hw1 solu 355

# hw1 solu 355 - Solutions to Problems 2-7 Cross multiplying,...

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Unformatted text preview: Solutions to Problems 2-7 Cross multiplying, (s 6 +7s 5 +3s 4 +2s 3 +s 2 +3)C(s) = (s 5 +2s 4 +4s 3 +s 2 +3)R(s). Taking the inverse Laplace transform assuming zero initial conditions, d 6 c dt 6 + 7 d 5 c dt 5 + 3 d 4 c dt 4 + 2 d 3 c dt 3 + d 2 c dt 2 + 3 c = d 5 r dt 5 + 2 d 4 r dt 4 + 4 d 3 r dt 3 + d 2 r dt 2 + 3 r . 10. s 4 2 s 3 5 s 2 s C The transfer function is ( s ) R ( s ) = g14 g14 g14 g14 1 s 5 3 s 4 2 s 3 4 s 2 5 s g14 g14 g14 g14 g14 2 . Cross multiplying, (s 5 +3s 4 +2s 3 +4s 2 +5s+2)C(s) = (s 4 +2s 3 +5s 2 +s+1)R(s). Taking the inverse Laplace transform assuming zero initial conditions, d 5 c dc 5 + 3 d 4 c dt 4 + 2 d 3 c dt 3 + 4 d 2 c dt 2 + 5 dc dt + 2 c = dr d 4 r dt 4 + 2 d 3 r dt 3 + 5 d 2 r dt 2 + dt + r . Substituting r(t) = t 3 , d 5 c dc 5 + 3 d 4 c dt 4 + 2 d 3 c dt 3 + 4 d 2 c dt 2 + 5 dc dt + 2 c = 18 g71 (t) + (36 + 90 t + 9 t 2 + 3t 3 ) u( t ). 11. Taking the Laplace transform of the differential equation, s 2 X(s)-s+1+2sX(s)-2+3x(s)=R(s). Collecting terms, (s 2 +2s+3)X(s) = R(s)+s+1. R ( s ) s 2 s g14 1 s 2 Solving for X(s), X(s) = g14 2 s g14 3 + 2 s g14 g14 3 . The block diagram is then, 12. Program: 'Factored' Gzpk=zpk([-15 -26 -72],[0 -55 -56 roots([1 5 30])' roots([1 27 52])'],5) 'Polynomial' Gp=tf(Gzpk) Computer response: ans = Factored Zero/pole/gain: 5 (s+15) (s+26) (s+72) ---------------------------------------------------- s (s+55) (s+56) (s+24.91) (s+2.087) (s^2 + 5s + 30) Solutions to Problems 2-11 b. Thevenizing, 1 s V o ( s ) V i ( s ) Using voltage division, V o ( s ) g32 V i ( s ) 2 g32 1 2 s 2 g14 s g14 2 . Thus, 1 2 g14 s g14 1 s 17. a. Writing mesh equations (s+1)I 1 (s) – I 2 (s) = V i (s) -I 1 (s) + (s+2)I 2 (s) = 0 But, I 1 (s) = (s+2)I 2 (s). Substituting this in the first equation yields, (s+1)(s+2)I 2 (s) – I 2 (s) = V i (s) or I 2 (s)/V i (s) = 1/(s 2 + 3s + 1)...
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## This note was uploaded on 11/07/2011 for the course MEM 355 taught by Professor Ani during the Fall '08 term at UPenn.

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hw1 solu 355 - Solutions to Problems 2-7 Cross multiplying,...

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