Low-Pass Filters: Commentary on a Remark by Feynman

TL;DR

This paper clarifies Feynman's remark about the positive real part of the limiting impedance in an infinite reactive ladder, providing a simple convergence argument and energy propagation analysis.

Contribution

It offers a straightforward explanation for the convergence of impedance in an infinite reactive ladder and analyzes the finite energy propagation speed.

Findings

The impedance sequence converges geometrically.

The limiting impedance has a positive real part.

Energy propagates at a finite speed in the ladder.

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.