Congruency-Based Hysteresis Model

for Transient Simulation

 

The present website contains a Demo version of a static hysteresis model that is an extension of the history-dependent hysteresis model described in [14]. The necessity for such an extension is described in Introduction.

This site is also supplied by a Demo version of the Classical Preisach model (CPM) with a Lorentzian PDF.

The authors of this Demo (S.E.Zirka and Y.I.Moroz) are with the Dnepropetrovsk National University, Ukraine

49050, Dnepropetrovsk, Naukova str. 13, Ukraine, e-mail: zirka@email.dp.ua

 

I. Introduction

 

The reliability of modeling and the quality of design of any device containing a magnetic core is greatly influenced by the adequacy of the hysteresis model employed and, in particular, by its ability to reproduce the magnetization history. This is especially true with respect to modern electrical machines and other equipment whose laminated cores are magnetized by nonsinusoidal voltage [1], [2]. Such an excitation causes complex shapes of the magnetization curves and their qualitative differences in different layers of the conducting ferromagnetic sheet (at the nodes of a finite-difference grid or finite elements corresponding to separate layers). It is important to note that the dynamic magnetization of such layers is described by employing static B(H) curves. In the case where excess losses should be taken into account, this can be done by introducing a time lag of the induction B behind the applied field H [2]. This transforms the static hysteresis model into a dynamic one and, remarkably, the static (basic) component of the dynamic hysteresis model obtained may be of any type and nature (history-dependent or history-independent, Preisach or non-Preisach). Thus, it is of fundamental importance to develop a quasi-static procedure, which allows one to predict the behavior of a ferromagnetic material (its induction or magnetization ) corresponding to an arbitrary change in the field . It is well known that this behavior is determined by the prehistory of the process, where every sheet layer (the node of the grid) has its own history. These nodal histories must be traced and updated when solving the boundary value problem (BVP) describing the magnetization transient. Because of its multiple recalls, such a procedure should be as simple as possible but also sufficiently accurate.

In the present website we present a history-dependent hysteresis model (HDHM) based on direct use of experimental first-order reversal curves only (the major loop branches belong to their family). This means that any first-order reversal curve is reproduced exactly by such a model, and the prediction algorithm is engaged starting from the construction of second-order reversal curves. Despite the multiplicity of hysteresis models developed to date, there is no adequate model possessing these qualities. The HDHM with the same input data most often used for studying the transient and steady-state regimes is the classical Preisach model (CPM) [3]. Its intrinsic properties are the return-point memory and the specific H-congruency of the produced curves (all branches starting from turning points with the same field H have the same shape, independent of history). It is well known that real materials do not obey the H-congruency of the CPM. The same is also true for the M-congruency of the model [4], the skew congruency of the moving model [5] or the nonlinear congruency of the product model [6]. As a result, even the modeled major loop and first-order reversal curves differ from experimental ones, and some nonphysical behavior of the calculated curves can arise. The most unwanted feature is a negative slope of the reversal curve, (examples can be found in [7, Fig. 6]), that leads to instability of the magnetodynamic equations solved.

The known iterative procedures aimed at the elimination of the nonphysical negative values of the Preisach distribution function, PDF, (described in [8] for the moving model, and in [9], [10] for the CPM), seem to be quite complex and lead to the mentioned difference between calculated and experimental first-order curves. It should be noticed that first-order reversal curves are reproduced exactly by models [11], [12] which are based on the concept of equal vertical chords [13]. However the implementation of this concept requires the measurement of first- and second-order reversal curves, and what is more important, it does not guarantee that transition curves with negative slopes do not occur starting from the third order reversal curves. This was not noticed by the authors of the concept, but becomes evident in the course of solving the BVP with complex magnetic trajectories.

The approach where negative slopes are impossible by definition is the direct use of experimental magnetization curves without their reconstruction. It is instructive, in this connection, to analyze the congruency based technique proposed in [3, formulas (1.75), (1.76)] and its modification suggested in [12, formulas (4), (5)]. Because exclusively experimental first-order curves are used in such techniques, they can be referred to as CPM-E. As such the CPM-E approach used in [3] and [12] is an uncritical extension of the H-congruency that is valid for the CPM to a real material which does not possess this property. As a result, the CPM-E [3] leads to a discontinuity of the magnetic trajectories, whereas the CPM-E [12] gives unclosed minor loops.

In the present model we renew the concept of the congruency by constructing any second and higher order reversal curve, whose overall dimensions (ODs) DH and DB are always known in the HDHM, by using internal segments of the experimental first-order reversal curves with the required ODs. The question of how to choose or approximate the specific segment arises since the number of such segments (transplants) belonging to the experimental or interpolated first-order curves is infinite. It has been proposed in [14] to close a minor loop by using a transplant which is the nearest to the formed loop in the B- (or M-) direction. In fact, this was an attempt to perfect the B-congruency observed by Madelung [15] who noticed a resemblance of the transition curves starting from turning points with the same induction B. In the model proposed we expand the transplant search region to the whole B-H(or M-H) plane. This allows us to change the shape of the reversal curve, and consequently the area enclosed by the minor loop. Such an increase in the model degrees of freedom enables one to carry out a fine adjustment of the model, taking into account, for example, the second order reversal curves (if available), and varying, in such a way, the energy losses calculated in the course of transient analysis.These details will be described in a full-length paper written in coauthorship with Prof. A. J. Moses and Dr. P. Marketos (Cardiff University, UK).

 

 

II. Magnetization regularities reproduced by the model

 

Despite the wide variety of characteristics of different magnetic media, some general regularities are observed in their magnetization processes. These regularities have been described in the early twentieth century [15] and are known as Madelungs rules (see e.g. [16]).

Considering the hysteresis curves in Fig. 1 (constructed with the proposed HDHM), the experimentally established magnetization rules [15] can be stated as follows.

Fig. 1. Transition curves illustrating Madelungs rules (the major static loop is scanned from [17]).

 

1) The path of any first-order reversal curve is uniquely determined by the coordinates of the reversal point, from which this curve emanates.

2) If any point 3 of curve 2-3-1 becomes a new reversal point, then curve 3-4-2 originating at point 3 returns to the initial point 2 (return-point memory).

3) If point 4 of curve 3-4-2 becomes the newest reversal point and the transition curve 4-3 extends beyond the point 3, it will pass along the section 3-1 of curve 2-3-1, as if the previous closed loop 3-4-3 did not exist at all (wiping-out property).

4) All transition curves which start from the reversal points having the same ordinates and sign of induction (magnetization) increment can be brought into coincidence by their parallel motion along the H-axis (B-congruency of the reversal curves).

5) If some point 5 of the initial magnetization curve becomes a reversal point, then the transition curve which starts from point 5 reaches point 5' which is symmetrical with respect to point 5 about the origin 0. At the same time, rules 2 and 3 remain in force here as everywhere (this is illustrated by the closed loops 6-7-6 and 8-9-8). (This rule has been formulated in [18] as a supplement to the regularities described by Madelung and can be useful when modeling a transient originating from the demagnetized state).

 

The proposed model reproduces rules 1, 2, 3 immediately. To reproduce the rule 4, a demagnetizing ac procedure should be first carried out (it can be performed by using the Demo proposed).

 

 

III. Input (Material) Data Required

 

The model is equally suited for both the B(H) and M(H) dependencies. It is impossible for a model to be accurate without using first-order reversal curves of a given material. Being the unique characteristics of a ferromagnet, these curves should be obtained by experiment. In the case when experimental first-order curves are not available, they can be generated by means of the procedure suggested in Appendix to [14] or even drawn by hand (this need only be done once for a given material).

The model imposes no restrictions both on the shape of the major loop and on the number of first-order curves employed, although 10-15 curves are still desirable to provide an engineering accuracy.

On this website, we post the following 5 sets of input data for both particulate recording media and soft crystalline alloys.

 

1.      Data file Mayer-89.txt contains the experimental data of the g-Fe2O3material scanned from [17]. To avoid sharp turns in the broken experimental curves [17], we have smoothed these curves by inserting some extra points and interpolating them by cubic splines.

2.      Data file Dupre-98.txt contains the experimental data of non-oriented steel. The major loop of the material has been obtained by exploiting a standard Epstein frame. First-order reversal curves of the material have been constructed by using the procedure described in [14].

3.      Data file Dupre-99.txt contains the experimental data of another non-oriented steel. Both the major loop and first-order curves have been obtained with the Epstein frame.

4.      Data file Wolf-NO.txt contains the experimental data of non-oriented steel. Both the major loop and first-order curves have been obtained with an Epstein frame.

5.      Data file Wolf-GO.txt contains the experimental data of grain-oriented steel. Both the major loop and first-order curves have been obtained with the Epstein frame.

 

 

IV. How to Install the Demo

 

1) Download the Demo.zip archive from the site. It contains an executable file of the Demo code (Demo.exe) and five data (material) files (Mayer-89.txt, Dupre-98.txt, Dupre-99.txt, Wolf-NO.txt, Wolf-GO.txt).

2) UnZip the Demo.zip with WinZip into a directory. No further installation is required.

3) If necessary, the Demo version of static (rate independent) CPM can be also installed. With this purpose, download the Preisach-demo.zip archive from the site. It contains an executable file of the Demo CPM code (Preisach-demo.exe). UnZip the Preisach-demo.zip with WinZip into a directory. No further installation is required.

 

 

V. How to Use the Demo

 

(Short Manual of April 29, 2003)

 

This program works under Microsoft Windows (95 or higher) in the 16-bit screen modes (for example 256 colors, High colors, but not True colors).

 

0) Execute the Demo.exe.

 

1) To initiate modeling, select the menu item:

File Open data file....

 

2) Choose a specific material data among the following data files:

2.1. ''Mayer-89.txt''.

2.2. ''Dupre-98.txt''.

2.3. ''Dupre-99.txt''.

2.4. ''Wolf-NO.txt''.

2.5. ''Wolf-GO.txt''.

 

3) Use the left mouse button to set the H-coordinate of the next reversal (the mouse pointer should be within the frame of the plot, in this case current H- and B-coordinates are shown at the bottom of the frame).

Use the right mouse button to construct the corresponding transition curve.

 

4) To set the next reversal point numerically, you may also use:

Menu item State Input Hn..., press Execute when ready.

 

5) To save the trajectory obtained in a text file (this option is unavailable in the Demo version):

Menu item File Save results Input a file name in the window.

 

6) Using this program, you may also simulate a demagnetizing AC procedure with up to 900 reversal points:

Menu item State Input Hn..., set 0 in the Hn input window, select item Gradual spiral approach to Hn, and set the number of cycles (<451), press Execute when ready.

Note: It may require about a minute to complete the procedure for a large number of reversals.

 

7) To study in more detail a fragment of the trajectory obtained, you can change the scales of both the H- and B-axes:

Menu item State Axes, set the values, press OK.

 

8) Default setting of the initial state is the negative saturation (Hs). In the case when the initial state of the positive saturation (+Hs) is desirable, use immediately after opening a data file:

Menu item State Settings, choose Ho=Hs, OK.

 

9) To initiate modeling with another material:

Menu item File Close workspace, and initiate the simulation beginning from item 1 above.

 

The manual for the Preisach-demo.exe is the same as above with the exception of item 2, where the value of the conventional coercivity should be entered instead of the file name.

 

 

VI. Results and Discussion

 

The use of the Demo in modeling complex trajectories is illustrated by Fig. 2, where experimental [3], [17] and calculated curves are almost indistinguishable despite the trajectory prediction is in the highest degree arduous in the square-loop material considered.

Fig. 2. Comparison of the experimental transition curves from [3] (solid lines) with the curves predicted by the proposed model (dashed lines)

 

 

The possibilities of the history-dependent model are also illustrated by an uninterrupted four-phase process shown in Fig. 3. Its first phase (a demagnetizing AC procedure) emanates from the negative saturation and completes at the origin. The second phase originates from the demagnetized state and represents the continuous magnetization up to the positive saturation (reversal point S). The trajectory showing this phase, successively passes through all the turning points in the first quadrant, and the closed minor loops obtained are wiped out from the model memory in accordance with Madelung rule 3.

The third phase of the process is a transition from point S to point 1 of the remanent coercivity. The latter point is characterized by first-order reversal curve 1-S passing through the origin (this final phase of the process is shown in Fig. 3 by dashed line). As seen from Fig. 3, first-order curve 1-S traced during the forth phase of the process substantially differs from the initial curve which is a locus of the minor loops extrema obtained in the course of the demagnetizing procedure. It is clear that the initial curve constructed with the HD model would depend on the preceding demagnetizing procedure. However, the difference between the initial curves following the demagnetization with large numbers of reversals (say 50 and 100) is negligible small. This allows us to bring these reversals into the model memory (the same history in each nodal memory) and use them in evaluating the transient when a symmetric steady-state dynamic loop is required.

Fig. 3. Uninterrupted process constructed by means of the proposed model

 

If the field reversal occurs on the ascending branch of the imaginary initial curve formed by the minor loop extrema in the first quadrant, then corresponding reversal curve returns to the point of the previous reversal which is situated on the ascending branch of the imaginary initial curve and, by virtue of the minor loop symmetry, is symmetric to the current reversal point about the origin. Consequently, rule 5 is performed automatically after demagnetizing process with a large number of reversals if the model satisfies rules 2 and 3. This is illustrated by Fig. 1 where uninterrupted curve 0-5-5'-6-7-8-9-8-6-1' has been built after the demagnetizing procedure like that shown in Fig. 3 but with 120 reversals instead of 20 ones.

The model proposed can be incorporated into a transient simulator [1, 2, 19, 20], where boundary value problem for a conducting ferromagnetic lamination is solved. As an example of the problem where both the transient and steady-state regimes can not be evaluated without hysteresis modeling, we consider the dynamic process in a single lamination subjected to the PWM voltage supply (the fundamental frequency f=50 Hz, the carrier frequency fc=1 kHz, the modulation index m=0.45). Static hysteresis loop of the magnetic material (non-oriented steel with the thickness d=0.51 mm and the conductivity s=2.63106 S/m) is shown in Fig. 4 by dashed lines. Before applying the excitation, the demagnetizing ac procedure consisting of 100 reversals has been simulated. The first three periods of the start-up transient calculated by a finite-difference scheme [19] are shown in Fig. 4 by solid line.

Fig. 4. Dynamic magnetization curves calculated for PWM and cosine voltage excitations

 

 

In the present website we have restricted our attention to the model possessing the wiping-out (deletion) property. The conventional definition of this property results in calculated minor loops immediately stable after the first traversal. In real media some number of traversals may be required to reach the static loops, so accommodation properties of the media should be taken into account for such materials. A model including accommodation can be developed as an extension of the nonaccommodating model described above. Our numerical experiments in this direction are illustrated by an uninterrupted curve shown in Fig. 5.

Fig. 5. Uninterrupted process constructed with accommodating model

 

The process shown in Fig. 5 includes 5 separate cyclings of the field. It is interesting that during one of them (cycling 5 consists of 10 reversals) the accommodation does not show itself. A detail experimental investigation of the accommodation behavior of real media is needed to complete this model.

 

Acknowledgment

 

We are grateful to Dr. L.Dupre (Ghent University, Belgium) for providing experimental data of the non-oriented steels (materials 2 and 3 above). Experimental data of the non-oriented and grain-oriented steels (materials 4 and 5 above) were obtained by the courtesy of Prof. A.J. Moses and Dr. P. Marketos (Cardiff University, UK).

 

 

REFERENCES

 

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