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CHAP. 12]



For the RLC series circuit, the Y-locus, with ! as the variable, may be determined by writing Y ¼ G þ jB ¼ whence

R R2 þ X 2

1 R  jX ¼ R þ jX R2 þ X 2 X B¼ 2 R þ 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 þ B2 ¼ or G þB ¼ R 2R 2R which is the circle shown in Fig. 12-30. and ! ¼ !h .

Note the points on the locus corresponding to ! ¼ !l , ! ¼ !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 ¼ 0 does not automatically minimize G2 þ B2 . The separation of

Fig. 12-31