mathematical modeling of dynamic induction motor with bearing fault

7/26/2019 mathematical modeling of dynamic induction motor with bearing fault

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Ashish Kamal * et al. / (IJITR) INTERNATIONAL JOURNAL OF INNOVATIVE TECHNOLOGY AND RESEARCH

Volume No. 1, Issue No. 4, June - July 2013, 336 - 340

ISSN 2320 5547 @ 2013http://www.ijitr.comAll rights Reserved. Page | 336

Mathematical Modelling of Dynamic Induction

Motor and Performance Analysis with Bearing Fault

ASHISH KAMALDepartment of Electrical, Madan

Mohan Malaviya Engineering CollegeGorakhpur,(UP), INDIA

V.K.GIRIDepartment of Electrical, Madan

Mohan Malaviya Engineering CollegeGorakhpur,(UP), INDIA

ABSTRACT: In this paper the modelling of 3 phase induction motor having bearing fault is being carried out with the

help of dq0 axis transformation. The theory of axis transformation is widely used to create such model because it

reduces the complexities of time-varying variables. In the present work, a step by step Simulink implementation of an

induction machine using dq0 axis transformations of the stator and rotor variables in the arbitrary reference frame

has been carried out. For this purpose, the relevant equations are derived and stated at the beginning, and then a

generalized model of a three-phase induction motor is developed. The implementation of developed mathematical

dynamic model of induction motor has been done using MATLAB -R2011a Simulink block set. Lastly, the

performance analysis of induction motor with bearing fault has been presented.

Keywords- 3-Phase Induction motor; Bearing Fault; Mathematical Modeling; MATLAB; Simulation

I. INTRODUCTION

In adjustable speed drives, the machine are

normally constitutes an element within feedback

loop and therefore its transients behaviour has to be

taken into consideration [1]. High -performance

drive control, such as vector control and field-

oriented control, is based on dynamic d-q model of

the machine. The machine model can be described

by the differential equation with time-varying

mutual inductance; This model can be seen as quite

complex due to continuous change in the position

of rotor with respect to stator. Therefore to remove

this complexity the dynamic d-q model is used in

present work [3].

Fig. 1: Ideal 2 Pole 3 Phase Induction Motor

Fig. 1 shows idealised three phase induction motor

having 2 poles and winding on stator and rotor are

concentrated types. The three phase winding are

either in star (Y) or in Delta () form are

distributed sinusoidally and embedded in slots. In

case of squirrel cage machine, the rotor is shortedby the end rings [4]. If we neglected the effect of

slots and space harmonics and consider an ideal

winding distribution then the 3 phase stator supply

is capable to create a rotating magnetic field whose

speed is considered as synchronous speed.

II. DYNAMIC D-Q MODELLING OF

THREE PHASE INDUCTION MOTOR

The 3 phase ac machine can be represented by an

equivalent two phase machine as shown in fig. 2.

Fig. 2: Two Winding Representation of 3 Phase

Induction Motor

Fig. 3: Stationary and Rotating Axes

Where, ds-q

scorrespond to stator direct and

quadrature axis and dr-q

rcorrespond to rotor direct

and quadrature axis.


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Let us assume that the ds-q

sare oriented at

angle as shown above. The voltagesdsV and

sqsV

can be resolved into as-b

s-c

scomponents and can be

represented in matrix form as

sos

sds

s

qs

cs

bs

as

V

V

V

1)120sin()120cos(

1)120sin()120cos(1sincos

V

VV

(1)

The corresponding inverse relation can be written

as

cs

bs

as

sos

sds

sqs

V

V

V

5.5.5.

)120sin()120sin(sin

)120cos()120cos(cos

3

2

V

V

V

(2)

Here, sosV shows the zero sequence component

which may or may not be present. The current andflux can be transformed by similar equation [7].

So far, we have transformed the three axes variable

into two axis variable now we have to transform

two axes stationary variables into two axes rotating

variables, for that consider the fig.3 [8]. In fig.3, wehave considered d

e-q

eas the rotating reference

frame which is rotating with speed e with

respect to ds-q

saxes, we can relate e and e as

e = e t (3)

The two phase ds-q

swinding are transformed into

the hypothetical winding mounted on de-q

eaxes.

The voltages on ds-q

saxes can be transformed into

de-q

eframe by following equations,

esdse

sqs

eqs sinVcosVV (4)

esdse

sqs

eds cosVsinVV (5)

We can transform back again the rotating frame

into stationary frame by using following equation,

edseqs

s

qssinVcosVV (6)

edseqsdscosVsinVV (7)

Since, we have considered that)120tcos(VVand)120tcos(VV),tcos(VV mcsmbsmas

therefore, variable in stationary reference frame and

rotating reference frame can be written as,

)tcos(VV emsqs (8)

)tsin(VV emsds (9)

cosVV mqs (10)

sinVV mds (11)

From equations (8) and (9) it can be seen thatsds

sqs VandV are balanced two phase voltage of equal

peak values and the latter /2 angle phase lead with

respect to other component.The equation (10) and (11) shows that, sinusoidal

variable in stationary frame appears as a d.c.quantities in a synchronously rotating reference

frame. This is an important derivation. One thing is

important to note that the stator variables are not

necessarily balanced sinusoidal wave. In fact, they

may be any arbitrary time function.

We can write the following stator circuit equationin stationary frame as [5],

sqs

sqss

sqs

dt

diRV (12)

sds

sdss

sds

dt

diRV (13)

Where,sqs and

sds are q-axis and d-axis stator

flux linkage respectively.

When these equation are converted to de-q

eframe,

the following equation can be written

dseqsqssqsdt

diRV (14)

qsedsdssds dt

diRV (15)

The last term in equation (14) and (15) can be

defined as speed e.m.f due to rotation of axes, that

is, when e =0, the equation reverts to stationary

form. It is important to note here that flux linkage

in de

and qe

axis induces e.m.f in qe

and de

axes

respectively, with /2 angle (lead).

When all the variable and parameter are referred to

the stator, since the rotor actually moves at

speed r , the d-q axis fixed on rotor moves at a

speed re relative to synchronously rotating

frame. Therefore, in de-qe frame, the rotor equation

can be modified as,

drrqrqrrqr )(dt

diRV (16)

qrrdrdrrdr )(dt

diRV (17)


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Fig. 4: de q

eDynamic Model Equivalent Circuit

In fig. 4, the flux linkage expression in terms ofcurrent can be written as follow

)ii(LiL qrqsmqss1qs (18)

)ii(LiL qrqsmqrr1qr (19)

)ii(L qrqsmqm (20)

)ii(LiL drdsmdss1ds (21)

)ii(LiL drdsmdrr1dr (22)

)ii(L drdsmdm (23)

Putting equation (18) & (21) in equation (14) and

(15),

drmedsseqrmqsssqs iLiLisLi)sLR(V (24)

qrmeqssedrmdsssds iLiLisLi)sLR(V (25)

Similarly putting equation (19) & (22) in equation

(16) & (17), we get

dsmredrrreqsmqrrrqr iL)(iL)(isLi)sLR(V (26)

qsmreqrrredsmqrrrdr iL)(iL)(isLi)sLR(V (27)

The development of torque by the interaction of

air-gap flux and rotor MMF is discussed before.

Here, it will be expressed in more general form

relating the d-q component if variable.

drqmqrdm

ii2

P

2

3T

(28)

From equations (24), (25), (26), (27), and (28)

gives the complete model of electro-mechanical

dynamics of induction machine in synchronous

frame [6].

III. EFFECT OF BEARING DAMAGE ON

INDUCTION MOTOR

If we consider damage in either in outer race or in

inner race of the bearing then rotor shifts toward

the line joining centre of bore and the point of

damage, and from the figure 5, it can be seen that

the air-gap is no longer remains uniform[11].

Fig. 5: Schematic Diagram of Static Eccentricity

Therefore, air-gap at any point P is given by,

cos*a)( (29)

Since, permeance of the air-gap is inversely

proportional to the air-gap length. Therefore,

)cos*1(

1

cos*a

1

)(

1ec

(30)

The Fourier series expansion of equation (30),

gives the harmonics contents in the permeance due

to static eccentricity and given by,

......)cos(1

10ec (31)

This can be also written as

sei0iec icossese(32)

Where ise =1,2,3, the value of magnitude ofharmonics content can be found from

2i

2

i

1

112

se

sei

se

(33)

the permeance of a slotted rotor and stator can be

expressed as,

tRiSiRicos)t,( rrtstrtstrtiist,rt strt (34)

MMF produced by stator and rotor winding

comprised of space and time harmonics.


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1i iri

1i i

1it

r r

r1rrrr

s s

ssss

tpnsipicosiF

tipicosiF)t,(F

(35)

Now the flux density distribution in the air-gap is

given as the product of the permeance and the mmf.

Hence, combining equations (34) and (35) to get

complete expression of flux density distribution as

riiMM

siiM

M

tMcosB

tMcosB)t,(B

rtrtrtirti

rti

rti

stststisti

stisti

(36)

Where,

rnsarri

sarri

saecsri

saecsri

mpsi2Ri

i2Ri

pi2iSiRinpM

pi2iSiRimpM

rt

st

st

rt

IV. SIMULATION MODEL , TEST RESULT

AND DISSCUSSION

On the basis of mathematical equation following

simulation model has been designed in Simulink.

Two bearing damages; 5mm and 10mm are testedon simulated model of 5.4 HP, 3 phase induction

motor with following specification:

HP= 3 LV =400V f= 50Hz sR =0.435

sXl =0.004rR =0.816

TXl =0.002

J=0.089Kgm2

mX =0 .0 6931

rpm=1430 P=4

Fig. 6: Simulation Model of 5.4 Hp Motor

It can be seen from fig.6. that the 3-phase supply is

given to B 1 ,where, it is converted into dynamic d-

q voltage variable. In block B 2 and B 3 separate

fluxes of stator and rotor are produced. Now, the

output of block B 2 and B 3 is given to block B 4,

where both fluxes interacts each other. The output

of block B 4 is again fed back to B 2 and B 3, afterwhich, the influenced rotor and stator current are

taken into consideration. The input in block B 6 isproduced by the interaction of electromagnetic

torque and load torque.

Fig. 7: Rotor and Stator Current of Healthy

Induction Motor

Fig. 7. shows the behaviour of stator and rotor

current of healthy induction motor. The transient

state of stator and rotor current is complete in

around in 0.75 sec and after that steady state

reaches.

Fig. 8: Electromagnetic Torque of Healthy

Induction Motor

Fig. 8 shows the behaviour of electromagnetic

torque. The transient state of electromagnetictorque ends after 0.85 sec.


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Fig. 9: Stator and Rotor Current with Load

Torque 15N and Coefficient of Friction 0.1

Fig. 10: Electromagnetic torque with Load

Torque 15N and Coefficient of Friction 0.1

As the bearing fault occurs in the induction

machine the transient state of induction machineincreases. It becomes around 1.2 sec for stator and

rotor current and 1.75 sec for electromagnetic

torque. It can be seen from fig. 9 and fig. 10. As the

severity of the bearing damage increases thetransient time also increases. If the harmonic

analysis of either stator or rotor current spectrum is

performed then we find that a particular harmonic

component is observed for certain degree of

damage.

The comparisons of different parameter of the

induction motor are shown in table 1.

HIM IMBF IMBF

LT (N-m) 11.9 15 20

ck 0 0.1 0.2

sI (A) 8.66 20.15 36.88

rI (A) 7.7 20.5 36.77

eT (N-m) 14.2 33.33 45.9

Where, ck = Coefficient of Friction, HIM= Healthy

Induction Motor, and IMBF= Induction Motor with

Bearing Fault

It can be seen from the table 1, that as the bearing

damage is getting severe, the magnitude of stator

and rotor current also increased. The

electromagnetic torque is also increases.

V. CONCLUSION

In this paper, an implementation and dynamic

modelling of a three-phase induction motor using

Matlab/Simulink are presented in a step-by-step

manner, by using differential equation. It can be

observed that how bearing damage affects the

performances of induction motor. The model was

tested by two different bearing damage of induction

motors. The two bearing fault in machines have

given a significant response in terms of stator and

rotor current. From the simulation results of

presented work it may be concluded that the wave

shape of stator and rotor changes as the percentage

of severity of damage increases. The harmonicanalyses further justify the presented result.

REFERENCES

[1] P. C. Krause, O. Wasynczuk, S. D. SudhoffAnalysis of E Machinery and Drive

Systems,Wiley&Sons,Inc. IEEE Publication

Second Edition, 2002.

[2] P.C. Krause and C. H. Thomas, Simulation

of Symmetrical Induction Machinery, IEEE

Transaction on Power apparatus and Systems,Vol. 84, pp. 1038-1053, November 1965.

[3] M. L. de Aguiar, M. M. Cad, The concept of

c applied to the modeling, Power Engineering

of Society inductio Winter Meeting, pp. 387

391, 2000 .

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Electromechanical Energy Conversion,John

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[5] S. Wade, M. W. Dunnigan, B. W. Wil

induction machine vector control with rotor

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on Power Electronics, vol. 12, No. 3, pp. 495

506, May 1997.

[6] H. C. Stanley, An Analysis of the Induction

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57(Supplement), pp. 751-755, 1938.

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[8] D. S. Brereton, D. G. Lewis, and C. G.

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[9] G. Kron, Equivalent Circuits of Electric

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[11] Ashish kamal and V.K.Giri Detection ofBearing Fault in 3 Phase Induction Motor

Using Wavelet, VSRD Journal, pp. 199-206, May 2013.

mathematical modeling of dynamic induction motor with bearing fault

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