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EE2010_Gp2_Chap2_1011s2 - Laplace Transform Motivation...

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Laplace Transform Motivation Definition Rules Inverse LT Example What is Laplace Transform ? Mathematical tool used for solving linear ordinary differential equation. Difficult Problem Solution of Difficult Problem Easy Problem Solution of Easy Problem Solve easy problem Laplace Transform Inverse Laplace Transform Problem is simplified by converting a differential equation into an algebraic equation. Differentiation Multiplication Integration Division I One of the important applications of Laplace Transform is in the analysis and characterisation of LTI systems I Laplace Transform enables us to analyse system behaviour without having to solve differential equations. I Transform from time-domain to complex frequency domain ( s -domain). I Express differential equations as transfer functions I Describe time domain response using poles and zeros EE2010 Systems & Control Notes WW Tan Chapter 2 Page 1
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Laplace Transform Motivation Definition Rules Inverse LT Example Laplace Transform : Definition Laplace transform belongs to a class of integral transform defined by T { f ( t ) } = Z -∞ K ( s , t ) f ( t ) d t K ( s , t ) is called the kernel of the transform. For Laplace transform, K ( s , t ) = 0 for t < 0 - e - st for t 0 L{ f ( t ) } = Z 0 - e - st f ( t ) dt Lower limit of integration is 0 - , instead of -∞ , because the system is assumed to be Linear Time Invariant (LTI) i.e. input signals are applied at t = 0. Consequently, any changes in the signal when t 0 - are ignored. EE2010 Systems & Control Notes WW Tan Chapter 2 Page 2
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Laplace Transform Motivation Definition Rules Inverse LT Example By convention, we shall write Laplace : L{ f ( t ) } = F ( s ) Inverse Laplace : L - 1 { F ( s ) } = f ( t ) Useful Laplace Transform pairs : f ( t ) F ( s ) δ ( t ) 1 U ( t ) 1 s tU ( t ) 1 s 2 e - at U ( t ) 1 s + a [sin at ] U ( t ) a s 2 + a 2 [cos at ] U ( t ) s s 2 + a 2 f ( t ) F ( s ) [ e - at sin bt ] U ( t ) b ( s + a ) 2 + b 2 [ e - at cos bt ] U ( t ) s + a ( s + a ) 2 + b 2 te - at U ( t ) 1 ( s + a ) 2 EE2010 Systems & Control Notes WW Tan Chapter 2 Page 3
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Laplace Transform Motivation Definition Rules Inverse LT Example Laplace Transform Rules I Linearity : L{ α f ( t ) + β g ( t ) } = α L{ f ( t ) } + β L{ g ( t ) } where α, β = constants I Transform of Derivatives : L{ f 0 ( t ) } = sF ( s ) - f (0 - )
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