Bearing Fault Detection of IPMSMs using Zoom FFT

1. Introduction

Permanent magnet synchronous motors (PMSMs) are used in a variety of applications. Fig. 1 shows a generalized structure of an IPMSM. Because the rotor of the PMSM is a permanent magnet, it does not require an external power source to generate the flux. This helps minimize power consumption and improve system efficiency [1, 2]. Thus, PMSMs compare well to other motors. As PMSM usage increases, motor failures are also expected. Accordingly, several researchers have conducted motor reliability studies. The results of several studies show that bearing faults cause more than 40% of all machine failures [3]. Thus, it is critical to monitor the condition of bearings in electrical machines.

Fig. 1.Equivalent model of IPMSMs

In many studies, a popular approach has been to measure bearing condition using a vibration sensor. This is because bearing faults cause radial movement between the stator and rotor, and bearing fault characteristics are reflected in the machine vibration [4]. However, an accelerometer or vibration velocity transducer has to be attached to the bearing or bearing support housing directly. Further, additional system cost is incurred because of the additional vibration sensor.

As a result, monitoring bearing condition through the stator current has been studied for many decades. Many researchers have determined that motor vibration frequencies are reflected in the stator current [5-7]. Therefore, monitoring the stator current provides information about bearing faults. In general, current sensors are already installed in many industries and therefore, there is no need to install additional sensors to monitor the stator current. Thus, monitoring the stator current to detect bearing faults provides economic and implementation benefits.

Frequency analysis of the stator current has to be performed to detect frequencies through bearing faults. The traditional frequency analysis method is the fast Fourier transform(FFT). The FFT can provide frequency information via a simplified Fourier transform algorithm. However, there are several frequencies in the stator current and it is difficult to differentiate between these frequencies and those of the bearing faults [8]. Thus, it is necessary to increase the resolution of the FFT to detect the bearing fault frequencies. Because the resolution of the FFT depends on the sampling frequency, when the number of samples is the same, FFT-based methods require a large amount of data and high computational costs to increase the resolution. However, it is not always possible to satisfy the requirement of extensive data because of hardware limitations.

The short time Fourier transform (STFT) can be used to achieve robustness and simplicity to overcome the limitations of the FFT [9]. However, the STFT is dependent on the type and the length of the window, and therefore, a tradeoff between time and frequency resolution is needed.

In fact, because the frequency of the bearing fault is present in specific frequency bins, and the high frequency resolution and narrow frequency band are needed, it is more efficient to expand the frequency region of interest. Therefore, the zoom FFT (ZFFT) algorithm has been studied to decrease the amount of data and computational cost [10, 11]. The ZFFT is the preferred method to magnify the frequency band centered on an arbitrary frequency. Therefore, the ZFFT can increase frequency resolution while keeping the same sampling frequency and number of samples.

This paper presents an efficient method to detect bearing fault by monitoring the stator current using ZFFT analysis. To generate bearing faults and scrutinize the validity of the ZFFT, equipment for generating oscillation is used. Because the ZFFT is efficient regarding the computational cost, the proposed method can give effective way to find out the bearing fault.

 

2. Stator Current Spectral Analysis

There are several reasons that can cause bearing faults. For example, an installation problem can cause the bearing to be unbalanced on the shaft or in the housing. Fig. 2 shows four types of bearing misalignments. Further, a prolonged period of motor drive leads to abrasion in bearings that produce motor vibration. Since ball bearings support the rotor, any bearing fault or unbalance causes a radial movement to the rotor and the stator.

Fig. 2.Four types of rolling-element bearing misalignment

The relationship between the bearing vibration and the stator current can be described by the air gap eccentricity. Because of the air gap eccentricity, anomalies are produced in the air gap flux density and the predictable frequencies are reflected in the stator current. These predictable frequencies, fbng , are given by [3]

where m = 1,2,3,···, ff is the fundamental frequency and fcf is one of the characteristic bearing fault frequencies.

The characteristic frequencies for a bearing fault depend on bearing dimensions as shown in Fig. 3. These frequencies can be described by three practicable forms [3]. First, the ball spin frequency is produced by the rotation of each ball. Since the ball makes contact with the inner and outer raceway upon each rotation, the vibration frequency in the stator current is twice the spin frequency and is given by

Fig. 3.Ball bearing dimensions

where PD is the bearing pitch diameter, BD is the ball diameter, β is the contact angle of the balls on the races and fmr is the mechanical rotor speed in Hertz.

Next, the outer and inner raceway frequencies need to be identified. These frequencies are produced when each ball touches a defect in the outer or inner raceway. If the number of balls is n, the vibration occurs n-times in the total cycle. Therefore, the outer and inner raceway frequencies are described as

The three practicable forms of bearing fault frequency — (2), (3), and (4) — require bearing dimensions to calculate the exact result. However, the characteristic frequencies can be approximated with forms where bearing dimensions are unnecessary [3].

These simplified equations can be used for bearings between six and twelve balls and allow users to detect bearing faults without accurate understanding about bearing dimensions. Therefore, bearing fault analysis can be performed easily by monitoring the stator current.

 

3. Basic Theory of ZFFT

3.1 Theory of ZFFT

As stated in section II, the characteristic bearing fault frequency is predictable. For frequency analysis of the stator current, FFT-based methods can be implemented. However, the magnitude of these frequencies is much less compared to that of other frequencies, such as a cyclical load torque variation and the fundamental frequency [8]. In addition, because the bearing fault frequencies are already known, a specific frequency band is only necessary to detect the bearing fault. Accordingly, a high-resolution frequency analysis technique is required to distinguish these frequencies.

To increase resolution in the FFT, the sampling time and the number of samples need to be increased. However, these methods can lead to an increase in the computational cost. Further, if we only need the particular frequency region and high resolution, FFT-based methods require unnecessarily excessive calculations. The ZFFT algorithm has been studied taking this into consideration.

The ZFFT is a useful algorithm to expand the narrow frequency band centered on any frequency. This is closely related to multi-rate filtering and is the same compared to the single sideband modulation [12]. In real applications on DSP-based systems, memory limitations exist and the size of matrices in the FFT prevents augmenting the number of samples. Compared with FFT, the ZFFT algorithm can provide a direct reduction of sampling frequency while maintaining the same sampling time and number of samples, thus increasing the frequency resolution.

A process of the ZFFT algorithm is shown in Fig. 4. Let us consider the system of N samples and the sampling frequency, fsamp, to compare the resolution between the ZFFT and the FFT.

Fig. 4.Block diagram of the ZFFT process

First, because the FFT is calculated on the condition of the power of two, the input signal is sampled at N = 2n. Then, the frequency resolution is Δf = fsamp / N . The frequency transition is performed to expand a narrow span centered on interested frequency fc. This process can be conducted by multiplying the real number by cos(2π fct) and the complex number by sin(2π fct). This is expressed as

The signal x[k] is obtained after the process of sampling the signal. The complex signal after the frequency transition is filtered by a low-pass filter with a bandwidth Bp and [fc, fc+Bp] is the frequency band of interest.

Second, the filter output signal xL[k] is undersampled. This process is called decimation. In this process, the signal xL[k] is extracted for the Mth time where M is a decimation factor. Owing to the Mth time sampling, the number of samples reduces to N /M and the sampling frequency also reduces to fsamp /M. Therefore, the frequency resolution is Δf = (fsamp / M ) / ( N /M ) M N M and this is the same frequency resolution of that of the first process.

Finally, the decimation signal xM[k] is filled with zeros at the end of this signal in the time domain. This process is called zero padding. The number of zeros that fills the matrix of xM[k] is the number of signals reduced in the decimation process, that is N − ( N/M ) and can be described as

where K is the zero padding factor.

If zeros of the zero padding process are filled with the end of a matrix xM [k] in the time domain, zeros of K are inserted in the intervals of each sample in the frequency domain. As a result, the number of inserted zeros does not affect the sampling, the sampling frequency is fsamp / M identically and the frequency resolution is Δf = (fsamp / M / N). Thus, it is evident that the ZFFT increases the resolution M times more than the FFT.

When the ZFFT and FFT have the same frequency resolution, NZFFT and NFFT required to the ZFFT and FFT are calculated as

The computation (C) comparison between the ZFFT and FFT is represented as

Depending on the value of M, CZFFT can become smaller than CFFT.

3.2 Application of Equipment for Bearing Fault

Bearing faults are classified into single-point defects and generalized roughness faults. Single-point defects are defined as a single, localized defect on an otherwise relatively undamaged bearing surface, for example, a pit or spall [7]. Under single-point defects, the characteristic bearing fault frequencies are produced depending on which surface of the bearing contains the faults as described in section II.

Generalized roughness faults are a type of fault where the condition of a bearing surface has degraded considerably over a large area and has become rough, irregular, or deformed, for example the overall surface roughness produced by a contamination or loss of lubricant [7]. These generalized roughness faults do not exhibit the normal frequency characteristic as that of the characteristic bearing fault frequencies. Since they are difficult to identify, many studies are being conducted to identify these faults.

Therefore, in this study, the equipment for producing bearing faults are installed to prove the validity of the ZFFT algorithm for detecting bearing faults under single-point defects. Fig. 5 shows a model of the equipment for producing bearing faults. As shown in Fig. 5, four bars are installed at the axis of rotation of a motor and an obstacle is raised under the bars so that harmonic frequencies are produced when the motor rotates. The mechanical rotation frequency is based on the motor speed in rpm as

Fig. 5.Model of the bearing fault equipment

where P is the number of poles.

The harmonic frequencies of the line current produced by an obstacle are the quadruple frequency of the mechanical rotation frequency calculated in (11) and the multiple harmonics of the quadruple frequency as described in (12) and (13).

where fosc is the quadruple frequency of the mechanical rotation frequency.

The harmonic frequencies of the line current produced by an obstacle in (12) have the same form as the characteristic bearing fault frequencies generated by a single-point defect described from (1) to (4). Thus, the validity of the ZFFT algorithm for detecting bearing faults under single-point defects can be proved through the equipment.

In this study, the ZFFT algorithm is conducted via the q-axis current in a synchronous reference frame. Because oscillation is produced by the axis of rotation and the torque is related to the q-axis current, the quadruple frequency of the mechanical rotation frequency can be detected by the q-axis current without considering the mechanical rotation frequency.

 

4. Experimental Results

The experiment is implemented using a digital signal processor (DSP) TMS320F28335. The model of the equipment for producing bearing faults is realized as Fig. 6. To test the proposed method for bearing fault detection, bearing fault signals are generated from the equipment as the motor rotates at 50 rpm. Line currents of the motor are measured by current sensors to monitor the harmonic frequencies and the amplitude spectrum is calculated to show the result of the FFT and ZFFT. The exact parameters of the bearing fault signals are shown in Table 1.

Fig. 6.Experiment setup of the bearing fault equipment

Table 1.Motor specifications

From (12) and (13), the quadruple frequency of the mechanical rotation frequency is 10/3 Hz and its first harmonic frequency is 20/3 Hz. Because the frequency resolution is 0.25 Hz/sample, the quadruple frequency of the mechanical rotation frequency and its first harmonic frequency will appear at the 13th and 26th samples, respectively.

Since ordinal numbers of the bearing fault signal can be calculated, bearing faults can be detected. However, the magnitude of bearing fault frequency is unknown and it is possible that unrelated noises exist at the corresponding bearing fault frequency. Therefore, when an ordinal number of the bearing fault frequency is derived from (12) and (13), the proposed method monitors at a corresponding ordinal number and calculates the average value of its magnitude during a set period.

Fig. 7 shows the waveform of the FFT. Experiment parameters are the same as that in Table 1 to compare the waveform with the ZFFT. The FFT is conducted using an a-phase current and the process of frequency transition is applied. The quadruple frequency of the mechanical rotation frequency appears at the 3rd sample and its harmonic frequencies appear at the 7th and 10th samples. Because the frequency resolution of the FFT is 1 Hz/sample, the quadruple frequency and its harmonic frequencies are calculated at 3 Hz, 7 Hz, and 10 Hz. Therefore, the errors of each frequency are approximately 0.33 Hz, 0.33 Hz, and 0 Hz. This is because the 10th sample, 30/3 Hz can be divided by the frequency resolution which is 1 Hz/sample and the other samples, 10/3 Hz and 20/3 Hz, cannot be divided.

Fig. 7.FFT spectrum of a-phase current

Fig. 8 shows the waveform of the ZFFT. The ZFFT is also conducted for an a-phase current. The quadruple frequency of the mechanical rotation frequency is shown at the 13th sample and its harmonic frequencies appear at the 26th and 40th samples. Because the frequency resolution is 4 Hz/sample, the quadruple frequency and its harmonic frequencies are each calculated at 13/4 Hz, 26/4 Hz, and 40/4 Hz. The errors of each frequency are approximately 0.08 Hz, 0.16 Hz, and 0 Hz. Therefore, the ZFFT is more accurate than the FFT when the number of samples and the sampling frequency are the same.

Fig. 8.ZFFT spectrum of a-phase current

Fig. 9 shows waveforms of the q-axis current and its frequency analysis on an oscilloscope. Because of the quadruple frequency of the mechanical rotation frequency by the equipment for producing bearing faults, oscillation is shown in the q-axis current. The expected harmonic frequencies that are the quadruple frequency and its multiple harmonic frequencies from (12) and (13) can be detected by FFT with an oscilloscope and the magnitude of multiple harmonic frequencies decreases as the frequency of those increases.

Fig. 10 shows waveforms of the a-phase current and ZFFT of the q-axis current. The sampling period shows the number of total samples, which is 512 in this experiment. In the waveform of the ZFFT, multiple harmonic frequencies appear as described above. The first frequency of the ZFFT is the quadruple frequency of the mechanical rotation frequency and its ordinal number is 13. The other frequencies are the harmonic frequencies of the quadruple frequency. The ordinal numbers of each frequency are 26, 40, and 52, and those frequencies calculated by frequency resolution are 20/3 Hz, 30/3 Hz, and 40/3 Hz. Therefore, frequency analysis of the ZFFT is accurate.

Fig. 11 shows waveforms of the a-phase current under no-load and load conditions and the ZFFT of the q-axis current for the healthy machine. The a-phase current is constant and there are no frequencies except noise in the ZFFT spectrum.

Fig. 12 shows waveforms of the a-phase current under no-load and load conditions and the ZFFT of the q-axis current for the faulty machine. The a-phase current changes with the motor oscillation and there are harmonic frequencies at 10/3 Hz, 20/3 Hz, and 30/3 Hz. The amplitude of the ZFFT under load condition for the faulty machine is smaller than that of the ZFFT under no load condition. This is because the fault signals in the q-axis current are related to torque and the load is applied in the opposite direction.

Table 2 shows the computational time of the ZFFT and FFT measured at the experiment. The ZFFT and FFT are the same frequency resolution. Since M of the ZFFT is 4, the ZFFT has the short computational time four times longer than the FFT.

Table 2.Computational time of ZFFT and FFT

 

5. Conclusion

This paper proposes the bearing faults detection method using ZFFT algorithm. The ZFFT algorithm is very suitable to detect the bearing faults because it analyzes frequencies in narrow frequency band with high frequency resolution which corresponds to requirements for detecting the bearing faults. Additionally, in comparison with the FFT algorithm, the ZFFT algorithm needs smaller data and computational cost which are shown in experimental results. Therefore, the proposed method using ZFFT algorithm can be applied with motor control algorithms at a DSP unit.

In experiment, the equipment for producing bearing faults is designed to prove the validity of the ZFFT algorithm. The harmonic frequencies caused by the equipment have the same form as characteristic bearing fault frequencies. According to the experimental results, we know that the proposed method exactly detects these fault frequencies and those results shows validity of the proposed bearing faults detection.