Midsol08 - MAE 143A Signals and Systems Prof M Krstic MIDTERM NAME SOLUTIONS • Closed book One sheet(both sides of handwritten notes allowed •

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Unformatted text preview: 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. 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 R 3 R 2 + R 3 bracketrightBig bracketleftBigg v 1 v 2 bracketrightBigg +...
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This note was uploaded on 04/30/2011 for the course MAE MAE 143b taught by Professor Callafon during the Winter '09 term at UCSD.

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Midsol08 - MAE 143A Signals and Systems Prof M Krstic MIDTERM NAME SOLUTIONS • Closed book One sheet(both sides of handwritten notes allowed •

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