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ORIGINAL RESEARCH PAPER

Measuring Spectral Dielectric Properties Using Gated Time Domain Transmission Measurements

^a Department of Hydrology and Water Resources, University of Arizona, Tucson AZ 85721
^b Sheffield Centre for Earth Observation Science, The University of Sheffield, Sheffield S3 7RH, England
^* Corresponding author (chawn{at}hwr.arizona.edu).

Received 17 September 2002.

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
A method to measure the frequency-dependent dielectric permittivity of simple materials based on a time domain transmission technique is described. A vector network analyzer (VNA) was connected to a twin-rod transmission line via a coaxial cable. The complex dielectric permittivity was found from the difference in phase and magnitude between a reference line surrounded by air and the same line surrounded by the substance of interest. The spectral response showed periodic variations in the dielectric permittivity as a result of multiple reflections in the experimental setup. These multiple reflections can be removed by using a time domain gate that selects only the primary transmission and filters out any subsequent reflections. It is essential that the apparatus be designed so that the first reflection is well separated from the primary transmission. This requires a long transmission line and a long coaxial cable. However, if the transmission line is too long, excessive conductive or dielectric losses make it hard to detect the primary transmission. The application of the gated time domain transmission technique to measure the frequency-dependent dielectric permittivity of water, ethanol, sand and saturated sand is demonstrated. This method does not have the typical limitations on sample volume. In addition, it does not require the assumptions necessary in previous time domain spectroscopy methods applied to open transmission lines where a probe model is used in conjunction with simple Debye relaxation and/or inverse methods.

Abbreviations: EM, electromagnetic • FDS, frequency domain spectroscopy • RMSE, root mean squared error • TDR, time domain reflectometry • TDS, time domain spectroscopy • VNA, vector network analyzer

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
BASIN-SCALE WATER BALANCE studies, especially those conducted in remote areas, must rely on remotely sensed soil moisture and plant canopy water fluxes. Measurements of the dielectric permittivity of the canopy at L-band (1.4 GHz) or lower frequencies would be particularly useful for inferring water content and water flux (Lee et al., 2002a,b). However, although dielectric mixing models have been developed to predict the permittivity of a canopy on the basis of measurements of the dielectric permittivities of the canopy constituents (e.g., air, stems, seeds, heads, and stocks) (e.g., Ulaby et al., 1986), to date no direct measurements of the permittivity of a canopy have been reported. The reasons for this are twofold. First, methods that use closed transmission lines or wave guides are destructive, and therefore the radiation regime and the water status of the canopy sample changes once it is removed from its environment for measurement. Second, the sample volumes of some of the dielectric spectroscopy techniques that use coaxial end probes are too small to apply at the canopy scale. Other common closed wave guide methods would require the construction of large, expensive coaxial lines or wave guides. The objective of this study is to examine the feasibility of an open transmission line dielectric spectroscopy technique that would be amenable to determining the frequency-dependent dielectric permittivity of sparse heterogeneous materials such as a vegetation canopy or a dry soil.

Dielectric spectroscopy can be performed by (i) measuring the S parameters at single frequencies over the frequency band of interest, known as frequency domain spectroscopy (FDS), or (ii) performing a Fourier transformation of the measured time domain response of the medium to an input signal, known as time domain spectroscopy (TDS). Both FDS and TDS measurements are typically made using closed sample holders that give rise to known modes of electromagnetic (EM) propagation over a limited range of frequencies. For example, measurements have been made with coaxial sample holders in transmission (Friel and Or, 1999; Matzler, 1998) and with coaxial sample holders or end probes in reflection (Colpitts, 1993; El-Rayes and Ulaby, 1987; Peplinski et al., 1995). Attempts have also been made to use open transmission lines to determine the frequency domain dielectric properties of materials as discussed further below (Heimovaara, 1994, Heimovaara et al., 1996; de Winter et al., 1996; Friel and Or, 1999).

Time domain reflectometry (TDR) is used widely within soil science and hydrology. Time domain reflectometry can be performed using fast rise-time square waves or impulses as input signals. Common transmission lines used for TDR measurement include coaxial and two-, three- and seven-rod configurations (Zegelin et al., 1989; Heimovaara, 1993, 1994; Heimovaara et al., 1996). Time domain spectroscopy can be performed by analyzing TDR waveforms via Fourier transformation (Heimovaara, 1994; Heimovaara et al., 1996; de Winter et al., 1996; Friel and Or, 1999). Generally, this requires that the transmission line characteristics of the probe be known. Typically, the transmission line characteristics of parallel three- or seven-rod TDR probes are assumed to be equivalent to those of a coaxial probe and the transmission line properties of this well-characterized probe type are adopted. For example, Heimovaara (1994) and Heimovaara et al. (1996) use a seven-rod probe, which approximates a coaxial cross section at frequencies below 250 MHz. To extend the TDS analysis to higher frequencies, they assume Debye (Debye, 1929) dielectric relaxation and use inversion techniques. The surface of the sum of squared residuals used in this inversion is found to have the shape of a long trough, making parameter estimation difficult and dependent on the initial guess (Heimovaara et al., 1996). De Winter et al. (1996) found Debye parameters within 10% of published values for homogeneous liquids known to have simple relaxation spectra, except ethanol, which showed an error of approximately 50% for all three Debye parameters.

The application of TDS analysis to broad-band TDR measurements faces two primary limitations. First, the information content of a square wave input, as indicated by the modulus of the Fourier transform, decreases as one over the frequency (Heimovaara, 1994; Heimovaara et al., 1996; Cicero, 1996; de Winter et al., 1996; Friel and Or, 1999). For example, the input signal used by Heimovaara et al. (1996) and de Winter et al. (1996) has a maximum response at the lowest frequency and a minimum response (55% of the maximum) at 1.0 GHz, in the frequency range of interest. As a result, the signal/noise ratio is relatively poor at high frequencies. Second, a relaxation spectrum must be assumed. This can give rise to errors in TDS analyses if the chosen model does not account for the combination of one or more of the following effects: polar relaxation of free water, Maxwell–Wagner loss, the dielectric effects of ionic conductivity, the frequency-dependent dielectric permittivities of bound water, and changes in the bound water fraction with textural class (Peplinski et al., 1995) or temperature (Or and Wraith, 1999; Wraith and Or, 1999; Colpitts, 1993). As a result of these phenomena, the relaxation frequency of bound water can range from 320 MHz to 20 GHz (Boyarskii and Tikhonov, 1998), that of dry sand is about 270 MHz (Matzler, 1998), and clay has relaxation frequencies of 30 MHz and 2.6 to 6.8 GHz, (Rinaldi and Francisca, 1999). Physical chemists have concluded from other observations and on theoretical grounds that porous media commonly exhibit non-Debye dielectric relaxation (Feldman et al., 2002; Leyderman and Qu, 2000; Druchinin, 2000), suggesting that a method of TDS interpretation that does not rely on an assumed dielectric relaxation spectrum is more generally applicable.

In this study, we explored a direct method for determination of the dielectric spectra of materials. The method uses open transmission lines to allow for measurement in situ in canopies or soils. In addition, the method employs an impulse source, which focuses the measurement sensitivity at higher frequencies than a typical square wave source. Finally, the method does not require the use of inverse methods and, therefore, does not rely on an assumed relaxation model. The objective of this investigation was to determine whether this method shows promise for dielectric spectroscopy in vegetation canopies and soils. Further refinements of the apparatus will be necessary to optimize the method.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
In typical TDR applications, an effective dielectric permittivity, _eff, is determined from the one-way travel time of a broad band EM pulse, t_TDR (s), as

[1]

where L is the length of the travel path through the sample (m) and c is the speed of light in a vacuum (m s^-1). For low loss materials, the effective dielectric permittivity is nearly equal to the real component of the complex dielectric permittivity. In saline soils, this assumption can lead to significant errors in the measured soil water content (e.g., Sun et al., 2000). However, it may be possible to use the loss information contained in TDR waveforms to account for these conductivity effects (Topp et al., 2000).

Measuring the Time Domain Response of a Sample Using a Vector Network Analyzer
Vector network analyzers are used to measure transmission and reflection properties (or S parameters) of a transmission line in the frequency domain. In this study, a series of single frequency signals was produced by the VNA and propagated along a twin-rod transmission line. At each frequency, the VNA calculates the complex transmission coefficient (S₂₁) from the magnitudes and phases of the transmitted and incident signals. These frequency domain measurements can be used to determine directly the frequency-dependent complex dielectric permittivity. However, each frequency domain measurement includes the effects of the connecting cables and connectors as well as the sample of interest. In contrast, time domain measurements allow for separation of reflections in time, and isolated analysis of key characteristic reflections. This minimizes the effects of multiple and spurious reflections. However, it is difficult to extract frequency-dependent dielectric permittivity directly from time domain measurements (Friel and Or, 1999). This paper presents a method whereby a time domain waveform is gated to eliminate multiple reflections, then transformed into the frequency domain to determine the frequency-dependent dielectric properties.

Calculation of the Complex Components of the Spectral Dielectric Permittivity in the Frequency Domain
The travel time can be calculated for each single frequency from the phase and magnitude of the transmitted pulse. The travel time in air, t_air (ns), is assumed to be equal to that in free space and can be calculated from the phase of a pulse that has traveled along the transmission line through air, _air:

[2]

where is the angular frequency (Hz). The difference in the travel time along a transmission line in air, t_air, compared with that along a transmission line within a sample, t, is given by

[3]

where ' is the difference between the phase of a pulse transmitted through air and that through the sample. The travel time, t, of the pulse in the medium can be calculated by combining Eq. [1] and [2]:

[4]

Note that this travel time is frequency dependent, whereas t_TDR is not. That is, t_TDR is effectively a weighted average of the frequency-dependent travel times in Eq. [4] where the weighting depends on the dielectric properties of the medium and the power spectrum of the input signal.

Frequency domain measurements contain additional information that can be used to calculate the attenuation of the signal, leading to a direct estimate of the imaginary component of the complex dielectric permittivity. The attenuation per unit length, , is given by (Ida, 2000):

[5]

where T_air, and T_sample, are the magnitudes of the transmission coefficient measured in air and in the sample, respectively, at a specific frequency (). The phase per unit length (ß) is proportional to the travel time:

[6]

The real part of the permittivity (^'_r), at a specific frequency, is then given by (Ida, 2000):

[7]

The magnitude of the permittivity, , at a specific frequency, can be calculated as (Ida, 2000)

[8]

Note that for low loss conditions is very small and Eq. [7] and [8] both reduce to Eq. [1], with t_TDR replaced by t. The imaginary part of the dielectric permittivity, ^''_r, at a specific frequency, can be found using the real part (Eq. [7]), the magnitude (Eq. [8]), and the Pythagorean theorem:

[9]

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
The VNA used in this study (HP 8752C Network Analyzer, Hewlett-Packard, Santa Rosa, CA) was used for measurements in the frequency range of 300 kHz to 1.5 GHz at 401 equally spaced points. Measurements were made in transmission mode along a twin-rod transmission line that was attached to the VNA via coaxial cables and terminal blocks at both ends. The transmission line comprised two steel rods 3 mm in diameter, 92 cm long, and separated by 2.2 cm. Relatively long rods were used to minimize the interference between primary and secondary arrivals, as discussed below. For this initial examination, the coaxial cables were attached to the transmission lines with inexpensive terminal blocks via a y-junction. The inner conductor was stripped and inserted into one end of the terminal block, and the outer conductor was twisted and inserted into the other end. The rods were inserted into the opposite side of the terminal block. The rods were installed vertically along the axis of a 10-cm-diameter, 92-cm-tall PVC column (Fig. 1). The bottom y-junction, tubing for introducing water into the column, and the rods were fixed at the base of the column in epoxy. The epoxy was poured into the column with the rods and column held in place with a clamp stand until the terminal block, cables, and 1 cm of the bottom of the rods was buried. The epoxy was allowed to cure for a week before measurements were performed. A lid with holes drilled into it was used to hold the rods in place as measurements were made.

View larger version (81K): [in this window] [in a new window]   Fig. 1. Photograph of the measurement setup.

Initially, the transmission coefficient was measured in an air-filled column as a function of frequency. These reference air measurements were used to calculate ' and T_air, for use in Eq. [4] thru [6]. Deionized water was added to the column. Measuring the mass of the column with the added water and subtracting the mass of the empty column determined the mass of water added. The height of the water column was determined by dividing the water mass by the cross-sectional area of the column and by the density of water. This height was then used for L in Eq. [2] through [9] to calculate the dielectric permittivity of the water. The complex transmission coefficient (S₂₁) was then measured with the VNA for the bandwidth of interest. These measurements were used along with those in air to calculate complex permittivities using Eq. [2] through [9]. Then more water was added to the column and the analysis was repeated. Measurements were made with 28, 41, 52, 65, 80, and 92 cm of water. These procedures were followed using ethanol, dry sand, and water-saturated sand. All substances were added from the top except for the water in the case of the saturated sand. In this case, water was added from the bottom through a tube set into the epoxy base until water started to pond slightly on the surface of the sand.

	RESULTS AND DISCUSSION

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
Figure 2 shows the complex dielectric permittivity as a function of frequency calculated using Eq. [2] through [9] and measurements made in the column filled with water to a height of 66 cm. The real and imaginary parts of the complex dielectric permittivity are compared with the Debye relaxation spectrum reported in the literature (Sato et al., 1999). Two major discrepancies are apparent:

1. The measured values show strong dependence of the real part of the dielectric permittivity on frequency, especially below 0.2 GHz and above 0.8 GHz.
   
2. Between 0.2 and 0.8 GHz there are many small, frequent oscillations.
   

View larger version (20K): [in this window] [in a new window]   Fig. 2. Complex components of the dielectric permittivity as a function of frequency for water before the application of the gate compared with published data (Sato et al., 1999).

View larger version (24K): [in this window] [in a new window]   Fig. 3. Magnitude of the complex transmission coefficient before application of the gate in the time domain for air and water. The dark shaded area indicates the region accepted by the gate for water and the light shaded area that for air.

[10]

where N_p is the total number of discrete frequencies sampled by the VNA, n is the frequency sample index, and is given by

[11]

where t is the gate width in the time domain (ns), f is the frequency (GHz), and f_c is the central frequency of the bandwidth (GHz). The frequency increment is defined by N_p equal steps in the value of f over the bandwidth of interest. This gate is applied in the frequency domain with prior knowledge of its effects in the time domain as the Fourier Transform of the sinc function {i.e., [sin()/} is a step function of finite width.

Figure 3 shows the region of application of the gate in the time domain (shaded). Within this region the gate maintains the shape of the time domain response and evaluates any other signal outside this region as zero. The frequency-dependent dielectric permittivity determined from the gated response using Eq. [3] through [9] should be free of most effects of internal reflections. However, it should be noted that as a result of the gating, the frequency-dependent dielectric permittivity cannot be determined for the entire range of frequencies measured. It should also be noted that the length of the rods used for measurement must be long enough to give adequate separation between the first arriving energy and secondary reflections from within the apparatus. Overlapping of primary and secondary arrivals makes separation with time domain gating impossible. Similarly, the air gap above the medium and the height of the medium above the base of the column should be large enough to allow for separation of the first arriving energy from the secondary reflections from the air–medium interface. Equation [1] can be used to determine an appropriate rod length that reduces the need for high quality connectors by separating the first arrival from secondary reflections. Separation of the first arriving energy and later reflections that originate at the instrument can be achieved through the use of longer connecting cables.

To demonstrate the range of frequencies for which the dielectric permittivity can be determined using a gated response, a gate of varying width was applied to a frequency-independent dielectric permittivity. The range for which the dielectric permittivity was within 1% of the known value was taken to define the valid measurement range. Figure 4 shows that the applicable frequency range is a complex function of gate width. The applicable range is 0.48 to 1.03 GHz for a 1.0-ns-wide gate. Gates of 12 ns and wider have applicable frequency ranges of 0.1 to 1.4 GHz. The minimum gate width recommended by Hewlett-Packard is 1 ns.

View larger version (28K): [in this window] [in a new window]   Fig. 4. Lower and upper limits of the frequency range where calculations of the complex dielectric permittivity are valid as a function of gate width. The vertical lines represent the gate widths used in the current study.

Measurements of Complex Dielectric Permittivity for Simple Substances
Figure 5 shows the time domain responses (left column) as a function of water height. The transmitted signal becomes more attenuated as the depth of the water increases. In the case of the 92-cm column of water, all secondary reflections are attenuated. The reflection from the air–water interface follows the first arrival more closely as the water level increases because the two-way distance from the end of the transmission line to the interface has decreased. In the longest column of water, the end of the twin-rod line and water surface closely coincided with the impedance mismatch between the coaxial line and the twin-rod transmission line. As a result, there was no additional reflection from the air-water interface.

View larger version (41K): [in this window] [in a new window]   Fig. 5. The left-hand column shows the time domain spectrum of water and air before application of the gate for a column of water of increasing height. The dark shaded area indicates the region accepted by the gate for water and the light shaded area that for air. The right-hand column shows the complex components of the dielectric permittivity of water as a function of frequency before and after the application of the gate compared with published values denoted S99 (Sato et al., 1999) for a column of water of increasing height.

View larger version (49K): [in this window] [in a new window]   Fig. 6. The left-hand column shows the time domain spectrum before the application of the gate for dry sand, saturated sand, and ethanol. The dark shaded area indicates the region accepted by the gate for the substance under test and the light shaded area that for air. The right-hand column shows the complex components of the dielectric permittivity of water as a function of frequency before and after the application of the gate compared with published values for dry sand, saturated sand, and ethanol.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 
A method is shown whereby the time domain response to an impulse input is filtered to separate the first arriving energy from secondary reflections. The filtered first arriving energy is converted to the frequency domain using a fast Fourier transform to enable the use of standard equations for the determination of the frequency-dependent complex dielectric permittivity. The results demonstrate that the gated impulse time domain method has promise for measuring the frequency-dependent dielectric permittivity with accuracy that is comparable to other accepted methods. This new approach has advantages for measuring in complex, heterogeneous media in that it can use long, open transmission lines and returns reasonable results even using inexpensive transmission line components. This is especially useful for field applications in vegetation canopies, which may require the installation of many probes. Furthermore, the impulse input focuses measurement sensitivity in the bandwidth of interest. Finally, analysis does not require any assumptions regarding the transmission line properties or the dielectric relaxation spectrum. This is of particular importance for heterogeneous media such as soils and vegetation canopies.

Despite the promise of this method, several issues still must be addressed to determine whether it is suitable for widespread use. First, application-specific requirements to ensure separation of first and later arrivals may limit the use of the method in some media, especially dispersive media with high electrical conductivity. Second, like other accepted methods, discrepancies between measured and known dielectric permittivities must be explained and, if possible, corrected. The results may be improved by using better quality cables and by employing a calibration procedure.

	ACKNOWLEDGMENTS

 
Primary support for Dr. Eleanor Burke, while preparing this paper came from NOAA project NA96GP0412. During the preliminary work for this paper, Chawn Harlow and Dr. Eleanor Burke were visiting scientists at ESSC, the University of Reading, UK. This material is based upon work supported by SAHRA (Sustainability of semi-Arid Hydrology and Riparian Areas) under the STC program of the National Science Foundation, Agreement No. EAR-9876800.

	REFERENCES

TOP ABSTRACT INTRODUCTION THEORY MATERIALS AND METHODS RESULTS AND DISCUSSION CONCLUSIONS REFERENCES

 

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This Article Abstract Figures Only Full Text (PDF) Free Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via Google Scholar Google Scholar Articles by Harlow, R. C. Articles by Shuttleworth, W. J. Search for Related Content PubMed Articles by Harlow, R. C. Articles by Shuttleworth, W. J. GeoRef GeoRef Citation Agricola Articles by Harlow, R. C. Articles by Shuttleworth, W. J. Related Collections Time Domain Reflectometry, TDR Plant Analysis