Michael Packer^{1}, Mark Fromhold^{1}, and Richard Bowtell^{2}

^{1}School of Physics and Astronomy, University of Nottingham, Nottingham, United Kingdom, ^{2}Sir Peter Mansfield Imaging Centre, University of Nottingham, Nottingham, United Kingdom

Shim coils can be formed from combinations of small numbers of simple “building block” coils (loops and saddles). In a seminal 1984 paper, Romeo and Hoult, showed how to combine building blocks to produce coils that generate a given spherical harmonic, whilst eliminating contamination from as many unwanted spherical harmonic terms as possible. By applying updated numerical optimisation approaches to this problem, we show how to produce improved zonal shim coils. Designs of Z2, Z3 and Z4 coils formed from three pairs of loops, which are shorter than previously reported coils while having large regions of conformity are described.

$$B_{z}\left(\rho,z\right)=\frac{2\mu_0 NI}{\pi a}\int_{0}^{\infty}\mathrm{d}k \ k\begin{bmatrix}\cos\left(\frac{kz}{a}\right)\cos\left(\frac{kd}{a}\right) \\\sin\left(\frac{kz}{a}\right)\sin\left(\frac{kd}{a}\right) \end{bmatrix}I_{0}\left(\frac{k\rho}{a}\right)K_{1}(k) \>\>\>\>\> [1],$$.

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 $$$N_p$$$

Here we designed zonal coils with three pairs of loops, whose performance was compared with previously described coils formed from two loop pairs

1. Romeo, F. and D.I. Hoult, Magnet Field Profiling - Analysis and Correcting Coil Design. Magnetic Resonance in Medicine, 1984. 1(1): p. 44-65.

2. de Graaf, R.A. and C. Juchem, B-0 Shimming Technology, in Magnetic Resonance Technology: Hardware and System Component Design, A.G. Webb, Editor. 2016. p. 166-207.

3. Forbes, L.K. and S. Crozier, Novel target-field method for designing shielded biplanar shim and gradient coils. Ieee Transactions on Magnetics, 2004. 40(4): p. 1929-1938.

4. Poole, M. and R. Bowtell, Novel gradient coils designed using a boundary element method. Concepts in Magnetic Resonance Part B-Magnetic Resonance Engineering, 2007. 31B(3): p. 162-175.

**Figure 2** (a,b) Z2 coil configuration for a 3-loop pair and previously described coil^{1}, respectively, showing number of turns and current direction (left) and loop position $$$d_p$$$ (right). All other specifications are detailed in Fig. 5. (c,d) Contour plots of the normalised magnetic field generated by the coils (a,b), respectively, where the desired harmonic is unity at $$$z/a=1$$$. (e,f) Field profile along the $$$z/a$$$-axis with dotted lines showing the region of 5% conformity while $$$\pm d_{3,2}$$$ markers indicate coil length for the configurations (a,b) respectively.

**Figure 5** Table detailing coil specifications showing number of turns, $$$N_p$$$, loop positions, $$$d_p$$$, efficiency, $$$\eta$$$, efficiency scaled by total number of turns in the coil, and z-extent of the region of conformity (<5% deviation) in units of $$$a$$$, for designs previously described by Romeo and Hoult^{1 }and optimised 3-loop-pair coils.