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| {{Linear analog electronic filter|filter1=hide|filter2=hide}}
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| An '''LC circuit''', also called a '''resonant circuit''', '''tank circuit''', or '''tuned circuit''', consists of an [[inductor]], represented by the letter L, and a [[capacitor]], represented by the letter C. When connected together, they can act as an electrical [[resonator]], an electrical analogue of a [[tuning fork]], storing energy oscillating at the circuit's [[resonant frequency]].
| | If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]] |
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| LC circuits are used either for generating signals at a particular frequency, or picking out a signal at a particular frequency from a more complex signal. They are key components in many electronic devices, particularly radio equipment, used in circuits such as [[Electronic oscillator|oscillators]], [[Electronic filter|filters]], [[Tuner (electronics)|tuners]] and [[frequency mixer]]s.
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| An LC circuit is an idealized model since it assumes there is no dissipation of energy due to [[electrical resistance|resistance]]. For a model incorporating resistance see [[RLC circuit]]. The purpose of an LC circuit is to oscillate with minimal [[damping]], and for this reason their resistance is made as low as possible. While no practical circuit is without losses, it is nonetheless instructive to study this pure form to gain a good understanding.
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| | :<math forcemathmode="mathml">E=mc^2</math> |
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| ==Operation== | | <!--'''PNG''' (currently default in production) |
| [[File:lc circuit.svg|200px|right|LC circuit diagram]]
| | :<math forcemathmode="png">E=mc^2</math> |
| An LC circuit can store [[electrical energy]] oscillating at its [[resonant frequency]]. A capacitor stores energy in the [[electric field]] between its plates, depending on the [[voltage]] across it, and an inductor stores energy in its [[magnetic field]], depending on the [[Electric current|current]] through it.
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| If a charged capacitor is connected across an inductor, charge will start to flow through the inductor, building up a magnetic field around it and reducing the voltage on the capacitor. Eventually all the charge on the capacitor will be gone and the voltage across it will reach zero. However, the current will continue, because inductors resist changes in current. The energy to keep it flowing is extracted from the magnetic field, which will begin to decline. The current will begin to charge the capacitor with a voltage of opposite polarity to its original charge. When the magnetic field is completely dissipated the current will stop and the charge will again be stored in the capacitor, with the opposite polarity as before. Then the cycle will continue in a similar way, with the current flowing in the opposite direction through the inductor.
| | '''source''' |
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| The charge flows back and forth between the plates of the capacitor, through the inductor. The energy oscillates back and forth between the capacitor and the inductor until (if not replenished by power from an external circuit) internal [[Electrical resistance|resistance]] makes the oscillations die out. Its action, known mathematically as a [[harmonic oscillator]], is similar to a [[pendulum]] swinging back and forth, or water sloshing back and forth in a tank. For this reason the circuit is also called a '''tank circuit'''. The oscillation frequency is determined by the capacitance and inductance values. In typical tuned circuits in electronic equipment the oscillations are very fast, thousands to millions of times per second.
| | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
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| == Time domain solution == | | ==Demos== |
| By [[Kirchhoff's voltage law]], the voltage across the capacitor, ''V''<sub>C</sub>, plus the voltage across the inductor, ''V''<sub>L</sub> must equal zero:
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| ::<math>V _{C} + V_{L} = 0.\,</math> | | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| Likewise, by [[Kirchhoff's current law]], the current through the capacitor equals the current through the inductor:
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| ::<math>i_{C} = i_{L} .\,</math> | | * accessibility: |
| | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
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| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| From the constitutive relations for the circuit elements, we also know that
| | ==Test pages == |
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| ::<math>V _{L}(t) = L \frac{di_{L}}{dt}\,</math> | | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
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| ::<math>i_{C}(t) = C \frac{dV_{C}}{dt}.\,</math>
| | ==Bug reporting== |
| | | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
| Rearranging and substituting gives the second order [[differential equation]]
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| ::<math>\frac{d ^{2}i(t)}{dt^{2}} + \frac{1}{LC} i(t) = 0.\,</math>
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| The parameter ω, the [[radian frequency]], is defined as: ω = (''LC'')<sup>−1/2</sup>. Using this can simplify the differential equation
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| ::<math>\frac{d ^{2}i(t)}{dt^{2}} + \omega^ {2} i(t) = 0.\,</math>
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| The associated polynomial is ''s''<sup>2</sup> +ω<sup>2</sup> = 0, thus
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| ::<math>s = +j \omega\,</math>
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| or
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| ::<math>s = -j \omega\,</math>
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| ::::where ''j'' is the [[imaginary unit]].
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| Thus, the complete solution to the differential equation is
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| ::<math>i(t) = Ae ^{+j \omega t} + Be ^{-j \omega t}\,</math>
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| and can be solved for ''A'' and ''B'' by considering the initial conditions.
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| Since the exponential is [[complex numbers|complex]], the solution represents a sinusoidal [[alternating current]].
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| If the initial conditions are such that ''A'' = ''B'', then we can use [[Euler's formula]] to obtain a real [[sinusoid]] with [[amplitude]] 2''A'' and [[angular frequency]] ω = (''LC'')<sup>−1/2</sup>.
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| Thus, the resulting solution becomes:
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| ::<math>i(t) = 2 A \cos(\omega t).\, </math>
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| The initial conditions that would satisfy this result are:
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| ::<math>i(t=0) = 2 A\,</math>
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| and
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| ::<math>\frac{di}{dt}(t=0) = 0.\,</math>
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| == Resonance effect ==
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| The resonance effect occurs when inductive and capacitive [[Reactance (electronics)|reactance]]s are equal in magnitude. The frequency at which this equality holds for the particular circuit is called the resonant frequency.
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| The [[Electrical resonance|resonant frequency]] of the LC circuit is
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| ::<math>\omega = \sqrt{1 \over LC}</math>
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| where '''L''' is the [[inductance]] in [[Henry (unit)|henries]], and '''C''' is the [[capacitance]] in [[farad]]s. The [[angular frequency]] <math>\omega\,</math> has units of [[radian]]s per second.
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| The equivalent frequency in units of [[hertz]] is
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| ::<math>f = { \omega \over 2 \pi } = {1 \over {2 \pi \sqrt{LC}}}. </math>
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| LC circuits are often used as filters; the L/C ratio is one of the factors that determines their [[Q factor|"Q"]] and so [[electronic selectivity|selectivity]]. For a series resonant circuit with a given resistance, the higher the inductance and the lower the capacitance, the narrower the filter [[Bandwidth (signal processing)|bandwidth]]. For a parallel resonant circuit the opposite applies. Positive feedback around the [[tuned circuit]] ("regeneration") can also increase selectivity (see [[positive feedback|Q multiplier]] and [[Regenerative circuit]]).
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| [[Tuned radio frequency receiver|Stagger tuning]] can provide an acceptably wide audio bandwidth, yet good selectivity.
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| ==Series LC circuit== | |
| === Resonance ===
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| Here L and C are connected in series. Inductive reactance magnitude (<math>\scriptstyle X_L\,</math>) increases as frequency increases while [[reactance (electronics)#Capacitive reactance|capacitive reactance]] magnitude (<math>\scriptstyle X_C\,</math>) decreases with the increase in frequency. At a particular frequency these two reactances are equal in magnitude but opposite in sign. The frequency at which this happens is the resonant frequency (<math>\scriptstyle f_r\,</math>) for the given circuit.
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| Hence, at <math>\scriptstyle f_r\,</math> :
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| :<math>X_L = -X_C\,</math>
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| :<math>{\omega {L}} = {{1} \over {\omega} {C}}\,</math>
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| Converting angular frequency into hertz we get
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| :<math>{2 \pi fL} = {1 \over {2 \pi fC}}</math>
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| Here ''f'' is the resonant frequency. Then rearranging,
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| :<math>f = {1 \over {2 \pi \sqrt{LC}}} </math>
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| In a series configuration, ''X''<sub>C</sub> and ''X''<sub>L</sub> cancel each other out. In real, rather than idealised components the current is opposed, mostly by the resistance of the coil windings. Thus, the current supplied to a series resonant circuit is a maximum at resonance.
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| * At ''f''<sub>r</sub>, current is maximum. Circuit impedance is minimum. In this state a circuit is called an ''acceptor circuit''.
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| * Below ''f''<sub>r</sub>, <math>\scriptstyle X_L \;\ll\; (-X_C)\,</math>. Hence circuit is capacitive.
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| * Above ''f''<sub>r</sub>, <math>\scriptstyle X_L \;\gg\; (-X_C)\,</math>. Hence circuit is inductive.
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| ===Impedance===
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| First consider the [[Electrical impedance|impedance]] of the series LC circuit. The total impedance is given by the sum of the inductive and capacitive impedances:
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| ::<math>Z = Z_{L} + Z_{C}</math>
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| By writing the inductive impedance as ''Z''<sub>L</sub> = ''j''ω''L'' and capacitive impedance as ''Z''<sub>C</sub> = (''j''ω''C'')<sup>−1</sup> and substituting we have
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| ::<math>Z = j \omega L + \frac{1}{j{\omega C}}</math> .
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| Writing this expression under a common denominator gives
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| ::<math>Z = \frac{(\omega^{2} L C - 1)j}{\omega C}</math> .
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| The numerator implies that if ω<sup>2</sup>''LC'' = 1 the total impedance Z will be zero and otherwise non-zero. Therefore the series LC circuit, when connected in series with a load, will act as a [[band-pass filter]] having zero impedance at the resonant frequency of the LC circuit.
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| ==Parallel LC circuit==
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| === Resonance ===
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| Here a coil (L) and capacitor (C) are connected in parallel with an AC power supply. Let R be the internal resistance of the coil. When X<sub>L</sub> equals X<sub>C</sub>, the reactive branch currents are equal and opposite. Hence they cancel out each other to give minimum current in the main line. Since total current is minimum, in this state the total impedance is maximum.
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| Resonant frequency given by: <math>f = {1 \over {2 \pi \sqrt{LC}}} </math>.
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| Note that any reactive branch current is not minimum at resonance, but each is given separately by dividing source voltage (V) by reactance (Z). Hence I=V/Z, as per [[Ohm's law]].
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| * At ''f''<sub>r</sub>, line current is minimum. Total impedance is maximum. In this state a circuit is called a ''rejector circuit''.
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| * Below ''f''<sub>r</sub>, circuit is inductive.
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| * Above ''f''<sub>r</sub>,circuit is capacitive.
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| === Impedance ===
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| The same analysis may be applied to the parallel LC circuit. The total impedance is then given by:
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| ::<math>Z = \frac{Z_{L}Z_{C}}{Z_{L} + Z_{C}}</math>
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| and after substitution of <math>\scriptstyle Z_{L}</math> and <math>\scriptstyle Z_{C}</math> and simplification, gives
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| ::<math>Z=\frac{-j \omega L}{\omega^{2}LC-1}</math>.
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| Note that
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| ::<math> \lim_{\omega^{2}LC \to 1}Z = \infty </math>
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| but for all other values of <math>\scriptstyle \omega^{2} L C</math> the impedance is finite (and therefore less than infinity). The parallel LC circuit connected in series with a load will act as [[band-stop filter]] having infinite impedance at the resonant frequency of the LC circuit. The parallel LC circuit connected in parallel with a load will act as [[band-pass filter]].
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| == Applications ==
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| === Applications of resonance effect ===
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| # Most common application is '''tuning'''. For example, when we tune a radio to a particular station, the LC circuits are set at resonance for that particular [[Carrier wave|carrier frequency]].
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| # A series resonant circuit provides voltage magnification.
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| # A parallel resonant circuit provides current magnification.
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| # A parallel resonant circuit can be used as load impedance in output circuits of RF amplifiers. Due to high impedance, the gain of amplifier is maximum at resonant frequency.
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| # Both parallel and series resonant circuits are used in induction heating.
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| LC circuits behave as electronic [[resonators]], which are a key component in many applications:
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| *[[Electronic amplifier|Amplifiers]]
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| *[[Oscillators]]
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| *[[electronic filter|Filters]]
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| *[[Tuner (electronics)|Tuners]]
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| *[[Frequency mixer|Mixers]]
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| *[[Foster-Seeley discriminator]]
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| *[[Contactless card]]s
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| *[[Graphics tablet]]s
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| *[[Electronic article surveillance|Electronic Article Surveillance]] (Security Tags).
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| == History ==
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| The first evidence that a capacitor and inductor could produce electrical oscillations was discovered in 1826 by French scientist [[Felix Savary]].<ref name="Blanchard">{{cite journal
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| | last = Blanchard
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| | first = Julian
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| | authorlink =
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| | coauthors =
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| | title = The History of Electrical Resonance
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| | journal = Bell System Technical Journal
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| | volume = 20
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| | issue = 4
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| | pages = 415-
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| | publisher = American Telephone & Telegraph Co.
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| | location = USA
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| | date = October 1941
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| | url = http://www.alcatel-lucent.com/bstj/vol20-1941/articles/bstj20-4-415.pdf
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| | issn =
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| | doi =
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| | id =
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| | accessdate = 2011-03-29}}</ref><ref>{{cite journal
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| | last = Savary
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| | first = Felix
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| | authorlink =
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| | coauthors =
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| | title = Memoirs sur l'Aimentation
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| | journal = Annales de Chimie et de Physique
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| | volume = 34
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| | issue =
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| | pages = 5–37
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| | publisher = Masson
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| | location = Paris
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| | year = 1827
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| | url =
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| | issn =
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| | doi =
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| | id =
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| | accessdate = }}</ref> He found that when a [[Leyden jar]] 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. American physicist [[Joseph Henry]] repeated Savary's experiment in 1842 and came to the same conclusion, apparently independently.<ref name="Kimball">{{cite book
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| | last = Kimball
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| | first = Arthur Lalanne
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| | authorlink =
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| | coauthors =
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| | title = A College Text-book of Physics, 2nd Ed.
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| | publisher = Henry Hold and Co.
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| | year = 1917
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| | location = New York
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| | pages = 516–517
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| | url = http://books.google.com/books?id=CwmgAAAAMAAJ&pg=PA516#v=onepage&q&f=false
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| | doi =
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| | id =
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| | isbn = }}</ref><ref name="Huurdeman">{{cite book
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| | last = Huurdeman
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| | first = Anton A.
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| | authorlink =
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| | coauthors =
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| | title = The worldwide history of telecommunications
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| | publisher = Wiley-IEEE
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| | year = 2003
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| | location = USA
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| | pages = 199–200
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| | url = http://books.google.com/books?id=SnjGRDVIUL4C&pg=PA200
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| | doi =
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| | id =
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| | isbn = 0-471-20505-2}}</ref> British scientist [[William Thomson]] (Lord Kelvin) in 1853 showed mathematically that the discharge of a Leyden jar through an inductance should be oscillatory, and derived its resonant frequency.<ref name="Blanchard" /><ref name="Kimball" /><ref name="Huurdeman" /> British radio researcher [[Oliver Lodge]], 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.<ref name="Kimball" /> In 1857, German physicist [[Berend Wilhelm Feddersen]] photographed the spark produced by a resonant Leyden jar circuit in a rotating mirror, providing visible evidence of the oscillations.<ref name="Blanchard" /><ref name="Kimball" /><ref name="Huurdeman" /> In 1868, Scottish physicist [[James Clerk Maxwell]] 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.<ref name="Blanchard" /> The first example of an electrical [[resonance]] curve was published in 1887 by German physicist [[Heinrich Hertz]] 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.<ref name="Blanchard" />
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| One of the first demonstrations of [[resonance]] between tuned circuits was Lodge's "syntonic jars" experiment around 1889.<ref name="Blanchard" /><ref name="Kimball" /> 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 circuits were adjusted to resonance. Lodge and some English scientists preferred the term "''syntony''" for this effect, but the term "''resonance''" eventually stuck.<ref name="Blanchard" /> The first practical use for LC circuits was in the 1890s in [[spark-gap transmitter|spark-gap radio transmitters]] to allow the receiver and transmitter to be tuned to the same frequency. 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 Italian radio pioneer [[Guglielmo Marconi]].<ref name="Blanchard" />
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| == See also ==
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| *[[RL circuit]]
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| *[[RC circuit]]
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| *[[RLC circuit]]
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| == References ==
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| {{Reflist}}
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| == External links ==
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| {{Wikibooks|Circuit Idea|How do We Create Sinusoidal Oscillations?}}
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| * [http://www.opamp-electronics.com/tutorials/an_electric_pendulum_2_06_01.htm An electric pendulum] by Tony Kuphaldt is a classical story about the operation of LC tank
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| * [http://www.tpub.com/neets/book9/34d.htm How the parallel-LC circuit stores energy] is another excellent LC resource.
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| {{clear}}
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| {{Refimprove|date=March 2009}}
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| [[Category:Analog circuits]]
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| [[Category:Electronic filter topology]]
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| [[ar:رنان مستحث ومكثف]]
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| [[be-x-old:Вагальны контур]]
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| [[bg:Трептящ кръг]]
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| [[ca:Circuit LC]]
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| [[cs:Rezonanční obvod]]
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| [[es:Circuito LC]]
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| [[fr:Circuit LC]]
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| [[lv:Svārstību kontūrs]]
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| [[ml:ടാങ്ക് പരിപഥം]]
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| [[nl:LC-kring]]
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| [[ja:LC回路]]
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| [[pl:Obwód rezonansowy LC]]
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| [[pt:Circuito LC]]
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| [[simple:LC circuit]]
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| [[sk:Rezonančný obvod]]
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| [[sl:Nihajni krog]]
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| [[fi:Värähtelypiiri]]
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| [[sv:Elektromagnetisk svängningskrets]]
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| [[zh:LC电路]]
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