CN109946630B - Disc type gradient coil design method capable of minimizing maximum temperature value - Google Patents

Disc type gradient coil design method capable of minimizing maximum temperature value Download PDF

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CN109946630B
CN109946630B CN201910264419.0A CN201910264419A CN109946630B CN 109946630 B CN109946630 B CN 109946630B CN 201910264419 A CN201910264419 A CN 201910264419A CN 109946630 B CN109946630 B CN 109946630B
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gradient coil
coil
heat transfer
copper plate
transfer coefficient
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CN109946630A (en
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李霞
胥祯浩
刘晓芳
徐文龙
李铁强
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China Jiliang University
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Abstract

The invention discloses a design method of a disc type gradient coil for minimizing a maximum temperature value, which comprises the following steps: 1) constructing a disc type gradient coil model; 2) obtaining a thermal equation; 3) are respectively provided withObtaining a heat transfer coefficient, a convection heat transfer coefficient and a radiation heat transfer coefficient to obtain a total heat transfer coefficient ht; 4) equation of pairs
Figure DSA0000181325380000011
Fourier analysis is carried out to obtain T under a polar coordinate system*A temperature expression at point (r, θ); 5) a disk gradient coil that minimizes the maximum temperature value. The gradient coil is designed by minimizing the maximum temperature value, the maximum temperature value of the coil is reduced, and the imaging performance of the gradient coil is improved.

Description

Disc type gradient coil design method capable of minimizing maximum temperature value
Technical Field
The invention relates to the technical field of magnetic resonance imaging, in particular to a design method of a disk type gradient coil with a minimized maximum temperature value.
Background
Magnetic resonance imaging is a technique for imaging based on the phenomenon of nuclear magnetic resonance. The gradient coil is an important component of a magnetic resonance imaging system, is used for generating a magnetic field which changes linearly along three orthogonal directions in a space in an imaging area, and is respectively applied to slice selection, phase encoding and frequency encoding so as to provide a positioning basis for image reconstruction.
In a magnetic resonance imaging system, the performance of the gradient coils determines the image resolution. The design of the gradient coil has design indexes such as eddy current, magnetic field energy storage and the like besides linearity. When the gradient coil is operated, local heat generated in the coil due to the resistance characteristics of the coil material can cause image distortion, especially when the gradient field strength is required to be relatively large. Therefore, when designing the gradient coil, the temperature is an important parameter, and how to reduce the maximum temperature value can improve the performance of magnetic resonance imaging.
Disclosure of Invention
In view of the above, an object of the present invention is to provide a method for designing a disk gradient coil that minimizes a maximum temperature value and reduces the maximum temperature value, so as to solve the technical problem of image distortion caused by an excessively high temperature in the prior art.
The technical scheme of the invention is to provide a disc type gradient coil design method for minimizing the maximum temperature value, which comprises the following steps:
1) constructing a disc type gradient coil model which comprises two disc type copper plates with the radius of R and the thickness of w; the main magnetic field direction is the z-axis direction, and the centers of two gradient coil copper plates are positioned at z ═ z0The current densities of an upper coil copper plate and a lower coil copper plate are the same; the gradient coil copper plate is embedded in the epoxy resin layer with the same radius, so that the thickness of the epoxy resin layer on the upper half part of the gradient coil copper plate is wuThe following thickness is wdThe copper plate of the lower half part is opposite to the copper plate of the gradient coil of the upper half part;
2) the internal energy of the coil includes ohmic heat generated in the copper plate by the current passing through the resistive material, conductive heat in the z-direction passing through the insulating epoxy layer, and heat convected and radiated to the surroundings of the coil, and the heat equation is obtained as
Figure BSA0000181325400000021
Wherein, T*＝T-Tenv(K) Is the temperature difference between the copper plate and the surrounding environment, ρd(kg/m3) Density of copper plate, ch(J/kg/K) is specific heat of copper plate, Kc(W/m/K) and ρr(Ω m) is the electrical conductivity and resistivity of the copper plate, respectively; j (A/m) is the current density vector, htIs the heat transfer coefficient;
3) respectively obtaining a heat transfer coefficient, a convection heat transfer coefficient and a radiation heat transfer coefficient to obtain a total heat transfer coefficient ht; to obtain a steady state solution to thermal equation (1), let the right end of the equation be zero, then the equation is rewritten as
Figure BSA0000181325400000022
It is further written in the form of,
Figure BSA0000181325400000023
wherein the content of the first and second substances,
Figure BSA0000181325400000024
f＝T*，u＝J·J；
4) equation of pairs
Figure BSA0000181325400000025
Fourier analysis is carried out to obtain T under a polar coordinate system*The temperature expression at point (r, θ) is:
Figure BSA0000181325400000026
5) gradient coil design to minimize maximum temperature values: according to the symmetry of the gradient coil, points are taken on the spherical surface of an 1/4 imaging area, 1/4 of a coil plane is taken for grid division, Nf parts are divided in an angle direction [0, pi/2 ] interval, Nr parts are divided in the radius direction, and temperature calculation is carried out;
the design for minimizing the temperature peak of the gradient coil is classified as the minimum and maximum optimization problem, and is a nonlinear optimization problem, namely
min max T*(r，θ)
Constraint conditions are as follows:
Figure BSA0000181325400000031
wherein, epsilon is 0.05, which is the given gradient magnetic field nonlinearity error limit.
The nonlinear constraint of the above equation can be written as a linear constraint:
AX≤(1+ε)Bzdes
-AX≤(ε-1)Bzdes(ii) a A is a coefficient matrix, X is an independent variable column vector, and Bzdes is an ideal target magnetic field column vector;
and solving to obtain X, then obtaining a current density expression, and dispersing by using a flow function technology to obtain a gradient coil shape with the minimized maximum temperature value.
Optionally, before step 5), a step of designing a gradient coil with minimum power consumption is set:
in a polar coordinate system, the current density has only r and theta components, because the current density can be expanded into a fourier series,
Figure BSA0000181325400000032
Figure BSA0000181325400000033
Figure BSA0000181325400000034
wherein N is the number of Fourier series, an(N is 1: N) is a coefficient to be solved; stream function SzFor obtaining a wound form of the coil,
Figure BSA0000181325400000035
according to the Biot-Savart law, obtaining the z-component of the magnetic field of the space point (x, y, z),
Figure BSA0000181325400000036
wherein
Figure BSA0000181325400000041
In the above formula, the first and second carbon atoms are,
Figure BSA0000181325400000042
selecting M points in an imaging area, and establishing an objective function:
Γ＝Φ+λΠ；
phi is the sum of squares of errors between calculated values of M points and an ideal value, lambda is a regularization factor, pi is a penalty function, and power consumption, energy storage and the like can be expressed. For controlling the temperature distribution, written here as an expression for power consumption
Figure BSA0000181325400000043
Minimizing the objective function to obtain a linear system of equations written in matrix form
(A+λP)X＝b；
Wherein X ═ { X ═ X1，x2，...xN}TFor the column vector of Fourier coefficients to be solved, xj＝aj(j＝1：N)；A＝αTα, the elements of the matrix α being Kij(i＝1：M；j＝1：N)；b＝αTBzdes，BzdesIs an element of
Figure BSA0000181325400000044
GxFor the purpose of the gradient field strength,
Figure BSA0000181325400000045
is the x-axis coordinate of the ith point. Element P of matrix Pij(i, j ═ 1: N) is
Figure BSA0000181325400000046
Obtaining the current density of the coil after obtaining the solution of the equation (A + lambda P) X ═ b, and obtaining the temperature distribution of the coil according to the temperature expression; the current density of the coil is used in the calculation of the gradient coil design that minimizes the maximum temperature value; the initial value of X in the gradient coil design that minimizes the maximum temperature value is given by the solution of the least power consuming gradient coil.
Optionally, the fourier analysis includes the specific steps of: suppose that the frequency domain (u, v) corresponds to the time domain (x, y) and has
Figure BSA0000181325400000051
In pair type
Figure BSA0000181325400000052
Is transformed to obtain
Figure BSA0000181325400000053
Wherein F and U are Fourier transforms of F and U, respectively.
According to the following expression
Figure BSA0000181325400000054
Wherein, Fq{ } is the frequency-domain to time-domain Hankel transform, K0() Is a second type of modified zero order bessel function.
Obtaining an expression of f according to the convolution theorem
Figure BSA0000181325400000055
Thus, in a polar coordinate system, T*The temperature at point (r, θ) is expressed as
Figure BSA0000181325400000056
Optionally, the specific steps of obtaining the total heat transfer coefficient ht include: assuming that the heat loss of the upper part is quThe temperature of the copper plate central layer is TcHas a small area AcThe corresponding upper plane area is AuAt a temperature of TuAs shown in fig. 2. Let z > z0+w/2+wuTemperature of the part is TenvThermal conductivity of epoxy resin plate is kf(W/m/K), convective heat transfer coefficient is hu(W/m2K); the fourier law is used to determine the conductive heat transfer rate,
Figure BSA0000181325400000057
the heat and z ═ z0+w/2+wuPlanar convective heat transfer is relevant. According to Newton's law of cooling, there are
qu＝huAu(Tu-Tenv)；
To obtain
Figure BSA0000181325400000061
By the same token, the result is obtained by0-w/2-wdHeat loss from the underside of the plane. Finally, a heat loss expression of conduction and convection is obtained
Figure BSA0000181325400000062
Wherein h isdIs obtained by z ═ z0-w/2-wdPlanar heat transfer coefficient, AdIs the area corresponding to the lower plane of the epoxy resin plate. Handle Au＝AcAnd Ad＝AcBringing in to obtain
Figure BSA0000181325400000063
By the Stefan-Boltzmann law, z ═ z can be obtained0-w/2-wdAnd z ═ z0+w/2+wuPlanar radiant heat losses
Figure BSA0000181325400000064
Wherein sigma is 5.67 multiplied by 10-8(W/m2/K4) Is the Stefan-Boltzmann constant, εuAnd εdRespectively, the radiance of both surfaces.
Linearizing the above formula to obtain
qrod＝hr(Tc-Tenv)(εuAu+εdAd)；
Wherein the radiant heat transfer coefficient
Figure BSA0000181325400000065
From hrCan be seen by the expression of (b), hrThe value of (d) is temperature dependent, so we will choose an appropriate value depending on the temperature range of the gradient coil; binding of Au＝AcAnd Ad＝AcThe total heat loss due to cooling is obtained
qcooling＝ht(Tc-Tenv)Ac；
Wherein the total heat transfer coefficient
Figure BSA0000181325400000066
Compared with the prior art, the method of the invention has the following advantages: the gradient coil is designed by minimizing the maximum temperature value, the maximum temperature value of the coil is reduced, and the imaging performance of the gradient coil is improved.
Drawings
FIG. 1 is a schematic diagram of a design model of a disk gradient coil of the present invention;
FIG. 2 is a schematic cross-sectional view of a temperature calculation of a disk gradient coil;
FIG. 3 is a schematic diagram of a minimum power consumption coil shape;
fig. 4 shows hd-hu-10 (W/m)2A disc gradient coil shape schematic of minimized maximum temperature value of/K);
fig. 5 shows hd-hu-100 (W/m)2A disc gradient coil shape schematic of minimized maximum temperature value of/K);
fig. 6 shows hd-hu-1000 (W/m)2A disc gradient coil shape schematic of minimized maximum temperature value of/K);
fig. 7 shows hd-hu-10 (W/m)2A comparison graph of temperature distribution before and after optimization of the/K);
fig. 8 shows hd-hu-100 (W/m)2A comparison graph of temperature distribution before and after optimization of the/K);
FIG. 9 shows hd ═ hu＝1000(W/m2A comparison of the temperature distribution before and after optimization of/K).
Detailed Description
Preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, but the present invention is not limited to only these embodiments. The invention is intended to cover alternatives, modifications, equivalents and alternatives which may be included within the spirit and scope of the invention.
In the following description of the preferred embodiments of the present invention, specific details are set forth in order to provide a thorough understanding of the present invention, and it will be apparent to those skilled in the art that the present invention may be practiced without these specific details.
The invention is described in more detail in the following paragraphs by way of example with reference to the accompanying drawings. It should be noted that the drawings are in simplified form and are not to precise scale, which is only used for convenience and clarity to assist in describing the embodiments of the present invention.
Referring to FIG. 1, a model of a disk gradient coil is illustrated. The model is shown in fig. 1 and was designed using an x-gradient coil as an example. The gradient coil is two disk-shaped copper plates with the radius of R and the thickness of w. The main magnetic field direction is the z-axis direction, and the coil copper plate is positioned at z ═ z0Plane, current density is the same. The gradient coil copper plate is embedded in the epoxy resin layer with the same radius, taking the copper plate of the upper half as an example, the thickness of the epoxy resin layer on the copper plate is assumed to be wuThe following thickness is wdThe lower half of the copper plate is opposite to the upper half.
In this model, the internal energy of the coil includes ohmic heat generated by the current through the resistive material at the copper plate, conductive heat in the z-direction through the insulating epoxy plate, and heat that is convected and radiated to the surroundings of the coil. This thermal equation can be written as
Figure BSA0000181325400000081
Wherein, T*＝T-Tenv(K) Is the temperature difference between the copper plate and the surrounding environment, ρd(kg/m3) The density of the copper plate is shown by the density,ch(J/kg/K) is specific heat of copper plate, Kc(W/m/K) and ρr(Ω m) is the electrical conductivity and resistivity of the copper plate, respectively. J (A/m) is the current density vector, htIs the heat transfer coefficient. H is given belowtThe derivation process of (1).
First, assume that the heat loss of the upper part is quThe temperature of the copper plate central layer is TcHas a small area AcThe corresponding upper plane area is AuAt a temperature of TuAs shown in fig. 2. Let z > z0+w/2+wuTemperature of the part is TenvThermal conductivity of epoxy resin plate is kf(W/m/K), convective heat transfer coefficient is hu(W/m2K) is added. We use fourier's law to determine the conductive heat transfer rate,
Figure BSA0000181325400000082
the heat and z ═ z0+w/2+wuPlanar convective heat transfer is relevant. According to Newton's law of cooling, there are
qu＝huAu(Tu-Tenv) (3)
From the formulae (2) and (3), we can obtain
Figure BSA0000181325400000091
By the same token, the result is obtained by0-w/2-wdHeat loss from the underside of the plane. Finally, a heat loss expression of conduction and convection is obtained
Figure BSA0000181325400000092
Wherein h isdIs obtained by z ═ z0-w/2-wdPlanar heat transfer coefficient, AdIs the area corresponding to the lower plane of the epoxy resin plate. Handle Au＝AcAnd Ad＝AcCarry over into (5) to obtain
Figure BSA0000181325400000093
By the Stefan-Boltzmann law, z ═ z can be obtained0-w/2-wdAnd z ═ z0+w/2+wuPlanar radiant heat losses
Figure BSA0000181325400000094
Wherein sigma is 5.67 multiplied by 10-8(W/m2/K4) Is the Stefan-Boltzmann constant, εuAnd εdRespectively, the radiance of both surfaces.
Linearizing the formula (7) to obtain
qrod＝hr(Tc-Tenv)(εuAu+εdAd) (8)
Wherein the radiant heat transfer coefficient
Figure BSA0000181325400000095
We analyzed the radiation pattern in a similar way as we analyzed convection. However, from hrCan be seen by the expression of (b), hrThe value of (d) is temperature dependent, so we will choose a suitable value depending on the temperature range of the gradient coil. According to formulae (6) and (8), in combination with Au＝AcAnd Ad＝AcThe total heat loss due to cooling is obtained
qcooling＝ht(Tc-Tenv)Ac (9)
Wherein the total heat transfer coefficient
Figure BSA0000181325400000101
To obtain a steady state solution of temperature equation (1), let the right end of the equation be zero, the equation can be written as
Figure BSA0000181325400000102
The above formula can be further written in the form of,
Figure BSA0000181325400000103
wherein the content of the first and second substances,
Figure BSA0000181325400000104
f＝T*，u＝J·J。
fourier analysis:
equation (11) can be analyzed by fourier transform. Suppose that the frequency domain (u, v) corresponds to the time domain (x, y) and has
Figure BSA0000181325400000105
By transforming the formula (11), the
Figure BSA0000181325400000106
Wherein F and U are Fourier transforms of F and U, respectively.
According to the following expression
Figure BSA0000181325400000107
Wherein, Fq{ } is the frequency-domain to time-domain Hankel transform, K0() Is a second type of modified zero order bessel function.
From the expressions (13) and (14), an expression of f is obtained according to the convolution theorem
Figure BSA0000181325400000111
Thus, in a polar coordinate system, T*The temperature at point (r, θ) is expressed as
Figure BSA0000181325400000112
Design of gradient coil with minimum power consumption:
in a polar coordinate system, the current density has only r and theta components, because the current density can be expanded into a fourier series,
Figure BSA0000181325400000113
Figure BSA0000181325400000114
Figure BSA0000181325400000115
wherein N is the number of Fourier series, an(N is 1: N) is a coefficient to be solved; stream function SzFor obtaining a wound form of the coil,
Figure BSA0000181325400000116
according to the Biot-Savart law, obtaining the z-component of the magnetic field of the space point (x, y, z),
Figure BSA0000181325400000117
wherein
Figure BSA0000181325400000121
In the above formula, the first and second carbon atoms are,
Figure BSA0000181325400000122
selecting M points in an imaging area, and establishing an objective function
Γ＝Φ+λΠ (23)
Phi is the sum of squares of errors between calculated values of M points and an ideal value, lambda is a regularization factor, pi is a penalty function, and power consumption, energy storage and the like can be expressed. For controlling the temperature distribution, written here as an expression for power consumption
Figure BSA0000181325400000123
Minimizing the objective function to obtain a linear system of equations written in matrix form
(A+λP)X＝b (25)
Wherein X ═ { X ═ X1，x2，...xN}TFor the column vector of Fourier coefficients to be solved, xj＝aj(j＝1：N)；A＝αTα, element K of matrix αij(i 1: M; j 1: N) is obtained by bringing the coordinates of the target points into formula (22); b ═ alphaTBzdes，BzdesIs an element of
Figure BSA0000181325400000124
GxFor the purpose of the gradient field strength,
Figure BSA0000181325400000125
is the x-axis coordinate of the ith point. Element P of matrix Pij(i, j ═ 1: N) is
Figure BSA0000181325400000126
After obtaining the solution of equation (25), we can obtain the current density of the coil, and can obtain the temperature distribution of the coil according to equation (16), and discretize the coil according to equation (19).
Gradient coil design to minimize maximum temperature values:
the sphere of the imaging region is taken at 1/4 according to the symmetry of the gradient coils. And (3) carrying out grid division on 1/4 of a coil plane, dividing 100 parts in an angle direction [0, pi/2 ] interval and 100 parts in a radius direction, and carrying out temperature calculation.
The design to minimize gradient coil temperature peaks is classified as a min-max optimization problem, which is a non-linear optimization problem.
min max T*(r，θ) (27)
Constraint conditions are as follows:
Figure BSA0000181325400000131
wherein, epsilon is 0.05, which is the given gradient magnetic field nonlinearity error limit.
To reduce complexity, the nonlinear constraint of the above equation can be written as a linear constraint:
AX≤(1+ε)Bzdes
-AX≤(ε-1)Bzdes (29)
to reduce the number of iterations, the initial value of X is given by the solution of the least power consuming gradient coil.
The computational model dimensions for this example are: the imaging region radius rdsv is 0.19m and the two coil spacings 2z0 are 0.5 m. The coil radius R is 0.43m, the thickness w is 1.8mm, and the thickness wu of the epoxy resin plate is 2 mm. Given gradient field intensity Gx of 12.5mT/m and copper plate conductivity kc401(W/m/K), resistivity ρr＝1.68×10-8(Ω m). Emissivity epsilonu＝εdCoefficient of radiative heat transfer hr7.8, the thermal conductivity of the epoxy board is kf0.6 (W/m/K). By varying the cooling mode, i.e. by varying the convective heat transfer coefficient hu＝hd. This example is for hu＝hd＝10(W/m2/K)，hu＝hd＝100(W/m2/K)，hu＝hd＝1000(W/m2K) three cases were analyzed.
Coil shape with minimum power consumption and three casesAs shown in fig. 3-6. Wherein the coil shape of the minimum power consumption is shown in FIG. 3, hu＝hd＝10(W/m2/K)，hu＝hd＝100(W/m2/K)，hu＝hd＝1000(W/m2K) three cases are shown in fig. 4-6. Fig. 7-9 are temperature profiles before and after optimization for three cases. Fig. 7 illustrates hd-hu-10 (W/m)2In the case of/K), the maximum temperature was decreased from 163.99 (. degree. C.) to 57.42 (. degree. C.), and the power consumption was decreased from 5862.86(W) to 2217.84 (W). Fig. 8 illustrates hd-hu-100 (W/m)2K), hd-hu-100 (W/m)2/K), maximum temperature is reduced from 67.34 (DEG C) to 22.40 (DEG C), and power consumption is reduced from 5862.86(W) to 2234.44 (W). Fig. 9 illustrates hd-hu-10 (W/m)2K), hd-hu-1000 (W/m)2/K), maximum temperature is reduced from 37.72 (DEG C) to 12.06 (DEG C), and power consumption is reduced from 5862.86(W) to 2259.33 (W). It can be found that after the maximum temperature value is minimized, the maximum temperature value is greatly reduced and the power consumption is reduced on the premise of satisfying the linearity.
Although the embodiments have been described and illustrated separately, it will be apparent to those skilled in the art that some common techniques may be substituted and integrated between the embodiments, and reference may be made to one of the embodiments not explicitly described, or to another embodiment described.
The above-described embodiments do not limit the scope of the present invention. Any modification, equivalent replacement, and improvement made within the spirit and principle of the above-described embodiments should be included in the protection scope of the technical solution.

Claims (4)

1. A design method of a disk gradient coil for minimizing a maximum temperature value includes the following steps:
1) constructing a disc type gradient coil model which comprises two disc type copper plates with the radius of R and the thickness of w; the main magnetic field direction is the z-axis direction, and the centers of two gradient coil copper plates are positioned at z ═ z0The current densities of an upper coil copper plate and a lower coil copper plate are the same; the gradient coil copper plate is embedded in the epoxy resin layer with the same radiusThe thickness of the epoxy resin layer on the copper plate of the gradient coil in the upper half part is wuThe following thickness is wdThe copper plate of the lower half part is opposite to the copper plate of the gradient coil of the upper half part;
2) the internal energy of the coil includes ohmic heat generated in the copper plate by the current passing through the resistive material, conductive heat in the z-direction passing through the insulating epoxy layer, and heat convected and radiated to the surroundings of the coil, and the heat equation is obtained as
Figure FSB0000192642150000011
Wherein, T*＝T-Tenv(K) Is the temperature difference between the copper plate and the surrounding environment, ρdDensity of copper plate, chSpecific heat of copper plate, kcAnd ρrThe conductivity and resistivity of the copper plate respectively; j is the current density vector, htIs the heat transfer coefficient;
3) respectively obtaining a heat transfer coefficient, a convection heat transfer coefficient and a radiation heat transfer coefficient to obtain a total heat transfer coefficient ht; to obtain a steady state solution to thermal equation (1), let the right end of the equation be zero, then the equation is rewritten as
Figure FSB0000192642150000012
It is further written in the form of,
Figure FSB0000192642150000013
wherein the content of the first and second substances,
Figure FSB0000192642150000014
f＝T*，u＝J·J；
4) equation of pairs
Figure FSB0000192642150000015
Fourier analysis is carried out to obtain T under a polar coordinate system*The temperature expression at point (r, θ) is:
Figure FSB0000192642150000016
5) gradient coil design to minimize maximum temperature values: according to the symmetry of the gradient coil, points are taken on the spherical surface of an 1/4 imaging area, 1/4 of a coil plane is taken for grid division, Nf parts are divided in an angle direction [0, pi/2 ] interval, Nr parts are divided in the radius direction, and temperature calculation is carried out;
the design for minimizing the temperature peak of the gradient coil is classified as the minimum and maximum optimization problem, and is a nonlinear optimization problem, namely
min max T*(r，θ)
Constraint conditions are as follows:
Figure FSB0000192642150000021
wherein epsilon is 0.05, which is the nonlinear error limit of the given gradient magnetic field;
the nonlinear constraint of the above equation can be written as a linear constraint:
Figure FSB0000192642150000022
a is a coefficient matrix, X is an independent variable column vector, and Bzdes is an ideal target magnetic field column vector;
and solving to obtain X, then obtaining a current density expression, and dispersing by using a flow function technology to obtain a gradient coil shape with the minimized maximum temperature value.
2. The disc gradient coil design method of minimizing maximum temperature values of claim 1, characterized in that: setting a minimum power consumption gradient coil design step before the step 5):
in a polar coordinate system, the current density has only r and theta components, because the current density is expanded in the form of a fourier series,
Figure FSB0000192642150000023
Figure FSB0000192642150000024
Figure FSB0000192642150000025
wherein N is the number of Fourier series, anFor the coefficients to be found, n is 1: n; stream function SzFor obtaining a wound form of the coil,
Figure FSB0000192642150000031
according to the Biot-Savart law, obtaining the z-component of the magnetic field of the space point (x, y, z),
Figure FSB0000192642150000032
wherein
Figure FSB0000192642150000033
In the above formula, the first and second carbon atoms are,
Figure FSB0000192642150000034
selecting M points in an imaging area, and establishing an objective function
Γ＝Φ+λ∏；
Phi is the sum of squares of errors of calculated values of M points and ideal values, lambda is a regularization factor, and pi is a penalty function and represents power consumption or energy storage; for controlling the temperature distribution, written here as an expression for power consumption
Figure FSB0000192642150000035
Minimizing the objective function to obtain a linear equation set written in the form of a matrix
(A+λP)X＝b；
Wherein X ═ { X ═ X1，x2，...xN}TFor the column vector of Fourier coefficients to be solved, xj＝ajWherein j is 1: n; a ═ αTα, the elements of the matrix α being KijWherein i is 1: m; j is 1: n; b ═ alphaTBzdes，BzdesIs an element of
Figure FSB0000192642150000036
Wherein i is 1: m, GxFor the purpose of the gradient field strength,
Figure FSB0000192642150000037
is the x-axis coordinate of the ith point; element P of matrix PijThe formula is shown in the specification, wherein i, j is 1: n is added to the reaction solution to form a reaction solution,
Figure FSB0000192642150000038
obtaining the current density of the coil after obtaining the solution of the equation (A + lambda P) X ═ b, and obtaining the temperature distribution of the coil according to the temperature expression; the current density of the coil is used in the calculation of the gradient coil design that minimizes the maximum temperature value; the initial value of X in the gradient coil design that minimizes the maximum temperature value is given by the solution of the least power consuming gradient coil.
3. The disc gradient coil design method of minimizing maximum temperature values according to claim 2, characterized in that: the Fourier analysis comprises the following specific steps: suppose that the frequency domain (u, v) corresponds to the time domain (x, y) and has
Figure FSB0000192642150000041
In pair type
Figure FSB0000192642150000042
Is transformed to obtain
Figure FSB0000192642150000043
Wherein F and U are Fourier transform of F and U respectively;
according to the following expression
Figure FSB0000192642150000044
Wherein, Fq{ } is the frequency-domain to time-domain Hankel transform, K0() Is a second type of modified zero order bessel function;
obtaining an expression of f according to the convolution theorem
Figure FSB0000192642150000045
Thus, in a polar coordinate system, T*The temperature at point (r, θ) is expressed as
Figure FSB0000192642150000046
4. The disc gradient coil design method of minimizing maximum temperature values of claim 3, characterized in that: the specific steps for obtaining the total heat transfer coefficient ht are as follows: assuming that the heat loss of the upper part is quThe temperature of the copper plate central layer is TcHas a small area AcThe corresponding upper plane area is AuAt a temperature of TuLet z > z0+w/2+wuTemperature of the part is TenvThermal conductivity of epoxy resin plate is kfThe convective heat transfer coefficient is hu(ii) a The fourier law is used to determine the conductive heat transfer rate,
Figure FSB0000192642150000051
the heat and z ═ z0+w/2+wuPlanar convective heat transfer concerns; according to Newton's law of cooling, there are
qu＝huAu(Tu-Tenv)；
To obtain
Figure FSB0000192642150000052
By the same token, the product is obtained by0-w/2-wdHeat loss below the plane; finally, a heat loss expression of conduction and convection is obtained
Figure FSB0000192642150000053
Wherein h isdIs obtained by z ═ z0-w/2-wdPlanar heat transfer coefficient, AdIs the area corresponding to the lower plane of the epoxy resin plate; handle Au＝AcAnd Ad＝AcBringing in to obtain
Figure FSB0000192642150000054
Obtaining z as z by Stefan-Boltzmann law0-w/2-wdAnd z ═ z0+w/2+wuPlanar radiant heat losses
Figure FSB0000192642150000055
Wherein sigma is 5.67 multiplied by 10-8Is the Stefan-Boltzmann constant in W/m2/K4，εuAnd εdEmissivity of the two surfaces, respectively;
linearizing the above formula to obtain
qrad＝hr(Tc-Tenv)(εuAu+εdAd)；
Wherein the radiant heat transfer coefficient
Figure FSB0000192642150000056
From hrIs shown in the expression ofrThe value of (d) depends on the temperature; binding of Au＝AcAnd Ad＝AcThe total heat loss due to cooling is obtained:
qcooling＝ht(Tc-Tenv)Ac；
wherein the total heat transfer coefficient
Figure FSB0000192642150000061
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