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Laplace Transform

Laplace Transform. ABE425 Engineering Measurement Systems. Agenda. Why do we need a Laplace Transform? Definition Laplace Transform Laplace Transform of common functions Unit step function Ramp function Exponential function Cosine/Sine Impulse function (dirac delta)

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Laplace Transform

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  1. Laplace Transform ABE425 Engineering Measurement Systems

  2. Agenda • Why do we need a Laplace Transform? • Definition Laplace Transform • Laplace Transform of common functions • Unit step function • Ramp function • Exponential function • Cosine/Sine • Impulse function (dirac delta) • Laplace Transform of Differentiation/Integration • Application: Step response of an Electric Dynamic system

  3. What is the Laplace Transform used for? Time domain s-domain (Laplace Domain) • The Laplace Transform turns a differential equation in the time domain into an algebraic equation in the s domain • By inverse transformation we obtain the solution in the time domain • Allows us to write control schemes, such as PID control, in algebraic equations (function of s) • Definition

  4. Example of an electric dynamic system i output input How do we find out what happens to the output if we kick it (give it a known input)? Laplace transform!!

  5. Laplace transform of unit step function Definition Laplace Transform

  6. Laplace transform of exponential function

  7. Laplace transform of operations

  8. Laplace transform of differentiation operation Remember: Product rule Integration by parts

  9. Laplace transform is a linear operation

  10. Example of an electric dynamic system i Capacitor voltage: Kirchhoff Voltage Law: In input-output notation <- differential equation

  11. Example of an electric dynamic system cont. Use Laplace Transform to study this dynamic system for instance the response to a step input

  12. The unit of RC is second! • Ohm’s Law • Capacitor equation • Current is charge per second • Unit of RC = second

  13. Translate the dynamic system to Laplace domain using unit step function as input

  14. How do we transform this back to time domain? Partial Fraction Expansion

  15. How do we transform this back to time domain? Output for = 20 s

  16. Laplace transform of other functions

  17. Laplace transform of ramp function Integration by parts

  18. Laplace transform of cosine function

  19. Laplace transform of sine function

  20. Laplace transform of impulse (Dirac d distribution) Writing as McLaurin series Writing as McLaurin series

  21. Laplace transform of impulse (Dirac d distribution)

  22. Laplace Transform of a function shifted in time

  23. Check Laplace Transform of differentiation operation Example Is this correct?

  24. Laplace Transforms of common functions

  25. The End

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