# Impedance of the infinitely long circuit

Is there a way to find impedance of the infinite circuit like this?

It will be much simpler if one has either parallel or series combination but here is both.

and treat the 'tail' as another copy of the circuit itself. This means that the impedance $Z$ of the whole circuit must satisfy
$$Z=2Z_1+\frac{1}{\frac{1}{Z_2}+\frac{1}{Z}}.$$
This gives a quadratic equation in $Z$ which is easy enough to solve.
• @EmilioPisanty ... and the challenge I've periodically pursued is to devise an estimator that can take input/output measurements of $Z$ and estimate the parameters $Z_1$ and $Z_2$. Have you seen this done? – docscience Jun 17 '15 at 14:59