Calibration Method and Verification

Material and method

The instrument errors due to imperfect amplitude calibration were examined for a Campbell Scientific TDR100 and a Tektronix 1502C. Three measurements were taken in which the front panel connector was open, shorted, and terminated by a 50 Ω block, respectively. To further demonstrate that the instrument error is not related to cable resistance, the three measurements were repeated with a 10 m RG58 cable connected to the TDR devices.

To experimentally investigate the effect of the imperfect amplitude calibration, TDR EC measurements were made on 7 NaCl solutions, with EC varying in the low EC range from 0 to 0.04 Sm^-1. The low EC range was used to clearly illustrate the effect of instrument error due to imperfect amplitude calibration. The measurements were conducted using a TDR probe (10-cm two rod probe with conductors 4 mm in diameter and 20 mm in spacing) connected to a Campbell Scientific TDR100 via a 2m-long RG-58 cable. The electrical conductivity of each electrolytic solution was measured independently with a standard EC meter (YSI-32 Yellow Spring Int. Inc., Yellow Spring, OH). When determining the Rcable using Eq. [4-8b], the measurements were performed by shorting the cable end with a short wire. The steady state responses were recorded near the end of the TDR pulse to better approximate the steady state. This is in fact mandatory for measurements in the high EC range or for the short-circuited probe. The computation of TDR EC involves Eq. [4-8], and Eq. [4-9]

successively for the series resistors model and Eq. [2-43] and Eq. [2-47] for the Castiglione-Shouse method. To calculate the TDR EC using Eq. [2-43], the probe constant β is first obtained using least square fitting of TDR EC to EC measurements made with the conventional conductivity meter. TDR EC measurements of electrolytic solutions were repeated using a 20m-long RG-58 cable to show the effect of cable resistance on the fitted

probe constants.

Results and discussions

A TDR device may output voltage (e.g. a Tektronix 1502C) or reflection coefficient (e.g.

a Campbell Scientific TDR100). To determine EC, v∞/ vS in Eq. [4-5] or ρ′∞ in Eq. [4-8a]

should be known. The source step voltage vS0 of a voltage-output TDR device is simply equal to the v_∞ when the TDR probe is open in air, which then serves as the reference voltage for computing the electrical conductivity. The original TDR waveform v(t) of a voltage-output TDR device is often converted to ρ(t), in which the incident step vi is determined by using a 50-Ω cable or terminating block as the impedance reference and for amplitude calibration.

Fig. 4-15a shows a group of TDR waveforms ρ(t) from the 1502C device, in which the front panel and a 10-m lead cable are shorted, open, and terminated with a nominal 50-Ω terminating block. The front panel terminated with a 50-Ω terminating block was used for amplitude calibration such that its reflection coefficient at long times is equal to 0.0. In this case, the steady-state reflection coefficient ρ∞ is 0.995 for the probe open in air, regardless of the cable length. It is not precisely 1.0 due to imperfect match between the source impedance and the terminating block. This reflection coefficient corresponds to 0.0045 dSm^-1 for the TDR probe used in this study, a small EC error in the condition of zero EC. The ρ∞ for the shorted front panel not being -1.0 is attributed to some internal resistance. The ρ∞ for a shorted cable increases as cable length increases. The amount of increase in ρ∞ due to cable resistance reduces nonlinearly with increasing ρ∞ and vanishes at ρ∞ = 1. Applying the calibration equation Eq. [4-10], the corrected reflection coefficient ρ′∞(t) can be obtained as shown in Fig. 4-15b. The corrected ρ′∞ becomes 1.0 in the condition of zero EC. The degree of imperfect match between the source impedance and the terminating block is indicated by

ρ′_∞ =0.0025 in Fig. 4-15b. A TDR device that outputs reflection coefficient uses the nominal 50-Ω internal cable as the impedance reference and for amplitude calibration. Analogous to Fig. 4-15, Fig. 4-16 shows a group of TDR waveforms ρ(t) from the TDR100 device. The mismatch between the reference impedance and source impedance in the TDR100 is more significant, leading to ρ∞ = 0.950 (corresponding to EC = 0.046 dSm^-1) for the open front panel and ρ∞ = 0.961 for the probe open in air as shown in Fig. 4-16a. The amplitude calibration error in a TDR100 seems to depend on whether the front panel is connected to a cable, a phenomenon which may be related to the fringing field of the open front panel. The apparent error can be corrected by applying the calibration equation Eq. [4-10], as shown in Fig. 4-16b.

Fig. 4-17 shows the results of several TDR EC measurements in the low EC range using the TDR100 device with 2 m and 20 m of RG-58 lead cable. The probe constant Kp was fitted as described before. Table 4-3 lists the fitted Kp using the Castiglione-Shouse method and the series resistors model. The percentage errors between the TDR EC measurements and conductivity meter EC measurements are listed in Table 4-4. As shown in Fig. 4-17, the Castiglione-Shouse method inherently corrects the instrument error and provides accurate TDR EC measurements when the probe constants are fitted. But the fitted probe constant Kp

varies with cable length, as shown in Table 4-3. The fitted probe constant decreases as cable resistance increases, as also suggested in Fig. 4-14.

If the measured reflection coefficient is not corrected for instrument error, the actual reflection coefficient is underestimated especially in low EC range as shown in Fig. 4-13a.

This will have an effect on the estimated EC using the series resistors model. Depending on the EC data range, the fitted Kp is lower than the actual Kp to some degree. As a consequence, the TDR EC by the series resistors model overestimates at lower EC and underestimates at higher EC, as shown in Fig. 4-17a and Table 4-4. This experimental result exactly agrees with

the theory illustrated in Fig. 4-13c. The results of the series resistors model with TDR100 reflection coefficient corrected by Eq. [4-10] are shown in Fig. 4-17b and Table 4-4. The large percentage error for the lowest EC in Table 4-4 can be attributed to the conductivity meter resolution and TDR quantization resolution. Except for the lowest EC (0.00039 Sm^-1), both the corrected series resistors model and the Castiglione-Shouse method give TDR EC measurements in precise agreement with that measured by the conventional EC meter. But the fitted probe constant Kp can be considered independent of the cable length only in the case of the series resistors model, as shown in Table 4-3.

To accurately determine TDR EC, both the instrument error due to imperfect amplitude calibration and cable resistance should be properly addressed. The instrument error results in an underestimation of reflection coefficient, which linearly decreases with decreasing reflection coefficient and vanishes at reflection coefficient = -1.0. In contrast, the effect of cable resistance leads to overestimation of reflection coefficient, which nonlinearly decreases with increasing reflection coefficient and vanishes at reflection coefficient = 1.0. The combined effect of instrument error and cable resistance on the steady-state reflection coefficient is nonlinear, so the Castiglione-Shouse method is incorrect, although the error can be compensated by adjusting the probe constant. The series resistors model is theoretically sound and precise if the reflection coefficient is properly calibrated to account for the instrument error.

The instrument error can be calibrated by the steady-state reflection coefficient at the zero-EC condition, while the cable resistance can be determined by the steady-state reflection coefficient when the probe is short-circuited. A calibration equation is derived to correct the measured reflection coefficient for instrument error. The corrected reflection coefficient can then be used in the series resistors model (Eq. [4-8]) for reduction of electrical conductivity considering the effect of cable resistance. To keep the usual practice and simplicity, the effect

of instrument error and cable resistance can be addressed in one step. An equation replacing the Castiglione-Shouse equation is suggested here:

( )( )

where ρ∞,Scale is the scaled reflection coefficient to be used in the usual Giese-Tiemann equation, ρ is the steady-state reflection coefficient of the sample under measurement, ρ∞,air is the steady-state reflection coefficient when the probe is open in air, and ρ_∞,SC is the steady-state reflection coefficient when the probe is short-circuited.

Table 4-3 Fitted Kp (m^-1)from laboratory measurements using a Campbell Scientific TDR100

Series Resistors Cable length (m) Castiglione-Shouse

Uncorrected Corrected^†

2 8.93 7.58 8.93

20 8.78 7.56 8.92

†Corrected using Eq. [4-10]

Table 4-4 Percentage errors between the TDR EC measurements and conductivity meter EC measurements

Series Resistors (Uncorrected)

Series Resistors (Corrected)

Castiglione-Shouse Error (%)

σYSI(Sm^-1) 2m 20m 2m 20m 2m 20m

0.00039 818.81 849.88 -6.23 68.29 -6.09 68.28 0.00529 46.64 47.07 -1.92 1.01 -1.78 1.01 0.01183 13.41 14.19 -0.80 0.93 -0.66 0.93 0.01525 7.80 7.64 -0.12 0.24 0.03 0.23 0.02014 2.14 2.45 -0.65 0.00 -0.51 0.00 0.03003 -2.55 -2.73 0.11 -0.07 0.25 -0.07 0.04015 -5.22 -5.02 0.18 0.33 0.33 0.33

10⁰ 10¹ 10² 10³

Fig. 4-15 (a) Original TDR waveforms from a Tektronix 1502C and (b) the associated corrected waveforms using calibration equation Eq. [4-10].

10⁰ 10¹ 10² 10³

Fig. 4-16 (a) Original TDR waveforms from a Campbell Scientific TDR100 and (b) the associated corrected waveforms using calibration equation Eq. [4-10].

0 0.01 0.02 0.03 0.04 0.05 0

0.01 0.02 0.03 0.04 0.05

σYSI, S m^-1 σ TDR, S m-1

1:1 line

Series Resistors(2m) Series Resistors(20m) Castiglione - Shouse(2m) Castiglione - Shouse(20m)

0 0.01 0.02 0.03 0.04 0.05

0 0.01 0.02 0.03 0.04 0.05

σ_YSI, S m^-1 σ TDR, S m-1

1:1 line

Corrected series Resistors(2m) Corrected series Resistors(20m)

Fig. 4-17 TDR EC measurements made by a Campbell Scientific TDR100 (a) without reflection coefficient calibration and (b) with reflection coefficient calibration.

(a)

(b)

5 TDR Probe Design

There are two main aspects of the TDR probe design that will be discussed in this chapter: the effects of probe rods configuration and boundary condition, and the measurement sensitivity of TDR travel time and EC measurement.

As mentioned in Chapter 2, the spatial EM density distribution has been investigated by analytical solution and numerical approaches for common probe configurations in the context of spatial sample volume. However, these early studies do not apply to some TDR probe types.

Advanced 3D EM field simulation software will be demonstrated to be useful for investigation of probe rod configuration and boundary effect.

Moreover, measurement sensitivity which is defined as the change of the measurement data due to variations of physical parameter is investigated to provide guidelines for probe design.

A development of a new type of TDR probe, TDR penetrometer, is included in this chapter to meet need of in-depth characterization of soil or soil-water mixtures. Due to the complexity of TDR penetrometer, this study will provide calibration methods for the TDR penetrometer and experimentally evaluate its measurement performance.