Chapter 2 - EEL4657 Dr Haniph Latchman Chapter 2 Solutions 1 L h d 2 y dt 2 i-→ s 2 Y s 2 Derive the Laplace Transform pairs from Table 2.1 page

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Unformatted text preview: EEL4657 - Dr. Haniph Latchman Chapter 2 Solutions 1. L h d 2 y dt 2 i-→ s 2 Y ( s ) 2. Derive the Laplace Transform pairs from Table 2.1, page 33 of the text **Note that the Laplace Equation is given as: X ( s ) = R ∞- x ( t ) e- st dt (a) x ( t ) = δ ( t ) X ( s ) = Z ∞- δ ( t ) e- st (since R- δ ( t ) dt = 1) X ( s ) = Z- δ ( t ) e- st dt X ( s ) = 1 · e- s · = 1 L [ δ ( t )] = 1 (b) x ( t ) = u ( t ) X ( s ) = Z ∞-∞ u ( t ) e- st dt X ( s ) = Z ∞ 1 · e- st dt X ( s ) = Z ∞-∞ u ( t ) e- st dt X ( s ) =- 1 s e- st ∞ X ( s ) = 1 s e- s ( ∞ ) + 1 s e- s (0) L [ u ( t )] = 1 s 1 (c) x ( t ) = e- at u ( t ) X ( s ) = Z ∞ e- at · e- st dt X ( s ) = Z ∞ e- t ( s + a ) dt X ( s ) = 1- ( s + a ) e- t ( s + a ) ∞ X ( s ) = 1- ( s + a ) e-∞ ( s + a ) + 1- ( s + a ) e 0( s + a ) X ( s ) = 1 s + a L [ e- at u ( t )] = 1 s + a (d) x ( t ) = cos ( ωt ) u ( t ) X ( s ) = Z ∞ 1 2 e jωt e- st dt + Z ∞ 1 2 e- jωt e- st dt X ( s ) = 1 2 Z ∞ e- t ( s- jω ) dt + 1 2 Z ∞ e- t ( s + jω ) dt X ( s ) =- e- t ( s- jω )- 2( s- jω )- e t ( s + jω ) 2( s + jω ) ∞ X ( s ) = 1- 2( s- jω ) + 1 2( s + jω ) X ( s ) = s + jω + s- jω 2( s- jω )( s + jω ) X ( s ) = s s 2 + ω 2 L [ cos ( ωt ) u ( t )] = s s 2 + ω 2 2 (e) x ( t...
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This note was uploaded on 02/01/2012 for the course EEL 4657 taught by Professor Latchman during the Fall '10 term at University of Florida.

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Chapter 2 - EEL4657 Dr Haniph Latchman Chapter 2 Solutions 1 L h d 2 y dt 2 i-→ s 2 Y s 2 Derive the Laplace Transform pairs from Table 2.1 page

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