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[CHAP. 12

the resonance and minimum-admittance frequencies is governed by the Q of the coil. Higher Qind corresponds to lower values of R. It is seen from Fig. 12-31(b) that low R results in a larger semicircle, which when combined with the YC -locus, gives a higher !a and a lower minimum-admittance frequency. When Qind  10, the two frequencies may be taken as coincident. The case of the two-branch RC and RL circuit shown in Fig. 12-32(a) can be examined by adding the admittance loci of the two branches. For fixed V ¼ V 08, this amounts to adding the loci of the two branch currents. Consider C variable without limit, and R1 , R2 , L, and ! constant. Then current IL is fixed as shown in Fig. 12-32(b). The semicircular locus of IC is added to IL to result in the locus of IT . Resonance of the circuit corresponds to T ¼ 0. This may occur for two values of the real, positive parameter C [the case illustrated in Fig. 12.32(b)], for one value, or for no value—depending on the number of real positive roots of the equation Im YT ðCÞ ¼ 0.

Fig. 12-32



The frequency scale of a filter may be changed by adjusting the values of its inductors and capacitors. Here we summarize the method (see also Section 8.10). Inductors and capacitors affect the frequency behavior of circuits through L! and C!; that is, always as a product of element values and the frequency. Dividing inductor and capacitor values in a circuit by a factor k will scale-up the !-axis of the frequency response by a factor k. For example, a 1-mH inductor operating at 1 kHz has the same impedance as a 1-mH inductor operating at 1 MHz. Similarly, a 1-mF capacitor at 1 MHz behaves similar to a 1-nF capacitor at 1 GHz. This is called frequency scaling and is a useful property of linear circuits. The following two examples illustrate its application in filter design. EXAMPLE 12.17 The network function of the circuit of Fig. 8-42 with R ¼ 2 k , C ¼ 10 nF, and R2 ¼ R1 is HðsÞ ¼

V2 2 ¼  2   V1 s s þ þ1 !0 !0

where !0 ¼ 50, 000 rad/s (see Examples 8.14 and 8.15). This is a low-pass filter with the cutoff frequency at !0 . By using a 1-nF capacitor, !0 ¼ 500,000 and the frequency response is scaled up by a factor of 10. EXAMPLE 12.18 A voltage source is connected to the terminals of a series RLC circuit. The phasor current is I ¼ Y  V, where YðsÞ ¼

Cs LCs2 þ RCs þ 1

pffiffiffiffiffiffiffi This is a bandpass function with a peak of the resonance pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi frequency of !0 ¼ 1= LC . Changing L and C to L=k and C=k (a reduction factor of k) changes 1= LC to k= LC and the new resonance frequency is increased to k!0 . You may verify the shift in frequency at which the current reaches its maximum by direct evaluation of Yð j!Þ for the following two cases: (a) L ¼ 1 mH, C ¼ 10 nF, !0 ¼ 106 rad/s; (b) L ¼ 10 mH, C ¼ 100 nF, !0 ¼ 105 rad/s.