Unurbileg.D, MASM

unurbileg@masm.gov.mn

APPLICATION OF TWO COMPARATORS for

DISSEMINATION of E2 WEIGHT

BY USING SUBDIVISION METHOD

APMP 2014, TCM meeting,

Daejeon , Korea

22 Sep 2014

For the determination of the conventional mass, in the calibration of

weights of the highest accuracy classes, the subdivision method

and its variants are widely used

The subdivision method has following advantages;

• it minimizes the use or wear of standards;

• it produces a set of data which provides important statistical

information about the measurement and performance of

comparator.

• it offers redundancy of data

However :

For subdivision calibration beside the number of comparison, mass

comparator’s resolution and maximum loading level required by the

design is crucial.

In this respect, due to limited weighing range of mass comparators

available in mass laboratory of MASM two mass comparators with

different readability were used for the second decade of subdivision

comparison (i.e from 100 g to 10 g) of entire 1 kg – 1 mg set

against one reference

INTRODUCTION

INTRODUCTION

E2 (1mg - 50 kg)

E1 (20 kg)

E2 (1 mg -10 kg)

E1 (1mg – 10 kg)

(1,2,2,5) x 10n kg where:

n – is positive negative integer

Design for 1 kg – 100 g decade

Weighing design for the 1 kg – 100 g decade

5 unknown weight, 12 equation, 100* g is being check standard

Applied

comparator

C1000S

d=2 mg

Sp ~ 5 mg

1)( XXV T

12

11

10

9

8

7

6

5

4

3

2

1

*100

100

*200

200

500

1000

110000

110000

111000

111000

110100

110100

001100

001100

101110

011110

101111

011111

y

y

y

y

y

y

y

y

y

y

y

y

g

g

g

g

g

X

MATRIX NOTATION

YMX

variance-covariance matrix of

Sum (diag) =0.26

>>

Sum (off diag) =0.04

Almost orthogonal..

Normal equation

MASS COMPARATOR

Sartorius C1000S

Max=1 kg, Min=100 g, d=2 mg, Sp ~ 5 mg

Used since 2001

Mass comparators in mass lab of MASM

Sartorius CC 50, Max=50 g, d=1 mg,

Sp ~ 2.5 mg

Used since 2001

Design for 100 g – 10 g decade

Weighing design for the 100 g – 10 g decade

7 unknown weights, 10 equations, 10* g is being check standard

CC 50

d=1 mg, Sp ~ 2 mg

Applied

comparators

C1000S

d=2 mg

Sp ~ 5 mg

ggg

gg

ggg

MX

*101020

2050

50100100

110000001110000011010000

00110000011110000111010000001100000011100000110100000011

'

'

YMX

MATRIX NOTATION

is

*2/12/1 RsYWMXW weighing equations

need to be weighed

prior to regression.

mass differencies with

from three STTS cycles

where - W is weight matrix:

mgY

)00086.0(01799.0

)00099.0(030986.0

)00144.0(01266.0

)000764.0(01916.0

)0225.0(034156.0

)00076.0(0588.0

)00099.0(00549.0

)0045.0(00983.0

)00133.0(055.0

)0025.0(0625.0

WEIGHT MATRIX

nis

wi

i .....1,

2

0

W= (wi)

n

i is12

2

01

1 mg0003486.00

*2/1 XXW

*2/1 YYW

*** RsYMX

AFTER WEIGHTING

84417.084417.0000000

31973.031973.031973.000000

31973.031973.0031973.00000

0028146.028146.00000

106315.00106315.0106315.0106315.0000

008561.008561.008561.0008561.000

0000120528.0120528.000

000012051.012051.012051.00

000006489.006489.0006489.0

0000001286.01286.0

*X

11000000

11100000

11010000

00110000

01111000

01110100

00001100

00001110

00001101

00000011

X

mgY

01799.0030986.001266.001916.0034156.0

0588.000549.000983.0055.00625.0

mgY

01168.00009.000773.000563.000154.000191.0

00052.000396.000106.0

00935.0

*

Weighing equation is transformed

-6

-5

-4

-3

-2

-1

0

1

2

3

4

0 50 100

mg

residuals st deviation*** YMXRs

g g

is

gM

9999951.90000132.100000396.200000206.200000204.500000147.500000377.100999979.99

RESULTS

**1** )( YXXXM TT

Solution matrix

variance-covariance matrix of weighted matrix X*

1**2 )( XXSVMT

261076.1 mgVM

mgS 0004039.0DF

Rs

S

n

i

1

2*

2

)(

16.1/ 0 S

mg0003486.00

internal consistency of observed weighting

results or absence of systematic errors.

with degree of freedom 23

Unknown

masses, g

type A, mg ref wt, mg balance, mg

Combined

st. uncertainty,

mg

Expanded

uncertainty, mg,

k=2

100.0000377 0.0026 0.0093 0.0012 0.010 0.019

50.0000147 0.0009 0.00465 0.0012 0.005 0.010

50.0000204 0.0009 0.00465 0.0012 0.005 0.010

20.0000206 0.0007 0.00186 0.0012 0.002 0.005

20.0000396 0.0007 0.00186 0.0012 0.002 0.005

10.0000132 0.00006 0.00093 0.0012 0.002 0.003

9.9999951 0.00006 0.00093 0.0012 0.002 0.003

jM 2/1)( jjjA VMu crjjr uhMu )( )( jba Mu )( jc Mu

UNCERTAINTY BUDGET

kMuU c )(

2/122 )()( ressenjba uuMu

)( jA Mu - is square of diagonal elements of VM matrix

- is comparator’s of smallest accuracy

)( jb Mu - is assumed to be neglected because mass

standards of a set have same density.

Nominal mass

(g)

Subdivision method Direct method

d (mg ) U, (mg) d (mg ) U, (mg)

100 0.0377 0.019 0.0152 0.051

50 0.0147 0.010 0.0146 0.021

20 0.0206 0.005 0.0187 0.010

20* 0.0396 0.005 -0.00052 0.010

10 0.0132 0.003 0.001 0.007

SUMMARY

The comparison results obtained for the E2 weights by

subdivision method for unknown method are better than

those with direct comparison.

It is possible to use two comparators in one decade and its

variance and covariance matrix is dependent from individual

comparison’s standard deviations.

In case of one comparator variance- covariance matrix is

fixed one, it can be orthogonal.

Necessitates of placing group of weights on balance pan

cause poor eccentricity characteristic of mass comparator

THANK YOU FOR YOUR

ATTENTION