Inverse Laplace Transforms

1

• Inverse Laplace Transforms

2
Rational Functions

• Function ????????????????? inverse Laplace
  transform ?????????????? ???????????????
  (rational function) ??????????? (numerator N(s)
  ) ????????? (denominator D(s))
  ????????????????????
• Rational function ????????????????
• Proper rational function ??????????? s
  ?????????????????????? ???????? s ?????????
  ??????????????? inverse Laplace transform
  ????????
• Improper rational function ??????????? s
  ???????????????????????????????????????? s
  ????????? ??????????????????????????????
  ?????????????? proper rational function ????????

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3
Proper Rational Functions

• ?????????????? proper rational function
  ????????????? inverse Laplace transform ???
  ??????????????????????? ??????????????????? D(s)
  ????????? ?????????????????????????? 3 ??? ??????
• Simple poles ????????????????? D(s) ??????
• Complex Poles ????????????????? D(s) ??????
• Repeated poles ????????????? simples ???? complex
  poles ????????? ????

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Simples Poles

• ???
• ????????
• ???
• ????????????? Method 1 ??? Example 15.9 ???? 693
  ????????????????????????? ?? Method 2 ??? Example
  15.9 ???? 694 ??????

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Complex Poles

• ???
• ????????
• ???????? k1 ???????????????????????? simple pole
  ???????????? B ??? C ?????????????????
  ????????????????????????? (?? Method 2 ???
  Example 15.11 ???? 696) ?????????? s ????????????
  ???? 0 ??? 1 ?????????????????????? B ??? C (??
  Method 1 ??? Example 15.11 ???? 696)

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Repeated Poles

• ???? simples poles ??? complex poles
  ??????????????? repeated poles ???
  ??????????????????????????? simple poles
  ?????????????????
• ???
• ????????
• ???????? k2, k3, .. , kn ?????????????????????????
  ????? simple poles

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Repeated Poles (???)

• ???????? A ??? B ???????????? (?? Method 1 ???
  Example 15.10 ???? 695)
• ?????????????????????????????? (?? method 2 ???
  Example 15.10 ???? 695)

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8
Long Division

• ????????????? ?????????????????????????? improper
  rational function ??????? proper rational
  function
• ????????????????? ?????????????????? polynomial
  ?????? ????
• ???????? improper rational function
  ???????????????????????????

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Long Division (???)
? Quotient

• ?????????????? order ??????????????
  ?????????????????? ?????????? Remainder

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