296

FREQUENCY RESPONSE, FILTERS, AND RESONANCE

[CHAP. 12

Fig. 12-37 (b) For the parallel combination with applied voltage vðtÞ, 1 1 2 q Ws ¼ LIL2 þ 2 2C C dWs di q ¼ LiL L þ C iC ¼ vðiL þ iC Þ ¼ 0 dt dt C

and If v ¼ 0, then qC ¼ 0 and

iL ¼ IL max ¼ 

giving

Wsjv¼0 ¼

Vmax !L

2 Vmax 2L!2

If iL þ iC ¼ 0, then (see Fig. 12-38) iL ¼ iC ¼ 0 and qC ¼ CVmax , giving 2 Ws jiL þiC ¼0 ¼ 12 CVmax

Fig. 12-38 ( Therefore

Ws max ¼

2 =2L!2 Vmax

ð! !a Þ

2 =2 CVmax

ð!  !a Þ

2 Vmax =R!.

The energy dissipated per cycle in R is Wd ¼  Ws max R=L! ¼ Q ¼ 2 !CR Wd

12.7

Consequently,  ð! !a Þ ð!  !a Þ

A three-element series circuit contains R ¼ 10 , L ¼ 5 mH, and C ¼ 12:5 mF. Plot the magnitude and angle of Z as functions of ! for values of ! from 0.8 !0 through 1:2 !0 . pﬃﬃﬃﬃﬃﬃﬃ !0 ¼ 1= LC ¼ 4000 rad/s.

At !0 ,

XL ¼ ð4000Þð5  103 Þ ¼ 20

XC ¼

1 ¼ 20

ð4000Þð12:5  106 Þ

Z ¼ 10 þ jðXL  XC Þ ¼ 10 þ j0

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An