Page 300

CHAP. 12]

12.15

FREQUENCY RESPONSE, FILTERS, AND RESONANCE

289

SERIES-PARALLEL CONVERSIONS

It is often convenient in the analysis of circuits to convert the series RL to the parallel form (see Fig. 12-26). Given Rs , Ls , and the operating frequency !, the elements Rp , Lp of the equivalent parallel circuit are determined by equating the admittances Ys ¼

Rs  j!Ls R2s þ ð!Ls Þ2

and

Yp ¼

1 1 þ Rp j!Lp

Fig. 12-26

The results are "

 # !Ls 2 Rp ¼ Rs 1 þ ¼ Rs ð1 þ Q2s Þ Rs "   #   Rs 2 1 Lp ¼ Ls 1 þ ¼ Ls 1 þ 2 !Ls Qs 

If Qs  10, Rp  Rs Q2s and Lp  Ls . There are times when the RC circuit in either form should be converted to the other form (see Fig. 12-27). Equating either the impedances or the admittances, one finds

Fig. 12-27

Rs ¼

Rp 2

¼

Rp 1 þ Q2p #

1 þ ð!Cp Rp Þ " ! 1 1 Cs ¼ Cp 1 þ ¼ Cp 1 þ 2 Qp ð!Cp Rp Þ2

as the parallel-to-series transformation, and  Rp ¼ Rs 1 þ

 1 ¼ Rs ð1 þ Q2s Þ ð!Cs Rs Þ2 Cs Cs Cp ¼ ¼ 1 þ ð!Cs Rs Þ2 1 þ ð1=Qs Þ2

as the series-to-parallel transformation.

Again, the equivalence depends on the operating frequency.

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