|5p - \ir- Prz-t—^tr^- "L I B RAHY OF THE UN IVER.SITY Of ILLINOIS 510.84- UGr no. 355-360 The person charging this material is re- sponsible for its return on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. UNIVERSITY OF ILLINOIS LIBRARY AT URBANACHAMPAIGN NOV 1 1971 OCT 1 3 Re Co L161— O-1096 Digitized by the Internet Archive in 2013 http://archive.org/details/modelingofdomain358harm ??1<.U, REPORT NO. 358 COO-1018-1192 MODELING OF DOMAIN GROWTH ACTIVITY IN POLYCRYSTALLINE FERRITES by RICHARD PAUL HARMS JAN 1 4 1970 November , 1969 THE UBRMW OF THE IWWERSW »f ill Report No. 358 MODELING OF DOMAIN GROWTH ACTIVITY IN POLYCRYSTALLINE FERRITES by RICHARD PAUL HARMS November , 1969 Department of Computer Science University of Illinois Urbana, Illinois 6l801 * Supported in part by Contract Number U.S. AEC AT(ll-l)-10l8 and submitted in partial fulfillment for the degree of Master of Science in Electrical Engineering, at the University of Illinois, November, 1969. Ill ACKNOWLEDGMENT The author wishes to thank his advisor, Professor Sylvian R. Ray, for his advice, guidance, and patience in the preparation of this thesis. The author also wishes to thank Mrs. Mariam Coleman for typing the final draft of the thesis and Mr. Stanley Zundo for preparing the figures included in the thesis. iv TABLE OF CONTENTS Page INTRODUCTION 1 1 . PROPERTIES OF FERRITE MATERIALS 2 1 . 1 Polycrystalline Ferrites 2 1 . 2 Domains and Bloch Walls 3 1. 3 Domain Wall Motion h 2 . MODELS OF DOMAIN WALL MOTION 7 2 . 1 Menyuk and Goodenough 7 2 . 2 Haynes ' Model 8 2 . 3 Lindsey ' s Model 10 2.U Comparison of the Models 12 3. FLUX REVERSAL SIMULATION 18 3-1 Introduction 18 3 . 2 Simulation Details 19 CONCLUSION 28 LIST OF REFERENCES 29 APPENDIX 30 LIST OF FIGURES Figure Page 1 . 1 Example of Domain Walls h 2 . 1 Experimental Full Switching Curve lk _ t 2 2.2 Plot of te vs. t 15 3 2.3 Plot of t 2 e -t vs. t 16 3.1 Simulation Grid Edge Effects 20 3 . 2 Simulated Full Switching Curve 2k 3 . 3 Simulated Partial Switching Curve 26 3.^ Experimental Partial Switching Curve 27 INTRODUCTION Because of the numerous uses of ferrite materials in computer technology, especially in the coincident current memory area, much work has been done in an attempt to explain the observed behavior of the materials in terms of their physical properties. These attempts have taken the form of mathematical equations which predict what the voltage output waveform of a ferrite specimen will be under specified conditions of drive. The problem here will be to produce the observed voltage output waveform of a ferrite core fully switched from an initially saturated condi- tion by simulating the domain growth activity on a microscopic scale. It is hoped that this will provide some assistance in determining which of the models of the flux reversal process that have been proposed is more nearly correct . 1. PROPERTIES OF FERRITE MATERIALS 1.1 Polycrystalline Ferrites Ferrites are ceramic ferromagnetic materials with the general chemical composition YO*FE ? 0_ where Y is a divalent metal such as iron, manganese, magnesium, nickel, zinc, cadmium, cobalt, copper, aluminum, or a mixture of these. The ferrites crystalize into a cubic lattice structure which can be thought of as being made up of the closest possible packing of layers of oxygen ions with the metallic ions fit in the spaces left between the oxygen ions. The metallic ions form bonds with the oxygen ions so that the otherwise free electrons of the metallic ions are not readily available for electrical conduction, thus eliminating the eddy current losses which are found in other magnetic materials. This is of significance here since, as a result of the high resistivity, the maximum rate of change of magnetization, which normally is limited by the eddy currents, can be large. Polycrystalline samples of ferrites are prepared by a sintering process comprised basically of the following operations. The metal oxides and other compounds which are to form the ferrite are mixed and wet-milled, usually in a ball mill. The dried powder is prefired at a temperature of about 1000 C in order to produce the initial chemical reaction. The powder, after another milling and the addition of a binder, is pressed or extruded into the required shape. These products are then sintered at a temperature between 1200 and 1^00 C, the exact temperature depending on the properties desired in the ferrites. In the sintering process the gas atmosphere in the furnace plays an important role, since it determines the degree of oxidation, which has an important influence on the magnetic properties. It is in the sintering process that the crystal structure and its size and the porosity of the ferrite are determined. The longer the sintering, the larger the crystals and the more dense the ferrite. Typically, polycrystalline ferrites have a porosity of from 0.05 to 0.1. The voids existing in these ferrites have a large effect on the magnetic properties, since if the void is large enough, it can approximate a crystalline surface and thus be a source of the mobile domain walls which will be discussed later. In a polycrystalline material, the directions of easy magnetization are not necessarily the same from grain to grain. However, since the ferrite materials in question have cubic structures, so that any given direction cannot be far from one of the three possible directions of easy magnetization, we can assume that the ferrites are essentially isotropic with regard to ease of magnetization. Later, when we consider the effects of magnetic fields on the ferrite materials, we will also assume that the sample is in the form of a toroid or a long rod parallel to the applied field. 1.2 Domains and Bloch Walls Because of energy considerations, it is usually more advantageous for a ferromagnetic body, rather than to be uniformly magnetized, to divide itself into a number of regions of uniform magnetization, in which the magnetization vectors are parallel to a preferred direction, such that the demagnetizing fields, and thus the energy, are as small as possible. The resulting regions, or domains, of uniform magnetization are called "Weiss domains". These uniformly magnetized domains are separated by a thin layer in which the direction of the magnetization gradually changes from one direction to another. This transition boundary is known as a "Bloch wall . Figure 1.1 Example of Domain Walls The change in angle of the magnetization vectors from one domain to another adjacent domain can be used to characterize the domain walls. In Figure 1.1, for example, there are four 90 domain walls and one 180 domain wall. Typically the walls have a small but non-negligible width (in the order of o 100 A). 1. 3 Domain Wall Motion There are basically two ways the magnetization of a ferrite sample can be changed. The first is by the rotation of all the magnetization vectors simultaneously by the externally applied field. The second is by the motion of domain walls in the sample. Flux changes due to rotation are the result of the torque caused by the magnetic field intensity, H, acting on the magnetization, M, according to the relation T = M x H. For a small field, only individual atomic spins are reversibly rotated. This is a small effect. To cause an irreversible rotation requires a much larger field. It has been estimated that the minimum field required is approximately four times the threshold field for materials such as are considered here. However, the usual applications of these materials, such as in core memories, requires fields smaller than this. Thus we can assume the changes in magnetization are primarily caused by domain wall motion. Goodenough (195*0 also asserts that the principal cause of flux change may be assumed to be the motion of 180 domain walls if the specimen does not have a special geometry or orientation of its axes of easy magnetization which would energetically favor the creation of many domains at right angles to the applied field. Domains of reverse magnetization are formed at lattice imperfections. Among the imperfections which can serve as nucleating centers for new domains are granular inclusions, lammellar precipitates, grain boundaries, voids, and the crystalline surface. Goodenough (195*+) has made a theoretical study of these imperfections and found that within the range of magnetic field strengths which are used in switch-core and coincident current memory applications, the grain boundaries are the primary source of mobile 180 domain walls. An externally applied magnetic field will try to alter the Weiss domain structure and, if it is strong enough, will cause the Weiss domains to vanish by eliminating the Bloch walls. This is accomplished by a movement of the Bloch wall. The magnetic field exerts a pressure on a Bloch wall which pushes the wall in such a direction that the domain with the magnetization in the same sense as the applied field becomes larger. The wall will not be entirely free to move, however. With no exter- nally applied field, a particular configuration of the Weiss domain structure will establish itself in a crystal. That configuration corresponds to a minimum of free energy. This means that the walls are bound with a certain stiffness to their positions of equilibrium. This also means that when a Bloch wall is moving under the influence of a driving field and this driving field is removed, the wall will come to rest in such a way as to minimize the free energy. Thus the change in magnetization of a sample of polycrystalline ferrite can be thought of as follows: starting with an initial configuration of Weiss domains, a magnetic field larger than the threshold field is applied causing the existing Bloch walls to move and new walls to be created in such a way as to increase the magnetization of the sample parallel to the applied field. A particular wall continues to move until either it meets another wall moving toward it, which annihilates both walls, or the driving field is removed and the walls come to rest in a new configuration. 2. MODELS OF DOMAIN WALL MOTION 2.1 Menyuk and Goodenough Menyuk and Goodenough (1955) laid much of the ground work for the investigations that have followed of the flux switching processes in f errites . They proposed that the magnetization reversal in polycrystalline ferrites is primarily due to the nucleation and growth of 180 Bloch walls. The walls took the form of ellipsoids with large eccentricities. The first result they obtained was the equation of motion for the domain walls 6 l , (2.1) dt b m o where 6 is the viscous-damping parameter for domain wall motion, r is the length of the domain semiminor axis, I is the spontaneous magnetization, H is the (constant) driving field, m ° ' H is the threshold (maximum) field strength at which the wall velocity is zero, and 6 is the angle between the magnetization vector in the domain and the direction of the driving field H . m This relation assumes an ideally square input pulse and so is valid only for input pulses whose rise time is much less than the total switching time. For the time rate of change of flux due to irreversible wall motion, Menyuk and Goodenough obtained the expression 2 3 d$ _ 16tt I dt . " ~ F ( ) ( H _ H ) (2.2) irr 3 mo where F() is defined as 9A /8, where A is the cross-sectional area of c c the growing domains in the material with reversed magnetization. 8 Menyuk and Goodenough went on to explain that with a constant drive applied to the material, F() determines the shape of the flux switching curve. Since F() is a function of the wall positions, and the walls are assumed to accelerate to a constant average velocity in a time which is much smaller than the switching time, F() is a maximum when the area of the domain walls is a maximum. The wall surrounding a growing domain continues to move outward until it meets a similarly moving wall from another domain and is annihilated. Because of the random distribution of the growing domains throughout the material, F() decreases to zero with a Gaussian-like tail at larger values of r. 2.2 Haynes' Model Haynes (1958) took the preceding work of Menyuk and Goodenough and extended the model using certain assumptions. Explicitly these assumptions were : (1) Only irreversible wall motion will be treated, (2) The nucleation field strength equals the threshold H , (3) The number of nucleated domains is independent of (H-H o ), (U) The nucleation process and finite minimum size of a domain are neglected, and (5) 3A /3 is a function of domain wall position and not c of the velocity d/dt. Further, Haynes assumed that the nucleating centers are distributed at random throughout the core volume, with an expectation of q nucleating centers per unit volume. This gives the Poisson distribution, P(n,V) =M!e-' V , (2.3) as the probability of occurrence of n nucleating centers in a volume V. Also, 3 3 3 a constant k is defined, k : UiTq/3A, so that qV = k s , where X is the ratio of the semiminor to semimajor axes, and s is the length of the semiminor axis. An expression is then derived for the rate of change of cross- sectional area as a function of s (8A /3). This expression is substi- c tuted into Equation 2.2 and integrated to obtain the result $ = $ (l - 2e~ qV ) (2.U) s where $ is the magnetic flux, and $ is the remanent value of the flux. If the flux is normalized with respect to $ by defining $ x = — , $ s we have (2.5) x = 1 - 2e~ qV = 1 - 2exp[-k 3 3 ] (2.6) Differentiating this expression and using Equations 2.1 and 2.2 the result is obtained 6kl . fj = -j~^ (H - H Q )(1 - x)(- In ^=-£ ) ' (2 - T) Assuming a constant current excitation producing an applied field H , the output waveform exhibits the characteristic shape with a distinct peak value and peaking time. From Equation 2.7 it can be shown that a peak -2/3 1/3 occurs at x = 1 - 2e or at k = (2/3) • Integrating Equation 2.1 P P eith H = H and initial conditions = at t = 0, gives m k = 2(1 /B)k(H - H )t (2.8) s mo 10 where the peaking time is t = (2/3) B/2I k(H - H ). (2.9) ps mo Normalizing the time with respect to t , i.e., t, = t/t , and substituting jr p ' 1 p' D into Equation 2.6 and differentiating we obtain f* = ht 2 ± exp[-(2/3)t^] (2.10) which is a normalized response function representing all cases of constant current excitation. This equation will he used subsequently in comparing the various models. 2.3 Lindsey's Model Lindsey (1959) also took the results of Menyuk and Goodenough and extended them. Instead of assuming the growing domains were elliptical with large eccentricity, Lindsey assumed the domains were expanding cylinders. The centers from which these domains started were scattered at random through- out the material. At time t = 0, all the domains were assumed to be of zero radius, at which time a field, H , was applied. The domains then expanded with radial velocity proportional to (H - H ), where H is the coercive force. Lindsey considered only a cross-section of the material perpen- dicular to the field, so that the domains became circles. He also assumed that the effect of not having the direction of easy magnetization parallel to the applied field was small and thus could be ignored. In his words: "...the only effect of having the direction of easy magnetization not parallel to the field would be to reduce the velocity of the Bloch walls, and this would not effect the shape of the output waveform basically." 11 With these assumptions, Lindsey proceeded to calculate the function F(r) in Equation 2.2. If r is the radius of a domain, v the velocity at which the walls move outward [proportional to (H - H )], and A is the area m c of the domain, then fr " f^* r2) " 2TO t ■ 2 ™ ■ <2 - n) However, this does not take into account the possibility that part of the domain wall might have "been annihilated through coalescence with other domains by the time the domain reaches a radius of r. Consider an element of domain wall subtending an angle d8 at the center of the domain. If this element has not intersected a second domain by the time the radius of the domains is r, it implies that the center of a second domain cannot lie within a circle of radius r centered on the element d6 . For, if the center of a second domain had been in this circle, the element d6 would not have gotten this far. Thus the probability that this element reaches a radius r is the probability of finding no other domain center within this circle of radius r. Since the centers are scattered at random, this is also the probability of finding a center in any circle of radius r. Let p be this probability. Then p also represents the proportion of such elements which reach a radius r. In this manner, then, Lindsey derived the following expression, ||= 2Ttrvp , (2.12) which represents the rate of change of the area enclosed by the circles, divided by the number of circles. The quantity p was then determined. If p is the density of domain centers, then it can be shown that 12 2 p = exp[-TTr p]. (2.13) Combining Equations 2.12 and 2.13 we have j A Q — = 2irrv exp[-iTr p] (2.lU) CI u But v is proportional to (H - H ), and the rate of change of the flux density is proportional to dA/dt, thus we have — « (H - H )r expt-ur p] (2.15) dt m c Comparing this equation with Equation 2.2, it can he seen that the required distribution function is given by F(r) = Kr exp[-TTr 2 p] (2.l6) where K is a constant. If a constant applied field H is assumed, so that v is constant, we have r = vt, and Equation 2.1 4 becomes H = 27TV 2 t exp[-uv 2 t 2 p] (2.17) and so f£ = k (H - H ) 2 t expI-k_(H - H ) 2 t 2 J (2.18) dt 1 m c 2m c = k H 2 t exp[-k 2 H 2 t 2 ] (2.19) where k.. and k~ are constants. 2.4 Comparison of the Models We have seen that for constant current excitation Haynes obtained & - ktl expl-ftf] (2.20) as the normalized response function. Rewriting this we have f|= k 3 t 2 expt-k^t 3 ] (2.21) 13 where k and k, are constants. Thus the model using ellipsoids with large eccentricities as the shape of the growing domains of reverse magnetization 3 2 -t results in a voltage output function of the form t e Using circular domains in the flux switching process, Lindsey obtained the result ||= k x H 2 t exp[-k 2 H 2 t 2 ] . (2.22) Combining the constants, we obtain || = kjt exp[-k^t 2 ] . (2.23) Thus for Lindsey' s model we have a voltage output function of the form Comparing Figure 2.1, which is an experimental voltage output 2 3 -t 2 -t waveform, with Figures 2.2 and 2.3, which are plots of te and t e respectively, it can be seen that Figure 2.2 corresponds more closely with the experimental waveform than does Figure 2.3. If the switching time, t , is defined as the time required for the output waveform to fall to 10$ of its peak value, and the peaking time, t , has the obvious definition, then com- ir paring the values of t /t given on the respective Figures, it can be seen s p that Figures 2.1 and 2.2 agree more closely than Figures 2.1 and 2.3. Thus we see that there is some evidence of similarity between the results obtained from Lindsey' s model and the results observed experimentally. On the exper- imental waveform, the region below 50ns no longer follows the predicted curve because of reversible effects not considered in this treatment. Ik > u o 3 •H xi o -p •H 3 c LU (auj) aanindiAiv 15 cvi II OJ {£> CO O OJ CO o -p CM -P I -P Cm O P o H Oh CM CM a- 3 * 16 CO II Q. OJ U> oo oi o -p o H CO OJ o ro (Q CD o ,1-M IT Further corroboration comes from Ray (1961) who calculates a general flux switching equation for the case of constant current excitation. The equation is g|- 2nK n t n_1 exp[-(Kt) n ] , (2.2*0 where K is a constant, x is the normalized flux, and n > 1. Using experimental data, Ray calculates n to be from 1.8 to 2.2 for the range of drives normally used in memory applications, the range to which we have limited ourselves. This again would correspond to Lindsey's model. Thus it would seem that the flux switching process can be modeled fairly closely using growing domains which are circular. An attempt then will be made to simulate the switching process using this model. 18 3. FLUX REVERSAL SIMULATION 3. 1 Introduction The following assumptions on the flux reversal process are basic to the simulation: (1) The ferrite material is essentially isotropic with respect to the ease of magnetization. (2) The changes in magnetization are primarily caused by the irreversible motion of 180 Bloch walls. (3) The material is in the form of a toroid or long rod parallel to the applied field. (h) The applied field is a step function going from zero to some constant value at t = 0. (5) The material is initially (i.e., at t = 0) magnetically saturated in a direction such as to diametrically oppose the field applied at t = . (6) The nucleating centers are randomly scattered throughout the material. (T) All domains are nucleated at t = and have an initial radius of zero. (8) The domain walls accelerate to a constant velocity in a negligible amount of time. Using the model of cylindrical domains, and the assumptions given above, a program was written for the 360/75 in FORTRAN IV to simulate the flux switching process and output a rate of change of flux curve. 19 3.2 Simulation Details In the simulation only a typical cross-section of ferrite material was considered. This was accomplished by performing the calculations on a TOO x TOO unit grid, considered to be perpendicular to the long axis of the domains, so that the problem was reduced to one of considering expanding circles . The nucleating centers for the domains were scattered over the grid by using a random number generator with a uniform distribution for both coordinates of each point. The number of nucleating centers used was 88. This number was arrived at empirically as the approximate minimum number of centers needed to insure that the output waveform was fairly smooth. The flux reversal process was limited to this grid without the need for considering edge effects by viewing the opposite edges of the square as coinciding. That is, any domain wall leaving the grid on the right, was introduced onto the grid on the left, and similarly for the top and bottom. For example, in Figure 3.1, the portion of the domain wall represented as the arc MN is considered, for calculating purposes, to exist at the left side of the grid as arc M'N' centered at A'. Similarly, for the domain centered at B, the arc PQ is considered to be at the lower right as arc P'Q' centered at B' , the arc QR is considered to be at the lower left as arc Q'R' centered at B" , and the arc RS is considered to be at the upper left as arc R'S' centered at B" ' , while the arc SP remains at the original location. If A is the area and r the radius of one of the domains, then dA d / 2x dr _, /-, , \ _=_ (7Tr ) = gTrr— = Cv , (3.1) 20 ■f QD + CD CO -P o 0) in W 0) bO 33 nd •H O a o •H -P 1 •H CXI Sn bO •H [in 21 where C is the circumference and v is the wall velocity of the domain. As a domain grows, it will begin to coalesce with nearby similarly growing domains, Once this happens, it is no longer the complete circumference of the domain which is contributing to the change in area, dA/dt. The only contribution now comes from that portion of the circumference which has intersected no other domain. If we let C be that portion of the circumference, we have dA dt = C'v . (3.2) This relation holds for all the domains on the grid. If is the total flux in the cross-section and A the total area of all of the domains , then d$ dt t=t. dt (3.3) t=t. or df dt t=t. = K^T dt = KvC^ t=t. (3.*0 t=t. where K is a constant and C' is the total length of all the domain t=t. circumferences which have not been intersected at t = t 1 . Since v is also a constant over time and over all domains, we have = K'C' n d$ dt t=t 1 (3.5) t=t. This gives us the rate of change of flux as a function of the total length of the domain walls which have not been intersected at a given time t, . For the simulation, the domain wall velocity was chosen to be one unit (on the TOO x TOO unit grid) per unit of time. This is a valid choice 22 since any other value of velocity would only change the output by a scale factor and would not affect the shape of the waveform. This means that the domain radius and the time always have the same numerical value, i.e., they both start at zero and are incremented by one for each successive step of the simulation. The simulation proceeds, then, as follows. Starting at zero, the time and the domain radius take on successive integral values. At each value, the length of the circumference that had not yet been intersected is determined for each domain. These lengths are then totaled giving the value of C^ at that point in the flux reversal. Taking the constant K' to be unity, which again involves only a scale factor at the output, this value of C' is output as the current value of d$/dt. The time and the domain radius are then incremented, and the process is repeated. This continues until the calculated value of d$/dt falls below a predetermined level. The results of the simulation are given in Table 3.1 and Figure 3.2. It can be seen that Figure 3.2 is the same shape as the experimental waveform of Figure 2.1. Here, again, the value of t /t given in Figure 3.2 corresponds s p more closely to the values in Figures 2.1 and 2.2 than to the value in Figure 2.3. An attempt was also made to simulate the partial switching process, that is, switching from an initially unsaturated state. This is accomplished by reversing the applied field at some point before the switching process is completed. This causes the existing domain wall segments to reverse their direction while new circular walls, growing outward, are created again at the nucleating centers. These two sets of walls move toward each other until they meet and all of the collapsing wall segments and an equal amount of the 23 Rate of Rate of Rate of Change Change Change Time of Flux Time of Flux Time of Flux t d4>/dt t d$/dt t d*/dt 3U 9652 67 3198 1 552.9 35 9579 68 3007 2 1106 36 9I+68 69 2885 3 1659 37 9361 70 2786 i+ 2212 38 9181+ 71 2689 5 2759 39 8926 72 2571 6 3290 Uo 8681+ 73 21+73 T 3828 1+1 850I+ 7U 23ll+ 8 1+33*+ 1+2 8319 75 2201+ 9 U811 1+3 820I+ 76 2067 10 5267 1+1+ 8033 77 19I+8 11 5715 1+5 7766 78 1822 12 61U8 1+6 7635 79 1711+ 13 65I+1 1+7 71+99 80 1618 lA 6886 1+8 7196 81 11+98 15 r J22k 1+9 69I+O 82 131+6 16 7568 50 6712 83 1186 IT 7871 51 61+11 81+ 1011+ 18 8167 52 6109 85 896.3 19 81+35 53 5835 86 768.5 20 8701 51+ 5595 87 626.6 21 8927 55 5363 88 5I+0.I 22 9165 56 5189 89 1+53.8 23 935^ 57 1+97^ 90 395-8 2U 9501 58 1+767 91 371.9 25 9612 59 1+566 92 31+8.5 26 9706 60 1+1+20 93 321+.8 27 9lhh 61 1+288 91+ 302.2 28 9772 62 1+092 95 280.5 29 9771 63 3922 96 21+1+.5 30 9777 61+ 3705 97 179.1 31 9788 65 3521+ 98 131.0 32 9797 66 3362 99 97.29 33 9753 Table 3.1 Full Switching Results CD cvi _. -<* 2U I- oo = UJ 10 00 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 0> 00 N (0 in *■ fO CVI (siiNn) 3arundwv 25 growing walls are annihilated. The remaining wall segments complete their growth and eventual annihilation. The results of this simulation, which utilized the same basic program as the full switching process, are given in Figure 3.3, along with an experimentally obtained curve in Figure 3.*+. The results are unsatis- factory in that there is a discontinuity at t = 90 which does not appear on the experimental curve. This discontinuity results from all of the collapsing wall segments meeting the growing domains and being annihilated simultaneously. The two sets of walls meet simultaneously because all of the walls have the same velocity, and the collapsing wall segments retain their shape as arcs of the original domain walls. o. CVJ< V ^ o 1 UNIT! II — ' UJ o 5 2 h- o ~~-~ o 00 o N o (0 o m o •H -P u PL, -p c 0) a •H CD ft •H 28 CONCLUSION For the case of a ferrite core fully switched from an initially saturated state, simulating the domain growth activity on a microscopic scale leads to results which coincide with the observed experimental results. These results also support Lindsey's contention that the flux switching process can be modeled using circular domains, rather than elliptical domains of large eccentricity. The simulation of a ferrite core partially switched from an initially unsaturated state leads to a discontinuity in the output waveform which does not occur experimentally. To avoid this discontinuity, the simulation process would have had to allow the collapsing walls to change their shape contin- uously as they collapsed. This was beyond the model's capability since it handled the walls purely from the standpoint of their center and radius. LIST OF REFERENCES Goodenough, J.B., (August 195*0, "A Theory of Domain Creation and Coercive Force in Polycrystalline Ferromagnetics", Phys . Rev., Vol. 95, No. k. Haynes, M.K., (March 1958), "Model for Nonlinear Flux Reversals of Square- Loop Polycrystalline Magnetic Cores", Jour, of Appl. Phys., Vol. 29, No. 3. Lindsey, C.H., (September 1959), "The Square-Loop Ferrite Core as a Circuit Element", Proc. of the Inst, of Elec. Engrs . , Vol. 106, Part C, No. 10, Menyuk, N. and Goodenough, J.B., (January 1955), "Magnetic Materials for Digital-Computer Components I. A Theory of Flux Reversal in Poly- crystalline Ferromagnetics", Jour, of Appl Phys., Vol. 26, No. 1. Nitzan, D. , (September 1965), "Computation of Flux Switching in Magnetic Circuits", IEEE Transactions on Magnetics, Vol. MAG-1, No. 3. Nitzan, D. , (December 1966), "Models for Elastic and Inelastic Flux Switching", IEEE Transactions on Magnetics, Vol. MAG-2, No. h. Ray, S.R. , (September 1961), "Engineering Model of a Partial Switching Effect in Ferrite Cores", PhD. Thesis, University of Illinois, Digital Computer Laboratory Report No. 111. Smit, J. and Wijn, H.P.J. , (1959), "Ferrites", John Wiley and Sons, New York. Soohoo, R.F., (i960), "Theory and Application of Ferrites", Prentice- Hall, Inc. , New Jersey. 30 APPENDIX The appendix contains the listing of the program used to simulate the full switching process. 31 C MAIN PROGRAM DIMENSION CTRLST(100,2) DIMENSION XCOORD(i+0),YCOORD( i +0),SDIST( i +0),THETA(t+0,2) DIMENSION BUFFPC10) COMMON CTRLST, NUMCTR, XLIM, DENSE, Ml, SDIST, XCOORD, II, XC COMMON YC, RADIUS, THETA, NUMTH, TWOPI, TOTBDY, YCOORD, DIST COMMON LTIME COMMON IZ IB=1 LTIME=0 TOTBDY=0. XLIM=700. DENSER 1.80 t +lE-4 TWOPI=6. 2831853 CALL CENTER DO 2 1=1, NUMCTR WRITE(6,90XCTRLST(I,J),J=1,2) 90 FORMAT (1H ,2E11.*0 2 CONTINUE GO TO 15 5 TOTBDY=0. RADIUS=LTIME 27 IZ=1 29 DO 10 Ml=l, NUMCTR CALL NEIGHB CALL CIRC I CALL SORT CALL BOUNDC 10 CONTINUE 15 BUFFP(IB)=TOTBDY IFCIB- 10)35, 20,35 20 WRITE(6,30)LTIME,(BUFFP(I), 1 = 1, IB) 30 FORMAT (1H0, 16, 10(1X, E10.*O) IB=0 35 IB=IB+1 40 IF(TOTBDY-100.)60,60,50 50 LTIME=LTIME+1 GO TO 5 60 IF(LTIME)70, 50,70 70 IF(IB-1)20,80,20 80 CALL EXIT END C CENTER GENERATOR SUBROUTINE SUBROUTINE CENTER DIMENSION CTRLST(100,2) DIMENSION XCOORD(40),YCOORDC i +0),SDIST( i +0),THETA(40,2) COMMON CTRLST, NUMCTR, XLIM, DENSE, Ml, SDIST, XCOORD, II, XC COMMON YC, RADIUS, THETA, NUMTH, TWOPI, TOTBDY, YCOORD, DIST COMMON LTIME NUMCTR=DENSE*XLIM**2 DO 10 1 1=1, NUMCTR CTRLSTO 1, 1)=XLIM*RAN3Z(0) 32 CTRLST(I1,2)=XLIM*RAN3Z(0) 10 CONTINUE RETURN END C FIND NEAREST NEIGHBORS SUBROUTINE SUBROUTINE NEIGHB DIMENSION CTRLST(100,2) DIMENSION XCOORD(40),YCOORD(40),SDIST(40),THETA(40,2) DIMENSION ADJACEC100, 180) COMMON CTRLST, NUMCTR, XLIM, DENSE, Ml, SDIST, XCOORD, II, XC COMMON YC, RADIUS, THETA, NUMTH, TWOPI, TOTBDY, YCOORD, DIST COMMON LTIME IF(LTIME-1)130,5,110 5 DO 10 11=1,40 10 SDIST(I1)=2.*XLIM DO 90 1 1=1, NUMCTR IF(I1-M1)20,90,20 20 CALL PROXIM DO 45 12=1,40 IF(DIST-SDIST(I2))45,40,40 40 IF(I2-1)75, 90,75 45 CONTINUE 50 13=1 60 IF(I3-I2)70,80,70 70 SDIST(I3)=SDIST(I3+1) XCOORD(I3)=XCOORD(I3+l) YCOORDC I 3)=YCOORD( 13+1) 13=13+1 GO TO 60 75 12=12-1 GO TO 50 80 SDIST(I3)=DIST XCOORD(I3)=XC YCOORDC I 3)=YC 90 CONTINUE DO 100 13=1,40 14=13-1 ADJACE(M1,3*I4+1)=XC00RD(I3) ADJACE(M1,3*I4+2)=YC00RD(I3) 100 ADJACE(M1,3*I4+3)=SDIST(I3) GO TO 130 110 DO 120 13=1,40 14=13-1 XCOORDC I 3)=ADJACE(M1, 3* 14+1 ) YCOORD(I3)=ADJACE(Ml,3*l4+2) 120 SDIST(I3)=ADJACE(M1, 3*14+3) 130 RETURN END C FIND THE CLOSEST IMAGE SUBROUTINE SUBROUTINE PROXIM DIMENSION CTRLST(100,2) DIMENSION XCOORD(40),YCOORD(40),SDIST(40),THETA(40,2) 33 COITION CTRLST, NUMCTR, XLIM, CENSE, Ml, SDIST, XCOORD, II, XC COMMON YC, RADIUS, THETA, NUMTH, TWOPI, TOTBDY, YCOORD, DIST COMMON LTIME CD I FF=CTRLST( 11,1 )-CTRLST(Ml , 1 ) B=XLIM/2. IF(ABS(CDIFF)-B)10, 10,20 10 XC=CTRLST(I1,1) GO TO 50 20 I F(ABS(CDIFF+XLIM)-B)30, 30,^+0 30 XC=CTRLST(I1,1)+XLIM GO TO 50 40 XC=CTRLST(I1,1)-XLIM 50 CDI FF=CTRLST( 1 1, 2)-CTRLST(Ml, 2) I F(ABS(CDI FF)-B)60, 60, 70 60 YC=CTRLST(I1,2) GO TO 100 70 IF(ABS(CDIFF+XLIM)-B)80,80,90 80 YC=CTRLST(I1,2)+XLIM GO TO 100 90 YC=CTRLST(I1,2)-XLIM 100 DIST=SQRT(CCTRLST(M1,1)-XC)**2+(CTRLST(M1,2)-YC)**2) RETURN END C FIND CIRCLE INTERSECTION POINTS SUBROUTINE SUBROUTINE CIRC I DIMENSION CTRLST(100,2) DIMENSION XCOORDC i +0),YCOORDC40),SDIST(40),THETAC i f0,2) COMMON CTRLST, NUMCTR, XLIM, DENSE, Ml, SDIST, XCOORD, II, XC COMMON YC, RADIUS, THETA, NUMTH, TWOPI, TOTBDY, YCOORD, DIST COMMON LTIME COMMON IZ NUMTH=0 DO 20 11=1,40 IF(SDIST(I1)-2.*RADIUS)10,20,20 10 DX=XCOORD(I 1)-CTRLST(M1, 1) DY=YCOORD(Il)-CTRLST(Ml,2) GAMMA=ATAN2(DY, DX) IF(GAMMA)11,12,12 1 1 GAMMA=GAMMA+TWOP I 12 PHI=ARCOS(SDISTCll)/C2.*RADIUS)) NUMTH=NUMTH+1 THETACNUMTH, 1)=GAMMA-PHI THETACNUMTH, 2)=GAMMA+PHI 20 CONTINUE GO TO (40,30), IZ 30 WRITE(6,31)M1 31 FORMAT(1H0,3HM1=,I3) DO 39 1=1, NUMTH WRITE(6,32)(THETA(I,N),N=1,2) 32 FORMAT(2E12.4) 39 CONTINUE 40 RETURN END 3U C SORT THE INTERSECTION POINTS SUBROUTINE SUBROUTINE SORT DIMENSION CTRLST(100,2) DIMENSION XCOORD(40),YCOORD(40),SDISTa+0),THETA(40,2) DIMENSION THETAL(120),NWT(120) COMMON CTRLST, NUMCTR, XLIM, DENSE, Ml, SDIST, XCOORD,- II, XC COMMON YC, RADIUS, THETA, NUMTH, TWOPI, TOTBDY, YCOORD, DIST COMMON LTIME COMMON IZ DO 50 1 1=1, NUMTH IF(THETA(Il,l))10,40,i+0 10 THETA(I1,1)=THETA(I1,1)+TW0PI 30 THETA(I1,2)=THETA(I1,2)+TW0PI 40 THETAL(2*I1-1)=THETA(I1,D NWT(2*I1-1)=I1 THETAL(2*I 1)=THETA(I 1,2) NWT(2*I1)=-I1 50 CONTINUE 51 IF(NUMTH-1)110, 110,52 52 LIM=2*NUMTH-1 55 INT=1 DO 70 11=1, LIM IFCTHETALO 1+D-THETALO D)60, 70, 70 60 TEMP=THETAL(I1+1) THETAL(I1+1)=THETAL(U) THETAL(I1)=TEMP NTEMP=NWT(I1+1) NWT(I1+1)=NWT(I1) NWT(I1)=NTEMP INT=I1 70 CONTINUE IFCINT-D80, 80,75 75 L1M=INT-1 GO TO 55 80 GO TO C85,76),IZ 76 WRITE(6,77) 77 FORMAT(1H0, 12HSORTED TABLE) DO 79 1 1=1, NUMTH I=NUMTH+I1 WR I TE(6, 78)THETAL( 1 1), NWT( 1 1 ), THETALC I ), NWT( I ) 78 FORMATC2CE12.4, 110, 20X)) 79 CONTINUE 85 LIM=2*NUMTH INT=1 NUMTH=0 NACC=0 DO 100 11=1, LIM NACC=NACC+NWT(I1) IF(NACC)90,90,100 90 NUMTH=NUMTH+1 THETACNUMTH, 1)=THETAL(INT) THETA(NUMTH, 2 )=THETAL( 1 1 ) INT=I1+1 NACC=0 GO TO 100 L00 CONTINUE GO TO (101, 111), IZ 111 WRITE(6,112) 112 FORMAT(1H0, 15HEND THETA TABLE) DO 114 1 1=1, NUMTH WRITE(6,113)(THETA(I1, I), 1=1,2) 113 FORMAT (2E 12. 4) 114 CONTINUE 101 DO 130 1 2=1, NUMTH DO 129 13=1,2 TEMP=THETA(I2, I3)+TWOPI IF(THETA(NUMTH,2)-TEMP)131, 120,120 120 THETACI2, I3)=0. 129 CONTINUE 130 CONTINUE 131 IF(I3-2)140, 132,132 132 THETA(NUMTH,2)=TWOPI 140 IF(I2-1)110,110,141 141 I4=NUMTH-I2+1 DO 142 15=1,14 16=12+15-1 THETA(I5,1)=THETA(I6,1) THETAC 15,2 )=THETA( 16,2) 142 CONTINUE NUMTH=I4 110 RETURN END C CALCULATE LENGTH OF BOUNDARY SUBROUTINE SUBROUTINE BOUNDC DIMENSION CTRLST(100,2) DIMENSION XCOORD(40),YCOORD(40),SDIST(40),THETA(40,2) COMMON CTRLST, NUMCTR, XLIM, DENSE, Ml, SDIST, XCOORD, II, XC COMMON YC, RADIUS, THETA, NUMTH, TWOPI, TOTBDY, YCOORD, DIST COMMON LTIME COMMON IZ ANGLE=0. IF(NUMTH)30,21,10 10 DO 20 11=1, NUMTH 20 ANGLE =ANGLE+THETA(I 1,2)-THETA(I 1, 1) BDY=RAD I US* (TWOPI -ANGLE) GO TO 23 21 BDY=RADIUS*TWOPI 23 GO TO (31,24), IZ 24 WRITE(6,25)M1,BDY 25 FORMAT(1H0, 3HM1=, I3,3X,4HBDY=,E12.5) WRITE(6,26)NUMTH 26 FORMAT(1H0,6HNUMTH=,I3) IF(NUMTH)30,31,27 27 DO 28 1 1=1, NUMTH 36 28 WRITE(6,29XTHETA(I1, I), 1 = 1,2) 29 F0RMAT(2E12.*O 3 1 TOTBDY=TOTBDY+BDY 30 RETURN END Form AEC 427 (6/68) AECM 3201 U.S. ATOMIC ENERGY COMMISSION UNIVERSITY-TYPE CONTRACTOR'S RECOMMENDATION FOR DISPOSITION OF SCIEUTIF C AND TECHNICAL DOCUMENT ( See Instructions on Reverse Side ) 1. AEC REPORT NO C00-10l8-1192-Report No. 358 2. TITLE Modeling of Domain Growth Activity in Polycrystalline Ferrites 1 TYPE OF DOCUMENT (Check one): Qa. Scientific and technical report l~1 b. Conference paper not to be published in a journal: Title of conference Date of conference Exact location of conference Sponsoring organization □ c. Other (Specify) 4. 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