# Laplace Transforms - boun.edu.tr

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)

## Recently Viewed Presentations

• TopPage. When's your Birthday? How old are you? Who do you live with? Do you have a pet? Where do you live? About you. What's your name? Tell me about your day?
• Chapter 7 - Data Link Control Protocols Eighth Edition by William Stallings Lecture slides by Lawrie Brown Data Link Control Protocols "Great and enlightened one," said Ten-teh, as soon as his stupor was lifted, "has this person delivered his message...
• Important ideas from this video include:Image sharing services are a universal and powerful way of communicating all over the world.Digital images are just data (lots of data) composed of layers of abstraction: pixels, RGB, binary.The RGB color scheme is composed...
• Genesis has a generalized search technique which can be extended to support new policies. The problem is still hard, but Genesis is not tailor-made for any specific type of policy. <N> In the vein of program synthesis approaches, Genesis uses...
• Asymptotic complexity focuses on behavior for large n and is independent of any computer / coding trick. But you can "abuse" it to be misled about trade-offs. Example: n. 1/10 vs. log . n. Asymptotically . n. 1/10 grows more...
• The stress response evolved as a survival mechanism in humans, and is most commonly referred to as the fight or flight response. In healthy adults and children this response can be activated immediately when threat is perceived and "turned off"...
• Acids taste sour, will change the color of an acid-base indicator, and can be strong or weak electrolytes in aqueous solution. Citrus fruits contain citric acid. Tea contains tannic acid.
• Short Term Goals (w/m) Hopefully at the end of 2007 you and your owner/manager had the opportunity to build or layout a business plan for 2008.