REPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if.

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1 Tranform Mhod and Calculu of Svral Variabl H7, p Lcurr: Armin Halilovic KTH, Campu Haning armin@dkh, wwwdkh/armin REPETITION bfor h am PART, Tranform Mhod Laplac ranform: L Driv h formula : a L[ a a f ] F a b L[ f a θ a] F c L[ ] d L [ f ] F L Find h Laplac ranform F if a f in b f c f θ d f τ dτ τ f in τ dτ τ co7 f f inτ dτ f *in τ dτ g f τ dτ h Anwr: a b c d f g h L Find f for h givn F b 7 7 f c g d h Anwr: a f co 7 b in 7 c 7 f in θ d

2 g co θ h ϑ L U h Laplac ranform o olv h following diffrnial quaion a, b,, Soluion a Th Laplac ranform of h quaion a i Uing h iniial condiion and implifing giv Hnc w hav [L8] Anwr a b,, Th Laplac ranform of h quaion b i! ' Uing h iniial condiion and implifing giv

3 Hnc w hav! Anwr b L U h Laplac ranform o olv h following diffrnial quaion a, b in, Anwr: a b co in L U h Laplac ranform o olv h following diffrnial quaion,,,, Anwr: a b in L7 Solv h following ingral quaion τ co τ dτ Anwr: a in co b in L8 Solv h following ingrodiffrnial quaion b τ τ dτ τ dτ, o

4 Anwr: L U h Laplac ranform o olv h givn m of diffrnial quaion, Soluion: X A B c D B parial fracion panion w can wri X which ild X, Hnc coh, B parial fracion: and inh L U h Laplac ranform o olv h givn m of diffrnial quaion a b co,, co,, Anwr: a, b in, L U h Laplac ranform o olv h givn m of diffrnial quaion find and, Soluion: Afr ranforming ach quaion w obain X X *

5 or X X ** Mulipling h fir quaion of ** b and adding o h cond quaion ild Subiuing in h cond quaion of ** giv X X X X Now and X impl and Anwr, L A coninuou-im dnamical m ranform an inpu ignal ino an oupu ignal Th rlaion bwn and i rprnd b h quaion blow Find h ranfr funcion H and h funcion pol, and drmin if h m i abl Juif our anwr a b c d f Soluion: a W aum ha and,and appl Laplac ranform o h quaion X X Hnc H W find pol ingularii from h quaion

6 , No ha if h ranfr funcion H i raional hn: Th m i abl All pol i all ingularii li in R < C* In our ca h condiion C* i aifid and h m i abl Anwr: a Th ranfr funcion i H Pol ar, Th m i abl Anwr: b Th ranfr funcion i H Pol ar, Th m i unabl For R > Anwr: c Th ranfr funcion i H Pol ar, Th m i unabl For R > Anwr: d Th ranfr funcion i H Pol ar, Th m i unabl For R > Anwr: Th ranfr funcion i H Pol ar i Th m i abl All pol aif R < i R < and R < Anwr: f Th ranfr funcion i H Pol ar i, i Th m i unabl For R >, i L A coninuou-im dnamical m ranform an inpu ignal ino an oupu ignal Th rlaion bwn and i rprnd b h quaion blow Find h ranfr funcion H and h funcion pol, and drmin if h m i abl Juif our anwr a b Soluion a W aum ha, and ranform h quaion: X X X Hnc h ranfr funcion H Pol:, Th m i unabl if R k > for a la on pol k In our ca R > Th m i unabl Anwr a Th m i unabl bcau R >

7 b W aum ha, and ranform h quaion: X X X Hnc h ranfr funcion H Pol: i, i R R < Anwr b Th m i abl bcau R < for all pol k L A coninuou-im dnamical m ranform an inpu ignal ino an oupu ignal Th rlaion bwn and i rprnd b h following quaion q a Find h ranfr funcion H and h impul rpon h b Drmin if h m i abl c For h inpu ignal find h oupu ignal which aifi a Soluion: W aum onl for h quion a ha, appl Laplac ranform o h quaion and olv for X X X Hnc H Uing invr ranform w find h Anwr: a Th ranfr funcion i H Th impul rpon i h b Th ranfr funcion i raional wih on pol No ha : No ha if h ranfr funcion H i raional hn: Th m i abl All pol i all ingularii li in R < Tha i no ru in our ca R > and hrfor h m i unabl Anwr: b unabl c Soluion: W ubiu Anwr: c

8 L Drmin h currn i in a ingl loop L-R-C circui whn L hnr, R ohm, C farad, i, q and h imprd volag u vol Soluion: In a ingl loop L-R-C, Kirchhoff cond law a ha h um of h volag drop acro an inducor, rior, and capacior i qual o h imprd volag u I i known ha di h volag drop acro h inducor L d h volag drop acro h rior Ri q and h volag drop acro h rior q i d C C whr i i h currn Th charg of h capacior i dnod b q Currn i rlad o dq charg b i d Adding h volag drop and quaing h um o h imprd volag u ild di L Ri q i d u d C In our ca q which ild h govrning quaion di L Ri i d u d C q Afr ubiuing L, R and C w obain di i i d q d Th Laplac ranform of h quaion q i I I I I I i

9 Anwr: i