1861-5252/ c© 2012 TSSD Transactions onSystems, Signals & DevicesVol. 7, No. 4, pp.311-325

Analysis of the Parameters of a Lossy

Coaxial Cable for Cable Fault Location

Q. Shi,, U. Troltzsch and O. Kanoun

Chair for Measurement and Sensor Technology (MST)Chemnitz University of Technology, Chemnitz, Germany.

Abstract This paper presents an estimation of the per-unit-length param-eters of a lossy cable. A RLGC distributed element model forthe lossy transmission line parameters of a coaxial cable includ-ing frequency dependent filtering effect is used in this study toevaluate reflectometry responses of the cable systems. The sim-ulated results of this model are compared with the measuredresults of a coaxial cable using impedance spectroscopy to showthe accuracy in frequency domain. This lossy transmission linemodel then is solved in the time domain to accurately locate thefaults. The simulated results are compared with the measuredresults using time domain reflectometry. Finally this model isused to simulate branched cable networks.

Keywords: Transmission line modeling, lossy coaxial cable, time domainreflectometry, impedance spectroscopy, cable fault location anddetection, branched cable networks.

1. Introduction

The growing use of the cables in the field of communication systems,power distribution systems and vehicles etc., has caused an increasedneed for analyzing cable parameters and localization of the cable faults[1–6]. Modeling and simulation of transmission lines has been used toevaluate reflectometry response or data transmission capability for avariety of applications, especially estimation of very large scale inte-grated circuits. So modeling and simulation of transmission line plays avery important role to estimate Electro Magnetic Compatibility, evalu-

312 Q. Shi et al.

ate communication for safety and reliability and detect and locate thecable faults.

There are several emerging approaches for the detection and locationof cable fault [7–10]. Generally, a high-frequency signal is send down thecable. The reflected signal includes information about changes of cableimpedance and can be therefore used to detect open and short circuits.Furthermore different techniques are available for detecting frays, jointsand other small anomalies. The nature of the incident signal is usedto distinguish each type of reflectometry: Time domain reflectometry(TDR) uses a pulse or half sine signal, Frequency domain reflectometry(FDR) uses a set of stepped sine waves, Ultra wide-band (UWB) basedTDR or Time-frequency Domain (TFDR) called uses a linearly modu-lated chirp signal with a Gaussian envelope, and sequence time domainreflectometry (STDR) uses a pseudo-noise (PN) code [1–10]. The re-flectometry response itself is not self-sufficient to detect and locate thedefects in the wire. We need to solve the inverse problem, which is todeduce knowledge about the faults from the response at the input of theline [9]. Therefore the transmission line model is required to simulatethe response of the transmission line.

In this paper we proposed an efficient method for the extraction ofparameters of the cable, detection and localization of faults in cablenetworks with transmission line model using time domain reflectometryand impedance spectroscopy. Different types of the faults (open, short,small changes of the cable impedance) are considered in this work. In thiswork we use the model based on telegrapher’s equations where per-unit-length electrical parameters matrices R′, L′, G′, and C′ are computedby a finite element method.

2. Transmission line model

Fig.1 illustrates the cross section of a simple coaxial transmission line.The inner conductor has a radius a. The outer conductor (shield) hasan inner radius b and thickness ∆ and the outer radius of the shieldis c. Both of the conductors have the same electrical conductivity σ.The cable interior is filled with a lossless dielectric having a relativepermittivity εrel. The magnetic permeability is assumed to be that offree space µ0. The relative permittivity is assumed to be frequencyindependent.

Analysis of the parameters of a lossy coaxial cable 313

(a) (b)

Fig. 1. (a) Cross section of the coaxial transmission line, and (b) its per-unit-length equivalent circuit

The per-unit-length line outer inductance parameter L′ is given by [11]:

L′out =

µ0

2πln

(

b

a

)

(H/m) (1)

The per-unit-length line capacitance parameter C′ can be described bythe following equation [11]:

C′ =2πε0εrel

ln(

ba

) (F/m) (2)

When the conductors of the coaxial line are finitely conducting, therewill be additional per-unit-length impedance elements in the transmis-sion line model that take into account both the magnetic flux penetrationinto the conductors and the resistive loss [11]. For the inner conductorwith radius a, the per-unit-length impedance is given by the followingequation [13]:

Z ′a(ω) =

η

2πa

[

J0(γa)

J1(γa)

]

(Ω/m) (3)

where J0 and J1 are modified Bessel functions of order zero and one.The Term η is the wave impedance in the lossy conductor, and if thedisplacement current in the conductor is neglected, this term is:

η ≈√

jωµ0

σ(4)

The Term γ is given as:γ =

√

jωσµ0 (5)

The per-unit-length impedance of the outer shield is derived by followingequation [13]:

Z ′b(ω) =

η

2πb

[

J0 (γb)K1 (γc) + J1 (γc)K0 (γb)

J1 (γc)K1 (γb)− J1 (γb)K1 (γc)

]

(6)

314 Q. Shi et al.

where K0 and K1 are the modified Bessel functions of the second kind,and c is the outer radius of the shield.

The transmission line model composes of discrete resistors, inductors,capacitors and conductance. A length z of transmission line can concep-tually be divided into an infinite number of increments of length ∆z(dz)such that series and shunt R′, L′, G′ and C′ are given as shown in Fig.1. Each of the parameters R′, L′ and G′ is frequency dependent. Forexample, R′ and L′ will change in value due to skin effect and proximityeffect. G′ will change in value due to frequency dependent dielectric loss[11–16]. VSrc is the voltage of signal generator. Rg is the inside resistorof signal generator. ZL is the load impedance. From equations (1)–(6)we can calculate the shunt R′(Ω/m), L′(H/m), G′(S/m) and C′(F/m):

R′ = ℜe Z ′a(ω) + Z ′

b(ω) (Ω/m) (7)

L′ =ℑm Z ′

a(ω) + Z ′b(ω)

ω+ L′

out(H/m) (8)

C′ =2πε0εrel

ln(

ba

) (F/m) (9)

ℜe and ℑm represent the real and the imaginary parts.

In this work we consider the cable has a non dispersive dielectric.That means, the imaginary part of the complex permittivity ε′′ is zeroand the conductance G′ is zero:

G′ =πωε

′′

ln(

ba

) (S/m) = 0 (10)

The frequency dependency of the characteristic Impedance of a Trans-mission Line (TL) Z0 can be described by the following equations:

Z0 =

√

R′ + jωL′

jωC′(11)

The propagation constant of the transmission line with attenuation con-stant α and phase constant β is:

γc = α+ jβ =√

(R′ + jωL′) (G′ + jωC′) (12)

The impedance of a cable with length z and a certain load impedanceZL is:

Zin(z) = Z0ZL + Z0 tanh (γcz)

Z0 + ZL tanh (γcz)(13)

Analysis of the parameters of a lossy coaxial cable 315

For a lossless transmission line, it can be shown:

Zin(z) = Z0ZL + jZ0 tan (βz)

Z0 + jZL tan (βz)(14)

3. Comparison of simulated and measured data

In this work we use a typical coaxial cable RG58 C/U having followingparameters: a = 0.50mm; b = 1.475mm; c = 1.700mm; εrel = 2.25;σ = 59.225 × 106S/m. The simulated results are compared with themeasured data in the time and frequency domain separately.

3.1 Frequency domain comparisons

When the load impedance ZL is zero, and from (13), we have:

Zst = Z0 tan (γcz) (15)

The cable’s impedance with the open circuit can be given by formula:

Zop =Z0

tan (γcz)(16)

Solving equations (12), (15) and (16) results in:

γc =√

ZstZop = R0 + jX0 (17)

Z0 =

√

Zst

Zop

(18)

Finally, R′, L′, G′ and C′ of the transmission line model are determinedas:

R′ = ℜe Z0γc (19)

L′ =ℑm Z0γc

ω(20)

G′ = ℜe

γcZ0

(21)

C′ =ℑm

γc

Z0

ω(22)

In this study we use HP 4294A to measure the impedance spectrum ofRG 58 C/U with the length 0.9 meter in frequency domain from 1 kHzto 110 MHz. Because of the resonance the simulated and measured data

316 Q. Shi et al.

for the R′, L′, G′, and C′ are only compared in the frequency domainfrom 1 kHz to 2 MHz. This bandwidth is used because the cable withthe length of 0.9 m was chosen and the resonance occurs outside theused frequency range (> 2MHz).

Figure 2 presents the shunt R′ using the transmission line model (−)and the measured data (·). The transition from the low frequency re-sistance to the high frequency resistance arising from the skin depth inthe conductor is clear. Figure 3 shows us the comparison of inductanceof transmission line by simulated and measured data. Because of theskin effect the value of inductance of the coaxial cable descends from lowfrequency to high frequency. From Fig.4 and Fig.5 you can see that thevalue of G′ and C′ are not frequency dependent.

A close examination of the results in Fig.2 - Fig.9 shows that themaximal error of the total inductance of the transmission line is lessthan 1% over the entire frequency range. This simulation has the bestaccuracy than the others. For the shunt resistance and capacitance ofthe transmission line, the maximal error is about 16%. In this work thedielectric is considered as non dispersive. But the Fig.4 shows that thevalue of conductance of the transmission line changed ruleless along thezero line with the increasing of the frequency.

Fig. 2. Comparison between simulated and measured data for shunt R′

Fig. 3. Comparison between simulated and measured data for shunt L′

Analysis of the parameters of a lossy coaxial cable 317

Fig. 4. Comparison between simulated and measured data for shunt G′

Fig. 5. Comparison between simulated and measured data for shunt C′

Fig. 6. Real part of cable’s impedance with open circuit

Fig. 7. Imaginary part of cable’s impedance with open circuit

318 Q. Shi et al.

Fig. 8. Real part of cable’s impedance with short circuit

Fig. 9. Imaginary part of cable’s impedance with short circuit

3.2 Time domain comparisons

It is useful to compare the transmission line model and measureddata in time domain when they are applied to cable fault detection andlocalization. We use time domain reflectometry (TDR)to analyze thistransmission line in time domain. TDR is the most popular techniqueto locate cable faults (Fig.10). The TDR method works by sending ashort rectangular voltage step or a pulse down the cable. The wave canbe reflected whenever a signal traveling in a cable line encounters animpedance discontinuity (Fig.11, Fig.12, Fig.13). The amount of signalreflected back is calculated by the reflection coefficient:

ρ =ZL − Z0

ZL + Z0= r · ej·φ (23)

where Z0 is the characteristic impedance of the cable and ZL is theimpedance at the discontinuity (Fig. 1). ρ is the reflection coefficient.The type of the faults can be estimated from the amplitude of the re-flected signal. For example, the cable has an open circuit then ρ is 1, ashort circuit then ρ is -1 and a matching then ρ is 0 by lossless trans-mission line. In the cases of short circuit, open circuit, or matching, the

Analysis of the parameters of a lossy coaxial cable 319

reflection coefficient ρ is real .However, in the cases of water, joints, andgauge changes, the reflection coefficient is complex [17].

Fig. 10. Block diagram of TDR sys-tem

Fig. 11. Transmission line withload resistor RL

In this work firstly a coaxial cable RG 58 C/U with the length of0.9 meter is used for comparison between simulated and measured data.Then the branched network is used to analyze this matrix based trans-mission line model in time domain. In this work a signal generatorAWG615 is used to generate a pulse voltage.

Fig. 12. Comparison of simulated and measured signals for the TDR response(l = 0.9m,RL = 20kΩ)

Fig. 13. Comparison of simulated and measured signals for the TDR response(l = 0.9m,RL = 0Ω)

The characteristic impedance of transmission line Z0 and propaga-tion constant of transmission line can be used to compute the spectral

320 Q. Shi et al.

responses of the load voltages at each impedance changes in the cable.Once it is accomplished, the spectra can be transformed into the timedomain using Digital Fourier Transformation (DFT).

Fig. 14. Network topology

Fig. 15. Comparison of simulated and measured signals for the TDR responseof Fig. 14

The distance d between a reflection point and the injection point isgiven by:

d =υ · t2

(24)

where υ is the frequency dependent propagation velocity of the signalinto the cable and t is the time interval between the incident signal andthe reflected signal. The main problem in cables is to define exactly thepropagation velocity υ which depends on the frequency, on wire proper-ties and on the mode injection of the signal. To calculate the frequencydependent propagation velocity of the signal, we begin with the phaseconstant β (rad/meter) as given below in terms of the parameters R′, L′,G′ and C′. Angular frequency ω (rad/sec) divided by β yields the fre-quency dependent propagation velocity υ of the signal into the cable. Inpractical cable application, neglecting losses, the propagation velocity ofthe cable is generally defined as a percentage of the speed of light c andcan be calculated by transmission line model or measured. For a cablethe propagation velocity υ of the differential mode can be approximated,neglecting losses and frequency dependent variation in the internal di-electric insulation by (26), where εrel is the relative permittivity of the

Analysis of the parameters of a lossy coaxial cable 321

cable dielectric and c is the light propagation velocity c = 3 × 108 m/s.This approximation used in (26) leads to a small error which does notalter significantly the defect localization for cables at high frequency [17].

β = ℑmγc (25)

=

√

1

2·[

√

(R′2 + ω2 · L′2) · (G′2 + ω2 · C′2)−R′ ·G′ + ω2 · L′ · C′

]

υ =ω

β∼= 1√

L′ · C′=

c√εrel

(26)

Solving equations (24) and (26) results in:

d =c · t

2 · √εrel(27)

With the equation we can get the Fig. 16, Fig. 17 and Fig. 18.

Fig. 16. Comparison of simulated and measured signals for the TDR response(l = 0.9m,RL = 20kΩ)

Fig. 17. Comparison of simulated and measured signals for the TDR response(l = 0.9m,RL = 0Ω)

322 Q. Shi et al.

Fig. 18. Comparison of simulated and measured signals for the TDR responsein Fig.14

4. Conclusion

This paper has described a frequency dependent model for a lossycoaxial transmission line. Bessel functions are used to calculate induc-tance and resistance of the inner and outer conductor. Each of these cir-cuits contains resistance and inductance elements that represent the lowand high frequency behavior of the line. This model has been discussedin the context of a coaxial cable. The simulated results of this modelare compared with the measured results using impedance spectroscopyto show the accuracy in frequency domain. This lossy transmission linemodel is then solved in the time domain to accurately locate the faults.The simulated results are compared with the measured results using timedomain reflectometry. Finally this model is used to establish branchedcable networks and analyze the reflectometry responses of the branchednetwork of lossy wires. This model can be used to evaluate reflectom-etry signatures and communication systems made up of branched wirenetworks and this method naturally is useful for the localization anddetection of cable faults. It can also be adapted to a grid of wires suchas may be used in interconnected sensor networks.

References

[1] P. Boets, T. Bostoen, L. Van Biesen and T. Pollet. Preprocessing of Signalsfor Single-Ended Subscriber Line Testing. IEEE Trans. on Instrumentationand Measurement, 55:1509–1518, 2006.

[2] Y. C. Chung, C. Furse and J. Pruitt. Application of phase detectionfrequency domain reflectometry for locating faults in an F-18 flight controlharness IEEE Trans. on Electromagnetic Compatibility, 47:327–334, 2005.

Analysis of the parameters of a lossy coaxial cable 323

[3] C. Buccella, M. Feliziani and G. Manzi. Detection and localization of de-fects in shielded cables by time-domain measurements with UWB pulseinjection and clean algorithm postprocessing. IEEE Trans. on Electro-magnetic Compatibility, 46:597–605, 2004.

[4] A. Carravetta, M. D’Arco and N. Pasquino. A ground monitoring systembased on TDR tests. Intrumentation and Measurement Technology Conf.,I2MTC ’09, :244–248, 2009.

[5] C. R. Sharma, C. Furse and R. R. Harrison. Low Power CMOS-Sensor forlocating faluts in aging aircraft wiring. IEEE Sensors Journal, 7:43–50,2007.

[6] P. Crapse, J. Wang, Y.-J. Shin, R. Dougal, T. Mai, J. Molnar and T. Lan.Design of optimized reference signal for Joint Time-Frequency DomainReflectometry-based wiring diagnostics. Int. Conf. on Autotestcon, :195–201, 2008.

[7] H. Shinagawa, T. Suzuki, M. Noda Y. Shimura, S. Enoki and T. Mizuno.Theoretical Analysis of AC Resistance in Coil Using Magnetoplated Wire.IEEE Trans. on Magnetics, 45:3251–3259, 2009.

[8] L. Hayden and V. Tripathi. Characterization and modeling of multipleline interconnections from time domain measurements. IEEE Trans. onMicrowave Theory and Techniques, 42:1737–1743, 1994.

[9] M. Smail, L. Pichon, M. Olivas F. Auzanneau and M. Lambert. Detectionof Defects in Wiring Networks Using Time Domain Reflectometry. IEEETrans. on Magnetics, 46:2998–3001, 2010.

[10] K. Siebert, H. Gunther S. Frei and W. Mickisch. Modeling of FrequencyDependent Losses of Transmission Lines with VHDL-AMS in Time Do-main. 20th Int. Zurich Symp. on Electromagnetic Compatibility, 7:313–316, 2009.

[11] R. P. Clayton. Analysis of Multiconductor Transmission Lines. NewYork,n Wiley, 2007.

[12] M. Lee, B. Kramer C.-C. Chen and J. Volakis. Distributed Lumped Loadsand Lossy Transmission Line Model for Wideband Spiral Antenna Minia-turization and Characterization. IEEE Trans. on Antennas and Propaga-tion, 55:2671–2678, 2007.

[13] S. A. Schelkunoff. The electromagnetic theory of coaxial transmissionlines and cylindrical shields. Bell Syst. Tech. J., :532–579, 1934.

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[16] S. Haykin. Digital Communication. New York, Wiley, 1988.

[17] Q. Shi,, U. Troltzsch and O. Kanoun Detection and localization of cablefaults by time and frequency domain. 7th International Multi-Conferenceon measurements Systems Signals and Devices, :1–6, 2010.

Biographies

Qinghai Shi received the Diploma degree in electrical

engineering from the Dresden University of Technology,

Dresden, Germany, in 2008. Since 2009 he has been a re-

search assistant and PhD student with the Chair for Mea-

surement and Sensor Technology at the Chemnitz Univer-

sity of Technology, Germany. His research concerns wire

fault detection and location with time domain reflectome-

try, impedance spectroscopy and network analyzer, mod-

eling of the transmission line, network topology, bio-impedance measurement,

design of sensor systems and methods for signal processing.

Uwe Troeltzsch since 2008 is the leader of the impe-

dance spectroscopy group at the Chair for Measurement

and Sensor Technology, Chemnitz University of Technol-

ogy. His main research interests are modeling of tech-

nical, biological and chemical systems and methods for

signal processing. After studying electrical engineering,

from 2002 to 2005 he was research assistant at the Uni-

versitat der Bundeswehr in Munich, Germany where he

worked on model-based diagnosis of secondary batteries. For his dissertation

in 2005 he received a research award from the Universitat der Bundeswehr.

From 2005 to 2008 he was an airforce officer in the Federal Armed Forces

Technical Command in Cologne.

Olfa Kanoun is since 2007 university professor for mea-

surement and sensor technology at Chemnitz University of

Technology, Germany. She studied electrical engineering

and information technology at the Technical University

in Munich from 1989 to 1996, where she specialized in

the field of electronics. During her PhD at the University

of the Bundeswehr in Munich she developed a novel cal-

ibration free temperature measurement method and was

awarded in 2001 by the Commission of Professors in Measurement Technology

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