Lab No PH-35 Porland Sa Univriy A. La Roa Tranfr funcion and h Laplac ranformaion. 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 c( ) 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 c(). Th dynamic bhavior of a phyical ym i ypically dcribd by diffrnial (and/or ingral) quaion: For a givn inpu ignal r(), h quaion nd o b olvd in ordr o find c(). 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. Tranform Problm in ranform pac R() Algbraic opraion Difficul oluion Eair oluion Soluion of original problm c ( ) 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
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 ()
Exampl. Afr an xrnal xciaion ha oppd, a ignal rpon (mpraur, for xampl) from a ym dcay xponnially. L find ou how uch a 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 rpon of a ym i govrnd by df diffrnial quaion. W would lik o know hn, how h fir drivaiv funcion a wll d f a h cond drivaiv funcion ranform by a Laplac ranformaion. For impliciy, df d f l u h noaion: = f and = f. If F = L ( f ) l valua L (f ). L ( f ' ) f ' ( ) - f ( ) - o f ( ) - f ( ) f ( ) - f ( ) [ L ( f ) ] f ( ) F() Laplac ranformaion of (3) h fir drivaiv funcion of f. Typically, on procd puing h iniial condiion qual o zro. (Th iuaion wih iniial condiion diffrn han zro can b addd in a para implr procdur). Thu,
L ( f ' ) F() Laplac ranformaion of h drivaiv of f (3) wih h iniial condiion qual o zro If F = L ( f ) valua L (f ) L ( f ") f " ( ) - f ' ( ) f ( ) F() Laplac ranformaion of h (4) cond drivaiv of f Typically, on procd puing h iniial condiion qual o zro. (Th iuaion wih iniial condiion diffrn han zro can b addd in a para implr procdur). Thu, L ( f ") F() Laplac ranformaion of h cond drivaiv (4) of f wih h iniial condiion qual o zro Exampl. Somim h ignal rpon from 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) du. If F = L ( f ) and g ( ) f ( u) du, valua L (g) L ( ) (g) g - [ f ( u) 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 () 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, () G() R() () ( 3) ( ) ( 6) (8) In a ypical procdur, whn poibl, () i r-wrin in h following form: K K K3 (), (9) 6 wih K, K, K 3, o b rmind. Tha i, w will b looking for K, K, K 3 uch ha, ( 3) ( ) ( 6) K K K3 () 6 Noic h following raighforward mhod o find K : From (9), ( ) Uing (8), K K3 () K ( ) ( ) 6 ( 3) ( 6) K K3 K ( ) ( 6 ) Taking h limi whn -, (- 3) (- 6) (- ) K K3 K ( ) ( ) 6.8 = K Tha i, K ( ) () lim.8 Similarly K ( 6) () lim 6 -.3 K 3 ( 6) () lim -.5
Thu, ().8 -.3 -.5 () 6 Now, uing () w idnify h im dpndn funcion h individual Laplac ranform com from, c( ).8 -.3-6.5- Anwr. Rcapiulaing h proc, r( ) - Original problm r( ) Difficul oluion Sym Diffrnial Eq Ingral Eq. c( ).8 -.5 -.3 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 ().8 -.3 6 -.5 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. 4. ELETRIAL SYSTEMS L analyz h hr baic lmn R, and L individually. L I I () b h Laplac ranform of h currn 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 () I() V c () I() () Dfining h capacianc impdanc, Z c (3) w can xpr, V c () = Z c I () (4) Inducor v L () = d L i ) uing (3)' ( V L () I L () (5) Dfining h inducor impdanc, Z L w can xpr, L (6) V L () = Z L I () (7) Rior v R () = R i () V R () R () (8) I Tim domain Laplac domain Analyi of a impl lag nwork Applying Kirchoff law in h im-domain and in h Laplac -domain
v in () R i q v ou () Fig. 7 Low pa filr in h im domain. 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 () V ou () R I() I() R I() (9) I() () From (9) and () V ou () R V in / Vou G() () V R in V in ( ) Inpu ignal G ( ) R V ou ( ) Oupu ignal Fig. 8 Tranfr funcion (in h Laplac domain) of a low pa filr. Analyi of a impl lag nwork Mhod uing h complx impdanc in h Laplac domain
V in () R V ou () q I() Z V in () = ( R Z ) Vin() () ( R Z ) I I() On h ohr hand, V ou () Z Z Uing xprion (9), Z c V ou ( ) V G() V ou in I() V in () ( R Z, ( ) ( ) V ( R in ) ) R R V in ( )