*2.2. 6PIM Model in z*<sup>1</sup> − *z*<sup>2</sup> *Subspace*

The 6PIM model in the *z*<sup>1</sup> − *z*<sup>2</sup> subspace behaves as a passive resistor–inductor (*R*–*L*) circuit as:

$$
\begin{bmatrix} V\_{sz\_1} \\ V\_{sz\_2} \end{bmatrix} = \begin{bmatrix} R\_s + \rho L\_{ls} & 0 \\ 0 & R\_s + \rho L\_{ls} \end{bmatrix} \begin{bmatrix} I\_{sz\_1} \\ I\_{sz\_2} \end{bmatrix} \tag{3}
$$

where, *Lls* is the stator leakage inductance. In this paper, 6PIM is applied with two isolated neutral points, with which this structure prevents the zero sequence currents. Hence, the *o*<sup>1</sup> − *o*<sup>2</sup> components can be neglected.

#### **3. Conventional DTC of 6PIM**

In a six-phase voltage source inverter (VSI), there are 2<sup>6</sup> = 64 switching states. Each state produces a voltage space vector (defined as *Vk*) in the *α* − *β* or *z*<sup>1</sup> − *z*<sup>2</sup> subspaces, shown in Figure 1. As can be seen, there are 12 large (e.g., 48), 12 single medium (e.g., 57), 24 double medium (e.g., 53), 12 small (e.g., 54), and 4 null voltage vectors. The block diagram of conventional DTC is shown in Figure 2.

The stator flux in this approach is obtained as:

$$
\overline{\Psi}\_s = \int \left( \overline{V}\_s - R\_s \overline{I}\_s \right) dt \tag{4}
$$

The electromagnetic torque can be calculated using the stator flux and current as:

$$T\_{\mathfrak{e}} = 1.5 \, P(\overline{\mathbf{V}}\_{\mathfrak{s}} \cdot \overline{\mathbf{I}}\_{\mathfrak{s}}^{\*}) \tag{5}$$

where, *P* is the number of pole pairs. The reference values of the stator flux and electromagnetic torque are compared with the estimated ones, and the errors are applied to the hysteresis controller. The outputs of the hysteresis regulators denote the signs of torque and flux change. In order to minimize the errors, the optimum vector is selected through ST, which is tabulated in Table 1.

**Figure 2.** Conventional DTC block diagram.

In this table, *k* is the number of the sector. *Vk* is the applied voltage vector to the inverter, which is defined as binary numbers in switching states of VSI as in Table 2.

**Table 1.** ST of conventional DTC.


**Table 2.** Selected vectors in ST of conventional DTC.


In the conventional DTC, only the large voltage vectors in the *α* − *β* subspace are applied to the 6PIM to maximize the utilization of the dc-link. From Equation (4), it can be seen that the stator flux variations and the applied voltage vectors have the same direction. Hence, the changes in the stator flux depend on the applied voltage vectors. Compared to the stator time constant, the rotor time constant is very large. Therefore, the rotor flux linkage changes are negligible and it can be assumed constant during short transients [22]. By the application of the active voltage vectors, stator flux linkage vector will be moved away from rotor flux linkage vector and the angle between them will be greater. This leads to changes in torque according to Equation (5).

#### **4. Harmonic Currents Reduction by Duty Cycle Control Strategy**

From Figure 1, it can be seen that each voltage vector in the *α* − *β* subspace has a corresponding vector in the *z*<sup>1</sup> − *z*<sup>2</sup> subspace with different position and magnitude. It is recommended to make the average volt-second outcome in the *z*<sup>1</sup> − *z*<sup>2</sup> subspace near to zero. Accordingly, two voltage vectors have been applied to the inverter in each sampling period. Active voltage vectors should be in a same direction (in order to have high effect on torque) and their correspondents in *z*<sup>1</sup> − *z*<sup>2</sup> subspace should be in an opposite direction (in order to have less losses). Therefore, the selected vectors will produce high outcome in *α* − *β* subspace and low outcome in the *z*<sup>1</sup> − *z*<sup>2</sup> subspace.

The applied voltage vectors in the *α* − *β* subspace are expressed as:

$$\begin{cases} \begin{array}{c} V\_{M\_{x-\beta}} = \frac{\sqrt{2}}{3} V\_{dc} \\ V\_{L\_{x-\beta}} = \frac{\sqrt{6} + \sqrt{2}}{6} V\_{dc} \end{array} \tag{6} \end{cases} \tag{6}$$

where, *VM<sup>α</sup>*−*<sup>β</sup>* and *VL<sup>α</sup>*−*<sup>β</sup>* are single-medium and large voltage vectors in the *α* − *β* subspace. A suitable duty ratio is calculated as:

$$\left\{ \begin{array}{c} \left| V\_{M\_{\rm a-\beta}} T\_{M\_{\rm a-\beta}} \right| = \left| V\_{L\_{\rm a-\beta}} T\_{L\_{\rm a-\beta}} \right| \\\ T\_{M\rm a-\beta} + T\_{L\_{\rm a-\beta}} = T\_{\rm s} \end{array} \Rightarrow \left\{ \begin{array}{c} T\_{L\_{\rm a-\beta}} = 0.73 \ T\_{\rm s} \\\ T\_{M\_{\rm a-\beta}} = 0.27 \ T\_{\rm s} \end{array} \right. \tag{7}$$

where, *TL<sup>α</sup>*−*<sup>β</sup>* , and *TM<sup>α</sup>*−*<sup>β</sup>* are the duration of the large and single-medium voltage vectors application in the *α* − *β* subspace, and *Ts* is the sampling period. In this way, the losses in the *z*<sup>1</sup> − *z*<sup>2</sup> subspace are reduced strikingly, while the reduction in the electromagnetic components is subtle. For instance, vectors number 48 and 57 have the same direction in the *α* − *β* subspace and the opposite direction in the *z*<sup>1</sup> − *z*<sup>2</sup> subspace. These voltage vectors are applied to the motor as illustrated in Figure 3, where *gn* is the number of legs in six-phase VSI, and *Kv* is the duty ratio defined as:

**Figure 3.** Switching pattern.

If *gn* = 1 the upper switch is on and the lower switch is off. On the contrary, when *gn* = 0, the lower switch becomes on and the upper switch turns off. The compound of these vectors in each sampling period is named virtual vector, shown in Figure 4.

**Figure 4.** Virtual vectors in α-β subspace.

Figure 3 shows that in two legs (among the six legs) of the inverter, the switches' status has been changed. Therefore, by this method has more switching frequency against DTC. This increase in switching frequency is less than twice. The vectors in the *α* − *β* subspace are replaced by virtual vectors in ST shown in Table 2.

#### **5. Proposed Control Algorithm for the 6PIM Drive**

Using FOC framework [23], 6PIM's mathematical equations are transformed to the synchronous reference frame (*d* − *q*), which creates possibility of decoupled control of the torque and flux as a permanent-magnet separated-excitation *dc* motor. In this reference frame, the stator flux vector is located on *d*-axis which is shown in Figure 5.

**Figure 5.** 6PIM structure and stator flux vector in 3rd sector and applied inverter voltage vectors.

Orthogonal currents of the 6PIM are mapped into the synchronous reference frame using Park transformation based on the flux vector position, which is achieved by field orientation process. For the 6PIM control, *iq* is a torque -and *id* is a flux- producing components. Hence, the equations of the torque and flux are related to stator currents in the *d* − *q* frame as follows:

$$\begin{cases} \quad T\_c \propto i\_q \\\quad \lambda\_s \propto i\_d \end{cases} \tag{9}$$

In order to decrease the torque ripples in the 6PIM, a new approach is employed by modifying the ST's inputs. Since the inputs of the ST in the classical ST-DTC are the errors between command and actual values of the electromagnetic torque and the stator flux, it seems effective to use the errors between the set and actual values of the *iq*, *id*, instead. In order to redesign the ST-DTC method to use these inputs, the inputs of ST in the conventional DTC are used as inputs for PI regulators. The outputs of PI regulators are the command values of the currents in the *d* − *q* axis. Replacing Δ*Te*, Δ*φ<sup>s</sup>* by Δ*iq*, Δ*id* in ST-DTC, respectively, the proposed method provides better inputs to the same ST presented in Table 1 which leads to a better performance in 6PIM. Table 3 shows that through defining virtual vectors, the ST is the same with conventional DTC with different inputs.

**Table 3.** The switching table of the proposed scheme.


Δ*iq* and Δ*id* imply that changing the signs of the *iq* and *id* is required. If *iq* needs to be increased, then Δ*iq* = 1. If there is no *iq* requirement, then Δ*iq* = 0. Also, Δ*iq* = −1 denotes the decrease of *iq*. All the states are defined as the same for the notation of Δ*id* . These digital output signals of the hysteresis controllers are described as:

$$\begin{array}{ll}\Delta\_{i\_q} = 1 & \textit{if} \quad i\_q \le i\_q \star | \textit{hystresis} \text{ band}|\\ & \Delta\_{i\_q} = 0 & \textit{if} \quad i\_q = i\_q \star\\ \Delta\_{i\_q} = -1 & \textit{if} \quad i\_q \ge i\_q \star | \textit{hystresis} \text{ band}|\end{array} \tag{10}$$

Similarly, for the changes required for the d-axis of the stator current, Δ*id* is described as:

$$\begin{array}{ll}\Delta\_{\dot{i}\_d} = 1 & \text{if} \quad \dot{i}\_d \le \dot{i}\_d \text{\*}-|hystress \text{is band}|\\\Delta\_{\dot{i}\_d} = 0 & \text{if} \quad \dot{i}\_d \ge \dot{i}\_d \text{\*}+|hystress \text{is band}|\end{array} \tag{11}$$

The block diagram of the proposed control strategy is shown in Figure 6. To concurrently achieve low Total Harmonics Distortion )THD( of the motor currents and low torque ripples, the proposed vector control scheme is synthesized with the duty cycle control strategy. The ST applies two voltage vectors in each sampling period in order to eliminate *z*<sup>1</sup> − *z*<sup>2</sup> subspace components. In comparison with the conventional DTC, the switching frequency of the proposed scheme is increased (less than twice according to Figure 3.) because two voltage vectors are applied in each sampling period. In contrary, both harmonic currents and torque ripples are reduced. Moreover, the proposed scheme has fast dynamic response, similar to the conventional DTC, and does not need any PWM modulator that creates complexity and time delay.

**Figure 6.** Block diagram of the proposed control scheme for 6PIM.

#### **6. Simulation Results**

The proposed and duty cycle control methods are simulated in MATLAB/Simulink. The sampling period of both methods are set to 100 μs. All the parameters are assumed to be constant, although each of them can be changed under the thermal effect, which is not within the scope of this essay. The simulations are carried out based on real specifications for 6PIM. The 6PIM parameters are specified in Table 4.


**Table 4.** 6PIM parameters.

The simulation results for the duty cycle and the proposed DTC strategies under load change from 0 to about 3.5 Nm at *t* = 0.5 s, speed command of 100 rad/s, and flux command of 0.5 Wb are shown in Figures 7 and 8, a speed torque and stator flux reference signals are shown with red dashed lines.

**Figure 7.** Load change condition of the 6PIM controlled by the duty cycle control strategy (**a**) and proposed method (**b**).

**Figure 8.** Speed direction change condition of the 6PIM controlled by the duty cycle control strategy (**a**) and proposed method (**b**).

As it can be seen from the both simulations (Figures 7 and 8), the torque ripples and stator flux fluctuations of proposed method is lower in compared to duty cycle control strategy. However, due to additional PI controller is used for stator flux, Equation (4), running time is higher.

#### **7. Experimental Setup**

In addition to the simulations, the performance of the proposed method is validated experimentally. Figure 9 shows the experimental setup, which contains the 6PIM and its coupled load, the main processor, two three-phase VSIs, current and voltage transducers, shaft encoder, and single-phase bridge rectifier.

The applied processor used in the driver is an eZDSP F28335 based on the floating point TMS320F28335 chip. The motor speed is measured by an Autonics incremental shaft encoder (Autonics, Busan, South Korea) mechanically coupled to the 6PIM with resolution of 2500 P/R. LEM LTS6np current transducers are implemented to measure all the phases' currents in order to be used in the estimation and the control processes. The DC-link voltage is also measured using *LV* 25 − *p* voltage transducer. A DC generator is applied as load machine and a PCI-1716 data acquisition card (DAQ, Advantech, Milpitas, CA, USA) as an A/D converter. A 700-W, 24-stator slots three-phase squirrel-cage

induction motor, which has been rewound to construct a 4-pole asymmetrical 6PIM is also tested for the proposed method performance. The MATLAB/Embedded Coder is used to generate usable code for the code composer studio development environment. The digital motor control and *IQmath* libraries along with *IQ17* data type are employed. The sampling period is set to *Ts* = 100 μs with a dead-time of 2 μs.

**Figure 9.** Experimental setup.

To demonstrate the torque ripples reduction precisely, the torque figures are shown within a short time frame in Figure 10.

**Figure 10.** The torque response of the 6PIM in steady state with 4 Nm load, derived by (**a**) duty cycle control strategy; (**b**) proposed method.

The experimental results of the duty cycle control strategy and the proposed method under the load changing from 0 to about 3.5 N/m are shown in Figure 11. In this test, the speed and flux commands are 100 rad/s and 0.5 Wb, respectively. The provided tests illustrate the alleviation of torque ripples in the proposed method compared with the duty cycle control strategy.

**Figure 11.** Load injection experiment of 6PIM, controlled by duty cycle control strategy (**a**) and proposed method (**b**).

In Table 5, the torque ripples in the no load condition are investigated to show the differences between the proposed method and the conventional DTC and the duty cycle control-based DTC strategies. It is clear that the torque ripples for the 6PIM is effectively decreased for the proposed method in comparison with other two methods. Furthermore, as it is seen from current THD in Figure 12, the low order harmonics of the stator currents, especially the fifth and seventh harmonics, are considerably reduced for the proposed control method.

**Table 5.** Torque ripples in three different conditions of 6PIM driving by three different methods.

**Figure 12.** Current THD of the conventional DTC and proposed method.

#### **8. Conclusions**

In this paper the performance of the 6PIM was improved by a new vector control strategy. Using a new set of inputs for the ST in DTC method and applying the duty cycle control strategy leads to decrease in both torque ripples and harmonic currents. From a complexity viewpoint, the proposed technique falls between the DTC and FOC. This method is more simple compared with FOC strategy due to the absence of PWM algorithm, and has a fast dynamic similar to DTC strategy. The main limitation of the proposed technique is variable switching frequency compared with FOC. The effectiveness of the proposed control strategy was confirmed using both simulation and experimental tests.

**Author Contributions:** Methodology, validation and formal analysis, H.H.; writing—original draft H.H., A.T., M.H.H.; writing—review & editing, A.R., T.V. and A.K.; supervision, A.B.

**Funding:** This work was supported by the Estonian Research Council under Grants PUT1260.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

$$T = \frac{1}{\sqrt{3}} \begin{bmatrix} 1 & \cos(4\gamma) & \cos(8\gamma) & \cos(\gamma) & \cos(5\gamma) & \cos(9\gamma) \\ 1 & \sin(4\gamma) & \sin(8\gamma) & \sin(\gamma) & \sin(5\gamma) & \sin(9\gamma) \\ 1 & \cos(8\gamma) & \cos(4\gamma) & \cos(5\gamma) & \cos(\gamma) & \cos(9\gamma) \\ 1 & \sin(8\gamma) & \sin(4\gamma) & \sin(5\gamma) & \sin(\gamma) & \sin(9\gamma) \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 1 \end{bmatrix}$$

where, *γ* = *α* = *<sup>π</sup>* 6

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
