IEC 62539:2007

INTERNATIONAL IEC
STANDARD 62539
First edition
2007-07
IEEE 930
Guide for the statistical analysis of electrical
insulation breakdown data
Reference number
IEC 62539(E):2007
IEEE Std 930-2004
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INTERNATIONAL IEC
STANDARD 62539
First edition
2007-07
IEEE 930
Guide for the statistical analysis of electrical
insulation breakdown data
© IEEE 2007   Copyright - all rights reserved
IEEE is a registered trademark in the U.S. Patent & Trademark Office, owned by the Institute of Electrical and Electronics Engineers, Inc.
No part of this publication may be reproduced or utilized in any form or by any means, electronic or
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МеждународнаяЭлектротехническаяКомиссия

–2 – IEC 62539:2007(E)
IEEE 930-2004(E)
CONTENTS
FOREWORD .4
IEEE Introduction .7
1. Scope. 8
2. References. 8
3. Steps required for analysis of breakdown data . 9
3.1 Data acquisition . 9
3.2 Characterizing data using a probability function . 10
3.3 Hypothesis testing.11
4. Probability distributions for failure data.12
4.1 The Weibull distribution.12
4.2 The Gumbel distribution.13
4.3 The lognormal distribution .13
4.4 Mixed distributions .13
4.5 Other terminology.14
5. Testing the adequacy of a distribution .14
5.1 Weibull probability data .14
5.2 Use of probability paper for the three-parameter Weibull distribution .15
5.3 The shape of a distribution plotted on Weibull probability paper .16
5.4 A simple technique for testing the adequacy of the Weibull distribution .16
6. Graphical estimates of Weibull parameters . 17
7. Computational techniques for Weibull parameter estimation . 17
7.1 Larger data sets . 17
7.2 Smaller data sets . 18
8. Estimation of Weibull percentiles. 19
9. Estimation of confidence intervals for the Weibull function. 19
9.1 Graphical procedure for complete and censored data. 20
9.2 Plotting confidence limits . 21
10. Estimation of the parameter and their confidence limits of the log-normal function. 21
10.1 Estimation of lognormal parameters. 21
10.2 Estimation of confidence intervals of log-normal parameters. 22
11. Comparison tests. 22
11.1 Simplified method to compare percentiles of Weibull distributions . 23
12. Estimating Weibull parameters for a system using data from specimens . 23
Published by IEC under licence from IEEE. © 2004 IEEE. All rights reserved.

IEEE 930-2004(E)
Annex A (informative) Least squares regression. 24
Annex B (informative) Bibliography. 48
Annex C (informative) List of participants. 49
Published by IEC under licence from IEEE. © 2004 IEEE. All rights reserved.

– 4 – IEC 62539:2007(E)
IEEE 930-2004(E)
INTERNATIONAL ELECTROTECHNICAL COMMISSION
___________
GUIDE FOR THE STATISTICAL ANALYSIS OF ELECTRICAL INSULATION
BREAKDOWN DATA
FOREWORD
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IEEE 930-2004(E)
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– 6 – IEC 62539:2007(E)
IEEE 930-2004(E)
IEEE Guide for the Statistical
Analysis of Electrical Insulation
Breakdown Data
Sponsor
Statistical Technical Committee
of the
IEEE Dielectrics and Electrical Insulation Society
Approved 29 March 2005
American National Standards Institute
Approved 23 September 2004
IEEE-SA Standards Board
Abstract: This guide describes, with examples, statistical methods to analyze times to break down
and breakdown voltage data obtained from electrical testing of solid insulating materials, for
purposes including characterization of the system, comparison with another insulator system, and
prediction of the probability of breakdown at given times or voltages.
Keywords: breakdown voltage and time, Gumbel, Lognormal distributions, statistical methods,
statistical confidence limits, Weibull
Published by IEC under licence from IEEE. © 2004 IEEE. All rights reserved.

IEEE 930-2004(E)
IEEE Introduction
This introduction in not part of IEEE Std 930-2004, IEEE Guide for the Statistical Analysis of Electrical Insulation
Breakdown Data.
Endurance and strength of insulation systems and materials subjected to electrical stress may be tested using
constant stress tests in which times to breakdown are measured for a number of test specimens, and
progressive stress tests in which breakdown voltages may be measured. In either case it will be found that a
different result is obtained for each specimen and that, for given test conditions, the data obtained may be
represented by a statistical distribution.
Failure of solid insulation can be mostly described by extreme-value statistics, such as the Weibull and
Gumbel distributions, but, historically, also the lognormal function has been used. Methods for determining
whether data fit to either of these distributions, graphical and computer-based techniques for estimating the
most likely parameters of the distributions, computer-based techniques for estimating statistical confidence
intervals, and techniques for comparing data sets and some case studies are addressed in this guide.
Notice to users
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– 8 – IEC 62539:2007(E)
IEEE 930-2004(E)
GUIDE FOR THE STATISTICAL
ANALYSIS OF ELECTRICAL INSULATION
BREAKDOWN DATA
1. Scope
Electrical insulation systems and materials may be tested using constant stress tests in which times to break-
down are measured for a number of test specimens, and progressive stress tests in which breakdown
voltages may be measured. In either case, it will be found that a different result is obtained for each speci-
men and that, for given test conditions, the data obtained may be represented by a statistical distribution.
This guide describes, with examples, statistical methods to analyze such data.
The purpose of this guide is to define statistical methods to analyze times to breakdown and breakdown
voltage data obtained from electrical testing of solid insulating materials, for purposes including
characterization of the system, comparison with another insulator system, and prediction of the probability
of breakdown at given times or voltages.
Methods are given for analyzing complete data sets and also censored data sets in which not all the speci-
mens broke down. The guide includes methods, with examples, for determining whether the data is a good
fit to the distribution, graphical and computer-based techniques for estimating the most likely parameters of
the distribution, computer-based techniques for estimating statistical confidence intervals, and techniques
for comparing data sets and some case studies. The methods of analysis are fully described for the Weibull
distribution. Some methods are also presented for the Gumbel and lognormal distributions. All the examples
of computer-based techniques used in this guide may be downloaded from the following web site “http://
grouper.ieee.org/groups/930/IEEEGuide.xls.” Methods to ascertain the short time withstand voltage or oper-
ating voltage of an insulation system are not presented in this guide. Mathematical techniques contained in
this guide may not apply directly to the estimation of equipment life.
2. References
The following publications may be used when applicable in conjunction with this guide. When the following
standards are superseded by an approved revision, the revision shall apply.
ASTM D149-97a(2004) Standard Test Method for Dielectric Breakdown Voltage and Dielectric Strength of
Solid Electrical Insulating Materials at Commercial Power Frequencies.
ASTM publications are available from the American Society for Testing and Materials, 100 Barr Harbor Drive, West Conshohocken,
PA 19428-2959, USA (http://www.astm.org/).
Published by IEC under licence from IEEE. © 2004 IEEE. All rights reserved.

IEEE 930-2004(E)
BS 2918-2, Methods of test for electric strength of solid insulating materials.
IEC 60243 series, Electrical strength of insulating materials—Test Methods—Part 1: Tests at power
frequencies.
3. Steps required for analysis of breakdown data
3.1 Data acquisition
3.1.1 Commonly used testing techniques
There are two commonly used breakdown tests for electrical insulation: constant stress tests and progressive
stress tests. In these tests a number of identical specimens are subjected to identical test regimes intended to
cause electrical breakdown. In constant stress tests the same voltage is applied to each specimen (they are
often tested in parallel) and the times to breakdown are measured. The times to breakdown may be widely
distributed with the longest time often being more than two orders of magnitude that of the shortest. In pro-
gressive stress tests an increasing voltage is applied to each specimen, usually breakdown voltages are
measured. The voltage may be increased continuously with time or in small steps. Other protocols, for
example impulse testing, may also be used. Breakdown voltages may be much less widely distributed with
the highest voltage sometimes only being 2% more than the lowest voltage.
Various international standards, e.g., BS 2918-2 and IEC 60243 series, give appropriate experimental procedures
for constant and progressive stress tests. This guide is intended to provide a more rigorous treatment for the
breakdown data obtained in this way.
3.1.2 Other data
Breakdown data may also be available from other sources; for example, times to breakdown of the insula-
tion in service may be available. Such data is generally much more difficult to analyze since the history of
each failed insulator may not be the same (see 3.1.4), particularly as units that failed will have been replaced.
It may also be unclear how many such insulation systems are in service and hence what proportion of them
have failed. The techniques described in this guide are, nevertheless, appropriate for such data provided suf-
ficient care is exercised in their application.
3.1.3 Data requirements
The number of data points required depends upon the number of parameters that describes the distribution
and the confidence demanded in the results. If possible, failure data on at least ten specimens should be
obtained; serious errors may result with less than five specimens (see also 3.2.2).
If all the specimens break down, the data is referred to as complete. In some cases, not all the specimens
break down, the data is then referred to as censored. Censored data may be encountered in constant stress
tests where the data are analyzed or the test is terminated before all the specimens break down. Censored
data can also occur with progressive stress tests where the power supply has insufficient voltage capability
to break down all the samples. In these cases, the data associated with a single group of specimens, those
with the highest strength, are not known and the data set is said to be singly censored. Data may also be
progressively censored. In this case, specimens may be withdrawn (or their data discounted) at any time or
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– 10 – IEC 62539:2007(E)
IEEE 930-2004(E)
voltage; such data are often referred to as “suspended.” This may be the case where specimen breakdown is
due to a spurious mechanism such as termination failure or flashover or where the specimen is deliberately
withdrawn for alternative analysis. Censoring can occur by plan or by accident in many insulation tests and
it is essential that this is taken into account in the data analysis. Less confidence can be placed in the analysis
of a censored data set than in a complete set of data with the same number of specimens. If possible censored
data sets should include at least ten (non-censored) data points and at least 30% of the specimens should
have broken down.
3.1.4 Practical precautions in data capture
Specimens should, as far as possible, be identical, have the same history prior to testing, and be tested under
the same conditions. In measuring the breakdown characteristics of materials it should be noted that the
breakdown field (kilovolt per millimeter) is usually dependent upon the rate of voltage rise, specimen
thickness, electrode material, configuration and method of attachment, temperature, area, and frequency if
an alternating voltage is applied. Other factors such as humidity and specimen age may also be important.
With insulating systems such as cables and bushings, surface and interfacial partial discharges must be min-
imized and stress enhancements due to protrusions, contaminants and voids are likely to reduce breakdown
strengths considerably.
The scope of this guide is limited to ac voltage testing, but the techniques may be applied to other failure
tests (such as impulse or dc testing) with care. Knowledge of the failure mechanism may be required in order
to establish the appropriate parameters to be measured. In pulse energized dc systems, for example, it may
be more appropriate to measure the number of pulses to breakdown than the dc time to failure. Precautions
in data capture are described more fully elsewhere, e.g., Abernethy [B1].
3.2 Characterizing data using a probability function
3.2.1 Types of failure distribution
Failure data, such as that described in breakdown of electrical insulation, may be represented in a histogram
form as numbers of specimens failed in consecutive periods. For example, the times to breakdown of poly-
mer coated wires subject to constant ac stress are shown in Figure A.1 as a histogram. The mean and
standard deviation of this data set is easily found using a scientific calculator and the corresponding Normal
probability density function can be superimposed on the histogram. Whilst the Normal is probably the best
known and its parameters (the mean and standard deviation) are easily calculated; it is not usually appropri-
ate to electrical breakdown data. For example, it can be seen in Figure A.1 that its shape is rather different to
the histogram. In particular the Normal distribution has a finite probability of failure at (physically impossi-
ble) negative times. An important step in analyzing breakdown data is the selection of an appropriate
distribution.
Distributions for electrical breakdown include the Weibull, Gumbel, and lognormal. The most common for
solid insulation is the Weibull and is the main distribution described in this guide. It is found to have wide
applicability and is a type of extreme value distribution in which the system fails when the weakest link
fails. The Gumbel distribution, another extreme value distribution, may have applicability in breakdown
involving percolation, in liquids and in cases where fault sites such as voids are exponentially distributed.
The effect of the size of test specimens (thickness, area, volume) on life or breakdown voltage can be mod-
eled using extreme value distributions. The lognormal distribution may be useful where specimens break
This is also known as “right” censored data since specimens beyond a certain time or voltage are not tested. It is possible to have “left”
censored data but this does not usually occur in electrical breakdown testing. In this guide, “singly” censored data always refers to
“right” censoring.
To convert this unit value from kV/mm to kV/inch multiply the value in kV/mm value by 25.4.
The numbers in brackets correspond to those of the bibliography in Annex B.
Published by IEC under licence from IEEE. © 2004 IEEE. All rights reserved.

IEEE 930-2004(E)
down due to unrelated causes or mechanisms. The lognormal distribution may be closely approximated by
the Weibull distribution.
The previous distributions may be described in terms of two parameters (as the normal distribution is
described in terms of the mean and standard deviation). To give more generality, however, a third parameter
may be included corresponding to a time before, or a voltage below, which a specimen will not break down.
In some cases two or more mechanisms may be operative, this may necessitate combining two or more dis-
tributions functions.
Mathematical descriptions of these distributions are given in Clause 4.
3.2.2 Testing the adequacy of a distribution
Having chosen a distribution to represent a set of breakdown data, it is necessary to check that the distribu-
tion is adequate for this purpose. It was seen in 3.2.1 that, although the parameters of a Normal distribution
could be found for a given set of data, this did not imply that the distribution was an adequate representation
(e.g., Figure A.1). The most common technique to test the adequacy of the distribution is to plot data points
on special probability paper associated with the distribution in question. Such paper is available for all the
distributions thus far mentioned. A good fit to a distribution will result in a straight line plot (5.1 and 5.2).
Statistical techniques are also available for assessing the adequacy of a distribution; a simple technique is
given in 5.4.
3.2.3 Estimating parameters and confidence limits
Probability plots can also be used for graphical estimation of the parameters of the distribution (Clause 6)
but this is not recommended; more accurate computation techniques are readily available (Clause 7).
The parameters obtained from all such techniques are only estimates because the measured data points are
randomly distributed according to a given failure mechanism. For example, if 100 experiments were per-
formed each with ten specimens, the analysis of each of the 100 experiments would give 100 estimates for
the parameters of the probability distribution each of which are slightly different. In such a case, it may be
possible to state with (for example) 90% confidence that the true value of the given parameter lies between
the fifth largest and fifth smallest value obtained. It is common to calculate (9.1), for each parameter esti-
mate, a statistical confidence interval that encloses the true parameter with high probability. In general, the
more specimens tested, the narrower the confidence interval. Enough specimens should be tested so as to
obtain sufficiently narrow confidence intervals for practical purposes. If the confidence intervals are calcu-
lated to be adequate before all the specimens have failed, the test could be aborted.
If an experiment is poorly performed, for example, if the applied voltage is not held constant in a constant
stress test, the statistical confidence intervals are inaccurate. Statistical confidence intervals are valid there-
fore only for identically tested specimens. If the variation in testing conditions is known it may be possible
to estimate confidence intervals, but this is beyond the scope of this guide.
3.3 Hypothesis testing
The estimation of the parameters (and confidence intervals) of the distribution describing an insulating spec-
imen or system may be required for a number of reasons, including:
— Reporting the characteristics of the insulating system following a manufacturing development.
— Testing of a batch of insulating systems and comparing them to another batch for quality control or
for development.
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– 12 – IEC 62539:2007(E)
IEEE 930-2004(E)
— Estimating whether early failures in the system are due to a mechanism likely to cause failure in the
remaining parts of the system.
— Estimating equipment life.
— Establishing operating conditions.
Examples of some of these processes are given as case studies in this guide (Clause 11).
4. Probability distributions for failure data
A brief introduction to these distributions has been given in this clause.
4.1 The Weibull distribution
The expression for the cumulative density function for the two-parameter Weibull distribution is shown in
Equation (1):
β
⎧⎫
t
⎛⎞
Ft(,;α β) =1e– xp – --- (1)
⎨⎬
⎝⎠
α
⎩⎭
where:
t is the measured variable, usually time to break down or the breakdown voltage,
F(t) is the probability of failure at a voltage or time less than or equal to t. For tests with large numbers
of specimens, this is approximately the proportion of specimens broken down by time or voltage, t.
α is the scale parameter and is positive, and
β is the shape parameter and is positive.
The probability of failure F(t) is zero at t = 0. The probability of failure rises continuously as t increases. As
the time or voltage increases, the probability of failure approaches certainty, that is, F(∞) = 1.
The scale parameter α represents the time (or voltage) for which the failure probability is 0.632 (that is  1 –
1/e where e is the exponential constant). It is analogous to the mean of the Normal distribution (e.g.,
Cochran and Snedecor [B2]). The units of α are the same as t, that is, voltage, electric stress, time, number
of cycles to failure etc.
The shape parameter β is a measure of the range of the failure times or voltages. The larger β is, the smaller
is the range of breakdown voltages or times. It is analogous to the inverse of the standard deviation of the
Normal distribution, Cochran and Snedecor [B2].
The two-parameter Weibull distribution of Equation (1) is a special case of the three-parameter Weibull dis-
tribution that has the cumulative distribution function shown in Equation (2).
β
⎧⎫
t– γ
⎛⎞
Ft() = 1– exp – ---------- ;t ≥ γ (2)
⎨⎬
⎝⎠
α
⎩⎭
0;t < γ
The additional term γ is called the location parameter. F(t) = 0 for t = γ, that is the probability of failure for t
< γ is zero.
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4.2 The Gumbel distribution
A cumulative Gumbel distribution function is given by Equation (3).
⎧⎫
tu–
F ()t =1e– xp –exp ---------- ;–∞ ≤ t+ ∞ (3)
⎨⎬
G
b
⎩⎭
where:
u is the location parameter and may have any value, and
b is the scale parameter and is positive.
The Gumbel distribution is asymmetrical and can have a physically impossible finite probability of break-
down for t < 0. This distribution is also called the smallest extreme-value (that is, weakest link) distribution.
If t is voltage, then the units of u and b are also voltage.
The Gumbel distribution is closely related to the Weibull distribution. That is, if t has a Weibull distribution
then y = ln(t) has a Gumbel distribution where: u = ln(α) and b = 1/β. Estimation techniques for one distribu-
tion (Gumbel or Weibull) apply to the other if this transformation is utilized.
4.3 The lognormal distribution
The lognormal distribution has sometimes been used to represent failure data from insulation systems, but it
has not been used nearly as often as the extreme-value distributions in 4.1 and 4.2. However, since this prob-
ability distribution is a simple logarithmic transformation of the well-known Normal distribution, methods
for data analysis are available in all standard statistical references. The probability density function of the
lognormal distribution is shown in Equation (4).
⎧⎫
1 ()z– μ
f ()z = --------------exp –------------------ (4)
⎨⎬
ln
σ 2π⎩⎭2σ
where:
z = log (t),
μ = logarithmic mean, and
σ = logarithmic standard deviation.
The cumulative density function is the integral of the above. There is no closed-form equation for the inte-
gral. Values of the distribution are in Cochran and Snedecor [B2] and Natrella [B12] or can be obtained
from statistical calculators or computer programs.
4.4 Mixed distributions
It is not uncommon to find that more than one breakdown mechanism is operative in a given specimen. The
probability that such a specimen survives to a given value or voltage or time t is 1 – F(t). If the probability of
failure due to mechanism 1 is F (t) and due to mechanism 2 is F (t), then the probability of survival is
1 2
1–Ft() =[]1– F ()t ×[]1– F ()t (5)
1 2
If both may be described by the two-parameter Weibull distribution, then we have
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β β
1 2
⎧⎫
t t
⎛⎞ ⎛⎞
Ft() = 1 – exp – ------ + ------ (6)
⎨⎬
⎝⎠ ⎝⎠
α α
⎩⎭1 2
Other forms of mixed distributions are also possible. A more detailed description can be found in Fischer
[B5].
4.5 Other terminology
In this guide, true values of parameters are represented by symbols (e.g., α), estimated values by “hatted”
symbols (e.g., αˆ ), and upper and lower confidence bounds by subscripts u and l (e.g., α and α ). Cumula-
u l
tive density functions are in upper case [e.g., F(t)] whereas probability density functions are in lower case
[e.g., f(t)]. The number of specimens is designated as n with the number broken down as r (r is less than n
for censored tests, r = n for complete tests).
5. Testing the adequacy of a distribution
5.1 Weibull probability data
Data distributed according to the two-parameter Weibull function should form a reasonably straight line
when plotted on Weibull probability paper. A sample probability paper is shown in Figure A.2 (the data
plotted on this paper is referred to in Clause 6). The measured data is plotted on the horizontal axis, which is
scaled logarithmically. The probability of breakdown is plotted on the vertical axis, which is also highly
non-linear.
5.1.1 Estimating plotting positions for complete data
To use this probability paper, place the n breakdown times or voltages in order from smallest to largest and
assign them a rank from i = 1 to i = n. An example of this from progressive stress testing of latex film is
shown in Table A.1.
A good, simple, approximation for the most likely probability of failure is found in Ross [B14]:
i– 0.44
Fi(,n) ≈------------------- × 100% (7)
n+ 0.25
The Weibull example data in Table A.1 are plotted in Figure A.3. In this case, there were ten specimens (n =
10) and all of them broke down so the data is “complete.” The data follows a reasonably straight line and it
is therefore reasonable to assume that they are distributed according to the Weibull function. (The line repre-
senting the Weibull relationship was plotted using the procedure in Clause 7.)
Some random deviations from a straight line may be expected. If, however, there is a consistent departure
from a straight line (for example curvature or a cusp) then another distribution may fit the data better (see
5.3). Probability papers for the Gumbel and lognormal distributions are also available. The probability of
failure for these graphs is estimated in exactly the same way.
7 th
The plotting position on the horizontal axis, X , of the i data point, x , is such that X α log x . The plotting position on the vertical
i i i i
th
axis, Y , of the probability of failure corresponding to the i data point, F(x ), is such that Y = log{–ln[1 – F(x )]}.
i i i i
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5.1.2 Estimating plotting positions for singly censored data
Table A.2 presents an example of singly censored data from constant stress tests on epoxy resin specimens;
these are plotted in Figure A.4. (Again, the line representing the Weibull relationship was plotted using the
procedure in Clause 7.) The test was stopped at 144.9 hours and so only seven of the nine specimens broke
down; the final 2 had still not broken down
...