3.3 Sensor Characteristics: Accuracy, Calibration, and Dynamics
Key Takeaways
- Repeatability is same-direction scatter under the same conditions; hysteresis is the upscale versus downscale difference at the same input.
- An accuracy stated as percent of span is a nearly fixed engineering-unit error; at low operating points that error is a much larger percent of reading.
- Turndown is typically URL divided by calibrated span; rangeability is the max/min ratio while still meeting stated accuracy—do not swap the terms.
- Five-point as-found/as-left calibration exposes linearity and hysteresis; TUR traditionally expects the standard to be about four times more accurate than the unit under test.
- First-order sensor lag reaches about 63% of a step at t = τ; square-root extraction on a DP flow loop must exist in only one place.
Why Characteristics Questions Are Almost Pure Measurement Language
Measurement spec 1.E tests whether you can read an instrument datasheet the way NCEES writes stems: accuracy, precision, repeatability, hysteresis, dead band, drift, turndown, rangeability, response time, and calibration language. The arithmetic is small. The traps are vocabulary and basis—especially percent of span versus percent of reading.
Static Characteristics
Accuracy is closeness to the true (reference) value. Manufacturers bundle bias and random contributions the way the datasheet defines them; you must use that definition, not a private one. Precision is scatter only: a precise transmitter can be precisely wrong if it has a large bias.
Repeatability is agreement among consecutive measurements of the same value, approached from the same direction, under the same conditions. Hysteresis is the difference between upscale and downscale output at the same input. If the stem says “increasing versus decreasing,” it is hysteresis, not repeatability. Calling hysteresis “repeatability” is one of the most reliable 1.E traps.
Dead band is a finite input range that produces no detectable output change—mechanical backlash, packing friction, or digital resolution. Hysteresis can exist with a narrow dead band; dead band can exist with little loop-shaped hysteresis. Drift is a slow change with time at constant input (aging, contamination, reference-cell consumption on an electrochemical head).
Temperature effects are usually split into zero shift and span shift with ambient or process temperature. They are often specified as percent of span per 10 °C, which again hurts you at the low end of a turned-down calibration. A “0.1% accurate” cell that you calibrated to 10% of URL still carries span-basis temperature error referenced to that small span—or worse, referenced to URL, depending on the datasheet. Read the basis.
Range, Span, Turndown, and Rangeability
Lower range value (LRV) and upper range value (URV) are the calibrated 0% and 100% points. Span = URV − LRV. Upper range limit (URL) is the maximum URV the sensor can be calibrated to.
Turndown in transmitter catalogs is commonly URL / calibrated span. A 0–1000 inH2O URL cell calibrated 0–100 inH2O is 10:1 turndown. Accuracy stated as percent of span then becomes a larger fraction of the live signal as you operate near LRV.
Rangeability is the ratio of maximum to minimum value at which the device still meets a stated accuracy—classic flowmeter language. A meter with 10:1 rangeability is accurate from 10% to 100% of maximum flow. Control-valve people also say rangeability for inherent flow turndown. The exam mix-up is treating turndown (how far you calibrated down from URL) as if it were rangeability (how far the physics stay accurate), or the reverse. A DP transmitter with huge turndown still has miserable percent-of-reading error at 5% of span if accuracy is percent of span. That is not “10:1 rangeability.”
Dynamics: Tau, Dead Time, and Reliability
A first-order sensor has time constant τ (tau): after a step, the output reaches about 63.2% of the final value at t = τ, about 86% at 2τ, and about 95% at 3τ. Dead time (θ) is pure delay before any response—transport in a sample line, a long thermowell, a programmable-logic-controller scan. Combined, many PE models use G(s) = K e^{−θs} / (τs + 1).
If the process time constant is 2 s and the thermowell-plus-transmitter τ is 8 s, the controller’s process variable is dominated by sensor lag, not by the process. Fast cascade secondaries and compressor antisurge loops fail this way: the measurement was specified for accuracy, not for bandwidth. Two first-order lags in series are not a single first-order lag; the PV is slower and more S-shaped than either τ alone.
Reliability on this spec is qualitative plus simple availability thinking. More components in series (thermowell, isolator, wireless hop) lower measurement availability. Voting can raise safety integrity and can also raise spurious-trip rate. A drifting electrochemical cell can be highly repeatable this morning and still be inaccurate.
Calibration Practice
Zero and span set LRV and URV. A five-point calibration at 0, 25, 50, 75, and 100% of span, often upscale and downscale, reveals linearity, hysteresis, and dead band. As-found is the error before adjustment—what the process has been living with. As-left is after adjustment. If as-found already exceeds the plant tolerance, the last campaign’s product may have been off-spec even if you leave the transmitter perfect.
Test uncertainty ratio (TUR) is the traditional rule that the calibration standard should be several times more accurate than the unit under test (UUT). 4:1 is the number most PE stems still expect. ISO/IEC 17025 laboratories may use 2:1 plus guard banding; you are not being asked to recite an accreditation clause, only to refuse a “standard” that is no better than the transmitter.
Square-root DP flow is the configuration trap. Volumetric flow Q is proportional to √ΔP. If the transmitter is linear in differential pressure, 12 mA (50% of the 4–20 mA span) is 50% of DP and about 70.7% of full-scale flow. If square-root extraction is enabled in the transmitter, 12 mA is 50% of flow. If both the transmitter and the distributed control system extract, you double-extract and the indication is wrong. If you apply a linear five-point pressure standard to a square-root transmitter and force the mA to be linear with DP, you have calibrated out the extraction you needed—or you have fought a correct nonlinear table. Extract in one place, document it, and calibrate against that basis.
Worked Example A: Combined Error and Percent of Span Versus Percent of Reading
A gauge-pressure transmitter is calibrated 0–200 psi. The manufacturer states:
- reference accuracy ±0.25% of span
- ambient temperature effect ±0.15% of span
- additional ±0.10% of reading for process-temperature effect
Treat the contributions as independent and combine them by root-sum-square (RSS).
Span-basis terms, in engineering units at any pressure:
- 0.25% × 200 psi = 0.50 psi
- 0.15% × 200 psi = 0.30 psi
At a normal operating pressure of 40 psi, the reading-basis term is 0.10% × 40 = 0.040 psi.
Combined error E = √(0.50² + 0.30² + 0.040²) = √(0.25 + 0.09 + 0.0016) ≈ 0.583 psi.
- As percent of span: 0.583 / 200 = 0.29% of span
- As percent of reading at 40 psi: 0.583 / 40 = 1.46% of reading
Same hardware, same RSS, different story at 20% of the calibrated range. If the stem asks for uncertainty at a stated operating point, convert every term to engineering units first, RSS, then convert to the basis requested. Never report 0.25% of span as 0.25% of reading at 40 psi.
Worked Example B: First-Order Sensor Lag Versus the Process
A temperature loop uses a thermowell assembly with τ_s = 12 s and negligible dead time. The fluid process is approximately first-order with τ_p = 3 s. A step in jacket steam changes true fluid temperature with the 3 s lag, but the transmitter output is a first-order lag of 12 s driven by that already-lagging tip temperature—two lags in series. At t = 12 s after the steam step, the sensor has reached only about 63% of the temperature that exists at its tip, and the tip itself has not finished tracking the bulk fluid. A PID controller tuned as if PV were τ = 3 s will overshoot. Specify a faster well, a reduced-diameter element, or a better location. Do not “tune through” a 12 s measurement on a 3 s process and call the overshoot a process problem.
Exam Traps
- Calling hysteresis “repeatability.”
- Mixing turndown (URL versus calibrated span) with rangeability (max/min while meeting accuracy).
- Quoting percent of span as if it were percent of reading at a low operating point.
- Calibrating or configuring a square-root DP transmitter without accounting for extraction, or extracting twice.
- Ignoring sensor τ when the process is faster than the measurement.
| Characteristic | How PE stems usually phrase it |
|---|---|
| Accuracy | Deviation from the true or reference value; often percent of span |
| Precision | Scatter or tight grouping; true value not required |
| Repeatability | Same direction, same conditions, consecutive readings |
| Hysteresis | Upscale versus downscale at the same input |
| Dead band | No output change until the input moves by X |
| Drift | Change over hours or weeks at constant input |
| Turndown | URL / calibrated span, or “calibrated down from URL” |
| Rangeability | Max/min flow or value while still meeting accuracy |
| Time constant τ | 63% of a step; first-order lag |
| Dead time θ | No response for θ seconds; transport delay |
| TUR | Standard about four times more accurate than the UUT |
A transmitter calibrated 0–200 inH2O has a reference accuracy of ±0.50% of span. At an operating point of 40 inH2O, that reference accuracy expressed as percent of reading is which value?
A five-point calibration shows that at 50% input, the upscale output and the downscale output differ by 0.30% of span. What characteristic is that difference?
A differential-pressure flow transmitter has square-root extraction enabled, and the DCS analog input is also configured to square-root extract the same 4–20 mA signal. What is the most likely result?