# Comparative Study and Overview of Field-Oriented Control Techniques for Six-Phase PMSMs
## 3 Control in a One Orthogonal System  
Furthermore, for a better overview, the abbreviation \"1 d-q\" is established in this paper as one orthogonal system. This approach is also known as the two-controller method, where various modulation techniques might be applied, as shown in Section 4. With an inclusion of higher harmonic control, it is known as the vector space decomposition (\(VSD\)) method [5; 11; 12; 13; 14; 15; 16; 25; 26; 27; 28; 29; 30; 31]. An approach of the 1 d-q field-oriented control, specified to a six-phase system, is shown in Figure 4. The whole control structure is shaped similarly to a standard one for three-phase machine control. Parallel operation of current loops is not required, since the structure contains just one set of flux/torque\(-\)producing components for the whole six-phase system. On the other hand, transition sections between the six-phases and one orthogonal system in both direct and inverse directions depend on the functions specified for a number of phases in a controlled machine. Figure 3: Axis orientation in a 2 d-q system. Park transformations between an orthogonal rotor reference frame d-q and a stationary \(\alpha-\beta\) reference frame are the same as those used in the standard field-oriented control of the three-phase PMSM. In this case, just the one orthogonal \(\alpha-\beta\) system can be recognized, where \(\alpha\) is aligned to phase \(A\), and \(\beta\) leads \(\alpha\) by \(90^{\circ}\), as depicted in Figure 5. Moreover, just one set of d-q stator currents/voltages can be observed. \[i_{\text{d}} =i_{\alpha}\cos(\vartheta_{r})+i_{\beta}\sin(\vartheta_{r}) \tag{10}\] \[i_{\text{q}} =i_{\beta}\ \cos(\vartheta_{r})-i_{\alpha}\sin(\vartheta_{r})\] (11) \[V_{\alpha} =V_{\text{d}}\cos(\vartheta_{r})-V_{\text{q}}\sin(\vartheta_{r})\] (12) \[V_{\beta} =V_{\text{q}}\cos(\vartheta_{r})+V_{\text{d}}\sin(\vartheta_{r}) \tag{13}\]  
Clark transformations in inverse and direct directions were reconstructed under the rule of the generalized Park transformation with a zero position reference frame (Figure 5). This approach can be used for any number of phases according to the following known parameters: angles between phases, number of phases, and an arbitrary parameter taken as the reference frame position of the orthogonal \(\alpha-\beta\) system. With the use of the six-phase machine where at least one three-phase star is a convectional, balanced, symmetrical system with 120\({}^{\circ}\) between phases, an advantage can be taken into an account. This first system can be calculated using the standard three-phase Clark transformations, and the other phases can be added into transformation equations according to their angles to the reference frame. The transformation between the \(n\)-phase and the orthogonal system is always recalculated by coefficients of a proportion. For example, with \(k_{d}=k_{q}=2/3\) for three-phase machines, the transformed system is power noninvariant (an amplitude invariant), and the same applies to the 2 d-q system, as described in the previous section. Transformations to the 1 d-q system with \(k_{d}=k_{q}=2/3\) will lead to a system behavior with double gain, thus neither the power nor the amplitude is invariant. With the requirements of amplitude\(-\)invariant behavior, the coefficients of proportion \(k_{d}=k_{q}=2/6\) were used [32]. The proposed transformation can be used for a variable angle (\(\vartheta_{shift}\)) between two symmetrical three-phase systems. For the machine examined in this paper, \(\vartheta_{shift}=-30^{\circ}\). \[V_{A}=V_{\alpha} \tag{14}\]  
\[V_{B}=\frac{1}{2}\left(-V_{\alpha}+\sqrt{3}V_{\beta}\right) \tag{15}\]  
\[V_{C}=\frac{1}{2}\left(-V_{\alpha}-\sqrt{3}V_{\beta}\right) \tag{16}\]  
Figure 3:0  
Figure 3:1  
Figure 3:2 Field-oriented control structure 1 d-q. Figure 3:3  
Figure 3:4  
Figure 3:5