# Detection of Stator Winding Faults in PMSMs Based on Second Harmonics of Phase Instantaneous Reactive Powers

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## Abstract

**:**

## 1. Introduction

## 2. Reactive Power Signature Analysis

#### 2.1. Instantaneous Reactive Power Definition

_{abc}, are defined to be the cross-product of the voltage and current vectors, shown as:

_{a}|, |q

_{b}| and |q

_{c}|, are investigated, in order to show a different phase IRP behavior among the three phases when an ITSC happens in only one of them.

#### 2.2. ITSC Characterization

#### 2.2.1. Case of Healthy PMSMs

_{i}(t) in healthy case as:

#### 2.2.2. Case of Faulty PMSMs

_{abc}represent the phase shift caused by the ITSC, and V

_{abc}represent the amplitudes of the phase voltages after the ITSC. Then, the phase currents and voltages with ITSC are expressed as:

_{n3}and φ

_{nv}

_{3}, n = a, b, c, represent the amplitudes and the phase angles of the third harmonic voltages, respectively; I

_{n}

_{3}and, φ

_{ni}

_{3}, n = a, b, c, represent the amplitudes and the phase angles of the third harmonic currents, respectively. According to the definition in (3), the second, fourth and sixth harmonics will come out in the spectra of the phase IRPs, and by substituting (9) and (10) into (3), the phase IRPs with ITSC is derived as:

_{n}

_{0}is the dc component; q

_{n}

_{2cos}(and q

_{n}

_{2sin}), q

_{n}

_{4cos}(and q

_{n}

_{4sin}), and, q

_{n}

_{6cos}(and q

_{n}

_{6sin}) represent the second, fourth, and sixth harmonics in the phase IRPs, respectively; n = a, b, c. Neglecting higher harmonics in q

_{n}(t), since the low harmonics are relevant to the proposed indicator, the dc component and second harmonics are shown as:

_{xi1}and φ

_{xv1}(x = a, b, c) represent the phase angle of currents and the phase shift in voltages caused by the ITSC, respectively, shown as:

## 3. Fault Model and Simulation

#### 3.1. Voltage Equations of Faulty PMSM

_{s}), self and mutual inductance (L and M) and the back-EMFs (e

_{abc}). The ITSC is introduced by shorting a portion of healthy windings with a resistor (r

_{f}), which represents the insulation degradation between two shorted turns. Phase A is split into two separate windings, denoted as a

_{h}for the healthy part and a

_{f}for the shorted area. A circulating current, i

_{f}, is generated in the short-circuit path. The voltage equations for a PMSM with ITSC is given as [2]:

_{abc}is the phase voltage; i

_{abc}is the phase current; r

_{af}is the resistance of the shorted turns; L

_{abc}is the three-phase self-inductance; M

_{ab}, M

_{ac}and M

_{bc}are the mutual inductances between phase A-B, phase B-C and phase A-C, respectively; L

_{af}is the self-inductance of the shorted turns; M

_{afb}and M

_{afc}are the mutual inductances between the shorted turns and phase B and phase C, respectively; M

_{ahaf}is the mutual inductance between the healthy turns of phase A and shorted turns; e

_{af}is the back-EMF of the shorted turns; λ

_{pm}is the flux linkage of the permanent magnet; ω

_{e}is the electrical angular velocity; θ

_{e}is the electrical angular. The number of turns involved in the ITSC is denoted as N

_{f}and the total number of turns per phase is N

_{total}. The ratio is defined as follows:

#### 3.2. Simulation Results

#### 3.3. Fault Feature Analysis

_{d}was set to zero. Figure 3 shows the phase currents and the circulating current, the neutral current and the phase IRPs before and after introducing ITSC at t = 2 s.

_{f}, in the short-circuit path, and a small current in the same phase of i

_{f}was introduced to the neutral conductor due to the fault. It was clear that the ITSC also caused dc component changes and the second harmonic pulsations in the three-phase IRPs. Moreover, the second harmonic amplitude of q

_{a}was smaller than that of q

_{b}and q

_{c}, which was reasonable because more disturbance was generated in the faulty phase (phase A) compared to the other two healthy phases. According to (3), the pulsations in q

_{b}and q

_{c}should be larger than that in q

_{a}.

_{x3}, I

_{x3}, x = a, b, c), but also the unbalances of the fundamental components (V

_{x}, x = a, b, c) caused by ITSC. Moreover, the indicators of q

_{b}and q

_{c}are greater than q

_{a}, which provides a method to locate the faulty phase.

#### 3.4. ITSCs Detection at Different Operating Conditions

_{a}indicator is always below that of q

_{b}and q

_{c}, which makes it available for ITSC location. In Figure 5a, the amplitude of q

_{a}also behaves similarly, always below that of q

_{b}and q

_{c}. However, the difference at low speeds is not obvious.

_{b}and q

_{c}are 79.9% and 105.3%, respectively, which are very high and caused by the small dc components of q

_{b}and q

_{c}.

_{a}indicator is always below that of q

_{b}and q

_{c}at the whole torque range.

## 4. Experimental Validation

#### 4.1. Fault Feature Analysis in Experiments

_{d}was set to zero. Figure 10 shows the phase currents, and phase IRPs before and after introducing ITSC at t = 11.74 s. The slight increase in phase currents can be observed after introducing ITSC, which is mainly due to the braking torque caused by the circulating current in the shorted turns, and the phase currents have to be increased to match the load torque. The braking torque due to the fault increases as the number of shorted turns and the amplitude of the circulating current increase [25]. In addition, the ITSC also caused dc component variations and more pulsations in the phase IRPs. The increased pulsations turn out to be mainly the second harmonics. It is clear that the phase IRPs contain harmonic components even in the healthy condition, which are caused by the inherent unbalances in the machine and setup. The three-phase IRPs have the same waveform because no neutral is connected.

#### 4.2. Performance of Proposed Indicator

_{L}= 3 N∙m), and with varying load torque (from 0 to 21 N∙m with a step of 3) at fixed speed (ω

_{m}= 400 r/min). During the tests, a short-circuit resistor r

_{f}= 0.2 Ω was added to limit the circulating current i

_{f}and avoid motor failure. The amplitudes of the second harmonic IRPs and the proposed indicator under different speeds and different torques are presented in Figure 12 and Figure 13, respectively, where the healthy case and faulty case are marked with H and F, respectively. Since |q

_{a}|, |q

_{b}| and |q

_{c}| are the same, only |q

_{a}| is shown in the results.

_{a}increased from 0.283 to 0.533 after ITSC at 100 r/min, which means the fault can be detected theoretically, the amplitude of the second harmonic q

_{a}itself is relatively small at low speed, and is not credible for fault detection due to its low signal-to-noise ratio. In this situation, the disturbance in the motor system is more likely to cause misdiagnosis. Compared with amplitude, the indicator has a better performance at low speed. The indicator increased from 38.6% to 48.6% after ITSC at 100 r/min, as shown in Figure 12b, and this high percentage value and its increase make it credible for fault detection due to the high signal-to-noise ratio. As the speed increases from 100 to 800 r/min, both the amplitude and percentage of second harmonic q

_{a}increase due to the increased circulating current caused by the higher back EMF of shorted turns at higher speed, and it can be seen that the proposed indicator is reliable for ITSC detection within the rated motor speed, as shown in Figure 12b.

_{a}increase significantly after ITSC, as shown in Figure 13. Since the circulating current increases with the stator currents, the amplitude of second harmonic q

_{a}increases with torque, as shown in Figure 13a. However, the percentage decreases with torque, as shown in Figure 13b, because the amplitude of second harmonic q

_{a}is more sensitive to speed than to torque [17], and the dc component of q

_{a}increases significantly with torque. Similarly to the case at low speed in Figure 12a, the amplitude of second harmonic q

_{a}at zero torque is relatively small, which means the fault detection based on amplitude is not reliable enough. As a contrast, the percentage at zero torque is high due to the low dc component of q

_{a}, increasing from 71.84% to 120.97% after ITSC. When the motor is fully loaded at 21 N∙m, the percentage increased from 12.58% to 19.84% after ITSC. Within the rated torque range, the proposed indicator values both before and after ITSC are higher than 10%, which makes it reliable for ITSC detection in PMSM.

_{x}and μ

_{x}are the standard deviation and the mean of amplitude or percentage of second harmonic q

_{a}. This means that small CV represents higher robustness.

_{a}after ITSC are 0.953 and 0.364, respectively. Therefore, the proposed indicator is more robust with respect to speed than the amplitude itself. When the motor is lightly loaded (below 9 N∙m), the percentage is high due to the low dc component of q

_{a}, which results in low robustness of the proposed indicator to torque. From 9 to 21 N∙m, the CVs of the amplitude and percentage after ITSC are 0.377 and 0.238, respectively. In this case, the proposed indicator is more robust to torque than the amplitude itself.

#### 4.3. Online Fault Detection Algorithm

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 3.**The phase currents and circulating current (

**top**), the neutral current (

**middle**) and the phase IRPs (

**bottom**) before and after ITSC. Operating conditions: machine speed ω

_{m}= 300 r/min and load torque T

_{L}= 10 N∙m. Fault severity: Δ = 0.01, r

_{f}= 50 mΩ.

**Figure 4.**The spectra of currents (

**a**), voltages (

**b**) and phase IRPs (

**c**) in healthy and faulty cases. The same operating conditions and fault severity as that in Figure 3.

**Figure 5.**The amplitudes of the second harmonics in IRPs (

**a**) and the proposed indicators (

**b**), with ITSC, at different speeds. T

_{L}was fixed at 10 N∙m.

**Figure 6.**The amplitudes of the second harmonics in IRPs (

**a**) and the proposed indicators (

**b**), with ITSC, at different torques. ω

_{m}was fixed at 100 r/min.

**Figure 7.**The phase IRPs before and after ITSC, without neutral conductor. Operating conditions: machine speed ω

_{m}= 300 r/min and load torque T

_{L}= 10 N∙m. Fault severity: Δ = 0.01, r

_{f}= 50 mΩ.

**Figure 10.**The phase currents and circulating current (

**top**), the neutral current (

**middle**) and the phase IRPs (

**bottom**) before and after ITSC. Operating conditions: machine speed ω

_{m}= 500 r/min and load torque T

_{L}= 10 N∙m. Fault severity: Δ = 0. 1, r

_{f}= 0.2 Ω.

**Figure 11.**The spectra of currents (

**a**), voltages (

**b**) and phase IRPs (

**c**) in healthy and faulty cases. The same operating conditions and fault severity as Figure 10.

**Figure 12.**The amplitudes of the second harmonics in IRPs (

**a**) and the proposed indicators (

**b**), with ITSC, at different speeds. T

_{L}was fixed at 3 N∙m.

**Figure 13.**The amplitudes of the second harmonics in IRPs (

**a**) and the proposed indicators (

**b**), with ITSC, at different torques. ω

_{m}was fixed at 400 r/min.

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

Number of phases | 3 | Number of Slots/Poles | 12/10 |

Rated phase current I_{rms} | 18 A | Turns of Phase N_{total} | 300 |

Rated line voltage V_{rms} | 480 V | Phase Resistance r_{s} | 1.5 Ω |

Base speed | 800 r/min | Phase Inductance L_{s} | 112 mH |

Magnet flux λ_{pm} | 0.23 Wb | Mutual Inductance M_{s} | ≈0 mH |

Currents | $\frac{{3}^{\mathrm{r}\mathrm{d}}}{{1}^{\mathrm{s}\mathrm{t}}}\times 100$ | Voltages | $\frac{{3}^{\mathrm{r}\mathrm{d}}}{{1}^{\mathrm{s}\mathrm{t}}}\times 100$ | IRPs | $\frac{{2}^{\mathrm{r}\mathrm{d}}}{\mathrm{d}\mathrm{c}}\times 100$ |
---|---|---|---|---|---|

i_{a} | 0.19 | v_{a} | 0.5 | q_{a} | 0.76 |

i_{b} | 0.18 | v_{b} | 0.51 | q_{b} | 2.17 |

i_{c} | 0.18 | v_{c} | 0.53 | q_{c} | 2.68 |

Currents | $\frac{{3}^{\mathrm{r}\mathrm{d}}}{{1}^{\mathrm{s}\mathrm{t}}}\times 100$ | Voltages | $\frac{{3}^{\mathrm{r}\mathrm{d}}}{{1}^{\mathrm{s}\mathrm{t}}}\times 100$ | IRPs | $\frac{{2}^{\mathrm{r}\mathrm{d}}}{\mathrm{d}\mathrm{c}}\times 100$ |
---|---|---|---|---|---|

i_{a} | 1.92 | v_{a} | 0.48 | q_{a} | 35.96 |

i_{b} | 3.04 | v_{b} | 0.73 | q_{b} | 35.96 |

i_{c} | 1.88 | v_{c} | 0.61 | q_{c} | 35.96 |

Detection Methods | Signal-to-Noise Ratio | Robustness to Speed | Capability of Fault Location |
---|---|---|---|

Method based on the third harmonic stator voltages | Low | Low | √ |

Method based on the third harmonic stator currents | Low | Low | √ |

Method based on the second harmonic IRPs | Medium | Low | × |

The proposed methods | High | High | √ |

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## Share and Cite

**MDPI and ACS Style**

Huang, S.; Bi, Z.; Sun, Z.; Aggarwal, A.; Huang, X.; Wu, L.; Niu, F.
Detection of Stator Winding Faults in PMSMs Based on Second Harmonics of Phase Instantaneous Reactive Powers. *Energies* **2022**, *15*, 3248.
https://doi.org/10.3390/en15093248

**AMA Style**

Huang S, Bi Z, Sun Z, Aggarwal A, Huang X, Wu L, Niu F.
Detection of Stator Winding Faults in PMSMs Based on Second Harmonics of Phase Instantaneous Reactive Powers. *Energies*. 2022; 15(9):3248.
https://doi.org/10.3390/en15093248

**Chicago/Turabian Style**

Huang, Shaopo, Zhenguo Bi, Zhaojia Sun, Anmol Aggarwal, Xiaoyan Huang, Lijian Wu, and Feng Niu.
2022. "Detection of Stator Winding Faults in PMSMs Based on Second Harmonics of Phase Instantaneous Reactive Powers" *Energies* 15, no. 9: 3248.
https://doi.org/10.3390/en15093248