CN107315890B - Electromagnetic adsorption force calculation method considering air gap layer influence - Google Patents

Electromagnetic adsorption force calculation method considering air gap layer influence Download PDF

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CN107315890B
CN107315890B CN201710565851.4A CN201710565851A CN107315890B CN 107315890 B CN107315890 B CN 107315890B CN 201710565851 A CN201710565851 A CN 201710565851A CN 107315890 B CN107315890 B CN 107315890B
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江鹏
张群
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Intesim Dalian Co ltd
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Abstract

The invention discloses an electromagnetic adsorption force calculation method considering the influence of an air gap layer, which comprises the following steps: 1. establishing an electromagnetic field model containing an air gap layer between adjacent objects; 2. establishing a physical relation which is satisfied between adjacent objects at two sides of the air gap layer, and obtaining an equivalent integral form corresponding to a control equation containing constraint conditions; 3. dispersing the equivalent integral form by adopting a finite element method, and obtaining a corresponding finite element equation; 4. solving the finite element equation obtained in the step 3, and obtaining the electromagnetic field distribution corresponding to the air gap layer according to the numerical solution of the finite element equation and the established physical relationship; 5. and obtaining the electromagnetic adsorption force between the adjacent objects by adopting an electromagnetic force calculation method according to the electromagnetic field distribution corresponding to the air gap layer. The invention fully considers the influence of the air gap on the electromagnetic force and has better precision and efficiency.

Description

Electromagnetic adsorption force calculation method considering air gap layer influence
Technical Field
The invention relates to the field of electronic and electrical and the field of numerical simulation methods of computer aided engineering, in particular to a method for calculating electromagnetic adsorption force under the condition that air gaps exist among objects; the technology can be used for solving the problem of calculating the electromagnetic force of electronic and electrical equipment with an air gap layer, such as the problem of calculating the electrostatic adsorption force of an electrostatic chuck-wafer in a semiconductor process, the problem of calculating the magnetic field adsorption force in a motor and an electromagnetic valve and the like.
Background
Electromagnetic adsorption devices have important industrial applications in the field of electronics and electronics. As in the semiconductor industry, electrostatic chucks, which are electrostatic chucking devices, are often used to hold wafers; in the electrical field, electromagnetic valves controlled by electromagnetic force are often used to control the opening and closing operations of important devices. It can therefore be seen that it is very important to accurately calculate the electromagnetic force in an electrical and electronic device. At present, the main approach of numerical calculation of electromagnetic force is to calculate the electromagnetic field distribution of the device and the surrounding air by using numerical methods such as finite elements, boundary elements and the like, and then calculate the electromagnetic force applied to the device by using electromagnetic force calculation methods such as Maxwell stress-tension method, virtual work principle method and the like.
The following method is adopted: and (3) subdividing a very fine grid at an air gap, and calculating the electromagnetic adsorption force borne by the component by using electromagnetic force calculation methods such as a Maxwell stress-strain method, a virtual work principle method and the like. However, the method has the disadvantage that when the air gap layer is thin, the method will require the generation of a large-scale grid at the air gap, resulting in a large amount of calculation.
Ren and Cendes propose a shell unit for calculating the magnetic field adsorption force between contacting objects (hereinafter referred to as magnetic contact force) detailed in Ren Z, Cendes Z. Shell elements for the calculation of magnetic forces IEEE traces Mag.2001,37: 3171-.
In recent years, Choi et al proposed a virtual air gap method [ see in detail Choi HS, Parkhi, &lTtTtransfer = L "&gTtL &/T gTtE SH. concept of virtual air gap and identity application for the Electrical transformation. IEEE transfer Mag.2006,42:663 666] which method first assumes that inserting a virtual air gap of infinite small thickness between two contacting objects does not change the Magnetic field distribution in the model, then in the result post-processing deduces the Magnetic field strength in the air gap according to the Magnetic field boundary conditions, finally, which method found the Magnetic Contact force [ detail Seo JH, Chxwell stress tension method or virtual work principle ] to find the Magnetic Contact force [ detail H HS, Ch.525. 60. the Magnetic field strength of Magnetic Contact force and the Magnetic field equivalent Contact force, 50, and the Magnetic field equivalent Contact force of Magnetic field distribution when the Magnetic field Contact force is considered by the Magnetic field Contact force and Magnetic field equivalent Contact force equivalent Contact method is not considered by the IEEE Contact field simulation model, 60 H.60. 60. the Magnetic field equivalent Contact force is considered by the Magnetic field equivalent Contact force model, the Magnetic field equivalent method, the Magnetic field equivalent Contact force model, the Magnetic field equivalent Contact method is considered as well as the Magnetic field equivalent Contact force equivalent method when the air gap Contact force model is considered to the Magnetic field equivalent Contact force equivalent.
Disclosure of Invention
In view of the defects of the prior art, the invention aims to provide an electromagnetic adsorption force calculation method suitable for a layer containing air gaps, which fully considers the influence of the air gaps on electromagnetic force and has better precision and efficiency.
In order to achieve the purpose, the technical scheme of the invention is as follows:
an electromagnetic adsorption force calculation method considering influence of an air gap layer, comprising the steps of:
step 1, establishing an electromagnetic field model containing an air gap layer between adjacent objects;
step 2, establishing a physical relation which is satisfied between adjacent objects at two sides of the air gap layer, introducing the established physical relation as a constraint condition into a control equation corresponding to the electromagnetic field model in the step 1, and obtaining an equivalent integral form corresponding to the control equation containing the constraint condition;
step 3, dispersing the equivalent integral form obtained in the step 2 by adopting a finite element method, and obtaining a finite element equation corresponding to a control equation containing constraint conditions;
step 4, solving the finite element equation obtained in the step 3, and obtaining the electromagnetic field distribution corresponding to the air gap layer according to the numerical solution of the finite element equation and the established physical relationship;
and 5, obtaining the electromagnetic adsorption force between adjacent objects by adopting an electromagnetic force calculation method according to the electromagnetic field distribution corresponding to the air gap layer.
Further, the step 1 comprises: when a geometric model of an electromagnetic field solving area is established, an air gap layer omega between adjacent objects is not consideredgEstablishing a geometric model, wherein the obtained geometric model only comprises the area of the adjacent object and is marked as a main domain omegamAnd the slave domain omegasAnd the surfaces on both sides of the air gap layer are respectively marked asm＝Ωm∩ΩgAnds＝Ωs∩Ωgand uniformly expressing the electric field or magnetic field control equation corresponding to the obtained geometric model as that in the domain omega-omegam∪ΩsThe internal requirements are as follows:
Figure GDA0001409009490000031
and satisfies at the respective boundaries:
Figure GDA0001409009490000032
if equation (1) represents the electric field equation, then
Figure GDA0001409009490000033
For electric scale potential, α represents isotropic dielectric constant, which is generally much larger than air dielectric constant, f represents charge excitation, if equation (2) represents magnetic field equation
Figure GDA0001409009490000034
For magnetic scale potential, α denotes isotropic permeability, generally much greater than air permeability, f has the following form:
f＝-divαHsource(3)
(3) in the formula, HsourceIt should satisfy:
curlHsource＝Jsource(4)
(4) j in (1)sourceRepresenting the excitation current density.
Further, the step 2 comprises:
firstly, establishing a region omega according to the boundary condition of an electromagnetic fieldm、ΩsAnd ΩgThe relationship between the electromagnetic field quantities, namely the air gap boundary condition:
Figure GDA0001409009490000041
Figure GDA0001409009490000042
wherein
Figure GDA0001409009490000043
Respectively representmUpper edge of surface omegamOuter normal and tangential flux densities;
Figure GDA0001409009490000044
respectively representsUpper edge of surface omegasOuter normal and tangential flux densities; q. q.sgDenotes the air gap layer omegagMid-edge omegamThe flux density in the outer normal direction;
secondly, the air gap boundary condition (5a) is described by adopting an electric scale potential or a magnetic scale potential, and the following conditions are provided:
at the boundarymAt the position of the air compressor, the air compressor is started,
Figure GDA0001409009490000045
expressed as:
Figure GDA0001409009490000046
at the boundarysAt the position of the air compressor, the air compressor is started,
Figure GDA0001409009490000047
expressed as:
Figure GDA0001409009490000048
in the formulae (6) and (7), nmTo representmOmega on the surfacemOuter normal direction, nsTo representsOmega on the surfacesExternal normal direction and their direction are exactly opposite αmAnd αsRespectively represent the main domain omegamAnd the slave domain omegasThe material parameters of (a);
Figure GDA0001409009490000049
and
Figure GDA00014090094900000410
respectively represent the main domain omegamAnd the slave domain omegasα scalar potential of0Representing an air material parameter; d0Represents the thickness of the air gap layer; h is0Represents the equivalent resistance or magnetoresistance of the air gap layer;
finally at Ω respectivelymAnd ΩsIntroducing a heuristic function wmAnd wsAnd applying a Green formula to obtain equivalent integral forms of (1), (6) and (7):
Figure GDA00014090094900000411
Figure GDA00014090094900000412
further, the step 3 is discrete by using a finite element method, which includes:
step 31, for region omegamAnd ΩsDivide the grid and note Ni(i ═ 1,2 … n) is the node basis function of the standard finite element space, phi ═ phi [ phi ], [ phi ] phi ═ phi [, phi ] phi1,φ2…φn]TA node degree of freedom that is a scalar potential, and expressing the scalar potential in the solution domain as the product of the node basis function and the node degree of freedom:
Figure GDA0001409009490000051
and will omegamThe degree of freedom is divided into two parts, namely corresponding to omega respectivelym/mAndmrespectively, of upper degrees of freedom
Figure GDA0001409009490000052
And
Figure GDA0001409009490000053
will omegasThe degree of freedom is divided into two parts, namely corresponding to omega respectivelys/sAndsrespectively, degree of freedom of
Figure GDA0001409009490000054
And
Figure GDA0001409009490000055
then phi is expressed as:
Figure GDA0001409009490000056
step 32, mixing10) And (11) and wm,s＝Ni(i ═ 1,2 … n) is substituted into (8) and (9), and a finite element equation containing the air gap boundary condition, namely a finite element equation corresponding to the control equation containing the constraint condition, is obtained:
Figure GDA0001409009490000057
will be (12) in
Figure GDA0001409009490000058
Is uniformly expressed as KmmThen, the following component expression forms are provided:
Figure GDA0001409009490000059
will be (12) in
Figure GDA00014090094900000510
Is uniformly expressed as KssThen, the following component expression forms are provided:
Figure GDA00014090094900000511
(13) and (14) in the formula, n2To represent
Figure GDA00014090094900000512
The largest subscript of (a) is,
load vector in right end of equal sign of formula (12)
Figure GDA00014090094900000513
Is uniformly expressed as fmThen, the following component expression forms are provided:
Figure GDA00014090094900000514
load vector in right end of equal sign of formula (12)
Figure GDA00014090094900000515
Is uniformly expressed as fsThen, the following component expression forms are provided:
Figure GDA0001409009490000061
due to the formula (12) in
Figure GDA0001409009490000062
Is from (8) and (9)mAndsthe boundary on the surface is integrated and is uniformly expressed as NThen, the following component expression forms are provided:
Figure GDA0001409009490000063
(17) in the formula, n1Is that
Figure GDA0001409009490000064
The minimum subscript of (a); n is3Is that
Figure GDA0001409009490000065
Maximum superscript of (1).
Further, the step 4 comprises:
numerical solution of scalar potential of neighboring cells at air gap according to the electromagnetic field model
Figure GDA0001409009490000066
And (10) obtaining
Figure GDA0001409009490000067
And
Figure GDA0001409009490000068
then, according to the formula (6), the flux density q in the air gap is calculatedgAnd the field intensity is calculated according to the formula:
Figure GDA0001409009490000069
further, in the electromagnetic force calculating method in step 5, the electromagnetic absorption force applied to the object is calculated by using a Maxwell stress tensor method, and then the ith component of the electromagnetic absorption force is obtained by integrating the Maxwell stress tensor T on the closed surface S of the objectijExpressed as:
Figure GDA00014090094900000610
(19) in the formula, an Einstein (Einstein) summation convention is adopted for the vector indexes;ijis a Kronecker function (Kronecker) notation; u. ofiAnd ujRepresenting components of the electric or magnetic field strength, respectively.
Compared with the prior art, the invention has the beneficial effects that:
the invention fully considers the influence of the air gap on the electromagnetic force, and has better precision and efficiency; specifically, compared with the traditional air gap mesh generation method, the method does not need to generate very fine meshes at the air gaps, so that the calculation scale is reduced, and the calculation efficiency is improved; secondly, compared with the shell unit method, no additional unit is required to be constructed, the processing process is simple, and the calculation efficiency is higher; thirdly, it takes into account the influence of the air gap on the electromagnetic force compared to the virtual air gap method, and thus has a higher calculation accuracy, especially when the equivalent resistance or reluctance of the air gap layer occupies a non-negligible proportion in the equivalent circuit or magnetic circuit.
Drawings
FIG. 1 is an electromagnetic field model of a layer containing air gaps;
FIG. 2 is a schematic diagram of adjacent cells at the air gap;
FIG. 3 is a schematic diagram of a parallel plate capacitor
FIG. 4 is a finite element model of a parallel plate capacitor;
FIG. 5 is a graph illustrating the electrostatic clamping force experienced by a dielectric as a function of thickness;
FIG. 6 is a schematic diagram of a finite element model a of the magnetic field holding force of the C-shaped core;
FIG. 7 is a schematic view of a finite element model b of the magnetic field holding force of the rectangular iron core;
FIG. 8 is a graph of magnetic field clamping force at an air gap as a function of thickness in a finite element model a;
FIG. 9 is a graph of magnetic field clamping force at an air gap as a function of thickness in a finite element model b;
FIG. 10 is a flow chart of the method of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention clearer, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all embodiments of the present invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The invention discloses a method for calculating electromagnetic adsorption force, which comprises the following steps:
step 1, establishing an electromagnetic field model containing an air gap layer between adjacent objects; when modeling, without loss of generality, we only consider the three-dimensional model and only consider the problem of electromagnetic attraction between two objects, since the gap layer omega between adjacent objects is not consideredgThe geometric model is built so that the built model contains only the region omega of two adjacent objectsmAnd ΩsIs denoted as the main domain omegamAnd the slave domain omegasAs shown in FIG. 1, and the surfaces on both sides of the air gap layer are respectively marked asm＝Ωm∩ΩgAnds＝Ωs∩Ωgand uniformly expressing the electric field or magnetic field control equation corresponding to the obtained geometric model as that in the domain omega-omegam∪ΩsThe internal requirements are as follows:
Figure GDA0001409009490000071
and satisfies at the respective boundaries:
Figure GDA0001409009490000072
if equation (1) represents the electric field equation, then
Figure GDA0001409009490000073
α denotes the isotropic dielectric constant, f denotes the charge excitation, for the electric scalar potential, if equation (2) denotes the magnetic field equation
Figure GDA0001409009490000074
For magnetic scale potential, α denotes isotropic permeability, f has the form:
f＝-divαHsource(3)
(3) in the formula, HsourceIt should satisfy:
curlHsource＝Jsource(4)
(4) j in (1)sourceRepresenting the excitation current density.
(4) H in (1)sourceIt can be determined by the Biot-Swart law, and any non-substance understanding satisfying the formula (4) can be selected.
Step 2, establishing a physical relation which is satisfied between adjacent objects at two sides of the air gap layer, introducing the established physical relation as a constraint condition into a control equation corresponding to the electromagnetic field model in the step 1, and obtaining an equivalent integral form corresponding to the control equation containing the constraint condition; further, when the air gap between adjacent objects is very thin, and the adjacent object is omegamOr ΩsGap layer omegagAccording to the boundary conditions of the electromagnetic field, omega, when the relative material parameters are largemOr ΩsThe field at the outer surface is nearly perpendicular to the outer surface and the electric flux or magnetic flux density can be considered approximately along the thickness of the air gap (i.e., perpendicular to the air gap thickness)mAnds) And if the size is not changed, the step 2 comprises the following steps: firstly, establishing a region omega according to the boundary condition of an electromagnetic fieldm、ΩsAnd ΩgThe relationship between the electromagnetic field quantities, namely the air gap boundary condition:
Figure GDA0001409009490000081
Figure GDA0001409009490000082
wherein
Figure GDA0001409009490000083
Respectively representmUpper edge of surface omegamOuter normal and tangential flux densities;
Figure GDA0001409009490000084
respectively representsUpper edge of surface omegasOuter normal and tangential flux densities; q. q.sgDenotes the air gap layer omegagMid-edge omegamThe flux density in the outer normal direction;
secondly, the air gap boundary condition (5a) is described by adopting an electric scale potential or a magnetic scale potential, and the following conditions are provided:
at the boundarymAt the position of the air compressor, the air compressor is started,
Figure GDA0001409009490000085
expressed as:
Figure GDA0001409009490000086
at the boundarysAt the position of the air compressor, the air compressor is started,
Figure GDA0001409009490000087
expressed as:
Figure GDA0001409009490000088
in the formulae (6) and (7), nmTo representmOmega on the surfacemOuter normal direction, nsTo representsOmega on the surfacesExternal normal direction and their direction are exactly opposite αmAnd αsRespectively represent the main domain omegamAnd the slave domain omegasThe material parameters of (a);
Figure GDA0001409009490000099
and
Figure GDA00014090094900000910
respectively represent the main domain omegamAnd the slave domain omegasα scalar potential of0Representing an air material parameter; d0Represents the thickness of the air gap layer; h is0Represents the equivalent resistance or magnetoresistance of the air gap layer;
finally at Ω respectivelymAnd ΩsIntroducing a heuristic function wmAnd wsAnd applying a Green formula to obtain equivalent integral forms of (1), (6) and (7):
Figure GDA0001409009490000091
Figure GDA0001409009490000092
step 3, dispersing the equivalent integral form obtained in the step 2 by adopting a finite element method, and obtaining a finite element equation corresponding to a control equation containing constraint conditions; the specific step 3 comprises: firstly, introducing the air gap boundary condition into a control equation by establishing the equivalent weak form of the control equation containing the air gap boundary condition; and then dispersing the equivalent weak form by adopting a finite element method to obtain a numerical model containing the boundary condition of the air gap. Specifically, the step 3 adopts a finite element method for discretization, and includes: step 31, for region omegamAnd ΩsMeshing (since no air gap layer omega is establishedgSo that it is not necessary to split omegagAnd is generally divided into tetrahedral or hexahedral meshes in the three-dimensional model), and note Ni(i ═ 1,2 … n) is the node basis function of the standard finite element space, phi ═ phi [ phi ], [ phi ] phi ═ phi [, phi ] phi1,φ2…φn]TA node degree of freedom that is a scalar potential, and expressing the scalar potential in the solution domain as the product of the node basis function and the node degree of freedom:
Figure GDA0001409009490000093
and will omegamThe degree of freedom is divided into two parts, namely corresponding to omega respectivelym/mAndmrespectively, of upper degrees of freedom
Figure GDA0001409009490000094
And
Figure GDA0001409009490000095
will omegasThe degree of freedom is divided into two parts, namely corresponding to omega respectivelys/sAndsrespectively, degree of freedom of
Figure GDA0001409009490000096
And
Figure GDA0001409009490000097
then phi is expressed as:
Figure GDA0001409009490000098
step 32, adding the formulae (10) and (11) and wm,s＝Ni(i ═ 1,2 … n) is substituted into (8) and (9), and a finite element equation containing the air gap boundary condition, namely a finite element equation corresponding to the control equation containing the constraint condition, is obtained:
Figure GDA0001409009490000101
will be (12) in
Figure GDA0001409009490000102
Is uniformly expressed as KmmThen, the following component expression forms are provided:
Figure GDA0001409009490000103
will be (12) in
Figure GDA0001409009490000104
Is uniformly expressed as KssTool for measuring the height of a workpieceThe following components are expressed:
Figure GDA0001409009490000105
(13) and (14) in the formula, n2To represent
Figure GDA0001409009490000106
The largest subscript of (a) is,
load vector in right end of equal sign of formula (12)
Figure GDA0001409009490000107
Is uniformly expressed as fmThen, the following component expression forms are provided:
Figure GDA0001409009490000108
load vector in right end of equal sign of formula (12)
Figure GDA0001409009490000109
Is uniformly expressed as fsThen, the following component expression forms are provided:
Figure GDA00014090094900001010
due to the formula (12) in
Figure GDA00014090094900001011
Is from (8) and (9)mAndsthe boundary on the surface is integrated and is uniformly expressed as NThen, the following component expression forms are provided:
Figure GDA00014090094900001012
(17) in the formula, n1Is that
Figure GDA00014090094900001013
The minimum subscript of (a); n is3Is that
Figure GDA00014090094900001014
Maximum superscript of (1).
Step 4, solving the finite element equation obtained in the step 3, and obtaining the electromagnetic field distribution corresponding to the air gap layer according to the numerical solution of the finite element equation and the established physical relationship; the method specifically comprises the step of obtaining an electromagnetic field distribution corresponding to the air gap layer by utilizing an air gap boundary condition (5) in a field quantity form or air gap boundary conditions (6) and (7) in a scalar form after obtaining a numerical solution. Preferably, the step 4 comprises: finite element equation (12) is solved to obtain a numerical solution for the electromagnetic field model, such as the cells adjacent to the air gap in the three-dimensional, two-dimensional and one-dimensional models shown in FIG. 2, which belong to ΩmAnd ΩsThe dotted line represents the central plane between adjacent cells; numerical solution of scalar potential of neighboring cells at air gap according to the electromagnetic field model
Figure GDA00014090094900001015
And (10) obtaining
Figure GDA00014090094900001016
And
Figure GDA00014090094900001017
then, according to the formula (6), the flux density q in the air gap is calculatedgAnd the field intensity is calculated according to the formula:
Figure GDA0001409009490000111
and 5, obtaining the electromagnetic adsorption force between adjacent objects by adopting an electromagnetic force calculation method according to the electromagnetic field distribution corresponding to the air gap layer. Further, according to the electromagnetic field in the air gap layer, in the electromagnetic force calculating method in step 5, the electromagnetic adsorption force applied to the object is calculated by adopting a Maxwell stress-tension method, and then the ith component of the electromagnetic adsorption force is obtained by integrating the Maxwell stress tensor T on the closed surface S of the objectijExpressed as:
Figure GDA0001409009490000112
(19) in the formula, Einstein summation convention is adopted for vector indexes;ijis a Kronecker symbol; u. ofiAnd ujRespectively, representing field strength components.
The following two practical cases are used to illustrate the operation steps of the method of the present invention, and the advantages of the method of the present invention are illustrated by comparing the calculation results with those of other methods.
Case 1: the specific case description: the model in this case is a parallel plate capacitor filled with a dielectric having a relative dielectric constant of 5, but having an air gap layer (air) of small thickness at the central plane of the dielectric and dividing the dielectric into two parts each having a thickness of 10mm and a cross-sectional area of 0.01m2The voltage A of the upper electrode is the voltage of the electrode, and the voltage B of the lower electrode is 0V; when the electrodes are energized, the resulting electrostatic force attracts the two portions of the dielectric. This electrostatic adsorption mechanism, also known as the Coulomb effect, has been widely used in the electrical field.
The method comprises the following specific implementation steps of firstly establishing an electromagnetic field model with an air gap layer corresponding to the parallel plate capacitor, as shown in FIG. 3, wherein the model only comprises two parts of dielectric bodies separated by the air gap layer, describing geometric dimensions by referring to a case, dividing a grid by the dielectric bodies based on the steps 2 and 3, as shown in FIG. 4, setting solving information such as dielectric constants, cell types and voltage boundary conditions for the grid, specifying the position of the air gap layer in a finite element model, and setting an equivalent resistance coefficient of the air gap layer, and finally obtaining the electrostatic adsorption force suffered by the dielectric bodies in the model based on the steps 4 and 5, wherein the advantages of the method are that the method is calculated by using a virtual analysis method [ Choi HS, Park IH &, ] lTTT transformation = L &lTt &/T &g &. E.E.C.E.C of virtual air gap access and adhesives, and the results of the air gap thickness of the method are calculated by using a virtual analysis method of the air gap thickness of the invention, namely the air gap layer thickness of the invention is greater than the virtual analysis method of the air gap layer, the method of the invention, the air gap thickness of the invention, the invention is calculated by using the method of the invention which is not more than the method of the invention shown in the invention of air gap of the invention of the.
Case 2: the specific case description: this case contains two finite element analysis models a and b of the magnetic field clamping force, as shown in FIGS. 6 and 7, respectively; in the model a, the current density in the annular coil is 10000A/m2A C-shaped iron core connected end to end passes through the toroidal coil, and an air gap layer (air) with a small thickness is arranged at the joint of the C-shaped iron core and the toroidal coil, as shown by a dotted line in fig. 6; unlike the model a, the core in the model b is an elongated rectangular parallelepiped block, and the relative permeability of the core is set to 10000, which is very large.
The method comprises the steps of firstly establishing an electromagnetic field model which comprises air gap layers and corresponds to a and b and does not comprise the air gap layer parts in the model, dividing a grid based on the model in steps 2 and 3, as shown in FIGS. 6 and 7, setting solving information such as permeability, cell types, boundary conditions and current excitation, simultaneously specifying the positions of the air gap layers in the model a and b and setting an equivalent Magnetic reluctance coefficient of the air gap layers, and finally obtaining the Magnetic field adsorption force applied to the air gaps in the model a and b based on steps 4 and 5. the application has the advantages that the method respectively adopts the documents [ choice HS, Park IH, &TtTtT transition = & "&gL &/lTtT g SH. concentrate of virtual air gap and adsorption for use of the documents for calculating IEEE model g.2006,42:663 666] and [ JH ] for Magnetic field adsorption, and gap induction, the method is superior to the Magnetic field adsorption method obtained by using the air gap adsorption force calculation method 60, the air gap adsorption method 60 and the air gap adsorption force calculation method is not superior to the air gap adsorption method when the air gap adsorption force calculation result is larger than the virtual adsorption force calculation method shown in the document 60 μ 0001. the invention, the air gap adsorption method is not larger than the air gap adsorption method shown in the specification when the thickness calculation method 60.
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art should be considered to be within the technical scope of the present invention, and the technical solutions and the inventive concepts thereof according to the present invention should be equivalent or changed within the scope of the present invention.

Claims (4)

1. An electromagnetic adsorption force calculation method considering influence of an air gap layer, comprising the steps of:
step 1, establishing an electromagnetic field model containing an air gap layer between adjacent objects;
step 2, establishing a physical relation which is satisfied between adjacent objects at two sides of the air gap layer, introducing the established physical relation as a constraint condition into a control equation corresponding to the electromagnetic field model in the step 1, and obtaining an equivalent integral form corresponding to the control equation containing the constraint condition;
the step 2 comprises the following steps:
firstly, establishing a region omega according to the boundary condition of an electromagnetic fieldm、ΩsAnd ΩgThe relationship between the electromagnetic field quantities, namely the air gap boundary condition:
Figure FDA0002484806870000011
Figure FDA0002484806870000012
wherein
Figure FDA0002484806870000013
Respectively representmUpper edge of surface omegamOuter normal direction andtangential flux density;
Figure FDA0002484806870000014
respectively representsUpper edge of surface omegasOuter normal and tangential flux densities; q. q.sgDenotes the air gap layer omegagMid-edge omegamThe flux density in the outer normal direction;
secondly, the air gap boundary condition (5a) is described by adopting an electric scale potential or a magnetic scale potential, and the following conditions are provided:
at the boundarymAt the position of the air compressor, the air compressor is started,
Figure FDA0002484806870000015
expressed as:
Figure FDA0002484806870000016
at the boundarysAt the position of the air compressor, the air compressor is started,
Figure FDA0002484806870000017
expressed as:
Figure FDA0002484806870000018
in the formulae (6) and (7), nmTo representmOmega on the surfacemOuter normal direction, nsTo representsOmega on the surfacesExternal normal direction and their direction are exactly opposite αmAnd αsRespectively represent the main domain omegamAnd the slave domain omegasThe material parameters of (a);
Figure FDA0002484806870000019
and
Figure FDA00024848068700000110
respectively represent the main domain omegamAnd the slave domain omegasα scalar potential of0Representing an air material parameter; d0Represents the thickness of the air gap layer; h is0Representing equivalent electricity of the air-gap layerResistance or magnetoresistance;
finally at Ω respectivelymAnd ΩsIntroducing a heuristic function wmAnd wsAnd applying a Green formula to obtain equivalent integral forms of (1), (6) and (7):
Figure FDA0002484806870000021
Figure FDA0002484806870000022
step 3, dispersing the equivalent integral form obtained in the step 2 by adopting a finite element method, and obtaining a finite element equation corresponding to a control equation containing constraint conditions;
the step 3 adopts a finite element method for discretization, and comprises the following steps:
step 31, for region omegamAnd ΩsDivide the grid and note Ni(i ═ 1,2 … n) is the node basis function of the standard finite element space, phi ═ phi [ phi ], [ phi ] phi ═ phi [, phi ] phi1,φ2…φn]TA node degree of freedom that is a scalar potential, and expressing the scalar potential in the solution domain as the product of the node basis function and the node degree of freedom:
Figure FDA0002484806870000023
and will omegamThe degree of freedom is divided into two parts, namely corresponding to omega respectivelym/mAndmthe upper degrees of freedom are respectively phii mAnd
Figure FDA00024848068700000211
will omegasThe degree of freedom is divided into two parts, namely corresponding to omega respectivelys/sAndsupper degree of freedom respectively phii sAnd
Figure FDA00024848068700000210
thus will phiExpressed as:
Figure FDA0002484806870000029
step 32, adding the formulae (10) and (11) and wm,s＝Ni(i ═ 1,2 … n) is substituted into (8) and (9), and a finite element equation containing the air gap boundary condition, namely a finite element equation corresponding to the control equation containing the constraint condition, is obtained:
Figure FDA0002484806870000024
will be (12) in
Figure FDA0002484806870000025
Is uniformly expressed as KmmThen, the following component expression forms are provided:
Figure FDA0002484806870000026
will be (12) in
Figure FDA0002484806870000027
Is uniformly expressed as KssThen, the following component expression forms are provided:
Figure FDA0002484806870000028
(13) and (14) in the formula, n2To represent
Figure FDA00024848068700000311
The largest subscript of (a) is,
the load vector f in the right end of the equation (12) equal signi m、
Figure FDA00024848068700000312
Is uniformly expressed as fmThen, the following component expression forms are provided:
Figure FDA0002484806870000031
the load vector f in the right end of the equation (12) equal signi s、
Figure FDA00024848068700000313
Is uniformly expressed as fsThen, the following component expression forms are provided:
Figure FDA0002484806870000032
due to the formula (12) in
Figure FDA0002484806870000033
Is from (8) and (9)mAndsthe boundary on the surface is integrated and is uniformly expressed as NThen, the following component expression forms are provided:
Figure FDA0002484806870000034
(17) in the formula, n1Is that
Figure FDA0002484806870000035
The minimum subscript of (a); n is3Is that
Figure FDA0002484806870000036
Maximum superscript of (1);
step 4, solving the finite element equation obtained in the step 3, and obtaining the electromagnetic field distribution corresponding to the air gap layer according to the numerical solution of the finite element equation and the established physical relationship;
and 5, obtaining the electromagnetic adsorption force between adjacent objects by adopting an electromagnetic force calculation method according to the electromagnetic field distribution corresponding to the air gap layer.
2. The electromagnetic adsorbability calculation method according to claim 1, wherein:
the step 1 comprises the following steps: when a geometric model of an electromagnetic field solving area is established, an air gap layer omega between adjacent objects is not consideredgEstablishing a geometric model, wherein the obtained geometric model only comprises the area of the adjacent object and is marked as a main domain omegamAnd the slave domain omegasAnd the surfaces on both sides of the air gap layer are respectively marked asm＝Ωm∩ΩgAnds＝Ωs∩Ωgand uniformly expressing the electric field or magnetic field control equation corresponding to the obtained geometric model as that in the domain omega-omegam∪ΩsThe internal requirements are as follows:
Figure FDA0002484806870000037
and satisfies at the respective boundaries:
Figure FDA0002484806870000038
if equation (1) represents the electric field equation, then
Figure FDA0002484806870000039
α denotes the isotropic dielectric constant, f denotes the charge excitation, for the electric scalar potential, if equation (2) denotes the magnetic field equation
Figure FDA00024848068700000310
For magnetic scale potential, α denotes isotropic permeability, f has the form:
f＝-divαHsource(3)
(3) in the formula, HsourceIt should satisfy:
curlHsource＝Jsource(4)
(4) j in (1)sourceRepresenting the excitation current density.
3. The electromagnetic adsorbability calculation method according to claim 1, wherein: the step 4 comprises the following steps:
numerical solution of scalar potential of neighboring cells at air gap according to the electromagnetic field model
Figure FDA0002484806870000045
And (10) obtaining
Figure FDA0002484806870000044
And
Figure FDA0002484806870000043
then, according to the formula (6), the flux density q in the air gap is calculatedgAnd the field intensity is calculated according to the formula:
Figure FDA0002484806870000041
4. the electromagnetic adsorbability calculation method according to claim 1, wherein:
in the step 5, the electromagnetic force calculation method is to calculate the electromagnetic adsorption force applied to the object by adopting a Maxwell stress tensor method, and then the ith component of the electromagnetic adsorption force is obtained by integrating the Maxwell stress tensor T on the closed surface S of the objectijExpressed as:
Figure FDA0002484806870000042
(19) in the formula, Einstein summation convention is adopted for vector indexes;ijis a Kronecker symbol; u. ofiAnd ujRespectively, representing field strength components.
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