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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Accurate measurement of moisture content is a prime requirement in hydrological, geophysical and biogeochemical research as well as for material characterization and process control. Within these areas, accurate measurements of the surface area and bound water content is becoming increasingly important for providing answers to many fundamental questions ranging from characterization of cotton fiber maturity, to accurate characterization of soil water content in soil water conservation research to bio-plant water utilization to chemical reactions and diffusions of ionic species across membranes in cells as well as in the dense suspensions that occur in surface films. One promising technique to address the increasing demands for higher accuracy water content measurements is utilization of electrical permittivity characterization of materials. This technique has enjoyed a strong following in the soil-science and geological community through measurements of apparent permittivity via time-domain-reflectometry (TDR) as well in many process control applications. Recent research however, is indicating a need to increase the accuracy beyond that available from traditional TDR. The most logical pathway then becomes a transition from TDR based measurements to network analyzer measurements of absolute permittivity that will remove the adverse effects that high surface area soils and conductivity impart onto the measurements of apparent permittivity in traditional TDR applications.

This research examines an observed experimental error for the coaxial probe, from which the modern TDR probe originated, which is hypothesized to be due to fringe capacitance. The research provides an experimental and theoretical basis for the cause of the error and provides a technique by which to correct the system to remove this source of error. To test this theory, a Poisson model of a coaxial cell was formulated to calculate the effective theoretical extra length caused by the fringe capacitance which is then used to correct the experimental results such that experimental measurements utilizing differing coaxial cell diameters and probe lengths, upon correction with the Poisson model derived correction factor, all produce the same results thereby lending support and for an augmented measurement technique for measurement of absolute permittivity.

Frequency domain analysis of soils, cotton lint, biological cells and media is rapidly gaining appreciation due to the ability to provide a true measurement of permittivity as opposed to an apparent permittivity that TDR analysis in the time domain provides. One of the driving factors behind this new trend is due to the recognition that in saline and high clay content soils, that the conductive soils dielectric loss has a profound impact on the measured apparent permittivity which causes large errors especially when temperature effects are taken into consideration.

Recent research [

Another similar research path [

In comparing the two approaches taken by Clarkson and Kraft [

Objectives: Derive a technique for the absolute measurement of permittivity from coaxial cells and,

show the impact of fringe capacitance on experimental measurements,

develop a theoretical model and predict the magnitude of the hypothesized error due to fringe- capacitance,

provide experimental results that quantify the magnitude of the fringe capacitance error,

show the correlation between the predicted error to the experimental error,

develop a correction solution for removal of this error from coaxial and TDR probe measurements.

We note that for propagation of a free-space plane wave, in a source-less region, that is directed only in the z direction, the form of the wave propagation can be shown to have the form of

ε′: = real, dielectric constant, term of the complex permittivity (F/m).

ε″: = imaginary, loss, term of the complex permittivity (F/m).

γ: = propagation coefficient (1/m).

α: = attenuation factor of the propagation coefficient (nepers/m).

β: = phase delay factor of the propagation coefficient (rads/m).

σ: = conductivity factor of the propagation coefficient (S/m).

μ: = material permeability (H/m).

ω: = omega (rads/s).

e: = the exponential transcendental number = 2.71828…

tan δ: = loss tangent definition as ratio of real to imaginary part of displacement current.

Of note is that while for free-space wave propagation,

Utilizing a power series closed form solution to an infinite series, provides an exact solution to the measured response of the coaxial cell that is due to the impedance miss-match between the coaxial cable and the interconnecting cable which can be used to model the measured reflection coefficient, Γ_{measured}, to that of the desired measurement of the free space propagation constant γ, which is required for determination of the material’s true permittivity as shown in

Of particular interest to this research is that even when the measured permittivity is corrected for the multiple reflections, per use of

In the interest of obtaining guidance into the levels of expected accuracy that can be obtained by the proposed miss-match impedance correction protocol in a non-radiating condition, provided by

Brass was chosen for the coaxial cell as Kraft [

Noting that for accurate utilization of a Network Analyzer, a major requirement is for the instrument and interconnecting cables and connectors to be calibrated out of the system. This calibration requirement however causes some difficulties in performing direct comparisons between the two like-coaxial cells, as ideally one would like to use the same calibration for both cells in order to avoid the calibration from obscuring the impact of the differences. In order to achieve this single calibration/dual use condition, a close fitting drop-in insert, designed to reduce the outer diameter, was machined to allow for direct comparison of two probe geometries without the need for changing connectors and the probe structure, thereby avoiding the need for a recalibration thereby enhancing the accuracy of the comparison.

The outer diameter, 50.67 mm, of the large coaxial cell was chosen for similar dimensions to industry standard 20 cm TDR probes, with a similar center conductor diameter of 4.67 mm. The smaller diameter insert provided a reduced 17.96 mm outer diameter, which was chosen to give a 50 Ω impedance when the coaxial cell was filled with dry sand, with an estimated relative permittivity of ε_{r} = 2.85. Further in an effort to increase the confidence of the obtained results, multiple internal probes, the current carrying member of the inner diameter of the coaxial cell, were all machined at the same diameter at different lengths, for comparison of experimental results to the theoretically predicted values for delays that are a direct function of the path-length provided by the length of the center conductor that is exposed to the material under investigation.

In the interest of restricting the internal reflections to only those of the model of

For the network analyzer calibration, referenced earlier, the research used an open-short-load protocol to move the reference plane to the location of the short. Some experimentation on calibration for this system, lead to the realization that a choice had to be made between using either an in-house built shorting element that provides the short at the correct location, thereby establishing the reference plane correctly; or to utilize a high-quality commercial short, designed specifically for calibrations, that would inadvertently put the calibration plane in the wrong location, thereby leading to a phase error that would have to be corrected via a model. Experiments suggested that for the highest accuracy work, a well designed in-house constructed short made from an identical connector to the one used in manufacturing the coaxial cell, provided the best results.

We note that due to the impedance miss-match issues, when dealing with materials with a wide range of permittivity’s, it becomes very difficult to separate multiple reflection errors from other mitigating factors. However, there is a special resonance frequency that is free from multiple reflection errors that can be used to examine additional errors that are not due to impedance miss-match. The use of this technique, quarter-wave impedance transformer, to remove the impedance miss-match error is what led to the discovery of the additional error that is hypothesized to be created by fringe capacitance emanating off the center conductor’s end-point (tip).

As a first comparison into the validity of the miss-match impedance correction technique, we note the work by Heimovarra [

For the case where the system is almost matched at the ¼ wavelength line length, the discontinuities between impedances are small and the product Γ_{1} Γ_{3} ≪ 1, which effectively reduces the denominator in

Further noting that by definition at the ¼ wavelength the following condition is true (assume very low loss condition).

Thus at the ¼ wavelength frequency equation, for the non-radiating conditions _{3}=1

Noting that in the matched condition,

Which leads to

Thus, when the phase of the reflected wave is delayed by π radians, the frequency of this occurrence corresponds to the ¼ wavelength matched condition. Further noting that at the matched ¼ wavelength condition, Zc is equal to Zo which is equal to the coaxial cell impedance,

Re-arranging

For the purposes of relating this theory to experimentally derived measurements, some basic test cases utilizing low loss materials were investigated. For the low loss, Γ = 0, test cases and for true TEM lines at ¼ wavelength, which occurs at the λ/4 phase delay which indicates the resonant frequency with matched condition, provides a direct measure of the propagation coefficient γ, as shown in

Rearranging 11 provides _{r} relative permittivity due to the two wave propagation caused by the open-circuit reflection off the end of the probe.

Of particular note, is that

ρ := current density

V:= Voltage (electric potential)

∇ := Gradient Operator

ε := Permittivity of medium.

Noting from Collin [

To find the fringe capacitance from the Poisson model, our protocol was to find the total capacitance for each simulation run, by means of the relation of cell’s capacitance being equal to the charge on the conductors divided by the voltage between the conductors, C = Q/V. By performing multiple simulations as the length of the center conductor was reduced, the results of several of the models at progressively shorter center conductors provides an estimate of the fringe capacitance which is found from the intercept on the graph of capacitance

To examine the predicted δ–length, or δγ, experimentally, several center conductors of varying lengths, shown in

Given the Poisson model, along with qualitative experimental evidence, suggests a unique length extension δλ is required to correct for the extra fringe capacitance in coaxial cell type measurements, or TDR soil probes; therefore we propose a new variant for

δλ:= effective wavelength extension of the center probe due to fringe capacitance.

To utlize

L_{probe} := physical length of the probe (m).

In practice, it was found that the experimentally derived lengths provided the following estimates for a required length extension when tested with air as the dielectric; small OD cell δλ = 0.25 cm and large OD cell’s δλ = 0.49 cm, that were similar to those predicted by the Poisson model. In practice it is suggested that experimentally derived δλ length extensions should be used as they are deemed to be more accurate than those provided via a Poisson simulation model.

In the next phase of the experiment, of interest was to examine if the length extensions from _{r} = 2.5 or so, in order to preserve reasonable geometries for the inner and outer diameter of the coaxial cell. To accommodate this work, a dry sand was chosen for this phase of the experimentation, which was estimated by utilizing a network analyzer and the following techniques; free space through transmission, through transmission coaxial line, frequency domain reflectance network analysis of 20 cm TDR probe and finally standard TDR utilizing an HP 54120b with 12 GHz TDR reflectance head. The survey of all these techniques, for estimation of the sand’s true permittivity, produced a range of permittivity’s ranging from ε_{r} ≈ 2.4–3.2. Taking an estimate in the middle of the range, a sleeve was manufactured for the coaxial cell, such that with the sleeve removed, the coaxial cell provided one large outer diameter configuration, and with the sleeve inserted, provided a much smaller outer diameter that was designed to provide a matched condition when the cell was filled with the dry sand. Thus, the system was configured to avoid the miss-match impedance condition when the small outer diameter coaxial cell probe was filled with the dry sand, thereby providing an accurate estimate of the true permittivity of the dry sand as there would be minimal reflections allowing for a minimal error associated with the direct use of

In performing the experiment, the small OD coaxial cell was loaded with dry sand and the ¼ wavelength resonant frequency was found to be 1081 MHz, which with _{r} = 2.87. For comparison, the ¼ wave frequency location as predicted for the large diameter coaxial cell with permittivity of 2.87 suggests the ¼ wave resonant frequency, with large cell’s δλ correction also provided by the air test, should occur at 1,024.1 MHz while experimental results were found to occur at 1031.3 MHz. This translates to an agreement between the large cell’s prediction of the ε_{r} = 2.91 compared to the smaller matched cell’s permittivity measurement of ε_{r} = 2.87. Conversely, without correction, the ¼ wave resonant frequency at the same ε′ = 2.87, is predicted to occur at 1164 MHz. Alternatively, using the measured resonance frequency of 1031 MHz along with the probe’s physical length of 3.8 cm, leads to the erroneous prediction of permittivity of ε′ = 3.65. Noting the densely packed cotton has a permittivity range from ε′ = 1.80 to ε′ = 2.00, and uncontaminated oil at a nominal permittivity of 2.35, it becomes clear that for applications with low permittivity materials and at frequencies where the probe length is limited, necessary to avoid radiation conditions, the correction for fringe capacitance is critical for obtaining accurate answers that are independent upon probe length.

The results of this research indicate that the hypothesis that fringe capacitance creates an artificial length extension to the coaxial cell’s center conductor is valid and is critically important for obtaining accurate measurements of permittivity in applications where the permittivity values are low and the length of the probes are limited due to the need to avoid placing the probe into radiation condition where the probe will effectively leak energy due to its length transition approaching the ¼ wave resonance location where it becomes a suitable antenna.

The magnitude of the fringe capacitance error, for a 3.8 cm probe, was found to cause a material with a permittivity of e’ = 2.87, to be incorrectly measured at e’ = 3.60 if the fringe capacitance was not accounted for. Furthermore we note that as the fringe capacitance acts as an effective probe lengthener, the error magnitude is dependent upon the probe length. As such, in transferring measurements from one laboratory to another that is likely using a different probe length, the only way to ensure transferability of the measurements is to account for the fringe capacitance.

The experimental results, and the Poisson model, suggest that this artificial length is not dependent upon permittivity but rather is a geometry factor and can therefore be used to correct and remove this artificial length once it’s been measured, regardless of the permittivity of the material. It is noted that the need for length correction is particularly important in drier soils and other media of low permittivity, such as cotton moisture sensing and water contamination of oils, and as the need for increased accuracy pushes the measurement into higher frequencies which limits the probe length. The technique outlined herein suggests a promising and simple method for quantifying an extra δγ pseudo correction length to account for the fringe capacitance.

Reflection/transmission map detail of resultant waveform from combination of multiple reflections from both the leading edge, undesired, and probe end, desired measurement, in TDR/FDR probes, or coaxial cells, due to impedance miss-match between inter-connecting coaxial cable’s impedance, of Zo, to the soil-probe impedance of Z1. Note: Hatched area indicates soil or other material under test.

Machined brass coaxial cells based on commercial N to UHF (RF) adapter. System utilizes an insert based center probe that allows for changing the center probe, thereby providing a center-probe length change for the system, while maintaining the original calibration that removes the effects of the instrument, interfacing cable and the RF adapter, while preserving the original system calibration. On the right is the large brass insert that provides a similar means to maintain the original system calibration while providing the means to alter the outer diameter of the coaxial cell, hence altering the impedance, of the coaxial cell. This system was designed with the center insert installed, to provide a near perfect 50 Ω match for the coaxial cell when filled with dry sand ε_{r} = 2.85.

Poisson cross-sectional model of coaxial cell with terminated center conductor, which is the notched area at the bottom of the screen. The color pictorial is of the electric-potential inside a cross-section of a coaxial, or TDR probe with symmetry boundary conditions applied along the bottom edge (not including the notched section which represents the center conductor and has an imposed Voltage). The left and top edge of the figure have a zero voltage boundary condition representing the conductors being tied to ground and the far right edge was set to a floating boundary condition to represent an open-ended coaxial cell in a homogenous media.

Comparison of the capacitance as calculated from the Poisson cross-sectional model of coaxial cell with terminated center conductor, as a function of the center conductor’s probe length.