JOURNAL OF APPLIED PHYSICS

VOLUME 87, NUMBER 9

1 MAY 2000

Quasistatic measurements for hysteresis modeling Marc De Wulf,a) Luc Dupre´, and Jan Melkebeek Department of Electrical Power Engineering, University of Gent, Belgium

In order to validate and test the correctness of hysteresis models used in the simulation of soft magnetic materials 共electrical steel兲, one has to dispose of experimental data describing the hysteresis behavior of the material e.g., measurements under slow time varying excitation conditions in order to reduce the dynamic effects to a strict minimum. Based on measured loops 共1–5 Hz兲 on nonoriented 3% SiFe alloys, and starting from the statistical loss theory of Bertotti, where the total applied field is subdivided into a hysteresis, a classical and an excess field, one can obtain the dc loop of the material by eliminating the remaining dynamic effects in the lamination. A comparison between three types of measurements is made, namely, hysteresis loops obtained by extrapolating from dynamic measurements using the statistical loss theory, second loops measured under slow time varying induction 共0,1 Hz兲—the influence of the used frequency 共0,1–5 Hz兲 and the used excitation wave form 共constant dH/dt...constant dH/dt兲 is examined—and third measured loops obtained from a static 共ballistic-like兲 method. An improvement of the quasistatic measurement procedure is proposed, and implemented in the data-acquisition software for the determination of the material Everett function. © 2000 American Institute of Physics. 关S0021-8979共00兲20608-7兴

I. INTRODUCTION

ings. A Kepco power amplifier and a PC with an acquisition card complete the measurement apparatus. The power amplifier can be used in voltage mode or in current mode. The current mode enables a feed forward control of the magnetic field, whereas in voltage mode an induction control can be achieved by considering the equivalent circuit of the primary side of the Epstein frame and by estimating the output voltage of the power amplifier using the time derivative of the desired induction wave form, the cross-section S and the number of windings n

One of the key points in using hysteresis models within finite element software for the design and optimization of electric machinery is the identification of the model parameters of the used material in the application. When using the Preisach model, the material is classically described by its Preisach distribution function,1 which on its turn can be deduced from the Everett function of the material. The determination of this Everett function is based on a series of measured quasistatic loops.2 In this work, different aspects on measuring these ‘‘nearly’’ static loops are investigated. First, the influence of the used frequency 共0,1–5 Hz兲 and the influence of the used excitation wave form 共constant dH/dt, and constant dB/dt兲 is explained. Also, the loops obtained with a slow time varying excitation are compared to loops measured with a static, ballistic-like method. As can be expected, the latter method results in a lower loss value. Second, it is shown that based on the statistical loss theory of Bertotti,3 where the total applied field is subdivided into a hysteresis field H hyst a classical field H cl and an excess field H exc , one can obtain a dc hysteresis loop starting from a measurement at ‘‘higher’’ frequency 共e.g., 5 Hz兲 and by extracting a dynamic correction field from the applied magnetic field. Indeed, at low magnetization rates, both the classical and the excess field may be written proportionally to the time derivative of the induction. In other words, the correction field H corr(t) is proportional to the secondary voltage, which gives the rate of induction change in the material.

␯ out共 t 兲 ⫽nS

dB 共 t 兲. dt

共1兲

However, for quasistatic measurements, using the amplifier in voltage mode results in negligible output voltage 共because of the time derivation兲 and thus in noisy and inaccurate measurements. A way to overcome this problem and to obtain workable voltages is to increase nS, in particular by increasing the number of windings on the primary coil of the Epstein frame 共a few thousand or more兲. The more obvious method to perform very low frequency measurements is to use the power amplifier in current mode. As the obtained induction in this case depends on the material under test, a feedback control algorithm is necessary. This periodic wave form feedback algorithm consists in a combination of two consequent wave form updates. The first action is a PI control feedback 共in fact a linear rescale of the output current wave form兲 in order to acquire the desired induction peak value. Second, a transformation according to Eq. 共2兲 controls the induction form.4

II. EXPERIMENTAL PROCEDURE

H u 关 i 兴 ⫽H p 关 j 兴 with ᭙i 共 i⫽0..n 兲 ,᭚ j:B m 关 j 兴 ⫽B d 关 i 兴 .

The experimental setup is based on an traditional Epstein frame with both on primary and on secondary side 700 wind-

The indices u, p, m, and d, respectively, stand for the updated output wave form, the previous output wave form, the measured induction, and the desired induction. One or two iterations suffice for the induction control.

a兲

Electronic mail: [email protected]

0021-8979/2000/87(9)/5239/3/$17.00

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共2兲

© 2000 American Institute of Physics

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J. Appl. Phys., Vol. 87, No. 9, 1 May 2000

after the transient effects have disappeared. We obtain in this way by stages the static hysteresis curve. The corresponding calculated losses are about 5% lower than the losses that result from the slow time varying excitation 共0,1 Hz under constant dB/dt兲 with the same induction peak level.

III. LOSS SEPARATION AT LOW MAGNETIZATION RATES

Following the statistical loss theory of Bertotti, the total applied field can be written as H tot共 t 兲 ⫽H hyst共 t 兲 ⫹H cl共 t 兲 ⫹H ext共 t 兲 .

FIG. 1. Importance of measurements under constant dB/dt.

The influence of the used frequency and the used excitation control is shown in the following graphs. Figure 1 displays 2 Hz measurements, one with controlled constant variation of the induction 共constant dB/dt兲 and the other under constant field variation 共constant dH/dt兲. The measurement under constant dH/dt shows a more or less sharply peaked secondary voltage—depending on the material quality—which results in an increase of the area enclosed by the magnetization loop. The secondary voltage of the measurement under constant dB/dt is a rectangular signal, distorted with noise mainly caused by the Barkhausen jumps during the magnetization process. A detailed investigation of Barkhausen noise experiments, in order to clarify the relationship between the magnetization process and the macroscopic magnetic properties can be found in Ref. 5. In Fig. 2, the measured losses are shown for a frequency range of 0,1–5 Hz. Again, the measurements under constant dH/dt reveal to lead to an increase of the measured loss. A comparison is made between measured losses obtained from a quasistatic measurement 共0,1 Hz under constant dB/dt兲 and measurements with static or dc magnetization 共ballistic-like method兲. In practice, with the latter type of measurement, the magnetic field is varied stepwise or its polarity reversed and the change in flux density is measured

共3兲

For low magnetization rates 共no skin effect in the lamination兲, the description of the classical Foucault field H cl(t), representing a background field acting uniformly in the slab cross-section, is independent of the material magnetization law B(H), ␴ is the electrical conductivity, and d is the thickness of the lamination: H cl共 t 兲 ⫽

␴ d 2 dB 共 t 兲. 12 dt

共4兲

According to Ref. 6 the excess field is given by H exc⫽

n 0V 0 2

冉冑

1⫹

冊

4 ␴ GS dB ⫺1 . n 20 V 0 dt

共5兲

G is a geometrical constant, S is the cross-section, n 0 and V 0 are phenomenological parameters characterizing a given material. Indeed, from low to high magnetizing frequencies, the excess field passes from a linear to a square-root dependence on dB/dt. For low magnetization rates, the excess field can be written as H exc共 t 兲 ⫽

␴ GS dB 共 t 兲. n 0 dt

共6兲

Considering Eqs. 共3兲, 共4兲 and 共6兲 one can propose a correction field to obtain the hysteresis field H hyst , which is frequency independent by definition, from the total applied field Eq. 共7兲. H corr共 t 兲 ⫽k

␴ d 2 ␴ GS dB ⫹ . 共 t 兲 with k⫽k cl⫹k exc⫽ dt 12 n0

共7兲

The parameter k can be identified using two measured loops at different frequencies, e.g., a 0,1 Hz loop under constant dB/dt and a 1 Hz loop under constant dH/dt. In order to facilitate notation, quantities are written from now on as function of the instantaneous induction b. Then, Eq. 共7兲 applied to the two measured loops becomes H 1 Hz共 b 兲 ⫺H 0,1 Hz共 b 兲 ⫽H corr共 b 兲 ⫽k

FIG. 2. Measured losses with variable frequency and excitation control.

dB 共 b 兲. dt 1 Hz

共8兲

The upper part of Fig. 3 shows the left and right side of Eq. 共8兲 from which k⫽0,074 is obtained. Figure 3 also shows the 0,1 and 1 Hz measured loops and the 1 Hz corrected one, i.e., the loop calculated from the 1 Hz measurement and from which the correction field according to Eq. 共8兲 is subtracted.

De Wulf, Dupre´, and Melkebeek

J. Appl. Phys., Vol. 87, No. 9, 1 May 2000

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FIG. 4. Identification of n 0 and V 0 .

Once the parameter k is identified, quasistatic BH loops can be calculated according to Eq. 共8兲 from measured loops at elevated frequencies 共e.g., 1 or 5 Hz, without any form of induction feedback兲 using k and the measured secondary voltage. IV. RESULTS AND CONCLUSIONS FIG. 3. Identification of k using a QS loop and a 1 Hz loop.

In order to determine the relative influence of the classical field H cl(b) or the excess field H exc(b) on the correction field H corr(b), the parameter k is also calculated directly from the right side of Eq. 共7兲. Given the numeric data of the sample: ␴ ⫽2,03e6 S/m; d⫽0,5e-3 m; G⫽0,1357; S ⫽4,43e-5 m2; n 0 ⫽381,6, and V 0 ⫽0,046 A/m, we obtain that k cl⫽0,042 and k exc⫽0,032. This not only validates the proposed correction method, but also shows the relative importance of both the classical field and the excess field. The values of the material parameters n 0 and V 0 are identified using measured losses with 共1T peak兲 sinusoidal induction and with frequency ranging from 0, 1 to 100 Hz. Equation 共5兲 is rewritten for sinusoidal induction P exc ⫽2B 共 冑n 20 V 20 ⫹16␴ GSV 0 B f ⫺n 0 V 0 兲 . f

共9兲

The low and high frequency limits of this expression, namely P exc 16␴ GS 2 B f ; LF limit ⫽ f n0

Different possibilities for the measurement of static properties of a given material, free from all dynamic influences such as eddy currents, relaxation and so on, have been compared. The use of 共e.g., 1 Hz兲 straight forward triangular current feed in order to obtain ‘‘hysteresis’’ data of a given material should be avoided. However, if this raw data is corrected using the scheme presented above, one obtains BH loops as is they were measured at 0, 1 Hz under constant dB/dt. The computation of the correction field requires one parameter k which can be deduced from two measurements. A true static method yields a slight lower measured loss than a measurement with slow time varying excitation, however, this static method is not suitable to be implemented in an automated measurement apparatus. The correction method, combined with an increase of the used frequency has been programmed in the data-acquisition software for the automated measurement of quasistatic loops of the material. The increase of the used frequency 共2 Hz兲 eliminates drift problems caused by integration, also, the determination of the Everett function of the material 共based on a series of consequent quasistatic loops with increasing amplitude兲 is less time consuming. F. Preisach, Z. Phys. 94, 277 共1935兲. D. Philips, L. Dupre´, J. Cnops, and J. Melkebeek, J. Magn. Magn. Mater. 133, 540 共1994兲. 3 G. Bertotti, IEEE Trans. Magn. 28, 621 共1988兲. 4 G. Bertotti, F. Fiorillo, and M. Pasquale, IEEE Trans. Magn. 29, 3496 共1993兲. 5 G. Bertotti, F. Fiorillo, and A. M. Rietto, IEEE Trans. Magn. 20, 1481 共1984兲. 6 G. Bertotti, Hysteresis in Magnetism 共Academic, New York, 1998兲. 1 2

⫽8 冑␴ GSV 0 B 3/2 f 1/2; HF limit,

共10兲

combined with the plot of ( P totPcl)/ f vs f or f 共Fig. 4兲 permit the separate identification of the material parameters n 0 and V 0 by considering the slope of the loss curve in the appropriate region. 1/2