netic field excited by an infinitely long current filament passing. h u g h the hole of an iron torus. To the best knowledge of the authors, this is the first 3D analytical ...
IEEE TRANSACTIONS ON MAGNETICS,VOL. 28, NO. 5, SEPTEMBER 1992
I
2799
MAGNETIC FIELD AROUND AN IRON TORUS Miklos Gyimesi and Doug Lavers University of Toronto, Department of Elecnical Engineering. Toronto, ON, Canada M5S-lA4, Tel :(416) 978-6842
Absrrucr - The paper presents a semi analytical solution of the magnetic field excited by an infinitely long current filament passing h u g h the hole of an iron torus. To the best knowledge of the authors, this is the first 3D analytical solution for a bounded multiply connected iron region. The solution can be of use when testing 3D magnetostatic software. The solution is expanded by toroidal functions which are calculated by efficient algorithms.
1. INTRODUCTION In the past few years, we have witnessed a tremendous development in the area of numerical electromagnetic field computation. As the range of the solvable problems increases, the testing and verification of such a softwarebecomes more and more difficult due, among other reasons, to the shortage of known test examples. For 3D magnetostatic calculation, the reduced potential formulation seems to be advantageous. However, it suffers h m cancellation errors 111. The total potential formulation avoids cancellation but it is inapplicable for multiply connected regions. The jumping periodic boundary condition succeeds [2], but it requires nonconventional FEM code. The general potential formulation can solve this problem by standad FEM software [31. In any case, there is a demand for an analytical solution to test the numerical results. To the best knowledge of the authors, there has been no analytical solution published for a bounded multiply connected iron region when the total current passing through the hole differs from zero. Driven by the motive to fill this gap, the authors developed a semi analytical solution of the magnetic field excited be an infinitely long current filament passing through the hole of an iron torus. (Fig. 1 .) 2. TOROIDAL COORDINATE SYSTEM The toroidal wordinate system was first deviced by Neumann in 1864 [1 11 then later independently by Riemann [141. The theory of the system and related functions was fmhg developed by Hicks [12] , &he [13]. Niven [14], Basset [15] in the previous and by prasad [17], Hobson [HI, Snow [7]. and MacRoben [19] in this century. Lowan [6] assembled a table of these functions. The basis of the toroidal coordinate system is two mutually orthogonal systems of Circles, as shown in Fig. 2. Turned around the z axis, the circles become orthogonal tori and spherical bowls. All bowls are in contact with the polar circle which is generated by the rotation of points, A or B. The relationship between the middle, polar and contour radii of any torus is the following : rm2 = rp2 + R , ~ The first, q. and the second, 8, coordinates give the position in the &an plane and the third coordinate, 0 is the azimuthal angle. 51 and 8 are de6ned by PA rl = log PA2 + PB2 - AB2 2 PA PB q is zero on the z axis and tends to infinity approaching the polar circle. 8 values of zero and x correspond to the outer and inner part separated by the polar circle of the equatorial plane, respectively. The points above and below the equatorial plane have a 8 value smaller and larger than 1c, respectively. The relationship between the polar and ~ N atoroidal l coordinates and their unit vectors, as well as the metric coefficients, g 1 , g 2 , g 3, and the Laplacian can be expressed by the following formulae: cos (e) =
Rc2 = r m 2 - rP
r - r, cos (yr) = Rc
e,, = - cos (v)e, - sin (v)e, q = - sin (yr) e, + cos (yr) e,
Since the common separation of variables fails, a more general method, called R-separation [5] should be applied to obtain particular solutions of the Laplace equation. The functions obtained by this procedure are r e f w d as toroidal functions. The general solution can be given by the linear combination of the toroidal functions.
3. SEMI ANALYTICAL SOLUTION According to the reduced potential formulation, the magnetic field can be decomposed into a source field, H,, excited by the currmt filament, and a potential field :
H = €I +s grudV The potential, V , can be expanded into a series by the toroidal functions,Pn-1/2'" and Qn-1/2"' 151. Vi =
Vo =
C C
Cn" Qn-l/~"'(ch (11)) Pn-la"(Ch (I~o))Fn'"
(1)
Cn" Pn-l/~'"(ch01)) Qn-la"(Ch (Ilo)) Fn"'
(2)
-4
cos (ne)cos (me)
Fnm =
qo = RC
where the subscripts i ahd o stand for inside and outside the torus, respectively, Cnmare the unknown coefficients, and qo is the coordinate of the iron torus.
0018-9464/92$03.000 1992 IEEE
2800
.
The potential identically satisfies the Laplace equation. Since it is continuous across the iron air boundary, the tangential component of the magnetic field, H, is also continuous. There remains to satisfy the continuity of the normal component of the flux density, B. The normal dinction of the iron surface coincides with the e,, . Therefore
~ o q ( H+ s grdvo) = Pi%(Hs + g r d v , ) Rearranging
d dq
b q g r d Vo - Pi%gr&Vi = (Pi - ~ l o h H s Substituting (1) and (2) provides
C cnm
fnm(P) = PO
--Pn(ch
e,, grdfnom- Pi % grdfmm Ol))Qn-l/2"'(ch
(Ilo)) Fnm
fkm(P)= Qn-1/2'"(~h (II))Pn-1/2"'(~h(Ilo)) Fnm
G (PI = (
~
-8 Po) e,, HAP)
This equation should hold for any point, P, on the torus surface. Its satisfaction is prescribed pointwise. The number of matching points
can be identical or greater than the number of unknowns. Experience showed that the best results can be achieved if the number of matching points is about twice as large as the number of unknown coefficients [4]. The non-quadratic linear equation system is solved by standard least square technique. 4. CALCULATION OF THE TOROIDAL FUNCTIONS For the computation of the toroidal functions, the following formulae w e n used. (3)-(6) and (11)-(14) can be found in the literature [6,7,8,9]. (7)-(9),(15) and (16) have not been published yet. P
(n - m
+ 1) !',,+Im
- =~2 e -~q n ~cel ( e - ? , 1 , 1 , 1 )
= (2n + 1) z Prim - (n + m )P,,-lm
(16)
First the key values are computed by (1)-(8). This is followed by the calculation of the zero ordered values by the recurrence scheme (9). Then first ordered values are generated by (11). Finally, the higher ordered values are determined by (10). The derivatives are computed by (12)-(14). Although this recurrence is unstable for Q, the first 5 orders and 10 degrees agree with 5 digit precision with those listed in Lowan's tables [51. The generalized complete elliptic integrals were evaluated by the efficientalgorithm given in [lo].
fnm(p)= W)
f,om(P) = Pn-1/2"'(~h
= Pn'(ch (q))
(3)
(11)
5. NUMERICAL RESULTS The investigated arrangement is shown on Fig. 1. For an excitation of 1 A and toms relative permeability 1O00, the computed results in the x -y plane are given in Table 1.
6. REFERENCES [I] J. Simkin, C.W. Trowbridge : "On the use of total scalar potential in the numerical solution of field problems in electromagnetics." Int. J. N u n Meth. Eng. Vol. 14. pp. 423-440, 1975.
[2]
s. Pissanetzky : "The new version of the finite element 3D magnetostatics program MAGNUS." Computational Electromagnetics, pp. 121-132 (1986)
[3] M.Gyimesi, D.Lavers : "Generalized Potential Formulation for 3D Magnetostatic Problems." Submitted paper to the IEEE Trans. on Magnetics. [4] M.Gyimesi : "Inverse Field Calculation for Structure Positioning." IEEE Trans on MAG, V01.27.No.5,pp.41704173,Sept.1991. [5] P.Moon, D.E.Spencer : Field Theory Handbook Berlin, Springer-Verlag. 1961. [6] A.N.Lowan : Tables of Associated Legendre Functions. Mathematical Tables Project by National Bureau of Standards. New York, Columbia University Press. 1945. [7] Ch. Snow : Hypergeomemc and Legendre functions with application to integral equations of potential theory. Applied mathematics series. Washington. U.S. Govt. Print. Off. 1948 [8] I.S.Gradshteyn, 1.M.Ryzhik : Table of Integrals, Series and Roducts. New York, Academic Press. 1965. [9] M.Abramowitz, 1.Stegun : Handbook of Mathematical Functions. New York, Dover Publications. 1965. [lo] W.H.Press, B.P.Flannery, S.A.Teukolsky, W.T.Vetterling : Numerical Recipes. New York, Cambridge University Press. 1986. [ l l ] K.G.Neumann : Theorie der Elekaicitats und Warmeverteilung in einem Ringe. Halle Verlag der Buchhandlung des Waisenhauses. 1864. [12] W.M.Hicks : "OnToroidal Functions." Royal Society of London - Philosophical Transactions. Vol. 172. pp. 609-652. 1881. [13] E.Heine : Handbuch der Kugelfuncionen. I S . Berlin, Druck und Verlag von Reimer. 1881. [14] G.F.B.Riemann : Gesammelte mathematische Werke. 1,II. Leipzig, Druck und Verlag von Teubner. 1892. [15] W.D.Niven : "Harmonics of a Ring." Proc. of the London Math. SOC. Vol. XXIV. pp.373-388. 1892-3.
2801
[la] A.B.Basset : "On Toroidal Functions." American J. of Math. Vol. 15. pp.287-302 1893. [17] G. Prasad :A treatise on Spherical Harmonics and h e Functions of Bessel and Lame. 1,II. Benares City (India), Mahamandal Press. 1930. [18] E.W.Hobson :The Theory of Spherical and Ellipsoidal Hannonics. London, Cambridge University Ress. 1931. [19] T.M.MacRobert : Spherical Harmonics. International Series of Monograph in Pure and Applied Mathematics. Vol. 98. New York,Pergamon Press. 1967. A B c
position [ml X
-10.0
I I
Y 0.0
flux density [nano Tesla] Bx By 0.0 I -19.86
id A
1
C'
Figure 1. Evaluationpoints around the inm m
zE P
0.0 0.0 0.0 0 .0 ..
0.0 0.0 0.0 0.0
-1.0 -1.5
-2.0 -2.5 -3.0 -4.0 -5.0 -10.0
200.0 133331. 99999. 79999. 66.7 50.0
40.0 20.0
-48.43 -14.73 -4.00 0.36 2.37 0.62 0.14 -0.08
b C
d e f g
h i
figure 2. orthogonalbowls and tom
m
