Higher-Order Spectral Analysis of Stray Flux Signals for Faults Detection in Induction Motors

Stray flux has been proved to detect the stator winding insulation failure. The main difference for fault detection respect to the use of flow signals consists of the type of sensor to be used and the position where the measurement is taken. One way to detect this type of failure is by applying a load and not load test, that provides significant similarities, even if the stator current gives interesting diagnostic information only when the motor is loaded.

Considering power supply harmonics, it is possible to easily detect the stator winding faults in the low-frequency range of the flux spectrum. One of the most used methods to detect this type of failure is the time-frequency analysis based on the signal spectrogram since this analysis generates important amplitude components in specific harmonics [32].

It is estimated that approximately 35–55 percent of all cases of failures in electrical machines are Bearing Fault Detection, so this type of failure is investigated very frequently in the monitoring and diagnosis of electrical machines. Approximately 40 percent of these are caused by the inappropriate use of lubricants. Some indicators that reveal that the nature of this type of failure lies in the increase in temperature and mechanical vibrations and the increase in the acoustic noise level generated by the machine [14, 24, 31]. Some experimental measurements were carried out using a current probe and by means of different flux probes, positioned in different positions [14, 23]. The comparison is conducted to find the main advantages which are the simplicity and the flexibility of the custom flux probe with its amplification and filtering stage depend on the relative position used for the experiments [47].

Motor eccentricity failures are permissible up to 10 percent, and they do not have a significant influence on the characteristics of the motor and its useful life. However, a high level of eccentricity can cause a magnetic imbalance and therefore an increase in noise and vibrations [71]. Experimental results have revealed the potential of a simple search coil for the detection and the distinction of the accurate eccentricity nature even in the presence of similar mechanical faults [53]. Dynamic eccentricity produces low-frequency air gap flux components. However, they can be observed in stator current only under mixed eccentricity. Moreover, detection of dynamic eccentricity in stator current around the principal slot harmonic (PSH) is only useful for some combination of pole pairs and rotor slots [53, 61, 62].

Under mixed eccentricity conditions, the stator currents contain the following frequencies [47]: (1)fecc=f±k(1−s)/pf,{f_{ecc}} = f \pm k(1 - s)/pf, where s is the machine slip. Since the frequencies related to the eccentricity and to the load torque overlap on the current sidebands, the frequencies provided by (1) are no longer enough for the diagnosis.

Induction motors are widely extended in industry, and among the defects which may appear, broken rotor bars represent 10 percent to 20 percent of the whole faults. This kind of failure does not cause an immediate breakdown, but it deteriorates the machine operation, decreasing its performance [7, 46]. For the detection of this type of failure, different algorithms based on the stray flux signal time-frequency analysis of the Fourier and wavelets transforms have been used. Each technique presents its particular advantages and drawbacks.

When a bar breakage occurs, a backward rotating magnetic field is generated due to the open-circuited bar. This creates an asymmetry in the rotor cage that is clearly reflected in the motors harmonic content [8, 15, 45]

The lower sideband harmonic leads to a torque (and speed) oscillation, which provokes the appearance of s, another harmonic in the stator current spectrum: the upper (or right) sideband harmonic given by 1 + 2s f. Moreover, the frequency modulation on the rotational frequency f, provoked by the speed oscillation, also leads to sideband harmonics in the vibration (and, accordingly, in the noise) spectrum: (2)fbb=frr±2ksf{f_{bb}} = {f_{rr}} \pm 2ksf

In this section, the definitions, properties, and the way of calculating higher-order statistics are introduced. Let X = {X(k)}, k = 0,±1,±2,... be a stationary random vector and its higher order moments exist, then: (3)mnX(τ1,τ2,…,τn−1)=E{X(k)X(k+τ1)⋯X(k+τn−1)}m_n^X({\tau _1},{\tau _2}, \ldots,{\tau _{n - 1}}) = E\{X(k)X(k + {\tau _1}) \cdots X(k + {\tau _{n - 1}})\} where E denotes the expected value operator [6, 37, 40], represents the n-th order moment of X, which depends only on the different temporary displacements τ₁,τ₂,...,τ_n−1, with τ_i = 0, ± 1,.... It can be seen that the second order moment m2X(τ1)m_2^X({\tau _1}) is the autocorrelation function of X, likewise m3X(τ1)m_3^X({\tau _1}) and m4X(τ1)m_4^X({\tau _1}) represent the third and fourth order moments, respectively. The cumulants are similar to the moments, the difference is that the moments of a random process are derived from the characteristic function of the random variable, while the cumulant generating function is defined as the logarithm of the characteristic function of the random variable.

The n-th order cumulant of X can be written as, see [37]: (4)cnX(τ1,…,τn−1)=mnX(τ1,…,τn−1)−mnG(τ1,…,τn−1), n=3,4,c_n^X({\tau _1}, \ldots,{\tau _{n - 1}}) = m_n^X({\tau _1}, \ldots,{\tau _{n - 1}}) - m_n^G({\tau _1}, \ldots,{\tau _{n - 1}}),\quad n = 3,4, where mnG(τ1,τ2,…,τn−1m_n^G({\tau _1},{\tau _2}, \ldots,{\tau _{n - 1}} is the n-order moment of a process with equivalent Gaussian distribution. with the same average value and autocorrelation function as X, the stationary random vector. From (4) it is evident that for a process following a Gaussian distribution, the cumulants of order greater or equal than 2 are null, since mnX(τ1,…,τn−1)m_n^X({\tau _1}, \ldots,{\tau _{n - 1}}) and mnG(τ1,…,τn−1)m_n^G({\tau _1}, \ldots,{\tau _{n - 1}}) are null, too [25]

Although fourth-order cumulants imply a considerable increase in calculation complexity, they are especially necessary when third-order cumulants are canceled in the case of symmetrically distributed processes, such as uniform distributed processes from [−a,a], with a ∈ ℝ, such as Laplace and Gaussian processes. Third-order cumulants are not canceled for processes whose probabilistic density function is not symmetric, such as exponential or Rayleigh processes, but can take extremely small values compared to the values presented by their fourth-order cumulants [60].

The n-th order spectrum of a stationary random vector X = {X(k)}, k = 0,±1,±2,... is defined as the multidimensional Fourier Transform F_k(·) of order n − 1 on the higher order statistical characteristics (moments and cumulants). The spectrum of the n-moment is defined as [25, 60]: (5)MnX(ω1,…,ωn−1)=Fn[mnX(τ1,…,τn−1)],M_n^X({\omega _1}, \ldots,{\omega _{n - 1}}) = {F_n}[m_n^X({\tau _1}, \ldots,{\tau _{n - 1}})], and, similarly, the spectrum of the n–cumulant is defined as (6)CnX(ω1,…,ωn−1)=Fn[mcnX(τ1,…,τn−1)],C_n^X({\omega _1}, \ldots,{\omega _{n - 1}}) = {F_n}[mc_n^X({\tau _1}, \ldots,{\tau _{n - 1}})], Note that the spectrum of the n-th order cumulant is also periodic with period 2π, that is: (7)CnX(ω1,…,ωn−1)=CnX(ω1+2π,…,ωn−1+2π),C_n^X({\omega _1}, \ldots,{\omega _{n - 1}}) = C_n^X({\omega _1} + 2\pi, \ldots,{\omega _{n - 1}} + 2\pi), Equation expression also reads as: (8)CnX(ω1,…,ωn−1)=1(2π)n−1∑τ1=−∞∞⋯∑τn−1=−∞∞cnX(τ1,…,τn−1)ej(ω1τ1+…+ωn−1τn−1)C_n^X({\omega _1}, \ldots,{\omega _{n - 1}}) = {1 \over {{{(2\pi)}^{n - 1}}}}\sum\limits_{{\tau _1} = - \infty}^\infty \cdots \sum\limits_{{\tau _{n - 1}} = - \infty}^\infty c_n^X({\tau _1}, \ldots,{\tau _{n - 1}}){e^{j({\omega _1}{\tau _1} + \ldots + {\omega _{n - 1}}{\tau _{n - 1}})}}

In particular, for n = 2 in (8), we have the power spectrum(9)C2X(ω)=12π∑τ=−∞∞c2X(τ)e−jωτ,C_2^X(\omega) = {1 \over {2\pi}}\sum\limits_{\tau = - \infty}^\infty c_2^X(\tau){e^{- j\omega \tau}}, where |ω| ≤ π and c2X(τ)c_2^X(\tau) represents the process covariance sequence, see [25, 60].

For n = 3 in (8), we have the bispectrum(10)C3X(ω1,ω2)=1(2π)2∑τ1=−∞∞∑τ"=−∞∞c3X(τ1,τ2)e−j(ω1τ1+ω2τ2),C_3^X({\omega _1},{\omega _2}) = {1 \over {{{(2\pi)}^2}}}\sum\limits_{{\tau _1} = - \infty}^\infty \sum\limits_{\tau '' = - \infty}^\infty c_3^X({\tau _1},{\tau _2}){e^{- j({\omega _1}{\tau _1} + {\omega _2}{\tau _2})}}, where |ω₁| ≤ π, |ω₂| ≤ π, and |ω₁ +ω₂| ≤ π and c3X(τ)c_3^X(\tau) represents the sequence of third order cumulants of X(k). The expression of the cumulant complies with the following symmetric relationships [60]: (11)c3X(τ1,τ2)=c3X(τ2,τ1)=c3X(−τ2,τ1−τ2)=c3X(τ2−τ1,−τ1)c_3^X({\tau _1},{\tau _2}) = c_3^X({\tau _2},{\tau _1}) = c_3^X(- {\tau _2},{\tau _1} - {\tau _2}) = c_3^X({\tau _2} - {\tau _1}, - {\tau _1})(12)=c3X(τ1−τ2,−τ2)=c3X(−τ1,τ2−τ1)=c3X(τ1,τ2) = c_3^X({\tau _1} - {\tau _2}, - {\tau _2}) = c_3^X(- {\tau _1},{\tau _2} - {\tau _1}) = c_3^X({\tau _1},{\tau _2})

From these relationships, we can get a division of the plane τ₁τ₂ in six regions and, consequently, upon knowing the third-order cumulant in any of these six regions (see Figure 3), we can reconstruct the complete sequence corresponding to the third-order cumulants. Note that each one of these regions contains its border. Thus, for example, sector I is an infinite region characterized by 0 eτ₂ ≤ τ₁. For non-stationary processes, these six regions of symmetry disappear. From these relationships and the definition of the spectrum of third-order cumulants, the following relationships in the two-dimensional frequency domain can be obtained [6, 25, 37] (13)C3X(ω1,ω2)=C3X(ω2,ω1)=C3X(−ω2,−ω1)=C3X(−ω1−ω2,ω2)C_3^X({\omega _1},{\omega _2}) = C_3^X({\omega _2},{\omega _1}) = C_3^X(- {\omega _2}, - {\omega _1}) = C_3^X(- {\omega _1} - {\omega _2},{\omega _2})(14)=C3X(ω1,−ω1−ω2)=C3X(−ω1−ω2,ω1)=C3X(ω2,ω1−ω2) = C_3^X({\omega _1}, - {\omega _1} - {\omega _2}) = C_3^X(- {\omega _1} - {\omega _2},{\omega _1}) = C_3^X({\omega _2},{\omega _1} - {\omega _2})

Fig. 3

Figure 3(b) shows the 12 symmetry regions of the bispectrum when real stochastic processes are considered and, similar to the temporal domain, the knowledge of the bispectrum in the triangular region It is enough for a total reconstruction of the bispectrum. Note that, in the frequency domain, the symmetry regions have a finite area and in general, the bispectrum takes complex values and, consequently, the phase information is preserved [60].

In the case n = 4, we get the trispectrum(15)C4X(ω1,ω2,ω3)=1(2π)3∑τ1=−∞∞∑τ2=−∞∞∑τ3=−∞∞c4X(τ1,τ2,τ3)e−j(ω1τ1+ω2τ2+ω3+tau3),C_4^X({\omega _1},{\omega _2},{\omega _3}) = {1 \over {{{(2\pi)}^3}}}\sum\limits_{{\tau _1} = - \infty}^\infty \sum\limits_{{\tau _2} = - \infty}^\infty \sum\limits_{{\tau _3} = - \infty}^\infty c_4^X({\tau _1},{\tau _2},{\tau _3}){e^{- j({\omega _1}{\tau _1} + {\omega _2}{\tau _2} + {\omega _3} + {tau}_3)}}, where |ω₁| ≤ π, |ω₂| ≤ π, |ω₃| ≤ π, and |ω₁ + ω₂ + ω₃| ≤ π, where c4x(τ1,τ2,τ3)c_4^x({\tau _1},{\tau _2},{\tau _3}) represents the sequence of fourth-order cumulants. By combining the definition of the trispectrum and the fourth-order cumulants, 96 regions of symmetry appear associated with a real process. From the spectra of higher-order cumulants, the expressions of their respective cumulants in the temporal domain can be retrieved, by applying the n-order inverse Fourier transform [60].

To illustrate in more detail the calculation of the higher-order spectra, we show several examples with periodic signals.

3.3.1

Cosine signal

First, let us consider the cosine signal (16)X=Akcos(2πfk+ϑk)X = {A_k}\cos (2\pi {f_k} + {\vartheta _k}) with amplitude A_k = 1, frequency f_k = 50Hz, and a phase υ_k = 0, discretely generated from a sampling frequency of f_s = 1000Hz and a duration of 1024 samples. Then the bispectrum for a 1024 sample data window is equivalent to discretely performing the Fourier transform of the third order cumulant of the signal.

In figures 4 (a) and (b), respectively shows the cosine signal and its frequency spectrum using the one-dimensional Fourier Transform (FT). Likewise, as a comparison mode, the bispectrum of the 50 Hz cosine signal (0.05 normalized to 1) it is shown in Figure 5, to illustrate that the fundamental frequency of the signal prevails; and also showing other frequency components that by means of the one-dimensional spectrum using Fourier, they do not appear, which gives the bispectrum a higher resolution from the spectral point of view, which can be useful for dissimilar applications to find linear combinations or not, of existing relations between frequencies, for the development of identifying spectral patterns.

Fig. 4

Apart from the fundamental frequency component 0.05 (50Hz) normalized to 1, there are other components as results of linear combinations of the fundamental frequency (16), since the calculation of bispectrum results in a two-dimensional frequency and a phase matrix, see Figure 5.

Fig. 5

On the other hand, although less used, the phase spectrum of the cosine signal of (16) is also shown. The phase spectrum contains an arrangement with the amplitude in degrees of the phase of the signal, that is taken as υ_k = π/4 in (16). Its corresponding phase spectrum is shown below in Figure 6. We also show the phase bispectrum, see Figure 7, where the linear phase of the cosine signal is resalted in the bispectrum phase matrix.

Fig. 6

Fig. 7

The bispectrum preserves the phase information of every spectral component, that is the phase of the original signaland it can be otained from the original signal as: (17)ϑC3X(ω1,ω2)=ϑX(ω1)+ϑX(ω2)+ϑX(ω1+ω2){\vartheta _{C_3^X}}({\omega _1},{\omega _2}) = {\vartheta _X}({\omega _1}) + {\vartheta _X}({\omega _2}) + {\vartheta _X}({\omega _1} + {\omega _2}) where ϑC3X(ω1,ω2){\vartheta _{C_3^X}}({\omega _1},{\omega _2}) represents the phases matrix of every spectral component. This relation also permits to retrieve the original phase of the signal.

3.3.2

Harmonic signal

Secondly, we have also experimented by calculating the bispectrum of a harmonic signal of the form: (18)x(t)=Akcos(2πfk+ϑk)+2Akcos(2π⋅3fk+ϑk)x(t) = {A_k}\cos (2\pi {f_k} + {\vartheta _k}) + 2{A_k}\cos (2\pi \cdot 3{f_k} + {\vartheta _k})

In Figures 8 (a) and (b), we show the harmonic signal in time and its frequency spectrum using the one-dimensional Fourier Transform (FT). In Figure 9, the bispectrum of the harmonic signal is shown. We notice that the calculation of the bispectrum preserves the fundamental frequencies of each signal in the sum of harmonics, analogous to the calculation of the one-dimensional FT. We also observe that since the bispectrum is absolutely summable, multiple frequency components appear as a result of the linear combinations of the bispectrum of both cosine signals individually.

Fig. 8

Fig. 9

We also show the phase bispectrum of the harmonic signal in Figure 10. It is more complex than the one of the cosine signal because the bispectrum is absolutely additive, and the phase of the 50Hz and 150Hz components are totally mixed.

Fig. 10