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BB 350 ‘ 28 September 1998 "1049: Name: . I Sec.: Test Form A ':I..:‘v__7Prot>lemi'1: 325 goings! ‘ TQ‘c‘o'nsider {hﬁ'ifur‘rcﬁon fit} shown in Figure 1. " fit) Figural. momma).  'jA “’1'.” ' .‘ 7.1I“IIb"l.f' r1 ' .4 Now. let g(t) = , a W version of ﬁt) from Figure 1. a! (5 points) . Sketch cm on the graph provided in Figure 2 and properly label the graph. E0) L 'L “3 4 S c.
F igurc 2. A blank graph for sketching g(t). I Wait JcMch (SL0 isa’hlme, exgcmdecl \Jersfon 0‘C
{3m} expanded b3a¥uc£oro¥ ’Z. '  '  mum.w,..,.m.:uﬂ r, ' ' ' WWW,
.m (Ahmadun ummWW bl (5 points} ‘ Sketch the individual component} of am on Figure 3 and verify that their su gives the same result as you found in {b}. gm ‘9'; .. ' . .. .. The {mm m0 {helihkee
‘ I Jeems isiAevL’cfcd’cp Figure 4. An LTI system with an input, [(0, and a zerostate response. y;(t). Now. use ch] as the input to the 5 am ' '
the system to am is hm. 9 LT! systgm shown m Figure 4. The zerostate tesponse of at» 350) F gm: 5. The LTI system with input, g(t), and zerostate response, y1(t). w“. amJajﬁu. again,— u ma.am.nm..'mr ﬁnk. mumwhkmmmhumlu. nmwmi; m aimnitsaawmgﬂnmmhmuhm.u l‘llulh“.¢lﬂdhsmdy t..mbu...:,..mu..‘ L m, . . , ,. u .u u. K. . i
. ‘ " '  c) (Spointsl Since the system is LTi. hit) can be expressed as a linear combination of timeshifted
' versions of y,(ti.__ Express y,{t} in terms _of the function mm. prop'ehtielszf ' ‘
= i (m r "HIM who) d) (10 points} is the signal sum in Figure 4 a newer signal. an energy signal or neither?
' If it is a power or energy signal, find the measure for hm. hm Ska : 0 MW) gigﬂa‘ “
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: .JE 2£1+L}Jt’
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: J. 3 Problem 2: [25 points! This problem contains four parts labeled 3) through d). i
Each part is completely independent of the
other parts and so each can be solved separately. Point value's for each part are indicated below. Consider the foll0wing:
Given a system with the input/output relationship ylti :2 flat  [H + y where.
ylt)  System output
ﬂtl  System input
and a. I}, & y are real numbers with the
constraint that a a: 0
unless Otherwise noted. {9 points} Using an appropriate proof. determine if the system is linear or nonlinear. If .—.  you determine the system to be nonlinear. suggest values for at, B, & 7 which would
make the system, linear. al z_....iul. r ~_4‘_.....—..Aa. .A bl (9 points) Using an appropriate proof, determine if the system is timevariant or time invariant. lfryouidetermine the system to be timevariant. suggest values for a, 13. & 1
which would make the system timeinvariant. _ cl (4 points) Suppose. at t  Ss. an input. ﬁt} is applied. Using at  1, and 4 0, what‘
must be the range on [i for the system to be causal? cl) (3 points) Suppose a. =1,D  0&1  1.
memoryless? Briefly explain your reasoning. 6% V 61m: 0?) 371 ﬁle) —% aptlvﬁlﬁ «was
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{aw ‘ smarterdame) ——= jeweledpm
W  V g = [it'ct(°.§*’f)tlu¥t.t°&'pﬂ +K
at Jaw weld“) ' ' Does this system have memory or is it 6) We): How) +6 511112: W 1311:) sag—the) : {Jaw—{0+}; W"H4,,(£3'21°I "(ﬁea) " "
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‘ ,5 W Problem 3: £25 gointsl In the following circuit. the switches are kept Open for a long time and then suddenly closed at
t = 0. m): ..... ._ m. . J.“ _.__.__ The ordinary differential equation relating the output, yttl, with the input, ﬁt], for t 2 0 is determined
to be ’ (R'fe+R’)gZ+iy(()=_Ld_f
,RJ dr 1. R, d
where.
R.= 20
R,= an
R3: 20
L = 0.1H . 3 a) (6 points} Find the initial inductor current. “0‘1. and the initial voltage. viO‘)
immediately after the switches are closed. bl (8 points] Find the zeroinput response of ym for t>0. cl (8 points} Find the zerdstate response of ylt} for t>0. d} (3 points) Is this system stable? Explain your answer. Problem 4: £25 Qointsl In the foll0wing circuit. the switch is kept open for a Ion 9 time and then Suddenly closed at
t = 0. . _..._...__~....—.u.—m..  The ordinary differential equation relating the output, yit}. with the input, ﬁt), for t 2 0 is determined to be . ' _
d2 +R. I l
_y+_y(;) =_._.f(()
d: CMz d: LC LC
where.
R, 1 0
R32 1 D
L r: 0.1 H
C  0.1 F
. . . .. dy . . . .
a] [6 pomtsl Fund the initial values yIO‘} and Immediately after the swutch as
closed. d! no. bl (8 points} Find the forced response of yttl.
cl (8 points) Find the natural response of ytt). d! (3 points! Is the total response. yttl. critieaﬂy damped. underdamped. or overdamped?
Why? ll ...
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 Fall '07
 SCHIANO,JEFFREYLDAS,ARNAB

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