RP-02 Laboratory: Determination of Mass Calibration Values and Related Uncertainties

N. Dupuis-Désormeaux, Senior Engineer, Gravimetry
May 2002

RP-02: Determination of Mass Calibration Values and Related Uncertainties, in PDF format 452 KB

Table of Contents

1.1 Calibration and Uncertainty Equations: Single Comparison

1.1.1 Absolute Mass Calibration
1.1.1.1 Mass Equation
1.1.1.2 Combined Uncertainty Equation for Absolute Mass
1.1.2 Conventional Mass Calibration
1.1.2.1 Mass Equation
1.1.2.2 Combined Uncertainty Equation for Conventional Mass

1.2 Calibration and Uncertainty Equations: Series of Intercomparisons

1.2.1 Absolute Mass Calibration
1.2.1.1 Mass Equation
1.2.1.2 Combined Uncertainty Equation for Absolute Mass
1.2.2 Conventional Mass Calibration
1.2.2.1 Mass Equation
1.2.2.2 Combined Uncertainty Equation for Conventional Mass

2. Components of Uncertainty

2.1 Uncertainty of the Reference or Lead Weight

2.1.1 Calibration of the CSL Working Standards
2.1.1.1 Absolute Mass
2.1.1.2 Conventional Mass
2.1.2 Calibration of District Standards and others
2.1.2.1 Absolute Mass
2.1.2.2 Conventional Mass
2.1.3 Tolerance Testing of Inspection Weights

2.2 Uncertainty of Air Density

2.3 Uncertainty of Density of Reference and of Weight Being Calibrated

2.3.1 CSL Working Standards
2.3.2 District Standards
2.3.3 Inspector Weight Kits (field working standards)

2.4 Uncertainty of the Process

2.4.1 Single Comparison and Tolerance Testing
2.4.2 Series of Intercomparisons

3. Uncertainty Multipliers

3.1 Density of reference and density of weight being calibrated

3.1.1 CSL Working Standards
3.1.2 District Standards
3.1.3 Inspector Weight Kits

3.2 Air Density

3.3 Conventional Air Density

3.4 Conventional Mass Density

3.5 Coefficients CM’P and CM’Q

3.6 Coefficients CijP and CijQ

4. Summary - At a Glance

5. References

Background

1.0 Definitions and Calibration Methods

1.1 Calibration and Uncertainty Equations: Single Comparison

1.2 Calibration and Uncertainty Equations: Series of Intercomparisons

The importance of quantifying measurement uncertainty has grown steadily with the pace of industrialization that followed the second World War. However, most companies and even most governments developed their own methods for calculating and expressing uncertainty without much consideration to international harmonization. In 1977, the Comité International de Poids et Mesures (CIPM) requested that the Bureau International de Poids et Mesures (BIPM) develop a document to harmonize methods of expression of uncertainty. In 1980, a BIPM work group developed the International Recommendation INC-1: Expression of Experimental Uncertainties. It was then decided that the responsibility for developing a detailed guide on how to express uncertainties according to INC-1 should be transferred to the International Standardization Organization (ISO). The BIPM/ISO Guide to the Expression of Uncertainty in Measurement, was first published in 1995 as ISBN 92-67-10188-9.

The methods and calculations proposed herein are in accordance with the Guide to the Expression of Uncertainty in Measurement, 1995. Please refer to the above document for any background information.

1.0 Definitions and Calibration Methods

Definitions

Absolute mass, X_abs

Absolute mass is obtained when weighing in vacuum. It is an intrinsic property of an artifact and is independent of gravity or buoyancy.

Effective mass, X_e = X_abs (1 - (ρ_at/ρ_X))

Effective mass is the absolute mass times the buoyancy factor (in parenthesis above). In other words, effective mass is less than the absolute mass by the mass of air that it displaces. (Note: ρ_at is the air density and ρ_x is the material density of the mass).

Conventional mass, X_c = X_abs (1 - (ρ_ac/ρ_X)) / (1 - (ρ_ac/ρ_c))

The conventional mass is by definition: “the mass X_c that a standard of density ρ_c equal to 8000 kg/m³ would require to have to balance a given mass X_abs of density ρ_x when this measurement is performed in an air density ρ_ac of 1.2 kg/m³ and a temperature of 20°C.” In other words:

X_c - ρ_ac v_c X_abs - ρ_ac v_x
X_c (1-(ρ_ac/ρ_c)) X_abs (1-(ρ_ac/ρ_x))
X_c X_abs (1-(ρ_ac/ρ_x)) / (1-(ρ_ac/ρ_c))

OIML R111(2001) definition 2.6 - Conventional Mass: For a weight taken at 20°C, the conventional mass is the mass of a reference weight of a density of 8 000 kg/m³ which it balances in air of a density of 1.2 kg/m³.

Please note that in the past, the term “apparent mass” was used to describe “conventional mass”.

Calibration methods used at the CSL

Single Comparison

Single comparison calibration refers to a method where the test mass and the reference are only compared once. This comparison is effected by using the substitution method?.

The difference in indications obtained with the substitution weighing does not represent the mass difference between the test mass and the reference. The following equations 1.1.1.1 and 1.1.2.1 show how the indicated difference is used to compute the associated mass difference.

Series of Intercomparisons

A calibration with series of intercomparisons refers to a situation where multiple masses are intercompared. Each comparison is performed by using the substitution method? on a group of masses. It is based on the method of Least Squares.

The absolute mass of a weight is therefore obtained by using 1.2.1.1, and its conventional mass is obtained by using 1.2.2.1.

• Substitution weighing is also called differential weighing or comparison weighing and refers to the procedure where a standard and a test mass are weighed consecutively on the same load receiving element and the difference in their indicated value forms the basis for the calibration.

The results obtained with single comparison weighing will have larger uncertainties than those obtained using series of intercomparisons. However, series of intercomparisons are much more time-consuming. Therefore, the choice of which method to use depends on if the limits of error (accuracy class) of the mass being calibrated can be met by performing single comparison weighing.

Single Comparison Weighing can be used instead of tolerance testing of the inspection weights. Series of Intercomparisons are used to calibrate masses with smaller limits of error (higher accuracy class) such as the CSL Working Standards and the District Standards.

1.1 Calibration and Uncertainty Equations: Single Comparison

- Example - Client’s Weight or Special Calibration of Inspection Weight

1.1.1 Absolute Mass Calibration

1.1.1.1 Mass Equation

When performing comparison weighing in effective mass, the following calibration equation holds:

m_t - ρ_at v_t = m_R - ρ_at v_R + D_t

v_R = volume of reference (standard mass) R
v_t = volume of unknown mass t

Since v_t = m_t /ρ_t and v_R = m_R /ρ_R , the above can be expressed as:

(1.1.1.1)

m_t = [ m_R (1 - (ρ_at/ρ_R)) + D_t ] / ( 1 - (ρ_at/ρ_t))

where D_t = corrected value for the difference between displayed value for unknown mass t and displayed value for reference (standard mass) R. D_t = SR(I_t - I_R), where SR is the sensitivity reciprocal at that nominal value, It is the indicated value for the test mass t and I_R is the indicated value for the reference (standard mass) R
m_t = absolute mass of unknown mass t
m_R = absolute mass of reference R - found on calibration certificate for reference R
ρ_t = density of unknown mass t (kg/m³) - usually provided by manufacturer; otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111 (1994) or Table 6 (2002) and knowing the material. See section 3.1
ρ_R = density of the reference (standard mass) R (kg/m³) - usually found on calibration certificate for reference (standard mass) R; otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111 (1994) or Table 6 (2002) and knowing the material . See section 3.1
ρ_at = air density at the time of the calibration (kg/m³) - the formula provided in section 3.2 can be used

1.1.1.2 Combined Uncertainty Equation for Absolute Mass

Note that rearranging the first equation in 1.1.1.1 above yields:

m_t = m_R - ρ_at v_R + ρ_at v_t + D_t

Pose f = m_t = m_R + ρ_at (v_t - v_R) + D_t

The function f is f(m_R, ρ_at, v_t, v_R, D_t). In other words: f is a function of m_R , ρ_at , v_t , v_R and D_t. By the Law of Propagation of Uncertainty (see section 5.1.2 of the ISO Guide to the Expression of Uncertainty in Measurement) and assuming no covariance, we have:

(u_combined(m_t))² =
(∂f/∂m_R(u(m_R)))² + (∂f/∂ρ_at (u(ρ_at)))² + (∂f/∂v_t (u(v_t)))² + (∂f/∂v_R (u(v_R)))² + (∂f/∂D_t(u(D_t)))²

Since ∂f/∂m_R ≈ 1
∂f/∂v_R ≈ -ρ_at
∂f/∂v_t ≈ ρ_at
∂f/∂ρ_at ≈ (v_t - v_R)
∂f/∂D_t ≈ 1

Please note that using the equation:

m_t (1-(ρ_at/ρ_t)) = (m_R - m_Rρ_at/ρ_R + D_t)
m_t = {1/(1-(ρ_at/ρ_t))} {m_R - m_Rρ_at/ρ_R + D_t} instead of equation “f” above, we have the following derivatives without approximation:

∂f/∂m_R = {ρ_t /(ρ_t -ρ_at)}{(ρ_R -ρ_at)/ρ_R}
∂f/∂D_t = {ρ_t /(ρ_t -ρ_at)}{1}
∂f/∂ρ_R = {ρ_t /(ρ_t -ρ_at)}{- ρ_at m_R/ρ_R²}
∂f/∂ρ_t = {ρ_t /(ρ_t -ρ_at)}{- ρ_at m_t/ρ_t²}
∂f/∂ρ_at = {ρ_t /(ρ_t -ρ_at)}{(m_R/ρ_R)((ρ_R -ρ_t)/(ρ_t -ρ_at) + D_t /(ρ_t -ρ_at)}

we have:

(u_combined(m_t))² ≈ [1(u(m_R))]²+[(v_t-v_R)(u(ρ_at))]²+ [ρ_at(u(v_t))]² + [-ρ_at(u(v_R))]² + [1(u(D_t))]²
≈ [u(m_R)]²+[(m_t/ρ_t - m_R/ρ_R)(u(ρ_at))]²+[ρ_at(u(v_t))]²+[-ρ_at(u(v_R))]₂+[u(D_t)]²

Note that, again by the Law of Propagation of Uncertainty, we obtain

(u(v_t))² = u²(m_t/ρ_t) = (∂(v_t)/∂m_t)² u²(m_t) + (∂(v_t)/∂ρ_t)² u²(ρ_t) = (1/ρ_t)² u²(m_t) +(-m_t/ρ_t²)² u²(ρ_t).

However, if the density determination is performed independently from the mass determination, we have: (u(v_t))² ≈ (∂(v_t)/∂ρ_t)² u²(ρ_t) ≈ (-m_t/ρ_t²)² u²(ρ_t) and similarly (u(v_R))² ≈ (-m_R/ρ_R²)² u²(ρ_R).

Hence,

(u_combined(m_t))² ≈ [1(u(m_R))]²+[(v_t-v_R)(u(ρ_at))]²+ [ρ_at(u(v_t))]² + [-ρ_at(u(v_R))]² + [1(u(D_t))]²
≈ [u(m_R)]² + [(m_t/ρ_t - m_R/ρ_R)× u(ρ_at)]² + [ρ_at(-m_t/ρ_t²)× u(ρ_t)]² + [-ρ_at(-m_R/ρ_R²)× u(ρ_R)]²+[u(D_t)]²

u² (m_t) ≈ u²(m_R) + (m_t/ρ_t-m_R/ρ_R)²u²(ρ_at) + ρ_at²(-m_t/ρ_t²)²u²(ρ_t) + ρ_at² (-m_R/ρ_R²)² u²(ρ_R) + u²(D_t)

Since ρ_at² (-m_R/ρ_R²)² u²(ρ_R) should already be included in u²(m_R), the combined standard uncertainty for the absolute mass mt is the square root of the following:

(1.1.1.2)

u² (m_t) ≈ u²(m_R) + (m_t/ρ_t-m_R/ρ_R)²u²(ρ_at) + ρ_at²(-m_t/ρ_t²)²u²(ρ_t) + u²(D_t)

where all terms are expressed in the same unit of mass

The u²(m_R) term used in the above equation should already contain the term ρ_at² (-m_R/ρ_R²)² u²(ρ_R) .

Please note that equation (1.1.1.2) without approximation is:

u²(m_t) = [ρ_t /(ρ_t-ρ_at)]² { u²(D_t) + [(ρ_R-ρ_at)/ρ_R]² u²(m_R) + [m_Rρ_at/(ρ_R)²]² u²(ρ_R)
+ [-m_tρ_at/(ρ_t)²]² u²(ρ_t) + [(m_R/ρ_R)(ρ_R-ρ_t)/(ρ_t-ρ_at) + D_t/(ρ_t-ρ_at)]² u²(ρ_at) }

where all terms are expressed in the same unit of mass

where

u(m_t) = combined standard uncertainty associated with the evaluation of the absolute mass of t
u(m_R) = combined standard uncertainty associated with the absolute mass of reference R - usually found on the calibration certificate for reference R. See section 2.1
u(ρ_at) = standard uncertainty associated with the estimation of the air density at the time of the calibration (kg/m³). See section 2.2
u(ρ_t) = standard uncertainty associated with the density of the unknown mass t (kg/m³) - usually provided by the manufacturer of mass t. A rectangular distribution estimate based on the limits provided in Table 3 of OIML Recommendation R111 (1994) can be used; i.e., ± (1340 kg/m³ ÷ 2% 3) = ± 380 (kg/m³) for Class F1, and a value of ± (4300 kg/m³ ÷ 2% 3) = ± 1240 (kg/m³) for Class F2. See section 2.3.
u(ρ_R) = standard uncertainty associated with the density of the reference (standard mass) R (kg/m³)) - usually included in u(m_R) or provided on the calibration certificate for reference R. A rectangular distribution estimate based on the limits provided in Table 3 of OIML Recommendation R111 (1994) can be used; e.g. , ± (400 kg/m³ ÷ 2% 3) = ± 115 (kg/m³) for Class E2. See section 2.3.
u(D_t) = standard process uncertainty associated with the calculated difference - the long-term process uncertainty of the procedure and equipment used in determining the difference between mass t and reference R. See section 2.4
m_t = absolute mass of unknown mass t
m_R = absolute mass of reference (standard mass) R
ρ_t = density of the unknown mass t (kg/m³) - usually provided by manufacturer; otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111 (1994) and knowing the material. See section 3.1
ρ_R = density of the reference R (kg/m³) - usually found on calibration certificate for reference R; otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111 (1994) and knowing the material. See section 3.1
ρ_at = air density at the time of the calibration (kg/m³) - the formula provided in section 3.2 can be used.

Note: The terms involved in the uncertainty equation 1.1.1.2 are described in detail in the following sections 2 and 3.

1.1.2 Conventional Mass Calibration

1.1.2.1 Mass Equation

The conversion from absolute to conventional mass is done with the definition below:

The conventional mass is “the mass m_ct that a standard of density ρ_c equal to 8000 kg/m³ would require to have to balance a given mass m_t of density ρ_t when this measurement is performed in an air density ρ_ac of 1.2 kg/m³ and a temperature of 20°C.” In other words:

m_ct - ρ_ac v_ct m_t - ρ_ac v_t
m_ct (1-(ρ_ac/ρ_c)) m_t (1-(ρ_ac/ρ_t))
m_ct (1- (ρ_ac/ρ_c)) / (1-(ρ_ac/ρ_t)) m_t

v_ct = volume of unknown mass t with a conventional density of 8000 kg/m³

Note that v_ct is equal to m_ct/ρ_ct and that ρ_ct = ρ_c. By rearranging the first expression in the definition, we get: m_ct - ρ_ac v_ct + ρ_ac v_t m_t. Similarly, m_cR - ρ_ac v_cR m_R - ρ_ac v_R can be rearranged as m_cR - ρ_ac v_cR + ρ_ac v_R m_R, or can be written as m_cR (1-(ρ_ac/ρ_cR)) m_R (1-(ρ_ac/ρ_R)).
v_cR = volume of reference (standard mass) R with a conventional density of 8000 kg/m³

Replacing m_t and m_R by their equivalence in conventional mass (using the last relationship in the box above) and inserting into f = m_t = m_R + ρ_at (v_t - v_R) + D_t (see section 1.1.1.1),
v_t = volume of unknown mass t with a density ρ_t
v_R = volume of reference R with a density ρ_R

we get:

m_t - ρ_at v_t = m_R - ρ_at v_R + D_t
[m_t] (1 - (ρ_at/ρ_t)) = m_R (1 - (ρ_at/ρ_R)) + D_t
[m_ct (1- (ρ_ac/ρ_c)) / (1-(ρ_ac/ρ_t))] (1 - (ρ_at/ρ_t)) = [m_cR (1 - (ρ_ac/ρ_c)) / (1- (ρ_ac/ρ_R))] (1 - (ρ_at/ρ_R)) + D_t
(m_ct (ρ_c - ρ_ac) (ρ_t - ρ_at) / ρ_c (ρ_t - ρ_ac)) = (m_cR (ρ_c - ρ_ac) (ρ_R - ρ_at) / ρ_c (ρ_R - ρ_ac)) + D_t
m_ct = ( (ρ_c - ρ_ac) (ρ_t - ρ_at) / ρ_c (ρ_t - ρ_ac) ) ^-1{ [m_cR (ρ_c - ρ_ac) (ρ_R - ρ_at) / ρ_c (ρ_R - ρ_ac)] + D_t }
m_ct = m_cR (ρ_R - ρ_at) (ρ_t - ρ_ac) / (ρ_t - ρ_at)(ρ_R - ρ_ac) + D_t ρ_c (ρ_t - ρ_ac) / (ρ_t - ρ_at)( ρ_c - ρ_ac)
m_ct = ( (1 - ρ_ac/ρ_t ) / (1 - ρ_at/ρ_t )) {(m_cR (1 - ρ_at/ρ_R ) / (1 - ρ_ac/ρ_R )) + D_t / (1 - ρ_ac/ρ_c )}

Therefore:

(1.1.2.1 A)

m_ct = ((ρ_t - ρ_ac) / (ρ_t - ρ_at)) { (m_cR (ρ_R - ρ_at)/(ρ_R - ρ_ac)) + (D_t ρ_c / ( ρ_c - ρ_ac)) }

where m_ct , m_cR and D_t are expressed in the same unit of mass

where

m_ct = conventional mass of unknown mass t
ρ_ac= conventional air density; equal by definition to 1.2 kg/m³. See section 3.3
m_t,R = absolute mass of object t, or reference R
ρ_ct = ρ_cR = ρ_c = conventional mass density; equal by definition to 8000 kg/m³. See section 3.4
ρ_at = air density at the time of the calibration (kg/m³) - the formula provided in section 3.2 can be used.
ρ_t = density of the unknown mass t (kg/m³) - provided by manufacturer; otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111 (1994) and knowing the material . See section 3.1
ρ_R = density of the reference R (kg/m³) - usually found on calibration certificate for reference R; otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111 (1994) and knowing the material. See section 3.1
m_cR = conventional mass of reference R
D_t = calculated difference between unknown mass t and reference R. See section 1.1.1
= equal by definition

Writing the above expression (1.1.2.1 A) in terms of m_R instead m_cR , we get:

m_t - ρ_at (v_t) = m_R - ρ_at (v_R) + D_t
[m_t] (1 - (ρ_at /ρ_t)) = m_R (1 - (ρ_at /ρ_R)) + D_t
[m_ct (1 - (ρ_ac/ρ_c)) / (1- (ρ_ac/ρ_t))] (1 - (ρ_at /ρ_t)) = m_R (1 - (ρ_at /ρ_R)) + D_t
m_ct ((ρ_c - ρ_ac) (ρ_t - ρ_at)) / (ρ_c (ρ_t - ρ_ac)) = m_R (1 - (ρ_at /ρ_R)) + D_t
m_ct = [(ρ_c (ρ_t - ρ_ac)) / ((ρ_c - ρ_ac) (ρ_t - ρ_at)) ] {m_R (1 - (ρ_at /ρ_R)) + D_t }

(1.1.2.1 B)

m_ct = [(ρ_c (ρ_t - ρ_ac)) / ((ρ_c - ρ_ac) (ρ_t - ρ_at)) ] {m_R (1 - (ρ_at /ρ_R)) + D_t }

where m_ct, m_R and D_t are expressed in the same unit of mass

where

m_ct = conventional mass of unknown mass t
ρ_ac = conventional air density; equal by definition to 1.2 kg/m³. See section 3.3
v_ct = volume of unknown mass t with a conventional density of 8000 kg/m³
m_t = absolute mass of object t
v_t = volume of unknown mass t with a density ρt
ρ_ct = ρ_cR = ρ_c = conventional mass density; equal by definition to 8000 kg/m³. See section 3.4
ρ_t = density of the object t. See section 3.1
ρ_at = air density at the time of the calibration (kg/m³) - see formula provided in section 3.2
m_R = absolute mass of reference R
ρ_R = density of the reference R. See section 3.1
v_R = volume of reference R with a density ρ_R
D_t = calculated difference between unknown mass t and reference R. See section 1.1.1

1.1.2.2 Combined Uncertainty Equation for Conventional Mass

First, let us look at equation (1.1.2.1 A).

Replacing m_t and m_R by their equivalence in conventional mass and inserting into the equation f = m_t = m_R + ρ_at (v_t - v_R) + D_t (see section 1.1.1.2), we get:

[m_t] = [m_R] + ρ_at (v_t - v_R) + D_t
[m_ct - m_ct (ρ_ac/ρ_c) + ρ_acv_t] = [m_cR - m_cR(ρ_ac/ρ_c) + ρ_acv_R] + ρ_at (v_t - v_R) + D_t
m_ct (1- (ρ_ac/ρ_c)) = m_cR (1- (ρ_ac/ρ_c)) + ρ_ac (v_R - v_t) - ρ_at (v_R - v_t) + D_t

Pose g = m_ct = m_cR + (1- (ρ_ac /ρ_c))^-1 {(ρ_ac - ρ_at )(v_R - v_t) + D_t }

Using the Law of Propagation of Uncertainty, the combined variance for mct assuming no covariance between the parameters is:

(u_combined(m^ct))² =
(∂g/∂m_cR (u(m_cR)))² + (∂g/∂ρ_ac(u(ρ_ac)))² + (∂g/∂ρ_c (u(ρ_c)))² + (∂g/∂ρ_at (u(ρ_at)))²
+ (∂g/∂v_t (u(v_t)))²+ (∂g/∂v_R (u(v_R)))² + (∂g/∂D_t (u(D_t)))²

By definition, u(ρ_c) and u(ρ_ac) are equal to zero; thus the above equation reduces to:

(u_combined(m_ct))² =
(∂g/∂m_cR (u(m_cR)))² + (∂g/∂ρ_at (u(ρ_at)))² + (∂g/∂v_t (u(v_t)))²
+ (∂g/∂v_R (u(v_R)))² + (∂g/∂D_t (u(D_t)))²

Recall that (u(v_t))² ≈ (-m_t/ρ_t²)² u²(ρ_t) and (u(v_R))² . (-m_R/ρ_R²)² u²(ρ_R). See section 1.1.1.2 for details.

Where ∂g/∂m_cR ≈ 1
∂g/∂ρ_at ≈ -(v_R - v_t)(1-(ρ_ac/ρ_c))^-1 = - (m_R/ρ_R - m_t/ρ_t)(1-(ρ_ac/ρ_c))^-1
∂g/∂v_t ≈ -(ρ_ac-ρ_at)(1-(ρ_ac/ρ_c))^-1
∂g/∂v_R ≈ (ρ_ac -ρ_at)(1-(ρ_ac/ρ_c))^-1
∂g/∂D_t ≈ (1-(ρ_ac/ρ_c))^-1

Please note that using the full equation 1.1.2.1.A. instead of “g”, we have the following derivatives without approximation:

Mg/MD_t = [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] [ρ_c/(ρ_c-ρ_ac)]
Mg/Mm_cR = [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] [(ρ_M’-ρ_at)/(ρ_M’-ρ_ac)]
Mg/Mρ_at = [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] [(m_ct (ρ_t-ρ_at)/(ρ_t-ρ_ac)²) - (m_cR/(ρ_R-ρ_ac))]
Mg/Mρ_R = [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] [m_cR(ρ_at-ρ_ac)/(ρ_R-ρ_ac)²]
Mg/Mρ_t = [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] [m_ct (ρ_ac-ρ_at)/(ρ_t-ρ_ac)²]

Thus

(u_combined(m_ct))² =
[ 1 × u(m_cR) ]² + [ - (m_R/ρ_R - m_t/ρ_t) × (1-(ρ_ac/ρ_c))^-1 × u(ρ_at) ]² + [ - (ρ_ac-ρ_at) × (1-(ρ_ac/ρ_c))^-1 × u(v_t) ]²
+ [ (ρ_ac -ρ_at) × (1-(ρ_ac/ρ_c))^-1 × u(v_R) ]²
+ [ (1-(ρ_ac/ρ_c))^-1 × u(D_t) ]²

(u_combined(m_ct))² =

u(m_cR)²
+ (1-(ρ_ac/ρ_c))^-2{[-(m_R/ρ_R - m_t/ρ_t) u(ρ_at)]² + [-(ρ_ac-ρ_at)u(v_t)]² + [(ρ_ac -ρ_at)u(v_R)]² + u(D_t)²}

Since (u(v_t))² ≈ (Mv_t/Mρ_t)² u²(ρ_t) ≈ (-m_t/ρ_t²)² u²(ρ_t) and
(u(v_R))² ≈ (Mv_R/Mρ_R)² u²(ρ_R) ≈ (-m_R/ρ_R²)² u²(ρ_R)

(u_combined(m_ct))² =
u(m_cR)²
+ (1-(ρ_ac/ρ_c))^-2 {[-(m_R/ρ_R - m_t/ρ_t) u(ρ_at)]² + [-(ρ_ac-ρ_at)(-m_t/ρ_t²) u(ρ_t) ]²
+ [ (ρ_ac -ρ_at)(-m_R/ρ_R²) u(ρ_R) ]² + u(D_t)² }

Again, replacing mt and m_R by their equivalence in conventional mass

(u_combined(m_ct))² ≈
u(m_cR)² + (1-(ρ_ac/ρ_c))^-2 {u(D_t)²}
+ (1-(ρ_ac/ρ_c))^-2 {- ([m_cR ρ_R (1-(ρ_ac/ρ_c))/(ρ_R-ρ_ac)] /ρ_R - [m_ct ρ_t (1-(ρ_ac/ρ_c))/(ρ_t-ρ_ac)] /ρ_t) u(ρ_at) }²
+ (1-(ρ_ac/ρ_c))^-2 {-(ρ_ac-ρ_at)(-[(m_ct ρ_t (1-(ρ_ac/ρ_c)) /(ρ_t-ρ_ac)] /ρ_t²) u(ρ_t) }²
+ (1-(ρ_ac/ρ_c))^-2 { (ρ_ac -ρ_at)(-[(m_cR ρ_R (1-(ρ_ac/ρ_c)) /(ρ_R-ρ_ac)]/ρ_R²) u(ρ_R)}²

(u_combined(m_ct))² ≈

u(m_cR)² + (1-(ρ_ac/ρ_c))^-2 {u(D_t)²}
+ (1-(ρ_ac/ρ_c))^-2 { (- (1-(ρ_ac/ρ_c)))² ( m_cR/(ρ_R-ρ_ac) - m_ct /(ρ_t-ρ_ac) )² u² (ρ_at)
+ (1-(ρ_ac/ρ_c))^-2 {(-(1-(ρ_ac/ρ_c)))² (ρ_ac-ρ_at)² (-[(m_ct /(ρ_t-ρ_ac)] /ρ_t)² u²(ρ_t)
+ (1-(ρ_ac/ρ_c))^-2 { (1-(ρ_ac/ρ_c))² (ρ_ac -ρ_at)² (-[(m_cR/(ρ_R-ρ_ac)]/ρ_R)² u² (ρ_R)

u²(m_ct) ≈ u²(m_cR)+ ρ_c²/(ρ_c - ρ_ac)² u²(D_t)+ [m_cR/(ρ_R - ρ_ac) - m_ct/(ρ_t - ρ_ac)] ²u²(ρ_at) +
+ (ρ_ac -ρ_at)² [ (m_cR ²u²(ρ_R))/ρ_R² (ρ_R - ρ_ac)² + (m_ct² u²(ρ_t)) / ρ_t²(ρ_t - ρ_ac)²]

Since (ρ_ac -ρ_at)² [ (m_cR ²u²(ρ_R))/ρ_R² (ρ_R - ρ_ac)²] should already be included in u²(m_R), the combined standard uncertainty for m_ct expressed in terms of m_cR is the square root of the following:

(1.1.2.2 A)

u₂₍m_ct) ≈ u₂(m_cR)+ ρ_c²/(ρ_c - ρ_ac)² u²(D_t)+ [m_cR/(ρ_R - ρ_ac) - m_ct/(ρ_t - ρ_ac)] ²u²(ρ_at) + (ρ_ac -ρ_at)² [(m_ct² u²(ρ_t)) / ρ_t² (ρ_t - ρ_ac)²]

where all terms are expressed in the same unit of mass

The u²(m_cR) term used in above equation should already contain the term (ρ_ac -ρ_at)² [ (m_cR ²u²(ρ_R))/ρ_R² (ρ_R - ρ_ac)²] .

Please note that equation 1.1.2.2 A without approximation is:

u²(m_ct) = [(ρ_t-ρ_ac)/(ρ_t-ρ_at)]² {[(ρ_M’-ρ_at)/(ρ_M’-ρ_ac)]² u²(m_cR)+ ρ_c²/(ρ_c-ρ_ac)² u²(D_t) + [m_cR(ρ_at-ρ_ac)/(ρ_R-ρ_ac)²]² u²(ρ_R)
+ [(m_ct (ρ_t-ρ_at)/(ρ_t-ρ_ac)²) - (m_cR/(ρ_R-ρ_ac))]² u²(ρ_at) + [m_ct (ρ_ac-ρ_at)/(ρ_t-ρ_ac)²]² u²(ρ_t)}

where

u(m_ct) = combined standard uncertainty associated with the evaluation of the conventional mass of t
u(m_cR ) = combined standard uncertainty associated with the conventional mass of reference R - usually found on the calibration certificate for reference R. If only u(m_R ) is known, the formula 1.1.2.2B should be used. See section 2.1.
u(ρat) = standard uncertainty associated with the estimation of the air density at the time of the calibration (kg/m³) - usually assumed to be ± 0.07 kg/m³ . See section 2.2
u(ρ_t) = standard uncertainty associated with the density of the unknown mass t (kg/m³) - usually provided by the manufacturer of mass t. Otherwise, a rectangular distribution estimate with the limits provided in Table3 of OIML Recommendation R111 can be used; i.e., ± (1340 kg/m³ ÷ 2/ 3) = ± 380 (kg/m³) for Class F1, and a value of ± (4300 kg/m³ ÷ 2/ 3) = ± 1240 (kg/m³) for Class F2 . See section 2.3
u(ρ_R) = standard uncertainty associated with the density of the reference R (kg/m³)) - usually included in u(mcR) or provided on the calibration certificate for reference R. A rectangular distribution estimate based on the limits provided in Table 3 of OIML Recommendation R111 (1994) can be used: ± (400 kg/m³ ÷ 2/ 3) = ± 115 (kg/m³) for Class E2 . See section 2.3
u(D_t) = standard uncertainty associated with the calculated difference - the long-term process uncertainty of the procedure and equipment used in determining the difference between mass t and reference R. See section 2.4
ρ_t = density of the unknown mass t (kg/m³) - usually provided by manufacturer; otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111 (1994) and knowing the material. See section 3.1
ρ_R = density of the reference R (kg/m³) - usually found on calibration certificate for the reference; otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111 (1994) and knowing the material. See section 3.1
ρ_at = air density at the time of the calibration (kg/m³) - the formula provided in section 3.2 can be used
ρ_ac = conventional air density; equal by definition to 1.2 kg/m³ . See Section 3.3
ρ_c = conventional mass density; equal by definition to 8000 kg/m³. See section 3.4
≈ = approximately equal

NOTE: The terms involved in the uncertainty equations 1.1.2.2A and the following 1.1.2.2.B are described in detail in sections 2 and 3.

Now let us look at equation (1.1.2.1 B). Replacing only m_t by its equivalence in conventional mass and inserting into the equation f = m_t = m_R + ρ_at (v_t - v_R) + D_t, we get:

[m_ct - m_ct (ρ_ac/ρ_c) + ρ_acv_t] = [m_R] + ρ_at (v_t - v_R) + D_t
m_ct (1-(ρ_ac/ρ_c)) = [m_R] + ρ_at (v_t - v_R) + D_t - ρ_acv_t
m_ct = (1-(ρ_ac/ρ_c)) ^-1{m_R + ρ_at (v_t - v_R) + D_t - ρ_acv_t}

Pose h = m_ct = (1-(ρ_ac /ρ_c))^-1 { m_R + ρ_at (v_t - v_R) - ρ_ac v_t + D_t }

Using the Law of Propagation of Uncertainty as in the previous section, and remembering that u(ρ_c) and u(ρ_ac) are equal to zero, the combined variance for mct, assuming no covariance between the parameters, is:

(u_combined(m_ct))² =
(∂h/∂m_R (u(m_R)))²+ (∂h/∂ρ_at (u(ρ_at)))₂+ (∂h/∂v_t (u(v_t)))²+ (∂h/∂v_R (u(v_R)))² + (∂h/∂D_t (u(D_t)))²

where

∂h/∂m_R = (1-(ρ_ac/ρ_c))^-1
∂h/∂ρ_at = -(v_R - vt)(1-(ρ_ac/ρ_c))^-1
∂h/∂v_t = -(ρ_ac-ρ_at)(1-(ρ_ac/ρ_c))^-1
∂h/∂v_R = -(ρ_at)(1-(ρ_ac/ρ_c))^-1
∂h/∂D_t = (1-(ρ_ac/ρ_c))^-1

(u_combined(m_ct))² =

((1-(ρ_ac/ρ_c))^-1 u(m_R))² + (-(v_R - v_t)(1-(ρ_ac/ρ_c))^-1 u(ρ_at)) ² +
+(-(ρ_ac-ρ_at)(1-(ρ_ac/ρ_c))^-1 u(v_t))² + (-(ρ_at)(1-(ρ_ac/ρ_c))^-1 u(v_R))² + ((1-(ρ_ac/ρ_c))^-1 u(D_t))²

(u_combined(m_ct))² =
(1-(ρ_ac/ρ_c))^-2 {u(m_R))² + (-(v_R - v_t)u(ρ_at))² + (-(ρ_ac-ρ_at)u(v_t))² + (-(ρ_at)u(v_R))² + (u(D_t))²}

Recall that (u(v_t))² ≈ (-m_t/ρ_t²)² u²(ρ_t) and (u(v_R))² ≈ (-m_R/ρ_R²)² u²(ρ_R). See section 1.1.1.2 for details.

(u_combined(m_ct))² =
(1-(ρ_ac/ρ_c))^-2 {u(m_R))² + (-(m_R/ρ_R - m_t/ρ_t)u(ρ_at))² + (-(ρ_ac-ρ_at)(-m_t/ρ_t²)u(ρ_t))²
+ (-(ρ_at)(-m_R/ρ_R²)u(ρ_R))² + (u(D_t))²}

u²(m_ct) ≈
(1-(ρ_ac/ρ_c))^-2 {u²(m_R) + (m_R/ρ_R - m_t/ρ_t)²u²(ρ_at) + (ρ_ac -ρ_at)² m_t² u²(ρ_t)/ρ_t⁴
+ ρ_at² m_R² u²(ρ_R)/ρ_R⁴ + u²(D_t)}

Since (1-(ρ_ac/ρ_c))^-2 {ρ_at² m_R² u²(ρ_R)/ρ_R⁴} should already be included in u²(m_R), the combined standard uncertainty for mct expressed in terms of m_R is the square root of the following:

(1.1.2.2 B)

u²(m_ct) ≈ (1-(ρ_ac/ρ_c))^-2 {u²(m_R) + (m_R/ρ_R - m_t/ρ_t)²u²(ρ_at) + (ρ_ac -ρ_at)² m_t² u²(ρ_t)/ρ_t⁴ + u²(D_t)}

where all terms are expressed in the same unit of mass

The u²(m_R) term used in equation above should contain the term ρ_at² m_R² u²(ρ_R)/ρ_R⁴.

Note: Some prefer to assume that the density of the object being measured is unimportant in conventional weighing. However, as can be seen by looking at the above expression, both the density of the test mass (ρ_t) and the uncertainty of this density (u(ρ_t)) are included in the equation for u(m_ct) and can only be neglected if certain conditions are met.

1.2 Calibration and Uncertainty Equations: Series of Intercomparisons

- Example: CSL Working Standards and District Standards

1.2.1 Absolute Mass Calibration

Since the CSL performs all series of intercomparisons in conventional mass, this section is theoretical and the reader can skip tosection 1.2.2.

1.2.1.1 Mass Equation

If we want to calibrate a series in absolute mass (instead of the usual conventional mass) we would use the same procedure and same Forms as presented in section 4 of RP-01: Laboratory Calibration Procedures for Standards of Mass; however, the calculated difference would have to be inserted in the following equation instead of into equation 1.2.2.1.

Recall from the above section 1.1.1.1 that, for single comparative weighing, the absolute mass equation (1.1.1.1) for a test mass ‘t’ is

m_t = [ D_t + m_R (1 - (ρ_at/ρ_R)) ] / ( 1 - (ρ_at/ρ_t))

The equivalent equation when performing a series of intercomparisons with the new Forms and using the associated C_ijP coefficients of the P matrix, is therefore:

(1.2.1.1)

M_t = [ ∑ C_ijP m_i + C_M’P M’(1 - (ρ_at/ρ_M’)) ] / ( 1 - (ρ_at/ρ_t))

M_t = absolute mass of object t
∑ = is the symbol representing the summation of the terms that follow it
C_ijP = the coefficients that appear in the P matrix for each mass and each m_i. These correspond to row i (masses) and column j (readings m_i) respectively. See section 3.6
m_i = comparative reading i obtained for each line of matrix A in the Forms - it is, for a given line ‘i’, the indicated difference in readings multiplied by the sensitivity reciprocal for that line.
C_M’P = the coefficient in the P matrix for mass M’ along the row corresponding to mass t. See section 3.5
M’ = absolute mass of reference (or lead weight) R
ρ_at = air density at the time of the calibration (kg/m³) - the formula provided in section 3.2 can be used
ρ_M’ = density of the reference (or lead weight) R (kg/m³) - usually found on calibration certificate for the reference (or lead weight); otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111 (1994) and knowing the material. See section 3.1
ρ_t = density of the unknown mass t (kg/m³) - usually provided by manufacturer; otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111(1994) and knowing the material. See section 3.1

Using the 1990 version of the Forms, the equivalent equation is:

(1.2.1.1_1990)

M_t = [ C_StS_t + C_M’ M’(1 - (ρ_at/ρ_M’)) ] / ( 1 - (ρ_at/ρ_t))

M_t = absolute mass of object t
C_St , C_M’ = the coefficients that appear before the S_t and M’ terms, respectively, in the Absolute error equations of the old (1990) Forms
S_t = the summation for mass t as described in the old Forms
M’ = absolute mass of reference (or lead weight) R
ρ_at = air density at the time of the calibration (kg/m³) - the formula provided in section 3.2 can be used
ρ_M’ = density of the reference (or lead weight) R (kg/m³) - usually found on calibration certificate for the reference (or lead weight); otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111(1994)
ρ_t = density of the unknown mass t (kg/m³) - usually provided by manufacturer; otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111(1994)

For example, the Error equation called (5) for mass M5 in the 1990 version of Form 1 should be modified with equation 1.2.1.1_1990 as follows:
(5) = S₅/100 + M should be replaced by equation 1.2.1.1_1990 such that:
(M5)(1-(ρ_at/ρ_M5)) = (1/100) S₅ + 1 M’ (1 - (ρ_at /ρ_M’))
(M5)= (1-(ρ_at/ρ_M5))-1 [S₅/100 + M’ (1 - (ρ_at /ρ_M’))]
Note that in this case, C_St = C_S5 = 1/100, C_M’ = 1 and M5 is the absolute mass of the weight called “5” in the series of intercomparisons.
Once M5 is calculated, the error is computed with the following:
M5 = N + (5), thus (5) = M5 - N, where N is the nominal value.

1.2.1.2 Combined Uncertainty Equation for Absolute Mass

Again, since the CSL performs all series of intercomparisons in conventional mass, this section is theoretical and the reader can skip tosection 1.2.2.

As discussed in 4.3.5 ofRP-01 Laboratory Calibration Procedures for Standards of Mass, the Variance and Uncertainty equations in the 1990 version of the Forms were incomplete and must be modified to account for the difference of the volumes of the masses being compared, for the long-term variance of the process instead of only the punctual variation, and for any covariances.

We obtain the uncertainty equation from equation 1.2.1.1 above, by again, using the Law of Propagation of Uncertainty.

Pose k = M_t = [ Σ C_ijP m_i + C_M’P M’(1 - (ρ_at/ρ_M’)) ] / ( 1 - (ρ_at/ρ_t))
Pose D_t = Σ C_ijP m_i

The combined variance for M_t assuming no covariance between the parameters is:

(u_combined(M_t))² =
(∂k/∂D_t (u(D_t)))² + (∂k/∂_M’(u(M’)))² + (∂k/∂ρ_at (u(ρ_at)))² + (∂k/∂ρ_M’ (u(ρ_M’)))²+ (∂k/∂ρ_t (u(ρ_t)))²

Performing all the derivatives, we get:

∂k/∂D_t = (ρ_t /(ρ_t-ρ_at))
∂k/∂_M’= (ρ_t /(ρ_t-ρ_at)) C_M’P (ρ_M’-ρ_at)/ρ_M’
∂k/∂ρ_at = (ρ_t /(ρ_t-ρ_at)²) [ (C_M’P M’(ρ_M’-ρ_t) / ρ_M’ ) + D_t ]
∂k/∂ρ_M’ = (ρ_t /(ρ_t-ρ_at)) (-ρ_atC_M’P M’/(ρ_M’²))
∂k/∂ρ_t = (ρ_t /(ρ_t-ρ_at))(-ρ_at M_t /(ρ_t)²)

Please note that equation 1.2.1.2 - the combined standard uncertainty - without approximation for the absolute mass Mt when a series of intercomparisons is performed (using the new Forms) is the square root of the following:

u²(M_t) = [ρ_t / (ρ_t-ρ_at) ]² {u²(D_t) + [C_M’P (ρ_M’-ρ_at)/ρ_M’ ]² u²(M’) + [(-ρ_atC^M’P M’)/(ρ^M’2)]² u²(ρ_M’)
+ [-ρ_atM_t/(ρ_t²)]² u²(ρ_t) + [(C_M’P M’(ρ_M’-ρ_t) /ρ_M’ (ρ_t-ρ_at)) + D_t /(ρ_t-ρ_at)]² u²(ρ_at)}

where all terms are expressed in the same unit of mass

Now, since ρ_t > ρ_at and ρ_M’ > ρ_at, then we can assume that (ρ_t-ρ_at) ≈ ρ_t and that (ρ_M’-ρ_at) ≈ ρ_M’. We therefore get the following approximate equation:

u²(M_t) ≈ [ρ_t /ρ_t ]² u²(D_t) + [ρ_t C_M’P (ρ_M’) / ρ_M’ (ρ_t)]² u²(M’)
+ [(-(ρ_at) C_M’P M’ρ_t ) /(ρ_M’² (ρ_t))] ² u²(ρ_M’) + [(-ρ_at) M_t / (ρ_t (ρ_t))]² u²(ρ_t)
+ (ρ_t/ (ρ_t)²) ² [ (C_M’P M’(ρ_M’ - ρ_t) / ρ_M’ ) + D_t ]² u²(ρ_at)

u²(M_t) ≈ = u²(D_t) + (C_M’P )² u²(M’)
+ [(-(ρ_at) C_M’P M’) /(ρ_M’²)] ² u²(ρ_M’) + [(-ρ_at) M_t /(ρ_t²)]² u²(ρ_t)
+ (1/ρ_t)² [ (C_M’P M’) - (C_M’P M’ ρ_t / ρ_M’ ) + D_t ]² u²(ρ_at)

u²(M_t) ≈ u²(D_t) + (C_M’P )² u²(M’)
+ [(-(ρ_at) C_M’P M’) /(ρ_M’²)] ² u²(ρ_M’) + [(-ρ_at) M_t /(ρ_t²)]² u²(ρ_t)
+ (1/ρ_t)² [ (C_M’P M’+ D_t) - (C_M’P M’ ρ_t / ρ_M’ ) ]² u²(ρ_at)

Since M_t = [ ∑ C_ijP m_i + C_M’P M’(1 - (ρ_at/ρ_M’)) ] / ( 1 - (ρ_at/ρ_t))
≈ [ ∑ C_ijP m_i + C_M’P M’] = [ D_t + C_M’P M’]

u²(M_t) ≈ u²(D_t) + (C_M’P )² u²(M’)
+ [(-(ρ_at) C_M’P M’) /(ρ_M’²)] ² u²(ρ_M’) + [(-ρ_at) M_t /(ρ_t²)]² u²(ρ_t)
+ (1/ρ_t)² [ (M_t) - (C_M’P M’ ρ_t / ρ_M’ ) ]² u²(ρ_at)

u²(M_t) ≈ u²(D_t) + (C_M’P )² u²(M’)
+ [(-(ρ_at) C_M’P M’) /(ρ_M’²)] ² u²(ρ_M’) + [(-ρ_at) M_t /(ρ_t²)]² u²(ρ_t)
+ (1/ρ_t) [ (M_t) - (C_M’P M’ ρ_t / ρ_M’ ) ]² u²(ρ_at)

u²(M_t) ≈ u²(D_t) + (C_M’P )² u²(M’)
+ [(-(ρ_at) C_M’P M’) /(ρ_M’²)] ² u²(ρ_M’) + [(-ρ_at) M_t /(ρ_t²)]² u²(ρ_t)
+ [M_t / ρ_t (C_M’P M’ ρ_t / ρ_t ρ_M’ ) ]² u²(ρ_at)

u²(M_t) ≈ u²(D_t) + (C_M’P )² u²(M’)
+ [(-(ρ_at) C_M’P M’) /(ρ_M’²)] ² u²(ρ_M’) + [(-ρ_at) M_t /(ρ_t²)]² u²(ρ_t)
+ [M_t / ρ_t (C_M’P M’ / ρ_M’ ) ]² u²(ρ_at)

Since (C_M’P)² = C_M’Q , finally we have that the combined standard uncertainty (with approximation) for the absolute mass Mt when a series of intercomparisons is performed (using the new Forms) is the square root of the following:

(1.2.1.2)

u²(M_t) ≈ u²(D_t) + C_M’Q u²(M’) + (M_t/ρ_t - C_M’P M’/ρ_M’)² u²(ρ_at) + ρ_at² (M_t²/ρ_t⁴) u²(ρ_t)

where all terms are expressed in the same unit of mass

The u₂(M’) term used in the above variance equation should already contain the term ρ_at² ((C_M’PM’)²/ρ_M’⁴) u²(ρ_M’); otherwise, it should be added.

where u²(D_t) is (see section 2.4.2) equal to:

Equation A

where

u(M_t) = combined standard uncertainty of the absolute mass of test mass t
u(D_t) = standard uncertainty of process (in the same units as u(M’)). See section 2.4.
C_M’Q = the coefficient in the Q matrix for mass M’ along the row corresponding to mass t. See section 3.5
u(M’)= combined standard uncertainty of the absolute mass of lead weight. See section 2.1
M_t = absolute mass of object t
ρ_t = density of the test mass (in kg/m³) - usually provided by the manufacturer; otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111 (1994) and knowing the material. See section 3.1
C_M’P = the coefficient in the P matrix for mass M’ along the row corresponding to mass t. See section 3.5
M’ = absolute mass of reference (or lead weight) R
ρ_M’ = density of the lead weight M’ (in kg/m³) - usually found on calibration certificate for the reference (lead weight); otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111(1994) and knowing the material. See section 3.1
u(ρ_at) = standard uncertainty associated with the estimation of the air density at the time of the calibration (in kg/m³) - usually assumed to be ± 0.07 kg/m³ for laboratory conditions. See section 2.2
ρ_at = air density at the time of the calibration (kg/m³) - the formula provided in section 3.2 can be used
u(ρ^t) = standard uncertainty associated with the density of the unknown mass t (in kg/m³) - usually provided by the manufacturer of mass t. Otherwise, a rectangular distribution estimate with the limits provided in Table3 of OIML Recommendation R111 can be used; i.e., ± (1340 kg/m³ ÷ 2% 3) = ± 380 kg/m³ for Class F1, and a value of ± (4300 kg/m³ ÷ 2% 3) = ± 1240 kg/m³ for Class F2. See section 2.3
u(ρ_M’) = standard uncertainty associated with the density of the reference M’ (kg/m³)) - usually included in u(M’) or provided on the calibration certificate for reference M’. See section 2.3
C_ijQ = the coefficients that appear in the Q matrix for each mass and each m_i. These correspond to row i (masses) and column j (readings mi) respectively.
u(i) = the uncertainty of the mass i due only to process (repeatability and reproducibility)
u²(i) = (S_i)_i² + u² (D_i) = σ_m² + u² (D_i)
u(j) = the uncertainty of the mass j due only to process (repeatability and reproducibility)
u²(j) = (S_i)_j² + u² (D_j) = σ_m² + u² (D_j)
σ_m² = the variance associated to the Form being used (this is a measure of the process repeatability)
u² (Di, j) = the base value of the Process Variance of the check standard of nominal value ‘i’ or ‘j’ (this is a measure of the process reproducibility)

Note: The terms involved in the uncertainty equation 1.2.1.2 are described in detail in the following sections 2 and 3.

Using the same reasoning as above, the combined standard uncertainty for the absolute mass M_t when a series of intercomparisons is performed using the 1990 version of the Forms is:

u²(M_t) ≈ u²(D_t) + (Cσ_M’) u²(M’) + [ρ_at² Cσ_M’ (M’)² / (ρ_M’⁴)] u²(ρ_M’) + [ρ_at² M_t² / (ρ_t⁴)] u²(ρ_t)
+ [ M_t /ρ_t - (C_M’ M’/ρ_M’ ) ]² u²(ρ_at)

Since (ρ_at² Cσ_M’ M’²/ρ_M’⁴) u²(ρ_M’) should already be included in u²(M’), the combined standard uncertainty for the absolute mass Mt when a series of intercomparisons is performed is the square root of the following:

(1.2.1.2_1990)

u²(M_t) ≈ u²(D_t) + Cσ_M’ u²(M’) + (M_t/ρ_t - C_M’ M’/ρ_M’)²u²(ρ_at) + ρ_at² [(M_t²/ρ_t⁴)u²(ρ_t)]

where all terms are expressed in the same unit of mass

Note: u²(D_t) would have to be calculated using the C _ijQ coefficients of the new forms.

Recall from section 4.3.5 of RP01: Laboratory Calibration Procedures for Standards of Mass (also see the following section 2.4) that u²(D_t) is equal to:

Equation A

where

1.2.2 Conventional Mass Calibration

The CSL Working Standards and the District Standards are calibrated with series of intercomparisons in conventional mass.

As discussed in 4.3.4 of RP-01 Laboratory Calibration Procedures for Standards of Mass, the Error Equations in the 1990 version of the Forms did not account for buoyancy. The 1990 equations have therefore been modified by replacing the Error equations by the following mass equation.

1.2.2.1 Mass Equation

Recall from section 1.2.1 that for an absolute mass calibration using a series of intercomparisons, we have equation (1.2.1.1):

M_t = [ Σ C_ijP m_i + C_M’P M’(1 - (ρ_at/ρ_M’)) ] / ( 1 - (ρ_at/ρ_t))

Once the absolute mass is known, the conventional mass can be computed by using the following relationships:

M_ct (1-(ρ_ac/ρ_c)) M_t (1-(ρ_ac/ρ_t))
M’_c (1-(ρ_ac/ρ_c)) M’ (1-(ρ_ac/ρ_M’))

Therefore, we have:

M_t= [ Σ C_ijP m_i + C_M’P M’(1 - (ρ_at/ρ_M’)) ] / ( 1 - (ρ_at/ρ_t))
M_ct (1-(ρ_ac/ρ_c)) / (1-(ρ_ac/ρ_t))= [ Σ C_ijP m_i + C_M’P M’(1 - (ρ_at/ρ_M’)) ] / ( 1 - (ρ_at/ρ_t))
M_ct =(1-(ρ_ac/ρ_t))[ Σ C_ijP m_i + C_M’P M’(1 - (ρ_at/ρ_M’)) ] / ( 1 - (ρ_at/ρ))
Mct = (([ρ_t-ρ_ac)/ρ_t) [ Σ C_ijP m_i + C_M’P M’(1 - (ρ_at/ρ_M’)) ] / ((ρ_t-ρ_at)/ρ_t) ((ρ_c-ρ_ac)/ρ_c)
M_ct = (ρ_t-ρ_ac) [ Σ C_ijP m_i + C_M’P M’(1 - (ρ_at / ρ_M’)) ]/ ((ρ_t-ρ_at)(ρ_c-ρ_ac)/ρ_c)
M_ct = (ρ_c/(ρ_c-ρ_ac)) (ρ_t-ρ_ac)/(ρ_t-ρ_at) [ Σ C_ijP m_i + C_M’P M’(1 - (ρ_at/ρ_M’)) ]
M_ct = (ρ_c/(ρ_c-ρ_ac)) (ρ_t-ρ_at) [ Σ C_ijp m_i + C_M'P M' ( ρM'-ρ_at)/ρ_M' ]
M_ct =((ρ_t - ρ_ac) / (ρ_t-ρ_at) [ Σ C_ijp m_i(ρ_c/ρ_c-ρ_ac)) + C_M'P M'((ρ_M'-ρ_at)/ρ_M') (ρ_c/ρ_c-ρ_ac)) ]
M_ct =((ρ_t - ρ_ac) / (ρ_t-ρ_at) { Σ C_ijp m_i(ρ_c/ρ_c-ρ_ac)) + C_M'P [ M'_c((ρ_c-ρ_ac)/ρ_c)(ρ_M’/(ρ_M’-ρ_ac))]((ρ_M’-ρ_at)/ρ_M’) (ρ_c/(ρ_c-ρ_ac))}
M_ct =((ρ_t - ρ_ac) / (ρ_t-ρ_at) { Σ C_ijp m_i(ρ_c/ρ_c-ρ_ac)) + C_M'P [ M'_c(ρ_M’ /(ρ_M’-ρ_ac)) (ρ_M’-ρ_at)/ρ_M’]
M_ct =((ρ_t - ρ_ac) / (ρ_t-ρ_at) { Σ C_ijp m_i(ρ_c/ρ_c-ρ_ac)) + ((ρ_M’-ρ_at) /(ρ_M’-ρ_ac)) C_M’P M’_c]
M_ct = (ρ_t-ρ_ac)/(ρ_t-ρ_at) { C_M’P M’_c (ρ_M’-ρ_at) /(ρ_M’-ρ_ac) + Σ C_ijP m_i ρ_c/(ρ_c-ρ_ac) }

We have:

M_ct = (ρ_t-ρ_ac)/(ρ_t-ρ_at) { C_M’P M’_c (ρ_M’-ρ_at) /(ρ_M’-ρ_ac) + ρ_c/(ρ_c-ρ_ac) Σ C_ijP m_i }

Therefore

(1.2.2.1)

M_ct = (ρ_t-ρ_ac)/(ρ_t-ρ_at) { (ρ_c/(_ρc-ρ_ac)) Σ C_ijP m_i + C_M’P M’_c (ρ_M’-ρ_at) /(ρ_M’-ρ_ac) }

where M_ct , M’_c and m_i are expressed in the same unit of mass

where

M_ct = conventional mass of unknown mass t
ρ_t = density of the unknown mass t (kg/m³) - provided by manufacturer; otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111(1994) and knowing the material. See section 3.1
ρ_ac= conventional air density; equal by definition to 1.2 kg/m³. See section 3.3
ρ_at = air density at the time of the calibration (kg/m³) - the formula provided in section 3.2 can be used
ρ_c = conventional mass density; equal by definition to 8000 kg/m³. See section 3.4
Σ = is the symbol representing the summation of the terms that follow it
C_ijP = the coefficients that appear in the P matrix for each mass and each mi. These correspond to row i (masses) and column j (readings m_i) respectively. See section 3.6
m_i = comparative reading i obtained for each line of matrix A in the Forms - it is, for a given line ‘i’, the indicated difference in readings multiplied by the sensitivity reciprocal for that line.
C_M’P = the coefficient in the P matrix for mass M’ along the row corresponding to mass t. See section 3.5
M’_c = conventional mass of reference or lead weight
ρ_M’ = density of the reference or lead weight (kg/m³) - usually found on calibration certificate for reference (lead weight) M’; otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111(1994) and knowing the material. See section 3.1

Note that in the above equation, M’_c must be used, not M’. Hence, if only the absolute mass of the reference (lead weight) M’ is known it must first be converted to a conventional mass M’_c using

M’_c M’ (1-(ρ_ac/ρ_M’)) / (1-(ρ_ac/ρ_c))

Recall from section 1.1.2.1 that, for single comparative weighing, equation (1.1.2.1 A) for conventional mass calibration is:

m_ct = ((ρ_t - ρ_ac) / (ρ_t - ρ_at)) { (ρ_c /( ρ_c-ρ_ac)) D_t + (m_cR (ρ_R - ρ_at)/(ρ_R - ρ_ac)) }

The similarities between equations 1.2.2.1 and 1.1.2.1 A can easily be noted.

Using the 1990 version of the Forms, the equivalent equation is:

(1.2.2.1_1990)

M_ct = ((ρ_t - ρ_ac) / (ρ_t - ρ_at)) { (C_M’ M’_c (ρ_M’ - ρ_at)/(ρ_M’ - ρ_ac)) + (C_St S_t ρ_c / ( ρ_c - ρ_ac)) }

where M_ct , M’_c and S_t are expressed in the same unit of mass

1.2.2.2 Combined Uncertainty Equation for Conventional Mass

As discussed in 4.3.5 of RP-01 Laboratory Calibration Procedures for Standards of Mass, the Variance and Uncertainty equations in the 1990 version of the Forms were incomplete and must be modified to account for the difference of the volumes of the masses being compared, for the long-term variance of the process instead of only the punctual variation, and for any covariances.

Using the same method as in section 1.2.1.2 and, by again, using the Law of Propagation of Uncertainty, we obtain the uncertainty equation from equation (1.2.2.1) above:

Pose L = M_ct = (ρ_t-ρ_ac)/(ρ_t-ρ_at) { (ρ_c/(ρ_c-ρ_ac)) Σ C_ijP m_i + C_M’P M’_c (ρ_M’-ρ_at) /(ρ_M’-ρ_ac) }
Pose D_t = Σ C_ijP m_i

The combined variance for Mct assuming no covariance between the parameters is:

(u_combined(M_ct))² =
(∂L/∂D_t (u(D_t)))²+(∂L/∂M’_c(u(M’_c)))² +(∂L/∂ρ_at (u(ρ_at)))² +(∂L/∂ρ_M’ (u(ρ_M’)))²+(∂L/∂ρ_t (u(ρ_t)))²

Performing all the derivatives, we get:

∂L/∂D_t = [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] [ρ_c / (ρ_c-ρ_ac)]
∂L/∂M’_c= [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] [C_M’P (ρ_M’-ρ_at) / (ρ_M’-ρ_ac)]
∂L/∂ρ_at = [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] [(M_ct /(ρ_t-ρ_ac)) - (C_M’P M’_c /(ρ_M’-ρ_ac))]
∂L/∂ρ_M’ = [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] [C_M’P M’_c(ρ_at-ρ_ac) / (ρ_M’-ρ_ac)²]
∂L/∂ρ_t = [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] [M_ct (ρ_ac-ρ_at) / (ρ_t-ρ_ac)²]

Please note that when performing ML/Mρ_at, we have:

∂L/∂ρ_at = [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] {[C_M’PM’_c (ρ_M’ - ρ_at)/(ρ_M’-ρ_ac)(ρ_t-ρ_at)] + [D_t ρ_c/(ρ_c-ρ_ac)(ρ_t-ρ_at)] - (C_M’P M’_c /(ρ_M’-ρ_ac))}
= [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] {[C_M’PM’_c (ρ_M’ - ρ_at)/(ρ_M’-ρ_ac)(ρ_t-ρ_at)] + [D_t ρ_c/(ρ_c-ρ_ac)(ρ_t-ρ_at)]} - {[(ρ_t-ρ_ac)/(ρ_t-ρ_at)] [C_M’P M’c /(ρ_M’-ρ_ac)]}
= [(1/(ρ_t-ρ_at))] [(ρ_t-ρ_ac)/(ρ_t-ρ_at)]{[C_M’PM’c (ρ_M’ - ρ_at)/(ρ_M’-ρ_ac)] + [D_t ρ_c/(ρ_c-ρ_ac)]} - {[(ρ_t-ρ_ac)/(ρ_t-ρ_at)] [C_M’P M’_c /(ρ_M’-ρ_ac)]}

But since M_ct = (ρ_t-ρ_ac)/(ρ_t-ρ_at) { (ρ_c/(ρ_c-ρ_ac)) D_t + C_M’P M’_c (ρ_M’-ρ_at) /(ρ_M’-ρ_ac) }, we have:
ML/Mρ_at = [(1/(ρ_t-ρ_at))] [M_ct ] - [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] [C_M’P M’_c /(ρ_M’-ρ_ac)]
= [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] {[M_ct /(ρ_t-ρ_ac)] - [C_M’P M’_c /(ρ_M’-ρ_ac)]}

Since (C_M’P)² = C_M’Q , equation 1.2.2.2. (without approximation) - combined standard uncertainty for the conventional mass M_ct when a series of intercomparisons is performed (using the new Forms) - is the square root of the following:

u²(M_ct) = [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] ² { [ρ_c/(ρ_c-ρ_ac)]² u²(D_t) + C_M’Q [(ρ_M’-ρ_at)/(ρ_M’-ρ_ac)] ² u²(M’_c)
+ [C_M’P M’_c(ρ_at-ρ_ac)/(ρ_M’-ρ_ac)²] ² u²(ρ_M’) + [M_ct (ρ_at-ρ_ac)/(ρ_t-ρ_ac)²] ² u²(ρ_t)
+ [(M_ct /(ρ_t-ρ_ac)) - (C_M’P M’_c /(ρ_M’-ρ_ac))] ² u²(ρ_at) }

where all terms are expressed in the same unit of mass

The u²(M’_c) term used in the above variance equation should already contain the term [C_M’P M’_c(ρ_at-ρ_ac)/(ρ_M’-ρ_ac)²] ² u²(ρ_M’); otherwise, it should be added.

In relative terms, this is:

u²_rel (M_ct) = (1/N_t)² [(ρ_t -ρ_ac)/(ρ_t -ρ_at)]² { [ρ_c/(ρ_c-ρ_ac)]² u²(D_t) + C_M’Q [(ρ_M’-ρ_at)/(ρ_M’-ρ_ac)]² u²(M’_c)
+ [C_M’P M’_c(ρ_at-ρ_ac)/(ρ_M’-ρ_ac)²]²u²(ρ_M’) + [M_ct (ρ_at-ρ_ac)/(ρ_t-ρ_ac)²]² u²(ρ_t)
+ [(M_ct /(ρ_t-ρ_ac)) - (C_M’P M’_c /(ρ_M’-ρ_ac))] ² u²(ρ_at) }

u²_rel (M_ct) = [(ρ_t -ρ_ac)/(ρ_t -ρ_at)]² {[ρ_c/(ρ_c-ρ_ac)]²(1/N_t)²u²(D_t) + C_M’Q(1/N_t)²[(ρ_M’-ρ_at)/(ρ_M’-ρ_ac)]² u²(M’_c) + (C_M’P M’_c/N_t)² [(ρ_at-ρ_ac)/(ρ_M’-ρ_ac)²] ² u²(ρ_M’) + (M_ct/N_t)² [ (ρ_at-ρ_ac)/(ρ_t-ρ_ac)²] ² u²(ρ_t)
+ [(M_ct /N_t(ρ_t-ρ_ac)) - (C_M’P M’_c /N_t(ρ_M’-ρ_ac))] ² u²(ρ_at) }

Since (C_M’PM_c’)/N_t 1 and M_ct/N_t 1, we have:

u²_rel (M_ct) [(ρ_t -ρ_ac)/(ρ_t -ρ_at)]² {[ρ_c/(ρ_c-ρ_ac)]²(1/N_t)²u²(D_t) + C_M’Q(1/N_t)²[(ρ_M’-ρ_at)/(ρ_M’-ρ_ac)]² u²(M’_c) +[(ρ_at-ρ_ac)/(ρ_M’-ρ_ac)²]²u²(ρ_M’)+[(ρ_at-ρ_ac)/(ρ_t-ρ_ac)²]²u²(ρ_t)
+[(1/(ρ_t-ρ_ac))-(1/(ρ_M’-ρ_ac))]²u²(ρ_at)}

If ρ_at ρ_ac, we have:

u²_rel (M_ct) [ρ_c/(ρ_c-ρ_ac)]²(1/N_t)²u²(D_t) + (C_M’P/N_t)² u²(M’_c) +[(1/(ρ_t-ρ_ac))-(1/(ρ_M’-ρ_ac))]²u²(ρ_at)

The combined variance associated with the determination of the conventional mass value of a test mass when using a series of intercomparisons is:

(1.2.2.2)

u²(M_ct) = [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] ² { [ρ_c/(ρ_c-ρ_ac)]² u²(D_t) + C_M’Q [(ρ_M’-ρ_at)/(ρ_M’-ρ_ac)] ² u²(M’_c)
+ [M_ct (ρ_at-ρ_ac)/(ρ_t-ρ_ac)²] ² u²(ρ_t) + [(M_ct /(ρ_t-ρ_ac)) - (C_M’P M’_c /(ρ_M’-ρ_ac))] ² u²(ρ_at) }

where all terms are expressed in the same unit of mass

(1.2.2.2_app.) approximate equation, if ρ_at = ρ_ac

u² (M_ct) [ρ_c/(ρ_c-ρ_ac)]² u²(D_t) + C_M’Q u²(M’_c) + [(M_ct /(ρ_t-ρ_ac)) - (C_M’P M’_c /(ρ_M’-ρ_ac))] ² u²(ρ_at)

(1.2.2.2_app rel.) approximate equation (relative terms)

u²_rel (M_ct) [ρ_c/(ρ_c-ρ_ac)]²(1/N_t)²u²(D_t) + (C_M’P/N_t)² u²(M’_c) +[(1/(ρ_t-ρ_ac))-(1/(ρ_M’-ρ_ac))]²u²(ρ_at)

where u²(D_t) is equal to:

Equation A

where

u (M_ct) = combined standard uncertainty of the conventional mass of test mass t (expressed in same units as u(D_t))
u_rel (M_ct) = relative combined standard uncertainty of the conventional mass of test mass t (dimensionless)
N_t = nominal value of the test mass (expressed in same units as u(D_t) and u(M_cR’)
C_M’Q = the coefficient in the Q matrix for mass M’ along the row corresponding to mass t . See section 3.5
u(M_c’) = combined standard uncertainty of the conventional mass of reference (lead weight) (expressed in same units as u(D_t) and N_t). See section 2.1
ρ_M’ = density of the reference (lead weight) M’ (in kg/m³) - usually found on calibration certificate for the reference (lead weight); otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111(1994) and knowing the material. See section 3.1
ρ_t = density of the test mass (in kg/m³) - usually provided by the manufacturer; otherwise, an estimate can be made by referring to Table 3 of OIML Recommendation R111(1994) and knowing the material . See section 3.1
ρ_ac = conventional air density; equal by definition to 1.2 kg/m³. See section 3.3
ρ_at = air density at the time of the calibration (kg/m³) - the formula provided in section 3.2 can be used
u(ρ_at) = standard uncertainty associated with the estimation of the air density at the time of the calibration (in kg/m³) - usually assumed to be ± 0.07 kg/m³ for laboratory conditions. See section 2.2
ρ_c = conventional mass density; equal by definition to 8000 kg/m³ . See section 3.4
u(ρ_t) = standard uncertainty associated with the density of the unknown mass t (in kg/m³) - usually provided by the manufacturer of mass t. Otherwise, a rectangular distribution estimate with the limits provided in Table3 of OIML Recommendation R111 can be used; i.e., ± (1340 kg/m³ ÷ 2% 3) = ± 380 kg/m³ for Class F1, and a value of ± (4300 kg/m³ ÷ 2% 3) = ± 1240 kg/m³for Class F2. See section 2.3
u(D_t) = standard uncertainty of process (in the same units as u(M_cR’) and N_t). See section 2.4
Σ = is the symbol representing the summation of the terms that follow it
C_ijP = the coefficients that appear in the P matrix for each mass and each mi. These correspond to row i (masses) and column j (readings m_i) respectively. See section 3.6
m_i = comparative reading i obtained for each line of matrix A in the Forms - it is, for a given line ‘i’, the indicated difference in readings multiplied by the sensitivity reciprocal for that line.
C_M’P = the coefficient in the P matrix for mass M’ along the row corresponding to mass t. See section 3.5
u(i) = the uncertainty of the mass i due only to process (repeatability and reproducibility)
u²(i) = (S_i)_i² + u² (D_i) = σ_m² + u² (D_i)
u(j) = the uncertainty of the mass j due only to process (repeatability and reproducibility)
u2(j) = (S_i)_j² + u² (D_j) = σ_m² + u² (D_j)
σm2 = the variance associated with the Form being used (this is a measure of the process repeatability)
u2 (Di, j) = the base value of the Process Variance of the check standard of nominal value ‘i’ or ‘j’ (this is a measure of the process reproducibility)

Note: In the 1990 Forms, only the (Punctual) Variance (σ_t ²) was used instead of equation 1.2.2.2 above and was expressed as: (σ_t ²) = C_i u²(m_i) + C_M’ (σ_M’)² = C_i (σ_m)² + C_M’ (σ_M’)². As discussed at the beginning of section 1.2.2.2, the uncertainty calculations of the 1990 forms were therefore incomplete.

Components of Uncertainty

2.1 Uncertainty of the Reference or Lead Weight

2.2 Uncertainty of Air Density

2.3 Uncertainty of Density of Reference and of Weight Being Calibrated

2.3.1 CSL Working Standards
2.3.2 District Standards
2.3.3 Inspector Weight Kits (field working standards)

2.4 Uncertainty of the Process

2.4.1 Single Comparison and tolérance Testing
2.4.2 Series of Intercomparisons

2.1 Uncertainty of the Reference or Lead Weight: u(m_R), u(M’), u(m_cR), u(M’_c)

The uncertainties u(m_R) or u(M’), and u(m_cR) or u(M’_c) are the combined standard uncertainties associated with, respectively, the absolute mass and the conventional mass of the reference R (Lead Weight). These values are usually found on the calibration certificate for a reference weight or, in the case of a Lead Weight composed of one or more weights from a previous calibration, the uncertainty is the value of the combined uncertainty obtained for those weights from the previous calibration.

2.1.1 Calibration of the CSL Working Standards

2.1.1.1 Absolute Mass

Since the CSL performs all series of intercomparisons in conventional mass, this section is theoretical and the reader can skip to 2.1.1.2.

When calibrating the Calibration Standards Laboratory (CSL) Working Standards in absolute mass, the Reference M’ will first (and subsequently, where it is also possible) be the Departmental Reference Kilogram MR-1(MR-2 or MR-3). The variance u²(M’) = σ²_M is the value (U_{c, absolute}/2)² where U_{c, absolute} is provided in the National Research Council of Canada (NRC) - Institute for National Measurement Standards (INMS) Report (calibration certificate). Values on the INMS Report are expressed as expanded uncertainties, unless otherwise stated, and that is why they must be divided by the coverage factor k=2 before being inserted in the equation for combined uncertainty.

2.1.1.2 Conventional Mass

The CSL Working Standards are calibrated with series of intercomparisons in conventional mass.

The CSL Working Standards are calibrated from the Departmental Reference MR-1 (MR-2 or MR-3) in conventional mass. When using equation 1.2.2.2 with the Departmental Reference as the Reference, the variance u²(M’_c) is the value (U_{c,conventional}/2)² where U_{c,conventional} is provided in the INMS Report. Values on the INMS Report are expressed as expanded uncertainties, unless otherwise stated, and must therefore be divided by the factor k=2 before being inserted in the equation for combined uncertainty.

When the calibration is performed against one or more Lead Weights, having been calibrated in a previous series, then u(M’_c) is therefore the value of u(M_ct) that was obtained for those masses by that previous series.

Multiples:

1st - 5 kg, 2 kg, 2' kg, 1 kg, M’
Form 7 with MR-1as the Reference M’

2nd - 20 kg, 10 kg, 10 b, 5b, M’
Form 11, with the 5 kg Working Standard as the Lead Weight and with borrowed weights of 10 kg and 5 kg

Submultiples:

1st - M’, 500 g, 200 g, 200' g, 100 g, (50 + 20 + 20' + 10) g
Form 8 with MR-1as the Reference M’

2nd - M’, 50 g, 20 g, 20' g, 10 g, (5 + 2 + 2' + 1) g
Form 8 with 100 gas the Lead Weight M’

3rd - M’, 5 g, 2 g, 2' g, 1 g, (500 + 200 + 200' + 100) mg
Form 8 with 10 gas the Lead Weight M’

4th - M’, 500 mg, 200 mg, 200' mg, 100 mg, (50 + 20 + 20' + 10) mg
Form 8 with 1gas the Lead Weight M’

5th - M’, 50 mg, 20 mg, 20' mg, 10 mg, (5 + 2 + 2' + 1) mg
Form 8 with 100 mgas the Lead Weight M’

6th - M’, 5 mg, 2 mg, 2' mg, 1 mg, 1 mg (repeat)
Form 8 with 10 mgas the Lead Weight M’

In other words, calibration of the multiples and submultiples of the CSL Working Standards is first performed using series involving MR-1 (MR-2 or MR-3) as the Reference. The remainder of the submultiples are then calibrated using, as Lead Weight(s), Working Standards that have just been calibrated in the previous series.

2.1.2 Calibration of District Standards and others

The choice of series of intercomparisons to use depends on the configuration of the District Standard set to be calibrated.

2.1.2.1 Absolute Mass

District Standards are calibrated with series of intercomparisons in conventional mass and the reader should skip to 2.1.2.2

District Standards could be calibrated in absolute mass by comparison to a group of CSL Working Standards. In this case, the value of the Reference M’ used in the calibration equation would have to be the absolute mass of a CSL Working Standard and u²(M’) would have to be the variance associated with the absolute mass of the Reference u²(Mt lead).

2.1.2.2 Conventional Mass

Equation 1.2.2.2 should be used when calculating the uncertainty of the conventional mass of District Standards. In this case, u(M’_c) is the uncertainty associated with the reference or lead weight. Therefore, when the reference is a Working Standard, u(M’_c) is the value of the combined uncertainty associated with the conventional mass found on the certificate for the reference. When the calibration is performed against one or more Lead Weights, having been calibrated in a previous series, then u(M’_c) is therefore the value of u(M_ct) that was obtained for those masses by that previous series.

2.1.3 tolérance Testing of Inspection Weights

Inspection Weights are not calibrated, they are tolérance tested. The convenience of tolérance testing is that the mass value and associated uncertainty are not calculated for the weight under test. However, to determine if tolérance testing is possible, we must first know both u(m_cR) of the standard and u(D_t). In particular, the reference weight shall be of a class at least one level above that of the mass being tolérance tested. In other words the tolérance of the reference weight must be smaller than one third that of the weight being tolérance tested.

We already have that:

U(m_cR) ≤ 1/3 [mpe (tolérance) of m_cR] as a condition for the reference weight to be acceptable.

- where U(m_cR) is the expanded uncertainty associated with the determination of the conventional mass of the reference. When calculating the expanded uncertainty, we use a coverage factor of k=2, which means that we multiply the combined uncertainty by two to get the expanded uncertainty. In other words: U(m_cR) = 2 u(m_cR)

Now, since the reference weight must be of at least one class higher than the weight being tolérance tested, we have that:

[mpe (tolérance) of m_cR] ≤ 1/3 [mpe (tolérance) of m_ct]

Therefore:

U(m_cR) ≤ 1/3 [mpe (tolérance) of m_cR]

U(m_cR) ≤ 1/3 [1/3 mpe (tolérance) of m_ct]

Since U(m_cR) = 2 u(m_cR), then
2u(m_cR) ≤ 1/9 [mpe (tolérance) of m_ct]
u(m_cR) ≤ 1/18 [mpe (tolérance) of m_ct]

Hence, in order to perform tolérance testing, the expanded uncertainty of the reference (standard mass) used must first be known to be of at least one class above that of the mass under test; in other words U(m_cR) = 2 u(m_cR) shall be less than 1/9 of the tolérance of the mass under test, which implies that u(m_cR) ≤ 1/18 [mpe (tolérance) of m_ct]

Although this section only addresses uncertainty of the reference, please recall that, as mentioned in section 1.7 of RP-01: Laboratory Calibration Procedures for Standards of Mass and in section 5. Performance Criteria of RP-03: Calibration Performance Monitoring, we know that for any mass being calibrated, we must have 3u(D_t) ≤ 1/3 mpe (tolérance) of that mass; thus, u(D^t) ≤ 1/9 tolérance of m_ct.

2.2 Uncertainty of Air Density: u(ρ_at)

To calculate the air density at the time of calibration, we can use the approximate equation provided by the Canada National Research Council (CNRC) as follows:

ρ_at (0.092695) + (1.18 x 10-5) P (in Pa) - (4.15 x 10-3) T (in°C) - (1.22 x10-4) H_rel (in %)

Inside:

The air density near sea level, at a temperature close to 20°C and humidity close to 50%RH, varies by no more than approximately ± 10% ρ_ac . In this case (ρ_ac - ρ_at) . 10%(1.2 kg/m³) = ± 0.12 kg/m³. If we assume a rectangular distribution, we have u(ρ_at) = ± (0.12 kg/m³) ÷ %3 = ± 0.07 kg/m³. This scenario corresponds to calibrations performed indoors. Therefore, if the air density indoors is not measured at the time of calibration, it can be assumed to be (1.2 ± 0.07) kg/m³ , and u(ρ_at) is ± 0.07 kg/m³.

Outside:

For calibrations performed outdoors, the air density for outside temperature and humidity may vary by as much as ± 24% ρac with the different seasons. This fact can be observed by looking at the following expected variations for different locations across the country:

temperature: -40°C to +40 °C
relative humidity: 20% to 100%
pressure: 90 kPa to 110 kPa

we get a maximum air density when H=20%, T= -40°C and P=110 kPa; ρ_at = 1.554 kg/m³
we get a minimum air density when H=100%, T= 40°C and P= 90 kPa ; ρ_at = 0.976 kg/m³

Hence, the maximum variation of ρat for an outdoor environment can be set at: ± (1.554 - 0.976) ÷ 2 = ± 0.29 kg/m³. If we treat this value as a rectangular distribution, we get u(ρ_at) = ± 0.29 kg/m³ ÷ √3 = ± 0.17 kg/m³. Therefore, if the air density outdoors is not measured when the calibration is performed outdoors, it can be assumed to be (1.2 ± 0.17) kg/m³ and u(ρ_at) is ± 0.17 kg/m³.

Inside: The value of the uncertainty of the air density to be used in all the uncertainty equations for calibrations performed inside, e.g., within the CSL Laboratories, is:
u(ρ_at) = ± 0.07 kg/m³.

Outside: The value of the uncertainty of the air density to be used in all the uncertainty equations for calibrations performed outside, e.g., railway test car calibrations, is:

u(ρ_at) = ± 0.17 kg/m³.

2.3 Uncertainty u(ρ_R) oru(ρ_M’) of Density of the Reference (or lead weight) and Uncertainty of Densityu(ρ_t) of Weight being Calibrated

2.3.1 CSL Working Standards

When calibrating in absolute mass or in conventional mass, an estimate of the densities of the test mass and of the reference (or lead weight) must be known, as well as their associated uncertainties.

When calibrating the CSL Working Standards with MR-1 (MR-2 or MR-3) as the Reference, the density ρ_R of the reference is known and its associated uncertainty u(ρ_R) is also provided, as both these values are included in the INMS Report of calibration. Again, values on the INMS Report are expressed as expanded uncertainties, unless otherwise stated, and they must be divided by the coverage factor k=2 before being inserted in the equation for combined uncertainty.

Please note that when calibrating the Working Standards using the Departmental Standard as the Reference, u(ρ_t) refers to the uncertainty of the density value for the Working Standard being calibrated. The value (or an estimate) of the density ρt and its associated uncertainty for the Working Standards should be supplied by the manufacturer of the weight set. Otherwise, a rectangular distribution estimate based on the limits in Table 3 of OIML Recommendation R111(1994) can be used: u(ρ_t) = ± (1340 kg/m³ ÷ 2√ 3) = ± 380 (kg/m³) for a class F1 weight.

In other words, when calibrating the Working Standards with MR-1 (MR-2 or MR-3) as the Reference, we shall use the figures:

u(ρ_R) = Udensity /2, where Udensity is provided in the INMS Report
u(ρ_t) = ± 380 kg/m³ or the value provided by the manufacturer of the set, whichever is greater

If an uncertainty for the density value is provided by the manufacturer but it is not defined if this value is an expanded or standard uncertainty, assume that the value represents limits of a rectangular distribution and divide the value by √ 3.

When calibrating the remainder of the submultiples using Working Standards of known values as the Lead Weights, then the density ρ_R of the reference (or lead weight) and the density ρ_t of the Working Standard being calibrated are equivalent. The same is true for the uncertainties of the density values u(ρ_R) and u(ρ_t).

For the purposes of calibration of the Working Standards using one or more Working Standard(s) of known value as the Lead Weight, we shall use:

u(ρ_R) = ± 380 kg/m³ or the value provided by the manufacturer of the set, whichever is greater
u(ρ_t) = ± 380 kg/m³ or the value provided by the manufacturer of the set, whichever is greater

Please recall that u(ρ_R) should not be accounted for twice as it should already be included in u(m_R), u(M'), u(m_cR) and u(M'_c).

2.3.2 District Standards

When calibrating the District Standards using Working Standards as References, the density ρ_R of the reference is the density of the Working Standard(s) and u(ρ_R) is the associated uncertainty. From the above 2.3.1, we have that u(ρ_R) = ± 380 kg/m³ or is the value provided by the manufacturer of the weight set.

The value (or an estimate) of the density ρt and its associated uncertainty for the District Standards should be supplied by the manufacturer of this weight set. Otherwise, a rectangular distribution estimate based on the limits in Table 3 of OIML Recommendation R111(1994) can be used. For a class F2 weight, this value is u(ρ_t) = ± (4300 kg/m³ ÷ 2√ 3) = ± 1240 (kg/m³).

For the purposes of calibration of the District Standards using Working Standards as References, we shall use:

u(ρ_R) = ± 380 kg/m³ or kg/m³ or the value provided by the manufacturer of the Working Standards set, whichever is greater
u(ρ_t) = ± 1240 kg/m³ or the value provided by the manufacturer of the District Standard set, whichever is greater

When calibrating the remainder of the submultiples using District Standards of known values as the Lead Weights, then the density ρ_R of the reference (or lead weight) and the density ρt of the District Standard being calibrated are equivalent. The same is true for the uncertainties of the density values u(ρ_R) and u(ρ_t).

For the purposes of calibration of the District Standards using one or more District Standard(s) of known value as the Lead Weight, we shall use:

u(ρ_R) = ± 1240 kg/m³ or the value provided by the manufacturer of the set, whichever is greater
u(ρ_t) = ± 1240 kg/m³ or the value provided by the manufacturer of the set, whichever is greater

Please recall that u(ρ_R) should not be accounted for twice as it should already be included in u(m_R), u(M’), u(m_cR) and u(M’_c).

2.3.3 Inspector Weight Kits

When calibrating or tolérance testing an inspector weight (field working standard) using a CSL working standard as a reference, we have that ρ_R is the density of the CSL working standard, and u(ρ_R) is the associated uncertainty on its density determination, ρt is the density of the field working standard, and u(ρ_t) is the associated uncertainty of its density determination.

The value (or an estimate) of the density and its associated uncertainty for the CSL working standards will be indicated on its calibration certificate.

Unless provided by the manufacturer of the field working standard, a rectangular distribution estimate based on the limits of the material densities can be used for u(ρ_t). In other words, if the material densities of section 2.4.5 of RP-01Field are respected, we have that, for class M1, M1-2 and M2 weights, the value of u(ρ_t) for the field working standard can be approximated as: ± (8400 - 6000) kg/m³ ÷ 2√ 3 = ± 693 kg/m³ or approximately ± 700 kg/m³. For classes M2-3 and M3 the value of u(ρ_t) can be approximated with: ± (8400 - 4400) kg/m³ ÷ 2√ 3 = ± 1155 kg/m³ or approximately ± 1200 kg/m³.

When calibrating or tolérance testing using CSL working standards, we have

u(ρ_R) = the value provided on the working standard certificate

u(ρ_t) M1, M1-2 and M2 weights * = ± 700 kg/m³ or the value provided by the manufacturer of the field inspection standard, whichever is greater

u(ρ_t) M2-3 and M3 weights = ± 1200 kg/m³ or the value provided by the manufacturer of the field inspection standard, whichever is greater

Please recall that u(ρ_R) should not be accounted for twice as is should already be included in u(m_R) or u(m_cR).

Note *: if unsure that the material densities have been respected, it is best to use the estimate of ± 1200 kg/m³.

2.4 Uncertainty of the Process u(D_t)

The Process Uncertainty u(D_t) is a function of two main components: short-term variance (S²_i) and long-term variance u²(LT_i). The short-term variance is based on the individual calibrations and the long-term variance is calculated by using the base value of the Process Variance u²(D_i) for the Check Standards of the same (or similar) value as the masses being calibrated.

The base value of the Process Variance of the Check Standards u²(D_i) is continuously monitored and adjusted by using RP-03: Mass Calibration Process Monitoring. It is downloaded directly into the calculation of u(D_t).

Therefore:

u²(D_t) = (S_i)² + u₂(LT_i)

2.4.1 Single Comparison and tolerance Testing

Please note that when performing single comparative weighing , only one value is obtained for this type of verification, hence a variance (S²_i) cannot be calculated and we use the readability of the device (its sensitivity or half the graduation) divided by /3 instead of Si. Further, as mentioned in the opening paragraph to this section, the base value of the Process Variance for the Check Standard of nominal value ‘i’ u²(D_i) is used as an estimate of u²(LT_i)

• tolerance Testing and Single Comparative Weighing (r = 1):

• Comparative Weighing with "r" Comparisons (r>1):

u²(D_t) = (S_ζ)² + u²(D_i)

where S_ζ is:

Readability = the sensitivity or half the graduation (division) of the device used, expressed in the same units as u²(D_t)
S_ζ = the standard deviation of the “r” comparisons, approximated with the range
Maximum (x) = the maximum value obtained over the “r” comparisons
minimum (x) = the minimum value obtained over the “r” comparisons
u²(D_i) = the base value of the Process Variance for the Check Standards of the same value as the mass ‘t’ being calibrated

2.4.2 Series of Intercomparisons

The process variance for a series of intercomparions is calculated as follows:

Equation C

u² (D_t) = the process variance (repeatability and reproducibility)
N_i,j = respectively, the number of rows and the number of columns in Q
C_ijQ = the coefficient in the Q matrix corresponding to row i and column j
u(i) = the uncertainty of the mass i due only to process (repeatability and reproducibility)
u²(i) = (S_i)_i² + u² (D_i) = σ_m² + u² (D_i)
u(j) = the uncertainty of the mass j due only to process (repeatability and reproducibility)
u²(j) = (S_i)_j² + u² (D_j) = σ_m² + u² (D_j)
σ_m² = the variance associated with the Form being used (this is a measure of the process repeatability)
u²(D_{i, j}) = the base value of the Process Variance for the Check Standard of nominal value ‘i’ or ‘j’ (this is a measure of the process reproducibility).

Therefore, the process variance u²(D_t) associated with a given test mass ‘t’ is obtained by summing all products along the row ‘i’ corresponding to ‘t’ in the Q matrix, where each C_ijQ coefficient is multiplied by the uncertainty of the parameter for the row and for the column. Each C_ijQ coefficient represents the covariance factor between masses i and j.

Please note that within the same series of intercomparisons u²(D_t) may be different for each weight involved since nominal values and equipment used may differ.

For example, the contribution by, say, mass (3) to the process variance for the system ‘f’ is:

Equation D

and represents that the process variance associated with mass (3) for the system of masses being calibrated is calculated by summing all products along the row for mass (3) in the Q matrix, where each coefficient is multiplied by the uncertainty of the parameter for the row and for the column. For Form 1, we have the following coefficients C_ijQ of the Q matrix:

"Q" Matrix		u(₅)	u(₃)	u(₂)	u(₁)	u(_Σ1)	u²(M')
u²(D₅)	u(₅)	0.5	0.15	0.10	0.05	0.05	1
u²(D₃)	u(₃)	0.15	0.22	0.03	-0.01	-0.01	0.36
u²(D₂)	u(₂)	0.10	0.03	0.22	0.01	0.01	0.16
u²(D₁)	u(₁)	0.05	-0.01	0.01	0.246	-0.086	0.04
u²(D_Σ1)	u(_Σ1)	0.05	-0.01	0.01	-0.086	0.246	0.04

which means that the contribution to the process variance by mass (3) is

u²(D₃)= 0.15u₍₃₎u₍₅₎ + 0.22u₍₃₎² + 0.03u₍₃₎u₍₂₎ - 0.01 u₍₃₎u₍₁₎ - 0.01 u₍₃₎u₍_Σ₁₎

where, for exemple

u₍₃₎ = the uncertainty of the mass (3) due only to process repeatability and reproducibility
u²(3) = (S_i)₃² + u²(D_i) = σ_m² + u²(D_i₌₃)
u₍₅₎ = the uncertainty of the mass (5) due only to process repeatability and reproducibility
u²(5) = (S_i)₅² + u²(D_j) = σ_m² + u²(D_j=₅)

where σ_m² = the variance implicit to the form being used for the series of intercomparisons. It is the same for all weights in a series but will vary every time the series is performed.
u²(D_i₌₃) = the base value of the Process Variance for the Check Standard of nominal value ‘3’
u²(D_j=5) = the base value of the Process Variance for the Check Standard of nominal value ‘5’

Recall from section 4.3.5 of RP-01: Laboratory Calibration Procedures for Standards of Mass that

σ_m² = vv / n-p

where:

n = the number of linear equations of condition set up
p = the number of true unknowns or number of independent normal equations
vv = Σ_i=1 (v_i )² is the summation of all residuals squared;

where:

v_i = | m_i - c_i| are the residuals between the observed and computed (theoretical) values, where mi are the readings obtained experimentally from the equations of part | 1 | and ci are the theoretical equivalents to mi calculated by replacing the values or the errors of each weight (from part | 2 |) into the equations of condition (part | 1 |).

The calculation of σ_m² is the same in these revised procedures as it was in th 1990 Forms

Note that u²(D_i,j) is the base value of the Process Variance for the Check Standard of the same nominal value as the mass ‘i’ (or j). These values are down-loaded directly by the calibration software and are obtained as described in RP-03: Mass Calibration Process Monitoring.

Technical Note:

The method used to calculate the process variance u²(D_t) for the new Forms respects the Law of Propagation of Uncertainties (see section 5.1.2 and 5.2.2 of the ISO Guide to the Expression of Uncertainty in Measurement) which says that the combined variance u_c²(y) associated with the result of a measurement is:

Equation E

which is the same as saying that:

Equation F

For calculation of the process variance, the (∂f/∂x_i)² terms are equivalent to the diagonal entries C_iiQ of the Q matrix and the terms(∂f/∂x_i*∂f/∂x_j) are the off-diagonal coefficients C_ijQ of the Q matrix. Further, the term u(x_i) is the same as u(i) referred to earlier as the uncertainty of the mass i due only to process (repeatability and reproducibility) and u(x_i, x_j) = u(x_i) u(x_j). Our equation is therefore:

Equation G

Uncertainty Multipliers

3.1 Density of reference and density of weight being calibrated

3.1.1 CSL Working Standards
3.1.2 District Standards
3.1.3 Inspector Weight Kits

3.2 Air Density

3.3 Conventional Air Density

3.4 Conventional Mass Density

3.5 Coefficients CM’P and CM’Q

3.6 Coefficients CijP and CijQ

3.1 Density ρ_R orρ_M’ of the reference (or lead weight) and density ρ_t of weight being calibrated

3.1.1 CSL Working Standards

When calibrating in absolute mass or in conventional mass, an estimate of the density of the test mass and of the reference (or lead weight) must be known.

When calibrating the CSL Working Standards with MR-1 (MR-2 or MR-3) as the reference, the density ρ_R of the reference is included in the INMS Report of calibration.

The density ρ_t for the Working Standards should be supplied by the manufacturer of the weight set.

In other words, when calibrating the Working Standards with MR-1(MR-2 or MR-3) as the Reference, we shall use:

ρ_R = density of MR-1 (MR-2 or MR-3) as provided in the INMS Report
ρ_t = value provided by the manufacturer of the set.

When calibrating the remainder of the submultiples using Working Standards of known values as the lead weights, then the density ρ_R of the lead weights and the density ρ_t of the Working Standard being calibrated are equivalent; we have that ρ_R = ρ_t

For the purposes of calibration of the Working Standards using a Working Standard(s) of known value as the Lead Weight(s), we shall use:

ρ_R = value provided by the manufacturer of the set
ρ_t = the value provided by the manufacturer of the set

3.1.2 District Standards

When calibrating the District Standards using Working Standards as References, the density ρ_R of the reference is the density of the Working Standards.

The value (or an estimate) of the density ρ_t for the District Standards should be supplied by the manufacturer of this weight set.

For the purposes of calibration of the District Standards using Working Standards as References, we shall use:

ρ_R = value provided by the manufacturer of the Working Standard set
ρ_t = value provided by the manufacturer of the District Standard set

When calibrating the remainder of the submultiples using District Standards of known values as the lead weights, the density ρ_R of the lead weights and the density ρ_t are equivalent. We have that ρ_R = ρ_t

For the purposes of calibration of the District Standards using a District Standard(s) of known value as the Lead Weight(s), we shall use:

ρ_R = value provided by the manufacturer of the set
ρ_t = value provided by the manufacturer of the set

3.1.3 Inspector Weight Kits

When tolerance testing inspector weight kits, it is not necessary to know the density of the weights. However, should single comparison calibration be performed on these weights, the density to use as ρ_R is the density of the Working Standards used in the calibration.

The density of the Inspector Weight kit masses should be provided by their manufacturer; otherwise, it should be assumed that these weights are of Class M1and an approximate value can be used by knowing the weight’s material.

For the purposes of single comparison calibration of the Inspector Weight Kits using Working Standards as References, we shall use:

ρ_R = value provided by the manufacturer of the Working Standard set
ρ_t = value provided by the manufacturer of the Inspector Weight Kits or an estimate

3.2 Air Density

As discussed in 2.2, the air density near sea level, at a temperature close to 20°C and humidity close to 50%RH, varies by no more than approximately ± 10% from the value 1.2 kg/m³.

Since we can’t really look at a worst case scenario, the value ρ_at = 1.2 kg/m³ is used when approximating. Should we wish to calculate the actual air density, the following equation (provided by the Canada National Research Council) can be used:

ρ_at . (0.092695) + (1.18 x 10-5) P (in Pa) - (4.15 x 10-3) T (in°C) - (1.22 x10-4) H_rel (in %)

In other words, the value of air density to be used in all the uncertainty equations for calibrations within the CSL Laboratories is: the true value calculated by using the NRC equation provided above or its approximation ρ_at = 1.2 kg/m³.

3.3 Conventional Air Density

The conventional air density is a theoretical value that has been set to ρ_ac = 1.2 kg/m³.

The value of conventional air density to be used in all the uncertainty equations is: ρ_ac = 1.2 kg/m³.

3.4 Conventional Mass Density

The conventional mass density is a theoretical value that has been set to ρ_c = 8000 kg/m³.

The value of conventional mass density to be used in all the uncertainty equations is: ρ_c = 8000 kg/m³.

3.5 Coefficient C_M’P and C_M’Q

The coefficient C_M’P and C_M’Q used in the calibration and uncertainty equations for when a series of intercomparisons is performed is the coefficient of the Reference mass in, respectively, the “P” matrix and the “Q” matrix, as found in the Calibration Forms.

The coefficient C_M’P and C_M’Q is the coefficient for the Reference found respectively, in the “P” matrix and the “Q” matrix along the line corresponding to mass “t” being calibrated.

3.6 Coefficients C_ijP and C_ijQ

The coefficients C_ijP and C_ijQ used in the calibration equation and in the calculation of u(D_t), for when a series of intercomparions is performed, are, respectively, the coefficients that appear in the “P” and the “Q” matrix.

Each C_ijP coefficient, corresponds to, along that row of the “P” Matrix, a given mass, and to, down that column, a given reading mi. Looking at the simplified “P” matrix corresponding to the revised version of Form 1, we have, for example that the C_ijP coefficient for mass (3) along column 5 would be C_(3)m5P and has the value 0.1.

"P" Matrix mass value	m₁	m₂	m₃	m₄	m₅	m₆	m₇	m₈	m₉	M'
(5)	0.5	0.25	0.25	-0.25	-0.25	0	0	0	0	1
(3)	0.15	0.25	0.2	0.05	0.1	-0.2	-0.2	-0.1	0	0.6
(2)	0.1	0.25	0.1	-0.1	0.15	0.2	0.2	-0.2	0	0.4
(1)	0.05	0	0.15	-0.1	-0.1	0.266	-0.07	0.15	-0.333	0.2
(Σ1)	0.05	0	0.15	0.1	-0.1	-0.066	0.266	0.15	0.333	0.2

Each C_ijQ coefficient, corresponds to, along that row of the “Q” Matrix, a given uncertainty for the mass indicated by the row, and that indicated by the column. For example, for a Form 1, we have the following coefficients C_ijQ of the “Q” matrix. Therefore the C_ijQ coefficient corresponding to u(3) and u(1) would be C_u(3)u(1)Q = -0.01 and the calculation of u²(D₃) would be:

u²(D₃) = C_u(3)u(5)Q *u₍₃₎u₍₅₎ + C_u(3)u(3)Q *u₍₃₎u₍₃₎ + C_u(3)u(2)Q *u₍₃₎u₍₂₎ - C_u(3)u(1)Q *u₍₃₎u₍₁₎ - C_u(3)u(_Σ_1)Q*u₍₃₎u₍₃₁₎

u²(D₃) = 0.15u₍₃₎u₍₅₎ + 0.22u₍₃₎² + 0.03u₍₃₎u₍₂₎ - 0.01 u₍₃₎u₍₁₎ - 0.01 u₍₃₎u₍₃₁₎

"Q" Matrix		u(₅)	u(₃)	u(₂)	u(₁)	u(_Σ1)	u²(M')
u²(D₅)	u(₅)	0.5	0.15	0.10	0.05	0.05	1
u²(D₃)	u(₃)	0.15	0.22	0.03	-0.01	-0.01	0.36
u²(D₂)	u(₂)	0.10	0.03	0.22	0.01	0.01	0.16
u²(D₁)	u(₁)	0.05	-0.01	0.01	0.246	-0.086	0.04
u²(D_Σ1)	u(_Σ1)	0.05	-0.01	0.01	-0.086	0.246	0.04

Summary - At a Glance

All terms are expressed in the same unit if mass

1.1 Single Comparison

1.1.1 Absolute Mass

(1.1.1.1) m_t = [ m_R (1 - (ρ_at/ρ_R)) + D_t ] / ( 1 - (ρ_at/ρ_t))

(1.1.1.2) u² (m_t) ≈ u²(m_R) + (m_t/ρ_t-m_R/ρ_R)²u²(ρ_at) + ρ_at²(-m_t/ρ_t²)²u²(ρ_t) + u²(D_t)

1.1.2 Conventional Mass

(1.1.2.1 A) m_ct = ((ρ_t - ρ_ac) / (ρ_t - ρ_at)) { (m_cR (ρ_R - ρ_at)/(ρ_R - ρ_ac)) + (D_t ρ_c / ( ρ_c - ρ_ac)) }

(1.1.2.1 B) m_ct = [(ρ_c (ρ_t - ρ_ac)) / ((ρ_c - ρ_ac) (ρ_t - ρ_at)) ] {m_R (1 - (ρ_at /ρ_R)) + D_t }

(1.1.2.2 A) u²(m_ct) ≈ u²(m_cR)+ ρ_c²/(ρ_c - ρ_ac)² u²(D_t)+ [m_cR/(ρ_R - ρ_ac) - m_ct/(ρ_t - ρ_ac)] ²u²(ρ_at) + (ρ_ac -ρ_at)² [(m_ct² u²(ρ_t))/ρ_t²(ρ_t - ρ_ac)²]

(1.1.2.2 B) u²(m_ct) ≈ (1-(ρ_ac/ρ_c))^-2 {u²(m_R) + (m_R/ρ_R - m_t/ρ_t)²u²(ρ_at) + (ρ_ac -ρ_at)² m_t² u²(ρ_t)/ρ_t⁴ + u²(D_t)}

1.2 Series of Intercomparisons

1.2.1 Absolute Mass

(1.2.1.1) M_t = [ Σ C _ijP m _i + C_M’P M’(1 - (ρ_at/ρ_M’)) ] / ( 1 - (ρ_at/ρ_t))

(1.2.1.1_1990) M_t = [ C_M’ M’(1 - (ρ_at/ρ_M’)) + C_StS_t ] / ( 1 - (ρ_at/ρ_t))

(1.2.1.2) u²(M_t) ≈ u²(D_t) + C_M’Q u²(M’) + (M_t/ρ_t - C_M’P M’/ρ_M’)² u²(ρ_at) + ρ_at² (M_t²/ρ_t⁴) u²(ρ_t)

(1.2.1.2_1990) u²(M_t) ≈ u²(D^t) + Cσ_M’ u²(M’) + (M_t/ρ_t - C_M’ M’/ρ_M’)²u²(ρ_at) + ρ_at² [(M_t²/ρ_t⁴)u²(ρ_t)]

1.2.2 Conventional Mass

(1.2.2.1) M_ct = (ρ_t-ρ_ac)/(ρ_t-ρ_at) { (ρ_c/(ρ_c-ρ_ac)) Σ C _ijP m _i + C_M’P M’_c (ρ_M’-ρ_at) /(ρ_M’-ρ_ac) }

(1.2.2.1_1990) M_ct = ((ρ_t - ρ_ac) / (ρ_t - ρ_at)) { (C_M’ M’_c (ρ_M’ - ρ_at)/(ρ_M’ - ρ_ac)) + (C_St S_t ρ_c / ( ρ_c - ρ_ac)) }

(1.2.2.2) u₂(M_ct) = [(ρ_t-ρ_ac)/(ρ_t-ρ_at)] ² { [ρ_c/(ρ_c-ρ_ac)]² u²(D_t) + C_M’Q [(ρ_M’-ρ_at)/(ρ_M’-ρ_ac)] ² u²(M’_c)

+ [M_ct (ρ_at-ρ_ac)/(ρ_t-ρ_ac)²] ² u²(ρ_t) + [(M_ct /(ρ_t-ρ_ac)) - (C_M’P M’_c /(ρ_M’-ρ_ac))] ² u²(ρ_at) }

(1.2.2.2_app.) approximate equation, if ρ_at ≈ ρ_ac

u² (M_ct) ≈ [ρ_c/(ρ_c-ρ_ac)]² u²(D_t) + C_M’Q u²(M’_c) + [(M_ct /(ρ_t-ρ_ac)) - (C_M’P M’_c /(ρ_M’-ρ_ac))] ² u²(ρ_at)

(1.2.2.2_app rel.) approximate equation (relative terms)

u²_rel (M_ct) ≈ [ρ_c/(ρ_c-ρ_ac)]²(1/N_t)²u²(D_t) + (C_M’P/N_t)² u²(M’_c) +[(1/(ρ_t-ρ_ac))-(1/(ρ_M’-ρ_ac))]²u²(ρ_at)

m_t , M_t = absolute mass of object t
m_R , M’ = absolute mass of reference (or lead weight) R
m_ct , M_ct = conventional mass of unknown mass t
m_cR , M’_c = conventional mass of reference (or lead weight) R
D_t = calculated difference between unknown mass t and reference (or lead weight) R
ρ_t = density of the unknown mass t (kg/m³) - usually provided by manufacturer
ρ_R, ρ_M’ = density of the reference (or lead weight) R (kg/m³) - usually found on calibration certificate for the reference (or lead weight)
ρ_at = air density at the time of the calibration (kg/m³) - the formula provided in section 3.2 can be used
ρ_ac = conventional air density; equal by definition to 1.2 kg/m³. See section 3.3
ρ_c = conventional mass density; equal by definition to 8000 kg/m³. See section 3.4
C_St , C_M’ = the coefficients that appear before the St and M terms, respectively, in the 1990 Error equations of the Forms
u(m_t), u(M_t) = combined standard uncertainty associated with the evaluation of the absolute mass of t
u(m_R), u(M’) = combined standard uncertainty associated with the absolute mass of reference (or lead weight) R - usually found on calibration certificate for reference (or lead weight) R.. See section 2.1
u(m_ct),u(M_ct) = combined standard uncertainty associated with the evaluation of the conventional mass of t
u(m_cR),u(M’_c)= combined standard uncertainty associated with the conventional mass of reference (or lead weight) R - usually found on the calibration certificate for reference (or lead weight) R.. See section 2.1
u(ρ_at) = standard uncertainty associated with the estimation of the air density at the time of the calibration (kg/m³) - usually assumed to be ± 0.07 kg/m³. See section 2.2
u(ρ_t) = standard uncertainty associated with the density of the unknown mass t (kg/m³) - usually provided by the manufacturer of mass t. See section 2.3
u(ρ_R), u(ρ_M’) = standard uncertainty associated with the density of the reference (or lead weight) R (kg/m³)) - usually included in u(m_R) or provided on the calibration certificate for reference (or lead weight) R.. See section 2.3
u(D_t) = standard uncertainty associated with the calculated difference. See section 2.4
(Cσ_m)²,(Cσ_M’)² = the coefficients appearing, respectively, in front of σ²_m and σ²_M in the 1990 Variance and Uncertainty equations in the Forms
≈ = approximately equal
Σ = is the symbol representing the summation of the terms that follow it
C_ijP = the coefficients that appear in the P matrix for each mass and each mi. These correspond to row i (masses) and column j (readings m_i) respectively. See section 3.6
m_i = comparative reading i obtained for each line of matrix A in the Forms - it is, for a given line ‘i’, the indicated difference in readings multiplied by the sensitivity reciprocal for that line.
C_M’P = the coefficient in the P matrix for mass M’ along the row corresponding to mass t. See section 3.5
C_M’Q = the coefficient in the Q matrix for mass M’ along the row corresponding to mass t . See section 3.5

References

1. Chapman, George D., Orthogonal Designs for the Calibration of Kilogram Submultiples, National Research Council of Canada, Report No. AMS-002 NRCC 25819, November 1993.

2. Debler, E., A Theoretical Approach to Measurement - Uncertainty of Mass, Conventional Mass and Force, OIML Bulletin, Volume XXXV, Number 4, October 1994.

3. Dupuis-Désormeaux, N., Procedure for Estimating Uncertainties in Mass Calibrations, Industry Canada, third edition, November 1994.

4. Joint International Committee ISO/IEC/OIML/BIPM- TAG-4, Guide to the Expression of Uncertainty in Measurement, first edition, June 1992.

5. OIML Recommendation R111 - Weights of classes E1, E2, F1, F2, M1, M2, M3, Edition 1994.

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RP-02 Laboratory: Determination of Mass Calibration Values and Related Uncertainties

Background

Components of Uncertainty

Uncertainty Multipliers

Summary - At a Glance

References