Influence of electrical steel sheet textures on their magnetization curves *

Transcription

1 ARCHIVES OF ELECTRICAL ENGINEERING VOL. 6(3), pp (013) DOI /aee Influence of electical steel sheet textues on thei magnetization cuves * WITOLD MAZGAJ, ADAM WARZECHA Institute of Electomechanical Enegy Convesion Cacow Univesity of Technology Waszawska 4, Kaków, Poland {pemazgaj/pewazec}@cyfonet.pl (Received: , evised: ) Abstact: The Goss textue is a chaacteistic featue of gain-oiented tansfome steel sheets. Geneato sheets, which ae poduced as non-oiented steel sheets, should have isotopic featues. Howeve, measuement esults of geneato sheets, confimed by cystallogaphic studies, indicate that these sheets ae chaacteized by cetain, quite significant anisotopy. The fist pupose of this pape is to pesent the influence of textues of geneato and tansfome steel sheets on thei magnetization chaacteistics. The second aim is to popose a method which takes into account the sheet textues in the calculations of magnetization cuves. In calculations of magnetization pocesses in electical steel sheets, models in which the plane of a sheet sample is divided into an assumed numbe of specified diections ae used. To each diection a cetain hysteesis loop, the so-called diection hysteesis, is assigned. The paametes of these diection hysteeses depend, among othe things, on the textue type in these steel sheets. This pape discusses the method which calculates the paametes of these diection hysteeses taking into account the given sheet textue. The poposed method gives a possibility of detemining the magnetization chaacteistics fo any diection of the field intensity changes. Key wods: electical steel sheets, hysteesis loop, magnetization chaacteistics, Goss textue, anisotopy 1. Intoduction Almost all electical steel sheets can be divided into two basic goups: geneato sheets and tansfome sheets. Geneato sheets ae used not only in the magnetic cicuits of otating machines but quite often they ae used in the coes of small powe tansfomes. Theefoe, these sheets should have isotopic featues in any diection on the sheet plane. Howeve, special magnetic measuements caied out by means of both the RSST devices and the * This is extended vesion of a pape which was pesented at the th Symposium on Electomagnetic Phenomena in Nonlinea Cicuits, Pula, Coatia,

2 46 W. Mazgaj, A. Wazecha Ach. Elect. Eng. Epstein fame showed that geneato steel sheets have cetain anisotopic featues. Cystallogaphic eseach, pefomed fo seveal typical geneato sheets with the use of the X-ay diffactomete, shows that an amount of sheet gains has a cetain textue. The volume pat of this textue is in the ange fom a few to a dozen o so pecent of the volume of the whole sample. Vaious technological pocedues in the manufactuing pocess of geneato sheets ae designed to achieve textue-fee mateials, but these sheets have not been poduced so fa on an industial scale. The well-known Goss textue is a chaacteistic of tansfome steel sheets which ae made as gain-oiented sheets. This textue allows us to obtain high values of the flux density in the olling diection. Obtaining of this textue in tansfome sheets is an intended action in the manufactuing pocess. In some places of tansfome coes the flux density is not paallel to the olling diection (fo example, in cones of coes). Then it causes the magnetization cuves to diffe fom the cuve measued fo the olling diection. The eason fo these diffeences is the fact that the cubic-shape ion cystals have thee axes of easy magnetization, so magnetic featues depend also on the field intensity diection [, 3]. Electical steel sheets can be magnetized in diffeent diections. Geneally, this efes to geneato sheets, although in tansfome sheets the magnetic flux can change its diection, which takes place in a cuent oveload. Thus, in many cases it is necessay to calculate the magnetic field distibution fo diffeent diections of the magnetic flux. This is possible using the appopiate model of the magnetization pocess which allows us to take into account the anisotopy popeties of electical steel sheets. In the papes [6, 7] the method of the magnetization modelling of soft magnetic mateials was pesented. This model can be applied to both otational and altenating magnetization. In the poposed model the plane of a sample of an anisotopic sheet is divided into an assumed numbe of specified diections. To each diection a cetain hysteesis loop, the so-called diection hysteesis, is assigned. These diection hysteeses ae descibed by such paametes as: satuation flux density, esidual flux density, and coecive foce. It is necessay to undeline that these paametes and diection hysteeses can not be measued and they diffe fom hysteesis of the electical sheet sample. The paametes of the diection hysteeses ae calculated on the basis of measuements of the limiting and some patial hysteesis loops. It is also necessay to take into account the distibution function of the gains in the sample of a given anisotopic mateial. The above-mentioned distibution function is diectly elated to the textue which quite often occus in geneato sheets.. Textues in electical steel sheets In the Goss textue, the majoity of ion gains (about 95-97%) ae situated with espect to the sheet plane as it is pesented in Figue 1. In cystallogaphy this textue is descibed as {110}<100>; this symbol shows which cystallogaphic planes and diections of ion cystals ae paallel to the sheet plane and the olling diection espectively [5]. Ion cystals have a cubic shape and each cystal has thee axes of easy magnetization, which ae descibed by the edges of the cube. One diection of these thee axes is paallel to the olling diection (RD), the othe two axes ae inclined to the sheet plane at an angle of 45. The chaacteistic

3 Vol. 6(013) Influence of electical steel sheet textues on thei magnetization cuves 47 size of the Goss textue is the angle of 54.7 o, which efes to the diection of the difficult magnetization. If the angle between the field intensity vecto and the easy magnetization diection of the sheet plane (axis [100]) is less than 54.7, the magnetic flux inceases along this diection of easy magnetization [4, 10]. In cases when tansfome sheets ae magnetized at angles geate than 54.7 o with espect to the olling diection, the flux density inceases mainly along the [010] and [001] axes. The pupose of pocedues in the manufactuing pocess of tansfome sheets is to stive fo such an oientation of gains so that all gains fom the Goss textue. In pactice, a small pecentage of the amount of gains is aanged in such a way that none of the easy magnetization axes is paallel to the olling diection. Fig. 1. Typical aangement of ion cystals in sheets with the Goss textue Geneato sheets ae poduced as non-oiented sheets, but quite often they have cetain anisotopic featues. Textue studies wee caied out fo the chosen typical geneato sheets with the thickness of 0.5 mm by means of the X-ay diffactomete 1. Duing these tests, the steeogaphic pojections ae caied out and the eflections fom the {100} o {110} plane of the given geneato sheet sample ae counted. Points with the same intensity of eflections ae connected by lines, and it enables us to obtain the so-called pole figues which make possible the desciption of the textue type in geneato sheets. The intensity of eflections in pole figues gives the possibility of estimation the volume of gains of a pivileged cystallogaphic oientation. Figue pesents the {100} type of pole figues of the M400-50A geneato sheets (made in Sweden), and Figue 3 shows the pole figues of the M400-50A sheet (made in Russia). In both cases, tests wee pefomed sepaately fo the suface laye and the inne laye of these selected sheets; the thickness of the layes was about 0.15 mm. The suface laye efes to the suface of the sheet distinct fom the inne laye which is situated inside of the sheet. Both layes wee obtained by appopiate mechanical - chemical teatment. The pole figues pesented in Figue show that a cetain amount of gains in the suface laye have the textue {011}<100>. It means that these gains ae situated similaly as in the 1 Textue studies wee made in the Institute of Non Feous Metals in Gliwice, Light Metal Division in Skawina (Poland).

4 48 W. Mazgaj, A. Wazecha Ach. Elect. Eng. Goss textue. Special softwae fo the analysis of pole figues, used in cystallogaphy, enable us to estimate the volumetic amount of this textue fo about 8% of the whole volume of the sheet sample. A small potion of gains have the weak textue {15}< 10>. In this case, none of the chaacteistic ion cystal planes is paallel to the sheet plane, and none of the easy magnetization axes is paallel to the olling diection. One can say that the aangement of these gains is athe simila to the so-called cubic textue [1]. In this textue two walls of cubic ion cystals ae paallel to the sheet plane, and then the given geneato sheet has the same magnetic popeties in two pependicula diections. In the inne laye of the M400-50A (Sweden) sheet the {001}<10> cystallogaphic oientation is pivileged and its volume pat is estimated at aound 6%. It means that these gains have the cubic textue but the angle between one of the easy magnetization axes and the olling diection is about 7 o. Othe gains, both in the suface and inne layes, ae aanged andomly. Fig.. Pole figues {100} of the M400-50A sheet (made in Sweden); Imax denotes the maximum value of eflection intensity; intensity scale is linea, RD olling diection, TD tansvese diection Fig. 3. Pole figues {100} of the M400-50A sheet (made in Russia); Imax denotes the maximum value of eflection intensity; intensity scale is linea, RD olling diection, TD tansvese diection Pole figues of the M400-50A sheet poduced in Russia (Fig. 3) show that in both suface and inne layes the cystallogaphic oientation {001} <10> is pivileged. The estimated

5 Vol. 6(013) Influence of electical steel sheet textues on thei magnetization cuves 49 volumetic pat of this textue in the suface laye is about 9%, and in the inne laye is about 19%. In this sheet the weak textue {15}< 10> also occus. Tests pefomed on samples of othe geneato sheets, e.g. M600-50A (Romania), M800-50A (Sweden), showed that the textues {001} <10> and {15}< 10> occu quite fequently. In some sheets the weak textue {111}< 11> can occu. This oientation means that the planes {111} ae paallel to the olling plane, and these planes ae detemined by thee points of a cubic-shaped ion cystal which do not belong to the same cystal wall. It is necessay to undeline that the volumetic amount of the pivileged cystallogaphic oientation does not exceed a dozen o so pecent of the total volume of the sheet sample. 3. Taking into consideation the geneato sheet textues It is known that the magnetization chaacteistics of the geneato sheets with anisotopic popeties depend on the diection of magnetization and on the sheet textue. The poposal of taking into account the anisotopy of gains was pesented in the pape [7]. In that pape, the plane of an anisotopic sheet sample was divided into n k specified diections. Due to the sheet anisotopy, a diffeent amount of ion gains was assigned to each specified diection. Although in geneato sheets seveal pivileged cystallogaphic oientations may exist, but the vast majoity of gains ae aanged andomly. Examples of gain distibutions ae pesented in Figue 4. Fig. 4. Examples of gain distibutions To each specified diection on the sheet plane the so-called diection hysteesis is assigned, which diffes fom the hysteesis of the whole anisotopic sheet sample. Theefoe, the paametes of the diection hysteeses like the satuation flux density b s, the esidual flux density b

6 430 W. Mazgaj, A. Wazecha Ach. Elect. Eng. and the coecive foce h c must be detemined. These paametes depend on the gain distibution function in the anisotopic sheet sample. If the given geneato sheet has a few textues then fo each oientation the individual gain distibution function d(k) should be detemined in the following fom (assuming 1 individual diections) d k) = d, d,..., d,..., d,, (1) ( 1 k 11 d1 whee d k denotes the elative amount of gains which ae assigned to the k-th diection. It is woth emphasizing that most of the d values ae equal to zeo because the gains of the given textues ae not associated with all diections. As peviously mentioned, the vast majoity of gains ae aanged andomly. Fo this pat of gains the distibution function has the same value fo all individual diections, and this value equals d = 1 V V ) 1, () ( 1 / whee V 1, V denote the elative volume pat (with espect to the whole volume of the given sheet sample) of the two given textues. If we assume that in the given sheet two textues occu, and d 1k, d k denote the value of the gain distibution function fo these two textues, then the condition 1 k= 1 ( 1k k d ) + 1d = 1 (3) must be fulfilled. The flux densities of the diection hysteeses each the satuation state b s if the field intensity inceases along one of individual diection. In this case the flux density vectos ae situated along the magnetic field intensity vecto, and the esultant flux density B s of the sheet sample is the algebaic sum of the satuation flux densities of all diection hysteeses. Then the satuation flux density b sk fo the given k-th diection equals b = d ) B, (4) sk ( 1k k s whee B s is the satuation flux density, detemined by measuements, of the anisotopic geneato sheet. If the field intensity is deceased fom the satuation state to zeo, the flux density vectos of the diection hysteeses lie along individual diections, and the length of these vectos ae equal to the esidual flux density b k of the diection hysteeses. The esidual flux densities of the diection hysteeses have diffeent values, and they depend on the gain distibution function. The esultant esidual flux density of the whole sheet sample is the vecto sum of the esidual flux densities of the individual diections, and the following elation can be witten as follows (assuming that the field intensity vaies along the diection of numbe 7 in Fig. 4) B = bcos 5α 3cos 4α cos 4α 1cos 5α, (5) whee B is the esidual flux density of the anisotopic sheet (detemined on the basis of

7 Vol. 6(013) Influence of electical steel sheet textues on thei magnetization cuves 431 measuements), b k denotes the esidual flux density in k-th diection, and α is an angle between two neighbouing diections. Unlike the case in which the textue is not distinguished, the esidual flux densities b k in the individual diections ae algebaic sums of the esidual flux densities elating to diffeent textues and of the esidual flux density of the goup of andomly aanged gains and b = b + ab, (6a) 1 b = b + ab, (6b) b = b + ab, (6c) b = b + ab, (6d) whee b 1k, b k denote the esidual flux densities in k-th diection fo the fist and second textue espectively, b epesents the esidual flux density in each diection elating to the gains which ae situated andomly, and a = cos7 o. The b 1 flux density does not change its value because it is assumed that the field intensity vaies along the diection 7 which is pependicula to the diection of numbe 1. The existence of the coefficient a of the esidual flux densities b in the last elationships equies some explanation. One of the easy magnetization axes (cube edges) of ion gains which ae distibuted andomly is always inclined to the sheet plane at an angle no geate than the 45 o. The esultant flux density vecto of these gains is not paallel to the sheet plane. It can be poved that this vecto is inclined to the sheet plane at an angle of about 7 o. Fo the fist textue the esidual flux density b 1k fo the k-th diection can be witten as N1k b 1k = bj N1k = bj N = bj Nd1k = bzd1k, (7) N whee b j is the esidual flux density of a gain, N 1k detemines the amount of gains of the fist textue which ae oiented in the k-th diection, N denotes the amount of all gains in the anisotopy sheet sample, d 1k is the value of the gain distibution function of the fist textue, and b z is a cetain auxiliay esidual flux density. Simila elationship as (7) can be witten fo the second textue and fo the andomly distibuted gains. In the latte case, the gain distibution function has the same value fo each diection. Taking into account the last elationship, the Equation (5) can be witten in the following fom + ( d 17 B 7 = b z [( d 1 + ad ) ( d + ad )cos 5α + ( d ad )cos 4α + ( d + ad )cos 4α ad )cos 5α ]. (8) Hence, the auxiliay esidual flux density b z equals

8 43 W. Mazgaj, A. Wazecha Ach. Elect. Eng. + ( d 17 b z 7 = B /[( d ad ) ( d + ad )cos 5α + ( d ad )cos 4α + ( d + ad )cos 4α ad )cos 5α ]. (9) As mentioned peviously, the esidual flux densities b k in the individual diections ae algebaic sums of the esidual flux densities elating to diffeent textues and of the esidual flux density of the goup of andomly aanged gains. Thus on the basis of elation (7) it may be stated that the esidual flux densities of the diection hysteeses ae equal to the expession b z (d 1k k + ad ). It is necessay to undeline that the esidual flux densities of the diection hysteeses ae diffeent fom the esidual flux density detemined on the basis of measuements of the whole sheet sample. In this case, they ae the pojections of the vecto B on individual diections. The coecive foce is a cetain field intensity value needed to move the domain walls though potential baies until the esultant flux density of the total sheet sample is equal to zeo [9]. The coecive foces of geneato and also tansfome steel sheets ae elatively low. Theefoe, otations of flux density vectos towads to the diection of the field intensity can be neglected. In ode to calculate the coecive foce h c of the diection hysteeses it is assumed that the cuves of diection hysteesis loops ae linea in a cetain ange. The coecive foces h c of the diection hysteeses do not depend diectly on the distibution function of ion gains in the geneato sheets, and they can be detemined on the basis of the elationship deived in [6, 7] h c b cos 5α 3 cos 4α cos 4α 1 cos 5α = H c, (10) b cos5α cos 4α cos 4α cos5α 3 whee H c denotes the coecive foce of an geneato sheet detemined by measuement along the olling diection, b k is the esidual flux density of the diection hysteesis in k-th diection. Calculations of diection hysteesis paametes with taking into account pivileged cystallogaphic oientations allow us to detemine magnetization cuves fo any diection. It is woth undelining that the esultant flux density in the sheet sample is the vecto sum of the flux densities of the specified diections [8]. The paametes of diection hysteeses wee detemined fo the geneato sheet M400-50A manufactued in Sweden (B s = 1.8 T at A/m, B = 1.05 T, and H c = 60 A/m) and fo the sheet M400-50A poduced in Russia (B s = 1.8 T at A/m, B = 0.85 T, H c = 85 A/m). The hysteesis loops fo the olling diection and tansvese diection fo both sheets wee detemined with the use of the otational magnetization model [6, 7] with taking into account the sheet textues. These hysteesis loops wee compaed with measued loops (Fig. 5, 6) Hysteesis loops wee measued in Laboatoy of Magnetic Measuements in Bochnia (Poland)

9 Vol. 6(013) Influence of electical steel sheet textues on thei magnetization cuves 433 Fig. 5. Hysteesis loops of the geneato sheet M400-50A (Sweden): a) fo the olling diection, b) fo the tansvese diection; continuous lines loops detemined by measuement, dashed lines calculated loops Fig. 6. Hysteesis loops of the geneato sheet M400-50A (Russia): a) fo the olling diection, b) fo the tansvese diection; continuous lines loops detemined by measuement, dashed lines calculated loops 4. Calculation of magnetization cuves in steel sheets with the Goss textue Taking into account the impact of the Goss textue in tansfome steel sheets on magnetization chaacteistics is simple than in the case of geneato sheets. In pactice, appoximately 95-97% of gains in tansfome sheets have the Goss textue. It means that one of the easy magnetization axes of ion gains is paallel to the sheet plane and to the olling diection. Without making a seious eo it can be assumed that the emaining pat of gains is distibuted unifomly with espect to othe diections. Thus, the value of the distibution function fo the olling diection anges fom 0.95 to 0.97 and fom to fo othe diections.

10 434 W. Mazgaj, A. Wazecha Ach. Elect. Eng. In cubic-shaped ion cystals, the magnetization pocess occus along this magnetization axis which is closest to the field intensity diection. Figue 7 shows, fo example, the flux density vectos in specified diections on the tansfome sheet plane when the field intensity vecto H is paallel to the olling diection RD. In this assumed case the angles between diections 1,, 3 and diections 11, 1 ae bigge than 54.7, so the magnetization pocess in gains assigned to the above-mentioned diections occus along the axes [010] and [001]. The field intensities fo individual diections can be witten as follows: o h = cos45, (11a) 1 H o h = cos45 cosα, (11b),1 H o h = cos45 cosα, (11c) 3,11 H h = cos3α, (11d) 4,10 H... h 7= H, (11e) whee indexes at h ae the numbes of individual diections, and α denotes, as befoe, the angle between two neighbouing diections. Fig. 7. Flux density vectos in a sheet sample with the Goss textue when the field intensity vecto is paallel to the olling diection RD The esidual flux density of the given sheet sample with the Goss textue is the vecto sum of the esidual flux densities in all specified diections, and it can be witten in the following fom (assuming that the field intensity changes along the diection 7)

11 Vol. 6(013) Influence of electical steel sheet textues on thei magnetization cuves 435 B = b 1 + ( b 4,10 cos 3α,1 5,9 cos α cos α 6,8 3,11 cos α) cos α , (1) whee B is the esidual flux density detemined by measuement of the whole tansfome sheet fo the olling diection, b k denotes the esidual flux density in the k-th diection. The flux density vectos of the diections 1,, 3, 11 and 1 fom the angles 45 with espect to the sheet plane. Thus, detemining the esultant flux density of the whole sheet sample with Goss textue, the length of these vectos must be multiplied by cos 45 o, additionally fo the diections and 1 by cosα, and fo diections 3 and 11 by cos α. Poceeding similaly as fo the geneato sheets the auxiliay esidual flux density b z can be expessed as b z = B /[ d + ( d cos 3α cos α cos α cos α) RD cos α +... ], (13) whee d RD is the value of the distibution function of gains fo the olling diection RD, d denotes the value of the distibution function fo emaining diections. Afte calculating the b z value on the basis of the last equation, the esidual flux densities of diection hysteeses can be detemined by multiplying the value b z by d RD o by d espectively. The vecto sum of flux densities of all diection hysteeses is equal to zeo if the field intensity in the whole sheet sample is equal to the coecive foce H c. This condition can be witten as b + ( b 1 4,10 1, cos 3α 5,9 cos α cos α 6,8 3,11 cos α) cos α = 0. (14) As it was assumed peviously, in a cetain ange the flux densities of the diection hysteeses ae linea functions with espect to the field intensity. Taking this into account and using the method descibed in [7], the coecive foce h c of the diection hysteeses is equal to whee L hc = H c, (15) M 1 4,10 L = b + ( b cos 3α,1 5,9 cos α cos α 6,8 3,11 cos α, cos α +... (15a) M = b 1 cos45 + ( b 4,10 cos3α,1 5,9 cos45 cosα cosα 6,8 3,11 cosα) cos45 cosα (15b)

12 436 W. Mazgaj, A. Wazecha Ach. Elect. Eng. One of the typical tansfome sheet with the Goss textue is the sheet M089-7N (poduced in Poland) with the following magnetic paametes: satuation flux density.0 T, esidual flux density and coecive foce fo the olling diection 1.4 T, 8 A/m espectively. Assuming that the textue degee is about 95%, the coecive foce h c of diection hysteeses equals 7.95 A/m, and the esidual flux densities b fo the olling and emaining diections ae 1.38 T and T espectively. Examples of the flux density changes in the given tansfome sheet M089-7N, which was magnetized in thee chaacteistic diections, ae shown in the Figue 8. Continuous lines epesent magnetization cuves calculated with taking into account the featues of the Goss textue, and dashed lines efe to the measued cuves. Continuous lines epesent magnetization cuves calculated using the model of the otational magnetization [7] with taking into account elations (11) (15), and dashed lines efe to the measued cuves. Fig. 8. Magnetization cuves of the tansfome sheet M089-7N; continuous lines calculated cuves, dashed lines measued cuves 5. Conclusions Changes of the flux density in electical steel sheets depend not only on the field intensity value but also on the textue types which can occu in these sheets. This is impotant if the diection of the field intensity is not paallel to the olling diection, especially duing the otational magnetization. The sheet textue is taken into account by an appopiate modification of the peviously defined gain distibution function. Consequently, the paametes of the diection hysteeses depend on the textue type. Howeve, it should be noted that the eo associated with the textue detemination can equal to a dozen o so pecent. Calculations of magnetization pocesses become moe complicated when the gains ae not aanged in a manne typical of the Goss textue o cubic textue, which quite often occus in some geneato sheets. In this case, the so-called Eule s angles ae helpful. These angles allow us to tansfom the co-odinates of points fom the co-odinate system which is connected

13 Vol. 6(013) Influence of electical steel sheet textues on thei magnetization cuves 437 with the olling and tansvese diections to the system which is assigned to the thee axes of cubic-shape ion cystals. It is necessay to undeline that fo the compehensive appoach to the magnetic field distibution the eddy cuents should be taken into account, because eddy cuents influence magnetic field distibution in electic steel sheets. Acknowledgments This pape was suppoted by eseach gant numbe DEC-011/01/B/ST7/04479 Modelling of nonlineaity, hysteesis, and anisotopy of magnetic coes in electomechanical convetes with otating magnetic field financed fom National Science Cente, (Poland) Refeences [1] Beckley P., Electical Steels fo Rotating Machines. Bell & Bain Ltd., Glasgow (00). [] Bozoth R.M., Feomagnetism. IEEE Pess, New Yok (1978). [3] Bailsfod F., Physical Pinciples of Magnetism. Butle & Tanne Ltd., London (1966). [4] Fioillo F., Dupé L.R., Appino C., Rietto A.M., Compehensive Model of Magnetization Cuve, Hysteesis Loops, and Losses in Any Diection in Gain-Oiented Fe-Si. IEEE Tansactions on Magnetics 38(3): (00). [5] Kelly A., Goves G.W., Cystallogaphy and cystal defects. Longman, London (1970). [6] Mazgaj W., Modelling of the otational magnetization pocess of soft feomagnetic mateials. Achives of Electical Engineeing LVII(3-4): (008). [7] Mazgaj W., Modelling of otational magnetization in anisotopic sheets. COMPEL: The Intenational Jounal fo Computation and Mathematics in Electical and Electonic Engineeing 30(3): (011). [8] Mayegoyz I.D., Mathematical Models of Hysteesis and Thei Applications, Elsevie Inc., New Yok (003). [9] Moish A.H., The Physical Pinciples of Magnetism. John Wiley and Sons, Inc., New Yok, NY (1965). [10] Pfützne H., Rotational Magnetization and Rotational Losses of Gain Oiented Silicon Steel Sheets Fundamental Aspects and Theoy. IEEE Tansactions on Magnetics 30(5): (1994).