CHAP. 12]

FREQUENCY RESPONSE, FILTERS, AND RESONANCE

295

Fig. 12-36 2 P ¼ Ieff R¼

2 Veff R jZin j2

2 shows that Pmax ¼ Veff =R, achieved at ! ¼ !0 , and that P ¼ 12 Pmax when jZin j2 ¼ 2R2 ; that is, when

!L 

1 ¼ R !C

or

!2 

R 1 ! ¼0 L LC

Corresponding to the upper sign, there is a single real positive root: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ  2 R R 1 þ ¼ 2338:3 rad=s or þ !h ¼ 2L 2L LC

fh ¼ 372:1 Hz

and corresponding to the lower sign, the single real positive root sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ  2 R R 1 þ ¼ 2138:3 rad=s or þ !l ¼  2L 2L LC

12.6

fl ¼ 340:3 Hz

Derive the Q of (a) the series RLC circuit, (b) the parallel RLC circuit. (a) In the time domain, the instantaneous stored energy in the circuit is given by 1 q2 Ws ¼ Li2 þ 2 2C For a maximum,   dWs di q dq di q ¼i L þ ¼ iðvL þ vC Þ ¼ 0 ¼ Li þ dt C dt dt C dt Thus, the maximum stored energy is Ws at i ¼ 0 or Ws at vL þ vC ¼ 0, whichever is the larger. Now the capacitor voltage, and therefore the charge, lags the current by 908; hence, i ¼ 0 implies q ¼ Qmax and  2 Q2 1 1 I I2 Ws ji¼0 ¼ max ¼ CVC2 max ¼ C max ¼ max2 2 2 2C !C 2C! On the other hand, vL þ vC ¼ 0 implies vL ¼ vC ¼ 0 and i ¼ Imax (see the phasor diagram, Fig. 1237), so that 2 Ws jvL þvC ¼0 ¼ 12 LImax

It follows that Ws max ¼

8 2 < Imax =2C!2

ð! !0 Þ

:

ð!  !0 Þ

2 LImax =2

2 R=!. The energy dissipated per cycle (in the resistor) is Wd ¼ Imax  W 1=!CR ð! !0 Þ Q ¼ 2 s max ¼ !L=R ð!  !0 Þ Wd

Consequently,

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An