Laplace Transform. Definition of Laplace Transform: f(t) that satisfies The Laplace transform of f(t) is defined as.
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1 Lplce Trfor The Lplce Trfor oe of he hecl ool for olvg ordry ler dfferel equo. - The hoogeeou equo d he prculr Iegrl re olved oe opero. - The Lplce rfor cover he ODE o lgerc eq. σ j ple do. I he pole o pule he lgerc eq. y ple lgerc rule -x o o he oluo he - do. The fl oluo oed y kg he vere Lplce rfor. Defo of Lplce Trfor: f h fe The Lplce rfor of f defed f σ e d < for fe rel σ L [ f ] F f e d referred o he Lplce operor, whch coplex vrle σ j ple
2 Lplce Trfor Ex Le f e u ep fuco h defed o hve vlue of uy for > d zero vlue for <. ow f u Fd Lplce Trfor of f Soluo F L[ f ] u e d L[ u ]
3 Lplce Trfor Ex Coder he expoel fuco f e ; Where co ow f e - Fd Lplce Trfor of f Soluo F L[ f ] e e d L[ e ]
4 Ivere Lplce Trfor Defo of Ivere Lplce Trfor: The Ivere Lplce rfor of F defed c j f L [ F ] F πj c j e d c rel co h greer h he rel pr of ll gulre of F Sgulry he po whch he vlue of fuco c o e evlued e.g. The ove pproch o fd Ivere Lplce rfor que coplced, prcce, we fd he vere Lplce rfor ug Look up le
5 Lplce Trfor Tle
6 Lplce Trfor Tle
7 Ipor Theore of he Lplce Trfor Mulplco y co L [ kf ] kf Where F he Lplce rfor of f Su d Dfferece L [ f ± f ] F ± F Where F d F he Lplce rfor of f d f 3 Dfferel d f L[ ] F f d I geerl, for hgher-order dervve, d L[ f ] d F f f... f
8 Lplce Trfor Tle 4 Iegro F L [ f τ dτ ] I geerl, for h-order egro, 5 Shf Te L... f d d... d τ τ F T L[ f T u T ] e F 6 Il-Vlue Theore l 7 Fl-Vlue Theore l f f l F If F lycl o he gry x d he rgh hlf of he -ple he, l F
9 Lplce Trfor Ex Coder he followg fuco, d fd edy e vlue of f F 5 ow F Fd f -> F Soluo 5/, he fl vlue heore co e ppled.
10 Ivere Lplce Trfor y Prl-Frco Expo Whe he Lplce rfor oluo of dfferel equo rol fuco, c e wre Q P P d Q polyol of, ued h he order of P greer h h of Q. The polyol Q y e wre P... Where,, re rel coeffce. The oluo of P or pole of he po whch o lyc re eher rel or coplex-cojuge pr, ple or ulple pole.
11 Prl-Frco Expo whe ll he pole of re ple... Q P Q Where -, -,, - re rel or gry uer. Applyg he prl frco expo echque,... The coeffce,,,,,, re deered y ulplyg oh de of Eq. By he fcor d he eg equl o Q P Q
12 Lplce Trfor Ex Coder he fuco ow Fd prl frco coeffce Soluo
13 Prl-Frco Expo whe oe pole of re ulple... r Q P Q The c e expded... If r of he pole of re decl or he pole of - of ulplcy r, r r A A A... - r er of ple pole r er of repeed pole... 3 Q P Q The coeffce,,, c e deered
14 Prl-Frco Expo whe oe pole of re ulple The coeffce for ulple-order pole c e deered A r [ ] r A A A d d r [ ] r r d! d [ ] r r d r r! d [ ] r
15 Lplce Trfor Ex Coder he fuco Fd prl frco coeffce Soluo ow A A A 3
16 Lplce Trfor Ex Coder he dfferel equo d y dy 3 y 5u d d dy The l codo y d d Soluo y 5 5e 3 e
17 Lplce Trfor Ex Coder he dfferel equo d y ζ dy y u d d The l codo y d dy d Soluo ζ < y ζ e ζ ζ φ φ rc ζ ζ
18 Trfer fuco Trfer fuco of ler e-vr ye defed he Lplce rfor of he pule repoe, wh ll he l codo e o zero. Le G deoe he rfer fuco of ye wh pu x d oupu y. The, he rfer fuco G defed G L[ g ] The rfer fuco G reled o he Lplce rfor of he pu d oupu hrough Y G Wh ll l codo e o zero, where Y L[ y ] L[ x ] Y G Trfer fuco defe he hecl opero h he euree ye perfor o pu o yeld he e repoe of he ye
19 h Order ordry ler dfferel equo wh co coeffce x d dx d x d d x d y d dy d y d d y d L L Trfer fuco To o he rfer fuco of he ler ye, we ply ke he Lplce rfor d ue zero l codo. Y L L The rfer fuco ewee x d y gve y Y G L L ] [ G L y
20 Trfer fuco For he fr-order ye: dy τ d y x Lplce Trfor: τ Y y Y τ Y y τ τ Where Y d Lplce Trfor of y d x Th c e rewre: Y G Q y Zero e repoe Zero pu repoe x Meuree ye y
21 Trfer fuco x y d dy d y d ζ The frequecy repoe of ye c e derved fro he rfer fuco y uued wh : [ ] / φ φ τ τ M G For he ecod-order ye: / / / / y y Y ζ ζ ζ Q G / / G ζ Trfer fuco [ ] [ ] / ] / / / / φ ζ ζ G Frequecy repoe ζ φ / / τ φ
22 Coupled ye Whe euree ye co of ore h oe rue, euree ye ehvor c ecoe ore coplced. x Meuree ye G y Meuree ye G y G Y G Y G G Y Equvle ye The overll rfer fuco of he coed ye he produc of he rfer fuco of ech ye
23 Coupled ye Ex P3.34 The oupu ge of fr-order rducer o e coeced o ecod-order dply ge devce. The rducer h kow e co of.4 d c evy of V/ o C, where he dply h vlue of evy, dpg ro, d url frequecy of V/V,.9, d 5 Hz, repecvely. Deere he edy repoe of h euree ye o pu gl of he fro T 5 68 o C. Soluo G order ye Y G d order ye Y G G τ / / ζ Equvle ye G G Y G G τ [ / ζ / ]
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