hw9soln - Purdue University, School of Electrical and...

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Purdue University, School of Electrical and Computer Engineering Prof. DeCarlo ECE 202 Linear Circuit Analysis II (Sp’09) Homework #9 Solution (33–36) Problem 33–34 (a) Find the Laplace transform of h ( t ) Given Impulse Response h ( t )[V ] h ( t )= Me at cos( ωt φ ) u ( t ) Laplace Transform Pair L{ M cos( φ ) u ( t ) } ± ²³ ´ f ( t ) = M s · cos( φ )+ ω · sin( φ ) s 2 + ω 2 Freq. Shift Property L µ e at · f ( t ) = F ( s + a ) Setting M cos( φ ) u ( t )to f ( t ), h ( t ) becomes h ( t e at · f ( t ) Using frequency shift property, H ( s L µ at cos( φ ) u ( t ) = M (cos φ ) s +( a (cos φ )+ ω (sin φ )) s 2 + ω 2 = ( M cos φ ) s a + ω tan φ ) ( s + a ) 2 + ω 2 (b) Determine M, ω ,and φ from the graph and datapoints 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -30 -20 -10 0 10 20 30 40 50 h(t) t in seconds datapoint 1 2 3 4 5 t 0 0.141187 0.2738 0.455347 0.587565 h ( t ) 41.0275 0 -20.3319 0 9.27 1/ 10
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Purdue University, School of Electrical and Computer Engineering Prof. DeCarlo From the ±rst four datapoints, we get the following equations: 41 . 0275 = M cos( φ )( e q 1 ) 0= Me at 2 cos( ωt 2 φ e q 2 ) 20 / 3319 = at 3 cos( 3 φ e q 3 ) at 4 cos( 4 φ e q 4 ) In (eq2) and (eq4) cosine terms should be zero to satisfy the condition. From (eq2) and (eq4) 2 φ = π 2 (eq2.1) 4 φ = 3 π 2 (eq4.1) Subtracting (eq4.1) from (eq2.1) and simplifying, ω = π ( t 4 t 2 ) = 10[rad/s] Now, to ±nd φ we use either (eq2.1) or (eq4.1) φ = 2 π 2 = 0 . 1589[rad/s] The values of M can be found using (eq1) and the value of a can be found using (eq3). M=41.5511 a=2.5 (c) Compute H ( s ) and construct pole zero plot Using part(a), H ( s )=41 . 0275 s +0 . 8972 ( s +2 . 5) 2 +10 2 z 1 = 0 . 8972 p 1 , 2 = 2 . 5 ± j 10 2/ 10
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Purdue University, School of Electrical and Computer Engineering Prof. DeCarlo -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 -10 -8 -6 -4 -2 0 2 4 6 8 10 Real Imaginary In the pole-zero plot above, all the poles and zeros lie in the left half complex plane.
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hw9soln - Purdue University, School of Electrical and...

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