Low-Pass Filters: Commentary on a Remark by Feynman

Abstract

In Feynman's lectures there is a remark about the limiting value of the impedance of an n-section ladder consisting of purely reactive elements (capacitances and inductances). The remark is that this limiting impedance $z=\lim_{n\to\infty}z_n$ has a positive real part. He notes that this is surprising since the real part of each $z_n$ is zero, therefore it is impossible for the limit to have a positive real part. A recent article in this journal offered an explanation of this paradox based on the fact that realistic impedances have a non-negative real part, but the authors noted that their argument was incomplete. We use the same physical idea, but give a simple argument which shows that the sequence $z_n$ converges like a geometric series. We also calculate the finite speed at which energy is propagated out into the infinite ladder.