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On the Importance of Incorporating Iron Losses in the Magnetic Field Solution of Electrical Machines Emad Dlala, Anouar Belahcen, and Antero Arkkio Department of Electrical Engineering, Aalto University School of Science and Technology, Espoo, Aalto FI-00076, Finland This paper studies the effects of iron losses on the magnetic field solution and evaluates their impacts on the overall performance of electrical machines. Because of the complications associated with the inclusion of iron losses into the magnetic field solution, the losses are usually omitted from the finite-element (FE) analysis while they are estimated in a post-processing stage. We conducted a comprehensive FE analysis to study how the iron losses affect the accuracy of the magnetic field solution and what kind of role the losses play in defining the behavior and operation of electrical machines. We found that the inclusion of iron losses in the FE field solution of rotating electrical machines is primarily important for predicting iron losses accurately. Other electrical and mechanical quantities, including input power, supply current, power factor, copper losses, and rotational speed are only slightly affected by iron losses. Therefore, we have proposed an empirical equation that can be used to correct the iron losses that are calculated from the post-processing of the FE solution. Index Terms—Eddy currents, electrical machines, finite-element method, hysteresis, iron loss, loss separation, magnetic field.
I. INTRODUCTION
II. METHODS OF ANALYSIS
NCORPORATING iron losses into the models of electrical machines may or may not play a vital role in characterizing the behavior and operation of the machine. Naturally, the geometric construction of an electrical device requires three-dimensional (3-D) modeling, but in certain applications, such as electrical machines, the problem can be well approximated using a 2-D model. Nowadays, electromagnetic modeling of electrical devices is more commonly conducted by finite-element (FE) techniques in which predicting iron losses is chiefly restricted to post-processing formulae that either apply empirical equations or statistical loss laws [1]. Such a common practice to estimate iron losses is widely acknowledged to be insufficient, and also, inevitably results in leaving out the influence of the losses on the field solution, and thus, on the overall performance of the electrical device. Recent advances in material modeling have, on the other hand, helped estimate iron losses accurately using macroscopic models that can imitate the magnetization behavior [2]. From an engineering point of view, modeling of the magnetization curves (the so-called - loops) can be considered as the most accurate and practical way to predict iron losses, including static hysteresis and dynamic eddy-current losses [3]–[7]. Such methods allow to incorporate the iron losses into the field solution and examine their significance. To do so, we propose to conduct a comparative analysis on four electrical machines using a simplified iron-loss model that predicts the magnetization curves and loop shapes [6]. The iron-loss model will first be incorporated into a 2-D time-stepping FE code and, then, be also implemented in a post-processing stage while using a lossless single-valued magnetization curve for the magnetic field computation. The principal aim of the paper is not to study the accuracy of the iron-loss models but rather to examine the effects of iron losses on the main, electrical and mechanical quantities of electrical machines.
The accurate solution of the magnetic field in a complicated geometry such as a rotating electrical machine requires the spatial discretization of the geometry. Here the magnetic field is approximated using a 2-D FE model coupled with an analytical model for the end-windings of the stator and rotor. The iron losses are incorporated into the FE solution through the constiand the tutive relation between the magnetic field strength magnetic flux density .
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Manuscript received December 14, 2009; accepted February 17, 2010. Current version published July 21, 2010. Corresponding author: E. Dlala (e-mail:
[email protected]). 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.2010.2044385
A. Governing Equations The constitutive nonlinear relation between and expressed by the fixed-point formulation as follows:
can be
(1) where is a magnetization-like quantity and is a reluctivitylike quantity that is calculated based on the method proposed in [8]. If a 2-D approach is performed, applying the magnetic vector with the fixed-point formulation (1) potential results in the following: (2) where and are the -components of the magnetic vector potential and the electric current density, respectively, as . The 2-D FE equations are coupled with the voltage equations of the stator windings. The rotor circuit is also included in the analysis. The skin effect is modeled in the rotor-cage but not in the stator conductors. The time dependance of the overall system of equations is handled by the Crank–Nicolson time-stepping scheme, the equation of motions being also simultaneously solved as the simulations have been made at constant torque [6], [9]. The choice of the magnetic characteristics in the - relation represented by (1) will define the inclusion or exclusion of the iron losses into the field solution.
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Relation When Including Iron Loss in FE
TABLE I MAIN PARAMETERS OF THE INDUCTION MACHINES STUDIED
The magnetic field strength in (1) is given as the sum of the , eddy-current, , and excess, magnetic hysteresis, fields
(3) (4) where is a function that accounts for the skin-effect and it is calculated as in [6]. The directional parameter is controlled by as . Here, the scalar value of represents the components of and and the same applies for in (4). and are to be identified from experiThe coefficients mental - loops. is calculated by the Mayergoyz model of vector hysteresis as
When the core losses are incorporated into the field solution, their effect will appear on the input power of the electrical machine and the power balance is applied for the average powers as follows: (8) When the core losses are not incorporated into the field solution, their effect will not appear and the power balance is applied as follows: (9)
(5) are projected over directions where the components separated by a constant angle and calculated as in [10]. is served by the classical Preisach model. The iron-loss model described above is referred to as the hybrid model. C.
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Relation When Neglecting Iron Loss in FE
The iron losses are inevitably neglected in the field solution when using a lossless single-valued nonlinear magnetization function by which only the effect of saturation is taken into account. To keep the saturation property consistent in the analysis, the Mayergoyz model (5) must also be applied here, but with a calculated by averaging the major single-valued function loop used in (5) (6) Combining (6) with (2) and (1) ensures that the iron losses are not included in the FE analysis. Nevertheless, the losses can still be calculated in a post-processing step applying the same hybrid model described in Section II.B. D. Core Losses and Power Balance The power balance will not only give insight into the results but also will examine the correctness of the numerical implementations of the models. In the two cases, when the iron losses are included and when they are not, the total core losses per unit volume are computed over a time period of the fundamental frequency using the Poynting vector theorem
(7)
is the input power, is the output power, is where is the average core loss. The methods the resistive loss, and for the determination and description of the powers , and can be found in [9]. III. RESULTS AND DISCUSSION The iron-loss model has been identified from dynamic loops obtained from experiments made under different flux patterns and frequencies. The identification procedure is carried out in the same manner detailed in [6], [7] in which also the models proved to give accurate results for the iron losses of electrical machines when compared with the experiments. The influence of iron losses on the field solution and machine’s performance were evaluated through the simulations of four induction machines of different sizes as described in Table I. The machines were run under different load conditions rising from 0% up to 120% of the rated torque, , and they were fed by a sinusoidal voltage supply and by a pulse-width modulated (PWM) voltage supply with a 1.5-kHz switching frequency. Here the effects of the PWM supply are not studied in detail, but a more comprehensive analysis can be found in [7]. The main quantities of Machine 2 and 4 computed at the rated torque both for the sinusoidal supply and PWM supply are shown in Table II. In the four machines, it is noted that the electrical and mechanical quantities such as stator current, input power, and rotational speed were slightly underestimated in the cases when the iron losses were omitted from the FE field solution, but the effect of iron losses starts to diminish for the bigger machines. Such an observation is expected because the machine requires more input power and current to compensate for the losses. The copper losses were underestimated when neglecting the iron losses in the field solution because the fundamental component of the current in the stator and the speed of the rotor (slip) were also underestimated (see Fig. 1). The core (iron) losses were the most affected quantity when they were included or when they were neglected in the field solution. To highlight this point more rigorously, the iron losses of the 37-kW induction motor (Fig. 2(a)) are plotted as a function of
DLALA et al.: ON THE IMPORTANCE OF INCORPORATING IRON LOSSES IN THE MAGNETIC FIELD SOLUTION
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Fig. 1. Stator current computed for the 37-kW induction motor at the rated torque when including and when neglecting iron losses in the FE solution.
Fig. 4. Flux-density waveform (a) tracked at a point in a stator tooth and hysteresis loops (b) and (c) tracked at a point in a root tooth of the 37-kW induction motor (Machine 2) simulated at the rated torque when including and when neglecting iron losses in the FE solution.
Fig. 2. Equipotential lines computed by the FE model at an instant of time. (a) 37-kW induction motor, (b) Test-device.
Fig. 3. Iron losses computed for the 37-kW induction motor (Machine 2) at different loading ratios when including and when neglecting iron losses in the FE solution. The empirical equation (10) was effective to correct the iron loss calculated in the post-processing stage and neglected in the FE solution.
the loading torque in Fig. 3. It is observed that in all the computations, the iron losses computed in the post-processing stage while neglected in the field solution are higher than the iron losses computed while included in the field solution, the lowest relative difference being 4% at no load and the highest 18% at 120% load. This phenomenon occurs because when they are included into the field solution, the iron losses work to damp the high-frequency harmonics, which increase with the increase of the loading torque (see Fig. 4). As a result of the damping, significant reduction in the eddy-current loss and excess loss are ensued; the hysteresis loss also experiences a minor decrease as a consequence of the shrinking of minor loops even though the
TABLE II NUMERICAL RESULTS OBTAINED BY THE FE MODELS WHEN INCLUDING AND WHEN NEGLECTING IRON LOSSES FOR MACHINE 2 AND 4. (A) MACHINE 2 (37-KW INDUCTION MOTOR), (B) MACHINE 4 (1250-KW INDUCTION MOTOR)
fundamental components of the flux densities remain constant or become slightly greater in the iron region. A. Determining the Role of the Air-Gap The air-gap plays a vital role in characterizing the behavior and operation of a rotating electrical machine and will therefore depreciate the impacts of iron losses. In the presence of an air-gap, an electrical machine constitutes a high reluctance magnetic circuit that draws significant amounts of reactive power, which considerably minimize the effects of iron losses. Therefore, we decided to conduct an analysis to assess the importance of iron losses in a test-device that has a conventional stator topology but no air-gap in it, as the rotor has not been punched out from the stator sheets (see Fig. 2(b) and [11] for more details). The operation principle of this device resembles an unloaded transformer, but by keeping the structure of the stator
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Fig. 5. Stator current for the test-device computed when including and when neglecting iron losses in the FE solution and compared with experiment.
for size of the machine and here was identified to be the 15-kW induction machine and for the 1250-kW induction machine. The coefficients for other machines of different rated output powers can be found by applying linear interpolation or extrapolation. The coefficient depends on the type of supply; for sinusoidal supply and for PWM supply . Using (10) with the proposed coefficients for and , it has been found that the maximum relative discrepancies found in the computations of core losses in the four machines simulated under different load conditions is less than 3% (see Fig. 3). The investigation was made by comparing the (corrected) core-loss results computed by post-processing the FE model using the hybrid model with the (actual) core-loss results computed with the FE model applying the hybrid model in the field solution. IV. CONCLUSION
identical to a rotating machine, we ensure that the distribution of the flux is similar to that of a real rotating machine. To support the numerical analysis, experimental results have been carried out on the 50-Hz, 400-V test-device using a sinusoidal supply. Thus, the device can also be considered as a validation tool for the proposed models. Fig. 5 shows the measured waveforms of the stator current compared with the computed ones in the two cases when the iron losses were included into the FE solution and when they were not. It is obvious here that the iron losses contribute significantly to the input power and the current drawn from the network. The results obtained by the FE model that includes the iron losses are in close correlation with the measured ones, proving the importance of including iron losses into the field solution in electrical devices that contain no air-gap such as transformers, especially under no-load conditions. B. Empirical Equation for the Correction of Iron Loss Based on the conducted analysis, we can deduce that neglecting the iron losses into the FE field solution of rotating electrical machines does not significantly affect the electrical and mechanical quantities of the machine, except the iron losses which will be always significantly overestimated. Therefore, because of the increase in the computation time and more importantly the complications related to the convergence resulting from taking the iron losses into account, choosing to apply a lossless single-valued magnetization curve for the field computation could be justified provided that the discrepancies of the iron losses are corrected. Based on extensive simulations conducted on induction machines of output powers ranging from 15 kW up to 1250 kW with loading conditions ranging from 0% up to 120% of the rated power, one can preliminary apply the following empirical formula for correcting the iron losses calculated posteriorly (10) where here represents the core losses computed from the post-processing of the FE model applying the lossless singlevalued curves and represents the corrected core losses. The coefficient is assumed to be linearly dependent on the
This paper has investigated the role of including or neglecting iron losses in the field solution of rotating electrical machines. As comprehended from the conducted analysis, the inclusion of iron losses in the FE field solution of rotating electrical machines is primarily important for predicting iron losses accurately, where the discrepancies can reach 15% at rated loads as a result of the neglection. Other electrical and mechanical quantities, including power factor, input power, electric current, and speed are marginally affected by iron losses. REFERENCES [1] F. Fiorillo and A. Novikov, “An improved approach to power losses in magnetic laminations under nonsinusoidal induction waveform,” IEEE Trans. Magn., vol. 26, no. 5, pp. 2904–2910, Sep. 1990. [2] O. Bottauscio, M. Chiampi, and D. Chiarabaglio, “Advanced model of laminated magnetic cores for two-dimensional field analysis,” IEEE Trans. Magn., vol. 36, no. 3, pp. 561–573, May 2000. [3] R. Van Keer, L. Dupre, and J. A. Melkebeek, “Computational methods for the evaluation of the electromagnetic losses in electrical machinery,” Arch. Comput. Methods Eng., vol. 5, no. 4, pp. 385–443, Dec. 1998. [4] J. C. Gyselinck, L. Dupre, L. Vandevelde, and J. A. Melkebeek, “Calculation of no-load induction motor core losses using the rate-dependent Preisach model,” IEEE Trans. Magn., vol. 34, no. 6, pp. 3876–3881, Nov. 1998. [5] O. Bottauscio, A. Canova, M. Chiampi, and M. Repetto, “Iron losses in electrical machines: Influence of different material models,” IEEE Trans. Magn., vol. 38, no. 2, pp. 805–808, Mar. 2002. [6] E. Dlala, “Comparison of models for estimating magnetic core losses in electrical machines using the finite-element method,” IEEE Trans. Magn., vol. 45, no. 2, pp. 716–725, Feb. 2009. [7] E. Dlala and A. Arkkio, “A general model for investigating the effects of the frequency converter on the magnetic iron losses of a squirrel-cage induction motor,” IEEE Trans. Magn., vol. 45, no. 9, pp. 3303–3315, Sep. 2009. [8] E. Dlala and A. Arkkio, “Analysis of the convergence of the fixed-point method used for solving nonlinear rotational magnetic field problem,” IEEE Trans. Magn., vol. 44, no. 4, pp. 473–478, Apr. 2008. [9] A. Arkkio, “Analysis of Induction Motors Based on the Numerical Solution of the Magnetic Field and Circuit Equations” Doctoral thesis, Electrical Engineering Series, Acta Polytechnica Scandinavica,, Espoo, Sep. 1987 [Online]. Available: http://lib.tkk.fi/Diss/198X/ isbn951226076X/, no. 59 [10] E. Dlala, A. Belahcen, K. Fonteyn, and M. Belkasim, “Improving loss properties of the Mayergoyz vector hysteresis model,” IEEE Trans. Magn., vol. 46, no. 3, pp. 918–924, Mar. 2010. [11] A. Belahcen, “Vibrations of rotating electrical machines due to magnetomechanical coupling and magnetostriction,” IEEE Trans. Magn., vol. 42, no. 4, pp. 971–974, Apr. 2006.
