examI_f01 - EE 350 EXAM I 20 September 2001 Last Name(Print...

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Unformatted text preview: EE 350 EXAM I 20 September 2001 Last Name (Print): L S Oi U'bl on S First Name (Print): ID number (Last 4 digits): Section: _ DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO 50 Weight —?- “- Test Form A INSTRUCTIONS 1. You have 2 hours to complete this exam. 2. Calculators are not allowed. 3. This is a closed book exam. You may use one 8.5” x 11” note sheet. 4. Solve each part of the problem in the space following the question. If yOu need more space, continue your solution on the reverse side labeling the page with the question number; for example, Problem 1.2 Continued. NO credit will be given to solutions that do not meet this requirement. 5. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers will not be accepted and a grade of ZERO will be assigned. 6. If you introduce a voltage or current in the analysis of a circuit, you must clearly label the voltage (current) in the circuit diagram and indicate the reference polarity (direction). If you fail to clearly define the voltages and currents used in your analysis, you will receive ZERO credit. 7. The quality of your analysis and evaluation is as important as your answers. Your reasoning must be precise and clear; your complete English sentences should convey what you are doing. To receive credit, you must show your work. Problem 1: (25 Points) 1.1 (8 points) The zero-state response of a signal-avenger with input f(t) is 1 t+t2 y(t)=t1+t2A_t1 fawn where t1 2 0 and t2 2 0 are constants. Is the system time-varying or time-invariant ? Justify your answer. Le": Jag-5 = 40:: 4-)) than t+-L2 ‘12-th I 3"“ ‘1 t,+-tzj ‘F-‘(flcat = fi+£23$Ct~fler t +3131 Now QQ/‘b f3: ‘C-‘T'J stchC t—T‘fi-t; a $2345.22,” ~I:-T+—I:-z. 95 3&4"): fizfie-fiih3itj (D) (2 fig «(Sums C DD 6! mg» C U) S\O\U a, (1:) = a 0547.).) ‘ 'lztm ext/NOW M” £7 ' he, 8 .{gm IS ______.________—————- (mi 80 '5 0"" 1.2 (9 points) In response to the inputs f1“) : €141: f2(t) = e_-74t, a linear time-invariant system yields the zero—state responses 11103) = 5 J «214‘ ya“) = —5 J 6‘7“, respectively. Find the zero-state response y3(t) for the input f3(t) = 4 cos(4t), and express y3(t) as a. real-valued sinusoidal function of time. 0 bserue. tl‘ 07b ‘145 _ vi- .205 +193 = 2% + e/ :1 ZCOS Qfi) and» 50 .9395) .=._ z—Ql-L—D + 24168. Because, the, ad” ‘15 Qmear; it)“; = refine) + lgzct) 1.3 (8 points) 0 (4 points) You are asked to find the product of the two polynomials Q1(A) : A3+2A2+A+1 Q20) = A4+3A+5 and to express your result as Q1(r\) Q20) = A7+as A5+as A5 +04 A4+a3 A3+az A2 +a1 A1 +ao- Write a complete MATLAB command for determining the coefficients ai. o (4 points) Suppose two vectors are defined in MATLAB as: H >>€D >>y [—110a1l; [2,1,2]; What does the MATLAB command >> 2 = a: . * y yield for z ? Problem 2: (25 points) A certain system has a characteristic equation whose roots are shown in Figure 1; note that two of the roots are real and identical. ImOL) ReOL) Figure 1: Location of the characteristic roots for the system considered in Problem 2.1 and 2.2. 2.1 (10 points) State the form of the homogeneous response of the system in terms of undetermined coefficients 0;. Each characteristic mode in your expression must be real-valued, that is, you cannot have terms of the form a em . 2.2 (5 points) Is the system in part 1 asymptotically stable, marginally stable, or unstable ? Justify your answer in one or two sentences. 2.3 (10 points) A system, different from the one considered in parts 1 and 2, is described by the ODE y(t) + a1 W) + a0 y(t) = 25 7r2 f(t). The zero-state unit-step response is characterized by a sinusoidal oscillation, with a period of 2 s, that exponentially decays to a steady-state value of four. Determine the following parameters: rad / sec] = _1T_[ — .5 In; [rad/sec] ( = If. [,5 I]: '21 a0 = 251122 3 . BGCW UQ/ [6pq‘l3 l5 WKE/ $6 '18 yr (3(5) 2' g 0 '3- ” O o _ o7 + W + W “ E :1 l a a II 2.7-“ _ 7. 77“ = 5;: “-1“ =9 l? l 70 L L: 2:" f =- l" '23: z" 2.5 U3!fl# 0" =_ Z’PWA fluxes _. 2/ r; 5.1: = 11—13:: qt " 7 Z Problem 3: (25 points) 3.1 (8 points) Derive an ODE that relates the input voltage f(t) to the output voltage y(t) in the circuit shown in Figure 2, and place your answer into standard form. 3.2 (8 points) Once again consider the circuit in Figure 3 and let f(t) = 2 + 4 u(t). 10 k9 Figure 3: Passive RC network. Determine the following: W) = l [V] i(0—) : 00' [mA] y(0+) = __l___[V] CH‘CUFE 03L t z 0‘ _ + (01" S 9-}, 15,0 J y {aacrosf aka/1 Q msw‘lsanewsd Owwit (A? ‘17 :6+ 3.3 (9 points) Consider a system characterized by the ODE W) + 3 W) = 4 f0), the input N) = 2 e‘3 t, and the initial condition y(0+) = -1- Determine the zero-state response y"(t), the zero-input response yn-(t), and the total response y(t) for t > 0. 6? (713 = ’A-r'b =0 :9 fight-f.) :. ce, Problem 4: (25 points) 4.1 (9 points) Assume that the operational amplifier in Figure 4 is ideal. Find the differential equation that relates the output voltage y(t) to the input voltage f(t), and place your answer into standard form. 11 4.2 (8 points) In the circuit shown in Figure 5 N) = sin(1")U(t)[V] 11(0") : 3 [V] «0-) = 5 [mA]. Determine the initial conditions W) = 3 [mA] 111 _ o .2. fl dt t=O+ s Figure 5: Network with input voltage f(t) and output current y(t). V(0_)=V(O'*) 3 3 V bled—use, uoHSq s2. Carma'b eta/7L “1:: o . MSJan'iDCMQoJ—iy Ukc r053 HG» Cafan fix 0:17 ‘5th V(O'T) 3 ——3-‘£~ 2: 3/» A flhfl. 50 gaff) 2 IKJL m9. Ob§eTu¢ thfle A A,t+_ .. \I" —. can» ‘ 0-24: - “a: am9~ So 3511?... s . 55—. db» _—: RC0” = ZMA toms S 12 13 4.3 (8 points) The following ODE describes a dynamic system W) + 8 W) + 16 W) = 9 f(t) + 16 N)- Suppose that f(t) :: e‘“ u(t), y(0+) : l, and y(0+) : 2, find y(t) for t Z 0. (9(3) 2 72‘ + 8% + )6 == Cfl+~0L .. t "‘it yum — CLQ q + C2“- ..zt “2b 2. (“ED 5'- 2‘ ’48 £05) 6 _) gr ’2'": 4. 162721: 3? *‘ 8W “" “7F '" "8"“ . .. v-Z‘b pw 5 $9- 2' ‘1': -l6 ez-t Hegxa 7- '16 > 14 15 ...
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