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

Question

anonymous · Answer

Something similar to this:

jamesj · Answer

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

anonymous · Answer

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

jamesj · Answer

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 (*)

anonymous · Answer

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

jamesj · Answer

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.