q2sols18085f04

# q2sols18085f04 - MIT OpenCourseWare http/ocw.mit.edu 18.085...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . -- 18.085 Quiz 2 November 3, 2004 Professor Strang Your name is: SOLUTIONS Grading 1. 2. 3. 1) (36 pts.) The 5 nodes in the network are at the corners of a square and the center. Node 5 is grounded so x 5 = 0. All 8 edges have conductances c = 1 so C = I . current source ⎡ ⎤ − 1 1 0 2 R 6 1 4 5 1 3 7 ? ? 2 5 R 8 3 4 f 1 = 3 z ⎢ ⎥ ⎢ ⎥ ⎢ − 1 0 1 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ − 1 0 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 − 1 1 0 ⎥ ⎢ ⎥ A 8 × 4 = ⎢ ⎥ ⎢ 0 − 1 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 − 1 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 − 1 0 ⎥ ⎣ ⎦ 0 0 − 1 (a) Fill in the 8 by 4 incidence matrix A (node 5 grounded). What is A T A ? Is A T A invertible ( YES ,NO)? ⎡ ⎤ 3 − 1 − 1 − 1 ⎢ ⎥ ⎢ ⎥ ⎢ − 1 3 − 1 0 ⎥ A T A = ⎢ ⎥ ⎢ ⎥ ⎢ − 1 − 1 4 − 1 ⎥ ⎣ ⎦ − 1 0 − 1 3 (b) How many independent solutions to A T y = 0? 4 . Write down one nonzero solution y . Ans. The upper left loop gives y = (1 , − 1 , , 1 , , , , 0) (c) The current source f 1 = 3 enters node 1 and exits at grounded node 5. In 2 by 2 block form (using A ), what are the 12 equations for the 8 currents y and the 4 potentials x ?...
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q2sols18085f04 - MIT OpenCourseWare http/ocw.mit.edu 18.085...

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