Analysis of Non-Sinusoidal Waveforms Part 2 Laplace Transform

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Noel Houston
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1 Aalyi o No-Siuoidal Wavorm Par Laplac raorm I h arlir cio, w lar ha h Fourir Sri may b wri i complx orm a ( ) C jω whr h Fourir coici C i giv by o o jωo C ( ) d o I h ymmrical orm, h Fourir ri i wri wih o /. h Fourir ri i wri or a priodic ucio wih priod, ad dicr rqucy compo ar obaid or h wavorm. W aw ha h udamal rqucy ω o i rlad o h priod by h xprio ω o π/. Now coidr h ollowig wavorm. () (a) () (b) () (c) Figur Priod o rpiio gradually icrad I igur (a), h priod o rpiio i qui mall ad i (b) omwha largr. Wavorm (c) could b coidrd a o whr h priod o rpiio ha b icrad up o iiiy. hu ay o-rpiiv wavorm may b coidrd a o which ha a priod, ad π h corrpodig udamal rqucy ω o ω. I i alo ha h Fourir coici C i h ymmrical xpoial ri C ( ) j ω o d C hory o lcriciy Aalyi o No-iuoidal Wavorm - Par J Luca Novmbr

2 h rquci ivolvd ar o logr dicr bu coiuou, o ha h gral rqucy ω o corrpod o Σ ω dω ω. hu or o-rpiiv ucio, h ollowig ca b wri. ω o dω C dc ω o ω ω d ω o π π hu h xprio or h complx Fourir Coici C bcom dc dω jω ( ).. d π dividig boh id by dω, hi may b wri a dc F( ω ) Diiio o h Fourir raorm jω ( ).. d dω π h origial ucio () i ow giv a jωo jω ( ) C dc. rom h diiio, dc F(ω).dω, o ha j ω ω ) F( ).. dω ( Fourir Ivr raorm h Fourir raorm xprio ad h Fourir Ivr raorm xprio oghr ar kow a h Fourir raorm Pair. I w muliply h Fourir raorm by a coa ad divid h Ivr raorm alo by h am coa, w would agai g a modiid raorm pair. I w xami h wo raorm xprio, w ha hy look vry imilar xcp ha hr i a dirc o a gaiv ig i h xpo ad a muliplyig acor o π. hu w could di a ymmrical raorm pair by uig a acor o I hi ca h Symmric Fourir raorm i did a dc jω F ( ω ) π ( ).. d dω π ad h corrpodig Symmric Ivr raorm i did a j ω ω ( ) F ( ).. dω π π. h Fourir raorm i uul i aalyig rai i lcrical circui, pcially whr h lm ar rqucy dpda. hory o lcriciy Aalyi o No-iuoidal Wavorm - Par J Luca Novmbr

3 Fourir Coiraorm A Fourir Coi raorm F (ω) may b did wh h o-rpiiv wavorm i v. () (-) o ha F ( ω ) ( ). coω d, ad π ( ) F ( ω). coω dω Fourir Siraorm A Fourir Si raorm F (ω) may b did wh h o-rpiiv wavorm i odd. () (-) o ha F ( ω ) ( ). iω d, ad π ( ) F ( ω). iω dω h Fourir raorm i omim xprd i rm o h um o a i ad coi ri, iad o h xpoial ri. whr [ A( ω).coω B( ω).iω ] ( ) dω A( ω) ( ). coω d, ad π B( ω) ( ). iω d π Wih ucio which ar o-rpiiv, ad do o dcay ill iii im (uch a h i wavorm or h coi wavorm), h Fourir Igral raorm may o b obaid. o avoid hi problm, wavorm which do o dcay may b ariicially dcayd by a xpoial acor o allow h igraio. h igrad rul i h xpoially magiid o corrc or h iiial dcay iroducd. Howvr, uch xpoial magiicaio ca alo magiy umrical rror. h Laplac raorm i did bad o hi ariicial dcay. Laplac raorm I obaiig h Laplac raorm, ay ucio () i iiially dcayd ariicially by a xpoial acor -σ, o ha h w ucio alway bcom igrabl. Howvr, h dcay would corrpod o a xpoial ri (rahr ha a dcay) wih gaiv im. h Laplac raorm i hu did oly or caual ucio (ucio ha ar caud ad hc ar o zro valu bor im zro). h Laplac raorm o a im ucio () i hu did a / [()] F() ( ) d whr σ j ω i h Laplac opraor hory o lcriciy Aalyi o No-iuoidal Wavorm - Par J Luca Novmbr 3

4 h Laplac opraor i alo coidrd a a complx rqucy. I w compar wih h Fourir raorm pair wih a muliplir o π, h h Laplac Ivr raorm ak h orm () π j σ j σ j F( ) d I i ha h orm o h raorm ha impliid rom ha o h Fourir raorm. Howvr, i i vry rarly ha h Ivr raorm i calculad i hi mar. I i grally obaid rom a kowldg o h raorm o commo ucio, grally oud i abulad orm. h Laplac raorm i vry uul i circui rai aalyi a i ca covr dirial quaio o liar algbraic quaio. po o a liar Paiv Bilaral Nwork Coidr a liar paiv bilaral wo-por work o which a xciaio () i giv a o por ad which cau om rpo r() a h ohr por. () Liar Paiv bilaral Nwork r() Figur rar Fucio I gral, h rpo r() would b rlad o h ipu () by a ordiary liar dirial quaio. r() F(p). () whr p d/d dirial opraor Coidr a xpoial xciaio ucio. i.. () r() F(p).. F() hu or a xpoial xciaio, h ym ha a rar ucio r()/() qual o F(). A ad arlir, ay o-rpiiv (or v rpiiv) ucio may b brok up io a ri o xpoial. h coici o h xpoial ar giv by h Laplac raorm. hu or ay ohr xciaio (), i h Laplac raorm () i coidrd, i will b rlad o h Laplac raorm () o h rpo r() by h rar ucio F(). hu or ay caual xciaio (), () F(). () O o h advaag o h Laplac raorm i ha i covr ordiary dirial quaio i o algbraic quaio, o ha h oluio i airly impl. h ivr raorm i h obaid o g h im rpo. L u ow coidr h Laplac raorm o om pcial caual ucio. Laplac raorm o Spcial Caual Fucio (a) Ui impul ucio δ() δ() h ui impul ha a valu a all valu o ohr ha a whr i ha a iii magiud. Alo h igral o h ui impul ucio ovr im i qual o. hory o lcriciy Aalyi o No-iuoidal Wavorm - Par J Luca Novmbr 4

5 / [δ()] δ ( ) d I h ui impul occur a i, rahr ha a, h h ucio i δ ( i ). / [δ( i )] δ ( - i ) d (b) Ui p ucio H() h ui p ha a valu or valu o < ad a valu o or >. / [H()] H ( ) d d I h ui p occur a i, rahr ha a, h h ucio i H( i ). / [H( i )] i H ( - i ) d (c) Caual xpoial ucio a. H() / [ a.h()] a. H ( ) d (d) Caual Siuoidal ucio i(ω φ).h() / [i(ωφ).h()] F() i( ω φ ). i( ω φ ). i d d ( a) i( ω φ ). H ( ) d F( ) d ω co( ω φ) d i a iφ ω ω co( ω φ) i( ω φ) d iφ ω ω coφ F( ) ( ω ). F(). i φ ω. co φ δ(- i ) i H() H(- i ) i a.h().iφ ω.coφ F ( ) wih φ o ad 9 o h ollowig ar obaid. ω ω ω / [i ω.h()], / [co ω.h()] ω hory o lcriciy Aalyi o No-iuoidal Wavorm - Par J Luca Novmbr 5

6 d ( ) () Laplac raorm o h caual drivaiv d d ( ) d / d d ( ) d d ( ) ( ).( ).. ( ) d () ( - ). d ( ) d /. F( ) ( ) ( ).. d I i worh oig ha ulik i h ca o h ordiary drivaiv, h raorm o h drivaiv alo kp iormaio abou h iiial codiio [i.. ( - )] () A xpoial muliplicaio o a i h im domai A xpoial muliplicaio o a i h im domai corrpod o a hi o a i h -domai. /[ a. ()] a ( a) ( ).. d ( ) ( ).. d F(-a) (g) A hi i h im domai A hi i a h im domai ( a).h( a), corrpod o a xpoial dcay i h -domai. /[(-a).h(-a)] ( a). H ( a).. d a ( a) ( a). H ( a).. d( a) a a a ( τ ). τ. dτ a ( τ ). τ. dτ -a. F() (h) For a priodic wavorm () wih priod ic (τ) or τ <. /[()] ( ).. d ( ).. d ( ).. d ( ).. d uig a chag o variabl, hi may b r-wri a ollow 3 /[()] ). (. d ( ).. d ( ).. d hory o lcriciy Aalyi o No-iuoidal Wavorm - Par J Luca Novmbr 6

7 Sic h ucio i priodic, () () ().. /[()] ( ).. d ( ).. d ( ).. d [ ] ( ).. d /[()] ( ).. d h raorm o ohr caual ucio may b imilarly obaid, ad h abl giv h Laplac raorm or h commo ucio. δ() ui impul H() ui p ramp -a xpoial dcay a a - -a ( a) ( a). -a -a - -b doubl xpoial b a ( a)( b) i ω i wav ω hory o lcriciy Aalyi o No-iuoidal Wavorm - Par J Luca Novmbr 7

8 ω coφ iφ i (ω φ) ω co ω coi wav ω rcagular pul h! ordr ramp ( > ) a ih a hyprbolic i a coh a hyprbolic coi a a. () b. () addiio a.f () b.f () d ( ) d ir drivaiv F() ( - ) d ( ) d h drivaiv F() j j j d d ( ) ( ) d dii igral F( ).() (-).() d F( ) d d F( ) d -α.() xpoial muliplir F(α) (-τ) hi -τ.f() priodic ucio (priod ) ( ( ).. ) d hory o lcriciy Aalyi o No-iuoidal Wavorm - Par J Luca Novmbr 8

9 rai Aalyi o Circui uig h Laplac raorm lcrical Circui ar uually govrd by liar dirial quaio. Sic drivaiv ad igral g covrd o muliplicaio ad diviio i h -domai, h oluio o circui quaio ca b covrd o h oluio o algbraic quaio. L u ir coidr h rpraio o h hr baic circui compo i Laplac raorm aalyi. (a) iiv lm i() I() I() v() V() V() v(). i() V(). I() i ( ) v( ) I ( ) V ( ) hu h rior may b rprd by a impdac o valu v i h -domai. (b) Iduciv lm L i() L I() L L. i( - ) V() I() L v() d i( ) v() L. d V() V() L. I() L.i( - ) i( ) i ( ) v( ). d i( ) I ( ). V ( ) L L hu h iducor may b rprd by a impdac o valu L ad ihr a ri volag ourc or a paralll curr ourc. h ourc rpr h iiial rgy ord i h iducor a im. I i o b od ha h iiial curr i( - ) appar i boh orm o h quaio ad ha o orm ca b obaid algbraically rom h ohr, wihou rorig o ay addiioal iormaio. (c) Capaciiv lm L v ( ) V() i() C I() C I() C i( v() v ( ) i( ). d v( C d v( ) i() C. d ) V() v( ) V ( ). I( ) C C. v( - ) I() C. I() C.v( - ) hu h capacior may b rprd by a impdac o valu ad ihr a ri C volag ourc or a paralll curr ourc. A wih h iducor h ourc rpr h iiial rgy ord i h capacior a im. hory o lcriciy Aalyi o No-iuoidal Wavorm - Par J Luca Novmbr 9

10 I addiio o Ohm Law ad Kircho law, Suprpoiio, hvi ad Noro horm may alo b applid o h raormd circui i h -domai. Uig h circui, ad h raorm o ourc volag ad/or curr, h ym rai could b obaid. You would by ow hav ralid ha hi mhod i much l diou ha h oluio o h dirial quaio o id h rai oluio ad h ubiuig h iiial ad ial codiio applicabl. xampl Fid h Laplac raorm o h ollowig wavorm. (a) (b) () () / Soluio (a) uig ir pricipl / [()] ( ).. d... d ( ).. d () i ω or < < / () lwhr d...( ) ( ). ( )..( ) ( ) [ ].[ ] 3 uig propri rom abl (hi i o alway poibl) () () h par o h ramp rom o ca b coidrd a h addiio o a poiiv ramp a, a gaiv ramp a ad a gaiv p o magiud a im. h rmaiig par o h wavorm ca b coidrd o b mad up o a gaiv p wavorm o magiud a, ad a poiiv p alo o magiud a. Suprpoiio o h wavorm will giv h rula wavorm. h will hav Laplac raorm which will add up a ollow. hory o lcriciy Aalyi o No-iuoidal Wavorm - Par J Luca Novmbr

11 / [()] [ ].[ ] 3 which i h idical rul ha wa obaid rom h ormal mhod. (b) () () / / W may alo work hi problm uig abl. h giv wavorm ca alo b coidrd a b buil up o a caual i wav arig a, ad a gaiv o ha wavorm arig a /. hu h raorm o h wavorm i giv by / [()] ω ω. ω xampl Drmi h rai volag apparig acro h capacior wh h wich i clod a A i ω im. Capacior C i iiially uchargd. Soluio h raormd circui i how. h capacior ha o b aociad wih a ourc a hr i o iiial charg (or volag) o h capacior. A. ω Uig poial dividr acio ω V ( ) ou C A. ω C C ω A ω A. α ω V ou ( ).., whr α C ω α ω C hi ca b pli up a ollow. A. ω. α α V ou ( ) ω α α ω Uig h abl, h ivr raorm i h giv a A. ω. α α α vou ( ) coω iω ω α ω C C v ou V ou () hory o lcriciy Aalyi o No-iuoidal Wavorm - Par J Luca Novmbr

12 xampl 3 I a ri LC circui, iiially h capacior i chargd o a volag o V o ad h iducor do o carry ay curr. A im, a p volag o magiud i applid o h ri combiaio. Drmi h rai volag acro L. Soluio V i() C I() C V.H() v() V() L L h circui i ir raormd o h Laplac domai. h volag ourc orm i lcd or h capaciac bcau h circui i a ri circui ad ha orm mak calculaio air. Sic hr i o iiial curr i h iducor, o ourc ha b aociad wih h iducor. I ( ) L ω LC V L C V() L.I() v( ) ( V ) co xampl 4 V L L C, V ( ) ( V ) LC V LC LC ω o ( V ) LC Figur how a circui which ha rachd ady a wih wich clod. I h wich S i opd a im, obai a xprio or h uig curr hrough h iducor. Ω S C µf L mh V Ω Soluio From poial dividr acio, udr ady a codiio, hal h upply volag will drop acro ad hal h volag acro. hror h volag acro h capacior iiially will b / 5 V, ad h iducor curr will b / 5 A. hory o lcriciy Aalyi o No-iuoidal Wavorm - Par J Luca Novmbr

13 raorm h circui o h Laplac domai. I() S No h dircio o h wo ourc. h corrpod o h dircio o h iiial volag acro h capacior ad h iiial curr hrough h iducor. No alo ha ic wich S i ow opd a, oly h ohr wo brach will bcom par o h circui hu I() ( 5) I() ( 5) 3.5 ( 5) 3.5 ( 5) 3.5 Fidig h ivr raorm rom h adard xprio, i() 5 co i 3.5 A hory o lcriciy Aalyi o No-iuoidal Wavorm - Par J Luca Novmbr 3

Part B: Transform Methods. Professor E. Ambikairajah UNSW, Australia

Part B: Transform Methods. Professor E. Ambikairajah UNSW, Australia Par B: rasform Mhods Profssor E. Ambikairaah UNSW, Ausralia Chapr : Fourir Rprsaio of Sigal. Fourir Sris. Fourir rasform.3 Ivrs Fourir rasform.4 Propris.4. Frqucy Shif.4. im Shif.4.3 Scalig.4.4 Diffriaio