Design method of three-axis high-uniform magnetic field coil based on improved multi-objective grey wolf optimization algorithm

Abstract

This invention presents a design method for a triaxial high-uniformity magnetic field coil based on an improved multi-objective gray wolf algorithm. The coil is idealized and modeled. The expression for the magnetic field generated by the coil is derived using the Biot-Savart law and the superposition property of magnetic fields. For the first time, the multi-objective gray wolf algorithm is directly used to optimize the coil parameters. The objective function is constructed based on the Taylor expansion coefficients of the magnetic field expression and the coil structural constraints. Compared to traditional equation solving or optimization algorithms using single-objective forms, which require manually setting weight parameters for each optimization objective, this method avoids dealing with unsolvable high-order equations or repeatedly experimenting to determine suitable parameters for each optimization objective, thus reducing computational difficulty and tedious calculations. Furthermore, this invention improves the initialization mechanism and convergence factor of the original multi-objective gray wolf algorithm, enhancing the quality of the final solution. Finally, a finished triaxial high-uniformity magnetic field coil is fabricated using flexible printed circuit technology. This invention marks the first time this algorithm has been used to design a triaxial uniform magnetic field coil.

Description

Technical Field

[0001] This invention relates to a design method for a triaxial high-uniformity magnetic field coil based on an improved multi-objective gray wolf algorithm, belonging to the technical field of designing sensors based on the magnetic field effect generated by current and optimizing sensor parameters using intelligent optimization algorithms. Background Technology

[0002] Common design methods for uniform magnetic field coils include the discrete conductor method (utilizing the positive relationship between current and magnetic field) and the target field method (utilizing the inverse relationship between current and magnetic field), as well as the finite element method. These methods all idealize the overall coil model; for example, the discrete conductor method treats the coil as a discrete, thicknessless conductor, while the target field method considers the coil surface as an ideal conductor. Furthermore, both the discrete conductor method and the target field method derive the magnetic field generated by the coil based on the Biot-Savart law.

[0003] The Gray Wolf Algorithm mimics the leadership hierarchy and hunting mechanism of gray wolves in nature. It employs four types of gray wolves—alpha, beta, delta, and omega—to represent the optimal, suboptimal, and other solutions in the parameter solution set during actual algorithm computation, respectively. Furthermore, it mathematically models the three main steps of a gray wolf's hunting: hunting, finding prey, surrounding and attacking prey. Compared to traditional algorithms such as particle swarm optimization and differential evolution used in designing uniform magnetic field coils, the Gray Wolf Algorithm exhibits superior performance, proven to outperform particle swarm optimization in both solution accuracy and convergence speed. For multi-objective Gray Wolf Algorithms, since it is difficult to find a solution optimal for all optimization objectives, alpha, beta, and delta are randomly selected from an externally archived set of solutions using a roulette wheel mechanism.

[0004] When the Gray Wolf Algorithm mathematically models the social hierarchy of wolves, the search (optimization) is guided by alpha, beta, and delta wolves, with omega wolves following them.

[0005] The wolf pack will first engage in hunting behavior, gradually approaching its prey (represented as the solution in the specific implementation of the algorithm):

[0006] The behavior of surrounding prey can be mathematically represented by equations (9) and (10):

[0007]

[0008]

[0009] in This represents the distance between the gray wolf and its prey, and t represents the current iteration number. Let be the position vector of the prey in the t-th iteration. Let represent the position vector of the gray wolf in the t-th iteration. It is the position vector of the gray wolf in iteration t+1. and These are coefficient vectors, used to represent the distance between the gray wolf and its prey, and the distance when determining the new position in the next iteration. The proportion it accounts for. and It can be expressed by equations (11) and (12):

[0010]

[0011] in It is an element iteration vector. The elements decrease linearly from 2 to 0 as the number of iterations increases. and It is a vector composed of randomly generated numbers between [0,1]. The elements of the vector take values ​​in the range [-α, α], where α is a fixed value. The elements decrease linearly from 2 to 0 as the number of iterations increases. The element values ​​will also decrease accordingly. This process allows Grey Wolf Optimization to simulate, in mathematical modeling, that it eventually tends towards the optimal solution.

[0012] However, in the decision space of the optimization problem, the gray wolf (search agent) does not know the precise location of the optimal solution (prey). During algorithm execution, the three optimal solutions are selected from the iterated solutions based on their degree of optimization and set as alpha, beta, and delta respectively. This forces other candidate solutions (i.e., omega) to update their positions based on the optimal search location. The formula for updating the position is shown in equation (13):

[0013]

[0014] in Let be the position vector corresponding to a certain omega at the (t+1)th iteration. These are temporary position vectors updated based on the positions of alpha, beta, and delta, respectively. Update the position based on these temporary position vectors.

[0015] The Grey Wolf algorithm changes The size of the [something] determines whether the search agent (the gray wolf participating in the hunting operation) is more inclined towards development or exploration behavior. When At this point, the algorithm tends to approach what is currently considered the optimal solution. However, this process may cause the algorithm to get trapped in local optima. When the value is greater than 1, the algorithm principle considers it to be the optimal solution in order to find the global optimal solution.

[0016] Compared to the single-objective Grey Wolf algorithm, the multi-objective Grey Wolf algorithm adds two new mechanisms: external archiving and adaptive grid. Since multi-objective problems involve multiple optimization objective functions, it is difficult to find a function that is optimal for all objectives. Therefore, only solutions that are better than all others on some objectives, but not worse than others, are stored in the external archive. During algorithm execution, the external archive is divided into a high-dimensional hypercube, the dimension of which is determined by the number of selected optimization objectives. When a newly added solution exceeds the archive's storage capacity, the algorithm activates the adaptive grid mechanism to expand the storage capacity to accommodate the newly added solution.

[0017] A search of existing literature revealed no instances of directly using the multi-objective gray wolf algorithm in coil design to transform coil optimization from equation solving into a nonlinear optimization problem, thereby reducing manual computation. Summary of the Invention

[0018] This invention idealizes the coil based on the discrete conductor method, derives the magnetic field B generated by the coil using the Biot-Savart law, and solves for the coil's position parameters using its Taylor expansion, seeking a solution that minimizes the absolute values ​​of the coefficients in the Taylor expansion. Since solving using equations is computationally complex at higher orders, this invention, for the first time, directly uses the multi-objective Grey Wolf algorithm to transform coil optimization from equation solving into a nonlinear optimization problem, reducing manual computation. By constructing the objective function and constraints, the coil parameters are obtained through multiple iterations. Compared to the tedious computational process of traditional equation solving, the coil design scheme based on the Grey Wolf algorithm reduces the computational load, allowing the designer to focus solely on deriving the magnetic field generated by the coil according to the Biot-Savart law.

[0019] The problem addressed by this invention is to provide a design method for a triaxial highly uniform magnetic field coil based on an improved multi-objective gray wolf algorithm. The triaxial uniform magnetic field coil designed using this method needs to cover the gas chamber containing the working material in an atomic sensor (such as rubidium-87 in a CPT magnetometer, CPT, Coherent Population Trapping, or Coherent Layout Trapping). Generally, the relative uniformity of the entire critical region (along the axis [-1R, 1R], where R is the coil radius) should be around 1% or lower. Therefore, it is necessary to design a triaxial (meeting the requirements for magnetic field vector measurement) highly uniform magnetic field coil that meets the uniformity requirements. Furthermore, the coil must have a large coil constant to ensure that it generates a geomagnetic field of magnitude (50000nT-60000nT) within the current range that the finished coil can withstand (common flexible circuit boards, with a conductor width of approximately 0.2mm, copper thickness of 1oz, and an ambient temperature of 25 degrees Celsius, can withstand a maximum current of 550mA). This is to facilitate the study of the performance of the CPT magnetometer in a geomagnetic field environment.

[0020] The technical solution of the present invention is as follows:

[0021] A design method for a triaxial highly uniform magnetic field coil based on an improved multi-objective gray wolf algorithm is characterized by the following steps:

[0022] Step 1: Create an ideal model of the coil;

[0023] Step 2: Determine the constraints and limitations for the miniaturization requirements of the sensor;

[0024] Step 3, determine the coil structure;

[0025] Step 4: Perform Taylor expansion of the magnetic field expression, including performing a single-component Taylor expansion of the axial coil in the triaxial coil and deriving the magnetic field of the axial coil using the Pi-Sar law, and performing a three-component Taylor expansion of the radial coil in the triaxial coil and deriving the magnetic field of the radial coil using the Pi-Sar law.

[0026] Step 5: Optimize parameters using the multi-objective gray wolf optimization algorithm, including constructing the axial coil optimization objective function using the linear weighted summation method for the axial coil, constructing the radial coil optimization objective function directly using the Taylor expansion term for the radial coil, and constructing the radial coil selection function based on the sum of the average relative uniformity and the proportion of uniform regions for the radial coil optimization objective function.

[0027] Step 6: For the axial coil optimization objective function and radial coil selection function, improve the initialization mechanism and convergence factor of the multi-objective gray wolf algorithm, including implementing the reverse learning initialization mechanism through Python, and changing the convergence mode of the gray wolf algorithm convergence factor α from linear mode to nonlinear mode;

[0028] Step 7: For the axial coil optimization objective function and the radial coil selection function, run the improved multi-objective gray wolf algorithm to obtain the axial coil optimization and radial coil optimization.

[0029] Step 8: Verify the uniformity using finite element simulation based on the optimization results;

[0030] Step 9: Determine whether the uniformity requirement is met. If not, return to step 7; if yes, proceed to step 10.

[0031] Step 10: Use FPC technology to process the finished coil and experimentally verify its uniformity.

[0032] Step 10 includes the following steps:

[0033] Step 10.1, construct the magnetic shielding bucket;

[0034] Step 10.2: Build an experimental platform inside the magnetically shielded barrel and place the coil on the platform;

[0035] Step 10.3: Use the current element NI-SMU to supply current to the coil so that it generates a magnetic field;

[0036] Step 10.4: Measure the actual magnetic field generated by the coil using a fluxgate magnetometer;

[0037] Step 10.5: Process the recorded magnetic field data using Python to calculate the actual relative uniformity and generate a chart.

[0038] Step 1 includes the following expression:

[0039]

[0040] Where B is the magnetic field generated by the coil derived using the Biot-Savart law for solving the magnetic field generated by an ideal, thin wire; μ0 is the permeability of free space; and I is the current flowing through the coil. For the current element on the energized coil, Let r be the vector connecting a current element at a point on the coil to any point in space. The model;

[0041] ε z B represents the relative uniformity of the magnetic field generated by the z-axis coil along the z-axis at a certain point on the z-axis. z (z) represents the magnitude of the axial magnetic field generated by the axial coil at point (0,0,z), B z (0) is the magnetic field at the origin of the z-axis;

[0042] ε x B represents the relative uniformity of the magnetic field generated by the x-axis coil along the x-axis at a point on the x-axis. x (x) represents the magnitude of the radial magnetic field produced by the radial coil at point (x,0,0), B x (0) is the magnetic field at the x-axis origin.

[0043] Relative uniformity is used to describe the uniformity of the magnetic field of a coil. The better the uniformity in a certain region, the lower the average relative uniformity of the corresponding region.

[0044] The coil structure in step 3 includes an axial coil and a radial coil. The axial coil is a z-axis uniform magnetic field coil composed of multiple pairs of coaxial, identical, and parallel circular coils. The radial coil is an x-axis or y-axis uniform magnetic field coil composed of multiple nested pairs of coaxial rectangular saddle-shaped coils.

[0045] Step 3 includes the following expression:

[0046]

[0047] in This is the solution vector of the axial coil. d1 is the distance from the position of the first pair of circular coils in the axial coil to the xy plane. The value of d1 is expressed as a multiple of R, where R is the coil radius. d2 to d8 follow the same pattern. Here, h1 is the solution vector of the radial coil, and h2 is the height of the first pair of rectangular saddle coils in the radial coil. The value of h1 is expressed as a multiple of R, and so on for h2 to h6. It is the angle subtended by the arc region of the first pair of rectangular saddle-shaped coils in the radial coil. to And so on.

[0048] Step 4 includes the following expression:

[0049]

[0050] Among them B z | (0,0,z) B represents the magnitude of the magnetic field generated by the coil along the z-axis at (0,0,z), where z is the z-axis coordinate of a point. z | (0,0,0) Let be the magnitude of the magnetic field generated by the coil along the z-axis at the origin. For B z The value of the j-th order partial derivative of z at (0,0,0) is R. j (z) represents an infinitesimal term in the Taylor expansion;

[0051] B i (x, y, z) represents the magnetic field components of magnetic field B along one of the x, y, or z axes, where x, y, and z are the x-axis, y-axis, and z-axis coordinates of a point. i (0,0,0) represents the magnitude of the magnetic field at the origin along a certain axis, m is the highest order of the Taylor expansion, and m1, m2, and m3 are the Taylor expansion orders corresponding to the x, y, and z components, respectively. It is a higher-order infinitesimal function, where n1, n2, and n3 correspond to the orders of the x, y, and z components in the partial derivatives, respectively.

[0052] I is the axial coil magnetic field, R is the coil radius, N is a positive integer representing the number of coil pairs, and μ0 is the free permeability. Let n be a unit vector along the z-axis. i It is the number of turns of the i-th pair of coils, where i is the sequence number, and d is the number of turns of the i-th pair of coils. i It is the distance from the position of the i-th pair of circular coils to the xy plane. Dia is the magnetic field of the radial coil at the origin, where Dia is the diameter of the coil. Let s be a unit vector along the x-axis. i Equals 1 plus the coil height h i The square of the ratio to the coil diameter Dia It is the angle of the circular arc segment of the rectangular saddle coil in the radial coil.

[0053] Step 5 includes the following expression:

[0054]

[0055] Where K p ω is the number of even-order numbers in the Taylor expansion, k is a positive integer, and ω is the number of even-order numbers in the Taylor expansion. k It is the coefficient of the k terms. For B z The 2k-th order partial derivative with respect to z at (0,0,0), B z It is the z-axis magnetic field, f penalty (x) is the penalty function, x is the candidate solution vector, and c is the uniformity index weight. θ represents the average relative uniformity of the key interval, M is the number of points in the key interval for calculating relative uniformity, w is the weight of the uniform region length index, which is usually negative, and θ is the length of the uniform region.

[0056] Step 6 includes the following expression:

[0057] x = a + (ba) * rand

[0058] x′=b-(xa),

[0059]

[0060] Where x is the initial solution, a is the upper bound of the coil parameter range, and b is the lower bound of the coil parameter range. rand is a random number between [0,1], x′ is the initial solution for reverse learning corresponding to x, α is the convergence factor of the improved multi-objective gray wolf algorithm, and t is the current iteration number of the algorithm. max The maximum number of iterations of the algorithm set by the designer.

[0061] The technical effects of this invention are as follows: This invention is based on an improved multi-objective gray wolf algorithm for designing triaxial high-uniformity magnetic field coils. The coil is idealized and modeled, and the expression for the magnetic field generated by the coil is derived using the Biot-Savart law and the superposition of magnetic fields. For the first time, the multi-objective gray wolf algorithm is directly used to optimize coil parameters, and the optimization objective function is constructed based on the Taylor expansion coefficients of the magnetic field expression and the coil structure constraints. Compared to traditional equation solving or optimization algorithms using single-objective forms, which require manually setting weight parameters for each optimization objective, this invention avoids the need to deal with unsolvable high-order equations or repeatedly experiment to determine appropriate parameters for each optimization objective, reducing computational difficulty and tedious calculations. Simultaneously, this invention improves the initialization mechanism and convergence factor of the original multi-objective gray wolf algorithm, improving the quality of the final solution. Finally, a finished triaxial high-uniformity magnetic field coil is fabricated using Flexible Printed Circuit (FPC) technology. This invention is the first to use this algorithm to design a triaxial uniform magnetic field coil.

[0062] Theoretically calculated average magnetic field relative uniformity along the axis [-1R, 1R] (R is the radius of the finished cylinder) of a triaxial high-uniformity magnetic field coil based on the improved multi-objective gray wolf algorithm is significantly higher than that of a common triaxial Helmholtz coil (average magnetic field relative uniformity 1.0618 × 10⁻⁶). -1 The reductions were 3 and 1 orders of magnitude, respectively, resulting in a significant improvement in uniformity compared to the Helmholtz coil. In contrast, the original Grey Wolf algorithm yielded an average relative uniformity of 2.9538 × 10⁻⁶ for both axial and radial coils in the same region. -4 and 2.9915×10 -2 Compared to the unmodified Grey Wolf algorithm, the design scheme proposed in this invention improves uniformity by 27.53% and 23.25%, respectively. Experiments show that the actual average relative uniformity of the magnetic field along the axis [-1R, 1R] for the axial and radial coils using the design scheme of this invention is 2.5646 × 10⁻⁶. -4 and 1.8×10 -2 The design achieves a high uniformity target of approximately 1% or less, meeting the established average relative uniformity objective. Compared to triaxial Helmholtz coils, this represents a significant improvement in uniformity, by three orders of magnitude and one order of magnitude, respectively. This demonstrates the effectiveness of the triaxial high-uniformity magnetic field coil design method proposed in this invention.

[0063] This invention is based on mature flexible circuit printing technology, resulting in lower manufacturing costs. Compared to traditional manually wound triaxial Helmholtz coils, it offers higher precision, better coaxiality, and a magnetic field uniformity that is 1-3 orders of magnitude higher. It can generate a geomagnetic field within the range of common current sources. This invention can be used for various miniature atomic sensors, particularly for calibration, magnetic compensation, and sensitivity enhancement of vector measurement sensors. Its applications are wide-ranging and it has significant practical value. Attached Figure Description

[0064] Figure 1 This is a flowchart illustrating the implementation of the triaxial high uniform magnetic field coil design method based on the improved multi-objective gray wolf algorithm of this invention. Figure 1 The process includes: Step 1, idealizing the coil; Step 2, determining constraints to meet sensor miniaturization requirements; Step 3, determining the coil structure; Step 4, performing a Taylor expansion of the magnetic field expression, including a single-component Taylor expansion for the axial coil in the triaxial coil, deriving the magnetic field of the axial coil using the Pi-Sarvassi-Bauer law, and a three-component Taylor expansion for the radial coil in the triaxial coil, deriving the magnetic field of the radial coil using the Pi-Sarvassi-Bauer law; Step 5, optimizing parameters using a multi-objective gray wolf optimization algorithm, including constructing the axial coil optimization objective function using a linear weighted summation method, constructing the radial coil optimization objective function directly using Taylor expansion terms, and constructing the radial coil optimization objective function based on the sum of the average relative uniformity and the proportion of uniform regions. Step 6: For the axial coil optimization objective function and the radial coil selection function, improve the initialization mechanism and convergence factor of the multi-objective gray wolf algorithm, including implementing the reverse learning initialization mechanism through Python and changing the convergence mode of the gray wolf algorithm convergence factor α from linear to nonlinear; Step 7: For the axial coil optimization objective function and the radial coil selection function, run the improved multi-objective gray wolf algorithm to obtain the axial coil optimization and radial coil optimization; Step 8: Verify the uniformity by performing finite element simulation based on the optimization results; Step 9: Determine whether the uniformity requirement is met. If not, return to step 7; if yes, proceed to step 10; Step 10: Use FPC technology to process the finished coil and experimentally verify the uniformity (FPC, Flexible Printed Circuit).

[0065] Figure 2 This is a schematic diagram of the axial and radial coil structures involved in the triaxial high uniform magnetic field coil design method based on the improved multi-objective gray wolf algorithm of this invention. Figure 2The left side of the image shows an axial coil, and the right side shows a radial coil. The axial coil is an axial (along the z-axis) uniform magnetic field coil composed of multiple pairs of coaxial, identical, and parallel circular coils. The radial coil is a radial (along the x and y axes) uniform magnetic field coil composed of multiple nested pairs of coaxial rectangular saddle-shaped coils. Figure 2 In this coordinate system, x, y, and z are the three axes (x-axis, y-axis, and z-axis), R is the coil radius, and d8 is the distance from the position of the 8th pair of circular coils forming the axial coil to the xy-plane. This distance is generally expressed as a multiple of the coil radius. The distance from the position of the i-th pair of circular coils to the xy-plane is denoted as d. i 'i' is the circular coil number. 'h' is the coil height, denoted as h_i, the height of the i-th pair of rectangular saddle coils forming the radial coil. i It is expressed as a multiple of the radius of the finished coil. It is the angle subtended by the circular arc segment of the rectangular saddle coil in the radial coil, and the angle subtended by the circular arc segment of the i-th pair of rectangular saddle coils is denoted as .

[0066] Figure 3 This is a flowchart illustrating the specific steps involved in implementing the triaxial high uniformity magnetic field coil design method based on the improved multi-objective gray wolf algorithm of this invention, including FPC technology for processing the finished coil and experimentally verifying its uniformity. Figure 3 1 represents the derivation of the magnetic field generated by the coil based on electromagnetic field theory. 2 represents the Taylor expansion of the magnetic field expression derived in 1, and the construction of a suitable optimization objective function for the improved Grey Wolf algorithm based on the coefficients of the expansion terms. 3 represents the improvement made to the original Grey Wolf algorithm. 4 represents the implementation of the algorithm program using Python and its execution. 5 represents the verification of the coil's uniformity corresponding to the parameters obtained in 4 using finite element simulation. 6 represents the fabrication of the finished coil using FPC technology and experimental verification. 6.1 represents the FPC fabrication steps. 6.2 represents the construction of a magnetic shielding barrel to isolate interference from other environmental magnetic fields. 6.3 represents the construction of an experimental platform and the placement of the coil on the platform. 6.4 represents the use of a NI-SMU (National Instruments-Source Measurement Unit) to provide current to the coil to generate a magnetic field. 6.5 represents the measurement of the actual magnetic field generated by the coil using a fluxgate magnetometer. 6.6 represents the processing of the recorded data using Python to calculate the actual relative uniformity and generate a graph for comparison with the relative uniformity of other traditional coils. Detailed Implementation

[0067] The following is in conjunction with the attached diagram ( Figures 1-3 The invention will be described in the following sections and examples.

[0068] Figure 1This is a flowchart illustrating the implementation of the triaxial high uniform magnetic field coil design method based on the improved multi-objective gray wolf algorithm of this invention. Figure 2 This is a schematic diagram of the axial and radial coil structures involved in the triaxial high uniform magnetic field coil design method based on the improved multi-objective gray wolf algorithm of this invention. Figure 3 This is a flowchart illustrating the specific steps involved in implementing the triaxial high-uniformity magnetic field coil design method based on the improved multi-objective gray wolf algorithm of this invention, including FPC technology for fabricating the coil and experimentally verifying its uniformity. (Reference) Figures 1 to 3 As shown, a design method for a triaxial high-uniformity magnetic field coil based on an improved multi-objective gray wolf algorithm includes the following steps: Step 1, idealizing the coil; Step 2, determining constraints to meet sensor miniaturization requirements; Step 3, determining the coil structure; Step 4, performing a Taylor expansion of the magnetic field expression, including a single-component Taylor expansion for the axial coil in the triaxial coil, deriving the magnetic field of the axial coil using the Pi-Sarvassler law, and a three-component Taylor expansion for the radial coil in the triaxial coil, deriving the magnetic field of the radial coil using the Pi-Sarvassler law; Step 5, optimizing parameters using the multi-objective gray wolf optimization algorithm, including constructing the axial coil optimization objective function using a linear weighted summation method, constructing the radial coil optimization objective function directly using the Taylor expansion term, and optimizing the radial coil optimization objective function according to... Step 6: Construct the radial coil selection function by summing the average relative uniformity and the proportion of uniform regions; Step 7: Improve the initialization mechanism and convergence factor of the multi-objective gray wolf algorithm for the axial coil optimization objective function and the radial coil selection function, including implementing the reverse learning initialization mechanism through Python and changing the convergence mode of the gray wolf algorithm convergence factor α from linear to nonlinear; Step 8: Run the improved multi-objective gray wolf algorithm for the axial coil optimization objective function and the radial coil selection function to obtain the axial coil optimization and radial coil optimization; Step 9: Verify the uniformity by performing finite element simulation based on the optimization results; Step 10: Determine whether the uniformity requirement is met. If not, return to step 7; if yes, proceed to step 11; Step 11: Process the finished coil using FPC technology and experimentally verify the uniformity.

[0069] Step 10 includes the following steps: Step 10.1, build a magnetic shielding barrel; Step 10.2, build an experimental platform inside the magnetic shielding barrel and place the coil on the platform; Step 10.3, use a current element NI-SMU to provide current to the coil to generate a magnetic field; Step 10.4, use a fluxgate magnetometer to measure the actual magnetic field generated by the coil; Step 10.5, process the recorded magnetic field data using Python to calculate the actual relative uniformity and generate a graph.

[0070] Step 1 includes the following expression:

[0071]

[0072] Where B is the magnetic field generated by the coil derived using the Biot-Savart law for solving the magnetic field generated by an ideal, thin wire; μ0 is the permeability of free space; and I is the current flowing through the coil. For the current element on the energized coil, Let r be the vector connecting a current element at a point on the coil to any point in space. The model;

[0073] ε represents the relative uniformity of the magnetic field generated by the z-axis coil along the z-axis at a certain point on the z-axis, B z (z) represents the magnitude of the axial magnetic field generated by the axial coil at point (0,0,z), B z (0) is the magnetic field at the origin of the z-axis; ε x B represents the relative uniformity of the magnetic field generated by the x-axis coil along the x-axis at a point on the x-axis. x (x) represents the magnitude of the radial magnetic field produced by the radial coil at point (x,0,0), B x (0) is the magnetic field at the x-axis origin.

[0074] The coil structure in step 3 includes an axial coil and a radial coil. The axial coil is a z-axis uniform magnetic field coil composed of multiple pairs of coaxial, identical, and parallel circular coils. The radial coil is an x-axis or y-axis uniform magnetic field coil composed of multiple nested pairs of coaxial rectangular saddle-shaped coils.

[0075] Step 3 includes the following expression:

[0076]

[0077] in This is the solution vector of the axial coil. d1 is the distance from the position of the first pair of circular coils in the axial coil to the xy plane. The value of d1 is expressed as a multiple of R, where R is the coil radius. d2 to d8 follow the same pattern. Here, h1 is the solution vector of the radial coil, and h2 is the height of the first pair of rectangular saddle coils in the radial coil. The value of h1 is expressed as a multiple of R, and so on for h2 to h6. It is the angle of the first pair of rectangular saddle coils in the radial coil. to And so on.

[0078] Step 4 includes the following expression:

[0079]

[0080]

[0081] Among them B z | (0,0,z)B represents the magnitude of the magnetic field generated by the coil along the z-axis at (0,0,z), where z is the z-axis coordinate of a point. z | (0,0,0) Let be the magnitude of the magnetic field generated by the coil along the z-axis at the origin. For B z The value of the j-th order partial derivative of z at (0,0,0) is R. j (z) represents an infinitesimal term in the Taylor expansion;

[0082] B i (x, y, z) represents the magnetic field components of magnetic field B along one of the x, y, or z axes, where x, y, and z are the x, y, and z coordinates of a point on the x, y, and z axes, respectively. i (0,0,0) represents the magnitude of the magnetic field at the origin along a certain axis, m is the highest order of the Taylor expansion, and m1, m2, and m3 are the Taylor expansion orders corresponding to the x, y, and z components, respectively. It is a higher-order infinitesimal function, where n1, n2, and n3 correspond to the orders of the x, y, and z components in the partial derivatives, respectively.

[0083] is the axial coil magnetic field, I is the current, R is the coil radius, N is a positive integer representing the number of coil pairs, and μ0 is the free permeability. Let n be a unit vector along the z-axis. i It is the number of turns of the i-th pair of coils, where i is the sequence number, and d is the number of turns of the i-th pair of coils. i It is the distance from the position of the i-th pair of circular coils to the xy plane. It is the magnetic field of the radial coil at the origin. This is a unit vector along the x-axis. Dia is the coil diameter, and s... i Equals 1 plus the coil height h i The square of the ratio to the coil diameter Dia It is the angle of the circular arc segment of the rectangular saddle coil in the radial coil.

[0084] Step 5 includes the following expression:

[0085]

[0086] Where K p ω is the number of even-order numbers in the Taylor expansion, k is a positive integer, and ω is the number of even-order numbers in the Taylor expansion. k It is the coefficient of the k terms. For B z The 2k-th order partial derivative with respect to z at (0,0,0), B z It is the z-axis magnetic field, f penalty (x) is the penalty function, x is the candidate solution vector, and c is the uniformity index weight. θ represents the average relative uniformity of the key interval, M is the number of points in the key interval for calculating relative uniformity, w is the weight of the uniform region length index, which is usually negative, and θ is the length of the uniform region.

[0087] Step 6 includes the following expression:

[0088] x = a + (ba) * rand

[0089] x′=b-(xa),

[0090]

[0091] Where x is the initial solution, a is the upper bound of the coil parameter range, and b is the lower bound of the coil parameter range. rand is a random number between [0,1], x′ is the initial solution for reverse learning corresponding to x, α is the convergence factor of the improved multi-objective gray wolf algorithm, and t is the current iteration number of the algorithm. max The maximum number of iterations of the algorithm set by the designer.

[0092] A design method for a triaxial highly uniform magnetic field coil based on an improved multi-objective gray wolf algorithm is characterized by the following steps:

[0093] (1) Ideal modeling of the entire coil

[0094] In practical coil design, the entire coil can be idealized and modeled (based on coil thickness). Therefore, the magnetic field B generated by the coil can be derived using the Biot-Savart law, which solves for the magnetic field generated by an ideal, thin conductor.

[0095]

[0096] Where μ0 is the permeability of the magnetic field in vacuum, which is a constant. I is the current through the coil. For the current element on the energized coil, Let r be the vector connecting a current element on a coil to any point in space. The modulus. For the derived expression for the magnitude of the magnetic field (magnetic flux), a single-component Taylor expansion (Equation 2) is performed on the axial coil, and a three-component Taylor expansion (Equation 3) is performed on the radial coil. The solutions are then obtained, and the set of coil parameters whose coefficients in the first N terms of the Taylor expansion are minimized (approaching zero) is the final coil parameter. The axial coil parameters are composed of the position of each coil (for simplicity, they are expressed as multiples of the radius of the final cylindrical coil). The radial coil parameters are composed of the height of each coil and the size of the angle of the arc portion.

[0097]

[0098] Among them, Bz | (0,0,z) B represents the magnitude of the magnetic field generated by the coil along the z-axis at (0,0,z), where z is the z-axis coordinate of a point. z | (0,0,z) The magnitude of the magnetic field generated by the coil along the z-axis at the origin. For B z The value of the partial derivative with respect to z at (0,0,0) is R. j (z) represents an infinitesimal term in the Taylor expansion. B i (x, y, z) represents the magnetic field components of magnetic field B along one of the x, y, or z axes, where x, y, and z are the x, y, and z coordinates of a point on the x, y, and z axes, respectively. i (0,0,0) represents the magnitude of the magnetic field at the origin along a certain axis, m is the highest order of the Taylor expansion, m1, m2, and m3 are the Taylor expansion orders corresponding to the x, y, and z components, respectively, and n1, n2, and n3 are the orders corresponding to the x, y, and z components in the partial derivatives, respectively. Generally, a higher expansion order results in better magnetic field uniformity from the coil, but also requires more computation when optimizing the parameters.

[0099] The uniformity of the coil is described by a relative uniformity ε, and its expression is shown in equations (4) and (5):

[0100]

[0101] Where, ε z ε represents the relative uniformity of the magnetic field generated along the z-axis by the z-axis coil (axial coil) at a point on the z-axis. x This represents the relative uniformity of the magnetic field generated by the x-axis coil (radial coil) along the x-axis at a point on the x-axis. Similarly, it can represent the relative uniformity of the magnetic field generated by the y-axis coil along the y-axis at a point on the y-axis. B z (z) represents the magnitude of the axial magnetic field generated by the axial coil at point (0,0,z), B x (x) represents the magnitude of the radial magnetic field generated by the radial coil at point (x,0,0). Since the relative uniformity ε describes the change in the magnetic field at other points relative to the magnetic field at the origin, a lower relative uniformity indicates that the magnetic field is more uniform within a certain range, and the overall uniformity is better. The increase in uniformity described in the abstract and the following text represents the decrease in the relative uniformity of the coil's magnetic field.

[0102] (2) Design for miniaturization and miniaturization of atomic sensors

[0103] Atomic sensors such as coherent population trapping (CPT) magnetometers, optical pumping (OP) magnetometers, spin-exchange relaxation-free (SERF) magnetometers, and nuclear magnetic resonance (NMR) gyroscopes all require highly uniform magnetic field coils, but their miniaturization requirements limit the coil's size. This invention decomposes the design problem of a triaxial uniform magnetic field coil into an axial coil parameter optimization problem and a radial coil parameter optimization problem. Since the y-axis coil can be obtained by selecting 90° from the x-axis coil, they share the same structural parameters (height, radius, and angle). Therefore, the parameters obtained after optimizing the x-axis coil can be applied to the y-axis simultaneously. Based on the structure of the atomic sensor, the overall coil is designed as a cylinder, and flexible printed circuit (FPC) technology is used to print the circuitry on a flexible circuit board, allowing the designed coil to be closely attached to a surface less than 100 cm thick. 3 The 3D-printed support surrounds the gas chamber containing the working material of the atomic sensor, providing a uniform magnetic field to the chamber. To ensure that the sensitivity of the atomic sensor is not affected by the non-uniform magnetic field generated by the coil, the target for coil design is that the relative uniformity of the target average magnetic field in the critical area should be around 1% or lower. Traditional triaxial uniform magnetic field coils are often triaxial Helmholtz coils, but their average relative uniformity of the magnetic field in the critical area (not less than the volume of the gas chamber, along the axis [-1R, 1R]) can only reach 10%. -1 The scale is large, and this structure can only be manually wound onto the coil frame, which introduces errors leading to poor coil coaxiality and further affecting its uniformity. This invention combines multiple pairs of coaxial, identical circular coils to form an axial coil, and multiple pairs of coaxial, rectangular saddle-shaped coils with the same radius of their arc portions to form a radial coil. The structure is as follows: Figure 2 Compared to Helmholtz coils, it generates a more uniform magnetic field and can be fabricated into cylindrical products using FPC technology (a cylindrical gas chamber is attached, with axial and radial coils attached to the cylindrical surface), reducing the weight and space occupied by the coil in the integrated atomic sensor (each copper layer is only 35μm thick, and the minimum volume of the coil designed in this invention can be less than 11.78cm). 3 This is beneficial for the miniaturization of atomic sensors.

[0104] (3) The multi-objective gray wolf algorithm was applied to the design of triaxial coils for the first time.

[0105] Traditional equation solving yields coil position parameters that minimize the coefficients of the Taylor expansion.

[0106] The problem, when the number of coil pairs is large, becomes a complex high-order equation, whose calculation is too complicated to find a suitable solution. Therefore, this invention uses an intelligent optimization algorithm to solve the coil parameters, transforming the coil optimization problem from a traditional equation-solving problem into a nonlinear optimization problem, avoiding complex equation-solving, and focusing on constructing the optimization objective function based on the coefficients of the Taylor expansion of the magnetic field expression. This is transformed into the algorithm iterating to obtain a set of solutions that minimize the optimization objective function value. Furthermore, constraints and penalty functions are added to meet the requirements of minimum line spacing in finished coil processing and the need to open a light-transmitting hole in the center of the coil to facilitate the entry of probe light (the technology required for optically pumped magnetometers) into the air chamber. When the results obtained by the algorithm iteration do not meet the constraints, a penalty value is added to the optimization objective function to filter candidate solutions that do not conform to the actual mechanical structure of the coil.

[0107] However, since the Taylor order of the expansion is usually high (typically greater than 4), the optimization objective of the optimization function is usually multiple. Because finding only the solution with the smallest coefficient value in one expansion term is often ineffective in coil optimization, it is necessary to find a set of solutions with better coefficient values ​​in multiple expansion terms when optimizing axial and radial coils. Therefore, the coil optimization problem is actually a multi-objective optimization problem. Traditional methods for coil design based on intelligent optimization algorithms typically use linear weighted summation to transform the multi-objective problem into a single-objective problem, and the algorithms used in these works are also designed based on single-objective problems. However, the linear weighted summation method for coil design requires determining a weight coefficient for each optimization objective before summing them as the final optimization objective function (the optimization objective function of a single-objective algorithm). Determining appropriate weights requires considerable effort and experience, and there are cases where the coil corresponding to the solution with the smallest optimization objective function may have better magnetic field uniformity across the entire air chamber region compared to other solutions, but the size of the uniform region may be smaller. Especially when designing radial coils, unlike axial coils, a single-component Taylor expansion cannot be used to transform the magnetic field expression from a complex integral form to a polynomial form. Therefore, a three-component expansion is required (the result obtained by single-component expansion is poor in the region along the non-axis; if single-component expansion cannot simplify the calculation, then a three-component expansion must be considered). Partial derivatives with respect to the three variables x, y, and z are required, resulting in more optimization objectives and a more complicated workload in determining the weighting coefficients.

[0108] Therefore, this invention is the first to use the multi-objective gray wolf algorithm, specifically designed for multi-objective problems, to design a triaxial uniform magnetic field coil. Compared to single-objective algorithms, it introduces an adaptive grid and an external archiving mechanism. Its algorithm is more complex. By directly using the multi-objective gray wolf algorithm, this invention eliminates the need to determine a suitable weighting coefficient for each optimization objective, significantly reducing the workload in the optimization stage. When constructing the optimization objective function of the multi-objective gray wolf algorithm, this invention comprehensively considers two key indicators: average uniformity and uniform region size. Furthermore, it adds a penalty function considering factors such as coil spacing, the need for a light aperture for the probe light, and the limitations imposed on coil volume during the miniaturization of the atomic sensor. When the parameters obtained by the algorithm do not meet the constraints of the coil structure, a penalty value is added to the corresponding objective function. This invention optimizes the position parameters of the axial and radial coils using either a linear weighted summation method or by directly employing a multi-objective mechanism, depending on the difficulty of optimization. When directly employing the multi-objective mechanism, since the multi-objective algorithm yields a set of solutions existing in an external archive rather than a single solution from a single-objective algorithm, but coil design only requires one solution, it is necessary to filter a series of solutions from the external archive to select a final solution. This invention innovatively incorporates a new screening function, which is the sum of the x-axis average relative uniformity and the size of the uniform region generated by the corresponding coil. This comprehensively considers both the average uniformity and the size of the uniform region—two key parameters.

[0109] This invention is the first to employ a multi-objective gray wolf algorithm to directly optimize coil parameters. The gray wolf algorithm mimics the leadership hierarchy and hunting mechanism of gray wolves in nature. It uses four types of gray wolves—alpha, beta, delta, and omega—to simulate the leadership hierarchy. Furthermore, it implements three main steps: hunting, finding prey, surrounding prey, and attacking prey. Compared to traditional coil optimization algorithms such as particle swarm optimization and differential evolution, the gray wolf algorithm exhibits superior performance, proven to outperform particle swarm optimization in both solution accuracy and convergence speed. In actual coil parameter optimization, the gray wolf algorithm sorts all candidate solutions that meet the constraints according to their objective function values. The solutions with the smallest objective function values ​​are selected and, from smallest to largest, designated as alpha (optimal solution), beta (second-best solution), and delta (third-best solution), i.e., the leaders. The remaining candidate solutions are considered omega (search agents). These search agents continuously update their positions based on the positions of alpha, beta, and delta, bringing them closer to the positions of these three solutions, until the algorithm reaches its maximum iteration count, or the objective function value corresponding to the new position (new coil parameters) is better than that of the leaders.

[0110] (4) Improvements to the original multi-objective gray wolf algorithm

[0111] After practical testing, it was found that although the original multi-objective gray wolf algorithm has advantages such as fast convergence speed, relatively simple structure, fewer controllable parameters, and strong robustness, it is prone to problems in actual operation, such as low convergence accuracy when facing complex problems, potentially slow convergence speed in later stages, and insufficient population diversity leading to getting trapped in local optima. Therefore, this invention improves the mechanism of the original gray wolf algorithm when optimizing the position parameters of the triaxial coil, improving its initialization and convergence mechanisms.

[0112] Since the quality of the initial population directly affects the convergence speed and solution accuracy of the algorithm, this invention first updates the initialization mechanism of the original Grey Wolf algorithm. A reverse learning mechanism is used to generate an initial set of candidate solution parameters. Compared to the original method of randomly generating initial candidate solution parameters within the coil position range, the reverse learning mechanism, while randomly generating a set of parameters, also generates a set of solutions that are centrally symmetric within the coil position range. This avoids the initial coil parameters being concentrated in a small range, preventing the algorithm from getting trapped in local optima and struggling to find a better solution. Furthermore, the reverse learning mechanism also filters the 2N initial candidate solutions, selecting the set of solutions with better objective function values ​​as the final set of initial candidate solution parameters. This allows the algorithm to converge to the position of the better solution more quickly. The parameters initialized by the reverse learning mechanism are shown in the following equation:

[0113] x = a + (ba) * rand (6)

[0114] x′=b-(xa) (7)

[0115] Where a is the upper bound of the coil parameter value range, and b is the lower bound of the coil parameter value range. rand is a random number between [0,1]. x is a generated initial solution, and x′ is the reverse learning initial solution corresponding to x.

[0116] In addition, to better balance the local convergence and global search performance of the Grey Wolf algorithm, the convergence method of the convergence factor α of the multi-objective Grey Wolf algorithm can be improved.

[0117] In running the optimized coil algorithm, this invention uses a non-linearly varying convergence factor α instead of the linearly varying convergence factor of the traditional Grey Wolf algorithm.

[0118]

[0119] Where α is the convergence factor of the improved multi-objective gray wolf algorithm, and t is the current iteration number of the algorithm. max The maximum number of iterations of the algorithm set by the designer.

[0120] When a non-linearly decreasing convergence factor is used, the decay of the convergence factor is slow in the early stage of the iteration, and in most cases it is greater than 1, which is beneficial for a large number of global searches; in the later stage of the iteration, the decay of the convergence factor increases, and it quickly enters the range of less than 1, which is beneficial for a large number of local searches.

[0121] Theoretical calculations, after optimization using the improved Grey Wolf algorithm, yielded average magnetic field relative uniformity of the axial and radial coils along the axis [-1R, 1R] (where R is the radius of the finished cylindrical coil), which were 2.1405 × 10⁻⁶. -4 and 2.2959×10 -2 Compared to the triaxial Helmholtz coils commonly used in miniature atomic sensors, the average magnetic field relative uniformity is 1.0618 × 10⁻⁶. -1 The improvements were reduced by three and one orders of magnitude, respectively. Furthermore, the axial coil uniformity obtained by the improved Grey Wolf algorithm was 27.53% higher than that obtained by the original Grey Wolf algorithm after parameter optimization. The radial coil uniformity in the critical region obtained by the improved Grey Wolf algorithm was 23.25% higher than that obtained by the original Grey Wolf algorithm.

[0122] (5) Precision machining of finished triaxial uniform magnetic field coils using FPC technology

[0123] The improved multi-objective gray wolf algorithm optimized coil position parameters of this invention have shown a significant improvement in uniformity compared to traditional triaxial Helmholtz coils, as verified by theoretical calculations. To achieve a triaxial uniform magnetic field coil (combining the designed axial and radial coils), FPC (Flexible Printed Circuit) technology can be used. Compared to the traditional method of manually winding wires onto the coil frame, FPC technology avoids the poor coaxiality (causing the actual magnetic field direction to deviate from the predicted direction) and errors between the actual and pre-designed positions caused by manual winding. To avoid mutual interference between the axial and radial coils, multi-layer PCBs are fabricated using FPC technology, placing the axial coil (corresponding to the z-axis) and the two radial coils (corresponding to the x and y axes) on different layers. Each axial and radial coil is independently connected to a current source. To achieve the designed coil structure (coaxial multiple pairs of circular coils and coaxial multiple pairs of nested rectangular saddle coils), both the axial and radial coils are connected in series with a straight wire. The axial coil wires are switched between layers vias in the four layers of the flexible circuit board to avoid crossing with the radial coil wires and affecting the magnetic field generated by the coil. In order to avoid interference from the magnetic field generated by the wire connected to the current source with the magnetic field generated by the axial coil and the radial coil, the current direction of the wire connected to the current source and the wire connected to the current source are opposite and symmetrical along the central axis of the cylindrical coil so as to cancel out the magnetic field generated by the wire connected to the current source and the wire connected to the current source.

[0124] After fabricating the finished coil using FPC technology, an experimental platform was built to verify that the average relative uniformity of the magnetic field along the axis [-1R, 1R] of the actual triaxial coil in both the axial and radial sections was 2.5646 × 10⁻⁶. -4 and 1.8×10 -2 It meets the pre-set design target of an average relative uniformity of around 1% or lower in key areas. Compared to triaxial Helmholtz coils, the uniformity is improved by 3 orders of magnitude and 1 order of magnitude.

[0125] This invention employs multiple pairs of coaxial, identical, and parallel circular coils to form an axial (along the z-axis) uniform magnetic field coil, and nests multiple pairs of coaxial rectangular saddle-shaped coils to form a radial (along the x and y axes) uniform magnetic field coil (e.g. Figure 2 (As shown). The magnetic field generated by the coil is derived using the Biot-Savart law, and the derivation result is then subjected to a Taylor expansion. For the axial coil, a single-component Taylor expansion is used according to equation (2) to simplify the calculation. The coefficients in the Taylor expansion are the magnetic field components B generated by the axial coil along the z-axis. z Regarding the partial derivative of z, for the radial coil, the single-component Taylor expansion cannot convert the complex integral into a polynomial form, and the optimization results of the single-component Taylor expansion along the non-axis region are poor. Therefore, a three-component Taylor expansion is performed according to equation (3). The coefficients in the Taylor expansion are the magnetic field components B generated by the radial coil along the x-axis. x Partial derivatives with respect to x, y, and z. Since the coil optimization problem is a multi-objective optimization problem (simultaneously minimizing the coefficients of multiple Taylor expansion terms), traditional coil design methods combining intelligent optimization algorithms often use a linear weighted summation method to transform the multi-objective problem into a single-objective problem. However, the linear weighted method requires determining appropriate weights for each optimization objective, and there is no unified standard for determining these weights, requiring extensive experimentation or empirical determination, which introduces a significant workload. The coil design scheme proposed in this invention uses a multi-objective gray wolf algorithm to obtain the parameters of the axial and radial coils. When optimizing the axial coil, since the expression for the axial magnetic field is relatively simple, a single-component Taylor expansion can be performed to obtain B... zFurthermore, converting the integral form into a polynomial form facilitates differentiation. Therefore, the axial coil optimization still employs linear weighted summation to construct the objective function. However, in deriving the radial magnetic field, due to the involvement of a three-component Taylor expansion, partial derivatives with respect to x, y, and z are required simultaneously, resulting in a large number of Taylor expansion coefficients. If linear weighted summation is still used, determining suitable coefficients for each optimization objective becomes extremely tedious. Therefore, the multi-objective Grey Wolf algorithm is directly used for optimization, with the objective function directly being the coefficients of each three-component Taylor expansion term. Moreover, this invention innovatively incorporates a screening function to address the situation where multi-objective algorithms cannot yield a unique deterministic solution (multi-objective optimization problems themselves struggle to find a solution optimal across all optimization objectives). For a set of solutions stored in an external archive, the screening function selects a parameter that best combines relative uniformity and the proportion of uniform regions as the final solution for radial coil optimization.

[0126] For the design problem of a triaxial uniform magnetic field, the solution vector of the axial coil can be expressed as: Where, d i Let be the distance from the position of the i-th pair of circular coils forming the axial coil to the xy plane, expressed as a multiple of the coil radius. The solution vector of the radial coil can also be expressed as... But at this time Among them, h i The height of the i-th pair of rectangular saddle-shaped coils that make up the radial coil is expressed as a multiple of the radius of the finished coil. Let be the angle corresponding to the arc segment of the i-th pair of rectangular saddle coils in the radial coil.

[0127] To address the slow convergence speed and susceptibility to local optima inherent in the original Grey Wolf Algorithm, this invention improves its initialization mechanism when designing coil parameters using the Grey Wolf Algorithm. This invention employs an initialization population strategy based on a reverse learning mechanism, enhancing the diversity of the initial population and preventing it from getting trapped in local optima. The parameter vector of a candidate solution in the initial population is... Assumption The upper bound of the range of values ​​for any element in the set is And the lower bound is The first step is when Parameters of a candidate solution are randomly generated between the given parameters. At the same time, generate and corresponding The position is Performing the above operations on all candidate solutions yields a current population size of 2N individuals, resulting in a more diverse initial position distribution within the population. The second step involves filtering these individual position vectors, selecting the top N individual position vectors with the optimal objective function values ​​as the initial position vector set for the entire gray wolf population. This improves the convergence speed of the gray wolf algorithm.

[0128] This invention improves the convergence mechanism of the gray wolf algorithm for coil design by using a non-linearly changing convergence factor instead of a linearly changing one. When a non-linearly decreasing convergence factor is used, the decay of the convergence factor is slow in the early stages of iteration. In most cases, a value greater than 1 is beneficial for performing large-scale global searches; the decay of the convergence factor increases in the later stages of iteration. Quickly entering a range less than 1 is beneficial for conducting a large number of local searches.

[0129] The optimized results are first compared with the theoretical uniformity of traditional coils in the source program, and the design effect of the algorithm is further verified by finite element method (FEM) simulation. Finally, the effectiveness of the triaxial coil product made using FPC technology is verified by building an experimental platform. The entire design process of this invention is as follows: Figure 3 As shown.

[0130] This method requires six steps to realize a triaxial uniform magnetic field coil based on the multi-objective gray wolf algorithm.

[0131] Step 1: Derive the magnetic field generated by the coil

[0132] This invention uses multiple pairs of coaxial circular coils of the same size as axial coils and multiple pairs of nested coaxial rectangular saddle coils as radial coils. Before constructing the objective function for optimization, it is necessary to derive the expression of the magnetic field generated by the coil based on the coil structure. When designing the coil, the discrete wire method considers the coil as an ideal wire with no thickness, so the magnitude of the magnetic field generated by the coil after energization can be used according to the Biot-Savart law. For circular coils, it is relatively easy to derive the magnetic field generated directly. For rectangular saddle coils, the principle of magnetic field superposition can be used to regard the magnetic field as the superposition of the magnetic fields generated by the energized straight wire and the circular arc part of the wire. The derived expression of the magnetic field along the z-axis of the axial coil is shown in (14). Similarly, the expression of the magnetic field along the x and y axes of the radial coil can be obtained. (15) To simplify the magnetic field expression of the obtained radial coil at the zero point.

[0133]

[0134] in is the axial coil magnetic field, I is the current, R is the coil radius, N is a positive integer, and μ0 is the free permeability. Let n be a unit vector along the z-axis. i It is the number of turns of the i-th pair of coils, where i is the sequence number, and d is the number of turns of the i-th pair of coils. i It is the distance from the position of the i-th pair of circular coils to the xy plane. Dia is the magnetic field of the radial coil at the origin, where Dia is the diameter of the coil. Let s be a unit vector along the x-axis. i Equals 1 plus the coil height h i The square of the ratio to the coil diameter Dia It is the angle of the circular arc segment of the rectangular saddle coil in the radial coil.

[0135] Step 2: Construct the optimization objective function of the multi-objective gray wolf algorithm

[0136] In the iterative process of intelligent optimization algorithms, an important indicator for evaluating the quality of a solution is its value in the objective function (also known as fitness in differential evolution algorithms). For the design problem of parameters for a uniform magnetic field coil, which is a typical minimization problem, the Grey Wolf algorithm will select candidate solutions with smaller values ​​in the objective function during the optimization process.

[0137] The expression for the magnetic field generated by the coil is expanded using Taylor expansion. For optimization of the axial coil, since B is derived using a single-component Taylor expansion... z Since it is in polynomial form, the process of finding partial derivatives is relatively simple. Therefore, we can directly use the linear summation method to transform the multi-objective optimization into a single-objective optimization, and then apply the formula to B. z Expand to the 16th order. The selected first 16 partial derivatives are linearly weighted and summed, resulting in the objective function of the Grey Wolf algorithm for optimizing the axial coil, as shown in equation (16). For radial coil optimization, due to the three-component Taylor expansion, B... x Since the integral form contains multiple parameters, calculating the partial derivatives is relatively complex. Therefore, the multi-objective gray wolf algorithm is directly used to optimize the radial coil parameters. The coefficient of each expansion term in the three-component Taylor expansion is regarded as the optimization objective function. And according to formula (18), the solution with better overall performance (comprehensive average relative uniformity, uniform region size) in the archived solution set is selected.

[0138]

[0139] Among them, K p Let k be the number of even-order Taylor series expansions, and k be one of the orders. For B z f is the 2kth-order partial derivative with respect to z at (0,0,0). x is a candidate solution vector obtained by the multi-objective gray wolf algorithm after a certain iteration. penalty This is a penalty function set for the constraints of the coil structure. Let be the average relative uniformity within the critical interval (along the axial direction [-1R, 1R], where R is the coil radius), M be the number of points in the critical interval used to calculate the relative uniformity, θ be the length of the uniform region (the region with a relative uniformity less than 1%), and is the ratio of the uniform region length to the critical region length. c is the weight of the average relative uniformity index in the screening function (18), and w is the weight of the uniform region length index in (18), which is usually negative. During screening, the parameter with the smallest value in (18) is usually selected as the final selected coil parameter.

[0140] Step 3: Improve the initialization mechanism and convergence factor of the multi-objective gray wolf algorithm

[0141] When writing the multi-objective Grey Wolf algorithm program, a new function `obl_init` was written to implement the reverse learning initialization mechanism using Python. The convergence method of the Grey Wolf algorithm's convergence factor α was also modified (from linear to nonlinear).

[0142] Step 4: Run the multi-objective gray wolf algorithm

[0143] This invention uses Python 3 to implement both a single-objective Grey Wolf algorithm (axial coil optimization) and a multi-objective Grey Wolf algorithm (radial coil optimization), with candidate solutions using ndarray as the data structure for recording coil parameters. The mathematical equations in the Grey Wolf algorithm, as well as the conversion between angles and radians, are implemented using the numpy and math modules. The magnetic field expression derived from the Biot-Savart law is also implemented using functions from the numpy module. Multiple iterations of the algorithm are represented by loops. The random module is used for random number generation. To verify the uniformity of the proposed design method, the algorithm program also compares the uniformity of the Lee-Whiting coil, the axial coil designed using the original Grey Wolf algorithm, and the axial coil designed in this invention; the uniformity of the single saddle coil, the radial coil designed using the original Grey Wolf algorithm, and the radial coil designed in this invention. The overall triaxial uniformity of a common triaxial Helmholtz coil and the coil designed in this invention is also compared. The comparison results are displayed using Python's matplotlib module for graph generation and the data computation software MATLAB.

[0144] Step 5: Finite element simulation verification

[0145] The Finite Element Method (FEM) is a common method for simulating and verifying the effectiveness of coil designs based on optimization algorithms. Its core idea is to divide a complex problem domain into many elements. Local equations are established on these elements, and then combined into global equations to approximate the solution of the entire problem domain. In the verification phase of this invention, the electromagnetic simulation software ANSYS MAXWELL 3D was used to simulate the electromagnetic properties of the coil. Since DC power is often used to power the coil in common micro-atomic sensors, the solution type of the simulation software was set to Magnetostatic.

[0146] Step Six: Use FPC technology to process the finished coil and experimentally verify its uniformity.

[0147] In the implementation stage, this invention fabricates a triaxial uniform magnetic field coil by printing coil wires onto a flexible circuit board. Flexible circuit boards, also known as flexible printed circuit boards, are highly reliable and highly flexible printed circuit boards made with polyester film or polyimide as the substrate, which can significantly reduce the volume and thickness occupied by coils in miniature atomic sensors. To avoid mutual interference between axial and radial coils, each coil is independently connected to a current source. To achieve the designed coil structure, both axial and radial coils are connected in series via a single wire. Axial and radial coils (y-axis coil, x-axis coil) require vias to prevent cross-contamination and interference between coils of different axes. The finished coil designed in this invention avoids interference from the magnetic field generated by the wires connecting to the current source, ensuring that the magnetic fields generated by the axial and radial coils cancel each other out by routing the current in the contact wires in opposite directions. Finally, the uniformity of the coil designed in this invention is experimentally verified using a method of DC power supply + magnetic shielding to reduce environmental magnetic field interference.

[0148] Contents not described in detail in this specification are prior art known to those skilled in the art. It is hereby indicated that the above description is intended to help those skilled in the art understand this invention, but does not limit the scope of protection of this invention. Any equivalent substitutions, modifications, improvements, and / or simplifications of the above descriptions that do not depart from the essential content of this invention fall within the scope of protection of this invention.

Claims

1. A design method for a triaxial high-uniformity magnetic field coil based on an improved multi-objective gray wolf algorithm, characterized in that, Includes the following steps: Step 1: Create an idealized model of the coil; Step 2: Determine the constraints and limitations for the miniaturization requirements of the sensor; Step 3, determine the coil structure; Step 4: Perform Taylor expansion of the magnetic field expression, including performing a single-component Taylor expansion of the axial coil in the triaxial coil and deriving the magnetic field of the axial coil using the Pi-Sar law, and performing a three-component Taylor expansion of the radial coil in the triaxial coil and deriving the magnetic field of the radial coil using the Pi-Sar law. Step 5: Optimize parameters using the multi-objective gray wolf optimization algorithm, including constructing the axial coil optimization objective function using the linear weighted summation method for the axial coil, constructing the radial coil optimization objective function directly using the Taylor expansion term for the radial coil, and constructing the radial coil selection function based on the sum of the average relative uniformity and the proportion of uniform regions for the radial coil optimization objective function. Step 6: For the axial coil optimization objective function and radial coil selection function, improve the initialization mechanism and convergence factor of the multi-objective gray wolf algorithm, including implementing the reverse learning initialization mechanism through Python, and changing the convergence mode of the gray wolf algorithm convergence factor α from linear mode to nonlinear mode; Step 7: For the axial coil optimization objective function and the radial coil selection function, run the improved multi-objective gray wolf algorithm to obtain the axial coil optimization and radial coil optimization. Step 8: Verify the uniformity using finite element simulation based on the optimization results; Step 9: Determine whether the uniformity requirement is met. If not, return to step 7; if yes, proceed to step 10. Step 10: Use FPC technology to process the finished coil and experimentally verify its uniformity.

2. The design method for a triaxial high-uniformity magnetic field coil based on the improved multi-objective gray wolf algorithm according to claim 1, characterized in that, Step 10 includes the following steps: Step 10.1, construct the magnetic shielding barrel; Step 10.2: Build an experimental platform inside the magnetically shielded barrel and place the coil on the platform; Step 10.3: Use the current element NI-SMU to supply current to the coil so that it generates a magnetic field; Step 10.4: Measure the actual magnetic field generated by the coil using a fluxgate magnetometer; Step 10.5: Process the recorded magnetic field data using Python to calculate the actual relative uniformity and generate a chart.

3. The design method for a triaxial high-uniformity magnetic field coil based on the improved multi-objective gray wolf algorithm according to claim 1, characterized in that, Step 1 includes the following expression: Where B is the magnetic field generated by the coil derived using the Biot-Savart law for solving the magnetic field generated by an ideal, thin wire; μ0 is the permeability of free space; and I is the current flowing through the coil. For the current element on the energized coil, Let r be the vector connecting a current element at a point on the coil to any point in space. The model; ε z B represents the relative uniformity of the magnetic field generated by the z-axis coil along the z-axis at a certain point on the z-axis. z (z) represents the magnitude of the axial magnetic field generated by the axial coil at point (0,0,z), B z (0) is the magnetic field at the origin of the z-axis; ε x B represents the relative uniformity of the magnetic field generated by the x-axis coil along the x-axis at a point on the x-axis. x (x) represents the magnitude of the radial magnetic field produced by the radial coil at point (x,0,0), B x (0) is the magnetic field at the x-axis origin.

4. The design method for a triaxial high-uniformity magnetic field coil based on the improved multi-objective gray wolf algorithm according to claim 1, characterized in that, The coil structure in step 3 includes an axial coil and a radial coil. The axial coil is a z-axis uniform magnetic field coil composed of multiple pairs of coaxial, identical, and parallel circular coils. The radial coil is an x-axis or y-axis uniform magnetic field coil composed of multiple nested pairs of coaxial rectangular saddle-shaped coils.

5. The design method for a triaxial high-uniformity magnetic field coil based on the improved multi-objective gray wolf algorithm according to claim 4, characterized in that, Step 3 includes the following expression: in This is the solution vector of the axial coil. d1 is the distance from the position of the first pair of circular coils in the axial coil to the xy plane. The value of d1 is expressed as a multiple of R, where R is the coil radius. d2 to d8 follow the same pattern. Here, h1 is the solution vector of the radial coil, and h2 is the height of the first pair of rectangular saddle coils in the radial coil. The value of h1 is expressed as a multiple of R, and so on for h2 to h6. It is the angle of the first pair of rectangular saddle coils in the radial coil. to And so on.

6. The design method for a triaxial high-uniformity magnetic field coil based on the improved multi-objective gray wolf algorithm according to claim 1, characterized in that, Step 4 includes the following expression: Among them B z | (0,0,z) B represents the magnitude of the magnetic field generated by the coil along the z-axis in the region (0,0,z), where z is the z-coordinate of a point on the z-axis. z | (0,0,0) The magnitude of the magnetic field along the z-axis at the origin. For B z The value of the j-th order partial derivative of z at (0,0,0) is R. j (z) represents an infinitesimal term in the Taylor expansion; B i (x, y, z) represents the magnetic field components of magnetic field B along one of the x, y, or z axes, where x, y, and z are the x, y, and z coordinates of a point on the x, y, and z axes, respectively. i (0,0,0) represents the magnitude of the magnetic field at the origin along a certain axis, m is the highest order of the Taylor expansion, and m1, m2, and m3 are the Taylor expansion orders corresponding to the x, y, and z components, respectively. It is a higher-order infinitesimal function, where n1, n2, and n3 correspond to the orders of the x, y, and z components in the partial derivatives, respectively. in is the axial coil magnetic field, I is the current, R is the coil radius, N is a positive integer representing the number of coil pairs, and μ0 is the free permeability. Let n be a unit vector along the z-axis. i It is the number of turns of the i-th pair of coils, where i is the sequence number, and d is the number of turns of the i-th pair of coils. i It is the distance from the position of the i-th pair of circular coils to the xy plane. Dia is the magnetic field of the radial coil at the origin, where Dia is the diameter of the coil. Let s be a unit vector along the x-axis. i Equals 1 plus the coil height h i The square of the ratio of the coil diameter Dia, It is the angle of the circular arc segment of the rectangular saddle coil in the radial coil.

7. The design method for a triaxial high-uniformity magnetic field coil based on the improved multi-objective gray wolf algorithm according to claim 1, characterized in that, Step 5 includes the following expression: Where K p ω is the number of even-order numbers in the Taylor expansion, k is a positive integer, and ω is the number of even-order numbers in the Taylor expansion. k It is the coefficient of the k terms. For B z The 2k-th order partial derivative with respect to z at (0,0,0), B z It is the z-axis magnetic field, f penalty (x) is the penalty function, x is the candidate solution vector, and c is the uniformity index weight. denoted as the average relative uniformity of the key interval, M is the number of points in the key interval for which the relative uniformity is calculated, w is the weight of the uniform region length index, and θ is the length of the uniform region.

8. The design method for a triaxial high-uniformity magnetic field coil based on the improved multi-objective gray wolf algorithm according to claim 1, characterized in that, Step 6 includes the following expression: x = a + (ba) * rand x′=b-(xa), Where x is the initial solution, a is the upper bound of the coil parameter range, b is the lower bound of the coil parameter range, rand is a random number between [0,1], x′ is the initial solution for reverse learning corresponding to x, α is the convergence factor of the improved multi-objective gray wolf algorithm, and t is the current iteration number of the algorithm. max This represents the maximum number of iterations.

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