# Monte Carlo simulations of the susceptometer method for the determination of magnetic properties of weights

24.06.2013

With a susceptometer both the volume magnetic susceptibility and the vertical component of the permanent magnetic polarization of a weight can be tested. Monte Carlo simulations of the susceptometer method were performed to evaluate the expanded measurement uncertainties of the magnetic properties for comparison measurements and intercomparisons.

In legal metrology the magnetic properties of a weight should not exceed the limits given in the international recommendation OIML R 111 in order to avoid magnetic distortion forces during the calibration of weights [1]. In practice a suitable procedure for testing the magnetic properties of weights is the susceptometer developed by Richard Davis [2]. Hereby the magnetic interaction forces between the object to be investigated and a strong permanent magnet are used to determine the magnetic volume susceptibility and the vertical component of the permanent magnetic polarization of the test weight.

In order to ascertain the expanded measurement uncertainties of the magnetic properties Monte Carlo simulations of the susceptometer method were performed with Microsoft Excel according to the „Guide to the Expression of Uncertainty in Measurement“ (supplement 1 and 2) [3, 4]. Rectangular, triangular, normal and scaled and shifted t-distributions were assigned to the probability density functions of more than 30 input quantities. The correlations between some input quantities were considered with the help of Gaussian copulas [5].

As an example the figure 1 shows the scaled frequency distributions of the volume magnetic susceptibility χ (left) and the vertical component of the permanent magnetic polarization µ0 · Mz (right) obtained from Monte Carlo simulations of the susceptometer method for a 2 g-weight. In this case the outer geometry of the weight was approximated by so-called “inner cylinders” [2]. The dashed vertical lines give the corresponding endpoints of the probabilistically symmetric 95.45 % coverage interval.

Figure 1: Scaled frequency distributions for the magnetic susceptibility χ (left) and the vertical component of the permanent magnetic polarization µ0 · Mz (right) obtained from Monte Carlo simulations of the susceptometer method for a 2 g-weight. The dashed vertical lines give the endpoints of the probabilistically symmetric 95.45 % coverage interval.

The results of the Monte Carlo simulations agree very well with the analytical uncertainty calculation developed in the WG 1.11 [6]. The full potential of the Monte Carlo simulations will be seen when they are used for the analysis of the intercomparison “Magnetic Properties of Weights” initiated by the German Calibration Service (DKD) and for the EURAMET Cooperation in Research (Ref. 1110) “Determination of Magnetic Properties of Mass Standards”. In both projects PTB will act as pilot laboratory.

### Literatur:

[1] OIML 2004 International recommendation R 111 – weights of classes E1 , E2 , F1 , F2 , M1 , M1-2 , M2 , M2-3 and M3 (Paris: International Organization of Legal Metrology (OIML))

[2] Davis R S 1995 Determining the magnetic properties of 1 kg mass standards. J. Res. Natl. Inst. Stand. Technol. 100 209-25

[3] JCGM (BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML) 2008 Evaluation of measurement data – Supplement 1 to the “Guide to the expression of uncertainty in measurement” – Propagation of distributions using a Monte Carlo method (Sèvres: International Bureau of Weights and Measures (BIPM))

[4] JCGM (BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP and OIML) 2011 Evaluation of measurement data – Supplement 2 to the “Guide to the expression of uncertainty in measurement” – Extension to any number of output quantities (Sèvres: International Bureau of Weights and Measures (BIPM))

[5] Possolo A 2010 Copulas for uncertainty analysis Metrologia 47 262-71

[6] Scholz F and Borys M 2006 Vergleichsmessungen zur Bestimmung der magnetischen Eigenschaften von Massenormalen mit einem Suszeptometer (PTB-MA-78) (Bremerhaven: Wirtschaftsverlag NW – Verlag für neue Wissenschaft GmbH)

### Contact person:

Frank Scholz, Dept. 1.1, WG 1.11, e-mail: frank.scholz@ptb.de