CN103969495A - Simulation measure method of linear antenna current - Google Patents

Abstract

The invention discloses a simulation measure method of a linear antenna current. Firstly, the length and the radius of an antenna and an impulse voltage at the center of the antenna are measured; the Helen integral equation of a physical model is derived according to Maxwell equations and electric field boundary conditions; the main strange part in an integral kernel is separated, and the Helen integral equation is simplified; the Helen integral equation is expanded by adopting the Chebyshev collocation method and regularity of an electric solution, and the integral in the expansion is calculated; simulation calculation is performed on currents of the linear antenna, and a current simulation measurement result of the linear antenna is output. Simulation measurement is performed on actual current distribution of the linear antenna by calculating the current distribution of the linear antenna, the simulation measure method can be applied to complex electromagnetic compatibility tests of large aircrafts, cost of a physical test is reduced, and the test period of the physical test is shortened.

Description

A kind of simulation measuring method of wire antenna electric current

Technical field

The present invention relates to the simulation measuring method of electromagnetic field, particularly a kind of simulation measuring method of wire antenna electromagnetic field.

Background technology

Airdreadnought complex structure, number of electronic devices is various, cable bundle network spreads all over whole aircraft, the design of complete machine cable bundle network structure is complicated, Electromagnetic Coupling Characteristic is measured difficulty, therefore the virtual design of airdreadnought complete machine cable bundle network electromagnetic coupled characteristic and the research of checking, have important practical value to the development of airdreadnought.At present, China also rests on the stage of testing and rectifying and improving according to experience to the Electro Magnetic Compatibility checking of aircraft, therefore the simulated measurement system of exploitation simulation aircraft complete machine electromagnetic compatible and electromagnetic protection ability, can reduce future aircraft and system delivery time, cut down physical testing cost, effectively shorten cycle of aircraft emc testing and rectification, and can the susceptibility of analytical engine loading system to High-Power Microwave, propose electromagnetic protection and reinforce requirement.

In complex electromagnetic fields research, dipole wire antenna is the simplest antenna form, is the elementary cell that forms all kinds of complex wire antennas, and the simulated measurement of its electric current is to build the basic problem that simulated measurement system will solve.For dipole wire antenna, the relation of its electromagnetic field can be represented by Helen's integral equation.Therefore the Research of electromagnetic field problem of antenna is finally summed up as the Solve problems of Helen's integral equation.But Helen's integral equation is Fredholm Linear Integral Equations of First Kind, it is more difficult directly trying to achieve its analytic solution, and particularly in Practical Project field, antenna pattern and boundary condition are more complicated, and it is almost impossible trying to achieve its analytic solution.Therefore, Helen's numerical solution of integral equation is very important for the simulated measurement of Electromagnetic Fields of Antenna.

Solving in the process of Helen's equation, because the accurate core of integral equation has singularity, can have a strong impact on the precision of numerical solution, must analyze it, therefore the singular part of core is converted into elliptic integral of the first kind by Schelkunoff, and point out unusual neighbouring part be can be long-pending Log unusual.Based on the work of Schelkunoff, the analytical approach of existing several accurate singular kernels is as follows: Pearson is by elliptic integral series development form analysis singular kernel, but the progression of Pearson is limited, and estimates at point larger away from initial point time error; Another analytical approach that Wang proposes is direct by accurate expression that comprises spheric Bessel function of accurate core generate, but the error ratio of the exhibition formula of Wang in the time that point approaches initial point is larger; Davies etc. calculate kernel function in conjunction with iterative algorithm and the compound trapezoidal integration formula of elliptic integral; Werner changes into exhibition formula on the working foundation of Wang to be convenient to calculate the form using more, and Bruno etc. have obtained a kind of effective analysis strategy in conjunction with this exhibition formula and trapezoidal integration; Although the strategy of Davies etc. and Bruno etc. is more effective, more complicated on calculating, is unfavorable for Project Realization.

In addition, current most emulation mode is in the regularity that does not take into full account solution for the current, directly utilize method of moment to solve integral equation, error is larger, though Bruno etc. have utilized the regularity of solution for the current and developed a kind of effective ways that solve integral equation that decompose based on new integral kernel.But error is easily introduced in the calculating of the method relative complex and some parameter, the calculating of for example parameter of Hunk function and the integral parameter that comprises weak singularity.

Summary of the invention

The present invention is directed in complex electromagnetic fields simulated measurement computational accuracy not high, and algorithm complexity is unfavorable for the problems such as Project Realization, proposed a kind of simulated measurement technology of elementary cell wire antenna electric current of complex electromagnetic fields.Analogue system based on this technology can be used in reference to conduction magnetic field compatibility test and purposively carry out, reduce the cost of physical testing, shorten test period, for the Electro Magnetic Compatibility checking and the certification/qualification test of seaworthiness accordance that realize airdreadnought provide design means effectively.

The technical solution adopted in the present invention is as follows:

Step 1: get a thin cylindrical dipole line, measure the length l of antenna, radius a, the pulse voltage V of center of antenna place ₀carry out feed, obtain the feed field E of its generation _in(z);

Step 2: by derive Helen's integral equation of this physical model of Maxwell equation group and electric field boundary condition;

Step 3: Helen's equation is carried out to abbreviation, concrete operations comprise: main singular part in integral kernel is separated, adopt the series expansion of complete elliptic integral of the first kind, the item number launching by predefined error upper limit control is controlled computational accuracy, and reduces the calculated amount of too much bringing because of the several expansion of level;

Step 4: adopt the regularity of Chebyshev collocation method (brief note is CCM) and solution for the current by Helen's equation expansion, not that very large situation adopts Gauss-Chebyshev integral method to calculate the integration in expansion to the ratio h/a of antenna, adopt the Gaussian integral method of grade gridding to calculate to the larger situation of the ratio h/a of antenna;

Step 5: the electric current of wire antenna is carried out to simulation calculation, the current simulations measurement result of output line antenna.

Helen's integral equation in described step 2 is:

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

Meet boundary condition

I(-h)＝I(h)＝0 (2)

Wherein

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

i is the electric current of antenna surface, and the half that h is length of antenna a days is line radius, for airborne characteristic impedance, μ, ∈ is respectively magnetic permeability and the specific inductive capacity of air, and λ is electromagnetic wavelength, k=2 π/λ;

Simplification process in described step 3 is as follows:

(1) integral kernel G (z) is expressed as two-part and, thereby main singular part (Section 1 in (3) formula) is separated:

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

Wherein G ^r(z) for removing remaining part after main singularity in core;

By (3) substitution of the expression formula of kernel function G (z) and do variable replace z '=ht ', z=ht, Helen's equation can turn to

$-\frac{h}{\mathrm{\πa}}{\∫}_{-1}^1\ln \left(\right|t-t^{\text{'}}\left|\right)I\left({\mathrm{ht}}^{\text{'}}\right){\mathrm{dt}}^{\text{'}}-\frac{h}{\mathrm{\πa}}{\∫}_{-1}^1\ln \left(h\right)I\left({\mathrm{ht}}^{\text{'}}\right){\mathrm{dt}}^{\text{'}}+{\∫}_{-1}^1{\mathrm{hG}}^R(\mathrm{ht}-{\mathrm{ht}}^{\text{'}})I\left({\mathrm{ht}}^{\text{'}}\right){\mathrm{dt}}^{\text{'}}=\frac{2\mathrm{\π}}{\mathrm{j\η}}V\left(t\right)---\left(4\right)$

Wherein $V\left(t\right)=C_1\cos \left(\mathrm{kht}\right)+C_2\sin \left(\mathrm{kht}\right)+h{\∫}_{-1}^1{E}_{\mathrm{in}}\left({\mathrm{ht}}^{\text{'}}\right)\sin \mathrm{kh}|t-t^{\text{'}}|{\mathrm{dt}}^{\text{'}},-1\≤t\≤1;$

Again by G in formula (3) ^rbe decomposed into

G ^R(z)＝G ^C(z)+G ^r(z),(5)

Wherein,

$G^C\left(z\right)=\frac{1}{\mathrm{a\π}}{\∫}_0^{\mathrm{\π}/2}\frac{e^{-j2\mathrm{ka}\sqrt{{(z/2a)}^2+{\sin }^{2\mathrm{\φ}}}}-1}{\sqrt{{(z/2a)}^2+{\sin }^2\mathrm{\φ}}}\mathrm{d\φ}---\left(6\right)$

It is a good function of slickness; And

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

Wherein $R_{\max }=\sqrt{{\left(z\right)}^2+{4a}^2},\mathrm{\κ}=\frac{2a}{R_{\max }}$ And ${\mathrm{\κ}}^{\text{'}}=\sqrt{1-{\mathrm{\κ}}^2};$ Work as z=0, $G^r\left(0\right)=\frac{\ln \left(8a\right)}{\mathrm{a\π}};$ Work as G ^r(z) while estimation by the front M item of formula (7), error be O (κ ' ^2M+1); But, when κ ' levels off to 1 while being tending towards infinite, error can increase gradually;

(2) propose a kind of high-efficiency high-accuracy and calculate G ^r(z) strategy, is had by another progression exhibition formula of complete elliptic integral of the first kind:

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

If G ^r(z) estimated by the front M item of formula (8), error is order can obtain high-precision calculative strategy is as follows: if z meets adopt the front M item of (7) to estimate; Not so, adopt the front M item of (8) to estimate, the G of this policy calculation ^r(z) error is to all | z| ∈ (0, ∞) sets up;

Carry out for convenience numerical solution, Helen's integral equation of this physical model carried out to equivalence distortion:

Make z=ht, z'=ht', Helen's equation turns to:

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

Make again x=arccost, y=arccost' ,-1≤t, there are t=cosx, t'=cosy in t'≤1 item; Being taken to Helen's equation just can obtain: $\underset{0}{\overset{\mathrm{\π}}{\∫}}I\left(h\cos y\right)G(h\cos x-h\cos y)\sin \mathrm{ydy}=\frac{2\mathrm{\π}}{\mathrm{j\ηh}}(C_1\cos \left(\mathrm{kh}\cos x\right)+V_0\sin k|h\cos x\left|\right);$

Note u (y)=I (hcosy) siny, full scale equation is reduced to

$\underset{0}{\overset{\mathrm{\π}}{\∫}}G(h\cos x-h\cos y)u\left(y\right)\mathrm{dy}=\frac{2\mathrm{\π}}{\mathrm{j\ηh}}(C_1\cos \left(\mathrm{kh}\cos x\right)+V_0\sin k|h\cos x\left|\right);$

The regularity of utilizing CCM method and electric current in described step 4 solves as follows the Helen's equation expansion process after abbreviation:

If electric current is expressed as follows:

$I\left({\mathrm{ht}}^{\text{'}}\right)=\mathrm{\ω}\left(t^{\text{'}}\right)\underset{n=0}{\overset{N-1}{\mathrm{\Σ}}}{I}_n{T}_n\left(t^{\text{'}}\right),t^{\text{'}}\∈(-\mathrm{1,1})---\left(9\right)$

Wherein ω (t ')=(1-t ' ²) ^-1/2, T _nn Chebyshev polynomials of the first kind,

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

And I _n, n=0 ..., N-1 is unknown parameter, and N is the number of Chebyshev's basis function, and N is chosen as positive integer;

With

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

As collocation point; Therefore the CCM that, solves Helen's equation is definite parameter { I _n: n=0 ..., N-1} sets up following formula:

$-\frac{h}{\mathrm{a\π}}{\∫}_{-1}^1\ln \left(\right|t_n-t^{\text{'}}\left|\right)I\left({\mathrm{ht}}^{\text{'}}\right){\mathrm{dt}}^{\text{'}}-\frac{h}{\mathrm{a\π}}{\∫}_{-1}^1\ln \left(h\right)I\left({\mathrm{ht}}^{\text{'}}\right){\mathrm{dt}}^{\text{'}}+\frac{h}{a}{\∫}_{-1}^1{\mathrm{aG}}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\text{'}})I\left({\mathrm{ht}}^{\text{'}}\right){\mathrm{dt}}^{\text{'}}=\frac{2\mathrm{\π}}{\mathrm{j\η}}V\left(t_n\right).---\left(12\right)$

Chebyshev polynomials have following special nature:

$-\frac{1}{\mathrm{\π}}{\∫}_{-1}^1\ln \left(\right|t-t^{\text{'}}\left|\right)\mathrm{\ω}\left(t^{\text{'}}\right)T_n\left(t^{\text{'}}\right){\mathrm{dt}}^{\text{'}}=\left\{\begin{array}{cc}\ln 2,& n=0,\\ \frac{1}{n}{T}_n\left(t\right),& n\≥1.\end{array}\right.---\left(13\right)$

This character makes taking Chebyshev polynomials as power, the special function that has a singular point of a class in Helen's equation series expansion is quadratured and is had analytic solution, thereby ensured the precision of numerical evaluation;

Preferably, as incident field E _inwhile being the even function of z, the distribution of current on antenna is about centrosymmetric, can accelerate CCM (CCM of note acceleration is sCCM) by omitting the odd function substrate of half, and not affect the precision of solution; Now the approximate expression of electric current is

$I\left({\mathrm{ht}}^{\text{'}}\right)=\mathrm{\ω}\left(t^{\text{'}}\right)\underset{n=0}{\overset{N-1}{\mathrm{\Σ}}}{I}_n{T}_{2n}\left(t^{\text{'}}\right),t^{\text{'}}\∈(-\mathrm{1,1})$

Its collocation point is

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

SCCM can further improve counting yield;

By the regularity of solution for the current, directly do not calculate electric current, but calculate because there is better slickness, adopt numerical solution can obtain more accurate ;

By (9) substitution of the expression formula of I (ht ') and in conjunction with special nature (13), formula (12) can be write as N × N matrix form:

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

Wherein

$Z_{\mathrm{nm}}=\left\{\begin{array}{cc}\ln 2-\ln h+{\∫}_{-1}^1\mathrm{\ω}\left(t^{\text{'}}\right)a{G}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\text{'}}){\mathrm{dt}}^{\text{'}},& m=0,\\ \frac{1}{m}{T}_m\left(t_n\right)+{\∫}_{-1}^1\mathrm{\ω}\left(t^{\text{'}}\right)a{G}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\text{'}})T_m\left(t^{\text{'}}\right){\mathrm{dt}}^{\text{'}},& m\≥1.\end{array}\right.---\left(15\right)$

And for n, m=0,1 ..., N-1;

In described step 4, when the ratio h/a of antenna is not while being very large, can utilize the integration of element in Gauss-Chebyshev integral method compute matrix Z on M-1 rank:

${\∫}_{-1}^1\mathrm{\ω}\left(t^{\text{'}}\right)a{G}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\text{'}})T_m\left(t^{\text{'}}\right){\mathrm{dt}}^{\text{'}}\≈\frac{\mathrm{\π}}{M}\underset{p=0}{\overset{M-1}{\mathrm{\Σ}}}a{G}^R({\mathrm{ht}}_n-{{\mathrm{ht}}^{\text{'}}}_p)T_m\left({t^{\text{'}}}_p\right)---\left(16\right)$

Wherein ${t^{\text{'}}}_p=\cos \frac{2p+1}{2M}\mathrm{\π}=\mathrm{0,1}...,M-1;$ Note $E={\left[E_{n,m}\right]}_{N\×M}=\left[{\∫}_{-1}^1\mathrm{\ω}\left(t^{\text{'}}\right)a{G}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\text{'}})T_m\left(t^{\text{'}}\right){\mathrm{dt}}^{\text{'}}\right],$ $G={\left[G_{n,m,}\right]}_{N\×M}=\left[a{G}^R({\mathrm{ht}}_n-{{\mathrm{ht}}^{\text{'}}}_m)\right],$ And $G={\left[V_{n,m,}\right]}_{M\×M}=\left[\cos \left(\frac{n(2m+1)}{2M}\mathrm{\π}\right)\right],$ Formula (16) can be converted into matrix form:

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

Formula (17) can be calculated by fast cosine transform; The front N row of E are the integration unit of the needs in discrete matrix Z; Thereby the complexity of calculating whole discrete matrix Z is by O (N ²mlnM) reduce to O (NMlnM);

Obtain Chebyshev polynomials expansion coefficient I _n, n=0,1 ..., N-1, obtains simulated current I by its substitution (9) formula, the current simulations measurement result of output line antenna.

Based on numerical experiment, as: in the time using N=64, require relative error≤1%, the ratio h/a ratio of antenna should be not more than 1 × 10 ³, be therefore less than 10 to antenna half is long with the ratio of antenna radius ³wire antenna, can utilize fast cosine transform design fast algorithm, thereby improve simulation efficiency;

In the time that the ratio of antenna is larger, because aG ^r(z) show approximate singularity at z=0, there will be very sharp-pointed summit, this character lost efficacy Gauss-Chebyshev integral method; In order to overcome this difficulty, adopt the Gaussian integral method based on grade gridding, this integral method has been proved to be can calculate weak unusual integration very effectively.

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

(1) by the distribution of current of calculating wire antenna, the actual current of wire antenna is distributed and carries out simulated measurement, be applied to the complicated emc testing of airdreadnought, can effectively instruct test, reduce cost and the test period of physical testing;

(2) singular part of the integral kernel of Helen's equation is separated, by complete elliptic integral of the first kind, singular part being carried out to progression exhibition launches, its calculating can be got front M item by prior given error and be estimated, not only effectively control computational accuracy, and reduced the calculated amount that the several expansion of level too much bring, thereby improve simulation efficiency;

(3) solve in the process of abbreviation Helen equation at use CCM, the situation larger to h/a, conventionally Gauss-Chebyshev integral method the integral accuracy adopting at singular point place can be had a strong impact on, in order to reduce the impact of singular point on error precision, adopt the Gaussian integral method of grade gridding to improve computational accuracy, reduced the error of measuring method;

(4) the present invention to the antenna of wire antenna half long with the ratio h/a of antenna radius be not very large situation, in ensureing computational accuracy, can adopt Gauss-Chebyshev integral method to calculate the element of integration in the discrete matrix of method of moment, to improve counting yield.Especially for h/a<10 ³wire antenna simulation efficiency higher can calculate by fast cosine transform because Helen's equation is carried out to the discrete matrix that Chebyshev polynomials launch to obtain under this condition, further reduced computation complexity, improved the efficiency of measuring method.

Brief description of the drawings

Fig. 1 dipole wire antenna physical model and correlation parameter

In figure, symbol description is as follows:

Z: the location variable of ordinate in rectangular coordinate system in space;

Δ z: the distance between dipole antenna;

L: the length of antenna;

A: the radius of antenna;

V ₀: the pulse voltage at center of antenna place;

E _in(z): the feed field that pulse voltage produces;

I (z): electric current corresponding to ordinate z place in rectangular coordinate system in space.

Embodiment

Below in conjunction with accompanying drawing, the present invention is described further.

Due to actual physics experiment test be the admittance at wire antenna point of excitation place, for convenient relatively, the Current calculation result of wire antenna is converted into admittance of equal value.

Step 1: get a thin cylindrical dipole line, measure radius a=7.002 × 10 of antenna ^-3(rice), the length of antenna is l (rice), its variation range is [0.3,1], wavelength X=1 (rice), center of antenna place driving voltage is 1 volt, the feed field E of its generation _in(z)=δ (z), as shown in Figure 1.

Step 2: obtaining Helen's integral equation is:

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

The electric current that wherein I is antenna surface, h=l/2, a is antenna radius, for airborne characteristic impedance, k=2 π.C ₁, C ₂be constant undetermined, its value provides detailed description in (27), (28) formula of specific embodiments step 4;

This integral equation meet boundary condition I (h)=I (h)=0, wherein the expression formula of integral kernel G (z) is:

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

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

Step 3: by integral kernel G (z) be expressed as two-part and:

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

Wherein G ^r(z) for removing remaining part after main singularity in core, it can be decomposed into again G ^r(z)=G ^c(z)+G ^r(z), $G^C\left(z\right)=\frac{1}{\mathrm{a\&pi;}}{\&Integral;}_0^{\mathrm{\&pi;}/2}\frac{e^{-j2\mathrm{ka}\sqrt{{(z/2a)}^2+{\sin }^{2\mathrm{\&phi;}}}}-1}{\sqrt{{(z/2a)}^2+{\sin }^2\mathrm{\&phi;}}}\mathrm{d\&phi;},$ It is a good function of slickness;

If z meets ?

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

Before adopting, M item is estimated its value, wherein M=192;

Otherwise

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

Before same employing, M item is estimated, M=192;

Wherein $R_{\max }=\sqrt{{\left(z\right)}^2+{4a}^2},\mathrm{\&kappa;}=\frac{2a}{R_{\max }}$ And ${\mathrm{\&kappa;}}^{\text{'}}=\sqrt{1-{\mathrm{\&kappa;}}^2};$ Work as z=0, $G^r\left(0\right)=\frac{\ln \left(8a\right)}{\mathrm{a\&pi;}};$

By the expression formula substitution of G (z), Helen's equation can be reduced to

$\underset{0}{\overset{\mathrm{\&pi;}}{\&Integral;}}G(h\cos x-h\cos y)u\left(y\right)\mathrm{dy}=\frac{2\mathrm{\&pi;}}{\mathrm{j\&eta;h}}(C_1\cos \left(\mathrm{kh}\cos x\right)+V_0\sin k|h\cos x\left|\right)$

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

Step 4: reometer is shown

$I\left({\mathrm{ht}}^{\text{'}}\right)=\mathrm{\&omega;}\left(t^{\text{'}}\right)\underset{n=0}{\overset{N-1}{\mathrm{\&Sigma;}}}{I}_n{T}_n\left(t^{\text{'}}\right),t^{\text{'}}\&Element;(-\mathrm{1,1}),---\left(21\right)$

Wherein N=64, ω (t ')=(1-t ' ²) ^-1/2, T _n(t ')=cos (narccost '), n>=0, t ' ∈ (1,1), (n=0,1 ..., N-1) be collocation point, I _n(n=0,1 ..., N-1) be calculative unknown parameter; The CCM that solves Helen's equation is definite parameter I _nfollowing formula is set up:

$-\frac{h}{\mathrm{a\&pi;}}{\&Integral;}_{-1}^1\ln \left(\right|t_n-t^{\text{'}}\left|\right)I\left({\mathrm{ht}}^{\text{'}}\right){\mathrm{dt}}^{\text{'}}-\frac{h}{\mathrm{a\&pi;}}{\&Integral;}_{-1}^1\ln \left(h\right)I\left({\mathrm{ht}}^{\text{'}}\right){\mathrm{dt}}^{\text{'}}+\frac{h}{a}{\&Integral;}_{-1}^1{\mathrm{aG}}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\text{'}})I\left({\mathrm{ht}}^{\text{'}}\right){\mathrm{dt}}^{\text{'}}=\frac{2\mathrm{\&pi;}}{\mathrm{j\&eta;}}V\left(t_n\right).---\left(22\right)$

(22) are write as to N × N matrix form:

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

Wherein

$Z_{\mathrm{nm}}=\left\{\begin{array}{cc}\ln 2-\ln h+{\&Integral;}_{-1}^1\mathrm{\&omega;}\left(t^{\text{'}}\right)a{G}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\text{'}}){\mathrm{dt}}^{\text{'}},& m=0,\\ \frac{1}{m}{T}_m\left(t_n\right)+{\&Integral;}_{-1}^1\mathrm{\&omega;}\left(t^{\text{'}}\right)a{G}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\text{'}})T_m\left(t^{\text{'}}\right){\mathrm{dt}}^{\text{'}},& m\&GreaterEqual;1.\end{array}\right.---\left(24\right)$

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

Utilize the integration of element in Gauss-Chebyshev integral method compute matrix Z on M-1 rank:

${\&Integral;}_{-1}^1\mathrm{\&omega;}\left(t^{\text{'}}\right)a{G}^R({\mathrm{ht}}_n-{\mathrm{ht}}^{\text{'}})T_m\left(t^{\text{'}}\right){\mathrm{dt}}^{\text{'}}\&ap;\frac{\mathrm{\&pi;}}{M}\underset{p=0}{\overset{M-1}{\mathrm{\&Sigma;}}}a{G}^R({\mathrm{ht}}_n-{{\mathrm{ht}}^{\text{'}}}_p)T_m\left({t^{\text{'}}}_p\right)---\left(25\right)$

Wherein can calculate this formula by fast cosine transform.

Intactly describe Helen's equation below and transform solving and the reconstruct of real current of the algebraic equation that obtains; By linear superposition, discrete equation solution can be expressed as:

$\left[I_n\right]=C_1\left[I_n^{\left(1\right)}\right]+C_2\left[I_n^{\left(2\right)}\right]+\left[I_n^{\left(3\right)}\right]---\left(26\right)$

Wherein $\left[Z_{\mathrm{nm}}\right]\left[I_n^{\left(i\right)}\right]=\left[V_n^{\left(i\right)}\right],V_n^{\left(1\right)}=\frac{2\mathrm{\&pi;}}{\mathrm{j\&eta;}}\cos {\mathrm{kht}}_n,V_n^{\left(2\right)}=\frac{2\mathrm{\&pi;}}{\mathrm{j\&eta;}}\sin {\mathrm{kht}}_n$ And

$V_n^{\left(3\right)}=\frac{2\mathrm{\&pi;h}}{\mathrm{j\&eta;}}{\&Integral;}_{-1}^1{E}_{\mathrm{in}}\left({\mathrm{ht}}^{\text{'}}\right)\sin \mathrm{kh}|t_n-t^{\text{'}}|{\mathrm{dt}}^{\text{'}},$ Constant C ₁, C ₂determined by terminal condition (2),

$C_1=-\frac{{\left[u_n\right]}^T\left[I_n^{\left(3\right)}\right]+{\left[v_n\right]}^T\left[I_n^{\left(3\right)}\right]}{2{\left[u_n\right]}^T\left[I_n^{\left(1\right)}\right]}---\left(27\right)$

$C_2=-\frac{{\left[u_n\right]}^T\left[I_n^{\left(3\right)}\right]+{\left[v_n\right]}^T\left[I_n^{\left(3\right)}\right]}{2{\left[u_n\right]}^T\left[I_n^{\left(2\right)}\right]}---\left(28\right)$

Wherein u _n=1 and v _n=(1) ⁿ, n=0 ..., N-1;

Electric current on antenna can be obtained by formula (21) and (26), for fear of occur divided by in formula (21), make variable and replace t=cos θ, reometer is shown:

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

Wherein,

and M _o=N-M _e;

Step 5: obtain Chebyshev polynomials expansion coefficient I according to (26) formula _n, n=0,1 ..., N-1, N=64, (21) formula that carries it into obtains simulated current I, and the current simulations measurement result of output line antenna is as shown in table 1, wherein G ₀represent the real part of admittance, B ₀represent the imaginary part of admittance.

Table 1 current simulations measurement result and actual measured results (unit: milli Siemens)

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

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

And

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

Wherein, G ₀represent the real part of admittance, B ₀represent the imaginary part of admittance, represent G ₀or B ₀simulation data result, σ _irepresent G ₀or B ₀actual measured results, K represents the number of Output rusults or measurement result, K=50 here, concrete error is listed in table 2:

The result of table 2 simulated measurement and the error of actual measurement data (unit: milli Siemens)

Error pattern	G 0	B 0
			Maximum error	0.4222	0.8544
Square error	0.1158	0.2631

The result of simulated measurement and the maximum error of actual measurement data and square error are all controlled in the magnitude of 1 milli Siemens as can be seen from Table 2, show that thus method and Physical Experiment test result that the present invention proposes are finely identical, there is actual engineering using value.

Claims (4)

1. a simulation measuring method for wire antenna electromagnetic field, comprises the steps:

Step 1: get a thin cylindrical dipole line, measure the length l of antenna, radius a, the pulse voltage V of center of antenna place ₀carry out feed, obtain the feed field E of its generation _in(z);

Step 2: by derive Helen's integral equation of this physical model of Maxwell equation group and electric field boundary condition;

Step 3: Helen's equation is carried out to abbreviation, concrete operations comprise: main singular part in integral kernel is separated, adopt the series expansion of complete elliptic integral of the first kind, the item number launching by predefined error upper limit control is controlled computational accuracy, and reduces the calculated amount of too much bringing because of the several expansion of level;

Step 4: adopt the regularity of Chebyshev collocation method (brief note is CCM) and solution for the current by Helen's equation expansion, not that very large situation adopts Gauss-Chebyshev integral method to calculate the integration in expansion to the ratio h/a of antenna, adopt the Gaussian integral method of grade gridding to calculate to the larger situation of the ratio h/a of antenna;

Step 5: the electric current of wire antenna is carried out to simulation calculation, the current simulations measurement result of output line antenna.

2. a kind of simulation measuring method of wire antenna electromagnetic field described in claim 1, is characterized in that:

Helen's integral equation in described step 2 is:

Meet boundary condition

I(-h)＝I(h)＝0 (2)

Wherein

i is the electric current of antenna surface, and the half that h is length of antenna a days is line radius, for airborne characteristic impedance, μ, ∈ is respectively magnetic permeability and the specific inductive capacity of air, and λ is electromagnetic wavelength, k=2 π/λ;

Simplification process in described step 3 is as follows:

(1) integral kernel G (z) is expressed as two-part and, thereby main singular part (Section 1 in (3) formula) is separated:

Wherein G ^r(z) for removing remaining part after main singularity in core;

By (3) substitution of the expression formula of kernel function G (z) and do variable replace z '=ht ', z=ht, Helen's equation can turn to

Wherein

Again by G in formula (3) ^rbe decomposed into

G ^R(z)＝G ^C(z)+G ^r(z),(5)

Wherein,

It is a good function of slickness; And

Wherein and work as z=0, work as G ^r(z) while estimation by the front M of formula (7), error be O (κ ' ^2M+1); But, when κ ' levels off to 1 while being tending towards infinite, error can increase gradually;

(2) propose a kind of high-efficiency high-accuracy and calculate the strategy of Gr (z), had by another progression exhibition formula of complete elliptic integral of the first kind:

If G ^r(z) estimated by the front M of formula (8), error is order can obtain high-precision calculative strategy is as follows: if z meets adopt the front M item of (7) to estimate; Not so, adopt the front M item of (8) to estimate, Gr (z) error of this policy calculation is to all | z| ∈ (0, ∞) sets up;

Carry out for convenience numerical solution, Helen's integral equation of this physical model carried out to equivalence distortion:

Make z=ht, z'=ht', Helen's equation turns to:

Make again x=arccost, y=arccost' ,-1≤t, there are t=cosx, t'=cosy in t'≤1 item; Being taken to Helen's equation just can obtain:

Note u (y)=I (hcosy) siny, full scale equation is reduced to

The regularity of utilizing CCM method and electric current in described step 4 solves as follows the Helen's equation expansion process after abbreviation:

If electric current is expressed as follows:

Wherein ω (t ')=(1-t ' ²) ^-1/2, T _nn Chebyshev polynomials of the first kind,

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

And I _n, n=0 ..., N-1 is unknown parameter, and N is the number of Chebyshev's basis function, and N is chosen as positive integer;

With

As collocation point; Therefore the CCM that, solves Helen's equation is definite parameter { I _n: n=0 ..., N-1} sets up following formula:

Chebyshev polynomials have following special nature:

By the regularity of solution for the current, directly do not calculate electric current, but calculate

By (9) substitution of the expression formula of I (ht ') and in conjunction with special nature (13), formula (12) can be write as N × N matrix form:

Wherein

And for n, m=0,1 ..., N-1;

In described step 4, when the ratio h/a of antenna is not while being very large, can utilize the integration of element in Gauss-Chebyshev integral method compute matrix Z on M-1 rank:

Wherein note and formula (16) can be converted into matrix form:

Formula (17) can be calculated by fast cosine transform; The front N row of E are the integration unit of the needs in discrete matrix Z; Thereby the complexity of calculating whole discrete matrix Z is by O (N ²mlnM) reduce to O (NMlnM);

Obtain Chebyshev polynomials expansion coefficient I _n, n=0,1 ..., N-1, obtains simulated current I by its substitution (9) formula, the current simulations measurement result of output line antenna.

3. a kind of simulation measuring method of wire antenna electromagnetic field described in claim 1 or 2, is characterized in that:

In step 3, as incident field E _inwhile being the even function of z, the distribution of current on antenna is about centrosymmetric, can accelerate CCM (CCM of note acceleration is sCCM) by omitting the odd function substrate of half, and not affect the precision of solution; Now the approximate expression of electric current is

Its collocation point is

。

4. a kind of simulation measuring method of wire antenna electromagnetic field described in claim 1 or 2, is characterized in that: the usable range of this method is that antenna half length is less than 10 with the ratio of antenna radius ³wire antenna.