True Hysteresis Research Articles

Cyclic behaviour modelling of additively manufactured Ti-6Al-4V lattice structures

The present work is concerned with the numerical modelling of the cyclic behaviour of Ti-6Al-4V lattice structures. In the study, diamond structures of titanium alloy produced by the additive laser powder bed fusion (LPBF) method with different degrees of relative density were used. Realistic geometric models of the studied mesostructures were generated using computed microtomography, taking into account the imperfections of the material resulting from the manufacturing process. The numerical calculations also took into account the actual material hardening curve in the elastic-plastic strain range. One of the achievements of this work is the numerical modelling of cyclic loading of realistic mesostructures with their imperfections. The areas of the mesostructures most susceptible to fatigue cracking have been identified and analysed. True hysteresis loops and values of local stress and strain amplitude were determined at the locations of highest stress concentration in cyclically loaded diamond structures. The main achievement of the present work was the modelling of the macroscopic fatigue life of the investigated mesostructures based on the true values of stress and strain at the locations most exposed to fatigue cracking. For this purpose, a stress criterion for fatigue cracking of Ti-6Al-4V lattice structures fabricated by the additive LPBF method was proposed.

Modeling and Adaptive Output Feedback Control of Butterfly-Like Hysteretic Nonlinear Systems With Creep and Their Applications

This paper proposes a novel robust adaptive piecewise- indirect inverse output- feedback control scheme to mitigate butterfly-like hysteresis with creep in smart-material actuators. Piecewise-indirect inverse indicates that it is not a true hysteresis and creep inverse compensatorbut an on-line decoupling mechanism to accessing the approximately actual control signal from the designed hysteretic and creeping temporary control signal. In addition, a new butterfly-like relay operator is proposed for the first time, leading to the construction of a new model of butterfly-like hysteresis with creep. Finally, the experimental results on the dielectric elastomer actuator motion control platform illustrate the effectiveness of the proposed output-feedback control scheme.

Hysteresis in a Three-Equation Model

This paper introduces two post-Keynesian hysteresis mechanisms into a standard textbook three-equation model. The mechanisms work through wage bargaining and price setting. Workers are assumed to change their wage aspirations when the actual wage differs from their target wage, and firms are assumed to change their mark-up norm when the actual profit share differs from their target share. These mechanisms do not themselves guarantee hysteresis. A pure inflation shock will create hysteresis even if expectations are anchored to the central bank’s inflation target. After a demand shock, if inflation expectations are not anchored, these mechanisms generate persistence but not true hysteresis. But if expectations are partially (as they seem to be) or fully anchored, a demand shock will have a permanent effect on output, employment, and the real wage, because in this case the central bank is not obligated to reflate as aggressively in order to manage expectations. Hysteresis effects may explain the absence of disinflation and the fall in the wage share in the aftermath of Global Financial Crisis.

Hysteresis of transient populations in absorbing-state systems

A nonequilibrium density matrix theory is used in order to explicitly describe the hysteresis interrelation between populations of nonstationary states in an absorbing multi-stage chain system in the one-particle approximation. As an illustrative example, we restrict ourselves to consideration of the 3-stage absorbing case for which we identify three types of the hysteresis; that is, the causal time dependent hysteresis with leaf-like and triangle-like closed loops, the hidden hysteresis with broken-line loops and the true hysteresis with open loops. Furthermore, we observe a common critical threshold for the hysteresis types and ascertain a reciprocal correspondence of this threshold as between the types as well with the experiment.

Reduction in energy dissipation rate with increased effective applied field

Dynamics of the response of type-II superconductors to a time-varying magnetic field can exhibit a rate-independent or rate-dependent hysteresis. An energy dissipation rate in a superconductor placed in a time-varying magnetic field depends on its wave form and type of hysteresis, which depends on temperature. The same wave form may reduce the energy dissipation rate in the case of true hysteresis, while it may increase the energy dissipation rate in the case of dynamic hysteresis compared with an energy dissipation rate in a pure sinusoidal field. We present experimental data which confirm the energy dissipation rate calculated using the critical state theory for the case of rate-independent hysteresis and limiting behavior in a normal state for the case of rate-dependent hysteresis.

Transport, Phase Reactions, and Hysteresis of Iron Fluoride and Oxyfluoride Conversion Electrode Materials for Lithium Batteries

Potentiostatic intermittent titration technique (PITT) was applied to FeF2, FeF3, and FeO0.67F1.33 to gain insight into the transport-related aspects of the conversion reaction by quantitative analysis of Li(+) diffusion and hysteresis. PITT derived diffusion coefficient measurements were benchmarked relative to values extracted by electrochemical impedance spectroscopy (EIS). A reverse-step PITT methodology was used to evaluate true hysteresis by eliminating nucleation induced overpotentials. This method evaluates the minimum potential hysteresis and allowed an accurate representation of the potential required to move conversion reactions forward at C/1000 rates in both lithiation and delithiation. The high resolution PITT data were also used to gain further insight into reaction mechanisms involved in the reversible conversion reactions. Physical evidence, based on pair distribution function (PDF) structural analysis, and electrochemical evidence are presented regarding a new step in the reaction during the rutile FeF2 reconversion reaction.

Quantification of Hysteresis and Transport in a Model FeF2 Conversion Compound

Research into fluoride based conversion materials has risen as a possible alternative to typical intercalation materials (eg, LiCoO2, LiFePO4) [1, 2]. Despite having higher capacity, one challenge both practically and in the scientific interpretation is the presence of a large voltage hysteresis in the lithiation-delithiation voltage profile [4]. Galvanostatic intermittence titration technique (GITT) has been used previously to identify the voltage hysteresis of these materials [5]. However, cycling in galvanostatic mode induces polarization that may affect the identification of a hysteresis that is activated by voltage overpotential. In addition, during relaxation, the voltage may not reach an equilibrium that is representative of the reaction itself. To eliminate polarization, potentiostatic intermittence titration technique (PITT) is used here-in to precisely control reactions through variation in voltage potential. Our group modified this technique in order to quantify the true hysteresis. In order to separate the hysteresis from initial “nucleation – like” activation potential present during the first lithiation and to identify the minimum voltage required to maintain conversion, we employed reversal PITT. For this technique, we commence with the same conditions as regular PITT, i.e. lowering the voltage in small 10mV voltage steps until a predefined capacity is reached. At that point (V1), we reverse the 10mV voltage step progression and monitor the current until the cell is unable to discharge 3mAh/g within 18 hours in a single step. Generally the current at this point is ~C/3000 and is labeled as “V3” in Fig. 1. Afterwards, regular PITT in the original direction is resumed. The voltage found at “V3” is determined to be the true reaction activating potential. Hysteresis is found by taking the difference of “V3” found in the lithiation and delithation direction at similar lithium contents (Lix). In our study, cells were lithiated 10mV steps with a 0.4mA/g (<C/1000) current cut-off until the following capacities are reached: 100, 198, 200, 290, 450, and 480 mAh/g. FeF2 was found to have a hysteresis of 1.28V as shown in Fig. 2. On lithiation, reversal PITT was initiated at x=0.70 in LixFeF2 (200mAh/g) in the middle of the conversion reaction at 1.85V. FeF2 was able to maintain this conversion until 2.05V. This large potential difference is consistent to the “nucleation – like” overpotential from galvanostatic cycling. [6]. The theoretical voltage for this conversion reaction FeF2 + 2Li → 2LiF + Fe is 2.66V. On delithiation, reversal PITT was initiated at 3.38V at a LixFeF2value similar to lithiation, and was only able to maintain delithiation for <50mV . The results indicate a voltage hysteresis on the delithiation reaction with little nucleation reaction induced polarization. Finally, to quantify the electronic component of the transport of converted FeF2, conductivity measurements were performed on a chemically lithiated FeF2 on an interdigitated electrode (IDE). Films of different thicknesses (50, 100, and 200nm) were deposited on a 250nm IDE so that the film sits in between the fingers of the IDE. They were allowed to soak in a 2.5M solution of n-butyllithium in hexane until the majority of the film has converted. DC polarization was performed at 20mV and 200mV and the decay in current was measured. As a comparison, electrochemical impedance spectroscopy (EIS) was also performed. Conductivity was determined from the contact surface area and the percentage of converted material which was determined by x-ray diffraction (XRD). These results will be discussed in the context of charge transport relative to other conversion materials. Acknowledgements This work is financially supported by the Northeastern Center for Chemical Energy Storage, an Energy Frontier Research Center, funded by the U.S. Department of Energy, Office of Basic Energy Sciences, under award contract DE-SC0001294. References F. Badway, F. Cosandey, N. Pereira, G. G. Amatucci, J. Electrochem. Soc., 150, A1318 (2003).G. G. Amatucci, N. Pereira, J. Fluorine Chem., 128, 243 (243).J. Cabana, L. Monconduit, D. Larcher, and M. R. Palacin, Adv. Energy Mater., 22, E170 (2010).R. Khatib, A. L. Dalverny, M. Saubanere, M., Gaberscek, and M. L. Doublet, J. Phys. Chem., 117, 837 (2013).N. Yamakawa, M. Jiang, B. Key, and C. P. Grey, C. P., J. Am. Chem. Soc., 131, 10525 (2009).

Sequential Pt(1 1 1) oxide formation in perchloric acid: An electrochemical study of surface species inter-conversion

A significant number of electrocatalytic reactions take place in a potential region in which the surface of platinum is partly covered by oxygenated species. Hence, the mechanism of the electrochemical formation and reduction of surface oxides is fundamental to understand how its properties can determine the reactivity of the catalyst. In this paper, the dynamic of the initial states of the electrochemical oxide growth on Pt(111), in the absence of anion specific adsorption and keeping surface stability, is studied by employing electrochemical techniques. Several species, such as chemisorbed hydroxide and oxygen, initial Pt oxide structures and even sub-surface oxygen, interconvert and interact. Their existences depend more strongly on the total surface coverage than on the electrode potential. Considering structural and thermodynamic data related to the identity of the electrochemical oxide film at platinum, the results suggest that Pt(111) oxide starts by water dissociation to OHads followed by its electrochemical oxidation to Oads at higher potentials. The latter process occurs as a phase transition through a nucleation and growth mechanism, until a stable adlayer is formed. In parallel, Oads transforms to an initial Pt oxide structure. Thus, there are at least two types of processes involved in the growth of the surface oxide layer (i.e. two states of oxide films), and the separation between anodic and cathodic j–E profiles reflects a true hysteresis arising from a change of state of the surface oxide film.

On the Reversibility of Sorption to Black Carbon: Distinguishing True Hysteresis from Artificial Hysteresis Caused by Dilution of a Competing Adsorbate

Sorption hysteresis in environmental sorbents has important implications for pollutant transport and bioavailability. We examined the reversibility of sorption of benzene, toluene, and nitrobenzene, both singly and in pairs, by wood charcoal. A previous study showed that these compounds compete for the same set of adsorption sites on the char. Single-solute sorption was weakly hysteretic at high concentrations. The finding of comparable irreversibility for these compounds was taken as evidence that hysteresis is true and caused by pore elasticity. Hysteresis in the presence of a competitor was weak at low cosolute concentration but became stronger as the cosolute concentration increased. We attribute the growing hysteresis with cosolute concentration to a thermodynamic "competitor dilution effect"--a heretofore-unrecognized cause of hysteresis in multi-solute systems when the competing solute is simultaneously diluted with the target solute in the desorption step. It arises because the target solute re-equilibrates from a sorption point where competition is relatively high, to a desorption point where competition is relatively low. Simulations based on Ideal Adsorbed Solution Theory, a thermodynamic competition model, support the hypothesis. The cosolute also causes an increase in the linearity of the target solute isotherm, also attributable to competition thermodynamics. The competitive dilution effect can play a role in pollutant behavior in real systems if competing substances, natural or anthropogenic, are diluted or degraded making the target less accessible with time.

An Isotope Exchange Technique to Assess Mechanisms of Sorption Hysteresis Applied to Naphthalene in Kerogenous Organic Matter

The sorption of organic compounds to natural sorbents is often found to show hysteresis. The objective of this study was to develop an experimental technique based on the use of 14C isotopes to distinguish hysteresis due to experimental artifacts from true hysteresis due to thermodynamically irreversible processes. The study was also designed to investigate causation of true hysteresis (irreversible sorption). The technique determines the rates and the degree of isotope exchange (IE) on equilibrated sorption and desorption points at different constant bulk chemical concentrations. The technique was applied to the sorption of naphthalene (NAPH) on Beulah-Zap lignite, a low rank reference coal composed mainly of kerogen. Sorption of bulk was found to be reversible below 10(-5) g L(-1), but irreversible above 10(-4) g L(-1). Complete isotope exchange on sorption and desorption points that defined an irreversible cycle demonstrated that hysteresis was true. A comparison of normalized uptake and release kinetics of labeled and bulk NAPH at different concentrations revealed slow structural deformation processes of the sorbent during bulk sorption and desorption. This is taken as corroborating evidence for the pore deformation hypothesis of hysteresis in which incoming sorbate molecules induce quasi-reversible changes in the organic matter that lead to different pathways for sorption and desorption. Although unable to rule it out completely, the data demonstrate that physical entrapment of sorbate molecules plays a minor, if any, role to the observed hysteresis in this system.

A Thermodynamically Based Method to Quantify True Sorption Hysteresis

Sorption of organic chemicals to soils and sediments often shows true hysteresis (i.e., nonsingularity of the sorption-desorption isotherm not attributable to known experimental artifacts). Since true sorption hysteresis is fundamentally important to contaminant fate, a way to quantify it is desirable. Previously proposed indices of hysteresis are empirical and usually depend on the isotherm model. True sorption hysteresis to synthetic and natural organic solids has been attributed to irreversible alteration of the solid during the sorption-desorption cycle. Given this mechanism, we propose the Thermodynamic Index of Irreversibility (TII) for quantifying hysteresis in soils where natural organic matter dominates the sorption process. The TII is based on the difference in free energy between the real desorption state and the hypothetical fully reversible state. The index is 0 for completely reversible systems and approaches 1 as the process tends toward complete irreversibility. It does not require any assumptions about the physical properties or molecular composition of the solid, and it does not depend on a specific equilibrium model. A sensitivity analysis of measurement errors provides general recommendations for the setup of sorption-desorption experiments. The TII was applied to sorption of 1,4-dichlorobenzene (DCB) to two high-organic soils, Pahokee peat (PP) and Amherst soil (AS), and a low-rank coal reference material, Beulah-Zap lignite (BZL). Common artificial causes of hysteresis were eliminated. Hysteresis was significant in the peat and the coal. The TII was clearly concentration dependent for both solids; it decreased with concentration for the peat, but increased with concentration for the coal. The TII allows quantification of hysteresis as a function of sorbate-sorbent combination, concentration, time, and other variables.

Demonstration of the "conditioning effect" in soil organic matter in support of a pore deformation mechanism for sorption hysteresis.

Hysteresis, or isotherm nonsingularity, is a confounding issue in sorption research that undermines the commonplace assumption of reversibility in environmental fate and effects models for organic compounds in soil media. Until now, a molecular-level mechanism for true hysteresis when the sorbate is retrievable, structurally intact, has not been forthcoming. We show here that two organic soils exhibit the "conditioning effect", which refers to the enhancement in sorption of a compound following brief exposure of the sorbent to high concentrations of the same or a similar compound. The conditioning effect has been used in support of a pore deformation mechanism for hysteresis in glassy polymers. By this mechanism, the sorbate causes irreversible changes in the structure of internal nanopores (holes) in the organic matrix upon its sorption. Trichloromethane was the test solute for dichloromethane-conditioned Pahokee soil (44.6% organic carbon), and chlorobenzene and 1,2,4-trichlorobenzene were the test solutes for benzene-conditioned Mount Pleasant silt loam (4.5% organic carbon). In each case, the isotherm of the test solute in the conditioned soil was shifted upward of, and was less linear than, the corresponding isotherm in the nonconditioned control. Application of the polymer-based Dual-Mode (partitioning-hole filling) Model shows an expansion of the hole domain as a result of conditioning. The memory of the conditioning effect persists for longer than 96 days at 21 degrees C but is lost upon heating the sample at 100 degrees C. A three-step (sorption-desorption-resorption) experiment demonstrated hysteresis followed by enhanced resorption, implying a mechanistic relationship between hysteresis and the conditioning effect. The results indicate that irreversible pore deformation is a mechanism for hysteresis in natural organic matter materials and suggest that slow matrix relaxation may contribute to the often-observed long-term resistance of some contaminants to desorption.

Thin-film lithium and lithium-ion batteries

Research over the last decade at Oak Ridge National Laboratory has led to the development of solid-state thin-film lithium and lithium-ion batteries. The batteries, which are less than 15 μm thick, have important applications in a variety of consumer and medical products, and they are useful research tools in characterizing the properties of lithium intercalation compounds in thin-film form. The batteries consist of cathodes that are crystalline or nanocrystalline oxide-based lithium intercalation compounds such as LiCoO 2 and LiMn 2O 4, and anodes of lithium metal, inorganic compounds such as silicon–tin oxynitrides, Sn 3N 4 and Zn 3N 2, or metal films such as Cu in which the anode is formed by lithium plating on the initial charge. The electrolyte is a glassy lithium phosphorus oxynitride (‘Lipon’). Cells with crystalline LiCoO 2 cathodes can deliver up to 30% of their maximum capacity between 4.2 and 3 V at discharge currents of 10 mA/cm 2, and at more moderate discharge–charge rates, the capacity decreases by negligible amounts over thousands of cycles. Thin films of crystalline lithium manganese oxide with the general composition Li 1+ x Mn 2− y O 4 exhibit on the initial charge significant capacity at 5 V and, depending on the deposition process, at 4.6 V as well, as a consequence of the manganese deficiency–lithium excess. The 5-V plateau is believed to be due to oxidation Mn of ions to valence states higher than +4 accompanied by a rearrangement of the lattice. The gap between the discharge–charge curves of cells with as-deposited nanocrystalline Li 1+ x Mn 2− y O 4 cathodes is due to a true hysteresis as opposed to a kinetically hindered relaxation observed with the highly crystalline films. This behavior was confirmed by observing classic scanning curves on charge and discharge at intermediate stages of insertion and extraction of Li + ions. Extended cycling of lithium cells with these cathodes at 25 and 100°C leads to grain growth and evolution of the charge–discharge profiles toward those characteristic of well crystallized films.

A causality algorithm for magnetic hysteresis of interacting systems

A causality algorithm is introduced to keep track of all history of interacting magnetic systems since saturation. It reflects the true hysteresis of the interacting system with ambient temperature. Though not entirely self-consistent, it satisfies the requirement of causality and is physically more realistic. Because it contains the evolution due to the interactions, thus the coercivity indeed shifts linearly with the coupling strength.

Calcium sensitivity is modified by contraction.

This paper summarizes three lines of experimental evidence showing that crossbridge interaction affects calcium sensitivity and probably also affects calcium binding. Evidence is presented that this is a true hysteresis, not just a slow approach to equilibrium. In barnacle single muscle fibers injected with aequorin to monitor intracellular Ca, a long duration stimulus under voltage clamp conditions can produce a long duration calcium transient and force record which both approach steady levels. However, if the stimulus is briefly elevated to transiently produce a higher force early in the contraction, the same steady state Ca level can eventually maintain a higher steady force. Thus Ca sensitivity is modified. In "skinned" barnacle muscle activated by Ca in the presence of buffered Ca, MgATP, and pH, force was measured in split, detergent treated fiber while it was transferred consecutively between solutions which were relaxing, submaximal contracting, maximal contracting, the same submaximal contracting, and finally relaxing. The submaximal contracting solution produced more force when stepping down in Ca concentration (as in relaxation) than when stepping up. Extending the time in the initial submaximal contracting solution did not result in more force. Thus, the force-pCa relationship shows marked hysteresis. The same phenomenon was seen in frog and mammalian muscle fibers. These experiments confirm the findings that contraction modifies Ca sensitivity. In the barnacle single muscle fiber preparation (under both voltage clamp and controlled length conditions), phasic (400 msec) depolarization leads to a calcium transient and a twitch contraction. Releasing the muscle to allow it to shorten rapidly during the declining phase of the calcium transient causes the force to fall and leads to extra Ca in the sarcoplasm. Rapidly stretching the muscle produces the opposite effect. The extra Ca probably comes from a myofilament Ca activating site. Thus, a length change (force change) affects Ca binding.

Reflectance Changes during Formation and Reduction of Oxide Films on Gold and Platinum: Corrections for “Double‐Layer Effects”

Hysteresis between the processes of formation and reduction of thin oxide films on Au and Pt have been investigated by ellipsometric (Δ) and relative reflectance measurements. The interpretation of results obtained by optical methods where thin films are in the course of formation towards a monolayer, or beyond, requires consideration of optical effects due to contributions from changing ion concentration and metal surface electron charge density in the double‐layer since substantial changes of optical parameters for reflectance at Au and Pt interfaces arise with changing electrode potential. Resulting changes of Δ and must therefore be corrected for continuing double‐layer effects as the ad‐layer is laid down or removed (in the reduction process). The theoretical basis for evaluation of double‐layer optical effects is reviewed but it is proposed that, at the present time, empirical correction procedures may be preferable. Three methods are examined and an optimum procedure is recommended. The procedure is applied to the evaluation of hysteresis effects in the formation and reduction of thin oxide films at Au and Pt where evaluation of the true hysteresis effects depends in an important way on a proper correction procedure for the double‐layer optical effects.

Metastable secondary structures in ribosomal RNA: A new method for analyzing the titration behavior of rRNA

AbstractThe pH titration behavior of E. coli rRNA in the acid range has been analyzed by combining spectrophotometric and potentiometric titration data. The “simplest” model for the system, which considers as possible reactions the protonation of adenine (A), cytosine (C), and guanine (G) residues along with the opening of A·U and G·C base pairs, does not adequately account for the titration properties. It is postulated that extra reactions may occur in addition to those in the “simplest” model, and a new analytical method was developed to deal with this situation. Our approach yields the ultraviolet spectral changes which accompany the extra reactions, from which the nature of these reactions can in principle be deduced. The calculations also give, at each pH, the extents of the extra reactions as well as the extents of those reactions which comprise the “simplest” model.We infer that in acidic RNA solutions of 0.1M ionic strength there occur at least two extra reactions, each of which involves G residues. We propose that in the pH range 6.0 ≥ pH ≥ 3.8 triple‐stranded helical sequences, presumably protonated G·C·G, are formed. These regions are replaced at lower pH by acid‐stable structures involving G·G and A·A base pairs. In solutions of lower ionic strength (I = 0.01M) no triple strands are formed, but G·G and A·A regions seem to develop even at pH values as high as 6.0. At I = 0.1M, an acid–base titration cycle between pH 7 and 2.8 is not reversible; rRNA shows true hysteresis behavior. We conclude that in ribosomal RNA's, which are generally G‐rich, guanine residues may participate in hitherto unpredicted conformations, some of which may be metastable while others are equilibrium structures.

Specific aggregation of poly(α‐L‐glutamic acid) and hysteresis effects in aqueous solutions. I. Influence of temperature‐dependent aggregation on the optical rotation of PGA

AbstractThe instability of aqueous solutions of poly(α‐L‐glutamic acid) (PGA) at low pH is due to two distinguishable phenomena: precipitation, favored above 40°C., and aggregation, favored below 20°C. The aggregated form of PGA can be isolated by gel permeation chromatography. Both aggregation and precipitation increase with decreasing pH, i.e., with decreasing ionization of the side chain carboxyl groups. Temperature‐induced aggregation and disaggregation give rise to a reproducible hysteresis loop which can be followed by optical rotation, light scattering, sedimentation, viscosity, and chromatography. Hysteresis has been observed with different PGA samples, and in several aqueous buffered or unbuffered solvents and organic‐aqueous solvent mixtures and in the pH range 4.1–4.5. Aggregation manifests itself as an increase in negative optical rotation in the visible and ultraviolet spectral range. The specific relation at 233 mμ is sensitive to aggregation and also reflects the hysteresis. Measurements of optical rotatory dispersion indicate that a0 reflects the hysteresis but b0 does not, the latter revealing only reversible changes with aggregation and disaggregation. The helix‐coil equilibrium is apparently unperturbed by aggregation, as is the thermal stability of the helix structure. For fully aggregated PGA it is estimated that a0 increases by about 300 degrees, which suggests that a0 may be a sensitive parameter to measure aggregation in other systems. The rate of aggregation increases with decreasing temperature. The disaggregation, upon heating, is more rapid. However, kinetics measurements have not yet been done. The temperature M at which all aggregates are disrupted increases with decreasing pH, but is independent of total PGA concentration, at constant pH. No molecular weight dependence of M was detected in the range 20–100 × 103. The shape and size of the hysteresis loop depends upon pH and molecular weight, which is interpreted as a dependence on the extent of aggregation. One branch of the loop, representing the helix–coil transition of isolated molecules, is reversible, while the others, representing the formation and disruption of the aggregates, are not. The system exhibits both ascending and descending scanning curves, which are typical of a true hysteresis.

On the theory and measurement of the magnetic properties of iron

The paper deals with the accepted methods of representing ferromagnetic properties and with the vector diagrams and equivalent electric circuits commonly employed to indicate the behaviour of iron-cored coils under a.c. excitation. The. point of view adopted is that such vector and circuit representations must not be interpreted literally when, for example, fundamental measurements of the properties of iron are undertaken by a.c. methods. For instance, the wattless component of no-load current of a transformer—frequently termed the magnetizing current—is shown to be in no sense a measure of the amplitude of the magnetizing force experienced by the core, even though, in the r.m.s. system of measurement, it must be used in assessing the so-called reactance of the winding. An inductance coefficient deduced from this reactance is an artificial quantity because it infers a permeability value which cannot be related with the true reversals curve of the core material at any point. It is demonstrated by experiment that the total no-load current (r.m.s.) is a closer measure of true peak magnetizing force, though this current needs to be corrected for eddy-current effects and, even then, has still a very limited application in fundamental investigations.Attention is directed towards precise methods of measurement of magnetic properties using alternating current, and a means of expressing results is suggested which avoids the worst anomalies of the r.m.s. equivalent circuit and vector diagrams. The mathematical treatment of the excitation ellipse associated with the latter is extended and its full implications are discussed. The relations between the ellipse and the true hysteresis figure are shown by an analysis of the harmonics of excitation. The true cycle of permeability is derived, thus enabling an expression to be formed which is general for all conditions in the core, so covering the cases of remanent magnetism and of excitation by reversed direct current such as is customary in ballistic measurements.The aim throughout the paper is to re-establish a clear outlook upon the elements of cyclic ferromagnetism, in the belief that the half-truths of the common a.c. vector treatments have an undue influence upon the engineer&apos;s mode of thought.

Errors in the magnetic testing of ring specimens

It is the usual practice to make the ratio of the radial thickness to the mean diameter of a ring about 1/10 to 1/15, so as to limit the error due to non-uniform flux distribution. The adoption of a ratio of this order is apparently based upon the work of M. G. Lloyd, who calculated the errors involved in the determination of the magnetization curve and of the hysteresis loss for various ratios of radial thickness to mean diameter. His calculations, however, were based on the assumption of constant permeability. It is shown in this paper that, when the variation of permeability is taken into account, the errors involved in the determination of the above-mentioned quantities are almost invariably either smaller or larger than those calculated by Lloyd.The paper discusses the influence of the ratio of radial thickness to mean diameter, and of reluctivity, upon:-(a) The relationship between the average flux density in the ring and the flux density corresponding to the average magnetizing force.(b) The relationship between the actual hysteresis loss in the ring and the hysteresis loss in the same volume of iron owing to the flux density corresponding to the average magnetizing force. The influence of the exponent of B upon this relationship is also considered.(c) The relationship between the true hysteresis coefficient and that determined from a test performed with (1) an alternating-current supply, (2) a ballistic galvanometer.Reference is made to the desirability—under certain conditions—of using a higher ratio of radial thickness to mean diameter than is customary, in order that any error due to mechanical strain produced by punching, etc., may be reduced.