Prediction of Rotor Slot Size Variations in Induction Motor Using Polynomial Chirplet Transform and Regression Algorithms

For constant monitoring of rotor slot in induction motor, average rotor slot variation (ARSV) prediction is proposed. The rotor slots expand due to thermal stress and high intensity magnetic flux. The magnetic flux with high intensity is created in rotor lamination sheet, because of stretching and curving magnetic flux. The surface of rotor slot exhibits thermal stress due to over current which in turn generates heat and transfers the heat into the rotor lamination surface. The magnetic stress-based rotor slot size variation is never monitored or measured before. In the proposed ARSV method, multimodal sensor signals such as Giant magnetoresistance, temperature, current, vibration and voltage acquire magnetic stress and temperature stress of rotor. The acquired signals are analyzed through polynomial chirplet transform to obtain energy band values of signal. The energy band values and microscopic camera image-based rotor slot variation values are used for ARSV prediction. The prediction of ARSV is computed using polynomial regression (PR) method. From experimental results, it is observed that ARSV is greater than 5% from the normal size which leads to extreme damage to the motor through vibration, sparking and harmonics. The prediction accuracy of ARSV is about 94.6% when compared to manual measurement of slots in rotor. Moreover, the prediction of ARSV avoids the major faults, namely eccentricity, overload, under voltage, unbalance voltage, induced rotor slot, rotor crack and rotor burn.


Introduction
Electrical drive systems and converter topologies are used in various applications such as the medical industry, water pumping station, cement factory, textile industry and chemical factory. The fault monitoring of induction motor during runtime is the major problem. Effective monitoring and predictive maintenance of motor increases productivity, life span, performance, safety and dependability. At the same time, it decreases cost and time. In IM, rotor is the vital part. Rotor is placed inside the stator, which cuts the magnetic flux and induces a voltage in the rotor. The induced voltage causes electrical current through the rotor bars and rotor rotation creates magnetic field around the rotor bars. Magnetic field of stator creates a force on the rotor bars and causes the rotor to rotate. Figure 1 depicts the direction of various parameters such as force, magnetic field, current and SC in IM.
Magnetic and thermal stresses are created on the laminated surface of rotor because of shaft bend, broken rotor, induced slot on rotor, high voltage, over current and high speed. Thermal and magnetic stresses are due to SC flux variation in rotor slots. The high load applied to IM produces over current in the rotor which causes excessive heat in the rotor bars and expands the size of rotor slots. Table  1 shows the various sensors used for induction motor fault diagnosis. Until now, the research on rotor slot size variation is not attempted. Spin test with sensors is evaluated for the rotor deformation due to mechanical stress. Simulation model analyzes the thermal stress, magnetic stress and deformation [1]. The rapid rise in the heat of motor causes non-uniform magnetic stress in motor [2]. The motor leakage current is used for motor fault detection [3]. In the stress study, the impact of permanent magnet (PM) rotor segmentation is examined. Segmented magnets and tensile stresses are due to the edging effect of the magnet [4]. Smart sensor measures the vibration and current to identify the faults in rotor. The motor vibration signal is utilized to analyze the motor speed and harmonics for the detection of rotor bar faults [5]. To reduce, the mechanical failure of rotor and to construct a compact rotor the length and diameter of the rotor is optimized [6]. Short-circuit fault in permanent magnet synchronous motor stator winding is monitored through stray magnetic field [7]. Limitations of mechanical design of machines are studied using surface-mounted magnets [8]. The inductance of wound rotor influences the various parts of induction motor. The influence of inductance is studied for their performance. The resulting inductance is compared with FEM and simulated inductance [9]. Optimization design of a PM rotor structure with different constraints such as critical speed and deformation stress is studied. [10]. Broken rotor bar failures of IM are analyzed through Taylor-Kalman filters [11]. IM starts with soft starters. Starting condition is assessed through three-dimensional analytical method and mechanical stress [12,13]. Mechanical stress on rotor is calculated using equivalent centroid method [14]. Mechanical strength and electromagnetic performances are analyzed in rotor by FEM [15,16]. Thermal stress and mechanical deformations on rotor are investigated using 3-dimensional FEM [17]. Stray flux from air gap is used for fault detection in motor through online [18]. Multilayer perceptron classifier is used in this article to identify the rotor faults of motor [19]. End ring faults in rotor are detected at the initial stage which increases the motor performance [20]. Deep learning-based rotor fault detects multiple faults in squirrel cage IM [21].
A. Problem identification Until now, faults in IM are diagnosed using UWB, current, sound, temperature, vibration and infrared sensors. However, defects such as rotor slot variation, sparking, leakage flux from rotor lamination need to be detected through novel method and algorithms. Moreover, magnetic stress produced on the rotor laminated surface is due to SC flux. Magnetic stress leads to high sparking and reduces leakage flux and shaft bend. Runtime monitoring of the stress generated by the magnetic effect on rotor laminated surface avoids burning of rotor and stator winding, high sparking and decreases leakage flux. Measuring of magnetic stress in rotor lamination is an important problem for researcher and needs to be focused with different approaches. The major problem for rotor lamination is to identify the appropriate sensor. Identification of appropriate location is needed for acquiring maximum rotor lamination flux in order to measure the stress caused by variation in rotor slot size. To solve the above problem, ARSV prediction is used in this research article.

B. Contributions of the proposed research
In induction motor, rotor faults arise due to magnetic stress and thermal stress, which is the root cause for all faults in rotor. Researches focus on various rotor faults such as BRB, 2-mm hole in rotor bar, rotor burn and rotor crack, whereas faults in rotor initially start with thermal stress. When thermal stress increases, magnetic stress is also increased. The magnetic stress arises from the rotor slots in the lamination sheet. The rotor slots size variation leads to SC flux and damages the stator which in turn reduces the torque of motor. The high magnetic and thermal stress creates crack in rotor that leads to broken rotor. However, researches focus only on rotor bar fault or rotor burn fault and never measured the magnetic stress in rotor, which is starting point for all faults in rotor. Traditional methods analyze this rotor slot variation only through FEM and never during runtime of motor. In this paper, rotor slot size variation is predicted in the runtime of motor with following contributions as follows.
To acquire SC magnetic flux from rotor through GMR sensor and analyze the GMR sensor signal for the behavior of SC flux, during different fault condition in motor such as To analyze the impact of rotor slot size due to different induced faults such as shaft bend, overload, broken rotor bar and frequent starting by multimodal sensor signals.
To predict ASSV through the energy band of SC magnetic flux signal obtain from polynomial chirplet transform, ORaDWT, TQWT and machine learning algorithms such as logistic regression (LR), multiple linear regression (MLR) and polynomial regression (PR). ARSV prediction from machine learning algorithm verified through ground truth measurement of rotor slots.

Materials and Methods
In this research article, the ARSV prediction is proposed with a novel technique. Multiple sensors are used for measuring SC on rotor. Current sensor (Model no: SCT013) is fixed in series of three phase AC power supply to measure the current. Voltage sensor (Model no: ZMPT101B) is fixed parallel to the input of AC power supply. The vibration sensor (Model no: ADXL335) detects the vibration during SC magnetic flux. The GMR sensor (Model No:AA002-02) is placed on motor drain to acquire magnetic flux from the rotor surface. The temperature sensor (Model No: LM335) is fixed at the end shield to measure thermal stress on rotor as shown in Fig. 2.
A. Sensor Signal acquisition Multimodal sensor signals are acquired and stored in the personal computer for ARSV prediction. The multimodal sensor signal is acquired using universal serial bus (USB) data acquisition (DAQ) that comprises data controller, digital Class-B amplifier and four analog channels. Data controller acquires sensor signals in parallel and stores them in the USB DAQ's buffer. Signals from the buffer are transferred to the personal computer. USB DAQ analog channels acquire and scan the signals at the rate of 1200 and 1300 samples per second, respectively. USB DAQ's supply voltage is + 3.6 V, and the resolution of analog channels is12 bits.
B. Microscopic images for manual measurement of rotor slot.
USB Digital Microscopic camera (Model No: microscope Smars® HD 1920 × 1080P) acquires rotor image of the induction motor. From the obtained rotor image, the size of rotor slot is measured. The microscopic camera has a magnification range of 50 × to 1000 × that enables excellent picture and high video recording quality up to 1920*1080P. In this paper, after every induced fault rotor is removed immediately and ARSV is calculated by the USB Digital Microscopic camera. The above method is the manual method for rotor slot size variation.

C. Evaluation of Magnetic Stress on Rotor Laminations
The stresses, which are caused by heating and magnetic effects on rotor lamination, are measured through multimodal sensor signals. SC magnetic flux is induced through various faults such as shaft bend, rotor burn, broken rotor bar, 2-mm holes on the lamination sheet of rotor. The supply of current in IM ranges from 1 to 12 Amps. Supply voltage of IM is varied from 180 to 415 V. IM rotating speed is varied from 1300 to 1600 rpm using the variable transformer. The mechanical load is varied from 0 to 18 kg by adjusting the spring balance which is connected with the brake drum of the IM. IM is induced with the above mentioned faults, and the sensor values are acquired using DAQ. After creating each and every fault in IM, the transforms such as ORaDWT, TQWT and PCT are used for obtaining energy band values from sensor signals. To measure rotor slot variation, the rotor is dismantled from the motor within 320 s and the rotor images are obtained through microscopic camera. In the obtained rotor image, pixel measurement method is utilized for measuring the size of rotor slot. From each faults, around 2750 signals are obtained from the multimodal sensor. Subsequently, microscopic camera images are taken to measure slot size variation through MATLAB software. Figure 3 shows the proposed methodology of ARSV prediction.

D. Energy band values of multimodal sensor signal from ORaDWT for ARSV prediction
ORaDWT is similar to Mallat's tree filter bank of DWT which is implemented using repeated 2-channel filter bank as illustrated in Fig. 4. ORaDWT has up samplers in order to filter low-pass signals. ORaDWT's dilation factor is q/p, where p and q are co-prime integers. ORaDWT bases with 1q/p2 are employed in this paper for greater frequency resolution than DWT. ORaDWT functions such as wavelet[ψ(t)] and scaling [ϕ(t)] need to satisfy the scaling relation of rational dilation as in Eqs. (1) and (2).
where g 0 (n) and h 0 (n) represent the filtering of wavelet function and scaling function, respectively. The merging of p, q, and s builds ORaDWT and never fulfilled the arbitrary conditions as indicated in Eq. (3) The multimodal sensor signals processed with ORaDWT provides the energy band values which are used for the ARSV prediction. Figure 5 shows the distribution of GMR sensor signal energy across subbands during overcurrent condition. Figure 6 shows ORaDWT-sampling of GMR sensor signal during over current condition.  The high oscillatory and transient parts of the signal are obtained by decomposing the corresponding signal using various parameters of TQWT. The parameters are the redundancy' r', Q-factor 'Q' and Levels (L). Q-factor is the calculation of total number of oscillations experienced by the wavelet. The sensor signals with transient impact experience low Q-factor. The different stages of redundancy in TQWT are denoted by the parameter 'r'. The total count of filter banks is specified by the parameter 'J' for L 3 shown in Fig. 7. The subbands of (L + 1) are obtained from the L value that is measured from the 1 st filter bank with high-pass and the 3 rd filter bank with low-pass.

E. Energy band values extraction from TQWT through multimodal sensor signal for ARSV prediction
The value of α and β can be determined from the 'Q' and 'r' as in Eq. (5) The acquired faulty signals contain the components of oscillation and transient. The original sensor signal is shown in Eq. (6) where 'x' represents the original sensor signal, x 1 is due to induced transient fault which deduces the features of faults. x 2 specifies the component of high oscillation which includes the noise from atmosphere. TQWT 1 and TQWT 2 denote the TQWT with a high and a low value of Q-factor as in Eq. (7) arg min The regularization of subband dependency and flexibility of TQWT is shown in Eq. (8). where (ω ij ) specifies the j th subband of TQWT. The x 1 and x 2 are gained from the determined values of ω 1 and ω 2 as indicated in Eq. (9).
The sensor signal acquired from the IM during overloaded condition has noise. As a result, the original signal representation is as in Eq. (10).
The signal is transformed into the following model as in Eq. (11) arg min where "Φ 2 " and "Φ 1 " denote the inverse TQWT with a low value and high value of Q-factor. Figure 8 shows the distribution of GMR signal energy for over current condition.

F. Energy band analysis using PCT for ARSV prediction
Chirplet transform (CT) illustrates the energy distribution signal in time frequency (TF). Furthermore, CT calculates nonlinear frequency modulation of the signal and the time frequency representation (TFR) can be smudged. This issue is rectified through frequency shifting and rotating operator in CT. In polynomial chirplet transform, polynomial The analytic signal of PCT is as in Eq. (12).
Equation (13) represents the rotating operator of frequency which revolves the non-stationary signal in TF plane and helps in filtering the harmonic signals. Equation (14) represents frequency shifting operator that shifts the frequency of the rotating harmonic signal in TF plane into IF trajectory location. PCT and (short time Fourier transform) STFT incorporate rotating and shifting operation in frequency of signal. The result of PCT transform is intense because of the addition of rotating and shifting operation. To fulfill the variation of the rapidly changing IF in PCT, the polynomial kernel function is used.
The variables of polynomial kernel function is estimated from the highest TF energy depending on the IF trajectory system. The least square approach is added into PCT in the IF trajectory calculation method and is iteratively estimated the maximum TF energy using a polynomial function. To extract the IF trajectory more accurately, fast path optimization (FPO) is applied. Figures 9 and 10 show the energy percentage for thermal signal of frequent starting and high load condition for short time duty cycle of motor.

Results and Discussion
Developmental setup for the proposed system is shown in Fig. 11. The ARSV is predicted for a three-phase IM. The sensors are mounted at various locations of three phase IM. GMR sensor is placed on motor drain, temperature sensor is fixed on end shield, voltage sensor is placed across the supply, current sensor is connected in series to the supply, and vibration sensor is fixed on the base of the motor. The sensors need 3.3 V to 5 V DC power supply to acquire the signal from the motor. The acquired sensor signals are stored in the computer using DAQ and SIGVIEW software. The obtained signals are processed with various transforms and machine learning techniques to predict the ARSV. The transformations and machine learning techniques are performed by MATLABR2021a version software in Intel3 and Intel5 workstation computer. ARSV is predicted from energy values of vibration, temperature, current, voltage and GMR sensor signals. ARSV is measured manually through the images of rotor slot obtained from microscopic camera. Figures 12 and 14 illustrate energy   Figure 13 shows the rotor slot size measurement. The slot size variation of rotor obtained by pixel measurement from microscopic image is utilized for the prediction of ARSV (Fig. 14). Figure 15 shows the distribution of GMR sensor signal energy band across subbands for broken rotor bar.
A. Magnetic and Thermal stress analysis through induced slot in rotor IM is induced with two slots, namely 8 and 2 mm, and their energy band values are shown in Figs. 16 and 17. After inserting a 2-mm hole in the rotor, the induction motor is driven for 5 h. After that, the SC magnetic flux is measured by GMR sensor and the rotor slot size variation is calculated using a microscope image after removing rotor slot from motor.  Initially, 2-mm rotor slot is invoked and examined under different faults such as phase reversal, frequent starting, shaft bend, broken rotor bar. The PCT energy bands are obtained for different sensor signals. The magnitudes of temperature and GMR sensor signals vary more than vibration signals for induced 2-mm slot. During steady rise in load, rotor with 8mm induced hole is analyzed for sub-band energy levels of SC flux signals. The percentage of PCT subband energy for GMR sensor signals influenced with 8-mm slot exhibits maximum voltage. The maximum expansion in slot size causes fluctuation in the energy band which results in higher SC flux. The magnitude of SC flux signals rises as the total count of bands increases for 2-and 8-mm generated slots. The magnitude of GMR and temperature signals amplitude increases as  Table  2 shows the energy band and ARSV values for various types of fault induced in induction motor.

B. Regression Algorithms for ARSV prediction
The multimodal sensor data like voltage, current, temperature, vibration, GMR and microscopic camera image values are tabulated for various induced faults such as overload, low voltage, over voltage, high speed, frequent starting, phase reversal. Then, these data are used for prediction through various machine learning algorithms. The various algorithms used for prediction are logistic regression, multiple linear regression and polynomial regression.
IM ARSV predicted values are categorized in Table 3. It shows the prediction value of ARSV for six types of IM with different conditions. During running condition, the slot size in rotor of the IM is evaluated in both load and noload conditions. The slot variation in rotor is estimated using the regression equations. High value of magnetic stress and thermal stress is created in IM when the rotor slot variation exceeds more than 5% of standard size. If the rotor slot size is not reduced by periodic maintenance, then the IM faults are increased which cause the IM to shut down.
Initially, LR method is applied to predict the ARSV of IM. The subband energy values, which are obtained through ORaDWT, are used for prediction of ARSV. The prediction accuracy of LR is 92%. Equation (15) shows ARSV prediction through LR.
Next, MLR-based method is used to predict the induction motor ARSV. The subband energy values that are obtained by TQWT are used for ARSV. Prediction and accuracy of MLR are about 93.4% which is greater than LR. Equation (16) shows ARSV prediction equation using MLR technique.
The LR and MLR method provide less accuracy in ARSV prediction. Therefore, polynomial regression method is applied for better accuracy of ARSV. PR uses the nth-degree polynomial to model the connection between independent variable (x) and dependent variable (y). PR provides the best evaluation for independent and dependent variables. The linear regression and multiple linear regressions are limited in its ability to find any relationship between reliable and undependable variables. As a result, linear regression representation has been changed into a polynomial regression model for ARSV prediction. The primary aim of regression analysis is to model the predicted value of a dependent parameter in the equation of an independent parameter. The polynomial regression is given as in Eq. (17) y a 0 + a 1 x 1 + a 2 x 2 1 + .... + a n x n 1 The PR regression equation for ARSV prediction is as in Eq. (18).
From the above equation, the ARSV prediction accuracy is about 94.6% which is higher than LR and MLR. Table 4 shows the estimated value, standard error value, t-statistic and p-value for various regression methods. In PR, the standard error value is very less compared to other methods. The R 2 value for the PR is 0.993 which is higher than LR and MLR. Figure 18 shows residual plot for LR, MLR and PR.

Conclusion
In this paper, three different methods such as prediction accuracy is 92%. In TQWT & MLR method, ARSV prediction is about 93.4%. The ARSV prediction through multimodal sensor data show excessive stress due to thermal and electromagnetic force on rotor that leads to rotor slot variation. The stresses produced due to various induction motor faults are inspected by multiple sensors. The microscopic camera image is used for ground verification of predicted ARSV. The PCT & PR-based ARSV prediction accuracy is about 94.6%. The proposed ARSV prediction method using PCT & PR reduces the maintenance, increases the life time and increases the performance and efficiency of the motor. The proposed research is extended by coating various nanomaterials on rotor lamination and using few more sensors such as UWB and acoustic for ARSV prediction.