A method for measuring fractional gradient of two-dimensional body magnetic field
By adopting the complex plane and fractional-order Taylor expansion method on the two-dimensional body, using an odd number of sampling points and a set of linear equations, the problem of too many measuring points in the two-dimensional fractional-order gradient measurement is solved, and fast and accurate fractional-order gradient measurement is achieved.
Patent Information
- Application Number
- CN202211566788.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2022-12-07
- Publication Date
- 2025-10-03
- Estimated Expiration
- 2042-12-07
AI Technical Summary
The existing fractional-order gradient measurement method requires a large number of measurement points when measuring the second body, which leads to long measurement time and difficulty in quickly obtaining accurate fractional-order derivatives.
A method based on the complex plane and fractional-order Taylor expansion is adopted. An odd number of sampling points are uniformly selected on the quadratic body. A complex vector is constructed and the fractional-order derivatives are solved through a system of linear equations. The passive and irrotational properties of the magnetic field are used to meet the analytical function conditions and reduce the number of measuring points.
It is possible to quickly obtain the measurement of the second-dimensional fractional gradient using a small number of measurement points, with an accuracy close to that of traditional methods, reducing measurement time and data requirements.
Smart Images
Figure CN116299083B_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the technical field of magnetic field gradient measurement, and in particular relates to a method for measuring the fractional gradient of a two-dimensional body magnetic field. Background Art
[0002] Magnetic field gradient measurement is based on the concept of derivatives, typically measuring the first-order derivative of the magnetic field in a specific direction. This measurement follows the definition of a derivative: measuring the magnetic field between two points, finding the difference, and then dividing it by the distance between them.
[0003] However, fractional magnetic gradient measurements face a wide variety of definitions, most of which involve integrals, limits, and special functions. Consequently, numerical methods are often used to calculate fractional gradients, requiring extensive measurement data to obtain accurate results. Unlike magnetic field gradients, which can be easily obtained on-site using simple methods, fractional gradient measurements cannot be performed. Therefore, if the number of measurement points required for fractional gradients can be reduced, measurement time can be shortened, and the required fractional derivatives can be quickly obtained, facilitating on-site analysis.
[0004] In summary, existing fractional gradient measurement methods, based on numerical algorithms, are not specific to the object being measured; they only require sufficient data to calculate its fractional derivative. However, accurate values can only be calculated when the data sampling rate is sufficiently dense and the number of sampling points is sufficient. Therefore, there is an urgent need to develop a method specifically for measuring the fractional derivative of a two-dimensional magnetic field, capable of measuring the fractional gradient of a two-dimensional object using a small number of measurement points. Summary of the Invention
[0005] The purpose of the present invention is to provide a method for measuring two-dimensional fractional-order magnetic gradients to solve the problem that the existing fractional-order gradient measurement method requires a large number of measurement points when applied to two-dimensional measurements.
[0006] The purpose of the present invention is achieved through the following technical solutions:
[0007] A method for measuring the fractional gradient of a two-dimensional body magnetic field comprises the following steps:
[0008] A. Determine the direction of the second body;
[0009] B. Establish the complex plane;
[0010] C. Determine the starting and ending points of fractional derivatives on the complex plane;
[0011] D. Determine the survey line circle;
[0012] E. Take an odd number of sampling positions on the survey line circle, measure the two magnetic field components along the axis at the sampling positions, and construct a complex vector;
[0013] F, construct coefficient matrix;
[0014] G. Solve the system of equations using a complex vector as a constant term and extract the fractional derivatives.
[0015] Furthermore, step B is specifically as follows: establish a complex plane xOy on the plane perpendicular to the two-dimensional body, determine the real axis and the imaginary axis, the cross section of the two-dimensional body is S, and the origin O of the complex plane must not be located in the space where S of the two-dimensional body is located.
[0016] Furthermore, step C is specifically as follows: the starting point of the fractional derivative is the origin O, and the end point is the center a.
[0017] Furthermore, step D is specifically as follows: the positioning plate is located on the complex plane, a measuring line circle is drawn on the positioning plate, which passes through the origin of the coordinate system and has its center on the real axis, and the fluxgate slots are evenly distributed on the measuring line circle for placing fluxgate sensors.
[0018] Furthermore, step E is specifically as follows: uniformly select K points w on the survey line circle k As sampling points, take K = 7 and measure the magnetic field components B along the real axis and imaginary axis at each sampling point x and B y , at each measuring point w k The complex number f is constructed using the measured magnetic field components in two directions. k =B y，k +iB x，k , forming a complex vector F.
[0019] Furthermore, in step F, a coefficient matrix is constructed The elements in the matrix C satisfy n is determined by the number of sampling points K,
[0020] Furthermore, in step G, the vector F and the coefficient matrix C are respectively used as the constant term and the coefficient matrix of the linear equation group, the vector of unknown numbers is denoted as D, and the linear equation group CD=F is solved to obtain the value D of the vector of unknown numbers.
[0021] Furthermore, the elements in the unknown vector D form a sequence of increasing derivative orders, The element at the middle of the vector This is the required fractional derivative.
[0022] Compared with the prior art, the present invention has the following beneficial effects:
[0023] The fractional-order gradient measurement method of the two-dimensional body magnetic field of the present invention is based on the fractional-order Taylor expansion, which integrates the passive and irrotational properties of the magnetic field. It can measure the fractional-order gradient of the two-dimensional body with a small number of measuring points and achieve an accuracy close to that of traditional methods. The measurement method of the present invention can quickly obtain the fractional-order gradient of the two-dimensional body on a specified path. BRIEF DESCRIPTION OF THE DRAWINGS
[0024] In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the following briefly introduces the drawings required for use in the embodiments. It should be understood that the following drawings only illustrate certain embodiments of the present invention and therefore should not be regarded as limiting the scope. For ordinary technicians in this field, other relevant drawings can be obtained based on these drawings without paying any creative work.
[0025] Figure 1-Figure 2 Schematic diagram of the sampling process;
[0026] Figure 3 Flowchart of the steps of the method for measuring the fractional gradient of the two-dimensional body magnetic field of the present invention. DETAILED DESCRIPTION
[0027] The present invention will be further described below in conjunction with embodiment:
[0028] The present invention will be further described in detail below with reference to the accompanying drawings and examples. It will be understood that the specific embodiments described herein are intended only to illustrate the present invention and are not intended to limit the present invention. It should also be noted that, for ease of description, the accompanying drawings only illustrate portions relevant to the present invention, not all structures.
[0029] It should be noted that similar reference numerals and letters represent similar items in the following drawings. Therefore, once an item is defined in one drawing, it does not need to be further defined or explained in subsequent drawings. At the same time, in the description of the present invention, the terms "first", "second", etc. are used only to distinguish the description and should not be understood as indicating or implying relative importance.
[0030] The present invention is based on the fractional-order Taylor expansion on the complex plane, which can be expressed as follows: For the analytical function f(z), the fractional-order derivative can be expressed as The series is expressed as:
[0031]
[0032] in, represents the (α+n)th order fractional derivative from complex point b to a, with α∈(0,1). z lies on the circle |za|=|ba|, and z≠b. The resulting fractional gradient is the fractional derivative of two mutually perpendicular magnetic field components along the coordinate axis with respect to a straight line path.
[0033] The entire measurement is performed on a complex plane perpendicular to the dihedral. Based on the premise that the fractional Taylor expansion is valid, the sampling points should be uniformly selected along a circular trajectory in the complex plane. The specific location of the circular trajectory is determined by the starting and ending points of the required linear path of the fractional gradient. Due to the inherent connection between the fractional gradient and the Fourier transform, the number of sampling points should be an odd number. The sampling points should be evenly distributed to reduce the condition number of the equation system.
[0034] The fractional Taylor expansion presupposes that the function being expanded is an analytic function, but the magnetic field in the space where the two-dimensional body resides does not naturally satisfy this condition. By leveraging the properties of the magnetic field being passive and irrotational, combined with the fact that the magnetic field of the two-dimensional body is two-dimensional, the magnetic field components measured at each measuring point are combined into complex numbers according to certain rules to ensure that they meet the conditions of an analytic function. The final fractional gradient is obtained by solving a system of linear equations. The coefficient matrix of the system of equations is calculated from the coefficients of the fractional Taylor series, and the constants of the system of equations are composed of complex numbers formed by combining the measurement results.
[0035] Specifically, the magnetic field component B generated by the second body S is measured. x , B y The fractional derivative of order α (α∈(0,1)) on the line from the origin O to the center a
[0036] First, create a complex plane xOy on a plane perpendicular to the two-dimensional solid and determine the real and imaginary axes. The cross section of the two-dimensional solid is S. The origin of the complex plane O must not be located in the space where the two-dimensional solid s is located.
[0037] Next, the starting point and end point of the fractional derivative are determined on the complex plane. The starting point of the fractional derivative is the origin O, and the end point is the center a.
[0038] Then, with the end point a as the center, determine the circle C passing through the starting point O: |wa| = |a|.
[0039] Next, evenly select K (K is an odd number) points w on the circle k As sampling points, K=7 is usually appropriate, and the magnetic field components B along the real axis and imaginary axis at each sampling point are measured. x and B y At each measuring point w k The complex number f is constructed using the measured magnetic field components in two directions. k =B y，k +iB x，k , forming a complex vector F.
[0040] Next, construct the coefficient matrix The elements in the matrix C satisfy n is determined by the number of sampling points K,
[0041] Then take the vector F and the coefficient matrix C as the constant term and coefficient matrix of the linear equation group respectively, denote the vector of unknowns as D, and solve the linear equation group CD=F to obtain the value D of the vector of unknowns.
[0042] Finally, the elements in the unknown vector D form a sequence of increasing derivative orders. In the present invention, they are The element at the middle of the vector This is the required fractional derivative.
[0043] Example 1
[0044] The purpose of this embodiment is to calculate the two long sides l of the current-carrying coil z = 10m generated magnetic field component B x , B y On the line from the origin to the center of the circle a = 0.25m fractional derivative. Since l z >>l y , the two long sides l z Considered as an ideal two-dimensional body.
[0045] 1. Coordinate system such as Figure 1 As shown. The rectangular coil is located in the zOy plane, with the long side l z Symmetric about the z axis, the short side l y Symmetrical about the y-axis. A current of I = 14.4A is passed through the coil, and the direction of the current is as follows Figure 1 As shown in .
[0046] 2. Take the xOy plane as the complex plane, with the real axis aligned with the x-axis and the imaginary axis aligned with the y-axis. The positioning plate is located on this complex plane.
[0047] 3. Determine the starting and ending points of the fractional derivative on the complex plane.
[0048] 4. Draw a measuring circle C on the positioning plate. It passes through the origin of the coordinate system and its center is on the real axis (x-axis). The coordinate of the center on the complex plane is a = 0.25m. Seven fluxgate slots are evenly distributed on the measuring circle C. Their coordinates on the complex plane are w k (k=1, 2, 3, 4, 5, 6, 7), used to place fluxgate sensors.
[0049] 5. Place the fluxgate on w in sequence k At the point where the magnetic field B is measured x (w k ) and B y (w k ) components. The magnetic field components measured at each position are organized into complex numbers f(wk )=B y (w k )+iB x (w k ), uniformly recorded as complex vector F
[0050]
[0051] 6. According to the formula Set the fluxgate position w k Substitute the derivative order α=1 / 2 and calculate the coefficient matrix
[0052]
[0053] The unknown vector is recorded as the fractional derivative vector D
[0054]
[0055] The fourth term in the middle is the required fractional derivative
[0056] 7. Solve the linear equation system CD=F to get the value of D. The final result is
[0057]
[0058] The fourth term d4 in D is the desired derivative
[0059] Note that the above are only preferred embodiments of the present invention and the technical principles employed. Those skilled in the art will understand that the present invention is not limited to the specific embodiments described herein, and that various obvious changes, readjustments, and substitutions can be made by those skilled in the art without departing from the scope of protection of the present invention. Therefore, although the present invention has been described in detail through the above embodiments, the present invention is not limited to the above embodiments and may include many other equivalent embodiments without departing from the concept of the present invention. The scope of the present invention is determined by the scope of the appended claims.
Claims
1. A method for measuring the fractional gradient of a two-dimensional body magnetic field, characterized in that: The following steps are involved: A. Determine the direction of the second body; B. Establish the complex plane; C. Determine the starting and ending points of fractional derivatives on the complex plane; D. Determine the survey line circle; E. Take an odd number of sampling positions on the survey line circle, measure the two magnetic field components along the axis at the sampling positions, and construct a complex vector; F, construct coefficient matrix; G. Solve the system of equations using a complex vector as a constant term and extract the fractional derivatives.
2. The method for measuring the fractional gradient of a two-dimensional body magnetic field according to claim 1, wherein: Step B is specifically as follows: establish a complex plane xOy on the plane perpendicular to the two-dimensional body, determine the real axis and the imaginary axis, the cross section of the two-dimensional body is S, and the origin O of the complex plane must not be located in the space where S of the two-dimensional body is located.
3. The method for measuring the fractional gradient of a two-dimensional body magnetic field according to claim 2, wherein: Step C is specifically as follows: the starting point of the fractional derivative is the origin O, and the end point is the center a of the circle.
4. The method for measuring the fractional gradient of a two-dimensional body magnetic field according to claim 1, wherein: Step D is specifically as follows: the positioning plate is located on the complex plane, a measuring line circle is drawn on the positioning plate, which passes through the origin of the coordinate system and has its center on the real axis, and gate slots are evenly distributed on the measuring line circle for placing fluxgate sensors.
5. The method for measuring the fractional gradient of a two-dimensional body magnetic field according to claim 1, wherein: Step E, specifically: uniformly select K points w on the survey line circle k As sampling points, take K = 7 and measure the magnetic field components B along the real axis and imaginary axis at each sampling point x and B y , at each measuring point w k The complex number f is constructed using the measured magnetic field components in two directions. k =B y,k +iB x,k , forming a complex vector F.
6. The method for measuring the fractional gradient of a two-dimensional body magnetic field according to claim 5, wherein: Step F, construct coefficient matrix The elements in the matrix C satisfy n is determined by the number of sampling points K, 7. The method for measuring the fractional gradient of a two-dimensional body magnetic field according to claim 1, wherein: In step G, the vector F and the coefficient matrix C are respectively used as the constant term and the coefficient matrix of the linear equation group, the vector of unknown numbers is denoted as D, and the linear equation group CD=F is solved to obtain the value D of the vector of unknown numbers.
8. The method for measuring the fractional gradient of a two-dimensional body magnetic field according to claim 7, wherein: The elements in the unknown vector D form a sequence with increasing order of derivative, The element at the middle of the vector This is the required fractional derivative.