Fractal AC circuits and propagating waves on fractals

TL;DR

This paper generalizes the analysis of AC circuits to fractal structures, revealing that fractal circuits can exhibit positive real impedance components and enabling wave propagation analysis on fractals.

Contribution

It extends Feynman's ladder circuit analysis to fractal circuits, introducing complex impedance measures and wave propagation insights on fractal geometries.

Findings

Impedances can have positive real parts despite purely imaginary components.

Provides a framework for analyzing wave propagation on fractals.

Generalizes the resistance metric to complex impedances on fractals.

Abstract

We extend Feynman's analysis of the infinite ladder AC circuit to fractal AC circuits. We show that the characteristic impedances can have positive real part even though all the individual impedances inside the circuit are purely imaginary. This provides a physical setting for analyzing wave propagation of signals on fractals, by analogy with the Telegrapher's Equation, and generalizes the real resistance metric on a fractal, which provides a measure of distance on a fractal, to complex impedances.