Analytical core loss models for electrical steel in power electronic applications

Dennis Kampen1, Michael Owzareck2 Stefan Beyer3 Nejila Parspour4, Stefan Schmitt1

1BLOCK Transformatoren-Elektronik GmbH, Verden, Germany 2University of Braunschweig, Germany, 3University of applied science, Hannover, Germany 4University of Stuttgart, Germany

dennis.kampen@block-trafo.de m.owzareck@tu-braunschweig.de stefan.beyer@fh-hannover.de parspour@iew.uni-stuttgart.de stefan.schmitt@block-trafo.de

Abstract- This paper shows different state of the art analytical iron loss models and a modified model, which combines the modified Bertotti model with the form factor of Grtzer. A loss comparison under sinusoidal and non-sinusoidal voltage excitation in the frequency range from 50Hz to 1000Hz without minor loops is shown. Measurements show that the accuracy of some models is high, independent from the shape of the voltage and easy to apply for engineers designing inductive components in power electronic applications.

I. INTRODUCTION

RON LOSSES in grain oriented and non-grain oriented electrical steel where approximated by different loss

models. The challenge hereby is the nonlinearity of the soft magnetic material. Losses depend on frequency, flux density, orientation of the flux in the steel sheets, mechanical handling of the steel sheets [1], temperature and many other influences. To characterize the losses at different frequencies and flux density levels, various loss models where developed, e.g. Steinmetz [2], Jordan [3] or Bertotti [4]. In the last decades original models where expanded to achieve better adaptability for sinusoidal [5], [6] and also for non sinusoidal voltage excitation, e.g. [7], [8], [9], [10], [11], [12] and [13]. The loss models where developed for different soft magnetic materials, i.e. ferrite, soft magnetic components (SMC) or electrical steel.

The scope of this paper is to compare the empirical iron loss models and the iron loss models based on loss separation and a modified model with loss measurements of non oriented electrical steel. Hysteresis models are explicitly not considered. The measurements are carried out in an Epstein frame from frequencies 50Hz to 1000Hz at different flux density levels under sinusoidal and non-sinusoidal voltage excitation.

II. LOSS MODELS

For describing iron losses there are essentially three approaches.

The first approach uses hysteresis models, which describe the static material characteristics by means of macroscopic energy considerations [14]. For finite element analysis and for cases where the time dependent behavior of the flux density is known an inverted hysteresis model was described

in [15]. Dynamic behavior of hysteresis models was implemented in [16] by superposition of the field strength shares. Considering the energy densities, the losses can be approximated. Also the Preisach model is among the presented hysteresis models [17]. Loss prediction under arbitrary induction with minor loops was proposed in [18].

The second approach are empirical solutions like the Steinmetz equation [2] and the third approach is the loss separation, where the losses are separated e.g. into eddy current losses, hysteresis losses and anomalous losses [19].

Although state of the art iron loss computation is mainly in the hysteresis models, the numerical calculation complexity is very high. In many applications, easy and fast calculation methods are sufficiently accurate and favored. Therefore this evaluation concentrates only on the second and the third approach, because these models are very simple and easy to use. The models which will be compared to each other will be briefly explained here.

A. Steinmetz

The most known empirical loss model for iron losses is the Steinmetz equation (1). The coefficients Cm, and can be determined by approximation techniques, where 1<

B. Jordan With the Jordan equation, loss separation into eddy current

losses pe and hysteresis losses ph was introduced [3].

( )( )f 2 2

fe h e Bp k f B k f B

= + (4)

According to classical eddy current theory the coefficient

ke can be determined by physical parameters of the steel sheet [19].

2 2fe sheet

efe 6

dk

=

(5)

However, it is suggested here to determine the coefficients

by approximation techniques.

C. Grtzer Grtzer expanded the Jordan equation with a form factor

for the eddy current losses to reach a better approximation for non sinusoidal signals [9].

( )( ) 2f 2 2arbfe h e

sin

Bp k f B k f B

= + (6)

( )

( )

2

0rms

ave

0

1

1

T

T

u t dtTU

Uu t dt

T

= =

(7)

The form factor, given by the is the ratio of RMS value

Urms and rectified value Uave, is defined with respect to the exciting voltages in contrast to the flux densities as the flux density and its distribution are not readily accessible for measurements. Moreover, the voltage form factor is a suitable indicator for the eddy current losses as experimental data shows a square dependency of the eddy current loss on the voltage RMS value. D. Bertotti

Because previous loss models calculated fewer eddy current losses than measured, Bertotti proposed the anomalous or anomalous loss term with the coefficient ka and expanded the Jordan model [4].

( )1,52 2 2

fe h e a

= + + p k f B k f B k f B

(8)

The anomalous loss term considers additional eddy current

losses due to domain wall shifting.

E. Fiorillo Model An expansion of the Bertotti model to fit also for non-sinusoidal flux density was proposed in [21]. Linear addition

of the Fourier components of the eddy current loss term is proposed and the anomalous loss term is obtained by integrating the derivation of the flux density over one period, (9).

( ) ( )

1,5T2 2

fe h 1 20

1 = + +

n

dB tp p f A n B A dt

T dt

(9)

where

2 2 2fe sheet

1fe 6

=

d fA

(10)

2 fe 0 fe= A G V A (11)

For sinusoidal voltage excitation the anomalous loss term

transforms to

( )1,5

fe 0 fe8, 76 = ap G V A f B (12)

For triangle voltage excitation the anomalous loss term

transforms to

( )1,5

fe 0 fe9, 051 = ap G V A f B (13)

For rectangle voltage excitation the anomalous loss term

transforms to

( )1,5

fe 0 fe8 = ap G V A f B (14)

F. Modified Jordan Model with variable coefficients

A modification of the Jordan equation with higher accuracy for high frequencies and different flux density levels was proposed in [5].

( ) ( )2 2 2fe stat d yn p k B f B k B f B= + (15)

The coefficients for eddy current losses and hysteresis losses where changed to static, kstat, and dynamic, kdyn, loss coefficients, which are functions depending on flux density.

( ) 3 2stat s tat,3 s tat,2 s tat,1 stat,0 k B k B k B k B k= + + + (16) ( ) 3 2dyn dyn,3 dyn,2 dyn,1 dyn,0 k B k B k B k B k= + + + (17)

With the variable coefficients a higher degree of freedom is reached in comparison to previous models.

The dynamic coefficient represents the classical and anomalous losses, whereas the static coefficient represents the hysteresis losses.

G. Expanded modified Jordan Model

In [10], [11] and [12] a method is described to expand the

110

modified Jordan model so that it is valid also for non sinusoidal voltage excitation. Like the Grtzer model, the RMS and the rectified values of the signal in comparison to a sinusoidal signal is needed.

2 2

fe h ep p p = + (18) where:

ave

ave ,1

UU

=

(19)

rms

rm s ,1

UU

=

(20)

The difference to Grtzer is that for the dynamic term the

ratio between the RMS value of the signal and the RMS value of the fundamental frequency component is used instead of the form factor, because it is stated that:

2

rms~ep U (21)

The static losses of non sinusoidal voltage signals the ratio between the rectified value of the signal and the rectified value of the fundamental frequency component is added, because it is stated that:

2

ave~hp Uf

(22)

In this paper we qualify the models according to DIN EN

60404-2 in an Epstein frame. The measurements are done at the same peak flux density levels. That means that at the same peak flux density the hysteresis losses are equal for all voltage wave forms without minor loops. Thus to compare this theory with the other models, this model is applied to the modified Jordan model with variable coefficients and following equation is obtained:

( ) ( )2 2 2 2fe stat dyn p k B f B k B f B= + (23) H. Modified Bertotti Model with variable coefficients

Variable coefficients where proposed also for the Bertotti model for higher frequencies and flux density levels [21].

( )( ) ( ) ( ) ( ) 1,5f 2 2fe h e a Bp k f B k B f B k B f B= + +

(24)

Like the modified Jordan model the modified Bertotti

model uses eddy current and anomalous loss coefficients which itself are third-degree polynomial functions of the flux density. The difference is that the hysteresis coefficient kh is constant.

In analogy to the classical Jordan model the loss exponent is a function of the maximal flux density:

( ) 3 23 2 1 0 f B B B B = = + + + (25) I. New combined model

The experience with all previous loss models for non sinusoidal voltage is that the measurements cannot be reproduced by the models with better accuracy than 20% at all flux density levels and frequencies. Therefore a new combined formula is proposed here. The modified Bertotti Model with variable coefficients is extended by the form factor of Grtzer:

2 1,5

arb arbfe h e a

sin sin

p p p p

= + + (26)

For sinusoidal voltage excitations the model is reduced to

( ) ( ) ( ) ( )1,52 2 2

fe h e a p k B f B k B f B k B f B= + + (27)

The new model differs from the modified Bertotti model

essentially in the calculation of the hysteresis losses. The loss exponent is considered to be constant, =2, and the hysteresis loss coefficient depends on flux density like the eddy current and anomalous loss coefficients. Thus all three loss coefficients are third-degree polynomial functions of the flux density.

To be valid also for non sinusoidal voltage excitations, the model was expanded by the form factor of Grtzer.

Grtzer originally assumed that only the classical eddy currents have a dependence on the shape of the voltage wave form. In the new combined model it is assumed that also the anomalous losses have this dependence because of dynamic domain wall movements.

Grtzers considerations start with the transformer main equation for sinusoidal voltages:

rms fe

22

U B f N A=

(28)

Equation (27) is expanded by the form factor to fit for all voltage wave forms. The RMS value of the voltage is then a function of the form factor and the transformer main equation transforms to

rms fe 4U B f N A = (29)

Substituting equation (27) and (29) following

proportionality is obtained.

2 2

e~ ~p B and 1,5 1,5a ~ ~p B (30)

111

e2 .

p const = and a

1,5 .p const =

(31)

Thus anomalous losses not only have proportionality to the

flux density but also to the form factor. Because of the constant relation in (31) also the relation in

(32) applies.

e,sin e,arb

2 2sin arb

p p =

and a,sin a,arb1,5 1,5

sin arb

p p =

(32)

The new model is therefore valid for any voltage wave

forms without minor loops. Minor loops would cause hysteresis losses which also the new model is not able to consider. This is also the cause, why the new combined model has no additional scaling factor for the hysteresis loss component for non sinusoidal voltage excitations in comparison to the expanded modified Jordan model.

The variation of the hysteresis loss coefficient with the maximum value of the magnetic flux density can be explained by the non linearity of the static loop and therefore also by the variation of the relative permeability with the flux density level.

In [5] the following analytic expression was found for the variation of the static coefficient:

irr

statfe

HkB

=

(33)

In (33) there is a reciprocal dependence between kstat and

the maximum value of the magnetic flux density. In the saturation region the static loss coefficient remains constant because the flux density cannot be increased further.

Physically, between the beginning of the area of the magnetization where only reversible processes occur and the saturation area where it comes to reversible rotary processes, there is the area of the non reversible processes. This area is responsible for the hysteresis losses. In this area irreversible domain wall movements and irreversible rotary processes occur, because the domain walls cannot return in her original orientation when switching off the external magnetic field.

The variation of the eddy current loss coefficient with the

maximum value of the magnetic flux density can be explained by the induced counter flux densities. Assuming that Jordans loss separation is valid for the eddy current loss expression from (5), some boundary conditions must be considered. First the thickness of the steel sheet must be considerably smaller than the width of the sheet. Secondly the equation is valid for the classical eddy current losses only for a sinusoidal homogeneous flux density distribution in the cross section of the sheet. Eddy currents generate magnetic fields in the steel sheets which counteract according to the rule of Lenz the main flux and the magnetic flux density is weakened.

dyn stat B B<

(34)

To maintain the main flux density level, the magnetic field

strength and therefore the current must be increased. Thus the area of the hysteresis loop and the losses increase too, Fig 1.

The round shape of the hysteresis loop at higher

frequencies is caused by the phase displacement between flux density and magnetic field strength. The physical reason for that is that the applied external field is not propagated within the core at once but dampened by eddy currents. The anomalous loss coefficient in this model varies significantly due to displacement effects, the skin depth B [23]

( )fe

B0 a ,f B f

=

(35)

and the variation according to the flux density. In the saturation region all domains are oriented in the direction of the external field and therefore no domain walls exist anymore.

sat alim 0

B BP

(36)

III. COMPARISON BETWEEN LOSS MODELS AND MEASUREMENTS

A. Measurement Setup All measurements where made with M400-50A non

oriented electrical steel sheets in an Epstein frame according to DIN EN 60404-2.

The flux density was adjusted using the rectified value of the measured voltage in formula (37).

m

fe

4

uB

f N A=

(37)

B. Coefficient fitting for the models

The exponents and of the Steinmetz equation where fitted as follows. With the loss measurement of two measurement points at two different flux density levels with

Fig. 1. Dependence of the magnetization cycles on frequency of non-grainoriented electric steel M400-50A

-1.5-1

-0.50

0.51

1.5

-500 -300 -100 100 300 500

B (T)

H (A/m)

50 Hz

250Hz

500Hz

1000Hz

112

constant frequency, can be determined.

( ) ( )( ) ( )

fe,1 fe,2

1 2

log log log log

=

p p

B B

(38)

With the loss measurement of two measurement points at

two different frequencies with constant flux density, can be determined.

( ) ( )( ) ( )

fe,1 fe,2

1 2

log loglog log

=

p pf f

(39)

The Steinmetz coefficient mC can now be determined using

equation (1).

fe

m pC

f B=

(40)

The loss coefficient fitting for the Jordan model and also

for the Grtzer model is based on the loss separation method described in [22].

The loss coefficient fitting for the Bertotti model is based on method described in [21].

For the loss coefficient fitting of the Fiorillo model the losses were measured at 5Hz for the hysteresis loss term. For the classical eddy current loss term the Fourier analysis of the flux density was made and for the anomalous loss term the derivation of the flux density was integrated and the factor A2 was determined by optimization techniques.

The loss coefficient fitting for the modified Jordan model is based on method described in [5].

The loss coefficient fitting for the modified Bertotti model is based on method described in [6].

The loss coefficient fitting for the new combined model was determined by the following approach:

First the specific losses of equation (27) are divided by the frequency to get the (loss) energy density.

( ) ( ) ( )Fe

fe

2 2 0,5 1,5h e a

pwf

k B B k B f B k B f B

=

= + +

(41)

A different notation for (41) is used:

( ) ( ) ( )( )

22 1,5fe e a

2h

w k B B f k B B f

k B B

= +

+

(42)

If now the loss energy density in dependence of the square

root of the frequency at a constant flux density is plotted, this plot can be approximated by a second order polynomial

function by least square methods.

( )2

few a f b f c= + + (43)

The three loss coefficients follow directly from:

( )e 2f ak B B= = ; ( )a 1,5f bk B B= = ; ( ) 2f h ck B B= =

(44)

In this investigation the coefficient approximation is done

at different flux densities. The results are plotted and a third-order polynomial fitting curve is approximated to the results.

C. Sinusoidal voltage excitation

In Fig 2 Fig. 8 the error between the measured losses under sinusoidal voltage excitation and the approximated loss model curves is shown.

Fig. 3. Relative error of the Jordan model/ Grtzer model in comparison tothe measurements under sinusoidal voltage excitation.

-20-10

010203040

0 0.5 1 1.5

rela

tive

erro

r / %

Ba (T)

50Hz

200Hz

500Hz

1000Hz

Fig. 2. Relative error of the Steinmetz model in comparison to themeasurements under sinusoidal voltage excitation.

-100-80-60-40-20

02040

0 0.5 1 1.5

rela

tive

erro

r / %

Ba (T)

50Hz

200Hz

500Hz

1000Hz

113

The relative error of the Steinmetz model is very high. This

is caused by the incapability of the model to consider eddy current effects separately and therefore by less possible degree of freedom. Thus also the modified Steinmetz equation for non sinusoidal voltage excitation is not followed up here anymore, although measurements were made.

The Jordan, the Bertotti and the Fiorillo model show smaller relative errors than the Steinmetz model, but the errors could not be kept smaller than 32% with the Jordan model, -52% with the Bertotti model and -22% with the Fiorillo model. This is caused by the static coefficients.

The modified/ expanded modified Jordan model, the modified Bertotti model with variable coefficients and the new combined model in comparison to the other models show the smallest relative errors of all models. The difference of the modified Jordan model is kept within +10 and -11% and the difference of the modified Bertotti model is kept within +14 and -6%. The new combined model has a relative error of +15% and -6%.

D. Triangle voltage excitation

In Fig 9 Fig. 12 the error between the measured losses under triangular voltage excitation and the approximated loss model curves is shown. The Steinmetz, Jordan, and Bertotti models are not shown, because the models are not adaptable for other voltage excitations except sinusoidal voltages. Thus only the Grtzer model, the Fiorillo model and the expanded modified Jordan model is compared with the new model.

Fig. 9. Relative error of the Grtzer model in comparison to themeasurements under triangular voltage excitation.

-20-10

010203040

0 0.5 1 1.5

rela

tive

erro

r / %

Ba (T)

50Hz

200Hz

500Hz

1000Hz

Fig. 8. Relative error of the new combined model in comparison to themeasurements under sinusoidal voltage excitation.

-10-505

101520

0 0.5 1 1.5

rela

tive

erro

r / %

Ba / T

50Hz

200Hz

500Hz

1000Hz

Fig. 7. Relative error of the Bertotti model with variable coefficients incomparison to the measurements under sinusoidal voltage excitation.

-10

-5

0

5

10

15

0 0.5 1 1.5

rela

tive

erro

r / %

Ba (T)

50Hz

200Hz

500Hz

1000Hz

Fig. 6. Relative error of the modified/ expanded modified Jordan model incomparison to the measurements under sinusoidal voltage excitation.

-20

-10

0

10

20

0 0.5 1 1.5

rela

tive

erro

r / %

Ba (T)

50Hz

200Hz

500Hz

1000Hz

Fig. 5. Relative error of the Fiorillo model in comparison to themeasurements under sinusoidal voltage excitation.

-30-20-10

0102030

0 0.5 1 1.5

rela

tive

erro

r / %

Ba (T)

50Hz

200Hz

500Hz

1000Hz

Fig. 4. Relative error of the Bertotti model in comparison to themeasurements under sinusoidal voltage excitation.

-60

-40

-20

0

20

40

0 0.5 1 1.5

rela

tive

erro

r / %

Ba (T)

50Hz

200Hz

500Hz

1000Hz

114

With the Grtzer model a maximum relative error of +40%

and -11% was reached. The form factor is very helpful to fit the losses more accurate but the Grtzer model is based on the Jordan model and therefore there is a subsequent error from the sinusoidal excitation results. The same conclusion fits for the Fiorillo model, which has relative errors between +15% and -39%.

The error of the expanded modified Jordan model is between +16% and -14%, which is a higher error than at sinusoidal voltage excitation.

The new combined model in comparison to the other models shows very small relative errors. The difference is kept within +12 and -8%, which is in the same range like with sinusoidal voltage excitation.

E. Rectangle voltage excitation

In Fig 13 Fig. 16 the error between the measured losses under rectangular voltage excitation and the approximated loss model curves is shown. Like under triangular voltage excitation only the Grtzer, the Fiorillo and the expanded

modified Jordan model is compared with the new model.

With rectangle voltage excitation the Grtzer model has

relative errors of maximum +30% and -20%. The error of the Fiorillo model is between +17% and -22%. The error of the expanded modified Jordan model is between +2% and -18%.

Fig. 16. Relative error of the new combined model with variable coefficientsin comparison to the measurements under rectangle voltage excitation.

-15-10-505

1015

0 0.5 1 1.5

rela

tive

erro

r / %

Ba (T)

50Hz

200Hz

500Hz

1000Hz

Fig. 15. Relative error of the expanded modified Jordan model with variablecoefficients in comparison to the measurements under rectangle voltageexcitation.

-20

-10

0

10

0 0.5 1 1.5

rela

tive

erro

r / %

Ba (T)

50Hz

200Hz

500Hz

1000Hz

Fig. 14. Relative error of the Fiorillo model in comparison to themeasurements under rectangle voltage excitation.

-30-20-10

0102030

0 0.5 1 1.5re

lativ

e er

ror /

%Ba (T)

50Hz

200Hz

500Hz

1000Hz

Fig. 13. Relative error of the Grtzer model in comparison to themeasurements under rectangle voltage excitation.

-30-20-10

010203040

0 0.5 1 1.5

rela

tive

erro

r / %

Ba (T)

50Hz

200Hz

500Hz

1000Hz

Fig. 12. Relative error of the new combined model with variable coefficientsin comparison to the measurements under triangular voltage excitation.

-10

-5

0

5

10

15

0 0.5 1 1.5

rela

tive

erro

r / %

Ba (T)

50Hz

200Hz

500Hz

1000Hz

Fig. 11. Relative error of the expanded modified Jordan model with variablecoefficients in comparison to the measurements under triangular voltageexcitation.

-20

-10

0

10

20

0 0.5 1 1.5

rela

tive

erro

r / %

Ba (T)

50Hz

200Hz

500Hz

1000Hz

Fig. 10. Relative error of the Fiorillo model in comparison to themeasurements under triangular voltage excitation.

-50-40-30-20-10

0102030

0 0.5 1 1.5

rela

tive

erro

r / %

Ba (T)

50Hz

200Hz

500Hz

1000Hz

115

In comparison to Grtzer, Fiorillo and the expanded modified Jordan model the new combined formula reaches errors of maximum only +13% and -10% with even lower average errors.

IV. CONCLUSION

In this paper various analytical iron loss models are briefly explained. Only empirical and loss separation models are considered. A new combined model based on the Bertotti model with variable coefficients and the form factor of the Grtzer model is proposed. All models where fitted to measured loss curves of non oriented electrical steel M400-50A in an Epstein frame. With the Steinmetz model the losses could not be approximated with smaller maximum error than +28% and -80% with sinusoidal voltage excitation. The Jordan the Bertotti and the Fiorillo model showed also very high maximum errors at sinusoidal voltage excitation. The best fitting for sinusoidal voltages was reached with the modified Jordan and the modified Bertotti model with variable coefficients and the new combined model.

The Steinmetz, Jordan, and Bertotti models are not considered for non sinusoidal voltage excitation, because the models are not adaptable for other voltage wave forms. Thus only the Grtzer, the Fiorillo and the expanded modified Jordan model are compared with the new model.

With triangle and rectangle voltage excitation the new combined model showed minimal relative errors in comparison to Grtzer, Fiorillo or the expanded modified Jordan model. This evaluation is made for voltage excitations between 50Hz and 1000Hz and signals without minor loops. The measurements show that the accuracy of some models is high, independent from the shape of the voltage and easy to apply for engineers designing inductive components in power electronic applications.

For higher frequencies and minor loops further investigations are required or state of the art hysteresis models must be considered.

V. REFERENCES

[1] Miyagi, D.; Aoki, Y.; Nakano, M.; Takahashi, N.; , "Effect of Compressive Stress in Thickness Direction on Iron Losses of Nonoriented Electrical Steel Sheet," Magnetics, IEEE Transactions on , vol.46, no.6, pp.2040-2043, June 2010

[2] Steinmetz, Chas.P.; , "On the law of hysteresis," Proceedings of the IEEE , vol.72, no.2, pp. 197- 221, Feb. 1984

[3] H. Jordan, Die ferromagnetischen Konstanten fr schwache Wechselfelder, Elektr. Nach. Techn., 1924.

[4] Bertotti, G.; , "General properties of power losses in soft ferromagnetic materials ," Magnetics, IEEE Transactions on , vol.24, no.1, pp.621-630, Jan 1988

[5] Popescu, M.; Miller, T.; Ionel, D.M.; Dellinger, S.J.; Heidemann, R.; , "On the Physical Basis of Power Losses in Laminated Steel and Minimum-Effort Modeling in an Industrial Design Environment," Industry Applications Conference, 2007. 42nd IAS Annual Meeting. Conference Record of the 2007 IEEE , vol., no., pp.60-66, 23-27 Sept. 2007

TABLE I LIST OF SYMBOLS

Symbol Quantity Unit

Afe magnetic cross section m2 Steinmetz coefficients - B magnetic flux density, magnetic induction T Steinmetz coefficients - Cm Steinmetz coefficients W/kg dsheet thickness of the steel sheet m B penetration depth m f frequency Hz feq equivalent sinusoidal frequency Hz frp repetition frequency Hz G dimensionless coefficient - Hirr irreversible magnetic field strength A/m ka anomalous eddy current loss coefficient Am3/(kgs0,5) kdyn dynamic loss coefficient Am4/(kgV) ke classical eddy current loss coefficient Am4/(kgV) kh hysteresis loss coefficient Am4/(kgVs) kstat static loss coefficient Am4/(kgVs) Fe specific conductivity of the steel sheet S/m

0 magnetic field constant - a relative amplitude permeability Vs/Am N number of Windings - n number of harmonics - pa specific anomalous eddy current losses W/kg ph specific hysteresis losses W/kg pfe specific magnetic iron losses W/kg

fe specific density of the steel sheet kg/m3 T periodic time s u voltage V V0

parameter for local distribution of coercitive force

-

wfe specific magnetic iron losses density Ws/kg

arb form factor of an arbitrary signal - sin form factor of a sinusoidal signal -

116

[6] Ionel, D.M.; Popescu, M.; Dellinger, S.J.; Miller, T.J.E.; Heideman, R.J.; McGilp, M.I.; , "On the variation with flux and frequency of the core loss coefficients in electrical machines," Industry Applications, IEEE Transactions on , vol.42, no.3, pp. 658- 667, May-June 2006

[7] Albach, M.; Durbaum, T.; Brockmeyer, A.; , "Calculating core losses in transformers for arbitrary magnetizing currents a comparison of different approaches," Power Electronics Specialists Conference, 1996. PESC '96 Record., 27th Annual IEEE , vol.2, no., pp.1463-1468 vol.2, 23-27 Jun 1996

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[9] D. Grtzer, D.:Ummagnetisierungsverluste weichmagnetischer Werkstoffe bei nichtsinusfrmiger Aussteuerung, Zeitschrift angewandte. Phys. 32 (1971), pp. 241-246

[10] Popescu, M.; Ionel, D.M.; Boglietti, A.; Cavagnino, A.; Cossar, C.; McGilp, M.I.; , "A General Model for Estimating the Laminated Steel Losses Under PWM Voltage Supply," Industry Applications, IEEE Transactions on , vol.46, no.4, pp.1389-1396, July-Aug. 2010

[11] Boglietti, A.; Cavagnino, A.; Ionel, D.M.; Popescu, M.; Staton, D.A.; Vaschetto, S.; , "A General Model to Predict the Iron Losses in PWM Inverter-Fed Induction Motors," Industry Applications, IEEE Transactions on , vol.46, no.5, pp.1882-1890, Sept.-Oct. 2010

[12] Boglietti, A.; Cavagnino, A.; Lazzari, M.; , "Fast Method for the Iron Loss Prediction in Inverter-Fed Induction Motors," Industry Applications, IEEE Transactions on , vol.46, no.2, pp.806-811, March-april 2010

[13] 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. 29042910, Sep. 1990.

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[15] Sadowski, N.; Batistela, N.J.; Bastos, J.P.A.; Lajoie-Mazenc, M.; , "An inverse Jiles-Atherton model to take into account hysteresis in time-stepping finite-element calculations," Magnetics, IEEE Transactions on , vol.38, no.2, pp.797-800, Mar 2002

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