Design of biplanar shim coils for ultra-low field MRI

Abstract

Ultra-low field magnetic resonance imaging (MRI) is a popular research topic in the field of mobile medical imaging. We construct a 50 mT ultra-low field MRI system for real-time monitoring of intracranial bleeding. The region of interest is a spherical space with a diameter of 200 mm, which is sufficiently large for a brain tissue. The homogeneity of the magnetic field determines the signal-to-noise ratio of MRI directly. First, the distribution of the main magnetic field is measured and expanded by using orthogonal spherical harmonic functions. Second, a harmonic coefficient method is used to design active shim coils of the first two orders, with each coil generating a specific harmonic magnetic field. Lastly, the contour distribution of the stream function is utilized to obtain the winding of the shim coils. Results show that each shim coil can generate a pure harmonic magnetic field within an error of 5%. The magnetic field uniformity is reduced from 137 ppm to 28 ppm by these shim coils.

Keywords

Active shim coils magnetic resonance imaging ultra-low field

1. Introduction

Intracranial bleeding [1] occurs when a blood vessel within the skull ruptures or leaks. This condition can result from physical trauma (during head injury) or nontraumatic causes (during hemorrhagic stroke), such as a ruptured aneurysm. Real-time monitoring [2] at the bedside and early indication of changes in the lesion area play key roles in rescue and treatment. A monitoring system for bedside use must be lightweight and easy to move. High-field magnetic resonance imaging (MRI) cannot satisfy these requirements given the large volume and high cost of this equipment. By contrast, an ultra-low field MRI system [3] is made of permanent magnets and does not demand a cooling system. Therefore, an ultra-low field MRI is appropriate for its mobility, which would be a point-of-care diagnostics tool in the future. An ultra-low field magnetic resonance device has high requirements for magnetic field homogeneity considering its low main magnetic field value (less than 50 mT), thereby affecting the signal-to-noise ratio (SNR) of images directly [4]. Therefore, shimming is essential.

Two major shimming methods, namely, passive and active, can be used [5]. Passive shimming [6,7] is a method of field correction that involves using ferromagnetic materials called shims. This method is a primary approach. We reduce the magnetic field uniformity from 236 ppm to 137 ppm by placing shims in a regular pattern at specific locations along the inner bore of the magnet. However, the magnetic field is significantly changed by the shims in the ultra-low field MRI system. The magnetic field uniformity cannot be reduced to a number of ppm with passive shimming only. By contrast, active shimming [8–10] uses shim coils with minimal DCs to uniform the magnetic field. Turner [11] proposed a target-field method to obtain the current distribution from a target field. Forbes and Crozier [12–14] designed biplanar active shim coils by constraining the distribution of current and selecting the regularization function. Liu [15–17] determined the relationship between current distribution and magnetic field. Corresponding integral equations have been derived according to the Biot–Savart law by setting target-field points and B _z values to calculate current densities. Xia [18] demonstrated that biplanar shim coils can generate all harmonic fields through several specific operations and used physical experiments to validate the proposed approach.

However, no research on shim coils in ultra-low field MRI is found. In this study, the distribution of the main magnetic field is measured first and then expanded by using orthogonal spherical harmonic functions. Second, the harmonic coefficient method based on the target-field method is used to design active shim coils of the first two orders, with each coil generating a specific harmonic magnetic field. Lastly, the contour distribution of a stream function is utilized to obtain the winding of the shim coils.

Fig. 1.

Structure diagram of a biplanar magnet.

2. Magnetic field analysis

A diagram of the biplanar magnet structure is illustrated in Fig. 1. The magnets and shim coils are distributed in two planes. In the biplanar MRI system, the main magnetic field B ₀ is selected in the z-axis direction. In the region of interest (ROI), B _z is larger than B _x and B _y, so B _x and B _y can be ignored. Based on relevant electromagnetic field knowledge, B _z in the rectangular coordinate system is written as [19] $\begin{eqnarray}\displaystyle B_{z}\text{} & =\text{} & \displaystyle A_{00}+2A_{10}z+3A_{11}x+3B_{11}y+3A_{20}[z^{2}-(x^{2}+y^{2})/2]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼12A_{21}xz+12B_{21}yz+15A_{22}(x^{2}-y^{2})+15B_{22}(2xy)+\cdots ∼.\end{eqnarray}$ (1)

B _z consists of constant magnetic field A ₀₀ and the first-, second-, and high-order components of x, y, and z. Table 1 lists the harmonic expressions of the magnetic field in spherical and rectangular coordinates. The harmonic magnetic field is necessary to offset by the opposite field that is generated by the corresponding shim coil.

Table 1

Harmonic expressions of a magnetic field

Order n	Degree m	Coefficient	Rectangular coordinate	Spherical coordinate
0	0	A₀₀	1	1
1	0	2A₁₀	z	rcosθ
1	+1	3A₁₁	x	rsinθcos𝜙
1	−1	3B₁₁	y	rsinθsin𝜙
2	0	3A₂₀	z ² − (x ² + y ² )∕2	r ² (3cos² θ −1)/2
2	+1	12A₂₁	xz	r ² sinθcosθcos𝜙
2	−1	12B₂₁	yz	r ² sinθcosθsin𝜙
2	+2	15A₂₂	x ² − y ²	r ² sin² θ cos2𝜙
2	−2	15B₂₂	2xy	r ² sin² θ sin2𝜙
…	…	…	…	…

Fig. 2.

Measurement system of the magnetic field map. CuSO₄ ⋅ 5H₂O samples are located on the disk, where the angles between the z positive axis and the sample are 18°, 30°, 42°, 54°, 66°, 78°, and 90°. These samples are distributed at different heights.

Fig. 3.

Original uniformity map of the magnetic field before shimming.

The magnetic field intensity is approximately 50 mT, and the ROI is a spherical space of 200 mm diameter. The confirmation that the maximum and minimum values of the magnetic field must appear on the surface of the sphere is theoretically feasible [20]. Therefore, we only measure the magnetic field on the sphere. Figure 2 depicts a magnetic field measurement system that consists of biplanar magnets, a magnetic field measurement disk, a computer, an NMR spectrometer (Magritek, Kea2, New Zealand), and an RF power amplifier (TOMCO, BT00500 ALPHA-S, Australia). The CPMG echo signal of the small sample is measured. The magnetic field is obtained by B = f∕𝛾 (𝛾 is the gyromagnetic ratio). The magnetic field intensity at different positions of the upper hemispherical surface is measured by rotating the measuring disk around the z-axis every 30°. This condition is the same for the other half. The magnetic field values are substituted into Eq. (1), and the first two-order formula of the measured magnetic field is obtained, as presented in Table 2. Figure 3 demonstrates the uniformity map of the measured magnetic field in the ROI in which we define D = (B − B ₀) ×10⁶∕B ₀, where B refers to the measured magnetic field intensity, and B ₀ refers to 50 mT.

Table 2

Harmonic coefficients of the measured magnetic field of the first two orders. The unit of B _z is mT

Harmonics	Coefficient
1	50.08
z	−6.1645 ×10⁻⁷
x	−1.6614 ×10⁻⁷
y	−2.3530 ×10⁻⁶
z ² − (x ² + y ²)∕2	−1.7070 ×10⁻⁷
xz	9.6979 ×10⁻⁸
yz	−1.3391 ×10⁻⁷
x ² − y ²	−6.0774 ×10⁻⁹
2 xy	$-2.1237\times10^{-8}$

Fig. 4.

Geometric structure of biplanar shim coils with field point R (r, θ, 𝜑) and source point R′ (f, 𝛼, 𝜑).

3. Method

We use the harmonic coefficient method on the basis of the target-field method to design active shim coils of the first two orders. The geometry structure of a biplanar shim coil is exhibited in Fig. 4. The distribution of the two coils is at z = ±a, and the maximum radius of the winding area is 𝜌_a. The continuous current density J (𝜌, 𝜓) is distributed in the winding area and can be written as tangential and radial components J = J _𝜓 e _𝜓 + J _𝜌 e _𝜌. e _𝜓 and e _𝜌 correspond to the unit vectors of the 𝜓 and 𝜌 direction. J _𝜓 and J _𝜌 satisfy the current continuity. $\begin{eqnarray}\frac{\partial }{\partial {\rho}}({\rho}J_{{\rho}})+\frac{\partial J_{{\psi}}}{\partial {\psi}}=0.\end{eqnarray}$ (2) The stream functions S (𝜌, 𝜓) can be introduced [21] and used to describe the current density distribution. The contour distribution of the stream functions can represent the actual current winding, and the difference between the stream functions of the adjacent contour lines is the actual current value. $\begin{eqnarray}I=\frac{S_{\max }-S_{\min }}{N},\end{eqnarray}$ (3) where S _min refers to the minimum value of the stream functions, and S _max represents the maximum value on the plane. The number of contour lines is N, and the current in each winding line equals I. Therefore, J _𝜓 and J _𝜌 are described by S (𝜌, 𝜓) as $\begin{eqnarray}J_{{\rho}}=\frac{\partial S({\rho},{\psi})}{{\rho}\partial {\psi}},\quad J_{{\psi}}=-\frac{\partial S({\rho},{\psi})}{\partial {\rho}}.\end{eqnarray}$ (4)

For the coil plane, z = ±a = fcos𝛼, and 𝜌 = fsin𝛼. The stream functions can be set as [20] $\begin{eqnarray}S^{\pm }({\rho},{\psi})=(\pm 1)^{l+k}\cos k{\psi}\mathop{\sum }_{q=1}^{Q}U_{q}s_{q}({\rho}),\end{eqnarray}$ (5) where l and k represent the order and degree of the coil, respectively. “±” represents the upper and lower planes. The expansion base of the flow function s _q(𝜌) can be arbitrary, such as the simplest trigonometric function $s_{q}({\rho})=\sin \frac{q{\pi}{\rho}}{{\rho}_{a}}$ . Thus, the two current density components can be written as $\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle J_{{\psi}}^{\pm }=(\pm 1)^{l+k}\cos k{\psi}\mathop{\sum }_{q=1}^{Q}U_{q}j_{{\psi},q}({\rho})\\ \displaystyle J_{{\rho}}^{\pm }=(\pm 1)^{l+k}\sin k{\psi}\mathop{\sum }_{q=1}^{Q}U_{q}j_{{\rho},q}({\rho}),\end{array}\right.\end{eqnarray}$ (6) where j _{𝜓, q}(𝜌) and j _{𝜌, q}(𝜌) are defined as $\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle j_{{\psi},q}({\rho})=-\frac{\partial s_{q}({\rho})}{\partial {\rho}}\\ \displaystyle j_{{\rho},q}({\rho})=-\frac{k}{{\rho}}s_{q}({\rho}).\end{array}\right.\end{eqnarray}$ (7) U _q is the Fourier expansion coefficient of current. The current distribution in the two planes is the same, and the only difference is the current direction. The magnetic field of the coil has an even (odd) symmetry on the z = 0 plane, and the current direction on both sides is the same (opposite) when l + k is even (odd). In the space where r < f _min = a, the Green function can be expanded as $\begin{eqnarray}\frac{1}{|r-r\prime |}=\frac{1}{f}\mathop{\sum }_{n=0}^{\infty }\mathop{\sum }_{m=0}^{n}{\varepsilon}_{m}\frac{(n-m)!}{(n+m)!}P_{nm}(\cos {\alpha})\left(\frac{r}{f}\right)^{n}P_{nm}(\cos {\theta})\cos [m({\phi}-{\psi})],\end{eqnarray}$ (8) where ${\varepsilon}_{m}=\left\{\begin{array}{@{}ll@{}}1 & m=0\\ 2 & m\neq 0\end{array}\right.$ . If m = 0, then P _n0 = P _n. The joint Legendre function is axisymmetric. According to the Biot–Savart law and $\frac{r-r^{\prime }}{(r-r^{\prime })^{3}}=-{\nabla}\frac{1}{|r-r^{\prime }|}$ , the magnetic field can be written as $\begin{eqnarray}dB=\frac{{\mu}_{0}}{4{\pi}}{\nabla}\frac{1}{|r-r\prime |}\times Jd{\sigma}\prime ,\end{eqnarray}$ (9) where dσ′ = 𝜌d𝜌d𝜓 is the area element of a source point. We only focus on B _z, that is, $\begin{eqnarray}\displaystyle dB_{z}\text{} & =\text{} & \displaystyle \frac{{\mu}_{0}}{4{\pi}}d{\sigma}\prime \mathop{\sum }_{n=0}^{\infty }\mathop{\sum }_{m=0}^{n}{\varepsilon}_{m}\frac{(n-m)!}{(n+m)!}\frac{r^{n}}{f^{n+2}}P_{nm}\bigg[(n+1)J_{{\psi}}\sin {\alpha}P_{nm}(\cos {\alpha})\cos m({\phi}-{\psi})\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -∼J_{{\psi}}\cos {\alpha}\frac{\partial P_{nm}(\cos {\alpha})}{\partial {\alpha}}\cos m({\phi}-{\psi})+mJ_{{\rho}}\frac{P_{nm}(\cos {\alpha})}{\sin {\alpha}}\sin m({\phi}-{\psi})\bigg].\end{eqnarray}$ (10) We define that $\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle C_{nm}=\frac{(n-m)!}{(n+m)!}\frac{1}{f^{n+2}}\\[12.0pt] \displaystyle C_{nm}^{{\rho}}=m\frac{P_{nm}(\cos {\alpha})}{\sin {\alpha}}\\[12.0pt] \displaystyle C_{nm}^{{\psi}}=\frac{1}{\sin {\alpha}}[(n+1)P_{nm}(\cos {\alpha})-(n-m+1)\cos {\alpha}P_{n+1,m}(\cos {\alpha})]\\[12.0pt] \displaystyle D\prime _{nm}=\frac{{\mu}_{0}}{4{\pi}}d{\sigma}\prime {\varepsilon}_{m}C_{nm}\left(C_{nm}^{{\psi}}J_{{\psi}}\cos m{\psi}-C_{nm}^{{\rho}}J_{{\rho}}\sin m{\psi},\right.\\[12.0pt] \displaystyle \qquad \left.C_{nm}^{{\psi}}J_{{\psi}}\sin m{\psi}+C_{nm}^{{\rho}}J_{{\rho}}\cos m{\psi}\right).\end{array}\right.\end{eqnarray}$ (11) The integral of Eq. (10) is obtained as $\begin{eqnarray}B_{z}=\mathop{\sum }_{n=0}^{\infty }\mathop{\sum }_{m=0}^{n}D_{nm}r^{n}P_{nm}(\cos {\theta})\left(\begin{array}{@{}c@{}}\cos m{\phi}\\ \sin m{\phi}\end{array}\right),\end{eqnarray}$ (12) where $\begin{eqnarray}D_{nm}=\frac{{\mu}_{0}}{2}\int _{0}^{{\rho}_{a}}{\rho}d{\rho}C_{nm}\mathop{\sum }_{q=1}^{Q}U_{q}\left(\begin{array}{@{}c@{}}[1+(-1)^{n+m+l+k}]{\delta}_{k,m}(C_{nm}^{{\psi}}j_{{\psi},q}-C_{nm}^{{\rho}}j_{{\rho},q})\\ 0\end{array}\right)^{T}.\end{eqnarray}$ (13) For the shim coil of order l and degree k, only the harmonic field of order n and degree m can be generated, that is, $\begin{eqnarray}r^{n}P_{nm}(\cos {\theta})\cos m{\phi}.\end{eqnarray}$ (14) n and m should be satisfied by using the following formulas: $\begin{eqnarray}\left\{\begin{array}{@{}l@{}}m=k\\ n=k+(l+k)∼\text{mod}∼2+2(i-1),\quad (i=1,2,\ldots ).\end{array}\right.\end{eqnarray}$ (15) A coefficient matrix D can be obtained by writing the specific harmonics, integrating the current density component, and extracting the current coefficient U _q. The matrix element is $\begin{eqnarray}D(i,q)={\mu}\int _{0}^{{\rho}_{a}}{\rho}d{\rho}C_{nk}\left(C_{nk}^{{\rho}}\frac{k}{{\rho}}s_{q}({\rho})-C_{nk}^{{\psi}}\frac{\partial s_{q}({\rho})}{\partial {\rho}}\right).\end{eqnarray}$ (16) Therefore, the magnetic field generated by the shim coil of order l and degree k can be expressed as $\begin{eqnarray}B_{z}=\mathop{\sum }_{n=1}^{\infty }\mathop{\sum }_{q=1}^{Q}U_{q}D(i,q)r^{n}P_{nk}(\cos {\theta})\cos k{\phi}.\end{eqnarray}$ (17) We can set the harmonic coefficient vector C = [C ₁, C ₂, …, C _N]^T to solve the current coefficient matrix U = [U ₁, U ₂, …, U _Q]^T. In the harmonic coefficient vector, C _i =δ _{l, n} b _nm (where b _nm is the harmonic field coefficient of order n and degree m), and N should meet the condition of Q ≥ N ≥ (l − k)∕2 + 1. n =1 represents a linear gradient field, which can be used for space gradient coding and linear shimming. n ≥2 represents a high-order homogenous harmonic field. The matrix equation is obtained as $\begin{eqnarray}D_{N\times Q}U_{Q\times 1}=C_{N\times 1}.\end{eqnarray}$ (18) The current coefficient vector U can be obtained directly through matrix inversion. $\begin{eqnarray}U_{Q\times 1}=D_{N\times Q}^{-1}C_{N\times 1}\end{eqnarray}$ (19) U is substituted to the flow function in Eq. (5), and the final coil distribution can be obtained by using the isoline method for the stream function.

Fig. 5.

Structures and magnetic field distributions of z ² − (x ² + y ²)∕2, y, and xz shim coils. (A) Structure of the z ² − (x ² + y ²)∕2 shim coil. The arrows indicate the direction of the current. This setting also applies to (D) and (G). (B) PCB of the z ² − (x ² + y ²)∕2 shim coil. (C) Magnetic field distribution of the z ² − (x ² + y ²)∕2 shim coil in the ROI. (D) Structure of the y shim coil. (E) PCB of the y shim coil. (F) Magnetic field distribution of the y shim coil in the ROI. (G) Structure of the xz shim coil. (H) PCB of the xz shim coil. (I) Magnetic field distribution of the xz shim coil in the ROI.

4. Result

The winding area is distributed in z = ±130 mm, which has a maximum radius of 𝜌_a = 220 mm. The ROI is a sphere that is located at (0,0,0) with a diameter of 200 mm. We consider z ² − (x ² + y ²)∕2 shim coil an example to describe the design process and result of a biplanar shim coil. In this case, l = 2, k = 0, m = 2, n = 2 i −2, and i = 1, 2, …, N. The shim coil generates T ₀₀, T ₂₀, T ₄₀, and T ₆₀ harmonic fields when N is set to 4; T ₂₀ is the target harmonic field. Therefore, C should be [0, b ₂₀, 0, 0]^T. The coil structure is calculated in accordance with the above-mentioned design method. The z ² − (x ² + y ²)∕2 shim coil is composed of several concentric circle windings and has different radii but the same current directions. The coil magnetic field has an even symmetry for the z = 0 plane, and the upper and lower planar coil structures are exactly the same because l + k = 2 is even. The structures of the z ² − (x ² + y ²)∕2, y, and xz shim coils, the printed circuit board (PCB), and the corresponding magnetic field distributions with 5 A current are displayed in Fig. 5. We position every two shim coil on one PCB to reduce space. X shim coil can be obtained by rotating y shim coil 90°, and yz shim coil can be obtained by rotating xz shim coil 90°.

The coil structure is slightly modified to be a closed-loop circuit before fabrication. Short copper wires are required to connect the circles, which will bias the magnetic field distribution. We establish a real coil structure in ANSOFT and analyze its magnetic field purity to assess the deviation. The analysis results are summarized in Table 3. For example, if the z shim coil is the ideal coil, the coefficient of z harmonic (called the main coefficient) will not be zero, whereas the coefficients of the other harmonics will be zero. Table 3 presents that the coils are not pure. However, the coefficients of the other harmonics are smaller than the main coefficient. Hence, the impurity is only 4.98%.

Table 3
Impurities of a real shim coil structure

Harmonics

Shim coil z x y z ² − (x ² + y ² )∕2 xz yz x ² − y ² 2xy Max impurity (%)

z 3621 42 3 0 0 0 0 68 1.88

x −35 2453 54 2 121 87 7 3 4.93

y 35 143 2429 74 −25 117 0 2 4.82

z ² − (x ² + y ²)∕2 30 53 68 1366 −1 0 −4 1 4.98

xz −5 16 46 1 3171 120 0 1 3.78

yz −55 −12 −25 0 −119 3173 0 −1 3.75

x ² − y ² 20 −38 55 1 0 0 2233 0 2.46

2xy 23 42 −10 0 1 0 0 4254 0.98

	Harmonics
z	3621	42	3	0	0	0	0	68	1.88
x	−35	2453	54	2	121	87	7	3	4.93
y	35	143	2429	74	−25	117	0	2	4.82
z ² − (x ² + y ²)∕2	30	53	68	1366	−1	0	−4	1	4.98
xz	−5	16	46	1	3171	120	0	1	3.78
yz	−55	−12	−25	0	−119	3173	0	−1	3.75
x ² − y ²	20	−38	55	1	0	0	2233	0	2.46
2xy	23	42	−10	0	1	0	0	4254	0.98

Fig. 6.

System for measuring the magnetic field distributions of shim coils.

Fig. 7.

Comparison of the measured and simulated magnetic fields of the z ² − (x ² + y ²)∕2 shim coil in the ROI. (A) Comparison on the x-axis. (B) Comparison on the z-axis. The red dotted lines indicate the measured distribution of the magnetic field that is generated by the coil; the blue solid line represents the simulation result.

The shim coils are fabricated on a 1.6 mm-thick fiber-reinforced 4 round board (250 mm diameter) using a coated Cu. The copper thickness is 70 μm. The magnetic field distribution of B _z is measured by using an FW Bell 8030 gauss meter, and the coil current is supplied by using Agilent 6654A DC power, as illustrated in Fig. 6. The shim coils cannot be driven by the actual shim current given the limitation of the measuring range of the gauss meter. Here, we use the amplified current (5 A) to analyze the magnetic field distribution. Figure 7 presents the magnetic fields of the z ² − (x ² + y ²)∕2 shim coil on the x- and z-axes. The measured and ideal magnetic fields have the same trend and numerical range. The small error is caused by additional copper wires.

Fig. 8.

System for measuring the magnetic field uniformity.

Fig. 9.

Spectrum, FID signal and magnetic field uniformity map before and after shimming. (A) Comparison of spectra. (B) Comparison of FID signal. (C) Magnetic field uniformity map after shimming.

We measure the magnetic field uniformity in the ROI by using a Magritek Kea2 spectrometer (New Zealand), as depicted in Fig. 8. The experimental sample, which is a sphere with a radius of 100 mm, is a mixed solution of 3.6 g/L of CuSO₄ and 1.955 g/L of NaCl. The current of the coils is supplied by an MXB-8-RCU multichannel shim power amplifier. The RF is 2.17 MHz. A CPMG ADD sequence is used to measure the transverse relaxation time T2 of the sample, and the spectrum of the echo signal is obtained. Table 4 lists the shim result. Figure 9 demonstrates the spectrum, Free Induction Decay (FID) signal and magnetic field uniformity map before and after shimming. The magnetic field uniformity is significantly reduced from 137 ppm to 28 ppm.

Table 4

Results of shimming

Shim coil	Current (A)	Voltage (V)	Uniformity (ppm)
/	/	/	137
z ² − (x ² + y ²)∕2	1.05	2.09	102
yz	0.74	2.43	80
z	1.24	2.94	72
xz	0.21	0.69	51
x	0.83	0.84	34
y	0.18	0.37	28

5. Conclusion

This study is conducted on the basis of the research on ultra-low field magnetic resonance brain imaging. The harmonic coefficient method based on the target-field method is used to design biplanar active shim coils, which are expected to improve the static magnetic field uniformity to ensure the SNR. First, the distribution of the main magnetic field is measured and expanded by orthogonal spherical harmonic functions. Second, we design active shim coils of the first two orders. Each coil can generate a specific harmonic magnetic field. Lastly, the contour distribution of the stream function is utilized to obtain the winding of the shim coils. The results show that each shim coil can generate a pure harmonic magnetic field, and the error of the other harmonic magnetic fields is up to 5%. The magnetic field uniformity is reduced from 137 ppm to 28 ppm with these shim coils, thereby indicating that the design is successful and effective. However, improvements remain essential. First, using multiple shim coils consumes the gap or bore space of the magnet. Second, the current when a shim coil works alone is different from that when the shim coil works with other shim coils. These ideas are the points of our future work.

Footnotes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 51677008, 51377182, 51707028, and 11647098), the National Key Basic Research and Development Program (973∼Project) (No. 2014CB541602), and the Fundamental Research Funds for the Central Universities (No. 106112017CDJQJ158834).

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