A MODEL FOR INDUCTION MOTORS WITH NON-UNIFORM AIR-GAP

Transcription

1 Latin Ameican Applied Reseach 35:77-82 (25) A MODEL FOR INDUCION MOORS WIH NON-UNIFORM AIR-GAP G. BOSSIO, C. DE ANGELO, J. SOLSONA, G.O. GARCÍA and M.I. VALLA # Gupo de Electónica Aplicada, Facultad de Ingenieía, Univ. Nac. de Río Cuato, Ruta Nac. #36 Km. 6, X584BYA Río Cuato, Agentina gbossio@ing.unc.edu.a; cdeangelo@ieee.og; g.gacia@ieee.og Instituto de Investigaciones en Ingenieía Eléctica "Alfedo Desages", Dto. de Ingenieía Eléctica y de Computadoas, Univ. Nac. del Su, Avda. Alem 253, B8CPB Bahía Blanca, Agentina jsolsona@uns.edu.a # Laboatoio de Electónica Industial, Contol e Instumentación (LEICI), Univ. Nac. de La Plata, CC 9, 9 La Plata, Agentina m.i.valla@ieee.og Abstact Equations to calculate inductances of induction motos, consideing non-unifom ai-gap, ae poposed. he analyzed ai-gap vaiations ae static and dynamic eccenticity and stato slots. he equations fo inductance calculation, obtained fom the modified winding functions and the enegy stoed in the ai-gap, allow consideing the effect of oto ba skewing. Expeimental esults that validate the poposed method ae pesented. Keywods Ai-Gap eccenticity, Induction Motos, Modeling, Winding Function Appoach. I. INRODUCION he conventional d-q model of an induction moto (IM) is based on the assumption that the stato phase windings and the squiel cage can be modeled as theephase winding sets, sinusoidally distibuted. his implies that the hamonics of the winding distibution ae neglected in the analysis of the machine. A model based on the geomety and winding distibution, having no estictions egading its symmety, is moe suitable fo the moto analysis and simulation unde asymmety and fault conditions. In oliyat et al. (99), a multiple-coupled cicuit machine model and a method to calculate the mutual inductances, known as "Winding Function Appoach" (WFA), was pesented. his model was used fo the analysis of concentated winding IM in adjustable speed dive applications. A detailed depiction of the pocedue, needed to implement such model, togethe with simulation esults fo an IM, was pesented in Luo et al. (995). By means of this model, all the hamonics his wok was suppoted by Univesidad Nacional de Río Cuato, Univesidad Nacional del Su, Univesidad Nacional de La Plata, ANPCy and CONICE. 77 of the spatial winding distibution ae taken into account, with no estictions concening the symmety of both the stato windings and the oto bas. Hence, this model has been applied in the analysis of asymmety and fault conditions in IM. he WFA has also been used in the analysis of a five-phase eluctance moto oliyat et al. (992). In Luo et al. (995), oliyat and Lipo (995), Milimonfaed et al. (999), Joksimovic and Penman (2) and Nandi and oliyat (22), the WFA was used to analyze IM faults such as shoting, opening and abnomal connections of the stato phase cicuits, as well as boken oto bas and cacked oto end ings. he analysis of static and dynamic eccenticity effects using the cited model is pesented in oliyat et al. (996) and Joksimovic et al. (2). he authos of these woks calculate the inductances using the equations pesented in Luo et al. (995), which do not take ai-gap vaiations into account. Fom this analysis, the mutual inductances between stato phase and oto loops (L s ) ae diffeent fom those between the oto loops and stato phase (L s ), and it is difficult to find a physical meaning fo this asymmety. In Al-Nuaim and oliyat (998), a modification of this method, consideing ai-gap eccenticity, is poposed and used in the analysis of dynamic eccenticity in a synchonous machine. he new method was called "Modified Winding Function Appoach" (MWFA), and it has been applied to analyze static, dynamic, and mixed eccenticity in IM by Nandi et al. (997), Nandi et al. (2) and Nandi et al. (22). In the pevious woks, based on WFA o MWFA, the machine analysis is caied out assuming unifomity down the axial length of the moto. hat is, without skew and with unifom ai-gap along the oto. In Joksimovic et al. (999), the skew effect on the inductances is analyzed using the equations developed

2 Latin Ameican Applied Reseach 35:77-82 (25) fo the analysis of the machine with axial unifomity. his extension, howeve, does not allow the analysis of adial and axial ai-gap non-unifomity effects. In Baakat et al. (2), the effect of stato and oto slots, as well as static eccenticity, is analyzed. he pemeance function, which includes slot effects, is deived fom simulation using finite elements. he equations fo inductance calculation allow consideing adial and axial non-unifomity effects of both ai-gap and windings. Howeve, the effects of such nonunifomities on the magnetomotive-foce distibution ae not consideed. In Bossio et al. (22), the equations fo the calculation of the machine winding inductances, consideing winding and ai-gap adial and axial nonunifomities, ae developed. hese equations ae obtained fom linked flux and applied to the analysis of static ai-gap eccenticity effects. In this wok, expeimental esults and those obtained fom the model ae simila except fo the oto slot effects, not modeled. In the pesent wok, the equations fo IM inductance calculation ae deived fom the stoed magnetic enegy. hese equations ae applied to the analysis of ai-gap eccenticity effects consideing skew and stato slots. he stato slot modeling allows obtaining a geat similitude with the expeimental esults. II. INDUCION MOOR MODEL Consideing an m-stato-cicuit, n-oto-ba IM, the cage can be viewed as n identical and equally spaced oto loops (Luo et al., 995). Voltage equations fo the IM can be witten in vecto-matix fom as follows: whee, d s Vs = Rs I s + λ, dt () d V = R I + λ, dt (2) s s s V s = v v2... v m, (3) [ ] V... =, (4) s s s I s = i i2... i m, (5) I = i i2... i n, (6) and the stato and oto flux linkages ae given by, λs = Lss Is + Ls I, (7) λ = Ls Is + L I, (8) whee L ss is an m*m matix containing the stato self and mutual inductances, L is an n*n matix also containing the oto self and mutual inductances, L s is an m*n matix composed of the mutual inductances between the stato phases and the oto loops, L s is an n*m matix composed of the mutual inductances between the oto loops and the stato phases, and Ls = L s. he mechanical equations fo the machine ae, d ω = ( e l), dt J (9) l dθ = ω, () dt whee θ is the oto position, ω is the angula speed, J l is the oto-load inetia and l is the load toque. Speed and position dependant viscosity can be included into the load toque. he machine electomagnetic toque, e, can be obtained fom, Wco e =. () θ ( Is, I constant) When satuation is neglected, the magnetic co-enegy can be expessed as the enegy stoed in the magnetic cicuits, W co = IsLssIs + IsLsI + ILsIs + ILI (2) he pecise knowledge of the inductances making up the matices in (7) and (8) is essential fo the analysis and simulation of the IM. he following sections pesent a method fo the calculation of these inductances, consideing ai-gap non-unifomities due to oto and stato slots and ai-gap eccenticity. Skew effects can also be included in inductance calculation with this method. III. INDUCANCE EQUAIONS DERIVED FROM SORED ENERGY he MMF in the ai-gap, Fx ( φ, z, θ ), poduced by a cuent i x flowing in any coil x, is given by (Bossio et al., 22), Fx ( φ, z, = Nx( φ, z, ix, (3) whee Nx ( φ, z, θ ) is the Modified Winding Function (MWF), and can be obtained as, ( ) Nx( φ, z, = nx φ, z, θ 2π L n (,, ) (,, ), x z g z dz d φ θ φ θ φ 2 πl g ( φ, z, θ ) (4) whee θ is the oto position, φ and z ae the angula and axial position of an ai-gap abitay point, 78

3 G. BOSSIO, C. DE ANGELO, J. SOLSONA, G.O. GARCÍA, M.I. VALLA espectively. n(, z, ) Distibution, g (, z, ) function and g ( z ) φ θ is the Winding Spatial φ θ is the invese of the ai-gap φ,, θ is thei aveage value. he MWF poposed in Bossio et al. (22) allows consideing a non-unifom geometic distibution of the windings and otos bas down the moto axial length, e.g. skew. It is also possible to model the oto eccenticity by means of the ai-gap function, g( φ, z, θ ), which has no estictions about axial nonunifomity. Fo the induction machine, the MWF can be defined fo each stato winding and each oto loop composed of two consecutive bas. Witing the magnetomotive foce in the ai-gap as a function of the MWF, the equations fo the moto inductance calculation can be deived fom the stoed magnetic enegy as follows. he enegy stoed in the ai-gap by two windings A and B, is given by (Lipo, 996): WAB = LAB ia ib = A B dv 2 2 B H, (5) v whee B A is the magnetic flux density poduced by a cuent I A flowing in any coil A, H B is the magnetic field intensity poduced by a cuent I B flowing in any coil B and v is the ai-gap volume. Witing the volume integal in cylindical coodinates and consideing that B A and H B ae in adial diection (adial flux), yields L 2π e( φ, z, θ) W B z H z dd dz = ( φ,, ( φ,, φ. AB A B 2 ( φ, z, i (6) If B A (φ, z, θ ) and H B (φ, z, θ ) ae constant in the adial diection, L 2π WAB = m g(, z, ) BA (, z, ) HB (, z, ) d dz, 2 φ θ φ θ φ θ φ (7) whee and ( ) g φ, z, θ = e i, (8) e + i m = cte. (9) 2 hen substituting FB ( φ, z, θ ) HB( φ, z, θ) = g φ, z, θ, (2) B A F φ θ µ (, z, ) ( ) ( φ, z, θ ) ( φ, z, = A g, (2) in (7) yields (,, ) (,, ) g( φ z L 2π FA z FB z W = AB m d dz 2 φ θ φ θ µ φ,,. (22) Replacing, in the pevious equation, F A and F B by Eq. (3) L 2π = (,, ) (,, ) 2 NA φ z θ NB φ z θ µ φ g( φ, z,, W i i d dz then AB m A B (,, ) (,, ) g( φ, z, L 2 A B LAB m µ dφdz (23) π = N φ z θ N φ z θ. (24) Fom this equation it is possible to calculate the stato coil and oto loop mutual and self inductances. his is a function of winding distibution and ai-gap geomety. Futhemoe, the equation shows that using the poposed MWF and the obtained equations fo the inductance calculation, the inductance L BA esults equal to the inductance L AB. his equality does not depend on eithe the winding distibution o the ai-gap. he same equations fo the inductance calculation can be obtained by using linked flux as shown in Bossio et al. (22). IV. INDUCANCE CALCULAIONS he moto inductance calculation, using the equations obtained in the pevious section, is pesented. Ba skewing effects, as well as those fom oto slots and static ai-gap eccenticity, ae consideed in inductance calculation. As an example, the mutual inductances between a stato phase and a oto loop fo the moto, whose paametes appea in the appendix, ae shown. A. Skew Roto ba skewing is modeled in the oto loop spatial distibution function, n( φ, z, θ ). If the ai-gap vaiation due to oto slots is not included in the ai-gap function, g ( φ, z, θ ), then the aveage effect can be included in the Cate coefficient (Lipo, 996). Figue ( shows the inductance fo no skew wheeas Fig. ( shows the coesponding inductance when a skewing of one stato slot peiod is consideed. Fom the obtained esults, no changes in the mutual inductance magnitude ae obseved. Howeve, its shape gets smoothe in pesence of skew, mainly in the egion whee the inductance vaies. In the case of ai-gap unifomity, the equations used fo inductance calculation can be educed to the equations pesented in Joksimovic et al. (999). 79

4 Latin Ameican Applied Reseach 35:77-82 (25).5.5 Mutual Inductance, LAR (mh) Mutual Inductance, LAR (mh) Position, θ (ad) Figue. Calculated mutual inductance (thick line) and its deivative (thin line) of stato phase A and oto loop, L AR. without skew, with skew. B. Stato Slot Modeling he ai-gap vaiation, due to the stato slots, was modeled consideing the distibution of flux lines on the slots, as poposed in Lipo (996). Figue 2 shows this flux line distibution on the slots. Regading this flux lines distibution, the ai-gap in a slot ises linealy to the cente of the slot and then dops up to its nominal value, g, at the othe slot side, as shown in Fig. 2. Figue 3 shows the effect of the stato slots on the mutual inductance between the oto loop and the stato phase A. Figue 3 shows the inductance when skew is not pesent, wheeas Fig. 3 shows the coesponding inductance when skew bas ae consideed. he effect of stato slots is bigge fo the not skewed moto and it seems bette to be appeciated in the inductance deivative. he oto-slot skewing poduces a significant eduction of this hamonics as it can be seen in Fig. 3. he esults fo the no-skewed moto ae simila to the obtained, using finite elements, in Nandi et al. (22). g g Stato Roto Figue 2. Flux lines distibution on the ai-gap, aigap function. φ Positon, θ (ad) Figue 3. Calculated mutual inductance (thick line) and its deivative (thin line) of stato phase A and oto loop, L AR, including the effect of stato slots. without skew, with skew. C. Ai-Gap Eccenticity Modeling As it was stated in Nandi et al. (22), thee ae two types of ai-gap eccenticity: static and dynamic. In pactice, these eccenticities appea as a mixed fom. he ai-gap function fo mixed eccenticity can be epesented by, g( φ, θ ) = g ( es cosφ ed cos( φ θ ) ), (25) whee e s and e d ae static and dynamic eccenticity amounts, espectively. his equation is used in (4) and (24) fo the IM self and mutual inductance calculation. he combination of ai-gap eccenticity and stato slot effects can also be analyzed. Fo this pupose, the incease of the ai-gap function due to oto slots is added to the ai-gap function (25). Fo static eccenticity, self and mutual inductances of the stato windings, L ss, ae constant wheeas those of the oto loops, L, change with oto position. Fo dynamic eccenticity, on the othe hand, the stato self and mutual inductances ae oto position functions. Unlike them, self and mutual inductances of the oto loops ae constant, since they do not expeiment any aigap change when the oto tuns. As an example, the mutual inductance of stato phase A and oto loop, L AR, with a 5% static eccenticity, without skew and with skew, is shown in Fig. 4 ( and in Fig. 4 (, espectively. his figues shows that the static eccenticity poduces a significant change in the mutual inductance due to the peiodic vaiation of the ai-gap in font of the oto loop. Fo the moto without skew, Fig. 4 (, the eccenticity inceases the hamonics poduced by the stato slots mainly when the oto loop is in the minimal ai-gap egion. he skew smoothes this inductance vaiation due to static eccenticity, educing the stato slots hamonics amplitude. his is evident in the inductance deivative (thin line). Fo static, dynamic and mixed ai-gap eccenticities, unifom down the axial length and without oto ba and stato winding skew, the equations used fo the 8

5 G. BOSSIO, C. DE ANGELO, J. SOLSONA, G.O. GARCÍA, M.I. VALLA inductance calculation can be educed to the ones used in Nandi et al. (2) and Nandi et al. (22). his is the case shown in Fig. 4 (, whee the esult obtained fom the poposed method is the same as that obtained fom the MWFA. In addition, the poposed method allows consideing skew and ai-gap eccenticity combined effects, as shown in Fig. 4 (. his poposal also allows modeling the non-unifom ai-gap eccenticity down the axial length of the IM. Mutual Inductance, LAR (mh) Position, θ (ad) Figue 4. Calculated mutual inductance (thick line) and its deivative (thin line) of stato phase A and oto loop, L AR as a function of oto position, θ. with 5% of static eccenticity and without skew, with 5% static eccenticity and skew. V. EXPERIMENAL RESULS In ode to validate the poposed method, the mutual inductance between a oto loop and a stato phase was measued on an IM consisting of a standad stato and two special otos. he standad stato coesponds to a 5.5 KW-38 V-Fame 32S machine with all its phase coils in seies connection. he IM paametes wee used to calculate the inductances in Section IV. wo special no-ba otos wee constucted, one with and the othe with no skew. A special winding was placed along two consecutive oto slots. A -Hz sinusoidal voltage was applied in a stato phase to calculate the inductance between this phase and the oto winding. he voltage induced in the oto winding was measued as a oto position function. he inductance was calculated fom the atio between the induced and the applied voltage, L ( R AR ( ) LAA VA V θ =, (26) whee L AA is the stato phase self inductance, peviously calculated by means of stato voltage and cuent measuements. he obtained inductance, efeed to one tun on the oto, is shown in Fig. 5. Figue 5 ( shows the measued mutual inductance fo the non-skewed oto (thick line) and its calculated deivative (thin line). his inductance is vey simila, in amplitude and shape, to that calculated using the poposed method, shown in Fig. 3 (. As fo the measued and the calculated using the poposed method inductances, a small 48-cycle-peoto-evolution hamonic component appeas, which can clealy be seen in the inductance deivative. his hamonic is geneated by the ai-gap vaiation poduced by the stato slots. Including the stato slots in the model, not only allows to have a good appoximation in the inductance calculation but also in its deivative, impoving the authos pevious poposal (Bossio et al., 22). Figue 5 ( shows the mutual inductance measued in a machine with skewed-oto bas (thick line) and its deivative (thin line). he hamonic component, poduced by the stato slots, is educed due to skew. As fo inductance calculation using the poposed method, expeimental esults show that the mutual inductance magnitude does not change but its shape gets smoothe with skew, mainly in the egion whee the inductance vaies. his effect is bette appeciated in the inductance deivative. VI. CONCLUSIONS In this pape, a method to calculate the inductances of an induction moto with non-unifom ai-gap was poposed. his method allows taking adial and axial asymmeties such as skew and ai-gap non-unifom eccenticity down the machine axial length into account. As an example, skew, ai-gap eccenticity and stato slot effects on the mutual inductances between stato phases and oto loops wee shown. Expeimental esults that validate the poposed method wee also pesented. Mutual Inductance, LAR (mh) Position, θ (ad) Figue 5. Measued mutual inductance (thick line) and its deivative (thin line) of stato phase A and oto loop, L AR as a function of oto position, θ. without skew, with skew. 8

6 Latin Ameican Applied Reseach 35:77-82 (25) APPENDIX Moto data: Nominal powe: 5.5 Kw. Pai of poles: 2. Stato winding: 67 tuns pe coil, 2 coil pe goup, 4 goup pe phase, seies connection, step ::2. Numbe of stato slots: 48. Numbe of oto slots: 4. Ai-gap (g ):.45 mm. Stato length (L):. m. Ai-gap aveage adius ( m ):.75 m. REFERENCES Al-Nuaim, N.A. and H.A. oliyat, A Novel Method fo Modeling Dynamic Ai-Gap Eccenticity in Synchonous Machines Based on Modified Winding Function heoy, IEEE ansactions on Enegy Convesion, 3, (998). Baakat, G., G. Houdouin, B. Dakyo and E. Destobbelee, An Impoved Method fo Dynamic Simulation of Ai-Gap Eccenticity in Induction Machines, IEEE Intenational Symposium on Diagnostics fo Electical Machines, Powe Electonics and Dives (SDEMPED'), Gado, Italy, (2). Bossio, G., C. De Angelo, J. Solsona, G. Gacía and M. I. Valla, A 2D- Model of the Induction Moto: An Extension of the Modified Winding Function Appoach, 28th Annual Confeence of the IEEE Industial Electonics Society IECON 22, Sevilla, Spain, (22). Joksimovic, M.G., D.M. Duovic and A.B. Obadovic, Skew and Linea Rise of MMF Acoss Slot Modeling-Winding Function Appoach, IEEE ansactions on Enegy Convesion, 4, (999). Joksimovic, M.G. and J. Penman, he Detection of Inte-un Shot Cicuits in the Stato Windings of Opeating Motos, IEEE ansactions on Industial Electonics, 47, (2). Joksimovic, M.G., D.M. Duovic, J. Penman and N. Athu, Dynamic Simulation of Dynamic Eccenticity in Induction Machines-Winding Function Appoach, IEEE ansactions on Enegy Convesion, 5, (2). Lipo,., Intoduction to AC Machine Design, Wisconsin Powe Electonics Reseach Cente, Univesity of Wisconsin, (996). Luo, X., Y. Liao, H. oliyat, A. El-Antably and.a. Lipo, Multiple Coupled Cicuit Modeling of Induction Machines, IEEE ansactions on Industy Applications, 3, 3-38 (995). Milimonfaed, J., H. Meshgin Kelk, A. De Minassians, S. Nandi, and H.A. oliyat, A Novel Appoach fo Boken Roto Ba Detection in Cage Induction Motos, IEEE ansactions on Industy Applications, 35, -6 (999). Nandi, S., H. oliyat and A. Palos, Pefomance Analysis of A Single Phase Induction Moto Unde Eccentic Conditions, IEEE Industy Applications Society Annual Meeting (IAS'97), New Oleans, USA, 74-8 (997). Nandi, S., S. Ahmed, and H.A. oliyat, Detection of Roto Slot and Othe Eccenticity Related Hamonics in a hee Phase Induction Moto with Diffeent Roto Cages, IEEE ansactions on Enegy Convesion, 6, (2). Nandi, S., and H.A. oliyat, Novel Fequency- Domain-Based echnique to Detect Stato Intetun Faults in Induction Machines Using Stato-Induced Voltages Afte Switch-Off, IEEE ansactions on Industy Applications, 38, - 9 (22). Nandi, S., R. Bhaadwaj and H. oliyat, Pefomance Analysis of a hee Phase Induction Moto Unde Mixed Eccenticity Condition, IEEE ansactions on Enegy Convesion, 7, (22). oliyat, H.A.,.A. Lipo and J.C. White, Analysis of a Concentated Winding Induction Machine fo Adjustable Speed Dive Applications- Pat I (Moto Analysis), IEEE ansaction on Enegy Convesion, 6, (99). oliyat, H.A., L. Xue, and.a. Lipo, A Five Phase Reluctance Moto with High Specific oque, IEEE ansaction on Industy Applications, 28, (992). oliyat, H.A. and.a. Lipo, ansient Analysis of Cage Induction Machines Unde Stato, Roto Ba and End Ring Faults, IEEE ansactions on Enegy Convesion,, (995). oliyat, H., M. Aeffen and A. Palos, A Method fo Dynamic Simulation of Ai-Gap Eccenticity in Induction Machines, IEEE ansactions on Industy Applications, 32, 9-98 (996). Received: Octobe 8, 23. Accepted: Octobe 27, 24. Recommended by Guest Editos C. E. D Attellis and S. Ponce. 82