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Unformatted text preview: Last Name (Print): Final Exam EE 350 Continuous Time Linear Systems
April 30, 2001 Sdhliocn . First Name (Print): ID. number (Last 4 digits) : Official Section number : Do not turn this page until you are told to do so Problem Score Weight
25
25
25
25 Total 100 Test form A
Instructions 1. You have 1 hour and 50 minutes to complete this exam. 2. Calculators are not permitted. 3. This is a closed book exam. You are allowed one sheet of 8.5” x 11” of paper for notes as mentioned
in the syllabus. 4. Solve each part of the problem in the space following the question. Place your ﬁnal answer in the box
when provided. If you need more space, continue your solution on the reverse side labeling the page
with the question number; for example, “Question 4.2) Continued”. NO credit will be given to 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 shall convey what you are doing. To receive credit,
you must show work. Problem 1: (25 Points) 1.1 (11 points) Consider the active ﬁlter shown in Figure 1. Assume that the operational ampliﬁer is
ideal. Figure 1: Active ﬁlter 1. (4 points) Draw an equivalent circuit at high frequency, and by inspection, ﬁnd the high frequency gain. I V
c —) she/d CAM/nut 13%) = o. 29‘ 2:2.  (3c)
Ht) 9 [k ‘4 h.?. gain/1 = O 2. (4 points) Draw an equivalent circuit at d.c., and by inspection, ﬁnd the d.c. gain.
c —’ opm oinau’r «4%) = ‘FLH 2.0. 152. (M33!) .
dc. clown = 1. 3. (3 points) Based on your answers in Parts 1 and 2, what type of ﬁlter does the circuit in Figure 1
represent? Highpass, lowpass, bandpass, bandstop? ’Tﬁj. oi/Lwii WP/CUYYJMld a, /0wa Film. 1.2 (14 points) Once again consider the active ﬁlter shown in Figure 2 below. I
 
I
29 29 W
+
vx~\/VV‘ i
f(t) IF I y“) Figure 2: Active ﬁlter 1. (6 points) Assuming that the operational ampliﬁer is ideal, write a node equation relating the node
voltage V1 (.9) to F(s) and Y(s). R R Use ==> F‘er + \/V96 + scR(Y—~vx)=o V = SCRY—lY + F SCR+2. YL5)(2S+4)+F(S)
2.3 + 2. (1) . (9) 3. (4 points) Using the results from Parts 1 and 2, determine the transfer function H (s) = Y(s)/F(s) of
the active ﬁlter. Substitu’m m WW (2), NS) = W5) (2%) + [2(3) (Z/SWHZHZ) (23+1)(2s+2) Problem 2: (25 Points) 2.1 (16 points) A feedback system shown in Figure 3 has an output Y(s), a command input F(s), and a
and [3 are adjustable parameters. F(s) Y(s) Figure 3: A feedback system block diagram for Problem 2.1 o (10 points) Derive the overall transfer function H (s) = gég of the feedback system in Figure 3 in
terms of a and ﬂ. To receive credit you must put your answer in the form bms’" + bm_13m_1 + + b18 + b0
3" + an_1s"—1 + + als +ao ' He) + 1/s+2 W5)
‘ ) p (é)
Gis ‘7 H(s) = 1+olp(.sl.)
,, z .1:
\I s+o<p
Y(S) T6) = (5)
3+2. I
'0?
I)
.4
‘63
hi
I!
A
m
+
51
I";
7/?
I
in
V *9 5 ~——# A o (6 points) If the parameters a and ﬂ are set as a=0.5, [3:1, is the overall system bounded—input, bounded—output (BIBO) stable? Explain. (x z: 0.5 , P=1
s H(S)= ‘ = ’
3
Sl~£s +0. 8(‘93:
2
ohm ova s = O :1 LHP
r :2 qé 2.2 (9 points) A ramp input f (t) = tu(t) is applied to the LTI system Whose transfer function H (s) is
given by 3(3s2 — l) .92 + 33 + 2’ compute the initial value y(0+) and the ﬁnal value ysa = limHoo y(t) of the zerostate response y(t). H(8) = «3(0“) = Kim sws) = 2m s(—1;H(3)> 3—)“) 9—900 . 2
saw s"+3s+2
. . 2.
«33 = 210’" syn): Km: 33 “1 = _L
S 8‘90 5"" 31+33+2 2 T [mew m 3=—2,~1 ELHP / Problem 3: (25 Points)
3.1 (15 points) An LTI system has the transfer function 8 H“) = s2 + 1103 + 1000' Construct the Bode magnitude and phase plots using the semilog graphs provided.
In order to receive credit: 0 In both magnitude and phase plots, indicate each term separately using dash lines. 0 Indicate the slope of the straightline segments and corner frequencies of the ﬁnal magnitude and phase
plots. Hm») = J‘d/iooo (saws!) o ZOXOg IKI 3 2o z: “‘60 3.2 (10 points) Consider another LTI system whose frequency response magnitude and phase plots are given
in Figure 4. If the input f (t) = [13 + 2cos(10t)]u(t) is applied to this system, determine the steadystate
response y(t). Bode Diagrams Phase (deg); Magnitude (d8) I a I 10— 1o 10 2 10
Frequency (rad/sec) Figure 4: Bode Plot for problem 3.2 Rh 2 [13 + 2. cos(1o\3)]u(£) 1‘ T 01.0.. u :10 ‘) HUG) = d.c. = ~00 d3 = o 4 ‘0
2) H = “200% 445° = 76 0 r
Wt) = 13~H%) + 2’H(j10)’COSUOt+K~HlJIO)) H 0.2 cos (wt +45") 11 Problem 4: (25 Points) 4.1 (11 points) Consider a certain LTI system whose transfer function has the
pole—zero pattern shown in Figure 5. lm
>5  1 2
4 15L 0 >Re
i ......... “1 Figure 5: A polezero map of the system in Problem 4.1 o (6 points) If the d.c. gain of the system H (0) = 1, determine the transfer function H (s) of the system. yous I S = ﬂij
we I 3 = 2
ms) = K (3‘2) _ KCS'Z)
' .  ______,_..
(3+1+3)(S+1~3) 32+23+2
H(o) :: K(~2) =1 glK =~1 2 o (5 points) Determine the damping ratio of the system. Is the system underdamped, overdamped or
critically damped? sz+2s +2 m 2. 2.
S + 23w”: +00." um=fi Omd. 32—3”: U
2% Ex!» Umolm, down/not 12 4.2 (14 points) The result from Matlab for generating a partial fraction expansion of a transfer function H (s) = %% of a certain LTI system is shown below. >> [R, P, K] = residue([1,0,2], [1,2, 2]) R = —1.0000 — 1.0000i
—1.0000 + 1.00001’
—1.0000 + 1.0000i
—1.0000 — 1.0000i
K = 1 P o (4 points) Write an ordinary differential equation relating the output y(t) and the input f (t) of the
system. l
H = S + z sz+23+2 o (10 points) Determine the impulse response h(t) of the system. Hls) = ~1”J‘ + "1+J_ + 1 ( F/zorm (mail/at)
s+1~3 9+1+j
j(iss°) jogs“)
= J3 1 4
. + 5" + 1
s+1~3 S+i+j ~t 25 2. cos (t —135°) wt) + at) 13 (continued) A lbw/n Minx WWOOC ; 9.
H6) : 8 +2 : ~28
gz+zs+2 3L+2s+2
‘23 H
+ (s+:)2+1 [1 1+ "209“) + 2{ 7 )
(3+1)"+1 lady) = 5 W62 603(k) + 2 Sfm(Jc))‘1/L({') 14 ...
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 Fall '07
 SCHIANO,JEFFREYLDAS,ARNAB

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