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

6-12 Passive, Active, and Digital Filtersis rational positive real, the degree of which cannot exceed that of W 1 (s), being at least two or four degreesless than that of Z(s), whereA 2 (s) ¼ p 4 s 2 þ js 0 j 2B 2 (s) ¼ p 2 sC 2 (s) ¼ p 3 sD 2 (s) ¼ p 1 s 2 þ js 0 j 2(6:50a)(6:50b)(6:50c)(6:50d)and {p 1 , p 2 , p 3 , p 4 } is the index set assigned to the point s 0 by the positive-real function W 1 (s). Solving forW 1 (s) in Equation 6.49 givesW 1 (s) ¼ A 2(s)W 2 (s) þ B 2 (s)C 2 (s)W 2 (s) þ D 2 (s)(6:51)which can be realized as the input impedance of a two-port network N 2 characterized by the transmissionmatrixT 2 (s) ¼ A 2(s) B 2 (s)C 2 (s) D 2 (s)(6:52)terminated in W 2 (s), as depicted in Figure 6.13.Consider the cascade connection of the two-port N 1 of Figure 6.3 and N 2 of Figure 6.13 terminated inW 2 (s), as shown in Figure 6.14. The transmission matrix T(s) of the overall two-port network N is simplythe product of the transmission matrices of the individual two-ports:T(s) ¼ T 1 (s)T 2 (s) (6:53)W 1 (s)N 2W 2 (s)FIGURE 6.13Realization of the impedance function W 1 (s).NZ(s) N 1N 2W 2 (s)W 1 (s)FIGURE 6.14 Cascade connection of two-port networks N 1 and N 2 .