Introduction

Shim coil systems form an essential component of human MRI scanners, pre-clinical MR systems and NMR spectrometers^1,2. They are also key for any magnetic field generating system in which high field homogeneity is required. In most cases a shim system comprises a set of multiple different coils wound on a cylindrical former, each generating a spatially-varying field that corresponds to a spherical harmonic function. Although shim coils can be formed from distributed wire-paths using methods originally developed for MRI gradient coil design^3,4, in many cases shim coils are still formed from small numbers of simple “building block” coils (loops and saddles), since such coils are simpler to fabricate and bring together into a shim set. Methods for combining these building block coils to produce shim coils that generate a given spherical harmonic, whilst eliminating contamination from as many unwanted spherical harmonic terms as possible were described in a seminal paper¹ by Romeo and Hoult. The resulting zonal and tesseral coil designs have been widely used. Here we revisit the analysis used in designing these coils and demonstrate that using current numerical optimisation approaches it is possible to produce improved zonal coil designs with small numbers of building blocks.

Methods

We consider coils composed of pairs of wire loops separated by distance $$$2d$$$ on the surface of a cylinder of radius $$$a$$$ (Fig. 1). It is well known that the z-component of the magnetic field from such an arrangement is given by
$$B_{z}\left(\rho,z\right)=\frac{2\mu_0 NI}{\pi a}\int_{0}^{\infty}\mathrm{d}k \ k\begin{bmatrix}\cos\left(\frac{kz}{a}\right)\cos\left(\frac{kd}{a}\right) \\\sin\left(\frac{kz}{a}\right)\sin\left(\frac{kd}{a}\right) \end{bmatrix}I_{0}\left(\frac{k\rho}{a}\right)K_{1}(k) \>\>\>\>\> [1],$$.

where $$$ρ$$$ and $$$z$$$ are cylindrical co-ordinates, $$$N$$$ and $$$I$$$ represent the number of turns and current, and the upper and lower terms represent the cases where the current in the two loops flows in the same or opposite senses. Forming a Taylor series about the origin, we can decompose the field into a series of weighted spherical harmonic terms;

$$B_{z}\left(\rho,z\right)=\frac{2\mu_0NI}{\pi a}\begin{bmatrix}b_0(\frac{d}{a})+\left(\frac{\rho^2}{4a^2}-\frac{z^2}{2a^2}\right)b_2(\frac{d}{a})+\cdot\cdot\cdot\\\frac{z}{a}b_1(\frac{d}{a})+\left(\frac{z\rho^2}{4a^3}-\frac{z^3}{6a^3}\right)b_3(\frac{d}{a})+\cdot\cdot\cdot \end{bmatrix} \>\>\>\>\> [2]$$

where
$$b_{2n}(\chi)=(-1)^n\frac{\partial^{2n+1}}{\partial \chi^{2n+1}}\Bigg(\frac{\pi\chi}{2\left(1+\chi^2\right)^{\frac{1}{2}}}\Bigg) \>\>\>\>\> [4].$$
and
$$b_{2n-1}(\chi)=(-1)^n\frac{\partial^{2n}}{\partial \chi^{2n}}\Bigg(\frac{\pi\chi}{2\left(1+\chi^2\right)^{\frac{1}{2}}}\Bigg) \>\>\>\>\> [3] $$

By combining together $$$P$$$ loop-pairs and choosing appropriate values of the number of turns $$$N_p$$$ and separation $$$d_p$$$ for each pair, $$$P$$$ spherical harmonics can potentially be eliminated from the field, producing a coil that predominately generates the desired spherical harmonics - i.e. we ensure that $$$\sum_{p=1}^P N_pb_{l}\left(\frac{d_p}{a}\right)=0$$$ for an appropriate set of $$$l$$$-values. In general there are multiple combinations of $$$N_p$$$ and $$$d_p$$$ values which will null the unwanted harmonics, and an optimal choice from the set of solutions can be made so as to produce a coil of limited length with maximum efficiency per unit resistance and a reasonable number of turns per loop.

Here we designed zonal coils with three pairs of loops, whose performance was compared with previously described coils formed from two loop pairs¹. The addition of one loop pair allowed the nulling of three,rather than two, unwanted spherical harmonics and the additional degrees of freedom were used to design shorter Z2, Z3 and Z4 coils (with a length to diameter ratio less than 1.4) and larger regions over which the generated field follows the desired spatial variation compared with previously described coils¹.

Results

Figure 2a shows the number of turns and loop positions in the Z2 coil formed from three loop pairs, which was designed to null Z0, Z4 and Z6 terms. A minus sign on the number of turns indicates that the current circulates in the opposite sense. Figure 2c, shows the contours of the field generated by the coil, scaled so that $$$B_z=1$$$ at $$$z=a$$$ and $$$\rho=0$$$. Figure 2e shows the field variation with $$$z$$$ on axis (i.e. when $$$\rho=0$$$). The dotted lines mark the extent of the region within which the generated field deviates by less than 5% from the desired form. The positions of the outermost loops ($$$d_3$$$) are also indicated on this plot. Figures 2b, d and f, show similar plots for a previously described Z2 coil¹, designed to null just the Z0 and Z4 terms. The shorter length of the 3-loop-pair coil and the greater extent of the region over which it generates a pure Z2-field is evident. Figures 3 and 4 show similar plots for Z3 and Z4 coils. Figure 5 details the $$$d$$$-values, number of turns per loop, extent of the region of field conformity on axis, efficiency and efficiency scaled by the total number of turns, for each of the coils.

Discussion

The coil lengths are reduced by 20%, 20% and 60% for the Z2, Z3 and Z4 three-loop-pair coils, compared to the corresponding two-loop-pair coils¹. The extent of the region of field conformity is also increased by 20% for the new Z2 and Z3 coils. The comparison for the Z4 coil is harder since the previously described coil produces a small Z0 contribution (Figure 4f), but it is notable that the extent of the region of conformity of the new coil design is 60% of the coil length.

Conclusions

By applying current numerical optimisation approaches to the analysis of shim coils formed from simple building blocks, shorter shim coils with larger regions of conformity have been designed. The focus here is on zonal coils, but similar approaches can be used for tesseral coils.

Acknowledgements

This work is supported by the UKRI EPSRC UK National Quantum Technology Hub in Sensing and Timing - EP/T001046/1.