When the motor appears to have an eccentric weakness, its topology will become asymmetric. When the strange fault model of the engine is built in Flux, the intact geometric construction of the PMSM needs to be contrived. In this paper, 10% and 20% static eccentricity fault models of permanent magnet synchronous motors are built in Flux. As shown in Fig. 2a, the coordinates Or and Os of the stator center of PMSM and the rotor axis are set as (0.06,0) and (0,0), respectively, so that the distance between them is 0.06 mm (uniform air gap length is 0.6 mm). In this way, the motor model with a 10% static eccentricity fault is obtained Fig. 1b shows the setting method of the PMSM model for 20% eccentric fault. That is, the fixed and rotor axes are set as (0.12,0) and (0.0), respectively, so that the distance between the fixed and rotor axes is 0.12 mm. Then in mechanical properties, set the center of rotation of the rotor to (0.0) so that the center of rotation of the rotor coincides with the axis of the rotor. Thus, the PMSM model with 10% and 20% static eccentricity faults is obtained.

Due to its high energy density and low price, neodymium boron (NdFeB) is commonly used to make permanent magnets for permanent magnet synchronous motors. However, it also has the disadvantages that the magnetization intensity is greatly affected by working conditions, and the demagnetization is irreversible. The demagnetization failure of PMSM is usually uniform demagnetization of permanent magnets.

In Flux, it can be described by “Remanent flux density” and “Relative permeability” to the type of permanent magnet and the residual magnetic of permanent magnet materials Br, the slope of the magnetic curve is defined. In this paper, three magnetic curves are defined respectively, which are the magnetic curves of ordinary (He), 25% (De25), and 50% (De50) demagnetized materials. An PMSM model with demagnetization faults of different degrees in permanent magnets on opposite pairs is established.

As shown in Fig. 2, yellow, blue, and red lines in three different colors represent normal, 25% demagnetization, and 50% demagnetization magnetic profile curves of permanent magnets. Their remanence Br can be set to 1.2, 0.9, and 0.6 T, respectively, and their permeability can be set to 1.05. Figs. 3a and 3b respectively represent the sub-densities of permanent magnets with 25% and 50% demagnetization. It can be seen from the figure that the more profound the demagnetization degree of permanent magnets, the more sparse the magnetic densities.

In this paper, finite element models of different types of PMSM established in Altair Flux were imported into a vector control system in MATLAB for co-simulation. Current strategies used in the vector control system id=0 and using the space vector pulse width modulation (SVPWM) motor power supply voltage vector modulation algorithm. Fig. 4 shows the PMSM finite element model in a loop simulation system co-simulated by AltairFlux and MATLAB-Simulink [23].

Fig. 5 respectively shows the stator three-phase current simulation results of 25% demagnetization, 50% demagnetization, and normal PMSM at a rated speed of 4000 rpm and load 0 N. Figs. 5a–5c are the simulation results of stator currents of three types of motors, respectively. It can be seen from the figure that demagnetization of the engine will increase the time for the motor to enter the steady state. Fig. 5d shows the speed variation of the three motors. It can be seen that demagnetizing failure will increase the time required for the motor speed to reach a steady state. Fig. 5e shows the electromagnetic torque comparison. In the same way, the greater the demagnetizing intensity, the longer the adjustment time required for the motor to reach a steady state and the smaller the electromagnetic torque. In summary, demagnetizing failure is equivalent to increasing the engine load, thus increasing the adjustment time for the motor to reach a steady state.

Fig. 6 shows the simulation results of typical, 10% eccentric, and 20% strange permanent magnet synchronous motor when the given speed is 4000 rpm, and the load is 0 N.m. Figs. 6a–6c are the simulation results of stator currents of three types of motors, respectively, and (d) and (e) are the comparison of three motor speeds and electromagnetic torques. It can be seen that pure static eccentricity fault has no apparent influence on the stator current, rotational speed, and electromagnetic torque of PMSM [24].

FFT is a standard signal processing method that can quickly realize signal transformation from a time domain to a frequency domain. Discrete fourier transform (DFT) is the most basic method, digital signal processing of the input signal by frequency fs interval sampling after n times, such as DFT processing of n discrete points get n the plural. Each complex number represents the amplitude and phase information of the frequency component with fs/n as the interval, to obtain the spectrum information of the input signal [25].

For a time-domain discrete signal X(n) with length n, the DFT transformation result is X(k), then:

(1)X(k)=DFT{x(n)}=∑n=0N−1x(n)∗e−j2πNnk,k=0,1,…,N−1

If the following definition is made:

(2)WN=e−j2πN

(3)WNnk=e−j2πNnk

Then X(k) can be expressed as:

(4)X(k)=DFT{x(n)}=∑n=0N−1x(n)∗WNnk,k=0,1,…,N−1

FFT mainly uses periodicity, Symmetry, and scaling of rotation factor to carry out butterfly iterative calculation of DFT to reduce the computation amount of DFT, but its basic properties are still similar to DFT.

In the PMSM fault model built by AltairFlux and MATLAB, the simulation model collected 8000 data points in the time domain signal of stator current at 0.25 s, and the sampling frequency was 32 kHz. The spectrum diagram of the stator three-phase currents of the standard motor, the 25% and 50% demagnetization fault motor, and the 10% and 20% static eccentricity fault motor obtained through the co-simulation is shown in Fig. 7. Since bilateral spectra are symmetrical, only unilateral spectra can be analyzed.

Because high-dimensional features often contain more data features, this paper first upgraded the frequency domain signal of motor stator current into the two-dimensional gray image, then produced a data set and imported it into CNN [26]. The scattering of data sample values within the range of gray values can be deemed as a probability problem. Because of the significant number theorem, on the condition the amount of data is large enough, the data will obey the Gaussian distribution. Therefore, the mapping function of data scattering into the gray range is set as a one-dimensional Gaussian distribution function in this paper, as shown in the formula (5). Fig. 8 shows the process of image conversion based on the autocorrelation matrix [27].

(5)f(xij)=255×12πσe(xij−u)2σ2

where xij are elements in the ith row and the Jth column of the autocorrelation matrix; μ is the mean value of the original data; σ² is the variance of the original data.

If P(i,j) it represents the intensity of pixels in the generated row I and column j and S(i,1) represents the I-th data point in the original frequency domain signal, then:

(6)P(i,j)=ST(1,j)∗S(i,1)+f(xij)

Therefore, the pixels in the I-th row and J-th column of the grayscale image generated by the image conversion method based on the autocorrelation matrix reflect the intensity of the I-th data point in the original data, and the generated grayscale map is symmetrical about its diagonal.

In this paper, an image conversion method based on an autocorrelation matrix is used to convert the frequency domain signal of motor stator current into a gray-level image. In the spectrum diagram, 500 data points are sampled every 30 data points and dimensionally upgraded. The way of data interception is shown in Fig. 9:

There are 8000 data points in the time domain signal of single-phase current of different fault motors obtained by co-simulation, and 8000 data points in the bilateral spectrum obtained by FFT. By converting the two-sided range of the three-phase current into the gray image, the original gray image of each type of fault is 793. After data enhancement, 20% of the grayscale images are randomly selected as the test set, and the data set information is shown in Table 2 and the grayscale images converted from frequency domain signals of PMSM with different fault types are shown in Fig. 10.

The training of CNN generally requires large data sets, and reading grayscale images in data sets one by one will occupy a lot of computing resources. In the TensorFlow framework, sample images can be made into TFRecord files to compress the data set into binary coding. During training, it can be decoded and then imported into CNN for training. This method will significantly save computing resources.

GCNN network parameters are the optimal hyperparameters determined by the grid search method. The parameters are set reasonably, and the diagnostic accuracy of different faults is close to the maximum accuracy allowed by the theory.

This article adopts a step decay learning rate, and the selected decay learning rate function is:

(11)k=k0∗a[nm]

Among them, k0 is the learning rate at the beginning of GCNN training; a is the attenuation coefficient of GCNN; n indicates how many rounds GCNN has trained so far; m is a constant applied to adjust the variation of GCNN learning rate; [nm] stands for rounding; k stands for the current learning rate of GCNN.

The initial learning rate is 0.01, the attenuation coefficient a is 0.85, and 30 rounds of training with a m of 5 can result in faster and better training results. The learning rate of GCNN during the exercise is shown in Fig. 15 [43]:

GCNN adopts the cross entropy that can be adopted to estimate the contrast between the theoretical value and the actual value to evaluate its loss function. The smaller the cross entropy value is, the closer the two probability distributions are, and it is described as formula (12) [44]:

(12)C=−1n∑x[y∗lna+(1−y)∗ln(1−a)]

In formula (12), x stands for the sample, n stands for the total amount of models, y stands for the theoretical value, and it stands for the actual value.

The other model parameters in GCNN, except for ASPP, are shown in Table 3, and the model parameters of ASPP are shown in Table 4, where k is the size of the ordinary convolution kernel, c is the number of characteristic output channels, d is the expansion rate of cavity convolution kernel, p is the filling mode, r is the type of activation function, and b is the offset.

The produced data set was imported into the built GCNN model for training. The training results are shown in Fig. 16. The accuracy of the test set is 99.08%, and the loss function is 0.0035. The diagnostic accuracy of De25, De50, He, Se10, and Se20 motors is 99.80%, 99.60%, 99.70%, 98.10%, and 98.20%, respectively, which has high diagnostic accuracy. In addition, cross-validation was used to repeat the experiment many times to enhance the reliability of training results and avoid the interference of random factors on experimental results. After many experiments, it is found that the average accuracy of the five-fold cross-validation is the highest. The verification results of GCNN are shown in Table 5. The difference between the average and experimental results obtained by random segmentation data is only 0.08%, which can fully prove the reliability of the experimental results.

To demonstrate the advantage of the GCNN, this paper conducts comparison experiments with MCNN [27], Bi-LSTM [45], DBN [46], and RNN [47], respectively, and the experimental results are shown in Fig. 17. The optimal hyperparameters of other comparison algorithms are determined by the grid search method, the parameters are set reasonably, and the diagnostic accuracy of different faults is close to the maximum accuracy allowed by the theory. The data set used is the same data set as GCNN, and both are self-made data sets. It can be seen that the accuracy rates of MCNN, Bi-LSTM, DBN, and RNN on the test set are 97.96%, 90.11%, 94.42%, and 91.78%, respectively, which are much lower than 99.00% of GCNN; the loss functions of the four comparison algorithms are 0.099, 0.242, 0.168, and 0.231, respectively, which are higher than that of GCNN. The comparison experiments thoroughly verify the superiority of GCNN.

In order to verify the anti-interference ability and robustness of the model proposed in this paper against noise, Gaussian noise, black pepper and salt noise, white pepper and salt noise, and black pepper and salt mixed noise with white pepper and salt were added to the self-made data set, respectively. The gray images after adding noise are shown in Figs. 18–21. After adding noise, different network fault diagnosis accuracy rates are shown in Table 6, and the parameters of each network for comparison experiment are determined as the optimal parameters by the grid search method.

It can be seen from the results of noise interference experiments that the fault diagnosis accuracy of the proposed GCNN network for different types of permanent magnet synchronous motors has not changed much in the face of noise interference such as Gaussian noise, black pepper and salt noise, white pepper and salt noise, black and white pepper, and salt noise, and is always maintained within 0.5%. The ability of anti-noise interference and robustness are obviously better than the other four networks.

Grad-CAM of Nos. 1, 17, 33, 49, 65, 81, 97, 113, 116 gray images of 25% and 50% demagnetization faults were selected for analysis. The Grad-CAM is shown in Figs. 22 and 23.

The analysis of the Grad-CAM of 25% demagnetization and 50%PMSM gray image shows that the fault characteristics of demagnetization faults of different degrees are the same. There are 16 fault characteristics of demagnetization faults of PMSM. They are 140, 280, 440 Hz, 2.88, 3.44, 5.72, 6.68, 7.12, 7.48, 8, 9.68, 10, 12, 12.32, 12.64, 14.96 kHz; It is marked on the spectrum diagram of De25 and De50 fault PMSM, as shown in Fig. 24.

Similarly, Grad-CAM of Nos. 1, 17, 33, 49, 65, 81, 97, 113, and 116 gray images with 10% and 20% static eccentricity faults are analyzed. The gradient-type activation thermal maps (Grad-CAM) are shown in Figs. 25 and 26.

Through the analysis of the Grad-CAM thermal map, it can be found that there are 21 fault features in Se10, were 180 Hz, 2, 3.04, 3.84, 4.16, 4.64, 4.96, 5.44, 5.76, 6.76, 8, 9.8, 10.88, 11.72, 12.24, 13.32, 14.16, 14.64, 14.88, 15.24, 15.72 kHz, it is marked on the PMSM spectrum diagram of Se10 and Se20 faults, as shown in Fig. 27.