KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Simulation)-Chapter_3

# KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Simulation)-Chapter_3

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1 Chapter 3 Laplace Transforms Important analytical method for solving linear ordinary differential equations. - Application to nonlinear ODEs? Must linearize first. Laplace transforms play a key role in important process control concepts and techniques. - Examples: Transfer functions Frequency response Control system design Stability analysis

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2 Chapter 3 [ ] ( 29 0 ( ) ( ) (3-1) st F s f t f t e dt i - = = L Definition The Laplace transform of a function, f ( t ), is defined as where F ( s ) is the symbol for the Laplace transform, L is the Laplace transform operator, and f ( t ) is some function of time, t . Note : The L operator transforms a time domain function f ( t ) into an s domain function, F ( s ). s is a complex variable : s = a + bj , 1 j = -
3 Chapter 3 Inverse Laplace Transform, L By definition, the inverse Laplace transform operator, L -1 , converts an s -domain function back to the corresponding time domain function: ( 29 ( 29 1 f t F s - = L Important Properties: Both L and L -1 are linear operators . Thus, ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 (3-3) ax t by t a x t b y t aX s bY s + = + = + L L L

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4 Chapter 3 where: - x ( t ) and y ( t ) are arbitrary functions - a and b are constants - ( 29 ( 29 ( 29 ( 29 X s x t Y s y t = L = L and Similarly, ( 29 ( 29 ( 29 ( 29 1 aX s bY s ax t by t - + = + L
5 Chapter 3 Laplace Transforms of Common Functions 1. Constant Function Let f ( t ) = a (a constant). Then from the definition of the Laplace transform in (3-1), ( 29 0 0 0 (3-4) st st a a a a ae dt e s s s i i - - = = - = - - = L

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6 Chapter 3 1. Step Function The unit step function is widely used in the analysis of process control problems. It is defined as:
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KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Simulation)-Chapter_3

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