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

The Current Generalized Immittance Converter Biquads 14-15whereD(s) ¼ Y 11 Y 12 (Y 1 þ Y 2 ) þ Y 12 (Y 3 þ Y 4 )Y 7 Y 8 =Y 9 þ (Y 5 þ Y 6 )Y 7 Y 10 (14:9)The admittances Y 1 –Y 12 can be selected in many ways and any stable second-order transfer function canbe realized. For the purposes of this chapter, Y 1 –Y 10 are taken to be purely conductive, while Y 11 and Y 12are purely capacitive, that is,Y 1 Y 10 ¼ G 1 G 10Y 11 ¼ sC 1 , Y 12 ¼ sC 2(14:10)Any rational and stable transfer function can be expressed as a product of second-order transferfunctions of the formT(s) ¼ a 2s 2 þ a 1 s þ a 0b 2 s 2 þ b 1 s þ b 0(14:11)where a 1 ¼ a 2 ¼ 0, a 0 ¼ a 1 ¼ 0, a 0 ¼ a 2 ¼ 0, or a 1 ¼ 0, for an LP, HP, BP, or N section, respectively. Thesesection can be realized by choosing the G i ’s (i ¼ 1–10) properly in Equation 14.8. By comparingEquations 14.8 through 14.11, circuits 1–4 in Table 14.3 can be obtained.All-pass transfer functions can be realized by setting a 2 ¼ b 2 , a 1 ¼ b 1 , a 0 ¼ b 0 ; these can be obtainedfrom circuit 5 of Table 14.3.It can be easily shown that this biquad possesses similar excellent low sensitivity properties andstability during activation as those of the 2-OA CGIC biquad, given in Section 14.4.14.10 Design and Tuning Procedure of the 3-OA CGIC BiquadSeveral degrees of freedom exist in choosing element values, as shown in Table 14.3. These are used tosatisfy the constraints of the given design, namely, those of low sensitivity, reduced dependence on theTABLE 14.3Element Identification for Realizing the Most Commonly Used Transfer FunctionsCircuits Number G 1 G 2 G 3 G 4 G 5 G 6 Transfer Function Remarksa G5G7G10C1C2(G5 þ G6)G7G10þC1C21 0 0T 3 ¼1 þ G2G7s 2 G 2 þ s G4G7G8C1G92a0 0s 2 G 1 1 þ G4G9T 1 ¼s 2 (G 1 þ G 2 ) þ s G4G7G8C1G9þ G6G7G10C1C2LPHP3 0a G3G7G8C1G9(G3 þ G4)G7G8C1G9s 1 þ G2G70 T 3 ¼s 2 G 2 þ sþ G6G7G10C1C2BP45aa0aT 1 ¼ G 1 1 þ G 4G 9s 2 G 10 0 T 3 ¼s G1G4G8þC1G9s 2 (G 1 þ G 2 ) þ s G4G7G8s 2 þ G5G7G10C1C2G1s 2 (G 1 þ G 2 ) þ s G4G7G8C1G9 1 þ G2 G5G7G10C1G9G7 C1C2þ G5G7G10C1C2(G5 þ G6)G7G10þC1C2NNonminimumphase bNotes: Y 7 Y 10 ¼ G 7 G 10 always.Y 11 sC 1 , Y 12 ¼ sC 2 .a These elements can be set to zero.b For all-pass G 2 ¼ 0, G 7 ¼ G 1 .