An accurate model of squirrel cage induction machines under stator faults

Mathematics and Computers in Simulation 63(2003)377–391An accurate model of squirrel cage inductionmachines under stator faultsV .Devanneaux a ,∗,B.Dagues a ,J.Faucher a ,G.Barakat baLaboratoire d’Électrotechnique et d’Électronique Industrielle,UnitéMixte de Recherche INPT-ENSEEIHT/CNRS,BP 7122,2,rue Charles Camichel,31071Toulouse Cedex 7,FrancebGroupe de Recherche en Électrotechnique et Automatique du Havre,Universitédu Havre,BP 540,25,rue Philippe Lebon,76058LE Havre Cedex,FranceAbstractA substantial portion of squirrel cage induction motor faults are stator related.An accurate transient model of squirrel cage induction machines under stator faults is presented here.A coupled magnetic circuits approach is used and very few restrictive assumptions are made.All parameters are calculated from the actual geometry and winding layout of the machines rather than from transformed or equivalent variables.The detailed depiction of the procedure needed to implement such an accurate model with simulation results is the subject of this paper.©2003IMACS.Published by Elsevier B.V .All rights reserved.Keywords:Inductance;Induction machines;Power system faults;Simulation;Squirrel cage motors;Windings1.IntroductionRotating electrical machines play a very important role in the world’s industry.Among them,the three-phase squirrel cage induction motor is frequently used because of its relatively simple,robust construction and its low price.Besides,there is also a strong industrial demand for reliable and safe operation of rotating machines.Faults and failures of critical electromechanical parts can indeed lead to excessive downtimes and generate costs of millions of dollars in reduced output,emergency maintenance and lost revenues.This is why industry is interested in adopting monitoring and diagnosis techniques to assess and evaluate electrical machines condition.Even if induction machines are known as reliable,they can be submitted to external and internal stresses,and degradation can occur in their electrical and mechanical parts [1].Stator faults constitute a substantial portion of the faults related to squirrel cage induction motors [2].These faults (turn-to-turn,coil-to-coil and phase-to-phase short circuits,phase breakdown,ground fault)occur primarily due to the thermal,electrical,mechanical and environmental stresses that the stator windings have to undergo during their life cycle.∗Corresponding author.E-mail address:vincent.devanneaux@leei.enseeiht.fr (V .Devanneaux).0378-4754/$30.00©2003IMACS.Published by Elsevier B.V .All rights reserved.doi:10.1016/S0378-4754(03)00083-1378V.Devanneaux et al./Mathematics and Computers in Simulation63(2003)377–391People who work on monitoring and diagnosis of squirrel cage induction machines often need an accurate model to predict performances and/or to extract faults signatures(on electromagnetic torque, stator currents,mechanical vibrations,etc.).However,model accuracy and computation time represent two opposite criteria.Conventional model obtained with the Park transformation,for instance,is based on restrictive assumptions and does not require lots of computation time.On the contrary,model obtained with thefinite elements method is based on minimal assumptions and requires lots of computation time. There is a real need to establish an alternative model which offers a good balance between accuracy and computation time.This is the aim of the present paper where an accurate transient model of squirrel cage induction machines under stator faults is presented.A coupled magnetic circuits approach[3]is used and very few restrictive assumptions are made.Parameters are calculated from the actual geometry and winding layout of the machines rather than from transformed or equivalent variables.The detailed depiction of the procedure needed to implement such an accurate model with simulation results is the subject of this paper.2.Minimal assumptionsConsider a three-phase squirrel cage induction machine having m stator and n rotor slots.Stator windings are,in afirst analysis,viewed as a set of independent stator coils which can be later connected in any fashion to form the stator phases.The rotor cage is viewed as n identical and equally spaced rotor coils(plus one end-ring coil)which can be later connected to form the correct rotor bars/end-rings configuration.The model is based on a coupled magnetic circuits approach and all space harmonics are taken into account without any restrictions concerning general symmetry of the machine.Moreover,the following assumptions are made:•Permeability of iron is infinite.•Rotor bars are insulated.•Skin effect is not taken into account.•Air-gap is not necessarily smooth.•Rotor bars skewing is taken into account.3.“Native”system equationsNotations used in this section are rather classical and self-explanatory;they will not then be deeply explained in detail.It is worth noticing that[I s]and[J r]are,respectively,composed of three stator phase currents and of n+1rotor loop currents as illustrated in Fig.1;the electromagnetic torque C em is obtained from the magnetic coenergy W co.The following“native”system equations can be written for the three-phase squirrel cage induction machine with n rotor bars:[V s]=[R s]·[I s]+d[Φs]d t(1)[V r]=[R r]·[J r]+d[Φr]d t(2)[Φs]=[L s]·[I s]+[M sr]·[J r](3)V .Devanneaux et al./Mathematics and Computers in Simulation 63(2003)377–391379[Φr ]=[M sr ]t ·[I s ]+[L r ]·[J r ](4)where[V s ]=[v sa v sb v sc ]t (5)[V r ]=[00...00]t (6)[I s ]=[i sa i sb i sc ]t(7)[J r ]=[j r1j r2...j r n j rcc ]t (8)andW co =12I sJ r t ·L s M srM t srL r· I sJ r(9)C em =∂W co∂θ(10)Jd ωd t=C em −C r ,d θd t=ω(11)With application of Kirchoff’s laws on the circuit described in Fig.1,the form of [R r ]rotor resistance matrix and [L r ]rotor inductance matrix can be easily derived.It is important to notice that the n +1rotor loops are coupled to each other and to stator phases.However,the end-ring loop does not couple with the stator phases and couples with the rotor loops only through the end-ring segment leakageinductanceFig.1.Equivalent circuits of stator phases and squirrel cage rotor.380V.Devanneaux et al./Mathematics and Computers in Simulation63(2003)377–391L fcc and the end-ring segment resistance R cc.It is also apparent that the calculation of all magnetising and mutual inductances as defined in the previous equations is the key to successful simulation of the squirrel cage induction machine.4.Calculation of inductancesThe reader willfind a complete demonstration of the formulae given in this section in[4].Main results are only here outlined;they give nevertheless the general guidelines for the calculation of inductances. In the following expressions,subscript‘i’stands for‘s’(stator)or‘r’(rotor);and subscripts‘x’and‘y’are,respectively,the slot numbers occupied by a⊗and oriented coil.4.1.Turns function and winding function of a coilAs mentioned in Section2,stator and rotor windings are in afirst analysis entirely divided into elemen-tary and independent coils.A coil always occupies two slots whose opening isεi and is composed of N iturns.The pitch between two consecutive slots is notedτi.A function n ix−y (ϕ,θ)called“turns function”is then introduced in order to completely describe the considered coil(ϕis a particular angular position along the stator inner surface andθis the angular position of the rotor).Fig.2illustrates turns function of a stator and a rotor coil.A function N ix−y (ϕ,θ)called“winding function”is also introduced;it represents the MMF of the air-gapproduced by a unit currentflowing in the considered coil,and is defined by:N ix−y (ϕ,θ)=n ix−y(ϕ,θ)−12π e−12πn ix−y(ϕ,θ)e−1(ϕ,θ)dϕ(12)where e1(ϕ,θ)is the air-gap function of the machine.4.2.Magnetising and mutual inductance of two coilsCalculation of theflux linking two coils i x−y and i x −y makes it possible to express the following generalised inductance formula:L ix−y/i x −y (θ)=µ0rL2πn ix−y(ϕ,θ)N ix −y(ϕ,θ)e−1(ϕ,θ)dϕ(13)where r is the average radius of air-gap and L the length of the stack.With(12)and(13),the inductancesbetween all stator coils and between all rotor loops can be calculated.Stator coils are then connected(series and/or parallel)in order to form stator phases;corresponding inductances are calculated from thosealready obtained.The air-gap is not considered as being smooth;stator and rotor slots openings effectsare introduced in e1(ϕ,θ).Rotor bars are often uniformly skewed in squirrel cage induction machines. Inductances have been calculated for unskewed rotor bars;ifγis a given angle of skewing,it is possible tocorrect inductances in order to take into account rotor bars skewing.Fig.3represents mutual inductancesbetween stator phases and thefirst rotor loop(no slots openings and no skewing here for simplicity—seemachine parameters in Appendix A).Finally,stator and rotor leakage inductances are calculated fromdesign data[5].V.Devanneaux et al./Mathematics and Computers in Simulation63(2003)377–391381Fig.2.Turns function of a stator and a rotor coil.5.Simulation resultsA complete methodology has been presented in order to determine parameters of Section3equations.These parameters(mostly inductances)are prepared and computed off-line.The“native”system equationsmust be transformed in order to take into account the wye or delta connection of the supplied machine.Theproblem is then entirely transformed into a state formulation˙X=f(X,U,t)in order to be numerically solved.A native C++program is used to perform calculation of inductances and simulate transient andsteady-state behaviour of the squirrel cage induction machine.This program is open and can be interfacedwith commercial simulation ing the derived model,a simulation study has been conductedon a“small”two-pole pairs squirrel cage induction machine whose parameters are given in Appendix A.The acceleration transient of the motor from rest under sinusoidal excitation and under a constant loadtorque of5N m is simulated.Figs.4–6represent,respectively,the electromagnetic torque,the speed andthe stator currents of the supplied machine during acceleration.382V .Devanneaux et al./Mathematics and Computers in Simulation 63(2003)377–391-0.00025-0.0002-0.00015-0.0001-5e-0505e-050.00010.000150.00020.000250123456I n d u c t a n c e (H )Position (rad )M sar1M sbr1M scr1Fig.3.Stator/rotor mutual inductances.-1001020304050607000.20.40.60.81 1.2 1.4T o r q u e (N .m )Time (s)TorqueFig.4.Electromagnetic torque.V .Devanneaux et al./Mathematics and Computers in Simulation 63(2003)377–391383-2002040608010012014016000.20.40.60.81 1.2 1.4S p e e d (r a d /s )Time (s)SpeedFig.5.Motor speed.-25-20-15-10-5051015202500.20.40.60.81 1.2 1.4C u r r e n t (A )Time (s)i sa i sb i scFig.6.Stator currents.384V .Devanneaux et al./Mathematics and Computers in Simulation 63(2003)377–391-1001020304050607000.20.40.60.81 1.2 1.4T o r q u e (N .m )Time (s)TorqueFig.7.Electromagnetic torque (phase breakdown at t =0.5s).6.Stator faultsIn this section,some stator faults cases are evaluated.The inclusion of a sudden phase breakdown and a turn-to-turn short circuit into this model is derived.Simulation results including stator faults cases are given for the same machine and the same experimental conditions as described in the previous sections.6.1.Phase breakdownThere is no need to explicitly change the topology of the equivalent circuit of stator phases in Fig.1to take into account a phase breakdown.Inserting a higher resistance in the injured phase during the simulation is suf ficient.For instance,the breakdown of phase ‘a ’is simulated.Figs.7–9represent,respectively,the electromagnetic torque,the speed and the stator currents of the supplied machine during acceleration,with a breakdown of phase ‘a ’at t =0.5s.It is worth noticing that the stator currents are then transformed into two-phase currents,involving some deep torque and speed oscillations.There is also a risk of further degradation because of the higher amplitude of currents in the two remaining phases.6.2.Turn-to-turn short circuit6.2.1.Elementary considerationsConsider a simple case where an elementary coil A –X has five turns.This elementary coil which is represented in Fig.10occupies two stator slots.A short circuit occurs between the contact points a1andV .Devanneaux et al./Mathematics and Computers in Simulation 63(2003)377–391385-2002040608010012014016000.20.40.60.81 1.2 1.4S p e e d (r a d /s )Time (s )SpeedFig.8.Motor speed (phase breakdown at t =0.5s).-5-4-3-2-10123450.460.480.50.520.54C u r r e n t (A )Time (s )i sa i sb i scFig.9.Stator currents (phase breakdown at t =0.5s).386V.Devanneaux et al./Mathematics and Computers in Simulation63(2003)377–391A'X'Fig.10.Turn-to-turn short circuit into an elementary coil.a2.There is then four effective turns in series and one additional short circuit turn(represented in bold). Therefore,when a short circuit occurs,the effective coil has less turns and then less MMF;there is also creation of a new short circuit loop which is magnetically coupled with other circuits.Turn-to-turn short circuits with more turns can be analysed in a similar manner.6.2.2.New“native”system equationsWith the information provided above,a new equivalent circuit of stator phases under a turn-to-turn short circuit is required.This one is represented in Fig.11.Two new stator branches‘sd’(short circuit turns)and‘scontact’(short circuit occurs between the contact points a1and a2)are added.R sd represents the effective resistance of the short circuit turns (usually small)and R scontact represents the resistance of the contact branch where the short circuit appears (usually high in the healthy case,zero in the sudden short circuit case).This last parameter offers a great flexibility to simulate incipient or sudden faults.It is also worth noticing that the stator branch‘sd’is magnetically coupled with other circuits,while the stator branch‘scontact’is not.Calculation of new inductances is then required.Some new“native”system equations are then associated with this topology:[V s]=[R s]·[I s]+d[Φs]d t(14)[V r]=[R r]·[J r]+d[Φr]d t(15)[Φs]=[L s]·[I s]+[M sr]·[J r](16)R R STATORR scontactsdi scontactFig.11.New equivalent circuit of stator phases under a turn-to-turn short circuit.[Φr ]=[M sr ]t ·[I s ]+[L r ]·[J r ](17)where[V s ]=[v sa v sb v sc v sd v scontact ]t (18)[V r ]=[00···00]t(19)[I s ]=[i sa i sb i sc i sd i scontact ]t (20)[J r ]=[j r1j r2···j r n j rcc ]t (21)andW co =12 I s J rt ·L s M sr M tsrL r· I sJ r(22)C em =∂W co ∂θ(23)Jd ωd t=C em −C r ,d θd t=ω(24)6.2.3.Simulation resultsConsider a case where 22turns of the first elementary coil (slots 1–12)of phase ‘a ’are shorted.It means that approximately 6%of the turns of phase ‘a ’are shorted (344effective turns left).R sa and R sd are then,respectively,6.4and 0.4 .Fig.12illustrates the modi fied mutual inductances between stator phases and the first rotor loop.Figs.13–15represent,respectively,the electromagnetic torque,the speed and the stator currents of phase ‘a ’during acceleration,with a sudden short circuit at t =0.5s.Before the short circuit,R scontact is high enough in order to prevent any current from flowing into the ‘scontact ’branch.The motor operates then at its nominal state.At t =0.5s,R scontact is set to zero in-0.00025-0.0002-0.00015-0.0001-5e-0505e-050.00010.000150.00020.000250.00030123456I n d u c t a n c e (H )Position (rad )M sar1M sbr1M scr1M sdr1Fig.12.Stator/rotor mutual inductances under a turn-to-turn short circuit.-1001020304050607000.20.40.60.81 1.2 1.4T o r q u e (N .m )Time (s )TorqueFig.13.Electromagnetic torque (turn-to-turn short circuit at t =0.5s).-2002040608010012014016000.20.40.60.81 1.2 1.4S p e e d (r a d /s )Time (s )SpeedFig.14.Motor speed (turn-to-turn short circuit at t =0.5s).-60-40-202040600.460.480.50.520.54C u r r e n t (A )Time (s )i sa i sdi scontactFig.15.Stator currents (turn-to-turn short circuit at t =0.5s).order to introduce a sudden short circuit.Some torque and speed oscillations are introduced by the fault. Current i sdflowing into the short circuit turns rises to an extremely high value.There is then a risk of further degradation(coil-to-coil,phase-to-phase short circuits,ground fault).A complete failure of the squirrel cage induction machine is possible.7.ConclusionsAn accurate transient model of squirrel cage induction machines under stator faults has been presented in this paper.This model is directly established on the actual geometry and winding layout of the machines (no transformed or equivalent variables).Model parameters determination mostly consists in inductances calculation;general guidelines have been given in order to calculate these inductances.The inclusion of a sudden phase breakdown and a turn-to-turn short circuit into the model has been studied.A simulation study has been conducted on a squirrel cage induction motor under these stator faults;effects on mechan-ical and electrical quantities have been investigated.It is worth noticing that many other faults can be studied with this model[6,7]because of its accuracy and its versatility.Appendix A.Machine parameters•Wye connection.•V oltage:400V.•Frequency:50Hz.•Stator winding topology:•Turns per coil:61.•Number of stator slots:36.•Number of rotor bars:32.•Unskewed rotor bars.•Rotor inertia:0.0066kg m2.•Length of stator stack:0.127m.•Average air-gap radius:0.046m.•Corrected air-gap length:0.56mm.•Phase resistance:6.8 .•Rotor bar resistance:1.3515×10−4 .•Rotor end-ring segment resistance:1.393×10−6 .•Rotor bar leakage inductance:0.295×10−6H.•Rotor end-ring segment leakage inductance:5.9×10−9H.References[1]A.H.Bonnett,G.C.Soukup,Cause and analysis of stator and rotor failures in three-phase squirrel-cage induction motors,IEEE Trans.Ind.Appl.28(4)(1992)921–937.[2]P.F.Albrecht,J.C.Appiarius,R.M.McCoy,E.L.Owen,D.K.Sharma,Assessment of the reliability of motors in utilityapplications—updated,IEEE Trans.Energy Convers.EC-1(1)(1986)39–46.[3]X.Luo,Y.Liao,H.A.Toliyat,A.El-Antably,T.A.Lipo,Multiple coupled circuit modeling of induction machines,IEEETrans.Ind.Appl.31(2)(1995)311–318.[4]V.Devanneaux,H.Kabbaj,B.Dagues,J.Faucher,A versatile model of squirrel cage induction machines for design,monitoring and diagnosis purposes,in:Proceedings of the Ninth European Conference on Power Electronics and Applications, Graz,Austria,2001(CD-ROM).[5]A.Foggia,Méthodes de calcul des inductances de fuites,TraitéGénieélectrique D3440,Techniques de l’Ingénieur,February1999.[6]V.Devanneaux,H.Kabbaj,B.Dagues,J.Faucher,An accurate model of squirrel cage induction machines under rotor faults,in:Proceedings of the International Conference on Electrical Machines and Systems,Shenyang,China,2001(CD-ROM).[7]V.Devanneaux,H.Kabbaj,B.Dagues,J.Faucher,An accurate model of squirrel cage induction machines under static,dynamic or mixed eccentricity,in:Proceedings of the Third International Symposium on Diagnostics for Electrical Machines, Power Electronics and Drives,Grado,Italy,2001,pp.121–126.。