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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. closedbook exam. You are allowed to use both sides of a handwritten 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 zerostate 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 (Itbl) 4. (8 points) Is the system linear timeinvariant (LTI)? In order to receive credit, you must justify your
answer using a single sentence. 3e ( $152 #3553 ¢ bane2,)— gamma), the, {754210 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 zerostate 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‘KS 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 zerostate unitstep 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: ‘ . QUC ; C. ’—
(«(‘53’ 0: 0U: 3. (15 points) Consider a ﬁrstorder 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 zeroinput response for t 2 0. CHM: 7\+6 so 5—) &%1H=b= c..e 4:20 é 6+: (b) (6 points) Calculate the zerostate 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= fm: mm + ’2’2(0*)+ 4:50 «:3er + 3m
d‘F/(lt = KAI/J": 4 Jada.» +25%“ CD +
Branch hluhmsbm '61P mill/day; ((113 : C(o‘l') 4 'LL" 3 y'l'CJJ'C, (Z) \ Unminochna, ilb) 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 zerostate 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 steadystate value of y(t), y“ = limtdm y(t)? Justify your
answer in a short sentence. In sbaajﬂnybfﬁa 1‘0 ”190ch af/LQaYS a5 a. Thof'l" dMoQ— 50 JJ 30V 2. (13 points) Consider a different secondorder 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 Aplane 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 MWei— = : 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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 Fall '07
 SCHIANO,JEFFREYLDAS,ARNAB

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