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Frequency versus Time Domain Simulation of Non-

uniform TDR System

Xiong Yu1

1Case Western Reserve University, Department of Civil Engineering, 10900 Euclid Ave., Bingham 210, Cleveland,

OH, U.S.A., 216-368-6247

Abstract Time Domain Reflectometry (TDR) is

finding increased applications for Civil Engineering field

instrumentation. New applications are being identifiedbased on innovative design of TDR sensor system.Numerical simulation plays an important role for guiding

and evaluation of these new designs. This paper discussesthe implementation of numerical simulation of frequency

domain and time domain models of non-uniform TDR

system. The frequency domain model is based on theconcept of impedance transform, which can easilyincorporate the dispersive characteristics of construction

materials. The time domain simulation is carried out byFDTD method. Results of simulation based on both

frequency domain model and time domain FDTD methodare compared to identify the relative merits of these

numerical methods. The simulation of TDR systemresponses establish the basis for aiding TDR sensor systemdesign and will be incorporated into the inversion

procedures to be investigated in the future.

I. INTRODUCTION

Originally developed in Electrical Engineering for

detection of discontinuities in the cable or electriccircuits, Time Domain Reflectometry (TDR) is finding

increased applications in other disciplines as well. In

Civil Engineering, it has been applied for evaluating

construction materials (such as compaction control of

soils [1] and indicating the chemical process in concrete

[2]), monitoring the deformation of structure and

infrastructure [3], and studying the contamination

transportation in soils [4]. Compared with other

techniques, TDR has the advantages such as safe, fast,

cost-effective, and easily automated. Characterization of

material properties is an important aspect among these

applications. Using a fast rising electrical pulse of

picoseconds provided by current TDR electronics, TDR

signal contains broad band information of material andstructural responses from a single measurement.

Innovative design of new TDR sensor as well as

development of cost-effective high performance TDR

system are important for discerning new TDR

applications. For both these purpose, computer aided

design is advantageous in that it provides realistic

prediction of system performance that is much faster and

cheaper than repeated hardware design.

In response to special field conditions, TDR sensor

systems designed for Civil Engineering field

instrumentation are generally not perfectly matched

system, although the assumption of a matched system is

frequently used. A model for unmatched TDR system

more realistically describes its behaviors.

This paper describes the implementation of numerical

simulations for the frequency domain model and time

domain model of unmatched (non-uniform) TDR system.

The frequency model is obtained by spectra-domain

solution of a cascaded non-uniform transmission line

equation. The time domain solution of TDR system

response is obtained by Finite Difference Time Domain

(FDTD) method. The results of frequency domain andtime domain methods are compared with actual

measurements on reference materials. The relative merits

of these two approaches are identified by comparison.

This work on the implementation of numerical

simulations will be further refined to aid TDR sensor

design and will be incorporated in the process of

inversion analysis for material and structure behaviors.

II. BACKGROUND

Time Domain Reflectometry is finding increased

applications in Civil Engineering. Characterization of

material behaviors and structure responses are important

topics for these applications. The successful introductionof TDR for construction materials is largely attributed to

the pioneering work by Topp et al. [5], which designed a

robust TDR sensor and established a universal

equation between TDR measured apparent dielectric

constant and volumetric water content. From this, the

water content can be determined from TDR measured

dielectric constant. The high sensitivity of TDR to water

content is attributed to the much larger dielectric constant

of water (around 81) than those of air (around 1) or soil

solids (around 3 to 5). Another important progress in this

aspect is made by Dalton et al. [6], who developed an

approach to obtain soil electrical conductivity from TDR

signal. The information of electrical conductivity has

been explored for estimation of pore fluid characteristicsand soil salinity, which are critical information for

disciplines such as agricultural and environmental

engineering.

The use of TDR for material characterization works by

generating a small-magnitude electromagnetic field

excitation and measures the material responses. Dipoles

(originated from the molecular geometry to charges of

the particles) oscillate under the excitation. The overall

response of the material is dependent on the excitation

frequencies and is described by the dielectric spectra.

Certain ions drift freely under the electrical field and the

macroscopic phenomena correspond to electrical

conductivity.

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Figure 1 shows the schema of a field TDR system. It

consists of TDR electronics, collection cable, conversion

head, and measurement probe. There are two distinctive

reflections while electrical pulse generated by TDR

electronics travels through soil specimen under test due

to mismatch in electrical impedance. The first reflectiontakes place at the air/soil interface, since soil generally

has distinctive dielectric properties compared with air.

The second reflection takes place at the end of the

measurement probe due to the boundary conditions. A

typical TDR signal measured in soil is shown in Fig. 1.

The information of dielectric constant and electrical

conductivity can be obtained simultaneously from the

TDR signal.

A distinctive characteristics of Civil Engineering

construction materials is that they generally cover a wide

distribution of particle sizes. It is mandatory for TDR

sensor to provide sufficient sampling volume to obtain

accurate measurements (ASTM D2216). This

consequently exerts constraints on the design of TDR

sensor and measurement system. Most frequently it is

very hard to design a totally matched TDR sensor thatsimultaneously meets the requirements for

instrumentation on soils. Most of the TDR systems for

Civil Engineering field instrumentation are, strictly

speaking, un-matched system. This means reflections

will take place beside at the desired locations (the air/soil

interface and at the end of the probe). This can be seen

from the delicate changes in the details of the TDR

signal (Fig. 1). The responses of these system are more

accurately described by non-uniform transmission line

model. The model of TDR system for material or

structure applications is composed of cascaded uniform

sections. Solution of these models can be achieved in

either the time-domain or the frequency-domain.

III.THEORETICAL BASIS

A. Frequency Domain Model for Non-uniform TDR

System

Under the assumption of TEM mode, the frequency

domain model for the electromagnetic wave propagation

in non-uniform TDR system is obtained by spectra-

domain representation of transmission line equation [7].

To analyze a cascaded transmission line model for

TDR system, where the line is divided into different

sections, each of which has uniform dielectric and

electrical properties, a useful concept is impedance

transform. Impedance transform is used to obtain the

lumped equivalent complex resistance. In the impedance

transform, the equivalent impedance at a location is

obtained from the load impedance and the characteristic

impedance of the connection lines in a bottom-up

fashion. The process continues to the connection point

where the voltage representing TDR signal is sampled by

the TDR electronics (Eq. 1 and Fig. 2).

nnLnc

nnncL

ncnin

Lnin

lZZ

lZZZzZ

ZzZ

tanh

tanh)(

)(

,

,

,1

(1)

In Eq.(1), n is the propagation constant which is

decided by the material properties and thus frequencydependent material behaviors can be incorporated, ln is

the length of the section.

By using impedance transform, the analysis of TDR

system is similar to that of a static circuit, albeit the

quantities are represented as complex numbers in the

frequency domain. The predicted sampling voltage of

the TDR signal can be obtained from the final equivalent

circuit on the right side of Fig. 2 as:

s

ins

in VZZ

ZV

)0(

)0()0(

(2)

In Eq. (1) the term Vs is the frequency domain

representation of source voltage. Zs is the internal

impedance of the TDR electronics, which is around 50

ohms. The voltage, V(0), in Eq. (2), represents the

Fig. 1. Up: Schematic plot of TDR measurement system;

Below: Typical TDR signal and information utilized

-1.25

-0.75

-0.25

0.25

0.75

1.25

0 1 2 3 4 5 6 7 8

Scaled Distance (m)

RelativeVoltage(V)

1

1

f

sb

V

V

CEC

2

p

aa

L

LK

Apparent Length,La

-1.25

-0.75

-0.25

0.25

0.75

1.25

0 1 2 3 4 5 6 7 8

Scaled Distance (m)

RelativeVoltage(V)

1

1

f

sb

V

V

CEC

2

p

a

aL

LK

-1.25

-0.75

-0.25

0.25

0.75

1.25

0 1 2 3 4 5 6 7 8

Scaled Distance (m)

RelativeVoltage(V)

-1.25

-0.75

-0.25

0.25

0.75

1.25

0 1 2 3 4 5 6 7 8

Scaled Distance (m)

RelativeVoltage(V)

1

1

f

sb

V

V

CEC

2

p

aa

L

LK

Apparent Length,La

V/2

L

z

0

ZL(l

n)Z

L(l

n-1)

s

Zs

V0

ZL(0)

s

Zs

V0

s

Zs

V0

ZL(l

n-1)

L

z

0

z

0

ZL(l

n)Z

L(l

n-1)

s

Zs

V0

ZL(l

n)Z

L(l

n-1) ZL(ln)ZL(ln-1)

s

Zs

V0

sss

Zs

V0

ZL(0)

s

Zs

V0

ZL(0)

s

ZsZ

L(0)Z

L(0)

sss

Zs

V0

s

Zs

V0

ZL(l

n-1)

s

Zs

V0

sss

Zs

V0

ZL(l

n-1)Z

L(l

n-1)

Simplification of

the equivalentcircuit for the non-

uniform

transmission line

model for TDR

system in bottom-

up fashion

Front Panel, Cable,, Measurement Probe

Fig. 2 Equivalent circuit model for spectral solution

of non-uniform transmission line

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sampled voltage by TDR in the frequency domain, which

can be transformed into a time domain TDR signal by

inverse FFT.

B. Time Domain Solutions of Non-Uniform TDR

System

The time domain description of cascaded non-uniform

TDR system is by the Telegraph equation[7], which can

be solved using FDTD method [8-9].

IV. NUMERICAL IMPLEMENTATION AND COMPARISONS

The code of the frequency domain model for non-

uniform TDR system has been developed. The

procedures for the prediction of system responses

include: a) obtain the spectrum of input signal by FFT; b)

obtain the lumped equivalent impedance at the input end

from Eq. (1). c) determine the spectrum of sampled

voltage by Eq. (2). d) obtain the time domain responses

of the TDR system by inverse FFT. In the second step,the electrical characteristics (impedance) of the

components in the transmission path (such as cable,

connector, and probe) are obtained from its standard

specifications, geometry or calibration test.

The predicted responses of the TDR system on know

liquid (water) versus those of actual measured signals are

shown in Fig. 3. The dielectric behavior of water is

described by Debyes model with the parameters from

literature. From the figure it can be seen that the

frequency domain model closely describes the actual

measured signals.

Fig. 3. Measured and simulated signals for TDRmeasurement in de-ionized water

The impedance transform makes it easy to implement alayer peeling procedure, which removes unmatched

sections in the TDR system. Figure 4 shows comparison

of original measured signal in soil versus those where the

components such as cable, connector are removed. A

calibration process is also incorporated to remove the

accumulation of system error. This resulted in a much

cleaner peeled signal (Fig. 4).

The scheme of finite difference time domain (FDTD)

for non-uniform TDR system is currently being

implemented. The results of this approach will be

compared with that of frequency domain method. The

comparison will emphasis on the speed of computation

and the relative merits for incorporated into inversion

procedure.

V. CONCLUSION

Accurate simulation of TDR system responses

provides important guidance for design of new TDR

sensor system as well as for inversion analysis for

material or structure behaviors. This work introduces the

development of numerical simulation of non-uniform

TDR system based on cascaded non-uniform

transmission line model. Further work on the

implementation of FDTD method and comparison of

relative advantages will be continued and reported.

REFERENCES

[1] Yu, X. and Drnevich, V.P. (2004). "Soil water content anddry density by time domain reflectometry", Journal ofGeotechnical and Geoenvironmental Engineering, Vol.130, No. 9, pp.922-934.

[2] Hager, N.E. and Domszy R.C. (2004). "Monitoring of

cement hydration by broadband time-domain-reflectometry dielectric spectroscopy", Journal of Applied

Physics, Vol. 96, No.9, pp.5117-5128[3] Su, M.B. and Chen, Y.J. (2000). "TDR monitoring of

infrastructure systems", Journal of Infrastructure Systems,Vol. 6, No.2, pp.67-72.

[4] Elrick, D.E., R.G. Kachanoski, E.A. Pringle and A.L.Ward. 1992. "Parameter estimates of solute transportmodels based on time domain reflectometrymeasurements", Soil Sci. Soc. Am. J. 56:1663-1666.

[5] Topp, G. C., Davis, J. L. and Annan, A. P. (1980)."Electromagnetic determination of soil water content andelectrical conductivity measurement using time domainreflectometry", Water Resources Research, Vol. 16,

pp.574-582.[6] Dalton, F.N., Herkelrath, W.N., Rawlins, D.S., and

Rhoades, J.D. (1984). "Time Domain Reflectometry :simutaneous measurement of soil water content andelectrical conductivity with a single probe", Science, Vol.224, pp.989-990.

[7] Ramo, S., Whinnery, J. R. and Duzer, V. (1994),Theodore, Fields and waves in communication electronics,

New York : Wiley.[8] Sekine, T., Kobayashi, K., and Kokokawa, S. (2002),

"Transient analysis of non-uniform transmission line usingthe finite difference time domain method", Electronics andCommunications in Japan, Vol. 85, No.2, pp.1062-1070.

[9] Trakadas, P.T. and Capsalis, C.N. (2001). "Validation of amodified FDTD method on non-uniform transmissionline", Progress in Electromagnetics Research, Pier 31,

pp.311-329.

Fig. 4. A TDR signal for soil and the peeled signal

before and after calibration

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