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EE-202/445, 5/1/09 9-1 © R. A. DeCarlo BIQUADRATICS AND STATE SPACE REALIZATIONS I. Introduction 1. The biquadratic transfer function is simply a transfer function having a second order numerator and a second order denominator: H ( s ) = b 0 s 2 + b 1 s + b 2 s 2 + a 1 s + a 2 = K s 2 + ! z Q z s + ! z 2 s 2 + ! p Q p s + ! p 2 Exercise . Determine K, ω z , Q z , ω p , and Q p in terms of b 0 , b 1 , b 2 , a 1 , a 2 . Exercise . Show that the gain of the biquadratic frequency response is given by G( ! ) = 20 log 10 (K) + 10 log 10 ! z 2 ! 2 2 + ! z 2 Q z 2 ! 2 – 10 log 10 ! p 2 ! 2 2 + ! p 2 Q p 2 ! 2 and that the phase is given by ! ( " ) = tan –1 " z " Q z " z 2 " 2 – tan –1 " p 2 " Q p " p 2 " 2 where π must be added if K is negative.

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EE-202/445, 5/1/09 9-2 © R. A. DeCarlo 2. For active filter realization, circuit transfer functions often normalize ω p to 1 via frequency scaling: s --> ω p s, i.e., H cir ( s ) = K s 2 + ! z ! p " # \$ % & Q z s + ! z 2 ! p 2 s 2 + 1 Q p s + 1 = K s 2 + ! ! z Q z s + ! ! z 2 s 2 + 1 Q p s + 1 Remark : It is this form that is often used in normalized active filter realization. One then frequency and magnitude scales to achieve the proper circuit characteristics. Exercise . Work through the details of this normalization.
EE-202/445, 5/1/09 9-3 © R. A. DeCarlo II. Controllable Canonical 4 OP AMP State Space Realization of the Biquadratic TF: A Block Diagram Development 1. Recall again the biquadratic structure with a 1 > 0 and a 2 > 0: H ( s ) = V out V in = b 0 s 2 + b

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