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Heretofore, the frequency response of a network has been exhibited by plotting separately the magnitude and the angle of a suitable network function against frequency !. This same information can be presented in a single plot: one nds the curve (locus diagram) in the complex plane traced by the point representing the network function as ! varies from 0 to 1. In this section we shall discuss locus diagrams for the input impedance or the input admittance; in some cases the variable will not be !, but another parameter (such as resistance R). For the series RL circuit, Fig. 12-28(a) shows the Z-locus when !L is xed and R is variable; Fig. 1228(b) shows the Z-locus when R is xed and L or ! is variable; and Fig. 12-28(c) shows the Y-locus when R is xed and L or ! is variable. This last locus is obtained from Y 1 1 q tan 1 !L=R R j!L R2 !L 2
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Fig. 12-28
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Note that for !L 0, Y 1=R 08; and for !L ! 1, Y ! 0 908. When !L R, 1 Y p 458 R 2 A few other points will con rm the semicircular locus, with the center at 1/2R and the radius 1/2R. Either Fig. 12-28(b) or 12-28(c) gives the frequency response of the circuit. A parallel RC circuit has the Y- and Z-loci shown in Fig. 12-29; these are derived from Y 1 j!C R and R Z q tan 1 !CR 1 !CR 2
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For the RLC series circuit, the Y-locus, with ! as the variable, may be determined by writing Y G jB whence G R R2 X 2 1 R jX R jX R2 X 2 X B 2 R X2
Both G and B depend on ! via X. Eliminating X between the two expressions yields the equation of the locus in the form    2 G 1 2 2 1 G2 B2 or G B R 2R 2R which is the circle shown in Fig. 12-30. and ! !h . Note the points on the locus corresponding to ! !l , ! !0 ,
Fig. 12-30
For the practical tank circuit examined in Section 12.14, the Y-locus may be constructed by combining the C-branch locus and the RL-branch locus. To illustrate the addition, the points corresponding to frequencies !1 < !2 < !3 are marked on the individual loci and on the sum, shown in Fig. 12-31(c). It is seen that jYjmin occurs at a frequency greater than !a ; that is, the resonance is highimpedance but not maximum-impedance. This comes about because G varies with ! (see Section 12.14), and varies in such a way that forcing B 0 does not automatically minimize G2 B2 . The separation of
Fig. 12-31
FREQUENCY RESPONSE, FILTERS, AND RESONANCE
[CHAP. 12
the resonance and minimum-admittance frequencies is governed by the Q of the coil. Higher Qind corresponds to lower values of R. It is seen from Fig. 12-31(b) that low R results in a larger semicircle, which when combined with the YC -locus, gives a higher !a and a lower minimum-admittance frequency. When Qind ! 10, the two frequencies may be taken as coincident. The case of the two-branch RC and RL circuit shown in Fig. 12-32(a) can be examined by adding the admittance loci of the two branches. For xed V V 08, this amounts to adding the loci of the two branch currents. Consider C variable without limit, and R1 , R2 , L, and ! constant. Then current IL is xed as shown in Fig. 12-32(b). The semicircular locus of IC is added to IL to result in the locus of IT . Resonance of the circuit corresponds to T 0. This may occur for two values of the real, positive parameter C [the case illustrated in Fig. 12.32(b)], for one value, or for no value depending on the number of real positive roots of the equation Im YT C 0.
Fig. 12-32
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