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A#15_Solutions

# A#15_Solutions - Assignment#15 Solutions — 13.1...

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Unformatted text preview: Assignment #15 - Solutions — 13.1 ECSE—Mlt) Signals & Systems - Fall 2006 Fri i1/03/06 1(20). Consider the RLC circuit below. + + R L 96(5) I C W) - O o .. The differential equation governing the input and output relationship is given by LCQaRCﬁi-iayq dt dz‘ Si“ a(5). Write down the characteristic equation (poles) . Le. 'l" 359 “a £15? 1,592+ (acute-Q (DE Fags-rt we } b(5) Evaluate the exact 1110(0)] at a) 2w” 2 J Grad/sec. Express answer in terms of some or all of R, L, C. y 5;! * “‘“ am: e :1 gram: h stat Gee-Eon») ﬁt; %+§ ﬁg; 5 gm. ya l: & w=w:”==~ it C(10) Suppose that 1221000th115. Given the following Bode magnitude plot, ﬁnd the values of L, and C. Note that the value of 1H0" a))E is exactly 3dB at 0:10 rad/sec. Assignment #15 w Solutions - pl ECSE~2410 Signals & Systems - Fall 2006 Fri 11/93/06 2(20). Using the properties and transform Tables (no integrations of the deﬁnition), ﬁnd X(s) , the Laplace transform for the following signals: (is a(5)- xawzmamava] = msiwauiealwﬂs: 4% m. 2%. w; §Ci~e ) E z ‘06) am : cos(4m){u<r+ 2) - u<r — 2)}. Z: {Gama mm} nwgcam? saw-l 50% ’3 ﬁr“: 3+3 :7?“ L w.“ w<tﬁr®+ttﬁ> twgifi'ﬁ' “*CQS 95.7%) "was Ceséﬁwéi-szled-éﬁlwCQSL‘fH‘Cé'2-3MV'E3 } s- \$ as MS ﬁn g 1 5&3? awaa “"‘ manta *gaeo {x m MELMM 231‘“ W .1. 0(5). x68) 1 gm [11(1) .— u( t- 4)] -.-..—._ 53 “EH .. a _g,§ +1341 hﬂémﬂﬂig es «3:. . ._ a: —-3 4+ "‘ magma; ﬁle, it pail-*qu :2 ~52. "-‘i’S l l 52-95 __ £53) 3‘“ Si, g - §+\$ n Sal-“'5 (1(5). Find the Laplace transform of x(r) shown at right. Assignment #15 w Solutions w p.3 ECSE-2410 Signals & Systems — Fail 2006 Fri 11/03/06 3(15). Using the properties and transform Table (no integrations of the deﬁnition), ﬁnd x(t) , the inverse Lapiace transform for the following signals: 51(5). X(S) = S (2+5? 1—6—23 33(5). X(s)= 1+3 we te+ mmia w W “=6 W) §~H Assignment #15 — Solutions - 13.4 ECSE-2410 Signals & Systems — Fall 2006 Fri 11/03/06 K 52 + s(1+ 0.25a)+ 0.02m 4(5). Given H (s) = , ﬁnd a so that H(5) has a poie at s=—5. Assignment #15 — Solutions - p.5 ECSE~2410 Signals & Systems - F213 2006 Fri 11/03/06 50.0). The transfer (system) function of a seconéwrder LTI system is given by Y(s) = a): 1'0) H s = . If the ole-zero dia am of H s is () X(5) S2 +2§mns+w3 p g () s—plane e100). Calculate the system step response, 3(1). Hgg} .3 5 1— gas-3 ligan mg a£+ Egg...» gi- f SLé’i-eﬁ"? 3} g gauge-i g *3“ _ ENC m7 gmﬁgz+es+s}mgsa+es Sﬂg+zf+ll S 6\$+292+i h: _§ﬁB ? snifﬁf. mi” Suﬁ 2&8” 5+2 “ 2 Fa; 5 Q-rg’ii—i 9‘ Cgi-gl-t“ @EJEH b(£0). Use Matlab to piot your solution in (a) Inciude your name in the title of your plot and hand in your Matlab code with your plot. \$68 mt page» Assignment #15 u Solutions - p.6 ECSE-24EO Sigials & Systems - Fall 2006 Fri RIOS/06 5(20). The transfer (system) function of a second-order LTE system is given by jet) 1 3-D one 1300) Use Matlab to plot your solution in (a). step response F[0:.{)1 :4]; y: 1-exp{-2.*E}.*eos(t)—2 *exp(-2. *t).*sin(t); piotttay); grid; titleCstep response') Note. Since the poies are complex conjugates, then you wouid expect to see overshoot. Actually the response does overshoot its steady-state vakue of I, as shown in the blowuup plot below. note overshoot 101 mmmm ««me—v—z—ﬁ~*we»~w¢m——~i—g—1w—m t=[0:.01:4}; 3 g 5 E g 5 : W 1»exp(w2.*t). *cos(t)-2 *exp(—2. *t). *Sin(t); 1.008 -------- --------- --------- ------- ~~~~~~~~~ iiiiiii 777777777 777777777 777777777 w ptotﬁay); grid; titleCnote overshoot) axis([2 4 .99 1.0g) 0' 99 _____memw§_ﬁﬁ_i—i—_i#, sssss ............ 2 2.2 2.4 2.6 2.8 3 3,2 3.4 3.6 3.8 4 Assignment #15 m Solutions - p.7 ECSE-2410 Signals & Systems - Fall 2006 Fri 11/03/06 6(20) 2m) Y(s) (21) Express damping factor 4 in terms of constant K, i.e., g" = {(K) . P0165: 32 +s+K:0. Coefﬁcients of poiynomiai give us: 2&0“ =1 and (of = K . Solving, 4' = W = ww— (b)Plot ; =;(K) for \$1194. damping factor vs. K K=[0.25:O,E:4]; zeta=0.5./sqrt(K); p10t(k,zeta) grid xlabelCK‘); ylabeICzeta’); titleCdamping factor vs. K‘) Assignment #15 — Solutions - {3.8 ECSE-24EO Signals & Systems « F31} 2006 Fri 11/03/06 6(20) continued. 2.5 (0) Plot step response for K m 2.5 . (Use Matlab command, step(num,den).) 11(3) w S + S + . Step Response :l‘ System: sys Amylitude Time (sec) numm25; den=[1 I 2.5]; step(num,den) grid (d) Use. plot in (c) to determioe % overshoot. Using the data cursor on the plot, we can see that the overshoot is 35%. The theoretical overshoot is 35.09% occurring at z = 2.22 , so the MATLAB plot is pretty ciose. ...
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A#15_Solutions - Assignment#15 Solutions — 13.1...

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