Page 308

CHAP. 12]

FREQUENCY RESPONSE, FILTERS, AND RESONANCE

297

The values of the reactances at other frequencies are readily obtained. A tabulation of reactances and impedances appear in Fig. 12-39(a), and Fig. 12-39(b) shows the required plots. !

XL

XC

3200

16

25

10  j9

13:4 428

3600

18

22.2

10  j4:2

10:8 22:88

4000

20

20

10

10 08

4400

22

18.2

10 þ j3:8

10:7 20:88

4800

24

16.7

10 þ j7:3

12:4 36:28

Z

(a)

Fig. 12-39

12.8

Show that !0 ¼

pffiffiffiffiffiffiffiffiffiffi !l !h for the series RLC circuit.

By the results of Problem 12.5, 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 10sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2  2 R 1 R R 1 R 1  A@ þ A¼ ¼ !20 !l !h ¼ @ þ þ 2L LC 2L 2L LC 2L LC

12.9

Compute the quality factor of an RLC series circuit, with R ¼ 20 , L ¼ 50 mH, and C ¼ 1 mF, using (a) Q ¼ !0 L=R, (b) Q ¼ 1=!0 CR, and (c) Q ¼ !0 =. 1 !0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 4472 rad=s 0:05  106 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2 R R 1 R R 1 þ ¼ 4276:6 rad=s !h ¼ þ ¼ 4676:6 rad=s !l ¼  þ þ 2L 2L LC 2L 2L LC and  ¼ !h  !l ¼ 400 rad/s. ðaÞ ðbÞ ðcÞ

!0 L 4472ð0:050Þ ¼ ¼ 11:2 R 20 1 1 Q¼ ¼ ¼ 11:2 !0 CR 4472ð106 Þ20 Q¼

!0 4472 ¼ 11:2 ¼ 400 

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