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, electrical properties of the medium, and probe characteristics. In the case of negligible cable resistance, Fig. 4-6 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 (Δt). The reflection voltage at a very long time (8.2×10^-6 sec,

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∞ = 1) in less than 10 multiple reflection within the probe, a time often used to represent the steady state in practice.

For the 12-cm probe, Fig. 4-7 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 reflection coefficient approaches the steady state strongly depends on the EC, as also suggested by Fig. 4-2. Two extreme cases, probe open in air (i.e. EC =0) and probe with conductors shorted together (EC = ∞), are shown in Fig. 4-7a and Fig. 4-7c. Fig. 4-7b 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 over 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). 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 much longer time to reach the steady state even with small cable resistance, as shown in Fig. 4-7a. Hence, cautions should be taken when determining the cable resistance from the TDR measurement of short-circuited probe using Eq. [2-46].

Four approaches may be used to determine the TDR EC from the steady state response:

(a) using the series resistor model with cable resistance directly measured by the short-circuited probe and probe constant fitted to calibration tests, (b) using the series resistor model with both cable resistance and probe constant fitted to calibration tests, (c) using the

Castglione-Shouse method with actual probe constant or calibrated with a very short cable, and (d) using the Castiglione-Shouse method with probe constant fitted to calibration tests.

Fig. 4-8 reveals the effect of recording time on estimated EC using 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 with 0.02 S/m spacing. The fitted probe constant is the probe constant that results in minimum least square error between estimated EC and actual EC in the fitting range. It coincides with the theoretical probe constant only when the series resisters model is used and the recording time is representative of the steady state. As shown in Fig. 4-8, the estimated EC by the series resisters model eventually converges to the true value, but the rate of convergence depends on calibration method, cable length, and EC. The results by fitting both probe constant and cable resistance (Fig. 4-8b) increase the estimation accuracy slightly for each recording time, but the convergence trend is similar to that by fitting only the probe constant with cable resistance directly measured by the short-circuited probe (Fig. 4-8a). 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. However, unlike what Fig. 4-7b may suggest, 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 probe constant and cable resistance. As shown in Fig. 4-7, TDR response approaches to 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 conductivity. 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 (Δt)

of roundtrip travel time in the lead cable (tcable). Except for the case of very short lead cable, accurate estimation of EC can be obtained with recording time as long as TDR equipment can serve, regardless of the fitting range for probe constant. The characteristic impedance of the lead cable increases with increasing cable length, giving rise to multiple reflections within lead cable, as shown in Fig. 4-2a. The convergence of EC estimation is governed by multiple reflections in the sensing probe for short lead cable, while it becomes dominated by multiple reflections in the lead cable for 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 3 multiple reflections within the lead cable as shown in this case; however, it urgently suggest that take recording time as long as pulse can provide as measuring EC. Errors found in the literature using the series resister model with cable resistance directly measured by the short-circuited probe may be explained by the time effect, imperfect shorting element, or wrong acquisition program.

The effect of recording time on the Castiglione-Shouse method is shown in Fig. 4-8 (c,d) and Fig. 4-9(c,d) for comparison. If the probe constant is fitted (Fig. 4-8d and Fig. 4-9d), 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. 4-8c and Fig. 4-9c), it takes much longer time for the estimated EC by the Castiglione-Shouse method to become invariant with time. When the recording time is greater than 6tcable, the estimated 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. 4-4.

In other words, the effect of recording time, expressed as multiples of roundtrip travel time in the lead cable, on the estimated probe constant β using series resisters model, and

Castiglione-Shouse method is illustrated in Fig. 4-10. The probe constants β estimated by resistors method converge regardless the cable length as recording time is greater than 4tcable, while the probe constants β estimated by Castiglione-Shouse method are not the same due to cable length, and it means that probe constants β estimated by Castiglione-Shouse is not consistent with actual probe constant.

2 3 4 5 6 7 8 9 10 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) .

10¹ 10² 10³

Fig. 4-7 Recording time required to reach the steady state for probes (a) short-circuited, (b) in water of two electrical conductivities, and (c) open in air.

10¹ 10² 10³

1.5 (c) Castiglione - Shouse, β determined

σ t / σ true

Fig. 4-8 The effect of recording time, expressed as the time that includes multiples of roundtrip travel time in the probe section, on the estimated EC using series resisters model

with (a) Rcable measured and β fitted, (b) Rcable and β fitted, (c) Castiglione-Shouse method with actual β determined, and (d) Castiglione-Shouse method with β fitted.

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

1.5 (c) Castiglione - Shouse, _β determined

σ t / σ true

Fig. 4-9 The effect of recording time, expressed as multiples of roundtrip travel time in the lead cable, on the estimated EC using series resisters model with (a) Rcable measured and β fitted, (b) Rcable and β fitted, (c) Castiglione-Shouse method with actual β determined, and (d)

Castiglione-Shouse method with β fitted.

2 3 4 5 6 7 8 9 10 0.05

0.1 0.15 0.2

β

Rcable measured, β fitted

2 3 4 5 6 7 8 9 10

0.05 0.1 0.15 0.2

t / t

cable

β

Castiglione - Shouse, _β fitted

1 m 30 m 100 m

σ =0.02 Sm^-1

σ =0.02 Sm^-1 α_R = 19.8 sec^-0.5; Z

p = 300 Ω

Probe length = 0.12m

Fig. 4-10 The effect of recording time, expressed as multiples of roundtrip travel time in the lead cable, on the estimated probe constant β using (a) series resisters model, and (b)

Castiglione-Shouse method