Solenoid coil magnetic field analysis method for axial gauss meter calibration

Abstract

The invention provides a solenoid coil magnetic field analysis method aiming at axial gauss meter calibration. The solenoid coil magnetic field analysis method comprises the following steps: S1, introducing a steady current into a limited-length solenoid coil with multiple layers of windings and constructing a cylindrical coordinate system; s2, cutting the electrified winding into directed current elements, and obtaining first coordinate data of a point where the current elements are located and second coordinate data of any spatial point; s3, obtaining size data of the limited-length solenoid coil, obtaining a current direction vector expression of the current infinitesimal according to the first coordinate data, and obtaining a vector diameter vector expression from a point where the current infinitesimal is located to any spatial point according to the first coordinate data and the second coordinate data; and S4, performing vector product operation on the current direction vector expression and the vector diameter vector expression, and combining the magnetic field quantity expression and the size data to obtain magnetic field components of the magnetic field vector in each direction. The method has the beneficial effect that the whole space vector magnetic field in the limited-length solenoid can be accurately calculated.

Description

Technical Field

[0001] This invention relates to the technical field of on-site calibration of axial gaussmeters, and more specifically, to a method for analyzing the magnetic field of a solenoid coil for axial gaussmeter calibration. Background Technology

[0002] Magnetic field scanning equipment is a key instrument for characterizing the magnetic properties of high-performance rare-earth permanent magnet materials. The accuracy of its core component—the axial gaussmeter (i.e., the axial magnetic field probe)—directly determines the overall measurement reliability of the equipment. To ensure accurate measurements, the axial gaussmeter needs to be calibrated regularly. Currently, calibration mainly relies on providing a known, uniform standard magnetic field in a laboratory environment using a standard magnetic field source (such as an electromagnet or a long solenoid coil).

[0003] Among them, solenoid coils are widely used as standard sources of axial magnetic fields due to their simple structure and unidirectional magnetic field. Traditional calibration theories and design methods are based on the ideal model of an "infinitely long solenoid," which assumes the existence of a uniform magnetic field region near the central axis of the solenoid, whose magnetic induction intensity can be expressed by the formula... Simplified calculations are performed. However, in practical applications, to meet the requirements of on-site calibration for equipment portability and probe installation space, calibration devices must be developed towards miniaturization and finite length. Once the length of the solenoid is limited, its internal magnetic field distribution will deviate significantly from the ideal model, the edge effect will be sharply enhanced, the uniform magnetic field region on the axis will be shortened, and a non-negligible radial magnetic field component will be generated in the non-axial region (i.e., the space actually occupied by the probe).

[0004] Current technologies lack rapid and accurate methods for calculating the internal spatial magnetic field distribution of finite-length, multi-layered winding solenoids. Traditional simplified formulas are only suitable for rough estimations of the axis center point, while significant errors occur in critical edge regions and the space outside the axis, making them unsuitable for accurately defining uniform magnetic field areas. This often leads to reliance on experience or costly trial and error in the design of practical calibration systems, resulting in long design cycles, high costs, and difficulty in guaranteeing calibration accuracy and reliability. Therefore, developing a method capable of accurately calculating the full-space vector magnetic field inside a finite-length solenoid has become an urgent need to overcome the technical bottleneck of axial gaussmeter field calibration and to realize the design of miniaturized standard magnetic field sources. Summary of the Invention

[0005] The technical problem to be solved by this invention is how to accurately calculate the full-space vector magnetic field inside a finite-length solenoid. To overcome the defects of the above-mentioned prior art (or related art), this invention provides a method for analyzing the magnetic field of a solenoid coil for axial gaussmeter calibration.

[0006] This invention provides a method for analyzing the magnetic field of a solenoid coil during axial gaussmeter calibration, comprising the following steps: Step S1: A steady current is passed through a finite-length solenoid coil with multiple windings, and a cylindrical coordinate system is constructed within the finite-length solenoid coil. Step S2: Based on Biot-Savart's law and the principle of magnetic field vector superposition, the energized winding in the finite-length solenoid coil is cut into directed current elements, and the first coordinate data of the point where the current micro-element is located and the second coordinate data of any point in space are obtained. Step S3: Obtain the size data of the finite-length solenoid coil; obtain the current direction vector expression of the current micro-element based on the first coordinate data; and obtain the radius vector expression from the point where the current micro-element is located to any point in the space based on the first coordinate data and the second coordinate data. Step S4: Perform vector product operation on the current direction vector expression and the radius vector expression, and combine them with the pre-constructed magnetic field quantity expression and the size data to obtain the magnetic field components of the magnetic field vector in each direction of the finite-length solenoid coil as the magnetic field analysis result.

[0007] The present invention provides a method for analyzing the magnetic field of a solenoid coil during axial gaussmeter calibration, which has the following advantages compared with existing technologies: This invention establishes a precise numerical calculation strategy based on Biot-Savart's law and the principle of magnetic field vector superposition. This strategy enables the quantitative calculation of the magnetic field components at any point in space inside a finite-length, multi-layered solenoid coil. This solves the problem of large calculation errors in the edge and off-axis regions of the traditional "infinite-length solenoid" ideal model in actual miniaturization design. The method of this invention can accurately predict the magnetic field distribution, uniform region range, and magnetic field vector direction inside the solenoid coil during the design stage. This provides a reliable theoretical basis and data support for the subsequent miniaturization and lightweight solenoid structure design, thereby fundamentally improving the accuracy of the standard magnetic field source used for calibration.

[0008] In one possible implementation, in step S1, the cylindrical coordinate system is obtained by taking the center point of the finite-length solenoid coil as the origin, the axial direction of the finite-length solenoid coil as the z-axis, the radial direction of the finite-length solenoid coil as the ρ-axis, and the counterclockwise direction around the z-axis as the φ-axis.

[0009] Compared with existing technologies, the above technical solution can fully utilize the axisymmetric characteristics of the solenoid coil structure by setting the center point and axis of the solenoid coil as the origin and z-axis of the cylindrical coordinate system, respectively. This coordinate system establishment method simplifies the subsequent magnetic field vector calculation because the magnetic field is symmetrical in the circumferential direction, thereby significantly reducing complexity and computational load and improving computational efficiency.

[0010] In one possible implementation, in step S2, the original coordinate data of the point where the current element is located in the cylindrical coordinate system is obtained, and then the original coordinate data is converted into the first coordinate data in the rectangular coordinate system based on the Cartesian coordinate system with the point where the current element is located at the origin.

[0011] Compared with existing technologies, the above-mentioned technical solution can uniformly transform the coordinates of the current element in the cylindrical coordinate system to the rectangular coordinate system with itself as the origin for vector operations, which conforms to the vector operation rules of Biot-Savart's law. This transformation ensures the accuracy and mathematical rigor of the calculation of the current direction vector and the radius vector, and is the basis for subsequent precise vector integration.

[0012] In one possible implementation, in step S3, the expression for the direction vector of the current element is obtained by the following calculation formula: in, Indicates the direction of current The differential; express Components on the x-axis; This represents the radius of the current loop represented by the first coordinate data; This represents the angle between the radius of the current element and the x-axis; express The differential; Represents the first unit vector; express Components on the y-axis; This represents the second unit vector.

[0013] In one possible implementation, in step S3, the position vector expression is obtained using the following calculation formula: in, Represents the infinitesimal point of current. to any point in the space The radius vector; This indicates the x-axis coordinate of the second coordinate data in a rectangular coordinate system; This indicates the y-axis coordinate of the second coordinate data in a rectangular coordinate system; This represents the z-axis coordinate of the second coordinate data in a Cartesian coordinate system; This represents the z-axis coordinate of the first coordinate data in a Cartesian coordinate system; This represents the third unit vector.

[0014] In one possible implementation, before performing step S4, the expression for the magnetic field quantity of the infinitesimal current excitation is constructed using the following calculation formula: in, express The differential; Indicates the permeability of free space; This represents the current intensity of a steady current.

[0015] In one possible implementation, in step S4, the magnetic field components in each direction are obtained using the following calculation formula: in, This represents the magnetic field component along the x-axis in the Cartesian coordinate system; This represents the equivalent current density; Indicates the total length of the solenoid; Indicates the inner diameter of the solenoid; Indicates the thickness of the solenoid; express The differential; express The differential; This represents the radial coordinates of any point P in the space within the cylindrical coordinate system. This represents the azimuth coordinates of any point P in the space within the cylindrical coordinate system. This represents the magnetic field component along the y-axis in the Cartesian coordinate system; The magnetic field component in the z-axis direction is represented in both Cartesian and cylindrical coordinate systems.

[0016] In one possible implementation, after performing step S4, the method further includes: Based on the axisymmetric properties of the magnetic field, the azimuth coordinates of any point in space are set to 0 so that the point in space lies in the xOz plane of the cylindrical coordinate system, and then the magnetic field components in each direction are analyzed.

[0017] Compared with existing technologies, the above-mentioned technical solution can make full use of the symmetry of the magnetic field distribution to simplify the calculation. By setting the spatial point in the xOz plane, the magnetic field component in the y-axis direction can be made to be zero, thereby reducing the number of variables that need to be calculated and significantly improving the calculation efficiency. Especially when performing a large number of numerical integrations, this simplification can save a lot of computing resources and time, while ensuring the correctness of the calculation results. Attached Figure Description

[0018] Figure 1 This is a flowchart of the steps of the present invention; Figure 2 This is a schematic diagram of the Biot-Savart law of the present invention; Figure 3 This is a structural model diagram of the finite-length solenoid coil of the present invention; Figure 4 This is a schematic diagram of the magnetic field calculation current element of the finite-length solenoid coil of the present invention; Figure 5 This is a finite element analysis model and mesh generation diagram of the finite-length solenoid coil of the present invention; Figure 6 This is a schematic diagram of the simulation results of the magnetic field distribution of the finite-length solenoid coil of the present invention; Figure 7 This is a schematic diagram showing the magnetic field distribution along the axis and symmetry plane of the finite-length solenoid coil of the present invention. Detailed Implementation

[0019] First, those skilled in the art should understand that these embodiments are merely illustrative of the technical principles of the present invention and are not intended to limit the scope of protection of the present invention. Those skilled in the art can make adjustments as needed to adapt to specific application scenarios.

[0020] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0021] See Figure 1 This invention discloses a method for analyzing the magnetic field of a solenoid coil during axial gaussmeter calibration, comprising the following steps: Step S1: A steady current is passed through a finite-length solenoid coil with multiple windings, and a cylindrical coordinate system is constructed within the finite-length solenoid coil. Step S2: Based on Biot-Savart's law and the principle of magnetic field vector superposition, the energized winding in the finite-length solenoid coil is cut into directed current elements, and the first coordinate data of the point where the current element is located and the second coordinate data of any point in space are obtained. Step S3: Obtain the dimensional data of the finite-length solenoid coil; obtain the current direction vector expression of the current element based on the first coordinate data; and obtain the position vector expression between the point where the current element is located and any point in space based on the first coordinate data and the second coordinate data. Step S4: Perform vector product operation on the current direction vector expression and the radius vector expression, and combine them with the pre-constructed magnetic field quantity expression and size data to obtain the magnetic field components of the magnetic field vector in each direction in the finite-length solenoid coil as the magnetic field analysis result.

[0022] In this embodiment of the invention, according to Biot-Savart's law, when a steady current flows through a finite-length solenoid coil, a static magnetic field will be excited around the coil. The strength and direction of this static magnetic field are determined by the magnitude, direction, and position of the current element. Figure 2 As shown, it consists of a current element. and vector diameter Determined magnetic field quantity The magnetic field quantity conforms to the right-hand rule. The expression for the magnetic field quantity is as follows: in, express The differential; Indicates the permeability of free space; This represents the current intensity of a steady current.

[0023] In this embodiment of the invention, in step S1, a cylindrical coordinate system is obtained by taking the center point of the finite-length solenoid coil as the origin, the axial direction of the finite-length solenoid coil as the z-axis, the radial direction of the finite-length solenoid coil as the ρ-axis, and the counterclockwise direction around the z-axis as the φ-axis. In step S2, the original coordinate data of the point where the current element is located in the cylindrical coordinate system is obtained. Then, based on the rectangular coordinate system with the point where the current element is located at the origin, the original coordinate data is converted into the first coordinate data in the rectangular coordinate system.

[0024] In this embodiment of the invention, a finite-length solenoid coil with multiple layers of winding, such as... Figure 3 As shown, the magnetic field of a finite-length solenoid coil can be calculated using Biot-Savart's law and the principle of magnetic field vector superposition. The energized winding is divided into directed current elements, and the magnetic field element at any point in space is calculated. This is then integrated to obtain the magnetic field. Analysis... Figure 3As can be seen from the structure, the magnetic field integral is along the circumferential direction, radial direction, and axial direction of the finite-length solenoid coil, respectively. The current infinitesimal element analysis is as follows: Figure 4 As shown, the point where the current element is located The original coordinate data in the cylindrical coordinate system is The first coordinate data in the corresponding rectangular coordinate system is The second coordinate data of any point P in space in the rectangular coordinate system is: .

[0025] In this embodiment of the invention, the magnetic field quantity expression and Figure 4 It can be seen that the current element that excites the magnetic field consists of two parts: the magnitude of the current and the direction of the current. The direction of the current is determined by... It is determined that the magnitude of the current is determined by the point. The area of ​​the infinitesimal element at that point It was decided that the number of winding turns would be... N Current intensity is Inner diameter is Thickness is , length is In a finite-length solenoid coil, the equivalent current density on the longitudinal section is: The magnitude of the infinitesimal current is ,exist Figure 4 In the middle, directed line segment The expression for the current direction vector is as follows: in, Indicates the direction of current The differential; express The component in the x-axis direction; This represents the radius of the current loop represented by the first coordinate data; This represents the angle between the radius at the infinitesimal current element and the x-axis. express The differential; Represents the first unit vector; express The component in the y-axis direction; Represents the second unit vector; The above formula neglects the helix angle of the winding inside a finite-length solenoid coil, therefore the directed line segment The direction is tangent to the current loop and has no axial component. According to magnetic field theory, the error in magnetic field strength caused by the helix angle is proportional to the square of the helix angle in radians. Therefore, with a fixed inner diameter of the solenoid... Based on this, selecting a smaller wire diameter for winding can effectively improve the accuracy of calculations.

[0026] In this embodiment of the invention, Figure 4 Medium current micro element to any point in space The position vector can be determined by its coordinates, and its position vector expression is as follows: in, Represents the infinitesimal point of current. to any point in space The radius vector; This indicates the x-axis coordinate of the second coordinate data in a rectangular coordinate system; This indicates the y-axis coordinate of the second coordinate data in a rectangular coordinate system; This represents the z-axis coordinate of the second coordinate data in a rectangular coordinate system; This represents the z-axis coordinate of the first coordinate data in a rectangular coordinate system. Represents the third unit vector; The modulus of its position vector expression is shown below: Then, by multiplying the current direction vector expression and the position vector expression, we can obtain: Subsequently, in the cylindrical coordinate system, any point in space The coordinate transformation is as follows: The magnitude of the position vector expression and the vector product of the current direction vector expression and the position vector expression can be rewritten as: Therefore, by combining the expression for magnetic field quantity, the directional components of the spatial magnetic field vector inside a finite-length solenoid coil can be calculated as follows: in, This represents the magnetic field component along the x-axis in the Cartesian coordinate system. Represents the equivalent current density; Indicates the length of the solenoid; Indicates the inner diameter of the solenoid; Indicates the thickness of the solenoid; express The differential; express The differential; This represents the radial coordinate of any point P in space on the cylindrical coordinate system. This represents the azimuth coordinates of any point P in space on a cylindrical coordinate system. This represents the magnetic field component along the y-axis in the Cartesian coordinate system. This represents the magnetic field component along the z-axis in Cartesian and cylindrical coordinate systems.

[0027] In this embodiment of the invention, based on the symmetry of the finite-length solenoid coil structure, the magnetic field components in the cylindrical coordinate system have the following relationship: It can be seen that the magnetic field also has axial symmetry and no component in the circumferential direction. Therefore, it is only necessary to know the magnetic field distribution in the xOz plane or yOz plane. After executing step S4, the following steps are also included: Based on the axisymmetric properties of the magnetic field, the azimuth coordinates of any point in space are set to 0 so that the point is in the xOz plane of the cylindrical coordinate system. Then, the magnetic field components in each direction are analyzed. Specific order , The magnetic field components are shown below: According to the above formula, we can calculate... Figure 3 The distribution of the magnetic field inside a finite-length solenoid coil is shown. To obtain an analytical solution for the coil's magnetic field, calculations using infinite series are commonly used. However, for engineers, infinite series calculations lack concrete physical meaning and are difficult to understand. Moreover, from a computational accuracy perspective, numerical calculation methods based on the Biot-Savart theorem can achieve higher accuracy. The above accuracy fully meets the requirements for magnetic field probe calibration. In this embodiment, an integral calculation program can be developed to solve the above formula numerically, thereby improving calculation efficiency.

[0028] In this embodiment of the invention, existing software is used to perform finite element simulation analysis on the magnetic field, which further visualizes the magnetic field parameters, reveals the relationship between the coil structure and the magnetic field parameters and their distribution characteristics, and provides data support for the selection of magnetic field analysis and calibration methods. Finite element method (FEM) simulation of magnetic fields divides the computational space into multiple computational regions, thus requiring extensive calculations to obtain accurate results. This process places high demands on computer performance. While ensuring computational accuracy, reasonable simplification based on the symmetry of the computational model is an effective way to improve computational efficiency. In this embodiment, the overall structure of the cylindrical finite-length solenoid coil exhibits axisymmetry. Based on the theoretical analysis above, its magnetic field distribution also exhibits axisymmetry. Therefore, establishing a two-dimensional model in the rz plane of the cylindrical coordinate system for simulation calculations can obtain the necessary computational accuracy while significantly improving computational efficiency. Furthermore, in the rz plane, the magnetic field of the solenoid is even-symmetric with respect to the z-axis and odd-symmetric with respect to the r-axis. Based on this characteristic, only one-quarter of the two-dimensional model of the finite-length solenoid coil needs to be calculated to obtain its entire magnetic field characteristics. Figure 5 (a) is a two-dimensional model of a finite-length solenoid coil established in existing software, where the z-axis is an odd-symmetric boundary and the x-axis (r-axis) is an even-symmetric boundary. After simplification, the computational load is reduced to one-quarter of that of the complete two-dimensional model. To improve computational accuracy, the coil cross-section providing the excitation current is set to a finer mesh in the mesh generation; simultaneously, the meshes of the axis and interface are subdivided to facilitate the analysis of the magnetic field distribution on the symmetry plane, such as... Figure 5 As shown in (b); Since there is no magnetic material in the finite-length solenoid coil structure, and no eddy currents exist in the static magnetic field solver, all parts except the coil itself are assumed to be vacuum. Therefore, in the finite element simulation of the magnetic field of the finite-length solenoid coil, based on the established two-dimensional model and simulation parameter settings, the magnetic field distribution of the finite-length solenoid coil after simulation calculation is as follows: Figure 6 As shown, Figure 6 The magnetic field strength distribution shown in (a) reveals a smaller gradient in the central region and a larger gradient at the ends of the finite-length solenoid coil, indicating higher uniformity in the central region, which is consistent with the theoretical analysis results above. Figure 6 (b) shows that the magnetic field lines are consistent with the theoretical calculation results, demonstrating the consistency of the magnetic field vector in the central region of the finite-length solenoid coil and the divergence of the magnetic field at the end. To further quantify and analyze the magnetic field distribution within a finite-length solenoid coil, theoretical analysis and simulation data were compared for the magnetic fields along the coil's axis and on its transverse symmetry plane. Figure 7 As shown, Figure 7 (a) shows the magnetic field distribution along the axis from the center to the edge. Based on the theoretical analysis above, the radial magnetic field along the axis is zero, and the axial magnetic field decreases towards the edge. Figure 7 As can be seen, the theoretical analysis data and simulation data of the axial magnetic field are highly consistent and almost identical. Figure 7(b) shows the variation of the magnetic field along the radial direction at the transverse interface of a finite-length solenoid coil. Due to the symmetry of the magnetic field, the radial magnetic field at the interface is also zero. Both theoretical calculations and simulation data show that the magnitude of the magnetic field increases with the increase of the radius. Table 1 below compares the data at three different locations inside the solenoid, showing the consistency between theoretical calculations and simulation analysis. Table 1 is shown below: Table 1. Comparison of Simulation and Theoretical Calculation Results of Magnetic Field for Finite-Length Solenoid Coils As can be seen from Table 1, the difference between the theoretical calculation and the simulation results is less than 0.05%, indicating that the method of the present invention has good accuracy.

[0029] In the description of this invention, the references to "one embodiment," "some embodiments," "in this embodiment," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0030] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for analyzing the magnetic field of a solenoid coil during axial gaussmeter calibration, characterized in that, Includes the following steps: Step S1: A steady current is passed through a finite-length solenoid coil with multiple windings, and a cylindrical coordinate system is constructed within the finite-length solenoid coil. Step S2: Based on Biot-Savart's law and the principle of magnetic field vector superposition, the energized winding in the finite-length solenoid coil is cut into directed current elements, and the first coordinate data of the point where the current micro-element is located and the second coordinate data of any point in space are obtained. Step S3: Obtain the size data of the finite-length solenoid coil; obtain the current direction vector expression of the current micro-element based on the first coordinate data; and obtain the radius vector expression from the point where the current micro-element is located to any point in the space based on the first coordinate data and the second coordinate data. Step S4: Perform vector product operation on the current direction vector expression and the radius vector expression, and combine them with the pre-constructed magnetic field quantity expression and the size data to obtain the magnetic field components of the magnetic field vector in each direction of the finite-length solenoid coil as the magnetic field analysis result.

2. The method for analyzing the magnetic field of a solenoid coil according to claim 1, characterized in that, In step S1, the cylindrical coordinate system is obtained by taking the center point of the finite-length solenoid coil as the origin, the axial direction of the finite-length solenoid coil as the z-axis, the radial direction of the finite-length solenoid coil as the ρ-axis, and the counterclockwise direction around the z-axis as the φ-axis.

3. The method for analyzing the magnetic field of a solenoid coil according to claim 1, characterized in that, In step S2, the original coordinate data of the point where the current element is located in the cylindrical coordinate system is obtained, and then the original coordinate data is converted into the first coordinate data in the rectangular coordinate system based on the rectangular coordinate system with the point where the current element is located at the origin.

4. The method for analyzing the magnetic field of a solenoid coil according to claim 3, characterized in that, In step S3, the direction vector expression of the current element is obtained through the following calculation formula: in, Indicates the direction of current The differential; - express Components on the x-axis; This represents the radius of the current loop represented by the first coordinate data; This represents the angle between the radius of the current element and the x-axis; express The differential; Represents the first unit vector; express Components on the y-axis; This represents the second unit vector.

5. The method for analyzing the magnetic field of a solenoid coil according to claim 4, characterized in that, In step S3, the position vector expression is obtained using the following calculation formula: in, Represents the infinitesimal point of current. to any point in the space The radius vector; This indicates the x-axis coordinate of the second coordinate data in a rectangular coordinate system; This indicates the y-axis coordinate of the second coordinate data in a rectangular coordinate system; This represents the z-axis coordinate of the second coordinate data in a Cartesian coordinate system; This represents the z-axis coordinate of the first coordinate data in a Cartesian coordinate system; This represents the third unit vector.

6. The method for analyzing the magnetic field of a solenoid coil according to claim 5, characterized in that, Before performing step S4, the magnetic field vector expression for the current infinitesimal element excitation is constructed using the following calculation formula: in, express The differential; Indicates the permeability of free space; This represents the current intensity of a steady current.

7. The method for analyzing the magnetic field of a solenoid coil according to claim 6, characterized in that, In step S4, the magnetic field components in each direction are obtained using the following calculation formula: in, The magnetic field component in the x-axis direction is represented; This represents the equivalent current density; Indicates the total length of the solenoid; Indicates the inner diameter of the solenoid; Indicates the thickness of the solenoid; express The differential; express The differential; This represents the radial coordinates of any point P in the space within the cylindrical coordinate system. This represents the azimuth coordinates of any point P in the space within the cylindrical coordinate system. This represents the magnetic field component along the y-axis in the Cartesian coordinate system; The magnetic field component in the z-axis direction is represented in both Cartesian and cylindrical coordinate systems.

8. The method for analyzing the magnetic field of a solenoid coil according to claim 7, characterized in that, After performing step S4, the process further includes: Based on the axisymmetric properties of the magnetic field, the azimuth coordinates of any point in space are set to 0 so that the point in space lies in the xOz plane of the cylindrical coordinate system, and then the magnetic field components in each direction are analyzed.

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