Core loss prediction combining physical models with numerical field ...

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Journal of Magnetism and Magnetic Materials 133 (1994) 647-650

journal of magnetism and magnetic materials

Core loss prediction combining physical models with numerical field analysis G. Bertotti a A. Canova b, M. Chiampi b,., D. Chiarabaglio a, F. Fiorillo a, A.M. Rietto

a

a Istituto Elettrotecnico Nazionale Galileo Ferraris and GNSM-INFM, Torino, Italy, b Dipartimento di Ingegneria Elettrica Industriale del Politecnico, Torino, Italy

Abstract

An improved procedure for calculating iron losses in electrical machine cores is presented. It is based on physical models and experiments on losses in magnetic laminations, under one- and two-dimensional fields, and exploits a finite element computation of the flux distribution in the core. Physical modelling relies on the basic concept of loss separation, extended to the case of vectorial magnetic flux with generic elliptical loci. Starting from a theoretical formulation of power losses under unidirectional fields and generic induction waveform and its extension to the case of elliptical flux, general expressions are derived for the hysteresis, excess and classical loss components in two dimensions. Quasi-static and 50 Hz total losses under alternating sinusoidal flux and pure rotational flux are the sole experimental data needed for a complete loss prediction. In the present work, two different types of nonoriented FeSi 3.2% laminations are considered, which are assumed to be assembled into a model three-phase motor core. By means of a 2D finite element analysis, the distribution of magnetic field and induction in the core is obtained for different values of the supply current and the loss calculation is carried out. A comparison with standard loss calculation methods points to the detrimental role of two-dimensional fluxes, although this may not be fully appreciated in conventional 50 Hz induction motors.

I. Introduction

Increasingly powerful and precise numerical methods for the calculation of field and flux distribution in rotating electrical machines have been developed in recent times, but their use in loss problems, hindered by lack of physical insight in the magnetisation process, relies in general on a number of empirical assumptions and simplifications. For instance, magnetic flux is very often considered to be unidirectional and sinusoidal everywhere in the core, in contrast with the experiments [1] and the very same results of the field calculations. But machine design accuracy is increasingly re-

* Corresponding author. Tel: +39-11-5647142; Fax: +39-115647199.

quired, as motor driving systems evolve, higher working frequencies are attained and the general need for efficient use of energy is emphasised. Efforts have therefore been made in the recent literature to refine the loss calculation in the cores, by taking into account either the contribution from harmonics [2-4] or the role of two-dimensional fluxes [5]. Failure to recognise the different effects of flux distortion on the classical and excess loss components, theoretically predicted and experimentally verified in magnetic laminations [6-7], and the need for many specific data under two dimensional fields are among the major drawbacks of some recent calculation methods. In the present paper a general predictive approach to power losses in rotating machine cores is discussed. It is based on independent formulations for the hysteresis, excess, and classical loss components under generalised elliptical flux, with non-sinusoidal compo-

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G. Bertotti et al. /Journal of Magnetism and Magnetic Materials 133 (1994) 647-650

648

nents. In the present form the theory is specifically suited to the practical important case of nonoriented lamination cores. It requires prior knowledge of a reduced set of experimental loss data on the material, namely the static and 50 Hz energy losses under alternating sinusoidal flux and pure rotational flux. An application is described in the following, concerning a model stator core of a three-phase four-pole motor, whose magnetic flux distribution is calculated by means of a finite element method. For each element of the mesh, the flux locus, of generic elliptical shape, and the time dependence of its components along major and minor axes of the ellipse are determined. The associated loss is then computed, for different values of the supply current, and integrated over the whole stator core. The so obtained results, compared with those predicted on the basis of the conventional approach, point to enhanced loss contribution from the stator regions characterised by the presence of two-dimensional fluxes. Although the loss estimated by the refined approach on the 50 Hz supplied model core is less than 10% higher than the value predicted by the standard calculation, an important difference can be anticipated to occur at higher frequencies.

2. The loss model

P~I(IM) = P ~ ( I M ) X [1 +C(rh(IM) -- 1)].

In this section we derive an expression for the power loss in non-oriented laminations when the mag-

2.o

i

netisation vector I describes, over a complete period, an elliptical locus. The amplitudes of major and minor semi-axes are defined as I M and 1m, respectively. As a starting point, the concept of loss separation, where the total power loss P is defined as the sum of the hysteresis (Ph), excess (Pc) and classical (Pc) loss components is considered. These three components are independently calculated, for the case of elliptical flux, in terms of loss data determined by experiments under 50 Hz sinusoidal alternating flux and pure rotational flux. Let us therefore introduce P~(IM) and P~(IM), the measured alternating and rotational hysteresis losses at peak magnetisation I M, and their ratio rh(I M) -r a --Ph(IM)/Ph(IM). Fig. 1 shows an example of the experimental behavior of r h vs. I M, as determined on a typical nonoriented FeSi lamination, r h is a monotonically decreasing function of I M and attains a value 1 at I M = 1.5 T, a standard result in nonoriented alloys [8]. Analysis of literature experiments [9] permits one to assume that in the elliptical case the hysteresis loss p~l will change in a linear fashion with c = Im/l M, reaching the extreme values P~ and P~ for c = 0 (Phe l /Pha = 1) and c = 1 (Phe l /Pha = P~/P~ = rh) , respectively. Recalling that flux waveform distortion does not affect the hysteresis loss (provided there are no minor loops) [6], we can write in a very general way

~,,

1

re

1.5

rhea,

~

The calculation of the excess loss component under elliptical flux Peel(IM) can be carried out along similar lines, duly taking into account the role of harmonics. To this end, we generalise the case where the two orthogonal flux components are sinusoidal (perfect elliptical locus). Assuming, on the same experimental grounds, the previous linear dependence of loss on c, we write the excess elliptical loss at the exciting frequency f as

pel(IM,f) = p a ( I M , f ) X [1 +c(re(IM) -- 1)] 1.0

.\

0.5

0.o -~

0,0

0.'5

~.'0

IM(T)

~)5

2.0

Fig. 1. Experimental ratio between rotational and alternating loss components vs. peak magnetization I M in a 0.33 mm thick nonoriented FeSi 3.2% lamination. The hysteresis loss ratio rh(IM )= P(a(IM)/P~(IM) (full points) and the excess loss one re(/M) = P~(IM)/P~(IM) (open points) follow a very similar decreasing trend with I M. Above 1M = 1.65 T the curves are induced by extrapolation of the experimental data. The ratio between rotational and alternating classical losses r e is assumed to be 2 everywhere.

(1)

(2)

where P~(I M, f ) is the excess alternating loss under sinusoidal flux at peak magnetisation I M and re(/M) = P~(IM)/P~(IM) is the ratio between the rotational and alternating sinusoidal excess losses, re(/M) is independent of f and can be determined by conventional alternating and rotational loss separation experiments [10]. The measurement of static and 50 Hz total losses will suffice to achieve such separation. Fig. 1 provides an example of the dependence of r e on IM, as obtained in one of the investigated nonoriented FeSi laminations. Eq. (2) can be extended to the case where the two flux components are distorted. Such distortion will reflect in general into an irregular elliptical locus (see Fig. 2) and the ratio c will be taken here as the one pertaining to the axes of the best fitting ellipse. In order to generalise Eq. (2) we need to express pa(l M, f ) under

G. Bertotti et al. /Journal of Magnetismand MagneticMaterials 133 (1994) 647-650

649

We are thus left with the problem of determining the two-dimensional classical loss component pcel(IM, f). To this end, the standard one-dimensional formulation (see for instance Ref. [6]) is extended to the two-dimensional case, according to the equation

P:'(IM,f)=(trd2/12) ×ffo'/f[IZx(t)+[y~(t)] at,

(5) with ix(t) and [y(t) the components of the instantaneous magnetisation derivative along two orthogonal axes. p~l can alternatively be expressed in terms of the harmonic content of L ( t ) and Iy(t), as

1.1

Pel¢I~,

M,J! ~'' =

(trTr2f2d2/6) × En2(12n + I#,) n

( n = odd),

with Ixn (Iyn) the peak amplitude of the nth order x (y) component. We are now in a position to calculate, by means of the above equations, the total power loss under generalised elliptical flux locus and arbitrary frequency P¢J(IM, f ) = P~,I(IM, f) + P21(IM, f) + P~I(IM, f) in each of the elementary regions of the motor core defined by the finite element procedure.

-1 .I 1.5

•

1.5

S ........

:

" ! Ti

)'~ ~i' ~

;

~

-.g--'

:'

0

Ix~T}

1.5

Fig. 2. (a) Instantaneous flux distribution in the core of the model three-phase four-pole motor. The computation domain is covered by a mesh of 4500 nodes and 6500 elements. (b) Flux loci computed at different points of the stator core, for a 5 A supply current and a 0.49 mm thick FeSi 3.2% nonoriented lamination. The tooth axis is at 45° to the coordinate axes.

distorted flux in terms of the corresponding quantity under sinusoidal flux. We rely for this on the theoretical approach developed in Ref. [6], where the following general equation has been derived:

pa(IM,f )

= ke(IM) X f£1/f

I [(t)l 3/2 dt,

(3)

with k¢(I M) a function depending on intrinsic and structural material properties. Eq. (3), applied to the particular case of sinusoidal induction, reduces to

pea(IM,f)

= 8.67 × ke(IM) ×

(IMf) 3/2.

(6)

(4)

As previously stressed, a conventional loss separation under sinusoidal induction, providing through Eq. (4) the quantity k~(IM), permits one to calculate P:(IM, f ) in Eq. (3), whatever the I(t) waveshape, and consequently the elliptical excess loss under distorted flux P~(I M, f) by means of Eq. (2).

3. Core loss computation A model induction motor core has been considered, made of a four-pole stator and a laminated slotless rotor, assumed to rotate at synchronous speed. With this simplified configuration the high frequency flux pulsations localised at stator teeth [11] are avoided, while keeping the usual field and flux distribution in the core. The field problem is described in terms of the vector potential A, by means of the equations curl n = J ,

H = ((curl A),

(7)

together with homogeneous Dirichlet boundary conditions. In Eq. (7) J is the source current density and the inverse relation H = ((B) is derived from the experimental magnetisation curve, neglecting hysteresis. The study is made for a 50 Hz supplied system, with sinusoidal three-phase balanced exciting currents. Eq. (7) is solved under 2D approximation by means of a finite element technique, employing truncated Fourier series and solving the field problem for each harmonic component. The study is carried out over a half portion of the motor, exploiting the symmetry of the system. A fine mesh of about 4000 nodes and 6500 elements is used in this domain. An example of calculated instantaneous flux distribution for the present model core, made of nonoriented laminations, is shown in Fig. 2a. Two different types of FeSi 3.2% laminations, 0.33 and

G. Bertotti et al. /Journal of Magnetism and Magnetic Materials 133 (1994) 647-650

650

Table 1 50 Hz total stator losses computed from the conventional model and the present model. The calculation is made for three different peak values of the supply current ip and two different types of FeSi 3.2% laminations, 0.33 mm and 0.49 mm thick. Pas and Pel are predicted by the conventional and the refined model, respectively ip(A) d = 0.49 mm 10 5 3 d = 0.33 mm 10 5 3

~s(W)

~l(W)

68 48.9 21.3

70.7 52.1 22.9

58.5 42.2 18.2

60.6 43.1 19.1

0.49 mm thick, have been considered. Alternating power losses have been measured vs. I M (0.25 T-1.65 T) on toroidal samples (90 mm o.d., 75 mm i.d.). An electronic wattmeter, providing tightly controlled sinusoidal induction, has been employed to achieve loss separation in the frequency range 1-50 Hz. The hysteresis loss component has been obtained either by extrapolating the P / f curve to f = 0 or by means of a calibrated hysteresigraph (LDJ 5500H). The measurement of rotational losses has been carried out, over the same (IM, f ) domain, on suitably prepared disks (maximum diameter 90 ram) using a built-on-purpose wattmeter [12]. In this way the basic quantities employed in the calculation of losses under generalised elliptical flux (P~(IM), rh(IM), P~(IM,f), re(IM)) are obtained. It should be stressed that the excess alternating and rotational power loss components (P~,P~) exhibit the same f3/2 behavior, which implies that re(/M) is independent of f. The experimental rc(l M) and rh(I M) vs. I M curves for the 0.33 mm thick lamination are shown in Fig. 1. The loss computation has been carried out for each iron element of the mesh applying Eqs. (1), (2), and (5), having calculated the time dep e n d e n c e of the components of [(t) along the axes of the related elliptical locus. Fig. 2b provides an example of calculated loci at three significant points of the stator core (0.49 mm thick laminations, peak amplitude of the supply current 5 A). It can be noticed that the

region behind the tooth is the seat of a conspicuous bidimensional flux. The 50 Hz loss figure computed with the present two-dimensional model has been compared with the one provided by the conventional model, where I(t) is assumed in each element to be sinusoidal and unidirectional, with peak amplitude I M. A substantial difference between the loss values predicted by the two models can be verified in the regions where the flux has an important bidimensional character. We obtain, for example, P = 2.51 W / k g and P = 3.04 W / kg around point R in Fig. 2b (IM = 1.38 T, I m = 0.82 T), using the conventional approach and the present theory, respectively. The loss figures computed over the whole stator core show, however, less remarkable differences, as put in evidence by the data presented in Table 1. The present model predicts in fact substantially higher losses than the conventional model at working frequencies higher than 50 Hz, due to increasingly important contributions from one- and two-dimensional classical losses.

References [1] A.J. Moses and G.S. Radley, J. Magn. Magn. Mater. 19 (1980) 60. [2] G.R. Slemon and X. Liy, IEEE Trans. Magn. 26 (1990) 1653. [3] M.K. Jamil, P. Baldassarri and N.A. Demerdash, IEEE Trans. Magn. 28 (1992) 2820. [4] K. Atallah, Z.Q. Zhu and D. Howe, IEEE Trans. Magn. 28 (1992) 2997. [5] H.W. Lorenzen and R. Nuscheler, IEEE Trans. Magn. 24 (1988) 1972. [6] F. Fiorillo and A. Novikov, IEEE Trans. Magn. 26 (1990) 2904. [7] F. Fiorillo and A. Novikov, IEEE Trans. Magn. 26 (1990) 2559. [8] F. Fiorillo and A.M. Rietto, J. Appl. Phys. 73 (1993) 6615. [9] J. Sievert, J. Magn. Magn. Mater. 112 (1992) 50. [10] F. Fiorillo, PTB Bericht E-43 (1992) 11. [11] G. Bertotti, A. Boglietti, M. Chiampi, D. Chiarabaglio, F. Fiorillo and M. Lazzari, IEEE Trans. Magn. 27 (1991) 5007. [12] F. Fiorillo and A.M. Rietto, IEEE Trans. Magn. 24 (1988) 1960.