Model for Nonlinear Flux Reversals of SquareLoop Polycrystalline MagneticCoresM. K. Haynes Citation: Journal of Applied Physics 29, 472 (1958); doi: 10.1063/1.1723184 View online: http://dx.doi.org/10.1063/1.1723184 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/29/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in SquareLoop Polycrystalline Garnets with and without Magnetic Field Heat Treatment J. Appl. Phys. 35, 2934 (1964); 10.1063/1.1713132 Compositional Requirements of SquareLoop Memory Core Ferrites J. Appl. Phys. 33, 3054 (1962); 10.1063/1.1728565 Rotational Model of Flux Reversal in Square Loop Soft Ferromagnets J. Appl. Phys. 29, 283 (1958); 10.1063/1.1723099 Rotational Model of Flux Reversal in Square Loop Ferrites J. Appl. Phys. 28, 1011 (1957); 10.1063/1.1722897 Influence of Pulsed Magnetic Fields on the Reversal of Magnetization in SquareLoop Metallic Tapes J. Appl. Phys. 26, 1318 (1955); 10.1063/1.1721902

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JOURNAL OF APPLIEb PHYSICS VOLUME 29, NUMBER 3 MARCH, 1958

Model for Nonlinear Flux Reversals of Square-Loop Polycrystalline Magnetic Cores

M. K. HAYNES

International Business Machines Corporation, Research Center, Poughkeepsie, New York

The theory of the reversal wave forms of polycrystalline magnetic cores proposed by Goodenough is based on the growth of ellipsoidal domains of reverse magnetization which originate from nucleating centers at grain boundaries. The present model extends this theory with the assumption that the nucleating centers are randomly distributed throughout the core volume. Radial field variations are neglected.

The rate of change of reversed magnetization area of the irreversibly moving and colliding walls is calcu­lated as a function of their position, starting from a Poisson distribution of nucleating centers, and then con­verted to a function of flux. The equation derived for rate of flux change is

dx = 4.82(H _Ho)(1_xl(_ln1-X)i dt S", 2 '

where S", is the switching coefficient, H 0 is the threshold field, H is the applied field, and x is the ratio of flux density to retentivity.

This nonlinear differential equation is solved for the simple case of constant current drive. For more complicated circuit conditions, solutions of the core equations require numerical methods. Programs have been written for the IBM Type 704 DPM to calculate the behavior of circuits containing many such cores interacting and switching together. Results obtained check reasonably well with experimental evidence.

INTRODUCTION

N UMEROUS attempts have been made to analyze the performance of magnetic devices by tech­

niques such as fitting curves to hysteresis loops, or broken straight line approximations. Generally, their degree of success has been dependent on the non­linearity involved and on the frequency range of interest. With the advent of large-scale digital com­puters, extremely nonlinear magnetic materials have been developed and used at high reversal rates. Owing to the lack of an accurate model for the nonlinearity and time dependence, previous methods are inadequate for device analysis and design.

A domain wall motion model has been proposed by Goodenough.1 This paper extends his model to obtain equations from which the behavior of magnetic core circuits can be determined.

RANDOM NUCLEATING CENTER DISTRIBUTION

Goodenough's theory assumes that flux reversal occurs by the motion, collision, and annihilation of walls bounding domains of reverse magnetization. These domains are nucleated at grain boundaries. According to this theory the equation of wall motion is

d(s) (3-= 2I8(H-Ho) (cosO),

dt (1)

where s is the semiminor axis of an ellipsoidal domain, {3 is the damping parameter, Is is the saturation mag­netization, H is the instantaneous applied field, H 0 is the threshold field at which (d(s)/dt) = 0, 8 is the angle between applied field and the magnetization, and A, the ratio of semiminor to semimajor axes, is very much

less than 1. The rate of change of flux is

d<I> d(s) aAc 16n-J.2 aAc -=87rls(cos8)- -=--(cosO)2(H-Ho)-, (2) dt dt a(s) (3 a(s)

where A c is the total area of reversed magnetization in the core cross section, and aA j a(s) is yet to be evaluated.

Let us assume that (1) only irreversible wall motion will be treated, (2) the nucleation field strength equals the threshold Ho, (3) the number of nucleated domains is independent of (H-Ho), (4) the nucleation process and finite minimum size of a domain are neglected, (5) variation of H with radial position within a core is neglected, and (6) aAja(s) is a function of domain wall position (s) and not of the velocity d(s)/dt.

Most important, assume that the nucleating centers are distributed both individually and collectively at random throughout the core volume, with an expecta­tion of q nucleating centers per unit volume. This ex­presses the absence of any information concerning the locations either of grain boundaries, or of nu­cleating centers on grain boundaries. Then the Poisson distribution,

(qv)n P(n,v)=---e-qV ,

n!

is the probability of occurrence of n nucleating centers in a volume V. For an ellipsoid, let k3=.4rrq/3X, so that qV=k3SS.

Consider a small subvolume v with a center point p. For a given value (s)', a domain wall may intersect v only if the nucleating center from which the wall originated is located somewhere on the surface of an ellipsoid of semiminor axis (s)' described symmetrically about p, and only if no other nucleating center is located

1 N. Menyuk and]. B. Goodenough,]. Appl. Phys. 26, 8 (1955). within that ellipsoid. Therefore, the probability of a 472

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FLUX REVERSALS OF POLYCRYSTALLINE MAGNETIC CORES 473

domain wall moving through v within an increment D.{s) is equal to the probability, P(NNC in D.(s», that the nearest nucleating center to p is within an ellipsoidal volume increment D. V determined by D.{s). This com­pound probability is equal to the product of the probability, pea, V), of occurrence of zero nucleating centers within V, times the probability, P(l,D. V), of occurrence of exactly one nucleating center within D. V. Therefore,

P(NNC in il(s») = P(O, V)· P(l,il V) = e-qv . gil V.

Assume the total volume V T of the core to be com­posed of M equal subvolumes 'Vi, O:S;i:S;M, each of volume D.v. A subvolume Vi will contribute an amount D.'V to the change, il V R, of volume of reversed mag­netization during il{s) only if a domain wall moves through 'Vi during il(s). The expected contribution of Vi is then

where A T is the total cross-sectional area of the core and 1= V T! AT is the mean path length. Then as D.(s) approaches zero,

(lAc (D.Ac) dV -= Lim --=ATe-qVq-. (3) a{s) A(S)-..Q D.(s) d(s)

Substituting in Eq. (2) and integrating, with initial conditions cI>o= -(cosO)cI>.= -4JrATI.{cosO) at Vo=O, gives

cI>= (cosO)cI>.(l- 2e-qV).

Normalizing with respect to cI>.(cosO) defines the new variable

x=cI>!cI>.(cosO)= 1-2e-qV = 1-2e-k'exp-k3(s)a, (4)

and in terms of x, Eq. (2) becomes

dx 6kI. ( l-X)i -=-(cosO} (H-Ho) (l-x) -In- . dt ~ 2

SOLUTIONS

(5)

The remaining material constants in Eq. (5) can be conveniently evaluated in terms of experimental data. Assume a constant current excitation source producing an applied field, H m' The output voltage wave form then exhibits a characteristic shape with a distinct peak value and peaking time. Examination of Eq. (5) shows that a peak occurs at Xp= 1-2e-f or at k{s)p= (2/3)i. Integration of Eq. (1), with H=Hm and initial condi­tions (s)=O at t=O, gives

k(s) = 2(I.!~)k<cosO)(H ... - Ho)t;

and the peaking time is

t1'= (2/3)t~!2I.k(cosO)(H m-Ho).

If time is normalized with respect to tp , and t1 is defined as t/tp, then k(s) = (2/3)ttl' Substitution in Eq. (4) and differentiation gives

dx/dt l = 4tl2 exp- (2/3)tI3, (6)

which represents all cases of constant current excitation as a single normalized response function.

From Eq. (6) the switching time is approximately t.= 1.84tp • The switching coefficient Sw= (II m- H o)t. can now be introduced, so that Eq. (5) becomes

dx 4.82 (1-X)1 -=-(lI-Ho)(l-x) -In- . dt Sw 2

(7)

Although solutions are possible for certain simple forms of excitation, solutions in closed form are in general not possible. Interconnected circuits, giving rise to simultaneous differential equations, are typically incapable of analytic solution because of the nonlinearity of Eq. (7).

Systems of nonlinear differential equations may be solved by numerical methods with calculating ma­chinery.2 Accordingly, a subroutine for Eq. (7) has been written and incorporated in a circuit simulation program for the IBM Type 704. To date, this program has been successfully used to simulate the operation of a diodeless8 magnetic core shift register using polycrystalline ferrite cores.

By extension of these methods, any type of inter­connected circuit using "square-loop" magnetic cores can be analyzed, if all other components can be repre­sented by appropriate differential equations.

EVALUATION

Predictions of constant current response, hysteresis loop shape, and shift register wave forms have been made and compared with experimental data. While the predictions are quite close, they do not agree in exact detail. The discrepancies are felt to be related to the assumptions previously introduced. Major defects appear to be the need for inclusion of: (1) nucleation effects and the finite size of a nucleated domain, (2) reversible wall motions and rotations which occur during change of applied field, (3) the variation of applied field with radial position, especially for fields near H o, and (4) the variation of the threshold Ho with the flux state of the core.

Reversible effects are presently represented by fixed linear inductance, but should include the damping factor and variation of wall area as in aAc/iJ(s). A laminar approach to radial effects has been used, but is not perfectly adequate in all cases.

2 H. H. Anderson and J. R. Johnson, Trans. Am. lnst. Elec. Engrs. 75, Part I, 569 (1956).

3 L. A. Russell, lnst. Radio Engrs. Nat!. Cony. Record 5, Part 4, 106 (1957).

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474 M. K. BAYNES

Gyorgy and Rogers4 have presented data in conflict with this theory, derived from an interrupted pulse ex­periment. However, a careful study of interrupted pulse wave forms on an IBM Mg-Mn ferrite core has shown no serious discrepancies, and none that cannot be explained by the four defects mentioned above.

4 E. M. Gyorgy and J. L. Rogers, Conference on Magnetism and Magnetic Materials, Boston (1956), p. 637.

SUMMARY

Goodenough's wall-motion model has been extended by assuming random distribution of nucleating centers, and as a result a nonlinear differential equation has been derived for the irreversible flux reversal of square­loop polycrystalline cores. Results obtained from this equation check reasonably well with experimental evidence.

JOURNAL OF APPLIED PHYSICS VOLUME 29, NUMBER 3 MARCH, '958

Temperature Dependence of Microwave Permeabilities for Polycrystalline Ferrite and Gamet Materials

J. NEMARICH AND J. C. CACHERIS

Diamond Ordnance Fuze Laboratories, Washington, D. C.

Measurements of the microwave permeabilities ILII and ILl = (p.2_ 1(2) IlL have been made at a frequency of about 9200 Mc using a cavity perturbation technique. Samples in the form of thin slabs were used. Both the static magnetic field and the temperature were varied. The temperature ranged from 25°C to 150°C and the internal magnetic fields were varied from zero to 1500 oersteds. A general qualitative agreement between theory and experiment is obtained for one of the ferrite materials. Both ILl and ILII were found to increase with increasing temperature for the range measured.

INTRODUCTION

A N unsaturated ferrimagnetic medium has a perme­ability representable by a modified Polder tensor

of the form'

-jK J.L o

0] o , J.LII

where a static magnetic field is assumed in the z-direc­tion. The operation ·of most microwave ferrite devices is based on the properties of the components of this tensor permeability. Analysis of these devices often reveals that it is desirable to know the behavior of J.Ll = (J.L2- K.2)/J.L and J.LII as a function of the dc magnetic field.2 In addition microwave devices are frequently operated at elevated temperatures because of ambient conditions or power dissipation in the material. Meas­urements were therefore made of J.Ll and J.LII as functions of both temperature and static magnetic field for two frequently used types of microwave ferrites and an experimental polycrystalline yttrium iron garnet. The de magnetic field was varied from zero to values some­what below gyromagnetic resonance and the tempera­tures ranged from 25° to 150°C. The normal magnetiza­tion curves of one of the ferrites at 25° and 150°C were used to compute theoretical curves for f-Ll' The extent of the agreement of these theoretical curves with the experimental curves is discussed.

1 G. Rado, Phys. Rev. 89, 529 (1953). 2 Ferrite Issue, Proc. Inst. Radio Engrs. 44, 1229 (1956).

EXPERIMENTAL PROCEDURES

Microwave cavity perturbation techniques are well established for obtaining the components of ferrite tensor permeabilities.3 ,4 The technique can also be used to measure J.Ll and J.LII when the sample is in the form of a thin slab in a rectangular cavity." This latter method was used for the measurements reported here.

A brass transmission type cavity was constructed that was resonant in the TE'02 mode at about 9200 Mc. The dimensions of the cavity were 1.800 in. long, 0.900-in. wide, and 0.400-in. high. Samples of three different materials were prepared in the form of thin slabs with dimensions about 0.005 in. XO.900 in. XO.400 in.

For each measurement the slab was cemented against a sidewall of the cavity. Both the static and microwave magnetic fields were parallel to the broad face of the slab. The result of a Bethe-Schwinger perturbation calculation for the relative frequency shift of the cavity when the sample is introduced in this manner is

The dc -and:microwave magnetic fields are assumed mutually perpendicular for this case. The quantities

3 J. O. Artman and P. E. Tannenwald, J. App\. Phys. 26 1124 (1955). '

4 Spencer, LeCraw, and Reggia, Proc. Inst. Radio Engrs. 44 788 (1956). '

6 Jones, Cacheris, and Morrison, Proc. Inst. Radio Engrs. 44 1431 (1956). '

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