Effect of temperature on the impulse life expectancyof crosslinked polythene

A.E. Davies, B.Eng. (Tech.), Ph.D., C.Eng., Mem. I.E.E.E., M.I.E.E., andB.M. Weedy, B.Sc. (Eng.), Ph.D., C.Eng, F.I.E.E.

Indexing terms: Insulators and insulation, Measurement and measuring

Abstract: The Weibull statistical method is widely used to predict the failure of electrical insulation in termsof stress and time (or, in this case, the number of applied impulses) for a given temperature. In the presentwork the probability of breakdown has been determined for flat-disc samples of crosslinked polythene,subject to repeated application of impulse voltages for various stresses and temperatures. The results obtainedindicate that an extension of the Weibull equation to include temperature is valid. Hence the probability offailure P for a given number of impulses n with the sample at stress E and temperature Tis given by P{E, n,7) = 1 — exp [ - (E/Eo )b X (n/n0 f X (T/Tof], where Eo, n0 and To refer to a probability value of 63.2%.

List of symbols

a, b, c = exponents related to number of impulses, stress andtemperature, respectively

E = electric stressn = number of impulses to failureP = probability of failure (cumulative value of the num-

ber of samples broken down as a proportion of thetotal number of samples for a given stress, tempera-ture and number of impulses)

T = temperature (absolute)0 — temperature (Celsius)

1 Introduction

When assessing the test results for the breakdown of electricalinsulation, either in the form of samples or real apparatus, theuse of a statistical method is often required. This is particularlythe case when dealing with extruded polymeric high-voltage-cable insulation, in which defects and inhomogeneities arerandomly distributed throughout the volume of the insulation.It has been found that the method due to Weibull [1] is themost applicable, and this has been widely used for severalyears for AC breakdown. The method is based on the premisethat the failure of the whole system depends on the failure ofthe weakest part.

The application of the Weibull method to the life of insu-lation, with the repeated application of impulse voltages hasreceived attention [ 2 - 6 ] , but much less than for AC break-down. In the case of repeated impulses on samples of constanttemperature, the Weibull relationship takes the form

P(E,n) = 1

where P is the probability of failure and stress E with theapplication of n impulses (n replaces the time of applicationin the AC case), and Eo and n0 refer to the stress and numberof impulses related to a probability of breakdown of 63.2%.

Some authorities introduce a threshold value of stressbelow which breakdown is impossible, although it is extremelydifficult to evaluate this stress. In practice, the absence of sucha threshold is no disadvantage, and discussion related tostresses tending to zero, requiring an infinite number ofimpulses for breakdown, is irrelevant to practical purposes.

Paper 2166A, first received 8th April and in revised form 25th June1982The authors are with the Department of Electrical Engineering, TheUniversity, Southampton SO9 SNH EnglandIEEPROC, Vol. 129, Pt. A, No. 8, NOVEMBER 1982

From the basic Weibull relationship, it can be shown that:

1In In

\-P(E,n)= b\nE-b\nE0

The plot of the left-hand side of this equation against ln(«),for a given E, results in a straight line, or alternatively the plotof the left-hand side against E for a given n. Hence the ex-ponents a and b can be evaluated. It is usual to use specialgraph paper for the Weibull plots, on which the ordinate interms of /Ms scaled to the value

In In-P(E, n)

thus avoiding repeated evaluation of this function.It is well known [7, 8] that the impulse strength of elec-

trical insulation decreases with increase in temperature. Inparticular, this has been investigated for polymer insulation[9]. The Weibull characteristics are usually related to theoperating temperature and, frequently with tests on cables,related to load cycles. It would be very useful to includetemperature as a variable in the Weibull expression, so thatthe probability of failure can be predicted for any stress, num-ber of impulses and temperature. In the present work, it isproposed to test experimentally the validity of a Weibullequation of the form:

<•>

Experimentation

The crosslinked polythene samples were in disc form, ofdiameter 37 mm and thickness of 6 mm. They were moulded25 at a time in a jig, subjected to a high pressure and heatedby steam to 175°C for vulcanisation. The basic polythenematerial used was Unifos HFBB 4201 natural.

The discs were moulded with a hemispherical indent onone side into which the high-voltage spherical electrode wasfitted, good contact being ensured by the use of a conductingmedium. The effective sample thickness between the high andlow-voltage spherical electrodes was 0.5 mm. The electrodeassembly and sample were immersed in a glass container filledwith silicone oil. The earth and temperature sensor (therm-istor) leads were brought out through the bottom of the con-tainer, which was wrapped with heater tapes. The current tothe tapes was controlled by a preset constant temperaturedevice in conjunction with the thermistor situated close to the

0143- 702X/82/08060 7 + 04 $01.50/0 607

sample. Surge voltages were derived from a conventionalimpulse generator, the wave having a rise time of 1.25 //s and ahalf-value fall time of 47/is.

3 Results

The probability P of sample breakdown is the cumulative valueof the number of samples broken down as a proportion of thetotal number of samples tested (50 for each condition), for agiven number of impulse applications. The determination of Pis illustrated by the hypothetical example shown in Table 1,in which m,- is the number of samples which have failed fora given number of impulses n, Pt is the failure probability forthat number of impulses, and P is the cumulative probabilitywhich includes the total number of samples which have failedup to and including the number in question, in terms of thesample population. The first row of Table 1 indicates thatthree samples failed after one impulse on each, giving a valuefor Pt of 3/50, i.e. 0.06. As n is 1, the value of /Ms also 0.06.

Table 1: Determination of P

0.060.040.00.02

0.060.10.10.12

2 3 U 5 6 810 20 3040 6080 200 500number of impulses n

Fig. 1 Weibull plot - temperature 20° C

a Stress 220 kV mm' 1 , b stress 200 kV mm" 1 , c stress 160 kV mm"From eqn. 1,

absolute temperature: temperature in celsius

99.999

90

=1 70

t 50

HE 3 00 2 0

X>2 10a

1 3° 2

I 1u

0.50.30.20.11 2 3 A 5 6 8 10 20 30 50 80100 200 300

number of impulses nFig. 2 Weibull plot - temperature 55°C

a Stress 175 kVmm"1, b stress lSOkVmm"1 , c stress UOkVinm'1

From eqn. 1,absolute temperature; temperature in Celsius

In the second row, two samples have failed with the applicationof two impulses of each giving a Pt of 2/50, i.e. 0.04, and acumulative probability of P of 0.06 + 0.04, i.e. 0.01. TheTable proceeds with ascending numbers of impulses until all50 samples have failed. This was repeated for each stress andtemperature, and, in all, 450 samples were used.

The Weibull plots for stresses in the range 220 kV mm"l to1 h 2 d 3 f ° °

p gn S V are shown in Figs. 1, 2 and 3 for 20°C, 55°Cand 90°C, respectively. It is seen that the plots compromisetwo linear relationships of different slope.

99.999

90: 70

50

r 30

i 2 °: 10

j- 53

1 2

1! 0.5

0.30.2

0.11 2 3 A 56 8 10 20 30 50 80100 200 300

number of impulses n

Fig. 3 Weibull plot - temperature 90°C

a Stress 160kVmm"' , b stress HOkVmm"1 ,c stress 125 kVmm"1

From eqn. 1,absolute temperature; temperature in Celsius

4 Discussion

The graphs of ln[ln{l/l —P{n, E, T)}] against ln(«) exhibittwo distinct linear portions. This has also been reported [2]for XLPE insulation at ambient temperature. In Reference2 it was suggested that the linear portion for the lower valuesof n relates to manufacturing defects in the sample, and thepart for the high values of n describes the effects of thedistributed inhomogeneities in the material. However, there isno physical evidence to support this hypothesis. It wouldbe prudent to assess insulation on the first (low n) linearportion of the graph, in view of the number of impulseslikely to be experienced by a cable in service. In the presentwork, the parameters for both linear parts will be quoted.

Eqn. 1 may be rewritten

In In1

\-P{E,n, T)= b In E — b In Eo

+ aln(n)-aln(no) + c\nT-c\nTo (2)

For a given value of E and T, eqn. 1 may be written

1In In

l-P(E,n, T)— a ln(«) + constant (3)

Hence, from the graphs in Figs. 1, 2 and 3, the value of a isobtained.

For a given temperature and a probability of failure of63.2% from eqn. 2,

b In E + a ln(«) = constant

or

In E = ln(rt) + constantb

(4)

Hence ajb is determined from the plot of hi E against ln(«),

608 IEEPROC, Vol. 129, Pt. A, No. 8, NOVEMBER 1982

for each of the temperatures used. Knowing a and a/b, b isdetermined. It should be noted that, because of the twodistinct linear parts obtained for the plot of eqn. 3, three setsof two graphs are obtained for hi E against ln(rt), as shown inFig. 4.

300

200

10010 100 1000

Fig. 4 Stress - number impulses for 63.2% probability of failure

a 20°C,6 55°C,c 90°Cx x x initial slope on Weibull ploto o o second slope on Weibull plot

The sets of values of a and b for each temperature are givenin Table 2.

Table 2: Values for exponents

Temperature, K a. J b,

293328363

Temperature inCelsius

0.80.80.8

0.50.50.5

988

655

Values of a and b as above

151515

1.8

9.49 49.4

1 2

The graphs in Fig. 4 relate E and n for a 63.2% value forP and for given temperatures. Each curve involves combinationsof values for Eo and n0 such that P is 63.2%. If a particularvalue is taken for Eo, in this case 160 kV mm"*, there is aunique value of nQ for a given temperature. The values of aand b are independent of the choice of EQ.

1000

100

10

absolute temperature

Celsius

100 200 300 400 500temperature, K

10 20 30 40temperature,°C

50 60 70 8090

Fig. 5 Number of impulses (n) against temperature

Stress 160kVmm"'x x x initial slope of Weibull plot• • • second slope of Weibull plot

Again from eqn. 2, for a given stress (Eo = 160kVmm *)and 63.2% probability,

ln(«) = — In T+ constanta (7)

and this is plotted in Fig. 5.From this the value of the exponent c is determined and

shown in Table. 2.Fig. 5 relates n and T for P equal to 63.2%, and Eo equal to

160kVmrrfx. n and T are interdependent and, for a given n0

(in this case 148 is used), To is defined (300 K).As there are two linear portions of the Weibull plot two

values of the exponents a, b and c exist. The values of a and bare rounded off to one significant figure, as is usual in lifetesting. For example, for a at 55°C the following values wereobtained:

175kVmmlSOkVmnT1

UOkVmm"1

0.770.80.83

0.450.450.54

These were rounded to 0.8 and 0.5. Because of the narrowrange of temperature obtained, expecially when expressed inabsolute temperature, the two graphs of ln(«) against In Tcorresponding to the two breakdown regimes can be rep-resented by one straight line, as shown in Fig. 5. It can be seenin Table 2 that the two exponents for the temperature termare relatively high.

In view of the narrow temperature range of interest inpractice, i.e. ambient temperature to 90°C, it would be ofinterest to consider the use of temperature in Celsius insteadof absolute values; even though the various theoretical modelsfor breakdown involve the latter. The result of consideringCelsius is shown in Fig. 5 and these graphs result in values forcl and c2 of 1.8 (associated with a 0O of 22°C) and 1.2(27°C) for the lower n and higher n regimes, respectively. Itshould be remembered that the use of Celsius has no theoreticalbasis, this fact being emphasised by the fact that at 0°C thewhole probability expression becomes meaningless.

Using the exponent values shown in Table 2, in eqn. 1probability values have been calculated and the resultingWeibull plots for 220kVmm"1, 20°C, nSkVmm'1, 55°C,and ^ S k V m m " 1 , 90°C, are shown in Figs. 1, 2 and 3. Theabove has been performed for c exponents related to tempera-ture, in Kelvin and also in Celsius, and reasonable agreementwith the original test results is obtained with both units.

The results presented relate to an electrode-sample arrange-ment which is unsymmetric. It is the intention of the presentwork to investigate the effect of temperature on XLPE in aparticular sample form, for a particular polarity of impulsevoltage. Further work is required to obtain a fresh set ofexponents for surges of the opposite polarity. It would beunwise to suggest that the exponent values indicated in Table2 would obtain for XLPE extruded cables. However, it wouldbe reasonable to assume that the extension of the Weibullequation to include temperature is valid for cables.

5 Conclusions

The extension of the Weibull equation, for the impulse life ofcrosslinked polythene insulation to include variation in tem-perature, appears justified. For the sample arrangement tested,the equation takes the form:

' n

148P(E,n,T) = 1-exp - — TTn\ ™160 300

The two values for each exponent relate to the two distinctlinear portions of the Weibull plot, the subscript 1 referring to

IEEPROC, Vol. 129, Pt. A, No. 8, NOVEMBER 1982 609

low values of n, and the subscript 2 to higher values. Theexponents are given in Table 2.

6 Acknowledgments

The authors wish to thank Pirelli General PLC for the facilitiesmade available for sample manufacture.

7 References

1 WEIBULL, W.: 'A statistical distribution function of wide appli-cation', /. Appl. Mech., 1951, 18, pp. 293-297

2 BAGIN, V.V., VAZHOV, V.F., SOTNIKOV, V.G., and USHKOV,V.Y.A.: 'Study of the lifetime of polythene insulation under im-pulsive voltage, Elektrichestvo, 1977, 5, pp. 79-81

3 KANEKO, R., and SUGIYAMA, K.: 'Statistical considerations onimpulse breakdown characteristics of cross-linked polyethyleneinsulated cables',IEEE Tram., 1975, PAS-94, pp. 367-374

4 OCCHINI, E.: 'A statistical approach to the discussion of thedielectric strength in electric cables', ibid., 1971, PAS-90, pp.2671-2682

5 McKEAN, A.L.: 'Breakdown mechanism studies in cross-linkedpolyethylene cables', ibid., 1976, PAS-95, pp. 253-260

6 TSUMOTO, M., and IWATA, M.: 'An application of Weibulldistribution to impulse breakdown of cross-linked polyethylenepower cables', Fujikura Tech. Rev. 1975, 7, pp. 19-22

7 BLYTHE, A.R.: 'Electrical properties of polymers' (CambridgeUniversity Press, 1979)

8 GRANEAU, P.: 'Underground power transmission' (J. Wiley &Sons, 1979)

9 LAWSON, W.G.: 'Effects of temperature and techniques of measure-ment on the intrinsic electric strength of polythene', Proc. IEE,1966, 113,(1), pp. 197-202

610 IEE PROC, Vol. 129, Pt. A, No. 8, NOVEMBER 1982