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3015_Discussion_12_Solutions

3015_Discussion_12_Solutions - University of Minnesota Dept...

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University of Minnesota Dept. of Electrical and Computer Engineering EE3015 – Signals and Systems Discussion Session #12: Laplace Transform 1. The following differential equation represents the dynamics of an LTI system: ). ( 3 ) ( 2 ) ( ) ( 2 2 t x t y dt t dy dt t y d = - + Where ) ( 3 ) ( 2 t u e t x t - = is the input and ) ( t y is the output, with initial conditions: 1 ) 0 ( = - y and 1 ) ( 0 = - = t dt t dy Find the natural and forced responses. The Unilateral Laplace Transform of the equation above is given by: ) ( 3 ) ( 2 ) 0 ( ) ( ) ( ) 0 ( ) ( 0 2 s X s Y y s sY dt t dy sy s Y s t = - - + - - - = - - We group terms that multiply the input ) ( s X and the initial conditions: ( ) ) 0 ( ) ( ) 0 ( ) ( 3 2 ) ( 0 2 - = - + + + = - + - y dt t dy sy s X s s s Y t 2 ) 0 ( ) ( ) 0 ( 2 ) ( 3 ) ( 2 0 2 - + + + + - + = - = - - s s y dt t dy sy s s s X s Y t Forced resp. Natural resp. ) ( s Y f ) ( s Y n The Laplace transform of ) ( s X is given by ) 2 ( 3 ) ( + = s s X . The forced response is given by 2 2 ) 2 ( 2 1 ) 2 )( 1 ( 9 ) ( + + + + - = + - = s C s B s A s s s Y f
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The residues A, B, C may be determined by placing the terms in the expansion over a common denominator: ) 1 ( ) 2 ( ) 1 ( ) 1 ( ) 2 ( ) 1 )( 2 ( ) 1 ( ) 2 ( ) 2 ( ) 2 ( 2 1 2 2 2 2 2 - + - + - + - + + - + + = + + + + - s s s C s s s s B s s s A s C s B s A Equating the coefficient of each power of s in the numerator on the right hand side of this equation, we obtain a system of three equations in the three unknowns A, B, C: C B A C B A B A - - + + + = = = 2 4 4 9 0 0 Solving these equations, we obtain: A
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