CN116266209B - Calculation method for double-sided medium loaded parallel plate waveguide electrostatic field - Google Patents
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Abstract
The invention provides a calculation method for a double-sided medium loaded parallel plate waveguide electrostatic field, and belongs to the field of high-power microwave component micro discharge. The method comprises the steps of solving a spectral domain green function through a Fourier transform method, solving the green function by adopting a Filon numerical integration method due to the complexity and the oscillation of the integrated function, carrying out surface integration on the green function according to the surface charge density of a medium layer by utilizing the superposition principle of the electric potential calculation of point charges, thus obtaining the electric potential generated by random electric charge distribution on the surface of the medium layer, and obtaining the corresponding electric field strength through a numerical difference method. In the field of micro-discharge of high-power microwave components, the calculation method of the electrostatic field can be expanded to research the multiplication effect of the double-sided medium loaded parallel plate waveguide.
Description
Technical Field
The invention provides a calculation method for a double-sided medium loaded parallel plate waveguide electrostatic field, and belongs to the field of high-power microwave component micro discharge.
Background
A dielectric loaded waveguide is a fundamental structural unit of an integrated optical system and its components. It mainly plays roles of limiting, transmitting and coupling light waves. Circular waveguides (optical fibers) and planar waveguides can be classified into two general categories according to the cross-sectional shape. A major consideration in integrated optics is planar waveguides. The simplest planar waveguide consists of a thin film, a substrate, and a planar medium covering three layers, the refractive indices of which are not the same.
The secondary electron multiplication effect, defined as free electron resonance multiplication and even avalanche discharge excited by frequent collision of primary electrons with the device wall in the vacuum device, is a complex physical process, and related research relates to knowledge in multiple disciplines such as solid physics, surface science, vacuum physics, materials science and the like, and is always a hotspot and difficulty problem in the research fields of high-power microwave sources, high-energy particle accelerators and spacecraft loading microwave components. In the field of aerospace vehicle component research, the secondary electron multiplication effect of high power components of satellite systems, also known as microdischarge, has been observed in several high power microwave systems, such as particle accelerators, high power klystrons, and satellite payloads.
Computing the electrostatic field in a dielectric loading waveguide due to any charge distribution on the dielectric layer is a very interesting problem in the aerospace industry due to the lack of rigorous research on the micro-discharge effects that occur in dielectric loading waveguide based microwave devices in satellite airborne equipment. When the double-sided medium loading parallel plate waveguide performs electron multiplication, electrons emitted by the dielectric surface charge positive charges of the dielectric material, and electrons absorbed by the dielectric surface charge negative charges of the dielectric material, so that both dielectric surfaces of the double-sided medium loading parallel plate waveguide have charge accumulation, and the two dielectric surfaces are combined to generate an electrostatic field, and the electrostatic field influences an electron multiplication effect by changing the motion trail of the electrons.
Much work has been done to investigate the electrostatic field that appears across the rf breakdown dielectric window, but not to the electrostatic field that is generated by the arbitrarily distributed charge on the dielectric layer surface in a double-sided dielectric-loaded parallel plate waveguide.
Disclosure of Invention
The invention aims to: because the electrostatic field is regarded as uniform distribution in a calculation space based on the assumption that the charges on the surface of the medium are uniformly accumulated, and the electrostatic field distribution in the micro-discharge process of the double-sided medium loading parallel plate waveguide is not suitable for solving, the calculation method of the electrostatic field of the double-sided medium loading parallel plate waveguide is provided, and the electrostatic field distribution of any charges on the surface of the double-sided medium loading parallel plate waveguide accumulated in a vacuum area is accurately solved by a numerical method.
The invention is realized by the following technical scheme:
an electrostatic field calculation method for accumulating any charges on the surface of a double-sided medium loading parallel plate waveguide in a vacuum area comprises the following seven steps:
(I) Establishing Laplace equation according to electrostatic field Green function
Assuming that space is extended wirelessly in the x, z direction, a unit charge Q of any point r ' = (x ',0, z ') on the parallel plate dielectric layer is established i Is a laplace function of (c);
(II) Fourier inverse transformation of spectral domain Green's function
According to the Dirac functions delta (x-x '), delta (z-z') of the planar Laplace operator, the spectral domain green function is obtained by Fourier transformation of the green function, and the green function is subjected to Fourier inverse transformation along the x axis and the z axis;
(III) reduction of the Laplace equation
Simplifying the Laplacian equation in (I) according to the Fourier inverse transformation of the Dirac function and the spectral domain green function;
(IV) solving the expression of the spectral domain green function in the vacuum region by combining the boundary conditions
Because the relative dielectric constants in the simplified Laplace equation (III) are related to the region, we divide the whole region into three regions, namely a top dielectric layer, a vacuum region and a bottom dielectric layer, and divide the spectral domain Green function into three equations to obtain six variables. Two boundary conditions are derived from the reduced Laplace equation and four other boundary conditions are derived due to the continuity of the potential change. Substituting six boundary conditions into three equations of the spectral domain green's function, so as to solve the expression of the spectral domain green's function in a vacuum area;
(V) solving the Green's function by Filon's integration method
Substituting the spectral domain green function of the vacuum region In (IV) into the expression of the Fourier inverse transformation in (II) can find that the calculation complexity is very high, so that different numerical integration techniques are needed to be used for solving, a Filon integration method is adopted for solving due to the oscillation of the integrated function, and the precision of the integration method is higher than that of a common Simpson method;
(VI) calculating the static electric field of arbitrary Charge distribution
According to the surface charge density rho (x ', z') of the dielectric layer, the green function obtained in the step (V) can be subjected to surface integration, so that potential generated due to any charge distribution on the surface of the dielectric can be obtained, and then the electric field intensity of a vacuum area can be further obtained through numerical difference;
(VII) Convergence analysis of parameters in Green's function
Since the green's function obtained in (V) is computationally complex, the convergence of the individual parameters should be considered.
The invention has the following advantages:
(1) Many works have studied the electrostatic field that appears on the radio frequency breakdown dielectric window, but have not studied the electrostatic field that is produced because of charge deposition in the double-sided dielectric loading parallel plate waveguide, the invention has proposed a calculation method of the double-sided dielectric loading parallel plate waveguide electrostatic field;
(2) Because the electrostatic field is regarded as uniform distribution in a calculation space based on the assumption that the charges on the surface of the medium are uniformly accumulated, the method is not suitable for solving the electrostatic field distribution in the micro discharge process of the double-sided medium loading parallel plate waveguide, and the method accurately solves the electrostatic field distribution of any charges on the surface of the double-sided medium loading parallel plate waveguide accumulated in a vacuum area through a numerical method.
Drawings
FIG. 1 is a schematic diagram of a two-sided dielectric-loaded parallel plate waveguide;
fig. 2 shows the ratio of h=1.96 mm, h=1.2 mm, ε r In the case of =2.25, x-x '=0, z-z' =0, y=h/50 and n=256, k x Or k z Is a convergence of (2);
FIG. 3 is N 1 、N 2 、N 3 、N 4 (N 1 :x-x’＝0,z-z’＝0,y＝H/50；N 2 :x-x’＝0.2mm,z-z’＝0.2mm,y＝H/50；N 3 :x-x’＝0,z-z’＝0,y＝H；N 4 :x-x’＝0.2mm,z-z’＝0.2mm,y＝H)；
Fig. 4 is a comparison of G and Φ in the case where x-x '=0, z-z' =0;
fig. 5 is a comparison of G and Φ of z-z '=0, z-z' =0.2 mm and z-z '=0, z-z' =0.2 mm when y=h/50;
FIG. 6 shows normalized normal electric field strength E of charge distribution in the whole region y /E max Distribution;
FIG. 7 shows normalized normal electric field intensity E of the whole area charge Gaussian distribution y /E max Distribution.
Detailed Description
In order to make the technical scheme of the invention better understood by those skilled in the art, the following description is further provided with a calculation method of the double-sided medium loading parallel plate waveguide electrostatic field by referring to the accompanying drawings of the specification and the specific embodiments.
An electrostatic field calculation method for accumulating any charges on the surface of a double-sided medium loading parallel plate waveguide in a vacuum area comprises the following six steps:
(I) Establishing Laplace equation according to electrostatic field Green function
FIG. 1 is a schematic diagram of a dual-sided dielectric-loaded parallel plate waveguide of interest in which the dielectric layer has a relative permittivity ε r The thickness is H, and the height of the vacuum area is H. The aim is to calculate the potential at the observation point r= (x, y, z), which potential in a rectangular coordinate system will be derived below, given that the space is infinitely extended in the x, z-axis direction, due to the point charge on the dielectric layer at r ' = (x ',0, z ').
Unit charge Q at any point r ' = (x ',0, z ') on parallel plate dielectric layer i The Laplace function of (1) is:
where G is a Green's function, ε 0 Is the dielectric constant in vacuum.
(II) Fourier inverse transformation of spectral domain Green's function
According to dirac functions δ (x-x '), δ (z-z ') of the planar laplace operator, the spectral domain green's function is obtained from the fourier transform of the green's function, and the spectral domain green's function is fourier-inverse transformed along the x-axis and z-axis as shown in formula (2):
wherein in formula (VI)Green's function, k, as spectral domain x And k z The spectral fourier variations in the x-axis and z-axis directions, respectively.
(III) reduction of the Laplace equation
Simplifying the Laplacian equation in the formula (1) according to the Fourier inverse transformation of the Dirac function and the spectral domain Green function:
wherein the method comprises the steps of
(IV) solving the expression of the spectral domain green function in the vacuum region by combining the boundary conditions
The relative dielectric constant in equation (3) is related to the area, so we can define the top dielectric layer, the vacuum area and the bottom dielectric layer as three areas, respectively designated as G A 、G B 、G C A zone.
From equation (3), two other boundary conditions (7), (8) can be derived
According to the continuity of the potential change, four boundary condition formulas (9), (10), (11) and (12) can be obtained, and G of different areas can be solved by substituting different boundary conditions:
substituting six boundary conditions into different areasThus, the spectral domain green function equation (13) of the vacuum region can be derived:
(V) solving the Green's function by Filon's integration method
Substituting formula (13) into formula (2) can find that the calculation complexity is very high, so that different numerical integration techniques are needed to solve, and because of the oscillation of the integrated function, the integration method adopts Filon integration to solve, and the accuracy of the integration method is higher than that of a common Simpson method;
the formula (2) is converted into the form of formula (14), and since the integral form has symmetry about x and y, only the formulas in brackets on the right need to be studied. Filon integration the integration in brackets of equation (14) is rewritten as equation (15) and the range of y values is divided into n segments to solve for the Green function G.
A＝(p 2 +psin2p/2-2sin 2 p)/p 3 (16)
B＝2{p(1+cos 2 p)-sin2p}/p 3 (17)
D＝4(sinp-pcosp)/p 3 (18)
Therein A, B, D, C o 、C e The expression of (C) is shown as the formula (16-20) o And C e The sum of odd cosine term and even cosine term; l=a/n and p=k 2 I, since the green function G converges when the upper integral limit is taken to a certain value, where a is k x Or k z The upper integral limit of infinity can be replaced with a, n being the number of divisions.
(VI) calculating the static electric field of arbitrary Charge distribution
According to the charge density rho (x ', z') of the surface of the dielectric layer, the green function can be subjected to area integral solution, so that the electric potential generated by any charge distribution on the surface of the dielectric layer can be obtained, and the electric field intensity of the vacuum area can be further obtained through numerical difference, as shown in formula (21).
(VII) Convergence analysis of parameters in Green's function
Because of the high computational complexity of equation (2), the convergence of the individual parameters should be considered. Consider k first x Or k z Wherein the dimensions of the double-sided dielectric-loaded parallel plate waveguide are selected as follows: h=1.96 mm, h=1.2mm、ε r =2.25, x-x '=0, z-z' =0, y=h/50 and n=256. As in fig. 2, when a is greater than 1×10 5 When G converges.
The segmentation number n in the Flion integral is then subjected to a convergence analysis, in which case H, H, ε are taken into account r The parameters are unchanged, we discuss by changing the x-x ', z-z', y parameters, and we take four parameter points N as follows 1 、N 2 、N 3 、N 4 (N 1 :x-x’＝0、z-z’＝0、y＝H/50；N 2 :x-x’＝0.2mm、z-z’＝0.2mm、y＝H/50；N 3 :x-x’＝0、z-z’＝0、y＝H；N 4 X-x '=0.2 mm, z-z' =0.2 mm, y=h), G of four parameter points with respect to the division number n is plotted in the figure. As in FIG. 3, when n>At 256, G reaches convergence.
To verify the feasibility of formula (13), two infinitely homogeneous media (. Epsilon.) were used r ) The electric potential generated by the point charges in the vacuum region therebetween is used as a reference, as in the formula (22). H, h, epsilon of double-sided dielectric-loaded parallel-plate waveguide r The parameter size is unchanged. Case 1: x-x '=0, z-z' =0, as shown in fig. 4. Case 2: y=h/50, z-z '=0 or z-z' =0.2 mm, as shown in fig. 5. It can be seen from fig. 4 and 5 that the closer the observation point is to the point charge, the smaller the difference between G and Φ.
(VIII) an electrostatic field simulating the even and Gaussian distribution of the surface charge of the Medium
Let us assume that a charge sheet having a side length of 11.8mm is placed on two dielectric layers, and that the point charges are set to be uniformly distributed and gaussian distributed with a minimum pitch of 10 μm, and that the observation range of each point charge is taken as x-x '=0.4 mm, z-z' =0.4 mm. The normalized normal electric field intensity E of the medium surface charge uniform distribution and Gaussian distribution in the whole x-z-y area is calculated through simulation y /E max As in fig. 6 and 7. It can be seen that the farther the observation point is from the dielectric surface, the smaller the electric field strength, E y Symmetric about the x-axis and z-axis.Meanwhile, the closer the observation point is to the center in FIG. 7, the electric field strength E y The larger.
The invention adopts a Fourier transform method to deduce a spectral domain electrostatic field Green function expression generated by medium surface charge accumulation in a double-sided medium loading parallel plate waveguide; due to the complexity and the oscillation of the integrated function, the Filon numerical integration method is adopted, so that the integration precision is effectively improved; researching a numerical difference method to accurately obtain the distribution of electrostatic fields; carrying out convergence analysis on the green function, and verifying the feasibility of the green function; the electrostatic field obtained by solving can be applied to tracking of the motion state of the electron cloud in the microdischarge simulation.
The present invention has been described in detail with reference to the above embodiments, which are not intended to limit the invention, but rather to the examples, and any modifications, equivalent substitutions, improvements, etc. within the spirit and principle of the present invention should be included in the protection scope of the present invention.
Claims (1)
1. The calculation method of the electrostatic field of the arbitrary charge on the surface of the double-sided medium loading parallel plate waveguide accumulated in the vacuum area is characterized by precisely solving the electrostatic field distribution of the arbitrary charge on the surface of the double-sided medium loading parallel plate waveguide accumulated in the vacuum area by a numerical method, and at least comprises the following steps:
(I) Establishing Laplace equation according to electrostatic field Green function
Assuming that space is extended wirelessly in the x, z direction, a unit charge Q of any point r ' = (x ',0, z ') on the parallel plate dielectric layer is established i Is a laplace function of (c):where G is a Green's function, ε 0 Is the dielectric constant in vacuum;
(II) Fourier inverse transformation of spectral domain Green's function
The spectral domain green function is derived from the Fourier transform of the green function based on the Dirac functions delta (x-x '), delta (z-z') of the planar Laplace operator, and the spectral domain green function is shifted along the x-axis and the z-axisPerforming an inverse Fourier transform, e.g. Wherein->Green's function, k, as spectral domain x And k z Spectral fourier variables in the x-axis and z-axis directions, respectively;
(III) reduction of the Laplace equation
Simplifying the Laplacian equation in (I) according to the Fourier inverse transformation of the Dirac function and the spectral domain green function;
(IV) solving the expression of the spectral domain green function in the vacuum region by combining the boundary conditions
Because the relative dielectric constants in the simplified Laplace equation are related to the area, the whole area is divided into three areas, namely a top dielectric layer, a vacuum area and a bottom dielectric layer, the spectral domain green function is also divided into three equations to obtain six variables, two boundary conditions are obtained by the simplified Laplace equation, four other boundary conditions are obtained due to the continuity of potential change, and the six boundary conditions are substituted into the three equations of the spectral domain green function, so that the expression of the spectral domain green function in the vacuum area is solved;
(V) solving the Green's function by Filon's integration method
Solving the formula obtained by the Fourier inverse transform in (II)Conversion to +.>The Filon integration method is used to integrate the formula +.> The integral in the right bracket is rewritten as +.> And dividing the range of y values into n segments to solve the green function G, where a= (p 2 +p sin2p/2-2sin 2 p)/p 3 、B＝2{p(1+cos 2 p)-sin2p}/p 3 、D＝4(sin p-p cos p)/p 3 、 C o And C e Is the sum of the odd cosine term and the even cosine term, l=a/n and p=k, respectively 2 /l，k x And k z Spectral fourier variables in x-axis and z-axis directions, respectively, a being k x Or k z N is the number of divisions;
(VI) calculating the static electric field of arbitrary Charge distribution
According to the surface charge density rho (x ', z') of the dielectric layer, carrying out surface integration on the green function obtained in the step (V), thus obtaining the electric potential generated by any charge distribution on the dielectric surface, and further obtaining the electric field intensity of the vacuum area through numerical value difference;
(VII) Convergence analysis of parameters in Green's function
Since the green's function obtained in (V) is computationally complex, the convergence of the individual parameters should be considered.
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| CN112505437A (en) * | 2020-11-25 | 2021-03-16 | 湘潭大学 | Method for researching micro-discharge of double-sided dielectric loaded microwave component |
| CN112800710A (en) * | 2021-02-19 | 2021-05-14 | 清华大学 | Multi-medium Green function pre-drawing method for extracting interconnection capacitance of integrated circuit |
| CN113705064A (en) * | 2021-07-27 | 2021-11-26 | 西安交通大学 | Electromagnetic-electrostatic hybrid simulation method and system for micro-discharge of microwave component |
| WO2022123500A1 (en) * | 2020-12-09 | 2022-06-16 | University Of Manitoba | Spectral differential equation approximation method for mixed potential green's functions in multilayered media |
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| CN112505437A (en) * | 2020-11-25 | 2021-03-16 | 湘潭大学 | Method for researching micro-discharge of double-sided dielectric loaded microwave component |
| WO2022123500A1 (en) * | 2020-12-09 | 2022-06-16 | University Of Manitoba | Spectral differential equation approximation method for mixed potential green's functions in multilayered media |
| CN112800710A (en) * | 2021-02-19 | 2021-05-14 | 清华大学 | Multi-medium Green function pre-drawing method for extracting interconnection capacitance of integrated circuit |
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