3 TDR Dielectric Permittivity Analysis and Influence Factors
3.2 Dielectric Spectrum Analysis
3.2.2 Frequency Domain Phase Velocity Method
3.2.2.2 Proof of Concept
The synthetic TDR measurement system is composed of a TDR device, a RG-58 lead cable, and a sensing waveguide. The transmission line parameters listed in Table 3-1 were used, except that the geometric impedance Zp of probe is set as 50 Ω to ensure negative first reflection for all cases. Tap water modeled by the Cole-Cole equation was used as one of the testing materials (as listed in Table 3-2). To modeling dielectric dispersion of soils, a four-component dielectric mixing model (Lin, 2003b) referred to Eq. [2-33] was used in this study. The associated parameters of four-component dielectric mixing are listed in Table 3-4.
Time interval dt = 2.69×10-11 sec and time window 0.5N dt = 8192×40 dt = 8.8×10-6 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 18 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 before onset of the next step pulse.
Table 3-4 Volumetric mixing parameters
Volumetric Mixing Parameters Range Reference value Soil physical parameters
volumetric water content θ, % 5 ~ 40 5 & 40
volumetric soil content θs, % 60 60
effective specific surface As, m2g-1 50 ~ 400 200 Dielectric parameters of air
constant ε 1 1
Dielectric parameters of soil particles
constant ε 4.7 4.7
Dielectric parameters of free water
static value εdc 80 80
high frequency value ε∞ 4.22 4.22
Relaxation frequency frel, GHz 17.4 17.4
Conductivity σfw, S m-1 0 ~ 0.5 0.02
Dielectric parameters of bound water
static value εdc 80 80
high frequency value ε∞ 4.22 4.22
Relaxation frequency frel, kHz 9 9
Conductivity σbw, S m-1 5 5
Empirical Parameter
Fitting Parameter α 0.5 0.5
A typical waveform for the tap water is shown in Fig. 3-18. This simple case (with electrical conductivity = 0.02 S/m) is firstly applied to verify the principle of TDR frequency domain phase velocity method.
Fig. 3-20(a) shows the phase angle of the cross-spectral density (Δφ ) from two characteristic signals before unwrapping, and Fig. 3-20(b) shows the result after unwrapping with comparison to the theoretical values. Fig. 3-20 (c) compares the measured frequency domain phase velocity (Vph) with the theoretical values obtained from Eq. [2-37]. Both Fig.
3-20(b) and (c) show that measured Δφ and Vph are in good agreement with the theoretical values in the frequency range from 0.1 GHz to the upper bound of TDR frequency bandwidth (1.5 GHz). The disagreement at frequency below 0.1 GHz may be resulted from leakage due to truncations of the two characteristic signals. Fortunately, the frequency range where the
frequency domain phase velocity method works happens to be where dielectric spectroscopy does not perform well. Therefore, these two techniques seem to be in good complement.
The effect of EC on the apparent dielectric constant has been fully discussed in Chapter 3.
Due to the EC effect, apparent dielectric constants estimated by the single tangent method and derivative method are increasingly overestimated as EC increases. Similarly, simulations were conducted to investigate the EC effect on the TDR frequency domain phase velocity method.
Fig. 3-21(a) shows the error percentage of the phase angle of the cross-spectral density (Δφ) and Fig. 3-21 (b) shows the error percentage of the phase velocity compared with theoretical values with a variety of EC values. This result indicates that the TDR frequency domain phase velocity method is practically not affected by the EC at the frequency range from 0.1 GHz to 1GHz for the tap water case.
107 108 109 1010 -5
0 5
Δφ, rad (before unwrapping)
107 108 109 1010
100 102 104
Δφ, rad
107 108 109 1010
2 2.5 3 3.5
4x 107
V ph, m/s
Frequency Theoretical
Measured
Fig. 3-20 (a) The phase angle of the cross-spectral density (Δφ) of two characteristic signals before unwrapping, (b) the results after unwrapping compared with theoretical values, and (c)
the measured frequency domain phase velocity (Vph) compared with the theoretical values
107 108 109 1010 -10
-5 0 5 10
Δφ error, %
107 108 109 1010
-10 -5 0 5 10
V ph, error, %
Frequency
σ = 0.01 S m-1 σ = 0.02 S m-1 σ = 0.04 S m-1 0 S m-1
0 S m-1
Fig. 3-21 (a) The error percentage of phase angle of the average cross-spectral density (Δφ) and (b) the error percentage of phase velocity compared with the theoretical one as a variety
of EC.
The four-component dielectric mixing model (Lin, 2003b), as shown in Eq. [2-33], was further used to examine the feasibility of the TDR frequency domain phase velocity method on soils. The modeled dielectric dispersion (in term of the apparent dielectric constant Ka
using Eq. [2-38]) due to soil-water interaction (in cases with effective specific surface As = 50, 200, and 400 m2g-1) and soil water content θ using four-component dielectric mixing model is shown in Fig. 3-22, in which frequencies near 1 GHz are least affected by soil type (As) and considered as the optimal frequency for water content measurement (Lin, 2003b). The
(a)
(b)
apparent dielectric constants are accordingly determined by the frequency domain phase velocity method at 1GHz for a variety of soil water content, as shown in Fig. 3-24. Also shown in Fig. 3-24 are the theoretical values of the volumetric mixing model and the apparent dielectric constant of single tangent method from the simulated waveforms. The measured values by the frequency domain phase velocity method agree well with the theoretical values, which shows invariance with soil types. On the contrary, the apparent dielectric constant of the single tangent method depends on the soil type, especially for high As. This result inspires a new approach for soil water content estimation because that the dielectric constant estimated by the frequency domain phase velocity method is less influenced by low-frequency dispersion and provide a actual dielectric constant in particular frequency range, whereas the dielectric constant estimated by the travel time analysis method are greatly effected by low-frequency dispersion and lacks actual physical meaning.
The frequency domain phase velocity method is further examined for the effects of electrical conductivity and cable length, which cause series problems for the tangent line method in dispersive materials, as discussed in Chapter 3. Fig. 3-24 shows the estimated apparent dielectric constant (Ka) at 1GHz from the frequency domain phase velocity analysis for As = 200, and the apparent dielectric constants estimated by the single tangent method as affected by EC of free water (σfw). The results from frequency domain phase velocity method at 1GHz remains relatively constant in the entire EC range. The dielectric constants estimated by the single tangent method, on the other hand, show dependency on the EC in high soil water content. Furthermore, Fig. 3-25 shows that dielectric constants estimated by single tangent method are dramatically influenced by the cable length, as also shown in Fig. 3-4. The dielectric constants estimated by the frequency domain phase velocity method, however, still remain relatively regardless of the cable length.
When measuring soil water contents using the travel time analysis, the same water content
may measure different apparent dielectric constant due to different electrical conductivity (e.g.
from water salinity), cable length, and dielectric dispersion (e.g. from soil texture). The soil water contents estimated by the frequency domain phase velocity method at 1GHz are less affected by the aforementioned factors. Therefore, the frequency domain phase velocity method not only provides good estimations of dielectric permittivity at high frequency, it also shows great promise of providing a universal correlation with soil water content. Laboratorial tests are suggested to further verify the feasibility of the frequency domain phase velocity method.
107 108 109 1010
0 1 2 3 4 5 6 7 8 9 10
sqrt(K a)
Frequency
As = 50 m2g-1 As = 200 m2g-1 As = 400 m2g-1
θ = 40 %
θ = 5 %
Fig. 3-22 The synthetic dielectric dispersion due to soil-water interaction and soil water content using four-component dielectric mixing model
0 10 20 30 40 50 2
2.5 3 3.5 4 4.5 5
sqrt(K a, 1GHz)
θ, %
□As=50
OAs=200
＊As=400
Single Tangent (As = 50 and 200) Single Tangent
(As = 400)
Theoretical set
Fig. 3-23 The estimated frequency domain phase velocity at 1GHz in term of dielectric constant (Ka) as affected by a variety of soil water content and soil type
0 0.1 0.2 0.3 0.4 0.5 1
2 3 4 5 6 7
sqrt(K a)
σfw, S/m
Measured Single tangent Theoretical
θ = 40%
θ = 5%
As = 200
Fig. 3-24 The estimated apparent dielectric constant (Ka) at 1GHz from frequency domain phase velocity analysis for As = 200 and the apparent dielectric constants estimated by the
single tangent method as affected by EC of free water (σfw) and soil water content
10 20 30 40 50 1
2 3 4 5 6 7
sqrt(K a)
Cable length, m Measured
Single tangent Theoretical
θ = 40%
θ = 5%
As = 200
Fig. 3-25 The estimated apparent dielectric constant (Ka) at 1GHz from frequency domain phase velocity analysis for As = 200 and the apparent dielectric constants estimated by the
single tangent method as affected by cable length and soil water content