FREQUENCY RESPONSE, FILTERS, AND RESONANCE
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
1 1 þ Rp j!Lp
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
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.