**3. The Preisach hysteresis model**

The Preisach model [31] is the most common and probably the most important model to represent magnetic hysteresis phenomenona in the literature. It was originally proposed by the physicist F. Preisach [10] in 1935 in the context of ferromagnetism and later the formalism was more broadly generalized to describe hysteretic behaviors in different fields [6, 8]. Nowadays, it is recognized as a fundamental tool for describing hysteretic systems with complex behaviors.

To describe the hysteresis, the classical Preisach model assumes that each particle of the material has an associated elementary hysteresis operator, called *relay operator*, *Preisach Hysteresis Model – Some Applications in Electrical Engineering DOI: http://dx.doi.org/10.5772/intechopen.99590*

which characterizes its state. Thus, the system can be modeled as a sum of elementary relays, whose calculation can be performed in parallel, weighted by a distribution function *μ*, called *Preisach density function*, determining, in a certain sense, the local influence of each relay on the global model. Then, the relationship between the input *u t*ð Þ and the output *w t*ð Þ is given by means of the integral:

$$w(t) = \iint\_{\rho\_2 \ge \rho\_1} \chi\_{\rho\_1 \rho\_2}(u(t), \xi) \,\mu(\rho\_1, \rho\_2) \,\mathrm{d}\rho\_1 \mathrm{d}\rho\_2,\tag{3}$$

where *μ* is the weight function, *ξ* contains the information about the "initial state" of every point of the domain and *γρ*1*ρ*<sup>2</sup> is the relay. The function *μ* is a non-negative function with compact support in the Preisach half-plane with *ρ*<sup>1</sup> ≥*ρ*<sup>2</sup> that identifies the system. In practice, it can be analytically approximated by, e.g., Lorentzian or Gaussian distributions (see [9]) or, as it will be explained in Section 3.3, estimated from experimental measurements using the so-called *Everett function* (see [8]). Depending on the properties of these relays and the relationships between them, two types of Preisach models are defined: the *classical (or static) Preisach model* based on the rate-independent relay, and the *dynamic Preisach model* introduced in [14] based on the dynamic relay.

#### **3.1 The static or rate-independent scalar Preisach operator**

In this model, the elementary relay operator is represented by a rectangular loop in the input–output plane with "up" and "down" switching values (see **Figure 2** for an example). Each elemental relay, here denoted by *hρ*, is associated to a point *<sup>ρ</sup>*<sup>≔</sup> *<sup>ρ</sup>*1, *<sup>ρ</sup>*<sup>2</sup> ð Þ<sup>∈</sup> <sup>2</sup> , with *ρ*<sup>1</sup> <*ρ*2, and, for any *ρ*, it has two states: "up" (*h<sup>ρ</sup>* ¼ 1) and "down" (*h<sup>ρ</sup>* ¼ �1) and two switching thresholds: *ρ*<sup>2</sup> is the switch-up threshold, and *ρ*<sup>1</sup> is the switch-down one.

Formally, for any continuous function *u* ∈*C*ð Þ ½ � 0, *T* and initial condition *ξ*∈f g 1, �1 , *hρ*ð Þ *u*, *ξ* is a real function defined in the time interval 0, ½ � *T* such that,

$$h\_{\rho}(u,\xi)(\mathbf{0}) \coloneqq \begin{cases} -\mathbf{1} & \text{if } \ u(\mathbf{0}) \le \rho\_1, \\ \quad \xi & \text{if } \ \rho\_1 < u(\mathbf{0}) < \rho\_2, \\ & \mathbf{1} & \text{if } \ u(\mathbf{0}) \ge \rho\_2. \end{cases} \tag{4}$$

**Figure 2.** *Scalar relay.*

Then, for any *t* ∈ð � 0, *T* , we consider the set *Xu*ð Þ*t* ≔ *τ* ∈ð � 0, *t* : *u*ð Þ¼ *τ ρ*<sup>1</sup> f g *or ρ*<sup>2</sup> . This set keeps account of the previous time instants in which *u* presents the thresholds *ρ*<sup>1</sup> or *ρ*2. Next, we define

$$h\_{\rho}(u,\xi)(t) \coloneqq \begin{cases} h\_{\rho}(u,\xi)(0) & \text{if } X\_{u}(t) = \mathcal{Q}, \\ -1 & \text{if } X\_{u}(t) \neq \mathcal{Q} \\ \mathbf{1} & \text{if } X\_{u}(t) \neq \mathcal{Q} \end{cases} \quad \text{and} \quad u(\max X\_{u}(t)) = \rho\_{1}, \tag{5}$$

If *u*ð Þ¼ 0 0, then the following initial condition can be considered

$$h\_{\rho}(u,\xi)(\mathbf{0}) \coloneqq \begin{cases} -\mathbf{1} & \text{if } \rho\_1 + \rho\_2 > \mathbf{0}, \\ \mathbf{1} & \text{if } \rho\_1 + \rho\_2 < \mathbf{0}. \end{cases} \tag{6}$$

When working with ferromagnetic materials, this initial configuration results in zero magnetic induction; thus, the material is often said to be "demagnetized" or in a "virginal" state. We remark that, in this situation, *h<sup>ρ</sup>* can only take values �1 and it changes instantaneously from its last value depending on the previous evolution of the system; more precisely, when *u* reaches the threshold *ρ*<sup>2</sup> from below, it "switches-up" to value 1, and when it attains *ρ*<sup>1</sup> from above, it "switches-down" to �1. Therefore, *h<sup>ρ</sup>* is not a local in time mapping: *h<sup>ρ</sup> u*, *ξρ* � �ð Þ*<sup>t</sup>* not only depends on *u t*ð Þ but on its past history. The *classical Preisach operator* F*<sup>S</sup>* is then defined as

$$w(t) = [\mathcal{F}\_{\mathbb{S}}(u,\xi)](t) = \int\_{T} \left[ h\_{\rho}(u,\xi(\rho)) \right](t) p(\rho) d\rho,\tag{7}$$

where we recall that variable *ξ ρ*ð Þ contains all the information of the initial state at each point of the system and *p* >0 denotes the density function.

The integral in (7) is calculated in the so-called *Preisach triangle* (see **Figure 3**, right):

$$\mathcal{T} \coloneqq \{ \rho = (\rho\_1, \rho\_2) \in \mathbb{R}^2 \, : \, -\rho\_0 \le \rho\_1 \le \rho\_2 \le \rho\_0 \}, \tag{8}$$

being �*ρ*0, *ρ*<sup>0</sup> ð Þ the thresholds setting the minimum and maximum values of *u t*ð Þ. By using this triangle, it is possible to provide a geometric interpretation of the Preisach operator. The following is a summary of the most important aspects. For an extended description, the reader can consult Refs. [8, 23]. For a given *u* ∈*C*ð Þ ½ � 0, *T* and any time *t*, the triangle is split into two sets (one of them possibly empty): *S*<sup>þ</sup> *<sup>u</sup>* ð Þ*t* and *S*� *<sup>u</sup>* ð Þ*t*

**Figure 3.** *An arbitrary input u t*ð Þ *(left) and its corresponding map on the Preisach triangle (right).*

*Preisach Hysteresis Model – Some Applications in Electrical Engineering DOI: http://dx.doi.org/10.5772/intechopen.99590*

**Figure 4.** *Staircase line Lu*ð Þ*t moving right to left (left) and moving up (right).*

containing the points *ρ*1, *ρ*<sup>2</sup> ð Þ for which the associated relays *hρ*ð Þ *u* are positive or negative, respectively, i.e.,

$$\mathcal{S}\_{\mathfrak{u}}^{-}(t) = \left\{ \rho \in \mathcal{T} : \left[ h\_{\rho}(\mathfrak{u}) \right](t) = -\mathbf{1} \right\} \quad \text{and} \quad \mathcal{S}\_{\mathfrak{u}}^{+}(t) = \left\{ \rho \in \mathcal{T} : \left[ h\_{\rho}(\mathfrak{u}) \right](t) = \mathbf{1} \right\} \tag{9}$$

(see **Figure 3**). The interface *Lu*ð Þ*t* between the two sets is a staircase line with vertices having coordinates *ρ*1, *ρ*<sup>2</sup> ð Þ respectively coinciding with the local minimum and maximum values of *u* at previous time instants. At time *t*, *Lu*ð Þ*t* intersects the line *ρ*<sup>1</sup> ¼ *ρ*<sup>2</sup> at ð Þ *u t*ð Þ, *u t*ð Þ . *Lu*ð Þ*t* moves up as *u t*ð Þ increases and from right to left as *u t*ð Þ decreases (see **Figure 4**).

Then, the Preisach operator (7) can be equivalently expressed as

$$w(t) = \int\_{S\_u^+(t)} p(\rho)d\rho - \int\_{S\_u^-(t)} p(\rho)d\rho = 2\int\_{S\_u^+(t)} p(\rho)d\rho - \int\_{\mathcal{T}} p(\rho)d\rho,\tag{10}$$

which can be readily calculated once *S*<sup>þ</sup> *<sup>u</sup>* ð Þ*t* is obtained. Since the second integral in (10) is constant in time, the computation of the output *w t*ð Þ depends on the effective computation of the first integral.

From (10), is clear that *w t*ð Þ depends on the shape of the interface *Lu*ð Þ*t* and the latter is determined by the extremal values of *u t*ð Þ, at previous instants of time. Notice that not all extremal input values are needed. In fact, considering the dependence on *Lu*ð Þ*t* , the Preisach operator exhibits a *wiping-out property*. This means that every time the input reaches a local maximum *u t*ð Þ, *Lu*ð Þ*t* erases ("wipes out") the previous vertices of the staircase having a *ρ*<sup>2</sup> value lower than the value *u t*ð Þ. Similarly, when an input reaches a local minimum *u t*ð Þ, the memory curve wipes out all previous vertices with a *ρ*<sup>1</sup> greater than the value of *u t*ð Þ. Therefore, only dominant extreme values contribute to the model, while all the other local extreme of the input are eliminated. **Figure 3** illustrates this in a particular case.

From the description above, we conclude that three basic steps characterize the model application:


• The value of *w t*ð Þ is obtained by computing the integral of *p* in the domain *S*<sup>þ</sup> *<sup>u</sup>* ð Þ*t* .

The first and third steps, namely, the identification of the Preisach function and the output calculation are difficult issues. To obtain an efficient procedure for the computation of *w t*ð Þ, we use, as usual, the so-called Everett function identified with the First-Order Reversal Curves (FORC).

#### **3.2 The dynamic or rate-dependent Preisach operator**

In [14, 32], Bertotti introduces a rate-dependent generalization of the classical Preisach model, aiming to take into account the rate of change of the input *u*. This new operator, termed as *dynamic* Preisach operator, overcomes some limitations of the classical model, in particular, the fact that the frequency of the input was not reflected in the shape of the hysteresis diagram.

Contrary to the classical relay operator, where only the two states �1 are possible, the dynamic relay can attain all the intermediate values in the interval ½ � �1, 1 , switching at a finite velocity which is assumed to be proportional to the difference between *u t*ð Þ and the threshold values *ρ*<sup>1</sup> and *ρ*<sup>2</sup> (see **Figure 5**, right panel). The proportionality factor is a parameter *k* depending on the material.

In a formal manner, and inspired by [14], for a fixed *<sup>ρ</sup>* <sup>¼</sup> *<sup>ρ</sup>*1, *<sup>ρ</sup>*<sup>2</sup> ð Þ<sup>∈</sup> <sup>2</sup> , *ρ*<sup>1</sup> < *ρ*2, we define the dynamic relay operator *ηρ* such that, for any *u* and *ξ*∈½ � �1, 1 , *ηρ*ð Þ *u*, *ξ* : ½ �! � 0, *T* ½ � 1, 1 is the unique function *y* such that *y t*ð Þ∈ ½ � �1, 1 be the solution of the (nonlinear) Cauchy problem:

$$\frac{d\boldsymbol{y}}{dt}(t) = \boldsymbol{F}(t, \boldsymbol{y}(t)) \coloneqq \begin{cases} k(\boldsymbol{u}(t) - \rho\_2)^+ & -k(\boldsymbol{u}(t) - \rho\_1)^- \quad \text{if} \quad -1 < \boldsymbol{y}(t) < \mathbf{1}, \\ 0 & \text{if} \quad \boldsymbol{y}(t) = -\mathbf{1} \text{ and } \boldsymbol{u}(t) \le \rho\_2, \\ k(\boldsymbol{u}(t) - \rho\_2) & \text{if} \quad \boldsymbol{y}(t) = -\mathbf{1} \text{ and } \boldsymbol{u}(t) \ge \rho\_2, \\ k(\boldsymbol{u}(t) - \rho\_1) & \text{if} \quad \boldsymbol{y}(t) = \mathbf{1} \text{ and } \boldsymbol{u}(t) \le \rho\_1, \\ 0 & \text{if} \quad \boldsymbol{y}(t) = \mathbf{1} \text{ and } \boldsymbol{u}(t) \ge \rho\_1, \end{cases} \tag{11}$$
  $\boldsymbol{y}(0) = \boldsymbol{\xi}.$ 

In the expressions above, the standard notations *x*<sup>þ</sup> ¼ max f g *x*, 0 and *x*� ¼ max f g �*x*, 0 , have been used; thus *x* ¼ *x*<sup>þ</sup> � *x*�. We remark that in this dynamic model the initial state *ξ* can attain all the values in the interval ½ � �1, 1 . Moreover,

#### **Figure 5.**

*Input function u t*ðÞ¼ 150 sin 2ð Þ *πft (left) for a fixed frequency f = 20 Hz. A dynamic relay and its slope are presented for switching values ρ*1, *ρ*<sup>2</sup> ð Þ¼ ð Þ 50, 100 *and initial state ξ* ¼ �1*. The center panel shows the relay with slope k* <sup>¼</sup> <sup>50</sup> *whereas the right panel shows the corresponding diagram u*, *<sup>∂</sup>ηρ=∂<sup>t</sup>* � �*. Black arrows represent increasing values of u while blue arrows represent decreasing values.*

*Preisach Hysteresis Model – Some Applications in Electrical Engineering DOI: http://dx.doi.org/10.5772/intechopen.99590*

definition (11) is consistent with that given in [2] because cases third and fourth in (11) cannot occur. Nevertheless, if one wants to perform a proper mathematical analysis, all cases must be considered. This mathematical analysis is out of the scope of the chapter; the interested readers are referred to reference [17].

Next, we consider two examples aiming to give an idea of the dynamic curve ð Þ *u t*ð Þ, *y t*ð Þ . For the first one, illustrated in **Figure 5**, a sinusoidal input *u t*ðÞ¼ 150 sin 2ð Þ *πft* is considered for frequency *f* ¼ 20 Hz and initial condition *ξ* ¼ �1. This figure also shows the curve ð Þ *u t*ð Þ, *y t*ð Þ parameterized with respect to the time variable when the relay *ηρ* is characterized by switching values *ρ*1, *ρ*<sup>2</sup> ð Þ¼ ð Þ 50, 100 and slope *<sup>k</sup>* <sup>¼</sup> 50. The right panel shows the slope *<sup>∂</sup>ηρ=∂t*, which represents Eq. (11). From the diagram it can be seen that when *ηρ* reaches value 1 or �1, the derivative is again equal to zero, as in the interval between switching values *ρ*1, *ρ*<sup>2</sup> ð Þ. The second example, illustrated in **Figure 6**, shows the dynamic relay when different frequencies of the input and slopes *k* are considered. From the center panel, it can be seen that the dynamic relay is rate-dependent. On the other and, the right panel shows curve ð Þ *u t*ð Þ, *y t*ð Þ for slopes *<sup>k</sup>* <sup>¼</sup> <sup>10</sup><sup>2</sup> (dash-dotted line), *<sup>k</sup>* <sup>¼</sup> <sup>10</sup><sup>4</sup> (dashed line) and *<sup>k</sup>* <sup>¼</sup> <sup>10</sup><sup>8</sup> (solid line). Notice that the solid line is an approximation to the discontinuous static relay of the classical Preisach model (see **Figure 2**).

From previous considerations, the dynamic Preisach operator F *<sup>D</sup>* can be defined as

$$w(t) = [\mathcal{F}\_D(u, \xi)](t) = \int\_{\mathcal{T}} \left[ \eta\_{\rho}(u, \xi(\rho)) \right](t) p(\rho) \, d\rho. \tag{13}$$

**Figure 6.**

*On the left panel, input function u t*ðÞ¼ 150 sin 2ð Þþ *πft* 75*. The center graph shows the u*ð Þ , *y curve corresponding to the dynamic relays with k* ¼ 50 *and initial state ξ* ¼ �1 *for frequencies f=50, 500, 5000 Hz (dashed, dashdotted and solid line, respectively). On the right panel, the u*ð Þ , *y curve is depicted for f= 5000 Hz and k* ¼ 102, 104, 108 *(dash-dotted, dashed and solid line), respectively.*

#### **Figure 7.**

*Input function u t*ð Þ *(left) defined in [0, 0.0045]. The isolines represent the corresponding dynamic relay values ηρ with k* ¼ 50 *and the classical relay hρ, at t* ¼ 0*:*003 *(center) and t* ¼ 0*:*0045 *(right).*

Notice that, if *ηρ* is replaced by *h<sup>ρ</sup>* the classical, rate-independent, Preisach model is obtained. **Figure 7** shows the classical and dynamic relay configurations with respect to the input *u*. The Preisach triangle is characterized by *ρ*<sup>0</sup> ¼ 300, a constant *k* ¼ 50 and the demagnetized state as initial condition.

#### **3.3 Computation of the Preisach operator based on Everett function**

In [8], Mayergoyz developed an approach for the computation of the Preisach model that does not require the Preisach density function *p* but *the Everett function*, which describes the effect of *p* on the hysteresis operator. The Everett function is obtained from the First-Order Reversal curves by a procedure described below. A FORC diagram is generated from a class of minor hysteresis loops referred to as firstorder reversal curves. A FORC branch *Fρ*<sup>0</sup> <sup>2</sup> is associated to the threshold *ρ*<sup>0</sup> 2. The input *u* rises up from the "reset" state (every relay is in the "down" state, i.e., *S*� *<sup>u</sup>* ðÞ¼ *t* T and *u* ¼ �*ρ*0). The output value in the *ρ*<sup>0</sup> <sup>2</sup> point is called *wρ*<sup>0</sup> <sup>2</sup> (inversion point). Then, *u* is brought back to �*ρ*0. The branch *Fρ*2<sup>0</sup> is drawn by taking the output value *wρ*<sup>0</sup> 1,*ρ*<sup>0</sup> <sup>2</sup> for any value *u* ¼ *ρ*<sup>1</sup><sup>0</sup> as is shown in **Figure 8** (left). The branch ends for *u* ¼ �*ρ*0, when *S*� *<sup>u</sup>* ðÞ¼ *t* T . The Everett function is defined in [8] as.

$$E\left(\rho\_1', \rho\_2'\right) \coloneqq \frac{w\_{\rho\_2'} - w\_{\rho\_2', \rho\_1'}}{2},\tag{14}$$

which is half of the output variation along the FORC branch starting in *ρ*<sup>0</sup> 2. The Everett function and the Preisach function are related by the following integral:

$$E(\rho\_1, \rho\_2) = \int\_{T(\rho\_1, \rho\_2)} p(\rho) d\rho \qquad \forall (\rho\_1, \rho\_2) \in \mathcal{T},\tag{15}$$

where the integration domain T *ρ*1, *ρ*<sup>2</sup> ð Þ is the triangle highlighted in **Figure 8** (right). It should be noted that the integral on the triangle T *ρ*1, *ρ*<sup>2</sup> ð Þ is a function of its upperleft corner and that *E ρ*1, *ρ*<sup>2</sup> ð Þ¼ 0 if *ρ*<sup>1</sup> ¼ *ρ*<sup>2</sup> (T *ρ*1, *ρ*<sup>2</sup> ð Þ degenerates in a single point). The introduction of the Everett integral simplifies the computation of the first integral on the right hand side of (10). First of all, we subdivide *S*<sup>þ</sup> *<sup>u</sup>* ð Þ*t* into *k* trapezoids *Qk*ð Þ*t* so that *S*<sup>þ</sup> *<sup>u</sup>* ¼ ∪*kQk*. Moreover, each trapezoid can be represented as the set difference of two triangles T *mk*�<sup>1</sup> ð Þ , *Mk* and T ð Þ *mk*, *Mk* :

**Figure 8.** *Left – First order reversal curves (solid line) and major loop (dashed line). Right – Triangle* T *ρ*1, *ρ*<sup>2</sup> ð Þ*.*

*Preisach Hysteresis Model – Some Applications in Electrical Engineering DOI: http://dx.doi.org/10.5772/intechopen.99590*

$$\int\_{Q\_k(t)} p(\rho)d\rho = \int\_{\mathcal{T}(\mathfrak{m}\_{k-1}, \mathcal{M}\_k)} p(\rho)d\rho - \int\_{\mathcal{T}(\mathfrak{m}\_k, \mathcal{M}\_k)} p(\rho)d\rho. \tag{16}$$

Taking into account that the integral on a generic triangle T *ρ*1, *ρ*<sup>2</sup> ð Þ can be expressed by (15), the Preisach integral (10) becomes

$$w(t) = 2\sum\_{k=1} (E(m\_{k-1}, M\_k) - E(m\_k, M\_k)) - E(-\rho\_0, \rho\_0),\tag{17}$$

where *<sup>E</sup>* �*ρ*0, *<sup>ρ</sup>*<sup>0</sup> ð Þ¼ <sup>Ð</sup> <sup>T</sup> *p*ð Þ*ρ dρ*. Form the previous procedure we obtain the following results. First, the output can be computed by using a simple linear combination of the *E* values in the memory state points represented by the vertices on *S*<sup>þ</sup> *<sup>u</sup>* . Second, the value of the Preisach density *p* is not required, since the identification of *E* in the domain T is sufficient to apply the model. The identification of *E* is simple and it is defined by a repeatable and reliable procedure based on experimental data (FORC branch).
