b09hwk5sol - ECE2311 HOMEWORK 5 B09 Name ECE Box Number Due...

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Unformatted text preview: ECE2311 HOMEWORK # 5 B09 Name : ECE Box Number: Due date: This assignment is due Thursday, December 10, bring to class or to the ECE Main Office by 3PM (for full credit). Make a photocopy for your reference for fast comparison to solution set. Put all answers on the lines provided! Attach any work sheets with a staple. Problem 1 ~ Laplace Transform Integral In this problem you will find the Laplace transforms of the functions given below and identify the ROC for each as described in class and in the textbook. Show all work of integration (you can check your answers with the table method if you like, but this part must show that you used the actual integration procedure.) 1a) x05) : u(t) — u(t —— 2) 00 “L \ 2 ’XA) : .ch~ -1: DZ 7/+)€“§er(*"": erstlt 3 e g 6’ -eo ' ~s -s 5" 0 o -13 L3: Tu Roc is~w<m§834w 3 0r, W Mliwe S—«p'M ma he? I s: c: fiw'rh oQam‘ls‘mMEsoldfla MHQMUQ yaw. (a; (lg/{ad mm ”4‘0 imri’lmok ww3q$.\ / / / ‘19 9/ Ali T? I c241“ 9:0 is 412+ <{ugm‘ “Kw “Q‘Lm lVe 1'; 26 ~1J 3 (~90 5 “7" = 01 1‘20 -— I" 1 v8 2 -_....._._... f v X(S) Z—S——-—————for ROG given by $00 4 &l 5‘) d 9’0 0r /’ 64/31/wa 1’ 1b) W) = 3te'2‘u(t) 0 O 61* , eaf i8 Fl/‘m ”4%ng M115 mil ‘ZW ’66 A“: V Z—(f“q g0) ( J: Ob r 9° _ 9+1) 3? , f .- m; ~3— '('Jc’—1——- : wee-‘5“) e (“z—H (2+?) ”(“75 “ ‘ 0 Hz (3H,)"- 0 (F Msw-z mm M karmgqowtfip Q “6 3' X(8) = ”'2‘ :- _________ for ROC given by ’39 {S} > ’a 1c) 1(f)=(63‘~ P")u(t) “’3 00 X8) 2 3* J==i -54» .- S—ijv , §+\ -k u? ‘eié’ate‘ C ,8( )fi 0 o / 0 amfl (146:) > 4 \ \ % I’m/“<1 VZL(r)>,\7 ’1? Measrwa 1:33 7 :3; : ‘W+"”'3‘ .. i! (I v3){5+ f) (puma-M L..- . X(s)=—L—)——"3 (5+’_>______" forROCgivenby ’Zem >3 Problem 2 w Laplace Transform Tables In this problem you will find the Laplace transforms of the functions given below using Laplace transforms in Table 4.1 in the book and the basic Laplace Transform properties found in Table 4.2. Show any work required to obtain these solutions 2a) 33(t) : (3"(““0)u(t —> to) be - \ 6 M94” §+\ "5+0 ~5£0 6 M8): §+t 2b): L't( ) — 2f€ u(t — 2) + 36(t — “L“ A u (132 it 2“. + i» 9 (1(6) Wk? lJec/ ‘foamona M19 Lumget;:€2.€+ 1/ 1 z 4: 2Q 26 t e H) +26 a“ é‘ ’ 4- 2C) $(t) : [877n(w0t) + (3—275 + 1] 21(0) 409 --\—v ___L_ 4" _-____. 7, 1 g +2, X(s> : 5_W_ 2d) 1(t) : ejthlC—Qtu(fl “U h ..,,..._.+A~ : 342451-25” ’ WEU'FE) SUV!) 9)\,‘ (‘4) \L) / J—[G )Wo‘é” d 6”ij ‘m‘fl‘a a; ‘hua ’L’b—rt‘f‘I‘J J“ 21.“: ~5vu)+; 2(;+5U0)_}02 :3 Ms) : __’ 01w 9 awn-z) H fl \4 1 x S‘ \. o.» \W 4. ‘ {an gm) +| ' ‘ - WWW—.— + ' ‘ «We “ {S IL‘IDH/(jr “0+3 J (3*JL‘J0)(3 4"Jmif'LJ H N 7? 1(91 + LU; we + Z(€+Quu Problem 3 * Inverse Laplace Transforms Use Partial Fraction Decomposition, Laplace Transforms from Table 4.1 and the basic properties of the Laplace Transform in Table 4.2 to find the following inverse unilateral transforms: Provide the intermediate answers as requested. 3a) X(5) _ 2s+6 2 (ft/{l _ s2+5s+6 w," ‘ The partial fraction decomposition of X (.3) is given by: — S'fZ 2.\ ms): 3+?— The £’1X(s) is given by A 7 N i .Z,‘(: _ 21+ W): 2 e a (4—) 3b) X(S> : (5+2)R’25 s(s+1)2 The partial fraction decomposition of X (9) is given by: Km: g7: ~ '1‘ 2 U (H01 _ (5 +1 ') C” iii?” X (9) : The £V1X(s) is given by g <~w~=> Luke) ‘1 - __..1_. i (”@563 Jae—mm M» ms wax» Problem 4 — Solving Differential Equations with Laplace Transforms Solve the following differential equations with the given initial conditions. Show the intermediate answers. 4a.) (D2 + 4D + 4)y(t) = (D +1)x(t) with y(0_) = 2, 39(0‘) 2 1 and x(t) : e"tu(t) B») 4?me WM) 6....» j...“ ‘34.; / z “ _: - m;«_ xv»\MM-w—‘wwx _ g r W” “ *3; 3 * 9? a) + “WA draw) HV/s) : mg.“ +535. ‘ 25 " I 21 WW *"1...»-..._',. I ” 5’ W?8 “”0“” - pl (3+5) web—M ” +1 2' "““m 2% Partial Frac. Decomp. of Y(s) = {grzjz 10 4b) (D2 + GB + 25)y(t) : (D + 2)a:(t) with y(0_) : 1,3](0‘) = 1 and $05) = 25u(t) 2 5 M (+3 4"“! ”gm-«- ) L 175”) ' 5:350”? ‘~ 202) 4,” 63‘”, (5) - b 2) (9") Jr '2‘; \I {5) = gflgq. 1.13: ~ ------- 1- ‘ s ”Hm 4 £ng -. 6 rt ’9 : 25 +9” (3‘+6§ * Lay/(.5) : 3+ 32 4.333 s K * y» , 7‘ . ‘w ~ J 3+ w ' « Avg—1%.!) 7/33 5( f ' Q) 1, ”5 3:-“me 3 , a m+w+ a5) 1 ‘V 11 S ,. “523 +92) 7/ 2 3-3 2" g- k f a an“ .. «w ) j; - Z + 2...: .. 2 Wmaéam 3 WWW ‘_ . w , Partial Frac. Decomp. of Y(s) = 5' ”‘ 5w” 'J X j“+'-{J ' ’ \ (“3‘“‘5'3": z; > (*I'l'j)‘t "MM #8:“ 237% + ('21.+ --‘ 6’ “(a ZI-ICJ+(-‘l coséfé’) +13 CM/‘H’Jv [a 11 §:€—Z+(&$9m (W£)’4ca§{ll£))ju_gr) 12 Problem 5 e Matlab Partial Fraction Decomposition Look at the example C41 in the textbook on pages 355—356 to get an idea about how to use the residue function in Matlab to obtain the partial fraction decomposition of a Laplace Transform function. (a) Find the Partial Fraction Decomposition of the function in Problem 3a above using the residue function. Attach your print out. (Why not verify your answer using this?) (b) Find the Partial Fraction Decomposition of the function in Problem 3b above using the residue function. Attach your print out. Problem 6 — Extra Credit Get 10 extra points by attaching a Matlab print out that shows complete time function solution :I:(t) for Problem 3a using the Matlab symbolic toolbox functions as illustrated in example C42 on page 356 of the textbook. Caution: not every computer on campus with Matlab has the symbolic toolbox (the sym function will fail) - you may have to try more than one. 13 ...
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