IEEE TRANSACTIONS ON MAGNETICS, VOL. 51, NO. 3, MARCH 2015 8101004 End-Region Leakage Fluxes and Losses Analysis of Cage Induction Motors Using 3-D Finite-Element Method Jalal Cheaytani 1,2 , Abdelkader Benabou 1 , Abdelkader Tounzi 1 , Maxime Dessoude 2 , Loïc Chevallier 1 , and Thomas Henneron 1 1 Lille Laboratory of Electrical Engineering and Power Electronics, University of Lille1, Villeneuve d’Ascq 59655, France 2 Département THEMIS, EDF Research and Development, Clamart Cedex 92141, France The stray load losses (SLLs) in electrical machines represent a nonnegligible contribution of the total losses and a key point for an accurate evaluation of the energy efﬁciency of considered device. In this paper, one aspect of these SLLs, the end-region leakage ﬂuxes and losses, is investigated and considered for the case of a high-power cage induction motor. The study is performed at locked rotor, no-load, and rated load conditions using a 3-D ﬁnite-element modeling approach. The inﬂuence of the leakage ﬂux on the end-region conductive parts of the motor is analyzed together with the eddy current loss calculation. Finally, the SLLs are calculated and compared with the experimental measurements based on the IEEE standard 112-method B test. Index Terms—Eddy currents, end-region leakage ﬂuxes, ﬁnite-element (FE) method, induction motor, stray load losses (SLLs). I. I NTRODUCTION A LARGE percentage of the electrical demands in the industry is linked to the use of induction electrical machines. Thus, in the context of energy saving, efﬁciency classes have been deﬁned for these machines. Therefore, it has become essential to estimate their losses and efﬁciency right from the design step to fulﬁll the standard requirements. In induction machines, the losses can be classiﬁed into mechanical, primary copper, secondary copper, iron, and stray load losses (SLLs). Many studies have been focused on how to estimate the end-region leakage ﬂuxes and losses of turbo generator [1] or induction machines [2]. Nevertheless, none of these works was interested in the inﬂuence of the end-region losses on the SLLs. As deﬁned in [3], SLLs represent the difference, at load, between on the one hand, the total losses in the electrical machine and, on the other hand, the sum of friction and windage, stator I 2 R, rotor I 2 R, and core losses. These SLL can be nonnegligible and are considered as a key point for an accurate evaluation of the machine efﬁciency. Consequently, several efforts based on experimental approaches, analytical, or 2-D ﬁnite-element (FE) models have been made to identify and evaluate the SLL components [4], but without accounting for the end-region losses. This paper improves the SLL studies by considering the end-region leakage ﬂuxes effects using a 3-D FE model, in the case of a high-power nonskewed cage induction motor. The study is performed at no-load, locked rotor, and rated load conditions. In the ﬁrst part, the mathematical model iron and SLLs models are described along with the studied machine. The sec- ond part is devoted to the calculation of the end-region losses. Simulation results are given in the last part showing no-load iron losses and SLLs compared with experimental results. Manuscript received May 21, 2014; revised September 9, 2014; accepted October 21, 2014. Date of current version April 22, 2015. Corresponding author: J. Cheaytani (e-mail: jalal.cheaytani@ed.univ-lille1.fr). Color versions of one or more of the ﬁgures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identiﬁer 10.1109/TMAG.2014.2365355 II. MODELING APPROACH A. Mathematical Model In this paper, the numerical study is achieved using a 3-D FE method based on the electrical A-ϕ formulation to account for the eddy currents in the squirrel cage bars curl  1 μ curlA  + σ  ∂ A ∂ t + gradϕ  = J (1) where A is the magnetic vector potential, such as B = curl(A), ϕ is the electric scalar potential, μ is the magnetic permeability, σ is the electric conductivity, and J is the source current density. The permeability μ is described by the nonlinear and single-valued magnetic behavior law of the iron core proposed in [5]. B. Iron Loss Model To determine the iron losses, an approach based on Bertotti’s iron loss decomposition into three contributions [6] is used in this paper W i = W h + W cl. + W exc (2) where W i , W h , W cl. , and W exc are, respectively, the total iron, quasi-static hysteresis, and classic eddy current and excess losses. The iron losses are calculated through an analytical postprocessing procedure, implemented in a time-stepping FE calculation code. The time evolution of the magnetic ﬁeld is stored, once the steady state is reached, for every element of the mesh where the iron losses are computed according to the following equations: W h = k h f  ΔB 2  α (3) W cl = k cl 1 T  T 0  dB(t ) dt  2 dt (4) W exc = k exc 1 T  T 0     dB(t ) dt     1.5 dt (5) where f is the frequency, B is the peak-to-peak magnetic ﬂux density, T is the period of the waveform, and k h , α, 0018-9464 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.