lab07_sol_su09

# lab07_sol_su09 - ECE 220 Section 051 Problem Lab Number 7...

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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( ωt ) Solution: s- a ( s- a ) 2 + ω 2 (b) e- at sin( ωt ) Solution: ω ( s- a ) 2 + ω 2 using the known transform pairs...
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lab07_sol_su09 - ECE 220 Section 051 Problem Lab Number 7...

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