A New Tool for Accurate S-Parameters Measurements
and Permittivity Reconstruction
Ainhoa G. Gorriti and Evert C. Slob
Abstract—In this paper, we present a tool for permittivity
re-construction. The tool consists of a coaxial transmission line and a novel analytical reconstruction technique. The line is designed for accurate permittivity measurements of soil samples. It allows for single or double phase flow. The full S-parameter matrix is mod-eled with transmission line theory. We show that for accurate mea-surements each component of the tool needs careful calibration. We give a method to compute the sensitivity of these measurements to different materials, and we show the accuracy in the determination of the sample permittivity from the measurements based on devia-tions of the forward model compared to measured results as a func-tion of the error in the sample permittivity. We demonstrate that a maximum error of less than 1% is obtained for the possible permit-tivity reconstruction from these measurements for wavelengths less than five times the sample holder length. The novel reconstruction technique is based in the propagation matrices method. It consists in rewriting the transmission line model in matrix notation and re-lating the S-parameters to the total reflection and transmission of the line seen as a two-port network. Thanks to this novel approach, the reconstruction of both permittivity and permeability can be done directly from the measured S-parameters of the line. Due to experimental instabilities the errors are considerable, and we must restrict ourselves to the reconstruction of permittivity. Results on an air sample are within the predicted error of 1%. Computing relative errors for an ethanol sample is not possible due to temper-ature and purity difference between our experiment and published results; nevertheless, the reconstructed permittivity of ethanol fol-lows the Debye model.
Index Terms—Calibration, coaxial transmission line, microwave
measurements, permittivity, soil measurements.
I. INTRODUCTION
A
S GEOPHYSICISTS, we are interested in the imaging ca-pabilities of electromagnetic (EM) waves. The goal is to obtain an image of the subsurface of the Earth from its response to EM fields. It is then essential to know how the materials that are commonly found in the subsurface of the Earth react to ap-plied EM fields. These materials are composed of several con-stituents, and different phases coexist (solid, liquid, and gas). They are very complex, and many different parameters influ-ence their electromagnetic properties. Therefore, to study these properties, convenient and controlled experiments have to be performed, taking special care in their accuracy and reliability.This paper presents the design and calibration of a tool to study the response of these materials to an applied EM field. The
Manuscript received October 7, 2004; revised March 7, 2005. This work was supported by the Netherlands Organization of Scientific Research (NWO) under Contract 809.62.013.
The authors are with the Department of Geotechnology, Technical University of Delft, 2600GA Delft, The Netherlands (e-mail: a.g.gorriti@citg.tudelft.nl).
Digital Object Identifier 10.1109/TGRS.2005.851163
tool consists of an experimental setup together with a novel an-alytical reconstruction technique. We have restricted our study to the complex permittivity because most materials of geo-physical interest are nonmagnetic , while the con-ductivity is incorporated in the complex permittivity. Yet, we show the reconstruction is able to compute both properties. Permittivity is usually measured from very low frequencies up to the gigahertz region, where it becomes constant for most natural materials. This is the range of interest for many field ap-plications. On the high-frequency regime, the relation between the permittivity and the measured quantities is highly nonlinear for reasonably sized sample holders, and complex setups have to be implemented. It is common to place the material in a sec-tion of a coaxial transmission line, known as the sample holder [1], [2], in case of loose sands, or in one of coaxial-circular wave guide [3], [4] in case of rigid cores. Then, the S-parameters of the whole setup or tool are measured with a network analyzer.
The reconstruction of permittivity from measured S-parame-ters can be done analytically or via an optimization procedure. Analytical methods compute the electrical permittivity with an-alytical expressions involving the S-parameters. Until now there were no straightforward methods: the S-parameters had to be compensated, in one way or an other [1], [5]–[8]. Then different reconstruction formulas were used: the Nicolson–Ross, Weir, and Stuchly–Matuszewsky equations [9]–[11]. In this paper, we develop a novel method, the Propagation Matrices Method. It reconstructs the permittivity directly from the measured S-pa-rameters of the tool, by inverting for the effects of the transition units.
In this paper, we show the effect of the different calibration steps on the accuracy of the measurements. It is shown how errors in estimated permittivity result in a model error that is larger than the measurement error, which leads to the maximum error of less than 1% of the possible permittivity reconstruction from these measurements for wavelengths less than five times the sample holder length. We introduce the novel propagation matrices method and the permittivity reconstruction of two calibrating materials, air and ethanol. The experimental results corroborate our prediction, and the reconstruction does achieve a 1% error.
II. EXPERIMENTALSETUP A. Design
The ultimate goal of our study is to accurately determine the permittivity of soils at different degrees of saturation with dif-ferent fluids. For that purpose, the probe has been designed such that it is possible to determine the full S-parameter matrix of the system and allowing two-phase flow through the sample to 0196-2892/$20.00 © 2005 IEEE
1728 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 43, NO. 8, AUGUST 2005
Fig. 1. Coaxial transmission line probe for permittivity measurements of porous media and its sectioning. SH: sample holder, TU: transition unit, L: to the left of SH, R: to the right of SH, C: connector.
study the relationship between and fluid saturation. The ge-ometry and size of the probe ensure that only the transverse EM (TEM) mode propagates along the line, and, therefore, it can be described with transmission line theory. The sample holder was chosen to be of a representative volume (10 cm long and 3 cm in diameter) and allowing for simultaneous capillary pressure measurements. It is gold plated to ensure low energy loss in the line conductors.
The probe (see Fig. 1) consists of three main sections: two transitions units and the sample holder. Both transition units are composed of three sections: a conical part, a cylindrical part, and fluid distributor. They can be dismounted, enabling separate measurements of the two transition units together for high-ac-curacy calibration measurements. The units are Teflon filled and long enough to prevent any higher order mode generated from reaching the measurement plane. These modes can generate at the interfaces of the transition units, but due to the geometry and diameter of the line they are evanescent. The conical part of the transition unit eliminates the impedance jump between the cable connection and the sample holder such that the generated higher order modes are negligible. The fluid distributors can be connected to four inlets and four outlets of liquids, producing a reasonably homogeneous flow through the sample. When the line is completely filled with Teflon the impedance throughout is 50 .
B. Measurement Characteristics
The probe can be connected to the two ports of an S-param-eter test set, and the full S-paramS-param-eter matrix of the network is measured with a network analyzer, controlled by a personal computer. Any influence from the cables is compensated by doing a full two-port calibration, so that the measurement plane is moved to the end of the cables. Once the cables are calibrated, the three main bodies of the probe are assembled. First one tran-sition unit to the sample holder, which is then filled with the ma-terial under study and finally the other transition unit. In case of flow experiments, convenient filters are placed at the interfaces between the sample holder and the transition units. These three pieces are tightly screwed together. Then the cables are screwed to the end connectors at both transition units. These are delicate processes and a good connection is essential to obtain accurate measurement.
Fig. 2. Standard deviation for ten groups of two measurements for the real and imaginary parts of(S ) and (S ).
The tool components were designed such that the transition units could be compensated by doing a “user-defined” calibra-tion [12]. The process is equivalent to the calibracalibra-tion of the ca-bles but the standards were specially designed and constructed to match the transition units characteristics. This calibration at-tains the best results since it removes any undesirable effects from the transition units. Unfortunately, it was impossible to obtain a proper short circuit, and we could not calibrate them experimentally. Instead, we fine tuned the forward model of the tool with multiple measurements of the different components. We refer to it as the calibration of the model. Note that it is not an experimental compensation of the transition units but a cali-bration of the model with measurements.
To determine the maximum measurement accuracy, or actual dynamic range, we performed a statistical analysis on one hun-dred measurements for different materials. We determined that the measurements are very stable. For our study, single measure-ments are accurate enough, as they have three significant num-bers, which is the maximum number of significant digits given by the noise floor of the measurements. This is clear from Fig. 2 where, as an example, the standard deviation of ten groups of two measurements each have been plotted for the real and imag-inary parts of the reflection and transmission when the sample holder is filled with air. For the whole frequency range the difference between any two measurements is of the
order of .
III. FORWARDMODEL ANDCALIBRATION A. Representation as Propagation Matrices
In physics it is common to represent the down-going and up-coming wave fields in a layered medium in matrix notation; for a seismics example see [13]. This technique is known as the propagation matrices method [14]. It can be applied to any medium composed of layers of different properties where waves are propagating, such as a succession of sections of a coaxial transmission line filled with different dielectrics. In essence, it consists in writing the traditional transmission line formulation
Fig. 3. Transmission line representation of a multisectional line, such as our tool, and its simplified three-section model. (see the Appendix) in matrix form and relating the
S-parame-ters to the total reflection and transmission of the line seen as a two-port network.
Maxwell’s equations applied to the specific case of transmis-sion lines can be expressed in terms of voltage and current . Their solution in between two interfaces and , placed at
and at respectively, can be written as
(1) (2) where is the propagation constant of section given by (A5) and is the impedance given by (A4). At any th interface, the fields must be continuous, and this can be expressed in a propagation matrices notation as
(3) where (4) and (5) (6) Then, for a multisectional line (see Fig. 3) it is easy to relate the fields at the first interface to those at the last one
via the expression
(7) Now, we can group its elements into three main sections with four interfaces, as shown in Fig. 3.
The sample holder is in its central part
(8)
where stands for the sample holder. Anything at its left is in-cluded in and at its right in
(9)
(10) and (7) can now be rewritten as
(11) But, and are related to the S-parameters of the line, and that is what effectively the network analyzer is measuring. Then, when an electromagnetic wave impinges from Port 1, (11) transforms into
(12) and when it does from Port 2
(13) Note, that we can keep the same formulation for this case, if we use a mirror image of the transmission line. So cor-responds to the sections between Port 2 and the sample holder, represented now by . And the sections from the sample holder to Port 1 are included in . Notice that , when it is filled with a homogenous material, because on both cases it is composed of only the sample holder. To avoid writing mirror equations and since the properties derived for the reflection and transmission coefficients of one specific pair are shared by the other pair, from now on, we will use a to represent both re-flections, or , and a for the transmissions, or . The transmission coefficients are always equal to each other: , since the path the waves travel is the same, but the reflection coefficients are equal only when the
multi-sectional line is perfectly symmetric: .
This way, (12) and (13) can be rewritten as
1730 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 43, NO. 8, AUGUST 2005
Expressing the full reflection/transmission of TEM waves along a transmission line in a propagation matrices formulation enables us to compute the electrical permittivity and magnetic permeability of any sample filling the sample holder. We show this method in Section IV.
The parameters that control the presented forward model are the impedance and propagation constant of each section, given by (A4) and (A5). The whole system is then determined by the dimensions of each section, the electromagnetic properties of the materials filling the sections and the frequency of operation. Although theoretically everything is known except for the per-mittivity of the material in the sample holder, in practice, it is necessary to calibrate the tool. Before calibration, direct com-parison of the model with actual measurements reveal signif-icant differences between both. To reduce this difference, we spent quite some effort in the calibration of each component and the final tuning of the model.
B. Calibration
The calibration of the tool consists in fine tuning the pa-rameters that control the forward model. The model parame-ters of the whole probe are obtained from measurements in-stead of using the numbers from the specs with which it was constructed. Comparing the measurements to the model we were able to find the optimum parameters that describe the be-havior of our probe. For that purpose, two main test materials were used: air and Teflon. In the comparison of measurements and model we looked at all the reflection and transmission coefficients in their complex form, real-imaginary or ampli-tude-phase. Initially it was always the case that reflection surements were much better modeled than transmission mea-surements. We modeled the conical sections of the transition units as a single cylinder and the fluid distributors, although formed by a hollow teflon disk, are represented by a solid teflon disk. Comparison of the measurements with the different model assumptions justify these choices.
In the calibration, we followed a step-by-step procedure, looking at every component and parameter that could improve our model. We started with the simplest model and geomet-rical data taken from the technical specifications of the probe, changing one parameter at a time. Here we list the calibration steps, and in Fig. 4 we show the improvement of each step by plotting the absolute difference between the model and the mea-surement for every step when the probe is filled with air. Since the initial difference between the model and the measured data is bigger for transmission than for reflection, the advantage of this calibration shows more clearly in the transmission data. The final error in the modeled transmission data is much smaller over the whole frequency range than the modeled reflection data.
Step 1) Fitting geometrical parameters: Most geometrical parameters of the probe (length and radii of the sec-tions) can be measured independently and others have to be extracted from their technical specifica-tions or from independent measurements. However, these may vary due to construction, or compression when the probe is mounted, or expansions-contrac-tions due to temperature changes. Adjusting these parameters proved to be crucial, and resulted in a
Fig. 4. Difference between the measured and modeled reflection and trans-mission coefficients for an air sample, at the different steps of the calibration of the tool. (Solid line) Before calibration. (Dashed line) Step 1. (Dotted line) Step 2+ 3.
TABLE I
GEOMETRICALPARAMETERS FOR THETOOL
general reduction in both reflection and transmis-sion differences. Their values are listed in Table I. Although the transition units can be modeled as symmetric their geometrical parameters vary some milliliters from their technical specifications. Step 2) Permittivity of Teflon: Over the working frequency
range of the network analyzer, the relative permit-tivity of Teflon is reported in literature to be in be-tween 2 and 2.1. The minimum difference bebe-tween the model and the measurement was found at 2.05. We could have used a measured permittivity for Teflon per frequency, but this only complicates the computing and does not improve the accuracy of the model. The deviations of the measured permittivity for Teflon from the mean value of 2.05 are so small that the model is insensitive to them.
Step 3) Losses: Including the conduction losses into the model did not account for all the losses present. A lossless ideal line has a unitary scattering matrix (see [15]), but our tool deviates linearly from ide-alness. This deviation could only be accounted for
with a loss in the propagation
coefficients of the conduction along the transition units. By doing so the resistance term of (A8) is put to zero transforming (A5) into
(15) This step represents a noticeable improvement, spe-cially for frequencies above 1.5 GHz, of the model presented in [16], where we treated the losses as only conduction losses.
Fig. 5. (Solid line) Measured reflection and transmission coefficients for an ethanol sample, (dashed line) its calibrated model, and (dotted line) their difference.
Fig. 4 shows that the difference between the model and the measurement, for an air sample, has been reduced significantly, thanks to the calibration of the model itself. Even though the maximum difference is bigger than the precision of the mea-surements by two orders of magnitude, the obtained accuracy is to our knowledge still better than reported in the literature. The calibrated forward model is tested regularly with air. If the difference shown in Fig. 4 increases, the parameters controlling the forward model are recalibrated. Nevertheless, we have found this calibration very stable over a broad period of time.
In Fig. 5, we have plotted the reflection and transmission co-efficients for measured data of ethanol, the model for these data and their difference. The permittivity of ethanol is well defined from its Debye parameters, [17], and it can be used in the for-ward model. The difference between the measured and mod-eled transmission is of the order of the measurement precision. Changing the permittivity value by 1% results in a noticeable increase in the difference between data and model results over the whole frequency range.
C. Sensitivity of the Model
To study the sensitivity of our model, we performed experi-ments with modeled and measured data. It is possible to deter-mine the accuracy in the permittivity if the difference between the model and the measurements is known. From the accuracy of the measurement itself and the differences in the model results as a function of permittivity values used in the model, conclu-sions can be drawn about the maximum error of the possible re-constructed permittivity based on the measurements and the de-rived and calibrated model. In Fig. 6, we show the result of this analysis. The plot shows the maximum error in permittivity, as a function of normalized wavelength, that gives a change in the model result, which is just observable such that it leads to an in-creased difference between data and model result above the un-certainty threshold due to the limited accuracy of the measure-ment itself. For the uncertainty threshold we use the maximum error in the data, which leads to the worst result scenario. It is plotted as a function of normalized wavelength, because the res-onances at large wavelengths are expected to prohibit the
accu-Fig. 6. Maximum error in electric permittivity, which does not lead to observable model error as a function of normalized wavelength.
rate permittivity reconstruction ability. These resonances are re-lated to the ratio of wavelength and sample holder length, rather than frequency.
With an air filled sample, the normalized wavelength of 60 is equivalent to a frequency of 50 MHz. It is clear that if we would allow a maximum error of 5% in the reconstructed permittivity, we can not use the set up below 250 MHz for air. Of course the lower bound of the usable frequency goes down proportional to the inverse square root of the permittivity of the material filling the sample holder. So for most soil samples, varying in per-mittivity from 4–25 (dry to wet), the lowest usable frequency would be 125 and 50 MHz, respectively. For lower frequencies, other techniques should be used. For frequencies higher than twice the lowest usable frequency the permittivity can be recon-structed with an error below 1%. The smallest error that we are able to achieve is 0.1%, which is due to the limited accuracy of the measurements.
IV. INVERSION
Now it is time to reconstruct the EM properties from the mea-sured S-parameters. The EM properties of the material under study are contained within the matrix of (14). can be cal-culated from (5), (6), and (8) as
(16) Now, we can rewrite (14) into a more convenient form as
(17) where
(18) Substituting into (17) and eliminating the exponential terms, we find a solution for the impedance of the sample under test as
1732 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 43, NO. 8, AUGUST 2005
If, instead, the impedance terms are eliminated, a solution in terms of the exponentials is found
(20)
Naming , we find and provide
an expression for the propagation factor as
(21)
The conditions , , are physical
con-straints, cf. (5) and (6); hence the sign for the acosh function is determined.
The electric permittivity and the magnetic permeability of the sample is related to the impedance and propagation factor via (A4) and (A5). We then find two solutions
the impedance method (22) the propagation method (23) with and defined in (19) and (21), respectively.
In Maxwell’s theory and appear coupled in character-istic combinations and therefore, any method trying to measure the permittivity will also be able to measure the permeability, specially in their coupled forms. Equations (22) and (23) are a perfect example. The first one represents the impedance of the medium and the second one the inverse of the velocity at which EM waves propagate. The coupling can be a drawback since the accurate determination of both properties is not always pos-sible. From the expressions (22) and (23), it is clear, that the impedance method is more prone to be unstable than the propa-gation method. In fact, is experimentally unstable and can produce numerical instabilities for low loss materials. For those materials, the imaginary part of is close to zero which causes the imaginary part of the to be poorly determined. Therefore, any solution including will suffer from instability, making the method less accurate, and any solution including should be revised in the case of low loss materials.
Nonetheless, we are interested in nonmagnetic soils and we can eliminate from the solution, making the propagation ma-trices method more robust. In that sense, we compute the ve-locity of propagation from (23) and assume the magnetic per-meability to be that of free space so we can accurately determine the permittivity of a sample, from
(24) Before showing results from actual experimental data we would like to highlight the fact that broad frequency band mea-surements of standing waves present destructive interference, at resonant frequencies.
A. Resonant Frequencies
For low-loss materials of constant
permit-tivity over the frequency range, the off-diagonal terms of the
Fig. 7. Measured (full tool: solid line) and modeled (sample holder: dotted line)jS j for (top) air and (bottom) alumina samples.
matrix that relate the input and output voltages [(8)] are zero,
since at those frequencies, and therefore
and .
This property is used by resonant methods to reconstruct the permittivity of a sample at fixed frequency points very accurately. However, broadband measurements using resonant methods are almost impossible, since for each frequency the dimensions of the resonator have to be changed. They also require small samples, specially for high-loss materials, which can be another disadvantage.
At resonant frequencies, the accuracy of the measurements is very poor and the inversion fails to reconstruct the permittivity of the sample properly. Nevertheless, we can use the resonant frequencies to calculate the real part of the permittivity from the resonances of the line. This is the first time, to our knowl-edge, that this technique is applied in a transmission line, as it is usually reserved for resonant cavities.
The line is in resonance when , while the prop-agation constant of nonmagnetic materials can be written as , so that the resonant frequencies occur when
for (25)
They are, thus, periodic and depend on the length of the sample holder and the permittivity of the material. Since is a complex quantity and can depend on the frequency, we have to distinguish between different cases.
For materials whose relative permittivity is not a function of frequency and it is real , the reflection coefficient will go to zero at periodic intervals, and its permittivity over the whole frequency range can be computed from these single frequency points as
with for (26)
In Fig. 7. we have plotted the amplitude of the measured for an air sample and for an alumina sample, placed in our tool (solid line), together with the model (dotted line) for those ma-terials considering only the sample holder.
TABLE II
RESONANTFREQUENCIES, PERIOD ANDCOMPUTEDPERMITTIVITY FOR ANAIR ANDALUMINASAMPLE
From the figure, it is clear that the resonances of the tool de-pend only on the material placed in the sample holder. The mea-sured reflection of the tool goes to zero at the same frequencies as the model considering only the sample holder.
For these samples, resonant frequencies and computed permittivities are listed in Table II.
The resonances are periodic within 1% error for both air and alumina. This indicates that their permittivities are not depen-dent on frequency over the frequency range of interest and that the reconstructed permittivities are valid over the whole fre-quency range. The accuracy of the method is remarkable.
The presence of a constant small loss in the material does not change the location of the minima; it only varies the amplitude of these: decreasing amplitude with increasing frequency. So if a reflection pattern of periodic minima of decreasing amplitude is encountered, we can be certain that the material’s permittivity is constant over the frequency range and that its loss is small but not negligible.
For materials with a frequency-dependent permittivity, the mentioned method of computing permittivity from resonant fre-quencies for the whole frequency range is not valid since their occurrence is not periodic. It can still be used to obtain the real part of the permittivity at certain frequency points from the broadband measurement, where the reflection coefficient has a minimum. This could be handy, since the accuracy of the mea-surements is not so good in the vicinity of these minima so that the reconstruction of permittivity around them can suffer from high inaccuracy, and via resonant methods we can determine quite accurately the value real part of the permittivity for those resonant frequencies.
V. RESULTS
To test the performance of the propagation matrices method, we reconstructed the permittivity of two standard materials: one whose permittivity is constant (air) and another one that is fre-quency dependent (ethanol).
A. Air
In Fig. 8, we have plotted the standard and the reconstructed electrical permittivity for an air sample with the
propaga-tion method, for both pairs of measurements and
.
Fig. 8. Standard and PM reconstructed permittivity for an air sample.
The reconstructed permittivity presents three significant fea-tures: around 500 MHz on both the real and the imaginary parts, and below that frequency, the accuracy of the pair worsens. Below 250 MHz, both solutions start to diverge. And at 1.5 GHz, there is a clear jump that corresponds to a resonant fre-quency. At these frequencies, the reflection coefficient is zero, and its accuracy is very poor, failing the reconstruction of per-mittivity. These features were already predicted from the mea-surements characteristics, although the experimental accuracy threshold of 1% is lower (500 MHz) than predicted (650 MHz). From 500 MHz to 3 GHz, the error of the real and imaginary parts of both solutions is always between 1%, as predicted. The 5% threshold is set below 50 MHz and the 2% around 250 MHz.
A result within 1% for air, for such a broad band, is a great achievement. It is also a confirmation of the accurate reconstruc-tions we expect for materials of unknown permittivity, since the air result is a measure for the errors introduced by the tool. B. Ethanol
Applying the same method to ethanol (99.9%), a polar liquid, whose permittivity varies along our frequency range, we find the reconstructed permittivity (dashed line) plotted in Fig. 9 for
the pair and Fig. 10 for . The fitted Debye
model is also shown (dotted line).
The reconstructed permittivities are very well captured by the frequency-dependent Debye model. They present three signifi-cant features: around 300 MHz, the imaginary part of devi-ates from the Debye fitted model. Below 50 MHz, the solution starts to diverge. And at high frequencies, above 1.5 GHz the influence of noise is clear. These frequency limits correspond to the ones predicted in Section III-C for ethanol. Above 1.5 GHz, the signal is no longer transmitted through the sample, and the network analyzer has difficulties determining the phase of the transmission coefficient, boosting up the inaccuracies in the re-constructed permittivity; however, the general trend is captured. The fitted Debye model and the reconstructed permittivities differ less than 1% between 300 MHz and 2 GHz. If the Debye
1734 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 43, NO. 8, AUGUST 2005
Fig. 9. PM reconstructed permittivity (dashed line) of ethanol, for the pair [S ; S ] and (dotted line) the fitted Debye model.
Fig. 10. PM reconstructed permittivity of (dashed line) ethanol for the pair [S ; S ] and (dotted line) the fitted Debye model.
TABLE III
EXPERIMENTAL ANDFITTEDDEBYEPARAMETERS. DATA FORETHANOL
TAKENFROM[17]AND[18]. HASBEENMEASURED IN THELABORATORY
WITH ACONDUCTIVITYMETERINSITU. THEFITTEDDEBYEPARAMETERS
HAVEBEENOPTIMIZED TO THERECONSTRUCTEDPERMITTIVITIES
parameters are changed by 1% this difference increases between two and three times. The fitted Debye parameters are listed in Table III.
Note that the fitted parameters are within the error bounds of the van Gemert values. The conductivities have been measured in the laboratory. The fitted conductivities are orders the magni-tude bigger, but the conductivity of ethanol is too small relative to the imaginary part of the permittivity and therefore the opti-mization is not sensitive to conductivity changes. Conductivities
100 times bigger than the one measured in the laboratory lead to changes, above the measurement accuracy, in the imaginary part of the permittivity for frequencies smaller than 300 MHz. Our fitted Debye and also resemble the values of [18], while is slightly different but it can be explained because of the temperature difference between both experiments. We have shown that the combination of the propagation matrices and the propagation method reconstructs satisfactorily the permittivity of nonfrequency and frequency-dependent nonmagnetic mate-rials without a priori information. It shows frequency thresh-olds predicted from measurement characteristics. It is simple to program and very fast.
VI. CONCLUSION
A coaxial transmission line for the measurement of per-mittivity from 300 kHz to 3 GHz has been designed and constructed. The forward model representing the reflection and transmission along the line is in very good agreement with the measured data after a profound calibration. Relative changes in the permittivity in the order of 1% can be detected over a wide frequency band up to 3 GHz, while the lowest usable frequency depends on the permittivity of the material filling the sample holder. We have shown that this design, together with the forward model and the hierarchical calibration procedure, allows for permittivity reconstruction with an error bound in the order of 1% and down to 0.1%.
These types of setups can be modeled with transmission line theory. The propagation matrices method consists in rewriting the transmission line model in matrix notation and relating the S-parameters to the total reflection and transmission of the line seen as a two-port network. This novel approach enables us to reconstruct and directly from the measured S-parame-ters of the line. Due to experimental instabilities when recon-structing for both properties, the errors are considerable, and we must restrict ourselves to the reconstruction of permittivity with the propagation method. Results on an air sample are within the predicted error of 1%. Computing relative errors for an ethanol sample is not possible due to temperature and purity difference between our experiment and published results; nevertheless, the reconstructed permittivity of ethanol follows the Debye model.
APPENDIX
The standard transmission line theory for a multisection line was used to model the probe. This theory can be applied when only the TEM mode is propagating. It includes conduction losses between the lines and resistance losses due to nonperfect conducting lines. It is known that the diagonal scattering param-eters are the same, when measured at the reference planes of a perfectly symmetric line. They are equal to the global reflection coefficient including all internal reflection and transmission effects. The off-diagonal elements of the S-parameter matrix are always equal to each other and to the global transmission coefficient , including all internal reflection and transmission effects. So that
where the global reflection and transmission coefficients are given by the expressions
(A2) (A3)
where is the length of the section
and is the local reflection coefficient given by the impedance contrast between two adjacent sections
(A4) where is the impedance of section . The propagation con-stant is given by
(A5) For a coaxial transmission line with inner radius and outer radius , we can describe the inductance, capacitance, and re-sistance in terms of the magnetic permeability and the complex electric permittivity, , of the material filling the line, the resistance (where is the electric conductivity for stationary currents), and the cross-sectional geometry of the line (A6) (A7)
(A8) (A9) where is the magnetic permeability, is the free space per-mittivity, and are the real and imaginary parts of the complex frequency-dependent relative permittivity.
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Ainhoa G. Gorriti was born in Madrid, Spain, in
1974. She received the M.Sc. degree in physics from the Universidad Complutense de Madrid, Madrid, and the Ph.D. degree in geophysics from the Technical University of Delft (TU Delft), Delft, The Netherlands, in 1999 and 2004, respectively. She worked on the electromagnetic characterization of sands with heterogeneous fluid distribution using FDR.
She is currently performing her postdoctoral re-search at the Dietz Laboratory in the Department of Geotechnology, TU Delft. Her main research interests are the electomagnetic characterization of geophysical materials and its applications, and the propaga-tion of waves in heterogeneous materials.
Evert C. Slob was born in Veldhoven, The
Nether-lands, in 1962. He received the M.Sc. degree in mining and petroleum engineering and the Ph.D. degree (cum laude) in technical sciences from the Technical University of Delft (TU Delft), Delft, The Netherlands, in 1989 and 1994, respectively.
In 1994 he was a Visiting Scientist at Schlum-berger–Doll Research, Ridgefield, CT. In 1995, he joined the Department of Geotechnology, TU Delft, where he is now an Associate Professor. His current research interests are advanced imaging and inversion techniques, heterogeneity determination, and soil characterization including the study of fundamental relations between geological and electro-magnetic properties, of soil and rocks, and their scale dependency.
