Determining resistor values in a mesh.

[Striplar]

Don't be fooled, this problem at first appears trivial, and then impossible.
Before you ask, there is no way to take out any of the resistors from the mesh or to know their values by their markings. The project is commercially sensitive so I'm afraid the discussion is going to have to be restricted to the example given. In the real world problem, the mesh will be very much larger but the resistor values will all be within +/-20% of each other. I've exaggerated the values in the example to make it easier to conceptualise where the currents are flowing.
The problem is simply this. If we know the voltages at all of the nodes, and we know the voltage across the mesh and, crucially, the current flowing from the voltage source, how can the resistor values be calculated?
I've shown the resistor values that were used to calculate the node voltages so that I know the correct solution to the problem. You have to imagine that you don't know any of the resistor values, these at what I'm trying to deduce.

Here are a few thoughts....
Node or mesh analysis can't provide enough equations for the number of unknown resistors.
If you were to allow the values of the bridging resistors to become infinite, multiple solutions are possible, but this won't be the case in practice.
My guess is that you can't solve this using classical methods because those methods can't accommodate those infinite values and discount them.
Once you add the constraint that no resistor can have an infinite value, I think there is only one solution, but I have no idea how to find it.
Any ideas?

**broken link removed**

 

[goldsmith]

Node or mesh analysis can't provide enough equations for the number of unknown resistors.

Hi Striplar
Why you think so ? because you'll have 5 equations which seems enough !
Did you ever tried to write them ?

By the way is that a homework ? :wink:
Best Wishes
Goldsmith

 

[Striplar]

There are 8 resistor values to find and only 5 equations. I think you may have misunderstood the question. I've shown an example solution with the resistor values in place, but you have to imagine that you don't know any of them. All you know is the information that I've indicated in the text below the circuit diagram. This circuit was drawn in a Spice simulation package to give me some sample voltages to work from with a known solution ie I decided the resistor values in this case. You have to remove those resistor values from the diagram and then see if you can determine them from the remaining information. We know the voltage source is 10volts and in the real world version of this problem, I'll be able to measure the amount of current that's providing.
Now you can see why it isn't so easy. This is a commercial job and I'm a Mechanical Engineer so it's not my primary discipline. Performing a node or mesh analysis is easy enough but I don't see how you can produce 8 equations from that. There are 5 nodes and 5 loops so that's not going to work. I've written them out on a slightly simpler mesh just to prove the point.
I think this needs an approach along the lines of Finite Element Analysis, but I first have to convince myself that I'm not chasing rainbows and that there is a unique solution with for the example shown first.

 

[The Electrician]

I was able to find another set of values that satisfy the 4 criteria:

R1 = 208.80858468280957; R2 = 133.72931408370076; R4 =232.67397559746723; R5 = 789.5680897947801;
R7 = 446.6046060851423; R8 = 240.2470482079432; R10 = 101.31105474335516; R12 = 336.099432067319;

 

[Striplar]

You are indeed a clever guy, this is disappointing but nonetheless useful. You have provided absolute proof that these circuits do not provide the unique solutions that I had hoped for, but at least that saves me wasting time on a false assumption.
This raises some interesting questions though, not least of which is how you managed to find that solution, I'd love to know!
The second question is how many solutions are there that satisfy those criteria? The reason I ask this, is because in the real world problem I'm trying to solve, the resistor values will actually be very close to each other, as of course will the node voltages. Perhaps I could home in on the actual solution by knowing which solution to pick if there are several.
I can't thank you enough for taking the time to prove this.

 

[The Electrician]

It's obvious since 4 equations completely specify the network that it won't be possible to find a unique solution for the 8 resistors. I attempted to find a symbolic solution using Mathematica as an aid. I assumed that it would be possible to arbitrarily select some resistors, and then the rest would be determined by the 4 constraints. I did get an apparent symbolic solution from Mathematica but it was very long and cumbersome. I thought that it might be more fruitful to use numerical techniques to explore the problem.

Edit: I should add that when I say "4 equations completely specify the network", I am referring only to the terminal behavior. I don't mean that the exact 8 resistors are determined.

Fooling around with the numerical solver, it's obvious that there likely are an infinite number of solutions. I can choose particular values of the various resistors (within limits) or I can constrain the resistors to lie in certain ranges. For example, I chose R1=390 and R2=270 and got this solution:

R1 -> 390., R2 -> 270., R4 -> 137.102, R5 -> 214.712, R7 -> 506.952,
R8 -> 257.904, R10 -> 270.863, R12 -> 306.14

Or, choosing R1=400 and R2=250 I get this solution:

R1 -> 400., R2 -> 250., R4 -> 139.471, R5 -> 242.545, R7 -> 470.149, \
R8 -> 234.989, R10 -> 176.531, R12 -> 266.606

Or, again, this solution:

R1 -> 410., R2 -> 240., R4 -> 135.624, R5 -> 250.323, R7 -> 493.538,
R8 -> 235.881, R10 -> 144.306, R12 -> 241.431

Et cetera, et cetera.

If you're interested in the details of how I did this, ask.

 

[Striplar]

That's very useful because I can now see that I need to look at this problem in a different way. I certainly would be interested in the details of how you did this, because it might help me to make a better guess at the resistor values, or at lease see how one resistor value influences another.
Many thanks for spending time to do this, it's much appreciated.

 

[The Electrician]

Here are some images from Mathematica showing the procedure I used. Interestingly, the final search shown is with constraints keeping the resistors within 10% of each other. I couldn't get a solution with the node voltages of the circuit from the first post, but if I made those node voltages more nearly equal I got a good solution.

- - - Updated - - -

Here are two more searches. The first one attempts to keep the resistors within 10% of each other. With node voltages of 4.8, 5.0 and 5.2, a good solution is not found. We can see this because the minimum value of the error function found is .00135723 which is much larger than the typical error of something times 10^-13 when a nearly exact solution is found.

The second search attempts to find a solution with node voltages of 4.85, 5.0 and 5.15. This time a nearly exact solution was found.

If the search tried to constrain the resistors to be within 20% of each other, rather than 10%, a good solution might be found with node voltages over a wider spread.

 

[Striplar]

Ok, I can't pretend to understand that without spending a serious amount of time on it. I've never used Mathematica, and It's not my best subject I'm afraid.
In the real world, the resistor values will probably be within 10% of each other and the mesh size is likely to be somewhere in the region of 5 cells in one direction and 50 in the other. Do you think this size of matrix could be solved in a matter of a second or two if all of the voltages were known?
The production process this would be incorporated in needs to be done quickly so some compromises in how many cells the model comprises may have to be made if the analysis takes too long. The manufacturing process is attempting to make all these resistors of equal value, so what we're looking at are manufacturing tolerances. This process only needs to be done once.
I'd need to see if Mathematica could be integrated within a Labview environment to take measured values for all those voltages and the current. I don't know if Labview has it's own Maths package that would be capable of performing these calculations for itself. These are all things that I'll need to explore.