Explanation Given data: The input to a system is unit step function at t = 0 . x ( t ) = u ( t ) (1) The output response of the system is, y ( t ) = [ 6 + 0.75 e − 3 t − e − 2 t ( 3 cos ( 4 t ) + 4.5 sin ( 4 t ) ) ] u ( t ) . (2) Formula used: Write a general expression to calculate the transfer function of the system. H ( s ) = Y ( s ) X ( s ) (3) Here, Y ( s ) is the output response of the system in s − domain , and X ( s ) is the input excitation of the system in s − domain . Calculation: Apply Laplace transform for equation (1) to find X ( s ) . X ( s ) = 1 s { ∵ ℒ ( u ( t ) ) = 1 s } The equation (2) can also be written as, y ( t ) = [ 6 u ( t ) + 0.75 e − 3 t u ( t ) − 3 e − 2 t cos ( 4 t ) u ( t ) + 4.5 e − 2 t sin ( 4 t ) ] (4) Apply Laplace transform for equation (4) to find Y ( s ) . Y ( s ) = [ 6 ( 1 s ) + 0.75 ( 1 s + 3 ) − 3 ( s + 2 ( s + 2 ) 2 + 4 2 ) + 4.5 ( 4 ( s + 2 ) 2 + 4 2 ) ] { ∵ ℒ ( e − a t cos ( ω 0 t ) u ( t ) ) = s + a ( s + a ) 2 + ω 0 2 , ℒ ( u ( t ) ) = 1 s ℒ ( e − a t sin ( ω 0 t ) u ( t ) ) = ω 0 ( s + a ) 2 + ω 0 2 , ℒ ( e − a t u ( t ) ) = 1 s + a } Simplify the above equation to find Y ( s ) . Y ( s ) = [ 6 s + 0.75 s + 3 − 3 ( s + 2 ) ( s + 2 ) 2 + 4 2 + 18 ( s + 2 ) 2 + 4 2 ] = [ 6 s + 1

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ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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