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Published online 29 June 2007
Published in Soil Sci Soc Am J 71:1278-1287 (2007)
DOI: 10.2136/sssaj2006.0383
© 2007 Soil Science Society of America
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SOIL PHYSICS

Accurate Time Domain Reflectometry Measurement of Electrical Conductivity Accounting for Cable Resistance and Recording Time

C.-P. Lin*, C.-C. Chung and S.-H. Tang

Dep. of Civil Engineering, National Chiao Tung Univ., 1001 Ta-Hsueh Rd., Hsinchu, Taiwan

* Corresponding author (cplin{at}mail.nctu.edu.tw).


   ABSTRACT
TOP
 NOTES
 ABSTRACT
INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Methods accounting for cable resistance in time domain reflectometry (TDR) based electrical conductivity measurements remain controversial, and the effect of TDR recording time has been underrated when long cables are used. A comprehensive full waveform model and the direct current (DC) analysis were used to show the correct method for taking cable resistance into account and guidelines for selecting proper recording time. The Castiglione–Shouse scaling method was found to be incorrect because the effect of cable resistance on the steady-state reflection coefficient is nonlinear. To account for cable resistance, the series resistors model is theoretically sound and should be used. The characteristic impedance of the lead cable has a frequency-dependent increase due to cable resistance, resulting in a rising step pulse and multiple reflections within the cable section. Hence, reaching the steady state takes much longer time than conventionally thought when long cables are used, in particular at very low and very high electrical conductivities. To determine the electrical conductivity accurately, the recording time should be taken after 10 multiple reflections within the probe and three multiple reflections within the lead cable.

Abbreviations: DC, direct current • EC, electrical conductivity • TDR, time domain electrical conductivity


   INTRODUCTION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
The bulk electrical conductivity (EC) of a soil is an important physical parameter for salinity assessment (Rhoades et al., 1989), studying solute transport (Kachanoski et al., 1992; Ward et al., 1994; Vanclooster et al., 1995), and correlating with hydraulic conductivity (Mualem and Friedman, 1991; Friedman and Seaton, 1998; Purvance and Andricevic, 2000). Contaminants also influence soil EC as they change the electrical properties of the pore fluid (Campanella and Weemees, 1990); however, soil water content plays an important role in these problems as well. Due to the ability to measure dielectric permittivity, which in turn can be used to estimate soil water content, and electrical conductivity in the same sampling volume, it is advantageous to measure soil EC based on TDR rather than the conventional DC resistivity method.

Time domain reflectometry is based on transmitting an electromagnetic pulse into a coaxial cable connected to a sensing waveguide and watching for reflections of this transmission due to impedance mismatches at the start and end of the sensing waveguide. The round-trip travel time in the sensing waveguide is related to the dielectric constant and the signal attenuation is associated with the EC of the material surrounding the sensing waveguide. In soil science, early attempts to measure soil EC with TDR used the magnitudes of first reflections from the start and end of the probe (Dalton et al., 1984; Topp et al., 1988; Zegelin et al., 1989), whose locations were somewhat arbitrary due to frequency-dependent attenuation. Later studies replaced the magnitude of the first end reflection with the steady-state reflection magnitude in the algorithm for calculating EC (Yanuka et al., 1988; Zegelin et al., 1989). These early algorithms suffered from several oversimplified assumptions, including the neglect of cable resistance, dielectric dispersion, and multiple reflections in a conductive medium. Topp et al. (1988) and Zegelin et al. (1989) presented the Giese–Tiemann method obtained from the thin sample theory (Giese and Tiemann, 1975) as an alternative method for EC measurement. The applicability of the thin sample theory was not ascertained but the experimental results indicated that it gives more reliable estimates than other methods. Nadler et al. (1991) rediscovered the Giese–Tiemann method, as pointed out by Heimovaara (1992) and Baker and Spaans (1993). Since then, the Giese–Tiemann method has become the standard equation for calculating EC from TDR measurements. At low frequency, as is the case for DC conductivity measurement, the thin sample theory is justified and the effects of dielectric dispersion and multiple reflections can be neglected. The cable resistance is not taken into account in the Giese-Tiemann method, however, and this assumption may become invalid when long cables are used in the field.

Heimovaara et al. (1995) observed that TDR EC measurements were increasingly underestimated as EC increased above 200 mS m–1. These errors were attributed to neglecting series resistance of the cable, connectors, and cable tester as a parameter in the EC calculation. They suggested modeling the coaxial cable and the sample as two resistors in series and the Giese–Tiemann method was modified accordingly. Calibration parameters involved in calculating the TDR EC include the geometric factor (probe constant) and the cable resistance (including resistance loss in connectors and cable tester). These parameters may be calibrated using least square fitting of TDR EC measurements in solutions of different concentrations to EC measurements made with a conventional conductivity meter. To expedite calibrations for probes of different lengths, Reece (1998) proposed a method that measures cable resistance directly. Unexplained differences in EC accuracy between the calibration method and the direct measurement method for cable resistance was observed, however, in Heimovaara et al. (1995) and Huisman and Bouten (1999). They suggested that the series resistors theory may be slightly incomplete and the fitting procedure corrects the deviation from theory. More recently, Castiglione and Shouse (2003) demonstrated, both theoretically and experimentally, that the formulation based on the series resistors model is incorrect; however, their disturbing arguments, while seeming logical at a first glance, were in fact troubled by wrong assumptions and incorrect data. The assumption that the steady-state voltage varies exponentially along the cable (their Eq. [17]) and the data (their Fig. 5b) showing the effect of cable resistance on TDR waveforms are not correct. In light of the wrongly claimed insufficiency of the series resistors model, they presented an intuitive method, in which the measured steady-state reflection coefficients are linearly scaled between –1.0 and 1.0 with respect to the range expanded by the measurements in air (EC = 0) and under the short-circuited condition (EC ={infty}) before applying the Giese–Tiemann method. Despite the lack of a theoretical basis, the effect of cable resistance is inactivated through this linear scaling process and the method is becoming widely accepted (e.g., Robinson et al., 2003). It should be pointed out, however, that the effect of cable resistance on the steady-state reflection coefficient has never been proven to be linear.


Figure 5
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  		Fig. 5. The estimated electrical conductivity ({sigma}est) using the fitted probe constant (ß) in three different methods compared with the numerically controlled true electrical conductivity ({sigma}true).

 
The EC measurement by TDR, as easy as it may seem, remains a controversial issue, particularly when long cables are used. It should be noted that TDR is a high-frequency measurement technique with frequency ranging from kiloHertz to gigaHertz. The pulse length (i.e., duration of a single step pulse) is in the order of several microseconds. When TDR is used for determining DC electrical conductivity, it can only work for cases where the time required to reach the steady state is less than the pulse length. The time required to reach steady state strongly depends on the cable resistance. An arbitrary "long" time is usually used without close examination of its legitimacy. No work has been done on the effect of recording time, particularly in the context of long cables. In this study, a comprehensive full waveform analysis and the DC analysis were conducted in a well-parameterized manner. The full waveform analysis was used to examine the theoretical validity of the series resistors model and Castiglione–Shouse method, and to investigate the effect of recording time on these methods. It will be shown that the series resistors model is theoretically sound; the unexplained observations and disputes in the literature may be explained by the time effect.


   THEORY
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Full-Waveform Analysis
While dielectric spectroscopy requires the full waveform, only the steady-state reflection magnitude is needed for determining EC. With the capability of modeling the full TDR waveform, however, the theoretical validity of DC methods can be objectively examined. The effect of recording time on the DC analysis can also be investigated numerically.

The wave phenomena in a TDR measurement include multiple reflection, dielectric dispersion, and attenuation due to conductive loss and cable resistance. Wave propagation models for TDR have been formulated in various forms (Feng et al., 1999; Lin, 2003), in which multiple reflections, dielectric dispersion, and conductive loss are taken into account. Neglecting cable resistance in these models was justified by the short lead cable used. Cable resistance becomes an important issue in practice, however, when long cables are used. To complete the TDR mathematical model, Lin and Tang (2007) formulated a resistance correction factor (A) within the modeling framework proposed by Lin (2003). The frequency-dependent resistance correction factor is put in a different form here to express the individual contributions of geometric impedance and surface resistivity of conductors.

The behavior of electromagnetic wave propagation in the frequency domain can be characterized by the propagation constant ({gamma}) and the characteristic impedance (Zc). The propagation constant controls the velocity and attenuation of electromagnetic wave propagation and the characteristic impedance controls the magnitude of reflection. The{gamma} and Zc, taking into account the cable resistance, can be written as (Lin and Tang, 2007)

Formula 1A[1a]

Formula 1B[1b]

Formula 1C[1c]
where c is the speed of light, {varepsilon}r* = {varepsilon}r – j{sigma}/(2{pi}f{varepsilon}0) is the complex dielectric permittivity (including the effect of dielectric permittivity {varepsilon}r and electrical conductivity {sigma}, in which {varepsilon}0 is the dielectric permittivity of free space), Zp is the geometric impedance (characteristic impedance in air), A is the (per-unit-length) resistance correction factor, j is the complex unit, {eta}0 = {surd}(µ0/{varepsilon}0) ~ 120{pi} is the intrinsic impedance of free space (in which {alpha}0 is the magnetic permeability of free space), {alpha}R (s–0.5) is the resistance loss factor (a function of the cross-sectional geometry and surface resistivity due to skin effect), and f is the frequency. If cable resistance is ignored (i.e., {alpha}R = 0), A becomes 1.0 and {gamma} and Zc have expressions identical to the nonresistance formulations (Feng et al., 1999; Lin, 2003). Each uniform section of a transmission line is characterized by its length, cross-sectional geometry, dielectric property, and cable resistance. These properties are parameterized by the length (L), geometric impedance (Zp), dielectric permittivity ({varepsilon}r*), and resistance loss factor ({alpha}R), as shown in Fig. 1a. Once these parameters are known or calibrated, TDR waveforms can be simulated using Eq. [1] and the modeling framework proposed by Lin (2003). The propagation constants and characteristic impedances of each uniform section are first determined by Eq. [1]. As illustrated in Fig. 1a, the input impedance at z = 0, i.e., Zin(0), represents the total impedance of the entire nonuniform transmission line. It can be derived recursively from the characteristic impedance and the propagation constant of each uniform section, starting from the terminal impedance ZL:

Formula 2[2]
where Zc,i, {gamma}i, and li, are the characteristic impedance, propagation constant, and length of each section, respectively, and ZL is the terminal impedance. A typical TDR measurement system uses an open loop (ZL = {infty}). The frequency response of the TDR sampling voltage V(0) can then be written in terms of the input impedance as

Formula 3[3]
where V(0) is the Fourier transform of the TDR waveform (vt); Vs is the Fourier transform of the TDR step input; Zs is the source impedance of the TDR instrument (typically Zs = 50 {Omega}), Zin(0) is the input impedance at z = 0, and H = Zin(0)/[Zin(0) + Zs] is the transfer function of the TDR response. The TDR waveform is the inverse Fourier transform of V(0). Inversion for transmission line parameters or material properties can be done based on this full waveform modeling.


Figure 1
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  		Fig. 1. (a) The multisection transmission line model of the time domain reflectometry (TDR) measurement system, in which each uniform section is characterized by the geometric impedance (Zp), resistance loss factor ({alpha}R), complex dielectric permittivity ({varepsilon}r*), and waveguide length (L). The transmission line is driven by a source voltage (Vs) with a source impedance (Zs) and terminated in a load (ZL). The input impedance (Zin) is defined as the ratio of line voltage (V) to the line current (I). (b) The associated direct current circuit model, in which Rs is the inner resistance (equal to the source impedance Zs), Rcable is the cable resistance, R is sample resistance, vs is the source voltage (in time domain), and v{infty} is the TDR steady-state voltage. (c) A typical TDR waveform showing definition of reflection coefficient ({rho}), where t0 is the roundtrip travel time in the probe section, and tcable is the roundtrip travel time in the cable section.

 
Direct Current Circuit Analysis
From basic circuit theory, the transmission line can be modeled as a lumped circuit when the wavelength is significantly greater than the electrical length. At zero frequency, the lumped circuit is shown in Fig. 1b, equivalent to the assumptions made by Heimovaara et al. (1995) and Reece (1998). The DC lumped circuit model includes the voltage source vs (double of the pulse step v0), the inner resistance Rs (equal to the source impedance Zs), and cable resistance Rcable (in fact, the combined series resistance of probe, cable, connector, and cable tester) and soil sample resistance R. The steady-state reflection voltage can be derived from circuit theory as

Formula 4[4]
where the sample resistance is related to the EC by

Formula 5[5]
in which Kp is a geometric factor. Substituting Eq. [5] into Eq. [4] and noting {rho}{infty} = (v{infty} – v0)/v0, in which v0 = 2vs since the source impedance is typically designed to be identical to the characteristic impedance of the connected transmission line, as shown in Lin (2003), the EC of the sample can be derived as a function of the steady-state reflection coefficient {rho}{infty}:

Formula 6[6]
where ß (=Kp/Rs) is a probe constant and k is the correction factor for cable resistance, called the cable correction factor (to be distinguished from the per-unit-length resistance correction factor A in Eq. [1]). The term Rcable/Rs(1 – {rho}{infty})/(1 + {rho}{infty}) is ≤1 (which can be proved by substituting Rcable from Eq. [11]), so the cable correction factor k can also be written as a power series:

Formula 7[7]

It should be noted that the cable correction factor k depends not only on Rcable but also on the EC of the sample, since it is a function of {rho}{infty}. The effect of cable resistance increases with increasing EC (i.e., as {rho}{infty} decreases).

To correctly determine the EC from a TDR measurement, both the probe constant ß and cable resistance Rcable need to be known. Giese and Tiemann (1975) analytically derived the TDR EC from transmission line and thin sample theory in the case of lossless cable (i.e., Rcable = 0):

Formula 8[8]
where {varepsilon}0 is the dielectric permittivity of free space, c is the speed of light, Zp is the geometric impedance of the probe, Zs is the source impedance, and L is the probe length. As Rcable approaches zero, the cable correction factor k in Eq. [7] becomes unity and Eq. [6] has an expression equivalent to the Giese–Tiemann equation (Eq. [8]). Equating Eq. [8] to Eq. [6] with Rcable = 0, the probe constant can be found as

Formula 9[9]
where the only unknown value in practice is the probe geometric impedance Zp, which can be analytically determined for coaxial and various multiconductor probes (Ball, 2002). The cable resistance depends on {alpha}R (a cable property), the cable length, and the geometric impedance. Their relationship is induced from full waveform simulations as

Formula 10[10]
where {alpha}R,i, Zp,i and Li are the resistance loss factor, geometric impedance, and transmission line length for each uniform section. The unknown values in Eq. [10] are {alpha}R,i and Zp,i.

Although the probe constant ß and cable resistance Rcable can be determined analytically from Eq. [9] and [10], Zp,i and {alpha}R,i are typically not known a priori. Using the full waveform propagation model, Zp,i and {alpha}R,i can be obtained (calibrated) from an inverse analysis of a single measurement in a sample with known electrical properties (such as air or pure water). Alternatively and more practically, Rcable can be directly determined from a measurement on a sample with known R, as suggested by Reece (1998). In the limiting case of a sample with R = 0 (i.e., TDR waveguide probe whose conductors are shorted together), the Rcable can be determined as

Formula 11[11]
where {rho}{infty},SC is the steady-state reflection coefficient of the measurement in which the conductors are shorted together. With known Rcable, the probe constant ß can be obtained using Eq. [6] and at least one calibration test in a salt solution with known EC.

The series resistors model should be theoretically sound according to the well-established circuit theory. Castiglione and Shouse (2003) presented an alternative approach for taking cable resistance into account, however, in which the steady-state reflection coefficients are linearly scaled between –1.0 and 1.0 with respect to the range expanded by the measurements in air (EC = 0) and the short-circuited condition (EC = {infty}):

Formula 12[12]
where {rho}scaled is the TDR measurement corrected for cable resistance by the scaling process; {rho}open and {rho}short are the reflection coefficients with the probe in open air and short-circuited, respectively. The value of {rho}scaled represents the TDR measurement as if there is no cable resistance, so the Giese–Tiemann equation (Eq. [8]) can be used for calculating the EC. Castiglione and Shouse (2003) claimed that the series resistors model is incorrect and Eq. [12] leads to better agreement with experimental results. It should be pointed out, however, that the scaling process is linear while the effect of cable resistance on the steady-state reflection coefficient will be shown to be nonlinear. We examined both the series resistors model and the Castiglione–Shouse method using full waveform analysis and experiments.


   MATERIALS AND METHODS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
The ability of the TDR wave propagation model to capture the resistance effect was first verified by several TDR measurements with a 30-m RG58A/U cable. The TDR measurements were made by attaching the TDR probe (12-cm two-rod probe with conductors 3 mm in diameter with 20-mm spacing) to a Campbell Scientific TDR 100 (Campbell Scientific, Logan, UT) via the 30-m-long lead cable and a SDMX multiplexer. Any uniform transmission line section can be parameterized by the length (L), geometric impedance (Zp), dielectric permittivity ({varepsilon}r*), and resistance loss factor ({alpha}R). One of the three parameters (L, Zp, or {varepsilon}r*) needs to be known so that the other two parameters and {alpha}R can be calibrated from a measured TDR waveform (Lin and Tang, 2007). With known lengths, the transmission line parameters (Zp, {varepsilon}r*, and {alpha}R) of the lead cable and multiplexer section were calibrated by a measurement with the lead cable open ended. The transmission line parameters (Zp, L, and {alpha}R) of the TDR probe were then calibrated by a measurement with the probe immersed in deionized water, whose dielectric property is known. Using the calibrated transmission line parameters, TDR waveforms were simulated and compared with measured waveforms for the probe in open air, immersed in tap water, and short-circuited. Time interval {Delta}t = 2.5 x 10–11 s and number of data points N = 65,536 were used in the numerical simulations (for details, see Lin and Tang, 2007). The resulting effective time window 0.5N{Delta}t = 8192 (40{Delta}t) = 8.2 x 10–6 s is slightly greater than the pulse length of 7 x 10–6 s in a TDR 100. The corresponding Nyquist frequency and frequency resolution are 20 GHz and 60 kHz, respectively. The Nyquist frequency is well above the frequency bandwidth of the TDR 100 and the long time window ensures that a steady state is obtained.

Using the verified TDR wave propagation model, the theoretical validity of the series resistors model and the Castiglione–Shouse method can be examined. The EC was numerically controlled and compared with that estimated from the synthetic waveforms using the Giese–Tiemann method, series resistors model, and Castiglione–Shouse method. The time window used for these numerical simulations was excessively large to ensure that a steady state was obtained and DC analysis was examined. As will be seen, the cable resistance can have a great effect on how the reflection approaches the steady state. Intermediate reflection plateaus at long times may be mistakenly taken as the steady-state reflection coefficient. The effect of recording time on the series resistors model and the Castiglione–Shouse method was investigated through a parametric study. Factors considered include lead cable length, probe length, probe impedance, and electrical properties of the material being tested. The simulation parameters used in the parametric study are listed in Tables 1 and 2. The resistance loss factor ({alpha}R) of the waveguide was set as 0.0 for all cases, since it has a negligible effect on the TDR waveform due to the short probe length.


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  		Table 1. Simulation parameters.

 

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  		Table 2. Cole–Cole{dagger} parameters for materials used in numerical simulations.

 
The numerical findings were verified by experimental data. Time domain reflectometry measurements were made on seven NaCl electrolytic solutions, with {sigma} varying from 0 to 0.15 S m–1, using the 30-m RG58A/U cable and 12-cm two-rod probe. The EC was measured independently with a standard EC meter (YSI-32, Yellow Springs Inc., Yellow Springs, OH). When directly determining Rcable using Eq. [11], the measurements were performed by shorting the cable end with a short wire. The resistance in the probe section was found to be negligible from Eq. [10] and a theoretical {alpha}R value computed from the probe geometry and conductor property. The cross-section of the probe is much larger than that of the coaxial cable. Shorting the probe end with a wire may introduce extra resistance. We suggest shorting the cable end with a short wire or the probe end with a metal plate.


   RESULTS AND DISCUSSION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Effect of Cable Resistance on Time Domain Reflectometry Waveforms
The effect of cable resistance on TDR waveforms is illustrated by TDR measurements with a 30-m RG58A/U cable and modeled by the full waveform analysis. The characteristics of the lead cable (Zp = 77.5 {Omega}, {varepsilon}r* = 1.95, and {alpha}R = 19.8 s–0.5) were backcalculated from the measured waveform with the lead cable open ended, while the characteristics of the probe (Zp = 290 {Omega}, L = 0.126 m, and {alpha}R = 153 s–0.5) were obtained from a measurement with the probe immersed in deionized water. Figure 2a shows the measured and predicted waveforms using the backcalculated parameters for the probe in open air, immersed in tap water, and short-circuited. The full waveform analysis takes into account the multiple reflections, dielectric dispersion, and attenuation due to conductive loss and cable resistance altogether. The excellent match between the measured and predicted waveforms validates the TDR wave propagation model and the calibration by full-waveform inversion. The predicted waveforms in which cable resistance is ignored are also shown in Fig. 2a for comparison. Of most importance to EC measurements is how cable resistance affects the steady-state response. As depicted in Fig. 2a, cable resistance gives rise to an increase in the steady-state response, causing an underestimation of EC if cable resistance is not taken into account. The amount of increase in the steady-state response depends on the EC, with no increase when EC = 0 (i.e., probe in open air) and maximum increase when EC = {infty}. Therefore, the TDR EC measurements are increasingly underestimated as EC increases, as also observed by Heimovaara et al. (1995) and Reece (1998). This monotonic behavior is different from that revealed by Castiglione and Shouse (2003) in their Fig. 5b, reproduced in Fig. 2b for comparison. The reflection coefficient in air (i.e., EC = 0) should be 1.0 regardless of the lead cable length, as also suggested by Eq. [6]. The data shown in Castiglione and Shouse (2003) seems abnormal. The error was probably caused by the data acquisition program, and was overlooked due to the misconception that the long-time reflection coefficient is reduced in absolute value due to cable attenuation (i.e., a positive long-time reflection coefficient decreases at low EC, while a negative long-time reflection coefficient increases at high EC, as shown in Fig. 2b).


Figure 2
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  		Fig. 2. Effect of cable resistance on time domain reflectometry (TDR) waveforms for a variety of electrical conductivities ({sigma}): (a) measured TDR waveforms compared with that predicted by the full waveform model in this study; (b) measured TDR waveforms in Fig. 5b of Castiglione and Shouse (2003).

 
In addition to the steady-state response, it is also interesting to note how cable resistance affects the time required to reach the steady state. The characteristic impedance of the cable used is actually 55 {Omega}, not precisely 50 {Omega}. The unmatched cable gives rise to multiple reflections within the cable section, as can be observed from the reflections around 560 ns in Fig. 2a. Even if the cable has a nominal characteristic impedance perfectly matched with the source impedance of the TDR device (typically 50 {Omega}), the characteristic impedance of the cable is in fact a function of frequency and cable resistance, as suggested in Eq. [1]. This is evidenced by the rising step pulse, as shown in Fig. 2a and illustrated in Fig. 1. Therefore, the multiple reflections within the cable section are inevitable. The magnitude of the multiple reflections within the cable depends not only on cable resistance but also on the EC. It is most prominent when the probe is in open air or shorted. The rising plateau of the step pulse and the rise time of the reflected pulse increase as {alpha}R or cable length increases. Hence, it takes a much longer time to reach steady state for long cables. The reflection coefficient beyond 400 ns may be mistakenly taken as the steady state if the waveform is not recorded long enough, as shown in Fig. 2a. This problem has been overlooked and may have a significant effect on TDR EC measurements.

Theoretical Assessment of Direct Current Analysis Methods (without Time Error)
Using the verified TDR wave propagation model, the theoretical validity of the series resistors model and the Castiglione–Shouse method can be examined. A very long time (8.2 x 10–6 s) was used in the numerical simulations to ensure that the assessment is performed under the true steady-state responses. The deficiency of the scaling process proposed by Castiglione and Shouse (2003) is illustrated in Fig. 3. To enhance visual illustration, a long RG-58 cable (200 m) was used for the numerical simulation. The steady-state reflection coefficient with the 200-m RG-58 cable ({alpha}R = 19.8 s–0.5) is plotted against that without cable loss ({alpha}R = 0 s–0.5), as shown by the solid line in Fig. 3. This curve is not a linear line and the scaled line by applying Eq. [11] is a nonlinear line rather than the 1:1 linear line. This disparity reveals that the Castiglione–Shouse method is correct only for EC = 0 and EC = {infty}, since the effect of cable resistance on the steady-state reflection coefficient is nonlinear while the scaling process is linear.


Figure 3
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  		Fig. 3. Illustration of the nonlinear relationship between the steady-state reflection coefficient with 200-m RG-58 cable and that without cable resistance, in which {rho}scaled is the scaled reflection coefficient by the Castiglione–Shouse method (Eq. [12]).

 
In Fig. 4, the electrical conductivity in the measurement system was numerically controlled and compared with that estimated from the synthetic waveforms using three different DC analysis methods. The result shows that the series resistor model is theoretically correct (if the true steady-state response is obtained), while the Giese–Tiemann method and Castiglione–Shouse methods result in underestimation and overestimation, respectively. The overestimation by the Castiglione–Shouse method linearly increases with EC, while the underestimation by the Giese–Tiemann method nonlinearly increases with EC. In Fig. 4, the probe constant ß is determined by Eq. [9], which is only a function of probe geometry and independent of cable resistance. If the probe constant ß is obtained using least square fitting of TDR EC measurements in salt solutions of different concentrations to conductivity measurements made with a conventional conductivity meter, the result becomes that shown in Fig. 5. The linear overestimation by the Castiglione–Shouse method is completely compensated for by the fitted probe constant, while the nonlinear underestimation by the Giese–Tiemann method is only minimized in a least square sense, resulting in slight overestimation at low EC and underestimation at high EC in the fitting range. It should be noted that the fitted probe constant depends not only on the probe geometry but also on the cable resistance. Hence, probes with the same probe geometry but different cable length should be individually calibrated when the Castiglione–Shouse method or the Giese–Tiemann method are used. This is not very practical for field monitoring with many probes. In practice, the series resistors model should be used. It has a unique probe constant for each type of probe, and the cable resistance can be easily determined by Eq. [11] without further calibrations.


Figure 4
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  		Fig. 4. The estimated electrical conductivity ({sigma}est) using the actual probe constant in three different methods compared with the numerically controlled true electrical conductivity ({sigma}true).

 
Effect of Recording Time
The assessment of DC analysis methods assumes that steady state is obtained. In practice, an arbitrary "long" time is usually assumed for the steady state without close examination of its legitimacy. The parametric study shows that the time required to reach the steady state depends on the cable resistance, the electrical properties of the medium, and probe characteristics. In the case of negligible cable resistance, Fig. 6 shows how EC, probe characteristics, and dielectric permittivity affect the time required to reach the steady state. The recording time is expressed as the time that includes multiples of roundtrip travel time in the probe section (t0). The reflection voltage at a very long time (8.2 x 10–6 s, slightly greater than the pulse length of 7 x 10–6 s in a TDR 100) was used to represent v{infty}. The time required to reach the steady state increases with decreasing EC, decreasing characteristic impedance, and increasing dielectric constant. But without cable resistance, reflection coefficients all converge to the steady state (vt/v{infty} = 1) in fewer than 10 multiple reflections within the probe, a time often used to represent the steady state in practice.


Figure 6
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  		Fig. 6. Examples showing how (a) electrical conductivity {sigma}, (b) geometric impedance Zp and length L, and (c) dielectric permittivity affect the time required to reach the steady state, with time expressed as the time that includes multiples of roundtrip travel time in the probe section (t0).

 
For the 12-cm probe, Fig. 7 shows the effect of recording time for different lengths of RG58 cable and electrical conductivities. The time required to reach the steady state increases with cable resistance. But the way the reflection coefficient approaches the steady state strongly depends on the EC, as also suggested by Fig. 2. Two extreme cases, the probe in open air (EC = 0) and the probe with conductors shorted together (EC = {infty}), are shown in Fig. 7a and 7c. Figure 7b shows the results for two electrical conductivities in between the two extreme cases. At high EC, the ratio vt/v{infty} decreases monotonically and gradually approaches the steady state, while at low EC, vt/v{infty} increases slightly above 1.0 and then quickly approaches the steady state. The medium EC is least affected by the recording time. The definition of "high," "medium," and "low" EC here means EC that results in reflection coefficient near –1.0, 0, and 1.0, respectively. This property depends on the probe characteristics (i.e., geometric impedance and probe length), as can be inferred from Eq. [9]. For example, the EC may be considered "high" for a long probe but is considered "medium" for a short probe. When the waveguide is short-circuited, it takes a much longer time to reach the steady state even with small cable resistance, as shown in Fig. 7a. Hence, cautions should be taken when determining the cable resistance from the TDR measurement of a short-circuited probe using Eq. [11].


Figure 7
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  		Fig. 7. Recording time required for the voltage (vt) to reach steady state (v{infty}) for probes that are (a) short-circuited, (b) in water of two electrical conductivities, and (c) in open air.

 
Four approaches may be used to determine the TDR EC from the steady-state response: (i) using the series resistor model with cable resistance directly measured by the short-circuited probe (Eq. [11]) and a probe constant fitted to calibration tests; (ii) using the series resistor model with both cable resistance and the probe constant fitted to calibration tests; (iii) using the Castglione–Shouse method with an actual probe constant determined by Eq. [9] or calibrated with a very short cable; and (iv) using the Castiglione–Shouse method with a probe constant fitted to calibration tests. Figure 8 reveals the effect of recording time on estimated EC using these four different approaches, in which the estimated EC of any recording time is expressed as {sigma}t. In this illustration, calibrations were performed with EC ranging from 0 to 0.2 S m–1 with 0.02 S m–1 spacing. The fitted probe constant is the probe constant that results in the minimum least square error between estimated and actual EC in the fitting range. It coincides with the theoretical probe constant only when the series resistors model is used and the recording time is representative of the steady state. As shown in Fig. 8, the estimated EC by the series resistors model eventually converges to the true value, but the rate of convergence depends on the calibration method, the cable length, and the EC. The results for fitting both the probe constant and cable resistance (Fig. 8b) increase the estimation accuracy slightly for each recording time, but the convergence trend is similar to that for fitting only the probe constant, with cable resistance directly measured by the short-circuited probe (Fig. 8a). The time window required to have accurate estimation of EC increases with cable length, as expected, and is generally less than that required to reach the steady state due to the fitted probe constant. Unlike what Fig. 7b may suggest, however, high EC converges to the true value faster than low EC does. This is due to the fact that TDR EC measurements are affected by the recording time not only when making measurements but also when fitting the probe constant and cable resistance. As shown in Fig. 7, the TDR response approaches the steady state in different ways for different electrical conductivities. Depending on the fitting range and data sampling, the fitted probe constant may work in favor of some electrical conductivities. But of most importance is how to obtain accurate estimation for all electrical conductivities. The recording time is expressed as the time that includes multiples of roundtrip travel time in the probe section (t0) in Fig. 8. The same result is plotted in Fig. 9 with recording time expressed as multiples of roundtrip travel time in the lead cable (tcable). Except for the case of a very short lead cable, accurate estimation of EC can be obtained with a recording time greater than 3tcable, regardless of the fitting range for the probe constant. The characteristic impedance of the lead cable increases with increasing cable length, giving rise to multiple reflections within the lead cable, as shown in Fig. 2a. The convergence of EC estimation is governed by multiple reflections in the sensing probe for a short lead cable, while it becomes dominated by multiple reflections in the lead cable for a long lead cable. A simple guideline for selecting an appropriate recording time can be drawn from the parametric study. To determine the EC accurately, the recording time should be taken after 10 multiple reflections within the probe and three multiple reflections within the lead cable. Errors found in the literature using the series resistor model with cable resistance directly measured by the short-circuited probe may be explained by the time effect, an imperfect shorting element, or the wrong acquisition program.


Figure 8
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  		Fig. 8. The effect of recording time (t), expressed as the time that includes multiples of roundtrip travel time in the probe section (t0), on the estimated electrical conductivity ({sigma}t) using the series resistors model with (a) cable resistance Rcable measured and probe constant ß fitted, and (b) Rcable and ß fitted, or using the Castiglione–Shouse method with (c) actual ß determined, and (d) ß fitted.

 

Figure 9
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  		Fig. 9. The effect of recording time (t), expressed as multiples of roundtrip travel time in the lead cable (tcable), on the estimated electrical conductivity ({sigma}t) using the series resistors model with (a) cable resistance Rcable measured and probe constant ß fitted, (b) Rcable and ß fitted, or using the Castiglione–Shouse method with (c) actual ß determined, and (d) ß fitted.

 
The effect of recording time on the Castiglione–Shouse method is shown in Fig. 8c, 8d, 9c and 9d for comparison. If the probe constant is fitted (Fig. 8d and 9d), the estimated EC by the Castiglione–Shouse method also converges to the true value with reduced time effect. But if the actual probe constant is determined and used (Fig. 8c and 9c), it takes a much longer time for the estimated EC by the Castiglione–Shouse method to become invariant with time. When the recording time is >6tcable, the estimated EC still gradually decreases with time. The asymptotic value overestimates the EC. The overestimation increases with cable length and the asymptotic {sigma}t/{sigma}true is independent of the EC, as also suggested in Fig. 4.

Experimental Verifications
To further verify the numerical findings, a few TDR measurements were made on NaCl electrolytic solutions, with {sigma} varying from 0 to 0.15 S m–1, using the 30-m RG58A/U cable and 12-cm two-rod probe. The TDR measurements were interpreted by the Giese–Tiemann method, Castiglione–Shouse method, and the series resistors model with cable resistance directly measured by the short-circuited probe. The steady-state responses were recorded at the time around 4.5tcable that includes 80 multiple reflections within the probe, satisfying the criteria for the steady state. The same data were used for calibrating the probe constant. Figure 10 compares the TDR EC with that measured by a conventional EC meter. The results are in good agreement with that found in Fig. 4 and 5. When the probe constant is fitted, both the series resistors model and the Castiglione–Shouse method provide accurate EC measurements in the full EC range, while the Giese–Tiemann method slightly overestimates at low EC and underestimates at high EC in the fitting range. The fitted probe constants are equal to the actual probe constant when the lead cable is very short. For long lead cables, the fitted probe constant is identical to the actual one only in the series resistors model. If the actual probe constant is used, linear overestimation by the Castiglione–Shouse method and nonlinear underestimation by the Giese–Tiemann method are obvious, agreeing well with the numerical findings.


Figure 10
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  		Fig. 10. Electrical conductivity measured by time domain reflectometry ({sigma}TDR) compared with that measured by a YSI conductivity meter ({sigma}YSI) using three different models with the probe constant ß measured or fitted.

 

   CONCLUSIONS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
Cable resistance and recording time are important factors in TDR EC measurements when long lead cables are used. In this study, a rigorous full waveform analysis and the DC analysis were used to show the correct method for taking cable resistance into account and guidelines for selecting the proper recording time.

At EC = 0, the steady-state response is not affected by the cable resistance. But as EC increases, cable resistance gives rise to a growing increase in the steady-state response. Hence, the TDR EC measurements are increasingly underestimated by the Giese–Tiemann method as EC increases. This effect of cable resistance can be precisely captured and taken into account by the series resistors model, which is theoretically sound according to the well-established circuit theory and verified by the full waveform analysis. The alternative Castiglione–Shouse method, in which the measured steady-state reflection coefficients are linearly scaled between –1.0 and 1.0 with respect to the range expanded by the measurements in air (EC = 0) and under the short-circuited condition (EC = {infty}), on the other hand, was shown to be incorrect. This can be explained by the fact that the effect of cable resistance on the steady-state reflection coefficient is nonlinear while the scaling process is linear. The error using the Castiglione–Shouse method may be completely compensated for if the probe constant ß is obtained using least square fitting of TDR EC measurements to known EC values or to EC measurements made with a conventional conductivity meter. The fitted probe constant then becomes a function of cable length (resistance).

The cable resistance affects not only the steady-state response but also the time required to approach the steady state. The characteristic impedance of the lead cable has a frequency-dependent increase due to cable resistance, resulting in a rising step pulse and multiple reflections within the cable section. Hence, it takes a much longer time than conventionally thought to reach the steady state when long cables are used, in particular at very low and very high EC. To determine the electrical conductivity accurately, the recording time should be taken after 10 multiple reflections within the probe and three multiple reflections within the lead cable.


   ACKNOWLEDGMENTS
 
This research is supported in part by the National Science Council of ROC under Contract no. 94-2211-E-009-044 and the MOU-ATU program at National Chiao Tung University. This support is greatly appreciated.


   NOTES
TOP
 NOTES
ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Permission for printing and for reprinting the material contained herein has been obtained by the publisher.

Received for publication November 7, 2006.


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




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