## 3 Random SVPWM  
The power circuit structure of a two-level VSI supplying an IM is also depicted in Figure 4a. SVPWM is a digital technique based on the representation of three reference voltages by a single space vector. For the two-level inverter, eight possible combinations of power switch states yield eight vectors whose two vectors are inactive because they are identically zero. The six non-zero vectors in the plane (\(\alpha\), \(\beta\)) are given in the form of a standard hexagon whose origin is aligned to the zero vectors, as shown in Figure 4b. The general expressions for reference voltages of amplitude \(\mathrm{V_{m}}\) are given by the system of Equations (7):  
\[\left\{ \begin{array}{l}\mathrm{V_{Aref}}=\mathrm{V_{m}}\sin(2\pi\mathrm{Ft} )\\ \mathrm{V_{Bref}}=\mathrm{V_{m}}\sin(2\pi\mathrm{Ft}-\frac{2\pi}{3})\\ \mathrm{V_{Cref}}=\mathrm{V_{m}}\sin(2\pi\mathrm{Ft}-\frac{4\pi}{3})\end{array} \right. \tag{7}\]  
where \(\mathrm{V_{m}}=\frac{\mathrm{i}\mathrm{m_{V_{c}}}}{2}\) is the amplitude of the sine wave, and \(\mathrm{i_{m}}\) is the modulation index. Figure 4: (**a**) Structure of IM fed by VSI, (**b**) space vector diagram. Vector V\({}_{0}\) is applied during a duty cycle noted d\({}_{0}\), and the two adjacent active vectors V\({}_{1}\) and V\({}_{2}\) are applied, respectively, during the cyclic ratios d\({}_{1}\) and d\({}_{2}\). The general expressions for the duty cycles d\({}_{0}\), d\({}_{1}\) and d\({}_{2}\) are given by:  
\[\left\{ \begin{array}{l}\mathrm{d}_{1}=\frac{\sqrt{3}\mathrm{V}_{\mathrm{ref} }}{\mathrm{E}}\sin\left(\mathrm{s}\frac{\pi}{3}-\mathrm{\theta}\right)\\ \mathrm{d}_{2}=\frac{\sqrt{3}\mathrm{V}_{\mathrm{ref}}}{\mathrm{E}}\sin\left( \mathrm{\theta}-(\mathrm{s}-1)\frac{\pi}{3}\right)\\ \mathrm{d}_{0}=1-(\mathrm{d}_{1}+\mathrm{d}_{2})\end{array} \right. \tag{8}\]  
where \(\mathrm{\theta}\) represents the instantaneous angle, s is the sector, and V\({}_{\mathrm{ref}}\) is the reference space vector. The SVPWM algorithm subdivides sectors and transforms the reference vector into active vectors, which is demonstrated in Figure 4. The SVPWM strategy is performed by rotating the space vector. Thus, the control signals are elaborated by determining the sector number and calculating the duty cycles. The triangular carrier wave for the fixed SVPWM algorithm widely adopts the up-down counting mode of the timer. During the PWM modulation process, the modulation wave and the carrier wave are compared at a fixed switching frequency. The three-phase inverter requires three switching signals: S11, S21 and S31. The switching state distribution where the reference voltage V\({}_{\mathrm{ref}}\) is positioned in the first sector is explained in Figure 5.  
The durations of switching signals for the SVPWM control are obtained and computed using the following relations:  
\[\left\{ \begin{array}{l}\mathrm{T}_{0}=\mathrm{T}_{6}=\frac{\mathrm{d}_{0}}{ 2}\\ \mathrm{T}_{1}=\mathrm{T}_{5}=\frac{\mathrm{d}_{1}}{2}\\ \mathrm{T}_{2}=\mathrm{T}_{4}=\frac{\mathrm{d}_{2}}{2}\\ \mathrm{T}_{3}=2\times\mathrm{T}_{6}=2\times\mathrm{T}_{0}=\frac{\mathrm{d}_{0} }{2}\end{array} \right. \tag{9}\]  
Figure 6 displays the general case of a switching signal q for a three-phase VSI, which can be characterized by three parameters: the switching period \(T\), the duty cycle \(d\) and the delay report \(\delta\). Figure 5: Switching state distribution at sector 1 for SVPWM control. Figure 6: Parameters of switching signal for three-phase VSI. Furthermore, three advanced random SVPWM algorithms can be obtained, such as RZV_SVPWM, RPP_SVPWM, and RSF_SVPWM. The RZV_SVPWM control consists of making period T and delay report \(\delta\) deterministic and duty cycle d random [10; 11; 12]. Figure 7 displays the switching state distribution at sector 1 for RZV_SVPWM control. The zero voltage vectors randomize the zero voltage vectors V\({}_{7}\) and V\({}_{0}\).