| Electronics > Beginners |
| Is it possible to know all voltages/amps in a grid or matrix of resistors? |
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| rstofer:
--- 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 --- Now you are putting limits on things happening at infinity. Infinity isn't a countable number, spooky stuff happens out there and the fact that you can't measure it with a bench DMM just means it is best left for theoreticians. The closest EEs get to infinity is in RF using the Earth as a ground plane. Infinity is the biggest number you can imagine, plus one! And that just takes care of the integers. There are an infinite number of real numbers between any two integers. And there are an infinite number of integers. Kind of like infinity squared! It's just spooky! When current flows, it does not, contrary to popular conjecture, take the path of least resistance. It takes ALL of the paths. Even the high resistance paths will have some current flow. The low resistance paths will just have a high percentage of the current flow. So, yes, more parallel paths means lower resistance. The way to get a handle on this is to start with 1 resistor and solve Kirchoff's Laws using Ohm's Law. Add another resistor in series and solve again. Add a resistor in parallel and solve again. Try 4 resistors in a square, try 12 resistors in a cube and then stop. The process is understood and further examples are just more of the same with uglier arithmetic. But you really need a handle on Kirchhoff's Laws for mesh and nodal analysis. You can't even analyze simple op amp circuits without understanding Kirchhoff's Current Law. It'a a simple law: The sum of all currents entering or leaving a node total to zero. Current can't pile up at a node. At ever more complicated levels, math is required every semester of the 4 year EE undergraduate program. In the upper division, math is masquerading in classes like Field Theory (Maxwell's Equations) or Control Systems (Laplace Transforms) or Signal Processing (Fourier Transforms) and this is AFTER taking Calc I, Calc II, Calc III and Differential Equations in the lower division. Electronics, as a profession, is a math intense way to make a living. The good news about today is the tools we have for doing the math. Getting by with a slide rule in the pre-HP35 calculator days made things even more difficult. I remember when nearly every electronics text book had tables for trigonometric functions and logarithms in the back. We actually had to use tables to perform trig problems! |
| Beamin:
--- Quote from: Kirr on June 07, 2018, 08:36:26 am ---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). --- End quote --- What was the value of the individual resistors? 1k ohms? When I said each is connected to three resistors I meant per lead 6 resistors total like a carbon latus with single bonds. You could use this math to solve graphene (-C=C- etc etc) and other conductivity of nano sheets. You could model out what the resistance is between the bonds and scale it up into ohms to show a life size graphene sheet and would be a great way to teach organic conductors. You could dope by include N and P atoms with different value resistors and geometry. |
| T3sl4co1l:
Graphene doesn't work, because any given electron zipping around in the field of atoms experiences dozens of atoms simultaneously. The electron is a wave confined by a boundary condition; that it ever has resistance is an accident brought on by defects and impurities, encountered with low probability over long distances -- in other words, in the thermodynamic limit. In other other words, it doesn't help to reason about quantum systems from the smallest possible unit, stacked up. There are long range order, and effects, that pull in later. You have to run an integral over the ensemble -- quantum statistical mechanics. A notoriously hard problem, because to calculate that integral, you must account for, not just the effect of each differential unit upon itself, but upon its neighbors as well! Think about taking the frequency response of an amplifier, by looking at just one tiny little blip. You get good information about the high frequency response, but the low frequency response is a slow and minuscule 'tail' that is, at best, inconvenient to read in this way. How much long-term information can you infer from a very short blip? Not much. Much better, use a holistic method, like frequency domain (Fourier) analysis, to find the properties overall. (In QM, this is called k-space.) The equivalent to this principle in linear electronics is: if you cascade a bunch of AC filters, you must account for the interactions between them, which in general won't be well-behaved if you pick everything randomly. In this case, we can use the transmission ('T' or 'ABCD') matrix to solve the problem: a two-port filter is only two zero-dimensional connections, so we don't have to worry about waves or distance or phase or any of that, we simply get four complex numbers describing the input, transmission and output properties of the block. There are six standard, equivalent matrix types used to describe these things, but the handy part about the T-matrix is, you simply multiply them together, and that's your total answer. Suppose you could derive the electron wave function's T-matrix for graphene, for one very small differential unit; now suppose you can cascade infinity of them in all directions. Well, it's not going to be a two-port, because if nothing else, there are two directions to integrate over. Maybe a four-port would work? Unsure. It's... complicated. ;D Tim |
| Kirr:
--- Quote from: Beamin on June 07, 2018, 05:49:51 pm --- --- Quote from: Kirr on June 07, 2018, 08:36:26 am ---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). --- End quote --- What was the value of the individual resistors? 1k ohms? --- End quote --- 1 Ohm. (You're free to use any other values that you like, including non-identical). |
| The Electrician:
--- Quote from: T3sl4co1l on June 07, 2018, 05:16:32 am ---(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 --- End quote --- Mathematica thinks the sum is not 2: :) |
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