Power dissipation in fractal AC circuits

TL;DR

This paper extends Feynman's analysis to fractal AC circuits, demonstrating that such circuits can have positive real characteristic impedances and providing algorithms for energy distribution analysis.

Contribution

It introduces a novel extension of impedance analysis to fractal AC circuits using self-similarity, expanding prior work on resistance networks.

Findings

Fractal circuits with purely imaginary impedances can have positive real characteristic impedances.

Algorithms are developed for analyzing energy distribution in these fractal circuits.

The work generalizes previous studies on self-similar resistance networks.

Abstract

We extend Feynman's analysis of an infinite ladder circuit to fractal circuits, providing examples in which fractal circuits constructed with purely imaginary impedances can have characteristic impedances with positive real part. Using (weak) self-similarity of our fractal structures, we provide algorithms for studying the equilibrium distribution of energy on these circuits. This extends the analysis of self-similar resistance networks introduced by Fukushima, Kigami, Kusuoka, and more recently studied by Strichartz et al.