Hysteresis model for magnetic materials using the Jiles-Atherton model
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1 Hysteresis odel for agnetic aterials using the Jiles-Atherton odel Predrag Petrovic Technical faculty Svetog Save Cacak, Nebojsa itrovic Technical faculty Svetog Save Cacak, ilorad Stevovic Technical faculty Svetog Save Cacak, Predrag Pejovic Electrotechnical faculty Bulevar Revolucije Belgrade, Abstract This paper presents a hysteresis odel of agnetic aterials, which is based on the theory of ferroagnetic hysteresis. Results obtained by nuerical siulation of hysteresis curves of the ferroagnetic, hard agnetic core d isotropic aterials have been given. The atheatical odel consists of only nine paraeters. The hysteresis curve obtained is in good agreeent with the curve defined for these aterials in the PSpice progra packet. The proposed ethod for paraeter deterination enables siulation of y agnetic aterial, which has a characteristic R, Z or F-shaped hysteresis curve. 1. Introduction The PSpice progra packet for the siulation of electronic circuits is widely used in the design of electronic devices. The Jiles-Atherton odel [1] for defining the hysteresis curves of ferroagnetic aterials is built into this packet. However, ost users of this progra packet are not failiar with odeling hysteresis curves d they have probles when defining paraeters of the odel. A description of possibilities for the alysis of coponents with a ferroagnetic core is included in the instruction ual [2], but it only gives recoendations to the user on how to deterine the five paraeters related to agnetic properties of the aterial: The PSpice progra packet includes coplete data on only one agnetic soft aterial: the n-zn ferrite-3c8 (with the so-called R-round shaped hysteresis curve d for a series of coon core shapes. Stdard catalogues of agnetic aterials contain data on only 5 paraeters (we need 9. The reaining four paraeters have to be additionally deterined. Three paraeters available in catalogues c be used: r - residual agnetization, H C - coercive field d χ i - starting agnetic susceptibility. It c be concluded that the procedure for deterining these paraeters is not singular, as a syste of three equations with four unknowns need to be solved. The result obtained exclusively depends on the user skill in attaining satisfactory agreeent with the real hysteresis curve using ultiple paraeter fitting. The error that c be ade using this ethod c seriously influence validity of the PSpice alysis, especially when alyzing hard agnetic aterials. A satisfactory solution of this proble requires knowledge of the Jiles-Atherton odel, which itself deds isotropy of the alyzed aterial d R- shaped hysteresis curve. However, agnetic aterials with Z or F-shaped hysteresis curves are often used, which requires certain odification of the odel presented here [3]. Input data for the n-zn ferrite-3c8 has been used for evaluation of the nuerical siulation developed here [4], while agnetic characteristics obtained have been copared with corresponding characteristics read fro the hysteresis curve of this aterial obtained using paraeters given in the PSpice progra packet. 2. Theoretical odel The basis of the theoretical alysis of ferroagnetic aterials is founded on the fact that the effective agnetic field (H C in each individual doain is obtained as the su of the external agnetic field (H d the influence of surrounding agnetic doains: He = H + α (1 Where: - agnetization of the aterial d α ALPHA. The hysteresis curve represents the dependence of agnetization on the agnetic field in the case when the pinning effect of the doain wall is disregarded; i.e. it represents the iniu doain energy. It c be deterined using Lengevins function
2 for copletely isotropic aterials L(H C [5] in the for of: He a ( H = s L( He = s coth a He (2 Where a A. However, in a real aterial, the oveent of doain walls is prevented (grain boundaries, grain inhoogeneneities, dislocations, non-agnetic inclusions, regions of inhoogeneous strain, which is defined as the pinning effect of a doain wall. This effect is active until the agnetic potential becoes high enough to cause wall oveent, until the next pinning center is fored. oveent of the doain wall c be reversible d irreversible. Irreversible oveent of the doain wall is described by the following equation: dirr irr = (3 dh δ k α ( irr μo Where: irr - irreversible agnetization coponent, k - a coefficient which depends on the energy required for doain wall oveent, μ - agnetic pereability of vacuu, δ= ±1 (+ denotes the direction of the increase of the agnetic field d - denotes the decrease of the agnetic field d finally, paraeter K is defined as: K k/μ. If bending of the doain wall is assued, the reversible coponent of agnetization rev c be described by the equation: ( = c (4 rev irr Where c C. The total agnetization is obtained as the su of: = + (5 irr rev So, using eqs. (3, (4 d (5 the expression for differential agnetic susceptibility χ is obtained as: d ( 1 c χ = = + c d irr (6 dh K δ α ( irr dh Variables which c be read fro the hysteresis curve d starting agnetization curve ( r, H C,, H, χ i... do not appear in the given equations, so further odel developent is required for their solution. The atheatical odel for hard agnetic devices akes use of the doain coupling paraeter α of the theory of ferroagnetic hysteresis. The strong coupling of the doain walls is considered as the ain reason of the hysteresis of hard agnetic aterials. Keeping the odel as siple as possible eq. (4 is neglected. Therefore, (1 is replaced by other hysteresis using power functions. Furtherore, the feedback loop in the differential equation is drooped too. The atheatical odel consist only of one nonlinear algebraic d one linear ordinary differential equation. H / s + α = + f ( / s (7 Kc s Rμ where g ( / s, f ( / s = g( / s < (8 g ( H β H β1 / $ $ s = + 1 s s d if sign dh, = 1d dt d = if sign dh (9, = 1d μ dh dt sign dh, elsewere K dt The paraeters of the odel are K c, α, s, R μ, H, H 1, β, β 1, d K. H, β are obtained by a curve fit of the second d H 1, β 1, of the first quadrt. The additional paraeter K c is redundt. It was included in the odel to ake the paraeter deterination easier. The existing isotropic odel equations of hysteresis in agnetic aterials have been extended to include the effects of isotropy d texture. This has proved particularly iportt as the odel is been used to describe increasing rge of agnetic aterials in which isotropy plays a significt role. Anisotropy d texture in polycrystalline agnetic aterials c be adequately described by odifying the equation for the hysteretic curve to account for these effects. To include isotropy effects into the odel, the isotropy energy ust be incorporated into the total energy of the oents. Following the developent of the generalized hysteretic function described previously [6]: iso = e E/ kbt all oents s E/ kbt e all oents cosθ (1 Where θ is the gle between the direction of the agnetic oent d the direction of the applied field, d E = μ ( H+ α + E iso (11 And E iso is the isotropy energy, which depends on the isotropy structure of aterial. In the case of cubic isotropy,
3 E iso = K1 cos i cos j i j θ θ (12 With the noral convention on sybols. In the present description, we only used the first isotropy coefficient, because this approxiation is in ost case sufficient to provide accurate description of the different agnetization curves along different directions. To incorporate the texture effect into the odel, we introduce texture coefficient t, which is a statistical evaluation of the fraction of the textured portion of the aterial. Th the hysteretic agnetization c be given as: = t + ( t iso iso 1 (13 Where iso is given by (1 d is the isotropic hysteretic agnetization. The general equation of hysteresis that has been described previously c now be solved with the incorporation of the isotropy d textured hysteretic agnetization given above to obtain the agnetization curves along particular directions. 3. Nuerical odel First, the equation for deterining the paraeter of reversible oveent of the doain wall - c, is defined. This paraeter c be deterined using the starting differential noral susceptibility χ in : χ in d ( c = 1 = dh H= = K δ α + c d dh (14 Where the fact that agnetization is copletely reversible for the starting agnetization curve = rev (i.e. d irr /dh is used. If eq. (2 d its derivative (for = are introduced in eq. (14, d the developent of the coth(h/a function for H is used, the following is obtained: c χ s in = a in, E.g. c = 3 χ. (15 3 a s The starting differential hysteresis susceptibility χ i defines a connection between paraeter a d α: d s χ i = lih = dh 3 (16 a α s So that: s a = 1 + α (17 3 χ i This equation is used as the connection between paraeters a d α. Differential susceptibility in the coercive point χ Hc, representing axial differential susceptibility on the whole agnetization curve, i.e. χ Hc =χ ax is used for deterining paraeter K. If eq. (6 is used to define the coercive point H = +H C d =, d it is trsfored, paraeter K is obtained in the following for: ( HC irr ( HC K = d ( HC d irr HC c dh ( χ ax dh (18 + α [ ( H ( H ] C irr C Eqs. (4 d (5 are used to deterine variables irr d d irr /dh in the coercive point as: c irr ( HC = ( HC, (19 1 c dirr ( HC 1 c d ( HC = χ ax dh 1 c 1 c dh If these expressions are introduced into eq.(18, the final expression used for the calculation of the pinning paraeter of the doain wall, K is obtained: ( HC 1 K = α + (2 1 c χ ax c d HC c ( 1 1 c dh A siilar procedure is repeated for the residual point (=+ r d H = using eqs. in the following order: (6 to define the so-called differential residual susceptibility χ r, then (4 d (5, then expressions for irr d d irr /dh are defined in this point, d the following is obtained: K r = ( r + (21 α c c d ( r χ r dh The axial excitation point ( = + d H = H c also be used. If this point is close to the saturation point, the noral d hysteresis differential susceptibility are alost the sae (χ n χ =χ d agnetization is practically equal to the irreversible value ( = irr. Differential susceptibility in the point of axial excitation c be defined using eq.(6 as: χ = K α d ( H + c dh ( H ( ( H d irr ( H dh (22
4 The approxiations ade in this point enable: dirr ( H d( H d ( H = = (23 dh dh dh And after exchging into (22 d trsforations: K χ = ( H (24. α χ The procedure used to solve the nuerical odel Having in ind that the equations used for defining paraeters of the Jiles-Atherton odel c only be iplicitly expressed as a function of the paraeter being calculated d the reaining odel paraeters, the procedure used for deterining these paraeters ust be iterative. Coparisons of values obtained in successive iterations are ade, d if there is no significt chge (for exaple: the deviation is saller th ξ=.1 the procedure is copleted, d the solutions obtained are treated as final values. As input values we use 1 paraeters (table 1. The progra was written using the atheatica environent, as it requires the nuerical solution of paraeter differential equations, which are supported by this progra packet. The paraeter of reversible oveent of the doain wall - c c be directly calculated using the starting noral susceptibility (eq. (15. The starting value of the averaging paraeter of the agnetic field is defined as α=.1, as this value is ost often found for all isotropic aterials, though other value (of the sae order of agnitude c be defined. The shape paraeter of the hysteresis curve - a c be initially calculated using eq. (17, while K is defined as. Calculation of the pinning paraeter of the doain wall K using eq. (2, followed by deterination of paraeter α using eq. (21 is perfored in a progra loop (which is exited when the difference between calculated paraeter values in two successive loops is less th ξ. Paraeter a is obtained in the sae loop using eq. (24. In this case it is necessary to ake a correct approxiation of hyperbolic functions. In this concrete case developing this function using Bernoulli nubers approxiates the coth function. This was later proved to be a correct ove, as this function easily diverges, thus disrupting the coplete calculation. By noralizing its arguent, we have retained the arguent value close to zero. The cubic equation obtained in this way was later solved using the atheatica 3.1 progra packet. At the sae tie, investigation of the nature of solutions obtained d selection of real ones were essential. The procedure for calculating paraeters K, a d α is repeated the necessary nuber of ties until the defined accuracy is attained, though the developed progra showed that results obtained converge to the final result in a sall nuber of steps (2-5. Negative values were also obtained as final paraeter values, which is noral, having in ind the nature of the hysteresis curve d the oveent direction on the curve, when the sign of coefficients used in the calculation is chged. This is why the absolute value of the calculated paraeter was taken in soe cases. This did not have y influence of further accuracy of the procedure. Fig. 1 shows the hysteresis curve for the n-zn ferrite - 3C8 obtained by introducing odel paraeters (calculated using our progra packet into the PSpice progra packet. Fig. 2 shows the hysteresis curve obtained using odel paraeters defined in the PSpice progra packet for the sae aterial. 5.K -5.K B(k1 H(k1 Fig. 1 Starting agnetization curve d hysteresis curve for the n-zn ferrite - 3C8 obtained using odel paraeter values (calculated by our progra in the PSpice packet. 5.K -5.K B(k1 H(k1 Fig. 2 Starting agnetization curve d hysteresis curve for the n-zn ferrite 3C8 obtained using odel paraeter values defined in the PSpice progra packet Good agreeent of both curves c be seen. agnetic characteristics read fro these curves are presented d copared with input values in table 1. We should p p
5 ention that the PSpice progra also gives the dependence of agnetic induction B on the agnetic field strength H, which is the case when experiental hysteresis curve is obtained. As these values do not figure in presented odel equations (1-24, their connection with odel variables is as follows: B( H ( H = H (25 μ o χ = μr 1 ; χ μr (26 Where μ r - the relative agnetic pereability (for soft agnetic aterials χ μ r. The unit for agnetic induction in the PSpice progra packet is Gauss - G (1G = 1-4 T, while Oersted - Oe (1Oe A/ is used as the agnetic field strength unit. Table 1 Input values of agnetic characteristics for the n-zn ferrite - 3C8, their values read fro the curve obtained using calculated odel paraeter (Fig. 1 d values obtained using odel paraeters defined in the PSpice progra packet input values wrote fro wrote fro Fig.2 Fig.2 B S (T χ i χ in H (A/ B (T χ B r (T χ r H c (A/ χ ax For hard agnetic core, due to the fact that one paraeter is redundt, we chose for the feedback paraeter a value α=1 without y loss of generality. R μ is obtained by a curve fit of the initial agnetization curve. The paraeters H 1, β 1 d H, β are deterined such that they approxiate the first d second quadrt, respectively. Fitting the second quadrt, initial guess for Kc, K, d H1 starts the coputation. Typical start values are K c =1...1, K=1, d H 1 =. For a atheatical treatent is hence replaced by in (7. We ded that the second ter of g(. is neglectible copared to the first in the second quadrt. Collects the linear part of (7, the curve fit is effected by selecting two points on deagnetization curve of the second quadrt, e.g. (H 1 =, 1 = r, where r is the reence agnetization d (H 2 =-H c /2, 2 =(H 2. The paraeters β, H are approxiately given by β ( c( H1 1 ( c( H 2 2 ln( 1 ln( 2 (, ln, ln, $ c H H 1 1 β 1 s H / s (, = + α $ c H H K c s R μ s If the slope of (7 becoes singular chges abruptly fro the upper to the lower brch d vice versa. The agnetization for which this singularity occurs is denoted as Hc. The field strength at Hc is therefore approxiately H c. The coercivity is nearly given by α 1 / R μ ( β H / s 1 c β H$ (28 H K H β H c H c $ c c ( α 1 / R μ s s After choosing a better guess for K c fro (28, the procedure is repeated aking use of (27. K is deterined by approxiating the slope of the agnetization curve near the corcivety aking use of (9. Now β 1 d H 1 are coputed by fitting the agnetization curve in the first quadrt. The paraeters ust be chosen such that they do not disturb the curve fit of the second quadrt, e.g., the second ter of g( / s in (7 ust be negligible for < r. We require that for (H 1 =, 1 = r β β (27 H$ 1 r 1 s H$ = k << 1 (29 β r s e.g., k=1-2. The paraeters β 1, H 1 are calculated by the sae reasoning as before by (27 replacing β by β 1 d H by H 1. The extended odel has been used to fit easured curves of Nd 2 Fe 14 B aterials, which have uniaxial isotropy. The odeled d easured curves are shown in Fig.3 d 4. The odeled curves show good agreeent with the experiental results. For the isotropic saple, the field was applied along its hard axis. The odeled texture level is t=.32, which es statistically 32 per cent of the saple was oriented along a specific direction d the rest was rdoly oriented.
6 Figure 3: odeled d easured curves of isotropic Nd 2 Fe 14 B aterial (s=61ka/; k=75ka/; a=42ka/; alpha=1.5; c=.6 Figure 5: Siulation curve of Alnico 2 core (β =1.2; H =12; K=8; K c =45; R μ =1; α=1 3 Figure 4: odeled d easured curves of textured Nd 2 Fe 14 B aterial (s=61ka/; k=75ka/; a=42ka/; alpha=2.3; c=.6; Ka=4266J/cu; t=.32; easy axis =1; field axis=1 Fig.5 d 6 copares easured d siulated hysteresis curve of Alnico 2 d Alnico 5 alloys. The paraeters were coputed using (27 d (28. The deviation of the easured to the siulated values is not significt for all two core aterials. Figure 6: Siulation curve of Alnico 5 core (β =16; H =216; K=2; K c =6; R μ =1; α=1 3 It c be noted that this algorith required a significt nuber of input data, not given by y producers of agnetic aterials, which necessitated the developent of a procedure for their calculation. We are able to experientally record the hysteresis curve d starting agnetization curve for y agnetic aterial. These curves are input into the atlab progra packet, which is used to deterine values of input data required by the iterative procedure. The required slope coefficients are obtained using slope values for two near points. The value of saturation induction B s c be calculated (predicted using the axial induction B value obtained fro the experiental curve, as the converging point. 5. Conclusion The proble of deterining paraeters of the Jiles- Atherton odel for R, Z, F-shaped hysteresis curves of agnetic aterials was alyzed in this paper. The algorith was tested at the siulation level, using the PSpice progra packet. Results obtained showed good agreeent with expected values. The presented algorith is of adaptive type in accordce with type of agnetic aterials. Expsion of this progra to classes
7 of isotropy (i.e. for Z d F-shaped hysteresis curves d hard agnetic aterials is ade, as corresponding physical-atheatical odels have been developed [3,6]. This enable the calculation of paraeters required in the alysis of coplex electronic circuits using the PSpice progra packet for a wide class of agnetic aterials. The proposed algorith suitable for on-line easureents. Finally, we obtained results which uch better then known results. References [1] D.C.Jiles d D.L.Atherton, Theory of Ferroagnetic Hysteresis, Journ. agn. agn. at. vol. 61, pp.48-6, [2]PSPICE User s Guide, icrosi Corp. Version 4.5., 1991 [3] A.Raesh, D.C.Jiles d J..Roderick, A odel of Anisotropic Anhysteretic agnetization, IEEE Trs. on agnetics AG vol. 32(5, pp , [4] D.C.Jiles, J.B.Thoelke, d.k.devine, Nuerical Deterination of Hysteresis Paraeters Using the Theory of Ferroagnetic Hysteresis, IEEE Trs. on agnetics AG vol. 28(1, pp.27-35, [5].Lengevin, Ann. de Che. et Phys. vol. 5, pp. 7, 195. [6] D.C.Jiles, A.Raesh, Y.Shi d X.Fg, Application of the Anisotropic Extension of the Theory of Hysteresis to the agnetization Curves of Crystalline d Textured agnetic aterials, IEEE Trs. on agnetics AG vol. 33(5, pp , 1997.
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