*Before you read this, I suggest you read* posts 17.44, 18.20 and 18.22.

The picture above shows a resistor (of resistance *R*, post 17.44), a capacitor (of capacitance *C*, post 18.20) and an inductor (of inductance *L*, post 18.22) connected together; I have used standard symbols to represent these components. There is a potential difference *V* (post 17.44) between P and P’, so that a current *I* flows through the three components.

So, how does *I* depend upon *V*? If *V* does not change with time, the answer is *I* = *V/R* (post 17.44). But if *V* is not constant, *C* and *L* also affect the magnitude of *I* and introduce a phase difference between *I* and *V* (posts 18.20 and 18.22). The combined effect of *C*, *L* and *R* is called impedance; since it affects the phase of *I*, it is a complex number (post 18.16) represented in the Argand plane by the picture below, for the reasons given in posts 18.20 and 18.22.

From this picture, we see that

Here, *i* represents the square root of -1 (post 18.16) and *ω* represents the angular frequency of the oscillating potential difference; *ω* is related to frequency, *f*, by *f* = 2π*ω* (post 18.11).

The modulus of *Z *(post 18.16) is then given by

and

Equation 2 shows that *I *has its maximum value when |*Z*| is as small as possible. Since *R *has a fixed value, this is when

that is when

(To obtain this result we need to do a bit of algebra; multiply every term in equation 3 by *ωC*, add 1 to both sides of the equation, divide both sides by *LC* and take the square root of both sides to get the result). The result is the value of *ω* that maximises the response of the system – we call it the *resonant angular frequency*, defined by

This is exactly he same idea as *resonance* in a mechanical system described in post 18.8. Since frequency, *f*, is related to *ω* by *ω* = 2π*f* (post 17.12), we define the resonant frequency of the system by

The graph above shows how *I* depends on *ω* (for a fixed value of *V*) for a range of *ω* values. Of course, *I* has its maximum value when *ω* = *ω*_{0}. Notice that, at low values of *ω*, *I* doesn’t change very much. This is because *V* isn’t changing very much, so the value of *I* is determined mostly by the value of *R*. When *ω* is much greater than *ω*_{0}, *I* is almost equal to zero. This enables a combination of an inductor and a capacitor to be chosen to filter out high frequency signals in an electrical system.

Why are the blue, green and red curves in the graph different? Each represents a different *sharpness* of the resonance: red is the sharpest shown and blue is the least sharp. The sharpness depends on the *quality factor* or *Q factor* of the resonance, where *Q* is defined by

The higher the value of *Q*, the sharper the resonance peak. The appendix shows how to calculate the dependence of *I* on *ω*/*ω*_{0} for different values of *Q*.

By changing the value of *L* or *C*, we can select the frequency of signal to which the system is most sensitive – this is called *tuning* the system. A conventional radio receiver has a circuit with a fixed value of *L* and a value of *C* that can be changed. By changing the value of *C* we can select the resonant frequency to correspond to the transmission frequency of the radio station we wish to listen to. Of course, the system needs to have a sufficiently high *Q *factor so that we don’t receive interference from signals with similar frequencies.

So capacitors and inductors enable us to remove high frequency components from an electrical signal (see post 18.14) and to tune systems to respond preferentially to a chosen frequency.

__Related posts__

18.14 Wave shapes – Fourier series

18.12 Vibrating strings

16.14 Aliasing

__Appendix__

This appendix explains how the graph in this post was calculated. From equations 4 and 6

Substituting these results into equation 2 and defining x = *ω*/*ω*_{0} gives the result that

which enables *I* to be plotted against *x*, for different values of *Q*, for a fixed value of *V*.