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data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Resson%C3%A0ncia_el%C3%A8ctrica" title="Ressonància elèctrica – Catalan" lang="ca" hreflang="ca" data-title="Ressonància elèctrica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs badge-Q70893996 mw-list-item" title=""><a href="https://cs.wikipedia.org/wiki/Elektrick%C3%A1_rezonance" title="Elektrická rezonance – Czech" lang="cs" hreflang="cs" data-title="Elektrická rezonance" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Resonancia_el%C3%A9ctrica" title="Resonancia eléctrica – Spanish" lang="es" hreflang="es" data-title="Resonancia eléctrica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</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%AA%D8%B4%D8%AF%DB%8C%D8%AF" title="مدار تشدید – Persian" lang="fa" hreflang="fa" data-title="مدار تشدید" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Elektri%C4%8Dna_rezonancija" title="Električna rezonancija – Croatian" lang="hr" hreflang="hr" data-title="Električna rezonancija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a 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class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Elektrisk_resonans" title="Elektrisk resonans – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Elektrisk resonans" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Resson%C3%A2ncia_el%C3%A9trica" title="Ressonância elétrica – Portuguese" lang="pt" hreflang="pt" data-title="Ressonância elétrica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Elektrick%C3%A1_rezonancia" title="Elektrická rezonancia – Slovak" lang="sk" hreflang="sk" data-title="Elektrická rezonancia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B0_%D1%80%D0%B5%D0%B7%D0%BE%D0%BD%D0%B0%D0%BD%D1%86%D0%B0" title="Електрична резонанца – Serbian" lang="sr" hreflang="sr" data-title="Електрична резонанца" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Elektri%C4%8Dna_rezonancija" title="Električna rezonancija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Električna rezonancija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Canceling impedances at a particular frequency</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Teslova_c%C3%ADvka_v_provozu.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Teslova_c%C3%ADvka_v_provozu.JPG/290px-Teslova_c%C3%ADvka_v_provozu.JPG" decoding="async" width="290" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/23/Teslova_c%C3%ADvka_v_provozu.JPG/435px-Teslova_c%C3%ADvka_v_provozu.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/23/Teslova_c%C3%ADvka_v_provozu.JPG/580px-Teslova_c%C3%ADvka_v_provozu.JPG 2x" data-file-width="3207" data-file-height="2602" /></a><figcaption>Resonant circuits can generate very high voltages. A <a href="/wiki/Tesla_coil" title="Tesla coil">tesla coil</a> is a high-Q resonant circuit.</figcaption></figure> <p><b>Electrical resonance</b> occurs in an <a href="/wiki/Electrical_network" title="Electrical network">electric circuit</a> at a particular <i><a href="/wiki/Resonance" title="Resonance">resonant frequency</a></i> when the <a href="/wiki/Electrical_impedance" title="Electrical impedance">impedances</a> or <a href="/wiki/Admittance" title="Admittance">admittances</a> of circuit elements cancel each other. In some circuits, this happens when the impedance between the input and output of the circuit is almost zero and the <a href="/wiki/Transfer_function" title="Transfer function">transfer function</a> is close to one.<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><p>Resonant circuits exhibit ringing and can generate higher voltages or currents than are fed into them. They are widely used in <a href="/wiki/Wireless" title="Wireless">wireless</a> (<a href="/wiki/Radio" title="Radio">radio</a>) transmission for both transmission and reception. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="LC_circuits">LC circuits</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electrical_resonance&action=edit&section=1" title="Edit section: LC circuits"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/LC_circuit" title="LC circuit">LC circuit</a> and <a href="/wiki/RLC_circuit" title="RLC circuit">RLC circuit</a></div> <p>Resonance of a circuit involving <a href="/wiki/Capacitor" title="Capacitor">capacitors</a> and <a href="/wiki/Inductor" title="Inductor">inductors</a> occurs because the collapsing magnetic field of the inductor generates an electric current in its windings that charges the capacitor, and then the discharging capacitor provides an electric current that builds the magnetic field in the inductor. This process is repeated continually. An analogy is a mechanical <a href="/wiki/Pendulum" title="Pendulum">pendulum</a>, and both are a form of <a href="/wiki/Simple_harmonic_oscillator" class="mw-redirect" title="Simple harmonic oscillator">simple harmonic oscillator</a>. </p><p>At resonance, the series <a href="/wiki/Electrical_impedance" title="Electrical impedance">impedance</a> of the LC circuit is at a minimum and the parallel impedance is at maximum. Resonance is used for <a href="/wiki/Tuner_(radio)" title="Tuner (radio)">tuning</a> and <a href="/wiki/Filter_(signal_processing)" title="Filter (signal processing)">filtering</a>, because it occurs at a particular <a href="/wiki/Frequency" title="Frequency">frequency</a> for given values of <a href="/wiki/Inductance" title="Inductance">inductance</a> and <a href="/wiki/Capacitance" title="Capacitance">capacitance</a>. It can be detrimental to the operation of <a href="/wiki/Telecommunications" title="Telecommunications">communications</a> circuits by causing unwanted sustained and transient oscillations that may cause <a href="/wiki/Noise" title="Noise">noise</a>, signal <a href="/wiki/Distortion" title="Distortion">distortion</a>, and damage to circuit elements. </p><p>Parallel resonance or near-to-resonance circuits can be used to prevent the waste of electrical energy, which would otherwise occur while the inductor built its field or the capacitor charged and discharged. As an example, asynchronous motors waste inductive current while synchronous ones waste capacitive current. The use of the two types in parallel makes the inductor feed the capacitor, and <i>vice versa</i>, maintaining the same resonant current in the circuit, and converting all the current into useful work. </p><p>Since the inductive <a href="/wiki/Electrical_reactance" title="Electrical reactance">reactance</a> and the capacitive reactance are of equal magnitude, </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}{\omega C}}~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mi>L</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ω</mi> <mi>C</mi> </mrow> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega L={\frac {1}{\omega C}}~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f2a6ef641c0b44088f24d2fce351debb8474ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.756ex; height:5.343ex;" alt="{\displaystyle \omega L={\frac {1}{\omega C}}~}"></span>,</dd></dl> <p>so </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ={\frac {1}{\sqrt {LC\,}}}~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>L</mi> <mi>C</mi> <mspace width="thinmathspace" /> </msqrt> </mfrac> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\frac {1}{\sqrt {LC\,}}}~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ab0695e7897642113e22ab54251142a904a7abf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.633ex; height:6.176ex;" alt="{\displaystyle \omega ={\frac {1}{\sqrt {LC\,}}}~}"></span>,</dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =2\pi f\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =2\pi f\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d75a1511d1fd022069b6b9b4363afc756abaf9c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.704ex; height:2.509ex;" alt="{\displaystyle \omega =2\pi f\,}"></span>, in which <span class="texhtml mvar" style="font-style:italic;">f</span> is the resonance frequency in <a href="/wiki/Hertz" title="Hertz">hertz</a>, <span class="texhtml mvar" style="font-style:italic;">L</span> is the inductance in <a href="/wiki/Henry_(unit)" title="Henry (unit)">henries</a>, and <span class="texhtml mvar" style="font-style:italic;">C</span> is the capacitance in <a href="/wiki/Farad" title="Farad">farads</a>, when standard <a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI units</a> are used. </p><p>The quality of the resonance (how long it will ring when excited) is determined by its <a href="/wiki/Q_factor" title="Q factor"><span class="texhtml mvar" style="font-style:italic;">Q</span> factor</a>, which is a function of resistance: <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={\tfrac {1}{R}}{\sqrt {{\tfrac {L}{C}}\,}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mi>R</mi> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>L</mi> <mi>C</mi> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> </msqrt> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q={\tfrac {1}{R}}{\sqrt {{\tfrac {L}{C}}\,}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/453acd0f998e6f6bf2ec84028039ff01fbd18005" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.203ex; height:4.843ex;" alt="{\displaystyle Q={\tfrac {1}{R}}{\sqrt {{\tfrac {L}{C}}\,}}\,}"></span>. An idealized, lossless <a href="/wiki/LC_circuit" title="LC circuit"><span class="texhtml mvar" style="font-style:italic;">LC</span> circuit</a> has infinite <span class="texhtml mvar" style="font-style:italic;">Q</span>, but all actual circuits have some resistance and finite <span class="texhtml mvar" style="font-style:italic;">Q</span>, and are usually approximated more realistically by an <a href="/wiki/RLC_circuit" title="RLC circuit"><span class="texhtml mvar" style="font-style:italic;">RLC</span> circuit</a>. </p> <div class="mw-heading mw-heading2"><h2 id="RLC_circuit">RLC circuit</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electrical_resonance&action=edit&section=2" title="Edit section: RLC circuit"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/RLC_circuit" title="RLC circuit">RLC circuit</a></div> <p>An <b>RLC circuit</b> (or <b>LCR circuit</b>) is an <a href="/wiki/Electrical_circuit" class="mw-redirect" title="Electrical circuit">electrical circuit</a> consisting of a <a href="/wiki/Resistor" title="Resistor">resistor</a>, an inductor, and a capacitor, connected in series or in parallel. The RLC part of the name is due to those letters being the usual electrical symbols for <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a>, <a href="/wiki/Inductance" title="Inductance">inductance</a> and <a href="/wiki/Capacitance" title="Capacitance">capacitance</a> respectively. 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> similarly to an <a href="/wiki/LC_circuit" title="LC circuit">LC circuit</a>. The main difference stemming from the presence of the resistor is that any oscillation induced in the circuit decays over time if it is not kept going by a source. This effect of the resistor is called <a href="/wiki/Damping" title="Damping">damping</a>. The presence of the resistance also reduces the peak resonant frequency of <i>damped</i> oscillation, although the resonant frequency for <i>driven</i> oscillations remains the same as an LC circuit. Some resistance is unavoidable in real circuits, even if a resistor is not specifically included as a separate component. A pure LC circuit is an ideal that exists only in <a href="/wiki/Theory" title="Theory">theory</a>. </p><p>There are many applications for this circuit. It is used in many different types of <a href="/wiki/Electronic_oscillator" title="Electronic oscillator">oscillator circuits</a>. An important application is for <a href="/wiki/Tuner_(electronics)" class="mw-redirect" title="Tuner (electronics)">tuning</a>, such as in <a href="/wiki/Receiver_(radio)" class="mw-redirect" title="Receiver (radio)">radio receivers</a> or <a href="/wiki/Television_set" title="Television set">television sets</a>, where they are used to select a narrow range of frequencies from the 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 <a href="/wiki/Band-pass_filter" title="Band-pass filter">band-pass filtering</a>. 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 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> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electrical_resonance&action=edit&section=3" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A series <span class="texhtml">RLC</span> circuit has resistance of 4 Ω, and inductance of 500 mH, and a variable capacitance. Supply voltage is 100 V alternating at 50 Hz. At resonance <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_{\mathsf {L}}~=~X_{\mathsf {C}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~X_{\mathsf {L}}~=~X_{\mathsf {C}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04dc83723fdb69dffa1a7507a901f76a100c343c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.322ex; height:2.509ex;" alt="{\displaystyle ~X_{\mathsf {L}}~=~X_{\mathsf {C}}~.}"></span> The capacitance required to give series resonance is calculated as: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{\mathsf {C}}~=~X_{\mathsf {L}}~=~2\pi fL~=~2\pi \times 50\ {\mathsf {Hz}}\times 0.5\ {\mathsf {H}}~=~157.1{\text{ Ω}}~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>L</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mn>2</mn> <mi>π</mi> <mo>×</mo> <mn>50</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">H</mi> <mi mathvariant="sans-serif">z</mi> </mrow> </mrow> <mo>×</mo> <mn>0.5</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">H</mi> </mrow> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mn>157.1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> Ω</mtext> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{\mathsf {C}}~=~X_{\mathsf {L}}~=~2\pi fL~=~2\pi \times 50\ {\mathsf {Hz}}\times 0.5\ {\mathsf {H}}~=~157.1{\text{ Ω}}~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c3ac82399aea1e9302e449403c857b50073d9d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:55.721ex; height:2.509ex;" alt="{\displaystyle X_{\mathsf {C}}~=~X_{\mathsf {L}}~=~2\pi fL~=~2\pi \times 50\ {\mathsf {Hz}}\times 0.5\ {\mathsf {H}}~=~157.1{\text{ Ω}}~}"></span></dd></dl> <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}{\;2\pi fX_{\mathsf {C}}\;}}~=~{\frac {1}{\;6.2832\times 50\ {\mathsf {Hz}}\times 157.1{\text{ Ω}}\;}}=20.3{\text{ μF}}~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thickmathspace" /> <mn>2</mn> <mi>π</mi> <mi>f</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </msub> <mspace width="thickmathspace" /> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mspace width="thickmathspace" /> <mn>6.2832</mn> <mo>×</mo> <mn>50</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">H</mi> <mi mathvariant="sans-serif">z</mi> </mrow> </mrow> <mo>×</mo> <mn>157.1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> Ω</mtext> </mrow> <mspace width="thickmathspace" /> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>20.3</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> μF</mtext> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C~=~{\frac {1}{\;2\pi fX_{\mathsf {C}}\;}}~=~{\frac {1}{\;6.2832\times 50\ {\mathsf {Hz}}\times 157.1{\text{ Ω}}\;}}=20.3{\text{ μF}}~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2500bf25152012204b05beea722fffd43675eca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:58.09ex; height:5.676ex;" alt="{\displaystyle C~=~{\frac {1}{\;2\pi fX_{\mathsf {C}}\;}}~=~{\frac {1}{\;6.2832\times 50\ {\mathsf {Hz}}\times 157.1{\text{ Ω}}\;}}=20.3{\text{ μF}}~}"></span></dd></dl> <p>Resonance voltages across the inductor and the capacitor, <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_{\mathsf {L}}~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~V_{\mathsf {L}}~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/558f7da06a36d44abfc932325aec04d7cf092ae1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.64ex; height:2.509ex;" alt="{\displaystyle ~V_{\mathsf {L}}~}"></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_{\mathsf {C}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ~V_{\mathsf {C}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fb1f9467c174674824bc942acfd8e742b23efcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.218ex; height:2.509ex;" alt="{\displaystyle ~V_{\mathsf {C}}}"></span>, will be: </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\;}{Z}}~=~{\frac {\;100\ {\mathsf {V}}\;}{4{\text{ Ω}}}}~=~25\ {\mathsf {A}}~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <mi>V</mi> <mspace width="thickmathspace" /> </mrow> <mi>Z</mi> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mspace width="thickmathspace" /> <mn>100</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">V</mi> </mrow> </mrow> <mspace width="thickmathspace" /> </mrow> <mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> Ω</mtext> </mrow> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mn>25</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I~=~{\frac {\;V\;}{Z}}~=~{\frac {\;100\ {\mathsf {V}}\;}{4{\text{ Ω}}}}~=~25\ {\mathsf {A}}~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/048f1929dfc9a0f1d127ce2cf172489eb7ea73dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:30.646ex; height:5.509ex;" alt="{\displaystyle I~=~{\frac {\;V\;}{Z}}~=~{\frac {\;100\ {\mathsf {V}}\;}{4{\text{ Ω}}}}~=~25\ {\mathsf {A}}~}"></span></dd></dl> <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_{\mathsf {L}}~=~V_{\mathsf {C}}~=~IX_{\mathsf {L}}~=~25\ {\mathsf {A}}\times 157.1{\text{ Ω}}~=~3927.5\ {\mathsf {V}}~.}"> <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="sans-serif">L</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">C</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mi>I</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> </msub> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mn>25</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">A</mi> </mrow> </mrow> <mo>×</mo> <mn>157.1</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> Ω</mtext> </mrow> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <mn>3927.5</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">V</mi> </mrow> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathsf {L}}~=~V_{\mathsf {C}}~=~IX_{\mathsf {L}}~=~25\ {\mathsf {A}}\times 157.1{\text{ Ω}}~=~3927.5\ {\mathsf {V}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12265cfbac8be74eac774986e054e506eeed2703" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:51.044ex; height:2.509ex;" alt="{\displaystyle V_{\mathsf {L}}~=~V_{\mathsf {C}}~=~IX_{\mathsf {L}}~=~25\ {\mathsf {A}}\times 157.1{\text{ Ω}}~=~3927.5\ {\mathsf {V}}~.}"></span></dd></dl> <p>As shown in this example, when the series <span class="texhtml">RLC</span> circuit is at resonance, the magnitudes of the voltages across the inductor and capacitor can become many times larger than the supply voltage. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electrical_resonance&action=edit&section=4" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Antiresonance" title="Antiresonance">Antiresonance</a></li> <li><a href="/wiki/Antenna_theory" class="mw-redirect" title="Antenna theory">Antenna theory</a></li> <li><a href="/wiki/Cavity_resonator" class="mw-redirect" title="Cavity resonator">Cavity resonator</a></li> <li><a href="/wiki/Electronic_filter" title="Electronic filter">Electronic filter</a></li> <li><a href="/wiki/Resonant_inductive_coupling" title="Resonant inductive coupling">Resonant energy transfer</a> - wireless energy transmission between two resonant coils</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electrical_resonance&action=edit&section=5" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output 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.cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/electric/serres.html">"Resonant RLC Circuits"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Resonant+RLC+Circuits&rft_id=http%3A%2F%2Fhyperphysics.phy-astr.gsu.edu%2Fhbase%2Felectric%2Fserres.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resonance" class="Z3988"></span></span> </li> </ol></div></div> <p><style data-mw-deduplicate="TemplateStyles:r1041539562">.mw-parser-output .citation{word-wrap:break-word}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}</style><span class="citation 1037C"><span class="noviewer" typeof="mw:File"><span><img alt="Public Domain" src="//upload.wikimedia.org/wikipedia/en/thumb/6/62/PD-icon.svg/12px-PD-icon.svg.png" decoding="async" width="12" height="12" class="mw-file-element" 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