3.2. Derivation of Laplace Transforms of Simple Functions

Transcription

1 3. aplac Tarform 3. PE TRNSFORM wid rag of girig ym ar modld mahmaically by uig diffrial quaio. I gral, h diffrial quaio of h ordr ym i wri: d y( a d d d y( dy( a a y( f( (3. d Which i alo ow a a liar diffrial quaio, if h coffici a, a,, a ar o fucio of y(. Soluio of h diffrial quaio wih dicoiuou ipu or of highr ha cod ordr i laboriou by h claical mhod. To implify or ymaiz h oluio of diffrial quaio, h aplac Traform (T mhod i ud xivly. 3.. Dfiiio of h aplac Traform Th dirc aplac raform F( of a fucio of im ( i giv by. f( f( d F( (3. whr ( i a horhad oaio for h aplac igral. complx variabl i rfrd o a h aplac opraor ad ha wo compo =j, a ral ad imagiary compo. Figur 3. illura h complx -pla, i which ay arbirary poi = i dfid by h = j. j Th rvr proc of fidig h im fucio ( from h F( i calld h ivr aplac raform - F( = ( ( Drivaio of aplac Traform of Simpl Fucio Exampl 3.. Ui-p fucio ( Figur 3. U ( Figur 3. (3.4 ( U if (3.4 if Figur 3. 38

2 U ( U ( d 3. aplac Tarform d a (3.5 a if Exampl 3. Expoial fucio (Figur 3.3 U ( if (3.6 U ( Figur 3.3 Exampl 3.3. Ramp fucio (Figur 3.4 U r ( U ( U ( (a a d a d (3.7 U ( r if if (3.8 Figur 3.4 U ( for u ad dv d. Thi, w i hav igrad du d v U ( d d ; r r by ad par. By uig b udv uv a b a b a vdu (3.9 Exampl 3.4. oidr h iuoidal fucio f( = co aplac raform for iuoidal fucio ar drmid, i rm of Eulr' raform jx jx jx jx i x ; co x (3. j 39

3 3. aplac Tarform co co j ( j j co d ( j ( j d ( j d j j (3 3.3 aplac Traform Thorm pplicaio of h aplac raform i may iac ar implifid by uilizaio of h propri of h raform. Th propri ar prd by h followig horm: Thorm. Muliplicaio by a oa b a coa ad F( b h aplac raform of f(. Th [f(] = F( (3. Thorm. Sum ad Diffrc F ( ad F ( b h aplac raform of f ( ad f (, rpcivly. Th [f ( ± f (] = F (S ± F (S (3.3 Thorm 3. Diffriaio F( b h aplac raform of f(, ad l f( b h limi of f( a approach. Th aplac raform of h im drivaiv of f( i [df(/d] = F(S - lim f ( = F( - f( I gral, for highr-ordr drivaio of f(, d f ( d F( lim f ( d f ( d f ( d d = F( f (o f (o f ( Whr f (o do h h -ordr drivaiv of f ( Thorm 4. Igraio ( wih rpc o, valuad a o. Th aplac raform of h fir igral of f ( wih rpc o im i h aplac raform of f ( dividd by ; ha i, f ( d For h-ordr igraio, F( o f( dτd d d F( (3.4 (3.5 (3.6 4

4 3. aplac Tarform Thorm 5. Shif i Tim Th aplac raform of f ( dlayd by im T i qual o h aplac raform f ( T muliplid by f ( Tu T ; ha i, T F( Whr u T do h ui-p fucio ha i hifd o h righ by T. (3.7 Thorm 6. Fial-Valu Thorm If h aplac raform of f ( i F(, ad if F( i aalyic o h imagiary axi ad i h righ half of h -pla, h lim f( lim F( (3.8 Th fial-valu horm i vry uful for h aalyi ad dig of corol ym, ic i giv h fial valu of a im fucio by owig h bhavior of i a lac raform a =. Th fial-valu horm i o valid if F( coai ay pol who ral par i zro or poiiv, which i quival o h aalyic rquirm of F( i h righhalf pla a ad i h horm. Th followig xampl illura h car ha mu b a i applyig hi horm. Exampl 3.5. oidr h fucio F(=5/( + + Sic F( i aalyic o h imagiary axi ad i h righ-half -pla, h fial-valu horm may by applid. Uig Eq. (3.7, w hav lim f( lim F( lim 5/ 5/ Exampl 3.6. oidr h fucio F(=/ + Which i h aplac raform of f(= i. Sic h fucio F( ha wo pol o h imagiary axi of h -pla, h fial valu horm cao b applid i hi ca. Thorm 7. Iiial valu horm lim f( lim F( (3.9 Thorm 8. omplx Shifig Th aplac raform of f ( muliplid by, whr i a coa, i qual o h aplac raform F(, wih rplacd by ; ha i f( = F( (3. Thorm 9. Ral ovoluio (omplx Muliplicaio F ( ad F ( b h aplac raform of f ( ad f (, rpcivly, ad f ( =, f ( =, for < ; h F F f *f f f d (3. Whr h ymbol "*" do covoluio i h im domai. Equaio (. how ha muliplicaio of wo raformd fucio i h complx - domai i quival o h covoluio of wo corrpodig ral fucio f ( ad f ( i h -domai. Th ivr aplac raform of h produc of wo fucio i h -domai i o qual o h produc of h wo corrpodig ral fucio i h -domai; ha i, i gral, - F (F ( f ( f ( 4

5 3. aplac Tarform 4 aplac Traform Tabl Tim fucio f( aplac raform F( ζ ζ α α α α α β α α α ( ( 7. ( ζ ; ζ a ζ i( ζ ( 6. ζ ; ζ i ζ 5. ( co 4. co 3. i. α ( α (. α ( (α α. α ( α ( α 9. α ( ( α 8. β α( ( β (α ; ( α β 7. α (! 6. α ( 5.! fucio p ui ( u. fucio dla δ(. ( ( ( α ( α ( α ( α ( β α( ( α (! α (!

6 3. aplac Tarform 3.4. Ivr aplac Traform by Parial-Fracio Expaio Wh h aplac Traform oluio of a diffrial fucio i a raioal fucio i ca b wri a G( Q( P( (3. Whr Q( ad P( ar polyomial of. I i aumd ha h ordr of P( i grar ha ha of Q(. Th polyomial P( my b wri P( = +a - - +a a +a Whr a, a,... a - ar ral coffici. Th roo of P( = ar calld h pol of ym. Th roo of Q( = ar h zro of ym. Th mhod of parial-fracio xpaio will ow b giv for h ca of impl pol, mulipl-ordr pol, ad complx cojuga pol of G(.. G( ha impl pol. If all h pol of G( ar impl ad ral, Eq. ( 3. ca b wri a Q( Q( G (3.3 P( ( ( ( 3...(...( whr.... pplyig h parial-fracio xpaio, i wri G( Th coffici (=,,... i drmid a (3.4 Q( ( P( ( ( - g( Q( ( ( Exampl 3.7 oidr h fucio 5 3 G( ( ( ( 3 - (3.5 (3.6 (3.7 which i wri i h parial-fracio xpadd form: 3 G( 3 (3.8 Th coffici,, ad 3 ar drmid a follow: 5( 3 (G( ((3 5( 3 (G( 7 ((3 5( 3 3 3(3G( 6 3 (3(3 Thu Eq. (3.8 bcom 7 6 G( ; 3 = -; = -; 3 = -3 ; g(=

7 3. aplac Tarform. G( ha mulipl - ordr pol. If r of h pol of G( ar idical, G( i wri G Q( P( ( Q( r (...( r ( i ; i =,,..., -r (3.9 =- i i a mulipl pol ordr-r. G( ca b xpdd a (-r rm of impl pol G( 3 3 r r (3.3 ir ( i r i(r ( i r i(r ( i r (r rm of rpad pol Th (-r coffici,,,..., (-r which corrpod impl pol,may b valuad by h mhod dcribd by Eq.(3.5. Th coffici ha corrpod o h mulipl ordr pol ar dfid a follow: ( G( ir i i r i(r i G( d (3.3 d i d r i(r G(! d i i r i G( i r d i(r - r (r! d I hi ca, Ivr aplac raform i drmid a r- r i i g( ir i(r (r -! (r!! i 3 i Exampl 3.8 Fid ivr aplac raform of h fucio G( 3 ( ( 3 By uig Eq. 3.3 (3.3 44

8 3 ( 3 ( 3. aplac Tarform ( ( G 3 Th coffici ar 3 ( 3 G( ( G( 3 3 d d 3 ( G( From Eq.(3.3 h Ivr aplac raform ; d! d 3 ( G( g( = ; 3 3 g( 3 3. G( ha impl complx-cojuga pol Th parial-fracio xpaio of Eq. (3.4 i alo valid for impl complx-cojuga pol. Sic complx-cojuga pol ar mor difficul o hadl ad ar of pcial ir i corol ym udi, hy drv pcial ram hr. Exampl 3.9 oidr h fucio ha dcrib a cod-ordr ym G( ζ (3.33 u aum ha h valu of (dampig raio ad w (aural frqucy ar uch ha h pol of G( ar complx. For < h pol of ym ar drmid a j Suppo, a ad Th compl parial- fracio xpaio of Eq. (3.33 i K K G( (3.34 a j a j Th coffici i Eq. (3.34 ar drmid a K ( a jg( a j j K ( a jg( a j j G( j a j a j (

9 3. aplac Tarform Taig h ivr aplac raform o boh id of Eq. (3.35 giv a j j ζ g( ( i ζ (3.36 j ζ 3.5. pplicaio of aplac Traform o h Soluio of iar Diffrial Equaio iar ordiary diffrial quaio ca b olvd by h aplac raform. Th procdur i oulid a follow:. Traform h diffrial quaio o h - domai uig h aplac raform abl. Maipula h raformd algbraic quaio ad olv for h oupu variabl 3. Prform parial-fracio xpaio 4. Obai h ivr aplac raform from h aplac raform abl. u illura h mhod by vral xampl. Exampl 3.. oidr h diffrial quaio d y( dy( 3 y( 5u ( (3.37 d d Whr u ( i h ui-p fucio. Th iiial codiio ar y( ad ( y ( dy( /d. To olv h diffrial quaio, w fir a h aplac raform o boh id of Eq. (3.37: Y( y( y ( (o 3Y( 3Y( Y( 5/ Subiuig h valu of h iiial codiio io Eq. (3.37 ad olvig for Y(, w g Y( 5 5 (3.38 ( 3 ( ( Equaio (3.38 i xpdd by parial fracio xpaio o giv Y( ( Taig h ivr aplac raform, w g h compl oluio Y( = a Dirc aplac Traform 3.5. Malab fil of aplac raform = PE(F i h aplac raform of h calar ym F wih dfaul idpd variabl. Th dfaul rur i a fucio of. If F = F(, h PE rur a fucio of : = (. = PE(F,w,z ma a fucio of z iad of h dfaul (igraio wih rpc o w. Exampl: ym a w x laplac(^5 rur /^6 46

10 3. aplac Tarform laplac(xp(a* laplac(i(w*x, laplac(co(x*w,w, laplac(x^ym(3/, laplac(diff(ym('f(' rur /(-a rur w/(^+w^ rur /(^+x^ rur 3/4*pi^(//^(5/ rur laplac(f(,,*-f( b Ivr aplac raform F = IPE( i h ivr aplac raform of h calar ym wih dfaul idpd variabl. Th dfaul rur i a fucio of. If = (, h IPE rur a fucio of x: F = F(x. F = IPE(,y ma F a fucio of y iad of h dfaul : F = IPE(,y,x ma F a fucio of x iad of h dfaul : Exampl: ym w x y ilaplac(/(- ilaplac(/(^+ ilaplac(^(-ym(5/,x ilaplac(y/(y^ + w^,y,x ilaplac(ym('laplac(f(x,x,',,x rur xp( rur i(x rur 4/3/pi^(/*x^(3/ rur co(w*x rur F(x 3.6 Impul Rpo ad Trafr Fucio of Sym impora ad fir p i h aalyi ad dig corol ym i h mahmaical modlig. Th claical way of modlig liar im-ivaria ym i o u rafr fucio o rpr ipu-oupu rlaio bw h variabl. O way o dfi h rafr fucio i o u impul rpo, which i dfid i h followig. oidr liar im-ivaria ym ( Figur 3.5. Impul rpo g( i dfid a h oupu wh h ipu i ui-impul fucio (. Th rpo Y( of ym i drmid by h u of covoluio igral Y ( U( g( d (3.39 Uig h covoluio propry of h aplac raform, w hav U( g( d U(G( =Y( (3.4 Trafr fucio G( of h ym i drmid by U( Y( ma of aplac raform g(= G(. I gral g( rafr fucio G( i rlad o h aplac raform of h ipu U( ad h oupu Y( hrough h followig rlaio: G( = Y(/U( wih all h Figur 3.5 iiial codiio ar o zro. u coidr ha h ipu-oupu rlaio of TIS i dcribd by h followig h ordr diffrial quaio wih coa ral coffici: 47

11 3. aplac Tarform dy ( dy ( a d d m m d u( d u( b m b m m m d d a dy( a y( d b du( b d u( (3.4 To obai h rafr fucio of a liar ym w implify a h aplac raform o boh id of h quaio ad aum zro iiial codiio. Th rul i: ( + a a +a Y( = (b m m- + b m- m b + b U( (3.4 Th rafr fucio G( bw U( ad Y( i giv by bm G( b a m m m b b a a Q( P( (3.43 Th propri of h rafr fucio ar ummarizd a follow:. Th rafr fucio i dfid oly for a liar-ivaria ym.. Th rafr fucio bw a ipu ad oupu variabl of a ym i dfid a a aplac raform of h impul rpo. lraivly, h rafr fucio i h raio of h aplac raform of h oupu o h aplac raform of h ipu. 3. ll iiial codiio of h ym ar o zro. 4. Th rafr fucio i idpd of h ipu of h ym. haracriic quaio. Th characriic quaio of a TIS i dfid a h quaio obaid by ig h domiaor polyomial of h rafr fucio o zro. Thu, from Eq. (3.43, h characriic quaio of h ym i: P( = + a a + a = (3.44 Th roo of h obaid from P( = ar calld h pol of h ym. Th roo obaid from Q( = ar calld h zro of ym. Exampl 3. Wha i h p rpo of a ym who rafr fucio ha a zro a, a pol a, ad a gai facor of? Th aplac raform of h oupu i giv by Y( = H( U(. Hr ( H(, U(, ( Y( ( y( Evaluaio h ivr raform of h parial fracio xpaio of Y( giv Exampl 3. Evalua h p rpo of a ym who rafr fucio i giv ( G( (.5( 4 48

12 3. aplac Tarform Th pol-zro map of h oupu i obaid by addig h pol ad zro of h ipu o h pol-zro map of h rafr fucio. Th oupu pol-zro map hrfor ha pol a, -,5 ad 4 ad a zro a a how blow. Pol of P( j S-pla * * * Zro of P( Figur 3.6 Pol du o h ipu Parial fracio xpaio for h oupu giv Whr B Y( B.5( (3.5 4( Exampl 3.3 Fid h rafr fucio ad valua h p rpo of h ym which ha a gai facor of 3 ad h pol-zro cofiguraio how i Figur 3.7. Th im rpo i hrfor y( B * Im j m * R * Figur 3.7 -j Th rafr fucio ha zro a - j ad pol a -3 ad a - j. Th rafr fucio ar hrfor 3( j( j G( ( 3( j( j 3( j( j Y( G(U( ( 3( j( j Expadig Y( io parial fracio yild 49

13 3. aplac Tarform ( 3 3 j j 4 Y Whr 3( j( j 3( j( j (( j 3 (7 j ( j( j( j 3( j( 3 (7 j ( j( j(j Evaluaig h ivr aplac raform, w hav y [ a j( j( ] [ 7] 8.3 Exampl 3.4. Fid h oupu of h ym how i Figur 3.8. U( = i4 ; g( =.5 U ( From h Tabl of aplac raform w hav 5 o U( co( ; g( Figur 3.8 y( 4 U( i 4 ; G(.5U (.5 ; Y( U(G( 4 ( 4 Taig h Ivr aplac raform from Y( w obai 8 y( = co Modlig of Elcrical Sym Uig aplac Traform Fir-Ordr Elcrical ircui Th mahmaical modl of a lcrical circui ca b obaid by applyig Ohm', o or boh of Kirhhoff, ad Faraday law. Baic law govrig lcrical circui ar Kirhhoff' law, which a:. Th algbraic um of h curr rig a od i qual o zro. I ohr word, h um of h curr rig a od i qual of curr lavig h am od.. Th algbraic um of h volag aroud ay loop i a lcrical circui i zro. Th volag drop acro a rior i giv by Ohm law, which a ha h volag V R drop acro a rior R i qual o h produc of h curr I ad i riac R I V R 5 R

14 3. aplac Tarform V R = RI Th volag drop acro a iducor i giv by Faraday law, which a ha h volag V drop acro a iducor i qual o h produc of h iducac ad h im ra of icra of curr di V d Th volag acro a capacior i i drmid a follow I V c V Id oidr h diffr lcrical circui. Exampl Fid h rafr fucio rial R- circui umig h circui i o loadd, i.. o curr flow hrough h oupu rmial. pplyig Kirhhof volag law o h ym ( Figur 3.9 a, w obai h followig quaio: I V V R V E di RI E d Th quaio of quaio (3.45 ad, aumig zro iiial codiio w obai RI( I( E( R givof mahmaical I modl of hcircui. Taig h aplac V ( (3.45 (3.46 raform (3.47 E V a Fig. 3.9 b Th oupu volag acro h iducor i V = di/d. Th curr hrough h iducor i hrfor I(=V /. Subiuig hi valu io h Eq. (3.46 giv R V V E V R ( ( ( ; ( ( E( Th rafr fucio of ym of ym i foud o b R V V E V R ( ( ( ; ( ( E( Th rafr fucio of ym of ym i foud o b 5

15 R V 3. aplac Tarform R ( V ( E(; V (( E( Th rafr fucio of ym of ym i foud o b V ( G( E( R For ui-p ipu, w hav rpo V ( ; R R R / - V ( V ( / Ui p rpo of ym how i Fig.3.5 (b ; whr τ Exampl.3.6. Srial R- circui pplyig Kirhhof volag law o h ym, w obai h followig quaio: R RI( V E( ; V Id ; V ( I( ; I( = V ( V ( RV ( E(; V ( R E( ad G( V ( E( R (3.48 R I g( / V ( E V R V a b Figur 3. Impul rpo of ym V ( (G( ; whr = R R / c Taig h Ivr aplac raform w dfi impul rpo g( ( Fig.3. b For p impul w hav (3.49 g( V ( / ; 5

16 3. aplac Tarform E( ; V ( Taig h Ivr aplac raform ( w dfi p rpo how i Figur 3. (b V ( / ; ( Scod-Ordr Elcrical ircui pplyig Kirchhof volag law o h ym how i Figur 3., w obai h R E V Fig.3. di( (3.5 RI( V c( E( followig quaio: d V ( I(d (3.5 Equaio (3.5 ad (3.5 giv a mahmaical modl of h ym. Taig h aplac raform of quaio (3.5 ad (3.5, aumig zro iiial codiio, w obai I( RI( I( E(; I( V ( I( V (; V ( RV ( V ( E( V ( G( E( R R ad Puig w hav G( (3.53 Th pol of hi ym ar, Whr i calld a dampig raio ad - aural frqucy of ym. 3.8 Tim-Rpo haracriic of Scod Ordr Sym Th oluio of h Eq.(3.53 G( Whr, dpd o whhr i mallr ha uiy, grar ha uiy, or qual o uiy (

17 3. aplac Tarform. Wh, h rpo of ym i aid o b udrdampd. For a ui-p ipu U(=/, oupu Y( of ym Y( ( Th Ivr aplac raform ( Tabl of aplac Traform y( i( a (3.55. Wh =, h roo ar ral ad qual = = -, ad ym i aid o b criically. For a ui-p ipu Y( ( y( ( Wh, h rpo i calld ovrdampd. For a ui-p ipu ( y( (3.57 family of curv corrpodig o hi quaio i how i Figur 3., whr abcia i h dimiol variabl y(. =. =.5..5 = = 5 Figur 3. Thi curv how ha h amou of ovrhoo dpd o h dampig raio. For h ovrdampd ad criically dampd ca, hr i o ovrhoo. For h udrdampd ca, <, h ym ocilla aroud h fial valu. Th ocillaio dcra wih im, ad h ym rpo approach h fial valu. Th pa ovrhoo for h udrdampd ym i fir ovrhoo. Th im p a which h pa ovrhoo occur, ca b drmid by diffriaig Y( from Eq.(3.55 wih rpc o im ad ig hi drivaiv o zro: 54

18 3. aplac Tarform dy i( d Thi drivaiv i zro a =,,,... Th pa ovrhoo occur a h fir valu afr zro, providd hr ar zro iiial codiio; hrfor p Irig hi valu of im i Eq. (3. giv h pa ovrhoo a Mp p xp( Th rai-rpo characriic for h cod ordr ym how a umbr of igifica paramr of ym. Th ovrdampd ym i low acig ad do o ocilla abou h fial poiio. For om applicaio h abc of ocillaio may b cary. For xampl, a lvaor cao b allowd o ocilla a ach op. Bu for ym whr a fa rpo i cary, h low rpo ca o b olrad. Th udrdampd ym rach h fial valu far ha h ovrdampd ym, bu h rpo ocilla abou hi fial valu. Th amou h prmiibl ovrhoo drmi h dirabl valu of h dampig raio. Th prformac of a ym may b valuad i rm of h followig quaii, a how, i Figur 3.3. p y( llowabl olrac -( or 5% p Figur 3.3. Pa ovrhoo p i h magiud of h larg ovrhoo ad of occur a h fir ovrhoo. Thi may b xprd i prc of h fial valu. Tim of maximum ovrhoo p i h im rquird rachig h maximum ovrhoo. 3. Tim of fir zro i h im rquird rachig a fial valu h fir im. I i of rfrrd o a duplicaig im 4. Sig im i h im rquird for h oupu rpo fir o rach ad hrafr rmai wihi a prcribd of h fial valu. ommo valu ud for ig im ar ad 5%. Th ig im for a % rror cririo i approximaly 4 im coa; for a 5% rror cririo, i i 3 im coa 55