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midsol08 - MAE 143A Signals and Systems Prof M Krstic...

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MAE 143A Signals and Systems Prof. M. Krstic MIDTERM July 15, 2008 NAME: SOLUTIONS Closed book. One sheet (both sides) of handwritten notes allowed. Present your reasoning and calculations clearly. Random or inconsistent etchings will not be graded. Write only on the paper provided. If you run out of space for a given problem, continue on the pages at the end of the set and indicate “Continued on page X.” The problems are not ordered by difficulty. Total points: 45. Time: 2 hours.
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Problem 1. (10 points) Find the state space representation for the following circuit: C 1 R 1 R 2 C 2 R 3 u y + + + + Solution: Denote the voltages on the capacitors by v 1 and v 2 . We have C 1 ˙ v 1 u v 1 v 2 R 1 = 0 C 2 ˙ v 2 u v 1 v 2 R 1 u v 2 R 2 + R 3 = 0 v 2 y R 2 + u y R 3 = 0 Solving for y , we obtain C 1 ˙ v 1 = 1 R 1 v 1 1 R 1 v 2 + 1 R 1 u C 2 ˙ v 2 = 1 R 1 v 1 parenleftbigg 1 R 1 + 1 R 2 + R 3 parenrightbigg ( v 2 u ) y = R 3 R 2 + R 3 v 2 + R 2 R 2 + R 3 u bracketleftBigg ˙ v 1 ˙ v 2 bracketrightBigg = 1 C 1 R 1 1 C 1 R 1 1 C 2 R 1 parenleftBig 1 C 2 R 1 + 1 C 2 ( R 2 + R 3 ) parenrightBig bracketleftBigg v 1 v 2 bracketrightBigg + 1 C 1 R 1 parenleftBig 1 C 2 R 1 + 1 C 2 ( R 2 + R 3 ) parenrightBig u y = bracketleftBig 0 R 3 R 2 + R 3 bracketrightBig bracketleftBigg v 1 v 2 bracketrightBigg + R 2 R 2 + R 3 u 2
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Problem 2. (10 points)
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