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Is it possible to know all voltages/amps in a grid or matrix of resistors?
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Beamin:
 
--- Quote from: rstofer on June 06, 2018, 04:40:10 pm ---Here is the solution for a finite grid and it is easily solved with MATLAB among other tools:
http://www.wiete.com.au/conferences/2wctee/papers/15-13-Zegarmistrz-P.pdf

The infinite grid is a little more difficult and hand-wavy because spooky stuff happens out at infinity.
http://www.mathpages.com/home/kmath668/kmath668.htm

Nothing new here, the finite grid is discussed in just about every EE101 class.

Sometimes the grid is wrapped around, like a sphere or just a cube
http://physicstasks.eu/285/the-resistance-of-a-wire-cube
http://www.rfcafe.com/miscellany/factoids/kirts-cogitations-256.htm

Really, these are just exercises in finding symmetry and creating the matrix.

--- End quote ---


The math goes over my head in those two papers. Need simpler examples With the infinite grid wouldn't the edges not matter at some point because the current would be so small and the resistance so high that effectively there would be no or unmeasurable (say your equipment goes down to 0.001 uV/uA) current flow out at those points? If it was all made of wires would the same thing happen or would it be the opposite with the net effect be a batter and better conductor as more connections were formed?
T3sl4co1l:
Yes, in this case the currents will go to zero at infinity, and the series converges.

However, proof of this is necessary.

Without proof, you may get very peculiar results.  An example is the infinite sum: 1+1+1+... = -1/2.

If the series of all partial sums converges, then the sum exists.  The partial sums of 1+1+1+... clearly do not, so we do not say that sum exists, under ordinary circumstances.  (However, the existence of such tricks, and its application to particle physics, makes it too good not to have a name -- in that case, we specify what kind of summation is being used to claim the existence of this relationship.)

For a grid of linear and positive-valued resistors, we expect a general 1/n^2 behavior at a distance, which will give a convergent series (e.g., the sum of 1/n^2 for n = 1, 2, ..., is exactly 2).  Proof of this fact, is, uh, left as an exercise for the student, yes. ;D

Tim
IanB:

--- Quote from: Beamin on June 07, 2018, 03:54:31 am ---The math goes over my head in those two papers. Need simpler examples. With the infinite grid wouldn't the edges not matter at some point because the current would be so small and the resistance so high that effectively there would be no or unmeasurable (say your equipment goes down to 0.001 uV/uA) current flow out at those points? If it was all made of wires would the same thing happen or would it be the opposite with the net effect be a batter and better conductor as more connections were formed?

--- End quote ---

But the examples are what you gave in the first post, and the math is what is required to solve those examples.

If the math goes over your head you will have to do what I do, which is to read it with fascination and accept that the details will take some effort to understand.

The questions you ask about what happens at infinity are very appropriate and reasonable questions and are discussed at some length in the second paper.
Kirr:
When googling, also try "resistor lattice" - it will find a lot of research papers on this topic.

For finite grids (and not too large), you can try my solver, e.g.: 5x5 grid.

The largest grid I solved so far with my solver was 251x250 (with terminals across a single central resistor). It took about 2 hours, in three runs - first simplifying a triangle (1/8 of the grid), then merging two triangles, then finally simplifying the rest. The resistance is ~0.50000868 Ohm (13 kB long number in rational form in hexadecimal).
CJay:
That's going to be some weird looking grid if each resistor is only connected to three others...
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