Guard-Banded Conformity Decision on a 4-20 mA Pressure Transmitter
A calibration cell verifies 0-10 bar, 4-20 mA pressure transmitters against a specification of plus or minus 0.5 percent of span before dispatch. The reference gauge is certified to plus or minus 0.05 percent of reading. The current is read on a 6.5-digit multimeter whose accuracy is plus or minus 0.02 mA, treated as a rectangular distribution. Ten repeat readings at an applied 5.000 bar give a mean of 11.940 mA with a sample standard deviation of 0.015 mA. Ideal output at 5.000 bar is 12.000 mA. Zero and span trims are available on the transmitter. The cell procedure requires a guard band: conformity may be declared only when the absolute error plus the expanded uncertainty U is less than or equal to the specification limit, with U at k = 2. The unit is due to ship against a purchase order allowing no more than 0.4 percent of span error, and the buyer's inspector will re-test it on a different day with a single reading.
1. Build the uncertainty budget in mA and in percent of span, compute the expanded uncertainty, and decide whether conformity can be declared at the 0.5 percent specification.
Read the worked answer
Span is 16 mA over 10 bar, i.e. 1.6 mA per bar.
Error: 11.940 - 12.000 = -0.060 mA = 0.060/16 = -0.375 percent of span.
| Source | Value | Standard uncertainty |
|---|---|---|
| Repeatability of the mean | s = 0.015 mA, n = 10 | 0.015/sqrt(10) = 0.0047 mA |
| Multimeter accuracy | 0.02 mA rectangular | 0.02/sqrt(3) = 0.0115 mA |
| Reference gauge | 0.05 percent of 5.000 bar = 0.0025 bar = 0.0040 mA, rectangular | 0.0023 mA |
Combined: sqrt(0.0047^2 + 0.0115^2 + 0.0023^2) = 0.0127 mA = 0.079 percent of span. U = 2 x 0.0127 = 0.025 mA = 0.159 percent of span.
Decision: 0.375 + 0.159 = 0.534 percent > 0.5 percent. Conformity cannot be declared. The unit is in the guard-band (indeterminate) zone: the error alone looks inside specification, but the measurement is not good enough to prove it. It also fails the buyer's 0.4 percent limit on the same arithmetic (0.534 > 0.4).
2. The technician proposes to trim the zero until the mean reads exactly 12.000 mA, then declare conformity and ship. Is that defensible? State what else must be checked first.
Read the worked answer
Partly defensible, and routinely done - but not as stated. A zero trim is a legitimate manufacturing adjustment; what is not defensible is trimming and then declaring conformity on the pre-trim data, or shipping without a fresh verification. After a zero trim the error at 5 bar becomes approximately zero, so 0 + 0.159 = 0.159 percent, comfortably inside both the 0.5 percent specification and the 0.4 percent purchase order.
What must be checked first: a 0.06 mA shift is 0.375 percent of span - large enough that it could be a genuine sensor offset (benign, trimmed out) or the visible symptom of a span or linearity error that a single-point check cannot reveal. Before trimming, the unit must be characterised at 0, 25, 50, 75 and 100 percent of span and the end-point line or best-fit straight line computed. If the unit is linear and merely offset, trim and pass. If the span is wrong, a zero trim will simply push the error to the top of the range.
After trimming: re-run the full five-point verification and re-issue the certificate against the post-adjustment data, with the guard band applied at each point. The buyer's inspector will measure a single reading on a different day; a single reading carries the full repeatability of 0.015 mA (0.094 percent of span), which is still inside 0.4 percent, so rejection risk is low provided the residual error after trim is a few thousandths of a mA.
3. The technician proposes to raise the sample size from 10 to 50 readings to tighten the guard band. Show numerically whether this changes the decision, and say where the effort should go instead.
Read the worked answer
No, it will not change the decision. With n = 50, the repeatability term becomes 0.015/sqrt(50) = 0.0021 mA.
Combined: sqrt(0.0021^2 + 0.0115^2 + 0.0023^2) = sqrt(4.4e-6 + 1.32e-4 + 5.3e-6) = 0.0120 mA. U = 0.024 mA = 0.150 percent of span.
Guard band check: 0.375 + 0.150 = 0.525 percent, still above 0.5 percent. Forty extra readings bought 0.009 percent and changed nothing.
Why: the multimeter's 0.02 mA accuracy term (0.0115 mA standard uncertainty) dominates the budget by a factor of five over repeatability. Averaging beats down random effects only; a rectangular Type B specification barely moves.
Where the effort should go:
- Read current across a precision shunt with a better instrument, or use a calibrator with a mA measurement accuracy of 0.005 mA. That alone drops the dominant term to 0.0029 mA, giving U = 0.012 mA = 0.075 percent and a guard-band total of 0.450 percent - a pass at both limits.
- Or trim the zero, which removes the 0.375 percent error outright and makes the uncertainty question moot.
The lesson for the procedure: review the uncertainty budget before increasing sample size; effort spent reducing a non-dominant term is wasted.