Beyond Phasors: Solving Non-Sinusoidal Electrical Circuits using Geometry

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.