Applications of the Frenet Frame to Electric Circuits

TL;DR

This paper introduces a geometric approach using the Frenet frame to analyze electrical quantities, unifying time- and frequency-domain frameworks and providing new insights into frequency dynamics.

Contribution

It presents a novel geometric framework based on the Frenet frame for analyzing electrical signals, linking differential geometry with circuit theory.

Findings

Unified framework for stationary and transient conditions

New interpretation of frequency and its derivatives

Introduction of the concept of torsional frequency

Abstract

The paper discusses the relationships between electrical quantities, such as voltages, currents, and frequency, and geometrical ones, namely curvature and torsion. The proposed approach is based on the Frenet frame utilized in differential geometry and provides a general framework for the definition of the time derivative of electrical quantities in stationary as well as transient conditions. As a byproduct, the proposed approach unifies and generalizes the time- and phasor-domain frameworks. Other noteworthy results are a new interpretation of the link between frequency and the time derivatives of voltage and current; and a definition of the rate of change of frequency that includes the novel concept of "torsional frequency." Several numerical examples based on balanced, unbalanced, harmonically-distorted and transient voltages illustrate the findings of the paper.