CZ2022253A3 - A method to determine complex permittivity - Google Patents

Abstract

The presented solution provides a method of determining the complex permittivity of the measured sample, which includes the following steps: a) determination of the complex permittivity of the calibration samples and the dielectric substrate of the transmission line; b) applying at least two calibration samples to a section of a transmission line containing a dielectric substrate on which electrical conductors are applied; c) measuring the dispersion parameters of the transmission line for each calibration sample using a circuit analyzer; d) determination of propagation constant and characteristic impedance in the transmission line section with the sample for one calibration sample using a numerical electromagnetic field simulator; e) determining the propagation constant in the section of the transmission line with the sample for the second calibration sample using the trace equation; f) providing field shape coefficients and ; g) application of the measured sample to the same transmission line; h) measurement of dispersion parameters of the transmission line with the measured sample using a circuit analyzer; i) determining the propagation constant in the section of the transmission line with the measured sample using the trace equation and selecting the correct solutions of the trace equation using a heuristic algorithm; j) calculation of the complex permittivity of the measured sample of the measured and calculated parameters.

Description

The method of determining the complex permittivity

Field of technology

The invention relates to a method of determining the broadband complex permittivity of samples. Determining the complex permittivity of materials is important for the development of electromagnetic technologies, such as wireless communication, radars, or for determining the electromagnetic properties of biomolecules and cells.

Current state of the art

Knowledge of the electromagnetic properties of biomolecules is important for understanding the interaction of electric fields with biosystems and for the development of new biomedical diagnostic and therapeutic methods. Electromagnetic fields at radio and microwave frequencies interact with biological systems mainly through the electric component of the field. Determining the permittivity of biomolecules is thus essential for determining the electromagnetic properties of proteins, cells and whole organisms. However, development in this area is hindered by the large sample volumes required for permittivity determination.

Permittivity is a measure of the electrical polarizability of a dielectric. A material having a high permittivity becomes more polarized in an applied electric field than a material with a low permittivity. Permittivity can also be viewed as a thermodynamic state function that can depend on the frequency, magnitude, and direction of the applied field. The SI unit of permittivity is the farad per meter (F/m).

The response of real materials to an applied electric field generally depends on the frequency of the field. This dependence reflects the fact that the polarization of the material does not change immediately when an electric field is applied, but changes gradually (ie with a delay) and only after its application. Thus, the polarization of the material is phase-delayed. For that reason, permittivity is often considered and expressed as a vector, as a complex function of the circular frequency of the applied field. The real part of this function corresponds to the polarization, and the complex part corresponds to the phase delay.

Currently, there are several types of methods for determining complex permittivity. Methods using a coaxial probe are mostly non-destructive, easy to implement, and allow measurements in a wide frequency range of 0.2 to 50 GHz. On the other hand, however, they require a relatively large sample volume, the sample must be homogeneous and isotropic, and it is very problematic to measure solid samples, and high accuracy cannot be achieved either. Methods based on free-space measurements are non-contact and therefore non-destructive, the frequency range can reach up to 325 GHz, but they are expensive and complicated to implement, and very large sample volumes are needed at lower frequencies. Methods using resonant cavities are more accurate and are satisfied with smaller sample volumes, but are not suitable for high-loss materials and only provide results for a single frequency. Methods using a transmission line are economically advantageous, work in a wide frequency range of 0.1 to 110 GHz, allow measuring even magnetic materials, and only a small amount of sample is needed for measurement. In transmission line methods, a precise sample shape is needed, and for low frequencies, a larger amount of sample is needed.

Different implementations of methods using a transmission line are described, for example, in US 20190377054, where, however, the upper usable frequency is only 5 GHz; and in Bao Xiue et al., IEEE Microwave and Wireless Components Letters 28, No. 4 (2018), 356-58 and Liu Song et al., IEEE Transactions on Microwave Theory and Techniques 65, no. 12 (2017): 5063-73, where the use of a microfluidic channel is necessary, and the measurement method is not precisely described in either article.

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The essence of the invention

The subject of the present invention is a method of determining the complex permittivity of a measured sample, which includes the following steps:

a) determination of the complex permittivity of the calibration samples and the dielectric substrate of the transmission line;

b) applying at least two calibration samples to a section of a transmission line containing a dielectric substrate on which electrical conductors are applied;

c) measuring the dispersion parameters of the transmission line for each calibration sample using a circuit analyzer;

d) determination of propagation constant and characteristic impedance in the transmission line section with the sample for one calibration sample using a numerical electromagnetic field simulator;

e) determining the propagation constant in the section of the transmission line with the sample for the second calibration sample using the trace equation;

f) providing the field shape coefficients Ka and Kb, with the advantage of determining the field shape coefficients Ka&Kb from the propagation constants of the calibration samples and the complex permittivity of the calibration samples and the dielectric substrate of the transmission line using the equations yf - y ² ₂ ^A ΙωΖ^-^y _K YÁia - Yi ^Ě 2A where yi is the propagation constant in the section of the transmission line with the first calibration sample, w is the propagation constant in the section of the transmission line with the second calibration sample, j is an imaginary number (f= -1), ω is the angular frequency (2π/), f is electromagnetic field frequency, ZRL=yi Zi, where Zi is the characteristic impedance of the transmission line, ¹ is the permittivity of the first ř* f* calibration sample, ² is the permittivity of the second calibration sample and ^s is the permittivity of the dielectric substrate;

g) application of the measured sample to the same transmission line;

h) measurement of dispersion parameters of the transmission line with the measured sample using a circuit analyzer;

i) determination of the propagation constant in the section of the transmission line with the measured sample using the trace equation and selection of correct solutions of the trace equation using a heuristic algorithm;

OF-*

j) calculation of the complex permittivity of the measured sample ^u from the propagation constant in the section of the transmission line with the measured sample, from the propagation constant in the section of the transmission line with one calibration sample and from the field shape coefficients using the equation

-2CZ 2022 - 253 A3 ___________Ml'______

K _A where Ka and Kb are the coefficients of the shape of the electric field, yi is the propagation constant in the section of the transmission line with one calibration sample, yu is the propagation constant in the section of the transmission line with the measured sample, ¹ is the permittivity of one calibration sample and ^s is the permittivity of the dielectric substrate.

Thus, the present invention provides a new method for obtaining complex permittivity in the frequency band of 1.5 GHz to 50 GHz (ie broadband complex permittivity) based on a transmission line, preferably a grounded coplanar waveguide. The main advantage of this method is the possibility to measure the dielectric properties of extremely small volumes of samples (on the order of hundreds of μΐ), approximately 20 times smaller than with commercially available methods. Another advantage is that the method does not require the use of a microfluidic channel. The accuracy of determining complex permittivity is comparable to existing commercially used methods. The method works independently of the manufacturer of the circuit analyzer.

Both calibration and measured samples are applied to the same section of the same transmission line. The advantage is that it does not depend on the state of the sample, the sample can be gaseous, liquid or solid.

A transmission line is defined as a structure that allows the propagation of an electromagnetic wave. The transmission line consists of a planar dielectric substrate, a center conductor and at least two ground conductors, the center conductor and the ground conductors being preferably located in one plane (i.e. co-planar transmission line) on one side of the dielectric substrate. Preferably, the transmission line is grounded by placing an additional grounding conductor on the other side of the dielectric substrate. The sides of the dielectric substrate here mean the opposite surfaces of the dielectric substrate.

The following text describes advantageous embodiments of the individual steps of the method according to the invention.

In steps a) to e), the values of the four inputs for method calibration are determined. These values shall be determined in a frequency band of at least 1.5 GHz to 50 GHz.

* *

The first input is ^{the és} complex permittivity of the dielectric substrate and the complex permittivity 6* f- * of the calibration samples, labeled according to their order as ¹ , ² , etc.

Complex permittivity is defined as ^e — G + JG , where is its real component and ^Ěi is its .2 imaginary component and aj is an imaginary number, i.e. J ^— . Complex permittivity values are determined from reference measurements, for example with a coaxial probe, or from values from the literature.

The second input is the scattering parameters (S-parameters) for all calibration samples. The values of the scattering parameters are measured using a circuit analyzer in the frequency range of at least 1.5 GHz to 50 GHz. A circuit analyzer is a commonly available device that measures the magnitude and phase of complex circuit quantities, including stray parameters. The basic structural elements of a circuit analyzer are a source, a microwave assembly (directional coupling elements, bridges, sampling receivers and mixers) and an analyzer.

Scatter parameters are dimensionless parameters and are defined using voltage waves. Dispersion parameters are always defined for a transmission line between two reference planes (at the input and output of the line) that are perpendicular to the direction of wave propagation along the line. The position of the reference planes in the direction of wave propagation is defined by the distance along the direction of wave propagation. Device gates, including the direction of wave propagation, are defined as all device inputs and outputs. For

-3 CZ 2022 - 253 A3 transmission line, which falls into the category of two-gate devices, the dispersion parameters are defined using voltage waves al and a2, which fall on the first and second gates of the line and voltage waves bl and b2, which are reflected from the first and second management gates. Each of the dispersion parameters Sil, S12, S21, S22 is a complex number and is defined in relation to voltage waves by the equation (1) bl\ ₌ (Sil S12x Z«/\ b21 ~ 8S21 S22) LzJ (1)

A huge advantage in this step is that it does not matter at what distance from the sample the reference planes are chosen. It is enough if they are chosen the same for measuring the scattering parameters of all samples.

Furthermore, the value of the third input is determined, which are the scattering parameters on the reference planes defining only the section of the transmission line with the first calibration sample. This input is obtained using a numerical electromagnetic field simulator. A numerical simulator is a computer program that numerically calculates the distribution and propagation of an electromagnetic field in an object through its spatial discretization. Numerical simulators are commonly available, examples are CST Microwave

Studio, COMSOL Multiphysics, ANSYS HFSS. The input for the numerical simulator is the dimensions and material properties (dielectric permittivity and electrical conductivity) of all parts of the transmission line. The value of the wave propagation constant in the line section with the first calibration sample is determined from the values of the scattering parameters. The propagation constant y is defined as Y ~ ^α +ίβ , where a [1/m] describes the wave attenuation and β [1/m] the phase velocity of wave propagation on the line. The value of the wave propagation constant in the section of the line with the first calibration sample, denoted as yi, is determined from equations (2), (3), (4)·.

cosh ^-J (ýA · D) n=----;---- (2) where Z is the length of the transmission line section with the first calibration sample and where the parameters A, D are obtained from the dispersion parameters Sil, S12, S21, S22 as:

(/ + ^)-(/-^) + ¾¾

2S„ (3) (/-^7)-(/+^2)+¾¾

2S _?1 (4)

Next, the value of the fourth input is determined, which is Zi - the characteristic impedance of the section of the transmission line with the first calibration sample. Characteristic impedance is defined as the ratio of voltage and current spread on the line. The characteristic impedance value is determined from a numerical simulator where the voltage value is obtained by numerical integration of the electric field and the current value is obtained by numerical integration of the magnetic field.

The advantage of the next steps is that it is no longer necessary to use a numerical simulator of the electromagnetic field to obtain all propagation constants and other results.

-4CZ 2022 - 253 A3

Next, the propagation constant y ₂ on the transmission line with the second calibration sample is obtained using trace equation (5)

Tr{M ₂ · Aíý ⁷ } = 2 · cosh(y ₇ · l) cosh(y ₂ · 0 — + · sinh(y ₇ · l) sinh(y ₂ · l), (5) where yi is the propagation constant on the transmission line with the first calibration sample, l is the length of the section of the transmission line with the first calibration sample, Tr denotes the trace operator, M ₂ is the matrix from the measurement with the second calibration sample, Mi is the matrix from the measurement with the first calibration sample, which are obtained from the measured of scattering parameters by equations (6), (7), (8), (9), (10) t — $ ²² *12 — č—' ^ύ 21 (8) _T _ $11 *21 ^— 7 7 ^ύ 21 (9) _ ^12^21 ^— $11^22 ⁱ 21 — ø >

²¹ (10)

A trace equation is an equation with one unknown y ₂ . The equation is transcendental (i.e. y ₂ cannot be explicitly expressed on one side) and therefore its solution can be found using the Newton-Raphson method.

The main advantage of the procedure is that the trace equation is sensitive only to changes in the scattering parameters, not to their absolute value. This procedure thus removes the influence of the feed transfer line (outside the reference planes) and all other transfer imperfections leading to and from the sample from the circuit analyzer. The method is therefore robust.

Next, the coefficients of the shape of the electric field K _s are determined. Kb, which, assuming that all the energy of the wave propagating along the transmission line is located in the region of the section with the sample only in the sample (field shape in the sample, described by Ka) and in the dielectric substrate (field shape in the dielectric substrate, described by Kb) (11) , (12):

Yi - Y ² 2 (Π)

-5CZ 2022 - 253 A3 _K vb'A - Ύ ² ι ^ε *2Α

Í^rl ^e s^ ^e *i ~ (12) where yi is the propagation constant in the section of the transmission line with the first calibration sample, y? is the propagation constant in the section of the transmission line with the second calibration sample, / is the imaginary number i ² = -Z ^J , ω is the angular frequency (2π/), / is the frequency, Zrl=Yi Zi, where Zi is the characteristic impedance of the transmission line, ¹ is the permittivity of the first calibration sample, ² is the permittivity of the second calibration sample and ^s is the permittivity of the dielectric substrate.

The advantage is that the above steps leading to obtaining K _t . Kb can be performed for a given transmission line only once after its production. These values can then be used for all measurements on this transmission line.

To determine the complex permittivity of the measured sample, the measured sample with unknown complex permittivity ^y is applied to a section of the same transmission line of the same length as for the calibration samples. Then the dispersion parameters of the transmission line with the measured sample are measured using a circuit analyzer. The variance parameters are converted to T-parameters that form the Z-* matrix

Mu from measurements with a measured sample of unknown complex permittivity ^u using equations (13), (14), (15), (16), (17) (13) „ _ ^l 12 —>

(14) hi —>

^ύ 21 (15) „ _ S/A ^— S]]S ₂ 2 hi —------ň<

(16) where (17)

Subsequently, yu - the propagation constant in the section of the transmission line with the measured sample is obtained from the trace equation using equation (18):

Τν{Μ _υ · Mj ¹ } = 2 · cosh(y ₇ · /) coshty^ l) — _|_ ÍŘ). sinh(y _; /) smh(y _[7 · i), (18)

-6CZ 2022 - 253 A3 where Mi and Mu are the matrices of the measured scattering parameters of one calibration sample and the measured sample, yi is the propagation constant of the line section with one calibration sample and l is the length of the transmission line section. The trace equation is transcendental (i.e., γυ cannot be explicitly expressed on one side), and therefore its solution can be found using the Newton-Raphson method.

The term "One calibration sample" means any of the calibration samples. However, all parameters refer to the same calibration sample.

The Newton-Raphson method is defined as a method that iteratively finds the root of an equation using the quotient of the function of the equation and its derivative. After finding γυ, this is inserted into the equation from which the permittivity ^{y of} the sample can be directly determined as (19) + K _a 6j] ' (~) ~ K _B e* (19) where Ka and Kb are the electric field shape coefficients, yi is propagation constant in the section of the transmission line with one calibration sample, γυ is the propagation constant in the section of the line with the measured sample, ¹ is the permittivity of one calibration sample and ^s is the permittivity of the substrate. The advantage of this procedure is that the permittivity samples expressed explicitly.

There are a theoretically infinite number of solutions to the trace equation for γυ because the equation contains trigonometric functions that are periodic in the complex space of γυ. A heuristic algorithm is used to determine the physically correct solution. The solution is found using the NewtonRaphson method, preferably twenty or more times with random initial conditions. Thus, a set of solutions is obtained and non-repeating solutions for γυ are selected from them. Subsequently, these solutions for γυ are ordered in order of magnitude based on their absolute complex number value. Only the first two solutions with the smallest absolute value are selected. From these two solutions for γυ, the permittivity of the measured sample ^u is then calculated Z-* as (20):

Ka (20) where Ka and Kb are the electric field shape coefficients, y> is the propagation constant in the section of the transmission e* f* line with one calibration sample, ¹ is the permittivity of one calibration sample and ^s is the permittivity of the dielectric substrate. Since two solutions of γυ were used, the result is two solutions of the complex permittivity of the measured sample. After that, the frequency intervals of the validity of the solution are determined as follows. The measured dispersion parameters of the entire transmission line (ie between the reference planes) with the calibration sample are inserted into (21), (22), (23):

(7+5^)-(7-¾¾¾¾ (21) (7-5^)-(7 + ¾¾¾¾

IS,, (22)

-7CZ 2022 - 253 A3 cosh VýA · ΰ) ^Y -----------Ϊ----------- (23)

Where l is the length of the sample-only transmission line section. This gives the propagation constants yu- and y2c for the line with the first and second calibration samples, which are then substituted into the trace equation (24):

R = 2- cosh(y _ic · Z) cosh(y _zc · í) - (— + —) · sinh(y _ic · ΐ) sinh(y _2C Γ) V2C Yic'

- Tr{M ₂ Mj ¹ }.

(24)

Frequency ranges where |R|> 100, (where |R| is defined as the absolute value of the real part of a complex number) are considered to be the range of invalid solutions. In the next procedure, permittivity values are used in the frequency range where |R| =< 100. Advantageously, the range of valid solutions will then expand heuristically.

In the frequency ranges with valid solutions, the imaginary permittivity component is then calculated ^for both the found solution with the smallest absolute value γυ and the solution with the second smallest absolute value γυ. A heuristic algorithm is then used to select a solution that leads to the correct values of the imaginary component of the permittivity.

The values of the real permittivity component are also calculated from the selected γυ solutions. Assuming the absence of resonance phenomena in the measured sample, the obtained frequency course is then interpolated by a function, which is the Havriliak-Negami dispersion function, for both the real and the imaginary component of the solution of the complex permittivity of the measured sample, by relation (25):

^{ω £h +} ieoa) ⁺ 2_in (/ + (ίωτη)“)4 (25) where ^E is the complex permittivity, ⁿ is the high-frequency permittivity limit, σι is the low-frequency conductivity of the sample, is the vacuum permittivity (constant with the value, 8.854 x 10 ¹² F/m) ω is the angular ______^Τΐ_______ frequency, the sum of individual dispersion terms ί ¹⁺ ( ^ιωτ η>“) ^/ϊ for effective n components of the sample, is the dielectric increment of the effective nth component of the sample, ^T is the characteristic relaxation time The nth sample components aa and β are the shape parameters of the given dispersion term. Subsequently, outliers are eliminated using an iterative procedure with a heuristic criterion based on the distance of the point from the interpolation function - preferably so that in the resulting interpolation of the permittivity of the measured sample, the farthest point from the interpolation function is found and removed, and the procedure is repeated until the distance of the most the farthest point of the permittivity of the measured sample from the interpolation function approximation does not reach the value 3. The output of this method is the frequency dependence of the complex permittivity described by the interpolation function based on the real data obtained by the previous procedure.

Thus, the present invention provides a new method for obtaining broadband complex permittivity based on a transmission line, preferably a grounded coplanar waveguide. The main advantage of this method is the possibility to measure the dielectric properties of extremely small volumes of samples (on the order of pL), approximately 20 times smaller than with previously known and commercially available methods. The accuracy of determining complex permittivity is comparable to existing commercially used methods. The method works independently of the manufacturer of the circuit analyzer.

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Clarification of drawings

Giant. 1 schematically shows a grounded coplanar transmission line.

Giant. 2 shows a comparison of the results of determining the complex permittivity by the method according to the invention (described in the embodiment example) in comparison with the method using a coaxial probe, for an aqueous solution of cysteine with a concentration of 50 mg/ml. (Re: real component; Im: imaginary component)

Giant. 3 shows the standard deviation for 5 waveforms of the determined complex permittivity for an aqueous solution of cysteine with a concentration of 50 mg/ml.

Giant. 4 shows a comparison of the results of determining the complex permittivity by the method according to the invention (described in the embodiment example) compared to the method using a coaxial probe, for an aqueous solution of alanine with a concentration of 100 mg/ml.

Giant. 5 shows the standard deviation for 5 courses of the determined complex permittivity for an aqueous solution of alanine with a concentration of 100 mg/ml.

Giant. 6 shows a comparison of the results of determining the complex permittivity by the method according to the invention (described in the embodiment example) in comparison with the method using a coaxial probe, for an aqueous solution of cysteine with a concentration of 100 mg/ml.

Giant. 7 shows the standard deviation for 5 courses of the determined complex permittivity for an aqueous solution of cysteine with a concentration of 100 mg/ml.

Giant. 8 shows a comparison of the results of determining the complex permittivity by the method according to the invention (described in the embodiment example) compared to the method using a coaxial probe, for an aqueous solution of alanine with a concentration of 150 mg/ml.

Giant. 9 shows the standard deviation for 5 waveforms of the determined complex permittivity for an aqueous solution of alanine with a concentration of 150 mg/ml.

Giant. 10 shows a comparison of the results of the determination of the complex permittivity by the method according to the invention (described in the exemplary embodiment) compared to the method using a coaxial probe, for an aqueous solution of cysteine with a concentration of 150 mg/ml.

Giant. 11 shows the standard deviation for 5 waveforms of the determined complex permittivity for an aqueous solution of cysteine with a concentration of 150 mg/ml.

Examples of implementation of the invention

Example 1

One type of transmission line is shown in Fig. 1. It is a coplanar grounded transmission line with an inverted center section, which contains conductive ground paths 1, signal conductive path 2, dielectric substrate 3, metallized holes 4 conductively connecting the conductive ground paths on both sides of the substrate, metallized holes 5 conductively connecting the signal conductive path on both sides of the substrate, and a non-conductive border 6 to define the place for the liquid sample. The samples for measurement were prepared by dissolving the amino acids L-alanine (purity 98.5%, P.Lab CZ, R30761) and L-cysteine (purity 98%, P.Lab CZ, C10502) in Millipore Q ultrapure water. The prepared concentrations were 50 mg/ml, 100 mg/ml and 150 mg/ml.

- 9 CZ 2022 - 253 A3

Water (Millipore Q) and an aqueous solution of L-alanine with a concentration of 50 mg/ml were used as calibration samples. Each sample was loaded onto a transfer line in a volume of 250 microliters. Between individual samples, the transfer line was always cleaned with isopropanol and dried with a stream of nitrogen. Each sample was measured 5 times at a temperature of 23 °C.

For the calibration samples, the complex permittivity was determined by the Agilent 85070E Dielectric Probe Kit open coaxial probe method (0.5 to 50 GHz, 201 frequency points, thin electrode). Calibration was performed with calibration standards air, short circuit (Preparation 030 from the Agilent 85070E kit) and ultrapure water. The resulting complex permittivity value was obtained by averaging 9 measurements (3 independent samples and each measured 3x).

The permittivity of the RO4350B dielectric substrate (508 pm) was defined as 3.74 + 0.03j over the entire frequency band.

For each sample, the dispersion parameters were measured in the frequency range of 1.5 to 50 GHz on a circuit analyzer ZVA67 (Rohde & Schwarz) with a power setting of 0 dBm, a bandwidth of 200 Hz, the number of points 1601 and a linear distribution of frequency points. The circuit analyzer was calibrated with a calibration set (see Havelka Daniel et al. Sensors and Actuators B: Chemical, no. 273 (2018), 62-69.) built on the Multiline TRL calibration technique (Thru passage, Reflect - reflection (short circuit was used ) and four Line - transmission lines with a length of 1 mm, 2 mm, 4 mm, 8 mm).

Furthermore, the values of scattering parameters and zi - the characteristic impedance of the section of the transmission line with the first calibration sample - water, were obtained from the CST Microwave Studio numerical simulator. A frequency solver, tetrahedral grid with adaptive mesh refinement with a boundary for all scattering parameters set to 0.01 with 1 check was used to calculate the scattering parameters in the simulator. The limit for the parameter kz/kO = 0.005 with 2 checks. The number of mesh refinement passes is a minimum of 3 and a maximum of 20. The excitation of the electromagnetic wave was realized using a wave water port and the boundary conditions were set to open end from all 6 sides.

The scattering parameters, determined from the numerical simulator, Sil, S12, S21, S22 were converted to parameters A, D using the equations (l+S _1} )fl-S ₂₂ )+S _i2 S ₂₁ A=------- ------------, „ _ Q ~ Su) ' Q + ^S ₂₂ ) + ^S 12 ^S 21

2S ₂₁

The value of the wave propagation constant in the section of the transmission line with the first calibration sample, denoted as γ, was determined from the equation:

cosh ^-/ (ýA D) /;=------1-----where l is the length of the transmission line section with the first calibration sample.

The measured dispersion parameters of the first and second calibration samples were then converted to T parameters (elements of the M matrix) using the following formulas:

-10 CZ 2022 - 253 A3

T _n , _ $22 — ~P~>

•>21 , _ ^S n 21 — T ^- ' ^b 21

T21

With 12^21 - ^11^22

Here T _]2

T21 T22.

where M„ is either the measurement for the first calibration sample (water) Mi or the measurement for the second calibration sample (alanine 50 mg/ml) M. calculated from the corresponding scattering parameters.

Next, we obtain the propagation constant /2 on the transfer line with the second calibration sample (alanine 50 mg/ml) using the trace equation

Tr{M ₂ · M7 ¹ ] = 2 · cosh(y _; · I) cosh(y ₂ ·/) — (— + —}· sinh(y ₇ · l) sinh(y ₂ · 0, where γι is a constant propagation on the transmission line with the first calibration sample (water), l is the length of the line section with the calibration sample, Tr is the trace operator, M is the matrix from the measurement with the second calibration sample, Mi is the matrix from the measurement with the first calibration sample, which we obtain from the measured of the scattering parameters as follows

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The trace equation is an equation with one unknown γ ₂ . The solution γ ₂ is found using the NewtonRaphson method implemented in the numerical solver vpasolve in the Matlab program (version R2021B). The solver is set to find solutions to γ ₂ in the range -10,000 to 10,000 for the real component, and from -Inf to +Inf for the imaginary component.

-11 CZ 2022 - 253 A3

Next, the coefficients of the shape of the electric field Ka, Kb are determined, which, assuming that all the energy of the wave propagating along the line is located only in the sample (the shape of the field Ka) and in the dielectric substrate (the shape of the field Kb):

Yi - Ύ ² 2

J(a)Z _rl (€j — ef) (26) „ Υ&Α-Υ&Α

ÍmZ _rl U _s (e] — ef) (27) where yi is the propagation constant in the section of the transmission line with the first calibration sample, y? is the propagation constant in the section of the transmission line with the second calibration sample, j is an imaginary number, ω is the angular frequency (2π/), f is the frequency, Z _M . y: Zi, where Zi is the characteristic e* r' impedance of the line, ¹ is the permittivity of the first calibration sample, ² is the permittivity of the second calibration sample, and ^s is the permittivity of the substrate.

Next, a measured sample with an unknown permittivity ^u is applied to a section of the transmission line of the same length as for the calibration samples. After that, the dispersion parameters of the entire line with the sample are measured using a circuit analyzer. The scattering parameters were converted to T-parameters, which , * form the matrix Mu from measurements with a measured sample of unknown permittivity ^υ

Subsequently, yu - propagation constant in the line section with the measured sample of unknown permittivity is obtained from the trace equation:

Tr{M _u · My ¹ } = 2 cosh(y ₇ · l) cosh(y _y · í) - 02 _|_ Πθ . smh(y _; · /) smh(y _[; · i), where Mi and Mu are matrices from the measured scattering parameters of the first calibration sample and the measured sample, yi is the propagation constant of the line section with the first calibration sample and al is the length of the line section.

-12 CZ 2022 - 253 A3

There are a theoretically infinite number of solutions to the trace equation for γυ because the equation contains trigonometric functions that are periodic in the complex space of γυ. This heuristic algorithm is used to determine the physically correct solution.

The solution γυ is found using the Newton-Raphson method implemented in the numerical solver vpasolve in the Matlab program (version R2021B). The solver is set to find y ₂ solutions in the range 0 to 2000 for the real component, and from -8000 to +4000 for the imaginary component.

The solution is found using the Newton-Raphson method with iterations hundreds of times with random initial conditions. Thus, a set of solutions is obtained and non-repeating solutions for γυ are selected from them. Subsequently, these solutions for γυ are ordered in order of magnitude based on their absolute complex number value. Only the first two solutions with the smallest absolute value are selected. From these two solutions for γυ, the permittivity of the measured sample ^u is then calculated as:

[Κ _Β 4 + Κ _Α ^]·(^) -K _B e* where Ka and Kb are the electric field shape coefficients, is the propagation constant in the line section e* f* with one sample, ¹ is the permittivity of one calibration sample and ^s is the permittivity of the substrate. Since two γυ solutions were used, the result is two permittivity solutions. After that, the frequency intervals of the validity of the solution are determined as follows. The measured dispersion parameters of the entire transmission line (i.e. between the reference planes) with the calibration sample are inserted into:

(i + s _n ) · (i - s ₂₂ ) + s _]2 s ₂₁ 2S ₂ , — S ₂₁ ) · (1 + S ₂₂ ) + s ₁₂ s ₂₁ 2S„ cosh ^-jF (ýri D) ^{r =} - ------and-------

This gives the approximate propagation constants y _ÍC and yu- for the entire line with the first and second calibration samples, which are inserted into the trace equation (24):

(y _jr y _7r \ ---+ — · sinh(y _7C · l) sinh(y _2C · l) - Tr(M ₂ · Aíý ⁷ }· Y2C Y1C'

The frequency range where |R|> 100, (where |R| is defined as the absolute value of the real part of a complex number) is considered to be the range of invalid solutions. In the next procedure, permittivity values are used in the frequency range where v |R| =< 100. In practice, the frequency range of invalid solutions is located in the middle of the measured frequency interval and divides the area of valid solutions into two parts - a low-frequency part (defined by the frequency interval f < fl) and a high-frequency part (defined by the frequency interval f > f2). fl is the frequency at which the function described by equation (24) reaches the value |R|= 100 when the calculation is carried out from the lowest frequency point 1.5 G z to higher frequency points. f2 is the frequency at which the function described by equation (24) reaches the value |R|= 100 when calculating from the highest frequency point of 50 GHz to lower frequency points. The fl value was then adjusted heuristically to 8.2 GHz and the f2 value to 18.2 GHz. The imaginary component of the permittivity ^u is then calculated both from the found solution with the smallest absolute value γυ and also from the solution with the second smallest absolute value γυ. This results in two frequency courses of the imaginary component ^u . Both waveforms are then interpolated with a 4th-order polynomial, and these interpolated waveforms are averaged to give the waveform marked here as P. Subsequently,

-13 CZ 2022 - 253 A3 in the region f < f 1 choose the solution γυ leading to the values of the imaginary component of the permittivity above the curve P and in the region f > f2 choose the solution γυ leading to the values of the imaginary component of the permittivity below the curve P. From the chosen solutions γυ the values of the real permittivity component are also calculated.

The obtained solution is then interpolated with the following function (which is obtained from equation (25) after substituting a = β = 1 and considering that there are n=2 components in the sample (1 - water, 2 - alanine, or cysteine) assuming negligible low-frequency conductivity of the sample σί), for the real and imaginary component of the solution of the permittivity of the measured sample:

Δ; Δ ₂ e* (ω) = + ——--1- ——-1 + ιωτ _; 1 + ιωτ ₂ where ^Ě is the complex permittivity, is the high-frequency permittivity limit, which is the angular frequency (2π/), f is the frequency of the electromagnetic field, Δ/ and A ₂ are the dielectric increments of the respective components of the sample (1 - water, 2 - alanine or cysteine), τ _; and v are the characteristic relaxation times of the respective sample components (1 - water, 2 - alanine or cysteine). In the resulting fit of the permittivity of the unknown sample, the farthest point from the fitting function is found and removed. The procedure is repeated until the distance of the furthest point of the permittivity of the unknown sample from the approximation of the interpolation function by the model reaches the value 3. The output of this method is then the course of the permittivity described by the interpolation function based on the data obtained by the previous procedure. The results for the measured samples are shown in the figures.

Claims (10)

1. The method of determining the complex permittivity of the measured sample, characterized by the fact that it contains the following steps: a) determination of the complex permittivity of the calibration samples and the dielectric substrate of the transmission line; b) application of at least two calibration samples to a section of a transmission line containing a dielectric substrate on which electrical conductors are applied; c) measuring the dispersion parameters of the transmission line for each calibration sample using a circuit analyzer; d) determination of propagation constant and characteristic impedance in the transmission line section with the sample for one calibration sample using a numerical electromagnetic field simulator; e) determining the propagation constant in the section of the transmission line with the sample for the second calibration sample using the trace equation; f) providing field shape coefficients K _t and Kb; g) application of the measured sample to the same transmission line; h) measurement of dispersion parameters of the transmission line with the measured sample using a circuit analyzer; i) determination of the propagation constant in the section of the transmission line with the measured sample using the trace equation and selection of correct solutions of the trace equation using a heuristic algorithm; j) calculation of the complex permittivity of the measured sample from the propagation constant in the section of the transmission line with the measured sample, from the propagation constant in the section of the transmission line with one calibration sample and from the field shape coefficients using the equation where Ka άΚβ are the electric field shape coefficients, y> is the propagation constant in the transmission line section with one calibration sample, ';'r jc is the propagation constant in the transmission line section with the measured sample, €*; is the permittivity of one calibration sample and is the permittivity of the dielectric substrate. 2. The method according to claim 1, characterized in that the field shape coefficients Ka and Kb are determined from the propagation constants of the calibration samples and the complex permittivity of the calibration samples and the dielectric substrate of the transmission line using the equations ^A _K Y2 ^ě 1a~Y1 ^ě 2A ja)Z _RL e^(e] — e ₂ ) where yi is the propagation constant in the section of the transmission line with the first calibration sample, yr is the propagation constant in the section of the transmission line with the second calibration sample, / is an imaginary number (j ² = -1), ω is angular frequency (2π/), f is the frequency of the electromagnetic field, Zrl =yi Zi , where Zi is the characteristic impedance of the section of the transmission line with the first calibration sample, C*i is the permittivity of the first calibration sample, €*2 is the permittivity of the second calibration sample ař' ýjc permittivity of the dielectric substrate. - 15 CZ 2022 - 253 A3 3. The method according to any one of the preceding claims, characterized in that the propagation constant in the section of the transmission line with the sample for one calibration sample is determined from the dispersion parameters on the reference planes defining only the section of the transmission line with this calibration sample, which are obtained using a numerical simulator electromagnetic field; and 5 the value of the wave propagation constant in the line section with this calibration sample, denoted as yi , is determined from the equations: cosh VaM · D) ---------------------------- _l -------------------- ------(28) where l is the length of the section of the transmission line with the first calibration sample and where the parameters A, D are 10 are obtained from the dispersion parameters Sil, S12, S21, S22 as: U + · (J - $22) + ^S 12 ^S 21 2S„ (29) Ji-s _n )fi+s ₂₂ ) + s ₁₂ s _2] 2S _? , (30) 4. The method according to any one of the preceding claims, characterized in that the propagation constant γ ₂ on the transmission line with the second calibration sample is determined using the trace equation Tr{M ₂ · Mf ⁷ } - 2 · cosh(y _; /) cosh(y ₂ · 0 - Qy + yy) · sinh(y ₂ Z) siiih(y ₂ · 0, (31) 15 where γ is the propagation constant on the transmission line with the first calibration sample, l is the length of the transmission line section with the first calibration sample, Tr denotes the trace operator, M ₂ is the matrix from the measurement with the second calibration sample, Mi is the matrix from the measurement with the first calibration sample , which are obtained from the measured scattering parameters by Eqs - 16CZ 2022 - 253 A3 , _ ^2.2 12 — 7 > ^d 21 (34) τ — ^ ^{11 1} 21 ^— 7—' *-*21 (35) 5. The method according to claim 4, characterized in that the solution of the trace equation is obtained using the NewtonRaphson method. 5 6. The method according to any one of the preceding claims, characterized in that the propagation constant in the section of the transmission line with the measured sample is obtained by converting the scattering parameters of the transmission line with the measured sample into T-parameters that form the Mu matrix from the measurement with the measured sample sample using Eqs - 17CZ 2022 - 253 A3 (37) τ - $ ²² 12 — F~> (38) where Til Tj ₂ Til T ₂ 2 (41) and subsequently from the trace equation yu ~ the propagation constant in the section of the transmission line with the measured 5 sample is obtained using equation (18): Ττ{Μ _υ · M] ¹ } = 2 · cosh(y ₇ · Z) cosh(y _y · l) — · sinh(y ₇ · Z) sinh(y, ₇ · Z), (42) where Mi and Mu are the matrices from the measured scattering parameters of one calibration sample and the measured sample, yi is the propagation constant of the line section with one calibration sample and al is the length of the transmission line section, while the solution of the trace equation is found using the Newton-Raphson 10 method and while to determine the physically correct solution a heuristic algorithm is used. 7. The method according to any one of the preceding claims, characterized in that the propagation constant yu in the section of the transmission line with the measured sample is substituted into the equation [^ ^bě s + Κί ⁶ ;] * ^— Ml / (43) where Ka and Kb are the coefficients of the shape of the electric field, yi is the propagation constant in the section of the transmission line with one calibration sample, yu is the propagation constant in the section of the line with the measured sample, C*i is the permittivity of one calibration sample and C *s is the permittivity of the substrate, and thus the complex permittivity C*u of the measured sample is obtained. - 18CZ 2022 - 253 A3 8. The method according to claim 6, characterized in that the heuristic algorithm for determining the frequency ranges of valid solutions of equation (42) is applied by inserting the measured dispersion parameters of the entire transmission line, i.e. the sections before and after the calibration sample, into the equation. _ (2 + · (2 — S ₂₂ ) + S] ₃ S ₃₁ ~ 2Š^ _} ' (44) (2-^)-(2+^) + 5^ 2S _3! (45) cosh ^-2 (\M ^r =--------Ϊ-------- (46) where l is the length of the transmission line section with the sample; this gives the propagation constants yu· ay ₂ c for the line with the first and second calibration samples, which are subsequently inserted into the trace equation (24)·. R = 2- coshÍKic 0 cosh(y _2C · /) - (— + —) · sinh(r _1C · /) sinh(^ _2C i) V2C Key? ⁷ - Tr{M ₂ · Mf ¹ }. (47) where the frequency ranges with valid solutions yu are those where |R|=< 100, where |R| is defined as the absolute value of the real part of a complex number. 9. The method according to any one of the preceding claims, characterized in that the final frequency course of the permittivity of the measured sample is obtained by choosing a physically correct solution of the propagation constant yu for each frequency point in all frequency ranges with valid results such that the trace solution equation (18) is obtained several times with random initial conditions of the Newton-Raphson method, and then the two solutions with the lowest absolute value are selected from the set of obtained solutions y and from them, the solution that leads to the correct values of the imaginary component of the permittivity is selected by a heuristic algorithm, and then from the selected solutions yu, the values of the real component of the permittivity are calculated, and then the obtained frequency course is interpolated with the function ^{ω Eft} + ⁺ 2-in (j + (ίωτ _π ) ^α )^ (48) where + jc the complex permittivity, C/, is highly -frequency limit of permittivity, σ is the low-frequency conductivity of the sample, What is the permittivity of the vacuum, ω is the angular frequency, the sum Σ„ of individual dispersion __ (1+(ίωτ„ ) ^tt )0 terms for the effective n components of the sample, A„ is the dielectric increment effective n - those - 19CZ 2022 - 253 A3 components of the measured sample, τ„ is the characteristic relaxation time of the nth component of the sample, and and &β are the shape parameters of the given dispersion term, and then outliers are iteratively removed. 10. The method according to claim 9, characterized in that the sample is an aqueous solution, and in that the heuristic algorithm for determining the final frequency course of the permittivity of the measured sample is applied by choosing a physically correct solution of the propagation constant yu for each frequency range such that that the solution of the trace equation (18) is obtained several times with random initial conditions of the Newton-Raphson method and then the two solutions with the lowest absolute value are selected from the set of obtained solutions y and from them the solution that leads to higher values of the imaginary component of the permittivity in the low-frequency part (f<f 1) and which leads to lower values of the imaginary component of the permittivity in the high-frequency part (l>f2), and then the values of the real component of the permittivity are calculated from the selected solutions yu, and then the obtained frequency course is interpolated with the function / * ( 49) where C* is the complex permittivity, Ch is the high-frequency permittivity limit, σι is the low-frequency conductivity of the sample, Co is the vacuum permittivity, ω is the angular frequency, the sum Σ„ of individual dispersion terms for the effective n components of the sample, A„ is the effective dielectric increment of the nth component of the measured sample, ⁿ is the characteristic relaxation time of the nth component of the sample, aaa β are the shape parameters of the given dispersion term, and then outliers are removed iteratively.