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Gauge block measurement (GUM H1)

An example from Appendix H1 of the GUM [1].

Code

.. literalinclude:: ../../examples/GUM_H1.py

Explanation

The measurand is the length of an end-gauge at 20\,^\circ\mathrm{C}. The measurement equation is [2]

l = l_\mathrm{s} + d - l_\mathrm{s}(\delta_\alpha \theta + \alpha_\mathrm{s} \delta_\theta) \;,

where

  • l_\mathrm{s} - the length of the standard
  • d - the difference in length between the standard and the end-gauge
  • \delta_\alpha - the difference between coefficients of thermal expansion for the standard and the end-gauge
  • \theta - the deviation in temperature from 20\,^\circ\mathrm{C}
  • \alpha_\mathrm{s} - the coefficient of thermal expansion for the standard
  • \delta_\theta - the temperature difference between the standard and the end-gauge

The calculation proceeds in stages. First, three inputs are defined:

  • the length difference measurement(d0) using the comparator, which is the arithmetic mean of several indications;
  • an estimate of comparator random errors (d1) and
  • an estimate of comparator systematic errors (d2).

These are used to define the intermediate result d

d0 = ureal(215,5.8,24,label='d0')
d1 = ureal(0.0,3.9,5,label='d1')
d2 = ureal(0.0,6.7,8,label='d2')

# Intermediate quantity 'd'
d = d0 + d1 + d2

Then terms are introduced to account for temperature variability and thermal properties of the gauge blocks.

In particular, the quantity \theta is defined in terms of two other input quantities

\theta = \bar{\theta} + \Delta

where

  • \bar{\theta} is the mean deviation of the test-bed temperature from 20\,^\circ\mathrm{C}
  • \Delta is a cyclical error in the test-bed temperature

In defining these inputs, functions :func:`type_b.uniform` and :func:`type_b.arcsine` convert the widths of particular error distributions into standard uncertainties [3].

alpha_s = ureal( 11.5E-6, type_b.uniform(2E-6), label='alpha_s' )
d_alpha = ureal(0.0, type_b.uniform(1E-6), 50, label='d_alpha')
d_theta = ureal(0.0, type_b.uniform(0.05), 2, label='d_theta')

theta_bar = ureal(-0.1,0.2, label='theta_bar')
Delta = ureal(0.0, type_b.arcsine(0.5), label='Delta' )

# Intermediate quantity 'theta'
theta = theta_bar + Delta

The length of the standard gauge block is given in a calibration report

l_s = ureal(5.0000623E7,25,18,label='ls')

two more intermediate results, representing thermal errors, are then

# two more intermediate steps
tmp1 = l_s * d_alpha * theta
tmp2 = l_s * alpha_s * d_theta

Finally, the length of the gauge block is evaluated

# Final equation for the measurement result
l = result( l_s + d - (tmp1 + tmp2), label='l')

The script then evaluates the measurement result

print( "Measurement result for l={}".format(l) )

which displays

Measurement result for l=50000838(32)

and the following commands display the components of uncertainty for l, due to each influence:

print("""
Components of uncertainty in l (nm)
-----------------------------------""")

for l_i,u_i in reporting.budget(l):
    print( "  {!s}: {:G}".format(l_i,u_i) )

The output is

Components of uncertainty in l (nm)
-----------------------------------
  ls: 25
  d_theta: 16.599
  d2: 6.7
  d0: 5.8
  d1: 3.9
  d_alpha: 2.88679
  alpha_s: 0
  theta_bar: 0
  Delta: 0

Footnotes

[1]BIPM and IEC and IFCC and ISO and IUPAC and IUPAP and OIML, Evaluation of measurement data - Guide to the expression of uncertainty in measurement JCGM 100:2008 (GUM 1995 with minor corrections), (2008) http://www.bipm.org/en/publications/guides/gum
[2]In fact, the GUM uses more terms to calculate the uncertainty than are defined: quantities d and \theta depend on more than one influence quantity.
[3]ureal creates a new uncertain real number. It takes a standard uncertainty as its second argument.