Probe with a metallic bowl termination to determine the electric material parameters from variable thicknesses

This paper discusses a new approach to determining the material electric parameters (dielectric constant and dissipation factor) through a technique based on identical material variation thicknesses. The copper bowl controls the material thicknesses and allows quick and safe measurements to get to the electromagnetic material information. This approach assumes that the discontinuities generated at the probe and bowl are identical and removed through mathematical operations. The material under test (MUT) gap thickness doesn’t exceed one millimeter. Some formulations have been developed to determine the discontinuity condensers and both capacitors of each thickness. Two proposed developments have been given to extract the complex effective and relative permittivities. A mathematical formula has been proposed to ensure the transition from effective to intrinsic parameters, where the MUT thicknesses and the probe’s outer diameter of the inner conductor are considered. A de-embedding operation is necessary, and the method is suitable but not limited to low-k. The coaxial probe is flat terminated, and the experimental validation has been made in the frequency ranges 1 MHz—2.2 GHz for FR-4 HTG-175 and 1 MHz—0.8 GHz for Alumina 99.5%. All MUTs are 5.0×5.0 cm2, except their thicknesses. The suggested approach is straightforward in sample insertion, simple in data acquisition and implementation, and precise in the final parameters’ results.


Introduction
Electromagnetic material characterization (EMMC) is a field of physics that is perpetually growing because of its importance.The literature is rich enough in the deep details of the methods involved [1,2].This domain is classified into two leading families: destructive or non-destructive [3,4], resonant or non-resonant [5], one or two accesses [6], broadband or narrowband [7][8][9], and distributed or lumped elements.However, these two large groups present six primary approaches or methods for accessing the test sample's intrinsic parameters.The material intrinsic parameters are the central column mapping the material's ability to react under the excitation of an electromagnetic wave.Thereby, the probe (in transmission or reflection) [7,10,11], free-space (antenna or ellipsometry) [12][13][14], cavity [15][16][17][18], parallel plates capacitor (MIM) [19][20][21], inductive (spiral approach) [22,23], and transmission line (stub, short-circuit, waveguide, etc) [24][25][26].All these methods are based on the perturbation [27][28][29] of the electromagnetic field lines in the insertion environment of the sample to be characterized.Improving access to material parameters at particular frequencies and the arrival of new materials are the continuous challenges of research in this science area.In that case, the design of new gadgets and their manufacture to satisfy consumers across industries specializing in different sectors become more and more desirable and challenging [11,30].This requires adapting the techniques to the new constraints by combining existing approaches and focusing on critical parameters of the material under test (MUT) [31][32][33][34].The six techniques mentioned above do not offer the same ease of sample insertion, simplicity and rapidity in the extraction implementation, details on the parameters extracted, and possibilities of coverage of study frequency range.MUT appears as a wafer (hard or soft), liquid (light or heavy), powder or seeds, and gas.The EMMC domain concerns activity sectors such as biomedical, food engineering, medical, telecommunication, microelectronics, building, aerospace, army, etc [35][36][37][38].The resonant cavity is the best method of extracting complex permittivity or permeability.Still, the narrowness of the swept frequency band limits it [23,39,40].In addition, the transmission line technique, offering a large frequency scan, is the most popular but limited by the accuracy of the parameters to be extracted.All this leads to the development of compromise principles to achieve the objectives.
In this context, a new approach is presented based on the simplicity of inserting the MUT, varying its thickness, determining the real capacitors related to the sample through a mathematical model, and overcoming the effects of discontinuities at the contact interfaces specimen-test fixture.As several publications work for [11][12][13], an open-coaxial-ended probe is used for wafer material instead of liquid.The novelty of the proposed approach is carried out by removing the effects of the interfaces because of the variation of the thickness [7,28], and the mathematical definition of the capacitor, allowing us to go back to the intrinsic parameters of the MUT.Scanning a frequency range from 1 MHz to 2.2 GHz with an Anritsu MS4642B 20 GHz vector network analyzer (VNA), the suggested approach has been experimentally validated with Alumina 99.5% and FR-4 HTG-175.The proposed method has been applied to dielectric wafers, and the test bench is suitable for wafers, not liquid, seed, and dust-type materials.Based on the MUT thickness variation, the proposed method helps determine and solve the discontinuity impacts on the dielectric parameter measurements.Thereby, the manufactured test device has several advantages for fast and reliable measurements, including simple sample cutting, easy to insert several times into the test cell, time-saving (in terms of insertion and measurement), fast data acquisition, and a reduced number of test fixtures (two for the coaxial method and one for the suggested method) to achieve thickness variation.One of the method's constraints is to ensure the flatness of the sample faces in contact with the different metallization sample holder plates.Additionally, the method is complex in its implementation.Nevertheless, it has excellent and encouraging results in the end.

The approach topology and methodology
This section presents the fixture's topology and the methodology for extracting the intrinsic parameters of the MUT in-depth.

Fixture description
The test cell cross-section is illustrated in figure 1, represented by an open-coaxial and a metallic bowl.The metal bowl can extend, and the MUT fits the fixture according to its thickness h 1 or h .The open-coaxial probe has l 0 length, the characteristic impedance Z , 0 and the propagation constant g .0 The rear of the bowl is movable, and its height, H, is fixed.The probe is the feedline, and the metal bowl is the MUT trapper.There are discontinuities at the trapper-feedline interface T , out and it's assumed to be the same for the same MUT.All measurements and data acquisition are made at the input interface T .
in Therefore, each design can be presented as an electric schema, as illustrated in figure 1.That electric model is defined through a mathematical model [41,42] explained in the next section.

Methodology and modeling formulation
The association of the reflection coefficient in two configurations (short-open circuit) allows the secondary feedline parameters to be determined [26,28].The open circuit and the short-circuit configurations of the probe are symbolized by G oc and G .
sc Let us note Z n the normalized impedance, which is often W 50 .Therefore, the characteristic impedance is given as follows: and the propagation constant is written as, The load admittances ( ) Y out g 1, at the end of a transmission line are expressed from the input admittance ( ) From the electric models, two load admittances, depending on the thicknesses h 1 and h , 2 are written as expressed below, ( where G, C, C d 1 and w are the conductance, capacitance, discontinuity capacitance, and pulsation, respectively.The equations ( 6) and ( 7) are associated with equation (3) equality.v, m and g symbolize vacuum, material under test, and global measurement in the case of h , 2 respectively.The following equation has to be used to solve the discontinuities' impacts.
From equations ( 9) and (10), it is written by ascertaining, ⎪ This is a system of three equations with three unknowns whose resolution leads to the following results, )

⎫ ⎬ ⎭
Because of the longitudinal increase in the thickness of the MUT, both capacitors are in series, and the second condenser may be determined as follows, ( Combining equations ( 8) and (10) through their division, allows to express the second admittance ( ) Y v m , 2 as given below, Developing the equation (20) shows several parts, including the loss and capacitor uncertainties ( )

⎫ ⎬ ⎭
This study highlights the sample dimension impacts after the mathematical formalism and determination of the discontinuity capacitors.At that stage, the effective complex permittivity model is given below, which is,

*
The gap between both thicknesses is as follows, The new approach relates to a quick measure of the electric material parameter in the frequency range up to 2.3 GHz when using high transversal material dimensions, around 5 5 cm 2 .However, the frequency range can be extended by reducing the material dimensions.Above all, the method is suitable for material whose thickness gap is equal to or less than one.The maximum acceptable material thickness is 2 mm.In that case, the complex relative permittivity is linked to the complex effective permittivity by, The equivalent condenser obtained in the case of h 2 is expressed as, , from the equation ( 29) and inserting it in the equation ( 8) with the approximation of ( ) ( ) , , it is found out that, ( ⎛ ⎝ ⎞ ⎠ This equation (30) highlights the presence of an additional coefficient, which must be solved by removing it from that expression.In this way, the real part of the complex effective permittivity becomes, as is given [45] below, and losses can be expressed as follows, Finally, associating equations (31) and (32), the second way to express the complex effective permittivity is, ( ) The conversion function that allows determining the complex relative permittivity is given by, where 'b' is a coefficient depending on Dh.As done previously, the corresponding loss tangent from the use of the equation (34) becomes, The ratio between capacitors' thicknesses h 1 and h 2 allows a frequency-dependent coefficient H to be obtained that provides information on the frequency band to be covered during the study.It also orients on the real surface where the electromagnetic wave responds to the outcome of its excitation.In that case, the effective surface differs from the actual surface, resulting from the physical dimensions.
( ) The factor 'a' can be found below according to the material under test (MUT) thickness.
( ) where, is the number of points and 'x' a factor, depending on Dh. = x 1 if Dh is less than 1, and

Measurement results and discussion
A specific bench test with two samples under test (SUTs) covering 1 MHz to 2.2 GHz in the frequency range has been used.That benchmark comprises a vectorial network analyzer (VNA), radiofrequency (RF) cables, an extended open-coaxial probe with a coaxial N-connector, and the test cell fixture shown in figure 4. At the same time, the apparatus cross-section illustration is presented in figure 2. The flatness of the extended connector is shown in figure 3. Figure 4 shows the extension connector design defining the open-coaxial probe and the inserted MUT with the possibility of changing its thickness.The Anritsu MS4642B 20GHz VNA RF equipment has been used to investigate the 1 MHz-2.2GHz frequency range.The two MUTs are FR-4 HTG-175 and Alumina 99.5% with variable thicknesses.To briefly describe the fixture: The operating principle of the test cell is essentially  based on the spreading rotating handle, which allows for trapping the MUT and gauging its thickness.They are two metallic flatness plates: fixed and movable, as depicted in figures 2 and 3.The MUT is inserted into the gap between both metallic plates.
Moreover, the movable frame ensures good contact with the specimen to be measured.The connector is extended through the moving plate.There is a hot point through which the electromagnetic excitation propagates to experience a disturbance by the presence of the MUT.Figures 3 and 4(a) illustrate the test cell performed.In contrast, figure 4(b) shows the assembly of the test bench in the presence of the wafer to be characterized.As previously stated, the electromagnetic characterization of materials consists of disturbing the lines of an environment's electric and/or magnetic fields by the presence of the element to be measured [28].This is what is done in this context.Figure 2 shows 'Z' the wafer's thickness while 'L' representing the connector's length.
The coefficients' specificities 'a' and 'b' are given in table 1 through their values, according to the equation applicability.It is observed that the bigger the thickness, the higher the coefficient's value.
Figure 5 allows us to understand the impacts at the interface contact between the open-coaxial probe and the MUT.This study permitted determining the capacitance discontinuity from the thickness change.That capacitance is neglected before the resonance.The resonance appears in an extensive frequency range, depending on the MUT. Figure 6 helps to fix the scanned frequency range to be validated and points out the possibility of amending results.
The test cell measurements, when vacuum-filled up the fixture (homogeneous state), taught that the frequency band borders may reach 2.5 GHz.But things are more straightforward when the MUT is inserted and the frequency range is defined.So, the FR-4 HTG-175 covers 1 MHz to 2.2 GHz while the Alumina 99.5% covers 1 MHz to 0.8 GHz.However, they are resonances at specific frequencies.These are caused by the MUT dimensions being separated from discontinuities.The lower the dielectric constant, the higher the studied frequency range.
A slight difference is observed in plotted figures 7 and 8 between the values resulting from both developed approaches.This is normal and conforms to the literature, which shows that the accuracy depends on the procedure used.4.30 1.60 b (equation ( 34)) 3.4 0.90 Figure 5 shows the possibility of extracting the discontinuity capacitances in both configurations, with and without the MUT.Around the resonance frequency, discontinuities impact the results and are negligible elsewhere.Thus, reducing the sample dimensions is necessary to widen the work or study frequency band.Figure 6 gives information on the electrical dimension [43,46,47] (unlike the known and fixed physical dimension), which is linked to the frequency and considers the field lines' behavior in the propagation environment.This capacitance ratio comes from the equation (36) and decreases with increasing thickness.It indicates the frequency band to be covered according to the MUT.Also, figure 6 shows that the capacitor depends on sample thickness.Figures 7-10 agree with the previous one in the work validity band.This band extends to 2.2 GHz for FR-4 and 0.8 GHz for alumina 99.5%, when considering 5.0 5.0 cm 2 as width and length.
Considering the dimensions used on the one hand and the results obtained on the other hand, this method makes it possible to predict the frequency range to be covered.Also, the results presented in figures 5-7 indicate the method's limit.This limitation is mainly due to the sample size, the MUT relative permittivity, and the fixture design [6].This significantly impacts the discontinuity created at the connector-MUT interface.Since the developed method is based on the second capacitor, the results are consistent with those published in the literature.Despite a slight discrepancy, which can be overlooked, the results presented in figures 7 and 9 have the same trend and confirm the reliability of the proposed method.In addition, this     method shows limits for materials of very low loss tangents in terms of loss extraction, as seen in table 2. This is observed in figures 8 and 10.However, this approach is adapted to thin-and thick-film materials.This makes it a good candidate for solving the problems of material characterization.A comparison of the results obtained with those of the supplier or manufacturer through table 2 shows that the approach using equations ( 27) and ( 28) is better enough.The ratio of capacitors in the absence of test samples is 0.618 for Alumina at 0.5 GHz and 0.781 for FR-4 at 1 GHz.Both values are different from the thickness ratios shown in table 1.Therefore, meaningful information on the feasibility and validity of the suggested approach is given in figures 5 and 6.

Conclusion
A new approach has been described and validated with Alumina 99.5% and FR-4 HTG-175 in a scan frequency range of 1 MHz to 2.2 GHz.The tested materials are squares of 5.0 5.0 cm 2 and have various thicknesses 0.79 mm and 1.524 mm) for FR-4 HTG-175 and (1.013 mm and 2.067 mm) for Alumina 99.5%.This suggested approach has consisted of MUT variation thicknesses to solve the impact of the discontinuity at the interface MUT-test cell.The possibility of expressing the discontinuity capacitor and the capacitance created by the MUT has been developed.Mathematical modeling has been developed through some equations, associating the MUT thickness variations to point out the origin of the discontinuity impacts and how to solve them.The final results show that the approach limits (regarding frequency range to scan) come from the MUT dimensions.Such an approach opens the way to bi-layer material characterization.The used fixture is a flatted open-coaxial probe terminated by two movable metallic conductors to tighten up the MUT and adjust its thickness.The deembedding principle has been used, and the approach considers lumped elements placed at the end of the probe, making the suggested approach a broadband and great candidate for bi-layer materials, which makes it a relevant method.The new approach is broadband and is suitable for extracting loss tangents up to - 10 . 3This approach is a good candidate for solving the EMMC problems and improving this ever-changing sector.´-4.485 @ 1.3 10 2 ´-9.47 @ 4.221 10 3 Method 2: equations (34) and (35) ´-4.556 @ 1.8 10 2 ´-9.573 @ 2.011 10 3

Figure 1 .
Figure 1.(a) Test cell topology and its equivalent electric circuit for h . 1 (b) Test cell topology and its equivalent electric circuit for h 2 .
depends on Dh.The loss tangent[44], coming from the equation(27) , is given below,

Figure 2 .
Figure 2. Cross-section of a movable plate test cell.

Figure 3 .
Figure 3.A rectangular open coaxial probe terminated by a flatness for wafers.

Figure 4 .
Figure 4. (a) A manufactured movable rectangular test cell fixture.(b) A movable rectangular test cell fixture with the MUT.

Figure 5 .
Figure 5. (a) Impacts of discontinuity capacitance in the presence and absence of Alumina 99.5%.(b) Impacts of discontinuity capacitance in the presence and absence of FR-4 HTG-175.

Figure 6 .
Figure 6.(a)  The ratio between vacuum capacitances according to the MUT thicknesses.(b) The ratio between MUT capacitances according to their thicknesses.

Figure 7 .
Figure 7. (a) The measured relative permittivity of Alumina 99.5% determined from two approaches.(b) The measured relative permittivity of FR-4 HTG-175 determined from two approaches.

Figure 8 .
Figure 8.(a) The measured loss tangent of Alumina 99.5% determined from two different approaches.(b) The measured loss tangent of FR-4 HTG-175 determined from two different approaches.

Figure 9 .
Figure 9. (a) Zoom of the Alumina 99.5% relative permittivity determined from two approaches.(b) Zoom of the FR-4 HTG-175 relative permittivity determined from two approaches.

Figure 10 .
Figure 10.(a) Zoom of the Alumina 99.5% loss tangent determined from two approaches.(b) Zoom of the FR-4 HTG-175 loss tangent determined from two approaches.

Table 1 .
Sample under test dimensions and equation coefficient specificities.

Table 2 .
Manufacturer and extracted comparison values according to the developed approach.