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

13-16 Passive, Active, and Digital FiltersG 6+–C 5G 3G 2G 4jω++V i–C 1+G 1V oσ––(a)FIGURE 13.20Second-order Cauer low-pass building block. (a) The circuitry. (b) The pole and zero locations.(b)This filter circuit has a voltage transfer ratioH(s) ¼ V o¼ s2 C 1 G 3 C 5 þ G 6 ðG 2 G 4 þ G 1 G 4 ÞV i s 2 C 1 G 3 C 5 þ sG 1 G 3 C 5 þ G 2 G 4 G 6s 2 þ a 43ð1 þ a 12 ÞT¼12 T 56s 2 þ s þ a 43T 11 T 12 T 56s 2 þ v 2 z¼s 2 þ sv p =Q p þ v 2 p(13:50)The magnitude v z of the zeros is always greater than the magnitude v p of the poles. Additionally, withoutany matching of elements, the real part of the zeros is always zero.The design can proceed in the following steps. First we predefine a 43 ¼ 1 and choose due to practicalconsiderations the values of the resistors R 3 ¼ R 4 . We also select the two capacitors C 1 and C 5 . Then,comparing the first coefficient in the denominator of H(s) in Equation 13.50, namely, T 11 ¼ Q p =v p withT 11 ¼ C 1 R 1 , we determine the resistorR 1 ¼Q pv p C 1(13:51)Further, comparing the last coefficients in the numerator and denominator yieldsv 2 zv 2 p¼ 1 þ a 12 (13:52)