examI_f03 - Test Form A 1 You hmtm We to comm this mm I 2 m...

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Unformatted text preview: Test Form A 1,. You hmtm We to comm this mm. , I 2. m. ii 1 mm mm You are allowed to me both sides of a. 8.5” by 11" note mm. 3. Coleman on not allowed. 4. Solution}: port of the problem in the space following the question. If you need more space, continue your edutim on the sever» side labeling the psge 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 my page; from this exam Loose papers will not be accepted and a grade of am will be assigned. \ 6. The quality of your analysis end evaluation is an important es your answers. ”Your reasoning must be preclse and deer; your comglete English sentences should convey whet you are doing. To receive credit, you must show your work. ' Problem 1: (25 Points) Figure 1 shows the zero-state response y1(t) of a. system to the input f1(t). Similarly, the zero-state response of the same system to the input f2(t) is y2(t). flu) y1(t) 4: LIJ any» 5° {it‘d-“2’" is Alana/Vito. Figure 1: The zero-state response of a. single system to two different inputs. 1. (10 points) What are the values of a and fl ? Observe. thee-E #20:): QyJC‘D— 12%,“:"13 “For - This means that {1063: Z‘F'H‘) —Z.F'(t—l).) a"; 5° «2": 23‘ (1;) Butt) : 2'5““) —- 25M“) 2. (5 points) Is the system instantaneous or dynamic ? Justify your answer in a single sentence. (fie Sachem kit‘amic becwsz 'H‘Q' mibflv'é ;,(t3 Gee/emas On Poul: volt/Q5 the— Input 'F‘Hz). (See Commant m Edy/‘2’ lam) 3. (10 points) A signal f(t) is represented as f(t) = 3 Z [u(t — 3n) — u.(t — a. — 3n)] where a is a. real-valued parameter in the range 0 < a. < 3. If possible, choose the value of a, within the specified range, so that f(t) is a, power signal with power P; = 3. If it is not possible, explain why. *9 ~ -' - LA. -aL-) Ht) = ”30:90: 30,- ace) — Buwc) 3 (4: 3 (134(3) 964.! —3) Problem 2: (25 points) 1. (9 points) Find the characteristic roots of the system represented by the ODE day 42y dy df 2E5+2F+2E—82£—2f(t) and plot the roots in the A—plane shown in Figure 2. Im( A) Re( A) Figure 2: A-plane. @005 $3.; 279+ 2.? so 9' 7\(’/\Z+ ZEN-+2) $0 7‘:“0 ‘ 7\7.’3= 'l ill“ 3 'li‘j 2. (4 points) Is the system in part 1 asymptotically stable, marginally stable, or unstable ? Justify your answer in a single sentence. Tjfie Sthem :5 mariqugg Sfiule, bQCou/S-e. “limLiaali333 {OJ wlule at. i7fi?=‘—-O. 3. (8 points) Consider another system represented by the ODE fi+4g+5y= 3f(t). If the zero-state unit-step response is oscillatory, what is the frequency of the oscillation in rad/ sec and what is the time constant that specifies the rate at which the oscillation decays ? If the zero-state unit-step response is not oscillatory, specify the time constant(s) associated with each natural mode. on): 79+” + 5‘ =0 7M1: ‘2 :t W = —2 r; 7k PaSponJe Is an €fipon€n£(u% agency/0&4 SIM/Smut. 7‘51 ‘ ‘Z‘t 3' "Pu/n ti'u/JL ‘[email protected]&Wnca 7" ogchlw‘lb‘lonz “Ema cans‘bqn'b 4 jecccy— '- 'C : 4. (4 points) You are just settling down to enjoy a cold one at the Horse 8!. Angel Bistro to celebrate your success at improving the Rocket Racer roller coaster at Kennywood Amusement Park. Based on your new design, the rise-time of the Rocket Racer to its maxium velocity of 200 mph was decreased to 5 seconds. Before you have a chance to take your first sip, David Salvia pulls up a chair and talks about his recent trip to Cedar Point where he rode a roller coaster with the velocity profile shown in Figure 3. Velocity Profile for the Vomit Comet at Cedar Point velocity (mph) 8 8 0 5 10 15 20 75 30 time (see) Figure 3: Velocity profile of the Vomit Comet at Cedar point. Given that David enjoys high speed roller coaster rides that reach their top velocity with the shortest possible rise-time, should he return to Cedar Point, or should he try the Rocket Racer at Kennywood ? Justify your answer. )0 ’Lo fl e—‘£!VM¢ “(TV ’X. 3,569 (Fl/om Had/(3.) rollov- coasber Vonn'b Com?)E 20¢,ka ROI-cc:- 75” z 558’ Dabs/‘9» Salute» Sludge {Iva "Zita Qpalge'l: gamer co‘é lLQnmaw/aogx Problem 3: (25 points) 1. (13 points) Find an ODE representation of the circuit shown in Figure 4 that has input voltage f(t) and output current y(t). Assume that the operational-amplifier' is ideal and place 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). N) Figure 4: Active RC operational amplifier circuit. k. KVL LOOP F}; —F(‘E) -_—_ ‘Lttl R -|- Vc if) 3 Ki (1:) 'l’ VC(O)+ fifyltyft O KvL- Loopb‘. Lam + zkam =0 9') M‘V'J'Wt) '2. as on ”)5 Substu‘lw‘hna llt) = -L}a(-t) 10450 the. -C\\r:-£- a (”@851 am , J.— H—a = - Hac+3K+ 0,103+ 45g waft “Arte ren +5in both $\£QS 2. (12 points) You have been recently hired by the National Security Agency (NSA) and your first assignment is to reverse engineer a dynamic system being developed by a terrorist group for a. weapon of mass destruction. Reports from a field agent reveal that the system is described by the ODE it + a 1103) = f“) and that the system is driven by the input N) = £3 + '7 e‘“. for t Z 0. However, the agent was unable obtain the values of the real-valued parameters a, [-3, 'y, and 6. Another agent observed that the zero-state response of the system to the input f(t) is y(t) = e e"31 + 9 + 6te‘“, but this agent was unable to obtain numerical vaules for the real-valued constants e and oi. Using the information available from both intelligence sources, determine the numerical values of the parameters a, ,6, 7, 6, e, and d), and place your answers in the table given below. 7‘ + a; =. O I"; {oliows From the cummurmao egaw’wn on»: a thw": }h('a = Que—fic-b {:20. 9mg- so alt) can be. pQY'LIDn?& 03 F3 ’ 5¢U=\'= Qe 4‘ + 7+61£e W” (,________—J ”/4 ham. mead; Paf‘bleluV‘ ‘°l"‘b‘”’b 3,005) g ebbtdvb y b if) Given the, 'CW‘M % the, purbtcvlcxr So (1...: 8 = ,5 = 3 lw'tlofl’ we, mag-E hon/L :0 tie, %’e—T0rS—b9\:be’ Sales-Elan) aCo) = e-f-‘I .' 2 Because, 3053 ‘5 (small so 9.: ‘7. To 9,1,9. {’45 7 Queue WW) .oto 13k, ODE: W + 39? =- 63? a => 6% ' [it + -34: _. 6+7€ 5P: 9+6t52I-3k Problem 4: (25 points) 1. (8 points) The RLC network in Figure 5 with output current y(t) is driven by the input voltage f(t) = Vl + Va u(t) where V1 and Va are real-valued constants. Determine the initial conditions y(0+) and y(o+) in terms of R, L, 0, V1, and V2. L ONO. f(t) C R Figure 5: RLC circuit with input voltage f(t) and output current y(t). Circw‘b 0-5"; 4-7;:01. ______________________________. + V1,“) — InJuc‘lzw- baiuua as on slior'b elm/r5 Ctkpauiar bohms qs dn open Chm/:6 Vcco-) 5 VI l l — - .- Changs— Imw-bmcous v m 9(0‘3 -' V¢(O D/R " vI/R the, (Wow/{700 J To 4...!1 9C°L 1CW4; Wee, thw‘a WW3:- Lj’fi. 0.0.51 So 2“}— : VL.(O‘I') USU/fl KVL: .9({\ == V1.06) + Vc It» um!- dt It =o+ L. ' wkere. @(O*) = V,+\/,__ mark Vc (0+3 2 V; (0‘) = V, beau/5e. voltuap. cannu'b cinema), Initqfl‘EH/kevufl; 0&4:va 0" cafla‘1éoc 9 2. (9 points) Another second-order system is represented by the ODE g—2y+§-y<t)=f<t). Determine the total response y(t) for t 2 0 given that f(t) = 10 + 17 e", y(0+) = 1, and g(o+) : 1. Q(7~)= 22—2) +:§I:;o 7‘57." Hi'vl-E‘I- :> +ITJ~J£ 51W): etCMsi-‘n ms.) t 20 Bacwse, O amoQ' 'l “are, not Nuts 'fi 63(3):?) yr L't'h =- a, +fé.e.—t. a? —. Zap? g‘fi? : 10+ [7&- Solvlnfl ‘pVY‘ 0C; an£ IS- yy-QLQS t “t "t ‘1: F764, 1 *7 fl; _ .1 —t -t: iii-rte. + $7109 ; We +10 =‘—> (x. =9 yr,(t3:$’+~ie't tgo 779%, £04399. Solv'ttun MUJ+ Sw'LcJQ the ("Feta-Q Canoa'éIo/u; _,. - 1: - guy) gut-3 +Ww) _ e (A(°$'—E+5Smi£)+ 9 + get aw): +1: ,4 +9 +7 :5 4=-II . -Il (7(0): +11/4-L5i #1:) 5:32. 10 3. (8 points) Write an m-file that determines the characteristic roots and calculates and plots the zero- state response for the system represented by the ODE .p r—~—'1 (D=+2D+1)(D’+10D+1oo)’(D‘+8D+4)y(t)=(D+3)(4+6e-*) L——\—] for 0 < t < 2 using a time vector containing 2000 points. F 070 DQJCl “Q. P q n9» Q P =1 C l) 3 j 3 Q z: Conv( Corn/(El) 2)]3} High-2,1003) Com! ([1) 10,100], f1) Y, Vj)>3 67° D‘l-‘pifia €06.25 09 CM. oLuymLey‘IS‘éic e ae‘irlon roots (a) 70 Dbflme, Elme— «.pr 91mg gamer-«2E2, Input 1': =‘- Qinstre. (‘0’ '2.J 7.000); —§- = L] + ekepr'fiLfij ”70 {Not bk. fQSu/é’ Plo'l'.‘ (t, -F) Xiuioei L‘ fume. Eseol’) aiatel (‘ Rmp I. 7&ng /) {mi/9' (' Zero-{bait aesponSe 'CUY‘ +2: ‘4‘!” 684{"é1 ) 11 ...
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