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Laplace Transform and Modeling in the Frequency Domain

Laplace Transform and Modeling in the Frequency Domain. The Laplace transform of a function f(t) is defined as: The inverse Laplace transform is defined as: where and the value of  is determined by the singularities of F(s) . And. Why Is Laplace Transform Useful ?.

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Laplace Transform and Modeling in the Frequency Domain

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  1. Laplace TransformandModeling in the Frequency Domain

  2. The Laplace transform of a function f(t) is defined as: • The inverse Laplace transform is defined as: where and the value of  is determined by the singularities of F(s). And

  3. Why Is Laplace Transform Useful ? • Model a linear time-invariant analog system as a transfer function. • In control theory, Laplace transform converts linear differential equations into algebraic equations. • This is much easier to manipulate and analyze.

  4. An Example • The Laplace transform of can be obtained by: Linearity property • These are useful properties:

  5. table_02_02 table_02_02

  6. Find the Laplace transform of f(t)=5u(t)+3e -2t. • Solution:

  7. Partial Fraction Expansion(Case 1: Roots of Denominator are Real and distinct)Find the inverse Laplace transform ofF(s)=5/(s2+3s+2). Solution:

  8. Exercise: Do example 2.3 of the textbook Laplace Transform solution of a differential equation

  9. Case 2: Roots of the Denominator are Real and Repeated

  10. Case 2: continue of the example F(s)=(2s+3)/(s3+2s2+s).

  11. Case 3: Roots of the Denominator are Complex.Example:F(s)=10/(s3+4s2+9s+10)

  12. The Transfer Function

  13. Example of the Transfer Function

  14. System Response from the Transfer Functions

  15. Electrical Network Transfer Functions • Resistance circuit: • Inductance circuit: • Capacitance circuit:

  16. Summary for Electrical Networks

  17. Electrical network (Example 2.6) Original network Transfer Function network Laplace Transform network

  18. Exercise: Do example 2.10 of the textbook • Excluding Operational Amplifiers

  19. fig_02_06 fig_02_06

  20. Transfer Functions for Mechanical Systems

  21. Exercise: Do example 2.17 of the text

  22. fig_02_17 fig_02_17

  23. fig_02_18 fig_02_18

  24. fig_02_19 fig_02_19

  25. Mechanical Rotational Systems • Moment of inertia: • Viscous friction: • Torsion:

  26. Model of a torsional pendulum (pendulum in clocks inside glass dome): Moment of inertia of pendulum bob denoted by J Friction between the bob and air by B Elastance of the brass suspension strip by K

  27. Transfer Functions for Systems with Gears

  28. fig_02_29 fig_02_29

  29. Electro-mechanical System Transfer Functions

  30. Equivalence between Mechanical and Electrical Systems

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