Science:Math Exam Resources/Courses/MATH307/April 2012/Question 04 (e)
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Question 04 (e)
Consider a network arranged in the shape of an octahedron as in this diagram:
The nodes have been labeled with the large numbers and 3 of the edges (resistors) have been given orientations and labeled with small numbers. Assume all resistances are equal to 1. Let D be the incidence matrix (for some choice of labeling and arrows for the remaining edges) and let L be the Laplacian.
(e) It seems reasonable to conjecture that the effective resistance between nodes 1 and 6 would remain unchanged if removed the resistors between 2 and 3, 3 and 4, 4 and 5, 5 and 2. Explain how you could use MATLAB/Octave to test this conjecture. Assume that L has been defined in MATLAB/Octave.
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Science:Math Exam Resources/Courses/MATH307/April 2012/Question 04 (e)/Hint 1
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To find the effective resistance between two nodes, you would have to swap the 1st and sixth columns to the 1 and 2 positions inside the Laplacian matrix. You would then break up this Laplacian matrix into a block matrix: where will be a 2x2 matrix.
You would then find the voltage to current map (A.K.A Schur’s complement), matrix S whose equation is:
Then, after solving for S, the effective resistance would be the reciprocal of the 1st entry of matrix S.
%nodes n and m position n = 1; m = 6; %find the size of the matrix L lenL = length(L); %swap nodes n and m to the 1st and 2nd position in the Laplacian matrix swap = [n, m, 1:(n-1), (n+1):(m-1), (m+1):lenL]; L = L(swap, swap); %compute matrix S which is the voltage to current map (Schur's complement). A = L((1:2), (1:2)); B = L((3:lenL), (1:2)); C = L((3:lenL), (3:lenL)); S = A - (B’ * C(-1) * B); %finds the effective resistance which is the first entry of S. r = 1/S(1,1);
You could run this for the 2 different Laplacian matrices to find the effective resistance for each and compare them to confirm the conjecture is correct.