Modeling of hysteresis loops in magnetic systems

MODELING OF HYSTERESIS LOOPS IN MAGNETIC SYSTEMS


O.T. Boltayev
Tashkent State Transport University


Introduction. Ferromagnetic electrotechnical materials are widely used in the production of electrical devices. When using these electrotechnical materials in practice, it is necessary to know their technical characteristics. There is a problem of calculating the power loss occurring in the core of electromagnetic devices, steel suspension structures, various ferromagnetic bodies and determining the distribution of the cumulative current and magnetic field in the ferromagnetic material when choosing the normal operating mode of this object. When solving problems at this level, it is necessary to take into account the nonlinearity of any ferromagnetic material. In particular, it is necessary to know the magnetic hysteresis surface of the used electrotechnical steel to perform the necessary calculations in practice. However, it is not always possible to find the required curves from suitable references. Therefore, several functions are used in the construction of the hysteresis surface of electrotechnical magnetic material. During the calculation, a convenient function (formula) is selected depending on the characteristics of the material and the accuracy level of the calculation result.
Main part. The following requirements are imposed on the approximating function (formula):

• 
  the approximating function gives a result close to the material hysteresis surface;
  
• 
  since the function is used in the process of differentiation, its derivative also gives a result close to the material hysteresis surface;
  
• 
  calculations performed on the approximating function should not be difficult;
  
• 
  the approximating function should be odd;
  
• 
  the approximating function includes the parameters of the universal and material hysteresis surface;
  
• 
  the function should consist of as few constants as possible.
  

The widely used Relay formula and arctangent functions are used to determine the magnetic hysteresis curve of various ferromagnetic materials.
In most cases, the following Relay formula is used to describe the magnetic hysteresis surface in areas of weak magnetic field ( :
(1)
where is – initial magnetic susceptibility; – Relay coefficient; – the strength of the magnetic field; – maximum strength of the magnetic field; - coercive force in the limit cycle.
In the given expression, the "+" sign is appropriate for the rising part of the hysteresis surface, and the "-" sign is appropriate for the falling part. Using this expression, the results of the generated hysteresis contour coincide only with the edge points of the real contour of the ferromagnetic material, and deviations are observed in the remaining parts. Therefore, trigonometric functions, harmonic series of Lissage figures, ellipses (Arkadev's method), Hooke's law, S-shaped curves (Akulov's method), line expansion methods, polar coordinates and other methods were used to describe the magnetic hysteresis surface. But these methods were not widely used in practice for various reasons.
An expression consisting of an arctangent function is most often used to describe the magnetic hysteresis surface and has the form:

(2)

where is – Hysteresis field can be a function of simultaneous or separate magnetic field strength or instantaneous value of material magnetization and their direction change; – correction function of magnetization or magnetic field strength; – The approximation parameters are determined using the point selection method.
In expression 2, we determine the approximation parameters by solving the system of equations formed when the following conditions are met:
1) ;
2) ;
3)
where is – saturation magnetization; – residual magnetization.

Putting the determined parameters into the 2nd expression, we create the following formula of magnetization:

(3)

here, the "-" sign is appropriate for the rising part of the hysteresis surface, and the "+" sign is appropriate for the falling part.
Expression 3 is called the Zatsepin approximation, and it is possible to describe only the outer limits of the magnetic hysteresis surface. Also, to describe the specific hysteresis loop using this expression, it is necessary to change the saturation magnetization, remanent magnetization, and coercive force to the maximum values of these quantities, and in this case, expression 3 has the following form:

But it is observed that the hysteresis surface approximated by this expression has jump-changing parts. This, in turn, causes the result to be erroneous.
Y.F. Panamaryov proposed to introduce the following expression as a correction function to eliminate the shortcomings of the Zatsepin approximation:

From the approximation parameters for expression 2
is the same as the coefficient determined when the condition is met. the coefficient is determined by differential magnetic susceptibility according to the conditions , and has the following form:

Substituting the expressions for the approximation parameters and the correction coefficient in expression 2, we form the following equation:

where the lower sign is appropriate for the rising part of the hysteresis surface, and the upper sign is appropriate for the falling part.
By inserting the condition into the last expression, the following expression for the magnetic hysteresis boundary surface is formed:

(4)

N.S. Akulov and B.A. Luchevskii proposed to include a non-standard expression for the correction function in expression 2 to describe the magnetic hysteresis surface of magnetoelectric materials. This expression is not widely used in practice due to the presence of residual magnetization, differential and reversible magnetic susceptibilities, as well as large errors in the experimental determination of the above-mentioned parameters.
M.A. Melguy proposed to introduce the following expression as a correction function to describe the magnetic hysteresis surface of magnetoelectric materials:

where is the coefficient determined by the condition .

Fig.1. Magnetic hysteresis contour calculated using expressions 4 and 5

( )

Among the proposed approximation methods for describing the hysteresis surface, it is possible to obtain the closest result to the experiment through the Panamaryov approximation. But the Melguy approximation, which is easy to determine the approximation parameters to achieve a relatively close result in a simple way, is widely used in practice.
Since the results of expressions 3 and 5 coincide with each other, it is sufficient to compare the results of expressions 4 and 5 with the experiment. Taking into account that the value of the coercive force of soft magnetic materials is in the range of 32-72 A/m, we compare the hysteresis surface for electrotechnical materials in this range with the experiment.
By comparing the results obtained using expression 4 with the experiment, we can see that the degree of accuracy of the magnetic hysteresis is several percent. In other cases, the error of calculation results is higher than 20%. It should be noted that when the hysteresis surface is calculated with the help of the 4th expression, an excess value of residual magnetization occurs. This condition should be taken into account when using the expression.

Fig.2. Magnetic hysteresis contour calculated using expressions 4 and 5
( )