COMPARISON OF A SIMPLE AND A DETAILED MODEL OF MAGNETIC
HYSTERESIS WITH MEASUREMENTS ON ELECTRICAL STEEL
Hanif Tavakoli 1, Dierk Bormann 2, David Ribbenfjärd 1, Göran Engdahl 1
1 Royal Institute of Technology, School of Electrical Engineering, Teknikringen 33 SE-100 44 Stockholm, Sweden, e-mail: [email protected] , [email protected] , [email protected] 2 ABB AB - Corporate Research, Power Technologies, Utvecklingsgränd 6, S-721 78 Västerås, Sweden, e-mail: [email protected]
Abstract - For efficient magnetic field calculations in electrical For comparison, we also report simulation results obtained machines, the hysteresis and losses in laminated electrical steel with a much more detailed model of the magnetic hysteresis, must be modeled in a simple and reliable way. In this paper, a eddy current and excess losses. frequency dependent complex permeability model and a more detailed model (describing hysteresis, classical eddy current II. THE COMPLEX-PERMEABILITY MODEL effects, and excess losses separately) are compared to single- sheet measurements. It is discussed under which circumstances the simple complex-µ model is an adequate substitute for the Reduced to its simplest terms, hysteresis introduces a time more detailed model . phase difference between B and H. B is assumed to lag H by a
constant angle θh called the hysteresis angle. In such a de- I. INTRODUCTION scription, harmonics introduced by saturation are ignored, and the hysteresis loop becomes an ellipse whose major axis is Recent research has resulted in detailed models of the mag- inclined by an angle θ relative to the H-axis. Using complex netic hysteresis and loss mechanisms in a wide frequency h ˆ ˆ range [1, 2, 3]. Although these models provide a good de- field components B and H , we can define a low-frequency scription of magnetic material properties or of simple reluc- complex permeability including hysteresis as tance circuits based on them, they are too demanding numeri- Bˆ µˆ = = µ µ e− jθh . (1) cally to be incorporated into a full-scale magnetic field simu- hHˆ 0 r lation of a realistic geometry, as with a FEM or FDM calcula- In addition to this, eddy currents in the lamination sheets tion tool. In other words, while such a detailed simulation of introduce frequency dependence. We briefly sketch the well- the H-B relation of a single or a few interacting cells is still known procedure [5] for deriving the effective frequency perfectly feasible, simulating thousands or ten thousands of dependent complex permeability. Faraday’s law them simultaneously may be inconvenient or impossible. ∂B ∇×E =− (2) Moreover, in many practical situations a detailed description ∂t is not required either. Often the goal is to obtain a good esti- and Ampere’s law mate of some local or global quantity containing much less ∂D ∇×H = J + , (3) information than the detailed local H-B relation, such as the ∂t local losses causing dangerous hot spots, or simply the total in combination with the constitutive relations losses in a machine relevant for cooling or economic reasons. J= σ E , D= ε E , B= µˆ H , (4) For such applications it is desirable to use a simple model of h and time-harmonic assumption lead to magnetic hysteresis and losses, which can easily be incorpo- 2 ˆ 2 ˆ rated in field calculation tools but which at the same time is ∇=Hα H =j ωµσh ( + j ωε ) H . (5) sufficiently close to reality, within the frequency range of For lower frequencies when wave propagation can be ignored interest for the specific application. Such a model is the de- 2 (i.e., σ>> ωε ), we have αˆ ≈ j ωµσˆh . scription of magnetic (meta-)materials by a suitable frequency dependent complex permeability, which is the most general x linear description of a local and isotropic H-B relation. If desired, it can easily be extended to a nonlocal and/or aniso- tropic H-B relation by turning µ from a scalar function of position into a distance dependent integral kernel and/or ten- H( x ) y sor, respectively [4]. 2b z ⊗ In this paper, we discuss to which extent the measurement results obtained with a single-sheet tester on strips of electri- cal steel can reliably be described by a simple complex per- Fig. 1. Laminate infinite in z direction, with a width in y direction much larger than its thickness 2b, exposed to a H field in z direction. H B meability function of frequency. Both the resulting - curves and the effective complex permeability are compared For analysis of the magnetic field in a laminate, the simple to the measured data at different frequencies. geometry illustrated in Fig. 1 is appropriate. The magnetic field is applied in the z direction, hence the only component The hysteresis curve for one particle is introduced by apply- of the magnetic field strength is H z which varies only in the x ing a “play operator” with “pinning strength” k (which will direction, H= H( x ) . In one dimension, eq. (5) reduces to determine the width of the hysteresis curve) on the anhyster- z z etic curve, see Fig. 2. ∂2 H z = αˆ 2 H , (6) Using a population of pseudo particles with different pinning ∂x2 z strengths allows to construct minor loops. We assign an indi- which has the general solution vidual pinning strength λi k to every pseudo particle, where k αˆx− α ˆ x Hxz ()= A1 e + A 2 e . (7) is the mean pinning strength, and the λi are dimensionless The field strength on the both sides of the laminate is as- numbers. The total magnetization is then given by a weighted sumed to be H0 . For the reason of symmetry the following superposition of the contributions from all pseudo particles condition is obtained (Fig. 3).
Hbz()= H z () − b = H 0 . (8) The final expression for the magnetic field strength then be- comes cosh (αˆ x) H( x ) = H . (9) z 0 cosh ()αˆb The effective, complex permeability of a lamination is given as the average magnetic flux density B in the laminate nor- Fig. 3. Weighted superposition of the contributions from pseudo particles malized to the surface magnetic field strength H , describes a minor loop; figure taken from [2]. 0 ˆ ′j ′′ µeff= µ eff − µ eff The expression b B1 tanh(αˆ b ) 2 πH χ = =∫ µˆhHz ( x )d x = µ ˆ h . (10) M( H )= M arctan (11) HHb2 αˆ b an s 0 0 −b π 2M s This expression accounts for the effect of hysteresis without is used for the anhysteretic magnetization, where M is the saturation, and the effect of eddy currents. We assume here s that additional (or “excess”) losses are either negligible or magnetization saturation and χ is the susceptibility at H = 0. have a similar frequency dependence so that they can be in- The total magnetization of the material is then given by ∞ corporated in the expression (10) for µˆeff . M= cMan() H + ∫ M an () PHλk ()()dς λ λ , (12) 0 III. THE DETAILED HYSTERESIS MODEL where c is a constant that governs the degree of reversibility, and the integral describes the hysteretic behavior (irreversible Later in this paper we will report some results obtained with a part). P is a play-operator with the pinning strength λ k , more detailed model of the magnetic hysteresis, eddy current λk i and excess losses, which is therefore described here in some and ς( λ ) is a weight function describing the density of the detail. pseudo particles. Finally, the magnetic flux density is ob-
The total hysteresis is a combination of three different phe- tained from B=µ0 ( H + M ) . nomena, namely, static hysteresis, eddy current effects and excess eddy currents. For the detailed hysteresis model, the Excess losses following approach has been used. The static hysteresis is Excess losses are caused by microscopic eddy currents in- modeled using Bergqvist’s lag model [6, 7], the classical eddy duced by local changes in flux density due to domain wall currents are modeled using Cauer circuits [9, 3], and the ex- movements. For our detailed model we use an approach de- cess losses are modeled using an approach by Bertotti [1]. scribed by Bertotti [1]. In this approach a number of active correlation regions are assumed randomly distributed in the Static hysteresis material. The correlation regions are connected to the micro- The lag model of static hysteresis starts from the idea that the structure of the material like grain size, crystallographic tex- magnetic material consists of a finite number of pseudo parti- tures and residual stresses. In Bertotti’s model, the resulting cles np , i.e., volume fractions with different magnetization. contribution to the magnetic field strength is given by The total magnetization is then a weighted sum of the indi- n V 42dσGbwB d B H =0 0 1 + − 1 sign , (13) vidual magnetization of all pseudo particles. excess 2n2 V dt d t 0 0 where w is the width of the laminate and 2 b, as before, its thickness. G is a parameter depending on the structure of the
magnetic domains. n0 is a phenomenological parameter re- lated to the number of active correlation regions when the
frequency approaches zero, whereas V0 determines to which extent micro-structural features affect the number of active
Fig. 2. Anhysteretic curve (left), play operator (middle), and resulting correlation regions. hysteresis curve (right); figure taken from [2]. The parameters n0 and V0 are by definition frequency inde- pendent, but they are expected in reality to depend on the amplitude of the B field [8]. Since the precise form of this dependence is unknown, their values are usually adjusted By inserting Eqs. (15) and (16) into (14) we get empirically for given amplitude. In the simulations reported A µ′′ = − . (18) here we use one set of (empirically determined) values, al- meas π H 2 though the amplitude of the B field slightly varies in the p measurements. Furthermore, from the relation µˆmeasH p= B p we obtain 2 B IV. THE MEASUREMENT SETUP µˆ 2= µ′ 2 + µ ′′ 2 = p , (19) meas()() meas meas Hp The magnetic measurements were carried out using a Single which implies Sheet Tester. It consists of two equal U-shaped yokes placed 2 face-to-face to each other (Fig. 4). The magnetic sheet to be B µ′=p − () µ ′′ 2 . (20) tested is placed between the yokes and most of the flux is measH meas forced through it due to its high permeability. For the meas- p urement of the flux in the test material a coil is surrounding Both µmeas′ and µmeas′′ are functions of frequency. the strip which is connected to a flux meter. The magnetic field strength is measured with a Hall probe placed close to 1.5 the surface of the sample and connected to a Tesla meter. A 1 sinusoidal H field was applied to the sample; the H and B field values were measured for 100 periods and numerically 0.5
filtered. Thereafter, the mean values at different phase angles la] s of the B and H fields were calculated. These values were then B 0 used in the paper. [Te B -0.5
-1
-1.5 -400 -300 -200 -100 0 100 200 300 400 H H [A/m] (a)
1
0.8
0.6
0.4
0.2 la] s
B 0
B [Te B -0.2
-0.4
-0.6 Fig. 4. Cross sectional view of our Single Sheet Tester. -0.8
-1 -400 -300 -200 -100 0 100 200 300 400 We approximate the measured H-B curve with a complex-µ H H [A/m] (b) ellipse characterized by the permeability µˆmeas by matching both its peak values H , B and its area A to the measured Fig. 5. H-B-curves from measurements (blue) and complex-µ model p p (green) with µˆ = µ′ − j µ ′′ , for (a) f = 50 Hz and (b) f = 400 Hz. results. This is of course appropriate as long as the shape of meas meas meas the measured H-B curve is close to an ellipse, i.e., if satura- Figure 5 compares the measured H-B curves with complex-µ tion effects are not too pronounced. The area A, which meas- ellipses, generated with the adapted µˆ at frequencies f = ures the power loss per cycle, is given by the integral meas T dH 50 Hz and 400 Hz. ABH=d = Bmeas d t , (14) ∫meas meas ∫ meas 0 dt µˆeff as defined in (10) is a function of frequency and of a where Bmeas and H meas are the time dependent measured B and 2 vector x = µ, θ , σ b containing the model parameters. It is H fields, respectively, and T is the duration of a period. If the ( r h ) measured magnetic field strength is assumed to vary sinusoi- adjusted to measured data by numerically minimizing the dally, expression N Ht()= Re H ejωt = H cos(ω t ) , (15) 2 meas( p) p ∑ µˆeff(x, ωi )− µ ˆ meas () ω i (21) then its derivative becomes i=1 with respect to x . µˆmeas ( ω i ) are the measured complex per- dHmeas ( t ) = − ωHp sin( ω t ) , (16) N dt meability values, defined by (18) and (20), at different frequencies ω= 2 π f , i= 1,..., N . We performed measure- and the measured magnetic flux density i i jωt ments at N = 9 different frequencies ranging from 50 Hz to B()Re t= µˆ H e meas( meas p ) 2 kHz (see Figs. 6 and 7 below), on a 100 mm x 3.2 mm strip jωt of the non-oriented magnetic material M600 with a thickness =Re((µmeas′ − j µ meas′′ ) H p e ) (17) of 2b = 0.5 mm . =Hp()µ meas′cos( ωµ t ) + meas′′ sin( ω t ) . f=100 1.5 V. RESULTS AND DISCUSSION
1 Since the measurement setup was quite sensitive to noise, the measurements had to be numerically filtered. Adjusting µˆeff 0.5
to the filtered data using Eq. (21), we obtain the following la] s 0 model parameter values: µr = 3366 ,θh = 0.477 rad , and B [Te B 2 6 σ b = 0.243 Sm, i.e., σ =3.89 × 10 S/m. The last value is -0.5 somewhat larger than the true dc conductivity of this material, 6 -1 σ dc =3.33 × 10 S/m, since we have included here excess losses in the classical phenomenological form (10). -1.5 In Figure 6, the real and imaginary parts of the measured -400 -300 -200 -100 0 100 200 300 400 H [A/m] complex permeability are compared at different frequencies with the adjusted µˆeff . f = 200 Hz 1.5
-3 x 10 1 4 µ' eff µ'' 0.5 3.5 eff µ' meas la] s
3 µ'' B 0 meas B [Te B
2.5 -0.5 ]
2 -1 [Vs/Am eff eff µ 1.5 -1.5 -400 -300 -200 -100 0 100 200 300 400 H [A/m]H 1
f = 400 Hz 0.5 1
0.8 0 0 2 4 6 8 10 10 10 10 10 0.6 frequency [Hz] 0.4 Fig. 6. Real and imaginary parts of the measured complex permeability (symbols) and of the fitted permeability function (curves). 0.2 la] s
B 0
[Te B -0.2
The agreement is quite satisfactory considering the simplicity -0.4 of the model, especially at higher frequencies. The deviation -0.6 ′′ ′′ between µmeas and µeff at the lowest frequencies is probably -0.8 due to saturation effects which are not properly taken into -1 -400 -300 -200 -100 0 100 200 300 400 account by the expression (10) for µˆ , see for instance the H [A/m] eff measurement at 50 Hz (Fig. 5(a)). The amplitude had to be f = 500 Hz chosen large enough for the signal not to be covered by noise. 1
Below the H-B hysteresis curves are shown for all measured 0.8 frequencies. Measurement, simple model, and detailed model 0.6 are represented by solid green lines, dashed blue lines and dotted red lines, respectively. 0.4 0.2
la] s
f = 50 Hz B 0 1.5 B [Te B -0.2
-0.4 1
-0.6
0.5 -0.8
la] -1
s -400 -300 -200 -100 0 100 200 300 400
B 0 H [A/m] B [Te B -0.5
-1
-1.5 -400 -300 -200 -100 0 100 200 300 400 H [A/m]H
f = 800 Hz both H and B amplitudes and magnetic losses in the whole 0.8 frequency range. This is illustrated in the Figure 7, where the 0.6 measured H-B curves are compared with the corresponding
0.4 complex-µ ellipses and the detailed model at different fre- quencies. As can be seen the simple model agrees very well 0.2 with the measurements as long as saturation is not too strong, la] s
B 0 which means for low amplitude fields and/or for frequencies B [Te B -0.2 higher than about 200 Hz.
-0.4 VI. CONCLUSIONS
-0.6 In this paper, a simple, frequency dependent complex-µ -0.8 -400 -300 -200 -100 0 100 200 300 400 model of magnetic core material has been developed and H [A/m] adjusted to measurements. Its real and imaginary parts were
f = 1000 Hz compared to measurements in a wide frequency range. The 0.8 agreement was found satisfactory, especially for higher fre-
0.6 quencies, which makes the complex-µ model a very conven- ient starting point for the estimation of flux distribution and 0.4 losses in complicated core geometries. 0.2 Furthermore, H-B curves from our measurements, the simple la]
s complex-µ model and the detailed hysteresis model were
B 0 compared for different frequencies. Again the results from the B [Te B -0.2 complex-µ model were found to agree well with measure-
-0.4 ments at higher frequencies. At low frequencies and high field amplitudes the complex-µ model deviates from measurements -0.6 and detailed hysteresis model, since it does not take saturation -0.8 -400 -300 -200 -100 0 100 200 300 400 effects properly into account. This is, however, not expected H [A/m] to affect its usefulness for loss estimation.
f = 1250 Hz REFERENCES 0.6
0.4 [1] G. Bertotti, “Hysteresis in Magnetism”, Academic Press, San Diego, 1998.
0.2 [2] D. Ribbenfjärd , “A lumped transformer model including core losses and winding impedances”, Licentiate Thesis in Electromagnetic Engineering, 0 Stockholm, Sweden, 2007. la] s [3] D. Ribbenfjärd, G. Engdahl, “Novel Method for Modelling of Dynamic
B [Te B -0.2 Hysteresis”, IEEE Transactions on Magnetics, Vol. 44, No. 6, pp. 854- 857, June 2008. -0.4 [4] J. D. Jackson, “Classical Electrodynamics”, third edition, John Wiley & Sons, Inc., New York, 1998. -0.6 [5] R. L. Stoll, “The analysis of eddy currents”, Clarendon press, Oxford, 1974. -0.8 [6] A. Bergqvist, “A phenomenological differential-relation-based vector -400 -300 -200 -100 0 100 200 300 400 hysteresis model”, Journal of Applied Physics, Vol. 75, No. 10, pages H [A/m]H 5484-5486, May 1994. [7] A. Bergqvist, “Magnetic vector hysteresis model with dry friction-like f = 2000 Hz pinning”, Physica B: Condensed Matter, Vol. 233, No. 4, pp. 342-347, 0.6 1997. [8] E. Barbisio, F. Fiorillo, C. Ragusa, “Predicting Loss in Magnetic Steels 0.4 Under Arbitrary Induction Waveform and With Minor Hysteresis Loops”, IEEE Transactions on Magnetics, Vol. 40, No. 4, pp. 1810- 0.2 1819, July 2004. [9] F. de León, A. Semlyen, “Complete Transformer Model for Electromag- 0 la]
s netic Transients”, IEEE Transactions on Power Delivery, Vol. 9, No. 1, B pp. 231-239, January 1994.
B [Te B -0.2
-0.4
-0.6
-0.8 -500 -400 -300 -200 -100 0 100 200 300 400 500 H [A/m]
Fig. 7. H-B curves from measurements (solid green line), detailed model (dotted red line) and complex-µ model (dashed blue line) with µ calculated from expression (10), at different frequencies ranging from 50 Hz to 2 kHz.
The above way of defining a “best fit” of ellipses to the more complicated H-B hysteresis relations approximately preserves