final_s04 - EE 350 Last Name EXAM IV 7 May 2004 Sol of 605...

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Unformatted text preview: EE 350 Last Name: EXAM IV 7 May 2004 Sol of] 605 First Name: ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO - - - 4 25 Test Form A ‘ Instructions 1. You have two hours to complete this exam. 2. This is a closed-book exam. You are allowed one 8.5” by 11” note sheet. 3. Calculators are not allowed. 4. 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. 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 should convey what you are doing. To receive credit, you must show your work. Problem 1: (25 Points) 1. (13 points) The switch in Figure 1 has been closed sufliciently long so that the voltages and currents in the circuit have reached steady-state values at time t = 0‘. At time t = 0 the switch is opened. Using Laplace transform analysis techniques, determine the output voltage y(t) for t 2 0 given that V, = 6 V, R = 1 Q, C = 2 F, and L : 2 H. Express your answer in terms of sine and cosine functions that do not include a phase angle; your answers must not contain complex exponential terms. No credit will be given for solutions using the classical approach covered on Exam 1. R t=0 Figure 1: The switch in the RLC circuit is opened at time t = 0. J 141': 1‘3an tso" #Co‘350V lanfl L(o-)= = 6H. For 't 20) S’fiamdlk clrfwf re/mboflfib‘éwn [‘5' USInfl. Va “513,0, (Q'IVuSIorI YLS) : " s c. 2. (12 points) Figure 2 shows the circuit diagram of a bandpass filter with input voltage f(t) and output current y(t). Assuming an ideal operational amplifier, determine the transfer function of the filter and place your answer in the standard form Y(8) _ bms’" +bm_18m_1 + "'+b18+bo H!) = _ () F09) s"+an_1s"'1+---+a18+ao R1 uh r Wt) vo f(t) 8 Figure 2: Active bandpass filter with input voltage f(t) and output current y(t). 4—— ' F653 (CL) V (s) = ____._..‘C' Pas) = + R‘ +3 JR\C,+l “Q amp/1 ver' 75 .. $9162 V-Hé) ('2) SC; fl — C finally" Vfimét Ohm‘s Law, YGSl » V°C3)/R3 3). pUJIna ebuw‘hons Cl) throaflll (3) 1-. I, *5 SR\CI+‘ filler? H0) #0 pram“ °° dich- c m opeanf) rebueny 3" fr 9.) ro‘Lag the h‘ .— H response, 04 kg? m GU 43m arc ya”) (95" bang/45$ walk”- Problem 2: (25 points) 1. (7 points) The feedback control system in Figure 3 has a command input F(s), an output Y(s), and a proportional controller with gain K. Determine the closed-loop transfer function and express your answer in the form Y(s) bms’" +b -13m'1 +---+b13+bo F(s) s"+an—1s"'1+~~+ais+ao Y(s) Figure 3: Feedback control system with command input F(s) and output Y(s). K—‘—— )c 665) :- $+l = M l—i-s pg: s(n+r) ‘H I: S __________ Lg; '- 6“) = 56 Hr) + l .5 _____,____E..——— f___________,_ 5MB 1" 2. (11 points) Another closed-loop system, different from the one considered in part 1, has the closed-loop transfer function Y(a) _ 300 F(a) _ 52+aa+a2+75’ where a is a real-valued parameter. o (6 points) For what value of a will the steady-state response of the closed-loop system to a unit-step input be equal to three ? o (5 points) For the value of 0: that you obtained, is the zero-state unit-step response underdamped, critically-damped, or overdamped ? We tie/2:9— Y5) : DC who : ‘3 Foals” 8 300 = 3- 2) “far 73’ :too :7 oozslr mz+7r IQ £0 @12— W 5c, .1 +S’or '5'. WWW”) t, [,6 3/50 the, 5 542m "V" _ steal -s'bwlse. res/Ionst ; is: Std-brag Obedi— pale roov‘EIOfl) "gr OC' d’;+5—: 51+§s+loo :0 For. a; ; +5 the, polo: \+ m ‘N 0) \J v \ a Q 80,2. 5 ’2‘: are, Complex, (ofldaaa'£2.§ “pg go Unt£’5’£€£ mygonsc :5 un/ércpamQQJ: 8 l 3. (7 points) In order to determine the partial fraction expansion of the transfer function Y(s)_ 23+1 H 2—— (s) F_(s) s +532+83+4’ the MatLab Command residue is used: [R, P, K] = residue(B, A); R = 1.0000 3.0000 -1.0000 P = -2.0000 -2.0000 -1.0000 K=EJ o (2 points) Specify the numeric value of the row vectors B and A. o (2 points) Write down the partial fraction expansion of H (s). O (3 points) Find h(t), the inverse Laplace transform of H (s). 6: E201”) ansz Halls-Jr, fl I 3 __ c H“) 2 5+2. + (Sn—)L S-H --24: —-b Ha -; e-ztuob) 4— 34:6 was) -e wit) Problem 3: (25 points) 1. (15 points) A system has the transfer function 105 .9 HM : (s + 10) (s + 1000)” Sketch the Bode magnitude and phase plots in Figure 4. In order to receive credit: 0 Use dashed lines to show the Bode plot of each term of the transfer function and a solid line to show the composite Bode plot. 0 Indicate the slope of the straight-line segments and corner frequencies of the final magnitude and phase plots. 0 Do not show the 3 dB corrections in the magnitude plot. 1+! 0- f—P‘on W dud - (d>w+’03(flw+loy) 10 24 1 (0 [a lo3 I01 Reg Figure 4: Semilog paper for constructing the Bode magnitude and phase plots. 11 Ltd] 32‘.) 2. (10 points) The Bode magnitude and phase response of a second-order LTI system with no zeros (m = 0) is shown in Figure 5. Find the steady-state response y(t) of the system for the input f(t) = 3 + cos(1o’t — 45°) + 106 cos(106t + 45°). gfl‘t) '= H9033 + "+9le Co$<joz+~qg° 4. XJKdIoo)) + \o‘ I H9105), (@0054; + 15° 4' 2;th )) m 6mg, Ina flt'éugae otan Fh‘om— HO‘L fl ‘ . H'(o sIXo" H90): -l ngo)MB—,-o J A d) 0‘ 1‘ #70" ngtooH- J- c-zOJex mil lug/ch "Io we gm mgzoepl mi No‘te tha£ fie, flrafh 1W5” the}? A‘iflflhjé) 7 OD 1063‘H0wevgf WQ’ d tJ the. ZADQ' orgér‘ “a In I) #Qéu-efl a g ‘ '£ HISO a£ fl fit .5 ’180 k I _ % ) ‘ V0 / q )0 how m wr‘fl? "0 Eras l2 Magnitude (dB) Phase (deg) Bode Diagram —1 0 10 10 1 3 1o2 10 Frequency (rad\sec) 1o 104 Figure 5: Bode magnitude (in dB) and phase (in degrees) plots for the second-order LTI system. 13 Problem 4: (25 points) 1. (12 points) A LTI system with 0 < C < 1 has the transfer function w: H = —.——, (a) 82 + 2Cwn s + w: o (6 points) Draw a. pole-zero diagram of the transfer function H (s), and clearly specify the real and imaginary parts of the pole locations in terms of C and wn. o (6 points) What is the distance between the origin of the s-plane (s = 0) and each pole '? Express your answer in terms of C and/ or w". ’L 5L4 ZPwnS + wn 90 5'32. : 4w : WWW: whith Because. 0 (‘F 4.5 “the, pate) cure Comp/ex— rogugabj suz = ’70Wn tint/NW" {Land the, ‘U‘o. 90?}:er Thea/em} 9,: Omis.3”+:m 13‘3": flvwaL+waPf9 (QR The O’t‘S'LaflCe fire”. 0f\ (0 b ‘e‘fle‘r cop‘f t; the Mafia/Va» “Pré “"7. 14 Fab. 73 Q 2. (13 points) A LTI system with input f (t) and output y(t) is represented by the ODE :7 + 20 y + 200 y(t) = 400 f(t). o (4 points) Find the transfer function H(s) : Y(s)/F(s) of the system and express your result in the form H“) __ Y(s) _ bms’" + bm_1s"‘"1 + - - - + (115+ (10 _F(5) s"+an_1s"‘1+---+als+ao ' o (5 points) Specify the impulse response function h(t) of the system. a (4 points) Suppose we need to experimentally determine the DC gain of the system. A unit-step input is applied to the system, and we wait T5 = 5 7' seconds to measure the steady-state response in order to calculate the DC gain. The parameter 'r is the largest time constant associated with the natural (response of the system. What is the numerical value of T5 '? $2"; 10?th Coflcpl'EW/‘S "’50 Zero anJ/ the 167[@C0/ trdnrfio/‘mi/ bofl JILJ% the, 006? 52 Tm Jr 2os ((s\ + 200 rm : woo PCS) Compl 2152, {he Spa/aw: .LHSX : 90" =— wca 1 (5 +tof +100 (S‘HD) + I0 Us”? #12. ‘lfit‘uS‘ch‘m (a Sr 3"“; Sing-'01 (1") “6—3 m2 15 ...
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