Dynamic Induction Machine Model Accounting for Stator and Rotor Slotting

1 Dynamic Induction Machine Model Accounting for Stator and Rotor Slotting Gojko Jokimović, Jakša Riger, Thoma Wolbank, Nedjeljko Perić, Mario Vašak, Goran Stojčić, Vinko Lešić Abtract A method for dynamic modelling of induction machine with a doubly lotted air gap i propoed and implemented for the cae of a cage induction motor. The decribed method i eaily extenible to wound rotor machine. A numerical decription of the air gap permeance i provided that take into account a lotted tator and rotor tructure a well a their mutual, time and pace dependant poition a a function of rotor rotation. The multiple coupled circuit model approach i ued with the modified winding function in order to calculate the inductance of all motor winding. The developed model i general in nature and could be ued for the analyi of different dynamic regime of induction machine, particularly different combination of tator and rotor lot number. Model validation i provided by tator current pectrum analyi of a tandard four pole induction motor with S=36 and R=32 lot. The experimental reult preented clearly upport thee finding. Index Term Induction machine, Slot permeance, Rotor lot harmonic, Principal lot harmonic, Modified Winding Function Approach. R I. INTRODUCTION otor lot harmonic (RSH) and their treatment can be conidered a an example of how the paage of time and development of technology change the perpective from which technical phenomena are viewed. A primary conideration of the induction motor deigner i the deign of a motor with good tarting capabilitie i.e. good tarting torque with a mall a poible tarting current, whilt endeavouring to deliver mooth and noiele operation at rated condition. Thi target i a valid today a it wa a century ago. In the firt intance, thi mean that the deigner mut make the choice of optimal tator and rotor lot number configuration, or more preciely, make a choice of optimum rotor lot (bar) number for an (ordinarily) predefined and fixed number of tator lot. Gojko Jokimović and Jakša Riger are with the Faculty of Electrical Engineering, Univerity of Montenegro, Podgorica, Montenegro (e-mail: joxo@ac.me, jaka@ac.me ). Thoma Wolbank and Goran Stojičić are with the Department of Energy Sitem and Electrical Drive, Vienna Univerity of Technology, Vienna, Autria (e-mail: thoma.wolbank@tuwien.ac.at, goran.tojicic@tuwien.ac.at). Nedjeljko Perić, Mario Vašak and Vinko Lešić are with the Faculty of Electrical Engineering and Computing, Univerity of Zagreb, Croatia (e-mail: nedjeljko.peric@fer.hr, mario.vaak@fer.hr, vinko.leic@fer.hr). During the previou century many rule for determining the optimum number of rotor bar for a given number of tator lot and pole pair number have been derived, [1], [2], [3]. More preciely, there are forbidden combination of the number of tator and rotor lot that coincide with the exitence of both RSH in the tator current pectrum. Therefore, intuitively or not, induction motor deigner alway tried to eliminate the chance of trong RSH occurrence. Today, however, the exitence of RSH i looked on more favourably than before, particularly from the perpective of enorle peed etimation, [4], [5], but alo from the condition monitoring and diagnoi point of view, [6]. Indeed it i even the cae that the election criteria for the commercially available motor can be predicated on the bai of pronounced RSH that are adequate for peed enorle application, [7]. Stator and rotor lot permeance i well defined and analyzed in a number of textbook and paper, from different point of view, [1]-[3], [8]. However, it i only recently the cae that thi effect ha been incorporated into the dynamic model of the induction machine, [9]. In [9] the air gap permeance wa decribed uing analytical expreion, i.e. the firt lot permeance harmonic. Thi paper propoe a method for dynamic modelling of the induction motor whilt taking into account a real profile of the double lotted air gap tructure. The invere air gap function i completely numerically decribed and the modified winding function approach (MWFA) i ued for the calculation of all of the motor winding inductance. The propoed model eaily take into account different tator and rotor lot number combination, different lot mouth width, and kewing of rotor bar, [1]. Thee poibilitie give a wide area of application for the model. II. AIR-GAP PERMEANCE MODELLING A i well known, the MWFA allow the calculation of the elf and mutual inductance of all the winding in the machine in the cae of non uniform air gap along the motor circumference, [11]. If the winding function of the winding A, the turn function of the winding B and the air gap permeance function P(θ) are known, then the mutual inductance between thee two winding i given by,

2 L AB 2 ( θ) N ( θ ) n ( θ ) θ = µ π rl P d (1) A A B B where r i the average radiu of the air gap, l i the length of the tack and θ i the angular poition of the rotor with repect to a tator reference (the mechanical angle). The mechanical angle θ A and θ B repreent the angular poition of winding A and B on tator or rotor from a referent poition. A a conequence of the tator and rotor lotting, the air gap length i.e. the air gap permeance function P(θ), become a function of the rotor poition. Thi function can be included in (1) on the bai that the total air gap length can be oberved a a um of two air gap that are eparated by the imaginary circle placed in the middle of the air gap between the tator and rotor a hown in Fig 1. Stator Rotor b( θ ) g ( θ ) B min g max θ b t B max g min=.5g b α.5g.5g Fig. 1. Stator lot geometry, magnetic flux denity and air gap length profile. Rotor i aumed mooth. The firt air gap i the air gap een from the tator ide, which i independent of rotor poition. The econd air gap i the air gap een from rotor ide which rotate with the rotor. The total air gap length i the um of thee two air gap at any intant in time. By rotating the rotor in a tep-by-tep fahion and integrating (1) the inductance of all the machine winding can be calculated. A linear drop of the air gap permeance i.e. a linear rie of air gap length i aumed under the lot, in accordance with the reult preented in [12] for the aturated lot bridge in cloed lot machine. If it i aumed that the rotor hown in Fig 1 i mooth then the following function for the air-gap length under a ingle tator lot pitch could be defined a,.5g g max g + θ, < θ.5( b / r) max.5( b / r).5g,.5( b / r) < θ α.5( b / r) g ( θ ) = p g max.5g.5g + ( θ α +.5( b / r) ),.5( b / r) α ( ).5 b / r < θ α (2) where b i the tator lot width and α =2π/S i the tator lot g θ θ pitch. The maximum value of the tator air-gap length i given approximately by [13], g max and 2u = g (3) 1+ u 2 2g 2 g 2 b b u = + 1+ (4) and the reultant air-gap length function along the tator circumference can be formed by S function which define air gap length under one tator lot pitch (2) given by: g = g g g g... g g (5) p p p p p p 1444444 24444443 S time Similarly, auming a mooth tator and taking into account the rotor lot, the reultant air-gap length function along the rotor circumference can be formed from R function under one rotor lot pitch given by: g = g g g g... g g (6) r rp rp rp rp rp rp 1444444 24444443 R time For known air gap function it i traightforward to define the invere air-gap function i.e. air gap permeance function. Of coure, it mut be conidered that the rotor part of the air gap rotate with the rotor. For a given rotor poition, the airgap permeance function i, P ( θ) 1 = g + g r During the formulation of the above function one hould bear in mind that the two air gap function hould be of the ame length. For example, if S=36 and R=32, the g and g r vector hould have 36 32=1152 or 234 or 3456 entrie etc. Fig 2 how the tator and rotor air-gap length, reultant air-gap length and the air-gap permeance function for the rotor poition defined by θ= rad (the poition where the 1 t tator lot and 1 t rotor lot are in oppoition). III. INDUCTANCE CALCULATION The inductance of all winding in the machine can be calculated uing the modified winding function approach a decribed in [11]. A) Stator winding elf and mutual inductance Stator winding inductance are calculated numerically, (7)

3 uing expreion (1). One rotor revolution can be dicretized in H tep hence dθ=2π/h. In rotating the rotor in a tep-bytep fahion, the air-gap permeance function i defined at each tep by (7), the tator winding function i defined, ( θ) = n( θ) ( θ) n( θ) P( θ) P N (8) look-up table by taking into account an appropriate pace hift between rotor loop. Thi tak can be olved in the ame manner a for tator inductance. However, the rotation of the rotor in a tep-by-tep fahion mean hifting not only the rotor part of air-gap function but alo the rotor turn or winding function. Fig 5 how the mutual inductance profile between phae A and firt rotor loop a well a it derivative. The ame profile are for other two tator phae, appropriately hifted in pace. Fig. 3. Phae winding elf inductance and it derivative, under one pole: S=36, R=32, b =.5α, b r=.5α r, g =.5mm. Fig 4 how the mutual inductance between phae A and B a well a derivative of thi profile a a function of rotor poition. The mutual inductance profile for phae A and C or B and C ha the ame profile but are hifted in pace. Fig. 2. Upide down: tator air-gap length, rotor air-gap length, reultant air-gap length and air-gap permeance function: θ=, S=36, R=32, b =.5α, b r=.5α r, g =.5mm. and the inductance i calculated by (1). In thi way, the elf and mutual inductance of the tator phae can be organized in look-up table. For every entry in the matrix of tator winding elf and mutual inductance, [L ] i a vector of length H, where H=n S R and n i an integer that repreent the number of angle ample at which inductance are calculated. A a reult of thi proce, the elf inductance profile for the tator phae winding A i obtained, a hown in Fig 3. The derivative of thi function, needed for electromagnetic torque calculation, i alo hown on the ame figure. B) Stator rotor mutual inductance In order that all element of the matrix [L r ] can be known, it i enough to know only three look up table: the mutual inductance between phae A and one rotor loop; the mutual inductance between phae B and the ame rotor loop and the mutual inductance between phae C and the ame rotor loop. All other mutual inductance can be obtained form thee three Fig. 4. Phae winding mutual inductance and it derivative, under one pole: S=36, R=32, b =.5α, b r=.5α r, g =.5mm. C) Rotor winding elf and mutual inductance The mot difficult tak in the modelling of an induction motor taking into account tator and rotor lot permeance i the calculation of rotor loop elf and mutual inductance. During the rotation of the rotor, the rotor loop experience a different picture of air gap profile at each dicrete intant in time. Moreover, in the general cae, the mutual-inductance between, for example, rotor loop 1 and rotor loop 2 have a different profile from the mutual inductance profile between any other two loop.

4 Fig. 5. Phae winding rotor loop mutual inductance and it derivative: S=36, R=32, b =.5α, b r=.5α r, g =.5mm. However, by uing the modified winding function approach thi tak can be olved and thee inductance can be numerically calculated. The firt tep i to form the turn function for all of the rotor loop. Then, for each tep, the air-gap permeance function, new winding function and inductance i defined a ha been previouly decribed. Thi i repeated for each tep in the tepby-tep rotor rotation. A the rotor ha a ignificant number of rotor loop thi proce of calculation could take ome time, however it i negligible in comparion with FEM technique. For example, for R=32, with H=36 32, the time for calculation of thi matrix i around 5 minute on a 2GHz Pentium IV PC running on Window XP Profeional with 256MB RAM. Uing the manner decribed above, the rotor elf and mutual inductance matrix [L rr ] become a three dimenional matrix with dimenion R R H, i.e. every entry of thi matrix become a vector (look-up table) of length H. Fig 6 how the elf inductance profile for firt rotor loop and their derivative. Self inductance of the other rotor loop are of the ame hape but with the appropriate pace hift. For other combination of tator and rotor lot number the rotor elf-inductance will have different profile. Fig 7 and 8 how the mutual inductance profile between the firt rotor loop and two other different rotor loop. A can be oberved, in the general cae the mutual inductance profile are different for each different rotor loop. Fig. 6. Self inductance of the 1 t rotor loop a a function of the rotor poition under one pole and it derivative: S=36,R=32,b =.5α,b r=.5α r, g =.5mm. Fig. 7. Mutual inductance between 1 t and 5 th rotor loop a a function of the rotor poition under one pole and it derivative: S=36, R=32, b =.5α, b r=.5α r, g =.5mm. Fig. 8. Mutual inductance between 1 t and 2 th rotor loop a a function of the rotor poition under one pole and it derivative: S=36, R=32, b =.5α, b r=.5α r, g =.5mm. At firt ight it look like R R different look-up table are needed; however, given the linear magnetic circuit mutual inductance between circuit 1 and 2 i the ame a for that between 2 and 1, and given that the elf inductance profile for different loop are the ame, only R R (R 1)(1+.5 R) look up table are needed (i.e. 497 for R=32 which i ignificantly lower than 32 32=124). IV. RESULTS AND DISCUSSION Uing the propoed model a four pole cage induction motor with S=36 lot and R=32 bar wa analyzed (the motor detail are given in the Appendix). In the tator current pectrum of thi motor only one of the rotor lot harmonic could exit: the upper rotor lot harmonic, [14]. Additionally, thi motor ha a number of tator and rotor lot uch that the lot permeance will alo have a very ignificant influence on the ame RSH amplitude becaue the fundamental tator magnetomotive force (MMF) wave through the lot permeance produce upper RSH. It might reaonably be expected that thi upper principal lot harmonic i very prominent in the tator current pectrum. Thee obervation are validated by the reult from the numerical model. Fig 9 how the tator current pectrum for an unloaded motor in three different cae: for a mooth air

5 gap; a lotted air gap with lot opening of 1% of the lot pitch, and for the cae of open lot on tator a well a on the rotor. The influence of the lot permeance i obviou and rather high. Interetingly, the influence of the lot permeance on the RSH amplitude i almot independent of the lot opening in the no-load regime. It hould be noted that the amplitude of the tator current (in thi regime it i the magnetizing current) rie with width of lot opening a could be expected. In the cae of loaded motor a hown in Fig 1, the opening of the lot ha an influence on the lot harmonic amplitude. In the cae of open lot the lot harmonic amplitude i 1% higher when compared with the mooth air gap cae. Experimental reult clearly upport the finding from the numerical model. Fig 11 how the tator current pectrum of a cage induction motor with S=36 lot, R=32 lot and p=2. The upper PSH in thi motor i the mot ignificant higher harmonic in the tator current pectrum, predominantly due to the lot permeance effect. Therefore, for thi motor, it can be concluded that lot permeance i the predominant caue of the exitence of rotor lot harmonic in the tator current pectrum. Fig. 11. Experimentally obtained tator line current pectrum: S=36, R=32, p=2 @ =5.25%. Only upper PSH exit at 88Hz a the mot prominent harmonic component in the pectra! V. CONCLUSION Fig. 9. Stator line current pectrum of the unloaded motor. Top: mooth air gap; Middle: lotted air gap (b =.1α, b r=.1α r); Bottom: lotted air gap (b =.5α, b r=.5α r). A method for the dynamic modelling of the induction motor taking into account a real profile of a double lotted air gap i preented in thi paper. The propoed method i implemented in the cae of a cage induction motor but can eaily be implemented for the cae of wound rotor motor a well. A numerically decribed air gap permeance take into account lotted tator and rotor tructure a well a their mutual, time and pace dependant poition a a function of rotor rotation. A multiple coupled circuit model and a modified winding function approach i ued in order to calculate the inductance of all motor winding. The model i general in nature and could be ued for the analyi of different dynamic regime of induction motor, particularly different combination of tator and rotor lot number. Model validation i provided through the tator current pectrum analyi of a tandard four pole induction motor with S=36 and R=32 lot. The preented experimental tator current pectrum clearly upport finding from the numerical model. ACKNOWLEDGEMENT Thi work wa upported by the European Commiion and the Republic of Montenegro under grant FP7-SEE-ERA.net PLUS ERA 8/1. Fig. 1. Stator line current pectrum of loaded motor, =4.32%. Top: mooth air gap; Middle: lotted air gap (b =.1α, b r=.1α r); Bottom: lotted air gap (b =.5α, b r=.5α r). APPENDIX Machine parameter: P r=3kw, Y, U=4V, I r=6.5a, n r=142rev/min, 5Hz, p=2, l=.15m, r=.5m, S=36 tator lot, R=32 rotor bar, w=15 turn per

6 coil, g =.5mm, J=.113 kgm 2, R (phae)=1.3ω, R b=2µω, R e=1µω, L b=1nh, L e=2nh, γ=2π/36 (angle of kewing of rotor bar). Winding cheme of phae A under one pair of pole: A 1-1-9'-2-1'-3-11'-2-12'-19-11'-18-1'-X 1 REFERENCES [1] W. Nürnberg, Die Aynchronmachine, Springer, Berlin, 1963. [2] P.L. Alger, The nature of induction machine, New York: Gordon and Breach, 1965. [3] I.Boldea, S.A.Naar, The induction machine handbook, CRC Pre 22. [4] K. D. Hurt, T. G. Habetler, "Senorle peed meaurement uing current harmonic pectral etimation in induction machine drive," IEEE Tran. on Power Electronic, Vol. 11, No.1, pp. 66-73, January 1996. [5] K. D. Hurt, T. G. Habetler, "A comparion of pectrum etimation technique for enorle peed detection in induction machine," IEEE Tran. on Indutry Application, Vol. 33, No.4, pp. 898-95, Jul/Aug 1997. [6] S. Nandy, H. A. Toliyat and L. Xiaodong, "Condition monitoring and fault diagnoi of electrical motor a review," IEEE Tran. on Energy Converion, vol. 2, No.4, pp.719-729, December 25. [7] S. Nandy, S. Ahmed, H. A. Toliyat and R. M. Bharadwaj, "Selection criteria of induction machine for peed-enorle drive application," IEEE Tran. on Indutry Application, vol. 39, No.3, pp.74-712, May/June 23. [8] M. H. Hee, "Air gap permeance in doubly-lotted aynchronou machine," IEEE Tran. on Energy Converion, vol. 7, pp.491-499, Sep.1992. [9] S. Nandy, "Modeling of induction machine including tator and rotor lot effect," IEEE Tran. on Indutry Application, vol. 4, pp.158-165, July/Augut 24. [1] G. Jokimović, M. Đurović, A. Obradović, Skew and Linear Rie of MMF Acro Slot Modeling Winding Function Approach, IEEE Tranaction on Energy Converion, vol. 14, pp.315-32, September 1999. [11] J. Faiz, I. Tabatabaei, "Extenion of winding function theory for nonuniform air gap in electric machinery," IEEE Tran. on Magnetic, vol. 38, pp.3654-3657, Nov. 22. [12] S.Williamon, Y.N. Feng, Slot-harmonic field in cloed-lot machine, IEEE Tran. on Indutry Application, vol. 44, No.4, pp.1165-1171, July/Augut 28. [13] T. Jokinen, V. Hrabovcova, Deign of rotating electrical machine, John Wiley and Son, 29. [14] G. Jokimović, Stator current harmonic in aturated cage and wound rotor induction motor, XIX International Conference on Electrical Machine, ICEM 21, Roma, Italy. BIOGRAPHIES G. Jokimović received the B.Sc. (Hon.), M.Sc., and Ph.D. degree in electrical engineering from the Univerity of Montenegro, Podgorica, Montenegro, in 1991, 1995, and 2, repectively. During 1997 1998, he wa a Viiting Reearch Fellow with the Department of Engineering, Univerity of Aberdeen, Scotland, U.K. During 21 22, he wa a Reearch Fellow of the Alexander von Humboldt Foundation with the Intitute of Electrical Energy Converion, Darmtadt Univerity of Technology, Darmtadt, Germany. He i currently a Full Profeor with the Department of Electrical Engineering, Univerity of Montenegro. Hi main reearch area include analyi of electrical machine, condition monitoring of electrical machine, and power electronic and control. He i the author of a few book and everal paper publihed in leading international journal. J. Riger received the B.Sc. degree in Electrical Power Engineering from Univerity of Montenegro, Podgorica, Montenegro in 211. He i currently a Project Aitant at the Department of Electrical Engineering, Univerity of Montenegro and working toward hi MSc degree. Hi pecial field of interet are modeling of cage rotor induction machine for fault detection and diagnoi purpoe. T. M. Wolbank received the doctoral degree and the Aociate Prof. degree from Vienna Univerity of Technology, Vienna, Autria, in 1996 and 24, repectively. Currently, he i with the Department of Energy Sytem and Electrical Drive, Vienna Univerity of Technology, Vienna, Autria. He ha coauthored ome 1 paper in refereed journal and international conference. Hi reearch interet include aliency baed enorle control of ac drive, dynamic propertie and condition monitoring of inverter-fed machine, tranient electrical behavior of ac machine, and motor drive and their component and controlling them by the ue of intelligent control algorithm. N. Perić, received the BSc, MSc and PhD degree in Electrical Engineering from the Faculty of Electrical Engineering and Computing (FER Zagreb), Univerity of Zagreb, Croatia, in 1973, 198, and 1989, repectively. From 1973 to 1993, he worked at the Intitute of Electrical Engineering of the Končar Corporation, Croatia, a an R&D Engineer, Head of the Poitioning Sytem Department, and Manager of the Automation Section. In 1993, he joined the Department of Control and Computer Engineering at FER Zagreb a an Aociate Profeor. He wa appointed a a Full Profeor in 1997 and he currently teache everal coure in automatic control. Hi current reearch interet are in the field of proce identification and advanced control technique, with a pecial focu on application with renewable energy ource. He erve a the Chairman of KoREMA, the Croatian Society for Communication, Computing, Electronic, Meaurement and Control. He i alo a member of everal international profeional aociation. He i a Fellow of Croatian Academy of Engineering. M. Vašak received the BSc and PhD degree in Electrical Engineering from the Faculty of Electrical Engineering and Computing (FER Zagreb), Univerity of Zagreb, in 23 and 27, repectively. He currently work at the Department of Control and Computer Engineering, FER Zagreb a a Senior Aitant. Hi main reearch interet are: identification and optimal control of hybrid ytem with application in automotive ytem, electrical drive, renewable energy ource and tranport ytem. G. Stojčić received the B.Sc. degree in Electrical Engineering and the M.Sc. degree in Power Engineering from Vienna Univerity of Technology, Vienna, Autria in 29 and 211, repectively. He i currently a Project Aitant at the Department of Energy Sytem and Electrical Drive, Vienna Univerity of Technology and working toward hi PhD degree. Hi pecial field of interet are fault detection and condition monitoring of inverter fed machine. V. Lešić received the B.Sc. degree at the Department of Electric Machine, Drive and Automation of the Faculty of Electrical Engineering and Computing (FER), Univerity of Zagreb with mater thei entitled Neural Network-baed Friction Compenation for Gantry Crane Control Sytem. He i currently a Reearch Aitant at the Department of Control and Computer Engineering, at Faculty of Electrical Engineering and Computing (FER), Univerity of Zagreb.