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Electrical resonance

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From Wikipedia, the free encyclopedia

Canceling impedances at a particular frequency

Resonant circuits can generate very high voltages. Atesla coil is a high-Q resonant circuit.

Electrical resonance occurs in anelectric circuit at a particularresonant frequency when theimpedances oradmittances 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 thetransfer function is close to one.^[1]

Resonant circuits exhibit ringing and can generate higher voltages or currents than are fed into them. They are widely used inwireless (radio) transmission for both transmission and reception.

LC circuits

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Main articles:LC circuit andRLC circuit

Resonance of a circuit involvingcapacitors andinductors 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 mechanicalpendulum, and both are a form ofsimple harmonic oscillator.

At resonance, the seriesimpedance of the LC circuit is at a minimum and the parallel impedance is at maximum. Resonance is used fortuning andfiltering, because it occurs at a particularfrequency for given values ofinductance andcapacitance. It can be detrimental to the operation ofcommunications circuits by causing unwanted sustained and transient oscillations that may causenoise, signaldistortion, and damage to circuit elements.

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, andvice versa, maintaining the same resonant current in the circuit, and converting all the current into useful work.

Since the inductivereactance and the capacitive reactance are of equal magnitude,

\omega L={\frac {1}{\omega C}}~

\omega ={\frac {1}{\sqrt {LC\,}}}~

where $\omega =2\pi f\,$ , in whichf is the resonance frequency inhertz,L is the inductance inhenries, andC is the capacitance infarads, when standardSI units are used.

The quality of the resonance (how long it will ring when excited) is determined by itsQ factor, which is a function of resistance: $Q={\tfrac {1}{R}}{\sqrt {{\tfrac {L}{C}}\,}}\,$ . An idealized, losslessLC circuit has infiniteQ, but all actual circuits have some resistance and finiteQ, and are usually approximated more realistically by anRLC circuit.

RLC circuit

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Main article:RLC circuit

AnRLC circuit (orLCR circuit) is anelectrical circuit consisting of aresistor, 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 forresistance,inductance andcapacitance respectively. The circuit forms aharmonic oscillator for current andresonates similarly to anLC circuit. 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 calleddamping. The presence of the resistance also reduces the peak resonant frequency ofdamped oscillation, although the resonant frequency fordriven 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 intheory.

There are many applications for this circuit. It is used in many different types ofoscillator circuits. An important application is fortuning, such as inradio receivers ortelevision sets, 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 aband-pass filter,band-stop filter,low-pass filter orhigh-pass filter. The tuning application, for instance, is an example ofband-pass filtering. The RLC filter is described as asecond-order circuit, meaning that any voltage or current in the circuit can be described by a second-orderdifferential equation in circuit analysis.

The three circuit elements can be combined in a number of differenttopologies. 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.

Example

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A seriesRLC 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 $~X_{\mathsf {L}}~=~X_{\mathsf {C}}~.$ The capacitance required to give series resonance is calculated as:

X_{\mathsf {C}}~=~X_{\mathsf {L}}~=~2\pi fL~=~2\pi \times 50\ {\mathsf {Hz}}\times 0.5\ {\mathsf {H}}~=~157.1{\text{ Ω}}~

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}}~

Resonance voltages across the inductor and the capacitor, $~V_{\mathsf {L}}~$ and $~V_{\mathsf {C}}$ , will be:

I~=~{\frac {\;V\;}{Z}}~=~{\frac {\;100\ {\mathsf {V}}\;}{4{\text{ Ω}}}}~=~25\ {\mathsf {A}}~

V_{\mathsf {L}}~=~V_{\mathsf {C}}~=~IX_{\mathsf {L}}~=~25\ {\mathsf {A}}\times 157.1{\text{ Ω}}~=~3927.5\ {\mathsf {V}}~.

As shown in this example, when the seriesRLC circuit is at resonance, the magnitudes of the voltages across the inductor and capacitor can become many times larger than the supply voltage.