examI_f98 - 1 BB 350 ‘ 28 September 1998"1049 Name I...

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Unformatted text preview: ' 1 BB 350 ‘ 28 September 1998 "1049: Name: . I Sec.: Test Form A -':I..:‘v__7Prot->lemi'1: 32-5 goings! ‘ TQ‘c‘o'n-sider {hfi'ifur‘rcfion fit} shown in Figure 1. " fit) Figural. momma). - 'jA “’1'.” ' .‘ 7.1I-“I-Ib"l.f' r1 ' .4 Now. let g(t) = , a W version of fit) from Figure 1. a! (5 points) . Sketch cm on the graph provided in Figure 2 and properly label the graph. E0) L 'L “3 4 S c. F igurc 2. A blank graph for sketching g(t). I Wait JcMch (SL0 isa’hlme, exgcmdecl \Jersfon 0‘C {3m} expanded b3a¥uc£oro¥ ’Z. ' - ' - mum.w,..,.m.:ufl r, ' ' ' WWW, .m- (Ahmadun ummWW bl (5 points} ‘ Sketch the individual component} of am on Figure 3 and verify that their su gives the same result as you found in {b}. gm ‘9'; .. -' .- .. .. The {mm m0 {helihkee ‘- I Jeems isiAevL’cfcd’c-p Figure 4. An LTI system with an input, [(0, and a zero-state response. y;(t). Now. use ch] as the input to the 5 am ' ' the system to am is hm. 9 LT! systgm shown m Figure 4. The zero-state tesponse of at» 350) F gm: 5. The LTI system with input, g(t), and zero-state response, y1(t). w“. amJajfiu. again,— u m|a.am.nm..'mr fink. mumwhkmmmhumlu. nmwmi; m aim-nitsaawmgflnmmhmuhm.u l‘llulh“.¢|lfldhsmdy t..mbu...:,..mu..‘ L m, . . , ,. u .u u. K. . i . ‘ " ' - c) (Spointsl Since the system is LTi. hit) can be expressed as a linear combination of time-shifted ' versions of y,(ti.__ Express y,{t} in terms _of the function mm. prop'ehtielszf ' ‘ = i (m r "HIM who) d) (10 points} is the signal sum in Figure 4 a newer signal. an energy signal or neither? ' If it is a power or energy signal, find the measure for hm. hm Ska : 0 MW) gigfla‘ “ Nfiéfi-flQAH-flruH-M __ __ A E%:.Slkt-1idic.. _ 5 ‘—‘| Munch § : .JE -2£1+L}Jt’ 3 1‘ : (9~l%+\z\:— (fig—8, +8) : J. 3 Problem 2: [25 points! This problem contains four parts labeled 3) through d). i Each part is completely independent of the other parts and so each can be solved separately. Point value's for each part are indicated below. Consider the foll0wing: Given a system with the input/output relationship ylti :2 flat - [H + y where. ylt) - System output fltl - System input and a. I}, & y are real numbers with the constraint that a a: 0 unless Otherwise noted. {9 points} Using an appropriate proof. determine if the system is linear or non-linear. If .—.-- - you determine the system to be non-linear. suggest values for at, B, & 7 which would make the system, linear. al z-_....iul. r ~-_4‘_.-....—-..Aa.- --.A bl (9 points) Using an appropriate proof, determine if the system is time-variant or time- invariant. lfryouidetermine the system to be time-variant. suggest values for a, 13. & 1 which would make the system time-invariant. --_ cl (4 points) Suppose. at t - -Ss. an input. fit} is applied. Using at - 1, and 4 0, what‘ must be the range on [i for the system to be causal? cl) (3 points) Suppose a. =1,D - 0&1 - 1. memoryless? Briefly explain your reasoning. 6% V 61m: 0?) 371 file) —% aptlvfilfi «was _ W +LI‘H a £(x+—/)+K' {aw ‘ smarter-dame) ——= jeweled-pm W - V g = [it'ct(°.§*’f)tlu¥t.t°&'pfl +K at Jaw weld“) ' ' Does this system have memory or is it 6) We): How) +6 511112: W 1311:) sag—the) : {Jaw—{0+}; W"H4,,(£3'21°I "(fie-a) " " .5-0 H I .. . .., “via-Fahd?) : fi(,<+:7$, £0) " .. .5 4- 3¢.m3fif) gflf—‘L—o)? Ifim 2—5315 (flu/W6 'g mp2: flats-meM O<=l ‘ ,5 W Problem 3: £25 gointsl In the following circuit. the switches are kept Open for a long time and then suddenly closed at t = 0. m): ..... ._ m. . J.“ _._-_.__ The ordinary differential equation relating the output, yttl, with the input, fit], for t 2 0 is determined to be ’ (R'fe+R’)gZ+iy(()=_Ld_f ,RJ dr 1. R, d where. R.= 20 R,= an R3:- 20 L =- 0.1H . 3 a) (6 points} Find the initial inductor current. “0‘1. and the initial voltage. viO‘) immediately after the switches are closed. bl (8 points] Find the zero-input response of ym for t>0. cl (8 points} Find the zerdstate response of ylt} for t>0. d} (3 points) Is this system stable? Explain your answer. Problem 4: £25 Qointsl In the foll0wing circuit. the switch is kept open for a Ion 9 time and then Suddenly closed at t = 0. . _..._...__~....—-.u.-—m..- - The ordinary differential equation relating the output, yit}. with the input, fit), for t 2 0 is determined to be .- ' _ d2 +R. I l _y+_y(;) =_._.f(() d: CMz d: LC LC where. R,- 1 0 R32 1 D L r: 0.1 H C - 0.1 F . . . .. dy . . . . a] [6 pomtsl Fund the initial values yIO‘} and Immediately after the swutch as closed. d! no. bl (8 points} Find the forced response of yttl. cl (8 points) Find the natural response of ytt). d! (3 points! Is the total response. yttl. critieafly damped. underdamped. or overdamped? Why? ll ...
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