A Generalized Two Axes Model of a Squirrel-Cage Induction Motor for Rotor Fault Diagnosis

Transcription

1 SEBIAN JOUNAL OF ELECTICAL ENGINEEING Vol. 5, No. 1, ay 2008, A Genealized Two Axe odel of a Squiel-Cage Induction oto fo oto Fault Diagnoi Sami Hamdani 1, Oma Touhami 2, achid Ibtiouen 2 Abtact: A genealized two axe model of a quiel-cage induction moto i developed. Thi model i baed on a winding function appoach and the coupled magnetic cicuit theoy and take into account the tato and the oto aymmetie due to fault. Thi pape peent a compute imulation and expeimental dynamic chaacteitic fo a healthy induction machine, machine with one boken ba and a machine with two boken ba. The eult illutate good ageement between both imulated and expeimental eult. Alo, the powe pectal denity PSD wa pefomed to obtain a tato cuent pectum. Keywod: Induction machine, Diagnoi, Powe pectal denity. 1 Intoduction ot electic moto failue inteupt a poce, educe poduction, and may damage othe elated machiney. In ome factoie, vey expenive cheduled maintenance i pefomed in ode to pevent udden moto failue. Theefoe, thee i a conideable demand to educe maintenance cot and pevent uncheduled downtime fo electical dive ytem, epecially induction moto [1]. Thu, finding efficient and eliable fault diagnotic method i extemely impotant. In liteatue, method baed on analytical model of moto ytem ae till the mot common choice fo condition monitoing of the induction machine. Some of thee method ae baed on a claical model which take into account the individual conducto in the oto cage uing -L eie cicuit, with cuent loop defined by two adjacent oto ba connected by potion of the end ing. Theefoe, fo N ba, thi appoach lead to a tanient model with a total of N + 3 diffeential equation. Such a model i quite complex and compute imulation become vey long [2,3]. Othe method, baed on the d-q model uppoe that both tato and oto winding ae ymmetic and the F i inuoidally ditibuted. In thi model, the equivalent inductance and 1 USTHB, faculté d électonique et infomatique, BP 32 El Alia Bab Ezzoua, Alge, Algeie; ham_ami@yahoo.f 2 eeach Laboatoy of Electotechnic, 10, Av Pateu El Haach, Algie, BP182, Algeia. 155

2 S. Hamdani, O. Touhami,. Ibtiouen eitance matix ae diagonal. Thi model i vey imple fo compute imulation but it can t eflect any winding aymmetie due to the tato o oto fault [4,5]. In thi pape, a new modelling method baed on a winding function appoach and the coupled magnetic cicuit theoy [6-8] which take into account the tato and the oto aymmetie due to the fault, i peented. The method uppoe that both eitance and inductance of the tato ae not equal. So, it i poible to intoduce any tato aymmety winding due to fault uch a the inte-tun hot cicuit. The oto i epeented by a N loop coupled to each othe and to the tato winding, though mutual inductance. Fo a healthy machine, the oto loop ae identical and have the ame paamete, but when a oto fault occu, ome loop ae affected. In thi condition, the equivalent eitance and inductance matix of the tato and the oto in the two axe model ae not diagonal, making thi model moe genealized than the claical d-q model. 2 Genealized Two Axe odel of the Induction achine The tato of the machine conit of thee inuoidally ditibuted winding, diplaced by 120, with N equivalent tun, but not neceaily having the ame eitance and inductance. The quiel cage contain N ba foming N identical loop. Each one conit of two adjacent oto ba connected by two end ing potion. So, fo a cage having N ba, thee ae 2N node and 3N banche. Theefoe, the cuent ditibution can be pecified in tem of N +1 independent oto cuent. Thee cuent compie of the N oto loop cuent i j (1 j N ) plu a ciculating cuent in one of the end ing i e a illutated by Fig. 1. Fo modelling of the induction machine, the following geneal aumption wee made: negligible atuation; unifom ai gap; tato winding inuoidally ditibuted; negligible inteba cuent. The voltage equation fo the tato winding can be witten a: d V = I + φ (1) dt whee 156

3 A Genealized Two Axe odel of a Squiel-Cage Induction oto fo oto... T V = [ v 1, v2, v3] i the tato voltage vecto; T I = [ I 1, I2, I3] i the tato cuent vecto; i an 3 by 3 eitance matix given by: = (2) T φ =φ [ 1, φ2, φ3] i a tato flux linkage vecto, witch i given by : φ = LI + I (3) The tato inductance matix L i ymmetic with contant element, it expeion i: L L = 21 L2 23 (4) L 3 The mutual inductance i a 3 by N matix coniting of mutual inductance between tato coil and oto loop, a follow: N 1 = N (5) N I = [ I1, I2,, IN ] i the oto cuent vecto ; each element I j epeent the cuent loop. A change of vaiable which fomulate tanfomation of the 3-phae tato vaiable and N oto vaiable to the abitay two axe efeence may be done by the ue of K and K tanfomation matixe, epectively given by: 2π 2π in ( θ) in θ in θ K = (6) 3 2π 2π co ( θ) co θ- co θ

4 S. Hamdani, O. Touhami,. Ibtiouen K ( 2j 1) pα p α in θ θ in θ θ = N pα p( 2j 1 ) α in in θ θ θ θ 2 2 whee θ=ω t, θ =ω t i the two axe efeence angula velocity, ω i the oto angula velocity. By the ue of K S, the tato voltage can be witten a: 1 d ( 1 V = K K I + K K φ ). (8) dt By development of thi expeion, it i poible to find the electic equation of the tato in the two axe efeence. Thi equation i given by: 0 1 d V = I +ω φ φ (9) dt whee dz i the equivalent eitance matix, it given by: d =. (10) q (7) Lb e Le b ij-1 Lb e e Le b ij Le e Le b Lb ie ij+1 b Lb whee Fig. 1 oto cage epeentation. 158

5 A Genealized Two Axe odel of a Squiel-Cage Induction oto fo oto... 1 d = ( ), q = 1+ ( ), (11) = ( 2 3 ). 2 3 In the ame manne, the flux equation can be witten a: φ = L I + I, (12) with: L 1 d L L = K L K =. (13) L Lq Fig. 2 Two axe efeence fame. Note that fo a healthy induction moto, the tato winding ae identical and have the ame eitance and inductance ( 1 = 2 = 3 =, L 1 = L2 = L3 = L ), the equivalent eitance and inductance matix and L will be diagonal: i the equivalent mutual inductance matix between the tato winding and the oto loop. Thi matix i given by: =, 0 (14) 0 L L =. 0 L (15)

6 S. Hamdani, O. Touhami,. Ibtiouen 1 d = K K =. (16) q Fom the oto cage equivalent cicuit epeented by Fig. 1, the electic equation of the j -th loop can be defined a: dφj 0= 2( b + e) ij bi( j 1) bi( j+ 1) +. (17) dt φ j i the flux coing the oto loop j, it epeent the um of the flux due to tato cuent and the flux due to the oto loop. The expeion of φ j i given by: N φ = ( L + 2( L + L )) i + + i L i L i i j p b e j jk k b ( j 1) b ( j+ 1) k k= 1 k= 1 k j 3. (18) Thu, the oto electic equation can be witten a: d 0 = I + φ (19) dt whee i the equivalent oto loop eitance matix, it expeion i given by : 2( b + e) b b b 2( b e) b 0 + =. (20) b 0 2( b + e) b and e ae: the ba eitance and the end-ing egment eitance, epectively. φ =φ [ 1, φ2,, φn ] epeent the oto loop flux vecto: φ = LI + I (21) whee L i a N N inductance matix of oto loop given by (22): Lp + 2( Lb + Le ) Lb Lb Lb Lp 2( Lb Le ) Lb + + L = Lb (22) Lb Lp + 2( Lb + Le ) 160

7 A Genealized Two Axe odel of a Squiel-Cage Induction oto fo oto... L b i the oto ba leakage inductance, e T = L i the end-ing egment inductance and. Tanfomation (6) and (7) applied to electical equation of the oto loop give: 0 1 d 0 = I + ( ω ω ) φ φ, d t (23) whee φ = L I + I, T = ( ) and: d =, (24) q L d L L =. (25) L Lq, L and epeent the oto eitance matix, the oto loop inductance matix and the coupling inductance matix between the oto loop and tato coil, epectively. Conideing (9) and (23), one can wite the geneal expeion of the quiel induction moto in the two axe efeence fame given in (26) a: V I φ d φ = + K +, (26) 0 I dt φ φ whee: 0 = (27) 0 0 ω 0 0 ω K = (28) ω ω ( ) 0 0 ( ω ω ) 0 φ I = L (29) φ I with: L L = (30) L 161

8 S. Hamdani, O. Touhami,. Ibtiouen In ode to obtain the healthiet poible imulation, it i convenient to avoid the deivative function and ue only integal function. Howeve, the tato and oto cuent can be obtained accoding to the flux by: I φ 1 = L (31) I φ Finally, ubtituting (31) into (26) yield: d φ V φ 1 = ( ) L + K. (32) dt φ 0 φ Note that the inductance matix L depend on the oto angle poition θ, it will be neceay to update and evee it at each calculation tep of the imulation. Fo the mechanical equation, it can be witten a: d p ω = ( T e T L) (33) dt J whee T L i the load toque; T e i the electomagnetic toque, it can be expeed a patial vaiation of the co-enegy with epect to the oto poition [5]. 1 T d 1 Te = ( K I) ( K I ). (34) dθ Equation (32-34) decibe the quiel induction moto model in the genealized two axe efeence fame. With thi model it will be poible to modify tato and oto paamete of the moto in ode to conide ome fault uch a: the inte-tun hot cicuit and the oto loop ba and end-ing default. 3 Calculation of the Inductance fo a Healthy Induction oto The inductance ae calculated by mean of a winding function. Accoding to thi theoy and fo a contant ai-gap, the mutual inductance between any two winding i and j i given by [6]: 2π μ0 ij = Ni ( θ, ϕ) N j ( θ, ϕ)dϕ g. (35) 0 Ni( θ, ϕ ) and N j( θ, ϕ ) ae the winding function of the i and j coil, epectively. They epeent the patial ditibution of the F along the aigap fo a unit cuent ciculating in the winding [5], [6]. Thi function depend on the patial poition of any point along the ai-gap defined by ϕ and the oto angula poition compaed to a given efeence defined by θ. 162

9 A Genealized Two Axe odel of a Squiel-Cage Induction oto fo oto... Conide the tato winding dipoition hown by Fig. 3. Taking a a efeence the magnetic axi of the fit phae, the nomalized tato winding function fo thi phae i: N N 1 = coϕ (36) 2 Subtituting (36) into (35), one can deduce the magnetizing inductance fo each tato coil a: 2 μ0 N π Lm = (37) 4g The total inductance of the tato coil i the um of the magnetizing inductance L m and the leakage inductance L f. Howeve, one can wite fo a ymmetic tato winding: L = L = L = L + L (38) f m whee l i the tack length, i the aveage adiu of the ai-gap, g i the adial ai-gap length and N i the numbe of tun of the tato coil. The mutual inductance of two tato coil i and j ( i j) i given by: ij Lm =. (39) 2 Fig. 3 Winding ditibution of the fit tato coil. Fig. 4 Winding ditibution of the oto loop. Fig. 4 how the function winding of a oto loop elated to a given oto poition. θ i and θ i+ 1 indicate the poition of the two ba which fom the oto loop. Thi function i defined by: α 0 ϕ θ i 2π α N i ( ϕ ) = 1 θi ϕ θi+ 1 (40) 2 π α θ i+ 1 ϕ 2 π 2π 163

10 S. Hamdani, O. Touhami,. Ibtiouen Subtituting (36) and (40) in (35), the mutual inductance between a tato winding j and a oto loop i i expeed by: p(2i 1) 2π ij = Lm co θ + ( j 1) 2 3, (41) whee: α 4in 2 Lm = L m. (42) πn The oto loop inductance L p and the mutual inductance between two oto loop ae epectively given by: μ0 α α Lp = 1, (43) g 2 π μ g 0 = 4 Calculation of the Inductance fo an Induction oto with N bc Ba Boken 2 α. (44) 2π A quiel cage induction machine with oto ba in default i epeented by Fig. 5. Conide N bc to be the numbe of adjacent boken ba. The winding function ditibution coeponding to the equivalent loop i hown in Fig. 6. Fo thi loop, calculation of the inductance led to the following expeion: μ0 α α Lp = ( Nbc + 1) (2Nbc + 1). (45) g 2 π The mutual inductance between thi loop and all othe loop not affected by the fault i: μ α = + g 2π 2 0 bc 1 ( N ). (46) In the ame manne, one can deduce the mutual inductance between a tato coil j and the faulty loop denoted i a: μ 0 p(2i+ Nbc 1) α 2π α id = N co ( j 1) in ( Nbc 1) g θ (47) 2 164

11 A Genealized Two Axe odel of a Squiel-Cage Induction oto fo oto... Fig. 5 epeentation of Nbc boken ba. Fig. 6 Winding ditibution fo N bc ba boken. 5 Simulation and Expeimental eult To imulate the quiel cage induction machine model, a compute pogam witten in ATLAB wa ued. The diffeential equation wee eolved uing the fouth ode unge-kutta method. Fit, the developed model wa ued to imulate a healthy and faulty 4kW quiel cage induction machine with the pecified paamete given in the appendix. To validate the popoed model, the expeimental etup epeented by Fig. 7 and 8 wa deigned and built. Thi tet etup conited of an indutial induction moto coupled to a DC geneato acting a a load and a data acquiition ytem. A hown in Fig. 9, two induction moto of the ame type but with diffeent fault wee available in addition to a healthy machine. The fit had one ba boken and the econd had two boken ba. The tato cuent and voltage wee meaued via LA-55P cuent eno and LV-25 voltage eno, epectively a hown in Fig. 10. Each meaued ignal wa imultaneouly ampled though channel of a 16 bit 200kHz PCI data acquiition (DAQ) boad and toed diectly into a dektop compute. Once the data acquiition wa complete, a atlab pogam wa ued to poce and analyze the data. 165

12 S. Hamdani, O. Touhami,. Ibtiouen Fig. 11, 12 and 13 how imulation and expeimental eult of the intantaneou tato cuent and peed duing tat -up fo the cae of a healthy machine, a machine with one boken ba and a machine with two boken ba. Thee eult wee obtained fo a inuoidal voltage upply with a contant load. The imulation and the expeimental eult ae not exactly the ame but a good ageement i noted between them. The eaon fo the diffeence wee aumed to be caued by the aumption ued in the model and light eo in etting paticula paamete value. Fo example, one of the aumption wa that the oto ba cuent i zeo if a ba i boken. In pactice, howeve, thee may be a cuent path though lamination between adjacent ba. Thi effect would lead to a deceae in the tato phae cuent modulation and theefoe could account fo the diffeence between the imulated and meaued tato phae cuent. Similaly, it i difficult to etimate exact value fo the model paamete (uch a ba eitance, elf-inductance and mutual-inductance). Eo in the etting of thee paamete could alo contibute to the diffeence between imulated and meaued eult. Fig. 7 Diagam of the expeimental etup. Fig. 8 Photo of the expeimental etup. 166

13 A Genealized Two Axe odel of a Squiel-Cage Induction oto fo oto... Fig. 9 oto unde tet. Fig. 10 Cuent and voltage eno boad. Fig. 11 Speed and cuent fo a healthy machine. Fig. 12 Speed and cuent fo a machine with one ba boken. 167

14 S. Hamdani, O. Touhami,. Ibtiouen Fig. 13 Speed and cuent fo a machine with two ba boken. Fig. 14 Powe pectal denity fo a healthy machine. Fig. 15 Powe pectal denity fo a machine with one ba boken. 168

15 A Genealized Two Axe odel of a Squiel-Cage Induction oto fo oto... Fig. 16 Powe pectal denity fo a machine with two ba boken. In ode to extact moe infomation fom the tato cuent wave, the powe pectal denity PSD wa ued to highlight the appeaance of (1 ± 2 k) f fequencie component nea the fundamental one. Fig. 14, 15 and 16 how imulation and expeimental tato cuent PSD value fo a healthy machine, a machine with one boken ba and a machine with two ba boken. One can notice the appeaance of hamonic component in the ight and left ideband of the fundamental with fequencie coeponding exactly to the mathematical elation (1 ± 2 k) f whee f i the fundamental fequency and i the lip. A hown, the fault-elated pectal component tend to gow with the extent of the fault. Fo expeimental eult the amplitude of the fit ight and left ideband i -52dB, -44 db and -36dB fo a healthy machine, a machine with one boken ba and a machine with two boken ba, epectively. In imulation eult, only the fundamental component i peent in the pectum of a healthy machine. Fo the two othe machine the amplitude of the ight and the left ideband i: -43dB and -50dB. 6 Concluion A genealized two axe appoach to quiel cage induction machine modelling ha been peented. The model i baed on a winding function appoach and the coupled magnetic cicuit theoy and take into account the tato and the oto aymmetie due to fault. The model i ued to imulate the behaviou of a healthy and faulty machine and it i validated by expeimental eult. The dynamic chaacteitic of induction moto unde diffeent condition obtained by compute imulation ae in good ageement with thoe obtained expeimentally. The pectum analyi of the tato cuent howed the 169

16 S. Hamdani, O. Touhami,. Ibtiouen exitence of hamonic component in the ideband of the fundamental with fequencie coeponding exactly to the elation (1 ± 2 k) f. 7 Appendix Squiel cage induction machine paamete: P n = 4kW ; V n = 220/380V ; ( Δ /Y); I n = 15.2/ 8.8A ; N = 1435t/mn ; p = 2 ; f = 50Hz ; coϕ = 0.83; n 2 J = 0.002kg m ; = 1.5Ω; L = 7mH ; L m = 0.55H ; L = 0.54H ; L = 0.28μH ; = μΩ; b = 5μΩ; L = 0.036μH ; e e b N = 28 ; = 70mm ; g = 0.28mm ; l = 120mm ; N = efeence [1]. Haji, H.A. Toliyat: Patten ecognition - A Technique fo Induction achine oto Boken Ba Detection, IEEE Tan. On Ene. Conv. Vol. 16, No. 2, Dec. 2001, pp [2] C.C.. Cunha,.O.C. Lya, B. C. Filho: Simulation and Analyi of Induction achine with oto Aymmetie, IEEE Tan. Ind. Appl., Vol. 41, No. 1, Jan/Feb 2005, pp [3] S.J. anola, J.A. Tegopoulo: Analyi of Squiel Cage Induction oto with Boken Ba and ing, IEEE Tan. on Ene. Conv., Vol. 14, No. 4, Dec. 1999, pp [4] Y. Zhao, T.A. Lipo: odeling and Contol of a ulti-phae Induction achine with Stuctual Unbalance, IEEE Tan. on Ene. Conv., Vol. 11, No. 3, Sep. 1996, pp [5] G. Didie, H. azik: Su la detection d un defaut au oto de moteu aynchone, 3EI, No. 27, Déc. 2001, pp [6] A.. unoz, T.A. Lipo: Complex Vecto odel of the Squiel-cage Induction achine Including Intantaneou oto Ba Cuent, IEEE Tan. Ind. Appl, Vol. 35, No. 6, Nov/Dec 1999, pp [7] X.L.Y. Liao, H.A. Toliyat, A.E. Antably, T.A. Lipo: ultiple Coupled Cicuit odeling of Induction achine, IEEE Tan. Ind. Appl, Vol. 31, No. 2, a/ap 1995, pp [8] H.A. Toliyat, T.A. Lipo: Tanient Analyi of Cage Induction achine Unde Stato, oto Ba and End ing Fault, IEEE Tan. on Ene. Conv., Vol. 10, No. 2, June 1995, pp