A General Model of the Laminated Steel Losses in Electric Motors with PWM Voltage Supply Dan Ionel Mircea Popescu C. Cossar M.I. McGilp Aldo Boglietti Andrea Cavagnino SPEED Laboratory, University of Glasgow Glasgow G1 8LT, UK AO Smith CTC Milwaukee, WI, USA Politecnico di Torino Dipartimento di Ingegneria Elettrica, 1019 Torino, ITALY Abstract A procedure is described for identifying a general mathematical model of core losses in ferromagnetic steel when the voltage supply is non-sinusoidal, i.e. PWM inverter-fed type. This model has a hysteresis loss multiplicative coefficient variable with frequency and induction and a combined coefficient for eddy-current and excess losses that is also variable with frequency and induction. The effect of the PWM supply voltage over the core losses is modeled using factors that depend on the average rectified and rms voltage values. Validation was performed on a number of different samples of non-grain oriented fully and semi-processed steel alloys. Index terms core loss, eddy-current loss, hysteresis loss, PWM voltage, ferromagnetic steel I. INTRODUCTION A successful electrical machine design optimization process requires the accurate prediction of core losses. Static converters are widely used as sources of supply voltage for electric motors. The non-sinusoidal waveform of the supply voltage will determine increased losses in the lamination steel. Thus, an accurate and efficient estimation of these losses represents a challenging task. For this purpose a large number of models and algorithms have been proposed by different authors, yet a definitive conclusion has not been reached. There are approaches proposed from the physical phenomenon point of view [1,,8,1] or from the engineering point of view [4-7,9,16,17]. It is widely accepted that the steel laminations change their properties not only due to the manufacturing process, but mainly due to the flux density and frequency level. This paper contributes to the continuous debate by proposing a general model of the laminated steel losses that allows the loss coefficients to vary with flux-density and frequency and employs an engineering relationship between the PWM and sinusoidal supply voltage. Practical experimental and numerical solutions are presented. II. CORE LOSSES MODELING UNDER SINUSOIDAL SUPPLY VOLTAGE The most employed model in the work published over the last decades is based on that developed by Bertotti [1] for laminations excited with sinusoidal magnetic flux, as in an Epstein test. According to this model, the specific iron losses are a summation of hysteresis, classical eddy-current and excess loss: α 1.5 1.5 wfe = kh f B + kef B + ka f B (1) The coefficients k h, k e k a and α are determined from the measured data for a certain frequency by using the least square method and are assumed to be constant. In practice the specific iron-loss for the same grade material varies in between coils and batches within acceptable manufacturing tolerances. A practical motor design approach, which accounts for such variations consists in building estimated curves by averaging large sets of experimental data on a per flux density and frequency basis. The results obtained using the model (1) illustrated the fact that the loss coefficients k h, k e k a need to be dependent of both flux density and frequency and that the usual approach of constant coefficients can lead to unpredictable and significant numerical errors. In order to identify the coefficients k h and α, further assumptions have to be made regarding their variation. Recently [3-4], a second order polynomial has been proposed for the α exponent. The experimental validation was performed on three materials: (a) fully-processed steel (M43) (b) semiprocessed steel (SPA) for which the an Epstein test frame was used for measurements and (c) a toroidal core that was built from a low grade semi-processed steel. The material properties are given in the Appendix. To account for coefficient variations the authors are using the model that extends previous research results [3,4] and allows the hysteresis loss coefficient to vary with frequency and induction while the classical loss and anomalous loss are grouped into an eddy-current loss term, where the corresponding coefficient varies with induction and frequency level: w = w + w Fe Hys EC Hys EC e (, ) (, ) ( ) w = k f B f B h w = k f B f B The authors have successfully fitted the model () to laminated steels typically employed in electric motor manufacturing, such as semi-processed and fully-processed materials. Representative numerical results will be tabulated in the full paper to provide a useful reference for the reader. Fig. 1-3 show the errors between test data and estimations using the model () with variable loss coefficients for all three materials. Note that the higher error values occur at low flux-density, while at high flux-density (> 1.T) the errors are within ±5%. As a first step to identify the values of the coefficients, (1) is divided by the product (f B ) resulting in: w Fe a b f fb = + (3) with: a= kh, b= ke (4) For any induction B at which measurements were taken, the coefficients of the above polynomial in f can be calculated by linear fitting, based on a minimum of two points [3,4]. From () and (3) the eddy-current coefficient k e is readily identifiable. ()
Fig. 1 Relative errors - the losses calculated by the model () with variable coefficients and Epstein measurements fully-processed steel M43 Fig. 4. Eddy-current loss coefficient k e variation with induction within a frequency range for CAL model fully-processed material M43 Fig. Relative errors - between the losses calculated by the model () with variable coefficients and Epstein measurements semi-processed steel SP Fig. 5. Eddy-current loss coefficient k e variation with induction within a frequency range for model () semi-processed material SP Fig. 3 Relative errors - low frequency estimated loss with model () and measurements on the toroid semi-processed steel sample This coefficient is independent of frequency if we consider a certain frequency domain for the core loss data, but unlike the conventional model (1), it exhibits a significant variation with the induction, especially for the semi-processed lamination steel material (See Fig.5). Several functions have been tried to describe the loss coefficients variation with the induction level. The lowest relative error values and simplicity of the implementation are provided by third order polynomials. Hence, we may employ for curve fitting of k e a polynomial of the form: 3 k ( B) = k B + k B + k B+ k (5) e e3 e e1 e0 Fig. 6. Eddy-current loss coefficient k e variation with induction for model () for the toroid semi-processed steel sample (frequency up to 00Hz) Depending on the frequency range of the core loss data considered for (), different curves for k e and k h as a function of the induction are obtained. In Figs. 4 5 the upper curves correspond to test data in a frequency range up to 400Hz, while the lower curves correspond to test data in a frequency range up to 000Hz. These results demonstrate that the eddy-current loss coefficient k e increases with the induction level and decreases with the frequency. Fig. 6 shows the variation of the eddy-current loss coefficient k e vs induction for frequencies up to 00Hz for the toroid semi-processed steel sample.
In the equation with the exponent of the induction set to be constant and equal to, a third order polynomial is employed for the coefficient k h that will have an induction variation of the form (See Fig.7-9): 3 kh( B) = kh3b + khb + kh1b+ kh0 (6) In the model (), for a considered frequency range, ke and kh are dependent only on B, which provides a more straight forward computation of the hysteresis and eddy-current loss components. Depending on the employed frequency range, the polynomial functions that describe the loss coefficients ke and kh will have different variations. It is important to note that the higher relative error for the estimated losses at lower flux-density levels may be significantly reduced if a higher polynomial function, i.e. 4 th order is used for the estimation of the hysteresis loss coefficient kh. In Figs. 7 8 the lower curves correspond to test data in a frequency range up to 400Hz, while the upper curves correspond to test data in a frequency range of up to 000Hz. These results suggest that the hysteresis loss coefficient k h will decrease with the induction level and increase with the frequency. Fig. 9 shows the variation of the hysteresis loss coefficient k h vs induction for frequencies up to 00Hz for the toroid semi-processed steel sample. As demonstrated in [3], the hysteresis loss coefficient kh may be determined experimentally as: H k h π = ρ irr V BP Where the irreversible field H irr can be identified as the positive field value at zero induction, Bp is the maximum (peak) value of induction in the hysteresis cycle and ρ V is the volumetric mass. At low frequency level the hysteresis loss is the dominant component. The validity of this model for sinusoidal supply voltage was demonstrated for frequencies up to khz [3]. A practical approach for the determination of k h and k e according to the empirical model (5) and (6) is described as follows: (a) measure the core losses corresponding to at least four induction values, i.e. 0.1T, 0.5T, 1T, 1.5T for a low frequency, i.e. 5Hz; (b) determine experimental k h with (7) for all four points; (c) compute the terms k h0, k h1, k h, k h3 from (6); (d) measure the core losses corresponding to four induction values, i.e. 0.5T, 1T, 1.5T, 1.9T for the mean frequency of the range of interest, e.g. 00Hz for a range up to 400Hz; (e) determine experimental k e for all four points with the expression: 1 π H irr ke = wfe f BP (8) f B ρ p V where H irr is the irreversible field-strength measured at step (a) for low frequency, but similar induction values; (f) compute the terms k e0, k e1, k e, k e3 from (5); (g) the loss coefficients are estimated with (5), (6) for any other induction level and may be easily applied to any further numerical method of computing the core losses in electrical motors. (h) the total core losses are computed with () (7) Fig. 7. Hysteresis loss coefficient k h variation with induction within a frequency range for model () fully-processed material M43 Fig. 8. Hysteresis loss coefficient k h variation with induction within a frequency range for model () semi-processed material SP Fig. 9 Hysteresis loss coefficient k h variation with induction for model () for the toroid semi-processed steel sample (frequency up to 00Hz) Figs. 10 13 show the variation of k e when both the induction and frequency vary and the algorithm described above is employed. Note that as previously estimated the variation with frequency of the eddy-current loss coefficient is less significant, while an increased induction level will determine an increased value of k e. We also observe that a fully-processed material will exhibit a lower degree of variation with induction and frequency as compared to the semi-processed material. Similar results for the toroid semiprocessed steel sample are not displayed here, as the experiments were performed for a limited frequency range. i.e. up to 00Hz.
III. CORE LOSSES MODELING UNDER PWM SUPPLY VOLTAGE Fig. 10 Eddy-currents loss coefficient k e variation with wide range induction and frequency as parameter fully-processed material M43 Fig. 11 Eddy-currents loss coefficient k e variation with wide range frequency and induction as parameter fully-processed material M43 Fig. 1 Eddy-currents loss coefficient k e variation with wide range induction and frequency as parameter semi-processed material SP Fig. 13 Eddy-currents loss coefficient k e variation with wide range frequency and induction as parameter semi-processed material SP In a wide range of electric motor applications, the supply voltage is non-sinusoidal. Consequently, both the static (hysteresis) and the rate-dependent (eddy-currents) effects in the lamination steel will exhibit an increased amount of losses. Several models for the core-losses with arbitrary supply voltage have been put forward [5-7, 1-14] in recent decades following the introduction of static converters, i.e. inverters or choppers. Most of these models [1-14] require a very detailed knowledge of the material physical properties and laborious mathematical models of the minor hysteresis loops. A simple and efficient engineering approach is detailed in [5,6] which shows that it is possible to model the variation of the core losses with the supply voltage, if the voltage characteristics are known. If the minor loops effect is neglected, it may be demonstrated [5] that the peak flux-density value is proportional to the average rectified supply voltage value V avg. Consequently, the hysteresis loss will vary with the average rectified voltage value. Similarly, as the eddycurrent loss depends on the rate of variation of the fluxdensity (db/dt), we may associate the eddy-current loss with the rms value of the supply voltage, V rms. Thus, the core-losses with an arbitrary voltage waveform may be expressed as: wfe = η whys + χ wec (9) Where the parameters η and χ may be computed as follows: Vavg η= V1 avg (10) Vrms χ= V1, rms In the above relations, the subscripts stand for: avg average rectified value, rms root means square value, 1 fundamental. Note that the exponent of the parameter related to the static hysteresis loss is variable and always set to be equal to for all frequencies. Two sets of experiments were performed for validating the core loss model described by (9) and (10): (I) The toroid semi-processed steel sample core was supplied from a PWM inverter with a fundamental frequency of the 50Hz, switching frequency of khz and a DC bus voltage of 30V. The electrical quantities have been recorded using a power analyser with a bandwidth of 800kHz. Table I summarizes the relative error values for the estimated core-losses in this case. The relative error obtained by employing eq. (9) is compared to the values obtained when using various methods described in [13-15]. Ref [15] and this paper use the same set of measured data, whereas for the methods described in [13,14] use a hysteresisgraph and the knowledge of the material electrical conductivity. It is worths highlighting that the computation overhead of the methods from [13] and [14] is significantly higher and that the relatively low errors for all methods is attributable to the low fundamental frequency used in the tests.
TABLE I RELATIVE ERRORS IN IRON LOSSES ESTIMATION UNDER PWM SUPPLY [15] RELATIVE ERRORS [%] INDUCTION [T] REF [13] REF [14] REF [15] EQ. (9) 0.311 0.91-1.4-5.18-10.73 0.49 5.15-11.91 -.83-6.45 0.716 0.91-6.51-3.90 -.8 0.951 -.34-6.09-5.88-3.05 1.185 -.69-3.53-6.98 -.51 1.414-3.97-3.41-3.70-3.8 1.506-3.06-3.06 -.58-1.45 Fig. 16 Estimated loss components for the toroid semi-processed steel sample with PWM voltage, 50Hz fundamental frequency. Fig. 14 Secondary induced PWM voltage waveform (fundamental flux-density = 1.414T, 50Hz) Fig. 15 Primary absorbed current waveform(fundamental fluxdensity = 1.414T, 50Hz) Figs. 14 and 15 show the recorded waveform of the induced PWM voltage in the toroidal winding and the absorbed primary current waveform respectively, for a fundamental flux-density value of 1.414T at 50Hz. Fig. 16 shows the estimated specific loss components (hysteresis and eddy-currents loss) in the toroidal winding for a 50Hz fundamental, khz switching PWM supply voltage. It should be observed that the dominant component is the hysteresis loss. This demonstrates that even under high switching frequency, the fundamental frequency will determine the loss segregation pattern. (II) The Epstein frame sample of the full-processed steel M43 and the semi-processed steel SPA were supplied from a PWM inverter with a variable fundamental frequency from 00Hz to 1000Hz, switching frequencies of 10kHz and 0kHz and a variable DC bus voltage up to 365V. The electrical quantities have been recorded using a power analyser with a bandwidth of 800kHz. If the ratio between the carrier frequency and the modulation frequency is high, the η coefficient can be considered equal to one, as verified in a high number of measurements on the threephase PWM inverter. As previously discussed no minor loops are present in the hysteresis cycle. As a consequence the increase in iron loss with a PWM supply is solely due to an increase in dynamic hysteresis (eddy-currents) loss. For a complete analysis of the core loss variation under PWM voltage supply, the inverter modulation index m was initially maintained constant to a value of 1. The computed value of the parameters χ and η were characterized by small variation around the values of 1.0 and 1 respectively. The experiments were repeated with a variable modulation index from 0.3 to 1. Fig. 17 shows the variation with the modulation index m of the parameter χ that models the dynamic core loss increase under PWM supply. As expected the parameter η that models the static core loss increase under similar conditions, was practically unchanged and a constant value of 1 was used. Thus, with a PWM voltage and constant modulation index, the computed core-losses are actually scaled from those obtained with sinusoidal supply, by using constant factors for the modeling of the increase in the dynamic (eddy-currents) and static (hysteresis) core loss components. The static hysteresis loss is basically unchanged for all frequencies, while the eddy-current loss is increased with a ratio that varies around 1. for all frequencies. Figs. 18 and 19 show the relative errors between the test data and the computation for constant modulation index m =1. A satisfactory agreement of test vs computed data can be observed for the whole range of frequencies and induction levels. The sudden increase of the errors for higher induction values in Fig. 18 is attributable to a practical difficulty in reading the information from the digital measurement devices. One may note that the relative error values are virtually constant for a frequency level up to 800Hz and inductions between 0.6T to 1.5T. Also for the same induction range, the error values are decreasing with frequency from 00Hz to 600Hz and starting to increase for
Fig. 17. Variation of parameter χ with PWM inverter modulation index m Fig. 0 Relative errors - losses calculated by the model (9) with variable variable modulation index m fully-processed steel M43 Fig. 18 Relative errors - losses calculated by the model (9) with variable constant modulation index m = 1 fully-processed steel M43 Fig. 19 Relative errors - losses calculated by the model (9) with variable constant modulation index m = 1 semi-processed steel SPA higher frequencies. The authors consider that this pattern may be associated with the minor hysteresis loop effects and the fundamental static hysteresis loss variation with frequency, i.e. this is the main core loss component for low and medium frequency. At high frequency the core losses are essentially of dynamic nature (eddy-currents). Thus, at low frequency model (9) underestimates the actual losses, while at high frequency it is overestimating the actual losses. A more detailed model for the hysteresis losses under PWM voltage supply would compromise the simplicity of the proposed method. Fig. 1 Relative errors - losses calculated by the model (9) with variable variable modulation index m semi-processed steel SPA Figs. 0 and 1 show the relative errors between the test data and the computation for variable modulation index. We note that the satisfactory agreement of test vs computed data is obtained only for induction level higher than 0.8T. This is partially attributable to the static hysteresis loss component that varies due to the minor loop effects. Also the low induction levels correlate with a low value of the modulation index m. As Fig. 17 shows, a low modulation index results in a higher value of the parameter χ. The investigations into better modeling of this parameter are part of on going further work. Finally, the obtained results with a PWM voltage supply can be considered very interesting, taking into account the considered fundamental frequency and flux density ranges and that the iron loss estimations have been done using average values for η and χ coefficients. IV. CONCLUSIONS A simple and effective general model for the core losses in steel laminations with PWM voltage supply is proposed. The model segregates the core losses in dynamic (eddy-current) and static hysteresis losses with loss coefficients that vary with the induction and frequency level. The loss increase due to the voltage harmonics is modeled through the ratio between the voltage values, root mean square and the fundamental or the average rectified
value and the average fundamental value. For a practical and efficient engineering approach, the core losses associated with the static hysteresis phenomenon [1,] may be considered constant for all the frequency range. At high frequency, the dominant core loss component is associated with the dynamic hysteresis (eddy-currents) phenomenon [1,]. The increase in this loss component depends on the frequency, induction and PWM inverter modulation index. Experimental validation shows a satisfactory agreement for a wide range of frequencies and induction levels, especially for the cases when the PWM inverter modulation index is maintained constant. Further work is being pursued for the improvement of the core loss model when the PWM inverter modulation index is variable. ACKNOWLEDGMENTS The authors would like to thank colleagues at A. O. Smith Corporation, especially Mr. R.J. Heideman, Mr. Ron Bartos and Mr. S..J. Dellinger, who supported this project aimed at improving the characterisation of electrical steel. APPENDIX MAIN CHARACTERISTICS OF SAMPLE MATERIALS Material Thickness Permeability at Loss @ 1.5T Density type 1.5T and 60Hz and 60Hz [in] [ ] [W/lb] [kg/m 3 ] M43 0.018 1387 1.88 7700 SP 0.0 3071 3.16 7850 Toroid 0.019 1180 3.81 7800 REFERENCES [1]. G. Bertotti, General properties of power losses in soft ferromagnetic materials, IEEE Trans. on MAG, vol. 4, no. 1, pp. 61 630, Jan. 1988 []. G. Bertotti, Hysteresis in magnetis, Academic Press, San Diego, 1998 [3]. D. Ionel, M. Popescu, S. J. Dellinger, T. J. E. Miller, R. J. Heideman, and M. I. McGilp, Computation of core losses in electrical machines using improved model for laminated steel, IEEE Transactions on Industry Appl. Vol. 43, No. 6, Nov/December 007, pp. 1554-1564 [4]. M. Popescu and D. M. Ionel, A best-fit model of power losses in cold rolled motor lamination steel operating in a wide range of frequency and induction, in IEEE Transactions on Magnetics. Vol. 43, No. 4, April 007, pp. 1753-1756 [5]. A. Boglietti, A. Cavagnino, M. Lazzari, and M. Pastorelli, Predicting iron losses in soft magnetic materials with arbitrary voltage supply: an engineering approach, IEEE Trans. on MAG, vol. 39, no., pp. 981 989, Mar. 003. [6]. A.Boglietti, A. Cavagnino, M. Lazzari, M. Pastorelli, Two simplified methods for the iron losses prediction in soft magnetic materials supplied by PWM inverter, Conf. Rec. IEEE- IEMDC 01, 17-0 June001, Boston, USA, pp. 391-395. [7]. A. Boglietti, M. Lazzari, M. Pastorelli, Iron losses prediction with PWM inverter supply using steel producer datasheets", Conf. Rec. IEEE-IAS'97 Annual Meeting, 5-9 October, New Orleans 1997, USA, pp. 83-88. [8]. D. Lin, P. Zhou, W. N. Fu, Z. Badics, and Z. J. Cendes, A dynamic core loss model for soft ferromagnetic and power ferrite materials in transient finite element analysis, IEEE Trans. on MAG, vol. 40, no., pp. 1318 131, Mar. 004. [9]. Y. Chen and P. Pillay, An improved formula for lamination core loss calculations in machines operating with high frequency and high flux density excitation, in IEEE 37th IAS Annual Meeting Conf. Rec., Pittsburgh, PA, Oct. 00, pp. 759 766. [10]. H. Domeki, Y. Ishihara, C. Kaido, Y. Kawase, S. Kitamura, T. Shimomura, N. Takahashi, T. Yamada, and K. Yamazaki, Investigation of benchmark model for estimating iron loss in rotating machine, IEEE Trans. on MAG, vol. 40, no., pp. 794 797, Mar. 004 [11]. I.D. Mayergoys, F.M. Abdel-Kader, The analytical calculation of eddy-current losses in steel laminations subjected to rotating magnetic fields IEEE Trans. on Magnetics, Vol. MAG-0, No. 5, September 1984, pp. 007-009 [1]. F. Fiorillo, A Novikov, Power losses under sinusoidal, trapezoidal and distorted induction waveform, IEEE Transactions on Magnetics, Vol. 6, Sep 1990, pp. 559-561 [13]. R. Kaczmarek, M. Amar, A general formula for prediction of iron losses under non-sinusoidal supply voltage waveform, IEEE Transactions on Magnetics, Vol.31, No.5, September 1995, pp. 505-509. [14]. E. Barbisio, F. Fiorillo, C. Ragusa: Predicting loss in magnetic steels under arbitrary induction waveform and with minor hysteresis loops IEEE Transactions on Magnetics Vol. 40, No. 4, Part I, July 004, pp. 1810-1819 [15]. A. Boglietti, A. Cavagnino: Iron Loss Prediction with PWM Supply: An Overview of Proposed Methods from an Engineering Application Point of View Conf. Rec. IEEE IAS Annual Meeting, 007, New Orleans, LA, pp. 81-88 [16]. Mthombeni, L.T.; Pillay, P.; Core losses in motor laminations exposed to high-frequency or nonsinusoidal excitation - Industry Applications, IEEE Transactions on, Vol. 40, No. 5, Sept.-Oct. 004, pp:135-133 [17]. Manyage, M. J.; Pillay, P. Low Voltage High Current PM Traction Motor Design Using Recent Core Loss Results Conf. Rec. IEEE IAS Annual Meeting, 007, New Orleans, LA pp:1560-1566