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Published online 18 July 2005
Published in Vadose Zone J 4:551-557 (2005)
DOI: 10.2136/vzj2004.0093
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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ORIGINAL RESEARCH

Single-Rod Probes for Time Domain Reflectometry

Sensitivity and Calibration

Mathis Nussbergera,*, Hansruedi Benedicktera, Werner Bächtolda, Hannes Flühlerb and Hans Wunderlib

a Lab. for Electromagnetic Fields and Microwave Electronics, Swiss Federal Inst. of Technology (ETHZ), 8092 Zürich, Switzerland
b Inst. of Terrestrial Ecology, Swiss Federal Inst. of Technology (ETHZ), 8092 Zürich, Switzerland
* Corresponding author (nussberger{at}ifh.ee.ethz.ch)

Received 21 June 2004.



   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 CONCLUSIONS
 APPENDIX:
REFERENCES
 
Time domain reflectometry (TDR) probes consisting of one conducting rod and a wave mode converter are an alternative configuration that overcomes some of the disadvantages of conventional probes. We examined four different single-rod probes (SRPs) and a two-rod probe for sensitivity to a small and a large conductive scatterer in their vicinity. The SRPs were assembled combining a small and large wave mode converter with an uncoated and coated rod. We found that the volume sampled by SRPs is larger and more symmetric than in the case of a two-rod probe of equal size. A comparison of the mode converters showed a higher loss for the smaller converter but only a small difference concerning the spatial sensitivity. Coating the conducting rod with a high dielectric constant material reduces the spatial sensitivity. One of the SRPs and the two-rod probe were calibrated in a sand tank (particle size 0.08–0.2 mm) with volumetric water content up to 0.35 m3 m–3. The calibration showed only small differences in the measured bulk dielectric constant between the single-rod and the two-rod probe. Based on this study the SRP is a promising new tool for improved TDR measurement of soil moisture.

Abbreviations: SRP, single-rod probe • TDR, time domain reflectometry


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
THEORY
 MATERIALS AND METHODS
 CONCLUSIONS
 APPENDIX:
REFERENCES
 
SOIL MOISTURE is often measured with TDR probes having two electrodes. Such probes suffer from two disadvantages: the sampled volume has a complicated shape and the probe is most sensitive between and immediately adjacent to the rods (Knight, 1992; Ferré et al., 1998; Robinson et al., 2003). The concept of a SRP is an attempt to alleviate these drawbacks (Oswald et al., 2004). A SRP consists of a single electrode and a mode converter that excites a guided wave propagating along the rod. The electrode can be coated with a dielectric material. The simple geometry of the SRP must result in a more symmetric measurement volume than for two-rod probes, and the size of the measurement volume may differ as well.


   THEORY
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
MATERIALS AND METHODS
 CONCLUSIONS
 APPENDIX:
REFERENCES
 
The theory of coated and uncoated single conductor wave guides in an atmospheric environment has been addressed extensively by Sommerfeld (1899), Harms (1907), and Goubau (1950). For uncoated wave guides, Oswald et al. (2004) adapted the analytical treatment to a medium with a homogeneous but arbitrary dielectric constant. We expanded this theory to coated wave guides in a soil-like environment (Appendix).

Single Conductor Wave Guides
From the theory, we can derive some general properties of single conductor wave guides valid for coated and uncoated electrodes. At a given frequency f and in an environment with a homogeneous relative dielectric constant {epsilon}r, a single conductor wave guide is fully characterized by kz, the number of full waves per meter along the rod. The longitudinal wave number kz is a result of a transcendental eigenvalue equation comprising f, {epsilon}r, the electrode radius re, the conductivity {sigma}e of the electrode, the dielectric constant of the coating {epsilon}rd (if any), and the coating thickness d (Eq. [A18] in the Appendix). For a chosen frequency f and for a given setup, kz has to be determined numerically.

Wave Velocity and Attenuation
The velocity of a harmonic wave on the guide is given by the real part of the wave number, v = {omega}/Re(kz), with the angular excitation frequency {omega} = 2{pi}f. Numerical calculations show that v does not change significantly with f in the TDR bandwidth (up to 6 GHz). For thin coatings (up to a tenth of the rod radius and small compared with the wavelength) and for uncoated probes, the relation between v and {epsilon}r becomes the same as the one known for two-rod probes v = c0/{surd}{epsilon}r, where c0 is the speed of light in a vacuum.

The wave loss is given by the imaginary part of the wave number, {alpha} = 8.686 Im(kz) (dB m–1). If we assume a low-loss coating or no coating, {alpha} is dominated by the soil loss tangent tan{delta}s = Im({epsilon}r)/Re({epsilon}r) and becomes the same as the loss for two-rod probes.

Electromagnetic Fields
The spatial sensitivity of the SRP is governed by the electromagnetic field profile of the guided wave. The analytical treatment presented in the appendix assumes the electrode of the SRP to be a section of an infinitely long rod embedded in a low-loss medium with a homogeneous dielectric constant {epsilon}r and a homogeneous permeability µ = µ0. Such a structure is able to support a transverse-magnetic mode with three nonzero field components: the azimutal magnetic component H{phi}, the radial electric component Er, and the z-directed electric component Ez (Fig. 1) . For all considered practical cases, the latter is three orders of magnitude smaller than the other components and is therefore neglected in our analysis of the electromagnetic power distribution around the rod. The Poynting vector S = E x H, which is the local power flux, is then given by the components Er and H{phi}, S = Er x H{phi}.



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  		Fig. 1. A single-rod probe exhibits a transverse-magnetic mode with nonzero field components Er, H{phi} (directed toward the reader), and Ez. The power flux S is mainly the cross product of the tangential magnetic field H{phi} and the radial electric field Er because Ez (not correctly scaled) is three orders of magnitude smaller than the other components.

 
Up to a characteristic radius rc, the field components H{phi} and Er decay at rates inversely proportional to the radial distance r from the wave guide. For r > rc an exponential law applies (Fig. 2) . Up to rc, a decay of the field components by 1/r means a decay of S with 1/r2. According to Kaden (1951) the power flux at r > rc through a plane {Omega} perpendicular to the guide is negligible, compared with the total power flux through {Omega}.



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  		Fig. 2. The magnitudes of the field strengths H{phi} and Er, normalized to their values at the surface of the wave guide, both decay inversely proportional to the radius up to rc. At r > rc they decrease exponentially. The vertical line indicates the approximation for rc explained in the text. Also shown is the magnitude of the power flux vector S, relative to its value at the surface of the wave guide. The graph is valid for the case of an uncoated aluminum electrode with 10-mm diameter in dry soil ({epsilon}r {approx} 3) and a signal frequency of 0.3 GHz.

 
The radius rc can be approximated by calculating the negative inverse of the imaginary part of the transverse wave number rc = –1/Im(kTs), with kTs = {surd} and the soil wave number ks = {omega}{surd}{epsilon}r/c0. In Table 1 we calculated rc for a homogeneous medium with dielectric constants corresponding to dry ({epsilon}r {approx} 3) and water-saturated ({epsilon}r {approx} 25) sand. Both calculations are given for an uncoated rod and a rod coated with a thin (d = 1.6 mm) dielectric layer with {epsilon}rd = 100. Aluminum rods were assumed ({sigma}c = 3.8 x 107 S m–1). Calculations for copper rods with {sigma}c = 5.8 x 107 S m–1 yield a 10% higher value for rc.


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  		Table 1. Calculated characteristic radii rc of uncoated and coated aluminum rods in dry ({epsilon}r {approx} 3) and wet ({epsilon}r {approx} 25) sand. The electrodes have a diameter of 10 mm. The coating is 1.6 mm thin and made of TiO2 with a relative dielectric constant {epsilon}rd = 100. The characteristic radius rc is not the effective radial field extent, as explained in the text.

 
The results presented in Table 1 show a distinct dependency of rc on f, {epsilon}r, and {epsilon}rd. In the case of the uncoated rod, however, rc is very large and S has already decreased to a negligible value at r << rc. For example, at 20 times the rod radius, S decreases to 0.25% of the corresponding value at the surface of the rod. No general answer can be given in the case of the coated rod, because rc is small. The sampled volume then depends on the frequency spectrum of the signal propagating on the wave guide, and on the dielectric structure of the medium.

Excitation of the Guided Wave
Common TDR measurement systems have coaxial, transverse electromagnetic mode inputs and outputs. A mode converter must be used to couple a coaxial transmission line to an SRP. One of the simplest converters is a cone-shaped horn where the outer conductor of the coaxial cable is gradually extended (Fig. 3) . Choosing the horn opening radius R has a trade-off: the launching loss and the reception loss at frequency f becomes minimal when R ≥ rc (Goubau, 1950). Thus, optimum conversion at low frequencies requires very large horns. For practical-sized horns a low-frequency cut-off must be accepted.



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  		Fig. 3. Outline of the linearly tapered horns used as mode converters. An air-filled small horn with a tapering angle of 15.1° and an opening radius of 45 mm has an attached PTFE disc that holds the conductor in place. The shell is made of a copper alloy and is 5 mm thick. A large horn with a tapering angle of 33.4° has an opening radius of 100 mm. The shell is made of a 0.5-mm-thick copper alloy sheet. A plastic fixation keeps the rod in place at the center of the horn. The horns are shown with an attached rod of length 300 mm.

 

   MATERIALS AND METHODS
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
CONCLUSIONS
 APPENDIX:
REFERENCES
 
Two-Way Conversion Loss and Cut-Off Frequency
Time domain reflectometry signals on SRPs are always attenuated by the two-way conversion loss comprising the launching and reception loss. We compared the two-way conversion loss of two different horn-shaped mode converters (Fig. 3). Two identical converters were attached to a copper alloy rod 10 mm in diameter and 1 m long, in air. The transmission loss of this setup is approximately equivalent to the two-way conversion loss of a one mode converter. The transmission loss was measured in the frequency range from 0 to 6 GHz, using an HP 8702 network analyzer (Fig. 4) .



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  		Fig. 4. Transmission loss of a 1-m-long copper alloy rod of diameter 10 mm (A) attached at both ends to the large horns with R = 100 mm, (B) attached to the small horns with R = 45 mm. The mean transmission loss in the frequency band 1 to 3 GHz is 3.3 dB for the large horns and 6.5 dB for the small horns. An attenuation of 10 dB occurs at 60 MHz for the large horns and at 195 MHz for the small horns. The horn types are specified in Fig. 3.

 
The results reveal a mean two-way conversion loss in the frequency band 1 to 3 GHz of 3.3 dB for the large horn and 6.5 dB for the small horn. The 10-dB attenuation limit occurs at 60 MHz for the large horn and at 195 MHz for the small horn. We found that the fraction of the power reflected at the converter input (the mismatch) above 400 MHz is below –7 dB for both horns. As expected from the theory, the two-way conversion loss of the smaller horn is higher than for the large horn.

Spatial Sensitivity
Two experiments were performed to compare the behavior of four different SRPs in the presence of conductive scattering objects in the sampling volume. Conductors correspond to a material with an infinite dielectric constant. A wave impinging on a metallic surface is completely reflected. A water-filled cavity with the same shape as the scattering objects we used would not exceed the amplitudes of the measured reflections.

The different SRPs were obtained by combining a coated or uncoated aluminum rod (Fig. 5) with the small or large horn (Fig. 3). We repeated the experiments with a conventional two-rod probe with rod thickness of 5 mm, rod spacing of 31 mm, and rod length of 300 mm. The square cross section of the aluminum SRPs was chosen because of difficulties in manufacturing a cylindrical coating. Preparatory measurements using an uncoated SRP with a square cross section (sidelength s) and an uncoated SRP with a round cross section (diameter s) yielded similar results.



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  		Fig. 5. Cross section of the three rod types used in the experiments and a side view of the coated rod. The calibration measurements were performed with a copper alloy electrode with re = 5 mm and 300 mm length. The sensitivity experiments were performed with an aluminum alloy electrode with side length of 11.1 mm and the same aluminum alloy electrode covered with TiO2 high-dielectric substrate slices 50 by 12.7 by 1.6 mm3 ({epsilon}rd = 100, TransTech Inc., Maryland, USA) tightly attached with conducting glue. Both rods were 300 mm long.

 
Lateral Stick as a Scatterer
In this experiment, the SRPs were mounted in a tank filled with dry quartz sand (Fig. 6) . As a scatterer, a steel stick with a diameter of 10 mm was inserted perpendicular to the probe axis and moved stepwise toward the rod. In the case of the two-rod probe, the stick was moved toward the center axis.



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  		Fig. 6. Setup for the first sensitivity experiment. The single-rod probe was mounted in a tank with dry sand (particle size of 0.08 to 0.2 mm). A 10-mm-diam. steel stick was inserted laterally toward the probe. We measured the reflection coefficient of the peak in the time-domain signal due to this scatterer as a function of the radial distance rs.

 
Using a HP 8753 vector network analyzer we measured the frequency response as a function of radial distance rs between the steel stick and the probe. Frequency ranged from 30 MHz to 5 GHz. With the built-in inverse Fourier transform function the measurement was converted to a time-domain signal (bandpass response). This procedure allowed us to obtain more amplitude-sensitive results than using a commercial TDR measurement device with low-pass step response. The peak due to the scatterer was identified in the trace (Fig. 7) , and the corresponding reflection coefficient {rho}(rs) was read. Some care had to be taken because of multiple reflections.



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  		Fig. 7. Sample trace (bandpass response) from the sensitivity experiment with the perpendicular stick. Clearly discernible are the reflection from the transition coaxial-cable to horn (A), the reflection from the end of the horn (B), the reflection from the scatterer (C), and the reflection from the end of the probe (D).

 
The results (Fig. 8) show that in all four cases the influence of the scatterer on the time-domain trace was not significant for rs > 70 mm. The SRPs with the coated rod measured about half the reflection amplitude of probes with the uncoated rod. The SRPs with different converters but the same rod type did not differ much regarding {rho}(rs). Compared with the SRPs, the two-rod probe showed a reflected amplitude that was significantly smaller.



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  		Fig. 8. Results from the sensitivity experiment with the perpendicular stick. Shown are uncoated rod and horn with R = 100 mm (A), uncoated rod and horn with R = 45 mm (B), coated rod with horn with R = 100 mm (C), coated rod with R = 45 mm (D), and two-rod probe (E).

 
Aperture as a Scatterer
A large scatterer was used in this second experiment. We cut a circular aperture of radius 80 mm into a quadratic aluminum sheet with an area of 1 m2 (Fig. 9) . The size of the opening was reduced by using rings, which were consecutively attached to the aluminum sheet until the aperture radius ra was 15 mm. The probe was mounted to pass through the center of the opening. We used the band-pass time domain function of the HP 8753 network analyzer, from 30 MHz to 5 GHz. The experiment was performed with the four SRPs and the dual-rod probe. Again, the peak due to the scatterer was identified in the trace, and the corresponding reflection coefficient {rho}(rs) was read.



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  		Fig. 9. Setup for the aperture experiment. A 1-mm-thick quadratic aluminum sheet of 1-m2 area has an opening of variable radius rh = 15 ... 80 mm in its center. The SRP is positioned in the center of the opening.

 
The results (Fig. 10) show that for all four SRPs the influence of the large scatterer was still visible at ra = 80 mm. There was no difference regarding the measured reflection amplitude between the coated SRP with the small horn and the coated SRP with the large horn. For the uncoated SRPs with different horns only a small difference was observed. The two-rod probe compared with the SRPs again showed a reflected amplitude that is significantly smaller.



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  		Fig. 10. Results from the experiment with the aperture of radius ra: uncoated rod and horn with R = 100 mm (A), uncoated rod and horn with R = 45 mm (B), coated rod and horn with R = 100 mm (C), coated rod with horn with R = 45 mm (D), two-rod probe (E).

 
Calibration of an SRP
Oswald et al. (2004) could not calibrate the SRP at higher water contents because of fast drainage problems. The probe, inserted from the top into the sand, did not reach the bottom of the tank, a layer into which considerable amounts of water escaped. To circumvent such problems we used a different calibration setup. The uncoated SRP with the small horn was mounted on the bottom pointing up in a tank with a 120-mm radius and 400-mm height (Fig. 11) . The tank was filled with moist sand of 0.08- to 0.2-mm particle size up to a few centimeters higher than the probe end. Drainage was still apparent, but no water could escape. We assumed that the SRP and the two-rod probe measure the length-weighted average water content if refractive index averaging is dominant (Schaap et al., 2003). The latter was supported by using a high bandwidth TDR device and an appropriate evaluation algorithm, which mainly tracks the higher frequency part of the signal. The Tektronix (Beaverton, OR) 11802 scope with an SD 24 TDR sampling head used for the measurements has a rise time of tr = 35 ps, corresponding to a theoretical bandwidth of 0.35/tr GHz (Strickland, 1970). The effective bandwidth is slightly lower because of cable losses. Due to the high-pass characteristic of the mode converter (Fig. 4), frequency components of the step signal below 200 MHz are not transmitted. The TDR curves obtained with an SRP (Fig. 12) therefore look different compared with usual two-rod probe curves. Nevertheless it is still possible to use the algorithm by Baker and Allmaras (1990) to determine the exact travel time of the signal. Using this approach, tangents are fitted both to the probe-head leading edge and the end reflection edge. They are intersected with tangents fitted to a certain range before these reflections. From the intersections we obtain the two-way travel time of the signal, including travel time through the head section (in this case the horn). Using inverse analysis of waveforms from dispersive media, Weerts et al. (2001) showed that this algorithm determines the velocity of the high frequency components of the signal. Preliminary measurements in water and air allowed us to determine the head-section offset for the SRP. From the two-way travel time we calculated the velocity of the signal and derived the apparent dielectric constant {epsilon}ra via the relation v = c0/{surd}{epsilon}r introduced in the Theory section. For each SRP measurement we collected a reference curve with the two-rod probe. The average volumetric water content was calculated for each case on the basis of the added sand and water. All calibration measurements were taken at room temperature with deionized water.



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  		Fig. 11. Setup for the calibration in the wet range of soils. The single-rod probe was mounted from below in a cylindrical tank with radius 120 mm; then the pass-through in the bottom was sealed. A temporarily mounted two-rod probe was used for reference measurements.

 


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  		Fig. 12. Calibration traces, truncated to the significant part for better display. From left to right: air, 0.0, 0.02, 0.05, 0.07, 0.09, 0.12, 0.15, 0.18, 0.24, 0.28, and 0.35 volumetric water content. The traces look different from those of usual two-rod probe curves because low-frequency components of the TDR step signal are not transmitted by the single-rod probe.

 
The apparent dielectric constants measured with the SRP match the reference values well (Table 2). The differences between the SRP and the two-rod measurements increase with a higher water content, although no systematic deviation could be observed. The slopes of TDR curves of wet sand are smoother than those of dry sand, which reduces accuracy in determining the end reflection point. In Fig. 13 the measured permittivities were compared with the values predicted by the model of Roth et al. (1990):

[1]
with the permittivity of the soil matric {epsilon}m = 4.5, the permittivity of water {epsilon}w = 80.2 at 20°C, the permittivity of soil air {epsilon}a = 1, the porosity {phi} = 0.35, and the fitting factor {alpha} = 0.46. The calculated values and the measured data are in good agreement, which corroborates the assumption that refractive index averaging is appropriate.


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  		Table 2. Results of the calibration measurements. The apparent dielectric constant {epsilon}ra of wetted sand was measured with a single-rod probe (SRP) and with a two-rod probe. The volumetric water content {theta} is indicated.

 


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  		Fig. 13. Results of the calibration measurements in the {theta} – {epsilon}ra plane. Dots correspond to the values measured with the SRP, circles correspond to the values obtained with the two-rod probe. The calibration curve by Eq. [1] is plotted as a solid line.

 

   CONCLUSIONS
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 CONCLUSIONS
APPENDIX:
REFERENCES
 
We found that all four SRP configurations had a considerably higher sensitivity to scattering objects in the vicinity than the two-rod probe. Following from the theory, the power distribution of an SRP-guided wave is circular symmetric perpendicular to the probe. An SRP therefore allows greater symmetry and sampled volume than a two-rod probe.

The experiments indicated a cylindrical sampled volume with radius below 120 mm (the radius of the tank used for the successful calibration measurements) for the SRPs used. This result agrees with theory. In the case of the uncoated rod, the power flux is below 0.3% at 20 times the rod radius (100 mm).

In our experiments we found that coating the rod with a layer of high dielectric material ({epsilon}rd = 100) reduces the radial sensitivity.

A large horn-shaped mode converter is in principle superior to a small one in terms of launching and receiving a guided wave. The cut-off frequency is lower, and the mode conversion in the TDR frequency range is better. However, our experiments showed only a small difference in spatial sensitivity between a horn with opening radius 100 mm compared with one with radius of 45 mm. This means that the extent of the guided wave is not significantly affected by the converter size. From a practical viewpoint one would prefer the smaller horn because of its compact size.

Because of these results we focused on the SRP with the uncoated rod and with the small horn, which we calibrated in the soil moisture range up to 0.35 m3 m–3. The calibration was performed with a sand–water mixture using the probe protruding from the bottom up into a tank. We measured an apparent dielectric constant close to the one obtained with the two-rod probe as reference.


   APPENDIX:
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 CONCLUSIONS
 APPENDIX:
REFERENCES
 
THEORY OF COATED CYLINDRICAL SINGLE CONDUCTOR WAVE GUIDES
For harmonic field vectors, for example, the electric field vector

[A1]
and nonmagnetic (µ = µ0) material with a complex dielectric constant {epsilon} = {epsilon}0{epsilon}r, the Maxwell equations relating the complex amplitude E to the complex amplitude of the magnetic field vector H are given as

[A2]

They can be rearranged to the Helmholtz equations:

[A3]
using the Laplace operator:

The Helmholtz equations have solutions of the form:

[A4]
which are traveling waves in the k direction with full wave number:

In case of a cylindrical coordinate system (z, r, {phi}) the Laplace operator {Delta} can be separated into transverse and z-directed components:

If we are only interested in a wave traveling in the z direction, for example, for the electric field vector

[A5]
and if we combine this with Eq. [A3] we obtain for the z component the following equation:

[A6]

With the transverse wave number kT = {surd}, the full wave number becomes k = {surd}. Analogously, we obtain for the z component of the magnetic field vector

[A7]
because

Hz != 0 -> Ez=0, nonzero values for Ez and Hz are mutually exclusive. The case Hz = 0 is called "transverse magnetic" mode (TM), Ez = 0 is called "transverse electric" mode (TE), and Hz = Ez = 0 is called "transverse electromagnetic" mode (TEM). A single-wire waveguide primarily conducts a TM mode (Stratton, 1941). We therefore proceed with Ez != 0, Hz = 0.

Because of the cylindrical symmetry of the guide, Ez can be separated in the form:

[A8]
Therefore Eq. [A6] can be decomposed into

[A9]
The expression {Phi} is a harmonic differential equation with solution:

[A10]
and the expression in R is a Bessel differential equation with the general solution:

[A11]

Therefore, the full solution is

[A12]
An, Bn, Cn, Dn are parameters, n is the mode order, (1,2) is the Hankel function of the first and the second kind.

The fundamental and the only mode of interest is n = 0. The higher modes, n > 0, suffer from severe attenuation (Stratton, 1941). Knowing Hz = 0 and the function for Ez, all field vector components can be calculated using the formulas given above, resulting in

[A13]

[A14]

[A15]
with the parameters A, a. These equations have to be formulated in each material domain because there are the metal rod (subscript "m"), the dielectric coating (subscript "d"), and the medium under test (i.e., the soil, subscript "s"). Each domain has its own parameter set and its own wave number (e.g., Ad, ad, and kTd for the dielectric coating). In general, the parameters of the wave in the coating are not the same as the parameters As, as of the wave in the medium. The transverse wave numbers kTd and kTs are therefore different. In contrast, from the boundary conditions given by the Maxwell equations for the transverse field components at the corresponding interfaces it follows that the longitudinal wave numbers are the same in both layers, kzd = kzs = kz.

Without introducing a significant error, the metal rod can be assumed to be a perfect conductor because the losses due to the skin effect are still small up to 10 GHz. This means Am = 0, am = 0. At the surface of the conducting rod, which shall be at radius rm, Ez must be zero which leads to

[A16]
for the parameters in the dielectric.

At infinite distance from the wave guide the field has to disappear, which implies as = 0. Finally, Ez and H{phi} have to be continuous at rd, the boundary of the dielectric and the soil:

[A17]

The unknown Ad and As disappear, an equation with the only unknown parameter kz remains:

[A18]
with

This equation is a transcendental eigenvalue equation for kz. It has an infinite number of roots, a primary one near ks and side roots that belong to asymmetric (leaky) modes. Asymmetric modes are highly damped according to (Stratton, 1941). The solution obtained for kz if the dielectric constant of the coating is lower than the permittivity of the surrounding soil is not a physical one, as it exhibits a higher velocity than the speed of light in the soil dielectric. The guiding effect is lost, and all energy is irradiated. For any practical case {epsilon}d > {epsilon}s, and thus {epsilon}rd > {epsilon}r, with {epsilon}rd = {epsilon}d/{epsilon}0 and {epsilon}r = {epsilon}s/{epsilon}0. The real part of the relative permittivity of natural soils is in the range of 2.5 ... 50 (Topp et al., 1980). Therefore, if a coating is used, {epsilon}rd must be higher than 50 to cover the range of the dielectric constant of common soils.

Using the boundary condition Ezd = Ezs at the interface of the two dielectrics we find Ad as a function of As. We now have the complete solution of the normalized field. The remaining free variable As represents the amplitude of the excitation.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 CONCLUSIONS
 APPENDIX:
REFERENCES
 




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Home page
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S. R. Evett and G. W. Parkin
Advances in Soil Water Content Sensing: The Continuing Maturation of Technology and Theory
Vadose Zone J., November 11, 2005; 4(4): 986 - 991.
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