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Presentation on laplace transforms

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Presentation on laplace transforms

  1. 1. Laplace TransformLaplace Transform Submitted By: Md. Al Imran Bhuyan ID: 143-19-1616 Department of ETE Daffodil International University Submitted to: Engr. Md. Zahirul Islam Senior Lecturer Department of ETE Daffodil International University
  2. 2. The French Newton Pierre-Simon Laplace  Developed mathematics in astronomy, physics, and statistics  Began work in calculus which led to the Laplace Transform  Focused later on celestial mechanics  One of the first scientists to suggest the existence of black holes
  3. 3. History of the Transform  Euler began looking at integrals as solutions to differential equations in the mid 1700’s:  Lagrange took this a step further while working on probability density functions and looked at forms of the following equation:  Finally, in 1785, Laplace began using a transformation to solve equations of finite differences which eventually lead to the current transform
  4. 4. Definition  The Laplace transform is a linear operator that switched a function f(t) to F(s).  Specifically:  Go from time argument with real input to a complex angular frequency input which is complex.
  5. 5. Laplace Transform TheoryLaplace Transform Theory •General Theory •Example
  6. 6. THE SYSTEM FUNCTION We know that the output y(t) of a continuous-time LTI system equals the convolution of the input x ( t ) with the impulse response h(t); that is, y(t)=x(t)*h(t) For Laplace, Y(s)=X(s)H(s) H(s)=Y(s)/X(s)
  7. 7. Properties of ROC of Laplace Transform The range variation of for which the Laplaceσ transform converges is called region of convergence. ROC doesn’t contains any poles If x(t) is absolutely integral and it is of finite duration, then ROC is entire s-plane. If x(t) is a right sided sequence then ROC : Re{s} > σo. If x(t) is a left sided sequence then ROC : Re{s} < σo. If x(t) is a two sided sequence then ROC is the
  8. 8. Real-Life ApplicationsReal-Life Applications Semiconductor mobilitySemiconductor mobility Call completion inCall completion in wireless networkswireless networks Vehicle vibrations onVehicle vibrations on compressed railscompressed rails Behavior of magneticBehavior of magnetic and electric fieldsand electric fields above the atmosphereabove the atmosphere
  9. 9. Ex. Semiconductor MobilityEx. Semiconductor Mobility MotivationMotivation  semiconductors are commonly madesemiconductors are commonly made with super lattices having layers ofwith super lattices having layers of differing compositionsdiffering compositions  need to determine properties ofneed to determine properties of carriers in each layercarriers in each layer concentration of electrons and holesconcentration of electrons and holes mobility of electrons and holesmobility of electrons and holes  conductivity tensor can be related toconductivity tensor can be related to Laplace transform of electron and holeLaplace transform of electron and hole densitiesdensities