# Performance Analysis of Direct Torque Controllers in Five-Phase Electrical Drives
## 1 Introduction
This is called the Clarke decoupled model of the electrical machine because a Clarke transformation (\(C_{5}\) shown below, with \(\vartheta=2\pi/5\)) is used to refer the rotor variables to the stator reference frame, leading to an invariant transformation of voltage and current magnitudes and allowing a considerable simplification of the machine model. \[[C_{5}]=\frac{2}{5}\cdot\begin{bmatrix}1&\cos(\vartheta)&\cos(2\vartheta)& \cos(3\vartheta)&\cos(4\vartheta)\\ 0&\sin(\vartheta)&\sin(2\vartheta)&\sin(3\vartheta)&\sin(4\vartheta)\\ 1&\cos(2\vartheta)&\cos(4\vartheta)&\cos(6\vartheta)&\cos(8\vartheta)\\ 0&\sin(2\vartheta)&\sin(4\vartheta)&\sin(6\vartheta)&\sin(8\vartheta)\\ \frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\\ \endparticular, the fundamental frequency together with the harmonics of order \(10\)\(k\pm 1\) (\(k=0\), \(1\), \(2\), etc.) are mapped in the \(\alpha-\beta\) plane, while the harmonics of order \(10\)\(k\pm 3\) are related to the plane \(x-y\). The homopolar component and harmonics of order \(5\)\(k\) are projected on the \(z\)-axis. In distributed winding induction machines, as the one used in our study, only the components of the \(\alpha-\beta\) plane are involved in torque production and a new transformation of variables is desirable (see Figure 3), moving the \(\alpha-\beta\) plane to a new rotating reference frame \(d-q\) (at synchronous speed \(\omega_{a}\)), where the components are not oscillating, are constant in steady state and vary only in transient state. The basis of this transformation is the \(P_{55}\) and \(P_{r5}\) operators,  
\[[P_{55}]= \begin{bmatrix}\cos(\theta_{a})&\sin(\theta_{a})&0&0&0\\ -\sin(\theta_{a})&\cos(\theta_{a})&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\end{bmatrix} \;[P_{r5}]= \begin{bmatrix}\cos(\delta)&\sin(\delta)&0& 0&0\\ -\sin(\delta)&\cos(\delta)&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\\ 0&0&0&0&1\end{bmatrix} \] \[\theta_{a}=\int_{0}^{t}\omega_{a}\cdot dt\] \[\delta=\theta_{a}-\theta=\int_{0}^{t}(\omega_{a}-\omega_{r}) \cdot dt=\int_{0}^{t}\omega_{sl}\cdot dt\]  
Figure 3: Reference frames for modeling the electrical machine. Figure 2: Mapping of the phase stator voltages of the two-level five-phase VSI in the \(\alpha-\beta\) (**left** graph) and \(x-y\) (**right** graph) planes. Giving the following mathematical representation of the system:  
\[ \begin{bmatrix}v_{sd}\\ v_{sq}\\ v_{sx}\\ v_{sy}\\ v_{sz}\end{bmatrix} =R_{s}\cdot \begin{bmatrix}i_{sd}\\ i_{sq}\\ i_{sq}\\ i_{sq}\\ i_{sq}\\ i_{sq}\end{bmatrix} +\frac{d}{dt} \begin{bmatrix}\lambda_{sd}\\ \lambda_{sq}\\ \lambda_{sq}\\ \lambda_{sx}\\ \lambda_{sy}\\ \lambda_{sz}\end{bmatrix} + \begin{bmatrix}0&-\omega_{a}&0&0&0\\ \omega_{a}&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{bmatrix} \cdot \begin{bmatrix}\lambda_{rd}\\ \lambda_{rq}\\ \lambda_{rx}\\ \lambda_{ry}\\ \lambda_{rz}\end{bmatrix} \]  
\[ \begin{bmatrix}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{bmatrix} =R_{r}\cdot \begin{bmatrix}i_{rd}\\ i_{rq}\\ i_{rx}\\ i_{ry}\\ i_{ry}\\ i_{rz}\end{bmatrix} +\frac{d}{dt} \begin{bmatrix}\lambda_{rd}\\ \lambda_{rq}\\ \lambda_{rx}\\ \lambda_{ry}\\ \lambda_{rz}\end{bmatrix} + \begin{bmatrix}0&-\omega_{sl}&0&0&0\\ \omega_{sl}&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{bmatrix} \cdot \begin{bmatrix}\lambda_{rd}\\ \lambda_{rq}\\ \lambda_{rx}\\ \lambda_{ry}\\ \lambda_{rz}\end{bmatrix} \]  
\[ \begin{bmatrix}\lambda_{sd}\\ \lambda_{sq}\\ \lambda_{sx}\\ \lambda_{sy}\\ \lambda_{sz}\end{bmatrix} = \begin{bmatrix}L_{s}&0&0&0&0&0\\ 0&L_{s}&0&0&0\\ 0&0&L_{ls}&0&0\\ 0&0&0&L_{ls}&0\\ 0&0&0&0&L_{ls}\end{bmatrix} \cdot \begin{bmatrix}i_{sd}\\ i_{sq}\\ i_{sx}\\ i_{sy}\\ i_{sz}\end{bmatrix} + \begin{bmatrix}L_{m}&0&0&0&0\\ 0&L_{m}&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{bmatrix} \cdot \begin{bmatrix}i_{rd}\\ i_{rq}\\ i_{rq}\\ i_{ry}\\ i_{rz}\end{bmatrix} \]  
\[ \begin{bmatrix}\lambda_{rd}\\ \lambda_{rq}\\ \lambda_{rx}\\ \lambda_{ry}\\ \lambda_{rz}\end{bmatrix} = \begin{bmatrix}L_{r}&0&0&0&0\\ 0&L_{r}&0&0&0\\ 0&0&L_{lr}&0&0\\ 0&0&0&0&L_{lr}\end{bmatrix} \cdot