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(LE/L, &) Major professor Date /?///f/I/7PZ/ ‘. 07639 ABSTRACT SYNTHESIS OF A STATE MODEL FOR HYSTERETIC DEVICES BY Artice M. Davis In its most general setting, the fundamental task which this thesis attempts to accomplish is an investigation of the application of a specific model to the characterization of hysteretic effects. More specifically, an algorithm is outlined for the synthesis of the model parameters from experimental measurements on an actual hysteretic device. In the course of development, a discrete state space repre- sentation is derived for the model. A simulation routine using this state description is then outlined. A more general continuous model is developed, and a state space description for it is advanced. Another simulation a1gorithm--in this case a continuous one—-is presented. AS a consequence of the work embodied here, a computer package is available for modeling hysteretic devices and for incor— porating them into larger networks for simulation purposes. SYNTHESIS OF A STATE MODEL FOR HYSTERETIC DEVICES BY Artice M3: Davis A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOC TOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1972 To Boojum ACKNOWLEDGMENTS It has been said that imitation is the sincerest form of flattery. If this be so then, in a sense, this thesis could be construed as a compliment to Dr. J. A. Resh. An attempt has certainly been made to assume a particular mental posture characteristic of Dr. Resh: an attitude which I term resilience of research. I am thankful to Dr. R. O. Barr, not only for his contribution to the current work, but also for valuable guidance during work prior to this. The other members of my committee, Dr. J. S. Frame, Dr. J. B. Kreer, and Dr. P. D. Fisher, provided occasional sorely needed injections of enthusiasm. A special word of appreciation is extended to my wife, Lalah. Indeed, if degrees were jointly held, half of this one would justly belong to her. ABSTRACT TABLE OF CONTENTS ACKNOWLEDGMENTS. . . . . . . . . . . . . . I. II. III. V. INTRODUCTION.............. 1.1 1.2 THE NNNNN 01990th The Problem . . . . . . . . Background . . . . . . . . . . . . . BEAM-ROD ANALOGY . . . . . . . . . The Basic Model . . . . . . . . . The Resh- Preisach Plane . . . . . . . Mathematical Development . . . . . . . The Process Model . . . . . Preliminary Results and Characteristics SYNTHESIS OF THE MODEL . . . . . . . . 3. l The Definition of the Problem . . . . . 3. 2 Experimental Measurements . . 3. 3 The Search for the Support of the Distribution Function . . . . . . . . . 3. 4 Synthesis Considerations . . . . . . . . 3. 5 Synthesis of the Distribution Functions , , , THE CONTINUOUS STATE MODEL . . . . . 4. 1 The Need for a Continuous Model . . . . . 4. Z The Continuous State Representation . . . . 4. 3 The Continuous Simulation Algorithm . . . CONCLUSIONS . . . . . . . . . . . . . . REFERENCES................. APPENDIX. THE PRACTICAL ASPECTS OF INTEGRATION Page ii 30 30 33 36 59 63 80 80 8O 86 95 98 100 CHAPTER I INTRODUCTION The phenomenon of hysteresis arises in a multiplicity of situ- ations in the physical realm. In such diverse fields as physiology, elasticity, electrical engineering and soil sciencel'4, hysteresis effects appear--and astonishingly exhibit the same class of charac- teristics. This uniformity is a good prognostication for an attempted generalization of a specific model to encompass a wider category. In the work presented here, the immediate tie is to the properties of ferromagnetic material. An underlying current of thought, however, tends toward hypothesizing that the characteristics of this particular process proffer a basis for the representation of more general types of hysteretic phenomena. 1. l. The Problem In most works purporting an explanation of hysteresis effects, one fact is conspicuous in its absence: the definition of the term it- self. Some of the notable features of ferromagnetic material--the shape of the saturation loop, minor loop behavior, and cyclic demag— netization—~are usually listed, but no attempt is made at a general definition. Thus, it is perhaps of benefit to attempt such a clarifica- tion at this point. The principle feature which distinguishes ferromagnetism, mechanical backlash, and the like from other effects is their static nature. Although rate dependent effects are present, they only I”. appear as perturbations to the static characteristics. The attempt here will be the clarification of the essential elements of hysteresis-- those present at extremely low rates of variation. The following definition could be phrased in terms of relations between pairs of variables associated with a multiterminal component or system, but it will be tacit here that the object under consideration is two-terminal. Consider, then, the oriented object of Figure l. 1. 1: Figure 1. 1. 1 X and Y are input and output time functions, and F is the operator associated with the object. Let X range over the space of testing functions such as the one shown in Figure 1. 1. Z. x(t)_ Figure l. 1. 2 In order to qualify as a testing function, a given input X must possess the following property: there must exist two adjacent time intervals (to, t1) and (t1, t2) such that X is monotone in opposite senses in the two intervals. to, t1, and t2 are arbitrary. Now let x and y be the restrictions of X and Y to (to, t2), and let ny be the consequent relation in RxR. Define x(to) = xa and x(t2) = xb, and let M be the maximum value of the magnitude of the slope of x on the interval (to, t2). Then: Definition: If there exist a pair of values xa and xb such that, for each associated testing function with finite M, ny is double valued; and if forX fixed outside (to, t2) ny approaches a limiting double valued rela— tion as M goes to zero, the object will be called hysteretic. Note that this proposed definition precludes such objects as inte- grators and RC circuits from being labeled hysteretic. Although double valued relations ny are produced, they are always dependent upon the exact nature of the input. The definition above character— izes objects which do not have this precise dependence. With the definition disposed of, more practical questions be- gin to arise. Do all replicas of a given object behave in the same manner? If so, can their behavior be generically characterized-- does there exist a model representative of them all? How catholic is that model with regard to the larger class of hysteretic objects? Finally, one is led to ask the very practical ”nuts and bolts" ques- tion: "C an the model parameters be determined by experimental measurements on the object?" Some of these questions have been answered. Empirical ob— servations indicate that all replicas do behave generally in like man- ner. Several models have been proposed, both for specific processes and for hysteresis in the abstract, although their collective merits are moot. The question of catholicity has not been adjudicated. A model for the ferromagnetic process has recently been proposedS, and this thesis attempts to answer the synthesis question for it. In addition, certain features of the model are developed farther-- features such as its state- space representation and simulation characteristic 5 . l. 2. Background The phenomenon of ferromagnetic hysteresis is usually treated by simply ignoring its existence. When it is not ignored, the proce— dure is almost always one involving piecewise linearization of the major loops. Although the area of circuit analysis has been under development since the very beginnings of electrical engineering, there is yet a paucity of theory pertaining to iron core inductors. Several interesting assaults have been launched against the bastion of the problem, but on the whole there has been a dearth of ”doings” in the area. In fact, the different approaches can be discussed seri- atim. These different methods are effectively enumerated by tag- ging them with names: Ewing, Weiss, Preisach, Volterra, Chua, and Resh. The first five will be synopsized in this section, and the sequel will be concerned with a full development of the last. Ewing's Modelé. Ewing's work adumbrated that of Weiss and a number of others. His main contribution consisted of the develop- ment of a physical analogy to the behavior of a ferromagnet. As indicated in Figure 1. Z. 1, his model consists of a number of small magnetic dipoles, pivoted and free to rotate in a plane. —>—>—>—>—> Figure l. 2. 1 In the absence of any external field, their mutual interaction aligns them in such fashion that the potential energy of the configuration is a minimum. Such a configuration is not necessarily unique--one such is illustrated in Figure 1. 2. 1. If an external field is now applied in some arbitrary direction, the dipoles begin to rotate in an elastic fashion about their axes. The term elastic is used in the sense that should the field be reduced, the rotation will exactly reverse. At some critical value of applied field, the assemblage becomes unstable, and an abrupt macroscopic change in configuration results. The change is, as intuitively ex— pected, nonreversible. This particular happenstance is the result of the interaction of the dipoles, and is in such a manner as to reduce the total energy of the configuration. As the applied field continues to increase, there will be a further elastic rotation of the dipoles and other irreversible changes of configuration. A final elastic rotation evinces itself as saturation is approached. If the external field is now reduced, the initial path will not be retraversed by consequence of the previous irreversible changes. The resulting hysteresis loop looks remarkably Similar to that observed for iron. The Ewing model results are depicted in Figure 1. Z. Z. Figure 1. Z. 2 As a physical experiment, Ewing's model is sirnple--the mathematics is more complex. Several investigators.7 have attempted the analysis of Ewing models in one, two and three dimensions. All tries proved fruitless for models whose number of component di- poles was other than trivially small. Another seeming disadvantage of the model is that when scaled to atomic dimensions it gives quanti- tatively incorrect results7. This is perhaps not such an important objection for, as one writer has pointed out8, the physical processes involved bear such close resemblance to those now known to hold for ferromagnetic action. The model remains valid as an analogy, albeit a mathematically intractable one. Weiss' Workg. Weiss' analysis was an attempt to describe the Ewing model in mathematical terms. As a prelude, Langevin10 had derived an expression for the magnetization of a paramagnetic material—-one consisting of magnetic dipoles having no interaction, but which are under the influence of thermal effects. The Ewing model, as mentioned above, is intractable. This complexity led Weiss to postulate as an approximation that each dipole experiences a total magnetic field made up of the external field and an additive component due to the effects of the other dipoles. This additive component became known as the ”molecular field”. Weiss made the assumption that it is proportional to, and collinear with, the total magnetization. Combining this with Langevin's previous results, he arrived at an expression for the magnetization as a function of applied field. Space here does not permit replication of the works of Langevin and Weiss, but the important features can be underscored. This attack led to the first analytical expression for ferromagnetic be- havior. It permitted calculation of thermal effects and led to the postulation of ferromagnetic domains. On the other hand, there were serious flaws. The assumption that the molecular field is collinear with the magnetization vector is too restrictive, and he allowed for no dependence on the crystalline structure of the material. Volterra's Theory”. With Volterra there was an effective shift of emphasis; he attempted to explain hysteretic action as a general phenomenon. Although Volterra seems to have been moti— vated primarily by the elastic behavior of solids, he postulated the validity of his theory on a more sweeping plane. He considered a physical system to be an operator defined on a suitable class of functions. He further restricted himself to linear operators of the special form t x(t) : Ky(t) +5 ¢(t,z)y(z)dz -00 where x(t) is the input function and y(t) the output. For ferromag- netic material, this assumes the form t H(t) : u;IB(t)+S ¢(t,z)B(z)dz -oo where the symbols have the usual significance. q>(t, z) is termed the hereditary kernel. These equations fall into the general category commonly given the name Volterra's Integral Equations of the Second Kind. The pioneering efforts of Volterra and Fredholm were directed toward solution of these equations for the output function in terms of the input. The innate disadvantage of this theory is the absence of any correlation between the describing equation and the physical mecha- nisms underlying the hysteretic process. It will be demonstrated later that there is a correspondence in form with ferromagnetism, but a very Pflfl assumption continues to mar its features. Preisach's Solutionlz. After its enunciation by Preisach, this theory went through a process of evolution, culminating in the work of Biorci and Pescettil3. From a practical viewpoint, their work has produced quite useful results in spite of the limitations to be alluded to later. It is perhaps worthwhile to note that the resulting model bears certain points of contact with the one currently being investigated, although the latter is more general in nature. The Preisach development begins with the assumption that the body of ferromagnetic material under consideration consists of an aggregate of infinitesimal volume elements, each characterized by the relation of Figure l. 2. 3. I is the magnetization of the infinitesi- mal volume experiencing a field intensity H. a and b are allowed to vary from one element to another, but 10 remains fixed. It must be true that a 2 b, since energy can only be dissipated in each element. In addition, no element has a 2 H3 or b S-HS, the saturation field field intensity. It is possible to characterize the model by a region 1 Figure 1. Z. 3 in the (a, b) plane, on which is defined a distribution function ¢(a,b). ¢(a,b)dadb, then, is the number of elements which possess (a,b) pairs lying in the rectangular region centered at (a, b) with sides of length da and db. The planar situation is shown in Figure 1. 2.4. Figure 1. Z. 4 It has been shown that all volumes with defining number pairs (a, b) belonging to AOB go to positive magnetization under a cyclically de- creasing H field. All others go to negative magnetization. If the distribution function 4) is known, it is possible to deter- mine the magnetization as a function of the applied field. Any continuous input function can be broken up into monotone segments, and these monotone segments define resulting regions in the plane. 10 The magnetization is then expressible as I(H) = 2155'; ¢(a,b)dadb Q where Is is the saturation value of I associated with HS, and Q is that region whose elements have undergone an odd number of re- versals. A procedure has been outlined for determining 4) from ex- perimental measurements--if certain restrictions are imposed. A more careful discussion of this theory is contained in reference 13. The Preisach theory has been quite useful to practically ori- ented investigators, but there are several flaws. Primary among them is the entirely irreversible nature of the process. This is a result of the square loop characterization of the elemental volumes. It is tempting to equate these volumes with ferromagnetic domains. They do not, however, exhibit such physical properties as elastic wall growth which domains are known to possess. Thus, the model must be considered as an abstract analogy to hysteretic behavior, and its application must be limited to materials in which the ir- reversible processes are dominant Chua's Methodl4. Chua, in the manner of Volterra, attempted to model the general phenomenon of hysteresis by mathematical ab- straction. He proposed the nonlinear differential equation g = g[x(t)- f(y(t))] for which he proceeded to demonstrate hysteretic behavior. In order that they be realizable, the functions g and f must be odd and invert- ible. Chua outlines a simple procedure for deriving f and g from measurements on the saturation loop. This implies that devices with identical saturation loops must exhibit identical minor loops. It 11 would appear from experimental observations that this is too sim- plistic a view. In addition to this, the model also implies that a complete set of state variables is provided by the instantaneous values of the in- put and output. Experiment indicates that, at least for ferromag— netic devices, there is a major loop passing through each point (x,y) inside the saturation loop. Thus, assumption of an initial state (x,y) should immediately result in the traversal of that loop. Simulation of the Chua model does not result in such closure--see Figure 1. Z. 5. The solution spirals inward and asymptotically approaches a steady state loop. Discussion of Previous Work. In all of the prior work, one fact is evident: with the exception of Chua, no investigator has ap— proached the subject from an engineering stance. In all cases, strong features were nullified to some extent by debilitating ones. An additional factor exhibited by all these models is the characteris- tic of abstract analogy to physical hysteretic behavior—-not necessarily a point of weakness, and one shared by the present model. The model to be considered next appears to obviate many of these short— comings. The state characterization techniques and synthesis procedures developed for it appear to extend to some of the previous models as well. fi\, CHAPTER II THE BEAM- ROD ANALOGY As mentioned previously, all of the hysteretic models to date have possessed inherent shortcomings. They have run the gamut from the abstract to the concrete, but there has been a common thread of correspondence, in the main, among them. The disappoint- ing feature is the lack of any sort of chronological evolution. In 1969, however, Resh5 delivered a paper describing a model which appears to consolidate the features of the previous models, while improving upon and generalizing their capabilities. This correspondence is 3 posteriori, and is deeply imbedded in the theoretical aspects of the model, but definitely exists. Hence, its more general nature can be shown. This section is devoted to the exposition of the basic model and its ancillary aspects. Following Resh's development, the basic model will be described; afterward its correspondence with the physics of ferromagnetism will be delimited. Some related con- ceptual and computational features will then be developed. 2.1. The Basic Model5 The essential feature of the model investigated here is its form as a mechanical analogy to the ferromagnetic process. Its configura- tion is illustrated in Figure 2. l. 1. It consists of a set of flexible beams attached to a rigid surface P and a rod R which moves parallel with the surface. There are two types of beam: one variety is 955 Figure 2. l. l engaged by the rod as it moves past the beam's equilibrium position from left to right, and the other kind is picked up as the rod moves past it from right to left. The former type of beam will be dubbed "right active" and the latter "left active”. The equilibrium posi- tions of the beams are distributed in some manner, and the drop-off positions and coefficients of elasticity are allowed to vary from one beam to another. While a given beam is engaged by the rod, its force versus flexure relation is assumed linear-~the beam merely behaves like a linear spring affixed to the rod. There are two important characteristics which should be noted at this point. The nature of the flexing process of an engaged beam is elastic and reversible until the drop- off point is attained. The other item of note is the autonomous nature of the process when the beam drop point is exceeded. This represents an energy dissi- pation activity which is decoupled from the input. These two features provide, on a macroscopic energy process level, a tight correspond- ence with solid state theory. The current status of ferromagnetic theory is such that a qualitative description of the magnetization curve is available. In the virgin, or undisturbed, state, a body of ferromagnetic material can be thought of as consisting of elementary dipoles. These dipoles are arranged such that all dipole moments in a small volume element, 15 or domain, have the same direction. The boundary of each volume, called a Bloch wall, consists of a layer of atoms several atoms thick. In the Bloch wall, the magnetization varies smoothly in direction from one side to the other, with parallelism being main- tained on each side with the contiguous domain magnetization. When an external field is applied and increased from zero, whichever of two adjacent domains has its magnetization vector closest in direction to the applied field begins to engage in territorial expansion at the expense of its neighbor. The atoms in the Bloch wall begin to rotate in order to align themselves more closely with those of the burgeoning domain; hence, with the applied field. This process is, of course, being replicated thousands of times through- out the mass of material. If the applied field continues to increase, a point is eventually reached such that less energy is required to maintain the configuration if those domains which have magnetizations more-or-less antiparallel with the applied field flip those magnetiza- tions by pi radians. This activity, when initiated, proceeds autono- mously, with energy being irretrievably lost by the concomitant eddy currents. All these reversals do not, of course, proceed in unison. They are, on the contrary, distributed with different critical values of field; this produces the discontinuities known as the Barkhausen effect. As the process continues, another type of activity is initiated: gross restructuring of individual domain geometry. This is also an irreversible stage in the process. Finally, as saturation is approached, all domains rotate their magnetization vectors elastically to pro— duce final alignment with the applied field. 16 The preceding discussion indicates that there are two process— es at work--one reversible, the other irreversible and autonomous when initiated. It is clear that the beam- rod analogy exhibits the same type of loss mechanisms. Thus, it is evident that its corres— pondence with ferromagnetic behavior is tight. It also seems alto- gether reasonable to postulate these mechanisms as being responsible for hysteretic effects in general. Indeed, it appears that such ef- fects have been observed in at least one other context-~the elastic behavior of solids”. Z. Z. The Resh- Preisach Plane A one-to— one correspondence can be set up between the model and a parameter plane with an associated distribution function. There are some points of similarity with the plane of Preisach's model, but the interpretation of the distribution function is different. This will become clear subsequently. Consider the plane sketched in Figure 2. 2. l. Figure 2. 2. 1 Each beam in the model can be represented as a point in the plane plus a number representing its coefficient of elasticity. If a beam in the model has equilibrium position ai, drop off point bi’ and .‘lvr l7 elasticity coefficient Ki’ it will be represented by a point (ai, bi) in the Resh—Preisach (or RP) plane. The point in the plane and the constant Ki suffice to characterize the beam. The rod position x can be represented as a point on the a axis. If the rod has increased its position monotonically from - co to x, the resultant force on it will be the sum of the forces due to those beams which have been engaged but not dropped. Mathematically, such beams can be de- scribed as those with ai < x and bi > x. In the plane, such beams occupy the shaded region of Figure 2.. 2. l. A moments reflection will reveal that the half-plane above the b = a line represents right— active beams, and the lower half-plane left- active ones. The preceding state of affairs is evidenced when the rod moves monotonically from — 00 to x. Suppose now that it increases mono- tonically from - 00 to x0 and then decreases from x0 to x < x0. The rod will then start to "back off" from those right- active beams still engaged at x0, and it will begin to acquire new left- active ones. This situation is sketched in Figure 2. 2. 2, with the shaded regions again representing active beams; that is, those which are engaged and delivering a component of force to the rod. Figure 2. 2. 2 18 Following this line of thought, one can break up any continuous input function into monotone segments, thus determining the region of active beams for a given input function at any instant of time. One of the most important effects which any model should ex- hibit is that of cyclic demagnetization. Resh demonstrated such an effect for the beam- rod model by assuming an input possessing a se- quence of adjacent intervals of monotonicity (of alternating senses), such as a sinusoid, whose peaks smoothly decreased in amplitude from infinity to zero. The resulting region of active beams is shown in Figure 2. 2. 3. In like fashion, one can demonstrate cyclic mag- netization effects, a virgin magnetization curve, and the cyclically demagnetized remagnetization curve. b Figure 2. 2. 3 Z. 3. Mathematical Development It is possible, then, to determine the region of active beams for any given input function and any instant of time. The primary item of interest, however, is the force versus displacement function. The past discussion has centered about intrinsically geometric fac- tors: the beam equilibrium positions, their drop— off points, and the position of the rod. At this juncture, the question of magnitudes must be answered. How does the distribution of coefficients of elasticity from one beam to another influence the force versus l9 displacement relationship? Before exploring the possible answers to this question, a detour will be made for the purpose of developing a continuous version of the model. In order to facilitate later investigation, it is essential that the model be generalized. Suppose the number of beams is allowed to approach infinity and each coefficient of elasticity allowed to tend to zero in such a way that the effective coefficient of elasticity per unit area. of planar surface remains constant. In this manner, one is led to consider a continuous model. A function 4) can then be de— fined on the plane such that c’p(a, b) da db is the effective elasticity coefficient of a beam at (a,b) equivalent to the mass of 4) in a rec— tangle with centroid at (a, b) and sides of length da and db. Con- sideration of the force due to one such element and summing over all such infinitesimal elements yields the force: f(x) : 55 (x - a)¢(a,b)dadb Q where Q is the active region as previously defined, but extended to the continuous case. The continuous model is appealing; for on a macroscopic level, an actual ferromagnet does exhibit smooth behavior--the discontinuous structure is of a "fine grain” nature. In addition, the continuous model provides an effective tool for both conceptual and computa- tional purposes. Although the beams in the model have lost their structural identity in the transition to the continuous model, the terminology will be retained. The background idea is that the model has a basically discrete character, but that the number of beams is so large that it is more convenient to think in terms of a continuum. 20 Returning now to the discrete model, it is evident that the total force on the rod is the sum of the forces due to the active beams: f(x) : Z (x- ai) - Ki Active Region where the parameters are the same as before. It is possible to re- write this as f(x) = Z SiKi(x - ai) where the sum is taken over the entire planar aggregate, and Si has the appearance of a state variable associated with the ith beam: 0; beam i unengaged Si(x) : l; beam i engaged The corresponding state—like variable of the entire model is then the Cartesian product of the individual ones, and it can be written as a vector §:[s s T 1, 2,...,sN] . The above expression is an input— output state relation and takes the form f(x) = STK(x 1- a) where; — [a1,a2,.. ,aN] ,—l = [1,1, ,1]T, and K1 0 . 0 K = .0 K1 . 0 0 0 K 21 With the input- output state relation disposed of, it becomes necessary to determine a state transition function or algorithm. This is necessary for the unique determination of the force in terms of the displacement. Although it is possible to go through an active beam pattern analysis for each monotone segment in an input func- tion, the nature of the state vector indicates the utility of a modular approach. Each beam can be analyzed for state transitions and then incorporated back into the aggregate to determine the overall state transition. The above plan of attack requires a state model for an indi- vidual beam. Since there are two types of beams, there are two varieties of state model. Some consideration of the beam- rod analogy reveals the efficacy of the following pair: Left— Active Right— Active Figure 2. 3. 1 As an example, consider a right- active beam. If the beam is un— flexed, Si is zero. If the rod position is greater than the equilibrium position of the beam, the state will remain the same. This will ob- tain until the rod position becomes less than the equilibrium position of the beam, in which case the state goes to ”enabled” and Si remains 22 zero. Then any motion of the rod back above the equilibrium posi- tion will cause the beam to enter the ”flexed" state, with Si 2 1. Similar verification can be made for the other transitions shown. In actuality, it is evident that Si is not a state variable. It is a function defined on the state space of beam i. The state of a beam can be written as a variable which assumes the values Flexed, E- flexed, or Enabled. If Si were invertible, it would also qualify as a state variable. Unfortunately, this is not the case, but the vector S serves essentially the same function as a state vector in the input- output state relation. The state vector of the model is a vector con- sisting of the entries Unflexed, Enabled, or Flexed in each position. At any rate, a complete state model is available for the finite beam discrete analogy. 2.. 4. The Process Model It is possible to derive a model for the ferromagnetic process which is a direct consequence of physical theory. Its very generality does not allow its use as a quantitative solution, but its conceptual power will be evident. It is introduced here to coalesce all the mod- els which have been discussed (including the bearn—rod analogy) and to provide a framework for their comparison. Consider the magnetization process for a uniform ferromag- netic toroid which is isotropic and homogeneous. If a current carrying conductor of small diameter is wound around the torus, the applied field intensity can be shown to be axial with respect to its cross section. In consequence of this, as well as the absence of physical boundaries, the magnetization is uniform. 23 The current carrying conductor produces a magnetic field intensity H = Ba/po, where Ba is the induction which would result if the torus were not present. The factor Ba (or H) can be considered as the motivating force for the ferromagnetic process. This disturb- ing influence causes the magnetic dipoles to begin to align themselves as outlined previously. This creates a resultant induction field which adds to Ba' This perturbed field creates further alignment, and so on. The resulting process-derived model is outlined in Figure 2. 4. 1. “o C(- Figure 2.. 4. 1 As indicated, the operator F is a functional relationship between the resultant induction field and the magnetization M. Note that this model is only valid if the magnetization can be characterized by a single number. In particular, if M is a spatial variant, the induc- tion field at a particular point depends upon the entire distribution of magnetization. Consider now the relation between this general model and those previously outlined. Suppose first that the operator F is given by t F (x(t)) = - S ¢(t,7)x(’r)d7 ’ -oo Z4 and GE 1. Then, M(t) t - 5-0:“,7) [Batu + no M(T)] d'r But B(t) = Ba(t) + no M(t), so t u;1[B(t) - 3am] = - X_Oo¢(t.7)B(7)d'r or finally, t H(t) = u: B(t) +5 ¢(t,T)B('T)d'r which is precisely Volterra's equation. Thus, Volterra's model is the special case which results when the operators F and G assume the special form indicated above. Now consider Chua's model. The proposed differential equa— 3%: = g[x(t)- £(y xm. In addition, the input- output relation is linear in the saturation region. This implies that no additional beams are acquired for [X] > xm. A farther feature is the fact that the relation does not consist of the horizontal axis (that is, the mmf, or force f, is nonzero) for [X] > xm. This implies that beams with infinite drops do exist in the model. These facts coupled with a few thought experiments imply the region of Figure 3. 3.13 for S(¢). 3. 4. Synthesis Considerations With the knowledge of the most general region possible for the support of (I), it becomes feasible to consider the more concrete problem of synthesizing the model parameters. As was indicated in Figure 3. 3. 13, it is convenient to think of 4) as representing the beam mass distribution in the square, and of two other functions r and s as representing the two linear distributions at :t 00. When considering the discrete model then, it is convenient to simply imag- ine these functions as being evaluated at a finite set of points. If these values are multiplied by an area weighting factor, there arises a discrete approximation to the continuous model. This is simply the inverse of the process used to make the transition from the discrete to the continuous model. In fact, if the continuous model 60 can be synthesized, it is possible to immediately make such an approximation--thereby allowing the Barkhausen effect to reappear. For the purpose of mitigating the synthesis problem as much as possible, it becomes desirable to farther simplify the model by introducing reasonable assumptions such as, for example, symme- tries of the various distribution functions. An added simplification is possible for the boundary distribution functions. To investigate this, it is necessary to consider the expression for the force con- tributed by the boundary. This is readily seen to be X x m g(x) = y (x- a)r(a)da+S (x- a)s(a)da -x X m The corresponding expression for the discrete model is: g(x) = Z (x- ai)ri +2 (x- ai)si a..<_x a.>x 1 1 where the ai are the equilibrium positions for the discrete model and ri and si are the elasticity coefficients of the top and bottom boundary beams. It is interesting to consider the derivation of these boundary distributions if g(x) is a prescribed function. Note that, on the con- trary, g(x) will not in general be known. The assumption is merely made for the purpose of deriving properties of the boundary distri- bution functions. Under this proviso, then, differentiating the first of the above expressions twice results in g”(X) = f(X) + S(X) The most striking feature here is one of redundancy: a single func— tion would suffice. If either r or s were identically zero, the re- sulting force component would be of constant sign (assuming the other 61 function is always _>. 0)--this conflicts with intuition and the under- lying physical mechanisms. In fact, it would imply a very strong asymmetry in the saturation loop. Hence, it appears that r and 5 should possess an alternation property--that is both are never zero simultaneously. A more explicit form will be presented later. Turning to the other simplification mentioned above, it is pertinent to inquire as to possible symmetry of the function <1). In measurements on actual devices, it is observed that should any se- quence of inputs be negated in sign, the corresponding output func- tion experiences a similar sign inversion. Since the boundary dis- tribution is memoryless, a little thought will convince one that this sign characteristic should also hold for the force due to the function <1). One symmetry condition which furnishes this behavior is given by ¢(—a,-b) = ¢(a,b) It is possible to use this symmetry condition to explicate the nature of the boundary distribution. If the saturation loop is swept out, the resulting forces can be written: f+(x) : g(x) + 6+(x) f-(X) = 80!) + 6"(X) where f+(x) and f-(x) are the saturation loop forces with x increasing and decreasing, respectively; g(x) is the force due to the boundary distribution; and 6+(x) and 6-(x) are the body contributions, also for x increasing and decreasing, respectively. Now the symmetry con- dition ¢(-a,-b) = ¢(a,b) implies that 6+(x) : - 6-(-x). This can be seen by inspection of Figure 3. 4. 1. Then, manipulation of the above pair of equations results in 62 ,//// W/ -X Z V/A Figure 3. 4.1 63 f+(x) + £’(-x) = g(x) + g(-x) In light of the above discussion of the ”oddness" of hysteretic be- havior, f+(x) = -f_(-x). Hence, g(x) = -g(-x), and the function g is odd. This implies the boundary distribution indicated in Figure 3. 4. 2. The basic characteristics of the model have been made con— crete, and the stage is set for expounding the method of synthesis of q) and r. This question will occupy center stage in the next section. 3. 5. Synthesis of the Distribution Functions There are a number of techniques which suggest themselves as answers to the synthesis question. The prominant characteristic with which the successful candidate must be endowed is practicality. A number of different methods were considered along the path toward selection of the one presented in the sequel, but all others suffered badly when extracted from the theoretical world. In truth, several seemed quite attractive when considered from the viewpoint of mere theoretical plausibility alone. Their downfall came at the hands of the ever—present demand for reasonable core storage and computation time on the digital machine. The one presently to be described possesses reasonable characteristics in those areas; it also provides excellent synthesis capabilities. One of the most reliable weapons in the numerical analyst's arsenal is the polynomial approximation. The usual usage is in providing an approximation to a scatter of data or to a more complex function. The main requirement which the data must meet is the possession of a compact support--if the fit is to approximate the entire data set. For instance, Weierstrass' Theorem in one 64 b r(a) ¢(a,b) r(- a) Figure 3. 4. 2 '|p 65 dimension is valid only for a closed interval of finite length. Of course, polynomial approximations are feasible in two dimensions, where the support must be a compact set. The importance of the previous effort in search of the support of 4) is now apparent. The function «1) is defined and nonzero only on the closed square (a com- pact set), and the function r is defined and nonzero only on a closed interval of finite length (another compact set). Thus, polynomial approximations would be practical if the functions 4) and r were known. But, this is precisely the synthesis question itself! How then can the possibility of polynomial approximations be exploited? Suppose the existence of such an approximation is as sumed--— that ¢ and r are continuous. For a given degree of approximation, the problem devolves to one of determining a given number of coef- ficients. A number of measurements are needed which will allow these coefficients to be determined. The question is, ”What kind and how many?" Of course, since a least squares solution is contem- plated, the answer to the latter part is, ”As many as feasible. " To acquire a grasp on the former requires a little more effort. With this objective in mind, consider the planar representation of Figure 3. 5.1(a). The square is partitioned into N2 equal sub- squares, and the interval (O,x ) into N/2 equal subintervals (by m construction N is even). The function (I) is assumed to be represented by discrete beams centered in the small squares, and r by beams centered in the subintervals. If x is allowed to increase from -00 to xj, 'then to decrease to Xi’ those beams in the shaded regions will be engaged. This measurement process is illustrated in Figure 3. 5. 1(b). There are (N (N- l) )/2 unknowns in the square and the same number of m -00 a) Partition of Support f b) Semimajor Loop Measurements Figure 3. 5.1 67 possible measurements. Note that those beams for which b = a have no effect and can be ignored. The boundary beams, represented in con- tinuous form by r, constitute an additional N/2 unknowns. Figures 3. 5. 2(a) and (b) represent a possible means for including an addi- tional N/2 measurements by way of the cyclically demagnetized remagnetization curve. The purpose of the preceding discussion is to motivate and heuristically justify the measurements to be performed for input to the continuous model synthesis procedure. There are NZ/Z measure- ments and N2/2 unknowns. It seems that these NZ/Z measurements would suffice to determine the discrete model parameters, but their independence has not been verified. Fortunately, the synthesis pro- cedure to be outlined does not require independence, although the supposition will be made that this type of measurements provides enough info rrnation for the continuous model. This, too, is non- critical, as will be brought to light later. Although the measurements have been decided upon, the ques— tion of synthesis is still open. To fill this hiatus, the continuous synthesis procedure will now be explored. Utilizing the previously stated assumption of continuity of both 4) and r, it is possible to hy- pothesize the existence of polynomials P(a,b) and Q(a) approximating 4) and r uniformly to an arbitrary degree. Suppose Nil N+1 P(a,b) = 2 Za..al-le-l 1.1 i=1 j=1 N+l 1-1 and Q(a) : Eb a 68 a) RP Plane b) CDRC: Cyclically Demagnetized Remagnetization Curve Figure 3. 5. 2 69 where the range of indices has been selected for ease of program- ming on the digital machine. Then, the resulting approximation f' to the force versus displacement characteristic can be found for any sequence of input variations. For the set of measurements outlined above, there are four requisite force expressions: the ascending saturation loop, that due to decreasing input after a monotone in- crease from -oo to some value xo (semimajor loop); the cyclically demagnetized remagnetization curve (CDRC); and the force in the saturation region (lxl Z xm). The last form is obtained from any of the others by inclusion of only the expression for the force due to the boundary distribution. The different cases will be explored one by one, but first the symmetry condition on 4) will be translated into a requirement on polynomial coefficients. The stipulation that ¢(-a,-b) = ¢(a,b) implies that N+1 N+1 Z a'ij[1- (4)1“) a1‘1b3'1 3 0 i=1 j=l Hence, aij : 0 if i+j is odd. This will, of course, significantly en- hance computation speed. The following expressions are derived from consideration of events in the Resh-Preisach plane. The Ascending Saturation Loop: x Xm f'(x) : S day (x — a)P(a,b)db -xrn x + sgn (x) r(x - a)Q(a sgn(x)) da 0 where the latter term is the boundary force, which can also be writ- ten in the less compact, but more intuitive form 70 (x — a)Q(a)da; x Z 0 o g(x) = o g(x- a)Q(-a)da; x<0 x Then, N+1 N+1 x x m '-l ._1 f'(x) = E 2 any day (x-a)a1 bJ db 1‘] -x x i: j:1 m i+j even N+1 x ._1 + Z bi. sgn(x)5> (x- a)(a sgn(x))1 da i=1 0 or N+1 N+1 N+1 l _ f (x) .. Z Z aij Hij(x) + Z biBi(x) i=1 j:1 i=1 i+j even where x X m ._1 .-1 H..(x) =§ day (x-a)a1 bJ db 13 -x x m and x . l Bi(x) : sgn(x)§ (x— a) 0, the lower right corner of Q+ will fall on this line. The regions Q+ and Q- 83 exhibit very regular features: in fact, they are horizontal— and vertical- edged polygons in the plane. As such, they are completely determined by their ”convex corners"; i. e. , those which protrude from the regions. The ensuing development will describe this repre— sentation, and will elaborate upon the state space of the model. In order to fix ideas, assume that the region under discussion is Q+, the region of engaged right-active beams. The properties of Q— are exactly analogous to those of Q+. The upper boundary of Q+ is a line segment with right endpoint at (x,xm). The left and right boundaries are vertical line segments whose endpoints will shortly be described. Perhaps the most lucid description of the region Q+ is a se- quential account of the manner in which it takes form under a proto- typic input waveform. A representative sequence of events is sketched in Figure 4. 2. 2. There are several important features of note. The convex edges, other than perhaps the sidecorners, reside at points in the plane given by (1 i’ r. 1), where {Ii} is the sequence of points within the interval (-xm,xm) where x(t) ”turns around" from left to right; that is, where x(t) changes from monotone decreasing to mono- tone increasing. {ri} is the sequence of right ”turn—around" points. The bottom left-most corner is given by (—xm, r1) if the first turn- around was on the right, and by (f 1’ r1) if the first turn- around was on the left. The bottom right-most corner is given by (x, r last)’ where rlast is the last turn- around on the right if x is decreasing, and by (x,x) if x is increasing. The principle result of such a description of the active beam regions is an analytical definition of the state of the model--it is 84 b b p—b X0 X0 :0 2 x0 x1 x0 a x1 x2 x0 a X1 X1 ——-1 —-¢--t X X X X xol/ /\ Ixo x2 ‘0 t t x / x1]\ 1 b b b l— L—— x x x0 X 0 X' X2 3 x22 x x43 x x0 x2 X0 X3 x0 X3 a X3 X2 x; ‘1 x1 x1 x1 , _.L —~x4 X X X *2 /\ A x /\ m K t V X3 t t Formation of Active Beam Pattern Under a Prototypic Input. Figure 4. 2. 2 85 difficult to manipulate patterns. The two sequences {1i} and { ri} above can be combined into one; namely, {xi} , the set of turn- around points of x(t). One other item must be available, as men- tioned above, for a viable state description. This quantity is the sign of the derivative of x, sgn(x(t) ). An alternative is to keep record of the nature of the last turn- around; that is, whether it was of the left or right kind. Hence, an element of the state space of the model is given by the pair (v,x), where v = sgn(x), and; = {xi} , the turn- around sequence. Now that the nature of the state space is clear, the only task remaining is the derivation of an algorithm for finding the state for any input function segment, given any initial state. Note that the state must yield to some rather obvious restrictions: there cannot be two successive turn- arounds of the same kind, and so forth. Suppose then that an initial state is given. This means specifically that the sign of the velocity at some time to is given, as well as the last left and right turn— arounds. Then, a little contemplation of such a diagram as Figure 4. 2. 2 indicates the following: 1. If x(t) exceeds either the last left or the last right turn— around, M are deleted from the sequence 32. The sequence is then reindexed so that the previous next- to-last pair is now the last pair (in indicial terms). 2. If x(t) changes sign at some time t', without hav- ing committed an excess as described in part one, x(t‘) is appropriately assigned as the new last element after the sequence is reindexed. 86 3. If x(t) exceeds either +x or -x , the sequence m m becomes vacuous, and the model is input- determined. Although a state transition function is not available in closed form, the above discussion provides a basis for a practical state transition algorithm. Its implementation will be considered in the next section. 4. 3. The Continuous Simulation Algorithm Once the nature of the state space has been brought to light, it is important to consider its practical usage. The most pressing need is for a simulation routine which will accept the model para- meters and deliver, for an arbitrary input, the output time function. This is important, both as a tool for checking the validity of a syn- thesized model and as an adjunct to other routines in a general net- work simulation package. In the digital machine, the time variable ranges over a dis- crete set; this implies a somewhat simpler method for finding turn- around points. In the continuous time case, a turn- around point is a value xi such that x(t) = xi, x(t) = 0, and 5t(t) 7! 0 for some t. In the discrete time case, a turn- around point is a value xi such that x(tk) : xi for some value of k and either 1. x(tk_1) < Xi’ x(tk+1) < xi or 2.x( >x,x( >x tk- 1) i tk+l) i No derivatives need be calculated. The direction of motion of x is available by storing the character of the last turn- around point. If this is done, it is only necessary to compare the two most current points. 87 The procedure for modifying the state was outlined in the pre— vious section, but there are some special cases. If the saturation loop is being traversed, there is only one active beam region. In this case, a turn- around indicates the inception of a new region. The theory still applies, but special care is required in its implementa- tion. The fact that this can occur as a consequence of either of two types of turn— arounds compounds the complexity. These happen- stances are provided for in a straight-forward fashion, but some care is required in the programming. The logical flow diagram for state transition determination in the simulation algorithm is pre- sented in Figure 4. 3. 1. Other details of the algorithm are straight- forward. Once the state is found, the output can be evaluated by solving the input- output relation for a given instantaneous value of the input. If the model under consideration is discrete, this involves the evaluation of a finite sum; if the model is continuous, a double inte— gral must be evaluated. This is the only modification required when changing the algorithm to accept one or the other--the state transi— tion algorithm remains the same. In the evaluation of the input- output state relation (in either case), two techniques can be used: the integral or sum can be taken over the entire active region at each iteration, or it can be calculated only over the regions of newly acquired and recently dropped beams. In the latter case, the process is more susceptible to accumulation of error. The former case is better from a numerical point of view, but is slower. The differential type of algorithm was selected for implementa— tion because of its speed. Figure 4. 3. 2 shows the nature of a typical 88 Start Calculate Left ls x Right Calculate A —-—1 _— Inside A Force ‘xmlxm Force Inside Modify as There Modify State F3, {Trainer ———§.r°,§1 State Vector e .twnfixclt ? 1g Vector None Calculate Calculate Force Is x Force Movmg Ri ht or eft? A Left Right A Two Two A means. end - . of iteration. Regions Regions . dif‘ d or One? or One? x 15 mo 1e ' and another iteration occurs. One Two One Two Is There a Is There Is There a Is There a Change of Change of Change of Change of State? State? State? State? Yes What Kind ? Turn xcess aroun odif 0 A State? State, Cale. Gale. Force Force A A State Transition Logic for Continuous Algorithm Figure 4. 3.1 89 change in the active beam region. Utilizing the discrete model for comparison with the discrete algorithm, the continuous version proved to be on the order of six to ten times faster for typical in- puts. No problem was encountered with error accumulation. The blackboard of past history is erased when operation enters the satur— ation region; but for long— duration inputs which do not drive the de- vice into saturation, error accumulation could become significant. The algorithm presented above is complete; that is, given the initial state and future values of the input, the future output can be determined. For the studies being described in this thesis, how— ever, it was not essential that the initial state be provided. All inputs utilized possessed the happy feature of starting in one of the saturation regions. In this region, the model is input determined, so no initial state was required. As soon as the operation enters the non-saturation zone, the algorithm commences to generate its own states. Hence, the flow diagram of Figure 4. 3. 1 does not pro- vide for inserting an initial state. One item of interest pertaining to operation of the algorithm is the fact that, theoretically, the sequence associated with the state can be of infinite extent. In View of the fact that machine storage is finite, something must be known about the nature of the input wave— form so that storage bounds can be set. In the absence of such knowledge, a reasonable amount of storage can be allocated, and a monitor can be set up to warn of imminent overflow. It is possible to summarize the advantages of the continuous algorithm by reiterating the fact that it subsumes the discrete model and allows for greater generality, both computationally and 9O &\\ -7 +: beams acquired -: beams dropped Change of Active Beam Region Figure 4. 3. 2 91 conceptually. Several model characteristics are plotted in Figures 4. 3. 3 and the following two in order that those assertions be sub- stantiated by graphical evidence. 92 Figure 4. 3. 3 Continuous Simulation of Finite Beam Analog 93 Continuous Simulation of Finite Beam Analog with a Large Number of Elements. Figure 4. 3. 4 94 Figure 4. 3. 5 Continuous Simulation of Polynomial Fit to Artificial Data. CHAPTER V C ONC LUSIONS In the course of the investigation described in these pages, a number of important advances were made. Beginning with Resh's basic model and continuing through the development of a rather com— plex, but efficient, simulation algorithm, a continuing path of pro- gress was traversed. That path has not halted in a cul- de-sac, but has branched out into a multiplicity of paths. Several seem to beckon with the promise of dramatic developments ahead. The View back along the path reveals the following concrete results. A discrete state model was developed for the original finite beam analog. It was used to more fully explore the capabilities of that model, as well as to derive the support of the characterizing beam elasticity density function. A continuous analog was derived by proceeding from the finite—bearn version. A synthesis algorithm was developed for the continuous model, and a set of experimental measurements was developed to support that procedure. Finally, a continuous state algorithm was developed which provides a fast, efficient simulation of either the discrete or the continuous analog. The mainstream of the work embodied here flowed along three channels: theoretical, laboratory experimental, and computer simu- lation experimental. Transition was made frequently from one area to another in order to bulwark an idea conceived in one with results from another. 95 II’ 96 As a result of the work done here, it is possible to experi— mentally characterize iron core inductors with either hard or soft cores; to synthesize the model parameters; and, utilizing the con- tinuous state algorithm, to incorporate these inductors into a larger network for simulation purposes. The techniques developed here are also directly applicable to the most important of the earlier models, that of Preisach. At the present stage of development, it is possible to glimpse a few outlines of developments along the paths ahead. As work has progressed, an intuitive idea has been taking form of the general nature of the distribution function (1). It should be non— negative definite and should peak near the b : a line. It will perhaps be pos— sible to utilize this characteristic to advantage. Another intuitive idea is the following: Magnetic dipoles, gas molecules, and the like, are distributed in number exponentially with energy. The function ¢(a, b) can be considered as that density function such that ¢(a,b)da db is the number of beams of unit elasticity coefficient in the elemental area da db. The maximum energy stored in one such beam is given by—é—(b — a)2. Hence, it is reasonable to assume that b-a o- -< ¢y(t) where the operator F is defined by t F(x,t) = §x(a)da VtZO o The LaPlace transform of the output time function y(t) is related to that of the input by 1 ms = g(xe) +y(0)) or assuming y(O) : 0, l Y(S) = §X(S) The integrator can thus be represented by a transfer function G(s) = 1/s. xme ms)»;- Y Now consider the practical aspect of realizing G(s). This is impos— sible to perform exactly, for the pole at the s—plane origin represents 100 101 an infinite dc gain. This fact implies that any attempt to integrate a variable will face practical obstacles. The obvious, and probably the best (in some sense) approxima- tion to G(s) is by a transfer function which consists of a simple pole perturbed from the origin by a slight amount. jw jw G(s) X G(s) Then, 1 S-6 /\ G(s) 2 Now 6 cannot be perfectly arbitrary, for the dc gain is now - —:—, and it must be real if G is to be realizable. The case where E > O is also ruled out because this would imply instability. Hence, G(s) can practically be approximated by the above G(s) for 6 < O, and the approximation converges to the ideal form as 6 —> 0—. Reverting to the time domain, the transform version of the input- output relation (8 - C)Y(S) = X(S) becomes %- 61' = X(t) Thus, the imperfect integrator can be modeled by: x(t) a Y(t) 102 It is found that in practice virtually all integrator circuits can be modeled by the above abstract form, or by the differential equa- tion d atWY = x Note that M = -€ :11, where A is the amplifier gain. The generality of this abstract integrator model can easily be seen by consideration of a number of examples. It should be observed that such examples tend to fall into two categories: those whose im— perfection is due to imperfect elements and those which contain ideal elements, but are imperfect for structural reasons. The addition of imperfect elements to the latter category merely exacerbate the s ituation . Example 1: ei(t) CI 60“) This circuit is an old friend, almost a party regular, because of its simplicity and passive elements. Summing currents at the out— put node results in 01‘ 103 If the substitutions y = eo and ei/RC = x are made, it is evident that l . E. This circuit falls into the second category, for it is imperfect even though the governing relation is of the form 3% + py : x with u, : the components are assumed ideal. A further disadvantage stems from the fact that as H -> O, x —> 0. Thus, even though the amplifier gain in the abstract model is increasing, the attenuation in the actual circuit is going up. Example 2: '— _ _ ’1 lr— I J“ l [Kl—— leaky capacitor ‘V i“) “' RIEI D II [E 2' 60m | l L. _ _ __ _J — In this circuit, if the capacitor were ideal, the integration would be perfect, for the terminal relation for a capacitor is t e(t) =—51(a)da + e(O). With this configuration, though, the capaci- o tor is "leaky”. Summing currents, de i(t) : C dt e +3 01' 22+;e . 1. dt RC 0 C which is in the general form with, once again, H :-1&;-. H —+ 0 and the capacitor becomes ideal. In this case, x : El and no AsR—soo, , input attenuation results as H —> 0. 104 Example 3: As a final example, consider the operational amplifier inte- grator with finite input impedance. ,Since the interest here is in very low frequency inputs, this is adequate allowance for non-ideal charac- teristics. Then, summing currents at Sj’ ei-es (1 es “in. +Cat‘eo‘es’ :5.- But,e =e /A, so S 0 e de 1 l l l l o n'(§+ri) Keo+c(1"§)dt -0 or 9:2 __1___._1+l . - i dt‘(A-1)C R R1 o“R Hence, the general equation is once more satisfied with: e. R+R. 1 1 Y-e’ Xurv Mm A is ordinarily negative, and as A —> — 00, H —> 0+. There is no in— crease in attenuation. Since operational amplifiers are readily available with extremely high gains, this configuration is very attrac- tive. Note, however, that it is of the second category, because if A is finite, H )5 0 even though Ri ~> oo. 105 Analysis of Error: Consideration of the general imperfect integrator equation d at + W = x once again yields the solution t y(t) = Sle'lla'a )x(a)do. o where the initial value of y is assumed to be zero. The departure from ideal is given by the multiplicative factor e_“(t-(1 ). If H is very small, this term will be close to unity over the range of inte- gration——if t is not too large. It is apparent that p is a direct mea- sure of the quality of the integrator. Now, suppose the above equation is integrated by parts. There results: t t a _ (t ) y(t) : yx(s)do —S‘ [‘8‘ x(¢)d T. Then, for t > T, T y(t) : S‘x(0')do + 6(t) o with 106 Thus, T y(t) : e-“(t-T) [S(T) + yx(o)d0] 0 Hence, the output can be written y(t) = y(T) e'u‘t’T’; t2 T This represents a decaying exponential with time constant p. There- fore, the decay time is a directly measurable criterion of integrator quality. The fact that the decay time is inversely proportional to u gives a clue to possible methods for improvement. The following one can be applied to any of the three previous examples, but the following discussion will center on the ”op- amp”. J! I\ s. ei 1' e e0 R s 1? sz F3 el-eS Be -eS eS R + R ”at“ “35):: but, es :eo/A, so de e. o E 2(A 14A _ _1 C(I'A) dt + 0, the above sum converges to the integral of x over [0, T]. Any offset in the A/D con- vertor will show up in a fashion analogous to that for the analog inte- grator. Static offset, though, can be compensated for by feeding in zero level and subtracting that digital number from all forthcoming samples. It would be difficult to analyze this configuration for its error, but one factor is clear: digital stores are not "leaky", so the equivalent ,1 would be infinite. . . ........ . ... . ...... .... .33 s. . ._ ......«w... : ... ”decanters“ .n // . . . \ ... . . .. , .... . . . . . . . . .. .. .. . $0.5... . . . . : . . .... ..fiuwn......,i. . . . . . . . . .. . , ._ . : : . .......,.....c...._... 1 . , . . . . . .1 . .. . . . .. . . . . . . . . . .... . .. . . . . . . . .. . . . . . . . . . . , ......ignunix.