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14
Theoretical Electrical Engineering and Electrophysics
UDC 621.3
V.M. Mikhailov, K.V. Chunikhin
doi: 10.20998/2074-272X.2017.6.06
TESTING OF NUMERICAL SOLUTION OF THE PROBLEM OF DETERMINING SOURCES OF MAGNETOSTATIC FIELD IN MAGNETIZED MEDIUM
Purpose. Testing of numerical solution algorithm for integral equation for calculation of plane meridian magnetostatic field source distribution at interfaces of piecewise homogeneous magnetized medium by means of electrostatic analogy. Methodology. The piecewise homogeneous medium consists of three regions with different magnetic permeabilities: the shell of arbitrary meridian section, external unlimited medium outside the shell, and the medium inside the shell. For testing external homogeneous magnetic field effect on spherical shell is considered. The analytical solution of this problem on the basis of electrostatic analogy from the solution of the problem uniform electrostatic field effect on dielectric shell is obtained. We have compared results of numerical solution of integral equation with the data obtained by means of analytical solution at the variation of magnetic permeabilities of regions of medium. Results. Integral equation and the algorithm of its numerical solution for calculation of .source field distribution at the boundaries of piecewise homogeneous medium is validated. Testing of integral equations correctness for calculation of fictitious magnetic charges distribution on axisymmetric boundaries of piecewise homogeneous magnetized medium and algorithms of their numerical solutions can be carried out by means of analytical solutions of problems of homogeneous electrostatic field effect analysis on piecewise homogeneous dielectric medium with central symmetry of boundaries - single-layer and multilayer spherical shells. In the case of spherical shell in wide range of values of the parameter lk, including close to ± 1, numerical solution of integral equation is stable, and relative error in calculating of fictitious magnetic charges surface density and magnetic field intensity inside the shell is from tenths of a percent up to several percent except for the cases of very small values of these quantities. Originality. The use analytical solutions for problems of calculation of external electrostatic field effect on piecewise homogeneous dielectric bodies for testing integral equations of magnetostatics and algorithms for their numerical solutions. Practical value. The described method of testing integral equations of magnetostatics and their numerical solutions can be used for calculation of magnetic fields of spacecraft control system electromagnets. References 12, tables 2, figures 3.
Key words: plane meridian magnetostatic field, piecewise homogeneous magnetized medium, integral equation, electrostatic analogy, fictitious magnetic charge.
Выполнена проверка правильности интегрального уравнения второго рода для расчета распределения источников плоскомеридианного магнитостатического поля на границах раздела кусочно-однородной намагничиваемой среды и его численного решения. Для этого использованы электростатическая аналогия и аналитическое решение задачи о воздействии однородного электростатического поля на сферическую диэлектрическую оболочку в кусочно-однородной диэлектрической среде. Подтверждена правильность интегрального уравнения и его численного решения при помощи аппроксимирующей системы алгебраических уравнений. Сделан анализ влияния магнитных проницаемостей однородных областей среды на распределение фиктивных магнитных зарядов на поверхностях и напряженность магнитного поля внутри сферической оболочки. Библ. 12, табл. 2, рис. 3.
Ключевые слова: плоскомеридианное магнитостатическое поле, кусочно-однородная намагничиваемая среда, интегральное уравнение, электростатическая аналогия, фиктивный магнитный заряд.
Introduction. For the calculation of magnetostatic fields in inhomogeneous magnetized media, the use of integral equations of the second kind with respect to the density of fictitious magnetic charges in the volume and on the interfaces of the sections of the medium is effective [1-3]. Integral equations are approximated on a spatial mesh by systems of algebraic equations of high order, which are solved using computers. As in the formulation of integral equations, and with their approximation, errors can be made, connected, for example, with inconsistencies in the directions of vectors, integrating on the elementary part of the computational domain with the singular point of the kernel of the integral equation.
The relevance of this work is due to the need to verify the correctness of the used algorithms and laborintensive computation procedures using tasks that have analytical (exact) solutions - testing. The number of such solutions in magnetostatics is relatively small. In the known papers, exact solutions of problems of calculating analogous physical fields are not fully utilized, giving preference to more accurate, in the opinion of the authors, numerical methods.
The goal of the work is use of electrostatic analogy for testing the algorithm for the numerical solution of the
integral equation for the surface density of fictitious magnetic charges at the interfaces of homogeneous regions of a piecewise homogeneous magnetized medium in the case of a plane meridian magnetostatic field.
Main equations and formulae. Let it be required to test the algorithm for solving a problem for a piecewise homogeneous medium consisting of three homogeneous regions with different constant absolute magnetic permeabilities fik (k = 1,3). The shell of an arbitrary meridian section (region 2) divides the unbounded environment into regions 1 and 3, respectively, outside and inside the shell (Fig. 1). In the particular case, region 3 is absent, i.e. there is an axisymmetric body in an unbounded medium, for example, the core of an electromagnet. Using the electrostatic analogy of the problem under consideration [1, 4-6], we represent the scalar potential ym of the magnetostatic field due to the magnetic properties of the medium in the form [3, 7, 8]:
9m (Q) = — i ^ l
am (M )rMK (k )
rdl,
M
(1)
- zm f + ^Q + rM f where Q, M e l are the point of observation and point
© V.M. Mikhailov, K.V. Chunikhin
with current coordinates, respectively; /u0 is the magnetic constant; om(M) is the surface density of fictitious magnetic charges; l, dlM are the total contour of the meridian section of the shell and its element with center at the point M, respectively; l = l1 +l2, A,2 are the outer and inner parts of the total contour, respectively; K(k) is the complete elliptic integral of the first kind of module k [9];
k = 2
rQrM
\(zQ - zM Y+^Q + rM f rq, rM and zq, zm are the radial and axial cylindrical coordinates of points Q and M.
The strength of the magnetic field due to the magnetic properties of the medium, and the resulting magnetic field are respectively equal [1]
and
Hm = - grad vn
H = H o + Hm :
(2)
(3)
cr„
Q)-—\°m (M)S(Q, M)dlM =2vohHon (Q), (4)
v J
where
S (Q, M ) = -
2J r
K (k )+-±r
k'2
'M
+ r,
Q a
2rM
kz -1
E(k )
v —m zQ -
COs(lr, nQ )+
(5)
+
^ V k)2 E(k )cos( * nQ )
elliptic integrals [9]; k' =
k2
—k =
Vk+1 -Vk
k = 1, 2;
Vk+1 + Vk
H0n(Q) is the normal projection of H 0.
Fig. 1. An axisymmetric shell in a piecewise homogeneous magnetized medium
The particular case of the problem under consideration. For testing, we consider a particular case of the problem described above - the effect of external constant homogeneous magnetic field directed along the axial coordinate z on the spherical shell in a piecewise homogeneous magnetized medium (Fig. 2). The meridian section of this shell is symmetric about the r axis, therefore, for the points M and M' with such symmetry om(M') = -om(M) and the domain of definition of om(M) is halved. We transform the integral equation (4) for this case to the form:
i(m (Q, M f-S (Q, M' Ym = l (6) = 2^o^kH 0 sm6>, where 8 is the spherical coordinate of elevation angle (Fig. 2).
k J
where H0 is the external magnetic field strength.
Following the idea of the method [i], we note that in order to perform calculations using formulae (1) - (3), it is necessary to find an unknown function om(Q), Q e l by solution of the integral equation
1r, 1z are the orts of cylindrical coordinates r and z; hq is the unit normal to the contour l in the point Q e l;
E(k), k is the complete elliptic integral of the second kind of the module k and the additional module of complete
Fig. 2. A spherical shell in a piecewise homogeneous magnetized medium
The total integration contour l in equation (6) consists of two halves l1 and l2 symmetric about the r-axis, located in the region z > 0. The functions S(Q, M) u S(Q, M') entering the kernel of this equation, we determine by formula (5), having adopted in it cos(lr,Hq)= cosd ,
cos(1z,
Q
1= sinö, ZM' =
ZM
rM' = rM . In addition, it
is necessary to take into account the change in the z-coordinates of the symmetric points M' in calculating the modulus k. After solving equation (6), the intensity of the homogeneous magnetic field at an arbitrary point Q inside the shell (region 3) is found using the formula that follows from (1) - (3):
1 r
Hi (Q) = Ho (M) X
1
8kvo (S1 - S1)
(7)
m >
where
S1 = (z
Q
ZM,
a
E (k).
(8)
The function S1 in the integrand of the second term (7) is determined by formula (8), replacing in it, as well as in the formula for determining k, the coordinate zm on zm.
k
X
X
X
r,
When forming the functions S(Q, M) and S(Q, M') entering the kernels of equations (4), (6), formulas were used to calculate the projections of the plane meridian electrostatic field from [3, 8]. The contour l was divided into N elementary regions with nodal points Mk at their center, which form a spatial mesh, and k = 1, N1 at
H, = - 9H 0
Cp1
±2. P1
+ 2 I-2c
f R ^
P2
V R1 / V
±2. Pi
-1
(11)
where R1, R2 are the radii of the boundary surfaces (Fig. 2);
Mk e l1 and k = N1 +1, N at Mk e l2 (Fig. 2). Equation (6) was transformed into a system of algebraic equations on a mesh using the quadrature formula of rectangles. The diagonal elements of the N*N matrix of this system of equations, corresponding to the elementary sections of the contour with the singular point of the kernel of equation (6), were determined by the method described in [10]. The system of algebraic equations was solved by a direct method based on the inversion of the matrix of the left-hand sides and the subsequent multiplication of the inverse matrix by the column vector of the right parts.
Analytical solution of a similar electrostatic problem of the action of an external homogeneous electric field on a dielectric spherical shell is known [11]. Using this solution, we obtain formulas for calculating the distribution of the surface density of fictitious magnetic charges on the boundary surfaces, as well as the intensity of the homogeneous magnetic field H inside the magnetized shell:
i ~ \
Bl = R
H0 + A
1 + c,
R
3 ^
, B2 = c
R A2;
,(R1,0) = Po
Ar (( - B2 )+h0 + a2
, R1 f
№,0) = Po
22Bi r3
+H, - A2
sin0 :
sin0 ; (9)
(10)
CA1 = A3/A2 + 2; cM2 = 1 "A3/A2 ; cM = c ^2/ cp.1;
A = HtcM1/3.
The values of am and H obtained by numerical solution of the integral equation (6) and calculations by the formula (7) will be called approximate, and using (9) - (11) - exact. *
Table 1, 2 show the values of am* = am / (m0H0) and H* = Hi / H0, respectively, and Fig. 3 shows the variation curves for am* vs 8 [0, n/2] on the boundary surfaces of the shell at jm1 = ^0, R2/R1 = 0.95 and the variation of ^2,3. The data in columns 1 are approximate, and in columns 2 - exact. For the data given in the numerators of columns 1 of Table 1, it was assumed that N = 80, in the denominators - 2160. The curves in Fig. 3 are built from the results of a numerical solution of equation (6) with N = 2160.
From Table 1, 2 it follows that in wide ranges of variation of the magnetic permeabilities ^2 h ^3 when the step of the spatial mesh is reduced the absolute discrepancies of the exact and approximate values of am* h H* are of the order of 10-3. In this case, the relative discrepancies vary from 0.1% to several percent, except for very small values of the calculated value.
Table 1
The values of the surface density of fictitious magnetic charges am on the surfaces of a spherical shell
The shell surface 9k, pag ^2 = 50^0, A1 = 0.961 = ß0
^3 = 10^0 ¿2 = -0.667 ¡x3 = 100^ ¿2 = 0.333 ¡x3 = 1000^ ¿2 = 0.905 ^2 = 50^0 ¿1 = 0.961 ¿2 = -0.961 ¡x2 = 500ß0 ¿1 =0.996 ¿2 = -0.996 00.99.99 20 0. -0 11 11 11
1 2 1 2 1 2 1 2 1 2 1 2
Outer 0.2945 0.7312 0.7490 0.8288 0.8350 0.8457 0.8498 0.6067 0.6383 0.7823 0.8362 0.8048 0.8617
0.7483 0.8348 0.8497 0.6370 0.8339 0.8594
0.6086 1.4404 1.4753 1.6326 1.6447 1.6658 1.6739 1.1951 1.2573 1.5412 1.6470 1.5855 1.6974
1.4739 1.6443 1.6736 1.2548 1.6426 1.6928
0.9228 2.0087 2.0572 2.2765 2.2935 2.3229 2.3342 1.6668 1.7532 2.1499 2.2967 2.2117 2.3670
2.0553 2.2929 2.3338 1.7497 2.2906 2.3605
1.2369 2.3812 2.4378 2.6977 2.7178 2.7525 2.7660 1.9765 2.0776 2.5502 2.7216 2.6236 2.8049
2.4355 2.7171 2.7656 2.0735 2.7144 2.7972
1.5510 2.5335 2.5798 2.8539 2.8762 2.9033 2.9272 2.1023 2.1986 2.7249 2.8802 2.8047 2.9683
2.5774 2.8754 2.9267 2.1943 2.8725 2.9602
Inner 0.2945 -0.0507 -0.0443 0.0103 0.0089 0.0208 0.0181 -0.1286 -0.1128 -0.0441 -0.0173 -0.0329 -0.0045
-0.0446 0.0090 0.0182 -0.1134 -0.0184 -0.0057
0.6086 -0.0999 -0.0872 0.0203 0.0176 0.0410 0.0357 -0.2533 -0.2222 -0.0868 -0.0340 -0.0647 -0.0089
-0.0878 0.0177 0.0359 -0.2234 -0.0362 -0.0112
0.9228 -0.1393 -0.1217 0.0282 0.0246 0.0572 0.0497 -0.3530 -0.3098 -0.1206 -0.0474 -0.0898 -0.0124
-0.1224 0.0247 0.0501 -0.3116 -0.0505 -0.0156
1.2369 -0.1646 -0.1442 0.0334 0.0291 0.0678 0.0590 -0.4174 -0.3671 -0.1410 -0.0562 -0.1044 -0.0147
-0.1450 0.0293 0.0593 -0.3692 -0.0598 -0.0185
1.5510 -0.1319 0.1526 0.0139 0.0308 0.0150 0.0624 -0.3902 0.3885 -0.0764 0.0595 -0.0349 0.0155
-0.1534 0.0310 0.0627 -0.3906 -0.0632 -0.0195
a
a
Tab le 2
The values of the magnetic field intensity H*
N ^2 = 50^0, = 0.961 = ^0
p.3 = 10^0 ¿2 = -0.667 ¡x3 = 100^0 ¿2 = 0.333 ^2 = 50^0 ¿1 = 0.961 ¿2 = -0.961 ¡x2 = 500^0 ¿1 =0.996 ¿2 = -0.996
1 2 1 2 1 2 1 2
80 0.2179 0.1908 0.0363 0.0308 0.4502 0.3965 0.1507
240 0.2006 0.0325 0.4159 0.0935
720 0.1941 0.0314 0.4031 0.0711
2160 0.1919 0.0310 0.3987 0.0634
A i ...—
—w
a,„ A
0.2 0.1 0 -0.1 -0.2
0.5 ii 4 L5 " .
—
0.5 -2-Ok. 1.5
——_L
Fig. 3. The distributions of am on the outer (a) and inner (b) surfaces of the spherical shell: for curve 1
= 100^0, ¡u3 = p.0, 2 - = 1000,«o, ¡u3 = p.0, 3 - p.2 = 10^0, ¡u3 = 5000^o
Increasing the accuracy of the numerical solution can also be achieved by using more precise quadrature formulas. From Table 1 and Fig. 3 it follows that the numerical solutions are stable for values of the parameter ¿k close to ±1. Thus, with the help of an analytical solution of a similar electrostatic problem, the correctness of the compilation of the original integral equation (4) and the algorithm for its numerical solution was confirmed. We note that analytical solutions of problems of the effect of external homogeneous electrostatic field on multilayer dielectric spherical shells are known [12], which can be used to test the algorithm for solving the integral equation in the case of multilayer axisymmetric magnetized shells.
The features of the variation of am* and H* with the variation of in the case of a spherical shell. The values of am* on the calculated part of the contour of the outer surface of the spherical shell (z > 0) are positive for all fik and vary insignificantly for large ¡i2. On the calculated part of the contour of the inner surface of the
shell for n3 < n2 (¿2 < 0), the values of am are negative, and for fi3 > fi2 (¿2 > 0) are positive (Table 1). Naturally, on the part of the contour symmetric about the r-axis, the signs of am are opposite.
At large fi2 > 100^0, the shell shields region 3, which results in small values of H* and am*. An increase in fi3 leads to an additional decrease in H* (Table 2).
The described features of the changes of am* and H * can also be useful in the analysis of the magnetostatic field in the case of axisymmetric shells and solid bodies of a different shape. Conclusions.
1. Testing the correctness of integral equations for calculating the distribution of fictitious magnetic charges on the axisymmetric boundaries of a piecewise homogeneous magnetized medium and the algorithms of their numerical solutions can be carried out using analytical solutions of problems of analyzing the action of homogeneous electrostatic field on a piecewise homogeneous dielectric medium with a central symmetry of boundaries - single-layer and multilayer spherical shells.
2. In the case of a spherical shell in a wide range of values of the parameter ¿k including those close to ± 1, the numerical solution of the integral equation is stable, and when reducing the mesh step, the relative error in calculating the surface density of fictitious magnetic charges and the magnetic field strength inside the shell is from the tenths percentages to a few percent, except for very small values of these values.
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Received 12.10.2017
V.M. Mikhailov1, Doctor of Technical Science, Professor, K.V. Chunikhin2, Postgraduate Student,
1 National Technical University «Kharkiv Polytechnic Institute», 2, Kyrpychova Str., Kharkiv, 61002, Ukraine,
phone +380 57 7076052,
e-mail: valery.m.mikhailov@gmail.com
2 State Institution «Institute of Technical Problems of Magnetism of the NAS of Ukraine»,
19, Industrialna Str., Kharkiv, 61106, Ukraine, phone +380 57 2992162, e-mail: kvchunikhin@gmail.com
How to cite this article:
Mikhailov V.M., Chunikhin K.V. Testing of numerical solution of the problem of determining sources of magnetostatic field in magnetized medium. Electrical engineering & electromechanics, 2017, no.6, pp. 42-46. doi: 10.20998/2074-272X.2017.6.06.
