# Calculating resistances in parallel

I was playing around with a question involving resistances in parallel. I realized that $n$ equal resistors (resistance $R$) in parallel have combined resistance $R/n$. I proved this using (mathematical) induction. Then I considered two resistors with ratio $1:2$ in parallel, e.g. a $3\Omega$ and $6\Omega$ resistor in parallel. Since $3=6/2$, we can view the $3\Omega$ resistor as two 6 ohm resistors in parallel. So the problem boils down to finding the resistance of three $6\Omega$ resistors in parallel, which is $6/3=2$.

So this law can be used to make resistances easy to calculate. Above we just easily calculated the combined resistance of a $3\Omega$ and $6\Omega$ resistor in parallel. Yes, we could also have done $(3*6)/(3+6)$, but I think what I did is much easier.

Question: Is this algorithm known/is there a general version of it?

• Yes, there is. If you download LTSpice from the linear.com website, you get a very nice numerical circuit simulator that uses it for free. – CuriousOne May 13 '15 at 20:30
• The inverse of resistance is conductance. And what you are doing is really equivalent to summing the conductances. In that sense it is known. – Floris May 13 '15 at 20:46
• @CuriousOne why do you always have to be so blunt and sarcastic? It doesn't help anyone (certainly not OP!) or contribute to anything. – Gonenc May 13 '15 at 22:16
• @gonenc I don't think CuriousOne's comment was intended to be sarcastic at all; (s)he merely provided an example of a known implementation which is accessible to the OP (although a linked URL would help). Any sarcasm was inferred, but not necessarily implied. – tok3rat0r May 13 '15 at 22:37
• @tok3rat0r look at the other comments by him/her. You'll probably see a general pattern from sarcasm to bluntness, from rudeness to utterly non-constructive "criticism". – Gonenc May 13 '15 at 22:42

$$R_{eff} = \frac{1}{\sum{\frac{1}{R_i}}}.$$
When the ratio between any of the resistors, $R_i$, is an irrational number it will not be possible to use the method you mentioned. For methods, which can be used to calculate something, the most robust method is usually preferred because it works in most, if not all cases. Even if every ratio between resistors can be written as factions of two integers, it might take more time finding those than it would to use the above equation. So the method you mentioned will only have limited application.