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Unformatted text preview: BB 350 ' EXAM II 14 March 2000 Name: Selv‘LionJ
ID#: Section: DO _NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO 1  Test Form B Important Guidelines 0 Answer each problem on the exam itself. 0 The quality of your written solutions is as important as your answers. Your reasoning must be precise and clear. Your grade Will reﬂect the
clarity of your presentation. b. Problem 1: (25 Points) A LTI system is described by the impulse response function Mt) = 2 e"(t+1)u(t+ 1). 1. (5 points) Sketch h(t). Is the system causal or noncausal ? Justify your answer in a. single
sentence. h (t) Tﬁe seai=8!” L5 OODCQUKSCIQ' inﬂow252. the. IMML reopens: q'uﬂo'Eloﬂ E) a. nuncausaQ—
S‘dnwa. 2. (5 points) Is the system BIBO stable ? Substantiate your result by showing the appropriate
calculations. For the .SvI‘tEM so be 5’60 sedate, CW“)in must beak In this Case
on ‘Q 69
j Itcetﬂt : Jze“tm‘ig ggja‘tJe :— zé'Ee’f *l ., 5
'3 Zé‘Eg‘méﬂq '52 BQ—Cuuse, j:lktt)1££ s z i; twat", 57%— xamiﬁm is 6160:1592. 90 ll 3. (15 points) Given the impulse response function
h(t) = 2 e(‘+1)u(t+ 1) and the input
m) = [um — no: — 1)], use the graphical convolution method to calculate the zerostate response y(t). In order to receive credit, clearly specify the regions of integration and, for each
region, provide a sketch offend h. No credit will be given if you quote a solution
from the Convolution Table (Table 2.1) provided in the text. 30:) = HtHMt) a [Homedale Raglan 3: '6'1'] 5,0 or" t _.
(See shah) “m3
“4(a) Mt r>JZ z; — 0 home. at) an}. Hc—r) 1:. Mt gm : Lo overlap when 't: .‘.’_"l. é! or l_¢_,i=$O Realm I 0 _é_‘l‘:+l hit1')
tit) There, L3 non'zoro warlap le‘lbw’eqﬂ {:{ﬂ
mil lanti "Car 0 ST 5+“ “c
tH {H '0: #lIH)  Ct“) 12‘“
WE = S‘FLcMHr—Clﬁt '—‘ j e"c 2e alt =29, it
 (tH)
3067126 (L—+r) ~Ex l 5+: 50
3 l I i __ (t 'TH) I
out)”, jﬂcﬂszclﬁt = Earle Get 2 2:4“7‘67: D (tm)
gm: 2e 40.» 11:30. Sum!" 4h?maté Problem 2: (25 points) 1. (8 points) The circuit shown in Figure 1 has an input current f(t) and an output current
b. y(t). Use appmpriate timedomain circuit analysis techniques to ﬁnd an ODE that relates
the input f(t) to the output y(t) (do not use phasor analysis techniques). If your analysis
requires the introduction of additional voltages and/or currents, clearly label
these variables in the ﬁgure below. y(t) Figure 1: RLC circuit with input f(t) and output y(t). KCL i ﬂcmgQa A __—. P :44: vce) + CJr + 55.5) _—_ He) . _ b g
L/ Sabs’bu'b/L'Irﬂ 'U'Q. brunch {PfaaLronsltp V'Ct)— LOTE. Prowﬂ A as new)
L J: + Lem? 1—0405) 2. (9 points) A circuit with input f(t) and output y(t) is described by the ODE dz'y E
03:2 +40 dt + 300 y(t) = 600 m). Find the zerostate response of the system for the input f(t) : u(t) (because the system is
strictly proper, the initial conditions 11(0) and 31(0) are both zero). 3L3: gnw) + g¢[b) M6456 th ‘Cdf‘unﬁ, £416le 33! i! ON. Con) an‘é) binQ
gorge1Q, rosfongg R“; the 'Furm (1E) :. 09¢ m'éo "HQ. 005. ﬂl'ehﬂj gift. VoggdP 300g? :600l =) 5i¢(t):l4=2 ‘FW'EZO T34. «P 5% 15k. a wt: WE— rasp case. in Le) is Jo. {mm/mg
WM '
g roJES % c Q “Ir/UL? (on Gym  77" + “103 + 300 r. (71 +lo)(7‘+3o) :0
=3 7, =. —:o) 71;: 3o
HS) " C 8.101: 4" CL e,"3°t "par 1‘: 30
n " 1 TJ‘ CanJ'iJ‘M'Es CI OVWL’ (2. are, cinooan $0 thrJé "3 ‘t‘
309 :: Cq 8.191: + 02,8 0 + Z t 70 b 3. (8 points) Another circuit exhibits the zerostate response
31(15): (6 e'101E — 2 €30 ‘) u(t) for the input f(t) = Determine the impulse response function h(t) of the circuit and
simplify any term containing an impulse function. Me) 2‘— 0‘3— —eoé'°f+6oe'3°t)u(4c) + (Ge—mazefaﬁ) 5m Problem 3: (25 points) 1. (5 points) A sinusoidal signal of frequency w = 1000 rad/ sec is represented by the phasor ‘ y = 8345" + e3135‘” Express y(t) in terms of a. single cosine function. 2. (10 points) A LTI system is described by the frequency function Y 50
HOW) 2 — F = (3m)2 + 0.1 (3w) + 100‘ Determine the sinusoidal steadystate response y(t) of the system for the input f(t) : 4 5111(10 t + 60°) u(t). 40:) = LI cos(:ut3o°) =9 A, 50 ?500
if; = “2'07 F = L” Figure 2 shows the magnitude (in dB) of the frequence response function H (3w) 2 37/13" as a.
function of angular frequency w for a. LTI circuit with input f (t) and output y(t). : 50 Magnitude of the Frequency Response Function 10‘ 102 1o3 10‘
frequency (redlsec) Figure 2: Magnitude of the frequency response function HUw) in dB. LI, 3. (3 points) At What frequency w (in rad/ sec) does the magnitude of the frequency response
function reach its maximum value. 71‘“ PM “arm‘s ’5 “0‘” “is 4. (3 points) What is the ratio of for the frequency speciﬁed in part 3 (do not specify
your anSWer in dB). A. I?" — , 9010;74' ‘2,
Zopoa‘olV/F’: tic) =9 Qualol IF "' > IO 3 l IF ‘0 5. (4 points) At what frequency w (in rad/sec) is the amplitude of the sinusoidal steadystate
response y(t) equal to the amplitude of the sinusoidal input f(t) '1' (“View > ZaQowli/F/WM “‘4 L/ 10 Problem 4: (25 points) 1. (15 points) Find the frequency response function for the circuit shown in Figure 3. Express
' ‘ your answer in the form HUw) = Z = bm Ow)” + bm~1 (Mm—1 + ‘ ' '51 (W) + be
P (30))“ + an—1(JW)"'1 +   '61 (1“) + “0 ' with the coefﬁcients a and I) expressed in terms of the components 01, Cz, and R. If your analysis requires the introduction of additional voltages and/or currents, clearly
label these variables in the ﬁgure below. y I. R + £0) c1 c2 y(t) Figure 3: RC circuit with input current f(t) and output voltage y(t). L/ 11 2. (10 points) An electrical network with input f(t) and output y(t) is described by the ODE (133; day dy dzf df Determine the frequency response function H ( gm) of the circuit. Place your solution in the
form HOW) = X : gm 0”)“ + bm—l (MVP1 + "'51 (1“) + 50
F (30))" + (In—1 (JW)““1 + ' ' "11 (30?) + an I to 312onwn +30 a + 209“) + {o 12...
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