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

Single-Amplifier Multiple-Feedback Filters 12-5TABLE 12.2Element Values for the MFB All-Pole RealizationsElement (Table 12.1) Low-pass Bandpass High-PassY 1G 1 ¼ H Gvp1 ¼ HC 1 ¼ H Q p 2v 2 p þ HY 2C 2 ¼v 2 G 2 ¼ 2v p Q p H G 2 ¼ v p (2 þ H)Q ppY 3 G 3 ¼ v p C 3 ¼ 1 C 3 ¼ 1Y 4 G 4 ¼ G 3 C 4 ¼ C 3 C 4 ¼ C 3Y 5C 5 ¼Q pv 2 p G 5 ¼ v p2v 2 p þ H2Q pv pG 5 ¼Q p (2 þ H)C 2 C 5 ¼ 1 (12:14)C 5 (G 1 þ G 3 þ G 4 ) ¼ v pQ p(12:15)G 3 G 4 ¼ v 2 p (12:16)These equations do not yield an equal-capacitor solution but can be solved for equal-resistor pairs.Hence, if G 1 ¼ G 3 ,G 1 ¼ G 3 ¼p ffiffiffiffiH(From Equation 12:13)G 4 ¼ p v2pffiffiffiffi (From Equation 12:16)HThen,C 5 ¼Q ppffiffiffiffiv p H ¼ 12H þ v 2 C 2pAn alternative solution for which G 3 ¼ G 4 is shown in Table 12.2, together with equal-capacitor designsfor the bandpass and high-pass cases.The conditions [4] for maximum Q p in the bandpass realizationp ffiffiffiffiffiffiffirequire C 3 ¼ C 4 and G 1 ¼ G 2 ¼ nG 5 ,where n is a real number. This yields a maximum Q p of n=2 , and requires that H ¼ vp Q p .Example 12.1Using the cascade approach, design a four-pole Butterworth bandpass filter having a Q of 5, a centerfrequency of 1.5 kHz, and midband gain of 20 dB. Assume that only 6800 pF capacitors are available.SolutionThe low-pass prototype is the second-order Butterworth characteristic having a dc gain of 10 (i.e., 20 dB).Thus,10T(s) ¼ ps 2 þffiffi2 s þ 1(i)