![RLC series](https://upload.wikimedia.org/wikipedia/commons/9/9f/RLC_series.png)
An RLC circuit is an
electrical circuit
An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, ...
consisting of a
resistor
A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
(R), an
inductor
An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
(L), and a
capacitor
A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals.
The effect of ...
(C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent components of this circuit, where the sequence of the components may vary from RLC.
The circuit forms a
harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'':
\v ...
for current, and
resonates in a manner similar to an
LC circuit
An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can ac ...
. Introducing the resistor increases the decay of these oscillations, which is also known as
damping
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
. The resistor also reduces the peak resonant frequency. Some resistance is unavoidable even if a resistor is not specifically included as a component.
RLC circuits have many applications as
oscillator circuits.
Radio receivers
In radio communications, a radio receiver, also known as a receiver, a wireless, or simply a radio, is an electronic device that receives radio waves and converts the information carried by them to a usable form. It is used with an antenna. The ...
and
television set
A television set or television receiver, more commonly called the television, TV, TV set, telly, tele, or tube, is a device that combines a tuner, display, and loudspeakers, for the purpose of viewing and hearing television broadcasts, or using ...
s use them for
tuning
Tuning can refer to:
Common uses
* Tuning, the process of tuning a tuned amplifier or other electronic component
* Musical tuning, musical systems of tuning, and the act of tuning an instrument or voice
** Guitar tunings
** Piano tuning, adjusti ...
to select a narrow frequency range from ambient radio waves. In this role, the circuit is often referred to as a tuned circuit. An RLC circuit can be used as a
band-pass filter
A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range.
Description
In electronics and signal processing, a filter is usually a two-por ...
,
band-stop filter
In 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. A notch filter is a ba ...
,
low-pass filter
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter des ...
or
high-pass filter
A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency d ...
. The tuning application, for instance, is an example of band-pass filtering. The RLC filter is described as a ''second-order'' circuit, meaning that any voltage or current in the circuit can be described by a second-order
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
in circuit analysis.
The three circuit elements, R, L and C, can be combined in a number of different
topologies
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
. 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.
Basic concepts
Resonance
An important property of this circuit is its ability to resonate at a specific frequency, the
resonance frequency
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillat ...
, . Frequencies are measured in units of
hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
. In this article,
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
, , is used because it is more mathematically convenient. This is measured in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s per second. They are related to each other by a simple proportion,
:
Resonance
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillatin ...
occurs because energy for this situation is stored in two different ways: in an electric field as the capacitor is charged and in a magnetic field as current flows through the inductor. Energy can be transferred from one to the other within the circuit and this can be oscillatory. A mechanical analogy is a weight suspended on a spring which will oscillate up and down when released. This is no passing metaphor; a weight on a spring is described by exactly the same second order differential equation as an RLC circuit and for all the properties of the one system there will be found an analogous property of the other. The mechanical property answering to the resistor in the circuit is friction in the spring–weight system. Friction will slowly bring any oscillation to a halt if there is no external force driving it. Likewise, the resistance in an RLC circuit will "damp" the oscillation, diminishing it with time if there is no driving AC power source in the circuit.
The resonance frequency is defined as the frequency at which the
impedance of the circuit is at a minimum. Equivalently, it can be defined as the frequency at which the impedance is purely real (that is, purely resistive). This occurs because the impedances of the inductor and capacitor at resonance are equal but of opposite sign and cancel out. Circuits where L and C are in parallel rather than series actually have a maximum impedance rather than a minimum impedance. For this reason they are often described as
antiresonators; it is still usual, however, to name the frequency at which this occurs as the resonance frequency.
Natural frequency
The resonance frequency is defined in terms of the impedance presented to a driving source. It is still possible for the circuit to carry on oscillating (for a time) after the driving source has been removed or it is subjected to a step in voltage (including a step down to zero). This is similar to the way that a tuning fork will carry on ringing after it has been struck, and the effect is often called ringing. This effect is the peak natural resonance frequency of the circuit and in general is not exactly the same as the driven resonance frequency, although the two will usually be quite close to each other. Various terms are used by different authors to distinguish the two, but resonance frequency unqualified usually means the driven resonance frequency. The driven frequency may be called the
undamped
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples in ...
resonance frequency or undamped natural frequency and the peak frequency may be called the damped resonance frequency or the damped natural frequency. The reason for this terminology is that the driven resonance frequency in a series or parallel resonant circuit has the value.
:
This is exactly the same as the resonance frequency of a lossless LC circuit – that is, one with no resistor present. The resonant frequency for a ''driven'' RLC circuit is the same as a circuit in which there is no damping, hence undamped resonant frequency. The resonant frequency peak amplitude, on the other hand, does depend on the value of the resistor and is described as the damped resonant frequency. A ''highly damped'' circuit will fail to resonate at all, when not driven. A circuit with a value of resistor that causes it to be just on the edge of ringing is called
critically damped
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples in ...
. Either side of critically damped are described as
underdamped
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
(ringing happens) and
overdamped
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
(ringing is suppressed).
Circuits with topologies more complex than straightforward series or parallel (some examples described later in the article) have a driven resonance frequency that deviates from
, and for those the undamped resonance frequency, damped resonance frequency and driven resonance frequency can all be different.
Damping
Damping
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
is caused by the resistance in the circuit. It determines whether or not the circuit will resonate naturally (that is, without a driving source). Circuits that will resonate in this way are described as underdamped and those that will not are overdamped. Damping attenuation (symbol ) is measured in
neper
The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. As ...
s per second. However, the unitless
damping factor
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples inc ...
(symbol , zeta) is often a more useful measure, which is related to by
:
The special case of is called ''critical damping'' and represents the case of a circuit that is just on the border of oscillation. It is the minimum damping that can be applied without causing oscillation.
Bandwidth
The resonance effect can be used for filtering, the rapid change in impedance near resonance can be used to pass or block signals close to the resonance frequency. Both band-pass and band-stop filters can be constructed and some filter circuits are shown later in the article. A key parameter in filter design is
bandwidth
Bandwidth commonly refers to:
* Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range
* Bandwidth (computing), the rate of data transfer, bit rate or thr ...
. The bandwidth is measured between the
cutoff frequencies, most frequently defined as the frequencies at which the power passed through the circuit has fallen to half the value passed at resonance. There are two of these half-power frequencies, one above, and one below the resonance frequency
:
where is the bandwidth, is the lower half-power frequency and is the upper half-power frequency. The bandwidth is related to attenuation by
:
where the units are radians per second and
neper
The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. As ...
s per second respectively. Other units may require a conversion factor. A more general measure of bandwidth is the fractional bandwidth, which expresses the bandwidth as a fraction of the resonance frequency and is given by
:
The fractional bandwidth is also often stated as a percentage. The damping of filter circuits is adjusted to result in the required bandwidth. A narrow band filter, such as a
notch filter
In 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. A notch filter is a ba ...
, requires low damping. A wide band filter requires high damping.
factor
The
factor is a widespread measure used to characterise resonators. It is defined as the peak energy stored in the circuit divided by the average energy dissipated in it per radian at resonance. Low- circuits are therefore damped and lossy and high- circuits are underdamped. is related to bandwidth; low- circuits are wide-band and high- circuits are narrow-band. In fact, it happens that is the inverse of fractional bandwidth
:
[
factor is directly proportional to ]selectivity
Selectivity may refer to:
Psychology and behaviour
* Choice, making a selection among options
* Discrimination, the ability to recognize differences
* Socioemotional selectivity theory, in social psychology
Engineering
* Selectivity (radio), a ...
, as the factor depends inversely on bandwidth.
For a series resonant circuit ( as shown below), the factor can be calculated as follows:
:[
where is the reactance ''either'' of or of at resonance, and
]
Scaled parameters
The parameters , , and are all scaled to . This means that circuits which have similar parameters share similar characteristics regardless of whether or not they are operating in the same frequency band.
The article next gives the analysis for the series RLC circuit in detail. Other configurations are not described in such detail, but the key differences from the series case are given. The general form of the differential equations given in the series circuit section are applicable to all second order circuits and can be used to describe the voltage or current in any element of each circuit.
Series circuit
In this circuit, the three components are all in series with the voltage source
A voltage source is a two-terminal device which can maintain a fixed voltage. An ideal voltage source can maintain the fixed voltage independent of the load resistance or the output current. However, a real-world voltage source cannot supply unli ...
. The governing differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
can be found by substituting into Kirchhoff's voltage law
Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchho ...
(KVL) the constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approx ...
for each of the three elements. From the KVL,
:
where , and are the voltages across , , and , respectively, and is the time-varying voltage from the source.
Substituting and into the equation above yields:
:
For the case where the source is an unchanging voltage, taking the time derivative and dividing by leads to the following second order differential equation:
:
This can usefully be expressed in a more generally applicable form:
:
and are both in units of angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
. is called the ''neper frequency'', or ''attenuation'', and is a measure of how fast the transient response
In electrical engineering and mechanical engineering, a transient response is the response of a system to a change from an equilibrium or a steady state. The transient response is not necessarily tied to abrupt events but to any event that affec ...
of the circuit will die away after the stimulus has been removed. Neper occurs in the name because the units can also be considered to be neper
The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. As ...
s per second, neper being a logarithmic unit of attenuation. is the angular resonance frequency.
For the case of the series RLC circuit these two parameters are given by:
:
A useful parameter is the ''damping factor'', , which is defined as the ratio of these two; although, sometimes is not used, and is referred to as ''damping factor'' instead; hence requiring careful specification of one's use of that term.
:
In the case of the series RLC circuit, the damping factor is given by
:
The value of the damping factor determines the type of transient that the circuit will exhibit.
Transient response
The differential equation has the characteristic equation,[Agarwal and Lang, p. 656.]
:
The roots of the equation in -domain are,[
:
The general solution of the differential equation is an exponential in either root or a linear superposition of both,
:
The coefficients and are determined by the ]boundary condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
s of the specific problem being analysed. That is, they are set by the values of the currents and voltages in the circuit at the onset of the transient and the presumed value they will settle to after infinite time. The differential equation for the circuit solves in three different ways depending on the value of . These are overdamped (), underdamped (), and critically damped ().
Overdamped response
The overdamped response () is
:
The overdamped response is a decay of the transient current without oscillation.
Underdamped response
The underdamped response () is[Nilsson and Riedel, p. 295.]
:
By applying standard trigonometric identities
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...
the two trigonometric functions may be expressed as a single sinusoid with phase shift,
:
The underdamped response is a decaying oscillation at frequency . The oscillation decays at a rate determined by the attenuation . The exponential in describes the envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a shor ...
of the oscillation. and (or and the phase shift in the second form) are arbitrary constants determined by boundary conditions. The frequency is given by[
:
This is called the damped resonance frequency or the damped natural frequency. It is the frequency the circuit will naturally oscillate at if not driven by an external source. The resonance frequency, , which is the frequency at which the circuit will resonate when driven by an external oscillation, may often be referred to as the undamped resonance frequency to distinguish it.
]
Critically damped response
The critically damped response () is
:
The critically damped response represents the circuit response that decays in the fastest possible time without going into oscillation. This consideration is important in control systems where it is required to reach the desired state as quickly as possible without overshooting. and are arbitrary constants determined by boundary conditions.
Laplace domain
The series RLC can be analyzed for both transient and steady AC state behavior using the Laplace transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform
In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
. If the voltage source above produces a waveform with Laplace-transformed (where is the complex frequency
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the compl ...
), the KVL
Kirchhoff's circuit laws are two equalities that deal with the current and potential difference (commonly known as voltage) in the lumped element model of electrical circuits. They were first described in 1845 by German physicist Gustav Kirchho ...
can be applied in the Laplace domain:
:
where is the Laplace-transformed current through all components. Solving for :
:
And rearranging, we have
:
Laplace admittance
Solving for the Laplace admittance
In electrical engineering, admittance is a measure of how easily a circuit or device will allow a current to flow. It is defined as the reciprocal of impedance, analogous to how conductance & resistance are defined. The SI unit of admittance ...
:
:
Simplifying using parameters and defined in the previous section, we have
:
Poles and zeros
The zeros of are those values of where :
:
The poles
Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...
of are those values of where . By the quadratic formula
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, gr ...
, we find
:
The poles of are identical to the roots and of the characteristic polynomial of the differential equation in the section above.
General solution
For an arbitrary , the solution obtained by inverse transform of is:
* In the underdamped case, :
*:
* In the critically damped case, :
*:
*In the overdamped case, :
*:
where , and and are the usual hyperbolic function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the u ...
s.
Sinusoidal steady state
Sinusoidal steady state is represented by letting , where is the imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
. Taking the magnitude of the above equation with this substitution:
:
and the current as a function of can be found from
:
There is a peak value of . The value of at this peak is, in this particular case, equal to the undamped natural resonance frequency:
:
From the frequency response of the current, the frequency response of the voltages across the various circuit elements can also be determined.
Parallel circuit
The properties of the parallel RLC circuit can be obtained from the duality relationship of electrical circuits and considering that the parallel RLC is the dual impedance
Dual impedance and dual network are terms used in electronic network analysis. The dual of an impedance Z is its reciprocal, or algebraic inverse Z'=\frac. For this reason the dual impedance is also called the inverse impedance. Another way of ...
of a series RLC. Considering this, it becomes clear that the differential equations describing this circuit are identical to the general form of those describing a series RLC.
For the parallel circuit, the attenuation is given by
:
and the damping factor is consequently
:
Likewise, the other scaled parameters, fractional bandwidth and are also reciprocals of each other. This means that a wide-band, low- circuit in one topology will become a narrow-band, high- circuit in the other topology when constructed from components with identical values. The fractional bandwidth and of the parallel circuit are given by
:
Notice that the formulas here are the reciprocals of the formulas for the series circuit, given above.
Frequency domain
The complex admittance of this circuit is given by adding up the admittances of the components:
:
The change from a series arrangement to a parallel arrangement results in the circuit having a peak in impedance at resonance rather than a minimum, so the circuit is an anti-resonator.
The graph opposite shows that there is a minimum in the frequency response of the current at the resonance frequency when the circuit is driven by a constant voltage. On the other hand, if driven by a constant current, there would be a maximum in the voltage which would follow the same curve as the current in the series circuit.
Other configurations
A series resistor with the inductor in a parallel LC circuit as shown in Figure 4 is a topology commonly encountered where there is a need to take into account the resistance of the coil winding and its self-capacitance. Parallel LC circuits are frequently used for bandpass filter
A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range.
Description
In electronics and signal processing, a filter is usually a two-port ...
ing and the is largely governed by this resistance. The resonant frequency of this circuit is
:
This is the resonant frequency of the circuit defined as the frequency at which the admittance has zero imaginary part. The frequency that appears in the generalised form of the characteristic equation (which is the same for this circuit as previously)
:
is not the same frequency. In this case it is the natural, ''undamped'' resonant frequency:
:
The frequency , at which the impedance magnitude is maximum, is given by
:
where is the quality factor
In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy los ...
of the coil. This can be well approximated by
:
Furthermore, the exact maximum impedance magnitude is given by
:
For values of this can be well approximated by
:
In the same vein, a resistor in parallel with the capacitor in a series LC circuit can be used to represent a capacitor with a lossy dielectric. This configuration is shown in Figure 5. The resonant frequency (frequency at which the impedance has zero imaginary part) in this case is given by
:
while the frequency at which the impedance magnitude is minimum is given by
:
where .
History
The first evidence that a capacitor could produce electrical oscillations was discovered in 1826 by French scientist Felix Savary
Felix may refer to:
* Felix (name), people and fictional characters with the name
Places
* Arabia Felix is the ancient Latin name of Yemen
* Felix, Spain, a municipality of the province Almería, in the autonomous community of Andalusia, S ...
. He found that when a Leyden jar
A Leyden jar (or Leiden jar, or archaically, sometimes Kleistian jar) is an electrical component that stores a high-voltage electric charge (from an external source) between electrical conductors on the inside and outside of a glass jar. It typi ...
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
Joseph Henry (December 17, 1797– May 13, 1878) was an American scientist who served as the first Secretary of the Smithsonian Institution. He was the secretary for the National Institute for the Promotion of Science, a precursor of the Smith ...
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
Sir Oliver Joseph Lodge, (12 June 1851 – 22 August 1940) was a British physicist and writer involved in the development of, and holder of key patents for, radio. He identified electromagnetic radiation independent of Heinrich Rudolf Hertz, H ...
, 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 Feddersen
Berend Wilhelm Feddersen (26 March 1832 in Schleswig – 1 July 1918 in Leipzig) was a
German physicist.
Biography
Feddersen studied chemistry and physics at the University of Göttingen, where he became member of Burschenschaft Hannovera ( ...
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 Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
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 resonance
Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillatin ...
curve was published in 1887 by German physicist Heinrich Hertz
Heinrich Rudolf Hertz ( ; ; 22 February 1857 – 1 January 1894) was a German physicist who first conclusively proved the existence of the electromagnetic waves predicted by James Clerk Maxwell's Maxwell's equations, equations of electrom ...
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 resonance 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 inductors 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 RLC circuits was in the 1890s in spark-gap radio transmitters to allow the receiver to be tuned to the transmitter. The first patent for a radio system that allowed tuning was filed by Lodge in 1897, although the first practical systems were invented in 1900 by Anglo Italian radio pioneer Guglielmo Marconi
Guglielmo Giovanni Maria Marconi, 1st Marquis of Marconi (; 25 April 187420 July 1937) was an Italians, Italian inventor and electrical engineering, electrical engineer, known for his creation of a practical radio wave-based Wireless telegrap ...
.
Applications
Variable tuned circuits
A very frequent use of these circuits is in the tuning circuits of analogue radios. Adjustable tuning is commonly achieved with a parallel plate variable capacitor
A variable capacitor is a capacitor whose capacitance may be intentionally and repeatedly changed mechanically or electronically. Variable capacitors are often used in L/C circuits to set the resonance frequency, e.g. to tune a radio (therefo ...
which allows the value of to be changed and tune to stations on different frequencies. For the IF stage in the radio where the tuning is preset in the factory, the more usual solution is an adjustable core in the inductor to adjust . In this design, the core (made of a high permeability material that has the effect of increasing inductance) is threaded so that it can be screwed further in, or screwed further out of the inductor winding as required.
Filters
In the filtering application, the resistor becomes the load that the filter is working into. The value of the damping factor is chosen based on the desired bandwidth of the filter. For a wider bandwidth, a larger value of the damping factor is required (and vice versa). The three components give the designer three degrees of freedom. Two of these are required to set the bandwidth and resonant frequency. The designer is still left with one which can be used to scale , and to convenient practical values. Alternatively, may be predetermined by the external circuitry which will use the last degree of freedom.
Low-pass filter
An RLC circuit can be used as a low-pass filter. The circuit configuration is shown in Figure 6. The corner frequency, that is, the frequency of the 3 dB point, is given by
:
This is also the bandwidth of the filter. The damping factor is given by
:
High-pass filter
A high-pass filter is shown in Figure 7. The corner frequency is the same as the low-pass filter:
:
The filter has a stop-band of this width.
Band-pass filter
A band-pass filter can be formed with an RLC circuit by either placing a series LC circuit in series with the load resistor or else by placing a parallel LC circuit in parallel with the load resistor. These arrangements are shown in Figures 8 and 9 respectively. The centre frequency is given by
:
and the bandwidth for the series circuit is[Kaiser, pp. 7.21–7.27.]
:
The shunt version of the circuit is intended to be driven by a high impedance source, that is, a constant current source. Under those conditions the bandwidth is
:
Band-stop filter
Figure 10 shows a band-stop filter formed by a series LC circuit in shunt across the load. Figure 11 is a band-stop filter formed by a parallel LC circuit in series with the load. The first case requires a high impedance source so that the current is diverted into the resonator when it becomes low impedance at resonance. The second case requires a low impedance source so that the voltage is dropped across the antiresonator when it becomes high impedance at resonance.[Kaiser, pp. 7.30–7.34.]
Oscillators
For applications in oscillator circuits, it is generally desirable to make the attenuation (or equivalently, the damping factor) as small as possible. In practice, this objective requires making the circuit's resistance as small as physically possible for a series circuit, or alternatively increasing to as much as possible for a parallel circuit. In either case, the RLC circuit becomes a good approximation to an ideal LC circuit
An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can ac ...
. However, for very low-attenuation circuits (high -factor), issues such as dielectric losses of coils and capacitors can become important.
In an oscillator circuit
:
or equivalently
:
As a result,
:
Voltage multiplier
In a series RLC circuit at resonance, the current is limited only by the resistance of the circuit
:
If is small, consisting only of the inductor winding resistance say, then this current will be large. It will drop a voltage across the inductor of
:
An equal magnitude voltage will also be seen across the capacitor but in antiphase to the inductor. If can be made sufficiently small, these voltages can be several times the input voltage. The voltage ratio is, in fact, the of the circuit,
:
A similar effect is observed with currents in the parallel circuit. Even though the circuit appears as high impedance to the external source, there is a large current circulating in the internal loop of the parallel inductor and capacitor.
Pulse discharge circuit
An overdamped series RLC circuit can be used as a pulse discharge circuit. Often it is useful to know the values of components that could be used to produce a waveform. This is described by the form
:
Such a circuit could consist of an energy storage capacitor, a load in the form of a resistance, some circuit inductance and a switch – all in series. The initial conditions are that the capacitor is at voltage, , and there is no current flowing in the inductor. If the inductance is known, then the remaining parameters are given by the following – capacitance:
:
resistance (total of circuit and load):
:
initial terminal voltage of capacitor:
:
Rearranging for the case where is known – capacitance:
:
inductance (total of circuit and load):
:
initial terminal voltage of capacitor:
:
See also
*RC circuit
A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC ci ...
*RL circuit
A resistor–inductor circuit (RL circuit), or RL filter or RL network, is an electric circuit composed of resistors and inductors driven by a voltage or current source. A first-order RL circuit is composed of one resistor and one inductor, eithe ...
*Linear circuit
A linear circuit is an electronic circuit which obeys the superposition principle. This means that the output of the circuit ''F(x)'' when a linear combination of signals ''ax1(t) + bx2(t)'' is applied to it is equal to the linear combination o ...
References
Bibliography
*
*
*
*
*
{{DEFAULTSORT:Rlc Circuit
Analog circuits
Electronic filter topology