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finals03

finals03 - Sample Final Exam Covering Chapters 1-9(finals03...

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Sample Final Exam Covering Chapters 1-9 (finals03) 1 Sample Final Exam (finals03) Covering Chapters 1-9 and part of Chapter 15 of Fundamentals of Signals & Systems Problem 1 (20 marks) Consider the causal op-amp circuit initially at rest depicted below. Its LTI circuit model with a voltage- controlled source is also given below. (a) [8 marks] Transform the circuit using the Laplace transform, and find the transfer function () Ao u t i n Hs V sVs = . Then, let the op-amp gain A →+∞ to obtain the ideal transfer function l im A A Hs H s →+∞ = . Answer: The transformed circuit is - + R 2 C R 1 L L vt in out x C R + - + - L R L in x Av t x + - + - in Vs x A x 1 Cs R 1 R 2 2 Ls 1

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Sample Final Exam Covering Chapters 1-9 (finals03) 2 There are two supernodes for which the nodal voltages are given by the source voltages. The remaining nodal equation is 1 22 11 () 0 in x x x Cs Vs Vs A RL s s −− += where 1 2 1 1 1 Cs s RLs s Cs RLCs Ls R == ++ and s s = + . Simplifying the above equation, we get: 2 1 1 1 1 (1 ) ( ) () 0 in x s A R L C sL s R s Vs  + + + −+ =   Thus, the transfer function between the input voltage and the node voltage is given by 2 2 1 1 2 2 2 2 1 1 2 2 ) ( ) ) ( ) ( ) x in s RLsR Ls AR L C s L s R R L s R L s L C s L s R R L s R L s + + + + + + + + . The transfer function between the input voltage and the output voltage is 2 2 2 1 1 2 2 ) ( ) ( ) out x A in in A V s AR L s R L s Hs RLsA = + + + The ideal transfer function is the limit as the op-amp gain tends to infinity: 2 1 2 2 2 2 2 1 22 11 1 1 21 1 ) () l im ) A A L RL R R H s L RL RLCs LLC s s R →∞ + + = (b) [5 marks] Assume that the circuit has a DC gain of 50 , one zero at 1 and two complex conjugate poles with 10 rd/s, 0.5 n ω ζ . Let 1 10 LH = . Find the values of the remaining circuit components 212 ,,, LRRC . Component values are obtained by setting 2 12 2 2 1 2 1 1 ) 1 50 0.01 0.1 1 ) L s LR s L L ss LCs s R + + =− which yields 0.2 , 100 , 0.2 , 0.001 R R C F = = (c) [7 marks] Give the frequency response of and sketch its Bode plot. Answer: Frequency response is 2 1 5 0 0.01( ) 0.1( ) 1 j Hj jj ωω + . Bode plot:
Sample Final Exam Covering Chapters 1-9 (finals03) 3 ω (log) (deg) -45 -90 -135 10 1 2 0 -1 45 3 90 135 -180 180 (log) (dB) -20 -40 -60 -1 20 40 60 () Hj 10 20log ( ) Gj

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Sample Final Exam Covering Chapters 1-9 (finals03) 4 Problem 2 (20 marks) Consider the causal differential system described by 2 2 () 22 ( ) 2 ( ) dyt d yt xt dt dt ++ = and with initial conditions (0 ) 1, (0 ) 2 dy y dt =− = . Suppose that this system is subjected to the unit step input signal x tu t = . (a) [8 marks] Find the system's damping ratio ζ and undamped natural frequency n ω . Give the transfer function of the system and specify its ROC. Sketch its pole-zero plot. Is the system stable? Justify. Ans: Let's take the unilateral Laplace transform on both sides of the differential equation. 2 (0 ) (0) 2 2 () 2 () dy ss s y s s y s s dt −−  + +=   YY Y X Collecting the terms containing Y s on the left-hand side and putting everything else on the right- hand side, we can solve for s Y .
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finals03 - Sample Final Exam Covering Chapters 1-9(finals03...

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