This paper presents a hybrid optimal design of bi-planar coils, which produce uniform fields and field gradients. It treats the coil design issue as a nonlinear programming problem, and seeks the best performance via the optimal algorithm. Target field method is firstly utilized to obtain the initial coefficients and ranges of searching areas in stream functions. Then the magnetic field relative error is defined as the fitness function for minimization, and the particle swarm optimization algorithm is adopted to optimize the above coefficients. It not only overcomes the dependence on the position distribution of target field points, but also reduces required computational dimensions without affecting the pursuit of the ultimate performance. Comparing this work with the traditional target field method, the absolute field error in the region of interest is reduced from 5.09% to 0.95% (Bx coil), 11.96% to 0.35% (Bz coil), 16.54% to 0.57% (dBx/dz coil), and 11.8% to 0.28% (dBz/dz coil). Numerical simulation and experimental results show the validity and significance of the proposed method. It has great improvement in both uniform fields and field gradients, which is beneficial to the perfect realization of nulling undesired magnetic fields.

A novel method to design highly uniform axial coils inside a closed magnetically shielded cylinder (MSC) is presented. The proposed method has two major advantages over existing methods. First, based on the coupling model of coil and MSC with finite permeability and finite thickness, the image method and target-field method (TFM) are introduced to provide a flexible design method for the coil inside the MSC. Second, a backpropagation neural network (BPNN) model and particle swarm optimization (PSO) algorithm are used to predict and optimize the coil parameters, thus reducing the magnetic field deviation caused by the use of discrete wires to approximate the continuous current density. Simulations indicate that this method reduces the magnetic field deviation within the target region from $2.63 \times 10^{-3}$ to $1.13 \times 10^{-4}$ compared with the coils designed using the TFM without consideration of the coupling effect. The magnetic field deviation is reduced from $1.42 \times 10^{-4}$ to $1.13 \times 10^{-4}$ compared with the coils designed in the ideal magnetic conductor. Moreover, the field deviation is reduced to $2.43 \times 10^{-5}$ after optimization using the BPNN and PSO methods. The experimental results also verify the effectiveness of the design method and its practicability in compensating for the residual field of the MSC. The method proposed in this research has major significance for the establishment of practical magnetoencephalography and magnetocardiography measurement systems.

This article presents a design method of cylindrical uniform field coil based on quasi-elliptic function fitting (EFF) method with multiple optimizations. The EFF method is a type of discrete wire techniques aiming at simplifying the coil structure. Combing with particle swarm optimization (PSO) algorithm, the optimized parameters of EFF coil are obtained for highly uniform magnetic field. To further improve the uniformity of magnetic field in the central space, a new regional points selection optimization method is proposed. The calculation results show that after PSO optimization, the absolute maximum relative error in the target region is optimized to 0.098% by EFF coil comparing with modified single saddle coil (3.08%), nest coil (2.4%), and target field method (TFM) coil (0.11%). After regional points selection optimization, the volume proportion of absolute magnetic field relative error less than $1\times 10^{-4}$ increases from 35.24% to 54.38% by EFF coil comparing with TFM coil. In addition, the simple EFF coil structure is verified to be easy to process and keep high performance through experiments. As the EFF coil’s high uniformity and simple structure, the new method has important significance for the miniaturization of atomic sensors.

An analytical model to calculate the magnetic field of triaxial uniform coils inside magnetically shielded cylinders (MSCs) composed of high-permeability materials is proposed. The proposed model offers two major advantages over existing methods. First, the model is not limited to the research on coaxial circular coils, but also can be used to analyze the magnetic field characteristics of saddle coils inside MSCs. Second, based on Green’s function expansion and the ferromagnetic boundary conditions, the image method is introduced to consider the reflection effect of the MSC end cap on the coil, thus enabling the coupling effect between the coil and the MSC to be analyzed. To verify the model’s correctness, simulations and experiments are performed separately. With regard to the magnetic field distributions of the saddle coil and the Lee-Whiting coil within a range of ± 80 mm along their axes, the results show that the maximum relative error between the analytical method and the finite element method results is 0.08% and 0.06%, respectively; the maximum relative error between the experimental values and the analytical values is 0.29% and 0.13%, respectively, thus confirming the validity of the analytical model. The results of this research have far-reaching implications for the research of quantum precision measurement devices and triaxial uniform field coils.

With the deepening research in atomic sensors, the extremely weak magnetic detection has shown great advantages in the biomedical field. To achieve high detection sensitivity, the uniform field coil is used in the magnetic detection system to maintain the uniform and stable working environment for the atomic sensors. This paper presents a design method of asymmetric cylindrical uniform field coil based on the boundary element method (BEM) and optimized by the particle swarm optimization algorithm (PSO). BEM could find optimal coil paths on arbitrary geometries, which is fit for the asymmetric configuration in an optically pumped magnetometer based magnetocardiogram (OPM-MCG) device. However, BEM used in the coil design would cause an ill-posed mathematical problem. A Tikhonov regularization combined with penalty function approach is proposed to solve the ill-posed problem and smooth the coil winding patterns, and the correction weights are optimized by PSO within constraints. The calculation results show that the absolute maximum relative error in the target region is optimized to 0.0124% by the proposed method comparing with target field method (TFM) coil (0.023%). Meanwhile, it illustrates that the high uniformity region is enlarged through the utilization of asymmetric structure. In addition, through experiments, the test performance of the designed coil is verified to be consistent with the theory. As the proposed method contributes a high performance uniform field coil on arbitrary structure, it has important significance in extremely weak magnetic field measuring devices such as magnetocardiogram and magnetoencephalogram.

This article proposes a novel design method for highly uniform radial coils in a magnetic shielding cylinder (MSC) with high permeability. It solves the problem that the uniformity of coils in the MSC decreases because of the coupling effect between the coils and high-permeability materials. The coupling effect between the MSC and coils is analyzed, and the analytical model of the magnetic field generated by saddle coils in an MSC is presented. Furthermore, through the extensive use of the Taylor expansion method, target field information, and particle swarm optimization (PSO) algorithm, the uniformity of target region is improved. The simulation results demonstrate that in the target region, compared with the six nested saddle coils without coupling design, the maximum magnetic-field relative error of the designed coil decreases from $1.05\,\, {}\times {}{10^{-3}} $ to $4.32\,\, {}\times {}{10^{-4}} $ , and the space proportion of magnetic-field relative error less than $1\,\, {}\times {}{10^{-4}} $ increases from 23.39% to 79.26%. The experimental results also have good consistency with the theoretical results, verifying the effectiveness of the proposed method. These results confirm that this method can provide a uniform field in an MSC for the background magnetic-field zeroing of a magnetocardiography (MCG) system.