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Unformatted text preview: EE 350 Last Name: EXAM IV 7 May 2004 Sol of] 605 First Name: ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO   
4 25 Test Form A ‘ Instructions 1. You have two hours to complete this exam. 2. This is a closedbook 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. (13 points) The switch in Figure 1 has been closed suﬂiciently long so that the voltages and currents in
the circuit have reached steadystate values at time t = 0‘. At time t = 0 the switch is opened. Using
Laplace transform analysis techniques, determine the output voltage y(t) for t 2 0 given that V, = 6 V,
R = 1 Q, C = 2 F, and L : 2 H. Express your answer in terms of sine and cosine functions that do
not include a phase angle; your answers must not contain complex exponential terms. No credit will
be given for solutions using the classical approach covered on Exam 1. R t=0 Figure 1: The switch in the RLC circuit is opened at time t = 0. J 141': 1‘3an tso" #Co‘350V lanﬂ L(o)= = 6H. For 't 20) S’ﬁamdlk clrfwf re/mboﬂﬁb‘éwn [‘5' USInﬂ. Va “513,0, (Q'IVuSIorI YLS) : " s c. 2. (12 points) Figure 2 shows the circuit diagram of a bandpass ﬁlter with input voltage f(t) and output current y(t). Assuming an ideal operational ampliﬁer, determine the transfer function of the ﬁlter and
place your answer in the standard form Y(8) _ bms’" +bm_18m_1 + "'+b18+bo H!) = _
() F09) s"+an_1s"'1++a18+ao R1 uh r Wt) vo f(t) 8 Figure 2: Active bandpass ﬁlter with input voltage f(t) and output current y(t). 4—— ' F653 (CL)
V (s) = ____._..‘C' Pas) =
+ R‘ +3 JR\C,+l “Q amp/1 ver' 75 .. $9162 VHé) ('2) SC;
ﬂ — C
ﬁnally" Vﬁmét Ohm‘s Law, YGSl » V°C3)/R3 3).
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1. I, *5 SR\CI+‘
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3" fr 9.) ro‘Lag the h‘ .— H
response, 04 kg? m GU 43m arc ya”) (95"
bang/45$ walk” Problem 2: (25 points) 1. (7 points) The feedback control system in Figure 3 has a command input F(s), an output Y(s), and
a proportional controller with gain K. Determine the closedloop transfer function and express your answer in the form
Y(s) bms’" +b 13m'1 ++b13+bo F(s) s"+an—1s"'1+~~+ais+ao Y(s) Figure 3: Feedback control system with command input F(s) and output Y(s). K—‘—— )c
665) : $+l = M
l—is pg: s(n+r) ‘H I: S __________
Lg; ' 6“) = 56 Hr) + l .5 _____,____E..———
f___________,_
5MB 1" 2. (11 points) Another closedloop system, different from the one considered in part 1, has the closedloop transfer function
Y(a) _ 300 F(a) _ 52+aa+a2+75’ where a is a realvalued parameter. o (6 points) For what value of a will the steadystate response of the closedloop system to a
unitstep input be equal to three ? o (5 points) For the value of 0: that you obtained, is the zerostate unitstep response underdamped,
criticallydamped, or overdamped ? We tie/2:9— Y5) : DC who : ‘3
Foals” 8 300 = 3 2) “far 73’ :too :7 oozslr
mz+7r IQ
£0 @12— W
5c, .1 +S’or '5'. WWW”) t, [,6 3/50
the, 5 542m "V" _ steal s'bwlse. res/Ionst ; is:
Stdbrag Obedi— pale roov‘EIOﬂ) "gr OC' d’;+5—: 51+§s+loo :0 For. a; ; +5 the, polo: \+
m
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are, Complex, (oﬂdaaa'£2.§ “pg go
Unt£’5’£€£ mygonsc :5 un/ércpamQQJ: 8 l 3. (7 points) In order to determine the partial fraction expansion of the transfer function Y(s)_ 23+1 H 2——
(s) F_(s) s +532+83+4’ the MatLab Command residue is used: [R, P, K] = residue(B, A); R = 1.0000
3.0000
1.0000 P = 2.0000
2.0000
1.0000 K=EJ o (2 points) Specify the numeric value of the row vectors B and A.
o (2 points) Write down the partial fraction expansion of H (s).
O (3 points) Find h(t), the inverse Laplace transform of H (s). 6: E201”) ansz HallsJr, ﬂ I 3 __ c
H“) 2 5+2. + (Sn—)L SH
24: —b
Ha ; eztuob) 4— 34:6 was) e wit) Problem 3: (25 points) 1. (15 points) A system has the transfer function 105 .9 HM : (s + 10) (s + 1000)” Sketch the Bode magnitude and phase plots in Figure 4. 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 Do not show the 3 dB corrections in the magnitude plot. 1+! 0 f—P‘on W
dud  (d>w+’03(ﬂw+loy) 10 24 1 (0 [a lo3 I01 Reg Figure 4: Semilog paper for constructing the Bode magnitude and phase plots.
11 Ltd] 32‘.) 2. (10 points) The Bode magnitude and phase response of a secondorder LTI system with no zeros
(m = 0) is shown in Figure 5. Find the steadystate response y(t) of the system for the input f(t) = 3 + cos(1o’t — 45°) + 106 cos(106t + 45°). gﬂ‘t) '= H9033 + "+9le Co$<joz+~qg° 4. XJKdIoo)) + \o‘ I H9105), (@0054; + 15° 4' 2;th )) m 6mg, Ina ﬂt'éugae otan Fh‘om— HO‘L ﬂ
‘ . H'(o sIXo"
H90): l ngo)MB—,o J A d) 0‘ 1‘ #70"
ngtooH J czOJex mil lug/ch "Io we gm mgzoepl mi No‘te tha£ ﬁe, ﬂrafh 1W5” the}? A‘iﬂﬂhjé) 7 OD 1063‘H0wevgf WQ’
d tJ the. ZADQ' orgér‘ “a In I) #Qéueﬂ
a g ‘ '£ HISO a£ ﬂ
ﬁt .5 ’180 k I _ % ) ‘ V0 / q )0 how m wr‘ﬂ? "0 Eras l2 Magnitude (dB) Phase (deg) Bode Diagram —1 0 10 10 1 3 1o2 10
Frequency (rad\sec) 1o 104 Figure 5: Bode magnitude (in dB) and phase (in degrees) plots for the secondorder LTI system. 13 Problem 4: (25 points) 1. (12 points) A LTI system with 0 < C < 1 has the transfer function w: H = —.——,
(a) 82 + 2Cwn s + w: o (6 points) Draw a. polezero diagram of the transfer function H (s), and clearly specify the real
and imaginary parts of the pole locations in terms of C and wn. o (6 points) What is the distance between the origin of the splane (s = 0) and each pole '? Express
your answer in terms of C and/ or w". ’L
5L4 ZPwnS + wn 90 5'32. : 4w : WWW: whith Because. 0 (‘F 4.5 “the, pate) cure Comp/ex— rogugabj suz = ’70Wn tint/NW" {Land the, ‘U‘o. 90?}:er Thea/em} 9,: Omis.3”+:m 13‘3": ﬂvwaL+waPf9 (QR
The O’t‘S'LaﬂCe ﬁre”. 0f\ (0 b ‘e‘ﬂe‘r cop‘f
t; the Maﬁa/Va» “Pré “"7. 14 Fab. 73 Q 2. (13 points) A LTI system with input f (t) and output y(t) is represented by the ODE
:7 + 20 y + 200 y(t) = 400 f(t). o (4 points) Find the transfer function H(s) : Y(s)/F(s) of the system and express your result in
the form H“) __ Y(s) _ bms’" + bm_1s"‘"1 +    + (115+ (10
_F(5) s"+an_1s"‘1++als+ao '
o (5 points) Specify the impulse response function h(t) of the system. a (4 points) Suppose we need to experimentally determine the DC gain of the system. A unitstep
input is applied to the system, and we wait T5 = 5 7' seconds to measure the steadystate response
in order to calculate the DC gain. The parameter 'r is the largest time constant associated with
the natural (response of the system. What is the numerical value of T5 '? $2"; 10?th Coﬂcpl'EW/‘S "’50 Zero anJ/ the 167[@C0/
trdnrﬁo/‘mi/ boﬂ JILJ% the, 006? 52 Tm Jr 2os ((s\ + 200 rm : woo PCS) Compl 2152, {he Spa/aw: .LHSX : 90" =— wca 1
(5 +tof +100 (S‘HD) + I0 Us”? #12. ‘lﬁt‘uS‘ch‘m (a Sr 3"“; Sing'01 (1") “6—3 m2 15 ...
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 SCHIANO,JEFFREYLDAS,ARNAB

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