Model Design of Inter-Turn Short Circuits in Internal Permanent Magnet Synchronous Motors and Application of Wavelet Transform for Fault Diagnosis

Abstract

The challenge in developing an AI deep learning model for motor health diagnosis is hampered by the lack of sufficient and representative datasets, leading to considerable time and resource consumption in research. Therefore, this paper focuses on the analysis of the second harmonic component fault characteristic induced by inter-turn short circuits (ITSCs) in phase voltages. First, it establishes a coil inter-turn short-circuit fault (ITSCF) model of the motor to identify the twice-frequency q-axis voltage error characteristics. Subsequently, it develops simulation programs by integrating control and fault models in MATLAB/Simulink/Simscape to observe and analyze the q-axis voltage and circulating current errors caused by the short circuit. Finally, a discrete wavelet transform method is established to analyze the q-axis synchronous reference frame voltage. By applying the energy-based method to extract the twice-frequency voltage error characteristics, the approach successfully detects the error features and confirms ITSCF in the motor. The contributions of this paper include not only the development of an ITSCF characteristic model for the motor but also the successful application of wavelet transform to effectively analyze the time-frequency characteristics of its signals. This approach can serve as a valuable reference for the design of deep learning models in future AI applications.

1. Introduction

The Interior Permanent Magnet Synchronous Motor (IPMSM) is characterized by a rotor structure featuring slots designed to embed the magnets. This configuration, where the magnets are encased by the rotor core, ensures high mechanical integrity and prevents magnet demagnetization, making it ideal for high-speed operation. Due to its high power density, efficiency, wide speed control range, low inertia, and excellent dynamic performance, this type of motor finds extensive applications in various fields, including industrial automation, electric vehicles, home appliances, and hydropower pumps. However, during operation, the IPMSM is subjected to various stresses, which may lead to potential faults. Although the initial symptoms of motor faults may be subtle, they can still result in increased energy consumption, reduced efficiency, and deteriorated performance, which may further cause overheating and insulation degradation or induce vibrations that reduce bearing life. Therefore, developing AI deep learning models for motor health diagnostics can not only enable preventive maintenance for IPMSMs but also provide early warning for equipment failures. This has become a critical research topic, driven by the advancement of smart manufacturing. The findings in [1] indicate that bearing-related failures and stator winding short circuits are the predominant causes of motor failures, accounting for 40% and 38% of all failures, respectively. These results highlight the significance of developing effective diagnostic techniques for these two types of failures.

Inter-turn short circuits (ITSCs) in stator windings, often triggered by environmental conditions or motor design flaws, can escalate into catastrophic failures such as ground faults or phase-to-phase short circuits. The significant circulating current induced by ITSCs results in an excessive temperature rise in the shorted turns, leading to accelerated insulation degradation. This cumulative effect can eventually cause insulation breakdown and catastrophic failures that may compromise the motor’s materials and structure. The early diagnosis of ITSC is crucial to prevent system failures [2]. Currently, extracting fault features from motor information is a primary approach for motor fault diagnosis, with Motor Current Signature Analysis (MCSA) [3] focusing on the Fourier spectrum of steady-state currents. However, the application of Fast Fourier Transform (FFT) results in the loss of time-related information, and traditional MCSA is not suitable for many industrial applications. Under conditions of continuous load variations, load vibrations, or voltage fluctuations [4,5,6], the Fourier spectrum of the motor current can be distorted, potentially leading to misdiagnosis of faults. To address these limitations, the Short-Time Fourier Transform (STFT) [7] can be employed to improve diagnostic accuracy. However, the choice of an appropriate window size, tailored to specific frequency components of the signal, is crucial. The fixed-length window of the STFT may not effectively capture varying frequency components, limiting its application to motors with very slow speed variations. The lack of sufficient and representative data poses a significant challenge in this research field, resulting in inefficient utilization of time and resources. Additionally, early warning signs are often subtle, making them difficult to capture effectively, and require prolonged testing under extreme operating conditions. The instability of analysis methods also complicates the establishment of deep learning models.

Therefore, this paper proposes a deep learning model design pre-processing concept for ITSCF in IPMSMs and innovatively applies wavelet transform to IPMSM fault diagnosis. Firstly, a motor ITSCF model is established to identify the second harmonic component fault characteristics. A simulation program combining MATLAB/Simulink/Simscape is developed to generate phase voltage and short-circuit coil circulating current waveforms during ITSCs. Next, the discrete wavelet transform (DWT) method is employed to perform wavelet transformation on the q-axis synchronous voltage and extract the second harmonic component fault characteristics based on energy principles. When a stator coil experiences a short circuit, discrete wavelet analysis of the stator voltage is used to extract specific frequencies associated with the ITSCF characteristics. The aim of this study is to improve the detection accuracy of motor fault features and effectively identify different types and severities of ITSCF. It provides a deep learning model for IPMSM health diagnosis, offering a more precise preprocessing design.

The simulation results demonstrate that the wavelet transform method exhibits high accuracy in fault detection. Based on these findings, this study provides the occurrence time of faults and their corresponding frequencies, accurately pinpointing the time of ITSC events and the resulting second harmonic component fault characteristics. This demonstrates the effectiveness of the proposed technique for detecting ITSCF in motors [8]. The contributions of this study include not only establishing a reference process for designing a deep learning feature model for motor ITSCs but also successfully applying wavelet transform to effectively analyze the time-frequency characteristics of the signals. Through the proposed method, motor fault features can be more accurately identified, thereby reducing the probability of incorrect diagnoses and improving detection efficiency. Moreover, this wavelet transform-based diagnostic technique effectively addresses non-stationary signals during motor operation, overcoming the limitations of traditional methods in terms of time and frequency resolution.

The contributions of this paper are as follows:

• Successfully established an IPMSM inter-turn short circuits fault feature extraction system framework, providing the data preprocessing required for designing deep learning models for motor failure analysis.
• Derived the physical model of the IPMSM ITSCF and successfully built simulation equations using MATLAB/Simulink/Simscape.
• Innovatively extracted ITSC fault features using wavelet transform, enabling the diagnosis of IPMSM ITSC failures.
• The proposed ITSC time-frequency features and ITSC energy level information can not only be used for motor fault diagnosis but also for determining the range of command failures based on energy characteristics.

2. Related Work

IPMSM fault diagnosis involves signal processing and analysis procedures that include sampling motor voltage and current and applying diagnostic methods. This section explores the issue from the perspective of failures and examines the feasibility of past fault diagnosis techniques.

2.1. The Study of IPMSM Failures

The failure of IPMSM in modern industrial automation can result in significant economic losses and safety risks. Disruptions to production processes caused by IPMSM failures can lead to decreased efficiency, increased maintenance costs, and potentially compromise the overall safety of industrial operations. With this concept, the early prevention of equipment damage and the effective motor fault detection and diagnosis have become critical research issues [9]. The establishment of motor failure models has become a critical application technology in the field of fault diagnosis. Data quality, derived from sensors mounted on motors and encompassing parameters such as current, voltage, speed, and vibration under normal and fault conditions [10], significantly influences the accuracy and reliability of the models used for motor fault diagnosis. During the data preprocessing phase, standardization and normalization are required to transform the data into a format suitable for model training. Since deep learning algorithms depend on stable and high-quality data models, their training relies on precise feature extraction and accurate data labeling. Therefore, through well-planned mathematical models [11], it is possible to develop deep learning models with high accuracy and reliability for motor fault diagnosis and prediction. Beyond data collection and feature processing for failure analysis, motor fault diagnosis can predict issues before they occur, thereby reducing unplanned equipment downtime and maintenance costs.

2.2. Diagnosis Method Analysis

In previous research, motor fault diagnosis has relied on monitoring stator current or voltage harmonics to identify the frequency components indicative of various faults. Time-domain, frequency-domain, and time-frequency-domain analyses have been commonly used for this purpose. A test motor for stator ITSCF is constructed, where a control system is implemented using DSP, and ITSCF monitoring is conducted based on the analysis of second-order harmonics in the q-axis current [12]. Reference [13] evaluates the current and voltage characteristics of PMSM’s ITSCF and applies Linear Discriminant Analysis (LDA) to improve diagnostic accuracy, distinguishing between different fault types such as static eccentricity and ITSCF, as well as estimating their severity. Additionally, zero-sequence voltage component (ZSVC) detection is another common method for fault diagnosis [14]. However, it requires a connection at the neutral point of the stator winding and is susceptible to disturbances from motor drives or other types of faults [15], increasing the difficulty of motor fault diagnosis.

In addition to the aforementioned methods, error diagnosis can also be performed using linear filtering techniques, such as the Kalman filter, which develops an appropriate model by identifying parameters specifically designed to indicate stator faults [16]. Alternatively, the Vold–Kalman filtering order tracking algorithm [17] can be used to track specific harmonics over a wide range of machine speeds and under different load conditions. In addition, some studies employ wavelet transform analysis, such as using complex wavelet analysis to examine radial [18], axial, and tangential vibrations for diagnostic purposes. For instance, ref. [19] utilizes the dual tree complex wavelet transform (DTCWT) for feature extraction, making it meaningful to model Probabilistic Neural Networks (PNNs) with wavelet methods. These approaches are often based on specific filters to isolate frequency bands that match the fault characteristic frequencies.

2.3. The Failure Models

The establishment of failure models is also a crucial aspect of fault diagnosis. Additionally, methods for modeling, such as using mathematical models to simulate fault models for motor failure detection [20] and employing physics-based back electromotive force (EMF) estimation for motor fault detection [21], are also analyzed. Additionally, state observers can generate specific vector residuals by decomposing current estimation errors, which are used for model-based detection strategies of stator inter-turn short circuits in induction motors [22]. Since the observer includes an adaptive scheme for rotor speed estimation, the proposed approach allows for online monitoring by measuring only the stator voltage and current. Aside from the aforementioned methods, parameterized models can also be used to simulate healthy and faulty SPMSM conditions to study the effects of stator ITSCF [23]. Failure models can also be studied under non-stationary conditions, including load or speed variations. A dynamic model of a faulted Surface-Mounted Permanent Magnet Synchronous Motor (SPMSM) was derived using a deformed flux model [24].

Reference [25] proposes the incorporation of salient pole modeling into the IPMSM model and develops an ITSCF model for IPMSM with series and parallel winding connections. This model is validated through finite element method simulations and experimental measurements. Reference [26] introduces modeling and detection methods for intermittent ITSCF in the stator end windings of PMSM. During the steady-state operation, fault-related information is contained in the stator current and reference voltage. The effectiveness of this method is confirmed through fault detection using wavelet transform, with simulation and experimental results validating the approach. References [27,28,29,30,31,32] focus on the application of rapidly developing machine learning techniques for diagnosing faults in motors, including the analysis of harmonics, voltage, and current signals.

Through the literature review, it is evident that wavelet transform can analyze signals to produce multiple time-frequency representations. Extracting features from fundamental information such as voltage and current has gradually become one of the mainstream methods for fault analysis, and it can help avoid issues related to motor signal measurement. Due to the high computational load of continuous wavelet transform and the fact that most data are now digitized, discrete wavelet transform is more suitable for motor fault diagnosis design compared to continuous wavelet transform. This study will apply discrete wavelet transform as the analytical method. In summary, motor fault diagnosis is a key technology for ensuring the stable operation of industrial automation systems. Through efficient data collection and processing, along with precise model development and evaluation, timely fault warning and effective preventive maintenance can be achieved, thereby enhancing the overall system performance and safety.

3. ITSCF Model and Feature

This section outlines the design concept of the IPMSM inter-turn short circuits fault feature extraction framework. It describes the motor failure analysis framework in detail, followed by the introduction of the IPMSM physical model. Detailed explanations of the IPMSM ITSC structure and the ITSCF judgment through voltage and current analysis are provided.

3.1. The Pre-Processing System Framework

Figure 1 illustrates the proposed innovative framework for the IPMSM ITSCF feature extraction system developed in this study. This framework aims to extract the ITSCF features in IPMSMs using wavelet transform, providing critical time-frequency analysis information necessary for the big data preprocessing required in the design of deep learning models. This system is capable of performing fault diagnosis analysis on IPMSMs in various application fields, including electric vehicles (EVs), fans (FANs), and pumps (PUMPs), based on operating conditions under different speed commands in IPMSM motors from these diverse applications. The system offers the flexibility to specify a particular range of speed commands or a comprehensive range, such as 1 to 6000 RPM. Based on the given speed command for the IPMSM operation, the framework generates corresponding voltage and current waveform information. Subsequently, a Voltage and Current Waveform Converter, as illustrated in the figure, is employed to transform and analyze these waveforms. Based on the specified speed command for the IPMSM, the framework generates corresponding voltage and current waveforms. These waveforms are transformed and analyzed using the Voltage and Current Waveform Converter depicted in the figure. The innovation of this study lies in utilizing wavelet transform for time-frequency conversion to further analyze and extract potential frequency bands and energy error information from the motor’s voltage and current signals.

The innovative features of this framework are illustrated in the WT process block diagram, where the voltage and current waveform information is simultaneously generated for multiple speed commands corresponding to the motor model. By employing the DWT across multiple frequency bands, this approach allows for a multi-scale decomposition of the signals. Based on computational requirements, each frequency level is processed to further analyze and extract characteristic frequencies of ISCF in the motor. The framework ultimately produces ITSCF time-frequency features and ITSCF energy level information, which can be used not only for motor fault diagnosis but also to determine the range of command failures based on energy percentages. The massive dataset of normal and abnormal waveforms generated by this framework can serve as a valuable reference for designing failure models for various types of IPMSMs. The innovative features of this framework are illustrated in the WT process block diagram, where the voltage and current waveform information is simultaneously generated for multiple speed commands corresponding to the motor model. By employing the DWT across multiple frequency bands, this approach allows for a multi-scale decomposition of the signals. Based on computational requirements, each frequency level is processed to further analyze and extract characteristic frequencies of ISCF in the motor. The framework ultimately produces ITSCF time-frequency features and ITSCF energy level information, which can be used not only for motor fault diagnosis but also to determine the range of command failures based on energy percentages. The massive dataset of normal and abnormal waveforms generated by this framework can serve as a well-defined reference for a preprocessed dataset used in deep learning analysis.

3.2. The Design of ITSCF Model

This research presents a design methodology for an ITSCF model [8], aimed at facilitating the development and application of deep learning models for motor fault diagnosis and analysis. Derive the d-q axis mathematical model for motor ITSCF from the three-phase ITSC mathematical model and compare the voltage characteristic differences between ITSC and healthy motors through the mathematical model. Subsequently, perform signal analysis and feature extraction using DWT as preprocessing steps, and convert the fault features into a format suitable for input into deep learning networks. The ITSC structure of the motor system model is shown in Figure 2, where N₁ represents the number of turns in the healthy coil, N₂ represents the number of turns in the short-circuited coil, and μ denotes the internal short-circuit ratio as described by Equation (1).

$$\mu ={\displaystyle \frac{N_2}{N_1+N_2}},$$

(1)

The ITSC model of the motor is given by Equation (2).

$${\mathit{v}}_{\mathit{a}\mathit{b}\mathit{c}\mathit{f}}={\mathit{r}}_{fs}{\mathit{i}}_{abcf}+{\mathit{L}}_{fs}\frac{\mathrm{d}}{\mathrm{d}t}\left({\mathit{i}}_{abcf}\right)+{\mathit{e}}_{abcf}+{\mathit{v}}_f,$$

(2)

The voltage and current with ITSC are represented by Equations (3) and (4), respectively Voltage with ITSC:

$${\mathit{v}}_{\mathit{a}\mathit{b}\mathit{c}\mathit{f}}={\left[\begin{array}{cc}\begin{array}{cc}{\mathit{v}}_{\mathit{a}}& {\mathit{v}}_{\mathit{b}}\end{array}& \begin{array}{cc}{\mathit{v}}_{\mathit{c}}& 0\end{array}\end{array}\right]}^{\mathbf{T}},$$

(3)

where the fourth item represents the shorted coil voltage, with a value of 0 V.

$${\mathit{i}}_{abcf}={ \left[\begin{array}{cc}\begin{array}{cc}{i}_a& i_b\end{array}& \begin{array}{cc}{i}_c& i_f\end{array}\end{array}\right]}^{\mathrm{T}},$$

(4)

where i_f refers to the shorted coil fault current; the resistance and inductance under ITSC are given by Equations (5) and (6). The ITSC resistance is expressed as follows:

$${\mathit{r}}_{fs}=\left[\begin{array}{cc}\begin{array}{cc}{ r}_s& 0\\ 0& { r}_s\end{array}& \begin{array}{cc}0& {-\mu r}_s\\ 0& 0\end{array}\\ \begin{array}{cc}0& 0\\ {\mu r}_s& 0\end{array}& \begin{array}{cc}{r}_s& 0\\ 0& -\mu r_s-r_f\end{array}\end{array}\right],$$

(5)

where μr_s represents the shorted coil resistance, and r_f is the external parallel resistance of the shorted coil. According to reference [8], when an internal short circuit occurs in phase a coil, the phase a current i_a still flows through the entire phase a coil, so the corresponding phase resistance remains r_s. However, the fault current i_f primarily constitutes a circulating current within the shorted coil, thus corresponding to a resistance value of μr_s. Additionally, the shorted coil may form a short circuit through other mediums; thus, r_f is introduced as a model parameter. This parameter influences the response of the short-circuit fault current i_f. The ITSC inductance is expressed as follows:

$${\mathit{L}}_{fs}=\left[\begin{array}{cc}\begin{array}{cc}L& M\\ M& L\end{array}& \begin{array}{cc}M& -\mu L\\ M& -\mu M\end{array}\\ \begin{array}{cc}M& M\\ \mu L& \mu M\end{array}& \begin{array}{cc}L& -\mu M\\ \mu M& {-\mu }^{2}L\end{array}\end{array}\right],$$

(6)

where μL represents the self-inductance of the phase a shorted coil, μM denotes the mutual inductance between the phase b and phase c coils with respect to the shorted coil, and μ²L refers to the self-inductance of the shorted coil. Similarly, based on reference [8], Figure 2, and the aforementioned resistance model, the phase a current i_a still flows through the entire phase a coil; thus, the corresponding phase inductance remains L. Considering the response of the shorted coil fault current i_f, the parameters μL, μM, and μ²L are defined to establish the inductance model for each phase coil in relation to the shorted coil. The back electromotive force with ITSC is represented by Equation (7).

$${\mathit{e}}_{abcf}=\left[\begin{array}{l}\begin{array}{c}{\lambda ’}_m{\omega }_r\cos ({\theta }_r)\\ {\lambda ’}_m{\omega }_r\cos ({\theta }_r-{\displaystyle \frac{2}{3}}\pi )\end{array}\\ \begin{array}{c}{\lambda ’}_m{\omega }_r\cos ({\theta }_r+{\displaystyle \frac{2}{3}}\pi )\\ {-\mu \lambda ’}_m{\omega }_r\cos ({\theta }_r)\end{array}\end{array}\right],$$

(7)

where the fourth item represents the back EMF of the shorted coil, presented using the per-unit concept. The initial voltage $v_0$ is expressed by Equation (8).

$$v_0=v_0{\left[\begin{array}{cc}\begin{array}{cc}1& 1\end{array}& \begin{array}{cc}1& 0\end{array}\end{array}\right]}^{\mathrm{T}},$$

(8)

By substituting the last term of each equation from Equations (3)–(8) into Equation (2), Equation (9) is obtained.

$$0=-\mu r_s{i}_a+\left(\mu r_s+r_f\right)i_f-2{\omega }_e\left(-\mu L{i}_a-\mu M{i}_b-\mu M{i}_c+{\mu }^{2}L{i}_f\right)+\left(-\mu Lp{i}_a-\mu Mp{i}_a-\mu Mp{i}_c+{\mu }^{2}Lp{i}_f\right)-e_f,$$

(9)

The three-phase currents can be obtained from the transformation matrix and are expressed by Equations (9)–(11).

$$i_a=\sqrt{{\displaystyle \frac{2}{3}}}\left[i_q\cos \left({\theta }_r\right)+i_d\sin \left({\theta }_r\right)\right]=\sqrt{{\displaystyle \frac{2}{3}}}I\sin \left({\theta }_r+\delta \right),$$

(10)

$$i_b=\sqrt{{\displaystyle \frac{2}{3}}}\left[i_q\cos \left({\theta }_r-{\displaystyle \frac{2}{3}}\pi \right)+i_d\sin \left({\theta }_r-{\displaystyle \frac{2}{3}}\pi \right)\right]=\sqrt{{\displaystyle \frac{2}{3}}}I\sin \left({\theta }_r-{\displaystyle \frac{2}{3}}\pi +\delta \right),$$

(11)

$$i_c=\sqrt{{\displaystyle \frac{2}{3}}}\left[i_q\cos \left({\theta }_r+{\displaystyle \frac{2}{3}}\pi \right)+i_d\sin \left({\theta }_r+{\displaystyle \frac{2}{3}}\pi \right)\right]=\sqrt{{\displaystyle \frac{2}{3}}}I\sin \left({\theta }_r+{\displaystyle \frac{2}{3}}\pi +\delta \right),$$

(12)

Due to the significant difference between μ² and the overall value, it is neglected. Subsequently, applying Equations (10)–(12), Equation (9) can be expressed as Equation (13).

$$\left(\mu r_s+r_f\right)i_f=-2{\omega }_r\left(\mu L\sqrt{{\displaystyle \frac{2}{3}}}I\sin \left(\theta +\delta \right)+\mu M\sqrt{{\displaystyle \frac{2}{3}}}I\sin \left(\theta -{\displaystyle \frac{2\pi }{3}}+\delta \right)+\mu M\sqrt{{\displaystyle \frac{2}{3}}}I\sin \left(\theta +{\displaystyle \frac{2\pi }{3}}+\delta \right)\right)+e_f$$

(13)

After rearranging and simplifying Equation (13), the result is obtained as follows:

$$i_f\cong -\sqrt{6}S{\omega }_r{L}_{2}I\cos \left(\theta -\delta \right)+S{\omega }_r\lambda \cos \left(\theta \right),$$

(14)

It can be concluded that the short-circuit circulating current i_f is a cosine function with amplitude proportional to the severity of the fault and the rotor speed ratio, as given by Equation (15) for S.

$$S={\displaystyle \frac{\mu }{r_f+\mu r_s}},$$

(15)

By combining the Clarke and Park transformations, the transformation matrix T_cp, as shown in Equation (16), is obtained. This matrix enables the conversion of a three-phase motor ITSC model into a d-q axis ITSC model.

$${\mathit{T}}_{cp}={\displaystyle \frac{2}{3}}\left[\begin{array}{ccc}-\sin (\theta )& -\sin (\theta -{\displaystyle \frac{2}{3}}\pi )& -\sin (\theta +{\displaystyle \frac{2}{3}}\pi )\\ \cos (\theta )& \cos (\theta -{\displaystyle \frac{2}{3}}\pi )& \cos (\theta +{\displaystyle \frac{2}{3}}\pi )\end{array}\right],$$

(16)

The d-q axis model can be obtained as shown in Equation (17).

$${\mathit{v}}_{qd}={\mathit{T}}_{cp}{\left[\begin{array}{ccc}{v}_a& v_b& v_c\end{array}\right]}^{\mathrm{T}}={\mathit{T}}_{cp}{\mathit{r}}_s{\mathit{T}}_{cp}^{-1}{\mathit{i}}_{abc}-{\mathit{T}}_{cp}\left[\begin{array}{c}\mu r_s{i}_f\\ 0\\ 0\end{array}\right]+{\mathit{T}}_{cp}{\mathit{L}}_s{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\left({\mathit{T}}_{cp}^{-1}{\mathit{i}}_{abc}\right)-{\mathit{T}}_{cp}\left[\begin{array}{c}\mu L\\ \mu M\\ \mu M\end{array}\right]{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{i}_f+{\mathit{T}}_{cp}{\mathit{e}}_{abcf},$$

(17)

After expanding the matrix, the result is obtained as follows:

$${\mathit{v}}_{qdf}=\left[\begin{array}{ccc}{r}_s& {\displaystyle \frac{3}{2}}{\omega }_r{L}_d& A\\ -{\displaystyle \frac{3}{2}}{\omega }_r{L}_q& r_S& B\end{array}\right]\left[\begin{array}{c}{i}_q\\ i_d\\ i_f\end{array}\right]+\left[\begin{array}{ccc}{\displaystyle \frac{3}{2}}{L}_q& 0& -\sqrt{{\displaystyle \frac{3}{2}}}\mu L_q\cos \left({\theta }_r\right)\\ 0& {\displaystyle \frac{3}{2}}{L}_d& -\sqrt{{\displaystyle \frac{3}{2}}}\mu L_d\sin \left({\theta }_r\right)\end{array}\right]\left[\begin{array}{c}{pi}_q\\ p{i}_d\\ {pi}_f\end{array}\right]=i_{qd}+i_{qd-ITSC},$$

(18)

and

$$A=-\sqrt{{\displaystyle \frac{3}{2}}}\mu r_s\cos \left({\theta }_r\right)+2{\omega }_r\sqrt{{\displaystyle \frac{3}{2}}}\mu L_2\sin \left({\theta }_r\right)+\sqrt{{\displaystyle \frac{3}{2}}}\mu {\omega }_e{L}_q\cos \left({\theta }_r\right),$$

(19)

$$B=-\sqrt{{\displaystyle \frac{3}{2}}}\mu r_s\sin \left({\theta }_r\right)+2{\omega }_r\sqrt{{\displaystyle \frac{3}{2}}}\mu L_2\cos \left({\theta }_r\right)+\sqrt{{\displaystyle \frac{3}{2}}}\mu {\omega }_e{L}_d\sin \left({\theta }_r\right),$$

(20)

where L_q = L₀ + L₂, L_d = L₀ − L₂.

By combining Equation (14) with Equation (18), the following result is obtained:

$$v_{qd\text{-}ITSC}=\left[\begin{array}{c}{\mathrm{v}}_{\mathrm{q}\text{-}\mathrm{ITSC}}\\ {\mathrm{v}}_{\mathrm{d}\text{-}\mathrm{ITSC}}\end{array}\right]=\left[\begin{array}{c}C\\ D\end{array}\right],$$

(21)

and

$$\begin{gathered} C=\mu {\omega }_e{L}_{2}SI\left(r_s\left(\cos \left(2\theta +\delta \right)+\cos \left(\delta \right)\right)-3L_2\left({\omega }_e\sin \left(2\theta -\delta \right)+{\omega }_e\sin \left(\delta \right)\right)+L_q\left({\displaystyle \frac{3}{2}}\cos \left(2\theta -\delta \right)+{\displaystyle \frac{3}{2}}\cos \left(\delta \right)-{\displaystyle \frac{3}{2}}{\omega }_e\sin \left(2\theta -\delta \right)+{\displaystyle \frac{3}{2}}{\omega }_e\sin \left(\delta \right)\right)\right) \\ +\mu {\omega }_{e}S\lambda \left(r_s\left(-\sqrt{{\displaystyle \frac{1}{6}}}\cos \left(2\theta \right)-\sqrt{{\displaystyle \frac{1}{6}}}\right)+\sqrt{{\displaystyle \frac{3}{2}}}{L}_2{\omega }_e\sin \left(2\theta \right)+L_q\left(-\sqrt{{\displaystyle \frac{3}{2}}}\cos \left(2\theta \right)-\sqrt{{\displaystyle \frac{3}{2}}}+\sqrt{{\displaystyle \frac{3}{8}}}{\omega }_e\sin \left(2\theta \right)\right)\right), \end{gathered}$$

(22)

$$\begin{gathered} D=\mu {\omega }_e{L}_{2}SI\left(r_s\left({\omega }_e\sin \left(2\theta +\delta \right)+\sin \left(\delta \right)\right)+L_2\left(-3{\omega }_e\sin \left(2\theta -\delta \right)-3{\omega }_e\sin \left(\delta \right)\right)+L_d\left({\displaystyle \frac{3}{2}}\sin \left(2\theta -\delta \right)+{\displaystyle \frac{3}{2}}\sin \left(\delta \right)+{\displaystyle \frac{3}{2}}{\omega }_e\cos \left(2\theta -\delta \right)-{\displaystyle \frac{3}{2}}{\omega }_e\cos \left(\delta \right)\right)\right) \\ +\mu {\omega }_{e}S\lambda \left(-\sqrt{{\displaystyle \frac{1}{6}}}{r}_s\sin \left(2\theta \right)+\sqrt{{\displaystyle \frac{3}{2}}}{L}_2{\omega }_e\sin \left(2\theta \right)+L_d\left(-\sqrt{{\displaystyle \frac{3}{8}}}\sin \left(2\theta \right)-\sqrt{{\displaystyle \frac{3}{8}}}{\omega }_e\cos \left(2\theta \right)+\sqrt{{\displaystyle \frac{3}{8}}}{\omega }_e\right)\right), \end{gathered}$$

(23)

By deriving the voltage characteristics from the above model, it can be observed from the previous equations that the voltage exhibits a second harmonic component characteristic, which can be used for ITSC fault detection. This model derives the fault characteristics of ITSC under ideal conditions, assuming a steady-state motor operating at a fixed speed with constant motor parameter and physical properties. Under non-stationary operating conditions, the error characteristic frequency will experience drift. In practice, motor operation generates losses, particularly in the short-circuited coil where the current i_f is substantial. This leads to increased resistance due to heating and potential core saturation, causing deviations from the theoretical results. These factors may hinder the effective observation of ITSCF.

4. Wavelet Transform

This section defines the innovative motor failure model process and introduces the DWT method. Subsequently, it presents an implementation of DWT. Finally, applying the inverse discrete wavelet transform (IDWT) process, the effectiveness of the results is verified for the diagnosis of IPMSM demagnetization.

4.1. Wavelet Transform Method

In the process of establishing motor fault models, the wavelet transform (WT) method [7] is effective in extracting various fault features from motor operation, enabling accurate fault diagnosis and model construction by transforming these features into both frequency and time domain information. The fundamental theory of wavelet analysis involves using a prototype function, called the mother wavelet, to generate a set of basis functions that allow for the multi-resolution analysis of signals. WTs are primarily categorized into continuous wavelet transform (CWT) and discrete wavelet transform (DWT). The main difference between them is that the CWT operates over all possible scales and translations, while the DWT uses only a specific subset of scales and translation values. Wavelets find applications in a wide range of fields, including discontinuity detection, compression, filtering, and frequency analysis. A continuous function ψ (t) is considered a mother wavelet if it has finite energy and zero mean. A zero integral indicates that the function is oscillatory, meaning its graph must exhibit an up-and-down oscillating wave pattern. Finite energy implies that the function’s energy is mostly confined within a finite time interval. Consequently, the area enclosed by ψ (t) must be relatively small, and the amplitude must decay rapidly to zero in both the positive and negative directions. Compared to sinusoidal waves, these functions can be considered as smaller wave shapes, hence the term “wavelet”.

• Continuous Wavelet Transform

The continuous wavelet basis functions are generated by applying dilation and translation operations to a single mother wavelet function ψ (t), as expressed in the following Equation (24):

$${\psi }_{a,b}={\displaystyle \frac{1}{\sqrt{a}}}\psi \left({\displaystyle \frac{t-b}{a}}\right),$$

(24)

In the formula, a represents the scale parameter and b represents the translation parameter. As a increases, the mother wavelet is stretched more, while a smaller a compresses the mother wavelet. Smaller values of b are suitable when the mother wavelet is narrow, as they reduce the displacement distance.

• Discrete Wavelet Transform

The DWT was introduced to simplify the computational complexity of the CWT. To address the redundancy and computational complexity inherent in the CWT, the DWT was developed by discretizing both the scale parameter a and the translation parameter b, thereby achieving a balance between theoretical completeness and practical implementation in the following Equation (25):

$${\psi }_{a,b}={\displaystyle \frac{1}{\sqrt{2^m}}}\psi \left({\displaystyle \frac{t-n}{2^m}}\right),$$

(25)

To perform approximate analysis, it is essential to select an appropriate level m for either a detailed or approximate further analysis, as represented by the DWT in Equation (26).

$${DWT}_f\left(m,n\right)={\displaystyle \frac{1}{\sqrt{2^m}}}{\sum }_{k}f\left(k\right){\psi }^{*}\left({\displaystyle \frac{k-n}{2^m}}\right),$$

(26)

Here, DWT_f (m, n) represents the discrete wavelet transform coefficients of the signal f(t), and k is the operation index. It is noteworthy that while the DWT is discrete in nature, it is essentially a transformation of a continuous signal, with discretization limited to the scale parameter a and translation parameter b.

The DWT can be considered a decomposition method that analyzes the original signal at multiple resolutions. By employing high-pass and low-pass filters, the DWT decomposes the original signal into high-frequency and low-frequency components. The signal is initially passed through high-pass and low-pass filters, followed by down-sampling by a factor of two to obtain the high-frequency and low-frequency coefficients, respectively. By applying Equation (27), which relates the sampling frequency f_e and fundamental frequency f_s, the number of levels n_ls in the discrete wavelet transform can be obtained.

$$n_{ls}=\mathrm{int}({\displaystyle \frac{\log \frac{f_e}{f_s}}{\log 2}}),$$

(27)

Wavelet functions, essentially bandpass filters, halve their bandwidth at each scale. To cover the entire spectrum, an infinite number of levels would be required. However, in practical applications, the number of analyzable levels is limited by the sampling quantity. Therefore, the scaling function (27) is employed to filter out the lowest level of the transform, ensuring the effective coverage of the entire spectrum.

In addition to performing the DWT, signal reconstruction is also involved. This process includes passing the signal through a low-pass filter and a high-pass filter to obtain the high-frequency and low-frequency components of the signal. The reconstruction of the original signal, known as the IDWT, involves applying the inverse transform filtering operation to recover the original signal. The reconstruction process, which is the inverse of the aforementioned decomposition process, involves using the obtained details from each level to reconstruct the original sequence. This is achieved by up-sampling the sequences and passing them through a series of n_ls two-band perfect reconstruction quadrature mirror filter banks (QMF). This process is known as the inverse discrete wavelet transform (IDWT). Since the motor voltage signal, after calculating the details of each level using DWT, cannot be directly used for fault feature analysis, IDWT is required to perform the inverse transform. Through up-sampling, the reconstructed waveforms at different frequency ranges are aligned to a consistent length, facilitating subsequent computations. Therefore, the IDWT comprises high-pass and low-pass filters and plays a crucial role in signal reconstruction. The hierarchical structure of the DWT process is illustrated in Figure 3.

The signal source, X(n), can be decomposed into a representation using wavelet functions as an orthonormal basis, as shown in Equation (28).

$$X\left(t\right)={\sum }_{k}c{A}_i\left(k\right){\varphi }_{j-i,k}+{\sum }_{k}c{D}_i\left(k\right)w_{j-k,k}+\dots +{\sum }_{k}c{D}_1\left(k\right)w_{j-k,k},$$

(28)

where the scaling function and wavelet function are represented by the following Equations (29) and (30), respectively.

$${\varphi }_{j,k}\left(t\right)=\sqrt{2^j}\varphi (2^{j}t-k),$$

(29)

$$w_{j,k}\left(t\right)=\sqrt{2^j}w(2^{j}t-k),$$

(30)

The coefficients cA_i and cD_i, representing the corresponding coefficients under the wavelet and scaling function bases, respectively, vary with the transformation level, denoted by i. The relationship between the DWT and IDWT coefficients is expressed by the following equation and illustrated by the process shown in Figure 4 and Figure 5, respectively.

• DWT (Analysis)

The first frequency range of decomposition:

$$c{A}_1\left(k\right)={\sum }_{n}g(n-2k)(n-2k)X\left(n\right),$$

(31)

$$c{D}_1\left(k\right)={\sum }_{n}h(n-2k)(n-2k)X\left(n\right),$$

(32)

The second frequency range of decomposition:

$$c{A}_2\left(k\right)={\sum }_{n}g(n-2k)(n-2k)c{A}_1\left(n\right),$$

(33)

$$c{D}_2\left(k\right)={\sum }_{n}h(n-2k)(n-2k)c{A}_1\left(n\right),$$

(34)

The n frequency range of decomposition:

$$c{A}_{n+1}\left(k\right)={\sum }_{n}g\left(n\right)(n-2k)c{A}_{i+1}\left(n\right),$$

(35)

$$c{D}_{n+1}\left(k\right)={\sum }_{n}h\left(n\right)\left(n-2k\right)c{A}_{i+1}\left(n\right),$$

(36)

• IDWT (Synthesis)

The first frequency range of reconstruction:

$$c{A}_1\left(n\right)={\sum }_{k}c{A}_2\left(k\right)g\left(n-2k\right)+{\sum }_{k}c{D}_2\left(k\right)h\left(n-2k\right),$$

(37)

The second frequency range of reconstruction:

$$c{A}_2\left(n\right)={\sum }_{k}c{A}_3\left(k\right)g\left(n-2k\right)+{\sum }_{k}c{D}_3\left(k\right)h\left(n-2k\right),$$

(38)

The n frequency range of reconstruction:

$$c{A}_i\left(n\right)={\sum }_{k}c{A}_{i+1}\left(k\right)g\left(n-2k\right)+{\sum }_{k}c{D}_{i+1}\left(k\right)h\left(n-2k\right),$$

(39)

In wavelet analysis, Daubechies wavelets, a class of orthogonal wavelets with compact support, are commonly used and are denoted by db_N, where N is an integer from 1 to 10. The filter bank consists of a high-pass filter and a low-pass filter. As the number of filter banks increases, the analysis accuracy improves, but the computational cost also rises. Based on the digital signal and frequency operation range of the motor voltage processed in this paper, and considering the computational load and the need for IDWT, this study adopts db45 as the mother-wavelet function.

Since the discrete wavelet transform is not effective in directly comparing error variation trends, the energy percentage for each frequency band is computed using the energy percentage Equation (40) to facilitate comparison [7].

$$E_i={\sum }_{i=1}^n{c}_{ij}^2/\left(\sqrt{{\sum }_i^n{\sum }_j^7{c}_{ij}^2}\right),$$

(40)

4.2. Implementation Process

As depicted in Figure 6, the initial step of the DWT process commences with data acquisition and preprocessing. Since the quality of the input data significantly influences the subsequent analysis, it is imperative to preprocess the data to mitigate the adverse effects of noise. The second step involves applying the DWT to obtain discrete wavelet coefficients. This step enables the decomposition of the input signal into multiple frequency bands, facilitating detailed analysis. Subsequently, the original waveform is reconstructed using the obtained discrete wavelet coefficients. While these coefficients provide valuable insights, especially at lower frequencies, they represent only a portion of the signal. To extract complete and meaningful features, the original signal within each frequency band is reconstructed, allowing for a comprehensive analysis at the corresponding resolution and capturing full-length characteristics rather than fragmented ones. Finally, the reconstructed signals are evaluated using a set of finite numerical features to assess the condition of the electrical equipment. To visualize the results, the energy ratio formula is employed.

5. Simulation Results

To validate the ITSCF system framework, this section used a simulation platform with the ITSCF model. Additionally, it explains the DWT analysis results of ITSCF in IPMSM.

5.1. Simulation Platform

In this section, based on the IPMSM ITSCF model established in Section 3, a motor control simulation is constructed using a hybrid combination of Simulink and Simscape in MATLAB. The overall structure of the speed control system is illustrated in Figure 7. The outer loop of the speed control system, implemented in Simulink, calculates the speed error by comparing the reference speed with the feedback speed. The resulting speed error is then fed into a speed PI controller to generate the q-axis current command, while the d-axis current is set to 0A. The inner loop calculates the d-q axis current errors by subtracting the feedback d-q axis currents from their respective commands. These error signals are then fed into d-q axis current PI controllers to generate the corresponding d-q axis voltage commands. These voltage commands are transformed from the d-q reference frame to the stationary abc frame using Park and Clarke transformations, and then applied to the three-phase motor model implemented in Simscape. The motor model produces various outputs, including currents, back EMF, flux linkages, torque, and speed. Finally, the three-phase currents and speed are fed back to close the loop between the outer and inner control loops. The complete motor simulation parameters are summarized in

Table 1. Simulation Parameters for IPMSM.

Quantity	Value	Units
Rated Voltage	48	V
Rated Current	9	A
Rated Power	400	W
Rated Torque	600	mNm
Max. Speed	6000	rpm
Poles (P)	8	pole
Stator Resistance (rs)	0.33	Ω
q-axis Inductance (Lq)	1.2606	mH
d-axis Inductance (Ld)	1.0045	mH
PM Flux Linkage (λm)	0.008625	Wb
Rotor Inertia (Jm)	0.0001	kg·m2
Damping (Bm)	0.0000484	N·m·s−1

.

Although Simulink can also be used to build an IPMSM model, it is based on standard three-phase balanced motor equations. Consequently, it cannot replicate the three-phase imbalance characteristics that arise when modeling the motor’s ITSCF. Consequently, to accurately simulate the IPMSM model under ITSCF conditions, a Simscape-based implementation is necessary. As illustrated in Figure 8, the three-phase voltages generated by the Simulink control loop are applied to the individual phase windings of the motor model constructed in Simscape. After passing through the inductance and resistance models, three-phase current responses are generated. The magnetomotive force and magnetic flux linkage produced by the currents are used to calculate the motor torque. Finally, the calculated torque is applied to the mechanical model of the motor rotor to obtain the rotor angle and speed responses, which are then fed back to the field-oriented control and speed control loops.

5.2. Simulation of the ITSCF Model

Based on the motor model in Figure 8, the a-phase ITSC architecture from Figure 2 is integrated into the Simscape block diagram, as shown in Figure 9. In this circuit, outlined in the red box in Figure 9 and detailed in Figure 10, V_af represents the voltage across the phase-a coil during ITSCF, while V_f denotes the ITSC voltage. The associated voltage drops include r_f for the parallel resistance of the short-circuited coil, μL for the ITSC inductance, μr_s for the ITSC resistance, and μM for the voltage generated by mutual inductance, as well as the induced back electromotive force (EMF) of the short-circuited coil, μλ^’_mω_rcos(θ_r). In this setup, r_f is set to represent the severity of ITSCFs and affects the magnitude of the circulating current generated in the ITSC. Additionally, the voltage generated by mutual inductance μM is calculated using the circuit model outlined in the orange box in Figure 9.

Figure 11 and Figure 12 show the voltage and current waveforms for the q-axis under conditions of 1500 rpm and 600 rpm, respectively, with r_f = 0 Ω and μ values of 0.3 and 0.7. It can be observed that both the q-axis voltage and current exhibit a second harmonic component response characteristic. Furthermore, as the internal short-circuit ratio μ increases, the responses of the q-axis voltage and current become more pronounced. The reason is that as μ increases, the back-EMF of the shorted coil also increases, leading to an increase in i_f. This causes the waveforms of V_q and i_q to include the voltage and current characteristics of the shorted coil. Furthermore, it is evident that when r_f is set to 0 Ω, the influence of ITSCF on the q-axis voltage and current is less pronounced. This study will further analyze the responses of V_q and i_q under different conditions of r_f regarding this phenomenon.

To investigate the influence of r_f on V_q and i_q, simulations are conducted at 1500 rpm and 600 rpm with a fixed μ of 0.7 and varying r_f values of 0 Ω, 10 Ω, and 100 Ω. The corresponding responses of V_q and i_q are presented in Figure 13 and Figure 14, respectively. By comparing the two sets of waveforms, it is observed that a larger r_f value corresponds to a less severe ITSC. This is attributed to the significant increase in voltage drop across r_f, which in turn leads to larger magnitudes of V_q and i_q. Conversely, a smaller r_f value results in a more severe ITSC, as the reduced voltage drop across r_f causes V_q and i_q to decrease. These results indicate that varying the value of r_f can manifest different levels of ITSCF severity. Specifically, as the short-circuit condition becomes more severe, the responses of V_q and i_q become less pronounced compared to the simulation results in reference [8], which showed a smaller V_q response for a larger r_f and a larger V_q response for a smaller r_f. However, the simulation results of this study exhibit an opposite trend. This discrepancy is attributed to the voltage drop caused by the interaction between the r_f and i_f currents, which is projected onto the phase voltage and subsequently transformed into the V_q component. The reason for this difference, aside from the variation in motor specifications and parameters, may also be due to whether the simulation program accounts for the three-phase imbalance caused by ITSC, which must also be considered. Therefore, this study employs MATLAB/Simscape for the simulation to accurately represent the three-phase voltage imbalance of the motor.

Further analysis of the responses of i_f with r_f set to 0 Ω and 10 Ω for different μ values is shown in Figure 15. Figure 15a presents the i_f response for r_f = 10 Ω. It is evident that the magnitude of i_f increases with increasing μ values, which is consistent with the characteristic that a more severe ITSC results in a more pronounced short-circuit circulating current i_f. Figure 15b depicts the i_f response for r_f = 0 Ω, which exhibits a significantly larger magnitude compared to the r_f = 10 Ω case, as expected. However, a counterintuitive phenomenon is observed: the magnitude of i_f decreases with increasing μ values, contradicting the conventional understanding that a more severe short circuit should correspond to a larger short-circuit circulating current i_f. For the case of r_f = 0 Ω, to further investigate the relationship between i_f and the voltage responses of various components in Figure 10, Figure 16 and Figure 17 were generated. Figure 16 reveals that the voltages across the ITSC resistance μr_s and inductance μL, as well as the back electromotive force μλ^’_mω_rcos(θ_r), scale proportionally with the severity of the ITSCF, as indicated by the scaling factor μ. Figure 17 reveals that v_af and i_a exhibit similar magnitudes, rather than the expected proportional relationship with μ. By integrating the analysis results of i_f in Figure 15b and the voltage responses of various components in Figure 16 and Figure 17, it is observed that under different μ conditions, the voltage drops across μr_s and μL, as well as the back-EMF μλ^’_mω_rcos(θ_r), exhibit proportional changes with μ. However, the magnitudes of V_af, i_a, and i_f do not change in proportion to the scaling of μ. Upon observing Figure 10, it can be inferred that the reason is that r_f and the shorted coil are configured in parallel, with i_a taking the path of least impedance, which is r_f (0 Ω), and thus not flowing through the shorted coil. Consequently, the responses of V_af, i_a, and i_f do not vary with μ.

From the results of the analysis mentioned above, it can be observed that when an ITSCF occurs, both V_q and i_q exhibit characteristics of a second harmonic component. However, when r_f = 0 Ω, although i_f is very large, the actual characteristics become less noticeable. This scenario can be considered the worst-case situation for ITSCF. Subsequent wavelet transform analysis will be conducted under this condition to verify its feasibility of using this approach for data preprocessing in AI deep learning models.

5.3. Wavelet Method Analysis Results

Applying Equation (27) to determine the number of layers n_ls for the DWT, with f_e = 10,000 Hz and f_s = 100 Hz, results in n_ls being 6.

Table 2. Frequency range of each coefficients at 1500 rpm.

DWT Coefficient	Symbol	Frequency Range
Detailed Coefficient of the First Range	D1	2500~5000 Hz
Detailed Coefficient of the Second Range	D2	1250~2500 Hz
Detailed Coefficient of the Third Range	D3	625~1250 Hz
Detailed Coefficient of the Fourth Range	D4	312~625 Hz
Detailed Coefficient of the Fifth Range	D5	156~312 Hz
Detailed Coefficient of the Sixth Range	D6	78~156 Hz
Approximate Coefficient of the Sixth Range	A6	0~78 Hz

presents the frequency range of each of the discrete wavelet coefficients. The ITSCF voltage data extracted from Figure 9 are subsequently fed into the DWT to acquire wavelet coefficients for each frequency range.

By employing the simulated data presented in Figure 9, the wavelet coefficients of V_q for various μ values within the ITSCF framework are extracted via wavelet transform. Subsequently, signal reconstruction is conducted based on these coefficients, and the results are depicted in Figure 18 and Figure 19. It can be observed that after performing the DWT and IDWT, the signals for each frequency range were analyzed. The reconstructed signal in the fifth range (156 Hz–312 Hz), which corresponds to twice the fundamental frequency (200 Hz), exhibited a significantly larger value compared to other frequency ranges. The analysis reveals a pronounced amplification of V_q in specific wavelet frequency components of ITSCF.

However, the DWT alone is insufficient for effectively comparing the trends of V_q variations in ITSC. Therefore, the values for each frequency band are converted into energy percentages using Equation (39), resulting in the bar charts shown in Figure 20 and Figure 21. The results demonstrated that a larger μ value was associated with a more significant second harmonic distortion, leading to a higher proportion of energy concentrated in the 156–312 Hz frequency range.

6. Experimental Results

Firstly, the drive test system of IPMSM is introduced in detail, including the composition of voltage and current data acquisition interfaces. The ITSCF setting conditions for verification are also described. Secondly, the health and ITSCF data results are analyzed using DWT and discussed. Finally, the diagnosis is performed using the energy percentage method and verified by FFT. Overall, the diagnostic results demonstrate the feasibility of the proposed DWT and energy percentage methods in practical applications.

6.1. Experimental Platform

The experimental platform for IPMSM ITSC DWT data diagnosis is illustrated in Figure 22. A 400 W IPMSM, with specifications and parameters listed in Table 1, is used as the test motor. A 1.1 kW load motor is coupled to the test motor via a coupling, forming a dynamometer testbed. For fault diagnosis, external acquisition interfaces and current sensors are utilized for acquiring voltage and current waveforms, as well as performing analog-to-digital conversion, with the sampling rate set to 10 kHz. This study primarily records the three-phase voltages of the motor. Each data set, representing a 1 min recording, comprises 600,000 data points.

The motor test conditions involve steady-state operation, with the rotational speeds set at 600 rpm and 1500 rpm, and the load set at 25% of the rated torque. The motor conditions are categorized into healthy and ITSC states, with the ITSC ratio set at μ = 0.3. To prevent the shorted coil from burning out, a parallel resistor, r_f =10 Ω, is used. The experiment first involves the acquisition and compilation of motor data, followed by filtering out noise from the input data. The three-phase voltage dataset is then utilized to calculate V_q, and the DWT is employed to obtain the wavelet coefficients of V_q. To acquire comprehensive feature information, the IDWT is employed to reconstruct the original waveforms across various frequency bands, which is a crucial aspect of this research. Finally, the novel energy percentage method is used to quantify the characteristics of each frequency band, visualizing the IDWT waveforms, which allows for an effective assessment of the motor’s condition.

6.2. DWT Results

As shown in Figure 23, the three-phase voltage data for both healthy and ITSCF conditions are obtained under test conditions of 1500 rpm and 25% load. The V_q is then derived using Equation (16). Despite Figure 23b, actual product testing reveals that the V_q waveform of ITSC motors exhibits significantly larger amplitude variations and a higher harmonic content compared to healthy motors, attributed to the impact of inter-turn short circuits. The V_q voltage data is analyzed using DWT, and the number of DWT levels is defined based on Equation (27), where f_e is 10,000 Hz and f_s is 100 Hz, resulting in a DWT level of n_ls = 6. The corresponding frequency ranges are shown in Table 2. The wavelet coefficients (details) obtained from the analysis exhibit varying temporal resolutions across different frequency bands, with higher frequency bands demonstrating finer temporal resolution. Furthermore, IDWT is used to reconstruct the waveforms at the resolution of each frequency range, as shown in Figure 24 and Figure 25, which display the IDWT-reconstructed signals for both healthy and ITSC conditions. In this experiment, an 8-pole test motor is used, meaning the twice-frequency at 1500 rpm is 200 Hz. As shown in Table 2, this falls within the D₅ frequency range. Although DWT and IDWT separated the information for each frequency range, Figure 24 and Figure 25 contain excessive data, making it difficult to directly observe error characteristics. To facilitate fault diagnosis, further quantification of the information in each frequency band is necessary.

6.3. ITSCF Diagnosis Results

In the practical fault diagnosis of IPMSM, this study employs an innovative energy percentage method (40) for the application diagnosis of actual motor products, aiming to expand the effective application of the research methodology. By calculating the proportion of energy from each frequency range and the reconstructed signals, visual indicators are obtained for error characteristic assessment, enabling the final results to reveal the differences between healthy motors and ITSCF. The energy percentage analysis results under steady-state operating conditions at 1500 rpm are shown in Figure 26. It can be observed that the D₅ value for ITSCF is higher than that of the healthy condition. The energy percentage data for each frequency range in Figure 26 are further organized in

Table 3. Frequency range of each coefficients at 1500 rpm, 25% load.

Frequency Range	Healthy Condition	ITSCF
2500~5000 Hz	5.0%	5.7%
1250~2500 Hz	5.0%	5.8%
625~1250 Hz	6.3%	8.3%
312~625 Hz	5.7%	13.1%
156~312 Hz	4.2%	15.3%
78~156 Hz	2.2%	11.7%
0~78 Hz	71.8%	40.1%

. The D₅ percentage for ITSCF is 15.3%, while the D₅ percentage for the healthy condition is 4.5%. This confirms the effective applicability of the DWT and IDWT methods proposed in this study for detecting ITSCF characteristics.

To validate the effectiveness and practicality of the proposed IPMSM inter-turn short-circuit fault feature extraction framework, a comparative analysis using FFT on V_q is conducted to demonstrate the superiority of the proposed method. As shown in Figure 27, the V_q component of the ITSC motor exhibits a second harmonic frequency of approximately 200 Hz, with an amplitude approximately 2.5 times that of the healthy motor. This characteristic second harmonic component can serve as a distinctive fault feature for ITSC diagnosis. Further analysis reveals that the synchronous V_q information can also show fault characteristics through FFT. However, in practical applications, due to issues such as rotational velocity errors and external environmental interference, directly observing the amplitude at specific FFT frequencies may lead to misjudgments. Therefore, it is confirmed that the method proposed in this study, which observes and analyzes the energy percentage within the DWT frequency ranges, is still superior to the application of FFT. Furthermore, the FFT analysis results corroborate the trends observed in the DWT analysis, thereby validating the effectiveness and field applicability of the proposed DWT-based approach for practical applications.

In addition to the aforementioned experimental comparisons, the experiment was also conducted at 600 rpm for comparative analysis. Under the test condition of 600 rpm, the twice-electrical frequency is approximately f_e = 80 Hz, with a sampling rate of f_s = 10 kHz. Given the 80 Hz characteristic frequency, which lies at the boundary between the A₆ and D₆ frequency bands, as illustrated in Table 2, it is imperative to adjust the sampling rate to prevent the feature from being dispersed across two different frequency bands. Therefore, the DWT frequency ranges for 600 rpm are calculated based on Equation (27), as shown in

Table 4. Frequency range of each coefficient at 600 rpm.

DWT Coefficient	Symbol	Frequency Range
Detailed Coefficient of the First Range	D1	7500~3750 Hz
Detailed Coefficient of the Second Range	D2	3750~1875 Hz
Detailed Coefficient of the Third Range	D3	1875~937 Hz
Detailed Coefficient of the Fourth Range	D4	937~468 Hz
Detailed Coefficient of the Fifth Range	D5	468~234 Hz
Detailed Coefficient of the Sixth Range	D6	234~117 Hz
Detailed Coefficient of the Seventh Range	D7	117~58 Hz
Approximate Coefficient of the Seventh Range	A7	58~0 Hz

. The energy percentage analysis results under steady-state operating conditions at 600 rpm are shown in Figure 28, where it can be directly observed that the D₇ value for ITSCF is higher than that for the healthy condition. The energy percentage data for each frequency band in Figure 26 is further organized in

Table 5. Frequency range of each coefficient at 600 rpm, 25% load.

Frequency Range	Healthy	ITSCF
7500~3750 Hz	5.3%	4.6%
3750~1875 Hz	6.6%	6.4%
1875~937 Hz	7.0%	5.1%
937~468 Hz	6.8%	6.3%
468~234 Hz	6.7%	8.3%
234~117 Hz	6.5%	9.9%
117~58 Hz	4.1%	10.2%
58~0 Hz	57.1%	41.2%

. The D₇ percentage for ITSCF is 10.2%, while the D₇ percentage for the healthy condition is 4.1%. This result is also consistent with the ITSCF characteristics derived in Section 3. As shown in Figure 29, the V_q component of the ITSC motor exhibits a second harmonic frequency of approximately 80 Hz, with an amplitude approximately three times that of the healthy motor. As shown in Table 3 and Table 5, the energy percentage of the second harmonic component (approximately 200 Hz at 1500 rpm and 80 Hz at 600 rpm) within the corresponding DWT frequency bands (156–315 Hz and 58–117 Hz, respectively) is significantly higher for ITSC motors compared to healthy motors. These results clearly demonstrate the characteristic second harmonic feature of ITSC faults and validate the effectiveness of applying DWT and energy percentage analysis for fault diagnosis.

7. Conclusions and Discussion

In this study, a simulation-based approach is employed to diagnose ITSCF in IPMSM using the DWT analysis of motor voltage. Initially, an error model for the internal phase-to-phase short circuit (ITSC) in the IPMSM windings was derived. The characteristic errors generated during inter-turn short circuits were calculated, and the ITSC simulation was performed using the integration of MATLAB/Simulink/Simscape. Subsequently, the ITSC simulation results were analyzed to understand the relationships among the relevant characteristics. Finally, the motor voltage under ITSC conditions was analyzed using wavelet transform to verify the proposed concept and methodology. The results demonstrate that specific frequencies corresponding to the characteristic features of internal short-circuit faults can be extracted from stator voltage using discrete wavelet transform when a short circuit occurs within the stator winding. The simulation results validate the proposed method, indicating that wavelet transform offers a relatively accurate detection technique. This study primarily focuses on data analysis during steady-state motor operation. Although DWT’s time-frequency analysis is well-suited for variable speed conditions, more sophisticated diagnostic tools, including CNNs and AEs, are required to extract meaningful information from the DWT data in these scenarios.

Furthermore, the proposed IPMSM ITSCF feature extraction system framework significantly contributes to the field of IPMSM diagnosis by improving the performance of condition-based monitoring and DWT-based techniques. In industrial applications, after capturing information such as voltage, current, and speed from the IPMSM on the production line, it can be transmitted to the upper-level controller, such as a web-based cloud server, via industrial buses such as EtherCAT, CANBus, and others. The wavelet transform method can be easily implemented on these devices, utilizing their computational resources to perform wavelet transform calculations, thereby enabling the diagnosis of ITSCF.

Moreover, the system offers valuable insights for dataset collection in the development of deep learning models aimed at identifying various motor faults. The findings of this study are expected to have a high degree of applicability, potentially extending to the diagnosis of other types of electrical machine faults. This method could also be applied to fault diagnosis in Surface Permanent Magnet Synchronous Motors (SPMSMs), Synchronous Reluctance Motors (SynRMs), and other similar systems. In the future, the research results of the IPMSM ITSCF feature extraction system framework could be used to address the extensive data preprocessing requirements for designing deep learning models and be applied to the field of AI-based health monitoring of industrial equipment.