CN101546347A - Parameter determination method of rectangular waveguide lowpass - Google Patents

Abstract

The invention discloses a method for determining parameters of a rectangular waveguide low-pass filter, which belongs to the technical field of electrical components. It mainly solves the design accuracy problem of rectangular waveguide low-pass filter. The process is as follows: firstly, the rectangular waveguide model is selected according to the electrical performance index of the rectangular waveguide low-pass filter, and the length of the rectangular waveguide and the structural parameters of the capacitor diaphragm are preliminarily determined; The hybrid genetic algorithm is used to solve the S' matrix of the rectangular waveguide capacitive diaphragm; again, the genetic algorithm is used to optimize, so that the S' matrix is approximately equal to the S matrix obtained by the generalized impedance transformation coefficient K, and finally the structural parameters of the diaphragm are determined; finally, the use of The S' matrix of the capacitive diaphragm corrects the waveguide length between the two diaphragms, and finally determines the cavity length of the rectangular waveguide. The invention has the advantages of accurate parameter determination and no need for repeated debugging, and can be used for the design of high-precision rectangular waveguide low-pass filters.

Description

Parameter Determination Method of Rectangular Waveguide Low-pass Filter

technical field

The invention belongs to the technical field of electrical components, in particular to a method for determining parameters of a rectangular waveguide low-pass filter, which is used to guide the design of a high-performance rectangular waveguide low-pass filter.

Background technique

In the design of rectangular waveguide low-pass filter, the low-pass prototype circuit is first obtained from the electrical performance index of the filter, and then the generalized impedance converter K is obtained by circuit transformation, and finally the impedance converter and branch components of the circuit are realized by waveguide microwave structure, including Coupling capacitor diaphragm design and transmission section waveguide design. For a rectangular waveguide low-pass filter that uses a capacitive diaphragm to achieve coupling, it is a very critical step to design the parameters of the capacitive diaphragm according to the generalized impedance transformation coefficient K.

Figure 1 is a schematic diagram of the structure of a rectangular waveguide low-pass filter, in which the capacitive diaphragm implements a parallel capacitance, and the waveguide section implements a series inductance.

Figure 2 shows the capacitive diaphragm structure and its equivalent circuit in a rectangular waveguide. The TE ₁₀ -mode electromagnetic wave transmitted in the rectangular waveguide excites higher-order modes at the discontinuous diaphragm. These higher-order modes are cut off in the waveguide and will be quickly attenuated not far from the diaphragm. . Because the TE ₁₀ -mode electric field has only y component, no x component, and the discontinuity is only in the y direction, and the x direction is continuous, so the electric force lines are distributed in the form of fringe electric fields on both sides of the diaphragm. In this way, the net electric energy is stored near the diaphragm, and the diaphragm is extremely thin and has no loss. Therefore, the discontinuous diaphragm can be equivalent to a lumped capacitance element, represented by the accommodation jB.

For the existing filter, the method of measuring the reflection coefficient is commonly used in engineering to determine the corresponding susceptance value. Connect the matched load at the end of the waveguide, then the reflection coefficient caused by the susceptance jB is

$\mathrm{<span\; class="notranslate">\; \&Gamma;</span>}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; Y</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; Y</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; jB</span>}<span\; class="notranslate">\; )</span>}{{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="Y"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">Y</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; Y</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; jB</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; jB</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="Y"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">Y</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; jB</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; j</span>}\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; B</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; j</span>}\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; B</span>}}}$

In the formula $\stackrel{\&OverBar;}{B}=\frac{\mathrm{<figure-callout\; id="0"\; label="B\; Y"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">B</figure-callout>}}{Y_{}}$ It is the normalized susceptance. Solve the above formula to get $j\stackrel{\&OverBar;}{B}=-\frac{2\mathrm{\&Gamma;}}{1+\mathrm{\&Gamma;}}.$

Wang Xinwen and Li Ping described this in detail in Chapter 2, Section 7 of Microwave Technology and Antennas. However, this method cannot give the relationship between susceptance and diaphragm size, and cannot be directly applied to filter design.

At present, when designing waveguide filters, there are mainly the following methods for calculating the susceptance of capacitive diaphragms:

1 Quasi-static field method

The quasi-static field method is a commonly used method to solve the waveguide discontinuity. The characteristic of the electrostatic field is that the working wavelength λ→∞, for the waveguide, although the working wavelength is not infinite. However, for the evanescent wave in the waveguide, there is λ→λ _c , and the higher the order of the evanescent wave, the smaller the cut-off wavelength λ _c , and there is λ》λ _c , so, for λ _c , it is like The working wavelength becomes longer, so the field of evanescent wave can be approximated by electrostatic field. Conformal transformation is a method to obtain quasi-static field solution directly from the static field problem, and a variety of functions can be used for transformation. Zhang Jun analyzed this in detail in Section 2 of Chapter 4 of the book "Discontinuity of Guided Waves". For the parallel-plate waveguide with a symmetrical capacitive diaphragm in Fig. 1(a), after the function $w=u+\mathrm{jv}=\arcsin \left[\csc \left(\frac{\mathrm{\&pi;d}}{2b}\right)\sin \frac{\mathrm{\&pi;}}{b}(\mathrm{the\; y}+\mathrm{jz})\right]$ Transform into a simple parallel plate waveguide, find the equivalent capacitance C, and then use B=ωC to find the parallel susceptance B of the equivalent network, and its normalized susceptance value is

$\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; B</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="B\; Y"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">B</figure-callout></span><figure-callout\; id="0"\; label="B\; Y"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">\; </figure-callout>}}{{\mathrm{<span\; class="notranslate">\; </span>}}_{}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; b</span>}}{{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; g</span>}}}\mathrm{<span\; class="notranslate">\; ln</span>}\mathrm{<span\; class="notranslate">\; csc</span>}<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;d</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; )</span>$

Where λ _g is the wavelength of the waveguide, b is the height of the waveguide, and d is the distance between the two diaphragms.

2 Variations

Variational method is a method to deal with the extremum value of the functional, and what is finally sought is the extremum function, so that the functional can obtain a maximum or minimum value. Its key theorem is the Euler-Lagrange equation, which corresponds to the critical point of the functional. In practical engineering problems, if an actual problem to be asked can be expressed as $J\left[\mathrm{the\; y}\right]={\&Integral;}_{x_0}^{x_1}f(x,\mathrm{the\; y},\mathrm{the\; y}\&prime;)\mathrm{dx},(\mathrm{the\; y}=\mathrm{the\; y}\left(x\right))$ The functional form of , then the required real value is the extremum value of the functional, which is also called the stable value. When solving the problem of waveguide discontinuity, the parameters of the equivalent network are often expressed as the integral formula of the unknown tangential electric field or current at the discontinuity, and the actual electric field or current is to make the integral formula take the extreme value and satisfy the The boundary conditions at the continuum, the parameters of the equivalent network can be determined by the variational method. Sangster AJ et al. published "Variational method for the analysis of waveguide coupling" in "Progress In Electromagnetics Research IEE", Volume 112, 1965. The waveguide coupling is analyzed by the variational method, and the first approximation for the normalized susceptance of the symmetric capacitive diaphragm is:

$\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; B</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="B\; Y"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">B</figure-callout></span><figure-callout\; id="0"\; label="B\; Y"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">\; </figure-callout>}}{{\mathrm{<span\; class="notranslate">\; </span>}}_{}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; b</span>}}{{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; g</span>}}}\mathrm{<span\; class="notranslate">\; ln</span>}\mathrm{<span\; class="notranslate">\; csc</span>}<span\; class="notranslate">\; (</span>\frac{\mathrm{<span\; class="notranslate">\; \&pi;d</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; )</span>$

3 pattern matching method

The mode matching method considers the high-order mode at the discontinuity, and treats the thick diaphragm part as a small waveguide, and the waveguide with a symmetrical capacitive diaphragm can be regarded as two waveguide double-sided steps and a section of length t A cascade of small waveguides, where t is the thickness of the diaphragm. In the analysis, the field in the research area is regarded as the superposition of infinite modes, and the scattering matrix on the interface can be obtained by using the boundary condition of the electric field and magnetic field continuous on the interface between the two areas and applying the orthogonality of the mode function . Mode matching is mainly used to analyze microwave devices with regular geometries. R.Safavi-Naini and R.H.Macphie. published in "IEEE Transactions on MTT", Volume 82, Issue 11, 1982, "Scattering atmospheric-to-rectangular waveguide junctions" using pattern matching to analyze waveguide discontinuity.

The current method has the following deficiencies:

1) The conformal transformation in the quasi-static method uses function transformation to solve the equivalent susceptance B, and does not take into account factors such as the thickness of the diaphragm and the inclination angle of processing. If the thickness and inclination factors are added, it is difficult to directly transform The function implements the transformation from a rectangular waveguide with a symmetrical capacitive diaphragm to a simple parallel-plate waveguide.

2) Variational method does not take factors such as diaphragm thickness and processing inclination angle into consideration.

3) Mode matching is mainly to analyze microwave devices with regular geometric structures, and there is no accurate analysis for irregular deformations such as inclination angles.

Contents of the invention

The object of the present invention is to avoid the disadvantages of the above existing methods, and provide a method for determining the parameters of the rectangular waveguide low-pass filter, so as to improve the design accuracy of the rectangular waveguide low-pass filter.

In order to achieve the above object, the parameter determination method of the rectangular waveguide low-pass filter of the present invention comprises the following steps: parameter pre-determining step: select the rectangular waveguide model according to the frequency requirement of the rectangular waveguide low-pass filter; Determine the length θ ₀ of the rectangular waveguide between the two capacitor diaphragms; according to the electrical performance index and circuit transformation of the rectangular waveguide low-pass filter, a low-pass prototype circuit containing only capacitive elements is obtained, and then the generalized impedance transformation coefficient K is obtained, Determine the capacitive reactance X of the parallel capacitor, and preliminarily determine the window width d, diaphragm thickness m, and processing inclination α of the capacitive diaphragm in the rectangular waveguide by checking the equivalent susceptance diagram loaded by the rectangular waveguide capacitance;

Diaphragm parameter optimization steps: According to the generalized impedance transformation coefficient K, determine the S matrix of the rectangular waveguide single-section capacitive diaphragm; for the initial determination of the window width d, diaphragm thickness m and processing inclination angle α of the rectangular waveguide capacitive diaphragm and waveguide The diaphragm cross-section polygon formed by the height b is subjected to Schwartz-Christopher inverse transformation, and the normalized equivalent susceptance B of the capacitive diaphragm is solved; the S′ matrix of the capacitive diaphragm is determined according to the equivalent susceptance B: ${\mathrm{<figure-callout\; id="11"\; label="S"\; filenames="A200810150928E00171.png,A200810150928E00172.png"\; state="\{\{state\}\}">S</figure-callout>}}_{11}^{\&prime;}=\frac{-j\stackrel{\&OverBar;}{B}}{2+j\stackrel{\&OverBar;}{B}},$ ${\mathrm{<figure-callout\; id="12"\; label="S"\; filenames="A200810150928E00172.png"\; state="\{\{state\}\}">S</figure-callout>}}_{12}^{\&prime;}=\frac{2}{2+j\stackrel{\&OverBar;}{B}};$ The genetic algorithm is used to optimize so that the S' matrix is approximately equal to the S matrix, and finally determine the window width d', diaphragm thickness m' and processing inclination α' of the single capacitor diaphragm;

Repeat the diaphragm parameter optimization step to determine the remaining capacitor diaphragm parameters of the rectangular waveguide low-pass filter;

The step of correcting the length of the waveguide cavity: through the S′ matrix of the capacitive diaphragm, the corrected waveguide length between the two diaphragms is:

${\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="g"\; filenames="A200810150928E00161.png,A200810150928E00171.png"\; state="\{\{state\}\}">g</figure-callout></span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; arctan</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; arctan</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 12</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>$

λ _g is the wavelength of the waveguide, θ ₀ is the length of the rectangular waveguide between the two capacitive diaphragms initially determined,

X is the normalized shunt reactance of the equivalent circuit of the capacitive diaphragm, namely $j\stackrel{\&OverBar;}{x}=\frac{2{\mathrm{<figure-callout\; id="12"\; label="S"\; filenames="A200810150928E00172.png"\; state="\{\{state\}\}">S</figure-callout>}}_{12}\&prime;}{{(1-{\mathrm{<figure-callout\; id="11"\; label="S"\; filenames="A200810150928E00171.png,A200810150928E00172.png"\; state="\{\{state\}\}">S</figure-callout>}}_{11}^{\&prime;})}^2-{\mathrm{<figure-callout\; id="12"\; label="S"\; filenames="A200810150928E00172.png"\; state="\{\{state\}\}">S</figure-callout>}}_{12}^{\&prime;2}};$

Repeat the step of modifying the length of the waveguide cavity to determine the length of the waveguide between the remaining two capacitive diaphragms of the rectangular waveguide low-pass filter, and complete the design of the entire rectangular waveguide low-pass filter.

Compared with the prior art, the present invention has the following advantages:

1) The present invention considers the dimensional tolerance and machining inclination in the process of diaphragm processing, and uses polygonal approximation to approximate the actual diaphragm contour as a polygon, which not only retains important information, but also improves the analysis speed.

2) When solving the conformal transformation of polygons, the improved genetic algorithm is used, which not only ensures the optimal result, but also improves the calculation speed.

3) The present invention considers the thickness and inclination factors of the rectangular waveguide capacitive diaphragm due to processing, and performs Schwartz-Christopher inverse transformation to solve the equivalent susceptance and S' matrix through the rectangular waveguide capacitive diaphragm, and optimizes it by genetic algorithm. The S' matrix is approximately equal to the S matrix obtained through the generalized impedance transformation coefficient K, and its structural parameters can be accurately determined.

4) The present invention can accurately analyze the influence of the diaphragm tolerance occurring in the processing process on the electrical performance.

The simulation test proves that the method of the invention can improve the accuracy of parameter design of the capacitive diaphragm of the rectangular waveguide low-pass filter and improve the performance of the whole filter.

Description of drawings

Fig. 1 is the structural representation of existing rectangular waveguide low-pass filter;

Fig. 2 is a schematic diagram of the existing waveguide capacitive diaphragm structure and equivalent circuit;

Fig. 3 is the equivalent circuit diagram of existing rectangular waveguide low-pass filter;

Fig. 4 is a schematic diagram of the process of Schwartz-Christopher inverse transformation in the present invention;

Fig. 5 is a flow chart of design steps of the present invention;

Fig. 6 is a flowchart of steps for optimizing and solving the inverse Schwartz-Christopher transformation in the present invention;

Fig. 7 is the comparison figure of the present invention to the calculation result of the single-section capacitor diaphragm and the simulation result of the single-section capacitor diaphragm;

Fig. 8 is a HFSS simulation characteristic diagram of a 2GHz rectangular waveguide low-pass filter designed by the method of the present invention.

Detailed ways:

With reference to Fig. 5, concrete steps of the present invention are as follows:

Step 1, input the electrical performance index of the rectangular waveguide low-pass filter.

The main electrical performance indicators of the rectangular waveguide low-pass filter include: cut-off frequency ω, maximum attenuation in the passband, minimum attenuation in the stopband and relative bandwidth BW, and these electrical performance indicators are input into the calculation program of the computer.

Step 2, the structural parameters of the rectangular waveguide low-pass filter are pre-determined.

Step 2.1, select the rectangular waveguide model according to the frequency requirements of the rectangular waveguide low-pass filter;

In step 2.2, according to the electrical performance index of the rectangular waveguide low-pass filter, the low-pass prototype circuit is designed by the calculation program, and the prototype circuit parameters are obtained: g ₀ ... g _n+1 , n is the circuit order;

Step 2.3, transform the low-pass prototype circuit into a low-pass prototype circuit containing only capacitive elements, through the formula $K_{01}={\mathrm{<figure-callout\; id="0"\; label="Z"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">Z</figure-callout>}}_0\sqrt{\frac{1}{\mathrm{\&omega;}{g}_0{\mathrm{<figure-callout\; id="1"\; label="g"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">g</figure-callout>}}_1}},$ $K_{i,i+1}=\frac{{\mathrm{<figure-callout\; id="0"\; label="Z"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">Z</figure-callout>}}_0}{\mathrm{\&omega;}}\sqrt{\frac{1}{g_i{g}_{i+1}}},$ $K_{\mathrm{no},\mathrm{no}+1}={\mathrm{<figure-callout\; id="0"\; label="Z"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">Z</figure-callout>}}_0\sqrt{\frac{1}{\mathrm{\&omega;}{g}_{\mathrm{no}}{g}_{\mathrm{no}+1}}},$ The generalized impedance transformation coefficient K is calculated, and Z ₀ is the waveguide characteristic impedance; according to the relative bandwidth BW of the rectangular waveguide low-pass filter, the formula $\mathrm{BW}=\frac{4{\mathrm{\&theta;}}_0}{\mathrm{\&pi;}},$ Calculate the length θ ₀ of the rectangular waveguide between the two capacitor diaphragms, as shown in Figure 3;

Step 2.4, according to the generalized impedance transformation coefficient K, through the formula $\stackrel{\&OverBar;}{x}=\frac{\stackrel{\&OverBar;}{K}}{1-{\stackrel{\&OverBar;}{K}}^2},$ K is the normalized K, and the capacitive reactance X of the normalized parallel capacitor is determined. By checking the equivalent susceptance diagram loaded by the rectangular waveguide capacitance, the window width d, diaphragm thickness m and Processing inclination α.

Step 3, according to the generalized impedance transformation coefficient K, determine the A matrix and S matrix of the single capacitor diaphragm.

According to the characteristics of the two-port microwave network, the A matrix and the S matrix of the single capacitor diaphragm are determined as:

$\mathrm{<span\; class="notranslate">\; A</span>}<span\; class="notranslate">\; =</span>\left[\begin{array}{cc}\mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; j</span>}\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; K</span>}}\\ \mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; /</span>\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; K</span>}}& \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right]<span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; =</span>\left[\begin{array}{cc}\frac{{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; K</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; K</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}& \mathrm{<span\; class="notranslate">\; j</span>}\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; K</span>}}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; K</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}\\ \mathrm{<span\; class="notranslate">\; j</span>}\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 2</span>}\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; K</span>}}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; K</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}& \frac{{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; K</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; K</span>}}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}\end{array}\right]<span\; class="notranslate">\; ,</span>$

In the formula $\stackrel{\&OverBar;}{K}=\frac{\mathrm{<figure-callout\; id="0"\; label="K\; Z"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">K</figure-callout>}}{Z_{}},$ ${\mathrm{<figure-callout\; id="11"\; label="S"\; filenames="A200810150928E00171.png,A200810150928E00172.png"\; state="\{\{state\}\}">S</figure-callout>}}_{11}=\frac{{\stackrel{\&OverBar;}{K}}^2-1}{1+{\stackrel{\&OverBar;}{K}}^2},$ ${\mathrm{<figure-callout\; id="12"\; label="S"\; filenames="A200810150928E00172.png"\; state="\{\{state\}\}">S</figure-callout>}}_{12}=j\frac{-2\stackrel{\&OverBar;}{K}}{1+{\stackrel{\&OverBar;}{K}}^2},$ j is the imaginary part.

Step 4: Carry out inverse Schwartz-Christopher transformation on the polygonal shape of the capacitor diaphragm section.

Referring to Fig. 4, the inverse Schwartz-Christopher transformation is performed on the diaphragm section polygon formed by initially determining the window width d, diaphragm thickness m, processing inclination angle α, and waveguide height b of the rectangular waveguide capacitive diaphragm as follows:

In step 4.1, the inverse Schwartz-Christopher transformation is used to transform the polygon in the complex plane z to the upper half plane of the complex plane, and transform the boundary of the polygon to the real axis, that is, the polygon plane ABCDZ ₀ in Fig. 4(a) Z ₂ Z ₃ Z ₄ is transformed into the upper half-complex plane A′B′C′D′T ₀ T ₂ T ₃ T ₄ in Fig. 4(b).

From the symmetry of polygons and the properties of conformal transformation, the coordinates of point T in the t-plane are symmetrical about the y-axis. When the ratio of the lengths of adjacent sides on the T plane is too large or too small, it may cause the strong peak phenomenon of the integrand and make the integrand partially non-integrable. Correctly inserting some virtual vertices can better solve this problem, and at the same time It can improve the accuracy and accelerate the speed of numerical integration and iterative convergence, such as adding virtual vertices Z ₁ and Z ₅ in Fig. 4(a). According to the polygonal conformal transformation rule, select T ₀ = 0, T ₁ = 1, T _n-1 = ∞, and obtain the transformation formula as:

$\mathrm{<span\; class="notranslate">\; Z</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="C"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">C</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{{<span\; class="notranslate">\; \&Integral;</span>}_{{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; T</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&theta;</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; T</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; T</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; T</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; /</span>\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; T</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; 3</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; T</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; 4</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&theta;</span>}}\mathrm{<span\; class="notranslate">\; dt</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="C"\; filenames="A200810150928E00161.png,A200810150928E00171.png"\; state="\{\{state\}\}">C</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ;</span>$

In step 4.2, the optimization model for calculating the value of T _j in the transformation formula Z is established as follows:

find T ₂ ，T ₃ ，T ₄ ，…，T _n-2

$\mathrm{<span\; class="notranslate">\; min</span>}\mathrm{<span\; class="notranslate">\; AIM</span>}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{<span\; class="notranslate">\; (</span>\frac{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{{\mathrm{<span\; class="notranslate">\; Y</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}}{{\mathrm{<span\; class="notranslate">\; Y</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}$

$\mathrm{<span\; class="notranslate">\; st</span>}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; Z</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="1"\; label="C"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">C</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{<span\; class="notranslate">\; \&Integral;</span>}_{{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}^{\mathrm{<span\; class="notranslate">\; T</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Pi;</span>}}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; T</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}^{<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}\mathrm{<span\; class="notranslate">\; dT</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="2"\; label="C"\; filenames="A200810150928E00161.png,A200810150928E00171.png"\; state="\{\{state\}\}">C</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}$

in:

$Y_j=\frac{{\mathrm{<figure-callout\; id="0"\; label="ZL\; j\; ZL"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">ZL</figure-callout>}}_j}{{\mathrm{ZL}}_{}}=\frac{|Z_{j+1}-Z_j|}{|Z_1-{\mathrm{<figure-callout\; id="0"\; label="Z"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">Z</figure-callout>}}_0|}=\frac{\underset{T_j}{\overset{T_{j+1}}{\&Integral;}}\left(\underset{i=0}{\overset{\mathrm{no}-1}{\mathrm{\&Pi;}}}{|T-T_i|}^{-v_i}\right)\mathrm{dT}}{\underset{T_0}{\overset{T_{j+1}}{\&Integral;}}\left(\underset{i=0}{\overset{\mathrm{no}=1}{\mathrm{\&Pi;}}}{|T-T_i|}^{-v_i}\right)\mathrm{dT}},$ where j = 0, 1, 2, ... n-3

${\stackrel{\&OverBar;}{Y}}_j=\frac{\underset{T_j}{\overset{T_{j+1}}{\&Integral;}}\left(\underset{i=0}{\overset{\mathrm{no}-1}{\mathrm{\&Pi;}}}{|T-\underset{j=0}{\overset{i-1}{\mathrm{\&Sigma;}}}\tilde{T}{L}_j|}^{-v_i}\right)\mathrm{dT}}{\underset{T_0}{\overset{T_{j+1}}{\&Integral;}}\left(\underset{i=0}{\overset{\mathrm{no}-1}{\mathrm{\&Pi;}}}{|T-\underset{j=0}{\overset{i-1}{\mathrm{\&Sigma;}}}\tilde{T}{L}_j|}^{-v_i}\right)\mathrm{dT}},$ where j = 0, 1, 2, ... n-3

In the formula, Z _i is the coordinates of each vertex of the polygon in the section of the capacitor diaphragm in the z plane,

T _i is the coordinates of each vertex of the capacitor diaphragm cross-section polygon in the t plane,

|Z _j+1 -Z _j | is the side length from Z _j to Z _j+1 on the Z plane, represented by ZL _j ,

|T _j+i -T _j | is the corresponding side length on the T plane, represented by TL _j ,

Y _j is the relative side length of the jth side represented by the vertex,

Y _j is the relative side length of the jth side represented by the side length;

In step 4.3, optimize the objective function AIM in the model so that it is less than the given accuracy requirement, so as to obtain the accurate values of all T _j , the optimization process is shown in Figure 6:

First, determine the unknown parameter set T _i , (i=2, 3,...n-2), C ₁ , C ₂ , fitness function AIM and constraint conditions, so that the optimization search is always in the feasible solution space run;

Secondly, according to the random selection rule of the genetic algorithm, the initial population is generated in the feasible solution area, and it is judged whether the fitness function satisfies the termination condition. , get the optimal solution as the initial point;

Finally, the optimal solution of the model is obtained by using the sequential quadratic programming method, namely T ₂ , T ₃ , T ₄ ,..., T _n-2 .

Step 5, solving the equivalent susceptance value and S' matrix.

In step 5.1, use the formula ω=μ+jv=arcsin(μ′+jv′) to transform the upper half complex plane of the t plane into a plate capacitor of the ω plane, that is, the upper half complex plane A′B in Fig. 4(b) 'C'D'T ₀ T ₁ T ₂ T ₃ T ₄ T ₅ is converted to the plate capacitor A"B"C"D"E"F"G"H"I" in Figure 4(c);

In step 5.2, substitute TA coordinates into the above formula to obtain ω _A =u _A +jv _A ;

Step 5.3, calculate the total capacitance C of the capacitor according to the width and length of the capacitor,

It can be seen from Figure 4(c) that the capacitance per unit width of the parallel plate between AD and AF $C_{\mathrm{AD}}=\mathrm{\&epsiv;}\frac{v_A}{\mathrm{\&pi;}},$

Since C _AD includes the parallel plate capacitance C _AB of section AB, set the coordinates of point A as y=—b/2, z=L/2, then

${\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; AB</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}\frac{\mathrm{<span\; class="notranslate">\; L</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; b</span>}}$

Therefore, since the capacitance introduced by the addition of the diaphragm should be C _AD -C _AB , the total capacitance C introduced by the diaphragm should be

$\mathrm{<span\; class="notranslate">\; C</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; AD</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; AB</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}<span\; class="notranslate">\; (</span>\frac{{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}}{\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; L</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ;</span>$

Step 5.4, calculate the equivalent susceptance according to the total current C $B=\mathrm{\&omega;\; C}=2\mathrm{\&omega;\&epsiv;}(\frac{v_A}{\mathrm{\&pi;}}-\frac{L}{2b})=\frac{4}{\mathrm{\&lambda;}}{Y}_0(v_A-\frac{\mathrm{L\&pi;}}{2b})$

Then use the equivalent characteristic admittance Y ₀ /b to normalize

$\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; B</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="0"\; label="B\; Y"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">B</figure-callout></span><figure-callout\; id="0"\; label="B\; Y"\; filenames="A200810150928E00133.png,A200810150928E00161.png"\; state="\{\{state\}\}">\; </figure-callout>}}{{\mathrm{<span\; class="notranslate">\; </span>}}_{}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 4</span>}\mathrm{<span\; class="notranslate">\; b</span>}}{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; A</span>}}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; L\&pi;</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; b</span>}}<span\; class="notranslate">\; )</span>$

In step 5.5, according to the normalized equivalent susceptance, the S′ matrix parameter of the single capacitor diaphragm is obtained as:

${\mathrm{<span\; class="notranslate">\; <figure-callout\; id="11"\; label="S"\; filenames="A200810150928E00171.png,A200810150928E00172.png"\; state="\{\{state\}\}">S</figure-callout></span>}}_{\mathrm{<span\; class="notranslate">\; 11</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\frac{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; j</span>}\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; B</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; j</span>}\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; B</span>}}}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 12</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; j</span>}\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; B</span>}}}<span\; class="notranslate">\; .</span>$

Step 6, optimizing and solving the structural parameters of the rectangular waveguide capacitive diaphragm.

In step 6.1, the optimization model is established as follows:

find d, m, α

min ΔSi ₁₁ , ΔSi ₁₂

in, $S_{i11}=\frac{{{\stackrel{\&OverBar;}{K}}_i}^2-1}{1+{{\stackrel{\&OverBar;}{K}}_i}^2},$ $S_{i12}=j\frac{-2{\stackrel{\&OverBar;}{K}}_i}{1+{{\stackrel{\&OverBar;}{K}}_i}^2},$ $S_{i11}^{\&prime;}=\frac{-j{\stackrel{\&OverBar;}{B}}_i}{2+j{\stackrel{\&OverBar;}{B}}_i},$ $S_{i12}^{\&prime;}=\frac{2}{2+j{\stackrel{\&OverBar;}{B}}_i},$

$\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 11</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 11</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 11</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; ,</span>$ $\mathrm{<span\; class="notranslate">\; \&Delta;</span>}{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 12</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 12</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 12</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>$

In the formula, d _i , _mi , α _i are the window width, film thickness and processing inclination angle of the i-th capacitive diaphragm respectively, _Bi is the normalized equivalent susceptance, ΔS _i11 , ΔS _i12 are the reflection coefficients S _i11 and transmission coefficient S _i12 error;

In step 6.2, the genetic algorithm is used to optimize so that the S′ matrix is approximately equal to the S matrix, and finally determine the window width d′, diaphragm thickness m′ and processing inclination α′ of the single capacitor diaphragm.

Step 7, repeating the Schwartz-Christopher inverse transformation of the cross-sectional polygon of the rectangular waveguide diaphragm to solve the equivalent susceptance and S' matrix steps and the steps of optimizing the structural parameters of the rectangular waveguide capacitive diaphragm to determine the rectangular waveguide low-pass filter All capacitor diaphragm parameters.

Step 8, correcting the length of the waveguide cavity.

In step 8.1, according to the S′ parameter of the capacitor diaphragm, the parallel reactance of the equivalent circuit is obtained:

$\mathrm{<span\; class="notranslate">\; j</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 2</span>}{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 12</span>}}^{<span\; class="notranslate">\; \&prime;</span>}}{{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; <figure-callout\; id="11"\; label="S\; i"\; filenames="A200810150928E00171.png,A200810150928E00172.png"\; state="\{\{state\}\}">S</figure-callout></span><figure-callout\; id="11"\; label="S\; i"\; filenames="A200810150928E00171.png,A200810150928E00172.png"\; state="\{\{state\}\}">\; </figure-callout>}}_{\mathrm{<span\; class="notranslate">\; </span>}}^{\mathrm{<span\; class="notranslate">i</span>}\mathrm{<span\; class="notranslate">\; 11</span>}}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 12</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}<span\; class="notranslate">\; ;</span>$

In step 8.2, the phase angle of the K converter is obtained according to the parallel reactance: φ _i =-arctan(2X _i );

In step 8.3, correct the electrical length as: ${\mathrm{\&theta;}}_k={\mathrm{\&theta;}}_0+{\mathrm{\&phi;}}_{i-1,i}+{\mathrm{\&phi;}}_{i,i+1}={\mathrm{\&theta;}}_0-\frac{1}{2}(\arctan \left(2{\stackrel{\&OverBar;}{x}}_{i-1,i}\right)+\arctan \left(2{\stackrel{\&OverBar;}{x}}_{i,i+1}\right));$

Step 8.4, modify the length of the waveguide cavity as: $L_i=\frac{{\mathrm{\&lambda;}}_{\mathrm{<figure-callout\; id="2"\; label="g"\; filenames="A200810150928E00161.png,A200810150928E00171.png"\; state="\{\{state\}\}">g</figure-callout>}}}{2\mathrm{\&pi;}}({\mathrm{\&theta;}}_0+{\mathrm{\&phi;}}_{i-1,i}+{\mathrm{\&phi;}}_{i,i+1}).$

Step 9, repeating the waveguide cavity length correction step to determine the waveguide cavity lengths between all two capacitive diaphragms of the rectangular waveguide low-pass filter, and complete the design of the entire rectangular waveguide low-pass filter.

The effectiveness of the present invention can be illustrated by simulation data:

1. Select the single-section rectangular waveguide capacitor diaphragm whose main mode frequency range is 1.4-2.8GHz, the thickness is 1mm, and the processing inclination angle is 1°, carry out the simulation test, and obtain the S matrix parameters of the inventive method and the S matrix of the traditional method respectively Parameters and S matrix parameters obtained through Ansoft HFSS simulation, as shown in Figure 7.

2. According to the electrical performance indicators: the cutoff frequency is 2.0GHz; the maximum attenuation in the passband is 0.2dB, and the attenuation is greater than 60dB at the stopband 3.5GHz, design a 2.0GHz waveguide low-pass filter, and use HFSS for simulation verification to get The amplitude-frequency characteristic curve shown in Figure 8.

In Fig. 7, the S matrix parameter of traditional method is compared with the S matrix parameter obtained by Ansoft HFSS simulation, S ₁₁ , S ₁₂ average errors are respectively -18.29% and 3.04%; Compared with the S matrix parameters obtained by Ansoft HFSS simulation, the average errors of S ₁₁ and S ₁₂ are 2.12% and -0.1% respectively. It can be seen that the coincidence of the transmission coefficient and the reflection coefficient obtained by the method of the present invention is more accurate than the traditional method, and the design accuracy is improved.

As can be seen from Fig. 8, the cut-off frequency of the filter designed by the method of the present invention is 2.0GHz, the reflection coefficient _S11 in the passband is lower than -20dB, and the attenuation in the stopband is greater than 60dB, which meets the design performance requirements. There is no need for repeated debugging, which improves the design efficiency.

Claims (4)

1. A method for determining parameters of a rectangular waveguide low-pass filter, comprising the steps of: A. Parameter pre-determined steps: Select the rectangular waveguide model according to the frequency requirements of the rectangular waveguide low-pass filter; according to the relative bandwidth BW of the rectangular waveguide low-pass filter, determine the length θ ₀ of the rectangular waveguide between the two capacitor diaphragms; according to the electrical performance index of the rectangular waveguide low-pass filter The low-pass prototype circuit containing only capacitive elements is obtained by summing circuit transformation, and then the generalized impedance transformation coefficient K is obtained, and the capacitive reactance X of the parallel capacitor is determined. By checking the equivalent susceptance diagram loaded by the rectangular waveguide capacitance, the internal capacitance of the rectangular waveguide is preliminarily determined Diaphragm window width d, diaphragm thickness m and processing inclination α: B. Diaphragm parameter optimization steps: According to the generalized impedance transformation coefficient K, determine the S matrix of the rectangular waveguide single-section capacitive diaphragm; The inverse Schwartz-Christopher transformation is performed on the diaphragm cross-section polygon formed by the preliminary determination of the window width d of the capacitive diaphragm in the rectangular waveguide, the diaphragm thickness m, the processing inclination angle α, and the waveguide height b, and the solution of the capacitive diaphragm Normalized equivalent susceptance B; Determine the S' matrix of the capacitive diaphragm according to the equivalent susceptance B: $S_{11}^{\&prime;}=\frac{-j\stackrel{\&OverBar;}{B}}{2+j\stackrel{\&OverBar;}{B}},$ $S_{12}^{\&prime;}=\frac{2}{2+j\stackrel{\&OverBar;}{B}};$ The genetic algorithm is used to optimize so that the S' matrix is approximately equal to the S matrix, and finally determine the window width d', diaphragm thickness m' and processing inclination α' of the single capacitor diaphragm; Repeat the diaphragm parameter optimization step to determine the remaining capacitor diaphragm parameters of the rectangular waveguide low-pass filter; C. Waveguide cavity length correction steps: Through the S' matrix of the capacitive diaphragm, the corrected waveguide length between the two diaphragms is: ${\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; g</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; \&pi;</span>}}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; \&theta;</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}<span\; class="notranslate">\; -</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; arctan</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; arctan</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{\mathrm{<span\; class="notranslate">\; x</span>}}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 12</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; \&Greater\; Equal;</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>$ λ _g is the wavelength of the waveguide, θ ₀ is the length of the rectangular waveguide between the two capacitive diaphragms initially determined, X is the parallel reactance of the equivalent circuit of the capacitive diaphragm, that is $j\stackrel{\&OverBar;}{x}=\frac{{2S}_{12}\&prime;}{{(1-S_{11}^{\&prime;})}^2-{\left(S_{12}^{\&prime;}\right)}^2};$ Repeat the waveguide cavity length correction step to determine the waveguide length between the remaining two capacitive diaphragms of the rectangular waveguide low-pass filter, and complete the design of the entire rectangular waveguide low-pass filter. 2. the method for determining the parameters of the rectangular waveguide low-pass filter according to claim 1, wherein the pair described in the diaphragm parameter optimization step is initially determined by the window width d of the capacitance diaphragm in the rectangular waveguide, the diaphragm thickness m and The diaphragm cross-section polygon formed by processing the inclination angle α and the waveguide height b is subjected to inverse Schwartz-Christopher transformation to solve the equivalent susceptance of the capacitor diaphragm, as follows: The first step is to use the hybrid optimization algorithm combining the genetic algorithm and the sequential quadratic programming method to inverse the Schwartz-Christopher transformation $Z=C_1{\&Integral;}_{T_0}^T\underset{i=0}{\overset{\mathrm{no}-1}{\mathrm{\&Pi;}}}{(T-T_i)}^{-v_i}\mathrm{dt}+C_2$ Perform an optimization solution to obtain the transformed coordinates of each vertex: T ₁ , T ₂ , T ₃ , T ₄ ,..., T _n-2 ; In the second step, the coordinates of each vertex are substituted into the Schwartz-Christopher inverse transformation formula Z, and in the complex plane, the polygonal plane is transformed into the upper half complex plane; The third step is to transform the upper half of the complex plane into a plate capacitor, and calculate the diaphragm capacitance and equivalent susceptance value according to the length and width of the plate capacitor. 3. the method for determining the parameters of the rectangular waveguide low-pass filter according to claim 2, wherein the Schwartz-Christopher inverse transform described in the first step $Z=C_1{\&Integral;}_{T_0}^T\underset{i=0}{\overset{\mathrm{no}-1}{\mathrm{\&Pi;}}}{(T-T_i)}^{-v_i}\mathrm{dt}+C_2$ To optimize the solution, proceed as follows: In the first step, the optimization model is established as: find T ₂ ，T ₃ ，T ₄ ，…，T _n-2 $\mathrm{<span\; class="notranslate">\; min</span>}\mathrm{<span\; class="notranslate">\; AIM</span>}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 3</span>}}{\mathrm{<span\; class="notranslate">\; \&Sigma;</span>}}}{<span\; class="notranslate">\; (</span>\frac{\stackrel{<span\; class="notranslate">\; \&OverBar;</span>}{{\mathrm{<span\; class="notranslate">\; Y</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}}{{\mathrm{<span\; class="notranslate">\; Y</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}$ $\mathrm{<span\; class="notranslate">\; st</span>}<span\; class="notranslate">\; .</span>\mathrm{<span\; class="notranslate">\; Z</span>}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}{<span\; class="notranslate">\; \&Integral;</span>}_{{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; 0</span>}}}^{\mathrm{<span\; class="notranslate">\; T</span>}}\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 0</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; \&Pi;</span>}}}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; T</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; )</span>}^{{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}}\mathrm{<span\; class="notranslate">\; dT</span>}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}$ in: $Y_j=\frac{{\mathrm{ZL}}_j}{{\mathrm{ZL}}_0}=\frac{|Z_{j+1}-Z_j|}{|Z_1-Z_0|}=\frac{\underset{T_j}{\overset{T_{j+1}}{\&Integral;}}\left(\underset{i=0}{\overset{\mathrm{no}-1}{\mathrm{\&Pi;}}}{|T-T_i|}^{{-v}_i}\right)\mathrm{dT}}{\underset{T_0}{\overset{T_{j+1}}{\&Integral;}}\left(\underset{i=0}{\overset{\mathrm{no}=1}{\mathrm{\&Pi;}}}{|T-T_i|}^{{-v}_i}\right)\mathrm{dT}},$ where j = 0, 1, 2, ... n-3 ${\stackrel{\&OverBar;}{Y}}_j=\frac{\underset{T_j}{\overset{T_{j+1}}{\&Integral;}}\left(\underset{i=0}{\overset{\mathrm{no}-1}{\mathrm{\&Pi;}}}{|T-\underset{j=0}{\overset{i-1}{\mathrm{\&Sigma;}}}\tilde{T}{L}_j|}^{{-v}_i}\right)\mathrm{dT}}{\underset{T_0}{\overset{T_{j+1}}{\&Integral;}}\left(\underset{i=0}{\overset{\mathrm{no}-1}{\mathrm{\&Pi;}}}{|T-\underset{j=0}{\overset{i-1}{\mathrm{\&Sigma;}}}\tilde{T}{L}_j|}^{{-v}_i}\right)\mathrm{dT}},$ where j = 0, 1, 2, ... n-3 In the formula, Z _i is the coordinates of each vertex of the polygon in the section of the capacitor diaphragm in the z plane, T _i is the coordinates of each vertex of the capacitor diaphragm cross-section polygon in the t plane, |Z _j+1 -Z _j | is the side length from Z _j to Z _j+1 on the Z plane, represented by ZL _j , |T _j+1 -T _j | is the corresponding side length on the T plane, represented by TL _j , Y _j is the relative side length of the jth side represented by the vertex, Y _j is the relative side length of the jth side represented by the side length; In the second step, the improved hybrid genetic algorithm is used to optimize and solve the model: First of all, the genetic algorithm is processed with constraints, that is, the method of narrowing the search space is adopted, so that the optimal search is always running in the feasible solution space, and the initial point is optimized and solved; Secondly, use the sequential quadratic programming method to optimize again to obtain the optimal solution of the model, namely: T ₂ , T ₃ ,..., T _n-2 ° 4. the method for determining the parameters of the rectangular waveguide low-pass filter according to claim 1, wherein the utilization of genetic algorithm optimization described in the diaphragm parameter optimization step makes S 'matrix approximately equal to the S matrix, and optimizes and solves by following optimization model ,Right now: find d _i ，m _i ，α _i min ΔS _i11 , ΔS _i12 in, $S_{i11}=\frac{{{\stackrel{\&OverBar;}{K}}_i}^2-1}{1+{{\stackrel{\&OverBar;}{K}}_i}^2},$ $S_{i12}=j\frac{-2{\stackrel{\&OverBar;}{K}}_i}{1+{{\stackrel{\&OverBar;}{K}}_i}^2},$ $S_{i11}^{\&prime;}=\frac{-j{\stackrel{\&OverBar;}{B}}_i}{2+j{\stackrel{\&OverBar;}{B}}_i},$ $S_{i12}^{\&prime;}=\frac{2}{2+j{\stackrel{\&OverBar;}{B}}_j},$ ${\mathrm{<span\; class="notranslate">\; \&Delta;S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 11</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 11</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 11</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; \&Delta;S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 12</span>}}<span\; class="notranslate">\; =</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 12</span>}}^{<span\; class="notranslate">\; \&prime;</span>}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; S</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; 12</span>}}<span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; ,</span>$ In the formula, d _i , m _i , α _i are the window width, diaphragm thickness and processing inclination angle of the i-th capacitive diaphragm respectively, B _j is the normalized equivalent susceptance, ΔS _i11 , ΔS _{i, 2} are respectively The error of reflection coefficient S _i11 and transmission coefficient S _i12 .

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