This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ECE 220, Section 051 Problem Lab Number 7 Solution Chapter 8, Laplace Transforms Part I. Fundamental Properties of Laplace Transforms 1. Use the basic definition of the Laplace transform to evaluate: f ( t ) = e a ( t t 1 ) u ( t t 1 ) Check using Laplace transform properties. Solution:(Using Basic Definition) F ( s ) = f ( t ) e st dt = e a ( t t 1 ) u ( t t 1 ) e st dt F ( s ) = e at 1 t 1 e t ( a + s ) dt = e at 1 ( s + a ) e t ( s + a ) t 1 F ( s ) = e at 1 ( s + a ) e t 1 ( s + a ) = e at 1 st 1 at 1 s + a = e st 1 s + a Solution:(Using Laplace Transform Properties) f ( t ) = e a ( t t 1 ) u ( t t 1 ) = g ( t t 1 ) Now, g ( t ) = e at u ( t ) So, G ( s ) = 1 s + a As f ( t ) = g ( t t 1 ), F ( s ) = e st 1 s + a 2. Use the basic definition of the Laplace transform to show that: L [ f ( t a ) u ( t a )] = F ( s ) e as where F ( s ) = L [ f ( t )] Check using Laplace transform properties. Solution:(Using Basic Definition) H ( s ) = h ( t ) e st dt = f ( t a ) u ( t a ) e st dt H ( s ) = a f ( t a ) e st e at e at dt = a f ( t a ) e st e as e as dt H ( s ) = e as a f ( t a ) e s ( t a ) dt = e as f ( t ) e s ( t ) dt = e as F ( s ) 1 Solution:(Using Laplace Transform Properties) h ( t ) = f ( t a ) u ( t a ) = g ( t a ) Now, g ( t ) = f ( t ) u ( t ) So, H ( s ) = e as G ( s ) H ( s ) = e as F ( s ) 3. Use the basic definition of the Laplace transform to show that: L [ e at f ( t )] = F ( s + a ) where F ( s ) = L [ f ( t )] Solution:(Using Basic Definition) H ( s ) = h ( t ) e st dt = e at f ( t ) e st dt H ( s ) = a f ( t ) e t ( s + a ) dt = f ( t ) e t ( s + a ) dt H ( s ) = F ( s + a ) 4. Use the results from the above problems to evaluate the Laplace transform of: (a) e at cos( t ) Solution: s a ( s a ) 2 + 2 (b) e at sin( t ) Solution: ( s a ) 2 + 2 using the known transform pairs...
View
Full
Document
 Summer '08
 NILSON

Click to edit the document details