CN113919191A - Analysis method for microwave low-pressure discharge - Google Patents

Analysis method for microwave low-pressure discharge Download PDF

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CN113919191A
CN113919191A CN202110988305.8A CN202110988305A CN113919191A CN 113919191 A CN113919191 A CN 113919191A CN 202110988305 A CN202110988305 A CN 202110988305A CN 113919191 A CN113919191 A CN 113919191A
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number density
electron
particle
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王瑞
张娜
崔万照
陈翔
陈国宇
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Xian Institute of Space Radio Technology
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Abstract

A microwave low-pressure discharge analysis method aims at the characteristics that microwave low-pressure discharge is sensitive to a field and the electron avalanche effect is obvious, the distribution of a microwave field is accurately calculated by adopting a time domain finite difference method, then the low-pressure discharge process in a microwave field environment is simulated by combining with a particle continuity equation fluid, the discharge analysis precision can be ensured by controlling the calculation precision of an electromagnetic field, and the problem of low calculation efficiency of the microwave low-pressure discharge when the number of charged particles is large is solved by a fluid algorithm.

Description

Analysis method for microwave low-pressure discharge
Technical Field
The invention relates to an analysis method of microwave low-pressure discharge, belonging to the technical field of space high-power microwave technical research.
Background
In order to realize functions of long-distance carrying, stable landing measurement and control, deep-space data long-distance transmission and the like of a spacecraft, a microwave component applied to a payload of the spacecraft requires wider bandwidth, higher element integration level and larger power capacity, which inevitably results in higher electric field density and smaller gap size in the microwave component. The accurate and efficient microwave low-pressure discharge analysis method provides a powerful analysis tool for the design of high-power microwave components in a spacecraft loading system. The low-pressure discharge effect is an important factor influencing the stability performance of the microwave component of the spacecraft, and the damage to the microwave component is usually serious, and once the damage occurs, the reflected power is absorbed by the microwave component, so that permanent damage is caused to the component.
Disclosure of Invention
The technical problem solved by the invention is as follows: aiming at the problem that a method for simulating and analyzing low-pressure discharge effect is lacked in the prior art, a microwave low-pressure discharge analysis method is provided.
The technical scheme for solving the technical problems is as follows:
an analysis method of microwave low-pressure discharge comprises the following steps:
(1) reading in the target model structure, setting initial value of electromagnetic field, gas temperature T, gas pressure P, boundary condition, and initial ion number density
Figure BDA0003231565250000011
Initial electron number density
Figure BDA0003231565250000012
The secondary electron emission coefficient of the structural material is gamma, the initial electron energy, the time step delta t and the simulation time, the particles comprise electrons, ions and excited particles, the lower corner marks are e, ion and exe respectively, and the set model structure is subjected to grid division;
(2) performing electromagnetic field propulsion according to a time domain finite difference method to obtain an electric field E and a magnetic field H at any position and time after grid division;
(3) solving the continuity equation of the particles to obtain the electron flux at t + delta t moment at any position
Figure BDA0003231565250000013
And the number density
Figure BDA0003231565250000014
Ion flux
Figure BDA0003231565250000015
And the number density
Figure BDA0003231565250000016
Flux of excited particles
Figure BDA0003231565250000017
And the number density
Figure BDA0003231565250000018
(4) The electron flux obtained according to the step (3)
Figure BDA0003231565250000021
And the number density
Figure BDA0003231565250000022
Solving to obtain the average energy of the electrons
Figure BDA0003231565250000023
Thereby obtaining the average temperature T of the electronse；
(5) Judging whether secondary electrons exist at the boundary according to the model structure material, and if so, judging according to the initial electron number density
Figure BDA0003231565250000024
Ion number density obtained in step (3)
Figure BDA0003231565250000025
And ion boundary velocity
Figure BDA0003231565250000026
Electron boundary velocity
Figure BDA0003231565250000027
Substituted on the boundaryThe electron continuity equation of (A) yields the boundary secondary electron flux
Figure BDA0003231565250000028
Then the boundary secondary electron flux is generated
Figure BDA0003231565250000029
Substituting the particle continuity equation in the step 1 to obtain the electron flux at the t + delta t moment at any position
Figure BDA00032315652500000210
And the number density
Figure BDA00032315652500000211
Otherwise, directly outputting the electron flux obtained in the step (3)
Figure BDA00032315652500000212
And the number density
Figure BDA00032315652500000213
In the step (1), the model is subjected to equidistant staggered grid division according to the input frequency of the electromagnetic field, main grid nodes, corresponding potential nodes and particle density nodes are set, and the grid division size is determined according to Courant stability conditions.
In the step (3), the chemical reaction rate constant k of each particle in the gas required for solving the continuity equation of the particle is obtainedrThe particle mobility mu is obtained by solving a steady state Boltzmann equation, and the diffusion coefficient D of the particles is obtained by an Einstein relational expression.
In the step (3), the discrete expression of the particle continuity equation is obtained by performing time and space integration on the continuity equation of the particle and grid dispersion:
Figure BDA00032315652500000214
in the formula, S is a source term of a continuity equation of each particle, and the expression is as follows:
Figure BDA00032315652500000215
wherein r represents the r-th reaction process, krDenotes a reaction rate constant, N, corresponding to the reactionrDenotes the number of species of particles participating in the r-th reaction process, nr,mDenotes the number density of the m-th particle participating in the r-th reaction process, wherein the number density of gas neutral molecules is nε＝P/kBT。srIs a symbolic function defined as:
Figure BDA0003231565250000031
in the formula, gammae、Γw、Γs、ΓnThe total flux of the particles on the e, w, s and n nodes at any time respectively. Particle flux density at time ttComprises the following steps:
Figure BDA00032315652500000321
wherein sgn (q) represents a sign of charge of the charged particles, and for neutral particles sgn (q) 0, for positive ions sgn (q) 1, and for negative ions and electrons sgn (q) 1, μ represents the mobility of the particles, EtIs the corresponding electric field at time t, ntThe corresponding particle number density at the time t;
decoupling each particle continuity equation by taking a value at t moment of a source item S in a discrete expression of the particle continuity equation, and further obtaining the electron flux at t + delta t moment at any position
Figure BDA0003231565250000032
And the number density
Figure BDA0003231565250000033
Ion flux
Figure BDA0003231565250000034
And the number density
Figure BDA0003231565250000035
Flux of excited particles
Figure BDA0003231565250000036
And the number density
Figure BDA0003231565250000037
In the step (4), the electron flux obtained according to the step 3
Figure BDA0003231565250000038
And the number density
Figure BDA0003231565250000039
The average energy of the electrons is obtained by
Figure BDA00032315652500000310
Figure BDA00032315652500000311
Calculating the average temperature T of the electronse：
Figure BDA00032315652500000312
In the formula, kBBoltzmann constant.
In the step (5), if the boundary is a material, it is determined that secondary electrons exist, and the number density of the secondary electrons is determined according to the initial number density of the secondary electrons
Figure BDA00032315652500000313
Ion number density obtained in step 3
Figure BDA00032315652500000314
And ion boundary velocityDegree of rotation
Figure BDA00032315652500000315
Electron boundary velocity
Figure BDA00032315652500000316
Substituting into the boundary electron continuity equation to obtain the boundary secondary electron flux
Figure BDA00032315652500000317
Figure BDA00032315652500000318
In the formula (I), the compound is shown in the specification,
Figure BDA00032315652500000319
is the velocity of the boundary of the ion,
Figure BDA00032315652500000320
velocity of ion heat
Figure BDA0003231565250000041
TionM is the average temperature of the ions, which is the same as the gas temperature TionIs the mass of the ion. Electron boundary velocity
Figure BDA0003231565250000042
Electron thermal velocity ve；thIs composed of
Figure BDA0003231565250000043
meIs the mass of the electrons. Then the boundary secondary electron flux is generated
Figure BDA0003231565250000044
Substituting the particle continuity equation in the step 1 to obtain the electron flux at the t + delta t moment at any position
Figure BDA0003231565250000045
And the number density
Figure BDA0003231565250000046
And (6) calculating the current formed by the charged particles according to the particle flux, feeding the current back to the electromagnetic field, judging whether the current exceeds a preset simulation time, repeating the steps (2) - (5) if the current does not exceed the preset simulation time, and ending the microwave low-pressure discharge analysis process if the current exceeds the preset simulation time and outputting a result.
Compared with the prior art, the invention has the advantages that:
(1) the analysis method for microwave low-pressure discharge provided by the invention is based on electromagnetic-fluid direct coupling modeling, realizes time domain finite difference combined fluid numerical modeling, considers the influence of charged particles in a plasma region formed by discharge on an electromagnetic field, not only accurately solves the electromagnetic field distribution, but also has higher calculation efficiency. Reference data are provided for the reliability design of the satellite, and the development and design cost of space microwave components is saved;
(2) aiming at the characteristics that microwave low-pressure discharge is sensitive to a field and the electron avalanche effect is obvious, the distribution of the microwave field is accurately calculated by adopting a time domain finite difference method, and the process of low-pressure discharge in the microwave field environment is simulated by combining a particle continuity equation fluid. The accuracy of discharge analysis is guaranteed by controlling the calculation accuracy of the electromagnetic field, the problem of low calculation efficiency when the number of charged particles is large is solved by a fluid algorithm, the secondary electron multiplication effect of the metal surface of the space microwave component is considered, and the whole process from low vacuum discharge to gas discharge of the microwave structure can be analyzed.
Drawings
FIG. 1 is a flow chart of microwave low-pressure discharge analysis provided by the present invention;
FIG. 2 is a schematic diagram of mesh generation control volume provided by the present invention
FIG. 3 is a diagram of a coaxial model architecture provided by the present invention;
FIG. 4 is a schematic diagram showing the average electron density and electron temperature with time when the electric field strength provided by the present invention is 1.5e 5V/m;
FIG. 5 is a schematic diagram showing the average electron density and the electron temperature with time when the electric field strength provided by the present invention is 5e 6V/m;
Detailed Description
An analysis method for microwave low-pressure discharge is based on electromagnetic-fluid direct coupling modeling, achieves time domain finite difference combined fluid numerical modeling, considers the influence of charged particles in a plasma region formed by discharge on an electromagnetic field, accurately solves electromagnetic field distribution and has high calculation efficiency. The method provides reference data for the reliability design of the satellite, saves the development and design cost of space microwave components, and comprises the following specific steps:
(1) reading in the target model structure, setting initial value of electromagnetic field, gas temperature T, gas pressure P, boundary condition, and initial ion number density
Figure BDA0003231565250000051
Initial electron number density
Figure BDA0003231565250000052
The secondary electron emission coefficient of the structural material is gamma, the initial electron energy, the time step delta t and the simulation time, the particles comprise electrons, ions and excited particles, the lower corner marks are e, ion and exe respectively, and the set model structure is subjected to grid division;
(2) performing electromagnetic field propulsion according to a time domain finite difference method to obtain an electric field E and a magnetic field H at any position and time after grid division;
(3) solving the continuity equation of the particles to obtain the electron flux at the t + delta t moment at any position
Figure BDA0003231565250000053
And the number density
Figure BDA0003231565250000054
Ion flux
Figure BDA0003231565250000055
And the number density
Figure BDA0003231565250000056
Flux of excited particles
Figure BDA0003231565250000057
And the number density
Figure BDA0003231565250000058
(4) The electron flux obtained according to the step (3)
Figure BDA0003231565250000059
And the number density
Figure BDA00032315652500000510
Solving to obtain the average energy of the electrons
Figure BDA00032315652500000511
Thereby obtaining the average temperature T of the electronse；
(5) Judging whether secondary electrons exist at the boundary according to the model structure material, and if so, judging according to the initial electron number density
Figure BDA00032315652500000512
Ion number density obtained in step (3)
Figure BDA00032315652500000513
And ion boundary velocity
Figure BDA00032315652500000514
Electron boundary velocity
Figure BDA00032315652500000515
Substituting into the electron continuity equation on the boundary to obtain the secondary electron flux on the boundary
Figure BDA00032315652500000516
Then the boundary secondary electron flux is generated
Figure BDA00032315652500000517
Substituting into the particle continuity method in step 1Obtaining the electron flux at t + delta t moment at any position
Figure BDA00032315652500000518
And the number density
Figure BDA00032315652500000519
Otherwise, directly outputting the electron flux obtained in the step (3)
Figure BDA00032315652500000520
And the number density
Figure BDA00032315652500000521
(6) And (3) calculating the current formed by the charged particles according to the particle flux, feeding the current back to the electromagnetic field, judging whether the preset simulation time is exceeded or not, if not, repeating the steps (2) - (5), and if the preset simulation time is exceeded, ending the microwave low-pressure discharge analysis process and outputting a result.
The following is further illustrated with reference to specific examples:
the specific steps are shown in figure 1:
(1) reading in the target model structure, setting initial value of electromagnetic field, gas temperature T, gas pressure P, boundary condition, and initial ion number density
Figure BDA0003231565250000061
Initial electron number density
Figure BDA0003231565250000062
The secondary electron emission coefficient of the structural material is gamma, the initial electron energy, the time step delta t, the simulation time, and the chemical reaction rate constant k of each particle in the gasrThe particle mobility mu is obtained by solving a steady state Boltzmann equation, and the diffusion coefficient D of the particles is obtained by an Einstein relational expression. The particles comprise electrons, ions and excited particles, the lower corner marks are e, ion and exe respectively, and the set model structure is subjected to grid division;
matched according to input electromagnetic field frequencyThe pattern is subjected to equidistant staggered meshing, as shown in FIG. 2, where the intersections between the solid lines
Figure BDA00032315652500000611
Called main grid node, corresponding to the node of scalar such as electric potential phi and particle density n, the corresponding control volume is composed of the adjacent control interface lines of the main grid node, as shown by the shaded part in fig. 2 (a). The dotted line is called a control interface line, and the intersection point "x" of the control interface line and the main grid line is applied to a node of vectors such as the electric field intensity E and the particle flux density Γ. The shaded portion in fig. 2(b) represents the volume controlled by node e.
(2) Performing electromagnetic field propulsion according to a time domain finite difference method to obtain an electric field E and a magnetic field H at any position and time after grid division;
(3) solving the continuity equation of the particles (non-electrons) to obtain the electron flux at the t + delta t moment at any position
Figure BDA0003231565250000063
And the number density
Figure BDA0003231565250000064
Ion flux
Figure BDA0003231565250000065
And the number density
Figure BDA0003231565250000066
Flux of excited particles
Figure BDA0003231565250000067
And the number density
Figure BDA0003231565250000068
And (3) performing time and space integration on a continuity equation of the particles and grid dispersion to obtain:
Figure BDA0003231565250000069
s is a source term of a continuity equation of each particle, and the expression is as follows:
Figure BDA00032315652500000610
wherein r represents the r-th reaction process, krDenotes a reaction rate constant, N, corresponding to the reactionrDenotes the number of species of particles participating in the r-th reaction process, nr,mDenotes the number density of the m-th particle participating in the r-th reaction process, wherein the number density of gas neutral molecules is nε＝P/kBT。srIs a symbolic function defined as:
Figure BDA0003231565250000071
Γe、Γw、Γs、Γnthe total flux of the particles on the e, w, s and n nodes at any time respectively.
Particle flux density at time ttIs composed of
Figure BDA00032315652500000719
Wherein sgn (q) represents a sign of charge of the charged particles, and for neutral particles sgn (q) 0, for positive ions sgn (q) 1, and for negative ions and electrons sgn (q) 1, μ represents the mobility of the particles, EtIs the corresponding electric field at time t, ntThe corresponding particle number density at time t.
Decoupling each particle continuity equation by taking a time t value of a source term S in the formula (1) so as to obtain the electron flux of any position at the time t + delta t
Figure BDA0003231565250000072
And the number density
Figure BDA0003231565250000073
Ion flux
Figure BDA0003231565250000074
And the number density
Figure BDA0003231565250000075
Flux of excited particles
Figure BDA0003231565250000076
And the number density
Figure BDA0003231565250000077
(4) Electron flux obtained according to step 3
Figure BDA0003231565250000078
And the number density
Figure BDA0003231565250000079
Solving equation (5) to obtain the average energy of electrons
Figure BDA00032315652500000710
Thereby obtaining the average temperature T of the electronse。
Figure BDA00032315652500000711
Figure BDA00032315652500000712
In the formula, kBIs the boltzmann constant, and is,
(5) and judging whether secondary electrons exist at the boundary or not according to the model structure material.
If the boundary is material, then considering that there is secondary electron, then according to the initial electron number density
Figure BDA00032315652500000713
Ion number density obtained in step 3
Figure BDA00032315652500000714
And ion boundary velocity
Figure BDA00032315652500000715
Electron boundary velocity
Figure BDA00032315652500000716
Substituting into the equation (7) of electron continuity on the boundary to obtain the secondary electron flux on the boundary
Figure BDA00032315652500000717
Figure BDA00032315652500000718
Velocity of ion boundary
Figure BDA00032315652500000810
Is composed of
Figure BDA0003231565250000081
Velocity of ion heat
Figure BDA0003231565250000082
TionM is the average temperature of the ions, which is the same as the gas temperature TionIs the mass of the ion. Electron boundary velocity
Figure BDA0003231565250000083
Electron thermal velocity ve；thIs composed of
Figure BDA0003231565250000084
meIs the mass of the electrons.
Then the boundary secondary electron flux is generated
Figure BDA0003231565250000085
Substituting the particle continuity equation in the step 1 to obtain the electron flux at the t + delta t moment at any position
Figure BDA0003231565250000086
And the number density
Figure BDA0003231565250000087
If the boundary is a port, the secondary electrons are not considered to exist, the particle continuity equation in the step 1 is directly solved, and the electron flux at the t + delta t moment at any position is obtained
Figure BDA0003231565250000088
And the number density
Figure BDA0003231565250000089
(6) And (3) calculating the current formed by the charged particles according to the particle flux, feeding the current back to the electromagnetic field, judging whether the preset simulation time is exceeded or not, if not, repeating the steps (2) - (5), and if the preset simulation time is exceeded, ending the microwave low-pressure discharge analysis process and outputting a result.
Therein, a coaxial model structure as shown in fig. 3 is established to simulate a low-pressure discharge process in the coaxial structure. The radius of an outer conductor of the coaxial structure is 3.55mm, the radius of an inner conductor of the coaxial structure is 1.54mm, the length of the coaxial structure is 10mm, the working frequency is 3.2GHz, and the simulation time is 100 ns. The background gas was a mixed gas of Ne (95%) and Xe (5%) and the gas pressure was 30 Torr.
As shown in FIG. 4, the electric field strength of 1.5X 10 was obtained by simulation5The discharge results of the coaxial structure shown in FIG. 3 at V/m are shown in FIG. 4(a) as the change in the average electron density with time under this condition, and in FIG. 4(b) as the change in the average electron temperature with time under this condition. The average electron density increased first and then decreased, and it was considered that no discharge occurred.
As shown in FIG. 5, the electric field strength of 5X 10 was obtained by simulation6The discharge results of the coaxial structure shown in FIG. 3 at V/m are shown in FIG. 5(a) as the change in the average electron density with time under this condition, and in FIG. 5(b) as the change in the average electron temperature with time under this condition. The development of the average electron density over time increases exponentially, indicating thatA discharge occurs.
Although the present invention has been described with reference to the preferred embodiments, it is not intended to limit the present invention, and those skilled in the art can make variations and modifications of the present invention without departing from the spirit and scope of the present invention by using the methods and technical contents disclosed above.
Those skilled in the art will appreciate that the details of the invention not described in detail in this specification are well within the skill of those in the art.

Claims (7)

1. A method for analyzing microwave low-pressure discharge is characterized by comprising the following steps:
(1) reading in the target model structure, setting initial value of electromagnetic field, gas temperature T, gas pressure P, boundary condition, and initial ion number density
Figure FDA0003231565240000011
Initial electron number density
Figure FDA0003231565240000012
The secondary electron emission coefficient of the structural material is gamma, the initial electron energy, the time step delta t and the simulation time, the particles comprise electrons, ions and excited particles, the lower corner marks are e, ion and exe respectively, and the set model structure is subjected to grid division;
(2) performing electromagnetic field propulsion according to a time domain finite difference method to obtain an electric field E and a magnetic field H at any position and time after grid division;
(3) solving the continuity equation of the particles to obtain the electron flux at t + delta t moment at any position
Figure FDA0003231565240000013
And the number density
Figure FDA0003231565240000014
Ion flux
Figure FDA0003231565240000015
And the number density
Figure FDA0003231565240000016
Flux of excited particles
Figure FDA0003231565240000017
And the number density
Figure FDA0003231565240000018
(4) The electron flux obtained according to the step (3)
Figure FDA0003231565240000019
And the number density
Figure FDA00032315652400000110
Solving to obtain the average energy of the electrons
Figure FDA00032315652400000111
Thereby obtaining the average temperature T of the electronse；
(5) Judging whether secondary electrons exist at the boundary according to the model structure material, and if so, judging according to the initial electron number density
Figure FDA00032315652400000112
Ion number density obtained in step (3)
Figure FDA00032315652400000113
And ion boundary velocity
Figure FDA00032315652400000114
Electron boundary velocity
Figure FDA00032315652400000115
Substituting into the electron continuity equation on the boundary to obtain the secondary electron flux on the boundary
Figure FDA00032315652400000116
Then the boundary secondary electron flux is generated
Figure FDA00032315652400000117
Substituting the particle continuity equation in the step 1 to obtain the electron flux at the t + delta t moment at any position
Figure FDA00032315652400000118
And the number density
Figure FDA00032315652400000119
Otherwise, directly outputting the electron flux obtained in the step (3)
Figure FDA00032315652400000120
And the number density
Figure FDA00032315652400000121
2. The method of claim 1, wherein the method comprises the following steps:
in the step (1), the model is subjected to equidistant staggered grid division according to the input frequency of the electromagnetic field, main grid nodes, corresponding potential nodes and particle density nodes are set, and the grid division size is determined according to Courant stability conditions.
3. The method of claim 1, wherein the method comprises the following steps:
in the step (3), the chemical reaction rate constant k of each particle in the gas required for solving the continuity equation of the particle is obtainedrThe particle mobility mu is obtained by solving a steady state Boltzmann equation, and the particle expansionThe dispersion coefficient D is obtained from the Einstein relationship.
4. The method of claim 1, wherein the method comprises the following steps:
in the step (3), the discrete expression of the particle continuity equation is obtained by performing time and space integration on the continuity equation of the particle and grid dispersion:
Figure FDA0003231565240000021
in the formula, S is a source term of a continuity equation of each particle, and the expression is as follows:
Figure FDA0003231565240000022
in the formula (I), the compound is shown in the specification,ris shown asrA reaction process, krDenotes a reaction rate constant, N, corresponding to the reactionrDenotes the number of species of particles participating in the r-th reaction process, nr,mDenotes the number density of the m-th particle participating in the r-th reaction process, wherein the number density of gas neutral molecules is nε＝P/kBT。srIs a symbolic function defined as:
Figure FDA0003231565240000023
in the formula, gammae、Γw、Γs、ΓnThe total flux of the particles on the e, w, s and n nodes at any time respectively. Particle flux density at time ttComprises the following steps:
Figure FDA0003231565240000024
wherein sgn (q) represents the sign of the charge of the charged particlesFor neutral particles sgn (q) 0, for positive ions sgn (q) 1, for negative ions and electrons sgn (q) 1, μ represents the mobility of the particles, EtIs the corresponding electric field at time t, ntThe corresponding particle number density at the time t;
decoupling each particle continuity equation by taking a value at t moment of a source item S in a discrete expression of the particle continuity equation, and further obtaining the electron flux at t + delta t moment at any position
Figure FDA0003231565240000025
And the number density
Figure FDA0003231565240000026
Ion flux
Figure FDA0003231565240000027
And the number density
Figure FDA0003231565240000028
Flux of excited particles
Figure FDA0003231565240000029
And the number density
Figure FDA00032315652400000210
5. The method of claim 1, wherein the method comprises the following steps:
in the step (4), the electron flux obtained according to the step 3
Figure FDA00032315652400000211
And the number density
Figure FDA00032315652400000212
The average energy of the electrons is obtained by
Figure FDA0003231565240000031
Figure FDA0003231565240000032
Calculating the average temperature T of the electronse：
Figure FDA0003231565240000033
In the formula, kBBoltzmann constant.
6. The method of claim 1, wherein the method comprises the following steps: in the step (5), if the boundary is a material, it is determined that secondary electrons exist, and the number density of the secondary electrons is determined according to the initial number density of the secondary electrons
Figure FDA0003231565240000034
Ion number density obtained in step 3
Figure FDA0003231565240000035
And ion boundary velocity
Figure FDA0003231565240000036
Electron boundary velocity
Figure FDA0003231565240000037
Substituting into the boundary electron continuity equation to obtain the boundary secondary electron flux
Figure FDA0003231565240000038
Figure FDA0003231565240000039
In the formula，
Figure FDA00032315652400000310
Is the velocity of the boundary of the ion,
Figure FDA00032315652400000311
velocity of ion heat
Figure FDA00032315652400000312
TionM is the average temperature of the ions, which is the same as the gas temperature TionIs the mass of the ion. Electron boundary velocity
Figure FDA00032315652400000313
Electron thermal velocity ve；thIs composed of
Figure FDA00032315652400000314
meIs the mass of the electrons. Then the boundary secondary electron flux is generated
Figure FDA00032315652400000315
Substituting the particle continuity equation in the step 1 to obtain the electron flux at the t + delta t moment at any position
Figure FDA00032315652400000316
And the number density
Figure 1
。
7. The method of claim 1, wherein the method comprises the following steps:
and (6) calculating the current formed by the charged particles according to the particle flux, feeding the current back to the electromagnetic field, judging whether the current exceeds a preset simulation time, repeating the steps (2) - (5) if the current does not exceed the preset simulation time, and ending the microwave low-pressure discharge analysis process if the current exceeds the preset simulation time and outputting a result.
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