IEC TR 62967:2018

IEC TR 62967 ®
Edition 1.0 2018-08
TECHNICAL
REPORT
Methods for calculating the main static performance indicators of transducers
and transmitters
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IEC TR 62967 ®
Edition 1.0 2018-08
TECHNICAL
REPORT
Methods for calculating the main static performance indicators of transducers

and transmitters
INTERNATIONAL
ELECTROTECHNICAL
COMMISSION
ICS 25.040.40 ISBN 978-2-8322-5980-1

– 2 – IEC TR 62967:2018  IEC 2018
CONTENTS
FOREWORD . 6
INTRODUCTION . 8
1 Scope . 9
2 Normative references . 9
3 Terms and definitions . 9
3.1 Basic terms . 10
3.1.3 Input terms . 10
3.1.4 Output terms . 10
3.2 Static calibration characteristics . 11
3.3 Definitions of static performance indicators . 11
4 Methods for calculating individual static performance indicators . 15
4.1 Establishment of static calibration characteristics . 15
4.1.1 General requirements for static calibration . 15
4.1.2 The calculation of static calibration characteristics . 15
4.2 Span (x ) . 16
FS
4.3 Full-span output (YFS) . 16
4.4 Resolution (Rx) . 16
4.5 Sensitivity (Si) . 17
4.6 Hysteresis (ξH) . 17
4.7 Repeatability (ξR) . 17
4.7.1 Calculating methods . 17
4.7.2 Determination of coverage factor . 18
4.7.3 Calculation of sample standard deviations . 18
4.8 Linearity (ξL) . 18
4.8.1 The general formula for calculating linearity . 18
4.8.2 Absolute linearity (ξL,ab) . 19
4.8.3 Terminal-based Linearity (ξL,te) . 19
4.8.4 Shifted-terminal-based Linearity (ξL,s,te) . 20
4.8.5 Zero-based linearity (ξL,ze) . 20
4.8.6 Front-terminal-based Linearity (ξL,f,te) . 21
4.8.7 Independent Linearity (ξL,in) . 21
4.8.8 Least-squares Linearity (ξL,ls) . 22
4.9 Conformity (ξC) . 23
4.9.1 The general formula for calculating conformity . 23
4.9.2 Absolute conformity (ξC,ab) . 23
4.9.3 Terminal-based conformity (ξC,te) . 24
4.9.4 Zero-based conformity (ξC,ze) . 24
4.9.5 Front-terminal-based conformity (ξC,f,te) . 24
4.9.6 Independent conformity (ξC,in) . 24
4.9.7 Least-squares conformity (ξC,ls) . 25
4.10 Drift and shift . 25
4.10.1 Zero drift (D0) . 25
4.10.2 Drift of upper-range-value output (Du) . 26
4.10.3 Thermal zero shift (γ). 26
4.10.4 Thermal shift of upper-range-value output (β) . 26
5 Methods for calculating combined static performance indicators . 27

5.1 Combined linearity and hysteresis (Linearity plus hysteresis) ξLH . 27
5.1.1 The general form of calculating formula . 27
5.1.2 The calculation of reference line . 27
5.2 Combined linearity, hysteresis and repeatability(ξLHR) . 27
5.2.1 The general form of calculating formula . 28
5.2.2 The alternative forms of the calculating formulas . 28
5.2.3 The method for calculating the working characteristics . 29
Annex A (informative) Methods and examples for calculating linearities . 31
A.1 Numerical examples for calculating zero-based linearity . 31
A.1.1 The general principle of calculation. 31
A.1.2 Solving for the first approximating straight line. 31
A.1.3 Solving for the second approximating straight line . 31
A.2 Numerical examples for calculating independent linearity . 32
A.2.1 The principle of a precise method . 32
A.2.2 The principle of the makeshift methods . 35
A.3 A comparison of the results of all kinds of linearities . 35
Annex B (informative) Methods and Examples for Calculating Conformities . 36
B.1 The general principle for calculating conformities . 36
B.1.1 Determining the degree of the fitting curves . 36
B.1.2 Choosing the number of the alternating points . 36
B.1.3 Determining the locations of the alternating points . 36
B.1.4 Finding the finally-successful alternating points . 36
B.2 Numerical examples for calculating conformities . 37
B.2.1 Solving for the terminal-based curve of the second degree and the
terminal-based conformity of the second degree . 37
B.2.2 Solving for the zero-based curve of the second degree and the zero-
based conformity of the second degree . 39
B.2.3 Solving for the front-terminal-based curve of the second degree and the
front-terminal-based conformity of the second degree . 40
B.2.4 Solving for the best curve of the second degree and the independent
conformity of the second degree . 41
B.2.5 Solving for the least-squares curve of the second degree and the least-
squares conformity of the second degree . 42
B.2.6 A rough principle guiding the choice of the theoretical curve . 43
Annex C (informative) Examples for calculating transducer individual and combined
performance indicators . 44
C.1 General principles of calculation . 44
C.2 Numerical examples . 44
C.2.1 Numerical example 1 . 44
C.2.1.4.7 Total uncertainty (linearity plus hysteresis plus repeatability) . 48
C.2.2 Numerical example 2 . 50
C.2.3 Numerical example 3 . 51
Annex D (informative) Examples for calculating transmitter individual and combined
performance indicators . 53
D.1 General principles of calculation . 53
D.2 Numerical example . 53
D.3 Calculation results . 53
Annex E (informative) The Pre-treatment of the Original Data . 56
E.1 The discovery of suspect and unreasonable data points. 56
E.2 The detection of suspect data points . 56

– 4 – IEC TR 62967:2018  IEC 2018
E.2.1 The general principle of statistical detection . 56
E.3 The Inspection of Unreasonable Data Points . 59
E.3.1 The Unreasonable Data Points . 59
E.3.2 Example 1 for Inspecting the Unreasonable Data Points . 60
E.3.3 Example 2 for Inspecting the Unreasonable Data Points . 60
Annex F (informative) The fundamentals for calculating transducer uncertainty . 62
F.1 Components of measurement uncertainty . 62
F.2 Combined uncertainty . 62
F.3 The combined uncertainty of a transducer . 62
F.4 The total uncertainty of a transducer at the ith calibration point . 62
F.5 The total uncertainty of a transducer . 63
Bibliography . 64

Figure 1 – Terminal-based Linearity . 21
Figure 2 – Zero-based Linearity . 21
Figure 3 – Front-terminal-based Linearity. 22
Figure 4 – Independent Linearity . 22
Figure 5 – Terminal-based conformity . 24
Figure 6 – Zero-based conformity . 24
Figure 7 – Front-terminal-based conformity . 25
Figure 8 – Independent conformity . 25
Figure 9 – The method of L(C)HR extreme-point envelope . 29
Figure A.1 – The transformed convex polygon . 33
Figure B.1 – The curve roughly drawn from the given data . 37
Figure C.1 – Deviation curves which are calculated relative to relevant best reference

lines of the first degree . 49
Figure C.2 – Deviation curves which are calculated relative to the working line of the
first degree . 49
Figure C.3 – Deviation curves which are calculated relative to relevant best reference
lines of the second degree . 51
Figure C.4 – Deviation curves which are calculated relative to the working line of the

second degree . 51
Figure D.1 – Deviation curves which are calculated relative to the given working
straight line . 54
Figure D.2 – Deviation curves which are calculated relative to the best reference
straight line . 55
Figure E.1 – Deviation curves which are calculated relative to the best working

straight line . 59
Figure E.2 – Deviation curves which are calculated relative to the best working
straight line . 61

Table 1 – Form to present reliability data with its data types . 18
Table A.1 . 31
Table A.2 . 31
Table A.3 . 32
Table A.4 . 32
Table A.5 . 34

Table B.1 . 37
Table B.2 . 38
Table B.3 . 39
Table B.4 . 40
Table B.5 . 40
Table B.6 . 42
Table B.7 . 43
Table C.1 – The original data obtained in the calibration . 44
Table C.2 – The intermediate results of calculation . 45
Table C.3 – Finding the extreme points n = 5 c = t 0.95 = 2.776 . 46
Table C.4 – The deviations from the best working line . 46
Table D.1 – The original data obtained in the calibration . 53
Table E.1 . 57
Table E.2 . 57
Table E.3 – The original data obtained in the calibration . 58
Table E.4 – A list of the computer-conducted inspection results for the unreasonable
data points . 60

– 6 – IEC TR 62967:2018  IEC 2018
INTERNATIONAL ELECTROTECHNICAL COMMISSION
____________
METHODS FOR CALCULATING THE MAIN STATIC PERFORMANCE
INDICATORS OF TRANSDUCERS AND TRANSMITTERS

FOREWORD
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data of a different kind from that which is normally published as an International Standard, for
example "state of the art".
IEC TR 62967, which is a technical report, has been prepared by subcommittee 65B:
Measurement and control devices of IEC technical committee 65: Industrial-process
measurement, control and automation.
The text of this International Standard is based on the following documents:
Enquiry draft Report on voting
65B/961/DTR 65B/1016/RVC
Full information on the voting for the approval of this technical report can be found in the
report on voting indicated in the above table.

This document has been drafted in accordance with the ISO/IEC Directives, Part 2.
The committee has decided that the contents of this document will remain unchanged until the
stability date indicated on the IEC website under "http://webstore.iec.ch" in the data related to
the specific document. At this date, the document will be
• reconfirmed,
• withdrawn,
• replaced by a revised edition, or
• amended.
A bilingual version of this publication may be issued at a later date.

– 8 – IEC TR 62967:2018  IEC 2018
INTRODUCTION
This technical report provides a comprehensive illustration of the methods for calculating the
main static performance indicators of transducers, transmitters and similar measuring devices.
First of all, in order to avoid any misunderstanding, we would like to review the commonly-
accepted definition of transducers and transmitters. Generally speaking, in a measurement
field, a transducer is a measuring device which converts the non-electrical quantity to be
measured into corresponding electrical quantity, while a transmitter is a kind of transducer
which is required to provide a previously-given linear output.
The common-in-use standards [01]-[06] listed in the relevant documents to be considered in
this report, are useful in evaluating the main static performance indicators of measuring
instruments and other similar devices. But the relevant descriptions of calculation methods in
standards [01]-[05] are not complete and adequate in many ways. This fact was clearly stated
in the Introduction of IEC 61298 [03].
On the whole, these publications [01]-[05] mainly contain relevant technical terms and
definitions. Since in essence, they are not standards which are dedicated solely to the
calculation of performance indicators, so they contain no or only very simple and inadequate
illustrations of the calculation methods. Moreover, as these contents have existed for about
tens of years, probably now is the time to make an all-round revision and improvement of
them. Since there are many static performance indicators that should be calculated and the
calculation methods can form a rather complete system. So it is better to create a separate
report or a separate standard.
For the main static performance indicators, the existing relevant IEC standards have only
theoretical definitions, but have no specific calculation methods. This does not mean that
these methods are too simple to mention. But on the contrary, some of them are too difficult
to be used in industry. Therefore, this report puts forward, improves and simplifies the existing
relevant calculation methods, may probably serve as a good basis on which to create a new
calculation-oriented IEC standard.
The report is intended for use by manufacturers to work out their factory-level test standards,
by users to make rigorous acceptance tests and wise applications, and by authorized
metrological establishments to verify the measuring device performance indicators of the
manufacturers or of the users.

Numbers in square brackets refer to the Bibliography.

METHODS FOR CALCULATING THE MAIN STATIC PERFORMANCE
INDICATORS OF TRANSDUCERS AND TRANSMITTERS

1 Scope
This Technical Report provides guidance on the assurance of reliability data of automation
devices. If the source of this data is calculation, guidance is given on how to specify the
methods used for this calculation. If the source is through observations, guidance is given on
how to describe these observations and their evaluations. If the source is the outcome of
laboratory tests, guidance is given on how to specify these tests and the conditions under
which they have been carried out.
This document defines the form to present the data.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their
content constitutes requirements 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 60050-300, International Electrotechnical Vocabulary – Electrical and electronic
measurements and measuring instruments
Part 311: General terms relating to measurements
Part 312: General terms relating to electrical measurements
Part 313: Types of electrical measuring instruments
Part 314: Specific terms according to the type of instrument
IEC 60050-351, International Electrotechnical Vocabulary – Part 351: Control technology
IEC 60770-1:1999 Transmitters for Use in Industrial-process Control Systems – Part 1:
Methods for Performance Evaluation
3 Terms and definitions
For the purposes of this document, the terms and definitions given in IEC 60050-300,
IEC 60050-351 and IEC 60770-1:1999, as well as the following apply.
ISO and IEC maintain terminological databases for use in standardization at the following
addresses:
• IEC Electropedia: available at http://www.electropedia.org/
• ISO Online browsing platform: available at http://www.iso.org/obp

– 10 – IEC TR 62967:2018  IEC 2018
3.1 Basic terms
3.1.1
static characteristics
relationship of the output of a transducer to its input, when the measurand is at the state of
stabilization or very slow variation
Note 1 to entry: There are many performance indicators under the title of static characteristics.
Note 2 to entry: Static performance indicators are applicable only under a given interval of temperature.
3.1.2
static calibration
process in which the static characteristics are obtained under given static conditions
3.1.3 Input terms
3.1.3.1
lower range-value
lowest value of the measurand
3.1.3.2
upper range-value
highest value of the measurand
3.1.3.3
measuring range
measuring region indicated by the upper and lowr range-values of the measurand
3.1.3.4
span
span, also called the full-span input, is the algebraic difference between the upper and lower
range-values of the measurand.
3.1.4 Output terms
3.1.4.1
zero-range-value output
output when measurand is at its zero range-value
3.1.4.2
lower-range-value output
output when measurand is at its lowest range-value
3.1.4.3
upper-range-value outpu
output when measurand is at its highest range-value
3.1.4.4
full-span output
algebraic difference between the upper-range-value output and lower-range-value output of a
device as defined by its working characteristics
3.1.5
linearity
closeness to which the output-input curve of a transducer approximates a straight line
Note 1 to entry: There should be no contribution of hysteresis and repeatability to linearity.

3.1.6
conformity
closeness to which the output-input curve of a transducer approximates a certain curve
Note 1 to entry: There should be no contribution of hysteresis and repeatability to conformity.
3.1.7
reference characteristics
straight line, curve or equation which is used as a reference or a contrast
Note 1 to entry: Under a certain application situation, reference characteristics can be accepted as the true
characteristics of a transducer.
Note 2 to entry: In this Technical report, reference characteristics are mainly used in the calculation of linearities,
conformities, and linearity (conformity) plus hysterresis.
3.1.8
working characteristics
output-input equation or curve, which is adopted as the true characteristics of a transducer
Note 1 to entry: Working Characteristics has taken into consideration the combined contribution of linearity
(conformity), hysteresis and repeatability.
3.1.9
utilization characteristics
relationship of the measurand to the output of a transducer
3.1.10
linear transducer
kind of transducers whose working characteristics are linear
3.1.11
non-linear transducer
kind of transducers whose working characteristics are non-linear
3.2 Static calibration characteristics
3.2.1
up-travel actual average characteristics
curve connecting all the arithmetic average points of a group of measured data at all the
calibration points in the up-travel
3.2.2
down-travel actual average characteristics
curve connecting all the arithmetic average points of a group of measured data at all the
calibration points in the down-travel
3.2.3
up-travel and down-travel actual average characteristics
curve connecting all the arithmetic average points of a group of measured data at all the
calibration points in the up- and down-travel
Note 1 to entry: It is also called the actual average characteristics (or curve) of a transducer.
3.3 Definitions of static performance indicators
3.3.1
resolution
smallest change in input that can cause observable change in output in the whole input span

– 12 – IEC TR 62967:2018  IEC 2018
3.3.2
sensitivity
ratio of output change to its corresponding input change
3.3.3
hysteresis
for the same input and in the whole input span, difference between the values of the down-
travel actual average characteristics and the up-travel actual average characteristics
3.3.4
repeatability
for a short time interval and in the same working condition, degree of scatterance of a group
of readings obtained when the input is approaching the same measuring point in the same
direction for a number of test cycles
3.3.5
linearity
maximum deviation of the actual average characteristics (curve) from the reference straight
line
Note 1 to entry: Linearity is expressed as a percentage of full-span output.
Note 2 to entry: According to different reference straight lines taken, there are different kinds of linearities, with
the following as the main ones.
Note 3 to entry: When expressed simply as linearity, it is assumed to be independent linearity.
Note 4 to entry: The choice of linearities depends on the application situations of transducers.
3.3.5.1
absolute linearity
also called theoretical linearity, it is calculated from the reference straight line or theoretical
straight line that can be specified before the calibration test is made
Note 1 to entry: Absolute linearity actually shows the linearity accuracy of a transducer and is quite different from
all the linearities that follow.
Note 2 to entry: Absolute linearity is exclusively used in transmitter applications.
3.3.5.2
terminal-based linearity
linearity calculated from the terminal-based straight line that is taken as the reference straight
line
Note 1 to entry: Terminal-based straight line coincides with the actual average characteristics (curve) at its upper
and lower limits.
Note 2 to entry: Terminal-based Linearity is easy to calculate, but its value is rather conserved.
3.3.5.3
shifted terminal-based linearity
linearity calculated from the shifted terminal-based straight line that is taken as the reference
straight line
Note 1 to entry: The shifted terminal-based straight line has the same slope as the terminal-based straight line
and can minimize its maximum deviation from the actual average characteristics (curve) by parallel shifting.
Note 2 to entry: If the device under test has a C-shaped actual average characteristics (curve), the shifted
terminal-based straight line will become the best straight line, or best line in short.
3.3.5.4
zero-based linearity
linearity calculated from the zero-based straight line that is taken as the reference straight
line
Note 1 to entry: Zero-based straight line goes through the theoretical zero point and can minimize its maximum
deviation from the actual average characteristics (curve) by changing its slope.
Note 2 to entry: Sometimes zero-based straight line is also called the forced zero-intersecting best straight line.
3.3.5.5
front-terminal-based linearity
linearity calculated from the front-terminal-based straight line that is taken as the reference
straight line
Note 1 to entry: The front-terminal-based straight line goes through the front end of the actual average
characteristics (curve) and can minimize its maximum deviation from the actual average characteristics (curve) by
changing its slope.
Note 2 to entry: Sometimes and in some references, the front-terminal-based straight line is also called the zero-
based straight line.
3.3.5.6
independent linearity
linearity calculated from the best straight line that is taken as the reference straight line
Note 1 to entry: The best straight line is a straight-line midway between the two parallel straight lines closest
together and enclosing the actual average characteristics (curve).
Note 2 to entry: The best straight line can minimize its maximum deviation from the actual average characteristics
(curve).
3.3.5.7
least-squares linearity
linearity calculated from the least-squares straight line which is adopted as the reference
straight line
Note 1 to entry: The least-squares straight line can guarantee that, the sum of the squares of the deviations of
the actual average characteristics (curve) from it, is a minimum.
3.3.6
conformity
maximum deviation of the actual average characteristics (curve) from the reference curve
Note 1 to entry: Conformity is expressed as a percentage of full-span output.
Note 2 to entry: According to different reference curves taken, there are different kinds of conformities, with the
following as the main ones.
Note 3 to entry: The reference curve is usually in the form of an algebraic polynomial of a certain degree.
Note 4 to entry: When expressed simply as conformity, it is assumed to be independent conformity.
Note 5 to entry: The choice of conformities depends on the application situations of transducers.
3.3.7
absolute conformity
also called theoretical conformity, it is calculated from the reference curve or theoretical curve
that can be specified before the calibration test is made
Note 1 to entry: Absolute conformity actually shows the conformity accuracy of a transducer and is quite different
from all the conformities that follow.
Note 2 to entry: The reference curve should be specified according to the application requirement of the
transducer.
3.3.7.1
terminal-based conformity
conformity calculated from the terminal-based curve that is taken as the reference curve
Note 1 to entry: Terminal-based curve coincides with the actual average characteristics (curve) at its upper and
lower limits and can minimize its maximum deviation from the actual average characteristics (curve).

– 14 – IEC TR 62967:2018  IEC 2018
3.3.7.2
zero-based conformity
conformity calculated from the zero-based curve that is taken as the reference curve
Note 1 to entry: Zero-based curve goes through the theoretical zero point and can minimize its maximum
deviation from the actual average characteristics (curve).
Note 2 to entry: Sometimes Zero-based curve is also called the forced zero-intersecting best curve.
3.3.7.3
front terminal-based conformity
linearity calculated from the front terminal-based curve that is taken as the reference curve
Note 1 to entry: The front terminal-based curve goes through the front end of the actual average characteristics
(curve) and can minimize its maximum deviation from the actual average characteristics (curve).
Note 2 to entry: Sometimes and in some references, the front terminal-based curve is also called the zero-based
curve.
3.3.7.4
independent conformity
conformity calculated from the best curve that is taken as the reference curve
Note 1 to entry: The best curve can minimize its maximum deviation from the actual average characteristics
(curve).
3.3.7.5
least-squares conformity
conformity calculated from the Least-squares curve that is taken as the reference curve
Note 1 to entry: The least-squares curve can guarantee that the sum of the squares of the deviations of the actual
average characteristics (curve) from it is a minimum.
3.3.8
combined linearity (conformity) and hysteresis
extreme value of the systematic error of a transducer
Note 1 to entry: This performance indicator shows the combined contribution, but not the pure addition, of
linearity (conformity) and hysteresis.
3.3.9
uncertainty
result of an evaluation that shows a zone in which the true values of the measurand lie under
specified operating conditions
Note 1 to entry: It is a parameter that can reasonably show the scatterance of the values of the measurand, and
also a parameter connecting with the measurement result.
Note 2 to entry: Uncertainty can more reasonably show the real picture of the total result of a measurement, both
qualitatively and quantitatively.
3.3.10
total uncertainty
combined linearity (conformity), hysteresis and repeatability, also called the primary
uncertainty, that is obtained from the static calibration under specified conditions and
calculated by using specified calculating methods based on the general principle of
measurement uncertainty
Note 1 to entry: In this Technical report, total uncertainty is the result of the combined contribution, but not the
pure addition, of linearity (conformity)，hysteresis and repeatability. This performance indicator is also called the
reference accuracy.
3.3.11
zero drift
undesired change in zero-range-value output over a specified period of time

Note 1 to entry: It is usually expressed as a percentage of full-span output.
3.3.12
drift of upper-range-value output
undesired change in upper-range-value output over a specified period of time
Note 1 to entry: It is usually expressed as a percentage of full-span output.
Note 2 to entry: For load cells sometimes this performance indicator is also called creep.
Note 3 to entry: If the specified period of time is very long, for example, several months or years, this
performance indicator is usually called long-term stability.
3.3.13
thermal zero shift
undesired change in zero-range-val
...