CN106980095A - A kind of Meta Materials electromagnetic parameter inversion method based on improvement K K algorithms - Google Patents

Abstract

The invention discloses a kind of inversion method based on the Meta Materials electromagnetic parameter for improving K K (Wayne Kramer Krona Buddhist nun lattice) algorithm, belong to technical field of measurement and test, the present invention solves the ambiguity in Meta Materials electromagnetic parameter refutation process relative to group delay method, imaginary part penalty method, the n values for testing initial frequency point need not be estimated, and cumbersome data processing need not be carried out, measuring accuracy is high.

Description

A kind of Meta Materials electromagnetic parameter inversion method based on improvement K-K algorithms

Technical field

The invention belongs to technical field of measurement and test, and in particular to a kind of Meta Materials electromagnetic parameter based on improvement K-K algorithms Inversion method.

Background technology

Meta Materials are a kind of new artificial materials, have unique physical characteristic to electromagnetic wave, such as negative index, negative Electromagnetic parameter etc., these unique physical characteristics make Meta Materials have very huge application value, such as terahertz in every field Hereby imaging, stealth material, Antenna Design etc..But Meta Materials are before commercial Application, often it is to be understood that its corresponding electromagnetic parameter, is situated between Electric constant and magnetic conductivity etc..Therefore, for having designed the Meta Materials of completion, its corresponding electromagnetism ginseng is quickly and accurately calculated Number turns into key.

Meta Materials electromagnetic parameter inverting Research Significance is great.Often using arrow during the electromagnetic parameter of test Meta Materials is surveyed at present Network Analyzer is measured, the S parameter of vector network analyzer two-port is drawn using vector network analyzer, then S parameter is carried out Inverting iteration, and then try to achieve the corresponding dielectric constant of Meta Materials and magnetic conductivity.In solution procedure, because the ambiguity of plural number, Often make the problem of result of calculation has multiple values, in multiple solution values, only one value is accurate, and will accurately from Accurate value is filtered out in multivalue result, is often difficult.

A kind of method of Meta Materials electromagnetic parameter inverting is described below.

The use free-space Method for such as scheming displaying tests the electromagnetic parameter of panel metamaterial.Electromagnetic wave in dielectric material one Multiple reflections and transmission, total reflected voltage V occur for face_RWith total projection voltage V_TWith incident voltage V_IThe ratio between be referred to as it is anti- Penetrate parameter S₁₁With configured transmission S₂₁, such as cross incident voltage be 1, then reflected voltage and projection voltage represent respectively reflection parameters and Configured transmission, as shown in figure 1, air is set with dielectric material boundary face A reflectance factor as Γ, dielectric material and air boundary face B reflectance factor is-Γ, and the thickness of dielectric material is set to the transmission coefficient of electromagnetic wave in d, medium for T, then can pushed away by Fig. 1 Obtain reflected voltage.

By (1), formula is obtained：

OrderIt can obtain

It can be obtained by equivalent Two-port netwerk theory

T=exp (- rd) (6)；

Wherein,

So as to：

Therefore only it is to be understood that the S parameter of dielectric material end face, by S parameter can in the hope of reflectance factor and transmission coefficient, And then can in the hope of dielectric material electromagnetic parameter ε_rAnd μ_r。

It can be obtained by transmission coefficient T=exp (- rd)

Plural number asks logarithm to have an ambiguity, r real part be it is accurate, but imaginary part with n change understand into it is periodic Change, therefore r has multiple values, because of r ambiguity, so as to cause μ_r、ε_rAlso there can be multiple values, this is also in the super material of inverting The reason for multiple values being produced during material electromagnetic parameter.In order to solve this ambiguity, group delay is usually used at present Slow method and imaginary part penalty method.Imaginary part penalty method is introduced herein：From formula (11), propagation constant r imaginary part and test frequency Should be between rate linear relation and for increase, week is presented with the increase of frequency for propagation constant r imaginary part as shown by the dash line in figure 2 Phase property changes, therefore can carry out imaginary part compensation, the imaginary part of the propagation constant r after compensation and survey using linear increasing function principle The relation such as Fig. 2 for trying frequency is shown in solid.

Imaginary part penalty method determines n method：

(1) initial testing frequency should be as low as possible, to ensure n=0；Really fixed pattern is original frequency：

(2) f when test frequency is raised_m＜ f_m+1If there is imag (r (f_m+1)) ＜ imag (r (f_m)), then propagation constant r Imaginary part there occurs cyclically-varying, therefore, frequency f_m+1Corresponding n values should add 1 i.e. (n=n+1), and measurement frequency is used and swept The mode of frequency so that the imaginary part of propagation constant changes in the range of π.

The ambiguity processing both at home and abroad to inversion result in Meta Materials electromagnetic parameter testing mainly uses following several at present The method of kind：

One is to use group delay method：It is the thickness of sample that the theoretical foundation of this method, which is electromagnetic wave by the time delay of sample, Function, the time delay that the time delay calculated when taking different numerical value by comparing n is obtained with measurement, so that it is determined that going out correct n values：

Calculate a series of ε when n takes values respectively first_r, then byDraw each The group delay of point, wherein, τ_gmIt is the ε drawn when making n=m_rSolution.The group delay immediately arrived at by measurement is by transmission coefficient T's PhaseSlope between frequencyTo determine correct n values, i.e. τ during n=k_gk-τ_g≈0.So as to obtain Dielectric constant, same method tries to achieve magnetic conductivity.

Two be to use imaginary part penalty method：As described above.

Calculating speed present in traditional Meta Materials electromagnetic parameter inverting is slow, it is necessary to carry out substantial amounts of data processing, and Imaginary part penalty method need to the n values of test frequency point according to relation estimate the problems such as.

The content of the invention

For above-mentioned technical problem present in prior art, the present invention proposes a kind of based on the super of improvement K-K algorithms The inversion method of material electromagnetic parameter, it is reasonable in design, the deficiencies in the prior art are overcome, with good effect.

To achieve these goals, the present invention is adopted the following technical scheme that：

A kind of inversion method based on the Meta Materials electromagnetic parameter for improving K-K algorithms, specifically includes following steps：

Step 1：Vector network analyzer is calibrated by traditional free-space Method；

Step 2：The reflection parameters S of specimen material is obtained with vector network analyzer₁₁With configured transmission S₂₁Two S parameters；

Step 3：Propagation constant γ of the electromagnetic wave in specimen material is solved, specific method is：

Wherein, Γ is reflectance factor, and T is transmission coefficient；

By (1), formula is obtained：

Wherein, S₁₁For reflection parameters,

Wherein, S₂₁For configured transmission,

OrderConvolution (2), formula (3), can be obtained：

Wherein Γ ＜ 1, formula (5) can be obtained by formula (4)：

It is theoretical by equivalent Two-port netwerk, it can obtain：

T=exp (- γ d) (7)；

Wherein, η_rFor the normalized characteristic impedance of specimen material, γ is propagation constant of the ripple in specimen material, and d is sample Product thickness；

Simultaneously because electromagnetic wave uses the pattern of quasi- TEM moulds during test, so specimen material normalized characteristic impedance η_rWith And propagation constant γ and permittivity ε_rAnd magnetic permeability μ_rBetween should meet following relation：

Wherein, γ₀For the aerial propagation constant of ripple；

Obtained by (6), (7) formula：

It can be pushed away by (8), (9), (10)：

The permittivity ε of specimen material is being sought it can be seen from above derivation_rWith magnetic permeability μ_rWhen, it is necessary to first obtain Ripple propagation constant γ in the sample and reflectance factor Γ, Γ can be uniquely determined by formula (4), and from formula (11), In γ solution procedure, because of the ambiguity of plural number (imaginary part has 2 π cyclically-varying) so that γ has multiple solutions：

Therefore the permittivity ε of specimen material is solved_rWith magnetic permeability μ_rWhen, because of γ ambiguity so that permittivity ε_rWith Magnetic permeability μ_rThere is also multiple values, therefore solving permittivity ε_rWith magnetic permeability μ_rBefore, multivalue first is gone to propagation constant γ Processing.

Multivalue is gone using the propagation constant γ based on improved K-K relation pairs ripple in the sample, following steps are specifically included：

Step 3.1：Set up the mathematical modeling of K-K relations：

For a linear passive receptance function α (ω)=α ' (ω)-j α " (ω), it is known that all incident frequencies of its imaginary part Value, can be in the hope of portion in fact, as shown in formula (14)；Similarly, it is known that the value of all incident frequencies of its real part, can be empty in the hope of it Portion, as shown in formula (15)：

α (s), α (ω) are respectively complex variable s, ω receptance function in formula；S is dummy variable；ω is circular frequency；α ' is sound The real part of function is answered, α " is the imaginary part of receptance function；For integration type (15) and (16), s=ω are its singular point, therefore integration takes master Value, is expressed as P, and P is met：

Step 3.2：Multivalue is gone to the propagation constant γ of ripple in the sample：

If propagation constant γ (ω)=α (ω)+j β (ω)；By the mathematical modeling of K-K relations, the imaginary part β of propagation constant (ω) and real part α (ω) can set up following relational expression：

Propagation constant γ real part can be tried to achieve according to formula (14)Uniquely determine, wherein T (ω) Uniquely obtained by formula (5), convolution (17) just can obtain the imaginary part of propagation constant：

That is propagation constantAnd it is unique true It is fixed.

Step 4：Solve permittivity ε_rWith magnetic permeability μ_r：

The S parameter of specimen material two-port in step 2, the reflectance factor Γ of specimen material is calculated by (4) formula, then The propagation constant γ that formula (18) is tried to achieve is brought into formula (12), (13), so as to just can accurately show that the dielectric of specimen material is normal Number μ_rWith magnetic conductivity ε_r。

The advantageous effects that the present invention is brought：

The present invention solves the ambiguity in Meta Materials electromagnetic parameter refutation process relative to group delay method, imaginary part penalty method and asked Topic, it is not necessary to estimate the n values for testing initial frequency point, and cumbersome data processing need not be carried out, measuring accuracy is high.

Brief description of the drawings

Fig. 1 is transmission and reflection schematic diagram of the specimen material to electromagnetic wave.

Fig. 2 is the relation schematic diagram between γ imaginary part and frequency.

Fig. 3 is the FB(flow block) of the inventive method.

Embodiment

Below in conjunction with the accompanying drawings and embodiment is described in further detail to the present invention：

Based on the Meta Materials electromagnetic parameter inversion method technical principle for improving K-K algorithms：

The present invention is broadly divided into four steps：

The first step：Vector network analyzer is calibrated by traditional free-space Method；

Second step：The S parameter of the two-port of specimen material (refering in particular to Meta Materials) is obtained with vector network analyzer；

3rd step, the propagation constant γ based on improved K-K relation pairs ripple in the sample goes multivalue.

(1) mathematical modeling of K-K relations：For a linear passive receptance function, it is known that all incident frequencies of its imaginary part Value, so that it may in the hope of portion in fact, similarly, it is known that real part can be in the hope of its imaginary part.

α (s), α (ω) are respectively complex variable s, ω receptance function in formula；S is that dummy variable, ω are circular frequency；α ' is sound The real part of function is answered, α " is the imaginary part of receptance function.For integration type (15) and (16), s=ω are its singular point, therefore integration master Value, is expressed as P,

If propagation constant γ (ω)=α (ω)+j β (ω)；Wherein α (ω) is that attenuation constant, β (ω) are phase-shift constant, it Represent real part, the imaginary part of propagation constant respectively；

By the mathematical modeling of K-K relations, the imaginary part β (ω) and real part α (ω) of propagation constant can set up following relation Formula：

According to the propagation constant γ tried to achieve real partUniquely determine, wherein T (ω) is by formula (5) uniquely obtain, convolution (17) just can obtain the imaginary part of propagation constant：

That is propagation constantAnd it is unique true It is fixed.

4th step, is finally inversed by the permittivity ε of specimen material_rAnd magnetic permeability μ_r。

The reflection parameters S of specimen material two-port in step 1₁₁With configured transmission S₂₁Two S parameters, pass through (4) formula meter The reflectance factor Γ of specimen material is calculated, then the propagation constant γ that formula (18) is tried to achieve is brought into formula (12), (13), so as to just may be used Accurately draw the dielectric constant μ of specimen material_rWith magnetic conductivity ε_r。

Certainly, described above is not limitation of the present invention, and the present invention is also not limited to the example above, this technology neck The variations, modifications, additions or substitutions that the technical staff in domain is made in the essential scope of the present invention, should also belong to the present invention's Protection domain.

Claims (1)

1. a kind of inversion method based on the Meta Materials electromagnetic parameter for improving K-K algorithms, it is characterised in that specifically include following step Suddenly：

Step 1：Vector network analyzer is calibrated by free-space Method；

Step 2：The reflection parameters S of specimen material is obtained with vector network analyzer₁₁With configured transmission S₂₁Two S parameters；

Step 3：Solve propagation constant γ of the electromagnetic wave in specimen material；

Specific method is：

$\mathrm{\Γ}-\left(\mathrm{\Γ}\right(1-{\mathrm{\Γ}}^2)T^2+{\mathrm{\Γ}}^3(1-{\mathrm{\Γ}}^2)T^4+\mathrm{...})=\mathrm{\Γ}-\frac{1-{\mathrm{\Γ}}^2}{\mathrm{\Γ}}\frac{{\left(\mathrm{\Γ}T\right)}^2}{1-{\left(\mathrm{\Γ}T\right)}^2}---\left(1\right);$

Wherein, Γ is reflectance factor, and T is transmission coefficient；

By (1), formula is obtained：

$S_{11}=\frac{\mathrm{\Γ}(1-T^2)}{1-{\mathrm{\Γ}}^2{T}^2}---\left(2\right);$

Wherein, S₁₁For reflection parameters,

$S_{21}=\frac{T(1-{\mathrm{\Γ}}^2)}{1-{\mathrm{\Γ}}^2{T}^2}---\left(3\right);$

Wherein, S₂₁For configured transmission,

OrderConvolution (2), formula (3), can be obtained：

$\mathrm{\Γ}=K\±\sqrt{K^2-1}---\left(4\right);$

Wherein Γ ＜ 1, formula (5) can be obtained by formula (4)：

$T=\frac{(S_{11}+S_{21})-\mathrm{\Γ}}{1-(S_{11}+S_{21})\mathrm{\Γ}}---\left(5\right);$

It is theoretical by equivalent Two-port netwerk, it can obtain：

$\mathrm{\Γ}=\frac{Z_c-Z_0}{Z_c+Z_0}=\frac{{\mathrm{\η}}_r-1}{{\mathrm{\η}}_r+1}---\left(6\right);$

T=exp (- γ d) (7)；

Wherein, η_rFor the normalized characteristic impedance of specimen material, γ is propagation constant of the ripple in specimen material, and d is that sample is thick Degree；

Simultaneously because electromagnetic wave uses the pattern of quasi- TEM moulds during test, so the normalized characteristic impedance η of specimen material_rAnd pass Broadcast constant γ and permittivity ε_rAnd magnetic permeability μ_rBetween should meet following relation：

${\mathrm{\η}}_r=\sqrt{\frac{{\mathrm{\μ}}_r}{{\mathrm{\ϵ}}_r}}---\left(8\right);$

$\mathrm{\γ}={\mathrm{\γ}}_0\sqrt{{\mathrm{\ϵ}}_r{\mathrm{\μ}}_r}---\left(9\right);$

Wherein, γ₀For the aerial propagation constant of ripple；

Obtained by (6), (7) formula：

${\mathrm{\η}}_r=\frac{1+\mathrm{\Γ}}{1-\mathrm{\Γ}}---\left(10\right);$

$\mathrm{\γ}=\frac{1}{d}ln\left(\frac{1}{T}\right)---\left(11\right);$

It can be pushed away by (8), (9), (10)：

${\mathrm{\μ}}_r=\frac{\mathrm{\γ}}{{\mathrm{\γ}}_0}\frac{1+\mathrm{\Γ}}{1-\mathrm{\Γ}}---\left(12\right);$

${\mathrm{\ϵ}}_r=\frac{\mathrm{\γ}}{{\mathrm{\γ}}_0}\frac{1-\mathrm{\Γ}}{1+\mathrm{\Γ}}---\left(13\right);$

From formula (11), in γ solution procedure, because the ambiguity of plural number is the cyclically-varying that its imaginary part has 2 π so that There are multiple solutions in γ：

Step 4：Multivalue is gone using the propagation constant γ based on improved K-K relation pairs ripple in the sample, following step is specifically included Suddenly：

Step 4.1：Set up the mathematical modeling of K-K relations；

For a linear passive receptance function α (ω)=α ' (ω)-j α " (ω), it is known that the value of all incident frequencies of its imaginary part, Can be in the hope of portion in fact, as shown in formula (14)；Similarly, it is known that the value of all incident frequencies of its real part, can be in the hope of its imaginary part, such as Shown in formula (15)：

${\mathrm{\α}}^{\′}\left(\mathrm{\ω}\right)-1=\frac{2}{\mathrm{\π}}P{\∫}_0^{\mathrm{\∞}}\frac{{\mathrm{s\α}}^{\′\′}\left(s\right)}{s^2-{\mathrm{\ω}}^2}ds---\left(15\right);$

${\mathrm{\α}}^{\′\′}\left(\mathrm{\ω}\right)-1=\frac{2\mathrm{\ω}}{\mathrm{\π}}P{\∫}_0^{\mathrm{\∞}}\frac{{\mathrm{\α}}^{\′}\left(s\right)-1}{s^2-{\mathrm{\ω}}^2}ds---\left(16\right);$

α (s), α (ω) are respectively complex variable s, ω receptance function in formula；S is dummy variable；ω is circular frequency；α ' is response letter Several real parts, α " is the imaginary part of receptance function；For integration type (15) and (16), s=ω are its singular point, therefore integration takes main value, P is expressed as, P is met：

Step 4.2：Multivalue is gone to the propagation constant γ of ripple in the sample；

If propagation constant γ (ω)=α (ω)+j β (ω), by the mathematical modeling of K-K relations, the imaginary part β (ω) of propagation constant and Real part α (ω) can set up following relational expression：

$\mathrm{\β}\left(\mathrm{\ω}\right)=1+\frac{2\mathrm{\ω}}{\mathrm{\π}}P{\∫}_0^{\mathrm{\∞}}\frac{\mathrm{\α}\left(\mathrm{\ω}\right)-1}{s^2-{\mathrm{\ω}}^2}ds---\left(17\right);$

Propagation constant γ real part can be tried to achieve according to formula (5), (14)Convolution (17) just can be obtained The imaginary part of propagation constant：

That is propagation constant

Step 5：Solve permittivity ε_rWith magnetic permeability μ_r；

According to formula (4), (12), (13), (18), the dielectric constant μ of specimen material just can be obtained_rWith magnetic conductivity ε_r。