finals04 - Sample Final Exam Covering Chapters 1-9...

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Sample Final Exam Covering Chapters 1-9 (finals04) 1 Sample Final Exam (finals03) Covering Chapters 1-9 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: - + C R 1 L R 2 vt in out x C R + - + - R L in x A x + - + - in Vs x A x 1 Cs R 1 2 R 1 Ls
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Sample Final Exam Covering Chapters 1-9 (finals04) 2 There are two supernodes for which the nodal voltages are given by the source voltages. The remaining nodal equation is 1 2 11 () 0 in x x x Cs Vs Vs A R RL s −− += where 1 2 1 1 1 Cs s RLs s Cs RLCs Ls R == ++ . Simplifying the above equation, we get: 2 1 1 21 1 2 (1 ) ( ) () 0 in x AR L C s L s R Vs RR L s R  + −+ =   Thus, the transfer function between the input voltage and the node voltage is given by 2 2 1 1 2 2 1 1 1 1 1 1 ) ( ) 1 ) ( ) x in R L C s L s R R R L C s L s R R L s = + + = + + . The transfer function between the input voltage and the output voltage is 2 1 1 1 1 1 ) ( ) out x A in in A V s AR L s Hs R L C s L s R R L s = + + The ideal transfer function is the limit as the op-amp gain tends to infinity: 1 2 2 2 1 211 1 1 () l im ) A A L R s H s L RRLCs RLs RR LCs s R →∞ = (b) [5 marks] Assume that the transfer function 1 s = has a DC gain of 50 , and that has one zero at 0 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 12 ,, R RC . Answer: DC gain of 1 2 1 1 1 ) LR L s s R is given by 50 −= . Component values are obtained by setting 1 2 2 2 1 1 1 50 0.01 0.1 1 ) L s R s L ss s R =−
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Sample Final Exam Covering Chapters 1-9 (finals04) 3 which yields 12 100 , 0.2 , 0.001 R RC F ⇔= =Ω= (c) [7 marks] Give the frequency response of () Hs and sketch its Bode plot. Answer: Frequency response is 2 5 0 0.01( ) 0.1( ) 1 j Hj jj ω ωω =− ++ . Bode plot: (log) (deg) -45 -90 -135 10 1 2 0 -1 45 3 90 135 -180 180 225 270 (log) (dB) -20 -40 -60 -1 20 40 60 10 20log ( ) Gj
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Sample Final Exam Covering Chapters 1-9 (finals04) 4 Problem 2 (20 marks) Consider the causal differential system described by its direct form realization shown below, 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] Write the differential equation of the system. 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.
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finals04 - Sample Final Exam Covering Chapters 1-9...

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