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Unformatted text preview: EE 350 EXAM I 20 September 2001
Last Name (Print): L S Oi U'bl on S First Name (Print): ID number (Last 4 digits): Section: _ DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO 50
Weight
—?
“ Test Form A
INSTRUCTIONS 1. You have 2 hours to complete this exam. 2. Calculators are not allowed. 3. This is a closed book exam. You may use one 8.5” x 11” note sheet. 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 1.2 Continued. NO credit will be given to solutions that do 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. If you introduce a voltage or current in the analysis of a circuit, you must clearly label the
voltage (current) in the circuit diagram and indicate the reference polarity (direction). If you
fail to clearly deﬁne the voltages and currents used in your analysis, you will receive ZERO
credit. 7. 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) 1.1 (8 points) The zerostate response of a signalavenger with input f(t) is 1 t+t2
y(t)=t1+t2A_t1 fawn where t1 2 0 and t2 2 0 are constants. Is the system timevarying or timeinvariant ? Justify your
answer. Le": Jag5 = 40:: 4)) than t+L2
‘12th I
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he, 8 .{gm IS ______.________—————
(mi 80 '5 0"" 1.2 (9 points) In response to the inputs
f1“) : €141:
f2(t) = e_74t, a linear timeinvariant system yields the zero—state responses 11103) = 5 J «214‘ ya“) = —5 J 6‘7“, respectively. Find the zerostate response y3(t) for the input
f3(t) = 4 cos(4t), and express y3(t) as a. realvalued sinusoidal function of time. 0 bserue. tl‘ 07b ‘145 _ vi
.205 +193 = 2% + e/ :1 ZCOS Qﬁ) and» 50
.9395) .=._ z—QlL—D + 24168. Because, the, ad” ‘15 Qmear;
it)“; = reﬁne) + lgzct) 1.3 (8 points) 0 (4 points) You are asked to ﬁnd the product of the two polynomials Q1(A) : A3+2A2+A+1
Q20) = A4+3A+5 and to express your result as Q1(r\) Q20) = A7+as A5+as A5 +04 A4+a3 A3+az A2 +a1 A1 +ao Write a complete MATLAB command for determining the coefﬁcients ai. o (4 points) Suppose two vectors are deﬁned in MATLAB as: H >>€D
>>y [—110a1l;
[2,1,2]; What does the MATLAB command
>> 2 = a: . * y yield for z ? Problem 2: (25 points) A certain system has a characteristic equation whose roots are shown in Figure 1; note that two of the roots
are real and identical. ImOL) ReOL) Figure 1: Location of the characteristic roots for the system considered in Problem 2.1 and 2.2. 2.1 (10 points) State the form of the homogeneous response of the system in terms of undetermined coefﬁcients 0;. Each characteristic mode in your expression must be realvalued, that is, you cannot
have terms of the form a em . 2.2 (5 points) Is the system in part 1 asymptotically stable, marginally stable, or unstable ? Justify your
answer in one or two sentences. 2.3 (10 points) A system, different from the one considered in parts 1 and 2, is described by the ODE y(t) + a1 W) + a0 y(t) = 25 7r2 f(t). The zerostate unitstep response is characterized by a sinusoidal oscillation, with a period of 2 s, that
exponentially decays to a steadystate value of four. Determine the following parameters: rad / sec] = _1T_[
— .5 In; [rad/sec] ( = If. [,5
I]: '21
a0 = 251122 3 . BGCW UQ/ [6pq‘l3 l5 WKE/ $6 '18 yr (3(5) 2' g 0 '3 ” O o _
o7 + W + W “ E
:1
l a
a
II 2.7“ _ 7.
77“ = 5;: “1“ =9 l? l 70
L L: 2:"
f = l" '23: z" 2.5
U3!ﬂ# 0" =_ Z’PWA ﬂuxes
_. 2/ r; 5.1: = 11—13::
qt " 7 Z Problem 3: (25 points) 3.1 (8 points) Derive an ODE that relates the input voltage f(t) to the output voltage y(t) in the circuit
shown in Figure 2, and place your answer into standard form. 3.2 (8 points) Once again consider the circuit in Figure 3 and let f(t) = 2 + 4 u(t). 10 k9 Figure 3: Passive RC network. Determine the following: W) = l [V]
i(0—) : 00' [mA]
y(0+) = __l___[V] CH‘CUFE 03L t z 0‘ _ + (01" S
9}, 15,0 J y {aacrosf aka/1 Q msw‘lsanewsd Owwit (A? ‘17 :6+ 3.3 (9 points) Consider a system characterized by the ODE W) + 3 W) = 4 f0), the input
N) = 2 e‘3 t,
and the initial condition
y(0+) = 1
Determine the zerostate response y"(t), the zeroinput response yn(t), and the total response y(t)
for t > 0. 6? (713 = ’Ar'b =0 :9 ﬁghtf.) :. ce, Problem 4: (25 points) 4.1 (9 points) Assume that the operational ampliﬁer in Figure 4 is ideal. Find the differential equation
that relates the output voltage y(t) to the input voltage f(t), and place your answer into standard
form. 11 4.2 (8 points) In the circuit shown in Figure 5 N) = sin(1")U(t)[V]
11(0") : 3 [V]
«0) = 5 [mA].
Determine the initial conditions
W) = 3 [mA]
111 _ o .2. ﬂ
dt t=O+ s Figure 5: Network with input voltage f(t) and output current y(t). V(0_)=V(O'*) 3 3 V bled—use, uoHSq s2. Carma'b eta/7L “1:: o .
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can» ‘ 024:  “a: am9~ So 3511?... s . 55—.
db» _—: RC0” = ZMA toms S 12 13 4.3 (8 points) The following ODE describes a dynamic system
W) + 8 W) + 16 W) = 9 f(t) + 16 N) Suppose that f(t) :: e‘“ u(t), y(0+) : l, and y(0+) : 2, ﬁnd y(t) for t Z 0. (9(3) 2 72‘ + 8% + )6 == Cﬂ+~0L .. t "‘it
yum — CLQ q + C2“
..zt “2b
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 SCHIANO,JEFFREYLDAS,ARNAB

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