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2. In this problem, we will consider the use of the active filter shown below as the loop filter in an analog PLL. IN VouT (a) Use the ideal model for the op-amp to show that the filter transfer function is F(s)Yot)ST2 +1 Vin(s) ST1 and express the time contants T1 and T2 in terms of R1, R2 and C. Sketch the magnitude Bode plot of the frequency response F(f) for the loop filter, assuming that > T2 Please note: that the loop filter transfer function F(s) is inverting, as indicated by the negative sign above. It turns out that this will be corrected by a subsequent inverting amplifier in the application described in Problem 3. Therefore, assume for the remainder of this problem that the loop transfer function is ST2 1 (b) Derive the closed-loop transfer function H(s), expressing your answer as a ratio of polynomials in s, with the coefficients in terms of the time constants , T2 and the loop gain K,Kd. Then determine each of the following in terms of the same parameters: i. the natural frequency wn ii, the dam ping factor ς. and iii. the quality factor Q (c) Sketch the magnitude Bode plot of the closed-loop frequency response H(f), assuming that > 1 /2

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