estimation of core losses under sinusoidal or nonsinusoidal induction by analysis of magnetization...

174 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 16, NO. 2, JUNE 2001

Estimation of Core Losses under Sinusoidalor Non-Sinusoidal Induction by Analysis

of Magnetization RateMarcelo S. Lancarotte and Aderbal de A. Penteado, Jr.

Abstract—This work deals with a method of analysis whichestimates the increase of core losses under sinusoidal and nonsinu-soidal inductions. The estimation can be carried out by correlatinginstantaneous magnetization power ( ) or the instantaneousmagnetic field ( ) as a function of the magnetization rate (dB/dt)and the induction level ( ). Experimentally, this correlation canbe obtained under triangular waveform induction tests. Predic-tions under sinusoidal and nonsinusoidal inductions were madein oriented and nonoriented FeSi steel sheets. The predictions,when compared to experimental results, show an error of up to3% for nonoriented, and of up to 5% for oriented materials. Also,this method allows the prediction of the loss loop shapes. Thelimitations and perspectives of the method are considered in thiswork.

Index Terms—Eddy currents, harmonic analysis, hysteresis,magnetic losses.

I. INTRODUCTION

T HE ESTIMATION of magnetic losses is an important stepduring electric machine design to determine the final tem-

perature. Standard methods, like the Epstein test frame [1], [2],furnish the average value of magnetic losses under sinusoidalinduction. Usually, in these standard tests, peak induction andfrequency are correlated to magnetic losses. Empiric equationsare the most efficient methods to estimate the magnetic lossesunder sinusoidal induction in electric machines.

Electronic drives are important devices used to control the en-ergy flow for electric machines and open new possibilities forapplication. However, this technology produces a high level ofharmonic distortions on magnetic induction and increases mag-netic losses. In this case, empiric expressions are less efficientand can produce faulty results.

The conventional analysis of magnetic losses proposes theseparation of the two most important components:

(1)

where is the eddy current loss and is the hysteresis loss.Equations for eddy currents in conventional analysis are easy

Manuscript received November 19, 1999; revised February 12, 2001. Thiswork was partially supported by FAPESP and PADCT/CNPq.

M. S. Lancarotte is with the Depto. de Física dos Materiais a Mecânica, In-stituto de Física da Universidade de São Paulo, Rua do Matão, Travessa R, no.187, São Paulo SP Brazil 05508-900 (e-mail: [email protected]).

A. de A. Penteado, Jr. is with the Depto. de Energia a AutomaçãoElétricas, Escola Politécnica da Universidade de São Paulo, Av. Prof. LucianoGualberto, Travessa 3, no. 158, São Paulo SP Brazil 05508-900 (e-mail:[email protected]).

Publisher Item Identifier S 0885-8969(01)04328-5.

Fig. 1. Domain wall and magnetic moment arrangements for various inductionlevels.

to solve by the electromagnetic theory. However, hysteresis losshas a nonlinear behavior.

Previous works have proposed an approximate method topredict the increase of magnetic losses under nonsinusoidalinduction, by adding the magnetic loss at each harmonicfrequency [3]. Concerning eddy currents, it is a natural method;however, it is controversial when dealing with hysteresis loss.In this case the phase of each harmonic frequency can not beneglected.

The ferromagnetism models came after the development ofelectric machines and conventional models. Only in 1949 did thefirst works regarding magnetic domain contribution to magneticlosses begin. Based on these models, ferromagnetic materialsunder high alternating induction levels have two different mag-netization processes. Domain wall motion is predominant forlow induction levels and rotation of magnetization is predom-inant for high induction levels. The two mechanisms originatemicrocurrents at a microstructural level and dissipate energy [4],[5]. However, each domain wall motion produces a “localizededdy loss.” If the number of walls changes during the magneti-zation process, the global eddy loss changes too.

The peak induction is another important factor. It deter-mines which magnetization process and domain walls will beactivated. The less dissipative domain walls move before otherdissipative ones. Fig. 1 shows a scheme of a loss loop indicating

0885–8969/01$10.00 © 2001 IEEE

LANCAROTTE AND PENTEADO: ESTIMATION OF CORE LOSSES UNDER SINUSOIDAL OR NON-SINUSOIDAL INDUCTION 175

Fig. 2. Microcurrents produced by domain wall motion.

Fig. 3. Diagram of the hysteresigraph with digital feedback.

domain patterns of the grains at each magnetization level. Theinternal area of the loss loop corresponds to the energy dissi-pated per cycle in J/m. Fig. 2 shows a scheme of microcurrentsproduced by domain wall motion. The magnetization processis much more complex than those presented in Figs. 1 and 2.Nevertheless, these models are sufficient to identify the originof magnetic losses.

During the development of electric motors with highenergetic permanent magnets and electronic drives, the estima-tion of the magnetic loss under trapezoidal waveform inductionwas studied. Previous works strongly stated the correlationbetween the magnetic loss and the rate of magnetization(dB/dt), in agreement with ferromagnetic models [6], [7].More recently, rotational losses and high frequency harmonicdistortions have been the subject of studies [8]. However,prediction methods are still rare.

This work proposes a new method of analysis of magneticlosses, based on ferromagnetism models. The results canimprove or validate analytical models to predict core losses inelectric machines under nonsinusoidal inductions.

II. EXPERIMENTAL APPARATUS

A hysteresigraph with digital feedback was developed at theLaboratório de Materiais Magnéticos do Instituto de Física daUniversidade de São Paulo. This hysteresigraph was used toinvestigate high permeability materials.

Usually, hysteresigraphs generate a sinusoidal magnetic field( ). When feedback is implemented, it is possible to generatesinusoidal induction ( ). Fig. 3 shows the schematic diagramof a hysteresigraph with digital feedback. An arbitrary func-tion generator and power amplifier ( ) are used to producethe magnetization current in the coil. The shunt ( ) gen-erates a proportional voltage to field. The analog to digitalconverter (A/D) of channel 2 is used to generate anfile in themicrocomputer. The voltage produced by is proportionalto the magnetization rate (dB/dt) and a second A/D converter

Fig. 4. Loss loops superimposed, under sinusoidal and nonsinusoidalinduction waveform.

Fig. 5. Magnetic induction versus time for sinusoidal and nonsinusoidalwaveforms.

(channel 1) is used for data acquisition. Numeric integration ofdB/dt to generate the file is made by a routine program. Thecomparison between the file and the desired function adjuststhe arbitrary waveform generator. This process is repeated untilthe convergence is obtained. This system can be used to generateany shape of waveform induction. Measuring with a hysteresi-graph has advantages over wattimetric systems. The instanta-neous and average magnetization power, the curve, andenergy the dissipated by cycle can be determined from the samedata. More details are presented in [9].

III. EXPERIMENTAL RESULTS

The first test was made with a toroidal sample of nonorientedFeSi at 60 Hz and 1.6 T peak induction (). This sample wastested under sinusoidal and nonsinusoidal induction and the re-sults are shown in Figs. 4–6. Fig. 4 shows the two loss loops su-perimposed, under sinusoidal and nonsinusoidal induction. Themagnetic losses were measured by the wattimetric method andby the loss loop area. The results are shown in Table I.

The systematic error of 2% between the two methods may becaused by errors in density value. This density value is used toconvert the dimensions from J/mto W/kg.

Analyzing loss loops is insufficient to demonstrate allthe complexity involved in magnetic losses. Fig. 5 shows


Fig. 6. Instantaneous power absorbed and returned during the magnetizationprocess.

TABLE IMAGNETIC LOSSMEASUREMENTSUNDER SINUSOIDAL AND NON-SINUSOIDAL

INDUCTION USING WATTIMETRIC AND LOSSLOOPAREA METHODS

the corresponding waveforms of the induction versus time.The corresponding waveforms of the instantaneous power ispresented in Fig. 6. These results show a close relationshipbetween the magnetization rate and the instantaneous magneti-zation power. Increasing the maximum magnetization rate from603 T/s to 920 T/s results in a variation of the instantaneousmagnetization power from 7 W/kg to 22 W/kg. The positivepart of the instantaneous magnetization power represents theaddition of dissipated power plus reactive power absorbedduring the magnetization process. The negative part shows thereactive power returning to the magnetizing source.

IV. PROPOSEDANALYSIS

Conventional analysis of magnetic losses as a function of fre-quency and peak induction has two problems. The magnetiza-tion rate changes for different peak inductions and the sinusoidalmagnetization does not have a constant dB/dt during the wholeloss loop. The aim of the next experimental procedure was toseparate the real contribution of magnetization rate on mag-netic losses. A new sample of nonoriented FeSi (3% Si) steelwith toroidal geometry was used to verify the proposed analysis.Magnetizing fields were applied to the sample in order to resultin trapezoidal waveforms with 377, 565, 754 and 943 T/s, atdifferent peak induction levels ( ). The trapezoidal waveformwas chosen because of its similarity to the sinusoidal shape.Each trapezoidal waveform is characterized by the time intervalof the ramp ( ), and the time interval ( ) in the peak inductionlevel ( ). The average of the dissipated power () was deter-mined by the energy loss per cycle ( ) divided into four

Fig. 7. Average of the dissipated power as a function of magnetization rate forvarious peak inductions (B ).

Fig. 8. Average of the dissipated power as a function of (dB/dt).

times , presented in Fig. 7. This value represents the averagepower required to move the domain walls or rotate the magneticmoments. We ascertained that the energy dissipated in a trape-zoidal induction was not affected by the. Thus, a triangularinduction may be used to characterize the sample without re-strictions. The triangular waveform increases the resolution ofthe experimental procedures and makes easier the convergenceof the digital feedback.

The average of the dissipated power () under trapezoidalor triangular induction is very consistent with ferromagneticmodels. A dissipative process occurs only during changes of theinduction level.

Conventional analysis assumes that the eddy loss is propor-tional to (dB/dt) . Fig. 8 shows the average of the dissipatedpower as a function of (dB/dt)using triangular inductions, with

and 1.4 T. The linear correlation between(dB/dt) and average of the dissipated power is remarkable.However, this correlation depends on the peak induction. Thissuggests that for higher peak induction values, more dissipativemagnetization processes are activated.

Simplified methods can be applied for sinusoidal, triangularand trapezoidal inductions, due to the well established magne-tization rates and peak inductions. Under nonsinusoidal induc-tions, it seems that it is more logical to apply the instantaneousanalysis.


Fig. 9. Induction waveform used to study the behavior of the instantaneousmagnetization power as a function of magnetization rates.

Fig. 10. Loss loops superimposed as a function of various compositions ofmagnetization rates.

To study the behavior of the instantaneous magnetizationpower as a function of dB/dt, an induction waveform with tworates of magnetization was implemented. Rates of 377 T/sand 943 T/s were used, and their proportions were variedfrom 0% to 100% of the peak induction. The point at whichthe magnetization rate changes, was denominated transitioninduction ( ). The waveform of the induction is presented inFig. 9.

Fig. 10 (top) exhibits the six experimental loss loopsmeasured with this waveform at different ratios. Loopswith 0% (top first) and 100% (top last) correspond topure triangular induction with 377 T/s and 943 T/s, respec-tively. The four inner loops were traced with two magnetizationrates. Fig. 10 (bottom) exhibits the superposition of theseloops. In this figure, the strong dependence of the magneticfield ( ) as a function of magnetization rate (dB/dt) can beclearly observed. These results suggest that dissipative poweris proportional to the velocity of rearrangement of the domainwalls. Thus, a prediction method correlating the magnetic field( ) as a function of dB/dt and the induction level () can beimplemented [equation (2)].

dB/dt (2)

Fig. 11. Loss loops superimposed under triangular induction from 99 T/s to943 T/s.

Finding an algebraic solution for equation (2) is not a trivialtask due to the nonlinear behavior of loss loops. A numericsolution seems to be the simplest and most precise method.

In order to test the efficiency of a numeric solution, severalloss loops under triangular waveform with different rates (99 T/sto 943 T/s) and the same were measured. Fig. 11 shows theloss loops superimposed. To predict the shape of loss loops withan arbitrary induction waveform, the value of was interpo-lated from these triangular inductions as a function of the mag-netization rates.

Loss loops under triangular induction have another importantadvantage for the analysis. The value of the magnetic field( ) is proportional to the instantaneous magnetization powerbecause the dB/dt is constant. It is possible to observe atwhich points of the loss loop the process is more dissipative.[Instantaneous magnetization power () is the product of themagnetization rate (dB/dt) and the magnetic field ()].

The value of magnetic losses is determined by the internalarea of the loss loop. This method allows the prediction of lossesand loss loops regardless of frequency, but only for the samepeak induction.

Several comparisons between prediction and experimentalresults were made from 40 to 120 Hz at 1.3 T peak induction.The results exhibit a typical error of 3% in sinusoidal andnonsinusoidal induction.

Fig. 12 shows an example of these tests. The predicted andthe experimental loss loops (at 60 Hz, T) undernonsinusoidal induction are presented. The error between theexperimental and the predicted internal areas was 2%.

V. EPSTEINFRAME IMPLEMENTATION

The application of the method in the Epstein frame was alogical sequence of this work. The Epstein frame is the stan-dard geometry used by manufacturers to measure the parametersfor electric machine design. Empiric equations can be used tocorrelate the Epstein tests to electric machines under sinusoidalinduction. Our aim was to verify if the same empiric equationscould be applied to predict magnetic losses under nonsinusoidalinduction.


Fig. 12. Predicted and experimental loss loops superimposed, 60 Hz,B =

1:3 T under nonsinusoidal induction.

Fig. 13. Loss loops superimposed under triangular induction from 48 T/s to1200 T/s,B = 1:5 T.

Oriented FeSi Steel

The same experimental procedure previously applied to thetoroidal sample was implemented in the Epstein frame. Anoriented sample of FeSi (3% Si) was submitted to triangularinduction from 48 T/s to 1200 T/s at 1.5 peak induction(Fig. 13). This sample exhibits a different behavior to thatobserved in the nonoriented material. In the oriented materialsthe metallurgical process optimizes the domain wall motionsand reduces the mechanism of rotation of magnetic moments.This fact produces a more square shaped loss loop with highermagnetic permeability.

We observed that magnetic losses increase in the four edges ofthe loss loops due to magnetization rate. A delay time between

and was observed (the peak induction is not coincidentalwith the peak magnetic field). This fact suggests the existenceof magnetic viscosity or eddy current decay that may affect theefficiency of the prediction. These effects are very subtle andvisible only in the best magnetic materials.

The proposed method of prediction was applied to this sampleusing Fig. 13 as a “map” of the magnetic field () as a functionof (dB/dt) and induction level ( ). Loss loops with sinusoidaland nonsinusoidal induction at frequencies from 30 to 100 Hzwere traced. The errors for sinusoidal prediction were typically

Fig. 14. Predicted and experimental loss loops superimposed, 60 Hz,B =

1:5 T and nonsinusoidal induction.

Fig. 15. Loss loops superimposed under triangular induction from 48 T/s to900 T/s.

2%. For nonsinusoidal induction, the error was of up to 4.8%,probably caused by delay times.

Fig. 14 shows an example of the prediction and experimentalloss loops obtained under a frequency of 60 Hz, Tand nonsinusoidal induction.

In this example, the prediction error is 4.8%. The error is high.However the loss loops are almost the same, as can be observedin Fig. 13.

Nonoriented FeSi Steel

Nonoriented FeSi steel is a very important material forelectric rotative machines. In these machines, a high inductionlevel is required during operation. We tried to implement thisprediction method in Epstein strips of nonoriented steel, usingthe same magnetic condition ( T) as that applied forthe oriented steel. Fig. 15 shows the loss loops under triangularinduction with magnetization rates from 48 to 900 T/s. Unfor-tunately the resolution of the experimental apparatus limitedthe precision of data. The peak of the magnetic field () ismuch higher than the coercive field, and higher resolution isnecessary to measure the sample appropriately. However, aninteresting effect could be observed. The peak induction levelis a very important parameter and determines which dissipative


processes are activated. At high induction levels it is supposedthat all dissipative processes have already been activated.

The nonoriented sample was submitted to triangularinduction at 300 T/s and peak induction of 1.4, 1.5 and 1.6 T.The A/D converter sensitivity was expanded to register thecentral area of the loss loops. In this case, the coercive fieldbehavior is not affected by the peak induction.

Thus, at a high level of induction, it is possible to applythe prediction method using only dB/dt, without consideringthe peak induction. Deeper studies are necessary to verify thecontribution of the magnetization rate to the strong increase ofmagnetic loss at high induction levels shown by manufacturercatalogs.

VI. CONCLUSION

Magnetic losses were analyzed under sinusoidal and non-sinusoidal induction on FeSi samples. The contribution of themagnetization rate to the magnetic losses was also analyzed.The results demonstrate the higher complexity of instantaneousdissipative processes than that expressed by conventionalanalyzes. The prediction of core losses by analyzing the instan-taneous magnetic losses as a function of the magnetization rateand induction level has advantages because it considers all ofthe nonlinearity of magnetic materials. Another advantage isthat it is possible to predict the magnetic behavior of a largerange of frequencies using only a six loss loops under triangularinduction.

At low induction levels, domain wall motion is thepredominant magnetization mechanism, and a simplifiedanalysis can be used. At high induction levels the domain wallstend to disappear.

The instantaneous dissipative processes are more intense inthe first and third quadrants, especially in nonoriented steel.

At the moment, this method is restricted to low orderharmonic components generally measured in large electric

machines. For small machines and PWM drives, reentrant lossloops can be produced, and an improved method is necessary.

The use of the hysteresigraph shows advantages over watti-metric methods, because the first allows the measurement of theinstantaneous dissipative power.

An improvement of the experimental apparatus will be nec-essary to implement higher induction level tests and generaterotational fields, so that experimental methods can approach theactual magnetic conditions of electric machines.

ACKNOWLEDGMENT

The authors would like to thank Dr. F. P. Missell from LM-MIFUSP, Dr. F. Landgraf and Dr. M. Emura from LMPMMIPT,and Dr. J. R. Cardoso, Dr. S. Nabeta from PEA-EPUSP for theirspecial contributions during the development of this work andK. Mahar for the text review.

REFERENCES

[1] ASTM A343, “Alternating current magnetic properties of materials atpower frequencies using the wattmeter ammeter voltmeter method and25 cm Epstein test frame,”.

[2] IEC 404-2, “Methods of measurement of magnetic electrical and phys-ical properties of magnetic sheet and strip,”.

[3] J. D. Laverset al., “Estimation of core losses when the flux waveformcontains the fundamental plus a single odd harmonic component,”IEEETrans. Mag., vol. 13, no. 5, pp. 1128–1130, 1977.

[4] B. D. Cullity, Introduction to Magnetic Materials: Addison Wesley,1972.

[5] H. J. Williamset al., “Studies of the propagation velocity of a ferromag-netic domain boundary,”Phys. Rev., vol. 80, pp. 1090–1094, 1950.

[6] P. Brissonneau and D. Lebouc, “Core losses with trapezoidal wave forminduction,” IEEE Trans. Mag., vol. 20, no. 5, pp. 1484–1486, 1984.

[7] T. Sato and Y. Sakaki, “Discussion on eddy current loss under squarewave voltage excitation,”IEEE Trans. Mag., vol. 24, no. 6, pp.2904–2906, 1988.

[8] A. J. Moses and N. Tutkun, “Investigation of power loss in woundtoroidal cores under PWM excitation,”IEEE Trans. Mag., vol. 33, no.5, pp. 3763–3765, 1997.

[9] M. S. Lancarotte, “Previsão das perdas magnéticas na presença de har-mônicos,” M.Sc. dissertation, EPUSP, 1999.

estimation of core losses under sinusoidal or nonsinusoidal induction by analysis of magnetization...

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