### Resonance

As an example of the phasor technique, let us consider the interesting phenomenon known as resonance. A circuit is said to be resonant if its impedance shows either a pronounced maximum or a pronounced minimum at one particular frequency, known as the resonant frequency. This effect can be quite dramatic. It is accompanied by a maximum of energy stored in the circuit, and large currents or voltages may appear. The analogous mechanical effect is quite familiar; the resonance of a plucked string or tuning fork is essentially the same phenomenon. Another mechanical analogy is the case of a child being pushed in a swing. The swing has its own natural frequency (that at which the child oscillates when not being pushed); this is the swing’s resonant frequency. If you give the swing periodic pushes at a frequency not equal to the swing’s resonant frequency, not much motion of the swing will occur on the average. This is because your pushes sometimes add to the motion of the swing (if you happen to push when the swing is moving away from you) and sometimes subtract from the motion (if you happen to push when the swing is coming toward you). However, if the frequency of your pushes is equal to the resonant frequency you can contrive to always push when the swing is going away from you. (Imagine yourself pushing a child in a swing. You will

see that you do exactly this.) In this case each push adds to the motion, which tends to become larger and larger.

The most usual type of resonant circuit contains one inductance and one capacitance. (In fact, each circuit must also contain some resistance, but this will be neglected for the moment.) The type known as a parallel resonant circuit is shown in above diagram. The impedance that appears between terminals A and B of this parallel combination is

Often one is interested in the absolute value |z|. (For example, if a current source i0 were connected across terminals A,B, the amplitude of the voltage between the terminals would be |v| = |i0|·|Z|.) The absolute value of the parallel impedance given in Eq. (5.33) is

This function of ω is graphed in Fig. 5.16(b). We see that the function |z| has a singularity at the frequency ωR = 1VLC; its value appears to become infinite when co ωR’ This frequency ωR given by

is called the resonant frequency of the circuit.

In a practical circuit the impedance would not actually become infinite at any frequency: actual physical circuit elements would contain some resistance in their wires. For example, the windings of the inductor may contain substantial resistance. If this resistance is taken into consideration, the circuit has the form shown in Fig. 5.17(a). This circuit has already been considered in Examples 5.19 and 5.20. There it was found that the impedance between terminals A and B is

and thus the absolute value of the impedance is

This quantity is plotted in Fig. 5.17(b) for several cases corresponding to the same resonant frequency but to different values of R. We shall define a dimensionless parameter Q, known as the quality factor of the circuit (or, more frequently, simply as its “Q”), according to the formula 3

Q = ωRL/R

The curves of Fig. 5.17(b) are designated by their values of Q. We see that when Q is large (that is, when R is small) the curves have a sharply peaked form which approaches that of Fig. 5.16(b) as Q approaches infinity. On the other hand, when Q = 1 the curve becomes broad, and in fact the maximum of |z| occurs at a frequency slightly less than ωR’ As the value of Q increases, the width of the peak becomes “narrower.” Let us define the bandwidth of the resonant circuit as the range of frequencies over which |z| is larger than 1√2 times its maximum value. This definition is illustrated by Fig. 5.18.

The bandwidth of a high-Q circuit is given by BW = ωR/Q.

Then it can be shown from Eq. (5.37) that if Q >> 1, the bandwidth is given by

The above definition of bandwidth makes quantitative our previous statement that larger values of Q imply a “narrower” resonance curve.

#### Example

A parallel-resonant circuit of the type shown in Fig. 5.17(a) is to be used as a “filter” to remove a signal of an unwanted frequency. The circuit contains two current sources of equal magnitude but with different frequencies, 1 MHz and 1.01 MHz, as shown in above diagram. Assume that WR for the resonant circuit equals 1 MHz, that Q > > 1, and that C = 3 x 10 – 10 F. Find the value of R which will cause the amplitude of the 1.01-MHz voltage at the voltmeter to be 3 dB less than that of the 1-MHz voltage.

#### Solution

As can be seen from Fig. 5.18, we need a circuit for which the bandwidth extends approximately from 0.99 MHz to 1.01 MHz, so that the 1.01-MHz signal will lie at one end of the bandwidth and be attenuated by 1√2, or 3 dB. Thus we wish

BW = 0.02 MHz. From Eq. (5.39) we have

Q ≅ ωR/BW = 2π(1) / 2π(0.02) = 50

The value of Q is related to that of R by Eq. (5.38). However, before Eq. (5.38) can be used we must find L. We can do so by using the formula for resonant frequency, Eq. (5.35):

L = 1/ω2RC = 1 / (2π)2 (106)2 . 3 x 10-5 H

Thus we find that an inductor with resistance of 10.6 n or less is required to make the filter.

Another important resonant circuit is the series resonant circuit, shown in above diagram. Assuming for the moment that R = 0, the impedance of this connection is

Z = jωL + 1 / jωC = 1 – ω2/ω2R/jωC

(a) A series resonant circuit. (b) For the case of Q = cc (R = 0), the impedance vanishes at the resonant frequency, as shown by the sol id curve. For comparison the dashed curve shows the case of Q = 3.

where ωR, as before, is defined by ωR = livre. We see that for the series resonant circuit the impedance decreases to zero when ω = ωR. The absolute value of Z, as given by Eq. (5.40), is graphed as the solid curve in Fig. 5.20(b) as a function of frequency. Of course, resistance must also be present in the series resonant circuit, and in practice this prevents the impedance from actually becoming zero. However, provided that Q, as defined in Eq. (5.38), is much greater than unity, the absolute value of the series impedance will still have a well defined minimum at the resonant frequency. For comparison, a curve with Q = 3 is also shown in Fig. 5.20(b).