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

Design of Resistively Terminated Networks 9-7The values of the final elements can also be calculated directly byL n ¼ 2R 2 sin p=2n(1 þ a)v c, n odd (9:38a)C n ¼2 sin p=2nR 2 (1 þ a)v c, n even (9:38b)A complete derivation of these formulas was first given by Bossé [1]. Hence we can calculate the elementvalues starting from either the first or the last element. When R 1 ¼ R 2 , formulas Equation 9.36 reduce toL 2m 1 ¼ 2R 1 sin g 4m 3v cC 2m ¼ 2 sin g 4m 1R 1 v c(9:39a)(9:39b)Example 9.1GivenR 1 ¼ 70 V, R 2 ¼ 200 V, v c ¼ 10 5 rad=s, n ¼ 4 (9:40)obtain a Butterworth LC ladder to meet these specifications.Since R 2 > R 1 , we choose the plus sign in Equation 9.32 and obtain r 11 (0) 0, a ¼ 0.833, andg m ¼ 22.5m. Thus, from Equations 9.35 and 9.36 the element values are found to beL 1 ¼2 70 sin 22:5¼ 3:2081 mH (9:41a)(1 0:833) 105 4 sin 22:5 sin 67:5 C 2 ¼L 1 ð1:69391:666 cos 45 ¼ 0:085456 mF (9:41b)Þ1010 4 sin 67:5 sin 112:5 L 3 ¼¼ 2:3587 mH (9:41c)C 2 ð1:69391:666 cos 90 Þ1010 4 sin 112:5 sin 157:5 C 4 ¼¼ 0:020877 mF (9:41d)L 3 ð1:69391:666 cos 135 Þ1010 Alternatively, C 4 can be computed directly from Equation 9.39b asC 4 ¼2 sin 22:5 ¼ 0:020877 mF (9:42)200 (1 þ 0:833) 105 The ladder network together with its termination is presented in Figure 9.3. This network possesses thefourth-order Butterworth transducer power-gain response with a DC gainK 4 ¼ 1 a 8 ¼ 0:7682 (9:43)