Mathematical Decomposition of Fault Signature using Multinomial Identity and Differentiating for Detection of Broken Rotor Bar in Induction Motor

1. Introduction

Squirrel cage induction motors (SQIM) are widely used in various industries [1,2]. In industries, where uninterrupted motor operation is necessary, timely detection of rotor faults is important, as these faults account for approximately 10% of all motor faults. This issue is especially critical in harsh environments, such as oil and gas environments, traction, and mines, where motor failures can result in heavy process losses [3,4].

Over the past few years, the use of induction motors in the electric vehicle (EV) industry has become increasingly widespread. The electric motor is a crucial component of an EV as it converts electric energy to mechanical energy [5]. Therefore, any failure in any of its components can directly impact the reliability of the powertrain and the safety of passengers [6,7]. As a result, it is crucial to develop an electric motor that enhances the efficiency and performance of EVs [8,9]. Among the various types of electric motors used in EVs, the induction motor (IM) is more effective and economical than others due to its reliability, simple mechanical design, and effective field-weakening characteristics [8].

To increase the reliability of IMs used in EVs, researchers have focused on different fault diagnosis methods in the past decade, in line with the growth of the EV industry. These methods are aimed at identifying and diagnosing faults in IMs before they lead to motor failure, and include techniques such as vibration analysis, Motor Current Signal Analysis (MCSA), and motor current signature analysis [8,10,11,12,13,14,15,16,17]. By implementing these fault diagnosis methods, the reliability of IMs can be enhanced, thereby improving the overall performance and safety of EVs.

A broken rotor bar (BRB) is a type of rotor fault that can cause damage to other parts of the motor, including the winding, bearings, and mechanical parts. In addition, BRBs can lead to torque and speed pulsation and create harmonics in the current, air-gap flux, and induced voltage of stray flux [3,18,19]. Therefore, given that the BRB of SQIM is a common type of rotor defect, this article will focus on the diagnosis of BRBs.

Generally, there are three groups of fault detection (FD) methods [20,21,22]:

• Signal-based methods
• Model-based methods [23]
• Knowledge-based methods [24]

Model-based approaches require a precise and accurate mathematical model of the motor, and knowledge-based approaches have low accuracy in transient conditions and require different and high-volume data for diagnosing different motors [25]. Therefore, signal-based methods are of great interest for fault detection in electric motors.

Any type of faults affects the general signals of the motor, such as current, torque, speed, and voltage, in a specific way and creates special harmonics in them. Most of the existing electrical condition monitoring and fault detection methods are based on the analysis of the measured output signals [3,26,27]. To detect the fault based on the signal, three steps are usually taken:

• acquiring a suitable signal to detect the fault,
• using the proper method to analyze the signal,
• determining the index to make the decision about the faulty or healthy condition.

According to articles analyzing various types of motor output signals, research on fault diagnosis includes MCSA, methods based on terminal voltage analysis, air gap torque analysis, impedance sequence component analysis, leakage flux measurement, infrared detection, stray flux detection, vibration analysis, thermal analysis, acoustic analysis, electromagnetic field interference, and more [28,29]. Among these methods, MCSA is utilized in this paper due to its non-destructive nature, compatibility with several signal processing tools, and being a spectral analysis method that is commonly used for online and remote monitoring of induction motors in industrial environments [18,30,31]. The following are some MCSA methods: frequency spectral analysis, current envelope analysis method, components of negative, positive, and zero current sequences, and Park’s vector representation of three-phase current and Clark transformation [4,27,32].

Although positive, negative, and zero sequence components of current or Park and Clark vector transformations can be used for MCSA, they require sampling of three-phase current [33], which increases the cost compared to single-phase current sampling, and there is no indicator to recognize the fault type in many cases. In the current envelope analysis method, there is a challenge to produce a suitable envelope without auxiliary processing tools, and it does not show clear results for fault detection. A common and simple frequency-domain method for detecting BRB is applying the FFT on the stator current and then analyzing the produced spectrum [3,25]. However, the sampled current is almost always accompanied by noise, and for motors that operate at small slips, the fault frequency is hidden in the shadow of the base frequency. This means that base frequency leakage can mask fault frequencies, making rotor fault detection difficult, and motor damage may progress to severe damage [34]. Therefore, direct application of frequency spectral analysis and FFT method on the stator current will not have a useful and usable result and requires initiatives and auxiliary processing tools.

In order to increase the slip and distinguish between the fault frequency and the base frequency, a rotor breakage detection test has been performed in [35], which is not in the steady state motor operation mode. In [36], a normalized energy operator in the frequency domain (FDEO) is proposed for BRB fault detection, and energy operators have also been used to diagnose the broken rotor bar in some other papers [37]. Among the different methods, those that can be practical in the condition of high current noise are preferred, and in order to reduce the cost, sampling the current of one phase is preferable. Therefore, in this paper, we sample the current of one phase of the stator and use the sum square identity along with methods to remove unnecessary frequency components from the current signal.

Different signal processing tools have been utilized in various studies. Typically, these tools can be categorized into three groups: time domain, frequency domain, and time-frequency domain [38]. Time domain indicators such as mean, median, peak, kurtosis, and skewness are less commonly used to detect BRB faults. Instead, some research studies have utilized time-frequency decomposition tools to identify fault components and harmonics [39,40]. For example, the Hilbert-Huang transform (HHT) is often used for analyzing non-linear and non-stationary signals, but identifying the signal with the fault frequency can be challenging [41]. Another commonly used time-frequency domain transform is the wavelet transform. In some applications, wavelet transform is used as a filter, and simple filters can also be effective. In some articles, wavelet transform has been used for time-frequency analysis of non-stationary signals [42]. In other cases, the short-time Fourier transform (STFT) has also been used [35], but it has lower ability compared to the wavelet transform.

Other methods belonging to the time-frequency signal processing tools include the adaptive slope transform (AST), the Multiple Signal Classification (MUSIC) algorithm, the Wigner-Ville distribution (WVD), and the Choi-William distribution [43,44]. While fault detection and evaluation in the time-frequency domain may require a skilled user with powerful pattern recognition techniques, it also takes more time for investigation and a relatively long start-up time.

In this paper, FFT is used for BRB fault detection due to its high speed and ease of analyzing the FFT spectrum results to identify the dominant frequency of the BRB. Furthermore, FFT is widely applied in other articles [45,46] and provides the possibility to define an index for the detection of the severity of the fault and its type.

The rest of the paper is structured as follows: In Section 2, the methodology used in this study is explained. Section 3 presents the benchmark and experimental results. Section 4 and Section 5 are dedicated to techniques for elevating the peak at the fault frequency, using derivation and convolution, respectively. The paper concludes with a summary in Section 6.

2. Methodology

The fault-related component obtained through direct FFT analysis of the current signal is often characterized by a much lower amplitude when compared to the fundamental frequency component (as illustrated in Figure 1). Furthermore, given that the fault frequency is usually very close to the fundamental frequency, the latter can mask the former, particularly during low-slip operations. In other words, within the FFT spectrum, the fault frequency spectrum can be concealed within the shadow of the fundamental frequency spectrum, thus rendering the fault undetectable [47]. To overcome this issue, it is crucial to attenuate the fundamental frequency component while magnifying the fault frequency component to distinguish the fault frequency in the spectrum. To this end, one possible solution is to calculate the difference between the fault frequency and the base frequency. In this context, we propose a mathematical method that is based on a particular identity, which we present in the following.

The frequencies resulting from the BRB fault [48] are as (1).

$$f_{brb}=(1\pm 2ks)f_s$$

(1)

where $f_{brb}$ is the fault frequency of broken rotor bar, $k=1,2,3,\dots$., s is slip and $f_s$ is the fundamental frequency. A fault detection method is proposed in which the fault signature concealed by the fundamental frequency component is detected using the trinomial expansion. To facilitate understanding of this approach, the identity of the trinomial expansion is presented in equation (2).

$${(a+b+c)}^2=a^2+b^2+c^2+2ab+2bc+2ca$$

(2)

where $a,b,c$ are either numbers or variables. In cases of squirrel-cage induction motor (SQIM) with rotor bar breakage, two sideband frequencies related to the rotor bar breakage will appear in the stator current signal spectrum, around the fundamental frequency ($f_s$). Specifically, referring to equation (1) with $k=1$, these frequencies are denoted by $f_2$ and $f_3$, and they are both situated in proximity to $f_s$. Let us consider a signal comprising three sinusoidal terms :

$$x\left(t\right)=a_{1}cos\left(2\pi f_{1}t\right)+a_{2}cos\left(2\pi f_{2}t\right)+a_{3}cos\left(2\pi f_{3}t\right)$$

(3)

Here $f_1=f_s$. It is known that $cos\left(2\pi (f_3-f_1)t\right)=cos\left(2\pi (f_1-f_2)t\right)$ and $a_2\cong a_3$. Using the trigonometric relations and neglecting the term of $2a_2{a}_{3}cos\left(2\pi f_{2}t\right)cos\left(2\pi f_{3}t\right)$ yields:

$$\begin{array}{cc}\hfill x^2\left(t\right)\cong & a_1^{2}co{s}^2\left(2\pi f_{1}t\right)+a_2^{2}co{s}^2\left(2\pi f_{2}t\right)+a_3^{2}co{s}^2\left(2\pi f_{3}t\right)+\hfill \\ \hfill & 2a_1{a}_{2}cos\left(2\pi (f_1-f_2)t\right)+a_1{a}_2\left[cos\right(2\pi \left(f_1+f_2)t\right)+\hfill \\ \hfill & cos\left(2\pi (f_1+f_3)t\right)]\hfill \end{array}$$

(4)

Thus, (4) has one component at DC and six terms with frequencies twice or nearly twice the base frequency ($2f_s$) and one component at the difference of the base frequency and the fault frequency or at $f_1-f_2$. Six components among them are filtered out, and only the component at the frequency of $f_1-f_2$ is used in the proposed fault detection method. Table 1 shows all of these components and indicates that which of them are going to be used in the proposed fault detection technique. Therefore, what remains is the following signal, which is used to diagnose the faulty motor:

$$\xi \left(t\right)=2a_1{a}_{2}cos\left(2\pi (f_1-f_2)t\right)$$

(5)

A pseudo code of the proposed technique is shown in Figure 2. By applying the trinomial expansion identity to the stator current signal and subsequently obtaining its corresponding Hilbert transform while removing unwanted frequencies, a resulting signal can be obtained that contains expression (5). The differential frequency of this expression can be obtained by using FFT transformation.

3. Benchmark and experimental results

The characteristics of the motor under examination are presented in Table 2. The experimental setup comprises a 3 kW AC generator, a 1.5 kW induction motor with broken rotor bars, and the corresponding load, a LA55-P/SP1 Hall effect current sensor and a DSP, TMS320F28379D and a control drive and a computer. The resolution level of the DSP analog to digital converter is 12 bits. A photo of the setup is illustrated in Figure 3.

The proposed methods were tested under three different operating conditions, including direct online start (DOL), direct torque control (DTC), and scalar control. Each operating condition was evaluated under three load levels: light, medium, and heavy (full) loads. The light load mode represented the condition where no electric load was connected to the generator. In the medium and full load modes, 600 watts and 1200 watts of electric loads were respectively connected to the generator. Accounting for the 80% efficiency of the generator, the load on the motor was 750 watts and 1500 watts for medium and full load modes respectively. The sampling frequency used was 10 kHz. The experimental setup was illustrated in Figure 4.

For instance, in direct online start (DOL) operation mode under the light load condition, the signal to be analyzed is shown in Figure 5. After removing the DC component and the component at frequencies around twice the base frequency ($f=2f_s$), the resulting signal is used for analysis. The rotor speed in this case is 1493 RPM, and the corresponding slip is $s=7/1500=0.467\%$. The results for three scenarios, including the healthy state, one broken rotor bar, and two broken rotor bars, are depicted in Figure 6.

Figure 7 illustrates the analysis results for the case of direct torque control (DTC) under the light load operation mode, where the rotor speed is 1490 RPM. The analysis was conducted for three scenarios, including the healthy rotor, one broken rotor bar, and two broken rotor bars, and the results are presented in Figure 7.

Figure 8 shows the results for the case of scalar control under the light load mode of operation. In this case the rotor speed is 1492 RPM and the situation of healthy motor, one broken rotor bar and two broken rotor bars are studied.

Table 3 and Figure 9 demonstrate that there is a strong correlation between the fault frequencies obtained from (1) and those obtained from the proposed method. Additionally, there is a relationship between the peak values and the number of broken rotor bars, as evidenced by the results presented.

4. Elevating the peak at the fault frequency using derivation

After differentiating Equation (5), the following equation is obtained, in which the amplitude is weighted by the difference between the fault frequency and the base frequency:

$$\frac{d\xi \left(t\right)}{dt}=-4\pi (f_1-f_2)a_1{a}_{2}sin\left(2\pi (f_1-f_2)t\right)$$

(6)

This technique elevates the peak at the fault frequency in comparison to the neighboring peaks at lower frequencies, making the fault signature more prominent and facilitating fault detection. The results for the DOL operating condition under full load, with one broken rotor bar, without frequency weighting and with frequency weighting through derivation, are depicted in Figure 10. Comparison of Figure 10 (fig9a) and (fig9b) reveals that the differentiating technique increases the ratio of the peak value at the fault frequency relative to the lower frequency region.

5. Elevating the peak at the fault frequency using convolution

The convolution of two time-discrete signals x and y is defined as follows [49]:

$$(x\ast y)\left[n\right]=\sum _{m=-\infty }^{+\infty }x\left[m\right]·y[n-m]$$

(7)

Based on the convolution theorem, convolution in the time domain corresponds to multiplication in the frequency domain [45,47,50].

$$\mathcal{F}\left((x\ast y)\left[n\right]\right)=X\left(\omega \right)·Y\left(\omega \right)$$

(8)

Therefore, by performing the convolution of the $\xi \left(t\right)$ and its derivative $\dot{\xi }\left(t\right)$ in the time domain while strengthening the fault component amplitude with frequency weighting in the time domain, the fault frequency amplitude is magnified in the frequency domain. Consequently, in the FFT spectrum, the amplitude of the fault frequency is amplified, while the amplitude of the frequency components around the fault frequency is weakened. The block diagram of the proposed method is shown in Figure 11.

By using this technique, the peak at the fault frequency in the FFT spectrum is more pronounced compared to neighboring peaks. This means that the fault signature becomes more visible in the spectrum with high resolution. To illustrate, Figure 12 depicts the fault frequency spectrum using the proposed method during Direct-On-Line (DOL) operation under full load conditions with a single Broken Rotor Bar (BRB). Figure 13 illustrates the fault frequency spectrum under light load mode and during Direct Torque Control (DTC) operation.

Figure 14 shows the scalar operation mode and light load condition to distinguish the severity of faults with one or two broken rotor bars from the healthy state. The figure illustrates that the proposed method accurately detects the state and severity of faults compared to the healthy state.

6. Conclusion

This article presents a diagnostic approach to identify and detect the broken rotor bar defect in squirrel cage induction motors using the identity of the square of sum, and the results are compared and analyzed. To validate the proposed methods, laboratory data obtained from three different operating conditions, including Direct-On-Line (DOL) start, Direct Torque Control (DTC), and scalar control, are examined, and the results demonstrate the effectiveness of the proposed methods in fault detection of induction motors.

Through the analysis of results and formulas, it is evident that the proposed method accurately identifies the broken rotor bar defect in squirrel cage induction motors. As the severity of the rotor bar breakage increases, the amplitude of the spectrum at the fault frequency also increases. Additionally, a frequency weighting tool is presented to enhance the clarity of observing the fault frequency in the resulting spectrum. The accuracy of the proposed method is shown to be higher with the frequency weighting method, which involves deriving the signal under spectrometry to obtain a frequency weighting factor.

In conclusion, this article presents a high-resolution broken rotor bar fault detection method, which has the potential to improve the performance of induction motors in various applications.