A method for improving the simulation speed of a cross-field diode physical quantity

Abstract

The application discloses a method for improving simulation speed of physical quantity of cross-field diode, relates to simulation technology of vacuum electronic device, and is proposed in view of the long simulation time in the prior art. Before solving the electric field force corresponding to each electron particle, the numbering is sorted in sequence from the cathode to the anode according to the positions of all the electron particles at the current time, the numbering is corresponding to the positions of the electron particles, and then the electric field force corresponding to each electron particle is solved according to the partial solution of the diode potential. The method has the advantages that the solving step of the electric field force corresponding to the partial solution of the diode potential in the cross-field diode is improved, the fast sorting algorithm is combined, the simulation speed is greatly improved compared with the general meshless numerical algorithm, and the method is suitable for the vacuum space of the cross-field diode in which the electron particles are injected from the cathode to the space with an arbitrary initial speed and an arbitrary magnetic field value.

Description

Technical Field

[0001] This invention relates to simulation technology for vacuum electronic devices, and more particularly to a method for improving the simulation speed of physical quantities of cross-field diodes. Background Technology

[0002] The space charge confinement effect has been widely discussed in recent years, affecting the performance of many electronic devices. In the one-dimensional non-relativistic case, the confinement current density of the vacuum gap in a diode is given by the classical CL law. Currently, the classical CL law has been extended to multidimensional models, various geometries, relativity, quantum states, field emission, and thermionic emission. In the classical CL law, the initial velocity of the electron particle injected from the cathode is zero.

[0003] like Figure 1 As shown, the spacing between the cross-field diodes is D, and a uniform magnetic field parallel to the cathode surface and perpendicular to the electric field is superimposed on the outside; hence, it is called a cross-field diode. The cathode is located at x = 0, with a potential value of φ. c =0; the anode is located at x=D, and the potential value is φ. a =V g A uniform magnetic field B is superimposed on the outside. Z All electron particles are injected from the cathode with the same initial velocity v0.

[0004] Given the cathode injection current density J in After the simulation time interval Δt, relying on the relationship q p =-J in Δt represents the charge of a single electron particle (the subscript p indicates a single electron particle). The negative sign indicates that the electron particle is negatively charged. An electron particle is a collection of many electrons, representing all electron particles emitted from the cathode within a time interval Δt. This treatment simplifies the simulation process and shortens the simulation time, as the number of electrons in a single simulation can be on the order of 10. 18 .

[0005] The potential distribution between the entire cross-field diodes still satisfies the one-dimensional Poisson equation, i.e., equation (1), and ρ(x) is the charge density distribution, which satisfies equation (2):

[0006]

[0007]

[0008] Where N is the number of electron particles in the vacuum gap of the cross-field diode at a certain moment.

[0009] Since the one-dimensional Poisson equation is a second-order non-homogeneous linear equation belonging to the second-order differential equations, its solution allows us to write the total potential distribution of the cross-field diode as:

[0010] φ(x)=φ h (x)+φ p (x) (3)

[0011] Where φ h (x) and φ p (x) represents the general solution and particular solution of the one-dimensional Poisson equation, respectively.

[0012] The general solution φ of the one-dimensional Poisson equation h (x) satisfies:

[0013] φ h (x)=Ax+B (4)

[0014] A and B in equation (4) are obtained based on the boundary conditions. According to equation (3), the potentials of the cathode and anode at both ends of the cross-field diode can be defined, satisfying the following equation:

[0015] φ(0)=φ h (0)+φ p (0)=A×0+B+φ p (0)=φ c (5)

[0016] φ(D)=φ h (D)+φ p (D)=A×D+B+φ p (D)=φ a (6)

[0017] Subtracting equation (6) from equation (5) yields A:

[0018]

[0019] Multiplying equation (6) by a / b and then subtracting it from equation (5), we get B:

[0020] B = φ c -φ p (0) (8)

[0021] Next, we will process the particular solution, which can be written as follows:

[0022]

[0023] Where G(x,x) k ) is a free-space Green's function that obeys the one-dimensional Poisson equation, specifically satisfying:

[0024]

[0025] Since it is assumed that each electron particle carries the same charge, q... pφ in equation (7) p (0) and φ p (D) From equation (9), we can obtain:

[0026]

[0027]

[0028] The potential of any electron particle in the vacuum gap of the cross-field diode can be obtained by equation (3). Since the electric field is the negative differential of the potential, the electric field at any electron particle can also be obtained. According to equation (13), the electric force on any electron particle in the vacuum gap of the cross-field diode can be obtained.

[0029] F e =q p E = q p ×(-dφ / dx) (13)

[0030] Due to the presence of an external magnetic field B Z Electron particles moving in the vacuum gap are subjected to Lorentz force from the magnetic field, as shown in equation (14).

[0031] F m =q p vB (14)

[0032] We are considering a one-dimensional space, specifically the motion of the electron particle along the horizontal axis. Therefore, we need to consider the total Lorentz force F acting on the electron particle. m Decompose the components by angle to obtain the component F along the horizontal axis. mx Therefore, for a certain electron particle in the vacuum gap of the cross-field diode, the actual force along the horizontal axis is as follows:

[0033] F t =F e +F mx (15)

[0034] The potential φ(x) in a cross-field diode is derived from the general solution φ h (x) and particular solution φ p (x) are superimposed, therefore the electric force F e It is essentially composed of two parts, as shown in equation (16).

[0035]

[0036] Where i represents the number of the electron particle in the vacuum gap.

[0037] Defects and shortcomings of existing technology:

[0038] 1. The numerical algorithm currently applicable to simulating the motion of electron particles in a cross-field diode requires that the initial energy of the injected electrons be only 0eV, 0.01eV, or 0.5eV.

[0039] 2. When the applied magnetic field is lower than the Hall cutoff magnetic field value, the theoretical analytical expression is limited to a portion of physical quantities under steady-state conditions.

[0040] 3. When the applied magnetic field is higher than the Hall cutoff magnetic field value, the limiting current density analytical formula is only applicable when the initial energy of the electron particle is zero.

[0041] 4. Existing meshless direct summation algorithms suffer from excessively long simulation times and low efficiency. If there are N electron particles in the vacuum gap, calculating the force on a single electron particle requires N-1 comparison calculations before summation. Currently, limited by computer hardware performance, the most time-consuming part of the simulation is the differential calculation required for each electron particle; for N electron particles, this requires (N-1) comparison calculations. N Secondly, this is the biggest technical bottleneck in simulation efficiency. Summary of the Invention

[0042] The purpose of this invention is to provide a method to improve the simulation speed of physical quantities of cross-field diodes, so as to solve the problems existing in the prior art.

[0043] The method for improving the simulation speed of physical quantities of cross-field diodes described in this invention involves numbering each electron particle according to the order in which electron particles are injected from the cathode;

[0044] Before solving for the electric force corresponding to each electron particle, the positions of all electron particles at the current moment are sorted according to the direction from cathode to anode, corresponding to the positions of the electron particles. Then, the electric force corresponding to each electron particle is solved based on the special solution of the diode potential.

[0045] Where, q p ε0 is the charge of each electron particle, N is the total number of electron particles, i is the number of the electron particle, and ε0 is the vacuum permittivity.

[0046] The sorting is performed in the form of pointers in the computer program.

[0047] The method for improving the simulation speed of physical quantities in cross-field diodes described in this invention has the advantage of improving the solution steps for the electric force corresponding to the special potential solution in cross-field diodes, and combining it with a fast sorting algorithm, resulting in a significant improvement in simulation speed compared to general meshless numerical algorithms. It is also applicable to the vacuum gap of a cross-field diode with electron particles injected from the cathode at arbitrary initial velocities into an arbitrary magnetic field value. Furthermore, the simulation results obtained using the meshless sorting and summing numerical algorithm are fitted with theoretical curves of some existing analytical formulas, demonstrating the applicability of the meshless sorting and summing algorithm in cross-field diodes from three perspectives. Attached Figure Description

[0048] Figure 1 This is a schematic diagram of the general structure of a cross-field diode.

[0049] Figure 2 This is a schematic diagram of the first round of electron particle sorting in this invention.

[0050] Figure 3 It is a theoretical curve showing the absolute value of the potential of the virtual cathode and its location.

[0051] Figure 4 This is a schematic diagram showing the fitting of the theoretical curves of the normalized electric field and potential under spatial constraints with numerical results.

[0052] Figure 5 This is a schematic diagram showing the fitting of the theoretical curve and numerical results for limiting the current density in a cross-field diode. Detailed Implementation

[0053] The method for improving the simulation speed of physical quantities in a cross-field diode, as described in this invention, can be implemented using MATLAB software. The entire simulation process of the motion of electron particles in a cross-field diode in one-dimensional space is as follows:

[0054] (1) Starting from the first simulation moment, the motion of electron particles in the cross-field diode is simulated and calculated;

[0055] (2) A new electron particle with non-zero initial energy is injected from the cathode of the cross-field diode into the vacuum gap;

[0056] (3) Based on the position and number of the electron particles in the vacuum gap of the cross-field diode at this time, the numbers of all electron particles are reordered in the direction from cathode to anode using the fast sorting algorithm.

[0057] (4) Based on the sorted number and the velocity of each electron particle at the current moment, the force situation of each electron particle in the vacuum gap of the cross-field diode in one dimension is obtained by using the meshless sorting and summation numerical algorithm, including electric force and Lorentz force.

[0058] (5) Based on the force situation of each electron particle in the vacuum gap of the cross-field diode at the current moment, the position and velocity of the electron particle at the next moment are obtained by using the fourth-order Runge-Kutta method;

[0059] (6) Since the electron particles will be absorbed by the cross-field diode when they reach the anode or return to the cathode, it is necessary to determine whether the electron particles have left the vacuum gap based on their position at the next moment. If so, their velocity is set to zero.

[0060] (7) Continue the simulation for the next moment, repeating steps (2) to (6) until the simulation time ends.

[0061] The electric field force on the electron particle is calculated by solving the special solution of the diode potential. Since each electron particle carries the same amount of charge, there is a lot of repetitive calculation in the whole calculation process. The innovation of this invention is to improve the solution to this problem and significantly shorten the simulation time.

[0062] The improved numerical algorithm for gridless sorting and summation targets the electric force corresponding to the particular solution of the electric potential, as shown in equation (17).

[0063]

[0064] in, By classifying and discussing, we can obtain:

[0065]

[0066] There are N electron particles in a vacuum gap. If these electron particles are numbered and ordered according to their positions from cathode to anode, then the number of electron particles x is... i The number of particles on the right is n R =Ni, electron particle x i The number of particles on the left is n L = i-1, then equation (16) can be rewritten as:

[0067]

[0068] Simplifying equation (19) yields:

[0069]

[0070] The electric field force corresponding to the particular solution of the electric potential is obtained using equation (20), which can significantly shorten the simulation time. The existing algorithm is a meshless direct summation numerical algorithm, while the improvement corresponding to equation (20) in this invention is a meshless sorted summation numerical algorithm.

[0071] Regarding the quicksort algorithm: sorting serves the purpose of formula (20). Only after sorting is completed can formula (20) be used for calculation. Using formula (20) can shorten the time. Table 3 below shows the time comparison.

[0072] This invention proposes an improved solution for the electric field force solution corresponding to the particular solution of the potential in a cross-field diode, which significantly reduces the simulation time. As an innovation of this invention, it is necessary to elaborate further.

[0073] As shown in equation (17), if there are N electron particles in the vacuum gap, the force on a certain electron particle needs to be calculated according to equation (17) by performing N-1 comparison calculations before summing.

[0074] The gridless sorting and summation algorithm, as shown in equation (20), reduces the computational load by only one calculation, whereas equation (17) originally required N-1 calculations before summation. However, it is important to note that before using equation (20), the numbers of all electron particles in the vacuum gap of the cross-field diode must be sorted from cathode to anode. The fast sorting algorithm used in this invention is as follows:

[0075] (1) Select one from the required set of data as the baseline; usually, the first one is sufficient.

[0076] (2) Place the data in this set that is less than the benchmark to the left of the benchmark, and the data that is greater than the benchmark to the right of the benchmark;

[0077] (3) Treat all the data to the left and right of the benchmark as two new data subsets, and repeat steps (1) and (2) for each of these two new data subsets.

[0078] (4) The sorting is completed when there is only one data in each data subset, in ascending order. In the actual physical scenario, this means that the numbers of all the electron particles in the vacuum gap of the cross-field diode are arranged in order from the cathode to the anode.

[0079] To illustrate more intuitively how to use the quicksort algorithm, let's take an example: Given the data set [23 45 171113], arr... Figure 2This demonstrates how to use the quicksort algorithm to sort this set of data in the first round. First, 23 is chosen as the pivot. Pointers i and j point to the leftmost and rightmost elements of the set arr, respectively (23 and 13). Pointer i moves from left to right, and pointer j moves from right to left. Second, when the number pointed to by pointer j is greater than the pivot, pointer j continues to move left until it encounters a number not greater than the pivot, arr[j], and fills the position arr[i]. Third, pointer i is moved. When the number pointed to by pointer i is not greater than the pivot, pointer i continues to move right until it encounters a number greater than the pivot, arr[i], and fills the position arr[j]. Fourth step, move pointers i and j alternately until pointers i and j meet, and fill the pivot number into arr[i]; this completes sorting steps (1) and (2). Next, sort the arrays to the left and right of the pivot number again using the above method (recursively) until each subset contains only one data, and then all sorting is complete. In the gridless sorting and summation algorithm, before using formula (20), it is necessary to quickly sort all the electron particles in the vacuum gap at this time. The time complexity of this part of the calculation is O(NlogN).

[0080] Applicability verification of the non-mesh sorting summation numerical algorithm:

[0081] First, the applicability is verified by examining the location and potential of the virtual cathode.

[0082] If the electron particles injected from the cathode have an initial velocity, they will induce the formation of a virtual cathode in the vacuum gap. It is called a virtual cathode because its characteristics are consistent with those of the cathode of a cross-field diode in the classical CL law: the potential is minimum and the electric field is zero. In this invention, physical quantities related to the virtual cathode are marked with an asterisk in the upper right corner, and the potential of the virtual cathode is denoted as -φ. * The position of the virtual cathode is denoted as x. * The theoretical analytical expressions for the normalized absolute value of the potential and the normalized position of the virtual cathode are shown in equations (21) and (22), respectively.

[0083]

[0084]

[0085] Among them, equations (21) and (22) This is the normalized parameter for the initial velocity.

[0086] In this verification, the normalized parameter β of the initial velocity of the electron particle injected from the cathode of the cross-field diode is uniformly selected in the interval [1e-7, 1.0], representing eleven values, which represent different incident initial velocities of the electron particle. This also proves that the numerical algorithm has universal applicability for any initial velocity. Since the background of the two analytical equations (21) and (22) is only for the motion of the electron particle under the electric field force in the cross-field diode, the magnetic field B parallel to the cathode needs to be set in this verification process. Z =0, in addition, the injected current density J in Slightly less than the limiting current density value, take J in =0.995×J SCL J SCL The limiting current density value is given when electron particles are injected from the cathode into a diode without a magnetic field, but the electron particles have an initial velocity.

[0087] Figure 3 The theoretical curves for the absolute potential and position of the virtual cathode, numerical solutions for different initial velocities, and the fitting relationship between the two are shown. (a) shows the variation of the normalized absolute potential of the virtual cathode with respect to the initial velocity parameter, and (b) shows the variation of the normalized position of the virtual cathode with respect to the initial velocity parameter. The solid line represents the theoretical curve, and the dots represent the numerical simulation results. When electron particles are injected from the cathode with different initial velocities, the absolute potential and position of the virtual cathode will differ.

[0088] pass Figure 3 The comparison between the theoretical curves and numerical solutions of the virtual cathode clearly shows that the simulation results corresponding to different normalized initial velocity parameters β all match the theoretical curves well. Therefore, from the first perspective, it is proven that the meshless sorting and summation numerical algorithm used is accurate and reliable, and applicable to electron particles with any injected initial velocity.

[0089] Second, the applicability is verified by examining the electric field and potential distributions.

[0090] Based on the same background conditions as the first verification, the injected current density J in Slightly less than the limiting current density, take J in =0.995×J SCL The electron particles injected from the cathode still have a certain initial velocity. In 2020, Lafleur proposed analytical formulas for electric field distribution and potential distribution, the specific contents of which are as follows: When the cross-field diode reaches the space charge confinement state, the electric field distribution in the vacuum gap of the cross-field diode satisfies formula (23), and the potential distribution satisfies formula (24).

[0091]

[0092]

[0093] Among them, E a =φ a / D, β is the normalized parameter for the initial velocity.

[0094] Figure 4 The normalized electric field and potential distributions are shown when the cross-field diode reaches the space-constrained state. (a) shows the normalized electric field distribution from cathode to anode, and (b) shows the normalized potential distribution from cathode to anode. The solid lines correspond to the theoretical distribution curves of equations (23) and (24), respectively, while the dashed lines represent the numerical simulation results under the same conditions. The curves from top to bottom near the anode in the electric field distribution diagram and from top to bottom in the potential distribution diagram both indicate that the initial velocity parameters of the electron particles are β = 1e-7 / 0.1 / 0.4 / 0.7 / 1.0, respectively. Figure 4 It can be clearly seen that the simulation results of the electric field and potential distribution in the vacuum gap of the cross-field diode almost coincide with the theoretical distribution curve. Only when β = 1.0 is the potential value near the virtual cathode obtained by the numerical algorithm slightly smaller than the theoretical value, but it is within the acceptable range for the numerical algorithm.

[0095] In this verification work, multiple β values ​​were uniformly selected, which once again proved that the meshless sorting and summation numerical algorithm used is accurate and reliable, and applicable to various cases where electron particles are injected from the cathode with arbitrary initial velocities.

[0096] Third, the applicability is verified by limiting the current density of the cross field.

[0097] The parameters for this verification work are: the electron particles are injected into the vacuum gap with an initial velocity, and the magnetic field value parallel to the cathode surface is B. Z ≠0, in addition, B H As shown in equation (25).

[0098]

[0099] Lau and Christenson, in 1993, addressed cross-field diodes with magnetic fields present, and B... Z / B H Within the range [0,1], the limiting current density J for this condition is... C A theoretical analytical formula was proposed. The specific theoretical content is as follows: When the initial velocity v0 of the electron particle injected from the cathode is zero, the confinement current density J in the cross-field diode... C With magnetic field B ZThe changes satisfy equation (26), where variable y1 follows equation (27); when the initial velocity of the electron particle injected from the cathode is v0≠0, the limiting current density J is... C With magnetic field B Z The changes satisfy equation (28), where variable y2 follows equation (29); there exists an inverse function h(y1 / y2) in equations (26) and (28), and the function x = h(y1 / y2) is the inverse function of function y1 / y2 = f(x), while f(x) follows equation (30).

[0100]

[0101]

[0102]

[0103]

[0104]

[0105] In the third verification work of this invention, the injection current density J was gradually increased or decreased at intervals of 0.5%. in The limiting current density J is determined by observing the trajectories of electron particles in the vacuum gap of the cross-field diode. C If, during the simulation, electron particles that were originally moving toward the anode of the cross-field diode return to the cathode, then the injected current density in this simulation is determined to exceed the limit current density value under that condition. Figure 5 The figure shows the fitting between the theoretical curve and the numerical results for limiting the current density in a cross-field diode. It can be clearly seen from the figure that both the black dots representing the numerical results and the curve representing the theoretical values ​​are well fitted.

[0106] Table 1

[0107]

[0108] Table 1 shows the error between the numerical results and the theoretical results for the constrained current density. It can be clearly seen from Table 1 that for any initial velocity value of the electron particle, the numerical simulation results for the constrained current density fit the theoretical analysis values ​​very well. This further demonstrates from a third perspective that the meshless sorting and summing numerical algorithm used is accurate in cross-field diodes and has universal applicability to the initial velocities of electron particles.

[0109] Fourth, the time advantage of the gridless sorting and summation algorithm.

[0110] Improving the meshless direct summation algorithm to a meshless sorting summation algorithm will significantly reduce simulation time and increase simulation speed, resulting in a practically significant improvement in the efficiency of analyzing and discussing simulation results.

[0111] Several simulation scenarios were selected for simulation, and the actual simulation time was used as the basis to demonstrate that the meshless sorting and summation algorithm does indeed shorten the simulation time compared to the meshless direct summation method. Specific simulation scenario parameters are shown in Table 2; simulation parameters not shown are kept consistent.

[0112] Table 2

[0113]

[0114] The computer used for simulation was an Intel i7-7700 processor, and the simulation software was MATLAB 2019a. Table 3 shows the time required for simulations 1 through 10 using both the meshless direct summation method and the meshless sorting summation method. The time percentages in Table 3 refer to the proportion of simulation time required by the meshless sorting summation algorithm to the time required by the meshless direct summation algorithm under the same simulation conditions.

[0115] Table 3

[0116]

[0117] For those skilled in the art, various other corresponding changes and modifications can be made based on the technical solutions and concepts described above, and all such changes and modifications should fall within the protection scope of the claims of this invention.

Claims

1. A method to improve the simulation speed of physical quantities of cross-field diodes, wherein each electron particle is numbered according to the time sequence of electron particle injection from the cathode; Its features are, Before solving for the electric force corresponding to each electron particle, the positions of all electron particles at the current moment are sorted according to the direction from cathode to anode, corresponding to the positions of the electron particles. Then, the electric force corresponding to each electron particle is solved based on the special solution of the diode potential. Where, q p ε0 is the charge of each electron particle, N is the total number of electron particles, i is the number of the electron particle, and ε0 is the vacuum permittivity.

2. The method for improving the simulation speed of physical quantities of cross-field diodes according to claim 1, characterized in that, The sorting is performed in the form of pointers in the computer program.

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