Bias, Linearity, Stability, and Hysteresis

Systematic Error and Measurement Bias

Error Decomposition: Systematic vs. Random

Every measurement error ($E$) consists of two distinct components:

$$E = x_{\text{indicated}} - x_{\text{true}} = E_{\text{systematic}} + E_{\text{random}}$$

• Random Error: Varies unpredictably in sign and magnitude between observations; cannot be eliminated, but can be reduced by averaging replicate measurements ($1/\sqrt{n}$).
• Systematic Error: Remains constant or varies in a predictable manner across replicate measurements under specified conditions. Cannot be reduced by averaging replicate measurements!

Measurement Bias (VIM 2.18)

Measurement bias is the estimate of a systematic error. It is defined quantitatively as the difference between the expectation of the measurement results (the mean of an infinite series of replicate observations $\bar{y}$) and an accepted reference quantity value ($y_{\text{ref}}$):

$$b = \bar{y} - y_{\text{ref}}$$

Corrected vs. Uncorrected Bias

Estimate and correct significant recognized systematic effects in the measurement model where appropriate, retaining uncertainty in those corrections. An observed error includes random effects and is not automatically an exactly known bias. Any intentionally uncorrected significant effect requires an appropriate evaluation and honest reporting; an uncertainty entry alone should not conceal a known offset.

$$y_{\text{corrected}} = y_{\text{indicated}} - b = y_{\text{indicated}} + C$$

Where $C = -b$ is the correction.

Corrections and residual uncertainty

Apply a significant estimated systematic correction when the model requires it and include the correction’s uncertainty. A correction changes the estimate; an uncertainty describes dispersion. Do not automatically add absolute bias to expanded uncertainty or replace a known correction with a guard band. If a result is intentionally reported uncorrected, use a documented method that accounts for the uncorrected effect and explains the reporting and decision consequences.

Linearity and Non-Linearity Evaluation Models

Concept of Linearity

Linearity is the property of a measuring system where the output indication is directly proportional to the input measurand across the entire specified operating range. In reality, all transducers exhibit physical departures from ideal proportionality due to material non-linearities, magnetic saturation, or geometric linkage kinematics.

Three Standard Linearity Models

Depending on the instrument standard (e.g., ASME, IEEE, ISA), non-linearity is calculated using one of three reference lines:

1. Endpoint (Terminal) Linearity:
   • The reference line connects the calibration point at zero (0% span) directly to the calibration point at full scale (100% span).
   • Maximum deviation $\Delta y_{\max}$ is evaluated against this fixed chord.
   • Trait: The line is fixed by its endpoints; noise there can strongly affect the residuals. It is not universally the largest possible linearity measure.
2. Best-Fit Straight Line (BFSL) / Independent Linearity:
   • The reference line is determined by linear least-squares regression across all calibration test points, optimizing both slope and intercept to minimize the sum of squared residuals:

$$\min \sum_{i=1}^N [y_i - (m x_i + c)]^2$$

• Trait: Least squares minimizes the sum of squared residuals, not the maximum absolute residual. A minimum-zone fit is a different optimization. No universal factor relates BFSL and endpoint linearity; use the datasheet’s definition.

1. Zero-Based Linearity:
   • The reference line is constrained to pass through the calibrated zero point ($c = 0$), with the slope optimized via least-squares across the remaining span.

Non-Linearity Calculation Formula

Non-linearity may be specified in engineering units, as a percentage of span or reading, or under another defined convention. Identify both the reference-line definition and denominator before comparing values.

$$\text{Non-Linearity (\% FS)} = \frac{\max\lvert y_{\text{actual}}-y_{\text{reference}}\rvert}{y_{\text{span}}} \times 100\%$$

---

Stability, Drift Rates, and Aging Physics

Metrological Stability (VIM 4.19)

Stability is the ability of a measuring instrument to maintain its metrological properties constant over time. It is quantified not as a single number, but by characterizing its inverse: drift.

Drift (VIM 4.21)

Drift is a continuous or incremental change of an indication over time, attributable to changes in the metrological properties of the measuring system, independent of any change in the measurand or external influence quantities.

• Zero Drift (Offset Drift): A parallel shift in the calibration curve where the zero intercept changes, but the slope remains constant. Caused by mechanical relaxation, thermal strain relief, or residual charge buildup.
• Span Drift (Gain Drift): A rotational change in the slope of the calibration curve where sensitivity changes over time. Caused by aging of reference zener diodes, magnet demagnetization, or spring alloy fatigue.

Physics of Aging in Standards

1. Zener voltage standards: References can change with aging, package stress, temperature history, and other influences. A logarithmic expression can be an empirical model for a particular history, but is not a universal law for all references. Validate any prediction and include residual drift uncertainty.
2. Liquid-in-Glass Thermometers: Glass bulbs undergo secular contraction over decades as amorphous glass molecules slowly settle into a relaxed amorphous structure, shrinking the bulb volume and inducing a progressive positive zero drift of several tenths of a degree Celsius.
3. Precision Standard Resistors: Manganin or Evanohm wire wound on ceramic bobbins experiences long-term resistance drift due to surface oxidation, moisture absorption, and microscopic wire work-hardening relief.

Optimizing Calibration Intervals via Drift Modeling

Historical calibration offsets can be plotted and modeled to estimate drift. Predictions carry uncertainty and may fail after shock, repair, or changed use. Review the interval using observed stability and risk; ILAC-G24/OIML D 10 describes review methods rather than guaranteeing a future limit crossing.

---

Hysteresis, Deadband, and Threshold

Hysteresis (VIM 4.20)

Hysteresis is the property of a measuring system whereby the output indication for a given input stimulus depends on the sequence of preceding input values—specifically, whether the operating point was approached from an increasing direction or a decreasing direction.

Physical Causes of Hysteresis

1. Mechanical Friction and Backlash: Play in gear teeth (such as the rack-and-pinion movement of dial calipers and Bourdon tube linkage pivots).
2. Viscoelastic Creep and Internal Molecular Friction: Non-ideal elasticity in metallic Bourdon tubes, diaphragm capsules, and load cell shear webs. When deformed, crystal grain boundaries undergo minute plastic shear, requiring finite relaxation time to return.
3. Magnetic Domain Wall Pinning (Barkhausen Effect): In electromagnetic cores (LVDTs, current transformers), magnetic domain walls become pinned on crystal dislocations, creating the classic B-H magnetic hysteresis loop.

Calibration Protocol for Hysteresis

To detect hysteresis, calibration procedures define suitable ascending and descending runs, for example at these five test levels (e.g., 0%, 25%, 50%, 75%, 100%, 75%, 50%, 25%, 0%):

$$H(x) = |y_{\text{descending}}(x) - y_{\text{ascending}}(x)|$$ $$\text{Maximum Hysteresis (\% FS)} = \frac{H_{\max}}{y_{\text{FS}}} \times 100\%$$

Operational Rule: Preserve the required approach direction when evaluating hysteresis. If a point is overshot, follow the method’s reset and re-approach instructions; do not silently substitute a descending observation for an ascending one.

Discrimination Threshold

The discrimination threshold describes the largest input change that still produces no detectable indication change under the stated conditions. Resolution instead concerns the smallest change producing a perceptible response. Noise, friction, input level, and how the change is applied can affect both. Distinguish this threshold from the bidirectional interval described by dead band. See VIM terminology.

Deadband

Deadband (dead zone) is the maximum interval through which an input quantity can be varied in both directions without producing a detectable change in the output indication.

Unlike hysteresis (where the output changes, but follows two separate curves), deadband represents a complete plateau of non-responsiveness, predominantly caused by mechanical clearance between intermeshing gears or software digital filtering algorithms.