CN103969495B

CN103969495B - A kind of simulation measuring method of wire antenna electric current

Info

Publication number: CN103969495B
Application number: CN201410218726.2A
Authority: CN
Inventors: 罗建书; 孙蕾; 王银坤; 陈祥玲
Original assignee: National University of Defense Technology
Current assignee: National University of Defense Technology
Priority date: 2014-05-22
Filing date: 2014-05-22
Publication date: 2016-07-06
Anticipated expiration: 2034-05-22
Also published as: CN103969495A

Abstract

The invention discloses a method for simulating and measuring the electromagnetic field of a wire antenna. First, the length and radius of the wire antenna and the pulse voltage at the center of the antenna are measured; the Heron integral equation of the physical model is deduced from Maxwell's equations and electric field boundary conditions; the integral The main singular part in the kernel is separated, and the Heron equation is simplified; the Chebyshev configuration method and the regularity of the current solution are used to expand the Heron equation, and the integral in the expansion is calculated; the current of the wire antenna is simulated and calculated, Outputs the current simulation measurement results for the wire antenna. The method simulates and measures the actual current distribution of the wire antenna by calculating the current distribution of the wire antenna, which can be applied to the complex electromagnetic compatibility test of large aircraft, reducing the cost and test cycle of physical testing.

Description

A Simulation Measurement Method of Wire Antenna Current

技术领域technical field

本发明涉及电磁场的仿真测量方法，特别涉及一种线天线电流的仿真测量方法。The invention relates to a simulation measurement method of an electromagnetic field, in particular to a simulation measurement method of a wire antenna current.

背景技术Background technique

大型航空器结构复杂，电子设备数量繁多，线缆束网络遍布整个航空器，整机线缆束网络结构设计复杂，电磁耦合特性测量困难，因此大型航空器整机线缆束网络电磁耦合特性的虚拟设计与验证的研究，对大型航空器的研制具有重要的实用价值。目前，我国对飞机的电磁兼容性验证还停留在依据经验进行测试和整改的阶段，因此开发模拟航空器整机电磁兼容性和电磁防护能力的仿真测量系统，可减少未来飞行器和系统的交付时间、削减物理测试成本、有效缩短飞机电磁兼容性测试和整改的周期，并可分析机载系统对高功率微波的敏感性，提出电磁防护加固要求。The structure of large aircraft is complex, the number of electronic equipment is large, the cable harness network is distributed throughout the aircraft, the structure design of the whole aircraft cable harness network is complex, and the measurement of electromagnetic coupling characteristics is difficult. Therefore, the virtual design of the electromagnetic coupling characteristics of the entire aircraft cable harness network and The verification research has important practical value for the development of large aircraft. At present, the electromagnetic compatibility verification of aircraft in my country is still in the stage of testing and rectification based on experience. Therefore, the development of a simulation measurement system that simulates the electromagnetic compatibility and electromagnetic protection capabilities of aircraft can reduce the delivery time of future aircraft and systems. Reduce the cost of physical testing, effectively shorten the cycle of aircraft electromagnetic compatibility testing and rectification, and analyze the sensitivity of airborne systems to high-power microwaves, and propose requirements for electromagnetic protection reinforcement.

在复杂电磁场研究中，偶极子线天线是最简单的天线形式，是构成各类复杂线天线的基本单元，其电流的仿真测量是搭建仿真测量系统要解决的基本问题。对于偶极子线天线，其电磁场的关系可由海伦积分方程表示。因此天线的电磁场研究问题最终归结为海伦积分方程的求解问题。但是海伦积分方程是第一类Fredholm积分方程，要直接求得其解析解是比较困难的，特别是在实际工程领域，天线形状和边界条件更为复杂，要求得其解析解几乎是不可能的。因此，海伦积分方程的数值解法对于天线电磁场的仿真测量极为重要。In the study of complex electromagnetic fields, the dipole wire antenna is the simplest antenna form and the basic unit of various complex wire antennas. The simulation measurement of its current is the basic problem to be solved in building a simulation measurement system. For a dipole wire antenna, the relationship of its electromagnetic field can be expressed by the Heron integral equation. Therefore, the research problem of the electromagnetic field of the antenna finally boils down to the solution of the Heron integral equation. However, the Heron integral equation is the first type of Fredholm integral equation, and it is difficult to directly obtain its analytical solution, especially in the field of practical engineering, where the antenna shape and boundary conditions are more complicated, and it is almost impossible to obtain its analytical solution . Therefore, the numerical solution of the Heron integral equation is extremely important for the simulation measurement of the antenna electromagnetic field.

在求解海伦方程的过程中，由于积分方程的精确核具有奇异性，会严重影响数值求解的精度，必须对其进行分析，因此Schelkunoff将核的奇异部分转化为第一类椭圆积分，并指出奇异的邻近部分是可积的Log奇异。基于Schelkunoff的工作，现有的几种精确奇异核的分析方法如下：Pearson将椭圆积分展成级数形式分析奇异核，但是Pearson的级数有限，并且估计在点远离原点时误差比较大；Wang提出的另一分析方法是直接将精确核展成一个包含球贝塞尔函数的精确表达式，但Wang的展式在点接近原点时的误差比较大；Davies等结合椭圆积分的迭代算法和复合梯形积分公式计算核函数；Werner在Wang的工作基础上将展式化成更加便于计算使用的形式，Bruno等结合这一展式和梯形积分得到了一种有效的分析策略；Davies等和Bruno等的策略虽然更有效，但在计算上比较复杂，不利于工程实现。In the process of solving the Heron equation, because the exact kernel of the integral equation has singularity, it will seriously affect the accuracy of the numerical solution, it must be analyzed, so Schelkunoff transformed the singular part of the kernel into the first kind of elliptic integral, and pointed out that the singularity The adjacent parts of are integrable Log singular. Based on Schelkunoff's work, several existing accurate singular kernel analysis methods are as follows: Pearson expands the elliptic integral into a series to analyze the singular kernel, but Pearson's series is limited, and it is estimated that the error is relatively large when the point is far away from the origin; Wang Another analysis method proposed is to directly expand the exact kernel into an exact expression containing spherical Bessel functions, but the error of Wang's expansion is relatively large when the point is close to the origin; Davies et al. combined the iterative algorithm of elliptic integral and the compound The trapezoidal integral formula calculates the kernel function; Werner transformed the expansion into a form that is more convenient for calculation and use based on Wang's work. Bruno et al. combined this expansion and trapezoidal integral to obtain an effective analysis strategy; Davies et al. and Bruno et al. Although the strategy is more effective, it is more complicated in calculation, which is not conducive to engineering implementation.

另外，目前大部分的仿真方法是在没有充分考虑电流解的正则性，直接利用矩量法求解积分方程，误差较大，Bruno等虽利用了电流解的正则性并发展了一种基于新的核函数分解的求解积分方程的有效方法。但是此方法相对复杂且有些参数的计算容易引入误差，例如汉克函数的参数以及包含弱奇异性的积分参数的计算。In addition, most of the current simulation methods do not fully consider the regularity of the current solution, and directly use the method of moments to solve the integral equation, with large errors. Although Bruno et al. have used the regularity of the current solution and developed a new method based on Kernel factorization is an efficient method for solving integral equations. However, this method is relatively complicated and the calculation of some parameters is easy to introduce errors, such as the calculation of the parameters of the Hank function and the integral parameters containing weak singularities.

发明内容Contents of the invention

本发明针对复杂电磁场仿真测量中计算精度不高，以及算法复杂不利于工程实现等问题，提出了一种复杂电磁场的最基本单元线天线电流的仿真测量技术。基于该技术的仿真系统可以用于指导电磁场兼容性测试有目的性地进行，减少物理测试的成本，缩短测试周期，为实现大型航空器的电磁兼容性验证和适航符合性认证/资格测试提供了有效地设计手段。The invention proposes a simulation measurement technology of the most basic element line antenna current of the complex electromagnetic field, aiming at the problems of low calculation accuracy in the simulation measurement of the complex electromagnetic field and the complexity of the algorithm which is not conducive to engineering realization. The simulation system based on this technology can be used to guide the electromagnetic field compatibility test to be carried out purposefully, reduce the cost of physical testing, shorten the test cycle, and provide a good foundation for the realization of electromagnetic compatibility verification and airworthiness compliance certification/qualification testing of large aircraft. Effectively design means.

本发明所采用的技术方案如下：The technical scheme adopted in the present invention is as follows:

步骤一：取一根细圆柱形偶极子线天线，测量天线的长度l，半径a，天线中心处用脉冲电压V₀进行馈电，获得其产生的馈电场E_in(z)；Step 1: Take a thin cylindrical dipole wire antenna, measure the length l of the antenna, the radius a, feed the center of the antenna with the pulse voltage V ₀ , and obtain the feed field E _in (z) generated by it;

步骤二：由麦克斯韦方程组和电场边界条件推导细圆柱形偶极子线天线的海伦积分方程；Step 2: Deduce the Heron integral equation of the thin cylindrical dipole wire antenna from Maxwell's equations and electric field boundary conditions;

步骤三：对海伦方程进行化简，具体操作包括：将核函数中主要奇异部分分离出来，采用第一类完全椭圆积分的级数展开，通过预先设定的误差上限控制展开的项数来控制计算精度，并减少因级数项展开过多带来的计算量；Step 3: Simplify the Heron equation. The specific operations include: separating the main singular part of the kernel function, using the series expansion of the complete elliptic integral of the first kind, and controlling the number of items expanded by the preset error upper limit Calculation accuracy, and reduce the amount of calculation caused by too much expansion of series items;

步骤四：采用切比雪夫配置法(简记为CCM)和电流解的正则性将海伦方程展开，对天线的比值h/a不是很大的情况采用高斯-切比雪夫积分法计算展开式中的积分，对天线的比值h/a比较大的情况采用分级网格的高斯型积分法计算；Step 4: Use the Chebyshev configuration method (abbreviated as CCM) and the regularity of the current solution to expand the Heron equation, and use the Gauss-Chebyshev integral method to calculate the expansion in the case where the ratio h/a of the antenna is not very large The integral of the antenna is calculated by the Gaussian integral method of the hierarchical grid when the ratio h/a of the antenna is relatively large;

步骤五：对线天线的电流进行仿真计算，输出线天线的电流仿真测量结果Step 5: Simulate the current of the wire antenna, and output the current simulation measurement results of the wire antenna

所述步骤二中的海伦积分方程为：The Heron integral equation in the step 2 is:

$\frac{j j η η}{22 π π} {&Integral; &Integral;}_{- - h h}^{h h} G G ((z z - - {z z}^{' '})) I I (({z z}^{' '})) {dz dz}^{' '} = = {C C}_{11} cos cos k k z z + + {C C}_{22} sin sin k k z z + + {&Integral; &Integral;}_{- - h h}^{h h} {E E.}_{i i n no} (({z z}^{' '})) sin sin k k | | z z - - {z z}^{' '} | | {dz dz}^{' '} - - - - - - ((11))$

满足边界条件meet the boundary conditions

I(-h)＝I(h)＝0(2)I(-h)=I(h)=0(2)

其中in

$G G ((z z)) = = \frac{11}{22 π π} {&Integral; &Integral;}_{00}^{22 π π} \frac{{e e}^{- - j j k k R R}}{R R} {dφ dφ}^{' '}$

I为天线表面的电流，h为天线长的一半，a为天线半径，为空气中的特性阻抗，分别为空气的磁导率和介电常数，λ为电磁波的波长，k＝2π/λ； I is the current on the surface of the antenna, h is half the length of the antenna, a is the radius of the antenna, is the characteristic impedance in air, are the magnetic permeability and permittivity of air respectively, λ is the wavelength of electromagnetic wave, k=2π/λ;

所述步骤三中的简化过程如下：The simplification process in the step 3 is as follows:

(1)将核函数G(z)表示为如下两部分的和，从而将主要奇异部分((3)式中的第一项)分离出来：(1) Express the kernel function G(z) as the sum of the following two parts, so as to separate the main singular part (the first item in (3) formula):

$G G ((z z)) = = - - \frac{11}{π π a a} l l n no ((| | z z | |)) + + {G G}^{R R} ((z z)) - - - - - - ((33))$

其中G^R(z)为核中去掉主要奇异性后余下部分；Among them, G ^R (z) is the remaining part after removing the main singularity in the kernel;

将核函数G(z)的表达式(3)代入(1)并作变量替换z′＝ht′，z＝ht，海伦方程可化为Substituting the expression (3) of the kernel function G(z) into (1) and making variable substitution z'=ht', z=ht, the Heron equation can be transformed into

$- - \frac{h h}{π π a a} {&Integral; &Integral;}_{- - 11}^{11} ln ln ((| | t t - - {t t}^{' '} | |)) I I (({ht ht}^{' '})) {dt dt}^{' '} - - \frac{h h}{π π a a} {&Integral; &Integral;}_{- - 11}^{11} ln ln ((h h)) I I (({ht ht}^{' '})) {dt dt}^{' '} + + {&Integral; &Integral;}_{- - 11}^{11} {hG hG}^{R R} ((h h t t - - {ht ht}^{' '})) I I (({ht ht}^{' '})) {dt dt}^{' '} = = \frac{22 π π}{j j η η} V V ((t t)) - - - - - - ((44))$

其中 $V (t) = C_{1} c o s (k h t) + C_{2} s i n (k h t) + h {&Integral;}_{- 1}^{1} E_{i n} ({ht}^{'}) \sin k h | t - t^{'} | {dt}^{'}, - 1 \leq t \leq 1;$ in $V (t) = C_{1} c o the s (k h t) + C_{2} the s i no (k h t) + h {&Integral;}_{- 1}^{1} {E.}_{i no} ({ht}^{'}) \sin k h | t - t^{'} | {dt}^{'}, - 1 \leq t \leq 1;$

再将式(3)中G^R分解为Then decompose G ^R in formula (3) into

G^R(z)＝G^C(z)+G^r(z)，(5)G ^R (z) = G ^C (z) + G ^r (z), (5)

其中，in,

${G G}^{C C} ((z z)) = = \frac{11}{a a π π} {&Integral; &Integral;}_{00}^{π π / / 22} \frac{{e e}^{- - j j 22 k k a a \sqrt{{((z z / / 22 a a))}^{22} + + {sin sin}^{22} φ φ}} - - 11}{\sqrt{{((z z / / 22 a a))}^{22} + + {sin sin}^{22} φ φ}} d d φ φ - - - - - - ((66))$

是一个光滑性很好的函数；以及is a smooth function; and

${G G}^{r r} ((z z)) = = \frac{11 - - κ κ}{a a π π} l l n no ((| | z z | |)) + + \frac{κ κ}{a a π π} l l n no \frac{88 a a}{κ κ} + + \frac{κ κ}{a a π π} {Σ Σ}_{n no = = 11}^{∞ ∞} {((\frac{((22 n no - - 11))!!!!}{((22 n no))!!!!}))}^{22} {κ κ}^{' ' 22 n no} ((l l n no \frac{44}{{κ κ}^{' '}} - - {Σ Σ}_{m m = = 11}^{n no} \frac{22}{((22 m m - - 11)) ((22 m m))})) - - - - - - ((77))$

其中 $R_{m a x} = \sqrt{{(z)}^{2} + 4 a^{2}},$ $κ = \frac{2 a}{R_{\max}}$ 以及 $κ^{'} = \sqrt{1 - κ^{2}};$ 当z＝0， $G^{r} (0) = \frac{\ln (8 a)}{a π};$ 当G^r(z)由式(7)前M项估计时，误差为O(κ′^2M+1)；然而，当κ′趋近于1即趋于无穷时，误差会逐渐增大；in $R_{m a x} = \sqrt{{(z)}^{2} + 4 a^{2}},$ $κ = \frac{2 a}{R_{\max}}$ as well as $κ^{'} = \sqrt{1 - κ^{2}};$ when z=0, $G^{r} (0) = \frac{\ln (8 a)}{a π};$ When G ^r (z) is estimated by the first M terms of formula (7), the error is O(κ′ ^2M+1 ); however, when κ′ approaches 1, that is When it tends to infinity, the error will gradually increase;

(2)提出一种高效高精度计算G^r(z)的策略，由第一类完全椭圆积分的另一级数展式有：(2) A strategy for calculating G ^r (z) with high efficiency and high precision is proposed. Another series expansion of the complete elliptic integral of the first kind is:

${G G}^{r r} ((z z)) = = \frac{11}{a a π π} l l n no ((| | z z | |)) + + \frac{11}{a a} \frac{κ κ}{11 + + {κ κ}^{' '}} + + \frac{11}{a a} \frac{κ κ}{11 + + {κ κ}^{' '}} {Σ Σ}_{n no = = 11}^{∞ ∞} {((\frac{((22 n no - - 11))!!!!}{((22 n no))!!!!}))}^{22} {((\frac{11 - - {κ κ}^{' '}}{11 + + {κ κ}^{' '}}))}^{22 n no} - - - - - - ((88))$

若G^r(z)由式(8)的前M项进行估计，误差为令可得则高精度的计算策略如下：若z满足则采用(7)的前M项估计；不然，则采用(8)的前M项估计，这一策略计算的G^r(z)误差为对所有|z|∈(0，∞)成立；If G ^r (z) is estimated by the first M terms of formula (8), the error is make Available The high-precision calculation strategy is as follows: If z satisfies Then use the first M item estimation in (7); otherwise, use the first M item estimation in (8), the error of G ^r (z) calculated by this strategy is holds for all |z|∈(0, ∞);

为方便进行数值求解，将细圆柱形偶极子线天线的海伦积分方程进行等价变形：In order to facilitate the numerical solution, the Heron integral equation of the thin cylindrical dipole wire antenna is equivalently deformed:

令z＝ht，z＝′ht′，则海伦方程化为：Let z=ht, z='ht', then Heron's equation can be transformed into:

${&Integral; &Integral;}_{- - 11}^{11} I I (({ht ht}^{' '})) G G ((h h t t - - {ht ht}^{' '})) {dt dt}^{' '} = = \frac{22 π π}{j j η η h h} (({C C}_{11} cos cos k k h h t t + + {V V}_{00} sin sin k k | | h h t t | |)),,$

再令x＝arccost，y＝arccost′，-1≤t，t′≤1则有t＝cosx，t′＝cosy；将其代入到海伦方程就可得到：Let x=arccost, y=arccost', -1≤t, t'≤1, then t=cosx, t'=cosy; substituting it into the Heron equation can be obtained:

${&Integral; &Integral;}_{00}^{π π} I I ((h h cos cos y the y)) G G ((h h cos cos x x - - h h cos cos y the y)) sin sin y the y d d y the y = = \frac{22 π π}{j j η η h h} (({C C}_{11} c c o o s the s ((k k h h cos cos x x)) + + {V V}_{00} sin sin k k | | h h cos cos x x | |));;$

记u(y)＝I(hcosy)siny，则原方程简化为Remember u(y)=I(hcosy)siny, then the original equation is simplified as

${&Integral; &Integral;}_{00}^{π π} G G ((h h cos cos x x - - h h cos cos y the y)) u u ((y the y)) d d y the y = = \frac{22 π π}{j j η η h h} (({C C}_{11} c c o o s the s ((k k h h cos cos x x)) + + {V V}_{00} sin sin k k | | h h cos cos x x | |));;$

所述步骤四中的利用CCM方法和电流的正则性求解将化简后的海伦方程展开过程如下：The process of expanding the simplified Heron equation by using the CCM method and the regularity of the current in the step 4 is as follows:

设电流如下表示为：Let the current be expressed as:

$I I (({ht ht}^{' '})) = = ω ω (({t t}^{' '})) {Σ Σ}_{n no = = 00}^{N N - - 11} {I I}_{n no} {T T}_{n no} (({t t}^{' '})),, {t t}^{' '} &Element; &Element; ((- - 11,, 11)) - - - - - - ((99))$

其中ω(t′)＝(1-t′²)^-1/2，T_n是第一类n次切比雪夫多项式，即Where ω(t′)=(1-t′ ² ) ^-1/2 , T _n is the nth degree Chebyshev polynomial of the first kind, namely

T_n(t′)＝cos(narccost′)，n≥0，t′∈(-1，1).(10)并且I_n，n＝0，…，N-1为未知参数，N为切比雪夫基函数的个数，N的选择为正整数；T _n (t′)=cos(narccost′), n≥0, t′∈(-1, 1).(10) and I _n , n=0,…, N-1 is unknown parameter, N is cut The number of Bishev basis functions, the choice of N is a positive integer;

用use

${t t}_{n no} = = c c o o s the s ((\frac{22 n no + + 11}{22 N N} π π)),, n no = = 00,, 11,, ... ...,, N N - - 11 - - - - - - ((1111))$

作为配置点；因此，求解海伦方程的CCM即为确定参数{I_n：n＝0，…，N-1}使得下式成立：As a configuration point; therefore, solving the CCM of the Heron equation is to determine the parameters {I _n : n=0,..., N-1} so that the following formula holds:

$- - \frac{h h}{π π a a} {&Integral; &Integral;}_{- - 11}^{11} ln ln ((| | {t t}_{n no} - - {t t}^{' '} | |)) I I (({ht ht}^{' '})) {dt dt}^{' '} - - \frac{h h}{π π a a} {&Integral; &Integral;}_{- - 11}^{11} ln ln ((h h)) I I (({ht ht}^{' '})) {dt dt}^{' '} + + \frac{h h}{a a} {&Integral; &Integral;}_{- - 11}^{11} {aG aG}^{R R} (({ht ht}_{n no} - - {ht ht}^{' '})) I I (({ht ht}^{' '})) {dt dt}^{' '} = = \frac{22 π π}{j j η η} V V (({t t}_{n no})) . . - - - - - - ((1212))$

切比雪夫多项式具有如下特殊性质：Chebyshev polynomials have the following special properties:

$- - \frac{11}{π π} {&Integral; &Integral;}_{- - 11}^{11} l l n no ((| | t t - - {t t}^{' '} | |)) ω ω (({t t}^{' '})) {T T}_{n no} (({t t}^{' '})) {dt dt}^{' '} = = \{\begin{matrix} ln ln 22,, & n no = = 00,, \\ \frac{11}{n no} {T T}_{n no} ((t t)),, & n no &GreaterEqual; &Greater Equal; 1. 1. \end{matrix} - - - - - - ((1313))$

该性质使得以切比雪夫多项式为权，对海伦方程级数展开式中一类特殊有奇异点的函数求积分具有解析解，从而保证了数值计算的精度；This property makes Chebyshev polynomials as the weight, and has an analytical solution to the integration of a special function with singular points in the series expansion of the Heron equation, thus ensuring the accuracy of numerical calculations;

优选的，当入射场E_in是z的偶函数时，天线上的电流分布是关于中心对称的，可以通过省略一半的奇函数基底加速CCM(记加速的CCM为sCCM)，而不影响解的精度；此时电流的近似表达式为Preferably, when the incident field E _in is an even function of z, the current distribution on the antenna is symmetrical about the center, and the CCM can be accelerated by omitting half of the odd function basis (record the accelerated CCM as sCCM), without affecting the solution Accuracy; the approximate expression of the current at this time is

$I I (({ht ht}^{′ ′})) = = ω ω (({t t}^{′ ′})) {Σ Σ}_{n no = = 00}^{N N - - 11} {I I}_{n no} {T T}_{22 n no} (({t t}^{′ ′})),, {t t}^{′ ′} &Element; &Element; ((- - 11,, 11))$

其配置点为Its configuration point is

${t t}_{n no} = = c c o o s the s \frac{44 n no + + 11}{44 N N} π π,, n no = = 00,, 11,, ... ...,, N N - - 1. 1.$

sCCM可以进一步提高计算效率；sCCM can further improve computational efficiency;

由电流解的正则性，不直接计算电流，而是计算因为具有更好光滑性，采用数值解法能得到更精确的 Due to the regularity of the current solution, the current is not directly calculated, but calculated because With better smoothness, the numerical solution can be used to obtain more accurate

将I(ht′)的表达式(9)代入式(12)，并结合特殊性质(13)，式(12)可以写成N×N矩阵形式：Substituting the expression (9) of I(ht′) into the formula (12) and combining the special property (13), the formula (12) can be written in the form of N×N matrix:

$\frac{h h}{a a} {[[{Z Z}_{n no m m}]]}_{N N \times \times N N} {[[{I I}_{n no}]]}_{N N} = = {[[{V V}_{n no}]]}_{N N} - - - - - - ((1414))$

其中in

${Z Z}_{n no m m} = = \{\begin{matrix} ln ln 22 - - ln ln h h + + {&Integral; &Integral;}_{- - 11}^{11} ω ω (({t t}^{' '})) {aG aG}^{R R} (({ht ht}_{n no} - - {ht ht}^{' '})) {dt dt}^{' '},, & m m = = 00 \\ \frac{11}{m m} {T T}_{m m} (({t t}_{n no})) + + {&Integral; &Integral;}_{- - 11}^{11} ω ω (({t t}^{' '})) {aG aG}^{R R} (({ht ht}_{n no} - - {ht ht}^{' '})) {T T}_{m m} (({t t}^{' '})) {dt dt}^{' '},, & m m &GreaterEqual; &Greater Equal; 1. 1. \end{matrix} - - - - - - ((1515))$

并且对于n，m＝0，1，…，N-1；and For n, m = 0, 1, ..., N-1;

所述步骤四中，当天线的比值h/a不是很大时，可以利用M-1阶的高斯-切比雪夫积分法计算矩阵Z中元素的积分：In the step 4, when the ratio h/a of the antenna is not very large, the integral of the elements in the matrix Z can be calculated using the M-1 order Gauss-Chebyshev integral method:

${&Integral; &Integral;}_{- - 11}^{11} ω ω (({t t}^{' '})) {aG aG}^{R R} (({ht ht}_{n no} - - {ht ht}^{' '})) {T T}_{m m} (({t t}^{' '})) {dt dt}^{' '} \approx \approx \frac{π π}{M m} {Σ Σ}_{p p = = 00}^{M m - - 11} {aG aG}^{R R} (({ht ht}_{n no} - - {ht ht}^{' '}_{p p})) {T T}_{m m} (({t t}^{' '}_{p p})) - - - - - - ((1616))$

其中 ${t^{'}}_{p} = c o s \frac{2 p + 1}{2 M} π, p = 0, 1 ..., M - 1;$ 记 $E = {[E_{n, m}]}_{N \times M} = [{&Integral;}_{- 1}^{1} ω (t^{'}) {aG}^{R} ({ht}_{n} - {ht}^{'}) T_{m} (t^{'}) {dt}^{'}],$ $G = {[G_{n, m}]}_{N \times M} = [{aG}^{R} ({ht}_{n} - {ht}^{'}_{m})],$ 且 $V = {[V_{n, m}]}_{M \times M} = [c o s (\frac{n (2 m + 1)}{2 M} π)],$ 式(16)可以转化为矩阵形式：in ${t^{'}}_{p} = c o the s \frac{2 p + 1}{2 m} π, p = 0, 1 ..., m - 1;$ remember $E. = {[{E.}_{no, m}]}_{N \times m} = [{&Integral;}_{- 1}^{1} ω (t^{'}) {aG}^{R} ({ht}_{no} - {ht}^{'}) T_{m} (t^{'}) {dt}^{'}],$ $G = {[G_{no, m}]}_{N \times m} = [{aG}^{R} ({ht}_{no} - {ht}^{'}_{m})],$ and $V = {[V_{no, m}]}_{m \times m} = [c o the s (\frac{no (2 m + 1)}{2 m} π)],$ Equation (16) can be transformed into a matrix form:

${E E.}^{T T} = = \frac{π π}{M m} V V \cdot &Center Dot; {G G}^{T T} - - - - - - ((1717))$

式(17)可以通过快速余弦变换进行计算；E的前N列即为离散矩阵Z中的需要的积分元；因而计算整个离散矩阵Z的复杂度由O(N²MlnM)降为O(NMlnM)；Equation (17) can be calculated by fast cosine transformation; the first N columns of E are the required integral elements in the discrete matrix Z; thus the complexity of calculating the entire discrete matrix Z is reduced from O(N ² MlnM) to O(NMlnM );

得到切比雪夫多项式展开系数I_n，n＝0，1，…，N-1，将其代入(9)式得到仿真电流I，输出线天线的电流仿真测量结果。Obtain Chebyshev polynomial expansion coefficient I _n , n=0, 1, .

基于数值实验，如：当使用N＝64时，要求相对误差≤1％，则天线的比值h/a比值应该不大于1×10³，因此对天线半长与天线半径之比小于10³的线天线，可以利用快速余弦变换设计快速算法，从而提高仿真效率；Based on numerical experiments, such as: when using N=64, the relative error is required to be ≤1%, then the ratio h/a ratio of the antenna should not be greater than 1×10 ³ , so the ratio of the half length of the antenna to the radius of the antenna is less than 10 ³ Wire antennas can use fast cosine transform to design fast algorithms, thereby improving simulation efficiency;

当天线的比值较大时，因为aG^R(z)在z＝0表现出近似的奇异性，会出现很尖锐的顶点，这一性质使得高斯-切比雪夫积分法失效；为了克服这一困难，采用基于分级网格的高斯型积分法，这一积分法已被证明能十分有效地计算弱奇异的积分。When the ratio of the antenna is large, because aG ^R (z) exhibits an approximate singularity at z=0, a very sharp apex will appear, this property makes the Gauss-Chebyshev integral method invalid; in order to overcome this difficulty , using a Gaussian-type integration method based on a hierarchical grid, which has been shown to be very efficient for weakly singular integrals.

与现有技术比，本发明具有如下优点：Compared with the prior art, the present invention has the following advantages:

(1)通过计算线天线的电流分布来对线天线的实际电流分布进行仿真测量，将其应用于大型航空器复杂电磁兼容性测试，可以有效地指导测试，减少物理测试的成本和测试周期；(1) Simulate and measure the actual current distribution of the wire antenna by calculating the current distribution of the wire antenna, and apply it to the complex electromagnetic compatibility test of large aircraft, which can effectively guide the test and reduce the cost and test cycle of the physical test;

(2)将海伦方程的核函数的奇异部分分离出来，由第一类完全椭圆积分将奇异部分进行级数展展开，其计算可以通过事先给定的误差取前M项估计，不仅有效地控制了计算精度，而且减少了级数项展开过多带来的计算量，从而提高了仿真效率；(2) The singular part of the kernel function of the Heron equation is separated, and the singular part is expanded by the first kind of complete elliptic integral, and its calculation can be estimated by taking the first M terms of the given error, which not only effectively controls It not only improves the calculation accuracy, but also reduces the amount of calculation caused by too much expansion of series items, thereby improving the simulation efficiency;

(3)在使用CCM求解化简海伦方程的过程中，对h/a比较大的情况，在奇异点处通常采用的高斯-切比雪夫积分法积分精度会受到严重影响，为了减少奇异点对误差精度的影响，采用分级网格的高斯型积分法提高计算精度，减小了测量方法的误差；(3) In the process of using CCM to solve the simplified Heron equation, when h/a is relatively large, the integration accuracy of the Gauss-Chebyshev integral method usually used at singular points will be seriously affected. In order to reduce the impact of singular points on Influenced by the error accuracy, the Gaussian integral method with hierarchical grids is used to improve the calculation accuracy and reduce the error of the measurement method;

(4)本发明对线天线的天线半长与天线半径之比h/a不是很大的情况，在保证计算精度的同时，可采用高斯-切比雪夫积分法计算矩量法的离散矩阵中的积分元素，以提高计算效率。特别对于h/a＜10³的线天线仿真效率更高，因为在这种条件下对海伦方程进行切比雪夫多项式展开得到的离散矩阵可以通过快速余弦变换计算得到，进一步降低了计算复杂度，提高了测量方法的效率。(4) In the case that the ratio h/a of the antenna half-length of the line antenna and the antenna radius is not very large, the Gauss-Chebyshev integral method can be used to calculate the discrete matrix of the method of moments while ensuring the calculation accuracy Integral elements of , to improve computational efficiency. Especially for the line antenna with h/a<10 ³ , the simulation efficiency is higher, because under this condition, the discrete matrix obtained by Chebyshev polynomial expansion of the Heron equation can be calculated by fast cosine transformation, which further reduces the computational complexity. The efficiency of the measurement method is improved.

附图说明Description of drawings

图1偶极子线天线物理模型及相关参数Figure 1 Dipole wire antenna physical model and related parameters

图中符号说明如下：The symbols in the figure are explained as follows:

z：空间直角坐标系中竖坐标的位置变量；z: the position variable of the vertical coordinate in the space Cartesian coordinate system;

Δz：偶极子线天线间的距离；Δz: distance between dipole wire antennas;

l：天线的长度；l: the length of the antenna;

a：天线的半径；a: Radius of the antenna;

V₀：天线中心处的脉冲电压；V ₀ : pulse voltage at the center of the antenna;

E_in(z)：脉冲电压产生的馈电场；E _in (z): the feeding field generated by the pulse voltage;

I(z)：空间直角坐标系中竖坐标z处对应的电流。I(z): The current corresponding to the vertical coordinate z in the spatial rectangular coordinate system.

具体实施方式detailed description

下面结合附图对本发明做进一步介绍。The present invention will be further introduced below in conjunction with the accompanying drawings.

由于实际物理实验测试的是线天线激励点处的导纳，为了方便比较，要将线天线的电流计算结果转化为等价导纳。Since the actual physical experiment tests the admittance at the excitation point of the wire antenna, for the convenience of comparison, the current calculation results of the wire antenna should be converted into equivalent admittance.

步骤一：取一根细圆柱形偶极子线天线，测量天线的半径a＝7.002×10^-3(米)，天线的长度为l(米)，其变化范围为[0.3，1]，波长λ＝1(米)，天线中心处激励电压是1伏特，其产生的馈电场E_in(z)＝δ(z)，如图1所示。Step 1: Take a thin cylindrical dipole wire antenna, measure the radius a=7.002×10 ^-3 (meter) of the antenna, the length of the antenna is l (meter), and its variation range is [0.3, 1], the wavelength λ=1 (meter), the excitation voltage at the center of the antenna is 1 volt, and the generated feeding field E _in (z)=δ(z), as shown in FIG. 1 .

步骤二：得到海伦积分方程为：Step 2: Get the Heron integral equation as:

$\frac{j j η η}{22 π π} {&Integral; &Integral;}_{- - h h}^{h h} G G ((z z - - {z z}^{' '})) I I (({z z}^{' '})) {dz dz}^{' '} = = {C C}_{11} cos cos k k z z + + {C C}_{22} sin sin k k z z + + {&Integral; &Integral;}_{- - h h}^{h h} {E E.}_{i i n no} (({z z}^{' '})) sin sin k k | | z z - - {z z}^{' '} | | {dz dz}^{' '} - - - - - - ((1818))$

其中I为天线表面的电流，h＝l/2，a为天线半径，η＝120π为空气中的特性阻抗，k＝2π。C₁，C₂是待定的常数，其值在具体实施方案步骤四的(27)、(28)式给出详细说明；Where I is the current on the surface of the antenna, h=l/2, a is the radius of the antenna, η=120π is the characteristic impedance in air, k=2π. C ₁ , C ₂ are undetermined constants, and their values are described in detail in the (27), (28) formulas of step 4 of the specific embodiment;

该积分方程满足边界条件I(-h)＝I(h)＝0，其中核函数G(z)的表达式为：The integral equation satisfies the boundary condition I(-h)=I(h)=0, where the expression of the kernel function G(z) is:

$G G ((z z)) = = \frac{11}{22 π π} {&Integral; &Integral;}_{00}^{22 π π} \frac{{e e}^{- - j j k k R R}}{R R} {dφ dφ}^{' '} - - - - - - ((1919))$

$R R = = \sqrt{{z z}^{22} + + 22 {a a}^{22} - - 22 {a a}^{22} {cosφ cosφ}^{' '}} . .$

步骤三：将核函数G(z)表示为两部分的和：Step 3: Express the kernel function G(z) as the sum of two parts:

$G G ((z z)) = = - - \frac{11}{π π a a} l l n no ((| | z z | |)) + + {G G}^{R R} ((z z)) - - - - - - ((2020))$

其中G^R(z)为核中去掉主要奇异性后余下部分，它又可以分解为G^R(z)＝G^C(z)+G^r(z)，Among them, G ^R (z) is the remaining part after removing the main singularity in the kernel, and it can be decomposed into G ^R (z)=G ^C (z)+G ^r (z),

$G^{C} (z) = \frac{1}{a π} {&Integral;}_{0}^{π / 2} \frac{e^{- j 2 k a \sqrt{{(z / 2 a)}^{2} + \sin^{2} φ}} - 1}{\sqrt{{(z / 2 a)}^{2} + \sin^{2} φ}} d φ,$ 是一个光滑性很好的函数； $G^{C} (z) = \frac{1}{a π} {&Integral;}_{0}^{π / 2} \frac{e^{- j 2 k a \sqrt{{(z / 2 a)}^{2} + \sin^{2} φ}} - 1}{\sqrt{{(z / 2 a)}^{2} + \sin^{2} φ}} d φ,$ is a smooth function;

若z满足 $κ^{'} \leq \sqrt{2} - 1,$ 则If z satisfies $κ^{'} \leq \sqrt{2} - 1,$ but

${G G}^{r r} ((z z)) = = \frac{11 - - κ κ}{a a π π} l l n no ((| | z z | |)) + + \frac{κ κ}{a a π π} l l n no \frac{88 a a}{κ κ} + + \frac{κ κ}{a a π π} {Σ Σ}_{n no = = 11}^{∞ ∞} {((\frac{((22 n no - - 11))!!!!}{((22 n no))!!!!}))}^{22} {κ κ}^{' ' 22 n no} ((l l n no \frac{44}{{κ κ}^{' '}} - - {Σ Σ}_{m m = = 11}^{n no} \frac{22}{((22 m m - - 11)) ((22 m m))}))$

采用前M项估计它的值，其中M＝192；Use the first M items to estimate its value, where M=192;

否则otherwise

${G G}^{r r} ((z z)) = = \frac{11}{a a π π} l l n no ((| | z z | |)) + + \frac{11}{a a} \frac{κ κ}{11 + + κ κ} + + \frac{11}{a a} \frac{κ κ}{11 + + κ κ} {Σ Σ}_{n no = = 11}^{∞ ∞} {((\frac{((22 n no - - 11))!!!!}{((22 n no))!!!!}))}^{22} {((\frac{11 - - κ κ}{11 + + κ κ}))}^{22 n no}$

同样采用前M项进行估计，M＝192；Also use the first M items to estimate, M=192;

其中 $R_{m a x} = \sqrt{{(z)}^{2} + 4 a^{2}},$ $κ = \frac{2 a}{R_{\max}}$ 以及 $κ^{'} = \sqrt{1 - κ^{2}},$ 当z＝0， $G^{r} (0) = \frac{\ln (8 a)}{a π};$ in $R_{m a x} = \sqrt{{(z)}^{2} + 4 a^{2}},$ $κ = \frac{2 a}{R_{\max}}$ as well as $κ^{'} = \sqrt{1 - κ^{2}},$ when z=0, $G^{r} (0) = \frac{\ln (8 a)}{a π};$

将G(z)的表达式代入，则海伦方程可以简化为Substituting the expression of G(z), the Heron equation can be simplified as

${&Integral; &Integral;}_{00}^{π π} G G ((h h cos cos x x - - h h cos cos y the y)) u u ((y the y)) d d y the y = = \frac{22 π π}{j j η η h h} (({C C}_{11} c c o o s the s ((k k h h cos cos x x)) + + {V V}_{00} sin sin k k | | h h cos cos x x | |))$

其中z＝ht，z′＝ht′，x＝arccost，y＝arccost′，-1≤t≤1，u(y)＝I(hcosy)siny，where z=ht, z'=ht', x=arccost, y=arccost', -1≤t≤1, u(y)=I(hcosy)siny,

步骤四：将电流表示为Step 4: Express the current as

$I I (({ht ht}^{' '})) = = ω ω (({t t}^{' '})) {Σ Σ}_{n no = = 00}^{N N - - 11} {I I}_{n no} {T T}_{n no} (({t t}^{' '})),, {t t}^{' '} &Element; &Element; ((- - 11,, 11)),, - - - - - - ((21 twenty one))$

其中N＝64，ω(t′)＝(1-t′2)^-1/2，T_n(t′)＝cos(narccost′)，n≥0，t′∈(-1，1)，(n＝0，1，…，N-1)是配置点，I_n(n＝0，1，…，N-1)为需要计算的未知参数；求解海伦方程的CCM即为确定参数I_n使得下式成立：where N=64, ω(t′)=(1-t′2) ^-1/2 , T _n (t′)=cos(narccost′), n≥0, t′∈(-1, 1), (n=0, 1, ..., N-1) is the configuration point, I _n (n = 0, 1, ..., N-1) is the unknown parameter to be calculated; the CCM for solving the Heron equation is the definite parameter I _n so that the following formula holds:

$- - \frac{h h}{a a π π} {&Integral; &Integral;}_{- - 11}^{11} ln ln ((| | {t t}_{n no} - - {t t}^{' '} | |)) I I (({ht ht}^{' '})) {dt dt}^{' '} - - \frac{h h}{π π a a} {&Integral; &Integral;}_{- - 11}^{11} ln ln ((h h)) I I (({ht ht}^{' '})) {dt dt}^{' '} + + \frac{h h}{a a} {&Integral; &Integral;}_{- - 11}^{11} {aG aG}^{R R} (({ht ht}_{n no} - - {ht ht}^{' '})) I I (({ht ht}^{' '})) {dt dt}^{' '} = = \frac{22 π π}{j j η η} V V (({t t}_{n no})) . . - - - - - - ((22 twenty two))$

将(22)写成N×N矩阵形式：Write (22) in N×N matrix form:

$\frac{h h}{a a} {[[{Z Z}_{n no m m}]]}_{N N \times \times N N} {[[{I I}_{n no}]]}_{N N} = = {[[{V V}_{n no}]]}_{N N} - - - - - - ((23 twenty three))$

其中in

${Z Z}_{n no m m} = = \{\begin{matrix} ln ln 22 - - ln ln h h + + {&Integral; &Integral;}_{- - 11}^{11} ω ω (({t t}^{' '})) {aG aG}^{R R} (({ht ht}_{n no} - - {ht ht}^{' '})) {dt dt}^{' '},, & m m = = 00,, \\ \frac{11}{m m} {T T}_{m m} (({t t}_{n no})) + + {&Integral; &Integral;}_{- - 11}^{11} ω ω (({t t}^{' '})) {aG aG}^{R R} (({ht ht}_{n no} - - {ht ht}^{' '})) {T T}_{m m} (({t t}^{' '})) {dt dt}^{' '},, & m m &GreaterEqual; &Greater Equal; 1. 1. \end{matrix} - - - - - - ((24 twenty four))$

并且 $V_{n} = \frac{2 π}{j η} V (t_{n}), (n, m = 0, 1, ..., N - 1);$ and $V_{no} = \frac{2 π}{j η} V (t_{no}), (no, m = 0, 1, ..., N - 1);$

利用M-1阶的高斯-切比雪夫积分法计算矩阵Z中元素的积分：Compute the integral of the elements in matrix Z using the Gauss-Chebyshev integral method of order M-1:

${&Integral; &Integral;}_{- - 11}^{11} ω ω (({t t}^{' '})) {aG aG}^{R R} (({ht ht}_{n no} - - {ht ht}^{' '})) {T T}_{m m} (({t t}^{' '})) {dt dt}^{' '} \approx \approx \frac{π π}{M m} {Σ Σ}_{p p = = 00}^{M m - - 11} {aG aG}^{R R} (({ht ht}_{n no} - - {ht ht}^{' '}_{p p})) {T T}_{m m} (({t t}^{' '}_{p p})) - - - - - - ((2525))$

其中可以通过快速余弦变换计算该式。in This equation can be calculated by fast cosine transform.

下面完整地描述海伦方程转化得到的代数方程的求解与真实电流的重构；由线性叠加性，离散方程的的解可以表示为：The solution of the algebraic equation obtained by the transformation of the Heron equation and the reconstruction of the real current are described completely below; from the linear superposition property, the solution of the discrete equation can be expressed as:

$[[{I I}_{n no}]] = = {C C}_{11} [[{I I}_{n no}^{((11))}]] + + {C C}_{22} [[{I I}_{n no}^{((22))}]] + + [[{I I}_{n no}^{((33))}]] - - - - - - ((2626))$

其中 $[Z_{n m}] [I_{n}^{(i)}] = [V_{n}^{(i)}], V_{n}^{(1)} = \frac{2 π}{j η} {coskht}_{n}, V_{n}^{(2)} = \frac{2 π}{j η} {sinkht}_{n}$ 以及in $[Z_{no m}] [I_{no}^{(i)}] = [V_{no}^{(i)}], V_{no}^{(1)} = \frac{2 π}{j η} {coskht}_{no}, V_{no}^{(2)} = \frac{2 π}{j η} {sinkht}_{no}$ as well as

$V_{n}^{(3)} = \frac{2 π h}{j η} {&Integral;}_{- 1}^{1} E_{i n} ({ht}^{'}) \sin k h | t_{n} - t^{'} | {dt}^{'},$ 常数C₁、C₂由终端条件(2)决定， $V_{no}^{(3)} = \frac{2 π h}{j η} {&Integral;}_{- 1}^{1} {E.}_{i no} ({ht}^{'}) \sin k h | t_{no} - t^{'} | {dt}^{'},$ The constants C ₁ and C ₂ are determined by the terminal condition (2),

${C C}_{11} = = - - \frac{{[[{u u}_{n no}]]}^{T T} [[{I I}_{n no}^{((33))}]] + + {[[{v v}_{n no}]]}^{T T} [[{I I}_{n no}^{((33))}]]}{22 {[[{u u}_{n no}]]}^{T T} [[{I I}_{n no}^{((11))}]]} - - - - - - ((2727))$

${C C}_{22} = = - - \frac{{[[{u u}_{n no}]]}^{T T} [[{I I}_{n no}^{((33))}]] - - {[[{v v}_{n no}]]}^{T T} [[{I I}_{n no}^{((33))}]]}{22 {[[{u u}_{n no}]]}^{T T} [[{I I}_{n no}^{((22))}]]} - - - - - - ((2828))$

其中u_n＝1的v_n＝(-1)ⁿ，n＝0，…，N-1；where u _n =1 v _n =(-1) ⁿ , n=0,...,N-1;

天线上的电流可以由式(21)和(26)得到，为了避免出现除以在式(21)中作变量替换t＝cosθ，将电流表示为：The current on the antenna can be obtained by equations (21) and (26), in order to avoid dividing by In formula (21), make variable substitution t=cosθ, and express the current as:

$I I ((h h c c o o s the s θ θ)) = = {Σ Σ}_{n no = = 11}^{N N - - 22} {\overset{~ ~}{I I}}_{n no} sin sin n no θ θ,, θ θ &Element; &Element; ((00,, π π)) - - - - - - ((2929))$

其中，in,

以及M_o＝N-M_e； and M _o = NM _e ;

步骤五：根据(26)式得到切比雪夫多项式展开系数I_n，n＝0，1，…，N-1，N＝64，将其代入(21)式得到仿真电流I，输出线天线的电流仿真测量结果，如表1所示，其中G₀表示导纳的实部，B₀表示导纳的虚部。Step 5: get Chebyshev polynomial expansion coefficient I _n according to formula (26), n=0, 1, ..., N-1, N=64, put it into formula (21) to get the simulated current I, the output line antenna The current simulation measurement results are shown in Table 1, where G ₀ represents the real part of the admittance, and B ₀ represents the imaginary part of the admittance.

表1电流仿真测量结果和实际测量结果(单位：毫西门子)Table 1 Current simulation measurement results and actual measurement results (unit: millisiemens)

若将仿真测量的结果和实际测量数据的最大误差和均方误差分别定义如下：If the maximum error and mean square error of the simulation measurement results and the actual measurement data are respectively defined as follows:

${e e}_{m m a a x x} = = \underset{i i = = 11,, ... ...,, K K}{m m a a x x} | | {\overset{~ ~}{σ σ}}_{i i} - - {σ σ}_{i i} | |$

以及as well as

${e e}_{M m S S E E.} = = \frac{11}{K K} \sqrt{{Σ Σ}_{i i = = 11}^{K K} {(({\overset{~ ~}{σ σ}}_{i i} - - {σ σ}_{i i}))}^{22}}$

其中，G₀表示导纳的实部，B₀表示导纳的虚部，表示G₀或B₀的仿真输出结果，σ_i表示G₀或B₀的实际测量结果，K表示输出结果或测量结果的个数，这里K＝50，具体误差在表2中列出：where G0 represents the real part of the _admittance , _B0 represents the imaginary part of the admittance, Indicates the simulation output result of G ₀ or B ₀ , σ _i indicates the actual measurement result of G ₀ or B ₀ , K indicates the number of output results or measurement results, where K=50, and the specific errors are listed in Table 2:

表2仿真测量的结果和实际测量数据的误差(单位：毫西门子)Table 2 The error between the results of the simulation measurement and the actual measurement data (unit: millisiemens)

误差类型error type G₀ G ₀ B₀ B ₀ 最大误差maximum error 0.42220.4222 0.85440.8544 均方误差mean square error 0.11580.1158 0.26310.2631

从表2可以看出仿真测量的结果和实际测量数据的最大误差和均方误差均控制在1毫西门子的量级内，由此表明本发明提出的方法与物理实验测试结果吻合得很好，具有实际的工程应用价值。As can be seen from Table 2, the maximum error and the mean square error of the result of the simulation measurement and the actual measurement data are all controlled in the order of magnitude of 1 millisiemens, thus showing that the method proposed by the present invention matches well with the physical experiment test results, It has practical engineering application value.

Claims

1. A simulation measurement method of wire antenna current, comprising the steps:

Step 1: Take a thin cylindrical dipole wire antenna, measure the length l and radius a of the antenna, and feed the center of the antenna with a pulse voltage V ₀ to obtain the feeding field E _in (z) generated by it, where z is the position variable of the vertical coordinate in the spatial Cartesian coordinate system;

Step 2: Deduce the Heron integral equation of the thin cylindrical dipole wire antenna from Maxwell's equations and electric field boundary conditions;

Step 3: Simplify the Heron equation. The specific operations include: separating the main singular part of the kernel function, using the series expansion of the complete elliptic integral of the first kind, and controlling the number of terms expanded by the preset error upper limit Calculation accuracy, and reduce the amount of calculation caused by too much expansion of series items;

Step 4: Use the Chebyshev configuration method, abbreviated as CCM, and expand the Heron equation with the regularity of the current solution. When the ratio h/a of the antenna is not very large, use the Gauss-Chebyshev integral method to calculate the expansion. The integral of the antenna is calculated by the Gaussian integral method of the hierarchical grid when the ratio h/a of the antenna is relatively large;

Step 5: Carry out simulation calculation on the current of the wire antenna, and output the current simulation measurement result of the wire antenna;

The Heron integral equation in the step 2 is:

meet the boundary conditions

I(-h)=I(h)=0(2)

in

I is the current on the surface of the antenna, h is half the length of the antenna, a is the radius of the antenna, is the characteristic impedance in the air, μ, ∈ are the magnetic permeability and permittivity of the air respectively, λ is the wavelength of the electromagnetic wave, k=2π/λ;

The simplification process in the step 3 is as follows:

(1) Express the kernel function G(z) as the sum of the following two parts, so as to separate the main singular part, that is, the first item in formula (3):

Among them, G ^R (z) is the remaining part after removing the main singularity in the kernel;

Substituting the expression (3) of the kernel function G(z) into (1) and making variable substitution z'=ht', z=ht, the Heron equation can be transformed into

in

Then decompose G ^R in formula (3) into

G ^R (z) = G ^C (z) + G ^r (z), (5)

in,

is a smooth function; and

in as well as when z=0, When G ^r (z) is estimated by the first M terms of equation (7), the error is O(κ ^′2M+1 ); however, when κ′ approaches 1, that is When it tends to infinity, the error will gradually increase;

(2) A strategy for calculating G ^r (z) with high efficiency and high precision is proposed. Another series expansion of the complete elliptic integral of the first kind is:

If G ^r (z) is estimated by the first M terms of formula (8), the error is make Available The high-precision calculation strategy is as follows: If z satisfies Then use the first M item estimation in (7); otherwise, use the first M item estimation in (8), the error of G ^r (z) calculated by this strategy is holds for all |z|∈(0, ∞);

In order to facilitate the numerical solution, the Heron integral equation of the thin cylindrical dipole wire antenna is equivalently deformed:

Let z=ht, z'=ht', then Heron's equation can be transformed into:

Let x=arccost, y=arccost', -1≤t, t'≤1, then t=cosx, t'=cosy; substituting it into the Heron equation can be obtained:

Remember u(y)=I(hcosy)siny, then the original equation is simplified as

The process of expanding the simplified Heron equation by using the CCM method and the regularity of the current in the step 4 is as follows:

Let the current be expressed as:

Where ω(t′)=(1-t ^′2 ) ^-1/2 , T _n is the nth degree Chebyshev polynomial of the first kind, namely

T _n (t')=cos(narccost'), n≥0, t'∈(-1,1).(10)

And I _n , n=0,..., N-1 are unknown parameters, N is the number of Chebyshev basis functions, and the choice of N is a positive integer;

use

As a configuration point; therefore, solving the CCM of the Heron equation is to determine the parameters {I _n : n=0,..., N-1} so that the following formula holds:

Chebyshev polynomials have the following special properties:

Due to the regularity of the current solution, the current is not directly calculated, but calculated

Substituting the expression (9) of I(ht′) into the formula (12) and combining the special property (13), the formula (12) can be written in the form of N×N matrix:

in

and For n, m=0,1,...,N-1;

In the step 4, when the ratio h/a of the antenna is not very large, the integral of the elements in the matrix Z can be calculated by using the M-1 order Gauss-Chebyshev integral method:

in remember and Equation (16) can be transformed into a matrix form:

Equation (17) can be calculated by fast cosine transformation; the first N columns of E are the required integral elements in the discrete matrix Z; thus the complexity of calculating the entire discrete matrix Z is reduced from O(N ² MlnM) to O(NMlnM );

The expansion coefficient I _n of the Chebyshev polynomial is obtained, n=0, 1, .

2. the simulation measurement method of a kind of wire antenna current described in claim 1, is characterized in that:

In step three, when the incident field E _in is an even function of z, the current distribution on the antenna is symmetrical about the center, and the CCM can be accelerated by omitting half of the odd function basis, and the accelerated CCM is denoted as sCCM without affecting the solution The accuracy; the approximate expression of the current at this time is

Its configuration point is

.

3. The simulation measurement method of a wire antenna current according to claim 1, characterized in that: the scope of application of the method is a wire antenna whose ratio of the half length of the antenna to the radius of the antenna is less than 10 ³ .