Calibration Examples: Real Sensors and Instruments¶

This page is a reference guide to the calibration examples in this chapter. Each example shows how to apply static calibration, error analysis, and uncertainty budgeting to a real measurement instrument.

Quick Reference: Sensor Calibration Workflows¶

Instrument	Key Error	Notebook	Best For Learning
LVDT (Linear Variable Differential Transformer)	Linearity, hysteresis	calibration-lvdt_calibration_example.md or calibration-lvdt_calibration_2.md	Linear calibration curve, typical electronics
Pressure Transducer	Hysteresis, non-linearity	calibration-pressure_calibration_example.md or calibration-calibration_error_analysis_pressure.md	Real data, systematic errors, uncertainty propagation
Micrometer	Resolution, repeatability, calibration blocks	calibration-micrometer_calibration.md	Mechanical instruments, gage block standards
Orifice Plate (Flow measurement)	Non-linear response	calibration-pressure_calibration_example.md (orifice section)	Application-specific calibration
Weight Scale (Wheeler method)	Zero error, hysteresis	calibration-weight_scale_example_Wheeler.md	Historical method, mechanical devices

How to Use These Examples¶

Choose your instrument type — find it in the table above
Run the recommended notebook — execute cells and observe:
How reference standards are applied
Hysteresis checks (up-scale vs down-scale)
Regression fits and residual analysis
Error quantification and Type B uncertainty sources
Final calibration curve and uncertainty budget
Adapt to your own instrument — the same workflow applies to thermometers, load cells, rotational encoders, etc.

Common Patterns Across All Examples¶

Every calibration follows this structure:

1. Reference Standard & Setup¶

Choose a reference with lower uncertainty than your test instrument
Document traceability and uncertainty of the reference
Example: calibrated gage blocks for mechanical instruments, pressure standard for transducers

2. Data Collection¶

Apply known input values across the full operational range
At each point: wait for equilibrium, record multiple readings (assess repeatability)
Hysteresis check: measure on both increasing and decreasing input to detect path-dependent errors
Example: temperature chamber holding 5 set points, pressure pump at 10 pressures

3. Initial Analysis & Error Detection¶

Plot output vs. input
Check for: zero offset, non-linearity, hysteresis, outliers
Compute repeatability (Type A) as standard deviation of repeated measurements
Example: LVDT output should be linear; pressure transducer might show 2% non-linearity

4. Fit a Calibration Curve¶

Linear regression for linear instruments
Polynomial or logarithmic fit for non-linear instruments
Report: equation, \(R^2\), standard error of regression
Example: \(y = 0.502x + 2.3\), \(u_\text{regression} = 0.1\) V

5. Quantify Systematic Errors (Type B sources)¶

Hysteresis error (\(u_h\)): max difference between up/down curves
Linearity error (\(u_L\)): max deviation from best-fit line
Zero error (\(u_z\)): offset when input = 0
Sensitivity uncertainty (\(u_K\)): confidence in the slope
Example: hysteresis = ±0.3%, linearity = ±0.5%

6. Build Uncertainty Budget¶

Combine Type A (repeatability) and Type B (systematic) sources
Use root-sum-square (RSS): \(u_c = \sqrt{u_A^2 + u_h^2 + u_L^2 + u_z^2 + \dots}\)
Compute expanded uncertainty: \(U = k \cdot u_c\) (typically \(k=2\) for 95% confidence)
Example: \(u_c = \pm 0.7\%\), \(U = \pm 1.4\%\)

Example-Specific Notes¶

LVDT Calibration¶

Why LVDT first? Excellent linearity, no hysteresis, clean data → easiest to learn from
What to notice: How gage blocks establish reference standard, data collection under different loading scenarios
See: calibration-lvdt_calibration_example.md

Pressure Transducer¶

Why challenging? Non-linear response (often log-scale at low pressures), noticeable hysteresis
What to notice: How to fit polynomial instead of linear, identifying and quantifying hysteresis error
See: calibration-pressure_calibration_example.md

Micrometer¶

Why practical? Mechanical instrument students will use in labs; familiar uncertainty sources (gage block tolerance, surface finish)
What to notice: How resolution and repeatability limit the usable precision; concept of "last-digit uncertainty"
See: calibration-micrometer_calibration.md

Non-linear Relations (Orifice, Polynomial Fits)¶

Why included? Many real sensors don't follow \(y = mx + b\); logarithmic pressure transducers, non-linear thermal sensors, etc.
What to notice: How to choose model (polynomial order, log vs linear); trade-off between complexity and accuracy
See: calibration-calibration_non_linear_relations.md

Weight Scale (Wheeler Method)¶

Why historical? Wheeler method is elegant and still used for some mechanical instruments
What to notice: How mechanical hysteresis is characterized; pure Type B error quantification without statistical fitting
See: calibration-weight_scale_example_Wheeler.md

Tips for Your Own Instrument¶

Start simple: If your instrument looks linear, use linear regression (LVDT example)
Don't over-fit: Resist the urge to fit 5th-order polynomials. A simple model with clear uncertainty is better than a complex one that overfits noise
Always measure hysteresis: Even if you think there's none, check up-scale vs down-scale. It often reveals mechanism (friction, compliance, temperature lag)
Propagate uncertainty into use: Once you have the calibration curve and its uncertainty, use it to predict how much margin/confidence you have in future measurements
Document everything: Which standard? What temperature? What humidity? A calibration is only useful if the conditions are repeatable