examII_s00 - BB 350 EXAM II 14 March 2000 Name Selv‘LionJ...

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Unformatted text preview: BB 350 ' EXAM II 14 March 2000 Name: Selv‘LionJ ID#: Section: DO _NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO 1 - Test Form B Important Guidelines 0 Answer each problem on the exam itself. 0 The quality of your written solutions is as important as your answers. Your reasoning must be precise and clear. Your grade Will reflect the clarity of your presentation. b. Problem 1: (25 Points) A LTI system is described by the impulse response function Mt) = 2 e"(t+1)u(t+ 1). 1. (5 points) Sketch h(t). Is the system causal or noncausal ? Justify your answer in a. single sentence. h (t) Tfie sea-i=8!” L5 OODCQUKSCIQ' inflow-252. the. IMML reopens: q'uflo'Elofl E) a. nuncausa-Q— S‘dnwa. 2. (5 points) Is the system BIBO stable ? Substantiate your result by showing the appropriate calculations. For the .Sv-I‘tEM so be 5’60 sedate, CW“)in must beak In this Case on ‘Q 69 j Itcetflt :- Jze“tm‘ig ggja‘tJe :— zé'Ee’f *l ., 5 '3 Zé‘E-g‘m--éflq '52 BQ—Cuuse, j:lktt)1££ s z i; twat", 57%— xamifim is 6160:1592. 90 ll 3. (15 points) Given the impulse response function h(t) = 2 e-(‘+1)u(t+ 1) and the input m) = [um — no: — 1)], use the graphical convolution method to calculate the zero-state response y(t). In order to receive credit, clearly specify the regions of integration and, for each region, provide a sketch offend h. No credit will be given if you quote a solution from the Convolution Table (Table 2.1) provided in the text. 30:) = HtHMt) a [Home-dale Raglan 3: '6'1'] 5,0 or" t _. (See shah)- “m3 “4(a) Mt -r>JZ z; —- 0 home. at) an}. Hc—r) 1:. Mt- gm : Lo overlap when 't: .‘.’-_"l. é! or -l_¢_-,-i=$O Realm I 0 _é_‘l‘:+l hit-1') tit) There, L3 non'zoro warlap le‘lbw’eqfl {:{fl mil lan-ti "Car 0 ST 5+“ “c t-H {-H '0: #l-IH) - Ct“) 12‘“ WE = S‘FLcMHr—Clfit '—‘- j e"c 2e alt =29, it - (t-H) 3067-126 (L—+r) ~Ex- -l 5+: 50 3 l I i __ -(t 'T-H) I out)”, jflcflszclfit = Earle Get 2 2:4“7‘67: D -(t-m) gm: 2e 40.» 11:30. Sum!" 4h?maté Problem 2: (25 points) 1. (8 points) The circuit shown in Figure 1 has an input current f(t) and an output current b. y(t). Use appmpriate time-domain circuit analysis techniques to find an ODE that relates the input f(t) to the output y(t) (do not use phasor analysis techniques). If your analysis requires the introduction of additional voltages and/or currents, clearly label these variables in the figure below. y(t) Figure 1: RLC circuit with input f(t) and output y(t). KCL i flcmgQa A __—. P- :44: vce) + CJr + 55.5) _—_ He) . _ b g L/ Sabs’bu'b/L'Irfl 'U'Q. brunch {PfaaLronsl-tp V'Ct)-— LOT-E. Prowfl A as new) L J: + Lem? 1—0405) 2. (9 points) A circuit with input f(t) and output y(t) is described by the ODE dz'y E 03:2 +40 dt + 300 y(t) = 600 m). Find the zero-state response of the system for the input f(t) : u(t) (because the system is strictly proper, the initial conditions 11(0) and 31(0) are both zero). 3L3:- gnw) + g¢[b) M6456 th ‘Cdf‘unfi, £416le 33! i! ON. Con) an‘é) bin-Q gorge-1Q, rosfongg R“; the 'Furm (1E) :. 09¢ m'éo "HQ. 005. fll'ehflj gift. Voggd-P 300g? :600-l =) 5i¢(t):l4=2 ‘FW'EZO T34. «P 5% 15k. a wt: WE— rasp case. in Le) is Jo. {mm/mg WM ' g roJES % c Q “Ir/UL? (on Gym - 77" + “103 + 300 r. (71 +lo)(7‘+3o) :0 =3 7, =. —:o) 71;: -3o HS) " C 8.101: 4" CL e,"3°t- "par 1‘: 30 n " 1 TJ‘ CanJ'iJ‘M'Es CI OVWL’ (2. are, cinooan $0 thrJé "3 ‘t‘ 309 :: Cq 8.191: + 02,8 0 + Z t 70 b 3. (8 points) Another circuit exhibits the zero-state response 31(15): (6 e'101E — 2 €30 ‘) u(t) for the input f(t) = Determine the impulse response function h(t) of the circuit and simplify any term containing an impulse function. Me) 2‘— 0‘3— —eoé'°f+6oe'3°t)u(4c) + (Ge—mazefafi) 5m Problem 3: (25 points) 1. (5 points) A sinusoidal signal of frequency w = 1000 rad/ sec is represented by the phasor ‘ y = 8345" + e3135‘” Express y(t) in terms of a. single cosine function. 2. (10 points) A LTI system is described by the frequency function Y 50 HOW) 2 —- F = (3m)2 + 0.1 (3w) + 100‘ Determine the sinusoidal steady-state response y(t) of the system for the input f(t) : 4 5111(10 t + 60°) u(t). 40:) = LI cos(:ut-3o°) =9 A, 50 -?500 if; = “2'07 F- = L” Figure 2 shows the magnitude (in dB) of the frequence response function H (3w) 2 37/13" as a. function of angular frequency w for a. LTI circuit with input f (t) and output y(t). : 50 Magnitude of the Frequency Response Function 10‘ 102 1o3 10‘ frequency (redlsec) Figure 2: Magnitude of the frequency response function HUw) in dB. LI, 3. (3 points) At What frequency w (in rad/ sec) does the magnitude of the frequency response function reach its maximum value. 71‘“ PM “arm‘s ’5 “0‘” “is 4. (3 points) What is the ratio of for the frequency specified in part 3 (do not specify your anSWer in dB). A. I?" -— , 9010;74' ‘2, Zopoa‘olV/F’: tic) =9 Qualol IF "' > IO 3 l IF ‘0 5. (4 points) At what frequency w (in rad/sec) is the amplitude of the sinusoidal steady-state response y(t) equal to the amplitude of the sinusoidal input f(t) '1' (“View --> ZaQowli/F/WM “‘4 L/ 10 Problem 4: (25 points) 1. (15 points) Find the frequency response function for the circuit shown in Figure 3. Express ' ‘ your answer in the form HUw) = Z = bm Ow)” + bm~1 (Mm—1 + ‘ ' '51 (W) + be P (30))“ + an—1(JW)"'1 + - - '61 (1“) + “0 ' with the coefficients a and I) expressed in terms of the components 01, Cz, and R. If your analysis requires the introduction of additional voltages and/or currents, clearly label these variables in the figure below. y I. R + £0) c1 c2 y(t) Figure 3: RC circuit with input current f(t) and output voltage y(t). L/ 11 2. (10 points) An electrical network with input f(t) and output y(t) is described by the ODE (133; day dy dzf df Determine the frequency response function H ( gm) of the circuit. Place your solution in the form HOW) = X : gm 0”)“ + bm—l (MVP1 + "'51 (1“) + 50 F (30))" + (In—1 (JW)““1 + ' ' "11 (30-?) + an I to 312-onwn +30 a + 209“) + {o 12 ...
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