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3876 IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998 Calculation of No-Load Induction Motor Core Losses Using the Rate-Dependent Preisach Model Johan J. C. Gyselinck, Luc R. L. Dupré, Lieven Vandevelde, and Jan A. A. Melkebeek, Senior Member, IEEE Abstract— In this paper the authors present a two-step algorithm for predicting the core losses in an electrical machine. As a first step, the flux patterns in the cross section of the machine are calculated by using a time stepped two-dimensional finite element (FE) model, neglecting hysteresis and eddy currents in the laminated core. The second step consists in enforcing the calculated tooth and yoke flux waveforms to a one-dimensional FE lamination model in which the variation along the thickness of the induction and of the induced eddy currents is considered. The hysteretic behavior of the ferromagnetic material is taken into account by means of a rate-dependent Preisach model. The outlined procedure is applied to a 3kW squirrel-cage induction motor with either open or closed rotor slots, the former yielding elevated flux harmonics. Computation results and measurements at no-load (phase currents, stator tooth flux, and total iron losses) are compared. Index Terms—Electric machines, finite element methods, hysteresis, induction motors, magnetic fields, magnetic losses. Fig. 1. Lamination model. I. INTRODUCTION E LECTRICAL steels are usually characterized under standard and well defined flux conditions, i.e., alternating and sinusoidal (50 or 60 Hz) flux using the Epstein frame. Straightforward extrapolation of Epstein loss data for predicting machine core losses is bound to produce a significant underestimation of the losses. Indeed, due to the slotting and—in case of inverter supply—due to the harmonic contents of the voltage supply, the induction may be considerably distorted. Furthermore, in some parts of the machine the induction is rotational rather than alternating . The discrepancy may be overcome by applying global empirical correction factors, as was the design practice for many decades, or by using a more physical approach, e.g., some kind of loss separation concept , . In  the rate-independent scalar Preisach hysteresis model embedded in a one-dimensional (1-D) lamination model is used for calculating the core losses in a switched reluctance motor. In this paper a similar method is applied to a squirrelcage induction machine, employing a rate-dependent Preisach Manuscript received December 31, 1996; revised April 1, 1998. This work was supported by the Inter-University Attraction Poles for fundamental research funded by the Belgian State and the Ministry of Economy of the Flemisch Government in collaboration with OCAS, the Research Centre of ARBED-SIDMAR-ALZ. The authors are with Laboratory for Electrical Machines and Power Electronics, Department of Electrical Power Engineering, University of Gent, B-9000 Gent, Belgium (e-mail:[email protected]). Publisher Item Identifier S 0018-9464(98)07331-2. model and calculating the flux pattern inside the machine core by means of the finite element (FE) method. II. PROPOSED METHOD A. Two-Dimensional FE Machine Model The heart of the machine model is a two-dimensional (2-D) FE representation of the cross section. The FE model may contain two distinct types of electrical conductors: stranded or filamented conductors on the one hand and massive conductors which may display skin effect on the other hand. These conductors are embedded in an electrical network so as to model voltage supply and the actual interconnection of coil sides in windings and bars in cages. The end effects due to end windings and end rings are accounted for by inserting resistances and inductances in the electrical network. Anisotropy, hysteresis, and eddy currents in the laminations are not considered. Magnetic saturation is introduced by means curve. of a single valued The coupled field-circuit system is time stepped by using efficient solving techniques . B. One-Dimensional FE Lamination Model The magnetic field in the machine, resulting from the 2 FE time stepped model, is approximated as a uniform and alternating flux in a number of flux tubes, judiciously chosen in the motor cross section. These flux waveforms are fed to a 0018–9464/98$10.00 1998 IEEE GYSELINCK et al.: CALCULATION OF NO-LOAD INDUCTION MOTOR CORE LOSSES 3877 1-D lamination model with thickness 2 , length , and width ( , ), as shown in Fig. 1. The variation of , the magnetic field the magnetic induction , and the induced current along the thickness of the lamination ( ) due to an alternating flux along the -direction is governed by the following differential equations: (1) (2) where is the electrical conductivity. The Preisach model, well described in literature (e.g., ), and . The supplies the hysteretic relationship between represents the relative Preisach distribution function density of the elementary dipoles with switching fields and and accurately describes the static hysteretic behavior of the material. At higher frequencies rate dependence should be included in the Preisach model in order to preserve a good agreement with measurements. The finite rate at which the dipoles switch can be defined by means of a single, frequency independent, parameter , . The Preisach function is obtained by differentiating twice . The latter has a simple the Everett function physical meaning and is directly (i.e., without fitting) identified by means of quasi-static measurements on a stacked ring core is obtained by fitting measured and . The parameter calculated dynamic hysteresis loops. Equation (1) is solved numerically using a FE spatial discretization and a finite difference time discretization. III. APPLICATION The iron losses calculation method presented in this paper has been applied to a three-phase 3 kW 4-pole induction motor. The stator has a single layer winding with three slots per pole and per phase. Two rotors are considered, the 32 unskewed slots of which are either open or closed. The motor, delta-connected and with either of the two 0.0004). rotors mounted, has been tested at no-load (slip Measurements have been carried out at a reduced sinusoidal voltage (185 V, 50 Hz instead of the rated 220 V, 50 Hz), because of the practical limitations of the present measuring set up for identifying the Everett function. Enforcing anti-periodicity conditions for both the magnetic field in the FE mesh (3034 nodes, 5306 triangular elements) and the electrical network voltages and currents, only one pole has been modeled. A flux plot at no-load is displayed in Fig. 2. The electrical network that represents one pole of the squirrel cage is depicted in Fig. 3. A. Measurements on the Motor and Calculation Results with the 2-D FE Model The stator phase currents and the line voltages have been measured by means of a data-acquisition system. A single turn Fig. 2. Flux plot at no load. Fig. 3. Electrical network for the squirrel cage. search coil has been fit around a stator tooth. Integration of the search coil voltage yields the tooth flux. The waveforms of measured and calculated stator phase currents and stator tooth flux are shown in Fig. 4 (for sake of brevity and clarity, the different waveforms have been phase shifted with respect to each other, as is also the case in Fig. 7). The “measured” iron losses are obtained by measuring or calculating the different components of the power balance. The results are listed in Table I. Multiplication of the three phase currents and voltages readily produces the electrical power input. The resistance of the stator windings was measured immediately after the tests, allowing an accurate calculation of the stator joule losses. The joule losses in the rotor cage are predicted using the 2-D FE model. The mechanical friction losses have been estimated from the rotor inertia and the instantaneous deceleration when interrupting the power supply. B. Identification of the Electrical Steel Some laminations (VH 800-65D) of the test motor have been punched and assembled to form a stacked ring core. Equilines of the measured Everett function are displayed in Fig. 5. A good agreement between calculated and measured iron losses, in the lamination model and in the ring core, respectively, has been observed in a frequency range up to 1 kHz and for arbitrary waveforms . 3878 IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998 Fig. 5. Measured Everett function E (H1 ; H2 ) . (a) Fig. 6. Flux tubes in stator tooth and yoke. C. Loss Density Calculation (b) Fig. 4. Measured and calculated waveforms of stator phase current and stator tooth flux. TABLE I POWER BALANCE YIELDING THE MEASURED IRON LOSSES 1) Stator: Fig. 6 shows the division of a stator tooth into six flux tubes, the flux passing through sections 1–7. The varying cross section and the slot leakage flux are thus accounted for. A yoke segment is split into four parallel flux tubes (sections 8–11), considering the lower induction levels toward the outer stator boundary. Distinction should further be made between stator teeth in between slots of the same phase belt (SPB, totaling 24 teeth) and teeth in between slots of different phase belts (DPB, 12 teeth), the latter displaying a higher but less distorted induction. As for the yoke segments, such a distinction was found not to be required. Some calculated induction waveforms (under the assumption of uniform flux) and hysteresis loops are depicted in GYSELINCK et al.: CALCULATION OF NO-LOAD INDUCTION MOTOR CORE LOSSES 3879 TABLE IV TOTAL IRON LOSSES Fig. 7. Calculated induction waveforms. TABLE II STATOR TOOTH IRON LOSS 3) Total Iron Losses: Table IV lists the total iron losses and the contribution of stator teeth, stator yoke segments, and—if present—rotor slot bridges. About one third of the stator iron volume is situated in the teeth (according to Fig. 6). In case of open rotor slots, the stator teeth produce 40% of the total iron losses due to the considerable harmonic distortion of the induction. In case of closed rotor slots, the average loss density in tooth and yoke do not differ so much. The computational cost of time stepping the 2-D machine model and the 1-D lamination model depends on the space and time discretization and on the number of periods (of 20 ms) to reach steady state. We use a 2-D first order triangular mesh with 3000 nodes and a 1-D second order mesh with 21 nodes, and 6 400 and 3 1000 time steps, respectively. The measured Preisach distribution function is stored in about 6000 plane. The 2-D calculation and each 1-D points in the half calculation take about 60 and 20 CPU minutes, respectively, on a Alpha Work station 200/166. D. Discussion TABLE III STATOR YOKE IRON LOSSES Figs. 7 and 8, respectively. The calculated iron loss densities are presented in Tables II and III. 2) Rotor: At no-load the iron losses in the rotor can be expected to be negligible. Indeed, the calculated loss density in the bulk of rotor teeth and rotor yoke is less than 0.02 W/kg. The ensuing total iron losses, less than 0.1 W, are negligible compared to the stator iron losses. However, in case of closed rotor slots, the slots bridges may contribute significantly to the iron losses. The magnetic induction in the bridges over the closed slots has a very large 900 Hz component and may be quite rotational. Application of the lamination model yields local loss densities ranging from 80 up to 500 W/kg for the eight bridges, yielding additional losses estimated at 5 W. The difference between measured and calculated iron losses (Tables I and IV) is 15% and 8% for open and closed rotor slots, respectively. This can be attributed to the different assumptions and simplifications implied by “measuring” the iron losses as described above and by using the presented motor, lamination, and material model, some of which are briefly discussed hereafter. 1) A Posteriori Inclusion of Hysteresis and Eddy Currents: The 2-D field calculations have been done with a single valued curve. Hysteresis and eddy currents have been accounted for a posteriori using the 1-D model. In  this two step approach is compared to the 2-D hysteretic case. The BH loops obtained with these two approaches differ to some extent, as can be seen in Fig. 9. However, the loss density is observed to differ only slightly. 2) Alternating versus Rotational Flux: An alternating flux has been assumed throughout the iron core. This is very questionable in the rotor slot bridges and in the vicinity of the stator tooth-yoke interface as can be seen in Fig. 10. This figure shows the B-loci in five points in the stator, the position of which is indicated in Fig. 6. Apparently, the induction in points 3 and 4 is significantly rotational, inconsistent with the assumption of alternating flux. Rotational flux causes an increase of the classical eddy current losses. As for the hysteresis losses, the ratio of rotational to alternating losses depends on the peak induction and may become smaller than one for peak inductions higher than 1.5 T . In  a 9% increase of induction motor iron losses due to rotational flux is reported. 3880 IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998 (a) (b) (c) Fig. 8. Calculated hysteresis loops. (Hs : magnetic field at the surface of the lamination, Ba : magnetic induction averaged out over the thickness of the lamination). 3) Machining and Assembling Laminations: The Preisach function has been obtained using a stacked ring core. Machining and assembling the laminations in motor core and ring core, respectively, may have considerable effect on the magnetic properties of the steel . The imperfect insulation of the laminations has been disregarded as well. IV. CONCLUSION Fig. 9. Two-dimensional BH loops with direct and a posteriori hysteresis, respectively. A two-step algorithm for predicting the iron losses in a squirrel cage induction motor has been proposed. The flux patterns in the cross section of the motor, calculated by means of the 2-D FE method, are approximated as an alternating and uniform flux in a number of flux tubes. The flux waveforms are fed to a 1-D lamination model incorporating the powerful rate-dependent Preisach hysteresis model. GYSELINCK et al.: CALCULATION OF NO-LOAD INDUCTION MOTOR CORE LOSSES 3881  G. Bertotti, “Dynamic generalization of the scalar Preisach model of hysteresis,” IEEE Trans. Magn., vol. 28, pp. 2599–2601, 1992.  L. Dupré, “Electromagnetic characterization of nonoriented electrical steel” (in Dutch), Ph.D. dissertation, Faculty of Applied Sciences, University of Gent, Belgium, 1995.  D. Philips, L. Dupré, and J. Melkebeek, “Magneto-dynamic computation using a rate-dependent Preisach model,” IEEE Trans. Magn., vol. 30, pp. 4377–4379, Nov. 1994.  R. Van Keer, L. Dupré, and J. Melkebeek, “Computational methods for the evaluation of the electromagnetic losses in electric machines,” Archives Computational Methods in Engineering, to be published. Johan J. C. Gyselinck graduated in electrical and mechanical engineering from the University of Gent, Belgium, in 1991. He joined the Laboratory for Electrical Machines and Power Electronics, Belgium, in 1993 as a Research Assistant. His research interests concern the finite element analysis of electrical machines. Fig. 10. B-loci in stator iron (SPB points defined in Fig. 6). Measurements and calculations have been done using two rotors with either open or closed rotor slots. The former displays an elevated tooth flux harmonic distortion. A good agreement between calculated stator currents and tooth fluxes has been observed in both cases. The iron losses calculation method presented in this paper has produced losses that are 15% and 8%, respectively, less than the measured losses. Luc R. L. Dupré graduated in electrical and mechanical engineering in 1989 and received the Ph.D. degree in 1995, both from the University of Gent, Belgium. He joined the Laboratory for Electrical Machines and Power Electronics, Belgium, in 1989 as a Research Assistant. His research interests mainly concern the finite element analysis of electrical machines, advanced modelling, and characterization of magnetic materials (electrical steels). Lieven Vandevelde graduated in electrical and mechanical engineering in 1992 and received the Ph.D. degree in 1997, both from the University of Gent, Belgium. He joined the Laboratory for Electrical Machines and Power Electronics, Belgium, in 1992 as a Research Assistant. His research interests mainly concern noise and vibrations of electrical machines, magnetic force computations, and finite element analysis of electrical machines. Dr. Vandevelde is a member of the Koninklijke Vlaamse Ingenieursvereniging (K.VIV). ACKNOWLEDGMENT The authors gratefully acknowledge Brook Hansen, Huddersfield, U.K., for supplying the motor. REFERENCES  R. Findlay, N. Stranges, and D. Mackay, “Losses due to rotational flux in three phase induction motors,” in Proc. 2nd Int. Workshop Electric Magn. Fields 1994, Leuven, Belgium, pp. 25–28.  G. Bertotti, A. Boglietti, M. Chiampi, D. Chiarabaglio, F. Fiorillo, and M. Lazzari, “An improved estimation of iron losses in rotating electrical machines,” IEEE Trans. Magn., vol. 27, pp. 5007–5009, Nov. 1991.  M. Jamil, P. Baldassari, and N. Demerdash, “No-load induction motor core losses using a combined finite element state space model,” IEEE Trans. Magn., vol. 28, pp. 2820–2822, Sept. 1992.  D. Philips and L. Dupré, “A method for calculating switched reluctance motor core losses,” in Proc. Int. Aegean Conf. Elect. Mach. Power Electron. 1992, Kusadasi, Turkey, May 27–29, 1992, pp. 124–129.  J. Gyselinck and J. Melkebeek, “Numerical methods for time stepping coupled field-circuit systems,” in Proc. ELECTRIMACS 1996, SaintNazaire, France, Sept. 17–19, 1996, vol. 1, pp. 227–234.  D. Philips, L. Dupré, J. Cnops, and J. Melkebeek, “The application of the Preisach model in magnetodynamics: Theoretical and practical aspects,” J. Magnetism Magn. Mater., vol. 133, pp. 540–543, 1994. Jan A. A. Melkebeek (M’82–SM’84) graduated in electrical and mechanical engineering in 1975, received the Doctor in Applied Sciences degree in 1980, and the “Doctor Habilitus” in electrical and electronical power technology in 1986, all from the University of Gent, Belgium. He was a Visiting Professor at the Université Nationale de Rwanda in Butare, Rwanda, Africa, in 1981 and a Visiting Assistant Professor at the University of Wisconsin, Madison, in 1982. Since 1987, he has been a Professor in Electrical Engineering (Electrical Machines and Power Electronics) at the Engineering Faculty of the University of Gent, as well as the Director of the Laboratory for Electrical Machines and Power Electronics (the former Laboratory for Industrial Electricity). Since 1993, he has also been the Head of the Department of Electrical Power Engineering. His teaching activities and research interests include electrical machines, power electronics, variable frequency drives, and control systems theory applied to electrical drives. Dr. Melkebeek is a member of the Koninklijke Vlaamse Ingenieursvereniging (K.VIV), a member of the board of the “Technologisch instituut” of the K.VIV, a member of the Koninklijke Belgische Vereniging van Elektrotechnici (KBVE-SRBE), a member of the Belgian Federation for Automatic Control (BIRA-IBRA), and a fellow of the IEE. He also serves as a member of the IEEE-IAS Electric Machines Committee and of the IEEE-PES Machine Theory Subcommittee.