Q_{L}{\sqrt {Q_{L}^{2}+2\ }}-\left(2Q_{L}^{2}+1\right)~}}\ }}={\frac {RQ_{L}}{\ {\sqrt {2{\sqrt {1+2/Q_{L}^{2}\ }}-\left(2+1/Q_{L}^{2}\right)~}}\ }}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b209b713541a98e2ff504a0a07fa2bc74834413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:80.695ex; height:10.176ex;" alt="{\displaystyle \ |Z|_{\mathrm {max} }={\frac {RQ_{L}^{2}}{\ {\sqrt {2Q_{L}{\sqrt {Q_{L}^{2}+2\ }}-\left(2Q_{L}^{2}+1\right)~}}\ }}={\frac {RQ_{L}}{\ {\sqrt {2{\sqrt {1+2/Q_{L}^{2}\ }}-\left(2+1/Q_{L}^{2}\right)~}}\ }}\ .}"></span></dd></dl> <p>For values of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ Q_{L}\gg 1\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>≫</mo> <mn>1</mn> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ Q_{L}\gg 1\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8b9c167f49d695b38c1dcfd07b2043c1a278ced" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.774ex; height:2.509ex;" alt="{\displaystyle \ Q_{L}\gg 1\ ,}"></span> this can be well approximated by<sup id="cite_ref-Cartwright_et_al._22-3" class="reference"><a href="#cite_note-Cartwright_et_al.-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </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 \ |Z|_{\mathrm {max} }\approx RQ_{L}^{2}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Z</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> </mrow> </msub> <mo>≈</mo> <mi>R</mi> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ |Z|_{\mathrm {max} }\approx RQ_{L}^{2}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/357e44b9252f7e5c641297c0bd8563fd0bcf3b17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.126ex; height:3.176ex;" alt="{\displaystyle \ |Z|_{\mathrm {max} }\approx RQ_{L}^{2}\ .}"></span></dd></dl> <figure typeof="mw:File/Thumb"><a href="/wiki/File:L_series_RC_parallel.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/L_series_RC_parallel.svg/160px-L_series_RC_parallel.svg.png" decoding="async" width="160" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/L_series_RC_parallel.svg/240px-L_series_RC_parallel.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/L_series_RC_parallel.svg/320px-L_series_RC_parallel.svg.png 2x" data-file-width="396" data-file-height="413" /></a><figcaption><b>Figure 5.</b> Parallel RC, series L circuit with resistance in parallel with the capacitor</figcaption></figure> <p>In the same vein, a resistor in parallel with the capacitor in a series LC circuit can be used to represent a capacitor with a lossy dielectric. This configuration is shown in Figure 5. The resonant frequency (frequency at which the impedance has zero imaginary part) in this case is given by<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </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 _{0}={\sqrt {{\frac {1}{\ LC\ }}-{\frac {1}{\ (RC)^{2}\ }}\ }}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>L</mi> <mi>C</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>R</mi> <mi>C</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \omega _{0}={\sqrt {{\frac {1}{\ LC\ }}-{\frac {1}{\ (RC)^{2}\ }}\ }}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0177cf39c048143138804de10e5fb6968296bc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:26.889ex; height:7.676ex;" alt="{\displaystyle \ \omega _{0}={\sqrt {{\frac {1}{\ LC\ }}-{\frac {1}{\ (RC)^{2}\ }}\ }}\ ,}"></span></dd></dl> <p>while the frequency <span class="texhtml"><i>ω</i><sub>m</sub></span> at which the impedance magnitude is minimum is given by </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 _{\mathrm {m} }=\omega '_{0}\ {\sqrt {-{\frac {1}{\ Q_{C}^{2}\ }}+{\sqrt {1+{\frac {2}{\ Q_{C}^{2}\ }}\ }}\ }}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>=</mo> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mo>′</mo> </msubsup> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mtext> </mtext> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mtext> </mtext> <msubsup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \omega _{\mathrm {m} }=\omega '_{0}\ {\sqrt {-{\frac {1}{\ Q_{C}^{2}\ }}+{\sqrt {1+{\frac {2}{\ Q_{C}^{2}\ }}\ }}\ }}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e15d297ca4636196fe3f29b0fd0b0d64cde80e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:36.258ex; height:8.343ex;" alt="{\displaystyle \ \omega _{\mathrm {m} }=\omega '_{0}\ {\sqrt {-{\frac {1}{\ Q_{C}^{2}\ }}+{\sqrt {1+{\frac {2}{\ Q_{C}^{2}\ }}\ }}\ }}\ ,}"></span></dd></dl> <p>where <span class="texhtml"><i>Q<sub>C</sub></i> = <i>ω′</i><sub>0</sub><i>RC</i></span>. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=21" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The first evidence that a capacitor could produce electrical oscillations was discovered in 1826 by French scientist <a href="/wiki/Felix_Savary" class="mw-redirect" title="Felix Savary">Felix Savary</a>.<sup id="cite_ref-Blanchard_24-0" class="reference"><a href="#cite_note-Blanchard-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> He found that when a <a href="/wiki/Leyden_jar" title="Leyden jar">Leyden jar</a> was discharged through a wire wound around an iron needle, sometimes the needle was left magnetized in one direction and sometimes in the opposite direction. He correctly deduced that this was caused by a damped oscillating discharge current in the wire, which reversed the magnetization of the needle back and forth until it was too small to have an effect, leaving the needle magnetized in a random direction. </p><p>American physicist <a href="/wiki/Joseph_Henry" title="Joseph Henry">Joseph Henry</a> repeated Savary's experiment in 1842 and came to the same conclusion, apparently independently.<sup id="cite_ref-Kimball_26-0" class="reference"><a href="#cite_note-Kimball-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Huurdeman_27-0" class="reference"><a href="#cite_note-Huurdeman-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> British scientist <a href="/wiki/William_Thomson,_1st_Baron_Kelvin" class="mw-redirect" title="William Thomson, 1st Baron Kelvin">William Thomson</a> (Lord Kelvin) in 1853 showed mathematically that the discharge of a Leyden jar through an inductance should be oscillatory, and derived its resonant frequency.<sup id="cite_ref-Blanchard_24-1" class="reference"><a href="#cite_note-Blanchard-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kimball_26-1" class="reference"><a href="#cite_note-Kimball-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Huurdeman_27-1" class="reference"><a href="#cite_note-Huurdeman-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p><p>British radio researcher <a href="/wiki/Oliver_Lodge" title="Oliver Lodge">Oliver Lodge</a>, by discharging a large battery of Leyden jars through a long wire, created a tuned circuit with its resonant frequency in the audio range, which produced a musical tone from the spark when it was discharged.<sup id="cite_ref-Kimball_26-2" class="reference"><a href="#cite_note-Kimball-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> In 1857, German physicist <a href="/wiki/Berend_Wilhelm_Feddersen" title="Berend Wilhelm Feddersen">Berend Wilhelm Feddersen</a> photographed the spark produced by a resonant Leyden jar circuit in a rotating mirror, providing visible evidence of the oscillations.<sup id="cite_ref-Blanchard_24-2" class="reference"><a href="#cite_note-Blanchard-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kimball_26-3" class="reference"><a href="#cite_note-Kimball-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Huurdeman_27-2" class="reference"><a href="#cite_note-Huurdeman-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> In 1868, Scottish physicist <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> calculated the effect of applying an alternating current to a circuit with inductance and capacitance, showing that the response is maximum at the resonant frequency.<sup id="cite_ref-Blanchard_24-3" class="reference"><a href="#cite_note-Blanchard-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>The first example of an electrical <a href="/wiki/Resonance" title="Resonance">resonance</a> curve was published in 1887 by German physicist <a href="/wiki/Heinrich_Hertz" title="Heinrich Hertz">Heinrich Hertz</a> in his pioneering paper on the discovery of radio waves, showing the length of spark obtainable from his spark-gap LC resonator detectors as a function of frequency.<sup id="cite_ref-Blanchard_24-4" class="reference"><a href="#cite_note-Blanchard-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>One of the first demonstrations of resonance between tuned circuits was Lodge's "syntonic jars" experiment around 1889<sup id="cite_ref-Blanchard_24-5" class="reference"><a href="#cite_note-Blanchard-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kimball_26-4" class="reference"><a href="#cite_note-Kimball-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> He placed two resonant circuits next to each other, each consisting of a Leyden jar connected to an adjustable one-turn coil with a spark gap. When a high voltage from an induction coil was applied to one tuned circuit, creating sparks and thus oscillating currents, sparks were excited in the other tuned circuit only when the inductors were adjusted to resonance. Lodge and some English scientists preferred the term "<i>syntony</i>" for this effect, but the term "<i>resonance</i>" eventually stuck.<sup id="cite_ref-Blanchard_24-6" class="reference"><a href="#cite_note-Blanchard-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p>The first practical use for RLC circuits was in the 1890s in <a href="/wiki/Spark-gap_transmitter" title="Spark-gap transmitter">spark-gap radio transmitters</a> to allow the receiver to be tuned to the transmitter. The first patent for a radio system that allowed tuning was filed by Lodge in 1897, although the first practical systems were invented in 1900 by Anglo Italian radio pioneer <a href="/wiki/Guglielmo_Marconi" title="Guglielmo Marconi">Guglielmo Marconi</a>.<sup id="cite_ref-Blanchard_24-7" class="reference"><a href="#cite_note-Blanchard-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=22" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Variable_tuned_circuits">Variable tuned circuits</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=23" title="Edit section: Variable tuned circuits"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A very frequent use of these circuits is in the tuning circuits of analogue radios. Adjustable tuning is commonly achieved with a parallel plate <a href="/wiki/Variable_capacitor" title="Variable capacitor">variable capacitor</a> which allows the value of <span class="texhtml mvar" style="font-style:italic;">C</span> to be changed and tune to stations on different frequencies. For the <a href="/wiki/Intermediate_frequency" title="Intermediate frequency">IF stage</a> in the radio where the tuning is preset in the factory, the more usual solution is an adjustable core in the inductor to adjust <span class="texhtml mvar" style="font-style:italic;">L</span>. In this design, the core (made of a high <a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">permeability</a> material that has the effect of increasing inductance) is threaded so that it can be screwed further in, or screwed further out of the inductor winding as required. </p> <div class="mw-heading mw-heading3"><h3 id="Filters">Filters</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=24" title="Edit section: Filters"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table style="float:right;"> <tbody><tr> <td><figure class="mw-default-size mw-halign-none" typeof="mw:File/Thumb"><a href="/wiki/File:RLC_low-pass.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/RLC_low-pass.svg/220px-RLC_low-pass.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7d/RLC_low-pass.svg/330px-RLC_low-pass.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7d/RLC_low-pass.svg/440px-RLC_low-pass.svg.png 2x" data-file-width="570" data-file-height="413" /></a><figcaption><b>Figure 6.</b> RLC circuit as a low-pass filter</figcaption></figure> </td> <td><figure class="mw-default-size mw-halign-none" typeof="mw:File/Thumb"><a href="/wiki/File:RLC_high-pass.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/RLC_high-pass.svg/220px-RLC_high-pass.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/RLC_high-pass.svg/330px-RLC_high-pass.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/RLC_high-pass.svg/440px-RLC_high-pass.svg.png 2x" data-file-width="570" data-file-height="413" /></a><figcaption><b>Figure 7.</b> RLC circuit as a high-pass filter</figcaption></figure> </td></tr> <tr> <td><figure class="mw-default-size mw-halign-none" typeof="mw:File/Thumb"><a href="/wiki/File:RLC_series_band-pass.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/RLC_series_band-pass.svg/220px-RLC_series_band-pass.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/RLC_series_band-pass.svg/330px-RLC_series_band-pass.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/RLC_series_band-pass.svg/440px-RLC_series_band-pass.svg.png 2x" data-file-width="570" data-file-height="413" /></a><figcaption><b>Figure 8.</b> RLC circuit as a series band-pass filter in series with the line</figcaption></figure> </td> <td><figure class="mw-default-size mw-halign-none" typeof="mw:File/Thumb"><a href="/wiki/File:RLC_parallel_band-pass.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/RLC_parallel_band-pass.svg/220px-RLC_parallel_band-pass.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/RLC_parallel_band-pass.svg/330px-RLC_parallel_band-pass.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/RLC_parallel_band-pass.svg/440px-RLC_parallel_band-pass.svg.png 2x" data-file-width="570" data-file-height="413" /></a><figcaption><b>Figure 9.</b> RLC circuit as a parallel band-pass filter in shunt across the line</figcaption></figure> </td></tr> <tr> <td><figure class="mw-default-size mw-halign-none" typeof="mw:File/Thumb"><a href="/wiki/File:RLC_series_band-stop.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/RLC_series_band-stop.svg/220px-RLC_series_band-stop.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/RLC_series_band-stop.svg/330px-RLC_series_band-stop.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/RLC_series_band-stop.svg/440px-RLC_series_band-stop.svg.png 2x" data-file-width="570" data-file-height="413" /></a><figcaption><b>Figure 10.</b> RLC circuit as a series band-stop filter in shunt across the line</figcaption></figure> </td> <td><figure class="mw-default-size mw-halign-none" typeof="mw:File/Thumb"><a href="/wiki/File:RLC_parallel_band-stop.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/RLC_parallel_band-stop.svg/220px-RLC_parallel_band-stop.svg.png" decoding="async" width="220" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/RLC_parallel_band-stop.svg/330px-RLC_parallel_band-stop.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/RLC_parallel_band-stop.svg/440px-RLC_parallel_band-stop.svg.png 2x" data-file-width="570" data-file-height="413" /></a><figcaption><b>Figure 11.</b> RLC circuit as a parallel band-stop filter in series with the line</figcaption></figure> </td></tr></tbody></table> <p>In the filtering application, the resistor becomes the load that the filter is working into. The value of the damping factor is chosen based on the desired bandwidth of the filter. For a wider bandwidth, a larger value of the damping factor is required (and vice versa). The three components give the designer three degrees of freedom. Two of these are required to set the bandwidth and resonant frequency. The designer is still left with one which can be used to scale <span class="texhtml mvar" style="font-style:italic;">R</span>, <span class="texhtml mvar" style="font-style:italic;">L</span> and <span class="texhtml mvar" style="font-style:italic;">C</span> to convenient practical values. Alternatively, <span class="texhtml mvar" style="font-style:italic;">R</span> may be predetermined by the external circuitry which will use the last degree of freedom. </p> <div class="mw-heading mw-heading4"><h4 id="Low-pass_filter">Low-pass filter</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=25" title="Edit section: Low-pass filter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An RLC circuit can be used as a low-pass filter. The circuit configuration is shown in Figure 6. The corner frequency, that is, the frequency of the 3 dB point, is given by </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 _{\mathrm {c} }={\frac {1}{\sqrt {LC}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>L</mi> <mi>C</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathrm {c} }={\frac {1}{\sqrt {LC}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eebc32a032381e8701f55693a5b44295c022f9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.662ex; height:6.176ex;" alt="{\displaystyle \omega _{\mathrm {c} }={\frac {1}{\sqrt {LC}}}\,.}"></span></dd></dl> <p>This is also the bandwidth of the filter. The damping factor is given by<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <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 \zeta ={\frac {1}{2R_{L}}}{\sqrt {\frac {L}{C}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>L</mi> <mi>C</mi> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta ={\frac {1}{2R_{L}}}{\sqrt {\frac {L}{C}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d95932c0f663ef43ea9fecd005dcb3b7fb1199" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.268ex; height:6.176ex;" alt="{\displaystyle \zeta ={\frac {1}{2R_{L}}}{\sqrt {\frac {L}{C}}}\,.}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="High-pass_filter">High-pass filter</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=26" title="Edit section: High-pass filter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A high-pass filter is shown in Figure 7. The corner frequency is the same as the low-pass filter: </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 _{\mathrm {c} }={\frac {1}{\sqrt {LC}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>L</mi> <mi>C</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathrm {c} }={\frac {1}{\sqrt {LC}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eebc32a032381e8701f55693a5b44295c022f9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.662ex; height:6.176ex;" alt="{\displaystyle \omega _{\mathrm {c} }={\frac {1}{\sqrt {LC}}}\,.}"></span></dd></dl> <p>The filter has a stop-band of this width.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Band-pass_filter">Band-pass filter</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=27" title="Edit section: Band-pass filter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A band-pass filter can be formed with an RLC circuit by either placing a series LC circuit in series with the load resistor or else by placing a parallel LC circuit in parallel with the load resistor. These arrangements are shown in Figures 8 and 9 respectively. The centre frequency is given by </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 _{\mathrm {c} }={\frac {1}{\sqrt {LC}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>L</mi> <mi>C</mi> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathrm {c} }={\frac {1}{\sqrt {LC}}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbc7624a6467c4d2ccc79f0f23287b9857d95136" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.662ex; height:6.176ex;" alt="{\displaystyle \omega _{\mathrm {c} }={\frac {1}{\sqrt {LC}}}\,,}"></span></dd></dl> <p>and the bandwidth for the series circuit is<sup id="cite_ref-Kaiser,_pp._7.21–7.27_30-0" class="reference"><a href="#cite_note-Kaiser,_pp._7.21–7.27-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </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 \Delta \omega ={\frac {R_{L}}{L}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mi>L</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \omega ={\frac {R_{L}}{L}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8e39f96331f86807719dbcc37dcb8b944111390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.466ex; height:5.176ex;" alt="{\displaystyle \Delta \omega ={\frac {R_{L}}{L}}\,.}"></span></dd></dl> <p>The shunt version of the circuit is intended to be driven by a high impedance source, that is, a constant current source. Under those conditions the bandwidth is<sup id="cite_ref-Kaiser,_pp._7.21–7.27_30-1" class="reference"><a href="#cite_note-Kaiser,_pp._7.21–7.27-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> </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 \Delta \omega ={\frac {1}{CR_{L}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>C</mi> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \omega ={\frac {1}{CR_{L}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3f9cb9c2de7263e61a98f26f3f91363395da2dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.232ex; height:5.676ex;" alt="{\displaystyle \Delta \omega ={\frac {1}{CR_{L}}}\,.}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Band-stop_filter">Band-stop filter</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=28" title="Edit section: Band-stop filter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Figure 10 shows a band-stop filter formed by a series LC circuit in shunt across the load. Figure 11 is a band-stop filter formed by a parallel LC circuit in series with the load. The first case requires a high impedance source so that the current is diverted into the resonator when it becomes low impedance at resonance. The second case requires a low impedance source so that the voltage is dropped across the antiresonator when it becomes high impedance at resonance.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Oscillators">Oscillators</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=29" title="Edit section: Oscillators"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For applications in oscillator circuits, it is generally desirable to make the attenuation (or equivalently, the damping factor) as small as possible. In practice, this objective requires making the circuit's resistance <span class="texhtml mvar" style="font-style:italic;">R</span> as small as physically possible for a series circuit, or alternatively increasing <span class="texhtml mvar" style="font-style:italic;">R</span> to as much as possible for a parallel circuit. In either case, the RLC circuit becomes a good approximation to an ideal <a href="/wiki/LC_circuit" title="LC circuit">LC circuit</a>. However, for very low-attenuation circuits (high <span class="texhtml mvar" style="font-style:italic;">Q</span>-factor), issues such as dielectric losses of coils and capacitors can become important. </p><p>In an oscillator circuit </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 \alpha \ll \omega _{0}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>≪</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \ll \omega _{0}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14951bfa1f00a84b496131ac48be5bb65b6e69f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.636ex; height:2.176ex;" alt="{\displaystyle \alpha \ll \omega _{0}\,,}"></span></dd></dl> <p>or equivalently </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 \zeta \ll 1\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ζ</mi> <mo>≪</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \zeta \ll 1\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a0dff23a61e8f3e1c42b3dd4790e0aa6634c137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.906ex; height:2.509ex;" alt="{\displaystyle \zeta \ll 1\,.}"></span></dd></dl> <p>As a result, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\mathrm {d} }\approx \omega _{0}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>≈</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathrm {d} }\approx \omega _{0}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19a471df19dda2a3a1a8f2662b29222404c40f57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.224ex; height:2.009ex;" alt="{\displaystyle \omega _{\mathrm {d} }\approx \omega _{0}\,.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Voltage_multiplier">Voltage multiplier</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=30" title="Edit section: Voltage multiplier"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a series RLC circuit at resonance, the current is limited only by the resistance of the circuit </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 I={\frac {V}{R}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <mi>R</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I={\frac {V}{R}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a951e7221b0876feb48aded3548d892394c4123" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.928ex; height:5.343ex;" alt="{\displaystyle I={\frac {V}{R}}\,.}"></span></dd></dl> <p>If <span class="texhtml mvar" style="font-style:italic;">R</span> is small, consisting only of the inductor winding resistance say, then this current will be large. It will drop a voltage across the inductor of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{L}={\frac {V}{R}}\omega _{0}L\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <mi>R</mi> </mfrac> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>L</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{L}={\frac {V}{R}}\omega _{0}L\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/489083619bb558ce82c542e6bed316bd220dd962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.545ex; height:5.343ex;" alt="{\displaystyle V_{L}={\frac {V}{R}}\omega _{0}L\,.}"></span></dd></dl> <p>An equal magnitude voltage will also be seen across the capacitor but in antiphase to the inductor. If <span class="texhtml mvar" style="font-style:italic;">R</span> can be made sufficiently small, these voltages can be several times the input voltage. The voltage ratio is, in fact, the <span class="texhtml mvar" style="font-style:italic;">Q</span> of the circuit, </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 {\frac {V_{L}}{V}}=Q\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mi>V</mi> </mfrac> </mrow> <mo>=</mo> <mi>Q</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {V_{L}}{V}}=Q\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a0309052ed07ee3260c4d0b78b7ac6434e68c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.514ex; height:5.343ex;" alt="{\displaystyle {\frac {V_{L}}{V}}=Q\,.}"></span></dd></dl> <p>A similar effect is observed with currents in the parallel circuit. Even though the circuit appears as high impedance to the external source, there is a large current circulating in the internal loop of the parallel inductor and capacitor. </p> <div class="mw-heading mw-heading3"><h3 id="Pulse_discharge_circuit">Pulse discharge circuit</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=31" title="Edit section: Pulse discharge circuit"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An overdamped series RLC circuit can be used as a pulse discharge circuit. Often it is useful to know the values of components that could be used to produce a waveform. This is described by the form </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 I(t)=I_{0}\left(\,e^{-\alpha \,t}-e^{-\beta \,t}\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> <mspace width="thinmathspace" /> <mi>t</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mspace width="thinmathspace" /> <mi>t</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(t)=I_{0}\left(\,e^{-\alpha \,t}-e^{-\beta \,t}\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84040140f89a155d559e2a09c9f95a2c9ba7c437" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.306ex; height:3.343ex;" alt="{\displaystyle I(t)=I_{0}\left(\,e^{-\alpha \,t}-e^{-\beta \,t}\right)\,.}"></span></dd></dl> <p>Such a circuit could consist of an energy storage capacitor, a load in the form of a resistance, some circuit inductance and a switch – all in series. The initial conditions are that the capacitor is at voltage, <span class="texhtml"><i>V</i><sub>0</sub></span>, and there is no current flowing in the inductor. If the inductance <span class="texhtml mvar" style="font-style:italic;">L</span> is known, then the remaining parameters are given by the following – capacitance: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C={\frac {1}{~L\,\alpha \,\beta \,~}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mi>L</mi> <mspace width="thinmathspace" /> <mi>α</mi> <mspace width="thinmathspace" /> <mi>β</mi> <mspace width="thinmathspace" /> <mtext> </mtext> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C={\frac {1}{~L\,\alpha \,\beta \,~}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4c7d6348cc97c6007fe5a36074c22673f0dcff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.46ex; height:5.676ex;" alt="{\displaystyle C={\frac {1}{~L\,\alpha \,\beta \,~}}\,,}"></span></dd></dl> <p>resistance (total of circuit and load): </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 R=L\,(\,\alpha +\beta \,)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>L</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mspace width="thinmathspace" /> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mspace width="thinmathspace" /> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=L\,(\,\alpha +\beta \,)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/161a21461ccefaa1f00a892b8559f9baae058714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.11ex; height:2.843ex;" alt="{\displaystyle R=L\,(\,\alpha +\beta \,)\,,}"></span></dd></dl> <p>initial terminal voltage of capacitor: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}=-I_{0}L\,\alpha \,\beta \,\left({\frac {1}{\beta }}-{\frac {1}{\alpha }}\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>L</mi> <mspace width="thinmathspace" /> <mi>α</mi> <mspace width="thinmathspace" /> <mi>β</mi> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>β</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{0}=-I_{0}L\,\alpha \,\beta \,\left({\frac {1}{\beta }}-{\frac {1}{\alpha }}\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb9e38ca6369c40d96a9697a3470bf42f226efb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.519ex; height:6.176ex;" alt="{\displaystyle V_{0}=-I_{0}L\,\alpha \,\beta \,\left({\frac {1}{\beta }}-{\frac {1}{\alpha }}\right)\,.}"></span></dd></dl> <p>Rearranging for the case where <span class="texhtml mvar" style="font-style:italic;">R</span> is known – capacitance: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C={\frac {~\alpha +\beta ~}{R\,\alpha \,\beta }}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mtext> </mtext> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mtext> </mtext> </mrow> <mrow> <mi>R</mi> <mspace width="thinmathspace" /> <mi>α</mi> <mspace width="thinmathspace" /> <mi>β</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C={\frac {~\alpha +\beta ~}{R\,\alpha \,\beta }}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bceba0f676b7e9cdca150b0acdb7bc3915711ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.556ex; height:5.843ex;" alt="{\displaystyle C={\frac {~\alpha +\beta ~}{R\,\alpha \,\beta }}\,,}"></span></dd></dl> <p>inductance (total of circuit and load): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L={\frac {R}{\,\alpha +\beta ~}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>R</mi> <mrow> <mspace width="thinmathspace" /> <mi>α</mi> <mo>+</mo> <mi>β</mi> <mtext> </mtext> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L={\frac {R}{\,\alpha +\beta ~}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff9bf465d998b8b48a8b185acbf30e14045c4aef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:13.179ex; height:5.676ex;" alt="{\displaystyle L={\frac {R}{\,\alpha +\beta ~}}\,,}"></span></dd></dl> <p>initial terminal voltage of capacitor: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{0}={\frac {\,-I_{0}R\,\alpha \,\beta ~}{\alpha +\beta }}\left({\frac {1}{\beta }}-{\frac {1}{\alpha }}\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thinmathspace" /> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>R</mi> <mspace width="thinmathspace" /> <mi>α</mi> <mspace width="thinmathspace" /> <mi>β</mi> <mtext> </mtext> </mrow> <mrow> <mi>α</mi> <mo>+</mo> <mi>β</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>β</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{0}={\frac {\,-I_{0}R\,\alpha \,\beta ~}{\alpha +\beta }}\left({\frac {1}{\beta }}-{\frac {1}{\alpha }}\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d787851d0b71b0c54615ba6a01f334e9e8ecffc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.117ex; height:6.176ex;" alt="{\displaystyle V_{0}={\frac {\,-I_{0}R\,\alpha \,\beta ~}{\alpha +\beta }}\left({\frac {1}{\beta }}-{\frac {1}{\alpha }}\right)\,.}"></span></dd></dl> <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=RLC_circuit&action=edit&section=32" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/RC_circuit" title="RC circuit">RC circuit</a></li> <li><a href="/wiki/RL_circuit" title="RL circuit">RL circuit</a></li> <li><a href="/wiki/Linear_circuit" title="Linear circuit">Linear circuit</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=33" title="Edit section: Footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"> RLC response to driving voltage occurs at the frequency for lossless oscillation, <span class="mwe-math-element"><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 {1}{\ 2\pi {\sqrt {LC\ }}\ }}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mtext> </mtext> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>L</mi> <mi>C</mi> <mtext> </mtext> </msqrt> </mrow> <mtext> </mtext> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\frac {1}{\ 2\pi {\sqrt {LC\ }}\ }}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3e6e2e53bba318f5fe9a897e863c6345c611313" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:12.166ex; height:6.176ex;" alt="{\displaystyle \ {\frac {1}{\ 2\pi {\sqrt {LC\ }}\ }}\ ,}"></span> even though loss resistance <span class="texhtml">R</span> may be present. The driven resonance does not occur at the damped free oscillation frequency, with a more complicated formula (see below) that produces a reduced value due to damping (<span class="texhtml">R</span>) which only applies to free oscillations (no driving signal).</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=34" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <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">Kaiser, pp. 7.71–7.72</span> </li> <li id="cite_note-Long-2004-04-15-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-Long-2004-04-15_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Long-2004-04-15_3-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Long-2004-04-15_3-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLong,_Steve2004" class="citation web cs1">Long, Steve (2004-04-15) [2002-01-17]. Rodwell, Mark (ed.). <a rel="nofollow" class="external text" href="http://www.ece.ucsb.edu/Faculty/rodwell/Classes/ece218b/notes/Resonators.pdf">"Resonant circuits – resonators and <span class="texhtml mvar" style="font-style:italic;">Q</span>"</a> <span class="cs1-format">(PDF)</span>. ECE145B / ECE 218B. <i>ece.ucsb.edu</i> (course notes). Electrical & Computer Engineering. Santa Barbara, CA: <a href="/wiki/University_of_California_Santa_Barbara" class="mw-redirect" title="University of California Santa Barbara">U.C. Santa Barbara</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2016-10-21</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ece.ucsb.edu&rft.atitle=Resonant+circuits+%E2%80%93+resonators+and+%3Cspan+class%3D%22texhtml+mvar%22+style%3D%22font-style%3Aitalic%3B%22%3EQ%3C%2Fspan%3E&rft.date=2004-04-15&rft.au=Long%2C+Steve&rft_id=http%3A%2F%2Fwww.ece.ucsb.edu%2FFaculty%2Frodwell%2FClasses%2Fece218b%2Fnotes%2FResonators.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARLC+circuit" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Nilsson and Riedel, p. 308.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Agarwal and Lang, p. 641.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Agarwal and Lang, p. 646.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Irwin, pp. 217–220.</span> </li> <li id="cite_note-Argawal656-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Argawal656_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Argawal656_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Agarwal and Lang, p. 656.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Nilsson and Riedel, pp. 287–288.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Irwin, p. 532.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Agarwal and Lang, p. 648.</span> </li> <li id="cite_note-Nilsson295-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-Nilsson295_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Nilsson295_12-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Nilsson and Riedel, p. 295.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Humar, pp. 223–224.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">Agarwal and Lang, p. 692.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Nilsson and Riedel, p. 303.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text">Irwin, p. 220.</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">This section is based on Example 4.2.13 from <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDebnathBhatta2007" class="citation book cs1">Debnath, Lokenath; Bhatta, Dambaru (2007). <i>Integral Transforms and Their Applications</i> (2nd ed.). Chapman & Hall/CRC. pp. 198–202. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-58488-575-7" title="Special:BookSources/978-1-58488-575-7"><bdi>978-1-58488-575-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Integral+Transforms+and+Their+Applications&rft.pages=198-202&rft.edition=2nd&rft.pub=Chapman+%26+Hall%2FCRC&rft.date=2007&rft.isbn=978-1-58488-575-7&rft.aulast=Debnath&rft.aufirst=Lokenath&rft.au=Bhatta%2C+Dambaru&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARLC+circuit" class="Z3988"></span> (Some notations have been changed to fit the rest of this article.)</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">Kumar and Kumar, <i>Electric Circuits & Networks</i>, p. 464.</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text">Nilsson and Riedel, p. 286.</span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text">Kaiser, pp. 5.26–5.27 </span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text">Agarwal & Lang, p. 805.</span> </li> <li id="cite_note-Cartwright_et_al.-22"><span class="mw-cite-backlink">^ <a href="#cite_ref-Cartwright_et_al._22-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Cartwright_et_al._22-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Cartwright_et_al._22-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Cartwright_et_al._22-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCartwrightJosephKaminsky2010" class="citation journal cs1">Cartwright, K.V.; Joseph, E.; Kaminsky, E.J. (2010). <a rel="nofollow" class="external text" href="http://tiij.org/issues/issues/winter2010/files/TIIJ%20fall-winter%202010-PDW2.pdf">"Finding the exact maximum impedance resonant frequency of a practical parallel resonant circuit without calculus"</a> <span class="cs1-format">(PDF)</span>. <i>The Technology Interface International Journal</i>. <b>11</b> (1): 26–34.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Technology+Interface+International+Journal&rft.atitle=Finding+the+exact+maximum+impedance+resonant+frequency+of+a+practical+parallel+resonant+circuit+without+calculus&rft.volume=11&rft.issue=1&rft.pages=26-34&rft.date=2010&rft.aulast=Cartwright&rft.aufirst=K.V.&rft.au=Joseph%2C+E.&rft.au=Kaminsky%2C+E.J.&rft_id=http%3A%2F%2Ftiij.org%2Fissues%2Fissues%2Fwinter2010%2Ffiles%2FTIIJ%2520fall-winter%25202010-PDW2.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARLC+circuit" class="Z3988"></span></span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text">Kaiser, pp. 5.25–5.26.</span> </li> <li id="cite_note-Blanchard-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-Blanchard_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Blanchard_24-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Blanchard_24-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Blanchard_24-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Blanchard_24-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Blanchard_24-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Blanchard_24-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-Blanchard_24-7"><sup><i><b>h</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlanchard1941" class="citation journal cs1">Blanchard, Julian (October 1941). <a rel="nofollow" class="external text" href="https://archive.org/stream/bstj20-4-415#page/n13/mode/2up">"The History of Electrical Resonance"</a>. <i>Bell System Technical Journal</i>. <b>20</b> (4). USA: AT&T: 415. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fj.1538-7305.1941.tb03608.x">10.1002/j.1538-7305.1941.tb03608.x</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:51669988">51669988</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2013-02-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bell+System+Technical+Journal&rft.atitle=The+History+of+Electrical+Resonance&rft.volume=20&rft.issue=4&rft.pages=415&rft.date=1941-10&rft_id=info%3Adoi%2F10.1002%2Fj.1538-7305.1941.tb03608.x&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A51669988%23id-name%3DS2CID&rft.aulast=Blanchard&rft.aufirst=Julian&rft_id=https%3A%2F%2Farchive.org%2Fstream%2Fbstj20-4-415%23page%2Fn13%2Fmode%2F2up&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARLC+circuit" class="Z3988"></span></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSavary1827" class="citation journal cs1">Savary, Felix (1827). "Memoirs sur l'Aimentation". <i>Annales de Chimie et de Physique</i>. <b>34</b>. Paris: Masson: 5–37.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+de+Chimie+et+de+Physique&rft.atitle=Memoirs+sur+l%27Aimentation&rft.volume=34&rft.pages=5-37&rft.date=1827&rft.aulast=Savary&rft.aufirst=Felix&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARLC+circuit" class="Z3988"></span></span> </li> <li id="cite_note-Kimball-26"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kimball_26-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kimball_26-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Kimball_26-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Kimball_26-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Kimball_26-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKimball1917" class="citation book cs1">Kimball, Arthur Lalanne (1917). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=CwmgAAAAMAAJ&pg=PA516"><i>A College Text-book of Physics</i></a> (2nd ed.). New York: Henry Hold. pp. 516–517.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+College+Text-book+of+Physics&rft.place=New+York&rft.pages=516-517&rft.edition=2nd&rft.pub=Henry+Hold&rft.date=1917&rft.aulast=Kimball&rft.aufirst=Arthur+Lalanne&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DCwmgAAAAMAAJ%26pg%3DPA516&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARLC+circuit" class="Z3988"></span></span> </li> <li id="cite_note-Huurdeman-27"><span class="mw-cite-backlink">^ <a href="#cite_ref-Huurdeman_27-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Huurdeman_27-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Huurdeman_27-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHuurdeman2003" class="citation book cs1">Huurdeman, Anton A. (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SnjGRDVIUL4C&pg=PA200"><i>The Worldwide History of Telecommunications</i></a>. USA: Wiley-IEEE. pp. 199–200. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-20505-2" title="Special:BookSources/0-471-20505-2"><bdi>0-471-20505-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Worldwide+History+of+Telecommunications&rft.place=USA&rft.pages=199-200&rft.pub=Wiley-IEEE&rft.date=2003&rft.isbn=0-471-20505-2&rft.aulast=Huurdeman&rft.aufirst=Anton+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSnjGRDVIUL4C%26pg%3DPA200&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARLC+circuit" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text">Kaiser, pp. 7.14–7.16.</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text">Kaiser, p. 7.21.</span> </li> <li id="cite_note-Kaiser,_pp._7.21–7.27-30"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kaiser,_pp._7.21–7.27_30-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kaiser,_pp._7.21–7.27_30-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Kaiser, pp. 7.21–7.27.</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">Kaiser, pp. 7.30–7.34.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=RLC_circuit&action=edit&section=35" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAgarwalLang2005" class="citation book cs1">Agarwal, Anant; Lang, Jeffrey H. (2005). <i>Foundations of Analog and Digital Electronic Circuits</i>. Morgan Kaufmann. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-55860-735-8" title="Special:BookSources/1-55860-735-8"><bdi>1-55860-735-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundations+of+Analog+and+Digital+Electronic+Circuits&rft.pub=Morgan+Kaufmann&rft.date=2005&rft.isbn=1-55860-735-8&rft.aulast=Agarwal&rft.aufirst=Anant&rft.au=Lang%2C+Jeffrey+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARLC+circuit" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHumar2002" class="citation book cs1">Humar, J. L. (2002). <i>Dynamics of Structures</i>. Taylor & Francis. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/90-5809-245-3" title="Special:BookSources/90-5809-245-3"><bdi>90-5809-245-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamics+of+Structures&rft.pub=Taylor+%26+Francis&rft.date=2002&rft.isbn=90-5809-245-3&rft.aulast=Humar&rft.aufirst=J.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARLC+circuit" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIrwin2006" class="citation book cs1">Irwin, J. David (2006). <i>Basic Engineering Circuit Analysis</i>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/7-302-13021-3" title="Special:BookSources/7-302-13021-3"><bdi>7-302-13021-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Engineering+Circuit+Analysis&rft.pub=Wiley&rft.date=2006&rft.isbn=7-302-13021-3&rft.aulast=Irwin&rft.aufirst=J.+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARLC+circuit" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaiser2004" class="citation book cs1">Kaiser, Kenneth L. (2004). <i>Electromagnetic Compatibility Handbook</i>. CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8493-2087-9" title="Special:BookSources/0-8493-2087-9"><bdi>0-8493-2087-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electromagnetic+Compatibility+Handbook&rft.pub=CRC+Press&rft.date=2004&rft.isbn=0-8493-2087-9&rft.aulast=Kaiser&rft.aufirst=Kenneth+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARLC+circuit" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNilssonRiedel2008" class="citation book cs1">Nilsson, James William; Riedel, Susan A. (2008). <i>Electric Circuits</i>. Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-198925-2" title="Special:BookSources/978-0-13-198925-2"><bdi>978-0-13-198925-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electric+Circuits&rft.pub=Prentice+Hall&rft.date=2008&rft.isbn=978-0-13-198925-2&rft.aulast=Nilsson&rft.aufirst=James+William&rft.au=Riedel%2C+Susan+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARLC+circuit" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q323477#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4166982-4">Germany</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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