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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 zerostate response y1(t) of a. system to the input f1(t). Similarly, the zerostate 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 zerostate response of a. single system to two different inputs. 1. (10 points) What are the values of a and ﬂ ? Observe. theeE #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. realvalued 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: Aplane. @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/Se.
“limLiaali333 {OJ wlule at. i7ﬁ?=‘—O. 3. (8 points) Consider another system represented by the ODE
ﬁ+4g+5y= 3f(t). If the zerostate unitstep 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 zerostate unitstep 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 risetime 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 risetime, 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 ROIcc: 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 operationalampliﬁer' 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 realvalued parameters a, [3, 'y, and
6. Another agent observed that the zerostate 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 realvalued 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—ﬁcb {: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, magE hon/L :0 tie, %’e—T0rS—b9\:be’ SalesElan) aCo) = ef‘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+6t52I3k 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 realvalued 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 05"; 47;:01.
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—  . Changs— Imwbmcous v m
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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 secondorder 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'vlE‘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 yyQLQS t “t "t ‘1: F764,
1 *7 ﬂ; _ .1
—t t:
iiirte. + $7109 ; We +10 =‘—> (x. =9 yr,(t3:$’+~ie't tgo 779%, £04399. Solv'ttun MUJ+ Sw'LcJQ the ("FetaQ Canoa'éIo/u;
_,.  1: 
guy) gut3 +Ww) _ e (A(°$'—E+5Smi£)+ 9 + get aw): +1: ,4 +9 +7 :5 4=II
. Il
(7(0): +11/4L5i #1:) 5:32. 10 3. (8 points) Write an mﬁle 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} High2,1003) 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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 Fall '07
 SCHIANO,JEFFREYLDAS,ARNAB

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