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The characterization of the dielectric properties of a material requires a measurement technique and its associated analysis method. In this work, the configuration involving two coaxial probes with a material for dielectric measurement between them is analyzed with a mode-matching approach. To that effect, two models with different complexity and particularities are proposed. It will be shown how convergence is sped up for accurate results by using a proper choice of higher-order modes along with a combination of perfect electric wall and perfect magnetic wall boundary conditions. It will also be shown how the frequency response is affected by the flange mounting size, which can be, rigorously and efficiently, taken into account with the same type of approach. This numerical study is validated through a wide range of simulations with reference values from another method, showing how the proposed approaches can be used for the broadband characterization of this well-known, but with a recent renewed interest from the research community, dielectric measurement setup.

The behavior of a material under the exposure to electromagnetic fields is determined by its dielectric properties. The accurate knowledge of these properties is essential in science and engineering, since they are used in a variety of industries [

More specifically, in microwave engineering [

Regardless of the specific area where the dielectric properties are required, the process to characterize them is based on two techniques: (i) a measurement technique extracting a parameter related directly or (more commonly) indirectly with the dielectric properties and (ii) an electromagnetic analysis technique (circuit-based or full-wave) modelling the measurement, i.e., providing estimations of the results of that measurement for a given material.

With respect to the measurement techniques, there is a plurality of options, such as those based on coaxial probes, transmission lines, free space methods, use of resonators (as in microwave filter characterization), etc. [

With respect to the analysis technique used in the dielectric characterization process, when the electromagnetic analysis is based on a full-wave approach, the results will be more precise, at the expense of a higher computational burden than simpler circuit analyses. This paper will use an accurate full-wave electromagnetic method based on mode-matching methods [

In fact, this kind of problem has received very recent attention for different types of materials and with different types of probes for the input/output [

The coaxial probe open at one end, whose basic schematic version is shown in Figure

Schematic of the one-port open coaxial used for dielectric measurement, showing the material under test (MUT) with its dielectric properties: (a) 3D schematic view; (b) longitudinal cut (revolution symmetry with respect to z).

From the electromagnetic point of view, the problem in Figure _{sample}) must be electrically large enough in order to confine the incoming electromagnetic field excited by the probe within the MUT limits. Within the same modelling, the metallic flange is assumed to be large enough for avoiding backward radiation.

The measurement technique for this case extracts the reflected electromagnetic wave at the one-port problem. The signal excited by the generator connected to the coaxial propagates through the transmission line and reaches the material, where it is partially reflected back. This reflection coefficient depends on the electromagnetic properties of the dielectric material. Thus, from this measurement, the values of the dielectric properties of the material in contact with the probe can be estimated. However, there is not a direct closed formula relating the

The two-port measurement is a method directly related to the coaxial probe [

Schematic of the two-port coaxial holder for dielectric measurement, showing the material under test (MUT): (a) 3D schematic view; (b) longitudinal cut (revolution symmetry with respect to z).

In this work, the two models shown in Figure

Schematic proposed models for the two-port characterization (additional details will be given in Figures

An alternative model B is shown in Figures

The proposed two models are analyzed with the mode-matching method, comprised of two main stages. First, the modes of all the waveguides involved in the problem are calculated. Then, the modal series to expand the fields in each waveguide are matched at each side of the step discontinuity, providing the Generalized Scattering Matrix (GSM) of each step. Finally, the GSMs of each step are cascaded [

The electromagnetic field inside homogeneous waveguides can be described in terms of transversal electromagnetic (TEM), transversal electric (TE), and transversal magnetic (TM) modes [

Cross-section of the type of homogenous waveguides used in this work: circular (with PEW and PMW boundary conditions) and circular coaxial.

For the coaxial waveguide, the TEM mode is also necessary to build the complete mode spectrum of the waveguide, and it can be easily computed analytically by solving the Laplace equation, where

It is important to note that the cutoff wavenumbers for the coaxial waveguide must be computed for each specific aspect ratio

As every waveguide in the two models of Figure

In the reference plane

Basic waveguide step between circular and coaxial waveguides for the models in Figure

(a) Dimensions defining the model A (with revolution symmetry): case A1, PEW; case A2, PMW; case A3, combination of PEW and PMW; (b) Block diagram of the analysis by mode-matching, showing the symmetry plane.

The mode-matching technique described in the previous section has been applied for the two proposed models in Figure

The test cases will be evaluated under the two models

Basic data for the test cases.

Item | Value |
---|---|

| 1.75 mm |

| 0.46 mm |

| 2.54 |

| 10-j0.1 ( |

| 5-j0.05 ( |

Sample Thickness ( | 0.2 mm/1.0 mm |

Frequency range | 100 MHz – 20 GHz |

The reference values are provided by the method from the

Model A is detailed in Figure

Since the problem has no angular variation, under the TEM excitation by the coaxial line, the series (

Magnitude (dB) of S_{11} and S_{21} for

In the simulation in Figure _{11} and S_{21} parameters is shown for different values of the radius_{sample} of the virtual circular waveguide between the input and output coaxial lines. The analysis starts with_{sample} equal to 20 times the outer radius of the input/output coaxial, which shows a ripple that starts to decrease when this circular waveguide radius is enlarged. In this case, when_{sample} increases, the results in both reflection and transmission are flatter and converge to the larger case of_{sample}=200

In order to overcome this issue, Figure _{sample} increases as well, now the results converge to those of the reference case obtained by DAK. Thus, the response improves the results obtained from the previous case, having a very good agreement. The phase is also shown in this graph, showing also a very good agreement. The phase reference is taken at the interface between the coaxial and the circular waveguide in all the cases.

Magnitude (dB) and phase (degrees) of S_{11} and S_{21} for

The same type of analysis is now done with the sample having a thickness of 1mm. From now on, the analysis will be always performed with TEM and _{sample} increases till achieving convergent results matching those of the DAK reference.

Magnitude (dB) and phase (degrees) of S_{11} and S_{21} for

In model A, we are expecting that the electromagnetic field at the lateral limit of the circular waveguide (i.e., for large_{11} and S_{21} has a very similar behavior to that of the case of model A1 with PEW in Figure _{sample}, which disappears when_{sample} is large enough.

Magnitude (dB) of S_{11} and S_{21} for

In all the simulations done so far, we have presented the results with a number of modes large enough to achieve convergent results with respect to the number of terms in the series (

The convergence has been verified with several analyses, varying the number of modes while keeping a fixed geometry. One of those analyses is shown in Figure _{coax} and in the large circular waveguide_{circ}.

Convergence analysis. Magnitude (dB) of S_{11} and S_{21} for_{sample}=200b).

Figure _{sample}=10_{sample}. In addition, since both boundary conditions in the circular waveguide are approximations of the actual measurement problem, the combination of both results could be also another suitable analysis model. Moreover, this type of strategy has also been used in [

Magnitude (dB) of S_{11} and S_{21} comparing models_{sample}=10

Following this rationale, Figure

Magnitude (dB) and phase (degrees) of S_{11} and S_{21} for

(a) Dimensions defining the model B (with revolution symmetry) taking the finite flanges into account. (b) Block diagram of the analysis by mode-matching, showing the symmetry plane.

The second model to compute the two-port problem is shown in Figure

From the mode-matching point of view, the new geometry involves two new virtual coaxial transmission lines at each side of the problem, associated with the incorporation of the finite flange with radius_{flange} and length _{o} and_{flange}, respectively, both with outer radius of_{sample}. They are filled by air.

This geometry involves a new step discontinuity between the two virtual air-filled coaxial lines associated with the outer part of the structure, as is shown in Figure _{flange} and_{sample}) and, at the other side, the larger circular waveguide of radius_{sample}. In this problem, two additional virtual ports appear, carrying the signal not confined within the MUT limits and not taken into account by model A. Once the GSM of the steps and bifurcations are computed, they are also easily cascaded.

For the following tests, in addition to the data in Table _{flange}=1mm, with a cover for the input/output coaxial of 0.5mm thickness, i.e.,_{o}=_{flange} =5.4_{sample} associated with the circular waveguide between 10_{sample}, the finite size of the flange is leading to results different from those of the DAK reference, regardless of the size of the virtual circular waveguide required in the mode-matching simulations. The DAK values corresponded to the typical geometry with a large enough flange and electromagnetic field fully confined within the MUT limits, which is coherent with results of model A.

Magnitude (dB) of S_{11} and S_{21} for_{sample} (thickness_{flange}=5.4

This effect of the size of the flange is further analyzed in Figure _{flange}. The ripple is now associated with_{flange} since these simulations have a fixed value of_{sample}. It is also important to note that additional terms in series (

Magnitude (dB) of S_{11} and S_{21} for_{sample}=50

In this case, since the new air-filled coaxial lines have a large size, many modes will be needed to approximate the field within them, with a high aspect ratio between the outer and inner conductor. Thus, their modes will have to be computed carefully to avoid instabilities related to the behavior of the

Figure _{flange} in Figure

Magnitude (dB) of S_{31} and S_{41} for_{sample}=50b).

Magnitude (dB) of S_{31} and S_{41} for_{sample}=50b).

The mathematical formulation of the mode-matching method applied to the two-port scattering problem arising in a material placed between two coaxial probes has been presented. This well-known configuration has received renewed attention in very recent years. In this line, this paper has developed two different approaches to characterize this dielectric measurement setup for broadband applications. The analysis of the problem with the MUT considering both PMW and PEW has been studied, while their combination has been shown to improve the efficiency of the method. In all cases, the use of higher-order modes in the coaxial waveguide at the MUT interface is essential to obtain fast and accurate results. The effect of the finite flanges has been also efficiently and rigorously assessed, proving that a flatter frequency response is achieved when the field is confined within the MUT, i.e., when large flanges are considered. Two setups have been analyzed and their results have been compared with the reference data provided by the DAK characterization method, showing excellent agreement between them.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported in part by the Spanish Government (Agencia Estatal de Investigación, Fondo Europeo de Desarrollo Regional) under Grant TEC2016-76070-C3-1/2-R (AEI/FEDER, UE) and in part by the program of Comunidad de Madrid under Grant S2013/ICE-3000. The authors want to acknowledge SPEAG company, Zurich, Switzerland, especially Pedro Crespo-Valero and Ferenc Muranyi, for their assistance with the DAK results.