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solutions_test2_s07_s002b

solutions_test2_s07_s002b - ECE 220 Exam#2 Spring 2007...

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Unformatted text preview: ECE 220 Exam #2 Spring 2007 Section 002 Student Name: 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) dﬁ— d, +v<r>=2e-3‘u<r> v<0>=2 W0): .. r“ a) Identify the type of damping. 11+ ﬁf l : O ”All; 111:3 ,7, 9— c. U Adf’fcluM‘Sg Ci b) Find the complementary solution, v (I), for t 2 0. DO NOT SOLVE for constants. Your answer should not contain any complex terms. , __ f3 .67» a9 -1 m3 t: -~ 5% (Jim—5V6 H a l. m ( {1‘9 e U VEM— Ce: e, * + w “w “ﬂawed _. ,1 ' 02 ' 3 ”f ._ Cga"?(ed( t+9l+ a » )uH) _. L l: x: .. : 2 C e, 1 ( K.) ‘1) K 35’& "V .9) M (1‘ ) c) Find the particular solution, vp (t) , for t 2 0. VPOf); A @‘3’tu (11) (Dual): % €ﬁe’7'3??? 7L. AH) e}%f+)+ Adﬁ—ZH} .= 2 e“ um "‘v ’3 a 3 t 9"“ Hi; 3'..- 63 a”) P 4. d) In the system represented by the differential equation below, select a value for the parameter b so that the system is overdamped. d2v(t) +1) dv(t) dtz dt b7._ 36> >0 b > 6 + 9v(t) = (8t)u (t) .f PROBLEM 2 (30) Consider the differential equation and the initial condition given below. Find the total solution for t 2 0. d2?) + 4v (t) = [6 + cos(4f)lu (t) 1 v(0) — E Useful identity: a cosa + b sina 2 Va 2 + [)2 cos(a — arctan %) 3 + ’- V; (Ti ):. C e 11 u ("H fl .1 {7+ "’2 A; 4"“ 3}?“ (‘i i z [(13% L1if i ' (ﬁt/((7')? 0 i' if A ("A {i i: I“) 5 i ‘2’ ,4“! ('04:: ,2 EX» £05 (11%, + a) u (—f) (aw/:0) —- 4 {35: of 17% ﬁdﬂlféliviwg 3 MS 55/ J‘ ”l” ahalﬂ" S (n7 75) M (4) .~' —— X“ .: '7 (["t 49313 (05i(4.4:.+9_.arc/m 77?) (“32> i) [ii : ——;—:__, 9 :- “I 1/3423, L/ i , rr ‘“ H) y“ (4 2v;'_ (35(«1-4: / J)“ P1 ﬁv‘i 2/ will: PROBLEM 3 (20) a. Give the phasor representation of the sinusoidal signal below. I v (t) = sin(2m +545) + 2003(2m 115:) N ’3'? , rr v a? -, v- '2 :2 2.87%---.a....+.,: ,. ”t 2 3n, :2. Jr; “W l ’1. - a / z E 1" {3: , 5 7 b. Consider the complex valued function, H (w) = j10 10+ja) Let this H ((0) represent the relationship between the phasors of the input signal and the output signal of a circuit in the following form. {/4 (w) =H(w).1}s (cu) Find the output signal v0 (t) , when the input signal is vs (I) =10sin(10t +Jr/ 4) -ﬂ/Z '3 \0 .. :l - 5 3 _ mum-«MW ~~ W4: * ~, 11/ H {10) r W45“) ‘“‘3 \TZ'”) ? W ll: ~«w... N 6‘3L AD 63 7 2-: ‘0 0(i0) :“ Mw—wf—ew‘ “i ’7: C: C 3 7 “Z PROBLEM 4 (15) Consider the differential equation and the initial condition given below. dv(t) + 2th (t) 2 (31‘ + 3)u (t) dt v(0)=3 tvvxt ’ 'ch ‘ t \ J ’ I a) IdentifythetypeofDE: Nan “Maven! V&(\/I/\C® (0 ﬂ /1 old? f b) Determine the values of the solution v(t) for t = [0, 0.1,0.2] using Euler’s method with a step size of h = 0.1 sec. Wt “0:110 .3 - 2—3 V W 3 (at + a) u H) in 3mm : ~2 13-33%; + M333'3‘m) 36+) 7/,[0l2l Vmu); 0+ Oll-3 + 7(0’2w; .3: _. la O/i ’ ’ Mr” PROBLEM 5 (15) The driving force and the particular solution pair given below corresponds to a first order system. Give a differential equation representation of this system. vs (t) = cos(10t)u(t) 1 v” (I) _ 10¢? cos(10t—%)u(t) TC" (/UJ, Sin (10%»5) + 01 fl; “3‘5 (mi? ”3;; : (05 “0‘9 __/ / VE— ? 0‘“ 2,. /(§ ; V /l Ewell?»- ,L +3—3i i\\:+§i© 1 if) a Z; 262 423» ...
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