Hysteresis

A system with hysteresis exhibits path-dependence, or "rate-independent memory". By contrast, consider a deterministic system with no hysteresis and no dynamics. In that case, we can predict the output of the system at some instant in time, given only the input to the system at that instant. If the system has hysteresis, then this is not the case; we can't predict the output without looking at the history of the input. In order to predict the output, we must look at the path that the input followed before it reached its current value. A system with hysteresis has memory.

Many physical systems naturally exhibit hysteresis. A piece of iron that is brought into a magnetic field retains some magnetization, even after the external magnetic field is removed. Once magnetized, the iron will stay magnetized indefinitely. In order to demagnetize the iron, it would be necessary to apply a magnetic field in the opposite direction. This effect is exploited commercially; for example, it provides the element of memory in a hard disk drive.

Human-designed systems will sometimes intentionally exhibit hysteresis. For example, consider a thermostat that controls a furnace. The furnace is either off or on, with nothing in between. The thermostat is a system; its input is the temperature, and its output is the furnace state. If we wish to maintain a temperature of 20 degrees, then we might set the thermostat to turn the furnace on when the temperature drops below 18 degrees, and turn it off when the temperature exceeds 22 degrees. This thermostat has hysteresis. Let us say that the temperature is 21 degrees. Given this information, we cannot predict whether the furnace will be on or off; it's not possible to predict the instantaneous output of the thermostat, knowing only its instantaneous input.

The term derives from an ancient Greek word υστέρησις, meaning "deficiency", or "lagging behind". It was coined by Sir James Alfred Ewing.

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Introduction

Hysteresis phenomena occur in magnetic and ferromagnetic materials, as well as in the elastic and electromagnetic behavior of materials, in which a lag occurs between the application and the removal of a force or field and its subsequent effect. Electric hysteresis occurs when applying a varying electric field, and elastic hysteresis occurs in response to a varying force. The term "hysteresis" is sometimes used in other fields, such as economics or biology. In such cases it describes a memory or lagging effect in which the order of previous events can influence the order of subsequent events.

The word "lag" above should not necessarily be interpreted as a time lag. After all, even relatively simple linear systems such as an electric circuit containing resistors and capacitors exhibit a time lag between the input and the output. For most hysteretic systems, there is a very short time scale when its dynamic behavior and various related time dependences are observed. In magnetism, for example, the dynamic processes occurring on this very short time scale have been referred to as Barkhausen jumps. If observations are carried out over very long periods, creep or slow relaxation typically toward true thermodynamic equilibrium (or other types of equilibria that depend on the nature of the system) can be noticed. When observations are carried out without regard for very swift dynamic phenomena or very slow relaxation phenomena, the system appears to display irreversible behavior whose rate is practically independent of the driving force rate. This rate-independent irreversible behavior is the key feature that distinguishes hysteresis from most other dynamic processes in many systems.

If the displacement of a system with hysteresis is plotted on a graph against the applied force, the resulting curve is in the form of a loop. In contrast, the curve for a system without hysteresis is a single, not necessarily straight, line. Although the hysteresis loop depends on the material's physical properties, there is no complete theoretical description that explains the phenomenon. The family of hysteresis loops, from the results of different applied varying voltages or forces, form a closed space in three dimensions, called the hysteroid.

Hysteresis was initially seen as problematic, but is now thought to be of great importance in technology. For instance, the properties of hysteresis are applied when constructing permanent memory for computers: hysteresis allows most superconductors to operate at the high currents needed to create strong magnetic fields. Hysteresis is also important in living systems. Many critical processes occurring in living (or dying) cells use hysteresis to help stabilize them against the various effects of random chemical fluctuations.

Some early work on describing hysteresis in mechanical systems was performed by James Clerk Maxwell. Subsequently, hysteresis models have received significant attention in the works of Preisach (Preisach model of hysteresis), Neel and Everett in connection with magnetism and absorption. A simple parametric description of various hysteresis loops may be found here ^[1] (with the model, substitution of rectangle, triangle or trapezoidal pulses instead of the harmonic functions also allows to built piecewise-linear hysteresis loops frequently used in discrete automatics). More formal mathematical theory of systems with hysteresis was developed in 1970s, by a group of Russian mathematicians, which was led by Mark Krasnosel'skii, one of the founders of nonlinear analysis. He suggested an investigation of hysteresis phenomena using the theory of nonlinear operators.

Informal definition

The phenomenon of hysteresis can conceptually be explained as follows. A system can be divided into subsystems or domains, much larger than an atomic volume but still microscopic. Such domains normally occur in ferroelectric and ferromagnetic systems, since individual dipoles tend to group with each other, forming a small isotropic region. Each of the system's domains can be shown to have a metastable state. The metastable domains can in turn have two or more substates. Such a metastable state fluctuates widely from domain to domain, but the average represents the configuration of lowest energy. The hysteresis is simply the sum of all domains, or the sum of all metastable states.

From the mathematical point of view hysteresis is rate independent memory. For the more formal, mathematical definitions see the bibliography.

Magnetic hysteresis

Hysteresis is well known in ferromagnetic materials. When an external magnetic field is applied to a ferromagnet, the atomic dipoles align themselves with the external field. Even when the external field is removed, part of the alignment will be retained: the material has become magnetized.

The relationship between magnetic field strength (H) and magnetic flux density (B) is not linear in such materials. If the relationship between the two is plotted for increasing levels of field strength, it will follow a curve up to a point where further increases in magnetic field strength will result in no further change in flux density. This condition is called magnetic saturation.

If the magnetic field is now reduced linearly, the plotted relationship will follow a different curve back towards zero field strength at which point it will be offset from the original curve by an amount called the remanent flux density or remanence.

If this relationship is plotted for all strengths of applied magnetic field the result is a sort of S- shaped loop. The 'thickness' of the middle bit of the S describes the amount of hysteresis, related to the coercivity of the material.

Its practical effects might be, for example, to cause a relay to be slow to release due to the remaining magnetic field continuing to attract the armature when the applied electric current to the operating coil is removed. 

This curve for a particular material influences the design of a magnetic circuit.

This is also a very important effect in magnetic tape and other magnetic storage media like hard disks. In these materials it would seem obvious to have one polarity represent a bit, say north for 1 and south for 0. However, if you want to change the storage from one to the other, the hysteresis effect requires you to know what was already there, because the needed field will be different in each case. In order to avoid this problem, recording systems first overdrive the entire system into a known state using a process known as bias. Analog magnetic recording also uses this technique. Different materials require different biasing, which is why there is a selector for this on the front of most cassette recorders.

In order to minimize this effect and the energy losses associated with it, ferromagnetic substances with low coercivity and low hysteresis loss are used, like permalloy.

In many applications small hysteresis loops are driven around points in the B-H plane. Loops near the origin have a higher µ. The smaller loops the more they have a soft magnetic (lengthy) shape. As a special case a damped AC field demagnetized any material.

Electrical hysteresis

Electrical hysteresis typically occurs in ferroelectric material, where domains of polarization contribute to the total polarization. Polarization is the electrical dipole moment (either C·m^-2 or C·m).

Liquid-solid phase transitions

Hysteresis manifests itself in state transitions when melting temperature and freezing temperature do not agree. For example, agar melts at 85 °C and solidifies from 32 to 40 °C. This is to say that once agar is melted at 85 degrees, it retains a liquid state until cooled to 40 degrees Celsius. Therefore, from the temperatures of 40 to 85 degrees Celsius, agar can be either solid or liquid, depending on which state it was before.

Contact angle hysteresis

The contact angle formed between a liquid and solid phase can be measured dynamically. When the maximum liquid volume is removed from the drop without the interfacial area decreasing the receding contact angle is thus measured. When volume is added to the maximum before the interfacial area increases, this is the advancing contact angle. The difference between the advancing and receding contact angles is referred to as the contact angle hysteresis.

Matric potential hysteresis

The relationship between matric water potential and water content is the basis of the water retention curve. Matric potential measurements (Ψ_m) are converted to volumetric water content (θ) measurements based on a site or soil specific calibration curve. Hysteresis is a source of water content measurement error. Matric potential hysteresis arises from differences in wetting behaviour causing dry medium to re-wet; that is, it depends on the saturation history of the porous medium. Hysteretic behaviour means that, for example, at a matric potential (Ψ_m) of 5kPa, the volumetric water content (θ) of a fine sandy soil matrix could be anything between 8% to 25% ^[2].

Tensiometers are directly influenced by this type of hysteresis. Two other types of sensors used to measure soil water matric potential are also influenced by hysteresis effects within the sensor itself. Resistance blocks, both nylon and gypsum based, measure matric potential as a function of electrical resistance. The relation between the sensor’s electrical resistance and sensor matric potential is hysteretic. Thermocouples measure matric potential as a function of heat dissipation. Hysteresis occurs because measured heat dissipation depends on sensor water content, and the sensor water content–matric potential relationship is hysteretic. As of 2002, only desorption curves are usually measured during calibration of soil moisture sensors. Despite the fact that it can be a source of significant error, the sensor specific effect of hysteresis is generally ignored.^[3]

Energy

When hysteresis occurs with extensive and intensive variables, the work done on the system is the area under the hysteresis graph.

Economics

Some economic systems show signs of hysteresis. For example, export performance is subject to strong hysteresis effects: it may take a big push (ie sizable changes in incentives) to start a country's exports, but once the transition is made, not much may be required to keep them going.

Another example is the notion that inflationary policy leads to a permanently higher 'natural' rate of unemployment (NAIRU), due to the proposition that inflationary expectations are 'sticky' downward because of wage rigidities and imperfections in the labour market.

Many economists also argue that unemployment itself is subject to hysteresis effects. Unemployment persistence is argued to arise from various factors that include demand deficiency and labour market institutions.

Hysteresis shows in game theory, for example, applied to quality, honesty or corruption. Slightly different initial conditions can lead to opposite results, stable "good" and "bad" equilibria.

Behavioral economists attempt to measure the utility gain from obtaining an item, and the utility loss from losing the same item. With great regularity, the utility loss is greater than the utility gain, meaning that if a person goes through a complete cycle of gaining and losing, the person may be worse off than if he or she had never received the initial gain.

User interface design

The field of user interface design has borrowed the term hysteresis to refer to times when the state of the user interface intentionally lags behind the apparent user input. For example, a menu that was drawn in response to a mouse-over event may remain on-screen for a brief moment after the mouse has moved out of the trigger region and the menu region. This allows the user to move the mouse directly to an item on the menu, even if part of that direct mouse path is outside of both the trigger region and the menu region. For instance, right-clicking on the desktop in most Windows interfaces will create a menu that exhibits this behavior.

Electronics

Hysteresis can be used to filter signals so that the output reacts slowly by taking recent history into account. For example, a thermostat controlling a heater may turn the heater on when the temperature drops below A degrees, but not turn it off until the temperature rises above B degrees. Thus the on/off output of the thermostat to the heater when the temperature is between A and B depends on the history of the temperature. This prevents rapid switching on and off as the temperature drifts around the set point.

A Schmitt trigger is a simple electronic circuit that also exhibits this property. Often, some amount of hysteresis is intentionally added to an electronic circuit (or digital algorithm) to prevent unwanted rapid switching. This and similar techniques are used to compensate for contact bounce in switches, or noise in an electrical signal.

A latching relay uses a solenoid to actuate a ratcheting motion that keeps the relay closed even if power to the relay is terminated.

Cell biology

Cells undergoing cell division exhibit hysteresis in that it takes a higher concentration of cyclins to switch them from G2 phase into mitosis than to stay in mitosis once begun. ^[4]

Neuroscience

The property by which some neurons do not return to their basal conditions from a stimulated condition immediately after removal of the stimulus is an example of hysteresis.

Respiratory Physiology

The Pressure vs Volume curve of inhalation is different to the Pressure vs Volume curve of exhalation, the difference being described as hysteresis. Lung volume at any given pressure during inhalation is less than the lung volume at any given pressure during exhalation ^[5].

Applications

Hysteresis represents states, and the characteristic curve shape is sometimes remiscent of a two-value state, also called a bistable state. The hysteresis curve really contains infinitely many states, but a simple application is to let the threshold regions (usually to the left and to the right) represent respectively the on and off states. In this way, the system can be regarded as bistable. Note that even if no external field is applied, the position of the hysteresis curve might change with time: it is not necessarily stationary; i.e. the system may not stay in the exact same state as it had previously. The system might need new energy transfer to be stationary.

The hysteresis effect can be used when connecting complex circuits with the so-called passive matrix addressing. This scheme is praised as a technique that can be used in modern nanoelectronics, electrochrome cells, memory effect, etc. In this scheme, shortcuts are made between adjacent components (see crosstalk) and the hysteresis helps to keep the components in a particular state while the other components change states. That is, one can address all rows at the same time instead of doing each individually.

In economics, hysteresis is used extensively in the area of Labour markets. According to theories based on hysteresis, Economic downturns (Recession) result in an individual becoming unemployed, losing his/her skills (commonly developed 'on the job'), demotivated/disillusioned, and employers may use time spent in unemployment as a screen. In times of an Economic upturn or 'boom', the workers affected will not share in the prosperity, remaining Long-Term Unemployed (>52 weeks). Hysteresis has been put forward as a possible explanation for the poor unemployment performance of many economies in the 1990s. Labour market reform, and/or strong economic growth, may not therefore aid this pool of long-term unemployed, and thus specific targeted training programs are presented as a possible policy solution.

In the field of audio electronics, a noise gate often implements hysteresis intentionally to prevent the gate from "chattering" when signals close to its threshold are applied.

Small vehicle suspensions using rubber (or other elastomers) can achieve the dual function of springing and damping because rubber, unlike metal springs, has pronounced hysteresis and does not return all the absorbed compression energy on the rebound. Mountain bikes have frequently made use of elastomer suspension, as did the original Mini car.

See also

• Remanence
• Hysteresivity
• Path dependence
• Backlash (engineering)
• Mark Krasnosel'skii and Alexei Pokrovskii, Systems with Hysteresis, Springer-Verlag, New York, 1989.
• Isaak D. Mayergoyz, Mathematical Models of Hysteresis and their Applications : Second Edition (Electromagnetism), Academic Press, 2003.
• The Science of Hysteresis (3-volume set), ed. by Isaak D. Mayergoyz, Giorgio Bertotti, Academic , 2005.

References

1. ^ Analytical model for the approximation of hysteresis loop and its application to the scanning tunneling microscope. Retrieved on 2007-06-05.
2. ^ Accuracy of capacitance soil moisture .... Retrieved on 2007-06-05.
3. ^ Scanlon, B. R., Andraski, B. J., and Bilskie, J. (2002). "Methods of soil analysis: Physical Methods: Miscellaneous methods for measuring matric or water potential" (PDF). Soil Science Society of America 4: 643–670. ISBN 0-89118-810-X. Retrieved on 2006-05-26.
4. ^ Joseph R. Pomerening, Eduardo D. Sontag, James E. Ferrell Jr (2003). "Building a cell cycle oscillator: hysteresis and bistability in the activation of Cdc2". Nature Cell Biology 5: 346-251.
5. ^ John B. West (2005). Respiratory physiology: the essentials. Hagerstown, MD: Lippincott Williams & Wilkins. ISBN 0-7817-5152-7.