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Methods for calculating the main static performance specifications of transducers
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GB/T 18459-2001: PDF in English (GBT 18459-2001) GB/T 18459-2001
NATIONAL STANDARD OF THE
PEOPLE’S REPUBLIC OF CHINA
ICS 17.020
N 05
Methods for calculating the main static performance
specifications of transducers
ISSUED ON: OCTOBER 08, 2001
IMPLEMENTED ON: MAY 01, 2002
Issued by: General Administration of Quality Supervision, Inspection and
Quarantine
Table of Contents
Foreword ... 3
1 Scope ... 4
2 Definitions ... 4
3 Calculation method for single static performance specification ... 10
4 Calculation methods for uncertainty and other comprehensive static
performance specifications ... 34
Annex A (Normative) General principle and calculation example of linearity
calculation ... 41
Annex B (Normative) General principle and calculation example of conformity
calculation ... 48
Annex C (Normative) Calculation examples for transducer sub-performance
specifications and comprehensive performance specifications ... 56
Annex D (Normative) Calculation example for transmitter sub-item performance
specifications and comprehensive performance specifications ... 69
Annex E (Normative) Inspection of transducer precision ... 73
Annex F (Informative) Pre-processing of raw data ... 75
Annex G (Informative) Basic principles for transducer uncertainty calculation
... 82
Annex H (Informative) Bibliography ... 85
Methods for calculating the main static performance
specifications of transducers
1 Scope
This Standard specifies the definitions and calculation methods of the main
static performance specifications of general transducers.
This Standard is applicable to the calculation of the main static performance
specifications of transducers during the process of development, production
and use. It is also applicable to the development and revision of product
standards for various transducers.
2 Definitions
For the purposes of this document, the following terms and definitions apply.
2.1 Basic terminologies
2.1.1 static characteristics
the relationship between output and input when measured under constant or
slow-change conditions
NOTES:
1 The static characteristics of the transducer include a variety of performance indicators that can be
determined by static calibration.
2 The static performance specifications of the transducer shall normally be marked with the applicable
temperature range.
2.1.2 static calibration
the process of obtaining static characteristics under specified static test
conditions
2.1.3 measuring range
the interval expressed by the maximum measured (measured upper limit) and
the smallest measured (lower measured limit) under the premise of ensuring
performance indicators
the transducer that working characteristics are represented by curve equation
2.2 Static calibration characteristics
2.2.1 up-travel actual average characteristics
the connection curve of the arithmetic mean point of a set of measured values
at each calibration point of the up-travel
2.2.2 down-travel actual average characteristics
the connection curve of the arithmetic mean point of a set of measured values
at each calibration point of the reverse stroke
2.2.3 up-travel and down-travel actual average characteristics
the connecting curve of the average point of the arithmetic mean of the positive
and down-travel strokes of each calibration point, also called the actual
characteristics (curve)
2.3 static performance specifications
2.3.1 resolution
the minimum input change that produces observable output changes over the
entire input range
2.3.2 sensitivity
the ratio of output change to corresponding input change
2.3.3 hysteresis
the difference between the positive and down-travel stroke outputs of the
transducer for the same input when the input is full-span
2.3.4 repeatability
the degree of dispersion between a set of measured outputs, in a short interval
of time, under the same working conditions, when the input volume changes
from the same direction to full scale, multiple times approaching and reaching
the same calibration point
2.3.5 linearity
the maximum deviation of the actual average characteristic curve of the positive
and down-travel strokes from the reference line, expressed as a percentage of
full-span output
the linearity that the reference line is the front terminal-based line
NOTES:
1 The front terminal-based line passes through the front terminal point of the actual characteristics of the
transducer. However, the maximum deviation of the transducer's actual characteristics shall be minimized
by changing the slope.
2 The front terminal-based line is called the zero-base line in some foreign standards and literature.
2.3.5.6 Independent linearity
the linearity that the reference line is the best straight line
NOTES:
1 The best straight line is the median line of two parallel lines that are closest to each other and can
accommodate the up-travel and down-travel actual average characteristics of the transducer.
2 The best straight line guarantees that the actual deviation of the transducer’s actual characteristics is
minimal.
2.3.5.7 least-squares linearity
the linearity that the reference line is the least square line
NOTE: The least-squares line shall ensure that the sum of the squares of the actual characteristics of the
transducer is minimal.
2.3.6 conformity
the maximum deviation of the curve of the up-travel and down-travel actual
average characteristics to the reference curve, expressed as a percentage of
full-span output
NOTES:
1 There are multiple compliances that vary with the reference curve.
2 Conformity shall be limited. Conformity without qualifiers means independent conformity.
2.3.6.1 absolute conformity
the conformity that the reference curve is the specified curve, also known as
theoretical conformity
NOTES:
1 The reference curve for absolute conformity is pre-defined. It reflects the conformity precision and is
absolutely different from the properties of several other degrees of conformity.
2.3.8 uncertainty
an evaluation result that characterizes the measured true value in a certain
range; it is a parameter that reasonably gives the dispersion of the measured
value, and it is also a parameter associated with the measurement result
NOTE: Uncertainty can more reasonably represent the nature of measurement results from both
qualitative and quantitative aspects.
2.3.9 total uncertainty
also known as basic uncertainty, an uncertainty obtained by static calibration
under specified conditions and according to the specified calculation method
NOTE: In this Standard, the total uncertainty is combined linearity, hysteresis and repeatability. It reflects
their joint role. It is not simply adding.
2.3.10 zero drift
zero output only changes with time within the specified time, usually expressed
as a percentage of full-span output
2.3.11 drift of output span
full-span output changes only over time within the specified time, usually
expressed as a percentage of full-span output
NOTE: If the prescribed assessment time is long, such as months to years, this indicator is often referred
to as long term stability.
2.3.12 thermal zero shift
zero output change caused by changes in ambient temperature, usually
expressed as a percentage of the full-span output of the unit temperature
2.3.13 thermal shift of output span
full-span output change that is due to changes in ambient temperature, usually
expressed as a percentage of the full-span output of the unit temperature
3 Calculation method for single static performance
specification
3.1 Establishment of static calibration characteristics
3.1.1 General requirements for static calibration
value of the difference between the maximum and minimum values of a set of
n measured values at the ith calibration point;
dR - range coefficient; it depends on the number of calibration cycles n, i.e. the
number of measurements at a calibration point or the sample size n. The
relationship between the range coefficient dR and the number of calibration
cycles is shown in Table 2.
Table 2
NOTES:
1 The range method is slightly simpler than the Bessel formula method. However, the calculated sample
standard deviation S is generally slightly larger.
2 When calculating S, if not specified, it refers to Bessel formula method. In case of dispute, the Bessel
formula shall prevail.
3.7.4 Selection of standard deviation of transducer sample
3.7.4.1 If the number of calibration points is m (usually m=5~11), 2m sample
standard deviation S can be calculated. This Standard stipulates that the largest
S (i.e. the maximum standard deviation Smax) is selected to participate in the
calculation of Equation 10 so as to obtain the repeatability of the transducer as
a single performance specification.
3.7.4.2 This Standard allows the user to perform an equal precision test on the
transducer being tested as an option according to the method of Annex E. The
variance at each measurement point of the equal precision transducer has the
same mathematical expectation. Therefore, the average variance can be used
instead of the variance at each measurement point. Therefore, if it is determined
that the transducer is an equal-precision sensor, the maximum standard
deviation Smax shall not be taken. It shall take the average standard deviation
Sav to calculate the repeatability. Sav is calculated as follow:
If the precision of the transducer being tested is not performed, or if the test
fails, the requirements of 3. 7. 4. 1 shall still be applied. That is, calculate
repeatability according to the requirements of unequal precision transducer.
NOTES:
- maximum and minimum input values of the transducer.
NOTES:
1 The calculation method for terminal-based line is simple and easy to implement with the bridge detection
circuit.
2 Compared with other linearities, the calculation results of terminal-based linearity is generally larger.
3.8.4 Shifted terminal-based linearity
See equation (16) for calculation.
As the shifted terminal-based equation of reference characteristics, see 3.8.3
for the calculation method of its terminal-based linear equation. See A.2.2.1 of
Annex A for the translation method.
NOTE: The shifted terminal-based linearity can be used to replace the independent linearity in some
occasions where the requirements are not too high.
3.8.5 Zero-based linearity
The principle of calculation method is shown in Figure 2. The calculation
formula is shown in equation (16).
By definition, a zero-based linear equation can be written as reference
characteristics:
where,
b - zero-base linear slope, i.e. the theoretical zero point of the transducer (x =
0, y = 0) and the slope of the line connecting the center of gravity of the smallest
maximum positive and negative deviation points.
For the calculation of the zero-based line, see Annex A, A1.
NOTE: The working characteristics of the transducer can be represented by a zero-based line. Its equation
is simple and easy to use.
3 The front terminal-based linearity is generally better than zero-based linearity. And it can make the
deviation near the zero point of the transducer smaller.
3.8.7 Independent linearity
The principle of calculation method is shown in Figure 4. The calculation
formula is shown in equation (16).
The calculation method of the best straight line is shown in Annex A, A2.
NOTES:
1 Independent linearity value is the smallest among various linearities. Independent linearity is used
whenever possible to accurately measure linearity.
2 If the transmitter has an adjustment means, it can adjust the translation and adjust the slope. Adjust the
best straight line to the specified straight line for the highest absolute linearity.
3 Linearity shall be added with qualifier. Linearity without qualifiers means independent linearity.
3.8.8 Least-squares linearity
See equation (16) for the calculation formula.
As the least-squares line of the reference characteristic, the sum of the squares
of the deviation of the actual characteristics of the transducer shall be minimized.
The least-squares linear equation is:
where,
Yls - theoretical output of the transducer;
a, b - intercept and slope of the least-squares line, respectively;
x - actual input of the transducer.
The intercept and slope of the least-squares line can be obtained by straight
line fitting of the actual characteristics of the transducer. The calculation formula
is as follow:
where,
- the maximum deviation of the actual characteristics curve of the
transducer from the reference curve;
- the total average characteristics value of the transducer at the ith
calibration point;
Yi - the reference characteristics value of the transducer at the ith calibration
point;
YFS - the full-span output of the transducer.
NOTES:
1 Example for calculation of :
(1) According to up-travel and down-travel actual average characteristics , use the best curve as
the reference curve to calculate the independent conformity.
(2) According to up-travel and down-travel actual average characteristics , use the working
characteristics curve as the reference curve to calculate the absolute conformity.
2 The results calculated by the second method described above shall contain the components of the
hysteresis and repeatability to varying degrees. Strictly speaking, it is not conformity.
3 If not stated, the conformity refers to the result of calculated according to the first method
above, that is, independent conformity.
4 In some applications, if necessary, it can also use a set of calibration data to calculate the conformity
without .
3.9.2 Absolute conformity
The calculation formula is shown in equation (23).
NOTES:
1 In several conformities, the absolute conformity requirement is the strictest.
2 If non-linear transducer is required to be interchangeable, absolute conformity shall be used.
3.9.3 Terminal-based conformity
See Figure 5 for the principle of calculation method. See equation (23) for
calculation formula.
If the thermal full-span output shift of the transducer is not linear with the
temperature interval, then ( T2 - T1 ) shall be divided into several small intervals.
And use equation (28) to calculate the β of each interval. Take the largest
absolute value of β.
4 Calculation methods for uncertainty and other
comprehensive static performance specifications
In static operation, linearity (conformity), hysteresis, and repeatability are often
referred to as transducer's sub-performance or individual performance
specification. And the different combinations of these specifications constitute
various comprehensive performance specifications. There is no mathematically
defined link between comprehensive performance specifications (such as total
uncertainty) and performance specifications for each sub-performance.
Take a linear transducer as an example (the algorithm of a non-linear
transducer is the same as that of a linear transducer). And it is based on the
principle of measuring the performance specification of the transducer by the
limit deviation. This Standard specifies the calculation methods for the following
comprehensive performance specifications.
4.1 Linearity plus hysteresis
4.1.1 General form of calculation formula
Linearity plus hysteresis is the limit of transducer system error. It is calculated
as follow:
where,
- the maximum deviation of up-travel actual average characteristics
and down-travel actual average characteristics to reference
line.
NOTES:
1 If not stated, the linearity plus hysteresis shall refer to the calculated results based on the up-travel
actual average characteristics and down-travel actual average characteristics of the transducer to its
reference line. Otherwise it shall indicate the type of reference line.
2 If the reference line is taken as transducer’s working characteristics line, the result of shall
contain repetitive components to varying degrees. It is not linearity plus hysteresis strictly speaking.
3 For non-linear transducer, refer to this section for calculation of conformity plus hysteresis based on the
same calculation principle.
4.1.2 Calculation of reference line
Use a best straight line to perform straight line fitting of transducer’s up-travel
actual average characteristics and down-travel actual average characteristics.
For specific practices, please refer to the relevant parts of Annex A, Annex B
and Annex C of this Standard.
4.2 Linearity plus hysteresis plus repeatability
In this Standard, it is also called the total uncertainty of the transducer
(corresponding to the original total precision, or the total error). It is to make that
under reference working conditions, the deviation of the actual characteristics
from its working characteristics is not exceeded by a specified confidence level.
When expressed as a percentage of the transducer's full-span output YFS, it is
called the relative total uncertainty of the transducer, which is often referred to
as total uncertainty.
4.2.1 General form of calculation formula
where,
Bi - limit of total system error at the ith calibration point, which can be obtained
separately by conventional non-statistical methods;
t0.95Si - limit of total accidental error at the ith calibration point; t0.95 is the inclusion
factor of 95% confidence in the t distribution; Si is the standard deviation of the
sample at the ith calibration point.
Equation (30) can also be expressed as:
where,
1 here is corresponding to Ur in Annex G;
2 If the limit-point envelope method of 4.2.3.2 is used to calculate , it shall be greatly simplified both
in terms of concept and calculation process.
3 For non-linear transducer, refer to this section and calculate the conformity plus hysteresis plus
repeatability according to the same calculation principle.
4.2.3 Calculation method for working characteristics
When calculating the linearity plus the hysteresis plus repeatability of the
transducer, the reference line shall be taken as the straight line of the working
characteristics of the transducer.
4.2.3.1 Selection and calculation principles
a) Consider the working characteristics of transducer.
b) It shall be beneficial to the use of transducer.
c) It shall help to reduce the value of the total uncertainty.
4.2.3.2 L(C)HR limit-point envelope method
In general, the total uncertainty of linear transducer depends on the combined
effects of linearity, hysteresis and repeatability. As shown in Fig. 9, at the xi-th
calibration point, the up-travel average point and the down-travel average
point are respectively determined. Then, according to 3.7.3, the sub-
sample standard deviation Su,i at the ith calibration point of the up-travel and the
sub-sample standard deviation Sd, i of the down-travel at the ith calibration point
are obtained. Subtract cSu,i from the average of the up-travel. Add cSd, i to the
average of the down-travel. Thus, two limit-points can be obtained at the xi-th
calibration point. The calculation formulae are as follow:
uncertainty of the transducer.
4.3 Other comprehensive static performance specifications and
characteristics
The following comprehensive static performance specifications are special
cases of transducer’s comprehensive static performance specifications. The
calculation method is simpler than the calculation method of the transducer’s
comprehensive static performance specifications, so only the calculation
principle is proposed.
4.3.1 Single-travel fixed-point transducer
For example, the fixed-point error of switches such as single-travel pressure
switch and temperature switch is only repeatability, and the calculation method
can refer to 4.2. At this time, the working characteristic of the transducer is a
fixed point. The single stroke is taken as an up-travel or a down-travel. The
working characteristics of the transducer shall be taken as the actual average
point of the selected up-travel or the actual average point of the down-travel.
4.3.2 Dual-travel fixed-point transducer
For example, the fixed-point error of a dual-travel pressure switch, temperature
switch, etc. is only hysteresis plus repeatability. Refer to 4.2 for its calculation
method. At this time, its working characteristics equation is: the measured
number (Y) = the measured amount (x). The working characteristics is a fixed
point, which shall be taken as the actual average point of the selected up-travel
and down-travel.
4.3.3 Linearity plus hysteresis plus repeatability of transmitter
The total uncertainty shall be calculated based on the straight line of its given
working characteristics. Refer to 4.2 and Annex D for its calculation method.
4.3.4 Transducer or transmitter with measured digital display
Its working characteristics equation is: the measured number (Y) = the
measured amount (x). Refer to 4.2 and Annex C for the calculation method of
its total uncertainty.
4.3.5 Utilization characteristics
Utilization characteristics equation: the measured amount (x) = function of
transducer output f (y). For linear transducer, the characteristics equation can
be derived directly from the working characteristics equation. For non-linear
transducer, generally only numerical solutions (such as Newton iteration) can
be used to calculate the working characteristics.
Annex A
(Normative)
General principle and calculation example of linearity calculation
A1 Calculation example for zero-based linearity
Try to calculate the zero-based line and zero-based linearity of a set of static
calibration test data (average of multiple measurements) listed in the table
below.
Table A1
A1.1 General calculation principle
The connection line between the desired theoretical zero point and the
minimum center of gravity of the smallest positive and negative deviation points
is not calculated at one time. It can only approach gradually. The so-called
minimum largest positive and negative deviation points mean that each
approximation shall produce a pair of largest positive and negative deviation
points. What it needs is only a pair of the minimum positive and negative
deviation points of a set of the largest positive and negative deviation points
obtained by each approximation (the characteristic is that the absolute values
of the largest positive and negative deviation of this pair of largest positive and
negative deviation points are equal and minimum).
A.1.2 Calculation of the first approximation line
The theoretical zero point and the rear terminal point line are used as the first
approximation line of the zero-based line. The equation is:
The deviation of each calibration point calculated by it is:
Table A2
Input x
Output y
A2.1.4 Lead a plumb-line from a vertex of a convex polygon to its opposite side
(a line parallel to the ordinate axis). After determining which two points in a set
of data points that the opposite side of the longest plumb-line in the convex
polygon is formed by the terminal-based line’s deviation points of the
connection line, which two points’ connection line makes the absolute value of
the largest positive, negative deviations equal through translation, the best
straight line shall be obtained. As clearly seen from Figure A1, the plumb-line
from the deviation point 4 to the deviation points 1 and 5 is the longest. This
corresponds to the longest plumb-line of the actual data point 4 to points 1, 5.
The length of the longest plumb-line of the actual data point is twice the
maximum deviation from the best straight line.
A2.1.5 The length of the plumb-line (expanded) of each vertex of the convex
polygon can be measured from the graph. It is also possible to accurately
calculate the length of each plumb-line without expansion. Therefore, set the
length of the plumb-line (unexpanded) determined by the connection of actual
data point 4 and the points 1 and 5 as . Its value can be calculated
by:
After substituting the corresponding value:
Similarly, the length of another plumb-line can be calculated:
After substituting the corresponding value:
A2.1.6 The length of the two plumb-lines accurately calculated above is
consistent with the case of the scaled line segment measured from the figure.
Line is the longest. In fact, it is twice the maximum deviation .
Finding the best straight line by translating the data points 1 and 5 is a bit
troublesome. In fact, the connection line of the center of gravity of points 1 and
4 and the center of gravity of points 4 and 5 is the best straight line. The
coordinates for center of gravity of points 1, 4 are:
Annex B
(Normative)
General principle and calculation example of conformity calculation
B1 General principle for conformity calculation
The reference curve for conformity is calculated by using the Chebyshev
interlaced point group principle. And use the improved Remez algorithm. The
specific principle of calculating the reference curve according to this principle
can be summarized into the following four parts.
B1.1 According to the shape of the curve formed by a set of test data points,
the function form or the degree of the polynomial of the fitting curve is selected
empirically or by other methods (for example, looking at the variation of the
adjacent difference quotients).
B1.2 If the reference curve equation has n unknown coefficients, then n+1
interlaced points shall be selected. When the mandatory reference curve
passes a specified point, the staggered point is reduced by one. But the total
number of interlaced points cannot be less than two.
B1.3 Orderly select the required number of interlaced points on the x-axis. In
the first approximation, n points can be selected substantially equidistantly,
preferably including the first and last points, to find a curve passing through the
n points. In the second and subsequent approximation, the interlaced point
group shall be selected according to the deviation and size of the previous set
of approximation curves. The deviation signal at each staggered point shall be
positive and negative alternate. The more the absolute value is the deviation
point, the more the priority shall be selected or swapped into the interlaced point
group. Zero deviation point can be regarded as the smallest positive deviation
point or negative deviation point.
B1.4 The process of finding the final correct interlaced point group is a process
that is gradually approached by continuous iteration. When the candidate
interlacing points sequentially and the symbols positive and negative alternately
obtain the same maximum deviation in a set of data, the interlaced point group
at this time is the correct interlaced point group. The maximum deviation on the
staggered point is for the same set of data. The minimum value that can be
achieved for the same reference curve function form and for the same
cons......
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Source: Above contents are excerpted from the PDF -- translated/reviewed by: www.chinesestandard.net / Wayne Zheng et al.
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