GB2213595A

GB2213595A - Magnetic field coils

Info

Publication number: GB2213595A
Application number: GB8828839A
Authority: GB
Inventors: Robert Turner
Original assignee: National Research Development Corp UK
Current assignee: National Research Development Corp UK
Priority date: 1987-12-11
Filing date: 1988-12-09
Publication date: 1989-08-16
Anticipated expiration: 2008-12-09
Also published as: DE3854149T2; JP3205553B2; EP0320285B1; DE3854149D1; EP0320285A1; GB8729037D0; JPH01264209A; GB2213595B; GB8828839D0

Description

1 IMPROVEMENTS IN OR RELATING TO ELECTRICAL COILS

The present invention relates to electrical coils intended to generate magnetic fields and more particularly to a method of designing coils with a minimum inductance to produce a desired magnetic field and to the coils when designed.

For many scientific purposes it is desirable to switch magnetic fields rapidly. For this reason coils generating these fields should have low inductance, since the switching, time depends on L/R, where L the inductance and R is the coil resistance.

Generally a specific variation of the magnetic field,,,enerated by the coil is required. Coil designers have usually been more concerned with ensuring a correct field distribution within the region of interest than with minimising the inductance, largely because calculation of the inductance was regarded as extremely difficult. The difficulty was such that the computation could not be performed in a systematic way, and hence it was not.possible in most cases to show analytically that the inductance obtained was actually the minimum consistent with the desired coil performance.

Recently, however, a new formulation of the magnetic fields generated by currents confined to cylindrical shells has led to an attractively simple expression for the inductance of an-y coil formed on a cylinder. This expression lends itself, in conjunction with a variant on the target field approach described in co-pending British Patent Application No. 8714434, to the design of coils which accurately generate specified fields within a volume of interest, and which can have the absolute minimum inductance consistent with their performance.

It is an object of the present invention to provide a method of designing coils with minimum inductance and to build coils to those designs.

The present invention therefore provides a method of designing a coil or a set of coils in which the conductors of the coil are positioned on one or more coaxial cylinders or parallel planes, in which the coil or set of coils have minimum inductance consistent with generating a given field, per unit current, at a finite set of specified points.

Preferably each point in the finite set of points is selected such that the field is specified within a known volume. In a particular embodiment a first number of the finite set of specified points are selected as being on the surface of an outer cylinder and the rest of the finite set of specified points are selected as being on the surface of an inner cylinder the field between the inner and outer cylindrical surfaces thereby being defined.

In a further embodiment a first number of the finite -set of specified points are selected as being on a first plane and the rest of the rest of the finite set of specified points are selected as being on a second Dlane the field between the first and second planes being thereby defined.

Embodiments of the present invention will now be described, by way of example with reference to the accompanying drawings in which:- Figure 1 shows a graph illustrating integrated current density for a field specified to be the same at 10 points along the axis at equal spacings of 1/10th of the coil radius;

Figure 2 shows the coil layout for a first specified coil design; Figure 3 shows the coil layout for a second specified coil design; Figure 4 shows a first typical coil designed according to the present method; 51 F ig u r e according to shows a second the present method, Figure 6 shows a first according to the present Figure 7 shows a according to the present THEORETICAL EXPLANATION Assume that the current -enerating the desired field is confined to the surface of a cylinder radius a. This assumption may be relaxed to include the case of current flowing in the wall of a thick cylinder, and the case of a screened coil (in which the coil is surrounded either by a thick conductive cylinder or by actively driven wire mimicking the effect of such a cylinder).

The current density is defined as J(r,O,z). -As in the article TURNER R and BOWLEY R.M. (1986) "Passive switching of switched magnetic field gradients", j. Phys.E.Sci.instru. 19,876 we define the Fourier transform of the current densities in the azimuthal and a-xiai directions as:typical coil designed truncated coil des-i--ned invention; and second truncated coil designed invention.

11.1 I m 1 0 iz (k) =:7IT 1 do Jim 7r 00 F 17 j m(k) = L <to e-lfno dz c o- 2 Tr I- IT Then the inductance of the - C0 ikz dz e j (0, Z), ( 1) --ikz r j ok. o, z j (2) cail may be written PO cl C0 C0 1 L dl( 1 jTn ( k) 12 1 a) IT (3) J2 M=-CC I-CO 0 m where I 'is the current in each turn of the coil. The axial component of the magnetic field is

Bz(r,0,z) m 'k)e ikz 1 k dki e Km(ka)lm(kr) Cc (4) For a screened coil, with screen radius b, the integrand' in each of these expressions (3) and (4) -must be multiplied by the factor S(a,b,k), where S(a,b,k) = 1 - 1 Im (ka) Rm(kb) j i Irn (kb) Km (ka) Def ining.1 normalised current write the inductance as (5) density f=j/I, we may I- 1 L (1k if M (IC 2 1 k a) K' ( ka) fil CO C- Cc d) in M and the coil efficiency as B'(r'O'Z) = -P',3 1 277 M I'l'dk fm(k)el Oe' ZKm(ka)lm(kr) nt -00 0' The desired magnetic field must finite set of points lving on the su generally of radius much less than a.

Bz(rn,On,7-n) = Bn where n=1,2 N.

(6) (7) be specified at rface of a cylinder, We write ( 8) We may now construct an expression involving Lagrange multipliers which enables the calculation of that current density distribution which minimises the coil inductance. We wish to minimise L subject to the condition (8). We define.a function U(f$(kM by 1 N L + -f n E 1 Xn (1n - 11z where B is r=(r On,z n n n- the desired field and B2 is the actual field at

Using functional differentiation, and assuming that f$ is either pure real or pure imaginary, we find the condition for a minimum in U:

t m 0 dIC 2uciZ12f (k'l'(ka)Km(lia) I-CO 1- 0 j m - i. 'Ulls 1 11 -l)n 277 This has the solution m h- fo(k) - imOn ikzn ke e t "xn Im (kr.) c e 4 7TaI, (ka) n=t 1 Km (ka) IM (ky. (10) n,] imOn ikzn (11) Subs-tituting back into (7) at the points at which the field is specified, we obtain

Bl -uo CO Y_ = - E k2dk 8772 M=-CC 1C0 where again n=1,2,,N; l=1,2 N K (ha N -- E X 1 (kr n m nlm(lir,') im(ka) n=t P-im(On+01)eikCZn+7-1) (12) This constitutes a set of N linear simultaneous equations in N unknowiis,. which may be solved easily using Gaussian elimination, to give a set of values for;k. The current density can then be evaluated explicitly using (11) and 1 id, ((D,Z) =:5 2 c -77 ill = - 00 cc imO e U(Z j m (k) dii (D 1 -CO z t ( 13) t 2h All that is now required to produce a practical coil design is to approximate the continuous current density thus derived by a set of discrete wire arcs each carrying a current I. The method for achieving this is described in copending U.K. Patent Applications Nos. 8622277 and 8714434; essentially each arc is placed along an equally spaced contour of the stream function in such a position that it carries the surface current from the areas on either side of it.

A similar analysis may be performed in the case of current confined to a plane or parallel planes. In the simplest useful case, here a purely azimuthal current is confined to two planes at z=+d, each of thickness 2c, the inductance can be written simply as:

L= 21 ' -aka -P- k k12 where the current transform of j(r) j 0 CK) == density j(k) ]c is now -- ú- J, ( k--)j'o (t) The axial magnetic field may be written as:

kdc (14) the Hankel (16) and again minimum inductance coil configurations may be derived. Of particular interest is a novel design for producing a uniform field, coils wound on parallel planes.

EXAMPLES consisting Of t10 pancake we 1,0 i 11 describe the three following examples of minimum inductance coils: a) solenoid, b) longitudinal gradient, and c) transverse gradient.

a) SOLENOID In this case only the m=0 term expressions (11) and (12) become f,g,( k) 1 k n 1 x 1 I.(kr.)eikzn = T77a -f 1 ( k a) and it is axis:

is needed. The (17) 11. = - 11 k 2clk T ' ( h. a t k eili(7. + 7.) ta - n 77 n-1 n 1 C" adequate to specify the field along the cylinder

1'n = 11z = DO n = 1,2, N - C19) Alternatively the field may tie specified on the central plane:

Bn =. B, = Bo n = 3,2, N (20) 4 t 1% -g- The example shown in Figure 1 shows the inte--rated current density distribution when the field is specified to be the same at 10 points along the axis, at enual sDacin-,:rs of 1/10th of the coil radius. Figure 2 shows the coil layout approximate calculated inductance homo!2eneit, when 40 turns of 4 mm thick wire are used to this current density. If the radius a=1m, the inductance of this coil is 0.042H: the of an Helmholtz pair giving the same axial and current efficiency is 0.74,13H. The field produced by this coill design is precisely uniform at the points specified, and the homogeneity is better than 1 -ppm for z<1.0a'and r<0. 6a.

b) LO-NIGITUDINAL GRADIENT COIL Here again only the m=O term is needed. A little care must be taken to ensure that the current density derived has the required anti-symmetry about z=O, the Fourier sine transform is needed. Usually longitudinal gradient coils are specified to produce a magnetic field which is linear in z, and independent of r and ou-, to a considerable fraction of the coil radius. The design shown in Figure 3) arises from a field specified as linear at 10 equally-spaced points along the axis, at separations 0.1a. It has an inductance of 3771pH, compared with 845OpH for the conventional Maxwell pair giving the same efficiency and linearity, when a=lm and the wire thickness = 4 mm.

c) TRA'SVERSE GRADIENT COIL To obtain a uniform transverse gradient only the m=+l terms are needed in the cylindrical harmonic expansion (4). This results in the following expressions for Bz(ri).

B 1 -p r) 1 4 TT 2 ---CO xtid F. (10 0 K N Ii(h-al n=l j k ( 7 11 4 Z 1) X c()5; (On+ cbi) k 21) i f (10 11 r n C. j 0 n e i h- zn (22) 4 17a a) ii= 1 A typical coil design found by solving these, now using a field specified at ten points at 0=0, r=0.01a and spaced by 0 - la in the z direction, is shown in Figure 4. A transverse gradient coil optimised for uniformity in the central plane, z=O, can also be derived, giving for example the design shown in Figure 5. This coil aives a transverse gradient uniform to 2% out to r=0.9a and z=+0.5a.

Small loops such as SL (Figure 5) can be eliminated in a systematic manner by multiplying the Fourier transform of current density (obtained by using procedures described hereinbefore) with the apodising function, -k2 t^. /2 e where t is a small distance chosen just large enough to eliminate the unnecessary undulations.

The resultant coil in general has a somwhat smaller inductance than the design prior to apodization, but this inductance is not the minimum consistent with producing exactly the required field at the selected points.

This simplified design sacrifices the desired field specification to a slight degree but leads to much simpler coil designs by eliminating small loops or rapid variation of turn spacing. The factor "t" has to be selected carefully, because as ?It.' is increased, conformity with the desired field specification at the selected points deteriorates. Typically t should be no more than a/10 (where a is the coil radius) which can give a coil producing the specified field at the selected points with an accuracy of 10 ppm.

z We have described and illustrated a design princilDie which enables the construction of. coils with the minimum inductance consistent with a given efficiency over a eiven volume. Of course all the coils shown have an actively-shielded counterpart, which may be derived simply by including the additional factor S(a,b,k) defined in equation (5). For purposes of rapid switching, used in magnetic resonance imaging, it is hard to imagine hoi.,; any further improvements in coil design strategy may be achieved.

as In some circumstances it is necessary to restrict the length of a cylindrical coil. Using the minimum inductance formulation it is possible to design coils which have the lowest inductance consistent with an efficiency specified at a given number of points, while at the same time the coil is confined within a specified axial length. This constraint, that the current density is zero beyond a certain distance from the plane z=O, is introduced as a further set of Lagrange multipliers into equation (9), which becomes N p U=L + 1 -B JJ: n(Bn z(rn,On,zn)) + lIlvpjO(Op'zp) n=l P=1 (23) where P is the number of points (OpjP) at which the current density is specified to be zero. We may write 00 00 m jf (OpjP) eimOP f dk eikz p j (k) 27r m---. _C>0 0 and hence, taldng the functional derivative of U as before, we obtain m R 9 0= -2goa2 f 0 (k) lm(ka) Km(ka) N p +IXn 27, k eimb eikzn Y(ka) Im(kin) + Ivp eimp eikzP n=l P=1 So N f' m (k) k - Y.Xn eimb eikzn Im(krn) 47ra n(ka) n=1 p + vp eim4p eikzp 2lioa21m(ka) Kmkka)p=l (24) (25) (_281 Using the argument presented earlier, we arive at a set of N+P simultaneous equations in the N+P unknownsX andyp which can again be solved by Gaussian elimination. The calculated current density is indeed zero at the points specified. In order to design an effective truncated coil it is only necessary to choose sufficiently many such points, sufficiently clo9sely spaced, that the current density between them and beyond them remains substantially zero. For reasonable choices of coil parmeters transverse gradient coil designs are easily derived, as shown in the Figures 6 and 7 which respectively show primary and screen coil designs to achieve the desired current density distributions.

C t-7

Claims

1. A method of designing a coil or a set of coils in which the conductors of the coil are positioned on one or more coaxial cylinders or parallel planes, in which the coil or set of coils have minimum inductance consistent with generating a given field, per unit current, at a finite set of specified points.

2. A method of designing a coil as claimed in claim 1 in which each point in the finite set of points is selected such that the field is specified within a known volume.

3. A method of designing a coil as claimed in claim 2 in which a first number of the finite set of specified points are selected as being on the surface of an outer cylinder and the rest of the finite set of specified points are selected as being on the surface of an inner cylinder the field between the inner and outer cylindrical surfaces thereby being defined.

4. A method of designing a coil as claimed in claim 2 in which a first number of the finite set of specified points are selected as being on a first plane and the rest of the finite set of specified points are selected as being on a second plane the field between the first and second planes being thereby defined.

5. A method of designing a coil as claimed in claim 3 in which the coil is specified by a desired current density evaluated by the following expression.

CO 77 in = (t jO CO'Z) t _. 1 c e ) d)i

6. A method of designin-g a coil as claimed in claim 4 in which a purely azimuthal current is confined to two planes at z=;d each of thickness 2c and in which the inductance is calculated from the following expression L =,7,:;zto -t-S kel kc 2.

-fk ( k c -1 11 ( l, - and wherein the current density j (k) is transform of j(r) such that ki) J0 the axial magnetic field being kCLk il(k J-)jp(k) is L-h(kc) kz- f kc

7. A solenoid coil in which the coil is defined b-,,,the expression:- the Hankel inductance L of the uo cl 2.0 m L = - - 1: dli 1 j 0 (k) 12 11 (k,,i) K' (1a) J2 M - -CO - cc in m wherein I is the current in each turn of the coil and in which the coil comprises a set of discrete wire arc each arc being placed along an equally spaced contour of the stream function in such a position that it carries the surface current from the areas on either side of it, the surface current density being defined by the expression 77 IR = - C0 1"0 c i in (D P- iliz iln (I0 di( -cc Published 1989 atThe Patent Oftlee, State House, 66.71 High Holborn, London WClR4TP. Further copies maybe obtained from The Patentofftm. Sales Branch, St. MLr_v Cray, Orpington, Rent BR5 3RD. Printed by Multiplex techniques ltd, St MaTY Cray, Kent, Con- 1/87