The harmonic errors in the field of an as-built magnet divide into purely axial variations (axisymmetric zonal harmonics, which are accompanied by radial variations dependent on the elevation в from the z axis, but independent of ф) and radial variations (tesseral harmonics, which depend on ф, where ф is the angle of azimuth in the x-y plane).

In order to compensate for the presence of various un­wanted harmonic errors in the center field of the as-built coils, additional coils capable of generating the opposite har­monics are applied to the magnet. For each set of n and m in the associated Legendre functions, a current array can be

Figure 7. A set of axial shim coils for harmonic correction up to B(3,0). These coils generate small harmonics of 4th order and higher.

designed in the form of a set of arcs of varying azimuthal extent and symmetry and with various positions and extents along the z axis. The magnitude of the harmonic field that an array generates can be controlled by the current. This is the principle of variable harmonic correction for both MRI and NMR magnets. (Correction by means of ferromagnetic shims is not variable.)

The shimming of the unwanted harmonics is a process in two independent parts. First, there is the design of as many sets of coils as are needed to generate the compensating har­monics. Second is the measurement of the actual field errors to determine the magnitudes of the various harmonic compo­nents and the application of currents to the previously de­signed coils to provide the compensation. In fact, because su­perconducting shims must be built into the magnet prior to installation in the cryostat and cooldown, the range of har­monic errors in the field of the as-built magnet must be largely anticipated. Typically it might be assumed that the level of harmonic error decreases by a factor of three for each unit increase in n or m. Therefore, as a rough guide it has been found that compensation of up to B(3,0) for the zonal harmonics and up to B(2,2) for the tesseral harmonics is satis­factory in most cases for the superconducting shims of small bore NMR magnets. There will also be a set of room tempera­ture shims in a high resolution NMR system. Those will com­pensate for errors typically up to B(6,0) and B(3,3) in many cases. Typically there may be up to 28, but exceptionally up to 45 independent shims in all. They will be constructed ac­cording to a different principle from the superconducting shims. The shimming of MRI magnets is accomplished by cur­rent shims, typically up to n = 3 and m = 2, and by ferromag­netic shims.

Superconducting Axial Shims

These will be simple circular coils combined in groups so as to generate a single harmonic only (13). Thus, a coil to generate B(3,0) must generate no B(1,0) nor B(5,0). Because the super­conducting shim coils need to generate only a small fraction of the field due to the main coil, they generally need only com­prise one to three layers of conductor. For that reason the harmonic sensitivities can be calculated directly from Eqs. (4) and (5). A set of axial shims providing correction of B(n,0) harmonics for n = 1 through 3 are shown in Fig. 7. Note that,
for a fixed linear current density, only the angles defining the start and end of each coil are needed, together, of course, with the current polarities, either side of the center plane of the magnet, odd for n = 1, 3, 5, . . . and even for 2, 4, 6, . . ..

The set of coils illustrated in Fig. 7 generate negligible har­monics above the third order, B(3,0). The individual coils of each harmonic group are connected in series in sets, there being in each set enough coils to generate the required axial harmonic but excluding, as far as is practical, those harmon­ics that are unwanted. Thus, in the figure, coils labeled 2 gen­erate second-order B(2,0) but no fourth order. However, they do generate higher orders. The first unwanted order is B(6,0) but that is small enough that it may be neglected. So also with all higher orders because the denominator in the expres­sions of Eqs. (4) and (5) strongly controls the magnitude of the harmonic. Also illustrated in the figure is the effect on harmonic generation of the angular position of a circular cur­rent loop. Each of the dashed lines lies at the zero position of an axial harmonic. Thus, at an angle of 70.1° from the z axis, the B(4,0) harmonic of a single loop is zero. Two loops car­rying currents of the same polarity and suitable magnitude may be located on either side of the 70.1° line to generate no fourth-order harmonic yet generate a significant second order harmonic. Similarly, a coil for the generation of only a first order axial harmonic is located on the line for zero third or­der. The zero first-order harmonic line is at 90°, the plane of symmetry. In order therefore to generate a third order with no first, two coils must be used, with opposing polarities. The coils are all mirrored about the plane of symmetry, but the current symmetries are odd for the odd harmonics and even for the even harmonics. The loops may be extended axially as multiturn coils while retaining the property of generating no axial harmonic of a chosen order, if the start and end angles subtended by the coils at the origin are suitably chosen.

Figure 8. Schematic of a set of radial shim coils for correction of a B(2,2) harmonic showing the positioning necessary to eliminate B(4,2) and B(4,4).


Superconducting Radial Shims

The radial shims are more complex than those for purely axial harmonics because the finite value of m requires a 2m – fold symmetry in the azimuthal distribution of current arcs, the polarity of current always reversing between juxtaposed arcs in one z plane (6,9). For instance, m = 2 requires four arcs, as shown in Fig. 8. However, as for m = 0, the set of current arcs shown in Fig. 8 will generate B(n, m), where n is 2, 4, 6, etc., or 1, 3, 5, etc., depending on even or odd current symmetry about the z = 0 plane. So, the positioning of the arcs along the z axis is again crucial to the elimination of at least one unwanted order, n. Fortunately, the azimuthal sym­metry generates unique values of the fundamental radial har­monic m. (Eight equal arcs cannot generate an m = 2 har­monic.) However, depending on the length of the arc, higher

The principles described earlier can be applied both in the design of shim coils and in the selection of main coil sets. A further observation from the zero harmonic lines of Fig. 7 is that the higher harmonics reverse sign at angles close to the plane of symmetry of the system. This implies that, to pro­duce single, high-order harmonics, coil positions close to the plane of symmetry must be chosen because the other coil loca­tions where the sign of the harmonic reverses are too far from the plane of symmetry to be usable; the coils lying a long way from the plane of symmetry generate weak high-order har­monics.

radial harmonics may be generated. For the shim coil config­uration of Fig. 8, the first unwanted radial harmonic is m =

6. The higher tesseral harmonics are much smaller than the fundamental because of the presence in the expression for the field of a term (r/r0)n. Generally, the arc length is chosen to eliminate the first higher-degree radial harmonic. As an ex­ample, if the arc length of each shim coil shown in Fig. 8 is 90° the B(6,6) harmonic disappears. The B(10,10) harmonic is negligible.

The superconducting shims are almost invariably placed around the outside of the main windings. Although the large radius reduces the effective strength of the harmonics they generate, the shim windings cannot usually be placed nearer to the center of the coil because of the value of winding space near the inner parts of the coil and because of the low critical current density of wires in that region due to the high field. A comprehensive treatment of shim coil design may be found in Refs. 6 and 9. Those references also include details of su­perconducting coil construction. It should be noted, however, that some expressions in Ref. 6 contain errors.

Ferromagnetic Shims

Ferromagnetic shimming is occasionally used in high field, small bore NMR magnets, but its principal use is in MRI magnets. It is in that application that it will be described. The principle invoked in this kind of shimming is different from that of shim coils. The shims now take the form of dis­crete pieces of ferromagnetic material placed in the bore of the magnet. Each piece of steel is subjected to an axial mag­netizing field at its position sufficient to saturate it. It then generates a field at a point in space that is a function of the mass of the shim and its saturation magnetization Bs with little dependence on its shape. For ease of example, a solid cylinder of steel will be assumed. The axis of the cylinder is in line with the field, as shown in Fig. 9. (In Fig. 9 the axis labeled z is that of the shim, not that of the MRI magnet itself. In fact, the shim will usually be placed at the inside surface of the bore of the MRI magnet.)

The field B, caused by the ferromagnetic shim, contains both axial and radial components. The axial component Bz is the correcting field required, and it adds arithmetically to the field of the magnet. The radial component adds vectorially to the field and produces negligible change in the magnitude of

Figure 9. Field vectors generated by a ferromagnetic shim in the bore of an MRI magnet. Bz adds arithmetically to the main field; Бг adds vectorially and so has negligible influence on the field.

the axial field. Therefore, only the axial component of the shim field must be calculated. If the saturation flux density of the shim is Bs, the axial shim field is given by


Bz = BsV[(2 — tan2 n)/(tan2 n + 1)5/2]/(4nz3)

where V is the volume of the shim and z and ^ are as shown in Fig. 9.

The practical application of ferromagnetic shims involves the measurement of the error fields at a number of points, and the computation of an influence matrix of the shim fields at the same points. The required volumes (or masses) of the shims are then determined by the inversion of a U, W matrix, where U is the number of field points and W is the number of shims. In an MRI magnet, the shims are steel washers (or equivalent) bolted to rails on the inside of the room tempera­ture bore of the cryostat. In the occasional ferromagnetic shimming of an NMR magnet, the shims are coupons of a magnetic foil pasted over the surface of a nonmagnetic tube inserted into the room temperature bore or, if the cryogenic arrangements allow, onto the thermal shield or helium bore tube. As in the design of the magnet, linear programming can be used to optimize the mass and positions of the ferromag­netic shims (e. g., to minimize the mass of material).

Resistive Electrical Shims

The field of an NMR magnet for high resolution spectroscopy must be shimmed to at least 10-9 over volumes as large as a 10 mm diameter cylinder of 20 mm length. If, as is usually the case, substantial inhomogeneity arises from high-order harmonics (n and m greater than 3), superconducting shims are of barely sufficient strength. This arises because of the large radius at which they are located, at least in NMR mag­nets (e. g., in the regions x and y of Fig. 6). In general, the magnitude of a harmonic component of field generated by a current element is given by



r" + 1 /r^ + 1

where r is the radius vector of the field point, and p0 is the radius vector of the source. Thus, the effectiveness of a remote source is small for large n.

In order to generate useful harmonic corrections in NMR magnets for large n and m, electrical shims are located in the warm bore of the cryostat. Although in older systems those electrical shims took the form of coils tailored to specific har­monics, modern systems use matrix shims. Essentially, the matrix shim set consists of a large number of small saddle coils mounted on the surface of a cylinder. The fields gener­ated by unit current in each of these coils form an influence matrix, similar to that of a set of steel shims. The influence matrix may be either the fields produced at a set of points within the magnet bore, or it may be the set of spherical har­monics produced by appropriate sets of the coils.

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