Resistor combinations: How many values using 1kohm resistors?
What analog designer hasn't had to derive a nonstandard resistor value by using series/parallel combinations of standard resistor values? In efforts to avoid a production trimpot tweak, we can use 0.1% resistors when we need precise voltage dividers. To obtain non-standard values, two or more resistors in series work well - the total resistance is their sum, which makes calculation easy. The significant value is the larger resistor, and the small value in series does the fine tuning.
Even for practical, everyday matters such as repairing an old vacuum tube electronic organ with an open wirewound heatsunk resistor, raiding the junkbox and soldering together a string of 5W resistors saves the day. And of course, with the heat dissipation spread out, these run much cooler than the original.
Parallel resistors are a bit harder to calculate in that RTOTAL = (R1 × R2) / (R1 + R2). Parallel works better for manually fine-tuning things like voltage regulators when hand-picking a large value resistor to solder in parallel with a small value. Not great for large-scale production, but OK for one-of-a-kind fixtures, and the elimination of an adjustable potentiometer means that in the future, nobody can mess with the calibration.
Martin Rowe threw this little challenge my way: "How many values can you make with only 1kΩ resistors?"
Well, it sounded like fun, and was something I had never studied before. I jumped into this with the intention of doing up to 10 of 1 kΩ resistors in various series/parallel arrangements. But by the time I got to just five resistors, the possible permutations were becoming overwhelming.
So how many permutations can be made from n of 1kΩ resistors?
The first obvious arrangements are all in series and all in parallel. When n = 2 resistors the possible permutations are two values (Figure 1).
When n = 3, there are four possible arrangements. A tip for quick calculation - when all resistors are the same value, their equivalent resistance is their value divided by the number in of parallel resistors. Break the circuit down into small pairs of series/parallel resistors to tally up the final equivalent resistance, working with pairs makes the parallel calculation much easier (Figure 2).
When n = 4, the number of permutations surges to nine (Figure 3). I think I have all variations covered, but if anyone can find one that I've missed let me know by leaving a comment.
The combination on the far right works out to be 1000Ω, same as a single resistor. The advantage is that by using four resistors, the power dissipation capability has also increased by a factor of four, and the voltage withstand doubles. This is a useful trick when you want to keep all your resistors surface-mount types.
Moving on up to n = 5 raises the permutation count to 23, assuming that I haven't overlooked any possibilities. Again, if you spot a missed arrangement, let me know in a comment.
At this point, I realized there was no hope of getting to my target of 10 resistors and still have paper large enough to draw them all. Maybe the math wizards here might have some idea how to calculate the number of possible permutations given n resistors. Go ahead, figure it out.
Is there an equation you can derive that will work for any number of series/parallel resistors?