FIELDS IN SQUARE HELMHOLTZ COILS

Transcription

1 Appl. sci. Res. Section B, Vol. 9 FIELDS IN SQUARE HELMHOLTZ COILS by R. D. STRATTAN and F. J. YOUNG Carnegie Institute of Technology, Pittsburgh, Pennsylvania, U.S.A. Summary The axial component of the magnetic field in square Helmholtz coils of rectangular cross-section is investigated. Expressions for the axial field are derived and simplified enough so that any specific case can be easily calculated with a small internally programmed digital computer. Plots of the field at the centre of the coils as a function of winding thickness with coil spacing as a parameter are presented. Curves of the variation of the field with distance from the centre are given. Spherical volumes in which the axial magnetic field varies 1% and 2% are computed for various configurations. The dimensions of the coils which hold the field variations to 1 ~o and 2~o over the maximum spherical volume are given. I. Introduction. The square Helmholtz pair shown in fig. 2 can be used to produce uniform magnetic fields in the vicinity of.17 z r Fig. 1. Square current filaments. the origin of the coordinates. In applications such as the rictometer where more than One Helmholtz pair is needed square Helmholtz

2 118 R. D. STRATTAN AND F. J. YOUNG coils are more compact and consume less power than ordinary Helmholtz pairs. In the case of /5 ~ 0.80 and ~ = 0.15 for the inside pair of a two phase-arrangement consisting of two Helmholtz pairs a circular set of coils require about 43% more copper to produce the same magnetic field at the same current density than a set comprising square coils. The copper losses are correspondingly greater for circular than square coils in arrangements where more than two Helmholtz pairs are required. 2. Calculation o[ the magnetic field. The field due to a coil system similar to the one shown in fig. 2 can be found at any point Fig. 2. Square coils filling a cubical. in space from Ampere's Jaw and integration over the winding cross-section. The final integral cannot be performed analytically but is easily evaluated with the aid of a digital computer and Simpson's rule. The only case calculated here is for the overall axial dimension equal to the transverse dimensions. However this method is applicable to any rectangular prism shape. The parameters of this coil system are ~, the dimensionless spacing between the coil sections, and fl, the dimensionless measure of the winding thickness. The gap of 2~L is placed between the coil sections to give a more uniform field and to provide better access to the interior of the coils. Only the axial field component is calculated, although the radial

3 FIELDS IN SQUARE HELMHOLTZ COILS 119 components can be calculated by the same method. First the axial field component due to the two current filaments shown in fig. 1 is found from Ampere's law. Then using this as the field due to the infinitesimal winding areas dd by dt, the total field is found by integrating over d from al to L and over t from fll to L. Rationalized MKS units are used in this paper. 54- e =0~ 4-6~=.1 i3.~=.2 _12 - ~ I DIMENSIONLESS WINDING THICKNESS ~B Fig. 3. The magnetic field at the origin as a function of coil spacing and winding thickness. The field dh at (x, y, z) due to the current elements dll and dl2 in fig. 1 is given by dn = I(d/1 ar~)/4~rl2 I(d12 ar~)/4~r2 2, at, ax(x -- Xl) ay(y d) az(z -- zl) rl 2 E(x - xl) 2 (y d)2 (z -- zl)21~ ' a,, = a~(x -- x2) av(y -- d) az(z -- 22) r2 2 [(x -- x2) 2 (y -- d) 2 (z -- z2)~ ~ Xl, x2, zl and z2 =- ± t in order to define the current filament. Taking only the y component of the field, t 4~H _f[ (x t) dzl T J~ [(xt) 2 (yd) 2 (z--zl)~l ~ --t (x t) dz2 [(xt) 2(y-d) 2(z-z2)21 ~

4 120 R. D. STRATTAN AND F. J. YOUNG (t -- x) dzl E(x - t)2 (y d)2 (z -- zl)2~ ~ (t -- x) dz2 ~(x -- t) 2 (y -- d) 2 (z- z2)2] t (z t) dxl [(x--xl) 2(yd) 2 (zt)2] t (z t) dx2 E(x--x2) 2 (y--d) ~ (zt)2] ~ (t -- z) dxl E(x - xl) z (y d)2 (z -- t)~? (t - z) dx2 } [(x--x2) 2(y-d) 2(z-t)2] ~ " This integral can be evaluated using the t dx f 1{ -t E(a -- x) 2 b 2 ca] ~ (b 2 c 2) [(a - (t- a) t)2 b2 c2]~ J If we define a quantity C,,j,k(x/t, y/t, z/t, d/t) as follows; c~,j,~ = { ' E1 (-1)~ x/t~2 fl/t (-i)j y/t? following identity: (t a) [(a t)2 b2 c2~ l I1 (-1)~z/t;~ [d/t (--1)~y/t;2 " [I (--1yx/t][a (--1)~z/tl I(1 (--1)ix~t) z (d/t (--l)jy/t) 2 (1 (--1)Xz/t)zJ ½ ' then the field due to the current filaments of fig. I is 4~H X 2~ X (l/t)c,,j,k (x/t, y/t, z/t, d/t). I i=1 j=l k=l The field due to the coil configuration shown in fig. 2 is then 1 L/t 4 ~ H 2 e e f f k = = Y~ 52 2~ d(t/l) d(d/t)cl,t,k(x/t, y/t, z/t, d/t). ~ ~ J L i=l j=l 1 ~L/t

5 FIELDS IN SQUARE HELMHOLTZ COILS 121 With the substitutions [(1 (--1)ix~t) 2 (1 (--1)kz/t)2J~ tan u = d/t (--1)Jy/t and sin u = w the integration over d/t reduces to the form f dw p2 we -- (l/p) arctan (w/p) ~',,,..,~=.2o E\ o < = J S - ~ o Z -4 <3 0 z u) x.-2.i--'--~.~.3,,~=.25 o ' SQUARE COIL ~ \ ---- CIRCULAR COIL ~ or.=.15, p=.80 \ (b) # 6 <'2" o 3-4. (c) ~'~ Fig. 4. Variation of axial field uniformity along the y-axis; (~) ~ = 0.75, (b) ~ = 0.80, (c) ~ = The result is L/t 2 2 f Ci,j,k d(d/t) = ~ alit m=l n= 1 Y~ (--1) m arctan i,m],~ (x/l, y/l, z/l, t/l) where for m = 1 and n = 1 i,j,k is defined as

6 122 R. D. STRATTAN AND F. J. YOUNG [t/l (--1)kz/LJ[o, (--1)~y/LJ [t/l (--1)*x/L][(t/L (--1)*x/L) 2 (t/l (--1)kz/L) ~ (o~ (-- 1)~y/L)2]~. For m : 2, the expression is the same except that ~ is replaced by one; and for n : 2, (--1)ix/L and (--1)kz/L are interchanged. The total field in the axial direction then becomes ~H _ Z Z X X Z (--1) mjarctan i,~:,'~(x/l,y/l,z/l, JL i=i i=i k=l ~=1 n=1 t/l) d(t/l). 0,I.2,~,4.5 # z u) x X o ~ -6, # z 0,I.2.3.at.5 -~ 1 l X-2- ~Z ~< lp Z # :o I I _.r d (b ~_4~ Fig. 5. Variation of axial field uniformity along the x-axis; (a) fl , (b) fl , (c) fl = This integral was evaluated numerically on the C.I.T. IBM 650 digital computer using arctangent and S imp o n's rule subroutines. (c)

7 FIELDS IN SQUARE HELMHOLTZ COILS Results and conclusions. The field at the centre of the coil system as a function of = and/3 is shown in fig. 3. For the ranges of and/5 shown the field at the centre is given very nearly by H = (1 -- ~)(1 --/5) JL amp-turns/meter. The percent deviation of the field from its value at the origin is given at points along the y and x axes by figs. 4 and 5 respectively for several values of dimensionless winding thickness..6 *5 me (-) <t Z -- m~ a# g.4 ~ /~ = ALONG X-AXIS 3E X = ALONG Y-AXIS (a).,5.k,, S AC,NG.6 a# ~N A, ALONG X- AXIS X- ALONG Y- AXIS (b).1o Fig. 6. The maximum distance from the origin for (a) 1%; or (b) 2~o field variation. Intersection of equal/5 curves yield values of = and/5 for spherical region of uniformity. Maximum spherical volumes occur for = = 0.185, /5 = 0.80 with radius ~ 0.38 L at 1% uniformity and 0.45 L at 2% uniformity. The maximum distance from the origin for 1% and 2% variation in the axial field as a function of = for a given/5 is shown in fig. 6.

8 124 FIELDS IN SQUARE HELMHOLTZ COILS The abscissae of the intersections of equal fl curves are the a's for maximum radius of a spherical volume of 1% or 2% uniformity. The maximum spherical volume occurs for fl , ~ with a radius for 1% variation of 0.38 L. The percentage variation of the axial field of a solenoid of length 2 R, outer radius R, inner radius 0.80 R with the windings omitted for 0.15 R on either side of the midplane is shown in fig. 4(b) for comparison. The field at the centre of the solenoid is H JR amp-turns/meter, while that of the cubical coil is H = 0.122JL amp-turns/meter for ~ = 0.15, fl = Received I7th October, 1960.