Fall07_Exam2soln

Fall07_Exam2soln - ECE 210/211 University of Illinois...

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Unformatted text preview: ECE 210/211 University of Illinois Analog Signal Processing Exam 2 Fall 2007 Levinson, Sarwate, Kudeki Thursday, October 18, 2007 — 7:00—8:15 PM éytahw . Name: Section: 9 AM 10 AM 1 PM (circle one) Class: ECE 210 ECE 211 (circle one) Please clearly PRINT your name IN CAPITAL LETTERS and circle your section in the boxes above. This is a closed book and closed notes exam. Calculators are not allowed. Please show all your work. Backs of pages may be used for scratch work if necessary. All answers should include units wherever appropriate. Problem 1 (25 points) Problem 2 (25 points) Problem 3 (25 points) Problem 4 (25 points) Total (100 points) Problem 1 (25 points) (a) [4 pts.] What is the polar form of the phasor V of voltage v(t) in the circuit shown? 3 cos 2t v(oe—v V: 3-)4 = 9' W3) 4sin2t4—yv'q 4 V=§fl> V (b) [5 pm] At what frequency to is the equivalent impedance Zeq of the circuit shown purely resistive? zeq R; 1009 L=1mH I IOuF = C. Zeq=IOOQat0J= 10.44% Tea . - c £241" ‘14- JwL+ jwc :K+j (“’Ldfiaié‘ zfi=fl=k§0fl ’E WL“;}E':O 5’.) w — LL 5 W I “W Problem 1 (continued) (c) [6 pts. ] Consider the function ; 260:.(3fi) /_\ fa) = eth + eJ3t + 2 + e—J2t+ e—j3t M (i) Is the function periodic? Explain your answer. = 2. +ZC»(Z€)+2¢0C3 9 1m (2+) Justification: cha‘“ u ()0; Wcfl’M/Hfh’ ‘4' ‘2, Q mrd‘d'e" ’3‘ Problem 1 (continued) (d) [10pts.] Consider the periodic function f(t) = 4 + cos(20t) + sin(30t) (i) What is the fundamental frequency 030 of f(t)? we 3 (*9 ¢0~0i /§¢<— '20 S ZWo/ 30 gawo (ii) What is the period T of f(t)? 60; i0 5’ (iii) Find the exponential Fourier coefficients F1, F2, and F3 of f(t). Problem 2 (25 points) Consider the circuit shown below. i(t) 4-9 I a 3 9 2cos 3t Va) 45 V=Z_.+)‘0 1H b (a) [4 pts.] Express the current i(t) as a phasor I in rectangular form. =n,zao..z._—L—~ by” a 2353’3‘5‘ 3J3 -xéw “1-31 -3 7’ 3 3 (b) [4 pts.] What is the angle between the phasor V and the phasor 1? AV: 0 LIV-434’ » AV» 4.: 7 O “(43):”‘50 . Q 5 WW Problem 2 (continued) (c) [7 pts.] A capacitor of C farads is placed between the points a and b as shown. What should the value of C be if it is desired that 4V = 11 ? -‘cs'zc gnu. cht’Jw J / I =. c+—L-——‘-.-- Yaw“??? )5 3”“ ._v +_Ll")\ ,JEC ski—— ‘ . \(efisg. if 3a,: =7 97%“? e ahmact'bMV-LH phqw %_J.;.A “thrush Th2 (“Mew Gallant»? «J t” 11»... Cafaww fin- AJJIHAJ C‘M‘“ - h - J. '4 A': A A -——-V ~ 2’)“ NW ZJSL+3"J; 3 ac. C=.L F oat/(Kr l P 3? g V. ’fkrmsk Problem 2 (continued) (d) [6 pts. ] Find the Thevenin voltage VT and Thevenin impedance ZT between the points a and b in the circuit shown below. j6Q 29 (e) [4 pts.] What is the matched impedance ZL for the circuit shown above and What is the available power Pav of the network? z 0 PM: iVTlZ , ‘l' 3135‘“, zL=2§k=ngQ 92436.7] gxz ‘6 Problem 3 (25 points) The circuit shown below has an input current is(t) and an output voltage Vo(t) across the l-ohm load resistor. 4/3 H 0 1 Q V0“) 1 The frequency response function is H(co) = 2 1 + 2 jco — 20) — j co 3 . (a) [5 pts. ] Find the amplitude response function lH(co)l . Simplify your answer as much as possible. on 3 i H( ) (vafimfl ‘5) ‘ l : _________.. ’ 1... +4u24+ v-deb H-w" (b) [5 pts.] The circuit acts as a filter. What kind of filter is it? o :— i H C ) l ' [Sirele one: High pass Band pasfl “40401—50 a4 weoo (c) [5pts.] Find ZH(co) asw——>0and03—+oo. 2) Law Wow; I'M“) : ——‘—'~ :9 4H6): b° AMP->0, 1H(w)—> 0° -1; Asco—éoo,zH(m)—> Clo" H (wean) ~.__:L—§— = '3’?” Problem 3 (continued) 1 Recall that H(co) = —————. 1+2joa—2co2 —jw3 (d) [5 pts. ] If is(t) = 2 + 3 cost, What is Vo(t)? (e)- [5 pts.] Suppose iS (t) = Z Inejnmot . What is the amplitude of the frequency component at n=—oo frequency nwo rad/s in Vo(t)? \lul Amplitude at ncoo rad/s = ____T———z—-, ,J l 1— n w. Problem 4 (25 points) Consider the periodic function f(t) plotted below: (a) [2 ptS.] Determine the period T. (b) [2 pts.] Determine the fundamental frequency (no. (DO = w a ; 31; = 2,; a 17 Md 1’ ‘1 Z 9“. .(c) [5 pts.] What is the exponential Fourier coefficient Fn for n = 0? Explain your answer. F0 F “11% D1. Ched‘vfla’l’ °.( HM. O Problem 4 (continued) The graph of f(t) is shown below for your convenience. . V- was ,1- J; .1- u~ ‘t—‘&\_.\_-——% ‘r '3'“ ‘l "in. 4‘ aw J“ 3“ . - . “ .- 2 ‘ ‘JE (~l)€’fl ‘ E d. J- 'C ‘l'ea a J. ( .Tr >: __l—-— pv ’- —' ’———"‘"’ v r. n "' ‘ a T emz ‘ 4 U" ‘* J 2”" W‘) O , (e) [3 pm] What is F_2? Explain your answer. F_2 : l' ggwc‘e’ {5 Wang.) fizz- "F2:k ALI“. _. ___L_.. (f) [3 ptS.] What is F n + RH for arbitrary n? Explain your answer. Fn + F_n = 0 _ 2p? lcwwfim : '31“?an Fifi—rfihz‘fln/ gocfecJ-odj oi. mount) 4m»: «‘MLV. Mwmam {WM .4 1L». tut-e. M ‘Ccm Moo-w. (5 3g; Fur F,“ = O ) MOL TLsn—e-c—Qvu-‘~ 04‘30. ...
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Fall07_Exam2soln - ECE 210/211 University of Illinois...

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