final_f98 - FINAL EXAM BB 350 16 December 1998 Name(Print...

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Unformatted text preview: FINAL EXAM BB 350 16 December 1998 Name (Print): ID#: Section: DO NOT TURN THIS PAGE'UNTIL‘YOU ARE. TOLD TO DO SO EXAM FORM A Instructions: 1. You have 1 hour and 50 minutes (110 minutes total) to complete this exam. 2. This test consists of six problems. The exam score is calculated by adding the five largest 3. 4. problem scores. This is a closed book exam. However, as told earlier, you are allowed 2 sheets of 8'/:" x 11" paper for notes, formulas, etc... Solve each part only in the space following each question. If you need more space, continue on the reverse side and write the question number; for example, Question 42) continued..; N0 credit will be given to a solution that does not meet this requirement. Box your answers where appropriate. DO NOT REMOVE ANY PAGES FROM THIS EXAM. Loose papers 'will NOT be accepted and a grade onERO will be assigned. Problem I: (20 points) With respect to the circuit in Figure l, the input is a voltage function {(0, applied at t=0. and the output is the current. y(t). The switch is at position A for a long time and then moved suddenly to position B at t=0. 120 {(t) Figure 1- Circuit for Problem I, all parts. Question 1.] (5 points) Determine the initial voltage. v‘(0‘). across the mpacitor. Solution: Question 1.2 (10 points) Write the complete expression for Y(s) alter the switch has been thrown. Identify which part of the expression pertains to the zero-input component and which part pertains to the zerOostatc component. DO NOT PERFORM THE INVERSE LAPLACE TRANSFORM. Because la) is arbitrary but assumed transfonnable. use F(s) as its Laplace transform. Solution: Question L3 (2 points) Wrile lIIc lmnsfcr function. H(s). aflcr the switch has been thrown. Solution: Question 1.4 (3 points) Determine the Circuit‘s lime-domain. ordinary differential equation system model for l 2 0. Solution: Problem 2: (20 points) The input, ('0), and the zero state response, y.,,(t), of 3 LT! wstem are shown in Figure 2. “0 yea) Figure 2 - The input and output of an LTI system. A. new input. f.(t) (which can be expressed as {(t - t;) + [(t - t;)) is shown in Figure 3. This new input is applied to the same LTl system from Figure 2. Figure 3 - A new input, Mt), which is applied to the LTI system in Figure 2. Question 2.! (10 points) Sketch and properly label y.(t). the zero state reSponse of the system to Mt), on the graph provided below in Figure 4. Solution: 3’10) Figure 4 — A graph for sketching yin). For questions 2.2 and 2.3, consider the following system: ' Figure 5. The system for Questions 2.2 and 2.3 Question 2.2 (5 points) Is the system, shown in Figure 5. causal or non-causal? Justify your answer. Solution: Question 2.3 (5 points) Is the system, shown in Figure 5. instantaneous or dynamic? Justify your answer. Solution: Problem 3: (20 points) Question 3.1 (10 points) This question has three sub-parts. labeled a). b) and c). In Figure 6, 0(5) is given as .H-l c: s 2 l0——-—-—-—-—-— () 32+lSJ+99 R(S) C(s) ~49 Figure 6 - Feedback system for Question 3.]. parts a). b). a c). Queuion 3.1:: (4 points) Find the closed-loop transfer function C(s)l'R(s). Simplify your expression into a mic of polynomials in 3. Solution: Question 3.15 (3 points) is the system, C(s)/R(s). BIBO stable? Justify your answer. Solution: Question 3.1:: (3 points) Find the damping ratio. 9 of the closed-loop system and state the type of reSp-onse (i.e.. critically damped. underdamped. overdamped. undamped. or negatively damped) the system will exhibit to a unit-step input. Solution: Question 3.2 ([0 points) Using the scmilog paper provided at the end of this exam. dmw the slmight-line approxiumtion of the Bode magnitude plot of (jw+ l) n =10——-——-—-—- Um) (jay): +lG(jw)+ too Use dashed lines for individual terms and a solid line to indicate the total re5ponse. Be sure to label the slopes of the straight line segments of the total re5ponsc curve. You do not need to apply correction factors at the break/comer frequencies. Problem 4: (20 points) Consider the periodic voltage waveform. v0). shown in Figure 7. Figure 7 - Periodic waveform, v(t). specified in Question 4.1. Question 4.1 (10 points) Specify clearly the symmetry (if any} of v(t) and find the trigonometric Fourier Series expression of v(t) taking A = 4 and T = 4 s. Specify clearly the values of the Fourier coefficients a. through a; and I): through b5. Solution: Question 4.2 (10 points) The Fourier series of input voltage \'.,(I) in Figure 8 is given as mm = 50 +ZIFOS-sin(nwal), where (do = 10/3 radsls nil Find only the third hannonic component ol'tllc oulpu! vollagc \',(1). lkfl vino) 100 ”F v.0) Figure 8 ~— Circuit for Question 4.2. Solution: Problem 5: (20 points) Consider the sampling operation in Figure 9. KO M) 5d!) Figure 9 - Mathematical representation of the sampling operation. Question 5.1 (10 points) Suppose {(I) - cos(20m) + sin(40nt). Determine the maximum sampling period. T. needed to avoid aliasing in the sampled specmim. Solution: Question 5.2 (10 points) Suppose the sampling rate is 60 Hz. Skclclt the sampled spcclrum. EU) over the interval -80 Hz 5 f S 80 Hz in Figure l0 below. Label the frequencies of each component in the Spectmnt and the areas of the impulses. Solution: FU) -60 -30 30 60 f (Hz) Figure 10 — Graph for the sketch of the sampled spectrum EU) Problem 6: (20 points) Consider the system shown in Figure l l, which implements single (lower) side band modulation. The Spectrum of 1110). M053). and the spectrum of H(m) is shown in Figure l2. Note that 2111} << 01‘. m(t) y. (I) Y0) Hilbert Tmnst‘onuer sin(mgt) Figure 11 - A system to implement single (lower) side band modulation. M(w) A Figure 12 — The spectrum of MG») and the spectrum of H(m). Question 6.1 (5 points) Sketch and properly label the spectra Y.(m) on the graph provided below in Figure 13. Solution: Figure 13 - Graph for the sketch of Y.(m). Question 6.2 (5 points) Sketch and properly label the spectra Mh(m) on the graph provided below in Figure 14. Solution: Figure 14 —'Graph for the sketch of the spectra M..(ro). Question 6.3 (5 points) Sketch and properly label the spectra Y;(m) on the graph provided below in Figure l5. Solution: Figure 15 — Graph for the sketch of the spectra View}. Question 6.4 (5 points) Sketch and properly label the single (lower) side band spectra Y(co) on the graph provided below in Figure l6. Solution: Figure 16 — Graph for the sketch of the spectra Y(co). 119,3 7 .U 4 .J 1. . ..,.:: 34.7.. VII! ..4 .LII Luna»? _ _. ._. _. _,. ._. v. .lJll H} H. w «’3... ..:. DJ .Zumd 6 .:*:Ju{ m23.u_>_3 a“ K .w..__U>U a. ..:E:h:L<CO .45..“ _ . m . H n . mi ...
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