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

Design of Resistively Terminated Networks 9-11In addition, the values of the last elements can also be computed directly by the equationsL n ¼C n ¼2R 2 sin p=2n, n odd (9:58a)ðsinh a þ sinh ^a Þv c2 sin p=2n, n even (9:58b)R 2 ðsinh a þ sinh ^a Þv cA formal proof of these formulas was first given by Takahasi [2]. Hence, we can calculate the elementvalues starting from either the first or the last element.Example 9.2GivenR 1 ¼ 150 V, R 2 ¼ 470 V, v c ¼ 10 8 p rad=s, n ¼ 4 (9:59)find a Chebyshev LC ladder network to meet these specifications with peak-to-peak ripple in thepassband not exceeding 1.5 dB.Since R 1 and R 2 are both specified, the minimum passband gain G min is fixed by Equation 9.53b as0 12470G min K 1n1 þ e 2 ¼ 1 B150C@470A¼ 0:7336 (9:60)150 þ 1For the 1.5 dB ripple in the passband, the corresponding ripple factor is given bye ¼p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10 0:15 1 ¼ 0:64229 (9:61)obtaining K 4 ¼ 1.036, which is too large for the network to be physically realizable. Thus, let K 4 ¼ 1, themaximum permissible value, and the corresponding ripple factor becomesWe next compute the quantitiesrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11e ¼ 1 ¼1 ¼ 0:6026 or 1:345 dB 1:5 dB (9:62)G min 0:7336g m ¼ mp2n ¼ 22:5m(9:63a)a ¼ 1 4 sinh 1 1¼ 0:32, ^a ¼ 0 (9:63b)0:6026f m ( sinh 0:32, 0) ¼ sinh 2 0:32 þ sin 2 g 2m ¼ 0:1059 þ sin 2 g 2mAppealing to formulas Equations 9.55 and 9.56, the element values are calculated as follows:L 1 ¼(9:63c)2 150 sin 22:5 ( sinh 0:32 sinh 0) 10 8 ¼ 1:123 mH (9:64a)p