Chip integrated micro dual-plane magnetic field coil design method

Abstract

A chip integrated micro dual-plane magnetic field coil design method is provided. First, the target field method is used for reverse design of the coil. The stream function is expanded by two-dimensional Fourier series. According to the relationship between the stream function and the current density, the current density expression is substituted into the Biot-Savart law to obtain a matrix equation. The regularization matrix is introduced to solve the problem of ill-conditioned solution. The coil structure is obtained by discretizing the stream function contour. Second, the intelligent algorithm is used to optimize the weight coefficient of the coil, so that the coil configuration with the best magnetic field uniformity is obtained under the premise of being processable. Third, the designed coil is simulated and analyzed by finite element method. The magnetic field uniformity and coil constant of the coil are analyzed. Finally, the micro machining process is used to complete the production of the magnetic field coil, and the chip integrated millimeter level high uniformity characteristic magnetic field coil design is realized.

Description

Technical Field

[0001] This invention belongs to the field of chip-level magnetic field coil design technology, specifically relating to a chip-integrated miniature dual-plane magnetic field coil design method. Background Technology

[0002] In recent years, with the in-depth research on quantum precision measurement theory and the continuous progress of engineering technology, various quantum sensors based on the principle of interaction between light, magnetic fields and atoms have flourished. Among them, the SERF atomic magnetometer, as a magnetic field sensor with the potential for high precision and small size, is of great significance for basic physics research, deep space and deep sea exploration, and biomagnetic measurement, and has become a research hotspot both domestically and internationally in recent years.

[0003] The SERF atomic magnetometer consists of several key components, including an atomic gas cell, a laser light source, a non-magnetic heating component for the gas cell, and a magnetic field coil. The magnetic field coil generates a uniform spatial magnetic field and serves as both the actuator for compensating for the residual magnetic field in the atomic magnetometer and the medium for generating the modulated magnetic field.

[0004] Current magnetometers are based on traditional optical devices, and the cost and size of these systems are nearing their limits. To further improve the spatial resolution of magnetic field measurements, the size of the magnetometer probe must be further reduced. Research on chip-scale magnetometers using micro / nano fabrication processes can further reduce the size of the magnetometer while effectively lowering costs. For chip-scale atomic magnetometers developed using micro / nano fabrication processes, the designed magnetic field coils must be integrated on-chip with micro / nano optical components such as miniature gas cells, micro / nano planar collimating lenses, and micro / nano polarizers. Therefore, when designing micro / nano magnetic field coils, it is crucial to fully consider factors such as the coil's size and whether its geometry can be integrated on-chip using micro / nano fabrication processes.

[0005] Therefore, in order to solve the above problems, this invention designs a millimeter-level dual-plane magnetic field coil for chip-based magnetometers. Theoretical calculations, parameter optimization, and simulation verification of the coil were carried out in sequence. Finally, the coil was fabricated using micro-nano fabrication technology, which further improves the magnetic field uniformity in the probe area of ​​the chip-based magnetometer. This invention also provides a new method for the design of on-chip integrated magnetic field coils for other chip-level microsystems. Summary of the Invention

[0006] This invention addresses the shortcomings of existing technologies by proposing a chip-integrated miniature dual-plane magnetic field coil design method. The feasibility of the coil is verified through theoretical calculations, parameter optimization, and simulation. Finally, the coil is fabricated using micro-nano fabrication technology, which improves the spatial uniformity of the chip-based magnetometer probe area. This invention also provides a new method for the design of on-chip integrated magnetic field coils for other chip-level microsystems.

[0007] The technical solution of the present invention is as follows:

[0008] A method for designing a chip-integrated miniature dual-plane magnetic field coil, characterized by comprising the following steps:

[0009] Step 1: The target field method is used to reverse engineer a planar coil with millimeter-scale dimensions, and the geometry and parameters of the coil are derived.

[0010] Step 2: The weighting coefficients of the coil are optimized using an optimization algorithm. The stream function is obtained using the optimized weighting coefficients. The stream function is discretized using contour lines to obtain the coil structure with optimized parameters.

[0011] Step 3: Verify the effect of weight coefficient optimization on the magnetic field uniformity of the target region through simulation. The target region is a cube region in the space where the two-plane magnetic field coils hold the center.

[0012] Step 4: Use micro-nano fabrication technology to complete the fabrication of the miniature dual-plane magnetic field coil, realizing on-chip integration of the coil.

[0013] The optimization algorithm in step 2 uses a particle swarm optimization (PSO) program to optimize the weight coefficients of the Tikhonov matrix. The PSO program includes the following steps:

[0014] Step 2a, set the maximum number of iterations;

[0015] Step 2b: Initialize the position and velocity limits, set the particle swarm size, and initialize position and velocity.

[0016] Step 2c: Calculate the fitness function for each particle and take the minimum value as the optimal solution for that individual particle.

[0017] Step 2d: Find the global optimal solution and record its location;

[0018] Step 2e: Update the position and velocity of the particles;

[0019] Step 2f: Has the maximum number of iterations been reached? If not, return to step 2c; if yes, proceed to step 2g.

[0020] Step 2g: Output the optimal solution.

[0021] Let the Tikhonov matrix be Γ, the weight coefficients of the Tikhonov matrix be λ, the target region magnetic induction intensity matrix be B, the matrix containing the side length L of the two-plane coil be A, and the product matrix of the coefficients of the two-dimensional Fourier series expansion of the stream function be w. Then, the following expression is obtained:

[0022] w = (A T A+λΓT Γ) -1 A T B

[0023] Substituting the particle's position and velocity information into the following formula, the updated position and velocity of the particle after each iteration are:

[0024]

[0025]

[0026] in It represents the velocity of the i-th particle in the t-th iteration, where t is the iteration number. It is the position of the i-th particle in the t-th iteration. This represents the velocity of the i-th particle in the (t+1)-th iteration. Let ω represent the position of the i-th particle in the (t+1)-th iteration, ω be the inertia weight, r1 and r2 be arbitrary numbers between 0 and 1, and C1 and C2 be learning factors. It is the optimal position for an individual. It is the globally optimal position.

[0027] Step 1 includes: For the design of a dual-plane coil, a coil current function is introduced, and the current function is expanded into a two-dimensional Fourier series in the XOY plane coordinate system; the parameters of the current function and the set values ​​of the magnetic induction intensity at each point in the target region are set respectively, and the current density is substituted into the Biot-Savart law to obtain the calculated value of the magnetic induction intensity in the target region. The calculated value and the set value of the magnetic field intensity at each point in the target region are made equal to construct a matrix equation. To address the ill-conditioned solution problem of the equation encountered in the design of finite-size coils, the Tikhonov regularization method is introduced to solve the ill-conditioned solution problem. The weight coefficients and penalty function of the regularization matrix are substituted into the error equation constructed using the least squares method, the differential of the error function is set to zero, and the matrix equation is solved to obtain the unknown parameters of the current function.

[0028] The fitness function in step 2c is represented by the maximum non-uniformity error of the dual-plane coil:

[0029]

[0030] Where ΔB is the non-uniformity error, B θ (x,y,z) represents the magnetic flux density of the magnetic field generated by the coil at any point in the target region, B θ (0,0,0) represents the magnetic flux density of the magnetic field generated by the coil at the center of the target region.

[0031] Step 3 includes: using COMSOL Multiphysics 6.0 finite element method (FEM) simulation software to solve for the uniformity of the magnetic field generated by the coil. A pair of square planar coils with side lengths of 10mm×10mm are placed parallel to each other with a spacing of 10mm. The uniformity of the magnetic field generated by the miniature dual-plane magnetic field coil in the target area with a volume of 4mm×4mm×4mm at the center position is calculated to be 4.6%, and the coil constant is 17.101nT / mA.

[0032] The micro-nano fabrication process in step 4 includes: designing the dual-plane magnetic field coil as two 10mm×10mm square planar coils, using Au, Ag, or Cu metal materials for the coil traces, determining the trace width and the shortest distance between the traces based on the coil configuration; using 500um SiO2 as the substrate layer and 400nm metal material as the traces; to avoid short circuits caused by trace crossings, each planar coil trace adopts a double-layer design, using 400nm thick SiO2 as the insulating layer between the two metal traces; and completing the coil fabrication through deposition, coating, exposure, development, etching, and stripping processes.

[0033] The technical effects of this invention are as follows: This invention discloses a chip-integrated micro dual-plane magnetic field coil design method. First, the coil is designed in reverse using the target field method. The stream function is expanded into a two-dimensional Fourier series. Based on the relationship between the stream function and the current density, the current density expression is substituted into the Biot-Savart law to obtain the matrix equation. A regularization matrix is ​​introduced to solve the ill-conditioned solution problem of the equation. The stream function contour lines are discretized to obtain the coil structure. Second, an intelligent algorithm is used to optimize the coil weight coefficients to obtain the coil configuration with optimal magnetic field uniformity while ensuring manufacturability. Then, the designed coil is subjected to finite element simulation analysis to analyze the coil magnetic field uniformity, coil constants, and other indicators. Finally, the magnetic field coil is fabricated using microfabrication technology, realizing the design of a chip-integrated, millimeter-level high-uniformity magnetic field coil. Attached Figure Description

[0034] Figure 1 This is a schematic flowchart illustrating the design method of a chip-integrated micro dual-plane magnetic field coil according to the present invention. Figure 1The process, between the start and end, includes: Step 1, selecting the stream function based on the coil type; Step 2, setting the stream function parameters and the target region magnetic field size; Step 3, optimizing the weight coefficients λ of the Tikhonov matrix (the Tikhonov matrix is ​​also known as the Tikhonov matrix); Step 4, obtaining the unknown parameter Qmn of the stream function; and Step 5, discretizing the stream function using the contour line method to obtain the coil structure. Step 3 includes: Step 3a, setting the maximum number of iterations; Step 3b, initializing the position and velocity limits, setting the particle swarm size, and initializing position and velocity; Step 3c, calculating the fitness function of each particle and taking the minimum value as the individual optimal solution; Step 3d, finding the global optimal solution and recording its position; Step 3e, updating the particle position and velocity; Step 3f, checking if the maximum number of iterations has been reached; if not, returning to Step 3c; if yes, proceeding to Step 3g; and Step 3g, outputting the optimal solution. Figure 1 The left side shows the overall process of coil design, and the right side shows the particle swarm algorithm program for optimizing weight coefficients.

[0035] Figure 2 Is adopted Figure 1 A schematic diagram of the routing shape of the miniature dual-plane magnetic field coil designed by the method of this invention. Figure 2 In the figure, X-Position(mm) represents the x-coordinate of each point in the coil plane, and Y-Position(mm) represents the y-coordinate of each point in the coil plane. Figure 2 The circular loop in the diagram represents the coil trace shape discretized from contour lines. During the operation of the coil, current in a specific direction needs to be passed through the circular loop. Figure 2 The mid-plane coil is square, with the edge positions being x = ±5mm and y = ±5mm.

[0036] Figure 3 Is adopted Figure 1 A schematic diagram of the placement structure of the miniature dual-plane magnetic field coil designed by the method of this invention. Figure 3 Two planar coils are placed parallel to each other with a 10mm gap (located at z = -5mm and z = 5mm respectively). The target area is a 4mm × 4mm × 4mm cube area in the center. Both planar coils are square (the edge positions are x = ±5mm and y = ±5mm respectively).

[0037] Figure 4 Is adopted Figure 1 A simulation diagram illustrating the uniformity of the magnetic field generated at various points in the three planes of the target region by the miniature dual-plane magnetic field coil designed by the method of this invention. Figure 4The target region is a 4mm × 4mm × 4mm cube with edges at x = ±2mm, y = ±2mm, and z = ±2mm. The target region has three planes: XOY, YOZ, and XOZ. Uniformity is defined as the relative deviation between the magnetic field strength at each point in the target region and the magnetic field strength at the center point. For all points in the region, the smaller the maximum relative deviation, the higher the magnetic field uniformity. As shown in the figure, the magnetic field uniformity within the three planes of this region is 3.78%, which is better than that of a Helmholtz coil of the same volume, demonstrating that the expected design effect has been achieved. Detailed Implementation

[0038] The following is in conjunction with the attached diagram ( Figures 1-4 The invention will be described in the following sections and examples.

[0039] Figure 1 This is a schematic flowchart illustrating the design method of a chip-integrated micro dual-plane magnetic field coil according to the present invention. Figure 2 Is adopted Figure 1 A schematic diagram of the routing shape of the miniature dual-plane magnetic field coil designed by the method of this invention. Figure 3 Is adopted Figure 1 A schematic diagram of the placement structure of the miniature dual-plane magnetic field coil designed by the method of this invention. Figure 4 Is adopted Figure 1 A simulation diagram illustrating the uniformity of the magnetic field generated at various points within the three planes of the target region by the miniature dual-plane magnetic field coil designed using the method of this invention. (Reference) Figures 1 to 4As shown, a chip-integrated miniature dual-plane magnetic field coil design method includes the following steps: Step 1, reverse design of a millimeter-sized planar coil using the target field method to derive the coil's geometry and parameters; Step 2, optimization of the coil's weighting coefficients using an optimization algorithm, obtaining the stream function using the optimized weighting coefficients, and discretizing the stream function using contour lines to obtain the optimized coil structure; Step 3, verification of the impact of weighting coefficient optimization on the magnetic field uniformity of the target region through simulation, wherein the target region is the cubic region at the center of the dual-plane magnetic field coil clamping space; Step 4, fabrication of the miniature dual-plane magnetic field coil using micro-nano fabrication technology to achieve on-chip integration of the coil. The optimization algorithm in step 2 uses a particle swarm optimization (PSO) program to optimize the weight coefficients of the Tikhonov matrix. The PSO program includes the following steps: Step 2a, setting the maximum number of iterations; Step 2b, initializing the position and velocity limits, setting the number of particles, and initializing the position and velocity; Step 2c, calculating the fitness function of each particle and taking the minimum value as the individual optimal solution; Step 2d, finding the global optimal solution and recording its position; Step 2e, updating the position and velocity of the particles; Step 2f, checking if the maximum number of iterations has been reached. If not, return to step 2c; if yes, proceed to step 2g; Step 2g, outputting the optimal solution.

[0040] Let the Tikhonov matrix be Γ, the weight coefficients of the Tikhonov matrix be λ, the target region magnetic induction intensity matrix be B, the matrix containing the side length L of the two-plane coil be A, and the product matrix of the coefficients of the two-dimensional Fourier series expansion of the stream function be w. Then, the following expression is obtained:

[0041] w = (A T A+λΓ T Γ) -1 A T B

[0042] Substituting the particle's position and velocity information into the following formula, the updated position and velocity of the particle after each iteration are:

[0043]

[0044]

[0045] in It represents the velocity of the i-th particle in the t-th iteration, where t is the iteration number. It is the position of the i-th particle in the t-th iteration. This represents the velocity of the i-th particle in the (t+1)-th iteration. Let ω represent the position of the i-th particle in the (t+1)-th iteration, ω be the inertia weight, r1 and r2 be arbitrary numbers between 0 and 1, and C1 and C2 be learning factors. It is the optimal position for an individual. It is the globally optimal position.

[0046] Step 1 includes: for the design of a dual-plane coil, introducing a coil current function, and performing a two-dimensional Fourier series expansion of the current function in the XOY plane coordinate system; setting the parameters of the current function and the set values ​​of the magnetic induction intensity at each point in the target region, substituting the current density into the Biot-Savart law to obtain the calculated value of the magnetic induction intensity in the target region, and constructing a matrix equation by making the calculated value and the set value of the magnetic field intensity at each point in the target region equal. To address the ill-conditioned solution problem encountered in finite-size coil design, the Tikhonov regularization method is introduced to solve the ill-conditioned solution problem. The weight coefficients and penalty function of the regularization matrix are substituted into the error equation constructed using the least squares method, the differential of the error function is set to zero, and the matrix equation is solved to obtain the unknown parameters of the current function. The fitness function in step 2c represents the maximum non-uniformity error of the dual-plane coil.

[0047]

[0048] Where ΔB is the non-uniformity error, B θ (x,y,z) represents the magnetic flux density of the magnetic field generated by the coil at any point in the target region, B θ (0,0,0) represents the magnetic flux density of the magnetic field generated by the coil at the center of the target region.

[0049] Step 3 includes: using COMSOL Multiphysics 6.0 finite element method (FEM) simulation software to solve for the uniformity of the magnetic field generated by the coil. A pair of square planar coils with side lengths of 10mm×10mm are placed parallel to each other with a spacing of 10mm. The uniformity of the magnetic field generated by the miniature dual-plane magnetic field coil in the target area with a volume of 4mm×4mm×4mm at the center position is calculated to be 4.6%, and the coil constant is 17.101nT / mA. The micro-nano fabrication process in step 4 includes: designing the dual-plane magnetic field coil as two 10mm×10mm square planar coils, using Au, Ag, or Cu metal materials for the coil traces, determining the trace width and the shortest distance between the traces based on the coil configuration; using 500um SiO2 as the substrate layer and 400nm metal material as the traces; to avoid short circuits caused by trace crossings, each planar coil trace adopts a double-layer design, using 400nm thick SiO2 as the insulating layer between the two metal traces; and completing the coil fabrication through deposition, coating, exposure, development, etching, and stripping processes.

[0050] A chip-integrated miniature dual-plane magnetic field coil design is presented. First, a miniature magnetic field coil suitable for chip microsystems is designed. The coil is reverse-engineered using the target field method to derive its geometry. Second, an optimization algorithm is used to optimize the weighting coefficients of the coil, and simulation is used to verify the impact of the weighting coefficient optimization on the magnetic field uniformity of the target region. Finally, the miniature magnetic field coil is fabricated using micro-nano fabrication technology to achieve on-chip integration of the coil.

[0051] The aforementioned chip-integrated micro dual-plane magnetic field coil reverse design theoretical calculation employs the target field method for coil reverse design. For dual-plane coil design, a coil current function is introduced, and this function is expanded into a two-dimensional Fourier series in the XOY plane coordinate system. By setting the current function parameters and the set values ​​of magnetic flux density at each point in the target region, the current density is substituted into the Biot-Savart law to obtain the calculated value of the magnetic flux density in the target region. The calculated and set values ​​of the magnetic flux density at each point in the target region are then made equal to construct a matrix equation. To address the ill-conditioned solution problem encountered in finite-size coil design, the Tikhonov regularization method is introduced. The weight coefficients and penalty function of the regularization matrix are substituted into the error equation constructed using the least squares method. The differential of the error function is set to zero, and the matrix equation is solved to obtain the unknown parameters of the current function. Finally, the current function is discretized using the contour method to obtain the coil structure.

[0052] The chip-integrated micro dual-plane magnetic field coil parameter optimization comprehensively considers the uniformity of the magnetic field generated by the coil and the coil's manufacturability to obtain the value range of the coil weight coefficient. An optimization algorithm is used to optimize the weight coefficient of the magnetic field coil, finding the optimal value within the acceptable range, and obtaining a coil design with optimal magnetic field uniformity under the premise of coil manufacturability.

[0053] The design and simulation of the chip-integrated micro dual-plane magnetic field coil is characterized by using COMSOL Multiphysics 6.0 finite element method (FEM) simulation software to solve for the uniformity of the magnetic field generated by the coil. A pair of square planar coils with side lengths of 10mm×10mm are placed parallel to each other with a spacing of 10mm. The uniformity of the magnetic field generated by the micro dual-plane magnetic field coil in the target area with a volume of 4mm×4mm×4mm at the center position is calculated to be 4.6%, and the coil constant is 17.101nT / mA.

[0054] The aforementioned chip-integrated micro-dual-plane magnetic field coil micro-nano fabrication process design is characterized by the dual-plane magnetic field coil being designed as two 10mm×10mm square planar coils. The coil traces can be made of metals such as Au, Ag, and Cu. The trace width and the shortest distance between traces are determined based on the coil configuration. A 500µm SiO2 substrate is used as the base layer, and a 400nm metal material is used as the trace. To avoid short circuits caused by trace crossings, each planar coil employs a double-layer design, using a 400nm thick SiO2 layer as the insulating layer between the two metal traces. The coil fabrication is completed through processes including deposition, coating, exposure, development, etching, and stripping.

[0055] This invention describes a miniature dual-plane magnetic field coil for chip microsystems and its design method. The invention provides a detailed derivation of the reverse design process for the optimized dual-plane coil, explains the simulation process of the designed miniature dual-plane coil, analyzes the simulation results, and details the MEMS fabrication process of the miniature dual-plane coil.

[0056] The technical solution adopted in this invention is as follows:

[0057] Under the constraint of limited coil size, the coil geometry was first calculated theoretically using reverse design methods. Then, the coil parameters were optimized using optimization algorithms, and the impact of parameter optimization on the magnetic field uniformity of the target area was verified through simulation. Finally, the micro-nano fabrication process was used to complete the fabrication of the micro magnetic field coil, realizing the on-chip integration of the coil.

[0058] The specific method is as follows:

[0059] (1) Theoretical Analysis of Reverse Design of Miniature Magnetic Field Coil

[0060] Treating the current on the surface of the biplane coil as a two-dimensional incompressible continuous fluid, let the current density be...

[0061]

[0062] in, It is the current density vector in space. Let u be the unit direction vector in space, u be the x-component of the current density, and v be the y-component of the current density.

[0063] Since streamlines are tangent to the flow direction at every point in the flow field, the fundamental differential equation of streamlines can be obtained:

[0064]

[0065] in, The differential of the unit direction vector is... Substituting, we get:

[0066] -udy+vdx=0

[0067] Where dx is the differential of the position variable in the x-direction, dy is the differential of the position variable in the y-direction, and the stream function of the streamline at a given time is... For a given time, the stream function is a constant. Taking the derivative, we can obtain

[0068]

[0069] dS(r) is the differential of the stream function. Let be the partial derivative of the stream function in the x-direction. Let be the partial derivative of the stream function in the y-direction, where the relationship between the stream function and the current density in each direction is:

[0070]

[0071]

[0072] Among them, J x It is the x-direction component of the current density, J y This represents the y-direction component of the current density. The current densities of the shimming coil in the x, y, and z directions share commonalities. Taking the shimming coil in the x-direction as an example, expanding the stream function into a two-dimensional Fourier series yields the current density in the y-direction:

[0073]

[0074] Where M is the order of the Fourier series expansion of the stream function in the x-direction, N is the order of the Fourier series expansion of the stream function in the y-direction, m is a positive integer less than or equal to M, n is a positive integer less than or equal to N, and Q mn This is the product of the coefficients of the two-dimensional Fourier series expansion of the stream function, where L is the side length of the biplane coil, x is the x-coordinate of a point in the coil plane, and y is the y-coordinate of the point in the coil plane. Substituting the current density expression into the Biot-Savart law, we can obtain the relationship between the current density and the magnetic field at any point in the target region:

[0075]

[0076] Among them, B x (x, y, z) represents the calculated magnetic flux density at a point in the target region, where x, y, and z are the three-axis coordinates of the point in the target region, μ0 is the free permeability (Henry / meter), and J is the magnetic flux density. y(x',y') represents the y-component of the current density at a point on the plane of the dual-plane coil, where x' and y' are the x-axis and y-axis coordinates of the midpoint of the dual-plane coil, respectively. z is the z-axis coordinate of the target region point, z' is the z-axis coordinate of the dual-plane coil, r is the radial vector from the target region point to the origin, r' is the radial vector from the current source on the coil surface to the origin, and dA' is the area element at any point on the dual-plane coil.

[0077] The target region is discretized into NUM points, where NUM is a positive integer. The magnetic field strength at each point is set according to the required magnetic field strength of the target region. Taking a uniform magnetic field in the x-direction as an example, it is sufficient to set the magnetic field strength in the x-direction of all points to the same value B. target That is, B x (x,y,z) is represented in matrix form, from which the following system of equations can be constructed to solve for the current density function:

[0078]

[0079]

[0080] Among them, U mn Let z1' be the z-axis coordinate of the left double-plane coil, z2' be the z-axis coordinate of the right double-plane coil, dx' be the differential of x', and dy' be the differential of y'.

[0081] The number of target points NUM is generally much larger than m·n. Solving the above system of equations will result in an overdetermined system of equations. Using the least squares method, the above equations are transformed into the form of an error function:

[0082]

[0083] Where E is the error function, x num ,y num ,z num Let B be the x-axis, y-axis, and z-axis coordinates of the num-th point in the target region, where num is a positive integer less than or equal to NUM. x (x num ,y num ,z num ) refers to B x The magnetic field strength generated by the coil at the num-th point in the target field will result in ill-conditioned solutions when the above equation is directly solved, leading to an inaccurate determination of Q. mn Not unique. By introducing the Tikhonov regularization method to restrict the above equation, we can obtain:

[0084]

[0085] Where E' is the constrained error function, Γ is the Tikhonov matrix, and λ is the weighting coefficient of the Tikhonov matrix. Γ can be solved using a penalty function. Using the curvature function of the biplane coil as the penalty function of the Tikhonov matrix to constrain the above equations not only solves the ill-conditioned solution problem but also prevents the coil structure from becoming overly complex, reducing the difficulty of subsequent coil manufacturing processes. The expression for solving the curvature function of the biplane coil is:

[0086]

[0087] in, Let the differential of the flow function of coil Bx at the infinitesimal area element dA' be given. Substituting the expression for solving the curvature function into the regularization equation, and calculating that the differential of the error function equals zero, we can obtain the following matrix:

[0088] w = (A T A+λΓ T Γ) -1 A T B

[0089] Where w is a variable containing the unknown Q mn A is a matrix containing the magnetic flux density values ​​of NUM points in the target region, B is a matrix containing the unknown Umn.

[0090] w = [Q] 11 Q 12 …Q MN ] T

[0091]

[0092]

[0093] Solving the w matrix yields the unknown parameter Q of the stream function. mn Q mn Substituting the stream function expression into the input yields the final expression for the stream function. Discretizing the stream function using contour lines allows us to obtain the coil winding structure.

[0094] (2) Optimization of weighting coefficients for miniature magnetic field coils

[0095] Calculations and analysis show that the smaller the weighting coefficient λ of the Tikhonov matrix, the more complex the coil winding and the greater the manufacturing difficulty; conversely, the larger λ is, the lower the magnetic field uniformity. Therefore, it is necessary to optimize the weighting coefficient λ using appropriate methods to ensure optimal magnetic field uniformity while maintaining manufacturability.

[0096] To address the above problem of optimizing the coil weight coefficient λ, this invention aims to achieve the optimal magnetic field uniformity within the acceptable complexity range of the coil, and utilizes a particle swarm optimization algorithm to optimize the weight coefficient λ. In the one-dimensional search region, there exist some particles... This represents the position of the i-th particle in the t-th iteration, and represents the direction of the particle's movement. This represents the velocity of the i-th particle in the t-th iteration, indicating how fast the particle moves.

[0097] First, set the maximum number of iterations for the algorithm to G. max The number of iterations can be set to values ​​such as 50 or 100, and the particle position limit range can be set to x. min To x max any number between x min and x max These represent the minimum and maximum values ​​of the particle's position, respectively. The particle's velocity is limited to a range of v. min to v max any number between, v min and v max These are the minimum and maximum values ​​of the particle velocity, respectively. The number of particles in the swarm is initialized to a positive integer pn, and the position of the i-th particle is initialized to... Initialize the velocity of the i-th particle as i is any positive integer between 0 and pn. The maximum deviation of the magnetic field in the target region is selected as the fitness function of the uniform coil. These fitness functions represent the maximum non-uniformity error of the dual-plane coil and can be expressed as:

[0098]

[0099] B θ (x,y,z) represents the magnetic flux density of the magnetic field generated by the coil at any point in the target region, B θ (0,0,0) represents the magnetic flux density of the magnetic field generated by the coil at the center of the target region. The fitness function of each particle is calculated, and the minimum value of all fitness functions in the history of each particle is taken as the individual optimal solution. A minimum value is then found among these individual optimal solutions, and this minimum value is compared with the historical global optimal solution. The minimum value is selected as the updated global optimal solution.

[0100] Substituting the particle's position and velocity information into the following formula, the updated position and velocity of the particle after each iteration are:

[0101]

[0102]

[0103] in, This represents the velocity of the i-th particle in the (t+1)-th iteration. Let ω represent the position of the i-th particle in the (t+1)-th iteration, ω be the inertia weight, r1 and r2 be any numbers between 0 and 1, C1 and C2 be the learning factors, and P be the position of the i-th particle in the (t+1)-th iteration. i t It is the optimal position for an individual. It is the globally optimal position.

[0104] After reaching the maximum number of iterations, the optimization algorithm terminates, outputting the particle position corresponding to the global optimal fitness as the weight coefficient λ. The optimized weight coefficient λ is then substituted into the solution to obtain the stream function. Finally, contour lines are used to discretize the stream function to obtain the parameter-optimized coil structure.

[0105] (3) Simulation of miniature magnetic field coils

[0106] Finite element simulation of the optimized coil was performed using COMSOL software. The designed planar coils with a side length of 10mm were placed parallel to each other with a 10mm interval. The 4mm×4mm×4mm area at the center of the double-planar coil was taken as the target area. The side current was input into the designed coil winding. The simulation results show that the uniformity of the magnetic field generated by the 10mm×10mm planar coil in the 4mm×4mm×4mm target area at the center of the double-planar coil is better than 4.6%, and the coil constant is 17.101nT / mA.

[0107] (4) Fabrication of miniature magnetic field coils

[0108] Micro / nano coil fabrication process:

[0109] First, a 400nm thick silicon dioxide thermal oxide layer is deposited on a 500um thick silicon dioxide substrate to facilitate subsequent etching.

[0110] Next, a SiO2 substrate is covered with photoresist, and the photoresist is then exposed, developed, and hard-baked using a mask. The exposed thermal oxide layer after the photoresist patterning is etched to obtain the embedded trench of the first metal layer.

[0111] Then, upper and lower metal coil traces with a thickness of 400nm are grown by magnetron sputtering, and the two metal layers are insulated by a 400nm thick SiO2 insulating layer.

[0112] Finally, a 500µm thick SiO2 layer is deposited as a protective layer using plasma-enhanced chemical vapor deposition (PECVD). The protective layer is then etched at the pads to expose the upper metal pads, and gold wire bonding is used to connect the electrodes at the pads.

[0113] 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 chip-integrated miniature dual-plane magnetic field coil, characterized in that, Includes the following steps: Step 1: The target field method is used to reverse engineer a planar coil with millimeter-scale dimensions, and the geometry and parameters of the coil are derived. Step 2: The weighting coefficients of the coil are optimized using an optimization algorithm. The stream function is obtained using the optimized weighting coefficients. The stream function is discretized using contour lines to obtain the coil structure with optimized parameters. Step 3: Verify the effect of weight coefficient optimization on the magnetic field uniformity of the target region through simulation. The target region is a cube region in the space where the two-plane magnetic field coils hold the center. Step 4: The micro dual-plane magnetic field coil is fabricated using micro-nano fabrication technology to achieve on-chip integration of the coil; The optimization algorithm in step 2 uses a particle swarm optimization (PSO) program to optimize the weight coefficients of the Tikhonov matrix. The PSO program includes the following steps: Step 2a, set the maximum number of iterations; Step 2b: Initialize the position and velocity limits, set the particle swarm size, and initialize position and velocity. Step 2c: Calculate the fitness function for each particle and take the minimum value as the optimal solution for that individual particle. Step 2d: Find the global optimal solution and record its location; Step 2e: Update the position and velocity of the particles; Step 2f: Has the maximum number of iterations been reached? If not, return to step 2c; if yes, proceed to step 2g. Step 2g: Output the optimal solution.

2. The chip-integrated miniature dual-plane magnetic field coil design method according to claim 1, characterized in that, The Tikhonov matrix is ​​set as Let the weight coefficients of the Tikhonov matrix be λ, the target region magnetic induction intensity matrix be B, the matrix containing the side length L of the two-plane coil be A, and the product matrix of the coefficients of the two-dimensional Fourier series expansion of the stream function be w. Then we get the following expression: ， Substituting the particle's position and velocity information into the following formula, the updated position and velocity of the particle after each iteration are: ， ， in It represents the velocity of the i-th particle in the t-th iteration, where t is the iteration number. It is the position of the i-th particle in the t-th iteration. This represents the velocity of the i-th particle in the (t+1)-th iteration. This represents the position of the i-th particle in the (t+1)-th iteration. It is inertial weight. and All are any numbers between 0 and 1. and All are learning factors. It is the optimal position for an individual. It is the globally optimal position.

3. The chip-integrated miniature dual-plane magnetic field coil design method according to claim 1, characterized in that, Step 1 includes: for the design of a dual-plane coil, introducing a coil current function and performing a two-dimensional Fourier series expansion of the current function in the XOY plane coordinate system; setting the current function parameters and the set values ​​of the magnetic induction intensity at each point in the target region, substituting the current density into the Biot-Savart law to obtain the calculated value of the magnetic induction intensity in the target region, and constructing a matrix equation by making the calculated value and the set value of the magnetic field intensity at each point in the target region equal; for the ill-conditioned solution problem of the equation encountered in the design of finite-size coils, introducing the Tikhonov regularization method to solve the ill-conditioned solution problem of the equation, substituting the weight coefficients and penalty function of the regularization matrix into the error equation constructed using the least squares method, setting the differential of the error function to zero, and solving the matrix equation to obtain the unknown parameters of the current function.

4. The chip-integrated miniature dual-plane magnetic field coil design method according to claim 1, characterized in that, The fitness function in step 2c is represented by the maximum non-uniformity error of the dual-plane coil: ， in It is non-uniformity error. This represents the magnetic flux density at which the coil generates a magnetic field at any point in the target region. This represents the magnetic flux density of the magnetic field generated by the coil at the center of the target region.

5. The chip-integrated miniature dual-plane magnetic field coil design method according to claim 1, characterized in that, Step 3 includes: using COMSOL Multiphysics 6.0 finite element method (FEM) simulation software to solve for the uniformity of the magnetic field generated by the coil. A pair of square planar coils with side lengths of 10mm×10mm are placed parallel to each other with a spacing of 10mm. The uniformity of the magnetic field generated by the miniature dual-plane magnetic field coil in the target area with a volume of 4mm×4mm×4mm at the center position is calculated to be 4.6%, and the coil constant is 17.101nT / mA.

6. The chip-integrated miniature dual-plane magnetic field coil design method according to claim 1, characterized in that, The micro-nano fabrication process in step 4 includes: designing the dual-plane magnetic field coil as two 10mm×10mm square planar coils, using Au, Ag, or Cu metal materials for the coil traces, determining the trace width and the shortest distance between the traces based on the coil configuration; using 500um SiO2 as the substrate layer and 400nm metal material as the traces; to avoid short circuits caused by trace crossings, each planar coil trace adopts a double-layer design, using 400nm thick SiO2 as the insulating layer between the two metal traces; and completing the coil fabrication through deposition, coating, exposure, development, etching, and stripping processes.

Images