Stator Inductance Fluctuation of Induction Motor as an ... - IEEE Xplore

IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 6, JUNE 2011

1775

Stator Inductance Fluctuation of Induction Motor as an Eccentricity Fault Index Jawad Faiz1 and Mansour Ojaghi2 Center of Excellence on Applied Electromagnetic Systems, School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran Department of Electrical Engineering, University of Zanjan, Zanjan, Iran A new index is introduced for eccentricity fault diagnosis in three-phase squirrel-cage induction motors. Basically, some physical quantities of the motors structure such as air gap length influence this index, while other operational factors including motor supply, control type, load level, and its fluctuation cannot affect the proposed index. The influence of various eccentricity faults upon this index is investigated and shows the segregation ability of different eccentricity types by the index. It is indicated that the change of load, speed, and saturation level can slightly change the index value. Experimental results confirm the simulation results. Index Terms—Eccentricity fault, fault diagnosis, induction motors, new index, stator inductances.

NOMENCLATURE

Modified winding function of circuit . Machine pole pairs.

Air gap flux density at the beginning of saturation.

Instantaneous active power.

Amplitude of fundamental harmonic of spatial air gap flux density.

Instantaneous reactive power.

Supply fundamental harmonic frequency.

Stator phase resistance.

Harmonic components in stator inductances due to saturation and static eccentricity .

Slip.

Air gap average radius.

Instantaneous spatial vector of stator voltage.

Rotor mechanical rotation speed in r.p.s. Uniform air gap length.

Rotor position angle in stator reference frame.

Inverse air gap function.

Free space permeability.

Instantaneous spatial vector of stator current.

Eccentricity degree.

Saturation factor.

Dynamic component of eccentricity.

Core length.

Static component of eccentricity.

Mutual inductance of circuits Stator phase

and .

Angle in stator stationary reference frame.

self-inductance.

Air gap flux density position angle.

Mutual-inductance between

and phases.

Minimum air gap position angle.

Mutual-inductance between

and phases.

Equivalent phase inductance of stator.

Angle at which rotation and stator axes are separated due to eccentricity.

Stator phase leakage inductance.

Rotor speed.

Stator inductance matrix.

Synchronous speed.

Harmonic components in stator inductances . due to mixed eccentricity

Angular frequency.

Turn function of circuit .

I. INTRODUCTION

Manuscript received July 06, 2010; revised December 27, 2010; accepted January 12, 2011. Date of publication January 20, 2011; date of current version May 25, 2011. Corresponding author: J. Faiz (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2011.2107562

S

TATISTICS shows that eccentricity is one of the commonest failure conditions in squirrel-cage induction motors (SCIM). It may be of static (SE), dynamic (DE), or mixed (ME) type. The eccentricity fault may be a result of wrong rotor/bearing positioning during motor assembly, bearing wear, misalignment of load and rotor axes, mechanical resonance in

0018-9464/$26.00 © 2011 IEEE

1776

critical speed, and asymmetry of the mechanical load. The eccentricity generates a force on the rotor that attempts to pull the rotor from the stator bore. It also causes excessive stress on the machine and greatly increases the bearing wear. Furthermore, the radial magnetic force due to eccentricity can act on the stator core and exposes the stator windings to unnecessary and harmful vibration [1]–[3]. In brief, the following indexes have been so far introduced for noninvasive eccentricity fault detection in three-phase SCIMs [4]: 1) Normalized splitting severity factor. 2) Park’s current vector. 3) High-frequency harmonic components of stator line current [3]. 4) Low-frequency harmonic components of stator line current [5]. 5) Gyration radius of developed torque [6]. 6) High- and low-frequency harmonic components of stator line current [7]. low-frequency and high-frequency harmonic 7) components of spatial vectors [8]. 8) Ratio of sum of stator current low-frequency components and no-load current [9]. 9) Sum of low frequency harmonics of stator current and voltage spatial vectors [10]. 10) Stator negative sequence current [11]. 11) Instantaneous power [12]. 12) Current, vibration spectrum and back-EMF of a search coil [13]. 13) Impulse method for flux density harmonics calculation in eccentric case [14]. 14) Effective air gap permeability [15]. Due to some open issues, the reliable identification and isolation of the fault is still under investigation. Insensitivity to operating conditions and reliable fault detection for drive connected motors are among these issues [2]. In this paper, a novel eccentricity index is introduced which is theoretically independent of operating conditions of the motor. Therefore, behavior of the new index against supply type (mains or drive), control type (openor closed-loop) and control method (scalar, vector, DTC, and FOC) is unchangeable. An ideal index has a direct-linear relationship with fault severity, while it has no dependency on the other conditions and quantities of the motor. Most of above-listed eccentricity indexes are based on the distortion and/or unbalance of the motor current and/or voltage signals arising from the fault. However, the distortion and unbalance of the motor current and voltage may be due to the electrical supply conditions and/or nature of mechanical load of the motor. So, those indexes are far from an ideal index, because the supply harmonics and unbalance and/or load changes and fluctuations may affect their values. To be near to an ideal index, we must search for a parameter which is directly dependent on some physical aspects of the motor, which is distortable by the fault. Such parameter may be evaluated using only physical quantities of the motor, regardless of the supply and/or load conditions. Basically, there are three electrical parameters that have this feature; they are resistances, capacitances, and inductances.


Their values are determined only by the physical structure of the motor. Among these parameters, inductance may be considered as an appropriate eccentricity fault index, because it has a direct relationship with air gap of the motor and its nonuniformity and fluctuation due to eccentricity. The choice of this index against other indexes is similar to the choice of distance relay for protection of transmission lines against current and voltage relays and, therefore, it may have similar advantages. Since the stator of SCIM is easily available, inductances of the stator are considered as an appropriate eccentricity fault index. Behavior of the proposed index against different types of eccentricities is studied using modified winding function (MWF) theory. Then, the effects of the load change, speed change (usually by drive) and saturation are investigated. To validate the theoretical results, some experimental results are given. Although these results have been obtained by a parameter estimation method, they agree well with the theoretical results. In order to use this index more effectively and noninvasively, a state estimation technique is practically necessary [16]. This technique must be able to determine the time variations of the stator inductances, or their linear combination, using only voltage and current signals of the motor. The authors are working towards establishing such state estimation technique. II. STATOR INDUCTANCES CALCULATION OF ECCENTRIC SCIM UNDER SATURATION Our investigation shows that the normal saturation of core (knee point vicinity) plays a key role in the correct functioning of the new index. The MWF theory offers all required means to calculate healthy and eccentric SCIM stator inductances as functions of rotor position as follows [1], [17]: (1) where or denotes a phase winding of the stator ( , , or ). Taking into account the saturation effect, the inverse air gap function can be expressed as follows [18], [19]: (2) where (3) (4) (5) Replacing

from (4) into (2),

will be as follows:

(6) where . In an eccentric SCIM the uniform air gap length disturbs and the resulting nonuniform air gap is denoted by [1]: (7)

FAIZ AND OJAGHI: STATOR INDUCTANCE FLUCTUATION OF INDUCTION MOTOR AS AN ECCENTRICITY FAULT INDEX

where

1777

TABLE I SPECIFICATIONS OF PROPOSED 11 kW THREE-PHASE SCIM

(8) (9) by (7) gives a general inverse air gap function, Replacing which includes both saturation and eccentricity effects:

(10) By replacing (10) in (1) and using numerical integration, it becomes feasible to calculate various self-/mutual-inductances of the stator windings under eccentricity fault and saturation condition. In [19], we have resolved the resultant indefinite integral and obtained analytic functions for the mentioned inductances, as we have done so for saturated healthy SCIM [20] and for eccentric unsaturated SCIM [1]. However, for the purpose of this paper, the numerical integration is sufficient. III. DIFFERENT ECCENTRICITIES IMPACT ON STATOR INDUCTANCE Using the equations adapted in the previous section, various inductances of stator under different eccentricity faults with and without saturation are evaluated in this section. In SCIMs under (air gap flux position) rotates with synchronous steady-state, speed and rotates with rotor speed. Thus, the instantaneous values of the two angular positions are as follows: (11) (12) where (13) (14) A required time interval (for example 100 s) is divided into small time steps (for example 1 ms), then and are calculated with this time step within the overall time interval. Thereafter, given , and , the time variation of various stator inductances are calculated using (1)–(10) and then, FFT is applied to determine the frequency spectra. A. Frequency Spectra of Inductances for Healthy Motor For and , frequency spectra of the inductances for a nonsaturated healthy SCIM were obtained. Specifications of the proposed 11 kW motor have been given in Table I. Under such conditions, the stator inductances must be constant and their spectra will have only one component at zero frequency. The reason is neglecting the slots effects in the application of WF and MWF theories. Fig. 1 shows the frequency spectra of three elements of stator inductance matrix . These spectra have been normalized based on their zero frequency component values. Fig. 2 presents the influence of the magnetic saturation upon the frequency spectrum of the inductances for , which is almost equal to its value in the rated supply condition [20]. As seen, there are components at 100 Hz frequency

Fig. 1. Normalized frequency spectra of self-inductance of phase a (L ) and mutual-inductances between phase a and other phases (L and L ), in nonsaturated healthy induction motor.

in addition to the zero frequency components. Inductance variations with this frequency arise from the variations of the reluctance of the windings flux path due to the saturation. Number of each complete variation of the above-mentioned reluctance over every rotation of the air gap flux density is equal to the motor poles number. So, it is expected that the total number of the complete reluctance variations per second be product of the poles number and air gap flux density rotation speed (in mechanical round per second): (15) is the variation frequency of the reluctance of the where windings path; therefore, the inductances variation frequency is due to the magnetic saturation effect, which arises from the fundamental harmonic component of the air gap flux density distribution around the air gap. The relative amplitude of

1778


Fig. 2. Normalized frequency spectra of self-inductance of phase a (L ) and mutual-inductances between phase a and other phases (L and L ), in satu= 1 25) healthy induction motor. rated (

K

:

L

Fig. 3. Normalized frequency spectrum of self-inductance of phase a ( nonsaturated induction motor with 50% SE.

component in Fig. 2 is 35.4 dB for and .

L

) and Fig. 4. Normalized frequency spectra of self-inductance of phase a ( and ), in a satmutual-inductances between phase a and other phases ( urated induction motor ( = 1 25) with 50% SE.

K

:

L

L

) in

and 19.2 dB for

B. Frequency Spectra of Inductances Under SE Fault In spite of nonuniform distribution of the air gap length around the rotor under SE fault, the air gap length is not time-dependent and does not vary by rotating rotor over different angles. It means that although the air gap length distribution is nonuniform, the stator windings are always faced with a fixed air gap distribution. Therefore, elements of matrix are constants. Considering , and (SE condition without saturation), frequency spectrum of has been evaluated and shown in Fig. 3. As seen, the frequency spectrum has only a zero frequency component. If saturation effect is to be taken into account by replacing , the frequency spectra will be as shown in Fig. 4. It means that in addition to component (100 Hz), its second and third multiples (200 and 300 Hz) also appear in the inductances spectra. Presence of the new harmonics is due to the SE fault which causes distortion of the spatial distribution of the flux density around the air gap and presence of its higher spatial harmonics (second, third, etc.) [19]. These spatial harmonics rotate with the same mechanical speed of the fundamental harmonic around the rotor, but the number of their complete fluctuations in the stator windings flux path reluctances, is larger corresponding to their harmonic order. So, the frequencies of the resultant fluctuations in the inductances are of the same order: (16)

L

) and Fig. 5. Normalized frequency spectra of self-inductance of phase a ( mutual-inductances between phase a and other phases ( and ), in a nonsaturated induction motor with 50% SE and 2% inherent DE.

L

L

A low degree of eccentricity, called the inherent eccentricity, occurs due to the manufacturing and fixing process of healthy SCIMs [3]. If inherent DE is taken to be 2%, the inductances spectra for nonsaturated condition will be as shown in Fig. 5. In this case, an ME fault exists. As seen in Fig. 5, several frequency components with equal intervals appear. Frequency of the lowest component (following the zero frequency component) is equal to the rotor speed ( in rps); frequencies of the other components are integer multiples of this frequency ( , ). By substituting (12) into (8), it will be clear that ME degree is a periodic function of . Thus, the harmonic components of (8) cause the corresponding components in inductances. Adding the value of saturation factor at the rated operating , inductances spectra will be as shown in voltage Fig. 6. In this figure, in addition to and components, other components at frequencies also are visible. component are indicated by arrows on spectrum. and components are indicated by arAlso, rows on and spectra, respectively. Therefore, some degree of saturation and/or a small degree of inherent DE suffices to detect SE by frequency components of the stator inductances.


Fig. 6. Normalized frequency spectra of self-inductance of phase a (L ) and mutual-inductances between phase a and other phases (L and L ), in a saturated induction motor ( = 1 25) with 50% SE and 2% inherent DE.

K

:

1779

L

) for Fig. 8. Normalized frequency spectra of self-inductance of phase a ( a nonsaturated induction motor with mixed eccentricity: dominant DE (upper), dominant SE (middle), or equal DE and SE (lower).

L

Fig. 9. Normalized frequency spectra of self-inductance of phase a ( ) for a saturated induction motor with mixed eccentricity: dominant DE (upper) or dominant SE (lower).

L

Fig. 7. Normalized frequency spectra of self-inductance of phase a ( ) for an induction motor with 50% DE: nonsaturated and no inherent SE (upper), nonsaturated and with inherent SE (middle), and saturated with inherent SE (lower).

C. Frequency Spectra of Inductances Under DE Fault In the previous subsection, it was observed that the same harmonic components with identical frequencies exist on the frequency spectra of the self- and mutual-inductances of phase due to the saturation and eccentricity faults. This is also true for other two phases. Knowing this, afterward only frequency spectrum of the self-inductance of phase will be presented. Fig. 7 shows the frequency spectrum of this inductance under 50% DE with and without saturation, with and without inherent SE. As seen, for a net DE with no saturation, there are components at frequencies which are integer multiple of . The reason is that by every rotation out of rotor axis, due to DE, the reluctance of every stator phase winding oscillates, while, the number of these oscillations is equal to the stator windings ). poles number (here Adding 2% SE to DE fault yields ME fault (no saturation). Thus as previous, components in inductance spectrum appear. Comparison of Figs. 7 and 5 shows that with higher DE degree more such components (sometime with larger amplitudes) are presented. Taking into account the saturation factor

leads to the presence of , and harmonics components [20]. Compared to Fig. 6, the only difference is that, here the amplitudes and number of and harmonics are larger. Therefore, DE in motor with and without saturation and/or inherent SE is detectable by the harmonics of the stator inductances. D. Frequency Spectra of Inductances Under ME Fault In Sections III-B and C, inductances frequency spectra and their appear-able components under ME were investigated; however, small inherent eccentricities are included and studied in details. Fig. 8 presents the frequency spectrum of under three ME cases with no saturation. In all cases, sum of SE and harmonic components can DE degrees is taken to be 70%. be observed in the three spectra of Fig. 8. When DE component dominates , a larger number of relevant harmonics with larger amplitudes is observable, and when ME is balanced , they place between and conditions. Fig. 9 shows two spectra for unbalanced ME in which the sat, and uration effect have been included. The components are obvious in the figure. Also for the dominant DE component case, there is larger number of and components in the spectrum and their amplitudes are larger.

1780


mf harmonics group for self-inductance of phase a versus SE and DE components at constant slip, speed and f.

Fig. 10. Variations of normalized amplitudes of saturation factor: (a) , (b) 2 , (c) 3 , and (d) 4

f

f

f

IV. INFLUENCE OF ECCENTRICITY DEGREE UPON HARMONICS OF STATOR INDUCTANCES In order to determine the fault degree, there must be a direct relationship between the proposed index and fault degree and it is more appropriate to have a linear relationship. In this section, relationship between the stator inductance relevant harmonics amplitudes and the degrees of SE and DE components of fault are studied. At this end, some more dominant harmonics are chosen which includes: , , and from group, , , from group, and , , and from group. Again is considered here. Fig. 10 the self-inductance of phase presents that how the normalized amplitudes of harmonics of group vary with SE and DE components ( and ) at constant slip, speed and saturation factor. As it is clear from this figure, the normalized amplitude of harmonic depends on the degree of each eccentricity components directly, and this dependency is almost linear for the eccentricity degrees larger than 0.1. The surface obtained for harmonic amplitude is symmet. Therefore, the exchange of and rical around plane values does not change the amplitude of this harmonic. Also, , the closer and values, the higher for a constant is the amplitude of the proposed harmonic; so in this condition, the amplitude tends to its peak value when . Harmonic with considerably lower amplitude has similar behavior by varying SE and DE degrees. However, amplitude dependency of on SE degree is low, and harmonic is almost harmonic independent of this degree. For , the two latter harmonics depend directly and almost linearly on DE degree, but for some lower degrees of eccentricity components, this dependency shows inverse behavior. It means that the amplitudes of the harmonics have increased by the decrease of the eccentricity

degree. By ignoring this inverse behavior over a small limited can be diagnosed using the amplitude range, DE degree of harmonic , then diagnosis of SE degree will be possible by referring to the amplitude of harmonic . Therefore, it seems that these two harmonics suffice to diagnose the eccentricity fault with the necessary details. Fig. 11 shows the normalized amplitudes of the harmonics group versus SE and DE degrees at constant slip, speed and saturation factor. It is obvious that harmonic has constant amplitude and so, it is independent of SE and DE degrees. In fact, the presence of this harmonic is only due to the saturation phenomenon within SCIM. But, normalized amplitudes of other two harmonics of this group, i.e., and depend on the degrees of the eccentricity components. However, their relationships are nonlinear and sometimes reversed; meanwhile amplitudes of the two latter harmonics are very low. So, it seems that the harmonics of this group are not useful in the eccentricity fault diagnosis. Fig. 12 exhibits the normalized amplitudes of the harmonics group versus SE and DE components of the eccentricity fault ( and ) at a constant slip, speed, and saturation factor. As seen, by varying degrees of the eccentricity components, two harmonics and have behaviors similar to that of or harmonics; of course with lower amplitudes. So, these two harmonics can be useful in the eccentricity fault diagnosis of SCIM. However, two other harmonics of this group, i.e., and , have considerably low amplitudes and for low degrees of SE and DE components, they show reverse behavior. Meanwhile, the normalized amplitude of has nonlinear variations with SE degree. Therefore, two latter harmonics are less important in the diagnosis of the eccentricity fault.


Fig. 11. Variations of normalized amplitudes of f , (b) f , and (c) f . saturation factor: (a) f

Fig. 12. Variations of normalized amplitudes of jf speed and saturation factor: (a) f 0 f , (b) f

1781

harmonics group for self-inductance of phase a versus SE and DE components at constant slip, speed and

6

+f

mf j harmonics group for self-inductance of phase a versus SE and DE components at constant slip, , (c) f 0 f , and (d) f f .

V. INFLUENCE OF LOAD, SPEED, AND SATURATION ON AMPLITUDES OF STATOR INDUCTANCE HARMONICS Influences of the load level, speed, and saturation on the inductances are estimated by slip change, synchronous speed change (via drive in reality), and saturation factor change respectively. Fig. 13 shows the influence of the slip and synchronous speed variations upon the normalized amplitudes

+

of the harmonics. As seen, two mentioned quantities have no effect on the normalized amplitudes of harmonics , and . But, the amplitude of harmonic has slightly changed at the no-load condition (slips close to 0); however, in the no-load case, amplitude of harmonic changes considerably. Therefore, only the no-load or low-load operation of induction motor may have considerable effect on some of the five proposed harmonics.

1782


Fig. 13. Variations of normalized amplitudes of elected harmonics of self-inductance of phase a versus slip and synchronous speed at constant saturation factor : and : ): (a) f , (b) f , (c) f , (d) f f , and (e) f f . for eccentric SCIM (

= 0 25

= 0 35

2

4

0

+

Figs. 14 and 15 present the influence of the saturation on the normalized amplitudes of the proposed harmonics. As shown are generated in Section III, harmonics group when both eccentricity fault and saturation are present. So, as shown in Fig. 15, amplitudes of the elected harmonics of are strongly affected by the saturation this group factor. Saturation effect on the mean reluctance of the stator windings flux path is the only reason for the change of the elected harmonics of group . Therefore, according to Fig. 14, sensitivity of these harmonics to the saturation factor is considerably low particularly harmonic which have the highest amplitude. By increasing the saturation factor from 1 to 1.25, the normalized amplitude of this harmonic decreases only about 2.6 dB. However, in practice the range of saturation factor change is lower. So, harmonic of the stator inductances is taken to be near an ideal index for eccentricity fault diagnosis. Fig. 14. Variations of normalized amplitudes of elected harmonics of group mfr of self-inductance of phase a versus saturation factor at constant slip and : and : ): (a) f , (b) synchronous speed for eccentric SCIM ( f , and (c) f .

2

4

= 0 25

= 0 35

VI. EXPERIMENTAL VERIFICATION When any frequency component of the elements of the stator inductance matrix is to be used as an eccentricity fault index, it


1783

Fig. 16. Time variation of stator equivalent phase inductance estimated using experimental data.

Fig. 15. Variations of normalized amplitudes of elected harmonics of group jf 6 mf j of self-inductance of phase a versus saturation factor at constant : and : ): slip and synchronous speed for eccentric SCIM ( 0 f and (b) f f . (a) f

+

= 0 25

= 0 35

is necessary to estimate the time variation of the elements or a linear combination of them during the motor operation. Equivais a linear combination lent phase inductance of the stator of the elements. In a symmetrical three-phase induction motor is (17) is a diagonal element of the stator inductance matrix where is a nondiagonal ele(including the leakage inductance) and ment of it. It is obvious that in a symmetrical three-phase induction motor, the diagonal elements of stator inductance matrix are identical and so are the nondiagonal elements. value, a method has been introduced in [21], To estimate which uses only the stator current and voltage signals. Here by eliminating the averaging part of that method, it is used as a state estimation method to determine the time variation of . At a constant supply frequency, the method in brief is as follows: first the instantaneous values of and quantities are evaluated using the relevant time samples of stator voltages and currents: (18) (19) where (20) (21) The instantaneous value of

is then estimated as follows: (22)

A SCIM with specifications given in Table I was tested in laboratory under healthy and various degrees of eccentricity fault, and its stator voltage and current signals were sampled and saved. The details of the test setup have been given in [12], obtained using the above [20]. A typical time variations of method and experimental data has been presented in Fig. 16. is a constant in a symmetrical motor; beside the above Since

Fig. 17. Normalized frequency spectrum of L calculated from recorded sig: and nals of voltage and current in laboratory for eccentric motor ( : : ) at over three successive frequency ranges.

= 0 254 slip = 2 06%

= 0 462

equations an averaging technique has been utilized in [21] to decrease the measurement and noise errors. However, many computations using laboratory recorded voltage and current signals show that the above instantaneous form has also enough accuobtains, frequency spectrum racy. When time variation of can be evaluated by FFT technique, as usual. of the actual Fig. 17 shows the frequency spectrum of motor under ME fault over successive frequency ranges, which was calculated as described above. Harmonics belonging to group have been marked by elliptic signs on the spectrum. Since the motor has been supplied by the rated voltage and has a deand gree of saturation, harmonics belonging to groups are also presented and marked by square and circle signs on the spectrum respectively. Therefore, capability of in representing all the frequency components related to the eccentricity and saturation has been demonstrated. Fig. 18 is similar to Fig. 17 and was obtained for the actual healthy motor. Magnetic saturation with inherent eccentricity in the actual healthy motor generate some of the harmonics of all three groups, which are marked by elliptic, square, and circle signs on the figure. Table II gives the frequencies and amplitudes of the marked harmonics of the two last figures. Due to small differences on the mains frequency and slip, the frequencies of the corresponding harmonics are not exactly the same in the figures. harmonics have As Table II indicates, amplitudes of the been increased between 3 to 6 dB in Fig. 18. This may be due to the reduced mains frequency, which causes the saturation factor to be increased. Thus, these harmonics are affected by saturation of the motor more than that of the eccentricity. However, and harmonics existing in Fig. 17 are many

1784


TABLE II FREQUENCIES AND AMPLITUDES OF HARMONIC COMPONENTS MARKED ON FIGS. 17 AND 18

Fig. 19. Normalized amplitude of harmonic f versus slip under two eccentricity fault cases: (a) = 0:462 and = 0:254 and (b) = 0:23 and = 0:38.

Fig. 18. Normalized frequency spectrum of L calculated from recorded signals of voltage and current in laboratory for healthy motor at slip = 2:35% over three successive frequency ranges.

not visible in Fig. 18, and those which are visible have much lower amplitudes. Therefore, amplitude of the latter harmonics has more dependency on the degree of eccentricity fault. Particularly concerning harmonic , which was introduced as the most effective index for eccentricity fault diagnosis previously, we observe a considerable change in its amplitude in the eccentric motor compared with the healthy motor. Fig. 19 shows the versus load (slip) under normalized amplitude of harmonic

two eccentricity fault cases, which have been obtained experimentally. As seen in both cases the curves are almost flat, as such that the difference between the minimum and maximum values of each curve is only about 2 dB. A slight deviation of the curves from the flat position may be due to its dependency on the motor saturation degree, which varies by varying the load level. In total, eccentricity case b is weaker than that of case a. So curve b is always below curve a, which implies a direct relationship between the fault severity and index amplitude. Howcannot discriminate and isolate SE and DE ever, harmonic components, a drawback which is common between most of the previous indexes, too. VII. CONCLUSION The induction motor’s stator inductances spectra, determined using a version of MWF theory which includes saturation effect, show that several harmonic components may appear on them due to various eccentricity fault types. Also, it was shown that


due to the presence of little inherent eccentricity components in actual motors, all types of eccentricity fault can be diagnosed using the amplitudes of the harmonics. It was seen that magnetic saturation increases the number of inductances harmonic components considerably. When DE component of fault dominates, number and amplitude of the harmonics are higher than when its SE component dominates. Among the harmonics, one has a frequency equal to the rotor speed in r.p.s. ( harmonic). This harmonic has the highest amplitude and the lowest frequency, so it is detectable more easily compared to the other related harmonics. There is a direct and rather linear relationship between amplitude of this harmonic with SE and DE degrees of fault. Also, load and speed changes have no impact on the amplitude of this harmonic, but saturation affects it slightly. Equivalent phase inductance of stator is a linear combination of self- and mutual-inductances of the stator windings, so it has the same harmonic content including eccentricity related harmonics. This was confirmed experimentally by estimating the equivalent phase inductance variations versus time using voltage and current signals recorded in laboratory and determining its spectrum by FFT. Experimental results show the negharmonic amplitude on the load; ligible dependency of the and also indicated its desirable dependency on the eccentricity degree. REFERENCES [1] J. Faiz and M. Ojaghi, “Unified winding function approach for dynamic simulation of different kinds of eccentricity faults in cage induction machines,” IET Elect. Power Appl., vol. 3, no. 5, pp. 461–470, 2009. [2] A. Bellini, F. Filippetti, C. Tassoni, and G. A. Capolino, “Advances in diagnostic techniques for induction machines,” IEEE Trans. Ind. Electron., vol. 55, no. 12, pp. 4109–4126, Dec. 2008. [3] S. Nandi, H. A. Toliyat, and L. Xiaodong, “Condition monitoring and fault diagnosis of electrical motors—A review,” IEEE Trans. Energy Convers., vol. 20, no. 4, pp. 719–729, Dec. 2005. [4] J. Faiz and M. Ojaghi, “Different indexes for eccentricity faults diagnosis in three-phase squirrel-cage induction motors: A review,” Mechatronics, Elsevier, vol. 19, no. 1, pp. 2–13, 2009. [5] A. M. Knight and S. P. Bertani, “Mechanical fault detection in a medium-sized induction motor using stator current monitoring,” IEEE Trans. Energy Convers., vol. 20, no. 4, pp. 753–760, Dec. 2005. [6] J. F. Bangura, R. J. Povinelli, N. A. O. Demerdash, and R. H. Brown, “Diagnostics of eccentricities and bar/end-ring connector breakages in polyphase induction motors through a combination of time-series data mining and time-stepping coupled FE-state-space techniques,” IEEE Trans. Ind. Appl., vol. 39, no. 4, pp. 1005–1013, Jul./Aug. 2003. [7] S. Nandi, R. M. Bharadwaj, and H. A. Toliyat, “Performance analysis of a three-phase induction motor under mixed eccentricity condition,” IEEE Trans. Energy Convers., vol. 17, no. 3, pp. 392–399, Sep. 2002. [8] H. Xianghui and T. G. Habetler, “Analysis of air gap eccentricity in closed-loop drive-connected induction motors,” in Proc. IEEE Int. Conf. Electric Machines and Drives, IEMDC’ 03, Jun. 2003, vol. 3, pp. 1443–1447. [9] S. M. A. Cruz, A. J. M. Cardoso, and H. A. Toliyat, “Diagnosis of stator, rotor and air gap eccentricity faults in three-phase induction motors based on the multiple reference frames theory,” in Proc. 38th IAS IEEE Ind. Appl. Annu. Meeting Conf., Oct. 2003, vol. 2, pp. 1340–1346. [10] H. Xianghui, T. G. Habetler, and R. G. Harley, “Detection of rotor eccentricity faults in closed-loop drive-connected induction motors using an artificial neural network,” in Proc. IEEE 35th Annu. Power Electronics Specialists Conf. (PESC 04), June 2004, vol. 2, pp. 913–918. [11] L. Wu, B. Lu, H. Xianghui, T. G. Habetler, and R. G. Harley, “Improved online condition monitoring using static eccentricity-induced negative sequence current information in induction machines,” in Proc. 32nd Annu. Conf. IEEE Industrial Electronics Soc. (IECON 2005), Nov. 2005, pp. 1737–1742.

1785

[12] J. Faiz and M. Ojaghi, “Instantaneous power harmonics as indexes for mixed eccentricity fault in mains-fed and open/closed-loop drive-connected squirrel-cage induction motors,” IEEE Trans. Ind. Electron., vol. 56, no. 11, pp. 4718–4726, Nov. 2009. [13] M. J. Kim et al., “Analysis of inverter-fed squirrel-cage induction motor during eccentric rotor motion using FEM,” IEEE Trans. Magn., vol. 45, no. 6, pp. 1538–1541, Jun. 2008. [14] A. Tenhunen, “Calculation of eccentricity harmonics of the sir gap flux density in induction machines by impulse method,” IEEE Trans. Magn., vol. 41, no. 5, pp. 1904–1907, May 2005. [15] U. Lundin and A. Wilfbrandt, “Method for modeling time-dependent nonuniform rotor/stator configurations in electrical machines,” IEEE Trans. Magn., vol. 45, no. 7, pp. 2976–2980, Jul. 2009. [16] D. J. Simon, Optimal State Estimation. New York: Wiley, 2006. [17] J. Faiz and I. Tabatabaei, “Extension of winding function theory for non-uniform air gap in electric machinery,” IEEE Trans. Magn., vol. 38, no. 6, pp. 3654–3657, Nov. 2002. [18] J. C. Moreira and T. A. Lipo, “Modeling of saturated ac machines including air gap flux harmonic components,” IEEE Trans. Ind. Appl., vol. 28, pp. 343–349, Mar./Apr. 1992. [19] M. Ojaghi, “Eccentricity Fault Diagnosis in Three-Phase Induction Motors Under Mains Voltage and DTC Drive Supply Modes,” (in Persian) Ph.D. thesis, School of Electrical and Computer Engineering, Univ. Tehran, Tehran, Iran, 2009. [20] M. Ojaghi and J. Faiz, “Extension to multiple coupled circuit modeling of induction machines to include variable degrees of saturation effects,” IEEE Trans. Magn., vol. 44, no. 11, pp. 4053–4056, Nov. 2008. [21] J. L. Zamora and A. Garcia-Cerrada, “Online estimation of the stator parameters in an induction motor using only voltage and current measurements,” IEEE Trans. Ind. Appl., vol. 36, pp. 805–816, May/Jun. 2000.

Jawad Faiz (M’90–SM’97) received the Master’s degrees in electrical engineering from the University of Tabriz, Iran, in 1975 graduating with First Class Honors. He received the Ph.D. degree in electrical engineering from the University of Newcastle upon Tyne, U.K., in 1988. Early in his career, he served as a faculty member in the University of Tabriz for ten years. After obtaining the Ph.D. degree, he rejoined the University of Tabriz where he held the position of Assistant Professor from 1988 to 1992, Associate Professor from 1992 to 1997, and has been a Professor since 1998. Since February 1999, he has been working as a Professor at the School of Electrical and Computer Engineering, Faculty of Engineering, University of Tehran. He is currently the Director of the Center of Excellence on Applied Electromagnetic systems. His teaching and research interests are switched reluctance and VR motors design, design, and modeling of electrical machines and drives, transformer modeling and design, and fault diagnosis in electrical machinery. He is the author of 158 papers in international journals and 166 papers in international conference proceedings. Dr. Faiz has received a number of awards, including the first basic research award from the Kharazmi International Festival in 2007, the silver Einstein medal for academic research from the UNESCO, the first rank medal in Research from the University of Tehran in 2006, and the Elite Professor Award from the Iran Ministry of Science, Research and Technology in 2004. He is a Senior Member of Power Engineering, Industry Applications, Power Electronics, Industrial Electronics, Education and Magnetics Societies of the IEEE. He has been a member of Iran Academy of Sciences since 1999.

Mansour Ojaghi was born in Zanjan, Iran, in 1970. He received the B.Sc. degree in electrical power engineering from Shahid Chamran University, Ahwaz, Iran, in 1993, the M.Sc. degree in electrical power engineering from the University of Tabriz, Tabriz, Iran, in 1997, and the Ph.D. degree from the School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran, in 2009. He joined Zanjan Regional Electricity Company as a member of Grid Technical Office and later became head of the office. He is now Assistant Professor in the Department of Electrical Engineering, University of Zanjan, Zanjan, Iran. He has published ten papers in international journals. His teaching and research interests are transformers modeling, fault diagnosis in induction motors, and protection relays.