298

FREQUENCY RESPONSE, FILTERS, AND RESONANCE

[CHAP. 12

12.10 A coil is represented by a series combination of L ¼ 50 mH and R ¼ 15 . Calculate the quality factor at (a) 10 kHz, (b) 50 kHz. Qcoil ¼

ðaÞ ðcÞ

!L 2ð10  103 Þð50  103 Þ ¼ ¼ 209 R 15   50 ¼ 1047 Qcoil ¼ 209 10

12.11 Convert the circuit constants of Problem 12.10 to the parallel form (a) at 10 kHz, (b) at 250 Hz. " Rp ¼ Rs 1 þ

ðaÞ

  # !Ls 2 ¼ Rs ½1 þ Q2s  ¼ 15½1 þ ð209Þ2  ¼ 655 k

Rs

or, since Qs 10, Rp  Rs Q2s ¼ 15ð209Þ2 ¼ 655 k .   1 Lp ¼ Ls 1 þ 2  Ls ¼ 50 mH Qs (b) At 250 Hz, 2ð250Þð50  103 Þ ¼ 5:24 15 2 Rp ¼ Rs ½1 þ Qs  ¼ 15½1 þ ð5:24Þ2  ¼ 426:9

    1 1 Lp ¼ Ls 1 þ 2 ¼ ð50  103 Þ 1 þ ¼ 51:8 mH Qs ð5:24Þ2 Qs ¼

Conversion of circuit elements from series to parallel can be carried out at a speciﬁc frequency, the equivalence holding only at that frequency. Note that in (b), where Qs < 10, Lp diﬀers signiﬁcantly from Ls .

12.12 For the circuit shown in Fig. 12-40, (a) obtain the voltage transfer function Hv ð!Þ, and (b) ﬁnd the frequency at which the function is real.

Fig. 12-40 (a) Let Z2 and Y2 represent the impedance and admittance of the R2 LC parallel tank. Hv ð!Þ ¼

¼

Z2 1 ¼ ¼ R1 þ Z2 1 þ R1 Y2

1  1 1 þ j!C þ R2 j!L

 1 þ R1

1   R1 1 þ jR1 !C  1þ R2 !L

(b) The transfer function is real when Y2 is real; that is, when 1 ! ¼ !a  pﬃﬃﬃﬃﬃﬃﬃ LC At ! ¼ !a , not only are jZ2 j and jHv j maximized, but jZin j ¼ jR1 þ Z2 | also is maximized (because R1 is real and positive—see the locus diagram, Fig. 12-41).

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An