3 TDR Dielectric Permittivity Analysis and Influence Factors
3.2 Dielectric Spectrum Analysis
3.2.1 Sensitivity Analysis and Reliability of Dielectric Spectroscopy
Lin (2003b) preformed dielectric spectroscopy based on realistic modeling of the multi-section TDR system. An example is shown in Fig. 3-8 for a silt loam. The measured data was obtained by directly solving the S11 function for the dielectric permittivity. A reliable result could only be obtained below 200 MHz. Inversion based on a dielectric dispersion model is more reliable but requires a good dispersion model, which is not always available for many composite materials.
Errors in high frequency are possibly affected by signal to noise ratio (SNR), imperfect TDR system calibration, and the fringing effect. The decreased SNR in high frequency range is due to lower high frequency energy of step input and greater signal loss at high frequencies in wave propagation. The calibration of the multi-section TDR measurement system is a delicate work. The connector and the probe head may have several mismatched that may not be perfectly considered in system calibration. Imperfect calibration of these mismatches may affect the dielectric spectroscopy in the high frequency range. The fringing effect occurs at the open end of the sensing waveguide, it may be treated as an equivalent end fringing capacitance Cf, as illustrated in Fig. 3-9. The equivalent end fringing capacitance causes a phase shift which results in the trace not being coincident with the open point after mathematical correction of equivalent extra line of length (Leo), as shown in Fig. 3-9. The end fringing capacitance is not difficult to be modeled and neglected in the current TDR model. This assumption may be another source of error for high frequency measurements.
Fringing effect can be avoided by using a probe with shorted end. de Loor et al. (1972) and Cereti et al. (2003) presented the TDR travel time analysis using shorted-end probe. On
an attempt to reduce error in the high frequency range, a shorted-end probe was also used to overcome the fringing effect in dielectric spectroscopy. Fig. 3-10 (Tang, 2007) displays the comparison of estimated dielectric spectrum of tap water from open-end and shorted-end coaxial probe. The shorted-end probe does not seem to improve the accuracy much in the high frequency range. On the other hand, great deviation form the theoretical values exist in the low frequency is observed. This result is unexpected and remains to be explained.
Since the shorted-end probe does not improve the estimated dielectric spectrum, an alternative approach which focuses on the sensitivity analysis is used to discover the source of error. The TDR scatter function (S11) is taken as the frequency response for the sensitivity analysis. The S11 is the reflection spectrum of the whole TDR system as shown in Eq.
[2-49b], thus it is influenced by several factors, including the dielectric constant of the material, length and impedance of the probe, and even the cable resistance. The investigation of S11 sensitivity is based on the TDR modeling whose basic parameters can be referred to the Table 3-1 and Table 3-2, thus the S11 can be estimated by these parameters using Eq. [2-49b]. Fig. 3-11 shows the abs(S11) response with different cable length as measuring the tap water and silt loam. Beyond 1MHz, the magnitude of S11 is decreased due to cable resistance.
To discuss the sensitivity of S11 to the measured dielectric permittivity and calibration parameters (probe length and Zp), the normalized sensitivity of S11 is formulated as:
[ ( ) ] ( ) [ ( ) ]
where m indicates the influence factors. Since there is no efficient way to derive the analytical formulation of sensitivity, the numerical derivative method is used in this study to obtain the
length and impedance of the probe, and the boundary condition of probe.
Fig. 3-12 to Fig. 3-16 are the normalized sensitivity of S11 due to εdc, ε∞, probe length L, and impedance Zp, respectively. Each of these figures has the S11 normalized sensitivity of tap water and silt loam in open and shorted boundary conditions. The results of all cases indicate that the sensitivity function of S11 in the shorted-end condition is much lower than in the open-end condition at frequency below 50 MHz. This may explain why the dielectric spectroscopy can yield reasonable result at low frequencies.
The shorted-end probe does not significantly improve the measurements in the high frequency range and yield poor results in the low frequency range. Therefore, the fringing effect is not the source for the large deviation in the high frequency range, and the shorted-end probe is not recommended to replace open-end probe. The large deviation in the high frequency should be attributed to low energy of signal is the high frequency range and imperfect calibration of the TDR system.
Fig. 3-8 Estimated frequency-dependent dielectric properties of a silt loam (after Lin, 2003b)
Fig. 3-9 Equivalent capacitance and extra length for fringing effect Zc
Zc
Zc
Cf
Zc
Leo
106 107 108 109
Fig. 3-10 Estimated dielectric spectrum of tap water from open-end and shorted-end coaxial probe (modified after Tang, 2007)
Fig. 3-11 TDR abs(S11) response with different cable length as measuring the (left) tap water and (right) silt loam
104 106 108 1010 1012
106 107 108 109 10-15
10-10 10-5 100
Normalized Sensitivity
106 107 108 109
10-15 10-10 10-5 100
Freq., Hz
Normalized Sensitivity
Open-end Shorted-end
(a)
(b)
Fig. 3-12 Normalized sensitivity of abs(S11) due to εdc as measuring the (a) tap water and (b) silt loam
106 107 108 109 10-15
10-10 10-5 100
Normalized Sensitivity
106 107 108 109
10-15 10-10 10-5 100
Freq., Hz
Normalized Sensitivity
Open-end Shorted-end
(a)
(b)
Fig. 3-13 Normalized sensitivity of abs(S11) due to ε∞ as measuring the (a) tap water and (b) silt loam
106 107 108 109 10-15
10-10 10-5 100
Normalized Sensitivity
106 107 108 109
10-15 10-10 10-5 100
Freq., Hz
Normalized Sensitivity
Open-end Shorted-end
(a)
(b)
Fig. 3-14 Normalized sensitivity of abs(S11) due to length of probe (L) as measuring the (a) tap water and (b) silt loam
106 107 108 109 10-15
10-10 10-5 100
Normalized Sensitivity
106 107 108 109
10-15 10-10 10-5 100
Freq., Hz
Normalized Sensitivity
Open-end
Shorted-end (a)
(b)
Fig. 3-15 Normalized sensitivity of abs(S11) due to impedance of probe (Zp) as measuring the (a) tap water and (b) silt loam