Comparison of the Different Reconstruction Techniques of Permittivity From S-Parameters

Comparison of the Different Reconstruction

Techniques of Permittivity From S-Parameters

Ainhoa G. Gorriti and Evert C. Slob

Abstract—In this paper, we compare analytical and optimizing reconstruction methods for the electrical permittivity of materials. The reconstruction is done from the measured S-parameters of a coaxial transmission line. We compare the different reconstruc-tions on materials of known electromagnetic properties, air, and ethanol. And, we conclude that two methods, one analytical (prop-agation matrices) and an optimization ( ₁norm), are the most ap-propriate for the reconstruction of permittivities, whether these are low-loss and constant over the frequency range or highly lossy and frequency dependent.

Index Terms—Compensation, optimization, permittivity recon-struction, propagation matrices (PM), S-parameters.

I. INTRODUCTION

D

ETERMINING the permittivity of soil and rock materials is a broad and complex problem. In the high-frequency regime, the relation between the electromagnetic (EM) proper-ties and the measured quantiproper-ties is highly nonlinear, making the reconstruction of these properties cumbersome and unstable. It is common to place the material in a section (sample holder) of a coaxial transmission line (see [1]–[6], and [7], among others) or in a coaxial-circular waveguide (see [8] and [9]) and mea-sure the S-parameters of the setup. The EM properties are re-constructed from these parameters. The nature of the problem allows to solve for both electrical permittivity and mag-netic permeability , while the conductivity is incorpo-rated in the complex permittivity. Yet, this paper deals only with since most materials of geophysical interest are nonmagnetic , and solving for both properties decreases the accu-racy of the result.

The reconstruction of can be classified into explicit and optimization methods.

A. Explicit Methods

The electrical permittivity is computed from explicit expres-sions involving the S-parameters of only the sample holder. There are three ways to obtain the S-parameters of the sample holder.

1) Phase Delay Correction: If a commercial fixture is used,

then moving the reference planes from the test ports to the sample interface is a matter of removing the phase delay, e.g., [5], since the empty line characteristic impedance matches the

Manuscript received December 7, 2004; revised June 23, 2005. This research was supported by the Netherlands Organization of Scientific Research 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.2005.854312

impedance of the instrument’s test ports. However, accuracy problems arise from the uncertainty of the sample location, and not every sample is suitable for these type of lines. Our tool is not a commercial fixture, and although is has been matched to 50 , this method could only be applied as an approximation because it does not take into account small impedance devia-tions from the match load. Therefore, we do not use it in the comparison presented in this paper.

2) Propagation Matrices Method: It is possible to compute

the permittivity from the measurements at the test ports via an-alytical expressions with the new technique we have developed, [10]. The acquisition of the S-parameters of the sample holder is done implicitly in the analytical expressions.

3) Compensation Methods: The contribution of unwanted

transition sections can be removed by compensation methods. Kruppa and Sodomsky [2] published the system of equations needed, and Freeman et al. [11], Shen [6], and Chew et al. [7] applied it, respectively, to two commercial fixtures and a large customized cell. It is based on a scattering matrix rep-resentation. They compensate for the transition sections by an error prone and tedious experimental calibration, and can only be applied in certain frequency-sample size combinations. Six short-circuit measurements and a through measurement of only the transition units connected together are needed to compen-sate and to characterize the scattering matrix of both transition units. In the compensation equations there is one term obtained as the inverse of a subtraction between two short-circuited mea-surements. For our particular tool, these measurements intro-duce resonances from the teflon filling of the transition units and propagate along the compensation, so that compensated param-eters with true measurements suffer from instability, with major deviance at resonant frequencies. Reconstructing permittivity from these parameters did not succeed. Previous authors suc-cessfully applied this method to their tools because they were air filled and the tool dimensions and the frequency spectrum were such that the resonances did not occur. Chew et al. [7] applied it in a broad-band modifying the compensation in the high-fre-quency range by using modeled short-circuited reflections. We will not consider this method in the comparison because of its frequency limitations.

Once the S-parameters of the sample holder are obtained, dif-ferent reconstruction formulas are used: [1], [3], [4], [12]. All of them are divergent for low-loss materials at frequencies cor-responding to integer multiples of one-half wavelength in the sample. However, in [13] we demonstrated that all the different equations are equivalent to one of the two fundamental solu-tions, and that the best result is given by the propagation method [(10), Section III-A].

Fig. 1. Coaxial Transmission line probe for permittivity measurements of porous media.

B. Optimization Methods

The EM properties of the sample can also be computed via an optimization procedure, minimizing a cost function involving the measured and modeled S-parameters of the whole tool where the material is placed. In principle, no compensation is needed.

This method was used with a coaxial-circular waveguide in [8] and [9]. The first applied a gradient technique to an norm, and the second a modified Newton method to an norm. Their sample sizes are relatively small, and the method results are in-accurate for low-loss samples as well as for low frequencies since the wavelength becomes too large compared to the sample size. In this paper, we use a quasi-Newton algorithm and apply it to an and norm.

It is ideal to use the most suitable method for each specific material and frequency range, but since that is almost unfeasible due to time constraints, in this paper, we compare analytical and optimization reconstructions of different materials to evaluate their merits.

II. EXPERIMENTALSETUP ANDPROCEDURE

The tool used is a customized coaxial transmission line to allow two-phase flow through the sample (Fig. 1). The geom-etry and size of the probe ensure that only the TEM mode prop-agates 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 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 a fluid distributor. They can be dismounted, enabling separate mea-surements of the two transition units together for high-accu-racy calibration measurements. The units are Teflon filled and long enough to prevent any higher order mode generated from reaching the measurement plane. The conical part of the transi-tion unit eliminates the impedance jump between the cable con-nection and the sample holder such that the generated higher order modes are negligible. The fluid distributors can be con-nected to four inlets and four outlets of liquids, producing a rea-sonably homogeneous flow through the sample. When the line is completely filled with Teflon the impedance throughout is 50 .

The tool can be connected to the two ports of an S-parameter test set, and the full S-parameter matrix of the network is mea-sured with a network analyzer, controlled by a personal com-puter. The frequency range of operation expands from 300 KHz to 3 GHz. 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. For a full two-port calibration, reflection measurements for an open, a load and a short con-nector on both cables are done, as well as thru-measurements and isolation. Once the cables are calibrated, the three main bodies of the probe are assembled. First one transition unit to the sample holder, which is then filled with the material under study and finally the other transition unit. In case of flow exper-iments, convenient filters are placed at the interfaces between the sample holder and the transition units. These three pieces are tightly screwed together. And the cables are screwed to the end connectors at both transition units. These are very delicate processes and a good connection is essential to obtain accurate measurement. From a statistical analysis on one hundred surements for different materials, we determined that the mea-surements are very stable and their precision can be improved by stacking, if needed. For our study, single measurements are accurate enough, as they have three significant numbers. For the whole frequency range the difference between any two measure-ments is of the order of .

The forward model of the tool was carefully calibrated for an optimum performance. The calibration is described in [10]. It significantly reduced the difference between modeled and mea-sured data, in some cases, down to measurement accuracy.

For test materials, we differentiated between non-frequency-and frequency-dependent materials to check the ability of the different methods in reconstructing distinct permittivity profiles. On one hand, materials that maintain a constant permittivity over the broad frequency range used in these experiments, also present a low-loss. To represent these type of materials we chose air. Air has a well-known relative permittivity of 1, which al-lows for a quantitative comparison of results. On the other hand, frequency-dependent materials are also highly dispersive. To represent these we chose ethanol, a polar liquid, whose per-mittivity is defined by Debye’s formula and its Debye param-eters [14]. These paramparam-eters depend on temperature and pu-rity. Reported Debye parameters are in many cases incomplete. Landolt- Börnstein [15] only lists the static parameter for dif-ferent temperatures, and the only complete set that was close in temperature conditions to our experiment (23 C) is the one measured in [16] (24 C). Unfortunately, in [16], the purity of the samples is not reported and the measured parameters have significant uncertainties. Therefore, an absolute compar-ison with standard values cannot be performed. To corroborate if our results are in agreement with the accepted model for polar liquids or not, once the permittivity of the sample was recon-structed per frequency with the appropriate method, we fitted a Debye model to the result. The Debye parameters were thus ob-tained from the optimization of the following cost function:

Fig. 2. Transmission line representation of a multisectional line, such as our tool, and its simplified three sections model.

where is the total number of frequencies, is the recon-structed permittivity, and is the Debye permittivity. In this manner, we fit a frequency-dependent model to the permit-tivity of ethanol computed without any frequency model, and as a reference we refer to the values given in [16].

III. RECONSTRUCTIONMETHODS

A. Explicit Methods: Propagation Matrices

This method allows to compute directly from the measured S-parameters at the measuring planes. There is no need to manu-ally compensate the S-parameters from the measuring planes to the interfaces of the sample as this is implicitly done in the ob-tained solution. In [10], we introduced the propagation matrices method as a representation of transmission lines and inverted for both the electric permittivity and the magnetic permeability . In essence, this representation is equivalent to the transfer scattering matrix representation used in microwave circuits [17]. Nevertheless, since this technique is of interest to geophysicists and they are used to relate the down and up going waves through different layers of soil, we will keep the propagation matrices notation. In this paper, we show this notation schematically and invert only for .

The measured S-parameters of a multisectional transmission line (see Fig. 2) can be expressed as

(2) where represents the propagation through the sample holder, whereas and represent the propagation at the left and right of the sample holder; for a detailed derivation see the Appendix. It is possible, to find two solutions for the relative permittivity of the sample . Rewriting (2) into a more convenient form as

(3) where

(4)

is given by (A8), and it is equal to

(5) For a lossless line (perfect conductors) and nonmagnetic sam-ples, the impedance and propagation of the sample holder are related to the relative permittivity of the sample through the following expressions:

where (6)

where

and is the angular frequency (7) Substituting (5) into (3) and eliminating the exponential terms, we find a solution for the impedance of the sample under test as

(8) If, instead, the impedance terms are eliminated, a solution in terms of the exponentials is found

(9) We find an expression for the propagation factor as

(10)

The conditions are physical

con-straints. Hence the sign for the acosh function is determined. The electric permittivity of the sample is related to the impedance and propagation factor via (6) and (7). We then find two solutions for

the impedance method

with (11)

the propagation method

with (12)

These two solutions allow to reconstruct, for the first time, the permittivity directly from the measurements at the reference planes of the tool. Previous techniques required a compensation of the measurements from the reference planes to the interfaces of the sample holder [1], [3]–[6], [12].

In the reviewed literature, there are at least six reported dif-ferent analytical equations to reconstruct the permittivity from

measurements of the total reflection and transmission coeffi-cients. However, as we proved in [13], these solutions are equal to either the impedance or the propagation methods.

From the expressions (8) and (10), it is clear, that the impedance method is more prone to be unstable than the propagation 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 deter-mined. 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. So, we can accurately determine the permittivity of a sample from the following equation:

(13) Note that depending on the pair of measurements chosen, or (see the Appendix), we can compute two different permittivities for the same sample. For homogenous samples these permittivities are equal within measurement error and for simplicity we only show results for the pair.

B. Optimization Methods

This technique does not use analytical expressions to recon-struct the permittivity; instead, it minimizes a cost function in-volving the measured and modeled S-parameters of the tool. The attainment of the S-parameters of the sample holder is not an issue, but the convergence of the method can become one. They are a must when only reflection or transmission is measured, since the analytical methods require always a combination of the two.

The optimization methods can be very varied in cost func-tions to minimize and the minimizing procedure is limited to nonlinear techniques. Belhadj-Tahar et al. [8] and Taherian et

al. [9] used this method with a coaxial-circular waveguide. The

first applied a gradient technique to an norm, and the second, a modified Newton method to an norm.

We have used a quasi-Newton algorithm implemented in the optimization toolbox of Matlab [18], as fminunc. It is an un-constrained nonlinear optimization routine that saves computing time by approximating the Hessian with an appropriate updating technique. It is limited to real numbers, and we bypassed the problem by splitting the output of the cost functions to their real and imaginary components. The tolerance was set to mea-surement accuracy and computing time varied with the cost functions.

The optimization is done per frequency and over the full spec-trum. An initial guess of the permittivity of the sample, for the first frequency point, is given, and the modeled S-param-eters are computed with the forward model presented in Section III-A. By minimizing a specific cost function, the op-timized permittivity is calculated and used as initial guess for the next frequency point. This is repeated until the permittivity of the sample has been reconstructed for the whole frequency range.

Fig. 3. Standard permittivity of air (solid line) and PM (dotted line) and OP (dashed line) reconstruction.

We have tested different cost functions to study their per-formance. With synthetic data all of the tested functions reproduced the permittivity introduced in the model, but with true measurements they varied in their performance. We tried normalized and non-normalized cost functions. The normalized ones always performed better than the non-normalized, and therefore we only show those ; and norms are given by the following equations:

(14)

(15)

We found that for high-loss materials, whose transmission is low, certain weighting is needed for convergence, because above certain frequencies, the sole reliability on reflection is not enough for the optimization to converge. The weighting consists in finding the proper reflection/transmission ratio that will pro-duce a convergent optimization. When the norm was con-sidered the solution diverged for high-loss materials even after weighting. We then show results only on the norm.

IV. RESULTS

In this section, we show the reconstruction of the permittivity of air and the fit to the reconstructed permittivity of ethanol with the propagation matrices (PM) method and that from optimizing (OP) an norm.

A. Non-Frequency-Dependent Materials: AIR

In Fig. 3, we have plotted the permittivity for air together with the PM and OP reconstruction.

The PM reconstructed permittivity presents two significant features: below 250 MHz the solution starts to diverge. And at 1.5 GHz there is a clear jump due to resonance. From 500 MHz to 3 GHz, the error of the real and imaginary parts is always

Fig. 4. PM reconstructed permittivity of ethanol (dashed line), for the pair [S ; S ], and the fitted Debye model (dotted line).

between %.1_{And a 5% threshold can be set below 50 MHz} and a 2% around 250 MHz.

As with the PM reconstruction, the OP reconstruction worsens below 500 MHz on both the real and the imaginary parts, and below 250 MHz the solution starts to diverge. At these frequencies, the wavelength becomes too great compared to the sample length, and the methods fail to reconstruct the permittivity of the sample. This time, the resonant frequency, at 1.5 GHz, is not seen in the OP reconstructed permittivity. From 500 MHz to 3 GHz, the accuracy of the solution is mostly of the order of 1% for the real part, while the imaginary part is within measurement accuracy. This result is very similar to that obtained with the PM reconstruction, slightly worse for the real part and better for the imaginary part.

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. Frequency-Dependent Materials: ETHANOL

We applied the same methods to ethanol (99.9% pure), a polar liquid, whose permittivity varies along our frequency range, and fitted the results to a Debye model. The PM results are presented in Fig. 4 and the OP results in Fig. 5.

The PM reconstructed permittivities are very well captured by the frequency-dependent Debye model. They present three significant features: around 300 MHz, the imaginary part of deviates from the Debye fitted model. Below 50 MHz the solu-tion starts to diverge. And at high-frequencies, above 1.5 GHz the influence of noise is clear. 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 coeffi-cient, boosting up the inaccuracies in the reconstructed permit-tivity; however, the general trend is captured. The fitted Debye 1_{The error given for the imaginary part of the permittivity should be} under-stood not as a relative error, but as an absolute error given by100 1 [(" ) 0 (" ) ]

Fig. 5. OP reconstructed permittivity of ethanol (dashed line) and the fitted Debye model (dotted line).

TABLE I

EXPERIMENTAL ANDFITTEDDEBYEPARAMETERS. DATA FORETHANOL

TAKENFROM[19]AND[16]. HASBEENMEASURED IN THELABORATORY

WITH ACONDUCTIVITYMETERINSITU. THEFITTEDDEBYEPARAMETERS

HAVEBEENOPTIMIZED TO THERECONSTRUCTEDPERMITTIVITIES

model and the reconstructed permittivities differ less than 1% between 300 MHz and 1.5 GHz. If the Debye parameters are changed by 1%, this difference increases between two and three times.

The OP reconstruction is slightly different to that obtained with the analytical methods. The solution holds up very well in the entire frequency range and only below 10 MHz it starts to diverge. At high frequencies, above 2 GHz, the influence of noise is clear. The fitted Debye model and the reconstructed permittivities differ less than 1% between 300 MHz and 1.5 GHz for the real part and from 500 MHz to 1.5 GHz for the imaginary part.

Table I shows the fitted Debye parameters to both reconstruc-tions, the propagation matrices and the optimization.

Note that the fitted parameters are within the error bounds of the van Gemert values. The conductivity given as a van Gemert parameters has been, in fact, measured in the laboratory. The fitted conductivities are orders of magnitude bigger, but the con-ductivity of ethanol is too small relative to the imaginary part of the permittivity and therefore the optimization is not sensitive to conductivity changes. Conductivities 100 times bigger than the one measured in the laboratory lead to changes, above the mea-surement accuracy, in the imaginary part of the permittivity for frequencies smaller than 300 MHz. Our fitted Debye and also resemble the values of [19], while is slightly different, but it can be explained because of the temperature difference between both experiments.

It is not conclusive whether the propagation matrices recon-struction is better than the optimization one or viceversa. The reconstruction of the real permittivity of air with the PM or the OP method is equivalent, while its imaginary part is better re-constructed by the PM method. The reconstruction of the real permittivity of ethanol is more robust when the OP method is used, but its imaginary part is arguably better captured by the PM method.

V. CONCLUSION

We can conclude that any method (PM or OP) is valid for the reconstruction of permittivities, whether they are low-loss and constant over the frequency range or highly lossy and frequency dependent. We could, maybe, argue that the optimization recon-struction is slightly better in its accuracy and that its solution is more stable for low frequencies.

It would be, then, most adequate to use both methods to ob-tain results in a broader spectrum. However, we rather use the propagation matrices method because the optimization is very time consuming and it is necessary to tune the cost function to each particular material, as the weighting needed for its conver-gence depends in the ratio of the amplitudes of reflection and transmission. On the contrary, the propagation matrices method is very fast and easy to program.

APPENDIX

The propagation matrices method consists in writing the tra-ditional transmission line formulation, in matrix form, and re-lating the S-parameters 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

(A1) (A2) where is the propagation constant of section , and is the impedance. At any th interface, the fields must be continuous, and this can be expressed in a propagation matrices notation as

(A3) where

(A4) (A5) (A6)

Then, for a multisectional line (see Fig. 2), it is easy to relate the fields at the first interface to those at the last one

via the expression

(A7)

Now, we can group its elements into three main sections with four interfaces, as shown in Fig. 2. The sample holder is in its central part ; anything at its left is included in and at its right in

where stands for the sample holder (A8) (A9)

(A10)

and (A7) can now be rewritten as

(A11) 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, (A11) transforms into

(A12) and when it does from Port 2

(A13) 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 . So that, by measuring the full S-pa-rameter matrix of the tool we can obtain two permittivities for the same sample. For homogeneous samples, these permittivi-ties are equal within error accuracy, as shown in [10].

ACKNOWLEDGMENT

The authors gratefully acknowledge the support of the NWO. REFERENCES

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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 electromagnetic 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.