CN107315890A - A kind of electromagnetic adsorption power computational methods for considering air gap layer influence - Google Patents
A kind of electromagnetic adsorption power computational methods for considering air gap layer influence Download PDFInfo
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Abstract
The invention discloses a kind of electromagnetic adsorption power computational methods for considering air gap layer influence, it includes：1st, the electromagnetic field model containing air gap layer between adjacent object is set up；2nd, the physical relation met between the adjacent object for setting up the air gap layer both sides, and obtain the corresponding equivalent integration forms of governing equation containing constraints；3rd, using the discrete equivalent integration forms of finite element method, and corresponding finite element equation is obtained；4th, the finite element equation obtained by solution procedure 3, according to the numerical solution for the finite element equation tried to achieve and the physical relation set up, obtains magnetic distribution corresponding with the air gap layer；5th, according to the corresponding magnetic distribution of the air gap layer, the electromagnetic adsorption power between adjacent object is tried to achieve using electromagnetic force computational methods.The present invention has taken into full account influence of the air gap to electromagnetic force, with preferable precision and efficiency.
Description
Technical field
The present invention relates to electric field and the numerical value emulation method field of computer-aided engineering, specifically one
Planting is used to calculate the method containing the electromagnetic adsorption power in the case of air gap between object；Technology of the present invention can be used for solving gassiness
The electrostatic adsorption force meter of electrostatic chuck-wafer in the electromagnetic force computational problem of the electronic electric equipment of gap layer, such as semiconductor technology
Magnetic field absorption affinity computational problem in calculation problem, motor and magnetic valve etc..
Background technology
Electromagnetic absorption device has important commercial Application in electric field.As in the semiconductor industry, as
The electrostatic chuck of electrostatic adsorption device, is commonly used for clamping wafer；In electric field, the magnetic valve controlled using electromagnetic force, often
It is used to control the switch motion of visual plant.Thus, it could be seen that it is very to calculate the electromagnetic force in electric device exactly
Important.For now, the main path of numerical computations electromagnetic force is, first using numerical methods such as finite element, boundary elements
Device and the magnetic distribution of surrounding air are calculated, the electromagnetism such as Maxwell stress tensors method, principle of virtual work method are then utilized
Electromagnetic force suffered by power computational methods computing device.
Such as use following methods：The subdivision very fine and closely woven grid at air gap, then using Maxwell stress tensors method,
The electromagnetic force such as principle of virtual work method computational methods calculate the electromagnetic adsorption power suffered by component.But be to work as air gap the drawbacks of this method
When layer is very thin, this method will need to generate grid in large scale at air gap, cause larger amount of calculation.
Or use following methods：Ren and Cendes proposes a kind of shell unit to calculate the magnetic field absorption affinity between contact object
(being hereinafter magnetic contact force) [refers to Ren Z, Cendes Z.Shell elements for the computation of
magnetic forces.IEEE Trans Mag.2001,37:3171-3174].This method is based on the principle of virtual work, utilizes part
Jacobi's partial derivative and seamed edge element method, magnetic field complementary energy partial derivative has been derived to shell unit, the computational methods of electromagnetic force are obtained.
Fu et al. also by this method be applied between permanent magnet and other objects contact when absorption affinity calculate [refer to Fu WN, Zhou P,
Lin D,et al.Magnetic force computation in permanent magnets using a local
energy coordinate derivative method.IEEE Trans Mag.2004,40:683-686].Fu et al. is also
In the FEM calculation that the shell unit is generalized to Three-Dimensional Magnetic contact force, and provide detailed implementation process and [refer to Fu WN, Ho
SL,Chen N.Application of shell element method to 3-D finite-element
computation of the force on one body in contact with others.IEEE Trans Mag.
2010,46:3893-3898 and Liu Huijuan, Fu is application of the agriculture thin shell elements method between contact object in electromagnetic force calculating
Electric Machines and Control, 2012,16 (8)：101-106].But be when two objects of contact have difference the drawbacks of this method
During magnetic conductivity, due to not having explicit field expression-form in shell unit, therefore this method will cause computational accuracy problem.This
Outside, due to introducing new shell unit, realizing program becomes complicated.
In addition, in recent years, Choi et al. proposes a kind of virtual air-gap method for calculating magnetic contact force and [refers to Choi HS, Park
IH,Lee SH.Concept of virtual air gap and its applications for force
calculation.IEEE Trans Mag.2006,42:663-666].This method is assumed initially that between two contact objects
The virtual air gap of one infinitely small thickness of insertion will not change Distribution of Magnetic Field in model, then according to magnetic field in result post processing
Boundary condition extrapolates the magnetic field intensity in air gap, is finally obtained using methods such as Maxwell stress tensors method or the principles of virtual work
Magnetic contact force [refers to Seo JH, Choi HS. Computation of Magnetic Contact Forces.IEEE
Trans Mag.2014,50:525-528, and Choi HS, Lee SH, Kim YS, et al.Implementation of
virtual work principle in virtual air gap.IEEE Trans Mag.2008,44:1286-1289]。
Yoo et al. is studied the electrostatic contact power of a Coulomb type electrostatic chuck, is compared him by calculating and is pointed out building
Have to consider that influence of the air gap to electrostatic contact power [refers to Yoo J, Choi JS, Hong SJ, et al. Finite in mould
element analysis of the attractive force on a Coulomb type electrostatic
chuck.International Conference on Electrical Machines and Systems.2007
October 8-11；Seoul,Korea]；The boundary condition post-processed based on result, he is it is also proposed that a kind of be similar to virtual air gap
The computational methods of method, as a result coincide very well with actual.Virtual air-gap method uses any numerical value in numerical modeling without considering
Which kind of material is method, be without the object for considering to contact with each other, such as iron block is contacted with iron block, iron block and permanent magnet contact etc.
[refer to Seo JH, Choi HS.Computation of Magnetic Contact Forces.IEEE Trans
Mag.2014,50:525-528].But be the drawbacks of this method in equivalent electric/magnetic circuit for being constituted of the contact gentle gap of object,
When equivalent electric/magnetic resistance of air gap in electro magnetic road can not ignore by proportion, electric/magnetic field point of the air gap for whole model
The influence of cloth will can not ignore.In this case, virtual air-gap method will appear from precision problem, and this is due to that this method assume that
Distribution of the air gap on physical field does not influence.Further, since this method is to utilize field boundary condition in result post processing
Extrapolate the electric field in air gap and magnetic field, and it is indirect calculate, therefore numerical error can be caused.
The content of the invention
In view of the defect that prior art is present, a kind of suitable for the electromagnetism containing air gap layer the invention aims to provide
Absorption affinity computational methods, it has taken into full account influence of the air gap to electromagnetic force, with preferable precision and efficiency.
To achieve these goals, technical scheme：
A kind of electromagnetic adsorption power computational methods for considering air gap layer influence, it is characterised in that comprise the following steps：
Step 1, set up the electromagnetic field model containing air gap layer between adjacent object；
Step 2, set up the physical relation met between the adjacent objects of the air gap layer both sides, and by the physics set up
Relation is incorporated into the governing equation corresponding to the electromagnetic field model of step 1 as constraints, obtains the control containing constraints
The corresponding equivalent integration forms of equation processed；
Step 3, the equivalent integration forms obtained using finite element method discrete step 2, and obtain containing constraints
Governing equation corresponding to finite element equation；
Finite element equation obtained by step 4, solution procedure 3, according to the numerical solution for the finite element equation tried to achieve and
The physical relation set up, obtains magnetic distribution corresponding with the air gap layer；
Step 5, according to the corresponding magnetic distribution of the air gap layer, tried to achieve using electromagnetic force computational methods between adjacent object
Electromagnetic adsorption power.
Further, the step 1 includes：When setting up the geometrical model of electromagnetic field domain, not to adjacent object
Between air gap layer ΩgGeometrical model is set up, gained geometrical model only includes the region of adjacent object, is designated as main domain ΩmWith from domain
Ωs, and the face of air gap layer both sides is designated as Γ respectivelym=Ωm∩ΩgAnd Γs=Ωs∩Ωg, the corresponding electric field of gained geometrical model
Or magnetic field governing equation is collectively expressed as, Ω=Ω in domainm∪ΩsIt is interior to meet：
And met on corresponding border：
If (1) formula represents electric field equation,For electric scalar potential, α represents isotropism dielectric constant, typically much deeper than sky
Gas dielectric constant, f represents charge drive；If (2) formula represents magnetic field equation,For magnetic scalar potential, α represents isotropism magnetic conductance
Rate, typically much deeper than air permeability, f have following form：
F=-div α Hsource (3)
(3) in formula, HsourceIt should meet：
curlHsource=Jsource (4)
(4) J insourceRepresent exciting current density.
Further, the step 2 includes：
First according to field boundary condition, region Ω is set upm、ΩsAnd ΩgBetween the relational expression of electromagnetic field amount be air gap
Boundary condition：
WhereinΓ is represented respectivelymAlong Ω on facemOuter normal direction and tangential flux density；Γ is represented respectivelys
Along Ω on facesOuter normal direction and tangential flux density；qgRepresent air gap layer ΩgIt is middle along ΩmThe flux density of outer normal direction；
Secondly air gap boundary condition (5a) is described using electric scalar potential or magnetic scalar potential, then had：
In border ΓmPlace,It is expressed as：
In border ΓsPlace,It is expressed as：
In formula (6) and (7), nmRepresent ΓmΩ on facemOuter normal direction, nsRepresent ΓsΩ on facesOuter normal direction, and they
Direction contrast；αmAnd αsMain domain Ω is represented respectivelymWith from domain ΩsMaterial parameter；WithMain domain Ω is represented respectivelymWith
From domain ΩsScalar potential；α0Represent air material parameter；d0Represent the thickness of air gap layer；h0Represent air gap layer equivalent resistance or
Magnetic resistance；
Finally respectively in ΩmAnd ΩsMiddle introducing tentative function wmAnd ws, and apply Green formula, obtain (1), (6),
(7) equivalent integration forms：
Further, the step 3 is discrete using finite element method progress, and it includes：
Step 31, to region ΩmAnd ΩsGrid division, and remember Ni(i=1,2 ... n) is the node in standard finite element space
Basic function, φ=[φ1,φ2…φn]TFor the degree of freedom on a node basis of scalar potential, and scalar potential in domain will be solved it is expressed as node base
The product form of function and the degree of freedom on a node basis：
And by ΩmOn the free degree be split as two parts, i.e., respectively correspond to Ωm/ΓmAnd ΓmOn the free degree be respectively
WithBy ΩsOn the free degree be split as two parts, i.e., respectively correspond to Ωs/ΓsAnd ΓsOn the free degree differenceWithIn
It is to be expressed as φ：
Step 32, by (10) and (11) formula and wm,s=Ni(i=1,2 ... n) are updated in (8) and (9), obtain gassiness
The finite element equation of gap boundary condition is the finite element equation corresponding to the governing equation containing constraints：
By in (12) formulaUnified representation is into Kmm, then with following component expression-form：
By in (12) formulaUnified representation is into Kss, then with following component expression-form：
(13) and in (14) formula, n2RepresentIn maximum subscript,
By the load vector in (12) formula equal sign right-hand memberUnified representation is into fm, then shape is reached with following subscale
Formula：
By the load vector in (12) formula equal sign right-hand memberUnified representation is into fs, then with following component expression-form：
Due in (12) formulaIt is the Γ from (8) and (9)mAnd ΓsBoundary integral on face is obtained, will
Its unified representation is into NΓΓ, then with following component expression-form：
(17) in formula, n1It isMinimum subscript；n3It isMaximum subscript.
Further, the step 4 includes：
According to the numerical solution of the scalar potential of adjacent unit at the air gap of the electromagnetic field model(10) formula is obtainedWithFurther according to formula (6), the flux density q in air gap is calculatedgAnd field strength, the field strength calculation formula is：
Further, electromagnetic force computational methods described in the step 5 are to calculate object using Maxwell stress tensors method
Suffered electromagnetic adsorption power, then the Maxwell stress on closing face S of the i-th component of electromagnetic adsorption power by integrating object
Tensor TijIt is expressed as：
(19) vector index is summed using Einstein (Einstein) in formula and arranged；δijFor Kronecker function
(Kronecker) symbol；uiAnd ujElectric field or the component of magnetic field intensity are represented respectively.
Compared with prior art, beneficial effects of the present invention：
The present invention has taken into full account influence of the air gap to electromagnetic force, with preferable precision and efficiency；Specifically, first its
Compared with the method for traditional air gap mesh generation, without the subdivision very fine and closely woven grid at air gap, so as to reduce calculating
Scale, improves computational efficiency；Second, it is compared with shell unit method, the extra unit without constructing, and processing procedure is simple,
Computational efficiency is higher；3rd, it is compared with virtual air-gap method, it is contemplated that influence of the air gap to electromagnetic force, thus has higher
Computational accuracy, especially when the equivalent resistance or magnetic resistance of air gap layer occupy the ratio that can not ignore in equivalent circuit or magnetic circuit
When.
Brief description of the drawings
Fig. 1 is the electromagnetic field model containing air gap layer；
Adjacent unit schematic diagram at Fig. 2 air gaps；
Fig. 3 plane-parallel capacitor schematic diagrames
The FEM model of Fig. 4 plane-parallel capacitors；
Electrostatic adsorption force suffered by Fig. 5 dielectrics with thickness change schematic diagram；
The FEM model a schematic diagrames of the magnetic field absorption affinity of Fig. 6 C-shaped iron cores；
The FEM model b schematic diagrames of the magnetic field absorption affinity of Fig. 7 pane shape iron cores；
Magnetic field absorption affinity in Fig. 8 FEM models a at air gap with thickness variation diagram；
Magnetic field absorption affinity in Fig. 9 FEM models b at air gap with thickness variation diagram；
The flow chart of Figure 10 the method for the invention.
Embodiment
To make the object, technical solutions and advantages of the present invention clearer, below in conjunction with attached in the embodiment of the present invention
Figure, technical scheme is clearly and completely described, it is clear that described embodiment is that a part of the invention is real
Apply example, rather than whole embodiments.Based on the embodiment in the present invention, those of ordinary skill in the art are not making creation
Property work under the premise of the every other embodiment that is obtained, belong to the scope of protection of the invention.
The computational methods of electromagnetic adsorption power of the present invention, it is comprised the steps of：
Step 1, set up the electromagnetic field model containing air gap layer between adjacent object；Modeling when, without loss of generality, here I
Only consider threedimensional model, and only consider the electromagnetic adsorption power problem between two objects, then due to not adjacent object
Air gap layer ΩgGeometrical model is set up, therefore the model set up only includes the region Ω of two adjacent objectsmAnd Ωs, it is designated as master
Domain ΩmWith from domain Ωs, as shown in figure 1, and the faces of air gap layer both sides is designated as Γ respectivelym=Ωm∩ΩgAnd Γs=Ωs∩Ωg, institute
Obtain the corresponding electric field of geometrical model or magnetic field governing equation is collectively expressed as, Ω=Ω in domainm∪ΩsIt is interior to meet：
And met on corresponding border：
If (1) formula represents electric field equation,For electric scalar potential, α represents isotropism dielectric constant, and f represents that electric charge swashs
Encourage；If (2) formula represents magnetic field equation,For magnetic scalar potential, α represents isotropism magnetic conductivity, and f has following form：
F=-div α Hsource (3)
(3) in formula, HsourceIt should meet：
curlHsource=Jsource (4)
(4) J insourceRepresent exciting current density.
(4) H insourceIt can be tried to achieve by Biot-Swart laws, the non-thing of any satisfaction (4) formula can also be selected
Understand.
Step 2, set up the physical relation met between the adjacent objects of the air gap layer both sides, and by the physics set up
Relation is incorporated into the governing equation corresponding to the electromagnetic field model of step 1 as constraints, obtains the control containing constraints
The corresponding equivalent integration forms of equation processed；Further, when the air gap layer between adjacent object is very thin, and adjacent object ΩmOr ΩsWith
Air gap layer ΩgMaterial parameter compared to it is larger when, according to field boundary condition, ΩmOr ΩsThe field of outer surface is almost vertical
In outer surface, and electric flux or magnetic density can be considered as approximately (i.e. perpendicular to Γ along air gap thickness directionm
And Γs), and size is constant, then the step 2 includes：First according to field boundary condition, region Ω is set upm、ΩsAnd Ωg
Between the relational expression of electromagnetic field amount be air gap boundary condition：
WhereinΓ is represented respectivelymAlong Ω on facemOuter normal direction and tangential flux density；Γ is represented respectivelys
Along Ω on facesOuter normal direction and tangential flux density；qgRepresent air gap layer ΩgIt is middle along ΩmThe flux density of outer normal direction；
Secondly air gap boundary condition (5a) is described using electric scalar potential or magnetic scalar potential, then had：
In border ΓmPlace,It is expressed as：
In border ΓsPlace,It is expressed as：
In formula (6) and (7), nmRepresent ΓmΩ on facemOuter normal direction, nsRepresent ΓsΩ on facesOuter normal direction, and they
Direction contrast；αmAnd αsMain domain Ω is represented respectivelymWith from domain ΩsMaterial parameter；WithMain domain Ω is represented respectivelymWith
From domain ΩsScalar potential；α0Represent air material parameter；d0Represent the thickness of air gap layer；h0Represent air gap layer equivalent resistance or
Magnetic resistance；
Finally respectively in ΩmAnd ΩsMiddle introducing tentative function wmAnd ws, and Green formula are applied, obtain (1), (6), (7)
Equivalent integration forms：
Step 3, the equivalent integration forms obtained using finite element method discrete step 2, and obtain containing constraints
Governing equation corresponding to finite element equation；The specific step 3 includes：First by setting up the perimeter strip containing air gap
The equivalent weak form of governing equation of part, air gap boundary condition is introduced into governing equation；Then the discrete institute of finite element method is used
Equivalent weak form is stated, the numerical model of the boundary condition containing air gap is obtained.Specifically, the step 3 is carried out using finite element method
Discrete, it includes：Step 31, to region ΩmAnd ΩsGrid division is not (due to setting up air gap layer ΩgGeometrical model, therefore
It is not required to subdivision ΩgFinite element grid, and be typically divided into tetrahedron or hexahedral mesh in the three-dimensional model), and remember Ni(i
=1,2 ... node basic function n) for standard finite element space, φ=[φ1,φ2…φn]TFor the degree of freedom on a node basis of scalar potential,
And will solve domain in scalar potential be expressed as node basic function and the product form of the degree of freedom on a node basis：
And by ΩmOn the free degree be split as two parts, i.e., respectively correspond to Ωm/ΓmAnd ΓmOn the free degree be respectively
WithBy ΩsOn the free degree be split as two parts, i.e., respectively correspond to Ωs/ΓsAnd ΓsOn the free degree differenceWithIn
It is to be expressed as φ：
Step 32, by (10) and (11) formula and wm,s=Ni(i=1,2 ... n) are updated in (8) and (9), obtain gassiness
The finite element equation of gap boundary condition is the finite element equation corresponding to the governing equation containing constraints：
By in (12) formulaUnified representation is into Kmm, then with following component expression-form：
By in (12) formulaUnified representation is into Kss, then with following component expression-form：
(13) and in (14) formula, n2RepresentIn maximum subscript,
By the load vector in (12) formula equal sign right-hand memberUnified representation is into fm, then shape is reached with following subscale
Formula：
By the load vector in (12) formula equal sign right-hand memberUnified representation is into fs, then with following component expression-form：
Due in (12) formulaIt is the Γ from (8) and (9)mAnd ΓsBoundary integral on face is obtained, will
Its unified representation is into NΓΓ, then with following component expression-form：
(17) in formula, n1It isMinimum subscript；n3It isMaximum subscript.
Finite element equation obtained by step 4, solution procedure 3, according to the numerical solution for the finite element equation tried to achieve and
The physical relation set up, obtains magnetic distribution corresponding with the air gap layer；It is specifically included in after acquisition numerical solution, institute
State the corresponding magnetic distribution of air gap layer and the air gap boundary condition (5) of form or the air gap perimeter strip of scalar form are measured using field
Part (6) and (7) are obtained.It is preferred that, the step 4 includes：Solving finite element equation (12) obtains the numerical value of electromagnetic field model
Adjacent unit at air gap in solution, three-dimensional as shown in Figure 2, two peacekeeping one-dimensional models, it is belonging respectively to ΩmAnd Ωs, dotted line represents
Median plane between adjacent cells；According to the numerical solution of the scalar potential of adjacent unit at the air gap of the electromagnetic field modelWith
(10) formula is obtainedWithFurther according to formula (6), the flux density q in air gap is calculatedgAnd field strength, the field strength calculation formula
For：
Step 5, according to the corresponding magnetic distribution of the air gap layer, tried to achieve using electromagnetic force computational methods between adjacent object
Electromagnetic adsorption power.Further, the electromagnetic field in air gap layer, electromagnetic force computational methods described in the step 5 are use
Maxwell stress tensors method calculates the electromagnetic adsorption power suffered by object, then the i-th component of electromagnetic adsorption power is by integrating object
Closing face S on Maxwell stress tensors TijIt is expressed as：
(19) vector index is summed using Einstein in formula and arranged；δijFor Kronecker symbols；uiAnd ujDifference table
Show field strength component.
Illustrate the operating procedure using the method for the invention below by two real cases, and by with other sides
The result of calculation contrast of method, to illustrate the advantage using the method for the invention.
Case 1：Concrete case is described：Model in the case is to be filled with phase in a plane-parallel capacitor, capacitor
It is 5 dielectric to dielectric constant, but has at dielectric median plane an air gap layer (air) of a thickness very little, and by electricity
Medium is divided into two parts, and the thickness of each part is 10mm, and dielectric cross-sectional area is 0.01m2, the voltage A of Top electrode
For the voltage of electrode, the voltage B of bottom electrode is 0V；After electrode is powered, the electrostatic force of generation can attract dielectric two portions
Point.This Electrostatic Absorption mechanism, is also Coulomb effects, has been widely used in electric field.
Specific implementation step：First, the electromagnetic field model containing air gap layer corresponding to the plane-parallel capacitor is set up, such as
Shown in Fig. 3, the model is only included by the dielectric of two separated parts of air gap layer, and physical dimension is described with reference to case；It is based on
Step 2,3 pairs of dielectric subdivision grids, as shown in figure 4, setting dielectric constant, cell type and voltage boundary condition to grid
The position of air gap layer is specified Deng solution information and in FEM model, and sets the equivalent resistance coefficient of air gap layer；Finally
Based on step 4,5 to obtain the electrostatic adsorption force in the model suffered by dielectric.This case advantage explanation：Text is respectively adopted in we
Offer [Choi HS, Park IH, Lee SH.Concept of virtual air gap and its applications
for force calculation.IEEE Trans Mag.2006,42:663-666] and [Yoo J, Choi JS, Hong
SJ, et al.Finite element analysis of the attractive force on a Coulomb type
electrostatic chuck.International Conference on Electrical Machines and
Systems.2007October 8-11；Seoul, Korea] in virtual air-gap method, the method for the invention and accurate
Analytic method, the electrostatic force changed calculating air gap thickness from 0.01 μm to 100 μm.Result of calculation is as shown in Figure 5.We can
To find, the calculating of the invention with accurate analytic method is consistent, and in air gap thickness increase, the size of electrostatic adsorption force is all
There is downward trend.By contrast, result is obtained always greater than accurate analysis result using virtual air-gap method, and not with thickness
Change, this is irrational.By the case, illustrate that the result of calculation of the present invention is better than the virtual air gap in above-mentioned document
Method.
Case 2：Concrete case is described：The case contains the finite element analysis model a and b of two magnetic field absorption affinities, point
Not as shown in Figure 6 and Figure 7；It is 10000A/m to have current density in a models, in loop coil2Constant excitation, a head and the tail phase
C-shaped iron core even passes through loop coil, and there is the air gap layer (air) of a thickness very little junction in C-shaped coil, such as Fig. 6
Shown in middle dotted line；From unlike a models, the iron core in b models is an elongated cuboid block, the relative magnetic permeability of iron core
Rate is very big, is set to 10000.
Implementation steps：First, set up and do not include air gap in the electromagnetic field model containing air gap layer corresponding to a and b, model
Layer segment；Based on step 2,3 pairs of model facetization grids, as shown in Figure 6 and Figure 7, and magnetic conductivity, cell type, perimeter strip are set
Part and current excitation etc. solve information；Specify the position of air gap layer in a and b in a model simultaneously, and set air gap layer etc.
Imitate reluctancy；It is finally based on step 4, the 5 magnetic field absorption affinities being subject to obtain in a and b models at air gap.This case advantage is said
It is bright：Document [Choi HS, Park IH, Lee SH.Concept of virtual air gap and are respectively adopted in we
its applications for force calculation.IEEE Trans Mag.2006,42:663-666] and [Seo
JH,Choi HS. Computation of Magnetic Contact Forces.IEEE Trans Mag.2014,50:
525-528] in virtual air-gap method and the inventive method calculate anaplasia of the air gap thickness from 0.0001 μm to 100 μm respectively
The magnetic field absorption affinity of change.Result of calculation is as shown in Figure 8 and Figure 9.We are it can be found that of the invention in air gap thickness increase, gas
The size of magnetic field absorption affinity at gap has downward trend.By contrast, result is obtained always greater than this hair using virtual air-gap method
Bright obtained result, and do not changed with thickness, this is irrational.By the case, illustrate the calculating of the present invention
As a result it is better than the virtual air-gap method in above-mentioned two document.
The foregoing is only a preferred embodiment of the present invention, but protection scope of the present invention be not limited thereto,
Any one skilled in the art the invention discloses technical scope in, technique according to the invention scheme and its
Inventive concept is subject to equivalent substitution or change, should all be included within the scope of the present invention.
Claims (6)
1. a kind of electromagnetic adsorption power computational methods for considering air gap layer influence, it is characterised in that comprise the following steps：
Step 1, set up the electromagnetic field model containing air gap layer between adjacent object；
Step 2, set up the physical relation met between the adjacent objects of the air gap layer both sides, and by the physical relation set up
As constraints, it is incorporated into the governing equation corresponding to the electromagnetic field model of step 1, obtains the controlling party containing constraints
The corresponding equivalent integration forms of journey；
Step 3, the equivalent integration forms obtained using finite element method discrete step 2, and obtain the control containing constraints
Finite element equation corresponding to equation processed；
Finite element equation obtained by step 4, solution procedure 3, according to the numerical solution for the finite element equation tried to achieve and is built
Vertical physical relation, obtains magnetic distribution corresponding with the air gap layer；
Step 5, according to the corresponding magnetic distribution of the air gap layer, try to achieve the electricity between adjacent object using electromagnetic force computational methods
Magnetic adsorbability.
2. electromagnetic adsorption power computational methods according to claim 1, it is characterised in that：
The step 1 includes：When setting up the geometrical model of electromagnetic field domain, not the air gap layer Ω adjacent objectgBuild
Vertical geometrical model, gained geometrical model only includes the region of adjacent object, is designated as main domain ΩmWith from domain Ωs, and air gap layer both sides
Face be designated as Γ respectivelym=Ωm∩ΩgAnd Γs=Ωs∩Ωg, the corresponding electric field of gained geometrical model or magnetic field governing equation are united
One is expressed as, Ω=Ω in domainm∪ΩsIt is interior to meet：
And met on corresponding border：
If (1) formula represents electric field equation,For electric scalar potential, α represents isotropism dielectric constant, and f represents charge drive；If
(2) formula represents magnetic field equation, thenFor magnetic scalar potential, α represents isotropism magnetic conductivity, and f has following form：
F=-div α Hsource (3)
(3) in formula, HsourceIt should meet：
curlHsource=Jsource (4)
(4) J insourceRepresent exciting current density.
3. electromagnetic adsorption power computational methods according to claim 1, it is characterised in that：
The step 2 includes：
First according to field boundary condition, region Ω is set upm、ΩsAnd ΩgBetween the relational expression of electromagnetic field amount be air gap border
Condition：
WhereinΓ is represented respectivelymAlong Ω on facemOuter normal direction and tangential flux density；Γ is represented respectivelysOn face
Along ΩsOuter normal direction and tangential flux density；qgRepresent air gap layer ΩgIt is middle along ΩmThe flux density of outer normal direction；
Secondly air gap boundary condition (5a) is described using electric scalar potential or magnetic scalar potential, then had：
In border ΓmPlace,It is expressed as：
In border ΓsPlace,It is expressed as：
In formula (6) and (7), nmRepresent ΓmΩ on facemOuter normal direction, nsRepresent ΓsΩ on facesOuter normal direction, and their side
To contrast；αmAnd αsMain domain Ω is represented respectivelymWith from domain ΩsMaterial parameter；WithMain domain Ω is represented respectivelymWith from domain
ΩsScalar potential；α0Represent air material parameter；d0Represent the thickness of air gap layer；h0Represent the equivalent resistance or magnetic of air gap layer
Resistance；
Finally respectively in ΩmAnd ΩsMiddle introducing tentative function wmAnd ws, and apply Green formula, obtain (1), (6), (7) etc.
Imitate integrated form：
4. electromagnetic adsorption power computational methods according to claim 1, it is characterised in that：
The step 3 is discrete using finite element method progress, and it includes：
Step 31, to region ΩmAnd ΩsGrid division, and remember Ni(i=1,2 ... n) is the node base letter in standard finite element space
Number, φ=[φ1,φ2…φn]TFor the degree of freedom on a node basis of scalar potential, and scalar potential in domain will be solved it is expressed as node basic function
With the product form of the degree of freedom on a node basis：
And by ΩmOn the free degree be split as two parts, i.e., respectively correspond to Ωm/ΓmAnd ΓmOn the free degree be respectivelyWithBy ΩsOn the free degree be split as two parts, i.e., respectively correspond to Ωs/ΓsAnd ΓsOn the free degree differenceWithIn
It is to be expressed as φ：
Step 32, by (10) and (11) formula and wm,s=Ni(i=1,2 ... n) are updated in (8) and (9), obtain border containing air gap
The finite element equation of condition is the finite element equation corresponding to the governing equation containing constraints：
By in (12) formulaUnified representation is into Kmm, then with following component expression-form：
By in (12) formulaUnified representation is into Kss, then with following component expression-form：
(13) and in (14) formula, n2RepresentIn maximum subscript,
By the load vector f in (12) formula equal sign right-hand memberi m、Unified representation is into fm, then with following component expression-form：
By the load vector f in (12) formula equal sign right-hand memberi s、Unified representation is into fs, then with following component expression-form：
Due in (12) formulaIt is the Γ from (8) and (9)mAnd ΓsBoundary integral on face is obtained, and is united
One is expressed as NΓΓ, then with following component expression-form：
(17) in formula, n1It isMinimum subscript；n3It isMaximum subscript.
5. electromagnetic adsorption power computational methods according to claim 1, it is characterised in that：The step 4 includes：
According to the numerical solution of the scalar potential of adjacent unit at the air gap of the electromagnetic field model(10) formula is obtainedWithFurther according to formula (6), the flux density q in air gap is calculatedgAnd field strength, the field strength calculation formula is：
6. electromagnetic adsorption power computational methods according to claim 1, it is characterised in that：
Electromagnetic force computational methods described in the step 5 are that the electromagnetism suffered by object is calculated using Maxwell stress tensors method
Absorption affinity, then the Maxwell stress tensors T on closing face S of the i-th component of electromagnetic adsorption power by integrating objectijIt is expressed as：
(19) vector index is summed using Einstein in formula and arranged；δijFor Kronecker symbols；uiAnd ujField is represented respectively
Strong component.
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