## 2 Principle of the Conventional and Improved FOC Strategy  
### Principle of the Conventional FOC Strategy  
The aim of the FOC strategy is to have an independent control over the couple and the flux similar to a separately excited DC machine [13; 14; 15]. To reach such decoupled control, a FOC algorithm is required to obtain the rotor flux angular position, to correctly align the stator correct vector. Hence, it is feasible to control torque and rotor flux in a DC machine control fashion, by acting on two separated stator current components: \(i_{sd}\) and \(i_{sq}\). Thanks to the FOC strategy, an asynchronous machine can be used in high-dynamic performance required where only a DC machine can be used  
After making the rotor field rotation (Figure 1) \(\hat{\psi_{rd}}=\hat{\psi_{r}}\) and \(\hat{\psi_{rq}}=0\). As presented in Equations (1) and (2) IM relations then becomes:  
\[\psi_{rd}=\frac{M}{1+pT_{r}}i_{sd}, \tag{1}\]  
\[T_{e}=n_{p}\frac{M}{L_{r}}\psi_{r}i_{sq}, \tag{2}\]  
Rotor field rotation is applied once the rotor flux angular position is known. Direct field oriented control (DFOC) and Indirect field oriented control (IFOC) are both employed to obtain this position. Manipulating Equations (3)-(7), \(\hat{\psi_{r}}\) and \(T_{e}^{{}^{\prime}}\) are estimated using \(i_{sd}\) and \(i_{sq}\), respectively, and are compared to \(\psi_{r}^{*}\) and \(T_{e}^{*}\). As shown in Figure 2, torque and flux are directly controlled in DFOC.  
\[v_{sd}=\bigg{[}(R_{s}+p\sigma\Delta_{s})\frac{(1+T_{r}p)}{\mathcal{M}}+\frac{ \mathcal{M}}{\Sigma_{r}p}\bigg{]}\hat{\psi_{rd}}-w_{s}\Delta_{s}\sigma i_{sq}, \tag{3}\]  
\[v_{sq}=(R_{s}+p\sigma\Delta_{s})\frac{T_{e}}{p\frac{M}{L_{r}}\hat{\psi_{rd}}} +w_{s}\Delta_{s}\sigma i_{sd}+w_{s}\frac{\mathcal{M}}{\Sigma_{r}}\psi_{r}, \tag{4}\]  
\[\hat{\psi_{r}}=\frac{M}{1+pT_{r}}i_{sd}, \tag{5}\]  
\[\theta_{s}=\int(n_{p}\Omega+\frac{M}{T_{r}}\frac{i_{sq}^{*}}{\psi_{rd}^{*}})dt T_{e}^{*}=n_{p}\frac{M}{L_{r}}\psi_{r}i_{sq}, \tag{6}\]  
Figure 2 presents the control strategy scheme of DFOC. This strategy is chosen for this work and will be named the FOC strategy.  
To secure decoupled control for torque and flux, compensation terms \(f_{emd}\) and \(f_{emq}\) Equations (7) and (8) are added to obtain d and q axes completely independently.  
\[f_{emd}=w_{s}\Delta_{s}\sigma i_{sq} \tag{7}\]  
\[f_{emq}=w_{s}\Delta_{s}\sigma i_{sq}+w_{s}\frac{\mathcal{M}}{\Sigma_{r}}\psi_ {r} \tag{8}\]  
Figure 1: Orientation of the rotor field to d axis of the (d,q) reference.  
Equation (2) presents how position \(\theta_{\text{s}}\) and voltages \(\
u_{\text{sq}}\), \(\
u_{\text{sd}}\) areaculculated. Using park position \(\theta_{\text{s}}\), \(\
u_{\text{sq}}\), and \(\
u_{\text{sd}}\) are changed and injected to the 2 L. The latter is controlled by SPWM. To obtain two voltage levels, SPWM is obtained by comparing a sinusoidal reference signal and triangular carrier. Thereby, the inverter's switches are controlled through pulses obtained by this comparison, as shown in Figure 3. The following Figure 4 shows an experimental result of the two level output voltage inverter implemented using dSPACE 1104 where we observed the two voltage levels.  
Figure 4: Output voltages of the two level inverter.  
Figure 3: Principle of the SPWM two level control strategy.  
Figure 2: DFOC strategy scheme of asynchronous machine using 2 L.