Uncertainty evaluation of slope coefficient of high precision displacement sensor
Vladimir Tudić
Polytechnic in Karlovac, Ivana Meštrovića 10, 47000 Karlovac, Croatia
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
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”.
2. DISPLACEMENT SENSOR
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:
(1)
Figure 2 presents slope coefficient of a typically LVDT micron repeatability sensor.
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 half-width 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 ones 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 () 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):
(2)
2)Uncertainty of length variation measurement results:
(3)
3)Uncertainty of results of gauge block flatness:
(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]:
(5)
(6)
(7)
During process of calibration sensor voltage hysteresis was in range . 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:
(8)
(9)
Square rot value of summary of all square uncertainties (5), (6), (7) and (9) will give value ofcombine combined uncertainty as proceed in [2]:
(10)
And the value of combined uncertainty:
(11)
With appropriate coverage factor the expanded uncertainty can be calculated:
(12)
Value of slope coefficient uncertainty can be expressed through calculation of displacement uncertainty. Slope coefficient equation (1) gives opportunity for substitution with , and with. 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:
(13)
With appropriate coverage factor the expanded uncertainty is:
(14)
Finally, expression for slope coefficient can be written with appropriate measuring uncertainty:
(15)
In words, parameter 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 .
Relative uncertainty if needed can be assessed through 4:
(16)
Another point of view recognize 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:
(17)
Coverage factor (two) multiplies value of combined voltage uncertainty and gives the value of expanded uncertainty:
(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 () exceed sensor voltage hysteresis
(range ). 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.199-201
6 Nyce, David S., Linear Position Sensors, Theory and Application, 2004, John Wiley and Sons, Hoboken, NJ, p. 96.