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Scientific Reports volume 14, Article number: 24650 (2024 ) Cite this article Refrigeration Controller
Currently, solenoids are extensively utilized in various research fields due to their flexibility of fabrication and high magnetic field strength. However, the internal magnetic field of the solenoid itself exhibits some non-uniformity defects, which limits its application in some domains. This article proposes a novel single winding tightly wound solenoid structure with an improved magnetic field uniformity. To optimize the magnetic field near the aperture of the conventional solenoid, an auxiliary solenoid with a gradually changing diameter is included at each end of the solenoid. By adjusting different parameters of the auxiliary solenoid, the edge effect of the solenoid was reduced, and its overall magnetic field uniformity was improved. The optimization of the auxiliary solenoids is achieved through GA-KAN network techniques. Finite element simulation results with the optimized parameters show that the proportion of uniform areas can be improved by more than five times. The research results are a reference for high-precision electromagnetic sensing applications based on solenoids.
Solenoids play a crucial role in electrical engineering due to their straightforward fabrication and high magnetic field strength. As a fundamental and widely used electromagnetic device, solenoid is rooted in the development of electromagnetic principles and engineering technology. It combines coil winding, the use of iron cores (if any), and modern materials science and manufacturing technology to form a multifunctional component that can convert electromagnetic energy efficiently and accurately. Adjusting the magnitude and direction of the current flowing into the solenoid allows a highly concentrated and directional magnetic field to be generated and controlled. This controllable magnetic field has a wide range of applications in medical equipment1, electronic transformers2, electromagnetic braking3,4, electromagnetic levitation, and so on. As an electromagnetic force generator5,6,7, solenoids can convert electrical energy into mechanical energy and achieve linear or rotational motion. This feature makes it a core equipment component such as electric motors, solenoid valves, electromagnetic brakes, and linear actuators. Solenoids are also used in various electromagnetic sensors and measuring instruments (e.g. current transformers, flux gate sensors, etc.) for non-contact measurement of physical quantities such as current and magnetic field strength, providing important data for power monitoring, geophysical exploration, material analysis, and other fields. In addition, solenoids also have many applications in fluid dynamics8,9. In summary, solenoid has became an indispensable basic component in modern industrial society, serving multiple important fields such as scientific research, energy, transportation, communication, and medical treatment. However, traditional single dense wound solenoids have some drawbacks, such as electromagnetic interference10,11,12, high losses13,14,15, large volume16, and edge effect problems17,18,19,20. It is worth noting that the edge effect of the solenoid becomes more pronounced with increasing length, which poses a challenge to its application, especially in current sensing systems21. To address these limitations, it is necessary to enhance the structure of the solenoid.
In recent years, thanks to the extensive application of magnetic navigation technology in fields such as medical equipment, robotics, and automation, many scholars have been attracted to research on improving the uniformity of magnetic fields. Reference22 describes a system that integrates square coils and Maxwell coils to achieve remote magnetic navigation of permanent magnet micro-robots without needing actual coil movement. In addition, the system can estimate the speed and acceleration of micro-robots based on their changes over time, which is crucial for feedback control systems in magnetic navigation. Reference23 proposed a control strategy based on orthogonal transformation, which can effectively adjust the posture and steering of the robot by controlling a single variable. This innovation has opened new avenues for enhancing the human-computer interaction experience. Reference24 introduces a three-axis square coil specifically designed for aviation applications. Reference25 presents a three-axis square coil structure with four coaxial magnetic shielding functions. To evaluate the effectiveness of limited magnetic shielding, multiple reflections theory is applied to establish an analytical model. The experimental results show that the design surpasses traditional systems in terms of magnetic shielding performance and maintains a high degree of accuracy. It can be seen that most research on improving magnetic field uniformity is based on Maxwell coils and Helmholtz coils, while for solenoids, researchers mainly optimize wire layout to enhance the solenoid and minimize resistance and electromagnetic radiation26,27,28,29,30. However, the solenoid structure has undeniable advantages compared to Maxwell coils and Helmholtz coils31,32,33,34. The magnetic field generated by the solenoid in the axial direction is more uniform, making it particularly suitable for placing slender samples. This uniformity is crucial for many physical experiments. Solenoids can generate relatively high magnetic field strength, especially along the axis. There is a good linear relationship between the current in the solenoid and the generated magnetic field, which means that the magnetic field strength can be adjusted by precisely controlling the current. In addition, the solenoid can also generate AC/DC magnetic fields, making it more flexible in experimental design.
This article proposes a single dense winding solenoid with an improved structure to meet the requirements of modern electrical equipment for high efficiency, compactness, and reliability. The traditional single densely wound solenoid is typically made of copper wire. It has a relatively simple layout, but it has some limitations and exhibits a noticeable boundary effect at the position of the solenoid mouth. This article adopts a new design method to enhance the structure of the solenoid. The improved solenoid structure is simulated using finite element analysis in COMSOL Multiphysics 5.6 software, and based on this simulation, an artificial GA-KAN network algorithm is utilized to optimize the structural parameters of the enhanced solenoid. The novel scheme weakened the edge effect greatly, and the solenoids excellent candidates for high-precision electromagnetic sensing applications.
Taking a single-layer densely wound copper wire solenoid as an example, assuming that the total number of turns of the solenoid is N, the DC current through the solenoid is I, the radius of the cross-section of the solenoid is a, and the length of the solenoid is 2b, the coil density of the solenoid is n = N/2b. In Fig. 1, a cylindrical coordinate system (r,φ,z) is established, with the geometric center of the solenoid as the origin of coordinates, and the transverse central axis of the solenoid is the z-axis. Among them, the direction of the magnetic field generated by the solenoid remains consistent with the positive direction of the z-axis. When energized, the line current on the solenoid wire is equal to the surface current along the solenoid. This article adopts the CGS unit system.
Schematic diagram of a single densely wound solenoid.
In as shown in Fig. 1a solenoid plane take any point Q (a,φ’,z’) and the solenoid tube in space any point P (r,φ,z). According to the Biot-Savart law, it is easy to obtain that the magnetic induction intensity at P is equal to35
Where \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{J_{S} }} (Q) = nI( - \sin ^{\prime}\varphi \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{i} + \cos ^{\prime}\varphi \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{j} )\) , \(D = \left| {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{R} (Q,P)} \right| = [r^{2} + a^{2} - 2ra\cos (\varphi - \varphi ^{\prime}) + (z - z^{\prime})^{2} ]^{{\frac{1}{2}}}\) , \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{R} (Q,P) = (r\cos j - a\cos j^{\prime})\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{i} + (r\cos i - a\cos i^{\prime})\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{j} + (z - z^{\prime})\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{k}.\)
In the above formulation, µ0 represents the vacuum permeability and D represents the vector path modulus. The axial component, radial component and circumferential component of the solenoid magnetic field can be obtained by formula (1). The axial component of the solenoid magnetic field is.
The radial component of the magnetic field of the solenoid is
Where \(f\left( g \right)=(\frac{2}{g} - g)K(g) - \frac{2}{g}E(g)\) , \({g_1}=\sqrt {\frac{{4ar}}{{{{\left( {r+a} \right)}^2}+{{\left( {z - {z_1}} \right)}^2}}}}\) and \({g_2}=\sqrt {\frac{{4ar}}{{{{\left( {r+a} \right)}^2}+{{\left( {z - {z_2}} \right)}^2}}}}\) .
The circumferential component of the magnetic field of the solenoid can be expressed as
Through Eq. (2) and Eq. (3), it is not difficult to obtain that the expressions of Br and Bz are independent functions only with respect to φ. Therefore, these two formulas can also be rewritten as Bz(r, z) and Br(r, z), which shows the central symmetry of the solenoid well. From Eq. (4), it can be seen that due to the use of a uniform cylindrical equivalent current in the established solenoid model, the value of the circular component in the solenoid magnetic field is zero. Although it is necessary to consider the axial magnetic field generated by the axial component of the current when the lateral spacing of the solenoid is relatively large in calculations, this solenoid type is rarely used in practical applications.
In addition to calculating the magnetic field magnitude of a single point, it is sometimes necessary to consider the magnetic field distribution of the entire space. Due to the symmetry of the solenoid set r = 0, it can be observed that the radial component of the magnetic field along the axis is zero, i.e. Br (0, z) = 0, and only the axial component needs to be considered. For ease of analysis, we use the geometric center of the solenoid as the coordinate origin, i.e. z2 = − z1 = b, and then simplify Eq. (2) to obtain the axial component of the magnetic field on the central axis:
Equation (6) describes the expression of the magnetic field when the solenoid is of infinite length. In this case, the magnetic field strength on the central axis of the solenoid is linear with the number of turns of the coil and the magnitude of the current. But in real life, there is no infinite screw tube. Nonetheless, it is very convenient to use this method to analyze the magnetic field of solenoids whose length of the analysis tube is larger than its cross section radius.
For the sake of discussion, the width ratio m is converted to m = b/a. Take current I = 2 A, coil density n = 1000, solenoid length 2b = 10 cm, radius a = 0.5 cm, 1.0 cm, 2.0 cm and 4.0 cm, and set r = 0. The size and distribution of Bz are shown in Fig. 2.
Distribution of Bz component on the axis when b is constant and a is changing.
It can be seen from Fig. 2 that when the length b of the solenoid remains constant, the axial magnetic field Bz gradually concentrates inside the solenoid as the radius a decreases, while the external magnetic field Bz gradually decreases. The smaller the radius a, the more stable the uniform magnetic field region (− L/2 < z < L/2), and the magnetic field outside the tube decays faster. The results show that the larger the ratio of length to diameter, the stronger the magnetic concentration capacity of the solenoid, the better the uniformity of the internal magnetic field. The uniformity here can be interpreted as being in(−L/2 < z < L/2). It is almost constant in the L/2 range, indicating good uniformity. L stands for the uniform magnetic field region of the solenoid.
It can be seen from Fig. 2 that the magnetic field strength at the position of the two ports of the solenoid is relatively uneven. Although the magnetic field variation at both ends of the solenoid can be reduced by reducing the ratio of the radius of the solenoid to the length of the solenoid, large diameter and short length solenoids are needed under some specific small components. At this time, the magnetic field change at both ends of the solenoid is more obvious than the overall change. It is necessary to seek an improved structure that enables the solenoid to obtain a more uniform magnetic field when the radius to length ratio is small.
The Taylor expansion of Eq. (5) can be obtained
According to Eq. (7), it is easy to see that in order to make the magnetic field deviation near point O smaller, the remainder of its expansion term should be smaller. It is necessary to introduce more adjustable parameters to make the remaining part of the expansion term smaller, thereby making the magnetic field of the solenoid more uniform. This article proposes an improved structure that adjusts the original magnetic field of the solenoid by introducing an auxiliary solenoid. The improved structure of the solenoid is shown in Fig. 3 below.
Compared with the ordinary solenoid, the improved solenoid has a better uniform region. The magnetic field finite element analysis of the improved solenoid is carried out in the following four aspects: the current size of the improved part I, the number of coils N, the length of the tube head L and the maximum radius of the improved structure R.
For the convenience of analysis and discussion, this article uses finite element analysis method to model and simulate the improved solenoid in COMSOL Multiphysics 5.6 software. When the number of solenoid turns is large, there are problems such as modeling difficulties and memory overflow caused by over-dense grid division. In order to make the simulation process more efficient, the solenoid structure is simplified into a cylinder structure for simulation in the geometric modeling process, and the grid is divided reasonably, and the superfine grid is used for the solenoid body, and the coarser grid is used for the air part. The grid division model is shown in Fig. 4.
Finite element simulation grid construction diagram.
In order to obtain the influence of the physical parameters of the improved structure on the magnetic field intensity, four groups of experiments were conducted respectively. In each group of experiments, while fixing the other three parameters, the relationship between the magnetic field uniform area (ECR) and the changed parameters when the relative deviation M was equal to 0.1%, 0.05% and 0.01% was recorded, as shown in Fig. 5. Among them, the magnetic field deviation rate M is the difference between the magnetic field strength at the center position of the solenoid and the magnetic field strength at any point in space, divided by the magnetic field strength at the center position of the solenoid.
The relationship between ECR and each parameter when the relative deviation M is 0.1%, 0.5% and 1%.
It can be seen from Fig. 5 that ECR under different deviations also increases with the increase of R. However, the changes between N, I and L variables and ECR are not linear, and there are extreme points in a certain interval, which indicate the optimized parameters. The optimum ECR increases with the increasing of the current and coils turns, and decreased with the increasing of the solenoid length. ECR is defined as the ratio of the area that satisfies the relative magnetic field deviation rate to the overall area of the solenoid.
Through the simulation experiment on the parameters of the improved solenoid, it is found that the improved structure has a more obvious adjustment effect on the magnetic field at both ends of the solenoid. Figure 6 shows the magnetic field intensity distribution of the improved solenoid under two extreme cases. It is not difficult to obtain a conclusion that there exists an optimal set of parameters P such that the magnetic field strength at both ends of the solenoid is close to the magnetic field strength in the central region of the solenoid. To find out the set of optimum parameters accurately, an intelligent optimization algorithm should be utilized.
Calculated magnetic field distribution for the modified solenoid in the extreme case (a, c), and the traditional solenoid (b).
Due to the increased complexity of the improved structure and the lack of clear linear relationships between parameters, it is difficult to obtain theoretical formulas for the structure. In order to find the optimal parameters for improving the structure, the Kolmogorov Arnold Networks (KAN) algorithm is utilized to train and validate the dataset collected from simulation experiments, and then the genetic algorithm is used to find the optimal solution for the obtained network. KAN is a novel neural network architecture inspired by the Kolmogorov Arnold representation theorem36. Unlike traditional neural networks that use fixed activation functions, KAN employs learnable activation functions at the edges of the network. This design allows each weight parameter in KAN to be replaced by a univariate function, which is typically parameterized in the form of spline functions, providing high flexibility and the ability to simulate complex functions with fewer parameters, enhancing the interpretability of the model. The specific structure is shown in Fig. 7.
The architecture of KAN revolves around an innovative concept: traditional weight parameters are replaced by univariate function parameters at the edges of the network. In KAN, each node aggregates the outputs of these functions without any nonlinear transformation. Splines are the core of KAN learning mechanisms, replacing traditional weight parameters commonly used in neural networks. The spline structure is detailed in Fig. 8. The flexibility of splines enables them to adaptively model complex relationships in data by adjusting their shape, thereby minimizing approximation errors and enhancing the network’s ability to learn subtle patterns from high-dimensional datasets.
The general formula for splines in KAN can be represented by B-splines:
Here, spline (x) represents the spline function. ci is the coefficient optimized during training, while Bi (x) is the B-spline basis function defined on the grid. The grid points define the intervals where each basis function Bi is active and significantly affects shape and smoothness. They can be regarded as hyperparameters that affect the accuracy of the network. More grids mean more control and higher accuracy, as well as the need to learn more parameters. During training, the ci parameters of these splines (coefficients of the basis function Bi (x)) are optimized to minimize the loss function, thereby adjusting the shape of the splines to best fit the training data. This type of optimization typically involves techniques such as gradient descent, where the spline parameters are updated at each iteration to reduce prediction errors.
Thanks to the locality of spline functions, KAN demonstrates the ability to retain learned information and adapt to new data without catastrophic forgetting. Unlike MLP that relies on global activation, which may inadvertently affect distant parts of the model, KAN only modifies a finite set of spline coefficients near each new sample. This centralized adjustment preserves the previous information stored in other parts of the spline. This article uses KAN network-based multivariate regression prediction (multi input single output) to construct a network model, and optimizes the constructed model. The specific implementation steps are as follows:
Kernel function selection and mapping: Use Gaussian kernel functions to map input data to high-dimensional space, capturing the nonlinear relationships of the data.
Kernel matrix calculation: Use kernel functions to calculate the similarity between input data and generate kernel matrices.
Hidden layer processing: In high-dimensional space, hidden layers use kernel and activation functions to perform nonlinear transformations on data, enhancing the expressive power of the model.
Output layer linear combination: The hidden layer outputs the final prediction result through linear combination. By using gradient descent to optimize the weights of the output layer, the model can accurately predict.
Adaptive learning: The model defines a loss function, updates weights using back- propagation algorithm, and prevents overfitting through L2 regularization to enhance the model’s generalization ability.
Model evaluation and prediction optimization: After training, the model uses test data for prediction optimization and evaluates its performance by calculating errors and other indicators.
Genetic algorithm search for optimal solution: Genetic algorithm (GA) is used to find a set of input feature values that maximize the output of the KAN network and output the set of feature values.
The optimization process is implemented on the platform of Matlab R2023b. Set 70% of the dataset as the training set, set the number of hidden layer neurons to 200, use current parameter I, auxiliary coil length parameter L, auxiliary coil turns parameter N, and maximum auxiliary coil radius R as the feature values of the sample dataset. And the output parameter is the ratio of ECR to the solenoid volume. Because it is to find the maximum output of the KAN network, the optimization objective function of the GA algorithm takes a negative value. The comparison and evaluation indicators of the predicted results after training are shown in Fig. 9.
Comparison of prediction results and evaluation indicators.
The parameter optimization results through genetic algorithm and KAN are: current parameter I = 4.3 A, auxiliary coil length parameter L = 10.38 mm, auxiliary coil turns parameter N = 39, and maximum auxiliary coil radius R = 43.54 mm. Finite element simulation of improved solenoid based on GA-KAN algorithm. The grid structure of the model is consistent with the simulation experiment, and the improved magnetic field distribution of the solenoid is shown in Fig. 10 (a). Figure 10 (b) and (c) respectively show the magnetic field distribution of Helmholtz coils and Maxwell coils of the same size, with the units in the legend being magnetic flux density modulus. The optimal parameter for the ratio of the coil radius of the Helmholtz coil to the distance between two coils is 0.544537,38. Under the premise of satisfying a magnetic field deviation of 0.1%, the ECR of the structure proposed in this paper is 44%, the ECR of the Helmholtz coil is 23%, and the ECR of the Maxwell coil is 31%. Moreover, when the coil radius is the same, the uniform magnetic field area of the structure proposed in this paper is significantly higher than that of the Helmholtz coil and the Maxwell coil. When there is a high requirement for magnetic field deviation rate, the structure mentioned in this article has more advantages.
Magnetic field distribution of (a) the improved solenoid, (b) the Helmholtz coil, and (c) the Maxwell coil.
Comparison of magnetic field distribution on the central axis of the solenoid.
The comparison of magnetic field distribution on the central axis of the improved solenoid structure and the single turn dense wound straight tube solenoid is shown in Fig. 11. The size and current parameters of the unimproved solenoid are consistent with the optimized structure, and the area that meets the specified magnetic field deviation rate is defined as a uniform magnetic field area. From the figure, it can be seen that the magnetic field distribution of the improved solenoid is significantly better than that of the ordinary solenoid, because the improved solenoid structure has a converging effect on the magnetic field at the edge of the solenoid. Compared with Helmholtz coils and Maxwell coils, the improved solenoid structure has a more uniform magnetic field distribution. Table 1 shows a comparison of uniform regions with relative magnetic field deviations of 1%, 0.5%, and 0.1%. From the table, it can be seen that the magnetic field uniformity of the new structure has been greatly improved, especially at a deviation rate of 0.1%, where the proportion of uniform regions has increased by more than 5 times.
Although increasing the ratio of the length to radius of the solenoid can also improve the magnetic field uniformity of the solenoid, this will greatly increase the size of the solenoid. As the length of the solenoid increases, the electromagnetic response speed of the solenoid slows down, and its capacitance also increases with the length, which is very unfavorable for some applications of the solenoid. Compared with changing the ratio of length to radius of the solenoid, the structure proposed in this paper has significant advantages in maintaining the performance of the solenoid.
This article proposes a new single turn tightly wound solenoid structure with an auxiliary solenoid, and uses finite element analysis method to conduct simulation experiments on the improved solenoid structure proposed in this article on COMSOL Multiphysics software. The effect of the improved structure on the magnetic field of the solenoid is studied by changing different parameter values, and the parameters of the new solenoid structure proposed in this article are optimized through GA-KAN algorithm. This article conducts research while keeping the size of the main solenoid unchanged. Through theoretical calculations and simulation results, the following conclusions are drawn:
The improved single bundle tightly wound solenoid proposed in this study can greatly improve the uniformity of the magnetic field and the uniformity of the region. In the case of a volume difference of 10%, the effective coverage obtained is more than twice that of a regular solenoid. In the case of higher magnetic field accuracy requirements, such as controlling the relative deviation at 0.1%, the effective coverage obtained can even be 5 times higher than that of a regular solenoid. In addition, by comparing the magnetic field distribution of different auxiliary coil parameters, it was found that the parameters of the main solenoid and auxiliary solenoid need to be matched to a certain extent to provide the best effective coverage.
Compared to reducing the edge effect of the solenoid by increasing the ratio of the length to radius of the solenoid, the new strategy has a smaller length increment, and this advantage becomes more significant as the radius of the solenoid increases, ensuring good electromagnetic response rate while improving the uniformity of the solenoid magnetic field.
The datasets generated and analyzed during the current study are available from the corresponding author on reasonable request.
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This work was funded by the Key Program of Natural Science Foundation of Anhui Provincial Education Department (No: KJ2020A0347) and the Open Research Fund of Anhui Key Laboratory of Detection Technology and Energy Saving Devices (No: JCKJ2022A02).
College of Integrated Circuits, Anhui Polytechnic University, Wuhu, 241000, China
Xuehua Zhu, Meng Xing, Juntao Ye, Xinyu Liu & Ziruo Ren
Anhui Engineering Research Center of Vehicle Display Integrated Systems, Anhui Polytechnic University, Wuhu, 241000, China
Xuehua Zhu, Meng Xing, Juntao Ye, Xinyu Liu & Ziruo Ren
Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education, Anhui Polytechnic University, Wuhu, 241000, China
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Xuehua Zhu and Meng Xing wrote the main manuscript text, Juntao Ye prepared Figs. 1, 2 and 3, Xinyu Liu prepared Figs. 4, 5 and 6 and Ziruo Ren prepared Figs. 7, 8, 9, 10 and 11. All authors reviewed the manuscript.
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Zhu, X., Xing, M., Ye, J. et al. Design and optimization of a novel solenoid with high magnetic uniformity. Sci Rep 14, 24650 (2024). https://doi.org/10.1038/s41598-024-76501-y
DOI: https://doi.org/10.1038/s41598-024-76501-y
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