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

Abstract

The invention discloses the simulation measuring method of a kind of wire antenna electromagnetic field, first measure length and the radius of wire antenna and the pulse voltage at center of antenna place；Derived by maxwell equation group and electric field boundary condition the Helen integral equation of this physical model；Singular part main in integral kernel is separated, Helen's equation is carried out abbreviation；Adopt the regularity of Chebyshev collocation method and solution for the current by Helen's equation expansion, the integration in expansion is calculated；The electric current of wire antenna is carried out simulation calculation, the current simulations measurement result of output lead antenna.The actual current distribution of wire antenna is carried out simulated measurement by calculating the CURRENT DISTRIBUTION of wire antenna by the method, it is possible to is applied to airdreadnought complexity emc testing, reduces cost and the test period of physical testing.

Description

Simulation measurement method for current of line antenna

Technical Field

The invention relates to a simulation measurement method of an electromagnetic field, in particular to a simulation measurement method of a line antenna current.

Background

The large aircraft has a complex structure, a large number of electronic devices, a cable bundle network is distributed all over the whole aircraft, the structure design of the whole aircraft cable bundle network is complex, and the measurement of the electromagnetic coupling characteristic is difficult, so that the research on the virtual design and verification of the electromagnetic coupling characteristic of the whole aircraft cable bundle network has important practical value for the development of the large aircraft. At present, the electromagnetic compatibility verification of the aircraft in China still stays at the stage of testing and rectifying according to experience, so that a simulation measurement system for simulating the electromagnetic compatibility and the electromagnetic protection capability of the whole aircraft is developed, the delivery time of the aircraft and the system in the future can be reduced, the physical testing cost is reduced, the period of the electromagnetic compatibility testing and rectifying of the aircraft is effectively shortened, the sensitivity of an airborne system to high-power microwaves can be analyzed, and the electromagnetic protection reinforcement requirement is put forward.

In the research of complex electromagnetic fields, dipole wire antennas are the simplest antenna forms and are basic units for forming various complex wire antennas, and the simulation measurement of current of the dipole wire antennas is the basic problem to be solved by building a simulation measurement system. For a dipole wire antenna, the relationship of its electromagnetic field can be represented by the Helen integral equation. Therefore, the electromagnetic field research problem of the antenna is finally solved as a solution problem of the Helen integral equation. However, the Helen integral equation is the first Fredholm integral equation, and it is difficult to directly obtain the analytic solution, especially in the field of practical engineering, the antenna shape and boundary conditions are more complicated, and it is almost impossible to obtain the analytic solution. Therefore, the numerical solution of the Helen integral equation is very important for the simulation measurement of the electromagnetic field of the antenna.

In the process of solving the Helen equation, since the precise kernel of the integral equation has singularity, which seriously affects the accuracy of numerical solution, and must be analyzed, Schelkunoff converts the singular part of the kernel into the first class of elliptic integrals and indicates that the neighborhood of the singularity is the integrable Log singularity. Based on the Schelkunoff work, several existing methods for analyzing the precise singular nucleus are as follows: pearson develops an elliptic integral into a series form analysis singular kernel, but the series of the Pearson is limited, and the estimated error is larger when the point is far away from the origin; another analysis method proposed by Wang is to directly develop the exact kernel into an exact expression containing the sobel function, but the error of Wang's development is larger when the point is close to the origin; davies and the like calculate a kernel function by combining an iterative algorithm of elliptic integral and a composite trapezoidal integral formula; werner converts the expansion into a form more convenient for calculation and use on the basis of Wang work, and Bruno and the like combine the expansion and the trapezoidal integral to obtain an effective analysis strategy; the strategies of Davies et al and Bruno et al, while more effective, are computationally complex and not conducive to engineering implementation.

In addition, most of the existing simulation methods directly use a moment method to solve an integral equation without fully considering the regularity of a current solution, and have large errors. However, this method is relatively complex and the calculation of some parameters, such as the parameters of the hank function and the calculation of integral parameters including weak singularities, is prone to introduce errors.

Disclosure of Invention

The invention provides a simulation measurement technology of antenna current of the most basic unit of a complex electromagnetic field, aiming at the problems of low calculation precision in simulation measurement of the complex electromagnetic field, complex algorithm, inconvenience for engineering realization and the like. The simulation system based on the technology can be used for guiding electromagnetic field compatibility testing to be purposefully carried out, reduces the cost of physical testing, shortens the testing period, and provides an effective design means for realizing electromagnetic compatibility verification and airworthiness conformity authentication/qualification testing of the large aircraft.

The technical scheme adopted by the invention is as follows:

the method comprises the following steps: taking a thin cylindrical dipole antenna, measuring the length l and radius a of the antenna, and applying a pulse voltage V to the center of the antenna₀Feeding to obtain a feed field E_in(z)；

Step two: deriving a Helen integral equation of the thin cylindrical dipole wire antenna according to the Maxwell equation set and the boundary condition of the electric field;

step three: simplifying the Helen equation, and the concrete operations comprise: separating out a main singular part in the kernel function, adopting series expansion of a first type of complete elliptic integral, controlling the calculation precision by controlling the number of expanded terms through a preset error upper limit, and reducing the calculation amount caused by excessive expansion of the series terms;

step four: expanding a Helen equation by adopting a Chebyshev Configuration Method (CCM) and the regularity of a current solution, calculating an integral in an expansion equation by adopting a Gaussian-Chebyshev integral method under the condition that the ratio h/a of the antenna is not very large, and calculating by adopting a Gaussian integral method of a hierarchical grid under the condition that the ratio h/a of the antenna is relatively large;

step five: the current of the line antenna is simulated and calculated, and the current simulation measurement result of the line antenna is output

The Helen integral equation in the second step is as follows:

$\frac{j\mathrm{\η}}{2\mathrm{\π}}{\∫}_{-h}^{h}G(z-z^{\′})I\left(z^{\′}\right){\mathrm{dz}}^{\′}=C_1\cos kz+C_2\sin kz+{\∫}_{-h}^h{E}_{in}\left(z^{\′}\right)\sin k\left|z-z^{\′}\right|{\mathrm{dz}}^{\′}---\left(1\right)$

satisfy the boundary conditions

I(-h)＝I(h)＝0(2)

Wherein

$G\left(z\right)=\frac{1}{2\mathrm{\π}}{\∫}_0^{2\mathrm{\π}}\frac{e^{-jkR}}{R}{\mathrm{d\φ}}^{\′}$

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 a characteristic impedance in the air and,respectively the magnetic permeability and the dielectric constant of air, wherein lambda is the wavelength of electromagnetic waves, and k is 2 pi/lambda;

the simplified process in the third step is as follows:

(1) the kernel function g (z) is expressed as the sum of two parts, thus separating the main singular part ((3) first term in equation):

$G\left(z\right)=-\frac{1}{\mathrm{\π}a}ln\left(\right|z\left|\right)+G^R\left(z\right)---\left(3\right)$

wherein G is^R(z) the remainder of the kernel after removal of the main singularities;

substituting expression (3) of kernel function g (z) into expression (1) and substituting z ' ht ', z ' ht, and the helen equation into

$-\frac{h}{\mathrm{\π}a}{\∫}_{-1}^1\ln \left(\right|t-t^{\′}\left|\right)I\left({\mathrm{ht}}^{\′}\right){\mathrm{dt}}^{\′}-\frac{h}{\mathrm{\π}a}{\∫}_{-1}^1\ln \left(h\right)I\left({\mathrm{ht}}^{\′}\right){\mathrm{dt}}^{\′}+{\∫}_{-1}^1{\mathrm{hG}}^R(ht-{\mathrm{ht}}^{\′})I\left({\mathrm{ht}}^{\′}\right){\mathrm{dt}}^{\′}=\frac{2\mathrm{\π}}{j\mathrm{\η}}V\left(t\right)---\left(4\right)$

Wherein $V\left(t\right)=C_{1}cos\left(kht\right)+C_{2}sin\left(kht\right)+h{\∫}_{-1}^1{E}_{in}\left({\mathrm{ht}}^{\′}\right)\sin kh\left|t-t^{\′}\right|{\mathrm{dt}}^{\′},-1\≤t\≤1;$

Then G in the formula (3)^RIs decomposed into

G^R(z)＝G^C(z)+G^r(z)，(5)

Wherein,

$G^C\left(z\right)=\frac{1}{a\mathrm{\π}}{\∫}_0^{\mathrm{\π}/2}\frac{e^{-j2ka\sqrt{{(z/2a)}^2+{\sin }^2\mathrm{\φ}}}-1}{\sqrt{{(z/2a)}^2+{\sin }^2\mathrm{\φ}}}d\mathrm{\φ}---\left(6\right)$

is a function with good smoothness; and

$G^r\left(z\right)=\frac{1-\mathrm{\κ}}{a\mathrm{\π}}ln\left(\right|z\left|\right)+\frac{\mathrm{\κ}}{a\mathrm{\π}}ln\frac{8a}{\mathrm{\κ}}+\frac{\mathrm{\κ}}{a\mathrm{\π}}\underset{n=1}{\overset{\mathrm{\∞}}{\Σ}}{\left(\frac{(2n-1)!!}{\left(2n\right)!!}\right)}^2{\mathrm{\κ}}^{\′2n}(ln\frac{4}{{\mathrm{\κ}}^{\′}}-\underset{m=1}{\overset{n}{\Σ}}\frac{2}{(2m-1)\left(2m\right)})---\left(7\right)$

wherein $R_{max}=\sqrt{{\left(z\right)}^2+4a^2},$ $\mathrm{\κ}=\frac{2a}{R_{\max }}$ And ${\mathrm{\κ}}^{\′}=\sqrt{1-{\mathrm{\κ}}^2};$ when z is equal to 0, the ratio of z, $G^r\left(0\right)=\frac{\ln \left(8a\right)}{a\mathrm{\π}};$ when G is^r(z) the error is O (κ 'when estimated from the pre-M term of formula (7)'^2M+1) (ii) a However, when κ' approaches 1, that isWhen the time tends to be infinite, the error is gradually increased;

(2) provides a high-efficiency high-precision calculation G^r(z) another series expansion of the first class of perfect elliptic integrals is:

$G^r\left(z\right)=\frac{1}{a\mathrm{\π}}ln\left(\right|z\left|\right)+\frac{1}{a}\frac{\mathrm{\κ}}{1+{\mathrm{\κ}}^{\′}}+\frac{1}{a}\frac{\mathrm{\κ}}{1+{\mathrm{\κ}}^{\′}}\underset{n=1}{\overset{\mathrm{\∞}}{\Σ}}{\left(\frac{(2n-1)!!}{\left(2n\right)!!}\right)}^2{\left(\frac{1-{\mathrm{\κ}}^{\′}}{1+{\mathrm{\κ}}^{\′}}\right)}^{2n}---\left(8\right)$

if G is^r(z) is estimated from the first M term of equation (8) with an error ofOrder toCan obtain the productThe high-precision calculation strategy is as follows: if z satisfiesThen the top M term estimate of (7) is used; otherwise, the top M term estimate of (8) is used, this strategy calculates G^r(z) error ofTrue for all | z | ∈ (0, ∞);

in order to conveniently carry out numerical solution, carrying out equivalent deformation on the Helen integral equation of the thin cylindrical dipole wire antenna:

let z be ht, z 'ht', the helron equation is:

$\underset{-1}{\overset{1}{\∫}}I\left({\mathrm{ht}}^{\′}\right)G(ht-{\mathrm{ht}}^{\′}){\mathrm{dt}}^{\′}=\frac{2\mathrm{\π}}{j\mathrm{\η}h}(C_1\cos kht+V_0\sin k|ht\left|\right),$

let x ═ arccos, y ═ arccos ', -1 ≤ t, t ═ cosx if t' ≤ 1, and t ═ cosy; substituting the obtained data into the Helen equation to obtain:

$\underset{0}{\overset{\mathrm{\π}}{\∫}}I\left(h\cos y\right)G(h\cos x-h\cos y)\sin ydy=\frac{2\mathrm{\π}}{j\mathrm{\η}h}\left(C_{1}cos\right(kh\cos x)+V_0\sin k|h\cos x\left|\right);$

the original equation is simplified to "i (hcasy)" siny

$\underset{0}{\overset{\mathrm{\π}}{\∫}}G(h\cos x-h\cos y)u\left(y\right)dy=\frac{2\mathrm{\π}}{j\mathrm{\η}h}\left(C_{1}cos\right(kh\cos x)+V_0\sin k|h\cos x\left|\right);$

In the fourth step, the simplified Helen equation expansion process by using the CCM method and the current regularization solution is as follows:

let the current be expressed as:

$I\left({\mathrm{ht}}^{\′}\right)=\mathrm{\ω}\left(t^{\′}\right)\underset{n=0}{\overset{N-1}{\Σ}}{I}_n{T}_n\left(t^{\′}\right),t^{\′}\∈(-1,1)---\left(9\right)$

wherein ω (t ') ═ 1-t'²)^-1/2，T_nIs an n-degree Chebyshev polynomial of the first kind, i.e.

T_n(t ') -cos (narccost '), n ≧ 0, t ' ∈ (-1, 1) · (10) and I_nN is 0, …, N-1 is an unknown parameter, N is the number of Chebyshev basis functions, and the selection of N is a positive integer;

by using

$t_n=cos\left(\frac{2n+1}{2N}\mathrm{\π}\right),n=0,1,...,N-1---\left(11\right)$

As a configuration point; therefore, CCM for solving the Helen equation is the determination parameter { I_n: n-0, …, N-1} makes the following:

$-\frac{h}{\mathrm{\π}a}{\∫}_{-1}^1\ln \left(\right|t_n-t^{\′}\left|\right)I\left({\mathrm{ht}}^{\′}\right){\mathrm{dt}}^{\′}-\frac{h}{\mathrm{\π}a}{\∫}_{-1}^1\ln \left(h\right)I\left({\mathrm{ht}}^{\′}\right){\mathrm{dt}}^{\′}+\frac{h}{a}{\∫}_{-1}^1{\mathrm{aG}}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\′})I\left({\mathrm{ht}}^{\′}\right){\mathrm{dt}}^{\′}=\frac{2\mathrm{\π}}{j\mathrm{\η}}V\left(t_n\right).---\left(12\right)$

chebyshev polynomials have the following special properties:

$-\frac{1}{\mathrm{\π}}{\∫}_{-1}^{1}ln\left(\right|t-t^{\′}\left|\right)\mathrm{\ω}\left(t^{\′}\right)T_n\left(t^{\′}\right){\mathrm{dt}}^{\′}=\left\{\begin{array}{cc}\ln 2,& n=0,\\ \frac{1}{n}{T}_n\left(t\right),& n\≥1.\end{array}\right.---\left(13\right)$

the property ensures that the integral of a special function with singular points in the Helen equation series expansion has an analytic solution by taking the Chebyshev polynomial as the weight, thereby ensuring the accuracy of numerical calculation;

preferably, when inRadiation field E_inWhen the current distribution on the antenna is symmetrical about the center when the current distribution is an even function of z, the CCM can be accelerated by omitting half of the odd function substrate (the accelerated CCM is denoted as sCCM), and the solution precision is not influenced; at this time, the approximate expression of the current is

$I\left({\mathrm{ht}}^{\mathrm{\′}}\right)=\mathrm{\ω}\left(t^{\mathrm{\′}}\right)\underset{n=0}{\overset{N-1}{\Σ}}{I}_n{T}_{2n}\left(t^{\mathrm{\′}}\right),t^{\mathrm{\′}}\∈(-1,1)$

The configuration point is

$t_n=cos\frac{4n+1}{4N}\mathrm{\π},n=0,1,...,N-1.$

The sCCM can further improve the calculation efficiency;

from the regularity of the current solution, the current is not calculated directly, but ratherBecause of the fact thatHas better smoothness and can obtain more accurate by adopting a numerical solution method

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

$\frac{h}{a}{\[Z_{nm}\]}_{N\×N}{\[I_n\]}_N={\[V_n\]}_N---\left(14\right)$

wherein

$Z_{nm}=\left\{\begin{array}{cc}\ln 2-\ln h+{\∫}_{-1}^1\mathrm{\ω}\left(t^{\′}\right){\mathrm{aG}}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\′}){\mathrm{dt}}^{\′},& m=0\\ \frac{1}{m}{T}_m\left(t_n\right)+{\∫}_{-1}^1\mathrm{\ω}\left(t^{\′}\right){\mathrm{aG}}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\′})T_m\left(t^{\′}\right){\mathrm{dt}}^{\′},& m\≥1.\end{array}\right.---\left(15\right)$

And isFor N, m is 0, 1, …, N-1;

in the fourth step, when the ratio h/a of the antenna is not very large, the integral of the element in the matrix Z can be calculated by using a gaussian-chebyshev integration method of order M-1:

${\∫}_{-1}^1\mathrm{\ω}\left(t^{\′}\right){\mathrm{aG}}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\′})T_m\left(t^{\′}\right){\mathrm{dt}}^{\′}\≈\frac{\mathrm{\π}}{M}\underset{p=0}{\overset{M-1}{\Σ}}{\mathrm{aG}}^R({\mathrm{ht}}_n-{{\mathrm{ht}}^{\′}}_p)T_m\left({t^{\′}}_p\right)---\left(16\right)$

wherein ${t^{\′}}_p=cos\frac{2p+1}{2M}\mathrm{\π},p=0,1...,M-1;$ Note the book $E={\[E_{n,m}\]}_{N\×M}=\[{\∫}_{-1}^1\mathrm{\ω}\left(t^{\′}\right){\mathrm{aG}}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\′})T_m\left(t^{\′}\right){\mathrm{dt}}^{\′}\],$ $G={\[G_{n,m}\]}_{N\×M}=\[{\mathrm{aG}}^R({\mathrm{ht}}_n-{{\mathrm{ht}}^{\′}}_m)\],$ And is $V={\[V_{n,m}\]}_{M\×M}=\[cos\left(\frac{n(2m+1)}{2M}\mathrm{\π}\right)\],$ Equation (16) can be converted to a matrix form:

$E^T=\frac{\mathrm{\π}}{M}V\·G^T---\left(17\right)$

equation (17) may be calculated by fast cosine transform; the first N columns of E are integral elements required in the discrete matrix Z; the complexity of calculating the entire discrete matrix Z is thus O (N)²MlnM) to o (nmlnm);

obtaining the Chebyshev polynomial expansion coefficient I_nAnd N is 0, 1, … and N-1, and the simulated current I is obtained by substituting the formula (9), and the current simulation measurement result of the antenna is output.

Based on numerical experiments, for example, when N is 64, the relative error is required to be less than or equal to 1%, the ratio h/a of the antenna should be not more than 1 × 10³So that the ratio of the antenna half-length to the antenna radius is less than 10³The line antenna can utilize fast cosine transform to design a fast algorithm, thereby improving the simulation efficiency;

when the ratio of the antenna is large, because of aG^R(z) exhibits approximate singularity at z-0, with sharp vertices, which makes the gaussian-chebyshev integration ineffective; to overcome this difficulty, a gaussian-type integration method based on a hierarchical grid has been used, which has proven to be very efficient for computing weak singular integrals.

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

(1) the current distribution of the line antenna is calculated to carry out simulation measurement on the actual current distribution of the line antenna, and the simulation measurement is applied to the complex electromagnetic compatibility test of the large aircraft, so that the test can be effectively guided, and the cost and the test period of the physical test are reduced;

(2) the singular part of the kernel function of the Helen equation is separated, the singular part is subjected to series expansion by the first-class complete elliptic integral, and the calculation can be estimated by taking M terms before the error given in advance, so that the calculation precision is effectively controlled, the calculated amount caused by excessive series term expansion is reduced, and the simulation efficiency is improved;

(3) in the process of solving the simplified Helen equation by using CCM, for the condition that h/a is larger, the integral precision of a Gaussian-Chebyshev integral method generally adopted at a singular point is seriously influenced, and in order to reduce the influence of the singular point on the error precision, the Gaussian integral method of a hierarchical grid is adopted to improve the calculation precision and reduce the error of a measurement method;

(4) the invention is not very large in the ratio h/a of the antenna half length of the line antenna to the antenna radius, and can adopt the Gaussian-Chebyshev integral method to calculate the integral elements in the discrete matrix of the moment method while ensuring the calculation precision so as to improve the calculation efficiency. In particular for h/a < 10³The simulation efficiency of the line antenna is higher, and the discrete matrix obtained by expanding the Chebyshev polynomial on the Helen equation under the condition can be obtained by fast cosine transform calculation, so that the calculation complexity is further reduced, and the efficiency of the measuring method is improved.

Drawings

FIG. 1 physical model of dipole wire antenna and associated parameters

The symbols in the figures are as follows:

z: position variable of vertical coordinate in the rectangular coordinate system of space;

Δ z: distance between dipole wire antennas;

l: the length of the antenna;

a: the radius of the antenna;

V₀: a pulse voltage at the center of the antenna;

E_in(z): a feed field generated by the pulse voltage;

i (z): and the current corresponding to the position of the vertical coordinate z in the space rectangular coordinate system.

Detailed Description

The invention will be further described with reference to the accompanying drawings.

Because the admittance at the excitation point of the line antenna is tested by an actual physical experiment, the current calculation result of the line antenna is converted into the equivalent admittance for comparison.

Step one, taking a thin cylindrical dipole antenna, and measuring the radius a of the antenna to be 7.002 × 10^-3(m), the length of the antenna is l (m), and the variation range is [0.3, 1 ]]Wavelength λ 1 (meter), excitation voltage 1 volt at the antenna center, and feed field E generated by it_in(z) ═ z, as shown in fig. 1.

Step two: the obtained Helen integral equation is:

$\frac{j\mathrm{\&eta;}}{2\mathrm{\&pi;}}{\&Integral;}_{-h}^{h}G(z-z^{\&prime;})I\left(z^{\&prime;}\right){\mathrm{dz}}^{\&prime;}=C_1\cos kz+C_2\sin kz+{\&Integral;}_{-h}^h{E}_{in}\left(z^{\&prime;}\right)\sin k\left|z-z^{\&prime;}\right|{\mathrm{dz}}^{\&prime;}---\left(18\right)$

wherein I is the current on the surface of the antenna, h is l/2, a is the radius of the antenna,η -120 pi is the characteristic impedance in air, k-2 pi-C₁，C₂Is a constant to be determined whose value is specified in the formulas (27) and (28) in step four of the embodiment;

the integral equation satisfies the boundary condition I (-h) ═ I (h) ═ 0, where the expression of the kernel function g (z) is:

$G\left(z\right)=\frac{1}{2\mathrm{\&pi;}}{\&Integral;}_0^{2\mathrm{\&pi;}}\frac{e^{-jkR}}{R}{\mathrm{d\&phi;}}^{\&prime;}---\left(19\right)$

$R=\sqrt{z^2+2a^2-2a^2{\mathrm{cos\&phi;}}^{\&prime;}}.$

step three: the kernel function g (z) is represented as the sum of two parts:

$G\left(z\right)=-\frac{1}{\mathrm{\&pi;}a}ln\left(\right|z\left|\right)+G^R\left(z\right)---\left(20\right)$

wherein G is^R(z) is the remainder of the kernel after removal of the major singularities, which can be decomposed into G^R(z)＝G^C(z)+G^r(z)，

$G^C\left(z\right)=\frac{1}{a\mathrm{\&pi;}}{\&Integral;}_0^{\mathrm{\&pi;}/2}\frac{e^{-j2ka\sqrt{{(z/2a)}^2+{\sin }^2\mathrm{\&phi;}}}-1}{\sqrt{{(z/2a)}^2+{\sin }^2\mathrm{\&phi;}}}d\mathrm{\&phi;},$ Is a function with good smoothness;

if z satisfies ${\mathrm{\&kappa;}}^{\&prime;}\&le;\sqrt{2}-1,$ Then

$G^r\left(z\right)=\frac{1-\mathrm{\&kappa;}}{a\mathrm{\&pi;}}ln\left(\right|z\left|\right)+\frac{\mathrm{\&kappa;}}{a\mathrm{\&pi;}}ln\frac{8a}{\mathrm{\&kappa;}}+\frac{\mathrm{\&kappa;}}{a\mathrm{\&pi;}}\underset{n=1}{\overset{\mathrm{\&infin;}}{\&Sigma;}}{\left(\frac{(2n-1)!!}{\left(2n\right)!!}\right)}^2{\mathrm{\&kappa;}}^{\&prime;2n}(ln\frac{4}{{\mathrm{\&kappa;}}^{\&prime;}}-\underset{m=1}{\overset{n}{\&Sigma;}}\frac{2}{(2m-1)\left(2m\right)})$

Estimate its value using the top M term, where M is 192;

otherwise

$G^r\left(z\right)=\frac{1}{a\mathrm{\&pi;}}ln\left(\right|z\left|\right)+\frac{1}{a}\frac{\mathrm{\&kappa;}}{1+\mathrm{\&kappa;}}+\frac{1}{a}\frac{\mathrm{\&kappa;}}{1+\mathrm{\&kappa;}}\underset{n=1}{\overset{\mathrm{\&infin;}}{\&Sigma;}}{\left(\frac{(2n-1)!!}{\left(2n\right)!!}\right)}^2{\left(\frac{1-\mathrm{\&kappa;}}{1+\mathrm{\&kappa;}}\right)}^{2n}$

The same estimation is carried out by adopting the former M term, wherein M is 192;

wherein $R_{max}=\sqrt{{\left(z\right)}^2+4a^2},$ $\mathrm{\&kappa;}=\frac{2a}{R_{\max }}$ And ${\mathrm{\&kappa;}}^{\&prime;}=\sqrt{1-{\mathrm{\&kappa;}}^2},$ when z is equal to 0, the ratio of z, $G^r\left(0\right)=\frac{\ln \left(8a\right)}{a\mathrm{\&pi;}};$

substituting the expressions of G (z), the Helen equation can be simplified to

$\underset{0}{\overset{\mathrm{\&pi;}}{\&Integral;}}G(h\cos x-h\cos y)u\left(y\right)dy=\frac{2\mathrm{\&pi;}}{j\mathrm{\&eta;}h}\left(C_{1}cos\right(kh\cos x)+V_0\sin k|h\cos x\left|\right)$

Wherein z is ht, z ' is ht ', x is arccos, y is arccos ', -1 ≦ t ≦ 1, u (y) i (hcosy) siny,

step four: expressing the current as

$I\left({\mathrm{ht}}^{\&prime;}\right)=\mathrm{\&omega;}\left(t^{\&prime;}\right)\underset{n=0}{\overset{N-1}{\&Sigma;}}{I}_n{T}_n\left(t^{\&prime;}\right),t^{\&prime;}\&Element;(-1,1),---\left(21\right)$

Where N is 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 an unknown parameter that needs to be calculated; CCM for solving Helen equation is determined parameter I_nSo that the following holds:

$-\frac{h}{a\mathrm{\&pi;}}{\&Integral;}_{-1}^1\ln \left(\right|t_n-t^{\&prime;}\left|\right)I\left({\mathrm{ht}}^{\&prime;}\right){\mathrm{dt}}^{\&prime;}-\frac{h}{\mathrm{\&pi;}a}{\&Integral;}_{-1}^1\ln \left(h\right)I\left({\mathrm{ht}}^{\&prime;}\right){\mathrm{dt}}^{\&prime;}+\frac{h}{a}{\&Integral;}_{-1}^1{\mathrm{aG}}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\&prime;})I\left({\mathrm{ht}}^{\&prime;}\right){\mathrm{dt}}^{\&prime;}=\frac{2\mathrm{\&pi;}}{j\mathrm{\&eta;}}V\left(t_n\right).---\left(22\right)$

write (22) in an nxn matrix form:

$\frac{h}{a}{\&lsqb;Z_{nm}\&rsqb;}_{N\&times;N}{\&lsqb;I_n\&rsqb;}_N={\&lsqb;V_n\&rsqb;}_N---\left(23\right)$

wherein

$Z_{nm}=\left\{\begin{array}{cc}\ln 2-\ln h+{\&Integral;}_{-1}^1\mathrm{\&omega;}\left(t^{\&prime;}\right){\mathrm{aG}}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\&prime;}){\mathrm{dt}}^{\&prime;},& m=0,\\ \frac{1}{m}{T}_m\left(t_n\right)+{\&Integral;}_{-1}^1\mathrm{\&omega;}\left(t^{\&prime;}\right){\mathrm{aG}}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\&prime;})T_m\left(t^{\&prime;}\right){\mathrm{dt}}^{\&prime;},& m\&GreaterEqual;1.\end{array}\right.---\left(24\right)$

And is $V_n=\frac{2\mathrm{\&pi;}}{j\mathrm{\&eta;}}V\left(t_n\right),(n,m=0,1,...,N-1);$

And (3) calculating the integral of the elements in the matrix Z by using an M-1 order Gauss-Chebyshev integral method:

${\&Integral;}_{-1}^1\mathrm{\&omega;}\left(t^{\&prime;}\right){\mathrm{aG}}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\&prime;})T_m\left(t^{\&prime;}\right){\mathrm{dt}}^{\&prime;}\&ap;\frac{\mathrm{\&pi;}}{M}\underset{p=0}{\overset{M-1}{\&Sigma;}}{\mathrm{aG}}^R({\mathrm{ht}}_n-{{\mathrm{ht}}^{\&prime;}}_p)T_m\left({t^{\&prime;}}_p\right)---\left(25\right)$

whereinThe equation can be calculated by a fast cosine transform.

The solution of an algebraic equation obtained by conversion of the Helen equation and the reconstruction of the real current are completely described below; from linear additivity, the solution of the discrete equation can be expressed as:

$\&lsqb;I_n\&rsqb;=C_1\&lsqb;I_n^{\left(1\right)}\&rsqb;+C_2\&lsqb;I_n^{\left(2\right)}\&rsqb;+\&lsqb;I_n^{\left(3\right)}\&rsqb;---\left(26\right)$

wherein $\&lsqb;Z_{nm}\&rsqb;\&lsqb;I_n^{\left(i\right)}\&rsqb;=\&lsqb;V_n^{\left(i\right)}\&rsqb;,V_n^{\left(1\right)}=\frac{2\mathrm{\&pi;}}{j\mathrm{\&eta;}}{\mathrm{coskht}}_n,V_n^{\left(2\right)}=\frac{2\mathrm{\&pi;}}{j\mathrm{\&eta;}}{\mathrm{sinkht}}_n$ And

$V_n^{\left(3\right)}=\frac{2\mathrm{\&pi;}h}{j\mathrm{\&eta;}}{\&Integral;}_{-1}^1{E}_{in}\left({\mathrm{ht}}^{\&prime;}\right)\sin kh\left|t_n-t^{\&prime;}\right|{\mathrm{dt}}^{\&prime;},$ constant C₁、C₂Is determined by the terminal conditions (2),

$C_1=-\frac{{\&lsqb;u_n\&rsqb;}^T\&lsqb;I_n^{\left(3\right)}\&rsqb;+{\&lsqb;v_n\&rsqb;}^T\&lsqb;I_n^{\left(3\right)}\&rsqb;}{2{\&lsqb;u_n\&rsqb;}^T\&lsqb;I_n^{\left(1\right)}\&rsqb;}---\left(27\right)$

$C_2=-\frac{{\&lsqb;u_n\&rsqb;}^T\&lsqb;I_n^{\left(3\right)}\&rsqb;-{\&lsqb;v_n\&rsqb;}^T\&lsqb;I_n^{\left(3\right)}\&rsqb;}{2{\&lsqb;u_n\&rsqb;}^T\&lsqb;I_n^{\left(2\right)}\&rsqb;}---\left(28\right)$

wherein u is_nV 1_n＝(-1)ⁿ，n＝0，…，N-1；

The current on the antenna can be obtained from equations (21) and (26) and is divided in order to avoid the occurrence ofIn equation (21), the current is represented by the variable t ═ cos θ:

$I\left(hcos\mathrm{\&theta;}\right)=\underset{n=1}{\overset{N-2}{\&Sigma;}}{\tilde{I}}_n\sin n\mathrm{\&theta;},\mathrm{\&theta;}\&Element;(0,\mathrm{\&pi;})---\left(29\right)$

wherein,

and M_o＝N-M_e；

Step five: obtaining the expansion coefficient I of the Chebyshev polynomial according to the formula (26)_nN is 0, 1, …, N-1, N is 64, which is substituted into equation (21) to obtain a simulated current I, which is a result of simulated measurement of the current of the output line antenna, as shown in table 1, where G is G₀Representing the real part of the admittance, B₀Representing the imaginary part of the admittance.

TABLE 1 Current simulation and actual measurements (unit: McSiemens)

If the maximum error and the mean square error of the simulation measurement result and the actual measurement data are respectively defined as follows:

$e_{max}=\underset{i=1,...,K}{max}\left|{\widetilde{\mathrm{\&sigma;}}}_i-{\mathrm{\&sigma;}}_i\right|$

and

$e_{MSE}=\frac{1}{K}\sqrt{\underset{i=1}{\overset{K}{\&Sigma;}}{({\widetilde{\mathrm{\&sigma;}}}_i-{\mathrm{\&sigma;}}_i)}^2}$

wherein G is₀Representing the real part of the admittance, B₀The imaginary part of the admittance is represented,represents G₀Or B₀Simulation of (2) output result, σ_iRepresents G₀Or B₀K denotes the number of output results or measurement results, where K is 50, and the specific error is listed in table 2:

TABLE 2 error of simulation measurement and actual measurement data (unit: milliSiemens)

Type of error	G0	B0
			Maximum error	0.4222	0.8544
Mean square error	0.1158	0.2631

From table 2, it can be seen that the maximum error and the mean square error of the simulation measurement result and the actual measurement data are both controlled within the magnitude of 1 millisiemens, thereby showing that the method provided by the invention is well matched with the physical experiment test result, and has practical engineering application value.

Claims (3)

1. A simulation measurement method of a line antenna current comprises the following steps:

the method comprises the following steps: taking a thin cylindrical dipole antenna, measuring the length l and radius a of the antenna, and applying a pulse voltage V to the center of the antenna₀Feeding to obtain a feed field E_in(z), wherein z is a position variable of a vertical coordinate in the rectangular spatial coordinate system;

step two: deriving a Helen integral equation of the thin cylindrical dipole wire antenna according to the Maxwell equation set and the boundary condition of the electric field;

step three: simplifying the Helen equation, and the concrete operations comprise: separating out a main singular part in the kernel function, adopting series expansion of a first type of complete elliptic integral, controlling the calculation precision by controlling the number of expanded terms through a preset error upper limit, and reducing the calculation amount caused by excessive expansion of the series terms;

step four: adopting a Chebyshev configuration method, namely CCM, and expanding a Helen equation with the regularity of current solution, adopting a Gaussian-Chebyshev integral method to calculate the integral in an expansion equation under the condition that the ratio h/a of the antenna is not very large, and adopting a Gaussian integral method of hierarchical grids to calculate under the condition that the ratio h/a of the antenna is relatively large;

step five: carrying out simulation calculation on the current of the line antenna, and outputting a current simulation measurement result of the line antenna;

the Helen integral equation in the second step is as follows:

satisfy the boundary conditions

I(-h)＝I(h)＝0(2)

Wherein

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, μ, ∈ are the permeability and dielectric constant of air, λ is the wavelength of electromagnetic waves, k is 2 pi/λ;

the simplified process in the third step is as follows:

(1) the kernel function g (z) is expressed as the sum of two parts, thereby separating the main singular part, i.e. the first term in equation (3):

wherein G is^R(z) the remainder of the kernel after removal of the main singularities;

substituting expression (3) of kernel function g (z) into expression (1) and substituting z ' ht ', z ' ht, and the helen equation into

Wherein

Then G in the formula (3)^RIs decomposed into

G^R(z)＝G^C(z)+G^r(z)，(5)

Wherein,

is a function with good smoothness; and

wherein Andwhen z is equal to 0, the ratio of z,when G is^r(z) the error is O (kappa) when estimated from the pre-M term of equation (7)^′2M+1) (ii) a However, when κ' approaches 1, that isWhen the time tends to be infinite, the error is gradually increased;

(2) provides a high-efficiency high-precision calculation G^r(z) another series expansion of the first class of perfect elliptic integrals is:

if G is^r(z) is estimated from the first M term of equation (8) with an error ofOrder toCan obtain the productThe high-precision calculation strategy is as follows: if z satisfiesThen the top M term estimate of (7) is used; otherwise, the top M term estimate of (8) is used, this strategy calculates G^r(z) error ofTrue for all | z | ∈ (0, ∞);

in order to conveniently carry out numerical solution, carrying out equivalent deformation on the Helen integral equation of the thin cylindrical dipole wire antenna:

let z be ht, z 'be ht', the helron equation is:

let x ═ arccos, y ═ arccos ', -1 ≤ t, t ═ cosx if t' ≤ 1, and t ═ cosy; substituting the obtained data into the Helen equation to obtain:

the original equation is simplified to "i (hcasy)" siny

In the fourth step, the simplified Helen equation expansion process by using the CCM method and the current regularization solution is as follows:

let the current be expressed as:

where ω (t') (1-t)^′2)^-1/2，T_nIs an n-degree Chebyshev polynomial of the first kind, i.e.

T_n(t′)＝cos(narccost′)，n≥0，t′∈(-1，1).(10)

And I_nN-1 is an unknown parameter, N is the number of chebyshev basis functions, and N is selected as a positive integer;

by using

As a configuration point; therefore, CCM for solving the Helen equation is the determination parameter { I_n: n-0, 1, N-1, such thatThe formula holds:

chebyshev polynomials have the following special properties:

from the regularity of the current solution, the current is not calculated directly, but rather

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

wherein

And isFor N, m ═ 0, 1, ·, N-1;

in the fourth step, when the ratio h/a of the antenna is not very large, the integral of the element in the matrix Z can be calculated by using a gaussian-chebyshev integration method of order M-1:

whereinNote the book And isEquation (16) can be converted to a matrix form:

equation (17) may be calculated by fast cosine transform; the first N columns of E are integral elements required in the discrete matrix Z; the complexity of calculating the entire discrete matrix Z is thus O (N)²MlnM) to o (nmlnm);

obtaining the Chebyshev polynomial expansion coefficient I_nAnd N is 0, 1, N-1, and the simulated current I is obtained by substituting the formula (9), and the current simulation measurement result of the antenna is output.

2. The method for simulating and measuring the current of the line antenna as claimed in claim 1, wherein:

in step three, when the incident field E_inWhen the current distribution on the antenna is symmetrical about the center when the current distribution is an even function of z, half of an odd function substrate can be omitted to accelerate CCM, and the accelerated CCM is written as sCCM without influencing the solution precision; at this time, the approximate expression of the current is

The configuration point is

。

3. The method for simulating and measuring the current of the line antenna as claimed in claim 1, wherein: the method can be used in a range that the ratio of the half length of the antenna to the radius of the antenna is less than 10³The line antenna of (1).