Proceedings of the International Conference on ENERGY and ENVIRONMENT TECHNOLOGIES and EQUIPMENT

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1 Proceedings o the International Conerence on ENERGY and ENVIRONMENT TECHNOLOGIES and EQUIPMENT Study regarding end winding inductance o three phase A.C. windings in a single layer OLIVIAN CHIVER, LIVIU PETREAN, LIVIU NEAMT and ZOLTAN ERDEI Electrical Engineering Department North University o Baia Mare Victor Babes 62/A, Baia Mare, Maramures ROMANIA olivian.chiver@ubm.ro Abstract: - This paper presents a study on the end winding inductance o stator windings used in the threephase A.C. machines. It reers to the windings in a single layer with coils ends in two and three plans respectively. End winding inductance is determined by numerical analysis based on inite elements method (FEM). Based on the analysis o more than 80 models designed by the authors, representing the stator o some asynchronous machines, a FEM-Analytic comparative analysis will be realized. Also, or one o the analyzed model, model that represents the stator o an asynchronous machine rom our laboratory, measurements are made in order to compare the results. Key-Words: - End Winding Inductance, FEM, Single Layer, A.C. Machines 1 Introduction The stator o the A.C. rotating electrical machines presents a winding, generally three-phase, which is the support or the currents that produce the rotating magnetic ield. Functionally, this winding includes two dierent regions. In the active region the useul energy is transerred between the stator and the rotor, and it corresponds axially to the length o the erromagnetic material. In this region, the coils sides are placed in the slots. Another region represents the end winding and the coils sides are placed in air and through them low the same currents as in the active region. In the active region, rotating magnetic ield can be considered plane, while in the end winding region rotating magnetic ield is three-dimensional. The magnetic ield produced by the coils ends represents a leakage ield, the corresponding inductance being named end winding inductance. The determination o this inductance as accurate as possible is very important because it inluences the starting current and starting torque. In the design phase, the leakage inductances in the active region are determined analytically with a satisactory accuracy, only end winding inductance is determined with less accuracy. This error is due to the complicated orm o the lux lines in the end winding region, which determines higher discrepancies between the simpliied model used to determine the analytical relations and the real orm o the magnetic ield. Improved analytical methods or end winding inductance determination, taking into account the 3D orm o coils ends, some o them taking in consideration also the eddy currents, have been presented in several papers [1], [2], [3], [4]. A more accurate determination o the end winding magnetic ield is possible using numerical methods. One o these, most utilized lately, is inite elements method (FEM) because it can solve complicated structures with reasonable assumptions and reliable results. Some papers that deal with the end winding magnetic ield and the method o determining the corresponding inductance based on FEM 3D can be speciied [2], [5], [6], [7], [8], [9], [10]. In order to determine experimentally the leakage inductance o stator winding in case o A.C. machines, the rotor is removed rom the stator. Also in case o simulations, the numerical model will be realized without the rotor. In order to separate the end winding inductance rom the total inductance, both 2D and 3D models are realized or the same machine, and the dierence between 3D and 2D total inductance gives the end winding inductance [6], [7], [8]. 2 FEM-Analytic comparative analysis In order to realize a comparative analysis between FEM and analytical values o the end winding inductance, a sotware developed by the authors or computer aided design o asynchronous machines has been used. The sotware has been realized in Visual Basic, because this program is recognized by ISSN: ISBN:

2 Proceedings o the International Conerence on ENERGY and ENVIRONMENT TECHNOLOGIES and EQUIPMENT the MagNet 6.25, the tool used or magnetic ield computation. The sotware has been created to allow the design computation o the asynchronous machines, and realizes the required 2D and 3D numerical models. The numerical models are realized on the basis o data that results rom the design computation or, in case o an existing machine, on the basis o user s data. 2.1 Numerical simulations The initially realized models corresponds to a hal o the machine, the stator winding having got the coils ends in two and three plans (Fig. 1 and Fig. 2). The numerical model also includes the carcass and an air volume that surrounds the machine and has got the length higher than the coils ends. In order to decrease the necessary time or analysis, rom the initial model only the part corresponding to a single pole has been selected. On the two radial aces Odd periodic boundary condition has been imposed (Fig. 3). On all other boundaries Flux tangential condition has been imposed. Fig.1. Model with end winding in two plans Fig.2. Model with end winding in three plans 2.2 Analytical computation o end winding inductance In the design phase, the end winding inductance has been determined using relation (1) or coils ends in two plans and (2) or coils ends in three plans respectively [11]. N 2 L = 1.34µ 0 ( l 0.64τ ) (1) p N 2 L = 0.94µ 0 ( l 0.64τ ) (2) p where L is the end winding inductance, µ 0 is the air magnetic permeability, N is the phase number o turns, p is the pole pairs number, l is the end coil length and τ represents the polar pitch. For each designed model, the end coil length has been determined in the same time with the 3D numerical model, in terms o the spatial coordinates. 2.3 The results o the analysis 2D and 3D numerical models have been carried out or more than 80 asynchronous machines. For these machines end winding inductance has been Fig.3. 3D numerical model with air volume and Odd Periodic boundary conditions Fig.4. A parameter determined both analytically and FEM. Finally the ratio o these values was computed and graphically represented in terms o distance A rom the stator s yoke to the plan where the ormation o rontal ends begins (Fig. 4). In case o 5.5 kw machine, with coil ends in two plans the variation o this ratio is shown in Fig. 5. ISSN: ISBN:

3 Proceedings o the International Conerence on ENERGY and ENVIRONMENT TECHNOLOGIES and EQUIPMENT Fig.5. Variation o L [FEM]/L [Analytic] ratio or 5.5 kw machine, coils ends in two plans Fig.6. Variation o L [FEM]/L [Analytic] ratio or 15 kw machine, coils ends in three plans In the ollows some aspects will be mentioned. In case o a machine with a certain power, i the stator winding is carried out or dierent values o distance A and all other geometrical parameter o coils ends are unchanged, the analytical value o end winding inductance diers rom FEM value as much as the distance is smaller. Also, or the same value o A, in case o higher power machines the dierence between FEM and Analytical values is more important, while or the same power the dierence is higher in case o several poles machine. The explanation o these results is the ollowing: analytical value o end winding inductance is a linear unction o parameter A, while the FEM value is not linear. For the smaller values o distance A the inluence o erromagnetic stator material on the end winding inductance is more important. This act is not relected in analytical relations. However can be noticed that analytical results diers rom FEM results up to 30-35% only or values o parameter A smaller than 15 mm in case o higher power machines. For the values o A used in practice (20-50 mm) in correlation with power o the machine, the average dierence between FEM and analytical values is generally smaller than 10%, only in a ew cases being up to 15%. Fig.7. Variation o L [FEM]/L [Analytic] ratio or the models with coils ends in two plans Fig.8. Variation o L [FEM]/L [Analytic] ratio or the models with coils ends in three plans In case o 15 kw machine with coils ends in three plans the variation o the same ratio is shown in Fig. 6. For all analyzed models with coils ends in two plans and in three plans respectively, the variation o L [FEM]/L [Analytic] average ratio in terms o A is shown in Fig.7 and Fig.8 respectively. 3 Practical measurements Measurements have been carried out or a stator o an asynchronous three-phase machine, the main data are presented in table 1 and the machine is shown in Fig. 9. In the rotor space a control coil has been placed (Inner coil in Fig. 9.), the coil span being equal with the polar pitch. The active sides o the coil are placed above the corresponding slots. The coil ends are connected to a multimeter to measure the rms value o the induced voltage. Table 1 Main data o test machine Parameter Value Rated power [kw] 0.37 Phase voltage [V] 230 Stator core length [mm] 75 Outer diameter o stator [mm] Inner diameter o stator [mm] 70 Pole pairs number 2 Stator slots number 36 Tooth width [mm] 2.75 Number o turns in a coil 133 Number o wires in parallel 1 ISSN: ISBN:

4 Proceedings o the International Conerence on ENERGY and ENVIRONMENT TECHNOLOGIES and EQUIPMENT To carry out the simulations, 2D and 3D numerical models have been made. 2D magnetostatic analysis allows inding out the magnetic vector potential distribution on the inner diameter o the stator. This distribution is shown in Fig.11 in case o phase current amplitude o 0.862A. Fig.9. The tested machine and two search coils Fig.10. Induced voltage as a unction o the stator phase current Another coil (End coil in Fig. 9.) is a closed loop along a coil end and near to the end o the stator. The voltages induced in the two coils have been measured or dierent values o stator currents (Fig.10). It can be noticed that removing the rotor rom inside the stator determines almost a linear behavior o the stator yoke in case o currents that do not exceed to much the nominal value. In terms o the induced voltage in the Inner coil the inductance corresponding to the useul magnetic lux rom the rotor space has been determined [12]: 1 U b Nk w = (3) ω I N where ω is the angular requency, U b is the induced voltage in the Inner coil, kw is the winding actor, I is the phase current and Nb is the turns number o the Inner coil. Analytically, L b inductance can be determined with relation [12]: 2 2 τ 15N k ( ) 1 w l+ 6 8 = 10 [ H ] (4) ω p where l is the stator length [cm], τ [cm] and is the current requency. The total leakage inductance represents the dierence between total inductance per phase and L b inductance. b Fig.11. Magnetic vector potential distribution o the stator inner diameter, I max =0,862 A Using a personal developed MATLAB program, the undamental component o the magnetic vector potential A 1 has been obtained and then L b inductance: 1 N = 2 A1lkw (5) 2I Nb The measured values are shown in table 2, where I phase represents the average value o the currents, U phase is the average value o the phases voltages, U b and U end are the induced voltage in the Inner coil and the End coil respectively and P is the absorbed power. The analytical, FEM and measured values are presented in table 3. Table 2 - Measured values I phase [A] U phase [V] U b [mv] U end [mv] P [W] Table 3 Inductances values Inductance Measured FEM Analytic L t [mh] L b [mh] L σ [mh] L [mh] ISSN: ISBN:

5 Proceedings o the International Conerence on ENERGY and ENVIRONMENT TECHNOLOGIES and EQUIPMENT In table 3 L t is the total inductance per phase when the rotor is removed and L σ is the total leakage inductance per phase. I relation (3) is used to determine end winding inductance in terms o the induced voltage in End coil, obtained value is 29.9 mh. Using FEM this value is mh, and analytically 26 mh. In this case the FEM/Analytic ratio is This value is expected since, in case o this machine, the distance A is 5mm. The smaller value obtained on the basis o the induced voltage in End coil is due to the act that not the entire magnetic lux produced by coils ends is closed trough this coil. A part o this ield is closed trough the stator, and does not induce voltage in the End coil. Thus the value o end winding inductance is higher than that determined using End coil induced voltage. Finally, the FEM value can be considered more accurate than other values reerred to in this paper. 4 Conclusion This paper presents a comparative analysis regarding end winding inductance o a single layer winding. Over 80 models were analyzed and the FEM results have been compared with analytical ones. Also, or a 0.37 kw asynchronous machine rom our laboratory, measurements have been carried out and again the values have been compared with FEM and analytical values. For L t and L σ inductances, the FEM values are close to the measured values. In conclusion, the FEM simulations allow the determination o the end winding inductance with enough accuracy. Also, in some cases, analytical results can be less accurate regarding this inductance. Reerences: [1] Ban D. Zarko D., Mandic I., Turbogenerator End Winding Leakage Inductance Calculation Using a 3-D Analytical Approach Based on the Solution o Neumann Integrals, Research Report, Wisconsin Electric Machines & Power Electronics Consortium, Iulie 2003; [2] Hsieh M. F., Hsu Y. C., Dorrell D. G., and Hu K. H., Investigation on end winding inductance in motor stator windings, IEEE Transactions on Magnetics, vol. 43, no. 6, June 2007; [3] Schramm A. and Gerling D., Analytical calculation o the end winding leakage inductance based on the solution o Neumann integral, IEEE International Symposium on Industrial Electronics (ISIE) 2005 Conerence, June 2005, Dubrovnik, Kroatien; [4] Williamson S., Mueller M. A., Induction motor end winding leakage reactance calculation using the Biot-Savart method, taking rotor currents into account, Proceedings o ICEM 90, Boston, August 1990; [5] Brahimi A. T., Foggia A., Meunier G., End winding reactance computation using a 3D inite element program, IEEE Transactions on Magnetics, vol. 29, no. 2, March 1993; [6] Chiver O., Micu E., Barz C., Stator winding leakage inductances determination using Finite Elements Method, 11 th International Conerence on Optimization o Electrical and Electronic Equipment OPTIM'08, Braşov, România, May 22-24, 2008; [7] Cox T., Eastham F., Proverbs J., End turn leakage reactance o concentrated modular winding stators, IEEE Transactions on Magnetics, vol. 44, no. 11, November 2008; [8] Lin R., Arkkio A., Calculation and analysis o stator end-winding leakage inductance o an induction machine, IEEE Transactions on Magnetics, vol. 45, no. 4, April 2009; [9] Y.B. Li, S.L. Ho, W.N. Fu and W.Y. Liu, An interpolative inite-element modeling and the process simulation o a large solid pole synchronous machine, IEEE Transactions on Magnetics, vol. 45, no. 10, October 2009; [10] Lin R., Haavisto A. and Arkkio A., Validation o a time-harmonic numerical model or solving magnetic ield in end region o a radial-lux machine, IEEE Transactions on Magnetics, vol. 45, no. 12, December 2009; [11] Cioc I., Nica C., Proiectarea maşinilor electrice, Ed. D. P., Bucureşti, 1994; [12] Drăgănescu O. Gh., Încercările maşinilor electrice rotative, Ed. Tehnică, Bucureşti, 1987; ISSN: ISBN: