hw5-key - EE 295 Homework 5 solutions 1 Readings 1 Chapters...

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EE 295 - Homework 5 solutions April 28, 2010 1 Readings 1. Chapters 8 and 9 of Bergen & Vittal 2 From the textbook (Bergen/Vittal) 2.1 Problem 8.2. To solve this problem we will want to follow the proceedure in example 8.1 (p.279). Ultimately we need to find the system transfer function: G g ( s ) = k v σ 1 + sσT 0 d 0 and then integrate this to find | V a | . To First we need to find K d and K q using Z = R + jX = 0 . 05 + j 0 . 8 and the values given in the book: K d = - ( X + X q ) ( r + R ) 2 + ( X + X 0 d )( X + X q ) = 1 . 5385 K q = ( r + R ) ( r + R ) 2 + ( X + X 0 d )( X + X q ) = 0 . 7692 Given these we can find | K | = ( K 2 d + K 2 q ) 1 / 2 = 1 . 7201 , | Z | = ( R 2 + X 2 ) 1 / 2 = 0 . 8016 , from which we can calculate k V and σ : k v = | K || Z | = 1 . 3788 σ = 1 1 - ( X d - X 0 d ) K d = - 4 . 3328 1
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Now we can calculate G g : G g ( s ) = | ˆ V a | ( s ) = k v σ 1 + sσT 0 d 0 = - 5 . 9741 1 - 17 . 3312 s = 0 . 3447 s - 0 . 0577 Now we can integrate to get | V a | ( t ) = 5 . 974( e 0 . 0577 t - 1) which is actually an unstable system. 2.2 Problem 8.4 To solve this one we need to find the closed-loop transfer function for the system. From my control systems text I know that for any feedback system: Y ( s ) X ( s ) = G ( s ) 1 + G ( s ) H ( s ) where G ( s ) and H ( s ) are the transfer functions for the system and the feedback, respectively. Since H ( s ) = 1 , the closed-loop transfer function is Y ( s ) X
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