ExamI_s04 - /A EE 350 EXAM I 9 February 2004 Last Name 5 915214 Q a First Name ID number(Last 4 digits Section DO NOT TURN THIS PAGE UNTIL YOU ARE

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Unformatted text preview: /A . EE 350 EXAM I 9 February 2004 Last Name: 5 915214;} Q a; First Name: ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Weight Score ‘ Test Form A Instructions 1. You have two hours to complete this exam. . This is a closed-book exam. You are allowed to use both sides of a 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. 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. (7 points) Consider the signal f(t) = 4 6"”. Is the signal f(t) an energy signal, a power signal, or neither ? If the signal is energy signal or power signal, calculate E; or Pf, respectively. °° °°-2H=| E; =j Flu-Dc“: = I6 e it _. °°-2.+. _ ‘2‘” ‘3ze££="e°“6 back/«Sie— TIRQ. Jams-l ~PH=3 ‘u an energy JTdnu/l E4 2 I6 is 'Clm'Le. 2. (8 points) The zero-state response y(t) of a system to an arbitrary input f(t) is 1/0) = e“f(t - 1)- Is the system time-invariant or time-varying ? Justify your anSWer. rflc sag-Lem L; filme—Invaruan‘lts °”&' ‘For Wm WM 9; t We 7. mate. mi— &,ca 2 e-bfitn) = e" —F(t - I -7) q n&- got Jr) : é-Ct—fl _(:C_L__, ,7) Be cause. a‘ (t) 15 a (1‘: *7} tho. ssEam is 'lrtm°'\’°-'I". 3. (10 points) The zero-state y(t) response of a LTI system to an input f(t) is W) = (1 - t) [11(1) - “(t - 1)]- Figure 1 shows the zero-state response y1(t) of the same system to another input f1(t). Find an expression for f1(t) in terms of f(t). { Hint: Start by sketching y(t) in Figure 1.} Y1(t) Figure 1: The zero-state response 3/103) of a LTI system to the input f1(t). P042, fha£ we. can exflrw g'H-A 09 JOE) -d-(1‘:"-‘)x a (t) ~d Cf}? yflt) 2 ate) '(7 H735) owl tl‘l sdxtem I) LTI) Problem 2: (25 points) 1. (10 points) A first-order system with a DC gain of ten is represented by the ODE 13(1) + 0 11(1) = 5N): where a and b are real-valued constants. Given that the zero-state unit-step response has a rise-time of0.]1n(81) sec, determine the numeric value of the constants a and b (simplify your answer so that it does not involve the in function). For a. {5'5 arng 5034:2014 with from-oJ 'lu‘ -c pix (to, fl In this 0:52 or a" = 2&ng :- ’0“ C9) 2 fl. (.4 ty- 0 ‘ 0 Z 2. (7 points) The characteristic roots for a LT] system with P(D) : 1 are located at —]O and —1. Suppose that you are assigned the task of reducing the rise—time of the zero-state unit-step response for the systemT and, that you are only allowed to change the location of one of the charcateristics roots. Which of the two characteristic roots should be changed, and should this root be moved to the left or the right along the real axis of the A-plane. ln order to receive credit, justisfy your answer. In (7s) (29. (79 the, QM» Far “- ware-'5'sz [AIL/'5’ the Y‘G—Sfefl-lz' Q" (t 5- C. Q W a- ) l \acr Acid-Hon “chode’W-WI 3‘4" pa ('EICV 1'" ouoer- ‘b @9055. the. nsz—‘E'MQJ we 02d, '59 "lch 'thQ routh “3E aQO/"vfleflw Nate. time tho, Mwfifié J \wtm to zero 0 the Maya... 0L6 (v.03) 3. (8 points) Write a MATLAB m-file that will plot the zero—state response of a system described by the ODE 1705) + 211(t)+ 21105) = P(D)f(t), where P(D) = 2D + 1, to the input 1'05) = e-1 sin(4t)u(t) over the interval 0 S t S 5 using a time vector with 250 points. Q = C") 2) 2-35 P = CZ, 113 1r. = “Pea .ac what-1:3; 8_ : ,Q..Snm Cfi-Ffit); xlwbel C.’ f‘WW— CSUQ') filabQ/l (J responseJ) Problem 3: (25 points) 1. (12 points) Find an ODE representation of the circuit shown in Figure 2 that has input voltage f(t) and output voltage y(t). Assume that the operational-amplifier is ideal and place‘your answer in standard form dny dn— 1y W+an—ldtn_l +“'+aay: m‘— +bm—l If you introduce a node voltage (mesh current) to determine the ODE, clearly indicate this variable in the circuit schematic and label its polarity (direction). Figure 2: Active operational amplifier circuit. 2. (13 points) Consider a first-order system with input f(t) and output y(t) that is represented by the ODE 31(1) + 631(1) = 91W)- The system is driven by the input f(t) : 26—32 for t 2 0, and y(0+) : 1. Determine - (6 points) the zero-input response yn-(t), o (6 points) the zero~state response yu(t), o (1 point) and total response y(t) for t 2 0. Zero - Inlet resf 9‘" be; (9’ 4 5d— =03 gCofi-st -e-t 9v»: fl-t-ézo —._-.> dyeing): (1.2. 1:20 ChooSe, C": I to Jp‘Ehg-Ei the, mrktaQr Cartoon-Eton, -6‘lr Zero- shade 3+ 6a :- lfie'ab J a(0"’)=0 am (-e) :. c‘e—Cb +- -- _ ~31: (hog; thcfl— «3 75 Q31? “’ m % a” (4:) —- at» 9 {ha okdrwéenyk 25¢ aa‘lsl-m) 11 Problem 4: (25 points) 1. (7 points) Derive an ODE that relates the input voltage to the output current y(t) for the circuit in Figure 3 and place your answer in standard form dny dn— 1y dm dm— 1f _ _ a : bm__ bm_ dt" +6" 1altn-l + +a y dtm + ldtm‘l +"'+baf. If you introduce a node voltage (mesh current) to determine the ODE, clearly indicate this variable in the circuit schematic and label its polarity (direction). R1 A «0 O C VARz d W) HON Figure 3: RLC circuit with input voltage f(t) and output current y(t). branch rah/liter)th Ar "have or: - = VA (,2) Lillfllnwfli‘ in ‘Fflvow % %' Suki—6.5%,? (2) Into (0 12 13 2. (9 points) A LTI system is characterized by the ODE (D2 + GB + 9) y(t) = (—4D + 23) f(t) where is the input and y(t) is the output. If y(0+) = 0, y(o+) : 5, and = cos(t)u(t), find y(t) for t Z 0. 67(7) =— 7i1+61+7 '=- 0+3)“ :9 7s, = ’22, =- —3 -+—- ~34: Masqea + cite, 1:20 ——-———o (' Lib + zen—(4;): - q gee-cw]; + 25cm: -. tum-t +28ca54: a? (t) 2 Acost .,. Osmi' '72-'20. (DL+ 6b +?)0n/P :2. C' ‘10 4-2.5) 4ft) §f+6jr+ye= (99 +655)wa + ass—cmm-t- =- k{Sm‘i‘s '1' 23605-6 WSM‘L 4' “(are .. gnaw-CIA: ‘iZ. :- 23 :7 HH- 4- 36 —- I“! :9 gnaw-66 '39 +Vb =2. => "29+ ‘3 5 9 86 ~64 = q r 2567-50 18:22.] Hfi-z-lw-Bs =8’=‘~> “3*: (Ii-Ea mt + am 305) :: arm-b) +Je (ab) 2 0.9. a (‘éYofl FMS" CR ach C; usmd’ ‘U‘Qa Iflt‘b‘Q’ C/‘vO 5 14 15 3. (9 points) Figure 4 shows the A-plane for three separate second-order systems for which P(D) = 1. o (3 points) State whether each system is unstable, marginally stable, or asymptotically stable (you may write your answer in the corresponding A—plane). ‘ o (3 points) Sketch the zero-input response for each of the three systems in the graph provided. 0 (3 points) If the system is asymptotically stable, state whether the zero—state unit-step response will be overdamped, underdamped, or critically damped (you may write your answer in the corresponding A-plane). Zero-Input Response exponentialla 0&6? I? -SH7U50|&- Re{}t} O t (on u a,on uni-T flea-ugh). feal Parts Im{ M Zero—Input Response C‘ + CZ—L Qmeam growth Re{)t} 0 t ' QM?) WWW (‘ob'lrs an (2) magma? ans 0 Im { A] - Zero-Input Response exp a men "(mo-Q- @ “Wt” _ M2) (902. root m th. Re{ A] 0 t Figure 4: Location of the characteristic roots for three separate systems. 16 ...
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This note was uploaded on 03/17/2008 for the course EE 350 taught by Professor Schiano,jeffreyldas,arnab during the Fall '07 term at Pennsylvania State University, University Park.

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ExamI_s04 - /A EE 350 EXAM I 9 February 2004 Last Name 5 915214 Q a First Name ID number(Last 4 digits Section DO NOT TURN THIS PAGE UNTIL YOU ARE

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