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1538 IEEE TRANSACTIONSON MAGNETICS,VOL. 29, NO. 2, MARCH 1993Constitutive Equation of Magnetic Materials and Magnetic Field Analysis

M. Enokizono and K. YukiDept. of Electrical and Electronic Engineering, Faculty of Engineering,Oita University, 700Dannoharu, Oita 870-11,JapanAbstract --- The conventional fleld analysismethodsneglect thephase relation between the magnetic flux density B and themagnetic fbld vector H under a rotating fleld. Magneticproperties havebeen measuredas scalar relationshipunderanalternatingmagnetic field, and only thescalar values havebeenapplied to analyze the two-dimensional magnetic fieldproblems. In thSs paper, the B- and H-values have beenmeasured as vector relationships under the influence of arotating fleld, using the two-dimenstonal magneticmeasurement apparatus. The magnetic properties arerepresented by a tensor exprsslon tm a function of magneticrelnctivitks This expression is then applied for the rotatingfleldanalysi&

I. INTRODUCTIONGenerally, rotational power loss, which is generated in the

T-joints of three-phase transformer cores and in rotatingmachines, is larger than the alternating power loss [l].Consequently, it is necessary to clarify the magneticproperties and iron loss values under rotationalmagnetization, and to design an optimum structure which hassmall rotational power loss.Recently, as a means of elucidating these problems, thefinite element method has been applied to magnetic fieldproblems. For this method of analysis, we need to use themagnetic properties of the core m aterials as input data. Themagnetic properties, which are obtained by conventionalmeasurement, are partially effective for an analysis ofalternating magnetic field problem, because, in this type ofmeasurement, the magnetic properties have been evaluatedsuch that the direction of the magnetic flux density B isparallel to that of the magnetic field vector I11 The obtainedcharacteristic values by such measurements, represent theproperties in a given direction only. Therefore theseproperties do not apply to problems under rotating fieldconditions. For analyzing rotating field problems, we wereable to measure the B nd H-values which have a vectorrelationship using the two-dimensional magneticmeasurement apparatus. In this paper, the twodimensionalmagnetic properties are formulated with the magneticreluctivities in a tensor form, and then this tensor expressionis applied to two-dimensional magnetic field analysis. Acomparison of the results obtained using the new method tothose of the conventional one is shown.

n. cbNSlTmrrrvE EQUATIONA. Permeability TensorWhen B is parallel to J3, the constitutive equation isManuscript eceivedon August 3,1992.

usually written as

where Xm is the magnetic susce ptibility, and c1, s the relativepermeability.Having a phase difference between B and H, heconstitutive equation is w ritten as

where, pjj is the permeability tensor. .In conventional field analysis, the magnetic properties inarbitrary directions are modeled by only using magneticproperties of the easy axis and its perpendicular direction. Inother words, off-diagonal terms of the tensor hj have beenassumed to be zero. However, this modeling can't suggest asubstantial characteristic of the magnetic material. Thereforethe permeability should be treated as a tensor.B. Two-Dimensional agnetic MeasurementWe have proposed a new m easurement technique in whichthe relationship between H and B is measured as a vectorrelation [2-51. We call these "twodimensional magneticmeasurements." The magnetic properties obtained by suchmeasurements are also called "two-dimensional magneticproperties."The structure of the two-dimensional magnetic measurement apparatus is shown in Fig. . The measurement systemis shown in Fig. 2, The sample used in this measurement isnon-oriented silicon steel sheet H30, and the excitingfrequency is 50Hz.The measurement conditions in the alternating magneticfield are as follows:

zcFig. 1. Two-dimensional measurementapparah~~.

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I 11 I + L l l l I IIFig. 2. Measurement ystem.

(a) Exciting direction is from 0 to 175 degrees;(b) Maximum m agnetic flux density is from 0.1 to 1.5T.are a s follows:(a) M aximum magnetic flux density is fiom 0.1 to 1.5T;(b) The axis ratio is from 0.1 to 1.0;(c) The inclination angle is0,30,45,6Dnd 90 degrees.where the ax i s ratio is the ratio of maximum flux density tominimum flux density, the inclination angle is the angle fromthe rolling direction to the direction of maximum fluxdensity. The rotating direction ofB nd H is clockwise.

The m easurement conditions in the rotating magnetic field

C . Numerical Modeling for Two-Dimens ionalM a g ne ti c P r o p e r t i e sIn th i s paper, we use the magnetic reluctivity v in place ofpermeability 1.1 As was stated previously, in the past, off-diagonal terms of the tensor have been assumed to be zero.In short, calculating a tensor from the results of theconventional measurement was impossible. First, weconsider the conventional expression of magnetic propertiesunder rotating flux. The expression is obtained from the dataof the parallel and the perpendicular directions to the rollingdirection under an alternating magnetic fie ld by using(3)

where, v, is the magnetic reluctivity in the rolling direction,vy is in the perpen dicular direction to the rolling direction.Next, consider the tensor expression of the magneticreluctivity in our study. The magnetic reluctivity tensor isrepresented by(4)

We introduce a new function defined by the followingequations.(yosrp "I(") b 2 = & 2+-B;" (5), a2 .Sin'COsrp By

where rpis the inclination angle, and a he axis ratio. Weassume that the elements of the tensor are'a function ofB m d , rp, and a Then the elements of the tensor are of aconstant value in a single loop. When it is premised on thesehypotheses, each element is calculated from thevarious conditions. Fig. 3 shows the magneticcurves when the axis ratio is 1by using the compexpression. As is evident from Fig. 3, the off-dvxy is nearly equal to -vyx in pure circular rotationalWhen the maximum flux density is l.OT, the relationshipsamong the off-diagonal terms, the axis ratio, and theinclination angle areshown in Figs. 4,5. In this figure, as hevalue of the axis ratio decreases,vxy ncreases sharply in therolling direction and vyx shows a sharp decrease along theperpendicular direction to the rolling direction.

0 1 2Square of magnetic flux density @]

x-200

1 1Squareof magnetic flux density p]

Fig.3. Magnetic reluctivitycurves in tensor form.

800

0?

A

Fig.4. Relation among ~ y , nd9.

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90- -2002x -4009 - 00 Here, the components of the magnetic flux density can bewritten a s

I Next, substituting (12) into (l l) , we obtainFig. 5. Relation among VYX a and q.

m. MAGNETIC FIELD ANALYSISA. FomularionIn the case when magnetic reluctivity is treated in a tensorform, the Poisson equation for the two-dimensional magneticfield problem including the offd iagon al terms is written as

- JO-Ac")\&**aAie i&( &)+$(vmG)-"( a x a Y a Y linear analysis using the finite element method.vv$)=-Jo.(6) From the minimization conditions of (13), we may carry out a

where A is the magnetic vector potential, and Jo the sourcecurrent density. The following equation is the functionalxtocorrespond to (6).B. Results andDiscussionThis analytical method was applied to the model of thetwo-dimensional measurement apparatus. The grade of thesample is equivalent to non-oriented silicon steel H30. Thedistribution of flux density in the sample is nearly uniform,because this apparatus has air gaps (0.1 mm) between thex j - 1 :HdB)dxdy-// JoAdxdy. (7) sample and each yoke.

The relationship between B and H is given by

Substituting (8) into (7), and rewriting with the values in eachelement, the following equation is obtained.

where the subscript Ye" shows points that compose anelement. Next, rewriting the first term of the right hand sidein (9), we get

Fig. 6. Flux distribution._ _ _ _ .calculated results with conventional expressionEquation (9) can be rewritten using the tensor v as follows: - calculated results with tensor expression

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Locus of B Locus ofH00 00 00000Locus ofB -ocus ofH--ig. 7. Locusof B-vector and H-vector.(a) caldatedresults with conventional expression(b) calculatedr d t s with tensor expression

F,FI>-:-;-tp j p , pScale: Scale: Scale X-direction:(~LI[rad]div.)(0.5fTjldiv.) (loO[A/m]/div.) Scale Y-direction:(~r/4[rad]/div.)

Pig. 8. E-loop, Sloopand phase difference.(a) measurement resuits(b)calculatedd t s ith conventional expression(c) calculatedr d t s ith tensor expression

The calculated results are shown in Figs. 6, 7 and 8. Fig. 6shows the m agnetic flux distribution. Fig. 7shows the locusof the Bv ecto r and H-vector. Fig. 8 shows the Bloop , theE-loop, and the phase difference between B nd H-vectors.The conditions used in this analysis are; The axis ratio is 0.5and the inclination angle is 45 degrees.On the measured rotational flux, the phase differencealways changed over at 0 degree. This change indicates thatB ags behind E The phase difference of the conventionalresult as shown in Fig. 8(b) has some angle of lag or leadwith the rotating direction. Therefore, in the conventionalexpression, it is found that the calculated results differcompletely from the measured results as shown in Fig. &a).On the other hand, the new calculated results as shown in Fig.

8(c) are in good agreement with the measured values. As aresult, it is shown that the two-dimensional magneticproperties can be expressed by using the tensor expression.Iv.CONCLUSIONS

In this paper, the constitutive equation of magneticmaterials and the associated magnetic field analysis havebeen presented. The results can be summarized as follows:1)The magnetic properties in the conven tional measurementare acceptable to analysis of the alternating magnetic fieldproblem. However these properties cant be applied toproblems with rotational magnetization.2)The off-diagonal terms of the magnetic reluctivity tensorwhen the axis ratio is 1O, are anti-symmetric.3)The calculated results with the tensor expression of theloop and ofthephasedifference are in good agreementwith the measured results.4) Not only the magnetic properties of anisotropic materialsbut also those of the non-oriented materia ls under rotatingmagnetic fields, should be dealt with using the tensor form.

REFEXENCIS[11F. Brailsford, Alternating Hysteresis Loss in ElectrolyticSheet Steel,

J. Inst. Elec. Engrs., vol. 84, pp. 399-407, 1939.[2] M. Enokimno, T. Suzuki and J. D. Sievert, MUsing Rotational Magnetic Loss MeasurementMag. n Japan., vol. 14,No. 2,pp. 455-458, 1990,Japan, vol. 6, No. 9,1991.Silicon Steel Sheet, IEEETrans. Magn, vol. 26, No. , pp1990 [INTERMAG90].[4] M. Enokimno,G. Shirakawa,T. Suzuki and J. Sievert,T\Ho-diMagnetic Fropefties of Silicon Steel Sheet, J.Applied Mag.

[3] M. Enokizono,T. Suzuki,J. Sievert and J. Xu, Rotational

V O ~ .15,NO. , pp. 265-270, 1991.[5]A.J. Mosesand T. Meydan, Results of H-coilRotational Lossin Soft Magnetic Materials,Oita,pp. 60-741992Massto Enokizonowas born in Oita, Japan, on FebHe is at present an associate professor at theand Electronic Engineering, faculty of EngineeOita, Japan, and he has been engagcomputational electromagnetic engineering, and the two-dimensional magnetic measuring methods, and researchworkon theelectrical machine. He is a member of the IEEE Magnetic Society.Kenji Yukiwas born in Oita, Japan, on Septembat present a second-year student of the masterEngineering, Oita university, Oita, Japan, andin improving the computational electromagnmagnetic field analysis method by usingmagnetic properties, and research work on the electricalmachine.