Laplace Transform 1 PDF

Umit Mo 5 Laplac Tantory Linan Tim Lnveutu ULTI) oro coe a elm L LT is p o w u l tool w tt Con dituwhal aau inte augebne ea Tran w - Convuuslon o me doi Slne into uouna domeln L...

Umit Mo 5 Laplac Tantory Linan Tim Lnveutu ULTI) oro coe a elm L LT is p o w u l tool w tt Con dituwhal aau inte augebne ea Tran w - Convuuslon o me doi Slne into uouna domeln LT dihuanhia auns CUL m t i m oomaiy aehbtc eals a im TG ()  T Lalau tramahorm of a tm Funchon ct) is dutin -st Lx)X)= |xey h u s u Complx vaiabla L u a l to s=6 tJ Two sidu or bilatua u n hemy lowi imt u changl to O o If then "on sidud or Unilatiul" |!xt]=X(t): zt)eat LT xCt) Xlt) paur sith X) xt) omy LT  Resion of mvyKm Roc Xls)-Sxte"at - S-planL T h t o f pims in the S plane for wich 'megral X9 Comv is t h giom S6tJ R+I Comveujenu (ROC) of th Laplau trauhunm C Bilatt pla tramsr ofa sigmac Ct exih xs) "at nite - Subshtuhm s: 6+ in the abou e C6ti)t Lxcy=X) = o«  tje iut F[xt Aboe x ls) is aindlicau tha basicady th Cnhmuw time Fow tramow cCt)et Th Loplaetramuhem a-xt t Fowi tranahom o st utie Exutenu aLoplsuatan x t t ) Should be Loninuw ingiu Closed intuvd 9CCt). e 6 t muut beabosdlu+y intaable e ct|dt coo x J 1cLt) Advange t L T s i g o l whidn o mt Conveunts inFT a Comveunt in UT ConvoluwHon in time domain Can be oPtaimedA by mulhplicahion in s dlmaur Dithwwhal be e" a sytem Can Convenbd into Simple alaubic a So LTI syttms be Can halned eai wima p l a tramatum Limitaiow hum pnL the S t Cannot be eatimatd Inttad on pole zco plot Com be dUlawm OS-jw w only for si moide sttasystiCUnai O Tmpule Signo t Set t= for t o fort#o L [ & t -X)=-[se -st e e Xls)=L - »(fr amg ) SCt L ROC is the enhr S-plan. i ROCAr s-plaun St gna ult) C t eutt) ult) for tro for t o X() x At st -3 s(u -slo) e S Xs) Lu - s-a4Ru> RC)o Th entis wgh hadM° t - plama Ramp Siana c t ) = tult) Ltult) = Xs)= x(t)t u v olt st -st L t -st s - (o - POC Rels)> -la ReCs)> Ralxpononiad Sianod eult Xls)L[eu)]-[xthut lt (s-t - (s-atn< A e - (S- (S) -(s- ,Xs) - S Re Cs-a) >o i-t Rels)>a ROC J S-plaae RoC e(s) a  AjO Il tt X(s) » Ruls)>-aa Sta jw S-plexp RLCs)- -a C -6 ROc - , for ulty Compl apmuntial signo t) e utt) t ) es t Xu) -o ' es t - s T ot dt X19 = [ E S - j E ) X19 S -j Sjw ROC isRLs) >0 pla ult) X) SH RoCels) >o Sine Cosima Siane Sinwt lt) £ COS t ult) Xs) fewt Cos wt ult) XCs) etiwt -st e ot ataur ee t ( *(5 +i+s-J SJ) (S+ju) S s2w2 RLCS) > eie. Rels)>o  sinsin wtul = ROCPets) >o jw Pels)o S-plan l - Prove tht t siqnady t)elt) xlt)z t hau th same XS) au) dtu on in Roc. Alse plot thai Rocs. So tt) -(Stat t -Uiyt» - (sta) o x(s) Sta.Roc Re(Sta) >o RLl) 7-  S - P l a n a Aj Roc Pels) - a Ct) ut ul-t tor tso.'X)= - x T a (Sta)t e -lra) J- S+a Re(s)> - s +e) Re(s)-a  Rpe-a S-Play 6 i n th Loplaeetrnsem ot th signad xCt» utd"uct) Lx -X) t e oUtt - ar + a -(S+)+1)t e (sAt At Convee COmv i A)>-4  - (s4)t5 x) s1) -LSt4) (o-+(o-1). -CS+1) S+I S+4 2S+5 Roc Rus) 71 5S't4 at Ct) eul+e ulty 4 Ayo Roc xs) e t(att - (s+2)t 2 s- Lonvuwy Comwu ) 7- PCs) 73 Conveyu H el) 3  S-plam la-jw RO Rals 3 Xla)=SEtuDt)4fG-Jt) e -GtJ (s- Co-1 +C0-) -(S+ -s-3) 2S POC et)>3  S - P w Xls) -CS+ Xtr) 2S 2t 4nd ihb Laut KU)=° - u-t) (S+2)t d -6 Coruy le