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Unformatted text preview: EE 350 ‘ EXAM I 3 October 2002 Last Name: Sol U'iZlO n S First Name:
ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Instruct ions 1. You have two hours to complete this exam. 2. This is a closed—book exam. You are allowed one 8.5” by 11” note sheet. 3. Calculators are not allowed. 4. 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. (9 points) The zerostate response y(t) of a. given system to the input
f(t) : 'u,(t — 2) y(t) = e—t Mt +1)— u<t — 1)]. o (3 points) Is the signal y(t) causal or noncausal ? Justify your answer in a short sentence. 30L) Sacwsa. #61:) #0 ‘Fer ﬂ ta»)
<7“) is mews40' o (3 points) Is the system causal or noncausal ? Justify your answer in a short sentence. ﬁe) 8:2.ch 'th maps dCt)
OCCMJ .bz'gmz ﬁe 1'an ‘ 'i: ‘FC'CD 7) “ff"e‘a) “the.
2. ﬁsher" is ﬂoﬂCawSaQ o (3 points) Is the system instantaneous or dynamic ? Justify your answer in a short sentence.
$¢cow5c 'He 85km 75 0‘” C‘KVSwOJ tAO— avifv't oi ﬁe. Vex/vb $1042 Qwrm knewioﬂyg #ﬁ‘éme
[AW—L7) afﬁx $70 the. 3&5?!” >3 Ogl‘aml.‘ . 2. (8 points) 0 (4 points) Is the input m) = w — 2) speciﬁed in part 1.1 a power signal, an energy signal, or neither ? If the signal is either an energy
signal or a power signal, ﬁnd Ef or Pf. °° z i “in > m s mm
E‘F 25.004: (‘5) £ = j: ‘ aneryy $.7nwL I + I’
Pg=ﬂmlf 2¥7/£)1£=)€<LA.#:SEJb :JZO; JTHl
T943 ~35 2 2 Taco T500 o (4 points) Is the zerostate response Mﬁ=€4M6+D—uU—DL speciﬁed in part 1.1 a power signal, an energy signal, or neither ? If the signal is either an energy
signal or a power signal, ﬁnd E, or Pu cr» l 2 _
E; :: §.iZH\Jé = 3‘.(’e“lygl_t —_—. X'e 2+0€£ 3. (8 points) Once again consider the LTI system in part 1.1) that produces the zerostate response y(t) = 6’t [u(t + 1) — u(t — 1)] for the input
N) = u(t — 2). In terms of y(t), ﬁnd an expression for the zerostate response y1(t) of the same system for the input
f1(t) shown in Figure 1. f1“) Ht) Figure 1: The input f1 (t) results in a zerostate response y1(t). We. Mei in Emma: ﬁlt) .46 av (um # way
out 'bme. shrcﬂxﬂ Info£5 ~Flt),  2%. +2.) 14 #(tw 4“ 6*) gas 2 ‘ “E” * "if—at + " *
4‘ — '2 I z .2 L Meﬂ Problem 2: (25 points) 1. (18 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) = 3 + 2e2t t 2 0. Im( A.) Figure 2: Location of the characteristic roots. 0 (7 points) State the form of the natural solution yn (t) for t Z 0. I + .
Ck4radzens'bic. TaitS 2 7i. '3 o) 7‘2 :‘2') as.) ‘1 : if 3501:) : Cl + (Zena: + e't (C5rost + 95:01:) #20 o (7 points) State the form of the particular solution yp(t) for t Z 0. Bacavse. ﬁg = 0 any. AL = ’g ~zL
grtt) .: 5,4: + cute 7529 o (4 points) Is the system unstable, marginally stable, or asymptotically stable ? Justify your
answer in a short sentence. f/lc. sayLen 75 mayﬁmy quélg because
7\, : o) wlmle, 32.1 73) an! 7H are. in ﬁe LAW, 2. (7 points) A certain LTI system is represented by the ODE
(D5 + 2D2 + 4D)y(t) : (2D — 1) f(t). Write a MATLAB mﬁle that o calculates the roots of the characteristic equation, and 0 displays the zerostate unit—step response over the interval 0 S t g 4 using 500 points. 70 59:th Poqu Q
Q : El) 0)&) Z) ‘1’ 0:];
P = EzJ ~12); 7a Fmﬂ rad(:5 % tic, cXmC’adzenj‘éf: 25w» (57/:
radio ( 6?) ‘70 ﬂat the Zeravrlsm'ée, vm'ASéeP r05fanJ€ ”Z; 7 mam/£2 ﬁma ValeW t :: 11(69ch (.0) ‘1) 50°); ,S'lsef (P) QJ't) Problem 3: (25 points) 1. (9 points) Find an ODE representation of the circuit shown in Figure 3 with input f(t) and output
y(t). If you introduce a node voltage (mesh current) to determine the ODE, clearly indicate this
variable in the circuit and label its polarity (direction). State your answer in the standard form dny dn—ly — dmf dm_1
+" +aay‘bmﬁ'7‘n‘4'bm—1 W+a"‘1dtn1 dtm_1 ++bof Figure 3: RL circuit with input voltage f(t) and output voltage y(t). KVLz'F:iR.\—a_$ C=——'£—;—T:—:—. 0) L
:, R R ——'
73:4 73—; mg: Mt 2. (7 points) The circuit in Figure 4 represents a single data line on a computer memory bus. The data
source is represented as an independent voltage source f(t) with source resistance R1 = 20 Q. The
bus is terminated at the receiving end by a termination resistance R; = 20 (Z and a stray capacitance
C,. The relationship between the input voltage f(t) and the output voltage y(t) of the bus is speciﬁed by the ODE
. R1 + R: _ 1
11+ R1320; y _ Rica f. (U)
Because of the short spacing between adjacent data pulses, it is necessary that the risetime at the receiving end be less than 5 ln(9) ns for a step change in the input voltage f (t) What is the maximum
value of stray capacitance C, that can be tolerated ? Figure 4: Representation of a computer data bus as at RC circuit. .. «,1? #L :1 ﬂ = .. R1+ﬂL
QC?) ~ 7‘ + RszCS :3 ’ R. “1C5 Choose fr ﬂitR2.
C6 5" A? 11.17.
“IO
4 5.0/51 )i 10‘ I10 " SX\0
(7) H00 3. (9 points) Consider a ﬁrstorder system whose output y(t) relates to its input f(t) via the ordinary differential equation
dy __
d—t + 4y— 2f(t) Suppose that the system is driven by the input ﬁt) 2 4+2e‘4t for t 2 0, and that the initial condition
is 1.40) = —
o (3 points) Calculate the zeroinput response for t 2 0. ~ ‘14:
(9(7): 7s+¢1 :0 =9 7\.‘"4 =? 3‘5”” z 0‘3 a"
‘I 4. 0 ~
yacéb) '—' ah ('b) +%(‘b) :— C\€ “£20 ylo}: ‘2. 3C1 o (5 points) Calculate the zerostate response for t Z 0.
a“ 1+.) = 9e” 9* t 2 0 as “5°“?
’ ‘l'b JvLs'b’izu‘LQ if m4}; ﬁe 00C) gechse’ 7/; ~be . ~‘H: ‘H’ o
' ewe” » at 43/" +%/ ”f" 10 Problem 4: (25 points) 1. (9 points) Derive an ODE that relates the input current f(t) to the output voltage y(t) for the circuit
shown in Figure 5. 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). 11 2. (9 points) A circuit, different from the one considered in part 1, is described by the ODE ﬁ+y=2ﬂﬂ Given the input f(t) = u(t) and the initial conditions y(0) = y(0) = 0, ﬁnd y(t) for t Z 0. 0(7)) == ?\7‘ + 7‘ =0 :9 7‘13“!) 7.110
t
gut)  0.62 + c; t 20
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y" 6; :3 2 =9 0 ‘i" 6' = 2‘
f P
2’? (t) :: Zt 7320 :— C4€~t 4’ (Z. + 2t tzo
C:
U59. 9(0) 2(7QD 3—6 “1:0 4“"‘Q C‘ 9' 2’
.. C =. —C2
(0)20 : CI+CZ ’9 l
9 ’ 7—) C‘zz 7) C2 "'2' 12 3. (7 points) A 12V battery is sitting in a hut on a deserted island somewhere in the Paciﬁc. The positive
terminal of the battery is connected to one end of a 1/(21r) uF capacitor in series with a 1/(21r) [AH
inductor. An earthquake in the Bonin Islands of Japan triggers a tsunami that crashes into the hut,
spilling salt water onto a rag that connects the other end of the series inductor/ capacitor combination
to the negative terminal of the battery so that a series RLC circuit is formed. The resulting oscillation
is detected by a nearby ship monitoring a radio beacon signal at 600 kHz. What is the resistance of the damp rag ? Post: 'bswmmi. Circuit: C: if“: L= 5,74%
VS (I. L K V; =‘l2.\/
cg WQ.
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 SCHIANO,JEFFREYLDAS,ARNAB

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