CC BY
79
UDC 621.3
doi: 10.20998/2074-272X.2018.6.07
K.V. Chunikhin
ON THE INFLUENCE OF THE LEVEL OF AN EXTERNAL MAGNETIC FIELD AND THE LENGTH ON THE MAGNETIC MOMENT OF CYLINDRICAL CORES
Purpose. Analysis of inhomogeneous magnetization of long cylindrical permalloy 50N cores by a uniform constant magnetic field and the influence of length and field level on their magnetic moment. Methodology. The magnetostatic field of a non-uniformly magnetized in a uniform magnetic field long cylindrical core of an electromagnet of a spacecraft control system is considered. To calculate this field, a transformation of the integral equation with respect to the density of fictitious magnetic charges, as well as an iterative algorithm for its numerical solution, are proposed. Results. The convergence of the algorithm and the fact that the magnetic moment of the core depends heavily on its length and the level of the external magnetic field is shown. We have made an analysis of the influence of the length of a permalloy 50N core in the entire range of the magnetization curve and the level of a uniform external magnetic field on the axial projection of the magnetic moment of the core. Originality. The use of an almost equal distribution of the axial projection of the resulting magnetic field in the cross sections of the greater part of the cylindrical core and its division into cylindrical elements can significantly reduce the order of the system of algebraic equations approximating the integral equation for the surface density of fictitious magnetic charges for its numerical solution. Practical value. Recommendations regarding the level of the external field created by the electromagnet coil, the increase of the magnetic moment in cases of long cores and the choice of the number of cylindrical elements depending on the length of the core are given. References 14, tables 1, figures 5.
Key words: electromagnet, spacecraft control system, non-uniformly magnetized core, integral equation, fictitious magnetic charge, magnetization curve, magnetic moment of the core.
Рассмотрено магнитостатическое поле неоднородно намагниченного в однородном магнитном поле длинного цилиндрического сердечника электромагнита системы управления космическим аппаратом. Для расчета этого поля предложены преобразование интегрального уравнения относительно плотности фиктивных магнитных зарядов, а также итерационный алгоритм его численного решения. Сделан анализ влияния длины сердечника из пермаллоя 50Н во всем диапазоне кривой намагничивания и уровня внешнего магнитного поля на осевую проекцию магнитного момента сердечника, а также даны практические рекомендации относительно уровня внешнего поля, создаваемого катушкой электромагнита, и увеличения магнитного момента в случаях длинных сердечников. Библ. 14, табл. 1, рис. 5.
Ключевые слова: электромагнит, система управления космическим аппаратом, неоднородно намагниченный сердечник, интегральное уравнение, фиктивный магнитный заряд, кривая намагничивания, магнитный момент сердечника.
Introduction. To control the spacecraft, DC electromagnets consisting of a coil and a long cylindrical core of a material with high magnetic permeability are used [1]. Dimensions, winding data and materials can be determined and selected based on the calculation of the magnetic field generated by the electromagnet. Such an electromagnet must have a certain magnetic moment, the main part of which is provided by the core [2]. In the well-known works, for the calculation of the magnetostatic field of the cores the methods of demagnetization coefficients [2, 3] and integral equations [1, 4, 5] are developed. To determine the demagnetization coefficients, it is necessary to carry out experiments, and numerical solutions of integral equations were obtained under the assumption of a constant magnetic permeability of the core material [1, 4]. In [5], the calculations were performed under the condition of relatively small changes in the magnetic permeability along the core volume.
The relevance of this paper lies in the fact that in well-known works the magnetization of cylindrical cores is insufficiently investigated in conditions of large changes in magnetic permeability and levels of an external magnetic field, which makes it difficult to design electromagnets for spacecraft control systems.
The goal of the work is analysis of inhomogeneous magnetization of long cylindrical permalloy 50N cores by a uniform constant magnetic field and the influence of length and field level on their magnetic moment.
Transformation of the original integral equation taking into account the peculiarities of the magnetization of the core. Consider a cylindrical core of length b and of radius R located in an unbounded nonmagnetic and nonconducting space coaxially with an external constant uniform magnetic field of strength H0 (Fig. 1).
tfo
-Mo kn0 hk -
rv- ."г _»P И So "Q -
J 0
b 2 Ы2
Fig. 1. Meridian section of the cylindrical core
The strength of the resulting magnetic field is represented as [6, 7]:
H = Ho + Hm, (1)
where Hm is the magnetic field strength due to the magnetic properties of the core («demagnetizing» field [2]).
© K.V. Chunikhin
51
The magnetic field H is plane-meridian, and the vector field Hm is potential and related to the scalar potential ym by
Hm =~ grad q>m .
(2)
%
Q = t J
am M >mK ( )
zQ - zM
}+(Q + rM }
rdl.
M
(3)
where l, dlM are the contour of the meridian section and its element with the center in the point M; Q, Me l are the observation point and the point with current coordinated; om(M is the surface density of fictitious magnetic charges; /u0 is the magnetic constant; K(k) is the complete elliptic integral of the first kind of module k [11];
k = 2
rQrM
y (zQ - zM }+(rQ + rM f
k = 1, No
having a length bk, and Zbk = b/2 (Fig. 1).
k=1
Such a replacement allows to neglect volume fictitious magnetic charges and limit the definition of am. In this case, the integral equation for am takes the form [12]:
am (Q)-—J*m (M )[S (Q, M)-S (Q, M ')]] =
n l (5)
= 2t0—H0n (q)
where l is the contour of the meridian section of the core in the first quarter of the coordinate plane zOr; l = l + + l2 + l3; li,2 is the contour of the side and end surfaces;
N0-1
l3 = Z l£ , lkC is the boundary between k and k + 1
k=1
cylindrical elements in the calculation region; H0n(Q) is the normal projection of H0 for Q e l;
(k-to )/(k + to I Q e l1 u l2;
(Mk- tk+1 )/(tk+tk+1), Q e l3;
M' is the point symmetric to point M relative to the r-axis. At Q e h the first term of the kernel of the equation (5)
k
S (Q, M) = -
M
rQ
k (k )+-LI rM+Q k 2 -1
k
2r
M
E (k )
at Q e l2 u l3 - S(Q, M) = ~ *M E(k)
(r.
u rQ rM k'
where E(k),
Using the electrostatic analogy [6, 8], we represent qm b in the following form [9, 10]:
rQ, rM and zQ, zM are the radial and axial cylindrical coordinates of points Q and M.
Since the core material is isotropic, the relationship between H and the magnetization J is determined by the known dependence
J = Sr (H) -1]H , (4)
where fir(H) is the relative magnetic permeability.
To take into account the inhomogeneity of magnetization, we replace the nonlinear magnetized medium of the core with a piecewise homogeneous medium, which consists of 2No homogeneous cylindrical elements with absolute magnetic permeability pk,
Nn
and
k is the complete elliptic integral of the second kind of modulus k and additional modulus of complete elliptic
integrals, k' = V1 -k2 [11]. The second term of the considered kernel S(Q, M') is determined by the same formulas as S(Q, M) if in them to replace the coordinates of the point M by M'.
The replacement of a non-uniformly magnetized core with a set of uniformly magnetized cylindrical elements was made on the basis of preliminary calculations, according to which at b/R > 16 the axial projection of H in the cross sections of the core along its entire length, except for small sections near the ends, is distributed almost uniformly. Considering this feature, as well as the well-known boundary condition about the jump of the normal projection of Hm on the boundary between two magnetized media [13], we assume that am is invariable for all l£. Simplified by this assumption, equation (5) takes the following form:
a
i(Q)-— jam (m )[s(Q, M)-S (Q, M ')]m
7T
l-l3
No -1
Z am(Mk)J[S(Q,M)-S(Q,M'] = (6)
k=1
= 2t0—H0n (Q)
The system of algebraic equations, with the help of which the integral equations are solved, for equation (6) has a much smaller order.
An iterative algorithm for the numerical solution of a transformed integral equation. The iteration algorithm cycle consists of the following main blocks. In the first block, for some initial values of ^k(0), we solve the integral equation (6). To do this, using the quadrature formula of rectangles, it was transformed into a system of algebraic equations of order N (N is the total number of nodes of the spatial mesh, N = Nx + N2 + N3, N is the number of nodes per l1, N2 - per l2, N3 - per l3, N3 = No -1). When calculating each integral of the sum in the third term of the left-hand side of (6), 100 nodes were taken, and to take into account the edge effect at the cylinder ends, a non-uniform mesh was used. The obtained system of algebraic equations was solved by a direct method based on inverting the matrix of the left-hand sides and further multiplying the inverse matrix by a column vector of the right-hand sides.
In the second block we find the radial and axial projections of H
Hr (Q) = Jam (M)Sr (Q,M)M , (7) 2t l
Hz(Q) = Ho +-tjam(M)Sz(Q,M)Im , (8) 2t l
and then the module of H at each point inside the core.
In formulas (7) and (8), the functions Sr(Q, M) and Sz(Q, M) are determined using the same expressions as the kernel of the integral equation (5), respectively, Q e A and Q e l2 u l3.
In the third block, we determine average over the volume of each cylindrical element the relative and absolute magnetic permeabilities
-(j)
2
Mkr = R2U
JrMiM)dSM, MkJ) = Mo Mkr , (9, 10)
-(J)
k Sk
where Sk is the meridian area of the k-th cylindrical element; j is the iteration number.
To determine ¡r, we use the magnetization curve of permalloy 50N [2]
J(H) = aH/(H + c), (11)
from which with the help of a known connection between B , H and J on the basis of the model of magnetization by molecular currents, we find
¡(H) = 1 + a/(H + c). (12)
In dependencies (11), (12), shown using the logarithmic scale in Fig. 2 (a, b), J, H are the modules of J, H ; a, c are the constants, a = 1.25-106 A/m, c = 40 A/m.
and return to the first block of the algorithm. We continue iterations until the condition
j^ - j < A, j = 1,2,..., nlt, k = 1N~o
Then we take = ¡i^
•o, (13)
where A is the specified discrepancy; nit is the iteration number.
10J, A/m 1
1
0.8 0.6 0.4 0.2 0
10
50
100
A/m
25000 20000 15000 10000 5000 0
10
50 100
...... A/m
Fig. 2. Dependencies J(H) - a and fir(H) - b for permalloy 50N
It is established that the iteration process converges to some values of ¡n^ for any ^k(0), for which we have a numerical solution of (6). As explained below, the magnetic moment of the core is determined by the axial
projection of the magnetization Jz. After determining am this projection can be calculated using formula (4), taking into account (8) and (12). The convergence of the iterations is illustrated by the curves in Fig. 3, constructed for the following initial data: R = 5 mm (this size is assumed to be the same in subsequent calculations); b = 330 mm; z* = z/b. The values of bk are hereinafter assumed to be the same. We see that with increasing No, the distributions of Jz converge. If it is necessary to clarify the influence of edge effects, the cylindrical elements at the edges of the core can be replaced by a set of ring elements. Note that in the work [5], when the integral equation regarding tangential projection of the magnetization was numerically solved in accordance with the recommendations of [14], the entire volume of the core was divided into ring elements.
Hr"/.
Fig. 3. Distribution of the axial projection of magnetization along the core axis at different No for H0 = 1646.66 A/m (curves 1-4) and H0 = 6586.62 A/m (curves 5-8); for curves 1 and 5, 2 and 6, 3 and 7, 4 and 8 values of No respectively equal to 1, 8, 15, 25
The influence of the level of the external magnetic field and the length of the core on its magnetic moment. Due to the axial symmetry of the field, the magnetic moment vector M of the core under consideration has only an axial projection in cylindrical coordinates
Mz = 4nJ rMJz M )Sm
(14)
where S is the part of the area of the meridian section in the positive half-plane z > 0.
Table 1 shows the values of Mz and the relative discrepancies £ between Mz at No = 25 (conventionally exact values) and Mz at smaller No. It follows from the above data that to ensure £ < 1% for all considered levels of H0 and values b = 80, 165 and 330 mm, No should be taken equal to 4, 8 and 8, respectively. Note that for a core with a length 80 mm with No = 1 the value of £ « (1 - 6) % (obviously, this is also true for cores of not very different length).
Fig. 4, 5 show the influence of the level of the external field and the core length on the value of Mz (values of H0 are marked with dots on the abscissa axis in Fig. 4, and the corresponding values of Mz are shown on the curves). Calculations show that at each point of the core, with increasing H0 the strength of the resulting field always increases, but the magnetization depends on which
part of the magnetization curve Hz falls into (Fig. 2). In section a of the fir(H) curve (Fig. 2,b), the magnetic permeability is maximum, which leads to large values of Hm and, as a consequence, small values of H. In the same part of the initial magnetization curve (Fig. 2,a), the latter corresponds to small values of J. At b = 80 mm, this leads to relatively small magnetic moments Mz = 0.5 ^ 2.8 A- m2 (Fig. 4, curve 1; Fig. 5).
Table 1
Values at different b, H0, No
M-, ,
A/m2 '
2.5 —Î 1.5 1
0.5 0
M .
A/m2 ' 25
20
15
10
5 0
b, mm; No H0, A/m
b/R 1646.66 3293.31 6586.62 9879.93
25 0.500 0.989 1.937 2.845
8 0.4998 0.9886 1.9342 2.8396
L, % 0.0392 0.0742 0.1348 0.1863
80; 16 4 0.499 0.986 1.926 2.822
L, % 0.150 0.294 0.562 0.803
1 0.495 0.970 1.862 2.686
L, % 1.006 1.985 3.838 5.587
25 3.090 5.927 10.839 13.043
15 3.088 5.921 10.825 13.026
L, % 0.050 0.094 0.135 0.127
165; 33 8 3.083 5.904 10.776 12.972
L, % 0.213 0.396 0.585 0.540
1 2.963 5.486 9.537 12.591
L, % 4.084 7.445 12.019 3.462
25 17.863 25.330 28.394 29.546
15 17.834 25.288 28.342 29.481
L, % 0.164 0.166 0.182 0.220
330; 66 8 17.765 25.186 28.202 29.331
L, % 0.551 0.568 0.677 0.728
1 15.796 24.538 31.252 31.964
L, % 11.571 3.126 10.067 8.184
1000
3000
5000
7000
1000
3000
5000
7000
9000
A/m
9000
A/m
Fig. 4. DependenciesM(H0) at No = 25: curve 1 - b = 80 mm, 2 - 165 mm, 3 - 330 mm
With an increase in the length of the core, we observe a significant increase in the magnetic moment (Fig. 4, curves 2, 3; Fig. 5), since the core points are magnetized either in all parts of the magnetization curve (b = 165 mm), or in sections c, d (b = 330 mm) with significantly larger J (Fig. 2,a). However, the growth of Mz of cores of greater length with increasing H0 slows down, since an increasing part of them is in a state of saturation. From Fig. 5 it follows that at b/R = (33 ^ 66), to achieve Mz < 18 A-m2, H0 = 1646.66 A/m is sufficient. For larger values of M, up to 25 A-m2, H0 = 3293.31 A/m is required.
A/-.
25
20 15 ¡0 5 0
I
/ 2 >
Jy
-
15 20
30
40
50
60
b/R
Fig. 5. DependenciesMz (b/R) at No = 25: curve 1 - H0 = 1646.66 A/m, 2 - 3293.31 A/m, 3 - 6586.62 A/m, 4 - 9879.93 A/m
Conclusions.
1. The choice of the sizes of the cylindrical cores of electromagnets of spacecraft control systems must be carried out on the basis of a given maximum value of the axial projection of the magnetic moment M2 and the magnetization curve of the core material.
2. The coil of an electromagnet should provide such levels of an external magnetic field at which the strength of the resulting magnetic field on the predominant part of the core is outside the saturation region of the magnetization curve and corresponds to a higher magnetization.
3. At R = 5 mm, cores with a relative length of b/R < 33 provide Mz < 13 A-m2. In cases of b/R > 33, an increase in Mz can be achieved by increasing b/R at certain levels of the external magnetic field, which do not lead to the saturation of a significant part of the core (Fig. 4, 5).
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Received 16.08.2018
K.V. Chunikhin, Postgraduate Student,
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:
Chunikhin K.V. On the influence of the level of an external magnetic field and the length on the magnetic moment of cylindrical cores. Electrical engineering & electromechanics, 2018, no.6, pp. 51-55. doi: 10.20998/2074-272X.2018.6.07.
