Sensor error budget builder

How it works

A sensor channel is only as accurate as the sum of its parts: the sensor, the amplifier, the reference and the converter. This tool lists each error source in the units its datasheet uses, converts them all to one basis at the reading you care about, and combines them as a worst case, by root sum square and as an expanded uncertainty, before and after calibration, against your accuracy target.

1. Convert every source to output units. Each source is converted to the units of the output at your chosen reading and temperature span. Percent of full scale and of reading scale with the full scale or the reading. Temperature coefficients are multiplied by the temperature span from the calibration temperature. Noise given as a density is multiplied by the square root of the noise bandwidth. One LSB is the ADC range divided by two to the power of the bits.% of FS: v / 100 x FS; % of reading: v / 100 x R; ppm per K of FS: v x 1e−6 x FS x dT; ppm per K of reading: v x 1e−6 x R x dT; LSB: v x range / 2^bits; density: v x sqrt(bandwidth)
2. Standard uncertainty of each source. A limit that is equally likely anywhere inside it is a bounded uniform source, and its standard uncertainty is the half-width divided by the square root of 3. A triangular limit divides by the square root of 6. A normal source is entered as one standard deviation, or as a limit at 2 or 3 standard deviations, and is divided to give one. This is the Type B evaluation of the Guide to the Expression of Uncertainty in Measurement.uniform: u = a / sqrt(3); triangular: u = a / sqrt(6); normal as 1 sd: u = s; normal limit at k sd: u = a / k
3. Worst case, root sum square and expanded uncertainty. The worst case adds every limit, as if all sources sit at their limits with the same sign. The root sum square of the limits adds them in quadrature, which assumes independent sources and is the usual datasheet shortcut. The combined standard uncertainty is the root sum square of the standard uncertainties, and the expanded uncertainty multiplies it by the coverage factor, 2 for about 95% coverage if the result is close to normal. A normal source takes its peak factor times its standard deviation as its worst-case limit.worst = sum of limits; RSS = sqrt(sum of limits^2); u_c = sqrt(sum of u^2); U = k x u_c
4. Before and after calibration. A source marked as calibrated out is removed in the after column, except for the share you say remains. The uncertainty of the calibration reference is added back as a source of its own, since a calibration can only be as good as what it is calibrated against. Offset, gain and, with enough points, non-linearity and temperature drift can be calibrated; noise, quantisation, hysteresis and long-term drift cannot.after calibration: calibrated sources x remaining share, plus the calibration reference (entered at k = 2, so u = limit / 2)
5. Margin against the target. The target is a share of full scale or of the reading. The margin is the target minus each combined figure in output units, shown as a pass or fail for the worst case, the root sum square and the expanded uncertainty. Contributions are sorted by size, with their share of the variance, so the largest sources to fix are first.margin = target − combined error; share of variance = u^2 / sum of u^2
6. What if the reading or temperature changes. Errors given as a share of full scale are a fixed amount, so they weigh more as a share of a small reading. The reading table shows the error as a share of the reading from 5% to 100% of full scale. The temperature table shows the expanded uncertainty as the temperature span grows from the calibration temperature.error as % of reading = combined error / reading x 100

Sources

The methods and default figures rest on these public sources. The wording here is our own; read the source for its full text.

• Evaluation of measurement data: Guide to the expression of uncertainty in measurement (JCGM 100:2008) (opens in a new tab), Joint Committee for Guides in Metrology (JCGM), published by the BIPM, September 2008. Type B evaluation with a rectangular distribution, combined standard uncertainty by root sum square, coverage factor and expanded uncertainty. Free to download from the BIPM.
• The ABCs of Analog-to-Digital Converters: How ADC Errors Affect System Performance (Tutorial 748) (opens in a new tab), Maxim Integrated, now Analog Devices, 22 July 2002. Root sum square against worst case, and how offset, gain, non-linearity, reference accuracy and temperature effects add up in a converter system.
• Voltage Reference Selection and Design Tips for Data Converters (SNAA320B) (opens in a new tab), Texas Instruments, January 2024. Converting error terms to ppm, combining them worst case and by root sum square, and the effect of calibration on a reference error budget.