CC BY
6
INVESTIGATION OF A VARIATIONAL METHOD FOR CALCULATING A STATIONARY MAGNETIC FIELD WITH ALLOWANCE FOR THE ERROR OF THE MAGNETIZATION CHARACTERISTIC
DOI: 10.36724/2072-8735-2020-14-5-56-61
Robert V. Harutyunyan,
Bauman Moscow state technical University, Moscow, Russia, Research on the topic of the article
rob57@mail ru was carried out with the support of the
RFBR grant 18-01-00204 "a"
Tigran R. Harutyunyan,
JSC " MIC "NPO mashinostroeniya", Moscow region, Reutov, Russia,
tigran_20l094@mail.ru Keywords: magnetic field calculation, the bilateral method,
Sergey A. Nekrasov, the error coefficient, ferromagnetics, permanent magnets,
South Russian state Polytechnic University (NPI), magnetic scalar and v^oi- potentia|s, the method of
Novocherkassk, Russia, nekrasoff_novoch@mail.ru Lagrange mMpliers
Bilateral methods for calculating the magnetic field characteristics of electrical systems containing ferromagnets and permanent magnets are considered. The methods are based on the application of the Lagrange multiplier method to the electromagnetic field equations in terms of the scalar potential. The study described in the article complements the results of the previous article of the authors devoted to a similar bilateral method based on the Pontryagin maximum principle. The approach based on the Lagrange multiplier method has the advantage that it is applicable for solving both dynamic and stationary problems with distributed parameters. The corresponding conjugate partial differential equations for different optimality criteria (for both uniform and standard metric) are obtained. The solution of the problem of calculation of bilateral estimates of the solution in the calculation of the static magnetic field in a ferromagnet placed in a third-party uniform magnetic field is considered. This method is also applicable for the calculation of the fields of permanent magnets, which requires taking into account the residual magnetization, the final width of the hysteresis loop. For this purpose, the corresponding ratios are given. It is assumed that the main source of error in the calculation is the approximate values of the magnetic permeability of the medium. The obtained results can also be used in solving direct and inverse problems for the system of ferromagnetic bodies and in test problems using other methods.
Information about authors:
Robert V. Harutyunyan, Bauman Moscow state technical University, Ph. D., associate Professor of the Department of computational mathematics and mathematical physics, Moscow, Russia
Tigran R. Harutyunyan, JSC " MIC "NPO mashinostroeniya", software engineer, Moscow region, Reutov
Sergey A. Nekrasov, South Russian state Polytechnic University (NPI), doctor of technical Sciences, Professor of the Department of applied mathematics, Novocherkassk, Russia
Для цитирования:
Арутюнян Р.В., Арутюнян Т.Р., Некрасов С.А. Исследование вариационного метода расчета стационарного магнитного поля с учетом погрешности характеристики намагничивания // T-Comm: Телекоммуникации и транспорт. 2020. Том 14. №5. С. 56-61.
For citation:
Harutyunyan R.V., Harutyunyan T.R., Nekrasov S.A. (2020) Investigation of a variational method for calculating a stationary magnetic field with allowance for the error of the magnetization characteristic. T-Comm, vol. 14, no.5, pр. 56-61. (in Russian)
Introduction
Calculation of electrical systems taking into account the irremediable error is one of the actual applied problems 11-5]. The traditional method that is widely used to account for the error of the coefficients of a mathematical model is the small parameter method [5-7]. The disadvantages of this method are due to the faet that it is asymptotic and not reliable enough, sinee its accuracy depends significantly on the error value of the original data. Another approach is to use interval analysis methods and two-way methods [9]. However, the corresponding methods are developed primarily for systems with concentrated parameters.
The article develops a corresponding variational approach for the case of distributed magnctostatie systems. Instead of the Pontryagin maximum principle, a more General and practical Lagrange method is used, which is applicable for both dynamic and stationary problems [10]. The advantages of the proposed method are greatest when taking into account the multidimensional functional error.
The article considers as an example a magnetic system consisting of a ferromagnetic material with approximately known characteristics. One of the actual problems of the theory of calculating the magnetic field of electrical devices is related to the influence of the width of the magnetic hysteresis loop of the material on the accuracy of calculating the magnetic field. Accounting for residual magnetization in the model allows calculating systems with permanent magnets [!-4j.
1. Method description
Let's consider the problem of calculating parameters and characteristics of a stationary magnetic field in a certain magnetic system [1-3]. Assume that a material with magnetic properties (generally speaking, non-uniform with nonlinear and anisotropic properties) Occupies volume V, Magnetization M is related to the induction B and strength of the magnetic field H according to the vector relation:
M = b/\i0 - H > where Ji(j is the magnetic permeability of the vacuum. In a non-magnetic medium B-\iuH , in a ferromagnet b - ji0 (/7 + M J ■ To model the field in a permanent magnet, it is necessary to take into account the residual magnetization MH [1]: M=xH + M„, where ^ = ji/p0-lis the magnetic susceptibility of the material, J-L is the specific magnetic permeability determined in the ferromagnetic region of the main magnetization curve = , and in air, insulation materials and tires ji = j.i(l, GN/m. Thus, the magnetic permeability
and susceptibility in the ferromagnetic region are not constants. For a stationary Held, the scalar magnetic potential q> is determined : a = - v (p . We assume that the potential is zero at infinity.
Consider the boundary value problem (BVP) for calculating a plane-parallel magnelostatic field in the XOY coordinate system (the ferromagnet has a prismatic shape, the field sources do not depend on the longitudinal z coordinate). We assume that the magnetic permeability of the volume of ferromagnetic material is several orders of magnitude higher than the magnetic permeabil-
ity of air. In this ease, the field can be found with very high accuracy using the asymptotic method described in [4]. The intensity in a non-magnetic medium can be represented as: H = where H^ = H„[x,y) is the intensity vector
thai takes into account the field sources , and is the scalar magnetic potential. The vector //„ ean be calculated, for example, using the Bio-Savard-Lap lace formula, etc. In a non - magnetic medium, the Laplace equation for the scalar magnetic potential V is solved under the zero boundary condition at the ferromagnetic boundary /-/t = 0 or u (where is the vector tan-
CT.T
OT
gent to the boundary of the region), taking into account the specified field sources in space. Let's consider an almost important case when H„ we can present it in the form U a =-Vvn ,
where V'cr (A', y) - some potential. An example is the problem
of finding the field of a ferromagnet placed in a constant third-party field. Then the boundary value problem in a non-magnetic medium has the form:
Av = 0, (x,>-) e CD: v = -v„, (*,>>) e T = 3D:
(U
where D is the region of the magnetic material, F is its boundary, CD is an addition to the set D + F (the region of the non-magnetic medium).
In the volume of a ferromagnetic body or permanent magnet, the boundary value problem of the second type is solved for the magnetic potential 11 at a given density of the magnetic flux through the volume boundary:
(2)
(3)
(4)
div(p(//) V«)- 0, (x,y)eD,
on on t/(0,0) = 0; H --Vu ,
in (3), the derivatives of potentials are written along the normal n to the boundary of the region D.
It is assumed that the origin of the coordinates belongs to the ferromagnet region. It is necessary to solve the recorded equations taking into account the error of setting the magnetization curve set by the corresponding implementation band:
M = p(//)e(M (//),H*(//)),
(5)
where (/■/) is the equation of the envelope of the line
from below, ¡4. = jj,+(H) - respectively of the envelope from above, the average value of the permeability is defined as = (/■/) +ph {¡¡))jl ■ Suppose you want to find a lower
bound for solving the short-circuit problem at a given point (Wo)in the reg'on D:
wt^oJ-^mm
(6)
(when searching for an estimate from above: V(l)-> max )>
(m ,»')
where the interval .„-) denotes the set on which the minimum is found - the band of possible implementations of the magnetization curves (5).
Y
To solve such problems, the Lagrange multiplier method is widely used [5, 9]. The Lagrange functional of ihe problem under consideration has the form:
+
dtl,
ji* = H0(|Vtt|)(l-O.5ro{[VM|)sign(VuVp)),
(7)
(when evaluating from below, the " - "sign in the formula changes to"+"), uj(//)- the relative error of setting the magnetization curve at the //»strength, p - the solution of the conjugate system of the Lagrange method, which is found by the standard method [5,91 and has the form:
div^fi' (|Vw|)V/?+ H 1 ^-(VwV£r)Vw j + . + 5(a:- , y - y0 ) - 5(.v, .y) = 0, (x, y) e D.
(8)
(9)
where n is the normal vector to the boundary of area D.
If you want to estimate a standard error instead of a uniform one, for example
IML
-> mm or
II lit
->■ max i
Htj(D) II "" '!;.(/)}
then, instead of (8), use the equation:
div|V(|VH|)V/7+/r'^-( VnVp)Vu j+Co(«-«c) = 0,(jc,y)eD'
where w° is the BVP solution when the average line of the band (5) corresponding to the arithmetic mean of the lower and upper estimates of magnetic permeability is selected as the magnetization characteristic. The coefficient C(1 can be chosen to be equal to one (it is used for matching the dimensions of the terms in the
equation). The numerical solution of this problem about the standard error is supposed to be considered in the following articles of the authors. Nevertheless, the equations of the corresponding BVP are given for methodological purposes to illustrate the broad possibilities of the proposed approach.
Using the known standard finite difference methods [6,71, the computational domain and differential equations are discretized {1)-(3), (8) and (9) we obtain a system of nonlinear equations:
+w,w(0,0) + jju(xv, v,v )ô{ jcv - ,y„ - y„ )dxsdys »
D
where »r, h>. — functional Lagrange multipliers, — numerical Lagrange multiplier, â — Delta function [6,7].
The first integral in this functional is converted to the form
r "N D
To solve the problem, according to the known method, the variation of the function u is calculated and the equation of extremals of this functional is found [6-7, 9].
The values corresponding to the lower (or upper) estimation of the solution are found from the condition of the extremum of the Lagrange functional: L ->■ m in (or L max) , for all
M(|Vi/[) of the admissible set (5). The extreme value in the top
estimation is reached if the magnetic characteristic is described by the expression
V* = i-Wj E vh = (xk,yh)e. T,
(m'(|VA|) V huh - ) * K = 0. K. y h >e r-
(10) (il)
(12)
<K ÍM* (|V a|) + H„ 1 ~{Vl,H!yi,Pl, ) VhUh j + (¡3)
5 * ( ■& - x*> y„ - y„ ) - K ( **. y„ >=0 - ( xu .yjjm D>,
M (I V,,»,!) Vj;„ +Hk< ^ i Vhu,yhPh ) VA,«„ j=0, (xlry„) e <l4)
where h is a parameter of the grid partition of the region, the index h marks the grid analogs of the corresponding continuous quantities and operators, and the boundary7 values of the grid functions. BVP (l)-(9) was approximated with the second order of accuracy on h both in the region and on the border.
The problem of the Lagrange method (l)-(9) is solved itera-tively:
1. Sets the initial approximation for the magnetization curve (usually the median line of the band (5)). The solution of the conjugate BVP is assumed to be zero at the beginning in area D.
2. The external and internal short circuits for calculating the magnetic Held are solved for a certain approximate solution of the conjugate problem (8)-(9) and the corresponding approximate value j.1 calculated according to the formula (7).
3. The conjugate BVP (8)-(9) is solved when the value is recalculated according to the results of item 2 p" and the approximate solution of the field BVP (1 )-{4) is re lined.
4. Values !/(.tu,j>0) are compared in two iterations. If the
specified accuracy is not reached, the iterations continue with the transition to step 2,
The computational process of solving such problems is described in detail and mathematically investigated in the article [10]. An example of an alternative solution method is an approach based on interval estimates [9].
2. Ail example of calculating the two-sided estimates for solutions of equations of magnetostatics in ferromagnetic
Consider the case of a ferromagnet in the form of a parallelepiped (Fig. 1), located in a uniform third-party magnctie field with an induction of B(, = 0.8 T, directed along the OY axis and inducing a field in a ferromagnet. BVP will lake the form (1) at
= —Ba /ji0 y. It is necessary to find two-way estimates of the
solution at a given point (*„,>;,) ih the region D, taking into account the known error of the magnetization characteristic set using the function ra(Zi) (see formula (7)). In calculations, the
W
For Fig. 3 for a unit on the vertical axis for induction graphs, the value £fraiis=2,55,. is taken, and on the horizontal axis, respectively, the value 7max=10ir The asymmetry of the graphs in Fig. 3 is related to the special feature of the Lagrange multiplier method-estimates are found at the top (bottom), which affects the symmetry of the solution of the problem.
Fig. 4. Graphs of the solution of the conjugate BVP after 10 iterations (in relative coordinates)
For Fig. 4 graphs of the solution of the conjugate BVP ¡n relative coordinates are given (the graphs are normalized to a value of 1500, approximately equal to the average absolute value of the function p for the region, the value ym^rLy is taken as a unit along the horizontal axis).Thc described method can be used for solving both stationaiy and dynamic problems, taking into account the error of the coefficients [10], but for non-stationary short circuits, a two-way method based on the Pontryagin maximum principle successfully competes with the Lagrange multiplier method [4].
Conclusion
A new two-way method is proposed for calculating the characteristics of the stationary magnetic field of electrical systems containing ferromagnets and (or) permanent magnets, taking into account the error of the magnetization characteristic. The method is based on applying the Lagrange multiplier method to magnetic field equations in terms of scalar potential. It is assumed that the main source of error in the calculation is the approximate values of the magnetic permeability of the medium. In comparison with the common small parameter method, the limit on the error value of the parameters and characteristics of the boundary value problem is much smaller, which greatly expands the range of problems to be solved. Although the article considered an example of a plane-parallel field, the idea and basic relations of the method remain unchanged for three-dimensional problems of the
magnetic field theory. The method is also applicable for solving non-stationary problems with an irremediable error. The corresponding problems include estimates of switching overvoltages, current surges when switching on circuits with nonlinear inductances.
The proposed method has advantages in solving problems in which it is necessary to find solution estimates in the vicinity of a finite number of singular points. As shown, it is possible in principle to estimate the standard error of the solution in the entire domain. Relevant examples are direct and inverse problems for calculating permanent magnets, and evaluating the effect of the hysteresis loop width. The method can be used to improve the reliability of results in design calculations of various electrical devices, as well as in problems of the theory of magnetic measurements.
1. Blokh Yu.l. (2012). Theoretical foundations of integrated magnetic prospecting. Moscow: MGAU. 160 p. From ihe site sigma3d.com
2. Harutyunyan R.V., Nekrasov S.A., Seredina P.B. (2018), Identification of the magnetization of permanent magnets based on the scalar magnetic potential method. !zv. universities. Electromechanics. Vol. 61. No. 6. P. 19-25. DOl: 10.17213/0136-3360-2018-6-19-2 5
3. Harutyunyan R.V., Nekrasov S.A., Sereduna P.B. (2019). Identification methods for magnetization of permanent magnets based on integral equations. Research and application examples. Izv. universities, Electromechanics. No. 1.
4. Harutyunyan T.R., Nekrasov S.A. (2019). Two-way method for calculating dynamic electrical systems with distributed parameters, taking into account the error of the initial data, higher educational. Electromechanics. Vol. 62. No. 2. P. 5-13.
5. Directory for automatic councils leniyu. (1987). Ed. AA. Krasovsky, Moscow: Science. 712 p.
6. Kom T., Korn T. (1978). Handbook of mathematics. Moscow: Science. 832 p.
7. Trenogin V.A, (1980). Functional analysis. Moscow: Science, 496 p.
8 . Vasilyev O.V. (1978). The principle of maximum Pontryagin in theory optimal systems with distributed parameters. Applied Mathematics. Novosibirsk. P, 109-138.
9. Rogalev A.N, (2004). Boundaries of sets of solutions of systems of ordinary differential equations with interval initial data. Computing technology. Vol. 9. No. I. P. 86-94.
10, Nekrasov S.A. (2018). Solution of Boundary Problems with Regard for Inherent Error on the Basis of the Lagrange Method. Automation and Remote Control. Vol. 79. Issue II. P. 2018-2032,
References
т
ИССЛЕДОВАНИЕ ВАРИАЦИОННОГО МЕТОДА РАСЧЕТА СТАЦИОНАРНОГО МАГНИТНОГО ПОЛЯ С УЧЕТОМ ПОГРЕШНОСТИ ХАРАКТЕРИСТИКИ НАМАГНИЧИВАНИЯ
Арутюнян Роберт Владимирович, Московский государственный технический университет имени Н.Э.Баумана, Москва, Россия,
rob57@mail.ru
Арутюнян Тигран Робертович, АО "ВПК "НПО машиностроения", Московская область, г. Реутов, Россия, tigran_20l094@mail.ru Некрасов Сергей Александрович, Южно-Российский государственный политехнический университет (НПИ),
г. Новочеркасск, Россия, nekrasoff_novoch@mail.ru
Аннотация
Рассмотрены двусторонние методы расчета характеристик магнитного поля электротехнических систем, содержащих ферромагнетики и постоянные магниты. Методы основаны на применении к уравнениям электромагнитного поля в терминах скалярного потенциала метода множителей Лагранжа. Исследование, описанное в статье, дополняет результаты предыдущей статьи авторов, посвященной двустороннему методу на основе принципа максимума Понтрягина. Подход на основе метода множителей Лагранжа имеет то преимущество, что он применим для решения как динамических, так и стационарных задач с распределенными параметрами. Получены соответствующие сопряженные уравнения в частных производных для разных критериев оптимальности (как для равномерной, так и среднеквадратической метрики). Рассмотрено решение задачи расчета двусторонних оценок решения при расчете статического магнитного поля в ферромагнетике, помещенном в стороннее равномерное магнитное поле. Данный метод применим и для расчета полей постоянных магнитов, для чего требуется учитывать остаточную намагниченность, конечную ширину петли гистерезиса. Для этой цели приведены соответствующие соотношения. Предполагается, что при расчете основным источником погрешности являются приближенные значения магнитной проницаемости среды. Полученные результаты могут использоваться также при решении прямых и обратных задач для системы ферромагнитных тел и в тестовых задачах при использовании других методов.
Ключевые слова: расчет магнитного поля, двусторонний метод, погрешность коэффициентов, ферромагнетики, постоянные магниты, скалярный и векторный магнитные потенциалы, метод множителей Лагранжа.
Литература
1. Блох Ю.И. Теоретические основы комплексной магниторазведки. М.: МГГА, 2012. 160 с. PDF. С сайта sigma3d.com.
2. Арутюнян Р.В., Некрасов С.А., Середина П.Б. Идентификация намагниченности постоянных магнитов на основе метода скалярного магнитного потенциала // Изв. вузов. Электромеханика. 2018. Т. 61, № 6. С. 19-25. DOI:I0.I72l3/0I36-3360-20l8-6-l9-25.
3. Арутюнян Р.В., Некрасов С.А., Середина П.Б. Методы идентификации намагниченности постоянных магнитов на основе интегральных уравнений. Исследование и примеры применения // Изв. вузов. Электромеханика. 2019. №1.
4. Арутюнян Т.Р., Некрасов С.А. Двусторонний метод расчета динамических электротехнических систем с распределенными параметрами с учетом погрешности исходных данных // Изв. вузов. Электромеханика. 2019. Т. 62. № 2. С. 5-13.
5. Справочник по автоматическому управлению / Под ред. А.А. Красовского. М.: Наука, 1987. 712 с.
6. Корн Г., Корн Т. Справочник по математике. М.: Наука, 1978. 832 с.
7. Треногин В.А. Функциональный анализ. М.: Наука, 1980. 496 с.
8. Васильев Ф.П. Численные методы решения экстремальных задач. М.: Наука, 1988. 552 с.
9. Рогалев А.Н. Границы множеств решений систем обыкновенных дифференциальных уравнений с интервальными начальными данными // Вычислительные технологии. 2004. Т. 9. № I. С. 86-94.
10. Nekrasov S.A. Solution of Boundary Problems with Regard for Inherent Error on the Basis of the Lagrange Method // Automation and Remote Control. November 2018. Vol. 79. Issue II, pp. 2018-2032.
Информация об авторах:
Арутюнян Роберт Владимирович, Московский государственный технический университет имени Н.Э.Баумана, к.ф.-м.н., доцент кафедры вычислительной математики математической физики, Москва, Россия
Арутюнян Тигран Робертович, АО "ВПК "НПО машиностроения", инженер - программист, Московская область, г. Реутов, Россия Некрасов Сергей Александрович, Южно-Российский государственный политехнический университет (НПИ), д.т.н., профессор кафедры прикладной математики, г. Новочеркасск, Россия
7ТТ
