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

10-18 Passive, Active, and Digital FiltersCase 2. RCv c 2 sin g 1 and L a1 < L. Under this situation, s 1 is nonzero and can be determined bythe formulas 1 ¼ 1 RC 1 þ 2 p ffiffi wp sinh32RCv c sin 2 g 1 þ sin g 33 sin g 1(10:98)wherep ¼ (RCv c 2 sin g 1 ) 2 sin g 39 sin g 1> 0 (10:99a)ðw ¼ 2RCv c sin 2 g 1 þ sin g 3 Þ54 sin 3 3(RCv c 2 sin gg 1 ) 2 sin g 1 sin g 31þ 2RCv c sin 2 i 2 R 2 C sin gg 1 þ sin g 332L sin g 1w ¼ sinh 1 wp ffiffi( p )3(10:99b)(10:99c)Using this value of s 1 , the DC gain K n is computed by Equation 10.90.Case 3. RCv c < 2 sin g 1 and L a2 L. Then, we haveK n ¼ 1(10:100a)s 1 ¼ 1 RC 1 RCv c> 02 sin g 1(10:100b)Case 4. RCv c < 2 sing 1 and L a2 < L. Then the desired value of s 1 can be computed by Equation10.98. Using this value of s 1 , the DC gain K n is computed by Equation 10.90.Example 10.5LetFrom Equation 10.92, we first computeSince L a1 > L andL a1 ¼R ¼ 100 V, C ¼ 100 pF, L ¼ 0:5 mF (10:101a)n ¼ 6, v c ¼ 10 8 rad=s (10:101b)100 sin 45 [(1 sin 15 ) 2 ¼ 1:84304 mHþ cos 2 15 ] 10 8 sin 15 (10:102)RCv c ¼ 1 > 2 sin 15 ¼ 0:517638 (10:103)Case 1 applies and the matching network can be realized as an LC ladder terminating in a resistor asshown in Figure 10.8b. With s 1 ¼ 0, the maximum attainable DC gain K 6 is from Equation 10.90