# Performance Analysis of Direct Torque Controllers in Five-Phase Electrical Drives
## 3 DTC in Five-Phase Drives
\begin{array}{l}T_{em}=\frac{5}{2}\cdot\frac{p\cdot k_{r}}{\sigma\cdot L_{s }}\cdot\left(\overset{\rightarrow}{\lambda}_{\text{ra}\beta}\times\overset{ \rightarrow}{\lambda}_{\text{sa}\beta}\right)=\frac{5}{2}\cdot\frac{p\cdot k_{ r}}{\sigma\cdot L_{s}}\cdot\left\|\overset{\rightarrow}{\lambda}_{\text{ra}\beta} \right\|\cdot\left\|\overset{\rightarrow}{\lambda}_{\text{sa}\beta}\right\| \cdot\sin\gamma\\ \sigma=1-\left(\frac{L_{n}^{2}}{L_{s}\cdot L_{r}}\right)\\ k_{r}=\frac{L_{n}}{L_{r}}\end{array} \]where \(\gamma\) is the angle of load (angle between the flux vectors of the stator and the rotor). Then, a change in \(\gamma\), obtained by applying a voltage vector, produces an increase or decrease in \(T_{em}\). Note that the rotor time constant is greater than the stator time constant, so it can be assumed that a slower variation of the rotor flux compared to the stator flux, and therefore the rotor flux, can be considered constant in a sampling time. Note also that there is an impact of a spatial voltage vector on the magnitude of the stator flux. Different results are obtained depending on the applied stator voltage (32 alternatives using the five-phases two-level VSI, being 30 active vectors and 2 null vectors; see Figure 4). In short, the applied voltage vector can be divided into a tangential and a radial component with respect to the flux. The tangential component produces a change in machine torque, increasing or decreasing the sine of the angle \(\gamma\), while the radial component modifies the magnitude of the stator flux, increasing or decreasing its modulus. Hence, the flux and the electromagnetic torque are controlled simultaneously using DTC. The application of the voltage vector controls the flux and electromagnetic torque and affects not only the \(\alpha-\beta\) plane, but also the plane \(x-y\), where losses and unwanted voltage and current harmonics are generated in the case study machine. Figures 2 and 4 show that 32 voltage vectors with 4 different magnitudes can be applied: 2 null vectors (0 volts applied), 10 short vectors (0.247 \(V_{dc}\) volts applied), 10 medium vectors (0.4 \(V_{dc}\) volts applied), and 10 long vectors (0.647 \(V_{dc}\) volts applied). Note also (see Figure 2) that switching states that represent long vectors in the \(\alpha-\beta\) plane, symbolize short vectors in the \(x-y\) plane (and vice versa), while switching states that generate medium vectors in the \(\alpha-\beta\) plane also cause medium vectors in the \(x-y\) plane. In addition, long and medium voltage vectors with the same direction in the \(\alpha-\beta\) subspace, are equivalent to medium and short vectors with opposite directions in the \(x-y\) subspace. The same happens with medium and short vectors in the \(\alpha-\beta\) plane, since they are equivalent to medium and long vectors with opposite directions in the plane \(x-y\). These geometrical characteristics make possible the definition of a kind of voltage vector, called virtual voltage vector or \(VV_{i}\)[23], which minimizes currents in the \(x-y\) plane. Each virtual vector is based on the application of two available voltage vectors (\(v1\) and \(v2\)) during adequate dwell time ratios (\(K_{v1}\) and \(K_{v2}\)) to generate zero average volts per second in the \(x-y\) subspace. It is then possible to define in each sector a long virtual vector (formed by a long and a medium vector in the \(\alpha-\beta\) plane) and a short virtual vector (formed by a medium and a short vector in the \(\alpha-\beta\) plane), as shown in the following equations and in Figures 5 and 6:  
\[VV_{Li} =v_{Long}\cdot K_{v1}+v_{Medium}\cdot K_{v2}\] \[VV_{Si} =v_{Medium}\cdot K_{v1}+v_{Short}\cdot K_{v2}\]  
Figure 4: Impact of voltage vectors on the stator flux and the load angle for the application of DTC. (a) Estimated phasor diagram. (b) Alternatives available using the 2-level 5-phase VSI.