IEC 61810-2:2011
(Main)Electromechanical elementary relays - Part 2: Reliability
Electromechanical elementary relays - Part 2: Reliability
IEC 61810-2:2011 covers test conditions and provisions for the evaluation of endurance tests using appropriate statistical methods to obtain reliability characteristics for relays. The main changes with respect to the previous edition are listed below: inclusion of both numerical and graphical methods for Weibull evaluation; establishment of full coherence with the second edition of the basic reliability standard IEC 61649.
This publication is to be read in conjunction with IEC 61649:2008.
Relais électromécaniques élémentaires - Partie 2: Fiabilité
La CEI 61810-2:2011 couvre les conditions d'essai et les dispositions pour l'évaluation des essais d'endurance utilisant les méthodes statistiques appropriées pour obtenir les caractéristiques de fiabilité pour relais. Les principales modifications portant sur cette précédente édition sont énumérées ci-dessous: inclusion de méthodes numériques et graphiques pour l'évaluation de Weibull; établissement d'une cohérence complète avec la deuxième édition de la norme de fiabilité de base CEI 61649.
Cette publication doit être lue conjointement avec la CEI 61649:2008.
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Standards Content (Sample)
IEC 61810-2 ®
Edition 2.0 2011-02
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
Electromechanical elementary relays –
Part 2: Reliability
Relais électromécaniques élémentaires –
Partie 2: Fiabilité
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IEC 61810-2 ®
Edition 2.0 2011-02
INTERNATIONAL
STANDARD
NORME
INTERNATIONALE
Electromechanical elementary relays –
Part 2: Reliability
Relais électromécaniques élémentaires –
Partie 2: Fiabilité
INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
COMMISSION
ELECTROTECHNIQUE
PRICE CODE
INTERNATIONALE
CODE PRIX V
ICS 29.120.70 ISBN 978-2-88912-376-6
– 2 – 61810-2 © IEC:2011
CONTENTS
FOREWORD . 3
INTRODUCTION . 5
1 Scope . 6
2 Normative references . 6
3 Terms and definitions . 7
4 General considerations . 9
5 Test conditions . 10
5.1 Test items . 10
5.2 Environmental conditions . 10
5.3 Operating conditions . 10
5.4 Test equipment . 11
6 Failure criteria . 11
7 Output data . 11
8 Analysis of output data . 11
9 Presentation of reliability measures . 12
Annex A (normative) Data analysis . 13
Annex B (informative) Example of numerical and graphical Weibull analysis . 22
Annex C (informative) Example of cumulative hazard plot . 26
Annex D (informative) Gamma function . 32
Bibliography . 33
Figure A.1 – An example of Weibull probability paper . 16
Figure A.2 – An example of cumulative hazard plotting paper . 18
Figure A.3 – Plotting of data points and drawing of a straight line . 18
Figure A.4 – Estimation of distribution parameters . 19
Figure B.1 – Weibull probability chart for the example. 24
Figure C.1 – Estimation of distribution parameters . 28
Figure C.2 – Cumulative hazard plots . 30
Table B.1 – Ranked failure data . 23
Table C.1 – Work sheet for cumulative hazard analysis . 26
Table C.2 – Example work sheet . 29
Table D.1 – Values of the gamma function . 32
61810-2 © IEC:2011 – 3 –
INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
ELECTROMECHANICAL ELEMENTARY RELAYS –
Part 2: Reliability
FOREWORD
1) The International Electrotechnical Commission (IEC) is a worldwide organization for standardization comprising
all national electrotechnical committees (IEC National Committees). The object of IEC is to promote
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8) Attention is drawn to the Normative references cited in this publication. Use of the referenced publications is
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patent rights. IEC shall not be held responsible for identifying any or all such patent rights.
International Standard IEC 61810-2 has been prepared by IEC technical committee 94: All-or-
nothing electrical relays.
This second edition cancels and replaces the first edition published in 2005. This edition
constitutes a technical revision.
The main changes with respect to the previous editions are listed below:
• inclusion of both numerical and graphical methods for Weibull evaluation;
• establishment of full coherence with the second edition of the basic reliability standard
IEC 61649;
• deletion of previous Annex A and Annex D since both annexes are contained in
IEC 61810-1.
– 4 – 61810-2 © IEC:2011
The text of this standard is based on the following documents:
FDIS Report on voting
94/316/FDIS 94/325/RVD
Full information on the voting for the approval of this standard can be found in the report on
voting indicated in the above table.
This publication has been drafted in accordance with the ISO/IEC Directives, Part 2.
A list of all parts of the IEC 61810 series can be found, under the general title
Electromechanical elementary relays, on the IEC website.
This International Standard is to be used in conjunction with IEC 61649:2008.
The committee has decided that the contents of this publication will remain unchanged until
the stability date indicated on the IEC web site under "http://webstore.iec.ch" in the data
related to the specific publication. At this date, the publication will be
• reconfirmed,
• withdrawn,
• replaced by a revised edition, or
• amended.
61810-2 © IEC:2011 – 5 –
INTRODUCTION
Within the IEC 61810 series of basic standards covering elementary electromechanical relays,
IEC 61810-2 is intended to give requirements and tests permitting the assessment of relay
reliability. All information concerning endurance tests for type testing have been included in
IEC 61810-1.
NOTE According to IEC 61810-1, a specified value for the electrical endurance under specific conditions (e.g.
contact load) is verified by testing 3 relays. None is allowed to fail. Within this IEC 61810-2, a prediction of the
reliability of a relay is performed using statistical evaluation of the measured cycles to failure of a larger number of
relays (generally 10 or more relays).
Recently the technical committee responsible for dependability (TC 56) has developed a new
edition of IEC 61649 dealing with Weibull distributed test data. This second edition contains
both numerical and graphical methods for the evaluation of Weibull-distributed data.
On the basis of this basic reliability standard, IEC 61810-2 was developed. It comprises test
conditions and an evaluation method to obtain relevant reliability measures for
electromechanical elementary relays. The life of relays as non-repairable items is primarily
determined by the number of operations. For this reason, the reliability is expressed in terms
of mean cycles to failure (MCTF).
Commonly, equipment reliability is calculated from mean time to failure (MTTF) figures. With
the knowledge of the frequency of operation (cycling rate) of the relay within an equipment, it
is possible to calculate an effective MTTF value for the relay in that application.
Such calculated MTTF values for relays can be used to calculate respective reliability,
probability of failure, and availability (e.g. MTBF (mean time between failures)) values for
equipment into which these relays are incorporated.
Generally it is not appropriate to state that a specific MCTF value is “high” or “low”. The
MCTF figures are used to make comparative evaluations between relays with different styles
of design or construction, and as an indication of product reliability under specific conditions.
– 6 – 61810-2 © IEC:2011
ELECTROMECHANICAL ELEMENTARY RELAYS –
Part 2: Reliability
1 Scope
This part of IEC 61810 covers test conditions and provisions for the evaluation of endurance
tests using appropriate statistical methods to obtain reliability characteristics for relays. It
should be used in conjunction with IEC 61649.
This International Standard applies to electromechanical elementary relays considered as
non-repaired items (i.e. items which are not repaired after failure), whenever a random
sample of items is subjected to a test of cycles to failure (CTF).
The lifetime of a relay is usually expressed in number of cycles. Therefore, whenever the
terms “time” or “duration” are used in IEC 61649, this term should be understood to mean
“cycles”. However, with a given frequency of operation, the number of cycles can be
transformed into respective times (e.g. times to failure (TTF)).
The failure criteria and the resulting characteristics of elementary relays describing their
reliability in normal use are specified in this standard. A relay failure occurs when the
specified failure criteria are met.
As the failure rate for elementary relays cannot be considered as constant, particularly due to
wear-out mechanisms, the times to failure of tested items typically show a Weibull
distribution. This standard provides both numerical and graphical methods to calculate
approximate values for the two-parameter Weibull distribution, as well as lower confidence
limits.
2 Normative references
The following referenced documents are indispensable for the application of this document.
For dated references, only the edition cited applies. For undated references, the latest edition
of the referenced document (including any amendments) applies.
IEC 60050-191:1990, International Electrotechnical Vocabulary (IEV) – Chapter 191:
Dependability and quality of service
IEC 60050-444:2002, International Electrotechnical Vocabulary (IEV) – Part 444: Elementary
relays
IEC 60300-3-5:2001, Dependability management – Part 3-5: Application guide – Reliability
test conditions and statistical test principles
IEC 61649:2008, Weibull analysis
IEC 61810-1:2008, Electromechanical elementary relays – Part 1: General requirements
61810-2 © IEC:2011 – 7 –
3 Terms and definitions
For the purposes of this document, the terms and definitions given in IEC 60050-191 and
IEC 60050-444, some of which are reproduced below, as well as the following, apply.
3.1
item
any component that can be individually considered
[IEC 60050-191:1990, 191-01-01, modified]
NOTE For the purpose of this standard, items are elementary relays.
3.2
non-repaired item
item which is not repaired after a failure
[IEC 60050-191:1990, 191-01-03, modified]
3.3
cycle
operation and subsequent release/reset
[IEC 60050-444:2002, 444-02-11]
3.4
frequency of operation
number of cycles per unit of time
[IEC 60050-444:2002, 444-02-12]
3.5
reliability
ability of an item to perform a required function under given conditions for a given number of
cycles or time interval
[IEC 60050-191:1990, 191-02-06, modified]
NOTE 1 It is generally assumed that the item is in a state to perform this required function at the beginning of the
time interval.
NOTE 2 The term “reliability” is also used as a measure of reliability performance (see IEC 60050-191:1990,
191-12-01).
3.6
reliability test
experiment carried out in order to measure, quantify or classify a reliability measure or
property of an item
[IEC 60300-3-5:2001, 3.1.27]
3.7
life test
test with the purpose of estimating, verifying or comparing the lifetime of the class of items
being tested
[IEC 60300-3-5:2001, 3.1.17, modified]
– 8 – 61810-2 © IEC:2011
3.8
cycles to failure
CTF
total number of cycles of an item, from the instant it is first put in an operating state until
failure
3.9
mean cycles to failure
MCTF
expectation of the number of cycles to failure
3.10
time to failure
TTF
total time duration of operating time of an item, from the instant it is first put in an operating
state until failure
[IEC 60050-191:1990, 191-10-02, modified]
3.11
mean time to failure
MTTF
expectation of the time to failure
[IEC 60050-191:1990, 191-12-07]
3.12
useful life
number of cycles or time duration until a certain percentage of items have failed
NOTE In this standard, this percentage is defined as 10 %.
3.13
failure
termination of the ability of an item to perform a required function
[IEC 60050-191:1990, 191-04-01, modified]
3.14
malfunction
single event when an item does not perform a required function
3.15
contact failure
occurrence of break and/or make malfunctions of a contact under test, exceeding a specified
number
3.16
failure criteria
set of rules used to decide whether an observed event constitutes a failure
[IEC 60300-3-5:2001, 3.1.10]
3.17
contact load category
classification of relay contacts dependent on wear-out mechanisms
NOTE Various contact load categories are defined in IEC 61810-1.
61810-2 © IEC:2011 – 9 –
4 General considerations
The provisions of this part of IEC 61810 are based on the relevant publications on
dependability. In particular, the following documents have been taken into account:
IEC 60050-191, IEC 60300-3-5 and IEC 61649.
The aim of reliability testing as given in this standard is to obtain objective and reproducible
data on reliability performance of elementary relays representative of standard production
quality. The tests described and the related statistical tools to gain reliability measures based
on the test results can be used for the estimation of such reliability measures, as well as for
the verification of stated measures.
NOTE 1 Examples for the application of reliability measurements are:
• establishment of reliability measures for a new relay type;
• comparison of relays with similar characteristics, but produced by different manufacturers;
• evaluation of the influence, on a relay, of different materials or different manufacturing solutions;
• comparison of a new relay with a relay which has already worked for a specific period of time;
• calculation of the reliability of an equipment or system incorporating one or more relays.
According to Clauses 8 and 9 of IEC 60300-3-5, for non-repaired items showing a non-
constant failure rate the Weibull model is the most appropriate statistical tool for evaluation of
reliability measures. This analysis procedure is described in IEC 61649.
Elementary relays within the scope of this standard are considered as non-repaired items.
They generally do not exhibit a constant failure rate but a failure rate increasing with time,
being tested until wear-out mechanisms become predominant. The cycles to failure of a
random sample of tested items typically show the Weibull distribution.
NOTE 2 In cases where no wear-out mechanisms prevail, random failures with constant failure rate can be
assumed. Then the shape parameter β of the Weibull distribution equals 1 and the reliability function becomes the
well-known exponential law. For relay tests where only very few failures (or even no failures at all) occur, the
WeiBayes approach of IEC 61649 might be appropriate. Another option may be the application of the sudden death
method described in Clause 13 of IEC 61649.
The statistical procedures of this standard are valid only when at least 10 relevant failures are
recorded.
Upon special agreement between manufacturer and user, the test may be performed with
even less than 10 relays, provided the uncertainty of the estimated Weibull parameters is
acceptable to them. In such a case the minimum number of tested relays shall be specified;
this number then replaces the minimum number of 10 relays wherever prescribed in this
standard. However, it shall be noted that this reduction of relay specimens is only acceptable
where the graphical methods of A.5.1 are applied. For the numerical method of A.5.2 at least
10 failures are required, since the maximum likelihood estimation (MLE) is a computational
method for larger sample sizes, i.e. when at least 10 relevant failures are recorded (see 9.3 of
IEC 61649).
The first step in the analysis of the recorded cycles to failure (CTF) of the tested relays is the
determination of the two distribution parameters of the Weibull distribution. In a second step,
the mean cycles to failure (MCTF) is calculated as a point estimate. In a third step, the useful
life is determined as the lower confidence limit of the number of cycles by which 10 % of the
relay population will have failed (B ).
With a given frequency of operation these reliability measures expressed in number of cycles
(MCTF) can be transformed into respective times (MTTF), see Annex B for an example.
The statistical procedures require some appropriate computing facility. Software for
evaluation of Weibull distributed data is commercially available on the market. Such software
– 10 – 61810-2 © IEC:2011
may be used for the purpose of this standard provided it shows equivalent results when the
data given in Annex B are used.
Since the number of cycles to failure highly depends on the specific set of test conditions
(particularly the electrical loading of the relay contacts), values for MCTF and useful life
derived from test data apply only to this set of test conditions, which have to be stated by the
manufacturer together with the reliability measures.
5 Test conditions
5.1 Test items
As a minimum of 10 failures need to be recorded to perform the analysis described in this
standard, 10 or more items (relays) should be submitted to the test. This allows the test to be
truncated when at least 10 relays have failed. When the test is truncated at a specific number
of cycles, all relays that have not yet failed may be considered to fail at that number of cycles
(worst case assumption). However, at least 70 % of the tested relays shall fail physically. This
allows the test to be carried out with 10 relays only, even when the test is truncated before all
relays have physically failed (with a minimum of 7 physical failures recorded).
The items shall be selected at random from the same production lot and shall be of identical
type and construction. No action is allowed on the test items from the time of sampling until
the test starts.
Where any particular burn-in procedure or reliability stress screening is employed by the
manufacturer prior to sampling, this shall apply to all production. The manufacturer shall
describe and declare such procedures, together with the test results.
Unless otherwise specified by the manufacturer, all contacts of each relay under test shall be
loaded as stated and monitored continuously during the test.
The test starts with all items and is stopped at some number of cycles. At that instant a
certain number of items (minimum: 10 items) have failed. The number of cycles to failure of
each of the failed items is recorded.
Items failed during the test are not replaced once they fail.
5.2 Environmental conditions
The testing environment shall be the same for all items.
– The items shall be mounted in the manner intended for normal service; in particular, relays
for mounting onto printed circuit-boards are tested in the horizontal position, unless
otherwise specified.
– The ambient temperature shall be as specified by the manufacturer.
– All other influence quantities shall comply with the values and tolerance ranges given in
Table 1 of IEC 61810-1, unless otherwise specified.
5.3 Operating conditions
The set of operating conditions
– rated coil voltage(s);
– coil suppression (if any);
– frequency of operation;
– duty factor;
61810-2 © IEC:2011 – 11 –
– contact load(s)
shall be as specified by the manufacturer.
Recommended values should be chosen from those given in Clause 5 of IEC 61810-1.
The test is performed on each contact load and each contact material as specified by the
manufacturer.
All specified devices (for example, protective or suppression circuits), if any, which are part of
the relay or stated by the manufacturer as necessary for particular contact loads, should be
operated during the test.
The contacts shall be continuously monitored to detect malfunctions to open and malfunctions
to close, as well as unintended bridging (simultaneous closure of make and break side of a
changeover contact).
The contacts are connected to the load(s) in accordance with Table 12 of IEC 61810-1 as
specified and indicated by the manufacturer.
5.4 Test equipment
The test circuit described in Annex C of IEC 61810-1 shall be used, unless otherwise
specified by the manufacturer and explicitly indicated in the test report.
6 Failure criteria
Whenever any contact of a relay under test fails to open or fails to close or exhibits
unintended bridging, this shall be considered as a malfunction.
Three severity levels are specified:
– severity A: the first detected malfunction is defined as a failure;
– severity B: the sixth detected malfunction or two consecutive malfunctions are defined as
a failure;
– severity C: as specified by the manufacturer.
The severity level used for the test shall be as prescribed by the manufacturer and stated in
the test report.
7 Output data
The data to be analysed consists of cycles to failure (CTF) for each of the items put on test.
These CTF values have to be known exactly. However, it is not necessary to gather the CTF
values for all items under test, as the test may be stopped before all items have failed,
provided at least 10 CTF values from different failed items are available.
8 Analysis of output data
The evaluation of the CTF values obtained during the test shall be carried out in accordance
with the procedures given in Annex A.
– 12 – 61810-2 © IEC:2011
9 Presentation of reliability measures
The basic reliability measures applicable to elementary relays as described in this standard
and obtained from the data analysis shall be provided.
However, since the values obtained for these reliability measures using the procedures of
Annex A depend to a great extent on the basic design characteristics of the relay, the test
conditions of Clause 5 and the failure criteria of Clause 6, the following information shall also
be provided together with the test results:
– relay type for which the results are valid:
a) contact material;
b) deviations from standard types (if any);
c) type of termination;
– set of operating conditions (see 5.3):
a) rated coil voltage(s);
b) coil suppression (if any);
c) frequency of operation;
d) duty factor;
e) contact load(s);
f) ambient conditions;
– test schematic selected (see Clause C.3 of IEC 61810-1, or test circuit details, if different
from the circuit described in Clause C.1 of IEC 61810-1);
– severity level (see Clause 6).
In addition basic data of the test and the related analysis (see Annex A) shall be given in the
test report:
– number of items (n) on test;
– number of failed items (r) registered during the test (minimum 10);
– time (given in number of cycles) when the test was stopped (T);
– confidence level, if other than 90 %.
The test results are applicable to the samples specifically tested and variants, as stipulated by
the manufacturer, provided that the relevant design characteristics remain the same.
NOTE Acceptable examples are coil variants with the same ampere-turns. Unacceptable examples are variants
with AC in place of DC coils, or different contact dynamics.
When test results for various operating conditions (for example, contact loads) are available,
they may be compiled as a family of curves or in suitable tables. However, it shall be ensured
that a sufficient number of points are determined when plotting such curves.
61810-2 © IEC:2011 – 13 –
Annex A
(normative)
Data analysis
A.1 General
This annex has been derived from the reliability standard IEC 61649:2008 with certain
modifications necessary to adopt the procedures to elementary relays. The distribution
considered in the reliability standard is of the Weibull type, which has been empirically
recognized to correspond to an appropriate data analysis for elementary relays.
The graphical method, as well as the numerical method are covered in IEC 61649. In addition,
not only the Weibull probability analysis but also the Weibull hazard analysis is taken up in
the graphical method. Here, Weibull hazard and Weibull probability analyses are applied to
complete and incomplete data, respectively. The latter is especially useful for the reliability
analysis of relays because many data sets obtained from life tests are incomplete (censored
tests).
NOTE 1 Incomplete data are the data sets obtained from the test after either a certain number of failures or a
certain number of cycles, when there are still items functioning, whereas complete data are the data sets without
censoring.
This annex deals with the Weibull probability plot and the Weibull hazard plot for the graphical
method based upon median rank regression (MRR) principles, and the maximum likelihood
estimation (MLE) for the numerical method in accordance with the provisions of IEC 61649.
When more in-depth information is required, IEC 61649 is to be consulted.
The concept “time” is to be understood as “cycles” in the case of relays. However, with a
given frequency of operation, the values indicated in numbers of cycles can be transformed
into respective times.
NOTE 2 Whereas the variable “time” (symbol: t) is used within IEC 61649, this standard therefore is based on the
variable “cycles” (symbol: c).
For the sake of consistency, the following symbols and equations are reproduced in
accordance with IEC 61649.
A.2 Abbreviations
CDF Cumulative distribution function
MRR Median rank regression
MLE Maximum likelihood estimation
MCTF Mean cycles to failure
PDF Probability density function
A.3 Symbols and definitions
The following symbols are used in this Annex A, and in both Annex B and Annex C. Auxiliary
constants and functions are defined in the text.
f(c) probability density function
F(c) cumulative distribution function (failure probability)
– 14 – 61810-2 © IEC:2011
h(c) hazard function (or instantaneous failure rate)
H(c) cumulative hazard function
R(c) reliability function of the Weibull distribution (survival probability)
B expected time at which 10 % of the population have failed
(10 % fractile of the lifetime)
c cycle – variable
ˆ
m mean cycles to failure (MCTF)
β Weibull shape parameter (indicating the rate of change of the instantaneous failure rate
with time)
η Weibull scale parameter or characteristic life (at which 63,2 % of the items have failed)
σ standard deviation
A.4 Weibull distribution
The fundamental Weibull formulae are defined as follows.
NOTE For more information, reference is made to IEC 61649.
The probability density function (PDF) of the Weibull distribution is:
β
c
β−1
−
c
η
f (c)=β e (A.1)
β
η
The cumulative distribution function (CDF), or the expected fraction failing at cycle c:
β
−(c /η)
F(c)= 1− e (A.2)
The reliability function R(c), or the expected fraction surviving at cycle c:
β
−(c /η )
R(c)=1− F(c)= e (A.3)
The hazard function (or instantaneous failure rate) h(c) is:
β−1
c
h(c)=β (A.4)
β
η
The cumulative hazard function H(c) is:
β
c
(A.5)
H(c)=
η
A.5 Procedure
A.5.1 Graphical methods
A.5.1.1 Overview
Graphical analysis is performed by plotting the data on a suitably designed Weibull probability
paper, fitting a straight line through the data, and estimating the distribution parameters (the
61810-2 © IEC:2011 – 15 –
shape parameter, and the characteristic life or scale parameter). Then the reliability
characteristics (i.e. MCTF, B value, and standard deviation) are calculated.
Graphical methods benefit from relatively straightforward processes and availability for data
with a mixture of failure modes. The fundamentals of the analysis and an outline of the
processes applied to Weibull probability and Weibull hazard plots are given in this clause.
A.5.1.2 Weibull probability plot
A.5.1.2.1 Ranking and plotting positions
To make the Weibull plot, rank the data from the lowest to the highest number of cycles to
failure (c ). This ranking will set up the plotting positions for the cycle (c), axis and the
i
ordinate, cumulative distribution function (F(c)), in percentage values.
F(c) is calculated by median rank regression (MRR).
An approximate value may be obtained using Benard’s approximation (see 7.2.1 of
IEC 61649:2008):
F(c ) = (i – 0,3) / (n + 0,4) % (A.6)
i
where
n is the number of tested items;
i is the ranked position of the data item.
Data points of (c ,F(c )) are plotted on the Weibull probability plotting paper.
i i
For details, see 7.2.1 and 7.2.2 of IEC 61649:2008.
A.5.1.2.2 Weibull probability plotting paper
The design of Weibull probability paper is shown below.
The equation (A.3) can be rewritten to the following equation:
β
(c /η )
= e (A.7)
1− F(c)
Taking normal logarithms of both sides of the equation (A.7) twice gives an equation of a
straight line as shown below:
lnln =β lnc−β lnη
(A.8)
1− F(c)
The equation is a straight line of the form y= ax+ b . Weibull paper is designed by plotting
the cumulative probability of failure using a log log reciprocal scale against c on a log scale.
When the equation is plotted as a function of ln(c), the slope of the straight line plotted in this
manner will be β, the shape parameter, i.e.
– 16 – 61810-2 © IEC:2011
y= ln ln (A.9)
1− F(c)
where
a = β;
x = ln (c);
b = – β ln (η ).
The scale parameter is obtained from b = – β ln (η ) as follows:
η = exp [– b /β] (A.10)
where b is the value of y when c is equal to 1, that is ln(c) = 0.
When data are following a Weibull distribution, those data plotted on a Weibull distribution
paper become a straight line. Figure A.1 shows a blank Weibull distribution paper.
In c
Number of operations at failure c
IEC 417/11
NOTE Partially rewritten on the basis of the paper published by JUSE PRESS
Figure A.1 – An example of Weibull probability paper
A.5.1.3 Hazard plot
A.5.1.3.1 Ranking and plotting positions
To perform the hazard plot, rank the data from the lowest to the highest number of cycles to
failure. This ranking will set up the plotting positions for the cycle (c), axis and the ordinate,
cumulative hazard value H(c), in percentage values. H(c) is calculated by hazard value h(c).
Data points of (c ,H(c )) are plotted on the cumulative hazard paper.
i i
F(c) (%)
In In
1 – F(c)
61810-2 © IEC:2011 – 17 –
For details, see 7.3 of IEC 61649:2008.
A.5.1.3.2 Cumulative hazard plotting paper
The design of cumulative hazard paper is shown below.
Taking natural logarithms of both sides of equation (A.5) gives:
In H(c) = β ln c – β ln η (A.11)
The equation is a straight line of the form y= ax+ b . Cumulative hazard paper is designed by
plotting the cumulative probability of failure using a log reciprocal scale against c on a log
scale. When the equation is plotted as a function of ln(c), the slope of the straight line plotted
in this manner will be β, the shape parameter, i.e.
y = ln H(c) (A.12)
where
a = β;
x = ln (c);
b = – β ln (η ).
The scale parameter is obtained from b = – β ln (η ) as follows:
η = exp [– b /β] (A.13)
where b is the value of y when c is equal to 1, that is ln(c) = 0.
When data points are following a cumulative hazard function, those data points plotted on a
cumulative hazard paper become a straight line. Figure A.2 shows a blank cumulative hazard
paper.
– 18 – 61810-2 © IEC:2011
In c
Number of operations at failure
c
IEC 418/11
NOTE Partially rewritten on the basis of the paper published by JUSE PRESS
Figure A.2 – An example of cumulative hazard plotting paper
A.5.1.4 Estimate values of distribution parameters and characteristics
Distribution parameters and characteristics in the Weibull probability plot and the hazard plot
are common.
Draw a straight line (that best fits the data) through the data points on the plotting paper
(Figure A.3).
100 0
1 c
IEC 419/11
Figure A.3 – Plotting of data points and drawing of a straight line
ˆ
1) The point estimate of the shape parameter, β
ˆ
β is derived from the slope a of the plotted straight line.
H(c) (%)
H(c) (%)
In H(c)
In H(c)
61810-2 © IEC:2011 – 19 –
A parallel line is drawn above the original plotted line, through the coordinate point
(ln c = 1, ln (H(c) = 0). The ordinate value of this point is equivalent to H(c) = 100 % (or
F(c) = 63,2 %).
ˆ
β is read from the value of lnH(c) corresponding to the cross point of this parallel line
and a vertical line through ln c = 0, as shown in Figure A.4.
In c
0 1
(Inc = 1, InH(c) = 0)
100 % 0
10,54 %
ˆ
–β
1 ˆ
η c
β ˆ
IEC 420/11
Figure A.4 – Estimation of distribution parameters
2) The point estimate of the scale parameter, ηˆ
ˆ is derived directly from the cross point of the original plotted line and a horizontal line
η
through H(c) = 100 % (or F(c) = 63,2 %) as shown in Figure A.4.
3) The point estimate of mean cycles to failure (MCTF), ˆ
m
ˆ
m is given by the following expression:
ˆ
ˆ ˆ
MCTF = m = η× Γ(1+1/β ) (A.14)
with ηˆ taken from step 2 above, and the gamma function value (Γ as defined e.g. in 2.56 of
ISO 3534-1:2006) obtained with a handy scientific calculator or a convenient gamma
functional table, respectively (see Annex D).
ˆ
4) The point estimate of standard deviation, σ
ˆ
σ is given by the following expression:
ˆ ˆ
ˆ ˆ ( ) ( ) (A.15)
σ=η× Γ 1+ 2 /β −Γ 1+1/β
ηˆ and the gamma function value are obtained in the same way as mentioned under
Where
step 3 above.
5) The point estimate of the fractile (10 %) of the cycles to failure, B
H(c)
In H(c)
– 20 – 61810-2 © IEC:2011
B is derived directly from the cross point of the original plotted straight line and a horizontal
line through F(c) = 10 % in the Weibull plot or H(c) = – ln 0,9 = 10,54 % in the hazard plot as
shown in Figure A.4.
A.5.1.5 Mixture of several failure modes
The Weibull probability plot or a hazard plot can result in a “dogleg curve”.
If the line is not straight, it is called “Dogleg Weibull”. This is caused by a mixture of more
than one failure mode, i.e. usually competitive failure modes.
When this occurs, a close examination of the failed items is the best way to separate the data
into different failure modes.
Suppose there is a data set of two kinds of failure modes (A and B). The first set should be
analyzed as A mode data only, suspending the B mode data. Consequently, the second set
would contain B mode data. These two sets of data can be used to predict the failure
distribution.
If this is done correctly, plotting the two separate data sets will result in straight lines. A
detailed description is shown in Annex G of IEC 61649:2008. In particular, it has to be noted
that at least 10 failures are required for each failure mode.
A.5.2 Numerical methods
A.5.2.1 Distribution parameters
Whereas the graphical method described in A.5.1 above applies to complete, single censored,
or multiple censored data, the numerical method of this subclause does not deal with
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