practice_problem_set3_solution

practice_problem_set3_solution - EEL3105 Fall2011...

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EEL 3105 Fall 2011 Practice Problem Set 3 Solution 1. Find Laplace transform of () 7 ( 4 ) 4 ( 2 ) 3s in (6 /4 ) ft Ut t t  Here U(t) is the unit step function, (t) is the unit impulse function. Answer: We use properties of Laplace transforms and linearity to calculate F(s). 4 2 22 2 4 2 2 (7 ( 4)) 7 ; (4 ( 2)) 4 ; sin(6 ) cos(6 ) 18 3 3sin(6 ) 3 3 4 2 2 2( 36) 36) 36 ( 36) 2 () 7 4 . ( 36) 2 s s s s e LT U t s LT t e tt s LT t LT ss s s es Fs e        2. Find inverse Laplace transform for the following functions: a. F(s) = 2 2 (2 9 ) s s b. F(s) = 2 2 2 9 ) s s e s Answer: Let’s start with part (a). We will break it into a partial fraction expansion of the form:
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22 2 0 2 2 2 ( 2 9) ( 2 9) 2 .2 / 9 . (2 9 ) 9 ) 9 ) 9 25 9 1 2 5 9 ) 9 9 9 ) s sA B s C ss s s s s s As s s Bs C s s s s s s s s s            All we need to do is to figure out the inverse LT of the second term: 2 2 (1 ) 7 9 )(1 )( 8 ) 5 ) 8 ) 2( 1) 7 8 8 ) 8 8 ) s sss s    We can now read off the inverse LT using the simple rules of LT: 2 1 2 ( 1 ) 1 7 8 9 ) 9 9 9 8 ) 8 8 ) 7 ( ) cos( 8 ) sin( 8 ), 0. 99 98 tt invLT invLT s s Ut e t e t t   Let’s now turn to part (b). We note that it is the same as part (a) except for the e 2s term multiplying it. Therefore the inverse LT will be the same as in part (a) but with a delay of 2 sec. Thus, the inverse LT of part b is ) ) 7 ( 2) cos( 8( 2)) sin( 8 ( 2)), 2, 0, 2 0 e t e t fo r t for t  3. Consider the differential equation 2 2 51 0 38 d y dy du yu dt dt dt Find the transfer function H(s)= () . Ys Us
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Answer: In finding the transfer function, we set initial conditions to zero. Taking LT of both sides, we get 2 2 () 5 () 10 () 3 () 8 () () 3 8 . 5 10 sY s Y s sU s U s Ys s Hs Us s s    4. Consider the differential equation 2 2 51 0 38 d y dy du yu dt dt dt  Find A, B, C for a state space representation of the form . dx Ax t Bu t dt yt Cxt  Answer: This is not so easy. Let us start with a simpler problem whose solution we know. Consider the system 2 2 0 .
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This note was uploaded on 12/27/2011 for the course EEL 3105 taught by Professor Boykins during the Fall '10 term at University of Florida.

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practice_problem_set3_solution - EEL3105 Fall2011...

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