## 2 Fourier Analysis in the Time and Complex Domains  
Fourier series, in many respects, offers a broader scope of applicability compared to the more familiar Taylor series from calculus. This is because the Fourier series can represent a wide range of discontinuous periodic functions, which often arise in real-world applications, whereas such functions may not always have a valid Taylor series expansion [4]. As an illustrative example, we can consider the pulse waveforms commonly generated by power electronic systems, as shown in Figure 1[5; 6; 7]. The Fourier series is expressed by the following relation [1; 2; 3; 4],  
\[u(t)=\sum_{v=1}^{\infty}\big{[}A_{v}\text{cos}\,\
u\omega t+B_{v}\text{sin}\, \
u\omega t\big{]}, \tag{1}\]  
Figure 1: Time waveforms of non-harmonic courses used in EE generated by PES compensator (**a**) and inverter (**b**). where \(A_{v},\) and \(B_{v}\) are amplitudes of harmonic functions, which are the functions of the Fourier coefficients \(a_{v},\) and \(b_{v}\), multiplied by the maximal value \(U_{m}\) or [8]  
\[u(t)=\sum\limits_{v=1}^{\infty}C_{v}\text{cos}(v\omega t+\delta_{v}),\] (1a) where \[C_{v}=\sqrt{A_{v}^{2}+B_{v}^{2}}\text{, }\tan\delta_{v}=\frac{B_{v}}{A_{v}}. \tag{1b}\]  
Current waveform under linear R-L load can be derived using complex conjugate amplitudes [6] applied on Equation (1a) for steady-state and zero initial conditions  
\[\begin{array}{c}i(t)=\sum\limits_{v=0}^{\infty}\frac{C_{v}\cos(v\omega t+ \delta_{v})}{R+dL}\bigg{]}_{d\equiv j\omega}\leftrightarrow\\ \quad\quad\quad\quad\quad  
Note: In the case, when the system is the second and higher order, the transient component can be calculated using Laplace or Laplace-Carson transform (\(\mathcal{L}\)-C), respectively, when the notation of Equation (1a) is [8]  
\[U(p)=\sum_{\
u=0}^{\infty}C_{\
u}\frac{p(\mathrm{pcos}\delta_{\
u}+\
u\omega \mathrm{sin}\,\delta_{\
u})}{p^{2}+\left(\
u\omega\right)^{2}}. \tag{4}\]  
The last Equation (4) can be used later for the transient Fourier analysis, but it will be significantly simplified which is one of the goals of this paper. Analytical Solutions of Transient Values Using Fourier Series Using the Laplace-Carson Transform and Complex Time Vectors Inside One Period  
Let us suppose a single electrical R-L circuit is supplied by harmonic voltage, as shown in Figure 2.  
In the case of harmonic sinusoidal supply, it is valid. \[u(t)=U_{m}\mathrm{sin}(\omega t+\delta)\text{ or } \tag{5a}\] \[u(t)=U_{m}\mathrm{cos}(\omega t+\delta), \tag{5b}\]  
where \(\delta\) is a voltage connection angle to the load circuit or/and the initial angle of the time vector in the complex Gaussian plane. Adding a two-position switch in Figure 2b, where one of the positions closes the circuit, and the other grounds the R-L load, it is possible to reproduce the voltage profile on the R-L load of Figure 1a. Both curves in Figure 1 can be represented by the Fourier series. Going back to the Fourier series, then using the relation Equations (5a) and (5b) we can write a projection in the complex domain  
\[\mathbf{U}(t)=\sum_{\
u=1}^{\infty}U_{m,\
u}\mathrm{e}^{i\left(\omega_{\
u}t+ \delta_{\
u}\right)},\] (6a) where the angle frequency \[\omega\] will be replaced by the frequency of the Figure 1:0 -th harmonic Figure 1:1, the phase displacement Figure 1:2 by Figure 1:3, and Figure 1:4 by Figure 1:5. After decomposing as a sum of the first and higher harmonic components, where the first term is a simple vector ( Figure 1:6 ) starting at Figure 1:7  rotated by an angle Figure 1:8, and the second one is the sum of the higher harmonic phas Figure 1:9, we obtain \[u(t)=\sum\limits_{v=1}^{\infty}C_{v}\text{cos}(v\omega t+\delta_{v}),\]0 (6b)  
\[u(t)=\sum\limits_{v=1}^{\infty}C_{v}\text{cos}(v\omega t+\delta_{v}),\]1 Time waveform of considered voltage (a), and electrical circuit (b). It is simpler to sum these two current vectors (right side of Equation (6b)) in a coordinate system fixed to the system's real axis i.e. to the fundamental voltage vector.