CN104182589A - Modeling method of electromagnetic rail launcher - Google Patents

Abstract

The invention relates to a modeling method of an electromagnetic rail launcher. The modeling method of the electromagnetic rail launcher simultaneously achieves matching of physical fields and armature speeds of a prototype and a model. According to the method, all the physical quantity similarity constants need to meet the conditions including that the geometric similarity constant is k; (2) the electromagnetism-related similarity constants include that the current density similarity constant is 1/k and meanwhile the conductivity similarity constant is 1/k without change of magnetic conductivity; (3) the heat-related similarity constants include that the thermal conductivity similarity constant is k, the unit volume heat capacity similarity constant is 1; the time and kinematics-related similarity constants include that the time similarity constant is k, the density and speed similarity constant is 1. The modeling method of the electromagnetic rail launcher achieves matching with kinematical equation analysis results and similarity of electric fields, and meanwhile, achieves matching of the physical fields of the armature speeds simultaneously, effectively simulates speed-associated physical phenomena, requires simple materials, is easy to implement and can guarantee similarity of temperature fields of a model launcher and a prototype launcher.

Description

A kind of modeling method of electromagnetic rail launching device

Technical field

The present invention relates to ELECTROMAGNETIC LAUNCH TECHNOLOGY field, especially relate to a kind of modeling method of electromagnetic rail launching device.

Background technology

Electromagnetic rail launching device main part, comprises that two parallel tracks contact armature composition good and that can be free to slide along track with track with one.Article two, after track rear end switches on power, two tracks and armature be as a part for circuit, conduction drive current.Armature under the promotion of Ampère force along track accelerated motion, thereby obtain high-speed.

In recent years, electromagnetic rail launching technology is with the fastest developing speed in ELECTROMAGNETIC LAUNCH TECHNOLOGY field, because of its application prospect and important military significance widely, is subject to the common concern of each military power.Current overall context is in tackling problems in key technologies stages such as lift-off technology, power technology, track damage control technologys.If directly studied as experiment porch using prototype transmitter, cost is high, the construction time is long, is unsuitable for carrying out a large amount of Basic Experiment Study, therefore at present most of research be all taking in small-bore model transmitter as research object.Scale model research has short, the advantage such as construction cost is low research cycle, but for the theory and technology means that obtain after deliberation have universality, need to ensure correctness and the feasibility of its modeling method.Modeling method refers to itself of not direct investigate natural phenomena or process, but uses a kind of method of studying with the model of these phenomenons or similar process.

For electromagnetic rail launching device, Chinese scholars has proposed some modeling methods, the document of delivering mainly comprises: " the One kind of scaling relations on electromechanical systems (a kind of scale-relation of Mechatronic Systems) " of " IEEE Transactions on Magnetics (electrical equipment and Electronic Engineering Association's magnetics proceedings) ", " EM gun scaling relationships (magnetic artillery scale-relation) ", " Energy partition and scaling issues in a railgun (energy distribution of electromagnetic railgun and contracting are than problem) ", " Scaling study in a capacitor-driven railgun (the scale reduction method research of capacitor drive electromagnetic railgun) " of " IEEE Transactions on Plasma Science (electrical equipment and Electronic Engineering Association's plasma science proceedings) " etc.The general character of these class methods is: in order to make magnetic field, temperature field and the stress field mirror image each other of prototype and model transmitter, except needs meeting geometric simulated condition, (establishing geometric similarity constant is k), and the affinity constant of electric current, time and materials mass density should meet respectively k, k ²and k ²; The method has been ignored the motional electromotive force in the law of electromagnetic induction in theoretical derivation.Under the method, the affinity constant of speed is 1/k, and in prototype transmitter, armature speed is the 1/k of model transmitter; If mate the speed of armature, its physical field is just no longer similar, the phenomenon that while therefore having caused physical field similar, armature speed can not be mated.The physical field of electromagnetic rail launching device and the damage of transmitter and life search are closely bound up; and speed is contacted directly with planing damage in electromagnetic rail launching device as launching one of technical indicator of most critical in class weapon, but above modeling method can not be realized the coupling of physical field and speed simultaneously.In addition,, according to the simulated condition in above modeling method, the mass density of material and the affinity constant of time are k ², this condition, in the time that geometric similarity constant value is larger, lacks realizability in reality.For example: geometric similarity constant is 3 o'clock, the mass density of time and materials need to be carried out modelling taking 9 as affinity constant.Due to above reason, above-mentioned electromagnetic rail launching device modeling method not yet obtains enough attention and application.

Summary of the invention

The modeling method that the object of the invention is to design a kind of novel electromagnetic rail launching device, addresses the above problem.

To achieve these goals, the technical solution used in the present invention is as follows:

A kind of modeling method of electromagnetic rail launching device:

Realize prototype and model simultaneously and comprise the physical field of electromagnetic field, temperature field, stress field and the coupling of armature speed, each physical quantity affinity constant need to meet following condition:

(1) geometric similarity constant:

The geometric configuration of prototype and model transmitter is similar, and yardstick exists k times of relation;

(2) the relevant affinity constant of electromagnetism:

The affinity constant of current density is 1/k, is 1/k with the affinity constant of seasonal conductivity, and magnetic permeability is constant;

(3) the relevant affinity constant of heat:

Thermal conductivity affinity constant is k, and the affinity constant of unit volume hot melt is 1;

(4) time and kinematic similarity constant:

Time similarity constant is k, and the affinity constant of density and speed is 1, and prototype is identical with speed with density in model.

In above physical quantity, as long as the affinity constant of geometric scale, current density, time, conductivity, magnetic permeability and density is definite, all the other physical quantity affinity constants are just determined voluntarily.

The system of equations of descriptive model electromagnetic rail launching device electromagnetic field, temperature field and stress field is as follows:

▽×B＝μj (24)

$\▿\×\frac{j}{\mathrm{\σ}}=-\frac{\∂B}{\∂t}+\▿\×(V\×B)---\left(25\right)$

$\▿\·K\▿T+\frac{j\·j}{\mathrm{\σ}}=C_p\frac{\∂T}{\∂t}---\left(26\right)$

$\▿\·S+j\×B=\mathrm{\ρ}\frac{\mathrm{dV}}{\mathrm{dt}}---\left(27\right)$

Wherein B is magnetic induction density, and j is current density, and V is armature speed, and S is stress, and T is temperature, and σ is conductivity, and μ is magnetic permeability, the mass density that ρ is material, C _pfor heat capacity per unit volume, K is its thermal conductivity, and the time is t;

Prototype electromagnetic rail launching device subscript for related physical quantity " ' " mark, physical field system of equations is as follows:

▽′×B′＝μ′j′ (28)

${\▿}^{\′}\×\frac{j^{\′}}{{\mathrm{\σ}}^{\′}}=-\frac{{\∂B}^{\′}}{{\∂t}^{\′}}+{\▿}^{\′}\×(V^{\′}\×B^{\′})---\left(29\right)$

${\▿}^{\′}\·K^{\′}{\▿}^{\′}{T}^{\′}+\frac{j^{\′}\·j^{\′}}{{\mathrm{\σ}}^{\′}}=C_p^{\′}\frac{{\∂T}^{\′}}{{\∂t}^{\′}}---\left(30\right)$

${\▿}^{\′}\·S^{\′}+j^{\′}\×B^{\′}={\mathrm{\ρ}}^{\′}\frac{{\mathrm{dV}}^{\′}}{{\mathrm{dt}}^{\′}}---\left(31\right)$

Suppose that prototype transmitter exists k times of relation: x '=kx with model transmitter on geometric scale, y '=ky, ▽ '=▽/k when z '=kz, even geometric similarity constant is C _l=k; Describe the affinity constant of other physical quantitys by similar method, and utilize affinity constant to pass through " similar conversion " the physical field equation of prototype is converted to:

$\frac{C_B}{C_l}\▿\×B=C_{\mathrm{\μ}}{C}_j\mathrm{\μj}---\left(32\right)$

$\frac{C_j}{C_l{C}_{\mathrm{\σ}}}\▿\×\frac{j}{\mathrm{\σ}}=-\frac{C_B}{C_t}\frac{\∂B}{\∂t}+\frac{C_V{C}_B}{C_l}\▿\×(V\×B)---\left(33\right)$

$\frac{C_K{C}_T}{{C_l}^2}\▿\·K\▿T+\frac{{C_j}^2}{C_{\mathrm{\σ}}}\frac{j\·j}{\mathrm{\σ}}=\frac{C_{C_o}{C}_T}{C_t}{C}_p\frac{\∂T}{\∂t}---\left(34\right)$

$\frac{C_s}{C_l}\▿\·S+C_j{C}_{B}j\×B=\frac{C_{\mathrm{\ρ}}{C}_v}{C_t}\mathrm{\ρ}\frac{\mathrm{dV}}{\mathrm{dt}}---\left(35\right)$

Can obtain by (9)～(12) formula:

$\frac{C_B}{C_l}=C_{\mathrm{\μ}}{C}_j---\left(36\right)$

$\frac{C_j}{C_l{C}_{\mathrm{\σ}}}=\frac{C_B}{C_t}=\frac{C_V{C}_B}{C_l}---\left(37\right)$

$\frac{C_K{C}_T}{{C_l}^2}=\frac{{C_j}^2}{C_{\mathrm{\σ}}}=\frac{C_{C_p}{C}_T}{C_t}---\left(38\right)$

$\frac{C_S}{C_l}=C_j{C}_B=\frac{C_{\mathrm{\ρ}}{C}_V}{C_t}---\left(39\right)$

Utilize (13)～(16) formula can obtain similar index formula:

$\frac{C_B}{C_l{C}_{\mathrm{\μ}}{C}_j}=1---\left(40\right)$

$\frac{C_j{C}_t}{C_l{C}_{\mathrm{\σ}}{C}_B}=1---\left(41\right)$

$\frac{C_j}{C_{\mathrm{\σ}}{C}_V{C}_B}=1---\left(42\right)$

$\frac{C_K{C}_T{C}_{\mathrm{\σ}}}{{C_l}^2{C_j}^2}=1---\left(43\right)$

$\frac{C_K{C}_t}{{C_l}^2{C}_{C_p}}=1---\left(44\right)$

$\frac{C_S}{C_l{C}_j{C}_B}=1---\left(45\right)$

$\frac{C_S{C}_t}{C_l{C}_{\mathrm{\ρ}}{C}_V}=1---\left(46\right)$

Magnetic field in model and prototype, stress field and temperature field are distributed and be mirror image, even B=B ', S=S ', T=T ', C in similar index formula _b=C _s=C _t=1; On the similar basis of physical field, the speed in coupling prototype and model, C _v=1, similar index formula becomes: C _μ=1, c _kc _σ=1, c _lc _j=1, C _ρ=1; Further can obtain the each physical quantity affinity constant with the statement of geometric similarity constant: C _μ=1, C _t=k, C _σ=1/k, C _k=k, c _j=1/k, C _ρ=1; While enforcement according to above condition, can realize this modeling method.

The invention discloses a kind of electromagnetic rail launching device modeling method, specifically comprise: (1) geometric similarity constant is k; (2) affinity constant of current density and conductivity is 1/k, and magnetic permeability is constant; (3) thermal conductivity affinity constant is k, and the affinity constant of heat capacity per unit volume is 1; (4) time similarity constant is k, and mass density affinity constant is 1.Can realize the coupling of the physical field such as electromagnetic field, stress field, temperature field and armature speed in prototype and model transmitter according to above condition simultaneously.This modeling method has avoided vast scale to contract than material parameter and time, has ensured its feasibility, has higher using value, can be used as the research of theoretical foundation guidance model.

The object of the invention is for overcoming above-mentioned existing methodical deficiency; a kind of effective and feasible electromagnetic rail launching device modeling method is provided; solve and in simulated condition, needed to carry out vast scale contracting than the deficiency of quality of materials density and time, can realize the coupling of armature speed simultaneously.

Affinity constant refers in the time of all corresponding point of a pair of similar phenomena and correspondence and engraves, the ratio that Physics amount all remains unchanged.

In order to realize the coupling of the physical field such as electromagnetic field, temperature field, stress field and armature speed in prototype and model simultaneously, each physical quantity affinity constant meets following condition:

(1) geometric similarity constant:

The geometric configuration of prototype and model transmitter is similar, and yardstick exists k times of relation;

(2) the relevant affinity constant of electromagnetism:

The affinity constant of current density is 1/k, is 1/k with the affinity constant of seasonal conductivity, and magnetic permeability is constant;

(3) the relevant affinity constant of heat:

Thermal conductivity affinity constant is k, and the affinity constant of unit volume hot melt is 1;

(4) time and kinematic similarity constant:

Time similarity constant is k, and the affinity constant of density and speed is 1, and prototype is identical with speed with density in model.

In above physical quantity, as long as the affinity constant of geometric scale, current density, time, conductivity, magnetic permeability, density is definite, all the other physical quantity affinity constants are just determined voluntarily.

Beneficial effect of the present invention can be summarized as follows:

(1) in theoretical derivation, considered the motional electromotive force in the law of electromagnetic induction, realized the result of analyzing with kinematical equation and mated;

(2), because the affinity constant of current density and conductivity is identical, on the similar basis of magnetic field, temperature field and stress field, can realize the similar of electric field simultaneously;

(3) this modeling method, meeting under the similar condition of physical field, because the time is identical with geometric similarity constant, can be realized the coupling of armature speed simultaneously, and analog rate relevant physical phenomena effectively comprises armature track contact interface damage phenomenon etc.;

(4) affinity constant of conductivity is 1/k, and not occurring need to be with k in similar other modeling method ²vast scale contracts than the condition of material parameter, and therefore material requested is simple, is easier to realize;

(5) thermal conductivity affinity constant is k, but within the utmost point short time of electromagnetic rail launching, can ignore the Heat Conduction Phenomenon of material, even if therefore do not meet this simulated condition, can ensure the similarity of model and prototype emitter temperature field.

In a word, the present invention can carry out more approaching real modelling by the scheme that is easier to realize.

Brief description of the drawings

Fig. 1 is electromagnetic rail launching device principle of work schematic diagram;

In figure: 1, armature; 2, track; 3, electric current; 4, the magnetic line of force.

Embodiment

In order to make technical matters solved by the invention, technical scheme and beneficial effect clearer, below in conjunction with drawings and Examples, the present invention is further elaborated.Should be appreciated that specific embodiment described herein, only in order to explain the present invention, is not intended to limit the present invention.

The modeling method of a kind of electromagnetic rail launching device as shown in Figure 1:

Realize prototype and model simultaneously and comprise the physical field of electromagnetic field, temperature field, stress field and the coupling of armature speed, each physical quantity affinity constant need to meet following condition:

(1) geometric similarity constant:

The geometric configuration of prototype and model transmitter is similar, and yardstick exists k times of relation;

(2) the relevant affinity constant of electromagnetism:

The affinity constant of current density is 1/k, is 1/k with the affinity constant of seasonal conductivity, and magnetic permeability is constant;

(3) the relevant affinity constant of heat:

Thermal conductivity affinity constant is k, and the affinity constant of unit volume hot melt is 1;

(4) time and kinematic similarity constant:

Time similarity constant is k, and the affinity constant of density and speed is 1, and prototype is identical with speed with density in model.

In the embodiment being more preferably, in above physical quantity, as long as the affinity constant of geometric scale, current density, time, conductivity, magnetic permeability and density is definite, all the other physical quantity affinity constants are just determined voluntarily.

In the embodiment being more preferably, the system of equations of descriptive model electromagnetic rail launching device electromagnetic field, temperature field and stress field is as follows:

▽×B＝μj (47)

$\▿\×\frac{j}{\mathrm{\σ}}=-\frac{\∂B}{\∂t}+\▿\×(V\×B)---\left(48\right)$

$\▿\·K\▿T+\frac{j\·j}{\mathrm{\σ}}=C_p\frac{\∂T}{\∂t}---\left(49\right)$

$\▿\·S+j\×B=\mathrm{\ρ}\frac{\mathrm{dV}}{\mathrm{dt}}---\left(50\right)$

Wherein B is magnetic induction density, and j is current density, and V is armature speed, and S is stress, and T is temperature, and σ is conductivity, and μ is magnetic permeability, the mass density that ρ is material, C _pfor heat capacity per unit volume, K is its thermal conductivity, and the time is t;

Prototype electromagnetic rail launching device subscript for related physical quantity " ' " mark, physical field system of equations is as follows:

▽′×B′＝μ′j′ (51)

${\▿}^{\′}\×\frac{j^{\′}}{{\mathrm{\σ}}^{\′}}=-\frac{{\∂B}^{\′}}{{\∂t}^{\′}}+{\▿}^{\′}\×(V^{\′}\×B^{\′})---\left(52\right)$

${\▿}^{\′}\·K^{\′}{\▿}^{\′}{T}^{\′}+\frac{j^{\′}\·j^{\′}}{{\mathrm{\σ}}^{\′}}=C_p^{\′}\frac{{\∂T}^{\′}}{{\∂t}^{\′}}---\left(53\right)$

${\▿}^{\′}\·S^{\′}+j^{\′}\×B^{\′}={\mathrm{\ρ}}^{\′}\frac{{\mathrm{dV}}^{\′}}{{\mathrm{dt}}^{\′}}---\left(54\right)$

Suppose that prototype transmitter exists k times of relation: x '=kx with model transmitter on geometric scale, y '=ky, ▽ '=▽/k when z '=kz, even geometric similarity constant is C _l=k; Describe the affinity constant of other physical quantitys by similar method, and utilize affinity constant to pass through " similar conversion " the physical field equation of prototype is converted to:

$\frac{C_B}{C_l}\▿\×B=C_{\mathrm{\μ}}{C}_j\mathrm{\μj}---\left(55\right)$

$\frac{C_j}{C_l{C}_{\mathrm{\σ}}}\▿\×\frac{j}{\mathrm{\σ}}=-\frac{C_B}{C_t}\frac{\∂B}{\∂t}+\frac{C_V{C}_B}{C_l}\▿\×(V\×B)---\left(56\right)$

$\frac{C_K{C}_T}{{C_l}^2}\▿\·K\▿T+\frac{{C_j}^2}{C_{\mathrm{\σ}}}\frac{j\·j}{\mathrm{\σ}}=\frac{C_{C_o}{C}_T}{C_t}{C}_p\frac{\∂T}{\∂t}---\left(57\right)$

$\frac{C_s}{C_l}\▿\·S+C_j{C}_{B}j\×B=\frac{C_{\mathrm{\ρ}}{C}_v}{C_t}\mathrm{\ρ}\frac{\mathrm{dV}}{\mathrm{dt}}---\left(58\right)$

Can obtain by (9)～(12) formula:

$\frac{C_B}{C_l}=C_{\mathrm{\μ}}{C}_j---\left(59\right)$

$\frac{C_j}{C_l{C}_{\mathrm{\σ}}}=\frac{C_B}{C_t}=\frac{C_V{C}_B}{C_l}---\left(60\right)$

$\frac{C_K{C}_T}{{C_l}^2}=\frac{{C_j}^2}{C_{\mathrm{\σ}}}=\frac{C_{C_p}{C}_T}{C_t}---\left(61\right)$

$\frac{C_S}{C_l}=C_j{C}_B=\frac{C_{\mathrm{\ρ}}{C}_V}{C_t}---\left(62\right)$

Utilize (13)～(16) formula can obtain similar index formula:

$\frac{C_B}{C_l{C}_{\mathrm{\μ}}{C}_j}=1---\left(63\right)$

$\frac{C_j{C}_t}{C_l{C}_{\mathrm{\σ}}{C}_B}=1---\left(64\right)$

$\frac{C_j}{C_{\mathrm{\σ}}{C}_V{C}_B}=1---\left(65\right)$

$\frac{C_K{C}_T{C}_{\mathrm{\σ}}}{{C_l}^2{C_j}^2}=1---\left(66\right)$

$\frac{C_K{C}_t}{{C_l}^2{C}_{C_p}}=1---\left(67\right)$

$\frac{C_S}{C_l{C}_j{C}_B}=1---\left(68\right)$

$\frac{C_S{C}_t}{C_l{C}_{\mathrm{\ρ}}{C}_V}=1---\left(69\right)$

Magnetic field in model and prototype, stress field and temperature field are distributed and be mirror image, even B=B ', S=S ', T=T ', C in similar index formula _b=C _s=C _t=1; On the similar basis of physical field, the speed in coupling prototype and model, C _v=1, similar index formula becomes: C _μ=1, c _kc _σ=1, c _lc _j=1, C _ρ=1; Further can obtain the each physical quantity affinity constant with the statement of geometric similarity constant: C _μ=1, C _t=k, C _σ=1/k, C _k=k, c _j=1/k, C _ρ=1; While enforcement according to above condition, can realize this modeling method.

Visible, the present invention has considered the motional electromotive force in the law of electromagnetic induction in theoretical derivation, has realized the result of analyzing with kinematical equation and has mated; Because the affinity constant of current density and conductivity is identical, on the similar basis of magnetic field, temperature field and stress field, can realize the similar of electric field simultaneously; This modeling method, meeting under the similar condition of physical field, because the time is identical with geometric similarity constant, can be realized the coupling of armature speed simultaneously, and analog rate relevant physical phenomena effectively comprises armature track contact interface damage phenomenon etc.; The affinity constant of conductivity is 1/k, and not occurring need to be with k in similar other modeling method ²vast scale contracts than the condition of material parameter, and therefore material requested is simple, is easier to realize; Thermal conductivity affinity constant is k, but within the utmost point short time of electromagnetic rail launching, can ignore the Heat Conduction Phenomenon of material, even if therefore do not meet this simulated condition, can ensure the similarity of model and prototype emitter temperature field.

In a word, the present invention can carry out more approaching real modelling by the scheme that is easier to realize.

More than by the detailed description of concrete and preferred embodiment the present invention; but those skilled in the art should be understood that; the present invention is not limited to the above embodiment; within the spirit and principles in the present invention all; any amendment of doing, be equal to replacement etc., within protection scope of the present invention all should be included in.

Claims (3)

1. a modeling method for electromagnetic rail launching device, is characterized in that:

Realize prototype and model simultaneously and comprise the physical field of electromagnetic field, temperature field, stress field and the coupling of armature speed, each physical quantity affinity constant need to meet following condition:

(1) geometric similarity constant:

The geometric configuration of prototype and model transmitter is similar, and yardstick exists k times of relation;

(2) the relevant affinity constant of electromagnetism:

The affinity constant of current density is 1/k, is 1/k with the affinity constant of seasonal conductivity, and magnetic permeability is constant;

(3) the relevant affinity constant of heat:

Thermal conductivity affinity constant is k, and the affinity constant of unit volume hot melt is 1;

(4) time and kinematic similarity constant:

Time similarity constant is k, and the affinity constant of density and speed is 1, and prototype is identical with speed with density in model.

2. the modeling method of electromagnetic rail launching device according to claim 1; it is characterized in that: in above physical quantity, as long as the affinity constant of geometric scale, current density, time, conductivity, magnetic permeability and density is definite, all the other physical quantity affinity constants are just determined voluntarily.

3. the modeling method of electromagnetic rail launching device according to claim 1, is characterized in that:

The system of equations of descriptive model electromagnetic rail launching device electromagnetic field, temperature field and stress field is as follows:

▽×B＝μj (1)

$\▿\×\frac{j}{\mathrm{\σ}}=-\frac{\∂B}{\∂t}+\▿\×(V\×B)---\left(2\right)$

$\▿\·K\▿T+\frac{j\·j}{\mathrm{\σ}}=C_p\frac{\∂T}{\∂t}---\left(3\right)$

$\▿\·S+j\×B=\mathrm{\ρ}\frac{\mathrm{dV}}{\mathrm{dt}}---\left(4\right)$

Wherein B is magnetic induction density, and j is current density, and V is armature speed, and S is stress, and T is temperature, and σ is conductivity, and μ is magnetic permeability, the mass density that ρ is material, C _pfor heat capacity per unit volume, K is its thermal conductivity, and the time is t;

Prototype electromagnetic rail launching device subscript for related physical quantity " ' " mark, physical field system of equations is as follows:

▽′×B′＝μ′j′ (5)

${\▿}^{\′}\×\frac{j^{\′}}{{\mathrm{\σ}}^{\′}}=-\frac{{\∂B}^{\′}}{{\∂t}^{\′}}+{\▿}^{\′}\×(V^{\′}\×B^{\′})---\left(6\right)$

${\▿}^{\′}\·K^{\′}{\▿}^{\′}{T}^{\′}+\frac{j^{\′}\·j^{\′}}{{\mathrm{\σ}}^{\′}}=C_p^{\′}\frac{{\∂T}^{\′}}{{\∂t}^{\′}}---\left(7\right)$

${\▿}^{\′}\·S^{\′}+j^{\′}\×B^{\′}={\mathrm{\ρ}}^{\′}\frac{{\mathrm{dV}}^{\′}}{{\mathrm{dt}}^{\′}}---\left(8\right)$

Suppose that prototype transmitter exists k times of relation: x '=kx with model transmitter on geometric scale, y '=ky, ▽ '=▽/k when z '=kz, even geometric similarity constant is C _l=k; Describe the affinity constant of other physical quantitys by similar method, and utilize affinity constant to pass through " similar conversion " the physical field equation of prototype is converted to:

$\frac{C_B}{C_l}\▿\×B=C_{\mathrm{\μ}}{C}_j\mathrm{\μj}---\left(9\right)$

$\frac{C_j}{C_l{C}_{\mathrm{\σ}}}\▿\×\frac{j}{\mathrm{\σ}}=-\frac{C_B}{C_t}\frac{\∂B}{\∂t}+\frac{C_V{C}_B}{C_l}\▿\×(V\×B)---\left(10\right)$

$\frac{C_K{C}_T}{{C_l}^2}\▿\·K\▿T+\frac{{C_j}^2}{C_{\mathrm{\σ}}}\frac{j\·j}{\mathrm{\σ}}=\frac{C_{C_o}{C}_T}{C_t}{C}_p\frac{\∂T}{\∂t}---\left(11\right)$

$\frac{C_s}{C_l}\▿\·S+C_j{C}_{B}j\×B=\frac{C_{\mathrm{\ρ}}{C}_v}{C_t}\mathrm{\ρ}\frac{\mathrm{dV}}{\mathrm{dt}}---\left(12\right)$

Can obtain by (9)～(12) formula:

$\frac{C_B}{C_l}=C_{\mathrm{\μ}}{C}_j---\left(13\right)$

$\frac{C_j}{C_l{C}_{\mathrm{\σ}}}=\frac{C_B}{C_t}=\frac{C_V{C}_B}{C_l}---\left(14\right)$

$\frac{C_K{C}_T}{{C_l}^2}=\frac{{C_j}^2}{C_{\mathrm{\σ}}}=\frac{C_{C_p}{C}_T}{C_t}---\left(15\right)$

$\frac{C_S}{C_l}=C_j{C}_B=\frac{C_{\mathrm{\ρ}}{C}_V}{C_t}---\left(16\right)$

Utilize (13)～(16) formula can obtain similar index formula:

$\frac{C_B}{C_l{C}_{\mathrm{\μ}}{C}_j}=1---\left(17\right)$

$\frac{C_j{C}_t}{C_l{C}_{\mathrm{\σ}}{C}_B}=1---\left(18\right)$

$\frac{C_j}{C_{\mathrm{\σ}}{C}_V{C}_B}=1---\left(19\right)$

$\frac{C_K{C}_T{C}_{\mathrm{\σ}}}{{C_l}^2{C_j}^2}=1---\left(20\right)$

$\frac{C_K{C}_t}{{C_l}^2{C}_{C_p}}=1---\left(21\right)$

$\frac{C_S}{C_l{C}_j{C}_B}=1---\left(22\right)$

$\frac{C_S{C}_t}{C_l{C}_{\mathrm{\ρ}}{C}_V}=1---\left(23\right)$

Magnetic field in model and prototype, stress field and temperature field are distributed and be mirror image, even B=B ', S=S ', T=T ', C in similar index formula _b=C _s=C _t=1; On the similar basis of physical field, the speed in coupling prototype and model, C _v=1, similar index formula becomes: C _μ=1, c _kc _σ=1, c _lc _j=1, C _ρ=1; Further can obtain the each physical quantity affinity constant with the statement of geometric similarity constant: C _μ=1, C _t=k, C _σ=1/k, C _k=k, c _j=1/k, C _ρ=1; While enforcement according to above condition, can realize this modeling method.