Signal Processing and Linear Systems-B.P.Lathi copy

Feedback systems are closed loop systems mainly used

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Unformatted text preview: t). 2 (i) e - 'u(t: (ii) e - 'u(t) ( iii) e - 4 ('-5)u(t_5) ( iv) e - 4 ('-5)u(t) ( v) e - 4'u(t-5) . 6 .3-8 J (t) + 88 2 + 5s + 7 83 H (s) = Solve t he f ollowing s imultaneous d ifferential e quations using t he L aplace t ransform, a ssuming a ll i nitial c onditions t o b e zero a nd t he i nput J (t) = u (t): ( a)(D + 3 J(t) + 6y(t) = 3 - + 7 - + 5 /(t) 2 = 25u(t) Solve t he d ifferential e quations i n P rob. 6.3-1 using t he L aplace t ransform. I n e ach case d etermine t he z ero-input a nd z ero-state c omponents of t he s olution. + 3)Yl(t) - 2Y2(t) = I (t) - 2Yl(t) + (2D + 4)Y2(t) = 0 ( b)(D + 2 )Yl(t) - (D + 1)Y2(t) = 0 - (D + l )Yl(t) + (2D + 1)Y2(t) = dy + 11 dt + 24y(t) (a) H(8) = y(O-) = 1 a nd J (t) = y(O-) FyOsrteemacht o f ;hefsyst~ms d escribed by t he following differential e quations find t he ranSler unctIon: ' 2 dy ( a) dt 2 = y(O-) = 0 a nd J (t) = u (t) = (D + l )J(t) i f y(O-) = 2, b y t aking t he t he L aplace t ransform o f loop e quations S U sing t he L aplace t ransform, solve t he following differential equations: + 3 D + 2) yet) = D J (t) ; : ~h:~:c~!t. in IFigd' ~6.3-4, t he s witch is in o pen p osition for a long t ime b efore , n I IS C ose mstantaneously. ( a) W rite l oop e quations ( in t ime d omain) for t 2: O. i~n~~~~!~: (~~t) a nd Y2(t) 6 .3-5 S tarting o nly w ith t he f act t hat oCt) { :=> 1, build P airs 2 t hrough lOb in T able 6.1, using various p roperties o f t he L aplace t ransform. H int: u (t) is t he i ntegral o f oCt), t u(t) is i ntegral o f u (t) [or second integral o f o(t)], a nd so on. ( a) (D2 v o(t) 2 H (S)=_8_ s2 + 9 e-')u(t) 6 464 C ontinuous-Time S ystem A nalysis U sing t he L aplace T ransform IH 3 -10 For a n LTIC s ystem w ith zero initial conditions (system initially in zero s tate), if a n i nput f (t) p roduces a n o utput yet), t hen show t hat: ( a) t he i nput d f / dt produces a n o utput d y/dt, a nd ( b) t he i nput J~ f er) d r produces a n o utput J~ y (r) dr. Hence, show t hat t he u nit s tep response of a system is a n i ntegral of t he impulse response; t hat is, J~ h( r ) dr. 465 P roblems IH f (l) ~ ~ YI ( I) 6 Y2 ( I) In In f (l) 0 (a) (b) + / (t) 1- IH In F ig. P 6.4-4 In IF Y1 ( I) F ig. P 6.4-1 IH 2H IF I OV In F ig. P 6.4-5 F ig. P 6.4-2 + 4H ~ y (l) f (l) L c / (1) In F ig. P 6.4-6 6 .4-4 ~ind t he l o?p ~urrents Yl (t) a nd Y2(t) for t I nput f (t) In Fig. P6.4-4b. 6 .4-5 F~ t he netw?r~ in Fig. P6.4-5 t he s witch is in a closed position for a long t ime before t - 0, when It IS o pened Instantaneously. Find YI(t) a nd v s(t) for t ~ O. 6.4-6 F ind~he o utput voltage vO(.t). for t ~ 0 for t he c ircuit in Fig. P6.4-6, if t he i nput f (t) - lOOu(t). T he s ystem IS In zero s tate initially. 6 .4-7 F ind t he o utput v oltage vo(t) for t he n etwork in Fig. P6.4-7 if t he i nitial conditions are tL(O) = 1 A a nd vc(O) = 3 V. (Hint: Use t he parallel form of initial condition generators. ) For t he netwo.r~ in F~g. P6.4-8, t he switch is in position a for a long time a nd t hen is m ?ved t o pOSitIOn b I...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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