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The French Newton Pierre-Simon Laplace

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. Real-Life Applications.

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The French Newton Pierre-Simon Laplace

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  1. The French NewtonPierre-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 FOW SS 2016

  2. Real-Life Applications Semiconductor mobility Call completion in wireless networks Vehicle vibrations on compressed rails Behavior of magnetic and electric fields above the atmosphere FOW SS 2016

  3. Introduction FOW SS 2016

  4. The Laplace Transform FOW SS 2016

  5. Limitations of FT and need of LT LT can be used to analyze large class of CT problems involving signals that are not absolutely integrable. So FT based problems can not be employed in this class of problems Using LT integrodifferential equations are transformed into simple algebraic equations and convolution in time domain becomes multiplication in s domain FT is evaluated on entire jω axis ,LT is evaluated on s plane (consideration in stability) The exponential factor (Complex Number) has the effect of forcing the signals to converge, hence LT can be applied to broader class of signals

  6. Region of Convergence (ROC) Properties of ROC: ROC Consists of strips parallel to jω axis. ROC does not contain any pole. If signal is time limited ROC is entire s plane. If signal is right sided , ROC is also right sided with σ> 0 If signal is left sided ROC is also left sided with σ< 0 A two sided signal has ROC given by a vertical strip of finite width in s plane.

  7. Relation between LT and FT LT = FT when σ = 0 FOW SS 2016

  8. LT of Periodic/aperiodic function FOW SS 2016

  9. LT of Periodic/aperiodic function FOW SS 2016

  10. LT of some standard signals FOW SS 2016

  11. Properties of LT FOW SS 2016

  12. LT evaluation using properties • The LT of a unit ramp function starting at t = a is ? • The LT of 3) The unilateral LT of x(t) = u(t) * tu(t) ? 4) The bilateral Laplace transform of the signal x(t) = 5) LT of x(t) = δ(at + b) ? FOW SS 2016

  13. LT evaluation using properties 6) LT of x(t) = • LT of x(t) = • LT of x(t) = • LT of x(t) = r(2t) FOW SS 2016

  14. Initial 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 Initial Value Theorem 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. FOW SS 2016

  15. Numerical Example: Given; Find f(0) FOW SS 2016

  16. 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 Final Value Theorem 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. FOW SS 2016

  17. Numerical Example: Final Value Theorem: Given: . Find FOW SS 2016

  18. Inverse Laplace Transform • L-1 s = ? • The inverse LT of X(s) = 3) The inverse LT of X(s) = FOW SS 2016

  19. Inverse Laplace Transformation Partial fraction Expansion. “Cover up Rule” A: B: FOW SS 2016

  20. Inverse Laplace Transformation A and B same as in previous problem. FOW SS 2016

  21. Stability and causality in s domain FOW SS 2016

  22. Stability and causality in s domain FOW SS 2016

  23. Stability and causality in s domain FOW SS 2016

  24. Laplace transforms are the primary tool used to solve DEs in control engineering. Applications of LT When initial conditions are zero: For non zero initial conditions FOW SS 2016

  25. Transfer Function • Definition • H(s) = Y(s) / X(s) • Relates the output of a linear system (or component) to its input • Describes how a linear system responds to an impulse • All linear operations allowed • Scaling, addition, multiplication H(s) X(s) Y(s) FOW SS 2016

  26. Example of Solution of an ODE • ODE w/initial conditions • Apply Laplace transform to each term • Solve for Y(s) • Apply partial fraction expansion • Apply inverse Laplace transform to each term FOW SS 2016

  27. Example R L v(t) C v(t) = R I(t) + 1/C I(t) dt + L di(t)/dt V(s) = [R I(s) + 1/(C s) I(s) + s L I(s)] Note: Ignore initial conditions FOW SS 2016

  28. Numerical on RLC circuit Step I: Apply KVL to entire circuit Step II : If initial conditions are not mentioned assume as zero. If initial conditions are given take them into account Step III : Find current i(t) from I(s) or asked parameter by taking Inverse LT FOW SS 2016

  29. Mail id : For future communication gngaikwad.scoe@sinhgad.edu ganesh1426@gmail.com FOW SS 2016

  30. Thank You FOW SS 2016

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