LaplaceTransform

# LaplaceTransform - Laplace Transform Linear Systems and...

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Laplace Transform Linear Systems and Signals Lectures 9 Dr. J. K. Aggarwal The University of Texas at Austin

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Continuous-time Systems/Circuits Example: + - x ( t ) R C y ( t ) () ( ) ( ) ( ) 1 1 ( ) ( ) dy t RC y t x t dt dy t y t x t dt RC RC Differential/Integral Equation + -
Continuous-time Systems/Circuits Time Domain Analysis (Discussed in the last chapter) Zero-input Response Zero-state Response Impulse Response Convolution

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Continuous-time Systems ...contd Frequency Domain Analysis (This chapter) The Laplace Transform Using Laplace Transform to solve problems Initial Value Theorem Final Value Theorem Transfer Function Relationship between Impulse response and Transfer Function
Laplace Transform Gives us a systematic way for relating time domain behavior of a circuit to its frequency domain behavior Converts integro-differential equations describing a circuit to a set of algebraic equations Considers transient behavior of circuits with multiple nodes and meshes, with initial conditions

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Using Laplace Transform The idea of using a transform is similar to using logarithms for multiplication A = BC Log A = Log B + Log C A = Antilog (Log B + Log C) The use of Laplace Transform is universal mechanical systems, electrical circuits and other systems as long as the behavior is described by ordinary differential equations.
Laplace Transform The general definition of Laplace Transform i.e. the Two-sided Transform: And Inverse Transform: ( ) ( ) st X s x t e dt 1 ( ) ( ) 2 cj st x t X s e ds j

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Laplace Transform But we shall consider a special case One-Sided Transform : We shall use the Table 4.1 (Lathi, page 344) to do inverse transform 0 ( ) ( ) st X s x t e dt
Laplace Transform: Definition t represents time s has the units of 1/t since the exponent must be dimensionless This is a one-sided, unilateral Laplace Transform i.e. it ignores negative values of t. Again, the definition of Laplace Transform is: 0 ( ) ( ) ( ) ( ) st L x t x t e dt X s L x t 0 ( ) ( ) ( ) st X s L x t x t e dt

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Properties of Laplace Transform Linearity of Laplace Transform: since Uniqueness of Laplace Transform: is a one-to-one relationship 1 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) L k x t k X s L x t x t X s X s 12 0 00 ( ) ( ) ( ) ( ) ( ) ( ) st st st x t x t e dt x t e dt x t e dt X s X s ( ) ( ) x t X s
Properties ...contd Question: Does the integral converge? We avoid functions like The integral converges for all the cases we shall consider! Question: What happens if the function is discontinuous at t=0 ? We choose the lower limit as 0-. In fact, we shall define the Laplace Transform as: 2 ,, tt t e etc 0 () st x t e dt

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Properties ...contd In taking Laplace Transform, it becomes important whether one chooses 0 - or 0 + Our choice is always 0 - for the lower limit! 0 ,0 at et 0, 0 t x (t) t 0 at e x (t) t Continuous at t=0 Discontinuous at t=0
Laplace Transform of the Unit Step 0 00 ( ) ( ) 1.

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## This note was uploaded on 02/08/2011 for the course EE 313 taught by Professor Cardwell during the Spring '07 term at University of Texas.

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LaplaceTransform - Laplace Transform Linear Systems and...

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