Laplace Transform PDF

Laplace Transforme it tood to solve Superficially just is a...

Laplace Transforme it tood to solve Superficially just is a differential equation time function from It transfers a domain a to frequencydomain Co slex The definition Seems ref - St Let (10) f(t) cas y ! 2 eg = (signal). f(t) Y = F(s] y > - = Time Laplace > - SY- SYOGY Domain Transf - 294370 : y" y - - by = 0 Calgebraic equ) y (0 = 2 y(0 = 1 Y(s) (Standard Solve four Diff equ) ↓ Invesse Transpor Gives yetdesired as domain We call it complex frequency because the frequency is in complex S = - + - represent- ↓ ation Consider the - example : cos(lot) w(- related ,0 Frequency The : 10 --St is 6 decaying behaviour bot decayingis  to Laplace transform astool a solve differential equation. Time Domain Schematic ↓equ It ↑Inverse Laplace FAlgebraic ) transform Complex Freg domain To be seem-laplace transform are related to Fortier Transform.  Actual Definition IT f(x) > - F(s) S = 6 + iw o or ↳ [fct] = F(s) = Yesfutide X & Generic LE + Y (+ Definition : Complex (Causality). Integral Units 's of are Jec (as of frequency] Jht) 1 Example = F(s) = the Stat first let ! plot us it lesty = X * ↑ Step function : ↑ (s) Valid - = -- for Res > O > it is t Of course valid sto for 52[ include questions covergence of Other twass form eg ↓ [ea+] +30 - Jetstat de o FIs) = & has diverges for Ss9 M In - This Integral deeper implication a a while !  Consider * modification of Step function -2 f &! ( = 2 > 0 tcX. * e-stdt What's ([fCt] = J a - at * Example : 1 [t] This is much the easily solved by introducing gamma function ↑ (x] = & e-t - do Cr] ↑ = · ↓ t that. by Integration Let s works it out doe to parts Judu = ora Judy t Let dv = - e v = th 9. -- test 100 Joane ↑ O g-t to goes = rapidly zco very & Dominant part  ↑ (n + 1 = R 9) - = than t · J = 2 T(n] · Similarly ↑ (n) = (n - D + (n - D - - TCn + D = n Cn-D + Ch - 1) · = n. Cn-D -.. nT(D ↑ # = n ! Formal M definition , 8 ! = T CD = 1 Note No restrictions were made on n back to the Coming * first question [t] J St L = - - to di ② Let st y = dt dy/s = = * site ! -8 ye dy =. (n+ 1 = - gn + I  Existance properties glaplace and transform should be Logic Existance some way : in the Integral related the * ↓ estf(t at e i E eg Valid foor only certain values of is There general is prescription a For a fC) function if If()sMest for large t - too some M + C Then corder ! f(t) of exponential is Note : e-St dow I Bring M => exist LT Integrand in unless flt) dominates : See aboveto Existance ~ if JC) is continuous example Theorem ! order of exponential F(u = LG ((t)] all defined for 870  # (s) = Test f (t)d + t (C s) [stMed - Eg ! here It is obvious from Note We that Inverse unique : assume a exist => f(t) = L"[Fes] We need to slowly more toward solving differential Question if F(x) ([f( ] egus : - - = + What is ↳ (5'] ↓ t'l] =st W dr Suda = un-Judu & = estfc-Just (t) E 5 Note : fft) is of exponential order : + -as t e-s f(t) >. > - - 0 : L[t'l] = - f(o) + Sh[fC]  Thus LT derivative of a o a function is expressed just interns of ↳T terms function a in of of origin a ations simple algebraic - equ how this could already You can visualize solving differential b in & e use will that equations. But we come to shoorthy. Similarly ↓ [f"(t]] can be computed simply ↓ [f" (x)] = - f'(02 + s)[5'(t)] [from above f'() Sy(01 3([f(tz] - - + + for Let us now come Laplace transform J Integral an We L FLS) know [fLAy = ↳ [Tycoudo] =? - just flor) dos & to g t N Region of Integration - t How do switch = - we it ! [et Ja > 9 tw ① - desty b) = = -. JE] = Ofcoal do Using this find = ] ↳ [sa] t S[s] = doo - - f↑ t Got don ea I dxl =  Now let's get everything together : y" - y' - 2y = 0 y'ld (0) 1 0. y = = both sides ! Laplace Transform Take f (We here). use linearity property ↓ - Multiplication Wort - I Y by i cat > - Shift Multiplication in other F(s - a) (delay)