WO2005081008A2

WO2005081008A2 - Magnetic field homogeneity control using a plurality of sub-coils connected in series and operating at various currents

Info

Publication number: WO2005081008A2
Application number: PCT/GB2005/000366
Authority: WO
Inventors: Ian Leitch Mcdougall; Peter Hanley
Original assignee: Oxford Instruments Plc
Priority date: 2004-02-12
Filing date: 2005-02-03
Publication date: 2005-09-01
Also published as: GB0615732D0; GB2427695B; GB2427695A; GB0403186D0; WO2005081008A3

Abstract

Magnetic resonance apparatus comprises a magnet for generating, in a working region, a magnetic field of sufficient homogeneity to permit magnetic resonance signals to be obtained from a target material in the working region when in use. The magnet has a plurality of coils (100, 101, 102...), wherein at least one of the plurality of coils comprises a plurality of subcoils (200, 201...) arranged adjacent one another. Each of the coils and subcoils are connected in series. A common current supply (105) is arranged to supply a common current to each coil and each subcoil of the magnet. At least one auxiliary current supply (400, 401...) is arranged to supply auxiliary currents to each of the subcoils and a controller is adapted to set the respective auxiliary currents within the subcoils, so as to control the effective geometry and/or position of the coil(s) formed from the subcoils.

Description

MAGNETIC RESONANCE APPARATUS AND METHOD

The present invention relates to magnetic resonance apparatus, particularly having a. relatively compact or "short" form and, for example, for use in medical imaging applications. The invention also relates to a method of using such apparatus and a method of selecting currents for use in the method. Magnetic resonance techniques are now well known and used widely for obtaining inforrmation from within the body of a subject such as a medical, patient. In medicine the techniq-ue provides invaluable information upon the internal structu e of a patient's body, without the need for more intervemtional measures. Typically, the apparatus ^required for use in this field is extremely expensive and larrge. This is because magnetic resonance techniques rely upon the provision of a "working region". It is from within this region, with the part of the patient of interest located within it, that the desired information is obtained. The working region requires a magnetic field of very high homogeneity and strength. A solenoi magnet provides a usefZul configuration for producing such a homogeneous region of magnetic field, particularly since the divergence of the field at the ends of the solenoid can be ensured to be distal from the working region in the centre . Therefore, in most magnetic resonance systems, the working" region is much smaller than the envelope of the surrounding equipment . This causes problems in terms of the physical volume occupied by trie equipment and therefore how it can be used, together with producing adverse psychological effects on patients (such as claustrophobia) . There is also an increasing interest in ^performing magnetic resonance techniques during surgery or other interventional procedures and this involves movement of either the subject or the equipment from a surgical- location to an imaging location and back again. It is desirrable that the distance between such locations is minimised. This is particularly the case for neurosurgery in whi_ch near- instantaneous imaging is needed (that is, obtaining- an image within less than one minute of leaving the surgical position) . There is therefore a need to produce mores compact apparatus, to alleviate psychological problems such as claustrophobia, and also to provide advantages in terms of minimising the distance between surgical and imaging locations . However, as a magnetic resonance system becomes shorter in axial length, the size of the useable workixig region reduces accordingly. It is extremely difficult to maintain the working region size and magnetic homogeneity whilst reducing the axial separation of the coils. As is known, for any magnetic resonance apparatus, any inhomogeneitd_es within the magnetic fields (which cannot be tolerated) are dealt with by positioning further coils (such as "shim" coils) at appropriate locations, these having magnetic fϊields to directly counter the effects of such inhomogeneiti-es . This becomes extremely difficult for compact magnets because of the high local field gradients that are inherent in such short systems. In order to deal with such gradients, the positioning of the coils becomes extremely critical. With a conventional approach, in a compact or short system, the engineering tolerances become limiting to the practical implementation of such coils, making a useful system unbuildable. In accordance with a first aspect of the prresent invention, we provide magnetic resonance apparatus comprising : - a magnet for generating, in a working region, a magnetic field of sufficient homogeneity to permit magnetic resonance signals to be obtained from a target material in the working region when in use, the magnet having a plurality of coils, wherein at least one of the plurality of coils comprises a plurality of subcoils arranged adjacent one another, wherein each of the coils and subcoils are connected in series; a common current supply arranged to supply a common current to each coil and each subcoil of the magnet; at least one auxiliary current supply arranged to supply auxiliary currents to each of the subcoils; and a controller adapted to set the respective auxiliary currents within the subcoils, so as to control the effective geometry and/or position of the coil(s) formed from the subcoils . In considering the problems associated with reducing the size of the apparatus with respect to that of the working region, so as to produce a "short" magnet, we have realised that it is advantageous to divide one or more of the coils into subcoils which can then be separately controlled so as to allow effective geometrical and/or positional changes in the coil (formed by the combined cooperative effect o£ such subcoils) . Specifically, we have realised that, with the -use of subcoils, the effective position and/or geometry of the coils can be subjected to minor modifications by the control of the current through the respective subcoils. This can be thought of as moving the effective centroid of the coil formed from the subcoils so as to produce the same magnetic field as a desired equivalent single coil . For example through control of the subcoils, the coil in question (formed JErom a combination of the subcoils) can be increased in effective radius, or moved in position along the coil axis (or a combination of both if sufficient numbers of subcoi Is are provided) . This solution overcomes the engineering tolerance difficulties since it is easier to control magnet currents to the required degree of accuracy than to build physical, coils with a very high position/geometry accuracy. It is advantageous for the adjacent subcoils to be positioned as close to one another as possible. The subcoils therefore preferably occupy substantially the same vol_ume as would a single coil with the same nominal function, although some additional space may be required due to the provision of insulation and connections. As will be understood, the invention is applicable to all magnetic resonance applications in which a magnetically homogeneous region is used to obtain magnetic resonance signals from a sample or subject. Such applications include MRI and NMR techniques . The subcoils are connected in series with the other coils of the magnet so that a common current (from the common current supply) flows through each of the coils and sutocoils. An additional current (which may be of the opposite polarity) is supplied to the subcoils from the one or more auxiliary current supplies. The controller is preferably operated in use to control the current supplied to each of the subcoils so as to reduce any magnetic inhomogeneities in the working volume resulting from errors in the geometry and/or the relative position of the coils with respect to a predetermined arrangement. Such errors arise to some extent in all practical implementations. The predetermined arrangement is therefore the desired arrangement for the operation of the system. The magnet may comprise a primary coil and two or more correction coils, at least each correction coil preferably being subdivided into subcoils. The primary coil may therefore take the form of an elongate solenoid, with correction coils provided for each end of the solenoid. Typically the magnet comprises at least one pair of coils arranged upon a common axis and positioned about a mid- plane of the apparatus, each coil within the pair(s) being separated from the mid-plane by a distance that is substantially equal to or less than half the respective coil radius. When equal to half the radius, this is the Helmholtz condition for the respective coil. It will be understood that the coils within a pair need not be of equal radius. Where a pair of coils are positioned at a distance from the mid-plane which is less than half their radius, then further coils may be used, positioned at a distance of greater than half their radius from the mid-plane, so as to cancel the resulting magnetic field gradients. Typically, a plurality of coil pairs are provided, each of these comprising various numbers of subcoils. Preferably at least three subcoils are provided within each coiZL of each pair, thereby allowing control of the effective geometry and or position of the coils in two substantially orthogonal directions. The subcoils need not be of the same physical size in cross section although preferably four s imilarly sized subcoils are provided so as to provide the ability to control the effective position/geometry in the two substantially orthogonal directions. Typically t e four coils are therefore arranged in a two-by-two matrix when viewed in cross-section. Further matrix arrangements such as two-by-three or three-by-three, and so on, arre also envisaged. The coils within each pair are normally positioned upon opposing sides of the working region which is bisected! by the mid-plane. The controller is provided so as to control the flow of current within the various subcoils. This normally amounts to setting the particular auxiliary currents (altho^~ugh the common current may also be set by the controller) since, once set to emulate the desired coil configuration, further modifications should not be required. This may be achieved by the provision of a separate auxiliary current supply for each of the subcoils (or at least the corresponding subcoils in each of the two coils of a pair) . The magnitude of the auxiliary currents is typically less than that of the common current. In superconducting magnet systems rather than resistive ones, superconducting switching devices arre also provided for each subcoil, so as to allow them to openrate in persistent mode, once the subcoil and coil currents ha-^ve been set. In accordance with a second aspect of the present invention we provide a method of controlling magnetic resonance apparatus, the apparatus comprising :- a magnet for generating, in a working region, a magnetic field of sufficient homogeneity to permit magnetic resonance signals to be obtained from a target material in the working region when in use, the magnet having a plurality of coils, wherein at least one of the plurality of coils comprises a plurality of subcoils arranged adjacent one another, wherein each of the coils and subcoils are connected in series; a common current supply arranged to supply a common current to each coil and each subcoil of the magnet; at least one auxiliary current supply arranged to supply auxiliary currents to each of the subcoils; and a controller adapted to set the respective auxiliary currents within the subcoils; the method comprising operating the controller so as to set the currents in at least the subcoils in order to control the effective geometry and/or position of the coil(s) formed from the subcoils. Preferably the method further comprises performing monitoring of the magnetic field produced by the magnet when in use. This may involve the use of small nuclear magnetic resonance samples, or a Hall Effect probe, and then plotting the field profiles for each coil and subcoil accordingly. The monitored magnetic field is then compared with a desired magnetic field configuration and the magnet currents are then set with the controller. In accordance with a third aspect of the present invention we provide a method of selecting currents to be applied to a plurality of subcoils in the method according to the second aspect of the invention, the subcoils having a predetermined arrangement so as to generate a resultant magnetic field that is substantially the same as that produced by an equivalent hoop of predetermined form, the method comprising :- a) calculating the magnetic field and magnetic field gradients produced by a predetermined current when applied to the hoop; b) for each subcoil, calculating the magnetic field and magnetic field gradients produced by selected currents when applied to the subcoils; c) selecting final currents for the subcoils by iteratively repeating step (b) , evaluating an error function representing the difference between the calculated values in steps (a) and (b) and selecting currents so as to minimise the error function. The third aspect of the invention therefore provides a method of deriving the final subcoil currents to be used by the controller in implementing the invention according to the second aspect . Some examples of magnetic resonance apparatus according to the present invention will now be described with reference to the accompanying drawings, in which: - Figure 1 shows the strength of even order magnetic field gradients for a current loop; Figure 2 shows the strength of odd order magnetic field gradients for a current loop; Figure 3 is a graph of a versus b/a at constant B₄/B₂; Figure 4 shows B₆ versus b/a; Figure 5 shows a versus b/a at constant B₄/B₂ for various K values; Figure 6 shows the field homogeneity for an example along the z axis; Figure 7 shows the field strength for the example along the z axis; Figure 8 shows a subcoil arrangement according to the invention; Figure 9 shows a first example with four coil pairs, each having four subcoils; Figure 10 shows the field homogeneity for the first example coil arrangement; Figure 11 shows the stray field contours for the first example arrangement; Figure 12 shows a second example with lower stresses; Figure 13 shows the subcoils of the second example; and Figure 14 shows a current supply and control system for either example. We first discuss the theoretical aspects to the invention using pairs of coils, and then provide examples based upon this theory. It will be appreciated however that the invention is not limited to pairs of coils. In order to produce a sufficiently homogeneous magnetic working region, for pairs of coils positioned symmetrically about a mid-plane (containing the co-ordinate origin) , it is necessary to seek to cancel 2nd and 4th order magnetic field gradients and to minimise the 6th order. The odd orders cancel due to the symmetry. All coils in this case are circular and are positioned in pairs at ±b along a common axis. Figure 1 and Figure 2 show the even and odd order axial magnetic field derivatives respectively for a single-turn circular coil. In Figures 1 and 2, the ordinates are the axial position in units of coil radius (b/a) and the abscissae are the field derivatives in units of amperes per metre per ampere, divided by the n-th power of the coil radius . It can be seen that for b/a < 0.5 (inside the Helmholtz line) the gradients vary rapidly with position. Also, for coils at different positions, their relative contributions of the different orders vary markedly, so that an attempt to correct an error due to one coil cannot be simply corrected by adjusting the currents in others. There is not necessarily a linear combination of strengths that will enable all the gradients to be cancelled. If we have a pair of coils whose ratios of B /B have the same value, and then choose their strengths to be such that their B s are equal and opposite, then B will cancel also. Note B₀ will not cancel (unless the coils are coincident) . By having several groups such as this we might get Bg to cancel also, or at least be sufficiently small. As will be understood, B_ns are the coefficients in a Taylor series describing the field profile on the common

B ( \z) ) = B_a0 + B,1 z +-₂| B₂2z² + -₃, B, 3z³ + ...

For a theoretical hoop of radius a and axial position b (with cross-sectional dimensions << a, b) , the z component of the magnetic field B(z) per unit current is:-

Explicit forms of B_n can be found by repeated differentiation:

*.M=^

Fortunately, as mentioned, symmetry enables the odd orders to be ignored. Working in units of radius, so that b = αx, we find:-

For given values, K, of this ratio, contours on the a x plane can be found, these being: -

This function is plotted for K = 30 and K = -30 in

Figure 3 . The graph is of a plotted against b/a . In Figure h h

3 , the singular points are at — = +0.297594 and — = ±0.5 α α

(Helmholtz) . The behaviour of B₆ is as shown in Figure 4.

In Figure 4, B₆ is plotted against b/a with units ^μ°/₇. The zero points are at - = ±0.21403985. / ^~ α Figure 5 is a graph of a versus b/a for constant B₄/B₂ and shows contours of constant K for a range of values . We can see that for negative values of K and x < 0.298, a varies relatively slowly with both x and K, but B₆ varies quite rapidly and changes sign. Coil pairs with zero B and

B will therefore produce little B₀ field (because it almost cancels) but may produce a strong B . Such a pair can therefore be used to cancel the Bg of a pair with positive K and 0.298 < x < 0.5, which also produce zero B and B₄ but also contribute a substantial net B₀. To choose values, we can use a spreadsheet to do the following: 1. Tabulate a against x for some positive value of K (e.g. 40), and calculate the corresponding values of B₀. .B₆;

2. Choose the smallest radius in the system (e.g. 0.5 metres) and take the hoop with the nearest value of a to this as

3. Apply the ampere-turns ratio for zero total

to the other hoops in the range of step 1 and calculate the total values of B 0...B6- '.

4. Choose a likely looking second coil from these, whose radius does not exceed the largest allowed radius in the system (e.g. 1.0 metres);

5. Tabulate a against x for some negative value of K (e.g. - 20) and calculate B ... etc;

6. Apply steps 2, 3 and 4 to this set, choosing a pair of coils whose B₆ is the same order-of-magnitude as that from step 4; and

7. Scale the second pair to cancel the total B₆. There is a wide choice of positive and negative values of K from which to choose, and the choice will eventually be determined by chosen optimisation criteria, for example, minimum ampere-metres/tesla. An example of this is given below :

Example 1

K = 40 αι = 0.506759 x = 0.36 bι=0.18243 N=-4.64736 B₀=l.677765 a₂ =0.744081 x = 0.40 b₂=0.29763 B₆=-33163.9

K = -20 ι = 0.499134 x = 0.25 bι=0.12478 N=-l.78588 B₀=0.39698 ₂ =0.637395 x = 0.21 b₂=0.13385 B₆=21242.8

Second ratio N' = -1.56119 Total strength B₀ = 1.058 A/m for half system Amp-metres A - /B ₀ = 40.49 This is summarised in Table 1 : -

Table 1

This produces field plots as shown in Figures 6 and 7. The above describes idealized thin hoops. However, for a practical system these must he translated into thick solenoid coils with realistic tolerances. First of all, we write down the errors in the gradients that we can tolerate. Assume we need a uniformity of δB as .1 ppm of 0.5 tesla ffi_ at a radius of 0 . 1 m, δB_n < n\ This gives

B₂ < 10 -4 B₄ < 0.12 Bg <360 We note that the variation of a gradient with position is SB_n = ~(n + \)B_n _ δb , so we can set a positional tolerance of røi B δb < - r which leads to tolerances for δb, in metres , as (« + ! " _u shown in Table 2

Table 2

In general, positional accuracy of several microns is required (although coil 2 is positioned on a zero of B₇ by charxce) . Also, the signs of the variations are different, which means that an attempt to correct one gradient by adjusting the strength (current) will lead to an increase in other gradients. For the same reason, it is likely to be very difficult to replace the elementary hoops with equivalent extended solenoid coils. It is then necessary to consider whether these tolerances can be coped with and whether an equivalent solenoid can be found, as is now discussed. A hoop is specified by a, b, I, giving:

A solenoid is specified by a_{l t} a₂ , b_lf b₂, J:

They would be equivalent if:-

This doesn't seem possible because the right-hand side has a form «/« α ^< +- α²x{z-b)² which is different from the left-hand

side. We now look for approximations to the expression for a solenoid with the objective of representing it by a hoop. Zf z-w<r which is true inside the Helmholtz line for a working region which also lies inside, then:-

α₂ μ₀J w ln(r) + -^-(w² -3wz+3z )+-^—(w⁴ -5w^sz + \0w²z² -I0w + 5z⁴) 4r² ' 16r^{4 ;} w=b.

d

Putting a₂ = ai +d (d<<aχ) and expanding in powers of γ- α.

1 1 α₂ - αL^«- αI_x [²y-^3ya+ --]

Ignoring higher orders, this gives:

Putting b = h — c and b, = b + c we obtain :

If we now compare this with the binomial expansion of the expression for the on-axis field of a hoop :

We can see that these are equivalents as c- 0 and 2cγJ -^ lla , as is to be expected. For non-zero c the two would only k>e equivalent if:-

and;

which are clearly only possible for σ=0. This analysis therefore shows that an equivalent solenoid cannot be found. The pr-oposed solution according to the invention is to represent a hoop by not one, but a number of subcoils. In this example, four subcoils are chosen as shown in Figure 8, so as to provide control freedom in the axial and radial directions . As shown, a 2-by-2 matrix of four subcoils is positioned centred about a distance "b" from the origin along a common "z" axis for ttie system, and having a centre radial distance "a" from this axis. Each of the subcoils has a radial dimension " d/2 " and an axial dimension "c" .

The advantages in doing this are: • The subcoils are smaller than the coil as a whole and therefore better approximate to hoops; • Errors can be cancelled; • In a practical implementation, adjustment of individual currents offers a way of compensating for inaccuracies. The problem is now to find the currents I , I₂, I₃ I₄ of the four suϊocoils 1, 2, 3, 4 centered at:-

which are in total equivalent to a hoop with 1₀ at (a, b) . The current can be found numerically, using one of the numerical optimisation methods known in the art. Using such a method we seek to minimise the error function :-

where B_n , ₀ ^^s the ^nth field derivative of the hoop we are trying to represent and B_n , i i-^s the derivative due to the ith coil, which in the numerical calculation can be a thick solenoid. This is given by repeated differentiation of _ \

B(z) = ere the current

densities are gi-ven by:- J, = x,J₀ where j^" ₀ is a realistic engineering value and the coil dimensions are:-

rounded, say, to the nearest millimetre. The minimisation is performed by adjusting the x_; about a mean value of unity. In the practical implementation, the design is realised by adjusting the currents in the four subcoils. We can also model the effects of tolerances by giving the subcoils displacements . Taking as an example the third subcoil, this has (for a 0.5 tesla system) the characteristics shown in Table 3.

Table 3

This can be replaced by four solenoids (subcoils) in a group whose nominal current density is 10⁸AzτT² and is 54mm wide and 54mm deep as shown in Table 4.

Table 4

If this is now displaced 1mm both radially and^' axially, representing a tolerancing error, and the optimisation repeated, we can obtain the parameters shown in Table 5.

Table 5

We have therefore corrected the effects of this substantial error by adjustment of the currents in the subcoils. In further modelling, the following errors shown in Table 6, were imposed on the same system, with similar results : -

Table 6

To summarise the above : -

1. A straightforward spreadsheet-based method can be used for short magnets, corrected to 8th order, approximated by hoops;

2. It is not possible to directly replace these hoops by thick-section solenoids;

3. Achievement of the uniformity requires dimensional accuracy in the range 1 to 10 ppm; 4. The hoops can be replaced by thick-section solenoids, for example divided axdally and radially into four subcoils, and by adjusting the relative currents in these subcoils they can be made to .represent the hoops to good accuracy (provided the current density is sufficiently high that the sections of the coils are not too big) ; 5. Dimensional errors can be corrected by adjusting the currents in the subcoils. Using this technique we now discuss the above example as a practical design for a multicoil magnet system. The same example as before was converted from hoops to thick solenoids, each divided into four subcoils of slightly different current densities. The result was reflected about the X-Y plane to provide the full eight-coil system (32 sub- elements) . The coil arrangement is shown in Figure 9. In Figure 9, coils with currents arranged in operation to flow in a first sense are labelled "A", and those with the currents flowing in the opposite sense are labelled "B" . The working region is indicated schematically at 100. Figure 10 shows a magnetic field homogeneity plot along the common ("z") axis for this arrangement. Figure 11 shows stray field contours at 1, 5 and lOmT . A number of assumptions were made in these calculations : - 1) Current density was specified to 1 in 10⁵ which should be achievable (lmA in 100A) ; 2) Dimensions were specified to μ_m which is not really practical. However, the coil sections have been rounded to 1mm and we have already shown that dimensional errors of the order of 1mm can be tolerated;

3) The half-length over the windings was 344mm which suggests a half-length over the cryostat of about 420mm; (a typical conventional length is 600 to 800mm) 4) The smallest inner winding radius was 472mm suggesting a clear bore of about 850 mm, flaring to about 1200mm at the ends ; 5) The working volume was about 250 mm diameter;

6) The maximum bursting hoop stress was 2.7 x 10⁸ pascals (no allowance for space factor or turn-to-turn support) and is generally considerably less;

7) The peak field on the windings was 0.82 tesla; 8) The 100 gauss stray-field footprint had a 3.5 m radius; 9) Conductor usage was 1.54 x 10⁷A-m; The implications of such a system are that it needs a large number (32) of superconducting switches and a complex protection circuit. It may therefore be most conveniently implemented with one main power supply and 32 subsidiary supplies, each providing say 10% of the main current for energisation. This can be set up for example by performing field plots for each of the 32 elements, with the field measurements accurate to 1 in 10⁵. Compatible shim and gradient coils can then be used. Further details relating to this system are set out in Appendix 1. Dimensions, field deviations, peak fields and stresses for this arrangement are also given in Appendix 2. Upon examination of the data, it can be seen that, although gradients are corrected to 8th order:, the stresses are rather high. In a second example, to reduce the stresses, each coil was eplaced by another of lower current density, using the optimising method to find new values of mean radius, mean axial position, winding height and winding length to produce the same field derivatives to 8th order as above at the specified current density, for each coil in turn. The resulting dimensions and stresses are shown in Appendix 3. The arrangement of the modified system is shown in Figure 12 again with the directions of current flow shown. A number of versions of this were then investigated: - • no rounding; • dimensions rounded to 0.1mm; • rounded to 0.1mm and currents adjusted; • dimensions rounded to 1mm; • rounded to 1mm and currents adjusted. The currents were adjusted using the numerical method, to find multiplying factors for each coil, having fixed the dimensions at the rounded values. Equal weighting was given to the field excursions due to all the even orders up to and including 6th, i.e. the. error function was:-

Table 7 details the results of this. It seems that tolerancing effects on the homogeneity can only partly be recovered by adjusting the coil currents. This was all done assuming a homogeneity volume ("working region") of 20cm diameter.

Table 7

The next stage was to sub-divide each coil into four sub-coils, in the way described earlier. The arrangement is shown in Figure 13 (with current flows as in Figure 12.) This used dimensions rounded to 1 mm (subcoils rounded to 0.5 mm) and succeeded in nearly recovering the homogeneity of the original. The dimensions of all ttie coils are given in Appendix 4. Table 8 provides further details : -

Table 8

To simulate the effect of tolerancing errors, coil 3 was given an axial displacement of 0.5 mm. The effect of this on the homogeneity i s shown Table 9.

Table 9

Adjusting trie currents in each of the four sub-coils in coils 3 and 7 allowed the homogeneity to be largely recovered, at least to within the scope of resistive bore- shims as shown in Table 10. Table 10

Co l 3 was chosen for this exercise because it has the greatest sensitivity to movement . The relevant coil data are given in Appendix 5. As a result of the above, the magnet is buildable if the current densities are reduced to deer-ease the stresses. Rounding errors in the coils dimensions can be accommodated if the technique of subdividing each coil into sub-coils is employed. However, this technique complicates the protection circuitrry, the power supplies and the energisation procedure. As mentioned, a modified control system is therefore needed to provide the extra control of the subcoils according to the invention. An example of such a system is shown in Figure 14. This shows a number of coils 100, 101, 102, 103... each connected in series and supplied with current by a main current supply 105. Referring to the coil 100, this comprises four serially connected superconducting subcoils 200, 201, 202, 203, each of these having a corresponding superconducting switch 300,

301, 302, 303, each switch being connected in parallel across the respective subcoil. As is also shown in Figure 14, four individual auxiliary current supplies 400, 401, 402, 403 are provided so as to control the individual currents passing through the respective subcoils in addition to the main current. Each subcoil is therefore provided with its own superconducting switch and means of adjusting the current through it, to a value different to that provided by the principal current from the main current supply 105. Because of mutual induction coupling, these must all be active and in control of the current while any of the other currents are adjusted.

In the appendices below, F_z is the total force on the coil in the z direction. Bmod is the total field strength.

Values in parentheses provide cylindrical polar co-ordinates of the position at which the value in question occurs [r, θ, z] with θ being arbitrary due to symmetry. "Current" values are current densities in amperes per square metre. J is also the current density- al,a2 are inner and outer winding radii. bl,b2 are .respective coil axial end positions.

"accuracy" is a parameter used by the field calculation in numerical integration. "element type" is another parameter used by the field calculation.

Appendix 1

# coil', i # from coil.1.4.out #3 coil 1 element 1 J -5.64113e+07 al 0.506759 a2 0.828259 bl 0.182433 b2 0.203933 Max Bmod » 1.981E-01 Tesla at [ 5.283E-01 3.600E+02 1.613E-02. Fz « -4.969E+04 Newtons Max hoop stress = -8.949E+06 Pa at [ 5.094E-01 2.700E+02 1.851E-01] Min hoop stress - -2.752E+07 Pa at [ 5.256E-01 2.700E+02 1.8S1E-01] coil 1 element 2 J -1.4054e+08 al 0.485259 a2 0.606759 bl 0.182433 b2 0.203933 Max Bmod = 1.227E-01 Tesla at [ 4.853E-01 3.600E+O2 1.613E-023 Fz = -5.319E+04 Hβvtons Max hoop stress « 7.108E+07 Pa at [ 4.879E-01 2.700E+02 1.851E-01] Min hoop stress - -2.712E+07 Pa at [ 5.041E-01 2.700E+02 2.012E-01] coil 1 element 3 al 0.485259 a2 0.606759 bl 0.160933 b2 0.182433 Max Bmod = 1.227E-01 Tesla at [ 4.853E-01 3.600E+02 1.613E-02] Fz = 1.568E+05 Nevtons Max hoop stress - 3.345E+07 Pa at [ 4.879E-01 9.000E+01 1.797E-01] Min hoop stress = 8.857E+06 Pa at [ 6.041E-01 9.000E+01 1.797E-01. coil 1 element 4 J -1.36678e+08 al 0.506759 a2 0.528259 bl 0.160933 b2 0.182433 Max Bmod = 1.981E-01 Tesla at [ 5.283E-01 3.600E+02 1.613E-02] Fz = 2.113E+05 Hewtons Max hoop stress = 3.049E+07 Pa at [ 6.094E-01 9.000E+01 1.636E-01] Min hoop stress = -6.72SE+07 Pa at [ 5.256E-01 2.700E+02 1.797E-01] # coil.2 # from coil.2.4. out #3 coil 2 element 1

J 8.73622e+07 al 0.744081 a2 0.790581 bl 0.297632 b2 0.344132

Max Bmod = 7.722E-01 Tesla at [ 7.441E-01 0.000E+O0 3.488E-02]

Fz = -1.091E+06 Newtons

Max hoop stress = 1.319E+07 Pa at [ 7.499E-01 2.7O0E+02 3.034E-01]

Min hoop stress = -1.387E+08 Pa at C 7.848E-01 2. "700E+02 3.034E-01] coil 2 element 2

J 1.12603e+08 al 0.697581 a2 0.744081 bl 0.297632 b2 0.344132

Max Bmod = 8.153E-01 Tesla at C 7.131E-01 3.60OE-M32 3.488E-02.

Fz - -1.178E+06 Newtons

Max hoop stress = 2.749E+08 Pa at [ 7.034E-01 2.7O0E+02 3.034E-01]

Min hoop stress = 5.484E+07 Pa at [ 7.383E-01 9.O00E+01 3.383E-013 coil 2 element 3

J 9.42493Θ+07 al 0.697581 a2 0.744081 bl 0.251132 b2 0.297632

Max Bmod = 8.161E-01 Tesla at [ 7.162E-01 0.0OOE- 3O 3.488E-02]

Fz = 1.683E+06 Hewtons

Max hoop stress = 2.281E+08 Pa at [ 7.034E-01 9. O00E+01 2.918E-01]

Min hoop stress = 7.690E+07 Pa at [ 7.383E-01 9. O00E+01 2.918E-01] coil 2 element 4 al 0.744081 a2 0.790581 bl 0.251132 b2 0.297632

Max Bmod = 7.722E-01 Tesla at [ 7.441E-01 0.0O0E→-00 3.488E-02]

Fz = 1.763E+06 Newtons

Max hoop stress «= 5.149E+07 Pa at t 7.499E-01 9. O00E+01 2.569E-01]

Min hoop stress = -1.782E+08 Pa at [ 7.848E-01 9 , 000E+01 2.918E-01]

# coil.3

S from coil.3.4.out #1 coil 3 element 1

J 9.14003e+07 al 0.499134 a2 0.526134 bl 0.124784 b2 0.151784

Max Bmod = 2.060E-01 Tesla at [ 5.261E-01 0.OOOE-+00 2.025E-02]

Fz = -2.824E+05 Newtons

Max hoop stress = -1.514E+07 Pa at [ 5.025E-01 9.OOOE+01 1.282E-01]

Min hoop stress = -7.469E+07 Pa at [ 5.228E-01 9.OOOE+01 1.282E-01] coil 3 element 2 J 1.07799e+08 al 0.472134 a2 0.499134 bl 0.124784 b2 0.151784 Max Bmod = 1.993E-01 Tesla at [ 4.721E-01 3.600E+02 2.02SE-023 Fz = -2.939E+05 Newtons

Max hoop stress « 7.388E+07 Pa at [ 4.755E-01 2.700E+02 1.282E-013 Min hoop stress = -1.019E+07 Pa at [ 4.958E-01 9. OOOE+01 1.484E-013 coil 3 element 3

J 9.54214e+07 al 0.472134 a2 0.499134 bl 0.0977835 b2 0.124784

Max Bmod = 1.993E-01 Tesla at [ 4.721E-01 3.600E+02 2.025E-023

Fz = 1.022E+05 Newtons

Max hoop stress = 6.636E+07 Pa at [ 4.755E-01 9. OOOE+01 1.214E-013

Min hoop stress « 6.713E+06 Pa at C 4.958E-01 2.700E+02 1.214E-013 coil 3 element 4

J 1.08103e+08 al 0.499134 a2 0.526134 bl 0.0977836 b2 0.124784

Max Bmod - 2.060E-01 Tesla at [ 5.261E-01 O.OOOE+00 2.025E-023

Fz - 1.033E+05 Newtons

Max hoop stress = -9.359E+06 Pa at [ 5.025E-01 9. OOOE+01- 1.012E-013

Min hoop stress = -9.230E+07 Pa at [ 5.228E-01 9.000E+OL 1 .214E-013

# coil.4

# from coil. .4.out #2 coil 4 element 1

J -8.69962e+07 al 0.637395 a20.673395 bl 0.133853 b20.169853

Max Bmod - 6.678E-01 Tesla at [ 6.734E-01 3.600E+02 2.7O0E-023

Fz " -8.579E+05 Newtons

Max hoop stress = -5.004E+07 Pa at [ 6.419E-01 9.000E+0Ϊ-. 1 .384E-013

Min hoop stress = -1.495E+08 Pa at [ 6.689E-01 2.700E+02 1.384E-013 coil 4 element 2

J -1.12948e+08 al 0.601395 a2 0.637395 bl 0.133853 b2 0.169853

Max Bmod = 3.938E-01 Tesla at [ 6.374E-01 O.O00E+00 2.7O0E-023

Fz = -8.974E+05 Newtons

Max hoop stress - 1.172E+08 Pa at [ 6.059E-01 2.700E+02 1.384E-013

Min hoop stress = -4.944E+07 Pa at [ 6.329E-01 9. OOOE+01- 1.654E-013 coil 4 element 3 J -9.40327e+07 al 0.601395 a2 0.637395 bl 0.0978529 b2 0.133853

Max Bmod = 3.938E-01 Tesla at [ 6.374E-01 O.OOOE+00 2.7O0E-023 Fz ■= 2.735E+05 NewtonsMax hoop stress - 9.902E+07 Pa at [ 6.059E-01 2.700E+02 1.294E-01] Min hoop stress = -8.563E+06 Pa at [ 6.329E-01 2.700E+02 1.294E-013 coil 4 element 4 J -1.1002e+08 al 0.637395 a2 0.673395 bl 0.0978529 b2 0. 133863

Max Bmod = 6.578E-01 Tesla at [ 6.734E-01 0.O00E+O0 2.7O0E-O23

Fz = 1.914E+05 Newtons

Max hoop stress = -3.281E+07 Pa at [ 6.419E-01 2.700E+02 1 .024E-013

Min hoop stress = -1.895E+08 Pa at [ 6.689E-01 9. OOOE+01 1 -294E-013

FIELD DERIVATIVES Tesla Metres Degrees partial derivatives w.r .t . Z range 1.000E-01 at R = O .OOOE+00, T = O .OOOE+00 , Z = O .OOOE+00 Tesla Metres Degrees order Br Btheta Bz Bmod

0 O.OOOOE+00 O.OOOOE+00 5.0001E-01 5.0001E-01

1 -O.OOOOE+00 -O.OOOOE+00 1.5289E-14 1.5289E-14

2 O.OOOOE+00 O.OOOOE+00 -2.5708E-04 -2.5708E-04

3 O.OOOOE+00 0.O00OE+0O -2.8564E-10 -2.8564E-10

4 -O.OOOOE+00 -O.OOOOE+00 7.2348E-01 7.2348E-01

5 O.OOOOE+00 0.OO00E+0O 5.2751E-06 5.2751E-06

6 0.0000E+00 O.OOOOE+00 -1.6949E+02 -1.6949E+02

7 0.000OE+00 O.OOOOE+OO -5.5879E-02 -S.5879E-02

8 -0.0000E+00 -O.OO00E+00 -3.11S5E+06 -3.1155E+06

Appendix 2

Cylindrical elements: element # 1 Current -6.2500E+08 al l.OOOOE+OO a2 1.0200E+00 bl 2.0000E-01 b2 2.2000E-01 accuracy 10 element type 0 element # 2 Current 1.8374E+09 al 6.9000E-01 a2 7.1000E-01 bl 1.4000E-01 b2 1.6000E-01 accuracy 10 element type 0 element # 3 Current -1.1278E+09 al 5.0000E-01 a2 5.2000E-01 bl 9.9170E-02 b2 1.1917E-01 accuracy 10 element type 0 element # 4 Current 2.5969E+08 al 4.0000E-01 a2 4.2000E-01 bl 7.7750E-02 b2 9.7750E-02 accuracy 10 element type 0 element # 5 Current -6.2500E+08 al l.OOOOE+OO a2 1.0200E+00 bl -2.2000E-01 b2 -2.0000E-01 accuracy 10 element type 0 element # 6 Current 1.8374E+09 al 6.9000E-01 a2 7.1000E-01 bl -1.6000E-01 b2 -1.4000E-01 accuracy 10 element type 0 element # 7 Current -1.1278E+09 al 5.0000E-01 a2 5.2000E-01 bl -1.1917E-01 b2 -9.9170E-02 accuracy 10 element type 0 element # 8 Current 2.5969E+08 al 4.0000E-01 a2 4.2000E-01 bl -9.7750E-02 b2 -7.7750E-02 accuracy 10 element type 0

FIELD DERIVATIVES Tesla Metres Degrees partial derivatives w.r. t. Z range l.OOOE-Ol at R = O.OOOE+00, T ■= 0, OOOE+00, Z = O.OOOE+00 Tesla Metres Degrees order Br Btheta Bz Bmod

0 0.0000E+00 O.OOOOE+00 1.9993E-01 1.9993E-01

1 O.OO00E+00 0.0000E+00 -2.3629E-14 -2.3529E-14

2 O.0O00E+O0 O.0OO0E+00 7.6387E-04 7.6387E-04

3 O.0000E+00 0.0000E+00 5.4374E-10 5.4374E-10

4 O.O00OE+00 O.0OO0E+00 -2.7787E-02 -2.7787E-02

6 O.0OO0E+00 0.0000E+00 -8.8221E-06 -8.8221E-06

6 O.OO00E+00 O.0O00E+00 4.4957E+00 4.49S7E+00

7 O.0O0OE+00 O.0OO0E+00 8.1200E-O2 8.1200E-02

8 O.OOOOE+00 O.0O00E+0O -5.4489E+06 ' -5.4489E+06

Peak fields for cylindrical element No. 1

Maximum and minimum fields in local coordinate system:

Max Br - 5.472E-02 Tesla at [ 1.003E+00 O.OOOE+00 1.750E-02]

Min Br = 7.639E-07 Tesla at [ 1.020E+00 3.600E+02 2.500E-07]

Max Btheta = 8.593E-33 Tesla at [ 1.O05E+OO 3.600E+02 1.750E— 023

Min Btheta - -1.354E-32 Tesla at [ l.OlOE+00 3.600E+02 1.500-E-023

Max Bz = -3.434E-01 Tesla at [ 1.020E+00 O.OOOE+00 1.750E-02U

Min Bz = -4.063E-01 Tesla at [ l.OOOE+00 O.OOOE+00 1.750E-02D

Max Bmod - 4.100E-01 Tesla at [ l.OOOE+00 3.600E+02 1.75OE-023

Peak fields for cylindrical element No. 2

Maximum and minimum fields in local coordinate system:

Max Br = -3.617E-06 Tesla at [ 7.100E-01 0.000E+O0 2.500E-073

Min Br - -2.617E-01 Tesla at [ 6.967E-01 3.600E+02 1.750E-02H

Max Btheta = 1.478E-16 Tesla at [ 6.967E-01 1.200E+02 2.500E-073

Min Btheta = -1.032E-16 Tesla at [ 7.056E-01 1.200E+02 2.50»:E-073

Max Bz = 6.915E-01 Tesla at [ 6.900E-01 3.600E+02 1.750E-023

Min Bz = 2.863E-01 Tesla at [ 7.100E-01 O.OOOE+00 1.7B0E-023

Max Bmod - 6.461E-01 Tesla at [ 6.900E-01 0.000E+00 1.750E-O23

Peak fields for cylindrical element No . 3

Maximum and minimum fields in local coordinate system:

Max Br = 2.933E-01 Tesla at [ 5.080E-01 3.600E+02 1.750E-023

Min Br = 3.915E-06 Tesla at [ 5.200E-01 0.000E+00 2.500E-073

Max Btheta - 5.816E-32 Teala at [ 5.040E-01 2.700E+02 1.750E-023

Min Btheta = -5.732E-32 Tesla at [ 5.020E-01 3.600E+02 1.25OE-023

Max Bz - 8.858E-01 Tesla at [ 6.200E-01 3.600E+02 1.750E-023

Min Bz = 5.357E-01 Tesla at [ 5.000E-01 3.600E+02 1.750E-02]

Max Bmod = 9.295E-01 Tasla at [ 5.200E-01 3.600E+02 1.750E-O23

Peak fields for cylindrical element No. 4

Maximum and minimum fields in local coordinate system:

Max Br - -3.915E-07 Tesla at [ 4.200E-01 3.600E+02 2.500E-07"3

Min Br •= -6.898E-02 Tesla at [ 4.000E-01 0.000E+00 1.750E-023 Max Btheta « 5.608E-17 Tesla at [ 4.145E-01 2.160E+02 1.500E-023 Min Btheta « -4.269E-17 Tesla at [ 4.182E-01 1.440E+02 1.500E-023 Max Bz - -2.094E-02 Tesla at [ 4.000E-01 3.600E+02 2.500E-073 Min Bz = -1.022E-01 Tesla at [ 4.200E-01 O.OOOE+00 1.750E-023 Max Bmod = 1.074E-01 Tesla at [ 4.200E-01 O.OOOE+00 1.750E-023

Forces for cylindrical element No. 1 in units of Newtons, Metres Degrees

Total forces in local coordinate system:

Fx = -3.650E-11 Fy ■= 1.091E-11 Fz => 1.077E+05 Newtons

Max hoop stress = 2.563E+09 Pa at [ 1.001E+00 9.000E+01 2.087E-013

Min hoop stress = -2.111E+09 Pa at [ 1.019E+00 2.700E+022.113E-01]

Forces for cylindrical element No. 2 in units of Newtons, Metres Degrees

Total forces in local coordinate system:

Fx = -1.159E-10 Fy = -6.112E-10 Fz = -5.640E+05 Newtons

Max hoop stress = 1.466E+10 Pa at C 6.912E-01 9.OOOE+01 1.487E-013

Min hoop stress = -1.335E+10 Pa at [ 7.088E-01 9.000E+01 1.513E-013

Forces for cylindrical element No. 3 in units of Newtons, Metres Degrees

Total forces in local coordinate system:

Fx - 5.487E-11 Fy - 5.821E-11 Fz - -1.446E+05 Newtons

Max hoop stress = 3.481E+09 Pa at [ 5.013E-01 9.OOOE+01 1.079E-01]

Min hoop stress - -4.270E+09 Pa at [ 5.187E-01 2.700E+02 1.1O4E-013

Forces for cylindrical element No. 4 in units of Newtons, Metres Degrees

Total forces in local coordinate system:

Fx - 4.645E-12 Fy » O.OOOE+00 Fz - -1.91OE+04 Newtons

Max hoop stress = 1.376E+08 Pa at [ 4.013E-01 2.700E+02 8.650E-023

Min hoop stress = -2.033E+08 Pa at [ 4.187E-01 2.700E+02 8.900E-023

Thu Sep 25 11:57:41 2003

Appendix 3

Cylindrical elements: element * 1 Current -1.0000E+08 al 9.8500E-01 a2 1.0350E+00 bl 1.8500E-01 b2 2.3500E-01 accuracy 10 element type 0 element # 2 Current 2.0000E+08 al 6.7000E-01 a27.3030E-01 bl 1.1970E-01 b2 1.8030E-01 accuracy 10 element type 0 element # 3 Current -2.0000E+08 al 4.8620E-01 a2 5.3380E-01 bl 8.S400E-02 b2 1.3290E-01 accuracy 10 element type 0 element # 4 Current 2.0000E+08 al 3.9860E-01 a2 4.2140E-01 bl 7.6300E-02 b2 9.9100E-02 accuracy 10 element type 0 element # 5 Current -1.0000E+08 al 9.8500E-01 a2 1.03EOE+00 bl -2.3500E-01 b2 -1.8500E-01 accuracy 10 element type 0 element # 6 Current 2.0000E+08 al 6.7000E-01 a2 7.3030E-01 bl -1.8030E-01 b2 -1.1970E-01 accuracy 10 element type 0 element # 7 Currant -2.0000E+08 al 4.8620E-01 a25.3380E-01 bl -1.3290E-01 b2 -8.5400E-02 accuracy 10 element type 0 element # 8 Current 2.0000E+08 al 3.9860E-01 a2 4.2140E-01 bl -9.9100E-02 b2 -7.6300E-02 accuracy 10 element type 0

Forces for cylindrical element No. 1 in units of Newtons, Metres Degrees

Total forces in local coordinate system:

Fx = -5.765E-13 Fy = 9.948E-14 Fz = 4.076E+03 Newtons

Max hoop stress - 6.497E+05 Pa at [ 9.8S1E-01 2.700E+02 1.881E-013

Min hoop stress = 6.528E+05 Pa at [ 1.032E+00 2.700E+02 2.319E-013

Forces for cylindrical element No. 2 in units of Newtons, Metres Degrees

Total forces in local coordinate system:

Fx - 4.634E-12 Fy = 4.036E-12 Fz - -2.877E+04 Newtons

Max hoop stress = -2.770E+O6 Pa at [ 7.265E-01 2.700E+02 1.765E-013

Min hoop stress * -3.926E+06 Pa at [ 6.738E-01 9.000E+01 1.23BE-01]

Forces for cylindrical element No. 3 in units of Newtons, Metres Degrees

Total forces in local coordinate system:

Fx - -1.084E-11 Fy - -3.O70E-12 Fz - 5.228E+04 Newtons

Max hoop stress - 1.739E+07 Pa at [ 4.892E-01 2.700E+02 8.837E-023

Min hoop stress = 9.014E+06 Pa at [ 5.308E-01 9.OOOE+01 1.299E-013

Forces for cylindrical element No. 4 in units of Newtons, Metres Degrees

Total forces in local coordinate system:

Fx ■ -1.941E-12 Fy - 8.185E-12 Fz - 1.364E-12 Newtons

Max hoop stress - 1.186E+08 Pa at [ 4.000E-01 2.700E+02 8.912E-02]

Min hoop stress - -1.023E+O8 Pa at C 4.200E-01 2.700E+02 8.912E-023

Fri Sep 26 10:09:47 2003

Appendix 4

# all-coils.1.text-data # coil 1 sub-coil 1 coil al 1.0095 a2 1.034 bl 0.209 b2 0.233 J -7.89666e+07 # coil 1 sub-coil 2 coil al 0.985 a2 1.0095 bl 0.209 b2 0.233 J -1.71197e+08 # coil 1 sub-coil 3 coil al 0.986 a2 1.0095 bl 0.185 b2 0.209

coil al 1.0095 a2 1.034 bl 0.185 b2 0.209 J -9.53S34e+07 # coil 2 sub-coil 1 coil al 0.6995 a2 0.729 bl 0.149 b2 0.178 J 1.89976e+08 # coil 2 sub-coil 2 coil al 0.67 a2 0.6995 bl 0.149 b2 0.178 J 2.36926e+08 # coil 2 sub-coil 3 coil al 0.67 a2 0.6995 bl 0.12 b2 0.149 # coil 2 sub-coil 4 coil al 0.6995 a2 0.729 bl 0.12 b2 0.149 J 2.9502e+08 # coil 3 sub-coil 1 coil al 0.51 a20.534 bl 0.109 b20.133 # coil 3 sub-coil 2 coil al 0.486 a20.51 bl 0.109 b20.133 J -2.18258e+08 # coil 3 sub-coil 3 coil al 0.486 a2 0.61 bl 0.086 b2 0.109 J -1.60133e+08 # coil 3 sub-coil 4 coil al 0.51 a20.534 bl 0.085 b2 0.109 J -2.41939e+08 # coil 4 sub-coil 1 coil al 0.4095 a2 0.42 bl 0.087

~b2^"0.098 J 2.39936e+08 # coil 4 sub-coil 2 coil al 0.399 a2 0.4095 bl 0.087 b2 0.098 J 2.85495e+08 # coil 4 sub-coil 3 coil al 0.399 a2 0.4096 bl 0.076 b2 0.087 J 1.68286e+08 # coil 4 sub-coil 4 coil al 0.4095 a2 0.42 bl 0.076 b2 0.087 # the other side ... # coil 5 sub-coil 1 coil al 1.0095 a2 1.034 b2 -0.209 bl -0.233 J -7.89565e+07 # coil 5 sub-coil 2 coil al 0.986 a2 1.0095 b2 -0.209 bl -0.233

J -1.71197e+08

# coil 6 sub-coil 3 coil al 0.985 a2 1.0095 b2 -0.185 bl -0.209

# coil 6 sub-coil 4 coil al 1.0095 a2 1.034 b2 -0.185 bl -0.209 J -9.53534e+07

# coil 6 sub-coil 1 coil al 0.6995 a20.729 b2 -0.149 bl -0.178 J 1.89975e+08

# coil 6 sub-coil 2 coil al 0.67 a20.6995 b2 -0.149 bl -0.178 J 2.36926e+08

# coil 6 sub-coil 3 coil al 0.67 a2 0.6995 b2 -0.12 bl -0.149 J 1.38166e+08

# coil 6 sub-coil 4 coil al 0.6995 a2 0.729 b2 -0.12 bl -0.149 J 2.9502e+08

# coil 7 sub-coil 1 coil al 0.51 a20.534 b2 -0.109 bl -0.133

J -1.63316e+08

# coil 7 sub-coil 2 coil al 0.486 a2 0.51 b2 -0.109 bl -0.133 # coil 7 sub-coil 3 coil al 0.486 a2 0.51 b2 -0.085 bl -0.109

J -1.60133e+08

# coil 7 sub-coil 4 coil al 0.51 a2 0.534 b2 -0.085 bl -0.109

# coil 8 sub-coil 1 coil al 0.4095 a2 0.42 b2 -0.087 bl -0.098

# coil 8 sub-coil 2 coil al 0.399 a2 0.4096 b2 -0.087 bl -0.098 J 2.85495e+08

# coil 8 aub-coil 3 coil al 0.399 a2 0.4095 b2 -0.076 bl -0.087 J 1.582866+08

# coil 8 sub-coil 4 coil al 0.409S a20.42 b2 -0.076 bl -0.087 J 2.15112e+08

Appendi 5 all-coils.2.text-data # coil 3 sub-coil 1 coil al 0.51 a20.534 bl 0.1095 b2 0.1335 # coil 3 sub-coil 2 coil al 0.486 a2 0.51 bl 0.1095 b2 0.1336 J -3.05886e+08 # coil 3 sub-coil 3 coil al 0.486 a2 0.51 bl 0.0855 b2 0.1095 J -1.461236+08 # coil 3 sub-coil 4 coil al 0.51 a2 0.534 bl 0.0855 b2 0.1096 J -2.302286+08 # coil 7 aub-coil 1 coil al 0.51 a2 0.634 b2 -0.109 bl -0.133 J -1.95675e+08 # coil 7 sub-coil 2 coil al 0.486 a2 0.51 b2 -0.109 bl -0.133 # coil 7 sub-coil 3 coil al 0.486 a20.61 b2 -0.085 bl -0.109 J -2.28865e+08 # coil 7 sub-coil 4 coil al 0.51 a2 0.534 b2 -0.085 bl -0.109 J -1.48732e+08

Claims

1. Magnetic resonance apparatus comprising :- a magnet for generating, in a working region, a magnetic field of sufficient homogeneity to permit magnetic resonance signals to be obtained from a target material in the working region when in use, the magnet having a plurality of coils, wherein at least one of the plurality of coils comprises a plurality of subcoils arranged adjacent one another, wherein each of the coils and subcoils are connected in series; a common current supply arranged to supply a common current to each coil and each subcoil of the magnet; at least one auxiliary current supply arranged to supply auxiliary currents to each of the subcoils; and a controller adapted to set the respective auxiliary currents within the subcoils, so as to control the effective geometry and/or position of the coil (s) formed from the subcoils .

2. Magnetic resonance apparatus according to claim 1, wherein the magnet comprises at least one pair of coils arranged upon a common axis about a mid-plane, each coil within the pair(s) being separated from the mid-plane by a distance that is substantially equal to or less than half the respective coil radius .

3. Magnetic resonance apparatus according to claim 2, wherein the coils within each pair are positioned upon opposing sides of the working region, wherein the mid-plane passes through the working region.

4. Magnetic resonance apparatus according to claim 2 or claim 3, wherein each of the coils within a pair has an equal number of sub-coils.

5. Magnetic resonance apparatus according to any of claims 2 to 4, wherein a plurality of coil pairs are provided, at least some of which comprise sub-coils.

6. Magnetic resonance apparatus according to any of claims 1 to 5 , wherein the magnet ^• comprises a primary coil and two correction coils, the correction coils each comprising subcoils.

7. Magnetic resonance apparatus according to claim 6, wherein the primary coil is a solenoid.

8. Magnetic resonance apparatus according to any of the preceding claims, wherein the controller is adapted in use to reduce any magnetic inhomogeneities in the working volume resulting from errors in the geometry and/or relative position of the coils with respect to a predetermined arrangement.

9. Magnetic resonance apparatus according to any of the preceding claims, wherein at least three subcoils are provided, so as to control the effective geometry and/or position of the coils in two substantially orthogonal directions.

10. Magnetic resonance apparatus according to claim 9, wherein four sub-coils are provided.

11. Magnetic resonance apparatus according to claim 10, wherein the subcoils are arranged in a two-by-two array when viewed in . cross-section.

12. Magnetic resonance apparatus according to any of the preceding claims wherein each sub-coil is provided with an auxiliary current supply.

13. Magnetic resonance apparatus according to claim 12, wherein the amount of current from the common current supply is greater than that supplied by the respective auxiliary current supplies.

14. Magnetic resonance apparatus according to any of the preceding claims wherein, when the subcoils are superconducting, each subcoil is provided with a switching device .

15. A method of controlling magnetic resonance apparatus, the apparatus comprising :- a magnet for generating, in a working region, a magnetic field of sufficient homogeneity to permit magnetic resonance signals to be obtained from a target material in the working region when in use, the magnet having a plurality of coils, wherein at least one of the plurality of coils comprises a plurality of subcoils arranged adjacent one another, wherein each of the coils and subcoils are connected in series; a common current supply arranged to supply a common current to each coil and each subcoil of the magnet; at least one auxiliary current supply arranged to supply auxiliary currents to each of the subcoils individually; and a controller adapted to set the respective auxiliary currents within the subcoils; the method comprising operating the controller so as to set the currents in at least the subcoils in order to control the effective geometry and/or position of the coil (s) formed from the subcoils.

16. A method according to claim 15, wherein the method further comprises, monitoring the magnetic field produced by the magnet when in use, comparing the monitored magnetic field with a desired magnetic field configuration, and modifying the magnetic field in accordance with the desired configuration by setting the current in at least the subcoils accordingly.

17. A method of selecting currents to be applied to a plurality of subcoils in a method according to claim 15 or claim 16, the subcoils having a predetermined arrangement so as to generate a resultant magnetic field that is substantially the same as that produced by an equivalent hoop of predetermined form, the method comprising :- a) calculating the magnetic field and magnetic field gradients produced by a predetermined current when applied to the hoop ; b) for each subcoil, calculating the magnetic field and magnetic field gradients produced by selected currents when applied to the subcoils; c) selecting final currents for the subcoils by iteratively repeating step (b) , evaluating an error function representing the difference between the calculated values in steps (a) and (b) and selecting currents so as to minimise the error function.

18. A method according to claim 17, wherein the error function is wherein B_{n >} ± is the nth

derivative of the magnetic field for the ith coil, B_n,o is the nth derivative of the magnetic field for the hoop and r₀ ⁿ is the radius of the hoop.

19. A method according to claim 17 or claim 18, wherein the magnetic field gradients in the z direction are given by repeated d ,i.f«feren oJ f^δ2 f^fl2 r²dr dw

— wi,t,h (r² +{z-w) Y respect to z, wherein the current densities are given by J_i = x,-J₀ where J₀ is a selected engineering value, wherein the coil dimensions are given by d - where Ar is the

aspect ratio of the coil c = Ar - — , and wherein the 2 minimisation is performed by adjusting the value of x_t about a mean value of unity.