# Performance Analysis of Direct Torque Controllers in Five-Phase Electrical Drives
## 1 Introduction
A graphical representation of the analyzed system is shown in Figure 1.  
The five-phase two-level VSI has \(2^{5}=32\) different switching states characterized by the switching vector \([S_{a}\ S_{b}\ S_{c}\ S_{d}\ S_{e}]^{T}\), with \(S_{k}=\{0,1\}\). Therefore, the phase voltages generated in the stator (subindex \(s\)), \([v_{sa}\ v_{sb}\ v_{sc}\ v_{sd}\ v_{sc}]^{T}\), can be defined as a function of the switching states as follows:  
\[\left[ \begin{array}{c}v_{sa}\\ v_{sb}\\ v_{sc}\\ v_{sd}\\ v_{se}\end{array} \right]=\frac{V_{dc}}{5}\cdot\left[ \begin{array}{cccccc}4&- 1&-1&-1&-1\\ -1&4&-1&-1&-1\\ -1&-1&4&-1&-1\\ -1&-1&-1&4&-1\\ -1&-1&-1&-1&4\end{array} \right]\cdot\left[ \begin{array}{c}S_{a}\\ S_{b}\\ S_{c}\\ S_{d}\\ S_{e}\end{array} \right]\]  
Note that the midpoint of the external DC power supply (\(V_{dc}\)) is assumed to be the ground of the electrical system and a balanced load is also considered, so the sum of all phase voltages must be equal to zero (\(v_{sa}+v_{sb}+v_{sc}+v_{sd}+v_{se}=0\)). The machine is modeled in the stationary reference frame (\(a\), \(b\), \(c\), \(d\), \(e\)) with a set of voltage equilibrium equations obtained from the electromagnetic circuits of the stator and rotor (subindexes \(s\) and \(r\), respectively), but then represented in two orthogonal planes, namely, \(\alpha-\beta\) and \(x-y\), plus the homopolar component \(z\) as follows:  
\[\left[ \begin{array}{c}v_{s\alpha}\\ v_{s\beta}\\ v_{sx}\\ v_{sy}\\ v_{sz}\end{array} \right]=R_{s}\cdot\left[ \begin{array}{c}i_{sa}\\ i_{s\beta}\\ i_{sx}\\ i_{sy}\\ i_{sz}\end{array} \right]+\frac{d}{dt}\left[ \begin{array}{c}\lambda_{sa}\\ \lambda_{s\beta}\\ \lambda_{sx}\\ \lambda_{sy}\\ \lambda_{sz}\end{array} \right]\]  
Figure 1: Schematic diagram of the case study. \[\left[ \begin{array}{c}\lambda_{s\kappa}\\ \lambda_{s\beta}\\ \lambda_{sx}\\ \lambda_{sy}\\ \lambda_{sz}\end{array} \right]=\left[ \begin{array}{cccc}L_{s}&0&0&0&0\\ 0&L_{s}&0&0&0\\ 0&0&L_{ls}&0&0\\ 0&0&0&L_{ls}&0\\ 0&0&0&0&L_{ls}\end{array} \right]\cdot\left[ \begin{array}{c}i_{s\kappa}\\ i_{s\beta}\\ i_{sx}\\ i_{sy}\\ i_{sz}\end{array} \right]+\left[ \begin{array}{c}L_{m}&0&0&0&0\\ 0&L_{m}&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{array} \right]\cdot\left[ \begin{array}{c}i_{r\kappa}\\ i_{r\beta}\\ i_{rx}\\ i_{ry}\end{array} \right]\]  
\[\left[ \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\end{array} \right]=R_{r}\cdot\left[ \begin{array}{c}i_{r\kappa}\\ i_{r\beta}\\ i_{rx}\\ i_{ry}\\ i_{ry}\end{array} \right]+\frac{d}{dt}\left[ \begin{array}{c}\lambda_{r\kappa}\\ \lambda_{r\beta}\\ \lambda_{r\gamma}\\ \lambda_{r\gamma}\\ \lambda_{r\gamma}\end{array} \right]+\left[ \begin{array}{c}0&\omega_{r}&0&0 &0\\ -\omega_{r}&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{array} \right]\cdot\left[ \begin{array}{c}\lambda_{r\kappa}\\ \lambda_{r\beta}\\ \lambda_{r\gamma}\\ \lambda_{r\gamma}\end{array} \right]\]  
\[\left[ \begin{array}{c}\lambda_{r\kappa}\\ \lambda_{r\beta}\\ \lambda_{r\gamma}\\ \lambda_{r\gamma}\\ \lambda_{r\gamma}\end{array} \right]=\left[ \begin{array}{cccc}L_{r}&0&0&0&0 \\ 0&L_{r}&0&0&0\\ 0&0&L_{lr}&0&0\\ 0&0&0&L_{lr}&0\\ 0&0&0&0&L_{lr}\end{array} \right]\cdot\left[ \begin{array}{c}i_{\kappa }\\ i_{r\beta}\\ i_{rx}\\ i_{ry}\\ i_{rz}\end{array} \right]+\left[ \begin{array}{c}L_{m}&0&0&0&0\\ 0&L_{m}&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\end{array} \right]\cdot\left[ \begin{array}{c}i_{s\kappa}\\ i_{s\beta}\\ i_{sx}\\ i_{sy}\\ i_{sz}\end{array} \right]\]  
where \(\lambda\) denotes flux variables, \(\omega_{r}\) is the electrical equivalent speed of the rotor, the resistances of the stator, and the rotor are \(R_{s}\) and \(R_{r}\), respectively, the mutual inductance is represented by \(L_{m}\), while \(L_{ls}\) and \(L_{lr}\) designate the leakage inductances of the stator and the rotor, respectively. Finally, \(L_{s}=L_{ls}+L_{m}\) and \(L_{r}=L_{lr}+L_{m}\) are called stator and rotor inductances.