Calibration — Introduction and Learning Goals¶

Short summary Calibration methods and error analysis for sensors: linear & nonlinear regression, hysteresis, sensitivity, and full calibration workflow.

Learning objectives - Perform linear regression for sensor calibration and compute confidence intervals. - Analyze hysteresis and nonlinearity errors. - Design calibration experiments and propagate calibration uncertainty into measurements.

Key concepts (brief) - Regression residuals, standard error, and calibration curve interpretation. - Hysteresis and repeatability characterization. - Sensitivity analysis and combining calibration with measurement uncertainty.

Prerequisites Basic regression, statistics, and familiarity with plotting in Python.

Ordered reading (suggested)¶

Follow this sequence to learn calibration from theory to practice. The order moves from foundational regression concepts, through systematic error characterization, to applied examples on real instruments.

Simulation of Calibration Errors — see how bias, noise, and hysteresis show up in a calibration curve.
Regression Analysis with Uncertainty — detailed regression analysis with uncertainty in slope and intercept.
Estimate \(a,b\) and also \(\Delta a\) and \(\Delta b\) — linear fit parameters and their uncertainties.
Sensitivity estimate example — how to estimate sensitivity coefficients from calibration data.
Sensitivity error example — identifying and quantifying hysteresis as a Type B uncertainty source.
Linearity error example — deviation from a straight line, quantified.
Hysteresis example — upscale vs downscale differences.
Hysteresis and regression analysis example — pressure-sensor example combining both.
Calibration of non-linear relations — polynomial and logarithmic fits when linearity fails.
Calibration of non-linear (logarithmic) function — log-log calibration workflow.
Weighing Scale Calibration (Wheeler and Ganji) — textbook scale example.
GUM Budgets: Orifice Flow Meter and Storage Tank — full GUM budgets for flow and volume.
Micrometer calibration using gage block — length calibration against a standard.
LVDT Calibration: Linearity, Hysteresis and Repeatability — LVDT worked example.
LVDT Calibration Curve and Its Uncertainty — LVDT curve with uncertainty bands.
Calibration and Uncertainty: Virtual Experiment, Abridged — short virtual pressure-calibration lab.
Calibration and Uncertainty: Virtual Experiment — full virtual experiment with complete analysis.
Uncertainty Budget Examples (SWGDRUG SD-3) — additional budget examples.
Calibration Examples: Real Sensors and Instruments — reference guide with links to all sensor calibration notebooks (LVDT, pressure, micrometer, orifice, weight scale). Choose examples matching your lab instruments.
Interactive Calibration Simulation — hands-on exploration of calibration errors.

Rationale: Students first master linear regression theory, then learn to identify and quantify systematic errors from the calibration process (hysteresis, linearity, repeatability). The consolidated examples guide at the end shows how to apply all these concepts to real sensors.

The Static Calibration Process¶

Static calibration is a procedure used to characterize a measurement system by applying a known, constant input value to the system and observing the resulting output value. The term "static" refers to the fact that the variables are held constant, meaning the input is not dependent on time.

Purpose¶

The primary purpose of static calibration is twofold:

Functional Relationship: To develop a functional relationship, or a correlation (\(y = f(x)\)), between the system's known input (\(x\)) and its output (\(y\)). This correlation is often determined using curve fitting techniques, such as regression analysis, on the calibration curve.
Systematic Error Identification: To identify and quantify systematic errors (biases) such as zero error, linearity error, and hysteresis.

Procedure Steps¶

To perform a static calibration, you generate a calibration curve by recording the output values (\(y_i\)) for a range of known input values (\(x_i\)).

Reference Standard: A standard (known value) that is traceable to national standards must be used. This reference instrument should have significantly lower uncertainty than the instrument being tested.
Controlled Points: The chamber or environment is set to a series of stable, discrete input points across the operational range.
Hysteresis Check: Measurements should be taken by varying the input value in both the increasing (upscale) and decreasing (downscale) directions to assess hysteresis.
Data Collection: At each stable point, the system is allowed to reach thermal equilibrium, and multiple readings are taken to assess repeatability.
Data Analysis: The data is plotted to create the calibration curve. This curve is used to determine key parameters like static sensitivity (the slope of the curve), and to quantify errors like linearity and hysteresis.

Error Analysis and Uncertainty Categorization¶

In metrology, an error is the quantifiable difference between an estimate and the "true value," which can sometimes be corrected. Uncertainty is the quantification of the doubt about that measurement result.

Systematic errors found during calibration, such as zero offset, can sometimes be corrected by adjustment. If the error cannot be corrected (e.g., hysteresis, remaining linearity deviation), the quantified range of that potential error must be included in the uncertainty budget as a Type B contribution.

The table below outlines common error types, how they are determined, and their subsequent role in uncertainty analysis:

Error Type	Description & Measurement Method	Systematic Error Correction?	Uncertainty Type/Source
Repeatability Error (\(u_{rep}\))	The variability or scatter found when the same input is applied repeatedly. Accounts for random variations. Measured: Calculated as the standard deviation (\(S_x\)) from repeated readings taken during the static calibration procedure.	No. This is a source of random error.	Type A. Calculated as the standard deviation of the mean (\(S_x / \sqrt{n}\)).
Static Sensitivity (\(K\))	The slope of the calibration curve at any given point. Measured: Defined as \(K(x_1) = (dy/dx)_{x=x_1}\).	Used to define the functional relationship for converting output (\(y\)) back to input (\(x\)).	Type B (The Sensitivity Error (\(u_K\)) is a statistical measure of random error in the estimate of the slope).
Zero Error (\(u_z\))	A vertical shift or offset from the true zero point of the instrument. Measured: Determined by checking the output when a zero input condition is applied.	Yes. If possible, the error is reduced by physically adjusting the output under a zero input condition.	Type B (The remaining, uncorrected zero uncertainty, often related to resolution).
Linearity Error (\(u_L\))	The deviation of the calibration curve from an expected straight line. Measured: Calculated as the maximum deviation between the actual output \(y(x)\) and the best-fit linear output \(y_L(x)\).	If the deviation is large, the correction may involve using a higher-order polynomial fit, which models the error.	Type B (If a linear fit is required, the linearity error bounds define a semi-range limit for a Type B rectangular distribution).
Hysteresis Error (\(u_h\))	The maximum difference in output value when the input is approached from increasing versus decreasing directions. Caused by friction or residual charge. Measured: Calculated as the maximum difference between upscale and downscale readings at the same input point across the full range.	No. Typically quantified, but not corrected. It is a systematic bias that must be factored into the uncertainty estimate.	Type B (The maximum hysteresis error defines the limit for a Type B uncertainty estimate, often expressed as a percentage of the full-scale output range).
Overall Instrument Error (\(u_c\))	The total quantified uncertainty of the instrument due to known effects (e.g., combining \(u_h\), \(u_L\), \(u_K\)). Measured: Combined using the Root-Sum-Squared (RSS) method: \(u_c = \sqrt{u_h^2 + u_L^2 + u_K^2 + \dots}\).	If the calibration shows a consistent offset or non-linearity, a correction can be applied to future readings.	Combined Standard Uncertainty (\(u_c\)).

What comes next?¶

Calibration tells you how to characterize and reduce uncertainty in your instruments — but instruments exist in the real world, where dynamic effects, noise, and signal acquisition choices matter. The Dynamic Signals chapter teaches you how measurement systems actually respond to inputs, and Signal Processing shows you frequency-domain tools to separate true signals from noise.