Theoretical Assessment of EC Measurement

Material and method

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. TDR measurements were made by attaching the TDR probe (12-cm two rod probe with conductors 3 mm in diameter and 20 mm in spacing) to a Campbell Scientific TDR 100 via the 30m-long lead cable and a SDMX multiplexer. Any uniform transmission line section can be parameterized by the length (L), geometric impedance (Zp), dielectric permittivity (εr*), and resistance loss factor (αR). One of the three parameters (L, Zp, εr*) needs to be known so that the other two parameters and αR can be calibrated from a measured TDR waveform (Lin and Tang, 2007).

With known lengths, the transmission line parameters (Zp, εr*, and αR) 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 αR) of the TDR probe were then calibrated by a measurement with the probe immersed in de-ionized water, whose dielectric property is known. Using the calibrated transmission line parameters, TDR waveforms were simulated and compared with measured waveforms for the probe open in air, immersed in tap water, and short-circuited. Time interval dt = 2.5×10^-11 sec and time window 0.5Ndt = 8192×40 dt = 8.2×10^-6 sec (slightly greater than the pulse length of 7×10^-6 sec in a TDR 100) were used in the numerical simulations. The corresponding Nyquist frequency and frequency resolution are 20 GHz and 60 kHz, respectively. The Nyquist frequency is well above the frequency bandwidth of TDR 100 and the long time window ensures that the steady state is obtained.

Using the verified TDR wave propagation model, the theoretical validity of the series resistors model and Castiglione-Shouse method can be examined. The electrical conductivity is numerically controlled and compared with that estimated from the synthetic

method. The time window used for above numerical simulations is excessively large to ensure that the steady state is obtained and DC analysis is 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 is investigated through a parametric study. Factors considered include lead cable length, probe length, probe impedance, and electrical properties of the material under test. The simulation parameters used in the parametric study are listed in Table 4-1 and Table 4-2. The resistance loss factor (αR) of the waveguide is set as 0.0 for all cases since it has a negligible effect on the TDR waveform due to the short probe length.

The numerical findings were verified by experimental data. Time domain reflectometry measurements were made on 7 NaCl electrolytic solutions, with σ varying from 0 to 0.15 S/m, using the 30-m RG58A/U cable and 12-cm two-rod probe. The electrical conductivity was measured independently with a standard EC meter (YSI-32 Yellow Spring Int. Inc., Yellow Spring, OH). When directly determining Rcable using Eq. [2-46], the measurements were performed by shorting the cable end with a short wire. The resistance in the probe section was found negligible from Eq. [2-45] and theoretical αR 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. It is suggested to shorten the cable end with a short wire or the probe end with a metal plate.

Table 4-1 Simulation parameters

Section Parameters Range

σ, S m^-1 0.005 ~ 0.2

εr Tap water, Ethanol alcohol, and Silt loam^†

Geometric impedance Zp , Ω 150 ~ 300

Length L, m 0.1 ~ 0.3

Waveguide

αR, sec^-0.5 0

σ, S m^-1 0

εr 1.95

Geometric impedance Zp , Ω 77.5

length, m 0 ~ 200

Lead cable

αR, sec^-0.5 0, 19.8

† Referring to the Cole-Cole parameters listed in Table 2

Table 4-2 Cole-Cole parameters for material used in numerical simulations

Material εdc ε_∞ f _rel ξ

Tap water 79.9 4.22 17*10⁹ 0.0125

Ethanol alcohol 25.2 4.5 0.78*10⁹ 0.0

Silt loam 26.0 18.0 0.2*10⁹ 0.01

Effect of Cable Resistance on TDR Waveforms

The effect of cable resistance on TDR waveform 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 Ω, εr* = 1.95, and αR = 19.8 sec^-0.5) were back calculated from the measured waveform with the lead cable open-ended, while the characteristics of the probe (Zp

= 290 Ω, L =0.126 m, and αR =153 sec^-0.5) were obtained from a measurement with the probe immersed in de-ionized water. Fig. 4-2(a) shows the measured waveforms and predicted waveforms using the back calculated parameters for the probe open in 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. 4-2(a) for comparison.

Of most importance to EC measurements is how cable resistance affects the steady state response. As depicted in Fig. 4-2(a), 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 open in air) and maximum increase when EC = ∞. 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. 5(b), reproduced in Fig. 4-2(b) 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. [4-3]. The data shown in Castiglione and Shouse (2003) seems abnormal. The error was most likely caused by the data acquisition program, and was overlooked due to the misconception that long-time reflection coefficient is reduced in absolute value due to cable attenuation (i.e. positive long-time reflection coefficient decreases at low EC, while negative long-time reflection coefficient increases at high EC, as shown in Fig. 4-2(b)).

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 Ω, not precisely 50 Ω. The unmatched cable gives rise to multiple reflections within the cable section, as can be observed from the reflections around 560 ns in

the source impedance of the TDR device (typically 50 Ω), the characteristic impedance of the cable is in fact a function of frequency and cable resistance as suggested in Eq. [2-51]. This is evidenced by the rising step pulse, as shown in Fig. 4-2(a) and illustrated in Fig. 4-1.

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 electrical conductivity. It is most prominent when the probe is open in air or shorted. The rising plateau of the step pulse and the rise time of the reflected pulse increase as αR or cable length increases. Hence, it takes 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. 4-2(a). This problem has been overlooked and may have significant effect on TDR EC measurements.

0 100 200 300 400 500 600 700 800 900 -1.5

-1 -0.5 0 0.5 1 1.5

Time, ns

ρ

Measured

Predicted (α_R = 19.8 sec^-0.5) Predicted (α_R = 0)

Open

σ = 0.046 S/m

Shorted (a)

Misjudged steady state

Fig. 4-2 Effect of cable resistance on TDR waveforms: (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).

(b)

Theoretical Assessment of DC 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 (8.2×10^-6 sec) 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. 4-3. To enhance visual illustration, a long RG-58 cable (200 m) was used for the numerical simulation. The steady state reflection coefficient with 200-m RG-58 cable (αR = 19.8 sec^-0.5) is plotted against that without cable loss (αR = 0 sec^-0.5), as shown by the solid line in Fig. 4-3. This curve is not a linear line and the scaled line by applying Eq. [2-46] 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 = ∞ since the effect of cable resistance on the steady state reflection coefficient is nonlinear while the scaling process is linear.

In Fig. 4-4, 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 method 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. 4-4, the probe constant β is only a function of probe geometry and independent of cable resistance. If the probe constant β 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. 4-5. 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 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 the probe geometry but also the cable resistance. Hence, probes with the same probe geometry but different cable length should be individually calibrated when the Castiglione-Shouse method and the Giese-Tiemann method are used. This is not very practical for field monitoring with many probes. In practice, the series resisters model should be used. It has a unique probe constant for each type of probe. The cable resistance can be easily determined by Eq. [2-46] without further calibrations.

-1 -0.5 0 0.5 1

-1 -0.5 0 0.5 1

ρ_∞ (α_R = 0) ρ ∞ (α R = 19.8 sec-0.5 )

1:1 line

ρ with α_R = 19.8 sec^-0.5 linear scale line

ρ_scaled

Fig. 4-3 Illustration of the nonlinear relationship between the steady state reflection coefficient with 200-m RG-58 cable and that without cable resistance.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

σ_true, S m^-1 σ est, S m-1

1:1

Giese - Tiemann Series Resistors Castiglione - Shouse

Zp = 300 Ω L = 0.12m Cable length = 200 m α_R=19.8 sec^-0.5

Fig. 4-4 The estimated EC using the actual probe constant in three different methods compared with the numerically-controlled true EC.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 -0.05

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

σtrue, S m^-1 σ est, S m-1

1:1

Giese-Tiemann, β fitted Series Resistors

Castiglione and shouse, β fitted

Zp = 300 Ω L = 0.12m Cable length = 200 m α_R=19.8 sec^-0.5

Fig. 4-5 The estimated EC using the fitted probe constant in three different methods compared with the numerically-controlled true EC.