1 / 31

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 ?.

Download Presentation

Laplace Transform and Modeling in the Frequency Domain

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 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

More Related