1509:
1854:
1247:
1255:
1651:
693:
Anhysteretic magnetization can be observed experimentally, when magnetic material is demagnetized under the influence of constant magnetic field. However, measurements of anhysteretic magnetization are very sophisticated due to the fact, that the fluxmeter has to keep accuracy of integration during
2072:
1058:
1504:{\displaystyle {\begin{aligned}E(1)&={\frac {H_{\text{e}}}{a}}\cos \theta -{\frac {K_{\text{an}}}{M_{\text{s}}\mu _{0}a}}\sin ^{2}(\psi -\theta )\\E(2)&={\frac {H_{\text{e}}}{a}}\cos \theta -{\frac {K_{\text{an}}}{M_{\text{s}}\mu _{0}a}}\sin ^{2}(\psi +\theta )\end{aligned}}}
966:
1849:{\displaystyle M_{\text{an}}^{\text{aniso}}=M_{\text{s}}{\frac {\displaystyle \int _{0}^{\pi }\!e^{\frac {E(1)+E(2)}{2}}\sin \theta \cos \theta \,d\theta }{\displaystyle \int _{0}^{\pi }\!e^{\frac {E(1)+E(2)}{2}}\sin \theta \,d\theta }}}
778:
3346:
Huang, Sy-Ruen; Chen, Hong-Tai; Wu, Chueh-Cheng; et al. (2012). "Distinguishing internal winding faults from inrush currents in power transformers using Jiles–Atherton model parameters based on correlation voefficient".
3474:
Szewczyk, R.; Salach, J.; Bienkowski, A.; et al. (2012). "Application of extended Jiles–Atherton model for modeling the magnetic characteristics of Fe41.5Co41.5Nb3Cu1B13 alloy in as-quenched and nanocrystalline State".
1913:
1242:{\displaystyle M_{\text{an}}^{\text{aniso}}=M_{\text{s}}{\frac {\displaystyle \int _{0}^{\pi }\!e^{E(1)+E(2)}\sin \theta \cos \theta \,d\theta }{\displaystyle \int _{0}^{\pi }\!e^{E(1)+E(2)}\sin \theta \,d\theta }}}
865:
1260:
2452:
2377:
1889:
1643:
1579:
1050:
1006:
1611:
818:
672:
39:
and D. L. Atherton. This is one of the most popular models of magnetic hysteresis. Its main advantage is the fact that this model enables connection with physical parameters of the
3021:"Extension of the Jiles–Atherton model for modelling the frequency dependence of magnetic characteristics of amorphous alloy cores for inductive components of electronic devices"
2257:
1547:
694:
the demagnetization process. As a result, experimental verification of the model of anhysteretic magnetization is possible only for materials with negligible hysteresis loop.
498:
200:
857:
623:
387:
227:
130:
2170:
2141:
2287:
2095:
323:
150:
527:
2500:
2412:
702:
2200:
2115:
576:
552:
445:
416:
351:
278:
250:
170:
99:
79:
2301:
Vectorized Jiles–Atherton model is constructed as the superposition of three scalar models one for each principal axis. This model is especially suitable for
3439:
Szewczyk, R.; Bienkowski, A. (2004). "Application of the energy-based model for the magnetoelastic properties of amorphous alloys for sensor applications".
2067:{\displaystyle {\frac {dM}{dH}}={\frac {1}{1+c}}{\frac {M_{\text{an}}-M}{\delta k-\alpha (M_{\text{an}}-M)}}+{\frac {c}{1+c}}{\frac {dM_{\text{an}}}{dH}}}
696:
Anhysteretic magnetization of typical magnetic material can be calculated as a weighted sum of isotropic and anisotropic anhysteretic magnetization:
1513:
It should be highlighted, that a typing mistake occurred in the original Ramesh et al. publication. As a result, for an isotropic material (where
2823:"Application of the anisotropic extension of the theory of hysteresis to the magnetization curves of crystalline and textured magnetic materials"
2964:
2514:
Since its introduction in 1984, Jiles–Atherton model was intensively developed. As a result, this model may be applied for the modeling of:
3551:
Moldovanu, B.O.; Moldovanu, C.; Moldovanu, A. (1996). "Computer simulation of the transient behaviour of a fluxgate magnetometric circuit".
3139:"Application of extended Jiles–Atherton model for modelling the influence of stresses on magnetic characteristics of the construction steel"
2455:
961:{\displaystyle M_{\text{an}}^{\text{iso}}=M_{\text{s}}\left(\coth \left({\frac {H_{\text{e}}}{a}}\right)-{\frac {a}{H_{\text{e}}}}\right)}
43:. Jiles–Atherton model enables calculation of minor and major hysteresis loops. The original Jiles–Atherton model is suitable only for
1613:
given by the
Langevin equation. Physical analysis leads to the conclusion that the equation for anisotropic anhysteretic magnetization
1020:
function. For this reason, integration has to be made numerically. In the original publication, anisotropic anhysteretic magnetization
3102:
Szewczyk, R.; Bienkowski, A. (2003). "Magnetoelastic
Villari effect in high-permeability Mn-Zn ferrites and modeling of this effect".
81:
of the magnetic material sample in Jiles–Atherton model is calculated in the following steps for each value of the magnetizing field
457:
Extension considering uniaxial anisotropy introduced by Ramesh et al. and corrected by
Szewczyk requires additional parameters:
2470:
2382:
2326:
1904:
253:
2572:
3641:
2592:
3178:
Szewczyk, R. (2006). "Modelling of the magnetic and magnetostrictive properties of high permeability Mn-Zn ferrites".
3301:"Dynamic Jiles–Atherton Model for Determining the Magnetic Power Loss at High Frequency in Permanent Magnet Machines"
2597:
47:. However, an extension of this model presented by Ramesh et al. and corrected by Szewczyk enables the modeling of
2425:
2350:
1862:
1616:
1552:
1023:
979:
2982:"Modelling the effects of eddy current losses on frequency dependent hysteresis in electrically conducting media"
1584:
791:
3256:
Szewczyk, R. (2007). "Extension of the model of the magnetic characteristics of anisotropic metallic glasses".
2761:"Validation of the anhysteretic magnetization model for soft magnetic materials with perpendicular anisotropy"
635:
2503:
3403:
1017:
1009:
825:
821:
2419:
2344:
2340:
The two most important computational problems connected with the Jiles–Atherton model were identified:
2302:
2711:
Ramesh, A.; Jiles, D. C.; Roderick, J. M. (1996). "A model of anisotropic anhysteretic magnetization".
2208:
1516:
3556:
3523:
3484:
3448:
3395:
3312:
3265:
3230:
3187:
3150:
3111:
3076:
3032:
2993:
2912:
2877:
2834:
2772:
2720:
2678:
2643:
3408:
2580:
476:
178:
32:
835:
601:
365:
205:
108:
3512:"Extended Jiles–Atherton model for modelling the magnetic characteristics of isotropic materials"
3421:
3364:
3328:
3281:
3203:
2944:
2850:
2522:
44:
2146:
2120:
2960:
2800:
2290:
2265:
829:
24:
2506:
are recommended. It was observed, that the best performing was 4-th order fixed step method.
2080:
308:
135:
3595:
3564:
3531:
3492:
3456:
3413:
3356:
3320:
3273:
3238:
3195:
3158:
3119:
3084:
3040:
3001:
2952:
2920:
2885:
2842:
2790:
2780:
2728:
2686:
2651:
2528:
20:
773:{\displaystyle M_{\text{an}}=(1-t)M_{\text{an}}^{\text{iso}}+tM_{\text{an}}^{\text{aniso}}}
512:
2903:
Szymanski, Grzegorz; Waszak, Michal (2004). "Vectorized Jiles–Atherton hysteresis model".
2476:
2388:
1892:
682:
678:
626:
3560:
3527:
3488:
3452:
3399:
3316:
3269:
3234:
3191:
3154:
3115:
3080:
3036:
2997:
2916:
2881:
2838:
2776:
2724:
2682:
2669:
Liorzou, F.; Phelps, B.; Atherton, D. L. (2000). "Macroscopic models of magnetization".
2647:
2795:
2760:
2185:
2100:
1013:
561:
537:
430:
401:
336:
263:
235:
155:
84:
64:
40:
3300:
3277:
3123:
3635:
3568:
3383:
3285:
2889:
59:
3584:"Computer simulation of the transient behaviour of a fluxgate magnetometric circuit"
3425:
3368:
3332:
3207:
2854:
3618:
2945:"Computational problems connected with Jiles–Atherton model of magnetic hysteresis"
2180:
2951:. Advances in Intelligent Systems and Computing. Vol. 267. pp. 275–283.
423:
Quantifies average energy required to break pinning site in the magnetic material
2956:
2924:
2818:
2576:
2334:
2330:
2322:
48:
36:
3460:
3360:
3163:
3138:
3045:
3020:
3622:
3536:
3511:
3496:
3324:
3221:
Deane, J.H.B. (1994). "Modeling the dynamics of nonlinear inductor circuits".
3199:
2459:
832:
connecting isotropic anhysteretic magnetization with effective magnetic field
257:
677:
This effective magnetic field is analogous to the Weiss mean field acting on
2868:
Jiles, D. C.; Atherton, D.L. (1986). "A model of ferromagnetic hysteresis".
1859:
In the corrected form, the model for anisotropic anhysteretic magnetization
3065:"Coupled magnetoelastic theory of magnetic and magnetostrictive hysteresis"
2804:
2634:
Jiles, D. C.; Atherton, D.L. (1984). "Theory of ferromagnetic hysteresis".
3386:(2008). "Energy-based hysteresis model for magnetostrictive transducers".
3600:
3583:
2518:
frequency dependence of magnetic hysteresis loop in conductive materials
1903:
In Jiles–Atherton model, M(H) dependence is given in form of following
3417:
3242:
3088:
3005:
2846:
2785:
2732:
2690:
3626:
3621:- open-source software for implementation of Jiles–Atherton model in
2655:
2542:
to consider changes of average energy required to break pinning site
2318:
2314:
3064:
2981:
2822:
2539:
to avoid unphysical states when reversible permeability is negative
629:
within the material may be calculated from the following equation:
3137:
Jackiewicz, D.; Szewczyk, R.; Salach, J.; Bieńkowski, A. (2014).
2949:
Recent
Advances in Automation, Robotics and Measuring Techniques
1581:
is not consistent with the isotropic anhysteretic magnetization
1549:), the presented form of anisotropic anhysteretic magnetization
2535:
Moreover, different corrections were implemented, especially:
289:
Original Jiles–Atherton model considers following parameters:
582:
Participation of anisotropic phase in the magnetic material
2575:, especially for models of inductive components, such as
358:
Quantifies domain walls density in the magnetic material
329:
Quantifies interdomain coupling in the magnetic material
2313:
The Jiles–Atherton model is implemented in JAmodel, a
2479:
2428:
2391:
2353:
2268:
2211:
2188:
2149:
2123:
2103:
2097:
depends on direction of changes of magnetizing field
2083:
1916:
1865:
1769:
1685:
1654:
1619:
1587:
1555:
1519:
1258:
1169:
1092:
1061:
1026:
982:
868:
838:
794:
705:
638:
604:
564:
540:
515:
479:
433:
404:
368:
339:
311:
266:
238:
208:
181:
158:
138:
111:
87:
67:
2551:Jiles–Atherton model may be applied for modeling:
2494:
2446:
2406:
2371:
2281:
2251:
2194:
2164:
2135:
2109:
2089:
2066:
1883:
1848:
1637:
1605:
1573:
1541:
1503:
1241:
1044:
1000:
960:
851:
812:
772:
666:
617:
570:
546:
521:
492:
439:
410:
381:
345:
317:
272:
244:
221:
194:
164:
144:
124:
93:
73:
1785:
1701:
1185:
1108:
1899:Magnetization as a function of magnetizing field
2938:
2936:
2934:
2176:Flux density as a function of magnetizing field
824:. In the case of isotropic magnetic materials,
132:is calculated considering interdomain coupling
2754:
2752:
2750:
2748:
2746:
2744:
2742:
2422:of the anisotropic anhysteretic magnetization
2347:of the anisotropic anhysteretic magnetization
3058:
3056:
1891:was confirmed experimentally for anisotropic
534:Angle between direction of magnetizing field
8:
2706:
2704:
2702:
2700:
2447:{\displaystyle M_{\text{an}}^{\text{aniso}}}
2372:{\displaystyle M_{\text{an}}^{\text{aniso}}}
1884:{\displaystyle M_{\text{an}}^{\text{aniso}}}
1638:{\displaystyle M_{\text{an}}^{\text{aniso}}}
1574:{\displaystyle M_{\text{an}}^{\text{aniso}}}
1045:{\displaystyle M_{\text{an}}^{\text{aniso}}}
1001:{\displaystyle M_{\text{an}}^{\text{aniso}}}
3553:Journal of Magnetism and Magnetic Materials
3441:Journal of Magnetism and Magnetic Materials
3104:Journal of Magnetism and Magnetic Materials
2870:Journal of Magnetism and Magnetic Materials
2629:
2627:
2625:
2623:
2621:
2619:
2617:
2615:
2613:
1645:has to be corrected to the following form:
202:is calculated for effective magnetic field
1606:{\displaystyle M_{\text{an}}^{\text{iso}}}
813:{\displaystyle M_{\text{an}}^{\text{iso}}}
3599:
3588:Serbian Journal of Electrical Engineering
3535:
3407:
3162:
3044:
2794:
2784:
2567:magnetic field sensors (e. g. fluxgates)
2478:
2438:
2433:
2427:
2390:
2363:
2358:
2352:
2273:
2267:
2231:
2210:
2187:
2148:
2122:
2102:
2082:
2047:
2037:
2019:
1998:
1965:
1958:
1940:
1917:
1915:
1875:
1870:
1864:
1836:
1790:
1779:
1774:
1761:
1706:
1695:
1690:
1683:
1677:
1664:
1659:
1653:
1629:
1624:
1618:
1597:
1592:
1586:
1565:
1560:
1554:
1524:
1518:
1473:
1457:
1447:
1436:
1430:
1407:
1401:
1354:
1338:
1328:
1317:
1311:
1288:
1282:
1259:
1257:
1229:
1190:
1179:
1174:
1161:
1113:
1102:
1097:
1090:
1084:
1071:
1066:
1060:
1036:
1031:
1025:
992:
987:
981:
945:
936:
918:
912:
891:
878:
873:
867:
843:
837:
804:
799:
793:
764:
759:
743:
738:
710:
704:
643:
637:
609:
603:
563:
539:
514:
484:
478:
432:
403:
373:
367:
338:
310:
265:
237:
213:
207:
186:
180:
157:
137:
116:
110:
86:
66:
2821:; Ramesh, A.; Shi, Y.; Fang, X. (1997).
459:
291:
2609:
1012:. However, in such a case, there is no
976:Anisotropic anhysteretic magnetization
667:{\displaystyle H_{\text{e}}=H+\alpha M}
589:Modelling the magnetic hysteresis loops
252:of the sample is calculated by solving
3619:Jiles–Atherton model for Octave/MATLAB
554:and direction of anisotropy easy axis
3299:Du, Ruoyang; Robertson, Paul (2015).
3258:Journal of Physics D: Applied Physics
788:Isotropic anhysteretic magnetization
394:Saturation magnetization of material
7:
280:(which is the source of hysteresis).
3019:Szewczyk, R.; Frydrych, P. (2010).
3063:Sablik, M.J.; Jiles, D.C. (1993).
2462:this quadrature is implemented as
1008:is also determined on the base of
505:Average anisotropy energy density
14:
2252:{\displaystyle B(H)=\mu _{0}M(H)}
2456:Gauss–Kronrod quadrature formula
1542:{\displaystyle K_{\text{an}}=0)}
2327:ordinary differential equations
2297:Vectorized Jiles–Atherton model
3477:IEEE Transactions on Magnetics
3388:IEEE Transactions on Magnetics
3349:IEEE Transactions on Magnetics
3305:IEEE Transactions on Magnetics
3223:IEEE Transactions on Magnetics
3069:IEEE Transactions on Magnetics
2986:IEEE Transactions on Magnetics
2827:IEEE Transactions on Magnetics
2713:IEEE Transactions on Magnetics
2671:IEEE Transactions on Magnetics
2489:
2483:
2471:ordinary differential equation
2401:
2395:
2383:ordinary differential equation
2246:
2240:
2221:
2215:
2010:
1991:
1905:ordinary differential equation
1817:
1811:
1802:
1796:
1733:
1727:
1718:
1712:
1536:
1494:
1482:
1391:
1385:
1375:
1363:
1272:
1266:
1215:
1209:
1200:
1194:
1138:
1132:
1123:
1117:
731:
719:
254:ordinary differential equation
1:
3124:10.1016/S0304-8853(02)00784-9
2573:electronic circuit simulation
2525:on magnetic hysteresis loops
2202:in the material is given as:
820:is determined on the base of
493:{\displaystyle K_{\text{an}}}
195:{\displaystyle M_{\text{an}}}
3569:10.1016/0304-8853(95)01101-3
3382:Calkins, F.T.; Smith, R.C.;
2957:10.1007/978-3-319-05353-0_27
2890:10.1016/0304-8853(86)90066-1
2593:Preisach model of hysteresis
852:{\displaystyle H_{\text{e}}}
618:{\displaystyle H_{\text{e}}}
451:Magnetization reversibility
382:{\displaystyle M_{\text{s}}}
256:taking into account sign of
222:{\displaystyle H_{\text{e}}}
125:{\displaystyle H_{\text{e}}}
16:Model of magnetic hysteresis
3278:10.1088/0022-3727/40/14/002
2925:10.1016/j.physb.2003.08.048
2571:It is also widely used for
2561:magnetostrictive actuators
2555:rotating electric machines
2531:of soft magnetic materials
175:anhysteretic magnetization
3658:
3461:10.1016/j.jmmm.2003.11.270
3361:10.1109/TPWRD.2011.2181543
3164:10.12693/aphyspola.126.392
3046:10.12693/aphyspola.118.782
2636:Journal of Applied Physics
2165:{\displaystyle \delta =-1}
689:Anhysteretic magnetization
35:was introduced in 1984 by
3537:10.12693/APhysPolA.113.67
3497:10.1109/TMAG.2011.2173562
3325:10.1109/TMAG.2014.2382594
3200:10.1007/s12043-006-0031-z
2136:{\displaystyle \delta =1}
598:Effective magnetic field
105:effective magnetic field
2309:Numerical implementation
2282:{\displaystyle \mu _{0}}
594:Effective magnetic field
3516:Acta Physica Polonica A
3143:Acta Physica Polonica A
3025:Acta Physica Polonica A
2564:magnetoelastic sensors
2090:{\displaystyle \delta }
318:{\displaystyle \alpha }
145:{\displaystyle \alpha }
2598:Stoner–Wohlfarth model
2496:
2448:
2408:
2373:
2325:algorithm for solving
2283:
2253:
2196:
2172:for decreasing field)
2166:
2143:for increasing field,
2137:
2111:
2091:
2068:
1885:
1850:
1639:
1607:
1575:
1543:
1505:
1243:
1046:
1018:Boltzmann distribution
1010:Boltzmann distribution
1002:
962:
853:
826:Boltzmann distribution
822:Boltzmann distribution
814:
774:
668:
619:
572:
548:
523:
494:
441:
412:
383:
347:
319:
274:
246:
223:
196:
166:
146:
126:
95:
75:
3510:Szewczyk, R. (2008).
2943:Szewczyk, R. (2014).
2759:Szewczyk, R. (2014).
2497:
2449:
2420:numerical integration
2409:
2374:
2345:numerical integration
2321:toolbox. It uses the
2303:finite element method
2284:
2254:
2197:
2167:
2138:
2112:
2092:
2069:
1886:
1851:
1640:
1608:
1576:
1544:
1506:
1244:
1047:
1003:
963:
854:
815:
775:
669:
620:
573:
549:
524:
522:{\displaystyle \psi }
495:
442:
413:
384:
348:
320:
275:
260:of magnetizing field
247:
224:
197:
167:
147:
127:
96:
76:
3601:10.2298/sjee0801021c
3582:Cundeva, S. (2008).
3555:. 157–158: 565–566.
2980:Jiles, D.C. (1994).
2495:{\displaystyle M(H)}
2477:
2426:
2407:{\displaystyle M(H)}
2389:
2351:
2266:
2209:
2186:
2147:
2121:
2101:
2081:
1914:
1863:
1652:
1617:
1585:
1553:
1517:
1256:
1059:
1024:
980:
866:
836:
792:
703:
636:
602:
562:
538:
513:
477:
431:
402:
366:
337:
309:
264:
236:
206:
179:
156:
136:
109:
85:
65:
51:magnetic materials.
29:Jiles–Atherton model
3642:Magnetic hysteresis
3561:1996JMMM..157..565M
3528:2008JMMM..320E1049S
3489:2012ITM....48.1389S
3453:2004JMMM..272..728S
3400:2000ITM....36..429C
3317:2015ITM....5182594D
3270:2007JPhD...40.4109S
3235:1994ITM....30.2795D
3192:2006Prama..67.1165S
3155:2014AcPPA.126..392J
3116:2003JMMM..254..284S
3081:1993ITM....29.2113S
3037:2010AcPPA.118..782S
2998:1994ITM....30.4326J
2917:2004PhyB..343...26S
2882:1986JMMM...61...48J
2839:1997ITM....33.3961J
2777:2014Mate....7.5109S
2725:1996ITM....32.4234R
2683:2000ITM....36..418L
2648:1984JAP....55.2115J
2558:power transformers
2510:Further development
2504:Runge–Kutta methods
2458:has to be used. In
2443:
2368:
1880:
1784:
1700:
1669:
1634:
1602:
1570:
1184:
1107:
1076:
1041:
997:
883:
809:
769:
748:
45:isotropic materials
33:magnetic hysteresis
2492:
2444:
2429:
2404:
2369:
2354:
2279:
2249:
2192:
2162:
2133:
2107:
2087:
2064:
1881:
1866:
1846:
1843:
1770:
1768:
1686:
1655:
1635:
1620:
1603:
1588:
1571:
1556:
1539:
1501:
1499:
1239:
1236:
1170:
1168:
1093:
1062:
1042:
1027:
998:
983:
958:
869:
849:
828:can be reduced to
810:
795:
770:
755:
734:
664:
615:
568:
544:
519:
490:
437:
408:
379:
343:
315:
270:
242:
219:
192:
162:
152:and magnetization
142:
122:
91:
71:
3418:10.1109/20.825804
3264:(14): 4109–4113.
3243:10.1109/20.312521
3089:10.1109/20.221036
3006:10.1109/20.334076
2966:978-3-319-05352-3
2847:10.1109/20.619629
2786:10.3390/ma7075109
2733:10.1109/20.539344
2691:10.1109/20.825802
2441:
2436:
2366:
2361:
2291:magnetic constant
2195:{\displaystyle B}
2110:{\displaystyle H}
2062:
2050:
2035:
2014:
2001:
1968:
1956:
1935:
1878:
1873:
1844:
1824:
1740:
1680:
1667:
1662:
1632:
1627:
1600:
1595:
1568:
1563:
1527:
1467:
1450:
1439:
1416:
1410:
1348:
1331:
1320:
1297:
1291:
1237:
1087:
1074:
1069:
1039:
1034:
995:
990:
951:
948:
927:
921:
894:
881:
876:
846:
830:Langevin function
807:
802:
767:
762:
746:
741:
713:
646:
612:
586:
585:
571:{\displaystyle t}
547:{\displaystyle H}
487:
455:
454:
440:{\displaystyle c}
411:{\displaystyle k}
376:
346:{\displaystyle a}
273:{\displaystyle H}
245:{\displaystyle M}
216:
189:
165:{\displaystyle M}
119:
94:{\displaystyle H}
74:{\displaystyle M}
41:magnetic material
25:materials science
3649:
3606:
3605:
3603:
3579:
3573:
3572:
3548:
3542:
3541:
3539:
3507:
3501:
3500:
3471:
3465:
3464:
3436:
3430:
3429:
3411:
3379:
3373:
3372:
3343:
3337:
3336:
3296:
3290:
3289:
3253:
3247:
3246:
3229:(5): 2795–2801.
3218:
3212:
3211:
3186:(6): 1165–1171.
3175:
3169:
3168:
3166:
3134:
3128:
3127:
3099:
3093:
3092:
3060:
3051:
3050:
3048:
3016:
3010:
3009:
2992:(6): 4326–4328.
2977:
2971:
2970:
2940:
2929:
2928:
2900:
2894:
2893:
2865:
2859:
2858:
2815:
2809:
2808:
2798:
2788:
2771:(7): 5109–5116.
2756:
2737:
2736:
2708:
2695:
2694:
2666:
2660:
2659:
2656:10.1063/1.333582
2631:
2529:magnetostriction
2502:dependence, the
2501:
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2013:
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2002:
1999:
1977:
1970:
1969:
1966:
1959:
1957:
1955:
1941:
1936:
1934:
1926:
1918:
1893:amorphous alloys
1890:
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1887:
1882:
1879:
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1855:
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928:
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768:
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742:
739:
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711:
679:magnetic moments
673:
671:
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665:
648:
647:
644:
627:magnetic moments
624:
622:
621:
616:
614:
613:
610:
577:
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569:
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100:
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80:
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72:
21:electromagnetism
3657:
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2698:
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2663:
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2512:
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2349:
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2269:
2264:
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2207:
2206:
2184:
2183:
2178:
2145:
2144:
2119:
2118:
2099:
2098:
2079:
2078:
2054:
2043:
2039:
2024:
1994:
1978:
1961:
1960:
1945:
1927:
1919:
1912:
1911:
1901:
1861:
1860:
1792:
1786:
1708:
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1673:
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1057:
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1022:
1021:
978:
977:
974:
941:
914:
908:
901:
897:
887:
864:
863:
839:
834:
833:
790:
789:
786:
706:
701:
700:
695:
691:
683:magnetic domain
639:
634:
633:
625:influencing on
605:
600:
599:
596:
591:
560:
559:
536:
535:
511:
510:
480:
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474:
429:
428:
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369:
364:
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335:
334:
307:
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287:
262:
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234:
233:
209:
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182:
177:
176:
154:
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134:
133:
112:
107:
106:
83:
82:
63:
62:
57:
17:
12:
11:
5:
3655:
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3630:
3629:
3614:
3613:External links
3611:
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3502:
3466:
3431:
3409:10.1.1.44.9747
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3291:
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3011:
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2911:(1–4): 26–29.
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2305:computations.
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1014:antiderivative
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956:
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935:
931:
926:
917:
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1479:
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1470:
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1454:
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1433:
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1418:
1413:
1404:
1398:
1396:
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1360:
1355:
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1344:
1339:
1335:
1325:
1314:
1308:
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1175:
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1165:
1162:
1158:
1155:
1152:
1149:
1146:
1143:
1135:
1129:
1126:
1120:
1114:
1110:
1103:
1098:
1094:
1081:
1077:
1063:
1055:
1054:
1053:
1052:is given as:
1028:
1019:
1015:
1011:
984:
971:
954:
942:
938:
933:
929:
924:
915:
909:
905:
902:
898:
888:
884:
870:
862:
861:
860:
840:
831:
827:
823:
796:
783:
756:
752:
749:
735:
728:
725:
722:
716:
707:
699:
698:
697:
688:
686:
684:
680:
661:
658:
655:
652:
649:
640:
632:
631:
630:
628:
606:
593:
588:
581:
579:
565:
558:
557:
541:
533:
530:
516:
509:
508:
504:
501:
481:
473:
472:
468:
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462:
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458:
450:
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434:
427:
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405:
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370:
362:
361:
357:
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340:
333:
332:
328:
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312:
305:
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300:
297:
294:
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290:
284:
267:
259:
255:
239:
231:
210:
183:
174:
159:
139:
113:
104:
103:
102:
88:
68:
61:
60:Magnetization
54:
52:
50:
46:
42:
38:
34:
30:
26:
22:
3594:(1): 21–30.
3591:
3587:
3577:
3552:
3546:
3519:
3515:
3505:
3480:
3476:
3469:
3444:
3440:
3434:
3391:
3387:
3384:Flatau, A.B.
3377:
3352:
3348:
3341:
3308:
3304:
3294:
3261:
3257:
3251:
3226:
3222:
3216:
3183:
3179:
3173:
3146:
3142:
3132:
3107:
3103:
3097:
3072:
3068:
3028:
3024:
3014:
2989:
2985:
2975:
2948:
2908:
2904:
2898:
2873:
2869:
2863:
2830:
2826:
2813:
2768:
2764:
2716:
2712:
2674:
2670:
2664:
2639:
2635:
2577:transformers
2570:
2550:
2547:Applications
2534:
2513:
2469:For solving
2468:
2463:
2417:
2381:solving the
2339:
2312:
2300:
2261:
2181:Flux density
2179:
2076:
1902:
1858:
1512:
1251:
975:
787:
692:
676:
597:
469:Description
456:
301:Description
288:
58:
28:
18:
3483:(4): 1389.
3447:: 728–730.
3110:: 284–286.
3075:(4): 2113.
2876:(1–2): 48.
2833:(5): 3961.
2819:Jiles, D.C.
2719:(5): 4234.
2642:(6): 2115.
2414:dependence.
2335:MIT license
2331:open-source
2323:Runge-Kutta
972:Anisotropic
49:anisotropic
37:David Jiles
3623:GNU Octave
3394:(2): 429.
3355:(2): 548.
3149:(1): 392.
3031:(5): 782.
2677:(2): 418.
2604:References
2466:function.
2460:GNU Octave
285:Parameters
258:derivative
55:Principles
3522:(1): 67.
3404:CiteSeerX
3286:121390902
2905:Physica B
2765:Materials
2333:is under
2271:μ
2229:μ
2157:−
2151:δ
2125:δ
2085:δ
2005:−
1989:α
1986:−
1980:δ
1972:−
1841:θ
1834:θ
1831:
1781:π
1772:∫
1766:θ
1759:θ
1756:
1750:θ
1747:
1697:π
1688:∫
1492:θ
1486:ψ
1480:
1455:μ
1428:−
1425:θ
1422:
1373:θ
1370:−
1367:ψ
1361:
1336:μ
1309:−
1306:θ
1303:
1234:θ
1227:θ
1224:
1181:π
1172:∫
1166:θ
1159:θ
1156:
1150:θ
1147:
1104:π
1095:∫
934:−
906:
784:Isotropic
726:−
681:within a
659:α
517:ψ
463:Parameter
313:α
295:Parameter
140:α
3636:Category
3426:16468218
3369:25854265
3333:30752050
3208:59468247
2855:38583653
2805:28788121
2587:See also
2523:stresses
2464:quadgk()
1016:for the
3557:Bibcode
3524:Bibcode
3485:Bibcode
3449:Bibcode
3396:Bibcode
3313:Bibcode
3266:Bibcode
3231:Bibcode
3188:Bibcode
3180:Pramana
3151:Bibcode
3112:Bibcode
3077:Bibcode
3033:Bibcode
2994:Bibcode
2913:Bibcode
2878:Bibcode
2835:Bibcode
2796:5455830
2773:Bibcode
2721:Bibcode
2679:Bibcode
2644:Bibcode
3627:Matlab
3424:
3406:
3367:
3331:
3284:
3206:
2963:
2853:
2803:
2793:
2581:chokes
2319:OCTAVE
2315:MATLAB
2262:where
2077:where
1252:where
27:, the
3422:S2CID
3365:S2CID
3329:S2CID
3282:S2CID
3204:S2CID
2851:S2CID
2440:aniso
2365:aniso
1877:aniso
1666:aniso
1631:aniso
1567:aniso
1073:aniso
1038:aniso
994:aniso
766:aniso
466:Units
298:Units
3625:and
2961:ISBN
2801:PMID
2473:for
2454:the
2418:For
2385:for
903:coth
23:and
3596:doi
3565:doi
3532:doi
3520:113
3493:doi
3457:doi
3445:272
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3321:doi
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3239:doi
3196:doi
3159:doi
3147:126
3120:doi
3108:254
3085:doi
3041:doi
3029:118
3002:doi
2953:doi
2921:doi
2909:343
2886:doi
2843:doi
2791:PMC
2781:doi
2729:doi
2687:doi
2652:doi
2579:or
2289:is
1828:sin
1753:cos
1744:sin
1599:iso
1471:sin
1419:cos
1352:sin
1300:cos
1221:sin
1153:cos
1144:sin
880:iso
806:iso
745:iso
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502:J/m
420:A/m
391:A/m
355:A/m
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