## 4 Discussion and Conclusions  
From the results shown in Figures 10b and 15, it is evident that during transient states, both the analyzed quantities exhibit a continuous spectrum of harmonic components, including the fundamental frequency. This implies that the electrical power, as a product of current and voltage, will contain distortion components, even in harmonic networks with linear loads (resistive, inductive, or both). Although this observation may seem unexpected, it is a significant contribution to practical applications. While classical Fourier analysis, including its application to transient signal analysis, may not be as effective as other methods, it still offers distinct advantages:  
* It provides insight into the harmonic content of the signal;
* It allows for the straightforward calculation of harmonic distortion [12]. The function being analyzed does not need to be analytically defined over the entire period and can be represented using methods like look-up tables. When performing analysis in the complex domain, regardless of the method used, it enables the calculation of the effective values of quantities and the average power components. This is especially useful, as it allows for time savings in repeated calculations--\(1/4\) of a time period for single-phase systems and \(1/6\) for three-phase systems. The Fourier transform also demonstrates that even with harmonic power supplies and linear time-invariant (LTI) loads, transient states generate a continuous spectrum of higher harmonics (Figures 10b, 14 and 15). In repeated or long-term START-STOP regimes, this could lead to excessive current stresses on system components, such as power transformers. Additionally, when combined with the Laplace or Laplace-Carson transforms, Fourier analysis becomes a powerful tool for solving transient phenomena, especially when addressing steady-state conditions within a single time period. This paper introduces two novel and effective approaches to transient analysis:  
* The combined use of Fourier and L-C transforms in the complex domain, which significantly simplifies the computation of the original function (Figures 3-6);
* The application of Fourier transforms for analyzing transient states in electrical systems, particularly in electrotechnical applications (Figures 10b, 14 and 15);
* The unique ability of this transformation to provide magnitude and phase frequency spectra, which are not easily obtained through other methods, demonstrates its utility [13]. In conclusion, this work presents a significant advancement in transient Fourier analysis, offering novel insights into the behavior of individual harmonics and their superposition in dynamic systems. The integration of complex time-domain analysis and the  
Figure 15: Magnitude spectrum \(|\mathbf{F}_{i}(j\omega)|\) of the decreasing sinusoidal function. innovative techniques presented here hold considerable potential for further developments in this field. Conceptualization, methodology, and supervision by [PERSON]; formal analysis, data curation, investigation, and validation by [PERSON] ([PERSON]) and [PERSON]; software and visualization by [PERSON]; resources, original draft preparation, review, and editing by [PERSON] and [PERSON]. ([PERSON]); and project administration and funding acquisition by [PERSON] All the authors have read and agreed to the published version of the manuscript. All authors have read and agreed to the published version of the manuscript. This research was funded by the Slovak Research and Development Agency, Bratislava, Slovakia, grant number APVV-22-0330, and the APC was funded by the Scientific Grant Agency of the Slovak Republic, Bratislava, Slovakia, grant number VEGA 1/0314/24. Not applicable. The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.