Harmonic distortion remains a critical issue in modern single-phase power systems, primarily due to the increasing penetration of nonlinear loads and power-electronic devices. Accurate estimation of harmonic content is essential for maintaining voltage quality, ensuring equipment reliability, and supporting overall grid stability. Existing numerical approaches, including discrete Fourier transform (DFT)-based methods, often face limitations such as high sensitivity to noise, spectral leakage, and increased computational load.

This study introduces a complex-analysis-based mathematical framework for harmonic distortion estimation, offering improved analytical insight and computational efficiency. The distorted voltage waveform is modeled as a complex-valued periodic function, enabling decomposition through Fourier series in the complex plane. Key tools from complex variables—such as contour integration, residue theorem, and analytic continuation—are employed to derive harmonic coefficients with enhanced precision. Linear algebraic formulations and numerical approximation methods from advanced engineering mathematics (including iterative solvers and error minimization techniques) further optimize coefficient extraction and reduce computational overhead.

Simulation results on multiple non-sinusoidal test signals demonstrate that the proposed method achieves higher accuracy in estimating harmonic magnitudes and phases, with up to an 18% improvement in convergence compared to conventional DFT-based approaches. The framework also exhibits strong noise tolerance, providing clearer differentiation between dominant and weak harmonic components.

Overall, this work offers a mathematically rigorous and efficient methodology for real-time harmonic analysis in power systems. The framework bridges complex analysis with power-engineering applications and has strong potential for integration into modern power quality monitoring and control systems.