Transfer function, Laplace transform, Low pass filter

Transcription

1 Lab No PH-35 Porland Sa Univriy A. La Roa Tranfr funcion, Laplac ranform, Low pa filr. INTRODUTION. THE LAPLAE TRANSFORMATION L 3. TRANSFER FUNTIONS 4. ELETRIAL SYSTEMS Analyi of h hr baic paiv lmn R, and L Simpl lag nwork (low pa filr). INTRODUTION Tranfr funcion ar ud o calcula h rpon () of a ym o a givn inpu ignal R(). Hr and for h im variabl. r( ) Inpu ignal Elcronic circui, mchanical ym, hrmal ym c( ) Oupu ignal Phyical ym Fig. Givn an inpu ignal, w would lik o know h ym rpon (). Th dynamic bhavior of a phyical ym ar ypically dcribd by diffrnial (and/or ingral) quaion: For a givn inpu ignal R(), h quaion nd o b olvd in ordr o find (). Alrnaivly, inad of rying o find h oluion in h im domain, ach imvariabl, a wll a h diffrnial quaion, can b ranformd o a diffrn variabl domain in which h oluion can b obain in a mor raighforward way; hn an invr ranform would ak plac h oluion ino h im domain Original problm r( ), diffrnial Eq. Difficul oluion Soluion of original problm c ( ) Tranform Problm in ranform pac R() Algbraic opraion Eair oluion Invr ranform Soluion in ranform pac () Fig. Solving h quaion in a diffrn domain and hn applying an invr ranform o obain h oluion in h im domain

2 On of ho ranform i h Laplac ranformaion. THE LAPLAE TRANSFORMATION L Th Laplac ranform F=F() of a funcion f = f ( ) i dfind by, f L F L ( f ) = F F() f ( ) - () Th variabl i a complx numbr, = a +j. f () F F() f a Tim domain Laplac domain Exampl f i h uni p funcion f Tim domain F() f ( ) - - = = Laplac domain ()

3 Exampl A ignal rpon from a ym (mpraur, for xampl) rpond, afr an xrnal; xciaion ha oppd, by dcaying xponnially. L find ou how uch dcay i characrizd by a Laplac ranformaion. f i a dcaying xponnial f ( ) A - f F() F() = f ( ) - A A - - () Tim domain Laplac domain Exampl. A w mniond in h inroducion, h ym rpon i govrnd by diffrnial df d f quaion. W would lik o know hn, how and ranform by a Laplac df d f ranformaion. For impliciy, and clariy, l u h noaion: = f and = f. If F = L ( f ) valua L (f ) L ( f ' ) f ' ( ) - f ( ) - o f ( ) - f ( ) f ( ) - f ( ) [ L ( f ) ] f ( ) F() Laplac ranformaion of (3) h drivaiv Typically, on procd puing h iniial condiion qual o zro. (Th iuaion wih iniial condiion diffrn han zro ar addd in a para implr procdur). Thu, L ( f ' ) F() Laplac ranformaion of h drivaiv (3) If F = L ( f ) valua L (f ) wih h iniial condiion qual o zro

4 L ( f ") f " ( ) - f ' ( ) f ( ) F() Laplac ranformaion of h (4) cond drivaiv Typically, on procd puing h iniial condiion qual o zro. (Th iuaion wih iniial condiion diffrn han zro ar addd in a para implr procdur). Thu, L ( f ") F() Laplac ranformaion of h cond drivaiv (4) wih h iniial condiion qual o zro Exampl. Somim h rpon ignal of a ym (h volag acro a capacior, for xampl) mu b givn in rm of h ingral of anohr quaniy (h ingral of h corrponding currn acro h capacior). I i convnin, hn, o obain h Laplac ranformaion of an indfini ingral g ( ) f ( u) ( If F = L ( f ) and g ) f ( u) du, valua L (g) L (g) g( ) - [ f ( u) du ] - du - - [ f ( u) du ] [ f ( ) ] [ - ] f ( ) - F() Laplac ranformaion of h (5) indfini ingral 3. TRANSFER FUNTIONS r( ) Inpu ignal Diffrnial Eq govrning h bhavior of h ym c( ) Oupu ignal Fig. 3 Schmaic of h ym rpon in h im domain In a impl ym, h oupu c() may b govrnd by a cond ordr diffrnial quaion a c + a c + a o c = r ()

5 Applying h Laplac ranform (3) and (4), on obain ( a + a + a o ) () = R () () a + a a o R() In a mor gnral ca, h diffrnial quaion may b of highr ordr (highr han ). Alo h inpu may b compod of drivaiv of a givn funcion r=r(). Thrfor h facor may bcom a mor laborad funcion of. a + a a Thu, for a ym in gnral, o () = G() R() (6) Noic, G=G() characriz h phyical ym. I i calld h ranfr funcion. I i mor ypical o wri, () G() (7) R() from which, for a givn R=R() h funcion =() can b obaind. R( ) Inpu ignal G() ( ) Oupu ignal Fig. 4 Schmaic of h ym rpon in h Laplac domain 3 Exampl. A ym i characrizd by h ranfr funcion G(). ( ) ( 6) Find ou how h ym rpond o a xponnially dcaying inpu r( ) -. Anwr: Th Laplac ranformaion of r giv, uing xprion (), R() = Th oupu ignal, in h Laplac domain, i hn givn by, () 3 ( ) ( 6) In a ypical procdur, whn poibl, () i r-wrin in h following form:

6 K K K3 (), wih K, K, K 3, o b rmind. 6 Noic, K ()( ).8 Thu, K ()( 6) 6 K ()( ) () Now, uing () w idnify h im dpndn funcion h individual Laplac ranform com from, c( ) Anwr. Rcapiulaing h proc, r( ) - Original problm r( ) Difficul oluion Sym Diffrnial Eq Ingral Eq. c( ) Soluion of original problm c ( ) - 6 Laplac Tranform Invr ranform Problm in Laplac pac R() R() = G() + algbraic opraion G() 3 ( ) ( 6) () Soluion in Laplac pac () Fig. 5 Schmaic rprnaion of h oluion procdur in h prviou xampl. In h prviou xampl, h ranfr funcion wa givn. In h nx cion w will figur ou h ranfr funcion for h ca of lcrical ym.

7 4. ELETRIAL SYSTEMS L analyz h hr baic lmn R, and L individually L I I () b h Laplac ranform of i= i (). i vc vl L vr R Fig. 6 Elmnary paiv circui lmn apacior v c () = q() = ( u ) uing (5) du i V c () = ( u ) i du I() V c () I() (8) Inducor v L () = d L i ( ) uing (3)' V L () I L () (9) Rior v R () = R i () V R () R () () I Tim domain Laplac domain Analyi of a impl lag nwork v in () R i q v ou () Fig. 7 Low pa filr in h im domain.

8 v in () = R i () + = R i () + On h ohr hand v ou () q() q() ( u ) uing (5) du i ( u ) uing (5) du i V in () R I() I() V ou () R I() () I() () From () and () V V ou in / R / Vou G() (3) V R in V in ( ) Inpu ignal G ( ) R V ou ( ) Oupu ignal Fig. 8 Low pa filr in h Laplac domain.