ME 322: Instrumentation Lecture 6

January 29, 2016Professor 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

– This assumes the deviations are equally distributed around best fit 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 – The best estimate of the measurand is

– The best statements of the confidence interval for the measured values are

• M = ± sM,R units (68%), • 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

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.

0 40.5328 6.881.0597 9.721.5617 12.482.0863 15.342.5295 17.831.9637 14.661.5483 12.350.9211 9.030.5216 6.83

0 4.010.5619 7.090.9595 9.181.4562 11.921.9927 14.842.6214 18.32.1092 15.431.6423 12.891.0696 9.860.5315 6.88

0 4.02

Standard Reading, hS

[in WC]

Transmitter Output, IT

[mA]

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

• Your data and confidence level may be different

𝑃 (−1.15<𝑧<1.15 )=𝐼 (1.15 )− 𝐼 (−1 .15 )= 𝐼 (1.15 )− [− 𝐼 (1 .15 ) ]=2 𝐼 (1 .15 )=2 ( 0.3749 )=75 %𝑥1 𝑥2

𝑓 (𝑥 )

I(z)

Interpretation of Measurement Question

• 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?

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, Sh,I

0.0065 in WC

68% Confidence Interval 1.2380 ± 0.0065 in WC (68%)Inverted Manufacturer's Transfer Function h = (0.1875 inWC/mA)IT - 0.75 inWC

Confidence Interval if not Calibrated (Unknown confidence Level)

1.2619 ± 0.0075 inWC

Standard Error of the Estimate of x given y

• Characterizes the horizontal spread that contains 68% of the data

• = , where = slope of best fit line

𝑠𝑦 , 𝑥𝑠𝑥, 𝑦

R or y

M or x

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/MECH322Instrumentation/Labs/Lab%2003%20PressureCalibration/Lab%20Index.htm

• Write abstract last: Objective, methods, results

End 2016

Table 1 Equipment Specifications

• 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.

Transmitter Model 616-1 Model 616-5Full Scale Range 3 inch-WC 40 inch-WCRelative Accuracy ±0.25% FS ±0.25% FSAbsolute Accuracy ±0.0075 inch-H2O ±0.1 inch-H2O

Manufacturer's Inverted Transfer Function h = (3 inch-H2O)(IT-4mA)/16mA h = (40 inch-H2O)(IT-4mA)/16mA

Pressure StandardFull Scale Range 25 mBar (10 in-WC) 350 mBar (141 in-WC)Relative Accuracy ±0.1% FS ±0.035% FSAbsolute Accuracy ±0.01 inch-H2O ±0.05 inch-H2O

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 0Miles Greiner

Lab Instructors:Josh McGuire, Şevki Çeşmeci, and Roberto Bejarano