Trasformata di Laplace

Transcript

1 Trafrata d aplac Nz d ba Agl Bcgla I t lar dcrtt da dll atatc l d dl tp cttt da qaz tgr-dffrzal d dffcl lz. S prfrc l dlla Trafrata d aplac [ latra f(t)=0 pr t<0 ] d dlla varabl cpla, ch trafra l qaz tgrdffrzal qaz algbrch. Igr (t) Rlz qaz tgr-dffrzal dll atatc d t Ucta (t) -trafrata -trafrata -attrafr Igr I() dll atatc d Rlz qaz algbrch Ucta U()

2 pratr -trafrata tablc a crrpdza bvca tra d partza d fz f(t) l d arrv d fz F() c I bl: = + j [f(t)] = F() - [F()] = f(t) -t f t f t dt F 0 S rcrd ch F() è a fz a valr cpl d varabl cpla: F(+j) = RF(+j)+ j IF(+j)= F(+j) j argf(+j) j Nr cpl j I[P] P P P R[P] arg[p] Crdat carta Crdat plar Il pt P pò r pr cd l crdat carta: ch l crdat plar: P = R[P] + j I[P] P = P j arg[p]

3 j Nr cpl j I[P] P P P R[P] arg[p] Paagg da crdat plar a crdat carta: R P P c argp IP P argp Paagg da crdat carta a crdat plar : I P P R P I P argp arctg R P D r cpl dc cgat ha la ta R I ppta Nr cpl Da qat dtt rcava: jα c j (pzal cpl frla d Npr) S t ch: j j c j ch jα c α α Pr praz r cpl: Pr ttraz cv ar la fra cartaa; Pr ltplcaz, dv, ptz radc cv ar la fra plar.

4 Prprtà dll trafrat d aplac Prdtt d f(t) pr a ctat Tra dlla lartà Tra dlla drvata Tra dll tgral Tra dl valr fal Tra dl valr zal Tra dlla tralaz f(t) f(t) f (t) + f (t) df(t) dt f(t) dt lf(t) t f(0) l f(t) t0 f(t-t) f(0)=0 pr t<t F() = [f(t)] [f(t)] [f (t)] + [f (t)] f(0) F() F() l F() 0 l F() -T F() Prcpal trafrat d aplac

5 x(t) Sta lar y(t) I lt ca l ta lar è rglat da qaz dffrzal rdar a cffct ctat dl tp [ c pr la fca ralzzabltà ] : - - d y(t) d y(t) d x(t) d x(t) a a -... a y(t) b b-... b0 x(t) - - dt dt dt dt Pr la lz ( tgraz) dll qaz.. la dtraz d a fz y(t) ch pr t 0 la ddf ccrr ccr: - cdz zal: - Il gal d gr: y(0 ) ; dy(t) dt x(t) pr t 0 ;... ; - d y(t) - dt t0 t0 Pr la lz dll qaz p cdrar paratat ctrbt dll cdz zal dl gal d gr,.. pò ttr la lz dll qaz dffrzal c a d d fz: - S pp x(t) = 0 dtra y 0 (t) [ vlz lbra dl ta ] - S ppg ll ttt l cdz zal dtra y (t) [ vlz frzata dl ta ] S tt cì la lz ttal: y(t) = y 0 (t) + y (t)

6 Sp t d ctrll atatc fa rfrt a t zalt qt,.. c ttt l cdz zal ll. C qta pt, trafrad cd aplac applcad l prprtà dall qaz dffrzal: - - d y(t) d y(t) d x(t) d x(t) a a -... a y(t) b b-... b0 x(t) - - dt dt dt dt tt: - - a Y() a - Y( )... a Y() b X() b- X()... b0 X() da c: - - a... a Y() b b... b X() a S qd: Y() b a b a b0... a X() S dfc fz d trafrt (f.d.t.) l rapprt: G() - b b-... b0 Y() - a a... a X() - a f.d.t. d ta è la fz dlla varabl ltplcad la qal pr la trafrata d aplac dll gr tt la trafrata d aplac dll vlz frzata Y (): X() G() Y ()

7 ANTITRAFORAZIONE DI APACE G() b a b a b... a 0 [ c pr la fca ralzzabltà] Il pl d grad al ratr att lz (ral / cpl cgat) all qaz: - b b -... b0 Tal lz ( z ) dtt zr dlla f.d.t. 0 Il pl d grad al datr att lz (ral / cpl cgat) all qaz: - a a -... a 0 0 Tal lz ( p ) dtt pl dlla f.d.t. a f.d.t. pò r cpta fratt plc: ( - z) ( - z )...( - z) G() ( - p ) ( - p )...( - p ) c = b Pr attrafrar cd aplac - [G()], dtg ca: - ha l pl plc (ral cpl cgat); - ha pl ltpl. I trab ca a l td dll vlpp fraz parzal td d Havd

8 ( - z ) ( - z )...( - z ) G()... ( - p ) ( - p )... ( - p ) ( - p ) ( - p ) Pl plc ral ll c rd pl -: G - p p ( - p ) Attrafrad la G() lla fra d a d fratt plc a - a t - p t p t p t [G ]... S t l qvalza: ' ( - p ) ( ) c = - /p τ = -/p τ è dtta ctat d tp carattrzza la rpta al grad d t dl rd Ep p = - p = - p 3 = -3 (5 3) 3 G() ( ) ( ) ( 3) ( ) ( ) ( 3) G a - p - a t - p G() ( ) ( ) ( 3) [G ] -t 7 -t 6-3t 7 6

9 Pr plctà cdra l d pl cpl cgat: p = + j p = - j S calcla rd: l rd pl -: I dftva: Pl plc cpl cgat S pò dtrar ch rd cgat. vlpp a d fratt plc è: G() G j ( - - j ) - p j j jv jv p -j ( - j ) v v arctg G() Attrafrad la fz tt: j ( j) t - j ( j ) t G() - ttd j j j t j t j - j t t - j t ] c t - [G j t j (t ) - j (t ) j α a -jα - a t c α

10 Pt - δ è la plaz atral δ è l cffct d rzat v - j jv j jv G() '' v - τ v - Cì a f.d.t. c a cppa d pl cpl cgat a la fra '' δ τ Paratr carattrtc d ta c d pl cpl cgat Rapprtaz d paratr carattrtc d tr dl rd j x x τ δ - δ

11 N.B. : Il cffct d rzat dv varar tra 0 δ <. Ifatt pr avr d pl cpl cgat l qaz d grad a datr: Δ δ dv avr l dtrat Δ < 0 δ 4 4δ 4 4 δ δ δ Ep G() G - p - p [G ( ) ( - j) ( j) ', j, j G(),69 ( j) ( j),5j,69 j j j t ] c t - [G j -j ( - j) ( j) -t ],69 ct,,j,5j,69,j,69,

12 N() G() ( - p Pl ltpl Pr plctà pp la f.d.t. c c pl d ltplctà h. Ach qta tplga d f.d.t. pò r a tt fra d a d fratt plc:... 3 h h 3 h ) ( - p) ( - p) ( - p) ( - p) C rd dat dall frl: d! d p h G() - - a t t Attrafrad: ( a) ( -)! p f(t) p t p t t! 3 t! p t... h h- p t t (h )! Ep G() (3 ) ( 5) ( 5) ( 5) G() - (3 ) ( 5) ( 5) ( 5) (3 ) ( 5) [G ] -5t ( 5) ( 5) 3t 3-5t - a t! - - a t