## 3 Transient Analysis of PES System Using Fourier Integral Transform
### Transient Analysis Using Fourier Transform (and Under Decreasing Cosine Function)  
Assuming an exponentially increasing back-emf voltage, such as during the start-up of an electric motor, and a constant harmonic input voltage, the voltage across the load (inductor) will follow a decreasing cosine function  
\[u_{L}(t)=U_{m}\text{cos}\omega_{1}t.\text{e}^{-\frac{t}{\tau_{rise}}}, \tag{28}\]  
where \(\tau_{rise}\) respects a rise in the back-emf voltage, as shown in Figure 13a (green). If we take \(u_{L}(t)\) according to (28), the corresponding load current of the inductor can be calculated, as shown in Figure 13b (red).  
Applying the Fourier transform on the transient voltage \(u_{L}(t)\), we can write [1]  
\[\mathbf{F}_{u}(j\omega)=U_{m}\int\limits_{0}^{\infty}e^{-at}\text{cos}\omega_{1}te^ {-j\omega t}\text{d}t, \tag{29}\]  
where \(a(=1/\tau_{rise})\) represents a damping of the circuit and \(\omega_{1}\) is an angular frequency of the input voltage.  
Figure 12: Time waveform of the load current and load and network voltages (**a**); the decomposition of the total current (**b**).  
Figure 13: Created a decreasing cosine function of the load voltage (**a**) and current (**b**).  
The infinitive integral \(\int(.)\) can be calculated by substitution or per partes methods with the result  
\[\int(.)=\frac{e^{-(a+j\omega)}}{\left[-(a+j\omega)\right]^{2}+\omega_{0}{}^{2}}(- (a+j\omega)\text{cos}\omega_{1}t+\omega_{0}\text{sin}\omega_{1}t), \tag{30}\]  
with use of the Euler formula as a substitution for \(\text{cos}\omega_{1}t\).  
So, after the establishment of integration boundaries  
\[\int\limits_{0}^{\infty}(.)=\left[0-\frac{-(a+j\omega)}{\left[-(a+j\omega) \right]^{2}+\omega_{1}{}^{2}}\right]=\left[\frac{(a+j\omega)}{\left(a+j\omega \right)^{2}+\omega_{1}{}^{2}}\right],\] (30a) thus \[\text{F}_{u}(j\omega)=U_{m}\frac{a+j\omega}{a+\omega_{1}{}^{2}-\omega^{2}+j2a \omega}.\] (31) The magnitude spectrum of \[\text{F}_{u}(j\omega)\] \[|\text{F}_{u}(j\omega)|=U_{m}\sqrt{\frac{a^{2}+\omega^{2}}{\left(a^{2}+\omega _{1}{}^{2}-\omega^{2}\right)^{2}+4a^{2}\omega^{2}}}\] (31a) The graphic interpretation of the magnitude spectra of \[|\text{F}_{u}(j\omega)|\] is shown in Figure 14 in the relative p.u. units (i.e., \[U_{m}\equiv 1\] ).  
The phase spectrum of \(\text{F}(j\omega)\) we can calculate as an arctan function of the share quotient of the imaginary and real part of the magnitude spectrum (\(|\text{F}_{u}(j\omega)|\))  
\[(\omega)=\tan^{-1}\frac{Im\{\text{F}_{u}(j\omega)\}}{Re\{\text{F}_{u}(j\omega )\}}. \tag{32}\]  
Similarly, as for \(u_{L}(t)\)--if we admit that the load current is decreasing the quasi sinusoidal function (Figure 15)--we can apply its Fourier transform  
\[\text{F}_{i}(j\omega)=I_{m}\int\limits_{0}^{\infty}e^{-at}[\sin(\omega_{1}t- \varphi_{L})+\text{sin}\varphi_{L}]e^{-j\omega t}\text{d}t, \tag{33}\]  
where the maximal value of the load current  
\[I_{m}=\frac{U_{m}}{L\sqrt{a^{2}+\omega_{1}^{2}}}, \tag{34}\]  
Figure 14: A decreasing cosine function and its voltage magnitude spectrum \(|\text{F}_{u}(j\omega)|\).  
and \(\varphi_{L}=\tan^{-1}\left(\frac{d}{\omega_{1}}\right)\) where \(a=1/\tau_{rise}\) as mentioned above.  
The magnitude spectrum of the current Fourier transform \(|\mathbf{F}_{i}(j\omega)|\) is depicted in Figure 15.