2.3 TDR Principle and Analysis
2.3.4 TDR Waveform Modeling
= −
short open
open sample
scale ρ ρ
ρ
ρ ρ [2-47]
where ρscaled is the TDR measurement corrected for cable resistance by the scaling process;
ρopen and ρshort are the reflection coefficients with the probe in open air and short-circuited, respectively. The value of ρscaled represents the TDR measurement as if there is no cable resistance, so the Giese–Tiemann equation (Eq. [2-43]) can be used for calculating the EC.
Castiglione and Shouse (2003) claimed that the series resistors model is incorrect and Eq.
[2-47] leads to better agreement with experimental results. It should be pointed out, however, that the scaling process is linear while the effect of cable resistance on the steady-state reflection coefficient will be shown to be nonlinear.
The TDR EC measurement is more straightforward, but methods accounting for the cable resistance remain controversial; and the effect of TDR recording time has been underrated when long cables are used. Therefore, one of objectives of this study attempts to develop a new model to show the correct method for taking account of cable resistance and guideline for selecting proper recording time.
2.3.4 TDR Waveform Modeling Principle of TDR waveform modeling
Heimovaara (1994a) firstly developed the modeling process of single-section TDR response in 1994. Subsequently, Feng et al. (1999) and Lin (2003a, and 2003b) provided the
electrical circuit theory to model the multi-section transmission line system, as shown in Fig.
2-12. The input impedance at the TDR sampler can be determined by the following eqautions:
where ZL is the terminated impedance of the system, which is equal to zero with shorted-end and infinity with open-end (e.g. in air), Zc,i and γi are the characteristic impedance and propagation constant of i section, respectively, and the li is length of each i section. The characteristic impedance and propagations constant are the phasor form and depend on frequency. They can be determined by Eq. [2-34] and Eq. [2-35]
Based on the concept of input impedance, the solution of the sampling voltage V(0) in frequency domain in Fig. 2-12 can be calculated according to the electric circuit theory as:
( ) ( )
where the V .and ZS are the voltage source and source impedance of the cable tester,
respectively. ZS is typically equal to 50 Ω, but it depends on the TDR cable tester. H is the system function of TDR, and S11 is called the scattering function, which can be seen as the reflection coefficient of the whole transmission line.
Fig. 2-13 shows a flow chart of overall modeling process, we first obtain the source impulse vs(t), and the transform vs(t) to Vs(f) by using Fast Fourier Transform (FFT). Then, the output response V(f) in the frequency domain is obtained by evaluating the product H(f) with Vs(f). Finally, using Inverse Fast Fourier Transform (IFFT) to restore the output response v(t) in time domain.
Since the voltage source vs(t) of the input function Vs(f) of the cable tester should be quantified before TDR modeling, several methods of treating the input function are available.
Heimovaara (2001) proposed an empirical input function involving the error function erf(t) as:
( ) [ ( ( ) ) ]
2
1 start
s
t t t erf
V + −
= α
[2-50]
where tstart is the beginning time of the input pulse rise, and α is related to the inverse of the rise time of the pulse. The parameters tstart and α can be estimated by fitting Eq. [2-50] to the measurement of the input pulse obtained by matching a 50 Ohms reference termination at cable tester.
Weerts et al., (2001) presented that the input function was chosen as the signal leaving the coaxial, and open and shorted-circuited voltages, Vopen and Vshort, were measured, respectively. By normalizing the absolute voltage values to the unity, the input signal is obtained from:
( )
) 2
( open short
s
V t V
V −
= [2-51]
Lin (1999) indicated that the input pulse can be generated from electric circuit model, thus the equation of the normalized input pulse was formulated as:
) exp(
1 )
( τ
ζ
− −
−
= t
t
Vs [2-52]
where τ= (rise time/αr) is a parameter correlated with the specification of step pulse rise time of TDR cable tester, ζ is the lag time of step pulse rise. The parameters αr and ζ are also estimated by fitting Eq. [2-52] to the measurement of the input pulse obtained by matching a 50 Ohms reference termination at cable tester.
As regard for the complexity for practical calibrations of input function, Mattei et al., (2006) examined aforementioned methods and proposed a new approach, in which the input function is derived form the signal reflected at the end of the coaxial cable disconnected form the TDR probe, and the response function coincides with the input function.
li z0=0
+
－ ZL
Source impedance
Vs
z1
l1 l n
#i #n
#1
z
Zin(z0)
Voltage
source V(0) (γi,Zc,i)
Zin(zi-1) Zin(zn-1)
zi-1 zi
Zs
~
zn-1 zn=L
Fig. 2-12 Equivalent circuit of a cascade of uniform section for TDR system. (after Lin,
2003a)
Fig. 2-13 The flow chart of the spectral algorithm (modified from Lin, 1999)
Comprehensive TDR modeling considering resistance effect
The transmission line parameters γ and Zc in Eqs. [2-34] and [2-35] were derived by neglecting the effect of resistance of the waveguide (transmission line) conductors. This is a typical assumption to simplify the derivation and be justified when cable length is not too long. However, when TDR measurements are used for field monitoring, long cable may be used. (Su, 1987; Heimovaara, 1993; Dowding et al., 2003).
Input Signal, vs(t)
Output Signal, v(t) Input Spectrum, Vs(f)
Do Loop V(f) =H(f)Vs(f) Output Spectrum, V(f)
t
f
f
t
Lin and Tang (2007) modified the TDR modeling process by considering the effect of resistance, and derived a complete form for the transmission line parameters. The characteristic impedance and propagation constant are parameterized as:
Z A intrinsic impedance of free space, and αR (sec-0.5) is the resistance loss factor (a function of the cross-sectional geometry and surface resistivity due to skin effect). If cable resistance is ignored (i.e. αR = 0), A becomes 1.0 and γ and Zc have expressions identical to the non-resistance formulations (Feng et al., 1999; Lin, 2003a).
Fig. 2-14 shows a TDR response under a long leading cable (30 m) and simulated TDR waveforms with/without consideration of cable resistance. The modeling with resistance effect can truly represent the dispersive characteristic in the reflected waveform when long cables are used.
This complete TDR model will be used to investigate the effect of cable resistance on the simplified analyses for determining apparent dielectric constant and electrical conductivity.
Sensitivity analysis of dielectric spectroscopy and development of new spectral analysis
Fig. 2-14 The comparison of TDR modeling with and without consideration of cable resistance (after Lin and Tang, 2007)