Explanation Given data: The transfer function of a system is, H ( s ) = 2 s s + 6 (1) The input response of a system is, x ( t ) = 45 e − 3 t u ( t ) (2) Formula used: Write an expression to calculate the transfer function of a system. H ( s ) = Y ( s ) X ( s ) (3) Rearrange equation (3) to find Y ( s ) . Y ( s ) = H ( s ) X ( s ) (4) Calculation: Apply Laplace transform for equation (2) to find X ( s ) . X ( s ) = 45 s + 3 { ∵ L ( e − a t u ( t ) ) = 1 s + a } Substitute 2 s s + 6 for H ( s ) , and 45 s + 3 for X ( s ) in equation (4) to find Y ( s ) . Y ( s ) = ( 2 s s + 6 ) ( 45 s + 3 ) = 90 s ( s + 3 ) ( s + 6 ) Taking partial fraction for above equation. Y ( s ) = 90 s ( s + 3 ) ( s + 6 ) = A s + 3 + B s + 6 (5) The equation (5) can also written as follows: 90 s = A ( s + 6 ) + B ( s + 3 ) (6) Substitute − 6 for s in equation (6) to find B . 90 ( − 6 ) = A ( − 6 + 6 ) + B ( − 6 + 3 ) − 540 = A ( 0 ) + B ( − 3 ) − 3 B = − 540 Simplify the above equation to find B . B = − 540 ( − 3 ) = 180 Substitute − 3 for s in equation (6) to find A . 90 ( − 3 ) = A ( − 3 + 6 ) + B ( − 3 + 3 ) − 270 = A ( 3 ) + B ( 0 ) 3 A = − 270 Simplify the above equation to find A

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ISBN: 9780078028229

Author: Charles K Alexander, Matthew Sadiku

Publisher: McGraw-Hill Education

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Find the output y ( t ) and the impulse response h ( t ) .

Solution Summary: The author explains how to calculate the transfer function of a system.

Find the output y(t) and the impulse response h(t).

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