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final_s03 - BB 350 EXAM IV 5 May 2003 Last Name S 0 VB OILS...

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Unformatted text preview: BB 350 EXAM IV 5 May 2003 Last Name: S 0' VB] OILS First Name: ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Test Form A Instruct ions 1. 2. 3. 4. You have two hours to complete this exam. This is a closed-book exam. You are allowed one 8.5” by 11” note sheet. Calculators are not allowed. Solve each part of the problem in the space following the question. If you need more space, continue your solution on the reverse side labeling the page with the question number, for example, “Problem 2.1) Continued.” No credit will be given to a solution that does not meet this requirement. . Do not remove any pages from this exam. Loose papers will not be accepted and a grade of zero will be assigned. . The quality of your analysis and evaluation is as important as your answers. Your reasoning must be precise and clear; your complete English sentences should convey what you are doing. To receive credit, you must show your work. Problem 1: (25 Points) 1. (12 points) Determine the system transfer function H (s) = Y(s)/ F (s) for the circuit shown in Figure 1. Assume that the operational amplifier is ideal, and express your final answer in the form: bms'" +b -1am'1 +---+bla+bo H3 = () s"+an—1s"‘1+---+a1s+aa C1 '/$C\ ” — 2. (13 points) The total response y(t) of a certain circuit has the unilateral Laplace transform 20::2 + 2009 + 1250 Y = _______ (8) 3(33 + 103 + 125) Determine y(t) for t _>_ 0. 'TJQ ‘Frao‘tlon i5 Strictly- Pro/)3” 0'6 n =3 > m =- Z. 'T/RO. deVaL'blc “9‘99!” yams 0" curl/"QC Cond’vcdofi‘ VQlQ {)cuf‘ 852‘; #51: Var-12.3” 7- —fi 10 Y(3) :: 43—- + if; 5 .57'+los+|Z§ 5-10 2052+ 2.066 + )250 A = 5/ iQs’L-t— (05 +05) 15:0 . Funny» 6 “0&- Q van”. theftth % 00¢ Erin/neg coemuéfl'é's. z. 2057’1- 2003 +1250 2 [0(5’1HOs—e-Izy) + 88 + CS 57': 20: to +5 :9 3:10 5‘: Zoo = loo + c, =-—> o :(00 100 YCS) 1‘- 10 + '0 "" b 5 (5415.)?- +100 K Problem 2: (25 points) Consider the feedback control system shown in Figure 2. Figure 2: Feedback control system with input F(s) and output Y(s). 1. (13 points) Determine the closed-loop transfer function Y(s)/F(s) for a = 2 and express your answer in the form Y(s) __ bmsm + bm_ls""'1 + - - - + bls + be F(s) — .s"+a,,,_1s"'1 +‘--+als+ao ' m 2. (12 points) Another feedback control system, different from the one considered in part 1, has the Y closed-loop transfer function Y(s) __ ' ’ 300 FM - (s+3) (a3+fi8+E4-+100)’ where fl is a real-valued design parameter chosen by the control engineer. (a) (6 points) For what value of 3 is the steady¥state response of the closed-loop system to a unit-step input equal to one-half; that is, firm...” y(t) = 0.5 ? (b) (6 points) For what value of 13 will the response of the closed-loop system to a unit-step input be a. sinusoid whose amplitude decays exponentially as 6‘". '(o) FPE) :- “('8‘ “'3 3’ H57 _ 300 YCS) ‘ (5+3) (5"+&s+,%i~rlo°) 5 . . .. 322—..— —= .L ya; :. Matt) == [guru .S —- (3) (LLH003 2- 155‘” 5-90 ‘1 7.. aim $L+ 6-5 4" flfmo?’ CS‘ré) + ’00 7 2. we, naofi %_.=2. or 6 Problem 3: (25 points) 1. (15 points) An electrical engineer designed the system shown in Figure 3 that has input F(s) and output Y(s). The engineer chose the transfer function C(41) so that the Bode magnitude and phase plot of the overall system transfer function H (s) = Y(a)/F(s) is given by Figure 4. Determine the transfer function C(a) that the engineer chose, and express you answer in the form: bms’" + bm_ls"“1 + - - - + by) + b0 03 = () s"+an_1s"'1+---+a1s+aa F(s> Y(s) Figure 3: The system with input F(s) and output Y(s). Y(s\ __ "" ‘ --—-—-”O)(’°”°) (3”“°*') s) ,. o Show“ F‘s) ’0 5/)° + l C( ” lo 0 4110 +1 L__,__._————————1 DC gain % éodfi [70-92. (£40, 204° .T/Qe. DC- aq'm swam [wafl )5 2'ng ‘q‘xeg “ml 5° ‘5‘“- D" gain at ccs) Int/5% be - - "Q, ’° [H'CDH 'QVevgo- W VO/Q— aIQ J=~IO circa €910: Save a» arena of. S~-looo Rina So C(J) Mu: “070% at 53’100 ___I , _____‘____... __L_.——. C653 ; s-r-Ioo / HQ‘Q/ L‘ng & by,» Maj w Er Jim/j 8 Figure 4: Bode magnitude and phase plot of the overall system transfer function H (s) = Y(.s)/F(s). 2. (10 points) Once again consider the system shown in Figure 3 that has the Bode magnitude and phase plot shown in Figure 4. An input f(t) that can be expressed as f(t) = C0 + i 01". cos(wnt + 0,1) 11:1 is applied to the system, and the resulting sinusoidal steady-state response y(t) is y(t) = 1 — cos(1000 t). k Specify the numeric value of the all non-zero coefficients Co, 0'", and 9,. in the expression for f(t). «u, = loco H(s) ha» a. Dc. gain ha. 20623) Co must be- 0°! . flab w::. loco rang/[5er ,1..— [CO li‘l’CduDll '3 "iO‘QS or 000 : —‘3§‘O 2S H'ch [wafliw c. ' t+ e — BI") “(S3 100 6.080000 VJ Q (a; (woo-L- + e) ’9 , , c. 0‘ J, 4/ 0.] +100“: (1006* ("‘15") gospel: Problem 4: (25 points) 1. (10 points) A LTI system is described the ODE {D2 + 2 D + 1} ya) = {D + 2}f(t)- Find the impulse response, h(t) of the system, and specify whether or not the system is BIBO stable (you must justify your answer to receive credit). 1“:- 00b represen-Ew-Ewn Fv-cm YCS) - ___§,'L7:__._, = 31—23—- “F-EES ’ 52.4.1544 CS-I—O” : I4 + 6 5+1 (rs-H)" 5: (5%?! Sal-2- : 1 51+ +| 5:4 5+7. .. A l (51—! '2— ’ -- ——' '4‘ 6’0 5-H 5:0 (5 +1)?“ 3:0 10 2. (15 points) Consider the operational amplifier circuit in Figure 5 virhere A and 1' are real-valued, positive parameters. Figure 5: Operational amplifier circuit with input f(t) and output y(t). o (10 points) Find the transfer function Y(s)/F(a) in terms of R1, R2, A, and 'r, and express your answer in the form Y(a) bms'" + bm_1a’"‘1 + - - - + bla + b0 __ _ _ . F(s) s"+an_1s" I+---+a1s+ao o (5 points) Is the circuit described by the transfer function Y(s)/F(s) BIBO stable ? In order to receive credit, you must justify your answer. , H5) — chi OM03 1%.; Q 1133 4 RI +021 2 ) Yls) = at Its) —z— V4453 L KL/L 3 (2) . ’9 k (3) Y“) / 5 'C-H vans) l VLZ Substm C0 “FLO CZ) Y6) = .3}. [Win 4%)] + ViCs) R442. {xiw'lnnuJ‘E Vania. €6uw‘i5lon 51+! YCQ -= .512; [7(5) -F65V\ 4' Trls) IL\+ 7. 11 12 ...
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