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Unformatted text preview: EE 350 EXAM IV 19 December 2003 Last Name: 8 Glut \ é) [LS First Name:
ID number (Last 4 digits): Section: Test Form A Instruct ions 1. You have two hours to complete this exam. 2. This is a closedbookexam. 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. (12 points) A system is described by the ODE
W) + 3 W) + 2 W) = 4 N) Using Laplace transform methods, determine the zero—input response 111, (t) and the zerostate response y,,(t) for
110’“) = 1
W) = —2
N) = Hum. No credit will be given for solutions using the classical approach covered on Exam L 3:. Ya) — 39‘ my) + 2 Y(s) = “l PCS) 1   '  «I
5. Yaw—Saw) 3C0) .
‘ s (0') 1" ‘0‘3+3&‘C°)
i52+3s +23YC5) '5 HHS) + a at S C0”)+ ' (0’) +3 (0")
Y(5 3 Li ‘ PCS) + '2. 35 +2.
> 52535” M ‘ f—‘__/
j {yay co] 1 ﬂaw]
7"“.— . ‘ “91—1 ‘2 _, A 4. L + ’<:¢_t\z. 2. (13 points) Figure 1 shows the small signal model of a FET ampliﬁer with input voltage f(t) and output voltage y(t). Determine the transfer function of the ampliﬁer and place your answer in the
standard form Y(a) _ bma’" + bm_1s’"‘1 + + bls + bo _F(s)_ a" +an—1s"‘ + +a15+ao vasﬁL‘jég, C0 l" £0 (2) 130 0. ’lm mute VC‘zS (J) Problem 2: (25 points) 1. (13 points) Find the transfer function Y(s)/R(s) of the feedback control system shown in Figure 2
and express your answer in the form Y(s) _ bms'“ +b 43"“ ++ b15+ b0
F03) 8"+an_19"‘1++als+ao ' H(s) : Y(s) Figure 2: Feedback control system with reference input R(s) and output Y(s).
(”MIL “Cram “2, Infloqe, tin/"t1 /_ ’
RC3) ' 5 53+! $Ll+sz+5 +51+S
1+ "" 5[JB+J+I) 53+!
____,_,_,_ 25V+51+2$ 2. (12 points) Another closedloop system, different from the one considered in part 1, has the closedloop
transfer function 0 (6 points) Is the closedloop system BIBO stable ? Show work and justify your answer in a single
sentence. VJ‘Q Poles % tie c/osqﬁagoyf 95hr” are, ﬂocwéﬂ
.1: the, was ¢ 52 +w+ ., :. CS+2JL=O get/dude. 50%“ Poles out, :7: the LHP (.2$~Z)J
tko, Sdeem‘ :5 3160' Sable; o (6 points) Suppose that r(t) = tu(t). Does y(t) approach a steadystate value ? It the answer is yes, what is the steadystate value of y(t). If the answer is no, explain why y(t) does not achieve
a. steadystate value. U52. the, ‘meQ‘W/VlVe, theorem H5)
«eon—SELV—(Jl ﬂo;\;/g‘x ill—‘2‘
a“  s—ao ”9 ’ 55° 5"+‘15+‘1 5
5'5 :7. Problem 3: (25 points) . 1. (16 points) A system has the transfer function _ ‘ (5+1)(s+100)2
H“)  W SketCh the 'Brode magnitude and phase plots in Figure 3. In order to receive credit: 0 Use dashed lines to show the Bode plot of each term of the transfer function and a solid line to
show the composite Bode plot. 0 Indicate the slope of the straightline segments and corner frequencies of the ﬁnal magnitude and
phase plots. 0 D9991? show the 3 dB corrections in' themagnitude plot. 10"l (5/100 1")?
10‘(5/;o+.)(5/;0W) Tje' {Inﬁfmlé/‘Q’ Q.I\(/€. ,7sz plo'b) % {ff/’75 W fémyl)
@ GHQ, /ﬂt[(Cové¢Q/~ % £44}ng Ema} an Wait: ab. ““993 MB Figure 3: Semilog paper for constructing the Bode magnitude and phase plots.
' 8 (£0.10 “3% out IO LWJ/x<) 2. (9 points) A secondorder LTI system with no zeros (m = 0) is characterized by the Bode magnitude
and phase plots in Figure 4. o (4 points) Specify the magnitude and sign of the DC gain and the high frequency gain. Express
the magnitudes as the ratio between the system output and input, rather than dB. bc. 5min élwowle 3V0o06 or 10075 saga/n
AHA/o) ' z. +130“) 'Y "°° ”Wk (reﬁne/L ‘H‘CJ )i O ”V“ “ﬁve”? gar}; ‘
'60'." 21H?» F70 ‘ . ‘ r0  o (5 points) Determine the damping ratio of the system and specify whether the zerostate unitstep
response is underdamped, criticallydamped or overdamped. Phase (deg); Magnitude (dB) Bode Diagrams 2 101 10
Frequency (rad/sec) Figure 4: Bode magnitude (in dB) and phase (in degrees) plots as function of frequency ( in rad/ sec)
for the secondorder LTI system. 10 10 Problem 4: (25 points) 1. (13 points) Just before the Thanksgiving holiday you and your faithful assistant Big Willy ﬁnish a
long series of experiments to determine the transfer function of a new product to be released before the
upcoming Christmas shopping season. After a long relaxing weekend you return to work and discover
that Big Willy’s attention to detail, or lack thereof, is going to cause you a major case of Monday
blues. Big Willy forgot to label the axes on the pole—zero map shown in Figure 5. Fortunately, you
had recorded in your laboratory notebook that the system has a DC gain of positive two, and that the
complexconjugate poles have a natural frequency of \/§ rad/sec and a dimensionlessdamping ratio of onehalf. Figure 5: Pole zeromap of the transfer function H (s). o (6 points) Specify the values of a. and b in the polezero map. { k iLL
The ("3” who. Y°l¢ pcu'v praguc¢5 ﬂ, 9/ «dz/‘0» «<— rm n
Jamar” In“, ,% ﬁg, +runsfer Ln; E10!“ ’2. (3+1, +fac)(s+by_a) = 5" + 21:: + bZI'a.
25: 420% => b= PWn = i"? sz at = s—a: — =ﬂ o (4 points) Specify the value of c in the polezero map. S+c.
”(53 =’ :
51+ 2b: +61%?" .S+<.
574 f§$ 4&— .— 6664v$e H'COW = 2. =— DC gong we Imago. c—IO. o (3 points) Specify the transfer function H (s) of the system in the form Y(s) _ bms'" + bm_1s"“1 +    + bls + be
F(8) _ s“+am_1s"'1++a18+ao ' H(s) = 11 2. (12 points) A secondorder LTI system has the impulse response function
Mt) = 30 r” sin(4t) u(t).
o (4 points) What is the DC gain of the system ? o (8 points) Is the zerostate unitstep response underdamped, critically damped, or overdamped ?  If the response is underdamped, determine the value of the natural frequency and the di
mensionless damping ratio. — If the response is either critically damped or overdamped, state the time constants associated
with the exponential decay of the transient response. ‘1
CS +231 +16 H65) 3 301 ezt51n(blt)uft)}: 30 _. [2.0 5" + H5 +20 m Dc. am a Moi = 6 12 ...
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 SCHIANO,JEFFREYLDAS,ARNAB

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