# Performance Analysis of Direct Torque Controllers in Five-Phase Electrical Drives

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- The fault-tolerant capability against a fault situation in the machine and/or the power converter, first presented in [2]. An n-phase machine can operate after one or several fault occurrences without any external equipment, as long as the number of healthy phases remains greater than or equal to three (assuming a single isolated neutral connection). Consequently, the system reliability is enhanced at the expense of a reduction in the post-fault electrical torque production.
- The capability of increasing the power density in healthy operation by injecting specific current harmonics, exposed in [3]. This is possible in certain multiphase machine configurations based on concentrated windings, where the lower current harmonic components can be used to increase the torque production.

## 2. The Case Study: Five-Phase Distributed Windings Induction Motor Drive Using a Conventional Two-Level VSI

^{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

_{se}]

^{T}, can be defined as a function of the switching states as follows:

_{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).

_{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. This is called the Clarke decoupled model of the electrical machine because a Clarke transformation (C

_{5}shown below, with ϑ = 2π/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.

_{5}by voltages in the (a, b, c, d, e) reference frame. The same happens with the stator current and flux, as well as with the rotor magnitudes (voltage, current, and flux).

_{a}S

_{b}S

_{c}S

_{d}S

_{e}]

^{T}expressed in binary logic (1 or 0), being S

_{a}and S

_{e}the most and the least significant bits, respectively. These vectors uniformly divide the space that they occupy in 10 sectors with a separation of π/5 between them. Likewise, active voltage vectors can be classified according to their magnitude in long (0.647 V

_{dc}), medium (0.4 V

_{dc}), and short (0.247 V

_{dc}) vectors. The switching states that generate long vectors in the α–β plane correspond to those that generate short vectors in the plane x–y and vice versa. The switching states corresponding to vectors of medium magnitude in the α–β plane, also generate medium vectors in the plane x–y. Null vectors are generated by the same switching states in both planes. This transformation allows for a detailed study of the harmonic components, since they are projected in certain planes. In particular, the fundamental frequency together with the harmonics of order 10 k ± 1 (k = 0, 1, 2, etc.) are mapped in the α–β plane, while the harmonics of order 10 k ± 3 are related to the plane x–y. The homopolar component and harmonics of order 5 k are projected on the z-axis.

_{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

_{s}

_{5}and P

_{r}

_{5}operators,

_{em}) can be obtained in different reference frames using the following equations, where n = 5 and p is the pair of poles of the electrical machine,

_{m}, the mechanical speed of the rotor shaft (ω

_{r}= p·ω

_{m}); T

_{L}, the load torque applied to the machine; J

_{m}, the rotational inertial constant; and B

_{m}, the friction coefficient of the rotor load bearings.

## 3. DTC in Five-Phase Drives

_{s}) depends on the voltage vector as follows:

_{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 γ, 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.

_{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 α–β plane, symbolize short vectors in the x–y plane (and vice versa), while switching states that generate medium vectors in the α–β plane also cause medium vectors in the x–y plane. In addition, long and medium voltage vectors with the same direction in the α–β 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 α–β 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

_{v}

_{1}and K

_{v}

_{2}) 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 α–β plane) and a short virtual vector (formed by a medium and a short vector in the α–β plane), as shown in the following equations and in Figure 5 and Figure 6:

_{25}and v

_{16}, respectively), which are in the same direction in the α–β subspace and are opposite in the subspace x–y. By selecting adequate values of K

_{v}

_{1}and K

_{v}

_{2}, it is obtained zero average volts-per-second in the x–y subspace (see Figure 5). Figure 6 shows all the virtual vectors in the α–β plane, VV

_{Li}being the long virtual vectors and VV

_{Si}the short ones. Furthermore, the dwell times of each vector in each sampling time T

_{s}to achieve the minimization of the x–y currents are K

_{v}

_{1}= 0.618 T

_{s}and K

_{v}

_{2}= 0.382 T

_{s}. However, it should be noted that even in the case where virtual voltage vectors are used, x–y currents are controlled with an open-loop strategy. Consequently, the machine must have low asymmetries and spatial harmonic content and/or high impedance in the x–y plane to effectively limit the circulation of x–y currents. It is important to note that the x–y components do not contribute to the torque in our case study machine, but they increase power losses in the electromechanical system. A good control practice is therefore to minimize the x–y components.

_{mth}the considered low-speed threshold. Selection of the applied VV is made using the lookup table shown in Table 1, as stated in [26,31,32].

## 4. Results and Discussion

_{s}*) is set to its rating value (0.4 Wb) during the experiments, the applied sampling frequency is fixed to 10 kHz and the hysteresis bands of the torque and flux regulators are programmed to be 1% of the rating values (this value was experimentally obtained for our test rig using a trial and error method). The maximum reference torque is set to 3.25 N·m, according to machine limits. The multiphase power converter is based on two conventional three-phase VSIs from SEMIKRON

^{®}(two SKS-22F modules in which five power legs are used). The DC link voltage is set to 300 V using an external DC power supply. The electronic control unit is based on a MSK28335 board and a Texas Instruments

^{®}TMS320F28335 digital signal processor. A digital encoder (GHM510296R/2500) and the enhanced quadrature encoder pulse peripheral of the DSP are used to measure the rotor mechanical speed ω

_{m}. The load torque (T

_{L}), which is demanded in the tests, is set by an independently controlled DC machine that is mechanically coupled to the five-phase machine. The experimental test rig is shown in Figure 8, where some photographs of the real system are included.

#### 4.1. Steady-State Operation

#### 4.2. Load Torque Rejection

#### 4.3. Step-Speed and Reverse Speed Tests

_{MAX}) is set to 3.25 N·m, being the stator flux constant and equal to the reference value.

#### 4.4. Fault-Tolerant Capability

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Mapping of the phase stator voltages of the two-level five-phase VSI in the α–β (

**left**graph) and x–y (

**right**graph) planes.

**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.

**Figure 9.**Experimental steady-state operation test where the reference speed is settled at 500 rpm and a load torque of 1 N·m is applied. Upper row: speed and torque responses. Second row: stator flux waveforms. Third row: current trajectories of the stator in the α–β and x–y planes. Last row: stator phase currents.

**Figure 10.**Experimental steady-state operation test where the reference speed is set at 500 rpm and a load torque of 2.75 N·m is applied. Upper row: speed and torque responses. Second row: stator flux waveforms. Third row: current trajectories of the stator in the α–β and x–y planes. Last row: stator phase currents.

**Figure 11.**Experimental response of the controlled system in a load torque rejection test. The reference speed is 500 rpm and a load torque is applied at 0.5 s. The upper row shows the speed and torque responses. The lower row shows the stator current waveform and modulus of the stator flux during the test.

**Figure 12.**Experimental response of the controlled system in a step speed test. The reference speed is changed from 0 to 500 rpm at 0.2 s. The upper row shows the speed and torque responses. The lower row shows the stator current waveform and modulus of the stator flux during the test.

**Figure 13.**Experimental response of the controlled system in a speed reversal test. The reference speed is changed from 500 to −500 rpm at 0.2 s. The upper row shows the speed and torque responses. The lower row shows the stator current waveform and modulus of the stator flux during the test.

**Figure 14.**Experimental response of the controlled system when stator phase a is open at t = 0.2 s and the system is controlled at 500 rpm. The upper row shows the speed response and the lower one shows the stator current waveforms corresponding to phases b, c, d, and e during the test.

**Figure 15.**Experimental response of the controlled system when stator phases a and b are open at t = 0.2 s and the system is controlled at 500 rpm. The upper row shows the speed response, and the lower one depicts the stator current waveforms corresponding to phases c, d, and e during the test.

**Figure 16.**Experimental response of the controlled system when stator phase a and c are open at t = 0.2 s and the system is controlled at 500 rpm. The upper row shows the speed response, and the lower one depicts the stator current waveforms corresponding to phases b, d, and e during the test.

dλ | dT | dω | Position of the Stator Flux (Sector) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||

+1 | +2 | +1 | VV_{L3} | VV_{L4} | VV_{L5} | VV_{L6} | VV_{L7} | VV_{L8} | VV_{L9} | VV_{L10} | VV_{L1} | VV_{L2} |

−1 | VV_{L2} | VV_{L3} | VV_{L4} | VV_{L5} | VV_{L6} | VV_{L7} | VV_{L8} | VV_{L9} | VV_{L10} | VV_{L1} | ||

+1 | +1 | VV_{S3} | VV_{S4} | VV_{S5} | VV_{S6} | VV_{S7} | VV_{S8} | VV_{S9} | VV_{S10} | VV_{S1} | VV_{S2} | |

−1 | VV_{S2} | VV_{S3} | VV_{S4} | VV_{S5} | VV_{S6} | VV_{S7} | VV_{S8} | VV_{S9} | VV_{S10} | VV_{S1} | ||

0 | +1 | v_{0} | v_{31} | v_{0} | v_{31} | v_{0} | v_{31} | v_{0} | v_{31} | v_{0} | v_{31} | |

−1 | v_{0} | v_{31} | v_{0} | v_{31} | v_{0} | v_{31} | v_{0} | v_{31} | v_{0} | v_{31} | ||

−1 | +1 | VV_{S9} | VV_{S10} | VV_{S1} | VV_{S2} | VV_{S3} | VV_{S4} | VV_{S5} | VV_{S6} | VV_{S7} | VV_{S8} | |

−1 | VV_{S10} | VV_{S1} | VV_{S2} | VV_{S3} | VV_{S4} | VV_{S5} | VV_{S6} | VV_{S7} | VV_{S8} | VV_{S9} | ||

−2 | +1 | VV_{L9} | VV_{L10} | VV_{L1} | VV_{L2} | VV_{L3} | VV_{L4} | VV_{L5} | VV_{L6} | VV_{L7} | VV_{L8} | |

−1 | VV_{L10} | VV_{L1} | VV_{L2} | VV_{L3} | VV_{L4} | VV_{L5} | VV_{L6} | VV_{L7} | VV_{L8} | VV_{L9} | ||

−1 | +2 | +1 | VV_{L4} | VV_{L5} | VV_{L6} | VV_{L7} | VV_{L8} | VV_{L9} | VV_{L10} | VV_{L1} | VV_{L2} | VV_{L3} |

−1 | VV_{L5} | VV_{L6} | VV_{L7} | VV_{L8} | VV_{L9} | VV_{L10} | VV_{L1} | VV_{L2} | VV_{L3} | VV_{L4} | ||

+1 | +1 | VV_{S4} | VV_{S5} | VV_{S6} | VV_{S7} | VV_{S8} | VV_{S9} | VV_{S10} | VV_{S1} | VV_{S2} | VV_{S3} | |

−1 | VV_{S5} | VV_{S6} | VV_{S7} | VV_{S8} | VV_{S9} | VV_{S10} | VV_{S1} | VV_{S2} | VV_{S3} | VV_{S4} | ||

0 | +1 | v_{31} | v_{0} | v_{31} | v_{0} | v_{31} | v_{0} | v_{31} | v_{0} | v_{31} | v_{0} | |

−1 | v_{31} | v_{0} | v_{31} | v_{0} | v_{31} | v_{0} | v_{31} | v_{0} | v_{31} | v_{0} | ||

−1 | +1 | VV_{S8} | VV_{S9} | VV_{S10} | VV_{S1} | VV_{S2} | VV_{S3} | VV_{S4} | VV_{S5} | VV_{S6} | VV_{S7} | |

−1 | VV_{S7} | VV_{S8} | VV_{S9} | VV_{S10} | VV_{S1} | VV_{S2} | VV_{S3} | VV_{S4} | VV_{S5} | VV_{S6} | ||

−2 | +1 | VV_{L8} | VV_{L9} | VV_{L10} | VV_{L1} | VV_{L2} | VV_{L3} | VV_{L4} | VV_{L5} | VV_{L6} | VV_{L7} | |

−1 | VV_{L7} | VV_{L8} | VV_{L9} | VV_{L10} | VV_{L1} | VV_{L2} | VV_{L3} | VV_{L4} | VV_{L5} | VV_{L6} |

Parameter | Value | Unit |
---|---|---|

Stator resistance, R_{s} | 12.85 | Ω |

Rotor resistance, R_{r} | 4.80 | Ω |

Stator leakage inductance, L_{ls} | 79.93 | mH |

Rotor leakage inductance, L_{lr} | 79.93 | mH |

Mutual inductance, M | 681.7 | mH |

Rotational inertia, J_{m} | 0.02 | kg-m^{2} |

Number of pairs of poles, p | 3 | - |

Closed-Loop System Performance | DTC |
---|---|

Speed tracking error when the fault appears | Negligible |

Torque tracking loss in control during the delay | No |

Robustness against fault detection delay | High |

Computational cost | Low |

Harmonic content in stator currents | High |

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**MDPI and ACS Style**

Bermúdez, M.; Barrero, F.; Martín, C.; Perales, M.
Performance Analysis of Direct Torque Controllers in Five-Phase Electrical Drives. *Appl. Sci.* **2021**, *11*, 11964.
https://doi.org/10.3390/app112411964

**AMA Style**

Bermúdez M, Barrero F, Martín C, Perales M.
Performance Analysis of Direct Torque Controllers in Five-Phase Electrical Drives. *Applied Sciences*. 2021; 11(24):11964.
https://doi.org/10.3390/app112411964

**Chicago/Turabian Style**

Bermúdez, Mario, Federico Barrero, Cristina Martín, and Manuel Perales.
2021. "Performance Analysis of Direct Torque Controllers in Five-Phase Electrical Drives" *Applied Sciences* 11, no. 24: 11964.
https://doi.org/10.3390/app112411964