**6. Design example of high homogeneous magnet**

A high homogeneous magnet system is the most important and expensive component in an MRI or NMR system. A superconducting magnet with the distribution of coils in single layer, two layers and even more layers is the best solution to achieve the high magnetic field strength and homogeneity requirements. The challenge for designing a MRI magnet system is to search the positions and sizes of coils to meet the field strength and homogeneity over the interesting volume and stray field limitation.

Superconducting Magnet Technology and Applications 99

(2)

(4)

*th* solenoid,

(1)

Minimum:

Minimum:

homogeneity level for design.

and easy to fabricate.

**6.2. Design cases** 

1

*i*

*I*

Subject to:

1

*i V* =

Subject to: 2 2

*6.2.1. 1.5 T actively shielded symmetric solenoid MRI* 

*i*

max

here, the *Ncoils* is the number of discretized solenoids, *Vi* is the volume of the *i*

*zstray rstray*

+ ≤

*B B Gauss*

/( )

≤ η

*I Ic B*

*Ncoils*

=

*i i*

0 0

(3)

*r I* +

m

*ax*

max( ) min( ) / ( )

− ≤

*zdsv zdsv zdsv*

*B B mean B H*

5

*Bzdsv*, *Bzstray* and *Brstray* are the magnetic field on DSV region and stray field ellipse region, respectively. *η* is the current margin, *B0* is the target field over the DSV and *H* is the

Many cases were studied by this hybrid algorithm, and the results show the method is flexible and efficient for the first LP stage which took about 5 minutes and the second NLP stage which took about 30 minutes, respectively. The resultant coil distributions were simple

The actively shielded symmetric MRI system with the length of 1.15 m, the central magnetic field of 1.5 T and the field quality of 10 ppm over 500 mm DSV and the radial inner and outer radius of 0.40m and 0.80 m, the stray field of 5 Gauss outside the scope of an ellipse with axial radius of 5 m and radial radius of 4 m, and the current margin of 0.8. The *Nr*, *Nz*, *Nd*, and *Ns* were set as 40, 40, 51 and 51, respectively. The current map by LP and the final actual coils sizes and positions are shown in Fig. 13. The coils distribution with two layers, the inner layer with four pairs of positive and one pair of negative current direction coils for

*A IB B A I Gauss*

− ≤ε

I

The current map with sparse nonzero clusters were calculated by first LP stage, the positions of nonzero clusters can be discretized into several solenoids with the size of inner radius (*rinner*), outer radius (*router*), and the z position of two ends (*zleft*, *zright*). Secondly, a NLP algorithm was built up, and the objective function and constraints of the algorithm are similar to the LP stage, and added current margin constraint into the algorithm which based on the maximum magnetic field within the superconducting coils and the relationship of critical current and magnetic field. The NLP mathematical model is formulated as following:

 ≤ ≤

\* \* 5

*d s*

*Nz Nr*

### **6.1. Mathematical model for a hybrid optimization algorithm**

The parameters of the system, including length, inner and outer radius of feasible region of coils, radius of DSV region and the axial and radial radius of 5 Gauss stray field line, are predetermined by designer based on the actual applications. Fig. 12 illustrates an example for the design of a symmetric solenoid magnet system. The required parameters of the feasible rectangular region for arrange coils are inner and outer radius (*rmin*, *rmax*) and the length (*L*), the interesting imaging volume is commonly sphere and the stray field is limited to smaller than 5 Gauss outside the scope of an ellipse. A hybrid optimization algorithm by combination of Linear Programming (LP) and Nonlinear Programming (NLP) was developed by IEECAS. This approach is very flexible and efficient for designing any symmetric and asymmetric solenoid magnet system with any filed distribution over any shape volume.

**Figure 12.** The region of feasible coils, interesting volume and stray field

Firstly, the feasible rectangular region can be meshed as 2-D continuous elements with *Nr* elements for radial direction and *Nz* elements for axial direction, and each element served as an ideal current loop. The surface of the sphere and the ellipse for homogenous filed and stray field limitation were evenly divided into *Nd* and *Ns* parts along the elevator from 0 to π, respectively. The field distribution at all target points including *Nd* and *Ns* points produced by all ideal current loops with unit current amplitude calculated, and the unit current field contribution matrices *Ad* and *As* were formed. A LP algorithm was built up with the objective functions totaling the volume of superconducting wires, the field distributions at all target points and the maximum current aptitude for all current elements were constrained. The LP mathematical model is formulated as following:

Superconducting Magnet Technology and Applications 99

$$\text{Minimum: } \sum\_{i=1}^{Nz + Nr} r\_i \Big| I\_i \Big| \tag{1}$$

$$\text{Subject to:} \begin{cases} \left| A\_d \stackrel{\*}{\*} I - B\_0 \right| \Big/ B\_0 \le \varepsilon\\ \left| A\_s \stackrel{\*}{\*} I \right| \le 5 Gauss\\ I \le \mathbf{I}\_{\max} \end{cases} \tag{2}$$

The current map with sparse nonzero clusters were calculated by first LP stage, the positions of nonzero clusters can be discretized into several solenoids with the size of inner radius (*rinner*), outer radius (*router*), and the z position of two ends (*zleft*, *zright*). Secondly, a NLP algorithm was built up, and the objective function and constraints of the algorithm are similar to the LP stage, and added current margin constraint into the algorithm which based on the maximum magnetic field within the superconducting coils and the relationship of critical current and magnetic field. The NLP mathematical model is formulated as following:

$$\text{Minimum: } \sum\_{i=1}^{\text{Nols}} V\_i \tag{3}$$

$$\text{Subject to:} \begin{cases} \left[ \max(B\_{zdsv}) - \min(B\_{zdsv}) \right] / mean(B\_{zdsv}) \le H\\ \sqrt{B\_{zstray}^2 + B\_{rstray}^2} \le 5 Gauss\\ I / \, Ic(B\_{\text{max}}) \le \mathfrak{n} \end{cases} \tag{4}$$

here, the *Ncoils* is the number of discretized solenoids, *Vi* is the volume of the *i th* solenoid, *Bzdsv*, *Bzstray* and *Brstray* are the magnetic field on DSV region and stray field ellipse region, respectively. *η* is the current margin, *B0* is the target field over the DSV and *H* is the homogeneity level for design.

Many cases were studied by this hybrid algorithm, and the results show the method is flexible and efficient for the first LP stage which took about 5 minutes and the second NLP stage which took about 30 minutes, respectively. The resultant coil distributions were simple and easy to fabricate.

### **6.2. Design cases**

98 Superconductors – Materials, Properties and Applications

the interesting volume and stray field limitation.

shape volume.

**6.1. Mathematical model for a hybrid optimization algorithm** 

**Figure 12.** The region of feasible coils, interesting volume and stray field

constrained. The LP mathematical model is formulated as following:

Firstly, the feasible rectangular region can be meshed as 2-D continuous elements with *Nr* elements for radial direction and *Nz* elements for axial direction, and each element served as an ideal current loop. The surface of the sphere and the ellipse for homogenous filed and stray field limitation were evenly divided into *Nd* and *Ns* parts along the elevator from 0 to π, respectively. The field distribution at all target points including *Nd* and *Ns* points produced by all ideal current loops with unit current amplitude calculated, and the unit current field contribution matrices *Ad* and *As* were formed. A LP algorithm was built up with the objective functions totaling the volume of superconducting wires, the field distributions at all target points and the maximum current aptitude for all current elements were

layer, two layers and even more layers is the best solution to achieve the high magnetic field strength and homogeneity requirements. The challenge for designing a MRI magnet system is to search the positions and sizes of coils to meet the field strength and homogeneity over

The parameters of the system, including length, inner and outer radius of feasible region of coils, radius of DSV region and the axial and radial radius of 5 Gauss stray field line, are predetermined by designer based on the actual applications. Fig. 12 illustrates an example for the design of a symmetric solenoid magnet system. The required parameters of the feasible rectangular region for arrange coils are inner and outer radius (*rmin*, *rmax*) and the length (*L*), the interesting imaging volume is commonly sphere and the stray field is limited to smaller than 5 Gauss outside the scope of an ellipse. A hybrid optimization algorithm by combination of Linear Programming (LP) and Nonlinear Programming (NLP) was developed by IEECAS. This approach is very flexible and efficient for designing any symmetric and asymmetric solenoid magnet system with any filed distribution over any

### *6.2.1. 1.5 T actively shielded symmetric solenoid MRI*

The actively shielded symmetric MRI system with the length of 1.15 m, the central magnetic field of 1.5 T and the field quality of 10 ppm over 500 mm DSV and the radial inner and outer radius of 0.40m and 0.80 m, the stray field of 5 Gauss outside the scope of an ellipse with axial radius of 5 m and radial radius of 4 m, and the current margin of 0.8. The *Nr*, *Nz*, *Nd*, and *Ns* were set as 40, 40, 51 and 51, respectively. The current map by LP and the final actual coils sizes and positions are shown in Fig. 13. The coils distribution with two layers, the inner layer with four pairs of positive and one pair of negative current direction coils for

producing the required magnetic strength and the homogeneity. The outer layer with a pair of negative current direction coils for reduces the stray field strength. The homogeneities and stray field distributions are shown in Fig. 14. The coils parameters are shown in Table I. The operating current density and actual current margin are 148 MA/m2 and 0.7546, the maximum magnetic field and hoop stress are 5.43 T and 145.16 MPa, respectively.

Superconducting Magnet Technology and Applications 101

0.2 0.4 0.6 0.8

The unshielded open biplanar MRI system has a central field strength of 1.0 T, inner and outer radius of 0.0 and 0.90m, lower and upper z positions for two ends of 0.40m and 0.45

0.4

r(m) z(m)

0.45

0.5 -1 0 1 x 104

I(A)

The *Nr*, *Nz*, *Nd*, and *Ns* were set as the same as the design of 1.5 T actively shielded MRI system. A quarter model current map for the LP stage and the coils distribution are shown in Fig. 15. The coils distributions with single layer have six coils, the largest coils with maximum field of 5.26 T are the outermost coils with positive current direction. The coils parameters are shown in Table II. The operating current density and actual current margin

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

r(m)

In this chapter we described the basic physical concepts of magnet design and the classification of magnet, then illustrated the magnet applications in the field of energy science, condensed physics, medical devices, scientific instruments and industry. Electromagnetic design is very important for the design of a magnet system with optimal coils distribution. Numerical methods are the best solution to designing complex field distribution magnet systems. A hybrid optimization algorithm combined with linear

*6.2.2. 1.0 T open biplanar MRI* 

m, and field quality of 15ppm over 450 mm DSV.

**Figure 15.** A quarter model current map and coils distribution

0.35

0.4

0.45

z(m )

0.5

**7. Conclusion** 

**Author details** 

are 150 MA/m2 and 0.74, the hoop stress is 128.2 MPa, respectively.

programming and nonlinear programming was presented and studied.

*Institute of Electrical Engineering, Chinese Academy of Sciences, China* 

Qiuliang Wang, Zhipeng Ni and Chunyan Cui

**Figure 13.** The current map and coils distribution of half model

**Figure 14.** The homogeneity over 500 mm DSV and stray field distribution
