Effect of Cable Resistance on Time Domain Refl ectometry Waveforms
The effect of cable resistance on TDR waveforms is illus-trated 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 s−0.5) were backcalculated 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 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 refl ections, dielectric dispersion, and attenuation due to conductive loss and cable resistance altogether. The excellent match between the mea-sured and predicted waveforms validates the TDR wave propa-gation 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 impor-tance to EC measurements is how cable resisimpor-tance 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 = ∞. 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 refl ection coef-fi cient 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 over-looked due to the misconception that the long-time refl ection coeffi cient is reduced in absolute value due to cable attenuation (i.e., a positive long-time refl ection coeffi cient decreases at low EC, while a negative long-time refl ection coeffi cient increases at high EC, as shown in Fig. 2b).
In addition to the steady-state response, it is also interest-ing 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 refl ections within the cable section, as can be observed from the refl ections 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 (typi-cally 50 Ω), 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 refl ec-tions within the cable section are inevitable. The magnitude of the multiple refl ections 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 refl ected pulse increase as R or cable length increases. Hence, it takes a much longer time to reach steady state for long cables. The refl ection coeffi cient 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 signifi cant effect on TDR EC measurements.
Theoretical Assessment of Direct Current Analysis Methods (without Time Error)
Using the verifi ed TDR wave propagation model, the theo-retical validity of the series resistors model and the Castiglione–
Shouse method can be examined. A very long time (8.2 × 10−6 s) was used in the numerical simulations to ensure that the assessment is performed under the true steadystate responses. The defi -ciency of the scaling process proposed by Castiglione and Shouse (2003) is illustrated in Fig. 3. To enhance visual illustration, a long Table 1. Simulation parameters.
Section Parameter Range
Waveguide
electrical conductivity (σ), S/m 0.005 ∼ 0.2
dielectric permittivity (εr) Tap water, ethanol, and silt loam†
geometric impedance (Zp), Ω 150 ∼ 300
† Referring to the Cole–Cole parameters listed in Table 2.
Table 2. Cole–Cole† parameters for materials used in numeri-cal simulations. where εr(f) is the complex dielectric permittivity, εdc is the di-electric constant at zero frequency, ε∞ is the dielectric constant at infi nite frequency, frel is the relaxation frequency, α is a param-eter characterizing a spread in the relaxation frequencies, j is the complex unity √(−1), and f is the frequency.
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RG-58 cable (200 m) was used for the numerical simulation. The steady-state refl ection coeffi cient with the 200-m RG-58 cable (αR = 19.8 s−0.5) is plotted against that without cable loss (αR = 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 = ∞, since the effect of cable resistance on the steady-state refl ection coeffi cient is nonlin-ear while the scaling process is linnonlin-ear.
In Fig. 4, the electrical conductivity in the measurement system was numerically con-trolled 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 overestima-tion 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 con-stant β is determined by Eq. [9], which is only a function of probe geometry and indepen-dent of cable resistance. If the probe constant β is obtained using least square fi tting of TDR EC measurements in salt solutions of differ-ent concdiffer-entrations to conductivity measure-ments 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 fi tted probe constant, while the non-linear underestimation by the Giese–Tiemann
Fig. 2. Effect of cable resistance on time domain refl ectometry (TDR) waveforms for a variety of electrical conductivities (σ): (a) measured TDR waveforms com-pared with that predicted by the full waveform model in this study; (b) mea-sured TDR waveforms in Fig. 5b of Castiglione and Shouse (2003).
Fig. 3. Illustration of the nonlinear relationship between the steady-state refl ection coeffi cient with 200-m RG-58 cable and that without cable resistance, in which ρscaled is the scaled refl ec-tion coeffi cient by the Castiglione–Shouse method (Eq. [12]).
Fig. 4. The estimated electrical conductivity (σest) using the actual probe constant in three different methods compared with the numerically controlled true electrical conductivity (σtrue).
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method is only minimized in a least square sense, resulting in slight overestimation at low EC and underestimation at high EC in the fi tting range. It should be noted that the fi tted probe constant depends not only on the probe geometry but also on the cable resistance. Hence, probes with the same probe geom-etry 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 fi eld 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.
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 exami-nation 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 permit-tivity 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 refl ection voltage at a very long time (8.2 × 10-6 s, slightly greater than the pulse length of 7 × 10-6 s in a TDR 100) was used to rep-resent v∞. The time required to reach the steady state increases with decreasing EC, decreasing characteristic impedance, and increasing dielectric constant. But without cable resistance, refl ection coeffi cients all converge to the steady state (vt/v∞
= 1) in fewer than 10 multiple refl ections within the probe, a time often used to represent the steady state in practice.
For the 12-cm probe, Fig. 7 shows the effect of recording time for different lengths of RG58 cable and electrical conduc-tivities. The time required to reach the steady state increases with cable resistance. But the way the refl ection coeffi cient
Fig. 5. The estimated electrical conductivity (σest) using the fi tted probe constant (β) in three different methods compared with the numerically controlled true electrical conductivity (σtrue).
Fig. 6. Examples showing how (a) electrical conductivity σ, (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 mul-tiples of roundtrip travel time in the probe section (t0).
Fig. 7. Recording time required for the voltage (vt) to reach steady state (v∞) for probes that are (a) short-circuited, (b) in water of two electrical conductivities, and (c) in open air.
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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
= ∞), 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∞ decreases monotonically and gradually approaches the steady state, while at low EC, vt/v∞ increases slightly above 1.0 and then quickly approaches the steady state. The medium EC is least affected by the record-ing time. The defi nition of “high,” “medium,” and “low” EC here means EC that results in refl ection coeffi cient 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 con-sidered “high” for a long probe but is concon-sidered “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].
Four approaches may be used to determine the TDR EC from the steady-state response: (i) using the series resis-tor model with cable resistance directly measured by the shorcircuited probe (Eq. [11]) and a probe constant fi t-ted to calibration tests; (ii) using the series resistor model with both cable resistance and the probe constant fi tted 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 fi tted to calibration tests. Figure 8 reveals the effect of record-ing time on estimated EC usrecord-ing these four different approaches, in which the estimated EC of any recording time is expressed as σt. 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 fi tted probe constant is the probe constant that results in the minimum least square error between estimated and actual EC in the fi tting 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 fi tting 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 fi tting 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 fi tted 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 fi tting 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 fi tting range and data sampling, the fi tted 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 fi tting range for the probe constant. The characteristic impedance of the lead cable increases with increasing cable length, giving rise to mul-tiple refl ections within the lead cable, as shown in Fig. 2a. The con-vergence of EC estimation is governed by multiple refl ections in the sensing probe for a short lead cable, while it becomes dominated by multiple refl ections 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 refl ections within the probe and three multiple refl ections within the lead cable. Errors
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 esti-mated electrical conductivity (σt) using the series resistors model with (a) cable resistance Rcable measured and probe constant β fi tted, and (b) Rcable and β fi tted, or using the Castiglione–Shouse method with (c) ac-tual β determined, and (d) β fi tted.
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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.
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 fi tted (Fig. 8d and 9d), the estimated EC by the Castiglione–Shouse method also con-verges 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 esti-mated EC still gradually decreases with time.
The asymptotic value overestimates the EC. The overestimation increases with cable length and the asymptotic σt/σtrue is independent of the EC, as also suggested in Fig. 4.
Experimental Verifi cations
To further verify the numerical fi ndings, a few TDR measurements were made on NaCl electrolytic solutions, with σ varying from 0 to 0.15 S m−1, using the 30-m RG58A/U cable and 12-cm two-rod probe. The TDR measure-ments 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 refl ections within the probe, satisfying the cri-teria 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 fi tted, 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 fi tting range. The fi tted probe constants are equal to the actual probe constant when the lead cable is very short. For long lead cables, the fi tted 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 fi ndings.
CONCLUSIONS
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.
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 (σt) using the series resistors model with (a) cable resistance Rcable measured and probe constant β fi t-ted, (b) Rcable and β fi tted, or using the Castiglione–Shouse method with (c) actual β determined, and (d) β fi tted.
Fig. 10. Electrical conductivity measured by time domain re-fl ectometry (σTDR) compared with that measured by a YSI conductivity meter (σYSI) using three different models with the probe constant β measured or fi tted.
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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 accord-ing to the well-established circuit theory and verifi ed by the full waveform analysis. The alternative Castiglione–Shouse method, in which the measured steadystate refl ection coeffi -cients are linearly scaled between −1.0 and 1.0 with respect to
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 accord-ing to the well-established circuit theory and verifi ed by the full waveform analysis. The alternative Castiglione–Shouse method, in which the measured steadystate refl ection coeffi -cients are linearly scaled between −1.0 and 1.0 with respect to