Force Proving Instrument Calibration Calculator

Calculate the uncertainty of an increasing-force calibration of a force proving instrument following the method of EURAMET cg-4 version 3.0 (February 2022), section 6.1 and the Annex A worked example. The calculator fits a quadratic interpolation curve to the mean deflections, combines eight uncertainty components at every force step and returns the expanded uncertainty U in N and the relative expanded uncertainty W in percent. No class is assigned.

JavaScript is disabled, so the Calculate button cannot run. The full method, every formula and the input definitions are written out below so the calculation can be reproduced by hand or in a spreadsheet.

Calibration data

Step	Force F (kN)	X1 at 0° (mV/V)	X2 at 0° (mV/V)	X3 at 120° (mV/V)	X5 at 240° (mV/V)
1
2
3
4
5
6
7
8
9
10

Creep reading at maximum force after 30 s, i30 (mV/V)

Creep reading at maximum force after 300 s, i300 (mV/V)

Absolute change of zero reading, z (mV/V)

Indicator resolution, r (mV/V)

Expanded uncertainty of applied force at k = 2, P (%)

Temperature coefficient, T (%/K)

Temperature range, ΔT (K)

The fields are pre-filled with the EURAMET cg-4 Annex A example; replace them with your own readings. The example does not state the resolution, so r = 0.00001 mV/V is a test choice (one increment of the last recorded digit, no zero fluctuation assumed).

Results

a0 (mV/V)—

a1 (mV/V per kN)—

a2 (mV/V per kN²)—

F (kN)	Xr (mV/V)	Xwr (mV/V)	Xa (mV/V)	wc (%)	U (N)	W (%)

These uncertainties apply only to this calibration, for increasing forces, under the conditions entered above. They are evaluated at the measured force steps only; no class is assigned and no value outside the measured force interval is reported.

How to use the calculator

Choose kN or N for force. Values you typed are kept exactly when you switch units and back.
Enter at least three distinct positive force steps. For each step enter the increasing deflections X1 and X2 at the 0° position, X3 at 120° and X5 at 240°, all in mV/V and corrected for the starting zero. Leave unused rows empty.
Enter the creep readings i30 and i300 at maximum force, the absolute change of zero reading z, the indicator resolution r, the expanded uncertainty P of the applied force (k = 2), the temperature coefficient T and the temperature range ΔT.
Press Calculate to obtain the fit coefficients and, at every force, Xr, Xwr, Xa, wc, U and W.

Method

At each force F the mean deflection over the rotated positions and the mean over the unrotated positions are

Xr = (X1 + X3 + X5) / 3 Xwr = (X1 + X2) / 2

A second-degree interpolation curve Xa = a0 + a1·F + a2·F² is fitted to Xr by least squares with F in kN. XN is Xr at the maximum force. The relative standard uncertainty components, in percent, are

w1 = P / 2 applied calibration force w2 = 100 · √( Σ(Xi − Xr)² / 6 ) / |Xr|, i = 1, 3, 5 reproducibility w3 = 100 · |X2 − X1| / |Xwr| / √3 repeatability w4 = 100 · r / ( √6 · |Xr| ) resolution w5 = 100 · |i300 − i30| / ( |XN| · √3 ) creep w6 = 100 · z / |XN| zero drift w7 = T · ΔT / ( 2 · √3 ) temperature w8 = 100 · |Xa − Xr| / |Xr| interpolation

They combine as wc = √(w1² + w2² + … + w8²), and the combined standard uncertainty in newtons is uc = wc · F · 1000 / 100. A straight line is fitted to uc against F by least squares and truncated below at the smallest uc of the measured points. The expanded uncertainty is U = 2 · max(smallest uc, fitted uc) in N, and the relative expanded uncertainty is W = 100 · U / (F · 1000) in percent. All arithmetic is carried at full precision; only the displayed values are rounded.

Input definitions and checks

Resolution r is one increment of the least significant count plus half the range of fluctuation of the reading at zero force.
The guide gives no general range for r, P, T or ΔT, so these are only required to be non-negative, as is z.
All inputs must be finite numbers, forces must be positive and distinct, the means Xr and Xwr must be non-zero, and the quadratic fit must be non-singular.

Frequently asked questions

Which uncertainty contributions does this calculator combine?

Eight relative standard uncertainty components in percent: applied calibration force (w1), reproducibility (w2), repeatability (w3), resolution (w4), creep (w5), zero drift (w6), temperature (w7) and interpolation from the quadratic fit (w8). They are combined in quadrature into wc.

How is the expanded uncertainty U obtained at each force?

The combined standard uncertainty uc = wc × F × 1000 / 100 in N is fitted as a straight line against force by least squares. The line is truncated below at the smallest uc of the measured points, and U = 2 × max(smallest uc, fitted uc). W = 100 × U / (F × 1000) in percent.

What value should be entered for resolution r?

Resolution is one increment of the least significant count of the indicator plus half the range of fluctuation of the reading at zero force, in mV/V.

Does the calculator assign a class or extrapolate?

No. It reports uncertainty only at the measured force steps for increasing forces and does not assign a class or evaluate forces outside the measured force interval.

This calculator implements the increasing-force uncertainty evaluation described in EURAMET cg-4 version 3.0, section 6.1 and Annex A. It is an aid to calculation and does not replace the guide, an accredited calibration procedure or the judgement of the responsible calibration laboratory. Check results against your own quality system before issuing a certificate.