Hysteresis

 Hysteresis is the dependence of the state of a system on its history. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field changed in the past. Plots of a single component of the moment often form a loop or hysteresis curve, where there are different values of one variable depending on the direction of change of another variable. This history dependence is the basis of memory in a hard disk drive and the remanence that retains a record of the Earth's magnetic field magnitude in the past. Hysteresis occurs in ferromagnetic and ferroelectric materials, as well as in the deformation of rubber bands and shape-memory alloys and many other natural phenomena. In natural systems it is often associated with irreversible thermodynamic change such as phase transitions and with internal friction; and dissipation is a common side effect.

Hysteresis can be found in physics, chemistry, engineering, biology, and economics. It is incorporated in many artificial systems: for example, in thermostats and Schmitt triggers, it prevents unwanted frequent switching.

Hysteresis can be a dynamic lag between an input and an output that disappears if the input is varied more slowly; this is known as rate-dependent hysteresis. However, phenomena such as the magnetic hysteresis loops are mainly rate-independent, which makes a durable memory possible.

Systems with hysteresis are nonlinear, and can be mathematically challenging to model. Some hysteretic models, such as the Preisach model (originally applied to ferromagnetism) and the Bouc–Wen model, attempt to capture general features of hysteresis; and there are also phenomenological models for particular phenomena such as the Jiles–Atherton model for ferromagnetism.

It is difficult to define hysteresis precisely. Isaak D. Mayergoyz wrote "..the very meaning of hysteresis varies from one area to another, from paper to paper and from author to author. As a result, a stringent mathematical definition of hysteresis is needed in order to avoid confusion and ambiguity

tymology and history

The term "hysteresis" is derived from ὑστέρησις, an Ancient Greek word meaning "deficiency" or "lagging behind". It was coined in 1881 by Sir James Alfred Ewing to describe the behaviour of magnetic materials.

Some early work on describing hysteresis in mechanical systems was performed by James Clerk Maxwell. Subsequently, hysteretic models have received significant attention in the works of Ferenc Preisach (Preisach model of hysteresis), Louis Néel and Douglas Hugh Everett in connection with magnetism and absorption. A more formal mathematical theory of systems with hysteresis was developed in the 1970s by a group of Russian mathematicians led by Mark Krasnosel'skii.

Types

Rate-dependent

One type of hysteresis is a lag between input and output. An example is a sinusoidal input X(t) that results in a sinusoidal output Y(t), but with a phase lag φ:

 it is mathematically equivalent to a transfer function in linear filter theory and analogue signal processing.

This kind of hysteresis is often referred to as rate-dependent hysteresis. If the input is reduced to zero, the output continues to respond for a finite time. This constitutes a memory of the past, but a limited one because it disappears as the output decays to zero. The phase lag depends on the frequency of the input, and goes to zero as the frequency decreases.

When rate-dependent hysteresis is due to dissipative effects like friction, it is associated with power loss.

Rate-independent

Systems with rate-independent hysteresis have a persistent memory of the past that remains after the transients have died out. The future development of such a system depends on the history of states visited, but does not fade as the events recede into the past. If an input variable X(t) cycles from X0 to X1 and back again, the output Y(t) may be Y0 initially but a different value Y2 upon return. The values of Y(t) depend on the path of values that X(t) passes through but not on the speed at which it traverses the path. Many authors restrict the term hysteresis to mean only rate-independent hysteresis.Hysteresis effects can be characterized using the Preisach model and the generalized Prandtl−Ishlinskii model.

Control systems

In control systems, hysteresis can be used to filter signals so that the output reacts less rapidly than it otherwise would by taking recent system history into account. For example, a thermostat controlling a heater may switch the heater on when the temperature drops below A, but not turn it off until the temperature rises above B. (For instance, if one wishes to maintain a temperature of 20 °C then one might set the thermostat to turn the heater on when the temperature drops to below 18 °C and off when the temperature exceeds 22 °C).

Similarly, a pressure switch can be designed to exhibit hysteresis, with pressure set-points substituted for temperature thresholds.

Electronic circuits

Often, some amount of hysteresis is intentionally added to an electronic circuit 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 Schmitt trigger is a simple electronic circuit that exhibits this property.

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

Hysteresis is essential to the workings of some memristors (circuit components which "remember" changes in the current passing through them by changing their resistance).

Hysteresis can be used when connecting arrays of elements such as nanoelectronics, electrochrome cells and memory effect devices using passive matrix addressing. 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. Thus, all rows can be addressed at the same time instead of individually.

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

User interface design

A hysteresis is sometimes intentionally added to computer algorithms. 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.

Aerodynamics

In aerodynamics, hysteresis can be observed when decreasing the angle of attack of a wing after stall, regarding the lift and drag coefficients. The angle of attack at which the flow on top of the wing reattaches is generally lower than the angle of attack at which the flow separates during the increase of the angle of attack.

Backlash

Moving parts within machines, such as the components of a gear train, normally have a small gap between them, to allow movement and lubrication. As a consequence of this gap, any reversal in direction of a drive part will not be passed on immediately to the driven part.This unwanted delay is normally kept as small as practicable, and is usually called backlash. The amount of backlash will increase with time as the surfaces of moving parts wear.

Elastic hysteresis

In the elastic hysteresis of rubber, the area in the centre of a hysteresis loop is the energy dissipated due to material internal friction.

Elastic hysteresis was one of the first types of hysteresis to be examined.

The effect can be demonstrated using a rubber band with weights attached to it. If the top of a rubber band is hung on a hook and small weights are attached to the bottom of the band one at a time, it will stretch and get longer. As more weights are loaded onto it, the band will continue to stretch because the force the weights are exerting on the band is increasing. When each weight is taken off, or unloaded, the band will contract as the force is reduced. As the weights are taken off, each weight that produced a specific length as it was loaded onto the band now contracts less, resulting in a slightly longer length as it is unloaded. This is because the band does not obey Hooke's law perfectly. The hysteresis loop of an idealized rubber band is shown in the figure.

In terms of force, the rubber band was harder to stretch when it was being loaded than when it was being unloaded. In terms of time, when the band is unloaded, the effect (the length) lagged behind the cause (the force of the weights) because the length has not yet reached the value it had for the same weight during the loading part of the cycle. In terms of energy, more energy was required during the loading than the unloading, the excess energy being dissipated as thermal energy.

Elastic hysteresis is more pronounced when the loading and unloading is done quickly than when it is done slowly. Some materials such as hard metals don't show elastic hysteresis under a moderate load, whereas other hard materials like granite and marble do. Materials such as rubber exhibit a high degree of elastic hysteresis.

When the intrinsic hysteresis of rubber is being measured, the material can be considered to behave like a gas. When a rubber band is stretched it heats up, and if it is suddenly released, it cools down perceptibly. These effects correspond to a large hysteresis from the thermal exchange with the environment and a smaller hysteresis due to internal friction within the rubber. This proper, intrinsic hysteresis can be measured only if the rubber band is thermally isolated.

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 made use of elastomer suspension, as did the original Mini car.

The primary cause of rolling resistance when a body (such as a ball, tire, or wheel) rolls on a surface is hysteresis. This is attributed to the viscoelastic characteristics of the material of the rolling body.

Contact angle hysteresis

The contact angle formed between a liquid and solid phase will exhibit a range of contact angles that are possible. There are two common methods for measuring this range of contact angles. The first method is referred to as the tilting base method. Once a drop is dispensed on the surface with the surface level, the surface is then tilted from 0° to 90°. As the drop is tilted, the downhill side will be in a state of imminent wetting while the uphill side will be in a state of imminent dewetting. As the tilt increases the downhill contact angle will increase and represents the advancing contact angle while the uphill side will decrease; this is the receding contact angle. The values for these angles just prior to the drop releasing will typically represent the advancing and receding contact angles. The difference between these two angles is the contact angle hysteresis.

The second method is often referred to as the add/remove volume method. 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. As with the tilt method, the difference between the advancing and receding contact angles is the contact angle hysteresis. Most researchers prefer the tilt method; the add/remove method requires that a tip or needle stay embedded in the drop which can affect the accuracy of the values, especially the receding contact angle.

Bubble shape hysteresis

The equilibrium shapes of bubbles expanding and contracting on capillaries (blunt needles) can exhibit hysteresis depending on the relative magnitude of the maximum capillary pressure to ambient pressure, and the relative magnitude of the bubble volume at the maximum capillary pressure to the dead volume in the system. The bubble shape hysteresis is a consequence of gas compressibility, which causes the bubbles to behave differently across expansion and contraction. During expansion, bubbles undergo large non equilibrium jumps in volume, while during contraction the bubbles are more stable and undergo a relatively smaller jump in volume resulting in an asymmetry across expansion and contraction. The bubble shape hysteresis is qualitatively similar to the adsorption hysteresis, and as in the contact angle hysteresis, the interfacial properties play an important role in bubble shape hysteresis.

The existence of the bubble shape hysteresis has important consequences in interfacial rheology experiments involving bubbles. As a result of the hysteresis, not all sizes of the bubbles can be formed on a capillary. Further the gas compressibility causing the hysteresis leads to unintended complications in the phase relation between the applied changes in interfacial area to the expected interfacial stresses. These difficulties can be avoided by designing experimental systems to avoid the bubble shape hysteresis.

Adsorption hysteresis

Hysteresis can also occur during physical adsorption processes. In this type of hysteresis, the quantity adsorbed is different when gas is being added than it is when being removed. The specific causes of adsorption hysteresis are still an active area of research, but it is linked to differences in the nucleation and evaporation mechanisms inside mesopores. These mechanisms are further complicated by effects such as cavitation and pore blocking.

In physical adsorption, hysteresis is evidence of mesoporosity-indeed, the definition of mesopores (2–50 nm) is associated with the appearance (50 nm) and disappearance (2 nm) of mesoporosity in nitrogen adsorption isotherms as a function of Kelvin radius. An adsorption isotherm showing hysteresis is said to be of Type IV (for a wetting adsorbate) or Type V (for a non-wetting adsorbate), and hysteresis loops themselves are classified according to how symmetric the loop is. Adsorption hysteresis loops also have the unusual property that it is possible to scan within a hysteresis loop by reversing the direction of adsorption while on a point on the loop. The resulting scans are called "crossing," "converging," or "returning," depending on the shape of the isotherm at this point.

Matric potential hysteresis

The relationship between matric water potential and water content is the basis of the water retention curve. Matric potential measurements (Ψm) 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 (Ψm) of 5 kPa, the volumetric water content (θ) of a fine sandy soil matrix could be anything between 8% and 25%.

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.