# Self-Sensing Six-Phase PMSM Drive Based on Back-EMF Measurement
## 2 Mathematical Model of Six-Phase PMSM  
The mathematical model of the six-phase PMSM can be constructed as the model of the dual three-phase machine (1) and (2) [15]. To each three-phase system corresponds one torque- and flux-productive subspace \(d1\)-\(q1\) and \(d2\)-\(q2\). Parameters \(L_{d1}\), \(L_{q1}\), \(L_{d2}\) and \(L_{q2}\) represent synchronous inductances of the first and the second three-phase windings, while \(M_{d12}\) and \(M_{q12}\) are mutual inductances between them. The size of the mutual coupling between partial three-phase systems in comparison to synchronous inductances is shown in (3) and (4), where \(M\) is the mutual inductance between two phases placed in the same position. \[ \begin{array}{c}\begin{bmatrix}V_{d1}\\ V_{q1}\end{bmatrix}=\begin{bmatrix}R_{s}&0\\ 0&R_{s}\end{bmatrix}\begin{bmatrix}i_{d1}\\ i_{q1}\end{bmatrix}+\begin{bmatrix}L_{d1}&0\\ 0&L_{q1}\end{bmatrix}\frac{d}{dt}\begin{bmatrix}i_{d1}\\ i_{q1}\end{bmatrix}+\begin{bmatrix}M_{d12}&0\\ 0&M_{q12}\end{bmatrix}\frac{d}{dt}\begin{bmatrix}i_{d2}\\ i_{q2}\end{bmatrix}\\ +\begin{bmatrix}0&-L_{q1}\\ L_{d1}&0\end{bmatrix}\omega_{e}\begin{bmatrix}i_{d1}\\ i_{q1}\end{bmatrix}+\begin{bmatrix}0&-M_{q12}\\ M_{d12}&0\end{bmatrix}\omega_{e}\begin{bmatrix}i_{d2}\\ i_{q2}\end{bmatrix}+\omega_{e}\begin{bmatrix}0\\ \psi_{PM}\end{bmatrix}\end{array} \end{array} \tag{1}\]  
\[ \begin{array}{c}\begin{bmatrix}V_{d2}\\ V_{q2}\end{bmatrix}=\begin{bmatrix}R_{s}&0\\ 0&R_{s}\end{bmatrix}\begin{bmatrix}i_{d2}\\ i_{q2}\end{bmatrix}+\begin{bmatrix}L_{d2}&0\\ 0&L_{q2}\end{bmatrix}\frac{d}{dt}\begin{bmatrix}i_{d2}\\ i_{q2}\end{bmatrix}+\begin{bmatrix}M_{d12}&0\\ 0&M_{q12}\end{bmatrix}\frac{d}{dt}\begin{bmatrix}i_{d1}\\ i_{q1}\end{bmatrix}\\ +\begin{bmatrix}0&-L_{q2}\\ L_{d2}&0\end{bmatrix}\omega_{e}\begin{bmatrix}i_{d2}\\ i_{q2}\end{bmatrix}+\begin{bmatrix}0&-M_{q12}\\ M_{d12}&0\end{bmatrix}\omega_{e}\begin{bmatrix}i_{d1}\\ i_{q1}\end{bmatrix}+\omega_{e}\begin{bmatrix}0\\ \psi_{PM}\end{bmatrix}\end{array} \end{array} \tag{2}\]  
\[L_{d1}=L_{q1}=L_{d2}=L_{q2}=L_{\sigma}+\frac{3}{2}M \tag{3}\]  
\[M_{d12}=M_{q12}=\frac{3}{2}M \tag{4}\]  
Another procedure for the construction of the mathematical model of the six-phase machine is the vector space decomposition (VSD) method [14]. The method represents the harmonics segregation in partial orthogonal subspaces. Only one subspace \(d\)-\(q\) is torque- and flux-productive (5), which covers the basic harmonic and harmonics of the order 12k \(\pm\) 1 (k = 1, 2, 3,... ). The additional subspace \(z1\)-\(z2\) is created (6) to cover all degrees of freedom in the six-phase machine. The subspace \(z1\)-\(z2\) includes harmonics of the order 6k \(\pm\) 1 (k = 1, 3, 5,... ) and represents the difference between the partial subspaces \(d1\)-\(q1\) and \(d2\)-\(q2\) defined in (1) and (2). \[ \begin{array}{c}\begin{bmatrix}V_{d}\\ V_{q}\end{bmatrix}=\begin{bmatrix}R_{s}&0\\ 0&R_{s}\end{bmatrix}\begin{bmatrix}i_{d}\\ i_{q}\end{bmatrix}+\begin{bmatrix}L_{d}&0\\ 0&L_{q}\end{bmatrix}\frac{d}{dt}\begin{bmatrix}i_{d}\\ i_{q}\end{bmatrix}+\begin{bmatrix}0&-L_{q}\\ L_{d}&0\end{bmatrix}\omega_{e}\begin{bmatrix}i_{d}\\ i_{q}\end{bmatrix}+\omega_{e}\begin{bmatrix}0\\ \psi_{PM}\end{bmatrix}\end{array} \end{array} \tag{5}\]  
\[ \begin{array}{c}\begin{bmatrix}V_{z1}\\ V_{z2}\end{bmatrix}=\begin{bmatrix}R_{s}&0\\ 0&R_{s}\end{bmatrix}\begin{bmatrix}i_{21}\\ i_{22}\end{bmatrix}+\begin{bmatrix}L_{\sigma}&0\\ 0&L_{\sigma}\end{bmatrix}\frac{d}{dt}\begin{bmatrix}i_{21}\\ i_{22}\end{bmatrix}\end{array}  \tag{6}\]The mutual coupling between two three-phase systems is not present in the vector space decomposition procedure, as the changes between partial systems are moved to the \(z1\)-\(z2\) subsystem. The value of the synchronous inductances \(L_{d}\) and \(L_{q}\) from the model (5) and (6) is defined in (7). Comparing dual three-phase and VSD mathematical models, the relation between inductances can be defined as shown in (8) and (9).