THE LAPLACE TRANSFORM LEARNING GOALS Diniion Th ranorm map a ncion o im ino a ncion o a complx variabl Two imporan inglariy ncion Th ni p and h ni impl Tranorm pair Baic abl wih commonly d ranorm Propri o h ranorm Thorm dcribing propri. Many o hm ar l a compaional ool Prorming h invr ranormaion By rricing anion o raional ncion on can impliy h invrion proc Convolion ingral Baic rl in ym analyi Iniial and Final val horm Ul rl rlaing im and -domain bhavior
ONE-SIDED LAPLACE TRANSFORM I will b ncary o conidr To inr niqn o h h ingral RoC σ jω a h ranorm on i lowr limi am wll dind or < A SUFFICIENT CONDITION FOR EXISTENCE OF LAPLACE TRANSFORM Tranorm xi or R{} σ > THE INVERSE TRANSFORM Conor ingral in h complx plan Evalaing h ingral can b qi im-conming. For hi raon w dvlop br procdr ha apply only o crain l cla o ncion
TWO SINGULARITY FUNCTIONS Uni p Imporan ncion in ym analyi Thi ncion ha drivaiv ha i zro vrywhr xcp a h origin. W will din a drivaiv or i For poiiv im ncion Uing qar pl o approxima an arbirary ncion Th narrowr h pl h br h approximaion Uing h ni p o bild ncion
Comping h ranorm o h ni p U T x x dx limt T x lim T U dx An xampl o Rgion o Convrgnc RoC Im T U limt σ jω U lim T σt jωt σ jω U ; R{ } > To impliy qion o RoC: A pcial cla o ncion RoC { : R{ } RoC }> σ In hi ca h RoC i a la hal a plan. And any linar combinaion o ch ignal will alo hav a RoC ha i a hal plan R{ } > Complx Plan R
THE IMPULSE FUNCTION Good modl or impac, lighning, and ohr wll known phnomna Th wo condiion ar no aibl or normal ncion Approximaion o h impl High i proporional o ara Rprnaion o h impl Siing or ampling propry o h impl Laplac ranorm For or h ingral i NOT dind In ordr o hav a valid ranorm or δ lowr limi i amd h
LEARNING BY DOING < π < π coπ π π > LEARNING EXAMPLE d Ingraion by par, dv d wih d d, v W will dvlop propri ha will prmi h drminaion o a larg nmbr o ranorm rom a mall abl o ranorm pair
Linariy Tim hiing Tim rncaion Mliplicaion by xponnial Mliplicaion by im Som propri will b provd and d a icin ool in h compaion o Laplac ranorm
LEARNING EXAMPLE Find h ranorm or F a d F a a a d a LINEARITY PROPERTY Homogniy Follow immdialy rom h linariy propri o h ingral APPLICATION Addiiviy Baic Tabl o Laplac Tranorm a jω jω jω L [ ] L[ ] a jω W dvlop propri ha xpand h abl and allow compaion o ranorm wiho ing h diniion F jω jω jω jω jω jω ω
Wih a imilar o linariy on how L[inω] ω ω Addiional nri or h abl LEARNING EXAMPLE Applicaion o Linariy X 4 4 LEARNING EXAMPLE Find h Laplac ranorm or x co π / x coπ /co in π /in Noic ha h ni p i no hown xplicily. Hnc and ar qivaln X coπ / in π / 9 9
MULTIPLICATION BY EXPONENTIAL LEARNING EXAMPLE a a L[ y ] y d y d a LEARNING EXAMPLE co Y a a X coπ /co4 in π /in 4 x a, b 4 coπ / in π 6 4 / 6 y co Y From abl y F Y Nw nri or h abl o ranorm pair
MULTIPLICATION BY TIME Dirniaion ndr an ingral Rmmbr ha w conidr h o b zro or x x <. Hnc ncion LEARNING EXAMPLE Find h r U o ranorm o d d n! n n L b h ni h ramp ncion d d By cciv applicaion h propry on how Thi rl, pl linariy, allow compaion o h ranorm o any polynomial LEARNING BY DOING x 6! X 4 p
TIME SHIFTING PROPERTY F F LEARNING EXAMPLE lwhr F
LEARNING EXTENSION FIND THE TRANSFORM FOR On can apply h im hiing propry i h im variabl alway appar a i appar in h argmn o h p. In hi ca a - On cold alo wri g And apply h im rncaion propry L g F [ g ] g L[ g ] Th wo propri ar only dirn rprnaion o h am rl
LEARNING EXAMPLE 6 6 F /6 in x π LEARNING EXAMPLE /6 in x π in 6 / 4 x θ π θ co in in co x θ θ 4 in 4 in 4 co 4 co 4 in 4 co X θ θ Uing im rncaion ] [ g F g L g g 6 ] [ 6 g L g ] [ g X L
A LEARNING EXTENSION Comp h Laplac ranorm o h ollowing ncion 4 4 5 4 5 4x 4 x B g G L[ ] a 4 a 4 4 x G a 4 C x co b X L[co b ] co b co bco b in bin b L[co b ] cob in b b b b X cob in b b b b
LEARNING EXTENSION lwhr 4 Comp h Laplac ranorm 4 4 4 4 4 4 4 4 F 4 4 4 4 d F diniion h Uing
PERFORMING THE INVERSE TRANSFORM Simpl, complx conjga pol FACT: Mo o h Laplac ranorm ha w nconr ar propr raional ncion o h orm Zro roo o nmraor Pol roo o dnominaor o m n NOWN: PARTIAL FRACTION EXPANSION I Q Q Q i a COPRIME acorizaion h dg Q n F i dnominaor wih i n P P ; Q Q i n, hn I m<n and h pol ar impl dg P < n i i C α α β Pol wih mlipliciy r Cβ α β... THE INVERSE TRANSFORM OF EACH PARTIAL FRACTION IS IMMEDIATE. WE ONLY NEED TO COMPUTE THE VARIOUS CONSTANTS
SIMPLE POLES / pi LEARNING EXAMPLE F 4 5 Wri h parial racion xpanion 4 F 4 5 Drmin h coicin rid 9 F 4 5 F 4 F 4 4 4 5 F 5 5 6 8 4 5 G h invr o ach rm and wri h inal anwr 9 6 8 4 5 5 Th p ncion i ncary o mak h ncion zro or < FORM o h invr ranorm 4 5 4
LEARNING EXTENSIONS 6 A F Find h invr ranorm Parial racion. Parial racion. Rid. Invr o ach rm F rid B F F F 6 6 F Formo olion : Invr o ach rm 5 5 4 Mak h ncion zro or <
COMPLEX CONJUGATE POLES θ α co β θ... USING QUADRATIC FACTORS Elr' coφ Idniy jφ jφ Q P [ α β ] C α α β Cβ α β... α α C co β C in β... Th wo orm ar qivaln! Avoid ing complx algbra. M drmin h coicin in dirn way
LEARNING EXAMPLE Y 4 5 Y 4 5 j j j j j * Y 4 j j 5 j 5 j Y j j j 5 5. 4 y 4.6co.678 Uing qadraic acor C C C Y 4 5 C j.6 5. 4 C 4 5.6 C C C C Alrnaiv way o drmin coicin : C For : 5C C C C 4 For : C C : 4C C C C For : C C C : 5C C 4 y C C co C in α co β θ... MUST radian in xponn.678 j
MULTIPLE POLES - L p n n! n / p p r Th mhod o idniicaion o coicin, or vn h mhod o lcing val o, may provid a convnin alrnaiv or h drminaion o h rid LEARNING EXAMPLE F 5 F 5 5 5 5 5 5 5 F 5 5 5 : For 4 d F : 5 5 5 r, j d : 5 For m dirnia 5 : 4 5 on mor im 5
LEARNING EXAMPLE F F d d F d d!! d d F d d 4 : 4 : 5 : : F Uing idniicaion o coicin
LEARNING EXTENSION Find h invr ranorm F Parial racion F Form o h invr Rid F alrnaivly F d F d
LEARNING EXTENSION Find h invr ranorm F Parial racion xpanion F Form o h invr Rid F F F d d F invr d d
CONVOLUTION INTEGRAL zro a rpon h Acally, or qaion h o olion a pariclar i ch ha, a ncion, hr xi Givn an ODE CLAIM:,...... h dx x x h y h b d d b y a d y d a d y d m m m n n n n n PROOF Shiing ncion im poiiv ar,, : I RESULT F F F ; > dx x y y Y x FIND EXAMPLE y y Y Y Y Y
LEARNING EXAMPLE Uing convolion o drmin a nwork rpon Nwork V H V S ncion V 5 5 5 5 Inp V S 5 v 5 λ 5 dλ v V S H V V H VS RESULT :I,, ar poiiv im ncion dλ 5λ 5 5 v [ ] ; For 5 5 λ F F F In gnral convolion i no an icin approach o drmin h op o a ym. B i can b a vry l ool in pcial ca
LEARNING EXAMPLE Thi xampl illra an idalizd modling approach and h o convolion a a ym imlaion ool. Thi lid how how on can obain a black box modl or a ym V in V o H Unknown linar ym rprnd in h Laplac domain Idal approach o modling Mar h impl rpon v o h x v in x dx v in V For any ohr inp on ha δ V o H V H, v In pracic, a good approximaion o an impl may b diicl, or impoibl o apply. Hnc w ry o mor nibl inp. in in o, h Th black box modl i a dcripion o h ym bad only on inp/op daa. Thr i no inormaion on wha i inid h box v in Uing h p rpon H Vin, Vo d H Vo h vo d Th impl rpon i h drivaiv o h p rpon o a ym Onc h impl rpon i obaind, h convolion can b valad nmrically
A CASE STUDY IN MODELING Unknown ym p rpon Compd impl rpon ini dirnc approximaion o drivaiv
T o h modl Th modl op h compd impl rpon and ampl o h inp ignal. Convolion ingral i valad nmrically y kt kt h kt x x dx k j h k j jt T
Daild viw o a gmn o h ignal howing bandpa acion DC and high rqncy ar rdcd in h op y kt kt h kt x x dx k j h k j jt T
INITIAL AND FINAL VALUE THEOREMS Th rl rla bhavior o a ncion in h im domain wih h bhavior o h Laplac ranorm in h -domain INITIAL VALUE THEOREM Am ha boh, d ranorm. Thn lim lim F, hav Laplac d d L[ ] F d And i h drivaiv i lim d L[ d ] ranormabl hn lim FINAL VALUE THEOREM Am ha boh, d ranorm and ha NOTE: lim lim lim F d, hav Laplac xi. Thn will xi i F ha pol wih ngaiv ralpar and a mo a ingl pol a d d d d d d F Taking limi a lim F
LEARNING EXAMPLE Givn F. Drmin h iniial and inal val or LEARNING EXTENSION Givn F. Drmin h iniial and inal val or Clarly, ha Laplac ranorm. And F - i alo dind. lim F lim lim F ha on pol a and h ohr hav ngaiv ral par. Th inal val horm can b applid. lim lim F lim lim 5 lim lim 4 NOTE:Comping h invr on 5 5 π co 4 g
On way o ing Laplac ranorm chniq in circi analyi h ollowing p:. Driv h dirnial qaion ha dcrib h nwork. Apply h ranorm a a ool o olv h dirnial qaion
LEARNING BY APPLICATION FIND i, > To ind h iniial condiion w h ady a ampion or < v R v L v S v v v v W will wri h qaion or i and olv i ing Laplac Tranorm For > S R L C, di L d v Ri C C v v v v S > R i x dx di i vc d C vc I I i L C On cold wri VL in h Laplac domain and kip h im domain i x dx I. L Circi in ady a or < V i 4 A; v C V 4 V Ω 4 4 I Rplac and rarrang 4 4 I j j * 4 I j j j j 4 j 4.6 7.57. 8. 4 j6 6 9 σ σ i co ω θ ω θ 8.4
LEARNING EXTENSION Aming h circi in ady a or drmin i, >. <, v R v L Eqaion or > 6 i di d Tranorming o h Laplac domain 6 I I i Nx w m drmin i Circi in ady a or < 6V i A Ω 6 I I I I i A; > Laplac