# Adaptive Scheme for Detecting Induction Motor Incipient Broken Bar Faults at Various Load and Inertia Conditions

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- ▪
- Proposing a novel scheme based on an analytical approach to detect and diagnose BBFs. Hence, the results are easy to interpret. Furthermore, no training process is required.
- ▪
- The proposed scheme does not need any settings instead; it employs an adaptive threshold to discriminate between healthy and faulty cases under different operating conditions.
- ▪
- The proposed scheme can effectively detect incipient BBFs and non-adjacent BBFs, representing a stumbling block to many other methods in the literature.
- ▪
- The proposed scheme can precisely detect BBFs under variable loading and different inertia conditions.
- ▪
- The proposed scheme is immune to high-level noise and independent from motor parameters.

## 2. Simulating Broken Bar Faults in Induction Motor

## 3. Methodology of the Proposed Scheme

## 4. Analyzing the Variation of Main Sideband Phase Angle under Different Conditions

#### 4.1. Main Sideband Phase Angle Variation under Healthy Conditions

#### 4.1.1. Under Inertia Changing

^{2}, respectively) to the maximum inertia that each motor can accelerate according to NEMA MG-1 (5.7 and 16.84 kg·m

^{2}, respectively). The effect of this variation on the phase angle of the main sideband components is recorded in Figure 4 and Figure 5 for Motor I and Motor II, respectively.

^{2}affects the phase angle of the left main sideband component to change from 123.8° to 21.6° $\left(\u2206{\theta}_{1}^{\u2033}={102.2}^{\xb0}\right)$ and the phase angle of the right main sideband component to change from 25.2° to 128.82° $\left(\u2206{\mathsf{\theta}}_{2}^{\u2033}=-{103.6}^{\xb0}\right).$ Thus, the increase in ${\theta}_{1}^{\u2033}$ is approximately equal to the reduction in ${\theta}_{2}^{\u2033}$, which matches with the representation of current components in Figure 3 and the derivation in the previous section. For any healthy condition, the variation of main sideband phase angle will always be small.

#### 4.1.2. Under Load Changing

#### 4.2. Main Sideband Phase Angle Variation under Fault Conditions

^{2}to 170° and 140° for non-adjacent 2 BBR at pole pitch distance and full load at 0.63 and 0.063 kg·m

^{2}, respectively, as will be illustrated in the results section.

## 5. Implementation of the Proposed Scheme

- ▪
- Data acquisition stage: It includes the sampling process of the current and voltage signal and storing samples to reach the required frequency resolutions.
- ▪
- Data processing stage: It includes data windowing block, FFT block, and localization of the main sideband component block to obtain the magnitude and angle of these sideband components.
- ▪
- Adaptive threshold determination and fault detection stage: It includes a method for adaptive threshold calculation to differentiate between healthy and BBFs conditions.
- ▪
- Severity index calculation stage: It provides a severity index to designate the severity of the fault.

#### 5.1. Data Acquisition Stage

#### 5.2. Data Processing Stage

#### 5.2.1. Signal Windowing Block

#### 5.2.2. Fast Fourier Transformer Block

#### 5.2.3. Localization of Main Side Band Components

- Step 1:
- Determine the current power frequency components that have the maximum magnitude $\left({m}_{f}\right)$ and its associated frequency $\left(f\right)$ from the two arrays, ${m}_{f}=\mathrm{max}\left[m\right]$ and $f=freq\left({m}_{f}\right)$.
- Step 2:
- Determine the frequency searching zone for the main side band components, which is limited by $\left[\left(1-4{s}_{max}\right)f\right]$ and $\left[\left(1+4{s}_{max}\right)f\right]$ where ${s}_{max}$ is the maximum slip at which the motor can operate and its associated magnitude $\left[{m}_{\left(1-4{s}_{max}\right)f},\cdots ,{m}_{\left(1+4{s}_{max}\right)f}\right]$.
- Step 3:
- Search for the local maxima in the magnitudes of the searching zone such that ${m}_{i-1}\le {m}_{i}\le {m}_{i+1}$ and then arrange them in descending order in$\left[{m}_{l}\right]$and their associated frequencies in $\left[{f}_{l}\right]$.
- Step 4:
- Ensure that the frequency of the first ${f}_{l}\left({m}_{l}\left(1\right)\right)$ and the second greatest local maxima ${f}_{l}\left({m}_{l}\left(j+1\right)\right),j=1,2,\dots $ is symmetrical around the power frequency otherwise; take the third greatest local maxima instead of the second one and check for this condition.
- Step 5:
- When the condition in Step 4 is fulfilled, set the following:$${I}_{new}={m}_{f},{\varphi}_{new}={\theta}_{v}\left(f\right)-\theta \left(f\right)$$$${f}_{{I}_{1new}^{\u2033}}=\mathrm{min}\left({f}_{l}\left({m}_{l}\left(1\right)\right),{f}_{l}\left({m}_{l}\left(j+1\right)\right)\right)$$$${I}_{1new}^{\u2033}=m({f}_{{I}_{1new}^{\u2033}}),{\theta}_{1new}^{\u2033}=\theta \left({f}_{{I}_{1new}^{\u2033}}\right)$$$${f}_{{I}_{2new}^{\u2033}}=\mathrm{max}\left({f}_{l}\left({m}_{l}\left(1\right)\right),{f}_{l}\left({m}_{l}\left(j+1\right)\right)\right)$$$${I}_{2new}^{\u2033}=m({f}_{{I}_{2new}^{\u2033}}),{\theta}_{2new}^{\u2033}=\theta \left({f}_{{I}_{2new}^{\u2033}}\right)$$

#### 5.3. Adaptive Threshold Determination and Fault Detection Stage

- Calculate ${I}_{r},{I}_{1r}^{\u2033},,{I}_{2r}^{\u2033},{\theta}_{1r}^{\u2033},{\theta}_{2r}^{\u2033}and{\phi}_{r}$ at a reference healthy condition for the motor to be monitored using FFT at any inertia and under any loading conditions.
- Calculate the reference values ${\beta}_{r}$ and ${I}_{1}^{o}{}_{r}$ from Equations (12) and (14). Only positive and real values of them will be accepted.
- The previous two steps are carried out once for the healthy motor in the commissioning phase.
- Calculate ${I}_{new},{I}_{1new}^{\u2033},{I}_{2new}^{\u2033},{\theta}_{1new}^{\u2033},{\theta}_{2new}^{\u2033}and{\phi}_{new}$ for the new current samples using FFT.
- Calculate the value of ${\beta}_{new}$ from Equations (12) and (14). Using ${I}_{1}^{o}{}_{r}$, which has been obtained from Step (2) to limit the variation to be in ${\beta}_{new}$ only. The value of ${\beta}_{new}$ that has the largest deviation from ${\beta}_{r}$ will be selected to provide a safety margin to avoid false diagnosis.
- Calculate the magnitudes of ${i}_{1\mathrm{mnew}}^{\u2034},{i}_{1\mathrm{mnew}}^{\prime},{i}_{2\mathrm{mnew}}^{\prime}and{i}_{2\mathrm{mnew}}$ using Equations (7), (5), (8) and (6), respectively, with the values of ${\beta}_{new}$ and ${i}_{1}^{o}{}_{r}$
- To study the magnitude variation effect of current components calculated in Step (5) on ${\theta}_{1}^{\u2033},{\theta}_{2}^{\u2033}$, the model introduced in Figure 2 is used to calculate ${\theta}_{1\mathrm{mnew}}^{\u2033},{\theta}_{2\mathrm{mnew}}^{\u2033}$ using the data obtained from Step (5).
- Calculate threshold value using Equation (15).
- For checking BBFs occurrence, calculate the angle difference using Equation (16).
- If the condition $diff\le threshold$ is satisfied, the healthy condition is confirmed otherwise a broken bar fault is detected.

#### 5.4. Severity Index Calculation Stage

- Calculate the values of ${i}_{1}^{o}{}_{\mathrm{mnew}}$ from Equations (12) and (14) using ${\beta}_{r}$ value, which has been obtained from Step (2) in the threshold determination module.
- Calculate the average of ${i}_{1}^{o}{}_{\mathrm{mnew}}$ values.
- Calculate the corrected current as follows, as ${i}_{1}^{o}{}_{\mathrm{mnew}}$ is directly proportional to ${I}_{new}$:$${i}_{1}^{o}{}_{mcorrected}={i}_{1}^{o}{}_{\mathrm{mnew}}\times \frac{{I}_{r}}{{I}_{new}}$$
- Calculate the severity index:$$severityindex=\frac{{i}_{1}^{o}{}_{mcorrected}}{{I}_{1}^{o}{}_{r}}$$

## 6. Testing Results for Proposed Scheme Performance

#### 6.1. Under Different System Inertia

^{2}) on both measured main sideband phase angles and adaptive threshold estimation under healthy conditions at full load conditions. The obtained results in the table confirmed such cases as healthy cases since the calculated difference is less than the adaptive estimated threshold.

^{2}). As obviously shown, different faults of BBR, 2 BBR and 3 BBR are accurately detected.

#### 6.2. Under Different Loading Conditions

#### 6.3. Under Different Fault Severity

^{2}, respectively. In addition, half BBR faults are detected at full loading, 50%, 25% and 10% loading as illustrated from Cases 3, 12, 15 and 18, respectively, as the measured differences were higher than the estimated adaptive threshold in such cases.

#### 6.4. In a Noisy Environment

#### 6.5. Faulty Severity Determination

#### 6.6. Validation of the Proposed Scheme Using Real Experimental Dataset

^{2}. The motor has the following characteristics: 400 V, 1.5 kW, 2-poles, 3.25 A, and 2860 rpm at full load. The phase current was captured by a current transformer and sampled with 5 kHz. Bars were drilled to represent a partial broken bar and one broken bar. The dataset has contained the following cases at full load and constant inertia:

- -
- Healthy;
- -
- One bar with 3 mm diameter hole;
- -
- One bar with two 3 mm diameter holes each;
- -
- One bar with two 4 mm diameter holes each (as illustrated in Figure 13a);
- -

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | Definition |

AI | Artificial Intelligence |

BBF | Broken Bar Fault |

BBR | Broken Bar |

EMD | Empirical Mode Decomposition |

EMF | Electromotive Forces |

ESPRIT | Estimation of Signal Parameters via Rotational Invariance Techniques |

FC | Fundamental Current Component |

FEM | Finite Element Method |

FFT | Fast Fourier Transform |

HBBR | Half-Broken Bar |

HHT | Hilbert–Huang transform |

MCSA | Motor Current Signature Analysis |

MUSIC | Multiple Signal Classification |

NEMA | National Electrical Manufacturers Association |

RLSH | Left-Side Harmonic |

RMS | Root Mean Square |

RSH | Right Sideband Harmonic |

SNR | Signal To Noise Ratio |

WT | Wavelet Transform |

${E}_{1-2s}$ and ${E}_{1+2s}$ | The RMS of electromotive force of the LSH and RSH respectively |

${e}_{1-2s}$$\mathrm{and}{e}_{1+2s}$ | The instantaneous of electromotive force of the LSH and RSH respectively |

$f$ | The supply frequency |

${f}_{r}$ | Rotor frequency component |

${I}_{a}$ | The stator current |

$J$ | System inertia |

$I,{I}_{l}$ and ${I}_{r}$ | The RMS of fundamental current, LSH and RSH components respectively |

${i}_{1}^{\prime},{i}_{2},{i}_{1}^{\u2034},{i}_{2}^{\prime},{i}_{2}^{\u2033}$ and ${i}_{1}^{\u2033}$ | Notations for different current components, where subscript (1) and (2) indicate components related to LSH and RSH respectively |

${i}_{1}^{o}$ | Initial left sideband current |

${i}_{1}^{o}{}_{mcorrected}$ | The corrected values of ${i}_{1}^{o}$ |

$P$ | Number of poles |

$s$ | Motor slip |

$\omega $ | Synchronous speed |

${Z}_{1-2s}$ and ${Z}_{1+2s}$ | The circuit impedance for LSH and RSH respectively |

$Z$ and $\varphi $ | The magnitude and the angle of the equivalent circuit impedance at supply frequency |

$\alpha ,{\alpha}_{l}$ and ${\alpha}_{r}$ | The phase angles of FC, LSH, and RSH |

${\alpha}_{\lambda}$ | The angle of the fundamental flux linkage |

$\beta $ | A parameter equals $\frac{3{P\lambda}^{2}}{8Js\omega \left|Z\right|}$ |

${\theta}_{1}^{\u2033}$ and ${\theta}_{2}^{\u2033}$ | The phase angle of the current components ${i}_{1}^{\u2033}\mathrm{and}{i}_{2}^{\u2033}$, respectively |

$\Delta \theta $ | The change in the angle $\theta $ |

$\lambda $ | The RMS value of the fundamental flux linkage |

${\lambda}_{fund}$ | The instantaneous value of the fundamental flux linkage |

${\dots}_{r}$ | The subscript$\left(r\right)$ denotes reference cases |

${\dots}_{new}$ | The subscript$\left(new\right)$ denotes new cases |

${\dots}_{\mathrm{mnew}}$ | $\mathrm{The}\mathrm{subscript}\left(\mathrm{mnew}\right)$ indicates the new case current components calculated using the model in Figure 2 |

${\dots}_{m}$ | The subscript (m) indicates the model indicates the model |

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**Figure 1.**Characteristics for modeling Motor I with 2 BBR. (

**a**) The magnetic flux lines distribution. (

**b**) Periodic oscillation in the stator current envelop. (

**c**) The main sideband frequency components for stator current (40–60 Hz). (

**d**) Phase angle of the main sideband components.

**Figure 2.**The effect of main sideband current components magnitude variation on their angles variation. (

**a**) The effect of right sideband component magnitude variation on $\u2206{\theta}_{2}^{\u2033}$. (

**b**) The effect of left sideband component magnitude variation on $\u2206{\theta}_{1}^{\u2033}$.

**Figure 3.**A phasor diagram representing the impact of inertia changes on ${i}_{1}^{\u2033}\mathrm{and}$ ${i}_{2}^{\u2033}$ current components under healthy conditions.

**Figure 4.**Motor I phase angle of left sideband and right sideband components for healthy conditions at inertia of; inertia of 0.063 kg·m

^{2}(*), inertia of 0.63 kg·m

^{2}(●), inertia of 0.93 kg·m

^{2}(×), and inertia of 5.7 kg·m

^{2}(+).

**Figure 5.**Motor II phase angle of left sideband and right sideband components for healthy conditions at; inertia of 16.84 kg·m

^{2}(+), and inertia of 0.041 kg·m

^{2}(*).

**Figure 6.**A phasor diagram representing the impact of load variation (from full loading to partial loading) on ${i}_{1}^{\u2033}and$ ${i}_{2}^{\u2033}$ current components under healthy conditions.

**Figure 7.**Motor I phase angle of left sideband, right sideband and fundamental components for healthy conditions at; 10% loading (■), 25% (+), 50% loading (×), and full loading (*).

**Figure 8.**A phasor diagram representing the impact of BBF on ${i}_{1}^{\u2033}and$ ${i}_{2}^{\u2033}$ current components.

**Figure 9.**Motor I vector representation of left sideband and right sideband components for; adjacent 2 BBR at inertia 0.63 kg·m

^{2}(●), non-adjacent 2 BBR at inertia 0.63 kg·m

^{2}(×), BBR at inertia 0.63 kg·m

^{2}(■), HBBR at 5.7 kg·m

^{2}(+), and healthy conditions at 0.063 kg·m

^{2}(*).

**Figure 10.**Motor II vector representation of left sideband and right sideband components for; inertia of 16.84 kg·m

^{2}(+), HBBR at 16.84 kg·m

^{2}(×), and HBBR at 0.042 kg·m

^{2}(∙).

Data | Motor I [4] | Motor II |
---|---|---|

Power | 11 kW | 11 kW |

Voltage (rms) | 380 V | 380 V |

No. of poles | 4 | 6 |

Rated slip | 2.9% | 2.3% |

Number of stator slots | 36 | 72 |

Number of rotor bars | 28 | 58 |

Number of turns | 27 | 6 |

Silicon steel material | M19_29G | M19_24G |

Case No. | Case Description | Performance of the Proposed Scheme | |||||
---|---|---|---|---|---|---|---|

Healthy/BBFs | Loading Condition (%) | Inertial Condition (kg·m ^{2}) | Estimated Threshold (°) | Calculated Phase Angle Difference (°) | Status | Severity Index | |

1 | Healthy | Full load | 5.7 | 49.97 | 8.62 | Healthy | -------- |

2 | BBR | Full load | 5.7 | 52.09 | 120 | Fault | 0.965 |

3 | Half BBR | Full load | 5.7 | 56.28 | 132.5 | Fault | 0.418 |

4 | Healthy | Full load | 0.625 | 22.58 | 0.91 | Healthy | -------- |

5 | Non-adjacent 2 BBR | Full load | 0.625 | 20.72 | 170.1 | Fault | 1.73 |

6 | 2 BBR | Full load | 0.625 | 23.27 | 155.1 | Fault | 2.126 |

7 | BBR | Full load | 0.625 | 23.74 | 145.30 | Fault | 1.034 |

8 | Healthy | Full load | 0.925 | 23.59 | 1.43 | Healthy | -------- |

9 | 3 BBR | Full load | 0.925 | 19.2085 | 104.5312 | Fault | 5.27 |

10 | Non-adjacent 2 BBR | Full load | 0.0625 | 19.99 | 141.4 | Fault | 2.26 |

11 | BBR | 50% | 0.0625 | 22.23 | 60.9 | Fault | 1.36 |

12 | Half BBR | 50% | 0.0625 | 24.56 | 43.44 | Fault | 1.030 |

13 | Healthy | 25% | 0.0625 | 26.42 | 2.72 | Healthy | -------- |

14 | BBR | 25% | 0.0625 | 24.16 | 50.6 | Fault | 1.09 |

15 | Half BBR | 25% | 0.0625 | 26.09 | 28.94 | Fault | 0.96 |

16 | Healthy | 10% | 0.0625 | 29.00 | 0.71 | Healthy | -------- |

17 | BBR | 10% | 0.0625 | 26.19 | 154.6 | Fault | 0.736 |

18 | Half BBR | 10% | 0.0625 | 27.81 | 102.38 | Fault | 0.708 |

19 | Healthy + 20 db Noise | Full load | 0.0625 | 21.2 | 9.75 | Healthy | -------- |

20 | Half BBR + 20 db Noise | Full load | 5.7 | 36.9 | 76.03 | Fault | 0.484 |

21 | Half BBR + 18.5 db Noise | 25% | 0.0625 | 20.67 | 34.74 | Fault | 1.96 |

Case No. | Case Description | Performance of the Proposed Scheme | ||||
---|---|---|---|---|---|---|

Healthy/BBFs | Loading Condition (%) | Inertial Condition (kg·m ^{2}) | Estimated Threshold (°) | Calculated Difference (°) | Status | |

1 | Healthy | Full load | 0.0410 | 64.08 | 5.85 | Healthy |

2 | Half BBR | Full load | 0.0410 | 48.49 | 169.67 | Fault |

3 | Half BBR | Full load | 16.84 | 48.56 | 60.58 | Fault |

4 | Healthy | 90% loading | 0.0410 | 49.57 | 3.13 | Healthy |

Case No. | Case Description | Performance of the Proposed Scheme | |||||
---|---|---|---|---|---|---|---|

Healthy/BBFs | Loading Condition (%) | Inertial Condition (kg·m^{2}) | Estimated Threshold (°) | Calculated Phase Angle Difference (°) | Status | Severity Index | |

1 | Healthy | Full load | 0.11 | 13.015 | 3.2 | Healthy | -------- |

2 | Healthy | Full load | 0.11 | 14.5 | 3.4 | Healthy | -------- |

3 | A bar with one 3 mm diameter hole | Full load | 0.11 | 18.5 | 77.94 | Fault | 0.78 |

4 | One bar with two 3 mm diameter holes each | Full load | 0.11 | 18.069 | 93.836 | Fault | 4.17 |

5 | One bar with two 4 mm diameter holes each | Full load | 0.11 | 18.071 | 160.78 | Fault | 5.02 |

6 | One full broken bar | Full load | 0.11 | 13.5 | 166.49 | Fault | 7.2 |

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## Share and Cite

**MDPI and ACS Style**

Atta, M.E.E.-D.; Ibrahim, D.K.; Gilany, M.; Zobaa, A.F.
Adaptive Scheme for Detecting Induction Motor Incipient Broken Bar Faults at Various Load and Inertia Conditions. *Sensors* **2022**, *22*, 365.
https://doi.org/10.3390/s22010365

**AMA Style**

Atta MEE-D, Ibrahim DK, Gilany M, Zobaa AF.
Adaptive Scheme for Detecting Induction Motor Incipient Broken Bar Faults at Various Load and Inertia Conditions. *Sensors*. 2022; 22(1):365.
https://doi.org/10.3390/s22010365

**Chicago/Turabian Style**

Atta, Mohamed Esam El-Dine, Doaa Khalil Ibrahim, Mahmoud Gilany, and Ahmed F. Zobaa.
2022. "Adaptive Scheme for Detecting Induction Motor Incipient Broken Bar Faults at Various Load and Inertia Conditions" *Sensors* 22, no. 1: 365.
https://doi.org/10.3390/s22010365