final_f03 - EE 350 EXAM IV 19 December 2003 Last Name 8...

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Unformatted text preview: EE 350 EXAM IV 19 December 2003 Last Name: 8 Glut \ é) [LS First Name: ID number (Last 4 digits): Section: Test Form A Instruct ions 1. You have two hours to complete this exam. 2. This is a closed-bookexam. 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. (12 points) A system is described by the ODE W) + 3 W) + 2 W) = 4 N)- Using Laplace transform methods, determine the zero—input response 111,- (t) and the zero-state response y,,(t) for 110’“) = 1 W) = —2 N) = Hum. No credit will be given for solutions using the classical approach covered on Exam L 3:. Ya) — 39‘ my) + 2 Y(s) = “l PCS) 1 - - ' - «I- 5. Yaw—Saw) 3C0) . ‘ s (0') 1" ‘0‘3+3&‘C°) i52+3s +23YC5) '5 HHS) + a at S C0”)+ ' (0’) +3 (0") Y(5 3 Li ‘ PCS) + '2. 35- +2. > 52-5-35”- M ‘ f—‘__/ j {ya-y co] 1 flaw] 7"“.— . ‘ “91—1 ‘2 _, A 4. L + ’<:¢_t\z. 2. (13 points) Figure 1 shows the small signal model of a FET amplifier with input voltage f(t) and output voltage y(t). Determine the transfer function of the amplifier and place your answer in the standard form Y(a) _ bma’" + bm_1s’"‘1 +- -+ bls + bo _F(s)_ a" +an—1s"‘ +- -+a15+ao vasfi-L‘jég, C0 l" £0 (2) 130 0. ’lm mute VC‘zS (J) Problem 2: (25 points) 1. (13 points) Find the transfer function Y(s)/R(s) of the feedback control system shown in Figure 2 and express your answer in the form Y(s) _ bms'“ +b 43"“ +---+ b15+ b0 F03) 8"+an_19"‘1+---+als+ao ' H(s) : Y(s) Figure 2: Feedback control system with reference input R(s) and output Y(s). (”MIL “Cram “2, Infloqe, tin/"t1 /_ ’ RC3) ' 5 53+! $Ll+sz+5 +51+S 1+ "" 5[JB+J+I) 53+! ____,_,_,_ 25V+51+2$ 2. (12 points) Another closed-loop system, different from the one considered in part 1, has the closed-loop transfer function 0 (6 points) Is the closed-loop system BIBO stable ? Show work and justify your answer in a single sentence. VJ‘Q Poles % tie c/osqfiagoyf 95hr” are, flocwéfl .1: the, was ¢ 52- +w+ ., :. CS+2JL=O get/dude. 50%“ Poles out, :7: the LHP (.2$~Z)J tko, Sdeem‘ :5 3160' Sable; o (6 points) Suppose that r(t) = tu(t). Does y(t) approach a steady-state value ? It the answer is yes, what is the steady-state value of y(t). If the answer is no, explain why y(t) does not achieve a. steady-state value. U52. the, ‘meQ‘W/VlV-e, theorem H5) «eon—SELV—(Jl -flo;\;/g‘x ill—‘2‘ a“ - s—ao ”9 ’ 55° 5"+‘15+‘1 5 5'5 :7. Problem 3: (25 points) . 1. (16 points) A system has the transfer function _ ‘ (5+1)(s+100)2 H“) - W SketCh the 'Brode magnitude and phase plots in Figure 3. 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 straight-line segments and corner frequencies of the final magnitude and phase plots. 0 D9991? show the 3 dB corrections in' the-magnitude plot. 10"l (5/100 1")?- 10‘(5/;o+.)(5/;0W) Tje' {Infifmlé/‘Q’ Q.I\(/€. ,7sz plo'b) % {ff/’75 W fémyl) @ GHQ, /flt[(Cové¢Q/~ % £44}ng Ema} an Wait: ab. ““993 MB Figure 3: Semilog paper for constructing the Bode magnitude and phase plots. ' 8 (£0.10 “3% out IO LWJ/x<) 2. (9 points) A second-order LTI system with no zeros (m = 0) is characterized by the Bode magnitude and phase plots in Figure 4. o (4 points) Specify the magnitude and sign of the DC gain and the high frequency gain. Express the magnitudes as the ratio between the system output and input, rather than dB. bc. 5min élwowle 3V0o06 or- 10075 saga/n AHA/o) ' z. +130“) 'Y "°° ”Wk (refine/L ‘H‘CJ )i O ”V“ “five”? gar}; ‘ '60'." 21H?» F70 ‘ .- -‘ r0 - o (5 points) Determine the damping ratio of the system and specify whether the zero-state unit-step response is underdamped, critically-damped or overdamped. Phase (deg); Magnitude (dB) Bode Diagrams 2 101 10 Frequency (rad/sec) Figure 4: Bode magnitude (in dB) and phase (in degrees) plots as function of frequency ( in rad/ sec) for the second-order LTI system. 10 10 Problem 4: (25 points) 1. (13 points) Just before the Thanksgiving holiday you and your faithful assistant Big Willy finish a long series of experiments to determine the transfer function of a new product to be released before the upcoming Christmas shopping season. After a long relaxing weekend you return to work and discover that Big Willy’s attention to detail, or lack thereof, is going to cause you a major case of Monday blues. Big Willy forgot to label the axes on the pole—zero map shown in Figure 5. Fortunately, you had recorded in your laboratory notebook that the system has a DC gain of positive two, and that the complex-conjugate poles have a natural frequency of \/§ rad/sec and a dimensionless-damping ratio of one-half. Figure 5: Pole zero-map of the transfer function H (s). o (6 points) Specify the values of a. and b in the pole-zero map. {- k iLL The ("3” who. Y°l¢ pcu'v- praguc¢5 fl, 9/ «dz/‘0» «<— rm n Jamar” In“, ,% fig, +runsfer Ln; E10!“ ’2. (3+1, +fac)(s+by_a) = 5"- + 21:: + bZ-I'a. 25: 420% => b= PWn = i"? sz- at = s—a: -— =fl o (4 points) Specify the value of c in the pole-zero map. S+c. ”(53 =’- : 51+ 2b: +61%?" .S+<. 57-4- f§$ 4&— .— 6664v$e H'COW = 2. =— DC gong we Imago. c—IO. o (3 points) Specify the transfer function H (s) of the system in the form Y(s) _ bms'" + bm_1s"“1 + - - - + bls + be F(8) _ s“+am_1s"'1+---+a18+ao ' H(s) = 11 2. (12 points) A second-order LTI system has the impulse response function Mt) = 30 r” sin(4t) u(t). o (4 points) What is the DC gain of the system ? o (8 points) Is the zero-state unit-step response underdamped, critically damped, or overdamped ? - If the response is underdamped, determine the value of the natural frequency and the di- mensionless damping ratio. — If the response is either critically damped or overdamped, state the time constants associated with the exponential decay of the transient response. ‘1 CS +231 +16 H65) 3 301 e-zt51n(blt)uft)}: 30 _. [2.0 5" + H5 +20 m Dc. am a Moi = 6 12 ...
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