**Subhasis Nandi Ph.D.**

*University of Victoria*

Generally speaking, most of the fault detection techniques used in real¬time fault detection in power systems are time-domain based. The over current, over voltage, earth fault, impedance relays, and so forth are mostly time-domain based. However, as far as detecting faults for electric machines are concerned, frequency-domain-based techniques, especially ones based on fast Fourier transforms (FFT) are very popular. Except for stator-related faults, most other faults can be reliably diagnosed using a spectrum analyzer provided the machines are operating under steady-state conditions for at least a reasonable period of time. For applications in which machines are made to operate under very frequently fluctuating load and speed conditions, traditional FFT has to be replaced with short time Fourier transforms (STFT), spectrograms, and other time-frequency analysis using wavelets and Wigner–Ville transforms. Usually the machine current, flux, mechanical vibration, torque, and speed signals are analyzed in frequency domain. High-resolution spectral techniques such as multiple signal clas¬sification (MUSIC), ROOTMUSIC, and higher-order spectral methods such as bispectrum and trispectrum have also been proposed by a few research¬ers. However, most of the popular frequency-domain-based techniques are based on fast Fourier transform of the line current generally known as motor current signature analysis (MCSA). Sometimes, when the frequen¬cies at which the detections are to be made are known, swept sine measure¬ments or the digital frequency locked loop technique (DFLL) are also used. This avoids lengthy computations while achieving good resolution.

Traditionally, in many countries, power engineers are not exposed even to the basic signal processing course, which is only taught to students in electronics and communication. Hence it will not be out of place to discuss a few basics of signal processing first, before going into actual fault diagnosis using signal processing.

A *continuous signal, x*(*t*), is one that is defined at any given point of time. Examples of such a signal are the line currents and line voltages of a motor that we can observe on an analog oscilloscope. The same signal data when acquired through a data acquisition system or seen through a digitizing oscilloscope becomes a *discrete signal, x*(*n*), which is nothing but the sampled version of the continuous signal at a regular time interval, *T*_{sp} [1]. The fre¬quency at which the sampling device works is *f*_{sp}. In general,

Tsp=1fsp(6.1)

x(n)=x(t)|t=nTsp(6.2)

Often one has to prefilter a signal to avoid *aliasing* (literally meaning “same name for one thing”) arising out of the sampling process. Unless proper care is taken in choosing the sampling frequency or the prefilter, one frequency component may be wrongly interpreted as another while trying to deter¬mine the frequency components present in a discrete signal. It must be noted, though, that sometimes aliasing can be used beneficially, too, such as in a stroboscope, a device used to measure speed, or for strengthening weak sig¬nals used for fault detection.

**Example 6.1**

Suppose we have a voltage signal given by *x*(*t*)=100sin(2π60*t*)+10sin(2π300*t*). Determine how the sampled version of this signal would look like when sampled by (a) a 200 Hz signal and (b) a 1000 Hz signal.

We have, using Equation (6.2)

X(n)=100sin(2π0.3n)+10sin(2π1.5n)=100sin(2π0.3n)+10sin(2πn+2π0.5n)=100sin(2π0.3n)+10sin(2π0.5n)

X(n)===100sin(2π0.3n)+10sin(2π1.5n)100sin(2π0.3n)+10sin(2πn+2π0.5n)100sin(2π0.3n)+10sin(2π0.5n) Now had the signal been

*x*(*t*)=100sin(2π60*t*)+10sin(2π100*t*), the result would have been the same, meaning that with 200 Hz sampling fre¬quency the 300 Hz signal can be misconstrued as a 100 Hz signal.However in this case

*x*(*n*)=100sin(2π0.06*n*)+10sin(2π0.3*n*) and the 300 Hz signal can be easily distinguished from a 100 Hz signal.

In order for proper signal reconstruction or interpretation after sampling, a continuous time signal *x*(*t*) has to be sampled at a rate greater than twice the maximum frequency contained in that signal. This result is actually one of the fundamental theorems in signal processing and is known as the *Shannon sampling theorem.*

A *continuous Fourier transform* is given by the following two formulas [1–3]:

X(jω)=∞∫−∞x(t)e−jωtdt(6.3)

x(t)=12π∞∫−∞X(jω)ejωtdω(6.4)

Equation (6.3) is known as the *analysis* or *forward* equation because it extracts the frequency information from the time-domain signal. Equation (6.4) is known as the *synthesis* or *inverse* equation because it creates the origi¬nal time-domain signal back from the spectral information.

A *discrete Fourier transform* (DFT), on the other hand, is given by [1–3]:

X(k)=N−1∑n=0x(n)e−j2πnkN,k=0,1,…., N−1(6.5)

x(n)=1NN−1∑k=0X(k)ej2πnkN,n=0,1,…., N−1(6.6)

with Equation (6.5) analogous to Equation (6.3) and Equation (6.4) analogous to Equation (6.6). It is also interesting to note that while both Equation (6.3) and Equation (6.4) are in continuous domain, both Equation (6.5) and Equation (6.6) are in discrete domain. It is also possible to write the analysis equation in the discrete form *but* the synthesis equation in the continuous form. In this case the equation set is known as *discrete-time Fourier transform*, meaning the discretization is done in time-domain only. The other alternative—that is, the equation set with the analysis equation in continuous form and the synthesis equation in discrete form—is the very well known form of the *Fourier series* that essentially expresses a periodic but continuous time in terms of discrete frequency components. Although it is not difficult to show the relationship between the four aforementioned kinds of transforms, the DFT is the most important for the fault diagnosis purpose. At this point in time it also becomes necessary to revisit the definition of FFT. A FFT is nothing but the collection of algorithms used for efficient computation of the DFT.

**Example 6.2: Compute the Continuous Fourier Transform of the Voltage Signal Given in Example 6.1**

Let us begin by computing the time-domain signal corresponding to the impulse signal in frequency domain given by

X(jω)=δ(ω−ω0)(6.7)

Using Equation (6.4)

x(t)=12π∞∫−∞δ(ω−ω0)ejωtdω=ejω0t2π∞∫−∞δ(ω−ω0)ejωtdω=ejω0t2π(6.8)

The signal in this case, using Euler’s identity can be written as

y(t)=100sin(2π60t)+10sin(2π300t)=1002j(ej2π60t−e−j2π60t)+102j(ej2π300t−e−j2π300t)(6.9)

Hence using Equation (6.7) and Equation (6.8)

Y(jω)=100πj(δ(ω−2π60t)−δ(ω+2π60t))+10πj(δ(ω−2π300t)−δ(ω+2π300t))(6.10)

Note that in Equation (6.10) for each frequency component there are two delta functions. Though their phases are opposite, their magnitudes are same. Normally one is interested in the magnitude and therefore it is suf¬ficient to have only one of the delta functions. Conventionally, only those lying on the right side of ω = 0 (that is, δ(ω − 2π60*t*) and δ(ω − 2π300*t*) in this example) are chosen.

Since one has to work with a finite data set in practice, DFT is the trans¬form to be used. If one could acquire a large set of steady-state data with minimal temporal variations, DFT signals would approach the true single line nature of the spectra as given by Equation (6.9) in a limiting sense. Unfortunately, many times, due to constraints such as speed of computa¬tion and memory, one has to use a limited data set. Significant improve¬ment, however, in the quality of the signal can be obtained by judicious choice of the *window function.* The limited data set is analogous to viewing something through a small window. Now if the glass pane on the window is not clear enough, details of whatever is viewed may not be distinct. The simplest window, as one could intuitively guess, is the so- called *rectan¬gular window*, embedded due to the very fact that the data set is limited in nature. The rectangular window has continuous spectra and as a result the power of the original signal data, instead of being concentrated at the points of interest, leaks out over the entire frequency range. This is called *spectral leakage.* Specialized window functions such as the Hanning win¬dow and Bartlett window are used to reduce spectral leakage. However, windows result in a loss of resolution. The only way to improve resolution is to increase *N* in Equation (6.5) and Equation (6.6). This can be achieved only by increasing the window length, meaning increasing the length of the data set. Increasing sampling frequency will not improve the resolution of the spectrum.

In practice, any data is bound to have some noise associated with it. As long as the noise is white (zero mean, unit variance), it can be easily mini¬mized by averaging several *power spectral density* (PSD) spectra as given by Equation (6.10)

X(kN)=1N|N∑n=0x(n)e−j2πnkN|2,k=0,1,….,N−1(6.11)

computed over small data segments. This is essentially computing the square of the magnitude of the FFT of the segments and then averaging them. In general this method is known as *nonparametric spectrum estimation* [1–3]. These segments can be either overlapping or nonoverlapping. For a given data set, the noise reduction is attained at the cost of frequency resolution and vice versa. Depending upon the type of averaging techniques or win¬dow used, the nonparametric power spectrum estimation may be known as, for example, the *periodogram, Bartlett, Welch*, or *Blackman-Tukey.*

**Example 6.3**

Suppose a 1-second data set of the signal given by

*x*(*t*)=100sin (2π60*t*)+2sin(2π63*t*)+2sin(2π57*t*)+10sin(2π180*t*)+*white noise.*The signal has been obtained with a sampling frequency of 3600 Hz. The*white noise*used is random numbers that vary between 0 and 1. From the machine diagnostic viewpoint such a signal would approximately represent the low frequency spectrum of an induc¬tion motor with broken rotor bars. Plot the FFT using all 3600 points with (i) a Rectangular and (ii) a Hanning window.Suppose a 10-second data set is used. Repeat (a).

Plot the PSD spectrum of the 10-second data set, with segment size of 12,000 data points and 10,000 overlapping data points between two segments. Use Hanning window.

Comparing the first two plots in Figure 6.1, the reduction of resolution with Hanning window is quite clear. However the spectral leakage is significantly reduced with this window. With a larger data set the effect of windowing is much less pronounced as can be seen in the next two plots. However, the frequency resolution has increased significantly with a larger data set. In the last plot the PSD spectrum shows significant reduction in noise with little sacrifice in resolution.

**FIGURE 6.1**

From top: FFT using rectangular window and 1 second of data, FFT using Hanning win¬dow and 1 second of data, FFT using rectangular window and 10 seconds of data, FFT using Hanning window and 10 seconds of data, PSD. All the plots have been normalized with respect to the 60 Hz component.

**FIGURE 6.2**

Zoomed spectra of signal in Example 6.3 around 60 Hz.

Sometimes it may be desirable to look closely at the narrow band of a spec¬trum, say around the 60 Hz frequency spectrum as described in Example 6.3. There exists a technique called *zoom FFT* [3,4] by which one can zoom on to the area of interest in a spectrum. This way one has to do far fewer numbers of FFT computations than would be required for the whole spec¬trum for a given resolution. To do this, the original collected data is shifted in frequency domain by multiplication with the sinusoid *e*^{j}^{ω1}^{t}, where ω_{1} is the lower limit of the band of interest. In the next step, the modulated signal is filtered with a low pass filter and then down sampled with a factor that essentially determines the “zoom.” Figure 6.2 shows such a zoomed spectra of the signal described in Example 6.3 with a zoom or decimation factor of 10. The original signal was collected using a frequency of 3600 Hz for 100 seconds. Without zooming one has to do FFT of 3600 × 100 = 360,000 samples. With zooming by a factor of 10 it is reduced to 36,000. The resolution remains same as 0.01 Hz.

The nonparametric form of spectrum estimation techniques discussed in the previous section is fairly simple, well understood and easy to compute. However, they suffer from the fact that improved resolution of the spectrum would entail long data records. Thus estimation of fault signals for motors running perpetually under transient modes, such as hoists and winches will be difficult. Also, since finite-length data records are used, spectral leakage effects would be present. This would tend to mask weak signals present in the data, particularly in the vicinity of a strong signal.

*Parametric* or *model-based power spectrum estimation* methods eliminate the need for window functions and as a result the associated spectral leakage and frequency resolution problems [3,5–7]. Thus they hold promise for appli¬cations where short data records are available due to time-variant or tran¬sient phenomena.

The parametric techniques essentially assume that the data sequence whose spectrum has to be analyzed is the output of a linear system charac¬terized by a rational transfer function in the discrete domain as

H(z)=B(z)A(z)=∑qk=0bkz−k1+∑pk=1akz−k(6.12)

with

X(z)=H(z)C(z)(6.13)

where *X*(*z*) is the *z* transform of the output data sequence *x*(*n*) to be analyzed and *C*(*z*) the *z* transform of the corresponding input data sequence *c*(*n*). Now if *c*(*n*) is a zero mean, unity variance white noise sequence, it is easy to show that

|X(jω)|2=|H(jω)|2(6.14)

It is very clear then that determining the sets {*a*_{k}}, {*b*_{k}} in Equation (6.12) are enough to estimate the spectrum of *x*(*n*).

Models such as those given by Equation (6.12) are generally known as *autoregressive-moving average* (ARMA) models. With *q = 0, b*_{0} *=* 1 it is known as an *autoregressive* (AR) model. Setting *A*(*z*)=1 makes it a *moving average* (MA) model. The AR model is most widely used because of its simple form and suitability for representing spectra with narrow peaks. One of the most important aspects of AR models is the selection of the order *p.* If *p* is too low, the spectrum is very smooth. Too high values of *p* may end up producing spurious low level peaks in the spectrum.

Numerous techniques to obtain these models are available in literature. Yule-Walker, Burg, and unconstrained least squares are some examples. In special cases, when the signal components are sinusoids corrupted by additive white noise, eigenvalue-based techniques such as MUSIC and ROOTMUSIC have also been found useful.

**FIGURE 6.3**

Spectrum of the signal described in Example 6.3 using Yule–Walker (top) and MUSIC (bottom) methods.

Figure 6.3 shows the spectral estimation carried out using the Yule-Walker and MUSIC method using MATLAB commands pyulear and pmusic. All 36,000 points were used. The order of the system was considered as four since there are four distinct sinusoidal signals. While the Yule–Walker method was just able to detect the 60 Hz component, the MUSIC was able to detect the 180 Hz signal as well. None were however able to detect the 57 and 63 Hz signals.

Higher-order spectra (HOS) based spectral analysis has received some attention with regards to detection of very weak harmonics under low signal-to-noise ratio (SNR) conditions [6,8,9]. Recently it has been reported to have been used in building a tool called statistical motor analysis in real time (SMART), a PC-based software implementing fault diagnosis using HOS.

The average of the PSD is a second-order spectral measure since it essen¬tially computes

P(ω)=E[X(jω)X*(jω)](6.15)

where X(ω) is same as the FT as described by Equation (6.5). *X** (*jω*) is the complex conjugate of *X(jω*) and *E*[] is the statistical expectation or average. The same definition can be extended to obtain higher-order spectra such as

B(jω1, jω2)=E[X(jω1)X(jω2)X*(jω1+jω2)](6.16)

T(jω1, jω2,jω3)=E[X(jω1)X(jω2)X(jω3)X*(jω1+jω2+jω3)](6.17)

Equation (6.15) and Equation (6.16) are known as bispectrum and trispectrum, respectively. A close look at Equation (6.15) and Equation (6.16) suggest that if certain *frequencies along with their sums* are present, their presence can be detected very easily even with low SNR. The principle of HOS can be shown using the following example.

**Example 6.4**

Find the bispectrum of (a) *x*(*t)* = cos(ω_{a}*t*) and (b) *x*(*t*) *=* cos(ω_{a}*t*) *+* cos(ω_{b}*t*) + cos{(ω_{a} + ω_{b})*t*} . Assume zero noise.

Following Example (6.2)

B(jω1, jω2)=π3[δ(ω1−ωa)+δ(ω1+ωa)][δ(ω1−ωb)+δ(ω1+ωb)][δ(ω1+ω2−ωa)+δ(ω1+ω2+ωa)]=0.

B(jω1, jω2)=π3[δ(ω1−ωa)+δ(ω1+ωa)][δ(ω1−ωb)+δ(ω1+ωb)][δ(ω1+ω2−ωa)+δ(ω1+ω2+ωa)]=0. This is because none of the impulses occur at the same frequency point.

In this case however,

B(jω1, jω2)π3[δ(ω1−ωa)+δ(ω1+ωa)+δ(ω1−ωb)+δ(ω1+ωb)+δ(ω1−ωa−ωb)+δ(ω1+ωa+ωb)][δ(ω2−ωa)+δ(ω2+ωa)+δ(ω2−ωb)+δ(ω2+ωb)+δ(ω2−ωa−ωb)+δ(ω2+ωa+ωb)][δ(ω1+ω2−ωa)+δ(ω1+ω2+ωa)+δ(ω1+ω2−ωb)+δ(ω1+ω2+ωb)+δ(ω1+ω2−ωa−ωb)+δ(ω1+ω2+ωa+ωb)]

B(jω1, jω2)π3[δ(ω1−ωa)+δ(ω1+ωa)+δ(ω1−ωb)+δ(ω1+ωb)+δ(ω1−ωa−ωb)+δ(ω1+ωa+ωb)][δ(ω2−ωa)+δ(ω2+ωa)+δ(ω2−ωb)+δ(ω2+ωb)+δ(ω2−ωa−ωb)+δ(ω2+ωa+ωb)][δ(ω1+ω2−ωa)+δ(ω1+ω2+ωa)+δ(ω1+ω2−ωb)+δ(ω1+ω2+ωb)+δ(ω1+ω2−ωa−ωb)+δ(ω1+ω2+ωa+ωb)]

Clearly here with ω_{1} = ω_{a}, ω_{2} = ω_{b} or ω_{1} = ω_{b}, ω_{2} = ω_{a} , *B*(*j*ω_{1}, *j*ω_{2}) is nonzero.

**FIGURE 6.4**

The FFT plot (top) and the bispectrum (bottom) plot of a signal with a very weak 60 Hz component.

The FFT (with Hanning window) and bispectrum of a signal with equal amplitude of *f*_{a} *=* 10Hz and *f*_{b} *=* 50Hz is given in Figure 6.4. None of them used any averaging technique. The signal also has a 60 Hz component whose amplitude is 0.01% of either ω_{a}or ω_{b}. The SNR is about −20 dB. It is very clear that the 60 Hz component can be identified much better from the bispectrum plot. It is shown by two largest peaks located at the grid points *f*_{1} = 10Hz, *f*_{2} *=* 50Hz and *f*_{1} *=* 10Hz, *f*_{2} *=* 50Hz . There are other minor peaks located at other grid points due to noise present in the signal. On the other hand, the 60 Hz signal is almost buried in the noise floor of the FFT output. It may be argued that the spectral quality of the FFT output can be improved by averaging. However it would mean increasing the computational over¬head also.

So far our attention was primarily focused on locating many spectral lines over a wide frequency range. However, many times if the frequency itself is roughly known or it varies only over a very narrow band under all operating conditions, then FFT analysis may not computationally be the best option.

The key to understanding the swept sine measurements or digital fre¬quency locked loop technique (DFLL) [10,11] lies in the evaluation of the integrals ∫2π0*nx* cos *mx dx*, ∫2π0*nx* sin *mx dx*, and ∫2π0*nx* sin *mx dx.* It is easy to show that they are all equal to π if *m=n* and 0 if *m*≠*n*. Essentially this means that if *m* denotes the frequency of interest in the signal then we can vary *n* over a narrow range to find the magnitude and location of *m* with great accuracy. Even the phase of the signal can be known. It is computed in the following way. If *f*(*t*) is the signal, then the following are computed at regular interval Aco in the range ω_{1} ≤ ω ≤ ω_{2} in which the frequency of inter¬est in *f*(*t*) is expected to lie in

a=T∫0f(t)cosωt dt(6.18)

b=T∫0f(t)sinωt dt(6.19)

M=√a2+b2(6.20)

P=tan−1ba(6.21)

*M* designates the magnitude of the signal and *P* the phase. The location at which the peak of *M* occurs gives the frequency of interest. The frequency resolution is 1Δω*T* spans over several cycles of the steady-state signal for improved detection. Also the products *f*(*t*)*cos*ω*t, f*(*t*)*sin*ω*t* have to be suitably low pass filtered before the integration. Figure 6.5 shows the detection of the 60 Hz component in a signal that has equal amplitude 120 Hz signal and 0 mean, 0.3 standard deviation white noise. The SNR is about 8 dB. The frequency resolution is 0.1 Hz. One second of data was used. To get similar resolution using FFT would require data collected over 10 seconds.

**FIGURE 6.5**

Detection of frequency signal using DFLL.

In the rest of the chapter, the diagnosis of the four most common faults encountered in electric machines, namely, the bearing faults, the stator inter-turn faults, the broken rotor bar faults, and the eccentricity faults, using fre¬quency-domain-based techniques will be discussed.

As stated earlier, bearing faults happen to be the most common cause of electric machine failure in industry. Also, bearings faults have been recently classified as *single-point defects* that produce predictable frequencies and *generalized roughness* that do not. The two most common ways to determine single-point defects of bearings are by mechanical vibration and current sig¬nature analysis [12,13]. Of these the mechanical vibration signal analysis are most popular and will be discussed first. The current signature analysis of bearings is comparatively new and seems to be a function of the mechanical vibration signals. They will be discussed later.

Most of the literature on fault detection of bearings deals with rolling-ele¬ment bearings [14–17]. Most common among them are the ball bearings. They consist typically of six to twelve balls inserted in a perforated cage in the form of a ring. The cage ensures uniform spacing and prevents mutual contact. The cage with the balls is held by an outer ring known as outer raceway and an inner ring known as inner raceway. The balls are lubricated with grease. Other types of rolling element bearings use cylinders instead of balls. Sometimes the ends of the bearings are sealed. Very large electric motors use sleeve (fluid-film) bearings. Magnetic bearings are also possible.

**FIGURE 6.6**

The different parts of a ball bearing (From Li B, M.-Y. Chow, Y. Tipusuwan, and James C. Hung, Neural-network-based motor rolling bearing fault diagnosis, IEEE Trans. on Industrial Electronics, pp. 1060-1069, vol. 47, no.5, Oct. 2000. With permission.)

The structure of the ball bearing is given in more detail in Figure 6.6. Let the cage, and outer and inner raceway velocities be *V*_{c}, *V*_{o}, and *V*_{i} respectively. These velocities essentially determine the different mechanical vibration frequencies associated with the cage, the ball, the outer raceway, the inner raceway, and the shaft. They are commonly known as the fundamental cage frequency (*F*_{C}), the ball rotational frequency (*F*_{B}), the ball pass outer raceway frequency (*F*_{BPO}), and the ball pass inner raceway frequency (*F*_{BPI}). All of them are a function of shaft rotational frequency (*F*_{S}). If *D*_{b} is the ball diameter and *D*_{c} is the bearing cage diameter, then following fundamental physics that relates angular velocity with linear velocity

FC=Vcrc=Vo+ViDc(6.22)

where *V*_{c} *=* (*V*_{o} + *V*_{i})/*2* and *r*_{c} = *D*_{c}/2. Also due to the contact angle θ, only a part of the *D*_{b} will contribute toward the frequencies *F*_{o} (correspond¬ing to *V*_{o}) and *F*_{i} (corresponding to *V*_{i}). Defining *r*_{i} = *r*_{c} − (*D*_{b} cos θ/2), *r*_{0} = *r*_{c} +(*D*_{b} cos θ/2), and with *V*_{o} = *F*_{o}*r*_{o}, *V*_{i} = *F*_{i}*r*_{i}, one could easily reduce Equation (6.22) to

FC=1Dc(FiDc−Dbcosθ2+FoDc+Dbcosθ2)(6.23)

Similarly *F*_{BPI} and *F*_{BPO}, which indicates the rate at which the balls pass a point on the track of the inner and the outer raceway, respectively, can be expressed as the product of the number of balls and absolute difference in velocity between the cage and the inner or the outer raceway.

Thus, using Equation (6.23)

FBPI=NB|FC−Fi|=NB2|(Fi−Fo)(1+DbcosθDc)|(6.24)

FBPO=NB|FC−Fo|=NB2|(Fi−Fo)(1−DbcosθDc)|(6.25)

Finally, *F*_{B}, which indicates the rate at which it rotates around its own axis, can be calculated as

FB=|(FC−Fi)rirb|=|(Fi−Fo)rorb|=Dc2Db|(Fi−Fo)(1−D2bcos2θD2c)|(6.26)

Since in a motor the outer raceway is tightly fixed to the static end bells of a motor, *F*_{o} *=* 0 . Similarly the inner raceway sits tightly on the rotor and rotates at the angular velocity *F*_{S}, and therefore *F*_{i} *= F*_{s}.

Thus Equations (6.23) to (6.26) can be written as

FC=FS2(1−DbcosθDc)(6.27)

FBPI=NB2FS(1+DbcosθDc)(6.28)

FBPO=NB2FS(1−DbcosθDc)(6.29)

FB=Dc2DbFS(1−D2bcos2θD2c)(6.30)

For some types of bearings Equation (6.28) and Equation (6.29) can be approximated as

FBPI=0.4NBFS(6.31)

FBPO=0.6NBFS(6.32)

In case of a single-point bearing defect, only one of the four characteristic frequencies given by Equations (6.27) to (6.30) would show up. The collision between the bearing defects at the point of contact sets shockwaves that excite the natural resonance frequencies of machines. These frequencies act as carriers to the fault signature frequencies given by Equations (6.27) to (6.30), which could be treated as baseband signals. If *f*_{c} is the carrier frequency and *f*_{b} is the baseband signal, then components such as *f*_{c}, *f*_{b}, *f*_{c} + *f*_{b}, *f*_{c} −*f*_{b} will be present. Since *f*_{c} + *f*_{b}, *f*_{c} − *f*_{b} are produced by *f*_{c}, *f*_{b} their phases are also sum and difference of *f*_{c} and *f*_{b} respectively. This type of interaction is known as quadratic phase coupling (QPC) and is best detected by the bispectrum as given by Equation (6.15) or by bicoherence, a normalized form of bispectrum. However, since *f*_{b} is also present, *f*_{c} can be erroneously detected as a sum of *f*_{b} and *f*_{c} −*f*_{b} if traditional bispectrum or bicoherence is used. Additionally, due to large mechanical damping at low frequencies, signals such as *f*_{b} can be significantly attenuated. Thus a mod¬ified bispectrum and bicoherence technique is proposed, where only the carrier, sum, and difference frequencies are included (ω_{b} = *2πf*_{b}; ω_{c} = *2πf*_{c})

B(jωc, jωb)=E[X{j(ωc+ωb)}X{j(ωc−ωb)}X*(jωc)X*(jωc)](6.33)

|b(jωc, jωb)|=|B(jωc, jωb)|2E{|X(jωc)X(jωc)|2}E[|X{j(ωc+ωb)}X{j(ωc−ωb)}|2](6.34)

It is easy to see from Equation (6.33) that zero phase angle is obtained when the carrier sum and difference frequencies are related, maximizing the expected value. Otherwise they are random, returning an expected value of zero. This technique has been able to clearly detect even incipient outer raceway faults, which could not be detected using standard power spectrum estimation of the mechanical vibration signal.

Detecting single-point faults in the inner raceway is more difficult because the fault moves in and out of the static load zone when the inner raceway is constantly moving. As a result, not only are the *F*_{BPI} frequencies modulated by the machine natural resonance frequencies but also the shaft rotational frequencies. Thus the fault frequencies occur in a group near the natural resonance frequency, each group containing several peaks separated by the shaft rotational frequency. The spacing from any peak in one group to another peak in another group can be given as

FSB=±FBPI+mFS, m=0, ±1, ±2…(6.35)

The fault-finding formula is now modified from Equation (6.33) and Equation (6.34) to (ω = *2πf* ; ω_{SB} = *2πf*_{SB})

B(jω)=E[X{j(ω+ωSB)}X{j(ω−ωSB)}X*(jω)X*(jω)](6.36)

|b(jω)|=|B(jω)|2E{|X(jω)X(jω|2}E[|X{j(ω+ωSB)}X{j(ω−ωSB)}|2](6.37)

The fault is now detected by counting the number of peaks for each value of *m*. It may be possible to extend this scheme for detection of the ball as well as cage defect.

However in the case of general roughness defect it is preferable to measure the root mean square (RMS) value of the mechanical vibration signal over a specified frequency range.

In general, the most appropriate measure to identify a fault varies from bear¬ing to bearing [18]. Mechanical vibration for constant low load levels (below 50%) can be detected as the fault progressed from incipient to advanced stages. For constant larger load levels (above 50%) the transitions could be random. Variable load can have significant effect on the fault development process. However, increased level of RMS value of the mechanical vibration signal seems to be an accurate estimator to diagnose advanced fault level.

Characteristic mechanical vibration frequencies described by Equations (6.27) to (6.30) can be seen in the line current spectrum of the motor also due to secondary effects [19,20]. Mechanical vibration causes radial displacement between stator and rotor, which can be treated as a combination of rotating eccentricities moving in clockwise and anticlockwise direction. This leads to the following frequencies in line current:

fbng=|fe±mfν|(6.38)

where *f*_{v} is one of the characteristic vibration frequencies and *f*_{e} the sup¬ply frequency. For example, brinelling is akin to single-point defects on both outer and inner raceways. Hence both *F*_{BPI} - and *F*_{BPO} -related frequencies show increase in the mechanical vibration spectrum. However in the current spectrum, the results are much less encouraging, especially for *F*_{BPI}. This sentiment was echoed again by Obaid et al. [12].

Recently efforts were also made to detect generalized roughness faults of bearings. Since it was observed that from mechanical vibration signal the RMS value of the vibration signal increases at the broadband level, it was felt that parametric spectral analysis of line current is a better approach to diagnose these faults. The popular, all pole, AR model is chosen for this purpose. Each time stator current is sampled the AR spectrum is estimated and stored. For each new spectral estimate, the mean spectral deviation (MSD) is computed. The MSD is the mean of the difference from each point in the spectral magnitude of the base line spectrum to the current spec¬trum. The MSD is computed and recorded every time the stator current is sampled, and the change in MSD is used as the fault index. Choosing MSD alone as the fault index usually works well for low motor loads. However for higher motor loads, the average value of several MSD readings gave better results [20].

The most important quality any scheme that detects stator faults must have is quickness of detection. Stator faults usually progress from incipient to a very advanced stage in a matter of seconds. Unless detected early enough, it might lead to fire, explosion, and even loss of personnel. Traditionally, stator faults are detected on-line by the negative sequence voltage or reduction of negative sequence impedance. However, voltage unbalance and machine asymmetries that also change the negative sequence current and impedance can cause misdiagnosis when faults involve only a few turns. While the machine asymmetries can be accounted for on a tempo¬ral basis, its robustness toward aging effects of a motor is yet to be vin¬dicated. Also, stator fault will, by its very nature, create some unbalance that cannot be measured by measuring the terminal conditions. Thus, till the present date, very low turn fault detection has remained as a major challenge to researchers.

There exists a few turns fault detection scheme utilizing frequency-domain-based techniques. They are not as popular as the schemes men¬tioned in the previous paragraph. However, most of them are relatively new concepts and further research is required for their improvement. They will be discussed next.

The earliest work on stator fault detection using external flux sensors was reported on by Penman et al. [21]. It was later experimentally proven by Penman et al. [22]. The basis of the stator fault detection using this approach lies in the fact that in an ideal machine, the axial flux of the machine is zero. In the presence of small machine inherent asymmetries they are still small. However, a stator fault causes large asymmetry and this produces compo¬nents such as

fss=(k±n(1−s)p)f(6.39)

in the axial flux, where *k* is the order of the time harmonic, *n* the order of the shorted coil space harmonic, *s* the slip, *p* the number of pole pairs, and *f* the supply frequency. Using *k* =1 and *n* =1,2 one could compute these frequen¬cies as 36.24, 48.12, 71.88, and 83.76 Hz. The last three of these were shown to increase under stator fault when tested on a 200 hp, 50 Hz, 8 pole slip-ring induction motor. A large coil with around 300 turns on a Plexiglas for¬mer was mounted concentrically around the shaft to detect the fault. Fault location was detected using four smaller symmetrically mounted coils of about 100 turns each on a plastic former and also mounted on the shaft.

Recently external flux sensors have been shown to diagnose stator faults even in variable speed drives using an 11 kW, 50 Hz, 4 pole squirrel-cage induction motor [23]. Some new frequency components as given by

fss=(γR(1−s)p±ν)f(6.40)

have been detected in the axial flux, where ν is the order of the time har¬monic, *R* is the number of rotor bars, γ = 0,1,2,3..., the order of rotor space harmonic, *s* the slip, *p* the number of pole pairs, and *f* the supply frequency. Similar frequency components are well known to be present in case of eccen¬tricity faults of induction motors. Therefore, confusion may arise as to what type of fault is being detected. Additionally, none of these schemes have been shown to be immune to voltage unbalance.

One of the earlier publications that discussed the line harmonic current increase due to stator faults also discussed components similar to Equation (6.39) and Equation (6.40) being found due to rotor faults in the line current [24]. According to Stavrou et al. [24], the stator current harmonics that are expected to vary due to stator inter-turn fault, and have their origins in stator current, are given by

fsc=(jrtR1−sp±2jsa±ist)f(6.41)

and those that have their origins in rotor current are given by

frc=((jrtR±k)1−sp±2jsa±irts)f(6.42)

Here *j*_{rt}, *j*_{sα}*, i*_{rt}, *i*_{st}, *k* are integers. The subscripts *sa,rt,st* are related to satura¬tion, rotor and stator.

Interestingly, the third harmonic in the line current was one of the com¬ponents that was shown to increase under fault in the study by Henao et al. [23] and was expected to show increase in the study by Stavrou et al. [24]. However, further investigation into the cause of this harmonic and related experimental results regarding its increase under stator fault have been completed [25–27]. Joksimovic and Penman showed that the negative sequence current interacts with the fundamental slip frequency current in the rotor conductors to produce torque pulsating at twice the line current frequency [25]. The consequent speed ripple caused flux density components at three times the line current frequency with respect to stator. This induced the third harmonic in line current. A more recent paper reports detection of third harmonic component in line current as a signature for stator fault [26]. It was attributed to the third harmonic present in the supply voltage and also to inherent machine asymmetry and voltage unbalance. While there is no doubt that the third harmonic voltages would manifest themselves in the line current of the machine, the degree to which these harmonics are nor¬mally present in line current are much larger than the voltages themselves [27]. It was further pointed out by Nandi that the fundamental frequency reverse rotating field (caused by fundamental frequency voltage unbalance and constructional asymmetry of the machine) interacts with the funda¬mental of the saturation-induced specific permeance function to produce the large third harmonic current, due to the presence of a matching pole pair associated with the third harmonic flux density component [27]. This can be clearly seen from the simulated plots in Figure 6.7.

**FIGURE 6.7**

Simulated line current in the b phase of a 2.2 kW, 4 pole, 460V, 60 Hz induction motor with saturation when healthy (top), with 5% voltage unbalance (middle), and with 5 turns’ fault in phase a. (From S. Nandi, “A detailed model of induction machines with saturation extend¬able for fault analysis,” IEEE Transactions on Industrial Application, vol. 40, no. 5, pp. 1302–1309, September/October 2004. With permission.)

The effect of voltage unbalance is naturally absent right from the moment it is switched off. However, due to the residual flux in the machine, current still flows in the rotor bars and also in the shorted coils in the stator. This fact has been utilized to detect the stator fault using rotor slot harmonics and later the more generic triplen-related harmonics [28]. According to Nandi and Toliyat [28], the voltage components induced by the shorted coil in the terminal voltage to detect is given by

fv=[k(R/p)±1]foff(6.43)

where *k*=1,2,3... and *f*_{off} is the frequency of the decaying stator voltage after switch-off and is proportional to the gradually diminishing speed of the induction machine. Figure 6.8 shows the simulated and Figure 6.9 the experimental results respectively for a 2.2 kW, 44 bar, 4 pole 60 Hz machine. The 23rd harmonic was already present in the spectrum due to the stator winding and the flux pole pair matching. Hence the 21st har¬monic was monitored. The faulty coil essentially produces all integral pole pairs and hence the 21st harmonic was induced in the voltage only under fault.

**FIGURE 6.8**

Simulated, normalized line voltage spectra of a 3 ph, 3 hp, 60 Hz, skewed, 44 bar, 4 pole induc¬tion motor under healthy (top), voltage unbalance (middle), and 5 turns fault in phase a (bot¬tom) at switch-off.

**FIGURE 6.9**

Experimental, normalized line voltage spectra of a 3 ph, 3 hp, 60 Hz, skewed, 44 bar, 4 pole induction motor under healthy (top), unbalanced (middle), and faulty with 5 turns’ fault in phase a (bottom) at switch-off.

It was shown later by Nandi [29] that the odd triplen harmonics also showed increase with stator faults due to the presence of residual saturation given by

fv=3nfoff, n=1, 3, 5…(6.44)

where *n =* 1,3,5.... In this case also the odd triplen harmonics are induced as the matching pole pairs are present only under fault. These harmonics are produced in the flux by the interaction of air-gap magnetomotive force (MMF) and saturation-related permeance components. Some simulation and experimental results are provided in Figure 6.10 and Figure 6.11. Detailed experimental results showed that faults with three turns and above under no load and two turns and above under full load could be unambiguously detected using this scheme.

**FIGURE 6.10**

Simulated line voltage (ab) spectrum at switch-off of a saturated induction machine under healthy (top), unbalanced (middle), and with 5 turns fault in phase a (bottom).

Similar harmonics show up in case of reluctance synchronous motors (RSMs) also [30]. Figure 6.12 shows the results for a 1.5 hp, 460V RSM.

Recently it was observed that due to the inherent asymmetry present in the field winding of the synchronous machines, harmonics of the form

fr=[k(1/p)±1]f, k=1, 2, 3…(6.45)

get induced in the field current. However, out of these only those harmonics not given by

fn=[n±1]f, n=1, 5, 7, 11, 13…(6.46)

**FIGURE 6.11**

Experimental line voltage (ab) spectrum at switch-off of a saturated induction machine under healthy (top), unbalanced (middle), and with 5 turns fault in phase a (bottom).

**FIGURE 6.12**

Experimental line voltage (ab) spectrum at switch-off of a RSM under unbalance (top) and with 5 turns fault in phase a (bottom).

can be considered, since harmonics given by Equation (6.46) can appear under balanced voltage (obtained with negative sign in Equation 6.46) as well as unbalanced voltage (obtained with positive sign in Equation 6.46) conditions. Since ideally the field coils can accept harmonics in the flux with only a certain number of pole pairs, the increase of the fault sig¬nature harmonics did not show sufficient increase for a low number of faulty turns and certain operating conditions. This is mostly due to the fact that these harmonics were induced due to inherent asymmetry of the field winding. However the rotor search coil, by construction, was capable of accepting harmonics of any integral pole pair number from the field current. Thus harmonics induced in the search coil can be used to detect even one turn fault under any operating condition [31]. Illustrative simula¬tion and experimental results are shown in Figure 6.13 and Figure 6.14. HB

**FIGURE 6.13**

The 150 Hz component in the field current of a 2 kW, 4 pole, 60 Hz, 208V, synchronous motor, under no-load (top), half-load (middle), and full-load (bottom), 0.8 lagging power factor condi¬tion (experimental). 150 Hz can be obtained by using k = 3, p = 2, f = 60 and the positive sign before 1 in Equation (6.45).

**FIGURE 6.14**

The 90Hz component in the rotor search-coil of a 2 kW, 4 pole, 60 Hz, 208V, synchronous motor, under no-load (top), half-load (middle) and full-load (bottom), 0.8 lagging power factor condi¬tion (experimental). 90 Hz can be obtained by using k = 3, p = 2, f = 60 and the negative sign before 1 in Equation (6.45).

implies healthy balance, HU is healthy unbalance, and T1-T4 implies one turn to four turns short.

During an inter-turn fault the stator has a shorted loop (can thus be treated as a single-phase winding) carrying current at supply frequency that gener¬ates two counter-rotating MMF waves [32]. The MMF produced by the asym¬metric stator carrying three-phase balanced voltage can be given as

Fsa=Asacos(kφ±ω1t+γ1)(6.47)

where *k =* 1, 2, 3... corresponds to space harmonic poles. Considering the spe¬cific permeance function (*P*_{0}) the flux density produced by this MMF, with respect to stator, can be given as

Bsa=AsaP0cos(kφ±ω1t+γ1)(6.48)

With respect to rotor, this flux density can be given as

Bra=AsaP0cos(kφ′+kωt±ω1t+γ1)(6.49)

Now substituting ω=(1−s)ω1p

Bra=AsaP0cos(kφ′+{kp(1−s)±1}ω1t+γ1)(6.50)

The term associated with *t* in Equation (6.50) gives the frequency compo¬nent *f*_{r} that can used for detection as

fr={kp(1−s)±1}f1(6.51)

For example, the frequencies that will be induced in the rotor circuit due to a fault in stator winding when a doubly fed induction generator (DFIG) is running at *s =* 0.25, *f*_{1} *=* 60 Hz, *p =* 2 and different values of *k* are expressed in Table 6.1 using Equation (6.51). As seen from Table 6.1, several frequencies can be induced as a result of the fault. Unfortunately, many of the compo¬nents given by Table 6.1 can be present even under healthy conditions and hence cannot be treated as reliable indicators of the fault.

Hence a detailed simulation study was conducted and compared with experimental results. Some of the very prominent components were 82.5 Hz for *k =* 1 and 127.5 Hz for *k =* 3, which arises due to asymmetry of the machine as can be seen from the simulated plots in Figure 6.15. Also the components related to *k* = 3 showed better promise as triplen-space-related harmonics seem more affected by asymmetry.

**TABLE 6.1**

Stator Fault Frequencies Induced in Rotor, s = 0.25

**FIGURE 6.15**

PSD of a simulated DFIG connected to balanced load with symmetrical rotor winding (top), symmetrical rotor winding subjected to 4-turn fault (middle), and asymmetrical rotor winding (1 reduced turn in one phase) subjected to 4-turn fault (bottom).

**FIGURE 6.16**

Simulation variation of fault spectra of rotor current frequency 127.5 Hz (k = 3) for slip 0.25 under no-load, half-load, and full-load, with varying fault severity. HB implies healthy bal¬anced load, UB implies unbalanced load, 1T-4T implies fault levels from 1–4 turns.

The model with one turn rotor asymmetry was further explored for unbal¬anced load and different fault levels. As can be seen from Figure 6.16, unbalanced load (10% on stator A phase) does not affect the result to a great extent. Also, the fault signature increased in proportion with the number of faulted turns.

The experimental results (Table 6.2) showed more consistent results for detection when rotor line current space vector was used rather than individ¬ual line current. The current space vector actually gave comparable results or even better results (at higher slip) with the rotor search coil.

The detection scheme was implemented on-line using the scheme shown in Figure 6.17. It worked quite reliably even down to two turns fault level, which can be detected within approximately 2 seconds (includes tripping signal to circuit breaker) (Figure 6.18). The scheme worked even under tran¬sient condition (Figure 6.19) quite reliably.

**TABLE 6.2**

Comparison of Signal-to-Noise Ratio (Given by the Difference between the Faulty and the Balanced Healthy Signature) for Fault Signature Frequency Component Power Level for Different Severity of Fault under Full-Load Condition

**FIGURE 6.17**

Schematic of experimental setup used to determine DFIG behavior under varying load, speed, fault severity, fault detection, and tripping.

**FIGURE 6.18**

Typical time of operation of the DSP-based fault detection device when using search coil volt¬age signature analysis. DFIG operating at slip = 0.25 at different loads.

**FIGURE 6.19**

Various signals of the fault detection scheme, DFIG operating at half-load during a speed change, fault severity is 2-turn, fault detection time is 1300 msec.

Unlike a stator inter-turn fault, the rotor bar fault is an *open circuit* fault. Also, unlike stator inter-turn fault, it often does not lead to a catastrophe within a short period of time. Either rotor bars or end-rings may be open circuited. However, since the bars are typically not insulated, bar breakage at the initial stages may not be detectable due to the presence of interbar currents [33]. Also this type of fault can be detected only under loaded condition, since under no-load the rotor current is almost zero. Although many techniques to detect these faults exist, unlike stator inter-turn faults, detection of signa¬ture frequency components in the line current is the most common way to detect these faults. Some of the different frequency-domain-based methods are discussed next.

When a rotor bar is broken, there is an increase in the current distribution in the two bars adjacent to the broken rotor bar [34]. This can be deemed as current flowing in a single-phase winding, and therefore the double revolv¬ing field theory used for the analysis of single-phase induction motors can be applied. The anomalous MMF produced by these two bars with a rotor bar broken between them can be expressed in rotor coordinates as

Fs=Fmcos(nx′±sωt)(6.52)

where *x*′ is the space angle with respect to rotor, ω is the supply frequency in rad./sec., *s* is the slip, *n =* 1, 2, 3….

MMF components as described by Equation (6.52) will induce voltages in the stator winding of a regular three-phase motor given by

vs=vmcos(nx+(n(1−sp)±s)ωt)(6.53)

only for *n*=*p*,5*p*,7*p*,11*p*,13*p*... as they alone can match the stator pole pairs. *x*′ is the space angle with respect to the stator and x=x′+(1−sp)ω