examIII_f98 - EE 350 Exam 3 23 November 1998 Name(Print ID...

Info icon This preview shows pages 1–15. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 8
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
Image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 12
Image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 14
Image of page 15
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: EE 350 Exam 3 23 November, 1998 Name: (Print) ID #: Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Test Form A Instructions: 1. This test consists of {our problems and each problem consists of either three or four parts. Solve each part only in the space following each question marked as Solution; if you need more space, continue on the reverse side and write the question number; for example, Question (4.2) gogg'nued..; N0 credit will be given to a solution that does not meet this requirement. 2. Box your answers wherever appropriate; failure to do so will result in reduced credit for the question. 3. Do NOT remove the pages under any circumstance. Loose papers will NOT be accepted as valid exam; a grade of ZERO will be assigned. Time for the exam is 75 minutes. You are allowed to use only a 8 112"x11" note as reference. GOOD LUCK .UIP Have a Nice Thanksgiving Problem 1: (25 points) For this problem assume that the operational amplifier is ideal and vc(0')=2V R=SQ C=lOpF Figure 1. Circuit for problem 1. (1.1) (5 points) Redraw the circuit, representing the sources and components with their complex- frequency domain equivalents. Solution: (Do not box your answer) EE BSOIExam 3I'Page l of 11 (1.2) (10 points) Find the zero—state response, stam. using Laplace analysis of the same circuit. Solution: EE BSOIExam 3IPage 2 of 11 (1.3) (10 points) Find the zero-input re5ponse, Yzmfl). using Laplace analysis of the same circuit. fit) a sin(lfl,flflcl t) u(t) C 3:0) _ v: (0 )- 2 v ' R-SQ C -= 10 ”F Solution: EE 350fExam 3I'Page 3 or 11 mm: (25 Points} Consider the system: f (t) + y (t) Proportional Gain [deal Integrator Plant Amplifier Sensor (2.1) (5 points) Express the closed-loop transfer function, H(s) - i—Esi-in terms of I-I,(s). H2(s), H3(s), s and Gc(s). Solution: EE 350lExam BlPage 4 of 11 (2.2) (5 points) Suppose the transfer function of the plant is Is the plant BIBO stable? Justify your answer. Solution: (2.3) (5 points) Write the transfer function of the ideal integrator. Solution: BE BSD/Exam BIPage 5 of 11 (2.4) (10 points) Suppose the overall transfer function was found to be K H(s)- 2 s +s+K—-2 determine the range of K for the system to have an overdamped response to a unit step input. Solution: EE350lExam SlPage 6 of 11 Problem 3: (25 points) NOTE: 1. u(t) is unit step function 2. Express the required transforms only as the ratio of two poiynomiais. as2 + bs + c Exam Ie: ————-—- p dss+esz+fs+g (3.1) (8 points) By direct integration find the Laplace transform of the function f (t) defined as f(t)=cosm 0st50.5 " = 0 elsewhere and specify the region of convergence. Solution: EE350!Exnm3/Page 7 of If (3.2) (8 points) Find the Laplace lmnsform of a signal g(t) = sin t u(t) -— sin I u(t-Jr) using standard table of transforms and properties. Solution: EEBSOIExam 3fPage 8 of 11 (3.3) (9 points) The impulse response of a linear time-invariant system is h(t) = exp(-t) u(t). Find the response y(t) of the system when an input {(t) = u(t-l) is applied to it using Laplace transforms. Solution: EE 350ffixam 3fPage 9 of l 1 mm: (25 points) This problem focuses on the concept of orthogonality. (4.1) (5 points) For an arbitrary set of signals, x,(t), x2(t). , xN(t) which exist in the interval [[0, to+T], state and explain the criteria for determining if the signals form an orthonormal set. Solution: (4.2) (5 points) In your own words, state the purpose of orthogonality. Solution: EE 350lExam 3/Page 10 of 11 (4.3) (15 points) Suppose you have the functions xl(t) and f (t) as shown below. Fmd the best-fit approximation of f(t) in terms of x,(t). i.e., find {0) 9‘ 0 X1“) x,(t) = 2 sin(ut) Solution: EE350IExam3lPagc 11 of 11 indefinite Integrals judv=uv—fvdu [name's = 1mm) ~ [Rama / . .1 f 1 . sm ax dz = —— cos n: cos a: d: = — sm a: a a .7: sin 2a.: 2 1- sin 20: sin azdx= §_ 4:: [cos urns—2+ 4a 1 zsin ax d1: = -——(sin ax - ax cos ax) 1 . 1 cos ardr: -2-(cos a: + ax sm as) a 2 1 2 2 a: sin a: d: = 1(211: sin a: + 2cos ax — a :1: cos ax) 2 1 2 2: cos a: d: = —3(an cos a: - 2sin a: + a 2: sin ax) sin (a -- b): sin (a + b): 2 2 sin b - -——-————-— - --—--—--—--——~ azsin zdz— 201—” 201+” a #6 cos (a — b): cos (a + b): 2 2 sin cos b a’ = — ~—-—-—-—-—-—-—- --—-————- b ” ‘ ’ [201—5) + 2(a+b) ° 5'5 . _ b . sm (a )1: sm (0 + b): a2 95 b2 2(a -— b) + 2(a +b) 1:“ dx— - —e° re" dx = -—-(a:c — l) 172:“ d1: _ {Ma -— 201: + 2) ecu: e J“sin b.1- dx— -- a2 + (12 (a sin bx - bcos bx) e” . cos b1: dz— .. a2 + b2(a cos b: + bsm bx) 2d l tan If 12+a a a 1 2 =—in(:.-:2 +a ) / f f f f f / [cos a: cos b: d; = f f f f f" f H: A Short Table of (Unilateral) Laplace Transforms m) PM 1 6(t) 1 2 u(t) l S . 3 in“) :15- n n! 4 t u(t) 3"“ 5 ("a“) a l A M 1 6 ts u(t) (3 A): r 7 mung) (s 1')“, s 80 cos bf 11(1) 3’ + b2 . b Sb sm 6! u(t) 2 + b: 5 9a 3““ cos bl 11(1) fia—p -gg . 6 9b C 51“ bt u(t) W _ rcosfia+ arcosflwbrsina 1°“ " “”0" + 9’ “(0 W 0.511” 0.511.” 105 "" (at 9 —.._. —. re cos( + )u(l:) a+a—jb+s+a+jb 10c 1'6““ cos (bt + 9) u(t) fi—ifi-‘ESB—H T = ’A'lccl-‘i'L-ZABa’ 9 = tan—l Arlee—pi: b= c—a , B — An .43 + B _¢¢ . 10d e [.4 cos bt + b sm 6!] u(t) ——_32 + 2“ + c a ‘ .-.1».14.x:l--‘.}.‘:'1'.J.;'$i'.i.L-'.""'29-h "JAM" It'd-P- -»«LALV-‘u' ‘ ‘ ' lumi’...-w._, The Laplace Transform Properties % Operation Ht) F(s) Addition f1(t)+ f2(t) . F1(a) + 5(3) Scalar multiplication kf(t) kF(.s) Time differentiation % _, sF(s) — f(0‘) d2 - (“if We) — afar) — no") (13)“ 3 2 _ _ . _ .. _ 3 .., F(s)—s 1(0) sum-1(0) ‘ 1 Time integration f(r) dr :Fh) o- wn: iron é ]_ ma: Time shift f(t -— to)u(t - to) F(s)e"“° to 2 0 Frequency shift f(t)e"°' F(s — so) dF Frequency utf (t) -%s}- differentiation Frequency integration I—E—Q f F (z) d: S . 1 a Scaling -. flat), at 2 0 :F (E) Time convolution 1'10) 1 f2(t) F1(s)F2(a) 1 Frequency convolution f1(t)f2(t) EaFfis) :- F2(a) Initial value f(0+) .1352; 317(5) (n > 111) Final value f(oo) limbsfls) (poles of 317(5) in LHP) m W...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern