A model of double star induction motors under rotor bar defect for diagnosis purpose

Hubert Razik† , Gaetan Didier†, Thierry Lubin†,C.R. da Silva‡, A.W. Mascarenhas‡, C.B. Jacobina‡, A.M.N. Lima‡ and E.R.C. da Silva‡

†Groupe de Recherches en Electrotechnique et Electronique de NancyGREEN - UHP - UMR - 7037

Universite Henri Poincare - Nancy 1 - BP 239F - 54506 Vandœuvre-les-Nancy, Cedex, FranceTel: +33-3-83-68-41-30 fax: +33-3-83-68-41-33

E-mail: Hubert.Razik@ieee.org‡Departamento de Engenharia Eletrica

Universidade Federal da ParaıbaCaixa Postal 10.105 - 58109-970 Campina Grande, PB,Brazil

Fax: +55-83-310-1015(1418)E-mail: edison@dee.ufcg.edu.br

Abstract— This paper investigates the modelling of a doublestar induction motor when this one operates under rotor fault. Aspecial attention is paid only to the rotor defect. The use of doublestar induction motors or six-phase induction motors is increasingand we can find it in high power process. Its main advantage liesin most reliability in case of a normal operating system. In orderto monitor a huge power process, we have to focus our attentionon signals given the state of health of an electromechanical axisin the whole. Consequently, an appropriated model is proposedallowing the quantification of the severity of the fault when rotorbars are defected. This is made thanks to the most widespreadapproach: the spectral analysis of the stator current using the wellknow method named Motor Current Signature Analysis (MCSA).

I. INTRODUCTION

In high power industry application drives, induction motorswith number of phases higher than three are commonly used.This kind of motor have the reputation of robustness and ofroughness. The major advantage of this type of motor is thelow cost and consequently its ability to operate under fieldcontrol. One can find this kind of motor in applications suchas the electrical ships, the pumps, etc.

Nonetheless, failures can appear [1]–[4]. So, we have tomonitor the process or the induction motor in order to limitthe dangerousness of the defective process and to limit thecost of the maintenance.

One advantage of double star induction motors in compar-ison with the classical three-phase induction motor, is in thenatural disappearance of some torque ripple frequencies [5]–[7]. Thus, the torque ripple is reduced as rotor losses.

An other advantage of the double star induction motor is thepower segmentation approach. This type of structure reduceseach power electronic device in multi-phase inverter system.By this way, the phase current is reduced in the motor as wellas in the inverter converter. Moreover, it is possible to ensurethe operating system not at its full nominal rate but lower ratewhen a damage occurs.

A disadvantage of the double star induction motor is theappearance of cumbersome currents when the motor is fed byan inverter converter [8]–[11]. Notwithstanding, applicationsuse multi-star induction machine and the cumbersome currentin this one is limited [12], [13].

Generally, high power induction motors have a squirrel-cagerotor. We propose to considered, in this study, the rotor asa secondary winding. However, we cannot omit that statordefect, rotor defects or bearings defects can appear at anytime. One of all the defects is the partial or full broken baror a broken part of the end-ring. Consequently, we focus ourattention in this paper only on the rotor failure of a doublestar asynchronous motor. We consider the rotor as a single starwinding. We propose a suitable model allowing to quantify theseverity of rotor defect using the MCSA approach.

II. A BRIEF REVIEW OF THE STATOR CURRENT SIGNATURE

One common way to monitor a process using an inductionmotor is the spectral analysis of the stator current. This oneinvolves with the research of specific sidebands around thesupply frequency and to extract their magnitude [14]–[16].A classical approach based on the Fast Fourier Transform(FFT) is used [17]–[20]. One other possibility is the SlidingDiscrete Fourier Transform (SDFT) analysis, or the Short TimeDiscrete Fourier Transform. These methods are adapted to theon-line analysis as the off-line analysis. In fact, an upgradedspectrum is given at each sample period. Another way is thetime frequency approach. It allows to get the time where thefault has appeared.

One can find in the literature that the broken bar faults,eccentricity, rotor asymmetry and shaft speed oscillation showup sideband frequencies [21]–[25]. The frequency componentsare associated with rotor faults and stator fault (short circuit).

A. Faults repartition

The rotor faults and stator faults are due to a stress or to acombination of stress [26], [27].

0-7803-9484-4/05/$20.00 ©2005 IEEE 197

For the stator, the effects come from a problem whichcan be: thermal (overload), electrical (insulation), mechanical(winding), environmental (agressive, . . . ).

For the rotor, the main effects are essentially due to aproblem which can be: thermal (overload), electromagnetic(magnetic force, . . . ), eccentricity (static or dynamic, . . . ),mechanical (bearing, . . . ), environmental (agressive, . . . ).

In case of three-phase induction motors, defects are clas-sified in 4 categories. The stator faults take place for almost40%, rotor faults for almost 30%, mechanical faults for almost20% and the rest for about 10%.

B. Broken Bars

Under rotor bar defect, the rotor currents produce forwardand backward rotating fields. This non-zero backward rotatingfield produces a first harmonic in the stator at the frequency{1 − 2} fs. This is the classical starting point of the classicalmotor theory in case of broken bar. The sidebands near thesupply frequency due to broken bars are lower and upperaround the supply frequency.{

flsb = {1 − 2s} fs

fhsb = {1 + 2s} fs(1)

Generally, sideband frequencies are defined by:

fbrb = {1 ± 2ks} fs (2)

where fs is the electrical supply frequency, k = 1,2,3,. . . , k ∈N and s denotes the per unit slip.

However, due to the winding harmonics, frequencies aregiven by the following relation.

fbrb ={

k

(1 − s

p/2

)± s

}fs (3)

where, in normal winding condition, 2k/p = 1, 11, 13 . . . ,k ∈ N, p is number of pole pairs. We can notice that termsof the harmonics are based on: 2k/p = 12n± 1 with n= 1, 2,3, . . . . These contribute to the air-gap flux. For the classicalthree-phase induction motor, the relation is: 2k/p = 6n ± 1with n= 1, 2, 3, . . . .

As the slip changes with the mechanical velocity (Fig. 1),a relationship is obtained:

s = 1 − fm

2fs/p(4)

where fm is the mechanical velocity.

III. MODELLING

The stator of the DSIM is wound with two stars spatiallydisplaced by a fixed electrical angle α [28]–[30]. Classically,the shift angle α is equal to 30o. The windings are shown infigure 2. In order to model the two stars induction motor, thefollowing general assumptions are to be considered: uniformair-gap, magnetic linearity (negligible saturation), no eddycurrents, stator windings identical, the two stars have the sameparameters, symmetrical magnetic circuit, negligible slottingeffect, no space harmonic.

Fig. 1. Layout of the double star induction motor

Fig. 2. Winding of the Double Star Induction Motor

With these assumptions, the equations describing this induc-tion machine with 2*3 stator phases and q rotor phases (Fig.2) are written in vector-matrix form as follows:

[v] = [R] [i] +d

dt[Ψ] (5)

[v] = [vs vr]t

[i] = [is ir]t

[Ψ] = [Ψs Ψr]t

(6)

[vs

vr

]=

[Rs 06x3

03x6 Rr

] [isir

]+

d

dt

[Ψs

Ψr

](7)

We suggest in this paper that the rotor is a three phasecircuits. Consequently, we consider the q phases rotor as athree-phase system. Thus, we have hereafter for both the statorand the rotor part:

vs = [ vsa1 vsb1 vsc1 vsa2 vsb2 vsc2 ]t

is = [ isa1 isb1 isc1 isa2 isb2 isc2 ]t

Ψs = [ λsa1 λsb1 λsc1 λsa2 λsb2 λsc2 ]t(8)

vr = [ vr1 vr2 vr3 ]t

ir = [ ir1 ir2 ir3 ]t

Ψr = [ λr1 λr2 λr3 ]t(9)

198

Lsrt = Lsr

cos(θr) cos(θr − ν) cos(θr + ν) cos(θr − α) cos(θr − α − ν) cos(θr − α + ν)

cos(θr + ν) cos(θr) cos(θr − ν) cos(θr − α + ν) cos(θr − α) cos(θr − α − ν)cos(θr − ν) cos(θr + ν) cos(θr) cos(θr − α − ν) cos(θr − α + ν) cos(θr − α)

(10)

Lss = Lms

1 cos(ν) cos(−ν) cos(α) cos(α + ν) cos(α − ν)cos(−ν) 1 cos(ν) cos(α − ν) cos(α) cos(α + ν)cos(ν) cos(−ν) 1 cos(α + ν) cos(α − ν) cos(α)cos(α) cos(α − ν) cos(α + ν) 1 cos(ν) cos(−ν)cos(α + ν) cos(α) cos(α − ν) cos(−ν) 1 cos(ν)cos(α − ν) cos(α + ν) cos(α) cos(ν) cos(−ν) 1

+ Lls.I6 (11)

Stator resistances and rotor resistances, with I the identitymatrix, are written as:

Rs = Rs I6×6 Rr = Rr I3×3 (12)

Let us define the angle ν between the two stator windingswhich is equal to 2π/3. The stator and rotor leakage induc-tance are Lls, Llr. The stator and rotor mutual inductancesare written as Lms, Lsr and Lmr (Table I).

[Ψs

Ψr

]=

[Lss Lsr

Lsrt Lrr

].

[isir

](13)

Lrr = Lmr

1 cos(ν) cos(−ν)

cos(−ν) 1 cos(ν)cos(ν) cos(−ν) 1

+ Llr.I3

(14)

The details of the two other inductance matrices Lsrt and

Lss can be found respectively in 10 and 11.

In order to develop a simple dynamic model, the trans-formation matrix Ts−1 is used to obtain a diagonal statorinductance matrix. We apply Ts−1 to the voltage and tothe flux equations. So, the original six-dimensional statorsystem is decomposed into three two-dimensional decoupledsubsystems. These are the usual (d − q) one, and a non-electromechanical energy conversion related one named (x −y), a zero sequence (o1− o2) one. The transformation matrix

for the stator part is given

Ts−1 =1√3

1 − 12 − 1

212

√3

2 00 − 1

2 −√

32

√3

212 −1

1 − 12 − 1

2 −√

32

12 0

0 −√

32 − 1

212

√3

2 −11 1 1 0 0 00 0 0 1 1 1

(15)

The transformation matrix for the rotor part is given

Tr−1 =

√23

1√2

1√2

1√2

cos(0) cos( 2π3 ) cos( 4π

3 )sin(0) sin(2π

3 ) sin(4π3 )

(16)

In order to transform the rotor variables into the stator refer-ence frame, the following rotation transformation is used:

P(−θ1) =

1 0 0

0 cos(−θ1) − sin(−θ1)0 sin(−θ1) cos(−θ1)

(17)

Consequently, the machine model is then simplified. Theelectromagnetic torque expression is now given by:

T = pM(issqi

srd − issdi

srq

)(18)

where p is the number of pole pairs. The superscript s denotesthe stator reference frame.

The voltage and current using the transformation is now inthe stator frame:

vs = [vssd vs

sq vsrd vs

rq vsx vs

y

]t

is = [issd issq isrd isrq isx isy]t (19)

The global equation of the induction motor is written in asuggested form allowing to show up a particular arrangement

vssd

vsrd

vssq

vsrq

vsx

vsy

=

Rs 0 0 0 0 0

0 Rr M•θ1 Lr

•θ1 0 0

0 0 Rs 0 0 0

−M•θ1 −Lr

•θ1 0 Rr 0 0

0 0 0 0 Rs 00 0 0 0 0 Rs

.

issd

isrd

issq

isrq

isxisy

+

Ls M 0 0 0 0M Lr 0 0 0 00 0 Ls M 0 00 0 M Lr 0 00 0 0 0 Lls 00 0 0 0 0 Lls

d

dt

issd

isrd

issq

isrq

isxisy

(20)

199

of the inductances part (Eq. 20). This form allows a simpleexpression of the inverse of the inductance’s matrix part.

As we put forward the relation of all the variables in thestator frame, one can see that only the variable in the (d −q) frame take a part in the energy conversion. The (x − y)variables does not take a part in the energy conversion andcontribute only on the increasing value of the resistive losses.The machine model in the (d− q) subspace can be written as

vssd

vsrd

vssq

vsrq

=

Rs 0 0 0

0 Rr M•θ1 Lr

•θ1

0 0 Rs 0

−M•θ1 −Lr

•θ1 0 Rr

issd

isrd

issq

isrq

+

Ls M 0 0M Lr 0 00 0 Ls M0 0 M Lr

d

dt

issd

isrd

issq

isrq

(21)

with•θ1 is the rotor velocity (ωr).

The inductances in previous equations come from the cal-culus using the transformation matrix Ts and Tr (Eq. 15 andEq. 16). Consequently, there expressions are given by:

Ls = Lls + 3LmsM = 3√

2Msr

Lr = Llr + 32Lmr

(22)

IV. A MODEL FOR DIAGNOSIS PURPOSE

In order to have a model when the rotor is defective, we takesome assumptions. As it is well known, one frequency appearsin the stator of the induction machine in case of rotor bardefect. As the rotor circuit is not symmetrical, 2 frequenciesappear. The first is at the slip frequency: +sfs which impliesa forward current (If ) in the rotor as a rotor rotating field,the second is at the slip frequency: −sfs which implies abackward current (Ib) in the rotor as a rotor rotating field (Fig.3). As the rotor angular speed is equal to: ωs(1 − s), and thestator field is at ωs, two torques are produced. The main torqueis constant and a ripple torque to the rotor defect appears atthe frequency: 2sfs. This double slip frequency characterizesthe unbalanced rotor. Consequently, if we put our attention onthe stator current, a line frequency appears at (1 − 2s)fs.

The model we propose to present is an extension of the worksuggested by Filippetti [31]. The stator current is composed

Fig. 3. Stator and rotor rotating fields with rotor bars defect

of the main current at the supply frequency added with a linecurrent (Il) at (1−2s)fs. The ratio of these two current givesinformation about the heath of the motor. As we want to havea light model, we consider again the rotor as a three-phasesystem. We affect the resistance of one phase in the rotor inorder to simulate a problem of the rotor bar(s). We considerthe rotor having N rotor bars which is larger than the numbern of broken bars. So, we increase the resistance of one phaseby adding a defective resistance ∆R.

Let us put forward that rotor is considered as a three-phase system composed of 3 circuits (phases). Each circuitis considered as N/3 resistance connected in parallel. Thus, anormal rotor resistance has the expression:

Rr ∝ Rb

N/3(23)

where n defective rotor bars induce N/3− n bars in parallel.So, the equivalent resistance is equal to:

Rrdefect ∝ Rb

N/3 − n(24)

Consequently, the increasing value of the rotor resistance isgiven by Rrdefect − Rr, it explains:

∆R = 3Rrn

N − 3n(25)

Considering the rotor variables to the stator frequency, andusing the current If and Ib of the rotor current, we obtain theequations for the induction motor under rotor defect in steadystate as follows:

Vs = (Rs + Lsωs)Is + Mωs(Is + If )0 = (Rs/(2s − 1) + Lsωs)Il + Mωs(Il + Ib)∆R/3s (If + Ib) = Mωs(Is + If ) + (Rr/s + Lrωs)If

∆R/3s (If + Ib) = Mωs(Il + If ) + (Rr/s + Lrωs)Ib

(26)The figure 4 represents these equations.Some simplifications can be make. The first one consists to

consider the current Il as greater than the magnetizing current.Consequently, we miss all the magnetizing inductances in theequivalent circuit. Moreover, another assumption is to considerthe slip s small. So the induction machine operates at steadystate and near the nominal rate. By this way, a new simpleequivalent circuit is represented in figure 5.

Fig. 4. Equivalent circuit in case of rotor defect

200

Fig. 5. Simplified equivalent circuit in case of rotor defect

As we are interested from the ratio between the main currentIs and the line current due to the rotor bar defect Il, we haveto evaluate the ratio Is/Il. Thus, an approximated expressionis:

Is

Il∼= ∆R/3s

Rr/s + ∆R/3s∼= ∆R

3Rr(27)

By the substitution in this expression the relation of ∆R, wefinally obtain:

Is

Il∼= n

N − 3n∼= n

N(28)

As we can see, this expression does not required the valueof the slip in order to determine if the motor operates underrotor defect or not. Nevertheless, the amplitude of the currentat (1 − 2s)fs is necessary.

Consequently, the monitoring of the induction motor can bemade through the evaluation of the ratio of the main currentand the left current due to the rotor defect. In case of anincreasing value of this ratio, we know that motor operatesunder a problem. This is the first step of the diagnosis. Thesecond step is to confirm the defect and to research the severityof the defect. This approach allows to schedule a maintenancein emergency or not.

Classically, the two frequencies close to the main frequencyare used to monitor the induction motor. Nonetheless, thecurrent fhsb, which is at (1 + 2s)fs frequency, is highlydepending of the equivalent rotor inertia. So, its amplitudedecreases when the inertia increases. But the current confirmesthe presence of a rotor problem. Thus, investigation arenecessary to make an effective diagnostic.

V. SIMULATION RESULTS

The Double Star Induction Motor studied is a 1.3 MWpower rate machine. This one is fed by a two three-phasebalanced voltage supply. We have restricted our study to thecase of rotor bars defect. In order to simulate the faulty bar, weincrease the value of the rotor resistance in one rotor phase.

We present in figure 6 the stator current of the inductionmotor in case of rotor defect. One can see the magnitude ofthis signal is modulated. It is a proof of the presence of rotorfault. Moreover, as we have above-mentioned, a torque rippleappears at twice the slip frequency. Thus, the rotor velocity incase of rotor fault have a modulation too. We have presentedthe velocity in the figure 7.

Stator current (A)

Time (s)

-600

-300

0

300

600

50 50.25 50.5 50.75 51

Fig. 6. Stator current in case of rotor defect

Rotor velocity (rad/s)

Time (s)

38.95

38.955

38.96

38.965

38.97

50 50.5 51 51.5 52 52.5

Fig. 7. Rotor velocity in case of rotor bar defect

Spectral density (dB)

Frequency (Hz)

|Il| (dB)

(1 − 2s)fs (1 + 2s)fs

|Is| (dB)

-80

-60

-40

-20

0

48 48.5 49 49.5 50 50.5 51 51.5 52

Fig. 8. Stator current spectral density

In order to monitor the process, we present in figure 8 thespectral result of the stator current using the FFT method. Aswe can see, two lines appear in this figure. The motor operatesat the nominal rate in our case. The ratio between the linecurrent Is and the current at the frequency (1 − s)fs (il) isapproximatively equal to 40 dB. Consequently, the ratio n/Nis approximatively equal to 1%. The slip is close to 0.78 %and the sidebands are located at the frequency of 49.22 Hzand 50.78 Hz respectively. The sampling time of the currentacquisition is 1 kHz and the length is a power of two (217).The frequency resolution of the spectrum is ∆f = 0.76mHz.The spectrum is obtained using a weighting function.

201

VI. CONCLUSION

This paper presents a dedicated model to the monitor of adouble star induction motor. According to the study, we areable to detect a faulty rotor thanks to the spectrum of the statorcurrent. Thanks to sidebands introduced in the main line bythe rotor fault, the monitoring current spectral analysis allowsto detect rotor defects. Moreover, we are able to estimate theseverity of the rotor defect. This diagnosis method is easierwhen the motor operates under nominal condition. Underno load, the rotor cage carries few currents. Consequently,lines appear again but the slip is near to zero. Thus, linesat the neighborhood of the main current are very difficult todistinguish to each others. Further works are to validate themwith experimental results using a double star induction motor.It needs a special induction motor allowing to have brokenbars. Developments are in progress to proof the effectivenessof this approach by experimental results.

VII. ACKNOWLEDGMENT

This research has been supported by the GREEN labora-tory at University Henri Poincare, Villers-les-Nancy, France,UFCG/Brazil and by CAPES-COFECUB/Brazil.

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TABLE I

SYMBOLS

Rs : Stator resistanceLls : Stator leakage inductanceLsr : Mutual inductanceLms : Magnetizing inductanceLlr : Rotor leakage inductanceLmr : Magnetizing inductanceRr : Rotor resistanceM : Magnetizing inductance referred to the statorLs : Stator inductanceLr : Rotor inductance referred to the stators : Slipp : number of pole pairs

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