Theory of line-start and inverter-fed induction motors

1. Introduction

Induction motors, which hire fundamental rules of electromagnetism in order to produce rotational or translational mechanical movement when supplied by an input electric power, are energy conversion apparatus. These electro mechanical devices are very well known for their smoothly run capability if a balanced input power supply, along with a symmetric motor structure, is used simultaneously. This is the reason why a balanced and symmetric three-phase motor supply system is usually preferred in industry due to its gentle and low noise.

However, a higher number of phases, despite its rather higher topological complexity, are also possible. Notably, the underlying idea of the operation of this kind of motor is basically not related to different possible structures used in various applications. It quite depends on the nature of the induction phenomenon which couples the stator and rotor magnetic fields. Regardless of the stator geometry, there are generally two types of induction motors as the following:

--wound rotor

--squirrel cage rotor.

The first type usually presents a less reliable structure in terms of mechanical stresses as well as the electrical deterioration of the brushes connecting rotor windings to the output terminals. The wound-rotor structure is relatively fragile compared to the second type, i.e. the squirrel cage rotor, while revealing a more complex topology that requires a relatively greater challenge for placing windings than casting rotor bars into the slots. On the other hand, the cage rotor presents an easy-to-build topology which merely requires a casting process to build complete bar and end-ring connections playing exactly the same role as brushes in a wound-rotor induction motor. Due to the robustness of a cage rotor, it is mostly utilized as a medium not only in industrial but also in domestic environment since the corresponding maintenance process is somehow less demanding regardless of the corresponding types of faults introduced by the presence of the cage. In addition, absence of brushes is a very important factor in choosing a cage rotor against a wound rotor.

The stator can be the same for both types of the rotors. It is generally a symmetric winding topology that encloses the supply current applied by the supply voltage with the goal of generating a rotating or translational magnetic field that induces currents into rotor windings or bars. On the other hand, thanks to the advances in power electronics devices, supply voltage can be precisely controlled in terms of both amplitude and frequency. As a result, many distinct operations of induction motors are observed and different electrical, magnetic and mechanical characteristics and motor profiles are obtained by means of controlling the frequency and amplitude of the supply. Moreover, any motor signature is subjected to change due to a fault occurrence. Thus, understanding the signatures of a healthy motor should be the very beginning step of any fault diagnosis process since faults introduce their own signatures which might be totally different. Three approaches including mathematical developments, simulation-based analysis and experimental measurements are used to comprehensively address induction motors behavior in terms of electrical, magnetic and mechanical quantities in the healthy case, but before moving forward, a structural analysis of a real induction motor is provided in the next section of this section to help beginners of the field to understand and find out how different parts are assembled. Then, we will step toward developing fundamental mathematical formulations. It should be noted that this section only focuses on the healthy motor operations, and further discussions in terms of different kinds of fault and the corresponding consequences are discussed in the next sections. Therefore, this section focuses on the analysis of the operating principals of healthy induction motors and addressing electrical and mechanical aspects of the motor by means of a preliminary linear analytical model. Then, the model is improved with the goal of incorporating a more comprehensive magnetic aspect by considering the winding distributions. Finally, a detailed harmonic analysis of the healthy induction motor which is a turning point of the study from a conventional one to an advanced investigation of the motor behavior is proposed.

Generally speaking, the healthy motor behavior in the ''line-start'' supply mode is targeted. Therefore, readers could potentially follow up future details in terms of more complex operations such as inverter-fed applications.

Otherwise, the advanced topics might be of a little bit vague if one does not have the proper background knowledge. All the motor-drive principles required for a better realization of motor behavior are investigated as well. As the starting point, the physical structure of the motor is first illustrated and analyzed in the following section.

Fig. 1 (a) Cut-view of an induction motor and (b) different rotor types

Fig. 2 (a) A 48-slot stator and (b) stator-winding layout, a three-phase D-connected stator with 36 slots

Fig. 3 Structure of a wound rotor and its connections with the brushes

2. Induction motor structure

This section provides a brief explanation of the main parts in Fig. 1 which illustrates a complete induction motor structure with two different wound and cage rotor.

The stator, as the combination of the nonmoving (stationary) parts supporting the motor, basically includes the magnetic circuit parts, and it carries the windings located inside the slots in a symmetric manner with the goal of generating the initial magneto-motive force (MMF) of the motor at its air gap level. The stator is usually made of insulated steel laminations sorted side by side to navigate the magnetic flux density produced by the windings through proper paths while reducing the magnetic losses inside the core material. Magnetic losses are generally divided into two distinct categories: the hysteresis and eddy losses which are related to the nonlinear magnetization phenomenon of laminations and the magnetically induced electrical currents in separate laminations, respectively. The three-phase stator windings are inserted into the slots of the stator and cover the inner circumference of the stator by a quasi-sinusoidal spatial distribution that guarantees a less spatial harmonic distortion of the motor quantities such as current, flux and electromagnetic torque (see Fig. 2(a)). Each winding consists of several series-connected coils, and each coil can be a solid wire or parallel stranded wires. Increasing the number of coils, as well as the number of turns, started in slot i and ended in slot j (see Fig. 2(b)), improves the torque of the motor while there should be a trade-off between the amount of the output torque and the increase in the winding resistance which, in turn, reduces the motor efficiency. It is noted that the standard routines ask to assign the letters (U, V and W) or (A, B and C) for the starting points of the three phases/windings and the letters (U ,V and W)or(A ,B and C ) for the end point of the corresponding windings. The terms ''phase'' and ''winding'' might be used interchangeably. The windings present a similar distribution but with a spatial shift in the neighboring slots. Each winding has two ends called ''terminal'' to which the supply voltage which produces the magnetic field is applied.

The three-phase stator windings can be Y or delta connected. The type of connection can be changed by changing the configuration of the connection of the six available terminals shown in Fig. 2. The amount of voltage which can be applied to the windings highly depends on the torque production capability of the motor as well as the quality of insulations. 230VD-400VY or 400VD-690VY are some of the most common voltage standards in industrial zones.

As shown in Fig. 1(b), the main difference between two types of the rotors, the wound and cage rotors, is related to the apparatus which carries the electrical current induced from the stator to the rotor. What plays this role in a wound rotor is a three-phase winding which is inserted symmetrically into the slots of the rotor while it is handled by the bars, normally made of aluminum, in a cage-rotor motor.

Fig. 3 illustrates the structure of a wound rotor along with the electrical connections of its terminals with the brushes. In fact, there would be no current flowing inside the rotor windings unless the rotor consists of an electrically short-circuited system. If the current is entirely zero in the rotor, the motor operation fails. Generally, the rotor windings possess a Y (star) connection, and only one terminal of each winding is available for purpose of resistance adjustment. This is a major task which is held to control the start-up torque by means of an adjustable resistor bank.

The resistor bank is connected to the rotor windings and changes the total resistance of the rotor. By increasing the resistance, the start-up torque increases as well. The symbols Ur, Vr and Wr represent the rotor phases/windings. Due to the weakness of the brushes and also presence of an additional electrical connection between the rotor and the equipment outside the motor, the wound rotor is not usually preferred in ordinary application. Instead, a squirrel cage rotor is mostly selected due to its solid structure and connection-free nature ( Fig. 4). In fact, the wound rotor is very fragile while the cage rotor reveals interesting robustness in a real-world application. However, the new technologies are beneficial of specific types of wound-rotor motors that are called doubly fed induction motor with the goal of controlling both the stator and rotor circuit currents by means of inverters. The only shortcoming of a cage rotor is its inability to provide an adjustable terminal resistance required for a strong start-up of the motor. Once a cage rotor is casted and built, its structure is totally enclosed inside the motor, and there is no access to it. However, it makes the structure more robust against stresses.

Fig. 4 Structure of a squirrel cage rotor.

The common feature of two types of rotors shown in Figures 2.3 and 2.4 is the skewed rotor slots which are intended to reduce the total harmonic distortion (THD) of the motor. Although the skew reduces also the fundamental motor torque component, it is always proposed to keep a specific level of skewing in order to have a smooth operation of the motor. Otherwise, the torque ripple increases. The torque ripple highly depends on the configuration of the windings, the number of slots and the number of poles. The number of poles is indeed an influential factor in deter mining the motor characteristics, particularly the supply frequency and also the output speed. This topic will be further discussed in the next sections of the current section. Like the stator, the rotor also consists of laminations (see Fig. 4) to reduce the magnetic (iron) losses. The end-rings shown in Fig. 4 play the same role as the brushes in a wound rotor. During the manufacturing process, both the bars and the end-rings which are located at the two ends of the rotor are casted together simultaneously to have an integrated unit of rotor electric circuit. There are also some other components, shown in Fig. 1, which deal with holding the entire motor and cooling it. These components are as follows:

--motor housing

--chassis

--bearings

--fans

--cooling fins.

The motor housing is the outer part which is electrically grounded by means of an electrical connection in order to prevent injuries. Mostly, the housing consists of the cooling blades, so-called cooling fins, through which the air generated by the cooling fan flows. Bearings are located at the two ends of the housing which holds the rotor.

They consist of inner and outer races in the middle of which the balls rotate while the rotor rotates. The bearings are the main sources of mechanical types of faults in induction motors. The eccentricity fault is a very good example of this kind of fault which is the result of an improper placement of the bearings or defect of balls. So the bearings must be monitored all the time to avoid future mechanical faults.

In order to provide a better demonstration of a rotor, an equivalent electric circuit that looks like a resistance network, as shown in Fig. 5, is used for a rotor with 28 slots/bars. The horizontal and vertical resistors represent the equivalent circuit of the end-ring and the bars, respectively. There are also inductances connected in series to each resistor. Therefore, the resistor must be replaced by resistive-inductive impedances. However, for the sake of conciseness, the resistors are used as the candidates to represent the impedances. The number of the vertical elements is the same as the number of bars and that of the horizontal elements should be equal to number of bars 1. The horizontal elements take care of short-circuiting the rotor bars so that the rotor current can be built-up. Any electrical disconnection of joints of the elements shown in Fig. 5 results in a rotor fault. For example, if the joint of one bar and one end-ring is broken, the bar current is zero. Consequently, the balanced operation of the motor is violated.

Fig. 5 Equivalent resistive network of the rotor

The discussed short-circuited equivalent network rotates around the inner surface of the stator with the same speed as that of the rotor. The movement across the stator windings induces electro-motive forces (EMF) in every bar or winding inserted inside the rotor slots. Consequently, an electric current is induced into the rotor conductors, so the electromagnetic torque is developed as a result of the interaction between the stator and rotor MMFs. This is the fundamental principle of the operation of induction motors and will be fully described mathematically in this section. The most critical part of the rotor circuit is the joint between the rotor bars and the end-rings which is produced during the casting process. This is exactly where the broken bar fault occurs. As there is no such connection in the wound rotor motor, the broken bar fault has no meaning in wound rotor case. Tbl. 1 characterizes the differences between the two types of induction motors.

Getting familiar with the fundamentals of the operation of induction motors is indeed a must-do step since any fault-related analysis will be useless unless readers have the idea of how motor quantities are in the healthy and normal condition.

Therefore, to conduct the analysis, a single-harmonic motor model is investigated here in this section. Then, a more advanced study on the principal slot harmonics of the motor is introduced. These two steps prepare the reader's mind for our future efforts in terms of how motor quantities might be affected by various faults.

The only required background is the concept of the electromagnetism.

Tbl. 1 Characteristics of wound- and cage-induction motors

Wound-rotor-induction motor

Slip ring and brushes make the construction much more complicated

Windings are the same in the stator and rotor sides

Rotor resistance is adjustable

Starting torque is improvable

It is rarely used

Lower efficiency due to the larger rotor copper losses

Cage-rotor-induction motor

Possess a very simple construction

Instead of windings, solid bars are replaced inside the rotor slots

Rotor resistance is fixed

Starting torque is fixed

Due to its simple structure and less maintenance requirements, it is mostly preferred

Higher efficiency due to smaller rotor copper losses

3. Line-start induction motor: linear and single harmonic model of a healthy motor

Analytical representation of induction motors is fundamentally initiated by assuming that the motor consists of the spatial harmonic components that are the result of an entirely sinusoidal winding distribution across the stator and rotor circumferences. Therefore, for a three-phase motor, one can assume that the rotor conductors, whether the wound rotor or the cage rotor, act as a balanced three phase configuration demonstrating three similar field vectors (see Fig. 6). The stator and the rotor are each identified by three identical windings (phases) which must be practically located 120 mechanically away from each other. This situation is shown in Fig. 6. With this intention in mind, one concentrated-at-one-point winding is ideally assigned to each stator and rotor winding. In the case of a cage rotor, it is for now assumed that the behavior can be equivalently modeled by the three-phase windings of the rotor (see Fig. 6).

The stator phase axes are fixed with respect to the origin of the rotation which is mostly assigned to the direction of the stator winding/phase A. However, the rotor windings rotate during the motor operations. Therefore, its corresponding angle is measured by means of the rotor mechanical angle (qm) which is equal to 1=p (qe), where p is the number of motor pole pairs and qe is the electrical angle of the rotor.

The next step is to make a couple of assumptions to eliminate the dependency on nonlinearity of the core material, thermal variations as well as the geometry.

--Motor is healthy.

--Nonlinearity of the stator and rotor core materials is neglected. Moreover, the core losses including the hysteresis and eddy losses are ignored.

--Every side of the windings and bars of each phase can be modeled as a concentrated-at-one-point winding. Therefore, the slotting effect is neglected. As a result, the air gap is smooth and uniform.

--Thermal variation is neglected.

--End-windings are the two ends of the coils outside the stator or rotor cores and their effects are neglected.

--Skewing effect is neglected. Otherwise, a 3D model must be developed.

Nevertheless, developing an analytical 3D model is a very advanced topic that cannot managed in a simple way in this section.

Fig. 6 Three-phase equivalent circuit of an induction motor

Fig. 7 Configuration of a line-start supply mode of an induction motor

Fig. 8 Balanced three-phase voltage system

These are the essential in producing a perfectly formulated analytical model of an induction motor which has been greatly used to analyze the healthy motor behavior in time domain. This also provides us with a promising tool for describing how an inverter-fed motor is and should be controlled.

The line-start motor model is the starting point of developing the analytical solutions to the analysis of motor operation. Therefore, a common line-start configuration is presented in Fig. 7. The line-start mode is the conventional approach of starting up and supplying an induction motor in industry. This network supplies a three-phase voltage system, shown in Fig. 8, which is formulated as follows.

(2.1)

vsa, vsb and vsc are the stator phase voltages as shown in Fig. 6. In a symmetric and balanced voltage system, all the phases exhibit the same amplitude but different zero-crossing points. The zero crossings determine the electrical phase shift of the three phases. 2 pi fs is the electrical angular frequency of the supply voltage, the so-called ws x fs [“w” is Greek letter omega, lower case ] is fixed for any line-start application while the amplitude of the voltages might be easily adjusted by means of an autotransformer which applies minor regulations to the voltages. An autotransformer, seen in Fig. 7, is the only interfering device that might be used to stabilize the supply voltage in order to make sure that all the phases have the same amplitudes (magnitude). It has nothing to do with the frequency of the supply network and only changes the voltage magnitude. Most of the time, transformers consist of an adjusting bottom by means of which all the three-phase voltages are changed simultaneously. According to this discussion, it is quite clear that why this kind of motor supply is called a ''line start'' supply mode. In fact, the motor is directly connected to a three-phase network (line) without any interconnection such as drives. Together with (2.1), the other stator quantities including the current and flux vectors are formulated as the following:

(2.2)

Likewise, a balanced three-phase system is assigned to the rotor windings (phases) (see (2.3)). The rotor currents and voltages are illustrated in Fig. 6. If a cage-rotor motor is investigated, the rotor voltages should be set to zero since the rotor circuit is always short-circuited by means of the end-rings. In the case of a wound rotor, the rotor voltage vector is equal to the voltage drop across the external resistance connected through the brushes to the rotor windings (see Fig. 3).

(2.3)

3.1 Flux equation

This section intends to deal with the linear electrical equations of an induction motor considering some assumptions related to the nonlinearity and thermal variation of the motor's materials. Focusing on the two main parts of the motor, the stator and the rotor, the input energy variation provided by the input three-phase network is split as follows:

(2.4)

In this section, the focus is on the electrical energy since the magnetic energy losses of the motor components, the stator and the rotor, are assumed to be zero. This is an acceptable assumption in the event that one does not look for a precise loss characterization of an electromagnetic system such as induction motors. On the other hand, the electrical qualities, the voltage and the current, are the mediums to develop any electrical energy in an induction motor. In view of the electrical energy, the time-dependent variation of the product of the voltage and current signals should be integrated over time.

(2.5)

Then the matrix form of the energy equation is used to express the total energy variation caused by all the windings of the stator and the rotor. Therefore, all the phases play a role in developing the energy of the system. On the other hand, the electrical energy losses, considering that the magnetic losses are equal to zero, are formulated as follows:

(2.6)

Fig. 9 Representation of mutual and leakage fluxes

In the case of any rotor bar breakage or open-circuit fault of the rotor windings, the corresponding equivalent resistance increases leading to an unbalanced resistance matrix which, in turn, produces undesirable harmonic components. For now, assume that all the phases possess a similar resistance value. Having ignored the magnetic losses, the electrical energy converted into the active electromagnetic energy, which contains both the required magnetizing and mechanical energy of the motor, is obtained as (2.7).

(2.7)

...where dW electrical Loss less electromagnetic energy change

By integrating (2.4-2.7), one can easily end up with the following equation representing the differential form of the Faraday's law:

dW electrical Is

(2.8)

By integrating the differential forms of the flux linkages expressed by (2.8), the total fluxes which cover the stator or rotor windings all over their circumferences are calculated. As a well-known fact, the total stator or rotor fluxes are made of two distinct types of flux lines, i.e. the mutual and leakage fluxes shown in Fig. 9.

The mutual flux is the part enclosing both the rotor and stator windings, and the leakage flux encloses only one side, whether the stator or the rotor. The underlying idea of the aforementioned definition is the way that the flux lines contribute to the stator-rotor magnetic coupling leading to the electromagnetic torque production of the motor. In other words, the mutual flux is the unique part of the flux that guarantees the coupling of stator and rotor components while the leakage flux serves as additional magnetic and electrical losses that lead to a higher temperature rise of the motor components. Therefore, it should be pointed out that

--The mutual flux: A flux that transforms power or energy from the primary winding (the stator) to the secondary winding (the rotor) and is produced by the coupled MMF of the stator and rotor windings. The concept of MMF will be discussed later in this section. This type of the flux should definitely pass across the motor air gap from the stator to reach the rotor and vice versa. In an aligned position of the two windings, the maximum mutual flux is achieved.

--The leakage flux: A flux that is confined to one of the current-holding windings. It might also partly pass into the air gap. However, it is inclined to the corresponding core to make a close path around its own source of production, one of the windings.

With this in mind, one realizes that each current-holding winding possess two flux components, namely the leakage flux and the mutual flux. The mutual fluxes are generally divided into two categories: the self-magnetizing and mutual magnetizing fluxes. The former one corresponds to the flux produced by one of the windings and encapsulated by the same winding. The latter corresponds to the encapsulated by two different windings.

A magnetic flux is defined as a combination of a current passing through a coil and a geometry-dependent quantity so-called inductance. The combination is normally in a form a product of the mentioned quantities, the current and the inductance, if a linear magnetic system is investigated. Moreover, the flux that crosses a winding might be composed of components built by different windings.

Therefore, depending on the number of windings, contributing to the flux linkage of a specific winding, different combinations of current-inductance are introduced.

For example, for an induction motor with three stator and three rotor windings, the flux linkage of one of the stator windings consist of six terms (see (2.10)) including the one related to the self-magnetizing and self-leakage inductance, and the remaining five terms that are related to the mutual magnetizing fluxes produced by the other windings. Flashing back to the main assumptions made at the beginning of this subsection, a linear model of the motor, as a result of which the magnetic flux can be written as the superposition of its governing terms (the leakage and mutual parts), is intended. If a nonlinear system is analyzed, the principle of superposition is not valid anymore. For a linear case, we can simply formulate the self-inductances as follows:

(2.9)

Considering a linear characteristic for the silicon steel material, the motor inductances would only depend on the physical geometry of the motor. In (2.9), the term ''magnitude'' has been used to denote that the real values of the mutual inductance vary as a function of the position of the two windings whose mutual inductances are calculated. Considering all these, the magnetic flux linkage of the stator winding

''A'' or ''a'' is formulated as follows:

(2.10)

where M_s l_sm and M_sr is a function of the configuration of the stator-rotor windings. It is noted that considering the inductance variations of the motor and its different possible compositions is a key point to understand the behavior of an eccentric motor since the inductance change is a kind of signature for this type of fault. Therefore, it is important to clearly understand the initial step in introducing the motor fundamentals. All the details provided in this section are sorted in a way that leads to a meaningful step-by-step realization of the characteristics of an induction motor which are highly significant in view of transition from healthy to faulty motors.

Going back to our main discussion, the mutual inductance of the winding (phase) A with itself is called ''self-magnetizing inductance'' and is equal to ''lsm'' while the magnitude of the mutual inductance of the phase A and other stator windings is a function of the position of the windings and equal to…

(2.11)

If the slotting effects and nonsinusoidal distribution of the windings come into play, the (2.11) is not valid anymore. Similarly, the terms which relate the stator windings to the rotor windings will be invalidated. All we have had so far is the con sequence of the sinusoidal distribution of the concentrated-at-one-point windings.

The rotation of the rotor is the compelling reason for the presence of the qe dependency of the stator-rotor mutual inductances. In this case, if the rotor rotates, the stator-rotor mutual inductances follow a sinusoidal alternation. Moreover, it is the electrical rotor angle which is incorporated into the formulations, not the mechanical one. However, wherever mechanical quantities are dealt with, the mechanical angle should be used correspondingly. Given the three-phase stator and rotor winding system, the matrix representation of the stator fluxes, i.e. (2.2), can be further extended as (2.12). Noticeably, the mutual and self-inductance matrices relate the stator and rotor current vectors to the fluxes such that the stator flux vector is a linear function of the stator currents while the dependency on the rotor current vectors exhibit a qe-associated function. Given a zero-stator current vector, the maximum flux enclosing the stator flux takes place when the stator and rotor windings are completely aligned. Additionally, the coupling is zero when they are located 90 mechanical or equivalently 180 electrical degrees away from each other. When a faulty condition such as the eccentricity or misalignment fault takes place, the M matrix (see (2.12)) which is the representation of the mutual inductance of the stator and the rotor should be modified according to the change in the length of motor air gap which is not a smooth and fixed quantity anymore.

(2.12)

As a result of mechanical deformations, the motor-inductance matrices, either Ls or M, will contain multi-harmonic sinusoidal components. Thus, a non-smooth variation of the motor fluxes should be expected. The saturation profile represented and modeled by nonlinear B-H curves of silicon steel materials is another factor that changes the inductance matrices. Thus, any factor that affect the motor saturation level brings up a nonlinear behavior of not only the inductance matrices but also all the resultant quantities such as the motor flux or current vectors. As another example, although broken rotor bars do not change the air gap length, they intro duce a highly saturated region close to the breakage. This leads to an unbalanced magnetic flux distribution that creates a nonsinusoidal inductance matrix. The concepts discussed above are only some of the hints to relate the motor fundamental parameters such as inductances to the fault diagnosis applications, and the corresponding detailed investigations will be discussed in the next sections. Correspondingly, the rotor flux equations are obtained as follows:

(2.13)

What is significant is the importance of the mutual inductances in studying the eccentricity fault while the self- and leakage inductances are not usually an appropriate index to realize the effect of the eccentric rotor. It is also interesting that depending on the topology of the winding connections, delta (D) or star (Y) connections, the stator-stator and rotor-rotor mutual inductances might be or might not be present in the motor flux. As a matter of fact, if a star (Y) connected and ground node-free stator and rotor-winding system are considered, the stator-stator mutual inductances will be zero. In the case where the three-phase currents of the stator and the rotor do not allow to have nonzero summation in a Y-connected system, one of the phase currents can be written as the negative of the sum of the other two currents, either in the stator or the rotor. Applying this situation to (2.10) and (2.13) causes the stator-stator and rotor-rotor matrices to have zero non diagonal components. In contrast, the stator-rotor inductance matrix remains intact.

This conveys the idea of using the mutual inductances as the tools for a consistent and connection-type free approach to analyze the motor behavior. Therefore, not only is it proposed to thoroughly understand the materials of this section, but it is also beneficial to become familiar with the advanced harmonic-included motor inductance analysis.

So far, the flux-inductance relevance has been demonstrated by means of a single harmonic approximation of the motor geometry and quantities. Now, it is time to propose a closed-form equation for the electromagnetic torque which takes care of rotating the motor shaft. This is exactly where several mechanical defects are imposed to the motor, especially to the moving parts such as the rotor shaft and bearings which hold the entire weight of the rotor. Bearing wear and shaft mis alignment which are all caused by mechanical deficiencies in the rotor are examples of the mentioned defects. It is also important to know that the movement can obviously worsen the situation. On the other hand, motor transients highly depend on the way that the electromagnetic torque is developed. In the first place, it is the motor geometry that affects the developed torque. Then, it is the input values and subsequently the resultant electrical and magnetic quantities, which are functions of the material characteristics and also the motor geometry that affects the electro magnetic torque. Therefore, it seems so much information can be extracted from the motor torque signal as well as the flux and the current. As a matter of fact, in Section 3, it is going to be mathematically proved that the electromagnetic torque contains unbelievably helpful information on the faults such as the broken bar and end-ring faults. Taking this brief discussion on the gravity of the torque signal into account, we try to present the single harmonic representation of the motor torque.

3.2 Electromagnetic torque equation

The torque is developed from the position-dependent variation of the mechanical energy and is applied to the rotor through the air gap. Therefore, the hint to calculate the torque is to find the variations of the mechanical energy in time. To this end, recall and combine (2.8), (2.12) and (2.13):

(2.14)

On the other hand, in a magnetically loss-less system, the stored magnetic energy variation is expressed as (2.15).

(2.15)

(2.16)

(2.17)

In addition, the electromagnetic torque is equal to the variations of the mechanical energy with respect to the angular position of the rotor.

(2.18)

There should also be a way to relate the developed electromagnetic torque to the motor speed, and the first-order mechanical differential equation of the motor handles this task.

(2.19)

Tbl. 2 Simulated motor data

Fig. 10 Impact of load variation in a line-start motor (a) current, (b) electromagnetic torque and (c) speed

T_load is the load torque that can be a constant or an oscillating load expressed in N m.

Oscillating load torques introduce a new challenge to the field of fault diagnosis of induction motors because the motor-harmonic components amplified by oscillating loads sometimes overlap the fault-related components. This leads to a mis understanding in terms of the severity of faults. The best example of the kind is the broken bar motor connected to an oscillating load with the oscillation frequency of 2sfs where s is the motor ''slip.'' In particular, broken bars or broken end-rings amplify the magnetic backward fields which have the same frequency pattern as that of the oscillating load presenting the same fashion of harmonic change in the motor quantities, specifically the motor current signal. Therefore, devising a strategy to discriminate between the fault and oscillating loads should be an inseparable part of the diagnosis process. However, a better understanding of the healthy motor behavior is still the best suggestion to tackle more complicated fault-related problems. Therefore, a helpful temporal study of the healthy motor quantities is provided herein. Through the following investigation, the requirements and concepts, such as the synchronous frequency or speed, the slip, the transient operation and the steady-state operation, etc., are fully discussed. To do so, a line-start supply mode of an induction motor with the data listed in Tbl. 2.2 is simulated, and the results are illustrated in Fig. 10.

Three motor signals are illustrated in detail in Fig. 10. The first signal is the motor current at three different load levels. It is important to note that whenever we talk about the motor load in this guide, we mean the motor torque, expressed in N m, connected to the motor shaft unless it is specified otherwise. If the mechanical losses are zero, this value should be equal to the electromagnetic torque. According to Fig. 10(a), the time required for the motor current to reach a steady-state point, beyond which the current magnitude does not change, greatly increases with the increase in the load level. Moreover, the amplitude of the motor steady-state current also increases by the load. The terms ''steady-state operation'' or sometimes called ''stationary operation'' is one of the key terms of this guide.

The fact that most of the fault indices are defined, formulated and extracted for the steady-state operating regime of the motor signals. Contrary to the steady-state operation, there is the ''transient operation,'' mostly defined as the start-up period of the motor from a zero speed to the steady-state point (see Fig. 10(b) and (c)). The transient period is exactly the same period in which the motor current possesses a larger speed value along with a swinging amplitude while the current amplitude becomes constant for a fixed load level (see Fig. 10(a)).

Since a single-harmonic model of the induction motor has been used so far, the fundamental frequency of the motor current is equal to that of the source, 50 Hz in this study. Accordingly, there should be a balanced three-phase current system with an electrical phase shift of 120 with respect to each other. Of course, the phase angles are different from that of the voltages because induction motors serve as highly inductive loads that produce a lagging phase angle. Given a healthy motor and that the higher order harmonic components are absent, the speed and the electromagnetic torque always follow a very smooth and non-oscillating trend which is simply expected from a linear and single harmonic model (see Fig. 10 and Fig. 10c). However, the transient operation is totally different. That is usually why most of the recent efforts in the field of diagnosis merely focus on the steady-state operation in which a well-behaved behavior is followed. The transient operation is somehow challenging specifically in the case of ''inverter-fed'' motors with a wholly nonsinusoidal supply voltage. Likewise, there is another strong excuse for not incorporating the transient which is the short-time period of the motor signal presence. In fact, the period in which the transient signals are present greatly affects the processing quality of the extracted signals such as the motor current. Besides, the diminishing nature of the fault-related components during the transient operation makes the process of extracting fault information a complex procedure. This is the compelling reason for insufficiency of the existing literature in terms of the diagnosis tools in the transient operations. Companies and researchers are markedly focused on the steady-state operation unless the under investigation motor is to perform in a highly stressful start-stop application. In high-power industrial applications, motors usually operate in a nonstop way unless a maintenance procedure is undertaken. Therefore, it is proposed to diagnose faults during start-up and prior to any aggressively growing fault level that normally happens in a lone-term steady-state operation.

Having focused on the discussion provided so far, the following guidelines are extracted.

--Any motor signal including the current, the flux, the electromagnetic torque and the speed consists of two main regimes, the transient and the steady-state operations.

--The transient operation reveals relatively large amplitude but a short-time presence of the motor signals.

--The motor signals possess a highly nonlinear time-varying behavior in the transient operation.

--In the steady-state operation, a smooth and favorably non-oscillating motor signal is observed in the case of a healthy motor. If faults occur, the situation becomes totally different, and a highly oscillating and non-smooth motor quantity should be detected depending on the fault type, the location and the severity.

--The smoothness of the healthy signals depends on the number of slots, the number of bars, the skewing angle, the saturation profile, etc.

--Increasing the load level increases the motor rise-time, the time that motor speed comes to a stable point.

--Increasing the load increases the developed electromagnetic torque. This makes sense because the equilibrium of (2.19) should be always valid.

--The transient torque and current levels are considerably larger than that of the steady-state one. This is why the motor starts up very fast. Sometimes, the start-up period is typically less than a second, depending on the motor size, but it is not larger than a couple of seconds in general.

--The steady-state current amplitude increases by increasing the load level.

--There are undershoots associated with the electromagnetic torque right before the rise-time of the motor speed.

--There are overshoots associated with the motor speed right before the rise-time.

--In line-start applications, the motor speed increases almost exponentially while it is linearly controlled in the inverter-fed applications.

--Undershoots and overshoots are easily controlled and eliminated if a well tuned drive is connected to the motor. If so, the amplitude of the transient parts of the motor current is constant in time. As a result, an almost constant electromagnetic torque is developed in the motor. This reduces the motor stresses, but it challenges the diagnosis process.

--Overload applications, larger than 100% of the full load are also possible.

--Note that the no-load operation does not mean a zero-current level. Normally, the no-load current possesses one-third, more or less, of the full-load current. This part of the motor current is called magnetizing current. Usually, it is very difficult, sometimes impossible, to diagnose the broken bar fault in no-load operation.

--Sometimes, the motor speed or load level changes from a value other than zero. Actually, the operation is not a zero-starting one, and the speed has a predetermined value, below or over the rated speed. The rated speed is the level at which the motor load is equal to the rated value. The term ''rated'' means that all the motor quantities including the voltage, the current, the torque and the speed are set at their nominal values. Therefore, the output power is equal to the rated value on the motor nomenclature plate. In this case, the voltage is definitely equal to the rated value.

It is important to note that the steady-state speed level decreases by increasing the load/torque level (see Fig. 10(c)). This arises from the fact that the rotor speed is NOT equal to the synchronous speed. This concept which is called the ''motor slip'' is thoroughly discussed below (see (2.20)-(2.27)).

(2.20)

(2.21)

(2.22)

(2.23)

(2.24)

(2.25)

(2.26)

(2.27)

The motor slip is, in fact, the most noticeable factor in induction motor operation such that in zero slip, there is no induced voltage in the rotor bars, and the corresponding currents are zero. As a result, there will be no field produced by the rotor for developing the electromagnetic torque. This is indeed why these kinds of machines are called ''induction motors'' since the main operating principle is based on the induced currents. Increasing the slip means decreasing the rotor mechanical speed and therefore increasing the induced voltage and current quantities in the rotor side. On the other hand, increasing the motor load increases the motor current which in turn causes an increase in the rotor bar currents. Hence, it is expected to have a larger slip value upon increasing the motor load and this is crucial for developing a higher developed electromagnetic torque. As a result of increasing the slip, according to (2.23) and assuming that the motor synchronous frequency is fixed, the rotor speed, nm, should decrease, and the same trend is observable in Fig. 10(c). A better realization of the mentioned points could be achieved by monitoring the steady-state operation of the motor quantities. The transient operation is a little bit challenging due to the fact that the frequency components of the motor quantities expect those which are related to the source increase gradually during the motor start-up from a zero speed to the steady-state mode. If the motor is subjected to a de-acceleration mode, the frequency components, specifically those of the rotor would represent a totally opposite behavior compared to the acceleration mode. It should be noted that as long as the supplied synchronous frequency is fixed (the case in any line-start application), the only load-dependent quantities would be those of the rotor only if an ideal single harmonic motor is tested or analyzed. Such kind of motors does not exist and all types of existing induction machines, whether operating in the motoring or generating modes, reveal some sort of undesirable harmonic components producing load- and so slip-dependent frequency components even in the stator side. All the claims provided in this section will be shortly addressed quantitatively or by means of the analytical solutions. However, since the main focus of this section is to provide the essential healthy motor information, it is preferred to skip the details of the harmonic components which might be used in fault diagnosis applications and switch to the main discussion on the principles of the motor operation. Hence, looking back at Fig. 6 in which the stator and rotor coils have been demonstrated as a set of concentrated-at-one-point windings, and assuming that the supply or synchronous electrical frequency is fs, and the rotor bars or windings electrical frequency is fr, the back-EMF equations in a single harmonic system can be expressed as follows:

(2.28)

Fig. 11 Electrical equivalent circuit of (a) stator and (b) rotor

Fig. 12 Electrical equivalent circuit for an induction motor

According to the theory of rotating fields, the three-phase stator currents with the electrical and spatial phase angle of 120 away from each other generate a rotating magnetic field with a fixed magnitude, but a changing position in space. The electrical frequency of rotation is equal to fs. Based on the Faraday's law, the rotating field passes across the rotor coils or bars and induces an EMF which is the main source of generating currents of the rotor conductors. Subsequently, the magnetic field of the rotor is generated. If the motor starts up from a zero speed, fr gradually decreases, so the corresponding EMF decreases also decreases. The flux linkage shown by l in (2.28) is the replacement for the pole cross section area multiplied by the air gap flux density. The magnetic quantities including the flux and the flux density will be further analyzed since these are actually the main driving forces of induction motors. For now, we assume that all the mentioned quantities including the voltage, the current, the flux-linkage, the back-EMF and the magnetic flux density are pure sinusoidal function of time and position of the rotor. The only difference is between the magnitudes as well as the phase angles compared to a common measure which is the voltage phase angle. Bearing this in mind and considering Fig. 6, every stator and rotor windings are modeled as series-connected inductances and resistances which represent the inductivity and the resistivity of the windings, respectively (see Fig. 11). For as much as the inherent operating principle of induction motor is based on induced magnetic fields, it is usually modeled as a rotating transformer in which the rotor side parameters depend on the motor slip.

In Fig. 11, Rc and Xm are the equivalent resistance of the core losses and the magnetizing inductance, respectively. Lr and Ls are the leakage part of the rotor and stator winding inductances. If the ratio of the effective number of turns neff Ns Nr is applied to the rotor circuit with the goal of referring the elements to the stator side, we will end up with the electrical circuit shown in Fig. 12 in which Rr n2 eff Rr, Xr0 n2 eff Xr0.

In Fig. 12, Vth and Zth are the Thevenin equivalents of the voltage source and the impedances on the stator side, respectively. By incorporating the Thevenin equivalents and solving the circuit for I1, (2.29) is obtained:

(2.29) So the absolute value of the active power transferred to the rotating mechanical parts of the motor can be calculated by the following equation:

Equality of the two Pm equations

(2.30) The number of phases, n, also contributes to the development of electromagnetic torque (see (2.30)). The only rotor component consuming the active power is Rr s

However, not all of its portion is transferable to the motion parts, and a part of it disappears through the copper losses consumed in Rr. Therefore, the active part of the rotor resistance would be equal to 1 s 1 Rr.

Fig. 13 Steady-state torque-speed profile of an induction motor

When it comes to the motor torque-speed profile analysis, (2.30) is exactly what is usually referred to. As seen, the above equation relates the torque to the voltage and the slip which together determine the operating point of the machine, of course in the steady-state operation, in both generating and motoring mode.

Fig. 13 illustrates the torque-speed profile of a line-start induction motor in the steady-state operation. The maximum torque and the points behind the maximum torque, the speed values ranging from zero to the corresponding point of the maximum torque, are called ''Unstable operating regions'' in which the motor does not usually operate unless transient operations are dealt with. On the other hand, the ''linear stable operating region'' is the most probable operating point of an induction motor. There is no shortage of agreement that an increasing motor load leads to a decrease in the motor speed and consequently the slip value. If the motor slip is equal to 1, the starting torque is obtained at the speed of zero. Moreover, the developed torque is zero at the synchronous speed.

Fig. 14 Transient torque-speed profile of an induction motor

Different from the demonstrated steady-state torque-speed profile in which the motor comes to a stable operating condition for every single pair of torque and speed, the transient behavior is somehow distinct particularly for the lower speed ranges (see Fig. 14). Although Figures 13 and 14 are similar in many ways, they are totally different representations of the motor behaviors. In Fig. 13, the effect of time has not been included while the second figure is obtained by sampling the torque and the speed at different time steps during a constant load of 50 N m.

However, the load is changed and the corresponding steady-state speeds are measured to obtain Fig. 13. A highly fluctuating torque component is observed at the very beginning instances of Fig. 14, i.e. a wide range of [ 500,500 ] N m is detected. This part of the torque profile, along with the following torque curve, forms the transient mode of the motor. Finally, the motor torque-speed profile comes to a stable point at around 50 N m ( Fig. 14) which is equal to the motor average torque in the steady-state operation. The corresponding speed is a little bit away from the synchronous speed and demonstrates the slip phenomenon. During the steady-state operation, the torque-speed profile is completely focused on one single point. There would only be a minor discrepancy caused by the inherent oscillations of the motor torque and speed components. The situation gets worse in case of the fault occurrence in which a bunch of undesirable fluctuations are added not only to the end-point of Fig. 14, shown as the ''Final steady-state torque'' but also to the transient operation. This issue will be further discussed when the fault diagnosis concepts are initiated in the next sections. The transient profile illustrated in Fig. 14 is obtained by simulating the single-harmonic motor model described in (2.1)-(2.19).

Fig. 15 General configuration of a closed-loop motor-drive system

A very small part of the motor speed goes over the synchronous speed, 1,500 rpm, in Fig. 14. This is related to the speed overshoot, also visible in Fig. 10, during the transition through the steady-state operation. The highly fluctuating transient regime makes it difficult, sometimes impossible, to extract proper fault indicators and that is why the fault diagnosis during the start-ups/stops is a rather new and intact field of study. However, this regime only matters when a large-scale machine with a considerably large start-up period, for example a couple of seconds, is under investigation. In the case of low- or medium-power applications or the applications in which the motor mostly operates in the steady-state mode, the transient analysis in not worth being analyzed. It is also interesting to know that incorporating the drive circuits into the fault diagnosis process brings up new challenges, specifically in the transient mode, since the motor behavior is not merely a function of the load any more. It is besides affected by other factors such as the variation of the voltage and frequency which in turn introduce new burdens to the problem. For instance, changing the motor frequency changes the magnitude, and the speed of the magnetic rotating field of the motor, and consequently all the quantities are affected. As a result, an extremely different motor behavior should be expected for inverter-fed applications. Nowadays, all the change in the motor voltage and frequency is handled by a control device which is called ''drive.'' This so-called drive device generally contains two main parts, namely the control unit and the inverter shown in Fig. 15. The control unit is somehow straightforward and takes care of translating the input commands in terms of the motor speed, torque or flux into the electrically applicable waveforms. The major responsibility of the control unit is to find an appropriate switching pattern applied to the inverter switches which modulate the DC bus voltage into the motor terminal voltages with a specific fundamental frequency component. Generally speaking, the whole drive circuit operates as a translator from a reference or command value to real supply voltages applied to the motor terminals. By means of this configuration, both the supply voltage and the frequency can be finely controlled and adjusted. There can be a speed feedback loop as well. If so, the corresponding configuration is called ''closed-loop control strategy.'' Otherwise, the strategy would be an ''open-loop'' one.

Both the methods are shown and discussed in the next subsection. It is noteworthy that the torque, the speed, the voltage, the current and also the flux could be controlled by drives. A closed-loop configuration operates better than an open-loop one since there is a direct control on the speed or even the torque. In Fig. 15, the terms w_ref and flux ref are the reference values of the speed and the flux, respectively.

Now, let us take a brief look at different drive types. Then, we will move toward the detailed analysis of their operating principles. The following strategies are essentially the most traceable techniques existing in the literature:

--Scalar Constant Voltage per Frequency Control (CV/F)

--Field-Oriented Control (FOC)

--Direct Torque Control (DTC).

Although the closed-loop implementation is also available, the first type is usually preferred for open-loop applications in which several motors should operate in parallel at the same time. The other two options are used in a closed-loop form in which a single motor is connected to the drive circuit. The CV/F strategy is more often used as a part of the other two general types in order to handle the acceleration or de-acceleration of the motor. The underlying idea of all the methods is based on controlling the torque and flux commands applied to the motor separately like what is done in DC motor control. Since quantities of induction machines are highly inter-related, unlike the DC machines, it is tried to first isolate the flux and torque components by any means, for example a mathematical transformation from a time-dependent to a time-independent space, and then control the quantities separately. After generating proper torque and flux commands, the values should be translated into the voltage signals and this is exactly where the switching pattern plays its role in the motor-drive system operation. In practice, there are many existing patterns each of which has its own advantages and disadvantages while the most frequent one is the space vector modulation (SVM) technique regardless of its different extensions. The switching pattern outputs the signals applied to the switches connecting the DC bus voltage to the motor terminal voltages. By means of the briefly discussed configuration, the speed and the torque can be independently controlled and adjusted depending on operating point characterized by application requirements.

There is a wide range of operating conditions in inverter-fed induction motors compared with the line-start one and this is the main difference between the line start and inverter-fed applications. In a line-start mode, the only controllable quantity is the amplitude of the motor voltage while the supply frequency, the speed and the motor torque can be easily controlled in the inverter-fed mode as well. This, in fact, makes the diagnosis process more troublesome since the number of factors influencing the fault indicators definitely increases. This is indeed the state of the art nowadays, and many researchers are trying to first realize how faulty motor quantities are affected if drives are incorporated and then find a novel solution or at least a complement for previous line-start-related diagnosis technique in order to outline the fault diagnosis of inverter-fed machines. Although attempts have been initiated couple of years ago, the idea is still new and worth dealing with.

Having discussed the major idea behind using drives and their applications in electrical motor industries, now it is time to go in depth of its principles, not very technical rules in terms of the structures or the design process of the drives, but a more comprehensive behavioral study of its effect on the motor quantities and how they should be translated into the drive circuit signals. In view of fault diagnosis procedures of electrical machines, what really matters is to process the motor signals, either the original or transformed signals, and extract the fault information out of the analysis. So we do not aim at working around different type of switches, various inverter topologies or switching techniques which might not be directly related to the condition monitoring of electrical machines. However, no one can deny that any of the mentioned factors might also have an impact on the quality of the process, but this is an absolutely uninvestigated problem, and there is no information on it.

4. Inverter-fed induction motors

4.1 Constant voltage per frequency strategy (CV/F)

The CV/F is one of the popular scalar methods of control in induction motors.

There are generally both the open-loop and closed-loop modes of this control strategy. It is preferred to start the discussion by targeting the open-loop regime.

The corresponding closed-loop strategy differs in only one feedback loop which is used to stabilize the motor speed at a reference value. The underlying idea comes from a very basic relationship as Motor flux L Voltage=frequency (see (2.28)).

If the number of turns and also the motor geometry is fixed, the flux is proportional to the supply voltage and inverse-proportional to the supply frequency. Fig. 16 shows the real variation trend of the motor voltage versus its frequency. There is a nonlinear initial curve located in [0 fcutt-off ] Hz related to the part of the voltage which should be added to the minimum voltage applied to the motor to compensate for the voltage drop across the stator winding resistance. The typical value should be around 10% of the rated voltage where the reference frequency or speed is zero.

Fig. 16 Stator voltage versus supply frequency

Then the voltage or rather equivalently and more technically the EMF should follow a linear direction toward the rated values with a constant voltage/frequency which represents the line slope. Practically speaking, when the voltage comes across the rated value, the rated frequency should also be satisfied. This is an extremely important requirement ensuring a rated flux at this point. Along the linear region, there is a perfectly tuned rated flux. Otherwise, the motor power density falls below its maximum capability if the slope is less than the rated. On the other hand, if the slope is larger than enough, the magnetic flux and consequently the magnetic losses increase dramatically. This is the reason that all the electrical machines are designed in a way to operate around the knee point of the magnetic saturation profile.

Fig. 17 (a) Torque-speed profile for different voltages in line-start mode and (b) torque-speed profile for different voltages and frequency in inverter-fed mode

Beyond the rated frequency, a very well-known operation, the so-called Flux weakening is present. Although the frequency can be freely increased up to a certain high value, there is a limitation on the motor voltage which should be restricted to the tolerable insulation limit. Otherwise, insulations would fail. Therefore, the voltage/frequency decreases by increasing the frequency beyond the rated value. In this operating mode, the motor definitely possesses a speed higher than that of a normal line-start mode. However, the speed or frequency cannot be raised infinitely due to the mechanical and magnetic losses limits. So there should always be trade off between the speed and the efficiency of the motor.

This simple but practical curve (see Fig. 16) determines the possible range of variation of the torque-speed of the induction motors controlled by inverters.

This curve includes the constant flux and flux weakening regions which are basically brought up by the primary torque-speed profile shown in Fig. 13 by assigning various voltage and frequency to (2.30) concerning the voltage limits of the motor. The same is done by changing the frequency and the voltage from very small values up to their rated and even larger values in the case of supply frequency. Then the torque-speed curves are plotted in Fig. 17(b). In addition, the effect of a variable voltage and fixed frequency in line-start mode is observed Fig. 17(a). As long as the synchronous frequency is fixed, the curves are not shifted toward the positive frequencies, and the change is only applied to the torque-axis which is proportional to voltage squared. This means that if the voltage becomes half of the rated value, the motor torque productivity is reduced by a factor of 4. This considerably affects the motor start-up quality due to the fact that the starting torque decreases as well. Decreasing the motor voltage in the line-start mode is also a reason for increasing the slip losses and this can undeniably be understood from the decreasing slope of the stable linear region of the profile (see Fig. 17(a)).

In the same manner, a comprehensive judgment in terms of the inverter-fed induction motor could be held herein as follows:

--The vertical line of the rated frequency defines two distinct operating modes, namely the constant flux (constant torque) and flux weakening (constant power) regions.

--Below the rated frequency, the maximum torque (breakdown torque) is fixed upon keeping the voltage/frequency ratio fixed. In fact, the voltage is kept increasing linearly as a function of the frequency up to the rated value which is required in the rated frequency point. In this case, the curves look like each other but with a specific frequency shift.

--The slope of the linear region of the curves is the same; hence, it is expected to have a close to similar slip losses in the constant torque region.

--Increasing the supply frequency beyond the rated values demands a decrease in the terminal voltage in order to follow the insulation safety. Accordingly, the motor flux and the torque productivity of the motor are reduced. An almost exponential reduction rate is normally observed for regular motors.

--Not only is there a frequency shift of the curves, but also the break-down torques gradually decrease by increasing the frequency.

--The variation observed in Fig. 17(b) explicitly explains the claim that torque and the speed can be adjusted independently if an inverter-fed application is targeted. Specifically, the constant flux region clearly reveals this fact.

As seen, the reference torque and speed can also play a role in determining the motor behavior based on which any fault diagnosis procedure is performed.

Therefore, it is not only the motor load affecting the detection process, but several other factors including the control strategy, the reference speed, the torque and the DC bus voltage might also have an impact on the process. This is the reason why we insist on getting familiar with the behavior of the motor quantities in both the line-start and inverter-fed cases and discriminating between them if needed.

Sometimes, instead of the motor original signals such as the current, their transformations used in drive circuits are utilized for the fault diagnosis purposes. On the other hand, these transformations are simply available in the most industrial drives such as ABB ACS800.

The CV/F control strategy is implemented based on intentionally following the inherent curves of the motor in terms of the voltage, the frequency, the torque and the flux with the goal of pertaining the safe and also possible motor limits. This is handled by means of the control strategy shown in Fig. 18. This is a demonstration of an open-loop strategy while a close-loop one is also possible and will be discussed soon.

The references could be the speed in RPM or the frequency in Hz. No matter which one is selected, the frequency control unit takes care of transforming the input into the electrical frequency, using (2.20)-(2.22). The voltage regulator keeps following the V/F curve illustrated in Fig. 16. What it does is to define the maximum terminal voltage, Vm, based on the voltage limits forced by the V/F curve.

To do so, the rated values should be definitely assigned to the drive. This is the first step that an operator needs to do when dealing with a drive. In the next step, the reference voltage commands are generated based on the reference frequency/ speed and are passed through the switching topology to generate the switching pulses. Finally, the terminal voltages of the inverter connected to a DC bus the motor runs. Since there is no feedback loop applied to the electrical or mechanical motor signals, this topology is inherently an open-loop control mode. It is not guaranteed to have a very small current or torque ripples in practice. Moreover, the motor speed is always smaller than the applied reference due to the fact that induction motor operates based on the slip phenomenon which causes a speed reduction in the rotor compared to the synchronous speed.

There is also a combination of an inductance and a capacitor between the converter and the inverter (see Fig. 18). This combination acts as a perfect filter as well as the DC bus maintaining an almost constant DC voltage on the bus.

The DC current flowing into this filter is sometimes referred to as a proper index of the broken bars fault in inverter-fed applications. Therefore, any part of the drive ranging from the control circuit to the DC bus might be affected by faults.

Like what has been mentioned before, there are several topologies to be hired at this point. However, just in the case of introducing the fundamentals and also providing industrial points of view, we will try to go through the SVM technique which is dominantly used in industry.

Fig. 18 Open-loop constant voltage per frequency drive

Fig. 19 A two-level three-phase inverter Tbl. 2.3 SVM switching states

4.2 Space vector modulation

SVM technique is the most popular switching topology in industry. The most interesting advantage of this technique is its capability to simply control the amplitude of higher-order harmonic components of the motor voltage, so the THD is some how controllable. It is normally based on determining the exact period in which switches should be on or off. Before going through analyzing SVM, we will have a look at a typical two-level three-phase inverter shown in Fig. 19.

Assuming that not both switches in one leg become on or off at the same time, there can be eight possible switching states for this circuit (see Tbl. 2.3). The DC bus voltage is often divided into two parts between the capacitors each containing half the total voltage, and the middle point of the capacitors is assigned as the neutral point. Based on this convention, the maximum and minimum phase-neutral voltages are Vdc/2 and Vdc/2, respectively.

According to the definitions, when all upper or lower switches of all the legs are off or on, the state is called ''zero vector.'' Otherwise, it is called ''active vector.'' The underlying idea is brought up by a vector representation of the voltage in the stationary reference frame. Given a group of switch states and phase voltages (see Fig. 19), the vector components of the vector representation are derived as follows:

(2.31)

(2.32)

(2.33)

As a well-known fact, there is a space vector of constant magnitude but rotating in real-imaginary plane if a set balanced three-phase network voltages is considered.

The angular frequency of the rotation is equal to ws 2 pi fs. The vector representation of (2.31) turns into zero if the zero vector states described in Tbl. 2.3 are met. Generally, the zero along with the nonzero vectors are demonstrated in a a-b plane illustrated in Fig. 20.

Fig. 20 Vector representation of the stator reference voltages

Assuming a voltage vector located in the first slice of Fig. 20, between V1 and V2, the voltage vector V can be modulated into the V1 and V2. This means that the vector V, the reference voltage, can be obtained by sum of two adjacent vectors, V1 and V2 each presenting a specific period of time, T1 and T2, within the switching period, Ts. ''T'' defines the duty cycles of the switches. If T1 T2 is less than half the switching cycle, the remaining period is handled by one of the zero vectors close to the vector V. The choice depends on the fact that transition from one state to the other one should be performed by changing only one leg of the inverter.

In the next half cycle, the switching order is reversed.

(2.34)

(2.35)

(2.36)

(2.37)

where n 1 : 6 defining the sector numbers.

Ultimately, the gate timings are shown in Fig. 21. All the curves are the same except in the phase shifts which obviously show different on/off periods of the switches. The x-axis represents the period of the reference voltage which is equal to 1/fref or s.

Fig. 21 Gate timings (duty cycles) of six switches

4.3 Analysis of motor behavior in open-loop CV/F mode

Having acquired the gate timings and applied them to the transistors (the switches), the line voltage is obtained. It is a PWM signal whose fundamental harmonic component is the same as the reference voltage calculated by the frequency control module implemented in the CV/F diagram (see Fig. 18). Compare Fig. 22 to one of the phase voltages in Fig. 8 which corresponds to a line start application. Assuming that the periods are the same, the two signals are totally different in shape and harmony. The aim of any PWM technique is to satisfy the possibility of different reference frequencies and fundamental voltages.

Fig. 22 (a) Time-domain line voltage in a symmetric SVM technique and (b) frequency-domain line voltage in a symmetric SVM technique

However, it ruins the quality of the signal in terms of the THD which is certainly larger in the case of an inverter-fed machine. In fact, higher-order harmonic components, which produce the spike-like signals and are present in the supply voltage introduce considerably higher saturation and loss levels. Furthermore, they are mostly a source of elimination of the fault components, especially those located in higher frequency bands, for example higher than twice or triple the fundamental component. This is one of the trickiest but sometimes useful parts of the inverter fed motor which facilitates the diagnosis process if utilized correctly. The discussion on harmonic components necessitates another important argument in terms of the Fourier Transform (FT) of the signals. It grounds some of the significant aspects of any diagnosis procedures based on which the harmonic components of the signals are analyzed and explored with the goal of finding proper fault indicators. A guideline on implementation and analysis of a Fourier Transform will be provided in section 7. For now, assuming an existing implemented algorithm, the FT of the line voltage is taken and provided in Fig. 22(b) within the range [0-10,000 ] Hz.

Accordingly, the following specifications are observed:

--There is an almost decreasing trend of amplitudes of the harmonic components upon increasing the frequency. This claims that the original time-domain signal contains a wide range of various-frequency sub-signals added to each other. However, the higher the component order toward the higher frequencies is, the smaller the amplitude will be.

--The largest amplitude corresponds to the fundamental component equal to 50 Hz in this study. This means that the frequency of the reference voltage applied as the input of the CV/F approach is equal to 50 Hz.

--The normalized amplitude of the largest component is equal to one, and the rest are less than 1. The spectrum has been normalized with respect to the fundamental frequency, 50 Hz.

--Although higher-order components exit in the supply voltage, they are not able to pass through the electromagnetic cycle of the motor because induction motors normally act as a low-pass filter which filters out the components with the order of larger than 20 to 25 times the fundamental component. This is due to the presence of a relatively large inductance produced by the windings.

Depending on the winding inductance, different harmonic orders might pass into the motor operation and can be observed by means of monitoring the signals other than the motor voltage. Let us focus on the fault-related components. They are usually low-frequency components detectable around the fundamental one. Therefore, it is expected that a group of strong signals exist in the motor magnetic, electrical and may be mechanical components. This conveys the idea of exploring and monitoring the motor frequency signature and check which components are sensitive or insensitive to the fault, the load and the speed, etc. Almost all the fault diagnosis approaches utilize the harmonic components and their variations as a tool to detect, determine and locate different faults.

--Fig. 22 includes a carrier/switching frequency ( fcarrier) equal to 1 kHz. The corresponding period, Ts, used in (2.36), is equal to 1/fcarrier. The larger the fcarrier is, the larger the number of pulses in Fig. 22(a) is. A higher-frequency resolution would also be achieved by increasing the carrier frequency.

--The amplitude of the voltage signal and the corresponding harmonic components directly depend on the DC bus voltage. Some researchers have claimed that changing the supply voltage amplitude or even the current does not change the normalized values; hence, they always rely on this normalized diagram in the diagnosis applications. We are going to show that this is a totally incorrect claim since the motor saturation level is definitely changed if the voltage or current are changed. Therefore, the fault components which are highly saturation-dependent undoubtedly change. This is a very demanding and at the same time challenging aspects being dealt with in this guide in detail.

So far, we have explained the route to a proper voltage extraction out of the applied reference value which is regularly the motor synchronous speed or frequency. Following the CV/F diagram, the line voltages are generated out of the DC bus voltage and then applied to the motor terminals. The synchronous frequency is equal to the reference frequency, so the rotor should rotate with a frequency smaller than the synchronous one in order to satisfy the motor slip. The motor quantities including the current, the torque and the speed are shown in Fig. 23. An acceleration unit can also be implemented prior to the reference value to control the motor transient start-up and make it a smoothly increasing regime. So the observed rattle behavior at the beginning of the curves shown in Fig. 23 will become smoother. This is actually the case in industrial drives, although herein a harsh start-up such as what happens in the line-start mode is practiced just in the case of preparing fair comparisons. Moreover, the speed overshoot can be completely eliminated if a proper start-up curve, usually a linear one, is applied to the system.

It is also important to note that a motor with different characteristics listed in Tbl. 2 is used here. This is a 11 kW#, 4-pole, 400 V induction motor which has 36 stator slots and 28 rotor bars. The complete list of specifications will be further provided in the Section 6 related to the finite element-based modeling of this motor. The simulated model used in this case is not a single harmonic model, but a more comprehensive one in which the higher-order spatial harmonics are incorporated as well. The model is based on the winding function theory (WTA) and will be discussed in the section corresponding to the simulation process of faulty motors. However, introducing new components to the model, a more nonsmooth and/or nonsinusoidal signal is observed in Fig. 23. The electromagnetic torque and the speed do not reveal a smooth steady-state performance and some obvious periodic oscillations are present which coupled with the previous discussion prove the existence of higher-order harmonic components in the motor-drive system. The components related to the switching phenomenon could be detected in the motor current, i.e. Fig. 23(a), in which the ripples are carried by the main signal. Not to mention that the positive slopes of the ripples are relevant to the positive voltage pulses and vice versa. The switching-related ripples are barely detectable in the speed or the torque since the high-frequency components are usually filtered out of the mechanical signals due to the low-pass nature of the mechanical loop of the induction motor. This concept will be further proved mathematically. If the afore mentioned acceleration block is added to the control strategy, the motor rise time is even controllable. As a technical issue, the amplitude of the ripples and so their improper effects on the motor operation can be reduced by increasing the switching frequency. Nevertheless, there should always be a trade-off between the switching frequency and the motor losses.

Fig. 23 (a) Current, (b) electromagnetic torque and (c) speed in CV/F mode without acceleration unit

In the case where a single harmonic model is used for simulating the CV/F mode, the torque and the speed would be smooth enough even if the switching frequency is very small. This statement brings up the significant effect of the additional spatial harmonic components used in the winding function theory by means of which the components produced by the nonsinusoidal winding distributions and also the slotting effects are taken into account. Therefore, it is claimed that the low frequency torque and speed periodic oscillations are not because of the switching phenomenon. The reason is the use of the accurate motor model. These low frequency components are so highly fault-sensitive that even a small level of the fault such as the bar breakage can change the amplitude of the oscillations.

Surprisingly, the fault-related components already exist in the motor natural frequencies and the fault occurrence merely changes their amplitude. However, some of the researchers claim, of course by mistake, that the fault occurrence is a reason to introduce new harmonic components. Maybe, in some particular and rare cases, they are right.

Previously, it was noted that the fault-related components possess a variable trend in any transient mode, and hence any factor influencing the transient operation certainly affects the fault-related components. Therefore, a new challenge is developed and that is the infinite possibility of the transient profiles if a user defined acceleration or de-acceleration adjustment is allowed. This matter has not so far been investigated deeply since the number of situations is not identifiable. So we highly recommend switching this case if a new research work is to be done. The analysis of steady-state analysis has been turned into a technology today and only minor improvements are suggested.

A closed-loop implementation of the CV/F control strategy is also possible by adding a closed speed loop and comparing it with the reference value. Doing so, the rotor speed approaches the reference value, and the synchronous speed goes beyond that due to the motor slip. This is handled by the very well-known slip compensation technique usually applied by a simple sum-loop to the system. It might also be noted that since there are more referable closed-loop strategies like the FOC and DTC modes, it is preferred to switch the discussion to analyzing them instead of working on the CV/F mode. The CV/F mode can also be a part of the strategies discussed later.

Fig. 24 DQ reference frame of an induction motor

4.4 Reference frame theory of induction motors

In an FOC method, some transformations are always applied to the three-phase motor quantities including the voltage, the current and the flux in order to eliminate the time-dependency and simply to control them as separate values like what is done in DC motors in terms of the torque and the flux. Actually, an abstract concept called the ''reference frame theory'' is introduced. In this theory, a three phase system is mapped onto a two-phase time-dependent or independent system.

Conventionally, the two phases are highlighted by D and Q notations. A general DQ representation of an induction motor is provided in Fig. 24. Instead of Wa, Wb and Wc which represent the three-phase system windings, Wqs, Wqr, Wds and Wdr are used to demonstrate the DQ-related windings/coils through the analysis. D and Q windings are fixed with respect to each other and the relative speed between the axes is zero. However, the DQ axis might or might not rotate depending on the user's choice. The speed of rotation can be chosen as zero, the synchronous speed, the rotor speed as well as a user-defined speed. Likewise, different reference frames called the stationary, synchronously rotating and rotor frames are achieved, respectively. By definition, the D-axis leads the Q-axis in a counter clockwise direction.

All motor variables must be projected to the DQ reference frame, using a mathematical transformation matrix, the so-called ''the Park's transformation matrix'' formulated as (2.38). The idea can also be extended to a system with larger number of phases.

Park's Transformation Matrix K

(2.38)

Transformed motor variables: xdq0 Kxabc

(2.39)

Applying the transformation and replacing the results into (2.1)-(2.19) leads to the following voltage balance equations with an arbitrary rotation speed of the DQ frame.

(2.40)

Certain simplifications in (2.40) are achieved if the term ''q'' is predefined. Here, the stationary reference frame for which pq 0, is discussed. The other two important cases including the rotor frame and also the synchronous frame are obtained by simply replacing the pq wr and ws, respectively.

(2.41)

The following features are valid in terms of the reference frames:

--In the stationary frame, all the DQ variables are time-dependent in both the transient and the steady-state modes. Therefore, the complexity of the analysis is somehow maintained.

--In the synchronous frame, any stator quantities such as voltage or current possess a constant value for the steady-state operation while the transient operation still reveals a fluctuating trend. The rotor quantities are also varied since they are projected to a frame with different rotating speed. The frequency of the electrical variables of the rotor should ideally be equal to that of the rotor in real world, sfs, of course in steady-state operation.

--In the rotor reference frame, the stator variables are also of an oscillating nature in both the transient and steady-state operations since they are projected to a frame whose rotating speed is equal to wr which is definitely different from the synchronous speed. In other words, the complexity of the analysis and also the level of time-dependency increases in the following order: the stationary frame, the rotor frame and the synchronous frame. Due to the variability of the existing alternatives, various control strategies are also available. However, a competent industrial drive should be able to control the motor with any possible topology.

Together with the FOC strategy, the reference frame theory helps control the motor flux and the speed separately. This is what really required in terms of a wide speed range control of induction motors. The flux must be independent of the motor speed in the constant flux region while it should be controlled as an exponential function of the speed for the flux weakening region. On the other hand, the variables of a three-phase motor are totally interrelated. So changing one parameter would change the others. The reference frame theory aims at decoupling the flux and speed control loops of the motor in a way that they are controlled somehow independently. The decoupling is not 100% accurate. However, it works well at least in terms of ordinary applications. The idea comes from the fact that D and Q components are literally independent proved by the inductance matrix shown in (2.40). However, there is always a cross saturation effect between the D and Q axes caused by the saturation of the silicon steel material. Thus, a saturation-based decoupling approach is sometimes applied to the control circuit to eliminate this effect as much as possible.

4.5 Field-oriented control of induction motors

The FOC of induction motors is fundamentally based on measuring the speed and the position of the motor by means of the equipment such as the encoder or by estimating analytically (see Fig. 25). This is the critical part of any closed loop application. If the speed and position estimation is handled analytically as describing the motor state space, the method would be called ''sensor-less closed loop control.'' Otherwise, it is a with-sensor solution. Normally, the sensor-less applications are less accurate in terms of control goals. Regardless of the type of the speed estimation, having measured the motor speed, it is compared to the reference speed value which is actually the input of the control process. This is the value which should be met by the control strategy. In fact, the rotor speed should precisely follow the reference value. Consequently, the synchronous speed should be a little bit larger than the reference speed in compensation for the slip. The other input is the reference value of id in the stator side. So it is likewise noted as ids in some papers. This parameter can be fixed or adjusted depending on the required flux level forced by the flux control block shown in Fig. 18. To do so, another block shown in Fig. 26, is usually added to the very beginning of the FOC circuit. The acceleration block takes care of smoothly increasing or decreasing the speed reference so that a very smooth start-up is obtained. This is finalized within the time period 0 to tacc. Then, a constant reference value equal to the desired one is applied to the motor. During the transient operation, the CV/F concept is utilized to maintain a constant rated flux if the motor speed should be kept below the rated value. Otherwise, by means of the exponential part of the flux controller, a flux value proportional to the motor speed is calculated and applied to the block which determines the value of id. It is obvious that the D-axis component of the motor current and consequently its voltage is related to the flux level. Indeed, it is the id which controls the flux level required for various operating speeds. This is where the CV/F concept contributes to the close-loop applications.

Fig. 25 Field-oriented control of induction motors

Fig. 26 Reference values of speed and id

On the other hand, there is a speed control loop separate from the flux control loop in any FOC control strategy. The input value is compared to the real rotor speed estimated or measured by the feedback and then passed through a linear PI controller to generate the reference iq value which is in relation with the motor torque.

Actually, iq is called the torque component of the DQ frame. In this level of control, the reference id and iq pass through a set of PI controller to generate the vd and vq (2.41) simply verify the applicability of the PI controllers. Basically, since the relation between (id, iq) and (vd, vq) is linear during the steady-state operation, a linear controller could be used to relate the voltages to the currents. The only challenging part is to find the appropriate proportional and integral coefficients (gains). The typical strategy is to choose the proportional gain 10 times larger than that of the integral one. The difference between the real and reference values of the currents producing the reference voltages, not the reference currents themselves.

vdref and vqref are the D and Q components of the required voltages for the requested speed and flux values. There is always a third voltage component, i.e. v0, as one of the outputs of the Park's transform. Meanwhile, this value should be ideally set to zero if a balanced three-phase system is desired. Then, according to the three-phase voltages, appropriate switching signals are forced to the inverter, and the motor terminal voltages are ready.

Now it is time to provide some clues in terms of the contribution of the FOC strategy in fault diagnosis procedures. As mentioned before, one of the biggest objections in the time-domain analysis of the faulty motor signals is the closeness of some of the fault indicators to the fundamental harmonic component of the motor variables such as the current. While applying the projection from the three phase system to a DQ frame, the dependency on the fundamental frequency will be eliminated if a proper reference frame is selected. Therefore, a useful alternative is to analyze the DQ components instead of the components in an ABC reference frame. This really matters in the case of a broken bar motor. Sometimes, the effect of the proportional and integral gains on the components of the ABC frame is not tractable. Nevertheless, the DQ components reveal a very well-behaved variation with respect to a change in the gain levels. The change in the gain levels is another significantly influential factor in any diagnosis process. Unfortunately, a few works has been reported in this field, so it is still an open area of research.

The other objection is the negative effect of the closed-loop control strategies on the fault indicators. For example, if an eccentric motor under an FOC drive is investigated, the torque or speed fluctuations caused by the fault will not be visible anymore if a very well-tuned drive is utilized. In case the fault goes beyond a specific level, the diagnosis process will be able to detect a mal-operation. This phenomenon makes the detection, determination and finally the diagnosis process way difficult and probably impossible at least in the case of a lower fault level.

The third and probably the most important challenge is the presence of two control loops each acting like a low-pass filter eliminating some of the fault-related components. This fact will be analyzed deeply by mathematical and experimental methods in the next sections. Moreover, different drive types and technologies reveal different band passes and also various effects on the variation rates of the fault indicators. This is an unaddressed issue in the literature, and there should be a proper investigation of the commonly used inverters and their impact on the diagnosis process. This is suggested as a new field of research for potential researchers in academia and even research and development centers of industries.

At this point, it is preferred to move toward investigating the next well-known closed-loop strategy called ''direct torque control (DTC)'' prior to providing the behavioral study of the motor quantities. The reason is that the corresponding behavior of both FOC and DTC approaches really look similar. Of course, there are differences which might not be considered as problems in fault diagnosis procedures. The main difference is the type of variables used to control the motor torque and the flux. Otherwise, the whole concept is the same. Depending on the control strategy, the type of variables which must be chosen for the diagnosis purposes should also be different because not all variables contributed in a FOC drive are simply available in a DTC drive. Therefore, in the fault analysis, the variable selection process might be the most demanding aspect instead of the major difference in how these two types of drives might operate on the motor.

Acknowledging the aforementioned points, the DTC strategy is studied; then a detailed behavioral analysis of the motor quantities in both ABC and DQ frames is provided.

4.6 Direct torque control of induction motors

Fig. 27 shows a generalized DTC strategy along with its fundamental elements including a PI controller, two hysteresis controllers, the torque/flux estimator, etc.

In DTC approach, there is no transformation of the motor variables from the ABC to DQ frame. It is basically grounded on the comparison of the real torque and flux values with the corresponding reference values. The reference values of the speed and the flux are obtained in the same way as that of the FOC drive. Then, the values are assigned to the inputs of the drive system. The inputs are compared to the real values estimated by the ''torque/flux estimator block'' in which the Park's trans form is implemented.

Fig. 27 Direct torque control of induction motors

The estimated torque and flux functions are formulated as follows:

(2.42)

(2.43)

The differences between the real and reference values, i.e. DTe and DLs, enters the hysteresis controllers which directly manage the torque and flux magnitudes. The controller corresponding to the flux and torque loops is a two-level and a three level controller, respectively represented as follows:

(2.44)

(2.45)

…dL and dTe are the maximum allowed range of variations of the flux and the torque, respectively. These two parameters are the maximum and minimum limits of the hysteresis controllers. The value of 1 in the torque controller means that the motor torque should increase. Similarly, the value of 1 corresponds to the torque reduction region, meaning that the real torque is larger than that of the reference value. If the real and the reference values are the same, there is no need to change the torque command and the corresponding voltage vector, so the output must be zero. Likewise, the flux controller consists of two states, 1 and 1. The state 1 is related to the case where the stator real flux is smaller than the reference value and vice versa. The combinations of two states of the flux controller and three states of the torque controller generate six different switching arrangements listed in Tbl. 2.4. ''N'' is the number of voltage sectors similar to that discussed in the SVM technique. The states are shown by a three-digit number ''xxx.'' If the first x from the left is equal to 1, then it means the upper switch of the first leg of the inverter must be turned on. If it is zero, the lower switch should be turned on. The second and third digits correspond to the second and third legs, respectively.

Tbl. 2.4 States of the switches in DTC strategy.

The switch states directly depend on the torque, the flux and the switching frequency cannot be controlled by the DTC method. The switching pattern used in this technology is not based on the SVM or even the sinusoidal PWM techniques.

These are the values in Tbl. 2.4 which define the switching pattern, and switching frequency is somehow random. This is one of the drawbacks of the DTC method. Hence, it is sometimes combined by a FOC-based technique to satisfy the fixed switching frequency requirement. Generally, the FOC and DTC strategies are different in the ways listed in Tbl. 2.5.

Following the description of the FOC and DTC strategies, a behavioral study of the motor quantities is offered herein to see how a closed-loop approach affects the motor variables. The simulated motor specifications are listed in Tbl. 6. A single harmonic model of the motor like what was previously discussed in this section is utilized. Here a very large motor of 150 kW is considered. The rated frequency is 60 Hz, and the number of poles is 2. Therefore, the rated synchronous speed is 3,600 rpm. However, the attempt is made to simulate the motor in a speed other than the synchronous speed in order to show the capability of the drive in controlling the motor.

==========================

Tbl. 5 Comparison of DTC and FOC control strategies

Comparison property

Dynamic response to torque

Coordinates reference frame

Low speed (<5% of nominal) behavior Controlled variables

Steady-state torque/current/flux ripple and distortion Parameter sensitivity, sensorless Parameter sensitivity, closed-loop Rotor position measurement

Current control

PWM modulator

Coordinate transformations

Switching frequency

Switching losses

Audible noise

Control tuning loops

Complexity/processing requirements Typical control cycle time

------------

DTC

Very fast

alpha, beta (stator)

Requires speed sensor for continuous braking

torque and stator flux

Low (requires high quality current sensors)

Stator resistance

d, q inductances, flux (near zero speed only)

Not required

Not required

Not required

Not required

Varies widely around average frequency

Lower (requires high-quality current sensors)

Spread spectrum sizzling noise

Speed (PID control)

Lower

10-30 ms

-------------

FOC

Fast

d, q (rotor) Good with position or speed sensor

rotor flux, torque current iq and rotor flux current id vector components Low

d, q inductances, rotor resistance d, q inductances, rotor resistance

Required (either sensor or estimation)

Required

Required

Required Constant y Low Constant frequency whistling noise

Speed (PID control), rotor flux control (PI), id and iq current controls (PI)

Higher

100-500 ms

=======================

Fig. 28 FOC controlled induction motor behavior (a) stator current, (b) torque, (c) speed, (d) stator DQ fluxes, (e) rotor DQ voltages and (f) rotor DQ currents

Tbl. 6 Simulated motor data

Here are the features and characteristics of the motor variables for a given speed and torque command applied to a FOC drive. The speed command is set at 2,000 rpm while the required torque at 0.

--For all closed-loop applications, maximum current and torque are applied to the motor in order to achieve as fast-speed response as possible. So even though the torque command is set at 0, the electromagnetic torque is a considerably large during the start-up (see Fig. 28(b)). The acceleration rate has been set at 2,000 rpm/s. It is usually an adjustable term in any industrial drive. It should not be essentially a linear curve like what is shown in Fig. 28(c). It can take any user-defined curve, of course if the inverter has enough potential to produce the required voltage and current. Actually, this is always the inverter limits which restrict the integrated motor-drive system operation. Therefore, beyond the inverter limits, no operation is available.

--The motor current and torque at start-up is easily adjustable by means of the current limiters implemented inside the PI controllers. The larger the limits are, the faster the response can be.

--The ripples caused by the switching are obviously seen in all the motor signals.

As far as the motor model is a single harmonic one, there should not be any ripple in the motor quantities. However, it is not the case here, and there exist ripples. Generally, the amplitude of the ripples can be reduced by increasing the switching frequency, but it cannot be reduced below a certain point due to the limits on the switching frequency. For normal applications, a maximum value of 20 kHz is assigned.

--Unlike the line-start mode, any other mode has an open-loop nature, the cur rent and torque magnitudes are fixed during the transient operation of the motor. This is the astounding feature of any closed-loop operation which allows the motor to start up as quickly as possible.

--The right subplot of Fig. 28(a) is a DQ representation of the stator current.

There are two semi-circle shapes demonstrating the transient and steady-state operations. The circle with the larger radius is of a transient nature, and the smaller one corresponds to the steady-state operation. Always, the DQ components of the motor variables produce circle-like locus assuming a balanced and healthy three-phase system. If faults occur, the locus will reshape depending on the fault level and type. So this is one of the great potentials of drive circuits in providing fault-friendly signals. It is noted that the system has been simulated in a rotor reference frame.

--In Fig. 28(c), the overshoot is also very small compared to the line-start mode. This value could be zero if a very well-tuned drive is used.

--During the transient operation, the variables, those of either the stator or the rotor, have sinusoidal nature. However, the rotor variables become constant during the steady-state operation since the motor was modeled in the rotor frame. The stator variables are still sinusoidal.

--The rotor voltages are set to zero deliberately (see Fig. 28(e)). In practice, there is a close to zero induced voltage on the bars.

--When the broken bar fault occurs, the voltage across the broken bar is not zero anymore and this can be considered as a fault indicator. However, measuring the bar voltage is not a generally applicable task in the industry.

--Depending on the drive and inverter types, different motor variables are directly controlled. For example, in simple DTC drive, there is closed loop applied to the motor speed and consequently the torque. Therefore, the torque and speed will finally come to a very smooth operation even in case a fault occurs. This is why the proper signal selection should be a part of the fault diagnosis process. Likewise, if a current source control inverter is used, the motor current would be stabilized regardless of the fault presence. In this case, the motor torque and speed might be fluctuating because of the fault. In the line-start mode, none of the mentioned challenges exists and all the motor quantities including the magnetic, electrical and mechanical ones are simply affected even with a very small fault level. So if an inverter-fed motor is investigated, more fault-sensitive processors should be hired. Otherwise, it is almost impossible to diagnose the fault.

--Due to the importance of the topic, it is noted again that the transient mode of faulty inverter-fed motors has not been addressed yet and more efforts are needed in the field. The point is that the transients matter only if a considerably high-power motor with the corresponding start-up of around a couple of seconds is under investigation.

So far in this section, an essentially required insight of the induction motor and its operation has been provided. Then, through mathematical and simulation-based developments, the challenges in terms of the behavioral study of a healthy and sometime the faulty motor were addressed. According to the discussion, the following are highlights of this section and the guidelines for the next sections as well, in terms of the fault diagnosis procedures.

--Distinct parts of an induction motor are subjected to different types of fault.

For example, the short-circuit fault occurs in the motor windings consisting of the insulation while the broken bar fault occurs in the rotor bars. Therefore, a comprehensive study on the types of fault and their target places should be done in the next sections. This is a must-do before going through the advanced topics.

--Equally important, the motor signals including the voltage, the current, the flux, the flux density, the torque, the speed and also the vibration might be affected in different ways upon the fault occurrence. Furthermore, the type of effect differs with the type of fault as well as the operating mode. Therefore, the process of selecting proper signals should be discussed deeply in order to get familiar with various practical situations and their requirements.

--So far, the type of the processor used to extract the fault information has been discussed. Depending on the fault features in time, frequency or even time frequency domains, various processors each revealing specifically useful aspects must be utilized. For example, for a given faulty steady-state operation, the FT and sometimes the time-domain processors are the most popular.

However, they lose accuracy when it comes to the transient analysis.

--Extracting proper indices, the fault indicators, is for sure an important step to take. So far, a lot of indices each dealing with a specific aspect of fault has been proposed. None of the existing indices is completely able to detect, determine and diagnose the fault. For example, assuming that the motor operates in a light- or no-load condition, the conventional FT processors might not be able to diagnose the fault unless a super high resolution of the spectrum is set. Instead, the Hilbert-Huang transform (HHT) which is of a time-frequency type easily detects the fault even in light loads. So being aware of different fault indicators and their pros and cons will be useful in introducing a competent fault diagnosis procedure.

--Unlike the researcher's claim based on which one single index or process is able to cover all the faulty detection conditions, it will be shown that an accurate diagnosis procedure is a matter of operating conditions, and the pro cess should be updated for any major change in the conditions. For instance, if the number of the rotor bars changes, one broken bar fault would have a different effect in terms of magnitude and variation rate of the corresponding fault indicators. Therefore, it is not a straightforward task as simply mentioned in the literature. In addition, any change in the shape of slots affects the pro cess. Even any change in the manufacturer of the drive might change the fault indicator which makes the detection a trickier problem. This aspect of the diagnosis task will be further addressed.

--If a research and development center intends to practice the fault diagnosis process, the experiments related to different types of faults come to mind first.

Thus, a correct implementation of the faults should be considered. This is the reason why we will devote a part of a section to this topic. Meanwhile, proper sensor selection, sampling process and finally index extraction are proposed.

--Sometimes, it is impossible to examine all the fault types and conditions, specifically in academic environments. Therefore, accurate motor and fault models become important. In this case, the finite element method (FEM), as well as the WFA, will be deeply challenged and analyzed. So a very useful basis will be provided for the academic researchers who suffer from lack of test rigs.

These are some significant aspects of this guide which differentiate it from the existing references. Actually, the goal is to merely focus on one single type of motors, i.e. the induction motor, and develop really useful and engineering ideas for the diagnosis process, instead of wandering around general topics of all types of motors without going through the details of materials.