System analysis in the time domain involves ( finding y(t)):

1. Chapter 7 The Laplace Transform Consider the following RC circuit ( System) System analysis in the time domain involves ( finding y(t)): Solving the differential equation OR Using the convolution integral Both Techniques can results in tedious (ممل)mathematical operation

2. Differential Equation Fourier Transform provided an alternative approach Inverse Back

7. 7.1 DEFINITIONS OF LAPLACE TRANSFORMS The Laplace transform variable is complex This show that the bilateral Laplace transform of a signal can be interpreted as the Fourier transform of that signal multiplied by an exponential function

8. The inverse Laplace transform where This transform is usually called, simply, the Laplace transform ,the subscript b dropped Laplace transform pairs

9. Unilateral (single sided) Laplace transform pairs Forward transform Inverse transform Complex domain Because of the difficulty in evaluating the complex inversion integral Simpler procedure to find the inverse of Laplace Transform (i.e f(t) ) will be presented later Using Properties and table of known transform ( similar to Fourier)

10. Laplace Transform for an impulse d(t) Assuming the lower limit is 0- The value of s make no difference Region of Convergence We also can drive it from the shifting in time property that will be discussed later

11. For this to converge then

12. We derive the Laplace transform of the exponential function

14. A short table of Laplace transforms is constructed from Examples 7.1 and 7.2 and is given as Table 7.2

15. We know derive the Laplace Transform (LT) for a cosine function Since

17. Proof Let s*= s+a

18. Example Since Then Similarly

19. Real Shifting Let

22. It is sometimes necessary to construct complex waveforms from simpler waveforms

23. (See example 7.3) Since (Shift in time property)

24. Transform of Derivatives

25. Proof Integrating by parts,

30. Proof

32. find the Laplace transforms of the following integrals:

33. Proof (See the book) Since

36. 7.6 Response of LTI Systems Transfer Functions Consider the following circuit We want a relation (an equation) between the input x(t) and output y(t) KVL

37. Writing the differential equation as

38. Real coefficients, non negative which results from system components R, L, C

39. are real, non negative which results from system components R, L, C In general, Now if we take the Laplace Transform of both side (Assuming Zero initial Conditions) We now define the transfer function H(s) ,

40. are real, non negative The roots of the polynomials N(s) , D(s) are either real or occur in complex conjugate The roots of N(s) are referred to as the zero of H(s) ( H(s) = 0 ) The roots of D(s) are referred to as the pole of H(s) ( H(s) = ± ∞ )

41. The Degree of N(s) ( which is related to input) must be less than or Equal of D(s) ( which is related to output)for the system to be Bounded-input, bounded-output (BIBO) Example : Using polynomial division , we obtain Now assume the input x(t) = u(t) (bounded input) We see that for finite bounded Input (i.e x(t) =u(t) ) We get an infinite (unbounded) output

42. The poles of H(s) must have real parts which are negative The poles must lie in the left half of the s-plan

44. Proof (See the book) ( It is very similar to the Fourier Transform Property )

45. The loop equation (KVL) for this circuit is given by If v(t) = 12u(t) ,find i(t) ?

46. v(t) = 12u(t) Now using the Laplace Table 7.2 to find i(t) The inverse transform, from Table 7.2

47. Source Transformation

48. Source Transformation

49. Example Using Laplace Transform Method find i(t)

50. KVL