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TWO-SIDED METHOD FOR CALCULATION OF DYNAMIC ELECTROTECHNICAL SYSTEMS WITH DISTRIBUTED PARAMETERS WITH GIVEN THE ERROR OF THE ORIGINAL DATA
DOI 10.24411/2072-8735-2018-10271
Tigran R. Harutyunyan,
Moscow, Russia, tigran 201094@ mail.ru
Keywords: calculation of magnetic field, two-way method, error of coefficients, ferromagnets, permanent magnets, scalar and vector magnetic potentials, maximum principle.
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 Pontryagin maximum principle to the electromagnetic field equations in terms of scalar and vector potentials. The solution procedure uses the transition from the differential formulation of boundary value problems of the magnetic field to the corresponding discrete-continuous in the form of a system of ordinary differential equations, to which the classical theory of the maximum principle applies. After finding the equations of the boundary value problem of the maximum principle, the inverse limit transition to the differential form is carried out by means of the grid step striving to zero. The corresponding conjugate partial differential equations for different optimality criteria are obtained. The solution of the problem of calculation of bilateral estimates of the solution in the calculation of the magnetic field in a ferromagnetic placed in an external equal 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.
In comparison with the used method of a small parameter, the limitations on the error of the parameters and characteristics of the boundary value problem are much smaller, which greatly expands the range of problems to be solved. Although an example of a plane-parallel field was considered in the article, the idea and the main relations of the method remain unchanged for the three-dimensional problems of the magnetic field theory. The proposed method has advantages in solving those problems in which it is required to find estimates of the solution in the neighborhood of a finite number of singular points. As shown, there is a fundamental possibility to estimate the mean square error of the solution in the entire region. The corresponding examples are the problems associated with switching overvoltages, current surges when switching circuits with nonlinear inductances, direct and inverse problems of calculating permanent magnets, when assessing the effect of the hysteresis loop width. The possibilities of modern numerical technique allow to find with the help of the considered methods also effective uniform estimates of the solution at large intervals. The method can be used to improve the reliability of the results in the design calculations of various electrical devices, as well as in the problems of the theory of magnetic measurements.
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.
The developed approach can also be successfully applied in solving a wide range of dynamic problems for systems with distributed parameters, including the theory of elasticity, thermal conductivity, piezoelectric oscillations, quantum mechanics and other fields.
Для цитирования:
Арутюнян Т.Р. Двусторонний метод расчета динамических электротехнических систем распределенными параметрами с учетом погрешности исходных данных // T-Comm: Телекоммуникации и транспорт. 2019. Том 13. №5. С. 49-55.
For citation:
Harutyunyan T.R. (2019). Two-sided method for calculation of dynamic electrotechnical systems with distributed parameters with given the error of the original data. T-Comm, vol. 13, no.5, pр. 49-55. (in Russian)
Introduction
One of the roost relevant in theoretical electrical engineering and in other applied sciences is the problem of adequately taking into account when solving problems of inaccuracy of initial data. This problem also it has important problems in processing the results of indirect measurements. In the article is given two-sided method of solution of the problem, based on the integration of the variational principles and bilateral methods. This method is applicable w the ease of systems with both concentrated and distributed parameters ¡1-12].
The traditional and the most common method, allowing under certain conditions to take into account the error of the model coefficients; is the method of small parameter (4-6J. Its prezision essentially depends on the error in the initial data. The disadvantage of this method is also the lack of guarantees of the reliability of the results of calculations. Another approach is the use of interval analysis methods. However, the corresponding methods are effective only in eases when the error of the model is sufficiently small
The approach described below is based on using the Pon-tryagin maximum principle | 4-12 J and assumes significantly less restrictions on the magnitude of unrecoverable error. The correct application of this approach involves the proof of the existence of an optimal solution, which is one of the actual problems of I he theory of the maximum principle [4-12].
Numerical methods are mainly used to solve the nonlinear boundary value problem of the maximum principle (BPPM), Biggest winnings when using the proposed method achieved by taking into account the functional error.
The article considers a magnetic system consisting of a ferromagnetic parallelepiped with known characteristics placed in a third-party magnetic Held as an example. One of the actual problems of the theory of calculation of the magnetic field of electrical systems is related to the influence of the magnetic hysteresis loop width of the material on the accuracy Of the magnetic Held calculation [2,3]. Taking into account the residua! magnetization in the model, it is possible to calculate systems with permanent magnets.
I. Method Description
Will consider the task of calculating parameters and characteristics non-stationary magnetic field in some magnetic system. Assume that material with magnetic properties (generally inho-mogeneous and anisotropic nonlinear properties) occupies a volume V. Heavens to formulate the tasks have to calculate the magnetic field in the space.
Мал well's equations in areas V has the form 111;
rot/7 = 5, го\Е = -дв/дгл (1)
div ß - 0. div£> = p.
(2)
where H - vector magnetic intensity, A / m; B - vector induces a magnetiction, T; E - the vector of the electric field, V / m; D - vector of electrical induction C/m3; 5 = ÖCu,p + Sl№:P -current density vector A An; 5«,.|> - vector known density (external) current; S,,,,,,, - density vector of eddy currents in tires and fcrromagnet; p - bulk density of electric charge, A/nv1;
t - time, sec.
For a stationary field , the scalar magnetic potential is determined: H =— Vu We assume that the potential at infinity is zero. Modeling of the field in the magnetic system with permanent magnets is usually carried out in terms of vector magnetization which first associated with the induction and magnetic
field vector according to the relation M = ¿'/f-i„ - H : where - magnetic permeability of vacuum, II / m. In a nonmagnetic environment B = ^xaH , in a ferromagnet B = f H + ~Mj .Within the scope of a permanent magnet is
required take into account residual magnetization A/„ [1]: M = iH + Mn, where x = %(^) " magnetic susceptibility of
the material. The relation^ -], where (.1 - specific
magnetic permeability of a substance thai determined by main magnetization curve ji = mBj , in air, insulation materials
and tires |.t = H / m.
Changes in the field in time will assume a sufficiently slow to satisfy the assumption of quasi-stationary field. Taking into account the above relations, the material equations of the medium can be written as:
B = li(B)H+ Br,D = EE,5 = yE + k,»l,, (3)
Where £ - specific electric permeability, f/m; y - specific electric conductivity, (different from zero in conductors) B, - residual induction.
Aii identity div rot F-0 that is valid for any vector function /■'( .v, y,z) is known, where .r, y, z - coordinates Cartesian coordinate system [5].
From the solenoid condition of the magnetic field
di v ¿J — 0 , so that the field B can be represented as B= rot A ,
where A - magnetic vector potential, Wb/m.
We shall consider practical importance the case when the magnetic field is plane-parallel with the induction vector lying inthe plane xay.
Then the vector potential canbe chosen in the form
A = .-((.*, v.i)e: where e-_ - the unit vectors of the Cartesian coordinate system.
According to the law of electromagnetic induction. It can be
linked the values A and E as follows:
- № - dA rotE-— — rotE = -rot —- ■ dt dt
It is known that for any scalar function U right rot( A + VO' j = rot.-i • consequently rotVi/ = 0. therefore, we can write
dt
where (J - U(.\\y,t)~ scalar electric potential, V.
Therefore, 8 = - y
dt
Of the Law of a total current should he:
rot — rot,4
\V>
- ôci)p — y
fdA — + VU
dt
Rewrite given expression in coordinate as :
- - dA - SA
roM = es--£y — t
dy dx
rot
fl
—i
M
H ra ' ISA" +JL (\ cMV
[B , M dx J dy a* syjj
Therefore
'eil SA
w
___| dfldA
' dx{ndx) l m 3y }j or
a
1 C'A
d +—
dx ^ j.( dx ) dy
'làf , H dy
- -Ô
dA eu
dû
Since U =&(x.vj), -- = 0 what hence, the equation
V 8z
of the vector of magnetic potential in the coil will have the form
d2A _ dA
— = -Ho '
82A
dx2
dvA
rmop T M-Q ' Y ' ~ "
It should be noted that in the conditions of low-frequency fields and the assumption of their quasi-stationarity, the eddy currents in the winding can beneglected. For the area occupied by magnetic materials, the equation
. M ty
- —-JL\ JL
dt dx {f-t dx J dy x
(4)
In isolation and air
Ö2A
2
Äo.
ôxT dy"
Another approach is to use scalar for non-magnetie regions. Magnetic potential (see below). The boundary conditions for the magnetic vector potential: conditions at the interface of media with different magnetic permeability have the form
i dA
[A] = 0.
Jul dn
= 0 ,
ability, (B ),i'*( B) - accordingly lower scores above and for the specific magnetic resistance V .
We will assume that the magnetic permeability of the volume ofa ferromagnetic material is several orders of magnitude higher than the magnetic permeability of air. In this case, the field can be found with a very high precision asymptotic method. As a rule, the values of induction (as opposed to tension) in a ferro-magnet and in its (non-magnetic) neighborhood have the same order of magnitude. The boundary conditions at the boundary of the media have thefortn (conditions the continuity of the components normal and tangentia! to the induction of tensions),
B,
[5„] = 0,[//r] = 0 ,Where
L m
= o. From the last equation,
where [,l] - size jump A when crossing the border of structural elements.
On the boundaries of the computational domain is, given the homogeneous Diriehlet condition A ~ 0.
Required to solve the equation, taking into account the recorded error of defining the magnetization curve, which I supposed to be in a band described by the equations:
where, (5) ,f.i* (/?) - functions corresponding to the lower and upper limits of the band for the values of magnetic perme-
because of the large difference of magnetic permeability values for the magnetic medium at the interface with high accuracy can be considered H = 0. Tension in (lie non-magnetic environment can
be ep resented as: H = Hil-V »where fi„ = Hc, {.r, y,t) -vector of intensity, taking into account the sources of the field, It - scalar magnetic potential. Vector Hei It can be calculated, for example, by the formula Biot-Savart-Laplace and the like. On this basis, the boundary problem of calculating a field in space can be divided into two: at non-magnetic medium solves the Laplace equation for scalar magnetic potential n
/
withzero boundary condition //. =0 (or — — H ) on ferro-
ck
magnetic boundary with the defined source fields in the space.
In tlie second phase boundary problem is solved in a magnetic medium at a predetermined boundary of its normal induction component (found on the first step of solving the problem in a nonmagnetic medium). The high accuracy of this method has been confirmed as the method of computing and analytical mode) problem [lj. Consider a practically important ease
when Ma can be represented in the form Ha Viv ,
where (x,y,t) some potential. An example is the task of
finding of the field of ferromagnetic, the premise of a permanent third-party field. Then the boundary value problem in a nonmagnetic medium has the form:
Ait = 0, U\ v) e CD\ u = -u^{x,y) gT = №, (6)
where D - area of magnetic material, T - its boundary, CD -addition to the set D + i (region non-magnetic medium). It is useful noted that boundary problem is generally speaking non-stationary: although lime is not included in the differential equation of the boundary problem (6), the boundary condition contains a function of time itLy (v, v,/) ■
in the volume of the ferromagnetic body or permanent magnet the second type boundary value problem is solved for slock vector magnetic potential at a given current density for the magnetic volume through the border:
dA dt
y— = div(v'(ß}V/i),<.\\_r)e D;/>0;
(7)
-- dA cA dA du a
ß = Bn = nv—-n — =—=-fi„— ,(*;>>)€ T; (8)
" .V n 1 i I U i '
ov dx ch dn
A(.v,y\t)=~ j „A.vv,y v,/)drH, (x,y) e T;
Bn
(9)
(H .,>'„ I
It should t>c noted that (7) is equivalent notation of (4). Having carried out the discretization of the computational domain and differential equations for spatial variables, we obtain the system ordinary differential equations:
Y ^ = div, (v (%) VlA). (x„,yk )eD,- (J Q)
■4/,= -'h,' ' Vh)e r 1,1
Where h - parameter of the grid partitioning of the region, index h labeled analogs grid corresponding continuous variables an
d operators, Ah - known boundary values of the grid function.
Suppose that we need to find an estimate below solutions of
a boundary value problem at a point {.Y,(, Vtl) at time t. :
Ai-Wm'r)-* mil1 • (M)
mm
I" v)
where (v ,v') is the interval denotes the set, where the minimum is finded - possible implementation bar of the magnetization curves.
To find such an estimate, we apply the Pontryagin maximum principle [4-12], then the estimate of the vector potential can be found as a solution to the boundary value problem of the maximum principle. The extreme value is reached if the magnetic character is described by the expression
v"-Vn(S){j~0.5^)sign(V;iV/>))y (12)
Where ra ( B) - relative error of the magnetization curve setting at induction 5, p - the solution of the conjugate system BPPM (cm. below). This expression v is by the standard method [4-!2] based on the maximization of the Hamiltonian of
the problem oil the interval (v~(tf),v' (/))). The task of finding estimates from above
max
(rV)
differs only in the value of the v* = V„ {B) (1 + 0.5m (B) sign (VAVp)).
The conjugate equation of CPM in the limit h form*.
y ^ = div iv* (B)Vp-r1 ^(VAVp)VA
coefficient
0 has the
(13)
(.r,v)EZ);0</</f; p(x,y,tf,) = 6(x-xi„y-y\[), (X,y)£ D: p( x. y, 0-0,(.v,.v)er;0<Kir
System (13) is a Cauchy problem, but its initial condition (second equation) is given at the right end of the interval of time considered [0, It means that when 1 ~ tp the solution of the adjoint system is a two-dimensional delta function. Equations (7) - (13) describe the CPM.
If we want to estimate not uniform, but RMS error, for example
A-AH
ll^fl)
nun,
that CPM appropriate adjoint equation has the form:
p(x,y,tt ) = 0, (,\\y) e D\
p{x,y,t) = O.(.v.y) e f;/ > 0;
Coeflicient Cij can be chosen equal toone(it serves for coordination of dimensions of summands in the equation). The numerical solution of this problem of the standard error is supposed to be considered in the subsequent articles of the authors. Nevertheless, the equations of the corresponding BPPM are presented for methodological purposes to illustrate the broad possibilities of the proposed approach.
If it is known from any additional considerations that the optimization problem (5)-(ll) has a solution, then the solution of the BPPM is optimal [4-12]. BPPM is solved iteratively:
I .Set an initial approximation for the magnetization curve (us ually middle line). Solution of the adjoint boundary value problem at the beginning assumed to be zero in the area D.
2. It is solved external and internal boundary value problems for the calculation of magnetic field at a certain approximate solution of the conjugate of (13) and the corresponding approximation constant value v' calculated according to the formula (12).
3. The boundary value problem is solved for the adjoint equation (13) with the value recalculated according to the results
of item 2 V and the refined approximate solution of the field boundary value problem (7) - (9).
4. Compare values A(xa, \\l7tr) on two iterations. If the
specified accuracy is not achieved, then the iterations continue by going to item 2,
Other options are possible stop criterion. For example, for higher reliability calculation the iterations is stopped if there is a match of the desired number of characters in the numerical values of approximate values of the vector potential field. Detailed calculation process of solving these problems is described mathematically in thesis [13].
2. Example of calculating two-sided estimates for solving equations electromagnetic fields in ferromagnetics
Consider the ease of a ferromagnet parallelepiped-shaped (Fig. I). In the initial state, it is considered fully demagnetized. At the initial moment of time /= 0 a uniform external magnetic iield appears with induction B{, — 0.8 T , directed along the axis OY , inducing Held in a ferromagnet. A boundary value problem takes the form (6) at //C1 = -5(l/(.t„ y.
Required to lind solutions sided estimates at a given point in space at time of interest in view of the known error of the magnetization characteristic, delined by using a function, and ni( B)
(sec formula (12)). In the calculations half size of sides cross -section of a parallelepiped of ferromagnetic material assumed to be one 4^1, ¿y-t (Fi& 0*
Below are the results of calculations finite difference method with the number partitions on the axes ferromagnetic region: nr, nt =5 .
References
Bmia= 3.75 B„ on the horizontal axis, respectively, the value ,)Wf= ¿y- Asymmelry of Ihe graphics in Fig. 3 is connected with the peculiarity of the BPPM - tere are estimates from above (below). which affects the symmetry of the solution of the problem.
Conclusion
A new method of calculating the two-way magnetic field characteristics unsteady electrical systems, containing the ferro-magnets and (or) permanent magnets with the magnetization characteristic error. The method is based on applying of Pontrjagin maximum principle for the estimate of the solution of the electromagnetic field equations in terms of scalar and Vector potentials. It is assumed, that the main source of error are the approximate values of the magnetic permeability of the medium. As compared with commonly used method of small parameter the limit on the magnitude of error parameters and characteristics of the boundary value problem much less, that greatly expands the range of solvable problems. Although the article was considered an example of plane field, the idea and the basic ratio method remain without change, and for three-dimensional problems theory of magnetic fields.
The proposed method has advantages for the cases, when it is required to find of the estimation of the solutions in the neighborhood of a finite number of singular points. As shown, it is possible in principle to estimate the RMS error of the solution in the entire region. Relevant examples is are challenges associated with the switching Overvoltages, catchys current when switched circuits with nonlinear inductance, direct and inverse prohlem of calculating the permanent magnets, when assessing the effect of the width of the hinge hysteresis. The possibilities of modern compute flax technology allow to find with the help of the considered methods are also effective uniform assessment solutions to large intervals. The method can be applied in order to improve the reliability of the results in design calculations of various electrotechnical devices, and also in problems of the theory of magnetic measurements.
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ДВУСТОРОННИЙ МЕТОД РАСЧЕТА ДИНАМИЧЕСКИХ ЭЛЕКТРОТЕХНИЧЕСКИХ СИСТЕМ РАСПРЕДЕЛЕННЫМИ ПАРАМЕТРАМИ С УЧЕТОМ ПОГРЕШНОСТИ ИСХОДНЫХ ДАННЫХ
Арутюнян Тигран Робертович, Москва, Россия, [email protected]
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 и векторного потенциалов принципа максимума Понтрягина. Процедура решения использует переход от дифференциальной постановки краевых задач магнитного поля к соответствующей дискретно-непрерывной в форме системы обыкновенных дифференциальных уравнений, к которой применима классическая теория принципа максимума. После нахождения уравнений краевой задачи принципа максимума осуществляется обратный предельный переход к дифференциальной форме при помощи устремления шага сетки к нулю. Получены соответствующие сопряженные уравнения в частных производных для разных критериев оптимальности. Рассмотрено решение задачи расчета двусторонних оценок решения при расчете 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. По сравнению с употребительным методом малого параметра ограничения на величину погрешности параметров и характеристик краевой задачи много меньше, что сильно расширяет круг решаемых задач. Хотя в статье была рассмотрен пример плоскопараллельного поля, идея и основные соотношения метода остаются без изменения и для трехмерных задач теории магнитного поля. Предложенный метод имеет преимущества при решении тех задач, в которых требуется находить оценки решения в окрестности конечного числа особых точек. Как показано, существует принципиальная возможность оценивать средне-квадратическую погрешность решения во всей области. Соответствующими примерами являются задачи, связанные с коммутационными перенапряжениями, бросками тока при включении цепей с нелинейными индуктивностями, прямые и обратные задачи расчета постоянных магнитов, при оценке влияния ширины петли гистерезиса. Возможности современной вычислительной техники позволяют находить при помощи рассмотренных методов также эффективные равномерные оценки решения на больших интервалах. Метод может применяться в целях повышения надежности результатов в проектных расчетах различных электротехнических устройств, а также в задачах теории магнитных измерений. Полученные результаты могут использоваться также при решении прямых и обратных задач для системы ферромагнитных тел и в тестовых задачах при использовании других методов. Разработанный подход может успешно применяться также при решении широкого круга динамических задач для систем с распределенными параметрам, в том числе в теории упругости, теплопроводности, пьезоэлектрических колебаний, квантовой механике и других областях.
Ключевые слова: расчет магнитного поля, двусторонний метод, погрешность коэффициентов, ферромагнетики, постоянные магниты, скалярный и векторный магнитные потенциалы, принцип максимума.
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Информация об авторе:
Арутюнян Тигран Робертович, инженер-математик, Москва, Россия
