UDC 538.3 DOI 10.52167/1609-1817-2022-121-2-274-281 L Kh Bazarov, I M Bedritsky, K K Jurayeva, M.S. Mirasadov “

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ҚазККА Хабаршысы № 2 (121), 2022 ISSN 1609-1817 (Print) The Bulletin of KazATC Вестник КазАТК № 2 (121), 2022 ISSN 2790-5802 (Online) DOI 10.52167/1609-1817 vestnik.alt.edu.kz

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UDC 538.3 DOI 10.52167/1609-1817-2022-121-2-274-281

L Kh Bazarov, I M Bedritsky, K K Jurayeva , M.S. Mirasadov

“Power Supply” Department, Tashkent State Transport University, Tashkent, Uzbekistan

Е-mail: [email protected]

APPROXIMATION OF MAGNETIZATION CURVES OF ELECTRICAL STEEL

Abstract. It is made a comparison of functions approximating the main magnetization curve of electrical steels, selected according to two main criteria: obtaining the simplest expressions in an analytical study and achieving a minimum calculation error in quantitative calculations of devices with ferromagnetic components. The cases are considered when ferromagnetic elements in devices operate at alternating current and at high values of magnetic induction, that is, they are in a mode close to saturation, while the phenomena of hysteresis are usually neglected, and the main magnetization curve is used as the magnetic characteristic of the ferromagnetic element, it is shown, that when the devices operate in a saturated mode, an incomplete polynomial of n - degree can be used for these purposes. It is proved that the errors when approximating by polynomials with degrees 9 and 11 give the smallest errors not exceeding 12 and 8% for components of devices based on cold-rolled steels, and errors not exceeding 13% when approximating by polynomials with degrees 3 and 5 for components of devices based on hot-rolled steels.

Keywords. Electrical steel, saturated mode, magnetic characteristic, magnetization curve, approximation, incomplete polynomial of odd degree, least squares method, approximation error, ferromagnetic elements.

Introduction.

The models of Giles-Atherton, Chan and others are most often used [3, 5, 6, 8, 12, 14, 15], in order to approximate the hysteresis loop when analyzing devices that have ferromagnetic elements in their design. However, if the ferromagnetic elements in these devices operate on alternating current and at high values of magnetic induction, which means they are in a mode close to saturation, then the phenomena of hysteresis are usually neglected. In this case, the main magnetization curve is used as the magnetic characteristic of the ferromagnetic element, which is approximated by a certain algebraic expression [1, 2, 4, 7, 9].

The choice of the approximating function depends on the nature of the problem being solved, the simplicity of transformations of analytical expressions and the accuracy of quantitative relationships. The requirement for simplicity of analytical transformations is decisive in a qualitative analysis of ferromagnetic devices.

Materials and Methods.

The aim of the study is to compare the functions for approximating the main magnetization curve of cold-rolled and hot-rolled electrical steels, selected according to two main criteria: obtaining the simplest expressions in the analytical study of devices with ferromagnetic elements and achieving the minimum calculation error in quantitative calculations.

We used ready-made cores made of laminated and twisted electrical steels as models for the study, the experimental magnetization curve of which was recorded on alternating current with a frequency of 50 Hz according to the method described in [9, 10, 11, 12].

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In particular, the method described in [9] for measuring the average values of voltages and the corresponding amplitude values of the magnetizing current was used to solve the problem. The diagram of the measuring unit is shown in Fig. 1.

Figure 1 - Setup diagram for finding the dependence B=f(H)

The measurements were carried out on samples of rod-type chokes of a twisted structure with a cross-sectional area of 0.0085 m² with a rolled thickness of 0.08 mm for steel E360 (3424) and 0.1 mm for steel E41 (1511). The amplitude of the induction and the amplitude of the magnetic field strength were calculated using the expressions

,

10 ; 4

1 2

4 l

w H I

f w S

B_m U^L _m  ^m

  _ (1) where w1 and w2 are the number of turns of the throttle windings; UL is the value of the rectified voltage on the winding w2, V, measured by a magnetoelectric voltmeter V2 at the output of the rectifier V; Im is the current amplitude in the winding w1, A; S is the cross – section of the core, sm²; l is the length of the middle line of the core, sm; f is the frequency of the supply network, Hz. The current amplitude Im was determined by measuring the voltage drop proportional to it at a known resistance R3 when it was powered from autotransformer AT. The specified voltage drop was compared with the constant voltage removed from the resistance of the divider R1, which was measured by a voltmeter V1. The moment of equality of these voltages is fixed by the oscilloscope O at the moment of disappearance of the rectified voltage pulse from VD, and the current Im was calculated by the formula

3

1

R

I_mU , (2) where U1 is the voltage measured by the voltmeter V1.

To approximate the dependence or where B is the induction, H is the magnetic field strength, the expressions chosen according to the criterion of the highest frequency of reference in literary sources were used:

 hyperbolic sine and tangent functions in the form and , where k, l, m, n are linear coefficients;

 polynomials up to the thirteenth degree inclusive in the form

where are linear

coefficients;

 incomplete polynomials in the form, where k is a linear coefficient, n is an exponent (an odd integer up to 13 inclusive).

Linear coefficients were calculated using the least squares method, the transition from nonlinear functions to linear ones was carried out using the appropriate substitutions [13, 16, 17, 19, 20, 22].

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276 Results.

The following approximating expressions were obtained for cores made of cold-rolled electrical steel grade E360 (3424) in the range of induction variation from 0 to 1.9 T:

- hyperbolic sine

- hyperbolic tangent ;

- polynomial of the thirteenth degree;

;

 incomplete polynomial of ninth degree

E-experiment; 1- ; 2 – ; 3 -

; 4- Figure 2 - Expressions for approximating the magnetization curve of steel E360 (3424)

Fig. 2 Shows the graphs of the dependence for steel E360 (3424), built on the basis of the above approximating expressions, and the main magnetization curve obtained experimentally. It can be seen from the graphs that all expressions are suitable for approximating the selected grade of cold-rolled electrical steel according to the accuracy criterion. However, expressions 1 and 2 are inconvenient for subsequent transformations, since expressions with hyperbolic functions are inconvenient for obtaining inverse dependencies (H versus B or B versus H). Usage of expression 3 leads to cumbersome analytical expressions. Obviously, the most compromise in terms of simplicity and accuracy is the approximation by an incomplete polynomial of the ninth degree (curve 4).

Let us investigate in more detail the approximation by a generalized polynomial of the form with an odd exponent. The objective of the study is to determine the values of k and n at which the minimum root-mean-square error is achieved. The value of the linear coefficient at fixed values of the exponent of the polynomial n = 3, 5, 7, 9, 11, 13 was found according to the well-known expression for the least squares method [16, 19, 20, 21], modified taking into account the passage of the approximating curve through the origin and replacement , where , leading to linearization of the approximating functions. Taking into

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account the indicated replacement, the expression for calculating the optimized value of k was transformed to the form

, (3) where is the number of experimental points on the magnetization curve; - point number; - experimental values, respectively, of magnetic induction and magnetic field strength at the -th point.

E-experiment;3- ; 5- ; 7- ;

9- ; 11- ;13- .

Figure 3 - Approximation of the magnetization curve of cold-rolled electrical steel E360 (3424) by an incomplete polynomial

1-experiment;3- ; 5- ; 7- ;

9- ; 11- .

Figure 4 - Approximation of the magnetization curve of hot-rolled electrical steel E41 (1511) by an incomplete polynomial

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In fig. 3 and fig. 4 shows the graphs of the dependencies , constructed according to the approximating incomplete polynomials indicated in the figure caption, the optimized linear coefficients k of which are calculated by expression (1) for cold-rolled steel E360 (3424) (Fig. 3) in the range of induction changes from 0 to 1, 8 T and for hot-rolled steel E41 (1511) (Fig. 4) in the range of induction change from 0 to 1 T. As follows from the graphs, the minimum approximation errors correspond to the use of incomplete polynomials of 9, 11 and 13 degrees for cold-rolled steel E360(3414) and incomplete polynomials of 5 and 7 degrees for hot-rolled steel E41(1511). Optimized coefficients k of polynomials are calculated using 20 points of the experimental dependence .

Discussion.

For an objective assessment of the errors, the nature of the change in the relative error of the approximation was investigated with a change in the magnetic field strength. The relative error of approximation for each of the experimental points was calculated by the expression

% 100 (%)  

i iA i

Н Н

 Н , (4)

where is the experimental value of the magnetic field strength at the -th point; - the value of the magnetic field strength, calculated by the expression for the approximating function.

3- ; 5- ; 7- ;

9- ; 11- .

Figure 5 - Incomplete polynomial approximation errors for cold-rolled electrical steel E360(3414)

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3- ; 5- ; 7- ;

9- ; 11- .

Figure 6 - Errors approximation by an incomplete polynomial for hot-rolled electrical steel E41 (1511)

Conclusion.

The curves of the dependence for incomplete polynomials of degrees from 3 to 11 for cold-rolled steel E360 (3424) are shown in Fig. 4. It can be seen from the graphs that the errors in the approximation by polynomials with degrees 9 and 11 give the smallest errors, not exceeding (12 and 8%, respectively), which can be considered acceptable when calculating the ferromagnetic components of devices based on cold-rolled steels.

The curves of the dependence for incomplete polynomials of degrees from 3 to 11 for hot-rolled steel E41 (1511) are shown in Fig. 5. It can be seen from the graphs that the errors in the approximation by polynomials with degrees 3 and 5 give the smallest errors, not exceeding 13%, which can be considered acceptable when calculating the ferromagnetic components of devices based on hot-rolled steels.

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Kamila Jurayeva, PhD, Associate Professor, Tashkent state transport university Tashkent, Uzbekistan, [email protected]

Laziz Bazarov, doctoral student, Tashkent state transport university Tashkent, Uzbekistan, [email protected]

Ivan Bedritskiy, Doctor of Technical Sciences, associate professor, Tashkent state transport university Tashkent, Uzbekistan, [email protected]

Mirkomil Mirasadov, assistant teacher, Tashkent state transport university Tashkent, Uzbekistan, [email protected]

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