ME 322: Instrumentation Lecture 6
January 30, 2015
Professor Miles Greiner
Review Calibration, Lab 3
Calculations, Plots and Tables
Announcements/Reminders
• HW 2 due Monday
– L3PP – Lab 3 preparation problem
• Create an Excel Spreadsheet to complete the tables,
plots and question in the Lab 3 instructions, using the
sample data on the Lab 3 website.
• Bring that spreadsheet to lab next week and use it for
your data.
• You can sign up for Extra credit Science
Olympiad until Wed, Feb 4, 2015
• HW 1 Comments
– Plot using Excel (not by hand)
– Hint: Do some summation calculations by hand to
be sure you know how it is done
Instrument Calibration (review)
• Measure instrument output (R) for a range of known
measurands (M, as measured by a reliable standard)
• Perform measurements for at least one cycle of
ascending and descending measurands
• Fit an algebraic equation to the R vs M data to get
instrument transfer function:
– Linear: R = aM + b
– Other: i.e. R = aM2 + bM + c, or …
• Find standard error of the estimate of R given M, sR,M
– 𝑠𝑦,𝑥 =
𝑛 (𝑦 −𝑎𝑥 −𝑏)2
𝑖
𝑖=1 𝑖
𝑛−2
– This assumes the deviations are the same for all values of M
How to Use the Calibration
• Invert transfer function
– If linear: M = (R-b)/a
• Find standard error of the estimate of M given R
– sM,R = sR,M/a
• For a given reading 𝑅
– The best estimate of the measurand is
• 𝑀 = (𝑅 − 𝑏)/𝑎
– The best statements of the confidence interval are
• M = 𝑀 + sM,R units (68%), or
• M = 𝑀 + 2sM,R units (95%), or …
What does Calibration do?
• Removes systematic bias (calibration) error
• Quantifies random errors
– imprecision, non-repeatability errors
– But does not remove them
• Quantifies user’s level of confidence in the
instrument
Manufacturers often state “accuracy”
• May include both imprecision and calibration drift
– Often not clearly defined
• This is one of the objectives of Lab 3
Table 1 Equipment Specifications
Model 616-5
Model 616-1
Transmitter
40 inch-WC
3 inch-WC
Full Scale Range
±0.25% FS
±0.25% FS
Relative Accuracy
±0.1 inch-H2O
±0.0075 inch-H2O
Absolute Accuracy
Manufacturer's Inverted
h = (3 inch-H2O)(I T-4mA)/16mA h = (40 inch-H2O)(IT-4mA)/16mA
Transfer Function
Pressure Standard
25 mBar (10 in-WC) 350 mBar (141 in-WC)
Full Scale Range
±0.035% FS
±0.1% FS
Relative Accuracy
±0.05 inch-H2O
±0.01 inch-H2O
Absolute Accuracy
• In your report you will use the first column, and only
one from the second and third columns
• The confidence levels for the transmitter accuracy is
not given by the manufacturer
– We will determine it in this experiment.
Standard
Reading, hS
[in WC]
Transmitter
Output, IT
[mA]
0
0.5328
1.0597
1.5617
2.0863
2.5295
1.9637
1.5483
0.9211
0.5216
0
0.5619
0.9595
1.4562
1.9927
2.6214
2.1092
1.6423
1.0696
0.5315
0
4
6.88
9.72
12.48
15.34
17.83
14.66
12.35
9.03
6.83
4.01
7.09
9.18
11.92
14.84
18.3
15.43
12.89
9.86
6.88
4.02
Table 2 Calibration Data
• This table shows two cycles of
ascending and descending
pressure calibration data.
• The transmitter current did not
return to 4.00 mA at the end of
the descending cycles.
Fig. 1 Measured Transfer Function
• For the sample data
– The measured transmitter current is consistently higher than that predicted by the manufacturerspecified transfer function.
– Standard errors of the estimates for the transmitter current for a given pressure heat is SI,h = 0.035
mA, and Sh,I = 0.0065 in-WC.
– The manufacturer-stated accuracy (0.0075 in-WC) for the transmitter is 1.15 times larger than Sh,I,
corresponding to a 75% confidence level.
• Your data may be different!
Fig. 2 Error in Manufacturer’s Transfer Function
• Error in the manufacturer-specified transfer function increases with
pressure
• Maximum error magnitude is 0.35 mA.
Fig. 3 Deviation from Linear Fit
• SI,h characterizes the deviations over the full range of hS
• Neither the ascending nor the deviations are generally positive or negative, which
suggests that hysteresis does not play a strong role in these measurements.
• There are no systematic deviations form the fit correlation, indicating the instrument
response is essentially linear.
This lecture demonstrates how to
• Format plot labels, borders, fonts,..
– Different symbols for ascending and descending
data
• Calculate standard error of estimate,
confidence level
• Write abstract last: Objective, methods, results
• Sample Data
• http://wolfweb.unr.edu/homepage/greiner/teaching/MECH3
22Instrumentation/Labs/Lab%2003%20PressureCalibration
/Lab%20Index.htm
Confidence Level of Manufacture-Stated
Uncertainty
• Find the probability a measurement is within
1.15 standard deviations of the mean
• Identify: Symmetric problem
• z1 = -1.15, z2 = 1.15
𝑃 −1.15 < 𝑧 < 1.15
= 𝐼 1.15 − 𝐼 −1.15
= 𝐼 1.15 − −𝐼 1.15
= 2𝐼 1.15 = 2 0.3749 = 75%
• Your confidence level may be different
Interpretation of Measurement Question
Transmitter Output, IT
10.73 mA
Inverted Measured Transfer Function
h = (0.1838 inWC/mA)IT - 0.7342 inWC
Standard Error of the Estimate of Pressure
Head for a Given Current, S h,I
68% Confidence Interval
Inverted Manufacturer's Transfer Function
Confidence Interval if not Calibrated
(Unknown confidence Level)
0.0065 in WC
1.2380 ± 0.0065 in WC
h = (0.1875 inWC/mA)IT - 0.75 inWC
1.2619 ± 0.0075 inWC
Abstract
• In this lab, a 3-inch-WC pressure transmitter was calibrated
using a pressure standard.
– The transmitter current IT was measured for a range of pressure
heads h, as measured by a pressure standard.
• The measured inverted-transfer-function was
– h = (0.1838 in-WC/mA)IT – (0.7335 in-WC),
– The 68%-confidence-level confidence-interval for this function is
± 0.0064 in-WC
• The manufacturer’s stated uncertainty is 0.0075 in-WC
– This is 1.15 time larger than the 68%-confidence-level interval,
which corresponds to a 75%-confidence-level
Lab 3
Static Calibration of Electronic Pressure
Transmitters
February 3, 2014
Group 0
Miles Greiner
Lab Instructors:
Josh McGuire, Şevki Çeşmeci, and Roberto Bejarano
Sxy= Standard error in X given Y
Syx
Sxy
Example of Hysteresis