Transcript

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Page 1 Setra 470 Pressure Transducer Tests

INSTRUMENT TEST REPORT NUMBER 622

Setra 470 Pressure Transducer Compliance Tests for A.W.S. Applications

Kent Gregory Position : PO2

Physics Laboratory, OEB 10th September, 1992

Authorisation

Jane Warne Senior Physicist Instruments and Laboratory for Director of Meteorology

24 pages including 2 appendices

TABLE OF CONTENTS

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1. INTRODUCTION 3 2. EXPERIMENTAL METHOD 4 2.1 Pressure Calibration Method 4 2.2 Sustained Temperature Test Method 5 2.3 Variable Temperature Test Method 5 3. RESULTS 6 3.1 Accuracy 6 3.2 Error Due to Non-Linearity 7 3.3 Error Due to Hysteresis 10 3.4 Response to Sustained Temperatures 11 3.5 Effect of Temperature on Accuracy 12 4. DISCUSSION 14 4.1 Accuracy at Room Temperature 14 4.2 Non-Linearity at Room Temperature 14 4.3 Hysteresis at Room Temperature 14 4.4 Accuracy at Various Temperatures 15 4.5 Stability with Time 15 5. RECOMMENDATIONS 16 6. REFERENCES 16 Appendix A 17 Appendix B 20 1. INTRODUCTION.

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The Physical Laboratories performed several tests on a Setra Pressure Transducer model 470 serial number 254649 (herein referred to as the Setra 470) to determine its suitability for use in an A.W.S. and to check its compliance to Bureau of Meteorology and factory specifications. The Setra 470 requires 5V DC at 90 mA to operate. Pressure readings are output digitally via a serial port. The claimed specifications [1] of the Setra 470 are presented in Table 1. Figures relating to thermal effects have been converted from the Fahrenheit scale to the Celsius scale and are valid for the 800 hPa to 1100 hPa pressure range. Setra Systems Inc. define Absolute Accuracy as the root sum square (R.S.S.) of non-linearity, hysteresis and non-repeatability but does not include errors due to temperature variability or instrument ageing. From correspondence with Setra Systems Inc. in Australia and the United States, the implied confidence interval used to derive the figures in Table 1 is 99% [7]. The thermal effects quoted in Table 1 are valid for the range -1°C to 43°C [1].

The uncertainty bounds quoted in Table 1 have been broadened to allow for uncertainty due to the reference barometer. These uncertainties appear in the middle column of Table 1. A R.S.S. of the reference barometer total uncertainty (±0.08 hPa) and the original Setra 470 absolute accuracy specification (± 0.06 hPa) yield the experimental specification (± 0.10 hPa). Also, the non-repeatability of the reference barometer during the term of the experiment was ± 0.06 hPa. This was also incorporated into Table 1 as per the absolute accuracy specification.

Characteristic Uncertainty due to Reference

Setra 470 Uncertainty

Experimental Uncertainty

Absolute Accuracy ± 0.08 hPa ±0.06 hPa ± 0.10 hPa at 21°C Non-linearity - ±0.04 hPa ± 0.04 hPa

Hysteresis - +0.03 hPa +0.03 hPa Non-repeatability ±0.06 hPa ±0.03 hPa ±0.07 hPa Thermal zero shift - ±0.0033 hPa/°C ±0.0033 hPa/°C

Thermal sensitivity shift - ±0.0016 hPa/°C ±0.0016 hPa/°C Stability - < ±0.15 hPa for 1

year < ±0.15 hPa for 1 year

Table 1. Specifications for Setra 470 [1].

The reference barometer has no hysteresis uncertainty, due to the measurement method. Any temperature or non-linearity characteristics of the reference barometer are removed after each measurement. 2. EXPERIMENTAL METHOD.

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The sequence of tests used to check the specifications of the Setra 470 is given in Table 2.

Test Test Type Temperature (°C) Pressure Range(hPa) Start Date End Date (a) Pressure

Calibration 24.5 ± 0.2 800 to 1050 7/3/91 7/3/91

(b) Pressure Calibration

25 ± 0.2 800 to 1050 13/3/91 13/3/91

(c) Sustained Temperature

40 1002 to 1017 14/3/91 26/3/91

(d) Pressure Calibration

40 ± 1 800 to 1050 28/3/91 28/3/91

(e) Sustained Temperature

23 993 to 1025 1/4/91 3/5/91

(f) Pressure Calibration

22.6 800 to 1050 13/5/91 13/5/91

(g) Variable Temperature

0 to 50 1011 7/6/91 12/6/91

Table 2. The sequence of tests performed on the Setra 470. The Reference Barometer was a Mechanism Ltd. digital aneroid barometer (serial number DA 670). It had a resolution of 0.01 hPa and an absolute accuracy of ± 0.08 hPa. This instrument is traceable to the W.M.O. Region V pressure standard (maintained by the Bureau of Meteorology) through direct comparisons at weekly intervals. The temperature of the air surrounding the reference barometer was measured every time a pressure measurement was made. Correction tables, based on pressure measured and air temperature, were used to correct the reference barometer readings for non-linearity and temperature effects. The pressure regulator was a Bell and Howell Pressure Volume Regulator (serial number 2491). The oven used to heat the Setra 470 was a Laboro (serial number CA1997) and a Muller McQuay refrigerator was used to cool the Setra 470 to temperatures below 25°C. 2.1 Pressure Calibration Method. The purpose of the pressure calibrations was to determine the absolute accuracy of the Setra 470 against the W.M.O. Region V pressure standard. Tests (a) and (b) were used to determine the hysteresis, non-repeatability and non-linearity uncertainties of the Setra 470. Tests (d) and (f) were carried out during and after the Setra 470 had been subjected to sustained high temperatures, to determine if any change in the Setra 470 performance occurred.

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The pressure regulator was connected via a 'T' piece to the reference barometer and the Setra 470 by lengths (less than 500 mm) of rubber tubing. The reference barometer always remained at room temperature. Using the Bell and Howell pressure regulator, the pressure in the line was decreased to 800 hPa. Both the Setra 470 and the reference barometer were allowed to stabilise for 60 seconds. After this time, the pressure indicated on the Setra 470 and the reference barometer were recorded. The pressure regulator was used to increase the pressure by 20 hPa. Again the sensors were allowed 60 seconds to stabilise and readings were taken. This continued up to a pressure of 1050 hPa. The process continued in the same manner, but the pressure was decreased in steps of 20 hPa. This test method was used in Tests (a), (b), (d) and (f). 2.2 Sustained Temperature Test Method. The purpose of Test (c) was to accelerate the ageing of the Setra 470 and to check its pressure response at elevated temperatures. Test (e) was designed to monitor how the Setra 470 recovered after being maintained at the elevated temperatures in Test (c). The Setra 470, the reference barometer and the ambient pressure of the laboratory were connected by lengths of tubing (less than 500 mm). The temperature of the Setra 470 was held within 1°C of the desired test temperature for the term of the experiment. The reference barometer always remained at room temperature. At 9 am and 4 pm a pair of readings from the reference barometer and the Setra 470 were taken. This test method applies to Tests (c) and (e). 2.3 Variable Temperature Test Method. A.W.S. pressure sensors can be exposed to a variety of temperatures every day. Test (g) was designed to examine the Setra 470's performance at various temperatures to check the claimed specifications regarding temperature effects. The Setra 470 was placed in an oven held at room temperature. The Setra 470 was allowed 25 minutes to stabilise at this temperature. A reading from the Setra 470 and the reference barometer was taken. The oven temperature was increased by 5°C and the Setra 470 was allowed another 25 minutes before readings were again taken. These steps were repeated until the oven reached a temperature of 50°C. The Setra 470 was removed from the oven and allowed to cool to room temperature for 60 minutes. The Setra 470 was then placed in a refrigerator held at room

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temperature. The temperature of the refrigerator was lowered by 5°C. The Setra 470 was allowed 30 minutes to come to thermal equilibrium inside the refrigerator. Readings were taken from the Setra 470 and the reference barometer in the same way as readings were taken in the oven. The temperature was lowered in steps of 5°C and the process repeated to a temperature of 0°C. This test method applies to Test (g).

3. RESULTS

The data from the tests is tabulated in Appendix B. Statistical proofs referenced in this section can be found in Appendix A. The difference between the Setra 470 and the reference barometer is expressed as a correction to 99% confidence. That is, the reference barometer reading minus the Setra 470 reading.

Since the confidence interval used by Setra Systems, Inc. is assumed to be 99% [7], all calculations and statistical proofs have been based on this confidence interval. 99% confidence intervals are also referred to as the random component of the total uncertainty. A correction cannot be applied to reduce this type of uncertainty. It describes the scatter of the data. Bias errors, in the context of this report, are the mean of the corrections. The bias is the mean difference between the Setra 470 and the reference barometer. The bias can be removed from the total uncertainty, leaving only the random uncertainty in the Setra 470 readings. The total uncertainty of a quantity, such as absolute accuracy or hysteresis, is expressed as a combination of bias and random uncertainties. Analysis has been performed over a large pressure range (800 hPa to 1050 hPa) and a smaller pressure range (930 hPa to 1030 hPa). The smaller pressure range was chosen for two distinct reasons. Firstly, it is the pressure range of greatest interest for A.W.S. applications. Secondly, the Setra 470 appears to perform better within this range and therefore may prove suitable for use in a restricted pressure range.

3.1 Accuracy Table 3 contains a summary of the data collected for Tests (a), (b), (d) and (f). Tests (a) and (b) produced similar corrections (see Statistical Proof 1 in Appendix A). Test (d) was conducted after the Setra 470 had been subjected to sustained high

Test Correction (hPa) Standard Deviation

Number of Data Points

99% Confidence Interval

(a) -0.18 0.040 38 ±0.10 hPa (b) -0.19 0.079 38 ±0.20 hPa

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(d) -0.46 0.063 80 ±0.16 hPa (f) -0.24 0.055 34 ±0.14 hPa

Table 3. Summary of pressure calibrations.

temperatures (13 days) and Test (f) after sustained room temperatures (33 days). Statistical Proof 1 shows that the correction from Test (f) is different to the corrections first found in Tests (a) and (b). Statistical Proof 2 shows that the mean correction has changed by 0.06 ± 0.03 hPa. Test (d) was a pressure calibration carried out at the high end of the temperature range. Table 3 shows that the variance of measurements in Test (d) is comparable to the other tests. However, the difference in correction between Test (d) and Tests (a) and (b) (pooled) is 0.28 ± 0.03 hPa (see Statistical Proof 2). This change is attributed to the affect of temperature on the instrument rather than random uncertainty. Setra's uncertainty specifications for measurements at 40°C was quoted at ± 0.163 hPa. Results of pressure calibrations in the restricted range of 930 hPa to 1030 hPa appear in Table 4. The corrections remained about the same as those in Table 3 but the uncertainty improved in Tests (b), (d) and (f) over this reduced range.

Test Correction (hPa) Standard Deviation

Number of Data Points

99% Confidence Interval

(a) -0.19 0.038 19 ±0.11 hPa (b) -0.20 0.048 18 ±0.14 hPa (d) -0.48 0.047 40 ±0.12 hPa (f) -0.26 0.027 14 ±0.08 hPa

Table 4. Summary of pressure calibrations over the reduced pressure range.

3.2 Error Due to Non-Linearity. Non-linearity error is the additional error obtained by assuming a sensor has a constant correction over a range, whereas its measured correction is not a constant. That is, it is the difference between the true behaviour of a sensor, represented by a function of the pressure, and the assumed constant behaviour of the sensor, represented by a constant. Non-linearity error is repeatable, but because the error is a function of pressure, it becomes cumbersome to deal with. Usually non-linearity errors are small and can be absorbed into the random error of the instrument without greatly affecting the total uncertainty. The difficulty in finding the non-linearity error of an instrument is that data also includes hysteresis and non-repeatability errors.

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Described below is the method used by the Physical Laboratories to derive a conservative estimate of the non-linearity error for the Setra 470. The method uses the data from Tests (a), (b), (d) and (f). The first task is to subtract the mean correction of each test from its data points. This has the effect of centring the data about the 0 hPa correction line. Next, a polynomial (in this case, a third order polynomial) is fitted to the entire data set. The standard deviation of this curve, which represents the non-linearity error, can now be calculated. This standard deviation does not contain substantial influences due to repeatability and hysteresis errors for the following reasons: (a). There were an equal number of data points taken during increasing and

decreasing pressures (an average of these data points reduces hysteresis error) (b). There were 4 data points taken at each pressure level (an average of these data

points reduces the repeatability error). The points plotted in Figure 1 are the data from Tests (a), (b), (d) and (f) over the entire pressure range. The standard deviation of the polynomial is 0.0288 hPa. Calculating a 99% confidence interval from this yields ±0.07 hPa. This is a conservative estimate of the non-linearity error. The data from Tests (a), (b), (d) and (f) were analysed separately and also in the reduced pressure range, the results of which can be found in Table 5. Details of the polynomials found to fit the data are tabulated in Appendix A. A plot of the data in the reduced pressure range appears in Figure 2.

750 800 850 900 950 1000 1050 1100 1150-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Reference Pressure (hPa)

Correction (hPa)

Figure 1. Non-linearity of the data from Tests (a), (b), (d) and (f), bias removed.

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920 940 960 980 1000 1020 1040-0.15

-0.1

-0.05

0

0.05

0.1

Reference pressure (hPa)

Correction (hPa)

Figure 2. Non-linearity of data from Tests (a), (b), (d) and (f) (reduced range), bias removed.

Range Data Variance of polynomial 99%C.I. No. of points

a 4.84 e-4 0.06 38 b 4.10 e-3 0.17 38

Full d 4.41 e-4 0.05 80 f 1.30 e-3 0.10 34 all data 6.76 e-4 0.07 190

Reduced all data 1.69 e-4 0.04 94

Table 5. Non-linearity results for individual and grouped data sets From the above table it can be seen that the performance of the Setra 470 with respect to non-linearity improves if used over the reduced range. The 99% confidence interval for the non-linearity in the reduced range is 0.04 hPa. 3.3 Error Due to Hysteresis. Hysteresis is defined as the mean difference in response between increasing the pressure and decreasing the pressure. To find the hysteresis of the Setra 470, comparisons made while increasing pressure were distinguished from those that were made while decreasing pressure. A third order polynomial was fitted to the increasing data set and the decreasing data set. The mean absolute difference between the two curves was calculated. From Section 3.2, the non-linearity was represented by a third order polynomial. Because the increasing and decreasing pressure curves are subtracted from each other, the influence of non-linearity is removed from the calculated hysteresis value.

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Data from Tests (a) and (b) were used to calculate the hysteresis using the method described above. An estimate of the 99% confidence intervals for the hysteresis has been generated from the pooled standard deviations of the data sets from their lines of best fit. Because this error estimate contains the non-repeatability error of the Setra 470, the estimate is larger than the true hysteresis error. Figure 3 is a plot of the data from Tests (a) and (b), with the polynomials fitted to the two sets of data points. Statistics appear in Table 6. A study was made for the reduced pressure range, a plot of the data appearing in Figure 4. Information regarding the polynomials fitted appears in Appendix A.

750 800 850 900 950 1000 1050 1100 1150-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

Reference Pressure (hPa)

Correction (hPa)

increasing pressure decreasing pressure

Figure 3. Tests (a) and (b) data with 3rd order polynomials fitted to the data points.

920 940 960 980 1000 1020 1040-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

Reference Pressure (hPa)

Correction (hPa)

increasing pressure decreasing pressure

Figure 4. Hysteresis for the reduced pressure range.

Data range Mean difference between fits Estimated 99% C.I. full 0.04 hPa 0.10 hPa

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reduced 0.02 hPa 0.09 hPa

Table 6. Performance of Setra 470 hysteresis over the two pressure ranges. 3.4 Response to Sustained Temperatures. Test (c) was performed at 40°C over a 13 day period. Six days later, the Setra 470 was removed from the oven and placed in ambient temperatures of 23°C. Test (e) was commenced and comparisons were made over a 33 day period. The data from Tests (c) and (e) are plotted in Figure 5. The mean of the corrections from Test (c), except the first two days (explanation below) is -0.41 hPa with a standard deviation of 0.044 hPa (14 degrees of freedom). The corrections from Test (e) appear to be decreasing in magnitude with time. A linear regression was performed on the data, the results of which can be found in Appendix A and is plotted on Figure 5.

0 10 20 30 40 50 60-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

Time (days)

Correction (hPa)

Figure 5. Graph of time versus correction for Tests (c) and (e). The first two readings in Test (c) and the first reading in Test (e) were made within hours of changing the temperature of the Setra 470's environment. All three readings differ significantly from the long term trend. This suggests that two thermal time constants were at play, one with a fast response and one with a slow response. The faster thermal time constant dominates the measurements for a period of hours after a temperature change, after which the slower thermal constant dominates. For an increase in environmental temperature, the Setra 470 will first give a more positive correction followed by a lower correction in the long term. The opposite occurs for a decrease in environmental temperature (see 3.5).

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3.5 Effect of Temperature on Accuracy Test (g) was the final test made on the Setra 470. The pressures measured during Test (g) ranged from 1010.12 hPa to 1012.78 hPa. A plot of Setra 470 temperature versus the associated correction is shown in Figure 6. The line in Figure 6 is a linear regression fitted to the data, the results of which can be found in Appendix A. The gradient of this line was (0.01 ± 0.003) hPa/°C. This implies that for every 10°C change in temperature, the correction to the Setra 470 reading changes by (0.1 ± 0.03) hPa.

-10 0 10 20 30 40 50 60-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Temperature (°C)

Correction (hPa)

Figure 6. Setra 470 Response to Ambient Pressure at various Temperatures.

-10 0 10 20 30 40 50 60-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Temperature (°C)

Correction (hPa)

Figure 7. Setra Systems, Inc. uncertainty bounds and data collected from Test (g).

In Table 1, Setra Systems, Inc. specified the effect of temperature on accuracy in terms of a thermal sensitivity shift and a thermal zero shift. If the total thermal effect is taken as the R.S.S. of these two effects, the additional uncertainty due to temperature is ±0.00373 hPa/°C. This increase in measurement uncertainty is taken to be for any

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measurement made when the Setra 470 was not at 21°C. The lines plotted on Figure 7 are the uncertainty bounds as interpreted above. Note that the results of Test (g) do not agree with the results found in Tests (c) and (e). That is, the mean correction found in Test (c) at 40°C was found to be -0.41 hPa but Test (g) suggests a correction of about 0 hPa. This was due to the measurements in Test (g) being made within half an hour of changing the temperature of the Setra 470's environment. Therefore, the faster thermal time constant was influencing the measurements. 4. DISCUSSION.

4.1 Accuracy at Room Temperature

The absolute accuracy stated in the Setra 470 specifications was not achieved when Tests (a), (b) and (f) were performed (see Table 6 below). These tests were performed within 4°C of the temperature used in the Setra Systems Inc. specification of absolute accuracy.

Data used in calculations Bias Error Random Error

Claimed Accuracy from Specifications 0 ±0.10 hPa Test (a) -0.18 ±0.10 hPa

Full range Test (b) -0.19 ±0.20 hPa Test (f) -0.24 ±0.14 hPa Test (a) -0.19 ±0.11 hPa

Reduced range Test (b) -0.20 ±0.14 hPa Test (f) -0.26 ±0.08 hPa

Table 6. Claimed accuracy compared to Physical Laboratory tests.

It can be seen from the above table that the Setra 470 did not perform within the manufacturer's specifications of absolute accuracy. 4.2 Non-linearity at Room Temperature Setra Systems Inc. claimed a non-linearity error of 0.036 hPa at the 99% confidence interval. From Section 3.2 it was found that the estimated non-linearity error was ±0.07 hPa, double the specifications. This value has been derived using a different method to that used by Setra Systems, Inc. The method used by Setra Systems Inc. was stated on the calibration certificate to comply with ISA.#S-37.1 (calibration sheet can be found in [7]).

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4.3 Hysteresis at Room Temperature The claimed hysteresis of the Setra 470 was 0.03 hPa. The mean hysteresis found by the Physical Laboratory was 0.04 hPa over the entire pressure range, with an estimated 99% confidence interval of 0.10 hPa. The claimed hysteresis specification is contained within the bounds of the calculated hysteresis value. The maximum hysteresis was found to occur at 860 hPa with a value of 0.11 hPa. 4.4 Accuracy at Various Temperatures The claimed uncertainty of the Setra 470 at 40°C was ±0.16 hPa at 99% confidence interval (assuming the uncertainty specification is derived using the method outlined in 3.5). The mean correction and random uncertainty at the 99% confidence interval of the Setra 470 at this temperature was found to be -0.46 hPa and 0.16 hPa respectively (see Statistical Proof 5). Therefore, the claimed uncertainty at 40°C was found to be much smaller than the uncertainty found in the tests. From Figure 7, 11 of the 22 comparisons at various temperatures did not fall within the bounds claimed by the specifications. Hence, the claimed specifications regarding temperature effects were not valid for this particular sensor. Further, there is a thermal effect (short term) that was not specified by the Setra 470 specification sheet. Its affect on measurements appears to be of the same order of magnitude as the long term thermal effect. In A.W.S. applications, where sensors are continuously exposed to a variety of temperatures, this uncertainty component is of major importance. 4.5 Stability with Time The Setra 470 specification quotes a maximum bias (correction) drift of ±0.06 hPa for 30 days. The test of longest duration, Test (e), was over a period of 33 days at room temperature with small pressure variations. A linear regression line was fitted to Test (e) data (see Statistical Proof 4). The gradient of this regression line was used to calculate the bias drift for the 30 day period. This resulted in a calculated drift of 0.14 ± 0.07 hPa (30 times the gradient for one day). This is larger than the quoted specifications. It was also disturbing to find that the Setra 470 took a long time to recover from elevated temperatures (see Figure 5). Readings from the Setra 470 seem to be affected by the temperatures the instrument was exposed to several days beforehand. With the large time constant identified in Test (e), the measurement uncertainty cannot be easily minimised. It is preferable to have an instrument with a shorter thermal time constant than a larger one. This is because the uncertainty can be minimised by calibrating the sensor across a temperature range. To reduce the uncertainty of

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measurements in an instrument with a large thermal time constant requires a temperature history of the environment and complicated correction formulae. 5. RECOMMENDATIONS.

Given the test results, the Setra 470 cannot be considered for A.W.S. applications. According to [5], the Setra 470 fails to meet the B.O.M. specifications of 0.50 hPa accuracy at all times on the grounds of the results of Test (d). In this test, the effect of prolonged operation at 40°C has resulted in 33 of the 80 corrections with magnitudes greater than or equal to 0.50 hPa. The performance of the Setra 470 in Tests (a) to (g) have shown the uncertainty was outside many of the manufacturer's specifications. The non-linearity, short term drift, performance at various temperatures and absolute accuracy specifications from [1] were found to be in disagreement with the results found by the Physical Laboratories for this particular Setra 470. It is recognised that the Setra 470 tested may not be representative of all Setra 470 units produced by Setra Systems, Inc. This report shows that the Bureau of Meteorology should be wary of specifications from all suppliers, and that testing one unit does not accurately represent the total population. 6. REFERENCES [1] Setra Systems, Inc. Model 470 Specification Sheet [2] Introduction to Statistics, 3rd Ed, R.E.Walpole, Page 311, Table 10.1, formula for t' [3] Linear regressions calculated using Lotus 123 version 3.1 [4] Introduction to Statistics, 3rd Ed, R.E.Walpole, page 258, formula for small sample confidence interval for u1 - u2 , where the variance of both sets is unknown [5] Bureau of Meteorology, Equipment Specification A2659, E.E.Jesson, page 19 [6] Grapher version 1.75, Golden Software, 1988 [7] Comparison of A.W.S. Transducers to Pressure Standards, Bureau of Meteorology Registry File, 25/3312/1

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Appendix A

Statistical Proof 1 Behren's Statistic can be used to find if there is a significant difference between the means of two data sets. The formula to calculate this statistic, t', is found in [2]. This was used to determine if the mean correction for Test (a) and Test (b) were the same. For these data, Behren's Statistic yields a value of 0.68. This is less than the 99% point in the Normal Distribution (=2.576). Therefore the means are equivalent. Behren's Statistic was used to determine if the mean corrections for Test (a) and Test (b) were the same as the mean correction from Test (f). The value of the statistic for Tests (a) and (f) and Tests (b) and (f) was found to be 5.2 and 3.2 respectively. These values are greater than the 99% point in the Normal Distribution (=2.576). Therefore the means are different. Statistical Proof 2 The difference and confidence interval between two finite sample data sets with unknown variances can be found using [4]. Variances and means from Tests (a) and (b) are pooled. The difference between the pooled data of Tests (a) and (b) and the data from Test (f) can now be calculated by substituting the following values; meana+b = -0.1862 vara+b(pooled) = 0.05960 meanf = -0.2415 varf = 0.003079 From [4], the mean has changed by 0.06 ± 0.03 hPa at the 99% confidence interval. The same test can be applied to calculate the difference between Tests (a) and (b) (pooled) and Test (d) at the elevated temperature. The equation from [4] yields a difference of 0.28 ± 0.03 hPa at 99% confidence interval. A software package [6] was used to fit third order polynomials to data in order to calculate the non-linearity of the Setra 470. The coefficients of the curves are tabulated in Table 7. The reference pressure is the independent variable (x) and the correction was the dependent variable (y).

Range Data Coefficients

x3 x2 x constant

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a 4.41156 e-8 -1.24478 e-4 0.116424 -36.0834 b 9.59992 e-8 -2.76223 e-4 0.263213 -83.0372

Full d 2.54470 e-8 -6.82155 e-5 0.0604138 -17.667 f 2.20178 e-8 -6.45401 e-5 0.0622974 -19.8037 all 4.32930 e-8 -1.22203 e-4 0.114239 -35.3579

Reduced all 6.11716 e-8 -1.69761 e-4 0.155945 -47.3849

Table 7. Details of the lines of best fit used in the non-linearity calculations. The same software package was used to fit third order polynomials to the data in the hysteresis calculations. The coefficients of the curves are tabulated in Table 8. The reference pressure was the independent variable (x) and the correction was the dependent variable (y). Also appearing in the same table is the standard deviation of the data from the polynomials fitted to them. Range Data Coefficients Standard

deviation x3 x2 x constant

Full a,b increase 1.01156 e-7 -2.875 e-4 0.270614 -84.4995 0.03886 a,b decrease 3.14977 e-8 -9.20216 e-5 0.0890456 -28.731 0.03546

Reduced a,b increase 1.62246 e-7 -4.61553 e-4 0.435816 -136.735 0.02866 a,b decrease -6.8624 e-8 2.20385 e-4 -0.234813 82.8369 0.04076

Table 8. Details of the lines of best fit for the hysteresis calculations.

A linear regression of the form

y = ax +b where y = the correction (the dependent variable) in hPa x = the time (the independent variable) in days a = the mean change in correction per day b = the mean initial correction was made on the 33 days of data in Test (e) using Lotus 123 [3]. The statistics for the regression appears in Table 9.

Regression Statistic Value coeff. of determination 0.361442

b -0.40869 Standard error of y 0.065909

a 0.004792

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Standard error of x 0.000949 No. of Observations 47 Degrees of Freedom 45

Table 9. Results of the linear regression on the data in Test (e).

From Table 9, the rate of change of the mean correction (y) per day (x) was 0.004792 hPa/day. The 99% confidence interval for this prediction of the daily drift is 2.576 times the standard error of the x coefficient, which is 0.00244 hPa/day. A linear regression of the form

y = ax +b where y = the correction (the dependent variable) in hPa x = the temperature (the independent variable) in °C a = the mean change in correction per °C b = the mean correction at 0°C was made on the data in Test (g) using Lotus 123 [3]. The statistics for the regression appears in Table 10.

Regression Statistic Value

coeff. of determination 0.82797 b 0.36119

Standard error of y 0.06901 a 0.00922

Standard error of x 0.00092 Number of Observations 23

Degrees of Freedom 21

Table 10. Results of the linear regression on the data in Test (g). From Table 10, the rate of change of the mean correction (y) per °C (x) was 0.00922 hPa/°C. The 99% confidence interval for this prediction of the drift due to temperature is 2.576 times the standard error of the x coefficient, which is ± 0.00237 hPa/°C.

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Appendix B: Data from the Tests

Setra 470 (hPa) Correction (hPa) Setra 470 (hPa) Correction (hPa) 795.06 -0.21 800.28 -0.27 818.79 -0.15 819.64 -0.12 839.46 -0.12 840.03 -0.04 859.39 -0.09 860.02 -0.02 878.68 -0.11 880.15 -0.03 899.68 -0.12 900.03 -0.04 920.84 -0.15 920.22 -0.09 940.73 -0.18 940.33 -0.13 950.64 -0.15 950.19 -0.11 960.33 -0.19 960.14 -0.14 969.65 -0.20 970.71 -0.22 980.60 -0.22 980.76 -0.22 990.30 -0.23 990.98 -0.24

1001.00 -0.22 1000.74 -0.16 1010.87 -0.22 1010.91 -0.24 1020.40 -0.22 1020.58 -0.26 1030.06 -0.21 1030.26 -0.25 1039.67 -0.23 1040.53 -0.31 1050.44 -0.15 1050.51 -0.27 1051.68 -0.19 1050.20 -0.29 1040.46 -0.18 1039.48 -0.24 1029.29 -0.25 1030.88 -0.22 1021.00 -0.10 1020.74 -0.24 1010.24 -0.19 1010.67 -0.22 999.96 -0.23 1000.97 -0.26 990.81 -0.19 990.89 -0.25 980.82 -0.16 980.70 -0.24 970.00 -0.20 970.88 -0.23 960.39 -0.20 960.58 -0.15 950.43 -0.17 950.24 -0.18 940.07 -0.13 940.91 -0.19 920.40 -0.16 920.23 -0.16 900.03 -0.19 900.81 -0.17 879.87 -0.18 880.67 -0.15 860.29 -0.17 860.51 -0.15 840.10 -0.23 841.16 -0.20 820.24 -0.20 820.97 -0.21 795.46 -0.20 803.48 -0.35

Test (a). Test (b).

Calibration at room temperature.

Time (days) Setra 470 hPa Correction hPa

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1.0 1016.98 -0.13 1.3 1014.48 -0.23 2.0 1008.72 -0.28 2.3 1016.51 -0.41 3.0 4.0 5.0 1010.31 -0.42 5.3 1008.47 -0.41 6.0 1008.14 -0.42 6.3 7.0 1012.10 -0.42 7.3 1009.79 -0.40 8.0 1013.05 -0.40 8.3 1012.20 -0.45 9.0 1012.52 -0.37 9.3 1013.55 -0.44 10.0 11.0 12.0 1010.25 -0.35 12.3 1006.75 -0.44 13.0 1002.55 -0.43 13.3 1003.65 -0.45

Test (c). Sustained temperature comparisons at 40°C.

Setra 470 hPa Correction hPa Setra 470 hPa Correction hPa

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801.28 -0.40 960.95 -0.56 802.32 -0.51 961.22 -0.46 810.44 -0.31 970.48 -0.40 811.78 -0.35 970.78 -0.50 820.38 -0.45 970.81 -0.53 820.41 -0.38 971.30 -0.52 820.80 -0.52 979.61 -0.47 821.00 -0.49 981.19 -0.45 840.00 -0.34 981.21 -0.48 840.61 -0.53 981.34 -0.44 840.96 -0.51 991.15 -0.55 841.02 -0.51 991.22 -0.52 860.14 -0.29 991.48 -0.52 860.55 -0.47 991.59 -0.48 860.56 -0.53 1000.70 -0.45 860.77 -0.46 1001.05 -0.53 880.60 -0.31 1001.19 -0.50 880.67 -0.54 1001.28 -0.51 880.99 -0.46 1010.66 -0.49 881.43 -0.50 1010.80 -0.45 900.32 -0.29 1011.19 -0.51 900.75 -0.49 1011.50 -0.45 900.84 -0.43 1014.82 -0.47 901.14 -0.44 1015.63 -0.52 920.20 -0.48 1017.27 -0.40 920.55 -0.39 1017.79 -0.40 920.61 -0.55 1020.80 -0.49 920.93 -0.47 1020.95 -0.52 940.29 -0.38 1021.40 -0.52 940.65 -0.53 1021.42 -0.45 940.91 -0.50 1030.40 -0.45 941.20 -0.44 1030.80 -0.48 950.50 -0.42 1030.87 -0.46 951.40 -0.49 1031.53 -0.48 951.68 -0.56 1040.73 -0.40 951.78 -0.50 1041.23 -0.50 960.69 -0.40 1041.26 -0.48 960.85 -0.50 1041.52 -0.49

1050.28 -0.45 1051.27 -0.34 1051.20 -0.48 1049.98 -0.40

Test 1(d) Calibration at 40°C.

Time (days)

Setra 470 hPa Correction hPa Time (days)

Setra 470 hPa Correction hPa

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1.0 1020.77 -0.55 17.0 1012.71 -0.26 1.3 1018.23 -0.43 17.3 998.07 -0.29 2.0 1022.17 -0.35 18.0 993.28 -0.25 2.3 1018.65 -0.44 18.3 1008.61 -0.22 3.0 1022.15 -0.33 19.0 1010.39 -0.24 3.3 1018.12 -0.48 19.3 1020.27 -0.26 4.0 1018.14 -0.28 22.0 1013.40 -0.23 4.3 1014.75 -0.41 23.0 1013.22 -0.23 5.0 1014.34 -0.29 23.3 1011.30 -0.24 5.3 1012.16 -0.37 24.0 1007.20 -0.26 8.0 1014.93 -0.43 24.3 1012.70 -0.22 8.3 1014.41 -0.51 25.0 1016.06 -0.27 9.0 1019.83 -0.31 25.3 1023.31 -0.29 9.3 1016.96 -0.43 26.0 1022.88 -0.29

10.0 1016.37 -0.38 26.3 1013.65 -0.36 10.3 1013.25 -0.35 29.0 1011.10 -0.31 11.0 1020.15 -0.37 30.0 1015.11 -0.31 11.3 1019.21 -0.40 30.3 1012.05 -0.29 12.0 1019.32 -0.41 31.0 1016.84 -0.30 12.3 1016.58 -0.41 31.3 1017.77 -0.30 15.0 1021.81 -0.35 32.0 1023.51 -0.30 16.0 1020.23 -0.36 32.3 1023.03 -0.34 16.3 1013.46 -0.19 33.0 1025.04 -0.28

33.3 1024.21 -0.25

Test 1(e). Recovery test over 33 days.

SETRA 470 hPa Correction hPa 798.56 -0.14

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820.42 -0.27 840.47 -0.17 860.20 -0.08 880.44 -0.14 900.25 -0.22 920.41 -0.24 940.33 -0.26 962.58 -0.24 979.90 -0.21 991.11 -0.29

1001.17 -0.23 1010.90 -0.31 1020.65 -0.29 1030.15 -0.28 1040.70 -0.30 1050.92 -0.32 1050.62 -0.32 1040.75 -0.30 1030.77 -0.23 1020.85 -0.25 1010.27 -0.28 1000.44 -0.28 991.02 -0.26 980.64 -0.24 959.34 -0.26 940.37 -0.25 920.55 -0.20 900.75 -0.22 880.50 -0.22 860.45 -0.21 840.55 -0.21 820.20 -0.31 800.78 -0.18

Test (f). Calibration at room temperature.

Temp. (°C) Setra hPa Correction hPa -2.0 1010.70 -0.37 0.3 1012.33 -0.27

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4.9 1012.48 -0.19 5.2 1010.73 -0.34 9.7 1010.60 -0.30 9.8 1012.78 -0.29

15.1 1010.67 -0.33 15.2 1012.26 -0.25 22.5 1012.19 -0.20 22.5 1011.22 -0.20 22.7 1011.19 -0.26 24.7 1011.16 -0.13 27.1 1011.02 -0.08 30.3 1011.37 -0.13 32.1 1010.76 0.03 35.2 1010.57 0.04 35.6 1011.36 -0.13 40.0 1011.10 0.06 40.1 1010.58 0.10 44.8 1011.01 0.02 45.2 1010.88 0.11 50.4 1010.74 0.05 50.5 1010.12 0.12

Test (g). Temperature variation test.


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