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A practical capacitor can be modeled by a parallel combination of R and C, as shown in Fig. 12.22. 2 2 The maximum stored energy is 1 CVmax and the energy dissipated per cycle is Vmax =R!. Thus, 2 Qcap !CR. The Q of the series RLC circuit is derived in Problem 12.6(a). It is usually applied at resonance, in which case it has the equivalent forms ! L 1 1 Q0 0 R !0 CR R r L C
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FREQUENCY RESPONSE, FILTERS, AND RESONANCE
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RLC PARALLEL CIRCUIT; PARALLEL RESONANCE Observe that V2 V1 . Under the open-circuit
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A parallel RLC network is shown in Fig. 12-23. condition, the input admittance is Yin !
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1 1 1 j!C R j!L Zin !
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The network will be in parallel resonance (or high-impedance resonance) when Yin ! , and thus Zin ! , is real (and so jYin ! j is a minimum and jZin ! j is a maximum); that is, when 1 ! !a  p LC p The symbol !a is now used to denote the quantity 1= LC in order to distinguish the resonance from a low-impedance resonance. Complex series-parallel networks may have several high-impedance resonant frequencies !a and several low-impedance resonant frequencies !0 . The normalized input impedance or Zin ! R 1   1 1 jR !C !L 1 !C 0 !L
is plotted (magnitude only) in Fig. 12-24. Half-power frequencies !l and !h are indicated on the plot. Analogous to series resonance, the bandwidth is given by !a Qa
where Qa , the quality factor of the parallel circuit at ! !a , has the equivalent expressions r R C !a RC R Qa !a L L See Problem 12.6(b).
FREQUENCY RESPONSE, FILTERS, AND RESONANCE
[CHAP. 12
Fig. 12-24
PRACTICAL LC PARALLEL CIRCUIT
A parallel LC tank circuit has frequency applications in electronics as a tuning or frequency selection device. While the capacitor may often be treated as pure C, the losses in the inductor should be included. A reasonable model for the practical tank is shown in Fig. 12-25. The input admittance is   1 R !L 2 j !C 2 Yin ! j!C R j!L R !L 2 R !L 2
Fig. 12-25
For resonance, ! L !a C 2 a R !a L 2 or 1 !a p LC r R2 C 1 L
At the resonant frequency, Yin !a RC=L and, from Section 12.11, the Q of the inductance at !a is Qind p If Qind ! 10, then !a % 1= LC and Zin !a 2 R % Qind The frequency response is similar to that of the parallel RLC circuit, except that the high-impedance resonance occurs at a lower frequency for low Qind . This becomes evident when the expression for !a above is rewritten as   1 1 !a p q LC 1 1=Q2 ind ! L a R r L 1 CR2
CHAP. 12]
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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 Rs j!Ls R2 !Ls 2 s and Yp 1 1 Rp j!Lp
Fig. 12-26
The results are  # !Ls 2 Rp Rs 1 Rs 1 Q2 s Rs "   #   Rs 2 1 Lp Ls 1 Ls 1 2 !Ls Qs  If Qs ! 10, Rp % Rs Q2 and Lp % Ls . s 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 nds "
Fig. 12-27
Rs
1 !Cp Rp " ! 1 1 Cs Cp 1 Cp 1 2 Qp !Cp Rp 2
Rp 1 Q2 p #
as the parallel-to-series transformation, and  Rp Rs 1
 1 Rs 1 Q2 s !Cs Rs 2 Cs Cs Cp 1 !Cs Rs 2 1 1=Qs 2 Again, the equivalence depends on the operating frequency.
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