RLC circuit

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.

A circuit with a value of resistor that causes it to be just on the edge of ringing is called critically damped.

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

into the equation above yields: For the case where the source is an unchanging voltage, taking the time derivative and dividing by L leads to the following second order differential equation: This can usefully be expressed in a more generally applicable form: α and ω0 are both in units of angular frequency.

α 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.

[8] The differential equation for the circuit solves in three different ways depending on the value of ζ.

B1 and B2 (or B3 and the phase shift φ in the second form) are arbitrary constants determined by boundary conditions.

This consideration is important in control systems where it is required to reach the desired state as quickly as possible without overshooting.

[15] The series RLC can be analyzed for both transient and steady AC state behavior using the Laplace transform.

By the quadratic formula, we find The poles of Y(s) are identical to the roots s1 and s2 of the characteristic polynomial of the differential equation in the section above.

For an arbitrary V(t), the solution obtained by inverse transform of I(s) is: where ωr = √α2 − ω02, and cosh and sinh are the usual hyperbolic functions.

Taking the magnitude of the above equation with this substitution: and the current as a function of ω can be found from There is a peak value of |I(jω)|.

For the parallel circuit, the attenuation α is given by[18] and the damping factor is consequently Likewise, the other scaled parameters, fractional bandwidth and Q are also reciprocals of each other.

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 filtering and the Q is largely governed by this resistance.

The first evidence that a capacitor could produce electrical oscillations was discovered in 1826 by French scientist Felix Savary.

American physicist Joseph Henry repeated Savary's experiment in 1842 and came to the same conclusion, apparently independently.

[25][26] 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.

[23][25][26] 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.

[25] In 1857, German physicist Berend Wilhelm Feddersen photographed the spark produced by a resonant Leyden jar circuit in a rotating mirror, providing visible evidence of the oscillations.

[23][25][26] In 1868, Scottish physicist James Clerk Maxwell 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.

[23] The first example of an electrical resonance 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.

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 L. 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.

For a wider bandwidth, a larger value of the damping factor is required (and vice versa).

Under those conditions the bandwidth is[29] 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.

[30] For applications in oscillator circuits, it is generally desirable to make the attenuation (or equivalently, the damping factor) as small as possible.

However, for very low-attenuation circuits (high Q-factor), issues such as dielectric losses of coils and capacitors can become important.

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.

If the inductance L 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 R is known – capacitance: inductance (total of circuit and load): initial terminal voltage of capacitor: