RU2789626C1 - Method for measuring the relative complex dielectric permittivity of a lossy material - Google Patents

Abstract

FIELD: measuring technology.

SUBSTANCE: invention relates to measuring technology, in particular to the measurement of ultra-high-frequency parameters of materials in free space. A method for measuring the relative complex permittivity of a flat sample of a material with dielectric losses in the frequency band of the microwave range, in which the dependence of the complex reflection coefficient is measured using a microwave meter when the electric field of the incident wave is parallel to the plane of incidence and the relative complex permittivity is determined. For each frequency of the microwave range, using the sweep method with a predetermined iteration step, the relative complex permittivity is calculated using the formula:

where i is an imaginary unit;

is an imaginary part of the dielectric constant;

is relative permittivity.

EFFECT: increase in the accuracy of measuring the frequency dependence of the relative complex permittivity of a flat sample material in a wide frequency band in free space.

1 cl, 3 dwg

Description

The invention relates to measuring technology, in particular to the measurement of microwave parameters of materials in free space.

There are known methods for measuring relative dielectric and magnetic permeability in a voluminous resonator at a fixed frequency, when the sample is measured in two positions: on the piston and on the stand (E.B. Zaltsman. Measure the parameters of magnetodiumlectrics and non -magnetized ferrites using a rectangular resonator on the wave of N ₁₀ . // Radio engineering, 1958, p.13, No. 10, p.76-80).

The disadvantage of the methods is the narrow band of the measurements. When measuring the parameters of paramagnetic and ferromagnetic materials, the measurement accuracy decreases, because the magnetic field of the material distorts the structure of the field in the resonator and the small perturbation method used in the resonant measurement method is not performed.

Waveguide and resonator methods for measuring the relative dielectric and magnetic permeability have high accuracy and sensitivity for measuring microwave parameters, but have a significant drawback - the narrow band of the measurements (V.N. Egorov. Resonance methods for studying dielectrics at microwave frequencies.// Instruments and experimental techniques. - 2007 , No. 2, p.5-38).

Known amplitude-phase devices that work on the "passage" and "reflection" of an electromagnetic wave in free space, used to measure the relative dielectric and magnetic permeability of materials (Devices for non-destructive testing of materials and products. Handbook in 2 books, book 1 under V. V. Klyuev, 2nd ed., revised and supplemented, M.: Mashinostroenie, 1986, pp. 220-224, 233-234, fig. 17 pp. 217-219).

Methods for determining the relative permittivity of a material by the power and phase of the transmitted and reflected waves are described (EA Vorobyov. Radio wave control of ship radio engineering structures and materials, L.: Sudostroenie, 1986, 81, p. 22).

Known methods for determining the relative permittivity of materials from the power and phase of the reflected wave from a plate located at the Brewster angle (AS USSR No. 1550436, class G01R 27/26, publ. , IPC G01B 15/02, G01R 27/26, published 03/27/2005, Bull. No. 9, Application for an invention 2003 106 528, published 09/10/2004).

The disadvantage of these methods is the requirement for high accuracy in the manufacture of flat samples of extended dimensions at the level of 0.005 mm, which is difficult to implement from a technological point of view for large sample sizes. High demands are also placed on the radio engineering quality of the measuring range and the accuracy of the measuring equipment.

With all the disadvantages inherent in the method of measuring the microwave parameters of a material in free space, its important advantage lies in the possibility of obtaining dependences of the relative permittivity on frequency in a wide band.

To obtain extended information about the material under study, it is necessary to measure its dielectric properties in a wide frequency band with a constant increase in the upper boundary frequencies, where the dimensions of resonant systems become comparable in wavelength, therefore, the direction of development of methods for studying the properties of materials is associated with the use of optical methods in free space in the radio range. , for which, with the creation of microwave meters in the form of broadband network analyzers, progress has been made in instrumentation.

At the same time, the known methods for measuring the microwave parameters of a material in free space have methodological shortcomings that do not allow simultaneous measurement of the relative permittivity and dielectric loss tangent of materials with high accuracy.

The closest technical solution is the method of measuring relative comprehensive dielectric permeability (a method of measuring relative complex dielectric permeability of the material with losses . Patent of the Russian Federation No. 2613810, MPC G01R 27/00, publ. 03/21/2017, Bul. No. 9).

In this method, when changing the angle of an incident electromagnetic linearly polarized wave with an electric field vector lying in the plane of incidence, an angle is found on a flat sample of material (TM wave), which is determined from the condition of perpendicularity between the directions of the transmitted and reflected waves, based on which the relationship is established between the Brewster angle and the electrodynamic parameters of the plate material in the form of the Brewster formula:

– угол, при котором выполняются условия закона Брюстера;Where

is the angle at which the conditions of Brewster's law are satisfied;

– относительная комплексная диэлектрическая проницаемость материала;

is the relative complex permittivity of the material;

– относительная диэлектрическая проницаемость;

is the relative permittivity;

– мнимая часть диэлектрической проницаемости;

is the imaginary part of the permittivity;

– тангенс угла диэлектрических потерь в материале.

is the dielectric loss tangent in the material.

As the authors found, formula (1) is valid only in the absence of losses in the material, but with an increase in the dielectric loss tangent, significant discrepancies are observed between the conditions for the fulfillment of Brewster's law, which reduces the accuracy of determining the relative complex permittivity.

The objective of the present invention is to improve the accuracy of measuring the frequency dependence of the relative complex permittivity of the material of a flat sample in a wide frequency band in free space.

The task is achieved by the fact that a method is proposed for measuring the complex relative permittivity of a flat sample of material with dielectric losses in the microwave frequency band, in which the dependence of the complex reflection coefficient is measured using a microwave meter, when the electric field of the incident wave is parallel to the plane of incidence and the relative complex value is determined. permittivity, characterized in that for each frequency of the microwave range, using the sweep method with a predetermined iteration step, the relative complex permittivity is calculated by the formula:

where i is the imaginary unit;

– мнимая часть диэлектрической проницаемости;

is the imaginary part of the permittivity;

– относительная диэлектрическая проницаемость,

is the relative permittivity,

найденного из измеренной угловой зависимости амплитуды коэффициента отражения Френеля и соответствующего расчетному минимуму отраженной волны по формуле:by solving two joint equations, the first of which is compiled for the angle value

found from the measured angular dependence of the amplitude of the reflection coefficient of the frane and corresponding to the calculated minimum of the reflected wave according to the formula:

– расчетная угловая зависимость комплексного коэффициента отраженной волны;Where:

is the calculated angular dependence of the complex coefficient of the reflected wave;

– параметр, соответствующий плоскому образцу толщиной d₂;

- parameter corresponding to a flat sample with a thickness d ₂ ;

– длина волны, f – частота измерений, C – скорость света;

is the wavelength, f is the measurement frequency, C is the speed of light;

α ₁ is the angle of incidence of the wave on a flat sample;

α ₂ - angle of refraction,

найденного из измеренной угловой зависимости фазы коэффициента отражения Френеля и соответствующего расчетной фазе, по формуле: and the second is composed for the value of the angle

found from the measured corner dependence of the phase of the reflection coefficient of the frane and the corresponding calculated phase, according to the formula:

и

To analyze the conditions of measurements of relative comprehensive dielectric permeability, the authors used the task of an inclined drop in an electromagnetic wave on an endless plate of a dielectric, in which, according to the calculated values of the amplitude and phase of the reflected wave within the framework of geometric optics, the relative complex dielectric permeability of the sample was determined. The parameters of the sample in the form of a plate are indicated in Fig. 1, on which between the first and third media with electrodynamic parameters

And

– относительные комплексные диэлектрические проницаемости сред,

– относительные комплексные магнитные проницаемости сред, располагается исследуемый плоский образец с относительной комплексной диэлектрической проницаемостью

и относительной комплексной магнитной проницаемостью

и геометрической толщиной d₂. Падающая волна изображена в виде луча, составляющего с перпендикуляром к границе пластины угол

а преломленная волна изображена в виде луча под углом α₂ к перпендикуляру. Where

are the relative complex permittivities of the media,

are the relative complex magnetic permeability of the media, the investigated flat sample is located with a relative complex permittivity

and relative complex magnetic permeability

and geometric thickness d ₂ . The incident wave is shown as a ray making an angle with the perpendicular to the plate boundary

and the refracted wave is shown as a beam at an angle α ₂ to the perpendicular.

The model for calculating the reflection coefficient for a problem within the framework of geometric optics is built on the basis of the matrix method for a flat sample (M. Born, E. Wolf. Fundamentals of Optics. 2nd ed. Literature, 1973, 720 pp.).

The calculated angular dependence of the complex coefficient of the reflected wave is:

Where:

– параметр, соответствующий плоскому образцу толщиной d₂,

- parameter corresponding to a flat sample with a thickness d ₂ ,

i - imaginary unit,

λ ₀ - wavelength,

;

;

α ₁ is the angle of incidence of the wave on a flat sample;

α ₂ is the angle of refraction.

The authors found that the left side of the formula is equal to the real angle, and the right side is equal to the tangent of the complex value, so it is only valid for the real value of the relative complex permittivity, that is, for a lossless material.

When an electromagnetic linearly polarized wave with an electric field vector lying in the plane of incidence is incident at the Brewster angle onto a plate of lossy material, the angle between the reflected and transmitted waves differs from 90° and increases with increasing dielectric and magnetic losses.

To determine the relative complex permittivity using the measured parameters of the reflected wave in the form of determining the angle corresponding to the Brewster angle, it is not enough.

Summarizing the results of theoretical and experimental studies and analyzing the mathematical formulas for determining the phase and amplitude of electromagnetic waves in the form:

it can be concluded that the phase jump of an electromagnetic wave by 180° in the region of the Brewster angle occurs when the condition of equality to zero of the real part of the amplitude is satisfied:

and the condition for the minimum amplitude when the absolute value of the amplitude is equal to zero:

is carried out only under the condition that the real and imaginary parts of the electromagnetic wave amplitude are equal to zero:

therefore, as can be seen from a comparison of conditions (5) and (7), in the general case, the points of the minimum amplitude and the phase jump by 180° should not coincide. But since the mathematical description of electromagnetic amplitudes is based on harmonic functions, it is obvious that for a medium with electromagnetic parameters with insignificant dielectric and magnetic losses, the coincidence of conditions (5) and (7) is quite often observed for frequency dependences and less often for angular dependences, as for both reflected and transmitted waves.

Therefore, the difference in the angular positions of the minimum and phase jump by 180° of the reflected wave in the region of the Brewster angle, observed in experiments, is not associated with deviations in the Fresnel laws, but in the difference in the conditions of their natural position, which can be seen when calculating the reflected waves within geometric optics when taking into account the complex values of the electrodynamic parameters of the medium.

Therefore, it is proposed to measure:

для условия минимума модуля коэффициента отражения:– angle of incidence α ₁ equal to the angle

for the condition of the minimum modulus of the reflection coefficient:

для условия равенства фазы коэффициента отражения: – angle of incidence α ₁ equal to the angle

for the condition of equality of the phase of the reflection coefficient:

используются для проведения расчетов относительной комплексной диэлектрической проницаемости в виде относительной диэлектрической проницаемости и тангенса угла диэлектрических потерь с использованием формулы (2).Incident angle data

are used to calculate the relative complex permittivity in the form of relative permittivity and dielectric loss tangent using formula (2).

Moreover, for the angle of Bruster, determined by the formula (1), the angles determined for conditions (8) and (9) coincide

для материала с

и с магнитной проницаемостью

с расчетами углов. The authors conducted a comparative calculation of the angle of Bruster according to the formula (1), depending on the increasing dielectric losses

for material with

and with magnetic permeability

with angle calculations.

соответствующий углу

и второму следствию закона Брюстера для каждого значения возрастающего тангенса угла диэлектрических потерь материала

при неизменном значении относительной диэлектрической проницаемости

при отсутствии относительной комплексной магнитной проницаемости

на длине волны λ₀=30 мм, и толщине пластины, равной

The numerical calculation of the formula (2) determined the angular dependence of the reflection coefficient module, according to which the angle with the minimum amplitude of the reflected wave from the condition was determined:

corresponding to the corner

and the second consequence of Brewster's law for each value of the increasing tangent of the dielectric loss angle of the material

at a constant value of the relative permittivity

in the absence of relative complex magnetic permeability

at a wavelength λ ₀ =30 mm, and a plate thickness equal to

толщиной в четверть волновой электрической толщины.The calculations of the angles corresponding to conditions (8) and (9), depending on the dielectric losses of the plate material, are shown in figure 2, which shows the dependence of the angles on the tangent of the dielectric loss angle, corresponding to: for curve 1, the minimum modulus of the reflection coefficient, for curve 2 - phase of the reflection coefficient equal to 90 degrees, curve 3 - for an angle equal to 90 degrees between the directions of the transmitted and reflected waves, when reflected from a plate with

a quarter of the electrical thickness of the wave.

From the comparison of the curves presented in figure 2 it can be seen that with increasing losses in the plate material, the difference between curves 1 and 2 increases, that is, the condition of coincidence, as in formula (1) between the angles at which the amplitude minimum is observed and at which the phase is equal to 90 deg, in the presence of losses in the material is not fulfilled and corresponds only to the condition for the plate material without losses.

Figure 2 also shows that the angle at which the direction of the reflected wave is observed, which is 90 degrees with the direction of the transmitted wave, with increasing losses in the plate material, is increasingly different from that calculated by formula (1) for the plate material without losses.

From the analysis of the obtained calculated dependences, the modeling of which did not go beyond the limits of geometric optics, it can be seen that deviations from the Fresnel laws are not observed, and the calculation results coincide with experimental observations, which indicates the inapplicability of the Brewster formula in the form (1) for determining the angles corresponding to its consequences.

Let us calculate the values of the Brewster angles using various criteria following from the Brewster law and compare them with the calculations using formulas (9) and (10) depending on the amount of plate material loss.

, толщиной в четверть волновой электрической толщины. Кривая 1 - это зависимость положения угла, соответствующего минимуму модуля амплитуды отраженной волны от тангенса угла диэлектрических потерь. Кривая 2 - это зависимость положения угла, соответствующего значению фазы отраженной волны, равной 90° от тангенса угла диэлектрических потерь. Кривая 3 - это зависимость положения угла Брюстера, рассчитываемого по более корректной формулеThe calculation results are shown in Fig. 3 for insert material with

, a quarter of the electrical thickness of the wave. Curve 1 is the dependence of the position of the angle corresponding to the minimum modulus of the amplitude of the reflected wave on the tangent of the dielectric loss angle. Curve 2 is the dependence of the position of the angle corresponding to the value of the phase of the reflected wave, equal to 90°, on the tangent of the dielectric loss angle. Curve 3 is the dependence of the position of the Brewster angle, calculated by the more correct formula

than formula (1), depending on the losses in the material or the tangent of the dielectric loss angle.

However, from FIG. It can be seen from Fig. 3 that the calculations using formula (10) do not coincide with either curve 1 or curve 2, and this difference grows with increasing losses in the material. Therefore, it is obvious that when determining the relative complex permittivity of lossy materials, it is necessary to calculate according to formula (2), as in the proposed technical solution.

The calculations were carried out using the Microsoft Visual Studio software platform of the FORTRAN 2008 standard using additional options for floating point numbers to reduce calculation errors when simulating the problem of oblique incidence of an electromagnetic wave on an infinite plate of a homogeneous dielectric material with losses.

Thus, the authors, by experimentally determining the position of the angles corresponding to the minimum of the angular dependence of the amplitude and phase, equal to 90° of the reflected wave, and then using these values to calculate the relative complex permittivity simultaneously corresponding to equation (2), increased the accuracy of its determination for materials with losses over a wide frequency band.

Claims (16)

A method for measuring the complex relative permittivity of a material with losses in the microwave frequency band, in which the dependence of the complex reflection coefficient is measured using a microwave meter, when the electric field of the incident wave is parallel to the plane of incidence and the relative complex permittivity is determined, characterized in that for each frequency In the microwave range, using the sweep method with a predetermined iteration step, the relative complex permittivity is calculated by the formula:

where i is the imaginary unit;

is the imaginary part of the permittivity;

is the relative permittivity, by solving two joint equations, the first of which is compiled for the angle value

found from the measured angular dependence of the amplitude of the reflection coefficient of the frane and corresponding to the calculated minimum of the reflected wave according to the formula: Where:

is the calculated angular dependence of the complex coefficient of the reflected wave;

- parameter corresponding to a flat sample with a thickness d ₂ ;

is the wavelength, f is the measurement frequency, C is the speed of light;

α ₁ is the angle of incidence of the wave on a flat sample; α ₂ - angle of refraction, and the second is composed for the value of the angle

found from the measured angular dependence of the phase of the Fresnel reflection coefficient and corresponding to the calculated phase, according to the formula:

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