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Unformatted text preview: EE 850 FINAL EXAM 14 December 2000 Last Name: 8 9' gift] at); First Name:
ID#: Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO 1 Test Form B Instructions 1. You have 2 hours to complete this exam. 2. This is a. closed book exam. You are allowed one heet of 8.5” X 11” of paper for notes and a.
calculator. 3. 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 4.2) Continued”. NO credit will be given to solution that does not meet
this requirement. 4. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted
and a grade of ZERO will be assigned. 5. 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. (4 points) Consider the electronic circuit shown in Figure 1. By inspection, state whether the circuit
implements a. lowpass ﬁlter, a. bandpass ﬁlter, a. notch ﬁlter, or a high pass ﬁlter. Justify your answer
with one or two sentences. f(t) Figure 1: Filter circuit. T/Ole. 90w Fre 5‘12”? ya in 75 Zero EQCGLVJ‘Q 9% DC. the Con/MI'EOP CL appéahj a5 an open CI’TUI  T’p‘a ar‘n the an, [ricer 1:; fro/orfronaz 75‘"
a f
the rw‘éco % {the IM/Qﬂeﬂfe, / K. aer C,_ 243‘ the grebuenﬁ /ﬂCV€c¢JeS/ £359. er/éﬁwxm/ C,
‘ 15:63
ﬂeecreaJeJ and C119. far/7 (ﬁcreaJaS. m I gualrﬁe‘lxvg, anagsls) the, Ir/éer arty/(é "flame/2‘55 o» ﬂlgt gas; act/{en 2. (8 points) Once again consider the electronic circuit shown in Figure 2. If the operational ampliﬁer is
ideal, then the transfer function of the circuit can be expressed as _Y(8) _ 5232+b18+bo H“) _ F(a) — s +ao Determine the parameters (:9, be, 61, and b; in terms of the component values R1, R3, 0;, and 02. f(t) V+C$) = if”. => \f+($) = S RLCL
{21 + ﬁle—«L J Fm sPLCLH ()5 H15 m ﬁof m. 0P 6LM/0 C01") Ltdurwﬁfun {‘5 an "an /}Ive)"bO/ d. mlall ﬂew ll YCs) J— Vii—L53
SCI (1+ RI )Mm => ﬂ=1+smq Cemhmm5_ tlm— ew" resvl'iﬁ atom} YCsE _ Wm .YCsl , “LC?— .(1+sR\c.)
Fax ’ PCS) V4453 ” SKICL“
.. z c VA c
' .S RIRLGICL 4" 822.. 2 2. 2..
’____________,__.. .sRLCz, +1 3. (13 points) The ﬁrstorder RC circuit in Figure 3 is driven by the input f(t) = 6u(—t) + 2u(t). 452 +
fa) 0 m 0.21: ya) Figure 3: Firstorder RC circuit.
0 (2 points) Determine the output voltage y(0“ o [5 points) Calculate the zerostate response y“ (t) for t 2 0. o (5 points) Find the zeroinput response y,,(t) for t 2 O.
o (1 point) Write an expression for the total response y(t) for t 2 0. 4t: 'l:=ov"J the. prGCIEVV‘ is an open crrcw'EJ (M‘Q So #055 = ?'Co"\ LR. :. 6n  In. 2. 6 V. Using, ox. .s owf‘CQ/ *5 Farmng ash: on, in. up. He) LIL
: Fﬂt) In the. S— ego/71am mate, thJ: ate") .2: th {w
uuHTL a.ch tub C“?“’“ 9.17 1%: =0‘. To 4..»‘0. the, aim—5% reaﬁonsa] .Se’é 51(o):_o_ p.
+
F6} é“ Y (3) 515' Fm ' F“)
C— : z
_ *5 RJ, g2 SCR H
05.3 JFCD: 2.0gth (1:20)) Rg‘yRU 619.2,?— we. OLE“?!
2. .: 25 z:
YZS($)'= 1:. .g.+ 3%;1 35:385.:0 Z
' .. 26“) :2.
B " sum )3»: Problem 2: (25 points) 1. (8 points) The closedloop system system shown in Figure 4 has an output Y(s), a command input
12(5), and a disturbance input 13(3). Using superposition, the Laplace transform of the output can be
expressed as m) = Ems) 1200+ Ems) ms), where Hw(s) is the transfer function from R(s) to Y(s) when 13(3) = 0, and Hinds) is the transfer
function from D(s) to Y(s) when R(s) = 0. Find 0 (4 points) H,..(s), and
o (4 points) Hyde). To receive credit, you must place your answer in the form bmem + b _1sm‘1 +  . 4);: + b0 H =
(a) s"+an_1s"“+a1s+ao controller ¢_ ®—> sensor gain Y(s) 2. (8 points) For a. different closedloop system, the transfer function from the command input R(s) to
the output error E(s) is given by E(s) = s 12(8) si+2s——3 Suppose that r(t) = u(t), determine £31086}
}
.. EC—‘ﬂ __ _____’§_________ L :.
5(5) ‘— 73) RC5) — 52425—3 ‘5‘ (Si3%?!) Becavm’. .SE LS3 has a. pole, (+0 )1) the, ﬁg fir40¢} the. ‘Clﬂu.’ Valve, Chauigm :5 (:94.— “qzilgggfe. we neeca 4:0
"Cumﬂa BUS) Mncg, then funk—2.. the ,anf «U f. “560: H __ 5+3 = __'11__.
A 3 C5+3)C$'13 3: .3
5(5) == —“ + "f‘
5+3 5! S" 1 Jq
3 = (smcsm i5“
_L_ J_.
E0) = —lqgl;3 + '1 5: 3. (9 points) A closedloop system with two independent control gains K1 and K3 has the closedloop transfer function
Y(s) _ 25
R(s) _ 5‘ + (K1 + 1):: + K2 Choose the gains K1 and K2 so that the following performance speciﬁcations are met. a The closedloop transfer function has a. DC gain of one, and o in response to a. unitstep input 1(t) = u(t), the closedloop response y(t) is underdamped with
a damping ratio of 0.7. The the gain (#9 YIN/p.05 is ﬁewen 17 ﬂu'é
3 m K; '>
5:0 the, 5C, gain ‘5 049.. TAQ. tranaén‘b PESfoﬂJ—D— ahurmcEQWS'btcs owe, £éQrm1;a£ y
the, Poles Croats # the OL’EKFMEQIG'eI‘c. Eém'éluﬂ )2 5" + (K,+ 05 + "2. «" D
L________1 L_.__1
LPWﬂ WnL wn :Kl'r—ZS" :1; wngg Z. fwn = [Q] +  :9 2: Z Fwn —! ($10.7) W0 : = 209.?) (S '1 Problem 3: (25 points) 1. (10 points) A LTI system has the transfer function 40.9 H =—.
(a) sz+5s+4 Construct the Bode magnitude and phase plots using the semilog graphs provided in Figure 5 (a
duplicate copy appears in Figure 6). \In order to receive credit: 0 In both your magnitude and phase plots, indicate each term separately using dashed lines. I Indicate the slope of the straight—line segments and corner frequencies of the ﬁnal magnitude and
phase plots. 0 Do not show the 3 dB corrections in the magnitude plot. Eecwse ﬁn pales al.79 both .2. the lace wigs t1CD} '05 c, ‘n Ll
H 53 : —————‘ _ tun». the. K0 n
C (St‘DCS'r‘n (51'1365/‘1 ‘4) (s): ﬂang So weCcm Smt :Mglsiw w = __‘°_£2’.__—_
“a” WuMMH) 10 Figure 5: Semilog paper for Bode magnitude and phase plots. 11 Figure 6: Semilog paper for Bode magnitude and phase plots.
12 2. (15 points) Consider a secondorder LTI system Whose frequency response magnitude and phase plots
are given in Figure 7. Magnitude (dB) Phase (deg) 1 1o" 10
Frequency (radlsec) Figure 7: Bode Plot for problem 3 part 2. 10 O (3 points) Find the DC gain and high frequency gain of the system. Express your answer in
terms of a ratio between the system output and inputs magnitudes, and not dB. "The. OY‘ DC awin ‘15 2009.9: The. high 'Fv‘eguenca, gain in use Of‘ 0. IO. 0 (4 points) Determine the damping ratio of the system. Is the system underdamped, overdamped or critically damped? Fr am the s \Le'hclﬁ Al‘XJJL)
So if? = to or P = 10
the 566mb." 75 unaQQRanmpeﬂ . 13 20 90510 g? 5.0: =7 2.0. AnJL
&,C‘duﬁe, l) o (3 points) Determine the system transfer function 11(5). 2.
K w
HCS3 = .___#———“——————;
52— 1_ szOS 'f'wﬂ
_ “9
The. DC anon KTIO, ‘F=c9.§') DUILO. Wn“5'£57e:_1 $0 e (5 points) Compute the sinusoidal steadystate response y(t) due to the input ﬂt) : 0.3 + 0.1 cos(5 t — 45°). (t) 2" 03 H 03 '9' HC 5')! COSCS4: _L.’SO+}L‘Hf 5—))
3 (0‘ f a"
PM tine, r5092. pm;
H’Liuw : ‘0
[Mann : $0428 or [00
ﬁngJ) *r—70°
Hag, So
#023 = 3 + {0 (213.5(5‘E —I3S°) 14 Problem 4: (25 points) In many cases a. mathematical model of a dynamic system can be obtained using fundamental equations,
such as Newton’s Law or KVL and KCL. In certain situations, where the theoretical model is complicated or
the physics of the process is poorly understood, it is convenient to obtain a model using experimental data.
In this problem you will obtain the transfer function based on the system’s transient response. Consider a
LTI system with input f(t) and output y(t). The zerostate response of this system to the unit step input
ﬁt) = u(t) is given by y(t) = 2u(t) — e"(1 — 4t)u(t). 1. (8 points) Find the transfer function 3(a) = Y(s)/F(s) of the system. Place your answer in the form bms’” + bm_1s'""1 +   bls+ be H =
(8) s“+an_1s“‘1+a1a+ao
30:3 = 2M9  {tunaA + life“? sects
z _ I + H
Y“) :1 “'5‘; (5+0 ($4.01
: 2.65 +1)?" " 59"") + his 3 sL—e 75 +2.
'——.__—rI—I——'_'"_'—"——'—_—__—‘ —_____,_,__,__—_—__
SCS+I)7— sC$+I)"
fluD sootD 4——> Pm =45— (QQ’L\nlLluﬂ) Hggdzj’éil: ﬂ 30* 15 2. (3 points) Is the system BIBO stable ? Justify your answer. HAS) ha; two Pole; of S'=‘“I' anL .50 {Lth
sds'bam F5 BIBO statute. (than, are, no. ales in the, We 4mm piqnet L 3. (3 points) What is the DC gain of the system ? =0 DC. 80m = M98 7—7— 4. (3 points) What is the high frequency gain of the system ? iii 'pr'a _, nc Tn __—_, :1
”d e a 3“ M“LN, ’ 5. (8 points) Using the transfer function 3(a), derive a linear differential equation that relates the output
y(t) to the input ﬁt). Y0) _ .37 + '75 + 2
F“) .sLr— 2.5 +1 gm + 2.209 yam = ﬁcar véca+ﬁéﬁ> 16 ...
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 SCHIANO,JEFFREYLDAS,ARNAB

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