Applications of the Frenet Frame to Electric Circuits
For researchers in electrical engineering, this work offers a novel geometric perspective on circuit analysis, but it is incremental as it applies existing mathematical tools to a known domain.
The paper introduces a general framework based on the Frenet frame from differential geometry to unify and generalize time- and phasor-domain analysis of electrical circuits, providing new interpretations of frequency and its rate of change. Numerical examples demonstrate the approach on balanced, unbalanced, harmonically-distorted, and transient voltages.
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.