Chapter 13 Laplace Transform Analysis
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1 Chapr aplac Tranorm naly
2 Chapr : Ouln
3 aplac ranorm
4 aplac Tranorm -doman phaor analy: x X σ m co ω φ x X X m φ x aplac ranorm: [ o ] d o d <
5 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.
6 On-Sdd Wavorm [ ] [ ] < > < > < un p : u u u
7 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
8 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..
9 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
10 Exampl.
11 Exampl. [ ] n c j j j j j j σ
12 Tranorm Propr nar combnaon: [ ] [ g ] [ Bg ] [ ] G BG
13 Tranorm Propr Mulplcaon by -a : [ ] [ ] a a
14 Tranorm Propr Mulplcaon by : d d d d d d [ ]
15 Tranorm Propr Tm dlay: [ ] g u g > <
16 Tranorm Propr Drnaon: [ ] d d [ ]
17 Tranorm Propr ngraon: [ ] d g d d g d dg d g λ λ λ λ
18 Exampl. D u u D D D < < ohrw
19 Exampl.4 [ ] [ ] [ ] [ ] [ ] [ ] n co n n co co co co n co n φ φ φ φ φ
20 Tabl.
21 Tabl.
22 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.
23 r-ordr Exampl v 6V [ ] V V v V v v [ ] V v
24 Scond-Ordr Exampl n > < v R
25 Scond Ordr Exampl Con. [ ] [ ] [ ] or no wchng / / / / / > C R C R R C V V R Cv v R Characrc polynomal
26 Tranorm nvron
27 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.
28 Ca : Dnc Ral Pol Havd' horm covr-up rul. K D N n n n n
29 Exampl.: nvron o a Thrd-Ordr uncon Havd horm Ω C H R C R C R / / / / 4 4
30 Exampl.: Mhod o Undrmnd Cocn 4 4 4
31 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
32 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
33 Exampl.6: nvron wh Complx Pol j K C B j K j K G G j j ± ± α
34 Exampl.6: Con. mhod o undrmnd cocn 67.4 co g j C B j B K C C C B B C B C B α co φ α K g m
35 Ca : Rpad Pol [ ] n j j j n n d d g G G doubl pol
36 Ca : Rpad Pol Con. [ ] [ ] [ ] d d d d g G!
37 Exampl.7: nvron wh a Trpl Pol lm
38 Tm Dlay nal-valu Thorm nal-valu Thorm
39 Tm Dlay [ ] [ ] u D N N g u g
40 Exampl.8: nvron wh Tm Dlay [ ] : xcaon u u y Y Y Y y u u x y y u u x
41 nal-valu Thorm lm rcly propr
42 nal-valu Thorm Con. [ ].. > u c c
43 nal-valu Thorm Con. [ ] [ ] [ ] lm ovr n lm lm / d d c c c c c c c c c
44 nal Slop [ ] lm known aumng nd nallop D D N D N
45 nal-valu Thorm magnary ax on h no pol orgn a h pol no mulpl RHP n h no pol lm [ ] lm d d
46 Exampl.9: Calculang nal and nal Valu rcly propr D N 9 9 lm rcly propr : lm 4 D D N
47 Exampl.9: Con. N 6 D rcly propr or nal valu chck pol locaon lm
48 Tranorm crcu analy
49 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.
50 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
51 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
52 Sp Rpon zro nal a by dnon x u > X / Y H N H P pol a h orgn
53 Exampl.: Sp Rpon Z V H H Z /6 V
54 Exampl.: Con.. lm lm H Sady a rpon naural rpon
55 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
56 Exampl.: Zro-Sa C Rpon rom g..9 N N N V H V v co8 Phaor analy
57 Exampl.: Con co co co V j H j H V N
58 Naural Rpon and orcd Rpon h orcd rpon o nd or u phaor analy ca : y X H Y x y H N
59 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
60 Zro-npu Rpon Th xcaon qual zro or bu h crcu conan ord nrgy a -. Thévnn/Noron quvaln crcu can b ablhd.
61 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
62 Zro-npu Rpon Con. [ ] Noron : Thvnn : V V V v
63 Exampl.: Calculang a Zro- npu Rpon 6 < DC ady a analy: 6 v C 6 V
64 Exampl.: Con. 4 co 6 co zro-npu rpon : K j K C B m φ α ω α α
65 Compl Rpon Compl rpon: compl rponzro-npu rpon zroa rpon wh cou ourc
66 Exampl.: Calculang a Compl Rpon V V v v V V v S C C 4 or DCady a analy nd < <
67 Exampl.: Con. 6.6 co /. 4 4 V v C B V V C C C Ovrhoo du o undrdampd
68 Chapr : Problm S
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