Download Uncertainty evaluation of slope coefficient of high precision

Uncertainty evaluation of slope coefficient of high precision
displacement sensor
Vladimir Tudić
Polytechnic in Karlovac, Ivana Meštrovića 10, 47000 Karlovac, Croatia
[email protected]
Abstract – Because of high precision displacement determination in many marine power aggregates governing
systems and other fine servomechanisms, uncertainty determination should be taken. Value of absolute
displacement of all precise movement parts in hydraulic servomechanisms of governing systems must be
controlled periodically. Uncertainty evaluation of displacement sensor and its sensibility cannot be avoided in
phase of its construction, production and implication. In process of uncertainty evaluation all contributions and
imperfections must be considered, but some with less influences in evaluating process because of clearly thesis
description are unimportant (elements temperature stability, electromagnetic influences). The purpose of thesis
is to calculate and predict maximal slope coefficient uncertainty and its affection to the measuring value in
similar cases. Scientific method, issued algorithm and calculation model performed in this paper can be used as
application in standard calibration protocols.
Key words – slope coefficient, high precision LVDT sensor, uncertainty
1. INTRODUCTION
2. DISPLACEMENT SENSOR
The high precision linear variable differential
transformer (LVDT) as a precise displacement
sensor is a type of electromechanical transformer
used
for
measuring
micrometer
linear
displacements in many applications e.g. precise
gauges and other fine governing servomechanisms,
[3]. An expression of the result of a measurement
concerned with calibration must include a statement
of the associated uncertainty of the measurement
result [1]. The uncertainty of the measurement
result is a parameter that characterizes the spread of
values that could reasonably be attributed to the
measuring value within a stated “level of
confidence” [2].
Slope coefficient is a unique parameter that
determines
transmission
characteristic
of
displacement sensor range, defined as proportion of
input and output sensor value. Input value of sensor
is displacement unit and the voltage signal is output
value. Entered sensor range in measurement
systems (in dos operated Lab View development
software applications only coefficient value can be
defined) computer unit converts the values to
displacement units again. In order that slope
coefficient determines sensor sensibility, in the
sensor calibration process special attention must
focused to slope coefficient validation. Uncertainty
evaluation and the methods for the estimation are
determined in accordance with the principle
published in national standards [2]. This paper also
provides guidance on the estimation of uncertainty
of
measurement
results
associated
with
requirements of ISO 5725- “Accuracy (trueness and
precision) of Measurement Methods and Results”.
The linear variable differential transformer has
three solenoid coils placed end-to-end around a
tube. The centre coil (marked with letter A in
Figure 1) is the primary, and the two outer coils are
the secondary (marked B). A cylindrical
ferromagnetic core, attached to the object whose
position is to be measured, slides along the axis of
the tube. An alternating current is driven through
the primary, causing a voltage to be induced in each
secondary proportional to its mutual inductance
with the primary, 6.
Figure 1. Linear variable differential transformer
As the core moves, these mutual inductances
change, causing the voltages induced in the
secondary coils to change. The coils are connected
in reverse series, so that the output voltage is the
difference (hence "differential") between the two
secondary voltages. When the core is in its central
position, equidistant between the two secondary
coils, equal but opposite voltages are induced in
these two coils, so the output voltage is zero.
2.1. Slope coefficient evaluation
During the process of calibration of
displacement sensor [5], slope coefficient can be
calculated with high accuracy, in specific needs.
Method comparison of displacement determination
between etalon values and sensor measurement
during calibration process and slope coefficient
calculation were preceded in National Calibration
System. Results reported in “Calibration
Certificate” were obtained by “Procedure for
calibrating gauge blocks by the method of
comparison”, named in [1]. Voltage values (marked
V in mV) were measured with precise voltmeter in
other to avoid transmission error after phase of
signal amplification but before standardization
(Data Acquisition).
Slope coefficient equation of proportion of
input and output sensor values is defined as:
cp 
d 10  10 3 m

 12.658227  10 3 mV 1 (1)
V 790  10 3 V
Figure 2 presents slope coefficient of a
typically LVDT micron repeatability sensor.
d/mm
10
V/mV

-790
790
-10
Figure 2. Sensor transmission characteristic.
2.2. Uncertainty determination
In theory, measurement uncertainty basically is
parameter, associated with the result of a
measurement, which characterizes the dispersion of
the values that could reasonably be attributed to the
particular quantity subject to measurement within a
stated “level of confidence”, [2].
In regard of measurement users, based on
significant experience commonly opinion suggests
that the slope coefficient uncertainty is the one of
“Major source of uncertainty” in precise
displacement measurements. Other sources of
sensor uncertainty are voltage drift, geometric
misalignment, thermal expansion, resolution, or
resistance. As a general role not many uncertainties
can be discovered in calibration routine, but sources
that are one-fourth or less of the largest source may
be considered as negligible.
The uncertainty parameter may be a standard
deviation (or a given multiple of it), or the halfwidth of an interval having stated level of
confidence. Uncertainty of all measurements
comprises many components. Some of these
components may be evaluated from the statistical
distribution of the results of series of measurement
and can be characterized by experimental standard
deviations. The other components are evaluated
from assumed probability distributions based on
experience or other information. Uncertainty
estimates can be obtained in one of two ways:
1) Type A uncertainty estimates is obtained
by the statistical analysis of data – for example,
repeatability may be estimated as the standard
deviation of a set of repeated measurements.
2) Type B uncertainty estimates are obtained
by other means, such as a finding the calibration
result uncertainty on a calibration certificate.
Uncertainty estimates can be based on ones
knowledge and experience, or on the laws of
physics or from knowledge about how
measurements behave.
Expand uncertainty given in calibration
certificates is a standard deviation (s) witch has
been multiplied by a number ( k  2 ) called the
“coverage factor”.
Normal or Gaussian distribution curve
represents the frequency with witch a particular
measurement result occurs in a repeated series of
measurement. In this case average measurement
result is zero, and the area under the curve between
–s and +s accounts for about 68% and in area –2s
and +2s is about 95%. That means that 95% of all
measurement results will be between the limits 2s.
Repeatability is precision under repeatability
conditions where independent results are obtained
with same method on identical items in the same
laboratory using the same equipment within short
intervals of time. Repeatability must be obtained as
a standard uncertainty due to limited resolution;
methods are listed in [1].
Combine combined uncertainty proceed from
squaring each one of the all uncertainties and
adding these values to one another and taking the
square rot of its sum. Finally, multiplying this value
by appropriate coverage factor the expanded
uncertainty is obtained.
2.3. Uncertainty contributors
Here are contributors that affect to the
displacement uncertainty and indirectly to slope
coefficient uncertainty. Values of expand
uncertainty are given in calibration certificate.
Typical estimation includes:
1) Uncertainty of gauge block (grade 0)
length measurement results (U in
micrometers, L in meters):
U E   0.08  1.2L; k  2; L  10mm
2) Uncertainty
of
length
measurement results:
(2)
variation
U f v   0.104m; k  2
(3)
3) Uncertainty of results of gauge block
flatness:
U f d   0.118m; k  2
(4)
4) Uncertainty of results of sensor hysteresis.
2.4. Combined and expand uncertainty
From the expanded uncertainty data from (2),
(3) and (4) and knowing each coverage factors,
combined uncertainty can be calculated in order to
[2]:
u E  
U E  0.092 m

 0.046m
k
2
u f v  
u f d  
(5)
U f v  0.104m

 0.052m (6)
k
2
U f d  0.118m

 0.059m (7)
k
2
During process of calibration sensor voltage
hysteresis was in range a  6 V . This value must
be divided with square root of 3 (distribution
assumed rectangular) and then multiplied with
value of slope coefficient which provides value of
hysteresis uncertainty:
u H  
u H  
6 10 6 V
3
a
3
 cP
12.658 10 3
(8)
m
 0.044 m (9)
V
Square rot value of summary of all square
uncertainties (5), (6), (7) and (9) will give value of
combine combined uncertainty as proceed in [2]:
2
uc d   u 2 E   u 2 f v   u 2 f d   u 2 H  (10)
uc d   0.101m
2
And the value of combined uncertainty:
uc d   0.101m
(11)
With appropriate coverage factor k  2 the
expanded uncertainty can be calculated:
U d   k  ud   0.202m
(12)
Value of slope coefficient uncertainty can be
expressed through calculation of displacement
uncertainty. Slope coefficient equation (1) gives
opportunity for substitution y with u c d  , and x
with V . In that case value of displacement
uncertainty will be distributed over range of output
sensor voltage and therefore deviation of slope
coefficient become transferable. Expressions for
slope coefficient uncertainty then become:
cp 
y
x
m
 V 
(13)
m
 u d   0.101m
u c c p    c
 0.127  10 6


V
790
m
V
V


With appropriate coverage factor k  2 the
expanded uncertainty is:
U c p   k  u c p   0.255  10 6
m
V
(14)
Finally, expression for slope coefficient can be
written with appropriate measuring uncertainty:
c p  12.658227  10 3  0.000255  10 3
m
(15)
V
In words, parameter c P is slope coefficient
evaluated in calibration process preceded in
National Calibration System with calculated value
of expanded uncertainty expressed at approximately
the 95% level of confidence using a coverage factor
k  2.
Relative uncertainty if needed can be assessed
through 4:
U c p 
cp
 2  10 5
(16)
Another point of view recognize u c d  as a
value in y-axis witch translates to x-axis as a value
of voltage uncertainty through slope coefficient
characteristic, mentioned in 4. Value of voltage
uncertainty in this case is:
uV  
uc d 
 8V
cp
(17)
Coverage factor (two) multiplies value of combined
voltage uncertainty and gives the value of expanded
uncertainty:
U V   k  uV   16V
(18)
This shown value is also on trail to discover the
stability and uncertainty of voltage output signal of
displacement sensor, but it is issue for some other
discussion.
3. CONCLUSION
A simple calculation has been developed which
determine value of slope coefficient uncertainty.
Estimated results of 0.101 micrometer value of
combined displacement uncertainty and 0.202
micrometer of expand uncertainty have common
frontier with sensor repeatability values (0.15
micrometers). Also, throw uncertainty calculation
estimated value of voltage expand uncertainty
( uV   16V ) exceed sensor voltage hysteresis
(range a  6 V ). The idea suggest itself that the
sensor repeatability values of 0.15 micrometers
cannot be taken seriously in this circumstances.
This stiffness can be sequently used to evaluate
measuring uncertainties during the process of
precise displacement measuring.
The method has diagnostic significance in the
contribution of the new issues and influence
components can be assessed.
The desired goal of thesis in this paper is to
give measurements user’s opportunity to express
this kind of uncertainty, which certainly affects
measurement results.
REFERENCES
[1] ISO 5725, “Accuracy (trueness and precision)
of Measurement Methods and Results - Part 1:
General Principles and Definitions”, No. 1,
January 1994
[2] DZNM - State office for standardization and
metrology, “ISO Guide for determination of
measurement uncertainty”, Zagreb, May 1995
[3] V. Bego, “Measurements in electro technique”,
Technical Book, Zagreb, October 1997
[4] M. Brezinšćak, “Measurements and calculation
in technique and science”, Technical Book,
Zagreb, June 1967
[5] M. DiSilvestro, T. Dietz, “Calibration of linear
displacement sensor for micromotion studies”,
Sensors for Industry Conference, 2004.
Proceedings the ISA/IEEE, IN, USA, pp.199201
6 Nyce, David S., Linear Position Sensors,
Theory and Application, 2004, John Wiley and
Sons, Hoboken, NJ, p. 96.