finals01 - Sample Final Exam Covering Chapters 1-9...

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Sample Final Exam Covering Chapters 1-9 (finals01) 1 Sample Final Exam (finals01) 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 is There are two supernodes for which the nodal voltages are given by the source voltages. The remaining nodal equation is C R 1 L R 2 - + 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 (finals01) 2 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 2 1 1 2 2 1 1 2 2 (1 ) ( ) () 0 in x A R Ls Vs  + + −+ =   Thus, the transfer function between the input voltage and the node voltage is given by 2 1 1 2 1 1 1 1 2 1 1 2 2 2 1 1 ) ( ) ) ( ) ( ) x in AR L s R L C s L s R RLsRLCs RLs A R RLs RLCs = + + + = + + + . The transfer function between the input voltage and the output voltage is 2 1 1 2 2 2 1 1 ) ( ) ( ) out x A in in A V s AR L s R L Cs L s R Hs R + = + + + The ideal transfer function is the limit as the op-amp gain tends to infinity: 2 1 21 2 22 11 1 1 1 2 2 2 1 2 ) () l im ) A A L LLC s s RL RLCs R H s L RL R R →∞ = + + (b) [5 marks] The circuit is used as a cascade equalizer for the system 2 1 50 0.01 0.1 1 s Gs ss + =− , that is, ()()0, Gj Hj dB ωω ω =∀ . Let 1 10 LH = . Find the values of the remaining circuit components 212 ,,, LRRC . Component values are obtained by setting 2 1 1 2 1 2 1 2 ) 0.01 0.1 1 0 .02 1 ) L LCs s LR Hs G s L sL s R = + + G j Hj m Vj in out system equalizer
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Sample Final Exam Covering Chapters 1-9 (finals01) 3 which yields 212 0.2 , 100 , 0.2 , 0.001 LH R R C F ⇔= =Ω =Ω= (c) [7 marks] Sketch the Bode plot of () Hs and Gs (magnitude only, both on the same plot). Magnitude Bode plot of 2 1 5 0 0.01( ) 0.1( ) 1 j Gj jj ω ωω + =− ++ and desired 2 0.01( ) 0.1( ) 1 ( ) 0.02 1 Hj j + : Problem 2 (20 marks) Digital Signal Generator A programmable digital signal generator generates a sinusoidal waveform by LTI filtering of a staircase approximation to a sine wave x t . (log) (dB) -20 -40 -60 10 1 2 0 -1 20 3 40 60 20 10 log ( ) 10 20log ( ) t A T T/2 T/6 -A T/3 2T/3 5T/6 -T/6 0.5A -0.5A x t
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Sample Final Exam Covering Chapters 1-9 (finals01) 4 (a) [9 marks] Find the Fourier series coefficients a k of the periodic signal x t () . Show that the even harmonics vanish. Express x t as a Fourier series. Answer: First of all, the average over one period is 0, so a 0 0 = . For k 0 , 2 2 2 22 2 0 36 23 6 2 2 0 6 0 1 T jk t T k T TT jk t jk t jk t T T T jk t jk t jk t T T j kt j j j ax t e d t T AAA e dt e dt e dt T e dt e dt e dt T AA ee d t π ππ −− = =− ++ +  +−   ∫∫ 3 2 63 2 0 0 6 222 2 sin sin sin 2 cos cos co 2 T T jk t jk t T T A dt e e dt T jA j A jA k t dt k t dt k t dt T T jA T j A T jA T Tk T T k T 
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This note was uploaded on 09/16/2011 for the course ECSE 361 taught by Professor Franciscodgaliana during the Winter '09 term at McGill.

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finals01 - Sample Final Exam Covering Chapters 1-9...

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