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Laplace transform

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Laplace transform

  1. 1. The Laplace Transform The University of Tennessee Electrical and Computer Engineering Department Knoxville, Tennessee wlg
  2. 2. The Laplace Transform The Laplace Transform of a function, f(t), is defined as; ∫ ∞ − == 0 )()()]([ dtetfsFtfL st The Inverse Laplace Transform is defined by ∫ ∞+ ∞− − == j j ts dsesF j tfsFL σ σ π )( 2 1 )()]([1 *notes Eq A Eq B
  3. 3. The Laplace Transform We generally do not use Eq B to take the inverse Laplace. However, this is the formal way that one would take the inverse. To use Eq B requires a background in the use of complex variables and the theory of residues. Fortunately, we can accomplish the same goal (that of taking the inverse Laplace) by using partial fraction expansion and recognizing transform pairs. *notes
  4. 4. The Laplace Transform Laplace Transform of the unit step. *notes |0 0 1 1)]([ ∞− ∞ − ∫ − == stst e s dtetuL s tuL 1 )]([ = The Laplace Transform of a unit step is: s 1
  5. 5. The Laplace Transform The Laplace transform of a unit impulse: Pictorially, the unit impulse appears as follows: 0 t0 f(t) δ(t – t0) Mathematically: δ(t – t0) = 0 t ≠ 0 *note 01)( 0 0 0 >=−∫ + − εδ ε ε dttt t t
  6. 6. The Laplace Transform The Laplace transform of a unit impulse: An important property of the unit impulse is a sifting or sampling property. The following is an important. ∫    >< << =− 2 1 2010 2010 0 ,0 )( )()( t t tttt ttttf dttttf δ
  7. 7. The Laplace Transform The Laplace transform of a unit impulse: In particular, if we let f(t) = δ(t) and take the Laplace 1)()]([ 0 0 === −− ∞ ∫ sst edtettL δδ
  8. 8. The Laplace Transform An important point to remember: )()( sFtf ⇔ The above is a statement that f(t) and F(s) are transform pairs. What this means is that for each f(t) there is a unique F(s) and for each F(s) there is a unique f(t). If we can remember the Pair relationships between approximately 10 of the Laplace transform pairs we can go a long way.
  9. 9. The Laplace Transform Building transform pairs: eL( ∫∫ ∞ +− ∞ −−− == e tasstatat dtedteetueL 0 )( 0 )]([ asas e tueL st at + = + − = ∞− − 1 )( )]([ |0 as tue at + ⇔− 1 )(A transform pair
  10. 10. The Laplace Transform Building transform pairs: ∫ ∞ − = 0 )]([ dttettuL st ∫ ∫ ∞ ∞ ∞ −= 0 0 0 | vduuvudv u = t dv = e-st dt 2 1 )( s ttu ⇔ A transform pair
  11. 11. The Laplace Transform Building transform pairs: 22 0 11 2 1 2 )( )][cos( ws s jwsjws dte ee wtL st jwtjwt + =       + − − = + = − ∞ − ∫ 22 )()cos( ws s tuwt + ⇔ A transform pair
  12. 12. The Laplace Transform Time Shift ∫ ∫ ∫ ∞ ∞ −−+− ∞ − = ∞→∞→→→ +==−= −=−− 0 0 )( )()( ,.,0, , )()]()([ dxexfedxexf SoxtasandxatAs axtanddtdxthenatxLet eatfatuatfL sxasaxs a st )()]()([ sFeatuatfL as− =−−
  13. 13. The Laplace Transform Frequency Shift ∫ ∫ ∞ +− ∞ −−− +== = 0 )( 0 )()( )]([)]([ asFdtetf dtetfetfeL tas statat )()]([ asFtfeL at +=−
  14. 14. The Laplace Transform Example: Using Frequency Shift Find the L[e-at cos(wt)] In this case, f(t) = cos(wt) so, 22 22 )( )( )( )( was as asFand ws s sF ++ + =+ + = 22 )()( )( )]cos([ was as wteL at ++ + =−
  15. 15. The Laplace Transform Time Integration: The property is: stst t st t e s vdtedv and dttfdudxxfuLet partsbyIntegrate dtedxxfdttfL −− − ∞∞ −== ==       =      ∫ ∫ ∫∫ 1 , )(,)( : )()( 0 0 00
  16. 16. The Laplace Transform Time Integration: Making these substitutions and carrying out The integration shows that )( 1 )( 1 )( 00 sF s dtetf s dttfL st = =      ∫∫ ∞ − ∞
  17. 17. The Laplace Transform Time Differentiation: If the L[f(t)] = F(s), we want to show: )0()(] )( [ fssF dt tdf L −= Integrate by parts: )(),( )( , tfvsotdfdt dt tdf dv anddtsedueu stst === −== −− *note
  18. 18. The Laplace Transform Time Differentiation: Making the previous substitutions gives, [ ] ∫ ∫ ∞ − ∞ −∞− +−= −−=      0 0 0 )()0(0 )()( | dtetfsf dtsetfetf dt df L st stst So we have shown: )0()( )( fssF dt tdf L −=   
  19. 19. The Laplace Transform Time Differentiation: We can extend the previous to show; )0(... )0(')0()( )( )0('')0(')0()( )( )0(')0()( )( )1( 21 23 3 3 2 2 2 − −− −− −−=      −−−=      −−=      n nnn n n f fsfssFs dt tdf L casegeneral fsffssFs dt tdf L fsfsFs dt tdf L
  20. 20. The Laplace Transform Transform Pairs: ____________________________________ )()( sFtf f(t) F(s) 1 2 ! 1 1 1 )( 1)( + − + n n st s n t s t as e s tu tδ
  21. 21. The Laplace Transform Transform Pairs: f(t) F(s) ( ) 22 22 1 2 )cos( )sin( )( ! 1 ws s wt ws w wt as n et as te n atn at + + + + + − −
  22. 22. The Laplace Transform Transform Pairs: f(t) F(s) 22 22 22 22 sincos )cos( cossin )sin( )( )cos( )( )sin( ws ws wt ws ws wt was as wte was w wte at at + − + + + + ++ + ++ − − θθ θ θθ θ Yes !
  23. 23. The Laplace Transform Common Transform Properties: f(t) F(s) )( 1 )( )( )( )0(...)0(')0()( )( )()( )([0),()( )(0),()( 0 1021 00 000 sF s df ds sdF ttf ffsfsfssFs dt tfd asFtfe ttfLetttutf sFetttuttf t nnnn n n at sot sot ∫ − −−−− + +≥− ≥−− −−− − − − λλ
  24. 24. The Laplace Transform Using Matlab with Laplace transform: Example Use Matlab to find the transform of t te 4− The following is written in italic to indicate Matlab code syms t,s laplace(t*exp(-4*t),t,s) ans = 1/(s+4)^2
  25. 25. The Laplace Transform Using Matlab with Laplace transform: Example Use Matlab to find the inverse transform of 19.12. )186)(3( )6( )( 2 prob sss ss sF +++ + = syms s t ilaplace(s*(s+6)/((s+3)*(s^2+6*s+18))) ans = -exp(-3*t)+2*exp(-3*t)*cos(3*t)
  26. 26. The Laplace Transform Theorem: Initial Value If the function f(t) and its first derivative are Laplace transformable and f(t) Has the Laplace transform F(s), and the exists, then)(lim ssF 0 )0()(lim)(lim →∞→ == ts ftfssF The utility of this theorem lies in not having to take the inverse of F(s) in order to find out the initial condition in the time domain. This is particularly useful in circuits and systems. Theorem: ∞→s Initial Value Theorem
  27. 27. The Laplace Transform Initial ValueTheorem:Example: Given; 22 5)1( )2( )( ++ + = s s sF Find f(0) 1 )26(2 2 lim 2512 2 lim 5)1( )2( lim)(lim)0( 2222 222 2 2 22 = ++ + =         +++ + = ++ + == sssss ssss ss ss s s sssFf ∞→s∞→s ∞→s ∞→s
  28. 28. The Laplace Transform Theorem: Final Value Theorem: If the function f(t) and its first derivative are Laplace transformable and f(t) has the Laplace transform F(s), and the exists, then)(lim ssF ∞→s )()(lim)(lim ∞== ftfssF 0→s ∞→t Again, the utility of this theorem lies in not having to take the inverse of F(s) in order to find out the final value of f(t) in the time domain. This is particularly useful in circuits and systems. Final Value Theorem
  29. 29. The Laplace Transform Final Value Theorem:Example: Given: [ ] ttesFnote s s sF t 3cos)( 3)2( 3)2( )( 21 22 22 − = ++ −+ = − Find )(∞f . [ ] 0 3)2( 3)2( lim)(lim)( 22 22 = ++ −+ ==∞ s s sssFf 0→s0→s

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