Yaohui Wang^{1}, Qiuliang Wang^{1,2}, Lei Guo^{3}, Zhifeng Chen^{4}, Zhongbiao Xu^{4}, Hongyi Qu^{1}, and Feng Liu^{3}

An improved method was proposed to design superconducting shim coil with smooth rounded corner, which can significantly ease the winding and fabrication, and augment the magnetic field accuracy. A quantitative comparison between the shim coil using the improved strategy and the conventional standard design shows clear advantage.

**Introduction**

**Methods**

The magnetic field in the imaging area of an MRI magnet can be expressed
as a summation of spherical harmonics series^{2}. $${B_z(x,y,z)=\begin{align}a_{00}&+a_{10}z&+a_{20}(2z^2-x^2-y^2)\\{}&+a_{11}x&+a_{21}xz\\{}&+b_{11}y&+b_{21}yz\\{}&{}&+a_{22}(x^2-y^2)\\{}&{}&+b_{22}xy\end{align}}$$ Superconducting shim coil is designed to produce a magnetic field
profile corresponds to a specific order of the spherical harmonics. For
tesseral coil design, saddle loop was typically used as the base geometry, as
shown in Fig. 1(a). Actually, when winding the saddle coil with superconducting
wires, a smooth connection is necessary from the arc section to the straight
section, forming a rounded corner, as illustrated in Fig. 1(b). The rounded
corner makes the magnetic field profile of the shim coil deviates from the
original design target.

It is necessary to eliminate the magnetic field deviation aroused by
rounded corner winding with an integral design strategy, namely taking the
corner directly into consideration during the design process. Here, we consider
the corner as a round/smooth connection and set its inner radius as 0.1 of the
arc length. If the current density of the quasi-saddle coil is *J*_{0}, a discrete-wire format
can be applied to compute the corresponding magnetic field4: $${B_z(r)=\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{N}{\frac{J_0s_idl_j\times(r-l_j)}{|r-l_j|}}}$$

where *s _{i}*
is the cross section of the wire and

**Results**

Taking $$${a_{11}}$$$ shim coil design as an example, the shim coil design with standard saddle coil and corresponding quasi-saddle pattern after winding was illustrated in Fig. 2(a). The radius of the shim coil is 0.277 m with a length no longer than 1.30 m, and the maximum magnetic field deviation constraint is 1%. Fig. 2(b) shows the maximum magnetic field deviation variation with respect to the increasing of the corner radius. Even though in the limit case where the radius is 0, the magnetic field deviation is still drifted from the designed 1% to 1.23%. The winding-induced magnetic field deviation is shown in Fig. 3, where Fig. 3(a) displays the magnetic field deviation of the original coil design and Fig. 3(b) presents that of the wound coil with corner radius 3 cm. It is obvious that the magnetic field area with identical homogeneity has reduced and the maximum magnetic field deviation reaches 1.85%.

The magnetic field deviations of a set of shim coils with a 3-cm winding corner after winding were listed in Table I. During the design with standard saddle geometry, all the coils have a maximum magnetic field deviation constraint 1% and length no longer than 1.30 m.

Based on the proposed design strategy, one- and two-order shim coils were designed, as presented in Fig. 4. All the coils have a maximum magnetic deviation constraint 1% and a corner radius to arc length ratio 0.1.

**Discussion**

**Conclusion**

1. F. Roméo and D. I. Hoult. Magnet field profiling: Analysis and correcting coil design. Magnetic Resonance in Medicine. 1984; 1: 44-65.

2. P. Konzbul and K. Sveda. Shim coils for NMR and MRI solenoid magnets. Measurement Science and Technology. 1995; 6: 1116.

3. Z. Ni, L. Li, G. Hu, C. Wen, X. Hu, F. Liu, et al. Design of Superconducting Shim Coils for a 400 MHz NMR Using Nonlinear Optimization Algorithm. IEEE Transactions on Applied Superconductivity. 2012; 22: 4900505.

4. Y. Wang, F. Liu, X. Zhou, and S. Crozier. Design of transverse head gradient coils using a layer-sharing scheme. Journal of Magnetic Resonance. 2017; 278: 88-95.