308

FREQUENCY RESPONSE, FILTERS, AND RESONANCE

[CHAP. 12

Fig. 12-64

12.41

Given HðsÞ ¼ ðs þ 1Þ=ðs2 þ 2s þp82Þ, ﬃﬃﬃﬃﬃ determine where jHð!Þj is at a maximum, its half-power bandwidth and quality factor. Ans: !0 ¼ 82  9 rad/s, ! ¼ 2 rad/s, Q ¼ 4:53

12.42

In a parallel RLC circuit R ¼ 10 k and L ¼ 20 mH. (a) Find C so that the circuit resonates at 1 MHz. Find the quality factor Q and the bandwidth in kHz. (b) Find the terminal voltage of the circuit if an AC current source of I ¼ 1 mA is applied to it at: (i) 1 MHz, (ii) 1.01 MHz, (iii) 1.006 MHz Ans: ðaÞ C ¼ 1:267 nF; Q ¼ 79:6; f ¼ 12:56 kHz; (b) V2 ¼ 10 V at 1 MHz, 5.34 V at 1.01 MHz, and 7.24 V at 1.006 MHz

12.43

A coil is modeled as a 50-mH inductor in series with a 5- resistor. Specify the value of a capacitor to be placed in series with the coil so that the circuit would resonate at 600 kHz. Find the quality factor Q and bandwidth f in kHz. Ans: C ¼ 1:4 nF; Q ¼ 37:7; f ¼ 15:9 kHz

12.44

The coil of Problem 12.43 placed in parallel with a capacitor C resonates at 600 kHz. Find C, quality factor Q, and bandwidth f in kHz. Hint: Find the equivalent parallel RLC circuit. Ans: C ¼ 1:4 nF; Q ¼ 37:7; f ¼ 15:9 kHz

12.45

The circuit in Fig. 12-65(a) is a third-order Butterworth low-pass ﬁlter. Find the network function, the magnitude of the frequency response, and its half-power cutoﬀ frequency !0 . Ans: HðsÞ ¼ 1=ðs3 þ 2s2 þ 2s þ 1Þ; jHð!Þj2 ¼ 1=ð1 þ !6 Þ; !0 ¼ 1 rad/s

Fig. 12-65

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An