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Iron losses in soft magnetic materials under periodic non-sinusoidal supply conditions
Physica B 275 (2000) 191}196
Iron losses in soft magnetic materials under periodic non-sinusoidal supply conditions O. Bottauscio!,*, M. Chiampi", D. Chiarabaglio! !Istituto Elettrotecnico Nazionale Galileo Ferraris, c.so M. d 'Azeglio, 42, 10125 Torino, Italy "Dipartimento di Ingegneria Elettrica Industriale, Politecnico di Torino, Italy
Abstract The paper investigates the role of low- and high-order harmonics, superposed to a 50 Hz fundamental magnetic #ux density waveform, on iron losses in NO Fe-Si laminations. The e!ect on the fundamental contributions of the losses (classical and hysteresis#excess items) is evaluated by means of a combined "nite element - dynamic hysteresis model of the ferromagnetic sheet. A comparison with the prediction of the statistical theory of losses is "nally presented and discussed. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Magnetic losses; Hysteresis model; Finite element model
1. Introduction The magnetic losses in electrical machines are usually a!ected by signi"cant distortions of local #ux density waveforms. The non-sinusoidal evolutions are produced by di!erent causes: (i) local saturations giving rise to harmonics whose order is usually less than 10, (ii) rotor and stator teeth which determine #ux ripple with harmonics having order of some tens, (iii) static electronic supplies, which involve Fourier components up to some hundred order. The presence of non-sinusoidal waveforms makes the prediction of iron losses very di$cult, also taking into account the complex magnetization mechanism which involves the loss separation into three contributions: classical, static and excess losses. Starting from these considerations, it becomes essential to accurately predict the e!ects on the magnetic losses due to the presence of harmonics. Experimental investigations [1,2] have provided interesting results, but, of course, they cannot give a general solution. Some analytical ap-
* Corresponding author. Tel.: #39-011-3919776; fax: #39011-6509471. E-mail address: [email protected] (O. Bottauscio)
proaches, based on loss separation analysis, have been also proposed [3]; however, they neglect the e!ects of the incomplete #ux penetration inside lamination and the presence of minor loops in the hysteresis cycles. Thus, the use of numerical techniques becomes necessary. The aim of this paper is the application of a numerical model to emphasize the in#uence of harmonic components on the magnetic power losses separated into their fundamental contributions. The analysis is developed, for the sake of clarity, in simple geometrical structures, such as magnetic laminations, in order to avoid the additional phenomena (rotational #uxes, moving parts, etc.) occurring in the cores of electrical machines. The numerical approach is based on the "nite element solution of the electromagnetic "eld problem inside laminations under time periodic supply conditions, taking into account the magnetic properties through the dynamic Preisach model of hysteresis (DPM). The investigation is developed by "xing the fundamental component of the magnetic #ux density and by superposing a given harmonic, having a stated amplitude with respect to the fundamental one. Therefore, the comparison is performed keeping the rms value of the magnetic #ux density "xed. The in#uence of several parameters (harmonic order, peak value of fundamental, ratio between harmonic and fundamental amplitude,
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 7 6 9 - 3
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O. Bottauscio et al. / Physica B 275 (2000) 191}196
phase angle) on the iron losses and on their di!erent contributions is analysed. Finally, a comparison with the predictions of statistical theory is presented.
where k and k are suitable constants depending on the 1 3 material. The hysteresis model is detailed in Ref. [6], where the identi"cation procedure is also presented. Knowning the "eld distribution, classical losses E are # estimated by:
2. Numerical model
`d@2 TJ2 dt dz, (6) p ~d@2 0 where ¹ is the period of the fundamental and J is the induced current density. Static E and excess E losses ) % are given from the area of local loops.
The electromagnetic "eld problem in an in"nite sheet having thickness d along the z direction, where a known magnetic #ux U(t) is #owing (x direction), is governed by the equation
P
p d@2 curlf(curlA)"!pAQ # AQ dz, (1) d ~d@2 where f(B)"H represents the relation between magnetic #ux density and "eld, p is the conductivity and A"(0, A(z), 0) is the vector potential with nonhomogeneous Dirichlet boundary conditions U U A(!d/2)"! , A(d/2)" . 2 2
(2)
In Eq. (1), due to symmetry considerations, the integral term of the right hand-side is zero. Problem (1), linearized using the H-version of "xed point (FP) technique, is developed in the harmonic domain to impose the steady-state periodic behaviour. Thus, for the mth harmonic at the ith iteration, the "eld equation becomes RR(m) R2A(m) ! M i~1 , l M i "jmupA(.) FP Rz2 i Rz M
(3)
where l is the FP coe$cient, u is the fundamental FP angular frequency and phasors A(m) and R(m) represent M i the residual M i the magnetic vector potential and term, respectively. After the "eld solution, the magnetic quantities are reported in the time domain through an inverse fast Fourier transform (FFT) and then the update of FP residual is computed, by the following relations: H (t)"R (t)#l B (t), i i~1 FP i R (t)"H (t)!l f~1(H (t)). (4) i i FP i The new FP residuals are "nally reported in the harmonic domain and "eld equations are updated for the next iteration. In Eq. (4), hysteretic relation f~1, which gives the B waveform from the time evolution of H, is handled employing a Preisach-type model. This model is based on the dynamic relation proposed by Bertotti [4] which introduces a di!erential law governing the material magnetization. The moving e!ect [5] is also included by correcting the applied magnetic "eld through the term: f (M)"k M#k M3, 1 3
(5)
P P
E " #
3. Discussion of the results The investigation has been performed on a NO Fe}Si 2.5 wt% lamination of thickness 0.65 mm, having an electrical conductivity of 2.37 MS/m, usually employed in electrical motor cores. The distorted induction waveforms consist of a fundamental 50 Hz sinusoid with a given harmonic component B(t)"B145[sin ut#k sin(nut#a)].
(7)
The ratio k between the amplitude Bn of the n-th harmonic component and the fundamental one B145 is kept "xed (k"0.1), while the harmonic order n and the phase angle a are varied (3)n)40 and 0)a)n). In such a way, the comparison is performed keeping the rms value of the magnetic #ux density "xed. For each considered B waveform, the classical and hysteresis#excess losses have been computed. In order to emphasize the harmonic e!ect, in the presentation of the results the values of each loss contribution have been divided by the corresponding sinusoidal ones (E /E ). The dia)!3. 4*/ gram of sinusoidal energy losses versus the induction level is shown in Fig. 1. In the following the results are discussed considering separately the in#uence of each parameter. 3.1. Inyuence of phase angle a The in#uence of phase angle on classical and hysteresis#excess losses has been analysed, varying the harmonic order and the amplitude of the fundamental induction value. The e!ect of phase angle is weak in the case of classical losses, as can be noted by Fig. 2: an appreciable dependence is found only in presence of the third harmonic. On the contrary, the in#uence of a on hysteresis#excess losses is signi"cant for harmonic orders lower than nine, as shown in Fig. 3. This behaviour is well explained considering that, mainly for low harmonic orders, the phase angle determines the peak value of B, as put in evidence by Fig. 4 which presents the waveforms of the mean magnetic #ux density and the corresponding dynamic loops in the case of n"3 and
O. Bottauscio et al. / Physica B 275 (2000) 191}196
193
Fig. 4. E!ect of phase angle a for n"3 and k"0.1: (a) magnetic #ux density waveforms; (b) dynamic loops. Fig. 1. Classical and hysteresis#excess energy losses per unit volume under sinusoidal supply.
the peak induction value and not on the rms value. The previous considerations hold for all the values of the fundamental amplitude. Increasing the harmonic order over n"15, angle a does not signi"cantly a!ect the peak #ux value and consequently its e!ect on hysteresis#excess losses becomes negligible. 3.2. Inyuence of harmonic order n
Fig. 2. Classical losses: behaviour of E /E versus phase )!3. 4*/ angle (k"0.1).
Fig. 3. Hysteresis#excess losses: behaviour of E /E ver)!3. 4*/ sus phase angle (k"0.1).
a equal to 0 and n, respectively. It is worth noting that, is some cases, the hysteresis#excess losses in the case of distorted waveforms can be lower than the sinusoidal ones, con"rming that these losses mainly depend on
The in#uence on the iron losses of the order n of the superposed harmonic has been investigated, varying the amplitude of the fundamental (from 0.5 up to 1.35 T). In Fig. 5, the ratio E /E for the classical losses is )!3. 4*/ plotted versus the parameter n; this quantity strongly increases at the increase of n, but it is practically independent of the amplitude of the magnetic #ux density. The e!ect on the hysteresis#excess losses is evidenced in Fig. 6. For n)9, taking into account the dependence on phase angle, the minimum and the maximum values of E /E are contemporary reported. These losses are )!3. 4*/ less in#uenced by the harmonic order in comparison with the classical ones, but they depend on the fundamental amplitude of the magnetic #ux density, as also shown in Fig. 7. The increase of total losses is put in evidence by the comparison of sinusoidal and distorted dynamic hysteresis loops, shown in Fig. 8 for two induction levels. As can be noted, an additional increment to the hysteresis losses is due to the presence of minor loops. The corresponding total losses, referred to the sinusoidal ones, are also plotted in Fig. 9 versus the harmonic order. All the considerations described for k"0.1 can be qualitatively repeated for other values of the ratio. As an example, in Fig. 10 the behaviour of hysteresis#excess losses versus the induction level are reported in the case of k"0.05.
4. Comparison with the statistical theory to iron losses It is interesting to compare the results computed by the FEM model with the ones provided by the statistical
194
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Fig. 8. Comparison between dynamic hysteresis loops under sinusoidal or distorted #ux waveforms (n"30): (a) B145"0.5 T, (b) B145"1.35 T Fig. 5. Classical losses: behaviour of E /E versus harmonic )!3. 4*/ order (k"0.1).
Fig. 6. Hysteresis#excess losses. Behaviour of E /E )!3. 4*/ versus harmonic order (k "0.1).
Fig. 9. Total losses. Behaviour of E /E versus harmonic )!3. 4*/ order (k"0.1).
the classical losses and the excess losses. Classical energy losses per unit volume in a lamination having thickness d and conductivity p are given by
P
pd2 T E " BQ 2(t) dt, (8) # 12 0 where ¹ is the period of the fundamental component of magnetic #ux density evolution B(t), which is supposed to be uniform in the cross section, neglecting skin e!ect. The excess energy losses per unit volume are deduced from
P
T DBQ (t)D3@2 dt, (9) 0 where G"0.1357 is a dimensionless coe$cient due to eddy current damping, S is the cross-sectional area of the lamination and < is a parameter with the dimension of 0 a magnetic "eld, characterizing the statistical distribution of the local coercive "eld. Static hysteresis energy losses per unit volume E are ) obtained by the area of the static hysteresis cycle; in absence of minor loops, the dependence of this component from the peak value of the magnetic #ux density E "JpG< S % 0
Fig. 7. Hysteresis#excess losses. Behaviour of E /E ver)!3. 4*/ sus B145 (k"0.1). For n"3 and 9, the minimum (dashed line) and the maximum (solid line) with respect to a are plotted.
approach of the electromagnetic phenomena inside ferromagnetic laminations [3]. The statistical theory provides an estimation of iron losses under non-sinusoidal #ux density waveform through the separation of losses into three fundamental components: the hysteresis losses,
O. Bottauscio et al. / Physica B 275 (2000) 191}196
Fig. 10. Hysteresis#excess losses. Behaviour of E /E ver)!3. 4*/ sus B145 (k"0.05). For n"3 and 9, the minimum (dashed line) and the maximum (solid line) with respect to a are plotted.
Fig. 11. Classical losses. Comparison between analytical and FEM model for B145"1.35 T.
waveform can be preliminary obtained from experiments. In presence of distorsions in B waveforms, which give rise to minor loops, empirical correction factors may be employed in simple situations [7]; anyway, since this correction cannot be straightforward generalized, in the following it will not be taken into account. The dependence of classical losses on the harmonic order is shown in Fig. 11 with B145"1.35 T, both for the analytical and the FEM model. For low-order harmonics, the prediction of the analytical method is good, while signi"cant discrepancies arise by increasing the harmonic order. Similar results have been found for other values of the fundamental amplitude. This behaviour can be explained by considering the spatial distribution of quantity J2 whose integral is proportional to the classical losses (see Fig. 12): the increase of the harmonic order produces a more signi"cant skin e!ect reducing the area interested by Joule losses. The behaviour of hysteresis#excess losses is more complex, because it is in#uenced by two e!ects con#icting each other. The skin e!ect tends to reduce both hysteresis and excess losses with respect to the analytical
195
Fig. 12. Spatial distribution of J2 along lamination thickness for the analytical and the FEM approach. The curves are normalised to the values on the boundary.
Fig. 13. Hysteresis#excess losses. Comparison between analytical and FEM model for two di!erent induction values: (a) B145"0.5 T, (b) B145"1.35 T.
model, while the presence of minor loops sensibly increases hysteresis losses. Besides, in a more accurate approximation, the parameter < which appears in rela0 tion (9) cannot be considered independent of the induction amplitude. The comparison between analytical and FEM model results shows that the analytical approach overestimates hysteresis#excess losses, for low induction levels (Fig. 13a); the increase of the induction level reverses the position of the two curves (Fig. 13b). This behaviour is put in evidence by Fig. 14, where the losses are plotted versus B145 for n"40.
5. Conclusions A numerical model based on "nite element method coupled with the dynamic Preisach model for the material modelling has been applied to the analysis of NO ferromagnetic sheets under distorted supply conditions, considering both low- and high-order harmonics superposed to a 50 Hz fundamental. The investigation has emphasized the in#uence of the shape of the #ux
196
O. Bottauscio et al. / Physica B 275 (2000) 191}196
esis#excess losses). The analysis has also provided evidence of the important role of skin e!ect and minor loops in the loss mechanism.
References
Fig. 14. Behaviour of hysteresis#excess losses versus B145 for n"40.
waveform on the magnetic losses, separately considering each fundamental contribution (classical and hyster-
[1] R. Kaczmarek, M. Amar, F. Protat, IEEE Trans. Magn. 32 (1996) 189. [2] A. Boglietti, M. Lazzari, M. Pastorelli, IEEE-IAS'97 Annual Meeting, New Orleans, 1997. [3] F. Fiorillo, A. Novikov, IEEE Trans. Magn. 26 (1990) 2904. [4] G. Bertotti, IEEE Trans. Magn. 28 (1992) 2599. [5] E. Della Torre, IEEE Trans. Audio 14 (1966) 86. [6] L. R. DupreH , O. Bottauscio, M. Chiampi, M. Repetto, J. Melkebeek, IEEE Trans. Magn. 35 (1999), in press. [7] J.D. Lavers, P.P. Biringer, H. Hollitscher, IEEE Trans. Magn. 14 (1978) 386.
Iron losses in soft magnetic materials under periodic non-sinusoidal supply conditions O. Bottauscio!,*, M. Chiampi", D. Chiarabaglio! !Istituto Elettrotecnico Nazionale Galileo Ferraris, c.so M. d 'Azeglio, 42, 10125 Torino, Italy "Dipartimento di Ingegneria Elettrica Industriale, Politecnico di Torino, Italy
Abstract The paper investigates the role of low- and high-order harmonics, superposed to a 50 Hz fundamental magnetic #ux density waveform, on iron losses in NO Fe-Si laminations. The e!ect on the fundamental contributions of the losses (classical and hysteresis#excess items) is evaluated by means of a combined "nite element - dynamic hysteresis model of the ferromagnetic sheet. A comparison with the prediction of the statistical theory of losses is "nally presented and discussed. ( 2000 Elsevier Science B.V. All rights reserved. Keywords: Magnetic losses; Hysteresis model; Finite element model
1. Introduction The magnetic losses in electrical machines are usually a!ected by signi"cant distortions of local #ux density waveforms. The non-sinusoidal evolutions are produced by di!erent causes: (i) local saturations giving rise to harmonics whose order is usually less than 10, (ii) rotor and stator teeth which determine #ux ripple with harmonics having order of some tens, (iii) static electronic supplies, which involve Fourier components up to some hundred order. The presence of non-sinusoidal waveforms makes the prediction of iron losses very di$cult, also taking into account the complex magnetization mechanism which involves the loss separation into three contributions: classical, static and excess losses. Starting from these considerations, it becomes essential to accurately predict the e!ects on the magnetic losses due to the presence of harmonics. Experimental investigations [1,2] have provided interesting results, but, of course, they cannot give a general solution. Some analytical ap-
* Corresponding author. Tel.: #39-011-3919776; fax: #39011-6509471. E-mail address: [email protected] (O. Bottauscio)
proaches, based on loss separation analysis, have been also proposed [3]; however, they neglect the e!ects of the incomplete #ux penetration inside lamination and the presence of minor loops in the hysteresis cycles. Thus, the use of numerical techniques becomes necessary. The aim of this paper is the application of a numerical model to emphasize the in#uence of harmonic components on the magnetic power losses separated into their fundamental contributions. The analysis is developed, for the sake of clarity, in simple geometrical structures, such as magnetic laminations, in order to avoid the additional phenomena (rotational #uxes, moving parts, etc.) occurring in the cores of electrical machines. The numerical approach is based on the "nite element solution of the electromagnetic "eld problem inside laminations under time periodic supply conditions, taking into account the magnetic properties through the dynamic Preisach model of hysteresis (DPM). The investigation is developed by "xing the fundamental component of the magnetic #ux density and by superposing a given harmonic, having a stated amplitude with respect to the fundamental one. Therefore, the comparison is performed keeping the rms value of the magnetic #ux density "xed. The in#uence of several parameters (harmonic order, peak value of fundamental, ratio between harmonic and fundamental amplitude,
0921-4526/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 0 7 6 9 - 3
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phase angle) on the iron losses and on their di!erent contributions is analysed. Finally, a comparison with the predictions of statistical theory is presented.
where k and k are suitable constants depending on the 1 3 material. The hysteresis model is detailed in Ref. [6], where the identi"cation procedure is also presented. Knowning the "eld distribution, classical losses E are # estimated by:
2. Numerical model
`d@2 TJ2 dt dz, (6) p ~d@2 0 where ¹ is the period of the fundamental and J is the induced current density. Static E and excess E losses ) % are given from the area of local loops.
The electromagnetic "eld problem in an in"nite sheet having thickness d along the z direction, where a known magnetic #ux U(t) is #owing (x direction), is governed by the equation
P
p d@2 curlf(curlA)"!pAQ # AQ dz, (1) d ~d@2 where f(B)"H represents the relation between magnetic #ux density and "eld, p is the conductivity and A"(0, A(z), 0) is the vector potential with nonhomogeneous Dirichlet boundary conditions U U A(!d/2)"! , A(d/2)" . 2 2
(2)
In Eq. (1), due to symmetry considerations, the integral term of the right hand-side is zero. Problem (1), linearized using the H-version of "xed point (FP) technique, is developed in the harmonic domain to impose the steady-state periodic behaviour. Thus, for the mth harmonic at the ith iteration, the "eld equation becomes RR(m) R2A(m) ! M i~1 , l M i "jmupA(.) FP Rz2 i Rz M
(3)
where l is the FP coe$cient, u is the fundamental FP angular frequency and phasors A(m) and R(m) represent M i the residual M i the magnetic vector potential and term, respectively. After the "eld solution, the magnetic quantities are reported in the time domain through an inverse fast Fourier transform (FFT) and then the update of FP residual is computed, by the following relations: H (t)"R (t)#l B (t), i i~1 FP i R (t)"H (t)!l f~1(H (t)). (4) i i FP i The new FP residuals are "nally reported in the harmonic domain and "eld equations are updated for the next iteration. In Eq. (4), hysteretic relation f~1, which gives the B waveform from the time evolution of H, is handled employing a Preisach-type model. This model is based on the dynamic relation proposed by Bertotti [4] which introduces a di!erential law governing the material magnetization. The moving e!ect [5] is also included by correcting the applied magnetic "eld through the term: f (M)"k M#k M3, 1 3
(5)
P P
E " #
3. Discussion of the results The investigation has been performed on a NO Fe}Si 2.5 wt% lamination of thickness 0.65 mm, having an electrical conductivity of 2.37 MS/m, usually employed in electrical motor cores. The distorted induction waveforms consist of a fundamental 50 Hz sinusoid with a given harmonic component B(t)"B145[sin ut#k sin(nut#a)].
(7)
The ratio k between the amplitude Bn of the n-th harmonic component and the fundamental one B145 is kept "xed (k"0.1), while the harmonic order n and the phase angle a are varied (3)n)40 and 0)a)n). In such a way, the comparison is performed keeping the rms value of the magnetic #ux density "xed. For each considered B waveform, the classical and hysteresis#excess losses have been computed. In order to emphasize the harmonic e!ect, in the presentation of the results the values of each loss contribution have been divided by the corresponding sinusoidal ones (E /E ). The dia)!3. 4*/ gram of sinusoidal energy losses versus the induction level is shown in Fig. 1. In the following the results are discussed considering separately the in#uence of each parameter. 3.1. Inyuence of phase angle a The in#uence of phase angle on classical and hysteresis#excess losses has been analysed, varying the harmonic order and the amplitude of the fundamental induction value. The e!ect of phase angle is weak in the case of classical losses, as can be noted by Fig. 2: an appreciable dependence is found only in presence of the third harmonic. On the contrary, the in#uence of a on hysteresis#excess losses is signi"cant for harmonic orders lower than nine, as shown in Fig. 3. This behaviour is well explained considering that, mainly for low harmonic orders, the phase angle determines the peak value of B, as put in evidence by Fig. 4 which presents the waveforms of the mean magnetic #ux density and the corresponding dynamic loops in the case of n"3 and
O. Bottauscio et al. / Physica B 275 (2000) 191}196
193
Fig. 4. E!ect of phase angle a for n"3 and k"0.1: (a) magnetic #ux density waveforms; (b) dynamic loops. Fig. 1. Classical and hysteresis#excess energy losses per unit volume under sinusoidal supply.
the peak induction value and not on the rms value. The previous considerations hold for all the values of the fundamental amplitude. Increasing the harmonic order over n"15, angle a does not signi"cantly a!ect the peak #ux value and consequently its e!ect on hysteresis#excess losses becomes negligible. 3.2. Inyuence of harmonic order n
Fig. 2. Classical losses: behaviour of E /E versus phase )!3. 4*/ angle (k"0.1).
Fig. 3. Hysteresis#excess losses: behaviour of E /E ver)!3. 4*/ sus phase angle (k"0.1).
a equal to 0 and n, respectively. It is worth noting that, is some cases, the hysteresis#excess losses in the case of distorted waveforms can be lower than the sinusoidal ones, con"rming that these losses mainly depend on
The in#uence on the iron losses of the order n of the superposed harmonic has been investigated, varying the amplitude of the fundamental (from 0.5 up to 1.35 T). In Fig. 5, the ratio E /E for the classical losses is )!3. 4*/ plotted versus the parameter n; this quantity strongly increases at the increase of n, but it is practically independent of the amplitude of the magnetic #ux density. The e!ect on the hysteresis#excess losses is evidenced in Fig. 6. For n)9, taking into account the dependence on phase angle, the minimum and the maximum values of E /E are contemporary reported. These losses are )!3. 4*/ less in#uenced by the harmonic order in comparison with the classical ones, but they depend on the fundamental amplitude of the magnetic #ux density, as also shown in Fig. 7. The increase of total losses is put in evidence by the comparison of sinusoidal and distorted dynamic hysteresis loops, shown in Fig. 8 for two induction levels. As can be noted, an additional increment to the hysteresis losses is due to the presence of minor loops. The corresponding total losses, referred to the sinusoidal ones, are also plotted in Fig. 9 versus the harmonic order. All the considerations described for k"0.1 can be qualitatively repeated for other values of the ratio. As an example, in Fig. 10 the behaviour of hysteresis#excess losses versus the induction level are reported in the case of k"0.05.
4. Comparison with the statistical theory to iron losses It is interesting to compare the results computed by the FEM model with the ones provided by the statistical
194
O. Bottauscio et al. / Physica B 275 (2000) 191}196
Fig. 8. Comparison between dynamic hysteresis loops under sinusoidal or distorted #ux waveforms (n"30): (a) B145"0.5 T, (b) B145"1.35 T Fig. 5. Classical losses: behaviour of E /E versus harmonic )!3. 4*/ order (k"0.1).
Fig. 6. Hysteresis#excess losses. Behaviour of E /E )!3. 4*/ versus harmonic order (k "0.1).
Fig. 9. Total losses. Behaviour of E /E versus harmonic )!3. 4*/ order (k"0.1).
the classical losses and the excess losses. Classical energy losses per unit volume in a lamination having thickness d and conductivity p are given by
P
pd2 T E " BQ 2(t) dt, (8) # 12 0 where ¹ is the period of the fundamental component of magnetic #ux density evolution B(t), which is supposed to be uniform in the cross section, neglecting skin e!ect. The excess energy losses per unit volume are deduced from
P
T DBQ (t)D3@2 dt, (9) 0 where G"0.1357 is a dimensionless coe$cient due to eddy current damping, S is the cross-sectional area of the lamination and < is a parameter with the dimension of 0 a magnetic "eld, characterizing the statistical distribution of the local coercive "eld. Static hysteresis energy losses per unit volume E are ) obtained by the area of the static hysteresis cycle; in absence of minor loops, the dependence of this component from the peak value of the magnetic #ux density E "JpG< S % 0
Fig. 7. Hysteresis#excess losses. Behaviour of E /E ver)!3. 4*/ sus B145 (k"0.1). For n"3 and 9, the minimum (dashed line) and the maximum (solid line) with respect to a are plotted.
approach of the electromagnetic phenomena inside ferromagnetic laminations [3]. The statistical theory provides an estimation of iron losses under non-sinusoidal #ux density waveform through the separation of losses into three fundamental components: the hysteresis losses,
O. Bottauscio et al. / Physica B 275 (2000) 191}196
Fig. 10. Hysteresis#excess losses. Behaviour of E /E ver)!3. 4*/ sus B145 (k"0.05). For n"3 and 9, the minimum (dashed line) and the maximum (solid line) with respect to a are plotted.
Fig. 11. Classical losses. Comparison between analytical and FEM model for B145"1.35 T.
waveform can be preliminary obtained from experiments. In presence of distorsions in B waveforms, which give rise to minor loops, empirical correction factors may be employed in simple situations [7]; anyway, since this correction cannot be straightforward generalized, in the following it will not be taken into account. The dependence of classical losses on the harmonic order is shown in Fig. 11 with B145"1.35 T, both for the analytical and the FEM model. For low-order harmonics, the prediction of the analytical method is good, while signi"cant discrepancies arise by increasing the harmonic order. Similar results have been found for other values of the fundamental amplitude. This behaviour can be explained by considering the spatial distribution of quantity J2 whose integral is proportional to the classical losses (see Fig. 12): the increase of the harmonic order produces a more signi"cant skin e!ect reducing the area interested by Joule losses. The behaviour of hysteresis#excess losses is more complex, because it is in#uenced by two e!ects con#icting each other. The skin e!ect tends to reduce both hysteresis and excess losses with respect to the analytical
195
Fig. 12. Spatial distribution of J2 along lamination thickness for the analytical and the FEM approach. The curves are normalised to the values on the boundary.
Fig. 13. Hysteresis#excess losses. Comparison between analytical and FEM model for two di!erent induction values: (a) B145"0.5 T, (b) B145"1.35 T.
model, while the presence of minor loops sensibly increases hysteresis losses. Besides, in a more accurate approximation, the parameter < which appears in rela0 tion (9) cannot be considered independent of the induction amplitude. The comparison between analytical and FEM model results shows that the analytical approach overestimates hysteresis#excess losses, for low induction levels (Fig. 13a); the increase of the induction level reverses the position of the two curves (Fig. 13b). This behaviour is put in evidence by Fig. 14, where the losses are plotted versus B145 for n"40.
5. Conclusions A numerical model based on "nite element method coupled with the dynamic Preisach model for the material modelling has been applied to the analysis of NO ferromagnetic sheets under distorted supply conditions, considering both low- and high-order harmonics superposed to a 50 Hz fundamental. The investigation has emphasized the in#uence of the shape of the #ux
196
O. Bottauscio et al. / Physica B 275 (2000) 191}196
esis#excess losses). The analysis has also provided evidence of the important role of skin e!ect and minor loops in the loss mechanism.
References
Fig. 14. Behaviour of hysteresis#excess losses versus B145 for n"40.
waveform on the magnetic losses, separately considering each fundamental contribution (classical and hyster-
[1] R. Kaczmarek, M. Amar, F. Protat, IEEE Trans. Magn. 32 (1996) 189. [2] A. Boglietti, M. Lazzari, M. Pastorelli, IEEE-IAS'97 Annual Meeting, New Orleans, 1997. [3] F. Fiorillo, A. Novikov, IEEE Trans. Magn. 26 (1990) 2904. [4] G. Bertotti, IEEE Trans. Magn. 28 (1992) 2599. [5] E. Della Torre, IEEE Trans. Audio 14 (1966) 86. [6] L. R. DupreH , O. Bottauscio, M. Chiampi, M. Repetto, J. Melkebeek, IEEE Trans. Magn. 35 (1999), in press. [7] J.D. Lavers, P.P. Biringer, H. Hollitscher, IEEE Trans. Magn. 14 (1978) 386.