What is Hysteresis? K. A. Morris∗ Hysteresis is a widely occuring phenomenon. It can be application and removal of tensile stress, I found found in a wide variety of natural and constructed systems. that during increment and decrement of the load Generally, a system is said to exhibit hysteresis when a char- equal values of load were associated with widely acteristic looping behaviour of the input-output graph is dis- different values of thermoelectric quality; the differ- played. These loops can be due to a variety of causes. Fur- ence being mainly of this character, that the changes thermore, the input-output graphs of periodic inputs at differ- of thermoelectric quality lagged behind the changes ent frequencies are generally identical. Existing definitions of stress. This lagging is, however, a static phe- of hysteresis are useful in different contexts but fail to fully nomenon, for it is sensibly unaffected by the speed characterize it. In this paper, a number of different situa- at which the load is changed; and again, when any tions exhibiting hysteresis are described and analyzed. The state of load is maintained constant, the thermoelec- applications described are: an electronic comparator, gene tric quality does not change with lapse of time... regulatory network, backlash, beam in a magnetic field, a Magnetic phenomena present many instances of a class of smart materials and inelastic springs. The common similar action- some of which will be described be- features of these widely varying situations are identified and low. Thus, when a magnetised piece of iron is al- summarized in a final section that includes a new definition ternately subjected to pull and relation of pull suffi- for hysteresis. ciently often to make the magnetic changes cyclic, these lag behind the changes of stress in much the same way as the changes of thermoelectric quality 1 Introduction do. I found it convenient to have a name for this Hysteresis occurs in many natural and constructed sys- peculiar action, and accordingly called it Hysteresis tems. (from nster˘ ew´ , to lag behind). The relay shown in Figures 1 is a simple example of a system exhibiting hysteretic behaviour. The relay in the This paragraph highlights two key aspects of hysteresis that Figure is centered at s with an offset of r. When the signal are still regarded as characteristic today: “lagging” and rate- u > s + r, the relay is at +1. As the input decreases, the independence. output R remains at +1 until the lower trigger point s − r is Lagging is still generally regarded as a key component reached. At this point the output switches to −1. When the of hysteresis. The online version of the Merriam-Webster signal is increasing, the output remains at −1 until u = s+r. dictionary defines hysteresis as + At this point, the output switches to 1. For inputs in the retardation of an effect when the forces acting upon s r u s r range − < < + the output can be +1 or −1, depending a body are changed (as if from viscosity or internal r on past history. Thus, the output may lag the input by 2 . friction) ; especially : a lagging in the values of This lag prevents devices such as thermostats from chattering resulting magnetization in a magnetic material (as as the temperature moves just above or below the setpoint. iron) due to a changing magnetizing force. [8] Hysteretic behaviour is also apparent in many other contexts, such as play in mechanical gears and smart materials [1–6]. In the simple relay shown in Figure 1, changes in the output Hysteresis also occurs naturally in a number of systems such lag changes in input. However, the output of a linear delay as genetic regulatory systems [2]. All these examples are system such as y(t) = u(t − 1) also lags the input. Such sys- discussed in more detail later in this paper. tems would not be described as hysteretic. The first mention of hysteresis appeared in an 1885 pa- The second property mentioned by Ewing, rate inde- per on magnetism [7]. Ewing wrote pendence, means that an input/output plot only depends on In testing the changes of thermoelectric quality the values of the input, but not the speed at which the in- which a stretched wire underwent when succes- put is changed. In a rate-independent system such as those sively loaded and unloaded so as to suffer alternate shown in Figures 1 and 2 the input/output plot with input u(t) = M sin(t) is identical to that with u(t) = M sin(10t). More formally, let a differentiable function j : R+ ! R+ be a time transformation if j(t) is increasing and satisfies ∗Dept. of Applied Mathematics, University of Waterloo, Waterloo, On- tario, Canada, ([email protected]). j(0) = 0 and limt!¥ j(t) = ¥. For any interval I ⊂ R+, let persist for a periodic input as the frequency of the input sig- Map(I) indicate the set of real-valued functions defined on nal approaches zero be regarded as hysteretic. This is useful I. as a test since it excludes systems (such as the linear system in Figure 3) where the looping is a purely rate-dependent Definition 1. [9] An operator G : U ⊂ Map(I) ! Map(I) phenomenon. However, in magnetic materials, magnetism is rate independent if for all time transformations j, and all can approach the single-valued anhysteretic curve as the rate inputs v 2 U, of change of the applied field approaches zero [17]. Since the response of a system to an input is affected by the his- (Gv) ◦ j = G(v ◦ j): tory of the current input (see again the simple relay in Figure 1), hysteretic systems are often described as having mem- ory. The state x in the familiar dynamical system description x˙(t) = f (x;t) encodes the memory of a system. Knowledge In [10, pg. 13] hysteresis is defined as rate independence of the current state, and the inputs is sufficient to determine combined with a memory effect, or dependence of previous the output. Consider a simple integrator: values of the input. True rate independence implies that the system is able x˙(t) = u(t) to transition arbitrarily quickly. For physical systems this y(t) = x(t): is an idealization. For instance, in the magnetic materials discussed in the above quote, changes in magnetization are controlled by a characteristic time constant. Furthermore, The output y is the integral of the input u and the state x stores thermal fluctuations means that magnetization can change the memory of the system, the current state of the integrator. even when the input doesn’t change. Thus, rate indepen- This linear system would not be described as hysteretic. dence in magnetic materials is an approximation valid when In this paper a number of common examples of systems thermal effects are low and the input does not change ex- that are said to be hysteretic are examined in detail: an elec- tremely fast. Similarly, it is not possible to vary the temper- tronic trigger, a biological switch, smart materials, mechani- ature of a shape memory alloy arbitrarily quickly, so from a cal play and inelastic springs. A model for a magneto-elastic practical point of view, the temperature-strain curves in these beam is also analysed and shown to display hysteretic be- materials typically display rate independence. Rate indepen- haviour. The common features of these systems are analysed dence/dependence in the context of various examples is dis- and used to formulate a definition of hysteresis. This paper cussed later in this paper. is not intended to be a review of the extensive literature on hysteretic systems. Some previous works that do provide re- Definition 2. [1, 10]. An operator G : U ⊂ Map(I) ! views are [1, 4, 10, 18]. Map(I) that is both causal and rate-independent is said to be a hysteresis operator. 2 Schmitt Trigger This definition has been quite useful in analysis of sys- A comparator or switch changes its output from some tems involving hysteresis - see for example, [12–15]. How- level y− to y+ when the input increases above a reference ever, there are a couple of difficulties with this approach. level. Conversely, the output is dropped from y+ to y− as the First, the definition is so general that it can include systems input drops below the reference level. A familiar example is which would not typically be regarded as hysteretic. A trivial a household thermostat where the furnace is turned on when example is y = u; another is y(t) = u(t − 1). Furthermore, as the room temperature falls below the set-point, and turned mentioned above, many systems regarded as hysteretic have off when the temperature rises above the setpoint. The prob- behaviour that is rate-independent at low input rates but rate- lem with a simple switch is that there is a tendency to os- dependent at high input rates. cillate about the setpoint as the input falls slightly above or Another typical characteristic of hysteretic systems is below the setpoint. Such oscillations can be caused by noise looping behaviour, illustrated in Figures 1 and 2. One stan- alone. For this reason, more complicated comparators with a dard text [1] writes: When speaking of hysteresis, one usually response similar to that of the simple relay shown in Figure refers to a relation between 2 scalar time-dependent quanti- 1 are used. ties that cannot be expressed in terms of a single value func- A common way to implement such a device is a circuit tion, but takes the form of loops.