ME 322: Instrumentation Lecture 6
January 29, 2016
Professor Miles Greiner
Review Calibration, Lab 3
Calculations, Plots and Tables
Announcements/Reminders
• HW 2 due Monday
– L3PP – Lab 3 preparation problem
• Bottom of Lab 3 Handout
• 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.
General Guidelines for HWs
• Use your ME 322 ID numbers and lab day, not name or UNR student ID
• Units and significant digits always!
• No hand-drawn plots! Starting from HW 3, whenever you are asked to plot
your data, use computer software, i.e. Excel, MatLab, etc.
– Include labels for both axes with the units. If necessary, include legends too.
– Make it look good!
• Show you work! Do not skip the steps and just write your final answer.
Whenever applicable, list your assumptions, write out your formulas and
work through to your final answer. If you use a Table or graph on your
solutions, give reference in your book, i.e from Table 6.3, z=-1.28.
• Be clear with your solutions and work neatly! If the grader needs to
spend more than 3 minutes to figure out what you wrote, you may not get
even partial credit.
• 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 equally distributed around
best fit for all values of 𝑥 (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 for the
measured values are
• M = 𝑀 ± sM,R units (68%),
• 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
Lab 3 Set-up and Procedure
• At each pressure level record hS (from Standard) and IT
(from DMM)
• Record data from at least two ascending and descending
pressure cycles, with at least 6 measurements in each
direction
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 manufacturer-specified transfer function.
• Your data and confidence level 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.
Standard errors of the estimates for the transmitter current for a given pressure head is
SI,h = 0.035 mA, and Sh,I = 0.0065 in-WC. (confidence level 68%)
The manufacturer stated error is 0.0075 in-WC (what is it’s confidence level, >68%)
Confidence Level of Manufacture-Stated
Uncertainty
• Number of standard deviations from the fit
– 0.0075/0.0065 = 1.15
• Find the probability a measurement is within 1.15
standard deviations of the mean
• Identify: Symmetric problem
• z1 = -1.15, z2 = 1.15
𝑓 𝑥
𝑃 −1.15 < 𝑧 < 1.15
𝑥1
= 𝐼 1.15 − 𝐼 −1.15
= 𝐼 1.15 − −𝐼 1.15
= 2𝐼 1.15 = 2 0.3749 = 75%
• Your data and confidence level may be different
𝑥2
I(z)
Interpretation of Measurement
Question
Transmitter Output, I T
10.73 mA
Inverted Measured Transfer Function
h = (0.1838 inWC/mA)I T - 0.7342 inWC
Standard Error of the Estimate of Pressure
Head for a Given Current, Sh,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 (68%)
h = (0.1875 inWC/mA)I T - 0.75 inWC
1.2619 ± 0.0075 inWC
• A measurement using your transmitter reads 10.73 mA
• Based on your calibration, find the confidence interval for
the actual pressure for which you have a 68% confidence
level.
• What confidence interval would you have gotten if you had
not calibrated the pressure gage?
Standard Error of the Estimate of x given y
𝑠𝑦,𝑥
𝑠𝑥,𝑦
R or y
M or x
• Characterizes the horizontal spread that contains
68% of the data
• 𝑠𝑥,𝑦 = 𝑠𝑦,𝑥 /𝑎, where 𝑎 = slope of best fit line
Demonstrate how to
• Create calibration, error and deviation
(ascending and descending) plots
• Calculate Standard error of the estimate
• Sample Data
• http://wolfweb.unr.edu/homepage/greiner/teaching/MECH3
22Instrumentation/Labs/Lab%2003%20PressureCalibration
/Lab%20Index.htm
• Write abstract last: Objective, methods, results
End 2016
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 level for the transmitter accuracy is
not given by the manufacturer
– We will determine it in this experiment.
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