Say I have to get a resistance \$R_{T}\$ and I have a bunch of resistors, \$R_1, R_2, \ldots, R_n\$.
Does there exist a tool, algorithm, or method that gets the resistance value closest to \$R_T\$ by connecting (one or more) of the resistors \$R_1, R_2, \ldots, R_n\$ in series, in parallel or a combination of those two? And how to connect those resistors to get that value? So not trial and error. 🙂
Thanks in advance.
Best Answer
It's not clear from your question whether you have a theoretical interest in the possibility of a solution, or a practical interest on the bench of finding some resistor values that will do your job.
To find the best possible combination of Rn to Rt would, I suspect, be a Travelling Salesman problem, NP complete and needing a brute force search for the global optimum. The structure of the problem is almost identical, it's a search of all possible combinations in a discrete space. Similarly it should admit to faster sub-optimal solutions by simulated annealing or similar.
If you are after a practical solution, then it's worth starting with an often overlooked caveat that can be used to limit just how close you attempt to get these resistors to be. Most 'cooking grade' resistors have tempcos of 100 to 200ppm. That is, if you assembled a resistor to an initial tolerance of 0.001%, you could not expect it to keep that precision with changes of temperature. When you buy a precision resistor to a specific value, you are paying most of the money for the 25 or 5ppm tempco that will keep it precise.
This is the non-optimum but fairly quick and rapidly terminating method of assembling special resistor values I use. I first attempt to get to within a few percent of my target value, above when dealing with low values, below when dealing with high values, by just randomly putting resistors in series or parallel. Then I measure the resistance, and calculate the shimming resistor value needed in series or parallel. As the needed resistor is at least an order of magnitude different (as I'm within a few % of target), the precision needed in the shim is an order of magnitude less than my overall precision.
I either have something close enough to that value, or I can construct one the same way, but it's an order of magnitude easier because of the relaxed precision. This rinse and repeat process gets easier each time, and will terminate in very few steps.