CN114764129A - Method for calculating central position of magnetic field in magnet - Google Patents
Method for calculating central position of magnetic field in magnet Download PDFInfo
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- CN114764129A CN114764129A CN202110036467.1A CN202110036467A CN114764129A CN 114764129 A CN114764129 A CN 114764129A CN 202110036467 A CN202110036467 A CN 202110036467A CN 114764129 A CN114764129 A CN 114764129A
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Abstract
The invention discloses a method for calculating the central position of a magnetic field in a magnet according to measured magnetic induction intensity values. The method mainly comprises the following steps: firstly, determining the direction of magnetic induction intensity to be measured, estimating the position of the center of a magnetic field in a magnet, and measuring the magnetic induction intensity values at a series of sampling points on a spherical surface with the radius of R by taking the position as the circle center; then, according to the measured magnetic induction intensity values at the sampling points, an interpolation function of the magnetic induction intensity is established; and finally, establishing a nonlinear optimization problem by taking the central position of the magnetic field as a variable to be optimized and the minimum mean square error of the magnetic induction intensity value at the sampling point on the spherical surface with the radius of R' as an optimization target and solving the problem. The method can be used for quickly and accurately positioning the magnetic field center of the magnet.
Description
Technical Field
The invention belongs to the field of research and design of magnets, and particularly relates to a method for calculating the central position of a magnetic field in a magnet.
Background
In Magnetic Resonance Imaging (MRI) systems, the homogeneity of the main magnetic field generated by a magnet is an important indicator that affects the quality of the image. One possible way to improve the homogeneity of the main magnetic field is to shim the magnetic field using a passive shimming method. In the shimming process, the magnetic field uniformity in the central area of the magnet needs to be accurately measured, and then shimming is guided according to the measured magnetic induction intensity. Prior to shimming, the problem of accurate positioning of the center of the magnetic field is involved. In particular for superconducting magnets, accurate positioning of the magnetic field center is particularly important. This is because the magnetic field center of the superconducting magnet coincides with the magnetic field center of the gradient coil. If the center positioning is not accurate, not only the measured magnetic induction intensity is larger, but also the magnetic field calculation is inaccurate due to the fact that the position of the shimming track deviates from the position of the magnetic field center, and accordingly shimming difficulty is increased. The positioning of the magnetic field center is less accurate by purely mechanical positioning. Some commercially available software for measuring a magnetic field includes a function of calculating the center position of the magnetic field based on the measured magnetic induction, but can perform positioning only in one direction. It is therefore necessary to develop a more accurate method.
Disclosure of Invention
The purpose of the invention is as follows: in order to overcome the defects in the prior art, the invention aims to provide a method for calculating the central position of a magnetic field in a magnet, so that the positioning precision of the magnetic field center is improved.
The technical scheme is as follows: in order to achieve the purpose of the invention, the invention adopts the following technical scheme:
a method of calculating a center of a magnetic field in a magnet, comprising:
the method comprises the following steps: determining the direction of a magnetic field to be measured, predicting the position of the center of the magnetic field in the magnet, establishing a coordinate system by taking the predicted position as an original point and the direction of the magnetic field to be measured as the direction of a z axis, and measuring the magnetic induction intensity at a series of sampling points on a spherical surface with the original point as the center of a circle and the radius of R;
step two: according to the magnetic induction intensity at each sampling point, establishing an interpolation function representing the relation of the magnetic induction intensity changing along with the coordinate;
step three: taking P sampling points on a spherical surface with an origin as a circle center and R' as a radius
Definition of
For the coordinate variables to be optimized, the following nonlinear optimization problem is established and solved:
x in the formula1、Y1、Z1、R1Is a non-negative real number and is used to define the pair x0，y0，z0The range of constraint of (a) to (b),
respectively represent coordinate points
The z component of the magnetic induction intensity is obtained by solving the nonlinear optimization problem
I.e. the magnetic field center coordinate vector within the magnet.
Further, in the first step, the measuring probes on the magnetic field measuring equipment are distributed on an arc surface with the radius of R and accord with Gaussian distribution, and the measuring equipment rotates around the axis at equal angle intervals for a circle to measure, so that the magnetic fields at all sampling points are obtained.
Further, in step two, any coordinate point
Z component of magnetic induction
The following interpolation formula is adopted for calculation:
in the above formula, N is the order of the interpolation function,
as a coordinate point
Spherical coordinates of (D), Pnm() Denotes an m-th Legendre polynomial of n-th order, anmAnd bnmThe harmonic coefficient is determined according to the measured magnetic induction intensity.
Further, a harmonic coefficient a is calculated from the measured magnetic field intensitynm，bnmM is more than or equal to 0 and less than or equal to N, and the formula of N is more than or equal to 0 and less than or equal to N is as follows:
in the above formula, K1 is the number of probes distributed on the arc surface with radius R in one measurement, K2 is the number of times of measurement when the measurement device rotates around the z axis for one circle at equal intervals,
the included angle theta between the arc surface of the measuring probe and the x axis in the ith measurementj、wjIs the jth integral point of the Gaussian integral of the K1 point in the interval of more than or equal to 0 and less than or equal to pi and the corresponding weight coefficient,
the vector coordinates of the jth probe position in the ith measurement,
as a coordinate point
The z-component of the magnetic induction.
Further, solving the nonlinear optimization problem in the third step by adopting an interior point method.
Further, R' is less than or equal to R.
Further, the measured magnet is a cylindrical superconducting magnet, and the direction of the magnetic field to be measured is the axial direction of the magnet.
Further, the magnet to be measured is a planar permanent magnet, and the direction of the magnetic field to be measured is the normal direction of the upper and lower polar plates in the permanent magnet.
Has the beneficial effects that: the invention has the advantages that: the invention provides a method for calculating the center of a magnetic field by using measured magnetic induction intensity, which is characterized in that the problem of positioning the center of the magnetic field is equivalent to a nonlinear optimization problem to solve, so that the center of a circle with the minimum mean square error of the magnetic field at a spherical sampling point is used as the center of the magnetic field, and the obtained result has higher precision.
Drawings
FIG. 1 is a flow chart of an algorithm for calculating the center of a magnetic field according to the present invention.
Fig. 2 is a distribution diagram of magnetic field measurement points in space.
Fig. 3 is a schematic diagram illustrating the definition of each coordinate parameter in the rectangular coordinate system and the spherical coordinate system.
Fig. 4 is a distribution diagram of the cross section of the magnet coil in space.
FIG. 5 is a graph of RMSE error at a sample point versus offset distance of the magnetic field center along the z-axis.
Detailed Description
The present invention is further illustrated by the following figures and specific examples, which are to be understood as illustrative only and not as limiting the scope of the invention, which is to be given the full breadth of the appended claims and any and all equivalent modifications thereof which may occur to those skilled in the art upon reading the present specification.
The method for calculating the central position of the magnetic field in the magnet according to the magnetic induction intensity disclosed by the embodiment of the invention is shown in figure 1, and comprises the following specific steps:
the method comprises the following steps: determining the direction of a magnetic field to be measured, predicting the position of the center of the magnetic field in the magnet, establishing a coordinate system by taking the predicted position as an original point and the direction of the magnetic field to be measured as the direction of a z axis, and measuring the magnetic induction intensity at a series of sampling points on a spherical surface with the original point as the center of a circle and the radius of R;
the superconducting magnet in superconducting magnetic resonance will be described as an example. In a superconducting magnetic resonance system, a magnet is of a cylindrical structure, and a magnet coil for generating a magnetic field is a plurality of solenoid coils connected in series. The magnetic field direction required for imaging is in the direction of the axis of the cylinder. We establish a coordinate system with the axis of the magnet center as the z-axis. An x-axis and a y-axis are established in a plane perpendicular to the z-axis. The directions of the x-axis and the y-axis can be defined according to the convenience of the research problem. Generally, the center of the magnetic field is located on and near the center of the central axis of the magnet. If the magnet to be measured is a planar permanent magnet, the direction of the magnetic field to be measured is the normal direction of the upper and lower pole plates in the permanent magnet, and the direction perpendicular to the pole plates is generally defined as the z direction. The method of the patent is more useful for superconducting magnets because in practical engineering, the permanent magnets are less likely to require precise calibration of the center of the magnetic field.
The existing magnetic field measuring equipment can conveniently measure the magnetic field of a plurality of sampling points at one time. For example, as many as 24 magnetic field measurement probes are integrated into a magnetic field camera of Metrolab, switzerland, and 24 sampling points on one arc can be measured at a time. The magnetic field camera is uniformly rotated for one circle around the central axis of the magnet, so that the high-precision magnetic field measurement of hundreds of sampling points on the whole spherical surface can be completed. This provides hardware support for calculating the magnetic field center from the measured magnetic induction. In magnetic field measurement devices, the probe is typically located at a gaussian integration point on the arc. The magnetic field measurement points thus follow a Gaussian distribution in the theta direction
The directions are evenly distributed as shown by the grid points in fig. 2. Angle theta in spherical coordinate system
The definition of the angle is shown in figure 3.
Step two: and establishing a magnetic field interpolation function in the area near the origin according to the magnetic field intensity at each measuring point. Any vector coordinate in space is
Magnetic field at the point of
Can be expressed as the following function:
it should be noted that, here
As a coordinate
The z-component of the magnetic induction of (a) is thus a scalar quantity.
According to the above formula, the calculation formula of the harmonic coefficient can be obtained as follows:
the integral expression above can be calculated numerically:
in the above formula, K1 is the number of probes distributed on the arc surface with radius R in one measurement, K2 is the number of times of measurement when the measurement device rotates around the z axis for one circle at equal intervals,
the included angle theta between the cambered surface where the measuring probe is positioned and the x axis in the ith measurementj、wjIs the jth integral point of the Gaussian integral of the K1 point in the interval of more than or equal to 0 and less than or equal to pi and the corresponding weight coefficient,
the vector coordinates for the jth probe position in the ith measurement,
as a coordinate point
The z-component of the magnetic induction. The calculation of the gaussian integral points and the weighting coefficients is introduced in many numerical integration books, and is not described herein again.
Step three: taking P sampling points on a spherical surface which takes an original point as a circle center and R' as a radius
Definition of
For the coordinate variables to be optimized, the following nonlinear optimization problem is established and solved:
x in the formula1、Y1、Z1、R1Is a non-negative real number and is used to define the pair x0，y0，z0The range of constraint of (a) to (b),
respectively represent coordinate points
The z component of the magnetic induction intensity is obtained by solving the nonlinear optimization problem
I.e. the magnetic field center coordinates within the magnet.
There are many methods for solving the non-linear problem, such as an internal penalty function method, and the like, and reference may be made to the book "optimization theory and algorithm" written in chen bao lin, and a detailed solution will not be described here.
In order to ensure the accuracy of the magnetic field calculation using the interpolation function, we set R 'to R'. Thus in the pair x0,y0,z0After the constraint is carried out, the coordinates of the sampling points can be ensured not to deviate from the spherical area with the radius of R seriously in the optimization process, so that the precision of the interpolation algorithm is ensured.
The method of calculating the center of the magnetic field in the magnet according to the present invention will be described below based on a specific example. In this example, we use the magnetic field at the sampling point calculated from the superconducting magnet coil parameters instead of the measured magnetic induction. The reason for this is that the offset value of the center of the magnetic field can be known in advance, facilitating the verification of the accuracy of the algorithm. The magnet coil dimensions in this example are shown in table 1. Because the coil is of a symmetrical construction, only coil dimensions with a positive z coordinate are given in the table. The coil numbered 4 in the table is a shield coil, and the other coils are main coils. Rmin, Rmax, Z1, Z2 in the table indicate the inside diameter, outside diameter, left side Z coordinate, right side Z coordinate of each solenoid coil. The coil operating current was 440.078553 a. The cross-sectional profile of the coil is shown in fig. 4.
TABLE 1 superconducting magnet coil size
| Number of | Rmin(m) | Rmax(m) | Z1(m) | Z2(m) |
| 1 | 0.500576 | 0.515576 | 0.023675 | 0.163675 |
| 2 | 0.5 | 0.518 | 0.224115 | 0.390115 |
| 3 | 0.5 | 0.527 | 0.522184 | 0.806184 |
| 4 | 0.845961 | 0.857961 | 0.399878 | 0.775878 |
Taking 24 × 24 as 576 sampling points on a spherical surface with the origin as the center and the diameter of 45cm, and using the coordinates of the sampling points
And (4) showing. All sampling points are shifted gradually along the z-axis, and the Root Mean Square Error (RMSE) of the magnetic field at all sampling points as a function of the shift distance is shown in fig. 5. Ppm in the figure means parts per million. The formula for calculating the root mean square error in the figure is:
in the above formula, the first and second carbon atoms are,
the vector is the circle center coordinate vector of the sphere where the sampling point is located, namely the offset of three coordinates of each sampling point.
As can be seen from fig. 5, the RMSE value of the magnetic field at the sampling point gradually increases with the offset distance. Similarly, shifting the sample points along the x-axis and the y-axis results in a similar curve.
Finally, we verified the accuracy of the algorithm in the present invention. In the above example, we offset the center of the spherical surface where the sampling point is located to (0.01m,0.01m,0.01m), and calculate the magnetic induction at each sampling point. Then, according to the magnetic induction intensity obtained by calculation, the method is adopted for optimization to obtain the magnetic field with the central position as
Note that the coordinates here are coordinates in a coordinate system established with the center of the sphere where the sample point is located as the origin. It can be seen that the algorithm of the present invention is capable of accurately calculating the magnetic field center of the magnet coilLocation.
Claims (8)
1. A method of calculating a position of a center of a magnetic field in a magnet, comprising:
the method comprises the following steps: determining the direction of a magnetic field to be measured, predicting the position of the center of the magnetic field in the magnet, establishing a coordinate system by taking the predicted position as an original point and the direction of the magnetic field to be measured as the direction of a z axis, and measuring the magnetic induction intensity at a series of sampling points on a spherical surface with the original point as the center of a circle and the radius of R;
step two: establishing an interpolation function representing the relation of the magnetic induction intensity changing along with the coordinate according to the magnetic induction intensity at each sampling point;
step three: taking P sampling points on a spherical surface with an origin as a circle center and R' as a radius
Definition of
For the coordinate variables to be optimized, the following nonlinear optimization problem is established and solved:
x in the formula1、Y1、Z1、R1Is a non-negative real number and is used to define the pair x0,y0,z0The range of the constraint of (2),
respectively represent coordinate points
The z component of the magnetic induction intensity is obtained by solving the nonlinear optimization problem
I.e. the magnetic field center coordinate vector within the magnet.
2. The method for calculating the central position of the magnetic field in the magnet according to claim 1, wherein in the first step, the measuring probes of the magnetic field measuring device are distributed on an arc surface with a radius of R and conform to gaussian distribution, and the measuring device rotates around the axis at equal angular intervals for one circle to obtain the magnetic field at all sampling points.
3. The method of claim 1, wherein in step two, any coordinate point is used to calculate the center position of the magnetic field in the magnet
Z component of magnetic induction
The following interpolation formula is adopted for calculation:
in the above formula, N is the order of the interpolation function,
as a coordinate point
Spherical coordinates of (D), Pnm() Denotes an m-th Legendre polynomial of n-th order, anmAnd bnmThe harmonic coefficient is determined according to the measured magnetic induction intensity.
4. A method of calculating the centre position of a magnetic field in a magnet as claimed in claim 3, characterised in that the harmonic coefficient a is calculated from the measured field strengthnm,bnmM is more than or equal to 0 and less than or equal to N, and N is more than or equal to 0 and less than or equal to N, the formula is as follows:
in the above formula, K1 is the number of probes distributed on the arc surface with radius R in one measurement, K2 is the number of times of measurement when the measurement device rotates around the z axis for one circle at equal intervals,
the included angle theta between the arc surface of the measuring probe and the x axis in the ith measurementj、wjIs the jth integral point of the Gaussian integral of the K1 point in the interval of more than or equal to 0 and less than or equal to pi and the corresponding weight coefficient,
the vector coordinates of the jth probe position in the ith measurement,
as coordinate points
The z-component of the magnetic induction.
5. The method for calculating the central position of the magnetic field in the magnet according to claim 1, wherein the nonlinear optimization problem in the third step is solved by adopting an interior point method.
6. The method of claim 1, wherein R' is ≦ R.
7. The method of calculating the center position of a magnetic field in a magnet according to claim 1, wherein the magnet to be measured is a cylindrical superconducting magnet, and the direction of the magnetic field to be measured is the axial direction of the magnet.
8. The method of claim 1, wherein the magnet to be measured is a planar permanent magnet, and the direction of the magnetic field to be measured is the normal direction of the upper and lower plates in the permanent magnet.
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