Beyond Phasors: Solving Non-Sinusoidal Electrical Circuits using Geometry

TL;DR

The paper addresses the fundamental limitation of traditional phasor analysis in non-sinusoidal, harmonic-rich AC circuits by introducing a complete Geometric Algebra (GA) framework. It defines a rotoflex operator, combining flextance (scaling) and rotance (rotation) to map multi-harmonic voltage and current directly in a $2N$-dimensional space, avoiding per-harmonic decomposition. The approach reproduces classical results for single-frequency cases and yields exact numerical agreement with per-harmonic methods for multi-harmonic cases, while providing clear geometric insight into power factor as the scalar part of the rotation. This GA-based method offers computational efficiency and a unified, interpretable representation of impedance across harmonics, with potential extensions to networks, machines, and high-frequency applications.

Abstract

Classical phasor analysis is fundamentally limited to sinusoidal single-frequency conditions, which poses challenges when working in the presence of harmonics. Furthermore, the conventional solution, which consists of decomposing signals using Fourier series and applying superposition, is a fragmented process that does not provide a unified solution in the frequency domain. This paper overcomes this limitation by introducing a complete and direct approach for multi-harmonic AC circuits using Geometric Algebra (GA). In this way, all non-sinusoidal voltage and current waveforms are represented as simple vectors in a $2N$-dimensional Euclidean space. The relationship between these vectors is characterized by a single and unified geometric transformation termed the \textit{rotoflex}. This operator elevates the concept of impedance from a set of complex numbers per frequency to a single multivector that holistically captures the circuit response, while unifying the magnitude scale (flextance) and phase rotation (rotance) across all harmonics. Thus, this work establishes GA as a structurally unified and efficient alternative to phasor analysis, providing a more rigorous foundation for electrical circuit analysis. The methodology is validated through case studies that demonstrate perfect numerical consistency with traditional methods and superior performance.

Figures (3)

• Figure 1: Rotation of a vector $\bm{b}$ to a vector $\bm{a}$ by an angle $\varphi$ within a plane $\hat{\bm{B}}$.
• Figure 2: The two fundamental topologies under study.
• Figure 3: Harmonic transformation for a series circuit. The input voltage vector $\bm{u}_h$ is transformed into an output current vector. A capacitive circuit applies a clockwise rotation $\varphi_{h, \text{cap}}$, while an inductive circuit applies a counter-clockwise rotation $\varphi_{h, \text{ind}}$.