# Comparative Study and Overview of Field-Oriented Control Techniques for Six-Phase PMSMs
## 2 Control in Two Orthogonal Systems  
For a better overview, the abbreviation \"2 d-q\" is established in this paper as two orthogonal systems. This approach is also known as dual, modular, or multiple three-phase control of \(n\times 3\)-phase machines (\(n=2,3,4,\dots\) ), as presented in [17; 18; 19; 20; 21; 22; 23]. As mentioned, three-phase field-oriented control is well known among engineers of electric drives, even for the control of two separated three-phase PMSMs by one microprocessor. Such an approach was adopted for the control of the six-phase machine, where a control structure was adapted to a common speed loop, as depicted in Figure 2.  
Two identical control structures were run in parallel, one for each three-phase system, where, in the second system, the transformation angle was shifted by \(-30^{\circ}\) in direct and inverse Park transformations. Similarly, an \(n\times 3\)-phase system can be controlled using functions intended for the control of conventional three-phase machines, limited by MCU power and a peripheral set. The point of connection for two parallel systems is in the speed loop, because the machine still contains just one shaft and produces one mechanical torque. In the 2 d-q control, all functions involved in the control of three-phase machines can be used, and the shape of the Clarke transformations or modulation function can remain unchanged.  
As a result of the 2 d-q control approach, two aligned sets of d-q currents/voltages in the rotor reference frame can be observed as two current loops run in parallel, controlled by separated, individual current controllers. Furthermore, two sets of \(\alpha-\beta\) orthogonal values in the stator reference frame can be recognized, where \(\alpha 1\) is aligned to phase \(A\), and \(\beta 1\) leads \(\alpha 1\) by \(90^{\circ}\), whereas the second orthogonal set \(\alpha 2\), \(\beta 2\) is shifted by \(-30^{\circ}\) from the first, as shown in Figure 3.  
Figure 1: (a) Six-phase setup; (b) machine phase configuration.  
Figure 2: Field-oriented control structure 2 d-q.  
\[A_{1},B_{1},C_{1}\ \to\ k=1\,,\ \vartheta_{shift}=0^{\circ}.\]  
\[A_{2},B_{2},C_{2}\ \to\ k=2\,,\ \vartheta_{shift}=-30^{\circ}.\]  
\[A_{2},B_{2},C_{2}\ \Leftrightarrow\ D,E,F.\]  
\[i_{ak}=i_{Ak} \tag{1}\]  
\[i_{\rm gk}\ =\frac{1}{\sqrt{3}}(i_{\rm gk}-i_{\rm Ck}) \tag{2}\]  
\[V_{Ak}=V_{ak} \tag{3}\]  
\[V_{Bk}=\frac{1}{2}\Bigl{(}-V_{ak}+\sqrt{3}V_{\rm gk}\Bigr{)} \tag{4}\]  
\[V_{Ck}=\frac{1}{2}\Bigl{(}-V_{ak}-\sqrt{3}V_{\rm gk}\Bigr{)} \tag{5}\]  
\[i_{\rm dk}=i_{ak}\cos\Bigl{(}\vartheta_{r}+\vartheta_{shift}\Bigr{)}+i_{\rm gk }\sin\Bigl{(}\vartheta_{r}+\vartheta_{shift}\Bigr{)} \tag{6}\]  
\[i_{\rm qk}=i_{\rm gk}\cos\Bigl{(}\vartheta_{r}+\vartheta_{shift}\Bigr{)}-i_{ \rm ak}\sin\Bigl{(}\vartheta_{r}+\vartheta_{shift}\Bigr{)} \tag{7}\]  
\[V_{ak}=V_{\rm dk}\cos\Bigl{(}\vartheta_{r}+\vartheta_{shift}\Bigr{)}-V_{\rm qk }\sin\Bigl{(}\vartheta_{r}+\vartheta_{shift}\Bigr{)} \tag{8}\]  
\[V_{\rm gk}=V_{\rm qk}\cos\Bigl{(}\vartheta_{r}+\vartheta_{shift}\Bigr{)}+V_{ \rm dk}\sin\Bigl{(}\vartheta_{r}+\vartheta_{shift}\Bigr{)} \tag{9}\]