Engineering Analysis المرحلة الثالثة Assit.Lec. Shaimaa Shukri

1. Engineering Analysis المرحلة الثالثة Assit.Lec. Shaimaa Shukri

2. First lecture

3. Laplace Transform Concepts and Applications

4. Laplace Transform Introduction

5. Laplace Transform How to find F(s) from f(t)?

6. Example If b>0, F(s) exists for Re{s}=0 where it reduces to F() Otherwise the Laplace transform F(s) doesn’t contain the Fourier transform F() as a subset

7. Existence for Laplace Transforms Let f(t) be a piecewise continuous on every finite interval on t  0 For some real constants k, p,T The Laplace transform of f(t) exists for all Re{s}>p

8. Since f(t) is piecewise continuous e-stf(t) is an integral over any finite interval on the t-axis

9. Example

10. Analyticits of the Laplace Transform Suppose f is piecewise continuous on [0,) |f(t)|<kept for tT where k, p and T are real The Laplace transform of fis analytic function in the right half-plane of Re{s}>p

11. Region of Convergence The range of values for the complex variable s for which F(s) converges In Laplace transform applications, the complex plane is referred as the s plane

12. Example

13. Linearity Modulation Properties of Laplace Transform

14. Modulation F(s) converges for Re{s}>k The modulation of the signal becomes Re{s}-Re{s0}>k for existence

15. Example

16. Example

17. Example Gamma Function relation Let st=x

18. Differentiation

20. Example

22. Example Find L{(t)} from u(t)