CHAP. 12]

FREQUENCY RESPONSE, FILTERS, AND RESONANCE

299

Fig. 12-41

12.13 Obtain the bandwidth  for the circuit of Fig. 12-40 and plot  against the parameter Rx 

R1 R2 R1 þ R2

Here, the half-power frequencies are determined by the condition jHv ð!Þj ¼ 0:707jHv jmax , or, from Problem 12.12(a),       1 R 1 R1 !C  or Rx !C  ¼ 1þ 1 ¼ 1 R2 !L !L But (see Section 12.13) this is just the equation for the half-power frequencies of an Rx LC parallel circuit. Hence, ¼

!a 1 ¼ Qa CRx

The hyperbolic graph is shown in Fig. 12-42.

Fig. 12-42

12.14 In the circuit of Fig. 12-40, let R1 ¼ R2 ¼ 2 k , L ¼ 10 mH, and C ¼ 40 nF. Find the resonant frequency and bandwidth, and compare with the results for R1 ¼ 0 (i.e., a pure parallel circuit). 1 !a ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 5  104 rad=s ð10  103 Þð40  109 Þ or fa ¼ 7958 Hz. With Rx ¼ 22 =4 ¼ 1 k , Problem 12.13 gives ¼

1 ¼ 2:5  104 rad=s ð40  109 Þð1  103 Þ

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An