Mathematics and Computers in Simulation 61 (2002) 1–15

Simulation and fault detection of three-phase induction motors

B. Lianga,∗, B.S. Payneb,1, A.D. Ballb, S.D. Iwnickiaa Department of Engineering and Technology, Manchester Metropolitan University, Manchester M1 5GD, UK

b School of Engineering, University of Manchester, Manchester M13 9PL, UK

Abstract

Computer simulation of electric motor operation is particularly useful for gaining an insight into their dynamicbehaviour and electro-mechanical interaction. A suitable model enables motor faults to be simulated and the changein corresponding parameters to be predicted without physical experimentation. This paper presents both a theoreticaland experimental analysis of asymmetric stator and rotor faults in induction machines. A three-phase induction motorwas simulated and operated under normal healthy operation, with one broken rotor bar and with voltage imbalancesbetween phases of supply. The results illustrate good agreement between both simulated and experimental results.© 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.

Keywords: Fault simulation; Mathematical models; Condition monitoring; Induction motors

1. Introduction

Induction motors are critical components in many industrial processes. In spite of their robustness theydo occasionally fail and their resulting unplanned downtime can prove very costly. Therefore, conditionmonitoring of electrical machines has received considerable attention in recent years. There are many waysto detect mechanical and electrical problems in induction motors, either directly or indirectly. Directly,many parameters can be monitored to provide useful indications of incipient faults. These parametersinclude case vibration and noise, stator phase current, air-gap or external magnetic flux density. Hargiset al. showed that mechanical defects are detectable through variations in the vibration of the stator core[1–6]. It has been suggested by Steele that per-phase current monitoring can provide similar indications.The advantages of per-phase current monitoring are firstly, it is easy to measure; secondly, the faultpatterns in the current signal are unique and thirdly, an estimation of the number of broken bars can bemade from the analysis[7–12]. Stephan et al.[13] used air-gap search coils to detect and estimate boththe peripheral and radial component of the leakage flux. An axial flux sensing method to detect rotorshort circuits and other malfunctions has also been described by Penman et al.[14]. Indirectly, modelling

∗ Corresponding author. Tel.:+44-161-2471667; fax:+44-161-2471663.E-mail addresses: b.liang@mmu.ac.uk (B. Liang), bradley.payne@man.ac.uk (B.S. Payne).

1 Tel.: +44-161-2754315.

0378-4754/02/$ – see front matter © 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.PII: S0378-4754(02)00064-2

2 B. Liang et al. / Mathematics and Computers in Simulation 61 (2002) 1–15

and simulation of electrical machine operation under healthy and faulty conditions will also provideuseful information for fault prediction and identification. Computer simulation of motor operation can beparticularly useful in gaining an insight into their dynamic behaviour and electro-mechanical interaction.With a suitable model, motor faults may be simulated and the change in corresponding parameters can besimulated. It has been found that an accurate motor simulation can be achieved with a general model andaccurate parameter selection, in addition to using a sophisticated model[15,16]. This can significantlyreduce the computer simulation time and make model-based condition monitoring more reliable andeasily achievable. Following the literature review, it has been found that this area has received only littleattention. Although a few papers exist on simulation and fault diagnosis of induction motors[17], noasymmetric stator and rotor fault simulation of induction motors have simultaneously been made to date.A review of literature also revealed that the variation in the rotor transient speed is thought to be insensitiveto fault conditions.

This paper presents both a theoretical and experimental analysis of asymmetric stator and rotor faultsin electric motors. A three-phase induction motor was simulated and operated under normal healthy

Fig. 1. Simulation of a healthy motor during start-up: (a) speed; (b) electromagnetic torque; (c) stator phase-A current; (d) statorphase-B current; (e) stator phase-C current.

B. Liang et al. / Mathematics and Computers in Simulation 61 (2002) 1–15 3

operation, with one broken rotor bar and with voltage imbalances on one phase of supply. In partic-ular, the rotor transient speed variation was been selected as one of the main parameters to be moni-tored.

2. Induction motor modelling and simulation

This research aimed to investigate whether induction motor operational simulations with general equa-tions could be used for condition monitoring and fault diagnosis of induction motors. Therefore, a theo-retical motor analysis was made based on generalised rotating field theory and by making the followingassumptions which are commonly regarded as appropriate[18]:

• The magnetic permeability of iron is considered to be infinite and the air-gap is very small andsmooth.

• The state of operation remains far from magnetic saturation.• The self-inductances and mutual-inductances between stator or rotor phases are constant.• Mutual-inductances between the stator and rotor windings are functions of the rotor position.• Space magnetic motive force (MMF) and flux profiles are considered to be sinusoidally distributed and

higher harmonics are negligible.

Fig. 2. Stator current waveforms from simulated and measured results during motor start-up.

4 B. Liang et al. / Mathematics and Computers in Simulation 61 (2002) 1–15

VsA

VsB

VsC

Vra

Vrb

Vrc

=

Rs + pLss pMs pMs pMsr cosθr pMsr cosθr1 pMsr cosθr2

pMs Rs + pLss pMs pMsr cosθr2 pMsr cosθr pMsr cosθr1

pMs pMs Rs + pLss pMsr cosθr1 pMsr cosθr2 pMsr cosθr

pMsr cosθr pMsr cosθr2 pMsr cosθr1 Rr + pLrr pMr pMr

pMsr cosθr1 pMsr cosθr pMsr cosθr2 pMr Rr + pLrr pMr

pMsr cosθr2 pMsr cosθr1 pMsr cosθr pMr pMr Rr + pLrr

IsA

IsB

IsC

Ira

Irb

Irc

(1)

hereIsA, IsB, IsC are the stator three-phase currents,Ira, Irb, Irc the rotor three-phase currents,Lss, Lrr

the stator and rotor self-inductances,Ms,Mr the stator or rotor phase mutual inductance,Msr thestator and rotor mutual inductance,p the differential operator,Rs, Rr the stator and rotor resistances,VsA, VsB, VsC the stator three-phase voltages,Vra, Vrb, Vrc the rotor three-phase voltages,θr the rotorangle (θr1 = θr + 2π/3, θr2 = θr + 4π/3).

• The rotor bars are electrically insulated as far as the iron surfaces are concerned.

2.1. Induction motor modelling

The necessary electrical model of the three-phase induction motor was obtained using well-docu-mented motor models[18]. The matrix form of the stator and rotor voltage equations is shown inEq. (1).

Fig. 3. Rotor speeds from simulated and measured results during motor start-up.

B. Liang et al. / Mathematics and Computers in Simulation 61 (2002) 1–15 5

The electromagnetic torqueTe is given by

Te = 32Msr(IqsIdr − IdsIqr) (2)

The mechanical model can be created by

Jdωr

dt= Te − Tl − Tf (3)

whereIds, Iqs are the statord, q phase currents,Idr, Iqr the rotord, q phase currents,J the inertia of therotor,Tl the applied load,Tf the frictional torque loss,ωr the rotor angular velocity.

Digital simulation was implemented in the MATLAB/SIMULINK software package.Fig. 1 presentsthe simulation results in rotor speed, electromagnetic torque and three-phase stator currents for a healthymotor. It can be observed that the rotor speed, electromagnetic torque and three-phase stator currentsbecome steady after initial fluctuation when the motor runs up.Figs. 2 and 3compare stator current androtor speed from simulated and measured results during motor start-up. They illustrate good agreementbetween simulated and measured results has been achieved. It therefore indicates the model and theselected parameters are appropriate.

Fig. 4. Simulation of 20 V drop in one phase of voltage supply: (a) speed; (b) electromagnetic torque; (c) stator phase-A current;(d) stator phase-B current; (e) stator phase-C current.

6 B. Liang et al. / Mathematics and Computers in Simulation 61 (2002) 1–15

Fig. 5. Comparison of simulated and measured rotor speed for 20 V drop in one phase of voltage supply.

Fig. 6. Measured three-phase stator currents for 20 V drop in one phase of voltage supply.

B. Liang et al. / Mathematics and Computers in Simulation 61 (2002) 1–15 7

In order to validate the simulation results an experimental induction motor test rig was developed forexperimentation. The test rig comprises a 3 kW induction motor, a loading DC motor and a resistor bankto dissipate the electrical energy generated. The instrumentation consisted of accelerometers, Hall effectcurrent transducers, a high resolution speed encoder (360 pulses per cycle), amplifiers, filters and a DIFA210 data acquisition system.

2.2. Simulation of imbalanced stator voltage supply faults

Phase imbalance faults of 20 and 40 V drops in one phase of supply were simulated. In order to simulateimbalanced supply faults little modification to the model was required. The stator voltage supply equationswere changed as follows:

VsA =√

2(E − E0) cosωt, VsB =√

2E cos(ωt − 2

3π), VsC =

√2E cos

(ωt + 2

3π)

(4)

hereE is the rated voltage,E0 = 20 or 40 V.The 20 V drop in one phase condition was simulated andFig. 4 illustrates the simulated rotor speed,

torque and three-phase stator current results. It can be observed that there is rotor speed and torque

Fig. 7. Simulation of 40 V drop in one phase of voltage supply: (a) speed; (b) electromagnetic torque; (c) stator phase-A current;(d) stator phase-B current; (e) stator phase-C current.

8 B. Liang et al. / Mathematics and Computers in Simulation 61 (2002) 1–15

Fig. 8. Comparison of simulated and measured rotor speed for 40 V drop in one phase of voltage supply.

ripple and there is a noticeable drop in the amplitude of the phase-A current. The ripple frequencyis at 100 Hz, which is two times the UK supply frequency of 50 Hz. This is the typical symptomof an asymmetric stator fault because the imbalanced stator phase voltages typically result in torquepulsing and rotor speed fluctuation[17]. Fig. 5 presents a comparison of the measured and simu-lated rotor speed. As expected, rotor speed fluctuation is clearly visible in the measured data fromthe test motor. Although both simulated and measured results are similar, the measured speed rippleis larger than the simulated speed ripple by about 30 rpm. This could be caused by approximationsused in the model as it only considers the fundamental frequency and non-saturation conditions.Fig. 6shows experimental three-phase stator currents and the corresponding current decrease in phase-A can beobserved.

Simulation of a 40 V drop in one phase of voltage supply is illustrated inFig. 7. It can be seenthat there are larger rotor speed and torque fluctuations (at 100 Hz) and a larger decrease in phase-Acurrent compared withFig. 4. Fig. 8 provides a comparison of measured and simulated rotor speed.As for the 20 V imbalance, a larger rotor speed ripple can also be seen in the measured results thanin the simulated results. Although the speed ripple difference between the measured and simulatedresults increases to about 50 rpm the frequency is still at 100 Hz.Fig. 9 represents the mea-sured three-phase stator currents in which a further reduction in the phase-A current can beobserved.

B. Liang et al. / Mathematics and Computers in Simulation 61 (2002) 1–15 9

Fig. 9. Measured three-phase stator currents for 40 V drop in one phase of voltage supply.

2.3. Simulation of a single broken rotor bar fault

In order to simulate a broken rotor bar, the mathematical motor model used for simulation of imbal-anced stator voltages was further modified with respect to each rotor bar’s resistance and inductanceeffects.Fig. 10presents a simple mesh model of a squirrel-cage rotor[16]. For a rotor havingNr bars andtwo end rings, there areNr + 2 loops and 2Nr nodes. This is becauseNr bars will haveNr loops. If thetwo end rings are considered as another two loops, there will beNr + 2 loops. The current distribution

Fig. 10. The rotor bar mesh model.

10 B. Liang et al. / Mathematics and Computers in Simulation 61 (2002) 1–15

can therefore be specified in terms ofNr + 2 independent rotor currents which flow in loops compris-ing of two adjacent rotor bars and two end ring segments that join them. Each rotor bar and end ringsegment is characterised by a resistanceRbn (or Ren) and an inductanceLrn,rn associated with the rotorloop. In order to get interaction between the stator and rotor the bar currentibar,n = irn+1 − irn �= 0under normal condition. If the bar is broken, thenibar,n = irn+1 − irn = 0, so thatirn+1 = irn.This means that a broken rotor bar can be simulated by forcing the current flowing in that particularbar to 0.

The following expression can be used to simplify the mathematical model for the voltage equations ofthe induction motor[18]:

V = RI + dψ

dt=

[R + dL

dt

]I + L

dI

dt(5)

hereV = [ VsA VsB VsC 0 0 · · · 0 ], I = [ IsA IsB IsC ir1 ir2 · · · irNr ie ],

Fig. 11. Simulation of a broken rotor bar fault: (a) speed; (b) electromagnetic torque; (c) stator phase-A current; (d) stator phase-Bcurrent; (e) stator phase-C current.

B. Liang et al. / Mathematics and Computers in Simulation 61 (2002) 1–15 11

and

R =

RsA 0 0 0 0 0 0 0

0 RsB 0 0 0 0 · · · 0

0 0 RsC 0 0 0 · · · 0

0 0 0 Rl −Rb1 · · · −Rbn Ren

0 0 0 −Rb1 Rl · · · · · · Ren

· · · · · · · · · · · · · · · · · · · · · · · ·0 0 0 −Rbn · · · · · · Rl Ren

0 0 0 Ren Ren Ren Ren Nr × Rbn

,

Fig. 12. Comparison of simulated and measured results in one stator phase current: (a) measured stator phase-A current; (b)simulated stator phase-A current.

12 B. Liang et al. / Mathematics and Computers in Simulation 61 (2002) 1–15

L =

LsA,sA MsA,sB MsA,sC Mr1,sA Mr2,sA · · · MrNr,sA 0

MsA,sB LsB,sB MsB,sC Mr1,sB Mr2,sB · · · MrNr,sB 0

MsA,sC MsB,sC LsC,sC Mr1,sC Mr2,sC · · · MrNr,sC 0

Mr1,sA Mr1,sB Mr1,sC Lr1,r1 Mr1,r2 · · · Mr1,rNr Len

Mr2,sA Mr2,sB Mr2,sC Mr1,r2 Lr2,r2 · · · Mr2,rNr Len

· · · · · · · · · · · · · · · · · · · · · Len

MrNr,sA MrNr,sB MrNr,sC Mr1,rNr Mr2,rNr · · · LrNr,rNr Len

0 0 0 Len Len · · · Len Nr × Len

hereirn is thenth rotor loop current,Len thenth end ring self-inductance,Nr the number of rotor bars,Rbn, Ren thenth bar and end ring resistance andψ the flux linkage.

The simulation results for a single broken rotor bar are presented inFig. 11. It shows that a low frequencyof 2sω (s is the rotor slip andω the supply frequency) modulation exists in rotor speed, electromagnetictorque and three-phase stator currents. This symptom is the typical fault indication for a broken rotor barfault [7]. Figs. 12 and 13provide a comparison of simulated and measured results for one phase statorcurrent and rotor speed. The measured results illustrate slight modulation in the stator per-phase current

Fig. 13. Comparison of simulated and measured rotor speeds for a broken rotor bar fault.

B. Liang et al. / Mathematics and Computers in Simulation 61 (2002) 1–15 13

Fig. 14. Rotor speed spectra for healthy and faulty conditions: (a) normal condition; (b) broken one rotor bar; (c) 20 V drop ofone phase voltage; (d) 40 V drop of one phase voltage.

and greater modulation in rotor speed with the introduction of this type of fault. These differences areassumed to be caused by the assumptions used in the model and by slight errors in setting particularparameter values. For example, one of assumptions was that the rotor bar current is 0 if a bar is broken.In practice, however, there may be a current path through the laminations between the adjacent bars. Thiseffect would lead to a decrease in the stator phase current modulation and therefore could account for thedifferences between the simulated and measured stator phase currents. Similarly, it is difficult to estimateexact values for the model parameters (such as bar resistance, self-inductance and mutual-inductance).Errors in the setting of these parameters could also contribute to the differences between simulated andmeasured results.

Finally, Fig. 14presents the measured rotor speed spectra for the induction motor under healthy andfaulty conditions. It is clearly seen that symptoms for both imbalanced stator voltage supply (2ω =100 Hz) and a broken rotor bar (2sω = 4.6 Hz) exist in rotor speed spectra.

3. Conclusions

This paper presented the successful simulation and condition monitoring of induction motors withtraditional electrical machine models. It has been demonstrated that a generalised motor model can beused to simulate induction motor faults to a high degree of accuracy.

14 B. Liang et al. / Mathematics and Computers in Simulation 61 (2002) 1–15

The simulated results showed that a clear typical fault symptom of 100 Hz ripple for asymmetricalstator voltage supply can be found in both electro-magnetic torque and rotor speed. The ripple amplitudeincreases with increasing asymmetric stator supply. Experimental results verified the simulations butshowed higher rotor speed ripple amplitudes when compared with the simulated results (with a 30–50 rpmerror).

Another modified model for asymmetric rotor faults (such as broken rotor bars) was also constructedand examined. The simulated results illustrated noticeable 2sω modulations in stator phase current,electromagnetic torque and rotor speed. The experimental results also showed the symptoms but withmuch less phase current modulation and higher fluctuations in rotor speed than the simulated results. Thereason for the phase current discrepancy may be related to the assumptions and approximations of themodified model for broken rotor bar.

The simulated and experimental results indicate that model-based fault detection and diagnosis is useful.The simulation showed that the dynamic characteristics of induction motors under different conditionscan be obtained purely by computer simulation. With a suitable model, motor faults may be simulatedand predicted without any experimental analysis.

References

[1] C. Hargis, B. Gaydon, K. Kamash, The detection of rotor defects in induction motors, in: Proceedings of the InternationalConference, Vol. 313, July 1982, IEE Publication, pp. 216–220.

[2] E. Erdelyi, P.A. Erie, Vibration modes of states of induction motors, Transactions of ASME, Paper A-28, 1956,pp. 39–45.

[3] I.W. Mayes, A.G. Steer, G.B. Thomas, The application of vibration monitoring for fault diagnosis in large turbo-generators,in: Proceedings of the Sixth Thermal Generation Specialists Meeting, Madrid, May 1981, pp. 567–575.

[4] R.G. Herbert, Computer techniques applied to the routine analysis of rundown vibration data for condition monitoring ofturbine-alternators, in: Proceedings of the International Conference on Condition Monitoring, Swansea, UK, April 1984,pp. 229–242.

[5] J.R. Cameron, W.T. Thompson, A.B. Row, Vibration and current monitoring for detecting air gap eccentricity in largeinduction motors, in: Proceedings of the International Conference on Electrical Machines, Design and Applications, Vol.254, September 1982, IEEE, London, pp. 173–179.

[6] J.H. Maxwell, Induction motor magnetic vibration, in: Proceedings of the Vibration Institute, Machinery VibrationMonitoring and Analysis Meeting, Houston, Texas, 1983, pp. 213–218.

[7] M.E. Steele, R.A. Ashen, L.G. Knight, An electrical method for condition monitoring of motors, in: Proceedings ofthe International Conference of Electrical Machines on Design and Application, July 1982, IEE, London, pp. 231–235.

[8] J.B. Kliman, J. Stein, Induction motor fault detection via passive current monitoring, in: Proceedings of the InternationalConference on Electrical Machinery, Cambridge, August 1990, pp. 13–17.

[9] J. Stefin, et al., Methods of motor current signature analysis, Electrical Mach. Power Syst. 20 (5) (1992) 463–474.[10] R.S. Randy, et al., Motor bearing damage detection using stator current monitoring, IEEE Trans. Ind. Appl. 31 (6) (1995).[11] David, R. Rankin, The industrial application of phase current analysis to detect rotor winding faults in squirrel cage induction

motors, Power Eng. J. 23 (2) (April 1995).[12] R.R. Schoen, An unsupervised, on-line system for induction motor fault detection using stator current monitoring, IEEE

Trans. Ind. Appl. 31 (6) (1995) 1281–1286.[13] J. Stephan, M. Bodson, J. Chiasson, Real-time estimation of the parameters and fluxes of induction motors, IEEE Trans.

Ind. Appl. 30 (3) (1994) 746–758.[14] Penman, J., Dye, M.A.N., Tit, A.J., Bryan W.E., Condition monitoring of electrical drives, in: Proceedings of the IEE, Vol.

133, Part B, No. 3, May 1986, pp. 142–148.

B. Liang et al. / Mathematics and Computers in Simulation 61 (2002) 1–15 15

[15] T.A. Lipo, A. Consoli, Modelling and simulation of induction motors with saturable leakage reactance, IEEE Trans. Ind.Appl. IA-20 (1984) 180–189.

[16] E. Ritchie, X. Deng, T. Jokinen, Dynamic model of three-phase squirrel cage induction motors with rotor faults, in:Proceedings of the ICEM, IEE Publication, 1994, pp. 694–698.

[17] M. Benbouzid, A review of induction motor signature analysis as a medium for fault detection, IEEE Trans. Ind. Electron.47 (5) (2000) 984–993.

[18] P. Vas, Electrical Machines and Drives: A Space-Vector Theory Approach, Clarendon Press, Oxford, 1988.