CN105718729B - The computational methods in magnetic field and inductance value in a kind of cylinder axial direction gradient coil design - Google Patents
The computational methods in magnetic field and inductance value in a kind of cylinder axial direction gradient coil design Download PDFInfo
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Abstract
The invention discloses the quick calculation methods in magnetic field and inductance value in a kind of cylinder axial direction gradient coil process of optimization, include the following steps：Before optimization algorithm starts, the distributed area by the axial coordinate of coil in positive axis is divided into N1Equal portions calculate the Maxwell's coil at each coordinate points in the magnetic field that each magnetic field sampled point generates；A variable storage is used in combination in the self-induction for calculating single annulus；The entire axial distributed area of coil is divided into N2Equal portions, mutual inductance of two annulus of calculating in each sampling interval；By all result of calculation variable storages, and in iteration according to it is at various discrete point as a result, by each sampled point magnetic field and inductance value indicated respectively with interpolating function.This method can save calculation amount to greatest extent compared with conventional method, improve calculating speed and save the calculating time so that efficient nonlinear optimization gradient coil algorithm is possibly realized.
Description
Technical field
The present invention relates to nuclear magnetic resonance technique field, more particularly to magnetic field and inductance value in a kind of axial gradient coil design
Quick calculation method.
Background technology
Gradient coil is the critical component of NMR system one of arranged side by side with superconducting magnet, radio-frequency coil.One core
In magnetic resonance system, including three gradient coils, provide the magnetic field with space coordinate linear change in three orthogonal directions respectively.
The gradient coil that two of which provides transverse gradients magnetic field is referred to as transverse coil, and the gradient coil for providing longitudinal gradient fields is known as indulging
To coil.For column type gradient coil, longitudinal coil is also referred to as axial coil.
There are a variety of methods for design gradient coil at present.A kind of method is non-linear using simulated annealing, genetic algorithm etc.
Algorithm designs.The advantages of this algorithm is to optimize nonlinear problem, thus in gradient coil design, optimised function
Range of choice is more extensive, and the performance of the gradient coil optimized is closer to true gradient coil performance.Especially in axial direction
When gradient coil design, this method is more advantageous.It, can be by the position of annulus since axial coil is all made of annulus
It is optimized directly as optimized variable.But nonlinear optimization algorithm there is a problem of one it is universal, exactly need very
More iterative steps.Such as simulated annealing, it usually needs hundreds thousand of or even up to a million ability optimizations of iteration are optimal
As a result.For gradient coil optimization problem, the parameter mainly optimized is magnetic field and the inductance value of imaging space.The two
The calculating of parameter directly affects optimization time and computational accuracy.Since the calculating of the two amounts is directed to some complicated letters
Number, the calculating time is longer, needs long time so as to cause nonlinear optimization.
Invention content
Goal of the invention：It, can be fast in a kind of optimization process of present invention offer in order to overcome the deficiencies in the prior art
Speed calculates the magnetic field in imaging region and the computational methods of coil inductance.
Technical solution：To achieve the above object, the present invention uses following steps：
Step 1：Using the cylinder center of gradient coil as origin, axis direction is that z-axis establishes coordinate system, determines M magnetic field
The coordinate of sampled point, it is assumed that the greatest axial length of gradient coil is 2Z；
Step 2：Axial section [0, Z] is divided into N at equal intervals1Equal portions, every section of length is Δ z=Z/N1, wherein Δ z≤
1cm, it is z to calculate separately coordinate pointsi=± i Δs z, i=1,2...N1Maxwell's coil adopted other than the axis and on axis
The magnetic field that sampling point generates, and it is M × N to store the result into dimension1Array in, wherein each Maxwell's coil correspond to two
The axial coordinate of annulus, two annulus is respectively ± z1, annular radii a, current direction is opposite；
Step 3：Calculate the self-induction L of single annuluss, a variable storage is used in combination；
Step 4：Axial section [0,2Z] is divided into N at equal intervals2Equal portions, every section of length is Δ d=2Z/N2, wherein Δ d
≤ 1cm, calculating axially spaced-apart are di=i Δs d, i=1,2...N2Two coils mutual inductance, and store the result into one
Dimension is N2Array in；
Step 5：During Optimized Iterative, the angular interval distance in the coordinate points and step 4 in step 2 is made
For interpolation point, it is located at the Maxwell's coil axial coordinate absolute value of other positions | z | each sampling when being [0, Z] interior arbitrary value
The mutual inductance interpolating function that the magnetic field of point and two angular interval d generate for two annulus of the mutual inductance of [0,2Z] interior arbitrary value
It indicates.Calculating speed can be significantly improved using the step, speed can be improved nearly ten times on the basis of original.
Wherein in step 2, the axial magnetic field B at the sampled point other than axiszIt is acquired using following formula：
Wherein μ0For magnetic conductivity, I is annulus current-carrying, and r is distance of the magnetic field sampled point to z-axis, and z is that magnetic field sampled point is axial
Coordinate, z1For the absolute value of the axial coordinate of Maxwell's coil；K, E are the first kind and elliptic integral of the second kind, parameter k1, k2
It is defined as follows：
Axial magnetic field B on axiszIt is acquired using following formula：
Single annulus self-induction L wherein in step 3sCalculation formula be：
Wherein raEquivalent redius when circle is equivalent to for coil section.
Wherein in step 4, the mutual inductance L of two annulusMIt is calculated using following formula：
Wherein d is the distance between two Circumferential coils, positive sign is taken when electric current is in the same direction before formula, when electric current is reversed before formula
Take negative sign.Compared with prior art, the formula for calculating mutual inductance is more succinct.
Wherein in step 5, the specific formula for calculation of interpolating function is as follows：
For zi+1≥z≥zi, the magnetic field B at m-th of sampled pointm(z) calculation formula is as follows：
For di+1≥d≥di, the calculation formula of mutual inductance L (d) is：
L (d)=GLi·(d-di)+L(di)
Wherein, L (d) is divided into the mutual inductance between the annulus of d between being two, andAndGLiIt calculates and stores before optimization algorithm starts.Compared with prior art, the method for mutual inductance is calculated more rapidly, formula
It is easier.
Advantageous effect：In the present invention, the interval between the coordinate range and two annulus where annulus is divided into
Several equal portions, magnetic field are calculated with value of the inductance at sampled point using analytic formula, and the value at other positions uses interpolating function
It calculates, and The present invention gives a kind of calculation formula of quick calculating magnetic field and inductance.Compared with prior art, energy of the present invention
Calculating speed is enough greatly improved, speed can be improved nearly ten times on the basis of original, so that efficient non-linear excellent
Change algorithm to be possibly realized.
Description of the drawings
Fig. 1 is the flow chart of the present invention；
Fig. 2 is the structural schematic diagram of gradient coil in embodiment；
Fig. 3 is the structural schematic diagram of Maxwell's coil in embodiment；
Fig. 4 is in embodiment using the gradient coil of simulated annealing design；
Fig. 5 be embodiment in gradient coil in 45cmDSV magnetic field with z coordinate change curve；
Fig. 6 is error of the magnetic field in embodiment in 45cmDSV using inventive algorithm.
Specific implementation mode
The present invention is further described below in conjunction with the accompanying drawings.
In superconducting MRI system, axial gradient coil is typically distributed across on one layer of cylindrical surface.Actual cylinder gradient coil
Generally can outside main coil additional layer shielded coil, annulus structure of the shielded coil equally by being distributed in one layer of periphery
At.Fig. 2 is typical active shielded gradient coil structure, and inner surface is main coil, and outer surface is shielded coil, and institute is wired
Circle is about axial centre odd symmetry.Wherein main coil has several Maxwell's coils composition with shielded coil.So-called Mike
This Wei coil is exactly coil pair identical by two radiuses, that the opposite annulus of current direction is constituted, as shown in Figure 3.Therefore, right
For axial coil, in the case where Maxwell's coil number determines, main purpose is to optimize the position of each annulus.One
As in the case of, optimize gradient coil constructed by object function it is as follows：
F=ω1E+ω2L
Wherein, E is the gradient field error in imaging region (DSV), and L is coil inductance.The wherein value of E and sampling in DSV
The axial magnetic field of point is related.As can be seen that the above-mentioned calculative amount of optimization problem is mainly axial magnetic field and inductance in DSV
Value.For the nonlinear optimization algorithm of simulated annealing, genetic algorithm one kind, the calculating of magnetic field and inductance is to be directly related to optimization
The efficiency of algorithm.The electric field in the present invention is sketched with inductance quick calculation method below.
The quick calculation method of magnetic field and inductance is as follows in the present invention：
Step 1：Using cylinder center as origin, axis direction is that z-axis establishes coordinate system, determines the seat of M magnetic field sampled point
Mark.It is assumed that the greatest axial length in cylinder axial direction coil design is 2Z；
Step 2：Axial section [0, Z] is divided into N at equal intervals1Equal portions, every section of length is Δ z=Z/N1, coordinates computed
Point is zi=± i Δs z, i=1,2...N1Maxwell's coil in the magnetic field that sampled point generates, and store the result into
Dimension is M × N1Array in；
Step 3：Calculate the self-induction L of single annuluss, a variable storage is used in combination；
Step 4：Axial section [0,2Z] is divided into N at equal intervals2Equal portions, every section of length is Δ d=2Z/N2, calculate axis
It is divided into d toi=i Δs d, i=1,2...N2Two Circumferential coils mutual inductance, and store the result into a dimension be N2
Array in；
Step 5：During Optimized Iterative, using the above sampled point as interpolation point, it is located at the Maxwell of other positions
Coil indicates the mutual inductance generated in two annulus at the magnetic field that sampled point generates and any distance interval with interpolating function.
In step 1, because axial coil is symmetrical structure mostly, Mike can be ensured by establishing coordinate system with cylinder center
Two annulus of this Wei coil are about coordinate axial symmetry.
In the present embodiment, in order to ensure computational accuracy, every section of length Δ z≤1cm, Δ d≤1cm.The value of Δ z, Δ d
It is related with the interpolating function of selection.Each section of length is shorter, then accurately calculates the rank in magnetic field and the interpolating function needed for inductance
Number is smaller, and the time of calculating magnetic field and inductance is shorter in optimization process.Because existing gradient coil, either main coil are also
It is shielded coil, axial length does not exceed 1m, if d=0.1mm, then storing the array dimension needed for mutual inductance value is
10000, it is the value of a very little for the memory of computer.
In the present embodiment, for each Maxwell's coil pair, it is assumed that the axial coordinate of two annulus is respectively ± z1,
Annular radii is a, then the axial magnetic field B at the interpolation point other than axiszIt is acquired using following formula：
Here μ0For magnetic conductivity, I is annulus current-carrying, and r is distance of the magnetic field sampled point to z-axis, and z is magnetic field sampled point
Axial coordinate, z1For the absolute value of the axial coordinate of Maxwell's coil；K, E are the first kind and elliptic integral of the second kind, parameter
k1, k2It is defined as follows：
Above-mentioned function relates generally to two ellptic integrals.There are some algorithms specifically for Gauss integration at present, compared to straight
Female connector is had higher efficiency with Biot Savart for mula.
Magnetic field B on axiszIt is acquired using following formula：
In the present embodiment, the self-induction of single annulus is calculated using formula below：
Wherein raEquivalent redius when circle is equivalent to for annulus cross section.The formula is the experience public affairs for calculating self-induction of loop
Formula.Especially for Circumferential coils, using the formula result of calculation and the result that is calculated using numerical integration very close to but having
There is higher computational efficiency.
In the present embodiment, the mutual inductance of two annulus is calculated using following formula：
Wherein d is the distance between two Circumferential coils.Electric current is in the same direction, and positive sign is taken before formula, reversely takes negative sign.The formula
Same includes only two ellptic integrals.
In above-mentioned steps five, magnetic field and mutual inductance that polynomial interopolation calculates any position annulus can be used.Order of a polynomial
Number is higher, then the discrete point quantity for reaching the required accuracy needs is fewer, and the memory that storage variable needs is also fewer, but every time repeatedly
It is more for the time of calculating magnetic field and inductance.Polynomial exponent number is lower, then the time of iterative calculation magnetic field and inductance gets over every time
It is few.Suitable exponent number can be selected using the technical staff of the algorithm in the present invention as the case may be.In the present embodiment, use
Linear interpolation function calculates the axial coordinate absolute value of Maxwell's coil | z | each sampled point when being [0, Z] interior arbitrary value
Magnetic field or the mutual inductance that two angular interval d are [0,2Z] interior arbitrary value.Specific formula is as follows：
For zi+1≥z≥zi, the magnetic field B at m-th of sampled pointm(z) calculation formula is as follows：
For di+1≥d≥di, the calculation formula of mutual inductance L (d) is：
L (d)=GLi(d-di)+L(di)
Wherein
GLiIt calculates and stores before optimization algorithm starts.
Relative to high-order interpolation, linear interpolation can save to greatest extent calculates the time.
In order to compare the performance of this method, one is provided in the present embodiment using the axial ladder of simulated annealing design single layer
Spend the example of coil.Maxwell's coil in coil is to being fixed as 14 pairs.Cylinder radius surface where coil is 380mm, ladder
Degree field strength is 68 μ T/m/A, the linearity 4.8%.By optimizing the gradient coil shape designed by the object function of front as schemed
Shown in 4.
It is limited for 10000 times with iteration, the time is only needed 35 seconds using the method in the present invention.And if not using this hair
Fast algorithm in bright then needs 327 seconds.As can be seen that using the fast algorithm in the present invention, the time saves 9.3 times,
Computational efficiency is substantially increased, the time has been saved.
Use the inductance that the interpolation formula in the present invention calculates for 302.584688497 μ H, using above-mentioned self-induction and mutual inductance
The inductance value that directly calculates of formula be 302.584688088 μ H.Its difference the 7th after decimal point, completely may be used by this error
To ignore.Secondly in imaging region magnetic field value, as shown in figure 5, being to take one on circumference that XOZ plane inside radius is r=225mm
Series of samples point, the relation curve of axial magnetic field and axial coordinate at sampled point.It is illustrated in figure 6 at sampled point using this
The difference between algorithm and exact algorithm in invention.As seen from Figure 6, the magnetic field that the algorithm in the present invention calculates has enough
Precision.
The above is only a preferred embodiment of the present invention, it should be pointed out that：For the ordinary skill people of the art
For member, various improvements and modifications may be made without departing from the principle of the present invention, these improvements and modifications are also answered
It is considered as protection scope of the present invention.
Claims (5)
1. the computational methods in magnetic field and inductance value in a kind of cylinder axial direction gradient coil design, which is characterized in that including walking as follows
Suddenly：
Step 1：Using the cylinder center of gradient coil as origin, axis direction is that z-axis establishes coordinate system, determines M magnetic field sampling
The coordinate of point, it is assumed that the greatest axial length of cylinder axial direction gradient coil is 2Z；
Step 2：Axial section [0, Z] is divided into N at equal intervals1Equal portions, every section of length is Δ z=Z/N1, wherein Δ z≤1cm,
It is z to calculate separately coordinate pointsi=± i Δs z, i=1,2...N1Maxwell's coil in the magnetic field that sampled point generates, and will
As a result it is M × N to be stored in dimension1Array in, wherein each Maxwell's coil correspond to two annulus, the axial direction of two annulus
Coordinate is respectively ± z1, annular radii a, current direction is opposite；
Step 3：Calculate the self-induction L of single annuluss, a variable storage is used in combination；
Step 4：Axial section [0,2Z] is divided into N at equal intervals2Equal portions, every section of length is Δ d=2Z/N2, wherein Δ d≤
1cm, calculating axially spaced-apart are di=i Δs d, i=1,2...N2Two coils mutual inductance, and store the result into a dimension
Number is N2Array in；
Step 5：During Optimized Iterative, by the angular interval distance in the coordinate points and step 4 in step 2 as slotting
It is worth point, is located at the Maxwell's coil axial coordinate absolute value of other positions | z | each sampled point when being [0, Z] interior arbitrary value
The mutual inductance interpolating function table that magnetic field and two angular interval d generate for two annulus of the mutual inductance of [0,2Z] interior arbitrary value
Show.
2. the computational methods in magnetic field and inductance value in a kind of cylinder axial direction gradient coil design according to claim 1,
It is characterized in that：In step 2, the axial magnetic field B at the sampled point other than axiszIt is acquired using following formula：
Wherein μ0For space permeability, I is annulus current-carrying, and r is distance of the magnetic field sampled point to z-axis, and z is that magnetic field sampled point is axial
Coordinate, z1For the absolute value of the axial coordinate of Maxwell's coil；K, E are the first kind and elliptic integral of the second kind, parameter k1, k2
It is defined as follows：
Axial magnetic field B on axiszIt is acquired using following formula：
3. the computational methods in magnetic field and inductance value in a kind of cylinder axial direction gradient coil design according to claim 2,
It is characterized in that：Single annulus self-induction L in step 3sCalculation formula be：
Wherein raEquivalent redius when circle is equivalent to for coil section.
4. the computational methods in magnetic field and inductance value in a kind of cylinder axial direction gradient coil design according to claim 2,
It is characterized in that：In step 4, the mutual inductance L of two annulusMIt is calculated using following formula：
Wherein d is the distance between two Circumferential coils, takes positive sign when electric current is in the same direction before formula, is taken before formula when electric current is reversed negative
Number.
5. the computational methods in magnetic field and inductance value in a kind of cylinder axial direction gradient coil design according to claim 1,
It is characterized in that：In step 5, the specific formula for calculation of interpolating function is as follows：
For zi+1≥z≥zi, the magnetic field B at m-th of sampled pointm(z) calculation formula is as follows：
Wherein, Bm(z) it is located at the Maxwell's coil of ± z for axial coordinate in the magnetic field that m-th of magnetic field sampled point generates, and
For di+1≥d≥di, the calculation formula of mutual inductance L (d) is：
L (d)=GLi·(d-di)+L(di)
Wherein, L (d) is divided into the mutual inductance between the annulus of d between being two, andAndGLi
It calculates and stores before optimization algorithm starts.
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| CN111523256B (en) * | 2020-07-06 | 2020-10-20 | 南京航空航天大学 | Mutual inductance calculation method of coaxial multi-coil related to non-ferromagnetic metal medium |
| CN112036017B (en) * | 2020-08-21 | 2022-09-23 | 河海大学 | Rapid calculation method for magnetic field of solenoid coil |
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