Laplace Transforms -

Laplace Transforms Very useful in the analysis and design of LTI systems. Operations of differentiation and integration are converted to algebraic equations of s. We can solve differential equations using Laplace transforms. Definition: For a time function f (t ) t 0, F(s) is: st F ( s ) f (t )e dt 0 Note that the time is integrated out. s has unit 1/Time s j For the transform to exist, the integral should be finite. It means that: lim f (t )e t 0 t

The inverse transform is given by : j 1 st f (t ) L F ( s ) F ( s ) e ds 2j j 1 This equation is rarely used. L-1 is usually determined using Partial Fraction Expansion (PFE). Example: f (t ) e at F ( s ) e at e st dt e ( s a )t dt 0

e ( s a )t sa 0 0 1 sa Step function : st Ae f (t ) A t 0 F ( s ) Ae st dt s 0 0 A s

A TABLE OF LAPLACE TRANSFORMS Properties Linearity : L k1 f1 (t ) k 2 f 2 (t ) k1 L f1 (t ) k 2 L f 2 (t ) Transform of a derivative: df (t ) L sF ( s ) f (0) dt Repeated application of the rule above : d n f (t ) n n 1 n 2 ( n 1) L s F ( s )

s f ( 0 ) s f ( 0 ) f ( 0) n dt Final Value Theorem Use the property of the derivative: st df (t ) L sF ( s ) f (0) dt

Let s 0 e 1 df dt f (t ) 0 lim f (t ) f (0) lim sF ( s ) f (0) t s 0 dt 0 lim f (t ) lim sF ( s ) t s 0 Very useful result States that the steady state behavior of f(t) is the same as the behavior of F(s) around s=0. Other Properties Integration : t F (s) L f ( )d

s 0 Frequency shift : Le at f (t ) F ( s a ) Time shift : L f (t T ) e Ts F (s)

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