# Topological properties of fractal Julia sets related to the signs and magnitudes of the real and reactive powers

## TL;DR

This paper explores the topological properties of fractal Julia sets in relation to the signs and magnitudes of real and reactive power in AC electrical systems, revealing fractal patterns that relate power characteristics to complex plane quadrants.

## Contribution

It introduces a novel application of fractal Julia sets to analyze the topological properties of complex power in electrical circuits, linking fractal patterns to power signs and magnitudes.

## Key findings

- Fractal diagrams exhibit distinct patterns based on power signs and magnitudes.
- Power relationships are reflected in the structure of Julia set fractals.
- Different power combinations produce unique topological features in the fractal diagrams.

## Abstract

In AC electrical systems, the power depends on the real power (P) due to resistive elements and the reactive power (Q) due to the inductive and capacitive elements, which are commonly studied by using phasor and scalar methods. Thus, this paper focuses on applying the fractal Julia sets to observe the topological properties related to the signs and magnitudes of the real and reactive powers consumed or supplied by an electrical circuit. To perform this, different power combinations were used to represent the fractal diagrams with an algorithm that considers the mathematical model of Julia sets. The study considers three type of loads: the first study considers the change of real power when the reactive power is fixed; the second study deals with the change of the reactive power when the real power is fixed; and finally, the third study contemplates that both real and reactive powers change. Furthermore, the fractal diagrams of the power in the four quadrants of the complex plane are studied to identify the topological properties that each sign and magnitude represent. A qualitative analysis of the diagrams helps to identify that the complex power loads present some fractal graphic patterns, with respect to the signs and magnitudes considered in the different quadrants of the complex planes. The diagrams represented in the complex planes save a relation in the forms and structure with other points studied, concluding that the power is related to other figures in other quadrants. Thus, this result allows a new study of the behavior of the power in an electrical circuit, showing a clear relation of the different fractal diagrams that the Julia sets obtained.