CN105718729A - Calculation method of magnetic field and inductance value in cylindrical surface axial gradient coil design - Google Patents

Abstract

The invention discloses a quick calculation method of a magnetic field and an inductance value in cylindrical surface axial gradient coil design. The quick calculation method comprises the following steps: before an optimization algorithm is begun, dividing the axial coordinate of the coil into N1 equal parts in the distribution intervals of a positive half axis, and calculating a magnetic field generated by a Maxwell coil on each coordinate point on each magnetic field sampling point; calculating the self-inductance of a single circular ring, and storing by one variable; dividing the whole axial distribution interval of the coil into N2 equal parts, and calculating the self-inductance of two circular rings in each sampling interval; and storing all calculation results by the variables, and expressing the magnetic field and the inductance value of each sampling point independently by an interpolation function according to a result on each discrete point during iteration. Compared with the traditional method, a calculated quantity can be saved to a high limit, calculation speed is quickened, calculation time is shortened, and an efficient nonlinear optimal gradient coil algorithm becomes possible.

Description

The computational methods of magnetic field and inductance value in the axial gradient coil design of a kind of cylinder

Technical field

The present invention relates to nuclear magnetic resonance technique field, particularly to the quick calculation method in magnetic field in a kind of axial gradient coil design Yu inductance value.

Background technology

Gradient coil is one of critical component of NMR system arranged side by side with superconducting magnet, radio-frequency coil.In one NMR system, comprise three gradient coils, provide the magnetic field with space coordinates linear change at three orthogonal directions respectively.Two of which provides the gradient coil in transverse gradients magnetic field to be called transverse coil, it is provided that the gradient coil of longitudinal gradient fields is called longitudinal coil.For column type gradient coil, longitudinal coil is also referred to as axial coil.

There is multiple method in design gradient coil at present.A kind of method is to adopt the design of the nonlinear algorithm such as simulated annealing, genetic algorithm.The advantage of this algorithm is to optimize nonlinear problem, thus when gradient coil design, and the range of choice of optimised function is more extensive, and the performance of the gradient coil optimized is closer to real gradient coil performance.Particularly when axial gradient coil design, this method has more advantage.Owing to axial coil is all made up of annulus, it is possible to the position of annulus is optimized directly as optimized variable.But there is a general problem in nonlinear optimization algorithm, it is simply that need very many iterative steps.Such as simulated annealing, it usually needs the result that hundreds thousand of even up to a million the ability optimizations of iteration are optimum.For gradient coil optimization problem, the main magnetic field that parameter is imaging space optimized and inductance value.The calculating of the two parameter directly affects optimization time and computational accuracy.Owing to the calculating of the two amount is directed to the function of some complexity, calculate the time longer, thus causing that nonlinear optimization needs long time.

Summary of the invention

Goal of the invention: in order to overcome the deficiencies in the prior art, the present invention provides a kind of computational methods optimizing magnetic field and the coil inductance that can quickly calculate in imaging region in process.

Technical scheme: for achieving the above object, the present invention adopts following steps:

Step one: with the cylinder center of gradient coil for initial point, axis direction is that z-axis sets up coordinate system, it is determined that the coordinate of M magnetic field sampled point, it is assumed that the greatest axial length of gradient coil is 2Z；

Step 2: axially interval [0, Z] N will be divided at equal intervals₁Equal portions, the length of every section is Δ z=Z/N₁, wherein Δ z≤1cm, coordinates computed point is z respectively_i=± i Δ z, i=1,2...N₁Maxwell's coil beyond axis and the magnetic field that produces of axis up-sampling point, and to store the result into dimension be M × N₁Array in, wherein corresponding two annulus of each Maxwell's coil, the axial coordinate of two annulus respectively ± z₁, annular radii is a, and the sense of current is contrary；

Step 3: calculate the self-induction L of single annulus_s, and by a variable storage；

Step 4: axially interval [0,2Z] N will be divided at equal intervals₂Equal portions, the length of every section is Δ d=2Z/N₂, wherein Δ d≤1cm, calculating axially spaced-apart is d_i=i Δ d, i=1,2...N₂The mutual inductance of two coils, and to store the result into a dimension be N₂Array in；

Step 5: in Optimized Iterative process, using the angular interval distance in the coordinate points in step 2 and step 4 as interpolation point, Maxwell's coil axial coordinate absolute value | the z | being positioned at other positions is [0, Z] interior arbitrary value time the magnetic field of each sampled point and the mutual inductance interpolating function that produces of two annulus of mutual inductance that two angular interval d are [0,2Z] interior arbitrary value represent.Adopt this step can significantly improve calculating speed, it is possible to speed is improved nearly ten times on original basis.

Wherein in step 2, the axial magnetic field B of the sample point beyond axis_zEmploying below equation is tried to achieve:

$\begin{array}{c}{B}_z=\frac{{\mathrm{\μ}}_0{\mathrm{Ik}}_1}{4\mathrm{\π}\sqrt{ar}}\[\left(\frac{a^2-r^2-{(z-z_1)}^2}{{(a-r)}^2+{(z-z_1)}^2}\right)E\left(k_1\right)+K\left(k_1\right)\]\\ -\frac{{\mathrm{\μ}}_0{\mathrm{Ik}}_2}{4\mathrm{\π}\sqrt{ar}}\[\left(\frac{a^2-r^2-{(z+z_1)}^2}{{(a-r)}^2+{(z+z_1)}^2}\right)E\left(k_2\right)+K\left(k_2\right)\]\end{array}$

Wherein μ₀For pcrmeability, I is annulus current-carrying, and r is the magnetic field sampled point distance to z-axis, and z is magnetic field sampled point axial coordinate, z₁Absolute value for the axial coordinate of Maxwell's coil；K, E are the first kind and elliptic integral of the second kind, parameter k₁, k₂Definition as follows:

$k_1=\sqrt{\frac{4ar}{{(a+r)}^2+{(z-z_1)}^2}},k_2=\sqrt{\frac{4ar}{{(a+r)}^2+{(z+z_1)}^2}}$

Axial magnetic field B on axis_zEmploying below equation is tried to achieve:

$B_z=\frac{{\mathrm{\μ}}_0{\mathrm{Ia}}^2}{2}\[\frac{1}{{({\left(z-z_1\right)}^2+a^2)}^{3/2}}-\frac{1}{{({\left(z+z_1\right)}^2+a^2)}^{3/2}}\].$

Wherein single annulus self-induction L in step 3_sComputing formula be:

$L_s={\mathrm{\μ}}_{0}a(ln\frac{a}{r_a}+0.32944154168)$

Wherein r_aIt is equivalent to equivalent redius during circle for coil section.

Wherein in step 4, the mutual inductance L of two annulus_MEmploying below equation calculates:

Wherein d is the distance between two Circumferential coils, electric current in the same direction time formula before take positive sign, take negative sign when electric current is reverse before formula.Compared with prior art, the formula calculating mutual inductance is more succinct.

Wherein in step 5, the specific formula for calculation of interpolating function is as follows:

For z_i+1≥z≥z_i, the magnetic field B of m-th sample point^mZ () computing formula is as follows:

$B^m\left(z\right)={\mathrm{GB}}_i^m\·(z-z_i)+B^m\left(z_i\right)$

Wherein, B^mZ () is positioned at ± magnetic field that produces at m-th magnetic field sampled point of the Maxwell's coil of z for axial coordinate, and ${\mathrm{GB}}_i^m=\frac{B^m\left(z_{i+1}\right)-B^m\left(z_i\right)}{\mathrm{\Δ}z},$

For d_i+1≥d≥d_i, the computing formula of mutual inductance L (d) is:

L (d)=GL_i·(d-d_i)+L(d_i)

Wherein, L (d) is the mutual inductance between two annulus being spaced apart d, andAndGL_iCalculated before optimized algorithm starts and store.Compared with prior art, the method calculating mutual inductance is more rapid, and formula is easier.

Beneficial effect: in the present invention, interval between the coordinate range at annulus place and two annulus is divided into some equal portions, magnetic field and inductance adopt analytic formula to calculate in the value of sample point, the value of other positions adopts interpolating function to calculate, and The present invention gives the computing formula of a kind of quick calculating magnetic field and inductance.Compared with prior art, the present invention can be greatly improved calculating speed, it is possible to speed is improved nearly ten times on original basis, so that efficient nonlinear optimization algorithm is possibly realized.

Accompanying drawing explanation

Fig. 1 is the flow chart of the present invention；

Fig. 2 is the structural representation of gradient coil in embodiment；

Fig. 3 is the structural representation of Maxwell's coil in embodiment；

Fig. 4 is the gradient coil adopting simulated annealing design in embodiment；

Fig. 5 be in embodiment gradient coil at the 45cmDSV internal magnetic field change curve with z coordinate；

Fig. 6 is the error that in embodiment, the magnetic field in 45cmDSV adopts inventive algorithm.

Detailed description of the invention

Below in conjunction with accompanying drawing, the present invention is further described.

In superconducting MRI system, axial gradient coil is typically distributed across on one layer of face of cylinder.Actual cylinder gradient coil generally can outside main coil additional layer shielded coil, shielded coil is made up of the annulus being distributed in one layer of periphery equally.Fig. 2 is typical active shielded gradient coil structure, and inner surface is served as theme circle, and outer surface is shielded coil, and all coils is about axial centre odd symmetry.Wherein main coil and shielded coil all have several Maxwell's coils to constitute.So-called Maxwell's coil, it is simply that identical by two radiuses, the coil pair that the annulus that current direction is contrary is constituted, as shown in Figure 3.Therefore, for axial coil, when Maxwell's coil number is determined, main purpose is to optimize the position of each annulus.Generally, the object function optimized constructed by gradient coil is as follows:

F=ω₁E+ω₂L

Wherein, E is the gradient fields error in imaging region (DSV), and L is coil inductance.Wherein the value of E is relevant with the axial magnetic field of sampled point in DSV.It can be seen that the calculative amount of above-mentioned optimization problem is mainly the axial magnetic field in DSV and inductance value.Calculating for the nonlinear optimization algorithm of simulated annealing, genetic algorithm one class, magnetic field and inductance is the efficiency being directly connected to optimized algorithm.Below the electric field in the present invention and inductance quick calculation method are sketched.

In the present invention, magnetic field is as follows with the quick calculation method of inductance:

Step one: with cylinder center for initial point, axis direction is that z-axis sets up coordinate system, it is determined that the coordinate of M magnetic field sampled point.Assuming that the greatest axial length in the axial coil design of cylinder is 2Z；

Step 2: axially interval [0, Z] N will be divided at equal intervals₁Equal portions, the length of every section is Δ z=Z/N₁, coordinates computed point is z_i=± i Δ z, i=1,2...N₁The Maxwell's coil magnetic field to producing at sampled point, and to store the result into dimension be M × N₁Array in；

Step 3: calculate the self-induction L of single annulus_s, and by a variable storage；

Step 4: axially interval [0,2Z] N will be divided at equal intervals₂Equal portions, the length of every section is Δ d=2Z/N₂, calculating axially spaced-apart is d_i=i Δ d, i=1,2...N₂The mutual inductance of two Circumferential coils, and to store the result into a dimension be N₂Array in；

Step 5: in Optimized Iterative process, using above sampled point as interpolation point, is positioned at the mutual inductance interpolating function that two annulus at the Maxwell's coil of other positions magnetic field to producing at sampled point and any distance interval produce and represents.

In step one, because axially coil is symmetrical structure mostly, sets up coordinate system with cylinder center and can ensure that two annulus of Maxwell's coil are symmetrical about coordinate axes.

In the present embodiment, in order to ensure computational accuracy, the length Δ z≤1cm of every section, Δ d≤1cm.Δ z, Δ d value relevant with the interpolating function chosen.The length of each section is more short, then accurately the exponent number of interpolating function needed for calculating magnetic field and inductance is more little, and in optimization process, calculating magnetic field is more short with the time of inductance.Because existing gradient coil, no matter being main coil or shielded coil, axial length is all not over 1m, if d=0.1mm, then the array dimension needed for storage mutual inductance value is 10000, is an only small value relative to the internal memory of computer.

In the present embodiment, for each Maxwell's coil pair, it is assumed that the axial coordinate of two annulus respectively ± z₁, annular radii is a, then the axial magnetic field B at the interpolation point place beyond axis_zEmploying below equation is tried to achieve:

$\begin{array}{c}{B}_z=\frac{{\mathrm{\μ}}_0{\mathrm{Ik}}_1}{4\mathrm{\π}\sqrt{ar}}\[\left(\frac{a^2-r^2-{(z-z_1)}^2}{{(a-r)}^2+{(z-z_1)}^2}\right)E\left(k_1\right)+K\left(k_1\right)\]\\ -\frac{{\mathrm{\μ}}_0{\mathrm{Ik}}_2}{4\mathrm{\π}\sqrt{ar}}\[\left(\frac{a^2-r^2-{(z+z_1)}^2}{{(a-r)}^2+{(z+z_1)}^2}\right)E\left(k_2\right)+K\left(k_2\right)\]\end{array}$

Here μ₀For pcrmeability, I is annulus current-carrying, and r is the magnetic field sampled point distance to z-axis, and z is the axial coordinate of magnetic field sampled point, z₁Absolute value for the axial coordinate of Maxwell's coil；K, E are the first kind and elliptic integral of the second kind, parameter k₁, k₂Definition as follows:

$k_1=\sqrt{\frac{4ar}{{(a+r)}^2+{(z-z_1)}^2}},k_2=\sqrt{\frac{4ar}{{(a+r)}^2+{(z+z_1)}^2}}$

Above-mentioned function relates generally to two ellptic integrals.There are some to be specifically designed for the algorithm of Gauss integration at present, compare and directly apply mechanically Biot Savart for mula there is higher efficiency.

Magnetic field B on axis_zEmploying below equation is tried to achieve:

$B_z=\frac{{\mathrm{\μ}}_0{\mathrm{Ia}}^2}{2}\[\frac{1}{{({\left(z-z_1\right)}^2+a^2)}^{3/2}}-\frac{1}{{({\left(z+z_1\right)}^2+a^2)}^{3/2}}\]$

In the present embodiment, the self-induction of single annulus adopts below equation to calculate:

$L_{nn}={\mathrm{\μ}}_{0}a(ln\frac{a}{r_a}+0.32944154168)$

Wherein r_aIt is equivalent to equivalent redius during circle for annulus cross section.This formula is the empirical equation calculating self-induction of loop.Especially for Circumferential coils, adopt this formula result of calculation and the result adopting numerical integration to calculate very close to, but there is higher computational efficiency.

In the present embodiment, the mutual inductance of two annulus adopts below equation to calculate:

Wherein d is the distance between two Circumferential coils.Electric current in the same direction, takes positive sign before formula, reversely take negative sign.This formula equally only comprises two ellptic integrals.

In above-mentioned steps five, polynomial interopolation can be adopted to calculate magnetic field and the mutual inductance of optional position annulus.Polynomial exponent number is more high, then the discrete point quantity reaching required precision needs is more few, and the internal memory that storage variable needs is also more few, but iterative computation magnetic field is more many with the time of inductance every time.Polynomial exponent number is more low, then iterative computation magnetic field is more few with the time of inductance every time.The technical staff adopting the algorithm in the present invention can select suitable exponent number as the case may be.In the present embodiment, the magnetic field of each sampled point when adopting linear interpolation function to calculate axial coordinate absolute value | z | of Maxwell's coil for [0, Z] interior arbitrary value, or the mutual inductance that two angular interval d are [0,2Z] interior arbitrary value.Concrete formula is as follows:

For z_i+1≥z≥z_i, the magnetic field B of m-th sample point^mZ () computing formula is as follows:

$B^m\left(z\right)={\mathrm{GB}}_i^m(z-z_i)+B\left(z_i\right)$

For d_i+1≥d≥d_i, the computing formula of mutual inductance L (d) is:

L (d)=GL_i(d-d_i)+L(d_i)

Wherein

${\mathrm{GB}}_i^m=\frac{B^m\left(z_{i+1}\right)-B^m\left(z_i\right)}{\mathrm{\Δ}z}$

${\mathrm{GL}}_i=\frac{L\left(d_{i+1}\right)-L\left(d_i\right)}{\mathrm{\Δ}d}$

GL_iCalculated before optimized algorithm starts and store.

Relative to high-order interpolation, linear interpolation can save the calculating time to greatest extent.

In order to compare the performance of the method, the present embodiment provides an example adopting simulated annealing design monolayer axial gradient coil.Maxwell's coil in coil is to being fixed as 14 pairs.The face of cylinder radius at coil place is 380mm, and gradient strength is 68 μ T/m/A, and the linearity is 4.8%.By optimizing gradient coil shape designed by object function above as shown in Figure 4.

It is limited for 10000 times with iteration, adopts the method in the present invention only to require time for 35 seconds.And if do not adopt the fast algorithm in the present invention, then need 327 seconds.It can be seen that the fast algorithm in the employing present invention, the time saves 9.3 times, substantially increases computational efficiency, has saved the time.

Adopting the inductance that the formula for interpolation in the present invention calculates is 302.584688497 μ H, and adopting the inductance value that the formula of above-mentioned self-induction and mutual inductance directly calculates is 302.584688088 μ H.Its difference after arithmetic point the 7th, this error can be ignored completely.Secondly the value of imaging region internal magnetic field, as it is shown in figure 5, be XOZ plane inside radius be that the circumference of r=225mm takes a series of sampled point, the axial magnetic field of sample point and the relation curve of axial coordinate.It is illustrated in figure 6 the difference between algorithm and exact algorithm that sample point adopts in the present invention.As seen from Figure 6, the magnetic field that the algorithm in the present invention calculates has enough precision.

The above is only the preferred embodiment of the present invention; it is noted that, for those skilled in the art; under the premise without departing from the principles of the invention, it is also possible to make some improvements and modifications, these improvements and modifications also should be regarded as protection scope of the present invention.

Claims (5)

1. the computational methods of magnetic field and inductance value in the axial gradient coil design of cylinder, it is characterised in that comprise the steps:

Step one: with the cylinder center of gradient coil for initial point, axis direction is that z-axis sets up coordinate system, it is determined that the coordinate of M magnetic field sampled point, it is assumed that the greatest axial length of the axial gradient coil of cylinder is 2Z；

Step 2: axially interval [0, Z] N will be divided at equal intervals₁Equal portions, the length of every section is Δ z=Z/N₁, wherein Δ z≤1cm, coordinates computed point is z respectively_i=± i Δ z, i=1,2...N₁The magnetic field that produces at sampled point of Maxwell's coil, and to store the result into dimension be M × N₁Array in, wherein corresponding two annulus of each Maxwell's coil, the axial coordinate of two annulus respectively ± z₁, annular radii is a, and the sense of current is contrary；

Step 3: calculate the self-induction L of single annulus_s, and by a variable storage；

Step 4: axially interval [0,2Z] N will be divided at equal intervals₂Equal portions, the length of every section is Δ d=2Z/N₂, wherein Δ d≤1cm, calculating axially spaced-apart is d_i=i Δ d, i=1,2...N₂The mutual inductance of two coils, and to store the result into a dimension be N₂Array in；

Step 5: in Optimized Iterative process, using the angular interval distance in the coordinate points in step 2 and step 4 as interpolation point, Maxwell's coil axial coordinate absolute value | the z | being positioned at other positions is [0, Z] interior arbitrary value time the magnetic field of each sampled point and the mutual inductance interpolating function that produces of two annulus of mutual inductance that two angular interval d are [0,2Z] interior arbitrary value represent.

2. the computational methods of magnetic field and inductance value in the axial gradient coil design of a kind of cylinder according to claim 1, it is characterised in that: in step 2, the axial magnetic field B of the sample point beyond axis_zEmploying below equation is tried to achieve:

$\begin{array}{c}{B}_z=\frac{{\mathrm{\μ}}_0{\mathrm{Ik}}_1}{4\mathrm{\π}\sqrt{ar}}\[\left(\frac{a^2-r^2-{(z-z_1)}^2}{{(a-r)}^2+{(z-z_1)}^2}\right)E\left(k_1\right)+K\left(k_1\right)\]\\ -\frac{{\mathrm{\μ}}_0{\mathrm{Ik}}_2}{4\mathrm{\π}\sqrt{ar}}\[\left(\frac{a^2-r^2-{(z+z_1)}^2}{{(a-r)}^2+{(z+z_1)}^2}\right)E\left(k_2\right)+K\left(k_2\right)\]\end{array}$

Wherein μ₀For pcrmeability, I is annulus current-carrying, and r is the magnetic field sampled point distance to z-axis, and z is magnetic field sampled point axial coordinate, z₁Absolute value for the axial coordinate of Maxwell's coil；K, E are the first kind and elliptic integral of the second kind, parameter k₁, k₂Definition as follows:

$k_1=\sqrt{\frac{4ar}{{(a+r)}^2+{(z-z_1)}^2}},k_2=\sqrt{\frac{4ar}{{(a+r)}^2+{(z+z_1)}^2}}$

Axial magnetic field B on axis_zEmploying below equation is tried to achieve:

$B_z=\frac{{\mathrm{\μ}}_0{\mathrm{Ia}}^2}{2}\[\frac{1}{{({\left(z+z_1\right)}^2+a^2)}^{3/2}}-\frac{1}{{({\left(z+z_1\right)}^2+a^2)}^{3/2}}\].$

3. the computational methods of magnetic field and inductance value in the axial gradient coil design of a kind of cylinder according to claim 1, it is characterised in that: single annulus self-induction L in step 3_sComputing formula be:

$L_s={\mathrm{\μ}}_{0}a(ln\frac{a}{r_a}+0.32944154168)$

Wherein r_aIt is equivalent to equivalent redius during circle for coil section.

4. the computational methods of magnetic field and inductance value in the axial gradient coil design of a kind of cylinder according to claim 1, it is characterised in that: in step 4, the mutual inductance L of two annulus_MEmploying below equation calculates:

Wherein d is the distance between two Circumferential coils, electric current in the same direction time formula before take positive sign, take negative sign when electric current is reverse before formula.

5. the computational methods of magnetic field and inductance value in the axial gradient coil design of a kind of cylinder according to claim 1, it is characterised in that: in step 5, the specific formula for calculation of interpolating function is as follows:

For z_i+1≥z≥z_i, the magnetic field B of m-th sample point^mZ () computing formula is as follows:

$B^m\left(z\right)={\mathrm{GB}}_i^m\·(z-z_i)+B^m\left(z_i\right)$

Wherein, B^mZ () is positioned at ± magnetic field that produces at m-th magnetic field sampled point of the Maxwell's coil of z for axial coordinate,

And ${\mathrm{GB}}_i^m=\frac{B^m\left(z_{i+1}\right)-B^m\left(z_i\right)}{\mathrm{\Δ}z},$

For d_i+1≥d≥d_i, the computing formula of mutual inductance L (d) is:

L (d)=GL_i·(d-d_i)+L(d_i)

Wherein, L (d) is the mutual inductance between two annulus being spaced apart d, andAndGL_iCalculated before optimized algorithm starts and store.