An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (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 for current, and resonates in a manner similar to an LC circuit. Introducing the resistor increases the decay of these oscillations, which is also known as damping. 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 and television sets use them for tuning 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, band-stop filter, low-pass filter or high-pass filter. 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 circuit analysis.
The three circuit elements, R, L and C, can be combined in a number of different topologies. 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, . Frequencies are measured in units of hertz. In this article, angular frequency, , is used because it is more mathematically convenient. This is measured in radians per second. They are related to each other by a simple proportion,
Resonance 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 resonant 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 resonant frequency 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 resonant 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 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.Kaiser, pp. 7.71–7.72
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. Either side of critically damped are described as underdamped (ringing happens) and overdamped (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 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 nepers per second. However, the unitless damping factor (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. 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 nepers 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, 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 and prone to amplitude extremes if driven at the resonant frequency. 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, 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. The governing differential equation can be found by substituting into Kirchhoff's voltage law (KVL) the constitutive equation for each of the three elements. From KVL,
where , and are the voltages across , , and , respectively, and is the voltage from the source. All are a function of time , since they may be time-varying.
Substituting and into the equation above yields:
For the case where the source is an unchanging voltage, taking the time derivative and dividing by (and reordering the first two terms and eliminating the 0 from taking the derivative of the constant 3rd term) 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. is called the neper frequency, or attenuation, and is a measure of how fast the transient response 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 nepers per second, neper being a logarithmic unit of attenuation. is the angular resonance frequency.Nilsson and Riedel, p. 308.
For the case of the series RLC circuit these two parameters are given by:Agarwal and Lang, p. 641.
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.Agarwal and Lang, p. 646.
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.Irwin, pp. 217–220.
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 conditions 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.Nilsson and Riedel, pp. 287–288. 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 () isIrwin, p. 532.
The overdamped response is a decay of the transient current without oscillation.Agarwal and Lang, p. 648.