18.23 Frequency response and resonance in electrical systems

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

fig 1

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.

fig 2

From this picture, we see that

eqn 1

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 = 2πω (post 18.11).

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

eqn 1a

and

eqn 2

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

eqn 3

that is when

eqn 3a

(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

eqn 4

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 = 2πf (post 17.12), we define the resonant frequency of the system by

eqn 5

fig 3

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 = ω0. 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 ω0, 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

eqn 6

The higher the value of Q, the sharper the resonance peak. The appendix shows how to calculate the dependence of I on ω/ω0 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

eqn A1

Substituting these results into equation 2 and defining x = ω/ω0 gives the result that

eqn A2

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

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