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

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

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The standard trick is to split off the circuit after the first link,

enter image description here

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.

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  • $\begingroup$ @EmilioPisanty A good reference that illustrates the infinite ladder network, and how to solve it as you have shown is Feynman's lectures on physics; I believe volume 2. Feynman calls it the Characteristic Impedance. What is interesting (to me) is that one starts off with a linear circuit, and by expanding to infinity one gets a nonlinear system. $\endgroup$ – docscience Jun 17 '15 at 14:54
  • $\begingroup$ @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? $\endgroup$ – docscience Jun 17 '15 at 14:59

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