202Biquad - EE-202/445, 5/1/09 9-1 R. A....

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Unformatted text preview: EE-202/445, 5/1/09 9-1 R. A. DeCarloBIQUADRATICS ANDSTATE 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)=bs2+b1s+b2s2+a1s+a2=Ks2+!zQzs+!z2s2+!pQps+!p2Exercise.Determine K, z, Qz, p, and Qpin terms of b, b1, b2, a1, a2. Exercise.Show that the gain of the biquadratic frequency response is given by G(!) = 20 log10(K) + 10 log10!z2!22+!z2Qz2!2 10 log10!p2!22+!p2Qp2!2and that the phase is given by !(") = tan1"z"Qz"z2"2 tan1"p2"Qp"p2"2where must be added if K is negative. EE-202/445, 5/1/09 9-2 R. A. DeCarlo2. For active filter realization, circuit transfer functions often normalize pto 1 via frequency scaling: s --> ps, i.e., Hcir(s)=Ks2+!z!p"#$%&Qzs+!z2!p2s2+1Qps+1=Ks2+!!zQzs+!!z2s2+1Qps+1Remark: 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. DeCarloII. Controllable Canonical 4 OP AMP State Space Realization of the Biquadratic TF: A Block Diagram Development 1. Recall again the biquadratic structure with a1> 0 and a2> 0: H(s)=VoutVin=bs2+b1s+b2s2+a1s+a2=bs2+b1s+b2d(s)where d(s)=s2+a1s+a2. Alternately H(s)=VoutVin=b2d(s)+b1sd(s)+bs2d(s)2. Consider auxiliary equation !...
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202Biquad - EE-202/445, 5/1/09 9-1 R. A....

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