final_s01

final_s01 - Last Name(Print Final Exam EE 350 Continuous...

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Unformatted text preview: Last Name (Print): Final Exam EE 350 Continuous Time Linear Systems April 30, 2001 Sdhliocn . First Name (Print): ID. number (Last 4 digits) : Official Section number : Do not turn this page until you are told to do so Problem Score Weight 25 25 25 25 Total 100 Test form A Instructions 1. You have 1 hour and 50 minutes to complete this exam. 2. Calculators are not permitted. 3. This is a closed book exam. You are allowed one sheet of 8.5” x 11” of paper for notes as mentioned in the syllabus. 4. Solve each part of the problem in the space following the question. Place your final answer in the box when provided. If you need more space, continue your solution on the reverse side labeling the page with the question number; for example, “Question 4.2) Continued”. NO credit will be given to solution that does not meet this requirement. 5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and a grade of ZERO will be assigned. 6. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be precise and clear; your complete English sentences shall convey what you are doing. To receive credit, you must show work. Problem 1: (25 Points) 1.1 (11 points) Consider the active filter shown in Figure 1. Assume that the operational amplifier is ideal. Figure 1: Active filter 1. (4 points) Draw an equivalent circuit at high frequency, and by inspection, find the high frequency gain. I V c —) she/d CAM/nut 13%) = o. 29‘ 2:2. - (3c) Ht) 9 [k ‘4 h.?. gain/1 = O- 2. (4 points) Draw an equivalent circuit at d.c., and by inspection, find the d.c. gain. c —’ opm oinau’r «4%) = ‘FLH 2.0. 152. (M33!) . dc. clown = 1. 3. (3 points) Based on your answers in Parts 1 and 2, what type of filter does the circuit in Figure 1 represent? Highpass, lowpass, bandpass, bandstop? ’Tfij. oi/Lwii WP/CUYYJMld a, /0wa Film. 1.2 (14 points) Once again consider the active filter shown in Figure 2 below. I | | I 29 29 W + vx~\/VV‘ i f(t) IF I y“) Figure 2: Active filter 1. (6 points) Assuming that the operational amplifier is ideal, write a node equation relating the node voltage V1 (.9) to F(s) and Y(s). R R Use ==> F‘er + \/-V96 + scR(Y—~vx)=o V = SCRY—l-Y + F- SCR+2. YL5)(2S+4)+F(S) 2.3 + 2. (1) . (9-) 3. (4 points) Using the results from Parts 1 and 2, determine the transfer function H (s) = Y(s)/F(s) of the active filter. Substitu’m m WW (2), NS) = W5) (2%) + [2(3) (Z/SWHZHZ) (23+1)(2s+2) Problem 2: (25 Points) 2.1 (16 points) A feedback system shown in Figure 3 has an output Y(s), a command input F(s), and a and [3 are adjustable parameters. F(s) Y(s) Figure 3: A feedback system block diagram for Problem 2.1 o (10 points) Derive the overall transfer function H (s) = g-ég of the feedback system in Figure 3 in terms of a and fl. To receive credit you must put your answer in the form bms’" + bm_13m_1 + + b18 + b0 3" + an_1s"—1 + + als +ao ' He) + 1/s+2 W5) ‘ ) p (é) Gis ‘7 H(s) = 1+olp(.sl.) ,, z .1: \I s+o<p Y(S) T6) = (5) 3+2. I '0? I) .4 ‘63 hi I! A m + 51 I"; 7/? I in V *9 5 ~——# A o (6 points) If the parameters a and fl are set as a=0.5, [3:1, is the overall system bounded—input, bounded—output (BIBO) stable? Explain. (x z: 0.5 , P=1 s H(S)= ‘ = ’ 3 Sl~£s +0. 8(‘9-3: 2 ohm ova s = O :1 LHP r :2 qé 2.2 (9 points) A ramp input f (t) = tu(t) is applied to the LTI system Whose transfer function H (s) is given by 3(3s2 -— l) .92 + 33 + 2’ compute the initial value y(0+) and the final value ysa = limHoo y(t) of the zero-state response y(t). H(8) = «3(0“) = Kim sws) = 2m s(—1;H(3)> 3—)“) 9—900 . 2 saw s"+3s+2 . . 2. «33 = 210’" syn): Km: 33 “1 = -_L S 8‘90 5"" 31+33+2 2 T [mew m 3=—2,~1 ELHP / Problem 3: (25 Points) 3.1 (15 points) An LTI system has the transfer function 8 H“) = s2 + 1103 + 1000' Construct the Bode magnitude and phase plots using the semilog graphs provided. In order to receive credit: 0 In both magnitude and phase plots, indicate each term separately using dash lines. 0 Indicate the slope of the straight-line segments and corner frequencies of the final magnitude and phase plots. Hm») = J‘d/iooo (saws!) o ZOXOg IKI 3 2o z: “‘60 3.2 (10 points) Consider another LTI system whose frequency response magnitude and phase plots are given in Figure 4. If the input f (t) = [13 + 2cos(10t)]u(t) is applied to this system, determine the steady-state response y(t). Bode Diagrams Phase (deg); Magnitude (d8) I a I 10— 1o 10 2 10 Frequency (rad/sec) Figure 4: Bode Plot for problem 3.2 Rh 2 [13 + 2. cos(1o\3)]u(£) 1‘ T 01.0.. u :10 ‘) HUG) = d.c. = ~00 d3 = o 4 ‘0 2) H = “200% 445° = 7-6- 0 r Wt) = 13~H%) + 2’H(j10)’COSUOt+K~HlJIO)) H 0.2 cos (wt +45") 11 Problem 4: (25 Points) 4.1 (11 points) Consider a certain LTI system whose transfer function has the pole—zero pattern shown in Figure 5. lm >5 --------- --1 2 4 15L 0 >Re i ......... “1 Figure 5: A pole-zero map of the system in Problem 4.1 o (6 points) If the d.c. gain of the system H (0) = 1, determine the transfer function H (s) of the system. yous I S = flij we I 3 = 2 ms) = K (3‘2) _ KCS'Z) ' . - ______,_.. (3+1+3)(S+1~3) 32+23+2 H(o) :: K(~2) =1 glK =~1| 2 o (5 points) Determine the damping ratio of the system. Is the system underdamped, overdamped or critically damped? sz+2s +2 m 2. 2. S + 23w”: +00." um=fi Omd. 32—3”: U 2% Ex!» Umolm, down/not 12 4.2 (14 points) The result from Matlab for generating a partial fraction expansion of a transfer function H (s) = %% of a certain LTI system is shown below. >> [R, P, K] = residue([1,0,2], [1,2, 2]) R = —1.0000 — 1.0000i —1.0000 + 1.00001’ —1.0000 + 1.0000i —1.0000 — 1.0000i K = 1 P o (4 points) Write an ordinary differential equation relating the output y(t) and the input f (t) of the system. l H = S + z sz+23+2 o (10 points) Determine the impulse response h(t) of the system. Hls) = ~1”J‘ + "1+J_ + 1 ( F/zorm (mail/at) s+1~3 9+1+j j(-iss°) jogs“) = J3 1 4 . + 5" + 1 s+1~3 S+i+j ~t 25 2. cos (t —135°) wt) + at) 13 (continued) A lbw/n Minx WWOOC ; 9. H6) : 8 +2 : ~28 gz+zs+2 3L+2s+2 ‘23 H + (s+:)2+1 [1 1+ "209“) + 2{ 7 ) (3+1)"+1 lady) = 5 W62 603(k) + 2 Sfm(Jc))‘1/L({') 14 ...
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final_s01 - Last Name(Print Final Exam EE 350 Continuous...

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