Induction motor faults: basics, developments and laboratory-scale implementation (part a)

__ 1 Introduction

In the previous sections, we devoted ourselves to the analysis of behavior of the healthy induction motor in different possible operating modes including transient and rated operations each having their own effects on the electrical, magnetic and mechanical quantities of the motor. In this section, we deal with an easy and comprehensible way of formulating and describing motor quantities by means of very useful mathematical equations. The goal is initially to see how motor quantities might vary in time and partly in frequency domains. However, the frequency domain is only mentioned as a black box while there would be a sufficient discussion in the next section corresponding to various time, frequency and time frequency processors. The materials provided earlier in the previous sections were interesting and, of course an essential step through understanding the induction motor faults and the corresponding fundamentals. By ''fundamentals,'' the causes, the effects and also any development related to the faults are targeted. Moreover, various types of the supply modes including the line-start, open-loop and closed loop applications were discussed in the previous section to wrap up what happens in industry in terms of the motor control strategies. So, a potential reader of this guide will get used to the terminologies and possible motor-drive interactions.

A general guideline based on which the motor operations are justified in different modes were provided as well. As the goal of this guide is not to discuss every single aspect of drive circuits and also the control strategies, they were explained in a way to cover a great portion of existing operations.

On the other hand, drive circuits are not directly related to the fault diagnosis procedure of induction motors. They mostly apply a kind of impact on the quantities of the faulty motor. It means that the drive and the inverter are not usually involved in the diagnosis procedure. However, they definitely affect the process in comparison to a line-start application. Having discussed and investigated the behavior of the healthy induction motor, now it is time to go through the theoretical and practical aspects of the faulty induction motor. This is a step-by-step process which illustrates and formulates various faults and their effects on the magnetic, electrical, mechanical and also thermal, of course if possible, quantities. Unfortunately, the fault diagnosis of electrical machines really suffers from lack of knowledge in terms of thermal analysis of induction motors. Therefore, the main focus will be on the analysis of other types of physics. In the meanwhile, you will sometimes find a subsection related to some specific aspects of thermal modeling or analysis of a faulty motor.

According to the early materials of Section 2, any induction motor, regardless of the type and the configuration, consists of two main parts, the stator and the rotor each having their own particular components. Depending on the location and the type of the component, various faults might occur. The magnetic, thermal, electrical and also mechanical stresses are the major causes of faults in induction motors. Generally, there are two types of faults as follows:

--Electrical faults: Any type of fault which has an electrical source is called an ''electrical fault.'' The most important and well-known kind of this type of fault is the ''short-circuit'' or ''interturn'' fault. This fault is common for all types of machines which include windings and coils. The main cause of the interturn fault is the deterioration or damage of winding insulations. Over voltages or over-currents are the main causes. In fact, if the voltage applied to the turns of windings goes over the tolerable thresholds of insulations, insulations fail. Over-current, which is the main cause of heat-up in the machine core and winding materials, is another factor in damaging insulations. As a result of this phenomenon, two or more coils in a winding or even between two windings are short-circuited, and an uncontrollable large current circulates through the windings without contributing to the main torque component of the motor. Therefore, it is only considered as a source of losses, unbalanced operation and failure in the symmetrical operation of the motor. This is why, this kind of unbalanced/asymmetrical operation is called ''fault.'' There are also other causes which lead to an unbalanced electrical operation of the motor quantities, mainly the magnetic flux density. The best examples are an unbalanced supply voltage, reversed-phase voltages as well as the gearbox fault. However, they have nothing to do with the internal structure of the motor. Therefore, they should not be considered as a type of fault assigned to the motor because the source is not located inside the motor. Unfortunately, some of research works take them as a kind of motor fault by mistake. In this guide, the term ''fault'' refers to a phenomenon which directly affects an interior motor component so that a change in the shape, the material or the motor operation takes place. Basically, if an external factor leads to a harsh motor operation, it does not necessarily damage the motor components, consequently, if the factor is removed, the motor will be back to its normal, symmetrical and balanced operation. Nonetheless, if the harsh conditions go on for a while, the external factor might lead to a destruction of the building blocks of the motor. This is exactly what we call the ''fault.'' Therefore, all the materials of this guide are valid as internal faults take place. This discussion necessitates the presence of an early fault diagnosis procedure usually called an ''incipient fault diagnosis'' which should be an essential part of any diagnosis task. Incipient diagnosis is an extremely important condition monitoring process existing in high-power applications. In fact, diagnosing the transition from a healthy state to a faulty one is always appreciate in industry.

--Mechanical faults: In this case, the main cause of the fault has a mechanical nature. The ''broken bar,'' ''broken end-ring'' and ''eccentricity'' faults are the best examples of the kind. There is no shortage of agreement that all the mechanical faults are caused by a corrosion, improper casting or unsuitable placement of the components. Sometimes, an internal defect of the motor materials also produces a mechanical fault. For instance, the bearing fault sometimes is a result of a partial breakage of the interior balls. The broken bar fault is generally assigned to the event of disjoining the end-rings and the bars in the rotor. As the bars are connected to the rings through a molded or casted area, they are likely subjected to a breakage if an overwhelming tension is applied to the joints. Unbalanced magnetic pulls (UMP) are mostly the main reasons. The UMP itself can be an output of a misaligned or eccentric rotor.

When a rotor center is not aligned with that of the stator, the motor air-gap experiences an UMP around the outer circumference of the rotor. As a result, the bars and the rings are impacted. When a broken bar motor comes across an eccentric rotor, it is expected to observe a higher level of the broken bar indicators. It is worth noting that any mechanical or even electrical fault eventually contributes to an improper magnetic and/or thermal tension and imbalance. Undoubtedly, the UMP is the best example of the magnetic imbalance. In terms of the thermal stress, the short-circuited turns or windings reveal a hot spot adjacent to the fault area and introduce the possibility of a molded silicon steel material due to the high temperatures around the faulty region. Hence, some specific types of fault faults might create another type of fault. Thus, any methodology leading to a fast, reliable and most importantly early detection of the fault is much respected.

In general, three major types of defects, namely the broken bars/end-rings, the eccentricity/misalignment and the short-circuit faults are going to be investigated in this guide in details. The materials of this section are sort in the way described below:

--the broken bars/end-ring fault

--the eccentricity/misalignment fault

--the short-circuit fault.

Each type will be analytically and experimentally dealt with. The broken bars fault is practically the most tackled and analytically addressed one, and the other two types somehow contain fewer databases in terms of analytical extensions and also the drive-connected problems. Nevertheless, the authors try to gather up every required detail for the purpose of providing the fundamentals of advanced knowledge of the field. To this end, it is preferred to first discuss the basics of all types of fault and then move toward some of the analytical demonstrations of the faulty motor behavior. Then, the experimental approaches corresponding to the laboratory-scale implementation of various faults, along with the required sampling and processing tools and equipment, are studied. Hence, this section seems to be interesting and, for sure, indispensable for almost all of the readers who are amateurs. If the readers already have the idea of how faults are examined, the current section might be skipped. Here are the main outlines of this section:

--to investigate the fundamental concepts of fault occurrence

--to study the time-domain effects of the faults on the motor quantities

--to provide some important analytical descriptions of the faulty motor behavior

--to provide the guidelines of the experimental implementation of various faults in a laboratory

--to explain how the faulty motor signals are measured and then processed experimentally

--to discuss various type of sensors such as the wireless apparatus which could be used during the sampling and measurement task

--to go through the advantages and disadvantages of the measurement techniques provided in this section and then introduce the most competent one.

As the starting point of the discussion, the broken bar fault is targeted as follows.

__ 2 Broken bar/end-ring fault in induction motors

Bars breakage usually takes place in applications which specifically require high-power motors, and a highly stressful environment normally exists.

In such conditions, oscillating loads, improper motor assembly, mechanical stresses, as well as the hysteresis stresses, might weaken the joints connecting the bars and the end-rings. Consequently, weakly connected bars might be disconnected from end-rings in one or both of the motor ends. This phenomenon, which partially or fully eliminates the current in the broken bar or bars, is known as the ''broken bars/end-rings'' (see Fig. 1). Sometimes, there can also be a crack or breakage in the end-ring, not in the location of the bar/end-ring joint, but somewhere between two bars. This type is usually caused by improper casting process during the motor construction. Equally important, the air bubbles located inside the joints or the materials could be a decisive factor in increasing the breakage likelihood. As it is conveyed by the name of this type of fault, it only affects the motors with a squirrel-cage rotor, and because the squirrel-cage motors are widely and prominently used in industry, the broken bars fault is very common.

Fig. 1 Broken bar fault in a squirrel-cage induction motor

Fig. 2 Equivalent network for a broken bar rotor

Fig. 3 Schematic field distribution in an induction motor with rotor broken bar

The term ''partially'' which was assigned to the broken bar fault in the previous statements reinforces the idea that there might be a partial or full broken bar or bars. What happens in the case of partial broken bar is that the connection between the bar and the end-ring still exists, although the breakage has already happened and caused to disconnect a part of the joint. As a result, the partial broken bars fault is introduced. Likewise, when there is a fully disconnected joint from which the bar current is not able to flow, the full broken bars fault is introduced.

Depending on the severity of the breakage, different levels of the partial broken bar are addressed and investigated in the literature. Moreover, more than one breakage might also be present at the same time. However, increasing the number of broken bars, whether partial or full breakage, does not necessarily mean that the effect of the fault on the motor quantities increases as well. It sometimes depends on another factor which is the location of the bars in the rotor across the rotor circumference.

So, a new influential factor is introduced as well; that is the fault location. It should be noted that increasing the severity of the breakage in the case of partial broken bar always amplifies the fault effect, while it is not always the case where the number of broken bars increases.

In Section 2, the rotor circuit was modeled and represented by an impedance network containing one impedance per bar or one end-ring (see Fig. 2.5). In the same fashion, the broken bar rotor is modeled in a way that the bar subjected to a full breakage is totally removed from the network. Instead, the current previously flowing in the broken bar finds a pathway through the adjacent bars leading to an increase in the bar currents close to the breakage, and the removed current is modeled by a current source in Fig. 2. The amount of the current source should be ideally equal to the bar current removed from the circuit. The main reason for the increase in the current level of the adjacent bars is the elimination of the armature reaction generated by the broken bar in a healthy case. The voltages applied to the adjacent bars increase; hence, their total current increases as well.

The amount of divergence forming a symmetric and healthy motor depends on the severity of the breakage. The more sever the joint is broken, the larger the asymmetry of the motor variables will be.

The first variable affected by the broken bar is the current space distribution.

Then, the corresponding magnetic field pattern is distorted. The development process of the broken bars fault is as follows:

Mechanical asymmetry ? Electrical distortion? Magnetic distortion? Thermal stress At the final stage of this fault, due to the dramatically increased magnetic saturation caused by the over-current phenomenon, the local losses and subsequently the local temperature increase as well. This is the general qualitative flow observed in any broken bar event.

Now, it is time to take a bit distance from the philosophical aspects of the broken bars fault and move along with the real-world happenings in terms of formulating this kind of fault. Without exception, if a symmetric, healthy and single harmonic motor is under investigation, the ideal magnitude of the back ward field should be zero. The term ''backward'' is assigned to a field which rotates in the opposite direction to that of the synchronous speed or frequency.

When an inherent asymmetry such as the nonsinusoidal spatial distribution of the bars exists, a very well-known backward field whose frequency is equal to sfs is produced (see Fig. 3, the clock-wise rotating sfs components). The other sfs component always exists in a motor.

According to the fundamentals of induction motor, the following relations hold:

Stator electrical frequency: fs Rotor rotating electrical frequency: fr 1 s fs Motor slip: s fs fr fs Rotor bar electrical frequency: sfs

(3.1)

Two magnetic fields, Bs and Br, which are spatially placed with a specific angle, q, with respect to each other produce the electromagnetic torque. In fact, in electrical motors, the electromagnetic torque is produced by cross product of two fields generated by the stator and the rotor. Like Bs, the forward magnetic field produced by the rotor rotates with the electrical frequency of fs fr sfs (3.2)

There is another backward field, Br_fault, which rotates in the direction opposite to the main rotor and stator fields. Therefore, the resultant electrical frequency in the stator reference frame is equal to fr sfs (1 2s) fs which is an indication of the armature reaction of the clock-wise rotating backward field on the stator variables including its current and back-EMF. This is actually the basis of investigating an asymmetry in the rotor. Not only the fault, but also the rotor inherent asymmetries produce a backward field causing additional harmonic components of the motor.

Thus, it is expected to observe the backward field in the case of a healthy motor as manufacturing process does not guarantee a hundred percent ideal rotor design.

Given the fact that even healthy motors consist of the backward fields, the fault occurrence only amplifies the magnitude of the corresponding field and has nothing to do with creating the frequency component. Most of the time, researchers make this big mistake that the broken bar/end-ring fault generates the backward sfs component, while this is a totally wrong notion. In reality, the broken bars fault only impacts the magnitudes of some existing harmonic components which are normally the outcomes of the motor structure.

By means of the overviewed concept and also the one related to the single harmonic motor model investigated in Section 2, one can develop the fundamental formulas of the induction motor with broken bars assuming a linear silicon steel material with a constant permeability of both the stator and the rotor. This makes the process of relating the magnetic field to their producing currents a simple one.

The fundamental component of the model reveals a synchronous frequency of fs in the stator. Therefore, the fundamental harmonic component of the air-gap flux density produced by the stator current is B fundamental Bmsin wst (3.3)

where Bm is the magnitude of the fundamental component of the flux density at the air gap. On the other hand, there is a flux density component with the frequency of (1 2s)fs produced by the rotor backward field . Hence, the corresponding flux density is formulated as the following:

B backward Bmbsin 1 2s wst aB backward (3.4)

where aB_backward is the electrical phase angle of the backward field. Bmb is the magnitude of backward magnetic flux density caused by rotor asymmetry. The assumption of a linear magnetic material leads to NI kB yR (3.5)

where I, N, k, y and R are the current, number of turns, constant coefficient and reluctance of the flux path, respectively. So the motor current is related to the magnetic flux density as follows:

(3.6)

As far as the material is linear, the superposition is applicable. Taking advantage of the superposition rule, each component of the magnetic flux density, the synchro nous and fault-related components, could be separately substituted in (3.4). If it is applied to the fault component, the following equation is achieved.

(3.7)

…where I sideband is a current component with the frequency and magnitude of (1-2s) fs and Im sideband, respectively. This component is the outcome of the backward rotating field. The term ''Sideband'' is assigned to this component because it is located within a specific frequency range around the fundamental component if the current spectrum is analyzed. For the moment, we do not intend to talk about the frequency-domain analysis and will only stick to the time-domain variations for justifications.

Suppose, there was an asymmetry in the rotor, whether a fault or inherent casting problem. Then, we ended up with the fact that describes the reason for the existence of a backward field and subsequently the corresponding (1-2s)fs components of the flux density and the current. This clearly proves that the sideband components are always present in a motor with asymmetries, regardless of the rotor fault occurrence. Nevertheless, the broken bar/end-ring fault, a kind of asymmetry in the rotor, helps the component get stronger compared to a healthy motor. Again, it should be noted that a healthy motor is different from an ideal motor in which the forward rotating fields are only observable.

Let us now flash back to the main discussion. Phase angle of faulty current component should be different from that of the flux density as nonlinear relation ship always holds in reality instead of a linear correspondence. This claim can be further discussed by the FFT analysis of both the current and the flux density and will be done in Section 4. For now, we do not miss the chance to continue with the analytical formulations which have been provided so far. Considering the fact that the electromagnetic torque is the main motive force to produce rotation, it is used here. The fundamental and fault-related components of the electromagnetic torque are expressed as follows:

(3.8)

where L fundamental, Lm, aL fundamental , I fundamental , Im, aI fundamental , T and p are the fundamental components of flux linkage, the amplitude of flux linkage, the phase angle of flux linkage, the fundamental current component, the amplitude of fundamental current component, the phase angle of fundamental current component, the developed electromagnetic torque and the number of poles, respectively. Here, the coupling between the fundamental flux linkage component and the stator current variables including the fundamental and sideband components are dealt with. The resultant torque consists of DC, 2wst,2swst and 1 s 2wst components. The DC component produces the average torque required for the main rotor operation. The terms 2wst,1 s 2wst are higher frequency components which are located around twice the fundamental synchronous frequency and are mostly subjected to being filtered out from the motor operating cycle due to the fact that the rotor plus load inertia is so high that the only the low-frequency components with the order of a couple of Hertz would remain in the torque. As a result, the remaining part which comes into considerations and might produce ripples in the motor torque is 2swst.

This is exactly the component which is the consequence of a rotor asymmetry such as the broken bar fault. As it is a linear function of the motor slip, it is located at very small frequencies, typically below 10 Hz, depending on the motor supply frequency and the slip level.

As a very interesting point, the motor torque and its harmonic components are controlled directly by means of a drive circuit. Therefore, any low-frequency fault related component which is located inside the pass band of the PI regulators is normally affected by the closed loop of the drive circuit. As a result, it is not likely to have a lower torque oscillation even in case of higher fault levels. In Section 8 which is related to the fault diagnosis of the broken bar motors, this issue will be discussed in details.

Now, consider the torque fluctuations caused by the 2swst component as follows:

(3.9)

On the other hand, the motor torque and speed are related to each other by the following equation given that the viscosity of the shaft is zero. Then, one comes up with the following equation which is demonstration of speed fluctuations caused by rotor fault.

(3.10)

J is the motor inertia and Dwr is the illustration of the motor speed fluctuations caused by the broken bars fault. It is obviously seen that the asymmetry in the rotor, for example the broken bar fault, brings the speed to oscillate with the frequency of 2sfs. This fluctuation happens over the average value of the speed. As the next step of the analysis, electromotive force (EMF) produced by the speed fluctuations is calculated and added to the fundamental component as follows:

(3.11)

… where wr average is the average speed of the motor. Equation (3.11) reveals a very interesting and significant aspect of the broken bars fault which is actually the production of a sideband component with the frequency of (1 2s)fs. It means that not only does a left sideband component of the fundamental current harmonic exist, but a right sideband component is also produced as a result of the motor-speed fluctuations. Both are the functions of the motor slip. These harmonic components are generally called the sideband components of the motor current shown by the pattern (1-2s) fs. Accordingly, if the speed is fixed by any means, for example, if it is connected to a closed-loop drive with a very well-refined speed control loop, there is no right sideband component, (1 2s) fs or it exists with a considerably small magnitude compared to the left sideband. In practice, eliminating speed fluctuations is impossible. Hence, the right sideband component always exists while its amplitudes would be so much sensitive.

Each sideband component appears as a current component in the stator windings with the same frequency of the mentioned pattern. In turn, the corresponding magnetic flux densities are produced at the air gap (see (3.12)). The corresponding magnetic flux density terms are named the fault component of the flux density as the source could be related to an asymmetry such as the broken bar phenomenon.

Taking this along with the Biot Savart law into account, two terms defining the right and left sideband components of the magnetic flux density are formulated as (3.12). In fact, hereafter, the terms ''right'' and ''left'' sideband components are used to address (1 2s) fs and (1 2s) fs elements in any motor variable, respectively. Accordingly, the left and right sideband components of the magnetic flux density are as follows:

(3.12)

where Bfault , Bmleft , Bmright , aBleft and aBright are the fault component of flux density, the amplitude of left sideband component, the amplitude of right sideband component, the phase angle of left sideband component and the phase angle of right sideband component, respectively. In a similar fashion the current components are formulated as follows:

(3.13)

where Ifault , Imleft , Imright , aIleft and aIright are the fault components of motor current, the amplitude of left sideband component, the amplitude of right sideband component, the phase angle of left sideband component and the phase angle of right sideband component, respectively. So the total motor current containing the fundamental and faulty components is as follows:

(3.14)

On the basis of the fact that the motor flux has been affected by the sideband components, (3.15) is conducted. It is the output of a fundamental electromagnetic rule based on which the product of a flux and flux-producing current results in an electromagnetic torque if the corresponding flux vectors make a specific angle. Therefore, as both the flux and the current are affected by the fault, the left and right sideband components contribute to the motor torque production as follows:

(3.15)

The above demonstration of the electromagnetic torque is the more comprehensive case compared to the simple representation in which all the flux and current signals were ideal and healthy. In the last illustration of the electromagnetic torque, (3.15), there are nine terms added up. By means of (3.16), the mentioned terms are separated into their fundamental terms expressing the corresponding frequencies.

(3.16)

Tbl. 3.1 lists the included frequency components in the electromagnetic torque equations.

Tbl. 3.1 Frequency components of the motor torque caused by the faulty current and flux signals

DC component corresponds to the average motor torque. The higher order harmonic components present themselves in the shape of fluctuations carried by the average value. The amplitude of fluctuations increases with the increase in fault level. As mentioned before and what will be proved in Section 4, the high-order harmonic components of the motor torque including 2fs,(1 s)2fs and (1 2s)2fs are usually filtered by the motor transfer function which is in practice a low-pass filter. So the only remaining components are those which are the product of the motor slip. If one takes electromagnetic torque spectrum, s/he will notice the higher harmonic components as well. The reason is that usually the motor mechanical torque is measured in which high-order harmonic components eliminate, depending on rotor and coupling inertias. The calculated electromagnetic torque will, of course, contain the high-order components.

Finally, the active power of the motor could be calculated mathematically by multiplying the torque and speed equations. There should certainly be an average value providing the motor with an active power to run. The active or sometimes the reactive power of the motor gives us the opportunity to discriminate between the fault-related 2sfs component and the one produced by an oscillating load. In fact, any load oscillating with the frequency of 2ksfs introduces some levels of change of the sideband components. Thus, they might be mistaken as the real broken bar fault if simply the motor current signature analysis (MCSA) is used. In this case, the motor power spectrum is usually preferred, of course in the steady-state regime, to find the principle source of sideband components change.

As shown in (3.11), the backward field whose frequency is equal to (1 2s)fs generates the right sideband component with the frequency of (1 2s)fs. This was revealed by substituting speed fluctuations caused by 2sfs component with the EMF equation (see (3.11)). If such a trend is repeated for the 4sfs component existing in motor torque equation, the corresponding torque and EMF will contain new components with the frequency of 4sfs and (1 4s)fs, respectively. On closer inspection, if one continues the process of mutation of the sideband components repeatedly, the following pattern is introduced and could be extracted by the frequency analysis of motor current and EMF.

Frequency pattern of the sideband components: 1 2ks fs (3.17) where k is an integer starting from 1. This pattern exists in the vicinity of the fundamental component of the current and the flux. However, it might be seen around the products of the fundamental components including the third, fifth, seventh product and so forth. Equation (3.17) prepares a very significant engineering basis for the starting point of the fault diagnosis procedure in terms of the broken bars fault. Actually, the amplitudes of these components are the key feature in the fault diagnosis procedure. The point is that the amplitudes are functions of so many influential factors including the fault severity, the fault location, the load level, the speed level, the supply mode, the motor voltage and the motor structure. Therefore, so many efforts have been made so far to find a meaningful trend of variations between the influential factors and the corresponding amplitude of indicator. The studies include the time, frequency and time-frequency processors which deal with stationary and non-stationary operations.

__ 2.1 Time-domain behavior of induction motors with broken bar/end-ring faults

Fig. 4 Line-start motor variables (a) current, (b) torque, (c) speed and (d) flux in full-load condition, one broken bar case

Fig. 5 Line-start motor variables (a) current, (b) torque, (c) speed and (d) flux in no-load condition, one broken bar case

A very useful analytical approach has been already provided for understanding what theoretically happens to the motor quantities due to the broken bar/end-ring fault occurrence. The statements should be completed by additional schematic illustration of the motor quantities such as current. Fig. 4 presents the current, torque, speed and flux signals for an induction motor with one broken bar.

The investigated motor is the same as the one with the power of 11 kW used in Section 2. This motor consists of 28 rotor bars along with 36 stator slots. The broken bar characteristics reflected in the motor variables are listed below.

Transient operation is always a tricky and challenging mode of diagnosis in every type of fault as the corresponding faulty signals are only present for a very short period of time. As a result, it is often impossible to track the frequency components dealing with the fault. Besides, the fault components have no fixed amplitude and frequency wise, and they reveal a dramatically changing trend. Thus, diagnosing the faults in the transient regime is still one of the trending aspects of the field. This is not merely restricted to the broken bars fault and is a matter of all the types.

On the other hand, although no/light-load operating condition might be very helpful in diagnosing the faults such as the eccentricity, it diminishes or at least reduces the possibility of detecting the broken bars fault. It is generally because of the masking effects of the fundamental harmonic component which is very close to the fault-related patterns, specifically if the motor current is the signal which is used for the diagnosis purposes. In the no/light-load condition, the slip value is so small that the sideband components almost stick to the fundamental component and they are hopelessly undetectable unless a frequency-domain analysis is hired ( Fig. 5).

The hints are provided here in terms of the theoretical and schematic behavioral study of an induction motor with one full broken bar. There are several other aspects associated with this problem such as:

--the location of multiple broken bars

--the partial broken bar fault

--the effect of various supply modes including the open- or closed-loop modes

--the effect of the drive reference parameters on the diagnosis procedure.

Having provided the fashion based on which the motor faults could be studied, let us move on to the next type of faults which corresponds to the rotor shaft and its displacement from the stator center. This is called the ''misalignment fault'' or generally the ''eccentricity fault.'' First, a set of mathematical developments are addressed; then a couple of illustrations are provided in terms of the behavioral study of the faulty motor.

Fig. 6 Induction motor schematic (a) 3D space and (b) 2D considering possible rotor displacements

__ 3 Eccentric/misaligned and bearing faults in induction motors

In induction motors, there are three physical dimensions x, y and z indicating the Cartesian axes in 3D space (see Fig. 6(a)). In addition, the 2D demonstration is also provided in Fig. 6(b) which illustrates the rotor symmetry center (Ar), the stator symmetry center (As) and the rotor rotation/whirling center (Aw). Besides, the Cartesian axes are highlighted in the following fashion:

--The x-axis is aligned with the spatial axis of the phase ''a'' of the stator.

--The y-axis is located exactly 90 mechanical degrees away, in a counter clock wise direction, from the x-axis.

--The z-axis is perpendicular to both of the x and y axes in the direction of the motor shaft.

In healthy motors all the three centers perfectly coincide with the spatial center of the stator, i.e., As, which is fixed in one point. This point is the origin of any mechanical displacement analysis. So the air-gap length, highlighted in Fig. 6(a), would be the same all over across the stator circumference and equals to g0.

The rotational movement of the rotor does not disturb this uniformity while a hundred percent match between the mentioned centers is impractical due to several imperfections such as the improper assembly or placement of the bearings. This is the reason that the bearing fault could also be a potential source of eccentricity fault.

This phenomenon could be easily detected even in the case of healthy and new brand motors. This type of eccentricity is called the inherent eccentricity. Other factors including the misalignment of the load and the shaft axes, as well as the mechanical stresses and imbalances, fortify the situation. Consequently, the eccentricity fault takes place, and the uniformity of the flux distribution in the air gap is distorted. This literally means that the air-gap length is not uniform across the stator circumference. Using a very acceptable approximation of the air-gap length variation, the following representation should always hold correctly [29-35 ]: g j; jm; r g0 1 rcos j jm (3.18)

where g0, r, j and jm are the uniform air-gap length in healthy condition, the ratio of the distance of the rotor and stator centers over g0 (the severity of the eccentricity fault), the mechanical angle of the rotating and nonrotating point is the stator reference frame and the angle separating the stator and rotor centers in the direction of the rotation, respectively (see Fig. 6(b)). It is obvious that the distance between the stator and rotor center could not exceed g0. As a result, r should geometrically remain within the range [0-1 ] if the rotor is locked. However, while the rotor rotates, the UMP, along with the centrifugal force, leads to the increase in the fault severity, so r might not go beyond 0.72 (proved by the experiments). In such situation, the rotor touches the inner surface of the stator at both ends while the motor operates.

The center of rotation of the rotor has not been taken into account yet and it has only been assumed that Aw is fixed. In fact, it is the position of Aw which defines the type of the eccentricity fault. If the center of rotation remains concentric with As, it is called the ''static eccentricity.'' The term static conveys the idea that although the air-gap length is nonuniform across the stator circumference, it is fixed and does not change with time. However, if the center of rotation is aligned with the stator center during any eccentric condition, it is called ''dynamic eccentricity'' fault. In this case, the rotor center rotates about the stator center with the same frequency as that of the rotor. As a result, the nonuniform air-gap length rotates by rotating the rotor, and the air-gap length is not fixed anymore. For example, jm which is the position of the minimum air-gap length, increases by the rotor speed as follows:

(3.19)
… where ad is the angle in the rotor reference frame along which Ar and As diverge and q is the angular position of the rotor with respect to the stator. As ad does not have any considerable change in the quantities expressed in the stator reference frame, it is neglected due to the sake of simplicity. In general, when the eccentricity fault occurs, both static and dynamic eccentricities could be detected, and a new type called ''mixed eccentricity fault'' is introduced. This means that the air-gap status changes regarding two aspects:

--the level of the eccentricity fault (r)

--the spatial distribution of the nonuniform air gap.

Considering this point, r and j are generally formulated as follows [36-38 ]:

(3.20)

(3.21)

where rs is the ratio of the distances of Aw and As from g0. rd and js are the ratio of the distances of Aw and Ar from g0 and the angle along which Aw and As diverge, respectively. The static and dynamic eccentricities are only some simpler forms of the mixed eccentricity fault where rd or rs are equal to zero. Therefore, rs defines the severity of the static eccentricity fault. Likewise, rd defines the severity of the dynamic eccentricity fault. If (3.20) and (3.21) are substituted with (3.18), the real value of the air-gap length is obtained as a function of the rotor position, the static eccentricity severity and also the dynamic eccentricity severity. The inverse function of (3.18) is called the inverse air-gap function or the permeance function which plays a vital role in analyzing the eccentricity fault as well as the misalignment fault which is a rather complex version of the eccentricity fault in which the motor shaft bends toward the z-axis shown in Fig. 6(a). This issue will be further investigated.

Depending on the values of rs and rd, one of the following three cases is possible:

(a) rd > rs In this case, the dynamic eccentricity dominates the static one. Using the equations represented so far, the air-gap function, along with its position, can be calculated upon changing the rotor position (see Fig. 7). This figure illustrates such calculation according to which the minimum air-gap length oscillates between two extreme points including g0(1 rd rs ) and g0 (1 rd rs) with respect to the rotor rotation. The frequency of jm is the same as that of the rotor rotation.

(b) rd rs

This situation is usually called a balanced mixed eccentricity fault. The corresponding variations of gmin and jm are shown in Fig. 8. As seen, gmin oscillates within the range g0(1 rd rs), and g0 and jm are limited to the [ p/2, p/2 ] period. Therefore, the minimum air-gap length is eliminated where the rotor position equals one of the odd factors of p. Again, the frequency of jm is the same as that of the rotor rotation.

(c) rd < rs

In the above situations, the static component of the eccentricity dominates the dynamic one, and the corresponding air-gap length looks like Fig. 9.

Accordingly, the minimum air-gap length oscillates between two extreme points including g0(1 rd rs) and g0(1 rd rs) with respect to the rotor rotation. The range of variations of jm is relatively small and it oscillates very close to js. The movement looks like a pendulous oscillation, and the corresponding frequency is again equal to the rotor frequency.

Fig. 7 Variation of the minimum air-gap length (up) and its position (down) with respect to the rotor position in mixed eccentricity fault (rd 0.4, rs 0.2 and js 0)

Fig. 8 Variation of the minimum air-gap length (up) and its position (down) with respect to the rotor position in balanced mixed eccentricity fault (rd 0.3, rs 0.3 and js 0)

Three general cases, namely a dominant dynamic, a dominant static, as well as a balanced mixed eccentricity fault, were discussed. Considering the dis cussed cases, the air-gap length affected by the eccentricity fault could have several different shapes each having its own effect on the motor quantities.

Fig. 9 Variation of the minimum air-gap length (up) and its position (down) with respect to the rotor position in mixed eccentricity fault (rd 0.2, rs 0.4 and js 0)

It should be mentioned that many assumptions are associated with the analysis such as:

--ignoring saturation effect

--ignoring slotting effect

--ignoring end winding effect

--the static or dynamic eccentricity level is applied to the rotor along the z-axis, and the rotor should be displaced from the origin. This means there is still symmetry in z-direction, and all the 2D analyses are still valid in 3D if the thermal issues are not of interest.

__ 3.1 Misalignment inclined rotor

The main focus has been already on a symmetric z-directional eccentricity fault which guarantees the symmetry in the third dimension of the motor. However, in a specific situation, an asymmetric distribution of the motor air-gap length along the z-axis might also be dealt with (see Fig. 10). Although Fig. 10 is a kind of exaggerated representation of the mentioned defect, it easily and clearly conveys the issue according to which the rotor axis Ar, inclines and takes a distance from As.

b is the amount of divergence from the concentric point, but it is totally different from a simple static eccentricity as the rotor ends move in the opposite directions, i.e., one goes up and the other falls down. So along the z-axis, the eccentricity severity of the fault is not constant anymore. In this case, we only refer to a static misaligned/inclined rotor to make sure that the analytical descriptions are easily extractable while a mixed misalignment fault should always take place in reality.

Fig. 10 Statically misaligned/ inclined rotor

Some of the most important causes are as follows:

--bearings wear caused by aging

--bent shafts

--swinging mechanical loads

--inherent assembly and manufacturing defects.

After a long motor operation, these factors result in a mechanical failure of the motor if a proper diagnostic and maintenance process is not held. The misaligned rotor not only harms the motor itself, but also produces an extensively fluctuating movement of loads leading to an improper operation outside the motor as well. This might be the most compelling evidence which necessitates the fault diagnosis procedures. Now, we would like to formulate the air-gap length in a misaligned motor, using the assumption that only the static form of the fault exists and there is no additional swing caused by a dynamic movement of the rotor center. In this case, there is an air-gap length distribution similar to (3.18) but with a variable r along the third dimension (z) of the motor. The air-gap length and consequently the corresponding permeance are the functions of b.

(3.22)

(3.23)

…where rs0 is the static eccentricity fault severity right in the middle of the rotor. Ls is the motor stack length, and z is the distance from the origin shown in Fig. 10.

According to the above equations, the air-gap length is not fixed in different motor cross-sections. For the future analysis, if the permeance function is required, (3.22) should be used but in an inverse form. Incorporating the dynamic eccentricity into the misalignment formulations should be so sufficient that you might not be able to find a suitable theoretical resource in this field. Therefore, it is preferred to skip the philosophical discussion of the eccentricity/misalignment fault at this point and switch to the mathematical and physical descriptions of the phenomenon by working on the impacted motor quantities including the magnetic flux density and the current. Then, we will end up with a pretty closed-form formula associated with the UMP existing in an eccentric induction motor. What is provided there is an intensive analytical representation of the eccentricity fault and its resultant components; then a few motor signals are illustrated to justify the effects in the time domain.

__ 3.2 Theoretical analysis of eccentric induction motor

The basis for providing the materials is introduced by the Ampere's law.

The main assumption here is the presence of a single-harmonic motor model based on which all the healthy motor signals only contain a single sinusoidal distribution.

So the stator current density should also follow the same rule. It is noted that the statement is true if the discussion is held in the steady-state mode. Considering this, the current density of the stator windings can be formulated as (3.24).

(3.24) where Jsm and j are the stator current density (in A/mm^2 ), the synchronous field position (in wst, t is time). jm is equal to pky where k is the inverse air-gap function, and y is the linear distance around the stator circumference. Using the Ampere's circuital law and also neglecting the angular component of the air-gap flux density which is practically a correct assumption, one will end up with the following equation.

(3.25)

where y; t is the permeance function described previously in this section. As seen, the magnetic flux density is related to the current density by an integral operator over the stator circumference. Reforming (3.18)-(3.21) returns, the following representation of the air-gap length demonstrating the components corresponding to the static and dynamic eccentricities:

(3.26)

It is assumed that the air-gap variation is a sinusoidal function if the stator and rotor slotting effects are not taken into account. Otherwise, the air-gap length must be equal to sum of many sinusoidal terms expressing the Fourier transform of the air gap function. This matter will be discussed later. It might also be noted that experiments and investigations show that the above assumption is correct in the case of a nonsalient pole machine. wr, rs, rd and g0 are rotor rotational speed, static eccentricity severity, dynamic eccentricity severity and average air-gap length in healthy condition, respectively. To avoid stator-rotor rub term (rs rd) < 1 must hold. Under small values of eccentricity fault, inverse air-gap function is approximated as follows:

(3.27)

In practice, if the eccentricity fault severity goes beyond 30%, rotor may rub the stator and probably fail to operate appropriately. By combining (3.24)-(3.27), one obtains:

(3.28)

Bs is the stator magnetic field observed at the air-gap level considering the presence of a mixed dynamic eccentricity fault. Multiplying the left cosine term by the terms residing inside the righter parentheses leads to an almost straightforward formulation of the magnetic flux density produced by the stator as follows:

(3.29)

(3.30)

There are five components associated with the calculated eccentricity-related magnetic flux density. From left to right of the left-hand side term, they are accordingly produced by the fundamental component, the static eccentricity and the dynamic eccentricity fault, respectively. It is clearly observable that the static eccentricity components, the ones with the magnitude of Bp 1 sm s m0Jsm 2kpg0 rs, merely depend on the functions of the synchronous frequency and have nothing to do with the rotor speed or position. On the other hand, the dynamic eccentricity fault related components, the ones with the magnitude of Bp 1 sm d m0Jsm 2kpg0 rd, are imposed to be the functions of the rotor position as well as the synchronous speed of the stator. This is another proof of the dependency of the motor quantities on the rotor rotation in the case of dynamic eccentricity fault. The developed and discussed formulation is a pure analytical practice with a lot of imprecise assumptions made just to allow us to extract a closed-form relationship. Therefore, although the magnitudes (see (3.30)) provide a useful common basis to compare the healthy and fault-related components, are not accurate in terms of the absolute values. However, the Bp 1 sm s and Bp 1 sm d are smaller than Bp sm m0Jsm kpg0 , proving the fact that the fundamental component still possesses the largest magnitude among the motor frequency components. As another perceivable fact, the fault-related components are indeed number-of-poles dependent. In fact, two faulty terms are separately assigned to each of the static and dynamic eccentricities. In the case of dynamic eccentricity, these terms are located within a specific frequency range, scaled by (ws wr), of the fundamental component.

So far, the air-gap field produced by the stator current, which in turn applies a specific level of the EMF to the rotor bars, has been evaluated. Inspecting a short circuited rotor circuit, the voltages induced by the mentioned EMF produce the bar currents flowing into the rotor circuits. The same principle is valid for a wound rotor. The point is that the calculations provided so far are based on a stator reference frame while the rotor-related calculations require to transfer the quantities from the stator to the rotor side. As a result, (3.29) should be mapped on the rotor reference frame using a wise substitution of ky wrt ky and wr 1 s p ws with (3.29). The stator magnetic flux density formulated in the stator reference frame is converted to (3.31).

(3.31)

A very clear interpretation of (3.31) is that the rotating frequency corresponding to the magnetic flux density is of course slip-dependent. This arises from the previous transformation from the stator to rotor reference frame. The induced EMF is derived by taking derivative of (3.31) with respect to time.

(3.32)

(3.33)

It is really important to realize that unlike the magnetic flux density, the amplitude of the induced EMF is the function of the motor slip, so increasing the motor slip should practically lead to an increase in the magnitude of the induced EMFs regardless of the fault type [44-46 ]. Thus, the rotor current must increase as well.

This is consistent with the fact that increasing the motor load requires a larger current flowing into the windings and bars as a larger slip is needed. Following the explanations, the rotor bar or winding currents are simply calculated by dividing the EMFs with a factor of rotor bar or winding resistance while neglecting the phase shift between the EMF and the current caused by the inductive nature of the rotor. Then, rotor bar current densities would be obtained as follows:

Equation (3.34) has been developed in rotor reference frame based on the stator supply frequency. a1, a2, a3, a4 and a5 are the phase angle of the rotor bar currents with respect to the induced EMF from the stator side. If it is assumed that the rotor inductance is very smaller than its resistance as rotor electric frequency is a fraction of rated frequency, the amplitude of harmonic components of bar current, i.e., Jrm p , Jrm(s) p 1 , Jrm(s) p 1 , Jrm(d) p 1 , Jrm(d) p 1 , could be simply obtained by dividing the induced EMFs by rotor bar resistance as follows:

where Rbar is the rotor bar resistance in the rotor reference frame. Generally, there are three current components rotating with the frequency of sws, 1 p 1 s p ws and 1 p 1 s p ws. Although this is a very rough approximation, it could be insightful.

The last two frequencies correspond to the static eccentricity fault while the dynamic eccentricity is of the same nature of the fundamental frequency. Rotor bar currents produce their own magnetic fields in the motor air gap and then induce the corresponding MMFs into the stator winding through being modulated by the air gap permeance similar to what has been previously done in (3.28). Working on the formulations and extracting the terms related to the non-supply-frequency components and also the p pole-pair components, we will end up with the following equations:

(3.37) From (3.36), it is seen that the rotor magnetic flux density waves caused by the static eccentricity are regulated with the dynamic component of the eccentricity and produce the sideband current components with the frequency pattern of ( fs fr).

The point is that the components listed in (3.36) do not contribute to the torque developing process as the number of poles is not equal to that of the stator, whereas (3.37) illustrates the torque-generating fault components containing the same frequency pattern as that of (3.36).

The ( fs fr) pattern is the very well-known index to diagnose the eccentricity fault; of course, in the case of a mixed eccentricity fault, both the static and dynamic types have some level of correlation with the mentioned pattern.

This pattern is also a function of the motor slip revealing a non-constant frequency position if the motor slip is changed by any means. Moreover, the corresponding magnitudes are load-dependent deduced from the rotor current densities in (3.37).

It should also be remembered that an UMP which results in additional noise or vibration is an output of any eccentricity fault. Nowadays, due to the lack of engineering importance, very little is known about the subject of noise in electrical machines. All motor designers treated noise mitigation as an art-like cookery. For example, they had a list of low-noise empirical rules and some forbidden combi nations of stator and rotor slots. All that was perfectly clear. However, these are some general laws that do not aim at explicitly describing the faulty motor behavior which is the main focus of the guide. Therefore, due to the significance of the pre-drawn topic, we are seeking to provide a satisfactory on the vibration and probably noise analysis throughout this guide. The starting point is to calculate the magnetic force between rotor and stator by means of the following commonly used approximation of the magnetic force.

(3.38)

where B radial is the radial component of the air-gap flux density all over the stator inner circumference. Considering a mixed eccentric rotor, 4 main flux components other than the fundamental one are present as the spatial harmonics of the flux density according to (3.29). It is worth noting that it is the interaction between the rotor and the stator flux densities which finally produces the net flux density at the air gap. So on the basis of the existing components shown in (3.29), the net flux density should also contain the same harmonics, but with different values of the magnitude and the phase angles. The general formulation should be as follows:

(3.39)

By replacing (3.39) with (3.38) and simplifying the equation, a closed-form expression of the radial forces at the motor air gap is obtained where the fault is present.

(3.40)

The most significant aspects in terms of the above expression are as follows:

--The minimum frequency included in the magnetic force is the same as that of the rotor mechanical quantities, i.e., fr.

--The maximum available frequency is equal to twice the synchronous frequency, for example if there is a supply frequency of 50 Hz, the largest observable frequency will be 100 Hz.

--There are also some medium frequency components including 2fs fr , 2fs fr ,2 fs fr ,2 fs fr and 2fr.

--Not all the frequency components would probably be detected in the vibration signal analysis of induction motors, and it actually depends on the motor structure, the mechanical damping factors, etc.

--Practically, relying on 2fs component to detect the fault is somehow a wrong technique as it cannot be clearly diagnosed most of the time.

According to the literature, the vibration signal is usually considered as one of the most interesting signals available throughout the sensors mounted in the motor body.

This is not only the case in induction motors; and vibration and subsequently noise analysis have been a matter of various investigations in terms of different types of machines. However, we do not aim at discussing the diagnosis procedures in this section; they will be dealt with in the next sections. For the time being, it seems terrific to have an insight into the real motor quantities including the current, the torque, the speed and even the motor flux when the eccentricity fault exists.

Therefore, we aim at providing finite element (FE) results of 28-bar, 36-slot and delta-connected induction motor to which 30% dynamic eccentricity fault is applied. A 2D FE analysis is conducted as a complete 3D one will be so time consuming that it might not be handled by means of low-power computational devices. A quasi-2D model in which several slices of one single motor in z-direction are modeled, paralleled and simulated at the same time is also an appropriate alternative of a 3D operation. However, there is still a high computational demand compared to 2D simulations. Considering these points, the motor is modeled in 2D. This is an acceptable approximation as long as the stack length is not small compared to the stator diameter. If so, the z-axis component of the motor quantities would be symmetric to that of the 2D simulations to a great extent.

Otherwise, the z-axis should be definitely modeled due to the fact that z-axis components play a vital role in motor operation. With this in mind, a group of results illustrated in Fig. 11 in terms of an eccentric motor are provided.

Fig. 11 shows the motor quantities in the no-load and full-load conditions.

The simulated motor has already been discussed in Fig. 4. The difference is that the skewing effect resulting in a smoother response as well as a larger motor rise time has not been considered in the FE-based represented plots in Fig. 11.

Therefore, it is expected to observe more fluctuations in terms of the variables illustrated in Fig. 11 compared to Figures 3.4 and 3.5. Significantly, skew effect is always used to reduce the magnitude of undesirable higher order spatial harmonic components caused by the slotting and nonsinusoidal distribution of winding effects while the magnitude of the fundamental component is reduced accordingly as well. So it is obvious that the motor torque production capability degrades by decreasing the fundamental magnetic field which leads to an increase in the motor rise time. These comments are made here to prepare minds for the next sections in which a comprehensive discussion on the analytical and FE-based modeling of the healthy and faulty induction motors is going to be included. So different aspects including the winding topology, the skewing effect, the material modeling, and the fault modeling will be further discussed. Now, take a look at the following features extracted from Fig. 11.

Fig. 11 Line-start motor variables (a) current, (b) torque, (c) speed, (d) flux and (e) force magnitude in full- and no-load condition, under a dynamic eccentricity of 30%

--Then again, the current decreases considerably by reducing the load. It gets more and more small up to the no-load current in which the steady-state cur rent signal diverges from a pure sinusoidal curve and turns into a flat top signal. The situation gets worse if the phase current of the motor is investigated. If a 3D model is used, the shape will be certainly more sinusoidal due to the fact that the skewing effect is incorporated. Otherwise, the uncomplicated model should always reveal higher signal ripple. However, a flat top no-load signal is expected even in practice. This is caused due to the presence of the harmonic components, which are powerful enough to change the shape of the signal other than the fundamental one. In view of the fault-related oscillations, the current envelope is not a good medium for reflecting the eccentricity fault as the effect is not clearly distinguishable. It, of course, leads to some sort of fluctuations, not very useful to follow up the fault.

--The motor rise-time is considerably larger than that of Figs 4 and 5.

This evidently proves the claim based on which the fundamental component of the magnetic flux is weakened if the skewing effect is applied. In Fig. 11, the skewing effect has been completely ignored.

--The time-domain representation of the motor signals, except that of the flux, clearly conducts the distinction of the no- and full-load condition in terms of the eccentricity fault and its effect on the motor. In contrast to the broken bar fault, connecting a larger load to the motor shaft lowers the impact of the eccentricity fault on the time-domain signals. In other words, increasing the load damps the oscillations caused by the eccentricity fault.

--The effect of the sideband components produced by the eccentric rotor and discussed throughout (3.24)-(3.39) is not properly distinguishable in time domain signals. This might be because of the smaller magnitude of the eccentricity fault compared to that of the broken bar fault. Notably, the side band components associated with the eccentricity fault are not generated or produced by the fault. They are already there in the motor structure while their magnitude is changed if a fault takes place.

--The transient mode is apparently the best candidate for demonstrating the differences between the no- and full-load conditions. The differences are less observable in the steady-state operation. According to the transient parts of the signals, the amplitude of the fluctuations in the no-load condition is more than that of the full-load condition. This conveys the fact that the motor load acts as a damping factor in terms of the mechanical faults. This is also true in terms of the steady-state analysis. However, the simulated FE model, due to the absence of the skewing effect, does not represent the idea.

--A very interesting plot is the one related to the magnetic force applied to the inner surface of the stator as shown in Fig. 11(e). The applied force of the no-load motor is obviously more fluctuating than that of the stator in the transient mode of operation. This is the net value of the applied force to the stator surface.

Fig. 12 Bearing structure

Generally, time-domain signals do not provide a reliable fault detection approach in terms of the dynamic eccentricity fault. They can only be considered as tools to detect a defection or improper operation. Nevertheless, some esoteric fault indicators such as the Gyration Radius (GR) address some unknown time-domain aspects of the eccentricity fault. However, these kinds of indicators are not popular anymore nowadays. A more reliable frequency-based diagnosis is usually preferred.

Having discussed and explained the eccentricity fault's commitment to the unsafe and unreliable performance of the motor, we now prefer to concisely talk about one of the major causes of eccentricity which is called ''bearing fault.'' Bearing fault itself produces additional noise and consequently vibration in the motor body. Depending on the location of fault, inner or outer race, various frequency patterns are provoked and used to detect the fault, but what is interesting is the contribution of this fault to the future eccentricity fault, and this is exactly what has been greatly focused in the literature.

__ 3.3 Bearing faults in induction motor

Bearings are indeed one of the most important parts of an induction motor in maintaining the reliability and safety of the machine performance. They should be checked constantly to make sure if the motor operation will not fail. What makes them highly significant is their mechanical operation and being subjected to higher levels of friction leading to wear and tear of their internal and external part. In fact, no rotation exits unless something holds the rotor concentric with respect to the stator and this crucial task is handled by bearings which not only take care of maintenance of a symmetrical operation but also tolerate the rotor weight which might run over hundreds of kilograms. The studies show a considerably high percentage, almost 40%, of contribution of bearing fault to induction motor failures.

Fig. 12 shows a general bearing structure which consists of the following main parts:

--the outer raceway

--the inner raceway

--the balls

--the cage.

More often, three types of faults are associated with bearings as follows:

--outer raceway defect

--inner raceway defect

--ball defect.

Noticeably, any bearing defect is of a mechanical nature which makes the motor vibrate due to the presence of a radially unbalanced force caused by the defect. This statement reminds the term UMP which was previously discussed in terms of an eccentric motor. So the final result of any bearing defect is somehow converted to an eccentricity type of fault based on which some principle harmonic components are used to detect the fault. Resulting from an eccentric-like fault, it is expected to observe a general frequency pattern of f bearing fs m fi;o

(3.41) where m 1, 2, 3, . . . and fi,o is a frequency characterizing and modulating bearing dimension into the motor variables. As a very well-known practice, the corresponding value is obtained as follows:

(3.42)

where Nb,fr,Db,Dc and b are the number of bearing balls, the mechanical rotor speed in Hz, the ball diameter, the bearing pitch diameter and the contact angle of the balls, respectively. The mentioned frequencies are the so-called ''characteristic frequencies.'' Although different bearings produce different characteristic frequencies, depending on the number of ball and their dimensions, as a rule of thumb, the corresponding values can be calculated as follows if the number of balls is between six and twelve.

(3.43)

Although the simplification is brief, it is certainly convincing and no more discussion seems to be necessary at this point. Just as a hint, vibration and noise analysis is one of the best approaches to detect this kind of fault.

__ 4 Short-circuit fault in induction motors

One of the phenomenon which has been carefully addressed so far is the problem of short-circuit fault in induction motors. Due to the dramatically wide application of induction motors in industry and also bring up the safety features among which the short-circuit fault relays or indicators are the most significant ones, we also aim at including this topic in the guide. This type of fault is interesting enough due to the fact that it might similarly happen in both rotor and stator if a wound rotor is targeted. Otherwise, it is restricted to stator side if a cage induction motor is analyzed. Although the wound-rotor motors are less requested, they are promising apparatuses to provide a higher starting torque respecting to a minimum starting current. However, in addition to the stator, the short-circuit fault can also take place in the rotor. The literature shows that the effect of the short-circuit fault in the stator is very higher than that of the rotor.

According to the investigations, almost 37% of the induction motor faults are related to its insulation failure leading to a short-circuit fault. As a result of this kind of failure, the current density of coils/windings runs over the rated values dangerously causing a hot spot area around and through the faulty coil/winding.

This is one of the major sources of degradation and also aging of motor coils/ windings.

In general, there are specific types of this fault including:

--turn-to-turn

--turn-to-ground

--phase-to-phase

--phase-to-ground.

Regardless of the type, the nature of the abovementioned faults is the same.

Actually, a considerably high current level circulates inside the windings without contributing to the torque production capability of the motor. So, it is considered as a kind of loss. The only difference is the severity of the fault which increases with the increase in the number of short-circuited turns as well as the decrease in the short-circuiting path resistance. Ignoring the fault and letting it progress might also cause an irreversible damage to the motor tank and bearings. Therefore, diagnosis of the short-circuit fault is of great importance.

The following are some of the main reasons of generating the fault.

--Thermal stresses: These are produced by thermal aging or over-loading.

Respecting to a 10 C of increase in temperature, insulation lifetime reduced by a factor of two. A better class insulation is recommended if the motor is going to be used in a highly stressful environment.

--Electrical stresses: This type of stress is usually classified into two general categories, namely the insulation breakdown and the partial discharge. When a relatively large voltage variation rate, i.e., dv=dt, is applied, insulations are subjected to a breakdown and destruction. This is usually the case where the voltage goes beyond 5 kV.

--Mechanical stresses: This kind of stress is predominantly caused by several start-stop operations of the motor causing frequent warm-up and cool-down of insulations. As a result, cracks are produced and become larger if start-stop operations are not avoided.

--Environmental stresses: External and polluting substances can also disturb the motor operation. For example, pollution might cause an improper thermal exchange between motor and surrounding environment and consequently it leads to motor temperature rise. This in turn increases the risk of electrical failure of coils/windings.

Many efforts have been already undertaken just in case of analyzing different aspects of short-circuit fault. We are going to focus more on the original fault related harmonic components which are the results of the short-circuit fault. like all the other types of faults, the natural frequencies which are used for detecting the short-circuit fault are present in the motor structure and are the functions of the number of poles, the number of slots as well as the saturation profile of the machine. The presence of the short-circuit fault only changes the amplitude of the harmonic components and it has nothing to do with creating new components.

This was also the case in previous types of faults including the broken bar and eccentricity faults. So, it is time to address a part of principle harmonics generated by geometrical placement of the rotor and stator slots and windings. These harmonic components are called principle slot harmonics (PSH) which are some times used to detect a specific type of fault. Both stator and rotor introduce their own PSHs.

It is very well known that induction machines, of course in healthy case, consists of a sort of MMF components formulated as (3.44):

(3.44)

where p is the number of pole pairs, ws is the fundamental angular frequency and m 6g 1, g 0, 1, 2, . . . . Assuming that the number of rotor loops is n, the MMF component corresponding to the first rotor loop is obtained in the rotor reference frame as follows:

(3.45)

where Irmax is the magnitude of the rotor bar current. Following the same fashion, the MMF component of the neighboring rotor loop is analytically derived as follows:

(3.46)

It is worth noting that the upper index of the summation is set at infinity due to the fact that the rotor loop MMFs looks like a spike which can only be modeled accurately by incorporating infinite terms of its Fourier transform. The total MMF produced by rotor is calculated by adding up the loop MMF.

(3.47)

On the basis of the derived closed-form representation of the rotor MMF waves, it is clearly observed that MMF waves are only present for the cases u p; u p ln and u p ln; l 1; 2; 3; ....As u can only take positive integers, it follows that only for u p and J ln p the MMF waves exist.

Therefore, in addition to the fundamental rotor harmonic component for u p which deals with the armature reaction of the fundamental component of the stator, the rest obtained by u ln p demonstrate the rotor slot harmonics (RSH).

Regarding the stator reference frame, the RSH is expressed as follows:

(3.48)

Coupled with previous formulation, the higher frequency MMF waves produced by higher order components are given by:

(3.49)

The corresponding frequency is the same, but the number of effective pole pairs differ. Multiplying the MMF components by the air-gap permeance, which is a con tent term in a healthy idea motor, returns the magnetic flux density wave which produces the EMFs and subsequently the currents in the stator. From (3.48) and (3.49), it is comprehensible that the stator EMFs and currents will only contain the additional slot frequencies 1 l n p 1 s fs which are now the time harmonic components extractable using a Fourier transform. As seen, the spatial components introduced by the slotting effect are eventually reflected into the time-domain signals.

Under short-circuited turns or phases, a new set of waves will apply changes to the stator MMFs described as:

(3.50)

Accordingly, there exist the MMF waves at all numbers of pole pairs and direction of rotations. One of the waves rotates with the same frequency of the fundamental component but in an opposite direction. If a normalized Fourier transform is used, no change will be observed in the current spectrum as it is always normalized to zero. However, there should be a change in the magnitude of the RSH. No new frequency should be added to the current spectrum if the short-circuit fault takes place in the stator. The fault only contributes to an increase in the RSHs.

In the first instance, consideration is given to the simplest case where the coil has only one turn from which it is possible to draw some important conclusions.

Fig. 13, however, shows one-phase group of three coils. It is assumed that interturn short-circuit arises between points a and b, as illustrated. It is clear that the circulating current has a closed path. From simple theory, it is clear that the path A-X can be expanded to two independent circuits. From Fig. 13, we can say that the phase current and the current which flows through the short circuited coil, produce opposite MMF. Therefore, interturn short-circuits have a cumulative effect in decreasing the MMF in the vicinity of the short-circuited turn(s). First, when a short-circuit occurs, the phase winding has less turns and less MMF. Second, the MMF of short-circuited part is opposite to the MMF of the phase winding. Clearly, interturn shorts with more turns can be analyzed in a similar manner.

In most commercially available induction motors, coils are insulated from one another in slots as well as in the end winding region. Therefore, the highest probability for the occurrence of interturn fault is between turns in the same coil.

Fig. 13 Short circuit between turns a and b.

Here, it is assumed that the interturn short-circuit is between two turns in the same coil, and that one-half of the coil is short-circuited; this means that approximately 8% of turns of one phase are short-circuited. Simulation was carried out for this condition.

As a consequence of the interturn short-circuit, the MMF of the phase winding in which interturn short circuit exists changes, as does the mutual inductance between that phase and all other circuits in the machine. In addition, a new ''phase,'' which we call the short-circuited phase D is introduced. It should be assumed that for modeling, this phase has no conductive contact with other phases, but it is mutually coupled with all other circuits on both the stator and rotor sides.

The currents in stator circuits and rotor loops are assumed independent.

The machine with the following specifications is analyzed using the numerical model based on multiple coupled circuit approach and winding function analysis.

3kW; 415V; D; 50Hz; p 3 six poles machine S 36 stator slots R 32 rotor bars Stator phase winding consists of 6 coils, 1 coil per pole with 77 turns in one coil, i.e., N 462 series turns per pole per phase. Stator phase A winding scheme is:

A 1 6 12 7 13 18 24 19 25 30 36 31

The connection diagram of the stator windings of the experimental machine is additionally shown in Fig. 14.

Fig. 14 Stator winding scheme of experimental motor. Interturn short-circuit occurs in coil placed in slots 1 and 6

--------------- The machine parameter:

Motor is loaded with 30 N m under the steady-state condition. Fault is made on such manner that 38 out of 77 turns is short-circuited in one stator phase coil, under one pole. We provide an interesting illustration of the healthy and faulty motors for different load (see Fig. 15) By referring to Fig. 15, it is observed that

--Increasing the load level leads to an increase in the motor current regardless of the short-circuit fault.

--The current reveals an increasing trend in both the transient and steady-state region upon the fault occurrence.

--The average developed electromagnetic torque is almost the same for the faulty and healthy conditions. However, the faulty motor contains some sort of low frequency harmonic components carried by the average value. This clear illustration of harmonic components in time domain is only achievable by the analytical models such as the winding function theory or the magnetic equivalent circuit. If an FE model shown in Fig. 11 is used for simulations, the torque ripple caused by slotting and saturation usually dominates the fault component unless the fault severity becomes very high.

--Similar oscillations are observed in the speed. Surprisingly, the average steady state speed decreases by the fault. This increases the power losses caused by the slip.

Fig. 15 Line-start motor variables (a) current, (b) torque, (c) speed, (d) flux and (e) phase D current in full- and no-load condition, healthy and faulty motor signals

--The phase D (short-circuited phase) current is considerably more than that of the rest of the windings. The current circulates and makes the thermal stress so worse that the other healthy insulations also fail to operate well.

--Not all the motor signals get larger by the fault. The phase D current marks down probably due to the increase in the armature reaction in the rotor side.

So far, a lot of contexts have been mentioned in terms of the motor harmonic components, the PSH, the fault-related harmonic components etc., without directly addressing how these frequency-domain signals might be observed or studied. A couple of spectra regarding the motor current signal are put forward without going in depth of how they are calculated to explain first what a spectrum is and second how different frequency analyses are performed by means of a frequency spectrum.

To this end, refer to Fig. 16.

Fig. 16 The line current spectrum (a) healthy and (b) faulty motor

The output of a Fourier transform is usually demonstrated by frequency and magnitude planes. Normally, the horizontal axis is an indication of frequency contents of the processed signal, for example the current and the vertical axis shows the corresponding power or the magnitude. Therefore,

--The synchronous frequency, i.e., 50 Hz for the tested motor, possesses the largest magnitude in the spectrum. That is why it is called the fundamental component (see Fig. 16(a)).

--The rest of the spectrum drops down the mentioned fundamental component in terms of the magnitude. This distinctly explains the lower interest of higher order components in forming the shape of the current which is the processed signal.

--The entire spectrum is an almost featureless curve representing negligible information in terms of the higher order components. This arises from the fact that the number of rotor bars and pole pairs are equal to 32 and 3, respectively.

So none of the principal RSH could be expected. In order that lower rotor slot harmonic exists in stator current spectrum, the following condition should be fulfilled:

(3.51)

--For p 3 and n 1, 2, . . . , the lower RSH (RL_RSH) is equal to 24, 42, . . . . On the other hand, the condition for the existence of the upper RSH is satisfied for rotors with the following number of bars.

(3.52)

--This leads to RU_PSH 12, 30, 48, . . . for a six-pole machine.

However, in a faulty motor, as a consequence of interturn short circuit, there are all harmonics of ''phase'' D MMF. In other words, in the following magnetic flux density waves,

(3.53)

The order of space harmonic n could be u 1/3, 2/3, 1, 4/3, . . . So in general, the RSH could arise in current spectrum ( Fig. 17).

For u 1/3 and l 1 lower rotor slot harmonic exists:

(3.54)

(3.55)

Fig. 17 MMF of ''phase'' D - as analyzed motor is motor with p 3, fundamental harmonic n 1 means six poles. In case of short circuit in one coil, it means that phase D also produces subharmonics

(3.56)

(3.57)

Therefore, in healthy machine with S 36, R 32 and p 3, none of RSHs exists, as it could be seen in Fig. 16(a). However, in case of interturn short-circuit, both of RSHs exist in the line current spectrum, i.e., Fig. 16(b), and they are very prominent. This effect could be additionally amplified by permeance harmonics waves:

(3.58)

(3.59)

(3.60)

In addition, new harmonic component in current spectrum appears due to the saturation phenomenon.

(3.61) now could attain 1 for u 1, u 2 1 2 1 (3.62) which means that new harmonic component at 150 Hz could appear as a result of the fault due to the following flux density wave:

(3.63)

However, this component does not exist in models in which the saturation is neglected. To validate the results from the dynamic model, an experimental investigation was conducted. The experiment was performed in the following manner. A standard commercially available motor was dismantled and isolation of the few turns from the same coil (in the end region) was mechanically injured, i.e., scratched. These spots were soldered conductors which were taken out from the motor. Short circuit was made between these conductors. Therefore, turns were shorted externally. By measuring EMF between conductors and having known winding details, we were able to conclude how many turns in one coil were shorted.

In the case of inrush current which was experienced particularly when two turns which are in the neighborhood (in the electrical sense) were shorted, the short circuit current was reduced by means of an externally placed resistor. In these cases, current was limited to the value of double-rated current for a short time during the experiment.

Fig. 18 shows the spectra of line current for a loaded machine for a healthy and a faulty condition, respectively. Fig. 18(a) shows that in a healthy machine, frequency components, the result of the saturation of magnetic material (150, 250, 350 Hz, etc.), exist. In the healthy condition, only the upper rotor slot harmonic is visible at 568 Hz (s 2.8%). From Fig. 18(b), it is clear that as it is predicted in the simulation model, the most significant changes arise at harmonic components of (1 lR(1 s)/p)fs. Now, the lower rotor slot harmonic (at 469 Hz) is prominent and the upper harmonic has risen. Moreover, the 150-Hz harmonic component is considerably higher under the fault condition.

Fig. 18 Current spectrum of a line start motor at slip 2.8% (a) healthy and (b) short-circuit fault

What has been already proposed in terms of PSHs existing in induction motors is actually the basis for future analysis not only in the case of short-circuit fault but also in the case of other types of faults. Furthermore, the appreciable practice of the frequency-domain analysis of the motor current was extracted and analyzed. What is not really deductible about faults in a time-domain analysis can be somehow easily detected, followed and analyzed by means of a frequency domain. The PSHs related to the short-circuit fault are the best examples of this kind and in the same fashion, the broken bar and eccentricity fault will be addressed shortly in the next sections. Considering the mentioned points, three apparently distinct but inherently correlated processing domains including time, frequency and time-frequency ones are going to be discussed in Sections 8-10. In the current section, the time-domain variations are emphasized more while the frequency analysis will be further studied and combined with the time-domain information to introduce a generalized solution to fault diagnosis challenges.

Measuring the magnetic, electrical, mechanical and partly thermal quantities of the motor, we will also go directly through the property of time and frequency components of every single motor variable to see how different faults apply changes to motor variables, how the fault information could probably be extracted and how the extracted information is used for a precise detection, determination and diagnosis procedure.

cont. to part b >>>