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Unformatted text preview: /A . EE 350 EXAM I 9 February 2004 Last Name: 5 915214;} Q a; First Name:
ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Weight Score ‘ Test Form A Instructions
1. You have two hours to complete this exam. . This is a closedbook exam. You are allowed to use both sides of a 8.5” by 11” note sheet. Calculators are not allowed. . 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. (7 points) Consider the signal
f(t) = 4 6"”. Is the signal f(t) an energy signal, a power signal, or neither ? If the signal is energy signal or power
signal, calculate E; or Pf, respectively. °° °°2H=
E; =j FluDc“: = I6 e it
_. °°2.+. _ ‘2‘”
‘3ze££="e°“6 back/«Sie— TIRQ. Jamsl ~PH=3 ‘u an energy JTdnu/l
E4 2 I6 is 'Clm'Le. 2. (8 points) The zerostate response y(t) of a system to an arbitrary input f(t) is 1/0) = e“f(t  1) Is the system timeinvariant or timevarying ? Justify your anSWer. rﬂc sagLem L; ﬁlme—Invaruan‘lts °”&'
‘For Wm WM 9; t We 7. mate. mi— &,ca 2 ebﬁtn) = e" —F(t  I 7) q n& got Jr) : éCt—ﬂ _(:C_L__, ,7) Be cause. a‘ (t) 15 a (1‘: *7} tho. ssEam is 'lrtm°'\’°'I". 3. (10 points) The zerostate y(t) response of a LTI system to an input f(t) is W) = (1  t) [11(1)  “(t  1)] Figure 1 shows the zerostate response y1(t) of the same system to another input f1(t). Find an
expression for f1(t) in terms of f(t). { Hint: Start by sketching y(t) in Figure 1.} Y1(t) Figure 1: The zerostate response 3/103) of a LTI system to the input f1(t). P042, fha£ we. can exﬂrw g'HA 09 JOE) d(1‘:"‘)x a (t) ~d Cf}? yﬂt) 2 ate) '(7 H735) owl tl‘l sdxtem I) LTI) Problem 2: (25 points) 1. (10 points) A ﬁrstorder system with a DC gain of ten is represented by the ODE 13(1) + 0 11(1) = 5N): where a and b are realvalued constants. Given that the zerostate unitstep response has a risetime
of0.]1n(81) sec, determine the numeric value of the constants a and b (simplify your answer so that
it does not involve the in function). For a. {5'5 arng 5034:2014 with fromoJ 'lu‘ c pix (to, fl In this 0:52 or a" = 2&ng : ’0“ C9) 2 ﬂ. (.4
ty 0 ‘ 0 Z 2. (7 points) The characteristic roots for a LT] system with P(D) : 1 are located at —]O and —1.
Suppose that you are assigned the task of reducing the rise—time of the zerostate unitstep response
for the systemT and, that you are only allowed to change the location of one of the charcateristics
roots. Which of the two characteristic roots should be changed, and should this root be moved to the
left or the right along the real axis of the Aplane. ln order to receive credit, justisfy your answer. In (7s) (29. (79 the, QM»
Far “ ware'5'sz [AIL/'5’ the Y‘G—Sfeﬂlz' Q" (t 5 C. Q W
a ) l \acr AcidHon “chode’WWI 3‘4" pa ('EICV 1'" ouoer ‘b @9055. the. nsz—‘E'MQJ we 02d, '59
"lch 'thQ routh “3E aQO/"vﬂeﬂw
Nate. time tho, Mwﬁﬁé J \wtm to zero
0 the Maya... 0L6 (v.03) 3. (8 points) Write a MATLAB mﬁle that will plot the zero—state response of a system described by the ODE
1705) + 211(t)+ 21105) = P(D)f(t), where P(D) = 2D + 1, to the input
1'05) = e1 sin(4t)u(t) over the interval 0 S t S 5 using a time vector with 250 points. Q = C") 2) 235
P = CZ, 113 1r. = “Pea .ac what1:3; 8_ : ,Q..Snm CﬁFﬁt); xlwbel C.’ f‘WW— CSUQ') ﬁlabQ/l (J responseJ) Problem 3: (25 points) 1. (12 points) Find an ODE representation of the circuit shown in Figure 2 that has input voltage f(t)
and output voltage y(t). Assume that the operationalampliﬁer is ideal and place‘your answer in standard form
dny dn— 1y W+an—ldtn_l +“'+aay: m‘— +bm—l 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). Figure 2: Active operational ampliﬁer circuit. 2. (13 points) Consider a ﬁrstorder system with input f(t) and output y(t) that is represented by the
ODE 31(1) + 631(1) = 91W)
The system is driven by the input f(t) : 26—32 for t 2 0, and y(0+) : 1. Determine  (6 points) the zeroinput response yn(t),
o (6 points) the zero~state response yu(t), o (1 point) and total response y(t) for t 2 0. Zero  Inlet resf 9‘" be; (9’ 4 5d— =03 gCoﬁst
et
9v»: ﬂtézo —._.> dyeing): (1.2. 1:20 ChooSe, C": I to Jp‘EhgEi the, mrktaQr CartoonEton, 6‘lr Zero shade 3+ 6a : lﬁe'ab J a(0"’)=0
am (e) :. c‘e—Cb +
 _ ~31: (hog; thcﬂ— «3 75 Q31? “’ m % a” (4:) — at» 9 {ha okdrwéenyk 25¢ aa‘lslm) 11 Problem 4: (25 points) 1. (7 points) Derive an ODE that relates the input voltage to the output current y(t) for the circuit
in Figure 3 and place your answer in standard form dny dn— 1y dm dm— 1f
_ _ a : bm__ bm_
dt" +6" 1altnl + +a y dtm + ldtm‘l +"'+baf. 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). R1 A «0 O C VARz d W) HON Figure 3: RLC circuit with input voltage f(t) and output current y(t). branch rah/liter)th Ar "have or:  = VA (,2) Lillﬂlnwﬂi‘ in ‘Fﬂvow % %' Suki—6.5%,? (2)
Into (0 12 13 2. (9 points) A LTI system is characterized by the ODE
(D2 + GB + 9) y(t) = (—4D + 23) f(t) where is the input and y(t) is the output. If y(0+) = 0, y(o+) : 5, and = cos(t)u(t), ﬁnd
y(t) for t Z 0. 67(7) =— 7i1+61+7 '= 0+3)“ :9 7s, = ’22, = —3 +— ~34:
Masqea + cite, 1:20 —————o (' Lib + zen—(4;):  q geecw]; + 25cm: . tumt +28ca54: a? (t) 2 Acost .,. Osmi' '72'20. (DL+ 6b +?)0n/P :2. C' ‘10 42.5) 4ft) §f+6jr+ye= (99 +655)wa + ass—cmmt = k{Sm‘i‘s '1' 236056 WSM‘L 4' “(are .. gnawCIA: ‘iZ.
: 23 :7 HH 4 36 — I“! :9
gnaw66 '39 +Vb =2. => "29+ ‘3 5 9 86 ~64 = q r 256750
18:22.] HﬁzlwBs =8’=‘~>
“3*: (IiEa mt + am 305) :: armb) +Je (ab) 2 0.9. a (‘éYoﬂ
FMS" CR ach C; usmd’ ‘U‘Qa Iﬂt‘b‘Q’ C/‘vO 5 14 15 3. (9 points) Figure 4 shows the Aplane for three separate secondorder systems for which P(D) = 1. o (3 points) State whether each system is unstable, marginally stable, or asymptotically stable
(you may write your answer in the corresponding A—plane). ‘ o (3 points) Sketch the zeroinput response for each of the three systems in the graph provided. 0 (3 points) If the system is asymptotically stable, state whether the zero—state unitstep response
will be overdamped, underdamped, or critically damped (you may write your answer in the
corresponding Aplane). ZeroInput Response exponentialla 0&6? I? SH7U50& Re{}t} O t (on u a,on uniT
ﬂeaugh). feal Parts Im{ M Zero—Input Response C‘ + CZ—L
Qmeam growth Re{)t} 0 t ' QM?) WWW (‘ob'lrs an (2) magma? ans 0 Im { A]  ZeroInput Response exp a men "(moQ @ “Wt” _ M2) (902. root m th. Re{ A] 0 t Figure 4: Location of the characteristic roots for three separate systems. 16 ...
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This note was uploaded on 03/17/2008 for the course EE 350 taught by Professor Schiano,jeffreyldas,arnab during the Fall '07 term at Pennsylvania State University, University Park.
 Fall '07
 SCHIANO,JEFFREYLDAS,ARNAB

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