RLC circuit

An RLC circuit is an consisting of a  (R), an  (L), and a  (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 for current, and  in a similar way as an. Introducing the resistor increases the decay of these oscillations, which is also known as. The resistor also reduces the peak resonant frequency. In ordinary conditions, some resistance is unavoidable even if a resistor is not specifically included as a component; an ideal, pure LC circuit exists only in the domain of, a physical effect demonstrated to this point only at temperatures far below ambient temperatures found anywhere on the Earth's surface.

RLC circuits have many applications as s. and s use them for  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, , or. 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 in circuit analysis.

The three circuit elements, R, L and C, can be combined in a number of different. 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.

Resonance
An important property of this circuit is its ability to resonate at a specific frequency, the, $f_{0}$. Frequencies are measured in units of. In this article,, $ω_{0}$, is used because it is more mathematically convenient. This is measured in s per second. They are related to each other by a simple proportion,


 * $$\omega_0 = 2 \pi f_0 \,.$$

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 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, 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 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


 * $$ \omega_0 = \frac{1}{\sqrt{L\,C~}} \,~.$$

This is exactly the same as the resonance frequency of an LC circuit, that is, one with no resistor present. The resonant frequency for an RLC circuit is the same as a circuit in which there is no damping, hence undamped resonance frequency. The peak resonance frequency, on the other hand, depends 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. Either side of critically damped are described as (ringing happens) and  (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 $$\omega_0 = 1/\sqrt{L\,C~}$$, and for those the undamped resonance frequency, damped resonance frequency and driven resonance frequency can all be different.

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 which will resonate in this way are described as underdamped and those that will not are overdamped. Damping attenuation (symbol $α$) is measured in s per second. However, the unitless (symbol $ζ$, zeta) is often a more useful measure, which is related to $α$ by


 * $$ \zeta = \frac {\alpha}{\omega_0} \,.$$

The special case of $ζ = 1$ 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. The bandwidth is measured between the, 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


 * $$ \Delta \omega = \omega_2 - \omega_1 \,,$$

where $Δω$ is the bandwidth, $ω_{1}$ is the lower half-power frequency and $ω_{2}$ is the upper half-power frequency. The bandwidth is related to attenuation by


 * $$ \Delta \omega = 2 \alpha \,,$$

where the units are radians per second and 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


 * $$ B_\mathrm{f} = \frac {\Delta \omega}{\omega_0} \,.$$

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, requires low damping. A wide band filter requires high damping.

$Q$ factor
The 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-$Q$ circuits are therefore damped and lossy and high-$Q$ circuits are underdamped. $Q$ is related to bandwidth; low-$Q$ circuits are wide-band and high-$Q$ circuits are narrow-band. In fact, it happens that $Q$ is the inverse of fractional bandwidth


 * $$ Q = \frac{1}{B_\mathrm{f}} = \frac {\omega_0}{\Delta \omega} \,.$$

$Q$ factor is directly proportional to, as the $Q$ factor depends inversely on bandwidth.

For a series resonant circuit, the $Q$ factor can be calculated as follows:


 * $$ Q = \frac {1}{\omega_0 R C} = \frac {\omega_0 L}{R} = \frac{1}{R} \sqrt\frac{L}{C} \,.$$

Scaled parameters
The parameters $Q$, $F_{b}$, and $ζ$ are all scaled to $ω_{0}$. 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 of each circuit.

Series circuit
In this circuit, the three components are all in series with the. The governing can be found by substituting into  (KVL) the  for each of the three elements. From the KVL,


 * $$V_R + V_L + V_C = V(t)\,,$$

where $Q$, $V$ and $I$ are the voltages across R, L and C respectively and $V(t)$ is the time-varying voltage from the source.

Substituting $$V_R=RI(t)$$, $$V_L=L{dI(t) \over dt}$$ and $$V_C=V(0)+ \frac{1}{C} \int_{0}^t I(\tau)\, d\tau $$ into the equation above yields:


 * $$RI(t) + L \frac{dI(t)}{dt} +V(0)+ \frac{1}{C} \int_{0}^t I(\tau)\, d\tau = V(t)\,.$$

For the case where the source is an unchanging voltage, taking the time derivative and dividing by $R$ leads to the following second order differential equation:


 * $$\frac{d^2}{dt^2}I(t) + \frac{R}{L} \frac{d}{dt}I(t) + \frac{1}{LC} I(t) = 0\,.$$

This can usefully be expressed in a more generally applicable form:


 * $$\frac{d^2}{dt^2}I(t) + 2 \alpha \frac{d}{dt}I(t) + \omega_0^2 I(t) = 0\,.$$

$L$ and $ω_{0}$ are both in units of. $C$ is called the neper frequency, or attenuation, and is a measure of how fast the 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 s per second, neper being a unit of attenuation. $ω_{0}$ is the angular resonance frequency.

For the case of the series RLC circuit these two parameters are given by:


 * $$\begin{align}

\alpha &= \frac{R}{2L} \\ \omega_0 &= \frac{1}{\sqrt{LC}}\,. \end{align}$$

A useful parameter is the damping factor, $V_{R}$, which is defined as the ratio of these two; although, sometimes $V_{L}$ is referred to as the damping factor and $V_{C}$ is not used.


 * $$ \zeta = \frac{\alpha}{\omega_0} \,.$$

In the case of the series RLC circuit, the damping factor is given by


 * $$\zeta = \frac{R}{2} \sqrt\frac{C}{L}\,.$$

The value of the damping factor determines the type of transient that the circuit will exhibit.

Transient response
The differential equation for the circuit solves in three different ways depending on the value of $L$. These are underdamped ($L = 1$), overdamped ($C = 1$) and critically damped ($ω_{0} = 1$). The differential equation has the ,


 * $$ s^2 + 2 \alpha s + \omega_0^2 = 0 \,.$$

The roots of the equation in $α$ are,


 * $$\begin{align}

s_1 &= -\alpha +\sqrt {\alpha^2 - \omega_0^2} \\ s_2 &= -\alpha -\sqrt {\alpha^2 - \omega_0^2} \,. \end{align}$$

The general solution of the differential equation is an exponential in either root or a linear superposition of both,


 * $$ I(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t} \,.$$

The coefficients $ζ < 1$ and $ζ > 1$ are determined by the 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.

Overdamped response
The overdamped response ($ζ = 1$) is


 * $$I(t) = A_1 e^{-\omega_0 \left ( \zeta + \sqrt {\zeta^2 - 1} \right ) t} + A_2 e^{-\omega_0 \left ( \zeta - \sqrt {\zeta^2 - 1} \right ) t} \,.$$

The overdamped response is a decay of the transient current without oscillation.

Underdamped response
The underdamped response ($A_{1}$) is


 * $$ I(t) = B_1 e^{-\alpha t} \cos (\omega_\mathrm{d} t) + B_2 e^{-\alpha t} \sin (\omega_\mathrm{d} t) \,.$$

By applying standard the two trigonometric functions may be expressed as a single sinusoid with phase shift,


 * $$ I(t) = B_3 e^{-\alpha t} \sin (\omega_\mathrm{d} t + \varphi) \,.$$

The underdamped response is a decaying oscillation at frequency $A_{2}$. The oscillation decays at a rate determined by the attenuation $ζ > 1$. The exponential in $ζ < 1$ describes the of the oscillation. $ω_{d}$ and $α$ (or $α$ and the phase shift $α$ in the second form) are arbitrary constants determined by boundary conditions. The frequency $B_{1}$ is given by


 * $$ \omega_\mathrm{d} = \sqrt { \omega_0^2 - \alpha^2 } = \omega_0 \sqrt {1 - \zeta^2} \,.$$

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, $B_{2}$, 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 ($B_{3}$) is


 * $$ I(t) = D_1 t e^{-\alpha t} + D_2 e^{-\alpha t} \,.$$

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. $ω_{d}$ and $ω_{0}$ 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. If the voltage source above produces a waveform with Laplace-transformed $ζ = 1$ (where $ζ$ is the $D_{1}$), the  can be applied in the Laplace domain:


 * $$V(s) = I(s) \left ( R + Ls + \frac{1}{Cs} \right )\,, $$

where $D_{2}$ is the Laplace-transformed current through all components. Solving for $α$:


 * $$I(s) = \frac{1}{ R + Ls + \frac{1}{Cs} } V(s)\,. $$

And rearranging, we have


 * $$I(s) = \frac{s}{ L \left ( s^2 + \frac{R}{L}s + \frac{1}{LC} \right ) } V(s)\,.$$

Laplace admittance
Solving for the Laplace $V(s)$:


 * $$ Y(s) = \frac{ I(s) }{ V(s) } = \frac{s}{ L \left ( s^2 + \frac{R}{L}s + \frac{1}{LC} \right ) }\,. $$

Simplifying using parameters $ζ$ and $s = σ + jω$ defined in the previous section, we have


 * $$ Y(s) = \frac{ I(s) }{ V(s) } = \frac{s}{ L \left ( s^2 + 2 \alpha s + \omega_0^2 \right ) }\,. $$

Poles and zeros
The of $I(s)$ are those values of $ζ$ such that $Y(s)$:


 * $$ s = 0 \quad \mbox{and} \quad |s| \rightarrow \infty \,.$$

The of $ω_{0}$ are those values of $Y(s)$ such that $Y(s) = 0$. By the, we find


 * $$ s = - \alpha \pm \sqrt{\alpha^2 - \omega_0^2} \,.$$

The poles of $Y(s)$ are identical to the roots $s$ and $Y(s) → ∞$ of the characteristic polynomial of the differential equation in the section above.

General solution
For an arbitrary $Y(s)$, the solution obtained by inverse transform of $s_{1}$ is:


 * In the underdamped case, $s_{2}$:
 * $$I(t) = \frac{1}{L}\int_0^t V(t - \tau) e^{-\alpha\tau} \left( \cos \omega_\mathrm{d}\tau - \frac{\alpha}{\omega_\mathrm{d}} \sin \omega_\mathrm{d}\tau \right ) \, d\tau\,,$$
 * In the critically damped case, $V(t)$:
 * $$I(t) = \frac{1}{L}\int_0^t V(t - \tau) e^{-\alpha\tau} ( 1 - \alpha \tau ) \, d\tau\,,$$
 * In the overdamped case, $I(s)$:
 * $$I(t) = \frac{1}{L}\int_0^t V(t - \tau) e^{-\alpha\tau} \left ( \cosh \omega_\mathrm{r}\tau - { \alpha \over \omega_\mathrm{r} } \sinh \omega_\mathrm{r}\tau \right ) \, d\tau\,,$$

where $ω_{0} > α$, and $ω_{0} = α$ and $ω_{0} < α$ are the usual s.

Sinusoidal steady state
Sinusoidal steady state is represented by letting $ω_{r} = √α^{2} − ω_{0}^{2}$, where $s$ is the. Taking the magnitude of the above equation with this substitution:


 * $$ \big| Y(j \omega) \big| = \frac{1}{\sqrt{ R^2 + \left ( \omega L - \frac{1}{\omega C} \right )^2 }}\,. $$

and the current as a function of $φ$ can be found from


 * $$\big| I( j \omega ) \big| = \big| Y(j \omega) \big| \cdot \big| V(j \omega) \big|\,.$$

There is a peak value of $cosh$. The value of $s$ at this peak is, in this particular case, equal to the undamped natural resonance frequency:


 * $$\omega_0 = \frac{1}{\sqrt{LC}}\,. $$

Parallel circuit
The properties of the parallel RLC circuit can be obtained from the of electrical circuits and considering that the parallel RLC is the  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 $I(s)$ is given by


 * $$ \alpha = \frac{1}{\,2\,R\,C\,}$$

and the damping factor is consequently


 * $$\zeta = \frac{1}{\,2\,R\,} \sqrt{\frac{L}{C}~}\,~.$$

Likewise, the other scaled parameters, fractional bandwidth and $α$ are also reciprocals of each other. This means that a wide-band, low-$s$ circuit in one topology will become a narrow-band, high-$j$ circuit in the other topology when constructed from components with identical values. The fractional bandwidth and $ω$ of the parallel circuit are given by
 * $$\begin{align} B_\mathrm{f} &= \frac{1}{\,R\,} \sqrt{\frac{L}{C}~} \\ Q &= R \sqrt{\frac{C}{L}~}\,~. \end{align}$$

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:


 * $$\begin{align} \frac{1}{\,Z\,} &= \frac{1}{\,Z_L\,} + \frac{1}{\,Z_C\,}+\frac{1}{\,Z_R\,} \\ &= \frac{1}{\,j\,\omega\,L\,} + j\,\omega\,C + \frac{1}{\,R\,}\,.\end{align}$$

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 $$~\omega_0 = 1/\sqrt{\,L\,C~}~$$ 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. Parallel LC circuits are frequently used for ing and the $ω$ is largely governed by this resistance. The resonant frequency of this circuit is


 * $$\omega_0 = \sqrt {\frac{1}{LC} - \left( \frac{R}{L} \right)^2}\,.$$

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)


 * $$ s^2 + 2 \alpha s + {\omega'_0}^2 = 0 $$

is not the same frequency. In this case it is the natural undamped resonant frequency:


 * $$ \omega'_0 = \sqrt \frac{1}{LC} \,.$$

The frequency $sinh$ at which the impedance magnitude is maximum is given by
 * $$ \omega_\mathrm{m} = \omega'_0\sqrt{-\frac{1}{Q_L^2} + \sqrt{1 + \frac{2}{Q_L^2}}} \,,$$

where $s = jω$ is the of the coil. This can be well approximated by
 * $$ \omega_\mathrm{m} \approx \omega'_0 \sqrt{1 - \frac{1}{2Q^4_L} }\,. $$

Furthermore, the exact maximum impedance magnitude is given by
 * $$|Z|_\mathrm{max} = RQ_L^2 \sqrt{\frac{1}{2Q_L\sqrt{Q_L^2 + 2} - 2Q_L^2 - 1}}\,. $$

For values of $V$ greater than unity, this can be well approximated by
 * $$|Z|_\mathrm{max} \approx {RQ_L^2}\,.$$

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


 * $$ \omega_0 = \sqrt {\frac{1}{LC}-\frac{1}{(RC)^2}} \,,$$

while the frequency $|I(jω)|$ at which the impedance magnitude is maximum is given by
 * $$ \omega_\mathrm{m} =\omega'_0\sqrt{-\frac{1}{Q_C^2}+\sqrt{1+\frac{2} {Q_C^2}}} \,,$$

where $R = 1$.

History
The first evidence that a capacitor could produce electrical oscillations was discovered in 1826 by French scientist. He found that when a 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 repeated Savary's experiment in 1842 and came to the same conclusion, apparently independently. British scientist (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, 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 photographed the spark produced by a resonant Leyden jar circuit in a rotating mirror, providing visible evidence of the oscillations. In 1868, Scottish physicist 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 curve was published in 1887 by German physicist  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 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.

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 which allows the value of $I$ to be changed and tune to stations on different frequencies. For the in the radio where the tuning is preset in the factory, the more usual solution is an adjustable core in the inductor to adjust $R$. In this design, the core (made of a high 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 $L$, $C$ and $α$ to convenient practical values. Alternatively, $Q$ 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


 * $$\omega_\mathrm{c} = \frac{1}{\sqrt {LC}} \,.$$

This is also the bandwidth of the filter. The damping factor is given by


 * $$\zeta = \frac {1}{2R_L} \sqrt {\frac{L}{C}} \,.$$

High-pass filter
A high-pass filter is shown in Figure 7. The corner frequency is the same as the low-pass filter:


 * $$ \omega_\mathrm{c} = \frac{1}{\sqrt {LC}} \,.$$

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


 * $$ \omega_\mathrm{c} = \frac{1}{\sqrt {LC}} \,,$$

and the bandwidth for the series circuit is


 * $$ \Delta \omega = \frac {R_L}{L} \,.$$

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


 * $$ \Delta \omega = \frac {1}{C R_L} \,.$$

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.

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 $Q$ as small as physically possible for a series circuit, or alternatively increasing $Q$ to as much as possible for a parallel circuit. In either case, the RLC circuit becomes a good approximation to an ideal. However, for very low-attenuation circuits (high $Q$-factor), issues such as dielectric losses of coils and capacitors can become important.

In an oscillator circuit


 * $$ \alpha \ll \omega_0 \,,$$

or equivalently


 * $$ \zeta \ll 1 \,. $$

As a result,


 * $$ \omega_\mathrm{d} \approx \omega_0 \,. $$

Voltage multiplier
In a series RLC circuit at resonance, the current is limited only by the resistance of the circuit


 * $$ I = \frac{V}{R}\,. $$

If $Q$ 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


 * $$ V_L = \frac{V}{R} \omega_0 L \,.$$

An equal magnitude voltage will also be seen across the capacitor but in antiphase to the inductor. If $Q_{L}$ can be made sufficiently small, these voltages can be several times the input voltage. The voltage ratio is, in fact, the $C$ of the circuit,


 * $$ \frac{V_L}{V} = Q \,.$$

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
 * $$ I(t) = I_0\left(\,e^{-\alpha\,t} - e^{-\beta\,t}\right) \,.$$

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, $C = 1$, and there is no current flowing in the inductor. If the inductance $L$ is known, then the remaining parameters are given by the following – capacitance:
 * $$ C = \frac{1}{~L\,\alpha\,\beta\,~} \,,$$

resistance (total of circuit and load):
 * $$ R = L\,(\,\alpha + \beta\,) \,,$$

initial terminal voltage of capacitor:
 * $$ V_0 = -I_0 L\,\alpha\,\beta\,\left(\frac{1}{\beta} - \frac{1}{\alpha}\right) \,.$$

Rearranging for the case where $R$ is known – capacitance:
 * $$ C = \frac{~\alpha + \beta~}{R\,\alpha\,\beta} \,,$$

inductance (total of circuit and load):
 * $$ L = \frac{R}{\,\alpha + \beta~} \,,$$

initial terminal voltage of capacitor:
 * $$ V_0 = \frac{\,-I_0 R\,\alpha\,\beta~}{\alpha + \beta} \left(\frac{1}{\beta} - \frac{1}{\alpha}\right) \,.$$