294

FREQUENCY RESPONSE, FILTERS, AND RESONANCE

and

L2 ¼

12.3

R1 50  103 pﬃﬃﬃ ¼ ¼ 91:9 mH 2fx 2ð50 3  106 Þ

1 pﬃﬃﬃ 1 þ ð50 3=f Þ2

0:90 ¼

ðbÞ

[CHAP. 12

or

f ¼ 179 MHz

A voltage divider, useful for high-frequency applications, can be made with two capacitors C1 and C2 in the generalized two-port network Fig. 12-2. Under open-circuit, ﬁnd C2 if C1 ¼ 0:01 mF and jHv j ¼ 0:20. From Table 12-1, Hv ¼

Hence,

Z2 ¼ Z1 þ Z2

0:20 ¼

1=j!C2 C1 ¼ 1 1 C1 þ C2 þ j!C1 j!C2

0:01 0:01 þ C2

or

C2 ¼ 0:04 mF

The voltage ratio is seen to be frequency-independent under open-circuit.

12.4

Find the frequency at which jHv j ¼ 0:50 for the low-pass RC network shown in Fig. 12-35. Hv ð!Þ ¼

1 1 þ jð!=!x Þ

where

!x 

1 R1 C2

Fig. 12-35

Then, and

12.5

1 from which 1 þ ð!=!x Þ2   pﬃﬃﬃ 1 !¼ 3 ¼ 8660 rad=s or R1 C2

pﬃﬃﬃ ! ¼ 3 !x

ð0:50Þ2 ¼

f ¼ 1378 Hz

For the series RLC circuit shown in Fig. 12-36, ﬁnd the resonant frequency !0 ¼ 2f0 . obtain the half-power frequencies and the bandwidth .   1 Zin ð!Þ ¼ R þ j !L  !C pﬃﬃﬃﬃﬃﬃﬃ At resonance, Zin ð!Þ ¼ R and !0 ¼ 1= LC . 1 !0 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 2236:1 rad=s 0:5ð0:4  106 Þ The power formula

f0 ¼

!0 ¼ 355:9 Hz 2

Also

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An