R

C

EL

a

ARRAA

KPTSAP

C

1

nttapc

0h

Electric Power Systems Research 99 (2013) 38– 44

Contents lists available at SciVerse ScienceDirect

Electric Power Systems Research

jou rn al h om epa ge: www.elsev ier .com/ locate /epsr

eview

haracteristics of partial discharge pulses propagation in shielded power cable

lhanafi Ouatah, Soufiane Megherfi, Karim Haroun, Youcef Zebboudj ∗

aboratoire de Génie Electrique, Université A. Mira de Béjaïa, 06000 Béjaïa, Algeria

r t i c l e i n f o

rticle history:eceived 3 July 2012eceived in revised form 19 January 2013ccepted 21 January 2013vailable online 6 March 2013

eywords:D discharge

a b s t r a c t

Partial discharges originates from defects in the insulation material of shielded power cable, joints andmachine coil. In cable the discharges induce small high frequency pulses which propagate to both cableends and the time domain reflectometry detection method of the PD pulse is suitable in this case. Inthis paper, a new experimental method has been introduced, it consist to inject at one end of the powercable one high frequency pulse generated by negative corona discharge in point-plane system. Under-standing high-frequency attenuation in transmission cable is relevant for PDs detection and widebandsignals distributions over such cable. We have realized in laboratory a high frequency pulse generator

richel pulseshielded power cablettenuation constantropagation length

and 50 m length of 30 kV XLPE power cable is used for the tests. The experimental results show that thepulse amplitude decreases and the pulse width increases with the propagated distance and this resultis in good agreement with the actual theoretical analysis based on the frequency attenuation. Impor-tant characteristics of PD pulse are successively determined at high frequency with maximum accuracypossible such as the wave speed, the relative permittivity, the characteristic impedance and the XLPEattenuation factor.

© 2013 Elsevier B.V. All rights reserved.

ontents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382. Propagation model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393. Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1. Pulses generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2. Constitution of the cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1. Trichel pulses waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.2. PD pulse propagation along power cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3. Length dependency of propagation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

. Introduction

Partial discharge (PD) measurements are a proven way to diag-ose power-cables. They are discharges that bridge only part ofhe insulation material in power cable or accessory and are indica-

can be altered or re-arranged by the propagation characteristics ofpropagation channel in the power cable.

Partial discharge in the XLPE insulation can generate frequenciesup to several hundred MHz [1]. So, the location of PD source islimited by high frequency attenuation of PD pulse as they propagate

ive for the local insulation conditions. A partial discharge induces short current pulse in the surrounding conductors, which willropagate to cable ends. Since we are only able to measure at theable ends in substations or ring main units, the measured PD pulses

∗ Corresponding author. Tel.: +213 34 21 43 33/34; fax: +213 34 21 60 98.E-mail address: yzebboudj@yahoo.fr (Y. Zebboudj).

378-7796/$ – see front matter © 2013 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.epsr.2013.01.012

through the cable [2]. The measurement and location of the defectas result by PDs are of significant importance for power systemreliability. In order to understand and predict the ageing of cableinsulation subject to these discharges, it is necessary to investigate

and understand the occurrence of PD signals and their propagationcharacteristics in XLPE cables [3].

Time domain reflectometry (TDR) is one of the diagnostic tech-niques that can be used to localize degraded sections of a power

E. Ouatah et al. / Electric Power Systems Research 99 (2013) 38– 44 39

cimip

tptTot

2

bsad

wopp

soa

I

w(n

Q

a

V

wi

V

w

V

T

V

wta

Fig. 2. Experimental assembly with the pulse generator.

and thermal data are shown in Table 2.

Fig. 1. Model of element dx of a two-conductor transmission line.

able. This technique is implemented by injecting a voltage pulsento the cable and measuring the returning reflections. On-line TDR

easurements are performed on the operating power cables bynjecting and detecting pulses through sensors [4] mounted on theower cable.

This paper gives an experimental study of pulse propaga-ion along XLPE cable by using the TDR technique. The injectedulse is generated in laboratory with a point-plane system. Whenhe corona occurs between non-uniform electrode arrangements,richel pulses [5] for which the rise time of some nanoseconds arebserved. They are able to propagate on a long distance along aransmission coaxial cable.

. Propagation model

The coaxial structure of a single-phase shielded power cable cane described by a normal two-wire transmission line model. Fig. 1hows the model of an element dx of a transmission line. The outputnd input voltage and current in frequency domain of this elementx are shown in Appendix A.

PD pulses tend to be either very narrow, symmetric pulses,hich can be modeled as Gaussian with a pulse width in order

f nanosecond range or somewhat large pulses asymmetric with aulse width of a few tens of nanoseconds [6]. The asymmetric PDulse is modeled as a sum of Gaussians [7].

A model of the attenuation theory is recently described by Ous-alah et al. [6,7] where the authors introduced an attenuation termf the PD signal. They start by assuming that the PD pulse is gener-ted by a Gaussian current propagating in the conductor:

(t) = I0 e−(t2/2�2) (1)

here I0 is the peak pulse current amplitude and the pulse widthfull width at half maximum) is 2.36 �. The partial discharge mag-itude of this pulse is given by:

=∫

I(t) dt = I0�√

2� = V0�√

2�

Z(2)

nd

0 = ZI0 = ZC

2I0 (3)

here V0 is the peak voltage and Zc is the cable characteristicmpedance. Thus the voltage waveform is:

(t) = V0 e−(t2/2�2) = QZC

2�√

2�e−(t2/2�2) (4)

here Q is the partial discharge magnitude (C).The Fourier transform of a Gaussian is a Gaussian:

(w) = QZ e−(1/2) ω2�2. (5)

hey multiply by the frequency dependent attenuation:

A(ω) = QZ e−(1/2) ω2�2 · e−ω˛L (6)

here ̨ is the attenuation factor, L is the distance propagated byhe PD pulse, t is the time, � characterizes the initial pulse widthnd ω = 2�f the frequency.

Fig. 3. Constitution of the cable.

3. Experimental set-up

3.1. Pulses generator

The experimental method aims to obtain partial dischargepulses in laboratory to inject into 30 kV XLPE shielded power cablein order to observe the propagated of this pulse. A pulse gener-ator based on corona discharge in point-plane system is realizedand shown in Fig. 2. Negative direct voltage, supplied by HV DCsource (1) (Phywe 0–±7 kV, 0.5 mA), is applied to the point (2) viaa 15 k� protection resistor (3) and an electrostatic voltmeter (4)(Phywe 0–7.5 kV) is used to control the high voltage V. The plane(5) of 150 mm diameter is set to earth via a circuit RC (6) and thesmooth point of radius curvature r = 100 �m is fixed with an insu-lating prop (7). The point-to-plane spacing is fixed at d = 4 mm anda cylindrical guard electrode (8) is used to protect the system fromexternal electromagnetic field. An oscilloscope Tektronics TDS 1012with interface TDS 2CMAX (9) is used to display and data acquisi-tion. The computer (10) connected to the oscilloscope via RS232bus (11) is introduced for data processing. The measured inceptioncorona voltage in negative polarity is Vi = 4.5 kV and the breakdownvoltage is Vb ≈ 6.5 kV. The corona regime is included between Vi andVb. At the negative corona voltage V = 5 kV we see clearly the directcurrent component and the pulse train component (Trichel pulses)of the corona discharge with the oscilloscope TDS 1012.

3.2. Constitution of the cable

To study the propagation of PD pulses in power cables, we used30 kV power cable. The used cable is constituted of elements pre-sented in Fig. 3. The geometric data are shown in Table 1, the electric

Table 1Geometry data of 30 kV XLPE power cable.

Geometry data (mm)

RAL a1 b a2 RCO c6.28 0.6 8.35 0.6 15.83 2.85

40 E. Ouatah et al. / Electric Power Systems Research 99 (2013) 38– 44

Table 2Electrical data of 30 kV XLPE power cable.

Electrical and thermal data

Dielectric losses factor (tg ı) 4 × 10−3

Electric conductivity (�) 10−12 (� cm)−1

Thermal conductivity (k) 0.286 W/m ◦CSpecific heat (cv) 2.08 J/cm3 ◦C

Ft

4

4

ewsatdtainT1aC1

4

tt(tn

FT

ig. 4. Shape of the regular Trichel pulse train obtained with R = 5 k� and a capaci-ance C = 20 pF.

. Results and discussion

.1. Trichel pulses waveform

The specificity of positive or negative DC discharge is the exist-nce of a drift zone of the ions which acts like an impedance in seriesith the zone of ionization and gives to the corona discharges their

tability [8]. In the case of a negative discharge, the positive ions cre-ted by ionization can return to cathode enough quickly, whereashe negative ions created by attachment of electrons migrate in therift zone where the field is weaker. The corona current createshrough the RC circuit a voltage u(t) with a pulse component whichre clearly observed with the oscilloscope when the point voltages higher than Vi like shown in Fig. 4. The pulse train observed inegative voltage is regular with a frequency in order of 100 kHz.he pulses have an asymmetric form with a time rise in order of0 ns and time decay depending on the values of the capacitance Cnd the resistance R. The pulse train is obtained with R = 5 k� and

= 20 pF and the time between two successive pulses is about of2 �s.

.2. PD pulse propagation along power cable

If we connect a coaxial cable of capacitance Cca with the resis-ance R, Fig. 5, the pulses can be propagate along the cable with a

otal reflection at the cable ends. The injected pulse signal at pointA) is transmitted into the coaxial cable and is totally reflected athe open end (B), the reflected pulse signal then returns to the con-ection point (A) and the pulse oscillates between the point (A) and

ig. 5. Coupling of the 30 kV shielded XLPE power cable with the pulse generator: (1) 30DS1012 with interface TDS2CMAX, (5) RS232 Bus, (6) acquisition computer.

Fig. 6. PD pulse propagation in 30 kV shielded XLPE cable of 130 m length (initialpulse and reflected pulse.

the point (B). For our investigation we used the cable presented inSection 3.2 with different length 30, 40, 50, 80, 130, 170 and 200 m.The per unit capacitance of the cable and the relative permittivitymeasured at low frequency f = 50 Hz with the Sheering bridge arerespectively Cl = 0.153 nF/m and εr = 2.63.

For example, with the cable of 130 m length, we have observedat the oscilloscope, Fig. 6, the initial PD pulse (point A, x = 0) and thereflected pulse at the end of the cable (point B, x = 130 m). The PDpulse are propagated a distance L = 2 × 130 m.

To observe the progression of PD pulse on a long distance, ashort length cable is suitable and the oscilloscope can unregisterall the reflected pulses at the cable ends. Indeed, we have observedwith the cable of 50 m length 16 successive reflected pulses whichcorrespond to a propagated distance L = 1600 m for the injected PDpulse as shown in Fig. 7 after data acquisition with the computer.The initial pulse (L = 0) is shown in Fig. 8 where we notice that theform is Gaussian (symmetric) of 28 MHz frequency. We notice alsoa small distortion of the signal due to the cable inductance. Thedistortion can be limited with small values of the resistance R inparallel with the cable.

The propagation velocity of PD signal can be calculated by thefollowing formula:

� = L

t(7)

where L = 2, l is the distance propagated by the pulse and l is thecable length, t is the time interval between the pulses (incidentand reflected). Two phenomena are also observed: the decreaseof the amplitude and the increase of the band width of the pulse.The propagation velocity of pulses signal can be measured in ashort cable with reasonable accuracy. But the time interval will befar more complicated and affected by many factors such as bandwidth of the measurement system, sampling rate, waveform oscil-

lation and overlapping, attenuation of high frequency components,background noise etc. Ideally the time interval should be measuredbetween the first and the second peak.

kV shielded XLPE power cable, (2) pulses generator, (3) resistance, (4) oscilloscope

E. Ouatah et al. / Electric Power Systems Research 99 (2013) 38– 44 41

ds of 3

p

ε

ls

r

w

ı

wpp

ia

l

l

l

s

•

•••

Fig. 7. Successive reflected pulses at the two en

The relative permittivity of the XLPE can be deduced from theropagation velocity of the PD pulse according to the formula:

r = c2

v2(8)

At high frequency the resistance r of a conductor of radius a andength l is given as a function of the material conductivity � and thekin effect ı:

(�) = 1�2�˛ı

(9)

ith:

(m) =√

1�or�f

(10)

here 0 is the vacuum permeability, r the material relativeermeability and f the signal frequency. The resistance is thus pro-ortional to the square root of the frequency.

The per unit length coaxial cable inductance lca is the sum of thenternal inductance lcai and the external inductance lcae and theyre given by the following expressions [9]:

cal (H/m) = oıAL

4�RAL+ oıCO

4�RCO(11)

cae (H/m) = o

2�ln

RCO

RAL(12)

ca (H/m) = lcal + lcae (13)

For 28 MHz frequency of the injected pulse in the 50 m lengthhielded power cable, we have obtained:

ıAL = 0.0154 mm, rAL = 2.184 � for the inner conductor of alu-

minum,ıCO = 0.0123 mm, rCO = 0.687 � for the outer conductor of copper,lcai = 0.323 × 10−9 H/m, lcae = 1.809 × 10−7 H/m, lca = 0.181 �H/m,Lca = 9.05 �H, Cca = 7.65 nF for the power cable.

Fig. 8. Gaussian form of the initial injected pulse in 3

0 kV shielded XLPE power cable of 50 m length.

where 0 = 4� × 10−7 H/m and r = 1 for both aluminum and cop-per. The resistances of both the inner conductor rAl and theouter conductor rCO can be neglected in compared with theinductive reactance (Lcaω) = 1593 � and the XLPE power cableinsulation resistance is very higher than the capacitive reactance(1/Ccaω) = 0.743 �. Thus, the characteristic impedance Zc of theshielded power cable is independent of the frequency and it canbe given by the simplifying expression:

Zc =√

Lca

Cca= 1

2�

√o

εoεrln

RCO

RAL(14)

The PD pulse amplitude can be also measured in function ofthe propagated distance L at constant frequency. Thus the relativeamplitude can be given as follow:

A(%) = A(x = L)A(x = 0)

(15)

The measured results are summarized in Table 3. Fig. 9 showsthe PD pulse propagated distance L in function of the time t, mea-sured with the oscilloscope. The deduced value of the propagatedvelocity is v = 1.73 × 108 m/s with an uncertainty of v/v = 0.2%. TheXLPE relative permittivity obtained is εr = 3 with an uncertainty�εr/εr = 7%. We notice that this value at high frequency f = 28 MHzis higher than the value obtained with the Schering bridge at lowfrequency f = 50 Hz. The characteristic impedance deduced frommeasurement is Zc = 32 � with an uncertainty �Zc/Zc = 3.6%.

4.3. Length dependency of propagation parameters

The frequency-dependent attenuation of power cable compli-cates PD location in several respects. In XLPE cable it’s simply the

sum of the power loss in the dielectric and dissipation in the semi-conductor layers. In our investigation we observe the progression ofone PD pulse of 28 MHz frequency and we can thus study the influ-ence of the propagated distance on the pulse amplitude. Fig. 10

0 kV shielded XLPE power cable of 50 m length.

42 E. Ouatah et al. / Electric Power Systems Research 99 (2013) 38– 44

Fig. 9. PD pulse propagated distance L as a function of the time t in 30 kV shielded XLPE power cable of 50 m length.

tion o

sgtpo

A

t(˛s

TE

Fig. 10. PD pulse relative amplitude A(%) as a func

hows the relative amplitude A (%) as a function of the propa-ated distance L. We notice that the pulse amplitude decrease withhe distance L and in the first order an exponential decay law asredicted by the equation (6) is found experimentally. We havebtained from the curve-fitting the following law:

(%) = 7.12 + 91.2 e−(L/122.37) (16)

We can identify with an acceptable precision the attenuation

erm -ω ̨ of the equation (6) with the term −1/122.37 of equation16) and then with f = 28 MHz we obtain the attenuation constant

= 4.64 × 10−11 Nepers s/m. This value is in good agreement withome of the earliest attenuation measurements from 1982 and most

able 3xperimental results with 50 m length shielded XLPE power cable, f = 28 MHz.

L (m) A (%) t (s) � (m/s) �r Zc (�)

0 100 – – – –100 42.42 5.83E−07 1.71E+08 3077 31.585200 26.46 1.15E−06 1.73E+08 3 31.988300 16.77 1.73E−06 1.73E+08 3 31.988400 16.20 2.30E−06 1.74E+08 2.97 32.149500 12.21 2.87E−06 1.74E+08 2.97 32.149600 11.07 3.46E−06 1.73E+08 3 31.988700 9.36 4.04E−06 1.73E+08 3 31.988800 7.08 4.59E−06 1.74E+08 2.97 32.149900 6.51 5.15E−06 1.75E+08 2.94 32.312

1000 5.94 5,74E−06 1.74E+08 2.97 32.1491100 5.94 6.33E−06 1.74E+08 2.97 32.1491200 5.94 6.90E−06 1.74E+08 2.97 32.1491300 5.37 7.55E−06 1.72E+08 3.04 31.7761400 5.37 8.09E−06 1.73E+08 3 31.9881500 5.37 8.68E−06 1.79E+08 3.15 31.2171600 4.80 9.19E−06 1.74E+08 2.97 32.149

Average values 1.73E+08 3.00 31.992

f the propagated distance L at 28 MHz frequency.

recent measurements of Lemke for XLPE cable [6]. The attenuationconstant measured is from 3.28 × 10−11 at 6.14 × 10−11 Nepers s/m.

The PD level (charge) can be calculated by the following formula:

Qi = Cca · V0,i (17)

where: Qi and V0,i are respectively the charge and the amplitude ofthe ith PD pulse, Cca = 7.65 nF is the 50 m length cable capacitance.

Fig. 11 shows the variation of PD charge with the propagateddistance. We notice that PD charge decreases like the relativeamplitude with the distance L (first order an exponential decaylaw).

With data acquisition we can show clearly the shape distor-tion of pulses after propagation along the shielded XLPE powercable of 50 m length due to the dispersion at high frequencies. Thewaveforms of the initial injected pulse and the reflected pulse afterhaving propagated a distance L = 100 m are represented in Fig. 12.For the first pulse, the rise time is 22 ns and the width is 36 ns.On the other hand, the second becomes a pulse with the risetimeof 28 ns and width of 70 ns. We notice that both the first and thesecond pulse keep the Gaussian waveform and their amplitude arerespectively 708 mV and 284 mV which prove that the signal atten-uates during its propagation along the cable. In 1982 Stone andBoggs [10] supposed that the losses at the high frequencies in thecables with screen are caused by the radial capacitive current. Inthe case of concentric neutral cables, there are additional losses

function of the number of neutral wires [11–13]. It proves that thelosses increase with the number of neutral wires [11], which is avery significant result in the design of shielded power cables. Themeasurement of the dielectric properties of the semiconductorscreens is often desirable to predict the losses at the high frequen-cies in cables [14].

E. Ouatah et al. / Electric Power Systems Research 99 (2013) 38– 44 43

Fig. 11. PD level as a function of the propagated distance L at 28 MHz frequency.

Fig. 12. PD pulse form after 100 m propagated distance.

the p

pwd

5

dccgfcobau

Fig. 13. PD pulse width PW as a function of

Fig. 13 shows the pulse width PW at (2/3), (1/2) and (1/3)ulse amplitude as a function of the propagated distance Lhere we notice that PW increase linearly with the propagatedistance.

. Conclusion

In this paper, an improved experimental method has beeneveloped to analyze the partial discharge pulses propagationharacteristics in shielded XLPE power cable. For better approa-hing the real case of PD pulses, we have realized in laboratory aenerator based on the point-plan system able to generate highrequency pulses similar to PD pulses occurring in shielded powerable. One PD high frequency pulse is injected at the one of ends

f 30 kV shielded XLPE power cable of 50 m length, it oscillatesetween the cable ends with a total reflection. 16 oscillations (gond return) has been clearly observed in oscilloscope which allows to analyze the pulse in the time domain reflectometry (TDR)up

ropagated distance L at 28 MHz frequency.

to 1600 m propagation before the quasi total attenuation of thesignal.

The experimental results show that the pulse amplitudedecreases and the pulse width increases with the propagated dis-tance. They are in good agreement with the actual theoreticalanalysis based on the frequency attenuation. Important character-istics of PD pulse on XLPE shielded power cable are successivelydetermined at high frequency with maximum accuracy possiblesuch as the propagated velocity, the relative permittivity, the char-acteristic impedance and the XLPE attenuation factor.

It should be noted that the results are obtained at 28 MHz fre-quency which permits the PD detection up to 1600 m. Beyond thisfrequency the propagated distance is gradually reduced and at lowfrequency the PD width is very large, so the detection becomes

more complicated. 30 MHZ frequency is the limit of PD detec-tion with the TDR method on a long length power cable. However,the attenuation factor obtained with our experimental method on1600 m propagated distance of the PD pulse is accurate.

4 er Syst

A

o

lew

V

I

wtdp

�

�

pat

Z

[

[

[

[loss from neutral wire – shield interaction, IEEE Electrical Insulation Magazine

4 E. Ouatah et al. / Electric Pow

ppendix A.

The output and input voltage and current in frequency domainf this element dx are related by:

∂

∂x

[V(x, w)

I(x, w)

]= −

[0 Z(ω)

Y(ω) 0

] [V(x, ω)

I(x, ω)

](A.1)

Where Z(ω) and Y(ω) represent respectively the per-unit-lengthine impedance and admittance. If we decouple the differentialquations, the plane wave solutions in terms of forward and back-ard propagating waves are obtained:

(x, ω) = V+ e−�(ω)x + V− e+�(ω)x (A.2)

(x, ω) = I+e−�(ω)x − I−e+�(ω)x (A.3)

here the + denotes the forward (in positive x-direction) and –he backward propagating wave. The effect of attenuation andispersion introduced by the cable is described by cable-specificropagation constant, given by:

(ω) =√

Y(ω)Z(ω) (A.4)

The propagation constant consists of real and imaginary parts:

(ω) = ˛(ω) + jˇ(ω) (A.5)

The real part ˛(ω) described the attenuation and the imaginaryart ˇ(ω) the delay and dispersion of the signal. The ratio of voltagend current of (forward and backward) propagating waves definedhe characteristic or wave impedance of the cable, it is given by:

c(ω) =√

Z(ω)Y(ω)

(A.6)[

ems Research 99 (2013) 38– 44

References

[1] S.A. Boggs, A. Pathak, P. Walter, Partial discharge XXII: high frequency attenu-ation in shielded solid dielectric power cable and implications thereof for PDlocation, IEEE Electrical Insulation Magazine 12 (1) (1996) 9.

[2] H.N.O.T.R. Blackburn, B.T. Phung, M. Vakilian1, M.S. Naderi1, H. Zhang, Investi-gation of high frequency signal propagation characteristics on HV XLPE Cables,in: 7th International Power Engineering Conference, vol. 2, 2005, p. 776.

[3] C. Pei, X. Jiang, F. Song, Propagation characteristics of partial discharge signals,International Journal of Emerging Electric Power Systems 11 (2010) 7.

[4] V. Dubickas, H. Edin, Couplers for on-line time domain reflectometry diag-nostics of power cables, in: Conference on Electrical Insulation and DielectricPhenomena, 2004, p. 210.

[5] G.W. Trichel, The mechanism of the negative point to plane corona near onset,Physical Review 54 (1938) 1078.

[6] N. Oussalah, Y. Zebboudj, S.A. Boggs, Analytic solutions for pulse propagationin shielded power cable for symmetric and asymmetric PD pulses, IEEE Transon Dielectrics and Electrical Insulation 14 (5) (2007) 1264.

[7] N. Oussalah, Y. Zebboudj, S.A. Boggs, PD pulse propagation in shielded powercable – implication for PD detection sensitivity, IEEE Electrical Insulation Mag-azine 23 (2007) 6.

[8] Y. Zebboudj, R. Iken, Positive corona inception in HVDC under variable air den-sity and humidity conditions, European Physical Journal: Applied Physics 10(2000) 211.

[9] H. Kaden, Wirbelströme und Schirmung in der Nachrichtentechnick, Spriger-Verlag, Berlin/Göttingen/Heidelberg, Germany, 1959.

10] G.C. Stone, S.A. Boggs, Annu. Rep. Conf. Elect. Insul. Dielect, Phenomena,National Academy of Science, Washington DC, 1982, 275.

11] C. Xu, L. Zhou, J.Y. Zhou, S.A. Boggs, High frequency properties of shielded powercables. Part 1: overview of mechanisms, IEEE Electrical Insulation Magazine 21(6) (2005) 24.

12] G. Mugala, R. Eriksson, U. Gafvert, P. Pettersson, Measurement technique forhigh novel characterization of semi-conducting materials in extruded cables,IEEE Transaction on Dielectrics and Electrical Insulation 11 (3) (2004) 471.

13] C. Xu, S.A. Boggs, High frequency properties of shielded power cables. Part 3:

23 (2) (2007) 12.14] C. Xu, S.A. Boggs, High frequency properties of shielded power cables. Part

2: sources of error in measuring shield dielectric properties, IEEE ElectricalInsulation Magazine 22 (1) (2005) 1.