anonymous 4 years ago Could someone explain, step-by-step, how to find the resistance of an infinite ladder circuit with resistors?

1. anonymous

Something similar to this:

2. JamesJ

This should help: http://www.crbond.com/papers/ent2-3.pdf

3. anonymous

Thanks, I've already been directed to that page before, however. I was wondering if you could explain what they wrote there: Z = R + R || Z

4. JamesJ

That's just a notational short cut for what comes next. If we added just one more resistor, and not in parallel, then the resistance would be Z + R where Z is the original resistance and R is the resistance of one resistor. If we added just one resister in parallel, the new resistance would be $\frac{RZ}{R+Z}$ Now, having added one more step in the ladder, the new resistance is $R + \frac{RZ}{R+Z}$ This must be equal to the original resistance; hence $Z = R + \frac{RZ}{R+Z} \ \ \ \ \ \ --- (*)$ The notation $Z = R + R || Z$ is just a short way of writing the equation (*)

5. anonymous

Oh, alright. It took me a bit, but I think I get it now. Thank you very much!

6. JamesJ

Sure. I haven't looked at this problem for a while and it's nice to see it again. I'm particularly happy to see the quantity $\frac{1 + \sqrt{5}}{2}$ in the answer. This is the Golden Ratio of mathematics and I'd forgotten that it turns up here as well.