Equivalent magnetic dipole method for designing gradient coils of the Halbach magnetic resonance device

Abstract

Based on the equivalent magnetic dipole theory, a method for designing gradient coils of the Halbach magnet is proposed. Minimizing the energy consumption of the gradient coil is set as the optimal object, and the variance between the calculated and expected magnetic fields is taken as the penalty function. In this way, a quadratic optimization problem that cannot easily satisfy the magnetic field constraint condition can be easily transformed into an unconstrained optimization problem. By using the lsqnonlin optimization toolbox of Matlab, the distribution of the stream function on the cylindrical surface is calculated, and the three gradient coils of the Halbach magnet are obtained according to the contour of the stream function. To validate the designed coils, the Biot-Savart law is applied to calculate the distribution of the magnetic field. The results show that the magnetic field distribution agrees well with the design object. The gradient coils are fabricated and the magnetic field distributions in the region of interest are measured. The measured data are almost similar with the calculated data, and the measured gradient uniformity of $x(y)$ and $z$ coils are 4.47% and 2.34%, respectively, which are less than 5% of the commercial gradient coil requirements.

Keywords

Equivalent magnetic dipole method gradient coils magnetic resonance Halbach magnet

1. Introduction

The Halbach magnet [1] performs well without an iron yoke and has the characteristics of a strong magnetic field, low weight, and small size. The magnetic field at the center of the magnet is homogeneous and has a low fabrication cost. Therefore, the Halbach magnet is an ideal portable NMR (nuclear magnetic resonance) device for analysis and imaging as well as presents a popular research direction in online NMR spectroscopy and imaging [2, 3, 4]. In this paper, the application of the equivalent magnetic dipole method for designing gradient coils is studied, and these coils will be used for the Halbach magnet with a special structure (i.e., the main magnetic field is in a radial direction and the gradient coils are placed on the cylindrical surface).

The design of gradient coils is an inverse problem of the electromagnetic field. These coils can be divided into discrete and distributed windings [5]. When discrete windings are designed, the shapes are pre-determined and the position and angle of the windings on the frame are adjusted to meet the given design object using certain algorithms, such as the conjugate gradient descent algorithm [6], simulated annealing algorithm [7, 8], neural network algorithm [9], and fuzzy membership functions algorithm [10]. Meanwhile, when distributed windings are designed, the shapes of the windings are not predefined, the desired magnetic field distribution in the region of interest (ROI) is known and set as a goal, and the structure of gradient coils can be obtained by reverse calculation. Moreover, this method, which shows better performance in designing coils [11], is easily constrained by inductor, energy consumption, and self-shielding conditions. The detailed solution methods include the matrix inversion method [12, 13], the stream function method [14, 15], the target field method [16], the harmonic coefficient method [17], and the equivalent magnetic dipole method (it is similar with the boundary element method) [18, 19]. Each magnet structure has a slightly difference of design method.

Given the radial main field direction of the Halbach magnet, the required magnetic field of the gradient coil differs from the field distribution of the conventional superconducting MRI and the C-type permanent magnet. Moreover, the coil needs to be distributed on a cylindrical surface. However, the above methods are only useful for magnets with an axial main field direction and for coils that are distributed on a flat surface. Meanwhile, the methods for designing gradient coils for the Halbach magnet are only limited to the equivalent magnetic dipole method [19] and the harmonic coefficient method [3]. The equivalent magnetic dipole method has been used by Lopez et al. to design shim coils for superconducting MRI and planar permanent magnets as well [20].

Figure 1.

Structure of the Halbach magnet. (A) 3D structure of the Halbach magnet. The gradient coils are distributed on a white plastic cylinder with a 75 mm diameter inside the Halbach magnet. The magnet is composed of three discrete Halbach rings with a radius of 125 mm and a gap of 12 mm. A single ring consists of 24 magnetic bars, and its main field is placed along the $x$ axis. (B) The real object of the Halbach magnet. Magnetic bars are installed inside the aluminum shell.

The gradient coils in this paper are designed based on the Halbach magnet shown in Fig. 1. This magnet is fabricated for measuring core porosity, and its main field lies along the $x$ -axis. To reduce the magnetization of minerals in the rock, the magnetic field is designed at only 47.89 mT, which corresponds to a magnetic resonance frequency of 2.039 MHz, and this kind of low field NMR has important research value because of its portability [21, 22]. In [19], an equivalent magnetic dipole method is applied via the quadratic programming algorithm, but the magnetic field constraint cannot be easily satisfied, the initial solution cannot be found, and the solution process is eventually interrupted. In this paper, an unconstrained optimization problem is constructed by transforming the magnetic field constraint as a penalty function to avoid the solution from being interrupted. The stream function on the cylinder can be calculated according to the expected magnetic field distribution in the ROI. Afterward, the distributed windings of gradient coils are obtained by contouring the stream function.

2. Equivalent magnetic dipole method

The gradient coil with the current can be treated as a number of current-carrying loops, which in turn can be regarded as magnetic dipoles when one loop is much too small for the area where the coil is located. As shown in Fig. 2, each rectangular arc block on the cylindrical wiring surface is supposed to be a magnetic dipole. Therefore, the magnetic field in the ROI is generated by these dipoles. The core question in this paper is how to derive the structures of gradient coils according to the given magnetic field of gradient distribution.

Figure 2.

Structure of the meshed cylindrical surface of coils. The surface is discretized to $Q$ magnetic dipoles, where dipole $q$ has a thickness of $h$ and arc length of $a$ . $S_{q}$ is the stream function value of element $q$ , and ${\bm{n}}_{q}$ denotes the outer normal direction vector.

2.1 Relationship between the stream function and equivalent magnetization

The magnetization of the single dipole is denoted as ${\bm{M}}({\bm{r}}^{\prime})$ , while its vector lies along the normal direction. Assuming that the equivalent magnetizing current flows in the thin volume of $h$ (the thickness of $h$ is almost negligible), based on relevant electromagnetic field knowledge, the equivalent current surface of the dipole is written as [18]

$\displaystyle{\bm{J}}_{S}({\bm{r}^{\prime}})\approx{\nabla}^{\prime}\times{\bm% {M}}({\bm{r}^{\prime}})h.$ (1)

The relation of current density ${\bm{J}}_{S}$ and stream function $S(r^{\prime})$ can be expressed as [15]

$\displaystyle{\bm{J}}_{S}({\bm{r}^{\prime}})=\nabla\times(S({\bm{r}^{\prime}})% \cdot{\bm{n}}({\bm{r}^{\prime}})),$ (2)

where ${\bm{n}}(r^{\prime})$ is the outward unit normal vector of the current surface.

Therefore, we can write

$\displaystyle{\bm{M}}({\bm{r}^{\prime}})=\frac{S({\bm{r}^{\prime}})\cdot{\bm{n% }}({\bm{r}^{\prime}})}{h}.$ (3)

The structure of the gradient coil can be directly obtained from the contour lines of stream function $S$ . Therefore, the key solution to the problem is to find the value distribution of stream function $S$ .

2.2 Calculation of the magnetic field in the ROI

The cylindrical surface where the gradient coil is located is divided into $Q_{c}$ columns along the circumference and $Q_{r}$ lines along the $z$ axis. A total of $Q=Q_{c}\times Q_{r}$ elements or magnetic dipoles are eventually obtained as shown in Fig. 2. These dipoles are sorted by line from left to right. The number difference between the left and right dipoles is set to 1, while the difference between the upper and lower dipoles is set to $Q$ ${}_{c}$ .

If $a<<|{\bm{r}}-{\bm{r}}^{\prime}|$ and $h<<a$ , then the equivalent magnetic moment produced by a magnetic dipole is computed as

$\displaystyle{\bm{m}}_{q}={\bm{M}}a^{2}h=a^{2}S_{q}{\bm{n}}_{q}.$ (4)

Equation (2) shows that the current distribution can be obtained by calculating the $S_{q}$ of each dipole. The magnetic field generated by the equivalent magnetic dipole moment ${\bm{m}}$ is calculated as [23]

$\displaystyle{\bm{B}}=\frac{\mu_{0}}{4\pi}({\bm{m}}\cdot\nabla)\nabla\frac{1}{% r}=\frac{\mu_{0}}{4\pi}\nabla\left({\bm{m}}\cdot\nabla\frac{1}{r}\right)=-% \frac{\mu_{0}}{4\pi}\nabla\left({\bm{m}}\cdot\frac{\bm{r}}{|r|^{3}}\right).$ (5)

By substituting Eq. (4) into Eq. (5), we can obtain the magnetic field expression of field point ( $x$ , $y$ , $z$ ) along the $x$ direction generated by $Q$ magnetic dipoles. This expression can be written as

$\displaystyle B_{x}(x,y,z)=\frac{\mu_{0}a^{2}}{4\pi}\sum\limits_{q=1}^{Q}{c_{x% ,q}S_{q}},$ (6)

where

$\displaystyle c_{x,q}=[(x-x_{q})^{2}+(y-y_{q})^{2}+(z-z_{q})^{2}]^{-\frac{5}{2% }}\times[2(x-x_{q})^{2}n_{x}-(y-y_{q})^{2}n_{x}-(z-z_{q})^{2}n_{x}+3(x-x_{q})(% y-y_{q})n_{y}+3(x-x_{q})(z-z_{q})n_{z}]$ (7)

and $n_{x}$ , $n_{y}$ , and $n_{z}$ are the components of the outward unit normal vector ${\bm{n}}_{q}$ along the $x$ , $y$ , and $z$ axes, respectively.

3. Optimization

If the design object only contains the gradient of the magnetic field distribution, then the structure of the coils may become too complicated and rough, which has problems of fabricating difficultly and high temperature with current, Therefore, the energy consumption must be minimized to obtain one optimal object. $J_{s,c}$ and $J_{s,r}$ represent the current components along the column and row of one dipole, respectively. When the elements are small enough, $J_{s,c}=(S_{q+1}-S_{q})/a$ and $J_{s,r}=(S_{q+Qc}-S_{q})/a$ . The energy consumption of a magnetic dipole can be computed as

$\displaystyle\Delta P=\frac{\rho}{h}(J_{s,r}^{2}+J_{s,c}^{2})dA=\frac{\rho}{ha% ^{2}}[(S_{q+1}-S_{q})^{2}+(S_{q+Q_{c}}-S_{q})^{2}]\propto[(S_{q+1}-S_{q})^{2}+% (S_{q+Q_{c}}-S_{q})^{2}]$ (8)

The total energy consumption can be expressed as

$\displaystyle P=\sum{[(S_{q+1}-S_{q})^{2}+(S_{q+Q_{c}}-S_{q})^{2}]}.$ (9)

In other words, if the energy $P$ is small, then the difference in the stream function values of the adjacent dipoles must be kept as small as possible. Its physical meaning is that the wiring on the adjacent elements must transit smoothly in order to decrease the temperature rise or the energy consumption. The gradient coils of the Halbach magnet must generate a magnetic field with a gradient distribution along the $x$ axis in the ROI, which is approximately the same as the expected field $B_{\textit{target}}$ . In this case, the energy consumption must also be small enough. The optimal mathematical model is constructed as

$\displaystyle\min:f=\sum{[(S_{q+1}-S_{q})^{2}+(S_{q+Q_{c}}-S_{q})^{2}]}$ (10) $\displaystyle st{\begin{array}[]{*{20}c}\hfil&{B_{x}-B_{\textit{target}}=0}% \hfill\\ \end{array}}$

According to the quadratic programming problem, the initial solution that satisfies the above constraints cannot be easily obtained, thereby interrupting the solution process. Therefore, the constraint problem can be transformed into an unconstrained optimization problem by constructing penalty function. The magnetic field constraint is taken as a penalty function multiplied by the penalty parameter $\lambda$ and added to the optimization function $f$ . Given that the penalty function is close to 0, the value of $\lambda$ must be extremely large to extrude the penalty term. The ROI is divided into $N$ field points, and the objective function is optimized as

$\displaystyle\min:F=\sum\limits_{q=1}^{Q}{[(S_{q+1}-S_{q})^{2}+(S_{q+Q_{c}}-S_% {q})^{2}]}+\lambda\sum\limits_{n=1}^{N}{(B_{x}-}B_{\textit{target}})^{2},$ (11)

where the latter term denotes the sum of squares of the difference between the magnetic field $B_{x}$ and the desired magnetic field $B_{\textit{target}}$ in the ROI, and $B_{x}$ is calculated using Eq. (6). The target field is changed as the designed requirements, such as for the $x$ gradient coil $B_{\textit{target}}=G_{x}x$ , for the $y$ gradient coil $B_{\textit{target}}=G_{y}y$ , and for the $z$ gradient coil $B_{\textit{target}}=G_{z}z$ .

To solve the optimization problem such as that in Eq. (11), the lsqnonlin toolbox of Matlab is used. This toolbox solves the nonlinear function of Eq. (12) as follows by using the least squares method

$\displaystyle\mathop{\min}\limits_{x}\left\|{f(x)}\right\|_{2}^{2}=\mathop{% \min}\limits_{x}(f_{1}(x)^{2}+f_{2}(x)^{2}+\cdots f_{n}(x)^{2}),$ (12)

where $x$ is a vector or matrix, and $f(x)$ is a function of the return vector or matrix. Equation (11) can be transformed into the form of least squares as

$\displaystyle\min:f(S_{q})=\sum\limits_{n=1}^{N}{\left({\sqrt{\sum\limits_{q=1% }^{Q}{[(S_{q+1}-S_{q})^{2}+(S_{q+Q_{c}}-S_{q})^{2}]}+\lambda(B_{x}-B_{\textit{% target}})^{2}}}\right)}^{2}.$ (13)

After solving the optimal values of $S_{q}$ , the contour lines of the stream function are drawn to show the structure of the coil.

The penalty parameter $\lambda$ must also be optimized. The gradient uniformity $\delta$ [4] is set as an optimization index, which is defined as

$\displaystyle\delta=\frac{1}{V}\mathop{\int\!\!\!\int\!\!\!\int}\limits_{\kern% -5.5ptV}\left[\frac{B-B_{\textit{target}}}{B_{\textit{target}}}\right]^{2}dV% \times 100\%.$ (14)

Figure 3.

Structures and magnetic field distributions of the $y$ and $z$ gradient coils. (A) Structure of the $y$ gradient coil. The solid and dashed lines indicate that the current must have a different direction, and the figure presents a plan view of the cylindrical gradient coil where the abscissa is the radius and the ordinate is the height. These settings apply to (C) as well. (B) Magnetic field distribution of the $y$ gradient coil in the ROI. (C) Structure of the $z$ gradient coil. (D) Magnetic field distribution of the $z$ gradient coil in the ROI.

When calculating $\delta$ , the magnetic field $B$ is the $B_{x}$ obtained from Eq. (6), and $B_{\textit{target}}$ represents the desired magnetic field that is set up above. The optimization works as follows: (1) Set a large initial value of $\lambda$ as 10 ${}^{10}$ . (2) Obtain the optimal solution of $S_{q}$ by solving Eq. (13). (3) Substitute $S_{q}$ into Eq. (6) to calculate $B_{x}$ , and then obtain the value of gradient uniformity $\delta$ . (4) Assume the exit condition to be $\delta<\varepsilon$ , where $\varepsilon$ is the required gradient uniformity. If this condition is satisfied, then the solution is interrupted and the current $\lambda$ is deemed as the best value. Otherwise, set $\lambda_{k+1}=a\lambda_{k}$ , and $a$ is usually set as 10 [24]. Afterward, repeat steps (2) to (4). The best $\lambda$ is eventually obtained, and the corresponding optimal stream function values are solved.

4. Experiments and results

The designed gradient coils are located on a cylinder with a radius of 37.5 mm and length of 235 mm. This cylinder is subdivided into 2500 elements, and the ROI is a sphere with a 20 mm radius and 100 target points. Using the above method, the structures of the $x$ , $y$ , and $z$ gradient coils can be obtained.

In the actual design, parameter $\varepsilon$ is set to 3% and then the value of $\lambda$ is optimized to be 10 ${}^{14}$ . Afterward, the $y$ and $z$ gradient coils are calculated. It’s found that when energy consumption is not set as the optimal object, the values of dissipation $P$ of $y$ and $z$ gradient coils are 1.02 $\times$ 10 ${}^{4}$ and 6.06 $\times$ 10 ${}^{4}$ respectively, which are calculated through removing the energy consumption term in front of Eq. (13). Nevertheless, the $P$ values of $y$ and $z$ gradient coils are reduced to 4.49 $\times$ 10 ${}^{3}$ and 9.12 $\times$ 10 ${}^{3}$ after optimization, only 44% and 15% of the dissipation data without optimization. In addition, the calculated coils have more gentle structures. So the relevant algorithm is valid.

Table 1
Performance parameters of the calculated and measured of gradient coils

Parameters	$y$ gradient coil	$z$ gradient coil
Calculated gradient $G$ (mTm ${}^{-1}$ A ${}^{-1})$	5.30	2.55
Calculated gradient uniformity $\delta$ (%)	1.12	0.98
Measured gradient $G$ (mTm ${}^{-1}$ A ${}^{-1})$	5.61	2.69
Measured gradient uniformity $\delta$ (%)	4.47	2.34

Figure 4.

(A) The PCB board of the $y$ gradient coil. (B) The PCB board of the $z$ gradient coil.

Figure 5.

System for measuring the magnetic field distributions of gradient coils.

The distributions of the magnetic field of the coils in the ROI are verified by applying the Biot-Savart law, and the $x$ gradient coil can be obtained by rotating the $y$ gradient coil by 45 ${}^{\circ}$ . The structures of the $y$ and $z$ gradient coils and the corresponding magnetic field distributions with 5 A current (consistent with the following measurement current) are shown in Fig. 3. The $y$ and $z$ gradient coils have the characteristics of the $y$ and $z$ gradients, respectively.

A flexible PCB board made of copper foil pasted on the substrate film and covered with an insulating layer is flexible, thin and easily processed. The above designed gradient coils are realized by using flexible PCB boards, and in this way the fabricated coils can be easily wound on a hollow plastic cylinder with a 75 mm diameter. The $y$ and $z$ gradient coils are shown in Fig. 4.

Using the FW.Bell8030 Gauss Meter made in the US, we measure the magnetic field distribution of $B_{x}$ with a current of 5 A of $y$ and $z$ gradient coils in the ROI. The measurement system is shown in Fig. 5. The current of the coil is supplied by Agilent 6654 A DC power, the Gauss Meter is controlled by the stepper motor to move inside the coil, and the magnetic field values are measured and transmitted to the computer. The diagrams of the magnetic field along the axial planes of $y$ and $z$ gradient coils in the ROI are shown in Fig. 6.

Figure 6.

Diagrams of the magnetic field distributions along the axial planes of $y$ and $z$ gradient coils in the ROI. (A) The magnetic field distribution in the XOY plane of the $y$ gradient coil. (B) Comparison of the measured and calculated magnetic fields of the $y$ gradient coil. (C) The magnetic field distribution in the XOZ plane of the $z$ gradient coil. (D) Comparison of the measured and calculated magnetic fields of the $z$ gradient coil.

As can be seen in Fig. 6, the measured magnetic field distributions are almost similar to the calculated ones. Therefore, the fabricated coils meet the design requirements. The calculated and measured performance parameters of the coils are shown in Table 1. The measured gradients of the $y$ and $z$ gradient coils are 4.8% and 5.5% different from their corresponding calculated gradients, respectively. Nevertheless, a relatively large difference in gradient uniformity is observed, which may be attributed to manufacturing tolerance and measurement error; however, the gradient uniformity of each coil satisfies the commercial requirement of less than 5% [5].

The magnetic field distributions of fabricated PCB coils are worse than those of designed coils because while drawing PCB board coils, the structure of the spiral plane is used to make sure that the current in the coil flows smoothly, thereby leading to the unavoidable asymmetry of coils. When the PCB boards are wound on the cylinder, a fabrication error may take place. However, the difference in the magnetic filed distributions of the measured and calculated gradients are within the acceptable error.

5. Conclusions

Given that the main magnetic field of the Halbach magnet lies along the radial direction, the methods for designing a gradient coil for superconducting nuclear magnetic resonance cannot be directly used. Therefore, based on the equivalent magnetic dipole method, energy consumption is proposed as the optimization object, and the quadratic problem with the magnetic field constraint is transformed into an unconstrained optimization problem that uses the difference between the calculated and expected magnetic fields as a penalty function. The lsqnonlin optimization toolbox of Matlab is used to solve the stream function that is distributed on the cylindrical coil. The structures of the gradient coils are then obtained by applying the contouring stream function. The proposed method is simple, efficient, general, and can be used for designing high-order shim coils for nuclear magnetic resonance.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 51677008, 51377182, 51707028 and 11647098) and the Fundamental Research Funds for the Central Universities (106112017CDJQJ158834).

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Equivalent magnetic dipole method for designing gradient coils of the Halbach magnetic resonance device

Abstract

Keywords

1. Introduction

Table 1 Performance parameters of the calculated and measured of gradient coils

Footnotes

Acknowledgments

References

Table 1
Performance parameters of the calculated and measured of gradient coils