An
LC circuit, also called a
resonant circuit or
tuned circuit, consists of an
inductorAn inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
, represented by the letter L, and a
capacitorA capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
, represented by the letter C. When connected together, they can act as an electrical
resonatorA resonator is a device or system that exhibits resonance or resonant behavior, that is, it naturally oscillates at some frequencies, called its resonant frequencies, with greater amplitude than at others. The oscillations in a resonator can be either electromagnetic or mechanical...
, an electrical analogue of a
tuning forkA tuning fork is an acoustic resonator in the form of a two-pronged fork with the prongs formed from a U-shaped bar of elastic metal . It resonates at a specific constant pitch when set vibrating by striking it against a surface or with an object, and emits a pure musical tone after waiting a...
, storing electrical energy oscillating at the circuit's resonant frequency.
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 applications such as oscillators,
filtersElectronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...
, tuners and frequency mixers.
An LC circuit is an idealized model since it assumes there is no dissipation of energy due to
resistanceThe electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
. For a model incorporating resistance see
RLC circuitAn RLC circuit is an electrical circuit consisting of a resistor, 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 resistance, inductance and capacitance respectively...
. The purpose of an LC circuit is to oscillate with minimal
dampingIn physics, damping is any effect that tends to reduce the amplitude of oscillations in an oscillatory system, particularly the harmonic oscillator.In mechanics, friction is one such damping effect...
, 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.
Operation
An LC circuit can store electrical energy oscillating at its resonant frequency. A capacitor stores energy in the
electric fieldIn physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
between its plates, depending on the
voltageVoltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
across it, and an inductor stores energy in its
magnetic fieldA magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
, depending on the
currentElectric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
through it.
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, and 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 begin again, with the current flowing in the opposite direction through the inductor.
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
resistanceThe electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
makes the oscillations die out. Its action, known mathematically as a
harmonic oscillatorIn classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: \vec F = -k \vec x \, where k is a positive constant....
, is similar to a
pendulumA pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position...
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 used. In typical tuned circuits in electronic equipment the oscillations are very fast, thousands to millions of times per second.
Time domain solution
By Kirchhoff's voltage law, the voltage across the capacitor,
VC, plus the voltage across the inductor,
VL must equal zero:
-
Likewise, by Kirchhoff's current law, the current through the capacitor equals the current through the inductor:
-
From the constitutive relations for the circuit elements, we also know that
-
and
-
Rearranging and substituting gives the second order
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
-
The parameter ω, the radian frequency, can be defined as: ω = (
LC)
−1/2. Using this can simplify the differential equation
-
The associated polynomial is
s2 +ω
2 = 0, thus
-
or
-
-
-
-
- where j is the imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...
.
Thus, the complete solution to the differential equation is
-
and can be solved for
A and
B by considering the initial conditions.
Since the exponential is complex, the solution represents a sinusoidal
alternating currentIn alternating current the movement of electric charge periodically reverses direction. In direct current , the flow of electric charge is only in one direction....
.
If the initial conditions are such that
A =
B, then we can use
Euler's formulaEuler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...
to obtain a real sinusoid with
amplitudeAmplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
2
A and
angular frequencyIn physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
ω = (
LC)
−1/2.
Thus, the resulting solution becomes:
-
The initial conditions that would satisfy this result are:
-
and
-
Resonance effect
The resonance effect occurs when inductive and capacitive reactances are equal in absolute value. The frequency at which this equality holds for the particular circuit is called the resonant frequency.
The
resonant frequencyElectrical resonance occurs in an electric circuit at a particular resonance frequency where the imaginary parts of circuit element impedances or admittances cancel each other...
of the LC circuit is
-
where
L is the
inductanceIn electromagnetism and electronics, inductance is the ability of an inductor to store energy in a magnetic field. Inductors generate an opposing voltage proportional to the rate of change in current in a circuit...
in henries, and
C is the
capacitanceIn electromagnetism and electronics, capacitance is the ability of a capacitor to store energy in an electric field. Capacitance is also a measure of the amount of electric potential energy stored for a given electric potential. A common form of energy storage device is a parallel-plate capacitor...
in
faradThe farad is the SI unit of capacitance. The unit is named after the English physicist Michael Faraday.- Definition :A farad is the charge in coulombs which a capacitor will accept for the potential across it to change 1 volt. A coulomb is 1 ampere second...
s. The
angular frequencyIn physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
has units of
radianRadian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s per second.
The equivalent frequency in units of
hertzThe hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....
is
-
LC circuits are often used as filters; the L/C ratio is one of the factors that determines their
"Q"In physics and engineering the quality factor or Q factor is a dimensionless parameter that describes how under-damped an oscillator or resonator is, or equivalently, characterizes a resonator's bandwidth relative to its center frequency....
and so
selectivitySelectivity is a measure of the performance of a radio receiver to respond only to the radio signal it is tuned to and reject other signals nearby in frequency, such as another broadcast on an adjacent channel....
. For a series resonant circuit with a given resistance, the higher the inductance and the lower the capacitance, the narrower the filter bandwidth. For a parallel resonant circuit the opposite applies. Positive feedback around the tuned circuit ("regeneration") can also increase selectivity (see
Q multiplierPositive feedback is a process in which the effects of a small disturbance on a system include an increase in the magnitude of the perturbation. That is, A produces more of B which in turn produces more of A. In contrast, a system that responds to a perturbation in a way that reduces its effect is...
and
Regenerative circuitThe regenerative circuit or "autodyne" allows an electronic signal to be amplified many times by the same vacuum tube or other active component such as a field effect transistor. It consists of an amplifying vacuum tube or transistor with its output connected to its input through a feedback...
).
Stagger tuningA tuned radio frequency receiver is a radio receiver that is usually composed of several tuned radio frequency amplifiers followed by circuits to detect and amplify the audio signal. Prevalent in the early 20th century, it can be difficult to operate because each stage must be individually tuned...
can provide an acceptably wide audio bandwidth, yet good selectivity.
Resonance
Here L and C are connected in series to an AC power supply. Inductive reactance magnitude (
) increases as frequency increases while capacitive reactance magnitude (
) 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 (
) for the given circuit.
Hence, at
:
Converting angular frequency into hertz we get
Here
f is the resonant frequency. Then rearranging,
In a series AC circuit,
XC and
XL cancel each other out. The only opposition to a current is coil resistance. Hence in series resonance the current is maximum at resonant frequency.
- At fr, current is maximum. Circuit impedance is minimum. In this state a circuit is called an acceptor circuit.
- Below fr,
. Hence circuit is capacitive.
- Above fr,
. Hence circuit is inductive.
Impedance
First consider the
impedanceElectrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
of the series LC circuit. The total impedance is given by the sum of the inductive and capacitive impedances:
-
By writing the inductive impedance as
ZL =
jω
L and capacitive impedance as
ZC = (
jω
C)
−1 and substituting we have
-
.
Writing this expression under a common denominator gives
-
.
The numerator implies that if ω
2LC = 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 filterA band-pass filter is a device that passes frequencies within a certain range and rejects frequencies outside that range.Optical band-pass filters are of common usage....
having zero impedance at the resonant frequency of the LC circuit.
Resonance
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
L equals X
C, 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.
Resonant frequency given by:
.
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 lawOhm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...
.
- At fr, line current is minimum. Total impedance is maximum. In this state a circuit is called a rejector circuit.
- Below fr, circuit is inductive.
- Above fr,circuit is capacitive.
Impedance
The same analysis may be applied to the parallel LC circuit. The total impedance is then given by:
-
and after substitution of
and
and simplification, gives
-
.
Note that
-
but for all other values of
the impedance is finite (and therefore less than infinity). Hence the parallel LC circuit connected in series with a load will act as
band-stop filterIn signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a specific range to very low levels. It is the opposite of a band-pass filter...
having infinite impedance at the resonant frequency of the LC circuit.
Applications of resonance effect
- 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 frequency
In telecommunications, a carrier wave or carrier is a waveform that is modulated with an input signal for the purpose of conveying information. This carrier wave is usually a much higher frequency than the input signal...
.
- A series resonant circuit provides voltage magnification.
- A parallel resonant circuit provides current magnification.
- 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.
- Both parallel and series resonant circuits are used in induction heating.
LC circuits behave as electronic resonators, which are a key component in many applications:
- Amplifiers
An electronic amplifier is a device for increasing the power of a signal.It does this by taking energy from a power supply and controlling the output to match the input signal shape but with a larger amplitude...
- Oscillators
- Filters
Electronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...
- Tuners
- Mixers
In electronics a mixer or frequency mixer is a nonlinear electrical circuit that creates new frequencies from two signals applied to it. In its most common application, two signals at frequencies f1 and f2 are applied to a mixer, and it produces new signals at the sum f1 + f2 and difference f1 -...
- Foster-Seeley discriminator
The Foster-Seeley discriminator is a common type of FM detector circuit, invented in 1936 by Dudley E. Foster and Stuart William Seeley. The circuit was envisioned for automatic frequency control of receivers, but also found application in demodulating an FM signal.It uses a tuned RF transformer...
- Contactless cards
- Graphics tablet
A graphics tablet is a computer input device that enables a user to hand-draw images and graphics, similar to the way a person draws images with a pencil and paper. These tablets may also be used to capture data or handwritten signatures...
s
- Electronic Article Surveillance
Electronic article surveillance is a technological method for preventing shoplifting from retail stores or pilferage of books from libraries. Special tags are fixed to merchandise or books. These tags are removed or deactivated by the clerks when the item is properly bought or checked out...
(Security Tags).
History
The first evidence that a capacitor and inductor could produce electrical oscillations was discovered in 1826 by French scientist
Felix SavaryFélix Savary, who was born on October 4, 1797 in Paris and died on July 15, 1841 in Estagel, was a French astronomer.He studied at the École Polytechnique, where he was later a professor of astronomy...
. He found that when a
Leyden jarA Leyden jar, or Leiden jar, is a device that "stores" static electricity between two electrodes on the inside and outside of a jar. It was invented independently by German cleric Ewald Georg von Kleist on 11 October 1745 and by Dutch scientist Pieter van Musschenbroek of Leiden in 1745–1746. The...
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 HenryJoseph Henry was an American scientist who served as the first Secretary of the Smithsonian Institution, as well as a founding member of the National Institute for the Promotion of Science, a precursor of the Smithsonian Institution. During his lifetime, he was highly regarded...
repeated Savary's experiment in 1842 and came to the same conclusion, apparently independently. 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. 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. In 1857 German physicist
Berend Wilhelm FeddersenBerend Wilhelm Feddersen was a German physicist.-Biography:Feddersen lived from 1858 as a private scholar in Leipzig. In 1859 he succeeded in experiments with the Leyden jar to prove that every single electric spark discharge composed of oscillations...
photographed the spark produced by a resonant Leyden jar circuit in a rotating mirror, providing visible evidence of the oscillations. In 1868 Scottish physicist
James Clerk MaxwellJames Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
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. The first example of an electrical
resonanceIn physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...
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.
One of the first demonstrations of
resonanceIn physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...
between tuned circuits was Lodge's "syntonic jars" experiment around 1889. 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. The first practical use for LC circuits was in the 1890s in
spark-gap radio transmittersA spark-gap transmitter is a device for generating radio frequency electromagnetic waves using a spark gap.These devices served as the transmitters for most wireless telegraphy systems for the first three decades of radio and the first demonstrations of practical radio were carried out using them...
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 MarconiGuglielmo Marconi was an Italian inventor, known as the father of long distance radio transmission and for his development of Marconi's law and a radio telegraph system. Marconi is often credited as the inventor of radio, and indeed he shared the 1909 Nobel Prize in Physics with Karl Ferdinand...
.
External links
An electric pendulum by Tony Kuphaldt is a classical story about the operation of LC tank
How the parallel-LC circuit stores energy is another excellent LC resource.