Why Laplace transforms?

Transcription

1 MAE4 Linar ircui Why Lalac ranform? Firordr R cc v v v KVL S R inananou for ach Subiu lmn rlaion v S Ordinary diffrnial quaion in rm of caacior volag Lalac ranform Solv Invr LT V u, v Ri, i A R V A _ v S dv d dv R v VAu d R[ V v ] V VA V R v V A / / R / R &, v V R # v R $ A * '! u % " i R v R v c Vol

2 Tim domain domain Linar cc Diffrnial quaion laical chniqu Lalac ranform omlx frquncy domain domain Lalac ranform L Algbraic quaion Algbraic chniqu Ron ignal Invr Lalac ranform L Ron ranform Th diagram commu Sam anwr whichvr way you go MAE4 Linar ircui 4

3 Lalac Tranform dfiniion Funcion f of im Picwi coninuou and xonnial ordr ò F f d f < b K limi i ud o caur ranin and diconinuii a i a comlx variabl jw Thr i a nd o worry abou rgion of convrgnc of h ingral Uni of ar c Hz A frquncy If f i vol am hn F i volcond amcond MAE4 Linar ircui

4 Lalac ranform xaml S funcion uni Havyid Funcion Afr Olivr Havyid 89 u ì, í î, for for < ³ Exonnial funcion Afr Olivr Exonnial 76 B 66 B Dla imul funcion d MAE4 Linar ircui 6

5 Lalac ranform xaml S funcion uni Havyid Funcion Afr Olivr Havyid 89 u F u d Exonnial funcion d σ jω Afr Olivr Exonnial 76 B 66 B σ jω ì, for í î, for if < ³ σ > Dla imul funcion d MAE4 Linar ircui 7

6 Lalac ranform xaml S funcion uni Havyid Funcion Afr Olivr Havyid 89 u F u d d σ σ jω jω ì, for í î, for if < ³ σ > Exonnial funcion Afr Olivr Exonnial 76 B 66 B F α d α d Dla imul funcion d α α α if σ > α MAE4 Linar ircui 8

7 Lalac ranform xaml S funcion uni Havyid Funcion Afr Olivr Havyid 89 u F u d d σ σ jω jω ì, for í î, for if < ³ σ > Exonnial funcion Afr Olivr Exonnial 76 B 66 B F α d α d Dla imul funcion d F MAE4 Linar ircui δ d for α α all α if σ > α 9

8 Lalac Tranform Pair Tabl Signal Wavform Tranform imul ram xonnial damd ram in coin damd in damd coin MAE4 Linar ircui δ u u α u α u in co α β u β u in α co β u β u α α β β β β α β α α β 4

9 Lalac Tranform Prori Linariy aboluly criical rory L Follow from h ingral dfiniion { Af Bf } AL{ f } BL{ } AF BF f Examl LAcoβ MAE4 Linar ircui 4

10 Lalac Tranform Prori Linariy aboluly criical rory L Follow from h ingral dfiniion { Af Bf } AL{ f } BL{ } AF BF f Examl $ LAcoβ L A ' & jβ jβ A % L jβ A jβ A A β jβ A L jβ MAE4 Linar ircui 4

11 Ingraion rory Lalac Tranform Prori & L% $ # f τ dτ "! F Proof L /. f, τ dτ ' f τ dτ * $ % & d MAE4 Linar ircui 4

12 44 MAE4 Linar ircui 44 Lalac Tranform Prori Ingraion rory Proof Dno o Ingra by ar F d f! " # $ % & τ τ L d d f d f $ % & ' *,. / τ τ τ τ L and, and, f d dy d dx d f y x τ τ $ $ % & ' ' $ $ % & ' ' d f d f d f τ τ τ τ L

13 Lalac Tranform Prori Diffrniaion Prory ' df L $ & # % d " F f Proof via ingraion by ar again " df % L# & $ d ' df d d F f Scond drivaiv [ f ] / d f /% d. df % df % df L' $ L' $ L' $ /& d /# & d, d * # & d # d F f f " f d MAE4 Linar ircui 4

14 Lalac Tranform Prori Gnral drivaiv formula " L d m f % # & $ d m ' m F m f m f f m Tranlaion rori domain ranlaion α L{ f } F α domain ranlaion L a { f a u a } F for a > MAE4 Linar ircui 46

15 Lalac Tranform Prori Iniial Valu Prory lim f lim F Final Valu Prory lim f lim F ava: Lalac ranform air do no alway handl diconinuii rorly Ofn g h avrag valu Iniial valu rory no good wih imul Final valu rory no good wih co, in c MAE4 Linar ircui 47

16 48 MAE4 Linar ircui 48 Raional Funcion W hall moly b daling wih LT which ar raional funcion raio of olynomial in i ar h ol and z i ar h zro of h funcion K i h cal facor or omim gain A ror raional funcion ha n³m A ricly ror raional funcion ha n>m An imror raional funcion ha n<m n m n n n n m m m m z z z K a a a a b b b b F

17 MAE4 Linar ircui A Lil omlx Analyi W ar daling wih linar cc Our Lalac Tranform will coni of raional funcion raio of olynomial in and xonnial li Th ari from dicr comonn rlaion of caacior and inducor h ind of inu ignal w aly S, imul, xonnial, inuoid, dlayd vrion of funcion Raional funcion hav a fini of dicr ol i an nir funcion and ha no ol anywhr To undrand linar cc ron you nd o loo a h ol hy drmin h xonnial mod in h ron circui variabl. Two ourc of ol: h cc n in h ron o Ic h inu ignal LT ol n in h forcd ron 49

18 Ridu a ol Funcion of a comlx variabl wih iolad, fini ordr ol hav ridu a h ol Siml ol: ridu lim a F a Mulil ol: ridu d m m! lim a d m a m F Th ridu i h c rm in h Laurn Sri Bundl comlx conjuga ol air ino condordr rm if you wan α jβ α jβ α α β [ ] bu you will nd o b carful Invr Lalac Tranform i a um of comlx xonnial For circui h anwr will b ral MAE4 Linar ircui

19 Invring Lalac Tranform in Pracic W hav a abl of invr LT Wri F a a arial fracion xanion F b m m b m m b b a n n a n n a a K z z z m n α Now aal o linariy o invr via h abl Surri! α α α α... α q q omuing h arial fracion xanion i b don by calculaing h ridu MAE4 Linar ircui

20 MAE4 Linar ircui Invring Lalac Tranform omu ridu a h ol Examl lim F a a!" # $% & lim! F m a m d m d a m

21 MAE4 Linar ircui Invring Lalac Tranform omu ridu a h ol Examl lim F a a!" # $% & lim! F m a m d m d a m lim lim ú ú û ù ê ê ë é d d lim! ú ú û ù ê ê ë é d d u ú ú û ù ê ê ë é L MAE4 Linar ircui

22 T&R, h d, Examl 9 Find h invr LT of F MAE4 Linar ircui 4

23 MAE4 Linar ircui T&R, h d, Examl 9 Find h invr LT of F * j j F π 4 lim lim j j j j F j j F 4 co 4 4 u u f j j j j!" # $% &!! " # $ $ % & π π π MAE4 Linar ircui

24 No Sricly Pror Lalac Tranform Find h invr LT of F MAE4 Linar ircui 6

25 7 MAE4 Linar ircui 7 No Sricly Pror Lalac Tranform Find h invr LT of onvr o olynomial lu ricly ror raional funcion U olynomial diviion Invr a normal F.. 4 F.. u d d f!" # $% & δ δ MAE4 Linar ircui

26 8 MAE4 Linar ircui 8 Mulil Pol Loo for arial fracion dcomoiion Equa li owr of o find cofficin Solv Kz K z K F Kz K

27 Rcall moivaing xaml for LT Firordr R cc v v v KVL S R inananou for ach Subiu lmn rlaion v S Ordinary diffrnial quaion in rm of caacior volag Lalac ranform Solv Invr LT V u, v Ri, i A MAE4 Linar ircui R V A _ v S dv d dv R v VAu d R[ V v ] V VA V R v V A / / R / R &, v V R # v R $ A * '! u % " R v R v c i Vol 9

28 V A _ v S An Alrnaiv Domain Aroach R v R v c i V A R _ V c I v _ Tranform h cc lmn rlaion Wor in domain dircly OK inc L i linar V I I V v v Imdanc ourc Admianc ourc KVL in Domain MAE4 Linar ircui RV Rv V V A 6

29 _ V A coβ MAE4 Linar ircui Timvarying inu Suo v S V A cob, wha han? v S KVL a bfor Solv R v R & v c i V R V VA β Rv VA R β _ R V v R A β V c R I v _ V V v A co A R R $ β θ v u! βr βr % # " 6

30 Lalac Tranform rca for cc Wha h big ida?. Loo a iniial condiion ron of cc du o caacior volag and inducor currn a im Mh or nodal analyi wih domain imdanc rianc or admianc conducanc Soluion of ODE drivn by hir iniial condiion Don in h domain uing Lalac Tranform. Loo a forcd ron of cc du o inu IS and IVS a funcion of im Inu and ouu ignal I O YV S or V O ZI S Th cc i a ym which convr inu ignal o ouu ignal. Linariy ay w add u ar and Th am a wih ODE MAE4 Linar ircui 6