Inverse Laplace Transform 1 Contour Integration Mathematical Topic

  • Slides: 27
Download presentation
Inverse Laplace Transform 1

Inverse Laplace Transform 1

 • Contour Integration – Mathematical Topic by itself • For rational transforms, no

• Contour Integration – Mathematical Topic by itself • For rational transforms, no multiple poles, order denominator > numerator polynomial, use partial fraction technique – a simpler approach. 2

Example 9. 9 Inverse Laplace Transform via Partial Fraction Technique. 3

Example 9. 9 Inverse Laplace Transform via Partial Fraction Technique. 3

Example 9. 9 First Term, using result from example 9. 1 4

Example 9. 9 First Term, using result from example 9. 1 4

Example 9. 9 Second Term, using result from example 9. 9 5

Example 9. 9 Second Term, using result from example 9. 9 5

Example 9. 9 Combining Result for 1 st. & 2 nd. Term. 6

Example 9. 9 Combining Result for 1 st. & 2 nd. Term. 6

Example 9. 10 Using result from example 9. 2 -2 -1 7

Example 9. 10 Using result from example 9. 2 -2 -1 7

Example 9. 11 Using result from example 9. 1 & 9. 2 -2 -1

Example 9. 11 Using result from example 9. 1 & 9. 2 -2 -1 8

Properties of Laplace Transform • • • Linearity Time Shifting in the s-domain Time

Properties of Laplace Transform • • • Linearity Time Shifting in the s-domain Time Scaling Conjugation Convolution Differentiation in the time-domain Differentiation in the s-domain Integration in the time-domain The initial- and final-value Theorems. 9

LT Property associated with Linearity 10

LT Property associated with Linearity 10

LT Properties continued 11

LT Properties continued 11

LT Property associated with Convolution 12

LT Property associated with Convolution 12

LT Properties continued 13

LT Properties continued 13

Laplace Transform Pairs • Refer to Table 9. 2 Page 692 in OWN for

Laplace Transform Pairs • Refer to Table 9. 2 Page 692 in OWN for Laplace Transform of Elementary Functions. • Table being derived using examples in the book and associated properties of Laplace transform. • Important application of LT is in the analysis and characterization of LTI systems using the property associated with convolution in time domain is equivalent to multiplication in the s-domain. 14

x(t) h(t) X(s) H(s) y(t)=x(t)*h(t) Y(s)=X(s)H(s) Laplace Transform of the impulse response of a

x(t) h(t) X(s) H(s) y(t)=x(t)*h(t) Y(s)=X(s)H(s) Laplace Transform of the impulse response of a system h(t) is H(s) which is known as the Transfer Function of System or System Function. 15

Causality • For a causal LTI system, h(t)=0 for t<0 i. e. right sided

Causality • For a causal LTI system, h(t)=0 for t<0 i. e. right sided signal. • Therefore ROC associated with system function for a causal LTI system is a right-half plane. • For a system with a rational system function, causality of system is equivalent to ROC being the right-half plane to the right of the rightmost pole. • Examples 9. 17, 9. 18, 9. 19 illustrate causality. 16

Stability • An LTI system is stable if its impulse response is absolutely integrable

Stability • An LTI system is stable if its impulse response is absolutely integrable i. e. its Fourier Transform converges. • An LTI system is stable if and only if the ROC of its system function H(s) includes the entire jw-axis [i. e. Re{s}=0] • A causal system with rational system function H(s) is stable if and only if all of the poles of H(s) lie in the left-half of the s-plane. 17

Stability for causal systems- All poles must be on left half of s=jw axis.

Stability for causal systems- All poles must be on left half of s=jw axis. 18

Solving Differential Equation using Laplace Transform 19

Solving Differential Equation using Laplace Transform 19

20

20

21

21

22

22

23

23

24

24

25

25

26

26

27

27