solprac1

# solprac1 - ECE360 PRACTICE EXAM#1 SOLUTIONS FALL 2007...

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ECE360 PRACTICE EXAM #1 – SOLUTIONS FALL 2007 Instructions : 1. Closed-book, closed-notes, open-mind exam. You may only use the exam objectives and the two tables of Laplace transforms distributed in class. 2. Work each problem on the exam booklet in the space provided. 3. Write neatly and clearly for partial credit. Cross out any material you do not want graded. 4. Leave one empty seat between each of you. Name : Problem 1 : /25 Problem 2 : /25 Problem 3 : /25 Problem 4 : /25 Total : /100 Theorems : Initial-Value Theorem : y (0) = lim s →∞ sY ( s ) Final-Value Theorem : y ( ) = lim s 0 sY ( s ) Mason’s Loop Gain Rule : T ( s ) = P 1 Δ 1 + ... + P n Δ n Δ 1

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Problem 1 (25 Points) Consider a dynamical system governed by the following second-order diﬀerential equation: d 2 y dt 2 + 5 dy dt + 6 y ( t ) = x ( t ) where x ( t ) is an input forcing function. (a) Derive the transfer function T ( s ) = Y ( s ) X ( s ) Solution : s 2 Y ( s ) + 5 sY ( s ) + 6 Y ( s ) = X ( s ) ( s 2 + 5 s + 6) Y ( s ) = X ( s ) T ( s ) = Y ( s ) X ( s ) = 1 s 2 + 5 s + 6 = 1 ( s + 2)( s + 3) (b) Find the steady-state output y ss = y ( ) for an impulse input x ( t ) = δ ( t ) = X ( s ) = 1. Y ( s ) = T ( s ) X ( s ) = 1 s 2 + 5 s + 6 × 1 = 1 s 2 + 5 s + 6 y ss = y ( ) = lim s 0 sY ( s ) = lim s 0 sT ( s ) X ( s ) = lim s 0 s × 1 s 2 + 5 s
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solprac1 - ECE360 PRACTICE EXAM#1 SOLUTIONS FALL 2007...

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