Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

Passive Cascade Synthesis 6-3not greater than that of Z(s). It was first pointed out by Richards that if k can be chosen so that the evenpart of Z(s) vanishes at k, then the degree of W(s) is at least two less than that of Z(s). Lets 0 ¼ s 0 þ jv 0 (6:7)be a point in the closed RHS. Then, according to the preceding theorem, the functionW _ 1(s) ¼ s 0Z(s) sZðs 0 Þs 0 Zs ð 0 Þ sZ(s)(6:8)is positive real if s 0 is positive real; and the functionW 1 (s) ¼ Zs ð 0 ÞW _ 1ðs 0 Þ s 0W _ 1(s) sW _ 1ðs 0 Þs 0 W _ 1ðs 0 Þ sW _ 1(s)(6:9)is positive real if s 0 is a positive-real constant and W _ 1(s) is a positive-real function. Substituting Equation6.8 in Equation 6.9 yieldsW 1 (s) ¼ D 1(s)Z(s) B 1 (s)C 1 (s)Z(s) þ A 1 (s)(6:10)whereA 1 (s) ¼ q 4 s 2 þ js 0 j 2B 1 (s) ¼ q 2 sC 1 (s) ¼ q 3 sD 1 (s) ¼ q 1 s 2 þ js 0 j 2(6:11a)(6:11b)(6:11c)(6:11d)q 1 ¼ R 0=s 0 X 0 =v 0R 0 =s 0 þ X 0 =v 02jZ 0 j 2q 2 ¼R 0 =s 0 þ X 0 =v 0q 3 ¼(6:12a)(6:12b)2R 0 =s 0 X 0 =v 0(6:12c)q 4 ¼ R 0=s 0 þ X 0 =v 0R 0 =s 0 X 0 =v 0¼ 1 q 1(6:12d)in whichZs ð 0 Þ ¼ R 0 þ jX 0 Z 0 (6:13)In the case v 0 ¼ 0, then X 0 =v 0 must be replaced by Z 0 (s 0 ):X 0! Z 0 ðs 0 Þ ¼ dZ(s)v 0 dsFor s ¼ 0 and R 0 ¼ 0, R 0 =s 0 is replaced by X 0 (v 0 ):R 0! X 0 ðv 0 Þ ¼ dZ(s)s 0 dss¼s0s¼jv0(6:14a)(6:14b)