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Published online 2 June 2005
Published in Soil Sci Soc Am J 69:983-989 (2005)
DOI: 10.2136/sssaj2004.0352
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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Soil Physics

Soil Dielectric Spectra from Vector Network Analyzer Data

S. D. Logsdon*

National Soil Tilth Lab., 2150 Pammel Dr., Ames, IA 50011

* Corresponding author (logsdon{at}nstl.gov)


   ABSTRACT
TOP
 NOTES
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 THEORY
 RESULTS AND DISCUSSION
 REFERENCES
 
For more than two decades, dielectric properties (i.e., permittivity) have been used for soil water content measurements, but unexpected results have been shown for saline soils or soils high in smectite clays. Permittivity is complex with real and imaginary components, and varies with frequency. The frequency dependence is due to dipole rotation and charge migration processes under alternating current. Dielectric relaxation refers to the reorienting of molecules after an electrical field is removed. The objective of this study was to develop a procedure for determining complex permittivity spectra for soils. Six soils with a range of mineralogies were preequilibrated at four duplicated water contents. They were packed into truncated coaxial cells, and the reflection-scattering parameter was measured for frequencies between 300 kHz to 3 GHz, although the primary calculation procedure was valid to <100 MHz. The calculated complex permittivity was difficult to use for deriving unique parameters because of overlapping influence of electrical conductivity and multiple dielectric relaxation processes that extended beyond the measured frequency range. The complex resistivity was easier to interpret, clearly showing one major relaxation process, except for the driest soils. The relaxation frequency determined from complex resistivity increased significantly as water content and electrical conductivity increased. For each soil, the square root of apparent permittivity, {epsilon}a1/2, significantly increased as frequency decreased and as water content increased. The {epsilon}a1/2 showed frequency dependence ranging from 1.1-fold per order of magnitude for Cecil soil to almost 2.5-fold for Ida soil. Developing procedures to extend the measured and calculated frequency range would enhance data analysis options.

Abbreviations: VNA, vector network analyzer


   INTRODUCTION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 THEORY
 RESULTS AND DISCUSSION
 REFERENCES
 
FOR MORE THAN TWO DECADES, dielectric properties have been used for soil water content measurements, but unexpected results have been shown for saline soils (Wyseure et al., 1997; Nadler et al., 1999; Seyfried and Murdock, 2004; Kelleners et al., 2004a, 2004b; Jones and Or, 2004) or soil high in smectite clays (Bridge et al., 1996; Wraith and Or, 1999; Logsdon, 2000).

Dielectric spectroscopy (frequency-dependent) is often used to characterize materials. The spectra are often described by relaxation processes. An applied electrical field causes both the movement of charge carriers and the alignment of dipolar molecules (Jonscher, 1996; Baker-Jarvis, 2000). When the electrical field is removed, the molecules reorient back to a more stable arrangement, with a time lag. The time lag of reorientation or relaxation varies as a function of frequency for alternating electrical fields.

The permittivity of free water (Fig. 1) is well defined (Stogryn, 1971). The dielectric relaxation frequency occurs at the peak in the imaginary component, which is the midpoint of the change in the real component. Water has a sharp dielectric relaxation, which means there is a large increase, and then a leveling off of the real permittivity at frequencies below the relaxation frequency. In the range of most dielectric measurements in soil (10 MHz to 1.5 GHz), there is very little change in the real component of permittivity free water. Also, the permittivity of dry soil and air do not change with frequency. However, when all are mixed together, there is often a large increase in real permittivity as the frequency decreases. The spectra become complicated because of overlapping relaxations and overlapping electrical conductivity contribution to the imaginary component of permittivity. The relaxations are due to interactions between the soil particles and water; that is, not all of the water behaves like free water.



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  		Fig. 1. Dielectric spectra and Cole–Cole plots for water at 25°C.

 
Material scientists have usually measured permittivity on very small samples that can be filled, cut, or packed into airlines (7-mm diam.), but this is much too small for soil analysis. Material scientists have depended on time domain measurements (converted to frequency domain by lumped capacitance approach; Cole et al., 1989; Berberian and King, 2002) because of the broad measurement frequency range of the technique. Hydrologists (West et al., 2003; Huisman et al., 2004) have favored the vector network analyzer (VNA) over complicated back-calculations from field TDR equipment (Feng et al., 1999; Lin, 2003), even though the frequency range is restricted for the VNA measurements. The VNA is a tool that determines complex electrical properties of materials, including the imaginary or loss component or angle. The vector component refers to the vector math used for complex numbers (as opposed to a scaler network analyzer that only measures the real component).

There are considerations for getting reliable and reproducible results from VNA data. Will an attachment be used with the VNA that is specifically designed for dielectric properties, and what are the attachment limitations for sample size and frequency range? Will homemade sample holders be used, and what form of data does the VNA generate? Will the measurements be reflection or transmission, and which calculation procedures will be used? What calibrations are necessary?

Frequency-dependent real permittivity has been measured in clay slurries and gels (Raythatha and Sen, 1986; Ishida and Makino, 1999; Ishida et al., 2000; Dudley et al., 2003), for humidified clays at low water contents (Calvet, 1972, 1975; Logsdon and Laird, 2002, 2003, 2004a, 2004b), and for soils (Anis and Jonscher, 1993; Arulanandan et al., 1973; Campbell, 1990; Arulanandan, 1991; Rinaldi and Fransisca, 1999). The limited soils database has not yet shown what useful information can be gained from the permittivity spectra. Sample preparation (disturbed vs. undisturbed, density, aggregate size, chemical alteration, and so on) likely has a strong influence on the results.

The objective of this study is to develop a procedure for determining complex permittivity spectra for soils.


   MATERIALS AND METHODS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
THEORY
 RESULTS AND DISCUSSION
 REFERENCES
 
Laboratory Soil Electrical Properties
The soils used are listed in Table 1. Relevant properties are that Ida soil is calcareous (which solubilizes at high water contents, adding to electrical conductivity), Weld soil is aridic and probably contains some soluble salts, Cecil soil has high levels of sand and is kaolinitic in the control section (subsoil), Palouse and Ida are loess soils with high amounts of silt, Weld and Okoboji soils are smectitic, and Okoboji soil has high levels of clay and organic matter.


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  		Table 1. Mean soil properties for the A horizon materials used in this study (from Soil Survey Staff, 2004).{dagger}

 
Each of the six soils was preequilibrated at four different water contents (air dry plus 7, 21, 28, or 35 g water per 140 g air-dry soil), so each of the duplicate samples for each water content was a separate packing. One set of duplicate samples for each soil across the four water contents was run on the same day, and with an attempt to get uniform bulk densities among these four packings.

The disturbed soil samples were packed into copper sample holders (Fig. 2) , scaled approximately as the truncated (or shielded) coaxial sample holders (Lawrence et al., 1989). The holder was soldered to a BNC connector, and a BNC connection was attached to the cable. The BNC connectors are inferior to other connectors, especially at high frequencies, but can be soldered for homemade sample holders. Copper was used to avoid corrections that might be necessary with steel probes (Kraft, 1987). Although a commercially-available open-ended dielectric attachment can be used with the VNA, it can only be used with finely ground material, and there are theoretical concerns (Grant et al., 1989) with this fringing dielectric approach.



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  		Fig. 2. Truncated coaxial sample holder used in this study.

 
A VNA (Model 8753E, Agilent Technologies, Inc., Palo Alto, CA) was used to make reflection measurements, that is, the S11 scattering parameter. The S11 is a scattering number that starts and stops at the same channel 1 (reflected back to the same channel). The VNA calibration setup had been done ahead of time, which incorporated the constants for the BNC calibration kit (Maury Microwave Corporation, Ontario, CA) open, short, and load, and was set to exclude the cable effects. The cable had been clamped into place to minimize movements that can affect the results. For each day of measurement, the VNA was specifically calibrated for that day (reflection) with open, short, and load. The S11 was measured as a sweep for 801 points between 300 kHz and 3 GHz on a log scale.

The hygroscopic water content was determined for each soil by first placing samples in a humidity chamber over distilled water (99% relative humidity) for 2 wk, then over MgNO3 (56% relative humidity) for 2 wk (Logsdon and Laird, 2002, 2004a, 2004b). The samples were weighed before and after oven drying for 24 h at 105°C, and the gravimetric water contents were converted to volumetric water contents by multiplying by the bulk densities.


   THEORY
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 THEORY
RESULTS AND DISCUSSION
 REFERENCES
 
The complex electrical properties of a material have both real and imaginary components:

[1]
in which {epsilon}* is the complex permittivity (relative to permittivity of vacuum), {epsilon}' is the real component, i is the square root of –1, and {epsilon}'' is the imaginary component. The imaginary component includes both relaxation and electrical conductivity contributions:

[2]
in which {sigma}dc (S m–1) is the direct current electrical conductivity, f (Hz) is the frequency, {epsilon}''R is the relaxation contribution to {epsilon}'', and {epsilon}v is the absolute dielectric of a vacuum (8.854 x 10–12 F m–1). The electrical properties may be expressed as the complex, frequency-dependent electrical conductivity, {sigma}*(f), which emphasizes the transfer of charge between charge carriers or movement of charge carriers; or they may be expressed as the complex frequency-dependent relative permittivity, {epsilon}*(f), which emphasizes molecular polarization and charge storage (Anis and Jonscher, 1993). The {sigma}*(f) and {epsilon}*(f) are interrelated:

[3]
The frequency dependence is often described by the Debye (1929) or Cole–Cole (Cole and Cole, 1941) models for one relaxation:

[4a]
and for two relaxations:

[4b]
in which {epsilon}s is the low frequency and {epsilon}{infty} is the high frequency of {epsilon}', fr is the relaxation frequency, and {alpha} is an exponent that describes the spread of the relaxation peak. If the relaxation is Debye, then {alpha} = 0, and the spread is small. These equations assume a certain underlying statistical distribution of relaxation frequencies, but the data may fit a relaxation equation and still have a different underlying distribution of relaxation frequencies (Hasted, 1973). Additional relaxation terms may be added similar to the two terms in Eq. [4b].

The {sigma}* can be converted to complex resistivity, {rho}*, given as

[5]
where {rho}' and {rho}'' are the real and imaginary components of the complex resistivity. Ruffet et al. (1991) express as a Cole–Cole response:

[6]
in which the exponent n, the direct current resistivity, {rho}dc (ohm m), and the characteristic frequency, fc, are fitted. The {rho}dc is converted to {sigma}dc = 1/{rho}dc.

Calculations
The S11 was converted to permittivity using the procedure of Logsdon and Laird (2002), who based their procedure on that of Campbell (1990), which, in turn, was based on that from Kraft (1987). First, the S11 is converted to complex impedance, Z* (ohm),

[7]
which is converted to {sigma}* (S m–1),

[8]
in which Zp (ohm) is impedance of the truncated coaxial sample holder, {epsilon}v is permittivity of a vacuum (8.854 x 10–12 F m–1), c is the speed of light (3 x 108 m s–1), and L (m) is the electrical length of the sample holder. Impedance of the sample holder, Zp, was calculated based on probe dimensions (= 55.8 ohm).


[9]
in which m (m) is the radius of the outer conductor, and n (m) is the radius of the inner conductor. Electrical conductivity was converted to complex permittivity (relative to permittivity of vacuum)

[10]

The complex spectra was converted to square root of apparent permittivity

[11]

For clays with high attenuation, reasonable data was achieved nearly to 1 GHz (Logsdon and Laird, 2004a, 2004b); however, for soils, this calculation procedure is rarely reliable for frequencies higher than 50 to 100 MHz (Kraft, 1987). To supplement results from the Kraft procedure described above, I used the quarter-wavelength procedure (Heimovaara et al., 1996) to determine the square root of the apparent permittivity at one or two high frequency points

[12]
in which n is the number of the quarter-wavelength frequency, fn, in the real component of S11. The quarter-wavelength frequencies were determined by examining the negative spikes in the real S11 spectra.

To calculate permittivity from either procedure, it is necessary to have an estimate of the electrical length of the sample holder and the impedance of the sample holder. Both can be back-fitted from expected S11 calculations (Heimovaara et al., 1996) using measured spectra for liquids with known properties. The fluids used were air (empty sample holder, {epsilon}* = 1 + i0), methanol (Fellner-Feldegg, 1969; Baker-Jarvis et al., 1998), isopropanol (Baker-Jarvis et al., 1998), and 0.02 M NaCl solution (to include a lossy liquid, Stogryn, 1971). Note for the liquids it was necessary to invert the sample holder and fill with liquid, whereas the sample holder pointed down when measuring the soils. The calibration was based on the quarter-wavelength frequencies, which depended on sample holder length (L) and dielectric properties of the liquid.


[13]
If 2c/(4f1) is plotted as a function of {epsilon}a1/2, the slope is L. Here, f1 is the first quarter wavelength frequency, and {epsilon}a1/2 is the square root of the apparent permittivity at that frequency.

The dielectric spectra can be fit with a series of relaxation equations (usually Debye or Cole–Cole) plus the electrical conductivity contribution to the imaginary component of permittivity. Derived parameters from a single relaxation process include {epsilon}s, {epsilon}{infty}, fr, and 1 – {alpha} (for Cole–Cole, Eq. [4a]). For most soils data (except sand), at least two overlapping relaxations need to be included (Eq. [4b]), and usually the spread term ({alpha} for Cole–Cole) must be included as well. To fit the large number of parameters from multiple relaxations, the data must cover a broad frequency range that contains nearly the full width of all the relaxations. Even then, local minimums rather than global minimums may result in improper fitted parameters (Heimovaara et al., 2004). For this study, there is only one example fitting of Cole–Cole (Eq. [4b]) for permittivity data to show that both relaxation processes were out of the range of measured data, hence, there was great uncertainty in the fitted values.

An alternative, but less physically satisfying, approach is to convert complex electrical conductivity spectra to complex resistivity spectra (Eq. [5]). If a relaxation process is apparent within the frequency range from the resistivity spectra (Eq. [6]), then no single relaxation process will be shown from the permittivity spectra, and vice versa (Ruffet et al., 1991). The characteristic frequency determined from resistivity spectra will not be the same as that determined from permittivity spectra. The resistivity spectra may show a relaxation process even for a small frequency range. All of the data were fit to Eq. [6].

Across all soils, {epsilon}a1/2 (Eq. [11]) at 10 MHz, 100 MHz, and 1 GHz was tested for significant correlation (P = 0.05) with {theta} and {sigma}dc. For each soil, {epsilon}a1/2 was tested for correlation with {theta} and f. At 10 MHz, {epsilon}'' was separated into relaxation and electrical conductivity contributions (Eq. [2]). (See Kelleners et al., 2005, for splitting {epsilon}'' was separated into relaxation and electrical conductivity contributions.)


   RESULTS AND DISCUSSION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 THEORY
 RESULTS AND DISCUSSION
REFERENCES
 
The four fluids used to determine electrical length, L, (Fig. 3) showed scatter, hence uncertainty, in the fitted L (0.0336 m). The assumed L inversely shifts the values of the calculated permittivities and electrical conductivities (Eq. [13]), but should not alter the fitted characteristic frequencies or the relative relationships among water contents or soils. Assuming an incorrect L would affect comparisons with values in the literature.



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  		Fig. 3. The electrical sample length, determined as the slope of 2c/(4f1) as a function of expected {epsilon}a1/2 (square root of apparent permittivity), in which c is the speed of light and f1 is the measured frequency of the first minimum in the real scattering parameter (S11) (quarter wavelength).

 
An example complex permittivity spectra for Weld soil at a water content of 0.346 m3 m–3 (Fig. 4) showed that the imaginary component did not reach a peak at the lowest measured frequency, and that the real component did not level off at either the low or high frequency end. This meant that there were uncertain relaxation processes that extended to higher and lower frequencies than the measured range (fitted values not reported). Also for frequencies below 106 Hz, there would likely be contribution from electrode polarization (Schwan, 1966, 1992) which allows charges to build up at the electrodes and increases both the real and imaginary permittivity.



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  		Fig. 4. Example spectra of real and imaginary permittivity for Weld soil at water content of 0.346 m3 m–3, after subtracting the electrical conductivity component from the imaginary permittivity. The {epsilon}hf is the real permittivity at the high frequency limit. The {sigma} = is the component due to electrical conductivity.

 
By contrast, the example Cole–Cole plot of resistivity (Fig. 5) was more enlightening because it highlighted a relaxation within the frequency range of measurement. The fit of Eq. [6] to the measured data was good at the low frequency end, but off at the high frequency end. A good fit at the low frequency end ensured a reliable estimate of {sigma}dc, whereas less accurate fit at the high frequency end would result in a less reliable estimate of n. The resistivity plots were similarly-shaped for all soils except for the lowest water contents (not shown); however, the magnitude and characteristic frequencies changed from sample to sample. The lowest water contents were exceptions because the relaxation process was off scale at the low frequency end, which resulted in less reliable estimates of {sigma}dc at low water contents. For the two smectitic soils (Weld and Okoboji soils) at the lowest water content, the permittivity spectra showed a relaxation process within the measured frequency range (Fig. 6) , though other relaxation processes were off scale for all soils.



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  		Fig. 5. Example resistivity spectra and Cole–Cole plots derived from the sample permittivity data (Weld at water content of 0.346 m3 m–3).

 


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  		Fig. 6. Example permittivity spectra at the lowest water content, with hygroscopic (bw) and free (fw) water contents given for the duplicate samples (separated by comma) of each soil. The Weld and Okoboji soils indicate a relaxation (bulge) within the measured frequency range (indicated by arrows).

 
The soil-specific mean hygroscopic water contents were 0.011, 0.045, 0.048, 0.054, 0.075, and 0.101 m3 m–3 for the soils Cecil, Miami, Palouse, Ida, Weld, and Okoboji. When hygroscopic water content was subtracted from total water content (to calculate free water content), there were no significant differences among the soils within each free water level at P = 0.05. The four mean free water content levels across soils were 0.028, 0.148, 0.279, and 0.393 m3 m–3. Because of the varied hygroscopic water contents, the total water contents at each level did vary among the soils. The reason for using a humidification chamber to determine hygroscopic water content was that ambient room humidity could vary, which could affect air dry water content. In this study, all samples were measured within a few weeks, and the unknown ambient humidity apparently did not vary significantly during this time frame. This resulted in a good relation between hygroscopic water content and air-dried water content. Ranges of bulk densities for each duplicate set of each soil were Cecil: 1.20 to 1.29 and 1.34 to 1.38 Mg m–3, Miami: 1.34 to 1.35 and 1.31 to 1.32 Mg m–3, Palouse: 1.16 to 1.18 and 1.17 to 1.19 Mg m–3, Ida: 1.26 to 1.27 and 1.26 to 1.27 Mg m–3, Weld: 1.19 to 1.20 and 1.15 to 1.20 Mg m–3, and Okoboji: 1.13 to 1.14 and 1.18 to 1.20 Mg m–3.

The {epsilon}a1/2 at 1 GHz was significantly correlated with water content (Table 2), but at 10 or 100 MHz was also significantly correlated with {sigma}dc. The {sigma}dc directly increased {epsilon}'' (Eq. [2]), which increased {epsilon}a1/2 (Eq. [11]). The contribution of {sigma}dc to {epsilon}'' was apparent from Fig. 7 , which shows the effect of water content, {theta}, and frequency, f, on {epsilon}' and {epsilon}'' (both relaxation and {sigma}dc). The derived factor fc was significantly correlated with both {theta} and {sigma}dc, but n was only significantly correlated with {theta}, and the r2 was only 0.61.


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  		Table 2. Correlation of measured {epsilon}a1/2 (square root of apparent permittivity) and derived parameters (the characteristic frequency, fc, in MHz, and the exponent, n) as a function of water content ({theta}, m3 m–3) and direct current electrical conductivity ({sigma}dc, mS m–1).

 


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  		Fig. 7. Effect of water content and electrical conductivity on the real and imaginary components of permittivity (relaxation and electrical conductivity contributions) at 10 MHz.

 
For all soils, {epsilon}a1/2 was significantly negatively correlated with log10(f) and significantly positively correlated with {theta} (Table 3). The {epsilon}a1/2 showed frequency dependence ranging from 1.1-fold per order of magnitude for Cecil soil to almost 2.5-fold for Ida soil.


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  		Table 3. Correlation/regression of {epsilon}a1/2 (square root of apparent permittivity) as a function of water content ({theta}, m3 m–3) and three frequencies (f, log base 10, for 107, 108, and 109 Hz).

 
In summary, permittivity is complex with real and imaginary components, which are frequency dependent. Parameters can be derived from the spectra. The complex permittivity was difficult to use for deriving unique fitted parameters because of multiple dielectric relaxation processes which extended beyond the measured frequency range. Extending the measured and calculated frequency range would improve data analysis. The complex resistivity was easier to interpret over the limited frequency range, clearly showing one major relaxation process, except for the driest samples. The greatest uncertainty was in the fitted electrical length, which is inversely related to the calculated permittivity.


   NOTES
TOP
 NOTES
ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 THEORY
 RESULTS AND DISCUSSION
 REFERENCES
 
Manufacturer's names are given for information only, and do not constitute endorsement by the USDA.

Received for publication November 9, 2004.


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




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