solutions_test2_yellow_s08

# solutions_test2_yellow_s08 - ECE 220 Exam#2 Spring 2008...

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Unformatted text preview: ECE 220 Exam #2 Spring 2008 Section 002 Student Name: K a i C \(c’ 0W) LAST NAME (PRINT) FIRST NAME (PRINT) (SIGNATURE) By signing I am stating that I have taken this exam in accordance with the NCSU honor code. Show all work — no credit for answer only PROBLEM 1 (20) Consider the differential equation and the initial conditions given below. d2v(t) + dv(t) dtz dt + 2w) = 4e_2’u(t) v(0) = 2 im) = o. a. Find the complementary solution. DO NOT SOLVE for constants. Your answer should not contain any complex terms. b. Find the Earticular solution. 0\) >§L+9l+izo A1,; 2 " i _' r J; \r \g gm 2-; a Jul : 9r): 4-» a u I A \w h m w i l r i ’1; (4' _~ ix C Q J a 2: 1 wk“ .4 (Ti w ya: “f ,e ’1 “A a — L 4’ “23%) «it mg?) t y: C 6 Z ( e 7’ + i ) U. } ﬂ “'1'” if if. ‘ i { Kl :1 «in»- {W m " 31“ \J? I: ‘1" > “f [ \fc H) Z 9 i ’7 k N H ' ~ ‘ k (4’) '” \M a ‘1’" VF ' A e u ‘ 7 / 9/4“ / “.14, a w x2: + / w/i' Jr LimAC’W‘ v 4%" F w {,1} w WA ,r A m a} Ye. [email protected]) Consider the 1 ferential equation and the initial condition given below. Find the total solution for t _>. 0. dv (t) dt + 8v (t) = sin(8t)u (t) 1 —I-6‘ Useful identity: a cosa +19 sin a 2 Va 2 + 192 cos (a —- arctan %) ”‘ ’ " H I ' 5‘ \l t B an (bush (,4) (30m gagingatt Ebwwwei (M) @ ., l a, \ “M g ,, . A l W v. H WY 3 “(2 r? 1'” :5 If“ ~~ .\_,.,-u» witle g ,....!M "5 wﬁ ' l i h! “ x! “a a M ‘ l_ﬂ .r : v” 1 M’""“ i x E j J j" x . l, 1 a “‘3‘ ~ 7 r t W " f i J C W "A i 351 J ,i, V a A; t ._... 2:} ‘ t , i 4 M W l ” wt" 1/ X l ' / I! g M é if: I i x M, ‘I ,t v. w ‘ 'x m.» V m»— g a y ' W" ‘1.” a .1. l J“; PROBLEM 3 (20) The relationship between the phasors of the input signal vs (t), and the output signal v0 (t) is given by the j“: - _e__VS (w) complex function {/0 (co) = . —a)+ 160 Find the output signal v0 (I) , when the input signal is vs (I) = 5005(2n'1 0t —Jr/ 4) ~105in(27rlOt +15/ 4) g ‘ "w “n ’ [Mk _ ‘ 2., i 7 11W av ’ .;_, a”; fit w. ,3 l 'i . V T” w”, {H 3 WW.» ,3 I; «M W "ll? 5‘) "‘7' E l A? a, 4/ , 3‘ 1‘” I“ ,9? V" ,. «4 my“ v ’—”\ I / a» “‘2 5 ‘ " a 3 » ‘2 " m o. a l w {*P .2 / : a a“ ’ ,v,‘ W T?“ r i ,1 y 3 ,. x 1 \ / 5%.. g a] Q L: ‘4 w. W ~~ ‘3 r g ’9 “W‘ a ‘ .f x3 3 if my I g a m.” a w i g i. z: m “a” 4' l A a n! 3 .' M er‘“ » m N am“, \ ‘ H PROBLEM 4 (20) Consider the differential equation and the initial conditions given below. a’v2 (t) + 2 dv(t) dz2 dt v(0) =1, {2(0) = 2 +v(t) : cos3 (2m)u(t) a) Identify the type of DE: b) Determine the value of the total solution v(t) for t = 0.2 using Euler’s method with a step size of h = 0.1 sec. x ‘ ‘ ‘ ar' non’IAO/wac'ewtouﬁi Cl) 2nd0(cic3’-’ Conrylan‘l' COQFFIUPA‘r/ [MC / . o (a) x.(+l ' *6) MG) 3 0 ‘ 7%) T 2 1 x10!) ' NH 'XLH) ,1 —z Xzﬁ) amount) 0 L++M ><M") O ( KIM + \A 3 X : 2m XLH 4%)] X10” + M ,l 2 K 2H) co 3 ( ) o 11 Moi) ‘ or O ' K + M r. I ' ” T ’ Z I ll Xz(0-ll 3; —( —2 ﬂ x(o,2):l1 + mom— 0 :135 PROBLEM 5 (10) For the differential equation and the initial condition given below, find and graph the particular solution. dV(‘)+2v(r)=u(t)—u(z-4) dz v(0)=1 ,r r ‘5 [/u eet(\\ 3 * x t" . «A r"; j W PROBLEM 6 (10) The driving force and the particular solution pair given below corresponds to a second order system. Give a differential equation representation of this system and identify the type of damping. vs (t) = 52e‘stu (t) + 5011 (t) vp (t) : 4e‘3‘u (t) + 5u (t) V ? l‘ y , V I 1 rm ,y. , t. y ? if 1“ 3, Wu ; r ,‘l u ‘ ﬂ *vw- ‘ /_ ‘ “4:1 ‘3 “w a s w y , Wt“ l 1 4 k 4» L «w ” “ a j 4” w '4 f i m, m A ; i M 15 f t a. «L, ~ i k” 4* V W“ b 3‘4 u/ f l ., I W ; ‘ Z - M ‘ if f v, 5' «Rufﬂiy. 2,2,“, ; “,2; t ” 3 I“; 23%" i x f . If ‘ 1 “M i .... m If am“?- e ' ~1me a W. l ,, Ls . J W m". ‘_ w.“ 4“. .. 3.2 MW f .49 if: , . 93‘! y j ’ vﬁ ( ' l x n um 5“ , ‘ , v 3.» L, g ...
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• Fall '08
• NILSON
• total solution, J W M

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