1 / 50

LAPLACE TRANSFORM

LAPLACE TRANSFORM. Prepared by Ertuğrul Eriş Reference textbook: Nilsson/Riedel. Updated: November 2011. LAPLACE TRANSFORM. WHY TRANSFORM PRO ORDINARY DIFFERENTIAL EQUATIONS/ ALGEBRIAC LINEAR EQUATIONS CON t-domain → s-domain→ t-domain OTHER KNOWN TRANSFORMS LOGARITMA FREQUENCY DOMANN.

manasa
Download Presentation

LAPLACE TRANSFORM

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. LAPLACE TRANSFORM Prepared by Ertuğrul Eriş Reference textbook: Nilsson/Riedel Updated: November 2011 Ertuğrul Eriş

  2. LAPLACE TRANSFORM • WHY TRANSFORM • PRO • ORDINARY DIFFERENTIAL EQUATIONS/ ALGEBRIAC LINEAR EQUATIONS • CON • t-domain→s-domain→ t-domain • OTHER KNOWN TRANSFORMS • LOGARITMA • FREQUENCY DOMANN Ertuğrul Eriş

  3. BASIC SIGNALS/FUNCTIONS IN ELECTRONIC SYSTEMS • Electrical signals: Analog, digital • Speech, image/video, light, x-ray, ultrasonography, • DA/AD converters • Fourier Transform • Sinusoidal signals • Sources • DC source + switch (unit function), discontinuous at the origin • Derivative • Dirac Delta function (Impulse) • integral • 0 • AC • Signals in linear circuits • DC voltage/current • AC voltage/current (Under damped, Sönümlü) • Exponential voltage/current Ertuğrul Eriş

  4. LAPLACE TRANSFORM DEFINITION-1 • Expectation • Differential equation becomes algebriac • definitions relations, KVL, KCL equations becomes algebriac • L {f(t)}= • F(s)=L {f(t)} • f(t)= L -1 {F(s)} Ertuğrul Eriş

  5. LAPLACE TRANSFORM DEFINITION-2 L {f(t)}= • Integration limits • Upper limit ∞ • Some functions laplace transform may not exsist • Lower limit (0), One-sided/unileteral • Physical reality • Lower limit at t=0 continuous/ discontinuous • t = 0- lower limit • t<0- accounted for initial conditions • From t = 0- to t=0+ integration: (0) • Exception: Dirac delta function • Functional/Operational transforms

  6. CONTINUOUS/DISCONTINUOUS AT ZERO t = 0- f(t)=1 lower limit t<0- , accounted for initial Conditions From t = 0- to t=0+ integral: (0) t = 0- f(t)=0 lower limit t<0- , accounted for initial Conditions From t = 0- to t=0+ integral: (0) Ertuğrul Eriş

  7. STEP (BASAMAK) FONCTION K=1 Birim basamak fonksiyonu Unit step function Unit step function (mathematical model): DC source + Switch Ertuğrul Eriş

  8. DISCONTINUITY(JUMP) OF STEP FUNCTION AT ZERO Theory*practise Ertuğrul Eriş

  9. STEP OCCURANCE SHIFT Could we express a «Pulse functıon» by using step functıons? Ertuğrul Eriş

  10. PIECEWISE LINEAR (KESİKLİ LİNEER) FUNCTIONS/STEP FUNCTIONS Ertuğrul Eriş

  11. DERIVATIVE AT DISCONTINUITY: A VARIABLE PARAMETER FUNCTION USED TO GENERATE IMPULSE (DIRAC DELTA) FONCTION Variable parameter function: As parameter approaches to zero; Amplitude approaches to infinity, The duration of the function approaches to zero, Area under the function is constant. Ertuğrul Eriş

  12. DERIVATIVE AT DISCONTINUITY: A VARIABLE PARAMETER FUNCTION USED TO GENERATE IMPULSE (DIRAC DELTA) FONCTION Variable parameter function: As parameter approaches to zero; Amplitude approaches to infinity, The duration of the function approaches to zero, Area under the function is constant. K:strenght Ertuğrul Eriş

  13. DIRAC DELTA (IMPULSE) FUNCTION δ(t) AND SIFTING (AYIRMA) PROPERTY Ertuğrul Eriş

  14. LAPLACE OF δ(t) L {f(t)}= Sifting property: L {δ(t)}= L {δ(t)}= 1 Ertuğrul Eriş

  15. LAPLACE OF THE DERIVATIVE OF δ’(t) L {δ’(t)}= s Genelleştirilmişi: L {δ(n)(t)}= sn L {f(t)}= Details is in Nilsson Ertuğrul Eriş

  16. UNIT STEP FUNCTION/DIRAC DELTA δ(t) FONCTION f(t)→u(t) ε →0 f’(t) →δ(t) ε →0 δ(t)= du(t)/d(t) Ertuğrul Eriş

  17. LAPLACE OF UNIT STEP FUNCTION L {f(t)}= L {u(t)}= 1/s F(s) Rational function! Ertuğrul Eriş

  18. LAPLACE OF e-at L {f(t)}= L {e - at}= 1/(s+a) F(s) Rational function! ! Ertuğrul Eriş

  19. LAPLACE OF SINUS L {f(t)}= L {sin ωt }= ω/(s2+ω2) L {cosωt }= s/(s2+ω2) F(s) Rational function! How to find Laplace of Cos(ωt+φ)? Ertuğrul Eriş

  20. LAPLACE OF RAMP FONCTION L {f(t)}= F(s) rational function! Ertuğrul Eriş

  21. LIST OF LAPLACE TRANSFORMS L {f(t)}= İmpuse δ(t) 1 Step u(t) 1/s Ramp t 1/(s2) Exponential e-at 1/(s+a) Sine sinωt ω/(s2+ω2) Cosine cosωt s/(s2+ω2) Damped Ramp te-at 1/(s+a)2 Damped sine e-at sinωt ω/((s+a)2+ω2) Damped cosine e-at cosωt (s+a)/((s+a)2+ω2) All of the F(s) functions are Rational function! Ertuğrul Eriş

  22. OPERATIONAL TRANSFORMS Kf(t) KF(s) f1(t)+f2(t)-f3(t) F1(s)+F2(s)-F3(s) df(t)/dt sF(s)-f(0-) d2f(t)/dt2 s2F(s)-sf(0-)-df(0-)/dt dnf(t)/dtn snF(s)- sn-1f(0-)-sn-2 df(0-)/dt -dfn-1(0-)/dtn-1 F(s)/s f(t-a)u(t-a), a>0 e-asF(s) e-atf(t) F(s+a) f(at), a>0 (1/a)F(s/a) tf(t) -dF(s)/ds tnf(t) (-1)n dnF(s)/dsn f(t)/t L {f(t)}= Note: Laplace transforms Differential equations to rational fonctions. Ertuğrul Eriş

  23. APPLYING THE LAPLACE TRANSFORM Ertuğrul Eriş

  24. INVERSE LAPLACE TRANSFORM-1 • Solutions in s-domain are rational fonctions • Proper rational n<m • Improper rational m<n • Partial fraction expansion (basit kesirler ) • Inverse Laplace Ertuğrul Eriş

  25. INVERSE LAPLACE TRANSFORM-2 u(t) 1/s e-at1/(s+a) sinωt ω/(s2+ω2) cosωt s/(s2+ω2) te-at1/(s+a)2 e-at sinωt ω/((s+a)2+ω2) e-atcosωt (s+a)/((s+a)2+ω2) • Poles of the rational function • real→exponential • Complex → damped sinusoidal • Imaginer →sinusoidal Ertuğrul Eriş

  26. INVERSE LAPLACE EXAMPLE, REAL ROOTS If this function were arelated to a circuit solution , does this circuit burn? Ertuğrul Eriş

  27. INVERSE LAPLACE EXAMPLE, REPEATED REAL ROOTS-1 If this function were related to a circuit solution , would this circuit burn? Ertuğrul Eriş

  28. INVERSE LAPLACE EXAMPLE, REAL ROOTS-2 If this function were related to a circuit solution , would this circuit burn? Ertuğrul Eriş

  29. INVERSE LAPLACE EXAMPLE, COMPLEX ROOTS-1 Complex conjugate roots MultipleComplex conjugate roots Note: In order to find the inverse laplace only «K» calculation required, no need to calculate K* What happens if the roots are imaginer (multiple case as well!)? Ertuğrul Eriş

  30. INVERSE LAPLACE EXAMPLE, COMPLEX ROOTS-2 If this function were related to a circuit solution , would this circuit burn? Ertuğrul Eriş

  31. INVERSE LAPLACE EXAMPLE, REPEATED COMPLEX ROOTS-3 If this function were related to a circuit solution , would this circuit burn? Ertuğrul Eriş

  32. IMPROPER RATIONAL FUNCTION • Numerator degree> denominator degree • After the division • Polynomial function+proper rational function • Inverse Laplace will have Dirac delta and its derivatives Ertuğrul Eriş

  33. IMPROPER RATIONAL FUNCTION EXAMPLE Ertuğrul Eriş

  34. POLES/ZEROS IN S-DOMAIN (KUTUP VE SIFIRLAR ) Ertuğrul Eriş

  35. INITIAL AND FINAL VALUE TEOREMS Ertuğrul Eriş

  36. EXAMPLE-1 Ertuğrul Eriş

  37. EXAMPLE-2 Ertuğrul Eriş

  38. EXAMPLE-3 Ertuğrul Eriş

  39. EXAMPLE-4 Ertuğrul Eriş

  40. EXAMPLE-5 Ertuğrul Eriş

  41. EXAMPLE-6 Ertuğrul Eriş

  42. EXAMPLE-7 Ertuğrul Eriş

  43. EXAMPLE-8 Ertuğrul Eriş

  44. EXAMPLE-9 Ertuğrul Eriş

  45. EXAMPLE-10 Ertuğrul Eriş

  46. PROGRAM DESIGN DEPT, PROGRAM G R A D U A T E S T U D E N T STUDENT P R OG R A M O U T C O M E S PROGRAM OUTCOMES P R OG R A M O U T C O M E S STATE, ENTREPRENEUR FIELD QALIFICATIONS EU/NATIONAL QUALIFICATIONS KNOWLEDGE SKILLS COMPETENCES NEWCOMERSTUDENT ORIENTIATION GOVERNANCE Std. questionnaire ALUMNI, PARENTS ORIENTIATION STUDENT PROFILE Std. questionnaire FACULTY NGO STUDENT, ??? CIRCICULUM ??? INTRERNAL CONSTITUENT Std. questionnaire EXTRERNAL CONSTITUENT EXTRERNAL CONSTITUENT REQUIREMENTS EU/NATIONAL FIELD QUALIFICATIONS PROGRAM OUTCOMES QUESTIONNAIRES QUALITY IMP. TOOLS GOAL: NATIONAL/INTERNATIONAL ACCREDITION

  47. BLOOM’S TAXONOMYANDERSON AND KRATHWOHL (2001) !!Listening !! Doesn’t exits in the original!!! http://www.learningandteaching.info/learning/bloomtax.htm Ertuğrul Eriş

  48. ULUSAL LİSANS YETERLİLİKLER ÇERÇEVESİ BLOOMS TAXONOMY Ertuğrul Eriş

  49. COURSE ASSESMENT MATRIX LEARNING OUTCOMES Devre Analizi İlk Ders

  50. ‘ABET’ ENGINEERING OUTCOMES Ertuğrul Eriş

More Related