examIII_s98 - BB 350 EXAM III 9 APRIL 1998 Name 50...

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Unformatted text preview: BB 350 EXAM III 9 APRIL 1998 Name: 50' u'i‘. [00-5 ID#: Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO _Problem -Weight Score Total 100 _ Test Form A This test consists of four problems. Answer each problem on the exam itself; if you use additional paper, repeat the identifying information above, and staple it to the rest of your exam when you hand it in. In order to receive credit, you must show all your work. Problem 1: (25 points) The block diagram of a speed control system for an electric train is shown below. controller am lifier train motor speed Y(s) 1. (10 points) Find the closed-loop transfer function Y(s)/R(s) in terms of the transfer functions 0(a) and 6(3) and the constant gains K and K5. 2. (8 points) Suppose that all components of the control system are specified except for the amplifier gain K , and that the resulting closed—loop transfer function is Y(s) _ K Iii—(5). — W' Determine the range of K for which the closed-loop system is BIBO stable. 3. (7 points) Using the system transfer function in part 2, and assuming K is set so that the system is BIBO stable, find an expression for the steady—state speed of the train in terms of the amplifier gain K when a unit step input r(t) = u(t) is applied to the closed—loop system. 1. Y“) = K recs) 6C9 Ems} — k5 no] meI + Ksk ccs\GCs\:) = scenes» Ms) For. (3‘60 étwblll'Lb-[J the palms S U’t’thlfl 2- m [apt {.qu pl «0.. The. » "For '1‘ H.) : (AC-b}! := F} fig So Y6) = K _1_ 51+ 55+ M6 5 (.[oSPQ— (cap 6% CM56 K "IS C‘IIOSBI') .56 s YLS) élgo Shut) Po‘es 'LLEL .02.“: qu42. 7‘1er mean! Problem 2: (25 points) The electrical network shown below has an input f(t) and an output y(t). 2 Q ‘ U 1. (10 points) Show that the transfer function of the network is Y(s) _sa+2s+1 F(s) _ sz+4s+4I In order to receive credit, you must show all work required to find H(s). 2. (5 points) What are the pole(s) and zero(s) of the transfer function H (a) ? 3. (10 points) Given f(t) : u(t), find the zero-state response y(t) using Laplace transform techniques. 1, 04% Kw. ,4: noflu 14 awe: \J YCs‘r PCs) MED. + fili— + ((5) = o + _’:_ 25+2 ; L .5 —J——~ — _L. -5— PCS) 4“ -";2- + 2542. + 1’5 " {z "' 1.: Y6) 5:1 2542. (s-HXSH) : = _.._..——--—-————-—‘ . --——--" : ___ H“) ch Lt: "’ "E's—Ti: + ‘ 2H7. (swat—H 4» 25+?- Problem 3: (25 points) 1. (10 points) Is the following signal f(t) = 2cos(31rt/8) + sin(3t/4) periodic (justify your answer !) '? If so, find the period. 2. (6.points) Consider the function f(t) : e-llw‘l' + 450}. (a) (2 points) Is the function f(t) odd, even, or neither even nor odd ? To receive credit, justify your answer. (b) (4 points) Express the function f(t) as the sum of an even function f,(t) and an odd function fa(t). Simplify your expressions for f,(t) and f,,(t) as much as possible. 3. (9 points) The output of a. rectifier circuit is a. periodic voltage y(t) whose period is T and trigonometric Fourier series coefficients are 2 an : — 1r _ —4 a" _ «(4112—1) 5,, = 0. Is the periodic signal y(t) even, odd, or neither even nor odd ? Justify your answer. I. -p('t +T) =5 2605 + 4- 5m (.374; .4. In anger- '5 ob‘lu‘m “F H: +7) -"'-—v ‘Fbr‘ um "bJ we. need- These. evade-has fin-U1" -;2:§n:-'T =’ W“ -%- 1—: 217-!” } be, Slmdlfiaqaslsdl T=gn 72%", ‘Furm r3 naflg’” 32-60.;de ‘5‘???— are. no [nigagps {115—0 I) méo that Sachs? fin = m m 3 J nd'l: '2 riot-0‘2" the. 5} {ml 430$) is 2cm 4,: mg = “9+ -FC--D 2. : eévsvcasw't +szqfiz) + ed $7025.44: — Zsmwfi) U 2 lam: golwwwt —-3 ¥e€t>= Com: O'Y‘ Z- ¥°6t3 = M). '2. r. 933‘" (Cong-I: + [gum/Q .. e‘JVS-aéqsuub «ZIMu/i) 2 £083) 1: ed- slnw-E __._.> 2:: g"; SI'IW'b 0v 1- CD on 2 an coscnwoa + Zion Sm (we) h=l 3 (“i-’3 7‘ a0 + “=1 ‘9 I _. .___._.. cos (magi) ’ 31- " i Z (1nz'l) n- rr nvl CQSCflwo‘L] u (sec 20 ___L__-- (a) (- at) a ,__ 3; —- LL 4510"!) g (3" 11- 11" n9! :1. d Ct), .1: ma- “ (m chef) Lno'é‘m the St n-Q at) D Problem 4: (25 points) Consider the periodic signal shown below where a = 3 and fl = 1. kg 0552 rue Una-fl; $60 is an Quen Milan [5 —5 —3 —1 1 3 5 1. (2 points) What is the period T of the signal ? 2. (6 points) Find the average value of the periodic signal 3. (12 points) Find the trigonometric Fourier series coefficients on. Simplify your expression for 0,,1 as much as possible. 4. (5 points) Find the trigonometric Fourier series coefficients b”. Simplify your expression for b" as much as possible. i, 143' Jim-kn in 1‘6 0'50"?) .172. Z I S?— 10' ...
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