# Performance Analysis of Direct Torque Controllers in Five-Phase Electrical Drives
## 3 DTC in Five-Phase Drives  
Direct Torque Control is a well-known strategy for three-phase electrical drives. It was presented in the mid-1980s by [PERSON] [15] and [PERSON] [16], showing fast flux and torque responses, as well as more robustness with respect to the variation of the electrical parameters of the machine and generating a high-torque/flux ripple and harmonic current content, compared to the more standard field-oriented control technique. The operating principle is based on an off-line look-up table, which is used to select the stator voltage to be applied to the machine. The selection is made taking into account the position of the flux vector and the stator flux and electromagnetic torque error signals, obtained from the difference between reference and estimated values and processed using hysteresis comparators. Another disadvantage of DTC that should be considered from the analysis of its operating principle is that it does not generate a constant switching frequency. In fact, this switching frequency is variable and depends on the operating point and the bandwidth of the hysteresis controllers. Note, however, that DTC schemes have been proposed to also be used with PI regulators and space vector PWM methods (see [17]), to compensate for the variable switching frequency and reduce the torque and flux ripple. DTC has been commercialized [18] and extended to the case of multiphase drives in recent times, considering different types of machines [19; 20], machine neutral connections [21; 22], and drives without speed sensors [23]. In the case of multiphase drives, since the controller has only two freedom degrees (stator flux and electromagnetic torque), there is no chance of regulating the current and voltage components in the orthogonal \(\alpha-\beta\) and \(x-y\) planes. In this sense, some DTC strategies have been developed that satisfy this additional requirement, controlling the current and voltage components in the \(\alpha-\beta\) plane while reducing at the same time the current and voltage components in the \(x-y\) plane. For example, in [24; 25], a modification of the traditional control scheme is proposed, performing a two-step search to minimize the effect of low-order harmonics. Alternatively, the use of virtual vectors has been suggested to reduce current distortion [23]. Some criteria have also been included in the selection process within the look-up table to improve its performance in the low-speed region and an optimization between the two zero vectors to minimize the average switching frequency obtained [26]. On the other hand, and based on the virtual vectors defined in [23], different DTC schemes are presented defining new virtual vectors and avoiding the use of the zero vector to reduce the common-mode voltage generated by the VSI in [27; 28], to improve open-phase fault operations in [29], or to avoid any reconfiguration of the controller when open-phase faults appear [30]. In our case example, the five-phase IM with distributed windings, the control goal is reduced to the \(\alpha-\beta\) plane (torque and flux regulation), while the \(x-y\) components are nullifying. The analysis is based on the machine model in the stationary reference frame, which defines the variation of the stator flux in the \(\alpha-\beta\) subspace using a vector notation as:  
\[\Delta\overset{\rightarrow}{\lambda}_{\text{sa}\beta}=\int_{0}^{t}\left( \overset{\rightarrow}{v}_{\text{sa}\beta}-R_{s}\cdot\overset{\rightarrow}{i }_{\text{sa}\beta}\right)\cdot dt\]  
If the voltage drop in the stator resistance is ignored for simplicity, it can be assumed that the variation in the modulus of the flux vector in a sampling period (\(T_{s}\)) depends on the voltage vector as follows:  
\[\Delta\overset{\rightarrow}{\lambda}_{\text{sa}\beta}\approx\overset{ \rightarrow}{v}_{\text{sa}\beta}\cdot T_{s}\]  
The electromagnetic torque behavior can be determined by the following expression:  
\[