Extraction of the Complex Relative Permittivity from the Characteristic Impedance of Transmission Line by Resolving Discontinuities

: This paper describes a material complex permittivity extraction technique based on four measurements of two identical coaxial (circular and rectangular) lines, distinguished by their lengths. The paper presents a combination of propagation parameters through mixing the eigenvalue principle and the lines’ characteristic impedance to improve the extraction techniques of intrinsic material parameters. However, the accuracy of some material parameters is insufﬁcient, as the discontinuities at the feedline–ideal line interface are not adequately solved. In these cases, a new formulation of the complex effective permittivity is suggested, associating the propagation constant and the characteristic impedance for a homogeneous structure. Next, uncertain errors that can negatively impact the method are removed from the mathematical expression. Then, a characteristic impedance expression is developed in the second stage to improve the mathematical formulation. Finally, a correction coefﬁcient in tune with reality and a polynomial function to amend the behavior of some of the curves are provided. The approach’s novelty lies in its ability to extract and correct the characteristic impedances despite discontinuity impedances at the ideal line–feedline interface. Several materials are tested with circular and/or rectangular coaxial ﬁxtures to conﬁrm the performance of the suggested method. The test cells are homogeneous, full, and long, at 80 mm and 100 mm (50 mm for the circular one). Determining the propagation constant from the eigenvalue of the wave cascading matrix (WCM) is a fundamental step in this method. Knowing the propagation constant helps to automatically compute a correction coefﬁcient that depends on the ﬁxture and the material being tested. Experimental validation is performed in the frequency range from some MHz to 10 GHz, 13.5 GHz, and 20 GHz, according to the tested material. Both test ﬁxtures are ﬁlled with the sample material, with a vacuum considered as a reference parameter. The method’s accuracy is better than 5% on the relative permittivity parameter throughout the frequency range. All the tested samples are compared with the results using the ﬁlled two-transmission-line technique (FTTL), using only the eigenvalue determination principle. The trapper cells are coaxially circular and rectangular.


Introduction
Advanced design systems have emerged out of the need to meet users' requirements in terms of comfort, safety, and protection, meeting these needs through the use of large amounts of information and strong communications. At the same time, the need for new technology keeps increasing. In this context, the integration of old and new materials, e.g., electric, magnetic, or magneto-electric materials, becomes an ever-increasing

The Approach Topology and Methodology
The transmission-line method is one of the most popular techniques because of its ability to be broadband and the simplicity of its implementation. It also has several types of structures (coaxial, microstrip, coplanar, etc.) and configurations [31,32], which contributes to its capacity to be used by many materials (wafers, liquid, powder, mud, etc.). Hence, the sample being tested cannot suffer from its insertion into a fixture cell if the right choice of structure has been made. However, at the same time, some limits appear because of the use thereof, especially when extracting loss tangents less than 5 × 10 −3 . Therefore, researchers continue to push through new and challenging barriers to improve the line technique's applicability. We try to contribute to this initiative by improving fundamental knowledge through the approach we present in this paper. Therefore, our aim is to set up a procedure Electronics 2022, 11, 4035 3 of 15 to extract electric and magnetic materials more accurately than what exists. This section deals with the topology and the methodology of extracting the intrinsic parameters of the material under test (MUT).

Topology of the Test Cell
Let us consider two identical waveguides (transmission lines) [33] that differ by their lengths, as shown in Figure 1 below. appear because of the use thereof, especially when extracting loss tangents less than 3 5 10 −  . Therefore, researchers continue to push through new and challenging barriers to improve the line technique's applicability. We try to contribute to this initiative by improving fundamental knowledge through the approach we present in this paper. Therefore, our aim is to set up a procedure to extract electric and magnetic materials more accurately than what exists. This section deals with the topology and the methodology of extracting the intrinsic parameters of the material under test (MUT).

Topology of the Test Cell
Let us consider two identical waveguides (transmission lines) [33] that differ by their lengths, as shown in Figure 1 below. l ), a feedline ( x l ) is placed to let the wave move from one side to another. To simplify the approach studied, both feedlines (connectors) are considered identical, while all conductors are the same [34]. Access ports 1 and 2 are used for the S-parameter data acquisitions during the measurement steps.

Mathematical Modeling: Methodology
The method is based on the combination of a transmission line's propagation constant and characteristic impedance to extract the complex relative permittivity of a dielectric material. On one side, developing and correcting the characteristic impedance before being amended through the automatic coefficient reduces the impact of uncertainties. On the other hand, the propagation constant is determined from the twoline eigenvalues. This last parameter is also amended as for the previous one, with the same objective. Therefore, the final definition of the material dielectric constant and loss tangent will be proposed. The Maxwell Equation developments link the electric waveguide length  to the relative part (for a homogeneous structure) of the propagation environment (area) or the MUT [32,35], as given in Equation (1): where ' r  and ' r  are the MUT's relative permittivity and permeability, respectively; l is the line's length;  is the phase constant, and f is the operating frequency. The electric length is given as follows when the ideal waveguide is filled with the vacuum: At each ideal line (l 1 and l 2 ), a feedline (l x ) is placed to let the wave move from one side to another. To simplify the approach studied, both feedlines (connectors) are considered identical, while all conductors are the same [34]. Access ports 1 and 2 are used for the S-parameter data acquisitions during the measurement steps.

Mathematical Modeling: Methodology
The method is based on the combination of a transmission line's propagation constant and characteristic impedance to extract the complex relative permittivity of a dielectric material. On one side, developing and correcting the characteristic impedance before being amended through the automatic coefficient reduces the impact of uncertainties. On the other hand, the propagation constant is determined from the two-line eigenvalues. This last parameter is also amended as for the previous one, with the same objective. Therefore, the final definition of the material dielectric constant and loss tangent will be proposed. The Maxwell Equation developments link the electric waveguide length θ to the relative part (for a homogeneous structure) of the propagation environment (area) or the MUT [32,35], as given in Equation (1): where ε r and µ r are the MUT's relative permittivity and permeability, respectively; l is the line's length; β is the phase constant, and f is the operating frequency. The electric length is given as follows when the ideal waveguide is filled with the vacuum: By substituting the intrinsic parameters from the Equation (1) and combining them with the Equation (2), it is proven that: At the same time, the characteristic structure impedances in vacuum (Z v ) and MUT (Z m ) configurations are linked as expressed in Equation (4): With the substitution of µ r from Equation (3) and into Equation (4), it appears that the relative permittivity is determined as: Due to Equation (5), the relative permeability becomes the following Equation: These Equations (1)-(6) are available when the MUT is perfect (does not have losses) and the test cell has perfect conductors. Otherwise, the dielectric and magnetic loss tangents can be determined after making some corrections highlighting some uncertainty coefficients that depend on the scanned frequency, as given below: and In that case, the dielectric loss tangent (tan δ d ) is obtained using the equation: and the magnetic loss tangent (tan δ m ) is given as follows: where α m and α v are the attenuation coefficients measured when the MUT is trapped inside the test cell and vacuum, respectively. In addition, Z m and Z v are the imaginary part of the complex characteristic impedances of the waveguide when the MUT and vacuum are trapped inside. Performing electromagnetic characterization is a priority in understanding any environment's electromagnetic wave propagation characteristics. Such characterization consists of disturbing field lines in a very different medium. One problem in this domain is solving the discontinuities at the trapper-feedline interface. Among many techniques, the transmission line technique becomes more important than the broadband technique [36,37], which uses several line configurations to reach its goals. Therefore, several studies have focused on the two-line approach, using the difference in the lines' lengths to solve the discontinuity effects [35,38]. According to electromagnetic theory, using the same material inside two identical waveguides through their variable lengths means the discontinuities are the same [39]. The wave cascading transfer matrix (WCM) [34] allowed us to extract the propagation constant by computing their eigenvalues [39][40][41]. The four T-matrices are given in the following [42,43] equations: and are the transfer T-matrices for vacuum (v) and MUT (m), respectively, written after data acquisitions in both transmission lines, long l 1 and l 2 . Moreover, using the CWM leads to: Let us consider that: The ideal transmission line has a length: where l 2 > l 1 , and its transfer T-matrix is written as follows: The resolution of Equations (13), (14) and (17) leads to: Equation (18) is written in both configurations as: The correction of this previous equation leads to: Figure 1 can be represented by the characteristic impedances Z v,m and the propagation constant, as seen in Figure 2.  The transfer T-matrix of Figure 2 can be written according to reference [44] as: Developing and simplifying Equations (36) and (37) gives the following results: where 0 50 Z = Ω, and "" x is the correction coefficient: and Figure 2. A waveguide of length l, represented by its secondary parameters and the source port's excited characteristic impedance, Z 0 .
The transfer T-matrix of Figure 2 can be written according to reference [44] as: with Developing and simplifying Equations (13) and (14) gives the following results: and where Z 0 = 50 Ω, and "x" is the correction coefficient: and As described earlier, the new proposed approach finds its origins and principle in Equations (13), (14), (25), and (26) before determining the relative permittivity through Equation (5) and the dielectric loss tangent given in Equation (9). Finally, the extracted results have been compared to those obtained with the eigenvalues of the filled two-lines principle (EFTLP). The eigenvalue's determination principle uses two primary mathematic relations from Equation (3) if the sample under test is either electric (that means µ r ≈ 1) or magnetic (that means ε r ≈ 1). Both equations are written as follows: Some equations, developed in [30,34], have shown how to take off the conductor dissipation factor or reduce its impact during data compilation and processing. Because of this, we can establish that the dielectric dissipation factor (DF) is inferred by: and for the magnetic loss tangent, by: The uncertainty impacts of the conductor DF are well-described in [45]. The characteristic impedance correction is made through the 3-degree polynomial function given as follows: where " f " is the frequency. This paper deals addresses chosen main points. These points are not necessarily the same for all tested materials. The coefficients a v,m 3 , a v,m 2 , a v,m 1 , and a v,m 0 are determined from four characteristic impedance measurement points from the material propagation parameters, as given in the following equation: Due to the new definition of Z v,m , all previous expressions including Z v,m are replaced by Z i v,m (c) .

Measurement Validations, Results, and Discussion
Two circular coaxial fixtures, manufacturer at Grenoble (France) were used for testing Q-Cell 5020 (ceramic powder), semolina and polenta (food), and aquarium sand (environment). In contrast, two other rectangle coaxial frames were used for PLA 100% and semi-flex 100% (3D-printing filament). As a result, the scanned frequency range was not the same for all of them. The validated experimental results were computed after measurements were taken with a vectorial network analyzer (VNA), Anritsu MS4642B. The Quasi-TEM (Q-TEM) mode was used to wave into the test cell. The phase constants and/or the electric lengths were linearized [46] to avoid the appearance of phase breaks, which were significant error sources. The five considered insulators were used through their specimens to extract the dielectric constant (DK) and DF material parameters. The rectangular and circular coaxial measurement fixtures are depicted in Figure 3.
The experimental results are depicted in Figures 4-10 for the six MUTs. We sketched results from the proposed approach and compared them to those from the eigenvalues principle to highlight its advantages and drawbacks. It can be seen that all relative permittivity and loss tangent plots had a similar trend. Similarly, differences in the results were not significant. This indicates that both Equations (5) and (29) are almost equal, but Equation (29) shows improved behavior. The slight difference is the use of the characteristic impedance (which does not depend on the length), which conceptually keeps the same value in the entire frequency range. At the same time, the loss tangent expressions given in Equations (9) and (30) depend on a factor that is not necessarily equal to a fixed number, as shown in Equation (9). The loss tangent, computed from Equation (9) not the same for all of them. The validated experimental results were computed after measurements were taken with a vectorial network analyzer (VNA), Anritsu MS4642B. The Quasi-TEM (Q-TEM) mode was used to wave into the test cell. The phase constants and/or the electric lengths were linearized [46] to avoid the appearance of phase breaks, which were significant error sources. The five considered insulators were used through their specimens to extract the dielectric constant (DK) and DF material parameters. The rectangular and circular coaxial measurement fixtures are depicted in Figure 3.           Aquarium Sand (Equation (5)) Aquarium Sand (Equation (29)) Polenta (Equation (5)) Polenta (Equation (29)) Aquarium Sand (Equation (9)) Aquarium Sand (Equation (30)) Polenta (Equation (9)) Polenta (Equation (30)    Aquarium Sand (Equation (5)) Aquarium Sand (Equation (29)) Polenta (Equation (5)) Polenta (Equation (29)) Aquarium Sand (Equation (9)) Aquarium Sand (Equation (30)) Polenta (Equation (9)) Polenta (Equation (30)   We sketched results from the proposed approach and compared them to those from the eigenvalues principle to highlight its advantages and drawbacks. It can be seen that all relative permittivity and loss tangent plots had a similar trend. Similarly, differences in the results were not significant. This indicates that both Equations (52) and (29) are almost equal, but Equation (29) shows improved behavior. The slight difference is the use of the characteristic impedance (which does not depend on the length), which conceptually keeps the same value in the entire frequency range. At the same time, the loss tangent expressions given in Equations (9) and (30) depend on a factor that is not necessarily equal to a fixed number, as shown in Equation (9). The loss tangent, computed from Equation (9), clearly depends on For the same test cell (circular coaxial), it can be seen that the relative permittivity and the correction coefficient X increased. However, at the same time, the a − coefficient decreased. As a consequence, it is observed in Table 1 that the correction coefficient X depends on the scanned frequency (in terms of the bandwidth and the considered number of points), the test cell, and its shape. For the same test cell (circular coaxial), it can be seen that the relative permittivity and the correction coefficient X increased. However, at the same time, the a-coefficient decreased. As a consequence, it is observed in Table 1 that the correction coefficient X depends on the scanned frequency (in terms of the bandwidth and the considered number of points), the test cell, and its shape. Scanned bandwidth frequency (GHz) 10 10 10 13.5 10 20 The relative permittivity uncertainty, using the eigenvalue principle, was presented in [45] by Equation (34) below: In Table 2, all plotted values are the medium values of the full results in the considered frequency range. It is noticed that ∆ε r is lower when using the eigenvalue principle than the newly developed approach. This is because the correction causes the gap observed in the final results in the propagation constant, but not in the characteristic impedance. For instance, taking off Z v in Equation (7), and according to the Brews equation for Q-TEM mode, as mentioned in [47]: where tan δ c is the conductor loss tangent. Then, Equation (7) can be rewritten as follows: In the absence of the MUT, Equation (7) becomes the following: The vacuum is a reference material with no dissipation factor (DF) or loss tangent, as described by Equation (36). It can be observed from Equation (37) that the uncertainty ∆ε r (c) depends on a coefficient and tan δ d . The corrected relative permittivity is at last given as: Table 3 compares the amended relative permittivity uncertainty from Equations (7) and (36). Table 3. Comparison of the uncertainty ∆ε r from the new approach definition when using the Equations (7) and (36).

Equation (7)
Equation ( Table 3 shows the excellent contribution of the new approach when Equation (36) is used. However, errors can be numerous when utilizing Equation (7). We finally note that the error levels in both techniques (use of the eigenvalue and the new approach) are identical. Moreover, the new approach has the advantage of using all materials (electric, magnetic, and magneto-electric). The measurements can be performed simultaneously with specific information from the characteristic impedance behavior.

Conclusions
We have modeled, developed, validated through experimental measures, and analyzed a new method that combines the secondary parameters of any waveguide. The propagation constant and the characteristic impedance constitute the transmission line's secondary parameters. The approach from the propagation constant extraction has been thoroughly presented using the two-line technique's eigenvalue principle and the characteristic impedance. Its enhancement through the correction of several parameters has Electronics 2022, 11, 4035 13 of 15 been detailed to reach the primary goal. We have pointed out the importance of making corrections, especially to the relative permittivity definition and the uncertainty parameter. Mathematical modeling expression has been critical to the broadband extraction of the microwave domain's dielectric constant (DK) and dielectric loss tangent (DF). The two test cell shapes (circular and rectangular coaxial) allowed for characterizing six sample materials: semolina, polenta, aquarium sand, semi-flex, PLA, and Q-Cell 5020. Some scanned a frequency range of 0.08-10 GHz, another reached 13.5 GHz, while the others covered an extensive frequency range up to 20 GHz. We observed differences in the scanned frequency, as the measurements were not taken simultaneously. Both fixtures were homogeneous, with lengths of 50 mm and 80 mm for the circular coaxial fixture and 80 mm and 100 mm for the rectangular coaxial fixture. The mathematical formulation helped to reduce the extraction parameter errors caused by the uncertainty term. This new proposed approach is a promising candidate for the use of any unknown material (electric, magnetic, or magneto-electric) with any two-conductor transmission line (stripline, coplanar, etc.). Funding: This research received no external funding.

Data Availability Statement:
No new data were created or analyzed in this study. Data sharing is not applicable to this article.