Learn the proper way to determine measurement uncertainty and evaluate it using the test accuracy ratio.
Engineers often struggle with the proper procedure to determine measurement uncertainty. The dilemma at the end of that analysis requires a decision as to the result. Did the item under scrutiny pass or fail? Is the measurement uncertainty tight enough to support the decision?
The procedure for any given problem includes identifying the sources of uncertainty, determining their individual contributions, combining them properly, and judging the result based on the application of the test accuracy ratio (TAR). TAR has long been included in the study and practice of metrology. The usual focus is on mechanical applications like checking dimensional conformity to ensure quality control. But did you know that TAR can also be applied to the evaluation of electronic circuits and systems?
The Measurement Problem
A few weeks ago I helped a former cubemate, Joe, understand the nature and characteristics of a manganin current shunt resistor.
See: Making Sense of Shunt Sensors: All Your Questions Answered
This afternoon, I looked up from my work to see he had reappeared at my Sundstrand Aerospace desk. Joe looked pale and distraught. Startled, I asked, “Joe, what’s wrong? Are you facing another divorce? Do you have more shunt problems?”
Joe replied, “No divorce. Your zealous tutorage led me to specify a Riedon RSH-150-100 busbar shunt. It’s rated current is 150A. Its operating current (2/3 of the rated current) is 100A. When the rated current flows through it, the shunt voltage will be 100mV.”
I was pleased. He had run through the highlights of our previous discussion. I did also wonder about “zealous tutorage” but decided to let that pass.
Joe continued: “The Unit Under Test (UUT) is a 270V to 28V Converter Regulator. Since the load is a straightforward resistive bank, low-side current sensing is used to keep the common-mode voltage small.” (Image (a).)
“It looks like the measurement system is well defined, so what’s the issue?” I asked.
Joe explained, “The customer—and therefore my manager—want us to guarantee we can measure the 60A ± 5% (60A ± 3A). The engineers specifying the DMM want to know how accurate (expensive) it needs to be. We are having a preliminary design review in two weeks and I’m supposed to present our confidence in our supposed 5%-accurate measurement. Egan in the Ghostbusters movie expressed it well: ‘I’m terrified beyond the capacity for rational thought.’”
I grinned and assured Joe we could arrive at the answer in just a few minutes.
The Overall Approach to Determine the Measurement Uncertainty
In the case of the current shunt, there are two sources of uncertainty: its tolerance (± 0.1%) and its temperature coefficient (15 ppm/^{o}C). The uncertainties should be expressed in like terms, which in this case is voltage in millivolts. By convention, both uncertainties follow a uniform (rectangular) probability distribution.
Let’s use the notation ± a_{tolerance}_{ }and ± a_{Temperature}, respectively, to designate the amplitudes of the uncertainties. Next, we determine the corresponding standard errors. We combine them. Our last step is to make a pass/fail decision with a defined confidence interval.
Finding the Uncertainty Due to Tolerance
The 150A shunt develops 100mV when the rated current flows. This gives us the mV/A shunt conversion.
The measurement of concern is 60A and (60A)(0.0006667mV/A) = 40mV. When 60A flows through the shunt, we have a shunt voltage (V_{sense}) of 40mV. Since the shunt resistance has a tolerance (Tol) of 0.1%, the shunt voltage will also have a tolerance of 0.1%.
The amplitude of the uncertainty (± a_{tolerance}) about the 40mV is easy to find.
The usual procedure is to assume the tolerance that has a rectangular (or uniform) probability distribution as shown in (b) of the image above. (This is the type of probability distribution that occurs when looking where raindrops land on a clothesline during a gentle spring rain. Every location or every value is equally likely to occur.) The standard error is determined by dividing by √3. The standard error for the tolerance is u_{j(tolerance)}.
Joe’s eyelids began to flutter, and I remarked, “Joe, I am showing you the how and not the why. We can refine and reach understanding later.”
Finding the Uncertainty Due to Temperature
The shunt manufacturer requires a shunt operating temperature range from 30^{o}C to 70^{o}C to maintain its accuracy. The shunt’s data sheet shows it has a temperature coefficient of resistance (TCR) of 15 ppm/^{o}C. The temperature coefficient contribution to the uncertainty is based on the temperature rise (T_{rise}) above the ambient temperature (T_{amb}).
The amplitude of the deviation due to temperature is ± a_{temperature}. The ambient temperature is taken to be 25^{o}C; the (worst-case) temperature rise is 70^{o}C. The amplitude variation due to temperature is calculated using the relationship as follows:
The distribution is again rectangular, so the standard error is calculated by dividing the deviation amplitude by √3. The standard error for the temperature is u_{j(temperature)}.
The two rectangular distributions are shown in image (c) below. The next job is to combine the two standard errors.
Finding the Combined Uncertainty
The two uncertainties u_{j(tolerance)} and u_{j(temperature)} are uncorrelated random variables and cannot be added or subtracted directly. Consequently, the root sum of squares (RSS) is used.
When the two uncorrelated random variables are combined, the resulting probability distribution follows a normal probability distribution. The combined standard error is the standard deviation of that normal (Gaussian) probability distribution, which is shown in image (d) below.
Determining the Confidence Interval
Next, we apply the coverage factor (k). The coverage factor is the number of standard deviations required on a normal probability distribution to give us the desired probability—and hence, confidence in our measurement tolerance. The interval is:
V_{nominal} ± ku_{j(combined}_{)}
We shall use a 99.73% confidence interval, which requires a coverage factor k = 3.00. In this instance we can state we are 99.73% certain the actual voltage is 40mV ± (3)(0.0279mV) or 40mV ± 0.0836mV.
Performing the TAR Analysis
The purpose of a TAR analysis is to ensure formally that we are measuring a quantity with sufficient accuracy to apply pass/fail criteria. When making a pass/fail decision, four possibilities can occur as indicated in the table:
PASS |
FAIL |
GOOD UNIT |
BAD UNIT |
BAD UNIT |
GOOD UNIT |
Passing a good unit or failing a bad unit are the desirable outcomes. However, failing a good unit means it must be examined and retested. It will pass its test and only expenses will increase. Passing a bad unit could be catastrophic. Equipment could be damaged, or personnel safety could be compromised.
TAR is defined as the ratio of specification limits to the measurement accuracy. Since the 1960s a TAR that is greater or equal to 4 has been used as the line of demarcation in the field of metrology. It works here too.
Had this been too stringent and we didn’t achieve a TAR of four or greater, a confidence interval of 95% as recommended by NIST may be used. In that case, the coverage factor k becomes 1.96.
Joe had taken copious notes and was eager to trot off and work on his presentation. No more laments from Ghostbusters past. I had a strong feeling I would receive a visit from the engineers specifying the DMM.
More from the Professor:
Making Sense of Shunt Sensors: All Your Questions Answered