Three-dimensional numerical modeling of a four-pin probe for soil water content.
Frequency domain (FD) technique is now accepted as an effective tool to measure soil water content (Dean et al. 1987; Gaskin and Miller 1996; Hilhorst 1998). A major advantage of FD technique over time domain reflectometry (TDR) is that the commercial FD sensors are relatively inexpensive and easy to operate; furthermore, the sensor geometry is very adaptable, facilitating the development of a variety of configurations (Robinson et al. 1998). However, the relationship between the output of the FD sensor and the soil water content is not straightforward and involves 3 stages from a theoretical standpoint. First is recognition that FD sensors actually react to the port impedance of the soil probes in electric circuits. The second stage is to establish the link between the port impedance of the soil probes and the average relative dielectric constant in the soil volume of interest. The third stage is to establish the relationship between the soil dielectric constant and soil water content (Kelleners et al. 2004). This relationship can be described separately using physical dielectric mixing models (e.g. Friedman 1998) or empirical models (e.g. Topp et al. 1980; Malicki et al. 1996). On the other hand, the relationship between the port impedance of the soil probe and the sensor output can be established with the help of electric circuit theory (e.g. Dean et al. 1987; Gaskin and Miller 1996; Hilhorst 1998). However, modelling the port impedance of soil probe is difficult and has not yet been solved effectively. Gaskin and Miller (1996) presented an impedance sensor using standing wave ratio principle, but an empirical approach was taken to calibrate the device because of the difficulties alluded to in modelling the probe's port impedance.
The objective of the study is to provide a powerful tool to model the port impedance of the soil probes, which not only takes the soil dielectric constant heterogeneity within the volume of influence into account, but also may be used to analyse and optimise the electrode geometry of soil probes and the operating frequency in FD methods. Previous studies have focused on quantifying the probe's performance in FD and TDR methods. Zegelin et al. (1989) described how the field distribution of 3- and 4-pin probes provided increasingly closer approximations to an ideal coaxial cell in which the equipotentials are concentric circles on the inner electrode based on quasi-static approximation. Thus, a multi-pin probe may be viewed as a segment of regular transmission line. Sun et al. (2005) gave an analytical solution to calculate the impedance of the multi-pin probe based on transmission-line theory. There were 3 limitations in applying this analytical solution: (i) the length of the pin was finite so that the end effect could not be ignored; (ii) they used an inequality to estimate the characteristic impedance of the multi-pin probe, and thus the analytical solution was still a rough approximation; (iii) the analytical solution based on transmission line was not able to take the soil heterogeneity within the volume of interest into account.
Compared with the analytical solution, numerical modelling is probably more relevant and practical for achieving the goal. By means of the numerical approaches, one can investigate any case without a restriction on the probe's geometry. Straub (1994) computed the admittance (reciprocal of impedance) of the electrodes of a soil probe using boundary element method. De Rosny et al. (2001) used a finite element method (FEM) to characterise a capacitance probe's response based on quasi-static approximation of electromagnetism. This capacitance probe operated at 38 MHz and had a central electrode together with an annular electrode. Bolvin et al. (2004) then described a 3-dimensional numerical modelling for the same probe as that of De Rosny et al. (2001). In these 2 studies, the models were validated by experiments and were used to analyse the influence of soil heterogeneity; however, the models did not appear to have the capability to analyse the effects of the operating frequency and geometry parameters on the port impedance of the soil probes.
This study was devised to create and validate a 3-dimensional numerical model with high frequency structure simulator (HFSS) for the 4-pin probes, and to attempt to analyse the characteristics of a 4-pin probe by FD methods. The 4-pin probes are a representative structure in FD and TDR methods, and convenient to highlight the characteristics of FD methods. HFSS is a general electromagnetic simulation software package and based on Maxwell's equations and FEM. A typical 3-dimensional electromagnetic distribution of a 4-pin probe is presented. The model was validated with 3 experiments under the aid of a network analyser. First, the experiment was performed using a series of fluids of known relative dielectric constants, then numerical simulations were carried out and confirmed by soil sample test with varying operating frequencies and probe lengths. At the same time, the effects of these parameters on FD methods were discussed based on the 4-pin probes.
Materials and methods
Fundamentals of HFSS
HFSS (Ansoft corporation 2003) is an interactive software package for analysing electromagnetic behaviours in terms of a given physical model. It can be used to treat most kinds of electromagnetic behaviours, such as open boundary problems, radiated near or far fields, characteristic impedances, and propagation constants. As general simulation software for 3-dimensional electromagnetic fields, HFSS is based on the following field equation derived from Maxwell's Equations:
[nabla] x (1/[[mu].sub.r][nabla] x [??]) - [k.sup.2.sub.0][[epsilon].sub.r][??] = 0 (1)
where E is a phasor representing an oscillating electric field, [k.sub.0] is the free-space wave number, [[epsilon].sub.r] the complex relative permittivity, and [[mu].sub.r] the complex relative permeability. The geometric model is automatically divided into a large number of tetrahedral elements and the collection of the tetrahedral elements is referred to as the finite element mesh. By integrating over volume, it gives:
[[integral].sub.v][[w.sub.n] x [nabla] x (1/[[mu].sub.r][nabla] x [??]) - [k.sup.2.sub.0][[epsilon].sub.r][W.sub.n][??]]dV = 0 (2)
where the basis functions, [W.sub.n], define 2 interpolation schemes between nodal locations in the overall mesh of the tetrahedral element. First, the 1st order tangential element basis function interpolates field values from both nodal values at vertices and on edges, which has 20 unknown per tetrahedral; second, the 0th order tangential element basis function makes use of nodal values at vertices only, which has 6 unknown per tetrahedral. The subscript n represents integration replicated for thousands of equations for n = l, 2 ..... N, and the intent is to obtain N equations with N unknowns for solution. Furthermore, using Green's and Divergence Theorems:
[[integral].sub.v][([nabla] x [W.sub.n]) x (1/[[mu].sub.r][nabla] x [??]) - [k.sup.2.sub.0][[epsilon].sub.r][W.sub.n][??]]dV = [[integral].sub.s](boundary term)dS (3)
Writing the E field as a summation of unknowns, [x.sub.m], times same basis functions used in generating the initial series of equations:
[??] = [N.summation over (m=1)][x.sub.m][W.sub.m] (4)
Rearranging Eqns 3 and 4 gives:
[N.summation over (m=1)][x.sub.m] x ([[integral].sub.v][([nabla] x [W.sub.n]) x (1/[[mu].sub.r][nabla] x [W.sub.m]) - [k.sup.2.sub.0][[epsilon].sub.r][W.sub.n][W.sub.m]]dV) = [[integral].sub.s](boundary term)dS (5)
Equation 5 has the basic matrix form Ax = B, where A is the basis functions and field equation, in a known N x N matrix, 'x' is the unknowns to be solved for, and B is the boundary term. In this way, the HFSS can transform Maxwell's equations into matrix equations that are solved using traditional numerical methods. Thus, the resulting Eqn 5 allows solution of the unknowns, [X.sub.m], to find E.
A typical 4-pin probe was used in this study. Figure 1 details the geometry of the probe. As the ground electrode, 3 outer pins were fixed to a metallic annulus apart with each angle of 120[degrees]. The inner pin was the excitation electrode. All pins were made of stainless steel. The 4 different lengths 8, 12, 15, and 20 cm were simulated and tested.
[FIGURE 1 OMITTED]
A network analyzer (model E5071B, Agilent, CA) was employed to measure the impedance parameters of the tested probes (Starr et al. 2000). This instrument can provide a broad range of measurement frequencies from 300 KHz to 8.5 GHz. For the purpose of validation, a series of solvents with known relative dielectric constants listed in Table 1 were used. Each liquid filled a glass vessel (height 12 cm, radius 3.5 cm). The relative dielectric constants of the liquids ranged from 2 to 81, representing the full range of soil water contents. The major advantage of using these liquids was to provide homogeneous conditions for reducing the influence of the air gaps (De Rosny et al. 2001; Topp et al. 2003; Bolvin et al. 2004). In each test, the measurements were replicated 6 times. In order to investigate the influences of the measurement frequency and pin length on the impedance of the probes, soil samples were prepared with different volumetric water contents ranging from 0 to 0.50[cm.sup.3]/[cm.sup.3]. The soil sample used was clay loam (sand 11%, silt 55%, clay 34%). The electrical conductivity of the soil samples was 0.22 mS/cm, so the real part of the port impedance of the probe was negligible for the situation encountered in the present study, and the imaginary part of the port impedance of the probe (Zp) was used for simulation and experiments.
HFSS version 9.0 was used in the study (Ansoft Corporation 2003). Three-dimensional modelling with HFSS comprised the following procedures. (1) The design was inserted into a project, and a physical model of the 4-pin probe was developed and material characteristics assigned (see Fig. 1). The cylinder was assigned its relative dielectric permittivity as a parameter with the range of 1-80 to represent a soil/liquid sample. (2) A driven modal was chosen as a solution type for calculating the port impedance of the 4-pin probe. (3) The wave port was assigned on the port surface of the 4-pin probe as excitation, and the radiation boundary on the surface of the soil/liquid sample cylinder chosen to represent an open boundary from which energy can radiate. The frequency of excitation was set to 100 MHz or as a sweep parameter to analyse its effect on the probe's port impedance. (4) To produce the optimal mesh, HFSS uses an iterative process, called an adaptive analysis, in which the mesh is automatically refined in critical regions. First, it generates a solution based on a coarse initial mesh. Then it refines the mesh in areas of high error density and generates a new solution. Stopping criteria are maximum adaptive times or error of the 2 successive mesh refinements ([DELTA]S). The effect of stopping criteria on the computed Zp is listed in Table 2. To save computation time, the maximum adaptive time and critical [DELTA]S were chosen as 7 and 0.01, respectively.
Results and discussion
Simulation of electric field distribution Here we employed HFSS to simulate the electric field distribution for the 4-pin probe with 8-cm pin length in a medium with relative dielectric constant equal to 25 as in a wet soil. It operated at 100 MHz. Figure 2a presents the electric field distributions on the horizontal plane, and Fig. 2b the vertical direction. In the vertical direction, most of the electric field intensity is similar, whereas it is greater near the lower end of the central electrode, which is regarded as the 'end effect' phenomenon. On the horizontal plane, the electric field distribution is approximately concentric circles centred on the inner electrode and the electric field intensity decreases gradually from the inner electrode, which confirms the field distribution of the 4-pin probe proposed by Zegelin et al. (1989).
[FIGURE 2 OMITTED]
Effect of relative dielectric constant
Figure 3 presents a comparison between the numerical analysis (solid line) and measured impedance of the probe (solid dot) with 8-cm pin length at the operating frequency 100MHz. As shown in Table 1, 11 solvents were used as various references of measured dielectric constants. All measured data of impedance were close to the numerical simulation. Statistically, the measured data fit the numerical results of HFSS with [R.sup.2] = 0.990. Moreover, Zp decreased dramatically as the relative dielectric constant varied from 1 to 10, whereas its changing tendency became flatter and flatter. The measurement accuracy of the impedance sensor will significantly deteriorate in the case where the relative dielectric constant of the medium is sufficiently high, e.g. >32, which is equivalent to volumetric soil water content of 0.50 [cm.sup.3]/[cm.sup.3], according to the Topp equation (Topp et al. 1980). This result might explain why the measurement range of most commercial FD sensors is limited in unsaturated soil for applicable and reasonable accuracy, in contrast to the TDR method. For TDR methods, a clear relationship for the travel time measured with circuit theory and dielectric constant of a medium exists:
[FIGURE 3 OMITTED]
T = 2L[square root of [[epsilon].sub.r]/c (6)
where T (s) is the travel time, L (cm) the length of the wave guide into the soil, [[epsilon].sub.r] the relative dielectric constant of the soil around the wave guide, and c (cm/s) the speed of light. From Eqn 6, TDR method is capable of measuring soil water content in the full range, and the more soil water content, the longer travel time, which makes measurement with TDR easier. On the other hand, it is relatively difficult to measure low soil water content because the short travel time (T) is not easily computed from the reflected wave.
Effect of operating frequencies
In the following tests, the numerical model and the measured data were used to investigate the effects of the operating frequencies and pin length on Zp of the tested probe. Figure 4 includes 4 parts corresponding to f = 50, 100, 150, 200 MHz, respectively. Here the pin length of the used probe was 8 cm. Firstly, Fig. 4 shows that all measured data agreed with the predications of the numerical model, with a correlation >0.95, and the higher the operating frequency, the lower the value of [R.sup.2]. The reason is probably the random influence of the distribution parameter at high frequencies.
[FIGURE 4 OMITTED]
The effect of the operation frequency on FD methods was highlighted. Generally, the higher the operation frequency, the smaller the influence of the salinity on the measurement of soil water content (Campbell 1990; Paltineanu and Starr 1997), so high operating frequency benefits the accuracy of FD sensors. However, as predicted by the numerical model, the measured data confirmed that Zp of the probe might be capacitive (negative) or inductive (positive) under relatively high operating frequencies. For instance, in Fig. 4 for f = 200 MHz, the predicted Zp was capacitive from the initial point to the volumetric water content of 0.40 [cm.sup.3]/[cm.sup.3], whereafter Zp was inductive. The measured data verified this, but at 0.37[cm.sup.3]/[cm.sup.3]. At this operating frequency, the same amplitude of the probe's Zp probably existed at 2 different soil water content points, and the FD sensors were not able to measure soil water content correctly. So, a too-high operating frequency was unsuitable for the FD method, which may be the limitation of FD methods and the reason why most commercial FD sensors operate at about 100 MHz.
[FIGURE 4 OMITTED]
Effect of pin length
Figure 5 also includes 4 parts corresponding to pin length L= 8, 12, 15, and 20 cm, respectively. Here the operating frequency was 100 MHz. All measured data in each part of Fig. 5 were close to the numerical predication, with correlation >0.970, which showed once more that the 3-dimensional numerical simulation of a 4-pin probe was able to represent the probe response. Similar to the effect of the operating frequency, the amplitude of the probe's Zp was not monotonic, nor did FD sensors measure soil water content correctly because the Zp might change from a negative to a positive value in Fig. 5 where L = 15 cm or L = 20 cm. So the probe with short pins is more suitable for FD method. The results may be used to explain why 6-cm probes are common, and probes longer than 10 cm are seldom used in commercial FD sensors. This characteristic of the FD method is different from the TDR method. From Eqn 6, the TDR travel time is linear with the length of the wave guide inserted in soil; moreover, the longer the probe, the more accurate is measurement of soil water content. A 15-cm probe is popular in commercial TDR sensors; however, probes shorter than 8 cm are seldom used in practice because the travel time is relative difficult to compute from the reflected wave.
[FIGURE 5 OMITTED]
A very good agreement between the measurements and the simulation of the 4-pin probe was observed in the 3 validated experiments. By comparison with TDR method, the influence of the relative dielectric constant of a medium, the operating frequency, and probe length on the probe's Zp was highlighted. The 3-dimensional model appears to be a meaningful tool to investigate a 4-pin probe in FD methods. When implementing the model in saline soil, the 3-dimensional model still has to be verified and the real part of the probe's port impedance should be contained to investigate the 4-pin probe more deeply. A typical 3-dimensional electric field distribution of a 4-pin probe was represented with the numerical model, which shows that the 3-dimensional numerical model is probably used to evaluate the impact of soil heterogeneity within the volume of influence of the 4-pin probes. The study was performed with HFSS based on Maxwell's equations and FEM. This opens the possibility of modelling the other geometry structures used in FD method with HFSS.
We acknowledge the financial support of the National Nature Science Foundation of China under project Nos. 30270775 and 30370823, and the Doctoral Program Foundation of Educational Ministry of China project No. 20030019012.
Manuscript received 22 August 2005, accepted 6 January 2005
Ansofi Corporation (2003) 'Anssoft HFSS v9.0.' (Ansoft Corporation: Pittsburgh, PA)
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D. Ma (A), Y. Sun (A,B), M. Wang (A), and Y. Gao (A)
(A) Research Center for Precision Agriculture, China Agricultural University, Box 63, East Campus, Beijing 100083, P.R. China.
(B) Corresponding author. Email: firstname.lastname@example.org
Table 1. Relative dielectric constants of air and liquids used (20[degrees]C) Medium Relative dielectric constant Air 1.0 Cyclohexane 2.0 Benzone 2.3 Ethyl acetate 6.3 2-Butanol 16.8 1-Butanol 17.6 2-Propanol 20.3 Ethanol 25.0 Ethanol: water (2:1) 43.6 Ethanol: water (1:1) 53.3 Deionised water 81.0 Table 2. Computed Zp ([OMEGA]) at operating frequency 100 MHz, assuming pin length 8 cm and uniform dielectric constant [[epsilon].sub.r] = 25 [DELTA]S Maximum adaptive times 3 5 7 9 11 0.02 -33.6 -37.0 -37.0 -37.0 0.01 -33.7 -37.0 -37.8 -37.8 -38.1 0.005 -33.7 -37.0 -37.8 -37.8
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|Author:||Ma, D.; Sun, Y.; Wang, M.; Gao, Y.|
|Publication:||Australian Journal of Soil Research|
|Date:||Mar 15, 2006|
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