Chapr Inroducion To Th aplac Tranorm Diniion o h aplac Tranorm - Th Sp & Impul uncion aplac Tranorm o pciic uncion 5 Opraional Tranorm Applying h aplac Tranorm 7 Invr Tranorm o Raional uncion 8 Pol and Zro o 9 Iniial- and inal-valu Thorm
Ovrviw aplac ranorm i a chniqu ha i paricularly uul in linar circui analyi whn: Conidring ranin rpon g wiching o circui wih mulipl nod and mh Th ourc ar mor complicad han h impl dc lvl ump Inroducing h concp o ranr uncion o analyz rquncy-dpndn inuoidal ady-a rpon Chapr,
y poin Wha i h diniion o h aplac ranorm? Wha ar h aplac ranorm o uni p, impul, xponnial, and inuoidal uncion? Wha ar h aplac ranorm o h drivaiv, ingral, hi, and caling o a uncion? How o prorm parial racion xpanion or a raional uncion and prorm h invr aplac ranorm?
Scion Diniion o h aplac Tranorm
Wha i aplac ranorm? Tranorming a ral uncion o ral variabl o a complx uncion o complx variabl : { } Th ingral will convrg ovr a porion o d C h -plan g R >, and or mo o h uncion xcp or ho o lil inr g i drmind by only or > - Thu w u i o prdic h rpon ar iniial condiion hav bn ablihd krnl 5
Scion, Th Sp and Impul uncion Diniion o uni p uncion u Diniion o impul uncion d aplac ranorm o d and d'
Th uni p uncion u u, or ;, or u can b approximad by h limi o a linar ramp uncion: 7
Rprnaion o im hi and rvral Tim hi: u a Tim rvral: u a 8
Exampl : A pul o ini widh Q: Expr h picwi linar uncion a uprpoiion o uncion y y y y i a i b i 9
Exampl or ach inrval, can b xprd a h produc o a linar uncion and a quar pul dirnc bwn wo p uncion or xampl, or < <, h corrponding linar and quar pul uncion ar: y, and p u u Th nir uncion can b rprnd by: y p y p y p
Th impul uncion d An idalizd mah rprnaion o harply pakd imulu: d,, d d ohrwi; d ha zro duraion, inini pak ; ampliud, uni ara rngh
d i h drivaiv o u u lim d lim u
Th iing propry o d-uncion Sampling o a = a > can b ormulad by ingral o im d-a: I can b ud in calculaing h aplac ranorm o a d-uncion: lim lim a d a a d a d a a a a a d d d or any, C d d d
Th drivaiv o d-uncion ara = d lim on-idd ara = / oal ara = d lim
5 lim lim lim lim lim d d d d aplac ranorm o d' lim d Evn d' ha wll dind aplac ranorm!
Scion aplac Tranorm o Spciic uncion
Eg Uni p uncion u u d, i R d Im R 7
Eg Singl-idd xponnial uncion a > Im a a a a d a d, i R a a -a R 8
9 Eg Sinuoidal uncion in in d d d
i o aplac ranorm pair Typ impul p ramp xponnial d u u u a a aplac ranorm o polynomial uncion:, n! n n
i o aplac ranorm pair Damping by xponnial dcay uncion cau a hi along h ral axi in h -domain in dampd in dampd ramp co coin in in Typ a u a u u u a a
Scion 5 Opraional Tranorm
Wha ar opraional ranorm? Opraional ranorm indica how h mahmaical opraion prormd on ihr or ar convrd ino h oppoi domain Uul in calculaing h aplac ranorm o a uncion g drivd by prorming om mah opraion on wih known
ir-ordr im drivaiv d d i - = d Eg d u, u, d u ingraion by par iniial condiion
5 Highr-ordr im drivaiv, g G g G g nh-ordr drivaiv: nd-ordr drivaiv: n n n n n iniial condiion iniial condiion
Tim ingral, ;, whr, v v u dx x u d v u d dx x dx x dx x dx x d dx x d v u v u dx x ini Th ormula i valid only i h uncion i ingrabl
Scaling a a a d; a l a d a, a a, i a Inuiivly, a largr valu o a corrpond o a narrowr uncion in h im domain bu a broadr uncion in h rquncy domain Th widh o a im h widh o /a i a conan indpndn o a 7
Tranlaion in h and domain Tranlaion in h im domain: a a u a, or a Tranlaion in h rquncy domain: a a Boh rlaion can b provn by chang o variabl o ingraion 8
Scion 7 Invr Tranorm o Raional uncion Diinc ral roo Diinc complx roo Rpad ral roo Rpad complx roo 5 Impropr raional uncion 9
Why only raional uncion? or linar, lumpd-paramr circui wih conan componn paramr, h -domain xprion or v, i ar alway raional uncion, i raio o wo polynomial: Gnral invr aplac ranorm: b b b a a a D N m m m m n n n n, i r i r d i involv wih complx ingral
How o calcula? I i a propr m > n raional uncion, h invr ranorm i calculad by parial racion xpanion, individual invr ranorm yp I i an impropr m n raional uncion, dcompo a h ummaion o a polynomial uncion and a propr raional uncion, which ar invr ranormd individually
Typ I: D ha diinc ral roo 8 8 5 9 8 95 8 5 9 ; 8 8 7, 8 8
Typ I: D ha diinc ral roo 8 7 8 7 8 8 7, 8 8 7 8 8 u Th circui i ovr-dampd
Typ II: D ha diinc complx roo 8 8,,,, : rooo, 5 D Conuga roo mu hav conuga coicin
5 Typ II: D ha diinc complx roo 5 co 8 R 8 8 8 8 8 8, 8 8 u u u u 5 Th circui how ovr-dampd or -ordr and undr-dampd characriic
Typ III: D ha rpad ral roo 5 5 5 5 5 high ordr 5 G 5 5 5 G 5 5 5 ;
7 Typ III: D ha rpad ral roo 5 5 5 5 5 5 5 5 5; 5 5 5 5 ; 5 5 5 5 G G G
8 Typ III: D ha rpad ral roo 5 5 5 5 5, 5 5 5 ; 5 5 5 5 5; 5 5 5 G G G
9 Typ III: D ha rpad ral roo 5 5 5!! 5 5 5 5 5 5 u Th circui how criically-dampd bhavior
Typ IV: D ha rpad complx roo 78 5 78 * * 8 78 ; 78 * * G G G high ordr
Typ IV: D ha rpad complx roo in co 9 co co 9 9 u u u c c c c c c c c Th circui how criically- and undr-dampd characriic
Uul ranormaion pair co co * * u u u a u a a a
Invr ranorm o impropr raional uncion 5 5 5 5 u d d d
Scion 8 Pol and Zro o
Diniion can b xprd a h raio o wo acord polynomial N/D Th roo o h dnominaor D ar calld pol and ar plod a X on h complx - plan Th roo o h numraor N ar calld zro and ar plod a o on h complx -plan 5
Exampl 8] 8][ [ ] ][ 5[ D N
y poin Wha i h diniion o h aplac ranorm? Wha ar h aplac ranorm o uni p, impul, xponnial, and inuoidal uncion? Wha ar h aplac ranorm o h drivaiv, ingral, hi, and caling o a uncion? How o prorm parial racion xpanion or a raional uncion and prorm h invr aplac ranorm? 7