examI_s03 - EE 350 EXAM I 10 February 2003 Last Name...

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Unformatted text preview: EE 350 EXAM I 10 February 2003 Last Name: Solutiong First Name: ID number (Last 4 digits): Section: ' DO”'NOT TURN THIS PAGE UNTIL YOU"ARE TOLD TO DO SO 12500 Test Form B Instructions 1. You havetWO hours to complete this exam. 2. This is a. closed-book exam. You are allowed to use both sides of a hand-written 8.5” by 11” note sheet. 3. Calculators are not allowed. 4. Solve each part of the problem in the space following the pquestion. 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) The zero-state responses of a single system to three different inputs, f1(t), f3(t), and f3(t), are denoted as y1(t), y; (t), and y3(t), respectively, and these signals are shown in Figure 1 on page 3. 1. (6 points) Which, if any, of the signals f1 (t), f2(t), f3(t), y1(t), y; (t), and y3(t) are causal? In order to receive credit, you must justify your answer using a single sentence. Only the “findig flit) ani j'lh) are (405423 the other SWM}, are ”one“; because, the; I’m/Q, flan‘EQro quveS ‘FW‘ t <0. 2. '(5 points), is the system causal? In order to receive credit, you must justify your answer using a single sentence. 7/942. 5(7 Sizer: 75 (95,) onSz 5L3 (vb) StartS More tha‘ ”7’4 Vb "5 ('5) :3 appilpfl . OOOCAVS¢£ ~56 W54- ‘U'O- Zero—rink 3. (6 points) Express f3(t) in terms of f1 (t) and f2(t) using only amplitude scaling and time shifting. $30k) _: ¥2(£‘Z) " “9| (It-bl) 4. (8 points) Is the system linear time-invariant (LTI)? In order to receive credit, you must justify your answer using a single sentence. 3e ( $152 #3553 ¢ bane-2,)— gamma), the, {75421-0 is not LT: Input Signal fi(t) Zero—State Response yi(t) -2 0 2 4 -2 0 2 4 time [sec] time [sec] Figure 1: The input fi(t) results in the zero-state response yi(t) for i = 1,2, and 3. Problem 2: (25 points) 1. (16 points) The location of the roots of the characteristic equation for a certain LTI system are shown in Figure 2. Suppose that the forcing function for the ODE representation is P(D)f(t) = 4e4t t > o. Figure 2: Location of the characteristic roots. (a) (6 points) State the form of the natural solution yn(t) for t Z 0. ‘H: Etna-'3 : C‘ + cze ‘bZO’ (b). (6 points) State the form of the particular solution yp(t) for t 2 0. Be. (M52, the. characfiemrb— ‘ic ear/ac'bw Q‘K-S the \‘00'6 9‘ : +‘1J 'H: (c) (4 points) Is the system unstable, marginally stable, or asymptotically stable? Justify your answer in a short sentence. 729— SdJ'bem L5 unJ flue, Bean/)2, tLQ. ‘ H (karmpéems'L-(c roof: 7\ :2 ‘1 7O anvqflh j} the. flick Clef“: 75 “05%“17/3- 2. (9 points) (a) (4 points) Consider a different LTI system that has a characteristic equation with roots located at A = —4, A = —2, and A = —8. Write a MATLAB m—file that computes Q(D) in the form Q(D) = D" + a,._1D"'1 + . - - + a1D + aaD° and stores the result in a vector Q. Q -.; ConV (LA!) ‘41] Zorn/(1:523) El)?])>j (b) (5 points) A certain LTI system is represented by the ODE g+2y+3y= f+2f- Write a MATLAB m—file that plots the zero-state unit-step response on the interval 0 S t S 2 using 600 points. i: 5 anspace. (0) 'L) 600)} F Z E511; Q = [1,273]; 5436‘) (P) Q) t) Problem 3: (25 points) 1. (5 points) Using source transformations, represent the circuit in Figure 3(A) using the network shown in Figure 3(B). Express g(t), R1, R2, and 01 in terms of f(t), R, and C. + N) R2 + R1 g0) y(t) C1 M) (A) (B) ‘ Figure 3: The RC circuit (A) and its equivalent network (B). V“, ”’7 I 2. (5 points) Find an ODE representation of the circuit in Figure 4 with input g(t) and output y(t) in terms of R1, R2 and 01 (do not replace g(t), R1, R2, and 01 in terms of f(t), R, and C) and express your answer in standard form. Figure 4: RC network with input g(t) and output y(t). (t— H») USma KVL: gufl :: RZU'Q 1—\d.(-b)) or “£7" E'fiL. Cl) “ranch Mahmsktp £14? 110 Capauuw: ‘ .- Q-UC ; C. ’— («(‘53’ 0: 0U: 3. (15 points) Consider a first-order system whose output y(t) relates to its input f(t) via the ordinary differential equation dy 71? Suppose that the system is driven by the input f(t) = 2 + 4t for t Z 0, and that the initial condition is y(0+) : 2. ‘1' 6y : 9f(t). (a) (6 points) Calculate the zero-input response for t 2 0. CHM: 7\+6 so 5—) &%1H=b= c..e 4:20 -é 6+: (b) (6 points) Calculate the zero-state response for t Z 0. - 6t 95 above) #5 (4;) :: C, Q t Z O, Beau/dz H9=Z+vg gran: [email protected]'b. +6+= 6+6ac’ ’2? 9‘) “—32 nhfl— SQ grit)” 2 ~6h 7 gait}: one +Z+6é t'0 5:) (1*‘2 (c) (3 points) Find an expression for the total response for t Z 0. a”): ggua+ 175/453 :: z + 615 1520. Problem 4: (25 points) 1. (12 points) Consider the RLC network in Figure 5 with input voltage f(t) and output voltage y(t). R c 1121+.) + -— (t) + C f(t) L YO) Figure 5: RLC circuit with input voltage f(t) and output voltage y(t). (a) (6 points) Derive an ODE that relates the input voltage f(t) to the output voltage y(t) and ex- press your answer in standard form. 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). - a |<VL= f-m: mm + ’2’2(0*)+ 4:50 «:3er + 3m d‘F/(l-t = KAI/J": 4- Jada.» +25%“ CD + Branch hluhmsbm '61P mill/day; ((113 : C(o‘l') 4- 'LL" 3 y'l'CJJ'C, (Z) \ Unminochna, il-b) M (D usmv, (Z) we) t J ‘ J—- «Mus : L: gm + ‘L‘c do”) + LC gay/r3026 +« #471? °’ ova/As + 15—: 5% + i: aw = at“r/ofltz (b) (2 points) Is the system strictly proper, proper, or improper? Justify your answer in a short sentence. fig, 5' g'laem l) I ropOY‘; because, m =’ ’7 7’ 2—. (c) (2 points) If f(t) = 5u(t), what is the initial value y(0+) of the zero-state solution? Justify your answer in one or two sentences. CCU-)1 ofl anflu 77216;) ’0‘] (FM Zero—Sixth veSpot‘Jea). fiacw lid"): {(6)5019 am} O‘clo‘”): m(o‘):ov) we marl: NW1- 3(o+) : 4(01') ‘ R .05") '- ’DZD'W :. 5 V (d) (2 points) If it exists, what is the steady-state value of y(t), y“ = limtdm y(t)? Justify your answer in a short sentence. In s-baajflnybffia 1‘0 ”190ch af/LQaYS a5 a. Thof'l" dMoQ— 50 JJ 30V 2. (13 points) Consider a different second-order system Q(D)y(t) = W) that has a characteristic equation with roots shown in Figure 6 and C : 3/5. Note that the distance from the origin of the A-plane to the complex root is 5. Im(}\.) Rem.) Figure 6: Location of the roots satisfying QO‘) 2 0. (a) (2 points) What is the numeric value of the natural frequency? g= W = @041er MM (b) (3 points) What is the numeric value of the damped frequency? - 3.8;"? was MW-ei— = : I-'(3/r31' ' 5" er 1’? (”AL : g I6/LS" =- if full/Sac. (c) (4 points) State the form of the natural response yn(t) for t Z 0. gulf) :— ;Pwnb—(H'Cas wit 4' Esther/1‘53 '15 2 o (d) (4 points) State the ODE representation of the system using numeric values for the coefficients of Q(D). Z. ’2. (3CD):- bL+ZPwnD+ W" = b+6b+2§ 10 ...
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