Chapr aplac Tranorm naly
Chapr : Ouln
aplac ranorm
aplac Tranorm -doman phaor analy: x X σ m co ω φ x X X m φ x aplac ranorm: [ o ] d o d <
aplac Tranorm Thr condon Unlaral on-dd aplac ranorm: aplac ranorm hold or. Prvou c ar ncludd n h nal condon a -. Exnc condon: lm σ σ > σ c. Th rulng ranorm a uncon o.
On-Sdd Wavorm [ ] [ ] < > < > < un p : u u u
Sucn Exnc Condon convrgnc :abca o ordr xponnal o rmann Th ngral n c lm σ σ σ σ σ > d d d d d d c
nvr aplac Tranorm nvr aplac ranorm or : [ ] c jω c jω d c > πj σ c nvr aplac ranorm uually don by paral racon xpanon wll b covrd n con..
Exampl. [ ] a a a σ lm lm a σ > [ ] a a a d d a a [ ] [ u ] No : a can bcomplx R [ a] σ > whn σ c > a
Exampl.
Exampl. [ ] n c j j j j j j σ
Tranorm Propr nar combnaon: [ ] [ g ] [ Bg ] [ ] G BG
Tranorm Propr Mulplcaon by -a : [ ] [ ] a a
Tranorm Propr Mulplcaon by : d d d d d d [ ]
Tranorm Propr Tm dlay: [ ] g u g > <
Tranorm Propr Drnaon: [ ] d d [ ]
Tranorm Propr ngraon: [ ] d g d d g d dg d g λ λ λ λ
Exampl. D u u D D D < < ohrw
Exampl.4 [ ] [ ] [ ] [ ] [ ] [ ] n co n n co co co co n co n φ φ φ φ φ
Tabl.
Tabl.
Solvng Drnal Equaon Th ranormaon auomacally ncorpora h nal condon zro-npu rpon. Tranormaon convr lnar drnal quaon o -doman algbrac quaon. Tranormaon mlar o h -doman phaor analy. Dnomnaor o h -doman uncon nclud h characrc polynomal. nvr ranormaon rqurd o oban h rulan m doman uncon.
r-ordr Exampl v 6V [ ] 7 6 7 6.. V V v V v v [ ] 9. 7 6 7 6 V v
Scond-Ordr Exampl n > < v R
Scond Ordr Exampl Con. [ ] [ ] [ ] or no wchng / / / / / > C R C R R C V V R Cv v R Characrc polynomal
Tranorm nvron
Paral-racon Expanon Paral-racon xpanon o a rcly propr raonal uncon: n m a a n n a n b b m m b m m b D N Thr ca wll b condrd: dnc ral pol complx pol and rpad pol.
Ca : Dnc Ral Pol Havd' horm covr-up rul. K D N n n n n
Exampl.: nvron o a Thrd-Ordr uncon Havd horm Ω C H R C R C R / / / / 4 4
Exampl.: Mhod o Undrmnd Cocn 4 4 4
Ca : Complx Pol n D ω α co φ α α α α ω α α ω α α α α φ α ± ± < K K K g K j K j K j K G G j j m j j j m j n
Ca : Complx Pol Con. [ ] K jk K C B j B K C B C B B C B G K K g K g m m m m m φ φ α ω α α ω α α α ω α α φ φ α φ α n co /. hn drmn compar cocn and nd n co co Or... Undrmnd cocn
Exampl.6: nvron wh Complx Pol 4 6 7 6 6 4 4 4 6 7 6 j K C B j K j K G G j j ± ± α
Exampl.6: Con. mhod o undrmnd cocn 67.4 co4 6 67.4 6 4 66 7 6 6 7 6 6 6 6 7 6 g j C B j B K C C C B B C B C B α co φ α K g m
Ca : Rpad Pol [ ] n j j j n n d d g G G doubl pol
Ca : Rpad Pol Con. [ ] [ ] [ ] d d d d g G!
Exampl.7: nvron wh a Trpl Pol 6 4 6 4 4 64 6 6 4 4 lm 4 6 4 4 4 4 6 4 4 4 4 4 4 4 4 4 4 4
Tm Dlay nal-valu Thorm nal-valu Thorm
Tm Dlay [ ] [ ] u D N N g u g
Exampl.8: nvron wh Tm Dlay [ ] 8 8 4 4 4 4 4 4 4 4 4 4 : xcaon u u y Y Y Y y u u x y y u u x
nal-valu Thorm lm rcly propr
nal-valu Thorm Con. [ ].. > u c c
nal-valu Thorm Con. [ ] [ ] [ ] lm ovr n lm lm / d d c c c c c c c c c
nal Slop [ ] lm known aumng nd nallop D D N D N
nal-valu Thorm magnary ax on h no pol orgn a h pol no mulpl RHP n h no pol lm [ ] lm d d
Exampl.9: Calculang nal and nal Valu rcly propr 8 9 8 6 D N 9 9 lm rcly propr 8 9 8 6 4 9 : lm 4 D D N
Exampl.9: Con. N 6 D 8 9 8 rcly propr or nal valu chck pol locaon lm
Tranorm crcu analy
Tranorm Crcu naly Gvn a crcu wh om nal a a - and an xcaon x arng a nd h rulng bhavor o any volag or currn y or. Zro-a rpon naural rpon orcd rpon zro-npu rpon and compl rpon.
Zro-Sa Rpon Zro-a rpon: a crcu wh no ord nrgy a -. - doman nwork uncon h am a a" "zro nal ] [ ] [ h ordr nwork or an n - a a a a b b b b X Y H Y d y d y Y y Y x b d dx b d x d b y a d dy a d y d a n n n n m m m m k k k m m m n n n
Zro-Sa Rpon [ ] polynomal characrc :.. oban dagram - doman Draw. Sp : P D N X P N H Y y X H Y X x X Y H X X H
Sp Rpon zro nal a by dnon x u > X / Y H N H P pol a h orgn
Exampl.: Sp Rpon Z V H H Z /6 V
Exampl.: Con.. lm lm.. 4 4.. 4 8 4 4 H Sady a rpon naural rpon
Zro-a C rpon co x X m x φ D X y m y m N N N X H Y X j H Y Y y y y y N P N Y Y P N N Y φ φ whr co Naural rpon rom Y orcd rpon rom phaor analy
Exampl.: Zro-Sa C Rpon rom g..9 N N N V H V v 4 4 64 4 4 64 4 4 64 co8 Phaor analy
Exampl.: Con. 8.9 7.7co 8.9 7.7 co8 8.9 7.7 co8 8.9 7.7 8 8.9.4 8 4 V j H j H V N
Naural Rpon and orcd Rpon h orcd rpon o nd or u phaor analy ca : y X H Y x y H N
Naural Rpon and orcd Rpon Con. ca : H phaor analy no applcabl whn xcaon rquncy h am a a naural rquncy u ranorm analy Y y naural mxd orcd
Zro-npu Rpon Th xcaon qual zro or bu h crcu conan ord nrgy a -. Thévnn/Noron quvaln crcu can b ablhd.
Zro-npu Rpon Con. [ ] Noron : Thvnn : C C C C C C C C C C C Cv CV C v V v C V Cv
Zro-npu Rpon Con. [ ] Noron : Thvnn : V V V v
Exampl.: Calculang a Zro- npu Rpon 6 < DC ady a analy: 6 v C 6 V
Exampl.: Con. 4 co 6 co 4 6 6 6 6 6 zro-npu rpon : K j K C B m φ α ω α α
Compl Rpon Compl rpon: compl rponzro-npu rpon zroa rpon wh cou ourc
Exampl.: Calculang a Compl Rpon V V v v V V v S C C 4 or DCady a analy nd < <
Exampl.: Con. 6.6 co8 8 8.. /. 4 4 V v C B V V C C C Ovrhoo du o undrdampd
Chapr : Problm S 6 6 8 8 4 46 7 8 6 6